Journal of Functional Analysis 256 (2009) 2747–2767 www.elsevier.com/locate/jfa
Composition operators induced by smooth self-maps of the real or complex unit balls ✩ Hyungwoon Koo a,∗ , Maofa Wang b a Department of Mathematics, Korea University, Seoul 136-713, Republic of Korea b School of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China
Received 18 July 2008; accepted 3 November 2008 Available online 28 November 2008 Communicated by N. Kalton
Abstract In this paper, we study the composition operator CΦ with a smooth but not necessarily holomorphic symbol Φ. A necessary and sufficient condition on Φ for CΦ to be bounded on holomorphic (respectively harmonic) weighted Bergman spaces of the unit ball in Cn (respectively Rn ) is given. The condition is a real version of Wogen’s condition for the holomorphic spaces, and a non-vanishing boundary Jacobian condition for the harmonic spaces. We also show certain jump phenomena on the weights for the target spaces for both the holomorphic and harmonic spaces. © 2008 Elsevier Inc. All rights reserved. Keywords: Composition operator; Smooth map; Bergman space; Unit ball; Carleson measure; Boundedness
1. Introduction and statement of results Let U n be the open unit ball centered at origin in Cn and write H (U n ) for the space of all holomorphic functions on U n , and let B n be the open unit ball centered at origin in Rn and write h(B n ) for the space of all harmonic functions on B n . For 0 < p < ∞ and α > −1, with Ω either p U n or B n , we let Lα (Ω) be the space of all measurable functions f on Ω such that ✩
Koo is partially supported by the KRF-2008-314-C00012 and Wang is partially supported by the NSF-10671147 and NSF-10571044 of China. * Corresponding author. E-mail addresses:
[email protected] (H. Koo),
[email protected] (M. Wang). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.002
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H. Koo, M. Wang / Journal of Functional Analysis 256 (2009) 2747–2767 p f Lp (Ω) α
=
f (z)p dVα (z) < ∞,
Ω
where dVα (z) = (1 − |z|2 )α dV (z) and dV is the normalized Lebesgue volume measure on Ω. p The weighted Bergman space Aα (U n ) is the space of all f ∈ H (U n ) for which f Lpα (U n ) < ∞, p and the weighted harmonic Bergman space bα (Ω) is the space of all f ∈ h(Ω) for which n f Lpα (Ω) < ∞, here we identify h(U ) with h(B 2n ) in the natural way. For the case α = 0, p p we will often write Ap (U n ) = A0 (U n ) and bp (Ω) = b0 (Ω) for simplification. n n With Ω either U or B , let Ψ be a map from Ω into itself, then Ψ induces the composition operator CΨ , defined by CΨ f = f ◦ Ψ. In this paper, we assume Ψ is a smooth but not necessarily holomorphic self-map of Ω and find a necessary and sufficient condition on Ψ such that
CΨ f (z)p dVα (z) C
Ω
f (z)p dVα (z)
(1.1)
Ω p
p
for some C > 0 and all f ∈ X, where X = Aα (U n ) when Ω = U n , and X = bα (B n ) when Ω = Bn. When Ψ is a holomorphic self-map of U n with n 2, W. Wogen has given a necessary and sufficient condition on Ψ ∈ C 3 (U n ) in [9] for CΨ to be bounded on Hardy spaces H p (U n ), p which recently has been generalized to weighted Bergman spaces Aα (U n ) in [5]. Recently, W. Wogen provided some geometric properties for bounded composition operators [10]. By the result of [5], it is straightforward to see that Ψ (z1 , z2 ) = (z1 , 0) induces a bounded operator CΨ p on Aα (U 2 ) for any 0 < p < ∞. But for the same map, the harmonic counterpart does not hold. 1 In fact, if we take fk (z) = |z−(1+1/k,0)| 2 , then a direct calculation shows that fk (z) is harmonic and belongs to b2 (U 2 ), but
lim
k→∞
CΨ fk L2 (U 2 ) fk b2 (U 2 )
= ∞.
This raises a natural question: p
What is the condition for (1.1) to hold with X = bα (B n )? If Ψ is not holomorphic, we cannot expect CΨ f to be holomorphic even f is, and CΨ f may not be harmonic even both Ψ and f are harmonic. Therefore, if we do not impose the analyticity condition on the symbol Ψ : Ω → Ω, then we lose the analyticity or the harmonicity of CΨ f , but we have much more flexibility for the choice of the symbol map Ψ . In this paper we find a simple necessary and sufficient condition for (1.1) to hold for all p p f ∈ bα (B n ) when Ψ ∈ C 2 (B n ), and the corresponding result for the holomorphic spaces Aα (U n ) 4 n when Ψ ∈ C (U ). Here we do not assume the analyticity of Ψ . More precisely, our first main result is for the harmonic spaces.
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Theorem 1.1. Let 0 < p < ∞, α > −1 and Ψ : B n → B n be a map with Ψ ∈ C 2 (B n ). Then there exists a constant C > 0 such that f ◦ Ψ Lpα (B n ) Cf bαp (B n ) p
for all f ∈ bα (B n ) if and only if JΨ (ζ ) = 0
for all ζ ∈ Ψ −1 ∂B n .
Here JΨ (ζ ) is the Jacobian of Ψ at ζ . Our second main result is the corresponding holomorphic version. Theorem 1.2. Let 0 < p < ∞, α > −1 and Φ : U n → U n be a map with Φ ∈ C 4 (U n ). Then there exists a constant C > 0 such that f ◦ ΦLpα (U n ) Cf Apα (U n ) p
for all f ∈ Aα (U n ) if and only if Φ satisfies: (1) Rank MΦη (ζ ) = 2 for all η, ζ ∈ ∂U n with Φ(ζ ) = η, and ζ Φη (ζ ) > D τ τ Φη (ζ ) for all η, ζ, τ ∈ ∂U n with Φ(ζ ) = η and τ ∈ ζ ⊥ . (2) D Here, Φη (z) = Φ(z), η is the Hermitian inner product of Φ(z) and η, MΦη (ζ ) is the real Jacobi matrix of Φη (·) at ζ , ζ ⊥ is the subspace of R2n which is orthogonal to (x1 , x2 , . . . , x2n ) ζ is the real directional derivative in the ζ direction with (x1 + ix2 , . . . , x2n−1 + ix2n ) = ζ , and D considered as a real vector in ∂B 2n . For more details see Section 4. As mentioned before when Φ is holomorphic, Theorem 1.2 is proved in [9] for Hardy spaces and in [5] for weighted Bergman spaces. In Section 4, we will show that our necessary and sufficient condition in Theorem 1.2 is equivalent to Wogen’s original condition in [9] and [5] for the holomorphic self-map Φ. Moreover, we show that there are jump phenomena in the optimal target spaces in our main p theorems: if the inducing self-map Ψ is smooth enough and CΨ does not map Aα (Ω) (respecp p p p tively bα (Ω)) into Lα (Ω), then it does not map Aα (Ω) (respectively bα (Ω)) into any larger p spaces Lβ (Ω) for all α < β < α + 0 . We show that 0 = min{1/4, α + 1} for holomorphic spaces, and 0 = min{1/2, α + 1} for the harmonic spaces, which all depend on the weight α. See Theorem 3.2 and Theorem 4.1 for more precise statements. This contrasts with that for the holomorphic spaces with holomorphic inducing symbols, where the jump is constant 1/4 (see [5]). We further provide examples which show these jumps are sharp. Since holomorphic functions are harmonic, the necessary and sufficient condition of Theorem 1.1 implies that of Theorem 1.2 for sufficiently smooth symbol, although they are not equivalent. Thus, the boundary Jacobian condition of Theorem 1.1 implies the real version of Wogen’s condition in Theorem 1.2, which also can be proved directly. One may consider the limiting case α = −1, the Hardy spaces. For Hardy spaces we need to replace f ◦ Ψ Lpα (Ω) by f ◦ Ψ Lp (∂Ω) , but f ◦ Ψ Lp (∂Ω) may not dominate the integrals over the sphere centered at the origin with radius 0 < r < 1 since f ◦ Ψ may be neither harmonic nor
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holomorphic. Note that the Hardy space norm of f ◦ Ψ is bounded by f ◦ Ψ Lp (∂Ω) if f ◦ Ψ is harmonic or holomorphic. For this reason, it is meaningless for the Hardy spaces if Ψ is not holomorphic and hence we do not include the limiting case. The rest of the paper is organized as follows: We state the well-known Carleson measure criteria in Section 2, which are main tools for our proofs of the main results. The proof of Theorem 1.1 together with the jump phenomenon is given in Section 3, and the proof of Theorem 1.2 together with the jump phenomenon is given in Section 4. In the sequel, we often use the same letter C, depending only on the allowed parameters, to denote various positive constants which may change at each occurrence. For non-negative quantities X and Y , we often write X Y or Y X if X is dominated by Y times some inessential positive constant. Also, we write X ≈ Y if X Y X. 2. Carleson measures For every ζ ∈ ∂U n and 0 < δ < 1, let S(ζ, δ) be the Carleson box on U n defined by S(ζ, δ) = z ∈ U n : 1 − z, ζ < δ . For a vector-valued function Φ : U n → U n which is continuous on U n , we have the following change of variables formula [4]: p |f ◦ Φ| dVα = |f |p dμα , Un
Un
where the Borel measure μα on U n is defined by μα (E) = Vα (Φ −1 (E)). Therefore, the usual Carleson measure type characterization also holds for the non-holomorphic Φ : U n → U n which is continuous on U n . Proposition 2.1. Let 0 < p < ∞ and α, β > −1. Suppose that Φ : U n → U n is a map which is continuous on U n . Define the Borel measure μβ on U n by μβ (E) = Vβ (Φ −1 (E)). Then, there p is some C > 0 such that for all f ∈ Aα (U n ) f ◦ ΦLp (U n ) Cf Apα (U n ) β
if and only if there exists C1 > 0 such that μβ S(ζ, δ) C1 δ n+1+α for all ζ ∈ ∂U n and 0 < δ < 1. For a proof of Proposition 2.1 we refer to [3]. When α = β, Proposition 2.1 is [3, Theorem 3.37] and the same proof works for α = β. There is a complete analogue of Proposition 2.1 for the harmonic setting. Fix an integer n 2 and let B n be the unit ball of Rn . For ζ ∈ ∂B n and 0 < δ < 1, let D(ζ, δ) be the Carleson box on B n defined by D(ζ, δ) = x ∈ B n : |ζ − x| < δ .
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The exactly same argument of the proof of Proposition 2.1 works for the following proposition if we just replace the Carleson boxes with ones defined with the Euclidian metric as above. For the necessary adjustments for the harmonic setting, see [1] and references therein. Proposition 2.2. Let 0 < p < ∞ and α, β > −1. Suppose that Ψ : B n → B n is a map which is continuous on B n . Define the Borel measure μβ on B n by μβ (E) = Vβ (Ψ −1 (E)). Then, there is p some C > 0 such that for all f ∈ bα (B n ) f ◦ Ψ Lp (B n ) Cf bαp (B n ) β
if and only if there exists C1 > 0 such that μβ D(ζ, δ) C1 δ n+α for all ζ ∈ ∂B n and 0 < δ < 1. We know from Propositions 2.1 and 2.2 that the inequalities above are independent of p, which means that if (1.1) holds for some p > 0, then it holds for every p > 0. So we can reduce the proofs of Theorems 1.1 and 1.2 to the case p = 2. 3. Harmonic case on the real unit ball Let Ψ : B n → B n be a smooth function on the unit ball in Rn with Ψ ∈ C 1 (B n ). We will say Ψ satisfies the non-vanishing boundary Jacobian condition if the following condition holds: JΨ (ζ ) = 0 for all ζ ∈ Ψ −1 ∂B n .
(3.1)
Here JΨ (ζ ) is the Jacobian of Ψ at ζ . In this section we will prove Theorem 1.1. First, we note that a similar boundary condition was used to characterize the boundedness of composition operators on the polydisk in [6] for holomorphic self-maps. The main idea for the sufficiency proof of Theorem 1.1 is from [6]. We restate the sufficiency part for an easy reference. Note that we only assume Ψ ∈ C 1 (B n ) for the sufficiency part, and Ψ ∈ C 2 (B n ) is assumed only for the necessity part. Theorem 3.1. Let 0 < p < ∞, α > −1 and Ψ ∈ C 1 (B n ) with Ψ (B n ) ⊂ B n . If Ψ satisfies the p p non-vanishing boundary Jacobian condition (3.1), then CΨ : bα (B n ) → Lα (B n ) is bounded. Proof. Let K = Ψ −1 (∂B n ). Then K ⊂ ∂B n . JΨ is continuous since Ψ ∈ C 1 (B n ), and thus JΨ (x) = 0 for all x in some neighborhood OK of K. Therefore, for each x ∈ K there exists a neighborhood Ox of x such that Ψ : Ox → Ψ (Ox ) is one-to-one and onto. We claim that Ψ −1 (η) is a finite subset of ∂B n for any η ∈ ∂B n . In fact, if there exists an infinite sequence of distinct points {ζj } ⊂ Ψ −1 (η), then there is a subsequence of {ζj }, which, we call it {ζj } again, converges to some ζ ∈ ∂B n . So Ψ (ζ ) = limj Ψ (ζj ) = η, and then ζ ∈ K. Since JΨ (ζ ) = 0, Ψ is locally invertible near ζ , which contradicts the fact that Ψ (ζj ) = η for all j . This proves the claim.
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Let η ∈ Ψ (∂B n ) ∩ ∂B n . Since Ψ −1 (η) is a finite set, there is a neighborhood Oη of η and k
some points {ζj }jη=1 ⊂ ∂B n and their neighborhoods {Oζ1 , . . . , Oζkη } such that Ψ : Oζj → Oη is one-to-one and onto for every j = 1, . . . , kη . This implies that if Oη is sufficiently small, then Ψ
−1
(Oη ) =
kη
Oζj ,
j =1
and {Oζj } are pairwise disjoint. Combining these with the compactness of Ψ (∂B n ) ∩ ∂B n we have sup kη : η ∈ Ψ ∂B n ∩ ∂B n < ∞. Since |JΨ | is bounded from below and above on any compact subset of OK , there are some positive constants c and C such that cVα (Oη ) Vα (Oζj ) CVα (Oη ),
j = 1, . . . , kη .
Therefore, if O is a sufficiently small open subset near Ψ (∂B n ) ∩ ∂B n , we have Vα Ψ −1 (O) Vα (O) for all α > −1, which completes the proof by Proposition 2.2.
2
The proof above can be modified for the holomorphic case to show that the non-vanishing boundary Jacobian condition is also a sufficient condition for the boundedness of CΦ in Theorem 1.2, and we leave it to the interested reader. Note that the non-vanishing boundary Jacobian condition is not a necessary condition for Theorem 1.2 by the example mentioned in the introduction. We now prove the necessity of Theorem 1.1 together with the jump phenomenon. Theorem 3.2. Let 0 < p < ∞, α > −1 and Ψ ∈ C 2 (B n ) with Ψ (B n ) ⊂ B n . If the non-vanishing p p boundary Jacobian condition (3.1) fails, then CΨ : bα (B n ) → Lα+m(α)− (B n ) is not bounded for any > 0, where m(α) = min{1/2, α + 1}. Proof. We prove this using Carleson measure characterization (Proposition 2.2). Suppose the boundary Jacobian condition (3.1) fails at some ζ ∈ Ψ −1 (∂B n ), and let Ψ (x) = (g1 (x), . . . , gn (x)), where x = (x1 , . . . , xn ) ∈ B n . Set gkj (x) =
∂gk (x), ∂xj
1 k n, 1 j n.
Let MΨ (x) be the Jacobi matrix of Ψ at x, that is, ⎛
g11 (x) ⎜ g21 (x) MΨ (x) = ⎜ ⎝ .. .
g12 (x) g22 (x) .. .
··· ··· .. .
gn1 (x)
gn2 (x)
···
⎞ g1n (x) g2n (x) ⎟ .. ⎟ ⎠. . gnn (x)
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First, observe that any orthogonal map Ψ0 of Rn satisfies (1.1), which, in fact, induces an isomep try CΨ0 on bα (B n ). By composing with orthogonal transformations of B n if necessary, we may assume that η = Ψ (ζ ) = ζ = e1 = (1, 0, . . . , 0). Let ekj = gkj (e1 ). Then by Taylor expansion of gk (x) we have gk (x) = δ1k + ek1 (x1 − 1) +
n
ekj xj + O |x − e1 |2
(3.2)
j =2
for each 1 k n. It follows that (x1 , . . . , xn ) ∈ Ψ −1 D(η, δ) for any 0 < δ < 1 if and only if 2 n n 2 ekj xj + O |x − e1 | < δ 2 . ek1 (x1 − 1) +
(3.3)
j =2
k=1
Since g1 (cos t, 0, . . . , 0, sin t, 0, . . . , 0) attains its maximum at t = 0, we see that from (3.2) e1j = 0 for j = 2, . . . , n.
(3.4)
We consider the case e11 = 0 and the case e11 = 0 separately. First, suppose e11 = 0. Since the rank of MΨ (ζ ) is less than or equal to (n − 1) by our hypothesis, there is a column, which is not the first column by (3.4), that is a linear combination of the remaining (n − 1) columns. Without loss of generality we may assume it is the last column, that is ⎞ ⎛ ⎞ ⎛e ⎞ ⎛ e1n e11 1(n−1) ⎜e ⎟ ⎜ e2(n−1) ⎟ ⎜ e2n ⎟ ⎟ + · · · + cn−1 ⎜ . ⎟ ⎜ . ⎟ = c1 ⎜ 21 (3.5) . ⎝ .. ⎠ ⎝ . ⎠ ⎝ .. ⎠ . en1 enn en(n−1) for some real constants cj where 1 j (n − 1). By (3.4) again we have c1 = 0, and thus (3.3) is equivalent to 2 n n−1 2 ekj (xj + cj xn ) + O |x − e1 | < δ 2 . ek1 (x1 − 1) + k=1
(3.6)
j =2
For c > 0, let Dδ,c be √ Dδ,c = (x1 , . . . , xn ) ∈ B n : 1 − x1 < δ, |xn | < c δ, |xj + cj xn | < δ for 2 j (n − 1) . By (3.6), we see that there is a constant C1 > 0 such that Dδ,c ⊂ Ψ −1 D(η, C1 δ) for sufficiently small c > 0.
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Since Vβ (Dδ,c ) ≈ δ n+β−1/2 , Vβ Ψ −1 D(η, C1 δ) δ n+α
1 implies β α + . 2
(3.7)
Next, for e11 = 0 there are two possibilities: either the first column is a linear combination of the other (n − 1) columns, or one of the last (n − 1) columns, which we may assume the last column, is a linear combination of the remaining (n − 1) columns. For the first case, we see that (3.3) is equivalent to n 2 n 2 ekj xj + cj (x1 − 1) + O |x − e1 | < δ 2 k=1 j =2
be for some real constants cj , 2 j n. For c > 0, let Dδ,c
√ = (x1 , . . . , xn ) ∈ B n : 1 − x1 < c δ, xj + cj (x1 − 1) < δ for 2 j n . Dδ,c ) ≈ δ n+β/2−1/2 and D ⊂ Ψ −1 (D(η, C δ)) for some C > 0 and From this we see that Vβ (Dδ,c 1 1 δ,c sufficiently small c > 0. Thus
Vβ Ψ −1 D(η, C1 δ) δ n+α
implies β 2α + 1.
(3.8)
For the second case, we see that (3.3) is equivalent to 2 n n−1 2 ekj (xj + cj xn ) + O |x − e1 | < δ 2 ek1 (x1 − 1) + c1 xn + k=1
j =2
be for other constants cj , 1 j n − 1. For c > 0, let Dδ,c
√ = (x1 , . . . , xn ) ∈ B n : (x1 − 1) + c1 xn < δ, |xn | c δ, |xj + cj xn | < δ Dδ,c for 2 j (n − 1) , ⊂ Ψ −1 (D(η, C δ)) for some C > 0 and sufficiently small c > 0. If c = 0, D is then Dδ,c 1 1 1 δ,c ) ≈ V (D ). So this completes the proof by (3.7) exactly Dδ,c . If c1 = 0, we see that Vβ (Dδ,c β δ,c and (3.8). 2 p
It follows from Theorem 1.1 that the boundedness of CΨ defined on bα (B n ) is independent of α when Ψ ∈ C 2 (B n ). But we know this is not a general phenomenon without the assumption of smoothness of Ψ (refer to [8]). While the jump on the optimal range spaces for the holomorphic symbol of the unit ball in Cn [5] is 1/4, the jump here is min{1/2, α + 1}. We give examples to show that this jump is sharp. First we give an example which shows the upper bound 1/2 is optimal for α −1/2.
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Example 3.3. Let Ψ : B n → B n be Ψ (x1 , x2 , . . . , xn ) = (0, x2 , . . . , xn ). For 0 < p < ∞ and p p p p α > −1, then CΨ : bα (B n ) → Lα (B n ) is not bounded, but CΨ : bα (B n ) → Lα+1/2 (B n ) is bounded. Proof. Note that JΨ (ζ ) = 0 for every ζ = (0, ζ2 , . . . , ζn ) ∈ ∂B n , which implies that p p CΨ : bα (B n ) → Lα (B n ) is not bounded by Theorem 3.2. p p We use Proposition 2.2 to show that CΨ : bα (B n ) → Lα+1/2 (B n ) is bounded, i.e., show that there is some C > 0 such that μα+1/2 D(ζ, δ) Cδ n+α
for all ζ ∈ ∂B n and 0 < δ < 1.
(3.9)
Let x = (x1 , . . . , xn ) = (x1 , x ) and ζ = (ζ1 , ζ ) ∈ ∂B n . Then Ψ −1 D(ζ, δ) = x ∈ B n : ⊂ x ∈ B n: Since |x1 |
1 − |x |2
ζ − (0, x ) < δ |ζ − x | < δ .
√ √ 2 1 − |x | and then Vα+1/2 x ∈ B n : |ζ − x | < δ ≈ δ n+α ,
which immediately completes the claim (3.9).
2
Next we give another example which shows the upper bound α + 1 is optimal for α −1/2. Example 3.4. Let Ψ (x, y) = (1 − (1 − x)2 − y 2 , y) with 0 < < 1/5. For 0 < p < ∞ p p p p and α > −1, then CΨ : bα (B 2 ) → Lα (B 2 ) is not bounded, but CΨ : bα (B 2 ) → L2α+1 (B 2 ) is bounded. Proof. First note that Ψ (x, y)2 1 − y 2 + ( y)2 1. Since Ψ −1 (∂B 2 ) = {e1 }, it is easy to see that CΨ : bα (B 2 ) → Lα (B 2 ) is not bounded by Theorem 3.2. p p To show the boundedness of CΨ : bα (B 2 ) → L2α+1 (B 2 ), we will verify the Carleson criterion for D(ζ, δ) with all ζ = (x0 , y0 ) ∈ ∂B 2 and 0 < δ < 1 by using Proposition 2.2. Without loss of generality, we may assume y0 is sufficiently small and y0 0. If 0 y0 4δ, then D(ζ, δ) ⊂ D(e1 , 16δ). Therefore, we have p
p
Ψ −1 D(ζ, δ) ⊂ (x, y) ∈ B 2 : (1 − x)2 < 16δ, |y| < 16δ . From this we have √
V2α+1 Ψ −1 D(ζ, δ) δ
δ
r 2α+1 dr ≈ δ 2+α . 0
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If y0 > 4δ, we first claim that (1 − x0 ) − y 2 δ for | y − y0 | δ. In fact, if it is not true, then (1 − x0 ) − δ > ( y)2 = ( y − y0 + y0 )2 y02 − 2y0 δ + δ 2 y02 /2, where the last inequality follows from the assumption that y0 > 4δ. This implies that (1 − x0 ) − δ > y02 /2 = 1 − x02 /2 (1 − x0 )/2 which is a contradiction, so the claim is proved. Now suppose Ψ (x, y) ∈ D(ζ, δ), then | y − y0 | < δ and (1 − x0 ) − y 2 − δ (1 − x)2 (1 − x0 ) − y 2 + δ. By the claim above, √ Ψ −1 D(ζ, δ) ⊂ (x, y) ∈ B 2 : | y − y0 | δ, (1 − x) 2 δ . Now the proof is completed by the same way as in the previous case. p
2 p
The following example shows that CΨ does not map bα (B 2 ) into Lβ (B 2 ) for any α, β > −1. p
p
Example 3.5. Let Ψ (x, y) = (1 − y 2 , y) with 0 < < 1, then CΨ : bα (B 2 ) → Lβ (B 2 ) is not bounded for any α, β > −1 and 0 < p < ∞. Proof. Note that Ψ (x, y)2 1 − y 2 + ( y)2 1 and Ψ −1 ∂B 2 = {e1 }. For any 0 < δ < 1, (x, y) ∈ B 2 : | y| δ, |x| < 1 ⊃ Ψ −1 D(e1 , δ) ⊃ (x, y) ∈ B 2 : | y| δ/2, |x| < 1 . Thus, we have Vβ Ψ −1 D(e1 , δ) ≈ δ. p
p
It is easy from Proposition 2.2 that the boundedness of CΨ : bα (B 2 ) → Lβ (B 2 ) must imply δ ≈ Vβ (Ψ −1 (D(e1 , δ))) σ 2+α . But it is impossible because of 2 + α > 1, which completes the proof. 2 p
We would like make a remark on the last example. It is well-known that CΦ : Aα (U n ) → is always bounded for any holomorphic symbol Φ : U n → U n (see [2] and [7]). The above example shows that this is not true for the Harmonic Bergman spaces with some nonanalytic symbols. Let Φ(z) = (1 − y12 + i y1 , 0, . . . , 0), then, by the same argument as the above p p proof of Example 3.5, we see that CΦ : Aα (U n ) → Aβ (U n ) is not bounded for any α, β > −1. p Aα+n−1 (U n )
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4. Holomorphic case on the complex unit ball Let Φ : U n → U n be a smooth function on the unit ball in Cn with Φ ∈ C 4 (U n ). In this section p we consider the boundedness of CΦ defined on Aα (U n ) for α > −1 and prove Theorem 1.2. For η ∈ ∂U n , let Φη (z) = Φ(z), η be the coordinate of Φ in the η direction, period. Here z, w is the usual Hermitian inner product ∗ ∗ ) ∈ ∂U n and f : U n → R with of z, w ∈ Cn . For any ζ = (x1∗ + ix2∗ , x3∗ + ix4∗ , . . . , x2n−1 + ix2n 1 n f ∈ C (U ), let ζ f (z) = D
2n
j f )(z) xj∗ (D
j =1
j denotes be the real directional derivative in the ζ direction viewed as a vector in R2n , where D ∂ n 1 the j th partial differentiation operator Dj = ∂xj . Note that Φη : U → U for any fixed η ∈ ∂U n .
Let MΦη (ζ ) be the (2 × 2n) real Jacobi matrix of Φη at ζ . We use ζ ⊥ to denote the (2n − 1)∗ ). We will say Φ dimensional real vector subspace of R2n which is orthogonal to (x1∗ , x2∗ , . . . , x2n satisfies the RW-condition (Wogen’s condition for real variables) if the following two conditions hold: (1)
Rank MΦη (ζ ) = 2 for all η, ζ ∈ ∂U n with Φ(ζ ) = η,
(2)
τ τ Φη (ζ ) for all η, ζ, τ ∈ ∂U n with Φ(ζ ) = η and τ ∈ ζ ⊥ . ζ Φη (ζ ) > D D
(4.1)
It is worth noticing that when Φ is a holomorphic self-map of U n and of class C 4 (U n ), this RW-condition is equivalent to the Wogen’s original condition: Dζ Φη (ζ ) > Dτ τ Φη (ζ )
for all η, ζ, τ ∈ ∂U n with Φ(ζ ) = η and ζ, τ = 0.
(4.2)
Here Dζ is the complex directional derivative in the ζ direction. To see this, without loss of generality, we suppose ζ = e1 = (1, 0, . . . , 0) with Φ(e1 ) = η ∈ ∂U n and let Φη (z) = F (z) + iG(z). Expand Φη at e1 by using Taylor’s theorem and Lemma 6.8 in [3], we have Φη (z) = 1 + a1 (z1 − 1) +
n
ajj zj2 /2 + O |1 − z1 |3/2
j =2
= 1 + a1 (x1 − 1) + + i a1 x 2 +
n j =2
n j =2
2 2 3/2 ajj x2j −1 − x2j /2 + O |1 − z1 |
ajj x2j −1 x2j
+ O |1 − z1 |3/2 .
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Here, we may assume ajj ∈ R for j = 2, . . . , n by a unitary transformation in (z2 , . . . , zn ) variables. From this we see that (1) of RW-condition is always true for holomorphic self-map Φ, ∂Φ ∂F since a1 = ∂z1η (e1 ) = ∂x (e1 ) > 0 by Lemma 6.8 of [3]. In addition, since F (z1 , 0, . . . , 0) = 1 F (cos t + i sin t, 0, . . . , 0) 1, we get 0 a1 (x1 − 1) + a11 (z1 − 1)2 /2 + O |1 − z1 |3 = a1 (x1 − 1) − a11 x22 /2 + O |1 − x1 |3/2 = [−a1 /2 − a11 /2]t 2 + O t 3 . 2F 2 (e ) = ∂ F2 (e1 ) since F (z1 , 0, . . . , 0) is ∂x12 1 ∂x2 holomorphic Φ : U n → U n with Φ(e1 ) ∈ ∂U n . There-
This implies a1 > − a11 . Notice that − a11 = − ∂ harmonic, and so a1 >
∂2F (e ) ∂x22 1
for any
fore, our RW-condition for holomorphic Φ is essentially a1 >
∂ 2F (e1 ) ∂xj2
(j = 3, . . . , 2n).
Since F (0, . . . , 0, zj , 0, . . . , 0) is harmonic, we have ajj =
∂ 2 Φη (e1 ) ∂zj2
=
for j = 2, . . . , n. Therefore, our RW-condition is equivalent to |ajj | < a1
∂2F (e1 ) 2 ∂x2j −1
2F 2 ∂x2j
= −∂
(e1 )
(j = 2, . . . , n).
This is exactly Wogen’s original condition (4.2), so the claim is proved. For the proof of Theorem 1.2 we split it into two parts. First, we prove the necessity together with the jump phenomenon. For notational convenience, for the rest of the paper we often use the real variable x = (x1 , x2 , . . . , x2n ) in place of the complex variable z = (x1 + ix2 , x3 + ix4 , . . . , x2n−1 + ix2n ) and identify ∂U n with ∂B 2n if necessary. Theorem 4.1. Let 0 < p < ∞, α > −1 and Φ ∈ C 4 (U n ) with Φ(U n ) ⊂ U n . If the RW-condition p p (4.1) fails, then CΦ : Aα (U n ) → Lα+m(α)− (U n ) is not bounded for any > 0, where m(α) = min{1/4, α + 1}. Proof. We prove this using Carleson measure characterization (Proposition 2.1). Suppose the RW-condition (4.1) fails at some ζ , where η = Φ(ζ ) ∈ ∂U n . By some unitary transformations we may assume ζ = e1 and let Φη (x) = Fη (x) + iGη (x). Note that by Taylor expansion of Φη (x) at e1 we have Φη (x) = 1 + a1 (x1 − 1) + a2 x2 + · · · + a2n x2n + i b1 (x1 − 1) + b2 x2 + · · · + b2n x2n + O |1 − x1 |2 + |x2 |2 + · · · + |x2n |2 . √ Since Fη ( 1 − t 2 , 0, . . . , 0, t, 0, . . . , 0) has its maximum at t = 0, we have a1 0,
aj = 0 (j = 2, . . . , 2n).
(4.3)
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First, suppose the condition (1) of (4.1) fails at ζ and a1 = 0. Then, Φη (x) = 1 + i b1 (x1 − 1) + b2 x2 + · · · + b2n x2n + O |1 − x1 |2 + |x2 |2 + · · · + |x2n |2 .
(4.4)
For δ ∈ (0, 1), let Ωδ := x ∈ U n : b1 (x1 − 1) + b2 x2 + · · · + b2n x2n < δ, |1 − x1 |2 + |x2 |2 + · · · + |x2n |2 < δ . By (4.4), we see that Ωδ ⊂ Φ −1 S(η, cδ)
(4.5)
for some c > 0. Now we estimate the volume Vβ (Ωδ ). If b2 = · · · = b2n = 0, then there is some c > 0 such that Ωδ ⊃ x ∈ U n : 1 − x1 cδ, |x2 |2 + · · · + |x2n |2 < cδ . A straightforward calculation shows that Vβ (Ωδ ) δ n+1+(β−1)/2 . Suppose bj = 0 for some j = 2, . . . , 2n. Without loss of generality we may assume that b2 = 0. Then, there is some c > 0 such that √ √ √ √ Ωδ ⊃ x ∈ U n : c δ < 1 − x1 < 2c δ, c δ < |xj | < 2c δ, j = 3, . . . , 2n, b1 (x1 − 1) + b2 x2 + · · · + b2n x2n < δ .
(4.6)
In fact, for any x in the above set on the right side, we have 1 b1 (x1 − 1) + b2 x2 + · · · + b2n x2n + b1 (x1 − 1) + b3 x3 + · · · + b2n x2n |b2 | √ 1 δ + 2c δ |b1 | + |b3 | + · · · + |bn | |b2 | |b2 | √ δ
|x2 |
for δ ∈ (0, 1). Then |1 − x1 |2 + |x2 |2 + · · · + |x2n |2 δ, that is, (4.6) is proved. Therefore, we have Vβ (Ωδ ) δ (β+1)/2+1+(2n−2)/2 = δ n+1+(β−1)/2 . From (4.5) we have Vβ Φ −1 S(η, cδ) δ n+1+(β−1)/2 .
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So Vβ Φ −1 S(η, cδ) δ n+1+α
implies β 2α + 1.
(4.7)
Second, suppose the condition (1) of (4.1) fails at ζ and a1 > 0. Then, we have b2 = · · · = b2n = 0 since the two rows of MΦη (ζ ) are linearly dependent and hence Φη (x) = 1 + (a1 + ib1 )(x1 − 1) + O |1 − x1 |2 + |x2 |2 + · · · + |x2n |2 . Therefore if x ∈ Ωδ := z ∈ U n : |1 − x1 | + |x2 |2 + · · · + |x2n |2 < δ for 0 < δ < 1, then Φ(x) ∈ S(η, cδ) for some c > 0, i.e., Ωδ ⊂ Φ −1 S(η, cδ) . Since Vβ (Ωδ ) ≈ δ n+β+3/4 , Vβ Φ −1 S(η, cδ) δ n+1+α
implies β α + 1/4.
(4.8)
Finally, suppose the condition (1) of (4.1) holds but the condition (2) fails at ζ (i.e., at e1 ). Note that a1 0 and a2 = · · · = an = 0 by (4.3). Since (1) holds, we see that a1 > 0.
(4.9)
τ τ Fη (e1 ) for some τ ∈ e⊥ ∩ ∂U n . Since (1) of (4.1) is independent Since (2) fails at e1 , a1 = D 1 of any unitary transformation, by some unitary transformation in (x2 , . . . , x2n ) variables we may assume 22 Fη (e1 ). a1 = D Since Fη
√
(4.10)
1 − s 2 − t 2 = 1 − (s 2 + t 2 )/2 + O(s 4 + t 4 ), by (4.3) we have
1 − s 2 − t 2 , s, t, 0, . . . , 0 = 1 + a1 1 − t 2 − s 2 − 1 + a22 s 2 + 2a23 st + a33 t 2 /2 + O |t| + |s| 1 − 1 − t 2 − s 2 + |s|3 + |t|3 = 1 − (a1 − a22 )s 2 /2 − (a1 − a33 )t 2 /2 + a23 st + O |s|3 + |t|3 .
Since a1 = a22 by (4.10), we have Fη
1 − s 2 − t 2 , s, t, 0, . . . , 0 = 1 − (a1 − a33 )t 2 /2 + a23 st + O |s|3 + |t|3 .
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√ By taking |t| = o(|s|) we see that a23 = 0, since Fη ( 1 − s 2 − t 2 , s, t, 0, . . . , 0) 1. By the same reason we have a2j = 0 (j = 3, . . . , 2n).
(4.11)
Since Fη
1 − s 2 , s, 0, . . . , 0 = 1 + (a222 /6 − a12 /2)s 3 + O s 4 1,
we see that a222 /6 − a12 /2 = 0.
(4.12)
Thus we have Φη 1 − s 2 , s, 0, . . . , 0 = 1 + O s 4 + i −b1 s 2 /2 + b2 s + b22 s 2 /2 + O s 3 . Since |Φη (x1 , x2 , 0, . . . , 0)| 1, we have b2 = 0.
(4.13)
Since (1) holds and a1 = 0 = a2 = · · · = a2n = b2 , we see that there is some j 3 such that bj = 0. So we may assume without loss of generality b3 = 0.
(4.14)
For notational convenience, we will often use the following notation: x = (x1 , . . . , x2n ) = (x1 , x ) = (x1 , x2 , x ) = (x1 , x2 , x3 , x ). With those notation, from Eq. (4.10) to (4.14), we have a22 2 a222 3 x2 + a12 (x1 − 1)x2 + x2 + O |1 − x1 |2 + |x2 |4 + |x |2 Fη (x) = 1 + a1 (x1 − 1) + 2 6 1 = 1 + (a1 + a12 x2 ) (x1 − 1) + x22 + O |1 − x1 |2 + |x2 |4 + |x |2 2 (a1 + a12 x2 ) 1 − x12 − x22 + O |1 − x1 |2 + |x2 |4 + |x |2 , =1− 2 and Gη (x) = b1 (x1 − 1) + b3 x3 +
bj xj + O |1 − x1 |2 + |x |2 .
j 4
For > 0 and 0 < δ < 1, let A( , δ) := x ∈ ∂U n : 1 − x12 − x22 < δ, |x2 | < δ 1/4 , |x3 | < δ 1/2 , |x | < δ 1/2 .
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Then, for x ∈ A( , δ) we have √ 1 − x1 1 − x12 δ + x22 δ + 2 δ, and A( , δ) ⊂ x ∈ ∂U n : 1 − Fη (x) Cδ
(4.15)
for some C > 0 and sufficiently small > 0. Also, for x ∈ A( , δ) we have Gη (x) − b3 x3 = O x 2 + x 2 + |x | . 2
3
Since Gη (x) = b3 x3 + (Gη (x) − b3 x3 ) and b3 = 0, by choosing > 0 sufficiently small, for each x ∈ A( , δ) there is x˜ = (x1 , x2 , x˜3 , x ) ∈ A( , δ) such that Gη (x) ˜ = 0. Let δ) := x: A( , ˜ x ∈ A( , δ) , and δ) . A(δ) := (y1 , x2 , y3 , x ) ∈ U n : |x1 − y1 | + |x˜3 − y3 | < δ, x˜ ∈ A( , Since Gη (x) ˜ = 0, by (4.15) we have A(δ) ⊂ Φ −1 S(η, Cδ) . Then Vβ Φ −1 S(η, Cδ) Vβ A(δ) δ β · δ · δ 1/4 · δ · δ (2n−3)/2 = δ n+β+3/4 . So Vβ Φ −1 S(η, Cδ) δ n+1+α
implies β α + 1/4.
This together with (4.7) and (4.8) completes the proof.
(4.16)
2 p
Note that Φ(z1 , . . . , zn ) = (z1 + z22 /2, 0, . . . , 0) induces a bounded operator CΦ : Aα (U n ) → p p CΦ : Aα (U n ) → Lα (U n ) is not bounded (see [5, Theorem 4.2]). Therefore, the following example shows that the jump in Theorem 4.1 is sharp. This example can be proved exactly as the same way as the proof of Example 3.4, and thus we omit its proof. p Lα+1/4 (U n ), but
Example 4.2. Let Φ(z1 , . . . , zn ) = (1 − (1 − x1 )2 − x22 + i x2 , z2 , . . . , zn ) with 0 < < 1/5. p p p Let 0 < p < ∞ and α > −1, then CΦ : Aα (U n ) → Lα (U n ) is not bounded, but CΨ : Aα (U n ) → p n L2α+1 (U ) is bounded.
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Now we suppose Φ : U n → U n with Φ ∈ C 4 (U n ) and satisfies the RW-condition (4.1). We will prove the sufficiency of Theorem 1.2 by checking the Carleson criterion via local analysis. First, we need several lemmas. For 0 < δ < 1, let K = Φ −1 ∂U n , (z, w) ∈ U n × U n : |z − ζ | < δ, w − Φ(ζ ) < δ , Kδ = ζ ∈K
Uδ = z ∈ U n : d(z, K) < δ , Wδ = z ∈ U n : d z, Φ(K) < δ .
(4.17)
Here, d(z, K) = minζ ∈K d(z, ζ ) and d(z, ζ ) = |1 − z, ζ |. Lemma 4.3. Let Φ : U n → U n with Φ ∈ C 2 (U n ) and satisfy the RW-condition (4.1). Then there exits δ0 > 0 such that the condition (4.1) also holds for every pair (ζ, η) ∈ Kδ0 . Proof. Fix any w ∈ U n . For any (x, z) ∈ ∂U n × U n , we define fw , gw and hw by fw (z) = Φ(z), w x Φ(z), w , gw (x, z) = D hw (x, z) =
sup τ ∈x ⊥ ∩∂U n
τ τ Φ(z), w , D
respectively. If zˆ = Φ(z), then the RW-condition is equivalent to (1)
Rank Mfζˆ (ζ ) = 2 for ζ ∈ K,
(2)
gζˆ (ζ, ζ ) > hζˆ (ζ, ζ )
and
for ζ ∈ K.
Note that fw (z), gw (x, z) and hw (x, z) are all continuous since Φ ∈ C 2 (U n ), and K is a compact subset of ∂U n . Thus, above two conditions also hold in some neighborhood O of K × Φ(K) in Uδ × Wδ , i.e., Rank Mfw (z) = 2 and gw (z, z) > hw (z, z)
for all (z, w) ∈ O.
Now choose δ0 > 0 sufficiently small so that Kδ0 ⊂ O. Then, the condition (4.1) holds on Kδ0 . 2 We also need certain estimates for the real part of 1 − Φη (z). For ζ ∈ ∂U n and 0 < δ < 1, define S(ζ, δ) by S(ζ, δ) = z ∈ U n : 1 − z, ζ < δ .
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Lemma 4.4. Let Φ : U n → U n with Φ ∈ C 3 (U n ) and satisfy the RW-condition (4.1). Then there exit δ0 > 0 and C > 0 such that if η ∈ Wδ0 ∩ ∂U n and ζ ∈ Uδ0 is a local minimum point for (1 − Φη (z)) with |η − Φ(ζ )| < δ0 , then ζ ∈ ∂U n and for all 0 < δ < δ0 Φ S(ζ, δ0 ) \ S(ζ, Cδ) ∩ S(η, δ) = ∅. Moreover, for |z − ζ | < δ0 , 1 − Φη (ζ ) + 1 − z, ζ ≈ 1 − Φη (z) .
(4.18)
Proof. We will choose δ0 small enough so that our local Taylor expansion holds with the uniform control over the coefficients up to the second order terms and the remainder terms, which is possible since Φ ∈ C 3 (U n ). Using Lemma 4.3, choose δ0 > 0 sufficiently small such that the condition (4.1) holds for all (ζ, η) ∈ Kδ0 . Fix η ∈ Wδ0 ∩ ∂U n and let ζ ∈ Uδ0 be a local minimum point for (1 − Φη (z)) with |η − Φ(ζ )| < δ0 . Then, (ζ, η) ∈ Kδ0 and so (4.1) holds for (ζ, η). By (1) of (4.1), Φη is an open map near ζ , which implies that ζ ∈ ∂U n . We may assume that ζ = e1 by some unitary transformation. Let Φη (z) = F (z) + iG(z). Since F (z) has a local maximum at e1 and |x |2 2(1 − x1 ), by the same argument for getting (4.3) we have F (z) = F (e1 ) + a1 (x1 − 1) +
2n 2n
aij xi xj /2 + O |1 − x1 |3/2 .
i=2 j =2
Then, after an orthogonal transformation in x -variables, we have F (z) = F (e1 ) + a1 (x1 − 1) +
2n
ajj xj2 /2 + O |1 − x1 |3/2 .
(4.19)
j =2
Since Φ ∈ C 3 (U n ) and (2) of (4.1) holds for all (ζ, η) ∈ Kδ0 , there exists > 0 independent of η and ζ such that a := max {ajj } < a1 − . 2j 2n
Thus we have 1 − F (z) 1 − F (e1 ) − a1 (x1 − 1) − a/2
2n
xj2 − O |1 − x1 |3/2
j =2
1 − F (e1 ) − a1 (x1 − 1) − a/2 1 − x12 − O |1 − x1 |3/2 1 − F (e1 ) − a1 (x1 − 1) − a(1 − x1 ) − O |1 − x1 |3/2 1 − F (e1 ) + (1 − x1 ) − O |1 − x1 |3/2 . On the other hand, since |x |2 2(1 − x1 ), from (4.19) it is straightforward to see that
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1 − F (z) 1 − F (e1 ) + C(1 − x1 ) + O |1 − x1 |3/2 for some constant C > 0, which completes the proof of (4.18). From (4.18) it is easy to see that Φ[S(ζ, δ0 ) \ S(ζ, Cδ)] ∩ S(η, δ) = ∅ for all 0 < δ < δ0 . More precisely, suppose z ∈ S(ζ, δ0 ) \ S(ζ, Cδ), then 1 − Φη (z) 1 − Φη (ζ ) + 1 − z, ζ 1 − z, ζ Cδ. Therefore, we can choose C > 0 such that Φ(z) ∈ / S(η, δ) if z ∈ S(ζ, δ0 ) \ S(ζ, Cδ).
2
We now prove a mapping property of Φ for the Carleson box on a local minimum point of (1 − Φη (z)). Instead of the usual Carleson box S(ζ, δ), we will use a certain “twisted” Carleson ˜ ζ ) are given in (4.20) in the proof of the box, S(ζ, δ). Precise definitions of S(ζ, δ) and d(z, following lemma. Lemma 4.5. Let Φ : U n → U n with Φ ∈ C 3 (U n ) and satisfy the RW-condition (4.1). Then, there exit δ0 > 0 and C > 0 such that if η ∈ Wδ0 ∩ ∂U n and ζ ∈ Uδ0 is a local minimum point for (1 − Φη (z)) with |η − Φ(ζ )| < δ0 , then for all 0 < δ < δ0 Φ S(ζ, δ0 ) \ S(ζ, Cδ) ∩ S(η, δ) = ∅. Moreover, 1 − Φη (z) C d Φ(ζ ), η + d(z, ˜ ζ) . Proof. As in the proof of Lemma 4.4, choose δ0 > 0 small enough so that condition (4.1) holds for all (ζ, η) ∈ Kδ0 . Fix η ∈ Wδ0 ∩ ∂U n and let ζ ∈ Uδ0 be a local minimum point for (1 − Φη (z)) with |η − Φ(ζ )| < δ0 . Then, ζ ∈ ∂U n and (4.1) holds for (ζ, η) by Lemma 4.4. We may assume that ζ = e1 and let a + ib = Φη (e1 ). By (4.19), we have Φη (z) = F (z) + iG(z) = a + a1 (x1 − 1) +
2n
ajj xj2 /2 + O |1 − x1 |3/2
j =2
+ i b + b1 (x1 − 1) +
2n
bj xj +
j =2
2n 2n
3/2 . bij xi xj /2 + O |1 − x1 |
i=2 j =2
˜ ζ ) and Define d(z, S(ζ, δ) by 2n 2n 2n ˜ ζ ) := (1 − x1 ) + b + b1 (x1 − 1) + d(z, bj xj + bij xi xj /2, ˜ ζ) δ . S(ζ, δ) := z ∈ U n : d(z,
j =2
Suppose Φ(z) ∈ S(η, δ) with |z − ζ | < δ0 , then by Lemma 4.4
i=2 j =2
(4.20)
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δ > 1 − F (z) C (1 − a) + (1 − x1 ) for some C > 0 independent of η. Therefore δ > G(z)
˜ ζ ) − (1 − x1 ) − O |1 − x1 |3/2 d(z, ˜ ζ ) − Cδ d(z,
for some C > 0. Therefore, there is C > 0 such that z ∈ S(ζ, C δ) if Φ(z) ∈ S(η, δ). This completes the proof of the first claim. For the second claim, note that 1 − Φη (z) 1 − Fη (z) + Gη (z) ˜ ζ ) + O |1 − x1 |3/2 . 1 − Fη (z) + d(z, Now the proof is complete by Lemma 4.4.
2
We are now ready to prove the sufficiency of Theorem 1.2. Note that we assumed Φ ∈ C 4 (U n ) for the necessity but we only assume Φ ∈ C 3 (U n ) for the sufficiency. The scheme here is similar to that of [9] by using Lemma 4.5. We include the proof for completeness. First, we restate the sufficiency part of Theorem 1.2 for an easy reference. Theorem 4.6. Let 0 < p < ∞, α > −1 and Φ ∈ C 4 (U n ) with Φ(U n ) ⊂ U n . If Φ satisfies the p p RW-condition (4.1), then CΦ : Aα (U n ) → Lα (U n ) is bounded. Proof. We complete the proof by verifying the Carleson condition μα S(η, δ) = O δ n+α+1 for all η ∈ ∂U n and 0 < δ < 1. Clearly it is enough to check this for all sufficiently small δ. Let m = max{|Φ(z)|: z ∈ U n \Uδ0 }, where δ0 is the number which satisfies Lemmas 4.4 and 4.5, then m < 1, and for δ 1 − m we have Φ −1 S(η, δ) ⊂ Uδ0 . Let t0 = min{δ0 , 1 − m}, we will verify the Carleson condition for all δ < t20 . If necessary, we will shrink δ0 > 0 further. Here K, Uδ and Wδ are defined in (4.17). It also suffices to consider η ∈ ∂U n which is close to Φ(K), i.e., η ∈ Wδ0 ∩ ∂U n , because the other case is trivial due to Φ −1 (S(η, t0 )) = ∅ for any η ∈ ∂U n \Wδ0 . Therefore, we may assume η ∈ Wδ0 ∩ ∂U n and δ < t20 . Let Oj be one of the components of Φ −1 (S(η, t0 )) which also intersects with Φ −1 (S(η, t0 /2)). Let ζj satisfy 1 − Φη (ζj ) = min 1 − Φη (z) , z∈Oj
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then the condition (4.1) holds at ζj by Lemma 4.4, and thus Φη is an open map at ζj if ζj ∈ U n . This implies ζj ∈ ∂U n since ζj is a local minimum point for (1 − Φη (z)). Next, we show that there is an upper bound M, which is independent of η ∈ Wδ0 ∩ ∂U n , on the number of the components of Φ −1 (S(η, t0 )) which also intersects with Φ −1 (S(η, t20 )). To see this, note that by the inequality of Lemma 4.5, there is c > 0 independent of η ∈ Wδ0 ∩ ∂U n such that Φ( S(ζj , ct0 )) ⊂ S(η, t0 ), i.e., S(ζj , ct0 ) ⊂ Oj by the connectivity of Oj . Also, from the definition (4.20) it is easy to see that S(ζj , ct0 ) ≈ t0n+α+1 . Vα This shows the number of these components has an upper bound M < ∞, since Mt0n+α+1 ≈ S(ζj , ct0 )) Vα (U n ), which is a constant. MVα ( Now fix such a component Oj as above. Then, by Lemmas 4.4 and 4.5, there is C > 0 independent of η such that S(ζj , Cδ). Oj ∩ Φ −1 S(η, δ) ⊂
(4.21)
By the definition of Oj , every point of Φ −1 (S(η, δ)) lies in one of the at most M components Oj , so S(ζj , Cδ) δ n+α+1 . 2 μα S(η, δ) MVα p
It also follows from Theorem 1.2 that the boundedness of CΦ defined on Aα (U n ) is independent of α when Φ is smooth enough, although this is not true for general case. Acknowledgments The authors wish to acknowledge the hospitality of SUNY at Albany, where this work was carried out. The authors also express their thanks to the referee for his/her useful suggestions and comments. References [1] B. Choe, H. Koo, H. Yi, Carleson type conditions and weighted inequalities for harmonic functions, Osaka J. Math. 39 (2002) 945–962. [2] J. Cima, W. Wogen, Unbounded composition operators on H 2 (B2 ), Proc. Amer. Math. Soc. 99 (1987) 477–483. [3] C. Cowen, B. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. [4] P. Halmos, Measure Theory, Springer-Verlag, New York, 1974. [5] H. Koo, W. Smith, Composition operators induced by smooth self maps of the unit ball in CN , J. Math. Anal. Appl. 329 (2007) 617–633. [6] H. Koo, M. Stessin, K. Zhu, Composition operators on the polydisc induced by smooth symbols, J. Funct. Anal. 254 (2008) 2911–2925. [7] B. MacCluer, P. Mercer, Composition operators between Hardy and weighted Bergman spaces on convex domains in C N , Proc. Amer. Math. Soc. 123 (1995) 2093–2102. [8] B. MacCluer, J. Shaprio, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (4) (1986) 878–906. [9] W. Wogen, The smooth mappings which preserve the Hardy space H 2 (Bn ), Oper. Theory Adv. Appl. 35 (1988) 249–267. [10] W. Wogen, On geometric properties of smooth maps which preserve H 2 (Bn ), Michigan Math. J. 54 (2006) 301– 306.
Journal of Functional Analysis 256 (2009) 2768–2779 www.elsevier.com/locate/jfa
A joint similarity problem on vector-valued Bergman spaces Olivia Constantin ∗ , Frédéric Jaëck Department of Mathematics, Trinity College Dublin, College Green, Dublin 2, Ireland Received 21 July 2008; accepted 17 November 2008 Available online 20 December 2008 Communicated by J. Bourgain
Abstract We investigate pairs of commuting Foias–Williams/Peller type operators acting on vector-valued weighted Bergman spaces. We prove that a commuting pair of such operators is jointly polynomially bounded if and only if it is similar to a pair of contractions, if and only if both operators are polynomially bounded. © 2008 Elsevier Inc. All rights reserved. Keywords: Hankel operators; Vector-valued Bergman spaces; Joint polynomial boundedness; Joint similarity to a pair of contractions
1. Introduction Given a separable Hilbert space H, denote by B(H) the set of bounded linear operators acting on H. An operator T ∈ B(H) is called similar to a contraction if it can be written as T = V −1 SV , where S, V ∈ B(H) with V invertible and S 1. Whether an operator T ∈ B(H) is similar to a contraction if and only if T is polynomially bounded (see Section 2 for the relevant definitions) was a long-standing open problem posed by Halmos [6] and finally solved by Pisier [12]. More precisely, Pisier found an example of a polynomially bounded operator that is not similar to a contraction, thus showing that the two concepts are not equivalent. * Corresponding author.
E-mail addresses:
[email protected] (O. Constantin),
[email protected] (F. Jaëck). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.022
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In previous attempts to find a counterexample, operators of the following type (sometimes called Foias–Williams/Peller type operators or Foguel–Hankel operators) RT =
S∗ 0
ΓT S
,
(1.1)
were first considered by Peller in [9]. Here RT is acting on the direct sum H 2 ⊕ H 2 , where H 2 is the usual scalar-valued Hardy space, S is the shift operator on H 2 , and ΓT is the Hankel operator with (analytic) symbol T . In a sequence of papers by Peller[10], Bourgain [4], Aleksandrov and Peller [1] it was shown that for this operator, polynomial boundedness is equivalent to similarity to a contraction, and hence there is no counterexample of this type. Pisier’s insight in finding a counterexample was to consider the same type of operator as in (1.1), but acting on the direct sum of Hardy spaces with values in an infinite dimensional Hilbert space. Ferguson and Petrovi´c [5] analyzed the corresponding problem for operators of type (1.1) acting on the direct sum of two scalar-valued standard weighted Bergman spaces. They found complete analogues of the Aleksandrov and Peller [1], Bourgain [4] results. One might expect to get an analogue of Pisier’s counterexample in the setting of vector-valued Bergman spaces. The vector-valued case was studied in [2], and, surprisingly, it was found that polynomial boundedness and similarity to a contraction are equivalent (even) in the case when RT is acting on the direct sum of two copies of standard weighted Bergman spaces with values in an infinite dimensional Hilbert space. This is in contrast to the situation in Hardy spaces. After Pisier’s solution to the similarity problem, a number of new problems appeared. In particular, it was reasonable to ask questions about a pair of (completely) polynomially bounded operators. Petrovi´c [11] showed that there exist commuting operators T1 , T2 such that each of them is polynomially bounded, but the product T1 T2 is not polynomially bounded, and hence the pair (T1 , T2 ) is not jointly polynomially bounded (see Section 2 for the definitions). Moreover, Pisier [13] constructed an example of two commuting operators T1 , T2 , each of which is similar to a contraction, but the pair (T1 , T2 ) is not jointly polynomially bounded. In [5], Ferguson and Petrovi´c considered also the joint similarity problem for a pair of commuting operators of type (1.1) acting on scalar-valued Bergman spaces. Two operators T1 , T2 ∈ B(H) are called jointly similar to a pair of contractions if there exist an invertible operator V ∈ B(H), and S1 , S2 ∈ B(H) with S1 , S2 1, such that T1 = V −1 S1 V , T2 = V −1 S2 V . Ferguson and Petrovi´c proved that a commuting pair of operators (RT1 , RT2 ) is jointly similar to a pair of contractions if and only if the pair is jointly polynomially bounded, if and only if each of RT1 , RT2 is polynomially bounded. In the present paper we investigate the commuting pair (RT1 , RT2 ), acting on standard weighted Bergman spaces with values in an infinite dimensional Hilbert space. We recover the results from the scalar case, more precisely, we obtain that (RT1 , RT2 ) is jointly similar to a pair of contractions if and only if the pair is jointly polynomially bounded, if and only if RT1 , RT2 are polynomially bounded. Moreover, we obtain a characterization of these properties in terms of the symbols T1 , T2 . The proofs in [5] involve certain factorization results, which are expressed as calculations of the projective tensor products of weighted Bergman spaces. Our approach is different and it also works in the vector-valued case. The paper is organized as follows. In Section 2 we present some definitions together with some relevant facts that we use throughout the paper. Section 3 starts by the analysis of the similarity
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problem for the pair (RT , R0 ), that is subsequently used to prove our main result (Corollary 3.2) concerning the more general pair (RT1 , RT2 ). 2. Preliminaries We start by presenting some definitions concerning the operators and the spaces that will be used in our further considerations. Given a separable Hilbert space H, let B(H) denote the bounded linear operators on H. Let d be a positive integer and denote by Dd the polydisc in Cd . A commuting d-tuple of operators (T1 , T2 , . . . , Td ), with Ti ∈ B(H) (1 i d), is said to be jointly polynomially bounded if there exists a positive constant k such that the following inequality holds p(T1 , T2 , . . . , Td ) k sup p(z), z∈Dd
for any analytic polynomial of d variables p. Note that this condition is equivalent to the boundedness of the representation π : A(Dd ) → B(H), with π(zi ) = Ti (1 i d), where A(Dd ) is the polydisc algebra. A commuting d-tuple (T1 , T2 , . . . , Td ) is said to be jointly completely polynomially bounded if there exists a constant k > 0 such that n
1 n
1 2 2 2 2 kP p (T , T , . . . , T )x , y x y , d ij 1 2 d i j i j Mn (A(D )) 1i,j n
(2.2)
j =1
i=1
whenever P = (pij )1i,j n is an n × n matrix of analytic polynomials of d variables, n = 1, 2, . . . , and {xi }ni=1 , {yj }nj=1 are vectors in H. Here P Mn (A(Dd )) = sup pij (z1 , . . . , zd ) M . n
z∈Dd
Denoting by P (T1 , T2 , . . . , Td ) the n × n matrix acting on n H, whose entries pij (T1 , . . . , Td ) are analytic polynomials of T1 , . . . , Td , we can rewrite (2.2) in the form P (T1 , . . . , Td ) kP
Mn (A(Dd )) ,
and hence (2.2) is equivalent to the complete boundedness of the map π defined above. For d = 1 in the previous definitions, we simply say that T1 is polynomially bounded, respectively completely polynomially bounded. Throughout this paper, we shall restrict our attention to the cases d = 1 and d = 2. A result by Paulsen [7,8] together with the existence of a unitary dilation for a contraction, respectively the existence of a commuting unitary dilation for a pair of commuting contractions (see [3]), yields: (a) an operator T is completely polynomially bounded if and only if T is similar to a contraction. (b) a pair of operators (T1 , T2 ) is jointly completely polynomially bounded if and only if it is jointly similar to a pair of contractions.
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Let dA denote the normalized area measure on the unit disc and for α > −1 let dμα (z) = (α + 1)(1 − |z|2 )α dA(z). We consider the standard weighted vector-valued Bergman spaces L2,α a (H) which consist of analytic functions x : D → H with x =
1 2 x(z)2 dμα (z) < ∞. H
(2.3)
D
We write L2,α a for the scalar space, i.e. when H = C. For an analytic operator-valued function T : D → B(H), the (little) Hankel operator ΓT is defined by means of the Hankel form ΓT x, y = lim
r→1
T (rz)x(r z¯ ), y(rz) dA(z),
D
where x, y are H-valued analytic functions in a disk of radius strictly larger than 1 (as it is well known these functions form a dense subset in L2,α a (H)). It turns out (see [2]) that ΓT extends to (H) if and only if a bounded linear operator on L2,α a sup 1 − |z|2 T (z) < ∞. z∈D
In the proof of our main result we shall make use of the next lemma, together with the wellknown formulas (2.5)–(2.6) included below for the sake of completeness. Lemma 2.1. (See [2].) Let γ 0 and let T : D → B(H) be an analytic operator-valued function satisfying supz∈D (1 − |z|2 )γ T (z) < ∞. Then the following equality holds
γ T (z)x(¯z), y(z) 1 − |z|2 dμα (z)
D
1 = α+γ +1
γ +1 y(z)
T (z)x(¯z), 1 − |z|2 dμα (z), z
(2.4)
D
for any x, y ∈ L2,α a (H) with y(0) = 0. For A ∈ B(Cn , H), the space of bounded linear operators from Cn to H, we denote by AB2 its Hilbert–Schmidt norm, that is AB2 =
n
1 2
Aek
2
k=1
where {ek }nk=1 is some orthonormal basis of Cn . On B(Cn ) we consider the usual operator norm and also the trace norm defined by
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A ∈ B Cn ,
Atr = tr |A| ,
where |A| = (A∗ A)1/2 . As is well known, the following inequalities hold. For A, B ∈ B(Cn ), we have tr(AB) A · Btr ,
(2.5)
and for T ∈ B(H), and X, Y ∈ B(Cn , H), we have ∗ Y T X XB Y B T . 2 2 tr
(2.6)
This last inequality is usually stated for operators acting on the same space, but the more general version stated above is a consequence of the classical one. In fact, if M is an n-dimensional subspace of H containing the range of X and U : M → Cn denotes a unitary operator, then ∗ Y T X = Y ∗ T U −1 U X Y ∗ T U −1 U XB 2 tr tr B2 ∗ ∗ XB2 Y T U −1 B = XB2 U −1 T ∗ Y B 2 2 −1 ∗ ∗ XB2 Y B2 U T XB2 Y B2 T . 3. Main results Let us first consider the pair of operators RT =
Mz∗ 0
ΓT Mz
,
R0 =
Mz∗ 0
Γ0 Mz
=
Mz∗ 0
0 Mz
,
2,α acting on the direct sum L2,α a (H) ⊕ La (H) where Mz denotes the operator of multiplication 2,α by z on La (H) and ΓT is defined above. Since ΓT Mz = Mz∗ ΓT , the operators RT and R0 commute, and the action of an analytic polynomial of two variables on the couple (RT , R0 ) is easily computed. Using induction, it is straightforward to show that, if p is such a polynomial, then ΔT (p) p(Mz∗ , Mz∗ ) , p(RT , R0 ) = 0 p(Mz , Mz )
where ΔT (p) = ΓT (∂z p)(Mz , Mz ). Now let P = (pij ) be an n × n matrix of analytic polynomials of two variables. Performing a change of basis (the so-called canonical shuffle) one obtains ∗ ∗ P (RT , R0 ) = (pij (Mz , Mz )) 0
(ΔT (pij )) . (pij (Mz , Mz ))
(3.7)
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Also, if P = (pij )1i,j n is an n × n matrix of analytic polynomials, we shall denote by P # the #) matrix with entries (pij 1i,j n , where # pij (z, w) = pij (¯z, w), ¯
1 i, j n.
Theorem 3.1. Let T : D → B(H) be a holomorphic operator-valued function with sup 1 − |z|2 T (z) < ∞. z∈D
Then the following are equivalent: (i) (ii) (iii) (iv)
The operator RT is polynomially bounded; The pair (RT , R0 ) is jointly polynomially bounded; The pair (RT , R0 ) is jointly completely polynomially bounded; supz∈D (1 − |z|2 )T
(z) < ∞.
Proof. The implications (iii) ⇒ (ii) ⇒ (i) are obvious. Also, the equivalence of (i) and (iv) was proved in [2]. Hence it is enough to prove (iv) ⇒ (iii). Throughout the proof C > 0 stands for a generic constant. Suppose T satisfies condition (iv). Since Mz is a contraction, both pairs (Mz∗ , Mz∗ ) and (Mz , Mz ) are jointly completely polynomially bounded. Then relation (3.7) implies that the pair (RT , R0 ) is jointly completely polynomially bounded if and only if the map 2,α ΔT is completely bounded from the bidisc algebra A(D2 ) to B(L2,α a (H), La (H)). In order to show that ΔT satisfies the above, we let n ∈ N and P = (pij )1i,j n be a matrix of analytic polynomials of two variables. Denote by ∂z P the matrix with entries (∂z pij )1i,j n . Let us first assume that z12 (∂z P )(z, z) is a matrix of analytic polynomials of one variable. The map ΔT is completely bounded if 1 1 2 2 2 2 CP x y Γ (∂ p )(M , M )x , y T z ij z z i j i j Mn (A)
(3.8)
1i,j n
for xi , yj ∈ L2,α a (H), n 1. Note that (iv) implies supz∈D T (z) < ∞, and supz∈D (1 − |z|2 )T (z) < ∞. Hence we can apply Lemma 2.1 twice to obtain
ΓT (∂z pij )(Mz , Mz )xi , yj
1i,j n
=
T (z)xi (¯z), yj (z) (∂z pij )(¯z, z¯ ) dμα (z)
1i,j n D
= =
1 α+1
1i,j n D
1 (α + 1)(α + 2)
(∂z pij )(¯z, z¯ ) 1 − |z|2 dμα (z) T (z)xi (¯z), yj (z) z¯
1i,j n D
2 (∂z pij )(¯z, z¯ ) T
(z)xi (¯z), yj (z) 1 − |z|2 dμα (z). 2 z¯
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For each z ∈ D we denote by T˜xy (z) the n × n matrix with entries 1 − |z|2 T
(z)xi (¯z), yj (z) i,j
and let
(∂z p ij )# (z, z) (1 − |z|2 ) # 2 (∂ P ) (z, z) = 1 − |z| . P˜ (z) = z z2 z2 1i,j n Then P˜ (z) =
z, z¯ ) 2 (∂z pij )(¯ . 1 − |z| z¯ 2 1i,j n
With these notations we have (∂z pij )(¯z, z¯ )
2 2 T (z)xi (¯z), yj (z) 1 − |z| dμα (z) = tr P˜ (z)T˜xy (z)t dμα (z). 2 z¯ 1i,j n D
D
We regard P˜ (z) and T˜xy (z) as operators acting on Cn . Using (2.5) we get tr P˜ (z)T˜xy (z)t dμα (z) P˜ (z)T˜xy (z) dμα (z) tr D
D
=
P˜ (z)T˜xy (z) dμα (z). tr
(3.9)
D
For any a, b ∈ Cn with a = b = 1, the function D2 (z, w) → P # (z, w)a, b is an analytic polynomial of two variables. Hence, by the Schwarz lemma, we deduce (∂z P )# (z, w)a, b 1 − |z|2 = ∂z P # (z, w)a, b 1 − |z|2 C sup P # (z, w)a, b z∈D
C sup P # (z, w)a, b z,w∈D
CP Mn (A) ,
z ∈ D.
Now first put z = w in the above relation, and then use the maximum modulus principle to deduce (1 − |z|2 ) # (∂z P ) (z, z)a, b CP Mn (A) , z2
z ∈ D.
Since a, b were arbitrarily chosen such that a = b = 1, we obtain P˜ (z) CP M (A) , n
z ∈ D.
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From this estimate and relation (3.9) we obtain tr P˜ (z)T˜xy (z)t dμα (z) CP M (A) T˜xy (z) dμα (z). n tr D
(3.10)
D
Consider the operator T˜xy (z) ∈ B(Cn ). For a, b ∈ Cn , we have
1 − |z|2 T
(z)xi (¯z), yj (z) ai b¯j
T˜xy (z)a, b =
1i,j n
2 = 1 − |z| T (z) ai xi (¯z) , bj yj (z) 1j n
1in
= 1 − |z|2 T
(z)Xa, Y b ,
(3.11)
where X, Y : Cn → H are the linear operators defined on the standard basis of Cn , {ei }ni=1 , by X(ei ) = xi (¯z),
Y (ej ) = yj (z),
1 i, j n.
Then from (3.11) we get T˜xy (z)a, b = 1 − |z|2 T
(z)Xa, Y b = Y ∗ 1 − |z|2 T
(z) Xa, b , where Y ∗ : H → C is the adjoint of Y . Now use (2.6) to obtain T˜xy (z) = Y ∗ 1 − |z|2 T
(z) X XB Y B 1 − |z|2 T
(z). 2 2 tr tr With these estimates we get from (3.10) and the Cauchy–Schwarz inequality tr P˜ (z)T˜xy (z)t dμα (z) D
CP Mn (A)
XB2 Y B2 1 − |z|2 T
(z) dμα (z)
D
1 1 2 2 C sup 1 − |z|2 T
(z) P Mn (A) xi 2 yj 2
z∈D
˜ M (A) Xn Y n , = CP n and hence (3.8) follows. If z12 (∂z P )(z, z) is not a matrix of analytic polynomials of one variable, we apply the above procedure to the matrix Q(z, w) = P (z, w) − L(z, w),
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where L(z, w) = ∂z P (0, 0)z + ∂w P (0, 0)w + ∂z2 P (0, 0)z2 + 2∂z ∂w P (0, 0)zw + ∂w2 P (0, 0)w 2 . Doing this we get ΔT (qij ) CQM (A) . n ij Taking into account s t ∂ ∂ P (0, 0) CP M (A) , z w n
s, t = 0, 1, 2, 3,
(3.12)
we deduce QMn (A) CP Mn (A) . Note also that ΔT (lij ) ΓT (∂z lij )(Mz , Mz ) CP M (A) , n ij ij where the last inequality follows by (3.12). So by the above we obtain ΔT (pij ) C ΔT (qij ) + ΔT (lij ) ij ij ij C QMn (A) + P Mn (A) CP Mn (A) , and the proof is complete.
2
Let us consider the more general pair of operators RX1 =
Mz∗
X1
0
Mz
,
RX 2 =
Mz∗
X2
0
Mz
,
(3.13)
where X1 , X2 ∈ B(L2,α a (H)). Note that the commutativity of RX1 and RX2 is equivalent to (X1 − X2 )Mz = Mz∗ (X1 − X2 ), that is X1 − X2 = ΓT is a Hankel operator. From now on we shall assume X1 − X2 = ΓT . If p is an analytic polynomial of two variables note that p(RX1 , RX2 ) =
p(Mz∗ , Mz∗ ) 0
δ(X1 ,X2 ) (p) , p(Mz , Mz )
where the map p → δ(X1 ,X2 ) (p) ∈ B(L2,α a (H)) is a derivation, that is, for any p, q, we have δ(pq) = p Mz∗ , Mz∗ δ(q) + δ(p)q(Mz , Mz ). Since Mz is a contraction, the pair (RX1 , RX2 ) is jointly (completely) polynomially bounded if and only if δ extends to a (completely) bounded map from A(D2 ) to B(L2,α a (H)). It turns out that the study of the joint (complete) polynomial boundedness for the pair (RX1 , RX2 ) reduces to studying the same property for the pairs (RΓT , R0 ) and (RX2 , RX2 ). This is expressed in the next result, whose proof is very similar to the proof presented in [5] for the scalar case (that is, dim H = 1). We include it for the sake of completeness.
O. Constantin, F. Jaëck / Journal of Functional Analysis 256 (2009) 2768–2779
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Proposition 3.1. Assume X1 , X2 ∈ B(L2,α a (H)) with ΓT = X1 − X2 , where ΓT is a bounded Hankel operator. Then δ(X1 ,X2 ) extends to a (completely) bounded map on A(D2 ) if and only if the derivations δ(ΓT ,0) and δ(X2 ,X2 ) are (completely) bounded on A(D2 ). Moreover δ(X1 ,X2 ) (f ) = δ(ΓT ,0) (f ) + δ(X2 ,X2 ) (f ),
(3.14)
for all f ∈ A(D2 ). Proof. We shall prove that (3.14) holds for all analytic polynomials p(z1 , z2 ). To this end, it is j enough to show that (3.14) holds for all monomials z1i z2 , for all integers i, j 1. By a straightforward calculation we obtain j −1
j −1
k=0
k=0
k j ∗ i+j −1−k S S ∗ X1 S i+j −1−k . X2 S k + δ(X1 ,X2 ) z1i z2 = Replacing X1 by ΓT + X2 in the above relation, we obtain j j j δ(X1 ,X2 ) z1i z2 = δ(ΓT ,0) z1i z2 + δ(X2 ,X2 ) z1i z2 , and hence (3.14) holds for all polynomials p(z1 , z2 ). Since (3.14) holds on a dense subset of A(D2 ), it is clear that if δ(ΓT ,0) and δ(X2 ,X2 ) are (completely) bounded on A(D2 ), then δ(X1 ,X2 ) is (completely) bounded on A(D2 ). On the other hand, if we suppose δ(X1 ,X2 ) is (completely) bounded on A(D2 ), then the pair (RX1 , RX2 ) is jointly (completely) polynomially bounded, and hence RX2 is (completely) polynomially bounded. This implies that the pair (RX2 , RX2 ) is jointly (completely) polynomially bounded, and hence the map δ(X2 ,X2 ) is (completely) bounded on A(D2 ). It now follows that δ(ΓT ,0) is (completely) bounded on A(D2 ), since δ(ΓT ,0) = δ(X1 ,X2 ) − δ(X2 ,X2 ) on a dense subset of A(D2 ). 2 The next corollary is an immediate consequence of Theorem 3.8 and Proposition 3.1. Corollary 3.1. Let RX1 , RX2 be commuting operators as in (3.13), such that the pair (RX1 , RX2 ) is jointly polynomially bounded. Then the following are equivalent: (i) RX1 is similar to a contraction; (ii) The pair (RX1 , RX2 ) is jointly similar to a pair of contractions; (iii) RX2 is similar to a contraction. Proof. Since the pair (RX1 , RX2 ) is jointly polynomially bounded, by Proposition 3.1 it follows that δ(ΓT ,0) is bounded on A(D2 ), where ΓT = X1 − X2 . Theorem 3.8 now implies that δ(ΓT ,0) is completely bounded on A(D2 ). From this together with δ(X1 ,X2 ) = δ(ΓT ,0) + δ(X2 ,X2 ) we obtain (ii) ⇔ (iii). The equivalence (i) ⇔ (ii) follows by the same argument applied to Γ−T = X2 − X1 . 2 We shall now state our main result
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Corollary 3.2. Let T1 , T2 : D → B(H) be holomorphic operator-valued functions with sup 1 − |z|2 T2 (z) < ∞. sup 1 − |z|2 T1 (z) < ∞, z∈D
z∈D
Then the following are equivalent: (i) (ii) (iii) (iv)
The operators RT1 , RT2 are polynomially bounded; The pair (RT1 , RT2 ) is jointly polynomially bounded; The pair (RT1 , RT2 ) is jointly similar to a pair of contractions; supz∈D (1 − |z|2 )T1
(z) < ∞ and supz∈D (1 − |z|2 )T2
(z) < ∞.
Proof. The implications (iii) ⇒ (ii) ⇒ (i) are obvious, while the equivalence (i) ⇔ (iv) was proven in [2]. To conclude, it is enough to prove (iv) ⇒ (iii). Assume (iv) holds and put T = T1 − T2 . Clearly T satisfies supz∈D (1 − |z|2 )T
(z) < ∞, and by Theorem 3.8 we deduce that the pair (RΓT , R0 ) is jointly completely polynomially bounded, or, equivalently, the map δ(ΓT ,0) is completely bounded on A(D2 ). On the other hand, the condition in (iv) on T2 implies RT2 is completely polynomially bounded (see [2]), and hence the pair (RT2 , RT2 ) is jointly completely polynomially bounded, or equivalently the map δ(ΓT2 ,ΓT2 ) is completely bounded on A(D2 ). By Proposition 3.1 we have δ(ΓT1 ,ΓT2 ) = δ(ΓT ,0) + δ(ΓT2 ,ΓT2 ) , which together with our considerations above implies (iii). 2 Remark 3.1. It was shown in [2] that RT is similar to a contraction, if and only if RT is polynomially bounded, if and only if RT is power bounded (i.e. supn1 RTn < ∞). In view of this result, the statements (i)–(iv) in Corollary 3.2 are actually equivalent to the statement: RT1 , RT2 are power bounded. Acknowledgment The authors are grateful for constructive suggestions made by the referee. References [1] A.B. Aleksandrov, V.V. Peller, Hankel operators and similarity to a contraction, Int. Math. Res. Not. 6 (1996) 263– 275. [2] A. Aleman, O. Constantin, Hankel operators on Bergman spaces and similarity to contractions, Int. Math. Res. Not. 35 (2004) 1785–1801. [3] T. Ando, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963) 88–90. [4] J. Bourgain, On the similarity problem for polynomially bounded operators on Hilbert space, Israel J. Math. 54 (1986) 227–241. [5] S.H. Ferguson, S. Petrovi´c, The joint similarity problem for weighted Bergman shifts, Proc. Edinb. Math. Soc. (2) 45 (1) (2002) 117–139. [6] P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970) 887–933. [7] V.I. Paulsen, Every completely polynomially bounded operator is similar to a contraction, J. Funct. Anal. 55 (1984) 1–17. [8] V.I. Paulsen, Completely bounded homomorphisms of operator algebras, Proc. Amer. Math. Soc. 92 (1984) 225– 228. [9] V.V. Peller, Estimates of functions of power bounded operators on Hilbert spaces, J. Operator Theory 7 (2) (1982) 341–372. [10] V.V. Peller, Estimates of Functions of Hilbert Space Operators, Similarity to a Contraction and Related Function Algebras, Lecture Notes in Math., vol. 1043, Springer-Verlag, Berlin, 1984, pp. 199–204.
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[11] S. Petrovi´c, Polynomially unbounded product of two polynomially bounded operators, Integral Equations Operator Theory 27 (1997) 473–477. [12] G. Pisier, A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc. 10 (1997) 351–369. [13] G. Pisier, Joint similarity problems and the generation of operator algebras with bounded length, Integral Equations Operator Theory 31 (1998) 353–370.
Journal of Functional Analysis 256 (2009) 2780–2814 www.elsevier.com/locate/jfa
A transfer principle for multivalued stochastic differential equations ✩ Jiagang Ren a,∗ , Siyan Xu b a School of Mathematics and Computational Science, Zhongshan University, Guangzhou,
Guangdong 510275, PR China b Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China
Received 21 July 2008; accepted 22 September 2008 Available online 10 October 2008 Communicated by Paul Malliavin
Abstract In this paper we prove a transfer principle for multivalued stochastic differential equations. © 2008 Elsevier Inc. All rights reserved. Keywords: Multivalued stochastic differential equation; Maximal monotone operator; Transfer principle
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . 3. Stroock–Varadhan approximation . . . . . 4. Weak solutions and martingale problems 5. Identification of the limit . . . . . . . . . . . 6. The transfer principle . . . . . . . . . . . . . 7. Uniformity in initial values . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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Supported by NSFC (Grant no. 10871215).
* Corresponding author.
E-mail addresses:
[email protected] (J. Ren),
[email protected] (S. Xu). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.09.016
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1. Introduction There have been intensive and extensive studies on the limit theorem for stochastic differential equations (see, e.g., [3,5,8,9,11,12]). According to [9, p. 205], the limit theorem makes it possible to transfer properties from ODE to SDE, and this transference method can provide simultaneously statements of stochastic propositions and their proofs. For example, the uniqueness and existence of solutions of SDE can be considered a direct consequence of the corresponding result for ODE combined with the limit theorem. Recently there has been an increasing interest in studying the following multivalued (Stratonovich) stochastic differential equation (MSDE in short):
dX(t) ∈ b X(t) dt + σ X(t) ◦ dw(t) − A X(t) dt, t ∈ R+ , X(0) = x0 ∈ D(A),
(1)
where ◦ denotes the Stratonovich integral, A is a maximal monotone operator, w(t) is a multidimensional standard Brownian motion, σ and b are maps from Rm to Rm and Rm × Rd , respectively (see [4,14] and references therein). The main difficulty in handling this equation comes from the high singularity of A, which is neither bounded nor continuous. The difficulty becomes even greater with the entering of the stochastic driving process, the Brownian motion. It is for this reason that the existence and uniqueness result is proved in [4] only under the additional assumption that the interior of the domain of A is non-empty (see also [14]), although its deterministic counterpart does not need this assumption. In this paper we aim at establishing a limit theorem for (1) and, as a by-product, give a new proof of Cépa’s existence and uniqueness result. 2. Preliminaries Given a multivalued operator A from Rm to Rm , define: D(A) := x ∈ Rm : A(x) = ∅ , Im(A) := A(x), x∈D(A)
Gr(A) := (x, y) ∈ R2m : x ∈ Rm , y ∈ A(x) . A−1 is defined by x ∈ A−1 (y) ⇔ y ∈ A(x). Definition 2.1. (See [1].) (1) A multivalued operator A is called monotone if y1 − y2 , x1 − x2 0,
∀(x1 , y1 ), (x2 , y2 ) ∈ Gr(A).
(2) A monotone operator A is called maximal monotone if (x1 , y1 ) ∈ Gr(A)
⇔
y1 − y2 , x1 − x2 0, ∀(x2 , y2 ) ∈ Gr(A) .
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We collect here some facts about the maximal monotone operator which will be needed in the sequel. For proofs we refer to [1]. Proposition 2.2. (1) For each x ∈ D(A), A(x) is a closed and convex subset of Rm . In particular, there is a unique y ∈ A(x) such that |y| = inf{|z|: z ∈ Ax}. A◦ (x) := y is called the minimal section of A, and we have x ∈ D(A)
⇔
◦ A (x) < +∞.
(2) The resolvent operator Jn := (1 + n1 A)−1 is single-valued and Lipschitz continuous with Lipschitz constant 1. Moreover, limn↑∞ Jn x = x for any x ∈ D(A). (3) The Yosida approximation An := n(1 − Jn ) is monotone and Lipschitz continuous with Lipschitz constant n. Moreover, as n ↑ ∞ An (x) → A◦ (x)
and An (x) ↑ A◦ (x)
if x ∈ D(A).
We will also need the following lemma. Lemma 2.3. If x ∈ / D(A), xn → x, then lim infAn (xn ) = ∞. n→∞
Proof. Suppose not. Then extracting a subsequence if necessary we may assume that {|An (xn )|} is bounded and lim An (xn ) = y.
n→∞
Hence lim
m,n→∞
xn − xm , An (xn ) − Am (xm ) = 0.
Thus by [2, Proposition 1.1] y ∈ A(x). In particular, x ∈ D(A).
2
The following two results are taken from [4]. Lemma 2.4. Let A be a multivalued maximal monotone operator, t → (X(t), K(t)) and t → (X (t), K (t)) be continuous functions with X(t), X (t) ∈ D(A), t → K(t), K (t) be of finite variation. Let (α, β) be continuous functions which satisfying α(t), β(t) ∈ Gr(A),
∀t 0.
If
X(t) − α(t), dK(t) − β(t) dt 0,
J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
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X (t) − α(t), dK (t) − β(t) dt 0,
then
X(t) − X (t), dK(t) − dK (t) 0.
Lemma 2.5. If Int(D(A)) = ∅, then there exists a ∈ Rm , γ > 0, μ 0, such that ∀x ∈ Rm , we have
x − a, An (x) γ An (x) − μ|x − a| − γ μ.
(2)
Let D be the space of all càdlàg functions from R+ to R. For f, g ∈ D define d(f, g) :=
∞
2
n=0
−n
n+1 1 ∧ f (t) − g(t) dt. n
Then (D, d) is Polish space and d is a metric corresponding to the Meyer–Zheng topology (see [7]) which is defined by the convergence in measure for the Lebesgue measure on R+ . The following tightness criterion is due to [10]. Theorem 2.6. Let {θn }n∈N∗ be a family of càdlàg increasing processes with θn (0) = 0. If for all t > 0 there exists a constant C(t) > 0 such that
sup E θn (t) C(t),
n∈N∗
then (θn )n∈N∗ is tight on D . W m := C(R+ , Rm ) will denote the space of continuous functions from R+ to Rm , equipped with compact uniform convergence topology. The following result, which links the convergence in W m and that in D , appears in [4], and is essentially proved in the language of Skorohod topology in [7]. Theorem 2.7. Suppose that for every n, xn ∈ W m , θn ∈ W 1 and θn is strictly increasing with θn (0) = 0. Set τn = θn−1 and yn (t) = xn (τn (t)). Suppose that there exists y ∈ W m , τ ∈ W 1 and θ ∈ D such that (yn , τn , θn ) → (y, τ, θ ) ∈ W m+1 × D,
n → ∞.
Define x(t) := y(θ (t)). Then a necessary and sufficient condition for (xn )n∈N ∗ to converge to x in W m is that y(s) = y(t) provided τ (s) = τ (t). Our basic probability space Ω will be the space of continuous functions from R+ to Rd . The generic point of Ω will be denote by ω and we set w(t, ω) := ω(t).
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Ω will be equipped with the compact uniform convergence topology and P will denote the standard Wiener measure under which t → w(t) is a standard Brownian motion starting from 0. Ft will denote the natural filtration on Ω: Ft := σ w(v), v < u P . u>t
Consider the multivalued Stratonovich stochastic differential equation (1) on Ω and recall the following definition due to [4]. Definition 2.8. A pair of continuous and Ft -adapted processes (X, K) is called a strong solution of (1) if (i) (ii) (iii) (iv)
X(0) = x0 and X(t) ∈ D(A) a.s.; K = {K(t), Ft ; t ∈ R+ } is of finite variation and K(0) = 0 a.s.; dX(t) = b(X(t)) dt + σ (X(t)) ◦ dw(t) − dK(t), t ∈ R+ , a.s.; for any continuous processes (α, β) satisfying α(t), β(t) ∈ Gr(A), ∀t ∈ R+ , the measure
X(t) − α(t), dK(t) − β(t) dt
is positive on R+ . Then we know by [4] that under the hypothesis (H) σ ∈ Cb2 , b ∈ Cb1 , (1) admits a unique solution which will be denoted by (X, K) in the sequel. For a function K of finite variation let |K|t denote its total variation on [0, t]. Then we have (see, e.g., [14]): Lemma 2.9. Let {Kn , n ∈ N} be a family of continuous functions of finite variation. Assume that (i) supn |Kn |t C(t) < ∞, ∀t; (ii) limn→∞ Kn = K ∈ W m . Then K is of finite variation. Moreover, if {fn , n ∈ N} is a family of continuous functions such that limn→∞ fn = f ∈ W m , then t lim
n→∞ s
fn (u), dKn (u) =
t
f (u), dK(u) ,
∀0 s t < +∞.
s
Throughout the paper, C denotes a positive constant whose value may vary from one place to another.
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3. Stroock–Varadhan approximation For every n, 0 t < ∞, consider the following ordinary differential equation:
X˙ n (t) = b Xn (t) + σ Xn (t) w˙ n (t) − An Xn (t) ,
(3)
Xn (0) = x0 ∈ D(A), where An is the Yosida approximation of A and
w˙ n (t) = 2n w tn+ − w(tn ) ,
tn+ =
[2n t] + 1 , 2n
tn =
[2n t] . 2n
Here [a] stands for the integer part of a. We denote the unique solution of (3) by Xn . Set t θn (t) :=
An Xn (u) du + t,
0
τn := θn−1 , t Kn (t) :=
An Xn (u) du,
0
t Mn (t) :=
σ Xn (u) w˙ n (u) du,
0
Yn (t) := Xn τn (t) , Hn (t) := Kn τn (t) , t wn (t) :=
w˙ n (u) du, 0
a (x) := ij
d
j
σki (x)σk (x),
k=1
αn (t) := σ Xn (tn ) w˙ n (t),
(σ σ )l,l i (x) :=
m ∂j σ il (x) σ j l (x), j =1
(Lf )(x) :=
m i=1
We now prove
d m m ∂ i 1 ij j σ (x) σ (x) ∂ f (x) + a (x)∂i ∂j f (x). bi (x) + i k ∂x j k 2 k=1 j =1
i,j =1
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Theorem 3.1. (τn , Hn , Mn , Yn , wn , θn )n∈N∗ is tight in W 3m+1 × Ω × D. Proof. The easy terms are τn , wn and Hn . In fact, we have, ∀0 s t < ∞, n ∈ N ∗ , τn (t) − τn (s) |t − s|,
(4)
τ n (t) Hn (t) − Hn (s) = An Xn (u) du τn (s)
t An (Yn (v)) = dv |An (Yn (v))| + 1 s
|t − s|,
(5)
2p
E wn (t) − wn (s) C|t − s|p .
(6)
and for all p 1,
The other terms are more delicate. First we look at θn . Let a be as in Lemma 2.5. By differential formula, ∀p 1, ∀0 s t < ∞, we have Xn (s) − a 2p = |x0 − a|2p s + 2p
Xn (u) − a 2p−2 Xn (u) − a, b Xn (u) du
0
s + 2p
Xn (u) − a 2p−2 Xn (u) − a, σ Xn (u) w˙ n (u) du
0
s − 2p
Xn (u) − a 2p−2 Xn (u) − a, An Xn (u) du.
0
Lemma 2.5 yields Xn (s) − a 2p C + Ct + C
s
Xn (u) − a 2p du
0
s +C
Xn (u) − a 2p−2 Xn (u) − a, σ Xn (u) w˙ n (u) du
0
s −C 0
Xn (u) − a 2p−2 An Xn (u) du,
(7)
J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
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where we have used the boundedness of b and Young’s inequality. Let s I := E sup f Xn (u) w˙ n (u) du , 0st
0
where f (Xn (u)) = |Xn (u) − a|2p−2 (Xn (u) − a)∗ σ (Xn (u)). Then s I E sup f Xn (un ) w˙ n (u) du 0st
0
s u ∂f (Xn (v)) + E sup w˙ n (u) dv du ∂v 0st
0 un
:= I1 + I2 .
(8)
Noticing that +
s
sn w˙ n (u) du =
0
ξu dw(u), 0
where u + n ∧s
f Xn (vn ) dv = 2n u+ n ∧ s − un f Xn (un ) .
ξu = 2 n un
We have by BDG inequality and the boundedness of σ tn+ I1 CE
1/2 |ξu | du 2
0
t CE
1/2 Xn (un ) − a 4p−2 du
0
CE
2p 1/2 sup Xn (s) − a 0st
t
1/2 2p−2 Xn (un ) − a du
0
2p 1 + CE E sup Xn (s) − a 2 0st
t 0
2p−2 Xn (un ) − a du
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2p 1 +C E sup Xn (s) − a 2 0st
t 2p ds + C(t), E sup Xn (u) − a 0
(9)
0us
where we have used the inequality |ab|
1 2 λa + λ−1 b2 2
(10)
for all constants a, b ∈ R and λ > 0. For I2 we have s u Xn (v) − a 2p−1 σ Xn (v) + Xn (v) − a 2p−2 σ Xn (v) I2 E sup 0st
0 un
× b Xn (v) + σ Xn (v) w˙ n (v) + An Xn (v) w˙ n (u) dv du s u 2p−1 E sup C Xn (v) − a + C C + C w˙ n (v) + An Xn (v) w˙ n (u) dv du 0st
0 un
t u CE
t u w˙ n (u) dv du + CE w˙ n (u)2 dv du
0 un
t u + CE
0 un
An Xn (v) w˙ n (u) dv du
0 un
t u + CE
2p−1 Xn (v) − a w˙ n (u) dv du
0 un
t u + CE
Xn (v) − a 2p−1 w˙ n (u)2 dv du
0 un
t u + CE
2p−1 Xn (v) − a An Xn (v) w˙ n (u) dv du
0 un
:= I21 + I22 + I23 + I24 + I25 + I26 .
(11)
It is easily seen that n
I21 C(t)2− 2
(12)
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2789
and I22 C(t).
(13)
Since |An (Xn (v))| n(1 + |Xn (v)|), we have I23 C(t)n2
− n2
t u + CnE
Xn (v) − a w˙ n (u) dv du
0 un
C(t)n2
− n2
t
u
E Xn (v) − a w˙ n (u) dv du
+ Cn 0 un
C(t)n2
− n2
t u + Cn
2p−1 2p 2p 1 2p 2p 2p−1 E w˙ n (u) E Xn (v) − a dv du
0 un
C(t)n2
− n2
+ Cn2
n 2
t u
2p 1 EXn (v) − a 2p dv du
0 un
C(t)n2
− n2
+ Cn2
− n2
t 2p du. E sup Xn (v) − a 0
(14)
0vu
Moreover, we have t u
2p−1
w˙ n (u) dv du E Xn (v) − a
I24 = C 0 un
t u C
2p 2p−1 2p 1 2p EXn (v) − a Ew˙ n (u) 2p dv du
0 un
C2
− n2
t 0
2p
n E sup Xn (v) − a du + C(t)2− 2 0vu
and t u I25 = C
2p−1
w˙ n (u)2 dv du E Xn (v) − a
0 un
t u C 0 un
2p 2p−1 4p 1 2p 2p dv du E Xn (v) − a E w˙ n (u)
(15)
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t 2p du + C(t). C E sup Xn (v) − a
(16)
0vu
0
Furthermore, t u I26 CnE
2p−1 Xn (v) − a C + C Xn (v) − a w˙ n (u) dv du
0 un
t u CnE
2p−1 Xn (v) − a w˙ n (u) dv du
0 un
t u + CnE
2p Xn (v) − a w˙ n (u) dv du
0 un n
Cn2− 2
t E
0vu
0
+ Cn2
−n
2p n du + C(t)n2− 2 sup Xn (v) − a
t
2p
E Xn (s) − a w˙ n (sn ) ds.
(17)
0
On the other hand, s 2p 2p 2p Xn (s) − a C Xn (sn ) − a + C b Xn (u) du sn
s 2p s 2p + C σ Xn (u) w˙ n (u) du + C An Xn (u) du sn
sn
2p 2p C Xn (sn ) − a + C2−2np + C2−2np w˙ n (sn ) 2p −2np
+ Cn 2
2p −n(2p−1)
s
+ Cn 2
Xn (u) − a 2p du.
sn
Hence it follows by Gronwall’s inequality, Xn (s) − a 2p exp Cn2p 2−2np C Xn (sn ) − a 2p 2p + Cn2p 2−2np + C2−2np w˙ n (sn ) . By the independence of w˙ n (sn ) and Xsn ,
(18)
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2p 2p
E Xn (s) − a w˙ n (sn ) C exp Cn2p 2−2np E Xn (sn ) − a w˙ n (sn )
+ C exp Cn2p 2−2np n2p 2−2np E w˙ n (sn ) 2p+1
+ C exp Cn2p 2−2np 2−2np E w˙ n (sn ) 2p C exp Cn2p 2−2np E Xn (sn ) − a E w˙ n (sn ) + C 2p n C2 2 E Xn (sn ) − a +C and 2p 2p 2p 2p
E Xn (s) − a w˙ n (sn ) C exp Cn2p 2−2np E Xn (sn ) − a w˙ n (sn ) 2p
+ C exp Cn2p 2−2np n2p 2−2np E w˙ n (sn ) 4p
+ C exp Cn2p 2−2np 2−2np E w˙ n (sn ) 2p 2p C exp Cn2p 2−2np E Xn (sn ) − a E w˙ n (sn ) +C
2p C2np E Xn (sn ) − a + C. So
n2
−n
t
2p
n E Xn (s) − a w˙ n (sn ) ds Cn2− 2
0
t 2p du + C(t). E sup Xn (v) − a 0
0vu
Therefore t 2p du + C(t). I26 C E sup Xn (v) − a 0
(19)
0vu
Combining (8), (9), (11)–(16), (19) gives t 2p 2p 1 I C E sup Xn (v) − a du + E sup Xn (s) − a + C(t). 2 0st 0vu 0
Hence E
2p sup Xn (s) − a C(t) + C 0st
t 2p du. E sup Xn (v) − a 0
0vu
By Gronwall’s inequality, E
2p C(t)eCt . sup Xn (s) − a 0st
(20)
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J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
In particular, taking p = 1, we obtain by (7) t E
An Xn (u) du C(t)eCt .
(21)
0
That is
sup E θn (t) C(t) < ∞,
n∈N∗
0 t < ∞.
(22)
Therefore, in virtue of Theorem 2.6, θn (t) is tight. Besides, we also have 2p t 2p
= E σ Xn (u) w˙ n (u) du E Mn (t) − Mn (s) s
2p t CE σ Xn (un ) w˙ n (u) du s
2p t + CE σ Xn (u) − σ Xn (un ) w˙ n (u) du s
2p t u j l ∂ il C(t − s) + CE σ Xn (v) b Xn (v) w˙ n (u) dv du ∂xj p
s un
2p t u l l,l l + CE (σ σ ) Xn (v) w˙ n (u)w˙ n (u) dv du s un
2p t u j l ∂ il + CE σ Xn (v) An Xn (v) w˙ n (u) dv du ∂xj s un
C(t − s) + C(t − s)2p 2p t u 2p−1 + C(t − s) E du An Xn (v) w˙ n (u) dv p
s
un
C(t − s)p + C(t − s)2p + Cn2p 2−2np (t − s)2p + Cn2p 2−n(2p−1) t u 2p 2p 2p−1 w˙ n (u) dv × (t − s) E du Xn (v) − a s
un
C(t − s)p + C(t − s)2p + Cn2p 2−2np (t − s)2p
J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
+ Cn2p 2−2np (t − s)2p−1 E
t
2793
2p 2p Xn (u) − a w˙ n (un ) du
s
C(t − s) + C(t − s)
2p
p
2p −2np
+ Cn 2
+ Cn2p 2−2np (t − s)2p t
(t − s)
2p−1
2p 2p
E Xn (u) − a w˙ n (un ) du.
s
By (20), 2p
E Mn (t) − Mn (s) C(t − s)p .
(23)
Furthermore, 2p
2p E Yn (t) − Yn (s) = E Xn τn (t) − Xn τn (s) 2p τ n (t) 2p
CE b Xn (u) du + CE Mn τn (t) − Mn τn (s) τn (s)
2p
+ CE Hn (t) − Hn (s) 2p
+ C(t − s)2p CE τn (t) − τn (s)
2p + CE Mn τn (t) − Mn τn (s) . Using |τn (t) − τn (s)| |t − s|, similarly to (23), we have 2p τ n (t) 2p
=E σ Xn (u) w˙ n (u) du E Mn τn (t) − Mn τn (s) τn (s)
2p τ n (t) CE σ Xn (un ) w˙ n (u) du τn (s)
2p τ n (t) + CE σ Xn (u) − σ Xn (un ) w˙ n (u) du τn (s)
C(t − s)p .
(24)
Hence 2p
E Yn (t) − Yn (s) C(t − s)p . Combining (4)–(6), (22), (23), (25) gives the desired tightness by Aldous’s theorem (see [6]).
(25) 2
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Denote by {Ln , n ∈ N ∗ } the distribution of (τn , Hn , Mn , Yn , wn , θn ) on W 3m+1 × Ω × D. Since W 3m+1 × Ω × D is a Polish space, by Prokhorov’s theorem and Proposition 3.1, there exist Lnk and probability L on W 3m+1 × Ω × D such that Lnk → L (k → ∞). To simplify the notation, we suppose that Ln → L (n → ∞). By Skorohod’s representation theorem, there exists ˆ Fˆ , Pˆ ) on which are defined random variables (τˆn , Hˆ n , Mˆ n , Yˆn , wˆ n , θˆn ), a probability space (Ω, ˆ ˆ ˆ ˆ (τˆ , H , M, Y , w, ˆ θ ) such that (τˆn , Hˆ n , Mˆ n , Yˆn , wˆ n , θˆn ) ∼ (τn , Hn , Mn , Yn , wn , θn ),
(26)
ˆ ˆ ˆ ˆ θ) ˆ = L, Pˆ (τˆ ,H ,M,Y ,w,
and as n → ∞, ˆ Yˆ , w, ˆ θˆ ) a.s. (τˆn , Hˆ n , Mˆ n , Yˆn , wˆ n , θˆn ) → (τˆ , Hˆ , M,
(27)
in W 3m+1 × Ω × D. Define Xˆ n (t) := Yˆn θˆn (t) ,
ˆ X(t) := Yˆ θˆ (t) ,
Kˆ n (t) := Hˆ n θˆn (t) ,
ˆ := Hˆ θˆ (t) . (28) K(t)
We now pass from the convergence of Yˆn to that of Xˆ n . According to Cépa in [4]. First, note that according to Theorem 2.7, this will be done if we prove the following Theorem 3.2. There exists Ωˆ 0 ∈ Fˆ , Pˆ (Ωˆ 0 ) = 1, such that for all ωˆ ∈ Ωˆ 0 , if there exist 0 s t < ∞ satisfying τˆ (s)(ω) ˆ = τˆ (t)(ω), ˆ then Yˆ (s)(ω) ˆ = Yˆ (t)(ω). ˆ Proof. We divide the proof into several steps. (A) We prove that there exists an Nˆ 1 with Pˆ (Nˆ 1 ) = 0 such that ˆ ω) ∈ D(A), X(t,
∀(t, ω) ∈ R+ × Nˆ 1c .
Since Xˆ is càdlàg, it is sufficient to prove that for every t, ˆ Pˆ X(t) ∈ D(A) = 1. Suppose not. Then there exists a t0 > 0 and Bˆ 0 ∈ Fˆ , such that Pˆ (Bˆ 0 ) > 0 and Xˆ t0 (ω) ˆ ∈ / D(A) for every ωˆ ∈ Bˆ 0 . Since Xˆ is right continuous, there exist δ > 0, Bˆ 1 ∈ Fˆ and Pˆ (Bˆ 1 ) > 0, such that Xˆ t (ω) ˆ ∈ / D(A) for every ωˆ ∈ Bˆ 1 and t ∈ [t0 ; t0 + δ]. By (21) and (26), ∀n ∈ N∗ , t 0 +δ
An Xˆ n (u) du C,
E t0
by Fatou’s lemma,
J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
t 0 +δ lim infAn Xˆ n (u) du d Pˆ C. n→∞
2795
(29)
Bˆ 1 t0
This means that lim infAn Xˆ n (u) < ∞, n
du d Pˆ -a.e. on Bˆ 1 × [t0 ; t0 + δ]. But by Lemma 2.3, lim infAn Xˆ n (u) = ∞ n→∞
on this set, a contradiction. (B) We prove that there exists Nˆ 2 ∈ Fˆ , Pˆ (Nˆ 2 ) = 0 such that ∀ωˆ ∈ Nˆ 2c , if there exists 0 s t < ∞ such that τˆ (s) = τˆ (t), then ∀x ∈ D(A), t
Yˆ (u) − x, d Hˆ (u) 0.
s
In fact, by (27), there exists Nˆ 2 ∈ Fˆ , Pˆ (Nˆ 2 ) = 0, such that ∀ω ∈ Ωˆ 1 := Nˆ 2c , ˆ Yˆ , w, (τˆn , Hˆ n , Mˆ n , Yˆn , wˆ n , θˆn ) → (τˆ , Hˆ , M, ˆ θˆ ), n → ∞. By pass to the limit, it suffices to prove (B) for x ∈ D(A). By (27) and (28), we have τ ˆn (t)
Xˆ n (u) − x, d Kˆ n (u) − An (x) du 0.
τˆn (s)
Let u = τˆn (v). By (28) and θˆn = τˆn−1 , we have t
Yˆn (u) − x, d Hˆ n (u)
s
t
Yˆn (u) − x, An (x) d τˆn (u).
s
Let n → ∞, by τˆ (s) = τˆ (t) t
Yˆn (u) − x, An (x) d τˆn (u) C sup Yˆn (u) + |x| A◦ (x)τˆn (t) − τˆn (s) 0ut s
Hence
C τˆn (t) − τˆn (s) → 0.
(30)
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J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
t
Yˆ (u) − x, d Hˆ (u) 0.
s
(C) We prove that there exists Ωˆ 2 ∈ Fˆ , Pˆ (Ωˆ 2 ) = 1 such that ∀ωˆ ∈ Ωˆ 2 and 0 s < t < ∞, if τˆ (s)(ω) ˆ = τˆ (t)(ω), ˆ then Yˆ (t)(ω) ˆ − Yˆ (s)(ω) ˆ = Hˆ (s)(ω) ˆ − Hˆ (t)(ω). ˆ Denote by Ωˆ (n) the subset of Ωˆ with Pˆ (Ωˆ (n) ) = 1 satisfying differential equation ˙ˆ n (t) dt − d Kˆ n (t), d Xˆ n (t) = b Xˆ n (t) dt + σ Xˆ n (t) w and let Ωˆ 2 = Ωˆ 1 ∩ ( n∈N ∗ Ωˆ (n) ) where Ωˆ 1 is defined in step (B). Then Pˆ (Ωˆ 2 ) = 1 and we have for all n ∈ N ∗ , 0 s < t < ∞ and ωˆ ∈ Ωˆ 2 : Yˆn (t) − Yˆn (s) =
τ ˆn (t)
b Xˆ n (u) du +
τˆn (s)
τ ˆn (t)
˙ˆ n (u) du − Hˆ n (t) − Hˆ n (s) σ Xˆ n (u) w
τˆn (s)
τ ˆn (t)
=
b Xˆ n (u) du + Mˆ n τˆn (t) − Mˆ n τˆn (s) − Hˆ n (t) − Hˆ n (s) .
τˆn (s)
Since τˆ (t) = τˆ (s) and τ ˆn (t) τ ˆn (t) b Xˆ n (u) du b Xˆ n (u) du τˆn (s)
τˆn (s)
C τˆn (t) − τˆn (s)
sup
1 + Xˆ n (u)
0uτˆn (t)
C τˆn (t) − τˆn (s) sup 1 + Yˆn (u) 0ut
C τˆn (t) − τˆn (s) , τˆ (t) we have τˆnn(s) b(Xˆ n (u)) du → 0 (n → ∞). By (27),
Mˆ n τˆn (t) − Mˆ n τˆn (s) → Mˆ τˆ (t) − Mˆ τˆ (s)
(n → ∞).
As τˆ (t) = τˆ (s), Mˆ n τˆn (t) − Mˆ n τˆn (s) → 0 (n → ∞). Therefore, if n → ∞ and τˆ (t) = τˆ (s), we have Yˆ (t) − Yˆ (s) = Hˆ (s) − Hˆ (t).
J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
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(D) We prove that Pˆ Yˆ (t) ∈ D(A); 0 t < ∞ = 1. Since Yˆ is continuous, it is sufficient to prove that Pˆ (Yˆ (t) ∈ D(A)) = 1 for all 0 t < ∞. ˆ > 0 and Yˆt0 (ω) ˆ ∈ / D(A) for Suppose that there exist 0 < t0 < ∞ and Bˆ ∈ Fˆ , such that Pˆ (B) ˆ Let S := sup{0 u t0 : Yˆ (u) ∈ D(A)}. Obviously Yˆ (0) ∈ D(A) and Yˆ (S) ∈ every ωˆ ∈ B. D(A) a.s. Then step (A) and the fact Yˆ (s) ∈ / D(A) for s ∈ ]S; t0 ], ωˆ ∈ Bˆ force τˆ (S) = τˆ (t0 ) ˆ Applying the step (B) on Ωˆ 2 ∩ B, ˆ we obtain on B. 2 1 ˆ Y (t0 ) − Yˆ (S) = 2
2 1 ˆ H (t0 ) − Hˆ (S) 2 t0
= Hˆ (u) − Hˆ (S), d Hˆ (u) S
t0 =−
Yˆ (u) − Yˆ (S), d Hˆ (u)
S
0. ˆ Hence Yˆ (t0 ) = Yˆ (S), which is impossible on B. ˆ (E) Finally we can finish the proof. Let Ω0 = Ωˆ 2 ∩ {Yˆ (t) ∈ D(A); t ∈ R+ }. Then Pˆ (Ωˆ 0 ) = 1. If τˆ (t) = τˆ (s), then 2 1 2 1 ˆ Y (t) − Yˆ (s) = Hˆ (t) − Hˆ (s) 2 2 t
= Hˆ (u) − Hˆ (s), d Hˆ (u) s
t =−
Yˆ (u) − Yˆ (s), d Yˆ (u)
s
0, ˆ = K(s). ˆ so Yˆ (s) = Yˆ (t) and moreover K(t) The proof is complete.
2
By Theorems 2.7 and 3.2, there is a subsequence {nk } such that ˆ K) ˆ (Xˆ nk , Kˆ nk ) → (X, in W 2m a.s.
(31)
Consequently, the image measure of (Xˆ nk , Kˆ nk ) has a weak limit μ in W 2m . Denote by μn the law of (Xn , Kn ). Since (Xˆ n , Kˆ n ) and (Xn , Kn ) are identically distributed, μ is a weak limit of {μn } in W 2m .
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J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
Starting from the very beginning with an arbitrary subsequence and repeating the above reasoning, we know that any subsequence has a weakly convergent sub-subsequence and we thus arrive at the following result. Theorem 3.3. {μn } is relatively compact in W 2m . Next we shall prove that the whole sequence {μn } converges weakly to a unique limit and we shall identify this limit. To do these, however, we need first to introduce the notions of weak solution and martingale problem associated to (1). 4. Weak solutions and martingale problems In this section we shall consider the following Itô MSDE
dX(t) ∈ σ (t, X) dw(t) + b(t, X) dt − A X(t) dt,
(32)
X(0) = x0 ∈ D(A),
where b : R+ × W m → Rm and σ : R+ × W m → Rm × Rd are B(R+ ) × B(W m )-measurable, and ∀t, b(t, ·) and σ (t, ·) are Bt (W m )-measurable. Parallel to the SDE case, we introduce the following definition. Definition 4.1. If there exists a filtered probability space (Ω, F , P , (Ft )t0 ) on which defined two continuous adapted processes X, K and a d-dimensional Brownian motion w such that (1) b(t, X· (ω)) ∈ L1loc (Rm ), σ (t, X· (ω)) ∈ L2loc (Rm ⊗ Rd ); (2) K is of finite variation and K(0) = 0; (3) For all t 0, t X(t) = x0 +
t σ (s, X) dw(s) +
0
b(s, X) ds − K(t) 0
(4) For every pair of Rm -valued continuous adapted process α, β verifying α(u), β(u) ∈ Gr(A), the measure
X(u) − α(u), dK(u) − β(u) du
is positive on R+ . Then we say that (32) admits a weak solution. Set a := σ σ ∗ .
a.s.;
J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
2799
For f ∈ Cb2 (Rm ), t ∈ R+ , w ∈ W m , set (Lf )(t, w) :=
m i=1
m 1 b (t, w)∂i f w(t) + a ij (t, w)∂i ∂j f w(t) . 2 i
i,j =1
The following result connects the weak solution to the so-called martingale problem. Theorem 4.2. A necessary and sufficient condition for the condition (3) in Definition 4.1 to admit a weak solution to (32) is that there exist a filtered probability space (Ω, F , P , (Ft )t0 ), an mdimensional continuous adapted process X and an m-dimensional continuous adapted process K of finite variation such that for all f ∈ Cb2 (Rm ) f Mt
:= f X(t) − f X(0) −
t
t (Lf )(s, X) ds +
0
∇f X(s) , dK(s)
0
is a continuous local martingale and the condition (4) in the above Definition 4.1 is satisfied. Proof. Let (X, K, w) be a weak solution to (32). By Itô’s formula, ∀f ∈ Cb2 (Rm ), t ∈ R+ , we have m f X(t) = f X(0) +
t
i=1 0
m t 1 i ∂i f X(s) dX (s) + ∂i ∂j f X(s) d M i , M j s 2
a.s.,
i,j =1 0
t where M(t) = 0 σ (s, X) dw(s). t Since [M i , M j ]t = 0 a ij (s, X) ds, we have f Mt
= f X(t) − f X(0) −
t
t (Lf )(s, X) ds +
0
=
m d t
∇f X(s) , dK(s)
0
σki (s, X)∂i f X(s) dw k (s)
a.s.
i=1 k=1 0 f
Hence {Mt , t ∈ R+ } is a continuous local martingale. On the other hand, if there exists (X, K) such that ∀f ∈ Cb2 (Rm ), M f is a continuous local martingale. ∀n ∈ N, let κn := inf{t; |X(t)| > n}. For i = 1, 2, . . . , m, choose f ∈ Cb2 (Rm ) such that f (x) = x i when |x| n. Then κ n ∧t
Mni (t) = X i (κn
∧ t) − X (0) −
dK i (s) ∈ M2c .
i
0
Since κn ↑ ∞ a.s., we have
κ n ∧t
b (s, X) ds +
i
0
2800
J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
t
1 b (s, X) ds − 2
M (t) = X (t) − X (0) − i
i
i
t
0
t
(σ σ )i (s, X) ds +
i
0
dK i (s) ∈ Mcloc . (33) 0
Similarly, for i, j = 1, 2, . . . , m, choosing f ∈ Cb2 (Rm ) such that f (x) = x i x j when |x| n, we see that t X (t)X (t) − X (0)X (0) − i
j
i
t b (s, X)X (s) ds −
j
i
j
0
t +
0
t X j (s) dK i (s) +
0
bj (s, X)X i (s) ds
t X i (s) dK j (s) −
0
a ij (s, X) ds ∈ Mcloc .
(34)
0
By (33), (34) and Itô’s formula, we have
i M , Mj =
t a ij (s, X) ds
a.s., t ∈ R+ , i, j = 1, . . . , m.
0
˜ of (Ω, F , P , F) and a d-dimension F˜ Brownian ˜ F˜ , P˜ , F) Hence there exists an extension (Ω, motion w, such that t M(t) =
σ (s, X) dw(s)
a.s., t ∈ R+ .
0
Therefore, t X(t) = X(0) +
t b(s, X) ds +
0
σ (s, X) dw(s) − K(t)
a.s.
0
Consequently (X, K, w) is a weak solution of Eq. (32).
2
Let V m := V : R+ → Rm , V (0) = 0, V is continuous and of finite variation on compacts . We have Proposition 4.3. V m is a Borel measurable set of W m and μ W m × V m = 1.
J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
2801
Proof. Set VTm,l := V ∈ V m : |V |T l . Then Vm=
∞ ∞
VTm,l .
T =1 l=1
Trivially each VTm,l is closed and thus Borel measurable and so is V m .
2
We endow V m with the filtration Bt V m := σ fA , A ∈ Bt (R+ ) , where fA is a map V m → Rm defined by fA (V ) := V (A). Theorem 4.4. A necessary and sufficient condition for the condition (3) in Definition 4.1 to admit a weak solution to (32) is that there exists a probability μ on (W m × V m , B(W m ) × B(V m ), (Bt (W m ) × Bt (V m ))t0 ) such that for all f ∈ Cb2 (Rm ), f Nt
:= f x(t) − f x(0) −
t
t Lf (s, x) ds +
0
∇f x(s) , dv(s)
0
is a μ-continuous local martingale. Here (x, v) denotes the generic point of W m × V m . Proof. If there exist (Ω, F , P , F) and (X, K) such that ∀f ∈ Cb2 (Rm ), M f be a continuous local martingale. Consider image (X., K.) : Ω → (W m , V m ) and μ(X,K) = P ◦ (X, K)−1 : the disf tribution of (X., K.) on the (W m × V m , B(W m ) × B(V m )). Then Nt is a Bt (W m ) × Bt (V m ) f f local martingale. On the other hand, if Nt is continuous local martingale, obviously Mt is P continuous local martingale. By Theorem 4.2, condition for (3) in Definition 4.1 holds. 2 5. Identification of the limit By Theorem 3.3, {μn } has a weak limit. We now prove: Theorem 5.1. Suppose that μ is a weak limit of {μn }. Then, under μ, f x(t) − f x(s) + is a martingale for all f ∈ Cb2 .
t
t
s
s
∇f x(u) , dv(u) −
Lf x(u) du
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Proof. By a density argument, it suffices to prove that t
μ E F · f x(t) − f x(s) = E F · Lf x(u) − ∇f x(u) , dv(u) du μ
s
for all f ∈ C0∞ (Rm ), 0 s < t, and bounded Bs (W m ) × Bs (V m ) measurable F : Ω → R. Clearly, it will suffice to do this when s and t have the form k/2N and F is bounded continuous, and Bs (W m ) × Bs (V m ) measurable. Observe that E
μn
F · f x(t) − f x(s) +
t
∇f x(u) , dv(u)
s
t
= Eμn F ·
t
∇x f x(u) , b x(u) du + EP F · ∇x f Xn (u) , αn (u) du
s
s
t
+ EP F ·
∇x f Xn (u) , M˙ n (u) − αn (u) du
s
t +E
P
∇x f Xn (u) , An Xn (u) du
s
= J1,n + J2,n + J3,n + J4,n . Obviously, J1,n → E
μ
t
F·
∇x f x(u) , b x(u) du ,
(35)
s
and t J4,n → E
P
∇x f X(u) , dK(u)
s
= E
t
F·
μ
∇x f x(u) , dv(u) .
(36)
s
Now we prove J2,n → E
μ
t F·
L0u f s
x(u) du ,
(37)
J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
2803
where 1 L0u = a ij (x) ∂ 2 /∂xi ∂xj . 2 Let H (x) denote the Hessian matrix of f . Since
EP αn (u)Bun W m × Bun V m = 0, we have J2,n = E
t F·
P
∇x f Xn (un ) , αn (u) du
s
+E
t F·
P
∇x f Xn (u) − ∇x f Xn (un ) , αn (u) du
s
=E
t F·
P
∇x f Xn (u) − ∇x f Xn (un ) , αn (u) du
s
t
= EP F ·
u du
s
t
+E
un
+ EP F ·
u du
s
un
t
u
F·
P
dv M˙ n (v), H Xn (v) αn (u)
du s
dv b Xn (v) , H Xn (v) αn (u)
dv −An Xn (v) , H Xn (v) αn (u)
un
= K1,n + K2,n + K3,n . Clearly, |K2,n | → 0, |K3,n | → 0 and K1,n = E
P
t F·
u du
s
+E
+E
P
dv αn (v), H Xn (v) αn (u)
un
t F·
P
u du
v dv
s
un
vn
t
u
v
F·
du s
dv un
j l ∂ il dr σ Xn (r) b Xn (r) w˙ n , H Xn (v) αn (u) ∂xj
l,l
dr (σ σ ) vn
Xn (r) w˙ nl w˙ nl , H Xn (v) αn (u)
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J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
t
u
− EP F ·
du s
v dv
un
vn
j l ∂ il dr σ Xn (r) An Xn (v) w˙ n , H Xn (v) αn (u) ∂xj
= K4,n + K5,n + K6,n + K7,n . Again, we have |K5,n | → 0, |K6,n | → 0, |K7,n | → 0 and that K4,n = E
t
u
F·
P
du s
+E
dv αn (v), H (Xn (un )αn (u)
un
t F·
P
u du
s
dv αn (v), (H Xn (v) − H Xn (un ) αn (u)
un
= K8,n + K9,n . Since |K9,n | → 0, it remains to examine K8,n . K8,n = 2 E
t F·
n P
du s
=2 E
∗ dv tr σ Xn (vn ) H Xn (un ) σ Xn (un )
un
t F·
n μn
u
u du
s
∗ dv tr σ x(vn ) H x(un ) σ x(un ) .
un
Since for bounded measurable functions ϕ and ψ on [s, t], t 2
n
u ϕ(u) du
s
1 ψ(v) dv → 2
un
t ϕ(u)ψ(u) du s
(see [12, Lemma 4.2]), using μn → μ, t
∗ 1 μ K8,n → E F · tr σ x(u) H x(u) σ x(u) du , 2 s
(37) is proved. Finally we treat J3,n and will prove: t
1 μ J3,n → E F · ∇x f x(u) , σ σ x(u) du . 2 s
We write
(38)
J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
t u
J3,n = EP F ·
2805
dv ∇x f Xn (u) , (σ σ )l,l Xn (v) w˙ nl w˙ nl
s un
+E
t u
F·
P
s un
t u
− EP F · s un
∂ il j l dv ∇x f Xn (u) , σ Xn (v) b Xn (v) w˙ n ∂xj ∂ il j dv ∇x f Xn (u) , σ Xn (v) An (v)w˙ nl ∂xj
= L1,n + L2,n + L3,n . Clearly, |L2,n | → 0 and |L3,n | → 0. Moreover,
t
L1,n = EP F ·
∇x f Xn (un ) , (σ σ )l,l Xn (un ) (u − un )w˙ nl w˙ nl
s
+E
t F·
P
+E
P
u du
s
un
t
u
F·
du s
dv ∇x f Xn (un ) , (σ σ )l,l Xn (v) − (σ σ )l,l Xn (un ) w˙ nl w˙ nl
dv ∇x f Xn (u) − ∇x f Xn (un ) , (σ σ )l,l Xn (v) w˙ nl w˙ nl
un
= L4,n + L5,n + L6,n . By |L5,n | → 0, |L6,n | → 0 and L4,n → 12 Eμ [F · the proof is completed. 2
t s
∇x f (x(u)), σ σ (x(u)) du], we get (38) and
Proposition 5.2. If (α, β) are continuous functions satisfying α(t), β(t) ∈ Gr(A),
∀t ∈ R+ ,
then the measure
x(t) − α(t), dv(t) − β(t) dt
is positive on R+ , μ-a.s. The proof can be done in an similar way to [4], so we leave the detail to the reader.
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J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
Now, instead of (1) we consider the following system ⎧ dX i (t) + Ai X(t) dt σ ij X(t) ◦ dw j (t) + bi X(t) dt, i = 1, 2, . . . , m, ⎪ ⎪ ⎪ ⎨ dX m+j (t) = dw j (t), ⎪ X i (0) = x0i , ⎪ ⎪ ⎩ X m+j (0) = 0.
(39)
Denote by νn the law of (wn , Xn , Kn ) in Ω × W m × V m . Then νn has a weak limit ν. Applying Theorem 5.1 and Proposition 5.2 to the above system we obtain Theorem 5.3. On (Ω × W m × V m , ν), t → w(t) is a Brownian motion and (X, K) is solution of the following multivalued Stratonovich SDE:
dX(t) + A X(t) dt b X(t) dt + σ X(t) ◦ dw(t), X(0) = x0 ∈ D(A).
(40)
It then follows from the uniqueness (in distribution) of (1) that the following corollary holds. Corollary 5.4. {νn } has a unique weak limit and thus the whole sequence converges. We shall denote this weak limit by ν. 6. The transfer principle Denote λn the law of (w, Xn , Kn ) in Ω × W m × V m . It is easily seen from the uniform convergence of wn to w that Theorem 6.1. w
λn −→ ν. Now can state the main results of the paper. Theorem 6.2. (Xn , Kn ) converges in W 2m to (X, K) in probability. Proof. We use an argument of Bismut [3]. Set Ω := Ω × W 2m . By the theorem above, for all bounded continuous function H on Ω ,
lim
n→∞ Ω
H (ω, x, v) dλn =
H (ω, x, v) dλ. Ω
J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
2807
That is lim
n→∞
H ω, Xn (ω), Kn (ω) dP =
Ω
H ω, X(ω), K(ω) dP .
Ω
In particular, if Ψ is a bounded continuous function on W 2m , Φ is a bounded continuous function on Ω, then lim
n→∞
Ψ Xn (ω), Kn (ω) Φ(ω) dP =
Ω
Ψ X(ω), K(ω) Φ(ω) dP .
Ω
Since the continuous functions on Ω are dense in L1 (Ω, P ), this convergence still holds for Φ ∈ L1 (Ω, P ). Therefore, taking Φ(ω) := Ψ (X(ω), K(ω)) gives lim
n→∞
Ψ Xn (ω), Kn (ω) Ψ X(ω), K(ω) dP =
Ω
Ψ 2 X(ω), K(ω) dP .
Ω
Consequently lim
n→∞
Ψ Xn (ω), Kn (ω) − Ψ X(ω), K(ω) 2 dP
Ω
=
Ψ 2 Xn (ω), Kn (ω) dP +
Ω
−2
Ψ 2 X(ω), K(ω) dP
Ω
Ψ Xn (ω), Kn (ω) Ψ X(ω), K(ω) dP
Ω
= 0. Hence for any bounded continuous function Ψ on W 2m , Ψ (Xn (ω), Kn (ω)) converge to Ψ (X(ω), K(ω)) in probability. Since W 2m is a Polish space, its topology can be defined by a countable family {Ψm } of bounded continuous functions on it. Consequently (Xn (ω), Kn (ω)) converges to (X(ω), K(ω)) in W 2m in probability. 2 7. Uniformity in initial values To stress the dependence of the solution on the initial values, we use (X(s, x), K(s, x)) and (Xn (s, x), Kn (s, x)) to denote the unique solutions of (1) and (3), respectively, with the initial value x. Theorem 7.1. For all T > 0 and all p 2, 2p C|x − y|2p , E sup X(t, x) − X(t, y) tT
(41)
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J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
2p C|x − y|2p , E sup K(t, x) − K(t, y)
(42)
tT
2p C|x − y|2p , E sup Xn (t, x) − Xn (t, y)
(43)
tT
2p C|x − y|2p . E sup Kn (t, x) − Kn (t, y)
(44)
tT
Proof. The first two inequalities are proved easily by using Itô calculus. Now we prove the third one. To simplify notations we omit the index n in Xn (t, x) and set σ (s, x) := σ Xn (s, x) ,
A(s, x) := An Xn (s, x) .
b(s, x) := b Xn (s, x) ,
Then we have t X(t, x) − X(t, y) = x − y +
σ (s, x) − σ (s, y) w˙ n (s) ds
0
t +
b(s, x) − b(s, y) ds −
0
t
A(s, x) − A(s, y) ds.
0
Using the monotonicity of An we obtain X(t, x) − X(t, y)2p = |x − y|2p t + 2p
X(s, x) − X(s, y)2p−2 X(s, x) − X(s, y), σ (s, x) − σ (s, y) w˙ n (s) ds
0
t + 2p
X(s, x) − X(s, y)2p−2 X(s, x) − X(s, y), b(s, x) − b(s, y) ds
0
t − 2p
X(s, x) − X(s, y)2p−2 X(s, x) − X(s, y), A(s, x) − A(s, y) ds
0
|x − y|2p t + 2p
X(s, x) − X(s, y)2p−2 X(s, x) − X(s, y), σ (s, x) − σ (s, y) w˙ n (s) ds
0
t + 2p 0
X(s, x) − X(s, y)2p−2 X(s, x) − X(s, y), b(s, x) − b(s, y) ds
(45)
J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
=: |x − y|2p + I1 (t) + I2 (t).
2809
(46)
Hence 2p |x − y|2p + E sup I1 (t) + E sup I2 (t) . E sup X(t, x) − X(t, y) tT
tT
(47)
tT
Set t αt :=
X(s, x) − X(s, y)2p−2 X(s, x) − X(s, y), σ (s, x) − σ (s, y) w˙ n (s) ds.
0
Then t αt =
t g(sn )w˙ n (s) ds +
0
s ds
0
g (u)w˙ n (u) du
sn
=: αt1 + αt2 ,
(48)
where 2p−2 ∗ g(u) := X(u, x) − X(u, y) X(u, x) − X(u, y) σ (u, x) − σ (u, y) . Since +
tn αt1 =
β(s) dw(s), 0
where s n+ ∧t
g(un ) du = 2n sn+ ∧ t − sn g(sn ),
β(s) := 2n sn
we have by BDG inequality E sup αt1 tT
Tn+ 1/2 2 CE β(u) du 0
T = CE 0
1/2 2 g(u) du
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J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
T CE
X(s, x) − X(s, y)4p ds
1/2
0
p CE sup X(u, x) − X(u, y)
T
uT
X(s, x) − X(s, y)2p ds
1/2
0
2p 1 E sup X(s, x) − X(s, y) +C 4p sT
T 2p ds, E supX(u, x) − X(u, y) 0
(49)
us
where we have used the inequality (10). A direct calculus gives 2p−4
g (u) = (2p − 2)X(u, x) − X(u, y) X(u, x) − X(u, y), σ (u, x) − σ (u, y) w˙ n (u) ∗ × X(u, x) − X(u, y) σ (u, x) − σ (u, y) 2p−4
X(u, x) − X(u, y), b(u, x) − b(u, y) + (2p − 2)X(u, x) − X(u, y) ∗ × X(u, x) − X(u, y) σ (u, x) − σ (u, y) 2p−4
X(u, x) − X(u, y), A(u, x) − A(u, y) − (2p − 2)X(u, x) − X(u, y) ∗ × X(u, x) − X(u, y) σ (u, x) − σ (u, y) 2p−2 σ (u, x) − σ (u, y) w˙ n (u) + b(u, x) − b(u, y) + X(u, x) − X(u, y) ∗ − A(u, x) + A(u, y) σ (u, x) − σ (u, y) 2p−2 ∗ X(u, x) − X(u, y) σ˜ (u), (50) + X(u, x) − X(u, y) where σ˜ is an m × d matrix whose (i, j )th component is given by
σ˜ ij (u) = (∇σij · σ ) X(u, x) − (∇σij · σ ) X(u, y) w˙ n (u) + (∇σij · b) X(u, x) − (∇σij · b) X(u, y) − (∇σij · An ) X(u, x) + (∇σij · An ) X(u, y) . Since w˙ n (v) is constant on [un , u+ n ), we have X(u, x) − X(u, y)
u = X(un , x) − X(un , y) + σ (v, x) − σ (v, y) w˙ n (v) dv un
u u + b(v, x) − b(v, y) dv + A(v, x) − A(v, y) dv un
un
(51)
J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
u
X(un , x) − X(un , y) + C n + w˙ n (v)
2811
X(v, x) − X(v, y) dv.
un
Gronwall’s lemma then gives X(u, x) − X(u, y) X(un , x) − X(un , y) exp 2−n C n + w˙ n (u) . Hence g (u) C X(u, x) − X(u, y)2p n + w˙ n (u)
2p C X(un , x) − X(un , y) exp 2−n C n + w˙ n (u) n + w˙ n (u) .
Obviously
E exp 2−n C n + w˙ n (u) n + w˙ n (u) w˙ n (u) C2n . Hence by the independence between X(un , x) − X(un , y) and w˙ n (u) we have E sup α2 (t) C
tT
T
s ds
2p
E X(un , x) − X(un , y) 2n du
sn
0
T C
2p
E X(sn , x) − X(sn , y) ds
0
T 2p C E supX(u, x) − X(u, y) ds.
(52)
us
0
Thus T 1 2p 2p E sup I1 (t) E sup X(t, x) − X(t, y) + C E supX(u, x) − X(u, y) ds. (53) 2 tT tT us
0
For I2 , by Lipschitz property we can obtain T 2p E sup I2 (t) C E supX(u, x) − X(u, y) ds. tT
Combining (47), (53), (54) gives
0
us
(54)
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J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
2p 2p 1 |x − y|2p + E sup X(t, x) − X(t, y) E sup X(t, x) − X(t, y) 2 tT tT T 2p ds, + C E supX(u, x) − X(u, y) us
0
from which (43) follows by Gronwall inequality. Finally, since K(t, x) − K(t, y) |x − y| + X(t, x) − X(t, y) t +C
t X(s, x) − X(s, y) ds + C σ (s, x) − σ (s, y) w˙ n (s) ds ,
0
(55)
0
we have 2p E sup K(t, x) − K(t, y) tT
t 2p σ (u, x) − σ (u, y) w˙ n (u) du + CE sup . tT
C|x − y|
2p
(56)
0
Write t
σ (u, x) − σ (u, y) w˙ n (u) du
0
t =
σ (un , x) − σ (un , y) w˙ n (u) du +
0
t
u σ˜ (v)w˙ n (v) dv,
du 0
un
where σ˜ (v) is defined in (51) and we have σ˜ (v) C X(un , x) − X(un , y) exp 2−n C n + w˙ n (u) n + w˙ n (u) . Thus repeating the same reasoning in treating the similar term in αt2 we see t 2p u E sup du σ˜ (v)w˙ n (v) dv tT
0
T C 0
un
2p EX(tn , x) − X(tn , y) dt
(57)
J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
T C
2813
2p sup EX(u, x) − X(u, y) dt ut
0
C|x − y|2p .
(58)
And trivially by BDG inequality t 2p E sup σ (un , x) − σ (un , y) w˙ n (u) du tT
0
T
2p
E σ (tn , x) − σ (tn , y) dt
0
T 2p E supX(u, x) − X(u, y) dt ut
0
C|x − y|2p .
(59)
2
Combining (56)–(59) gives (44).
Denote by C(R+ × D(A), Rm × Rm ) the space of continuous mappings from R+ × D(A) to × Rm , equipped with the topology of compact uniform convergence.
Rm
Theorem 7.2. There exists a subsequence {nk } such that (Xnk , Knk ) → (X, K)
(60)
in C(R+ × D(A), Rm × Rm ), almost surely. Proof. We fix positive numbers γ and T . For each positive integer k, choose a subset Dγ ,k of B(0, γ ) ∩ D(A) such that for every x ∈ D(A) there exists an x˜ ∈ Dγ ,k such that |x − x| ˜ 2−k , m where B(0, γ ) denotes the closed ball in R centered at the origin and with radius γ . Since for every x ∈ D(A), Xn (·, x) → X(·, x)
in W m
in probability, and since (see (20)) p sup E sup Xn (t, x) < ∞ n
tT
for all p, we deduce that for every k, there exists an nk such that for all x ∈ Dγ ,k 2p 2−kp E sup Xnk (t, x) − X(t, x) tT
(61)
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J. Ren, S. Xu / Journal of Functional Analysis 256 (2009) 2780–2814
and 2p E sup Knk (t, x) − K(t, x) 2−kp .
(62)
tT
Consequently for all x, y ∈ B(0, γ ) ∩ D(A), 2p C 2−kp + |x − y|2p E sup Xnk (t, x) − X(t, y)
(63)
2p E sup Knk (t, x) − K(t, y) C 2−kp + |x − y|2p .
(64)
tT
and
tT
Thus using the Kolmogorov’s continuity criterion in the same way as in [11] we obtain the desired convergence. 2 Remark 7.3. In [4] it is proved that (1) admits a unique strong solution provided that the coefficients σ and b are Lipschitz. Recently it is shown in [13] that in the one-dimensional case the solution can be obtained pathwisely by using the Doss approach. In the multi-dimensional case, however, the Doss approach does not work any more. An expected substitute for the Doss solution would be such a limit theorem. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13] [14]
J.-P. Aubin, A. Cellina, Differential Inclusions, Grundlehren Math. Wiss., vol. 264, Springer, Berlin, 1984. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. J. Bismut, Mécanique Aléatoire, Lecture Notes in Math., vol. 866, Springer, Berlin, 1981. E. Cépa, Équations différentielles stochastiques multivoques, in: Sémin. Probab. XXIX, in: Lecture Notes in Math., vol. 1613, Springer, Berlin, 1995, pp. 86–107. N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, second ed., Kodansha/NorthHolland, Tokyo/Amsterdam, 1989. J. Jacod, A.N. Shiryaev, Limit Theorems for Stochastic Processes, Springer, Berlin, 1987. T. Kurtz, Random time change and convergence in distribution under the Meyer–Zheng conditions, Ann. Probab. 19 (3) (1991) 1010–1034. P. Malliavin, Un principe de transfert et son application au calcul des variations, C. R. Acad. Sci. Paris Sér. A– B 284 (3) (1977) A187–A189. P. Malliavin, Stochastic Analysis, Grundlehren Math. Wiss., vol. 313, Springer, Berlin, 1997. P.A. Meyer, W.A. Zheng, Tightness criteria for laws of semimartingales, Ann. Inst. H. Poincaré Probab. Statist. 20 (1984) 353–372. J. Ren, Analyse quasi-sûre des équations différentielles stochastiques, Bull. Sci. Math. 114 (2) (1990) 187–213. D.W. Stroock, S.R.S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, in: Proc. Sixth Berkeley Sympos. Math. Statist. Probab., vol. 3, Univ. California Press, Berkeley, CA, 1972, pp. 333–359. S. Xu, Explicit solutions for multivalued stochastic differential equations, Statist. Probab. Lett. 78 (2008) 2281– 2292. X. Zhang, Skorohod problem and multivalued stochastic evolution equations in Banach spaces, Bull. Sci. Math. 131 (2) (2007) 175–217.
Journal of Functional Analysis 256 (2009) 2815–2841 www.elsevier.com/locate/jfa
On the classification of the coadjoint orbits of the Sobolev Bott–Virasoro group François Gay-Balmaz 1 Section de Mathématiques, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland Received 24 July 2008; accepted 5 December 2008 Available online 8 January 2009 Communicated by D. Voiculescu
Abstract The purpose of this paper is to give the classification of the Bott–Virasoro coadjoint orbits, with nonzero central charge, in the functional analytic setting of smooth Hilbert manifolds. The central object of the paper is thus the completion of the Bott–Virasoro group with respect to a Sobolev topology, giving rise to a smooth Hilbert manifold and topological group, called the Sobolev Bott–Virasoro group. As a consequence of this approach, analytic and geometric properties of the coadjoint orbits are studied. © 2008 Elsevier Inc. All rights reserved. Keywords: Bott–Virasoro group; Virasoro algebra; coadjoint orbit; Hilbert manifold; diffeomorphism group
1. Introduction The classification of the coadjoint orbits of the Virasoro algebra in the smooth category can be found in different formulations in [14,16,18,19] and others. [2] reformulate and complete these results; see also [4] and [5] for another approach. In this paper the classification problem of the coadjoint orbits of the Virasoro algebra is studied in the setting of smooth infinite dimensional Hilbert manifolds. The techniques are extensions of those in [2] to the Sobolev manifold category. The classical approach to the Bott–Virasoro group consists in seeing it as a one-dimensional C ∞ (S 1 ) of smooth orientation preserving diffeomorphisms of the central extension of the group D+ circle. This group can be endowed with the structure of a smooth infinite dimensional Fréchet E-mail address:
[email protected]. 1 Partially supported by a Swiss NSF grant.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.12.002
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F. Gay-Balmaz / Journal of Functional Analysis 256 (2009) 2815–2841
manifold, making it into a regular Lie group. Analysis on Fréchet manifolds can be made rigorous by using the convenient calculus of [15], for example. See [12] for an extensive study of the Bott– Virasoro group. Unfortunately, many fundamental results such as the implicit function theorem, valid on finite dimensional manifolds, do not generalize to the case of Fréchet manifolds, without appeal to the Nash–Moser theorem. Consequently, it has been of fundamental interest to consider completions of diffeomorphism groups, or more generally, of manifolds of maps, with respect to C k , Hölder, or Sobolev H s topologies, in order to obtain smooth Banach or Hilbert manifolds. This approach was successfully used, for example, in [7–11,13] to address different problems in mathematical physics and geometry. The present paper is one more step in this direction. It studies the classification and geometric properties of the Bott–Virasoro coadjoint orbits, in the setting of smooth Hilbert manifolds. More precisely, we consider the Sobolev Bott–Virasoro group, which is the H r -completion of the smooth Bott–Virasoro group, acting via the coadjoint action on the H s -completion of the regular part of its dual Lie algebra. We will see that the coadjoint action is well-defined provided r s + 3 and s > 1/2. When classifying the coadjoint orbits, the natural choice will appear to be r = s + 3. Using this rigorous functional analytic approach new results concerning coadjoint orbits are found. 2. Sobolev vector fields and diffeomorphisms on S 1 Consider the circle S 1 and its universal covering map x ∈ R → s = ei2πx ∈ S 1 . We denote by ∂ the smooth vector field on S 1 defined by ∂ ei2πx := i2πei2πx ∈ Tei2πx S 1 . Any vector field u ∈ X(S 1 ) on the circle can be identified with the Z-periodic function u : R → R, through the relation u ei2πx = u(x)∂ ei2πx . For any real number s 0 we have the equivalence u ∈ Xs S 1
⇐⇒
u ∈ HZs (R),
where Xs (S 1 ) denotes the Hilbert space of Sobolev class H s vector fields on S 1 and HZs (R) is defined by s (R) u is Z-periodic . HZs (R) := u ∈ Hloc s (R) denotes the functions which are locally in H s , that is, u ∈ H s (R) if and only if Here Hloc loc for each open bounded subset U ⊂ R, there exists v ∈ H s (U ) such that u = v on U . Equivalently we have
s (R) = u : R → R ϕu ∈ H s (R), ∀ϕ ∈ C0∞ (R) . Hloc
F. Gay-Balmaz / Journal of Functional Analysis 256 (2009) 2815–2841
2817
In terms of Fourier series, the space HZs (R) is characterized by HZs (R) =
an ei2πx |n|2s |an |2 < ∞, an = a−n .
n∈Z
n∈Z
Recall that for k 0 and s > 1/2 + k, the Sobolev embedding theorem states that we have the continuous inclusions H s (U ) ⊂ C k (U )
k and Xs S 1 ⊂ XC S 1 , k
for any bounded open subset U of R, where XC (S 1 ) denotes the space of C k vector fields on S 1 . Recall also that for s > 1/2 and r −s, pointwise multiplication on C ∞ (U ) extends to a continuous bilinear map H r (U ) × H s (U ) → H min{r,s} (U ),
(u, v) → uv;
(2.1)
see Corollary 9.7 of [17] for a proof. C 1 (S 1 ) the group of C 1 orientation preserving diffeomorphisms of the circle. Denote by D+ C 1 (S 1 ) determines, up to an additive constant m ∈ Z, a strictly increasing C 1 function Any g ∈ D+ g : R → R such that g(x + 1) = g(x) + 1 and g ei2πx = ei2πg(x) . r (S 1 ) the set of Sobolev class H r orientation preserving diffeoFor all r > 3/2, we denote by D+ 1 morphisms of S . The equality
1 r C1 1 D+ S = D+ S ∩ H r S1, S1 ,
(2.2)
proven in [7], implies the equivalence 1 r g ∈ D+ S
⇐⇒
r g ∈ D+ Z (R),
where r r D+ Z (R) := g ∈ Hloc (R) g > 0, g(x + 1) = g(x) + 1 . The relation between the derivatives of g and g is given by the equality T g ∂ ei2πx = g (x)∂ g ei2πx . r (S 1 ), r > 3/2, and Xs (S 1 ), s > 1/2, but their elements From now on, we will use the symbols D+ r (R) and H s (R). will be denoted by g and u as if they would belong to D+ Z Z r 1 Recall that for all r > 3/2, the set D+ (S ) can be endowed with the structure of a smooth r (S 1 ) is an open subset Hilbert manifold modeled on the Hilbert space Xr (S 1 ). More precisely, D+ r 1 1 r (S 1 ) is of the smooth Hilbert manifold H (S , S ). With respect to this manifold structure, D+
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F. Gay-Balmaz / Journal of Functional Analysis 256 (2009) 2815–2841
a topological group with smooth right translation. For r > 1/2 + k, k 1 we have the smooth Sobolev embedding 1 r Ck 1 S ⊂ D+ S , D+ k
C (S 1 ) denote the smooth Banach manifold of class C k orientation preserving diffeowhere D+ morphisms of S 1 . The reader can consult [7] and [8] for proofs and details concerning the Hilbert manifold structure of the group of Sobolev diffeomorphisms of a compact manifold.
3. The Sobolev Bott–Virasoro group The completion of the Bott–Virasoro group in the Sobolev topology H s is presented in this section. The Bott–Virasoro group is, up to isomorphism, the unique nontrivial central extension of the diffeomorphism group of the circle by R. This group, denoted by BVir is, as a set, the Cartesian product C∞ 1 S × R. BVir = D+ The group multiplication on BVir is defined by (g, α)(h, β) = g ◦ h, α + β + B(g, h) , ∞
∞
C (S 1 ) × D C (S 1 ) → R is the Bott–Thurston two-cocycle defined by where B : D+ +
1 log(g ◦ h)d log h . B(g, h) := 2 S1
The Lie algebra vir of BVir is called the Virasoro algebra and is given by the central extension of X(S 1 ) associated to the Gelfand–Fuchs two-cocycle
C(v, w) = v w . C : X S 1 × X S 1 → R, S1
Consequently, the Lie bracket is given by (v, a), (w, b) = v w − vw , C(v, w) . Identifying the dual Lie algebra of vir with itself (regular dual) through the L2 pairing, the coadjoint action reads Ad†(g,α) (u, c) = (g )2 (u ◦ g) + c S(g), c , where S is the Schwarzian derivative defined by S(g) :=
g 3 g 2 − . g 2 g
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We now show that for all r > 3/2 it is possible to complete the Bott–Virasoro group with respect to the Sobolev H r topology. To see this, it suffices to observe that for all r > 3/2, the Bott–Thurston two-cocycle is well defined as a map 1 1 r r B : D+ S × D+ S → R. r (S 1 ), we have log(g ◦ h) ∈ H r−1 and d log h ∈ H r−2 . Using the condition Indeed, for g, h ∈ D+ r > 3/2, we obtain that the L2 -pairing involved in the definition of B makes sense. Thus the H r completion of the Bott–Virasoro group is given by
1 r BVir r := D+ S ×R and is called the Sobolev Bott–Virasoro group. It is a smooth Hilbert manifold modeled on the r (S 1 ) is a codimension one submanifold of BVir r . Hilbert space virr := Xr (S 1 )×R. Note that D+ r (S 1 ) and Lemma 3.1 below, it is readily seen that BVir r is a topologUsing the properties of D+ ical group with smooth right translations R(g,α) . The tangent map to R(g,α) is given by
Vg ◦ h 1 . d log h T R(h,β) Vg , (α, v) = Vg ◦ h, α + β + B(g, h), v + 2 g ◦ h S1
For all s > 1/2 and r s + 3, the coadjoint representation extends to a continuous representation BVir r × virs → virs ,
Ad†(g,α) (u, c) = (g )2 (u ◦ g) + c S(g), c .
To see this, it suffices to note that, for all r s + 3, the map g → S(g) is continuous (and in fact smooth) as a map with values in Xs (S 1 ). This is a consequence of Lemma 3.1 below. Remark that for (g, α) fixed, (u, c) → Ad†(g,α) (u, c) is smooth since it is continuous and linear. However, for (u, c) fixed, the map (g, α) → Ad†(g,α) (u, c) is not even C 1 . The following lemma is a particular case of a result we will prove in Lemma 5.2. s (R), s > 1/2, such that f (x) > 0 for all x ∈ R. Then Lemma 3.1. Consider a function f in Hloc s we have 1/f ∈ Hloc (R).
R) 4. Hill’s equation and the conjugacy classes of SL(2,R In this section we recall from [16] and [18] the link between the Hill’s equation and the Bott– Virasoro coadjoint action and the fact that the monodromy is a coadjoint invariant. We adapt these results to the case of the smooth Hilbert manifold BVir r . In what follows we shall use the notations of [2]. For u ∈ Xs (S 1 ), s > 1/2, the Hill’s equation with potential u is the second order linear ODE defined by ψ + uψ = 0.
(4.1)
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Consider a vector solution Ψ := (ψ1 , ψ2 ), where ψi ∈ C 2 (R) are two independent normalized solutions of (4.1), that is, ψ1 ψ2 − ψ1 ψ2 = 1. Such a Ψ is called a normalized base of solutions associated to u. If Φ = (ϕ1 , ϕ2 ) is another normalized base of solutions, then there exists A ∈ SL(2, R) such that Ψ = ΦA. For example, for Ψ˜ (x) := Ψ (x + 1) we obtain the existence of MΨ ∈ SL(2, R), called the monodromy matrix of Ψ , such that Ψ˜ = Ψ MΨ . One can show that for all A ∈ SL(2, R) we have MΨ A = A−1 MΨ A, so we obtain a well-defined map M : Xs S 1 → SL(2, R)/conj,
u → M(u) := [MΨ ]conj ,
where SL(2, R)/conj denotes the set of conjugacy classes of SL(2, R), Ψ = (ψ1 , ψ2 ) are two independent normalized solutions of (4.1) with potential u, and [MΨ ]conj denotes the conjugacy class of MΨ . Note that the map M has the remarkable property to be invariant under the coadjoint action of the Bott–Virasoro group BVir r , r s + 3, for central charge c = 1/2, that is, M Ad†g (u, 1/2) = M(u),
1 r S . for all g ∈ D+
(4.2)
Indeed, one can compute that if ψ is a solution of (4.1) with potential u, then 1 ψ g := ψ ◦ g ∈ C 2 (R) g is a solution of (4.1) with potential Ad†g (u, 1/2) = (g )2 (u ◦ g) + g
1 S(g). 2
g
Since MΨ g = MΨ , where Ψ g := (ψ1 , ψ2 ), we obtain (4.2). Note that if Ψ is normalized, then Ψ g is normalized. Thus M induces a well-defined map r 1 S → SL(2, R)/conj, O → M(O) := M(u) M : Xs S 1 × {1/2} /D+ r (S 1 ) denotes the set of coadjoint orbits with central charge c = 1/2 where (Xs (S 1 ) × {1/2})/D+ and u is any element in the coadjoint orbit O. We will see that the natural choice for r is r = s +3. In this case we will denote by Os the coadjoint orbits of BVir s+3 . As we will see later, the map M is surjective and not injective.
F. Gay-Balmaz / Journal of Functional Analysis 256 (2009) 2815–2841
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We list below, for future use, the elements of SL(2, R)/conj. The terminology elliptic, hyperbolic, and parabolic corresponds to the three cases | Trace | < 2, | Trace | > 2 and | Trace | = 2, respectively. (i) An elliptic conjugacy class is represented by a matrix Ell± (α) of the form
cos α Ell± (α) = ± sin α
− sin α , cos α
α ∈ ]0, π[.
(ii) A hyperbolic conjugacy class is represented by a matrix Hyp± (β) of the form
eβ Hyp± (β) = ± 0
0 e−β
,
β > 0.
(iii) A parabolic conjugacy class is represented by a matrix Par± (ν) of the form
1 Par± (ν) = ± ν
0 , 1
ν ∈ {−1, 0, 1}.
The corresponding isotropy groups, with respect to conjugation in SL(2, R), are given by cos θ − sin θ SL(2, R)Ell± (α) = 0 θ < 2π , sin θ cos θ ζ 0 SL(2, R)Hyp± (β) = ζ = 0 , 0 1/ζ ±1 0 SL(2, R)Par± (ν) = γ ∈ R , for ν = ±1, γ ±1
SL(2, R)Par± (0) = SL(2, R). 5. Classification of the coadjoint orbits and isotropy groups In this section we obtain the classification of the Bott–Virasoro coadjoint orbits, associated to the coadjoint representation of the H r completion BVir r of the Bott–Virasoro group BVir. We show that the method used in [2] for the classification of the coadjoint orbits of BVir can be adapted to the case of the smooth Hilbert manifold BVir s+3 , s > 1/2 acting on the completed Virasoro algebra virs . This will use in a crucial way the two following results involving properties of Sobolev class H s functions. Lemma 5.1. Let u ∈ Xs (S 1 ), s > 1/2 be an H s vector field on S 1 , and consider a solution ψ of s+2 (R). the Hill equation with potential u. Then ψ ∈ Hloc Proof. Let ψ be a solution of ψ + uψ = 0 and U be an open and bounded subset of R. If 1/2 < s 3/2, then u is continuous, so we have ψ ∈ C 2 (R) and ψ ∈ C 2 (U ) ⊂ H s (U ), and we obtain ψ = −uψ ∈ H s (U ), by property (2.1).
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If 3/2 < s 4/2, then u is of class C 1 , so we have ψ ∈ C 3 (R) and ψ ∈ C 3 (U ) ⊂ H s (U ), and we obtain ψ = −uψ ∈ H s (U ). More generally, if 1/2 + k < s 1/2 + k + 1, then u is of class C k , so we have ψ ∈ C k+2 (R) and ψ ∈ C k+2 (U ) ⊂ H s (U ), and we obtain ψ = −uψ ∈ H s (U ). This proves that ψ ∈ H s+2 (U ) for all s > 1/2. 2 s (R), s > 1/2, and f ∈ C ∞ (R). Then f ◦ u ∈ H s (R). Lemma 5.2. Let u ∈ Hloc loc
Proof. It suffices to prove that for each open and bounded subset U of R we have f ◦ u ∈ H s (U ). We will show this first, by induction, for s = k ∈ N, k 1. Suppose that u ∈ H 1 (U ). By the Sobolev embedding theorem we have u ∈ C 0 (U ), so we obtain that f ◦ u ∈ C 0 (U ) ⊂ H 0 (U ). A candidate for the first distributional derivative is v := (f ◦ u)u ∈ H 0 (U ). Let us show that
(f ◦ u)ϕ = − vϕ, ∀ϕ ∈ C0∞ (U ). (5.1) U
U 1
H −→ u. By the Sobolev embedding Consider a sequence (un ) ⊂ C ∞ (U ) ∩ H 1 (U ) such that un − C0 C0 theorem we have un −−→ u, so we obtain f ◦ un −−→ f ◦ u and
(f ◦ un )ϕ → (f ◦ u)ϕ . U
U
On the other hand we have
(f ◦ un )u ϕ − (f ◦ u)u ϕ n U
U
(f ◦ un − f ◦ u)u ϕ +
n
U
f ◦ un − f ◦ uC 0
(f ◦ u) u − u ϕ n
U
u ϕ + f ◦ u n
U
C0
u − u ϕ n
U
2 2 f ◦ un − f ◦ uC 0 u n H 0 ϕ2H 0 + f ◦ uC 0 u n − u H 0 ϕ2H 0 ,
so we obtain that
U
(f ◦ un )u n ϕ →
(f ◦ u)u ϕ.
U
This proves formula (5.1). Fix k ∈ N and suppose that for each u ∈ H k (U ) we have f ◦ u ∈ H k (U ). Let u ∈ H k+1 (U ). By induction we know that f ◦ u ∈ H k (U ). A candidate for the k + 1 distributional derivative is given by
F. Gay-Balmaz / Journal of Functional Analysis 256 (2009) 2815–2841
w:=
k+1 (i) f ◦u
j1 ,...,ji 1, j1 +···+ji =k+1
i=1
= (f ◦ u)u(k+1) +
k+1
Aij1 ...ji u(j1 ) . . . u(ji )
(f (i) ◦ u)
j1 ,...,ji 1, j1 +···+ji =k+1
i=2
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Aij1 ...ji u(j1 ) . . . u(ji ) ,
where Aij1 ...ji are the constants such that the formula is true for the smooth case. The first term is in H 0 (U ), since f ◦ u ∈ H 1 (U ) ⊂ C 0 (U ) and u(k+1) ∈ H 0 (U ). In the second term, all the jl are strictly smaller than k + 1, so the u(jl ) are in H 1 (U ). Since f (i) ◦ u ∈ H 1 (U ), we obtain that the second term is in H 1 (U ), so w ∈ H 0 (U ). The same argument as before shows that w is the distributional derivative of (f ◦ u)(k) . s (R) with s > 1/2 and s ∈ / N. We now consider the case of u ∈ Hloc If 1/2 < s < 1, then the norm of the Hilbert space H s (U ) is given by
u2H s
= u2H 0
+ U U
|u(x) − u(y)|2 dx dy. |x − y|1+2s
For u ∈ H s (U ), by the Sobolev embedding theorem, we have f ◦ u ∈ C 0 (U ), so we obtain f ◦ u ∈ H 0 (U ). It remains to show that f ◦ uH s < ∞, this is a consequence of the inequality |f (u(x)) − f (u(y))| M|u(x) − u(y)| where M := max{f (y) | y ∈ u(U )}. If k < s < k + 1, k ∈ N, k 1, we have u ∈ H k (U ) so we know that f ◦ u ∈ H k (U ) and that the k order distributional derivative is given by (f ◦ u)
(k)
= (f ◦ u)u
(k)
k (i) f ◦u +
j1 ,...,ji 1, j1 +···+ji =k
i=2
Aij1 ...ji u(j1 ) . . . u(ji ) .
As before, the second term is in H 1 (U ) ⊂ H s−k (U ). We have f ◦ u ∈ H 1 (U ) and u(k) ∈ H s−k (U ), so by the property (2.1) we obtain that the first term is also in H s−k (U ). This proves that (f ◦ u)(k) ∈ H s−k (U ), so f ◦ u ∈ H s (U ). 2 Using these results, we can now proceed to the classification of the coadjoint orbits, by adapting the method used in [2]. 5.1. Coadjoint orbits with elliptic monodromy s+3 1 Let Os ∈ (Xs (S 1 ) × {1/2})/D+ (S ) be a Bott–Virasoro coadjoint orbit such that M(Os ) = [Ell± (α)], where α ∈ ]0, π[. Such an orbit always exists. It suffices to consider the orbits generated by the constant vector fields α 2 or (α + π)2 . Indeed, in this case a normalized base of solutions associated to α 2 is given by
Ψ (x) =
1 1 √ sin(αx), √ cos(αx) , α α
and we have MΨ = Ell+ (α). If we consider (α + π)2 we obtain MΨ = Ell− (α).
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For any u ∈ Os , there exists a normalized base of solutions Ψ such that MΨ = Ell± (α). Such a Ψ = (ψ1 , ψ2 ) is unique up to right multiplication by an element of SL(2, R)Ell± (α) . So the Z-periodic function Ru := ψ12 + ψ22 > 0 depends only on u and, by Lemma 5.1, belongs to the s+2 space Hloc (R). Moreover we have u=−
ψ1 ψ1 + ψ2 ψ2 ψ12 + ψ22
1 R u 1 R u 2 1 =− + + 2 2 Ru 4 Ru Ru
2 1 = gu au2 + S(gu ) 2 2 † = Adgu au , 1/2 , where
1 au := 0
dy >0 Ru (y)
1 and gu (x) := au
x 0
dy . Ru (y)
s+3 1 By Lemma 5.2, we have R1u ∈ HZs+2 (R), so we obtain that gu ∈ D+ (S ). A normalized base 2 of solutions associated to a is given by
Ψ (x) =
1 1 √ sin(ax), √ cos(ax) , a a
so we have MΨ = Ell± (a). Since we must have [MΨ ] = [Ell± (α)], we obtain that au = α + πn for some n = 0, 1, 2, . . . . Note that for a1 = a2 , the corresponding coadjoint orbits are disjoint. Indeed, since au does not depend on u in a given elliptic orbit, given a1 , a2 two constant vector fields in the same orbit we have aa1 = aa2 . Using that Rai = a1i we obtain aai = ai . This proves that a1 = a2 . To see that au does not depend on u in the orbit we consider a vector field v of s+3 1 the form v = Ad†g (u, 1/2), where g ∈ D+ (S ). Choosing Ψ associated to u such that MΨ = g g Ell± (α), we have MΨ g = MΨ and Rv = (ψ1 )2 + (ψ2 )2 = g1 Ru ◦ g, so by change of variable in the integral, we find av = au . We conclude that the coadjoint orbits with elliptic monodromy are exactly the coadjoint orbits generated by the vector fields ell(α, n) := (α + πn)2 ,
α ∈ ]0, π[, n = 0, 1, 2, . . . .
We will use the notation Ells (α, n) for the orbit generated by the group BVir s+3 acting on ell(α, n). Note that we have M(Ells (α, n)) = [Ell(−1)n (α)]. 5.2. Coadjoint orbits with hyperbolic monodromy s+3 1 Let Os ∈ (Xs (S 1 ) × {1/2})/D+ (S ) be a Bott–Virasoro coadjoint orbit such that M(Os ) = [Hyp± (β)], β > 0. For any vector field u ∈ Os , there exists a normalized base of solutions Ψ such that MΨ = Hyp± (β). Let N(u) = 0, 1, 2, . . . be the number of zeros of ψ2 on [0, 1[. Since
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Ψ is unique up to a right multiplication by an element of SL(2, R)Hyp± (β) , the map N is well defined, that is, it depends only on u. Moreover we have N Ad†g (u, 1/2) = N(u), g
indeed, the number of zeros is invariant under the transformation ψ2 → ψ2 . 5.2.1. The case N(u) = 0 s+3 1 (S ) such that M(Os ) = [Hyp± (β)], β > 0 and Consider Os ∈ (Xs (S 1 ) × {1/2})/D+ s N(u) = 0 for u ∈ O . Note that an orbit with M(Os ) = [Hyp+ (β)] always exists: it suffices to consider the orbit generated by the constant vector field −β 2 . Indeed in this case a normalized base of solutions associated to −β 2 is given by
Ψ (x) =
1 1 √ eβx , √ e−βx , 2β 2β
and we have MΨ = Hyp+ (β) and N(−β 2 ) = 0. For any u ∈ Os , consider a normalized base of solutions Ψ such that MΨ = Hyp± (β). Ψ is unique up to a right multiplication by an element of SL(2, R)Hyp± (β) . Let f :=
ψ1 , ψ2
s+2 so by Lemma 5.2 we have f ∈ Hloc (R). Moreover, since
f =
1 , ψ22
s+3 we obtain, again with Lemma 5.2, that f ∈ Hloc (R). Let
g :=
1 ln f, 2β
since f (x + 1) = e2β f (x) and f > 0, we have f > 0, so g is well defined and g(x + 1) = s+3 1 g(x) + 1. By Lemma 5.2 we obtain g ∈ D+ (S ). A direct computation shows that 1 S(g) = u + (g )2 β 2 , 2 or, equivalently, that u = Ad†g (−β 2 , 1/2). Thus, we see that for each β > 0, there exists exactly one hyperbolic orbit Os denoted by Hyps (β, 0) such that M(Os ) = [Hyp± (β)] and N(u) = 0 for u ∈ Os . A representative vector field is given by hyp(β, 0) := −β 2 , and we have M(Os ) = [Hyp+ (β)]. This also shows that it does not exist a hyperbolic orbit with N(u) = 0 and with monodromy Hyp− (β).
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5.2.2. The case N(u) = 0 s+3 1 Consider Os ∈ (Xs (S 1 ) × {1/2})/D+ (S ) such that M(Os ) = [Hyp± (β)], β > 0 and N(u) = n = 0 for u ∈ Os . Note that an orbit with M(Os ) = [Hyp(−1)n (β)] always exists: it suffices to consider the orbit generated by the smooth vector field hyp(β, n) := −β 2 − 2
(nπ)2 + β 2 (nπ)2 +3 2 , Fβ,n Fβ,n
where 2 β cos(nπx) . Fβ,n (x) = cos (nπx) + sin(nπx) + nπ
2
Indeed, in this case a normalized base of solutions associated to hyp(β, n) is given by
Ψ (x) =
eβx β e−βx cos(nπx) , cos(nπx) + sin(nπx) , nπFβ,n (x) 2nπ nπFβ,n (x)
and we have MΨ = Hyp(−1)n (β) and N(u) = n. We now show that for all u ∈ Os such that M(Os ) = [Hyp± (β)], β > 0 and N(Os ) = n = 0, s+3 1 there exists g ∈ D+ (S ) such that Ad†g (u, 1/2) = hyp(β, n). This is a consequence of the following lemma. Lemma 5.3. Let u, u ∈ Xs (S 1 ), s > 1/2, such that M(u) = M(u) = [Hyp± (β)], β > 0, and s+3 1 N(u) = N(u) = n. Then there exists g ∈ D+ (S ) such that Ad†g (u, 1/2) = u. Proof. Let Ψ = (ψ1 , ψ2 ) and Ψ = (ψ 1 , ψ 2 ) be normalized bases of solutions associated to u s+3 1 and u such that MΨ = MΨ = Hyp± (β). It suffices to show that there exists g ∈ D+ (S ) such g that Ψ = Ψ (see Section 4). Let 0 x1 < · · · < xn < 1 and 0 x 1 < · · · < x n < 1 denote the n zeros of ψ2 and ψ2 on [0, 1[. The other zeros are given and denoted by xmn+k = xk + m, m ∈ Z. Let f (x) :=
ψ1 (x) , ψ2 (x)
∀x = xk .
We have f (x + 1) = e2β f (x) and f (x) =
1 ψ22 (x)
> 0,
∀x = xk .
(5.2)
So f is of class C 3 away from the set of zeros and fk := f |]xk ,xk+1 [ : ]xk , xk+1 [ → R is a strictly increasing C 3 diffeomorphism. We denote by f and f k the corresponding functions associated to u.
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Define g : R → R, by g(xk ) := x k+d and g(x) := (f k+d )−1 (f (x)), ∀x ∈]xk , xk+1 [, where d is such that, for each k, the sign of ψ2 (xk ) and ψ2 (x k+d ) are the same. Note that g is continuous, C 3 away from the set of zeros and we have
1 g (x)
ψ 2 g(x) = ψ2 (x),
and f g(x) = f (x),
∀x = xk ,
∀x ∈ R.
By multiplication of the preceding two equalities we obtain
1 g (x)
ψ 1 g(x) = ψ1 (x),
∀x = xk .
For all x such that x = xk and ψ1 (x) = 0, we have: ψ 21 (g(x))
g (x) =
ψ12 (x)
.
Since ψ1 (xk ) = 0 by the wronskian condition, the previous expression is well defined and continuous near xk . This proves that g is a C 1 orientation preserving diffeomorphism. s+3 1 s+2 We still need to show that g ∈ D+ (S ). We have ψ2 ∈ Hloc (R). By the proof of Lemma 5.2, s+2 we know that 1/ψ2 is in H (]xk + δ, xk+1 − δ[), for any δ > 0 such that ]xk + δ, xk+1 − δ[ = ∅. Since s + 2 > 1/2, by property (2.1) we obtained fk =
1 ∈ H s+2 ]xk + δ, xk+1 − δ[ . 2 ψ2
Since fk : ]xk + δ, xk+1 − δ[ → ]fk (xk + δ), fk (xk+1 − δ)[ is of class H s+3 and is a C 1 diffeomorphism, we obtain that fk−1 ∈ H s+3 (]fk (xk + δ), fk (xk+1 − δ)[) by (2.2). The same is true for f k+d and its inverse. Since the composition of two H s+3 -diffeomorphisms is of class H s+3 we obtain that g ∈ H s+3 (]xk + δ, xk+1 − δ[). It remains to show that g is of class H s+3 near the zeros. Fix a zero xk of ψ2 . We will show that g ∈ H s+3 (]xk − ε, xk + ε[). Let hk (x) :=
ψ2 (x) , ψ1 (x)
∀x ∈ ]xk − ε, xk + ε[,
hk is well defined in a neighborhood of xk since ψ1 (xk ) = 0, because of the wronskian condition. We have hk (x + 1) = e−2β h(x) and h k (x) = −
1 ψ12 (x)
< 0,
∀x ∈ ]xk − ε, xk + ε[,
so, as before, hk : ]xk − ε, xk + ε[ → ]h(xk + ε), h(xk − ε)[ is a strictly decreasing H s+3 diffeomorphism. The corresponding function associated to u and to the zero x k+d is denoted by hk+d
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F. Gay-Balmaz / Journal of Functional Analysis 256 (2009) 2815–2841
and has the same properties. So (hk+d )−1 ◦ hk ∈ H s+3 (]xk − ε, xk + ε[). Using that hk =
1 fk
on ]xk , xk + ε[ and hk =
1 fk−1
on ]xk − ε, xk [,
we obtain g = (hk+d )−1 ◦ hk on ]xk − ε, xk + ε[. C 1 (S 1 ) ∩ H s+3 (S 1 , S 1 ) = D s+3 (S 1 ). So we have proven that g ∈ D+ +
2
Thus we have shown that for each β > 0 and n ∈ N ∪ {0}, there exists exactly one hyperbolic orbit Os denoted by Hyps (β, n) such that M(Os ) = [Hyp± (β)] and N(u) = n for u ∈ Os . Representant vector fields are given by hyp(β, 0) = −β 2 ,
hyp(β, n) = −β 2 − 2
and
(nπ)2 + β 2 (nπ)2 +3 2 Fβ,n Fβ,n
and we have M(Os ) = [Hyp(−1)n (β)]. 5.3. Coadjoint orbits with parabolic monodromy Recall that a parabolic conjugacy class is represented by a matrix Par± (ν) of the form
1 Par± (ν) = ± ν
0 , 1
ν ∈ {−1, 0, 1}.
5.3.1. The case ν = 0 s+3 1 (S ) be such that M(Os ) = [Par± (0)]. Such an orbit always Let Os ∈ (Xs (S 1 ) × {1/2})/D+ exists, it suffices to consider the orbit generated by the constant vector field (nπ)2 , n ∈ N. Indeed in this case a normalized base of solutions associated to (nπ)2 is given by
Ψ (x) =
1 1 sin(nπx), √ cos(nπx) , √ nπ nπ
and we have MΨ = Par(−1)n (0). For any u ∈ Os , there exists a normalized base of solutions Ψ such that MΨ = Par± (0). Such a Ψ = (ψ1 , ψ2 ) is unique up to a right multiplication by an element of SL(2, R)Par± (0) = SL(2, R). We will proceed as in the elliptic case. Define the Z-periodic function R := ψ12 + ψ22 > 0. Remark that here R depends on the choice of Ψ , but as in the elliptic case we find u=−
ψ1 ψ1 + ψ2 ψ2 ψ12
+ ψ22
= Ad†g a 2 , 1/2 ,
where
1 a := 0
1 dy > 0 and g(x) := R(y) a
x 0
dy . R(y)
F. Gay-Balmaz / Journal of Functional Analysis 256 (2009) 2815–2841
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s+3 1 As before we have g ∈ D+ (S ). A normalized base of solutions associated to a 2 is given by
Ψ (x) =
1 1 √ sin(ax), √ cos(ax) . a a
Since we must have [MΨ ] = [Par± (0)], we obtain that a = nπ for some n ∈ N. Note that for n1 = n2 , the corresponding coadjoint orbits associated to (n1 π)2 and (n2 π)2 are disjoint. These disjoint parabolic coadjoint orbits are denoted by Pars (0, n),
n ∈ N.
We have M(Pars (0, n)) = [Par(−1)n (0)]. The corresponding vector fields representing these orbits are given by par(0, n) := (nπ)2 ∈ Pars (0, n). 5.3.2. The case ν = ±1 s+3 1 Let Os ∈ (Xs (S 1 ) × {1/2})/D+ (S ) such that M(Os ) = [Par± (ν)], ν = ±1. s For any u ∈ O , there exists a normalized base of solutions Ψ such that MΨ = Par± (ν). Let N(u) ∈ N ∪ {0} be the number of zeros of ψ2 on [0, 1[. Since Ψ is unique up to right multiplication by an element of SL(2, R)Par± (ν) , the map N is well-defined, that is, it depends only on u. Moreover we have N Ad†g (u, 1/2) = N(u). 5.3.3. The case N(u) = 0 s+3 1 Consider Os ∈ (Xs (S 1 ) × {1/2})/D+ (S ) such that M(Os ) = [Par± (ν)], ν = ±1, and s N(u) = 0 for u ∈ O . Remark that an orbit with M(Os ) = [Par+ (1)] always exists: it suffices to consider the orbit generated by the constant vector field 0. Indeed in this case a normalized base of solutions associated to 0 is given by Ψ (x) = (x, 1), and we have MΨ = Par+ (1) and N(0) = 0. For any u ∈ Os , consider a normalized base of solutions Ψ such that MΨ = Par± (ν). Ψ is unique up to a right multiplication by an element of SL(2, R)Par± (ν) . As in the elliptic case, consider the function g :=
ψ1 , ψ2
then we have g =
1 >0 ψ22
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s+3 and g ∈ Hloc (R). Note also that g(x +1) = g(x)+ν. Using that g is strictly increasing we obtain s+3 1 (S ). Moreover, a simple computation that the case ν = −1 is impossible and that g ∈ D+ shows that
1 S(g) = u. 2 This proves that u = Ad†g (0, 1/2). Thus we see that there exists only one orbit Os such that M(Os ) = Par± (ν) and N (u) = 0. This orbit is denoted by Pars (1, 0), it is represented by the vector field par(1, 0) := 0 and we have M(Pars (1, 0)) = [Par+ (1)]. 5.3.4. The case N(u) = 0 s+3 1 Consider Os ∈ (Xs (S 1 ) × {1/2})/D+ (S ) such that M(Os ) = [Par± (ν)], ν = ±1, and s N(u) = n = 0 for u ∈ O . Remark that an orbit with M(Os ) = [Par(−1)n (ν)] always exists: it suffices to consider the orbit generated by the smooth vector field
par(ν, n) := π 2
ν 3n2 (1 + 2π ) 2n2 , − 2 Hν,n Hν,n
where Hν,n (x) = 1 +
ν sin2 (nπx). 2π
Indeed, in this case a normalized base of solutions associated to par(ν, n) is given by
Ψ (x) =
2 1 νx sin(nπx) − cos(nπx) , sin(nπx) , n 2πHν,n (x) 2πHν,n (x) 1
and we have MΨ = Par(−1)n (ν) and N(u) = n. Remark that Lemma 5.3 is still valid for u, u ∈ Xs (S 1 ), s > 1/2, such that M(u) = M(u) = [Par± (ν)] and N(u) = N(u) = n. So we obtain that for all u ∈ Os such that M(Os ) = [Par± (ν)] s+3 1 and N(Os ) = n = 0, there exists g ∈ D+ (S ) such that Ad†g (u, 1/2) = par(ν, n). Thus we have shown that for each ν = ±1 and n ∈ N, there exists exactly one hyperbolic orbit Os denoted by Pars (ν, n) such that M(Os ) = [Par± (ν)] and N(u) = n for u ∈ Os . Representative vector fields are given by
par(ν, n) := π 2
ν 3n2 (1 + 2π ) 2n2 − 2 Hν,n Hν,n
and we have M(Os ) = [Par(−1)n (ν)]. Note that setting n = 0 or ν = 0 in the representative vector fields par(ν, n), we recover the expression of par(1, 0) and par(0, n).
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Remark on other central charges. Strictly speaking, the representative vector fields given above should be written with the central charge c = 1/2. For example, the representative vector fields associated to the coadjoint orbit Ell(α, n), α ∈ ]0, π[, n = 0, 1, 2, . . . , and Par(0, n), n = 1, 2, . . . , should be written as (α + πn)2 , 1/2 and (πn)2 , 1/2 . Note that for other non-zero central charges c the coadjoint classification is the same, but then corresponding representative vector fields are given by 2c(α + πn)2 , c
or
2c(πn)2 , c .
5.4. Classification of isotropy groups r (S 1 ) Given (u, c) ∈ Xs (S 1 ) × R, c = 0, we denote by D+ (u,c) the isotropy group of (u, c) r (S 1 ), r > 3/2. We are here especially interested in the isotropy under the coadjoint action of D+ s+3 1 (S )(u,c) . Using the equality groups D+
1 1 r r D+ S (u,c) = D+ S ( u ,1) 2c 2
it suffices to compute the isotropy groups for momenta with central charge c = 1/2. Since all isotropy groups of the elements of an orbit are conjugate to a given one, it suffices to compute the isotropy group of the representative vector fields ell(α, n),
hyp(β, n),
and
par(ν, n).
The classification of isotropy groups is known in the case of the coadjoint action of the smooth C ∞ (S 1 ) on the smooth vector fields; see for example [2]. We are here indiffeomorphisms D+ s+3 1 C ∞ (S 1 ) terested in the coadjoint action of the Hilbert manifold D+ (S ), which contains D+ as a strict subgroup. Therefore it may happen that, for a smooth vector field u, the isotropy s+3 1 C ∞ (S 1 ) group D+ (S )(u,c) is strictly bigger than the isotropy group D+ (u,c) . In fact this is not the case, since one can use the arguments given in Appendices B and C of [2], replacing the smooth diffeomorphisms by Sobolev class H s+3 diffeomorphisms, and obtain that for u ∈ {ell(α, n), hyp(β, n), par(ν, n)} we have s+3 1 C∞ 1 S (u,1/2) = D+ S (u,1/2) . D+ More precisely, following [2], we get s+3 1 s+3 1 s+3 1 S (ell(α,n),1/2) = D+ S (hyp(β,0),1/2) = D+ S (par(1,0),1/2) = S 1 , D+
for all n = 0, 1, 2, . . . , α ∈ ]0, π[, and β > 0; and n s+3 1 1) , D+ S (par(0,n),1/2) = PSU(1,
n = 1, 2, . . . ,
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F. Gay-Balmaz / Journal of Functional Analysis 256 (2009) 2815–2841 n
1) denotes the n-cover of PSU(1, 1), acting on S 1 as where PSU(1,
e → ix
Aeinx + B
1
Beinx + A
n
where
A B
B A
∈ SU(1, 1).
s+3 1 s+3 1 (S )(hyp(β,n),1/2) and D+ (S )(par(ν,n),1/2) , ν = ±1, n = 1, 2, . . . , are The isotropy groups D+ isomorphic to R+ × Zn . The table below, gives the classification of the Bott–Virasoro coadjoint orbits, together with the corresponding monodromy, representative vector fields, and isotropy groups.
Coadjoint orbit O s
Monodromy M(O s )
Representative vector field
Isotropy group
Ells (α, n)
Ell(−1)n (α)
ell(α, n) = (α + π n)2
S1
Hyps (β, n)
Hyp(−1)n (β)
• n = 0: hyp(β, 0) = −β 2 • n = 0:
S1
hyp(β, n) = −β 2 − 2 Pars (ν, n)
Par(−1)n (ν)
(nπ )2 + β 2 (nπ )2 +3 2 Fβ,n F
• ν = 0, n = 0:
2 ν 3n 1 + 2π 2n2 par(ν, n) = π 2 − 2 Hν,n Hν,n • ν = 0, n = 0:
≈ R + × Zn
β,n
≈ R + × Zn
par(0, n) = (nπ )2
1) PSU(1,
• (ν, n) = (1, 0): par(1, 0) = 0
S1
n
The domain of the variables is (α, n) ∈ ]0, π[ × {0, 1, 2, . . .}, (β, n) ∈ ]0, ∞[ × {0, 1, 2, . . .}, (ν, n) ∈ {−1, 0, 1} × {0, 1, 2, . . .},
(ν, n) = (−1, 0)
and (ν, n) = (0, 0).
The space of coadjoint orbit of the Bott–Virasoro group can be illustrated as follows. 6. Geometric properties of the isotropy groups Recall that a C 1 map f : M → N between Banach manifolds is said to be a subimmersion if and only if for each m ∈ M there exists a neighborhood U of m in M, a Banach manifold P , a submersion s : U → P and an immersion i : P → N such that f |U = i ◦ s. It is known that, that f : M → N is a subimmersion if and only if the tangent bundle to the fibers ker(Tf ) is a subbundle of T M and for each m ∈ M, Tm f (Tm M) is closed and splits in Tm N (see Theorem 3.5.18 in [1]). The principal property of a subimmersion is that for each n ∈ N , the set Q := f −1 ({n}) is a submanifold of M and Tm Q = ker(Tm f ).
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Lemma 6.1. For u ∈ Xs+k (S 1 ), s > 1/2, k 1, and c = 0, the coadjoint action s+3 1 S → Xs S 1 , Ad†− (u, c) : D+
g → Ad†g (u, c) = (g )2 (u ◦ g) + c S(g)
is a subimmersion of class C k . The tangent map at g = id is given by Tid Ad†− (u, c) : Xs+3 S 1 → Xs S 1 , ξ → ad†ξ (u, c) := Tid Ad†− (u, c) (ξ ) = cξ + 2uξ + u ξ. Proof. By the definition of the topologies, the map s+3 1 S → Xs+2 S 1 , D+
g → g
is smooth. The map Xs+2 S 1 → Xs+2 S 1 ,
g → (g )2
is smooth by property (2.1). For u ∈ Xs+k (S 1 ), k 1, the map s+3 1 S → Xs S 1 , D+
g → αu (g) = u ◦ g
is of class C k by the α-lemma (see [7]). By property (2.1) we obtain that the map s+3 1 S → Xs S 1 , D+
g → (g )2 (u ◦ g)
is of class C k . We now show that the Schwarzian derivative s+3 1
S : D+
S
→ Xs S 1 ,
g → S(g)
is smooth. Let Xs (S 1 )>0 := {u ∈ X(S 1 ) | u(x) > 0}, s > 1/2. Then Xs (S 1 )>0 is an open subset of Xs (S 1 ). By the α-lemma, the map Xs S 1 >0 → Xs S 1 ,
f → αi (f ) = i ◦ f =
1 , f
where i : ]0, ∞[ → R, i(x) := x1 , is smooth. Thus, the Schwarzian derivative g → g
1 3 1 2 g − g 2 g
is smooth by property (2.1). Using the formula ad†ξ (u, c) := Tid Ad†− (u, c) (ξ ) = cξ + 2uξ + u ξ
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for the infinitesimal coadjoint action, we now show that ker(T Ad†− (u, c)) is a subbundle of s+3 1 s+3 1 T D+ (S ). For each g ∈ D+ (S ), by right-invariance, we have ker Tg Ad†− (u, c) = T Rg ker Te Ad†− (u, c) = T Rg ker ad†− (u, c) , where T Rg is the tangent map to the right composition of diffeomorphisms. So we obtain ker T Ad†− (u, c) =
s+3 1 g∈D+ S
T Rg ker ad†− (u, c) .
Since ker(T ad†− (u, c)) = {ξ ∈ Xs+3 (S 1 ) | cξ + 2uξ + u ξ = 0} is finite dimensional ( 3), we s+3 1 (S ), by Lemma 3 in Appendix A of [8]. obtain that ker(T Ad†− (u, c)) is a subbundle of T D+ s+3 1 We now show that for each g ∈ D+ (S ), the range of Tg Ad†− (u, c) is closed in Xs (S 1 ). Since we have s+3 1 S = Tg Ad†− (u, c) T Rg Xs+3 S 1 Tg Ad†− (u, c) Tg D+ = T(u,c) Ad†g ad†− (u, c) Xs+3 S 1 , it suffices to show that ad†− (u, c)(Xs+3 (S 1 )) is closed in Xs (S 1 ). By the classification of the coadjoint orbits presented in the previous section, we know that there exists a diffeomorphism s+3 1 (S ) such that (u, c) = Ad†g (u, c), where g ∈ D+ u ∈ ell(α, n)2c, hyp(β, n)2c, par(ν, n)2c , that is, u is a smooth vector field. By a general property of coadjoint actions we know that ad† − (u, c) = Ad†g −1 ◦ ad†− Ad†g (u, c) ◦ Adg −1 = Ad†g −1 ◦ ad†− (u, c) ◦ Adg −1 . Consider the linear map ad†− (u, c) : Xs+3 (S 1 ) → Xs (S 1 ), ξ → cξ + 2uξ + u ξ . Since c = 0 and u is smooth, the map ad†− (u, c) is a third order elliptic differential operator. So its range is closed in Xs (S 1 ). 2 This lemma immediately yields the following result. Theorem 6.2. For u ∈ Xs+k (S 1 ), s > 1/2, k 1, and c = 0, the coadjoint isotropy groups s+3 1 D+ S (u,c) s+3 1 are closed C k submanifolds of D+ (S ), and the tangent space at the identity is given by
s+3 1 Te D + S (u,c) = ξ ∈ Xs+3 S 1 cξ + 2uξ + u ξ = 0 .
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Remarks on the Lie group structure. The isotropy group of a vector field u ∈ Xs+k (S 1 ), s+3 1 (S ), but also a C k Lie group, since it is s > 1/2, k 1, is not only a C k submanifold of D+ n k 1 1) or R+ × Zn . Thus, the tangent C conjugated to a smooth Lie group, given by S , PSU(1, s+3 1 (S )(u,c) ) is a Lie algebra. In other words, there is no derivative loss in the Lie space Te (D+ bracket. This fact, that can be checked directly, is in some sense remarkable since the tangent s+3 1 (S )) to the whole diffeomorphism group is not a Lie algebra. This is of course space Te (D+ s+3 1 due to the fact that D+ (S ) itself is not a Lie group. 7. Coadjoint orbits are closed submanifolds From Lemma 6.1 and using for u a smooth representative vector field, we obtain that Ad†− (u, c) is a smooth subimmersion, so the induced map on the smooth manifold s+3 1 s+3 1 D+ (S )/D+ (S )(u,c) is a smooth injective immersion. The fact that the quotient space s+3 1 s+3 1 D+ (S )/D+ (S )(u,c) is a smooth manifold follows from Proposition 10 of Chapter III, §1, s+3 1 s+3 1 in [3]. Indeed, the action of the Lie group D+ (S )(u,c) on the Hilbert manifold D+ (S ), given by (α, g) → α ◦ g, is smooth and proper. Hence we obtain that every coadjoint orbit Os is a smooth immersed submanifold of Xs (S 1 ), s > 1/2. In order to show that the coadjoint orbits are closed in Xs (S 1 ), s > 1/2, we will use a complete coadjoint invariant, called the lifted monodromy, see [18] and [4], which takes values in the R) of SL(2, R). conjugacy classes of the universal cover SL(2, For u ∈ Xs (S 1 ), s > 1/2, consider the unique normalized base of solution Ψu = (ψu,1 , ψu,2 ) of the Hill’s equation such that
(0) ψu,1 (0) ψu,2 1 0 = . (7.1) ψu,1 (0) ψu,2 (0) 0 1 Note that its monodromy matrix is given by
ψu,1 (1) MΨ u = ψu,1 (1)
(1) ψu,2 ψu,2 (1)
∈ SL(2, R).
We will also need the following Wronskian map
ψu,1 (x) Wu : R → SL(2, R), Wu (x) := ψu,1 (x)
(x) ψu,2 ψu,2 (x)
=
Ψu (x) . Ψu (x)
R), Note that we have Wu (0) = I2 and Wu (1) = MΨu . Recall that the universal cover SL(2, can be identified with the set of homotopy classes of the Banach manifold P(I2 , SL(2, R)) := {c ∈ C 0 ([0, 1], SL(2, R)) | c(0) = I2 }, and that the quotient map R), P I2 , SL(2, R) → SL(2,
c → [c]hom
R) → is continuous (see [6] for example). The projection on SL(2, R) is given by Π : SL(2, SL(2, R), Π([c]hom ) = c(1). For example we have [Wu ]hom ∈ SL(2, R) and Π([Wu ]hom ) = MΨu . This explains why [Wu ]hom is called the lifted monodromy of u. The following theorem, adapted here to the H s case, can be found in [18] and [4]. We give below a detailed proof.
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Theorem 7.1. The lifted monodromy R)/conj, : Xs S 1 → SL(2, M
(u) := [Wu ]hom u → M , conj
is a complete coadjoint invariant, that is, for all u, v ∈ Xs (S 1 ) we have the equivalence Ous = Ovs
⇐⇒
(u) = M (v), M
where Ous and Ovs are the coadjoint orbits containing u and v respectively. Proof. We denote by p : R2 − {0} → RP1 and π : SL(2, R) → PSL(2, R) := SL(2, R)/Z2 the natural projections. The elements of PSL(2, R) will be denoted by [M] = π(M) = ±M, with M ∈ SL(2, R). Let the group SL(2, R) act on R2 − {0} by right multiplication, that is (M, (a, b)) → (a, b)M. Consequently, PSL(2, R) acts on RP1 by the natural induced action ([M], z) → p((a, b)M), where (a, b) ∈ p −1 (z). Denote by Imm(R, RP1 ) the set of all immersions f : R → RP1 of class C 1 . Up to a sign, there exists only one map fˆ = (fˆ1 , fˆ2 ) : R → R2 − {0} of class C 1 such that p ◦ fˆ = f and fˆ1 fˆ2 − fˆ1 fˆ2 = 1. Given f ∈ Imm(R, RP1 ), we can define the map f: R → PSL(2, R),
f(x) :=
fˆ1 (0) fˆ1 (0)
fˆ2 (0) −1 fˆ1 (x) fˆ2 (0) fˆ1 (x)
fˆ2 (x) . fˆ2 (x)
(7.2)
Note the three following important properties. (1) We have f(0) = [I2 ]. (2) Given u ∈ Xs (S 1 ), s > 1/2, we can define f := fu := p ◦Ψu , and we get fu ∈ Imm(R, RP1 ), fˆu = Ψu , and fu = π ◦ Wu . (3) If there exists [M] ∈ PSL(2, R) such that f (x + 1) = f (x)[M], then f(x + 1) = f(x)[M], so by (1) we obtain that f(1) = [M]. For example, we have fu (x + 1) = fu (x)[MΨu ]. Let u, v ∈ Xs (S 1 ) be in the same orbit, we will show that [[Wu ]hom ]conj = [[Wv ]hom ]conj . g s+3 1 (S ) such that v = Ad†g (u, 1/2). Since Ψv and Ψu are normalized bases There exists g ∈ D+ g of solutions associated to v, there exists A ∈ SL(2, R) such that Ψv = Ψu A. Therefore we obtain that fv = (fu ◦ g)[A]. Indeed,
g 1 fv (x) = p Ψv (x) = p Ψu (x)A = p Ψu (g(x))A g = p Ψu g(x) A = fu g(x) [A]. Let us show that [fu ]hom is conjugated to [(fu ◦ g)[A]]hom . First, we note that fu is homotopic to fu ◦ g. Indeed, it suffices to consider the homotopy H given by H : [0, 1] × [0, 1] → PSL(2, R),
H (t, x) := f u ◦ gt ,
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s+3 1 where gt is a curve in D+ (S ) such that g0 = Id and g1 = g. Observe that we have
u (x) H (0, x) = (f u ◦ Id)(x) = f H (1, x) = (f u ◦ g)(x) u (0) H (t, 0) = (f u ◦ gt )(0) = [I2 ] = f u (1). H (t, 1) = (f u ◦ gt )(1) = [Mψu ] = f For the third equality we used the property (1). For the last one we used property (3) and the fact that fu (x + 1) = fu (x)[MΨu ] implies (fu ◦ gt )(x + 1) = fu (gt (x) + 1) = (fu ◦ gt )(x)[MΨu ]. u (1). This proves that (f u ◦ gt )(1) = [Mψu ] = f R) to [f]hom . Indeed, conjuSecond, we note that [f (_ )[A]]hom is conjugated in PSL(2, −1 (_)[A] = gation by [A] is given by I[A] ([c]hom ) = [[A]c(_)[A] ]hom so by (7.2) we get f −1 [A] f (_)[A]. Thus we obtain I[A] ([f (_)[A]]hom ) = [f ]hom . Hence we have shown that [fu ]hom is conjugated to [fu ◦ g[A]]hom , that is, [π ◦ Wu ]hom is R) of PSL(2, R). Since PSL(2, R) = conjugated to [π ◦ Wv ]hom in the universal cover PSL(2, (v). SL(2, R), we obtain that [Wu ]hom is conjugated to [Wv ]hom in SL(2, R), that is, M(u) = M s s Conversely, suppose that M(u) = M(v), and let us show that Ou = Ov . Since M(u) = M(v), we must have one of the following three cases: u ∈ Ells (α, n), u ∈ Hyps (β, n),
v ∈ Ells (α, m)
or
v ∈ Hyps (β, m)
or
u ∈ Pars (ν, n),
v ∈ Pars (ν, m),
with m − n = 2k, k ∈ Z. Since the three cases are similar, we only treat the elliptic case. Without loss of generality, we can suppose that u = ell(α, n) = (α + nπ)2 and v = ell(α, n + 2k) = (α + (n + 2k)π)2 . We have
Wu (x) =
cos(α + nπ)x 1 α+nπ sin(α + nπ)x
−(α + nπ) sin(α + nπ)x cos(α + nπ)x
and
Wv (x) =
cos(α + (n + 2k)π)x 1 α+(n+2k)π sin(α + (n + 2k)π)x
−(α + (n + 2k)π) sin(α + (n + 2k)π)x cos(α + (n + 2k)π)x
,
which is conjugated to
W (x) =
cos(α + (n + 2k)π)x 1 α+nπ sin(α + (n + 2k)π)x
−(α + nπ) sin(α + (n + 2k)π)x cos(α + (n + 2k)π)x
.
Remark that Wu and W are curves in SL(2, R) with same endpoints. Since they are homotopic, we must have k = 0. This proves that m = n and then that Ous = Ovs . 2
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Lemma 7.2. The lifted monodromy R)/conj, M : Xs S 1 → SL(2, (u) = [Wu ]hom M conj is a continuous map. Proof. Since the maps R) Ψu ∈ C 1 [0, 1], R2 → Wu ∈ P I2 , SL(2, R) → [Wu ]hom ∈ SL(2, are continuous, it suffices to prove that the map u ∈ Xs (S 1 ) → Ψu ∈ C 1 ([0, 1], R2 ) is continuous, that is, for each continuous curve t → ut in Xs (S 1 ), the curve t → Ψut is continuous in C 1 ([0, 1], R2 ). Suppose that t ∈ K where K is a compact interval of R. Consider the map u : K × [0, 1] → R, defined by u(t, x) := ut (x). Using the Sobolev embedding theorem, one can show that u is continuous on K × [0, 1]. Consider the equation ψ (x) + u(t, x)ψ = 0 with parameter t. This equation is equivalent to
ψ ϕ
=
0 1 −u 0
ψ ϕ
.
The normalized base of solutions of this equation, verifying the initial conditions (7.1), is given by Ψut . Since u depends continuously on (t, x), by the theorem of continuous dependence of the solutions on a parameter, we obtain that for each x ∈ [0, 1] the map t → (ψut (x), ψu t (x)) is continuous. By compactness of [0, 1], this proves that t → ψut is continuous as a map with values in C 1 ([0, 1], R). 2 The following theorem shows that all coadjoint orbits, except for Pars (±1, n), n 1, are closed subsets of Xs (S 1 ). The proof uses in a crucial way the fact that the lifted monodromy is a continuous map and a complete coadjoint invariant. Theorem 7.3. Let Os be a coadjoint orbit such that Os = Pars (±1, n), n 1. Then Os is a closed subset of Xs (S 1 ). Proof. Let (un ) ⊂ Os be a sequence of vector fields converging to u ∈ Xs (S 1 ) in the H s topolR)/conj. Since M (u) in SL(2, is (un ) converges to M ogy. By Lemma 7.2, we know that M constant on the coadjoint orbits, we obtain that the sequence M(un ) has a constant value denoted (u) contains m. Since the topological space by m. So we obtain that each neighborhood of M R)/conj is not a Hausdorff space, we cannot immediately conclude that m = M (u). HowSL(2, ever, using the remark below concerning the topology of SL(2, R)/conj and the fact that m does (u). Using Theorem 7.1 we obtain that u is an not project to Par± (±1), we obtain that m = M s s element of O and therefore that O is closed. 2
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Fig. 1. The space of coadjoint orbits of BVir s+3 acting on Xs (S 1 ) for nonzero central charge. The “comb” together with the “floating points” is the space of coadjoint orbits. The vertical lines, each of which is parametrized by β > 0, and labeled by an integer n = 0, 1, 2, . . . , represent the hyperbolic orbits Hyps (β, n). The non-integer points on the horizontal axis represent the elliptic orbits Ells (α, n). The integer points represent those parabolic orbits Pars (ν, n) with ν = 0. The parabolic orbits Pars (ν, n) with ν = ±1 and n = 0, 1, 2, . . . are represented by the “floating points.” The open circle at n = 0, β = 0 is an empty point with no corresponding orbit. The groups under the orbit labels are the associated coadjoint isotropy subgroups. The points representing Pars (±1, n) can be separated from any other point, but the points representing Pars (0, n) contain in any neighborhood the two “floating points” next to it. The rest of the topology of the “comb” is that induced from the plane. All orbits are closed submanifolds of Xs (S 1 ) with the possible exception of Pars (±1, n) whose closure could intersect only Pars (0, n) for each n = 1, 2, 3, . . . . Figure adapted from [2].
(u) is not necessarily Note that if m projects to Par± (±1), in the previous proof, then M (u) can project to Par± (0), since each neighborhood of Par± (0) contains equal to m. Indeed, M Par± (±1). This shows that for a sequence (un ) ⊂ Par(±1, n) which converges to u ∈ Xs (S 1 ), we only know that u ∈ Par(±1, n) ∪ Par(0, n). This does not exclude that the orbit Par(±1, n) is also closed. Note that this argument shows that the orbit Par(1, 0) is closed. This is the only orbit among the “floating points” in Fig. 1 which is known to be closed. R)/conj. Remark on the topology of PSL(2,
Consider the trace map
Tr : SL(2, R) → R. We know that a matrix A is hyperbolic if | Tr(A)| > 2, is elliptic if | Tr(A)| < 2, and parabolic if | Tr(A)| = 2. This shows that the sets of hyperbolic and elliptic matrices are open subsets of SL(2, R) and that the set of parabolic matrices different from ±I2 is a submanifold of SL(2, R) of dimension 2. Indeed, the map Tr : SL(2, R) → R is a submersion at A if and only if A is not equal to ±I2 . We endow SL(2, R)/conj with the quotient topology. From the preceding considerations it follows that for each m, n ∈ SL(2, R)/conj, m = n, such that (m, n) = (Par+ (0), Par+ (±1)) and (m, n) = (Par− (0), Par− (±1)) we can find two disjoint open subsets, one containing m and the
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other containing n. If m = Par+ (0) and n = Par+ (±1), then we can find an open neighborhood of n which does not contain m, but all open neighborhoods of m contain n. This shows that SL(2, R)/conj, endowed with the quotient topology, is not a Hausdorff space. Remark that SL(2, R)/conj − {Par± (±1)} is Hausdorff. The same observations apply to the quotient space R)/conj, since these spaces are locally homeomorphic. PSL(2, In order to prove that the coadjoint orbits are submanifolds, we will need the following result (see Proposition 14(iii) in Chapter III of [3]). Proposition 7.4. Let Φ : G × M → M be a smooth action of a Banach Lie group G on a Banach manifold M. Let m ∈ M and suppose that the orbit map Φ m : G → M is a subimmersion, and that the orbit Orb(m) containing m is locally closed. If the topology of G is second countable, then Orb(m) is a submanifold of M and Tm Orb(m) = Im Te Φ m . s+3 1 (S ) and Φ = An inspection of the proof shows that this result is still valid if G = D+ s+3 1 † † Ad , even though D+ (S ) is not a Banach Lie group and Ad is not a smooth action. From Lemma 6.1 we know that the orbit map is a subimmersion, so it remains to show that the topology s+3 1 of D+ (S ) is second countable. From Theorem 5.1 in [8] we know that for a compact manifold M with volume form μ, Dr (M) is diffeomorphic to Dμr (M) × Vμr (M), where Dμr (M) is the manifold of volume preserving diffeomorphisms of M and Vμr (M) consist of H s -volume forms ν on M such that 1 M ν = M μ. Applying this result to the case of S with a volume form dx, we obtain that r (S 1 ) is diffeomorphic to S 1 × V r (S 1 ). In this case we have D+ dx
1 r Vdx S = f ∈ H s S 1 f > 0 and f = dx . S1
S1
r (S 1 ) is also S 1 is obviously second countable. As an open set in an affine Hilbert space, Vdx r (S 1 ) is second countable. second countable. This proves that D+ By 7.3 we know that the coadjoint orbits (except Pars (±1, n)) are closed, so they are locally closed. Thus, using 7.4, we obtain the following result.
Theorem 7.5. Let Os be a coadjoint orbit such that Os = Pars (±1, n). Then Os is a closed submanifold of Xs (S 1 ). The tangent space to Os at v := Ad†g (u, c), where u is a smooth representative vector field, is given by Tv Os = Im Tg Ad†− (u, c) . In particular, if v ∈ Os is in H s+1 we have Tv Os = cξ + 2vξ + v ξ ξ ∈ Xs+3 S 1 . References [1] R. Abraham, J.E. Marsden, T.S. Ratiu, Manifolds, Tensor Analysis, and Applications, Appl. Math. Sci., vol. 75, Springer-Verlag, 1988.
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[2] J. Balog, L. Fehér, L. Palla, Coadjoint orbits of the Virasoro algebra and the global Liouville equation, Internat. J. Modern Phys. A 13 (1998) 315–362. [3] N. Bourbaki, Groupes et Algèbres de Lie, Hermann, Paris, 1972, Chapitres 2 et 3. [4] J. Dai, Conjugacy classes, characters and coadjoint orbits of Diff† (S 1 ), PhD thesis, University of Arizona, 2000. [5] J. Dai, D. Pickrell, The orbit method and the Virasoro extension of Diff† (S 1 ), I. Orbital integrals, J. Geom. Phys. 44 (2003) 623–653. [6] J.J. Duistermaat, J.A.C. Kolk, Lie Groups, Universitext, Springer-Verlag, Berlin, 2000. [7] D.G. Ebin, The manifold of Riemannian metrics, Proc. Sympos. Pure Math. 15 (1968). [8] D.G. Ebin, J.E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970) 102–163. [9] A.E. Fischer, A.J. Tromba, On a purely “Riemannian” proof of the structure and dimension of the unramified moduli space of a compact Riemann surface, Math. Ann. 267 (3) (1984) 311–345. [10] F. Gay-Balmaz, Infinite dimensional geodesic flows and the universal Teichmüller space, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, 2008. [11] F. Gay-Balmaz, T.S. Ratiu, Group actions on chains of Banach manifolds and application to fluid dynamics, Ann. Global Anal. Geom. 31 (3) (2007) 287–328. [12] L. Guieu, C. Roger, L’algèbre et le groupe de Virasoro. Aspects géométriques et algébriques, généralisations, CRM Publications, Montreal, 2007. [13] J. Isenberg, J.E. Marsden, A slice theorem for the space of solutions of Einstein equations, Phys. Rep. 89 (1982) 179–222. [14] A.A. Kirillov, Infinite-dimensional Lie groups: Their orbits, invariants and representations, in: The Geometry of Moments, in: Lecture Notes in Math., vol. 970, Springer-Verlag, 1982, pp. 101–123. [15] A. Kriegl, P.W. Michor, The Convenient Setting of Global Analysis, Math. Surveys Monogr., vol. 53, Amer. Math. Soc., Providence, RI, 1997. [16] V.P. Lazutkin, T.F. Pankratova, Normal forms and versal deformations for Hill’s equations, Funct. Anal. Appl. 9 (4) (1975) 306–311. [17] R.S. Palais, Foundations of Global Non-Linear Analysis, W.A. Benjamin, Inc., New York–Amsterdam, 1968. [18] G. Segal, Unitary representations of some infinite-dimensional groups, Comm. Math. Phys. 80 (1981) 301–342. [19] E. Witten, Coadjoint orbits of the Virasoro group, Comm. Math. Phys. 114 (1) (1988) 1–53.
Journal of Functional Analysis 256 (2009) 2842–2866 www.elsevier.com/locate/jfa
Linearizing non-linear inverse problems and an application to inverse backscattering Plamen Stefanov a,1 , Gunther Uhlmann b,∗,2 a Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA b Department of Mathematics, University of Washington, Seattle, WA 98195, USA
Received 27 July 2008; accepted 20 October 2008 Available online 7 November 2008 Communicated by Paul Malliavin
Abstract We propose an abstract approach to prove local uniqueness and conditional Hölder stability to nonlinear inverse problems by linearization. The main condition is that, in addition to the injectivity of the linearization A, we need a stability estimate for A as well. That condition is satisfied in particular, if A∗ A is an elliptic pseudo-differential operator. We apply this scheme to show uniqueness and Hölder stability for the inverse backscattering problem for the acoustic equation near a constant sound speed. © 2008 Elsevier Inc. All rights reserved. Keywords: Linearization; Stability; Inverse backscattering
1. Introduction Non-linear inverse problems are often linearized, and reduced to the problem of the injectivity of their linearization. If that linearization is an injective map with a closed range, then this implies local uniqueness and Lipschitz stability, see Theorem 1 below. Most inverse problems in Mathematical Physics however are ill-posed and do not fall in this category. The mere fact that the linearization is injective, if it is, is not enough for local injectivity of the original problem when the closed range condition fails. The closed range condition is also equivalent to a linear * Corresponding author.
E-mail address:
[email protected] (G. Uhlmann). 1 Partly supported by NSF Grant DMS-0800428. 2 Partly supported by NSF and a Walker Family Endowed Professorship.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.017
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stability estimate, see (11) below. The main purpose of this work is to propose a systematic approach to treat inverse problems for which the closed range condition may still hold if we replace the original spaces with new ones satisfying interpolation estimates, see (11). A typical exam ple is to replace C k or H s by C k , respectively H s , with different k , s . A sufficient condition for the linear stability estimate, besides the injectivity of the linearization A, is A∗ A to be an elliptic pseudo-differential operator (ΨDO). The later is a standard consequence of the theory of elliptic ΨDOs and elliptic ΨDOs. Our main result in this direction is Theorem 2 that states, roughly speaking, that linearization plus an appropriate stability estimate of the linearized map imply local injectivity and a conditional Hölder stability for the non-linear map. We recently used this approach for studying boundary rigidity/lens rigidity questions for compact Riemannian manifolds with boundary [17,18,20]. Some of these ideas have been used before, on a case by case basis, for proving local uniqueness with or without a Hölder stability, see for example [6,10,11,15,16]. Our goal is to systematize this approach, and in particular to understand what makes (at least a large enough class of) ill-posed inverse problems ill posed, and in what cases one can get local uniqueness by linearization. In particular, we show why many severely ill-posed inverse problems cannot be treated this way — the linearization is unstable in any pair of Sobolev spaces. The most famous example of this type is Calderón’s problem [4,22,25]. We give an application to the inverse backscattering problem for the acoustic equation. In this problem, we are trying to determine the sound speed or, equivalently, the index of refraction, of a medium by measuring the scattering amplitude with the direction of incidence of a plane wave opposite to the direction of the reflected wave. In other words, we are measuring the echos produced by plane waves. This problem arises for instance in ultrasound tomography. We prove injectivity near constant sound speeds and we give a Hölder type conditional stability estimate, see Theorem 4. Conditional Lipschitz stability estimates in different norms were given in [26] based on the local uniqueness proof in [16]. In Section 2 we prove the main result — Theorem 2. In Section 3, we give sufficient conditions for the stability of the linearized inverse problem required by Theorem 2. Those conditions are based on standard elliptic ΨDO theory and Fredholm theory. Finally, in Section 4, we give the application to the inverse backscattering problem. 2. An abstract inverse problem We describe our abstract inverse problem now. Let A : B1 → B2 be a continuous non-linear map between two Banach spaces. Consider the “inverse problem”: Given h ∈ Ran(A), find f so that A(f ) = h.
(1)
Here, we study only the (local) uniqueness and stability questions for (1). Definition 1. We say that there is a weak local uniqueness for (1) near f0 ∈ B1 , if there exists a neighborhood of f0 so that for any f in that neighborhood that solves A(f ) = A(f0 ), we have f = f0 .
(2)
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Definition 2. We say that there is a strong local uniqueness for (1) near f0 ∈ B1 , if there exists a neighborhood of f0 so that for any f1 , f2 in that neighborhood with A(f1 ) = A(f2 ),
(3)
we have f1 = f2 . Besides uniqueness, we are often interested in stability estimates. By a classical argument, if A : U → V is injective, where U and V are open, and K ⊂ U is compact, then A−1 , restricted to A(K), is continuous. The compactness assumption is one of the ways ill-posedness of many (but not all) inverse problems manifests itself. The continuity of A−1 is viewed as stability, and such kind of results are called conditional stability. One of the central problems in Inverse Problems is to estimate the modulus of continuity of A−1 (restricted appropriately). In other words, we want to find a function φ(t), t → 0, so that φ → 0, as t → 0, and f1 − f2 B1 φ A(f1 ) − A(f2 )B 2
for all f1 , f2 in some neighborhood of a fixed f0 , and possibly also, restricted to a compact subset K of B1 (hence the stability is conditional). The estimate is of Hölder type, if one can choose φ(t) = Ct α , 0 < α 1, and of Lipschitz type, if φ(t) = Ct. We view in this paper inverse problems that allow Hölder estimates as “stable” (Lipschitz stability occurs rarely); while any other are viewed as “unstable.” Recall that for the Electric Impedance Tomography problem, see [4,22,25], φ(t) = C(log(1/t))−μ , μ > 0, see [2], that tends to 0, as t → 0, much slower than t α . The choice of the compact set K also matters. If it is “too small,” the conditional stability is trivial and not interesting. For example, if K is finite-dimensional, and A is smooth enough, one always has a Lipschitz stability estimate under the assumption that A has an injective differential, see below. If B1 is H s (M) or C k (M) with a compact set (or manifold) M, we are interested whether one can choose K to be H s1 (M) or C k1 (M) with s1 > s, or k1 > k. A subspace K consisting of real analytic functions only, for example, will be considered “too small.” 2.1. “Inverse problems” in Rn We start with the trivial case when B1 = Rn , B2 = Rm . Then the problem is reduced to solving (1) with A : Rn → Rm . Assume that A ∈ C 2 (A ∈ C 1 actually suffices, see Theorem 1 below). Let Ax0 be the differential of A at x = x0 . Then A(x) = A(x0 ) + Ax0 (x − x0 ) + Rx0 (x)
with Rx0 (x) Cx0 |x − x0 |2 ,
(4)
for x near x0 . Assume now that Ax0 is injective (which can happen only when m n). This immediately implies the estimate |h| C0 |Ax0 h|,
∀h ∈ Rn .
(5)
The easiest way to prove (5) is to note that one can choose 1/C0 to be the minimum of |Ah| for h on the (compact) unit sphere. Then the local uniqueness question about the number of possible solutions of A(x) = A(x0 ) near x = x0 can be answered as follows. We have
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|x − x0 |/C0 Ax0 (x − x0 ) A(x) − A(x0 ) + Rx0 (x) A(x) − A(x0 ) + Cx0 |x − x0 |2 . Therefore, if |x − x0 | (2C0 Cx0 )−1 , one gets |x − x0 | 2C0 A(x) − A(x0 ).
(6)
We therefore have local uniqueness and a Lipschitz stability estimate. Moreover, one can choose Cx0 to be locally independent of x0 , if A ∈ C 2 , and the same applies to C0 . Thus one gets that there exists a neighborhood U of x0 in Rn , so that for any x1 ∈ U , x2 ∈ U , the equality A(x1 ) = A(x2 ) implies x1 = x2 . Moreover, similarly to (6), one has |x1 − x2 | 2C0 A(x1 ) − A(x2 ),
x1 ∈ U, x2 ∈ U.
(7)
Note that B2 can be infinite-dimensional here, and the arguments do not change. In particular, if A ∈ C 1,μ , μ > 0, and Ax0 is injective, then one always has local uniqueness and the stability estimate (7) of Lipschitz type. If B1 is infinite-dimensional, then A|K has that property for any finite-dimensional K. 2.2. A well-posed inverse problem. The local injectivity theorem in Banach spaces We return to the case where B1,2 are finitely-dimensional Banach spaces. Consider now a truly infinitely-dimensional version of our abstract inverse problem under assumptions that make it well posed, and therefore, not typical. The main assumption is that Ax0 : B1 → Ax0 (B1 ) is an isomorphism, i.e., that the differential Ax0 is not only injective but also has a closed range. Then we have the following theorem that is closely related to the inverse function theorem and the implicit function theorems in Banach spaces (Ax0 is assumed to be an isomorphism from B1 to B2 then). Theorem 1. (Local injectivity theorem [1].) Let A : U ⊂ B1 → B2 be C 1 , with U f0 open, let Af0 be injective and let it have a closed range. Then there exists a neighborhood V ⊂ U of f0 , on which A is injective. Moreover, the inverse A−1 : A(V ) → U is Lipschitz continuous. This theorem shows that we have the same conclusions as in the finitely-dimensional case, with a Lipschitz stability of non-conditional type. We refer to [1] for the definition of the differential Af0 of A at a fixed f0 and for a definition of (a Fréchet) differentiable map. In particular, if the Gâteaux derivative (the directional derivative in all directions) exists in some open set and it is continuous there, then A is Fréchet differentiable as well and the two derivatives coincide. Then A is said to be C 1 . 2.3. An abstract ill-posed inverse problem We turn out attention now to an abstract inverse problem that may not be well posed. In order to study the inverse problem (1) by linearization, we assume that the differential Af0 of A exists at f = f0 with some f0 . Actually, we will assume something slightly stronger — a quadratic estimate on the remainder, that roughly speaking means that A is C 1,1 . Namely, we assume that
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A(f ) = A(f0 ) + Af0 (f − f0 ) + Rf0 (f ),
with Rf0 (f )B Cf0 f − f0 2B1 , 2
(8)
for f in some neighborhood in B1 of some f0 ∈ B1 . It may happen that we have the weaker estimate Rf (f ) ˜ Cf f − f0 2 0 0 ˜ B
B1
2
(9)
for f in some neighborhood in B˜1 of some f0 ∈ B˜1 , where B˜1 ⊂ B1 , B˜2 ⊃ B2 . Here and below, B˜1 ⊂ B1 , B˜2 ⊃ B2 mean also that · B1 C · B˜ , 1
· B˜ C · B2 . 2
(10)
In such case, we can simply replace B2 by B˜2 ; and B1 by B˜1 . In the boundary rigidity problem [18], actually B˜1 = C 1 (M) ⊃ B1 = C 2 (M), i.e., (8) is even stronger. Keeping B˜1 , B˜2 and assuming appropriate interpolation estimates would improve a bit the exponent μ1 μ2 in the Hölder exponent in Theorem 2 below but for the sake of simplicity, we will not pursue this. In other words, we have some freedom how to chose our spaces B1,2 , and we choose them in a way that makes (8) true. This choice might not be the optimal for the analysis of Af0 however. Assume now that Af0 is injective. This does not imply automatically a stability estimate of the type (5), of course. Assume in addition, that such an estimate holds in some norms, i.e., hB CAf0 hB , 1
2
∀h ∈ B1 ,
(11)
with some Banach spaces B2 ⊂ B2 , B1 ⊃ B1 . In applications, we want not only such B1 , B2 to exist (they always do, one can set B1 = B1 , Af0 hB = hB1 on Ran Af0 ) but we also want 2 they to be “reasonable” spaces independent of f0 , typically some H s or C k spaces. Then, if B1 = B1 , B2 = B2 one can apply Theorem 1 to prove local injectivity and a Lipschitz stability estimate of the type (7). The inverse problem is well-posed then. If the pairs of spaces in (8) and (11) cannot be chosen to be the same, and often that is the case, one can still prove local uniqueness and a stability estimate but of a Hölder and conditional type, if certain interpolation estimates are satisfied, as we show below. Then the inverse problem is ill-posed but we think of it as mildly ill-posed. Before formulating this as a formal statement, we will give an example where (11) does not hold (in any Sobolev spaces), and there is no local uniqueness. 2 |xk |2 . Example 1. Let l 2 be the Hilbert space of sequences x = {xk }∞ k=1 with norm x = 2 s 2s 2 The Sobolev spaces h are defined through the norms xhs = k |xk | . We think of the components of x as the Fourier coefficients of a 2π -periodic function, which explains why we call hs a Sobolev space. Define the non-linear map A by A(x) = Ex − (x, a)x, where a = {1/k},
E = diag e−k .
(12)
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Clearly, A(0) = 0, the linearization of A near x = 0 is A0 = E, and the latter is an injective map. Moreover, there is an inverse E −1 with a dense domain, but E −1 is unbounded as an operator from any Sobolev space to any other one, thus (11) does not hold in such spaces. Since [A(x)]k = e−k xk − xk xm /m, we get that x (k) = 0, . . . , ke−k , 0, . . . , where the non-zero entry is the kth one, is a solution to A(x) = 0. Therefore, in any neighborhood of 0 in l 2 there are infinitely many solutions of A(x) = 0 despite the fact that the linearization A0 is injective. Moreover, for any s, in any neighborhood of 0 in hs , there are still infinitely many solutions. Therefore, there is no local uniqueness in this case, neither weak nor strong, in any Sobolev space. One can still get local uniqueness for x∗ 1 by choosing an h∞ type of norm · ∗ with an exponential weight, namely x∗ = E −1 x. This however translates into a neighborhood of the origin of 2π -periodic functions that involves certain real analytic functions only. We think of such topology as “unreasonably restrictive.” Based on that example, one can consider the following map in L2 (Rn ) A(f ) = φ ∗ f − (f, a)f. ˆ ) = χ{k−1|ξ |k} (ξ )e−k|ξ | , where χK stands for the characteristic function of K. Choose φ(ξ Also, fix a ∈ L2 with |a(ξ ˆ )| (1 + |ξ |)−m for some m. Then f = 0 solves A(f ) = 0, and A has an injective differential at f = 0. Also, if fˆ is supported in {k − 1 |ξ | k}, and fˆ solves (fˆ, a) ˆ = (2π)n e−k|ξ | , such an f would also solve A(f ) = 0. This gives us a sequence of such solutions, converging exponentially fast to 0 in any H s . Our main abstract theorem for linearizing inverse problems with a “stable” linearization is the following. Theorem 2. Let A be as above, and assume (8), (11) with B1 ⊂ B1 , B2 ⊂ B2 as above. Assume also that there exist Banach spaces B2 ⊂ B2 , B1 ⊂ B1 so that Af0 continuously maps B1 into B2 and the following interpolation estimates hold μ
1−μ
uB2 CuB22 uB 2 , 2
1−μ
μ
hB1 ChB1 hB 1 , 1
1
μ1 , μ2 ∈ (0, 1], μ1 μ2 > 1/2. (13)
(a) For any L > 0 there exists > 0, so that for any f with f − f0 B1 ,
f B1 L,
(14)
one has the conditional stability estimate μ μ f − f0 B1 C(L)A(f ) − A(f0 )B1 2 , 2
C(L) = CL2−μ1 −μ2 .
(15)
In particular, there is a weak local uniqueness near f0 , i.e., if A(f ) = A(f0 ), then f = f0 .
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(b) Assume in addition that there is a Banach space K ⊂ B1 so that (11) holds for f0 replaced with f close enough to f0 in K, and Af : B1 → B2 is uniformly bounded for such f . Then there exists > 0, so that for any f1 , f2 with f1 − f0 K ,
f2 − f0 K ,
(16)
one has the conditional stability estimate μ μ f1 − f2 B1 C A(f1 ) − A(f2 )B1 2 . 2
(17)
In particular, there is a strong local uniqueness near f0 , i.e., if A(f1 ) = A(f2 ), then f1 = f2 . A theorem of a similar type has been proven by Pagani [10,11]. Remark 1. Typical choices of the Banach spaces above are C k or H s spaces. Interpolation estimates for those norms are well known. In Rn , for example, one has uH s CuαH1s1 uαH2s2 ,
s = s1 α1 + s2 α2 , α1 + α2 = 1, α1 0, α2 0.
(18)
This can be easily verified by using the Fourier transform. Similar estimates hold for Sobolev spaces and the C k spaces in domains/manifolds, see [24]. For example, if M is a compact manifold with boundary, then uC k (M) CuαC1k1 (M) uαC2k2 (M) ,
k = k1 α1 + k2 α2 , α1 + α2 = 1, α1 0, α2 0. (19)
Interpolation estimates that include an H s and a C k norm can be obtained by using Sobolev embedding theorems. Remark 2. Assume, for example, that all spaces above are Sobolev spaces. Then one can make the exponent μ1 μ2 in (15), (17) arbitrary close to 1, thus making the stability estimate “almost Lipschitz.” The price to pay for that is to assume that f is a priori bounded in H s with s 1, see (18). Alternatively, one may try to choose B1 , K close to B1 , which is the natural limit but that will decrease μ1 . Since μ1 > 1/(2μ2 ) 1/2, this puts a limit on how close B1 , K can be to B1 . If B1 = H s , B1 = H s , for example, s < s, then we will have at least B1 = H s with s so (s + s )/2 > s. Therefore, we cannot take s > s arbitrary close to s unless s = s. The choice of s in this case is further restricted by μ2 . Remark 3. The second inequality in (14) is a typical compactness condition, when we work with Sobolev spaces or C k spaces in a bounded domain or on a compact manifold. It may seem strange that there is no such explicit condition in (b). It is actually there, since (16) implies such a condition with B2 replaces by K. Actually, using interpolation estimates, one could write (14) in the form (16), and vice versa, with a slight change of the Banach spaces (i.e., small change of k in C k , etc.). We formulated (16) in a form different than (14) in order to avoid introducing yet another Banach space. What makes (15) and (17) conditional estimates is that in the conditions (14), respectively (16), there is at least one norm stronger that the norm in the left-hand side
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of (15), respectively (17). Note also that (15), (17) can be formulated with different choices of the norms. Proof of Theorem 2. We start with (a). By (8) and (10), Af (f − f0 ) A(f ) − A(f0 ) + Cf − f0 2 . 0 B1 B B 2
2
By (11) and the Hölder inequality (a + b)μ a μ + bμ for a 0, b 0, we get Af (f − f0 ) CL1−μ2 A(f ) − A(f0 )μ2 + f − f0 2μ2 . 0 B1 B B 2
2
1−μ
The constant CL1−μ2 comes from estimating the term Af0 (f − f0 )B 2 that is bounded by the 2 assumption on Af0 and by the second inequality in (14). By (11), and the second interpolation inequality in (13), μ μ 2μ μ f − f0 B1 CL2−μ1 −μ2 A(f ) − A(f0 )B1 2 + f − f0 B1 1 2 . 2
Since 2μ1 μ2 > 1, for 1 we get (15). Note that the condition on has the form (CL2−μ1 −μ2 )−1 , where C depends on A only. This proves (a). To prove (b) we note that the same proof can be applied under the assumptions of (b). Indeed, we start with Af (f2 − f1 ) A(f2 ) − A(f1 ) + Cf2 − f1 2 . 1 B1 B B 2
2
Then we proceed as above under the condition f1 − f0 K 1 1 that would guarantee (11) with f0 replaced by f1 . Then (17) holds if f2 − f1 B1 2 1. Those two conditions certainly hold if in (16) is small enough. 2 Remark 4. The variety of norms in the theorem above is typical. For example, consider the boundary rigidity problem, where one has to recover a Riemannian metric g on a compact manifold M, up to a group of isometries, from the distance function ρ(x, y) restricted to ∂M × ∂M, see e.g., [17,21]. Linearizing, we get a tensor tomography problem: recover a symmetric 2-tensor field f from its integrals If (γ ) =
fij γ (t) γ˙ i (t)γ˙ j (t) dt
(20)
along all maximal geodesics in M. Since this transform vanishes on potential tensors (symmetric covariant derivatives of vector fields v vanishing on ∂M), we restrict I on solenoidal (i.e., divergence free) tensors that form an orthogonal complement to potential ones. The role of Ag0 then is played by I related to g0 . We apply the adjoint operator I ∗ taken in 2 L spaces with respect to suitable measures. The role of Ag0 now is played by I ∗ I , and we made the following choices, see [21]:
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B1 = C 2 (M),
B1 = L2 (M),
B1 = H s (M),
s 1,
K = C k (M),
k s, (21)
B2 = L∞ (M1 ),
B2 = H˜ 2 (M1 ),
B2 = H s−1 (M1 ),
(22)
where M1 M, and H˜ 2 (M1 ) is an appropriate Hilbert space such that H 2 (M1 ) ⊂ H˜ 2 (M1 ) ⊂ H 1 (M1 ). Here and below, A B means that the closure of A is contained in the interior of B; and B A means that A B. There are other complications there, coming from the fact that we have to work with equivalence classes of isometric metrics. Then I ∗ I acting on symmetric tensors in H s (M), extended as zero to M1 , is not a bounded operator with values in H s−1 (M1 ) even though I ∗ I is a ΨDO of order −1. The reason is that f ∈ H s (M), extended as zero, belongs to H s (M1 ) only if the first s normal derivatives of f (assuming that s is integer) vanish on ∂M. This is the reason for introducing the space H˜ 2 . A stable recovery of the derivatives of the metric on ∂M from the boundary distance function is used in [21] to deal with this problem. As a result, we prove a Hölder type of stability estimate for the boundary rigidity problem, using ideas similar to those in the proof of Theorem 2. 3. Stability of linear inverse problems Most of the material in this section is based on the theory of elliptic ΨDOs and on the theory of Fredholm operators, see e.g. [23]. We give proofs for the sake of completeness of the exposition. Let A : B1 → B2 be a bounded linear map between two Banach spaces. Assume in addition, that A is injective. We are interested whether one has the stability estimate f B1 CAf B2
(23)
with some constant C > 0. It is well known, that if A is smoothing, and B1 and B2 are among C k or H s with finite k, or s, then there cannot be such a stability estimate. We will quickly review those arguments below. First, note that (23) is not equivalent to invertibility, because (the closure of) the range Ran A of A can be much smaller than B2 . One the other hand, one can replace B2 by B2 = Ran A, and then (23) is equivalent to the invertibility of A : B1 → B2 , as the next proposition shows. Proposition 1. (23) holds if and only if A is injective and Ran A is closed. Proof. Assume that Ran A is closed and A is injective. Then A : B1 → B2 has range that coincides with B2 . Since A is injective, then A−1 exists with domain the whole B2 . By the open mapping theorem, A−1 is bounded, hence (23) holds. Assume (23). Then the injectivity and the closedness of Ran A follow easily. 2 Proving that Ran A is closed is not always straightforward. Proving that it is not (when it is not), is usually easier. As an example, suppose that B1 = B2 = L2 (Ω), where Ω is a domain in Rn with smooth boundary, or Ω = Rn . If there exists s > 0 so that Af ∈ H s (Ω) for any f ∈ L2 (Ω), then (23) cannot be true. Indeed, if we know that A has a trivial cokernel (for example, because it is injective and self-adjoint, as the operator Nw below), then Ran A cannot be closed. In the general case, assume that Ran A is closed. Then the latter is a Banach space as
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well, and A is invertible on it. But A is compact, so we get that the identity on L2 (Ω) is compact, that is not true. On the other hand, (23) may still hold with B2 = H s (Ω), if Ran A is closed in H s (Ω). Take for example A = (1 − )−1 on L2 (Rn ), and s = 2. Then one can ask the more general question: is there a stability estimate of the type f L2 CAf H s
(24)
for some s > 0? If so, it will be a weaker substitute for (23), if the latter does not hold but it will still allow us to apply Theorem 2. For (24) to be true, A must map L2 into H s , therefore, this is the same question as before, with B2 = H s . Example 2 (The weighted X-ray transform). In Rn , let Iw be the weighted X-ray transform Iw f (x, θ ) =
w(x, θ )f (x + tθ ) dt,
x ∈ Rn , θ ∈ S n−1 ,
(25)
R
where w is a continuous function. Let us restrict Iw to functions supported in Ω, where Ω ⊂ Rn is a strictly convex domain with a smooth boundary. One can assume then that (x, θ ) ∈ ∂− SΩ, which means that x ∈ ∂Ω, and θ ∈ S n−1 points into Ω. We equip ∂− SΩ with the measure dΣ = |θ ·ν(x)| dSx dθ , where dSx is the measure on ∂Ω induced by the Euclidean measure in Rn , and dθ is the measure on S n−1 . Then one can show that Iw : L2 (Ω) → L2 (∂− SΩ, dΣ) [14], see also the remark following (28) below. Let Nw = Iw∗ Iw : L2 (Ω) → L2 (Ω)
(26)
be the “normal operator.” We can think of the lines through Ω as a manifold, and ∂− SΩ then is a global chart for them (singular at the lines tangent to ∂Ω). The measure dΣ remains invariant if we replace ∂Ω by another strictly convex surface encompassing Ω. Thus one can expect that Nw will not depend on ∂Ω. Indeed, on can easily get that (see [17]) Nw f (x) = cn
W (x, y)f (y) dy |x − y|n−1
(27)
with
x −y x−y x−y x −y w y, − + w¯ x, w y, . W (x, y) = w¯ x, − |x − y| |x − y| |x − y| |x − y|
(28)
Now we think of Nw as the operator defined by the formula above, with w(x, θ ) extended for all x. It is easy to see that Nw maps L2comp (Rn ) into L2loc (Rn ). In particular, Nw is bounded on L2 (Ω). That shows that Iw : L2 (Ω) → L2 (∂− SΩ, dΣ) is bounded. Let Ω1 Ω be another domain with the properties of Ω. Then Nw : L2 (Ω) → L2 (Ω1 ) is also bounded, and Iw is injective if and only if Nw : L2 (Ω) → L2 (Ω1 ) is injective, where one can replace Ω1 by Ω as well, see Lemma 1 below. If w ∈ C ∞ (Ω), then Nw is a ΨDO of order −1, as we will see below. Thus Nw maps L2 (Ω) into H 1 (Ω) and a stability estimate for Nw of the kind (24), with B1 = B2 = L2 (Ω), or B2 =
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L2 (Ω1 ), is not possible. If w ∈ C 1 (Ω) only, then one can still show that Nw maps L2 (Ω) into H 1 (Ω) using the theory of operators with singular kernels, see [8] for details. We then change B2 and replace it with H 1 (M1 ). The question we ask now is the following: assuming that Iw (and therefore Nw ) is injective for some class of w’s, what conditions on w would guarantee the stability estimate
f L2 (Ω) CNw f H 1 (Ω1 ) ,
∀f ∈ L2 (Ω).
(29)
We will show below that a sufficient condition for that is that Nw is elliptic, that can be easily formulated in terms of w. We return to the general case and the estimate (23). In the example above, if Nw is elliptic, then it is also a Fredholm operator. Recall that F : B1 → B2 is called Fredholm, if there exist bounded operators A : B1 → B2 and B : B2 → B1 so that AF − Id and FB − Id are compact. Then Ran F is necessarily closed, and F has a finitely-dimensional kernel Ker F and a finitely-dimensional cokernel Coker A = B2 / Ran A . Equivalently, F is Fredholm if its kernel and cokernel are finitely-dimensional. In case the inverse problem is over-determined, we cannot expect that A would have a finite cokernel, as in the example above. For this reason, if B2 is a Hilbert space, we will study whether N = A∗ A is a Fredholm operator, and the consequences of that. In the applied literature, applying A∗ is viewed a “back-projection” that, in inverse boundary value problems, returns our data form the boundary back to the domain. Another reason to study A∗ A instead of A is that, as in the example above, A might be a Fourier Integral Operator (FIO) but A∗ A is a ΨDO, thus easier to study. The latter is not alway true for any FIO, so some conditions are needed. In case of integrals along geodesics, see (20), this is guaranteed by the assumption of no conjugate points, see [17, Sections 3 and 5], where the injectivity of the exponential map is used to guarantee that the Schwartz kernel of A∗ A has singularities on the diagonal only. If the metric is Euclidean, there are no conjugate points, of course. Again the example above shows that we may need to restrict N to a subspace L of H in order to avoid working with ΨDOs on manifolds with boundary. Then F |L may not have finite cokernel anymore. For this reason, we will only study operators F that are upper semi-Fredholm, i.e., Ker F is finite-dimensional, and Ran F is closed. Equivalently, F is upper semi-Fredholm, if there exits a bounded left inverse A modulo compact operators (AF − Id is compact). Note that if N = A∗ A is Fredholm, then A itself is upper semiFredholm. We are not losing generality when replacing A by N = A∗ A as the next lemma shows. Note that also stability estimates for N of the type (29) and (31) below imply stability estimates for A of the type (23), (24) as well. Lemma 1. Let A : B1 → B2 be bounded, where B1,2 are Hilbert spaces, and let L ⊂ B1 be a (closed) subspace. The following statements are equivalent. (a) A : L → B2 is injective. (b) A∗ A : L → B1 is injective. (c) ΠL A∗ A : L → L is injective, where ΠL is the orthogonal projection to L.
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Proof. Let f ∈ L. Then ∗ A Af, f B = Af 2B2 .
(30)
1
This proves the implication (a) ⇒ (b). If Af = 0 with f ∈ L, then A∗ Af = 0 as well, therefore (b) ⇒ (a). The equivalence of (c) and (a) follows in the same way. Indeed, assume (a) and let ΠL A∗ Af = 0 with some f ∈ L. Then by (30), Af = 0, hence f = 0. Assume now (c) and let Af = 0 with f ∈ L. Apply ΠL A∗ to get ΠL A∗ Af = 0, hence f = 0. 2 Proposition 2. Let N : B1 → B2 be upper semi-Fredholm. Assume in addition, that N is injective on some (closed) subspace L ⊂ H. Then: (a) There exists C > 0, so that f B2 CNf B1 ,
∀f ∈ L.
(31)
(b) Estimate (31) remains true for any other bounded operator close enough to N with a uniform C > 0. Clearly, one can always take L = (Ker N )⊥ if B1 is a Hilbert space. The way we formulated the proposition above however is intended to underline the fact that if we can prove injectivity on some explicitly given subspace that is of interest to us (L2 (Ω) in the example above), then we automatically have stability. Proof of Proposition 2. The operator N : L → N (L) is bounded, injective, and has a closed range. By Proposition 1, (31) holds. This proves (a). To prove (b), it is enough to write ˜ B . f B1 /C Nf B2 (N − N˜ )f B + Nf 2 2
2
(32)
Remark 5. In the tensor tomography problems, see (20), one has to use a finer version of Proposition 2(b). Then N = Ng depends continuously on the metric g, and Ng : H 1 (M) → H˜ 2 (M1 ), where M1 M and tensors in M are alway extended as 0 outside M. The space H˜ 2 (M1 ) has the property that H 2 (M1 ) ⊂ H˜ 2 (M1 ) ⊂ H 1 (M1 ). On the other hand, H˜ 2 (M1 ) is “too large,” and Ran Ng is not closed there. It turns out, that one can construct a left parametrix Qg so that Qg Ng = Id +Kg , where Kg is a smoothing operator. However, Qg : H˜ 2 (M1 ) → H 1 (M) is not bounded; we can show only that Qg : H˜ 2 (M1 ) → L2 (M) is bounded. On the other hand, Qg Ng is bounded on H 1 (M), of course. This requires some modification of the arguments above, and the resulting estimate is f s L2 (M) CNg f H˜ 2 (M) ,
∀f ∈ H 1 (M),
with C > 0 locally uniform in g. Here f s is the orthogonal projection of f onto a certain space, called solenoidal tensors, where Ng is injective by assumption. Note that (32) does not apply with B1 = L2 (M) and B2 = H˜ 2 (M1 ) because Ng is not bounded as an operator between those two spaces. On the other hand, one cannot replace B1 by H 1 (M) because Ng (H 1 (M)) is not
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closed in H˜ 2 (M). For more details, and for the needed modifications in the arguments, we refer to [17,18]. The following proposition is well known, see e.g. [23]. We define Sobolev spaces on a compact manifold by fixing a finite atlas. We assume in the proposition below that N is of order −m 0 in view of applications to tomography problems. The order −m does not need to be non-positive, and one can study N as an operator mapping H s into H s+m for s not necessarily 0 by the usual technique — applying powers of 1 − to the left and right. Proposition 3. Let N ∈ Ψ −m (M), m 0 be an elliptic ΨDO on a compact manifold M. (a) Then for any l 0, one has f L2 (M) Cl Nf H m (M) + f H −l (M) .
(33)
(b) Assume in addition, that N is injective on some (closed) subspace L ⊂ L2 (M). Then there exists C > 0, so that f L2 (M) CNf H m (M) ,
∀f ∈ L.
(34)
Proof. Part (a) follows directly from the existence of a parametrix. Indeed, there is Q ∈ Ψ m so that QAf = f + Kf , where K has a smooth kernel. Note that for this proof it is only enough to have a parametrix Q of finite order l, i.e., to have K that maps L2 into H l . Then (33) follows directly. Note that the inclusion L2 → H −l is a compact map. Part (b) follows from Proposition 2 above. An alternative way to prove (b) is to notice that it follows directly from (a), by [23, Proposition 5.3.1]. 2 Recall that Ψ −m (M) is a Fréchet space with semi-norms pk (A), k = 1, 2, . . . , given by the semi-norms (they are actually norms) of its amplitude in the finitely many charts (Uj , χj ) of M. The norms pk are given by pk (a) = max j
sup
max
x∈U¯ j , ξ ∈Rn |α|+|β|+|γ |k
m+|γ | α β γ ∂ ∂ ∂ aj (x, y, ξ ), 1 + |ξ | x y ξ
where aj is the full symbol of A in the j th coordinate chart. Proposition 4. There exists k 1, depending only on dim M with the following property. Let A = A0 ∈ Ψ −m satisfy the assumptions of Proposition 3(b). Then (31) holds for any A ∈ Ψ −m close enough to A0 in the norm pk with a constant C independent of A. Proof. Note that Ψ −m equipped with the norm pk , k 2n + m + 1, is a subspace of the space of linear bounded operators mapping L2 (M) into H m (M), see [16, the appendix]. Now the proposition follows from Proposition 2(b). 2
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Remark 6. Proposition 3 can be applied to compact manifolds M with boundary; then we require the symbol to be defined and to satisfy the symbol estimates for x in the closed M. One can then extend it as an elliptic symbol to some extension of M without boundary, and treat L2 (M) as L. The same remark applies to Proposition 4 because on can choose the extension of the symbol in a way that is continuous in any fixed pk norm. Finally, we formulate a direct corollary of these statements. Corollary 1. Let A : L2 (M) → H, be a bounded linear operator, where M is a compact manifold without boundary, and H is a Hilbert space. Assume that N = A∗ A is an elliptic ΨDO of order −m 0. Assume also that A is injective on a (closed) subspace L of L2 (M). (a) Then N : L → L is onto, and f L2 (M) CNf H m (M) ,
∀f ∈ L.
(35)
(b) Estimate (35) remains true with an uniform C > 0 if N is replaced by any other operator close enough to N in the operator norm L2 (M) → H, and in particular for all ΨDOs in a small enough neighborhood of A in the norm pk , k 1. Proof. The only statement left to prove is that N : L → L is onto. This follows from the fact that N is injective and self-adjoint, hence it has a trivial cokernel. Next, since N is Fredholm, its range is closed. 2 In particular, by Lemma 1, Corollary 1(b) implies that A remains injective under a small perturbation (which also follows from the fact that A|L is upper semi-Fredholm). Also, (35) implies an estimate of the kind f CAf ∗ with a suitable norm · ∗ . Note that Corollary 1 can be applied to manifolds with boundary as in the example below. Example 3. We return to Example 2. Let Ω1 Ω as before. Assume that Nw = Iw∗ Iw is an elliptic ΨDO (of order −1) in Ω1 . To apply Corollary 1 directly, one can embed Ω1 into a compact manifold and extend Nw as an elliptic operator there in a way independent of w. Alternatively, on can work with the parametrix in Ω1 and eventually restrict to Ω. By (28), the principal symbol of N2 is given by (see [8,17,18] for a derivation in even more general cases) σp (Nw )(x, ξ ) = 2π
w(x, θ )2 δ(ξ · θ ) dθ,
S n−1
where δ is the Dirac delta function, and clearly, σp (Nw ) is homogeneous in ξ of order −1. The singularity at ξ = 0 can be cut-off. Since w is at least continuous, we see that the ellipticity assumption is equivalent to the following: ∀(x, ζ ) ∈ Ω × S n−1 , ∃θ ∈ S n−1 ,
θ ⊥ ζ, so that w(x, θ ) = 0.
(36)
In invariant terms, (x, ξ ) ∈ S ∗ Ω, while (x, θ ) ∈ SΩ. Another way to express the same is to say that
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N ∗ supp0 w ⊃ T ∗ Ω,
(37)
where supp0 w is the open set where w = 0. Under this condition, using the continuity of w in SΩ ∼ = Ω × S n−1 , one can get the same with Ω replaced by Ω1 . By Proposition 3, we then get f L2 (Ω) CNw f H 1 (Ω1 ) ,
(38)
under the assumption that Iw is injective on L2 (Ω). Since Nw is elliptic, it is enough to assume only that Iw is injective on C0∞ (Ω). Without that assumption, one has (33); and (38) on (Ker Iw )⊥ . To analyze the continuous dependence on w, let us note first that given k 0, one can construct a bounded extension operator extending functions in C k (Ω) to C k (Ω 1 ). With the aid of this operator, one can map a neighborhood of some w0 ∈ C k (Ω) to some neighborhood of an extended w0 in C k (Ω1 ). Then we can apply Corollary 1 to get the following. Theorem 3. Let w ∈ C 1 (Ω × S n−1 ) satisfy (36). Assume that Iw is injective on C0∞ (Ω). Then Nw : L2 (Ω) → L2 (Ω) is onto, and there exists a constant C > 0 so that f L2 (Ω) CNw f L2 (Ω1 ) ,
∀f ∈ L2 (Ω).
(39)
Moreover, (39) remains true under small C 1 (Ω) perturbations of w. The transform Iw is injective for a dense open set of weights satisfying (23), including real analytic ones, see [8]; but not for all positive smooth weights, see [3]. Without the injectivity assumption, (39) holds on (Ker Iw |C0∞ (Ω) )⊥ . Theorem 3 is a weaker version of the results in [8] and is presented here just to illustrate the method. Note that w ∈ C k with k = 1 is not enough to guarantee that operator convergence in the pk (N ) norm implies convergence in the norm N L2 (Ω)→H 1 (Ω1 ) , viewing N = Nw as a ΨDO. On the other hand, Nw is also an operator with singular kernel, see (28), and w ∈ C 1 is enough for L2 → H 1 continuity. We refer to [8] for details. We show finally that for an injective ΨDO, ellipticity is a necessary and sufficient condition for the estimate (35) in Corollary 1 to hold. Proposition 5. (a) Let N be a ΨDO of order −m 0 on a compact manifold M. Then (34) holds for any f ∈ L2 (M) if and only if N is injective and elliptic. (b) Let M be a compact manifold with boundary and let M1 be another compact manifold so that M1 M. Assume that N is a ΨDO of order −m 0 defined on some neighborhood of M. Then f L2 (M) CNf H m (M1 ) ,
∀f ∈ L2 (M)
(40)
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if and only if N is injective on L2 (M) and elliptic on π −1 (M); more precisely, inf lim inf |ξ |m p0 (x, ξ ) > 1/C,
x∈M |ξ |→∞
where p0 is the principal symbol of N . Proof. To prove (a), we will work locally near a fixed point on M. In fixed local coordinates near x0 , let p(x, ξ ) be the symbol of N , multiplied by a standard cut-off supported near some x0 . We will study p(x, D) in Rn first. Fix ξ0 = 0 and apply N to the normalized “coherent state” fλ (x) = (λ/π)n/4 eiλx·ξ0 −λ|x−x0 |
2 /2
.
Then 2 fˆλ (λξ ) = (λ/π)n/4 (2π/λ)n/2 e−iλ(ξ −ξ0 )·x0 −λ|ξ −ξ0 | /2 .
Then we get
p(x, D)fλ (x) = eiλx0 ·ξ0 (λ/π)n/4 (2πλ)−n/2
eiλ(x−x0 )·ξ −λ|ξ −ξ0 |
2 /2
p(x, λξ ) dλξ.
For any C > 0, the contribution to the integral above coming from integrating over |ξ −ξ0 | > 1/C is O(λ−N ), ∀N > 0. To estimate the effect of replacing p(x, λη) by p(x, λξ0 ) in the integral above, notice first that p(x, λξ ) − p(x, λξ0 ) = ζ (x, ξ, ξ0 , λ) · (ξ − ξ0 ) with ζ = O(1), therefore
|ξ − ξ0 |e−λ|ξ −ξ0 |
2 /2
dξ = O λ−n
(41)
|ξ −ξ0 |
(pass to polar coordinates rω = ξ − ξ0 ). Thus we get
p(x, D)fλ (x) = eiλx0 ·ξ0 (λ/2π)n/2 (λ/π)n/4
eiλ(x−x0 )·ξ −λ|ξ −ξ0 |
2 /2
p(x, λξ0 ) dξ + O λ−n
2 = eiλx0 ·ξ0 (λ/2π)n/2 (λ/π)n/4 p(x, λξ0 )(2π/λ)n/2 eiλ(x−x0 )·ξ0 −λ|x−x0 | /2 + O λ−n = p(x, λξ0 )fλ (x) + O λ−n/4 . Assume now that (34) holds. It is enough to consider the case m = 0. Then, choosing f = fλ , we get 1/C p(·, λξ0 )fλ (·) + O λ−n/4 .
(42)
The function φλ (x − x0 ) = |fλ (x)|2 = (λ/π)n/2 e−λ|x−x0 | is normalized so that it has L1 norm equal to 1. Therefore, 2
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p(·, λξ0 )fλ (·)2 =
2 2 φλ (x − x0 )p(x, λξ0 ) dx = p(x0 , λξ0 ) + O λ−n
(43)
by the argument already used above, see (41). Relations (42) and (43) show that p is elliptic at (x0 , ξ0 ). More precisely, we get p(x, ξ ) 1/C − C1 |ξ |−n/4 ,
∀(x, ξ ) ∈ T ∗ M,
(44)
where C > 0 is the constant C in (34), and C1 > 0 depends on p only. To relate N and p(x, D), note that in a fixed coordinates system near some x0 , R = N − p(x, D) is smoothing when acting on distributions supported near x0 . Consider then (N − p(x, D))χfλ , where χ is a standard cut-off near x0 . Then p(x, D)fλ = p(x, D)(1 − χ)fλ + p(x, D)χfλ = p(x, D)(1 − χ)fλ + (N − R)χfλ . Now, (1 − χ)fλ = O(e−λ/C ), N χfλ = O(λ−∞ ) as can be seen by integrating by parts and using the smoothness of the kernel of R. Therefore, N χfλ = p(x, D)fλ + O(λ−∞ ), and this completes the proof of (a). To prove (b), we proceed in the same way in a neighborhood of any x0 ∈ M int but we apply p(x, D) to χfλ instead to fλ , where χ is a standard cut-off near x0 supported in M. As x0 gets closer to ∂M, the derivatives of χ will have larger norms. This will affect the constant C1 in (44) that nay not be uniform in x. The conclusion (b) is still true, however. Note that if N has a polyhomogeneous symbol, then the sub-principal symbol must be uniformly bounded for x ∈ M, because we assume that N is a ΨDO in the larger M1 . Then C1 is actually uniformly bounded for x ∈ M. 2 3.1. Non-sharp linear stability estimates In view of the analysis in Section 2, we are also interested whether an estimate of the type (11), weaker than (35) can still be proven when N is not an elliptic ΨDO. In other words, what conditions would imply (35) in different H s spaces? Ellipticity of N is not necessary anymore, and for certain classes of hypoelliptic operators, one can still have (35) with a loss of one derivative, for example. We are not going to review those classes of operators. Without a proof, we will only mention that if the full symbol of N vanishes on some open conic set, (35) cannot hold in any Sobolev spaces. This is well known and used in tomography to determine subsets of lines/curves on which the X-ray transform can or cannot determine the function we integrate in a stable way, see e.g. [12]. In particular, in Example 2, if T ∗ \ N ∗ (supp w) contains an open conic set, see (37), then no H s1 → H s2 stability estimate of the type (38) is possible. 3.2. Injectivity of the differential Af0 In this section so far we were trying to find sufficient conditions that would guarantee a stability estimate under the assumption that the differential Af0 is known to be continuous. It is not the scope of this work to study in detail the typical approaches to prove that the differential is continuous. We will only briefly mention some of them: smallness of the coefficients is often used when the constant coefficient case is easy to study directly. Another technique is the use of the Analytic Fredholm Theorem [13] that, when can be applied, gives injectivity of Af for
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generic f ’s. When the unknown coefficient is in the principal symbol, as in the boundary rigidity question and related hyperbolic inverse problems [19], the calculus of the analytic ΨDOs can be applied, see e.g. [18]. 4. An example: the inverse back-scattering problem for the acoustic wave equation Consider the acoustic wave equation 2 ∂t − c2 (x) u = 0,
t ∈ R, x ∈ R3 ,
(45)
with a variable speed c(x) > 0 that equals 1 for large x, i.e., c(x) = 1 for |x| > ρ
(46)
with some ρ > 0. Let S(s, ω, θ ), where (s, ω, θ ) ∈ R × S 2 × S 2 , be the scattering kernel associated with c, see below. In [16] we showed that if c is close enough to 1 in the C 9 norm, then the back-scattering kernel S(s, −θ, θ ) determines c uniquely. We recall some facts about the time-dependent scattering theory for (45), see [16]. For simplicity, we work in space dimension n = 3. Given θ ∈ S 2 , we denote by u(t, x, θ ) the distorted plane wave defined as the solution of (45) satisfying u|t 0 = δ(t − x · θ ).
(47)
Then we set usc = u − δ(t − x · θ ). In the Lax–Phillips scattering theory [9], see also [5], the asymptotic wave profile u of u is defined by usc (s, ω, θ ) = lim (t + s)∂t usc t, (t + s)ω, θ , t→∞
where s ∈ R, ω ∈ S 2 . The scattering kernel S(s, ω, θ ) is given by S(s, ω, θ ) = −
1 u (s, ω, θ ). 2π sc
Then S(s − s, ω , ω) is the Schwartz kernel of R(S − Id)R−1 , where S is the scattering operator, and R is the Lax–Phillips translation representation. One can formulate this problem with stationary data in mind. The scattering amplitude a, that is also defined through the stationary scattering theory, is essentially just a Fourier transform (a stable and invertible operation) of S in the s variable, more precisely, iλ a(λ, ω, θ ) = 2π
e−isλ S(s, ω, θ ) ds,
(48)
see [16]. Set ω = −θ . Then supp S(·, −θ, θ ) ⊂ (−∞, 2ρ].
(49)
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Let Pθ be the plane x · θ = −ρ, and let “dist” be the distance function in the metric c−2 dx 2 . Fix T so that T > 2 max dist(Pθ , x); x ∈ B(0, ρ), θ ∈ S 2 − 2ρ, (50) where B(0, ρ) is the ball with center 0 and radius ρ, see also (61). We use the information contained in S for s −T only. We will denote by h the Heaviside function (the characteristic function of R+ ). Our data now is ST (s, −θ, θ ) := h(s + T )S(s, −θ, θ ). Instead of working with ST , we will work with its stationary analog aT defined by 2π aT (λ, −θ, θ ) := 3 e−isλ ST (s, ω, θ ) ds, λ > 0. iλ
(51)
(52)
The extra factor λ−2 there, compared to (48), is there for convenience. Clearly, λ2 aT = SˆT ∗ χˆ T , where the hat indicates Fourier transform with respect to s. Getting a stability estimate in terms of aT instead of a is even stronger because it uses less information. Since aT is analytic in λ, we can throw away any finite interval without losing information. Our next theorem shows that we are not losing stability, either. We fix λ0 0 and restrict λ to the interval λ λ0 . Theorem 4. There exist ε > 0, k 2, μ ∈ (0, 1) with the following property. For any two c, ˜ c satisfying (46) and c − 1C k + c˜ − 1C k ε,
(53)
and for any λ0 0, we have c˜ − cL∞ C
sup
μ (1 + λ)−2 (a˜ T − aT )(λ, −θ, θ ) ,
(54)
λ>λ0 ,θ∈S 2
where T is fixed so that it satisfies (50) with respect to both c˜ and c. Proof. We start with standard geometric optics arguments, see also [16, Proposition 3.1]. Let hj (t) = t j /j ! for t 0, and h(t) = 0 for t < 0; then hj = hj −1 , j 0, with the convention h0 = h, h−1 = δ. Proposition 6. If c ∈ C ∞ , and ε 1, then ∀N , u=
N
αj (x, θ )hj t − φ(x, θ ) + rN (t, x, θ )
(55)
j =−1
for |x| ρ, where φ solves the eikonal equation c2 (x)|∇φ|2 = 1,
φ|x·θ<−ρ = x · θ,
αj solve the corresponding transport equations and rN ∈ C N .
(56)
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2861
Proof. The construction of αj is standard, we will focus our attention on rN . The latter solves 2 ∂t − c2 (x) rN = c2 αN hN t − φ(x, θ ) ,
rN |t<−ρ = 0.
N +1 By standard energy estimates, rN ∈ Hloc , see also [16]. By Sobolev embedding theorems, we N −1 N get rN ∈ C . To prove that rN ∈ C , we apply this to rN +1 ; then αN +1 hN +1 (t − φ(x, θ )) + rN +1 ∈ C N . 2
Note that ρ + φ(x, θ ) = dist(Pθ , x). The expansion (55) is valid in B(0, ρ) if the eikonal equation has a smooth solution there. The latter is guaranteed by the following geometrical condition. Let γz,θ (t) be the geodesic in the metric c−2 dx 2 issued from z in the direction of θ . Then we want z ∈ Pθ , and t to be coordinates in B(0, ρ). The is certainly true if c is close enough to a constant in C 2 that we assume. The leading coefficient α−1 solves a homogeneous transport equation, and in particular, α−1 > 0. If c = 1, then α−1 = 1, αj = 0 for j 0, and φ = x · θ . Note that (55) remains true if c ∈ C k , k 2, but for some N = N (k), and N (k) 1 if k 2. k Moreover, {αj }N j =−1 depend continuously on c ∈ C . Therefore, under the assumptions of the theorem, for any fixed N , α−1 − 1C N Cε,
αj C N Cε,
j = 1 . . . N,
rN C N Cε,
(57)
if k 1. The C N norm of αj is taken in B(0, ρ) × S 2 , while the norm of rN is taken in for (x, θ ) ∈ B(0, ρ) × S 2 , and t in any fixed finite interval (under the assumption ε 1). Similarly, φ − x · θ C N Cε,
(58)
see also [16]. ˜ u˜ the scattering kernel S and the solution u Let c and c˜ be two sound speeds, and denote by S, related to c. ˜ The following formula is proven in [16]: (S˜ − S)(s, ω, θ ) =
1 3 ∂ 8π 2 s
−2 ˜ x, θ )u(−s − t, x, −ω) dt dx. c˜ − c−2 u(t,
(59)
The integral above makes sense even though u and u˜ are distributions. By the finite speed of propagation for (45), see e.g. [7], supp u(t, ·, θ ) ⊂ x; φ(x, θ ) t ,
supp u(−s − t, ·, θ ) ⊂ x; φ(x, θ ) −s − t . (60)
Therefore, for s −T , on the support of the integrand in (60), we have t T − φ(x, θ ),
−s − t T − φ(x, θ ),
see (60). This gives an upper bound for t and −s − t for |x| ρ. We want that upper bound to be so that the singular front of both waves involved in (48), with ω = −θ there, would pass through the whole ball B(0, ρ). Physically, this is equivalent to the requirement that we make our measurements for an interval of time so that the front of the wave u(t, x, θ ) would have enough
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time to pass through the whole B(0, ρ), and then return back a scattered signal. This explains the choice of T : we want T to be so that T − φ(x, θ ) > φ(x, θ ), ∀x ∈ B(0, ρ), i.e., T > 2 max φ(x, θ ); |x| ρ, |θ | = 1 . (61) This is equivalent to (50). Therefore, we study S restricted to −T s 2ρ (the upper bound is not a restriction actually, see (49)), and the length of that interval, assuming for a moment equality in (50), equals twice the length of the longest geodesic issued from some support plane Pθ = {x · θ = −ρ}, and staying in B(0, ρ). For x ∈ B(0, ρ), s −T , and t > T + ρ, we have that −s − t < −ρ, and then u(−s − t, x, θ ) = 0, see (60). Similarly, for x ∈ B(0, ρ), s −T , and −s − t > T + ρ, we have that t < −ρ, and then u(t, ˜ x, θ ) = 0. This shows that in the integral (59), the solutions u˜ and u can be replaced by the same solutions restricted to t T + ρ, and this will not change the left-hand side when s > −T . Let χ be a smooth function so that χ(t) = 1
for −ρ t T + ρ,
χ ∈ C0∞ (R).
(62)
Set uT = χ(t)u(t, x, θ),
(63)
and we define u˜ T in a similar way. Note that the subscript T in S and u has a different meaning. The cut-off for t < −ρ in (63) does not change u for x ∈ B(0, ρ) because u = 0 there. Then −2 1 3 ∂ (64) c˜ − c−2 u˜ T (t, x, θ )uT (−s − t, x, θ ) dt dx. (S˜T − ST )(s, −θ, θ ) = s 2 8π Take Fourier transform of the both sides to get −2 1 (a˜ T − aT )(λ, −θ, θ ) = − c˜ (x) − c−2 (x) v˜T (x, θ, λ)vT (x, θ, λ) dx, 4π where
(65)
vT (x, θ, λ) =
eiλt uT (t, x, θ ) dt.
(66)
Remark 7. Note that if we replace uT by u on the right-hand side above, we get the “distorted harmonic wave” solution v = eiλx·θ + vsc , where vsc is outgoing, i.e., it satisfies the Sommerfeld radiation conditions. In our case, vT is a convolution of v with the Fourier transform of a certain characteristic function. That convolution does not affect the high λ asymptotic of v, so it is not surprising that below we get the expansion familiar from the stationary theory. The advantage that we have here is that we do not need to estimate the contribution of large t’s, and instead of using estimates of the cut-off resolvent for non-trapping systems, we can just use the geometric optics expansion (55) that is more direct, easier to justify for c’s of finite smoothness, and easier to perturb near a fixed c. Furthermore, we get an even stronger result by discarding information (t 0) that is not needed. Linearizing (65), we get the following.
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Proposition 7. Let c, c˜ be two speeds satisfying the assumptions of Theorem 4. Then (a˜ T − aT )(λ, −θ, θ ) =
−2 c˜ (x) − c−2 (x) vT2 (x, θ, λ) dx + R(θ, λ)
(67)
where sup θ∈S 2 , λ>0
2 (1 + λ)−2 R(θ, λ) C c˜−2 − c−2 L∞ .
(68)
Proof. Let w be defined as u but satisfying w|t 0 = h2 (t − x · θ ), where, as above, h2 (s) = 2 1 3 s 2 /2, if s 0, and h2 (s) = 0, otherwise. Then h 2 = δ, u = ∂t w, and w ∈ Hloc , wt ∈ Hloc . The difference w˜ − w solves 2 ˜ ∂t − c2 (w˜ − w) = c˜2 − c2 w,
(w˜ − w)|t 0 = 0.
By standard energy estimates, in any compact set, w˜ t − wt L2x + w˜ − wHx1 C c˜2 − c2 L∞ . The constant C depends on c˜ and c and is easy to see that it remains uniformly bounded, if c is bounded in C 1 that follows from the hypothesis (53). Then we get by (66), (1 + λ)−2 v˜T − vT L2 C c˜2 − c2 L∞ . Compare (65) and (67) to get 2 sup (1 + λ)−2 R(θ, λ) C c˜−2 − c−2 L∞ ,
∀θ,
λ
where the L2 norm is taken in any compact.
2
We focus our attention now on the linear operator Af (λ, θ ) =
vT2 (x, θ, λ)f (x) dx,
f ∈ L2 B(0, ρ) .
(69)
The geometric optics expansion (55) yields 2 vT2 (x, θ, λ) = ei2λφ(x,θ) a−1 (x, θ, λ) + a0 (x, θ, λ) + · · · + aN (x, θ, λ) + RN (x, θ, λ) , λ → ∞,
(70)
where aj =
eiλt αj (x, θ )χ(t)hj t − φ(x, θ ) dt,
RN =
eiλt χ(t)rN (t, x, θ ) dt.
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Set η = 2λθ , and extend φ by homogeneity (of order 1) with respect to θ ; then 2λφ(x, θ ) = φ(x, η). The leading term is 2 2 = χ 2 φ(x, θ ) α−1 . a−1 We have that φ(x, θ ) is O(ε) close to x · η, see (58). The natural choice for the space containing Ran A is therefore L2 (Rnη ), that we will restrict to |η| > 2λ0 . In terms of the original variables, this space is isomorphic to L2 ((λ0 , ∞) × S 2 ; λ2 dλ dθ ). Clearly, if c ∈ C ∞ , (70) implies that the expression in the parentheses on the right-hand side is a classical elliptic symbol. For c ∈ C k , k 1, the symbol satisfies the symbol estimates up to a finite order m only, with m 1, when k 1. To prove this, we first show that aj is a symbol of order −j (of finite regularity). The properties of rN that we established do not prove that RN satisfies the symbol j α j α estimate |λj ∂λ ∂x,θ rN | Cλ−N but it is clear that it satisfies |∂λ ∂x,θ rN | Cλ−N . We can replace ∗ N by N + m now and this argument is enough to show that A A is a ΨDO with symbol of finite regularity m. This is summarized in the following. Proposition 8. Let c ∈ C k (Rn ) satisfy (46). Then vT x, η/|η|, |η|/2 = eiλφ(x,η) a(x, η) where φ and a belong to C N (B(0, ρ) × S 2 ) with N = N (k) → ∞, as k → ∞; a(x, η) is a formal elliptic symbol of order 0 satisfying the symbol estimates with all its derivatives up to order N . We now consider A as the operator given by (69) in the following spaces: A : B1 := L2 B(0, ρ) −→ B2 := L2 (λ0 , ∞) × S 2 ; λ2 dλ dθ .
(71)
Then
∗
e−iλ2(φ(x,θ)−φ(y,θ)) a(x, ¯ θ, λ)a(y, θ, λ)f (y) dy λ2 dλ dθ.
A Af (x) = S2
(72)
λ>λ0 |x|ρ
By a standard argument, define ξ = ξ(x, y, η) by 2λφ(x, θ ) − φ(y, θ ) = (x − y) · ξ , where η = 2λθ as above. The latter equation certainly has a solution for |x| ρ if c is close enough to a constant in C k , k 1, and that solution is close enough to ξ = η in C l with l 1 when k 1. Then we make a change of variables from η to ξ to get that A∗ A is a ΨDO of order 0 in a neighborhood of B(0, ρ), and it is clearly elliptic. To apply the results in Section 3, and especially Proposition 3(b), we have to show that A∗ A is injective. Set M = B(0, ρ1 ), where ρ1 > ρ is close enough to ρ so that the construction above remains valid on M. Then set L = L2 (B(0, ρ)). Assume first that c = 1. Then ∗
A Af (x) = S 2 λ>λ0 |x|ρ
e−i2λ(x−y)·θ f (x) dy λ2 dλ dθ = π 3 h |D| − 2λ0 f,
P. Stefanov, G. Uhlmann / Journal of Functional Analysis 256 (2009) 2842–2866
2865
where f is extended as zero outside B(0, ρ). It is easy to see that h(|D| − 2λ0 ) is a Fredholm injective operator on L2 (B(0, ρ)), see [15] for any λ0 0 (and it is the identity of λ0 = 0). Therefore, the conditions of Proposition 3(b) are satisfied for c = 1 and we get f L2 (B(0,ρ)) /C A∗ Af L2 (B(0,R)) Cf L2 (B(0,ρ))
(73)
with some R > ρ. This show in particular that A : B1 → B2 is bounded, depending continuously on c ∈ C k , k 1. That property is preserved if we increase ρ a bit. Therefore, A∗ : B2 → L2 (B(0, R)) is bounded a well, and also depends continuously on c ∈ C k , k 1. Thus f L2 (B(0,ρ)) CAf B2 .
(74)
This is the basic estimate that we need to apply Theorem 2. The spaces B1 and B2 are determined by the estimate in Proposition 7; set B1 = L∞ (B(0, ρ)), and let B2 be the Banach space determined by the norm on the left-hand side of (68). We need to show that the interpolation . Since estimates (13) hold with a proper choice of B1,2 f B1 = f L∞ (B(0,ρ)) Cε f H n/2+ε , ∀ε > 0, we use (18) to deduce that the second interpolation estimate in (13) holds with B1 = H s1 (B(0, ρ)), where n/2 + ε = s1 (1 − μ1 ). Note that we can make μ1 as close to 1 as we want by choosing s1 large enough. Next, we need to choose B2 so that the first interpolation estimate in (13) holds: 1−μ
μ
uB CuB22 uB 2 . 2
(75)
2
The following estimate follows from the Hölder inequality, and is actually a version of (18):
|u| λ dλ
2 −4
|u| λ
2 2
α
1−α |u| λ dλ 2 2s
dλ
,
(76)
with s = (1 + 2α)/(1 − α) for any 0 α < 1. We therefore define B2 to be the Hilbert space with a norm
u
B2
=
1/2 |u| λ
2 2s2
dλ
,
with s2 > 1, and then (75) holds with μ2 = (s2 − 1)/(s2 + 2). The condition μ1 μ2 > 1/2 is therefore equivalent to
s2 − 1 1 n > . 1− s2 + 2 2s1 2 This can always be achieved with s1 1, s2 1. The condition (16) is then guaranteed by (53), if k s1 . This completes the proof of Theorem 4.
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Acknowledgments The authors thank G. Alessandrini for pointing out the references [10,11]. References [1] R. Abraham, J.E. Marsden, T. Ratiu, Manifolds, Tensor Analysis, and Applications, second ed., Appl. Math. Sci., vol. 75, Springer-Verlag, New York, 1988. [2] G. Alessandrini, Open issues of stability for the inverse conductivity problem, J. Inverse Ill-Posed Probl. 15 (5) (2007) 451–460. [3] J. Boman, An example of nonuniqueness for a generalized Radon transform, J. Anal. Math. 61 (1993) 395–401. [4] A.P. Calderón, On an inverse boundary value problem, Comput. Appl. Math. 25 (2–3) (2006) 133–138. [5] J. Cooper, W. Strauss, Scattering of waves by periodically moving bodies, J. Funct. Anal. 47 (2) (1982) 180–229. [6] G. Eskin, Inverse scattering problem in anisotropic media, Comm. Math. Phys. 199 (2) (1998) 471–491. [7] L.C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998. [8] B. Frigyik, P. Stefanov, G. Uhlmann, The X-ray transform for a generic family of curves and weights, J. Geom. Anal. 18 (1) (2008) 81–97. [9] P.D. Lax, R.S. Phillips, Scattering Theory, second ed., Pure Appl. Math., vol. 26, Academic Press, Boston, MA, 1989, with appendices by C.S. Morawetz and G. Schmidt. [10] C.D. Pagani, Questions of stability for inverse problems, Rend. Sem. Mat. Fis. Milano 52 (1982) 599–608. [11] C.D. Pagani, Stability of a surface determined from measures of potential, SIAM J. Math. Anal. 17 (1) (1986) 169–181. [12] E.T. Quinto, Singularities of the X-ray transform and limited data tomography in R2 and R3 , SIAM J. Math. Anal. 24 (5) (1993) 1215–1225. [13] M. Reed, B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York, 1972. [14] V.A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse Ill-posed Probl. Ser. VSP, Utrecht, 1994. [15] P. Stefanov, Generic uniqueness for two inverse problems in potential scattering, Comm. Partial Differential Equations 17 (1–2) (1992) 55–68. [16] P. Stefanov, G. Uhlmann, Inverse backscattering for the acoustic equation, SIAM J. Math. Anal. 28 (5) (1997) 1191–1204. [17] P. Stefanov, G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J. 123 (3) (2004) 445–467. [18] P. Stefanov, G. Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc. 18 (4) (2005) 975–1003 (electronic). [19] P. Stefanov, G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not. 17 (17) (2005) 1047–1061. [20] P. Stefanov, G. Uhlmann, Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds, J. Differential Geom. (2007), in press. [21] P. Stefanov, G. Uhlmann, Integral geometry of tensor fields on a class of non-simple Riemannian manifolds, Amer. J. Math. 130 (1) (2008) 239–268. [22] J. Sylvester, G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1) (1987) 153–169. [23] M.E. Taylor, Partial Differential Equations. I, II, Appl. Math. Sci., vols. 115, 116, Springer-Verlag, New York, 1996, basic theory. [24] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Math. Library, vol. 18, North-Holland, Amsterdam, 1978. [25] G. Uhlmann, Commentary on Calderón’s paper (29) “On an inverse boundary value problem”, in: A. Bellow, C. Kenig, P. Malliavin (Eds.), Selected Papers of A.P. Calderón, Amer. Math. Soc., Providence, RI, 2008, pp. 623– 636. [26] J.-N. Wang, Stability estimate for an inverse acoustic backscattering problem, Inverse Problems 14 (1) (1998) 197– 207.
Journal of Functional Analysis 256 (2009) 2867–2893 www.elsevier.com/locate/jfa
Brownian survival and Lifshitz tail in perturbed lattice disorder Ryoki Fukushima Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan Received 30 July 2008; accepted 28 January 2009 Available online 24 February 2009 Communicated by Daniel W. Stroock
Abstract We consider the annealed asymptotics for the survival probability of Brownian motion among randomly distributed traps. The configuration of the traps is given by independent displacements of the lattice points. We determine the long time asymptotics of the logarithm of the survival probability up to a multiplicative constant. As applications, we show the Lifshitz tail effect of the density of states of the associated random Schrödinger operator and derive a quantitative estimate for the strength of intermittency in the parabolic Anderson problem. © 2009 Elsevier Inc. All rights reserved. Keywords: Brownian motion; Random media; Perturbed lattice; Random Schrödinger operators; Lifshitz tail; Random displacement model
1. Introduction We consider the annealed asymptotics for the survival probability of Brownian motion among randomly distributed traps. This problem for the Poissonian configuration of traps was firstly investigated by Donsker and Varadhan [3] and later by Sznitman [18] with generalizations on the shape of each trap, the diffusion coefficient, and the underlying space. Sznitman also generalized the configuration to some Gibbsian point processes in [20]. In this article, we discuss another model where the traps are attached around a randomly perturbed lattice. Namely, our process is the killed Brownian motion whose generator is E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.030
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1 Hξ = − + W ( · − q − ξq ), 2 d
(1.1)
q∈Z
where (ξq )q∈Zd is a collection of i.i.d. random vectors and W is a nonnegative function whose support is compact and has nonempty interior. We allow W to take the value ∞, which means imposing the Dirichlet boundary condition on {W = ∞}. If W ≡ ∞ on its support, the traps are said to be hard. The random potential in (1.1) is a model of the “Frenkel disorder” in solid state physics and is called the “random displacement model” in the theory of random Schrödinger operator. For such models with bounded displacements, there are some results concerning the spectral properties of the generator. Kirsch and Martinelli [10] discussed the existence of band gaps and Klopp [11] proved the spectral localization in a semi-classical limit. More recently, Baker, Loss and Stolz [1] studied which configuration minimizes the bottom of the spectrum of (1.1). On the other hand, there are few results when the displacements are unbounded, which is the object of this article. It seems important to allow unbounded displacements from the physical viewpoint. For example, it is natural to take the Gaussian distribution for the displacements to model the defects caused by self-diffusion. In the future paper [5], we will extend the investigation to non-compactly supported potentials and negative potentials. We will also discuss in [5] the one-dimensional result which is not discussed in the present article. There are at least three important aspects of the survival probability. The first is as the partition function for the Brownian motion conditioned to survive. Actually, some detailed studies on the surviving Brownian motion were developed after [18]. See e.g. [19] and [14] for path localization results. The second is as the Laplace transform of the density of states. It is well known that one can derive the asymptotic behavior of the density of states near the bottom of the spectrum from the survival asymptotics using an exponential Tauberian theorem. See e.g. [4,13,18] for this way of studies on the density of states. The last is as the solution of the parabolic Anderson problem. The quenched survival probability of the Brownian motion is expressed by a Feynman– Kac functional. From the expression, we can identify it with the solution of the heat equation associated with Hξ . Therefore, the annealed asymptotics of the survival probability gives the moment asymptotics of the solution. Now we describe the settings precisely. Let ((ξq )q∈Zd , Pθ ) (θ > 0) be Rd -valued i.i.d. random variables with the distribution Pθ (ξq ∈ dx) = N (d, θ ) exp −|x|θ dx,
(1.2)
where N (d, θ ) is the normalizing constant. Although our proof needs such an assumption only on the tail, we assume ξq to have the exact density (1.2) for simplicity. The parameter θ controls the strength of the disorder: large θ implies weak disorder and small θ implies the converse. Given random vectors, we define the perturbed lattice by ξ = q∈Zd δq+ξq and let V (·, ξ ) be the random potential in (1.1). We denote by Ξ the sample space of ξ , the space of simple pure point measures on Rd . We use the notation ((Bt )t0 , Px ) for the standard Brownian motion which is independent of ξ . The entrance time to a closed set F and the exit time from an open set U are denoted by HF and TU , respectively. Then the survival probability, our main object of this article, is described as follows: St = Eθ ⊗ E 0
t exp − V (Bs , ξ ) ds . 0
R. Fukushima / Journal of Functional Analysis 256 (2009) 2867–2893
2869
Intuitively, this quantity seems to decay exponentially since the traps are distributed almost uniformly in the space. However, the decay rate should be slower than the periodic case since large trap free regions caused by the disorder help the Brownian survival. We make a remark on the starting point of the Brownian motion before stating the results. Since our trap field is not Rd -translation invariant but Zd -shift invariant, the asymptotics of the survival probability may depend on the starting point. However, it will be clear from the proof that all the results stated in this article do not depend on the starting point. For this reason, we only consider the Brownian motion starting from the origin. We discuss the long time asymptotics of log St , instead of St itself, in this article. We introduce some notations for asymptotic behaviors to state the results. Definition 1. Let f and g are real-valued functions of a real variable and ∗ = 0 or ∞. Then f (x) g(x)
as x → ∗
means that there exists a constant C > 0 such that C −1 g(x) f (x) Cg(x)
(1.3)
when x is sufficiently close to ∗. Similarly, f (x) ∼ g(x)
as x → ∗
(1.4)
means that limx→∗ f (x)/g(x) = 1. We are now ready to state our main result. Theorem 1.1. For any θ > 0, we have log St
2+θ
−t 4+θ (log t)− 4+θ −t
d 2 +2θ d 2 +2d+2θ
θ
(d = 2), (d 3),
as t → ∞. Our result says that the survival probability decays faster than in the Poissonian case where log St ∼ −ct d/(d+2) (cf. [3]). This implicitly implies that the perturbed lattice is more ordered than the Poisson point process. Furthermore, we have the following simple but interesting observations. Remark 1.2 (Weak and strong disorder limits). Concerning the power of t in Theorem 1.1, we have the following: d 2 +2θ
1, which is the same as for the periodic traps. d 2 +2d+2θ d 2 +2θ d d+2 , which is the same as for the Poissonian traps. d 2 +2d+2θ
(i) As θ ∞, the power (ii) As θ 0, the power
On the other hand, a logarithmic correction remains in the case d = 2 and θ ↑ ∞, which we do not have for the periodic traps.
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This remark says that our model can be regarded as an interpolation between a perfect crystal and a completely disordered medium. We will also show the similar results concerning the convergence of point processes in Appendix A. Remark 1.3. The perturbed lattice has another interesting aspect in the two-dimensional case. Let ZC be the flat chaotic that is, the zero points of the Gaussian √ points (CAZP), analyticn zero ∞ is a collection of i.i.d. standard complex a z / n! where (a ) entire function fC (z) = ∞ n n=0 n=0 n Gaussian variables. Sodin and Tsirelson [17] proved that there exists a collection of random variables (ζq )q∈Z2 such that q∈Z2 δ√πq+ζq has the same distribution as ZC . Though (ζq )q∈Z2 is not an independent family, it is invariant under lattice shifts and the distribution of each ζq has a Gaussian upper bound for the tail. Therefore, our model with the parameter θ = 2 can be regarded as a toy model for the flat CAZP. Indeed, Sodin and Tsirelson called our model “the second toy model” in [17]. Let us briefly explain the construction of the article. We prove Theorem 1.1 in Section 2. Our strategy to prove the survival asymptotics is based on the idea in [18,21] rather than the one in [3]. The first step is a reduction to a certain variational problem. In this step, we use a coarse graining method which is a slightly altered version of Sznitman’s “method of enlargement of obstacles”. The second step is the analysis of the variational problem. However, we reverse the order and analyze the variational problem first since it gives the correct scale which we need in the coarse graining. In Section 3, we give two applications of the survival asymptotics. The first is the Lifshitz tail effect on the density of states of Hξ , which says that the spectrum of Hξ is exponentially thin around the bottom (cf. [12]). The second is a quantitative estimate for the strength of intermittency for the solution of the parabolic Anderson problem associated with Hξ . 2. Proof of the survival asymptotics 2.1. Rough procedure We explain the rough procedure of the proof in this section. First of all, we slightly modify the random potential as follows: V (x, ξ ) =
h · 1{ξ(C( q, ))1} 1C( q,L) (x) + ∞ · 1T c (x),
(2.1)
q∈Zd
where C(y, l) = y + [−l/2, l/2]d and T = (−t/2, t/2)d . This new potential bounds the original one from both above and below in T by taking small and varying h ∈ (0, ∞] and L > 0. Moreover, the restriction on T does not affect the results since P0 (TT t) decays exponentially in t. Therefore it is sufficient to prove the survival asymptotics for the modified potential (2.1). Hereafter we take , h, L = 1 so that V (x, ξ ) = 1supp V (·,ξ ) (x) almost everywhere in T , for simplicity. We start with the following obvious lower and upper bounds. Lower bound: Let S be the set of possible shapes of supp V (·, ξ )ξ ∈Ξ . Then,
t c St sup Pθ ξ U = 0 E0 exp − 1U (Bs ) ds ; TT > t . U ∈S
0
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Upper bound: By summing over U ∈ S, we obtain St
t c Pθ ξ U = 0 E0 exp − 1U (Bs ) ds
U ∈S
0
#S sup Pθ ξ U c = 0 E0
U ∈S
t exp − 1U (Bs ) ds ; TT > t . 0
Here we have #S < ∞ thanks to above modification and therefore the upper bound makes d sense. However, there still remains a problem since we have too many configurations: #S ∼ 2t . We shall remedy this situation by reducing #S to the small order using a coarse graining method. Once #S is shown to be negligible, the proof of the survival asymptotics is reduced to the analysis of the variational problem
t c (2.2) sup Pθ ξ U = 0 E0 exp − 1U (Bs ) ds ; TT > t . U ∈S
0
As we announced in the introduction, we shall analyze this variational problem in Section 2.2 and give the coarse graining scheme in Section 2.3. Finally, we shall patch them together in Section 2.4 to complete the proof. Remark 2.1. For log St with the above modified potential, we can derive a finer asymptotics than Theorem 1.1. We shall state this in Section 2.4 (Theorem 2.9) since it requires the notation defined in the proof. 2.2. Analysis of the variational problem In this section, we analyze the variational problem (2.2) and find the correct scale. Firstly, it is well known that the Brownian expectation part is controlled by the principal eigenvalue λ1 (U ) of the Dirichlet–Schrödinger operator −1/2 + 1U in T :
t exp − 1U (Bs ) ds ; TT > t ∼ −λ1 (U )t
log E0
as t → ∞,
0
for fixed U . Let us assume for the moment that this relation holds uniformly in U ∈ S. We will give a rigorous argument in Section 2.4. On the other hand, we use the following lemma to control the emptiness probability, that is, the probability of the perturbed lattice putting no point in a region.
Lemma 2.2. If {Uv }v>0 ⊂ S satisfies U c d(q, ∂Uv )θ dx/|Uvc | → ∞ as v → ∞, then we have v (2.3) log Pθ ξ Uvc = 0 ∼ − d(x, ∂Uv )θ dx as v → ∞, Uvc
where d(·,·) denotes the Euclidean distance.
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Proof. Let ∈ (0, 1) and U ∈ S be fixed. Note that |U c | < ∞ since U c is contained in T . For the upper bound, we consider the probability of a necessary condition: Pθ |ξq | > d(q, ∂U ) for all q ∈ U c ∩ Zd N (d, θ ) exp −|x|θ dx = q∈U c ∩Zd|x|>d(q,∂U )
=
∞
q∈U c ∩Zd
N (d, θ )r d−1 exp −r θ dr
σd d(q,∂U )
∞
(1 − )θ r θ−1 exp −(1 − )r θ dr
M1 ( )
q∈U c ∩Zd
= M1 ( )#U
d(q,∂U )
c ∩Zd
exp −(1 − )
d(q, ∂U )θ .
(2.4)
q∈U c ∩Zd
Here σd is the surface area of the unit sphere in Rd and M1 ( ) =
N (d, θ )σd sup r d−θ exp − r θ < ∞. (1 − )θ r>1/2
We can replace the sum in the last line of (2.4) by the integral by making M1 ( ) larger since θ θ sup d(x, ∂U ) dx − d(q, ∂U ) < ∞. U ∈S , q∈U c ∩Zd
C(q,1)
Therefore we obtain c |U c | θ Pθ ξ U = 0 M1 ( ) exp −(1 − ) d(x, ∂U ) dx Uc
for arbitrary > 0 and the upper bound follows. For the lower bound, we consider a sufficient condition:
Pθ ξ U c = 0
q∈U c ∩Zd
×
Pθ q + ξq ∈ a nearest C(q , 1) ⊂ U c
Pθ q + ξq ∈ C(q, 1)
q∈Zd ∩U ; d(q,∂U )M2
×
q∈Zd ∩U ; d(q,∂U )>M
Pθ q + ξq ∈ / Uc .
2
The first factor of the right-hand side is bounded below by
(2.5)
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c c0 (d, θ )|U | exp −
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d(q, ∂U )θ
q∈U c ∩Zd
exp − d(x, ∂U )θ dx − U c · log c0 (d, θ )
(2.6)
Uc
for some constant c0 (d, θ ) > 0. For instance, it suffices to take c0 (d, θ ) as N (d, θ ) ∧ 1 inf exp d(x1 , y1 )θ − d(x2 , y2 )θ ;
q, q ∈ Zd , x1 , x2 ∈ C(q, 1), y1 , y2 ∈ C(q , 1) .
Next, the second factor is bounded below by (3M2 )d |U c | Pθ q + ξq ∈ C(q, 1) ,
(2.7)
since we have #{q ∈ Zd ∩ U ; d(q, ∂U ) M2 } (3M2 )d |U c | by considering the M2 -neighborhood of each unit cube contained in U c . Before proceeding the estimate for the third factor, we recall that we have shown in (2.4) that Pθ q + ξq ∈ U c M1 ( ) exp −(1 − )d(q, ∂U )θ for any > 0 and q ∈ U . Now, if we pick some 0 ∈ (0, 1) (e.g. 0 = 1/2) and take M2 so large that M1 ( 0 ) exp −(1 − 0 )(M2 − 1)θ < 1, then the third factor is bounded below by
1 − M1 ( 0 ) exp −(1 − 0 )(n − 1)θ
nM2 q∈Zd ; n−1d(q,∂U )
(2n+1)d |U c | 1 − M1 ( 0 ) exp −(1 − 0 )(n − 1)θ
nM2
c |U | 1 − M1 ( )(2n + 1)d exp −(1 − 0 )(n − 1)θ ,
(2.8)
nM2
where we have used #{q ∈ Zd ; n − 1 d(q, ∂U ) < n} (2n + 1)d |U c | in the second line and the elementary inequality (1 − x)m 1 − mx (x ∈ [0, 1], m ∈ N) in the last line. Note that the infinite product in the third line is convergent. Combining (2.5)–(2.8), we obtain Pθ ξ U c = 0 exp − d(x, ∂U )θ dx − c0 (d, θ )U c , Uc
which shows the lower bound.
2
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Lemma 2.2 says that (2.3) holds for a large class of families in S. In fact, we shall prove in Proposition 2.11 that the family of sets which are relevant in our analysis satisfies the assumption of Lemma 2.2. If we assume that c (2.9) log Pθ ξ U = 0 = − d(x, ∂U )θ dx Uc
holds together with (2.2) for all U ∈ S, we can rewrite our variational problem as
t c log sup Pθ ξ U = 0 E0 exp − 1U (Bs ) ds ; TT > t U ∈S
0
∼ − inf λ1 (U )t + d(x, ∂U )θ dx . U ∈S
(2.10)
Uc
It is easy to see that the infimum of (2.10) is attained when U c is large for large t. Thus, it is convenient to introduce a scaling U = rUr by a factor r > 0. Under this scaling, the right-hand side of (2.10) takes the form r −2 d+θ θ − inf λ1 (Ur )tr + r d(x, ∂Ur ) dx Ur ∈ Sr
Urc
r d+θ −2 r θ inf λ1 (Ur ) + −2 d(x, ∂Ur ) dx . = −tr tr Ur ∈ Sr
(2.11)
Urc
Here Sr = {r −1 U ; U ∈ S} and λr1 (Ur ) is the principal eigenvalue of the scaled Dirichlet– Schrödinger operator −1/2 + r 2 1Ur in Tr = r −1 T . Let us summarize the status. We have shown that r d+θ log St ∼ −tr −2 inf λr1 (Ur ) + −2 d(x, ∂Ur )θ dx (2.12) tr Ur ∈ Sr Urc
for any r > 0 under the three assumptions: the first is on the coarse graining step (#S is negligible) and the second and third are that (2.2) and (2.3) respectively hold for U ∈ S in some uniform manners. The first one will be verified in Section 2.3 and the second and third ones in Section 2.4. Now, if we can find a scale r = r(t) for which the infimum in (2.12) stays bounded both above and below by positive constants as t → ∞, then tr −2 gives the asymptotic order of log St . It might seem natural to take r = t 1/(d+θ+2) , to satisfy r d+θ /tr −2 = 1, at the first sight. However, this scale gives a wrong magnitude t (d+θ)/(d+θ+2) . The key to finding the correct scale is that we
can easily decrease the value of the integral U c d(x, ∂Ur )θ dx. For instance, consider a domain r with many tiny holes C δ(r)q, r −1 (2.13) Urc = (−n, n)d \ q∈Zd
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with n ∈ N and δ(r) → 0 as r → ∞. Then we have d(x, ∂Ur )θ dx δ(r)θ , Urc
which goes to 0 as r → ∞. Although such holes generally increase the principal eigenvalue λr1 (Ur ), it is known that we can take δ(r) small to some extent while keeping the control of λr1 (Ur ). Indeed, Rauch and Taylor [15] proved that for this specific example with the hard traps (i.e. h = ∞), 1 (log r)− 2 (d = 2), δc (r) = (2.14) d−2 r− d (d 3), are the critical intervals in the following sense: (i) If limr→∞ δ(r)/δc (r) = 0, then limr→∞ λ1 (Ur ) = ∞. (ii) If limr→∞ δ(r)/δc (r) = ∞, then limr→∞ λ1 (Ur ) = λ1 (([−n, n]d )c ). Therefore, we have to take the scale r at least so large as to satisfy r d+θ /tr −2 = δc (r)−θ . Otherwise, we find that the infimum in (2.12) goes to zero as r → ∞, by considering the domain (2.13) with a large n and an appropriate δ(r). The next proposition, a generalization of the above criticality, shows that the infimum is actually bounded below for this choice of the scale. Proposition 2.3. There exists a function M2 ( ) → ∞ ( → 0) such that if Ur ⊂ Sr satisfies 1 d c # q ∈ Ur ∩ Z ; d(q, ∂Ur ) δc (r) < r d , (2.15) r then λr1 (Ur ) > M2 ( ). In particular, we have r −θ θ λ1 (Ur ) + δc (r) inf d(x, ∂Ur ) dx > 0. r1, Ur ∈Sr
Urc
Proof. We first recall that the principal eigenvalue can be expressed by the Dirichlet form 1 r λ1 (Ur ) = |∇ψr |2 (x) + r 2 1Ur (x)ψr2 (x) dx (2.16) 2 Tr
using the associated L2 -normalized eigenfunction ψr . Our basic strategy is estimating the righthand side by patching local estimates. For the local estimates, we use the following lemma. Lemma 2.4. There exists c1 (d) > 0 such that for any i ∈ Zd , C(y, 1r ) ⊂ C( δc (r)i, 2 δc (r)), > 0, and φ ∈ W 1,2 (C( δc (r)i, 2 δc (r))), we have 1 1 (2.17) |∇φ|2 (x) + r 2 1C(y, 1 ) (x)φ 2 (x) dx c1 (d) −d . 2 r 2 φ2 C( δc (r)i,2 δc (r))
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Proof. Using the scaling with the factor δc (r), we can bound the right-hand side of (2.17) below by −2 2 1 1 2 inf (x) + r δ (r) 1C(y, 1 ) (x)φ 2 (x) dx. δc (r) |∇φ| c r δc (r) 2 φ∈W 1,2 (C(i,2)) φ22 C(i,2)
Note that the infimum appearing in the above expression is the Neumann principal eigenvalue of the associated operator. The asymptotic behavior of the eigenvalue of this kind of operator has been studied thoroughly by Ben-Ari [2] (we also refer the reader to Taylor’s earlier work [22] for the case d 3). Our situation can be found in Theorem 1.3 of [2], which tells us 1 1,2 φ∈W (C(i,2)) φ22
inf
∼
2 1 |∇φ|2 (x) + r δc (r) 1C(y, 1 ) (x)φ 2 (x) dx r δc (r) 2
C(i,2)
c(2)(log(r δc (r)))−1 c(d)(r δc (r))2−d
(d = 2), (d 3).
Recalling the definition of δc (r), (2.17) follows immediately.
2
Now we show how to patch the local estimates. Let > 0 be small and I(r) be the collection of i ∈ Zd for which C( δc (r)i, δc (r)) intersects both Ur and Urc . Then, for large r, each C( δc (r)i, 2 δc (r)) (i ∈ I(r)) contains at least one 1/r-box ⊂ Ur . Therefore for all i ∈ I(r), we have
1 2 2 2 C( δc (r)i,2 δc (r)) 2 |∇ψr | (x) + r 1Ur (x)ψr (x) dx
c1 (d) −d (2.18) 2 (x) dx ψ C( δc (r)i,2 δc (r)) r by using Lemma 2.4 with φ = ψr |C( δc (r)i,2 δc (r)) . Moreover, since there exists m(d) ∈ N such that every x ∈ Rd is contained in at most m(d) different C( δc (r)i, 2 δc (r))’s, we find T
1 |∇ψr |2 (x) + r 2 1Ur (x)ψr2 (x) dx 2 −1
m(d)
i∈I (r) C( δ (r)i,2 δ (r)) c c
1 |∇ψr |2 (x) + r 2 1Ur (x)ψr2 (x) dx. 2
On the other hand, it is easy to see that q∈Urc ∩ 1r Zd ; d(q,∂Ur )< δc (r)
1 ⊂ C q, C δc (r)i, 2 δc (r) r i∈I (r)
for large r. From this and the assumption (2.15), it follows Tr \ C δc (r)i, 2 δc (r) i∈I (r)
(2.19)
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when r is sufficiently large. Therefore,
1 = ψr 22
i∈I (r) C( δ (r)i,2 δ (r)) c c
ψr2 (x) dx + ψr 2∞ .
(2.20)
We consider the case ψr ∞ −1/4 first. In this case, we have
m(d)−1 c1 (d) −d i∈I (r) C( δc (r)i,2 δc (r)) ψr2 (x) dx r
λ1 (Ur ) 2 1/2 i∈I (r) C( δc (r)i,2 δc (r)) ψr (x) dx + by substituting (2.19) and (2.20) into (2.16) and using (2.18). The right-hand side is greater than (2m(d))−1 c1 (d) −d when 1/4. Next, we consider the case ψr ∞ > −1/4 . This case is easier since we know the following L∞ -bound for the L2 -normalized eigenfunction (see e.g. (3.1.55) of [21]) ψr ∞ c2 (d)λr1 (Ur )d/4 , which gives λr1 (Ur ) c2 (d)−4/d −1/d . Combining the estimates in the two cases, we obtain λr1 (Ur )
−1 2m(d) c1 (d) −d ∧ c2 (d)−4/d −1/d
and the former part of the proposition is proven. From the former part, we find δc (r)−θ d(x, ∂Ur )θ dx → 0 as r → ∞ Urc
⇒
λr1 (Ur ) → ∞ as r → ∞ 2
and the latter part follows immediately.
This proposition tells us that the correct scale r should be 1 θ t 4+θ (log t) 8+2θ (d = 2), r= d (d 3), t d 2 +2d+2θ so that r d+θ /tr −2 ∼ δc (r)−θ as t → ∞ and thus (2.12) becomes −2 r −θ θ inf λ1 (Ur ) + δc (r) d(x, ∂Ur ) dx . log St ∼ −tr Ur ∈ Sr
Urc
For these scales, tr −2 actually gives the correct magnitudes tr
−2
=
(2.21)
2+θ
θ
t 4+θ (log t)− 4+θ t
d 2 +2θ d 2 +2d+2θ
(d = 2), (d 3).
(2.22)
(2.23)
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Remark 2.5. As is mentioned before, we have to assure that the infimum in (2.23) is also bounded above. It is possible to prove it here by considering the domain (2.13) with n = 1 and δ(r) = δc (r) but we postpone the discussion to Appendix B since we will have a slightly different variational problem after the coarse graining. 2.3. Coarse graining In this section, we give the coarse graining scheme which reduces the combinatorial complexity of configurations by replacing dense traps by a large box-shaped hard traps. Throughout this section, we are dealing with the scaled traps with the correct scale r in (2.22). The scaled configuration of points q δr −1 (q+ξq ) is denoted by ξ r . We take a positive number η ∈ (0, 1) so small as to satisfy θ 1 d −2 θ η2 + + η< ∧ (2.24) 2 d d 2 and let γ=
d − 2 2η + < 1. d d
We further introduce some notations concerning a dyadic decomposition of Rd . Let Ik be the collection of indices of the form k iı = (i0 , i1 , . . . , ik ) ∈ Zd × {0, 1}d . We associate to the above index iı a box: Ciı = qiı + 2−k [0, 1]d
where qiı = i0 + 2−1 i1 + · · · + 2−k ik .
For iı ∈ Ik and k k, we define the truncation [iı]k = (i0 , i1 , . . . , ik ). The notation iı iı means that iı is a truncation of iı . Finally, we introduce log r nβ (r) = β log 2 for β > 0 so that 2−nβ −1 < r −β 2−nβ . Now we give the precise definition of the “dense traps” in the first paragraph. Definition 2. We call Cq (q ∈ Zd ) a density box if all Ciı ’s (iı ∈ Inηγ , q iı) satisfy the following: for at least half of iı iı (iı ∈ Inγ ), qiı + 2−nγ −1 [0, 1]d contains a point of ξ r . The union of all density boxes is denoted by Dr (ξ ).
(2.25)
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Remark 2.6. We use 2−nγ −1 instead of 2−nγ in the definition to have separated traps. The role of this choice will be clear in the proof of Proposition 2.7 (see (2.30)). In [21], Sznitman defined density boxes in a different way and proved that they can be replaced by hard traps. We shall prove that our density set is a subset of Sznitman’s one to use the result in [21]. We start by recalling Sznitman’s definition of the density set and a result on the principal eigenvalue. For iı ∈ Ik , the skeleton of the traps is defined by √ Kiı = 2k B(x, d/r) . x∈Ciı ∩ supp ξ r
Sznitman defined the density box as follows: Definition 3. (See [21, pp. 150–152].) Ciı (iı ∈ Inγ ) is called a density box if the quantitative Wiener criterion: cap(K[iı]k ) δnγ (2.26) 1knγ
holds for some δ > 0. Here cap(·) denotes the capacity relative to 1 − /2 when d = 2 and −/2 when d 3. The union of all density boxes is denoted by Dr (ξ ). The next theorem enables us to replace the density boxes by hard traps without inducing a substantial upward shift of the principal eigenvalue. Spectral control. (See [21, Theorem 4.2.3].) There exists ρ > 0 such that for all M > 0 and sufficiently large r, (2.27) sup λr1 r −1 supp V (·, ξ ), Rr (ξ ) ∧ M − λr1 r −1 supp V (·, ξ ) ∧ M r −ρ , ξ ∈Ξ
where Rr (ξ ) = Tr \ Dr (ξ ) and λr1 (U, R) denotes the principal eigenvalue of Dirichlet– Schrödinger operator −1/2 + r 2 · 1U in R. As is announced before, we show the next proposition to apply this theorem to our density set. def
Proposition 2.7. Dr (ξ ) ⊂ Dr (ξ ). Accordingly, Rr (ξ ) = Tr \ Dr (ξ ) ⊃ Rr (ξ ). Proof. Let Cq be a density box. We check the quantitative Wiener criterion (2.26) for all iı q (iı ∈ Inγ ) by showing cap(K[iı]k ) c3 (d)
for all k nηγ .
To get the lower bound for the capacity, we use the following variational characterization: cap(K) = sup
−1 g(x, y)ν(dx)ν(dy) ; ν ∈ M1 (K) ,
(2.28)
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where M1 (K) denotes the set of probability measure supported on K and g(·,·) the Green function corresponding to 1 − /2 when d = 2 and to −/2 when d 3. By this expression, the proof of (2.28) is reduced to finding a νk ∈ M1 (K[iı]k ) which satisfies g(x, y)νk (dx)νk (dy) c3 (d)−1 (2.29) for each k nηγ . Now, note that (2.25) remains valid for [iı]k instead of iı ∈ Inηγ as long as k nηγ . Therefore for such k, we can find a collection of points xm ∈ qiım + 2−nγ −1 [0, 1]d ; iım ∈ Inγ are distinct 1mn ⊂ supp ξ r whose cardinality n 2d(nγ −k)−1 √ . We denote by em and capm respectively the equilibrium measure and the capacity of 2k B(xm , d/r) and let n em νk = n m=1 ∈ M1 (K[iı]k ). m=1 capm
g(x, y)em (dx)em (dy) = capm to Let us show that this νk satisfies (2.29). We use the fact obtain g(x, y)νk (dx)νk (dy) =
n
−2 capm
m=1
n
n
g(x, y)em (dx)em (dy) +
g(x, y)el (dx)em (dy)
l=m
m=1
−1 capm
+ const(d)
m=1
(2.30)
g(x, y) dx dy. (0,1)d ×(0,1)d
−2 for In the second inequality, we have implicitly used the fact that d(supp el , supp em ) 2−nγ sufficiently large r, which is due to our definition of the density set, to replace the sum l=m by the integral. Since the last integral in (2.30) is a constant depending only on d, it suffices for (2.29) to show that√ nm=1 capm → ∞ (r → ∞). If we recall that capm is just the capacity of a ball with radius 2k d/r, we find n m=1
capm
c4 (d = 2)(log(2−k r))−1 2d(nγ −k)−1 c4 (d)(2k /r)d−2 2d(nγ −k)−1
(d = 2), (d 3).
When d 3 and 1 k nηγ , the right-hand side is larger than c4 (d)r 2−d 2dnγ −2k−1 c4 (d)r 2−d+dγ −2ηγ /8 = c4 (d)r 2η(1−γ ) /8 → ∞ (r → ∞),
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as desired. Here we have used 2−nβ −1 < r −β 2−nβ for β > 0 in the first inequality. The case d = 2 can be treated by the same way and the proof of Proposition 2.7 is completed. 2 Now we turn on to the estimate for the number of non-density boxes in Tr . It is clear from the definition that the number should be very small. However, we need a quantitative estimate for the coarse graining to go well. We pick a positive parameter 2θ 2θ η, ∧1 χ ∈ 2η2 + d − 2 + d d
(2.31)
so that d(1 − ηγ ) + (1 − γ )θ + χ > d + d +χ
2θ , d
2θ . d
(2.32) (2.33)
It is easy to see from (2.24) that such a choice of χ is possible. Thanks to the relation (2.32), the right-hand side of the next proposition is d 2 +2θ 2θ o exp −r d+ d = o exp −t d 2 +2d+2θ .
Proposition 2.8. Pθ Rr (ξ ) r χ exp −c5 (d)r d(1−ηγ )+(1−γ )θ+χ . Proof. Throughout the proof, c5 (d) > 0 is a constant whose value may change line by line. We start with an estimate for the probability of Cq ⊂ Dr (ξ ). To this end, we consider the following necessary condition: there exists an iı q (iı ∈ Inηγ ) such that for a half of iı iı (iı ∈ Inγ ), / qiı + 2−nγ −1 [0, 1]d for all r −1 q ∈ qiı + 2−nγ −1 [0, 1]d . r −1 q + r −1 ξq ∈
(2.34)
Note that the events in the second line are independent in iı ∈ Inγ . Moreover, the probability of the each event is / r qiı + 2−nγ −1 [0, 1]d for all q ∈ r qiı + 2−nγ −1 [0, 1]d Pθ q + ξq ∈ θ d x, ∂ r 1−γ [0, 1]d dx 1 + o(1) exp − r 1−γ [0,1]d
exp −c5 (d)r (1−γ )(d+θ) , where we have used (2.4) for the first inequality. Therefore, summing over the choices of the indices iı and iı ’s in (2.34), we obtain
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Pθ Cq ⊂ Dr (ξ ) d(nγ −nηγ ) 2d(nγ −nηγ )−1 2 dnηγ exp −c5 (d)r (1−γ )(d+θ) 2 d(n −n )−1 γ ηγ 2 exp −c5 (d)r (1−γ )(d+θ)+dγ (1−η) for large r. In the second line, the first factor represents the choice of the index iı and the second factor the choice of the indices iı ’s. Since the event (2.34) itself is independent in q ∈ Zd , we have r χ χ Pθ Tr \ Dr (ξ ) r χ t dr exp −c5 (d)r (1−γ )(d+θ)+dγ (1−η) exp −c5 (d)r d(1−ηγ )+(1−γ )θ+χ , which is the desired estimate.
2
Now, let us bound the cardinality of Sr =
Rr (ξ ), r −1 supp V (·, ξ ) ∩ Rr (ξ ) ; ξ ∈ Ξ, Rr (ξ ) = ∅ is connected, Rr (ξ ) < r χ .
To this end, we first note that |Rr (ξ )|, the number of unit cubes contained in Rr (ξ ), varies from χ 1 to r χ . Secondly, we have at most (t/r)dr choices for the configuration of the unit cubes in d Rr (ξ ), for any given |Rr (ξ )| < r χ . Finally, there are at most 2r possible configurations of the traps inside each unit cube in Rr (ξ ). Therefore, we have χ d r χ #S r r χ (t/r)dr 2r = exp r d+χ log 2 1 + o(1)
d 2 +2θ θ = exp o t d 2 +2d+2θ (log t)− 4+θ ,
(2.35)
where the third line comes from the relation (2.33). 2.4. Patching estimates We complete the proof of survival asymptotics in this section. Throughout this section, we use the correct scale r in (2.22) and let > 0 denote an arbitrary small number. We introduce Mr =
inf
(Rr ,Ur )∈S r
λr1 (Ur , Rr ) + δc (r)−θ
θ d x, ∂(Rr \ Ur ) dx
Rr \Ur
to describe the asymptotics. We know infr1 Mr > 0 from Proposition 2.3 and we can also prove supr1 Mr < ∞ by substituting the punched domain (2.13) with n = 1 and δ(r) = δc (r) to Rr \ Ur . We postpone the proof of the latter fact to Appendix B. What we prove here is the following asymptotics which is finer than the results stated in Section 1.
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Theorem 2.9. Let r be as in (2.22). For modified potential (2.1) with , L, h = 1, we have 1 log St ∼ −Mr tr −2
as t → ∞.
Remark 2.10. The extensions of this theorem for other values of , L, h are routine with appropriate changes on the notation. Though we have this finer result only for the modified traps (2.1), it seems not so far from the original model at least in the case of hard traps. Indeed, for the hard traps, the modification is equivalent to discretizing the distribution of ξq as Pθ (ξq ∈ dx) = Ndisc (d, θ )
exp −|q|θ δq (dx).
q∈Zd
However, we still do not know whether lim supr→∞ Mr and lim infr→∞ Mr coincide or not. Upper bound: For any U ⊂ Rd , we have the following upper bound on the Feynman–Kac semigroup (see e.g. (3.1.9) of [21]): sup Ex x∈Rd
t d/2 exp −λ1 (U, T )t . exp − 1U (Bs ) ds ; TT > t c(d) 1 + λ1 (U, T )t 0
It follows from this estimate that
t exp − 1U (Bs ) ds ; TT > t c(d, ) exp −(1 − )λ1 (U, T )t ,
E0
0
where c(d, ) = supλ>0 c(d)(1 + λd/2 ) exp{− λ}. Thus, using Spectral control (2.42) and Proposition 2.8, we have St c(d, )Eθ exp −(1 − )λ1 supp V (·, ξ ) t c(d, )Eθ exp −(1 − ) λr1 r −1 supp V (·, ξ ), Rr (ξ ) ∧ Mr − r −ρ tr −2 ; Rr (ξ ) < r χ + Pθ Rr (ξ ) r χ c(d, )#S r sup Pθ ξ r (Rr \ Ur ) = 0 (Rr ,Ur )∈S r
d 2 +2θ × exp −(1 − ) λr1 (Ur , Rr ) ∧ Mr − r −ρ tr −2 + o exp −t d 2 +2d+2θ
(2.36)
for large t, where we have used the fact that the principal eigenvalue is the infimum of those over the connected components of the domain to assume Rr to be connected. Since the factor #S r (by (2.35)) and the second term is negligible compared with the results, we focus on the variational problem. In order to apply Lemma 2.2 to the emptiness probability term, we see that r(Rr \ Ur ) satisfies the assumption of the lemma when (Rr , Ur ) ∈ S r .
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Proposition 2.11. For any (Rr , Ur ) ∈ S r , let Wr = Rr \ Ur . Then we have
d(x, ∂Wr )θ dx c6 (d, θ )r −γ (θ+dη) |Wr |
Wr
for large r. In particular
lim
rWr
d(x, ∂(rWr ))θ dx
= ∞.
|rWr |
r→∞
Proof. By the definition of the density box, each Cq ⊂ Rr contains a Ciı (iı ∈ Inηγ ) such that def
half of {Ciı = qiı + 2−nγ −1 [1/4, 3/4]d }iıiı ∈Inγ do not intersect with Ur for large r. Therefore, the number of such qiı + 2−nγ −1 [1/4, 3/4]d in the whole Rr is larger than 2−d−1 2dnγ −dnηγ |Rr |. Since d(x, ∂Wr ) d(x, ∂Ciı ) for x ∈ Ciı , we can obtain the desired estimate as follows: d(x, ∂Wr ) dx 2 θ
−d−1 dnγ −dnηγ
2
|Rr |
θ d x, ∂Ciı dx
Ciı
Wr
c6 (d, θ )r dγ (1−η) r −γ (d+θ) |Wr |, where we have used the change of variables
θ d+θ d x, ∂Ciı dx = 2nγ −1
Ciı
θ d x, ∂ [1/4, 3/4]d dx
[1/4,3/4]d
for the second inequality. The latter claim follows immediately since (2.24) implies θ > γ (θ + dη). 2 Using the relation tr −2 = r d+θ δc (r)θ , Lemma 2.2, and Proposition 2.11 in (2.36), we obtain 1 λr1 (Ur , Rr ) ∧ Mr − r −ρ log S −(1 − ) inf t tr −2 (Rr ,Ur )∈S r θ + δc (r)−θ d x, ∂(Rr \ Ur ) dx 1 + o(1) Rr \Ur
−(1 − 2 )Mr for sufficiently large r. Since > 0 is arbitrary, the proof of the upper bound is completed.
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Lower bound: We start with the following obvious bound: log St log sup Pθ ξ r (Rr \ Ur ) = 0 (Rr ,Ur )∈S r
tr −2
r · 1Ur (Bs ) ds ; TRr > tr
× E0 exp −
2
−2
,
(2.37)
0
where S r =
−1 r q + Rr , r −1 q + Ur ; q ∈ Zd , (Rr , Ur ) ∈ S r .
The role of this extension of S r will be clear in the proof of Proposition 2.12. We first rewrite the emptiness probability term in the right-hand side of (2.37). Thanks to Proposition 2.11, we can use Lemma 2.2 to obtain θ log St log sup exp −r d+θ d x, ∂(Rr \ Ur ) dx 1 + o(1) (Rr ,Ur )∈S r
Rr \Ur
tr −2
r · 1Ur (Bs ) ds ; TRr > tr
× E0 exp −
2
−2
.
(2.38)
0
Next, we rewrite the Brownian motion part of (2.38). Though the result seems to be natural, the proof is rather complicated. Proposition 2.12. For sufficiently large t, we have 1 r −θ λ1 (Ur , Rr ) + δc (r) log St −(1 + ) inf tr −2 (Rr ,Ur )∈S r
θ d x, ∂(Rr \ Ur ) dx .
(2.39)
Rr \Ur
Proof. We give the proof for the case h = ∞ since we need a modified potential with h = ∞ to derive the lower bound for the original potential in Theorem 1.1. In this case, the second factor in the right-hand side of (2.38) is P0 (TRr \Ur > tr −2 ). The proof for the case h < ∞ is different but much simpler. We shall later explain how to adapt the following argument to that case (see Remark 2.14 below). Note first that the functional in the infimum in (2.39) is invariant under r −1 Zd -shift. If we also recall that S r contains only finite pairs of sets modulo r −1 Zd -shift, it follows that there exists (Rr∗ , Ur∗ ) ∈ S r which attains the infimum. We denote by p∗ (t, x, y) the transition kernel of the killed Brownian motion when exiting Rr∗ \ Ur∗ and by φ∗ the L1 -normalized positive eigenfunction corresponding to λr1 (Ur∗ , Rr∗ ). Since supp φ∗ ⊂ Rr∗ , there exists a box C(r −1 q, r −1 ) where φ∗ (x) dx r −d−χ . (2.40) C(r −1 q,r −1 )
We can assume q = 0 by the shift invariance and the extension of S r to S r . Then, it follows that C(0, r −1 ) ⊂ Rr∗ \ Ur∗ . We also have the following uniform upper bound on φ∗ ∞ .
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Lemma 2.13. φ∗ ∞ exp 2 sup Mr < ∞. r1
Proof. Since p∗ (t, x, y) is smaller than the usual heat kernel |x − y|2 1 , exp − p(t, x, y) = 2t (2πt)d/2 we have p∗ (1, ·, ·) < 1 and therefore φ∗ (x) = exp λr1 Ur∗ , Rr∗
p∗ (1, x, y)φ∗ (y) dy
Rr∗
< exp λr1 Ur∗ , Rr∗
φ∗ (y) dy
Rr∗
for all x ∈ Rr∗ . The rest is easy using supr1 Mr < ∞ and φ∗ 1 = 1.
2
From this lemma and the fact χ < 1 in (2.31), we see that the integral in (2.40) is not supported on the r −2 -neighborhood of the boundary:
φ∗ (x) dx φ∗ ∞ C 0, r −1 \ C 0, r −1 − r −2
C(0,r −1 )\C(0,r −1 −r −2 )
= o r −d−χ .
Therefore, we can discard it to find C(0,r −1 −r −2 )
1 φ∗ (x) dx r −d−χ . 2
(2.41)
Now, let pC (t, x, y) denote the transition kernel of the killed Brownian motion when exiting C(0, r −1 ). Clearly pC (t, x, y) p∗ (t, x, y) and we can also show that √ inf pC r −4 , 0, x const(d)r −2d exp − dr 2 /4 x∈C(0,r −1 −r −2 )
by using Theorem 2 and (6) in [23]. From these estimates for pC and the Chapman–Kolmogorov identity, we obtain P0 TRr∗ \Ur∗ > tr −2 −4 φ∗ (y) pC r , 0, x p∗ tr −2 − r −4 , x, y dy dx φ∗ ∞ C(0,r −1 −r −2 )
Rr∗
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φ∗ −1 ∞
inf
x∈C(0,r −1 −r −2 )
pC r −4 , 0, x exp −λr1 Ur∗ , Rr∗ tr −2
√ const(d, θ )r −d−χ−2d exp −λr1 Ur∗ , Rr∗ tr −2 − dr 2 /4 ,
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φ∗ (x) dx
C(0,r −1 −r −2 )
(2.42)
where we have used Lemma 2.13 and (2.41) in the last inequality. Finally, substituting (2.42) to (2.38) and recalling r d+θ /tr −2 = δc (r)−θ and r 2 = o(tr −2 ), we find ∗ ∗ 1 r −θ U + δ log S −(1 + ) λ , R (r) t c r r 1 tr −2
θ d x, ∂ Rr∗ \ Ur∗ dx
Rr∗ \Ur∗
for sufficiently large r. This completes the proof of Proposition 2.12.
2
Remark 2.14. For a modified potential with h < ∞, we need not discard the neighborhood of ∂C(0, r −1 ) and can proceed to (2.42) directly after Lemma 2.13. Then, the same argument works with C(0, r −1 − r −2 ) and pc (r −4 , 0, x) replaced by C(0, r −1 ) and e−h p(r −2 , 0, x) p∗ (r −2 , 0, x), respectively. Now, note that S r in the right-hand side of (2.39) can be replaced by S r since both terms in the infimum are invariant under r −1 Zd -shift. Therefore, the right-hand side of (2.39) equals −(1 + )Mr and the proof of the lower bound is completed. 3. Applications 3.1. Lifshitz tail In this section, we discuss the asymptotic behavior of the density of states of Hξ that is defined by the thermodynamic limit 1 δλD (Hξ in (−N,N )d ) (dλ). i N →∞ (2N )d
(dλ) = lim
i1
d d Here λD i (Hξ in (−N, N) ) is the ith smallest Dirichlet eigenvalue of Hξ in (−N, N ) . It is well known (see e.g. [9]) that the above limit exists in the sense of vague convergence and that its Laplace transform can be expressed as
∞ e 0
−tλ
− d2
(dλ) = (2πt)
Eθ ⊗ E x [0,1)d
t exp − V (Bs , ξ ) ds Bt = x dx 0
using Brownian bridge measure. As one expects from this expression, it is not difficult to see that the right-hand side admits essentially the same upper and lower bounds as St (see e.g. the
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discussion in [18]):
∞ log
e
−tλ
(dλ)
0
2+θ
−t 4+θ (log t)− 4+θ −t
θ
(d = 2),
d 2 +2θ d 2 +2d+2θ
(d 3),
as t → ∞. From these asymptotics and the exponential Tauberian theorem due to Kasahara [8], we find the following asymptotics for ([0, λ]). Corollary 3.1. For any θ > 0, we have log [0, λ]
θ
θ
−λ−1− 2 (log λ1 )− 2 −λ
− d2 − dθ
(d = 2), (d 3),
as λ → 0. This result says that the density of states is exponentially thin around the bottom of the spectrum, which is called “the Lifshitz tail effect” (cf. [12]). Moreover, we find similar phenomena as in Remark 1.2 for the power of λ. Namely, it approaches to d/2, the same power as in the Poissonian traps (cf. [13]), in the limit θ → 0 and to ∞ in the limit θ → ∞, which corresponds to the periodic traps where the density of states vanishes near the origin. 3.2. Intermittency We consider the initial value problem ∂ u(t, x) = Hξ u(t, x) ∂t
with u(0, ·) ≡ 1,
which is called the “parabolic Anderson problem”. The bounded solution uξ of this problem is known to be unique and admits Feynman–Kac representation (see e.g. Chapter 1 of [21]). Therefore, we can identify St with Eθ [uξ (t, 0)]. We analogously write the pth moment by St,p = Eθ [uξ (t, 0)p ]. Then, the solution uξ is said to be “intermittent” if 1/q
St, q
1/p
St, p
t→∞ − −−→ ∞ when p < q.
Intermittency is usually regarded as an evidence of the strong inhomogeneity of the solution field. Indeed, if one considers a function consisting of a few high peaks, its Lq -norm tends to be much larger than its Lp -norm for p < q. For more on intermittency, see for instance [6]. In our model, intermittency follows from Theorem 3.2(iii) of [6]. Although it is stated in the discrete setting, the proof of this part of the theorem works in the continuous setting as well. Our aim in this section is to prove the following quantitative estimate for the moment asymptotics. In particular, it follows that small θ implies strong intermittency.
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Corollary 3.2. Suppose that we have a modified potential (2.1). Then for any 1 p < q, 1/q
lim sup
log St,q
1/p
log St,p
t→∞
2 p d+θ+2 lim supr→∞ Mr . q lim infr→∞ Mr
(3.1)
Proof. We prove this result only for d 3 and the parameters , L, h = 1. For the twodimensional case, we have to care about the logarithmic correction but it is not difficult. The key to the proof is that we can prove t
−
d 2 +2θ d 2 +2d+2θ
log St,p ∼ −
inf
(Rr , Ur )∈S r
r −θ pλ1 (Ur , Rr ) + δc (r)
θ d x, ∂(Rr \ Ur ) dx
Rr \Ur
by exactly the same argument as for Theorem 2.9. Then, using the spatial scaling by the factor 2 p = p d/(d +2d+2θ) , we find that the right-hand side equals −p
d 2 +2θ d 2 +2d+2θ
inf
(Rr ,Ur )∈S rp
rp −θ λ1 Ur , Rr + δ(rp )
θ d x, ∂ Rr \ Ur dx .
Rr \Ur
Since the infimum in this expression is Mrp , we obtain 1/q
lim sup t→∞
and (3.1) follows.
log St,q
1/p
log St,p
2 p d+θ+2 Mrq ∼ q Mrp
as r → ∞,
2
Acknowledgments The author would like to thank professor Frédéric Klopp for drawing his attention to [1]. The problem discussed in Appendix A is suggested by professors Tokuzo Shiga, Nariyuki Minami, and Hiroshi Sugita. He appreciates them for the interesting problem. He is also grateful to the referees for careful reading of the article and constructive comments which substantially improved the exposition. Appendix A We discuss here weak convergences of the perturbed lattice as point processes. When we discuss weak convergence, we regard (Pθ )θ>0 as probability measures on Ξ equipped with the vague topology. Let P∞ denote the perturbed lattice with the perturbation variables distributed uniformly on B(0, 1) and P0 the Poisson point process with unit intensity. Theorem A.1. Pθ converges weakly to P∗ (∗ = 0 or ∞) as θ → ∗.
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Remark A.2. It will be clear from the proof that we can make the perturbed lattice converge to the perfect lattice as θ → ∞ by changing the distribution of ξq as θ Pθ (ξq ∈ dx) = N (d, θ ) exp − 1 + |x| dx. As we mentioned after (1.2), such a change does not affect the main results. Therefore, we see again that our model interpolates a perfect crystal and a completely disordered medium (cf. Remark 1.2). To prove Theorem A.1, we use the following result concerning the convergence of point processes (see Theorem 4.7 of [7]). Lemma A.3. Let (Pθ )θ∈[0,∞] be a family of probability measures on Ξ . Suppose that the following two conditions hold for any bounded Borel set B ⊂ Rd : (i) (ii)
lim Pθ ξ(B) = 0 = P∗ ξ(B) = 0 ,
θ→∗
lim sup Eθ ξ(B) E∗ ξ(B) . θ→∗
Then Pθ converges weakly to P∗ (∗ = 0 or ∞) as θ → ∗. Proof of Theorem A.1. We consider the limit θ → ∞ first. In this case, the law of each ξq converges to the uniform distribution on B(0, 1). Moreover, we have Pθ (q + ξq ∈ B) |B|N (d, θ ) exp −d(q, B)θ for any bounded B ⊂ Rd . This implies that the law of ξ(B) is essentially determined by finite ξq ’s when θ is large. From these facts, it is easy to verify the conditions (i) and (ii) in Lemma A.3 and we have desired convergence. Next, we turn to more subtle case θ → 0. We first verify the condition (i), that is, limθ→0 Pθ (ξ(B) = 0) = e−|B| . Let us take M > 0 so large that B ⊂ [−M, M]d . Then it follows that sup d x∈B, q ∈[−2M,2M] /
|x − q|θ − |q|θ 2θ M θ
for θ < 1 from the mean value theorem. Therefore, for any > 0, we have
1− <
B
exp{−|x − q|θ } dx <1+ |B| exp{−|q|θ }
(A.1)
for all q ∈ / [−2M, 2M]d when θ is sufficiently small. The right inequality in (A.1) gives us an upper bound
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1 − N (d, θ ) exp −|x − q|θ dx Pθ ξ(B) = 0 = q∈Zd
B
1 − (1 − )N (d, θ )|B| exp −|q|θ .
d q ∈[−2M,2M] /
Using 1 − a e−a in the right-hand side, we get lim sup Pθ ξ(B) = 0 θ→0
lim sup exp −(1 − )N (d, θ )|B| θ→0
θ exp −|q|
d q ∈[−2M,2M] /
= exp −(1 − )|B| .
(A.2)
θ In the third line, we have used the fact N (d, θ ) q ∈[−2M,2M] d exp{−|q| } → 1 (θ → 0), which / can be shown by the same way as (A.1). For the lower bound, we use the left inequality in (A.1) as follows: 1 − N (d, θ ) exp −|x − q|θ dx Pθ ξ(B) = 0 = q∈Zd
B
q∈[−2M,2M]
×
1 − N (d, θ )
exp −|x − q|θ dx
B
1 − (1 + )N (d, θ )|B| exp −|q|θ .
d q ∈[−2M,2M] /
Since N (d, θ ) → 0 (θ → 0), the first factor in the right-hand side goes to 1 and also sup (1 + )N(d, θ )|B| exp −|q|θ → 0
as θ → 0.
q∈Zd
Thus, we can use 1 − a e−(1+ )a , which is valid only for small a > 0, in the second factor and get lim inf Pθ ξ(B) = 0 θ→0 lim inf exp −(1 + )2 N (d, θ )|B| θ→0
= exp −(1 + )2 |B| .
exp −|q|θ
d q ∈[−2M,2M] /
(A.3)
Now that we have (A.2) and (A.3) for an arbitrary > 0, the condition (i) is verified. Next, we proceed to check the condition (ii), that is, lim supθ→0 Eθ [ξ(B)] |B|. Using the right inequality in (A.1), we find
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Eθ ξ(B) = Pθ (q + ξq ∈ B) q∈Zd
= N (d, θ )
exp −|x − q|θ dx
q∈Zd B −1
(1 − )
|B|N (d, θ )
θ d exp −|q| + (4M) .
d q ∈[−2M,2M] /
Since the right-hand side of this inequality goes to (1 − )−1 |B| as θ → 0, we are done.
2
Appendix B In this appendix, we prove the finiteness of sup Mr = sup r1
inf
r1 (Rr ,Ur )∈S r
λr1 (Ur , Rr ) + δc (r)−θ
θ d x, ∂(Rr \ Ur ) dx .
Rr \Ur
To this end, we have only to find a specific sequence {(Rr , Ur ) ∈ S r }r1 for which the functional in the infimum is bounded above. We take Rr = (−1, 1)d and Ur as in (2.13) with δ(r) = δc (r), the critical interval. This critical regime is called the “constant capacity regime” (see Section 3.2.B in [21]). Note that Rr is not a density box, with a slight abuse of notation, and thus (Rr , Ur ) ∈ S r . To be honest, we have to rearrange Ur a little so that each cubes being centered on r −1 Zd but we ignore this point for simplicity. Firstly, it is easy to see that
δc (r)−θ
θ d x, ∂(Rr \ Ur ) dx 1 as r → ∞.
Rr \Ur
Thus, it suffices to show that the principal Dirichlet eigenvalue λr1 (∅, Rr \ Ur ) λr1 (Ur , Rr ) is bounded above. This assertion is shown in Theorem 22.1 of [16] in the case d = 3. Since the same proof directly applies to all d 3, we restrict the discussion on d = 2. Let ψ be the L2 normalized principal Dirichlet eigenfunction in (−1, 1)d and φr (x) =
log |x − δc (r)q| − log(1/r) q∈Zd
log(δc (r)/2) − log(1/r)
+
∧ 1.
Then it follows from the direct calculation that for arbitrary small > 0, d inf φr (x); x ∈ (−1, 1) \ C δc (r)q, δc (r) → 1 q∈Zd
as r → ∞. Moreover, it is not difficult to show that both (∇ψ)φr 2 and ψ(∇φr )2 are bounded. We combine these three estimates to bound the right-hand side of
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λr1 (∅, Rr
1 \ Ur ) ψφr 22
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2 1 ∇(ψφr ) (x) dx 2
Ur
and get the desired result. References [1] J. Baker, M. Loss, G. Stolz, Minimizing the ground state energy of an electron in a randomly deformed lattice, Comm. Math. Phys. 283 (2) (2008) 397–415. [2] I. Ben-Ari, The asymptotic shift for the principal eigenvalue for second order elliptic operators on bounded domains under various boundary conditions in the presence of small obstacles, Israel J. Math. 169 (2009) 181–220. [3] M.D. Donsker, S.R.S. Varadhan, Asymptotics for the Wiener sausage, Comm. Pure Appl. Math. 28 (4) (1975) 525–565. [4] M. Fukushima, On the spectral distribution of a disordered system and the range of a random walk, Osaka J. Math. 11 (1974) 73–85. [5] R. Fukushima, N. Ueki, Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice, preprint. [6] J. Gärtner, S.A. Molchanov, Parabolic problems for the Anderson model. I. Intermittency and related topics, Comm. Math. Phys. 132 (3) (1990) 613–655. [7] O. Kallenberg, Random Measures, 4th ed., Akademie-Verlag, Berlin, 1986. [8] Y. Kasahara, Tauberian theorems of exponential type, J. Math. Kyoto Univ. 18 (2) (1978) 209–219. [9] W. Kirsch, F. Martinelli, On the density of states of Schrödinger operators with a random potential, J. Phys. A 15 (7) (1982) 2139–2156. [10] W. Kirsch, F. Martinelli, On the spectrum of Schrödinger operators with a random potential, Comm. Math. Phys. 85 (3) (1982) 329–350. [11] F. Klopp, Localization for semiclassical continuous random Schrödinger operators. II. The random displacement model, Helv. Phys. Acta 66 (7–8) (1993) 810–841. [12] I.M. Lifshitz, Energy spectrum structure and quantum states of disordered condensed systems, Soviet Phys. Uspekhi 7 (1965) 549–573. [13] S. Nakao, On the spectral distribution of the Schrödinger operator with random potential, Japan. J. Math. (N.S.) 3 (1) (1977) 111–139. [14] T. Povel, Confinement of Brownian motion among Poissonian obstacles in Rd , d 3, Probab. Theory Related Fields 114 (2) (1999) 177–205. [15] J. Rauch, M. Taylor, Potential and scattering theory on wildly perturbed domains, J. Funct. Anal. 18 (1975) 27–59. [16] B. Simon, Functional Integration and Quantum Physics, 2nd ed., AMS Chelsea Publishing, Providence, RI, 2005. [17] M. Sodin, B. Tsirelson, Random complex zeroes. II. Perturbed lattice, Israel J. Math. 152 (2006) 105–124. [18] A.-S. Sznitman, Lifschitz tail and Wiener sausage. I, II, J. Funct. Anal. 94 (2) (1990) 223–246, 247–272. [19] A.-S. Sznitman, On the confinement property of two-dimensional Brownian motion among Poissonian obstacles, Comm. Pure Appl. Math. 44 (8–9) (1991) 1137–1170. [20] A.-S. Sznitman, Brownian survival among Gibbsian traps, Ann. Probab. 21 (1) (1993) 490–508. [21] A.-S. Sznitman, Brownian Motion, Obstacles and Random Media, Springer Monogr. Math., Springer-Verlag, Berlin, 1998. [22] M.E. Taylor, Scattering length and perturbations of − by positive potentials, J. Math. Anal. Appl. 53 (2) (1976) 291–312. [23] M. van den Berg, A Gaussian lower bound for the Dirichlet heat kernel, Bull. London Math. Soc. 24 (5) (1992) 475–477.
Journal of Functional Analysis 256 (2009) 2894–2916 www.elsevier.com/locate/jfa
The reduced effect of a single scattering with a low-mass particle via a point interaction Jeremy Clark Katholieke Universiteit Leuven, Instituut voor Theoretische Fysica, Celestijnenlaan 200d, 3001 Heverlee, Belgium Received 1 August 2008; accepted 26 November 2008 Available online 19 December 2008 Communicated by J. Bourgain
Abstract In this article, we study a second-order expansion for the effect induced on a large quantum particle which undergoes a single scattering with a low-mass particle via a repulsive point interaction. We give an approximation with third-order error in λ to the map G → Tr2 [(I ⊗ρ)Sλ∗ (G⊗I )Sλ ], where G ∈ B(L2 (Rn )) is a heavy-particle observable, ρ ∈ B1 (Rn ) is the density matrix corresponding to the state of the light m is the mass ratio of the light particle to the heavy particle, S ∈ B(L2 (Rn ) ⊗ L2 (Rn )) is particle, λ = M λ the scattering matrix between the two particles due to a repulsive point interaction, and the trace is over the light-particle Hilbert space. The third-order error is bounded in operator norm for dimensions one and three using a weighted operator norm on G. © 2008 Elsevier Inc. All rights reserved. Keywords: Scattering operator; Point interaction
1. Introduction In theoretical physics, many derivations of decoherence models begin with an analysis of the effect on a test particle of a scattering with a single particle from a background gas [6,8,9]. A regime that the theorists have studied and which has generated interest in experimental physics [7] is when the test particle is much more massive than a single particle from the gas. Mathematical progress towards justifying the scattering assumption made in the physical literature in the regime where a test particle interacts with particles of comparatively low mass can E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.020
J. Clark / Journal of Functional Analysis 256 (2009) 2894–2916
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be found in [1,3,5]. In this article, we study a scattering map expressing the effect induced on a test particle of mass M by an interaction with a particle of mass m = λM, λ 1. The force interaction between the test particle and the gas particle is taken as a repulsive point potential. We work towards bounding the error (G, λ) in operator norm for G ∈ B(L2 (Rn )), n = 1, 3 of a second order approximation: Tr2 (I ⊗ ρ)S∗λ (G ⊗ I )Sλ = G + λM1 (G) + λ2 M2 (G) + (G, λ),
(1.1)
where ρ ∈ B1 (L2 (Rn )) is a density matrix (i.e. ρ 0 and Tr[ρ] = 1), G ∈ B(L2 (Rn )), Sλ ∈ B(L2 (Rn ) ⊗ L2 (Rn )) is the unitary scattering operator for a point interaction, and the partial trace is over the second component of the Hilbert space L2 (Rn ) ⊗ L2 (Rn ). M1 and M2 are linear maps acting on a dense subspace of B(L2 (Rn )) (M2 is unbounded). Our main result is that there exists a c > 0 such that for all ρ, G, and 0 λ (G, λ) cλ3 ρwtn Gwn , where · wn is a weighted operator norm of the form + G|X| Gwn = G + |X|G + Xi Pj G + GPj Xi + 0i,j d
|P |e1 G|P |e2 , e1 +e2 3
and and · wtn is a weighted trace norm which will depend on the dimension. In the above, X P are the vector of position and momentum operators respectively: (Xj f )(x) = xj f (x) and (Pj f )(x) = i( ∂x∂ j f )(x). Expressions of the type A∗ GB for unbounded operators A and B are identified with the kernel of the densely defined quadric form F (ψ1 ; ψ2 ) = Aψ1 | GBψ2 in the case that F is bounded. The scattering operator is defined as Sλ = (Ω + )∗ Ω − , where Ω ± = s-lim eitHtot e−itHkin t→±∞
(1.2)
are the Möller wave operators, and Hkin is the kinetic Hamiltonian and is the standard self-adjoint 1 1 extension of the sum of the Laplacians − 2M heavy − 2m light , while the total Hamiltonian Htot includes an additional repulsive point interaction between the particles. The definition of Htot is a little tricky for n > 1 since, in analogy to the Hamiltonian for a particle in a point potential [2], it cannot be defined as a perturbation of Hkin even in the sense of a quadratic form. Rather, it is 1 1 defined as a self-adjoint extension of − 2M heavy − 2m light with a special boundary condition. Going to center of mass coordinates, we can write 1 1 1 M +m heavy + light = cm + dis , 2M 2m 2(m + M) 2mM so that the special boundary condition will be placed on the displacement coordinate corresponding to dis and follows in analogy with that a single particle in a point potential as discussed
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in [2]. This also allows us to write down expressions for Sλ . Non-trivial point potentials in dimensions > 3 do not exist and the main result of our analysis is restricted to dimensions one and three. The first and second order expressions M1 (G) and M2 (G) respectively have the form M1 (G) = i[V1 , G] and 1 1 1 M2 (G) = i V2 + {A, P }, G + ϕ(G) − ϕ(I )G − Gϕ(I ), 2 2 2
(1.3)
j for j = 1, . . . , n are bounded real-valued functions of the operator X, where V1 , V2 , and (A) and ϕ is a completely positive map admitting a Kraus decomposition: ϕ(G) =
j
d k m∗j,k Gmj,k ,
(1.4)
R3
with the mj,k ’s being bounded multiplication operators in the X-basis. Notice that terms in (1.3) are reminiscent of the form of a Lindblad generator [10]. In [4] the results of this article are applied to the convergence of a quantum dynamical semigroup to a limiting form with generator including the terms (1.3). and ϕ are: The explicit forms for V1 , V2 , A, V1 = cn sn−1
A = cn sn−1 ϕ(G) = cn2 sn−2
| v1 |=| v2 |=k
d v1 d v2 ( v1 + v2 )∇T ρ( v1 , v2 )ei X(v1 −v2 ) ,
−1
(1.6)
d v1 d v2 ∇T ρ( v1 , v2 )ei X(v1 −v2 ) ,
dk |k|
(1.7)
| v1 |=| v2 |=k
−2 d k |k|
Rn
(1.5)
| v1 |=| v2 |=k
R+
d v1 d v2 ρ( v1 , v2 )ei X(v1 −v2 ) ,
dk |k|
dk |k|−1
R+
−1
R+
V2 = cn sn−1
d v1 d v2 ρ( v1 , v2 )ei X(−v1 +k) Ge−i X(−v2 +k) ,
(1.8)
| v1 |=| v2 |=|k|
where sn is surface area of a unit ball in Rn , cn is a constant arising form the scattering operator Sλ , ρ(k1 , k2 ) is the integral kernel of ρ, and ∇T is the gradient of weak derivatives in the diagonal direction which is formally (∇T ρ(k1 , k2 ))j = limh→0 h−1 (ρ(k1 + hej , k2 + h ej ) − ρ(k1 , k2 )). The integral kernel ρ(k1 , k2 ) is well defined since ρ is traceclass and hence Hilbert– Schmidt. In dimension one, the integrals |v1 |=|v2 |=k are replaced by discrete sums. In dimension two, V2 has an additional term due to the logarithm in (4.4) which we did not write down in the expression for V2 above. The multiplication operators mj,k are defined as
= cn sn−1 βj m (X) fj ( v ) e−i X(−v +k) , where ρ = j βj |fj fj | is the diagonalized dv j,k
| v |=|k|
and ϕ are bounded under certain norm restrictions on ρ, since, for example, form of ρ. V1 , V2 , A, n−2 V1 cn |P | ρ1 and ϕ = cn |P |n−2 ρ|P |n−2 1 .
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With the center of mass coordinate at the origin, the scattering operator S (neglecting the index λ until it is explained) acts identically on the center-of-mass component of L2 (R2n ) = L2 (Rn ) ⊗ L2 (Rn ) as (1.9) S = I + Icm ⊗ S(k) ⊗ |φ φ| , where the right copy of L2 (Rn ) corresponds to the displacement variable and is decomposed in the momentum basis into a radial and an angular component as L2 (R+ , r n−1 dr) ⊗ L2 (∂B1 (0)), 1 S(k) acts as a multiplication operator on the L2 (R+ ) component, and φ = (sn )− 2 1∂B1 (0) , is the normalized indicator function over the whole surface ∂B1 (0). We call S(k) the scattering coefficient, and it has the form Dim-1: Sα (k) =
Dim-2: −iα k
+ i 12 α
Sl (k) =
,
Dim-3: −iπ l −1
+γ
+ ln( k2 ) + i π2
,
Sl (k) =
−2ik , l −1 + ik
(1.10)
where α is a resonance parameter defined for the one-dimensional case, l is the scattering length in the two- and three-dimensional cases and γ ∼ 0.57721 is the Euler–Mascheroni constant. In the one-dimensional case a scattering length l is sometimes defined as the negative inverse of the resonance parameter α = μc2 , where c is the coupling constant of the interaction and μ is the h¯
mM λ relative mass m+M = M 1+λ . However, this contrasts with the two- and three-dimensional cases where the scattering length is proportional to the strength of the interaction. In the context of this article, where the point interaction is between a light and a heavy particle, we parameterize λ the resonance parameter as α = 1+λ α0 in the one-dimensional case and the scattering length as λ l = 1+λ l0 in the two- and three-dimensional cases for some fixed α0 and l. This corresponds to holding the strength of the interaction fixed. Thus Sλ and Sλ will be indexed by λ for the remainder of the article. There are two main obstacles in attempting to find a bound for the error (G, λ) from (1.1). The first obstacle is to find helpful expressions to facilitate making a Taylor expansion in λ of Tr2 [(I ⊗ ρ)S∗λ (G ⊗ I )Sλ ]. Writing Aλ = Sλ − I , then Tr2 (I ⊗ ρ)S∗λ (G ⊗ I )Sλ = G + Tr2 (I ⊗ ρ)A∗λ G + G Tr2 (I ⊗ ρ)Aλ + Tr2 (I ⊗ ρ)A∗λ (G ⊗ I )Aλ ,
and it turns out to be natural at all points of the analysis to approach the terms on the right individually. Propositions 2.2 and 2.3 are directed towards finding expressions for Tr2 (I ⊗ ρ)A∗λ and Tr2 (I ⊗ ρ)A∗λ (G ⊗ I )Aλ (1.11) respectively (since Tr2 [(I ⊗ ρ)Aλ ] is merely the adjoint of Tr2 [(I ⊗ ρ)A∗λ ]). The expressions we find in Propositions 2.2 and 2.3 are of the form
∗ ∗ ∗ dσ Uk,σ (1.12) Tr2 (I ⊗ ρ)Aλ G = d k fk,σ G, and Tr2 (I ⊗ ρ)A∗λ (G ⊗ I )Aλ =
Rn
Rn
d k
SOn ×SOn
SOn ∗ dσ1 dσ2 Uk,σ
1 ,λ
h∗k,σ
1 ,λ
Ghk,σ 2 ,λ Uk,σ 2 ,λ ,
(1.13)
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∗ for some unitaries U ∗ , Uk,σ 2 ,λ and some bounded operators gk,σ 2 ,λ , hk,σ 1 ,λ which are functions k,σ of the vector of momentum operators P . In general, we will have the problem that
Rn
d k
dσ gk,σ,λ =∞
and Rn
SOn
d k
dσ1 dσ2 hk,σ 1 ,λ hk,σ 2 ,λ = ∞,
SOn ×SOn
so the integrals of operators only have strong convergence. Propositions 3.1 and 3.3 make sense of the integrals of operators such as (1.12) and (1.13) that arise and give operator norm bounds for the limits. The basic pattern in the proof of Propositions 3.1 and 3.3 is an application of the simple inequalities in Propositions A.2 and A.3 in addition to intertwining relations that we have between the multiplication operators and the unitaries appearing in (1.12) and (1.13). Bounding the third order error of the expansions in λ of the strongly convergent integrals (1.12) and (1.13) brings up the second major obstacle. We will need to bound certain strongly convergent integrals for all λ in a neighborhood of zero. For small λ there will be unbounded expressions arising from the scattering coefficient Sλ (k) that will have contrasting properties between the one- and three-dimensional cases. For example, In the limit λ → 0, 1 λ Sλ (k) becomes increasingly peaked in absolute value at k ∼ 0 in the one-dimensional case. For the three-dimensional case, λ1 Sλ (k) becomes increasingly peaked at k = ∞. A difficulty with the two-dimensional case is the presence of the natural logarithm in the expression for Sλ (k) and the fact that λ1 S λ l0 (k) is not peaked at a fixed point as λ varies. The peak point does tend towards 1+λ k ∼ 0 as λ → 0, but it is unknown how to attain the necessary inequalities in this case. This article is organized as follows. Section 2 is concerned with proving Propositions 2.2 and 2.3 which give expressions for Tr2 [(I ⊗ ρ)A∗λ ] and Tr2 [(I ⊗ ρ)A∗λ (G ⊗ I )Aλ ]. In Section 3 we prove Propositions 3.1 and 3.3 which give the primary tools for bounding the integrals of operators which will arise in bounding the error term (G, λ) of our expansion (1.1). Section 4 contains the proof of Theorem 4.2 which is the main result of the article. This involves expanding the expressions in Propositions 2.2 and 2.3 that we found in Section 2 in λ and bounding the error. The difficult parts of the proof are characterized by using the Propositions 3.1 and 3.3 to translate unbounded expressions arising from the expansion of the scattering coefficient Sλ into conditions on G and ρ through the weighted norms Gwn and ρwtn being finite. Sections 2 and 3 apply to dimensions one through three (all dimensions where non-trivial point potentials exist), while Section 4 does not treat dimension two. 2. Finding useful expressions for a single scattering In this section, we will find expressions for Tr2 [ρA∗λ ] and Tr2 [ρA∗λ GAλ ]. For notational convenience, we will begin identifying I ⊗ ρ with ρ and G ⊗ I with G. Finding formulas for Tr2 [ρA∗λ ], Tr2 [ρA∗λ GAλ ] begins with writing Aλ = Sλ − I in a convenient way. Let f, g ∈ L2 (Rn × Rn ), where the first and second component of Rn × Rn correspond to the displacement and the center of mass coordinate, then
g | Aλ f = Rn
d K cm
∞ 0
Sλ (k) dk sn k n−1
∂Bk (0)
ˆ ˆ d k1 g( ¯ k1 , Kcm ) ∂Bk (0)
ˆ ˆ d k2 f (k2 , Kcm ) .
(2.1)
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The above formula gives a quadratic form representation of Aλ that involves integrating over a surface of 3n − 1 degrees of freedom rather than 4n, since it acts identically over the center-ofmass component of the Hilbert space and conserves energy for the complementary displacement coordinate. The integral kernel for Aλ in center-of-mass momentum coordinates can be formally expressed as Aλ (kdis,1 , Kcm,1 ; kdis,2 , Kcm,2 ) =
Sλ (|kdis,1 |) δ |kdis,1 | − |kdis,2 | δ(Kcm,1 − Kcm,2 ). n−1 sn |kdis,1 |
However, for instance, this does not work directly towards finding even a formal expression for ∗ Tr2 ρAλ G (K 1 , K 2 ) =
K 2 ), d k1 d k2 d K ρ(k1 , k2 )A∗λ (k2 , K 1 ; k1 , K)G( K,
(2.2)
where we have written down a formal equation between integral kernel entries Tr2 [ρA∗λ G], G, and A∗λ using momentum coordinates corresponding to the heavy particle and the light particle. In finding an expression for (2.2), it would be natural to have K 2 as a parameterizing variable since the expression above is just multiplication of G from the left by Tr2 [ρA∗λ ]. m cm = λ x + 1 X where x For λ = M , the center of mass coordinates are X − X, 1+λ 1+λ and xd = x and X are the position vectors of the particle with mass m and M. The corresponding momentum 1 λ The proposition below gives two quadratic coordinates are kd = 1+λ k − 1+λ K and K cm = k + K. form representations of Aλ using different parameterisations of the integration in (2.1). (2.3) is directed towards finding an expression for Tr2 [ρA∗λ ] and (2.4) is for Tr2 [ρA∗λ GAλ ]. The proof of the following proposition requires changes of integration. Proposition 2.1 (Quadratic form representations of Aλ ). Let f, g ∈ L2 (Rn × Rn ), then (1) First quadratic form representation
g | Aλ f =
d k d K
Rn
dσ Sλ |k|
SOn
K + (σ − I )k f σ k + λ(K + σ k), K . × g¯ k + λ(K + σ k),
(2.3)
(2) Second quadratic form representation
g | Al f =
d K 2 d k1
SOn
−1
dσ det(I + λσ )
I (k1 − λK2 ) Sλ I + λσ
(σ − I ) σ −I × g¯ k1 , K2 + (k1 − λK2 ) f k1 + (k1 − λK2 ), K2 , (2.4) 1 + λσ 1 + λσ where the total Haar measure on SOn is normalized to be 1 (and for dimension one, the integral over SOn is replaced by a sum over {+, −}).
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The proofs of Propositions 2.2 and 2.3 work by using the spectral decomposition of ρ, special cases of G, etc. so that the quadratic form representations (2.3) and (2.4) of Aλ can be applied. = f (p − k). Defining τk = ei k·X , recall that τk acts in the momentum basis as a shift: (τk f )(p) Proposition 2.2. Let ρ have continuous integral operator elements in momentum representation. Tr2 [ρA∗λ ] has the integral form B˜ λ∗ =
d k
Rn
, dσ τk τσ∗k pk,σ,λ S¯λ |k|
(2.5)
SOn
is a multiplication operator: where τa is a translation by a in the momentum P basis and pk,σ,λ = ρ (1 + λ)k + λP , (σ + λ)k + λP . pk,σ,λ Proof. The following equality holds: Tr2 ρA∗λ = Tr2 βj |fj fj |A∗λ = βj id ⊗ fj | A∗λ id ⊗ |fj , j
j
where the infinite sum on the right converges absolutely in the operator norm. If we take a partial
sum ρm = m β j =1 j |fj fj |, then using (2.3), m w id ⊗ fj | A∗λ id ⊗ |fj v j =1
=
m
d K 1 d K 2
d k S¯λ |k|
j =1Rn ×Rn
×
w( v K 2 + (σ − I )k . ¯ K 1 )fj k + λ(K 1 + σ k) dσ f¯j σ k + λ(K 1 + σ k)
This has the form w | [·]v , where [·] is given by
Rn
d kS¯λ (k)
(σ + λ)k + λK . dσ τσ∗k τk ρm (1 + λ)k + λK,
SOn
This converges in operator norm to the expression given by (2.5), since ρm → ρ in the trace norm and by the bound given in Corollary 3.2. 2 Tr2 [ρAλ ] has a similar integral representation by taking the adjoint. Now we will delve into = the form of Tr2 [ρA∗λ GAλ ]. In the following, the operator DA acts on f ∈ L2 (Rn ) as (DA f )(k) 1 | det(A)| 2 f (Ak) for an element A ∈ GLn (R).
J. Clark / Journal of Functional Analysis 256 (2009) 2894–2916
Proposition 2.3. Let written in the form
B˜ λ (G) =
2901
βj |fj fj | be the spectral decomposition of ρ. Tr2 [ρA∗λ GAλ ] can be
j
d k
j Rn
∗ ∗ ¯ dσ1 dσ2 Uk,σ 1 ,λ mj,k,σ 1 ,λ Sλ
SOn ×SOn
k − λP m × GSλ U , 1 + λ j,k,σ2 ,λ k,σ2 ,λ
k − λP 1+λ (2.6)
∗ where Uk,σ is a 2 ,λ = τk D 1+λσ τσ k ,. τσ k , τk , and D 1+λ act on the momentum basis and mj,k,σ,λ 1+λ 1+λσ function of the momentum operator P of the form 1 σ −I βj det(1 + λσ )− 2 fj k + (k − λP ) . I +λ
Proof. Eq. (2.4) tells us how Aλ acts as a quadratic form. In order to use (2.4), we will look at v | Tr2 [ρA∗λ GAλ ]w in the special case where G = G ⊗ I = |y y| ⊗ I is a one-dimensional projection tensored with the identity over the light-particle Hilbert space. Formally, this allows us to write βj v ⊗ fj A∗λ y ⊗ φl y ⊗ φl |Aλ |w ⊗ fj , v Tr2 ρA∗λ GAλ w = j
l
where (φm ) is some orthonormal basis over the light-particle Hilbert space allowing a representation of the identity operator as a sum of one-dimensional projections, and the spectral decomposition of ρ has been used. Once (2.4) has been applied, we build up to an expression (2.6), taking care with respect to the limits involved. By Corollary 3.4, the expression (2.6) defines a bounded completely positive map (c.p.m.). Since Tr2 [ρA∗λ GAλ ] defines a c.p.m. and agrees with (2.6) for one-dimensional orthogonal projections, it follows that the two expressions are equal on B(L2 (Rd )). This follows because c.p.m.’s are strongly continuous and the span of one-dimensional orthogonal projections is strongly dense. The following holds, where the right-hand side converges in the operator norm: βj id ⊗ fj | A∗λ GAλ id ⊗ |fj . Tr2 ρA∗λ GAλ = j
For G = |y y|, (id ⊗ fj |)A∗λ GAλ (id ⊗ |fj ) = ϕy,j (I ), where ϕy,j is the completely positive map such that for H ∈ B(H) ϕy,j (H ) = id ⊗ fj | A∗λ |y y| ⊗ H Aλ id ⊗ |fj .
Since ϕy,j is completely positive, ϕy,j ( m l=1 |φl φl |) converges strongly to ϕy,j (I ). ϕy,j (I ) is determined by its expectations v|ϕy (I )v , and moreover N N v ϕy,j (I )v = lim v ϕy,j |φm φm | v = lim φm | υv,j,y υv,j,y | φm
N →∞
= υv,j,y 2 = Rn
m=1
v,j,y (k), d k υ¯ v,j,y (k)υ
N →∞
m=1
(2.7)
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where υv,j,y is defined as the vector υv,j,y = ( y| ⊗ id)Aλ (|v ⊗ |fj ). Using (2.4), φm |υv,j,y
can be expressed as
I (k − λK) det(I + λσ )−1 dσ Sλ φm | υv,j,y = d K I + λσ SOn
fj k + σ − I (k − λK) v(K). y¯ K + σ − I (k − λK) × φ¯ m (k) I + λσ I + λσ
(2.8)
By (2.7), we can evaluate Tr2 [(|fj fj |)A∗λ (|y y| ⊗ I )Aλ ] = v|ϕy (I )v through expression ¯ v,j,y (k)υ v,j,y (k). Through (2.8) we have an a.e. defined expression for the values Rn d k υ Now, writing down n d kυ¯ v,j,y (k)υ v,j,y (k) using the expression for υv,j,y (k), the υv,j,y (k). R result can be viewed as an integral of operators acting from the left and the right on |y y|, followed by an evaluation v | (·)v . Using the intertwining relation: σ −I (k − λP ) , m(P )τk∗ D 1+λσ τσ k = τk∗ D 1+λσ τσ k m P − 1+λ 1+λ 1 + λσ for a function m(P ) of the momentum operators P and the fact that isometry for 0 λ < 1, the expression can be written: v ϕy,j (I )v = v|
Rn
d k
∗ dσ1 dσ2 Uk,σ
1
m∗ ,λ j,k,σ
SOn ×SOn
σ +λ I +λσ
−1
= σ I +λσ I +λσ
is an
¯ λ k − λP S 1+λ 1 ,λ
k − λP m × |y y| Sλ |v . U 1 + λ j,k,σ2 ,λ k,σ2 ,λ So ϕy,j (I ) = Tr2 [(|fj fj |)A∗λ (|v v|)Aλ ] agrees with the expression (2.6) for a fixed j and for G = |v v| for all v, and hence by our observation at the beginning of the proof, Tr2 [(|fj fj |)A∗λ GAλ ] is equal to the expression (2.6) for a single fixed j and all G ∈ B(L2 (Rn )). However, if we take the limit m → ∞ for ρm = m j =1 βj |fj fj |, then the expression (2.6) converges in the operator norm and Tr2 [ρm A∗λ GAλ ] converges to Tr2 [ρA∗λ GAλ ]. Hence we have equality for all trace class ρ. 2 ˜ ˜ ) Through the formula Tr2 [ρS∗λ GSλ ] = G+ B˜ ∗ G+GB˜ + B(G), it is clear that B˜ ∗ + B˜ = −B(I by plugging in G = I . However, it is not at all obvious that this equality takes place through the ˜ ), respectively, since the operators U expressions (2.5) and (2.6) for B˜ ∗ and B(I k,σ,λ appear only ˜ in form for B(I ). = Uk,σ,λ h( I1+λ It is convenient to notice the intertwining relation h(k − λP )Uk,σ,λ +λσ k − λP )). ˜ )g can be written: Let g ∈ L2 (Rn ), then gˆ = B(I g( ˆ p) =
d k
j Rn
SOn ×SOn
∗ dσ1 dσ2 Uk,σ
1
m∗ ,λ j,k,σ
1
(p) U k,σ ,λ 1 ,λ
I ∗ ∗ × |Sλ | Uk,σ (k − λp) Uk,σ (2.9) 1 ,λ (p) 1 ,λ g (p), 2 ,λ Uk,σ 1 ,λ mj,k,σ 1 ,λ Uk,σ I + λσ1 2
J. Clark / Journal of Functional Analysis 256 (2009) 2894–2916
where we have intertwined U ∗
k,σ1 ,λ
1
from the left to the right, and
σ1 − I − 12 ¯ (p) = βj det(I + λσ1 ) fj k + (k − λp) , I + λσ1 − 12 + (σ2 − I )(I + λσ1 )−1 (k − λp) ( p) = k m U β det(I + λσ ) f , j 2 j k,σ j, k,σ ,λ ,λ 2 1 ,λ
∗ Uk,σ ∗ Uk,σ
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m∗ ,λ Uk,σ 1 ,λ 1 ,λ j,k,σ 1
∗ Uk,σ
1 1 1+λ 2 I + λσ2 2 U g (p) = det det 1 ,λ k,σ1 ,λ I + λσ1 1+λ × g p + (σ1 − σ2 )(I + λσ1 )−1 (k − λp) . the resulting expression has only angular → k, ˜ = σ , and integrating out the other angular degrees of freedom yields B.
Making the change of variables dependance of
σ2 σ1−1
σ1 I +λσ1 (k − λp)
3. Bounding integrals of non-commuting operators Now we move on to proving Propositions 3.1 and 3.3 below which are proved in much greater generality than needed for this section, but they will serve as the principle tools in Section 4. To state these propositions we will need to generalize the concept of a multiplication operator. Let H1 , H2 be Hilbert spaces. Given a bounded function M : Rn → B(H1 , H2 ) we can construct an element M ∈ B(L2 (Rn )⊗H1 , L2 (Rn )⊗H2 ) using the equivalence L2 (Rn )⊗H1 ∼ = L2 (Rn , H1 ), 2 n where for f ∈ L (R ) ⊗ H1 , M(f)( x ) = M( x )f( x ). We will call these multiplication operators. Proposition 3.1. Define B : L2 (Rn ) ⊗ H1 → L2 (Rn ) ⊗ H2 , s.t.
B = d k dσ τk∗ τaσ k qk,σ , Rn
(3.1)
SOn
where qk,σ is a multiplication operator in the P basis of the form: qk,σ = nk,σ (P )η(x1,σ k + yσ P , x2,σ k + yσ P ), where η(k1 , k2 ) is continuous and defines a trace class integral operator on L2 (Rn ), aσ , x1,σ , 2 n 2 n x2,σ , yσ ∈ Mn (R), and nk,σ ∈ B(L (R ) ⊗ H1 , L (R ) ⊗ H2 ) is a multiplication operator. Let det(x1,σ + yσ (aσ − I )),
det x2,σ + yσ (aσ − I ) ,
be uniformly bounded from below by B(H1 , H2 ) satisfy the norm bound:
1 c
det(x1,σ ),
and det(x2,σ )
for some c > 0. Finally, let the family of maps nk,σ (K) ∈ sup nk,σ r. k,σ
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Then B is well defined as a strong limit and is bounded in operator norm by B crη1 . Proof. We check the conditions for Proposition A.2 (applied for integrals rather than sums). Due to the intertwining relations between the unitaries τ ∗ τaσ k and the multiplication operators qk,σ , k we will then have a bound from above by an integral of multiplication operators. We must show that 12 (G1 + G2 ) is bounded, where
G1 =
d k
dσ τk∗ τaσ k qk2 ,σ2 and G2 =
SOn
d k
dσ qk∗ ,σ τa∗ k τk . 1
1
σ
SOn
The integrand of G1 is the multiplication operator ∗ τ τ q = n (P )η(x1,σ k + yσ P , x2,σ k + yσ P ), k,σ k ak k,σ and the integrand of G2 is ∗ ∗ q τ τ = τ ∗ τ ∗ n (P )η(x1,σ k + yσ P , x2,σ k + yσ P )τ ∗ τ k k,σ k,σ aσ k k aσ k aσ k k = nk,σ (P + σ k − k) η(x1,σ k + yσ P , x2,σ k + yσ P )
(3.2)
where xj,σ = xj,σ + yσ (aσ − I ), and we have used that τk M(P ) = M(P − k)τk .
However since the operators in the integrand of G1 are all multiplication operators in P , bounding a sum on them in the operator norm can be computed as a supremum in the following way:
dσ nk,σ G1 sup d k (P ) η(x1,σ k + yσ P , x2,σ k + yσ P ) P
B(H1 )
SOn
sup nk,σ dσ η(x1,σ k + yσ P , x2,σ k + yσ P ) (3.3) dk (P ) B(H ) sup
P
1
P
SOn
A similar result holds for G2 . Now applying Lemma A.1 to (3.3) along with our conditions on x1,σ , x2,σ , and nk,σ (P ) we get the bound G1 rcη1 . 2 Corollary 3.2. The integral of operators (2.5) converges strongly to a bounded operator with 1 norm less than or equal to (1−λ) n ρ1 . The bound in the above corollary in not sharp, since in Proposition (2.2) we show that B˜ = ˜ ρ1 Sλ − I 2ρ, since Sλ is unitary. Tr2 [ρA∗λ ]. Thus B Proof. We apply Proposition (3.1) with nk,σ (P ) = Sλ (|k|), η = ρ, aσ = σ , x1,σ = 1 + λ, x2,σ = I + σ , and yσ = λ. |nk,σ (P )| 1, so we can take r = 1. All determinants involved are of
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operators of the form σ1 + λσ2 where σ1 , σ2 ∈ SOn , so these determinants have a lower bound of (1 − λ)n . Hence we can take c = (1 − λ)−n . 2 Proposition 3.3. Let G ∈ B(Hl ⊗ L2 (Rn ), Hr ⊗ L2 (Rn )), and ϕ : B(Hl ⊗ L2 (Rn ), Hr ⊗ L2 (Rn )) → B(Hl0 ⊗ L2 (Rn ), Hr0 ⊗ L2 (Rn )) has the form ϕ(G) =
j
d k
∗ ∗ dσ1 dσ2 Uk,σ 2, 2 Uk,σ hj,k,σ Ggj,k,σ 1
1
SOn ×SOn
∗ 2 n where Uk,σ acts on the L (R ) tensor as Uk,σ = τk Dbσ τa k , and hj,k,σ and gj,k,σ are multipliσ cation operators in P of the form:
hj,k,σ =n
(1) (1) (P )ηj (x1,σ k + x2,σ P ) j,k,σ
and gj,k,σ =n
(2) (2) (P )ηj (x1,σ k + x2,σ P ). j,k,σ
In the above, x1,σ , x2,σ , aσ ∈ Mn (R), bσ ∈ GLn (R), the family of operators n(1)
j,k,σ
in B(Hl , Hl0 ) and B(Hr , Hr0 ), respectively, and finally
(1) (2) ηj , ηj
and n(2)
j,k,σ
lie
∈ L2 (Rn ). We will require that
1 infdet x1,σ + x2,σ b−1 σ aσ − I σ c (1)
(2)
and supj,k,σ nj,k,σ nj,k,σ , supj,k,σ r. In this case, the integral of operator converges strongly to an operator ϕ(G) with the norm bound ϕ(G) cr 2 1 T1 1 + T2 1 G. 2 where T =
j
()
()
|ηj ηj | for = 1, 2 and · 1 is the trace norm.
Proof. We work towards showing the conditions of Proposition A.3 with sums replaced an integral-sum. We thus need to find bounds for the operator norms of
∗ 2 ∗ 2 dσ d k Uk,σ and dσ d k Uk,σ (3.4) . |g|j,k,σ Uk,σ |h|j,k,σ Uk,σ j SO n
j SO n
Rn
Rn
= |g|2 (P ) is a multiplication operator with elements in B(Hl0 , Hl0 ) or an element in j,k,σ j,k,σ B(L2 (Rn ) ⊗ Hl , L2 (Rn ) ⊗ Hl ). Conjugating with Uσ , we get only multiplication operators back: |g|2
−1 ∗ 2 = |g|2 bσ P + b−1 Uk,σ σ aσ − I k . |g|j,k,σ (P )Uk,σ j,k,σ With the calculations for bounding integrals of multiplication operators as in the proof of (3.1), we get the bound (1) 2 (2) 2 η + η G = 1 cr 2 T1 1 + T2 1 G. ϕ(G) 1 cr 2 j j 2 2 2 2 j
2
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Corollary 3.4. The integral of operators (2.6) converges strongly to a limit with operator norm 1 n ) . bounded by ρ1 G( 1−λ Proof. We apply Proposition 3.3 in the case where aσ = σ , bσ = (1) (2) −σ ) x2,σ = λ(II +λ , ηj = ηj = βj fj , and n In this case |n
(1) (P )| j,k,σ
x2,σ (b−1 σ aσ
I +λσ 1+λ
, x1,σ =
σ +λ 1+λ ,
(1) (2) − 12 . (P ) = nj,k,σ (P ) = det(1 + λσ ) j,k,σ
(2) − n2 , so we (P )| (1 − λ) j,k,σ σ (1+λ) σ (1+λ) −1 I +λσ and ( I +λσ ) 1, and hence
and |n
can take r = 1. Also
n x1,σ + − I) = | det( σI(1+λ) +λσ )| (1) = 1 independent of λ and σ , so we can take c = 1. Hence by (3.3), we have our conclusion with a bound ρ1 G(1 − λ)−n . 2
4. Reduced Born approximation with third-order error In this section, we will prove Theorem 4.2. To make mathematical expression more compact it will be helpful to have the dictionary below. In the following expressions λ, r ∈ R+ , σ ∈ SOn , P ∈ Rn . and k, Dictionary of vectors in Rn :
Dictionary of matrices in Mn (R):
(P ) = (1 + rλ)k + rλP , (1) ak,r,λ
(1) c1,σ,r,λ =
σ (1+λ) σ (1+λ)−λr(σ −I ) ,
(P ) = (2) vk,σ,λ
(2) c2,σ,r,λ =
−λr(1−λ) σ (1+λ)−λr(σ −I ) ,
(3) c3,σ,r,λ =
(1+λ)(r+(1−r)σ ) σ (1+λ)−λr(σ −I ) .
σ +λ σ −I 1+λ k − λ 1+λ P ,
(P ) = (3) vk,σ,r,λ (4) dk,λ (P ) =
σ (1+λ)−λr(σ −I ) −I k − λr σ1+λ P, 1+λ
1 λ 1+λ k − 1+λ P .
Now we will list some relations between the vectors. The significance of these relations will become apparent once we begin doing calculations. Relations 1 1 (R1) k + P = 1+rλ ak,r,λ (P ) + 1+rλ P, (P ) − λc3,σ,r,λ P , (R2) d (P ) = c1,σ,r,λ v k,λ
k,σ,r,λ
(P ) + c2,σ,r,λ P . (R3) k + P = c1,σ,r,λ vk,σ,r,λ In the proof of (4.2) the analysis is organized around the fact that certain expressions are bounded. In the limit λ → 0, expressions of the type λ1 S¯λ (·) will be a source of unboundedness, and ρ and G will have to be constrained in such a way as to compensate for this. The following expressions, defined for dimensions n = 1, 3, are uniformly bounded in P , k ∈ R, σ ∈ {+, −}, 0 r 1, and 0 λ: r, λ) = E1 (P , k,
ak,r,λ (P )|n−2 )−1 1 (δn,3 + | , S¯λ |k| λ 1 + |P |
(4.1)
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σ, r, λ) = E2 (P , k,
vk,σ,r,λ (P )|n−2 )−1 1 (δn,3 + | S¯λ dk,λ (P ) , λ 1 + |P |
λ) = E3 (P , k,
n−2 )−1 1 (δn,3 + |k| S¯λ dk,λ (P ) . λ 1 + |P |
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(4.2) (4.3)
Their boundedness can be seen by using (R1) to rewrite k in terms of ak,r,λ (P ) and P for E1 (P , k, r, λ), (R2) to write dk,λ (P ) in terms of vk,σ,r,λ (P ) and P for E2 (P , k, σ, r, λ), and λ), d (P ) explicitly defined in terms of k and P . for E3 (P , k, k,λ A second-order Taylor expansion of the scattering coefficients gives: Dim-1 Sλ (k) =
λ −iα 1+λ λ k + i 12 α 1+λ
∼ −λ(1 − λ)
iα λ2 α 2 − . k 2 k2
Dim-2 Sλ (k) =
−iπ 1+λ −1 λ l
+ γ + ln( k2 ) − i π2 k λ2 2 2 − π. ∼ −λ(1 − λ)iπl − iλ l γ + ln 2 2
Dim-3 Sλ (k) =
−2ik 1+λ −1 λ l
+ ik
∼ −λ(1 − λ)2ilk − 2λ2 l 2 k 2 .
We can summarize the above expressions as Sλ (k) ∼ −iλ(1 − λ)cn k n−2 −
k λ2 2 2(n−2) cn k , − δn,2 iλ2 l 2 γ + ln 2 2
where c1 = α, c2 = πl, and c3 = 2l. We will use the following simple lemma. K ∈ R3 . Lemma 4.1. Let k, K ∈ R and k, (1) We have the inequality 1
(k − λK)2 +
α2 4
2 λ2
K2 + α|k|
α2 4
2|K| + α , α|k|
(4.4)
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(2) and for dimension one the scattering coefficient satisfies Sλ k − λK λ 2|K| + α , 1+λ |k| (3) and Sλ k − λK − −iαλ λ2 |K| 2|K| + α , 1+λ |k| |k|2 (4) for dimension three, the scattering coefficient satisfies Sλ k − λK − −2ilλ|k| λ2 4l |k| + |K| . 1 + l|k| 1+λ (1 + λ)2 Proof. (1) follows by evaluating the critical points in λ. (2) and (3) follow with an application of (1). 2 Define the following weighted trace norm · wtn for the density matrices on the single reservoir particle Hilbert space ρ: ρwtn = ρ1 +
|P |n−2+ Xi , [Xj , ρ] 1i,j n
+
1
n n−2+ |P | Xj ρXj |P |n−2+ 1 + |P |2(n−2) ρ|P |2(n−2) 1 ,
(4.5)
j =1
where the sums in are over {0, 1} for dimension one and {−1, 0, 1} for dimension three. Notice the contrast between dimension n = 1 and n = 3 with respect to the weights applied in the norms for the absolute value of the momentum operators |P |. For n = 1, ρwtn will blow up if ρ has non-zero density of momenta near momentum zero, while for n = 3, ρwtn can blow up if the momentum density does not decay fast enough for large momenta. This difference in requirements for different dimensions can be seen also in the formulas (1.5)–(1.8). The norm ρwtn is not really asymmetric with respect to operators multiplying from the left and the right when ρ is self-adjoint. Theorem 4.2. Let (G, λ) be defined as in (1.1), then there exists a c s.t. for all density operators ρ ∈ B1 (L2 (Rn )), G ∈ B(L2 (Rn )), and 0 λ (G, λ) cλ3 ρwtn Gwn .
(4.6)
Proof. We will prove the result for density operators ρ with a twice continuously differentiable integral kernel ρ(k1 , k2 ) in the momentum representation, and a spectral decomposition ρ= ∞ j =1 λj |fj fj | of vectors fj (k) that are continuously differentiable in the momentum representation. Since such ρ are dense with respect to the · wtn , the result extends to all ρ with ρwtn < ∞. By (B.1), the V1 , V2 , A operators and the map ϕ are well defined for all ρ with ρwtn < ∞ and they vary continuously as a function of ρ with respect to the norm · wtn .
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Our challenge is to expand the expressions we found in Propositions 2.2 and 2.3 in λ, until we reach our second-order Taylor expansion while making sure that we only throw away terms which are bounded as in (4.6). We will organize our analysis using the expressions (4.1)–(4.3), in conjunction with Propositions 3.1 and 3.3 to effectively transfer the conditions for the boundedness of the differences in our expansions to conditions on ρ and G. Both of the expressions (2.5) and (2.6) have multiple sources of λ dependence. If we expand the expressions involving ρ and fj first for (2.5) and (2.6) respectively, then the resulting expressions left to expand will be summable in the operator norm and thus not require the heavy preparation involved with the use of Propositions 3.1 and 3.3. Breaking Tr2 [ρS∗λ GSλ ] into parts and dividing by λ we just need bound the differences 1 λ λ Tr2 [ρA∗λ ]G − iV1 + iλV2 + i {A, P } − ϕ(I ) G λ 2 2
and
1 Tr2 [ρA∗λ GA] − λϕ(G), λ
(4.7) (4.8)
where there is a similar expression to (4.7) for λ1 G Tr2 [ρAλ ]. We begin with (4.7), and will have to bound a sequence of intermediate differences. The main differences are the following: Difference 1
1 1 ∗ Tr2 ρA∗ G − d k ¯ dσ τk τσ k ρ(k, σ k) + λ(P + k)∇T ρ(k, σ k) Sλ |k| G λ λ . λ Rn
SOn
Difference 2
d k σ k) + λ(P + k)∇ T ρ(k, σ k) dσ τk τσ∗k ρ(k, Rn
×
SOn
1¯ 2−n + λ cn2 |k| 2(2−n) G. Sλ |k| − (1 − λ)cn |k| λ 2
Difference 3
d k σ k) + λ(P + k)∇ T ρ(k, σ k) dσ τk τσ∗k ρ(k, Rn
SOn
λ λ 2(2−n) λ 2−n G − iV1 + iλV2 + {A, P } − ϕ(I ) G + cn |k| × (1 − λ)cn |k| . 2 2 2
By the differentiability properties of the integral kernel ρ,
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σ k + λ(P + k) ρ k + λ(P + k), σ k) + λ(P + k)∇ T ρ(k, σ k) = ρ(k, ⊗2 + λ (P + k)
1
2
s ds
0
2 σ k + λ(P + k)r , dr ∇T⊗ ρ k + λ(P + k)r,
0
where ∇T⊗ g(x, y) is 2 tensor of derivatives with 2
⊗2 (∇T g)i (x + hej , y + hej ) − (∇T g)i (x, y) . ∇T g(x, y) (i,j ) = lim h→0 h The first difference can be rewritten as
1 λ
2
s ds
0
⊗2 δn,3 + a (P )n−2 dr d k dσ τk τσ∗k 1 + |P | (P + k) k,r,λ
0
Rn
SOn
2 × ∇T⊗ ρ ak,r,λ (P ), ak,r,σ,λ (P ) + (σ
− I )k E1 (P , k, r, λ)G .
2 Using (R1) and expanding the tensor: ( ak,r,λ (P ) + P )⊗ a single term has the form m 2−m (P )⊗ P ⊗ . Note that the order of the tensors does not matter in this situation, since ak,r,λ
the whole vector is in an inner product with ∇ ⊗ ρ, and partial derivatives commute. Now we apply Proposition 3.1 with a single term: 2
nk,σ
r, λ) = E1 (P , k,
qk,σ
1 1 + rλσ
2
m 2−m ( ak,r,λ (P )⊗ ⊗ P ⊗ )j,k , | a (P )|m |P |2−m
k,r,λ
m ⊗2 ∇ ρ n−2 |k| η = δn,3 + |k| , T j,k = nk,σ,λ η ak,r,λ (P ), ak,r,σ,λ (P ) + (σ − I )k .
2 Finally with (3.1) we get the bound λ2 C(δn,3 + |P |n−2 )|P |m (∇T⊗ ρ)j,k 1 |P |2−m (I + |P |)G, − ρ X). for some constant C. Note that ∇T ρ = i(Xρ The second difference can be bounded for dimension one using the inequality
2 2 3 1 S¯λ |k| − iα(1 − λ) − λα λ α , λ 2 3 |k| 2|k| |k| and for dimension three using the inequality 1 Sλ |k| − 2i(1 − λ)|k| − 2λl|k| 2 2λ2 l 2 |k| 3. λl
J. Clark / Journal of Functional Analysis 256 (2009) 2894–2916
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Finally, the last difference comes down to bounding the cross term:
2 ∗ n−2 2 2 2(n−2) dk G dσ τk τσ k λ(P + σ k)∇T ρ(k, σ k) λ cn |k| + λ cn |k| . Rn
SOn
The bound for the above term follows from (A.1) and that |P |n−2 ρ|P |n−2 1 .
k, k)| k| 2(n−2) = d kρ(
The λ1 G Tr2 [ρAλ ] is similarly analyzed so now we study (4.8). Again we have three main , Uk,σ,λ , and Sλ (|dk,λ differences. There is a λ dependence in mj,k,σ,λ (P )|). It is most convenient to begin expanding mj,k,σ,λ first. Difference 1
1 1 Tr2 ρA∗ GAλ − d k λ λ λ j
∗ dσ1 dσ2 Uk,σ
det(I + λσ1 )− 2 f¯j (k) 1
1 ,λ
SOn ×SOn
Rn
− 12 ¯ × Sλ dk,λ Uk,σ (P ) GSλ dk,λ (P ) fj (k) det(I + λσ2 ) 2 ,λ . Difference 2
d k j Rn
∗ dσ1 dσ2 Uk,σ
f¯j (σ1 k) 1 ,λ
1 λ−1 det(1 + λσ1 )− 2 S¯λ dk,λ (P ) G
SOn ×SOn
− 12 2 2(n−2) × Sλ dk,λ det(1 + λσ ( P ) ) − c | k| G f (σ k)U 2 j 2 2 ,λ . n k,σ Difference 3
2 2 2 |k| 2(n−2) λ c d k fj (k) n j
R3
∗ dσ1 dσ2 Uk,σ
∗ ∗ GUk,σ 2 ,λ − τσ k τk Gτk τσ2 k 1 ,λ 1
.
SOn ×SOn
Using the differentiability of fj ’s σ1 − 1 (k + P ) fj σ 1 k − λ 1+λ
1 σ1 − 1 σ1 − 1 (k + P ) dr ∇fj σ1 k + rλ (k + P ) . = fj (σ1 k) + λ 1+λ 1+λ 0
(4.9)
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The first difference
1 Tr2 ρA∗ GAλ − d k λ λ j Rn
∗ dσ1 dσ2 Uk,σ
det(1 + λσ1 )− 2 f¯j (k) 1
1 ,λ
SOn ×SOn
GS m × S¯λ dk,λ d ( P ) ( P ) (P )U λ j,k,σ2 ,λ 2 ,λ k,λ k,σ is less than
r λ2 cn2
dr d k j Rn
0
∗ dσ1 dσ2 Uk,σ
1
1 ,λ
det(1 + λσ1 )− 2
SOn ×SOn
∇ f¯j (vk,σ1 ,r,λ (P )) σ1 − I ) + c2,σ1 ,r,λ P cσ1 ,r,λ vk,r,σ ( P 1 ,λ I +λ (δn,3 + | vk,r,σ (P )|n−2 )−1 1 ,λ σ1 , r, λ) 1 + |P | G 1 + |P | × E2 (P , k, mj,k,σ 2 ,λ (P ) × E2 (P , k, σ2 , r, λ) Uk,σ 2 ,λ , n−2 −1 (δn,3 + | vk,r,σ ( P )| ) 2 ,λ ×
σ, r, λ) expressions and used (R3) to where we have rearranged to substitute in the E2 (P , k, rewrite k + P . Two applications of Proposition 3.3 corresponding to cσ1 ,r,λ vk,σ,r,λ (P ) and c2,σ1 ,r,λ P will give us our bound. For the cσ1 ,r,λ vk,σ,r,λ (P ) we use Proposition 3.3 with (1) ∇fj (k) n−2 |k| , = δn,3 + |k| ηj (k) n
(1) 1 (P ) = det(I j,k,σ
σ, r, λ)cσ1 ,r,λ + λσ1 )− 2 E2 (P , k,
hj,k,σ 1 =n
1
(1) (1) k,σ 1 ,λ (P ) , 1 ηj v j,k,σ
∇fj (k) , |∇fj (k)|
= δn,3 + |k| n−2 fj (k), ηj(2) (k) n
(2) 2 (P ) = det(I j,k,σ
gj,k,σ 2 =n
σ2 , r, λ), + λσ2 )− 2 E2 (P , k, 1
(2) (2) ηj vk,σ2 ,λ (P ) . j,k,σ2
Hence the term is bounded by a constant multiple of |P | δn,3 + |P |n−2 Xj ρXj δn,3 + |P |n−2 |P | λ 1 2
j
+ δn,3 + |P |n−2 ρ δn,3 + |P |n−2
1
1 + |P | G 1 + |P | .
J. Clark / Journal of Functional Analysis 256 (2009) 2894–2916
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The c2,σ1 ,r,λ P term is bounded by a constant multiple of δn,3 + |P |n−2 Xj ρXj δn,3 + |P |n−2 λ 1 2
j
+ δn,3 + |P |n−2 ρ δn,3 + |P |n−2
1
1 + |P | |P |G 1 + |P | .
The next intermediary difference has the form:
dk j Rn
1 ∗ S¯λ d (P ) dσ1 dσ2 Uk,λ,σ det(1 + λσ1 )− 2 f¯j (k) k,λ 1
SOn ×SOn
− 12 . ¯ d det(1 + λσ × G f vk,σ ( P ) − f (σ ( P ) ) U k) S 2 λ 2 2 ,λ k,λ k,λ,σ 1 Expanding f ( vk,σ 2 ,λ (P )) − f (σ2 k) as in (4.9), we can apply a similar analysis to the above, λ) rather than E3 (P , k, σ, r, λ). except that for the left-hand side we organize around E3 (P , k, j (σ2 k), the second difference is summable, and we do not need to prepare Due to f¯j (σ1 k)f any more applications of Proposition 3.3. We begin by bounding
2 λf cn d k j
Rn
SOn ×SOn
∗ dσ1 dσ2 Uk,σ
1
1 ,λ
det(1 + λσ1 )− 2
f¯j (σ1 k) 2−n |k|
|k|2−n ¯ × Sλ dk,λ Sλ dk,λ (P ) G (P ) − G λcn λcn 1 fj (σ2 k) . × det(1 + λσ2 )− 2 Uk,σ ,λ 2 2−n |k|
|k|2−n
We observe the inequality 2−n 2−n |k| |k| ¯ Sλ dk,λ (P ) G (P ) − G λc Sλ dk,λ λcn n 2−n |k| 1 ¯ Sλ dk,λ (P ) − i G |P | + I E3 (P , k, λ) cn λcn 2−n |k| + G Sλ dk,λ (P ) + i . λcn By (3) and (4) of (4.1), the right-hand side is bounded by a sum of terms proportional to r(n−2) |P |1 G(I + |P |)2 for r = 0, 1, 2, 1 , 2 = 0, 1. Bounding the above integral is then λ|k| routine and requires that |P |2(n−2) ρ|P |2(n−2) 1 . The last thing to do for the second difference 1 1 is expanding det(1 + λσ1 )− 2 and | det(1 + λσ2 )|− 2 , which does not pose much difficulty.
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For the third difference, we will need to work with the D 1+λσ terms 1+λ
∗ U
1 k,σ
∗ ∗ ∗ GUk,σ 2 ,λ − τσ k τk Gτk τσ2 k (D 1+λσ1 − I )G + G(D 1+λσ1 − I ) , ,λ 1
1+λ
1+λ
∗ ∗ since Uk,σ 2 ,λ , τσ k , and τk are unitary and D 1+λσ1 τk = τ 1+λσ k D 1+λσ1 . D 1+λσ satisfies the integral 1+λ
1+λ
relation
λ D 1+λσ = I + 1+λ
0
and hence
1+λ
1+λ
1 + sσ d log D 1+sσ P , X , ds 1+s ds 1+s
1 1 + sσ d (D 1+λσ − I )G sup log Pi Xj G. λ 1+λ 1+s 0sλ ds i,j
2 n−2 n−2 The
third difference is then bounded by a fixed constant multiple of λ |P | ρ|P | 1 × j |P |Xj G. 2
Acknowledgments I thank Bruno Nachtergaele for discussions on this manuscript. This work is partially funded from the Belgian Interuniversity Attraction Poles Programme P6/02. Also financial support has come from Graduate Student Research (GSR) fellowships funded by the National Science Foundation (NSF # DMS-0303316 and DMS-0605342). Appendix A. Hilbert spaces and operator inequalities Lemma A.1. Let η be a trace class operator on Rn with continuous integral kernel η(k1 , k2 ), A, A ∈ GLn (R), and a , a ∈ Rn , then
1 1 1 x + a , A x + a ) + η1 . d x η(A 2 | det(A)| | det(A )| Rn
Proof. Let ρ =
Rn
j
λj |fj gj |, then we have
x + a , A x + a ) = d x ρ(A
Rn
d x λj fj (A x + a )g¯ j (A x + a )
j
1 1 1 + ρ1 , 2 | det(A)| | det(A )|
(A.1)
where the inequality follows from 2ab a 2 + b2 for a, b ∈ R+ and completing the integration. The equality on the left-hand side of (A.1) is formal for a general integral kernel ρ(k1 , k2 ) which is defined only a.e. with respect to joint integration over k1 , k2 , but with our continuity condition it is well defined. 2
J. Clark / Journal of Functional Analysis 256 (2009) 2894–2916
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Proposition A.2. For n ∈ N, let An ∈ B(H) for a Hilbert space H and 1 |An | + A∗n 2 n be weakly convergent to a bounded operator with norm c. Then a bounded operator X with X c.
n An
is strongly convergent to
Proof. Let g ∈ H and with the polar decomposition [11] An = Un |An |, then taking a tail sum ∞ ∞ 1 1 An g = sup h Un |An | 2 |An | 2 g h2 =1 N
N
2
∞
sup
h2 =1 N
sup
h2 =1
|An | 12 Un h |An | 12 g 2 2 1
∞ ∗ h An h
2
1
∞ g |An |g
2
,
N
n=1
where the first and second inequalities follow by two different applications of the Cauchy– Schwartz The right side then tends to zero for large N by our assumptions on the
inequality. series n |An | and n |A∗n |. The operator norm bound can be seen from the same calculation with a sum over all n rather than a tail. 2 Proposition A.3.
∗
Let∗ An , Bn for n ∈ N be elements in B(H) for a Hilbert space H such that A A and n n n n Bn Bn converge weakly to bounded operators with norms less than c, then the sum ϕ(G) =
A∗n GBn
n
is strongly convergent to an operator with norm less than or equal to cG. The proof follows a similar pattern to that of Proposition A.2. Appendix B. Norm bounds for V1 , V2 , A and ϕ The following lemma shows that the limiting expressions (1.3) vary continuously with respect to the density operator ρ in the · wtn topology. It allows the limiting expression to be defined for all ρ with ρwtn < ∞ without the additional assumption that the integral kernel of ρ in the momentum representation in continuously differentiable. A somewhat weaker norm than · wtn would suffice.
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J. Clark / Journal of Functional Analysis 256 (2009) 2894–2916
and ϕ be defined as in (1.5)–(1.8) for a ρ ∈ B1 (Rn ), ρ 0, with conLemma B.1. Let V1 , V2 , A, tinuously differentiable integral kernel in the momentum representation, then there is a constant c > 0 such that for all ρ and j [Pj , Aj ], ϕ cρwtn . V1 , V2 , A, Proof. By an argument similar to (A.1) V1 cn |P |ρ 1 ,
cn A
|P |n−2 Xj ρ , 1 j
Pj |P |n−1 , [Xj , ρ] , V2 cn 1
ϕ cn2 |P |n−2 ρ|P |n−2 1 ,
[Pj , Aj ] cn Pj |P |n−2 , [Xj , ρ] , 1
j
where we have used that ϕ = ϕ(I ) since ϕ is a positive map. By ρ being self-adjoint, we have inequalities such as Pj |P |n−2 ρXj 1 Xj ρXj 1 + Pj |P |n−2 ρ|P |n−2 Pj , 1 1 2 k, k)| k| 2s = |P |s ρ|P |s 1 , and inand then that Pj |P |. Finally, since ρ is positive Rn d kρ( equalities of the form r |P | ρ|P |r ρ1 + |P |s ρ|P |s 1 1 follow, where r, s have the same sign and |r| |s|. Hence ρwtn bounds the expressions for V1 , [Pj , Aj ] and ϕ. 2 V2 , A, References [1] R. Adami, R. Figari, D. Finco, A. Teta, On the asymptotic dynamics of a quantum system composed by heavy and light particles, Comm. Math. Phys. 268 (2006) 819–852. [2] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, Springer-Verlag, Berlin, 1988. [3] C. Cacciapuoti, R. Carlone, R. Figari, Decoherence induced by scattering, a three-dimensional model, J. Phys. A 38 (2005) 4933–4946. [4] J. Clark, An infinite-temperature limit for a quantum scattering process, Rept. Math. Phys. (2009), in press, arXiv: 0801.0722. [5] D. Dürr, R. Figari, A. Teta, Decoherence in a two particle model, J. Math. Phys. 45 (2004) 1291–1309. [6] M.R. Gallis, G.N. Fleming, Environmental and spontaneous localization, Phys. Rev. A 42 (1990) 38–48. [7] L. Hackermüller, K. Hornberger, B. Brezger, A. Zeilinger, M. Arndt, Decoherence in a Talbot–Lau interferometer, the influence of molecular scattering, Appl. Phys. B 77 (2003) 781–787. [8] K. Hornberger, J. Sipe, Collisional decoherence reexamined, Phys. Rev. A 68 (2003) 012105. [9] E. Joos, H.D. Zeh, The emergence of classical properties through interaction with the environment, Z. Phys. B 59 (1985) 223–243. [10] G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48 (1976) 119–130. [11] M. Reed, B. Simon, Functional Analysis, Academic Press, 1980.
Journal of Functional Analysis 256 (2009) 2917–2943 www.elsevier.com/locate/jfa
Structure of derivations on various algebras of measurable operators for type I von Neumann algebras S. Albeverio a,1 , Sh.A. Ayupov b,∗ , K.K. Kudaybergenov c a Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D-53115 Bonn, Germany b Institute of Mathematics and information technologies, Uzbekistan Academy of Sciences, F. Hodjaev str. 29,
100125 Tashkent, Uzbekistan c Karakalpak State University, Ch. Abdirov str. 1, 742012 Nukus, Uzbekistan
Received 1 August 2008; accepted 11 November 2008 Available online 28 November 2008 Communicated by N. Kalton
Abstract Given a von Neumann algebra M denote by S(M) and LS(M) respectively the algebras of all measurable and locally measurable operators affiliated with M. For a faithful normal semi-finite trace τ on M let S(M, τ ) be the algebra of all τ -measurable operators from S(M). We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra M. In particular, we prove that if M is of type I∞ then every derivation on LS(M) (resp. S(M) and S(M, τ )) is inner. © 2008 Elsevier Inc. All rights reserved. Keywords: Von Neumann algebras; Noncommutative integration; Measurable operator; Locally measurable operator; τ -Measurable operator; Type I von Neumann algebra; Derivation; Inner derivation
Introduction Derivations on unbounded operator algebras, in particular on various algebras of measurable operators affiliated with von Neumann algebras, appear to be a very attractive special case of the general theory of unbounded derivations on operator algebras. The present paper continues the * Corresponding author.
E-mail addresses:
[email protected] (S. Albeverio),
[email protected] (Sh.A. Ayupov),
[email protected] (K.K. Kudaybergenov). 1 Member of: SFB 611, BiBoS; CERFIM (Locarno); Acc. Arch. (USI). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.003
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series of papers of the authors [1,2] devoted to the study and a description of derivations on the algebra LS(M) of locally measurable operators with respect to a von Neumann algebra M and on various subalgebras of LS(M). Let A be an algebra over the complex number. A linear operator D : A → A is called a derivation if it satisfies the identity D(xy) = D(x)y + xD(y) for all x, y ∈ A (Leibniz rule). Each element a ∈ A defines a derivation Da on A given as Da (x) = ax − xa, x ∈ A. Such derivations Da are said to be inner derivations. If the element a implementing the derivation Da on A, belongs to a larger algebra B, containing A (as a proper ideal as usual) then Da is called a spatial derivation. In the particular case where A is commutative, inner derivations are identically zero, i.e. trivial. One of the main problems in the theory of derivations is automatic innerness or spatialness of derivations and the existence of noninner derivations (in particular, nontrivial derivations on commutative algebras). On this way A.F. Ber, F.A. Sukochev, V.I. Chilin [3] obtained necessary and sufficient conditions for the existence of nontrivial derivations on commutative regular algebras. In particular they have proved that the algebra L0 (0, 1) of all (classes of equivalence of) complex measurable functions on the interval (0, 1) admits nontrivial derivations. Independently A.G. Kusraev [13] by means of Boolean-valued analysis has established necessary and sufficient conditions for the existence of nontrivial derivations and automorphisms on universally complete complex f -algebras. In particular he has also proved the existence of nontrivial derivations and automorphisms on L0 (0, 1). It is clear that these derivations are discontinuous in the measure topology, and therefore they are neither inner nor spatial. It seems that the existence of such pathological example of derivations deeply depends on the commutativity of the underlying von Neumann algebra M. In this connection the present authors have initiated the study of the above problems in the noncommutative case [1,2], by considering derivations on the algebra LS(M) of all locally measurable operators with respect to a semi-finite von Neumann algebra M and on various subalgebras of LS(M). Recently another approach to similar problems in the framework of type I AW ∗ -algebras has been outlined in [7]. The main result of the paper [2] states that if M is a type I von Neumann algebra, then every derivation D on LS(M) which is identically zero on the center Z of the von Neumann algebra M (i.e. which is Z-linear) is automatically inner, i.e. D(x) = ax − xa for an appropriate a ∈ LS(M). In [2, Example 3.8] we also gave a construction of noninner derivations Dδ on the algebra LS(M) for type Ifin von Neumann algebra M with nonatomic center Z, where δ is a nontrivial derivation on the algebra LS(Z) (i.e. on the center of LS(M)) which is isomorphic with the algebra L0 (Ω, Σ, μ) of all measurable functions on a nonatomic measure space (Ω, Σ, μ). The main idea of the mentioned construction is the following. Let A be a commutative algebra and let Mn (A) be the algebra of n × n matrices over A. If ei,j , i, j = 1, n, are the matrix units in Mn (A), then each element x ∈ Mn (A) has the form x=
n
λij eij ,
λi,j ∈ A,
i, j = 1, n.
i,j =1
Let δ : A → A be a derivation. Setting n n Dδ λij eij = δ(λij )eij i,j =1
i,j =1
(1)
S. Albeverio, et al. / Journal of Functional Analysis 256 (2009) 2917–2943
2919
we obtain a well-defined linear operator Dδ on the algebra Mn (A). Moreover Dδ is a derivation on the algebra Mn (A) and its restriction onto the center of the algebra Mn (A) coincides with the given δ. In [1] we have proved spatialness of all derivations on the noncommutative Arens algebra Lω (M, τ ) associated with an arbitrary von Neumann algebra M and a faithful normal semi-finite trace τ . Moreover if the trace τ is finite then every derivation on Lω (M, τ ) is inner. In the present paper we give a complete description of all derivations on the algebra LS(M) of all locally measurable operators affiliated with a type I von Neumann algebra M, and also on its subalgebras S(M)—of measurable operators and S(M, τ ) of τ -measurable operators, where τ is a faithful normal semi-finite trace on M. We prove that the above mentioned construction of derivations Dδ from [2] gives the general form of pathological derivations on these algebras and these exist only in the type Ifin case, while for type I∞ von Neumann algebras M all derivations on LS(M), S(M) and S(M, τ ) are inner. Moreover we prove that an arbitrary derivation D on each of these algebras can be uniquely decomposed into the sum D = Da + Dδ where the derivation Da is inner (for LS(M), S(M) and S(M, τ )) while the derivation Dδ is constructed in the above mentioned manner from a nontrivial derivation δ on the center of the corresponding algebra. In Section 1 we give necessary definition and preliminaries from the theory of measurable operators, Hilbert–Kaplansky modules and also prove some key results concerning the structure of the algebra of locally measurable operators affiliated with a type I von Neumann algebra. In Section 2 we describe derivations on the algebra LS(M) of all locally measurable operators for a type I von Neumann algebra M. Sections 3 and 4 are devoted to derivation respectively on the algebra S(M) of all measurable operators and on the algebra S(M, τ ) of all τ -measurable operators with respect to M, where M is a type I von Neumann algebra and τ is a faithful normal semi-finite trace on M. Finally, Section 5 contains an application of the above results to the description of the first cohomology group for the considered algebras. 1. Locally measurable operators affiliated with type I von Neumann algebras Let H be a complex Hilbert space and let B(H ) be the algebra of all bounded linear operators on H . Consider a von Neumann algebra M in B(H ) with the operator norm · M . Denote by P (M) the lattice of projections in M. A linear subspace D in H is said to be affiliated with M (denoted as DηM), if u(D) ⊂ D for every unitary u from the commutant M = y ∈ B(H ): xy = yx, ∀x ∈ M of the von Neumann algebra M. A linear operator x on H with the domain D(x) is said to be affiliated with M (denoted as xηM) if D(x)ηM and u(x(ξ )) = x(u(ξ )) for all ξ ∈ D(x). A linear subspace D in H is said to be strongly dense in H with respect to the von Neumann algebra M, if (1) DηM; (2) there exists a sequence of projections {pn }∞ n=1 in P (M) such that pn ↑ 1, pn (H ) ⊂ D and pn⊥ = 1 − pn is finite in M for all n ∈ N, where 1 is the identity in M.
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A closed linear operator x acting in the Hilbert space H is said to be measurable with respect to the von Neumann algebra M, if xηM and D(x) is strongly dense in H . Denote by S(M) the set of all measurable operators with respect to M. A closed linear operator x in H is said to be locally measurable with respect to the von Neumann algebra M, if xηM and there exists a sequence {zn }∞ n=1 of central projections in M such that zn ↑ 1 and zn x ∈ S(M) for all n ∈ N. It is well known [14] that the set LS(M) of all locally measurable operators with respect to M is a unital ∗-algebra when equipped with the algebraic operations of strong addition and multiplication and taking the adjoint of an operator. Let τ be a faithful normal semi-finite trace on M. We recall that a closed linear operator x is said to be τ -measurable with respect to the von Neumann algebra M, if xηM and D(x) is τ dense in H , i.e. D(x)ηM and given ε > 0 there exists a projection p ∈ M such that p(H ) ⊂ D(x) and τ (p ⊥ ) < ε. The set S(M, τ ) of all τ -measurable operators with respect to M is a solid ∗subalgebra in S(M) (see [15]). Consider the topology tτ of convergence in measure or measure topology on S(M, τ ), which is defined by the following neighborhoods of zero: V (ε, δ) = x ∈ S(M, τ ): ∃e ∈ P (M), τ e⊥ δ, xe ∈ M, xeM ε , where ε, δ are positive numbers. It is well known [15] that S(M, τ ) equipped with the measure topology is a complete metrizable topological ∗-algebra. Note that if the trace τ is a finite then S(M, τ ) = S(M) = LS(M). The following result describes one of the most important properties of the algebra LS(M) (see [14,16]). Proposition 1.1. Suppose that the von Neumann algebra M is the C ∗ -product of the von Neumann algebras Mi , i ∈ I , where I is an arbitrary set of indices, i.e. M=
Mi = {xi }i∈I : xi ∈ Mi , i ∈ I, sup xi Mi < ∞ i∈I
i∈I
∗ -norm {x } with coordinate-wise algebraic operations and involution and with the C i i∈I M = supi∈I xi Mi . Then the algebra LS(M) is ∗-isomorphic to the algebra i∈I LS(Mi ) (with the coordinate-wise operations and involution), i.e.
LS(M) ∼ =
LS(Mi )
i∈I
(∼ = denoting ∗-isomorphism of algebras). In particular, if M is a finite, then S(M) ∼ =
i∈I
S(Mi ).
S. Albeverio, et al. / Journal of Functional Analysis 256 (2009) 2917–2943
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It should be noted that similar isomorphisms are not valid in general for the algebras S(M), S(M, τ ) (see [14]). Proposition 1.1 implies that given any family {zi }i∈I of mutually orthogonal central projections in M with i∈I zi = 1 and a family of elements {xi }i∈I in LS(M) there existsa unique element x ∈ LS(M) such that zi x = zi xi for all i ∈ I . This element is denoted by x = i∈I zi xi . Further in this section we shall prove several crucial results concerning the properties of algebras of measurable operators for type I von Neumann algebras. In particular we present an alternative and shorter proof of the statement that the algebra of locally measurable operators in this case is isomorphic to the algebra of bounded operators acting on a Hilbert–Kaplansky module (cf. [2]). It is well known [18] that every commutative von Neumann algebra M is ∗-isomorphic to the algebra L∞ (Ω) = L∞ (Ω, Σ, μ) of all (classes of equivalence of) complex essentially bounded measurable functions on a measure space (Ω, Σ, μ) and in this case LS(M) = S(M) ∼ = L0 (Ω), 0 0 where L (Ω) = L (Ω, Σ, μ) the algebra of all (classes of equivalence of) complex measurable functions on (Ω, Σ, μ). Further we shall need the description of the centers of the algebras S(M) and S(M, τ ) for type I von Neumann algebras. It should be noted, that if M is a finite von Neumann algebra with a faithful normal semi-finite trace τ , then the restriction τZ of the trace τ onto the center Z of M is also semi-finite. Indeed by [19, Chapter V, Theorem 2.6] M admits the canonical center valued trace T : M → Z. It is known that T (x) = co{uxu∗ , u ∈ U } ∩ Z, where co{uxu∗ , u ∈ U } denotes the norm closure in M of the convex hull of the set {uxu∗ , u ∈ U } and U is the set of all unitaries from M. Therefore given any finite trace ρ (since it is norm-continuous and linear on M) one has ρ(T x) = ρ(x) for all x ∈ M. Given a normal semi-finite trace τ on M there exists a monotone increasing net {eα } of projection in M with τ (eα ) < ∞ and eα ↑ 1. The trace ρα (x) = τ (eα x), x 0, x ∈ M, is finite for any α and therefore for all x ∈ M, x 0, we have τ (T x) = limα τ (eα T x) = limα τ (eα x) = τ (x). Now given any projection z ∈ Z there exists a non-zero projection p ∈ M such that p z and τ (p) < ∞. Consider the element T (p) ∈ Z. From properties of T it follows that T (p) is a non-zero positive element in Z with τ (T (p)) = τ (p) < ∞ and T (p) T (z) = z. By the spectral theorem there exists a nonzero projection z0 in Z such that z0 λT (p) for an appropriate positive number λ. Therefore τ (z0 ) τ (λT (p)) = λτ (p) < ∞ and z0 λz, i.e. z0 z, and thus the restriction of τ onto Z is also semi-finite. Proposition 1.2. If M is finite von Neumann algebra of type I with the center Z and a faithful normal semi-finite trace τ , then Z(S(M)) = S(Z) and Z(S(M, τ )) = S(Z, τZ ), where τZ is the restriction of the trace τ on Z. Proof. Given x ∈ S(Z) take a sequence of orthogonal projections {zn } in Zsuch that zn x ∈ Z for all n. Since M is finite, one has that LS(M) = S(M) and therefore x = n zn x ∈ LS(M) = S(M), i.e. x ∈ Z(S(M)). ∞ Conversely, let x ∈ Z(S(M)), x 0 and let x = 0 λ dλ be it spectral resolution. Put z1 = e1 and zk = ek − ek−1 , k 2. Then {zk } is a family of mutually orthogonal central projections with z = 1. It is clear that z x ∈ Z for all k. Therefore x = k k k n zn x ∈ S(Z), and thus Z(S(M)) = S(Z). In a similar way we obtain that Z(S(M, τ )) = S(Z, τZ ). The proof is complete. 2
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Recall that M is a type I∞ if M is of type I and does not have non-zero finite central projections. Proposition 1.3. Let M be a type I∞ von Neumann algebra with the center Z. Then the centers of the algebras S(M) and S(M, τ ) coincide with Z. ∞ Proof. Suppose that z ∈ S(M), z 0, is a central element and let z = 0 λ deλ be its spectral resolution. Then eλ ∈ Z for all λ > 0. Assume that en⊥ = 0 for all n ∈ N. Since M is of type I∞ , Z does not contain non-zero finite projections. Thus en⊥ is infinite for all n ∈ N, which contradicts the condition z ∈ S(M). Therefore there exists n0 ∈ N such that en⊥ = 0 for all n n0 , i.e. z n0 1. This means that z ∈ Z, i.e. Z(S(M)) = Z. Similarly Z(S(M, τ )) = Z. The proof is complete. 2 Let M be a von Neumann algebra of type In (n ∈ N) with the center Z. Then M is ∗isomorphic to the algebra Mn (Z) of n × n matrices over Z (see [17, Theorem 2.3.3]). In this case the algebras S(M, τ ) and S(M) can be described in the following way. Proposition 1.4. Let M be a von Neumann algebra of type In , n ∈ N, with a faithful normal semi-finite trace τ and let Z(S(M, τ )) denote the center of the algebra S(M, τ ). Then S(M, τ ) ∼ = Mn (Z(S(M, τ ))). Proof. Let {eij : i, j ∈ 1, n} be matrix units in Mn (Z). Consider the ∗-subalgebra in S(M, τ ) generated by the set Z S(M, τ ) ∪ {eij : i, j ∈ 1, n}. This ∗-subalgebra consists of elements of the form n
λij eij ,
λi,j ∈ Z S(M, τ ) ,
i, j = 1, n
i,j =1
and it is ∗-isomorphic with Mn (Z(S(M, τ ))) ⊆ S(M, τ ). Since M is tτ -dense in S(M, τ ), it is sufficient to show that the subalgebra Mn (Z(S(M, τ ))) is closed in S(M, τ ) with respect to the topology tτ . The center Z(S(M, τ )) is tτ -closed in S(M, τ ) and therefore the subalgebra e11 Z S(M, τ ) e11 = Z S(M, τ ) e11 , is also tτ -closed in S(M, τ ). Consider a sequence xm = ni,j =1 λ(m) ij eij in Mn (Z(S(M, τ ))) such that xm → x ∈ S(M, τ ) (m)
in the topology tτ . Fixing k, l ∈ 1, n we have that e1k xm el1 → e1k xel1 . Since e1k xm el1 = λkl e11 (m) one has λkl e11 → e1k xel1 . The tτ -closedness of Z(S(M, τ ))e11 in S(M, τ ) implies that (m)
λkl e11 → λkl e11
(2)
for an appropriate λkl ∈ Z(S(M, τ )). Multiplying (2) by ek1 from the left side and by e1l from (m) the right-hand side we obtain that λkl ekl → λkl ekl . Therefore xm → ni,j =1 λij eij and thus n x = i,j =1 λij eij . This implies that S(M, τ ) ∼ = Mn (Z(S(M, τ ))). The proof is complete. 2
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Proposition 1.5. Let M be a von Neumann algebra of type In , n ∈ N, and let Z(S(M)) denote the center of S(M). Then S(M) ∼ = Mn (Z(S(M))). Proof. Let τ be a faithful normal semi-finite trace on M and consider a family {zα } of mutually orthogonal central projections in M with α zα = 1 and such that τα = τ |zα M is finite for every α (such family exists because M is of type In , n < ∞). Then M=
zα M.
α
Since each trace τα is finite one has S(zα M) = S(zα M, τα ) = Mn Z S(zα M, τα ) = Mn Z S(zα M) , i.e. S(zα M) = Mn Z S(zα M) . This and Proposition 1.1 imply that S(M) ∼ =
α
= Mn
S(zα M) =
Mn Z S(zα M)
α
zα Z S(M) = Mn Z S(M) ,
α
i.e. S(M) ∼ = Mn Z S(M) . The proof is complete.
2
The last proposition enables us to obtain the following important property of the algebra LS(M) in the case of type I von Neumann algebra M. Proposition 1.6. Let M be a type I von Neumann algebra. Then for any element x ∈ LS(M) thereexists a countable family of mutually orthogonal central projections {zk }k∈N in M such that k zk = 1 and zk x ∈ M for all k. Proof. Case 1. The algebra M has type In , n ∈ N. In this case LS(M) = S(M) and Proposin tion 1.5 implies that S(M) ∼ M (Z(S(M))). Consider x = = n i,j =1 λij eij ∈ Mn (Z(S(M))). Put ∞ n c = i,j =1 |λij |. Then c ∈ Z(S(M)) and if c = 0 λ dλ is its spectral resolution, put z1 = e1 and zk = ek − ek−1 , k 2. Then {zk } is the family of mutually orthogonal central projections with k zk = 1 and by definition zk c ∈ Z for all k. Therefore zk |λij | ∈ Z for every k ∈ N, 1 i, j n. Thus zk x ∈ M for all k.
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Case 2. M is a finite von Neumann algebra of type I. Then there exists a family {qn }n∈F , F ⊆ N, of central projections from M with supn∈F qn = 1 such that the algebra M is ∗isomorphic with the C ∗ -product of von Neumann algebras qn M of type In respectively, n ∈ F , i.e. qn M. M∼ = n∈F
By Proposition 1.1 we have that S(M) ∼ =
S(qn M).
n∈F
Take x = {xn }n∈F ∈ n∈F S(qn M). The case 1 implies that for every n ∈ F there exists a family {zn,m } of mutually orthogonal central projections with m zn,m = qn and zn,m xn ∈ qn M for all m ∈ N. In this casewe have the countable family {zn,k }(n,k)∈F ×N of mutually orthogonal central projections with (n,k)∈F ×N zn,k = 1 and zn,k x ∈ M for all (n, k) ∈ F × N. Case 3. M is an arbitrary von Neumann algebra of type I and x ∈ S(M). Without loss of generality we ∞may assume that x 0. Let x = 0 λ dλ be the spectral resolution of x. Since x ∈ S(M) by the definition there exists λ0 > 0 such that eλ⊥0 is a finite projection. Thus eλ⊥0 Meλ⊥0 is a finite von Neumann algebra of type I and xeλ⊥0 ∈ S(eλ⊥0 Meλ⊥0 ). From the case 2 we have that there exists a family of mutually orthog ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ } onal projections {zm m∈N in eλ0 Meλ0 such that m1 zm = eλ0 and zm xeλ0 ∈ eλ0 Meλ0 for all m ∈ N. Each central projection z in eλ⊥0 Meλ⊥0 has the form z = eλ⊥0 zeλ⊥0 for an appropriate central projection z ∈ M. Moreover passing if necessary to z(eλ⊥0 )z one may chose z with z z(eλ⊥0 ), = e⊥ z e⊥ , m ∈ N. Mutuwhere z(eλ⊥0 ) is the central cover of the projection eλ⊥0 in M. Let zm λ0 m λ0 ally orthogonality of the family {z } then implies the similar property of the corresponding {zm }. m Denote z0 = z(eλ⊥0 )⊥ . Then m0 zm = 1 and z0 x = z0 xeλ0 + z0 xeλ⊥0 = z0 xeλ0 ∈ M, zm x = zm xeλ0 + zm xeλ⊥0 = zm xeλ0 + zm xeλ⊥0 ∈ M
for all m ∈ N, i.e. zm x ∈ M for all m 0. Case 4. The general case, i.e. M is a type I von Neumann algebra and x ∈ LS(M). By the definition there exists a sequence {fn }n∈N of mutually orthogonal central projection with n fn = 1 for each n ∈ N there exists a seand fn x ∈ S(M) for all n ∈ N. Then the case 3 implies that quence {zn,m } of mutually orthogonal central projections with m zn,m = fn and zn,m xn ∈ fn M for all m ∈ N. Now wehave that {zn,k }(n,k)∈N×N is a countable family of mutually orthogonal central projections with (n,k)∈N×N zn,k = 1 and zn,k x ∈ M for all (n, k) ∈ N × N. The proof is complete. 2 Now let us recall some notions and results from the theory of Hilbert–Kaplansky modules. Let (Ω, Σ, μ) be a measure space and let H be a Hilbert space. A map s : Ω → H is said to be simple, if s(ω) = nk=1 χAk (ω)ck , where Ak ∈ Σ , Ai ∩ Aj = ∅, i = j , ck ∈ H ,
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k = 1, n, n ∈ N. A map u : Ω → H is said to be measurable, if for any A ∈ with μ(A) < ∞ there is a sequence (sn ) of simple maps such that sn (ω) − u(ω) → 0 almost everywhere on A. Let L(Ω, H ) be the set of all measurable maps from Ω into H , and let L0 (Ω, H ) denote the space of all equivalence classes with respect to the equality almost everywhere. Denote by uˆ the equivalence class from L0 (Ω, H ) which contains the measurable map u ∈ L(Ω, H ). Further we shall identify the element u ∈ L(Ω, H ) and the class u. ˆ Note that the function ω → u(ω) is measurable for any u ∈ L(Ω, H ). The equivalence class containing the function u(ω) is + v(ω), λuˆ = λ(ω)u(ω). denoted by u. ˆ For u, ˆ vˆ ∈ L0 (Ω, H ), λ ∈ L0 (Ω) put uˆ + vˆ = u(ω) 0 Equipped with the L (Ω)-valued inner product x, y = x(ω), y(ω) H , where ·,·H in the inner product in H , L0 (Ω, H ) becomes a Hilbert–Kaplansky module over L0 (Ω). The space L∞ (Ω, H ) = x ∈ L0 (Ω, H ): x, x ∈ L∞ (Ω) is a Hilbert–Kaplansky module over L∞ (Ω). It should be noted that L∞ (Ω, H ) is a Banach space with respect to the norm x∞ = 1 (x, x) 2 L∞ (Ω) . Let us show that if dim H = α then the Hilbert–Kaplansky module L∞ (Ω, H ) is αhomogeneous. Indeed, let {ϕi }i∈J be an orthonormal basis in H with the cardinality α, and consider the equivalence class ϕˆ i from L∞ (Ω, H ) containing the constant vector-function ω ∈ Ω → ϕi . From the definition of the inner-product it is clear that ϕˆi , ϕˆ j = δij 1, where δij is the Kroenecker symbol, 1 is the identity from L∞ (Ω). Let us show that if y ∈ L∞ (Ω, H ) and ϕˆi , y = 0 for all i ∈ J then y = 0. Put Sp{ϕˆ i } =
n
∞
λk ϕˆ ik : λk ∈ L (Ω), ik ∈ J, k = 1, n, n ∈ N .
k=1
Since the set of elements of the form nk=1 tk ϕik , where tk ∈ C, ik ∈ J , k = 1, n, n ∈ N, is norm dense in H , we have that inf{ψ − y: ψ ∈ Sp{ϕˆ i }} = 0, for each fixed y ∈ L∞ (Ω, H ). The set Sp{ϕˆi } is an L∞ (Ω)-submodule in L∞ (Ω, H ), and by [6, Proposition 4.1.5] there exists a sequence {ψk } in Sp{ϕˆ i } such that ψk − y ↓ 0, i.e. the set Sp{ϕˆ i } is (bo)-dence in L∞ (Ω, H ). Now let y ∈ L∞ (Ω, H ) be such an element that ϕˆi , y = 0 for all i ∈ J . Then ξ, y = 0 for all ξ ∈ Sp{ϕˆi }. Since the set Sp{ϕˆi } is (bo)-dense in L∞ (Ω, H ), we have that ξ, y = 0 for all ξ ∈ L∞ (Ω, H ). In particular y, y = 0, i.e. y = 0. Therefore {ϕˆ i }i∈J is an orthogonal basis in L∞ (Ω, H ) with the cardinality α, i.e. L∞ (Ω, H ) is α-homogeneous, where α = dim H .
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Denote by B(L0 (Ω, H )) the algebra of all L0 (Ω)-bounded L0 (Ω)-linear operators on L0 (Ω, H ) and by B(L∞ (Ω, H ))—the algebra of all L∞ (Ω)-bounded L∞ (Ω)-linear operators on L∞ (Ω, H ). In [4] it was proved that B(L0 (Ω, H )) is a C ∗ -algebra over L0 (Ω). Put B L0 (Ω, H )b = x ∈ B L0 (Ω, H ) : x ∈ L∞ (Ω) . Note that the correspondence x → x|L∞ (Ω,H ) gives a ∗-isomorphism between the ∗-algebras B(L0 (Ω, H )b and B(L∞ (Ω, H )). We further shall identify B(L0 (Ω, H )b ) with B(L∞ (Ω, H )), i.e. the operator x from B(L0 (Ω, H )b ) is identified with its restriction x|L∞ (Ω,H ) . Since L∞ (Ω, H ) is a Hilbert–Kaplansky module over L∞ (Ω), [9, Theorem 7] implies that B(L∞ (Ω, H )) is an AW ∗ -algebra of type I and its center is ∗-isomorphic with L∞ (Ω). If dim H = α, then L∞ (Ω, H ) is α-homogeneous and by [9, Theorem 7] the algebra B(L∞ (Ω, H )) has the type Iα . The center Z(B(L∞ (Ω, H ))) of this AW ∗ -algebra coincides with the von Neumann algebra L∞ (Ω) and thus by [10, Theorem 2] B(L∞ (Ω, H )) is also a von Neumann algebra. Thus for dim H = α we have that B(L∞ (Ω, H )) is a von Neumann algebra of type Iα . Now let M be a homogeneous von Neumann algebra of type Iα with the center L∞ (Ω). Since two von Neumann algebras of the same type Iα with isomorphic center are mutually ∗-isomorphic, it follows that the algebra M is ∗-isomorphic to the algebra B(L∞ (Ω, H )), where dim H = α. It is well known [19] that given any type I von Neumann algebra M, there exists a (cardinalindexed) system of central orthogonal projections (qα )α∈J ⊂ P(M) with α∈J qα = 1 such that qα M is a homogeneous von Neumann algebra of type Iα , i.e. qα M ∼ = B(L∞ (Ωα , Hα )) with ∗ dim Hα = α, and the algebra M is ∗-isomorphic to the C -product of the algebras Mα , i.e. M∼ =
Mα .
α∈J
Note that if L∞ (Ω) is the center of M then qα L∞ (Ω) ∼ = L∞ (Ωα ) for an appropriate Ωα , α ∈ J . Therefore L∞ (Ω) ∼ =
L∞ (Ωα ).
α∈J
The product α∈J
L∞ (Ωα , Hα )
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equipped with coordinate-wise algebraic operations and inner product becomes a Hilbert– Kaplansky module over L∞ (Ω). The product B L∞ (Ωα , Hα ) α∈J
equipped with coordinate-wise algebraic operations and involution becomes a ∗-algebra and moreover ∞ ∞ ∼ B L (Ωα , Hα ) = B L (Ωα , Hα ) . α∈J
α∈J
Indeed, take x ∈ B( α∈J L∞ (Ωα , Hα )). For each α define the operator xα on L∞ (Ωα , Hα ) by xα (ϕα ) = qα x(ϕα ), ϕα ∈ L∞ (Ωα , Hα ). Then {xα } ∈
B L∞ (Ωα , Hα ) α∈J
and the correspondence x → {xα } gives a ∗-homomorphism from B( α∈J L∞ (Ωα , Hα )) into α∈J B(L∞ (Ωα , Hα )). Conversely, consider B L∞ (Ωα , Hα ) . {xα } ∈ α∈J
Define the operator x on
α∈J
L∞ (Ωα , Hα ) by
x {ϕα } = xα (ϕα ) ,
{ϕα } ∈
L∞ (Ωα , Hα ).
α∈J
Then x ∈ B( α∈J L∞ (Ωα , Hα )) and therefore M∼ =
∞ ∞ ∼ B L (Ωα , Hα ) = B L (Ωα , Hα ) . α∈J
α∈J
The direct product
L0 (Ωα , Hα )
α∈J
equipped with the coordinate-wise algebraic operations and inner product forms a Hilbert– Kaplansky module over L0 (Ω) ∼ = α∈J L0 (Ωα ).
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The proof of the following proposition in [2] has a small gap, therefore here we shall give an alternative proof for this result. Proposition 1.7. If von Neumann algebra M is ∗-isomorphic with B( α∈J L∞ (Ωα , Hα )) then the algebra LS(M) is ∗-isomorphic with B( α∈J L0 (Ωα , Hα )). Proof. Let Φ be a ∗-isomorphism between M and B( α∈J L∞ (Ωα , Hα )). Take norm. Consider a family of mux ∈ B( α∈J L0 (Ωα , Hα )) and let x be its L0 (Ω)-valued ∞ (Ω) with zn = 1 such that zn x ∈ L∞ (Ω) for tually orthogonal projections {zn }n∈N in L all n ∈ N. Then zn x ∈ M for all n ∈ N and n zn Φ(zn x) ∈ LS(M). Put Ψ :x →
zn Φ(zn x).
n
It is clear that Ψ is a well-defined ∗-homomorphism from B( α∈J L0 (Ωα , Hα )) into LS(M). Since given any element x ∈ LS(M) there exists a sequence of mutually orthogonalcentral pro1.6) and x = n zn x, this jections {zn } in M such that zn x ∈ M for all n ∈ N (Proposition implies that Ψ is a ∗-isomorphism between LS(M) and B( α∈J L0 (Ωα , Hα )). The proof is complete. 2 It is known [4] that B( α∈J L0 (Ωα , Hα )) is a C ∗ -algebra over L0 (Ω) and therefore there exists a map · : LS(M) → L0 (Ω) such that for all x, y ∈ LS(M), λ ∈ L0 (Ω) one has x 0,
x = 0
⇔
x = 0;
λx = |λ|x; x + y x + y; xy xy; xx ∗ = x2 . This map · : LS(M) → L0 (Ω) is called the center-valued norm on LS(M). 2. Derivations on the algebra LS(M) In this section we shall give a complete description of derivations on the algebra LS(M) of all locally measurable operators affiliated with a type I von Neumann algebra M. It is clear that if a derivation D on LS(M) is inner then it is Z-linear, i.e. D(λx) = λD(x) for all λ ∈ Z, x ∈ LS(M), where Z is the center of the von Neumann algebra M. The following main result of [2] asserts that the converse is also true. Theorem 2.1. Let M be a type I von Neumann algebra with the center Z. Then every Z-linear derivation D on the algebra LS(M) is inner. Proof. (See [2, Theorem 3.2].)
2
We are now in position to consider arbitrary (non-Z-linear, in general) derivations on LS(M). The following simple but important remark is crucial in our further considerations.
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Remark 1. Let A be an algebra with the center Z and let D : A → A be a derivation. Given any x ∈ A and a central element λ ∈ Z we have D(λx) = D(λ)x + λD(x) and D(xλ) = D(x)λ + xD(λ). Since λx = xλ and λD(x) = D(x)λ, it follows that D(λ)x = xD(λ) for any λ ∈ A. This means that D(λ) ∈ Z, i.e. D(Z) ⊆ Z. Therefore given any derivation D on the algebra A we can consider its restriction δ : Z → Z. Now let M be a homogeneous von Neumann algebra of type In , n ∈ N, with the center Z. Then the algebra M is ∗-isomorphic with the algebra Mn (Z) of all n × n-matrices over Z, and the algebra LS(M) = S(M) is ∗-isomorphic with the algebra Mn (S(Z)) of all n × n matrices over S(Z), where S(Z) is the algebra of measurable operators for the commutative von Neumann algebra Z. The algebra LS(Z) = S(Z) is isomorphic to the algebra L0 (Ω) = L(Ω, Σ, μ) of all measurable functions on a measure space (see Section 2) and therefore it admits (in nonatomic cases) non-zero derivations (see [3,13]). Let δ : S(Z) → S(Z) be a derivation and Dδ be a derivation on the algebra Mn (S(Z)) defined by (1) in Introduction. The following lemma describes the structure of an arbitrary derivation on the algebra of locally measurable operators for homogeneous type In , n ∈ N, von Neumann algebras. Lemma 2.2. Let M be a homogenous von Neumann algebra of type In , n ∈ N. Every derivation D on the algebra LS(M) can be uniquely represented as a sum D = Da + D δ , where Da is an inner derivation implemented by an element a ∈ LS(M) while Dδ is the derivation of the form (1) generated by a derivation δ on the center of LS(M) identified with S(Z). Proof. Let D be an arbitrary derivation on the algebra LS(M) ∼ = Mn (S(Z)). Consider its restriction δ onto the center S(Z) of this algebra, and let Dδ be the derivation on the algebra Mn (S(Z)) constructed as in (1). Put D1 = D − Dδ . Given any λ ∈ S(Z) we have D1 (λ) = D(λ) − Dδ (λ) = D(λ) − D(λ) = 0, i.e. D1 is identically zero on S(Z). Therefore D1 is Z-linear and by Theorem 2.1 we obtain that D1 is inner derivation and thus D1 = Da for an appropriate a ∈ Mn (S(Z)). Therefore D = Da + Dδ . Suppose that D = Da1 + Dδ1 = Da2 + Dδ2 .
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Then Da1 − Da2 = Dδ2 − Dδ1 . Since Da1 − Da2 is identically zero on the center of the algebra Mn (S(Z)) this implies that Dδ2 − Dδ1 is also identically zero on the center of Mn (S(Z)). This means that δ1 = δ2 , and therefore Da1 = Da2 , i.e. the decomposition of D is unique. The proof is complete. 2 Now let M be an arbitrary finite von Neumann algebra of type I with the center Z. There exists a family {zn }n∈F , F ⊆ N, of central projections from M with supn∈F zn = 1 such that the algebra M is ∗-isomorphic with the C ∗ -product of von Neumann algebras zn M of type In respectively, n ∈ F , i.e. M∼ =
zn M.
n∈F
By Proposition 1.1 we have that LS(M) ∼ =
LS(zn M).
n∈F
Suppose that D is a derivation on LS(M), and δ is its restriction onto its center S(Z). Since δ maps each zn S(Z) ∼ = Z(LS(zn M)) into itself, δ generates a derivation δn on zn S(Z) for each n ∈ F . Let Dδn be the derivation on the matrix algebra Mn (zn Z(LS(M))) ∼ = LS(zn M) defined as in (1). Put Dδ {xn }n∈F = Dδn (xn ) ,
{xn }n∈F ∈ LS(M).
(3)
Then the map Dδ is a derivation on LS(M). Now Lemma 2.2 implies the following result: Lemma 2.3. Let M be a finite von Neumann algebra of type I. Each derivation D on the algebra LS(M) can be uniquely represented in the form D = Da + D δ , where Da is an inner derivation implemented by an element a ∈ LS(M), and Dδ is a derivation given as (3). In order to consider the case of type I∞ von Neumann algebra we need some auxiliary results concerning derivations on the algebra L0 (Ω) = L(Ω, Σ, μ). Recall that a net {λα } in L0 (Ω) (o)-converges to λ ∈ L0 (Ω) if there exists a net {ξα } monotone decreasing to zero such that |λα − λ| ξα for all α. Denote by ∇ the complete Boolean algebra of all idempotents from L0 (Ω), i.e. ∇ = {χ˜ A : A ∈ Σ}, where χ˜ A is the element from L0 (Ω) which contains the characteristic function of the set A. A partition of the unit in ∇ is a family (πα ) of orthogonal idempotents from ∇ such that α πα = 1.
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Lemma 2.4. Any derivation δ on the algebra L0 (Ω) commutes with the mixing operation on L0 (Ω), i.e. πα λα = πα δ(λα ) δ α
α
for an arbitrary family (λα ) ⊂ L0 (Ω) and any partition {πα } of the unit in ∇. Proof. Consider a family {λα } in L0 (Ω) and a partition of the unit {πα } in ∇ ⊂ L0 (Ω). Since δ(π) = 0 for any idempotent π ∈ ∇, we have δ(πα ) = 0 for all α and thus δ(πα λ) = πα δ(λ) for any λ ∈ L0 (Ω). Therefore for each πα0 from the given partition of the unit we have πα λα = δ πα0 πα λα = δ(πα0 λα0 ) = πα0 δ(λα0 ). πα0 δ α
α
By taking the sum over all α0 we obtain δ πα λα = πα δ(λα ). α
The proof is complete.
α
2
Recall [11] that a subset K ⊂ L0 (Ω) is called cyclic, if (uα )α∈J ⊂ K and for any partition of the unit (πα )α∈J in ∇. We need the following technical result.
α∈J
πα uα ∈ K for each family
Lemma 2.5. Let A be a cyclic subset in L0 (Ω). If the set πA is unbounded above for each non-zero π ∈ ∇, then given any n ∈ N there exists λn ∈ A such that λn n1. Proof. For fixed n ∈ N and an arbitrary λ ∈ A denote πλ = {q ∈ ∇: qλ qn}. Then πλ λ πλ n
(4)
πλ⊥ λ πλ⊥ n.
(5)
and
Put π0 =
{πλ : λ ∈ A}.
Since π0⊥ =
πλ⊥ : λ ∈ A
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from (5) we obtain π0⊥ λ π0⊥ n for all λ ∈ A, i.e. π0⊥ A is bounded above. By the assumption of lemma π0⊥ = 0, i.e. π0 =
{πλ : λ ∈ A} = 1.
By [20, p. 111, Theorem 4] there existsa partition of unit {πi } in ∇ such that for any πi there exists λi ∈ A with πi πλi . Put λn = i πi λi . Since A is a cyclic we have λn ∈ A. From (4) one has πλi λi πλi n for all i. Thus πi λi πi n for all i, therefore λn n1. The proof is complete. 2 Given an arbitrary derivation δ on L0 the element zδ = inf{π ∈ ∇: πδ = δ} is called the support of the derivation δ. Lemma 2.6. Given any nontrivial derivation δ : L0 (Ω) → L0 (Ω) there exist a sequence {λn }∞ n=1 in L∞ (Ω) with |λn | 1, n ∈ N such that δ(λn ) nzδ for all n ∈ N. Proof. Considering if necessary the algebra zδ L0 (Ω) instead L0 (Ω) and the derivation zδ δ instead δ, we may assume that zδ = 1. Put A = {δ(λ): λ ∈ L0 (Ω), |λ| 1} and let us prove that for any non-zero π ∈ ∇ the set πA is unbounded from above. Suppose that the set π{δ(λ): λ ∈ L0 (Ω), |λ| 1} is order bounded in L0 (Ω) for some π ∈ ∇, π = 0. Then πδ maps any uniformly convergent sequence in L∞ (Ω) to an (o)-convergent sequence in L0 (Ω). The algebra L∞ (Ω) coincides with the uniform closure of the linear span of idempotents from ∇. Since πδ is identically zero on ∇ it follows that πδ ≡ 0 commutes with the mixing operation and every element λ ∈ L0 (Ω) can be on L∞ (Ω). Since δ , where {λα } ⊂ L∞ (Ω), and {πα } is a partition of unit in ∇, we represented as λ = α πα λα have δ(λ) = δ( α πα λα ) = α πα δ(λα ) = 0, i.e. πδ ≡ 0 on L0 (Ω). This is contradicts with zδ = 1. This contradiction shows that the set π{δ(λ): λ ∈ L0 (Ω), |λ| 1} is not order bounded in L0 (Ω) for all π ∈ ∇, π = 0. Further, since δ commutes with the mixing operations and the set {λ: λ ∈ L0 , |λ| 1} is cyclic, the set {δ(λ): λ ∈ L0 (Ω), |λ| 1} is also cyclic. By Lemma 2.5 ∞ there exist a sequence {λn }∞ n=1 in L (Ω) with |λn | 1 such that |δ(λn )| n1, n ∈ N. The proof is complete. 2 Now we are in position to consider derivations on the algebra of locally measurable operators for type I∞ von Neumann algebras. Theorem 2.7. If M is a type I∞ von Neumann algebra, then any derivation on the algebra LS(M) is inner.
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Proof. Since M is of type I∞ there exists a sequence of mutually orthogonal and mutually equivalent abelian projections {pn }∞ n=1 in M with the central cover 1 (i.e. faithful projections). For any bounded sequence Λ = {λk } in Z define an operator xΛ by xΛ =
∞
λ k pk .
k=1
Then x Λ pn = pn x Λ = λ n pn .
(6)
Let D be a derivation on LS(M), and let δ be its restriction onto the center of LS(M), identified with L0 (Ω). Take any λ ∈ L0 (Ω) and n ∈ N. From the identity D(λpn ) = D(λ)pn + λD(pn ) multiplying it by pn from both sides we obtain pn D(λpn )pn = pn D(λ)pn + λpn D(pn )pn . Since pn is a projection, one has that pn D(pn )pn = 0, and since D(λ) = δ(λ) ∈ L0 (Ω), we have pn D(λpn )pn = δ(λ)pn .
(7)
Now from the identity D(xΛ pn ) = D(xΛ )pn + xΛ D(pn ), in view of (6) one has similarly pn D(λn pn )pn = pn D(xΛ )pn + λn pn D(pn )pn , i.e. pn D(λn pn )pn = pn D(xΛ )pn . Eqs. (7) and (8) imply pn D(xΛ )pn = δ(λn )pn . Further for the center-valued norm · on LS(M) (see Section 1) we have: pn D(xΛ )pn pn D(xΛ )pn = D(xΛ ) and δ(λn )pn = δ(λn ).
(8)
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Therefore D(xΛ ) δ(λn ) for any bounded sequence Λ = {λn } in Z. If we suppose that δ = 0 then π = zδ = 0. By Lemma 2.6 there exists a bounded sequence Λ = {λn } in Z such that δ(λn ) nπ for any n ∈ N. Thus D(xΛ ) nπ for all n ∈ N, i.e. π = 0—that is a contradiction. Therefore δ ≡ 0, i.e. D is identically zero on the center of LS(M), and therefore it is Z-linear. By Theorem 2.1 D is inner. The proof is complete. 2 We shall now consider derivations on the algebra LS(M) of locally measurable operators with respect to an arbitrary type I von Neumann algebra M. Let M be a type I von Neumann algebra. There exists a central projection z0 ∈ M such that (a) z0 M is a finite von Neumann algebra; (b) z0⊥ M is a von Neumann algebra of type I∞ . Consider a derivation D on LS(M) and let δ be its restriction onto its center Z(S). By Theorem 2.7 z0⊥ D is inner and thus we have z0⊥ δ ≡ 0, i.e. δ = z0 δ. Let Dδ be the derivation on z0 LS(M) defined as in (3) and consider its extension Dδ on LS(M) = z0 LS(M) ⊕ z0⊥ LS(M) which is defined as Dδ (x1 + x2 ) := Dδ (x1 ),
x1 ∈ z0 LS(M), x2 ∈ z0⊥ LS(M).
(9)
The following theorem is the main result of this section, and gives the general form of derivations on the algebra LS(M). Theorem 2.8. Let M be a type I von Neumann algebra. Each derivation D on LS(M) can be uniquely represented in the form D = D a + Dδ where Da is an inner derivation implemented by an element a ∈ LS(M), and Dδ is a derivation of the form (9), generated by a derivation δ on the center of LS(M). Proof. It immediately follows from Lemma 2.3 and Theorem 2.7.
2
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3. Derivations on the algebra S(M) In this section we describe derivations on the algebra S(M) of measurable operators affiliated with a type I von Neumann algebra M. Let M be a type I von Neumann algebra and let A be an arbitrary subalgebra of LS(M) containing M. Consider a derivation D : A → LS(M) and let us show that D can be extended to a derivation D˜ on the whole LS(M). Since M is a type I, for an arbitrary element x ∈ LS(M) there exists a sequence {zn } of mutually orthogonal central projections with n∈N zn = 1 and zn x ∈ M for all n ∈ N. Set ˜ D(x) =
zn D(zn x).
(10)
n1
Since every derivation D : A → LS(M) is identically zero on central projections of M, the equality (10) gives a well-defined derivation D˜ : LS(M) → LS(M) which coincides with D on A. In particular, if D is Z-linear on A, then D˜ is also Z-linear and by Theorem 2.1 the derivation D˜ is inner on LS(M) and therefore D is a spatial derivation on A, i.e. there exists an element a ∈ LS(M) such that D(x) = ax − xa for all x ∈ A. Therefore we obtain the following Theorem 3.1. Let M be a type I von Neumann algebra with the center Z, and let A be an arbitrary subalgebra in LS(M) containing M. Then any Z-linear derivation D : A → LS(M) is spatial and implemented by an element of LS(M). Corollary 3.2. Let M be a type I von Neumann algebra with the center Z and let D be a Z-linear derivation on S(M) or S(M, τ ). Then D is spatial and implemented by an element of LS(M). We are now in position to improve the last result by showing that in fact such derivations on S(M) and S(M, τ ) are inner. Let us start by the consideration of the type I∞ case. Let M be a type I∞ von Neumann algebra with the center Z identified with the algebra L∞ (Ω) and let ∇ be the Boolean algebra of projection from L∞ (Ω). Denote by St (∇) the set of all elements λ ∈ L0 (Ω) of the form λ = α πα tα , where {πα } is a partition of the unit in ∇, and {tα } ⊂ R (so-called step-functions). Suppose that a ∈ LS(M), a = a ∗ and consider the spectral family {eλ }λ∈R of the operator a. For λ ∈ St (∇), λ = α πα tα put eλ = α πα etα . Denote by P∞ (M) the family of all faithful projections p from M such that pMp is of type I∞ . Set Λ− = λ ∈ St (∇): eλ ∈ P∞ (M) and
Λ+ = λ ∈ St (∇): eλ⊥ ∈ P∞ (M) .
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Lemma 3.3. (a) Λ− = ∅ and Λ+ = ∅; (b) the set Λ+ (resp. Λ− ) is bounded from above (resp. from below); (c) if λ+ = sup Λ+ (resp. λ− = inf Λ− ) then λ ∈ Λ+ (resp. λ ∈ Λ− ) for all λ ∈ St (∇) with λ + ε1 λ+ (resp. λ − ε1 λ− ) for some ε > 0; (d) if λ+ ∈ L∞ (Ω) and λ− ∈ L∞ (Ω), then a ∈ S(M). Proof. (a) Take a sequence of projections {zn } from ∇ such that zn a ∈ M for all n ∈ N. Then for tn < − zn a M we have zn etn = 0 or zn et⊥n = zn for all n ∈ N. Therefore for λ = zn tn one has eλ⊥ = zn et⊥n = zn = 1, i.e. λ ∈ Λ+ and hence Λ+ = ∅. Similarly Λ− = ∅. (b) Suppose that the element λ = πα λα ∈ St (∇), satisfies the condition π0 λ π0 a + επ0 for an appropriate non-zero π0 ∈ ∇, where · is the center-valued norm on LS(M). Without loss of generality we may assume that π0 = πα for some α, i.e. πα tα πα a + επα . Then tα πα aM + ε and therefore πα etα = πα 1, i.e. πα et⊥α = 0. Since πα et⊥α = 0, we have z(et⊥α ) = 1 and so λ ∈ / Λ+ . Therefore Λ+ is bounded from above by the element a. Similarly the set Λ− is bounded from below by the element −a. (c) Put λ+ = sup Λ+ and λ− = inf Λ− . Take an element λ ∈ St (∇) such that λ + ε1 λ+ , where ε > 0. Suppose that eλ⊥ ∈ / P∞ (M). Then π0 eλ⊥ Meλ⊥ is a finite von Neumann algebra for some non-zero π0 ∈ ∇. Without loss of generality we may assume that π0 = πα for some α, i.e. πα et⊥α is a finite projection. Then πα et⊥ is finite for all t > tα . This means that πα λ+ πα tα . On the other hand multiplying by πα the unequality λ + ε1 λ+ we obtain that πα tα + πα ε πα λ+ . Therefore πα ε 0. This contradiction implies that λ ∈ Λ+ for all λ ∈ St (∇) with λ + ε1 λ+ . (d) Let λ+ , λ− ∈ L∞ (Ω). Take a number n ∈ N such that λ+ n1 and λ− −n1. Take a ⊥ is a finite projection and π ⊥ e⊥ is an infinite projection. largest element π ∈ ∇ such that πen+1 n+1 ⊥ For λ ∈ Λ+ put λ = πλ + π (n + 1). Then λ ∈ Λ+ and therefore λ λ+ . Hence π ⊥ λ π ⊥ λ+ , i.e π ⊥ (n + 1) π ⊥ λ+ . That contradicts the inequality λ+ n1. Therefore π = 1, i.e. ⊥ is a finite projection. Similarly e⊥ en+1 −(n+1) is a finite projection. Therefore a ∈ S(M). The proof is complete. 2 Lemma 3.4. If M is a type I∞ von Neumann algebra then every derivation D : M → S(M) has the form D(x) = ax − xa, for an appropriate a ∈ S(M).
x ∈ M,
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Proof. By the Remark 1 D maps the center Z of M into the center of S(M) which coincides with Z by Proposition 1.3, i.e. we obtain a derivation D on commutative von Neumann algebra Z. Therefore D|Z = 0. Thus D(λx) = D(λ)x + λD(x) = λD(x) for all λ ∈ Z, i.e. D is Z-linear. By Theorem 3.1 there exists an element a ∈ LS(M) such that D(x) = ax − xa for all x ∈ M. Let us prove that one can choose the element a from S(M). For x ∈ M we have (a + a ∗ )x − x(a + a ∗ ) = (ax − xa) − (ax ∗ − x ∗ a)∗ = D(x) − D(x ∗ )∗ ∈ S(M) and (a − a ∗ )x − x(a − a ∗ ) = D(x) + D(x ∗ )∗ ∈ S(M). This means that the elements a + a ∗ and a − a ∗ implement derivations from M into S(M). Since ∗ a−a ∗ a = a+a 2 + i 2i , it is sufficient to consider the case where a is a self-adjoint element. Consider the elements λ+ , λ− ∈ L0 defined in Lemma 3.3(c) and let us prove that λ+ , λ− ∈ L∞ (Ω). Lemma 3.3(c) implies that there exists an element λ1 ∈ Λ− such that − 14 λ− − λ1 − 18 . Since D(x) = (a − λ1 )x − x(a − λ1 ), replacing a by a − λ1 , we may assume that − 14 λ− − 18 . Then eε ∈ P∞ (M) for all ε > − 18 and eε is a finite for all ε < − 14 . In particular (e− 1 − e− 1 )M(e− 1 − e− 1 ) is of type I∞ , and moreover λ+ − 12 . 16
2
16
2
Suppose that λ+ ∈ / L∞ (Ω). Since λ+ − 12 , we have that λ+ is unbounded from above and thus passing if necessary to the subalgebra zM, where z is a non-zero central projection in M with zλ+ z, we may assume without loss of generality that λ+ 1. First let us consider the particular case where M is of type Iℵ0 , where ℵ0 is the countable cardinal number. Take an element λ0 ∈ St (∇) such that λ+ − 12 λ0 λ+ − 14 . By Lemma 3.3(c) we have eλ⊥0 ∈ P∞ (M). Since algebras eλ⊥0 Meλ⊥0 and (e− 1 − e− 1 )M(e− 1 − e− 1 ) are algebras 16 2 16 2 of type Iℵ0 , then there exists a sequences of pairwise equivalent and pairwise orthogonal abelian projections {fk }k∈N and {gk }k∈N such that fk = eλ⊥0 , gk = e− 1 − e− 1 . Since z(eλ⊥0 ) = 16 2 z(e− 1 − e− 1 ) = 1, then z(fk ) = z(gk ) = 1 for all k, and therefore fk ∼ gk for all k. Thus 16
2
the projections p1 = eλ⊥0 and p2 = e− 1 − e− 1 are equivalent. From λ0 eλ⊥0 aeλ⊥0 it follows 16 2 that λ0 p1 p1 ap1 . Since p1 Mp1 is of type Iℵ0 , the center of the algebra S(p1 Mp1 ) coincides / L∞ (Ω) and with the center of the algebra p1 Mp1 (Proposition 1.3). Due to the fact that λ0 ∈ z(p1 ) = 1, we see that λ0 p1 is an unbounded linear operator from LS(p1 Mp1 ) \ S(p1 Mp1 ). / S(p1 Mp1 ). Therefore ap1 = p1 ap1 ∈ Let u be a partial isometry in M such that uu∗ = p1 , u∗ u = p2 . Put p = p1 + p2 . Consider the derivation D1 from pMp into pS(M)p = S(pMp) defined as D1 (x) = pD(x)p,
x ∈ pMp.
This derivation is implemented by the element ap = pap, i.e. D1 (x) = apx − xap,
x ∈ pMp.
1 Since p2 = e− 1 − e− 1 then − 12 e− 1 (e− 1 − e− 1 )a(e− 1 − e− 1 ) − 16 e− 1 . Therefore 16 2 2 16 2 16 2 16 ap2 ∈ pMp, the element b = ap1 = ap − ap2 implements a derivation D2 from pMp into S(pMp).
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Since D2 (u + u∗ ) = b(u + u∗ ) − (u + u∗ )b, it follows that b(u + u∗ ) − (u + u∗ )b ∈ S(M). From up1 = p1 u∗ = 0 it follows that bu − u∗ b ∈ S(M). Multiplying this by u from the left side we obtain ubu − uu∗ b ∈ S(M). From ub = 0, uu∗ = p1 , it follows that p1 b ∈ S(M), i.e. / S(M). The contradiction shows that λ+ ∈ ap1 ∈ S(M). This contradicts the above relation ap1 ∈ L∞ (Ω). Now Lemma 3.3(d) implies that a ∈ S(M). Let us consider the case of general type I∞ von Neumann algebra M. Take an element λ0 ∈ St (∇) such that λ+ − 12 λ0 λ+ − 14 . Lemma 3.3(c) implies that eλ⊥0 ∈ P∞ (M). Consider projections p1 and p2 with the central cover 1 such that p1 eλ⊥0 , p2 e 1 and such that pi Mpi 4 are of type Iℵ0 , i = 1, 2. Put p = p1 + p2 . Consider the derivation Dp from pMp into pS(M)p defined as Dp (x) = pD(x)p,
x ∈ pMp.
Since pMp is of type Iℵ0 the above case implies that pap ∈ S(M) and therefore p1 ap1 ∈ S(M). On the other hand λ0 p1 p1 ap1 and λ0 p1 ∈ / S(M). From this contradiction it follows that λ+ ∈ L∞ (Ω). By Lemma 3.3(d) we obtain that a ∈ S(M). The proof is complete. 2 From the above results we obtain Lemma 3.5. Let M be a type I von Neumann algebra with the center Z. Then every Z-linear derivation D on the algebra S(M) is inner. In particular, if M is a type I∞ then every derivation on S(M) is inner. Now let M be an arbitrary type I von Neumann algebra and let z0 be the central projection in M such that z0 M is a finite von Neumann algebra and z0⊥ M is a von Neumann algebra of type I∞ . Consider a derivation D on S(M) and let δ be its restriction onto its center Z(S). By Lemma 3.5 the derivation z0⊥ D is inner and thus we have z0⊥ δ ≡ 0, i.e. δ = z0 δ. Since z0 M is a finite type I von Neumann algebra, we have that z0 LS(M) = z0 S(M). Let Dδ be the derivation on z0 S(M) = z0 LS(M) defined as in (3). Finally Lemmas 2.3 and 3.5 imply the following main result the present section. Theorem 3.6. Let M be a type I von Neumann algebra. Then every dereivation D on the algebra S(M) can be uniquely represented in the form D = Da + Dδ , where Da is inner and implemented by an element a ∈ S(M) and Dδ is the derivation of the form (3) generated by a derivation δ on the center of S(M). 4. Derivations on the algebra S(M, τ ) In this section we present a general form of derivations on the algebra S(M, τ ) of τ measurable operators affiliated with a type I von Neumann algebra M and a faithful normal semi-finite trace τ . Theorem 4.1. Let M be a type I von Neumann algebra with the center Z and a faithful normal semi-finite trace τ . Then every Z-linear derivation D on the algebra S(M, τ ) is inner. In particular, if M is a type I∞ then every derivation on S(M, τ ) is inner.
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Proof. By Theorem 3.1 D(x) = ax − xa for some a ∈ LS(M) and all x ∈ S(M, τ ). Let us show that the element a can be chosen from the algebra S(M, τ ). As in Lemma 3.3 we may assume that a = a ∗ . Case 1. M is a homogeneous type In , n ∈ N von Neumann algebra. Then LS(M) = S(M) ∼ = 0 (Ω)) can be a S(M) and M (L Mn (L0 (Ω)). By [12, Theorem 3.5] a ∗-isomorphism between n chosen such that the element a can be represented as a = ni=1 λi ei,i , where λi = λi ∈ L0 (Ω), · · · λn . i = 1, n, λ1 Put u = nj=1 ej,n−j +1 . Then Da (u) = au − ua =
n (λi − λn−i+1 )ei,n−i+1 i=1
and n Da (u) = (λi − λn−i+1 )en−i+1,i . ∗
i=1
n
Therefore Da (u)∗ Da (u) = i=1 (λi − λn−i+1 )2 ei,i , and thus |Da (u)| = Since λ1 · · · λn , we have
n
i=1 |λi
− λn−i+1 |ei,i ,
|λi − λn−i+1 | |λi − λ[ n+1 ] |
(11)
2
for all i ∈ 1, n. Denote b = ni=1 |λi − λ[ n+1 ] |ei,i . From (11) we obtain that |Da (u)| b, and thus b ∈ 2 S(M, τ ). [ n+1 ] Put v = i=12 ei,i − nj=[ n+1 ] ej,j . Then vb = a − λ[ n+1 ] 1 and vb ∈ S(M, τ ). Therefore 2
2
a − λ[ n+1 ] 1 ∈ S(M, τ ) and this element also implements the derivation Da . 2 Case 2. Let M be a finite type I von Neumann algebra. Then LS(M) = S(M) ∼ =
Mn L0 (Ωn ) ,
n∈F
where F ⊆ N. Therefore a = {an }, where an =
n
(n) (n) i=1 λi ei,i ,
(n) ei,j
(n)
and are the matrix units in Mn (L0 (Ωn )), i, j = 1, n, n ∈ F . For each n ∈ F consider the following elements in Mn (L0 (Ωn ) bn =
n (n) (n) λ − λ(n) n+1 ei,i i [
i=1
2
]
and [ n+1 2 ]
vn =
i=1
(n) ei,i −
n j =[ n+1 2 ]
(n) ej,j .
(n)
(n)
λ 1 · · · λn , λ i
∈ L0 (Ωn )
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Set b = {bn }n∈F and v = {vn }n∈F . Consider the element λ = {λ[ n+1 ] }n∈F ∈ L0 (Ω) ∼ = 2
L0 (Ωn ).
n∈F
Similar to the case 1 we obtain that a − λ1 = vb ∈ S(M, τ ). Case 3. M is a type I∞ von Neumann algebra. Since S(M, τ ) ⊆ S(M) by Lemma 3.4 there exists an element a ∈ S(M) such that D(x) = ax − xa for all x ∈ M. Let us show that a can be picked from the algebra S(M, τ ). Since a ∈ S(M), there exists λ ∈ R, λ > 0 such that f = e−λ ∨ eλ⊥ is a finite projection. Suppose that z0 ∈ Z is a central projection such that z0 gMg is a finite von Neumann algebra, ⊥ ∧ e = e − e . Then z 1 = z f + z g is a finite projection and thus z = where g = e−λ λ λ −λ 0 0 0 0 0. Therefore gMg is a type I∞ von Neumann algebra, in particular z(g) = 1. There exists a central projection z in M such that zf zg and z⊥ f ! z⊥ g. Since gMg is a type I∞ von ⊥ Neumann algebra, we have that z g = 0. From z(g) = 1 one has z⊥ = 0 and therefore f g. This means that there exists q g such that q ∼ f . Let u be a partial isometry in M such that uu∗ = q, u∗ u = f . Similar to Lemma 3.4 we obtain that uaf u − uu∗ af ∈ S(M, τ ) and af = a(e−λ ∨ eλ⊥ ) ∈ S(M, τ ). Therefore a ∈ S(M, τ ). The proof is complete. 2 Let N be a commutative von Neumann algebra, then N ∼ = L∞ (Ω) for an appropriate measure space (Ω, Σ, μ). It has been proved in [3,13] that the algebra LS(N ) = S(N ) ∼ = L0 (Ω) admits nontrivial derivations if and only if the measure space (Ω, Σ, μ) is not atomic. Let τ be a faithful normal semi-finite trace on the commutative von Neumann algebra N and suppose that the Boolean algebra P (N) of projections is not atomic. This means that there exists a projection z ∈ N with τ (z) < ∞ such that the Boolean algebra of projection in zN is continuous (i.e. has no atom). Since zS(N, τ ) = zS(N ) = S(zN ), the algebra zS(N, τ ) admits a nontrivial derivation δ. Putting δ0 (x) = δ(zx),
x ∈ S(N, τ ),
we obtain a nontrivial derivation δ0 on the algebra S(N, τ ). Therefore, we have that if a commutative von Neumann algebra N has a nonatomic Boolean algebra of projections then the algebra S(N, τ ) admits a non-zero derivation. Lemma 4.2. If N is a commutative von Neumann algebra with a faithful normal semi-finite trace τ and δ is a derivation on S(N, τ ) then τ (zδ ) < ∞, where zδ is the support of the derivation δ. Proof. Suppose the opposite, i.e. τ (zδ ) = ∞. Then there exists a sequence of mutually orthogonal projections zn ∈ N , n = 1, 2, . . . , with zn zδ , 1 τ (zn ) < ∞. For z = supn zn we have τ (z) = ∞. Since τ (zn ) < ∞ for all n = 1, 2, . . . , it follows that zn S(N, τ ) = zn S(N ) = S(zn N ). Define a derivation δn : S(zn N ) → S(zn N ) by δn (x) = zn δ(x),
x ∈ S(zn N ).
Since zδn = zn , Lemma 2.6 implies that for each n ∈ N there exists an element λn ∈ zn N such that |λn | zn and |δn (λn )| nzn .
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Then |λ| n1 zn 1 and therefore λ ∈ S(N, τ ). On the other hand δ(λ) = δ δn (λn ) δ = = λ z λ z δ(λ ) nzn , n n n n n =
Put λ =
n1 λn .
n1
n1
n1
n1
n1
i.e. |δ(λ)| n1 nzn . But τ (zn ) 1 for all n ∈ N, i.e. n1 nzn ∈ / S(N, τ ). Therefore δ(λ) ∈ / S(N, τ ). The contradiction shows that τ (zδ ) < ∞. The proof is complete. 2 Let M be a homogeneous von Neumann algebra of type In , n ∈ N, with the center Z and a faithful normal semi-finite trace τ . Then the algebra M is ∗-isomorphic with the algebra Mn (Z) of all n × n- matrices over Z, and the algebra S(M, τ ) is ∗-isomorphic with the algebra Mn (S(Z, τZ )) of all n × n matrices over S(Z, τZ ), where τZ is the restriction of the trace τ onto the center Z. Now let M be an arbitrary finite von Neumann algebra of type I with the center Z and let {zn }n∈F , F ⊆ N, be a family of central projections from M with supn∈F zn = 1 such that the algebra M is ∗-isomorphic with the C ∗ -product of von Neumann algebras zn M of type In respectively, n ∈ F , i.e. zn M. M∼ = n∈F
In this case we have that S(M, τ ) ⊆
S(zn M, τn ),
n∈F
where τn is the restriction of the trace τ onto zn M, n ∈ F . Suppose that D is a derivation on S(M, τ ), and let δ be its restriction onto the center S(Z, τZ ). Since δ maps each zn S(Z, τZ ) ∼ = Z(S(zn M, τn )) into itself, δ generates a derivation δn on zn S(Z, τZ ) for each n ∈ F . Let Dδn be the derivation on the matrix algebra Mn (zn Z(S(M, τ ))) ∼ = S(zn M, τn ) defined as in (1). Put (12) Dδ {xn }n∈F = Dδn (xn ) , {xn }n∈F ∈ S(M, τ ). By Lemma 4.2 τ (zδ ) < ∞, thus zδ S(M, τ ) = zδ S(M) ∼ = zδ
S(zn M) = zδ
n∈F
S(zn M, τn ),
n∈F
and therefore {Dδn (xn )} ∈ zδ S(M, τ ) for all {xn }n∈F ∈ S(M, τ ). Hence we obtain that the map Dδ is a derivation on S(M, τ ). Similar to Lemma 2.3 one can prove the following. Lemma 4.3. Let M be a finite von Neumann algebra of type I with a faithful normal semi-finite trace τ . Each derivation D on the algebra S(M, τ ) can be uniquely represented in the form D = D a + Dδ , where Da is an inner derivation implemented by an element a ∈ S(M, τ ), and Dδ is a derivation given as (10).
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Finally Theorem 4.1 and Lemma 4.3 imply the following main result the present section. Theorem 4.4. Let M be a type I von Neumann algebra with a faithful normal semi-finite trace τ . Then every derivation D on the algebra S(M, τ ) can be uniquely represented in the form D = Da + Dδ , where Da is inner and implemented by an element a ∈ S(M, τ ) and Dδ is the derivation of the form (12) generated by a derivation δ on the center of S(M, τ ). If we consider the measure topology tτ on the algebra S(M, τ ) (see Section 1) then it is clear that every non-zero derivation of the form Dδ is discontinuous in tτ . Therefore the above Theorem 4.4 implies Corollary 4.5. Let M be a type I von Neumann algebra with a faithful normal semi-finite trace τ . A derivation D on the algebra S(M, τ ) is inner if and only if it is continuous in the measure topology. 5. An application to the description of the first cohomology group Let A be an algebra. Denote by Der(A) the space of all derivations (in fact it is a Lie algebra with respect to the commutator), and denote by InDer(A) the subspace of all inner derivations on A (it is a Lie ideal in Der(A)). The factor-space H 1 (A) = Der(A)/InDer(A) is called the first (Hochschild) cohomology group of the algebra A (see [5]). It is clear that H 1 (A) measures how much the space of all derivations on A differs from the space on inner derivations. The following result shows that the first cohomology groups of the algebras LS(M), S(M) and S(M, τ ) are completely determined by the corresponding cohomology groups of their centers (cf. [3, Corollary 3.1]). Theorem 5.1. Let M be a type I von Neumann algebra with the center Z and a faithful normal semi-finite trace τ . Suppose that z0 is a central projection such that z0 M is a finite von Neumann algebra, and z0⊥ M is of type I∞ . Then (a) H 1 (LS(M)) = H 1 (S(M)) ∼ = H 1 (S(z0 Z)); 1 (S(z Z, τ )), where τ is the restriction of τ onto z Z. (b) H 1 (S(M, τ )) ∼ H = 0 0 0 0 Proof. It immediately follows from Theorems 2.8, 3.6 and 4.4.
2
Remark 2. In the algebra S(M, τ ) equipped with the measure topology tτ one can consider another possible cohomology theories. Similar to [8] consider the space Derc (A) of all continuous derivation on a topological algebra A and define the first cohomology group Hc1 (A) = Derc (A)/InDer(A). Under these notations the above results and Corollary 4.5 imply the following result (cf. [8, Theorem 4.4]).
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Corollary 5.2. Let M be a type I von Neumann algebra with the center Z and a faithful normal semi-finite trace τ . Consider the topological algebra S(M, τ ) equipped with the measure topology. Then Hc1 (S(M, τ )) = {0}. Acknowledgments The second and third named authors would like to acknowledge the hospitality of the, “Institut für Angewandte Mathematik,” Universität Bonn, Germany. This work is supported in part by the DFG 436 USB 113/10/0-1 project (Germany) and the Fundamental Research Foundation of the Uzbekistan Academy of Sciences. The authors are indebted to the reviewer for useful comments. References [1] S. Albeverio, Sh.A. Ayupov, K.K. Kudaybergenov, Non-commutative Arens algebras and their derivations, J. Func. Anal. 253 (2007) 287–302. [2] S. Albeverio, Sh.A. Ayupov, K.K. Kudaybergenov, Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra, Siberian Adv. Math. 18 (2008) 86–94. [3] A.F. Ber, V.I. Chilin, F.A. Sukochev, Non-trivial derivation on commutative regular algebras, Extracta Math. 21 (2006) 107–147. [4] V.I. Chilin, I.G. Ganiev, K.K. Kudaybergenov, The Gelfand–Naimark–Segal theorem for C ∗ -algebras over a ring measurable functions, Vladikavkaz. Math. J. 9 (2007) 16–22. [5] H.G. Dales, Banach Algebras and Automatic Continuity, Clarendon Press, Oxford, 2000. [6] A.E. Gutman, Banach bundles in the theory of lattice-normed spaces, in: Order-Compatible Linear Operators, in: Trudy Ins. Mat., vol. 29, Sobolev Institute Press, Novosibirsk, 1995, pp. 63–211 (in Russian), English transl. in Siberian Adv. Math. 3 (1993) 1–55 (Part I), Siberian Adv. Math. 3 (1993) 8–40 (Part II), Siberian Adv. Math. 4 (1994) 54–75 (Part III), Siberian Adv. Math. 6 (1996) 35–102 (Part IV). [7] A.E. Gutman, A.G. Kusraev, S.S. Kutateladze, The Wickstead problem, Sib. Electron. Math. Reports 5 (2008) 293–333. [8] R.V. Kadison, J.R. Ringrose, Cohomology of operator algebras. I. Type I von Neumann algebras, Acta Math. 126 (1971) 227–243. [9] I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953) 839–859. [10] I. Kaplansky, I. Kaplansky, Algebras of type I, Ann. of Math. (2) 56 (1952) 460–472. [11] A.G. Kusraev, Dominated Operators, Kluwer Academic Publishers, Dordrecht, 2000. [12] A.G. Kusraev, Cyclically compact operators in Banach spaces, Vladikavkaz. Math. J. 2 (2000) 10–23. [13] A.G. Kusraev, Automorphisms and derivations on a universally complete complex f -algebra, Siberian Math. J. 47 (2006) 77–85. [14] M.A. Muratov, V.I. Chilin, ∗-Algebras of unbounded operators affiliated with a von Neumann algebra, J. Math. Sci. 140 (2007) 445–451. [15] E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1975) 91–102. [16] K. Saito, On the algebra of measurable operators for a general AW ∗ -algebra, Tohoku Math. J. 23 (1971) 525–534. [17] S. Sakai, C∗ -Algebras and W∗ -Algebras, Springer-Verlag, 1971. [18] I. Segal, A non-commutative extension of abstract integration, Ann. of Math. 57 (1953) 401–457. [19] M. Takesaki, Theory of Operator Algebras, vol. 1, Springer, New York, 1979. [20] D.A. Vladimirov, Boolean Algebras, Nauka, Moscow, 1969 (in Russian).
Journal of Functional Analysis 256 (2009) 2944–2966 www.elsevier.com/locate/jfa
Mass transportation and rough curvature bounds for discrete spaces Anca-Iuliana Bonciocat a , Karl-Theodor Sturm b,∗ a Institute of Mathematics “S. Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania b Institute for Applied Mathematics, University of Bonn, Poppelsdorfer Allee 82/1, 53115 Bonn, Germany
Received 2 August 2008; accepted 16 January 2009 Available online 20 February 2009 Communicated by C. Villani
Abstract We introduce and study rough (approximate) lower curvature bounds for discrete spaces and for graphs. This notion agrees with the one introduced in [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press] and [K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65–131], in the sense that the metric measure space which is approximated by a sequence of discrete spaces with rough curvature K will have curvature K in the sense of [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press; K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65–131]. Moreover, in the converse direction, discretizations of metric measure spaces with curvature K will have rough curvature K. We apply our results to concrete examples of homogeneous planar graphs. © 2009 Elsevier Inc. All rights reserved. Keywords: Optimal transport; Ricci curvature; GH-limits; Graphs; Concentration of measure
1. Introduction We develop a notion of rough curvature bounds for discrete spaces, based on the concept of optimal mass transportation. These rough curvature bounds will depend on a real parameter h > 0, which should be considered as a natural length scale of the underlying discrete space or as * Corresponding author.
E-mail addresses:
[email protected] (A.-I. Bonciocat),
[email protected] (K.-T. Sturm). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.029
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the scale on which we have to look at the space. For a metric graph, for instance, this parameter equals the maximal length of its edges (times some constant). The approach presented here will follow the one from [12], where the second author introduced a notion of lower curvature bounds for metric measure spaces, which is based on the concept of mass transportation. A closely related theory has been developed independently by J. Lott and C. Villani in [8], see also [15]. Both these approaches required the Wasserstein space of probability measures (and thus in turn the underlying space) to be a geodesic space. Therefore, in the original form they will not apply to discrete spaces. Moreover, if we consider a graph, more precisely the union of the edges of a graph, as a metric space it will have no lower curvature bound in the sense of [12], since the vertices will be branch points of geodesics which destroy the K-convexity of the entropy. The modification to be presented here overcomes this difficulty in the following way: mass transportation and convexity properties of the relative entropy will be studied along h-geodesics. For instance, instead of midpoints of a given pair of points x0 , x1 we look at h-midpoints which are points y with d(x0 , y) 12 d(x0 , x1 ) + h and d(x1 , y) 12 d(x0 , x1 ) + h. Our first main result (Theorem 3.10) states that an arbitrary metric measure space (M, d, m) has curvature K (in the sense of [12]) provided it can be approximated by a sequence (Mh , dh , mh ) of (‘discrete’) metric measure spaces with h-Curv(M, d, m) Kh with Kh → K as h → 0. That is, this result allows to pass from discrete spaces to continuous limit spaces. Our second main result (Theorem 4.1) states that curvature bounds will also be preserved under the converse procedure: Given any metric space (M, d, m) with curvature K and any h > 0 we define standard discretizations (Mh , d, mh ) of (M, d, m) with D2 ((Mh , d, mh ), (M, d, m)) → 0 as h → 0 and with h-Curv(Mh , dh , mh ) K. Further, we apply our results to concrete examples. We prove (Theorem 5.3) that every homogeneous planar graph has h-curvature K where K is given in terms of the degree, the dual degree and the edge length. To be more precise,both the set M = V of vertices, equipped with the counting measure, as well as the union M = e∈E e of edges equipped with one-dimensional Lebesgue measure will be metric measure spaces with h-curvature K, where the metric is the one induced by the Riemannian distance of the 2-dimensional Riemannian manifold whose discretization will be our given graph. Our notion of h-curvature yields the precise value for K if we consider discretizations of hyperbolic spaces. In the final section we show that positive rough curvature bound implies a perturbed transportation cost inequality, weaker than what is usually called the Talagrand inequality. However, it still implies concentration of the reference measure m and exponential integrability of the Lipschitz functions with respect to m. An independent, alternative approach to generalized Ricci curvature bounds for discrete spaces—again based on optimal transportation—was presented by Yann Ollivier [10], see Remark 6.4. 2. Preliminaries Throughout this paper, a metric measure space will always be a triple (M, d, m) where (M, d) is a complete separable metric space and m is a measure on M (equipped with its Borel σ -algebra B(M)) which is locally finite in the sense that m(Br (x)) < ∞ for all x ∈ M and all sufficiently small r > 0. We say that the metric measure space (M, d, m) is normalized if m(M) = 1. Two metric measure spaces (M, d, m) and (M , d , m ) are called isomorphic iff there exists an isometry ψ : M0 → M0 between the supports M0 := supp[m] ⊂ M and M0 := supp[m ] ⊂ M
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such that ψ∗ m = m . The diameter of a metric measure space (M, d, m) will be the diameter of the metric space (supp[m], d). We shall use the notion of L2 -transportation distance D for two metric measure spaces (M, d, m) and (M , d , m ), as defined in [12]: D (M, d, m), (M , d , m ) = inf
1/2 ˆd2 (x, y) dq(x, y) ,
MM
where dˆ ranges over all couplings of d and d and q ranges over all couplings of m and m . Here a measure q on the product space M × M is a coupling of m and m if q(A × M ) = m(A) and q(M × A ) = m (A ) for all measurable A ⊂ M, A ⊂ M ; a pseudo-metric dˆ on the disjoint union M M is a coupling of d and d if dˆ (x, y) = d(x, y) and dˆ (x , y ) = d (x , y ) for all x, y ∈ supp[m] ⊂ M and all x , y ∈ supp[m ] ⊂ M . The L2 -transportation distance D defines a complete and separable length metric on the family 2 of all isomorphism classes of normalized metric measure spaces (M, d, m) for which M d (z, x)dm(x) < ∞ for some (hence all) z ∈ M. The notion of D-convergence is closely related to the one of measured Gromov–Hausdorff convergence introduced in [4]. Recall that a sequence of compact normalized metric measure spaces {(Mn , dn , mn )}n∈N converges in the sense of measured Gromov–Hausdorff convergence (briefly, mGH-converges) to a compact normalized metric measure space (M, d, m) iff there exist a sequence of numbers n 0 and a sequence of measurable maps fn : Mn → M such that for all x, y ∈ Mn , |d(fn (x), fn (y)) − dn (x, y)| n , for any x ∈ M there exists y ∈ Mn with d(fn (y), x) n and such that (fn )∗ mn → m weakly on M for n → ∞. According to Lemma 3.17 in [12], any mGH-convergent sequence of normalized metric measure spaces is also D-convergent; for any sequence of normalized compact metric measure spaces with full supports and with uniform bounds for the doubling constants and for the diameters the notion of mGH-convergence is equivalent to the one of D-convergence. It is easy to see that D((M, d, m), (M , d , m )) = inf dˆ W (ψ∗ m, ψ∗ m ) where the inf is taken ˆ ψ : M → Mˆ of the ˆ dˆ ) with isometric embeddings ψ : M0 → M, over all metric spaces (M, 0 supports M0 and M0 of m and m , respectively, and where dˆ W denotes the L2 -Wasserstein distance derived from the metric dˆ . Recall that for any metric space (M, d) the L2 -Wasserstein distance between two measures μ and ν on M is defined as dW (μ, ν) = inf
1/2 d2 (x, y) dq(x, y)
: q is a coupling of μ and ν ,
M×M
with the convention inf ∅ = ∞. For further details about the Wasserstein distance see the monograph [14]. We denote by P2 (M, d) the space of all probability measures ν which have finite second moments M d2 (o, x) dν(x) < ∞ for some (hence all) o ∈ M. For a given metric measure space (M, d, m) we put P2 (M, d, m) the space of all probability measures ν ∈ P2 (M, d) which are absolutely continuous w.r.t. m. If ν = ρ · m ∈ P2 (M, d, m) we consider the relative entropy of ν with respect to m defined by Ent(ν|m) := lim 0 {ρ>} ρ log ρ dm. We denote by P2∗ (M, d, m) the subspace of measures ν ∈ P2 (M, d, m) of finite entropy Ent(ν | m) < ∞.
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We recall here the definitions of the lower curvature bounds for metric measure spaces introduced in [12]: (i) A metric measure space (M, d, m) has curvature K for some number K ∈ R iff the relative entropy Ent(· | m) is weakly K-convex on P2∗ (M, d, m) in the sense that for each pair ν0 , ν1 ∈ P2∗ (M, d, m) there exists a geodesic Γ : [0, 1] → P2∗ (M, d, m) connecting ν0 and ν1 with Ent Γ (t) m (1 − t) Ent Γ (0) m + t Ent Γ (1) m −
K t (1 − t)d2W Γ (0), Γ (1) 2
(2.1)
for all t ∈ [0, 1]. (ii) The metric measure space (M, d, m) has curvature K in the lax sense iff for each > 0 and for each pair ν0 , ν1 ∈ P2∗ (M, d, m) there exists an -midpoint η ∈ P2∗ (M, d, m) of ν0 and ν1 with Ent(η | m)
1 K 1 Ent(ν0 | m) + Ent(ν1 | m) − d2W (ν0 , ν1 ) + . 2 2 8
(2.2)
Briefly, we shall write Curv(M, d, m) K, respectively Curvlax (M, d, m) K. Recall that in a given metric space (M, d) a point y is an -midpoint of x0 and x1 if d(xi , y) 1 1 2 d(x0 , x1 ) + for each i = 0, 1. We call y midpoint of x0 and x1 if d(xi , y) 2 d(x0 , x1 ) for i = 0, 1. 3. Rough curvature bounds for metric measure spaces In order to adapt the notion of curvature bound to other spaces then geodesic without branching we shall refer in this paper to a larger class of metric spaces: Definition 3.1. Let h > 0 be given. We say that a metric space (M, d) is h-rough geodesic iff for each pair of points x0 , x1 ∈ M and each t ∈ [0, 1] there exists a point xt ∈ M satisfying d(x0 , xt ) t d(x0 , x1 ) + h,
d(xt , x1 ) (1 − t)d(x0 , x1 ) + h.
(3.1)
The point xt will be referred to as the h-rough t-approximate point between x0 and x1 . The h-rough 1/2-approximate point is actually the h-midpoint of x0 and x1 . Example 3.2. (i) Any nonempty set X with the discrete metric d(x, y) = 0 for x = y and 1 for x = y is h-rough geodesic for any h 1/2. In this case, any point is an h-midpoint of any pair of distinct points. (ii) If > 0 then the space (Rn , d) with the metric d(x, y) = |x − y| ∧ is h-rough geodesic for h /2 (here | · | is the euclidian metric). (iii) For > 0 the space (Rn , d) with the metric d(x, y) = |x − y| + |x − y|2 is h-rough geodesic for each h /4.
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The above examples are somewhat pathological. We actually have in mind the more friendly examples of discrete spaces and some geodesic spaces with branch points, e.g. graphs, that do not have curvature bounds as defined in [12]. For a discrete h-rough geodesic metric space (M, d) one should think of h as a discretization size or “resolution” of M. In an h-geodesic space a pair of points x and y is not necessarily connected by a geodesic but by a chain of points x = x0 , x1 , . . . , xn = y having intermediate distance less then h/2. In the sequel we will use two types of perturbations of the Wasserstein distance, defined as follows: Definition 3.3. Let (M, d) be a metric space. For each h > 0 and any pair of measures ν0 , ν1 ∈ P2 (M, d) put
d±h W (ν0 , ν1 ) := inf
1/2 2 d(x0 , x1 ) ∓ h + dq(x0 , x1 ) : q coupling of ν0 and ν1 , (3.2)
where (·)+ denotes the positive part. Remark 3.4. According to Theorem 4.1 from [15] there exists a coupling for which the infimum in (3.2) is attaint. We will call it +h-optimal coupling (resp. −h-optimal coupling) of ν0 and ν1 . −h The two perturbations d+h W and dW are related to the Wasserstein distance dW in the following way:
Lemma 3.5. For any h > 0 we have +h (i) d+h W dW dW + h; −h (ii) dW dW dW + h.
Proof. (i) Let ν0 and ν1 be two probabilities in (M, d) and consider q an optimal coupling and q+h a +h-optimal coupling of them. Then d+h W (ν0 , ν1 ) =
1/2 2 d(x0 , x1 ) − h + dq+h (x0 , x1 )
1/2 2 d(x0 , x1 ) − h + dq(x0 , x1 )
1/2 d(x0 , x1 ) dq(x0 , x1 ) = dW (ν0 , ν1 ) 2
and dW (ν0 , ν1 ) =
(ii) Similar to (i).
1/2 1/2 2 d(x0 , x1 ) dq(x0 , x1 ) d(x0 , x1 ) dq+h (x0 , x1 ) 2
2
1/2 2 d(x0 , x1 ) − h + + h dq+h (x0 , x1 ) d+h W (ν0 , ν1 ) + h.
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With an elementary proof we have also a monotonicity property of d±h W in h: Lemma 3.6. Let 0 < h1 < h2 be arbitrarily given. Then for each pair of probabilities ν0 and ν1 −h2 1 (i) d−h W (ν0 , ν1 ) < dW (ν0 , ν1 );
+h2 +h1 1 (ii) d+h W (ν0 , ν1 ) dW (ν0 , ν1 ) and the inequality is strict if and only if dW (ν0 , ν1 ) > 0.
We introduce now the notion of rough lower curvature bound: Definition 3.7. We say that a metric measure space (M, d, m) has h-rough curvature K for some numbers h > 0 and K ∈ R iff for each pair ν0 , ν1 ∈ P2∗ (M, d, m) and for any t ∈ [0, 1] there exists an h-rough t-approximate point ηt ∈ P2∗ (M, d, m) between ν0 and ν1 satisfying Ent(ηt | m) (1 − t) Ent(ν0 | m) + t Ent(ν1 | m) −
K 2 t (1 − t)d±h W (ν0 , ν1 ) , 2
(3.3)
where the sign in d±h W (ν0 , ν1 ) is chosen ’+’ if K > 0 and ’−’ if K < 0. Briefly, we write in this case h-Curv(M, d, m) K. Remark 3.8. We could also choose two parameters in the above definition, h for the approximate midpoint and for the inequality (3.3). Having two parameters instead of one is not essentially useful for further results. One can always think of h ∨ in the definition of rough curvature bound, which is an approximate notion. Remark 3.9. (i) If (M, d, m) and (M , d , m ) are two isomorphic metric measure spaces and K ∈ R then h-Curv(M, d, m) K if and only if h-Curv(M , d , m ) K. (ii) If (M, d, m) is a metric measure space and α, β > 0 then h-Curv(M, d, m) K if and only if αh-Curv(M, α d, βm) αK2 , because Ent(ν | βm) = Ent(ν | m) − log β, (α · d)±h W (ν0 , ν1 ) = (ν , ν ) and for t ∈ [0, 1] η is h-rough t-approximate point between μ, ν with respect to α · d±h 0 1 t W dW if and only if ηt is αh-rough t-approximate point between μ, ν with respect to (α d)W . Theorem 3.10. Let (M, d, m) be a normalized metric measure space and {(Mh , dh , mh )}h>0 a family of normalized metric measure spaces with uniformly bounded diameter and with hCurv(Mh , dh , mh ) Kh for Kh → K as h → 0. If D → (M, d, m) (Mh , dh , mh ) −
as h → 0 then Curvlax (M, d, m) K. If in addition M is compact then Curv(M, d, m) K. Proof. Let {(Mh , dh , mh )}h>0 be a family of normalized discrete metric measure spaces. AsD → (M, d, m) as h → 0 and suph>0 diam(Mh , dh , mh ), sume that (Mh , dh , mh ) −
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diam(M, d, m) for some ∈ R. Now let > 0 and ν0 = ρ0 m, ν1 = ρ1 m ∈ P2∗ (M, d, m) be given. Choose R with sup Ent(νi | m) + i=0,1
|K| 2 2 + + 3|K|(2 + 3) R. 8 8
(3.4)
We have to deduce the existence of an -midpoint η which satisfies inequality (2.2). Choose 0 < h < with |Kh − K| < and 2 + 4 2 R . D (Mh , dh , mh ), (M, d, m) exp − 2
(3.5)
Like in Section 4.5 in [12], one can define the canonical maps Qh : P2 (M, d, m) → P2 (Mh , dh , mh ), and Qh : P2 (Mh , dh , mh ) → P2 (M, d, m) as follows. We consider qh a coupling of m and mh and dˆ h a coupling of d and dh such that
dˆ 2h (x, y) dqh (x, y) 2D2 (M, d, m), (Mh , dh , mh ) .
Let Qh and Qh be the disintegrations of qh w.r.t. mh and m, resp., that is dqh (x, y) = ˆ denote the m-essential supremum of the map Qh (y, dx) dmh (y) = Qh (x, dy) dm(x) and let x →
1/2 ˆd2h (x, y)Qh (x, dy) .
Mh
ˆ 2 . In our case For ν = ρm ∈ P2 (M, d, m) define Qh (ν) ∈ P2 (Mh , dh , mh ) by Qh (ν) := ρh mh where ρh (y) :=
ρ(x)Qh (y, dx).
M
The map Qh is defined similarly. Lemma 4.19 from [12] gives the following estimates: Ent Qh (ν) mh Ent(ν | m)
for all ν = ρm
(3.6)
and
d2W ν, Qh (ν)
ˆ 2 · Ent(ν | m) 2+ . − log D((M, d, m), (Mh , dh , mh ))
provided D((M, d, m), (Mh , dh , mh )) < 1. Analogous estimates hold for Qh . For our given ν0 = ρ0 m, ν1 = ρ1 m ∈ P2∗ (M, d, m) put νi,h := Qh (νi ) = ρi,h mh
(3.7)
A.-I. Bonciocat, K.-T. Sturm / Journal of Functional Analysis 256 (2009) 2944–2966
with ρi,h (y) = that
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ρi (x)Qh (y, dx) for i = 0, 1 and let ηh be an h-midpoint of ν0,h and ν1,h such
Ent(ηh | mh )
1 1 Kh δh h d (ν0,h , ν1,h )2 , Ent(ν0,h | mh ) + Ent(ν1,h | mh ) − 2 2 8 W
(3.8)
where δh is the sign of Kh . From (3.5)–(3.7) we conclude d2W (ν0 , ν0,h )
ˆ 2 · Ent(ν0 | m) 2+ − log D((M, d, m), (Mh , dh , mh )) 2 + 4 2 R 2 − log D((M, d, m), (Mh , dh , mh ))
and similarly d2W (ν1 , ν1,h ) 2 . If K < 0 we can suppose Kh < 0 too. From Lemma 3.5(ii) we have
2
2 d−h W (ν0,h , ν1,h ) dW (ν0,h , ν1,h ) + h
2 dW (ν0 , ν1 ) + 3 dW (ν0 , ν1 )2 + 6 + 9 2 ,
because dW (ν0 , ν1 ) . For K > 0 one can choose h small enough to ensure Kh > 0. Then Lemma 3.5(i) implies
dW (ν0 , ν1 )2 dW (ν0,h , ν1,h ) + 2
2
2 +h 2 2 d+h W (ν0,h , ν1,h ) + 3 dW (ν0 , ν1 ) + 6 + 9 .
In both cases the estimates above combined with (3.6), (3.8) and the fact that we chose h with −Kh < − K will imply Ent(ηh | mh )
1 1 K Ent(ν0 | m) + Ent(ν1 | m) − d2W (ν0 , ν1 ) + 2 2 8
(3.9)
with = [ 2 + 3|K|(2 + 3)]/8. The case K = 0 follows by the calculations above, depending on the sign of Kh . Finally, put η = Qh (ηh ). Then again by (3.5), the estimates given in Lemma 4.19 [12] for Qh and by the previous estimate (3.9) for Ent(ηh | mh ) we deduce d2W (ηh , η)
ˆ 2 · Ent(ηh | mh ) 2+ − log D((M, d, m), (Mh , dh , mh )) 2 + 4 2 R 2. − log D((M, d, m), (Mh , dh , mh ))
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For i = 0, 1 we have dW (η, νi ) 2 + dW (ηh , νi,h ) 2 + 12 dW (ν0,h , ν1,h ) + h 12 dW (ν0 , ν1 ) + 4. Hence, 1 sup dW (η, νi ) dW (ν0 , ν1 ) + 4, 2 i=0,1 i.e. η is a (4)-midpoint of ν0 and ν1 . Furthermore, by (3.6) Ent(η | m) Ent(ηh | mh ) 1 K 1 Ent(ν0 | m) + Ent(ν1 | m) − d2W (ν0 , ν1 ) + 2 2 8 with as above. This proves that Curvlax (M, d, m) K.
2
4. Discretizations of metric spaces Let (M, d, m) be a given metric measure space. For h > 0 let Mh be a discrete subset of M, say Mh = {xn : n ∈ N}, with M = ∞ i=1 BR (xi ), where R = R(h) 0 as h 0. If (M, d, m) finite number of points. Choose Ai ⊂ BR (xi ) has finite diameter then Mh might consist of a mutually disjoint with xi ∈ Ai , i = 1, 2, . . . , and ∞ i=1 Ai = M (e.g. one could choose a Voronoi tessellation) and consider the measure mh on Mh given by mh ({xi }) := m(Ai ), i = 1, 2, . . . . We call (Mh , d, mh ) a discretization of (M, d, m). Theorem 4.1. D (i) If m(M) < ∞ then (Mh , d, mh ) − → (M, d, m) as h → 0. (ii) If Curvlax (M, d, m) K with K = 0 then for each h > 0 and for each discretization (Mh , d, mh ) with R(h) < h/4 we have h-Curv(Mh , d, mh ) K. (iii) If Curv(M, d, m) K for some real number K then for each h > 0 and for each discretization (Mh , d, mh ) with R(h) h/4 we have h-Curv(Mh , d, mh ) K.
Proof. (i) The measure q =
∞
i=1 (m(Ai )δxi ) × (1Ai m)
is a coupling of mh and m, so
D (Mh , d, mh ), (M, d, m) 2
d2 (x, y) dq(x, y)
Mh ×M
=
∞
i=1
d2 (xi , y) dm(y)
m(Ai )
∞
Ai
2
m(Ai )
R(h) R(h) 2
2
i=1
∞
2 m(Ai )
i=1
= R(h)2 m(M)2 → 0 as h → 0. (ii) Fix h > 0 and consider a discretization (Mh , d, mh ) of (M, d, m) with R(h) < h/4. Let ν0h , ν1h ∈ P2∗ (Mh , d, mh ) be given; it is enough to make the proof for ν0h , ν1h with compact support.
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h 1 h Suppose then νih = ( nj=1 αi,j {x } )mh , i = 1, 2 (some of the αi,j can be zero). We take also an nj h 1 )m ∈ P ∗ (M, d, m) for i = 1, 2. Choose > 0 such arbitrary t ∈ [0, 1]. Put νi := ( j =1 αi,j Aj 2 that 4R(h) + h.
(4.1)
Since Curvlax (M, d, m) K for our given t ∈ [0, 1] there exists ηt ∈ P2∗ (M, d, m) an -rough t-approximate point between ν0 and ν1 such that Ent(ηt | m) (1 − t) Ent(ν0 | m) + t Ent(ν1 | m) −
K t (1 − t)d2W (ν0 , ν1 ) + . 2
(4.2)
We compute Ent(νi | m) =
n
h h αi,j log αi,j dm =
j =1A
n
h h αi,j log αi,j mh {xj } = Ent νih mh ,
(4.3)
j =1
j
for i = 0, 1. Denote ηth ({xj }) := ηt (Aj ), j = 1, 2, . . . , n. Suppose ηt = ρt · m. From Jensen’s inequality we get ∞ Ent ηth mh =
j =1
∞ j =1
Aj
ρt dm
m(Aj ) 1 m(Aj )
log
Aj
ρt dm
m(Aj )
mh {xj }
ρt log ρt dm mh {xj } = Ent(ηt | m),
Aj
which together with (4.2) and (4.3) implies K Ent ηth mh (1 − t) Ent ν0h mh + t Ent ν1h mh − t (1 − t)d2W (ν0 , ν1 ) + . (4.4) 2 Firstly, we consider the case K < 0. Let q h be a −2R(h)-optimal coupling of ν0h and ν1h . Then the formula qˆ :=
n
q h (xj , xk ) δ(xj ,xk ) ×
j,k=1
1Aj ×Ak m(Aj )m(Ak )
(m × m)
defines a measure on Mh × Mh × M × M which has marginals ν0h , ν1h , ν0 and ν1 . Moreover, the projection of qˆ on the first two factors is equal to q h . Therefore we have dW (ν0 , ν1 )2
d(x, y)2 d qˆ x h , y h , x, y
2 h h d x, x h + d x h , y h + d y h , y d qˆ x , y , x, y
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n q h ({(xj , xk )}) = m(Aj )m(Ak ) j,k=1
n
2
d(x, xj ) + d(xj , xk ) + d(xk , y)
dm(x) dm(y)
Aj ×Ak
2 −2R(h) h h 2 q h (xj , xk ) d(xj , xk ) + 2R(h) = dW (ν0 , ν1 ) ,
j,k=1
which together with (4.4) yields Ent ηth mh (1 − t) Ent ν0h mh + t Ent ν1h mh −
K −2R(h) h h 2 ν0 , ν1 + . t (1 − t)dW 2
(4.5)
In the case K > 0 we start with an optimal coupling q of ν0 and ν1 and we show that the measure q h :=
n
q(Aj × Ak )δ(xj ,xk )
j,k=1
is a coupling of ν0h and ν1h . Indeed, if A ⊂ Mh then we have in turn n
q(Aj × Ak )δ(xj ,xk ) (A × Mh ) =
j,k=1
n
q(Aj × Ak )δxj (A) =
n
q(Aj × M)δxj (A)
j =1
j,k=1
=
n
ν0 (Aj )δxj (A) =
j =1
n
ν0h {xj } δxj (A) = ν0h (A).
j =1
Since for any j, k = 1, 2, . . . , n and for arbitrary x ∈ Aj and y ∈ Ak we have (d(xj , xk ) − 2R(h))+ (d(xj , xk ) − d(x, xj ) − d(y, xk ))+ d(x, y) one can estimate: +2R(h) h h 2 ν0 , ν1
dW
n
2 q(Aj × Ak ) d(xj , xk ) − 2R(h) +
j,k=1
=
n
2
d(xj , xk ) − 2R(h) +
dq(x, y)
j,k=1A ×A j k
n
2
d(xj , xk ) − d(x, xj ) − d(y, xk ) +
dq(x, y)
j,k=1A ×A j k
n
j,k=1A ×A j k
d(x, y)2 dq(x, y) = M×M
d(x, y)2 dq(x, y) = dW (ν0 , ν1 )2 .
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Therefore from (4.4) we obtain Ent ηth mh (1 − t) Ent ν0h mh + t Ent ν1h mh −
K +2R(h) h h 2 ν0 , ν1 + . t (1 − t)dW 2
(4.6)
For sufficiently small we can get −
h h 2 K K ±2R(h) h h 2 ν0 , ν1 + − t (1 − t)d±h t (1 − t)dW W ν0 , ν1 2 2
(4.7)
and then (4.5), (4.6) yield K h h 2 Ent ηth mh (1 − t) Ent ν0h mh + t Ent ν1h mh − t (1 − t)d±h W ν0 , ν1 , 2
(4.8)
h h depending on the sign of K. The inequality (4.7) fails only when K > 0 and d+h W (ν0 , ν1 ) = 0, h h h h but in this case dW (ν0 , ν1 ) h and either η = ν0 or η = ν1 verifies directly the condition (3.3) from the definition of h-rough curvature bound for the discretization. The measure π = nj=1 (ηth ({xj })δxj × 1Aj ηt ) is a coupling of ηth and ηt , so
d2W ηth , ηt
d2 (x, y) dπ(x, y) R 2 (h),
Mh ×M
and similarly d2W (νih , νi ) R 2 (h) for i = 1, 2. Because ηt is an -rough t-approximate point between ν0 and ν1 we deduce
dW ηth , ν0h dW (ηt , ν0 ) + 2R(h) t dW (ν0 , ν1 ) + 2R(h) +
t dW ν0h , ν1h + 2R(h)(1 + t) +
and by a similar argument
dW ηth , ν1h (1 − t)dW ν0h , ν1h + 2R(h)(2 − t) + .
From (4.1) we conclude that ηh is an h-rough t-approximate point between ν0h and ν1h , which together with (4.8) proves that h-Curv(Mh , d, mh ) K. (iii) follows the same lines as (ii). 2 Example 4.2. from the norm | · |1 in Rn defined by |x|1 = (i) If we consider on Zn the metric d1 coming n n |x | and with the measure m = n i=1 i x∈Zn δx , then h-Curv(Z , d1 , mn ) 0 for any h 2n. (ii) The n-dimensional grid En having Zn as set of vertices, equipped with the graph distance and with the measure mn which is the 1-dimensional Lebesgue measure on the edges, has h-Curv(En , d1 , mn ) 0 for any h 2(n + 1).
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Proof. We use the following result: Lemma 4.3. (See [15].) Any finite dimensional Banach space equipped with the Lebesgue measure has curvature 0. We tile the space Rn with n-dimensional cubes of edge 1 centered in the vertices of the grid. The | · |1 -radius of the cells of the tessellation with such cubes is n/2. Therefore, claim (i) is a consequence of Theorem 4.1(iii) applied to the space (Rn , | · |1 , dx) and of Lemma 4.3. For the proof of (ii) we follow the same argument like in the proof of Theorem 4.1. In this case, we pass from a probability on the grid to a probability on Rn by averaging on each cube of the tessellation and scaling. Here one should take into account that for a cube C from the tiling n+1 , sup |x − y|1 : x ∈ C ∩ En , y ∈ C = 2 that provides the minimal h = 2(n + 1) starting from which h-Curv(En , d1 , mn ) 0.
2
Example 4.4. (i) Let G be the graph that tiles the euclidian plane with equilateral triangles of edge r. We endow G with the graph metric dG induced by the euclidian metric and with the 1-dimensional √ Lebesgue measure m on the edges. Then G has h-curvature 0 for any h 8r 3/3. (ii) The graph G that tiles the euclidian plane with regular hexagons of edge length r, equipped as usual with the graph metric dG and with the 1-dimensional measure m , has h-curvature 0 for any h 34r/3. Proof. Consider a Cartesian coordinate system in the euclidian plane with origin O and axes Ox and Oy. We equip R2 with the Banach norm · that has as unit ball the regular hexagon centered in O, having two opposite vertices on Ox and the edge length (measured in the euclidian metric) √ √ equal to 1. Explicitly (x, y) = max{ 2 3 3 |y|, |x| + 33 |y|} for any (x, y) in R2 . We denote by d the metric determined by this norm. (i) For the triangular tessellation we choose the origin O to be one of the vertices of the graph and two of the 6 edges emanating from O be along the Ox axis. The edges of the graph have length r in the euclidian metric. We see that dG (v1 , v2 ) = d(v1 , v2 ) for any two vertices v1 and v2 of the graph. In general for x, y ∈ G we have |dG (x, y) − d(x, y)| r. Then one can construct a coupling dˆ of dG and d by setting dˆ (v, x) := d(v, x) for v vertex of G and x ∈ R2 and dˆ (y, x) := infi=1,2 {dG (y, vi ) + d(vi , x))} if y ∈ G belongs to an edge with endpoints v1 , v2 and x ∈ R2 . By Lemma 4.3 Curv(R2 , d, λ) 0, where λ is the 2-dimensional Lebesgue measure. If we tile the plane with regular hexagons Aj , j ∈ N,√which have vertices in the centers of the triangles of the graph G, we have dˆ (y, x) 2r 3/3 for any y ∈ Aj ∩ G and x ∈ Aj . The proof of the h-curvature bound is a modification of the proof of Theorem 4.1. We start with ν0 , ν1 ∈ P2∗ (G, dG , m) with νi = ρi m, i = 0, 1, and we define νi :=
∞ j =1
1 λ(Aj )
G∩Aj
ρi dm 1Aj · λ ∈ P2∗ R2 , d, λ
for i = 0, 1.
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√ We have then dˆ W (νi , νi ) 2r 3/3. We consider ηt = ρ t · λ the geodesic that joints ν0 and ν1 , along which the convexity condition for the entropy on P2∗ (R2 , d, λ) is fulfilled and denote ηt :=
∞ j =1
1 m(G ∩ Aj )
ρ t dλ 1G∩Aj · m.
Aj
√ Then ηt is 8r 3/3-rough t-approximate point between ν0 and ν1 . From Jensen’s inequality we obtain Ent(ηt | m) Ent( ηt | λ) − log m(G ∩ A) + log λ(A) and Ent( νi | λ) Ent(νi | m) + log m(G ∩ A) − log λ(A) (observe that all sets Aj have the same Lebesgue measure λ(A) and all sets G ∩ Aj have the same measure m(G ∩ A)). Hence ηt satisfies Ent(ηt | m) (1 − t) Ent(ν0 | m) + t Ent(ν1 | m), √ and so we have proved h-Curv(G, dG , m) 0 for any h 8r 3/3. (ii) For the hexagonal tessellation let O be again one of the vertices of the graph and one of the 3 edges emanating from it be along the Oy axis. In this case we use the Banach norm · := 34 · on R2 and denote by d the associated metric. The length of the edges of the graph in the metric d is equal to 4r/3. We see that dG (v1 , v2 ) = d (v1 , v2 ) for any two vertices v1 , v2 with dG (v1 , v2 ) = 2kr, k ∈ N. In general |dG − d | r/3 on the set of vertices and |dG − d | r everywhere on G . One can construct then a coupling dˆ of dG + and d in the following way: Fix v0 = O. If v is a vertex of the graph with dG (v0 , v) = 2kr, k ∈ N then set dˆ (v, x) := d (v, x), x ∈ R2 . For y ∈ G with dG (v0 , y) = 2kr, k ∈ N define dˆ (y, x) := inf{dG (y, v) + d (v, x): v ∈ G , dG (v0 , v) = 2kr}. We tile the plane with equilateral triangles Bi , i ∈ N, with vertices in the centers of the hexagons of the graph. Then dˆ (y, x) 17r/6 for y ∈ Bi ∩ G , x ∈ Bi . By the same argument as for the triangular tiling we obtain h-Curv(G , dG , m ) 0 for any h 4 · 17r/6 = 34r/3. 2 5. Some remarks on homogeneous planar graphs We refer in the sequel to a special class of graphs. In general, a graph G is determined by the set of vertices V (G) and the set of edges E(G). In order to regard graphs as discrete analogues of 2-dimensional manifolds one has to specify also the set of faces F (G) and to impose the graph to be planar. A graph is planar if it can be drawn in a plane without graph edges crossing (i.e., it has graph crossing number 0). Only planar graphs have duals. The graphs we will be concerned with are connected and simple (with no self-loops and no multiple edges) and such that their dual graphs are also simple, therefore any two faces have at most one common edge and every face is bounded by a cycle. We consider in the following the (possibly infinite) homogeneous graph G(l, n, r) with vertices of constant degree l 3, with faces bounded by polygons with n 3 edges (thus n is the degree of all vertices in the dual graph) and such that all edges have the same length r > 0 (see Fig. 1). The following result is probably well-known, but since we did not find a reference we present here the easy proof.
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Fig. 1. G(7, 3, r).
Lemma 5.1. (i) If 1l + n1 < 12 then G(l, n, r) can be embedded into the 2-dimensional hyperbolic space with constant sectional curvature 2 cos2 ( π ) 1 . (5.1) K = − 2 arccosh 2 2 πn − 1 r sin ( l ) There are infinitely many choices of such l and n. In any case, the graph is unbounded. (ii) If 1l + n1 > 12 then G(l, n, r) is one of the five regular polyhedra (Tetrahedron, Octahedron, Cube, Icosahedron, Dodecahedron) and can be embedded into the 2-dimensional sphere with constant sectional curvature 2 cos2 ( πn ) 1 . (5.2) K = 2 arccos 2 2 π − 1 r sin ( l ) (iii) If 1l + n1 = 12 then G(l, n, r) can be embedded into the euclidian plane (K = 0). In this case there are exactly three cases corresponding to the 3 regular tessellations of the euclidian plane: the tessellation of triangles (l = 6, n = 3), of squares (l = n = 4), and of hexagons (l = 3, n = 6). Proof. Firstly we see that 2
cos2 ( πn ) sin2 ( πl )
−1>1
⇔
sin2
π π − 2 n
> sin2
π l
⇔
1 1 1 + < l n 2
hence in each case the expression that defines the curvature K makes sense.
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(i) For given l, n, r we construct the embedding in the following way: we start from an arbitrary point O of the 2-hyperbolic space with curvature K, denoted by HK,2 . From this point we construct n geodesic lines OA1 , OA2 , . . . , OAn of length √ 1 sinh ( −Kr) π , R := √ arcsinh sin l −K sin( 2π ) n
(5.3)
such that the inner angle between any two consecutive geodesics OAk , OAk+1 is 2π/n. We prove that A1 , A2 , . . . , An correspond to vertices of the given graph, and the geodesics A1 A2 , . . . , An−1 An , An A1 correspond isometrically to consecutive edges in G(l, n, r) that bound a regular n-polygon with edge-length r and all angles equal to 2π/ l. Let us denote by d the intrinsic metric on HK,2 . From the Cosine Rule for hyperbolic triangles applied to OA1 A2 and from (5.1) and (5.3) we have: √ √ √ 2π 2 2 cosh −K d(A1 , A2 ) = cosh ( −KR) − sinh ( −KR) cos n √ 2π = 1 + sinh2 ( −KR) 1 − cos n √ 2π sinh2 ( −Kr) 2 π 1 − cos sin =1+ 2 2π l n sin ( n ) √ cosh2 ( −Kr) − 1 2 π sin =1+ l 1 + cos( 2π n ) 2 cos2 ( πn ) sin2 ( πl ) 2 2 π −1 −1 =1+ 2 cos2 ( πn ) sin ( l ) =2
cos2 ( πn ) sin2 ( πl )
√ − 1 = cosh( −Kr),
so d(A1 , A2 ) = r and the same holds for all the other edges of the polygon. We apply now the Sine Rule for the hyperbolic triangle OA1 A2 and (5.3) in order to compute: sin (A1 ; O, A2 ) =
√ ) sin( 2π π , sinh( −KR) = sin √n l sinh( −Kr)
(5.4)
where (A1 ; O, A2 ) denotes the angle at A1 in the triangle OA1 A2 . This angle is less then π/2 because it is equal to (A2 ; O, A1 ) and in the hyperbolic triangles the sum of the angles of a triangle is less then π . Therefore (5.4) shows that all the angles of the polygon are equal to 2π/ l, so around each vertex one can construct other l − 1 polygons with n edges, congruent with the first one. We repeat the procedure with each of the vertices of the new polygons. In this way the whole space HK,2 can be tiled with regular polygons which are faces of the graph G(l, n, r). (ii), (iii) Since there is only a finite number of examples with well-known realizations, the claim can be verified directly. Alternatively, one can prove it like in the part (i) with appropriate
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interpretations of the hyperbolic sine as sine for positive curvature and as length for the euclidian plane. 2 Remark 5.2. The dual graph G(l, n, r)∗ = G(n, l, r ) is embedded into the 2-manifold of the same constant curvature as G(l, n, r), where the dual edge length is cos2 ( π ) cos2 ( π ) r := r · arccosh 2 2 πn − 1 arccosh 2 2 πl − 1 for K < 0, sin ( l ) sin ( n ) and with appropriate modifications for the other two cases. In each of the three cases from Lemma 5.1 the 2-manifold will be endowed with the intrinsic metric d and with the Riemannian volume vol. We equip G(l, n, r) with the metric d induced by the corresponding Riemannian metric and with the uniform measure m on the edges. We denote further by V(l, n, r) the set of vertices of the graph G(l, n, r) equipped with the same metric d inherited from the Riemannian manifold and with the counting measure m := v∈V δv . Theorem 5.3. For any numbers l, n 3 and for any r > 0 both metric measure spaces (V(l, n, r), d, m ) and (G(l, n, r), d, m) have h-curvature K for h r · C(l, n), where ⎧
cos2 ( π ) 2 ⎪ ⎪ − r12 arccosh 2 2 πn − 1 for 1l + n1 > 12 , ⎪ sin ( ) ⎪ l ⎨ π (5.5) K = 1 arccos2 cos2 ( n ) − 12 for 1l + n1 < 12 , π 2 2 ⎪ sin ( l ) ⎪ ⎪r ⎪ ⎩ 0 for 1l + n1 = 12 and C(l, n) = 4 · arcsinh( sin(1 π ) n
cos2 ( πn ) sin2 ( πl )
− 1)/ arccosh(2
cos2 ( πn ) sin2 ( πl )
− 1).
Proof. We look at V(l, n, r) and G(l, n, r) as subsets of the 2-manifold with constant curvature K (given by Lemma 5.1). We tile the manifold with the faces of the dual graph G(n, l, r ) having vertices in the centers of the faces of G(l, n, r) (the center O of the polygon with n edges in the proof of Lemma 5.1 becomes vertex of the dual). We make explicitly the calculations only in the hyperbolic case, the other two cases are similar. One can decompose the hyperbolic space as HK,2 = ∞ j =1 Fj , where {Fj }j are the faces of the dual graph, as described above. The curvature bound for the discrete space V(l, n, r) is then a consequence of the Theorem 4.1. For G := G(l, n, r) the proof of the curvature bound is a modification of the proof of Theorem 4.1. We start with ν0 , ν1 ∈ P2∗ (G(l, n, r), d, m) with νi = ρi · m, i = 0, 1, and define νi :=
∞ j =1
1 vol(Fj )
ρi dm 1Fj · vol ∈ P2∗ HK,2 , d, vol
for i = 0, 1.
G∩Fj
Now the place of R(h) from Theorem 4.1 is taken by R from the proof of Lemma 5.1(i), so dW (νi , νi ) R. One can express R only in terms of our initial data l, n and r as R = rC(l, n)/4,
t · vol the geodesic that with C(l, n) given in the statement of the theorem. We consider ηt = ρ
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joints ν0 and ν1 , along which one has the K-convexity for the entropy on HK,2 (Theorem 4.9 from [12]) and denote ηt :=
∞ j =1
1 m(G ∩ Fj )
ρ t d vol 1G∩Fj · m.
Fj
Then ηt is 4R-rough t-approximate point between ν0 and ν1 . From Jensen’s inequality ηt | vol) − log m(G ∩ F ) + log vol(F ) and Ent( νi | vol) we obtain Ent(ηt | m) Ent( Ent(νi | m) + log m(G ∩ F ) − log vol(F ) (observe that all faces Fj have the same volume vol(F ) and all sets G ∩ Fj have the same measure m(G ∩ F )). Hence, like in the proof of Theorem 4.1, ηt satisfies Ent(ηt | m) (1 − t) Ent(ν0 | m) + t Ent(ν1 | m) −
K −2R t (1 − t)dW (ν0 , ν1 )2 , 2
so we have proved h-Curv(G(l, n, r), d, m) K for any h 4R in the hyperbolic case (K < 0). 2 Remark 5.4. There are various notions of combinatorial curvature for graphs in the literature, see for instance [3,5,6]. The notion of curvature introduced by Gromov in [5] was used in studying hyperbolic groups. Later on it was modified and investigated by Higuchi [6] and other authors. Forman has introduced in [3] a different notion of combinatorial Ricci curvature for cell complexes. The graphs considered in the above mentioned works have neither specified metric, nor specified reference measure. In [6] the combinatorial curvature of a graph G is a map ΦG : V (G) → R that assigns to each m(x) 1 vertex x ∈ V (G) the number ΦG (x) = 1 − m(x) i=1 d(Fi ) , where m(x) is the degree of the 2 + vertex x, d(F ) is the number of edges of the cycle bounding a face F , and F1 , F2 , . . . , Fm(x) are the faces around the vertex x. The combinatorial curvature introduced in [5] is a map ∗ : F (G) → R, where the curvature Φ ∗ (F ) of a face F is given by the curvature Φ of ΦG G G the corresponding vertex in the dual graph. For the homogeneous graph G(l, n, r), the curvature of any vertex x is ΦG (x) = l( 1l + n1 − 12 ) and the curvature in the sense of Gromov [5] of any ∗ (F ) = n( 1 + 1 − 1 ). face F is ΦG l n 2 Note that the sign of the combinatorial curvature in both approaches above changes according to whether 1l + n1 is greater or less than 12 . Rather curiously, in our Theorem 5.3 the sign of the rough curvature bound changes in the same manner, although our notion of curvature applies to graphs that have a metric structure and a reference measure. For the moment we see no further links with the notions of combinatorial curvature mentioned here. 6. Perturbed transportation inequalities, concentration of measure and exponential integrability Let (M, d) be a metric space and m ∈ P2 (M, d) be a given probability measure. The measure m is said to satisfy a Talagrand inequality (or a transportation cost inequality) with constant K iff for all ν ∈ P2 (M, d) 2 Ent(ν | m) . (6.1) dW (ν, m) K
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Such an inequality was first proved by Talagrand in [13] for the canonical Gaussian measure on Rn . A positive rough curvature bound allows us to obtain a weaker inequality, in terms of the perturbation d+h W of the Wasserstein distance: Proposition 6.1 (“h-Talagrand inequality”). Assume that (M, d, m) is a metric measure space which has h-Curv(M, d, m) K for some numbers h > 0 and K > 0. Then for each ν ∈ P2 (M, d) we have 2 Ent(ν | m) +h . (6.2) dW (ν, m) K We will call (6.2) h-Talagrand inequality. Proof. Since we assumed that m is a probability measure, for any ν ∈ P2 (M, d) the entropy functional is nonnegative: Ent(ν | m) − log m(M) = 0, according to Lemma 4.1 from [12]. The curvature bound h-Curv(M, d, m) K implies that for the pair of measures ν and m and for each t ∈ [0, 1] there exists an h-rough t-approximate point ηt ∈ P2 (M, d) such that Ent(ηt | m) (1 − t) Ent(ν | m) −
K 2 t (1 − t)d+h W (ν, m) . 2
2 If Ent(ν | m) < K2 d+h W (ν, m) then there exists an > 0 such that Ent(ν | m) + < This together with (6.3) would imply
Ent(ηt | m) <
(6.3) K +h 2 2 dW (ν, m) .
K 2 (1 − t)2 d+h W (ν, m) − (1 − t) 2
for each t ∈ [0, 1]. We choose now t very close to 1, such that 0 < 1 − t < and 2 2 2 K(1 − t)2 d+h W (ν, m) < . This entails Ent(ηt | m) < − /2 < 0, in contradiction with the 2 fact that the entropy functional is nonnegative. Therefore Ent(ν | m) K2 d+h W (ν, m) , which is precisely our claim. 2 A Talagrand inequality for the measure m implies a concentration of measure inequality for m (see for instance [9]). For a given Borel set A ⊂ M denote the (open) r-neighborhood of A by Br (A) := {x ∈ M: d(x, A) < r} for r > 0. The concentration function of (M, d, m) is defined as
1 , r > 0. α(M,d,m) (r) := sup 1 − m Br (A) : A ∈ B(M), m(A) 2 We refer to [7] for further details on measure concentration. The following result shows that positive rough curvature bound implies a normal concentration inequality, via h-Talagrand inequality. Proposition 6.2. Let (M, d, m) be a metric measure space with h-Curv(M, d, m) K > 0 for some h > 0. Then there exists an r0 > 0 such that for all r r0 α(M,d,m) (r) e−Kr
2 /8
.
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2963
Proof. We follow essentially the argument of K. Marton used in [9] for obtaining concentration of measure out of a Talagrand inequality for the Wasserstein distance of order 1. Let A, B ∈ B(M) be given with m(A), m(B) > 0. Consider the conditional probabilities mA = m(· | A) and mB = m(· | B). For these measures the h-Talagrand inequality holds: d+h W (mA , m)
2 Ent(mA | m) , K
d+h W (mB , m)
2 Ent(mB | m) . K
(6.4)
Let qA and qB be the +h-optimal couplings of mA , m and mB , m respectively. According to [2], section 11.8, there exists a probability measure qˆ on M × M × M such that its projection on the first two factors is qA and the projection on the last two factors is qB . Then we have in turn +h d+h W (mA , m) + dW (m, mB )
2
d(x1 , x2 ) − h +
=
1/2 d q(x ˆ 1 , x2 , x2 )
M×M×M
+
2
d(x2 , x3 ) − h +
1/2 d q(x ˆ 1 , x2 , x2 )
M×M×M
1/2 2 d(x1 , x2 ) − h + + (d(x2 , x3 ) − h)+ d q(x ˆ 1 , x2 , x2 )
M×M×M
1/2 2 d(x1 , x2 ) + d(x2 , x3 ) − 2h + d q(x ˆ 1 , x2 , x2 )
M×M×M
2
d(x1 , x3 ) − 2h +
1/2 d q(x ˆ 1 , x2 , x2 ) .
M×M×M
Assume now that d(A, B) 2h. Since the projection on the first factor of qˆ is mA and the projection on the last factor is mB , the support of qˆ must be a subset of A × M × B, hence
1/2 2 d(x1 , x3 ) − 2h + d q(x ˆ 1 , x2 , x2 ) d(A, B) − 2h.
M×M×M
The above estimates together with (6.4) imply 2 Ent(mA | m) 2 Ent(mB | m) + d(A, B) − 2h K K 1 1 2 2 log + log . = K m(A) K m(B)
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If we choose now 2h r and for a given A ∈ B(M) we replace B by Br (A), we get
1 2 log + K m(A)
r − 2h Hence, for m(A)
1 2 log . K 1 − m(Br (A))
1 2
r − 2h Therefore whenever r 2
2 K
2 log 2 + K
1 2 log . K 1 − m(Br (A))
log 2 + 4h for instance we have 1 2 log , K 1 − m(Br (A))
r 2 or equivalently
2 1 − m Br (A) e−Kr /8 , which ends the proof.
2
In [1] it has been shown that a Talagrand type inequality implies exponential integrability of the Lipshitz functions. We prove further that an h-Talagrand inequality leads to the same conclusion. Theorem 6.3. Assume that (M, d) is a metric space and let h > 0 be given. If m is a probability measure on (M, d) that satisfies an h-Talagrand inequality of constant K > 0 then all Lipschitz functions are exponentially integrable. More precisely, for any Lipschitz function ϕ with ϕLip 1 and ϕ dm = 0 we have
t2
etϕ dm e 2K +ht ,
(6.5)
2 t 2 e dm exp t ϕ dm exp ϕLip + htϕLip . 2K
(6.6)
∀t > 0, M
or equivalently, for any Lipschitz function ϕ ∀t > 0,
tϕ
M
M
Proof. The proof we present here extends the one given in [1]. Let f be a probability density with f log f integrable with respect to m. The h-Talagrand inequality implies !2 1 t + f log f dm f log f dm K 2K t
! d+h W (f m, m) "
M
M
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for each t > 0. We consider now the Wasserstein distance of order 1 of two probability measures μ and ν d1W (μ, ν) := inf d(x0 , x1 ) dq(x0 , x1 ), M×M
where q ranges over all couplings of μ and ν. If q is a +h-optimal coupling of f m and m then by the Cauchy–Schwartz inequality, d+h W (f m, m) =
1/2 2 d(x0 , x1 ) − h + d q (x0 , x1 )
M×M
d(x0 , x1 ) − h + d q (x0 , x1 ) d1W (f m, m) − h.
M×M
The Kantorovich–Rubinstein theorem gives the following duality formula d1W (f m, m) =
ϕf dm −
sup
ϕLip 1
M
ϕ dm .
M
If ϕ is a Lipschitz function that satisfies the assumptions of the theorem (ϕLip 1 and ϕ dm = 0) then ϕf
dm d+h W (f m, m) + h
1 t + 2K t
M
f log f dm + h, M
which can be written as t2 tϕ − f dm f log f dm + ht. 2K M
M
This estimate should take place for any probability density f . Therefore one can take t2
f = etϕ− 2K
−1 t2 etϕ− 2K dm
M
in formula (6.7) and obtain
−1 t2 t2 t2 tϕ − etϕ− 2K dm etϕ− 2K dm 2K M
t2
etϕ− 2K M
M
t2
etϕ− 2K M
−1
t2 t2 tϕ − − log dm etϕ− 2K dm dm + ht. 2K M
(6.7)
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This yields t2 tϕ− 2K log e dm dm ht, M
that proves the claim (6.5). The general estimate (6.6) is a consequence of (6.5) applied to the function ψ = ϕ1Lip [ϕ − ϕ dm]. 2 Remark 6.4. In the continuous case, by formal calculus, the following two assertions are equivalent (see [11] for the case of Riemannian manifolds): (i) The entropy functional Ent(· | m) is weakly K-convex on P2 (M, d), in the sense of inequality (2.1); (ii) The gradient flow Φ : R+ × P2 (M, d) → P2 (M, d) with respect to Ent(· | m) satisfies dW Φ(t, μ), Φ(t, ν) e−Kt dW (μ, ν) ∀μ, ν ∈ P2 (M, d), ∀t 0. (6.8) The rough notion of curvature bound that we have introduced in this paper is a discrete version of (2.1), whereas the approach presented in [10] is a discrete form of (6.8). Both imply e.g. measure concentration, although in general there is no real overlap, since in the discrete case there is no direct relation between Markov chains and entropy functionals. References [1] S. Bobkov, F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal. 163 (1999) 1–28. [2] R.M. Dudley, Real Analysis and Probability, The Wadsworth Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1989, reprinted by Cambridge University Press, 2002. [3] R. Forman, Bochner’s method for cell complexes and combinatorial Ricci curvature, Discrete Comput. Geom. 29 (2003) 323–374. [4] K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987) 517–547. [5] M. Gromov, Hyperbolic groups, in: Essays in Group Theory, in: Math. Sci. Res. Inst. Publ., vol. 8, Springer, 1987, pp. 75–263, New York. [6] Y. Higuchi, Combinatorial curvature for planar graphs, J. Graph Theory 38 (2001) 220–229. [7] M. Ledoux, The Concentration of Measure Phenomenon, Math. Surveys Monogr., vol. 89, Amer. Math. Soc., 2001. [8] J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press. [9] K. Marton, A measure concentration inequality for contracting Markov chains, Geom. Funct. Anal. 6 (1997) 556– 571. [10] Y. Ollivier, Ricci curvature of Markov chains on metric spaces, J. Funct. Anal. 256 (2009) 810–864. [11] M.K. von Renesse, K.T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math. 58 (2005) 923–940. [12] K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65–131. [13] M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal. 6 (1996) 587–600. [14] C. Villani, Topics in Mass Transportation, Grad. Stud. Math., Amer. Math. Soc., 2003. [15] C. Villani, Optimal Transport: Old and New, Grundlehren Math. Wiss., vol. 338, Springer, 2008.
Journal of Functional Analysis 256 (2009) 2967–3034 www.elsevier.com/locate/jfa
The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems Guangcun Lu 1 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China Received 7 August 2008; accepted 2 January 2009 Available online 26 January 2009 Communicated by H. Brezis Dedicated to Professor Yiming Long on his 60th birthday
Abstract In this paper, the Conley conjecture, which was recently proved by Franks and Handel [J. Franks, M. Handel, Periodic points of Hamiltonian surface diffeomorphism, Geom. Topol. 7 (2003) 713–756] (for surfaces of positive genus), Hingston [N. Hingston, Subharmonic solutions of Hamiltonian equations on tori, Ann. Math., in press] (for tori) and Ginzburg [V.L. Ginzburg, The Conley conjecture, arXiv: math.SG/0610956v1] (for closed symplectically aspherical manifolds), is proved for C 1 -Hamiltonian systems on the cotangent bundle of a C 3 -smooth compact manifold M without boundary, of a time 1-periodic C 2 -smooth Hamiltonian H : R × T ∗ M → R which is strongly convex and has quadratic growth on the fibers. Namely, we show that such a Hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations. If H also satisfies H (−t, q, −p) = H (t, q, p) for any (t, q, p) ∈ R×T ∗ M, it is shown that the time-1-map of the Hamiltonian system (if exists) has infinitely many periodic points siting in the zero section of T ∗ M. If M is C 5 -smooth and dim M > 1, H is of C 4 class and independent of time t, then for any τ > 0 the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of τ such that any one of them cannot be obtained from others by iterations or rotations. These results are obtained by proving similar results for the Lagrangian system of the Fenchel transform of H , L : R × T M → R, which is proved to be strongly convex and to have quadratic growth in the velocities yet.
E-mail address:
[email protected]. 1 Partially supported by the NNSF 10671017 of China, the Program for New Century Excellent Talents of the Education
Ministry of China and Research Fund for the Doctoral Program Higher Education of China (Grant No. 200800270003). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.001
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© 2009 Elsevier Inc. All rights reserved. Keywords: Conley conjecture; Hamiltonian and Lagrangian system; Cotangent and tangent bundle; Periodic solutions; Variational methods; Morse index; Maslov-type index
Contents 1. 2.
Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maslov-type indices and Morse index . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. A review on Maslov-type indices . . . . . . . . . . . . . . . . . . . . . . . 2.2. Relations between Maslov-type indices and Morse indices . . . . . . 3. Iteration inequalities of the Morse index . . . . . . . . . . . . . . . . . . . . . . . 3.1. The case of general periodic solutions . . . . . . . . . . . . . . . . . . . . 3.2. The case of even periodic solutions . . . . . . . . . . . . . . . . . . . . . . 4. Critical modules under iteration maps . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. .............................................. 4.2. .............................................. 4.3. .............................................. 5. Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Proof of (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Proof of (ii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Proof of Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Proof of Theorem 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Proof of (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Proof of (ii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Questions and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Proof of Proposition A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. An inequality for C 1 -simplex in C 1 Riemannian–Hilbert manifolds References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction and main results Recently, a remarkable progress in Symplectic geometry and Hamiltonian dynamics is that the Conley conjecture [9,33] was proved by Franks and Handel [13] (for surfaces of positive genus, also see [19] for generalizations to Hamiltonian homeomorphisms), Hingston [17] (for tori) and Ginzburg [16] (for closed symplectically aspherical manifolds). See [13,16,19] and references therein for a detailed history and related studies. In this paper we always assume that M is an n-dimensional, connected C 3 -smooth compact manifold without boundary without special statements. For a time 1-periodic C 2 -smooth Hamiltonian H : R × T ∗ M → R, let XH be the Hamiltonian vector field of H with respect to the standard symplectic structure on T ∗ M, ωcan := −dq ∧ dp in local coordinates (q, p) of T ∗ M, that is, ω(XH (t, q, p), ξ ) = −dH (t, q, p)(ξ ) ∀ξ ∈ T(q,p) T ∗ M. Unlike the case of compact symplectic manifolds we only consider subharmonic solutions of the Hamiltonian equations x(t) ˙ = XH t, x(t)
(1.1)
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for C 2 -smooth Hamiltonians H : R × T ∗ M → R satisfying the following conditions (H1)–(H3): (H1) H (t + 1, q, p) = H (t, q, p) for all (t, q, p) ∈ R × T ∗ M. In any local coordinates (q1 , . . . , qn ), there exist constants 0 < C1 < C2 , depending on the local coordinates, such that (H2) C1 |u|2
∂2H ij ∂pi ∂pj
(t, q, p)ui uj C2 |u|2 ∀u = (u1 , . . . , un ) ∈ Rn ,
∂2H
2
H (t, q, p)| C2 (1 + |p|2 ). (H3) | ∂qi ∂pj (t, q, p)| C2 (1 + |p|), | ∂q∂ i ∂q j
A class of important examples of such Hamiltonians are Physical Hamiltonian (including 1-periodic potential and electromagnetic forces in time) of the form 2 1 H (t, q, p) = p − A(t, q) + V (t, q). 2
(1.2)
For C r -smooth Hamiltonians H : R × T ∗ M → R satisfying the conditions (H1)–(H3), r 2, by the inequality in the left side of the condition (H2), we can use the inverse Legendre transform to get a fiber-preserving C r−1 -diffeomorphism LH : R/Z × T ∗ M → R/Z × T M,
(t, q, p) → t, q, Dp H (t, q, p) ,
(1.3)
and a C r -smooth function L : R × T M → R: L(t, q, v) = max p, v − H (t, q, p) = p(t, q, v), v − H t, q, p(t, q, v) , p∈Tq M
(1.4)
where p = p(t, q, v) is a unique point determined by the equality v = Dp H (t, q, p). (See [12, Prop. 2.1.6].) By (1.4) we have (L1) L(t + 1, q, v) = L(t, q, v) for all (t, q, v) ∈ R × T M. It is easily checked that the corresponding L with the physical Hamiltonian in (1.2) is given by 1 L(t, q, v) = v 2 + A(t, q), v − V (t, q). 2 In Appendix A we shall prove Proposition A. Under the condition (H1), (H2) is equivalent to the following (L2) plus the third inequality in (L3), and (H2) + (H3) ⇔ (L2) + (L3). In any local coordinates (q1 , . . . , qn ), there exist constants 0 < c < C, depending on the local coordinates, such that (L2) (L3)
∂2L 2 n ij ∂vi ∂vj (t, q, v)ui uj c|u| ∀u = (u1 , . . . , un ) ∈ R , 2 2 L L (t, q, v)| C(1 + |v|2 ), | ∂q∂i ∂v (t, q, v)| C(1 + |v|), | ∂q∂i ∂q j j
2
L and | ∂v∂i ∂v (t, q, v)| C. j
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(One can also write these two conditions in the free coordinates, see [2, §2].) So Proposition A shows that the conditions (L2)–(L3) have the same properties as (H2)–(H3). (Note: we do not claim that the condition (H2) (resp. (H3)) is equivalent to (L2) (resp. (L3)).) By (L2), the Legendre transform produces the inverse of LH , (t, q, v) → t, q, Dv L(t, q, v) ,
LL : R/Z × T M → R/Z × T ∗ M,
(1.5)
and H and L are related by: H (t, q, p) = p, v(t, q, p) − L t, q, v(t, q, p) , where v = v(t, q, p) is a unique point determined by the equality p = Dv L(t, q, v). In this case, it is well known that a curve R → T ∗ M, t → x(t) = (γ (t), γ ∗ (t)) is a solution of (1.1) if and only if γ ∗ (t) = Dv L(t, γ (t), γ˙ (t)) ∀t ∈ R and γ is a solution of the Lagrangian system on M: d dt
∂L ∂ q˙i
−
∂L =0 ∂qi
(1.6)
in any local coordinates (q1 , . . . , qn ). Hence we only need to study the existence of infinitely many distinct integer periodic solutions of the system (1.6) under the assumptions (L1)–(L3). To describe our results we introduce the following notations and notions. For any T > 0, each map in C(R/T Z, M) represents a homotopy class of free loops in M. As topological spaces C(R/T Z, M) and C(R/Z, M) are always homeomorphic. For a homotopy class α of free loops in M, denote by C(R/T Z, M; α) the subset of maps in C(R/T Z, M) representing α. For k ∈ N, if we view γ ∈ C(R/T Z, M; α) as a T -periodic map γ : R → M, it is also viewed as a kT -periodic map from R to M and thus yields an element of C(R/kT Z, M), called the k th iteration of γ and denoted by γ k . This γ k ∈ C(R/kT Z, M) represents a free homotopy class in M, denoted by α k . So γ k ∈ C(R/kT Z, M; α k ). Note also that topological spaces C(R/T Z, M; α) and C(R/Z, M; α) are always homeomorphic yet. For m ∈ N let C m (R/T Z, M) denote the subset of all C m -loops γ : R/T Z → M. A periodic map γ : R → M is called reversible (or even) if γ (−t) = γ (t) for any t ∈ R. Note that such a map is always contractible! For γ ∈ C(R/T Z, M) we define rotations of γ via s ∈ R as maps s · γ : R → M defined by s · γ (t) = γ (t + s) for t ∈ R. Then s · γ ∈ C(R/T Z, M) and (s · γ )m = s · γ m for any s ∈ R and m ∈ N. We call the set m γ m∈N
s∈R resp. s · γ m m∈N
a T -periodic map tower (resp. T -periodic orbit tower) based on γ (a T -periodic map from R to M). A T1 -periodic map tower {γ1m }m∈N (resp. T1 -periodic orbit tower {s · γ1m }s∈R m∈N ) based on a T1 -periodic map γ1 : R → M is called distinct with {γ m }m∈N (resp. {s · γ m }m∈N ) if there is no τ -periodic map β : R → M such that γ = β p and γ1 = β q for some p, q ∈ N (resp. γ = s · β p and γ1 = s · β q for some p, q ∈ N and s, s ∈ R). When γ is contractible as a map from R/T Z to M, we call the T -periodic map tower {γ m }m∈N (resp. T -periodic orbit tower {s · γ m }s∈R m∈N ) contractible. For τ ∈ N, if γ : R → M is a τ -periodic solution of (1.6), we call the set {γ m }m∈N a τ -periodic solution tower of (1.6) based on γ . Two periodic solution towers of (1.6) are said to
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be distinct if they are distinct as periodic map towers. Furthermore, if s · γ is also a τ -periodic solution of (1.6) for any s ∈ R (for example, in the case L is independent of t), we call {s · γ m }s∈R m∈N a τ -periodic solution orbit tower of (1.6). When two periodic solution orbit towers are distinct as periodic orbit towers we call them distinct periodic solution orbit towers of (1.6) based on γ . Clearly, the existence of infinitely many distinct integer periodic solution towers (resp. solution orbit towers) of (1.6) implies that there exists an infinite sequence of integer periodic solutions of (1.6) such that each of them cannot be obtained from others by iterations (resp. iterations or rotations). The following is the first main result of this paper. Theorem 1.1. Let M be a C 3 -smooth compact n-dimensional manifold without boundary, and let C 2 -smooth map L : R × T M → R satisfy the conditions (L1)–(L3). Then: (i) Suppose that for a homotopy class α of free loops in M and an abelian group K the singular homology groups Hr (C(R/Z, M; α k ); K) have nonzero ranks for some integer r n and all k ∈ N. Then either for some l ∈ N there exist infinitely many distinct l-periodic solutions of (1.6) representing α l , or there exist infinitely many positive integers l1 < l2 < · · · , such that for each i ∈ N the system (1.6) has a periodic solution with minimal period li and representing α li . (ii) Suppose that for some abelian group K and integer r n the singular homology groups Hr (C(R/Z, M); K) have nonzero ranks. Then either for some l ∈ N there exist infinitely many distinct l-periodic solutions of (1.6), or there exist infinitely many positive integers l1 < l2 < · · · , such that for each i ∈ N the system (1.6) has a periodic solution with minimal period li . Let 0 denote the free homotopy class of contractible loops in M, i.e., C(R/Z, M; 0) consists of all contractible loops γ : R/Z → M. The obvious inclusion ı : M → C(R/Z, M; 0) and the evaluation EV : C(R/Z, M; 0) → M,
γ → γ (0)
satisfy EV ◦ ı = idM . It easily follows that ı∗ : Hk (M; Z2 ) → Hk C(R/Z, M; 0); Z2 is injective for any k ∈ N. Since Hn (M, Z2 ) = Z2 for n = dim M, we get rank Hn C(R/Z, M; 0); Z2 = 0.
(1.7)
Corollary 1.2. Let M be a C 3 -smooth compact n-dimensional manifold without boundary, and let C 2 -smooth map L : R × T M → R satisfy the conditions (L1)–(L3). Then the system (1.6) possesses infinitely many distinct contractible integer periodic solution towers. Remark 1.3. (1◦ ) When M has finite fundamental group, Benci [4] first proved that the system (1.6) has infinitely many distinct contractible 1-periodic solutions for C 2 -smooth Lagrangian L satisfying the conditions (L1)–(L3) and ∂L ∂L 2 ∂v L(t, q, v) C 1 + |v| ∂q (t, q, v) C 1 + |v| , i i
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in some local coordinates (q1 , . . . , qn ) for some constant C > 0. Recently, under weaker assumptions than (L1)–(L3), i.e. Tonelli conditions and (L5) below, Abbondandolo and Figalli [1, Cor. 3.2] showed that the system (1.6) has an infinite sequence of 1-periodic contractible solutions with diverging action and diverging Morse index. The key in [1,4] is the fact that the space of free loops in a compact simply connected manifold has infinitely many nonzero (co)homology groups with real coefficients [34]. A new technique in [1] is to modify their Tonelli Lagrangian L to one satisfying (L1)–(L3). (2◦ ) On n-dimensional torus T n , for the Lagrangian of the form 1 L(t, q, v) = gq (v, v) + U (t, q) 2
(1.8)
for all (t, q, v) ∈ R×T T n = R×T n ×Rn , where g is a C 3 -smooth Riemannian metric on T n and U ∈ C 3 (R/Z × T n , R) (such an L satisfies the conditions (L1)–(L3)), Yiming Long [23] proved that the system (1.6) possesses infinitely many distinct contractible integer periodic solution towers. We refer the reader to [23] and the references given there for the detailed history on the integer periodic solutions of the Lagrangian system. If L : R × T M → R also satisfies (L4) L(−t, q, −v) = L(t, q, v) for any (t, q, v) ∈ R × T M, we can improve Corollary 1.2 as follows. Theorem 1.4. Let M be a C 3 -smooth compact n-dimensional manifold without boundary, and let C 2 -smooth map L : R × T M → R satisfy the conditions (L1)–(L4). Then the system (1.6) possesses infinitely many distinct contractible integer periodic solution towers based on reversible periodic solutions. This result was proved by the author and Mingyan Wang [30] in the case that M = T n and that L has the form (1.8) and satisfies (L4), i.e. U (−t, q) = U (t, q) for any (t, q) ∈ R × T n . In particular, we have a generalization of [30, Th. 1.6]. Corollary 1.5. If L ∈ C 2 (T M, R) satisfies (L2)–(L4), then for any real number τ > 0, the following three claims have at least one to be true: • L has infinitely many critical points sitting in M = 0T M and thus the system (1.6) possesses infinitely many different constant solutions in M; • there exists some positive integer k such that the system (1.6) possesses infinitely many different nonconstant kτ -periodic solution orbit towers based on reversible periodic solutions of (1.6); • there exist infinitely many positive integers k1 < k2 < · · · , such that for each km the system (1.6) possesses a reversible periodic solution with minimal period km τ , m = 1, 2, . . . . When M = T n and L has the form (1.8) with real analytic g and nonconstant, autonomous and real analytic U , the author and Mingyan Wang [29] observed that suitably improving the
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arguments in [6] can give a simple proof of Corollary 1.5. It should also be noted that even if M is simply connected the methods in [1,4] cannot produce infinitely many reversible integer periodic solutions because the space of reversible loops in M can contract to the zero section of T M and therefore has no infinitely many nonzero Betti numbers. If L ∈ C 2 (T M, R) only satisfies (L2)–(L3), it is possible that two distinct solutions γ1 and γ2 obtained by Theorem 1.1 only differ by a rotation, i.e., γ1 (t) = γ2 (s + t) for some s ∈ R and any t ∈ R. However, we can combine the proof of Theorem 1.1 with the method in [26] to improve the results in Theorem 1.1 as follows: Theorem 1.6. Let M be a C 5 -smooth compact n-dimensional manifold without boundary, and let C 4 -smooth map L : T M → R satisfy the conditions (L2)–(L3). Then for any τ > 0 the following results hold: (i) Suppose that for a homotopy class α of free loops in M and an abelian group K the singular homology groups Hr (C(R/Z, M; α k ); K) have nonzero ranks for some integer r n and all k ∈ N. If either r n + 1 or r = n > 1, then either for some l ∈ N there exist infinitely many distinct periodic solution orbit towers based on lτ -periodic solutions of (1.6) representing α l , or there exist infinitely many positive integers l1 < l2 < · · · , such that for each i ∈ N the system (1.6) has a periodic solution orbit tower based on a periodic solution with minimal period li τ and representing α li . (ii) Suppose that the singular homology groups Hr (C(R/Z, M); K) have nonzero ranks for some integer r n and some abelian group K. If either r n + 1 or r = n > 1, then either for some l ∈ N there exist infinitely many distinct periodic solution orbit towers based on lτ -periodic solutions of (1.6), or there exist infinitely many positive integers l1 < l2 < · · · , such that for each i ∈ N the system (1.6) has a periodic solution orbit tower based on a periodic solution with minimal period li τ . By (1.7) we immediately get: Corollary 1.7. Let M be a C 5 -smooth compact manifold of dimension n > 1 and without boundary, and let C 4 -smooth map L : T M → R satisfy the conditions (L2)–(L3). Then for any τ > 0 the system (1.6) possesses infinitely many distinct periodic solution orbit towers based on contractible periodic solutions of integer multiple periods of τ . Clearly, when (L4) is satisfied Corollary 1.5 seems to be stronger than Corollary 1.7. If n = 1 and (L4) is not satisfied, we do not know whether Corollary 1.7 is still true. Moreover, the reason that we require higher smoothness in Theorem 1.6 and Corollary 1.7 is to assure that the normal bundle of a nonconstant periodic orbit is C 2 -smooth. When M = T n and L has the form (1.8) with flat g and autonomous U , Yiming Long and the author [26] developed the equivariant version of the arguments in [23] to prove Corollary 1.7. Even if g is not flat, the author and Mingyan Wang [30, Th. 1.6] also derived a stronger result than Corollary 1.7 in the case that M = T n . Campos and Tarallo [6] obtained a similar result provided that the metric g is real analytic, and that the potential U is autonomous, real analytic and nonconstant. Even if L = 12 g for a C 4 -Riemannian metric g on M, it seems that Theorem 1.6 or Corollary 1.7 cannot yield infinitely many geometrically distinct closed geodesics.
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Assume that L also satisfies (L5) For any (q, v) ∈ T M there exists a unique solution of (1.6), γ : R → M, such that (γ (0), γ˙ (0)) = (q, v). By [1, §2], this assumption can be satisfied if −∂t L(t, q, v) c 1 + Dv L(t, q, v)[v] − L(t, q, v) ∀(t, q, v) ∈ R × T M.
(1.9)
(Clearly, the left side may be replaced by const − ∂t L(t, q, v) since (L5) is also satisfied up to adding a constant to L. Moreover, that L satisfies (L1)–(L3) is equivalent to that the Fenchel transform H of L given by (1.4) satisfies the assumptions (H1)–(H3) below. In this case (1.9) is equivalent to (1.11) below. Hence (1.9) holds if L is independent of t as noted below (1.11).) Under the assumption (L5), we have an one-parameter family of C 1 -diffeomorphisms ΦLt ∈ Diff(T M) satisfying ΦLt (γ (0), γ˙ (0)) = (γ (t), γ˙ (t)). (See [12, Th. 2.6.5].) Following [23], the time-1-map ΦL = ΦL1 is called the Poincaré map of the system (1.6) corresponding to the Lagrangian function L. Every integer periodic solution γ of (1.6) gives a periodic point (γ (0), γ˙ (0)) of ΦL . If γ is even, then the periodic point (γ (0), γ˙ (0)) sits in the zero section 0T M of T M. So Corollary 1.2 and Theorem 1.4 yield the following Corollary 1.8. Let M be a C 3 -smooth compact n-dimensional manifold without boundary, and let C 2 -smooth map L : R × T M → R satisfy the conditions (L1)–(L3) and (L5). Then the Poincaré map ΦL has infinitely many distinct periodic points. Furthermore, if (L4) is also satisfied then the Poincaré map ΦL has infinitely many distinct periodic points sitting in the zero section 0T M of T M. If L is independent of t, for a periodic point (γ (0), γ˙ (0)) of ΦL generated by a τ -periodic solution γ , then all points of {(γ (s), γ˙ (s)) | s ∈ R} are periodic points of ΦL . We call such period points orbitally same. By remarks below (1.9), using Corollary 1.7 we can improve Corollary 1.8 as follows: Corollary 1.9. Let M be a C 5 -smooth compact manifold of dimension n > 1 and without boundary, and let C 4 -smooth map L : T M → R satisfy the conditions (L2)–(L3). Then the Poincaré map ΦL has infinitely many orbitally distinct periodic points. It is easily checked that the assumption (L4) is equivalent to the following: (H4) H (−t, q, −p) = H (t, q, p) for any (t, q, p) ∈ R × T ∗ M. In this case, v = v(t, q, p) uniquely determined by the equality p = Dv L(t, q, v) satisfies v(−t, q, −p) = −v(t, q, p) ∀(t, q, p) ∈ R × T ∗ M.
(1.10)
So if a solution γ : R → M of (1.6) satisfies γ (−t) = γ (t) ∀t ∈ R, then γ ∗ (−t) = −γ ∗ (t) for all t ∈ R. With the same way as the definition of solution towers and solution orbit towers to (1.6) we can define solution towers to (1.1), and solution orbit towers to (1.1) in the case H is independent of t.
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Then the Hamiltonian versions from Theorem 1.1 to Corollary 1.7 can be obtained immediately. For example, from Corollary 1.2, Theorem 1.4 and Corollary 1.7 we directly derive: Theorem 1.10. (1◦ ) Let M be a C 3 -smooth compact n-dimensional manifold without boundary, and let C 2 -smooth map H : R × T ∗ M → R satisfy the conditions (H1)–(H3). Then the system (1.1) possesses infinitely many distinct contractible integer periodic solution towers. Furthermore, if (H4) is also satisfied then the system (1.1) possesses infinitely many distinct contractible integer periodic solution towers based on periodic solutions with reversible projections to M. (2◦ ) Let M be a C 5 -smooth compact manifold of dimension n > 1 and without boundary, and let C 4 -smooth map H : R × T ∗ M → R satisfy the conditions (H2)–(H3). Then for any τ > 0 the system (1.1) has infinitely many distinct periodic solution orbit towers based on contractible periodic solutions of integer multiple periods of τ . Remark 1.11. If π1 (M) is finite, Cieliebak [8] showed that the system (1.1) has infinitely many contractible 1-periodic solutions (with unbounded actions) provided that H ∈ C ∞ (R/Z × T ∗ M, R) satisfies ∂ ] − H (t, q, p) h0 p 2 − h1 , (HC1) dH (t, q, p)[p ∂p 2
2
H H (HC2) | ∂p∂ i ∂p (t, q, p)| d and | ∂p∂ i ∂q (t, q, p)| d, j j
for all (t, q, p) ∈ R × T ∗ M, with respect to a suitable metric on the bundle T ∗ M → M and constants h0 > 0, h1 and d. Here q1 , . . . , qn , p1 , . . . , pn are coordinates on T ∗ M induced by geodesic normal coordinates q1 , . . . , qn on M. Recently, Abbondandolo and Figalli stated in [1, Remark 7.4] that the same result can be derived from [1, Th. 7.3] if the assumptions (HC1)–(HC2) are replaced by ∂ ] − H (t, q, p) a(|p|q ) for some function a : [0, ∞) → R with (HAF1) dH (t, q, p)[p ∂p lims→+∞ a(s) = +∞, (HAF2) H (t, q, p) h(|p|q ) for some function h : [0, ∞) → R with lims→+∞ h(s) s = +∞ and all (t, q, p) ∈ R × T ∗ M,
and (H5) below. Note that no convexity assumption on H was made in [1,8] and therefore that their results cannot be obtained from one on Lagrangian system via the Legendre transform. It is easily seen that the assumption (L5) is equivalent to the following: ˙ = XH (t, x(t)), x : R → M, (H5) For any (q, p) ∈ T ∗ M there exists a unique solution of x(t) such that x(0) = (q, p). The assumption can be satisfied under the following equivalent condition of (1.9): ∂t H (t, q, p) c 1 + H (t, q, p)
∀(t, q, p) ∈ R × T ∗ M,
(1.11)
see [1, p. 629]. Since (H2) implies that H is superlinear on the fibers of T ∗ M, (1.11) holds clearly if H is independent of time t. The condition (H5) guarantees that the global flow of
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XH exists on T ∗ M. Thus we have an one-parameter family of Hamiltonian diffeomorphisms ΨtH ∈ Ham(T ∗ M, ωcan ) satisfying ΨtH (γ (0), γ˙ ∗ (0)) = (γ (t), γ˙ ∗ (t)). As usual, the time-1-map Ψ H = Ψ1H is called the Poincaré map of the system (1.1) corresponding to the Hamiltonian function H . For each t ∈ R recall that the Legendre transform associated with Lt (·) = L(t, ·) is given by LLt : T M → T ∗ M,
(q, v) → q, Dv L(t, q, v) .
It is easy to check that ΨtH ◦ LL0 = LLt ◦ ΦLt
for any t ∈ R.
(1.12)
From this one immediately gets the following equivalent Hamiltonian versions of Corollaries 1.8 and 1.9. Theorem 1.12. (1◦ ) Let M be a C 3 -smooth compact n-dimensional manifold without boundary, and let C 2 -smooth map H : R × T ∗ M → R satisfy the conditions (H1)–(H3) and (H5). Then the Poincaré map Ψ H has infinitely many distinct periodic points. Furthermore, if (H4) is also satisfied then the Poincaré map Ψ H has infinitely many distinct periodic points sitting in the zero section 0T ∗ M of T ∗ M. (2◦ ) Let M be a C 5 -smooth compact manifold of dimension n > 1 and without boundary, and let C 4 -smooth map H : R × T ∗ M → R satisfy the conditions (H2)–(H3). Then the Poincaré map Ψ H has infinitely many orbitally distinct periodic points. (That is, any two do not sit the same Hamiltonian orbit.) Theorems 1.10, 1.12 may be viewed as a solution for the Conley conjecture for Hamiltonian systems on cotangent bundles, and Corollaries 1.8 and 1.9 may be viewed as confirm answers of Lagrangian systems analogue of the Conley conjecture for Hamiltonian systems. The main proof ideas come from [23]. We shall prove Theorems 1.1, 1.6 in the case r = n, and Theorem 1.4 by generalizing the variational arguments in [23,26,30] respectively. Some new ideas are needed because we do not lift to the universal cover space of M as done in [23,26,30] for the tori case. We also avoid using finite energy homologies used in [23,26,30]. Let us outline the variational setup and new ideas as follows. For τ > 0, let Sτ := R/τ Z = [s]τ [s]τ = s + τ Z, s ∈ R ,
and Eτ = W 1,2 (Sτ , M)
denote the space of all loops γ : Sτ → M of Sobolev class W 1,2 . For a homotopy class α of free loops in M, let Hτ (α),
Hτ = Hτ (0),
EHτ
respectively denote the subset of loops of Eτ representing α, that of all contractible loops in Eτ , and that of all reversible loops in Eτ . Then EHτ ⊂ Hτ . For integer m 2, if M is C m -smooth, all these spaces Eτ , Hτ (α) and EHτ have m−1 -smooth Hilbert manifold structure [18], and the tangent space of Eτ at γ is Tγ Eτ = C W 1,2 (γ ∗ T M). Moreover, any (C m−1 ) Riemannian metric ·,· on M induces a complete Riemannian metric on Eτ :
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τ ξ, η τ =
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ξ(t), η(t) γ (t) + ∇t ξ(t), ∇t η(t) γ (t) dt
0
∀γ ∈ Eτ , ξ, η ∈ Tγ Eτ = W 1,2 (γ ∗ T M).
(1.13)
Here ∇t denotes the covariant √ derivative in direction γ˙ with respect to the Levi-Civita connection ∇ of ·,· . Let ξ τ = ξ, ξ τ ∀ξ ∈ Tγ Eτ . Then the distance on Eτ induced by · τ is complete and also compatible with the manifold topology on Eτ . Consider the functional Lτ : Eτ → R,
τ Lτ (γ ) =
L t, γ (t), γ˙ (t) dt
∀γ ∈ Eτ .
(1.14)
0
For integer m 3, if M is C m -smooth and C m−1 -smooth L : R × T M → R satisfies the assumptions (L1)–(L3), then the functional Lτ is C 2 -smooth, bounded below, satisfies the Palais–Smale condition, and all critical points of it have finite Morse indexes and nullities (see [1, Prop. 4.1, 4.2] and [4]). By [12, Th. 3.7.2], all critical points of Lτ are all of class C m−1 and therefore correspond to all τ -periodic solutions of (1.6). Let LE τ denote the restriction of Lτ on EHτ . When L satisfies (L4), it is not hard to prove that a map γ : R → M is a τ -periodic even solution to (1.6) if and only if γ is a critical point of LE τ on EHτ , cf. [30, Lem. 1.7]. When we attempt to prove Theorem 1.1 by the method of [23], we first need to know how to relate the Morse index and nullity of a critical point γ ∈ Eτ of Lτ to those of the kth iteration γ k ∈ Ekτ as a critical point of Lkτ on Ekτ . Since we do not assume that M is orientable or γ is contractible, the bundle γ ∗ T M → Sτ might not be trivial. However, for the 2nd iteration γ 2 , the pullback bundle (γ 2 )∗ T M → S2τ is always trivial. Since our proof is indirect by assuming that the conclusion does not hold, the arguments can be reduced to the case that all τ -periodic solutions have trivial pullback bundles (as above Lemma 5.2). For such periodic solutions we can choose suitably coordinate charts around them on Ekτ so that the question is reduced to the case M = Rn as in Lemma 3.2. Hence we can get expected iteration inequalities as in Theorem 3.1. The second new idea is that under the assumption each Lkτ has only isolated critical points we show in Lemma 5.2 how to use an elementary arguments as above Corollary 1.2 and the Morse theory to get a non-minimal saddle point with nonzero nth critical module with Z2 -coefficient; the original method in [23, Lemma 4.1] is to use Lemma II.5.2 on the page 127 of [7] to arrive at this goal, which seems to be difficult for me generalizing it to manifolds. It is worth noting that we avoid using finite energy homologies used in [23,26,30]. That is based on an observation, that is, the composition (jkτ )∗ ◦ ψ∗k in (5.13) has a good decomposition (Jk )∗ ◦ (ψ k )∗ ◦ (I1 )∗ as in (5.15) such that for each ω ∈ Cn (Lτ , γ ; K), (I1 )∗ (ω) is a singular homology class of a C 1 -Hilbert manifold and hence has a C 1 -singular cycle representative. It is the final claim that allows us to use the singular homology to complete the remained arguments in Long’s method of [23]. A merit of this improvement is to reduce the smoothness of the Lagrangian L. That is, we only need to assume that L is of class C 2 . However, a new problem occurs, i.e. Θ˜ kτ in (4.12) is only a homeomorphism. It is very fortunate that α˜ kτ is also of class C 2 as noted at the end of proof of Theorem 5.1 (the generalized Morse lemma) on the page 44 of [7]. Using the image of Gromoll–Meyer of α˜ kτ (η) + β˜kτ (ξ ) under Θ˜ kτ , called topological Gromoll–Meyer, to replace a
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Gromoll–Meyer of L˜ kτ at γ˜ k , we construct topological Gromoll–Meyer pairs of Lτ at γ ∈ Hτ (α) and of Lkτ at γ k ∈ Hkτ (α k ), to satisfy Theorem 4.4 which is enough to complete our proof of Theorem 1.1. For the proof of Theorem 1.6 we need to complete more complex arguments as in Section 4.3. But the ideas are similar. The paper is organized as follows. Section 2 will review some basic facts concerning the Maslov-type indices and relations between them and Morse indexes. In Section 3 we give some iteration inequalities of the Morse indexes. Section 4 studies changes of the critical modules under iteration maps. In Sections 5–7, we give the proofs of Theorems 1.1, 1.4 and 1.6 respectively. Motivated by the second claim in Theorem 1.10(1◦ ), a more general question than the Conley’s conjecture and a program in progress are proposed in Section 8. In Appendix A we prove Proposition A and a key Lemma A.4, which is a generalization of [23, Lemma 2.3]. 2. Maslov-type indices and Morse index 2.1. A review on Maslov-type indices Let Sp(2n, R) = {M ∈ R2n×2n | M T J0 M = J0 }, where J0 =
0 −In In 0 .
For τ > 0, denoted by
Pτ (2n) = Ψ ∈ C [0, τ ], Sp(2n, R) Ψ (0) = I2n , Pτ∗ (2n) = Ψ ∈ Pτ (2n) det Ψ (τ ) − I2n = 0 . The paths in Pτ∗ (2n) are called nondegenerate. The Maslov-type index (or Conley–Zehnder index) theory for the paths in Pτ∗ (2n) was defined by [10,22,27,36]. Yiming Long [25] extended this theory to all paths in Pτ (2n). The Maslov-type index of a path Ψ ∈ Pτ (2n) is a pair of integers (iτ (Ψ ), ντ (Ψ )), where ντ (Ψ ) = dimR KerR Ψ (τ ) − I2n and iτ (Ψ ) = inf iτ (β) β ∈ Pτ∗ (2n) is sufficiently C 0 close to Ψ in Pτ (2n) with iτ (β) defined as in [10]. Clearly, the map iτ : Pτ (2n) → Z is lower semi-continuous. For any paths Ψk ∈ Pτ (2n), k = 0, 1, (iτ (Ψ0 ), ντ (Ψ0 )) = (iτ (Ψ1 ), ντ (Ψ1 )) if and only if there exists a homotopy Ψs , 0 s 1 from Ψ0 to Ψ1 in Pτ (2n) such that Ψs (0) = I2n and ντ (Ψs (τ )) ≡ ντ (Ψ0 ) for any s ∈ [0, 1]. For a < b and any path Ψ ∈ C([a, b], Sp(2n, R)), choose β ∈ P1 (2n) with β(1) = Ψ (a), and define φ ∈ P1 (2n) by φ(t) = β(2t) for 0 t 1/2, and φ(t) = Ψ a + (2t − 1)(b − a) for 1/2 t 1. It was showed in [25] that the difference i1 (φ) − i1 (β) only depends on Ψ , and was called the Maslov-type index of Ψ , denoted by (2.1) i Ψ, [a, b] := i1 (φ) − i1 (β). Clearly, i(Ψ, [0, 1]) = i1 (Ψ ) for any Ψ ∈ P1 (2n). Let (F, {·,·}) be the symplectic space with F = R2n ⊕ R2n and
{u, v} = J u, v ∀u, v ∈ F,
where J =
−J0 0
0 J0
.
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All vectors are understand as column vectors in this paper without special statements. Let Lag(F ) be the manifold of Lagrangian Grassmannian of (F, {·,·}), and μCLM be the Cappell– Lee–Miller index characterized by properties I–VI of [5, pp. 127–128]. There exists the following relation between μCLM and the index defined by (2.1), W, Gr(Ψ ), [a, b] − n, i Ψ, [a, b] = μCLM F
(2.2)
where W = {(x T , x T )T ∈ R4n | x ∈ R2n }. With U1 = {0} × Rn and U2 = Rn × {0}, two new Maslov-type indices for any path Ψ ∈ C([a, b], Sp(2n, R)) were defined in [28] as follows: U , Ψ U , [a, b] , μk Ψ, [a, b] = μCLM k k 2n R Let Ψ (b) =
A B CD
k = 1, 2.
(2.3)
, where A, B, C, D ∈ Rn×n . In terms of [28, (2.21)], define
ν1 Ψ, [a, b] = dim Ker(B)
and ν2 Ψ, [a, b] = dim Ker(C).
(2.4)
In particular, for Ψ ∈ Pτ (2n) and k = 1, 2 we denote by
μk,τ (Ψ ) = μk
τ τ Ψ, 0, and νk,τ (Ψ ) = νk Ψ, 0, . 2 2
(2.5)
Assumption B. (B1) Let B ∈ C(R, R2n×2n ) be a path of symmetric matrix which is τ -periodic in time t, i.e., B(t + τ ) = B(t) for any t ∈ R. 11 (t) B12 (t) n×n are even at t = 0 and τ/2, and (B2) Let B(t) = B B21 (t) B22 (t) , where B11 , B22 , t → R n×n are odd at t = 0 and τ/2. B12 , B21 , t → R Under the assumption (B1), let Ψ be the fundamental solution of the problem Ψ˙ (t) = J0 B(t)Ψ (t),
Ψ (0) = I2n .
(2.6)
By the classical Floquet theory, ντ (Ψ ) is the dimension of the solution space of the linear Hamiltonian system ˙ = J0 B(t)u(t) u(t)
and u(t + τ ) = u(t).
Similarly, under the assumptions (B1) and (B2), it was also shown in [28, Prop. 1.3] that ν1,τ (Ψ ) and ν2,τ (Ψ ) are the dimensions of the solution spaces of the following two problems respectively,
˙ = J0 B(t)u(t), u(t) u(t + τ ) = u(t),
u(−t) = N u(t),
˙ = J0 B(t)u(t), u(t) u(t + τ ) = u(t),
u(−t) = −N u(t),
where N = −I0 n I0n . Let (x1 , . . . , xn , y1 , . . . , yn ) denote the coordinates in R2n = Rn × Rn . De note by ω0 = nk=1 dxk ∧ dyk the standard symplectic structure on R2n , i.e. ω0 (u, v) = J0 u, v
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∀u, v ∈ R2n . Here ·,· is the standard inner product on R2n . Define H : R × R2n → R by H (t, u) = 12 B(t)u, u . Let XH be the corresponding Hamiltonian vector field defined by ω0 XH (t, u), v = −du H (t, u)(v).
(2.7)
Then XH (t, u) = J0 B(t)u for any u ∈ R2n . For Ψ ∈ Pτ (2n), extend the definition of Ψ to [0, +∞) by Ψ (t) = Ψ (t − j τ )Ψ (τ )j ,
∀j τ t (j + 1)τ, j ∈ N,
(2.8)
and define the mth iteration Ψ m of Ψ by Ψ m = Ψ |[0,mτ ] .
(2.9)
It was proved in [24, pp. 177–178] that the mean index per τ of Ψ ∈ Pτ (2n), imτ (Ψ m ) iˆτ (Ψ ) := lim m→+∞ m
(2.10)
always exists. Lemma 2.1. (i) For any Ψ ∈ Pτ (2n) it holds that max 0, miˆτ (Ψ ) − n imτ Ψ m miˆτ (Ψ ) + n − νmτ Ψ m ,
∀m ∈ N.
(ii) |μ1 (Ψ ) − μ2 (Ψ )| n for any Ψ ∈ Pτ (2n) with τ > 0. (iii) Under Assumption B, let Ψ : [0, +∞) → Sp(2n, R) be the fundamental solution of the problem (2.6). (It must satisfy (2.8).) Then μ1,mτ (Ψ |[0, mτ2 ] ) + μ2,mτ (Ψ |[0, mτ2 ] ) = imτ (Ψ |[0,mτ ] ) + n ∀m ∈ N
(2.11)
(or equivalently μ1 (Ψ, [0, mτ ]) + μ2 (Ψ, [0, mτ ]) = imτ (Ψ |[0,mτ ] ) + n ∀m ∈ N). Moreover, for k = 1, 2 the mean indices of Ψ per τ defined by μˆ k,τ (Ψ ) := lim
m→+∞
μk,mτ (Ψ |[0,mτ ] ) m
(2.12)
always exist and equal to 12 iˆτ (Ψ ). (i) comes from [20] or [24, p. 213, (17)], (ii) is [28, Th. 3.3], and (iii) is [28, Prop. C, Cor. 6.2] (precisely is derived from the proof of [28, Prop. C, Cor. 6.2]). It is easily checked that (i) implies |imτ − miτ | (m + 1)n for any m ∈ N. A similar inequality to the latter was also derived in [11, (12)] recently.
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2.2. Relations between Maslov-type indices and Morse indices Lemma 2.2. (See [21,35].). Let the Lagrangian L : R × R2n → R be given by 1 1 L(t, y, v) = P (t)v · v + Q(t)y · v + R(t)y · y, 2 2 where P , Q, R : R → Rn×n are C 1 -smooth and τ -periodic, R(t) = R(t)T , and each P (t) = P (t)T is also positive definite. The corresponding Lagrangian system is d dt
∂L ∂L (t, y, y) ˙ − (t, y, y) ˙ = (P y˙ + Qy)· − QT y˙ − Ry = 0. ∂v ∂y
(2.13)
Let y˜ be a critical point of the functional
τ fτ (y) =
L t, y(t), y(t) ˙ dt
0
on W 1,2 (Sτ , Rn ), and the second differential of fτ at it be given by
τ d fτ (y)(y, ˜ z) = 2
(P y˙ + Qy) · z˙ + QT y˙ · z + Ry · z dt.
0
The linearized system of (2.13) at y˜ is the Sturm system: −(P y˙ + Qy)· + QT y˙ + Ry = 0. Let
S(t) =
P (t)−1 −Q(t)T P (t)−1
−P (t)−1 Q(t) , Q(t)T P (t)−1 Q(t) − R(t)
(2.14)
and Ψ : [0, +∞) → Sp(2n, R) be the fundamental solution of the problem ˙ = J0 S(t)u u(t)
(2.15)
with Ψ (0) = I2n . Suppose that each P (t) is symmetric positive definite, and that each R(t) is symmetric. Then fτ at y˜ ∈ W 1,2 (Sτ , Rn ) has finite Morse index mτ (fτ , y) ˜ and nullity m0τ (fτ , y), ˜ and m− ˜ = iτ (Ψ ) τ (fτ , y)
and m0τ (fτ , y) ˜ = ντ (Ψ ).
(2.16)
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Remark 2.3. Since Lvv (t, y, v) = P (t) is invertible for every t, L has the Legendre transform H : R × R2n → R: H (t, x, y) = x · v(t, x, y) − L t, x, v(t, x, y) , where v(t, x, y) ∈ Rn is determined by Lv (t, y, v(t, x, y)) = x. Precisely, v(t, x, y) = P (t)−1 [x − Q(t)y] and 1 1 1 H (t, x, y) = P (t)−1 x · x − P (t)−1 x · Q(t)y + P (t)−1 Q(t)y · Q(t)y − R(t)y · y. 2 2 2 Then XH (t, x, y) = J0 S(t)u with u = (x T , y T )T , and u˜ = (x˜ T , y˜ T )T is a τ -periodic solution of (2.15). Let EW 1,2 Sτ , Rn = y ∈ W 1,2 Sτ , Rn y(−t) = y(t) ∀t ∈ R , OW 1,2 Sτ , Rn = y ∈ W 1,2 Sτ , Rn y(−t) = −y(t) ∀t ∈ R . Lemma 2.4. (See [30, Th. 3.4].) Under the assumptions of Lemma 2.2, suppose furthermore that ⎧ ⎨ P (t + τ ) = P (t) = P (t)T = P (−t) ∀t ∈ R, R(t + τ ) = R(t) = R(t)T = R(−t) ∀t ∈ R, ⎩ Q(t + τ ) = Q(t) = −Q(−t) ∀t ∈ R,
(2.17)
and thus L in Lemma 2.2 satisfies (L4). So the present S(t) in (2.14) also satisfies Assumption B. Let y˜ be a critical point of the restriction fτE of the functional fτ to EW 1,2 (Sτ , Rn ). (It is also a critical point of the functional fτ on W 1,2 (Sτ , Rn ) because fτ is even.) As in Lemma 2.1, let Ψ denote the fundamental solution of (2.15). Let + 0 − EW 1,2 Sτ , Rn = EW 1,2 Sτ , Rn ⊕ EW 1,2 Sτ , Rn ⊕ EW 1,2 Sτ , Rn , + 0 − OW 1,2 Sτ , Rn = OW 1,2 Sτ , Rn ⊕ OW 1,2 Sτ , Rn ⊕ OW 1,2 Sτ , Rn be respectively d 2 fτ (y)-orthogonal ˜ decompositions according to d 2 fτ (y) ˜ being positive, null, and negative definite. Then − E dim EW 1,2 Sτ , Rn = m− τ fτ , y˜ = μ1,τ (Ψ ), 0 dim EW 1,2 Sτ , Rn = m0τ fτE , y˜ = ν1,τ (Ψ ), − dim OW 1,2 Sτ , Rn = μ2,τ (Ψ ) − n,
(2.20)
ντ (Ψ ) = ν1,τ (Ψ ) + ν2,τ (Ψ ).
(2.21)
(2.18) (2.19)
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For conveniences we denote by − ˜ := dim OW 1,2 Sτ , Rn , m− 2,τ (fτ , y) 0 ˜ := dim OW 1,2 Sτ , Rn . m02,τ (fτ , y)
(2.22) (2.23)
Then under the assumptions of Lemma 2.4, Lemma 2.1(ii), (iii) and (2.21) become E n + m− (fτ , y) ˜ − m− τ fτ , y˜ n, 2,τ E − m− ˜ + m− ˜ τ fτ , y˜ = mτ (fτ , y), 2,τ (fτ , y) ˜ = m0τ fτE , y˜ + m02,τ (fτ , y). ˜ m0τ (fτ , y)
(2.24) (2.25) (2.26)
3. Iteration inequalities of the Morse index 3.1. The case of general periodic solutions In this subsection we always assume: M is C 3 -smooth, L is C 2 -smooth and satisfies (L1)–(L3). Let γ ∈ Eτ be a critical point of the functional Lτ on Eτ . It is a τ -periodic map from R to M. For each k ∈ N, γ : R → M is also kτ -periodic map and therefore determines an element in Ekτ , denoted by γ k for the sake of clearness. It is not difficult to see that γ k is a critical point of Lkτ on Ekτ . Let k m− kτ γ
and m0kτ γ k
denote the Morse index and nullity of Lkτ on Ekτ respectively. Note that 0 m0kτ γ k 2n ∀k ∈ N. (This can be derived from (2.16) and Lemma 3.2 below.) A natural question is how to estimate 0 0 k m− (γ k ) in terms of m− τ (γ ), mτ (γ ) and mkτ (γ ). The following theorem gives an answer. Theorem 3.1. For a critical point γ of Lτ on Eτ , assume that γ ∗ T M → Sτ is trivial. Then the mean Morse index k m− kτ (γ ) k→∞ k
m ˆ− τ (γ ) := lim
(3.1)
always exists, and it holds that − k max 0, k m ˆ− km ˆ τ (γ ) + n − m0kτ γ k ∀k ∈ N. τ (γ ) − n mkτ γ
(3.2)
2 ˆ− Consequently, for any critical point γ of Lτ on Eτ , m 2τ (γ ) exists and
2 2k − n m− km ˆ 2τ (γ ) + n − m02kτ γ 2k ∀k ∈ N max 0, k m ˆ− 2τ γ 2kτ γ because (γ 2 )∗ T M → S2τ is always trivial.
(3.3)
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Before proving this result it should be noted that the following special case is a direct consequence of Lemmas 2.1(i) and 2.2. Lemma 3.2. Under the assumptions of Lemma 2.2, for each k ∈ N, y˜ is also a kτ -periodic solution of (2.13), denoted by y˜ k . Then y˜ k is a critical point of the functional
kτ fkτ (y) =
L t, y(t), y(t) ˙ dt
0
on W 1,2 (Skτ , Rn ), and k m− ikτ (Ψ k ) ˆ kτ (fkτ , y˜ ) = lim = iτ (Ψ ), k→+∞ k→+∞ k k k ˆ− ˜ − n m− ˜ + n − m0kτ fkτ , y˜ k max 0, k m ˆ− τ (fτ , y) τ (fτ , y) kτ fkτ , y˜ k m
m ˆ− ˜ := lim τ (fτ , y)
(3.4) (3.5)
with 0 m0τ (fkτ , y˜ k ) 2n for any k ∈ N. This result was actually used in [23,26,30]. In the following we shall show that Theorem 3.1 can be reduced to the special case. Proof of Theorem 3.1. Step 1. Reduce to the case M = Rn . Let γ ∈ Eτ be a critical point γ of Lτ on Eτ with trivial pullback γ ∗ T M → Sτ . Take a C 2 -smooth loop γ0 : Sτ → M such that maxt d(γ (t), γ0 (t)) < ρ, where d and ρ are the distance and injectivity radius of M with respect to some chosen Riemannian metric on M respectively. (Actually we can choose γ0 = γ because γ0 is C 2 -smooth under the assumptions of this subsection.) Clearly, γ and γ0 are homotopic, and thus γ0∗ T M → Sτ is trivial too. Since γ0 is C 2 -smooth, we can choose a C 2 -smooth orthogonal trivialization Sτ × Rn → γ0∗ T M,
(t, q) → Φ(t)q.
(3.6)
It naturally leads to a smooth orthogonal trivialization of (γ0k )∗ T M for any k ∈ N, ∗ Skτ × Rn → γ0k T M,
(t, q) → Φ(t)q.
(3.7)
Let Bρn (0) denote an open ball in Rn centered at 0 with radius ρ. Then for each k ∈ N, we have a coordinate chart on Ekτ containing γ k , φkτ : W 1,2 Skτ , Bρn (0) → Ekτ ,
˜ . φkτ (α)(t) ˜ = expγ k (t) Φ(t)α(t) 0
(3.8)
Clearly, φkτ (α) ˜ has a period τ if and only if α˜ is actually τ -periodic. Thus we have a unique γ˜ ∈ W 1,2 (Sτ , Bρn (0)) such that φkτ (γ˜ k ) = γ k for any k ∈ N. Denote by the iteration maps
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ψ k : Eτ → Ekτ ,
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α → α k ,
ξ → ξ k , ψ k : Tα Eτ → Tα k Ekτ , ψ˜ k : W 1,2 Sτ , Rn → W 1,2 Skτ , Rn ,
α˜ → α˜ k .
It is easy to see that φkτ ◦ ψ˜ k = ψ k ◦ φτ
∀k ∈ N.
(3.9)
For any k ∈ N, set L˜ kτ : W 1,2 Skτ , Bρn (0) → R,
L˜ kτ = Lkτ ◦ φkτ .
(3.10)
−1 k (γ ) = ψ˜ k (γ˜ ) is a critical Then γ˜ = φτ−1 (γ ) is a critical point of L˜ τ , and therefore γ˜ k = φkτ point of L˜ kτ for any k ∈ N. Moreover, the Morse indexes and nullities of these critical points satisfy the relations:
k k = m− m− kτ γ˜ kτ γ
and m0kτ γ˜ k = m0kτ γ k ,
∀k ∈ N.
(3.11)
Viewing γ0 a τ -periodic map from R → M, consider the C 2 -smooth map (t, q) ˜ → expγ0 (t) Φ(t)q˜ .
Ξ : R × Bρn (0) → M,
(3.12)
Then Ξ (t + τ, q) ˜ = Ξ (t, q) ˜ for any (t, q) ˜ ∈ R × M. Clearly, ˜ = Ξ t, α(t) ˜ φkτ (α)(t)
and
d d ˙˜ φkτ (α) ˜ q= ˜ (t) = Ξ (t, q)| + dq˜ Ξ t, α(t) ˜ α(t) ˜ α(t) ˜ dt dt
(3.13) (3.14)
for any t ∈ R and α˜ ∈ W 1,2 (Skτ , Bρn (0)). Define L˜ : R × Bρn (0) × Rn → R by
d ˜ q, Ξ (t, q) ˜ + dq˜ Ξ (t, q)( ˜ v) ˜ . L(t, ˜ v) ˜ = L t, Ξ (t, q), ˜ dt
(3.15)
˜ + τ, q, ˜ q, Then L(t ˜ v) ˜ = L(t, ˜ v) ˜ ∀(t, q, ˜ v) ˜ ∈ R × Bρn (0) × Rn , and L˜ also satisfies the conditions (L2 )–(L3 ) (up to changing the constants). For α˜ ∈ W 1,2 (Skτ , Bρn (0)), by (3.10) we have L˜ kτ (α) ˜ = Lkτ φ k (α) ˜
kτ
d k φ (α) ˜ ˜ (t) dt = L t, φ k (α)(t), dt 0
kτ = 0
˙˜ L˜ t, α(t), ˜ α(t) dt.
(3.16)
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Therefore we may assume M = Rn . That is, by (3.11) we only need to prove k m− kτ (γ˜ ) exists, k→∞ k − k max 0, k m ˆ− km ˆ τ (γ˜ ) + n − m0kτ γ˜ k τ (γ˜ ) − n mkτ γ˜
m ˆ− τ (γ˜ ) := lim
(3.17) ∀k ∈ N.
(3.18)
Step 2. Reduce to the case of Lemma 3.2. Note that d L˜ τ (γ˜ )(ξ˜ ) =
τ
Dq˜ L˜ t, γ˜ (t), γ˙˜ (t) ξ˜ (t) + Dv˜ L˜ t, γ˜ (t), γ˙˜ (t) ξ˜ (t) dt
0
τ
=
d ˙ ˙ ˜ ˜ Dq˜ L t, γ˜ (t), γ˜ (t) − Dv˜ L t, γ˜ (t), γ˜ (t) · ξ˜ (t) dt dt
0
for any ξ˜ ∈ W 1,2 (Sτ , Rn ). Since d L˜ τ (γ˜ ) = 0, we have also d 2 L˜ τ (γ˜ )(ξ˜ , η) ˜ =
τ
˙˜ Dv˜ v˜ L˜ t, γ˜ (t), γ˙˜ (t) ξ˙˜ (t), η(t)
0
˙˜ + Dq˜ v˜ L˜ t, γ˜ (t), γ˙˜ (t) ξ˜ (t), η(t) ˜ + Dv˜ q˜ L˜ t, γ˜ (t), γ˙˜ (t) ξ˙˜ (t), η(t) ˜ dt + Dq˜ q˜ L˜ t, γ˜ (t), γ˙˜ (t) ξ˜ (t), η(t)
for any ξ˜ , η˜ ∈ W 1,2 (Sτ , Rn ). Set ⎫ Pˆ (t) = Dv˜ v˜ L˜ t, γ˜ (t), γ˙˜ (t) , ⎬ ˆ = Dq˜ v˜ L˜ t, γ˜ (t), γ˙˜ (t) , Q(t) ⎭ ˆ = Dq˜ q˜ L˜ t, γ˜ (t), γ˙˜ (t) R(t)
(3.19)
1 ˆ y˜ · v˜ + 1 R(t) ˆ y˜ · y. ˆ y, ˜ L(t, ˜ v) ˜ = Pˆ (t)v˜ · v˜ + Q(t) 2 2
(3.20)
and
Clearly, they satisfy the conditions of Lemma 2.2, and y˜ = 0 ∈ W 1,2 (Sτ , Rn ) is a critical point of the functional ˜ = fˆτ (y)
τ
˙˜ Lˆ t, y(t), ˜ y(t) dt
0
on W 1,2 (Sτ , Rn ). It is also easily checked that d 2 fˆτ (0)(ξ˜ , η) ˜ = d 2 L˜ τ (γ˜ )(ξ˜ , η) ˜
∀ξ˜ , η˜ ∈ W 1,2 Sτ , Rn .
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It follows that − k ˆ m− kτ (fkτ , 0) = mkτ γ˜
and m0kτ (fˆkτ , 0) = m0kτ γ˜ k ∀k ∈ N.
These and Lemma 3.2 together give the desired (3.17) and (3.18).
2
3.2. The case of even periodic solutions Let M and L be as in Section 3.1. But we also assume that L satisfies (L4). Note that the even periodic solutions are always contractible. Let LE kτ denote the restriction of Lkτ on EHkτ .
k As noted in the introduction, if γ ∈ EHτ is a critical point of LE τ on EHτ then γ is a critical point of Lkτ on Hkτ for each k ∈ N. Let
k m− 1,kτ γ
and m01,kτ γ k
0 k denote the Morse index and nullity of LE kτ on EHkτ respectively. Then 0 m1,kτ (γ ) 0 k mkτ (γ ) 2n for any k. We shall prove
Theorem 3.3. Let L satisfy the conditions (L1)–(L4). Then for any critical point γ of LE τ on EHτ , the mean Morse index m ˆ− 1,τ (γ ) := lim
k m− 1,kτ (γ )
k
k→∞
(3.21)
exists, and it holds that k m− + m01,kτ γ k n ∀k ∈ N if m ˆ− 1,kτ γ 1,τ (γ ) = 0.
(3.22)
Firstly, by (2.10) and (2.16) the mean Morse index k m− kτ (fkτ , y˜ ) k→∞ k
m ˆ− ˜ := lim τ (fτ , y)
(3.23)
exists and equals to iˆτ (Ψ ). Under the assumptions of Lemma 2.4, for each k ∈ N, y˜ k is a critical E of the functional f 1,2 (S , Rn ), and it follows from (2.12), point of the restriction fkτ kτ to EW kτ (2.18), (2.20) and (2.22) that E k E m− 1 − kτ (fkτ , y˜ ) = μˆ 1,τ (Ψ ) = m ˆ (fτ , y), f , y ˜ := lim ˜ m ˆ− τ τ k→+∞ k 2 τ
m ˆ− ˜ := lim 2,τ (fτ , y)
k→+∞
k m− 2,kτ (fkτ , y˜ )
k
1 − ˜ = μˆ 2,τ (Ψ ) = m ˆ (fτ , y). 2 τ
Moreover, by (2.25) and (2.26), for any k ∈ N it holds that − E − k k k m− 2,kτ fkτ , y˜ + mkτ fkτ , y˜ = mkτ fkτ , y˜ , E k , y˜ + m02,kτ fkτ , y˜ k . m0kτ fkτ , y˜ k = m0kτ fkτ
(3.24) (3.25)
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From these we derive that (3.5) becomes E − − E k k max 0, 2k m ˆ− τ fτ , y˜ − n m2,kτ fkτ , y˜ + mkτ fkτ , y˜ E E k 0 0 k 2k m ˆ− τ fτ , y˜ + n − mkτ fkτ , y˜ − m2,kτ fkτ , y˜
(3.26)
E ˜ = 0, then for any k ∈ N. In particular, if m ˆ− τ (fτ , y)
E k E k 0 m− kτ fkτ , y˜ + mkτ fkτ , y˜ n ∀k ∈ N
(3.27)
[30, Th. 3.7]. Proof of Theorem 3.3. Since γ is even we can still choose γ0 and Φ in (3.6) to be even, i.e. γ0 (−t) = γ0 (t) and Φ(−t) = Φ(t) for any t ∈ R. These imply d d d Ξ (−t, q) ˜ = − Ξ (s, q)| ˜ s=−t = Ξ (t, q). ˜ dt ds dt
Ξ (−t, q) ˜ = Ξ (t, q), ˜
(3.28)
It follows that the coordinate chart φkτ in (3.8) naturally restricts to a coordinate chart on EHkτ , E φkτ : EW 1,2 Skτ , Bρn (0) → EHkτ
(3.29)
which also satisfies E φkτ ◦ ψ˜ k = ψ k ◦ φτE
∀k ∈ N.
(3.30)
By (L4), (3.15) and (3.28) we have
˜ L(−t, q, ˜ −v) ˜ = L −t, Ξ (−t, q), ˜
d Ξ (−t, q) ˜ + dq˜ Ξ (−t, q)(− ˜ v) ˜ d(−t)
d ˜ − dq˜ Ξ (t, q)( ˜ v) ˜ = L −t, Ξ (t, q), ˜ − Ξ (−t, q) dt
d Ξ (−t, q) ˜ + dq˜ Ξ (t, q)( ˜ v) ˜ = L t, Ξ (t, q), ˜ dt
d Ξ (t, q) ˜ + dq˜ Ξ (t, q)( ˜ v) ˜ = L t, Ξ (t, q), ˜ dt
˜ q, = L(t, ˜ v). ˜
(3.31)
That is, L˜ also satisfies (L4). It follows that for any k ∈ N, the functional 1,2 L˜ E Skτ , Bρn (0) → R, kτ : EW
E E L˜ E kτ = Lkτ ◦ φkτ
(3.32)
is exactly the restriction of the functional L˜ kτ in (3.10) to EW 1,2 (Skτ , Bρn (0)). Hence the question is reduced to the case M = Rn again. That is, we only need to prove
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m ˆ− 1,τ (γ˜ ) := lim
k m− 1,kτ (γ˜ )
k k m− + m01,kτ γ˜ k n 1,kτ γ˜ k→∞
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exists,
(3.33)
∀k ∈ N if m ˆ− 1,τ (γ˜ ) = 0.
(3.34)
By (3.31) we have ˜ ˜ q, q, ˜ −v) ˜ = Dv˜ v˜ L(t, ˜ v), ˜ Dv˜ v˜ L(−t, ˜ ˜ q, Dq˜ v˜ L(−t, q, ˜ −v) ˜ = −Dq˜ v˜ L(t, ˜ v), ˜ ˜ ˜ q, q, ˜ −v) ˜ = Dq˜ q˜ L(t, ˜ v) ˜ Dq˜ q˜ L(−t, for any (t, q, ˜ v) ˜ ∈ R × Bρn (0) × Rn . Since γ˜ (−t) = γ˜ (t) and γ˙˜ (−t) = −γ˙˜ (t), it follows from this that Pˆ , Qˆ and Rˆ in (3.19) satisfy (2.17). For Lˆ in (3.20) and the functionals
E fˆkτ (y) ˜ :=
kτ
˙˜ Lˆ t, y(t), ˜ y(t) dt
0
on EW 1,2 (Skτ , Rn ), k = 1, 2, . . . , we have E − k ˆ m− kτ fkτ , 0 = m1,kτ γ˜
E and m0kτ fˆkτ , 0 = m01,kτ γ˜ k
∀k ∈ N.
(3.35)
By (3.24) and (3.27) we get ˆE E m− kτ (fkτ , 0) ˆ f , 0 := lim m ˆ− τ τ k→+∞ k
(3.36)
E E 0 ˆ ˆ m− kτ fkτ , 0 + mkτ fkτ , 0 n ∀k ∈ N.
(3.37)
ˆE exists, and if m ˆ− τ (fτ , 0) = 0,
Now (3.35)–(3.37) give (3.33) and (3.34), and therefore the desired (3.21) and (3.22).
2
4. Critical modules under iteration maps In this section we shall study relations of critical modules under iteration maps in three different cases. We first recall a few of notions. Let M be a C 2 Hilbert–Riemannian manifold and f ∈ C 1 (M, R) satisfies the Palais–Smale condition. Denote by K(f ) the set of critical points of f . Recall that a connected submanifold N of M is a critical submanifold of f if it is closed, consists entirely of critical points of f and f |N = constant. Let N ⊂ M be an isolated critical submanifold of f with f |N = c, and U be a neighborhood of N such that U ∩ K(f ) = N . For q ∈ N ∪ {0}, recall that the qth critical group with coefficient group K of f at N is defined by Cq (f, N; K) := Hq {f c} ∩ U, {f c} \ N ∩ U ; K .
(4.1)
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Hereafter H∗ (X, Y ; K) stands for the relative singular homology with the abelian coefficient group K without special statements. The group Cq (f, N; K) does not depend on a special choice of such neighborhoods U up to isomorphisms. There also exists another equivalent definition of critical groups, which is convenient in many situations. Let V : (M \ K(f )) → T M be a pseudo-gradient vector field for f on M. According to [7, pp. 48, 74] and [38, Def. 2.3] or [14], a pair of topological subspaces (W, W − ) of M is called a Gromoll–Meyer pair with respect to V for N , if (1) W is a closed neighborhood of N possessing the mean value property, i.e., ∀t1 < t2 , η(ti ) ∈ W , i = 1, 2, implies η(t) ∈ W for all t ∈ [t1 , t2 ], where η(t) is the decreasing flow with respect to V . And there exists > 0 such that W ∩ fc− = f −1 [c − , c) ∩ K(f ) = ∅, W ∩ K(f ) = N ; / W, ∀t > 0}; (2) the set W − = {p ∈ W | η(t, p) ∈ (3) W − is a piecewise submanifold, and the flow η is transversal to W − . By [7, p. 74] or [38, §2], there exists an (arbitrarily small) Gromoll–Meyer pair for N , (W, W − ), and for such a pair it holds that H∗ (W, W − ; K) ∼ = C∗ (f, N; K).
(4.2)
Hence H∗ (W, W − ; K) may be used to give an equivalent definition of C∗ (f, N; K). We need the following fact which seems to be obvious, but is often neglected. Lemma 4.1. Let M1 and M2 be C 2 Hilbert–Riemannian manifolds, and let Θ : M1 → M2 be a homeomorphism. Suppose that fi ∈ C 1 (Mi , R), i = 1, 2, satisfy the Palais–Smale condition and f1 = f2 ◦ Θ. Let N1 ⊂ M1 and N2 = Θ(N1 ) ⊂ M2 be isolated critical submanifolds of f1 and f2 respectively. Assume that (W1 , W1− ) is a Gromoll–Meyer pair of N1 of f1 . Then C∗ (f2 , N2 ; K) ∼ = H∗ Θ(W1 ), Θ W1− ; K though (Θ(W1 ), Θ(W1− )) is not necessarily a Gromoll–Meyer pair of N2 of f2 (because Θ is only a homeomorphism). Moreover, for c = f1 |N1 and > 0 it is clear that W1 , W1− ⊂ f1−1 [c − , c + ], f1−1 (c − ) implies (Θ(W1 ), Θ(W1− )) ⊂ (f2−1 [c − , c + ], f2−1 (c − )). Proof. Take a small open neighborhood U of N1 so that U ⊂ W1 . Since Θ({f1 c} ∩ U ) = {f2 c} ∩ U and Θ(({f1 c} \ N1 ) ∩ U ) = ({f2 c} \ N2 ) ∩ Θ(U ), we have isomorphisms Θ∗ : H∗ W1 , W1− ; K → H∗ Θ(W1 ), Θ W1− ; K , Θ∗ : H∗ {f1 c} ∩ U, {f1 c} \ N1 ∩ U ; K → H∗ {f2 c} ∩ Θ(U ), {f2 c} \ N2 ∩ Θ(U ); K = C∗ (f2 , N2 ; K). By (4.1) and (4.2), H∗ (W1 , W1− ; K) ∼ = H∗ ({f1 c} ∩ U, ({f1 c} \ N1 ) ∩ U ; K). The desired conclusion is obtained. 2
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It is this result that we may often treat (Θ(W1 ), Θ(W1− )) as a Gromoll–Meyer pair without special statements. For conveniences we call it a topological Gromoll–Meyer of f2 at N2 . The usual Gromoll–Meyer pair can be viewed the special case of it. Moreover, if Γ : M2 → M3 is a C 1 -diffeomorphism onto another C 2 Hilbert–Riemannian manifold M3 , then (Γ ◦ Θ(W1 ), Γ ◦ Θ(W1− )) is also a topological Gromoll–Meyer pair of f3 = f2 ◦ Γ −1 at N3 = Γ (N2 ). Eq. (4.2) and Lemma 4.1 show that the topological Gromoll–Meyer may be used to give an equivalent definition of the critical group. To understand the Note at the end of proof of Theorem 5.1 of [7, p. 44] we add a lemma, which is need in this paper. Lemma 4.2. Let Hi be Hilbert spaces with origins θi , i = 1, 2, 3. For ε > 0 let f ∈ C 2 (Bε (θ1 ) × Bε (θ2 ) × Bε (θ3 ), R). Assume that d3 f (x1 , θ2 , θ3 ) = 0 for x1 ∈ Bε (θ1 ) and that d32 f (θ1 , θ2 , θ3 ) : H3 → H3 is a Banach space isomorphism. Then there exist a small 0 < δ ε and C 1 -map h : Bδ (θ1 ) × Bδ (θ2 ) → H3 such that (i) d3 f (x1 , x2 , h(x1 , x2 )) = θ3 for all (x1 , x2 ) ∈ Bδ (θ1 ) × Bδ (θ2 ), (ii) g : Bδ (θ1 ) × Bδ (θ2 ) → R, (x1 , x2 ) → g(x1 , x2 ) = f (x1 , x2 , h(x1 , x2 )) is C 2 . Proof. Applying the implicit function theorem to the map d3 f : Bε (θ1 ) × Bε (θ2 ) × Bε (θ3 ) → H3 we get a 0 < δ ε and a C 1 -map h : Bδ (θ1 ) × Bδ (θ2 ) → H3 such that h(θ1 , θ2 ) = θ3 and d3 f x1 , x2 , h(x1 , x2 ) = 0
∀(x1 , x2 ) ∈ Bδ (θ1 ) × Bδ (θ2 ).
Set g(x1 , x2 ) = f (x1 , x2 , h(x1 , x2 )). Then dg(x1 , x2 ) = d(1,2) f x1 , x2 , h(x1 , x2 ) + d3 f x1 , x2 , h(x1 , x2 ) ◦ d(x1 ,x2 ) h(x1 , x2 ) = d(1,2) f x1 , x2 , h(x1 , x2 ) because d3 f (x1 , x2 , h(x1 , x2 )) = 0, where d(1,2) denotes the differential for the first two variables of f . Hence 2 f x1 , x2 , h(x1 , x2 ) + d3 d(1,2) f x1 , x2 , h(x1 , x2 ) ◦ d(x1 ,x2 ) h(x1 , x2 ). d 2 g(x1 , x2 ) = d(1,2) The desired claims are proved.
2
4.1. The arguments in this section are following Section 3 in [23]. However, since our arguments are on a Hilbert manifold, rather than Hilbert space, some new techniques are needed. The precise proofs are also given for reader’s convenience. In this subsection we always assume: M is C 3 -smooth, L is C 2 -smooth and satisfies (L1)–(L3). Lemma 4.3. Let γ ∈ Hτ (α) be an isolated critical point of the functional Lτ on Hτ (α) such that γ k is an isolated critical point of the functional Lkτ in Hkτ (α k ) for some k ∈ N. Suppose that
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γ ∗ T ∗ M → Sτ is trivial. Then there exist Gromoll–Meyer pairs (W (γ ), W (γ )− ) of Lτ at γ and (W (γ k ), W (γ k )− ) of Lkτ at γ k such that
− ψ k W (γ ) , ψ k W (γ )− ⊂ W γ k , W γ k .
(4.3)
Proof. For each j ∈ N, let φj τ : W 1,2 Sj τ , Bρn (0) → Hj τ α j
and L˜ j τ = Lj τ ◦ φj τ
(4.4)
as in (3.8) and (3.10). They satisfy (3.9), i.e. φj τ ◦ ψ˜ j = ψ j ◦ φτ ∀j ∈ N, where ψ j : Hτ (α) → Hj τ (α j ) and ψ˜ j : W 1,2 (Sτ , Rn ) → W 1,2 (Sj τ , Rn ) are the iteration maps. Let γ˜ = (φτ )−1 (γ ). Then φj τ (γ˜ j ) = γ j for any j ∈ N. Let · τ and · kτ denote the norms in W 1,2 (Sτ , Rn ) and W 1,2 (Skτ , Rn ) respectively. By the construction on page 49 of [7], we set 1,2 W˜ (γ˜ ) := L−1 Sτ , Rn λLτ (x) + x 2τ μ , τ [c − ε, c + ε] ∩ x ∈ W 1,2 Sτ , Rn λLτ (x) + x 2τ μ , W˜ (γ˜ )− := L−1 τ (c − ε) ∩ x ∈ W 1,2 Skτ , Rn λLkτ (y) + y 2kτ kμ , W˜ γ˜ k := L−1 kτ [kc − kε, kc + kε] ∩ y ∈ W − 1,2 Skτ , Rn λLkτ (y) + y 2kτ kμ , W˜ γ˜ k := L−1 kτ (kc − kε) ∩ y ∈ W where positive numbers λ, μ, ε and kλ, kμ, kε are such that the conditions as in (5.13)–(5.15) on page 49 of [7] hold. Then (W˜ (γ˜ ), W˜ (γ˜ )− ) and (W˜ (γ˜ k ), W˜ (γ˜ k )− ) are Gromoll–Meyer pairs of L˜ τ at γ˜ and of L˜ kτ at γ˜ k , and
− ψ˜ k W˜ (γ˜ ) , ψ˜ k W˜ (γ˜ )− ⊂ W˜ γ˜ k , W˜ γ˜ k .
(4.5)
W (γ ), W (γ )− := φ W˜ (γ˜ ) , φ W˜ (γ˜ )− , k − k k − τ k τ := φkτ W˜ γ˜ , φkτ W˜ γ˜ . W γ ,W γ
(4.6)
Define
Since φkτ ◦ ψ˜ k = ψ k ◦ φτ , (4.3) follows from (4.5).
2
When γ and γ k are isolated, according to the definition of critical groups in (4.1) it is easy to see that the iteration map ψ k : Hτ (α) → Hkτ (α k ) induces homomorphisms k ψ ∗ : C∗ (Lτ , γ ; K) → C∗ Lkτ , γ k ; K . Lemma 4.3 shows that the homomorphisms are still well defined when the critical groups C∗ (Lτ , γ ; K) and C∗ (Lkτ , γ k ; K) are defined by (4.2). Later similar cases are always understand in this way. Our purpose is to prove:
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Theorem 4.4. Let γ ∈ Hτ (α) be an isolated critical point of the functional Lτ on Hτ (α) such that γ ∗ T M → Sτ is trivial. Suppose that for some k ∈ N the iteration γ k is also an isolated critical point of the functional Lkτ in Hkτ (α k ), and k m− = m− τ (γ ) kτ γ
and m0kτ γ k = m0τ (γ ).
(4.7)
Then for c = Lτ (γ ) and any > 0 there exist topological Gromoll–Meyer pairs of Lτ at γ ∈ Hτ (α) and of Lkτ at γ k ∈ Hkτ (α k ), Wτ , Wτ− ⊂ (Lτ )−1 [c − , c + ], (Lτ )−1 (c − ) and − Wkτ , Wkτ ⊂ (Lkτ )−1 [kc − k, kc + k], (Lkτ )−1 (kc − k) , such that k − ψ (Wτ ), ψ k Wτ− ⊂ Wkτ , Wkτ
(4.8)
and that the homomorphism k − ψ ∗ : C∗ (Lτ , γ ; K) := H∗ Wτ , Wτ− ; K → C∗ Lkτ , γ k ; K := H∗ Wkτ , Wkτ ;K
(4.9)
is an isomorphism. Specially, (ψ 1 )∗ = id, and (ψ k )∗ ◦ (ψ l )∗ = (ψ kl )∗ if the iterations γ l and γ kl are also isolated, and kl − l − m− klτ γ = mlτ γ = mτ (γ ), m0klτ γ kl = m0lτ γ l = m0τ (γ ).
(4.10)
When M = Rn , this theorem was proved by [23, Th. 3.7]. We shall reduce the proof of Theorem 4.4 to that case. Using the chart in (4.4) let γ˜ = (φτ )−1 (γ ). Then γ˜ j = (φj τ )−1 (γ j ) for each j ∈ N. Then j γ˜ are isolated critical points of L˜ j τ = Lj τ ◦ φj τ in W 1,2 (Sj τ , Rn ), j = 1, k, l, kl. More0 j − j 0 − ˜ ˜ over, m− j τ (γ˜ ) = mτ (γ˜ ) and mkτ (γ˜ ) = mτ (γ˜ ) for j = k, l, kl. Let (W (γ˜ ), W (γ˜ ) ) and (W˜ (γ˜ k ), W˜ (γ˜ k )− ) be Gromoll–Meyer pairs of L˜ τ at γ˜ and of L˜ kτ at γ˜ k , satisfying (4.5). Define C∗ (L˜ τ , γ˜ ; K) = H∗ W˜ (γ˜ ), W˜ (γ˜ )− ; K , C∗ (Lτ , γ ; K) = H∗ W (γ ), W (γ )− ; K , − C∗ L˜ kτ , γ˜ k ; K = H∗ W˜ γ˜ k , W˜ γ˜ k ; K , − C∗ Lkτ , γ k ; K = H∗ W γ k , W˜ γ k ; K . Since φkτ ◦ ψ˜ k = ψ k ◦ φτ , we have (φkτ )∗ ◦ (ψ˜ k )∗ = (ψ k )∗ ◦ (φτ )∗ . Clearly, (φτ )∗ : C∗ (L˜ τ , γ˜ ; K) → C∗ (Lτ , γ ; K) and (φkτ )∗ : C∗ L˜ kτ , γ˜ k ; K → C∗ Lkτ , γ k ; K
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are isomorphisms. Hence we only need to prove that
ψ˜ k
∗
: C∗ (L˜ τ , γ˜ ; K) → C∗ L˜ kτ , γ˜ k ; K
(4.11)
is an isomorphism which maps generators to the generators. This is exactly one proved by [23, Th. 3.7]. Theorem 3.7 in [23] also gives that (ψ˜ 1 )∗ = id and (ψ˜ k )∗ ◦ (ψ˜ l )∗ = (ψ˜ kl )∗ . So other conclusions follow immediately. For later conveniences we outline the arguments therein. Let W 1,2 Skτ , Rn = M 0 (γ˜k ) ⊕ M(γ˜k )− ⊕ M(γ˜ )+ = M 0 (γ˜k ) ⊕ M(γ˜k )⊥ be the orthogonal decomposition of the space W 1,2 (Skτ , Rn ) according to the null, negative, and positive definiteness of the quadratic form L˜ kτ (γ˜ k ). The generalized Morse lemma [7, Th. 5.1, p. 44] yields a homeomorphism Θ˜ kτ from some open neighborhood U˜ kτ of 0 in W 1,2 (Skτ , Rn ) to Θ˜ kτ (U˜ kτ ) ⊂ W 1,2 (Skτ , Rn ) with Θ˜ kτ (0) = γ˜ k , and a map h˜ kτ ∈ C 1 (U˜ kτ ∩ M(γ˜ k )0 , M(γ˜ k )⊥ ) such that 1 L˜ kτ Θ˜ kτ (η + ξ ) = L˜ kτ γ˜ k + η + h˜ kτ (η) + L˜ kτ γ˜ k ξ, ξ ≡ α˜ kτ (η) + β˜kτ (ξ ) 2
(4.12)
for any η + ξ ∈ U˜ kτ ∩ (M(γ˜ k )0 ⊕ M(γ˜k )⊥ ). (Note: β˜kτ is C ∞ , α˜ kτ is C 2 as noted at the end of proof of Theorem 5.1 on the page 44 of [7]. Carefully checking the beginning proof therein one can easily derive this from Lemma 4.2.) It is easy to prove that ψ˜ k L˜ τ (x) = L˜ kτ ψ˜ k (x)
and ψ˜ k L˜ τ (x)ξ = L˜ kτ ψ˜ k (x) ψ˜ k (ξ )
(4.13)
for any τ, k ∈ N, x ∈ W 1,2 (Sτ , Bρn (0)) and ξ ∈ W 1,2 (Sτ , Rn ), and that α˜ kτ ψ˜ k (η) = k α(η) ˜
and β˜kτ ψ˜ k (ξ ) = k β˜τ (ξ )
(4.14)
for any η ∈ U˜ τ ∩ M 0 (γ˜ ) and ξ ∈ U˜ τ ∩ M ⊥ (γ˜ ). Lemma 4.5. (See [23, Lem. 3.2, 3.3].) The iteration map ψ˜ k : M ∗ (γ˜ ) → M ∗ (γ˜ k ) for ∗ = 0, −, + k − − − k ˜k is linear, continuous and injective. If m− kτ (γ˜ ) = mτ (γ˜ ), the map ψ : M (γ˜ ) → M (γ˜ ) is a 0 k 0 k 0 0 k ˜ linear diffeomorphism. If mkτ (γ˜ ) = mτ (γ˜ ), then the map ψ : M (γ˜ ) → M (γ˜ ) is a linear diffeomorphism, and U˜ kτ , the homeomorphism Θ˜ kτ and map h˜ kτ ∈ C 1 (U˜ kτ ∩ M(γ˜ k )0 , M(γ˜ k )⊥ ) are chosen to satisfy: U˜ kτ ∩ ψ˜ k W 1,2 Sτ , Rn = ψ˜ k (U˜ τ ), Θ˜ kτ ◦ ψ˜ k = ψ˜ k ◦ Θ˜ τ : U˜ τ → Θ˜ τ U˜ τ ∩ M 0 γ˜ k , h˜ kτ ψ˜ k (η) = ψ˜ k h˜ τ (η) ∀η ∈ U˜ τ ∩ M(γ˜ ).
(4.15) (4.16) (4.17)
Let (W0 , W0− ) and (W1 , W1− ) be Gromoll–Meyer pairs of α˜ τ and β˜τ at their origins respectively. By [23, Prop. 3.5. 2◦ ], (ψ˜ k (W0 ), ψ˜ k (W0− )) is a Gromoll–Meyer pair of α˜ kτ at the origin. The Gromoll–Meyer pair (W1 , W1− ) can also be chosen to satisfy
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k ψ˜ (W1 ), ψ˜ k W − ⊂ (V , V − )
(4.18)
1
for some Gromoll–Meyer pair (V , V − ) of β˜kτ at the origin. By [7, Lem. 5.1, p. 51] W0 × W1 , W0 × W1− ∪ W0− × W1 , k ψ˜ (W0 ) × V , ψ˜ k (W0 ) × V − ∪ ψ˜ k W0− × V
(4.19) (4.20)
are Gromoll–Meyer pairs of α˜ τ + β˜τ and α˜ kτ + β˜kτ at their origins respectively, and also satisfy k ˜ k W0 × W − ∪ W − × W1 × W ), ψ ψ˜ (W 0 1 1 0 k k ⊂ ψ˜ (W0 ) × V , ψ˜ (W0 ) × V − ∪ ψ˜ k W0− × V .
(4.21)
τ− := Θ˜ τ W0 × W1 , W0 × W − ∪ W − × W1 , τ , W W 1 0 kτ , W − := Θ˜ kτ ψ˜ k (W0 ) × V , ψ˜ k (W0 ) × V − ∪ ψ˜ k W − × V W
(4.22)
Note that
kτ
0
(4.23)
are topological Gromoll–Meyer pairs of L˜ τ at γ˜ and L˜ kτ at γ˜ k respectively. Let C∗ (α˜ τ + β˜τ , 0; K) := H∗ W0 × W1 , W0 × W1− ∪ W0− × W1 ; K , τ , W τ− ; K , C∗ (L˜ τ , 0; K) := H∗ W C∗ (α˜ kτ + β˜kτ , 0; K) := H∗ ψ˜ k (W0 ) × V , ψ˜ k (W0 ) × V − ∪ ψ˜ k W0− × V ; K , kτ , W − ; K . C∗ (L˜ kτ , 0; K) := H∗ W kτ We have the isomorphisms on critical modules, C∗ (α˜ τ + β˜τ , 0; K) ∼ = C∗ (L˜ τ , γ˜ ; K), (Θ˜ kτ )∗ : C∗ (α˜ kτ + β˜kτ , 0; K) ∼ = C∗ L˜ kτ , γ˜ k ; K . (Θ˜ τ )∗ :
By (4.21) we have a homomorphism k ψ˜ ∗ : C∗ (α˜ τ + β˜τ , 0; K) → C∗ (α˜ kτ + β˜kτ , 0; K).
(4.24)
Moreover, (4.16) and (4.21) show that − k τ ⊂ W kτ , W τ ), ψ˜ k W − ψ˜ (W kτ
(4.25)
and therefore the homomorphism k ψ˜ ∗ : C∗ (L˜ τ , 0; K) → C∗ (L˜ kτ , 0; K) satisfy k ψ˜ ∗ ◦ (Θ˜ τ )∗ = (Θ˜ kτ )∗ ◦ ψ˜ k ∗ .
(4.26)
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Hence the problem is reduced to prove: Lemma 4.6. The Gromoll–Meyer pairs (W1 , W1− ) and (V , V − ) in (4.18) can be chosen such that k ψ˜ ∗ : C∗ (α˜ τ + β˜τ , 0; K) → C∗ (α˜ kτ + β˜kτ , 0; K)
(4.27)
is an isomorphism. Proof. For j = 1, k, decompose ξ ∈ M(γ˜j )⊥ = M(γ˜j )− ⊕ M(γ˜j )+ into ξ = ξ − + ξ + and write β˜j τ (ξ ) = β˜j τ (ξ − ) + β˜j τ (ξ + ) = β˜j−τ (ξ − ) + β˜j+τ (ξ + ). Then β˜j−τ and β˜j+τ are negative and positive definite quadratic forms on M(γ˜j )− and M(γ˜j )+ with Morse indexes m− (γ˜ j ) and 0 respectively, j = 1, k. The (4.12)–(4.14) imply − ˜k − β˜kτ ψ (ξ ) = k β˜τ− (ξ − )
− ˜k + ψ (ξ ) = k β˜τ− (ξ + ) and β˜kτ
k − ˜k for any ξ − ∈ M − (γ˜ ) and ξ + ∈ M + (γ˜ ). Since m− kτ (γ˜ ) = mτ (γ˜ ), by Lemma 4.5 the map ψ : − − − k M (γ˜ ) → M (γ˜ ) is a linear diffeomorphism. Let (W11 , W11 ) be a Gromoll–Meyer pair of β˜τ− at the origin. Then
k ψ˜ (W11 ), ψ˜ k W −
(4.28)
11
− is a Gromoll–Meyer pair of β˜kτ at the origin. For δ > 0 sufficiently small, set
W12 := ξ + ∈ M(γ˜ )+ ξ + τ δ , − W12 := ξ + ∈ M(γ˜ )+ ξ + τ = δ , √ + V12 := ξ + ∈ M γ˜ k ξ + kτ kδ , √ + − := ξ + ∈ M γ˜ k ξ + kτ = kδ . V12 − − + It is easily checked that (W12 , W12 ) and (V12 , V12 ) are Gromoll–Meyer pairs of β˜τ+ and β˜kτ at their origins respectively, and that
k ψ˜ (W12 ), ψ˜ k W − ⊂ V12 , V − . 12
12
(4.29)
By [7, Lem. 5.1, p. 51], we may take − − ∪ W11 × W12 , W1 , W1− := W11 × W12 , W11 × W12 − ˜ k − (V , V − ) := ψ˜ k (W11 ) × V12 , ψ˜ k (W11 ) × V12 ∪ ψ W11 × V12 .
(4.30) (4.31)
Then (W0 × W1 , (W0 × W1− ) ∪ (W0− × W1 )) becomes (W, W − ), and C∗ (α˜ τ + β˜τ , 0; K) = H∗ (W, W − ; K),
(4.32)
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where W := W0 × W11 × W12 and − − W − := W0 × W11 × W12 ∪ W11 × W12 ∪ W0− × W11 × W12 .
(4.33)
Moreover, (ψ˜ k (W0 ) × V , (ψ˜ k (W0 ) × V − ) ∪ (ψ˜ k (W0− ) × V )) becomes (U, U − ), and C∗ (α˜ kτ + β˜kτ , 0; K) = H∗ (U, U − ; K),
(4.34)
where U = ψ˜ k (W0 ) × ψ˜ k (W11 ) × V12 and − ˜ k − U − = ψ˜ k (W0 ) × ψ˜ k (W11 ) × V12 ∪ ψ W11 × V12 ∪ ψ˜ k W0− × ψ˜ k (W11 ) × V12 .
(4.35)
Note that ψ˜ k (W ) = ψ˜ k (W0 ) × ψ˜ k (W11 ) × ψ˜ k (W12 ) and − k − ψ˜ k (W − ) = ψ˜ k (W0 ) × ψ˜ k (W11 ) × ψ˜ k W12 ∪ ψ˜ W11 × ψ˜ k (W12 ) ∪ ψ˜ k W0− × ψ˜ k (W11 ) × ψ˜ k (W12 ) .
(4.36)
Since ψ˜ k : M + (γ˜ ) → M + (γ˜ k ) is a linear, continuous and injection, by (4.29) and the con− − − structions of (V12 , V12 ) and (W12 , W12 ) it is readily checked that (ψ˜ k (W12 ), ψ˜ k (W12 )) is a − deformation retract of (V12 , V12 ). It follows that k ψ˜ (W ), ψ˜ k (W − ) ⊂ (U, U − ) is a deformation retract of (U, U − ). Hence k ψ˜ ∗ : H∗ (W, W − ; K) → H∗ (U, U − ; K) and therefore, by (4.32) and (4.34), the homomorphism (ψ˜ k )∗ in (4.27) is an isomorphism. We may also prove the conclusion as follows. By the arguments at the middle of [7, p. 51] we can use Künneth formula to arrive C∗ (α˜ τ + β˜τ , 0; K) = H∗ W0 , W0− ; K − − ⊗ H∗ W11 , W11 ; K ⊗ H∗ W12 , W12 ;K , C∗ (α˜ kτ + β˜kτ , 0; K) = H∗ ψ˜ k (W0 ), ψ˜ k W0− ; K − − ⊗ H∗ ψ˜ k (W11 ), ψ˜ k W11 ; K ⊗ H∗ V12 , V12 ;K . 0 k − k 0 Now m− kτ (γ˜ ) = mτ (γ˜ ) and mkτ (γ˜ ) = mτ (γ˜ ) imply that
k ψ˜ ∗ : H∗ W0 , W0− ; K → H∗ ψ˜ k (W0 ), ψ˜ k W0− ; K , − k − ;K ; K → H∗ ψ˜ k (W11 ), ψ˜ k W11 ψ˜ ∗ : H∗ W11 , W11
(4.37)
(4.38)
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− − are isomorphisms. Since (ψ˜ k (W12 ), ψ˜ k (W12 )) is a deformation retract of (V12 , V12 ) as above, it follows that
k − − ψ˜ ∗ : H∗ ψ˜ k (W12 ), ψ˜ k W12 ; K → H∗ V12 , V12 ;K is an isomorphism. By (4.37) and (4.38) we get the proof of Lemma 4.6.
(4.39)
2
τ , W τ− ) in (4.22) and (W kτ , W − ) in (4.23), where the Gromoll–Meyer pairs (W1 , W − ) For (W kτ 1 − and (V , V ) in (4.18) are also required to satisfy Lemma 4.6. Set
− τ τ ), φτ W Wτ , Wτ− := φτ (W
and
− − kτ ), φkτ W Wkτ , Wkτ := φkτ (W kτ .
− ) and that the Since φkτ ◦ ψ˜ k = ψ k ◦ φτ , by (4.25) we have (ψ k (Wτ ), ψ k (Wτ− )) ⊂ (Wkτ , Wkτ homomorphism
k − ψ ∗ : H∗ Wτ , Wτ− ; K → H∗ Wkτ , Wkτ ;K − is an isomorphism. Consequently, (Wτ , Wτ− ) and (Wkτ , Wkτ ) are desired topological Gromoll– Meyer pairs. The other conclusions are also easily proved. So Theorem 4.4 holds.
4.2. In this subsection we always assume: M is C 3 -smooth, L is C 2 -smooth and satisfies (L1)–(L4). Let γ ∈ EHτ be an isolated critical point of the functional LE τ on EHτ , and E φkτ : EW 1,2 Skτ , Bρn (0) → EHkτ
and L˜ E kτ = Lkτ ◦ φkτ
(4.40)
E ◦ψ ˜ k = ψ k ◦ φτE for any be as in (3.29) and (3.32) for each k ∈ N. They satisfy (3.30), i.e. φkτ k k ∈ N, where ψ : EHτ → EHkτ and
ψ˜ k : EW 1,2 Sτ , Rn → EW 1,2 Skτ , Rn E (γ˜ k ) = γ k for any k ∈ N. Suppose that are the iteration maps. Let γ˜ = (φτE )−1 (γ ) and thus φkτ γ k and therefore γ˜ k are also isolated. Denote by
k k ˜ ˜ k − Cq L˜ E kτ , γ˜ ; K = Hq W γ˜ E , W γ˜ E ; K k the critical module of L˜ E kτ at γ˜ via the relative singular homology with coefficients in K, where ˜E (W˜ (γ˜ k )E , W˜ (γ˜ k )− E ) is a Gromoll–Meyer pair via some pseudo-gradient vector field of Lkτ near k 1,2 n γ˜ in EW (Skτ , R ). Let
+ 0 ⊥ EW 1,2 Skτ , Rn = M 0 (γ˜k )E ⊕ M(γ˜k )− E ⊕ M(γ˜ )E = M (γ˜k )E ⊕ M(γ˜k )E be the orthogonal decomposition of the space EW 1,2 (Skτ , Rn ) according to the null, neg ative, and positive definiteness of the quadratic form (L˜ E kτ ) (γ˜ ). As above we can use the E E ˜ generalized Morse lemma to get a homeomorphism Θkτ from some open neighborhood U˜ kτ
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E (U ˜ E ) ⊂ EW 1,2 (Skτ , Rn ) with Θ˜ E (0) = γ˜ k , and a map h˜ E ∈ of 0 in EW 1,2 (Skτ , Rn ) to Θ˜ kτ kτ kτ kτ 0 E ⊥ 1 k k ˜ C (Ukτ ∩ M(γ˜ )E , M(γ˜ )E ) such that
E 1 E k E E ˜E k ˜E ˜ L˜ E L˜ kτ γ˜ ξ, ξ ≡ α˜ kτ (η) + β˜kτ (ξ ) kτ Θkτ (η + ξ ) = Lkτ γ˜ + η + hkτ (η) + 2 E ∩ (M(γ˜ k )0 ⊕ M(γ˜ )⊥ ), where β˜ E and α E are respectively C ∞ and C 2 as for any η + ξ ∈ U˜ kτ ˜ kτ k E E kτ E ˜ noted below (4.12). Then Θkτ induces isomorphisms on critical modules,
E Θ˜ kτ ∗ :
E E k C∗ α˜ kτ + β˜kτ , 0; K ∼ = C∗ L˜ E kτ , γ˜ ; K .
(4.41)
Note that k E k E − k W γ E , W − γ k E := φkτ W˜ γ˜ E , φkτ W˜ γ E
(4.42)
k is a Gromoll–Meyer pair of LE kτ at γ . Define the critical modules
k k − k γ E; K . C∗ L E kτ , γ ; K := H∗ W γ E , W
(4.43)
Then corresponding to Theorem 4.4 we have the following generalization of [30, Lemma 4.1]. Theorem 4.7. Let γ ∈ EHτ be an isolated critical point of the functional LE τ on EHτ . If the iteration γ k is also isolated for some k ∈ N, and k m− = m− 1,kτ γ 1,τ (γ )
and m01,kτ γ k = m01,τ (γ ),
E then for c = LE τ (γ ) and any > 0 there exist topological Gromoll–Meyer pairs of Lτ at γ ∈ E k EHτ and of Lkτ at γ ∈ EHkτ ,
−1 −1 Wτ , Wτ− ⊂ LE [c − , c + ], LE (c − ) and τ τ −1 −1 − ⊂ LE [kc − k, kc + k], LE (kc − k) , Wkτ , Wkτ kτ kτ such that k − ψ (Wτ ), ψ k Wτ− ⊂ Wkτ , Wkτ
(4.44)
and that the iteration map ψ k : EHτ → EHkτ induces isomorphisms k E k − − ψ ∗ : C∗ L E τ , γ ; K := H∗ Wτ , Wτ ; K → C∗ Lkτ , γ ; K := H∗ Wkτ , Wkτ ; K .
(4.45)
Specially, (ψ 1 )∗ = id, and (ψ k )∗ ◦ (ψ l )∗ = (ψ kl )∗ if the iterations γ l and γ kl are also isolated, and kl − l − m− 1,klτ γ = m1,lτ γ = m1,τ (γ ), m01,klτ γ kl = m01,lτ γ l = m01,τ (γ ).
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4.3. Let us consider the case L is independent t. In this subsection we always assume: M is C 5 -smooth, L is C 4 -smooth and satisfies (L1)–(L3). The goal is to generalize [26, Th. 2.5] to the present general case. However, unlike the last two cases we cannot choose a local coordinate chart around a critical orbit. For τ > 0, let Sτ := R/τ Z = {[s]τ | [s]τ = s + τ Z, s ∈ R} and the functional Lτ : Hτ (α) → R be still defined by (1.14). By [18, Chp. 2, §2.2], there exist equivariant and also isometric operations of Sτ -action on Hτ (α) and T Hτ (α): [s]τ · γ (t) = γ (s + t), ∀[s]τ ∈ Sτ , γ ∈ Hτ (α), [s]τ · ξ(t) = ξ(s + t), ∀[s]τ ∈ Sτ , ξ ∈ Tγ Hτ (α)
(4.46)
which are continuous, but not differentiable. Clearly, Lτ is invariant under this action. Since under our assumptions each critical point γ of Lτ is C 4 -smooth, by [15, p. 499], the orbit Sτ · γ is a C 3 -submanifold in Hτ (α). It is easily checked that Sτ · γ is a C 3 -smooth critical submanifold of Lτ . Seemingly, the theory of [38] cannot be applied to this case because the action of Sτ is only continuous. However, as pointed out in the second paragraph of [15, p. 500] this theory still holds since critical orbits are smooth and Sτ acts by isometries. For any k ∈ N, there is a natural k-fold cover ϕk from Skτ to Sτ defined by ϕk : [s]kτ → [s]τ .
(4.47)
It is easy to check that the Sτ -action on Hτ (α), the Skτ -action on Hkτ (α k ), and the kth iteration map ψ k defined above (3.9) satisfy: k k, [s]τ · γ = [s] · γ kτ Lkτ [s]kτ · γ k = kLτ [s]τ · γ = kLτ (γ )
(4.48)
for all γ ∈ Hτ (α), k ∈ N, and s ∈ R. Let γ ∈ Hτ (α) be a nonconstant critical point of Lτ with minimal period τ/m for some m ∈ N. Denote by O = Sτ · γ = Sτ/m · γ . It is a 1-dimensional C 3 -submanifold diffeomorphic to the circle. Let c = Lτ |O . Assume that O is isolated. We may take a neighborhood U of O such that K(Lτ ) ∩ U = O. By (4.1) we have critical group C∗ (Lτ , O; K) of Lτ at O. For every s ∈ [0, τ/m] the tangent space Ts·γ (Sτ · γ ) is R(s · γ )· , and the fiber N (O)s·γ at s · γ of the normal bundle N(O) of O is a subspace of codimension 1 which is orthogonal to (s · γ )· in Ts·γ Hτ (α), i.e. N (O)s·γ = ξ ∈ Ts·γ Hτ (α) ξ, (s · γ )· 1 = 0 . Since Hτ (α) is C 4 -smooth and O is a C 3 -smooth submanifold, N (O) is C 2 -smooth manifold.2 Notice that N(O) is invariant under the Sτ -actions in (4.20) and each [s]τ gives an isometric bundle map N (O) → N (O),
(z, v) → [s]τ · z, [s]τ · v .
(4.49)
Under the present case it is easily checked that Lτ satisfies Assumption 7.1 on the page 71 of [7], that is, there exists > 0 such that 2 This is the reason that we require higher smoothness of M and L.
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σ L τ (x) ∩ [−, ] \ {0} = ∅,
dim ker L τ (x) = constant
3001
(4.50)
for any x ∈ O. Then Lemma 7.4 of [7, p. 71] gives the orthogonal C 2 -smooth bundle decomposition N(O) = N (O)+ ⊕ N (O)− ⊕ N (O)0 ,
N (O)∗ = P∗ N (O)
(4.51)
for ∗ = +, −, 0. Here P∗ : N (O) → N (O)∗ , ∗ = +, 0, −, are orthogonal bundle projections. Each N(O)∗ is a C 2 -smooth submanifold. It is not hard to check that L τ and L τ satisfy L τ [s]τ · x = [s]τ · L τ (x)
and L τ [s]τ · x [s]τ · ξ = [s]τ · L τ (x)(ξ )
for all x ∈ Hτ (α), ξ ∈ Tx Hτ (α) and [s]τ ∈ Sτ . It follows that the bundle map (4.49) preserves the decomposition (4.51). In particular, we obtain 0 ∀x ∈ O, rank N (O)− , rank N (O)0 = m− τ (x), mτ (x) − 1 0 where m− τ (x) and mτ (x) are Morse index and nullity of Lτ at x respectively. Define
− mτ (O), m0τ (O) := rank N (O)− , rank N (O)0 .
(4.52)
Then − 0 mτ (O), m0τ (O) = m− τ (x), mτ (x) − 1
∀x ∈ O.
(4.53)
For a single point critical orbit O = {γ }, i.e., γ is constant, we define − 0 mτ (O), m0τ (O) := m− τ (γ ), mτ (γ ) .
(4.54)
Note that for sufficiently small ε > 0 the set N (O)(ε) := (y, v) ∈ N (O) y ∈ O, v 1 < ε is contained in an open neighborhood of the zero section of the tangent bundle T Hτ (α). By [18, Th. 1.3.7, p. 20] we have a C 2 -embedding from N (O)(ε) to an open neighborhood of the diagonal of Hτ (α) × Hτ (α), N (O)(ε) → Hτ (α) × Hτ (α),
(y, v) → (y, expy v),
where exp is the exponential map of the chosen Riemannian metric on M and (expy v)(t) = expy(t) v(t) ∀t ∈ R. This yields a C 2 -diffeomorphism from N (O)(ε) to an open neighborhood Qε (O) of O, Ψτ : N (O)(ε) → Qε (O),
Ψτ (y, v)(t) = expy(t) v(t) ∀t ∈ R.
(4.55)
(Note that it is not the exponential map of the Levi-Civita connection derived the Riemannian metric , τ on Hτ (α).) Clearly,
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Ψτ (y, 0) = y
∀y ∈ O
and Ψτ [s]τ · y, [s]τ · v = [s]τ · Ψτ (y, v)
(4.56)
for any (y, v) ∈ N(O)(ε) and [s]τ ∈ Sτ . It follows that Qε (O) is an Sτ -invariant neighborhood of O, and that Ψτ is Sτ -equivariant. We also require ε > 0 so small that Qε (O) contains no other critical orbit besides O, and that Ψτ ({y} × N (O)y (ε)) and O have a unique intersection point y (after identifying O with the zero section N (O)(ε)). Then Lτ ◦ Ψτ |N (O)y (ε) possesses y as an isolated critical point. Checking the proofs of Theorem 7.3 and Corollary 7.1 in [7, p. 72], and replacing f ◦ exp |ξx and expx φx therein by Lτ ◦ Ψτ |N (O)x (ε) and Ψτ |N (O)x (ε) ◦ φx for x ∈ O, one easily gets: Lemma 4.8. For sufficiently small 0 < < ε, there exist an Sτ -equivariant homeomorphism Φτ from N (O)() to an Sτ -invariant open neighborhood Ω (O) ⊂ Qε (O) of O, and a C 1 -map hτ : N (O)0 () → N (O)+ () ⊕ N (O)− () such that Lτ ◦ Φτ (y, v) =
1 P+ (y)v 2 − P− (y)v 2 + Lτ ◦ Ψτ y, P0 (y)v + hτ P0 (y)v 1 1 2
for (y, v) ∈ N(O)(), where P∗ is as in (4.51). Let N (O)⊥ () = N (O)+ () ⊕ N (O)− () and write v = v 0 + v ⊥ . Set 2 2 1 Ξτ y, v ⊥ = P+ (y)v 1 − P− (y)v 1 , 2 Υτ y, v 0 = Lτ ◦ Ψτ y, P0 (y)v + hτ P0 (y)v
(4.57)
for (y, v) ∈ N(O)(). Then define Fτ : N (O)() → R by Fτ (y, v) = Lτ ◦ Φτ (y, v) = Υτ y, v 0 + Ξτ y, v ⊥
(4.58)
for all (y, v) ∈ N(O)(). (Note: Though we require the higher smoothness of M and L we do not know whether or not Lτ has higher smoothness than order two unlike the special L considered in [23]. Hence from [7, Th. 7.3, p. 72] we can only get that Φτ is a homeomorphism. However, N(O)() is a C 2 -bundle3 and therefore both Ξτ and Υτ are C 2 .
(4.59)
By the local trivialization of N (O)() the final claim can be derived from Lemma 4.2 and the proofs of [7, Th. 5.1, p. 44] and [7, Th. 7.3, p. 72].) Clearly, both Υτ and Ξτ are also Sτ -invariant, and have the unique critical orbit O in N (O)⊥ (ε) and N (O)0 (ε) respectively. Since Fτ is C 2 -smooth, we can follow [38] to construct a Gromoll–Meyer pair of O as a critical submanifold of Fτ on N (O)(ε), W (O), W (O)− .
(4.60)
(Note that different from [38] the present Sτ -action on N (O)() is only continuous; but the arguments there can still be carried out due to the special property of our Sτ -action in (4.20) and 3 The requirements of the higher smoothness of M and L is used to assure this.
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the definition of Fτ .) In the present case, for any y ∈ O, Fτ |N (O)y () has a unique critical point y in N(O)y () (the fiber of disk bundle N (O)() at y), and − W (O)y , W (O)− y := W (O) ∩ N (O)y (), W (O) ∩ N (O)y ()
(4.61)
is a Gromoll–Meyer pair of Fτ |Ny (O)() at its isolated critical point y satisfying
− W (O)[s]τ ·y , W (O)− [s]τ ·y = [s]τ · W (O)y , [s]τ · W (O)y
(4.62)
for any [s]τ ∈ Sτ and y ∈ O. Clearly, (O), W (O)− := Φτ W (O) , Φτ W (O)− W
(4.63)
is a topological Gromoll–Meyer pair of Lτ at O, which is also Sτ -invariant. Define (O) , W (O)− ; K , C∗ (Lτ , O; K) := H∗ W C∗ (Fτ , O; K) := H∗ W (O), W (O)− ; K
(4.64) (4.65)
via the relative singular homology. Φτ induces an obvious isomorphism (Φτ )∗ :
C∗ (Lτ , O; K) ∼ = C∗ (Fτ , O; K).
(4.66)
Since the normal bundle N (O) is differentiably trivial, it follows from [38, (2.13), (2.14)] (cf. also the shifting theorem in [14] and [7]) that for any q ∈ {0} ∪ N, Cq (Fτ , O; K) ∼ =
q Cq−j (Fτ |N (O)y () , y; K) ⊗ Hj (Sτ ; K) j =0
∼ =
q Cq−j −m−τ (O) (Fτ |N (O)0y () , y; K) ⊗ Hj (Sτ ; K) j =0
∼ = Cq−1−m−τ (O) (Fτ |N (O)0y () , y; K)
∀y ∈ O.
Here Cq−1−m−τ (O) (Fτ |N (O)0y () , y; K) is independent of the choice of y ∈ O = Sτ · γ . Taking y = γ we obtain C∗ (Lτ , Sτ · γ ; K) ∼ = C∗−1−m−τ (Sτ ·γ ) (Fτ |N (Sτ ·γ )0γ () , γ ; K).
(4.67)
Suppose that ψ k (O) = Skτ · γ k is also an isolated critical orbit of the functional Lkτ on Hkτ (α k ) for some k ∈ N. Our purpose is to study the relations between critical groups C∗ (Lτ , O; K) and C∗ (Lkτ , ψ k (O); K). Let N (Skτ · γ k ) be the normal bundle of Skτ · γ k in Hkτ (α k ) and N Skτ · γ k (ε) = (y, v) ∈ N Skτ · γ k y ∈ Skτ · γ k , v 1 < ε .
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Corresponding to (4.51) there exist natural orthogonal bundle decompositions + − 0 N ψ k (O) = N ψ k (O) ⊕ N ψ k (O) ⊕ N ψ k (O) , + − 0 N ψ k (O) (ε) = N ψ k (O) (ε) ⊕ N ψ k (O) (ε) ⊕ N ψ k (O) (ε),
(4.68) (4.69)
where N(ψ k (O))+ (ε) = N (ψ k (O))(ε) ∩ N (ψ k (O))∗ for ∗ = +, −, 0. It is not hard to check that √ ψ k N (O)(ε) ⊂ N Skτ · γ k ( kε)
∗ and ψ k N (O)∗ (ε) ⊂ N Skτ · γ k
(4.70)
2 diffeomorphism for ∗ = +, 0, −. By √ shrinking ε > 0 we have also a C -smooth Sτ -equivariant k from N(Skτ · γ )( kε) to an Skτ -invariant open neighborhood Q√kε (Skτ · γ k ) of Skτ · γ k ,
√ Ψkτ : N Skτ · γ k ( kε) → Q√kε Skτ · γ k , Ψkτ (y, v)(t) = expy(t) v(t)
∀t ∈ R.
(4.71)
With the same arguments as above Lemma 4.8, by furthermore √ shrinking 0 < < ε, there exist an Skτ -equivariant homeomorphism Φkτ from N (ψ k (O))( k) to an Skτ -invariant open neighborhood Ω√k (ψ k (O)) ⊂ Q√kε (ψ k (O)) of ψ k (O), and a C 1 -map 0 √ + √ − √ hkτ : N ψ k (O) ( k) → N ψ k (O) ( k) ⊕ N ψ k (O) ( k) such that Lkτ ◦ Φkτ (y, v) = Υkτ y, v 0 + Ξkτ y, v ⊥ √ k (O))( k), for (y, v) ∈ N(ψ √ ⊕ N(ψ k (O))− ( k) and
where
(4.72)
√ √ v ⊥ ∈ N (ψ k (O))⊥ ( k) = N (ψ k (O))+ ( k)
Ξkτ y, v ⊥ = 12 v + 21 − v− 21 , Υkτ y, v 0 = Lτ ◦ Ψτ y, v 0 + h y, v 0
(4.73)
have the similar properties to (4.59). As in (4.58) we define an Skτ -invariant, C 2 -smooth function √ k Fkτ : N(ψ (O))( k) → R by Fkτ (y, v) = Lkτ Φkτ (y, v) = Υkτ y, v 0 + Ξkτ y, v ⊥ .
(4.74)
√ It has the unique critical orbit ψ k (O) in N (ψ k (O))( k). Note that (4.55) and (4.71) imply Ψkτ ◦ ψ k = ψ k ◦ Ψτ . As in [26, Prop. 2.3], we can suitably modify the proof of [23, Lem. 3.3] to get:
(4.75)
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Lemma 4.9. Suppose that m0kτ (ψ k (O)) = m0τ (O). Then: (i) The maps hτ and hkτ satisfy hkτ ψ k (p) = ψ k hτ (p) ,
∀p = (y, v) ∈ N (O)0 ().
(4.76)
(ii) The homeomorphisms Φτ and Φkτ satisfy Φkτ ◦ ψ k = ψ k ◦ Φτ
(4.77)
as maps from N (O)() to Hkτ (α k ). (iii) For q ∈ N(O)0 (), p ∈ N (O)⊥ (), there hold Υkτ ψ k (q) = kΥτ (q),
Ξkτ ψ k (p) = kΞτ (p).
(4.78)
Indeed, the key in the proof of [23, Lem. 3.3] is that the maps hτ and hkτ are uniquely determined by the implicit function theorem as showed in the proof of the Generalized Morse lemma [7, p. 44]. It follows from (4.78) that Fkτ ◦ ψ k = kFτ .
(4.79)
By the construction of√the Gromoll–Meyer pair in [38] we can construct such a pair of Fkτ at ψ k (O) on N (ψ k (O))( k), (W (ψ k (O)), W (ψ k (O)) such that k − ψ W (O) , ψ k W (O)− ⊂ W ψ k (O) , W ψ k (O)
(4.80)
for the pair (W (O), W (O)− ) in (4.60). Set k k ψ (O) , W ψ (O) − := Φkτ W ψ k (O) , Φkτ W ψ k (O) − , W
(4.81)
which is a topological Gromoll–Meyer pair, and k k ψ (O) , W ψ (O) − ; K , C∗ Lkτ , ψ k (O); K := H∗ W − C∗ Fkτ , ψ k (O); K := H∗ W ψ k (O) , W ψ k (O) ; K .
(4.82) (4.83)
It follows from (4.77) and (4.80) that k k k (O) , ψ k W (O)− ⊂ W ψ (O) , W ψ (O) − ψ W
(4.84)
and that ψ k induces homomorphisms k ψ ∗ : C∗ (Lτ , O; K) → C∗ Lkτ , ψ k (O); K , k ψ ∗ : C∗ (Fτ , O; K) → C∗ Fkτ , ψ k (O); K
(4.85) (4.86)
satisfying k ψ ∗ ◦ (Φτ )∗ = (Φkτ )∗ ◦ ψ k ∗
(4.87)
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because of (4.77). By (4.66) and the isomorphism (Φkτ )∗ :
C∗ Lkτ , ψ k (O); K ∼ = C∗ Fkτ , ψ k (O); K
(4.88)
we only need to prove: Lemma 4.10. The Gromoll–Meyer pairs in (4.80) can be chosen so that the homomorphism in (4.86) is an isomorphism provided that k − m− kτ ψ (O) = mτ (O)
and m0kτ ψ k (O) = m0τ (O).
(4.89)
Proof. By (4.58), (4.72) and (4.74) we have C∗ (F τ , O; K) = C∗(Υτ +Ξτ , O; K), C∗ Fkτ , ψ k (O); K = C∗ Υkτ + Ξkτ , ψ k (O); K .
(4.90)
We shall imitate the proof of Lemma 4.6 to prove that the homomorphism k ψ ∗ : C∗ (Υτ + Ξτ , O; K) → C∗ Υkτ + Ξkτ , ψ k (O); K
(4.91)
is an isomorphism. Let (W0 (O), W0− (O)) be a Gromoll–Meyer pair of Υτ at O ⊂ N (O)0 (). Since (4.89) implies √ that ψ k : N(O)0 () → N (ψ k (O))0 ( k) is a bundle isomorphism. Hence k ψ W0 (O) , ψ k W0− (O) √ is a Gromoll–Meyer pair of Υkτ at ψ k (O) ⊂ N (ψ k (O))0 ( k). For j = 1, k let us write N(ψ j (O))⊥ = N (ψ j (O))+ ⊕ N (ψ j (O))− and ⊥ + − N ψ j (O) ( j ) = N ψ j (O) ( j ) ⊕ N ψ j (O) ( j ), Ξj τ y, v ⊥ = Ξj+τ (y, v + ) + Ξj τ (y, v − ), v⊥ = v+ + v−. By (4.78), for p ∈ N (O)± (), there hold ± k ψ (p) = kΞτ± (p). Ξkτ
(4.92)
− (O)) be a Gromoll–Meyer pair of Ξτ− at O ⊂ N (O)− (). Then Let (W11 (O), W11
− k (O) ψ W11 (O) , ψ k W11
(4.93)
√ − at ψ k (O) ⊂ N (ψ k (O))− ( k) because (4.89) implies that is a Gromoll–Meyer pair of Ξkτ √ ψ k : N (O)− () → N (ψ k (O))− ( k) is a bundle isomorphism. For 0 < δ , set
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W12 := (y, v) ∈ N (O)+ () v τ δ , − W12 := (y, v) ∈ N (O)+ () v τ = δ , √ + V12 := (y, v) ∈ N ψ k (O) () v kτ kδ , √ + − := (y, v) ∈ N ψ k (O) () v kτ = kδ . V12 − − + ) (resp. (V12 , V12 )) is a Gromoll–Meyer pair of Ξτ+ (resp. Ξkτ ) at O ⊂ Then (W12 , W12 √ + k k + N(O) () (resp. ψ (O) ⊂ N (ψ (O)) ( k)), and that k − − ψ (W12 ), ψ k W12 ⊂ V12 , V12 . (4.94)
By Lemma 5.1 on the page 51 of [7], we may take W1 (O) := W11 (O) ⊕ W12 , − − W1− (O) := W11 (O) ⊕ W12 ∪ W11 (O) ⊕ W12 , V := ψ k W11 (O) ⊕ V12 , − k − ∪ ψ W11 (O) ⊕ V12 V − := ψ k W11 (O) ⊕ V12 and get a Gromoll–Meyer pair of Υτ + Ξτ at O ⊂ N (O)(), (W (O), W (O)− ), where W (O) := W0 (O) ⊕ W11 (O) ⊕ W12 , − − W − (O) := W0 (O) ⊕ W11 (O) ⊕ W12 ∪ W11 (O) ⊕ W12 ∪ W0− (O) ⊕ W11 (O) ⊕ W12 .
(4.96)
C∗ (Υτ + Ξτ , 0; K) = H∗ W (O), W − (O); K .
(4.97)
(4.95)
Therefore
√ Similarly, we have a Gromoll–Meyer pair of Υkτ + Ξkτ at ψ k (O) ⊂ N (ψ k (O))( k), (W (ψ k (O)), W (ψ k (O))− ), where W ψ k (O) := ψ k W0 (O) ⊕ ψ k W11 (O) ⊕ V12 , − k − ∪ ψ W11 (O) ⊕ V12 W − ψ k (O) := ψ k W0 (O) ⊕ ψ k W11 (O) ⊕ V12 ∪ ψ k W0− (O) ⊕ ψ k W11 (O) ⊕ V12 . It follows that C∗ Υkτ + Ξkτ , ψ k (O); K = H∗ W ψ k (O) , W − ψ k (O) ; K .
(4.98)
Note that ψ k (W (O)) = ψ k (W0 (O)) ⊕ ψ k (W11 (O)) ⊕ ψ k (W12 ) and − k − ∪ ψ W11 (O) ⊕ ψ k (W12 ) ψ k W − (O) = ψ k W0 (O) ⊕ ψ k W11 (O) ⊕ ψ k W12 ∪ ψ k W0− (O) ⊕ ψ k W11 (O) ⊕ ψ k (W12 ) . (4.99)
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Since ψ k : N + (O) → N + (ψ k (O)) is a continuous bundle injection, by (4.94) and the construc− − − tions of (V12 , V12 ) and (W12 , W12 ) above (4.94) it is readily checked that (ψ k (W12 ), ψ k (W12 )) − is a deformation retract of (V12 , V12 ). It follows that k ψ W (O) , ψ k W − (O) ⊂ W ψ k (O) , W − ψ k (O) is a deformation retract of (W (ψ k (O)), W − (ψ k (O))). Hence k ψ ∗ : H∗ W (O), W − (O); K → H∗ W ψ k (O) , W − ψ k (O) ; K is an isomorphism. Therefore, by (4.97) and (4.98), the homomorphism (ψ k )∗ in (4.91) is an isomorphism. Lemma 4.10 is proved. 2 When γ is constant, i.e. O = Sτ · γ is an isolated critical point, this case has been proved in Theorem 4.4. Combing this with Lemma 4.10, and (4.66) and (4.88) we get Theorem 4.11. For an isolated critical submanifold O = Sτ · γ of Lτ in Hτ (α), suppose that for some k ∈ N the critical submanifold ψ k (O) = Skτ · γ k of Lkτ in Hkτ (α k ) is also isolated, and 0 k − k 0 that (4.89) is satisfied, i.e. m− kτ (Skτ · γ ) = mτ (Sτ · γ ) and mkτ (Skτ · γ ) = mτ (Sτ · γ ). Then for c = Lτ |O and small > 0 there exist topological Gromoll–Meyer pairs of Lτ at O ⊂ Hτ (α) and of Lkτ at ψ k (O) ⊂ Hkτ (α k ) (O), W (O)− ⊂ (Lτ )−1 [c − , c + ], (Lτ )−1 (c − ) and W k k ψ (O) − ⊂ (Lkτ )−1 [kc − k, kc + k], (Lkτ )−1 (kc − k) , ψ (O) , W W such that k k k (O) , ψ k W (O)− ⊂ W ψ (O) , W ψ (O) − ψ W and that the iteration map ψ k : Hτ (α) → Hkτ (α k ) induces an isomorphism: (O), W (O)− ; K ψ∗k : C∗ (Lτ , O; K) := H∗ W k k ψ (O) , W ψ (O) − ; K . → C∗ Lkτ , ψ k (O); K := H∗ W Lemma 4.12. Suppose that Cq (Lτ , O; K) = 0 for O = Sτ · γ . Then q − 2n q − 1 − m0τ (O) m− τ (O) q − 1
(4.100)
if O is not a single point critical orbit, i.e. γ is not constant, and q − 2n q − m0τ (O) m− τ (O) q otherwise.
(4.101)
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Proof. If γ is not a constant solution, it follows from (4.66) and (4.67) that Cq−1−m−τ (O) (Fτ |N (O)0γ () , γ ; K) ∼ = Cq (Lτ , O; K) = 0.
(4.102)
Since γ is an isolated critical point of Fτ |N (O)0γ () in N (O)0γ () and N (O)0γ () has dimension m0τ (O), we get 0 0 0 q − 1 − m− τ (O) dim N (O)γ () = mτ (O).
(4.103)
By (4.53), m0τ (O) = m0τ (γ ) − 1 2n − 1. Eq. (4.100) easily follows from this and (4.103). If γ is a constant solution, i.e. O = {γ }, using the isomorphisms above (4.11) and (4.24) we derive Cq (α˜ τ + β˜τ , 0; K) ∼ = Cq (L˜ τ , γ˜ ; K) = 0,
where γ˜ = (φτ )−1 (γ ).
On the other hand, (3.11) and the shifting theorem ([14] and [7, p. 50]) imply Cq (ατ + βτ , 0; K) ∼ = Cq−m−τ (γ ) (α˜ τ , 0; K). Since α˜ τ is defined on a manifold of dimension m0τ (γ ) 2n, (4.101) follow immediately.
2
Lemma 4.13. Suppose that Cq (Lτ , O; K) = 0 for O = Sτ · γ . If either O is not a single point critical orbit and q > 1, or O is a single point critical orbit and q > 0, then each point in O is non-minimal saddle point. Proof. When O is a single point critical orbit and q > 0, the conclusion follows from [7, Ex. 1, p. 33]. Now assume that O is not a single point critical orbit and q > 1. For any y ∈ O, by (4.66) and the formula above (4.67) we have 0 = Cq (Fτ , O; K) ∼ =
q Cq−j (Fτ |N (O)y () , y; K) ⊗ Hj (Sτ ; K) j =0
∼ = Cq−1 (Fτ |N (O)y () , y; K). Since y is an isolated critical point of Fτ |N (O)y () and q − 1 > 0, we derive from [7, Ex. 1, p. 33] that y is a non-minimal saddle point of Fτ |N (O)y () . This implies that y is a non-minimal saddle point of Lτ on the submanifold Ψτ (N (O)y ()) ⊂ Hτ (α) (and therefore on Hτ (α)). 2 5. Proof of Theorem 1.1 5.1. Proof of (i) For any τ ∈ N, let Hτ (α k ) denote the Hilbert manifold of W 1,2 -loops γ : R/τ Z → M representing α k . Since Hr (C(R/τ Z, M; α k ); K) = Hr (C(R/Z, M; α k ); K) and the inclusion Hτ (α k ) → C(R/τ Z, M; α k ) is a homotopy equivalence, rank Hr Hτ α k ; K = 0 ∀τ, k ∈ N.
(5.1)
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By [4] the functional Lτ on the Hilbert manifold Hτ (α k ) is C 2 -smooth, bounded below, satisfies the Palais–Smale condition, and all critical points of it have finite Morse indexes and nullities. In particular, the critical set K(Lτ , α k ) of Lτ on Hτ (α k ) is nonempty because Lτ can attain the minimal value on Hτ (α k ). Clearly, for any τ, k ∈ N we may assume that each critical point of Lτ on Hτ (α k ) is isolated. By contradiction we make: Assumption F (α). (i) For any given integer k > 0, the system (1.6) only possesses finitely many distinct, k-periodic solutions representing α k , (ii) there exists an integer k0 > 1 such that for each integer k > k0 , any k-periodic solution γ˜ of the system (1.6) representing α k must be an iteration of some l-periodic solution γ of the system (1.6) representing α l with l k0 and k = ls for some s ∈ N. Under this assumption we have integer periodic solutions γˆi of the system (1.6) of period τi k0 and representing α τi , i = 1, . . . , p, such that for each integer k > k0 any integer k-periodic solution γ of the system (1.6) representing α k must be an iteration of some γˆi , i.e. τ/τ γ = γˆil for some l ∈ N with lτi = k. Set τ := k0 ! (the factorial of k0 ) and γi = γˆi i , i = 1, . . . , p. Then each γi is a τ -periodic solution of the system (1.6) representing α τ . We conclude Claim 5.1. For any k ∈ N, it holds that K Lkτ , α kτ = γjk 1 j p .
(5.2)
Proof. Let γ ∈ K(Lkτ , α kτ ). Since kτ > k0 , by (ii) in Assumption F (α) we have γ = γˆil for τ/τ some l ∈ N with lτi = kτ . Hence γ = γˆil = (γˆi )kτ/τi = (γˆi i )k = γik . 2 Since M is not assumed to be orientable, it is possible that the pullback bundle γj∗ T M → R/τ Z is not trivial. However, each 2-fold iteration γj2 , (γj2 )∗ T M → R/2τ Z is always trivial. Note that (5.2) implies k K L2kτ , α 2kτ = γj2 = γj2k 1 j p .
(5.3)
Hence replacing {γ1 · · · γp } by {γ12 · · · γp2 } we may assume: γj∗ T M → R/τ Z,
j = 1, . . . , p,
are all trivial.
(5.4)
Lemma 5.2. For each k ∈ N there exists γk ∈ K(Lkτ , α kτ ) such that Cr Lkτ , γk ; K = 0 and r − 2n r − m0kτ γk m− kτ γk r. Proof. Let c1 < · · · < cl be all critical values of Lτ , l p. Then kc1 < · · · < kcl are all critical values of Lkτ , k = 1, 2, . . . . In particular, inf Lkτ = kc1 because Lkτ is bounded below and satisfies the Palais–Smale condition. By (5.1), rank Hr (Hkτ (α kτ ); K) m for some m ∈ N. Recall that a subset of an abelian group is defined to be linearly independent if it satisfies the usual condition with integer coefficients, cf. [31, p. 87]. Take linearly independent elements of Hr (Hkτ (α kτ ); K), β1 , . . . , βm , and singular
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cycles Z1 , . . . , Zm of Hkτ (α kτ ) which represent them. Let S be a compact set containing the supports of Z1 , . . . , Zm . Then S ⊂ (Lkτ )b := {Lkτ b} for a sufficiently large regular value b > kcl . Note that Z1 , . . . , Zm are also singular cycles of (Lkτ )b , and that non-trivial K-linear combination of them cannot be homologous to zero in (Lkτ )b (otherwise the same combination is homologous to zero in Hkτ (α kτ )). Hence we get rank Hr (Lkτ )b ; K m > 0. Take the regular values of Lkτ , a0 < a1 < · · · < al = b such that kci ∈ (ai−1 , ai ), i = 1, . . . , l. By Theorem 4.2 of [7, p. 23],
Hr (Lkτ )ai , (Lkτ )ai−1 ; K ∼ =
Cr (Lkτ , z; K).
(5.5)
Lkτ (z)=kci ,d Lkτ (z)=0
Since each critical point has finite Morse index, it follows from the generalized Morse lemma that each group Cr (Lkτ , z; K) has finite rank, and therefore that rank Hr (Lkτ )ai , (Lkτ )ai−1 ; K < +∞,
i = 1, . . . , l.
By the arguments on the page 38 of [7] and the fact (b) on the page 87 of [31], for a triple Z ⊂ Y ⊂ X of topological spaces it holds that rank Hq (X, Z; K) rank Hq (X, Y ; K) + rank Hq (X, Y ; K) if these three numbers are finite. It follows that 0 < m rank Hr (Lkτ )b ; K = rank Hr (Lkτ )al , (Lkτ )a0 ; K
m
rank Hr (Lkτ )ai , (Lkτ )ai−1 ; K < +∞.
i=1
Hence rank Hr ((Lkτ )ai , (Lkτ )ai−1 ; K) 1 for some i. By (5.5) we get a γk ∈ K(Lkτ , α kτ ) such that rank Cr (Lkτ , γk ; K) = 0 and thus Cr (Lkτ , γk ; K) = 0. Noting (5.4), we can use the isomorphism above (4.11) to derive Cr L˜ kτ , γ˜k ; K = 0,
where γ˜k = (φkτ )−1 γk .
Replacing γ˜ k in (4.12) by γ˜k , and using the isomorphism above (4.24), (3.11) and the shifting theorem ([14] and [7, p. 50]) we get Cr−m− (γ ) (α˜ kτ , 0; K) ∼ = Cr (αkτ + βkτ , 0; K) ∼ = Cr L˜ kτ , γ˜k ; K = 0. kτ
k
Since α˜ kτ is defined on a manifold of dimension m0kτ (γk ) 2n, the desired inequalities follow immediately. 2
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Lemma 5.3. Without Assumption F (α), let γ be an isolated critical point of Lτ in Hτ (α τ ) such that γ ∗ T M → Sτ is trivial. For every integer q n + 1, let k(q, γ ) = 1 if m ˆ− τ (γ ) = 0, and − (γ ) = 0. Assume that γ k is also an isolated critical point of L for some k(q, γ ) = mˆq+n if m ˆ kτ − τ τ (γ ) integer k > k(q, γ ). Then Cq Lkτ , γ k ; K = 0.
(5.6)
Proof. Let φkτ : W 1,2 (Sτ , Bρn (0)) → Hkτ (α kτ ) be a coordinate chart on Hkτ (α kτ ) around γ k − 0 0 as in (3.8). Set γ˜ = (φτ )−1 (γ ). Then γ˜ k = (φkτ )−1 (γ k ) and m− τ (γ˜ ) = mτ (γ ), mτ (γ˜ ) = mτ (γ ) − − 0 0 k k k k and mkτ (γ˜ ) = mkτ (γ ) and mkτ (γ˜ ) = mkτ (γ ). As in the proof of Lemma 5.2, by the isomorphisms above (4.11) and (4.24) we have Cq Lkτ , γ k ; K ∼ = Cr L˜ kτ , γ˜ k ; K ∼ = Cq (α˜ kτ + β˜kτ , 0; K) ∼ = Cq−m− (γ k ) (α˜ kτ , 0; K). kτ
Here α˜ kτ is defined on a manifold of dimension m0kτ (γ k ) 2n. − 0 k k If m ˆ− τ (γ ) = 0, by (3.2) (or (3.18)) we have 0 mkτ (γ ) n − mkτ (γ ). Hence k q − n − m0kτ γ k 1 + m0kτ γ k . q − m− kτ γ This gives Cq−m− (γ k ) (α˜ kτ , 0; K) = 0. kτ
− k If m ˆ− ˆ− τ (γ ) > 0, by (3.2) (or (3.18)) we have k m τ (γ ) − n mkτ (γ ) and thus
k − q − m− q − km ˆ τ (γ ) − n = q + n − k m ˆ− τ (γ ) < 0 kτ γ if k >
q+n . m ˆ− τ (γ )
This also leads to Cq−m− (γ k ) (αkτ , 0; K) = 0. kτ
2
So we immediately get the following generalization of Lemma 4.2 in [23]. Corollary 5.4. Under Assumption F (α), for every integer q n + 1 there exists a constant k0 (q) > 0 such that for every integer k k0 (q) there holds Cq (Lkτ , y; K) = 0 ∀y ∈ K Lkτ , α kτ . Here k0 (q) = 1 if m ˆ− τ (γj ) = 0 for all 1 j p, and q + n − ˆ τ (γj ) = 0, 1 j p k0 (q) = 1 + max m m ˆ− τ (γj ) otherwise. ([s] denotes the largest integer less than or equal to s.)
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Indeed, by (5.2) we may assume y = γjk for some 1 j p. Then Lemma 5.3 yields the desired conclusion. Clearly, if r n + 1 then Lemmas 5.2 and 5.3 immediately give a contradiction. Theorem 1.1(i) is proved in this case. In the following we consider the case r = n. Under Assumption F (α) we apply Lemma 5.2 to all k ∈ {2m | m ∈ {0} ∪ N} to get an infinite subsequence Q of {2m | m ∈ {0} ∪ N}, some l ∈ N and a γ ∈ {γ1 , . . . , γp } such that − 0 0 kl l kl l Cn (Lklτ , γ kl ; Z2 ) = 0, m− klτ (γ ) = mlτ (γ ) and mklτ (γ ) = mlτ (γ ) for any k ∈ Q. In order to save notations we always assume l = 1 in the following. That is, we have γ k ∈ K(Lkτ , α kτ ) with k Cn L kτ , γ ; K = 0, (5.7) k = m− (γ ), m0kτ γ k = m0τ (γ ) m− τ kτ γ for any k ∈ Q. By Corollary 5.4 there exists k0 > 0 such that for any γ ∈ {γ1 , . . . , γp }, Cn+1 Lkτ , γ k ; K = 0 ∀k ∈ Q(k0 ) := {k ∈ Q | k k0 }.
(5.8)
To avoid the finite energy homology introduced and used in [23] we need to improve the proof and conclusions of Theorem 4.3 in [23]. Let c = Lτ (γ ). Take > 0 sufficiently small so that for each k ∈ N the interval [k(c − 3), k(c + 3)] contains a unique critical value kc of Lkτ on Hkτ (α kτ ), i.e. Lkτ K Lkτ , α kτ ∩ k(c − 3), k(c + 3) = {kc}. By Theorem 4.4, for each integer k ∈ Q we may choose topological Gromoll–Meyer pairs of Lτ at γ and Lkτ at γ k , (W (γ ), W (γ )− ) and (W (γ k ), W (γ k )− ), such that W (γ ), W (γ )− ⊂ (Lτ )−1 [c − 2, c + 2] , (Lτ )−1 (c − 2) , k k − W γ ,W γ ⊂ (Lkτ )−1 [kc − 2k, kc + 2k] , (Lkτ )−1 (kc − 2k) , − k ψ W (γ ) , ψ k W (γ )− ⊂ W γ k , W γ k and that the iteration map ψ k : Hτ (α) → Hkτ (α k ) induces isomorphisms k ψ ∗ : C∗ (Lτ , γ ; K) = H∗ W (γ ), W (γ )− ; K − → C∗ Lkτ , γ k ; K = H∗ W γ k , W γ k ; K . For j = 1, k, denote by the inclusions − j → (Lj τ )j (c+2) , (Lj τ )j (c−2) , h1 : W γ j , W γ j j h2 : (Lj τ )j (c+2) , (Lj τ )j (c−2) → (Lj τ )j (c+2) , (Lj τ )◦j (c−) , j h3 : (Lj τ )j (c+2) , (Lj τ )◦j (c−) → Hj τ , (Lj τ )◦j (c−) .
(5.9) (5.10) (5.11)
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Hereafter B ◦ denote the interior of B without special statements. The arguments above [23, Th. 4.3] show that − j j h2 ◦ h1 ∗ : H∗ W γ j , W γ j ; K → H∗ (Lj τ )j (c+2) , (Lj τ )◦j (c−) ; K , j h3 ∗ : H∗ (Lj τ )j (c+2) , (Lj τ )◦j (c−) ; K → H∗ Hj τ , (Lj τ )◦j (c−) ; K are monomorphisms on homology modules. For j = 1, k, we have also inclusions − → (Lj τ )−1 [j c − 2j , j c + 2j ] , (Lj τ )−1 (j c − 2j ) , Ij : W γ j , W γ j Jj : (Lj τ )−1 [j c − 2j , j c + 2j ] , (Lj τ )−1 (j c − 2j ) → Hj τ , (Lj τ )◦j c−j . It is clear that j
j
j
Jj ◦ Ij = h3 ◦ h2 ◦ h1 ,
j = 1, k.
(5.12)
By (5.11), we have also ψ k ◦ I1 = Ik ◦ ψ k as maps from (W (γ ), W (γ )− ) to ((Lkτ )−1 ([kc − 2k, kc + 2k]), (Lkτ )−1 (kc − 2k)). So we get the following result, which is a slightly strengthened version of [23, Th. 4.3] in the case M = T n. Proposition 5.5. Under Assumption F (α), there exist a periodic solution γ of (1.6) of integer period τ and representing α, a large integer k0 > 0, an infinite integer set Q containing 1, and a small > 0 having properties: For any k ∈ Q(k0 ) := {k ∈ Q | k k0 } there exist topological Gromoll–Meyer pairs (W (γ ), W (γ )− ) and (W (γ k ), W (γ k )− ) satisfying (5.9)–(5.11) such that for the inclusion − jkτ = hk3 ◦ hk2 ◦ hk1 : W γ k , W γ k → Hkτ α kτ , (Lkτ )◦k(c−) the following diagram holds: (ψ k )∗ Cn Lkτ , γ k ; K 0 = Cn (Lτ , γ ; K) −−−→ (j )∗ −−kτ−→ Hn Hkτ α kτ , (Lkτ )◦k(c−) ; K ≡ Hk ,
(5.13)
where c = Lτ (γ ), (ψ k )∗ is an isomorphism, and (jkτ )∗ is a monomorphism among the singular homology modules. In particular, if ω is a generator of Cn (Lτ , γ ; K) = Hn (W (γ ), W (γ )− ; K), then (jkτ )∗ ◦ ψ k ∗ (ω) = 0 in Hk , (jkτ )∗ ◦ ψ k ∗ (ω) = (Jk )∗ ◦ (Ik )∗ ◦ ψ k ∗ (ω) = (Jk )∗ ◦ ψ k ∗ ◦ (I1 )∗ (ω) It is (5.15) that helps us avoiding to use the finite energy homology.
(5.14) in Hk .
(5.15)
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The notion of a C 1 -smooth singular simplex in Hilbert manifolds can be defined as on page 252 of [31]. Proposition 5.6. For τ ∈ N, c ∈ R, > 0, q 0, and a C 1 -smooth q-simplex η : ( q , ∂ q ) → Hτ α τ , (Lτ )◦c− , there exists an integer k(η) > 0 such that for every integer k k(η), the q-simplex ηk ≡ ψ k (η) : ( q , ∂ q ) → Hkτ α kτ , (Lkτ )◦k(c−) is homotopic to a singular q-simplex ηk : ( q , ∂ q ) → (Lkτ )◦k(c−) , (Lkτ )◦k(c−)
(5.16)
with ηk = ηk on ∂ q and the homotopy fixes ηk |∂ q . This is an analogue of [3, Th. 1], firstly proved by Y. Long [23, Prop. 5.1] in the case M = T n . Proposition 5.1 in [23] actually gave stronger conclusions under weaker assumptions: If the q-simplex η above is only a finite energy one (C 1 -smooth simplex must be of finite energy), then the simplex ηk is finite energy homotopic to a finite energy q-simplex ηk . Hence Proposition 5.6 can be derived with the same reason as in [23, Prop. 5.1] as long as we generalize an inequality as done in Lemma A.4 of Appendix A. But we also give necessary details for the reader’s convenience. Proof of Proposition 5.6. Recall that for paths σ : [a1 , a2 ] → M and δ : [b1 , b2 ] → M with σ (a2 ) = δ(b1 ) one often define new paths σ −1 : [a1 , a2 ] → M by σ −1 (t) := σ (a2 + a1 − t) and σ ∗ δ : [a1 , a2 + b2 − b1 ] → M by σ ∗ δ|[a1 ,a2 ] = σ and σ ∗ δ(t) := δ(t − a2 + b1 )
for t ∈ [a2 , a2 + b2 − b1 ].
Given a C 1 -path ρ : [a, b] → Hτ (α τ ) and an integer k 3 we want to construct a path ρk : [a, b] → Hkτ (α kτ ) such that ρk (a) = ψ k ρ(a)
and ρk (b) = ψ k ρ(b) .
Define the initial point curve βρ of ρ by [a, b] → M,
s → βρ (s) = ρ(s)(0).
It is C 1 -smooth. Following [23, p. 460] and [3, p. 381], for 0 s (b − a)/k and 1 j k − 2 define • ρ˜k (a + s) = ρ(a)k−1 ∗ (βρ |[a,a+ks] ) ∗ ρ(a + ks) ∗ (βρ |[a,a+ks] )−1 , • ρ˜k (a + j (b − a)/k + s) = ρ(a)k−j −1 ∗ (βρ |[a,a+ks] ) ∗ ρ(a + ks) ∗ (βρ |[a,a+ks] ) ∗ ρ(b)j ∗ (βρ )−1 , • ρ˜k (b − (b − a)/k + s) = ρ(a + ks) ∗ (βρ |[a,a+ks] ) ∗ ρ(b)k−1 ∗ (βρ |[a,a+ks] )−1 .
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These are piecewise C 1 -smooth loops in M representing α k , and ρ(a) ˜ = ρ(a)k−1
and ρ(b) ˜ = ρ(b) ∗ βρ ∗ ρ(b)k−1 ∗ βρ−1 .
For each u ∈ [a, b], reparametrizing the loop ρ˜k (u) on R/kτ as in [23, p. 461] we get a piecewise C 1 -smooth loop ρk (u) ∈ Hkτ (α kτ ) and therefore a piecewise C 1 -smooth path ρk : [a, b] → Hkτ (α kτ ) with ρk (a) = ψ k (ρ(a)) = ρ(a)k and ρk (b) = ψ k (ρ(b)) = ρ(b)k . Replacing all the terms of powers of ρ(a) and ρ(b) by the constant point paths in the definition of ρ˜k above, we get a piecewise C 1 -smooth path βρ,k : [a, b] → Hτ (α). For s ∈ [a, b] and j = [k(s − a)/(b − a)], by the arguments of [23, p. 461], Lkτ ρk (s) = (k − j − 1)Lτ ρ(a) + j Lτ ρ(b) + Lτ βρ,k (s) (k − 1)M0 (ρ) + M1 (ρ) + 2M2 (ρ),
(5.17)
where M0 (ρ) = max{Lτ (ρ(a)), Lτ (ρ(b))}, M1 (ρ) = maxasb |Lτ (ρ(s))| and
b M2 (ρ) =
L s, βρ (s), β˙ρ (s) ds.
(5.18)
a
Note that (L3 ) implies L(t, q, v) C 1 + v 2
∀(t, q, v) ∈ R × T M
(5.19)
for some constant C > 0. Therefore it follows from Lemma A.4 that
b M2 (ρ) =
L s, βρ (s), β˙ρ (s) ds
a
b (b − a)C + C
β˙ρ (s) 2 ds (b − a)C + 1 + τ Cc(ρ). 2τ
a
This and (5.17) yield lim sup max
k→+∞
asb
1 Lkτ ρk (s) M0 (ρ). k
(5.20)
Next replacing [23, Lem. 2.3] by Lemma A.4, and almost repeating the reminder arguments of the proof of [23, Prop. 5.1], we can complete the proof of Proposition 5.6. 2 Lemma 5.7. (See [3, Lem. 1].) Let (X, A) be a pair of topological spaces and β a singular relative p-cycle of (X, A). Let Σ denote the set of singular simplices of β together with all their faces. Suppose to every σ ∈ Σ, σ : q → X, 0 q p, there is assigned a map P (σ ) : q × [0, 1] → X such that
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(i) (ii) (iii) (iv)
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P (σ )(z, 0) = σ (z) for z ∈ q , P (σ )(z, t) = σ (z) if σ ( q ) ⊂ A, P (σ )( q × {1}) ⊂ A, P (σ ) ◦ (eqi × id) = P (σ ◦ eqi ) for 0 i q.
Then the homology class [β] ∈ Hp (X, A) vanishes. For the class ω in (5.15), by the definition of I1 above (5.12) we have (I1 )∗ (ω) ∈ Hn (Lτ )−1 [c − 2, c + 2] , (Lτ )−1 (c − 2); K .
(5.21)
Since both (Lτ )−1 ([c −2, c +2]) and (Lτ )−1 (c −2) are at least C 2 -smooth Hilbert manifolds, we can choose a C 1 -smooth cycle representative σ of the class (I1 )∗ (ω). Denote by Σ(σ ) the set of all simplexes together with all their faces contained in σ . By [7, Ex. 1, p. 33] each γ k in (5.7) is a non-minimal saddle point of Lkτ on Hkτ (α kτ ). As in the proof of [23, Prop. 5.2] we can use Proposition 5.6 and Lemma A.4 to get the corresponding result without using the finite energy homology. Proposition 5.8. There exists a sufficiently large integer k(σ ) k0 such that for every integer k ∈ Q(k(σ )) and for every μ ∈ Σ(σ ) with μ : r → Hτ (α τ ) and 0 r n, there exists a homotopy P (ψ k (μ)) : r × [0, 1] → Hkτ (α kτ ) such that the properties (i) to (iv) in Lemma 5.7 hold for (X, A) = (Hkτ (α kτ ), (Lkτ )◦k(c−) ). It follows that the homology class (Jk )∗ ◦ (ψ k )∗ ◦ (I1 )∗ (ω) ∈ Hk vanishes. By (5.15), (jkτ )∗ ◦ (ψ k )∗ (ω) = 0 in Hk . This contradicts to (5.14). Therefore Assumption F (α) cannot hold. Theorem 1.1(i) is proved. 5.2. Proof of (ii) Since the inclusion Eτ → C(R/τ Z, M) is a homotopy equivalence, and therefore rank Hr (Eτ ; K) = 0 for all τ ∈ N. Consider the functional Lkτ on Ekτ . It has still a nonempty critical point set. Replace Assumption F (α) by Assumption F. (i) For any given integer k > 0, the system (1.6) only possesses finitely many distinct, k-periodic solutions, (ii) there exists an integer k0 > 1 such that for each integer k > k0 , any k-periodic solution γ˜ of the system (1.6) must be an iteration of some l-periodic solution γ of the system (1.6) with l k0 and k = ls for some s ∈ N. Then slightly modifying the proof of (i) above one can complete the proof. 6. Proof of Theorem 1.4 The proof is similar to that of Theorem 1.1. We only give the main points. Identifying R/τ Z = [− τ2 , τ2 ]/{− τ2 , τ2 }, let C(R/τ Z, M)e := x ∈ C(R/τ Z, M) x(−t) = x(t), −τ/2 t τ/2 .
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We have a contraction from C(R/τ Z, M)e to the subset of constant loops in C(R/τ Z, M)e which is identified with M: [0, 1] × C(R/τ Z, M)e → C(R/τ Z, M)e ,
(s, x) → xs ,
where xs (t) = x(st) for −τ/2 t τ/2. Since the inclusion C(R/τ Z, M)e → EHτ is also a homotopy equivalence, we get Hn (EHτ ; Z2 ) = Hn C(R/τ Z, M)e ; Z2 = Hn (M; Z2 ) = 0
(6.1)
for any τ > 0. Note that LE τ can always attain the minimal value on EHτ and therefore has a E nonempty critical set K(Lτ ). Under the conditions (L1)–(L4) we replace Assumption F (α) in Section 5 by Assumption FE. (i) For any given integer k > 0, the system (1.6) possesses only finitely many distinct reversible kτ -periodic solutions, (ii) there exists an integer k0 > 1 such that for each integer k > k0 , any reversible kτ -periodic solution γ˜ of the system (1.6) is an iteration of some reversible lτ -periodic solution γ of the system (1.6) with l k0 and k = ls for some s ∈ N. Under this assumption, as the arguments below Assumption F (α) we may get an integer τ ∈ N and finitely many reversible τ -periodic solutions of the system (1.6), γ1 · · · γp , such that for any k ∈ N every reversible kτ -periodic solution of the system (1.6) has form γjk for some 1 j p. Namely, k K LE kτ = γj 1 j p .
(6.2)
Using the same proof as one of Lemma 5.2 we may obtain: Lemma 6.1. Under Assumption FE, for each k ∈ N there exists a critical point γk of LE kτ such that Cn L E kτ , γk ; Z2 = 0 and
− n n − m01,kτ γk m− 1,kτ γk n.
(6.3)
Let k0 = 1 if m ˆ− 1,τ (γj ) = 0 for all 1 j p, and 3n + 2 − m ˆ (γ ) = 0, 1 j p k0 = 1 + max 1,τ j 2m ˆ− 1,τ (γj ) otherwise. Corresponding with Corollary 5.4 we have the following generalization of [30, Lem. 4.4]. Lemma 6.2. Under Assumption FE, for any integer number k k0 , every isolated critical point z of LE kτ has the trivial (n + 1)th critical module, i.e. Cn+1 LE kτ , z; K = 0.
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E in (3.29) let z˜ = (φ E )−1 (z). We only need to prove Proof. Using the chart φkτ kτ
Cn+1 L˜ E kτ , z˜ ; K = 0
∀k k0 .
(6.4)
Let z = γjk and thus z˜ = γ˜jk with γ˜j = (φτE )−1 (γj ). By (4.41), it follows from Shifting theorem [7, p. 50, Th. 5.4] and the Künneth formula that E E ∼ Cn+1 L˜ E kτ , z˜ ; K = Cn+1 αkτ + βkτ , 0; K E E ∼ , 0; G ⊗ Cm− (γ˜ k ) βkτ , 0; K = Cn+1−m− (γ˜ k ) αkτ 1,kτ j 1,mτ j E ∼ = Cn+1−m− (γ˜ k ) αkτ , 0; K ⊗ K 1,kτ j E ∼ , 0; K = Cn+1−m− (γ˜ k ) αkτ 1,kτ
j
E . If (6.4) does not hold, we because 0 is a nondegenerate critical point of quadratic function βkτ − k 0 k get that 0 n + 1 − m1,kτ (γ˜j ) m1,kτ (γ˜j ) because γ˜kτ is defined on a manifold of dimension m01,kτ (γ˜jk ). Note that
k − ˜E − k k m− 1,kτ γ˜j = mkτ Lkτ , γ˜j = m1,kτ γj , k k 0 m01,kτ γ˜jk = m0kτ L˜ E kτ , γ˜j = m1,kτ γj . We have E k E k − ˜E k 0 ˜ ˜ or m− 1,kτ Lkτ , γ˜j n + 1 m1,kτ Lkτ , γ˜j + m1,kτ Lkτ , γ˜j − k k 0 k m− 1,kτ γj n + 1 m1,kτ γj + m1,kτ γj
(6.5) (6.6)
for any k ∈ N. By (2.24) E k − ˜E − ˜E k k ˜ m− kτ Lkτ , γ˜j − 2n m2,kτ Lkτ , γ˜j mkτ Lkτ , γ˜j
∀k ∈ N.
Hence it follows from this, (3.26) and (6.5) that E − ˜ − ˜E − ˜E k k k ˜ 2k m ˆ− τ Lτ , γ˜j − n m2,kτ Lkτ , γ˜j + mkτ Lkτ , γ˜j 2mkτ Lkτ , γ˜j 2n + 2. 3n+2 ˜E Therefore, when m ˆ− ˆ− ], which contradicts to k k0 . τ (Lτ , γ˜j ) > 0, k [ 2m 1,τ (γj ) = m ˆ 1,τ (γj ) − − E ˜ ˆ τ (Lτ , γ˜j ) = 0, (3.22) and (6.6) also give a contradiction. The deWhen m ˆ 1,τ (γj ) = m sired (6.4) is proved. 2
Now as the arguments below Corollary 5.4, under Assumption FE we may use Lemma 6.1 to get an infinite subsequence Q of {2m | m ∈ {0} ∪ N} and a γ ∈ {γ1 , . . . , γp } such that Cn L E , γ k ; Z2 = 0, kτ k = m− (γ ), m− 1,kτ γ 1,τ
m01,kτ γ
k
= m01,τ (γ )
(6.7)
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for any k ∈ Q. By Lemma 6.2, for any x ∈ {γ1 , . . . , γp } we have also k Cn+1 LE kτ , x ; K = 0 ∀k ∈ Q(k0 ) := {k ∈ Q | k k0 }.
(6.8)
Then from Proposition 5.5 to the end of Section 5.1 we only need to make suitable replacements for some notations such as Hj τ (α j τ ), Lj τ by EHj τ , LE j τ for j = 1, k, and so on, and can complete the proof of Theorem 1.4. 7. Proof of Theorem 1.6 7.1. Proof of (i) Denote by KO(Lτ , α k ) the set of critical orbits of Lτ on Hτ (α k ). It is always nonempty because Lτ can attain the minimal value on Hτ (α k ). Clearly, we may assume that each critical orbit of Lτ on Hτ (α k ) is isolated for any k ∈ N. As in Section 5.1, by contradiction we assume: Assumption FT(α). (i) For any given integer k > 0, the system (1.6) only possesses finitely many distinct, kτ -periodic solution orbit towers based on kτ -periodic solutions of (1.6) representing α k , (ii) there exists an integer k0 > 1 such that for each integer k > k0 , any kτ -periodic solution γ˜ of the system (1.6) representing α k must be an iteration of some lτ -periodic solution γ of the system (1.6) representing α l with l k0 and k = lq for some q ∈ N. Under this assumption, there only exist finitely many periodic solution orbit towers k s∈R {s · γˆ1k }s∈R k∈N , . . . , {s · γˆp }k∈N of the system (1.6) such that • γˆi has period ki τ k0 τ and represents α ki for some ki ∈ N, i = 1, . . . , p; • for each integer k > k0 any kτ -periodic solution γ of the system (1.6) representing α k must be an iteration of some s · γˆi , i.e. γ = (s · γˆi )l = s · γˆil for some s ∈ R and l ∈ N with lki = k. m/k
Set m := k0 ! (the factorial of k0 ) and γi = γˆi i , i = 1, . . . , p. Then each γi is an mτ -periodic solution of the system (1.6) representing α m . We conclude Claim 7.1. For any k ∈ N, it holds that KO Lkmτ , α km = Skmτ · γjk 1 j p . Proof. Let γ ∈ K(Lkmτ , α km ). Since km > k0 , then γ = (s · γˆi )l for some s ∈ R and l ∈ N with m/k lki = km. Hence γ = s · γˆil = s · (γˆi )km/ki = s · (γˆi i )k = s · γik . 2 Hence replacing τ by mτ we may assume m = 1 below, i.e. ∀k ∈ N. KO Lkτ , α k = Skτ · γjk 1 j p As in Section 5.1 we can also assume: γj∗ T M → R/τ Z, j = 1, . . . , p, are all trivial.
(7.1)
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Lemma 7.2. For each k ∈ N there exists Ok ∈ KO(Lkτ , α k ) such that Cr (Lkτ , Ok ; K) = 0. Moreover, r − 2n r − 1 − m0kτ (Ok ) m− kτ (Ok ) r − 1 if Ok is not a single point critical orbit, (O ) r otherwise. and r − 2n r − m0kτ (Ok ) m− k kτ Proof. By Lemma 4.12 we only need to prove the first claim. The proof is similar to that of Lemma 5.2. Let kc1 < · · · < kcl be all critical values of Lkτ , l p, and inf Lkτ = kc1 , k = 1, 2, . . . . As in the proof of Lemma 5.2 we have a large regular value b of Lkτ such that rank Hr ((Lkτ )b ; K) > 0. Take the regular values of Lkτ , a0 < a1 < · · · < al = b such that kci ∈ (ai−1 , ai ), i = 1, . . . , l. Noting (7.1), by Theorem 2.1 of [38] or the proof of Lemma 4 of [15, p. 502], we get Hr (Lkτ )ai , (Lkτ )ai−1 ; K ∼ =
Cr Lkτ , Skτ · γjk ; K .
Lkτ (γjk )=kci
Since each critical point has finite Morse index, (4.67) implies that each critical group Cr (Lkτ , Skτ · γjk ; K) has finite rank. Almost repeating the proof of Lemma 5.2 we get some Skτ · γjk in KO(Lkτ , α k ) such that rank Cr (Lkτ , Skτ · γjk ; K) > 0 and thus rank Cr (Lkτ , Skτ · γjk ; K) = 0. 2 Corresponding to Corollary 5.4 we have Lemma 7.3. Under Assumption FT(α), for every integer q n + 1 there exists a constant k0 (q) > 0 such that Cq (Lkτ , Ok ; K) = 0 ˆ− for every integer k k0 (q) and Ok ∈ KO(Lkτ , α k ). Here k0 (q) = 1 if m r (γj ) = 0 for all 1 j p, and k0 (q) = 1 + max
q + n − ˆ r (γj ) = 0, 1 j p m m ˆ− r (γj )
otherwise. Proof. Let Ok = Skτ · γjk . If γj is constant, by the proof of Lemma 5.3 we have Cq (Lkτ , Ok ; K) = Cq Lkτ , γjk ; K = 0 ˆ− for any k > k(q, γj ), where k(q, γj ) = 1 if m τ (γj ) = 0, and k(q, γj ) =
q+n m ˆ− τ (γj )
if m ˆ− τ (γj ) = 0.
Suppose that γj is not a constant solution. If Cq (Lkτ , Ok ; K) = 0, Lemma 4.12 yields − k k 0 k m− kτ Skτ · γj q − 1 mkτ Skτ · γj + mkτ Skτ · γj .
(7.2)
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By (4.53) this becomes k k − k 0 m− kτ γj q − 1 mkτ γj + mkτ γj − 1.
(7.3)
If m ˆ τ (γj ) > 0, it follows from (7.3) and (3.2) that − k km ˆ− τ (γj ) − n mkτ γj q − 1
and therefore k
q+n−1 . m ˆ− r (γj )
This contradicts to k k0 (q). If m ˆ τ (γj ) = 0, by (3.2), k k 0 ∀k ∈ N. 0 m− kτ γj n − mkτ γj
It follows that k − k k 0 k 0 m− kτ Skτ · γj + mkτ Skτ · γj = mkτ γj + mkτ γj − 1 n − 1. k 0 k Since q n + 1, (7.2) implies that m− kτ (Skτ · γj ) + mkτ (Skτ · γj ) n. This also gives a contradiction. Lemma 7.3 is proved. 2
Clearly, Lemmas 7.2 and 7.3 imply Theorem 1.1(i) in the case r n + 1. In the following we consider the case r = n. Under Assumption FT(α) we apply Lemma 7.2 to all k ∈ {2m | m ∈ {0} ∪ N} to get an infinite subsequence Q of {2m | m ∈ {0} ∪ N}, some l ∈ N and a γ ∈ {γ1 , . . . , γp } such that Cn (Lklτ , − 0 0 kl l kl l Sklτ · γ kl ; K) = 0, m− klτ (Sklτ · γ ) = mlτ (Slτ · γ ) and mklτ (Sklτ · γ ) = mlτ (Slτ · γ ) for any k ∈ Q. As before we always assume l = 1 in the following. Then we have k Cn L kτ , Skτ · γ ; K = 0 and (7.4) k = m− (S · γ ), m0kτ Skτ · γ k = m0τ (Sτ · γ ) m− τ τ kτ Skτ · γ for any k ∈ Q. By Lemma 7.3 there exists k0 > 0 such that for any γ ∈ {γ1 , . . . , γp }, Cn+1 Lkτ , Skτ · γ k ; K = 0
∀k ∈ Q(k0 ) := {k ∈ Q | k k0 }.
(7.5)
Denote by O = Sτ · γ , and by c = Lτ (γ ) = Lτ (O). Under Assumption FT(α), as in Section 5.1 let us take ν > 0 sufficiently small so that for each k ∈ N the interval [k(c − 3ν), k(c + 3ν)] contains a unique critical value kc of Lkτ on Hkτ (α k ), i.e. Lkτ KO Lkτ , α k ∩ k(c − 3ν), k(c + 3ν) = {kc}. For any k ∈ Q, by Theorem 4.11, we may choose a topological Gromoll–Meyer pair of Lτ at (O), W (O)− ) satisfying O ⊂ Hτ (α), (W (O), W (O)− ⊂ (Lτ )−1 [c − 2ν, c + 2ν] , (Lτ )−1 (c − 2ν) , W and a topological Gromoll–Meyer pair of Lkτ at ψ k (O) ⊂ Hkτ (α k ), k k ψ (O) , W ψ (O) − W
(7.6)
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such that k k k (O) , ψ k W (O)− ⊂ W ψ (O) , W ψ (O) − and ψ W k k ψ (O) − ⊂ (Lkτ )−1 [kc − 2kν, kc + 2kν] , (Lkτ )−1 (kc − 2kν) ψ (O) , W W
(7.7) (7.8)
and that the iteration map ψ k : Hτ (α) → Hkτ (α k ) induces an isomorphism: (O), W (O)− ; K ψ∗k : C∗ (Lτ , O; K) := H∗ W k k ψ (O) , W ψ (O) − ; K . → C∗ Lkτ , ψ k (O); K := H∗ W Identifying ψ(O) = O, for j = 1, k, denote by the inclusions j j j ψ (O) − → (Lj τ )j (c+2ν) , (Lj τ )j (c−2ν) , ψ (O) , W h1 : W j h2 : (Lj τ )j (c+2ν) , (Lj τ )j (c−2ν) → (Lj τ )j (c+2ν) , (Lj τ )◦j (c−ν) , j h3 : (Lj τ )j (c+2ν) , (Lj τ )◦j (c−ν) → Hj τ , (Lj τ )◦j (c−ν) . As in Section 5.1 we have monomorphisms on homology modules, j j j j ψ (O) , W ψ (O) − ; K → H∗ (Lj τ )j (c+2ν) , (Lj τ )◦ h2 ◦ h1 ∗ : H∗ W j (c−ν) ; K , j h3 ∗ : H∗ (Lj τ )j (c+2ν) , (Lj τ )◦j (c−ν) ; K → H∗ Hj τ , (Lj τ )◦j (c−ν) ; K . Moreover, the inclusions j j ψ (O) , W ψ (O) − → (Lj τ )−1 [j c − 2j ν, j c + 2j ν] , (Lj τ )−1 (j c − 2j ν) , Ij : W Jj : (Lj τ )−1 [j c − 2j ν, j c + 2j ν] , (Lj τ )−1 (j c − 2j ν) → Hj τ , (Lj τ )◦j c−j ν satisfy j
j
j
Jj ◦ Ij = h3 ◦ h2 ◦ h1 ,
j = 1, k.
(7.9)
By (7.7), we have also ψ k ◦ I1 = Ik ◦ ψ k
(7.10)
(O), W (O)− ) to ((Lkτ )−1 ([kc − 2kν, kc + 2kν]), (Lkτ )−1 (kc − 2kν)). These as maps from (W yield the following corresponding result with Proposition 5.5. Proposition 7.4. Under Assumption FT(α), there exist a τ -periodic solution γ of (1.6) representing α, a large integer k0 > 0, an infinite integer set Q containing 1, and a small > 0 having properties: For the orbit O = Sτ · γ and any k ∈ Q(k0 ) := {k ∈ Q | k k0 } there exist (ψ k (O)), W (ψ k (O))− ) satisfying (O), W (O)− ) and (W topological Gromoll–Meyer pairs (W (7.6)–(7.8) such that for the inclusion k k ψ (O) , W ψ (O) − → Hkτ α k , (Lkτ )◦ jkτ = hk3 ◦ hk2 ◦ hk1 : W k(c−ν)
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the following diagram holds: ψ∗k 0 = Cn (Lτ , O; K) −−→ Cn Lkτ , ψ k (O); K (j )∗ −−kτ−→ Hn Hkτ α k , (Lkτ )◦k(c−ν) ; K ≡ Hk ,
(7.11)
where c = Lτ (γ ), ψ∗k is an isomorphism, and (jkτ )∗ is a monomorphism among the singular (O), W (O)− ; K), homology modules. In particular, if ω is a generator of Cn (Lτ , O; K) = Hn (W then (jkτ )∗ ◦ ψ k ∗ (ω) = 0 in Hk , (jkτ )∗ ◦ ψ k ∗ (ω) = (Jk )∗ ◦ (Ik )∗ ◦ ψ k ∗ (ω) = (Jk )∗ ◦ ψ k ∗ ◦ (I1 )∗ (ω)
(7.12) in Hk .
(7.13)
Now we can slightly modify the arguments from Proposition 5.6 to Proposition 5.8 to complete the proof of (i). The only place which should be noted is that for ψ k (O) in (7.11) Lemma 4.13 implies each point y ∈ ψ k (O) to be a non-minimum saddle point of Lkτ on Hkτ (α k ) in the case dim M = n > 1. 7.2. Proof of (ii) Proof of (ii) can be completed by the similar arguments as in Section 5.2. 8. Questions and remarks For a C 3 -smooth compact n-dimensional manifold M without boundary, and a C 2 -smooth map H : R × T ∗ M → R satisfying the conditions (H1)–(H5), we have shown in (1◦ ) of Theorem 1.12 that the Poincaré map Ψ H has infinitely many distinct periodic points sitting in the zero section 0T ∗ M of T ∗ M. Notice that the condition (H5) can be expressed as: H (t, x) = H (−t, τ0 (x)) ∀(t, x) ∈ R × M, where τ0 : T ∗ M → T ∗ M, (q, p) → (q, −p), is the standard anti-symplectic involution. So it is natural to consider the following question: Let (P , ω, τ ) be a real symplectic manifold with an anti-symplectic involution τ on (P , ω), i.e. τ ∗ ω = −ω and τ 2 = idP . A smooth time dependent Hamiltonian function H : R × P → R, (t, x) → H (t, x) = Ht (x) is said to be 1-periodic in time and symmetric if it satisfies Ht (x) = Ht+1 (x)
and H (t, x) = H −t, τ (x)
In this case, the Hamiltonian vector fields XHt satisfies XHt+1 (x) = XHt (x) = −dτ τ (x) XH−t τ (x)
∀(t, x) ∈ R × P .
for all (t, x) ∈ R × P .
If the global flow of x(t) ˙ = XHt x(t) exists, denoted by ΨtH , then it is obvious that H = ΨtH ◦ Ψ1H Ψt+1
∀t ∈ R,
−1 Ψ1H ◦ τ = τ ◦ Ψ1H .
(8.1)
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So each τ -invariant k-periodic point x0 , i.e. τ (x0 ) = x0 , of Ψ H = Ψ1H with k ∈ N yields a k-periodic contractible solution x(t) = ΨtH (x0 ) of (8.1) satisfying x(−t) = τ (x(t)) for all t ∈ R. Such a solution is called τ -reversible. By [35, p. 4] the fixed point set L := Fix(τ ) of τ is either empty or a Lagrange submanifold. It is natural to ask the following more general version of the Conley conjecture. Question 8.1. Suppose that L is nonempty and compact, and that (P , ω) satisfies some good condition (e.g. geometrically bounded for some J ∈ RJ (P , ω) := {J ∈ J (P , ω) | J ◦ dτ = −dτ ◦ J } and Riemannian metric μ on P ). Has the system (8.1) infinitely many distinct τ -reversible contractible periodic solutions of integer periods? Furthermore, if the flow ΨtH exists globally, has the Poincaré map Ψ H = Ψ1H infinitely many distinct periodic points sitting in L? Let P0 (H, τ ) denote the set of all contractible τ -reversible 1-periodic solutions of (8.1). Since the Conley conjecture came from the Arnold conjecture, Question 8.1 naturally suggests the following more general versions of the Arnold conjectures. Question 8.2. Under the assumptions of Question 8.1, P0 (H, τ ) CuplengthF (L) for F = Z, Z2 ? Moreover, ifL some nondegenerate assumptions for elements of P0 (H, τ ) are satisfied, P0 (H, τ ) dim k=0 bk (L, F)? This question is closely related to the Arnold–Givental conjecture. In order to study it we try to construct a real Floer homology F H∗ (P , ω, τ, H ) with P0 (H, τ ) under some nondegenerate assumptions for elements of P0 (H, τ ), which is expected to be isomorphic to H∗ (M). Moreover, 1 if L ∈ C 2 (R/Z × T M) satisfies (L1)–(L4) and the functional L(γ ) = 0 L(t, γ (t), γ˙ (t)) dt on EH1 has only nondegenerate critical points, then one can, as in [2, §2.2], construct a Morse complex CM∗ (L) whose homology is isomorphic to H∗ (M) as well. As in [2,32,37], it is also natural to construct an isomorphism between H F∗ (T ∗ M, ωcan , τ0 , H ) and H (CM∗ (L)) and to study different product operations in them. The author believes that the techniques developed in this paper are useful for one to generalize the results of multiple periodic solutions of some Lagrangian and Hamiltonian systems on the Euclidean space to manifolds. Acknowledgments I am greatly indebted to Professor Yiming Long for leading me to this question ten years ago. The author sincerely thanks Professors Le Calvez and C. Viterbo for organizing a seminar of symplectic dynamics at Beijing International Mathematics Center in May 2007, where my interest for this question was aroused again. He also sincerely thanks Professor Alberto Abbondandolo for some helps in understanding his paper. The results and outlines of proofs in this paper were reported in the workshop on Floer Theory and Symplectic Dynamics at CRM of University of Montreal, May 19–23, 2008. I would like to thank the organizers for their invitation, and CRM for hospitality. Finally, I sincerely thank Professor Kung-Ching Chang for his helps in correcting mistakes in the first draft, and the referees for pointing out a few of good improvement suggestions.
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Appendix A A.1. Proof of Proposition A The first claim is a direct consequence of the following (9.4). As to the second, since for each t ∈ R the functions Lt = L(t, ·) and Ht = H (t, ·) are Fenchel transformations of each other, we only need to prove that (H2)–(H3) can be satisfied under the assumptions (L2)–(L3). For conveniences we omit the time variable t. In any local coordinates (q1 , . . . , qn ), we write (q, v) = (q1 , . . . , qn , v1 , . . . , vn ). By definition of H we have
n ∂L ∂L H q, (q, v)vj . (q, v) = −L(q, v) + ∂v ∂vj
(9.1)
j =1
Differentiating both sides with respect to the variable vi we get
n ∂H j =1
∂pj
2 n ∂L ∂ L ∂ 2L q, (q, v) (q, v) = vj (q, v). ∂v ∂vi ∂vj ∂vi ∂vj j =1
2
L Since the matrix [ ∂v∂i ∂v (q, v)] is invertible, it follows that j
∂L ∂H q, (q, v) = vj . ∂pj ∂v
(9.2)
Let p = ∂L ∂v (q, v). Differentiating both sides of (9.1) with respect to the variable qi and using (9.2) we obtain n j =1
vj
∂ 2L ∂L (q, v) − (q, v) ∂qi ∂vj ∂qi
2
n ∂ L ∂H ∂H ∂L ∂L (q, v) + (q, v) = (q, v) q, q, ∂qi ∂v ∂pj ∂v ∂qi ∂vj j =1
n ∂L ∂ 2L ∂H q, (q, v) + = vj (q, v) ∂qi ∂v ∂qi ∂vj j =1
and hence
∂H ∂L ∂L q, (q, v) = − (q, v). ∂qi ∂v ∂qi
(9.3)
Differentiating both sides of (9.2) with respect to the variable vi yields n ∂ 2H ∂ 2L (q, p) (q, v) = δij , ∂pj ∂pk ∂vk ∂vi
i.e.
k=1
2 −1 ∂ 2H ∂ L (q, p) = (q, v) . ∂pi ∂pj ∂vi ∂vj
(9.4)
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Differentiating both sides of (9.2) with respect to the variable qi , and both sides of (9.3) with respect to the variable qj respectively, we arrive at
2 n ∂L ∂L ∂ L ∂ 2H ∂ 2H q, q, (q, v) + (q, v) (q, v) = 0, ∂pj ∂qi ∂v ∂pj ∂pk ∂v ∂vk ∂qi ∂ 2H ∂qi ∂qj
k=1
2 n ∂L ∂L ∂ L ∂ 2H ∂ 2L q, q, (q, v) + (q, v) (q, v) = − (q, v), ∂v ∂qi ∂pk ∂v ∂vk ∂qj ∂qi ∂qj k=1
or their equivalent expressions of matrixes,
2 2 ∂L ∂L ∂ 2H ∂ H ∂ L q, q, (q, v) + (q, v) (q, v) = 0, ∂pi ∂qj ∂v ∂pi ∂pj ∂v ∂vi ∂qj 2
2 t 2 2 ∂ H ∂L ∂L ∂ L ∂ H ∂ L q, q, (q, v) + (q, v) (q, v) = − (q, v) . ∂qi ∂qj ∂v ∂pi ∂qj ∂v ∂vi ∂qj ∂qi ∂qj
It follows from these that
∂ 2L (q, v) ∂qi ∂qj
t 2 −1 2 2 ∂L ∂ H ∂L ∂ H ∂L ∂ H q, q, q, (q, v) (q, v) (q, v) = ∂pi ∂qj ∂v ∂pi ∂pj ∂v ∂pi ∂qj ∂v
2 ∂L ∂ H q, (q, v) . (9.5) − ∂qi ∂qj ∂v
Finally, differentiating both sides of (9.3) with respect to the variable vj we get
2 n ∂L ∂ L ∂ 2H ∂ 2L q, (q, v) (q, v) = − (q, v), i.e. ∂qi ∂vj ∂qi ∂pk ∂v ∂vk ∂vj k=1
2 2 2 ∂L ∂ L ∂ L ∂ H q, (q, v) (q, v) = − (q, v) ∂qi ∂vj ∂qi ∂pj ∂v ∂vi ∂vj
2 −1 2 ∂L ∂L ∂ H ∂ H q, q, (q, v) (q, v) . =− ∂qi ∂pj ∂v ∂pi ∂pj ∂v
(9.6)
∂H Here the final equality is due to (9.4). Since p = ∂L ∂v (q, v) and v = ∂p (q, p), the desired conclusions will follow from (9.4)–(9.6). Indeed, by (9.4) it is easily seen that (L2) is equivalent to
(H2 )
∂2H ij ∂pi ∂pj
(t, q, p)ui uj 1c |u|2 ∀u = (u1 , . . . , un ) ∈ Rn .
Moreover, the three inequalities in (L3) have respectively the following equivalent versions in terms of matrix norms:
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2 ∂ L C 1 + |v|2 , (t, q, v) ∂q ∂q i j 2 ∂ L C. (t, q, v) ∂v ∂v i j
2 ∂ L C 1 + |v| (t, q, v) ∂q ∂v i
and
j
Then (L3) is equivalent to the following (H3 )
2 t 2 −1 2 2 ∂ H ∂ H ∂ H ∂ H (t, q, p) (t, q, p) − (t, q, p) ∂p ∂q (t, q, p) ∂pi ∂pj ∂pi ∂qj ∂qi ∂qj i j 2
∂H C 1 + (t, q, p) , ∂p 2
2 −1 ∂H ∂ H ∂ H ∂q ∂p (t, q, p) ∂p ∂p (t, q, p) C 1 + ∂p (t, q, p) , i j i j 2 −1 ∂ H C. ∂p ∂p (t, q, p) i j
and
∂H ∂H Here ∂H ∂p (t, q, p) = ( ∂p1 (t, q, p), . . . , ∂pn (t, q, p)), and |A| denotes the standard norm of n n matrix A ∈ Rn×n , i.e. |A| = ( i=1 j =1 aij2 )1/2 if A = (aij ).
Note that |A| = sup|x|=1 |(Ax, x)Rn | for any symmetric matrix A ∈ Rn×n , and |A| = sup|x|=1 (Ax, x)Rn if A is also positive definite, where (·,·)Rn is the standard inner product in Rn . As usual, for two symmetric positive matrixes A, B ∈ Rn×n , by “A B” we mean that (Ax, x)Rn (Bx, x)Rn for any x ∈ Rn . Then it is easily proved that 2 −1 ∂ H C (t, q, p) ∂p ∂p i
j
⇐⇒
∂ 2H 1 (t, q, p) In . ∂pi ∂pj C
(9.7)
This and (H2 ) yield 2 1 ∂ H 1 In (t, q, p) In . C ∂pi ∂pj c Lemma A.1. For a matrix B ∈ Rn×n and symmetric matrixes A, B ∈ Rn×n , suppose that there exist constants 0 < c < C and α 0 such that (i) C1 In A 1c In , (ii) |BA−1 | C(1 + α), (iii) |B t A−1 B − E| C(1 + α 2 ). Then it holds that |B|
C (1 + α) c
and |E|
2C 3 1 + α2 . + C 2 c
(9.8)
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Conversely, if (i) and (9.8) are satisfied, then −1 C 2 BA (1 + α) c Proof. By (i), |A|
1 c
and B t A−1 B − E
4C 3 + C 1 + α2 . 2 c
(9.9)
and |A−1 | C. Hence
C |B| = BA−1 A BA−1 |A| (1 + α), c t −1 t −1 t −1 |E| = B A B − E − B A B B A B − E + B t A−1 B C3 C 1 + α 2 + |B|2 A−1 C 1 + α 2 + 2 (1 + α)2 c 3 2C C 1 + α2 + 2 1 + α2 c
3 2C 1 + α2 . C+ 2 c Eq. (9.8) is proved. The “conversely” part is easily proved as well.
2
By this lemma we get immediately: Proposition A.2. In any local coordinates (q1 , . . . , qn ), the conditions (L2)–(L3) are equivalent to the fact that there exist constants 0 < C1 < C2 , depending on the local coordinates, such that
∂ 2H C1 I n (t, q, p) C2 In , ∂pi ∂pj 2
∂H ∂ H ∂q ∂p (t, q, p) C2 1 + ∂p (t, q, p) , i j 2 2
∂ H ∂H ∂q ∂q (t, q, p) C2 1 + ∂p (t, q, p) . i j For each (t, q) ∈ R/Z × M, since the function Tq∗ M → R, p → H (t, q, p) is strictly ¯ = 0. Recall convex, it has a unique minimal point p¯ = p(t, ¯ q). In particular, Dp H (t, q, p) that the diffeomorphism LH in (1.3) is the inverse of LL in (1.5), and that L(t, q, v) = p(t, q, v), v − H (t, q, p(t, q, v)), where p = p(t, q, v) is a unique point determined by the equality v = Dp H (t, q, p). It follows that t, q, p(t, ¯ q) ∈ R/Z × T ∗ M (t, q) ∈ R/Z × M = LH (R/Z × 0T M ) is a compact subset. So in any local coordinates (q1 , . . . , qn ), there exists a constant C3 > 0, depending on the local coordinates, such that the expression of p¯ = p(t, ¯ q) in the local coordinate (q1 , . . . , qn ), denoted by p¯ = (p¯ 1 , . . . , p¯ n ), satisfies |p| ¯ = (p¯ 1 , . . . , p¯ n ) C3 .
(9.10)
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By the mean value theorem we have 0 < θ = θ (t, q, p) < 1 such that ∂H ∂H ∂H = (t, q, p) (t, q, p) − (t, q, p) ¯ ∂p ∂p ∂p 2 ∂ H = t, q, θp + (1 − θ )p¯ (p − p) ¯ t . ∂qi ∂qj Since the first inequality in Proposition A.2 implies 2 ∂ H C1 |u| (t, q, p)u C2 |u| ∂pi ∂pj using (9.10) and the inequality ab 2ε a 2 +
1 2 2ε b
∀u = (u1 , . . . , un )t ∈ Rn ,
∀ε > 0 we easily get
∂H (t, q, p) C2 |p − p| ¯ ¯ C2 |p| + C2 C3 , C1 |p| − C1 C3 C1 |p − p| ∂p 2 ∂H C12 2 2 2 |p| − 2C1 C3 (t, q, p) 2C22 |p|2 + 2C22 C33 . 2 ∂p These two inequalities and Proposition A.2 lead to: In any local coordinates (q1 , . . . , qn ), the conditions (L2)–(L3) are equivalent to the fact that there exist constants 0 < c < C, depending on the local coordinates, such that 2 ∂ H cIn (t, q, p) CIn and ∂pi ∂pj 2 2 ∂ H ∂ H C 1 + |p|2 . C 1 + |p| , (t, q, p) (t, q, p) ∂q ∂q ∂q ∂p i j i j Proposition A is proved. A.2. An inequality for C 1 -simplex in C 1 Riemannian–Hilbert manifolds For every integer q 0 we denote by q the standard closed q-dimensional simplex in Rq with vertices e0 = 0, e1 , . . . , eq , i.e. 0 = {0} and q := (t1 , . . . , tq ) ∈ Rn0 t1 + · · · + tq 1 with q 1. For 1 i q denote by Fqi : q−1 → q the ith face. Let e(s) = (s, . . . , s) ∈ Rq with s ∈ [0, 1], eˆ = e(1/(q + 1)), and L be the straight line passing through e(0) and eˆ successively in Rq , i.e. L = {s eˆ | s ∈ R}. Then we have an orthogonal subspace decomposition Rq = Vq−1 × L, and each w ∈ q may be uniquely written as w = (v, s0 ) ∈ [Vq−1 × L] ∩ q . This (v, s) decomposition of the simplex q was introduced by Yiming Long on the page 447 of [23]. Denote by l(v) the intersection segment of q with the straight line passing through w and parallel to L,
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i.e. l(v) = {w + s eˆ ∈ q | s ∈ R} = {(v, s) | s1 s s2 } for some s1 s0 and s2 s0 . Clearly, √ each l(v) has length no more than q/2. 1 Let (M, , ) be a C Riemannian–Hilbert manifold and · be the induced Finsler metric. For φ ∈ C( q , M) and each w = (v, s0 ) ∈ [Vq−1 × L] ∩ q , following [23] we define φ˜ v : l(v) → M,
s → φ(v, s).
If φ ∈ C 1 ( q , M), i.e. φ can be extended into a C 1 -map from some open neighborhood of q in Rq to M, then there exists a constant c = c(φ) > 0 such that 2 ∂ φ(v, s) c(φ), ∂s
∀(v, s) ∈ q .
So for any (v, s) ∈ q we get 2 √
d φ˜ v (s) ds c(φ)Length l(v) q c(φ). ds 2
(9.11)
l(v)
Now consider the case M = Eτ = W 1,2 (Sτ , M) with the Riemannian metric given by (1.13). Using the local coordinate chart in (3.8) it is easy to prove Lemma A.3. For each t ∈ Sτ the evaluation map EVt : W 1,2 (Sτ , M) → M,
γ → γ (t),
is continuous and maps W 1,2 -curves in Eτ to W 1,2 -curves in M. Proof. We only need to prove the case M = Rn . Let [a, b] → γ (s) be a W 1,2 -curve in d γ (s) is a W 1,2 -vector field along γ (s). Since Tγ (s) W 1,2 (Sτ , Rn ) = W 1,2 (Sτ , Rn ). Then ξ(s) := ds 1,2 n 1,2 W (Sτ , R ), ξ(s) ∈ W (Sτ , Rn ) and γ (s + ) − γ (s) − ξ(s) = 0. lim 1,2 →0 W (Sτ ,Rn ) Carefully checking the proof of Proposition 1.2.1(ii) in [18, p. 9] one easily derives
η C 0
1+τ
η W 1,2 τ
∀η ∈ W 1,2 Sτ , Rn .
(9.12)
Hence we get γ (s + )(t) − γ (s)(t) =0 − ξ(s)(t) lim n →0 R uniformly in t. This means that [a, b] → M, s → EVt (γ (s)), is differentiable and d EVt γ (s) = ξ(s)(t) ds
at each s ∈ [a, b].
(9.13)
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Fix an > 0 such that γ (s + ) − γ (s) − ξ(s) 1,2 W (Sτ ,Rn )
1+τ . τ
By (9.12) we get 2 γ (s + )(t) − γ (s)(t) − ξ(s)(t) n 1 R
∀t ∈ R.
It follows that for any s ∈ [a, b], 2 2 ξ(s)(t)2 n 2 γ (s + )(t) − γ (s)(t) − ξ(s)(t) + γ (s + )(t) − γ (s)(t) n R Rn R 2 1 2 1 + 2 γ (s + )(t) − γ (s)(t)Rn 1+τ γ (s + ) − γ (s)2 1,2 . 2 1+ W (Sτ ,Rn ) τ 2 Here the final inequality is due to (9.12). Hence
b a
ξ(s)(t) 2Rn ds < +∞, and thus
b d 2 EVt γ (s) ds < +∞ ds n R
a
because of (9.13).
2
For a singular simplex σ from q to Eτ and every w = (v, s0 ) ∈ q , following [23] we define curves σv (s) = ! s → EVt ! σv (s)(t)
! σvt : l(v) → M,
(9.14)
σv0 the initial point curve. Suppose that σ ∈ C 1 ( q , Eτ ). Then ! σv ∈ for each t ∈ Sτ , and call ! 1 C (l(v), Eτ ), and by (9.11) there exists a positive constant c(σ ) such that 2
d ! σ (s) ds v l(v)
√
σv (s)∗ T M) W 1,2 (!
ds
q c(σ ) 2
(9.15)
d 1,2 (! for any (v, s) ∈ q , where ds ! σv (s) ∈ T! σv (s)∗ T M). Specially, by Lemma A.3 σv (s) Eτ = W t 1,2 we get each ! σv ∈ W (l(v), M) for any t. As in the proof of Proposition 1.2.1(ii) in [18, p. 9] one can easily derive that
ξ C 0 (γ ∗ T M)
1+τ
ξ W 1,2 (γ ∗ T M) τ
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for any γ ∈ W 1,2 (Sτ , M) and ξ ∈ W 1,2 (γ ∗ T M). Applying to γ = ! σv (s) and ξ = 2 d ! ds σv (s) 0
C (! σv (s)∗ T M)
2 d 1+τ ! σv (s) τ ds
W 1,2 (! σv (s)∗ T M)
.
d σv (s) ds !
we get (9.16)
Moreover, it follows from (9.13) and (9.14) that
d d t d ! σv (s) (t) = ! σv (s) = ! σv (s)(t) ∈ T! σv (s)(t) M ds ds ds
for all s ∈ [a, b] and t ∈ Sτ . Hence for any t ∈ Sτ , we can derive from (9.16) that d t 2 ! (s) σ ds v
T! σv (s)(t) M
2 d = (s) (t) ! σ ds v T! σv (s)(t) M
2
d ! σ max (s) (t) v t∈Sτ ds T! σv (s)(t) M 2 d = σv (s) ds ! 0 σv (s)∗ T M) C (! 2 1+τ d! σ (s) . v 1,2 τ ds σv (s)∗ T M) W (!
This and (9.15) together give the following generalization of [23, Lem. 2.3]. Lemma A.4. If σ ∈ C 1 ( q , Eτ ), for every w = (v, s0 ) ∈ q , it holds that
d 0 2 ! ds σv (s) l(v)
T! σ 0 (s) v
√ (1 + τ ) q c(σ ). ds 2τ M
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Journal of Functional Analysis 256 (2009) 3035–3054 www.elsevier.com/locate/jfa
Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality Anatolii Grinshpan a , Dmitry S. Kaliuzhnyi-Verbovetskyi a,∗ , Victor Vinnikov b , Hugo J. Woerdeman a,1 a Department of Mathematics, Drexel University, 3141 Chestnut St., Philadelphia, PA 19104, USA b Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
Received 8 August 2008; accepted 12 September 2008 Available online 4 October 2008 Communicated by D. Voiculescu
Abstract We obtain a decomposition for multivariable Schur-class functions on the unit polydisk which, to a certain extent, is analogous to Agler’s decomposition for functions from the Schur–Agler class. As a consequence, we show that d-tuples of commuting strict contractions obeying an additional positivity constraint satisfy the d-variable von Neumann inequality for an arbitrary operator-valued bounded analytic function on the polydisk. Also, this decomposition yields a necessary condition for solvability of the finite data Nevanlinna– Pick interpolation problem in the Schur class on the unit polydisk. © 2008 Elsevier Inc. All rights reserved. Keywords: Multivariable von Neumann inequality; Commuting contractions; Unitary dilation; Multivariable Schur class; Schur–Agler class; Scattering system; Nevanlinna–Pick interpolation problem
1. Introduction The classical von Neumann inequality [39] states that for a scalar-valued function f analytic and bounded on the unit disk D = {z ∈ C: |z| < 1} and a strict contraction T on a Hilbert space (i.e., T < 1), the operator norm of f (T ) is not greater than the sup-norm of f , * Corresponding author.
E-mail addresses:
[email protected] (A. Grinshpan),
[email protected] (D.S. Kaliuzhnyi-Verbovetskyi),
[email protected] (V. Vinnikov),
[email protected] (H.J. Woerdeman). 1 Partially supported by NSF grant DMS-0500678. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.09.012
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f (T ) supf (z).
(1.1)
z∈D
There are several proofs of this result (see, e.g., [32, Chapter 1]). The different proofs fall roughly in two categories: function-theoretic and operator-theoretic. One of the function-theoretic proofs is based on the properties of the Pick kernel Kf (z, w) =
1 − f (z)f (w) 1 − zw
associated with a function f from the Schur class, S, i.e., with an analytic function f : D → D. Such a kernel is positive, i.e., for any m ∈ N and z(1) , . . . , z(m) ∈ D the matrix [Kf (z(j ) , z(k) )]m j,k=1 is positive semidefinite. It is well known (see, e.g., [7]) that a positive kernel admits a factorization Kf (z, w) = h(z)h(w)∗ with a function h analytic on D, whose values are bounded linear operators (actually, functionals) from an auxiliary Hilbert space H to C. Thus, the identity 1 − f (z)f (w) = h(z) (1 − zw)IH h(w)∗
(1.2)
holds for all z, w ∈ D, and it is straightforward to compute (e.g., using Taylor series expansions centered at 0) that, for T a strict contraction on a Hilbert space K, one has the identity IK − f (T )f (T )∗ = h(T ) I − T T ∗ ⊗ IH h(T )∗ , the right-hand side of which is clearly positive semidefinite. Thus f (T ) 1. The implication (f ∈ S)
⇒
f (T ) 1 for an arbitrary strict contraction T
is equivalent to the von Neumann inequality. Operator-theoretic proofs are often based on the dilation theory for contractions. The key fact [36] is that every contraction T : K → K has a unitary dilation, in the sense that for some unitary U on a Hilbert space Kext containing K, T n = PK U n |K ,
n ∈ N,
where PK is the orthogonal projection onto K. For a unitary U , one obtains by the spectral theorem for unitary operators that f (rU ) supz∈D |f (z)|, 0 < r < 1, and (1.1) easily follows. Let us note that the von Neumann inequality and both proofs indicated extend naturally to the case where f is an operator-valued function. Both of the above approaches have been generalized to the two-variable case. In [4] Andô showed that a pair of commuting contractions (T1 , T2 ) acting on a Hilbert space K has a unitary dilation, i.e., a pair of commuting unitary operators (U1 , U2 ) on a common Hilbert space Kext containing K, such that T1 n1 T2 n2 = PK U1 n1 U2 n2 |K ,
n1 , n2 ∈ N.
With that it was shown that the two-variable von Neumann inequality
A. Grinshpan et al. / Journal of Functional Analysis 256 (2009) 3035–3054
f (T1 , T2 )
sup
3037
f (z1 , z2 )
(z1 ,z2 )∈D2
holds for any function f analytic and bounded on the bidisk D2 and any pair (T1 , T2 ) of commuting strict contractions. The function-theoretic approach to the d-variable von Neumann inequality was advanced in the work of Agler [2]. Let f be an analytic function on the polydisk Dd taking values in the space L(F , F∗ ) of bounded linear operators acting from a Hilbert space F to a Hilbert space F∗ . Given a Taylor series expansion of f , f (z) =
fn z n ,
z ∈ Dd
n∈ZN +
(we use the standard notation zn = z1n1 · · · zdnd for multi-powers), and a d-tuple T = (T1 , . . . , Td ) of commuting strict contractions on a Hilbert space K, one can define f (T ) =
T n ⊗ fn ∈ L(K ⊗ F , K ⊗ F∗ ).
n∈ZN +
Agler’s main result in [2] is that the following statements are equivalent: (i) The inequality f (T ) 1 holds for any d-tuple T of commuting strict contractions. (ii) There exist positive sesquianalytic L(F∗ )-valued kernels Kj (z, w) on Dd , j = 1, . . . , d, such that the decomposition IF∗ − f (z)f (w)∗ =
d (1 − zj wj )Kj (z, w)
(1.3)
j =1
holds for (z, w) ∈ Dd × Dd . (A L(X )-valued function K(z, w) on a set Λ × Λ is a positive kernel on Λ if for any m ∈ N and z(1) , . . . , z(m) ∈ Λ the block matrix [K(z(j ) , z(k) )]m j,k=1 represents a positive semidefinite operator on X m . Equivalently (see [7]), there exist an auxiliary Hilbert space H and a function H : Λ → L(H, X ) such that K(z, w) = H (z)H (w)∗ . Moreover, K(z, w) is sesquianalytic if and only if H is analytic on Λ.) We say that the function f analytic on Dd belongs to the Schur–Agler class SAd (F , F∗ ) if f satisfies any of the equivalent conditions (i), (ii) above. We will denote by Sd (F , F∗ ) the dvariable Schur class consisting of functions which are analytic on Dd and whose values there are contraction operators from L(F , F∗ ). It is clear that SAd (F , F∗ ) is a subclass of Sd (F , F∗ ). Due to the validity of von Neumann inequality in one and in two variables, the classes Sd (F , F∗ ) and SAd (F , F∗ ) coincide when d = 1, 2. In fact, the “if” direction in Agler’s theorem is obtained in the same way as the one-variable von Neumann inequality is obtained from decomposition (1.2). Thus, if one can construct Agler’s decomposition (1.3) for a contractive analytic function f on a bidisk bypassing Andô’s theorem, then one would obtain a function-theoretic proof of the von Neumann inequality in two variables. In [17] it was shown that using Rudin’s approximation [34, Theorem 5.2.5] it suffices to prove (1.3) for a rational inner function f . In [25] a 2-variable
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Christoffel–Darboux formula for stable polynomials is proved, which yields Agler’s decomposition for rational inner functions, and thus the von Neumann inequality is established in this way for any scalar-valued function from the bidisk algebra (see also [28] for some discussion). In [13] various Agler’s decompositions are obtained for an arbitrary operator-valued Schur-class function f on the bidisk D2 from the system realizations of f , which in turn are constructed from the scattering system associated with f . Therefore, the von Neumann inequality is proved in [13] for any f ∈ S2 (F , F∗ ). It is well known that for d variables, d 3, the von Neumann inequality fails in general. Counterexamples may be found in [19,26,38]. Also, the dilation theory fails to generalize directly to the d-variable case. (A d-tuple U = (U1 , . . . , Ud ) of commuting unitary operators acting on a common Hilbert space Kext is called a unitary dilation of a d-tuple T = (T1 , . . . , Td ) of commuting contractions acting on a common Hilbert space K if Kext ⊃ K and T n = PK U n |K , n ∈ Zd+ .) In [31] an example of three commuting contractions which have no unitary dilations is given. On the other hand, for several classes of d-tuples T = (T1 , . . . , Td ) of commuting contractions unitary dilations do exist (see, e.g., [5,6,16,23,24], [37, Section I.9]). Remark 1.1. In this paper we consider commuting unitary dilations only. More generally, one can define unitary dilations, which are not necessarily commuting, as follows. Let T = (T1 , . . . , Td ) be a d-tuple of (not necessarily commuting) contractions on a Hilbert space K. A d-tuple U = (U1 , . . . , Ud ) of unitary operators acting on a Hilbert space Kext is called a unitary dilation of T if Kext ⊃ K and Tj1 · · · Tjm = PK Uj1 · · · Ujm |K for all m ∈ N and all j1 , . . . , jm ∈ {1, . . . , d}. Such a unitary dilation of a d-tuple of contractions (in particular, of commuting contractions) always exists [37, Section 1.5] (see also [15]). In this paper we show that a class of d-tuples T = (T1 , . . . , Td ) of commuting strict contractions subject to certain inequalities satisfies the d-variable von Neumann inequality f (T ) sup f (z)
(1.4)
z∈Dd
for any bounded analytic operator-valued function f on Dd . As a consequence of Arveson’s ideas from [8,9] (see also [32, Corollary 4.9] for an explicit formulation) the latter is equivalent to the existence of a unitary dilation of T . We do not construct a unitary dilation explicitly, our method is rather function-theoretic. Exploring the scattering system associated with a function f from the d-variable Schur class Sd (F , F∗ ) we obtain the following result. Theorem 1.2. Let f ∈ Sd (F , F∗ ). Then for any p, q ∈ {1, . . . , d}, p < q, there exist positive I (z, w) and K II (z, w) on Dd such that L(F∗ )-valued sesquianalytic kernels Kpq pq IF∗ − f (z)f (w)∗ =
k =p
I (1 − zk wk )Kpq (z, w) +
II (1 − zk wk )Kpq (z, w)
(1.5)
k =q
holds for all (z, w) ∈ Dd × Dd . The decomposition (1.5) for a function from Sd (F , F∗ ) is somewhat analogous to Agler’s decomposition (1.3) for a function from SAd (F , F∗ ). Moreover, (1.5) coincides with (1.3) when
A. Grinshpan et al. / Journal of Functional Analysis 256 (2009) 3035–3054
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d = 2. If the evaluation of the right-hand side of (1.5) at a d-tuple T of commuting strict contractions in the sense of Agler’s hereditary calculus [2] (i.e., such that the adjoint operators Tj∗ appear to the right of all the operators Tk in every monomial in the power series expansion) gives a positive semidefinite operator, then f (T ) is a contraction, i.e., the von Neumann inequality holds for such a T . This observation naturally leads to the following definition. For a d-tuple T = (T1 , . . . , Td ) of commuting bounded linear operators on a common Hilbert space K and a vector β ∈ {0, 1}d , set β
ΔT :=
(−1)|α| T α T ∗ α ,
(1.6)
0αβ
where the inequality α β for α, β ∈ {0, 1}d means that αj βj , j = 1, . . . , d, |α| = α1 + · · · + αd , T α = T1α1 · · · Tdαd , and T ∗ = (T1∗ , . . . , Td∗ ). We will say that a d-tuple T = (T1 , . . . , Td ) of d , with some integers p and linear operators on a common Hilbert space belongs to the class Pp,q q satisfying 1 p < q d, if (i) T1 , . . . , Td commute; (ii) T1 , . . . , Td are contractions; e−e e−e (iii) the operators ΔT p and ΔT q are positive semidefinite, where e = (1, . . . , 1), ek = k th place
(0, . . . , 0,
1 , 0, . . . , 0) ∈ Zd+ .
3 if For example, a triple T = (T1 , T2 , T3 ) of commuting contractions belongs to P1,3 e−e3
ΔT
= IK − T1 T1∗ − T2 T2∗ + T1 T2 T2∗ T1∗ 0,
1 = IK − T2 T2∗ − T3 T3∗ + T2 T3 T3∗ T2∗ 0. Δe−e T
Note that if T is doubly commutative, i.e., each Tj commutes with each Tk and with each Tk∗ , one has k Δe−e = T
IK − Tj Tj∗ . j =k
The main result of the present paper is the following. d . Then T satisfies Theorem 1.3. Let T be a d-tuple of strict contractions from a class Pp,q the von Neumann inequality (1.4) for any bounded analytic operator-valued function f on Dd . Equivalently, T has a commuting unitary dilation.
The machinery used in our proofs is multi-evolution scattering systems and associated formal reproducing kernel Hilbert spaces [13,14]. This is reviewed in Section 2. The proofs of Theorems 1.2 and 1.3 are given in Section 3. In Section 4, we obtain as a corollary a necessary condition for the solvability of the finite-data Nevanlinna–Pick problem in the d-variable Schur class. We have included some discussion, examples, and connections in Section 5.
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2. Scattering systems and formal reproducing kernel Hilbert spaces In this section we recall a version of multi-evolution scattering systems as it appears in [13,14] (see also [12,18,27,35] for some earlier versions). A d-evolution scattering system is a collection S = (K, U, F , F∗ ), where K is a Hilbert space (the ambient space for the scattering system), U = (U1 , . . . , Ud ) is a d-tuple of commuting unitary operators on K (the evolutions of the system), and F and F∗ are wandering subspaces for U , i.e. U nF ⊥ F ,
U n F∗ ⊥ F∗ ,
n ∈ Zd , n = 0.
A multi-evolution scattering system S is said to be minimal if the smallest closed subspace of K containing F and F∗ and invariant for U and U ∗ is the whole space K, i.e., if the subspace
U nF + U n F∗
n∈Zd
n∈Zd
is dense in K. The subspaces W :=
U nF ,
W∗ =
U n F∗
n∈Zd \Zd+
n∈Zd+
are called the outgoing subspace and the incoming subspace, respectively: W is invariant for all the evolutions Uk , k = 1, . . . , d, which act on W as forward shifts, and W∗ is invariant for all Uk∗ , k = 1, . . . , d, so that the evolutions Uk act on W∗ as backward shifts. The system S is called causal if W ⊥ W∗ . In this case, the ambient space K has the orthogonal decomposition K = W∗ ⊕ V ⊕ W,
(2.1)
where V, the scattering subspace, is the orthogonal complement of W∗ ⊕ W in K. Let us note that in [13] a more general Ω-orthogonal scattering system is defined, where Zd+ in the definitions of the outgoing and incoming subspaces, of causality, and of the scattering subspace is replaced with a general shift-invariant sublattice Ω ⊆ Zd . Also, it was shown in [13, Proposition 3.1] that if a scattering system S is causal then S is Ω-orthogonal. Let S be a causal d-evolution scattering system. Define the Fourier representation operators Φ and Φ∗ from K to L2 (Td , F ) and to L2 (Td , F∗ ), respectively by Φ : h →
PF U −n h zn , n∈Zd
Φ∗ : h →
PF∗ U −n h zn , n∈Zd
where Td = {z ∈ Cd : |zk | = 1, k = 1, . . . , d} is the unit d-torus, and L2 (Td , X ) denotes the Lebesgue space of measurable norm-square integrable X -valued functions on Td , for a Hilbert space X . Then Φ is a coisometry with the initial space equal W and with ΦW = H 2 (Td , F ),
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while Φ∗ is a coisometry with the initial space equal W∗ and with Φ∗ W∗ = H 2 (Td , F∗ )⊥ . Here H 2 (Td , X ) denotes the Hardy space, i.e., the subspace in L2 (Td , X ) consisting of functions determined by boundary values of analytic functions on Dd . It turns out that the operator Φ∗ Φ ∗ maps H 2 (Td , F ) into H 2 (Td , F ∗ ), moreover it is a multiplication operator: Φ∗ Φ ∗ = MS . The corresponding multiplier S = SS , which is called the scattering function of S, belongs to the d-variable Schur class Sd (F , F∗ ). Conversely, given any f ∈ Sd (F , F∗ ), there is a minimal causal scattering system Sf with scattering function SSf = f . Such a scattering system is determined by f uniquely up to a unitary equivalence. One of equivalent ways to describe Sf is the de Branges–Rovnyak model (see [21,22] for the classical, i.e., single-variable case) which is defined as follows: f f f f f f f SdBR = KdBR , UdBR = UdBR,1 , . . . , UdBR,d , FdBR , FdBR∗ , where f
KdBR = im
I Mf∗
Mf I
1/2
⊂
L2 (Td , F∗ ) , L2 (Td , F )
with
I Mf∗
Mf I
1/2 h I , g Mf∗
Mf I
1/2
h g
f
KdBR
h h = Q , , g L2 (Td ,F∗ ⊕F ) g
I where Q is the orthogonal projection onto the orthogonal complement of ker M ∗ f
Mf 1/2 , I
f
UdBR,k = (Mzk ⊗ IF∗ ⊕F )|Kf , and where we set dBR
f
FdBR =
f F, I
f
FdBR∗ =
It is easily checked that the scattering function SSf
I F∗ . f∗
of the de Branges–Rovnyak model scatter-
dBR
f
ing system SdBR agrees with the original function f in the sense that f ∗ f SSf (z) = idBR∗ f (z) idBR , dBR
f
where idBR : F → FdBR = maps given by
f f F and idBR∗ : F∗ → FdBR∗ = fI∗ F∗ are unitary identification I
f idBR
f : e → e, I
f idBR∗
I : e∗ → e . f∗ ∗
In [14] the de Branges–Rovnyak model was interpreted in terms of formal reproducing kernel Hilbert spaces. Since not resorting to this formalism in full, we will only need a definition and some basic facts (for more details, see [11] and [14, Section 2]). Let H be aHilbert space whose elements are formal power series in d commuting indeterminates F (x) = n∈Zd Fn x n with the coefficients in a Hilbert space F , i.e., F ∈ F Jx ±1 K where F Jx ±1 K denotes the space of formal
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Laurent series in the commuting indeterminates x1 , . . . , xd and their inverses x1−1 , . . . , xd−1 . We say that H is a formal reproducing kernel Hilbert space if the linear functional F → Fn is continuous for each n ∈ Zd . In this case, for each n ∈ Zd there is a formal power series Kn ∈ L(F )Jx ±1 K such that, for each vector u ∈ F we have F, Kn uH = Fn , uF . We then define K(x, y) = n∈Zd Kn (x)y n ∈ L(F )Jx ±1 , y ±1 K, write H = H(K), and say that K(x, y) is the reproducing kernel for the formal reproducing kernel Hilbert space H(K). The following theorem characterizes the formal power series K ∈ L(F )Jx ±1 , y ±1 K which arise as the reproducing kernel for some formal reproducing kernel Hilbert space. Theorem 2.1. (See [11, Theorem 2.1].) Suppose that F is a Hilbert space and that we are given a formal power series K ∈ L(F )Jx ±1 , y ±1 K. Then the following are equivalent: (1) K is the reproducing kernel for a uniquely determined formal reproducing kernel Hilbert space H(K) of formal power series in the commuting variables x ±1 = (x1 , . . . , xd , x1−1 , . . . , xd−1 ) with coefficients in F . (2) There is an auxiliary Hilbert space H and a formal power series H ∈ L(H, F )Jx ±1 K so that K(x, y) = H (x)H (y)∗ , where we use the convention n ∗ x = x −n ,
H (y)∗ =
Hn∗ y −n
if H (x) =
n∈Zd
(3) K(x, y) =
n,n ∈Zd
Hn x n .
n∈Zd
Kn,n x n y n is a positive kernel in the sense that
Kn,n un , un F 0
n,n ∈Zd
for all finitely supported F -valued functions n → un on Zd . Moreover, in this case the formal reproducing kernel Hilbert space H(K) can be defined directly in terms of the formal power series H (x) appearing in condition (ii) by H(K) = H (x)h: h ∈ H with norm taken to be the pullback norm H (x)h
H(K)
= QhH ,
where Q is the orthogonal projection of H onto the orthogonal complement of the kernel of the map MH : H → F Jx ±1 K given by MH : h → H (x)h.
A. Grinshpan et al. / Journal of Functional Analysis 256 (2009) 3035–3054
3043
After identification F ∼ n∈Zd Fn x n of an L2 function with its Fourier series viewed as a f f f f f f formal Laurent series, the spaces KdBR , FdBR , FdBR∗ , WdBR , WdBR∗ , VdBR in the de Branges– Rovnyak model can be interpreted as formal reproducing kernel Hilbert spaces (see [14, Proposition 3.3]). We will be particularly interested in such an interpretation for the scattering subspace: f VdBR ∼ = H(KV f ), whose reproducing kernel is given by dBR
KV f
dBR
I − f (x)f (y)∗ = f (x)∗ − f (y)∗
f (y) − f (x) k (x, y), f (x)∗ f (y) − I Sz,+
(2.2)
where kSz,+ (x, y) =
x n y −n
n∈Zd+
is the formal Szeg˝o kernel. Let us remark that the formula (2.2) implies that the first component of any formal power series in the space H(KV f ) is “formal analytic,” i.e., is a formal power dBR
series supported on Zd+ , and thus the first component of the corresponding function from the f space VdBR is determined by boundary values of an analytic function on Dd . Accordingly, the (1, 1) entry of the kernel KV f , dBR
KV f 11 (x, y) = I − f (x)f (y)∗ kSz,+ (x, y)
(2.3)
dBR
is “formal sesquianalytic,” and therefore I − f (z)f (w)∗ KV f 11 (z, w) = I − f (z)f (w)∗ kSz,+ (z, w) = d dBR k=1 (1 − zk w k )
(2.4)
is a sesquianalytic positive kernel on Dd . This is a consequence of the fact that the positivity of a sesquianalytic kernel K(z, w) = n,n ∈Zd Kn,n zn wn is equivalent to the positivity of the + matrix of its coefficients [Kn,n ]n,n ∈Zd , because both kernels in (2.3) and in (2.4) have the same + matrix of coefficients. 3. Proofs 3.1. Proof of Theorem 1.2 Proof. Let f ∈ Sd (F , F∗ ), and let p = 1 and q = d. Consider the associated minimal causal d-evolution scattering system Sf = (K; U; F , F∗ ). Let Lmax be the maximal subspace of the scattering subspace V that is invariant for Ud : Lmax = h ∈ V: Udk h ∈ V for all k ∈ Z+ . Define Lmin = V Lmax . We will first prove that Lmin is invariant for U1 , . . . , Ud−1 (in a way analogous to the first part of the proof of Theorem 5.5 in [13]). In fact, we will only use the invariance of Lmin for U1 .
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Let h ∈ V. Observe that Udk h ∈ V for k 1 if and only if
Udk h ⊥
n∈Zd+ :
U nF ,
nd
since for every h ∈ V and n ∈ Zd+ such that nd k one has Udk h ⊥ U n F . Hence h ∈ Lmax if and only if h ⊥ Ud−k U n F , for every k 1. n∈Zd+ : nd
Thus Lmin = V Lmax = spank1 PV Ud−k
U nF ,
n∈Zd+ : nd
i.e., Lmin is the closure of the linear span of vectors of the form g = PV Ud−k U n fn , n∈Zd+ :
where k 1, and fn ∈ F are such that form (3.1), i.e., g = PV u where
nd
n∈Zd+ : nd
u = Ud−k
2
< ∞. Let g be a vector of the
U n fn .
n∈Zd+ : nd
Let us observe that u⊥W
=
U F . n
n∈Zd+
Then, according to the decomposition (2.1), one can write u = g + y, where n n y = PW∗ u = U f∗ n ∈ W ∗ U F∗ . = n∈Zd \Zd+
n∈Zd \Zd+
On the other hand, u ∈ Ud−k W ⊥ Ud−k W∗ , which implies that y ∈ Ud−k
U n F∗ .
n∈Zd+
We infer that f∗ n = 0 outside the set
(3.1)
n ∈ Zd : n1 , . . . , nd−1 0, −k nd < 0 ,
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i.e.,
y=
U n f∗ n = Ud−k
n∈Zd : n1 ,...,nd−1 0, −knd <0
U n f∗ n .
n∈Zd+ : nd
It follows that g has a form (3.1) if and only if
Ud−k
n∈Zd+ :
U n fn = g + Ud−k
n∈Zd+ :
nd
U n f∗ n ,
(3.2)
nd
with some f∗ n ∈ F∗ satisfying n∈Zd : nd
Ud−k
U n fn = Uj g + Ud−k
n∈Zd+ : nj >0, nd
U n f∗ n ,
n∈Zd+ : nj >0, nd
which has the same form as (3.2). Thus, Uj g ∈ Lmin . By linearity and continuity, the invariance of Lmin for the evolutions U1 , . . . , Ud−1 follows. Consider the Wold decompositions (see [37, Section 1.1]) of Lmin with respect to U1 , and of Lmax with respect to Ud : Lmin =
∞
U1n1 C1
Lmax =
⊕ R1 ,
n1 =0
∞
Udnd Cd
⊕ Rd ,
nd =0
where Cj is the wandering subspace and Rj is a reducing subspace for Uj , j = 1, d. We have V = Lmin ⊕ Lmax =
∞
U1n1 C1
⊕ R1 ⊕
n1 =0
∞
Udnd Cd
⊕ Rd .
nd =0 f
Let us now identify the scattering system Sf with the de Branges–Rovnyak model SdBR (see Section 2). In terms of the reproducing kernels, KV (x, y) = KLmin (x, y) + KLmax (x, y) =
∞
x1n1 y1−n1 KC1 (x, y) + KR1 (x, y) +
n1 =0
∞
n
−nd
xd d yd
KCd (x, y) + KRd (x, y),
nd =0 f
where, in particular, V is identified with VdBR and KV (x, y) is identified with KV f (x, y). Each dBR of the kernels KA (x, y), where A = V, C1 , Cd , R1 or Rd , is a 2 × 2 matrix. As observed in Section 2, the first component of the elements of the scattering subspace in the de Branges– Rovnyak model is formal analytic. Therefore (KA )11 (x, y) are all formal sesquianalytic positive kernels. Next, observe that U1 (R1 ) = R1 implies that x1 y1−1 KR1 (x, y) = KR1 (x, y), and thus
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x1 y1−1 (KR1 )11 (x, y) = (KR1 )11 (x, y). The fact that (KR1 )11 (x, y) is formal sesquianalytic implies that (KR1 )11 (x, y) = 0. Analogously, (KRd )11 (x, y) = 0. Thus we obtain that (KV )11 (x, y) =
∞
x1n1 y1−n1 (KC1 )11 (x, y) +
n1 =0
∞
xdnd yd−nd (KCd )11 (x, y).
nd =0
As in the last paragraph of Section 2, we write this relation for the formal sesquianalytic positive kernels as a relation for the sesquianalytic positive kernels on Dd , which in view of (2.4) becomes IF ∗ − f (z)f (w)∗ (KC1 )11 (z, w) (KCd )11 (z, w) = + , d 1 − z1 w1 1 − zd wd k=1 (1 − zk w k )
(z, w) ∈ Dd × Dd .
Thus we obtain d
IF∗ − f (z)f (w)∗ =
(1 − zk wk )(KC1 )11 (z, w) +
k=2
d−1
(1 − zk wk )(KCd )11 (z, w)
k=1
for all (z, w) ∈ Dd × Dd , which is the desired decomposition (1.5) for p = 1 and q = d, with I (z, w) = (K ) (z, w) and K II (z, w) = (K ) (z, w). K1d C1 11 Cd 11 1d The decomposition (1.5) for arbitrary p and q is obtained by a suitable renumbering the operators U1 , . . . , Ud . 2 Remark 3.1. In the case where f ∈ SAd (F , F∗ ), Theorem 1.2 follows immediately from the existence of Agler’s decomposition (1.3) for f . Indeed, (1.3) becomes (1.5) if we set, e.g., I Kpq (z, w) :=
Kq (z, w) , l =p, =q (1 − zl w l )
II := Kpq
j =q
Kj (z, w) . l =j, =q (1 − zl w l )
(3.3)
I (z, w) (In the case d = 2, the denominators in (3.3) must be regarded as unity.) The functions Kpq II d and Kpq (z, w) defined in (3.3) are positive sesquianalytic L(F∗ )-valued kernels on D due to the fact (see, e.g., [7,37]) that so are:
• the functions kl (z, w) := 1−z1l wl IF∗ , l = 1, . . . , d; • the product of a positive sesquianalytic L(F∗ )-valued kernel on Dd with any kl (z, w); • the sum of a finite number of positive sesquianalytic L(F∗ )-valued kernels on Dd . 3.2. Proof of Theorem 1.3 d for some p and q. Let f be an arbitrary Proof. Let T = (T1 , . . . , Td ) ∈ L(K)d be in the class Ppq L(F , F∗ )-valued function from the d-variable Schur class Sd (F , F∗ ). By Theorem 1.2, there are I (z, w) and K II (z, w) on Dd such that (1.5) positive L(F∗ )-valued sesquianalytic kernels Kpq pq I II and analytic functions H I and H II holds. Then (see [7]) there exist Hilbert spaces Hpq and Hpq pq pq I , F ) and L(HII , F ), respectively, such that the equalities on Dd which take values in L(Hpq ∗ ∗ pq I I I Kpq (z, w) = Hpq (z)Hpq (w)∗ ,
II II II Kpq (z, w) = Hpq (z)Hpq (w)∗
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hold for every (z, w) ∈ Dd × Dd . Using hereditary calculus (see [2]) we obtain from (1.5) that I e−e I IK⊗F∗ − f (T )f (T )∗ = Hpq Hpq (T )∗ (T ) ΔT p ⊗ IHpq I II e−e II ∗ + Hpq (T ) ΔT q ⊗ IHpq II Hpq (T ) , whose right-hand side is positive semidefinite. Then f (T ) 1. Therefore T satisfies the von Neumann inequality (1.4) for any bounded analytic operator-valued function f on Dd . As was mentioned in the paragraph preceding Theorem 1.2, the latter means that T has a unitary dilation. 2 4. Nevanlinna–Pick interpolation (j )
(j )
Let N be a positive integer, and z(j ) = (z1 , . . . , zd ), j = 1, . . . , N , points in the polydisk Dd . Let F and F∗ be Hilbert spaces, and let W (j ) ∈ L(F , F∗ ), j = 1, . . . , N , be given operators. The Nevanlinna–Pick interpolation problem with data (z(1) , . . . , z(N ) , W (1) , . . . , W (N ) ) consists of finding a function f ∈ Sd (F , F ∗ ) such that f (z(j ) ) = W (j ) , j = 1, . . . , N . It was shown in [1] (see also [3,10]) that a solution f exists in the Schur–Agler class SAd (F , F ∗ ) (k) if and only if there exist positive semidefinite block matrices [aij ]N i,j =1 with entries in L(F∗ ), k = 1, . . . , d, such that d N (k) N (i) (j ) N 1 − zk zk i,j =1 ◦ aij i,j =1 , IF∗ − W (i) W (j )∗ i,j =1 = k=1
where “◦” denotes the Schur (entrywise) multiplication of matrices. Using Theorem 1.2 we now obtain necessary conditions for the existence of a solution to the Nevanlinna–Pick interpolation problem in the d-variable Schur class. Theorem 4.1. If f ∈ Sd (F , F ∗ ) is a solution to the Nevanlinna–Pick interpolation problem with data (z(1) , . . . , z(N ) , W (1) , . . . , W (d) ), then for any integer p and q, 1 p < q d, there exist positive semidefinite matrices AIpq , AII pq with entries in L(F∗ ) such that N IF∗ − W (i) W (j )∗ i,j =1 N (i) (j ) 1 − zk zk = k =p
i,j =1
◦ AIpq +
k =q
N (i) (j ) 1 − zk zk
i,j =1
◦ AII pq .
(4.1)
Proof. Suppose that f ∈ Sd (F , F ∗ ) is a solution sought. By Theorem 1.2 we have that (1.5) I and K II . Set AI := (K I (z(i) , z(j ) ))N II holds for some positive kernels Kpq pq pq pq i,j =1 and Apq := II (z(i) , z(j ) ))N I II (Kpq i,j =1 . Clearly, Apq and Apq are positive semidefinite, and so equality (1.5) implies (4.1). 2 The following example shows that there exists a Nevanlinna–Pick problem which is solvable in the d-variable Schur class but not in the Schur–Agler class.
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Example 4.2. Let T = (T1 , . . . , Td ) be a d-tuple (d 3) of commuting diagonalizable N × N matrices, Tk < 1, and let p(z1 , . . . , zd ) be a scalar-valued polynomial with p∞ 1 such that the von Neumann inequality fails for T , p(T ) > 1. This choice is possible, for instance, by the Lotto–Steger perturbation [30] of the Kaijser–Varopoulos example from [38], or by the example in [26, Section 5]. Let Y be a N × N nonsingular matrix such that Tk = Y Δk Y −1 , where (1) (N ) Δk = diag(zk , . . . , zk ), k = 1, . . . , d. Consider the Nevanlinna–Pick data (j ) (j ) z(j ) = z1 , . . . , zd ∈ Dd ,
w (j ) = p z(j ) ∈ D,
j = 1, . . . , N.
The interpolation problem is clearly solvable in the class Sd (for instance, p is a solution). However, for any analytic function f (z) = n∈Zd cn zn on Dd satisfying f (z(j ) ) = w (j ) , + j = 1, . . . , N , one has f (T ) =
n∈Zd+
cn T n = Y
cn Δn Y −1
n∈Zd+
= Y diag f z(1) , . . . , f z(N ) Y −1 = Y diag w (1) , . . . , w (N ) Y −1 = Y diag p z(1) , . . . , p z(N ) Y −1 = Yp(Δ)Y −1 = p(T ). Hence f (T ) = p(T ) > 1, and the interpolation problem is not solvable in the Schur–Agler class SAd . The question of whether any of the conditions of Theorem 4.1 are sufficient for the solvability of the Nevanlinna–Pick problem in the d-variable Schur class is open. 5. Discussion 5.1. We present here one more decomposition of an analytic function f ∈ Sd (F , F∗ ) and introduce one more class of d-tuples T = (T1 , . . . , Td ) of commuting strict contractions on a common Hilbert space which satisfies the d-variable von Neumann inequality. These results are more evident and less involved. Theorem 5.1. Let f ∈ Sd (F , F∗ ). Then IF∗ − f (z)f (w)∗ =
d
(1 − zk wk )K(z, w),
(z, w) ∈ Dd × Dd ,
k=1
where K(z, w) is a positive L(F∗ )-valued sesquianalytic kernel on Dd . Proof. The function IF − f (z)f (w)∗ K(z, w) = ∗d k=1 (1 − zk wk )
(5.1)
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is a positive L(F∗ )-valued sesquianalytic kernel on Dd . This is an attribute of the de Branges– Rovnyak model, as we have seen in the last paragraph of Section 2. (In fact, this goes back to [29] where the positivity of an analogous kernel has been proved for a scalar-valued analytic function on Dd with positive real part. The proof can easily be extended to operator-valued functions. The positivity of the kernel K(z, w) is then obtained by applying the Cayley transform.) 2 We will say that a d-tuple T = (T1 , . . . , Td ) of linear operators on a common Hilbert space belongs to the class P d if (i) T1 , . . . , Td commute; (ii) T1 , . . . , Td are contractions; β (iii) the operator ΔeT is positive semidefinite, where ΔT is defined in (1.6) for any β ∈ {0, 1}d , in particular for e = (1, . . . , 1). Theorem 5.2. Let T be a d-tuple of strict contractions from the class P d . Then T satisfies the von Neumann inequality (1.4) for any bounded analytic operator-valued function f on Dd . Equivalently, T has a commuting unitary dilation. Proof. Let T = (T1 , . . . , Td ) ∈ L(K)d be a d-tuple of strict contractions from the class P d . Let f be an arbitrary L(F , F∗ )-valued function from the d-variable Schur class Sd (F , F∗ ). By Theorem 5.1, there is a positive L(F∗ )-valued sesquianalytic kernel K(z, w) on Dd such that (5.1) holds. Then (see [7]) there exist a Hilbert space H and an analytic function H on Dd which takes values in L(H, F∗ ), such that the equality K(z, w) = H (z)H (w)∗ holds for every (z, w) ∈ Dd × Dd . Using hereditary calculus (see [2]) we obtain from (5.1) that IK⊗F∗ − f (T )f (T )∗ = H (T ) ΔeT ⊗ IH H (T )∗ , holds. The right-hand side of this equality is clearly positive semidefinite. Then f (T ) 1. Therefore T satisfies the von Neumann inequality (1.4) for any bounded analytic operator-valued function f on Dd . As was mentioned in the paragraph preceding Theorem 1.2, the latter means that T has a unitary dilation. 2 5.2. The class P d is closely related to another class of d-tuples of commuting contractions, which was introduced and studied by Brehmer in [16] (see also [37, Section 1.9]). We will say that a d-tuple T = (T1 , . . . , Td ) of commuting bounded linear operators on a common Hilbert β space belongs to Brehmer’s class B d if ΔT is a positive semidefinite operator for every β ∈ e {0, 1}d . In particular, the inequalities ΔTk 0 mean that the operators Tk are contractions for k = 1, . . . , d. It was proved in [16] that T ∈ L(K)d belongs to the class B d if and only if T has a unitary dilation U ∈ L(Kext )d , where Kext ⊃ K, which is *-regular in the sense that +
−
T n T ∗ n = PK U n |K,
n ∈ Zd .
Here we use the notation n+ = max{n1 , 0}, . . . , max{nd , 0} ∈ Zd+ , n− = max{−n1 , 0}, . . . , max{−nd , 0} ∈ Zd+ ,
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so that n = n+ − n− . The class B d is clearly a subclass in P d . Moreover, B d = P d for d = 1, 2, and B d is a proper subclass in P d for any d 3. Example 5.3. Let d 3 and T = (T1 , . . . , Td ) a d-tuple of commuting contractions on a common Hilbert space such that Td commutes with all Tk and Tk∗ , k = 1, . . . , d − 1. Let T = (T1 , . . . , Td−1 ) and e = (1, . . . , 1) ∈ Zd−1 + . Then we have ΔeT = I − Td Td∗ ΔeT .
(5.2) e−ed
In fact, (5.2) holds whenever Td commutes with ΔeT . Suppose that T is such that ΔT is not positive semidefinite. For example, for Tk =
0 0
1 , 0
= ΔeT
k = 1, . . . , d − 1,
one has ΔeT
2−d = 0
0 0. 1
If, in addition, Td is a coisometry (e.g., if Td = I ), the identity (5.2) implies that ΔeT = 0. Therefore, T ∈ P d \ B d . In particular, it follows from the Brehmer theorem that T has no ∗ -regular unitary dilations. Still, T has a unitary dilation U = (U1 , . . . , U1 , Ud ) where the pair (U1 , Ud ) is a unitary dilation of a pair of commuting contractions (T1 , Td ), which exists by Andô’s theorem [4]. (In the case where Td = I the d-tuple T = (T1 , . . . , T1 , I ) has a unitary dilation U = (U1 , . . . , U1 , I ) where U1 is a unitary dilation of T1 which exists by the Sz.-Nagy dilation theorem [36].) Observe that, in Example 5.3, T ∈ / P d−1 and T is a (d − 1)-tuple of nonstrict contractions, however T has a unitary dilation (or, equivalently, satisfies the (d − 1)-variable von Neumann inequality for any polynomial with operator coefficients). Thus, both the condition T ∈ P d and the condition of strictness of contractions in Theorem 5.2 are not necessary for a d-tuple T of commuting contractions to have a unitary dilation (or, equivalently, to satisfy the d-variable von Neumann inequality for any polynomial with operator coefficients). On the other hand, one cannot relax the strictness condition in Theorem 5.2. E.g., for d = 4 one can modify Example 5.3 by choosing the triple T = (T1 , T2 , T3 ) of commuting contractions with no unitary dilations, say as in Parrott’s example [31], and T4 = I . Then T = (T1 , T2 , T3 , T4 ) ∈ P 4 , however T has no unitary dilations. Let a d-tuple T of commuting contractions have the following property (see [33]): (P) there exists ρ ∈ [0, 1) such that for every r ∈ [ρ, 1) one has rT ∈ P d , where rT = (rT1 , . . . , rTd ). Then T ∈ B d by [24, Theorem 4.4]. Some additional properties and generalizations of the class B d are studied in [20].
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5.3. Let d 3, and let p and q be integers such that 1 p < q d. We will say that a d-tuple T = (T1 , . . . , Td ) of commuting bounded linear operators on a common Hilbert space belongs to d if Δβ is a positive semidefinite operator for every β ∈ {0, 1}d such that β e − e the class Bpq p T or β e − eq . In particular, ΔeTk 0 for k = 1, . . . , d, so that all Tk are contractions. It was d has a unitary dilation (see also [6, Theorem 4.8] shown in [24, Theorem 3.1] that any T ∈ Bpq d is a subclass of P d . Moreover, for an alternative proof in the case d = 3). It is clear that Bpq pq d d d d Bpq = Ppq for d = 3, and Bpq is a proper subclass of Ppq for any d 4. Example 5.4. Let d 4. We modify here Example 5.3 as follows. Define T = (T1 , . . . , Td ) by
0 Tk = 0
1 , 0
k = p, = q,
1 0 and Tp = Tq = . 0 1
d \ B d . Still, T has a unitary dilation Then arguing as in Example 5.3 we obtain that T ∈ Ppq pq U = (U1 , . . . , Ud ) where Up = Uq = I and Uk = U0 is a unitary dilation of Tk , k = p, = q. d in our Theorem 1.3 is weaker than Example 5.4 shows that for d 4 the assumption T ∈ Ppq d in [24, Theorem 3.1]. However, our additional assumption of strictness the assumption T ∈ Bpq of contractions in Theorem 1.3 cannot be relaxed. E.g., for d = 5 one can modify Example 5.4 by choosing the triple of commuting contractions Tk , k = p, = q with no unitary dilations, e.g., 5 , however T has no as in Parrott’s example [31], and Tp = Tq = I . Then T = (T1 , . . . , T5 ) ∈ Ppq unitary dilations.
Example 5.5. Let d 3. Define T = (T1 , . . . , Td ) by Tk =
0 r , 0 0
k = 1, . . . , d,
√ where r is any real number such that 1/ d − 1 < r < 1. Then for any k one has
k Δe−e T
1 − r 2 (d − 1) = 0
0 0. 1
d for any Therefore, T is a d-tuple of commuting strict contractions which does not belong to Ppq p and q. On the other hand, T has a unitary dilation U = (U0 , . . . , U0 ), where U0 is a unitary d is not necessary for T to dilation of T1 = · · · = Td . Therefore, the assumption that T ∈ Ppq have a unitary dilation (equivalently, to satisfy the d-variable von Neumann inequality for any polynomial with operator coefficients).
5.4. We show here that there are d-tuples of commuting contractions which belong to all d and do not belong to P d . classes Ppq √ √ Example 5.6. Define Tk , k = 1, . . . , d, as in Example 5.5, however with 1/ d < r 1/ d − 1. Then for any k = 1, . . . , d,
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k Δe−e = T
1 − r 2 (d − 1) 0
0 0, 1
however
ΔeT
1 − r 2d = 0
0 0. 1
d and does not belong to P d . Thus, T belongs to all classes Ppq d . It is still unclear whether there are d-tuples in P d but in none of Ppq k 0, k = 1, . . . , d, 5.5. In Example 5.5 a d-tuple T does not satisfy any inequality Δe−e T while in Example 5.6 a d-tuple T satisfies all of these inequalities. These examples can be easily modified to have only part of the inequalities satisfied.
as in Example 5.5, and Tk = rI for k = l + 1, . . . , d, Example 5.7. Define Tk for k√= 1, . . . , l, √ where 2 l d − 1, with 1/ l < r 1/ l − 1. Then 2 e−ek 2 d−l 1 − r (l − 1) 0 0 for k = 1, . . . , l, ΔT = 1 − r 0 1 and 2 2 d−l−1 1 − r l k = 1 − r Δe−e T 0
0 0 for k = l + 1, . . . , d. 1
k 0 holds. The following example covers the case where only one of inequalities Δe−e T
Example 5.8. Let A = √
1 1 01
and let T1 =
√
rA, T2 =
√ √ −1 rA , T3 = 1 − r(A−I ). Then, A =
A−1 = 1+2 5 , and so T = (T1 , T2 , T3 ) is a triple of commuting contractions for 0 < r One easily checks that 0 1 − 4r + r 2 e−e ΔT 3 = 0 (1 − r)2
√ 3− 5 2 .
√ is positive semidefinite for 0 < r 2 − 3. At the same time, (−1)k r −r 2 e−e 0, k = 1, 2. ΔT k = (−1)k r 1 − r Acknowledgments Hugo Woerdeman wishes to thank Ben-Gurion University, where he was invited as a Faculty of Natural Sciences Distinguished Scientist Visitors Program Visiting Fellow, for its hospitality. This project was initiated with discussions at Ben-Gurion University. The authors would like to thank John McCarthy for suggesting a way to construct Example 4.2.
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Journal of Functional Analysis 256 (2009) 3055–3070 www.elsevier.com/locate/jfa
Property (T ) and strong property (T ) for unital C ∗ -algebras ✩ Chi-Wai Leung a , Chi-Keung Ng b,∗ a Department of Mathematics, The Chinese University of Hong Kong, Hong Kong b Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
Received 21 August 2008; accepted 7 January 2009 Available online 22 January 2009 Communicated by D. Voiculescu
Abstract In this paper we will give a thorough study of the notion of property (T ) for C ∗ -algebras (as introduced by M.B. Bekka) as well as a slightly stronger version of it, called “strong property (T )” (which is also an analogue of the corresponding concept in the case of discrete groups and type II1 -factors). More precisely, we will give some interesting equivalent formulations as well as some permanence properties for both property (T ) and strong property (T ). We will also relate them to certain (T )-type properties of the unitary group of the underlying C ∗ -algebra. © 2009 Elsevier Inc. All rights reserved. Keywords: Property (T ); Unital C ∗ -algebras; Hilbert bimodules
1. Introduction Property (T ) for locally compact groups was first defined by D. Kazhdan in [11] and was later extended to Hausdorff topological groups. In [12], property (T ) for a pair of groups H ⊆ G was ✩ This work is jointly supported by Hong Kong RGC Research Grant (2160255), Hong Kong RGC Direct Grant (2060319), the National Natural Science Foundation of China (10771106) and NCET-05-0219. * Corresponding author. E-mail addresses:
[email protected] (C.-W. Leung),
[email protected],
[email protected] (C.-K. Ng).
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.004
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introduced. This notion was proved to be very useful and was studied by many people (see e.g. [2,4,8,10–12,15]). In [6], A. Connes introduced the notion of property (T ) for type II1 -factors and this notion was then extended to von Neumann algebras in [7]. A discrete group G has property (T ) if and only if the von Neumann algebra generated by the left regular representation of G has property (T ) (this was first proved in [7] for ICC groups and was generalized by P. Jolissaint in [9] to general discrete groups). The notion of property (T ) for a pair of von Neumann algebras was defined by S. Popa in [14]. This notion was also proved to be very useful in the study of von Neumann algebras. Recently, M.B. Bekka introduced in [3] the interesting notion of property (T ) for a pair consisting of a unital C ∗ -algebra and a unital C ∗ -subalgebra. He showed that a countable discrete group G has property (T ) if and only if its full (or equivalently reduced) group C ∗ -algebra has property (T ). In [5], N.P. Brown did a study of property (T ) for C ∗ -algebras and showed that a nuclear unital C ∗ -algebra A has property (T ) if and only if A = B ⊕ C where B is finite dimensional and C admits no tracial state. The aim of this paper is to give a thorough study of property (T ) as well as a slightly stronger version called strong property (T ) for unital C ∗ -algebras. On our way, we will show that our stronger version is equally good (if not a better) candidate for the notion of property (T ) for a pair of unital C ∗ -algebras. The paper is organised as follows. In Section 2 we will give two simple and useful reformulations of both property (T ) and strong property (T ). In Section 3 we consider two Kazhdan constants tuA and tcA for a C ∗ -algebra A which are the analogues of the Kazhdan constant for locally compact groups (see [15]). We will show that A has property (T ) (respectively, strong property (T )) if and only if tcA > 0 (respectively, tuA > 0). Through them, we obtain some interesting reformulations of property (T ) and strong property (T ). In particular, we show that one can check property (T ) by looking at just one bimodule. We will also show that one can express the gap between property (T ) and strong property (T ) by another Kazhdan constant tsA . In Section 4 we obtain some permanence properties for property (T ) and strong property (T ), including quotients, direct sums, tensor products and crossed products. In Section 5 we will show that finite dimensional C ∗ -algebras have strong property (T ). Moreover, we show that a corresponding result of Bekka concerning relation between property (T ) of discrete groups and their group C ∗ -algebras as well as a corresponding result of Brown concerning amenable property (T ) C ∗ -algebras also holds for strong property (T ). In Section 6 we study the relation between property (T ) (as well as strong property (T )) of a unital C ∗ -algebra A and certain (T )-type properties of the unitary group of A. Let us first set the following notations that will be used throughout the whole paper. Notation 1.1. (1) A is a unital C ∗ -algebra and B ⊆ A is a C ∗ -subalgebra containing the identity of A. Set ADou := A ⊗max Aop (where Aop is the “opposite C ∗ -algebra” with a op bop = (ba)op ). (2) F(E) is the set of all non-empty finite subsets of a set E and S1 (X) is the unit sphere of a normed space X. (3) U (A) and S(A) are respectively the unitary group and the state space of A. (4) Bimod∗ (A) is the collection of unitary equivalence classes of unital Hilbert bimodules over A (or equivalently, non-degenerate representations of ADou ). For any H ∈ Bimod∗ (A), let H B := {ξ ∈ H : b · ξ = ξ · b for all b ∈ B}
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and PHB : H → H B be the orthogonal projection. Elements in H B are called central vectors for B. Moreover, for any (Q, β) ∈ F(A) × R+ , set VH (Q, β) := ξ ∈ S1 (H ): x · ξ − ξ · x < β for all x ∈ Q . Elements in VH (Q, β) are called (Q, β)-central unit vectors. On the other hand, a net of vectors (ξi )i∈I in S1 (H ) is called an almost central unit vector for A if a · ξi − ξi · a → 0 for any a ∈ A. (5) For any topological group G, we denote by Rep(G) the collection of all unitary equivalence classes of continuous unitary representations of G. If (π, H ) ∈ Rep(G), we let H G := ξ ∈ H : π(s)ξ = ξ for all s ∈ G and PHG : H → H G be the orthogonal projection. Furthermore, if F ∈ F(G) and > 0, we set Vπ (F, ) = ξ ∈ S1 (H ): π(t)ξ − ξ < for all t ∈ F . (6) For any (μ, H ), (ν, K) ∈ Rep(G), we write (μ, H ) (ν, K) if (μ, H ) is a subrepresentation of (ν, K). 2. Definitions and basic properties Let us first recall Bekka’s notion of property (T ) in [3]. The pair (A, B) is said to have property (T ) if there exist F ∈ F(A) and > 0 such that for any H ∈ Bimod∗ (A), if VH (F, ) = ∅, then H B = (0). In this case (F, ) is called a Kazhdan pair for (A, B). Moreover, A is said to have property (T ) if the pair (A, A) has property (T ). Note that Bekka’s definition comes from the original definition of property (T ) for groups (see e.g. [10, Definition 1.1(1)]). We will now give a slightly stronger version which comes from an equivalent form of property (T ) for groups (see [10, Theorem 1.2(b2)]). Note that the corresponding stronger version of property (T ) for type II1 -factor is also equivalent to property (T ) (see e.g. [7, Proposition 1]) but we do not know if it is the case for C ∗ -algebras. Definition 2.1. The pair (A, B) is said to have strong property (T ) if for any α > 0, there exist Q ∈ F(A) and β > 0 such that for any H ∈ Bimod∗ (A) and any ξ ∈ VH (Q, β), we have ξ − PHB (ξ ) < α. In this case (Q, β) is called a strong Kazhdan pair for (A, B, α). We say that A has strong property (T ) if (A, A) has such property. It is clear that if A has property (T ) (respectively, strong property (T )) then so has the pair (A, B). Moreover, by taking α < 1/2, we see that strong property (T ) implies property (T ). We will see later that strong property (T ) is an equally good (if not a better) candidate for the notion of property (T ) for a pair of C ∗ -algebras. Let us now give the following simple reformulation of property (T ) and strong property (T ) which will be useful in Section 6. Lemma 2.2. For any (Q, β) ∈ F(A) × R+ , there exists (Q , β ) ∈ F(U (A)) × R+ such that VH (Q , β ) ⊆ VH (Q, β) for any H ∈ Bimod∗ (A). Consequently, one can replace F(A) by F(U (A)) in the definitions of both property (T ) and strong property (T ).
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Proof. This lemma is clear if Q = {0}. Let Q\{0} = {x1 , . . . , xn } and M = max{x1 , . . . , xn }. For each k ∈ {1, . . . , n}, consider uk , vk ∈ U (A) such that 2xk = xk ((uk + u∗k ) + i(vk + vk∗ )). β , then VH (Q , β ) ⊆ If we take Q to be the set {u1 , u∗1 , v1 , v1∗ , . . . , un , u∗n , vn , vn∗ } and β = 2M ∗ VH (Q, β) for any H ∈ Bimod (A). 2 Before we give a second simple reformulation, we need to set some notations. Let S(D) and St (D) be respectively the sets of all states and the set of all tracial states on a C ∗ -algebra D. For any τ ∈ S(D) and any for τ and by Mτ,α the cardinal α, we denote by Mτ the GNS construction α-times direct sum, α Mτ , of Mτ (we use the convention that 0 Mτ = {0}). Definition 2.3. Let H :=
Mτ
and K :=
Mτ .
τ ∈St (A)
τ ∈S(ADou )
We call H and K the universal and the standard bimodules (over A) respectively. Moreover, a bimodule of the form τ ∈St (A) Mτ,ατ is called a quasi-standard bimodule. Proposition 2.4. (a) (A, B) has property (T ) if and only if for any H ∈ Bimod∗ (A), the existence of an almost central unit vector for A in H will imply that H B = {0}. (b) The following statement are equivalent. (i) (A, B) has strong property (T ). (ii) For any almost central unit vector (ξi )i∈I for A in any bimodule H ∈ Bimod∗ (A), we have ξi − PHB (ξi ) → 0. (iii) For any almost central unit vector (ξi )i∈I for A in H and any n ∈ N, there exists in ∈ I with B (ξ ) < 1/n. ξin − PH in Proof. (a) This part is well known. (b) It is clear that (i) ⇒ (ii) and (ii) ⇒ (iii). To obtain (iii) ⇒ (i), we suppose, on the contrary, that (A, B) does not have strong property (T ). Then one can find α0 > 0 such that for any i = (Q, β) ∈ I := F(A) × R+ , there exist Hi ∈ Bimod∗ (A) and ξi ∈ VHi (Q, β) with ξi − PHBi (ξi ) α0 . If Ki = A · ξi · A, then Hi = Ki ⊕ Ki⊥ and HiB = KiB ⊕ (Ki⊥ )B . As ξi ∈ Ki , we have ξi − P B (ξi ) = ξi − P B (ξi ) α0 . Ki
Hi
We set X := {Ki : i ∈ I } ⊆ Bimod∗ (A) and K0 := K∈X K. Since all bimodules in X are cyclic (as representations of ADou ) and any two elements in X are inequivalent, K0 is a Hilbert sub-bimodule of H. Moreover, each Ki is equivalent to a unique element in X and this gives a canonical Hilbert bimodule embedding Ψi : Ki → K0 . It is easy to check that (Ψi (ξi ))i∈I is an B (Ψ (ξ )) = ξ − P B (ξ ) α for almost central unit vector for A in H with Ψi (ξi ) − PH i i i 0 Ki i every i ∈ I (since Ψi (Ki ) is a direct summand of H). This contradicts statement (iii). 2 Since Proposition 2.4 is so fundamental to our discussions, we may use it without mentioning it explicitly throughout the whole paper.
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Remark 2.5. (a) Note that if A is separable, then in the above proposition, one can replace almost central unit vector by a sequence of unit vectors that is “almost central” for A. (b) In Proposition 2.4(b)(iii) we only need to check one bimodule (namely, the universal one) in order to verify strong property (T ). (c) One may wonder if it is possible to check whether a C ∗ -algebra has property (T ) by looking at its universal bimodule alone. However, this cannot be done using the original formulation of property (T ) because there exists a unital C ∗ -algebra A which does not have property (T ) but HA = (0) (i.e. A has a tracial state). Nevertheless, we will show in Theorem 3.4 below that it is possible to do so using an equivalent formulation of property (T ). 3. Kazhdan constants In this section, we will define and study some Kazhdan constants in the case when B = A. Let us start with the following lemma. Lemma 3.1. Let H ∈ Bimod∗ (A). If HC is the sub-bimodule generated by H A (called the centrally generated part of H ), then HC is a quasi-standard bimodule (Definition 2.3) and HC⊥ contains no non-zero central vector for A. Proof. Without loss of generality, we may assume that C := S1 (H A ) is non-empty. Let S := {A · ξ : ξ ∈ C} and M := {M ⊆ S: K ⊥ L for any K, L ∈ M}. By the Zorn’s lemma, there exists a maximal element M0 in M and we put H1 := K∈M0 K. Then clearly H1 ⊆ HC and H1⊥ contains no non-zero central vector for A. Together with the fact H A = H1A ⊕ (H1⊥ )A , this shows that H A = H1A ⊆ H1 and hence H1 = HC . Finally, for any A · ξ ∈ M0 with ξ ∈ C, the functional defined by τ (a) := aξ, ξ (a ∈ A) is a tracial state and A·ξ ∼ = Mτ . This completes the proof. 2 Suppose that H ∈ Bimod∗ (A) and K is a Hilbert subspace of H . For any Q ∈ F(A), we set
1/2 t A (Q; H, K) := inf x · ξ − ξ · x2 : ξ ∈ S1 (H K) x∈Q
(we use the convention that the infimum over the empty set is +∞). Lemma ∈ Bimod∗ (A) and K be a Hilbert subspace of H . Suppose that 3.2. Let Q ∈ F(A), H H = λ∈Λ Hλ such that K = λ∈Λ Kλ where Kλ := Hλ ∩ K. (a) If αλ isa cardinal for any λ ∈ Λ, and if we set H0 := λ∈Λ ( αλ Kλ ), then t A (Q; H, K)2 ζ 2
x∈Q
x · ζ − ζ · x2
λ∈Λ (
αλ
Hλ ) and K0 :=
(ζ ∈ H0 K0 ).
(3.1)
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(b) t A (Q; H, K) = infλ∈Λ t A (Q; Hλ , Kλ ). (c) (t A (Q; H, K))Q∈F(A) is an increasing net and limQ∈F(A) t A (Q; H, K) = 0 if and only if there exists an almost central unit vector for A in H K. In this case one can choose an almost central unit vector (ξi )i∈I for A such that for any i ∈ I , there exists λi ∈ Λ with ξi ∈ Hλi Kλi . Proof. For any ζ ∈ S1 (H0 K0 ), we have ζ = (ζi,λ )λ∈Λ;i∈αλ with ζi,λ ∈ Hλ Kλ ⊆ H K (a) and λ∈Λ i∈αλ ζi,λ 2 = 1. Thus, t (Q; H, K) A
2
λ∈Λ i∈αλ
2 ζi,λ ζi,λ ζi,λ x · ζ − ζ · x2 . x · ζ − ζ · x = i,λ i,λ 2
x∈Q
x∈Q
(b) Note that t A (Q; H, K) t A (Q; Hλ , Kλ ) for all λ ∈ Λ (as Hλ Kλ ⊆ H K). For any > 0, there exists ξ ∈ S1 (H K) such that x∈Q x · ξ − ξ · x2 t A (Q; H, K) + . Now, ξ = (ξλ )λ∈Λ with ξλ ∈ Hλ Kλ and λ∈Λ ξλ 2 = 1. A similar argument as part (a) implies that there exists λ0 ∈ Λ such that 2 x · ξλ0 − ξλ0 · x t A (Q; H, K) + . ξ ξ λ0 λ0
x∈Q
(c) It is clear that (t A (Q; H, K))Q∈F(A) is increasing and that t A (Q; H, K) = 0 for any Q ∈ F(A) if there exists an almost central unit vector for A in H K. Now, suppose that sup
inf t A (Q; Hλ , Kλ ) = lim t A (Q; H, K) = 0.
Q∈F(A) λ∈Λ
Q∈F(A)
Then for any Q ∈ F(A) and > 0, there exist λQ, ∈ Λ and ξQ, ∈ S1 (HλQ, KλQ, ) such that 2 2 x∈Q x · ξQ, − ξQ, · x < . It is easy to see that (ξQ, )(Q,)∈F(A)×R+ is an almost central unit vector for A. 2 Now, we define three Kazhdan constants: for any Q ∈ F(A), set
tuA (Q) := t A Q; H, HA ,
tcA (Q) := t A (Q; H, HC ),
tsA (Q) := t A Q; K, KA
(where H and K are the universal bimodule and the standard bimodule respectively) and tuA := sup tuA (Q), Q∈F(A)
tcA := sup tcA (Q) Q∈F(A)
as well as
tsA := sup tsA (Q). Q∈F(A)
Lemma 3.3. (a) For any H ∈ Bimod∗ (A), we have tuA (Q) t A (Q; H, H A ) and tcA (Q) t A (Q; H, HC ). If, in addition, H is quasi-standard, then tsA (Q) t A (Q; H, H A ). (b) tuA (Q) min{tcA (Q), tsA (Q)}.
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Proof. (a) There are cardinals ατ (τ ∈ S(ADou )) such that H ∼ = τ ∈S(ADou ) Mατ ,τ . For any ξ ∈ S1 (H H A ), we have ξ = (ξτ ) where ξτ ∈ Mατ ,τ MαAτ ,τ . By inequality (3.1), we have tuA (Q)2 ξτ 2
x · ξτ − ξτ · x2
x∈Q
and so
tuA (Q)2
x · ξτ − ξτ · x2 =
τ ∈S(ADou ) x∈Q
x · ξ − ξ · x2
x∈Q
(as τ ∈S(ADou ) ξτ 2 = ξ 2 = 1). Thus, we have tuA (Q) t A (Q; H, H A ). The arguments for the other two inequalities are similar. (b) tuA (Q) tcA (Q) because HA ⊆ HC and tuA (Q) tsA (Q) because of part (a). 2 Theorem 3.4. (a) The following statements are equivalent. (i) tuA > 0. (ii) A has strong property (T ). A (ξ ) = 0. (iii) There exists (Q, δ) ∈ F(A) × R+ such that for any ξ ∈ VH (Q, δ), we have PH (b) The following statements are equivalent. (i) tsA > 0. (ii) For any > 0, there exists (Q, δ) ∈ F(A) × R+ such that for any quasi-standard bimodule H and any ξ ∈ VH (Q, δ), we have ξ − PHA (ξ ) < . (iii) There exists (Q, δ) ∈ F(A) × R+ such that for any ξ ∈ VK (Q, δ), we have PKA (ξ ) = 0. (c) The following statements are equivalent. (i) (ii) (iii) (iv)
tcA > 0. A has property (T ). There is (Q, δ) ∈ F(A) × R+ such that VH (Q, δ) ∩ HC⊥ = ∅ for any H ∈ Bimod∗ (A). There exists (Q, δ) ∈ F(A) × R+ such that VH (Q, δ) ∩ HC⊥ = ∅.
(d) tuA > 0 if and only if both tcA > 0 and tsA > 0. Proof. (a) (i) ⇒ (ii). There exists Q ∈ F(A) with tuA (Q) > 0. Let m be the number of elements t A (Q)
in Q and δ = u√m . For any H ∈ Bimod∗ (A) and τ ∈ S(ADou ), there is a cardinal ατ such that H = τ ∈S(ADou ) Mατ ,τ (ατ can be zero). Pick any ξ ∈ VH (Q, δ) and consider ξ = ξ − PHA (ξ ) ∈ (H A )⊥ . Since ξ = (ζτ )τ ∈S(A) where ζτ ∈ (MαAτ ,τ )⊥ , we have, by inequality (3.1), ξ 2 tuA (Q)−2
x∈Q
x · ξ − ξ · x2 = tuA (Q)−2
x · ξ − ξ · x2 < 2 .
x∈Q
(ii) ⇒ (iii). By taking = 1/2, we see that statement (iii) holds.
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on the contrary that tuA (Q) = 0. Then there exists ξ ∈ S1 ((HA )⊥ ) with (iii) ⇒ (i). Suppose 2 < δ 2 . Hence, ξ ∈ V (Q, δ) and so P A (ξ ) = 0 which contradicts the fact x · ξ − ξ · x H x∈Q H that ξ ∈ (HA )⊥ . (b) The proof of this part is essentially the same as that of part (a) with H and tuA being replaced by K and tsA respectively. (c) (i) ⇒ (ii). Let Q ∈ F(A) such that tcA (Q) > 0. Suppose that A does not have property (T ). There exists H ∈ Bimod∗ (A) that contains an almost central unit vector (ξi ) for A but H A = {0}. Hence, HC⊥ = H , and
1/2 2 t (Q; H, HC ) = inf x · ξ − ξ · x : ξ ∈ S1 (H ) = 0. A
x∈Q
Now Lemma 3.3(a) gives the contradiction that tcA (Q) = 0. (ii) ⇒ (i). Suppose on the contrary that tcA (Q) = 0 for any Q ∈ F(A). There exists, by Lemma 3.2(c), an almost central unit vector for A in HC⊥ which contradicts the fact that A has property (T ) (because of Lemma 3.1). (i) ⇒ (iii). Let Q ∈ F(A) such that tcA (Q) > 0 and 0 < δ < elements in Q. By Lemma 3.3(a), we have
tcA (Q) √ m
tcA (Q) tcA (Q; H, HC ).
where m is the number of
Thus,
x · ζ − ζ · x2 tcA (Q) > mδ 2
x∈Q
for any ζ ∈ S1 (HC⊥ ). Suppose that there exists ξ ∈ VH (Q, δ) ∩ HC⊥ . Then mδ 2 < x∈Q x · ξ − ξ · x2 < mδ 2 which is absurd. (iii) ⇒ (iv). This is obvious. (iv) ⇒ (i). Suppose on the contrary that tcA (Q) = 0. Then there exists ξ ∈ S1 (HC⊥ ) with ⊥ 2 2 x∈Q x · ξ − ξ · x < δ which gives the contradiction that ξ ∈ VH (Q, δ) ∩ HC . A A A (d) If tu > 0, then ts > 0 and tc > 0 (by Lemma 3.3(b)). Conversely, suppose that tuA = 0. Then by Lemma 3.2(c), there exists an almost central unit vector (ξi )i∈I for A in
H HA = (H HC ) ⊕ HC HA . Let ηi ∈ H HC and ζi ∈ HC HA be the corresponding components of ξi . Then either ηi 0 or ζi 0. Therefore, by rescaling, there exists an almost central unit vector for A in either H HC or HC HA = HC HCA . In the first case we have tcA = 0 (by Lemma 3.2(c)). In the second case we have tsA supQ∈F(A) t A (Q; HC , HCA ) = 0 (by Lemma 3.3(a), Lemma 3.1 and Lemma 3.2(c)). 2 Part (a) of the above theorem tells us that in order to show that A has strong property (T ), it suffices to verify a weaker condition than that of Definition 2.1 for just the universal bimodule H. Remark 3.5. (a) The argument of Theorem 3.4(a), together with Lemma 3.3(a), tell us that for any Q ∈ F(A), δ > 0 and H ∈ Bimod∗ (A), if ξ ∈ VH (Q, δ), then
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√ tuA (Q)ξ − PHA (ξ ) < δ m (where m is the number of elements in Q). (b) The argument of Theorem 3.4(c), together with Lemma 3.3(a), tell us that if tcA (Q) > 0, t A (Q)
for any Q ∈ F(A), H ∈ Bimod∗ (A) and δ ∈ (0, c√m ), we have VH (Q, δ) ∩ HC⊥ = ∅. (c) The gap between property (T ) and strong property (T ) is represented by the gap between tcA and tuA or equivalently between HA and HC . Note that in the case of a locally compact group G, such a gap does not exist because the set of G-invariant vectors defines a subrepresentation. By Theorem 3.4 and Lemma 3.2(c), we have the following corollary. Corollary 3.6. (a) A has property (T ) (respectively, strong property (T )) if and only if there is no almost central unit vector for A in HC⊥ (respectively, in (HA )⊥ ). (b) A has strong property (T ) if and only if A has property (T ) and tsA > 0. Note that one can also obtain part (b) of the above corollary by using a similar argument as that of [7, Proposition 1]. 4. Some permanence properties In this section, we study the permanence properties for property (T ) and strong property (T ). First of all, we have the following lemma which implies that the quotient of any pair having property (T ) (respectively, strong property (T )) will have the same property. Since the proof is direct, we will omit it. Lemma 4.1. Let A1 and A2 be two unital C ∗ -algebras and let B1 ⊆ A1 and B2 ⊆ A1 be C ∗ subalgebras containing the identities of A1 and A2 respectively. Suppose that ϕ : A1 → A2 is a unital ∗-homomorphism such that B2 ⊆ ϕ(B1 ). If (A1 , B1 ) has property (T ) (respectively, strong property (T )), then so does (A2 , B2 ). Lemma 4.2. Let A1 , A2 , B1 and B2 be the same as in Lemma 4.1. If both (A1 , B1 ) and (A2 , B2 ) have property (T ) (respectively, strong property (T )), then so does (A1 ⊕ A2 , B1 ⊕ B2 ). Proof. The statement for property (T ) is well known and we will only show the case for strong property (T ). Suppose that H ∈ Bimod∗ (A1 ⊕ A2 ) and e = (1A1 , 0) ∈ A1 ⊕ A2 . Then H = 2 k,l=1 Hkl where Hkl is a non-degenerate Hilbert Ak -Al -bimodule. Suppose that (ξi )i∈I is an almost central unit vector in H for A1 ⊕ A2 and ξi = 2k,l=1 ξikl where ξikl ∈ Hkl . Then 12 2 21 2 ξ + ξ = e · ξi − ξi · e2 → 0. i
i
If ξi22 → 0, then we can assume that ξi11 > 1/2 (i ∈ I ), and (
ξi11 ) ξi11 i∈I
is an almost central
1 unit vector for A1 in H11 . In this case for any > 0, there is i0 ∈ I such that ξi11 − PHB11 (ξi11 ) < 2 (i i0 ), which implies that
ξj − P B1 ⊕B2 (ξj ) ξ 11 − P B1 ξ 11 2 + ξ 12 2 + ξ 21 2 + ξ 22 2 < H H11 i i i i i
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when j is large enough. The same conclusion holds if ξi11 → 0. We consider now the case when ξi11 0 and ξi22 0. There exist a constant κ > 0 as well as subnets (ξi11 ) k k∈J1 11 , ξ 22 κ for every k ∈ J and l ∈ J . One can show easily and (ξi22 ) such that ξ 1 2 ik il l l∈J2 that (
ξi11
k ξi11 k
)k∈J1 and (
ξi22 l
) ξi22 l∈J2
are almost central unit vectors for A1 and A2 in H11 and H22
l
respectively. Thus, for any > 0, one can find i0 ∈ I such that 11
ξ 11 ξ − P B1 ξ 11 < i0 , i0 H11 i0 2 and ξi12 2 + ξi21 2 < 0 0
2 2.
22
ξ 22 ξ − P B2 ξ 22 < i0 i0 H22 i0 2
Consequently,
ξi − P B1 ⊕B2 (ξi ) ξ 11 − P B1 ξ 11 2 + ξ 22 − P B1 ξ 11 2 + ξ 12 2 + ξ 21 2 < . 0 0 i0 i0 i0 i0 H H11 i0 H11 i0 In any case, (A1 ⊕ A2 , B1 ⊕ B2 ) has strong property (T ) because of Proposition 2.4(b)(iii).
2
Our next task is to consider tensor products and crossed products. Let us first recall the following useful terminology of co-rigidity from [3, Remark 19]. We will also introduce a stronger version of co-rigidity corresponding to strong property (T ). Definition 4.3. The pair (A, B) is said to be (a) co-rigid if there exists (Q, β) ∈ F(A) × R+ such that for any H ∈ Bimod∗ (A) with VH (Q, β) ∩ H B = ∅, we have H A = {0}, (b) strongly co-rigid if for any γ > 0, there exists (Q, δ) ∈ F(A) × R+ such that for any H ∈ Bimod∗ (A) and any ξ ∈ VH (Q, δ) ∩ H B , we have ξ − PHA (ξ ) < γ . The idea of the following result comes from [1, 2.3]. Proposition 4.4. Suppose that B has strong property (T ). (a) A has property (T ) if and only if (A, B) is co-rigid. (b) A has strong property (T ) if and only if (A, B) is strongly co-rigid. Proof. (a) The sufficiency is clear and we will only show the necessity. Let (Q, r) ∈ F(A) × R+ be the pair satisfying the condition in Definition 4.3(a). Suppose that (F, s) ∈ F(B) × R+ is the r strong Kazhdan’s pair for (B, B, α) where α = min{ 8M , 12 } and M = max{a: a ∈ Q}. Put E = r ∗ Q ∪ F and t = min{ 4 , s}. Assume that H ∈ Bimod (A) with ξ ∈ VH (E, t). As ξ ∈ VH (F, s), one has ξ − PHB (ξ ) < α and PHB (ξ ) 12 . If η = a · η − η · a
PHB (ξ ) , PHB (ξ )
then we have, for any a ∈ Q,
a · ξ − ξ · a + 2aξ − PHB (ξ )
Thus, η ∈ VH (Q, r) ∩ H B and H A = {0}.
PHB (ξ )
< 2t +
ra r. 2M
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(b) Again, we only need to show the necessity. For any > 0, let (Q, r) ∈ F(A) × R+ be the pair satisfying Definition 4.3(b) for γ = 2 . Take a strong Kazhdan’s pair (F, s) ∈ F(B) × R+ r for (B, B, α) where α = min{ 8M , 12 , 2 } and M = max{a: a ∈ Q}. If E and t are as in the argument of part (a), then for any ξ ∈ VH (E, t), we have η = implies that
PHB (ξ ) PHB (ξ )
∈ VH (Q, r) ∩ H B which
B B P (ξ ) − P A (ξ ) < PH (ξ ) H H 2 2
(note that H A ⊆ H B ). Since ξ − PHB (ξ ) < strong Kazhdan’s pair for (A, B, ). 2
2
as well, we see that (E, t) ∈ F(A) × R+ is a
We do not know whether B having property (T ) and (A, B) being co-rigid will imply that A has property (T ). If it is the case, then the statement in Theorem 4.5(a) below concerning property (T ) can be improved and a similar statement as Theorem 4.6 below for property (T ) will also hold. The first application of the above proposition is the following theorem. Notice that unlike the case of type II1 -factors (see [1, 2.5]), the fact that B ⊗max D having property (T ) (or strong property (T )) will not imply both B and D to have property (T ) (respectively, strong property (T )), but at least one of them have property (T ) (respectively, strong property (T )). Theorem 4.5. Let B and D be two unital C ∗ -algebras, A = B ⊗max D and A0 = B ⊗min D. (a) If B has strong property (T ) and D has property (T ) (respectively, strong property (T )), then A has property (T ) (respectively, strong property (T )). (b) If there is no almost central unit vector for D in any K ∈ Bimod∗ (D), then A has strong property (T ). (c) Suppose that there exists an almost central unit vector (ηj )j ∈J for D in some K ∈ Bimod∗ (D). If A0 has property (T ) (respectively, strong property (T )), then so does B. (d) If A0 has property (T ) (respectively, strong property (T )), then either B or D has property (T ) (respectively, strong property (T )). Proof. (a) We show the statement for strong property (T ) first. Suppose that α > 0 and (F, r) ∈ F(D) × R+ is the strong Kazhdan’s pair for (D, D, α). Let Q := 1 ⊗ F ∈ F(A) and H ∈ Bimod∗ (A). If H B = {0}, then H B ∈ Bimod∗ (D) under the canonical multiplications. For any ξ ∈ VH (Q, r) ∩ H B = VH B (F, r), we have ξ − P A (ξ ) ξ − P DB (ξ ) < γ H H (note that (H B )D = H A ). Thus, (A, B) is strongly co-rigid and we can apply Proposition 4.4(b). The proof for the case of property (T ) is similar. (b) In this case there is no almost central unit vector for A in any H ∈ Bimod∗ (A) and we can apply Proposition 2.4. (c) We will establish the statement for strong property (T ) and the statement for property (T ) is similar (and easier). Suppose H ∈ Bimod∗ (B) and there exists an almost central unit vector
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(ξi )i∈I for B in H . Then (ξi ⊗ ηj )(i,j )∈I ×J is an almost central unit vector for A0 in H ⊗ K. For any > 0, there exists, by Proposition 2.4(b)(ii), (i0 , j0 ) ∈ I × J such that ξi ⊗ ηj − P A0 (ξi ⊗ ηj ) < . 0 0 0 0 H ⊗K 0 Let ζ = PH ⊗K (ξi0 ⊗ ηj0 ) ∈ (H ⊗ K)A0 and ϕ ∈ K ∗ be defined by ϕ(η ) = η , ηj0 . For any b ∈ B and any ξ ∈ H , we have
A
b · (id ⊗ ϕ)(ζ ), ξ = (b ⊗ 1) · ζ, ξ ⊗ ηj0 = ζ · (b ⊗ 1), ξ ⊗ ηj0 = (id ⊗ ϕ)(ζ ) · b, ξ .
Therefore, (id ⊗ ϕ)(ζ ) ∈ H B and
ξi − (id ⊗ ϕ)(ζ ) = (id ⊗ ϕ) ξi ⊗ ηj − P A (ξi ⊗ ηj ) < . 0 0 0 0 0 H ⊗K This shows that B has strong property (T ) (by Proposition 2.4(b)(iii)). (d) If there is no almost central unit vector for D in any K ∈ Bimod∗ (D), then D has strong property (T ) (by definition). Otherwise, we can apply part (b). 2 Next, we will consider crossed product of C ∗ -algebras by actions of discrete groups. The idea of which comes from [1, 4.6]. Again, unlike the case of type II1 -factors, even if B ×α Γ has strong property (T ), this will not imply that Γ has property (T ) (notice that if α is trivial and any element in Bimod∗ (B) does not contain an almost central unit vector, then B ×α Γ will have strong property (T ) whether or not Γ have property (T )). Theorem 4.6. Let B be a unital C ∗ -algebra with an action α by a discrete group Γ and A = B ×α Γ . If Γ has property (T ), then (A, B) is strongly co-rigid. Consequently if B has strong property (T ) and Γ has property (T ), then A has strong property (T ) (and so does B ×α,r Γ ). Proof. As Γ has property (T ), for any > 0, there exists (F, δ) ∈ F(Γ ) × R+ such that for any (K, π) ∈ Rep(Γ ) and any η ∈ Vπ (F, δ), one has η − PKΓ (η) < (by [10, Theorem 1.2(b2)]). Let μ : B → A and u : Γ → A be the canonical maps. For any H ∈ Bimod∗ (A), we define a representation π : Γ → L(H B ) by π(t)ξ = ut · ξ · u∗t (t ∈ Γ, ξ ∈ H B ) which is well defined because for any b ∈ B,
μ(b) · π(t)ξ = ut μ αt −1 (b) · ξ · u∗t = ut · ξ · μ αt −1 (b) u∗t = π(t)ξ · μ(b). Moreover, it is easy to check that (H B )Γ = H A . Thus, if η ∈ VH (u(F ), δ) ∩ H B = Vπ (F, δ), then η − PHA (η) = η − PHΓ B (η) < . This shows that (A, B) is strongly co-rigid. The last statement follows from Proposition 4.4(b). 2 5. Some examples of strong property (T ) Our first example is finite dimensional C ∗ -algebras. It is easy to see that any element in Bimod∗ (Mn (C)) has a non-zero central vector and so Mn (C) has property (T ). In fact, Mn (C) also has strong property (T ) but a bit more argument is needed to establish this fact.
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Example 5.1. A = Mn (C) has a unique tracial state τ and hence K = Mτ . If tsA = 0, there exists an almost central unit vector in (MτA )⊥ for A (by Lemma 3.2(c)). As S1 ((MτA )⊥ ) is compact, this implies the existence of a central unit vector η for A in (MτA )⊥ but such η ∈ Mn (C) should be an element of C1 = MτA which is absurd. Now by Corollary 3.6(b) and Lemma 4.2, we see that any finite dimensional C ∗ -algebra has strong property (T ). By the argument of [3, Remark 17], we know that if A does not have tracial states, then there is no almost central unit vector for A in any H ∈ Bimod∗ (A). This, together with Proposition 2.4(b), give the following result (cf. [3, Remark 17]). Proposition 5.2. If A has no tracial states, then A has strong property (T ). If H is any Hilbert space, L(H ) has strong property (T ) (because of Example 5.1 and Proposition 5.2) and so if B ⊆ L(H ) is any unital C ∗ -subalgebra, then (L(H ), B) has strong property (T ). On the other hand, Proposition 5.2, together with Example 5.1, Lemma 4.2 and [5, 5.1], imply the following result. Proposition 5.3. Suppose that A is separable and amenable. Then A has strong property (T ) if and only if A = B ⊕ C where B is finite dimensional and C has no tracial states. We end this section with the following analogue of [3, Theorem 7]. Proposition 5.4. Let Γ be a countable discrete group and Λ be a subgroup of Γ . The following statements are equivalent. (i) (ii) (iii) (iv) (v)
(Γ, Λ) has property (T ). (C ∗ (Γ ), C ∗ (Λ)) has strong property (T ). (C ∗ (Γ ), C ∗ (Λ)) has property (T ). (Cr∗ (Γ ), Cr∗ (Λ)) has strong property (T ). (Cr∗ (Γ ), Cr∗ (Λ)) has property (T ).
Proof. It is clear that (ii) ⇒ (iii) ⇒ (v) and (ii) ⇒ (iv) ⇒ (v) (by Lemma 4.1). The implication (v) ⇒ (i) was proved in [3]. It remains to show that (i) ⇒ (ii). As (Γ, Λ) has property (T ), for any α > 0, there exists (Q, β) ∈ F(Γ ) × R+ such that for any unitary representation π : Γ → L(K) and any ξ ∈ Vπ (Q, β), one has ξ − PKΛ (ξ ) < α (by [10, Theorem 1.2(b2)]). Consider Γ ⊆ C ∗ (Γ ). For any H ∈ Bimod∗ (C ∗ (Γ )), one can define a unitary representation πH : Γ → L(H ) C ∗ (Λ) by πH (t)η = t · η · t −1 (η ∈ H ). If ξ ∈ VH (Q, β) ⊆ Vπ (Q, β), we have ξ − PH (ξ ) = ξ − PHΛ (ξ ) < α as required. 2 6. Property (T ) for the unitary group of a C ∗ -algebra It was shown in [13] that a unital C ∗ -algebra is amenable if and only if its unitary group under the weak topology is amenable. Motivated by this result as well as by Lemma 2.2, we study in this section the relation between property (T ) and strong property (T ) of a unital C ∗ -algebra A and certain (T )-type properties of its unitary group.
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Remark 6.1. We denote by ΦA : U (A) → U (ADou ) the group homomorphism u → u ⊗ (u∗ )op . Note that any K ∈ Bimod∗ (A) defines a non-degenerate ∗-representation μK of ADou and πK = μK ◦ ΦA is a unitary representation of U (A). The proof of the following result is more or less the same as the argument for the equivalence of (b2) and (b3) in [10]. Proposition 6.2. (a) (A, B) has property (T ) if and only if for any K ∈ Bimod∗ (A), the weak containment of the trivial representation 1U (A) of U (A) in the unitary representation πK will imply that πK |U (B) contains the trivial representation 1U (B) of U (B). (b) (A, B) has strong property (T ) if and only if for any net (fi )i∈I in S(ADou ) with (fi ◦ ΦA )i∈I converges pointwisely to 1U (A) , one has (fi ◦ ΦA |U (B) )i∈I converges uniformly to 1U (B) on U (B). Proof. (a) Note that for any K ∈ Bimod∗ (A) and any ξ ∈ S1 (K), we have t ∈ U (A) . t · ξ − ξ · t2 = 2 − 2 Re πK (t)ξ, ξ Now, this follows almost directly from Proposition 2.4(a). (b) (⇒). If (Hi , μi , ξi ) is the GNS representation of fi , then Hi ∈ Bimod∗ (A). For any > 0, let (Q, β) ∈ F(U (A)) × R+ be a strong Kazhdan’s pair for (A, B, /2) (see Lemma 2.2). By the assumption of (fi ), there exists i0 ∈ I such that for any i i0 , we have supu∈Q |fi (ΦA (u))−1| < β 2 /2. Thus,
u · ξi − ξi · u = μi u ⊗ (u∗ )op ξi − ξi = 2 Re 1 − fi ΦA (u) < β for any u ∈ Q and so ξi ∈ VHi (Q, β). Therefore, ξi − PHBi (ξi ) < /2 and for any v ∈ U (B), fi ΦA (v) − 1 = v · ξi · v ∗ − ξi , ξi v · ξi · v ∗ − v · P B (ξi ) · v ∗ + P B (ξi ) − ξi < . Hi
Hi
(⇐). Suppose on the contrary that (A, B) does not have strong property (T ). Let I := F(U (A)) × R+ . There exists α0 > 0 such that for any i = (F, ) ∈ I , one can find Hi ∈ Bimod∗ (A) and ξi ∈ VHi (F, ) with ξi − PHBi (ξi ) > α0 . For every such Hi , let πHi be as in Remark 6.1. By [10, 2.2], we see that there exists vi ∈ U (B) such that πH (vi )ξi − ξi > α0 i U (B)
(note that HiB = Hi
). On the other hand, for any i ∈ I , we define fi ∈ S(ADou ) by
fi a ⊗ bop = a · ξi · b, ξi (a, b ∈ A).
For any i = (Q, β) ∈ I , we have ξi ∈ VHi (Q, β) and thus, sup πHi (u)ξi − ξi < β.
u∈Q
(6.1)
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This shows that fi ◦ ΦA converges pointwisely to 1U (A) . Therefore, by the hypothesis, fi ◦ΦA |U (B) converges uniformly to 1U (B) on U (B) which contradicts with (6.1) (since it implies that Re(1 − ϕi (vi )) > α02 /2 for any i ∈ I ). 2 We will show in the following that A has property (T ) (respectively, strong property (T )) if and only if U (A) has some (T )-type property. Note that the argument of (iii) ⇒ (i) in part (b) of the following result is adapted from that of Proposition 16 of Chapter 1 of [8]. Theorem 6.3. Let SB(U (A)) := {(μ, H ) ∈ Rep(U (A)): (μ, H ) (πK , K) for some K ∈ Bimod∗ (A)} (for Rep(U (A)), we regard U (A) as a discrete group). (a) A has property (T ) if and only if there exists (F, ) ∈ F(U (A)) × R+ such that for any K ∈ Bimod∗ (A) with VπK (F, ) = ∅, we have K U (A) = {0}. (b) The following statements are equivalent. (i) A has strong property (T ). (ii) There exists (F, ) ∈ F(U (A)) × R+ such that for any (μ, H ) ∈ SB(U (A)) and ξ ∈ U (A) Vμ (F, ), we have PH (ξ ) = 0. (iii) There exists (F, ) ∈ F(U (A)) × R+ such that for any (μ, H ) ∈ SB(U (A)) with Vμ (F, ) = ∅, we have H U (A) = {0}. Proof. (a) This follows from Lemma 2.2. (b) (i) ⇒ (ii). Let (F, ) ∈ F(U (A)) × R+ be a strong Kazhdan’s pair for (A, A, 1/2) (see Lemma 2.2). Suppose that (μ, H ) ∈ SB(U (A)) and K ∈ Bimod∗ (A) such that (μ, H ) (πK , K). If ξ ∈ Vμ (F, ) ⊆ VK (F, ), then ξ − P A (ξ ) < 1/2. K As K = H ⊕ H ⊥ and both H and H ⊥ are invariant under πK , we have K A = K U (A) = H U (A) ⊕ U (A) U (A) (H ⊥ )U (A) . As ξ ∈ H , we know that PH ⊥ (ξ ) = 0 and PH (ξ ) = PKA (ξ ) = 0. (ii) ⇒ (iii). This implication is clear. (iii) ⇒ (i). Let (F, ) be the pair satisfying the hypothesis. For any 2 α > 0, we take ∗ A ⊥ U (A) )⊥ and μ = π | . Then β := α K H 2 . Suppose that K ∈ Bimod (A), H = (K ) = (K A (μ, H ) ∈ SB(U (A)). For any ξ ∈ VK (F, β), we have ξ = PK (ξ ) + η where η ∈ H . Assume that η > β and put ζ := η/η. As μ(v)ζ − ζ = πK (v)ξ − ξ /η < β/η <
(v ∈ F ),
we have ζ ∈ Vμ (F, ). Hence by the hypothesis, H contains a non-zero μ-invariant vector which contradicts the definition of H . Therefore, we must have η β/ and hence ξ − PKA (ξ ) < α. 2 Remark 6.4. (a) Let SB 0 (U (A)) = {(μ, H ) ∈ Rep(U (A)): (μ, H ) (πH , H)}. Then using the same argument, one can show that a similar statement as Theorem 6.3(b) holds when SB(U (A)) is replaced by SB 0 (U (A)). (b) Note that in the proof for (i) ⇒ (ii) in Theorem 6.3(b) we only need the existence of a strong Kazhdan’s pair for (A, A, 1/2).
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References [1] C. Anantharaman-Delaroche, On Connes’ property (T ) for von Neumann algebras, Math. Japon. 32 (1987) 337– 355. [2] M.B. Bekka, Kazhdan’s property (T ) for the unitary group of a separable Hilbert space, Geom. Funct. Anal. 13 (2003) 509–520. [3] M.B. Bekka, Property (T ) for C ∗ -algebras, Bull. London Math. Soc. 38 (2006) 857–867. [4] M.B. Bekka, A. Valette, Kazhdan’s property (T ) and amenable representations, Math. Z. 212 (1993) 293–299. [5] N.P. Brown, Kazhdan’s property (T ) and C ∗ -algebras, J. Funct. Anal. 240 (2006) 290–296. [6] A. Connes, Classification des facteurs, in: Operator Algebras and Applications, Part 2, Kingston, Ont., 1980, in: Proc. Sympos. Pure Math., vol. 38, 1982, pp. 43–109. [7] A. Connes, V. Jones, Property (T ) for von Neumann algebras, Bull. Lond. Math. Soc. 17 (1985) 57–62. [8] P. de la Harpe, A. Valette, La propriété (T ) de Kazhdan pour les groupes localement compacts, Astérisque 175 (1989). [9] P. Jolissaint, Property T for discrete groups in terms of their regular representation, Math. Ann. 297 (1993) 539–551. [10] P. Jolissaint, On property (T ) for pairs of topological groups, Enseign. Math. (2) 51 (2005) 31–45. [11] D. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal. Appl. 1 (1967) 63–65. [12] G. Margulis, Finitely-additive invariant measures on Euclidean spaces, Ergodic Theory Dynam. Systems 2 (1982) 383–396. [13] A.L.T. Paterson, Nuclear C ∗ -algebras have amenable unitary groups, Proc. Amer. Math. Soc. 114 (1992) 719–721. [14] S. Popa, On a class of type II1 factors with Betti numbers invariants, Ann. of Math. 163 (2006) 809–899. [15] A. Valette, Old and new about Kazhdan’s property (T ), in: Pitman Res. Notes Math. Ser., vol. 311, Longman, 1994, pp. 271–333.
Journal of Functional Analysis 256 (2009) 3071–3090 www.elsevier.com/locate/jfa
Parseval frames for ICC groups Dorin Ervin Dutkay a,1,∗ , Deguang Han a , Gabriel Picioroaga b a University of Central Florida, Department of Mathematics, 4000 Central Florida Blvd., PO Box 161364,
Orlando, FL 32816-1364, United States b Binghamton University, Department of Mathematical Sciences, Binghamton, NY 13902-6000, USA
Received 21 August 2008; accepted 21 November 2008 Available online 5 December 2008 Communicated by D. Voiculescu
Abstract We analyze Parseval frames generated by the action of an ICC group on a Hilbert space. We parametrize the set of all such Parseval frames by operators in the commutant of the corresponding representation. We characterize when two such frames are strongly disjoint. We prove an undersampling result showing that if the representation has a Parseval frame of equal norm vectors of norm √1 , the Hilbert space is spanned by N an orthonormal basis generated by a subgroup. As applications we obtain some sufficient conditions under which a unitary representation admits a Parseval frame which is spanned by a Riesz sequences generated by a subgroup. In particular, every subrepresentation of the left-regular representation of a free group has this property. Published by Elsevier Inc. Keywords: Parseval frames; Kadison–Singer problem; Undersampling; II 1 -factors
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2. General theory: A new parametrization theorem 3. Parseval frames and subgroups . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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* Corresponding author.
E-mail addresses:
[email protected] (D.E. Dutkay),
[email protected] (D. Han),
[email protected] (G. Picioroaga). 1 Research supported in part by a grant from the National Science Foundation DMS-0704191. 0022-1236/$ – see front matter Published by Elsevier Inc. doi:10.1016/j.jfa.2008.11.017
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1. Introduction Frames play a fundamental role in signal processing, image and data compression and sampling theory. They provide an alternative to orthonormal bases, and have the advantage of possessing a certain degree of redundancy which can be useful in applications, for example when data is lost during transmission. Also, frames can be better localized, a feature which lead to the success of Gabor frames and wavelet theory (see, e.g., [5]). The term “frame” was introduced by Duffin and Schaffer [6] in their study of non-harmonic Fourier series, and has generated important research areas and remarkable breakthroughs [21]. Recent results show that frames can provide a universal language in which many fundamental problems in pure mathematics can be formulated: the Kadison–Singer problem in operator algebras, the Bourgain–Tzafriri conjecture in Banach space theory, paving Töplitz operators in harmonic analysis and many others (see [3] for an excellent account). Definition 1.1. Let H be a Hilbert space. A family of vectors {ei | i ∈ I } in H is called a frame if there exist constants A, B > 0 such that for all f ∈ H, Af 2
f, ei 2 Bf 2 . i∈I
If A = B = 1, then {ei | i ∈ I } is called a Parseval frame. If we only require the right-hand inequality, then we say that {ei | i ∈ I } is a Bessel sequence. In their seminal paper by Han and Larson [13], operator theoretic foundations for frame theory and group representations where formulated. One of the key observations in their paper was that every Parseval frame is the orthogonal projection of an orthonormal basis. This lead them to the notion of disjointness of frames. Definition 1.2. Let {ei | i ∈ I }, {fi | i ∈ I } be two Parseval frames for the Hilbert spaces H1 and H2 respectively. The Parseval frames are called strongly disjoint if {ei ⊕ fi | i ∈ I } is a Parseval frame for H1 ⊕ H2 . Similarly, if we have {ei1 | i ∈ I }, . . . , {eiN | i ∈ I } Parseval frames for the Hilbert spaces Hi respectively, we say that these frames form an N -tuple of strongly disjoint Parseval frames if {ei1 ⊕ · · · ⊕ eiN | i ∈ I } is a Parseval frame for H1 ⊕ · · · ⊕ HN . The Parseval frames are called unitarily equivalent if there exists a unitary operator U : H1 → H2 such that U ei = fi for all i ∈ I . If the direct sum of two Parseval frames is an orthonormal basis, we say that one is the complement of the other, or one complements the other. Many properties of frames are encoded in the associated frame transform (or analysis operator). It is the operator that associates to a vector its coefficients in the given frame. Definition 1.3. Let E := {ei | i ∈ I } be a Bessel sequence in a Hilbert space H. The operator ΘE : H → l 2 (I ) defined by ΘE (f ) = f, ei i∈I
(f ∈ H)
is called the analysis operator or frame transform associated to E .
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Strong disjointness can be characterized in terms of frame transforms as follows: Proposition 1.4. (See [13].) Let E := {ei | i ∈ I } and F := {fi | i ∈ I } be two Parseval frames for the Hilbert spaces H1 and H2 respectively. Then E and F are unitarily equivalent iff the frame transforms ΘE and ΘF have the same range. The following affirmations are equivalent: (i) (ii) (iii) (iv)
The Parseval frames E and F are strongly disjoint. The frame transforms ΘE and ΘF have orthogonal ranges. ∗ Θ = 0. ΘF E For all v1 ∈ D1 , v2 ∈ D2 , where D1 , D2 are dense in H1 , H2 respectively, v1 , ei v2 , fi = 0. i∈I
In what follows we will call two Bessel sequences (with same index set) strongly disjoint if the range spaces of their analysis operators are orthogonal. In this paper we will be interested only in the Parseval frames generated by the action of an infinite-conjugacy-classes (ICC) group on a Hilbert space. The ICC property implies that the associated left-regular representation is a II 1 factor, and we can use the theory of factor von Neumann algebras [18]. Our purpose is to follow and further the results in [13]. After the Han–Larson paper, frames for abstract Abelian groups have been studied in [1,22] using Pontryagin duality. As we shall see, Parseval frames for ICC groups have quite different properties. Many strongly disjoint Parseval frames can be found in the same representation (Theorem 2.2), and there are undersampling results in some cases (Proposition 3.1 and Theorem 3.3). Frames for ICC groups fit into the more general theory of group-like unitary systems, a theory which has the Gabor (or Weyl–Heisenberg) frames as the central example, see [8,11,14] for details. Definition 1.5. Let G be a countable group. Let π be a unitary representation of G on the Hilbert space H. A vector ξ ∈ H is called a frame/Parseval frame/ONB vector for H (with the representation π ) iff {π(g)ξ | g ∈ G} is a frame/Parseval frame/ONB for H. Two Parseval frame vectors ξ, η for H are called unitarily equivalent/strongly disjoint if the corresponding Parseval frames {π(g)ξ | g ∈ G} and {π(g)η | g ∈ G} are unitarily equivalent/strongly disjoint. Similarly for an N -tuple of strongly disjoint Parseval frame vectors. As shown in [13], every representation of a group that has a Parseval frame vector is isomorphic to a sub-representation of the left-regular representation (Proposition 1.8). Definition 1.6. Let G be a countable group. The left-regular representation λ of G is defined on l 2 (G) by λ(g)ξ (h) = ξ g −1 h
ξ ∈ l 2 (G), h ∈ G, g ∈ G .
Equivalently, if δg , g ∈ G is the canonical orthonormal basis for l 2 (G), then λ(g)δh = δgh
(h, g ∈ G).
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The von Neumann algebra generated by the unitary operators λ(g), g ∈ G is denoted by L(G). The right-regular representation ρ of G is defined on l 2 (G) by ρ(g)ξ (h) = ξ(hg)
ξ ∈ l 2 (g), h ∈ G, g ∈ G .
Equivalently, ρ(g)δh = δhg −1
(h, g ∈ G).
The group G is called an ICC (infinite conjugacy classes) group if, for all g ∈ G, g = e, the set {hgh−1 | h ∈ G} is infinite. The commutant of the von Neumann algebra L(G) is the von Neumann algebra generated by the right-regular representation ρ. When the group is ICC, L(G) is a II 1 factor (see [18]). The paper is organized as follows: in Section 2, we study the general properties of Parseval frames for ICC groups. In Theorem 2.2 we show how a Parseval frame vector can be complemented by several other Parseval frame vectors for the same representation and a “remainder” Parseval frame vector for a subspace. While Lemma 2.3 characterizes strongly disjoint Parseval frame vectors in terms of their cyclic projections we present some new properties along the same lines in Proposition 2.5. We give a parametrization of Parseval frame vectors in Theorem 2.6. Another such parametrization was given in [13], see Theorem 1.9. The advantage of our parametrization is that it uses operators in the commutant in the representation, so it can be extended to all the vectors of the frame. Theorem 2.7 characterizes strong disjointness and unitary equivalence in terms of this parametrization. In Section 3 we will be interested in the relation between Parseval frames and subgroups. We will work with Parseval frame vectors that have norm-square equal to N1 with N ∈ N and we will assume that there exists a subgroup H of index N . We give two “undersampling” results: in Proposition 3.1, we show that in this situation, there exist orthonormal bases generated by the action of the subgroup H , and in Theorem 3.3 we show that, if in addition H is normal, then we can construct N strongly disjoint Parseval frame vectors such that, when undersampled to the subgroup, they form orthonormal bases (up to a multiplicative constant). We apply these results to the Feichtinger frame decomposition problem for free groups to get that every frame representation for free groups admits a frame that is a finite union of Riesz sequences. Definition 1.7. We will use the following notations: if A is a set of operators on a Hilbert space H, then A is the commutant of A, i.e., the set of all operators that commute with all the operators in A. A is the double-commutant, i.e., the commutant of A . By von Neumann’s double commutant theorem, A coincides with the von Neumann algebra generated by A. Two (orthogonal) projections p and q in a von Neumann algebra A are said to be equivalent (denoted by p ∼ q) if there exists an operator (partial isometry) u ∈ A such that uu∗ = p and u∗ u = q. A von Neumann algebra A is finite it there is no proper projection of I that is equivalent to I in A. We refer to [19] for more details and some properties about von Neumann algebras that will be used in the rest of the paper.
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If π is a representation of a group G on a Hilbert space H, and N ∈ N, then π N is the representation of G on HN defined by π N (g) = π(g) ⊕ · · · ⊕ π(g) (g ∈ G). N times
If p is a projection in π(G) , then πp is the representation of G on p H defined by (πp )(g) = p π(g)p
(g ∈ G).
If M is a von Neumann algebra with a trace, we will denote the trace by trM The next proposition proved by Han and Larson in [13] is the starting point for the theory of Parseval frames for groups. It shows that any Parseval frame generated by the representation of a group is in fact isomorphic to the projection of the canonical basis in the left-regular representation of the group. The isomorphism is in fact the frame transform, the projection is its range and it lies in the commutant of the left-regular representation. Proposition 1.8. (See [13].) Let G be a countable ICC group and let π : G → U(H) be a unitary representation of the group G on the Hilbert space H. Suppose ξ ∈ H is a Parseval frame vector for H. Then: (i) The frame transform Θξ is an isometric isomorphism between H and the subspace p l 2 (G), where p := Θξ Θξ∗ . (ii) The frame transform Θξ intertwines the representations π on H and λ on l 2 (G), i.e., Θξ π(g) = λ(g)Θξ , for all g ∈ G. The projection p := Θξ Θξ∗ commutes with λ(G). (iii) Θξ ξ = p δe , Θξ∗ δe = ξ . The trace of p in L(G) is trL(G) (p ) = ξ 2 . Thus the Parseval frame {π(g)ξ | g ∈ G} for H is unitarily equivalent to the Parseval frame {λ(g)p δe | g ∈ G} for p l 2 (G), via the frame transform Θξ . In [13], the authors proved that Parseval vectors can be parametrized by unitary operators in the algebra π(G) . Note that these operators are not in the commutant, and this has the drawback that one can not map the entire Parseval frame into the other. We will give an alternative parametrization, that uses operators in the commutant in Proposition 2.6. Theorem 1.9. (See [13].) Let G be a countable ICC group and let π : G → U(H) be a unitary representation of the group G on the Hilbert space H. Suppose ξ ∈ H is a Parseval frame vector for H. Then η ∈ H is a Parseval frame vector for H if and only if there exists a unitary u ∈ π(G) such that uξ = η. In particular η = ξ . 2. General theory: A new parametrization theorem We will consider an ICC group G and π : G → U(H) a unitary representation of the group on the Hilbert space H. We will assume that this representation has a Parseval frame vector ξ1 . The main goal of this section is to obtain an alternate parametrization of all the Parseval frame vectors by using operator vectors with entries in the commutant of π(G). Although the focus of
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this paper is on ICC groups, the new parametrization result works for other groups for which we will discuss the details at the end of this section. The norm-square of the vector ξ1 2 fits into one N times, and there may be some remainder 0 r ξ1 2 . The next theorem shows that we can complement the Parseval frame vector ξ1 by N − 1 Parseval frame vectors for H and one “remainder” Parseval frame vector for a subspace of H. Moreover, the complementing procedure works also if we start with several strongly disjoint Parseval frame vectors for H. Of course, if ξ1 2 = N1 with N ∈ N, the remainder Parseval vector is not needed and can be discarded. Our results have a simpler statement if the extra assumption ξ1 2 = N1 is added, and the remainders disappear. We recommend the reader do this for an easier understanding of the statements. However we sacrificed (part of) the aesthetics for generality. Lemma 2.1. (See [9,13].) Let G be a countable ICC group and let π : G → U(H) be a unitary representation of the group G on the Hilbert space H. Suppose ξ ∈ H is a Parseval frame vector for H and let pξ = Θξ Θξ∗ and p be a projection in L(G) . Then (i) p ∼ pξ in the von Neumann algebra L(G) if and only if there exists a Parseval frame vector η for H such that p = pη ; (ii) p is equivalent to a subprojection of pξ if and only if there exists a Parseval frame vector η for span{π(g)η: g ∈ G} such that p = pη ; Theorem 2.2. Let G be a countable ICC group and let π : G → U(H) be a unitary representation of the group G on the Hilbert space H. Suppose ξ1 ∈ H is a Parseval frame vector for H. Let N := ξ12 ∈ N. 1
(i) There exist ξ2 , . . . , ξN , ξN +1 ∈ H with the following properties: (a) ξ2 , . . . , ξN are Parseval frame vectors for H. (b) ξN +1 2 = 1 − N ξ1 2 =: r < ξ1 2 , and there exists a projection pr ∈ π(G) such that {π(g)ξN +1 | g ∈ G} is a Parseval frame for pr H, so ξN +1 is a Parseval frame vector for pr H with the representation π r := πpr . (c) ξ1 , . . . , ξN +1 is a strongly disjoint N + 1-tuple of Parseval frame vectors. (ii) If ξ1 , . . . , ξN +1 are as in (i) then there is no vector ξN +2 such that ξN +2 is a Parseval frame vector for some representation πN +2 of G, and ξN +2 is strongly disjoint from all ξi , i = 1, N + 1. (iii) If 1 M N and ξ1 , . . . , ξM is a strongly disjoint M-tuple of Parseval frame vectors for H, then there exist ξM+1 , . . . , ξN , ξN +1 such that the properties in (i) are satisfied. Proof. (i) Let p1 = Θξ1 Θξ∗1 . Since L(G) is a factor von Neumann algebra, we have that there , p exist mutually orthogonal projections p2 , . . . , pN N +1 in L(G) such that pi ∼ p1 for i =
N +1 2, . . . , N , pN +1 is equivalent to a subprojection of p1 and i=1 pi = I . By Lemma 2.1, there exist ξ2 , . . . , ξN , ξN +1 ∈ H such that ξ2 , . . . , ξN are Parseval frame vectors for H with pi = pξ i , and ξN +1 has the property that {π(g)ξN +1 | g ∈ G} is a Parseval frame for its closed linear , p span. Since p1 , . . . , pN N +1 are mutually orthogonal, we get that ξ1 , . . . , ξN +1 is a strongly
+1 disjoint N + 1-tuple of Parseval frame vectors. Moreover, from N i=1 pi = I we obtain that ξ1 ⊕ · · · ⊕ ξN ⊕ ξN +1 has norm one and hence it is an ONB vector for π N ⊕ π r , where π r = πpr and pr is the orthogonal projection onto span{π(g)ξN +1 | g ∈ G}.
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(ii) If ξN +2 is strongly disjoint from all ξi , i = 1, N , then ζ := (ξ1 ⊕ + · · · + ⊕ξN +1 ) ⊕ ξN +2 is a Parseval frame vector with ζ > 1, and this is impossible. (iii) This clearly is a more general form of (i), and the proof is exactly as in the proof of (i). 2 The following two more lemmas are needed in the proof of Theorem 2.6. Lemma 2.3. (See [15].) Let G be a countable group and let π : G → U(H) be a unitary representation of the group G on the Hilbert space H. Assume there exists a Parseval frame vector ξ1 for H. Suppose η1 , η2 are two Parseval frame vectors for some subspaces of H. Let pηi be the projection onto the subspace π(G) ηi , i = 1, 2. Then (i) The two Parseval frame vectors η1 , η2 are strongly disjoint if and only if projections pη1 , pη2 are orthogonal. (ii) The two Parseval frame vectors η1 , η2 are unitarily equivalent if and only if pη1 = pη2 . Lemma 2.4. (See [7].) Let G be a countable group and let π : G → U(H) be a unitary representation of the group G on the Hilbert space H. Assume there exists a Parseval frame vector ξ1 for H. Suppose η is a Parseval frame vector for a subspace of H. Then there exists a vector ζ such that η + ζ is a Parseval frame vector for H, and η and ζ are strongly disjoint Parseval frame vectors. Proposition 2.5. Let G be a countable ICC group and let π : G → U(H) be a unitary representation of the group G on the Hilbert space H. Assume that there exists a Parseval frame vector ξ1 for H. For ξ ∈ H, let pξ be the projection onto the subspace π(G) ξ . (i) If η is a Parseval frame vector for a subspace for H then trπ(G) (pη ) = η2 . (ii) If η is a Parseval frame vector for H and u is a unitary operator in π(G) , then the Parseval frame vector uη is strongly disjoint from η iff uη is orthogonal to the range of pη , and in this case pη ⊥ puη . (iii) Let N 1, N ∈ Z, and suppose ξ1 , . . . , ξN +1 is
a strongly disjoint N + 1-tuple of Parseval +1 2 frame vectors for some subspaces of H, with N i=1 ξi = 1. Then the projections pξi , i = 1, . . . , N are mutually orthogonal, and pξ1 + · · · + pξN+1 = 1. Proof. By Proposition 1.8, we can assume H = p l 2 (G) and π = λp for some projection p ∈ L(G) with trL(G) (p ) = ξ1 2 . Let pη be the projection onto the subspace π(G) η generated by the Parseval frame vector η. Then pη ∈ π(G) and by [18, Remark 2.2.5] we have dimπ(G) H =
trπ(G) (pη ) . trπ(G) (pη )
But, by [18, Proposition 2.2.6(vi)], dimπ(G) H = dimL(G)p p l 2 (G) = trL(G) (p ) dimL(G) l 2 (G) = ξ1 2 .
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On the other hand, using the uniqueness of the trace on factors, we have trL(G) (pη ) η2 = trπ(G) pη = trp L(G) p pη = . trL(G) (p ) ξ1 2 We used the fact that pη ∈ L(G) is the projection onto the subspace in H ⊂ l 2 (G) spanned by the Parseval frame {π(g)η = λ(g)η | g ∈ G}, and therefore, by Proposition 1.8, trL(G) (pη ) = η2 . From these equalities, (i) follows. To prove (ii), we use Theorem 1.9 to see that uη is a Parseval frame vector. By Lemma 2.3, the Parseval frame vectors are disjoint iff pη and puη are orthogonal. Thus one implication is trivial. If uη is orthogonal to the range of pη , then for all x , y ∈ π(G) we have x uη, y η = uη, x ∗ y η = 0 so puη is perpendicular to pη . For (iii), with Lemma 2.3, we have that pξi are mutually orthogonal. From (i), we have that trπ(G)
N +1
pi =
i=1
therefore
N +1 i=1
pi = 1.
N +1
ξi 2 = 1,
i=1
2
Now we are ready to parametrize the Parseval frame vectors for H. As we mentioned before, such a result was given in [13], see Theorem 1.9. The drawback of their result is that they are using operators in the von Neumann algebra π(G) itself, not in its commutant. We give here a parametrization that uses operators in the commutant. As Han and Larson proved in their paper, we cannot expect to use unitary operators in π(G) . Instead, we will use the N + 1 strongly disjoint Parseval frame vectors given by Theorem 2.2 and N + 1 operators in the commutant π(G) that satisfy the orthogonality relation (2.2) which in fact represents the norm-one property of a first row in a unitary matrix of operators. Theorem 2.6. Let G be a countable ICC group and let π : G → U(H) be a unitary representation of the group G on the Hilbert space H. Suppose ξi , i = 1, . . . , N + 1 is a strongly disjoint N + 1tuple of Parseval frame vectors as in Theorem 2.2(i). Let η ∈ H be another Parseval frame vector for a subspace of H, and let q be the projection onto this subspace. Then there exist unique u i ∈ π(G) , i = 1, . . . , N + 1, with q u i = u i , i = 1, . . . , N + 1, and u N +1 pr = u N +1 (where pr is the projection onto the span of the Parseval frame generated by the vector ξN +1 , as in Theorem 2.2) such that η = u 1 ξ1 + · · · + u N +1 ξN +1 .
(2.1)
Moreover N +1
u i u i ∗ = q .
(2.2)
i=1
Conversely, if the vector η is defined by (2.1) with u i ∈ L(G) , q u i = u i , i = 1, . . . , N + 1, and u N +1 pr = u N +1 satisfying (2.2), then η is a Parseval frame vector for H.
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Proof. We prove the theorem first for the case when q = 1, so η is a Parseval frame vector for the entire space H. By Theorem 2.2(iii), there exist η2 , . . . , ηN +1 ∈ H that together with η1 := η form a strongly disjoint N + 1-tuple of Parseval frame vectors as in Theorem 2.2(i). Then η1 ⊕ · · · ⊕ ηN +1 is an ONB vector for π N ⊕ πpr = λ. Then there exists a unitary u ∈ L(G) such that u (ξ1 ⊕ · · · ⊕ ξN +1 ) = η1 ⊕ · · · ⊕ ηN +1 . Let pi be the projections onto the ith component. We can identify pN +1 = pr , and in our case q = p1 . Let u i := p1 u pi . Then u i ∈ π(G) , u N +1 pr = u N +1 , and N +1
u i ξi
=
i=1
N +1
p1 u pi (ξ1
⊕ · · · ⊕ ξN +1 ) = p1 u
N +1
i=1
pi
(ξ1 ⊕ · · · ⊕ ξN +1 ) = η1 .
i=1
This proves (2.1).
+1 To prove uniqueness, suppose N i=1 vi ξi = 0 for some operators vi ∈ π(G) , with vN +1 pr =
N +1 vN +1 . Then, for all g ∈ G, π(g) i=1 vi ξi = 0. By Lemma 2.3, the vectors vi ξi are mutually orthogonal. Since π(g) is unitary, it follows that π(g)vi ξi = 0 for all i = 1, . . . , N + 1. Therefore vi π(g)ξi = 0 for all g ∈ G, i = 1, N + 1. But π(g)ξi span H for i = 1, . . . , N , and span pr H for i = N + 1. Therefore vi = 0 for all i = 1, . . . , N + 1. This implies the uniqueness. We check now (2.2). We have
q
= p1
= p1 u u ∗ p1
= p1 u
N +1
pi u ∗ p1 =
i=1
N +1
+1 ∗ N p1 u pi p1 u pi = u i u i ∗ .
i=1
i=1
This proves (2.2). For the converse, we can use [13, Proposition 2.21]. We include the details. Consider the frame transforms Θi for the Parseval frame vectors ξi , Θi defined on H for i = 1, . . . , N and on pr H for i = N + 1. We have, by Proposition 1.8, Θi∗ Θi = 1H for i = 1, . . . , N , ΘN∗ +1 ΘN +1 = 1pr H . ∗ Note that Θu i ξi = Θξi u ∗ i for i = 1, . . . , N + 1 and Θi Θj = 0 for i = j . So we have Θη∗ Θη
=
N +1 i=1
u i Θi∗
N +1 i=1
Θi u ∗ i
=
N +1 i,j =1
u i Θi∗ Θj u j∗ =
N +1
u i u i ∗ = I.
i=1
Hence η is a Parseval frame. In the case when q 1, we have q ∈ π(G) and, by Lemma 2.4, there exists a Parseval frame vector η˜ for H such that η = q η. ˜ Using the proof above we can find u˜ i ∈ π(G) such that
N +1 η˜ = i=1 u˜ i ξi and all the other properties. Then we can define u i := q u˜ i , i = 1, . . . , N + 1, and a simple computation shows that the required properties are satisfied. The proof of uniqueness and the converse in this case is analogous to the one provided for the case q = 1. 2 We can use the above parametrization result to characterize strongly disjoint (resp. unitary equivalent) Parseval frame vectors in terms of the given parametrization. Recall that two Parseval frame vectors η and ξ are unitary equivalent if and only if their analysis operators have the same range spaces, which in turn is equivalent to the condition that Θη Θη∗ = Θξ Θξ∗ .
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Theorem 2.7. Let G be a countable ICC group and let π : G → U(H) be a unitary representation of the group G on the Hilbert space H. Suppose ξi , i = 1, . . . , N + 1 is a strongly disjoint N + 1tuple of Parseval frame vectors as in Theorem 2.2(i). Let η, ζ ∈ H, be two Parseval frame vectors for some subspaces of H. Suppose η = u 1 ξ1 + · · · + u N +1 ξN +1 ,
ζ = v1 ξ1 + · · · + vN +1 ξN +1 ,
with u i , vi ∈ π(G) , i = 1, N + 1, u N +1 pr = u N +1 , vN +1 pr = vN +1 . Then we have
(i) η and ζ are strongly disjoint if and only if ∗ v1 u 1∗ + · · · + vN +1 uN +1 = 0.
(2.3)
(ii) η and ζ are unitary equivalent if and only if t t
∗ ∗ v1 , . . . , v N u1 , . . . , u N∗+1 u 1 , . . . , u N +1 = v1 ∗ , . . . , vN +1 +1 ,
(2.4)
where “t” represents the transpose of the row vector. Proof. (i) Let ψ, ψ ∈ H. Then, by Proposition 1.4, η and ξ are disjoint if and only if 0=
N +1 N +1 π(g)η, ψ π(g)ζ, ψ = ui ξi , ψ π(g) v j ξj , ψ π(g)
g∈G
=
N +1
g∈G
i=1
j =1
π(g)ξi , u i ∗ ψ π(g)ξj , vj ∗ ψ = (since ξi are mutually disjoint)
i,j =1 g∈G
=
N +1
π(g)ξi , u i ∗ ψ π(g)ξi , vi ∗ ψ
i=1 g∈G
∗ = since ξi is a Parseval frame vector, u N∗+1 ψ, vN +1 ψ ∈ pr H N +1 N +1 ∗ ∗ ∗ ui ψ, vi ψ = vi ui ψ, ψ . = i=1
i=1
Since ψ, ψ ∈ H are arbitrary, the proof of (i) is complete.
N +1 +1 ∗ ∗ (ii) Let Θi be the analysis operator for ξi . Then Θη = N i=1 Θi ui and Θζ = i=1 Θi vi . ∗ ∗ Then η and ζ are unitary equivalent if and only if Θη Θη = Θζ Θζ , i.e., N +1 i=1
Θi u i ∗ Θη∗ =
N +1
Θi vi ∗ Θζ∗ .
i=1
Since Θi have orthogonal range spaces, we have that the above equation holds if and only if Θi u i ∗ Θη∗ = Θi vi ∗ Θζ∗
(2.5)
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for all i = 1, . . . , N + 1. Applying Θi∗ to both sides of (2.5) and using that fact that Θi∗ Θi = I for ∗ ∗ i = 1, . . . , N , ΘN∗ +1 ΘN +1 = pr , u N +1 pr = u N +1 and vN +1 pr = vN +1 , we obtain that ui Θη = vi ∗ Θζ∗ for all i = 1, . . . , N + 1, i.e., N ∗ ui ui − vi ∗ vi Θi∗ = 0. i=1
Apply the above left-hand side operator to Θj (H) and use the fact Θi∗ Θj = 0 when i = j , we get u i ∗ u i − vi ∗ vi = 0 for all i, j . Hence we have t t
∗ ∗ v1 , . . . , v N u1 , . . . , u N∗+1 u 1 , . . . , u N +1 = v1 ∗ , . . . , vN +1 +1 . Conversely, if the above identity holds, then we clearly have N ∗ ui ui − vi ∗ vi Θi∗ = 0 i=1
for all i and so we have that (2.5) holds for all i. Therefore η and ζ are unitary equivalent
2
We conclude this section by pointing out that we also have similar results as Theorems 2.6 and 2.7 for frame representations of arbitrary countable groups. The following lemma replaces Theorem 2.2 for the general group case. Note that, unlike the ICC group case, here we can not require that ξ2 , . . . , ξN are Parseval frame vectors for H. Recall that the cyclic multiplicity for a subspace S of operators on H is the smallest cardinality k such that there exist vectors yi (i = 1, . . . , k) with the property span{Syi : S ∈ S, i = 1, . . . , k} = H. Lemma 2.8. (See [15].) Let π : G → U(H) be a unitary representation of a countable group G on the Hilbert space H such that π(G) has cyclic multiplicity N + 1 (here N could be ∞). Assume ξ1 is a Parseval frame vector for H. Then there exist ξi for i = 2, . . . , N + 1 with the properties: (i) {π(g)ξi : g ∈ G} is a Parseval frame for Mi := span{π(g)ξi : g ∈ G}; (ii) ξi (i = 1, . . . , N + 1) are mutually strongly disjoint; (iii) there is no non-zero Bessel vector which is strongly disjoint with all ξi . One fact we used in the proof of Theorem 2.6 is that two ONB vectors for a unitary representation are linked by a unitary operator in the commutant of the representation, i.e., all the ONB vectors are unitarily equivalent. However, as we have already mentioned before this is no longer true in general for non-ONB vectors. The following characterizes the representations that have this property, and it is needed in order to prove our new parametrization result (Theorem 2.10) for general countable groups. Lemma 2.9. (See [9].) Let π : G → U(H) be a unitary representation of a countable group G on the Hilbert space H and ξ be a Parseval frame vector for H. Then the following statements are equivalent:
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(i) there is no non-zero Bessel vector which is strongly disjoint with all ξ ; (ii) Pξ = Θξ Θξ∗ ∈ L(G) ∩ L(G) ; (iii) a vector η is a Parseval vector for H if and only if there exists a unitary u ∈ π(G) such that η = u ξ . Theorem 2.10. Let G be a countable group and let π : G → U(H) be a unitary representation of the group G on the Hilbert space H. Suppose ξi , i = 1, . . . , N + 1 are as in Lemma 2.8. (i) Let η ∈ H. Then η is a Parseval frame vector for H if and only if there exist u i ∈ π(G) , i = 1, . . . , N + 1, with u i pi = u i , i = 1, . . . , N + 1 (where pi is the projection onto the closed linear span of {π(g)ξi : g ∈ G}) such that η = u 1 ξ1 + · · · + u N +1 ξN +1
+1 ∗ and N i=1 ui ui = I . Moreover these ui s are unique. (ii) Let η, ζ ∈ H, be two Parseval frame vectors for H such that η = u 1 ξ1 + · · · + u N +1 ξN +1 ,
ζ = v1 ξ1 + · · · + vN +1 ξN +1 ,
with ui and vi satisfying the requirement as in (i). Then (a) η and ζ are strongly disjoint if and only if ∗ v1 u 1∗ + · · · + vN +1 uN +1 = 0,
and (b) η and ζ are unitary equivalent if and only if t t
∗ ∗ v1 , . . . , v N u1 , . . . , u N∗+1 u 1 , . . . , u N +1 = v1 ∗ , . . . , vN +1 +1 . Proof. We only give a sketch proof for the necessary part of (i). The rest is similar to the ICC group case. Let η is a Parseval frame vector for H. Then we have that pη ∼ pξ 1 in L(G) by Lemma 2.1. Since L(G) is a finite von Neumann algebra, we can find projections [19] qi (i = 2, . . . , N + 1) such that pη , qi (i = 2, . . . , N + 1) are mutually orthogonal and qi ∼ pξ i for i = 2, . . . , N + 1. From Lemma 2.1, there exist Parseval frame vectors ηi for Mi := span{π(g)ξi : g ∈ G} such that pη i = qi (i = 2, . . . , N + 1). Define unitary representation σ : G → U(K) (where K = H ⊕ M2 ⊕ · · · ⊕ MN +1 ) by σ (g) = π(g) ⊕ π(g)pξ2 ⊕ · · · ⊕ π(g)pξN+1 . Then both ξ˜ := ξ1 ⊕ · · · ⊕ ξN +1 and η˜ := η1 ⊕ · · · ⊕ ηN +1 are Parseval frame vectors for K. By Lemma 2.8(iii) we have that there is no non-zero Bessel vector that is strongly disjoint with all ξi . This implies that there is no non-zero Bessel vector that is strongly disjoint with ξ˜ . Thus, from Lemma 2.9, we get that there is a (unique) unitary operator u ∈ σ (G) such that η˜ = u ξ˜ . Let u i = u 1,i Pi (where [u 1,1 , . . . , u 1,N +1 ] is the first row vector of u ). Then it can be checked that η = u 1 ξ1 + · · · + u N +1 ξN +1 , and u i satisfy all the requirements listed in (i). 2
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We make a final remark that all the results in this section remain valid when unitary representation is replaced by projective unitary representations of countable groups. In particular, Theorems 2.6 and 2.7 remain true for Gabor unitary representations. The interested reader can check (cf. [7–12,15,16]) for definitions and recent developments about projective unitary representations and Gabor representations. 3. Parseval frames and subgroups In this section we will only be interested in Parseval frame vectors ξ that have ξ 2 = N1 . We will assume in addition that there is a subgroup H of G of index N . The next proposition shows that in this situation we can find orthonormal bases for H obtained by the action of the subgroup H . Proposition 3.1. Let G be a countable ICC group and let π : G → U(H) be a unitary representation of the group G on the Hilbert space H. Suppose ξ ∈ H is a Parseval frame vector for H with ξ 2 = N1 for some N ∈ N. Assume in addition that there exists an ICC subgroup H of √ index [G : H ] = N . Then there exists a Parseval frame vector η for H with the property that N {π(h)η | h ∈ H } is an orthonormal basis for H. Proof. By Theorem 1.9 we can assume that H = p l 2 (G) for some p ∈ L(G) with trL(G) (p ) = 1 2 N and π = λ restricted to p l (G). We claim that dimL(H ) H = 1. Using [18, Proposition 2.3.5, Example 2.3.3, Proposition 2.2.1] we have
dimL(H ) H = dimL(G) H · L(G) : L(H ) = N dimL(G) p l 2 (G) = N trL(G) (p ) dimL(G) l 2 (G) = 1. Thus (see [18, Chapter 2.2]) the Hilbert space H considered as a module over L(H ), with the representation π(h) = λ(h)p , h ∈ H , is isomorphic to the module l 2 (H ), i.e., there exists an isometric isomorphism Φ : H → l 2√ (H ) such that Φπ(h) = λ(h)Φ for all h ∈ H . Define η := √1 Φ −1 (δe ). Then N {π(h)η | h ∈ H } = Φ −1 {δh | h ∈ H } so it is an orthonorN mal basis for H. We check that {π(g)η | g ∈ G} is a Parseval frame for H. Let {a1 , . . . , aN√} be a complete set of representatives for the left-cosets {gH | g ∈ G}. Let v ∈ H. Then, since N {λ(h)η | h ∈ H } is an orthonormal basis, we have N N v, π(g)η 2 = v, π(ai h)η 2 = π(ai )∗ v, π(h)η 2 g∈G
i=1 h∈H
=
N 1 π(ai )∗ v 2 = 1. N
i=1 h∈H
2
i=1
Remark 3.2. The condition ξ 2 = N1 is essential. In other words, suppose η is a Parseval frame vector for the representation π of G on H. Let H be a subgroup of G of index N , and suppose the family {π(h)η | h ∈ H } is an orthogonal basis for the whole space H. Then ξ 2 = N1 .
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To see this, let g1 . . . gN ∈ G be representatives of the left-cosets of H in G. We have on one hand, using the orthogonal basis, for all x ∈ H: π(h)η, x 2 = η2 x2 . h∈H
On the other hand, using the Parseval frame
x2 =
N N π(gi )π(h)η, x 2 = π(h)η, π(gi )∗ x 2 i=1 h∈H
=
N
i=1
2 η2 π(gi )∗ x = Nx2 η2 ,
thus η2 =
i=1
1 . N
We saw in Theorem 2.2 that we can construct N strongly disjoint Parseval frame vectors for our Hilbert space H. We want to see if we can do this in such a way that, by undersampling with the subgroup H , we have orthonormal bases (up to a multiplicative constant), as in Proposition 3.1. We prove that this is possible in the case when H is normal and has an element of infinite order. Theorem 3.3. Let G be a countable ICC group and let π : G → U(H) be a unitary representation of the group G on the Hilbert space H. Suppose there exists a Parseval frame vector ξ ∈ H with ξ 2 = N1 , N ∈ Z. Assume in addition that H is a normal ICC subgroup of G with index [G : H ] = N , and H contains elements of infinite order. Then there exist a strongly disjoint √ N -tuple η1 , . . . , ηN of Parseval frame vectors for H such that for all i = 1, N , the family N {π(h)ηi | h ∈ H } is an orthonormal basis for H. Proof. By Proposition 1.8, we can assume that π is the restriction of the left-regular representation λ on p l 2 (G), where p is a projection in L(G) , with trL(G) (p ) = N1 . We will define some unitary operators ui on l 2 (G) that will help us build the frame vectors ηi from just one such frame vector η given by Proposition 3.1. Let ak , k = 0, N − 1 be a complete set of representatives for the cosets in G/H . Since H is normal, ak−1 H , k = 0, N − 1 forms a partition of G. We can take a0 = e. kj
Define the functions ϕj : G → C, ϕj (g) = e2πi N if g ∈ ak−1 H . Note that N −1
ϕi ak−1 g ϕj ak−1 g = 0 (g ∈ G, i = j ).
(3.1)
k=0
Indeed, if g = ar h for some r ∈ {0, . . . , N − 1} and h ∈ H , then ak−1 g, k = 0, N − 1 will lie in different sets of the partition {al−1 H }l=0,N −1 , because H is normal. Then N −1 k=0
−1 N (i−j )k ϕi ak−1 g ϕj ak−1 g = e2πi N = 0. k=0
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Also ϕi (gh−1 ) = ϕi (g), for all g ∈ G, h ∈ H , and, since H is normal ϕi (hg) = ϕi (g), i = 0, N − 1, g ∈ G, h ∈ H . Define the operators uj on l 2 (G) by uj δg := ϕj (g)δg , for all g ∈ G, j = 0, N − 1. Since uj maps an ONB to an ONB, it is a unitary operator on l 2 (G). Then for all g ∈ G, using (3.1), N −1
λ(ak )ui u∗j λ(ak )∗ δg =
k=0
N −1
ϕi ak−1 g ϕj ak−1 g δg = 0.
k=0
Also ui λ(h)δg = ϕi (hg)δgh = ϕi (g)δgh = λ(h)ui δg , so ui commutes with λ(h) for all i = 0, N − 1, h ∈ H . We want to compress the unitaries ui to a subspace p1 l 2 (G) for some well chosen projection p1 in L(G) . Take h0 ∈ H , such that hn0 = e for all n ∈ Z \ {0}. Then, with ρ the right-regular representation,
ρ(h0 )n δe , δe = δn =
zn dμ(z)
(n ∈ Z),
T
where μ is the Haar measure on T. Then let p1 be the spectral projection χE (ρ(h0 )), where E is a subset of T of measure N1 . We have trL(G) p1 = p1 δe , δe = χE ρ(h0 ) δe , δe =
χE dμ = T
1 . N
Also, for i = 0, N − 1, g ∈ G, ui ρ(h0 )δg = ui δgh−1 = ϕi gh−1 0 δgh−1 = ϕi (g)δgh−1 = ρ(h0 )ui δg , 0
0
0
so ui ρ(h0 ) = ρ(h0 )ui , and therefore p1 commutes with all ui , i = 0, N − 1. In addition, since p1 ∈ L(G) , it commutes with λ(g) for all g ∈ G. Then we compute for i = j N −1 ∗ λ(ak )p1 p1 ui p1 p1 uj p1 λ(ak )p1 δg = p1 λ(ak )ui u∗j λ(ak )∗ δg = 0.
N −1 k=0
k=0
The operators u˜ i := p1 ui p1 are unitary on p1 l 2 (G) because p1 commutes with ui . Also, commute with λ(h) on p1 l 2 (G). We will couple these results with the following lemma to finish the proof.
p1 ui p1
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Lemma 3.4. Let G be an ICC group with an ICC subgroup H of index [G : H ] = N . Let p1 be √ a projection in L(G) with trL(G) (p1 ) = N1 and let η ∈ p1 l 2 (G) such that N {λ(h)η | h ∈ H } is an orthonormal basis for p1 l 2 (G). Suppose u˜ i , i = 0, N − 1 are unitary operators on p1 l 2 (G) such that u˜ i commutes with λ(h) for all h ∈ H , and for some complete set of representatives a0 , . . . , aN −1 of the left-cosets in G/H , N −1
λ(ak )u˜ i u˜ ∗j λ(ak )∗ = 0 (i = j ).
k=0
Then the vectors u˜ i η0 , i = 0, . . . , N − 1 have the following properties: √ (i) N {λ(h)u˜ i η0 | h ∈ H } is an orthonormal basis for p1 l 2 (G) for all i = 0, . . . , N − 1. (ii) u˜ 0 η0 , . . . , u˜ N −1 η0 is a strongly disjoint N -tuple of Parseval frame vectors. Proof. Since u˜ i commutes with λ(h) for all h ∈ H , property (i) follows immediately from the hypothesis. This implies also that u˜ i η is a Parseval frame vector for p1 l 2 (G) (see the proof of Proposition 3.1). To check the strong disjointness, let Θi be the frame transform of the vector u˜ i η0 . Let Θ0H : p1 l 2 (G) → l 2 (H ) be the frame transform for the √1 -orthonormal basis {λ(h)η | h ∈ H }. ∗
Then Θ0H Θ0H =
1 N I.
N
We have for v ∈ p1 l 2 (G):
Θi (v) = v, λ(g)u˜ i η g∈G = v, λ(ak )λ(h)u˜ i η h∈H k=0,N −1 = u˜ ∗i λ(ak )∗ v, λ(h)η0 h∈H k=0,N −1 . Then for v, v ∈ p1 l 2 (G), −1 ∗ N u˜ i λ(ak )∗ v, λ(h)η u˜ ∗j λ(ak )∗ v, λ(h)η Θi (v), Θj (v ) = k=0 h∈H
=
N −1
Θ0H u˜ ∗i λ(ak )∗ v , Θ0H u˜ ∗j λ(ak )∗ v
k=0
=
N −1
∗ λ(ak )u˜ j Θ0H Θ0H u˜ ∗i λ(ak )∗ v, v = 0.
k=0
This proves that the frames are strongly disjoint.
2
Returning to the proof of the theorem, we √ see that we can apply Lemma 3.4. Let η be a Parseval frame vector in p1 l 2 (G) such that N {λ(h)η | h ∈ H } is an ONB for p1 l 2 (G). It can be obtained from Proposition 3.1. Then, using Lemma 3.4, we get that ηi := p1 ui p1 η form a √ strongly disjoint N -tuple of Parseval frames and N {λ(h)ηi | h ∈ H } are ONBs for p1 l 2 (G). Then we can move everything onto our space H, because trL(G) (p1 ) = trL(G) (p ) so the projections p1 and p are equivalent in L(G) and the representations λ on p1 l 2 (G) and π on H are equivalent. 2
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Remark 3.5. (i) There exist ICC groups with ICC subgroups of any finite index. For example, let FN the free group on N generators and p ∈ N. Also, let φ : FN → Zp a surjective group morphism. Then H := Ker φ is a (free, thus ICC) normal subgroup of FN of finite index. (ii) There exist ICC groups without finite index proper subgroups. For example, let F be the Thompson’s group and F its commutator. Both groups are ICC (see, e.g., [17]). Moreover, F is infinite and simple. By a classic group theoretical argument an infinite simple group cannot have finite index proper subgroups. Indeed, let G be infinite simple and H of finite index k in G. Then there exists a group morphism from G to the (finite) group of permutations of the set of right-cosets X := {Hgi | i = 1, . . . , k}. This is given by g → αg where αg (Hgi ) = Hgi g. Because G is infinite the kernel of the above morphism must be nontrivial. Moreover, because G is simple the kernel must be all of G, i.e., αgi (H ) = H = Hgi for all i = 1, . . . , k. Hence H is not proper. (iii) If G is ICC and H is a finite index subgroup of G then H is ICC. Indeed, let G = ∪kj =1 cj H , where cj are distinct left-cosets representatives. If for some h ∈ H the conjugacy class {ghg −1 | g ∈ H } is finite then the set {cj gh(cj g)−1 | g ∈ H, j = 1, . . . , k} is finite. Notice {chc−1 | c ∈ G} ⊂ {cj gh(cj g)−1 | g ∈ H, j = 1, · · · , k}. However, the conjugacy class of h in G is infinite as G is ICC. (iv) There exist ICC groups with all elements of finite order, e.g., the Burnside groups of large enough exponents (see [20]). Proposition 3.6. Let G be a countable ICC group and let π : G → U(H) be a unitary representation of the group G on the Hilbert space H. Suppose ξ ∈ H is a Parseval frame vector for H with ξ 2 = M N for some M, N ∈ N. Assume in addition that there exists a normal ICC subgroup H of N index [G : H ] = N such that H contains elements of infinite order. Then there exists K := M strongly disjoint Parseval frame vectors ηi for H, i = 1, . . . , K such that {π(h)ηi | h ∈ H } is an orthogonal family (in a subspace of H) for all i = 1, . . . , K. Proof. Consider a projection p in L(G) of trace trL(G) (p ) = N1 . Using Theorem 3.3 we can find ξ1 , . . . , ξN strongly disjoint Parseval frame vectors for H1/N := p l 2 (G) with the representation π1/N := p λ, such that for all i = 1, . . . , N , {π(h)ξi | h ∈ H } is an orthogonal basis for H1/N . M on HM with the strongly disjoint Parseval frame Then consider the representation π1/N 1/N vectors ηi := ξ(M−1)i+1 ⊕ · · · ⊕ ξMi , i = 1, . . . , K. M is equivalent to a subrepresentation of the left-regular Using Proposition 1.8 we have that π1/N representation, corresponding to a projection of trace ξ1 ⊕ · · · ⊕ ξM 2 = M N . But the same is true for the representation π on H. Therefore the two representations are equivalent and the vectors ηi can be mapped into H to obtain the conclusion. 2 Remark 3.7. Suppose the hypotheses of Theorem 3.3 are satisfied, with N 2. Then we can construct uncountably many inequivalent Parseval frame vectors η for H with the property that {πh(η) | h ∈ H } is a Riesz basis for H. To see this, use Theorem 3.3 to obtain strongly disjoint Parseval frame vectors η1 , . . . , ηN for H, such that {π(h)ηi | h ∈ H } is an orthogonal basis for H, for all i = 1, N .
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Then take α, β ∈ C with |α|2 + |β|2 = 1, and |α| = |β|. Using Theorem 2.6, we obtain that ηα,β := αη1 + βη2 is a Parseval frame vector for H. Since η1 and η2 generate orthogonal bases under the action of H , there is a unitary u ∈ π(H ) such that uη1 = η2 . Then ηα,β = (α + βu)η1 . Since u is unitary and |α| = |β| it follows that α + βu is invertible. Therefore {π(h)ηα,β | h ∈ H } = {(α + uβ)π(h)η1 | h ∈ H } is a Riesz basis for H. It remains to see when two such vectors ηα,β , ηα ,β are equivalent. Using Theorem 2.7 we see that this happens only if |α| = |α |, |β| = |β | and αβ = α β , i.e., (α, β) = c(α , β ) for some c ∈ C with |c| = 1. Since we can find uncountably many pairs (α, β) such that no two such pairs satisfy this condition, it follows that we can construct uncountably many inequivalent Parseval frame vectors ηα,β that satisfy the given conditions. Finally we discuss how our results fit in the recent effort on the Feichtinger’s frame decomposition conjecture. It was recently discovered (in particular, by Pete Casazza and his collaborators) that the famous intractible 1959 Kadison–Singer Problem in C∗ -algebras is equivalent to fundamental open problems in a dozen different areas of research in mathematics and engineering (cf. [3,4]). Particularly, the KS-problem is equivalent to the Feichtinger’s problem which asks whether every bounded frame (i.e., the norms of the vectors in the frame sequence are bounded from below) can be written as a finite union of Riesz sequences. Since this question is intractible in general, much of the effort has been focused on special classes of frames. One natural and interesting class to consider is the class of frames obtained by group representations [see open problems posted at the 2006 “The Kadison–Singer Problem” workshop]. Unfortunately, except for a very few cases (e.g., Gabor frames associated with rational lattices [2]) very little is known so far even for this special class. Particularly, it is unknown whether for every (frame) unitary representation we can always find one frame vector which is “Riesz sequence” decomposable. Therefore the results obtained in this section certainly addressed some aspects of the research effort in this direction. In particular, we have the following as a consequence of our main result. Proposition 3.8. Let G be a countable ICC group and assume that there exists an ICC subgroup H of index [G : H ] = N . Then (i) If p ∈ L(G) is any projection such that trL(G) (p ) N1 , then there exist a Parseval frame vector η for the subrepresentation π := L|p such that {π(g)η: g ∈ G} is a finite union of Riesz sequences. (ii) For any ONB vector for the left-regular representation L and any α such that 1 > α N1 , there exists a projection p ∈ L(G) such that trL(G) (p ) = α and {p L(g)ψ: g ∈ G} is a finite union of Riesz sequences. Proof. (i) Since L(G) is a factor von Neumann algebra, there exists a subprojection q of p such that trL(G) (q ) = N1 . By Proposition 3.1, there exists a Parseval frame vector, say η1 , for √ the representation π|q such that { N π(h)η1 : h ∈ H } is orthonormal. By Lemma 2.4, we can “dilate” η1 to a Parseval frame vector η for π . Let η2 = (p − q )η. Then for any sequence {ch }h∈H (finitely many of them are non-zero) we have 2 2 2 2 ch π(h)η = ch π(h)η1 + ch π(h)η2 ch π(h)η1 h∈H
h∈H
h∈H
h∈H
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since π(G)η1 and π(G)η2 are orthogonal. Thus {π(h)η: h ∈ H } is a Riesz sequence as {π(h)η1 : h ∈ H } is Riesz. (ii) Let r ∈ L(G) be a projection such that trL(G) (r ) = α. Then by part (i) there exists a Parseval frame vector ξ such that {σ (h)ξ : h ∈ H } is a Riesz sequence, where σ = L|r . We will show that there exists a projection p ∈ L(G) such that {p L(g)ψ: g ∈ G} and {σ (g)ξ : g ∈ G} are unitarily equivalent, and this will imply that {p L(h)ψ: h ∈ H } is a Riesz sequence. In fact, again by Lemma 2.4, there exists ONB vector ψ˜ for the left-regular representation L such that r ψ˜ = ξ . Since both ψ and ψ˜ are ONB vectors for L we have that there exists a unitary operator u ∈ L(G) such that ψ = u ψ˜ . Let p = u r u ∗ . Then p ∈ L(G) is a projection such that trL(G) (p ) = trL(G) (r ) = α. Moreover, p L(g)ψ = L(g)p ψ = L(g)(u r u ∗ )u ψ˜ = u L(g)r ψ˜ = u L(g)ξ = u σ (g)ξ for all g ∈ H . Hence {p L(g)ψ: g ∈ G} and {σ (g)ξ : g ∈ G} are unitarily equivalent, as claimed and so we completed the proof. 2 There are some interesting special cases. For example, as we mentioned in Remark 3.5 if G is a free group with more than one generator, then we can find Nk → ∞ such that there exist ICC subgroups Hk having the property [G : Hk ] = Nk . Thus we have the following corollary which for the free group case answered affirmatively one of two open problems posted by Deguang Han at the 2006 “The Kadison–Singer Problem” workshop. Corollary 3.9. Let G be a free group with more than one generator. Then (i) For any non-zero projection p ∈ L(G) , there exists a Parseval frame vector η for the subrepresentation π := L|p such that {π(g)η: g ∈ G} is a finite union of Riesz sequences. (ii) For any ONB vector for the left-regular representation L, and any α > 0, there exists a projection p ∈ L(G) such that trL(G) (p ) = α and {p L(g)ψ: g ∈ G} is a finite union of Riesz sequences. This sequence will be orthogonal when α = N1 for some N ∈ N. References [1] Akram Aldroubi, David Larson, Wai-Shing Tang, Eric Weber, Geometric aspects of frame representations of abelian groups, Trans. Amer. Math. Soc. 356 (12) (2004) 4767–4786 (electronic). [2] Peter G. Casazza, Ole Christensen, Alexander M. Lindner, Roman Vershynin, Frames and the Feichtinger conjecture, Proc. Amer. Math. Soc. 133 (4) (2005) 1025–1033 (electronic). [3] Peter G. Casazza, Matthew Fickus, Janet C. Tremain, Eric Weber, The Kadison–Singer problem in mathematics and engineering: A detailed account, in: Operator Theory, Operator Algebras, and Applications, in: Contemp. Math., vol. 414, Amer. Math. Soc., Providence, RI, 2006, pp. 299–355. [4] Peter G. Casazza, Janet Crandell Tremain, The Kadison–Singer problem in mathematics and engineering, Proc. Natl. Acad. Sci. USA 103 (7) (2006) 2032–2039 (electronic). [5] Ingrid Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. [6] R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341–366. [7] Jean-Pierre Gabardo, Deguang Han, Subspace Weyl–Heisenberg frames, J. Fourier Anal. Appl. 7 (4) (2001) 419– 433. [8] Jean-Pierre Gabardo, Deguang Han, Frame representations for group-like unitary operator systems, J. Operator Theory 49 (2) (2003) 223–244. [9] Jean-Pierre Gabardo, Deguang Han, The uniqueness of the dual of Weyl–Heisenberg subspace frames, Appl. Comput. Harmon. Anal. 17 (2) (2004) 226–240.
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[10] Karlheinz Gröchenig, Foundations of Time–Frequency Analysis, Appl. Numer. Harmon. Anal., Birkhäuser Boston Inc., Boston, MA, 2001. [11] Deguang Han, Tight frame approximation for multi-frames and super-frames, J. Approx. Theory 129 (1) (2004) 78–93. [12] Deguang Han, Frame representations and Parseval duals with applications to Gabor frames, Trans. Amer. Math. Soc. 360 (6) (2008) 3307–3326. [13] Deguang Han, David R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (697) (2000), x+94. [14] Deguang Han, D. Larson, Wandering vector multipliers for unitary groups, Trans. Amer. Math. Soc. 353 (8) (2001) 3347–3370 (electronic). [15] Deguang Han, David R. Larson, Frame duality properties for projective unitary representations, Bull. London Math. Soc. 147 (40) (2008) 685–695. [16] Christopher Heil, History and evolution of the density theorem for Gabor frames, J. Fourier Anal. Appl. 13 (2) (2007) 113–166. [17] Paul Jolissaint, Central sequences in the factor associated with Thompson’s group F , Ann. Inst. Fourier (Grenoble) 48 (4) (1998) 1093–1106. [18] V. Jones, V.S. Sunder, Introduction to Subfactors, London Math. Soc. Lecture Note Ser., vol. 234, Cambridge University Press, Cambridge, 1997. [19] Richard V. Kadison, John R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. II, Grad. Stud. Math., vol. 15, Amer. Math. Soc., Providence, RI, 1997, Elementary theory, reprint of the 1983 original. [20] A.Ju. Ol’šanski˘ı, On the question of the existence of an invariant mean on a group, Uspekhi Mat. Nauk 35 (4(214)) (1980) 199–200. [21] Joaquim Ortega-Cerdà, Kristian Seip, Fourier frames, Ann. of Math. (2) 155 (3) (2002) 789–806. [22] Wai Shing Tang, Eric Weber, Frame vectors for representations of abelian groups, Appl. Comput. Harmon. Anal. 20 (2) (2006) 283–297.
Journal of Functional Analysis 256 (2009) 3091–3105 www.elsevier.com/locate/jfa
Probabilistic solution of the American options Ali Süleyman Üstünel Telecom-Paristech (formerly ENST), Dept. Infres, 46, rue Barrault, 75013 Paris, France Received 10 October 2008; accepted 15 October 2008 Available online 6 November 2008 Communicated by Paul Malliavin
Abstract The existence and uniqueness of probabilistic solutions of variational inequalities for the general American options are proved under the hypothesis of hypoellipticity of the infinitesimal generator of the underlying diffusion process which represents the risky assets of the stock market with which the option is created. The main tool is an extension of the Itô formula which is valid for the tempered distributions on Rd and for nondegenerate Itô processes in the sense of the Malliavin calculus. © 2008 Elsevier Inc. All rights reserved. Keywords: Malliavin calculus; American options; Nondegenerate; Wiener functionals; Quasi-variational inequalities; Hypoellipticity; Sobolev spaces; Tempered distributions
1. Introduction The difficulty to justify the validity of the probabilistic solutions of the American options is well known. This is in fact due to the lack of regularity of the classical solutions of the variational inequalities (cf. [2]) which are satisfied by the value function which characterizes the Snell envelope (cf. [8] for a recent survey about this subject). In particular the value function is not twice differentiable hence the Itô formula is not applicable to apply the usual probabilistic techniques. In the case of Black and Scholes model, there are some results using extensions of the Itô formula for the Brownian motion, which, however, are of limited utility for more general cases. In this note we give hopefully more general results in the sense that the option is constructed by the assets which obey to a general, finite-dimensional stochastic differential equation with deterministic coefficients, i.e., a diffusion process. The basic hypothesis used is the nondegeneracy E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.016
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of this diffusion in the sense of the Malliavin calculus (cf. [9]): recall that an Rd -valued random variable F = (F1 , . . . , Fd ), defined on a Wiener space is called nondegenerate (cf. [9,15,16]) if it is infinitely Sobolev differentiable with respect to the Wiener measure and if the determinant of the inverse of the matrix ((∇Fi , ∇Fj )H : i, j d), where ∇ denotes the Sobolev derivative on the Wiener space, is in all the Lp -spaces with respect to the Wiener measure. In this case, the mapping f → f ◦ F , defined from the smooth functions on Rd to the space of smooth functions on the Wiener space extends continuously to a linear mapping, denoted as T → T (F ), T ∈ S (Rd ), from the tempered distributions S (Rd ) to the space of Meyer distributions on the Wiener space (cf. [9,15,16]). Similarly, if F is replaced with an Itô process whose components satisfy similar regularity properties, we obtain an Itô formula for T (Ft ) − T (Fs ), 0 < s t, where the stochastic integral should be treated as a distribution-valued Gaussian divergence and the absolutely continuous term is a Bochner integral concentrated in some negatively indexed Sobolev space. Moreover, if this latter term is a positive distribution, then the resulting integral is a Radon measure on the Wiener space due to a well-known result about the positive Meyer distributions on the Wiener space (cf. [1,12–16]). Having summarized the technical tools that we use, let us explain now the main results of the paper: for the uniqueness result we treat two different situations; namely the first one where the coefficients are time dependent and the variational inequality is interpreted as an evolutionary variational inequality in S (Rd ). The second one concerns the case where the coefficients are time-independent and we interpret it as an inequality in the space D (0, T ) ⊗ S (Rd ) with a boundary condition, which is of course more general than the first one. In both cases the operators are supposed only to be hypoelliptic; a hypothesis which is far more general than the ellipticity hypothesis used in [2]. The homogeneity in time permits us more generality since, in this case the time-component regularization by the mollifiers of the solution candidates preserve their property of being negative distributions, hence measures. The existence is studied in the last section using the similar techniques and we obtain as a by product some regularity results about the solution of the variational inequality. In particular, we realize there that even if the density of the underlying diffusion has zeros, there is still a solution on the open set which corresponds to the region of [0, T ] × Rd where the density is strictly positive. 2. Preliminaries and notation Let W be the classical Wiener space C([0, T ], Rn ) with the Wiener measure μ. The corresponding Cameron–Martin space is denoted by H . Recall that the injection H → W is compact and its adjoint is the natural injection W → H ⊂ L2 (μ). Since the translations of μ with the elements of H induce measures equivalent to μ, the Gâteaux derivative in H direction of the random variables is a closable operator on Lp (μ)spaces and this closure will be denoted by ∇, cf. for example [4,15,16]. The corresponding Sobolev spaces (the equivalence classes) of the real random variables will be denoted as Dp,k , where k ∈ N is the order of differentiability and p > 1 is the order of integrability. If the random variables are with values in some separable Hilbert space, say Φ, then we shall define similarly the corresponding Sobolev spaces and they are denoted as Dp,k (Φ), p > 1, k ∈ N. Since ∇ : Dp,k → Dp,k−1 (H ) is a continuous and linear operator its adjoint is a well-defined operator which we represent by δ. δ coincides with the Itô integral of the Lebesgue density of the adapted elements of Dp,k (H ) (cf. [15,16]). For any t 0 and measurable f : W → R+ , we note by
A.S. Üstünel / Journal of Functional Analysis 256 (2009) 3091–3105
Pt f (x) =
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f e−t x + 1 − e−2t y μ(dy),
W
it is well known that (Pt , t ∈ R+ ) is a hypercontractive semigroup on Lp (μ), p > 1, which is called the Ornstein–Uhlenbeck semigroup (cf. [4,15,16]). Its infinitesimal generator is denoted by −L and we call L the Ornstein–Uhlenbeck operator (sometimes called the number operator by the physicists). The norms defined by φ p,k = (I + L)k/2 φ Lp (μ)
(2.1)
are equivalent to the norms defined by the iterates of the Sobolev derivative ∇. This observation permits us to identify the duals of the space Dp,k (Φ), p > 1, k ∈ N, by Dq,−k (Φ ), with q −1 = 1 − p −1 , where the latter space is defined by replacing k in (2.1) by −k, this gives us the distribution spaces on the Wiener space W (in fact we can take as k any real number). An easy calculation shows that, formally, δ ◦ ∇ = L, and this permits us to extend the divergence and the derivative operators to the distributions as linear, continuous operators. In fact δ : Dq,k (H ⊗ Φ) → Dq,k−1 (Φ) and ∇ : Dq,k (Φ) → Dq,k−1 (H ⊗ Φ) continuously, for any q > 1 and k ∈ R, where H ⊗ Φ denotes the completed Hilbert–Schmidt tensor product (cf., for instance [9,15,16]). We shall denote by D(Φ) and D (Φ) respectively the sets D(Φ) =
Dp,k (Φ),
p>1,k∈N
and D (Φ) =
Dp,−k (Φ),
p>1,k∈N
where the former is equipped with the projective and the latter is equipped with the inductive limit topologies. A map F ∈ D(Rd ) is called nondegenerate if det γ ∈ p Lp (μ), where γ is the inverse of the matrix ((∇Fi , ∇Fj )H , i, j d) and (·,·)H denotes the scalar product in H . For such a map, it is well known that [9,15,16] the map f → f ◦ F from S(Rd ) → D has a linear, continuous extension to S (Rd ) → D , where S(Rd ) and S (Rd ) denote the space of rapidly decreasing functions and tempered distributions on Rd , respectively. In fact, due to the “polynomially increasing” character of the tempered distributions, the range of this extension is much smaller than D , in fact it is included in ˜ = D
Dp,−k .
p>1 k∈N
This notion has been extended in [13] and used to give an extension of the Itô formula as follows. Theorem 1. Assume that (Xt , t ∈ [0, T ]) is an Rd -valued non-degenerate Itô process with the decomposition dXt = bt dt + σt dWt
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where b ∈ Da (L2 ([0, T ]) ⊗ Rd ) and σ ∈ Da (L2 ([0, T ]) ⊗ Rd ⊗ Rn ), where the upper index means adapted to the Brownian filtration. Assume further that
a
1 (det γs )p ds ∈ L1 (μ),
(2.2)
ε j
for any p > 1, where γs is the inverse of the matrix ((∇Xti , ∇Xt )H ; i, j d). Then, for any T ∈ S (Rd ) and 0 < s < t 1, we have t T (Xt ) − T (Xs ) =
t Au T (Xu ) du +
s
∂T (Xu ), σu dWu ,
s
˜ and the second where Au = 12 ai,j (u)∂i,j + bi (u)∂i , the first integral is a Bochner integral in D one is the extended divergence operator explained above. Remark 1. The conditions under which the hypothesis (2.2) holds are extremely well studied in the literature, cf. [6,7]. Remark 2. The divergence operator acts as an isomorphism between the spaces Dap,k (H ) and Dp,k for any p > 1, k ∈ R, cf. [14]. Remark 3. We can extend the above result easily to the case where t → Tt is a continuous map of finite total variation from [0, T ] to S (Rd ) in the sense that, the mapping t → Tt , g is of finite total variation on [0, T ] for any g ∈ S(Rd ). In fact, the kernel theorem of A. Grothendieck implies that Tt can be represented as Tt =
∞
λi αi (t)Fi ,
i=1
where (λi ) ∈ l 1 , (αi ) is bounded in the total variation norm and (Fi ) is bounded in S (Rd ). Using this decomposition, it is straightforward to show that t T (t, Xt ) − T (s, Xs ) =
t Au T (u, Xu ) du +
s
t T (du, Xu ) +
s
∂T (u, Xu ), σu dWu .
s
where the second integral is defined as t T (du, Xu ) = s
∞
i=1
t λi
Fi (Xu ) dαi (u) s
and the right-hand side is independent of any particular representation of Tt . Integrals are concentrated in D .
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We can prove easily the following result using the technique described in [13,16]. Theorem 2. Assume that (lt , t ∈ [0, 1]) is an Itô process dlt = mt dt +
zti dWti ,
i
with m, zi ∈ Da (L2 [0, T ]), then we have t lt T (t, Xt ) − ls T (s, Xs ) =
t lu Au T (u, Xu ) du +
s
lu T (du, Xu ) s
t +
lu ∂T (u, Xu ), σu dWu +
s
T (u, Xu )mu du s
t +
t
T (u, Xu )
t zui dWui +
i
s
∂T (u, Xu ), σu zu du
s
where z = (z1 , . . . , zn ). An important feature of the distributions on the Wiener space is the notion of positivity: we say that S ∈ D is positive if for any positive ϕ ∈ D, we have S(ϕ) = S, ϕ 0. An important result about the positive distributions is the following (cf. [1,9,12,15,16]). Theorem 3. Assume that S is a positive distribution in D , then there exists a positive Radon measure νS on W such that
S, ϕ =
ϕ dνS , W
for any ϕ ∈ D ∩ Cb (W ). In particular, if a sequence (Sn ) of positive distributions converge to S weakly in D , then (νSn ) converges to νS in the weak topology of measures. Remark 4. In fact we can write, for any ϕ ∈ D ϕ˜ dνS ,
S, ϕ = W
where ϕ˜ denotes a redefinition of ϕ which is constructed using the capacities associated to the scale of Sobolev spaces (Dp,k , p > 1, k ∈ N), cf. [4,9].
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3. Uniqueness of the solution of parabolic variational inequality Assume that (Xts (x), 0 s t T ) is a diffusion process governed by an Rn -valued Wiener process (Wt , t ∈ [0, T ]). We assume that the diffusion has smooth, bounded drift and diffusion coefficients b(t, x), σ (t, x) defined on [0, T ] × Rd , with values in Rd and Rd ⊗ Rn , respectively, and we denote by At its infinitesimal generator. We shall assume that Xts is nondegenerate for any 0 s < t T , ∂/∂t + At is hypoelliptic and t
p det γvs dv ∈ L1 (μ)
s+ε s,j
for any 0 < s < t T and ε > 0, where γv is the inverse of the matrix ((∇Xvs,i , ∇Xv )H : i, j d). Suppose that f ∈ Cb (Rd ) and we shall study the following partial differential inequality whose solution will be denoted by u(t, x). Theorem 4. Assume that u ∈ Cb ([0, T ] × Rd ) such that t → u(t, ·), g is of finite total variation on [0, T ] for any g ∈ S(Rd ) and that it satisfies the following properties: ∂u + At u − ru 0, u f in [0, T ] × Rd , ∂t ∂u + At u − ru (f − u) = 0, ∂t
(3.3) (3.4)
U (T , x) = f (x),
(3.5)
where all the derivatives are taken in the sense of distributions, in particular the derivative with respect to t is taken using the C ∞ -functions of compact support in (0, T ). Then
u(t, x) = sup E f Xτt (x) exp − τ ∈Zt,T
τ r
s, Xst (x)
ds ,
t
where Zt,T denotes the set of all the stopping times with values in [t, T ] and r is a smooth function on [0, T ] × Rd . Proof. We shall prove the case t = 0. Let us denote by l the process defined as lt = t exp − 0 r(s, Xs ) ds. From Theorem 2, we have, for any ε > 0, t lt u(t, Xt ) − lε u(ε, Xε ) − ε
ls As u(s, Xs ) ds − (ru)(s, Xs ) ds + u(ds, Xs ) = Mtε
(3.6)
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where Mtε is a D -valued martingale difference, i.e., denoting by E[·|Fs ] the extension of the conditional expectation operator to D ,1 we have E[Mtε |Fs ] = Msε for any ε s t. Note also that Kt u = ∂u ∂t + At u − ru 0 hence its composition with Xt is a negative measure and this implies that the integral at the left-hand side of (3.6) is a negative distribution on the Wiener space. Consequently we have Mtε lt u(t, Xt ) − lε u(ε, Xε )
(3.7)
in D . For α > 0, let Pα be the Ornstein–Uhlenbeck semigroup and define Mtα,ε as Mtα,ε = Pα Mtε . Then (Mtα,ε , t ε) is a continuous martingale (in the ordinary sense). From the inequality (3.7), we have, for any τ ∈ Zε,T , Mτα,ε Pα lt u(t, Xt ) − lε u(ε, Xε ) t=τ . Taking the expectation of both sides, we get E lε u(ε, Xε ) E Pα lt u(t, Xt ) t=τ for any α > 0, hence we also have E lε u(ε, Xε ) E lτ u(τ, Xτ ) for any ε > 0 which is arbitrary, and finally we obtain u(0, x) E lτ u(τ, Xτ ) for any τ ∈ Z0,T . To show the reverse inequality let D = {(s, x): u(s, x) = f (x)} and define τx = inf s: s, Xs0,x ∈ D c . Since Kt is hypoelliptic, and since Kt u = 0 on the set D, u is smooth in D. If μ{τx = 0} = 1, from the continuity of u, we have , u(0, x) = f (x) = E lτx u τx , Xτ0,x x hence the supremum is attained in this case. If μ{τx = 0} > 0, then from the 0–1-law μ{τx = 0} = 1 and τx is predictable. Let (τn , n 1) a sequence of stopping times announcing τx . From the classical Itô formula, we have 1 Such an extension is licit since the conditional expectation operator commutes with the Ornstein–Uhlenbeck semigroup.
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τn lτn u(τn , Xτn ) − u(0, x) =
ls σ ∂u (s, Xs ) · dWs .
0
By the hypothesis the left-hand side is uniformly integrable with respect to n ∈ N, consequently we obtain u(0, x) = lim E lτn u(τn , Xτn ) = E lτ u(τ, Xτ ) n
2
hence τx realizes the supremum.
In the homogeneous case the finite variation property of the solution follows directly from the quasi-variational inequality: Theorem 5. Suppose that the infinitesimal generator At of the process (Xt ) is independent of t ∈ [0, T ] and denote it by A. In other words the process is homogeneous in time. Assume that u ∈ Cb ([0, T ] × Rd ) satisfies the following properties: ∂u + Au − ru 0, uf ∂t ∂u + Au − ru (f − u) = 0, ∂t
in [0, T ] × Rd ,
(3.8)
in [0, T ] × Rd ,
(3.9)
U (T , x) = f (x).
(3.10)
Then
u(t, x) = sup E f Xτt (x) exp − τ ∈Zt,T
τ r
s, Xst (x)
ds ,
t
where Zt,T denotes the set of all the stopping times with values in [t, T ] and r is a smooth function on [0, T ] × Rd . Remark 5. The relations (3.8) and (3.9) are to be understood in the weak sense. This means that for any g a C ∞ function of support in (0, T ) and γ ∈ S(Rd ), both of which are positive, we have
∂u + Au − ru, g ⊗ γ 0 ∂t
and
∂u + Au − ru (f − u), g ⊗ γ = 0. ∂t
Proof. As in the proof of the preceding theorem, we shall prove the equality for t = 0, then the general case follows easily. Let ρδ be a mollifier on R and let ηε be a family of positive smooth
A.S. Üstünel / Journal of Functional Analysis 256 (2009) 3091–3105
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functions on (0, T ), equal to unity on the interval [ε, T − ε], converging to the indicator function of [0, T ] pointwise. Define uδ,ε as uδ,ε = ρδ (ηε u). From the hypothesis the distribution ν defined by ν=
∂u + Au − ru ∂t
is a negative measure on (0, T ) × Rd . A simple calculation gives ∂uδ,ε + Auδ,ε − ruδ,ε = ρδ ηε u + ρδ (ηε ν) + ρδ (ηε ru) − ruδ,ε . ∂t As in the preceding theorem, we have from Theorem 2 t δ,ε
lt u
(t, Xt ) − la u
δ,ε
ls Ks uδ,ε (s, Xs ) ds = Mtδ,ε,a ,
(a, Xa ) − a
where M δ,ε,a is a D -martingale difference. Since ν is a negative measure, we get the following inequality in D : t Mtδ,ε,a
lt u
δ,ε
(t, Xt ) − la u
δ,ε
(a, Xa ) −
ρδ ηε u + ρδ (ηε ru) − ruδ,ε (s, Xs ) ds.
a
Let now (Pα , α 0) be the Ornstein–Uhlenbeck semigroup. Then (Pα Mtδ,ε,a , a t T ) is a real-valued martingale difference, consequently, we have 0 = E Pα Mtδ,ε,a t=τ
E Pα lt uδ,ε (t, Xt ) − la uδ,ε (a, Xa ) t −
ρδ ηε u + ρδ (ηε ru) − ruδ,ε (s, Xs ) ds
, t=τ
a
for any stopping time τ with values in [ε, T − ε]. By letting α → 0, we get by continuity
τ
0 E lτ u
δ,ε
(τ, Xτ ) − la u
δ,ε
(a, Xa ) − a
δ,ε ρδ ηε u + ρδ (ηε ru) − ru (s, Xs ) ds .
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Let us choose a > 0 and let then ε, δ → 0. Note that ηε → δ0 − δT (i.e., the Dirac measures at 0 and at T ), by the choice of a and by the weak convergence of measures and by the dominated convergence theorem, we obtain τ lim E
ε,δ→0
ρδ ηε u (s, Xs ) ds = 0.
a
Again from the dominated convergence theorem we have τ lim E
ε,δ→0
ρδ (ηε ru) − ruδ,ε (s, Xs ) ds = 0.
a
Consequently E la u(a, Xa ) E lτ u(τ, Xτ ) E lτ f (Xτ ) , for any stopping time τ with values in [a, T − a], since a > 0 is arbitrary, the same inequality holds also for any stopping time with values in [0, T ]; hence u(0, x) E lτ f (Xτ ) for any stopping time τ ∈ Z0,T and we obtain the first inequality: u(0, x) sup E lτ f (Xτ ) . τ ∈Z0,T
The proof of the reverse inequality is exactly the same that of Theorem 4 due to the hypoellipticity hypothesis. 2 4. Existence of the solutions In this section, under the hypothesis of the preceding section, we shall prove that the function defined by the Snell envelope (cf. [3]) of the American option satisfies the variational inequality (3.8) and the equality (3.9). We start with a lemma: Lemma 1. Assume that Z = (Zt , t ∈ [0, T ]) is a uniformly integrable, real-valued martingale on the Wiener space. Let Z κ = (Ztκ , t ∈ [0, T ]) be defined as Ztκ = Pκ Zt , where Pκ is the Ornstein–Uhlenbeck semigroup at the instant κ > 0. Then (Ztκ , t ∈ [0, T ]) is a uniformly integrable martingale with 1/2 1/2 E Z κ , Z κ T cE Z, Z T , where c is a constant independent of Z and κ. In particular, if Z has the representation
(4.11)
A.S. Üstünel / Journal of Functional Analysis 256 (2009) 3091–3105
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T ZT =
(ms , dWs ), 0
with m ∈ L1 (μ, H ) optional, then T Pκ ZT =
e−κ (Pκ ms , dWs ).
0
Proof. From Davis’ inequality (cf. [10]), we have 1/2 E Z κ , Z κ T c1 E sup Ztκ t∈[0,T ]
c1 E Pκ sup |Zt | = c1 E
t∈[0,T ]
sup |Zt |
t∈[0,T ]
1/2 cE Z, Z T . The second part is obvious from the inequality (4.11).
2
Theorem 6. Assume that (Xts ) is a hypoelliptic diffusion such that, for any ε > 0, T
p det γvs dv ∈ L1 (μ)
s+ε
for any p > 1. Let p(s, t; x, y), s < t, x, y ∈ Rd be the density of the law of Xts (x) and denote by S0,z the open set S0,z = (s, y) ∈ (0, T ) × Rd : s > 0, p(0, s; z, y) > 0 . Then, for any z ∈ Rd , u is a solution of the variational inequality (3.8)–(3.10) in D (S0,z ). If S0,z = (0, T ) × Rd for any z ∈ Rd , then u is a solution of the variational inequality (3.8)–(3.10) in D (0, T ) ⊗ D (Rd ). Proof. From the optimal stopping results, we know that u is a bounded, continuous function and t → u(t, x) is monotone, decreasing (cf. [3]). Moreover u(t, Xt )lt − u(0, x) = Mt + Bt is a supermartingale where Xt = Xt0 (x) and we denoted by M its martingale part and by B its continuous, decreasing process part. In particular dB × dμ defines a negative measure γ on [0, T ] × C([0, T ], Rd ). We can write u(ds, x) dx as the sum uac (s, x) ds dx + using (ds, x) dx
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where uac is defined as the absolutely continuous part of u and using is the singular part with respect to the Lebesgue measure ds on [0, T ]. We have, from the extended Itô formula, t u(t, Xt )lt − u(ε, Xε ) =
(As u − ru + uac )(s, Xs ) ds + using (ds, Xs ) ε
t +
(σ ∂u)(s, Xs ), dWs
ε
= Mtε + Btε , hence regularizing both parts by the Ornstein–Uhlenbeck semigroup, from Lemma 1, we get t Btε
=
(As u − ru + uac )(s, Xs ) ds + using (ds, Xs ), ε
t Mtε
=
(σ ∂u)(s, Xs ), dWs .
ε
Consequently, for any α ∈ D(0, T ) and φ ∈ D
T E φ α(s) dBs = α ⊗ φ dγ 0
=
α(s) (As u − ru + uac )(s, Xs ) ds + using (ds, Xs ), φ
(0,T )
and this quantity is negative for any α ∈ D+ (0, T ) and φ ∈ D+ . Let now 0 g ∈ S(Rd ) and assume that (ti , i m) is a partition of [0, T ]. Define ξm as ξm (t, w) =
1[ti ,ti+1 ] (t)g(Xti ).
i
Then it is immediate from the hypothesis about the diffusion process (Xt ) that (ξm , m 1) converges to (g(Xs )1[0,T ] (s), s ∈ [0, T ]) in D(Lp ([0, T ])) for any p 1 and (ξm (s, ·)) converges to g(Xs ) in D for any fixed s ∈ [0, T ] as the partition pace tends to zero. Let us represent u(ds, ·), using the kernel theorem (cf. [5,11]), as u(ds, ·) =
∞
λi Tk ⊗ αk ,
k=1
where (λk ) ∈ l 1 , (Tk ) ⊂ S (Rd ) is bounded and (αk ) is a sequence of measures on [0, T ], bounded in total variation norm. It follows then
A.S. Üstünel / Journal of Functional Analysis 256 (2009) 3091–3105
u(ds, Xs ) =
∞
3103
λi Tk (Xs ) αk (ds)
k=1
˜ Dp,−k for some k ∈ N and p > 1, in the projecand this some is convergent in V ([0, T ]) ⊗ tive topology, where V ([0, T ]) denotes the Banach space of measures on [0, T ] under the total variation norm. Since sup ξm (s, Xs )p,l sup g ◦ Xs p,l ,
s∈[0,T ]
s∈[0,T ]
uniformly in m ∈ N, for any p, l and since ξm (s, ·) − ξn (s, ·)
p,l
→0
as m, n → ∞ for any p, l and s ∈ [0, T ], we obtain lim
m→∞ (0,T )
δ(s) ξm (s, ·) − g(Xs ), u(ds, Xs ) = 0
for any δ ∈ D(0, T ) from the dominated convergence theorem. The above relation implies in particular that we have
α(s) (As u − ru + uac )(s, ·), p0,s g ds +
(0,T )
α(s) using (ds, ·), gp0,s 0,
(0,T )
with smooth, positive α and g, where the brackets in the integral correspond to the duality between D(Rd ) and D (Rd ). For the functions of support in (0, T ), we can replace the term
α(s) u(ds, Xs ), g(Xs )
(0,T )
by
∂ u(s, Xs ), g(Xs ) α(s) ∂s
(0,T )
where ∂/∂s denotes the derivative in D (0, T ). Since α and g are arbitrary, we obtain the inequality (3.8) in D (S0,x ). If S0,x = (0, T ) × Rd , then we have the inequality in the sense of distributions on (0, T ) × Rd . To complete the proof, let D be the set defined as D = (s, x) ∈ (0, T ) × Rd : u(s, x) = f (x) . Then we have
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T 1D c (s, Xs ) dBs = 0 0
almost surely (cf. [3]). Let C = −B, then for any smooth function η ∈ D(0, T ) ⊗ S(Rd ) such that η 1D c , we have T 0=E
1D c (s, Xs ) dCs 0
T E
η(s, Xs ) dCs 0
(As u − ru)(s, Xs ) ds + u(ds, Xs ), η(s, Xs )
=− (0,T )
0, where, the second equality follows from the estimates above. Hence As u − ru +
∂ u=0 ∂s
as a distribution on the set S0,x ∩ D c , by the hypoellipticity, the equality is everywhere on this set. If S0,x = (0, T ) × Rd , then we obtain the relation (3.9). 2 Remark 6. From the general theory, we can express the martingale part of (lt u(t, Xt ), t ∈ [0, T ]) as T (Hs , dWs ) 0
where H is an adapted process which is locally integrable. On the other hand, we have t Mtε
=
σ (s, Xs )∂u(s, Xs ), dWs
ε
where the right-hand side is to be interpreted in a negatively indexed Sobolev space on the Wiener space. Using Lemma 1, we obtain the identity Hs = σ (s, Xs )∂u(s, Xs ) ds × dμ-a.s., in particular we have
A.S. Üstünel / Journal of Functional Analysis 256 (2009) 3091–3105
T E
σ (s, Xs )∂u(s, Xs )2 ds
3105
1/2 < ∞.
0
References [1] H. Airault, P. Malliavin, Inte´gration géométrique sur l’espace de Wiener, Bull. Sci. Math. 112 (1) (1988) 3–52. [2] A. Bensoussan, J.L. Lions, Applications des inéquations variationnelles en contrôle stochastique, Dunod, Paris, 1978. [3] N. El Karoui, Les aspects probabilistes du contrôle stochastique, in: Lecture Notes in Math., vol. 876, Springer, Berlin, 1981, pp. 72–238. [4] D. Feyel, A. de La Pradelle, Capacités gaussiennes, Ann. Inst. Fourier (Grenoble) 41 (1) (1991) 49–76. [5] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955). [6] S. Kusuoka, D.W. Stroock, Applications of the Malliavin calculus. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1) (1985) 1–76. [7] S. Kusuoka, D.W. Stroock, Applications of the Malliavin calculus. III, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (2) (1987) 391–442. [8] D. Lamberton, Optimal stopping and American options, Daiwa Lecture Ser., Kyoto, 2008, in press. [9] P. Malliavin, Stochastic Analysis, Springer, Berlin, 1997. [10] P.-A. Meyer, Un cours sur les intégrales stochastiques, in: Séminaire de Probabilités X, in: Lecture Notes in Math., Springer, Berlin, 1976, pp. 246–354. [11] H.H. Schaefer, Topological Vector Space, Grad. Texts in Math., Springer, Berlin, 1970. [12] H. Sugita, Positive generalized Wiener functions and potential theory over abstract Wiener spaces, Osaka J. Math. 25 (3) (1988) 665–696. [13] A.S. Üstünel, Extension of the Itô Calculus via the Malliavin Calculus, Stochastics 23 (1988) 353–375. [14] A.S. Üstünel, Representation of the distributions on Wiener space and stochastic calculus of variations, J. Funct. Anal. 70 (1987) 126–139. [15] A.S. Üstünel, Introduction to Analysis on Wiener Space, Lecture Notes in Math., vol. 1610, Springer, Berlin, 1995. [16] A.S. Üstünel, Analysis on Wiener space and applications, electronic text at the site, http://www.finance-research. net/.
Journal of Functional Analysis 256 (2009) 3107–3142 www.elsevier.com/locate/jfa
Berezin–Toeplitz quantization and composition formulas Wolfram Bauer 1 Ernst-Moritz-Arndt-Universität Greifswald, Institut für Mathematik und Informatik, Jahnstrasse 15a, 17487 Greifswald, Germany Received 5 November 2007; accepted 1 October 2008
Communicated by L. Gross
Abstract Extending results in [L.A. Coburn, The measure algebra of the Heisenberg group, J. Funct. Anal. 161 (1999) 509–525; L.A. Coburn, On the Berezin–Toeplitz calculus, Proc. Amer. Math. Soc. 129 (11) (2001) 3331–3338] we derive composition formulas for Berezin–Toeplitz operators with i.g. unbounded symbols in the range of certain integral transforms. The question whether a finite product of Berezin–Toeplitz operators is an operator of this type again can be answered affirmatively in several cases, but there are also well-known counter examples. We explain some consequences of such formulas to C ∗ -algebras generated by Toeplitz operators. © 2008 Published by Elsevier Inc. Keywords: Berezin–Toeplitz operator; Heat equation; Berezin transform; Star product
1. Introduction Let H 2 (Cn , μ) be the Segal–Bargmann space of all entire functions on Cn which are square integrable with respect to some Gaussian measure μ, cf. [1]. Via the orthogonal projection P from L2 (Cn , μ) onto H 2 (Cn , μ) the class of Berezin–Toeplitz operators (shortly: Toeplitz operators) Tf on the Segal–Bargmann space arises naturally as Tf := P Mf where Mf denotes the pointwise multiplication by a suitable measurable, complex valued symbol f . E-mail address:
[email protected]. 1 Partially supported by the Grant-in-aid Scientific Research (C) No. 17540202, Japan Society for the Promotion of
Science and by an Emmy-Noether grant of Deutsche Forschungsgemeinschaft. 0022-1236/$ – see front matter © 2008 Published by Elsevier Inc. doi:10.1016/j.jfa.2008.10.002
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It is well known, cf. [15], that through the Bargmann transform β which maps L2 (Rn ) isometrically onto H 2 (Cn , μ) the operator Tf is unitary equivalent to a pseudo-differential operator Wσ (f ) on L2 (Rn ) in its Weyl form. In this correspondence the operator symbols f and σ (f ) are related via the heat equation. More precisely, with the usual identification R2n ∼ = Cn the symbol σ (f ) turns out to be the solution at a certain time evolving from the initial data f . The Toeplitz operators with polynomial symbols correspond to partial differential operators with polynomial coefficients and therefore form an algebra of unbounded Toeplitz operators. In general, if the polynomials are replaced by another symbol space S (e.g. S = L∞ (Cn )), it turns out that {Wσ (f ) : f ∈ S} only forms a linear subspace in the algebra of pseudo-differential operators. Therefore the question remains open, whether for fj ∈ S the operator product Tf1 Tf2 · · · Tfm is a Toeplitz operator again and what its symbol would be. It is equivalent to ask in which cases the product Wσ (f1 ) · · · Wσ (fm ) of pseudo-differential operators has a total symbol that arises as the solution of the heat equation with some initial data f at a certain fixed time t0 > 0 and to determine f . To obtain necessary and sufficient conditions in the most general situation of just measurable functions seems to be quite challenging since—roughly speaking—this would require a time reverse in the heat flow. The first approach to such questions in the case of two operators Tf and Tg seems to be [10] where the author has proved that for Fourier transforms f and g of compactly supported measures the product Tf Tg is a Toeplitz operator with symbol h. Furthermore, the function h can be calculated as a certain product h = f g in form of an infinite series involving higher order derivatives of f and g. Subsequently in [11], the corresponding product formula has been extended to polynomial symbols (i.e. unbounded operators) which turn into an associative unital algebra under the -product. There are various ways to extend the -product to spaces of real analytic or even smooth functions and to determine function -algebras. However, it is not clear which of these extensions lead to composition formulas for the corresponding Toeplitz operators. In the present paper we derive several spaces of smooth symbols S such that for f1 , f2 ∈ S the product f1 f2 exists and there is a composition formula Tf1 Tf2 = Tf1 f2 of Toeplitz operators. Since we are dealing with unbounded symbols in general we restrict ourselves to cases where the (unbounded) operators Tfj , j = 1, 2, act on a scale of Banach spaces by a finite order shift. Operators of such kind form an algebra L and the composition of unbounded operators is well defined. Theorem A below is a (simplified) version of Theorem 16 in the text and it provides a typical example of our results. Here Sym>0 (Cn ) is a space of measurable functions having certain growth at infinity (for a precise definition see (20)): Theorem A. Let f, g ∈ Sym>0 (Cn ). Denote by f˜ and g˜ the solutions of the heat equation with initial data f and g at a (suitable fixed) time t0 > 0, respectively. Then the product Tf˜ Tg˜ ∈ L is a Toeplitz operator with symbol f˜ g˜ ∈ Sym>0 (Cn ). It is not an easy task to fully characterize the range of the Berezin transform B which injectively maps L into the real analytic functions on Cn . Necessary conditions for a real analytic function to be the Berezin transform of a bounded operator have been given recently in [12] and the papers cited therein. Restricted to functions, B can be seen as the correspondence between the initial data and the solution of the heat equation at a certain time. Therefore Theorem A is a result on symbols in the range of B (see also [16,17]). As an integral transform on i.g. unbounded functions with an explicitly given inverse we employ the Fourier–Wiener transform W to derive composition formulas for (i.g. unbounded) Toeplitz operators similar to Theorem A. However, in some applications W may be easier to
W. Bauer / Journal of Functional Analysis 256 (2009) 3107–3142
3109
handle than B. In the case of bounded symbols and operators we also determine -algebras via the Fourier transform and multiple operator products can be treated. Let Tvo be the linear space of Toeplitz operators with symbols of vanishing oscillation at infinity. We write Cvmo for the C ∗ -algebra generated by Toeplitz operators with symbols of vanishing mean oscillation at infinity. As an application of Theorem A we show that the inclusion of quotient spaces Tvo /K → Cvmo /K is dense where K denotes the closed ideal of compact operators on H 2 (Cn , μ). The topics we deal with in this paper combine aspects of Fourier analysis, function theory, and operator theory on reproducing kernel Hilbert spaces and the methods we apply have sources from all these directions. In Section 2 we introduce our notations and prove an integral form of the -product in [11] on polynomials. As an application we derive extensions of this product to spaces of real analytic functions under some integrability conditions. We also extend the -product to a class of smooth functions by posing growth conditions on the sequence of higher derivatives in each point (i.e. local conditions). Typically, these conditions ensure real analyticity. In Section 3 we prove composition formulas for Toeplitz operators acting on a scale of Banach spaces by using various integral transforms. We explain a relation between the -product and a star product that arises in deformation quantization. Moreover, we give an application of our results to C ∗ -algebras generated by Toeplitz operators with symbols of vanishing mean oscillation. 2. Preliminaries In the present section we introduce some basic notations that will be used throughout the text. With k, n ∈ N we write P[z1 , z1 , . . . , zk , zk ] for the space of polynomials on Cnk having coefficients in C and in the complex variables zj = (z1,j , . . . , zn,j )t and zj = (z1,j , . . . , zn,j )t where j = 1, . . . , k. By zi,j we mean the complex conjugation of zi,j . For fixed m ∈ N0 we denote by Pm [z1 , z1 , . . . , zk , zk ] the polynomials of degree m. Not all variables have to appear, e.g. P[z1 , z2 ] would be the space of holomorphic polynomials of 2n variables zi,j where i = 1, . . . , n and j = 1, 2. For k complex-valued functions fl defined on spaces Xl and with xl ∈ Xl where l = 1, . . . , k we set f1 ⊗ · · · ⊗ fk (x1 , . . . , xk ) := f1 (x1 )f2 (x2 ) · · · fk (xk ). For any dimension q ∈ N let dv(w) = (i/2)q dw1 ∧ dw1 ∧ · · · ∧ dwq be the Lebesgue measure on Cq ∼ = R2q . Given t > 0 we denote by μt a normed Gaussian measure on Cq : 1 |w|2 dv(w). exp − dμt (w) := (tπ)q t For√u, v ∈ Cq with components uj and vj we write uv := u1 v1 + · · · + uq vq . Moreover, |u| := uu stands for the usual Euclidean norm on Cq . Let H 2 (Cq , μt ) be the Segal–Bargmann space of all μt -square integrable entire functions on q C (see [1]). It is well known that H 2 (Cq , μt ) is a closed subspace of L2 (Cq , μt ) having an q orthonormal basis [(t |α| α!)−1/2 zα : α ∈ N0 ] with respect to the L2 -inner product:
f, g t :=
f (z)g(z) dμt (z), Cq
f, g ∈ H 2 Cq , μt .
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We shortly write ·,· := ·,· 1 . Let · t and · := · 1 denote the norms induced by ·,· t and ·,· , respectively. The spaces H 2 (Cq , μt ) are equipped with a reproducing kernel zu . Kt (z, u) := exp t
Kt : Cq × Cq → C:
More precisely, the evaluation functionals for each z ∈ Cq are continuous and given by an integral with respect to Kt : for f ∈ H 2 (Cq , μt ), zu dμt (u). f (u) exp t
f (z) = Cq
(1)
Formula (1) will be used frequently in the calculations below, mostly without being cited. For a smooth function h ∈ C ∞ (Cn ) and α = (α1 , α2 ) ∈ Nn0 × Nn0 we shortly write for the higher order Wirtinger derivatives: h[α1 ,α2 ] :=
∂ |α| h . ∂zα1 ∂ z¯ α2
In the first part of the present paper an associative non-commutative product introduced in [10,11] is extended from polynomials to various function spaces. We recall the definition: Definition 1 (-Product). Let p, q ∈ P[z, z] and z ∈ Cn , then we set: [p q](z) :=
(−1)|γ | p [γ ,0] (z)q [0,γ ] (z). γ ! n
(2)
γ ∈N0
To p ∈ P[z1 , z1 ] we can assign a holomorphic polynomial A(p) on C2n by replacing z1 with an independent complex variable z2 := (z1,2 , . . . , zn,2 )t . Clearly, A defines an isomorphism between P[z1 , z1 ] and the space of holomorphic polynomials P[z1 , z2 ]. Via A and for P , Q ∈ P[z1 , z2 ] the -product on P[z1 , z1 ] defines a second product P A Q on P[z1 , z2 ]. With this notation we obtain a commutative diagram: P[z1 , z2 ] × P[z1 , z2 ] A
J
P[z1 , z2 , z3 , z4 ]
T :=R◦L
P[z1 , z2 ] where [J (P , Q)] := P ⊗ Q. Here the operator L is defined by L :=
(−1)|γ | ∂ 2|γ | γ ! ∂z1γ ∂z4γ n
γ ∈N0
W. Bauer / Journal of Functional Analysis 256 (2009) 3107–3142
3111
and for F ∈ P[z1 , z2 , z3 , z4 ] we set [RF ](z1 , z2 ) := F (z1 , z2 , z1 , z2 ). Applied to a fixed element, L involves a finite number of differentiations only. Moreover, it preserves the degree of polynomials. The composition T = R ◦ L can be rewritten as an integral operator in an explicit form. It is readily verified that the formal adjoint of the derivative ∂/∂zi,j for i = 1, . . . , n and j = 1, . . . , 4, with respect to the inner-product of H 2 (C4n , μ1 ) is given by ∂z∗i,j := zi,j − ∂/∂zi,j . Therefore, by using (1) and with the notation w := (z1 , z2 , z1 , z2 ) ∈ C4n it follows that [T F ] (z1 , z2 ) = LF, K1 (·, w) = F, L∗ K1 (·, w)
=:v
where with ∂u∗j := (∂u∗1,j , . . . , ∂u∗n,j ) for j = 1, . . . , 4, the integral kernel on the right-hand side has the form: (−1)|γ | γ γ ∗ ∂u∗4 ∂u∗1 exp z1 (u1 + u3 ) + z2 (u2 + u4 ) L K1 (·, w) (u1 , u2 , u3 , u4 ) = γ! n γ ∈N0
= exp −u1 u4 + z1 (u1 + u3 ) + z2 (u2 + u4 ) .
(3)
Using this factorization of A , one can write the -product in an integral form. Proposition 2. For z1 ∈ Cn and p, q ∈ P[z1 , z1 ], [p q](z1 ) =
[Ap ⊗ Aq](u1 , z1 , z1 , u4 )K(u1 , u4 ; z1 , z1 ) dμ1 (u1 , u4 )
(4)
C2n
and the kernel K is given by K(u1 , u4 ; z1 , z1 ) = exp{−u1 u4 + z1 u1 + z1 u4 }. Proof. Pulling back the A -product to P[z1 , z1 ] where in the integral below we write u := (u1 , u2 , u3 , u4 ) ∈ C4n gives: [p q](z1 ) = A−1 T (Ap ⊗ Aq) (z1 ) = Ap ⊗ Aq(u) exp −u1 u4 + z1 (u1 + u3 ) + z1 (u2 + u4 ) dμ1 (u). C4n
The integration over u2 and u3 can be carried out using the reproducing kernel property (1) and the assertion follows. 2 Remark 3. By a second application of (1), the -product (4) even can be reduced to an integration over Cn . One possible expression would be: [p q](z1 ) =
[Ap ⊗ Aq](u1 , z1 , z1 , z1 − u1 ) exp{z1 u1 } dμ1 (u1 ). Cn
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We write P∞ [z1 , z1 ] for the space of real analytic functions f defined on Cn , that can be written as a power series: γ f (z1 ) := cγ1 ,γ2 z11 z1 γ2 (5) γ1 ,γ2 ∈Nn0
with infinite radius of convergence. In the following we denote by H(Cn ) the Fréchet space of all entire functions on Cn equipped with the topology of uniformly compact convergence. Using the same notation as for polynomials, (5) defines a function Af ∈ H(C2n ) by γ γ [Af ](z1 , z2 ) := cγ1 ,γ2 z11 z22 . γ1 ,γ2 ∈Nn0
We define a sequence of normed subspaces Hk (Cn ) ⊂ H(Cn ) for k ∈ N by an exponential growth condition at infinity. With αk := 1/2 − 1/k consider 2 Hk Cn := f ∈ H Cn : f k := supf (z)e−αk |z| < ∞ .
(6)
The inclusion maps (Hk (Cn ), · k ) → H(Cn ) are continuous and since H(Cn ) is complete, (6) defines an increasing scale of Banach spaces. We set H∞ Cn := (7) Hk Cn ⊂ H 2 Cn , μ1 k∈N
and (7) is equipped with the topology of an inductive limit corresponding to the continuous maps Hk (Cn ) → Hk+1 (Cn ). Now, we can describe an extension of the -product to a suitable space of real analytic functions. To f ∈ P∞ [z1 , z1 ] we assign two maps Ji [f ] : Cn → H(Cn ), where i = 1, 2, given by J1 [f ](z) := [Af ](·, z) and J2 [f ](z) := [Af ](z, ·). Let Ji [f ](Cn ) denote the range of Ji [f ] and consider the spaces Si := f ∈ P∞ [z1 , z1 ]: ∃ k ∈ N such that Ji [f ] Cn ⊂ Hk Cn . Corollary 4. (4) extends to a bilinear map ˜ : S1 × S2 → P∞ [z1 , z1 ]. Proof. For (f, g) ∈ S1 × S2 choose k ∈ N such that J1 [f ](Cn ) and J2 [g](Cn ) are subsets of Hk (Cn ). Hence we can fix dz1 > 0 depending only on z1 ∈ Cn with [Af ](·, z1 ) + [Ag](z1 , ·) dz . 1 k k By the definition of the · k -norm it therefore follows: [Af ⊗ Ag](u1 , z1 , z1 , u4 ) dz exp αk |u1 |2 + |u4 |2 . 1 Note that the kernel K in Proposition 2 can be estimated by 2 2 K(u1 , u4 ; z1 , z1 ) exp Re(z1 u1 + z1 u4 ) + |u1 | + |u4 | . 2 2
(8)
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Since αk < 1/2 the integral on the right-hand side of (4) converges if p and q are replaced by f and g. 2 Example 5. Using Remark 3 with z1 = 0 and p ∈ P[z1 , z1 ] shows the integral formula: [p p](0) =
[Ap](u1 , 0)[Ap](0, −u1 ) dμ1 (u1 ). Cn
For f ∈ P∞ [z1 , z1 ] the conditions (∗): [Af ](·, 0), [Af ](0, ·) ∈ H 2 (Cn , μ1 ) would be sufficient to ensure the convergence of f f in z = 0. Note that (∗) is weaker than the assumption f ∈ S1 ∩ S 2 . A look at the initial definition (2) shows, that the -product may extend from polynomials to suitable spaces of smooth functions h ∈ C ∞ (Cn ). However, instead of a finite sum, one has to deal with an infinite series and convergence has to be ensured. Growth conditions at infinity of the above kind have to be replaced by suitable assumptions on the growth of the sequence of nth derivatives of h in each single point. Consider the function space F (Cn ) which by definition consists of smooth functions f ∈ ∞ C (Cn ) such that there are Cf (z) > 0 and Df (z) > 0 depending on z ∈ Cn with [α ,α ] f 1 2 (z) Cf (z) · Df (z)|α| for all α := (α1 , α2 ) ∈ Nn0 × Nn0 and z ∈ Cn . If Cf (z) and Df (z) can be chosen continuously, it follows from the Taylor formula that f is a real analytic function. Given an arbitrary set X ⊂ Cn a corresponding subspace FX (Cn ) ⊂ F (Cn ) is defined by FX Cn := f ∈ F Cn : sup Cf (z) + Df (z) < ∞ . z∈X
Clearly, P[z, z¯ ] ⊂ FX (Cn ) for compact sets X ⊂ Cn and z = (z1 , . . . , zn ). The -product (2) extends from P[z, z] to f, g ∈ F (Cn ) as an i.g. infinite and pointwise convergent series: [f g](z) =
(−1)|γ | f [γ ,0] (z) · g [0,γ ] (z). γ! n
(9)
γ ∈N0
Lemma 6. The series (9) converges in z ∈ Cn . With α = (α1 , α2 ) ∈ Nn0 × Nn0 : {f g}[α1 ,α2 ] (z) Cf (z)Cg (z)enDf (z)Dg (z) Df (z) + Dg (z) |α| .
(10)
For X ⊂ Cn all spaces FX (Cn ) are unital -algebras and with k ∈ N: [α1 ,α2 ] |α| k(k − 1) k 2 nDg (z) · kDg (z) . (z) Cg (z) exp (g g) · · · g
2 k times
(11)
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Proof. Clearly e ≡ 1 is the unit in FX (Cn ). For α = (α1 , α2 ) ∈ Nn0 × Nn0 and functions f, g ∈ F (Cn ): {f g}[α1 ,α2 ] (z) 1 n γ! γ ∈N0
ρ=(ρ1 ,ρ2 )α
α [γ +ρ1 ,ρ2 ] f (z) · g [α1 −ρ1 ,α2 +γ −ρ2 ] (z) ρ
|α| Cf (z) · Cg (z)enDf (z)·Dg (z) Df (z) + Dg (z) .
(12)
We prove (11) by induction with respect to k. The cases k = 1, 2 are clear from the definition and using the above calculation. The implication k → k + 1 follows by replacing f with g · · · g (k times) in (12). 2 In some of the examples below we prove associativity of the -product by using an integral form for multiple -products such as (34) in Lemma 21 or a relation to operator composition. Here we give some further examples of -algebras consisting of real analytic functions and containing elements which not necessarily are in the spaces Sj defined above Corollary 4. We fix a polynomial of degree m,
p(z) :=
aγ1 ,γ2 zγ1 z¯ γ2 ∈ Pm [z, z¯ ],
(13)
|(γ1 ,γ2 )|m
with
|(γ1 ,γ2 )|=m aγ1 ,γ2 z
γ1 z¯ γ2
= 0. Set λ(z) :=
Fp (z) :=
1 + |z|2 and assign to p the function
|aγ1 ,γ2 | · λ|(γ1 ,γ2 )| (z).
|(γ1 ,γ2 )|m
Nn0
With the definition c(p) := inf{c > 0: Fp (z) cλ(z)m , ∀z ∈ Cn } and for α = (α1 , α2 ) ∈ × Nn0 one readily verifies that c p [α1 ,α2 ] c(p)
(m!)2 (m − α1 )!(m − α2 )!
(14)
where m := (m, . . . , m) ∈ Nn0 . In the case of polynomials, where only a finite number of derivatives are not vanishing we can prove an estimate finer than (11). Let qk := p · · · p (a k-times product), then it holds: Proposition 7. For k ∈ N, c(qk ) 2mn(k−1) c(p)k {[(k − 1)m]!}n . Proof. The assertion is valid for k = 1. It follows from (14) that F
[α1 ,0]
qk
(z) c(qk )λ(z)km−|α1 |
km! (km − α1 )!
where km := (km, . . . , km) ∈ Nn0 and similarly for Fp[0,α2 ] (z). Using the definition of the -product,
W. Bauer / Journal of Functional Analysis 256 (2009) 3107–3142
Fqk+1 (z)
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1 F [γ ,0] (z) · Fp[0,γ ] (z) γ ! qk
|γ |m
c(qk )c(p)λ(z)(k+1)m
m km! γ (km − γ )!
|γ |m
c(qk )c(p)λ(z)(k+1)m
km! 2nm . (km − m)!
Now, the assertion follows by induction with respect to k.
2
We return to the case of non-polynomial functions and with p(z) as in (13) we consider the exponentials Ep (z) := ep(z) . Given α = (α1 , α2 ) ∈ Nn0 × Nn0 we set
Iα :=
n n (ρ1 , τ1 ), . . . , (ρ , τ ) : 0 = (ρj , τj ) ∈ N0 × N0 and (ρj , τj ) = α, ∈ N, |α| . j =1
Lemma 8. For α = (α1 , α2 ) ∈ Nn0 × Nn0 Ep[α1 ,α2 ] (z) = Ep (z)
p [ρ1 ,τ1 ] (z) · · · p [ρ ,τ ] (z).
(15)
Iα
Proof. We use induction with respect to |α| = |α1 | + |α2 |. In case |α| = 1 we can assume that |α1 | = 1 and α2 = 0. Then Iα = {(α)} and formula (15) is clear. As for the implication |α| → |α| + 1 we may assume that β = (β1 , α2 ) = α + e1 where e1 := (1, 0, . . . , 0) ∈ N2n 0 . Then
Ep[β1 ,α2 ] (z) = Ep (z)
Iα
p [ρ1 +δ1,k e1 ,τ1 ] (z) · · · p [ρ +δ ,k e1 ,τl ] (z)
k=1
+ p [ρ1 ,τ1 ] (z) · · · p [ρ ,τ ] (z)p [e1 ,0] (z) = Ep (z)
p [ρ1 ,τ1 ] (z) · · · p [ρ ,τ ] (z).
2
Iβ
Combining (14) and Lemma 8 it follows: Corollary 9. For α = (α1 , α2 ) ∈ Nn0 × Nn0 there is dp > 0 only depending on p ∈ Pm [z, z] where z = (z1 , . . . , zn ) ∈ Cn such that [α ,α ] E 1 2 (z) Ep (z) dp λ(z)m−1 |α| . p
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In particular, it follows that Ep ∈ FK (Cn ) for all compact sets K ⊂ Cn . Moreover, the kth -product can be estimated by k k(k − 1) 2 ndp λ(z)2(m−1) . (Ep Ep ) · · · Ep (z) Ep (z) exp
2 k times
Proof. Inserting (14) into (15) yields with m := (m, . . . , m) ∈ Nn0 : [α ,α ] E 1 2 (z) p Ep (z)λ(z)|α|(m−1) c p [ρ1 ,τ1 ] · · · c p [α ,τ ] Iα
|α| Ep (z)λ(z)|α|(m−1) c(p) + 1 ×
Iα
(m!)2 . (m − ρ1 )!(m − τ1 )! · · · (m − ρ )!(m − τ )!
The sum can be estimated by [m!e]2n|α| since it holds |α|. Therefore, with dp := [c(p) + 1][m!e]2n the first estimate follows. The second inequality is a direct consequence of (11). 2 In general, the functions Ep do not define elements in the spaces Sj , e.g. choose n = 1 and p(z) := z3 . Using Lemma 6 and Corollary 9 it follows that {Ep : p ∈ P[z, z]} generates a unital -sub-algebra of FK (Cn ) for all compact subsets K ⊂ Cn . 3. Composition formulas for Berezin–Toeplitz operators In Section 3 we embed the -product into a family of products t parametrized by the “time” t > 0 such that 1 = . Then we prove composition formulas of (i.g. unbounded) Toeplitz operators acting on a scale of Banach spaces. Some special cases have been treated before in [10,11]. Moreover, a relation to a star-product in deformation quantization (cf. [7,9,13]) is explained. We write M(Cn ) for the space of measurable complex-valued functions on Cn . As usual, ∞ (L (Cn ), · ∞ ) means the essentially bounded functions. For c ∈ R and generalizing the definition of H∞ (Cn ) before we set (16) Dc := f ∈ M(Cn ): ∃d > 0 such that f (z) d exp c|z|2 a.e. . We equip (16) with the norm f Dc := exp(−c| · |2 )f ∞ turning (Dc , · Dc ) into a Banach space. If c < 1/2t one can check that the embedding It : Dc , · Dc → L2 Cn , μt is continuous with a norm estimate It (1 − 2ct)−n/2 . For a fixed t > 0, any sequence (cj (t))j ∈N0 in R+ 0 which is strictly increasing in j ∈ N0 with c0 (t) = 0 and limj →∞ cj (t) = 1/2t defines an increasing scale of Banach spaces: L∞ Cn = Dc0 (t) ⊂ Dc1 (t) ⊂ · · · ⊂ Dt := Dcj (t) ⊂ L2 Cn , μt . j ∈N
(17)
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In order to avoid difficulties related to the composition of unbounded operators, we only consider such acting on a scale of type (17). For normed spaces X and Y let L(X, Y ) and L(X, Y ) be the linear and bounded linear operators from X to Y , respectively. As usual we set L(X) := L(X, X) and L(X) := L(X, X). With k, j ∈ N0 we define Lkj (Dcj (t) , Dcj +k (t) ) := A ∈ L Dt : A|Dc
j (t)
∈ L(Dcj (t) , Dcj +k (t) ) .
The operators acting on (17) of order-shift k are given by Ot (k) :=
Lkj (Dcj (t) , Dcj +k (t) ).
j ∈N0
Finally, the algebra of operators acting on (17) by a finite order-shift is defined as Lfos Dt := Ot (k). k∈N0
Let Pt be the orthogonal projection from L2 (Cn , μt ) onto H 2 (Cn , μt ). We write P := P1 and remark that Pt can be expressed as an integral operator [Pt g](z) = g, Kt (·, z) t where g ∈ L2 (Cn , μt ). From now on we choose the sequence (cj (t))j above in a certain way. Inductively, we define: c0 (t) := 0 and cj +1 (t) :=
1 4[t − t 2 cj (t)]
(18)
which has the explicit form cj (t) := (2t)−1 − (2tj + 2t)−1 . Thus (cj (t))j fulfills the above assumptions and it defines a scale (17) for each t > 0. We show that the projection Pt acts on (17) of order shift 1 (cf. [3,4]). Lemma 10. It holds Pt ∈ Ot (1). Moreover, for j ∈ N0 and g ∈ Dcj (t) , Pt g Dcj +1 (t)
1 g Dcj (t) . [1 − tcj (t)]n
(19)
Proof. It is sufficient to prove (19). Using the integral expression for Pt together with the transformation rule shows: [Pt g](z) g(x) exp z · x¯ dμt (x) t Cn
g Dcj (t) (tπ)n
Cn
1 Re z · x¯ − − cj (t) |x|2 dv(x) exp t t
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=
g Dcj (t) [1 − tcj (t)]n g Dcj (t) [1 − tcj (t)]n
Cn
exp
Re z · x¯ t − t 2 cj (t)
exp{cj +1 (t)|z|2 }.
dμ1 (x)
2
In order to define Toeplitz operators acting on the scale (17) we restrict ourselves to the following space of (i.g. unbounded) symbols: ∞ D1/j Sym>0 Cn :=
(20)
j =1
which clearly is a ∗-algebra under pointwise multiplication and with respect to the complex conjugation. Furthermore, for f ∈ Sym>0 (Cn ) and all t > 0 one easily checks that the pointwise multiplication Mf defines an element in Ot (1). Due to Lemma 10 it follows: Pt Mf ∈ Ot (2) ⊂ Lfos Dt and the restriction of this operator product to Dt ∩H 2 (Cn , μt ) is called Berezin–Toeplitz operator (shortly: Toeplitz operator) with symbol f . We denote the Berezin–Toeplitz operator by Tf(t) . In particular, since Lfos (Dt ) is an algebra, arbitrary products of Toeplitz operators with symbols in (20) are well defined. (t) The assignment f → Tf which maps functions to operators can be seen as a quantization. The existence of a reproducing kernel allows us to define a family of Berezin transforms (Bt )t>0 as some kind of “inverse quantizations”: For z, u ∈ Cn and t > 0 we write kzt ∈ Dc1 (t) for the normalized reproducing kernel of the space H 2 (Cn , μt ). Using (1) it follows: kzt (u) :=
Kt (u, z) u · z¯ |z|2 − . = exp Kt (·, z) t t 2t
The Berezin transform Bt , cf. [6], maps Lfos (Dt ) into the space C ω (Cn ) of real analytic functions on Cn . For A ∈ Lfos (Dt ) it is defined by Bt [A](z) := Akzt , kzt t .
(21)
Moreover, for f ∈ Sym>0 (Cn ) we write (t) f (t) (z) = Bt [f ] := Bt Tf (z) = f kzt , kzt t
(22)
and we call (22) the Berezin transform of f with respect to the time-parameter t. Remark 11. Restricted to functions one can compute that up to a constant Bt is the heat operator exp(t/4). In particular, it fulfills the semi-group property {f (s1 ) }(s2 ) = f (s1 +s2 ) with s1 , s2 > 0.
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3119
By intersecting the scale (17) with H 2 (Cn , μt ) we obtain a second scale of Banach spaces in H 2 (Cn , μt ): C∼ = Hc0 (t) ⊂ Hc1 (t) ⊂ · · · ⊂ Ht :=
Hcj (t) ⊂ H 2 Cn , μt
(23)
j ∈N
where we set Hcj (t) := Dcj (t) ∩ H 2 (Cn , μt ) equipped with the norm · Dcj (t) . Completely analogous to our definition above we can consider the algebra: Lfos Ht ⊂ L Ht of all operators that are acting on (23) by a finite order shift. Via (21) and using kzt ∈ Hc1 (t) for all z ∈ Cn the Berezin transform Bt [A] is defined for all operators A ∈ Lfos (Ht ). Clearly, the (t) algebra Lfos (Ht ) contains all Toeplitz operators Tf having symbols in Sym>0 (Cn ) and as a crucial property we remark: Lemma 12. The Berezin transform Bt is one-to-one on Lfos (Ht ). Proof. For fixed t > 0 and with the sequence (cj (t))j defined in (18) we choose numbers κj > 0 with j ∈ N0 such that cj (t) <
1 1 < cj +1 (t) < 2κj 2t
∀j ∈ N0 .
Corresponding to each κj we have a Hilbert space Hj := H 2 (Cn , μκj ) of entire functions. Clearly, the embedding (Hcj (t) , · Dcj (t) ) → Hj is well defined and continuous. With f ∈ Hj and z ∈ Cn it holds: 2 f (z) = f, Kκ (·, z) f κ exp |z| j j κj 2κj and one also has the continuous embedding Hj → (Hcj +1 (t) , · Dcj +1 (t) ). Therefore, we obtain the following scale of spaces with continuous embeddings: C = H0 ⊂ H0 ⊂ Hc1 (t) ⊂ · · · ⊂ Hcj (t) ⊂ Hj ⊂ Hcj +1 (t) ⊂ · · · ⊂ Ht . As a consequence, operators in Lfos (Ht ) also act on the scale of Hilbert spaces below: H0 ⊂ H1 ⊂ · · · ⊂ Hj ⊂ Hj +1 ⊂ · · · ⊂ H∞ :=
∞
Hj = Ht
(24)
j =0
by a finite order shift. Note, that X := span{kzt : z ∈ Cn } is a dense subspace of Hj for all j ∈ N0 . Let A ∈ Lfos (Ht ) and assume that for all z ∈ Cn : Bt [A](z) = Akzt , kzt t = 0
⇒
AKt (·, z), Kt (·, z) t = 0.
(25)
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For fixed j ∈ N0 the monomials [eγ (z) := (κj γ !)−1/2 zγ : γ ∈ Nn0 ] form an orthonormal basis of Hj . Moreover, we have: (j )
! " |γ | 1 uγ zγ zγ " # κj (j ) e (u). Kt (u, z) = = |γ | |γ | γ! γ n γ! t n t γ ∈N0
(26)
γ ∈N0
Since for all z ∈ Cn it holds 1 |¯zγ |2 κj|γ | |z|2 κj = exp{ } < ∞, t 2|γ | t2 n γ!
γ ∈N0
we find that the right-hand side of (26) considered as a function of u is convergent in Hj for all j ∈ N0 . By assumption, the operator A acts on the scale (24). Hence for each j ∈ N0 there is
j ∈ N0 such that A : Hj → H j is continuous:
AKt (·, z) =
zγ |γ | n t
γ ∈N0
! " |γ | "κ # j Aeγ(j ) ∈ H j . · γ!
In particular, the assignment Cn z → AKt (·, z) ∈ H j → H 2 Cn , μt defines an anti-holomorphic Hilbert-space-valued function. Hence the map Cn z → AKt (·, z), Kt (·, z) t is the restriction of an entire function in the variables z and w to the diagonal (z, z). Now, we can apply [14, Proposition (1.69)] and it follows from (25), that:
AKt (·, z), Kt (·, w) t = 0,
for all z, w ∈ Cn .
Since X is dense in H 2 (Cn , μt ) we have AKt (·, z) = 0 for all z ∈ Cn and therefore the restriction of A to X vanishes. Since A : Hj → H j is continuous and X ⊂ Hj is a dense subspace we conclude that the restriction of A to Hj vanishes for all j . This finally implies that A = 0 on $ Ht = ∞ H j =0 j . 2 Example 13. Let t = 1 and fix p = |γ1 |+|γ2 |k aγ1 ,γ2 zγ1 zγ2 ∈ Pk [z, z¯ ] where k ∈ N and z = (z1 , . . . , zn ). By a straightforward calculation the Berezin transform p (1) of p is a polynomial having the same principal part: p (1) (z) =
|γ1 |+|γ2 |=k
aγ1 ,γ2 zγ1 zγ2 + r(z),
where r ∈ Pk−1 [z, z].
(27)
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In particular, B1 is an isomorphism of P[z, z] leaving Pk [z, z] invariant. It follows from (27) that (I − B1 )k+1 identically vanishes on Pk [z, z]. Therefore, the inverse of B1 restricted to Pk [z, z] can be written as the sum: B1−1 =
j k k j (−1) B1 . (I − B1 )j =
j =0
j =0 =0
For ∈ N the function B1 p = p ( ) coincides with the solution of the heat equation on Cn with initial data p at the time t0 where t0 > 0 is fixed. Here 1 |u − z|2 dv(u). p(u) exp − p ( ) (z) = ( π)n
Cn
Restricted to Pk [z, z], the inverse B1−1 of the Berezin transform can be written as an integral operator with respect to the Lebesgue measure v having the kernel j k 1 j (−1) |u − z|2 . exp − Lk (u, z) = n
π
n
j =0 =0
For α = (α1 , α2 ) ∈ Nn0 × Nn0 and t > 0 consider Bα,t ∈ P|α| [u, u, z, z] defined by Bα,t (u, z) :=
γ α1 ,α2
(−1)|γ | α1 α2 (u − z)α2 −γ (u¯ − z¯ )α1 −γ . γ! |α|−|γ | t γ γ
For f ∈ Sym>0 (Cn ) it can be checked by a straightforward calculation that (t) [α1 ,α2 ] f (z) = f Bα,t (·, z)kzt , kzt t .
(28)
Now, we embed the -product in (9) into a family t of products parametrized by the time t > 0 such that = 1 . We define for all f, g ∈ C ∞ (Cn ) such that the series below converges, ∞ (−t)|γ | [γ ,0] f (z) · g [0,γ ] (z). [f t g](z) := γ! n
(29)
γ ∈N0
Lemma 14. For t, s > 0 and f, g ∈ Dc where 0 < c < 1/t − s/2t 2 (in particular that means t > 2s ) the product f (t) s g (t) exists and has an integral form (t) 2 2 f s g (t) (z) = e− t |z| f ⊗ g(w) exp qs,t (w, z) dμt (w) C2n
where w := (w1 , w2 ) ∈ Nn0 × Nn0 and qs,t (w, z) := −
s 2 (w1 − z)(w2 − z) + Re z(w1 + w2 ). 2 t t
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Proof. With γ ∈ Nn0 one has B(γ ,0),t (u, z) =
(u¯ − z¯ )γ t |γ |
and B(0,γ ),t (u, z) =
(u − z)γ . t |γ |
Therefore, it follows that (−s)|γ | (t) [γ ,0] (t) [0,γ ] f (t) f s g (t) (z) = (z) g (z) γ! n γ ∈N0
=
f ⊗ g(w)
γ ∈Nn0C2n
=
γ ∈Nn0C2n
2 (−s)|γ | (w1 − z)γ (w2 − z)γ t kz ⊗ kzt (w) dμt (w) |γ | |γ | γ! t t
γ √ √ (−s)|γ | w1 γ w2 f ⊗ g( tw1 + z, tw2 + z) dμ1 (w). γ! t |γ |
The series over the γ -dependent part is dominated by exp{s|w|2 /2t}. The first factor of the integrand in the last line can be estimated in w by d exp(ct|w|2 + ρ|w|) for suitable numbers d, ρ > 0. Hence a sufficient condition for the interchange of summation and integration is f, g ∈ Dc with 0 < c < 1/t − s/2t 2 . 2 Lemma 15. For s, t > 0 with t >
s 2
and for f, g ∈ Sym>0 (Cn ) it holds: f (t) s g (t) ∈ Sym>0 Cn .
In particular, Sym>0 (Cn ) is invariant under the Berezin transform Bt for all t > 0. Proof. Fix f, g ∈ Sym>0 (Cn ) and s, t > 0 with t > 2s . For 0 < ε 1 there is Cε > 0 such that for h ∈ {f, g}, h(z) Cε exp ε|z|2 .
(30)
With qs,t defined in Lemma 14 we use the decomposition Re qs,t (w1 , w2 , z) = ps,t (w1 , w2 , z) + rs,t (w2 , z) −
s 2 |z| t2
where ps,t and rs,t are the polynomials ps,t (w1 , w2 , z) :=
2 s Re w1 (z − w2 ) + Re{w1 z} 2 t t
and rs,t (w2 , z) =
2t + s Re{w2 z}. t2
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Using the integral form of f (t) s g (t) in Lemma 14 together with (30) one obtains the pointwise estimate: (t) f s g (t) (z) Cε − 2t+s |z|2 t2 e (tπ)2n
eps,t (w1 ,w2 ,z)+rs,t (w2 ,z)−(t
−1 −ε)|w|2
dv(w) = (∗).
C2n 1
After the transformation w1 → w1,ε := (t −1 − ε)− 2 w1 one has: (∗) =
Cε − 2t+s |z|2 e t2 n n n π t (1 − tε) ×
ers,t (w2 ,z)−(t
−1 −ε)|w |2 2
Cn
eps,t (w1,ε ,w2 ,z) dμ1 (w1 ) dv(w2 )
Cn
=
Cε − 2t+s |z|2 e t2 n n n π t (1 − tε)
×
e
rs,t (w2 ,z)+
1 | s (z−w2 )+z|2 −(t −1 −ε)|w2 |2 t−t 2 ε 2t
dv(w2 )
Cn 2
=
(2t+s)(s−2t+4t ε) 2 Cε |z| e 4t 3 (1−tε) n n n π t (1 − tε) (2t+s)(2t−2t 2 ε−s) 2 2 −s 2 Re(w2 z¯ )− 4t (1−tε) |w2 |2 4t 3 (1−tε) × e 2t 3 (1−tε) dv(w2 )
Cn 2
=
(2t+s)(s−2t+4t ε) 2 (4t 2 )n Cε |z| 4t 3 (1−tε) e [4t 2 (1 − tε)2 − s 2 ]n (2t+s)(2t−2t 2 ε−s) Re(w2 z) √ √ t 3 (1−tε) 4t 2 (1−tε)2 −s 2 dμ (w ) × e 1 2
Cn
=
(4t 2 )n Cε exp κs,t (ε)|z|2 2 n −s ]
[4t 2 (1 − tε)2
where κs,t (ε) :=
(2t + s) 4t 3 (1 − tε)[4t 2 (1 − tε)2
− s2] 2 × (2t + s) 2t − 2t 2 ε − s + 4t 2 (1 − tε)2 − s 2 s − 2t + 4t 2 ε .
One easily checks that κs,t (0) = 0 independently from t > f (t) s g (t) ∈
0<ε1
s 2
> 0 showing that
Dκs,t (ε) = Sym>0 Cn
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and the first assertion is proved. The invariance of Sym>0 (Cn ) under the Berezin transform follows now from f (t) = f (t) t 1. 2 3.1. Berezin transform and composition formulas Let t > 0. In the case of (suitable) measurable symbols m1 and m2 it is known that the product (t) (t) Tm1 Tm2 of two Toeplitz operators in general is not an operator of this type anymore, cf. the remark following (44) below. Moreover, given two bounded symbols b1 and b2 it may happen (t) (t) that the operator product Tb1 Tb2 is a (bounded) Toeplitz operator with unbounded symbol. These effects are highly related with the type of products on function spaces which we have studied so far. The aim of the next section is to determine spaces St of i.g. unbounded symbols which lead to composition formulas (t)
Ts(t) = Th Ts(t) 1 2
for s1 , s2 ∈ St and to clarify the relation between s1 , s2 and h. We study higher products of Toeplitz operators and give some applications of our results to Toeplitz C ∗ -algebras. In all the cases we consider here, St arises as the range of an integral transform on some suitable space. For certain classes of bounded or for polynomial symbols results of similar kind have been proved in [10,11] before and partly follow as special cases from our method. We start calculating the Berezin transform of a product of Toeplitz operators having symbols in the range of Bt . Let f, g ∈ Sym>0 (Cn ) and fix t > 2s > 0. By using the integral expressions of f (t) and g (t) one obtains: (s) (s) Bs Tf (t) Tg (t) (z) = f (t) Ps g (t) kzs , kzs s 2 2 = f ⊗ g(u3 , u4 )kut 1 (u3 ) kut 2 (u4 ) C2n C2n
× kzs (u2 )kzs (u1 )e
u1 u2 s
dμt (u3 , u4 ) dμs (u1 , u2 ).
(31)
In order to justify the interchanges of integration we need to check the existence of the above integral in a second step. Due to Tonelli’s theorem it is enough to ensure the existence of an iterated integral over the absolute value of the integrand and we can assume that f, g are positivevalued. By Lemma 15 it follows f (t) kzs , g (t) kzs ∈ Sym>0 (Cn ) for all z ∈ Cn . According to the calculation in Lemma 10 the assignment u1 u2 n G : C u1 → g (t) (u2 )kzs (u2 )e s dμs (u2 ) Cn
is an element in Dc2 (s) . Hence the product G · f (t) · kzs ∈ Dc3 (s) Since c3 (s) =
1 1 1 − < 2s 8s 2s
it follows that G · f (t) · kzs ∈ L1 (Cn , μs ) which guarantees the existence of (31). To reduce the order of integration in (31) note that
W. Bauer / Journal of Functional Analysis 256 (2009) 3107–3142
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t u u k (u3 )2 k s (u1 )e 1s 2 dμs (u1 ) u1 z
Cn
tn 1 |z|2 = (su3 + tz)(su3 + tu2 ) − . exp (s + t)n st (s + t) 2s
We insert the expression on the right-hand side above into (31) and integrate over the u2 dependent part: exp
Cn
2 1 (su3 + tz)u2 kut 2 (u4 ) kzs (u2 ) dμs (u2 ) s(s + t)
su3 + tz u4 su4 tz |z|2 tn + + + . exp − = (t + s)n 2s s[s + t] t t +s t +s
This now implies, that (1)
Kz,s,t (u3 , u4 ) t u u k (u3 )2 k t (u4 )2 k s (u2 )k s (u1 )e 1s 2 dμs (u1 , u2 ) := u1 u2 z z C2n
=
|z|2 t 2n e− s 2n (s + t) su3 + tz u4 su4 tz s|u3 |2 + tzu3 + + + . × exp t (t + s) s(s + t) t t +s t +s
On the other hand, in case of t > 2s > 0 and with f, g ∈ Sym>0 (Cn ) we determine Bs [f (t) s (t) g ]. The calculations in Lemma 15 together with Tonelli’s theorem shows that the integral below exists and one can interchange the order of integration. By using Lemma 14: Bs f (t) s g (t) (z) = f (t) s g (t) kzs , kzs s |z|2 = e− s f ⊗ g(u3 , u4 ) Cn C2n
×e
−
u u s (u3 −w)(u4 −w)+2 Re w( t3 + t4 + zs )− 2t |w|2 t2
dμt (u3 , u4 ) dμs (w) = (∗).
An integration over the variable w leads to (∗) = C2n (2)
(2)
f ⊗ g(u3 , u4 )Kz,s,t (u3 , u4 ) dμt (u3 , u4 )
where the integral kernel Kz,s,t is given by
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Kz,s,t (u3 , u4 ) 2
=e
− |z|s −
s u3 u4 t2
e
w( s+t 2 u3 + t
u3 z u4 z (s+t)2 s+t 2 t + s )+w( t 2 u4 + t + s )− t 2 s |w|
Cn
dv(w) . (πs)n
The integral can be calculated explicitly and one has (2)
Kz,s,t (u3 , u4 ) =
|z|2 t 2n e− s 2n (s + t) u3 tz su3 su4 tz su3 u4 u4 + + + + . × exp − 2 + t s + t s(s + t) t s+t s+t t
(1)
(2)
Comparing both the expressions shows Kz,s,t = Kz,s,t and it follows that (s) (s) Bs Tf (t) Tg (t) = Bs f (t) s g (t) . According to Lemma 12 the Berezin transform Bs is one-to-one on Lfos (Hs ). Though we have proved a composition formula for Berezin–Toeplitz operators. Theorem 16. With functions f, g ∈ Sym>0 (Cn ) and for t > (s)
(s)
(s)
Tf (t) Tg (t) = Tf (t)
sg
(t)
s 2
> 0 there is a composition formula:
.
Using the semi-group property {f (s1 ) }(s2 ) = f (s1 +s2 ) for s1 , s2 > 0 the map (s) St , s f (t) → Tf (t) ∈ Lfos Hs t> 2s
is multiplicative with respect to the s -product and composition of operators if we define the space of real analytic functions St by St := Bt {Sym>0 (Cn )}. The semi-group property of (Bt )t>0 implies that St2 ⊂ St1 for t2 t1 . Remark 17. There are close relations between the s -product and a star-product ∗s on C ∞ (Cn ) parametrized by s > 0 in deformation quantization which appears by a different approach (cf. [7–9,13]). Here ∗s fulfills as s ↓ 0: f ∗s g → f · g,
1 i [f ∗s g − g ∗s f ] → {f, g}. s 2π
The ∗s -product is defined on symbols which arise as Berezin transforms of bounded operators A, B on H 2 (Cn , μs ) as follows: Bs [A] ∗s Bs [B] := Bs [AB].
(32)
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By using an asymptotic expansion of Bs in the time parameter s > 0, (32) can be clued together into a star-product on C ∞ (Cn ) (cf. [13]). Fix bounded functions f, g ∈ L∞ (Cn ), then it follows from Theorem 16 with t > 3s 2, (s) (t−s) (s) (s) (s) f (t) ∗s g (t) = f (t−s) ∗s g = Bs Tf (t−s) Tg (t−s) (s) = f (t−s) s g (t−s) .
According to the first and second equality above the star product ∗s is defined on St = Bt {Sym>0 (Cn )} with t s. However, s does not converge on all smooth functions and due to our calculation in Lemma 14 it seems that even the expressions f (t−s) s g (t−s) do not exist for general bounded symbols f and g in the case where s < t < 32 s. As a corollary to Theorem 16 we obtain a proof of the following result which has been shown in [11] by different methods. (t)
(t)
(t)
Corollary 18. Let p, q ∈ P[z, z¯ ] where z = (z1 , . . . , zn ) and t > 0, then it holds Tp Tq = Tpt q . Proof. Due to Example 13—which generalizes to all t > 0—Bt defines an isomorphism of P[z, z] for t > 0. In particular, P[z, z] ⊂ Bt {Sym>0 (Cn )} and the assertion follows from Theorem 16. 2 3.2. Applications and examples We give applications of the results above to algebras of Toeplitz operators. We only consider the case t = 1 and we write f˜ := f (1) for the Berezin transform of a symbol f ∈ Sym>0 (Cn ). For n ∈ N let C(Cn ), Cb (Cn ) and C0 (Cn ) be the spaces of continuous functions, bounded continuous functions and continuous functions vanishing at infinity, respectively. The mean oscillation MO is given by MO : Sym>0 Cn → Sym>0 Cn :
% MO(f, ·) := |f |2 − |f˜|2 .
(33)
Using (33) we can define the space VMO(Cn ) of functions having vanishing mean oscillation at infinity (cf. [2]) by VMO Cn := f ∈ Cb Cn : lim MO(f, z) = 0 . z→∞
Furthermore, the oscillation of a symbol h ∈ Cb (Cn ) is the expression: Oscz (h) := sup h(z) − h(w): |z − w| < 1 which gives a bounded continuous function of z. Similarly we have the functions VO(Cn ) of vanishing oscillation at infinity: VO Cn := f ∈ Cb Cn : z → Oscz (f ) ∈ C0 Cn .
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It holds MO(f, ·) 0 for f ∈ Sym>0 (Cn ) and VMO(Cn ) is a linear space containing VO(Cn ) as a linear subspace. We consider bounded Toeplitz operators having symbols of vanishing oscillation and of vanishing mean oscillation: Lemma 19. Let f, g ∈ VO(Cn ). Then f˜ and all its derivatives are of vanishing oscillation. The following map is well defined: VO Cn × VO Cn → VO Cn : (f, g) → f˜ g. ˜ Proof. For γ ∈ Nn0 and f ∈ Cb (Cn ) it follows from (28) and a translation in the integral that {f˜}[γ ,0] (z) =
f (u + z)uγ dμ1 (u) Cn
and {& f }[γ ,0] = {f˜}[0,γ ] . Fix z1 , z2 ∈ Cn with |z1 − z2 | < 1 and R > 1, then [γ ,0] {f˜} (z1 ) − {f˜}[γ ,0] (z2 ) f (u + z1 ) − f (u + z2 )uγ dμ1 (u) Cn
γ ! sup Oscu+z1 (f ) + 2 f ∞ |u|R
γ u dμ1 (u).
|u|R
Let ε > 0 and fix R > 1 such that the second term on the right-hand side is dominated by ε. By the assumption on f the first term tends to zero as z1 → ∞. A similar argument applies to the higher derivatives with respect to z¯ and all “mixed derivatives.” Therefore, the Berezin transform f˜ and its derivatives belong to VO(Cn ). From our calculations in Lemma 14 it follows for f, g ∈ VO(Cn ), ˜ f (u1 + z)g(u2 + z)e−u1 u2 dμ1 (u1 , u2 ). [f g](z) ˜ = C2n
' Note that f˜ g˜ is bounded since C2n |e−u1 u2 | dμ1 (u) = 4n /3n is finite and f and g are bounded. With z1 , z2 ∈ Cn as above, u := (u1 , u2 ) ∈ C2n and R > 1 one has: [f˜ g](z ˜ 1 ) − [f˜ g](z ˜ 2 ) ) 4n ( n sup Oscu1 +z1 (f ) g ∞ + sup Oscu2 +z1 (g) f ∞ 3 |u1 |R |u2 |R −u u e 1 2 dμ1 (u). + 2 f ∞ g ∞ |u|R
Fix R 1 such that the second term on the right-hand side is dominated by ε. The first term tends to zero for z1 → ∞ and f˜ g˜ ∈ VO(Cn ). 2
W. Bauer / Journal of Functional Analysis 256 (2009) 3107–3142
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(1)
Example 20. For symbols f ∈ VMO(Cn ) we shortly write Tf := Tf . As it was shown in [2] the space VMO(Cn ) decomposes via f = f˜ + (f − f˜) into % 2 ∈ C Cn . VMO Cn = VO Cn + h ∈ Cb Cn : |h| 0 Let K be the ideal of compact operators in L(H 2 (Cn , μ1 )) and fix functions f, g ∈ %2 ∈ C0 (Cn ) (as for the proof of this fact see [5]) there VMO(Cn ). Because of Th ∈ K for |h| is K0 ∈ K such that Tf Tg = (Tf˜ + Tf −f˜ )(Tg˜ + Tg−g˜ ) = Tf˜ Tg˜ + K0 = Tf˜g˜ + K0 . From Lemma 19 it follows that f˜ g˜ ∈ VO(Cn ) ⊂ VMO(Cn ) and the process can be iterated. More precisely, for h ∈ VMO(Cn ) there is K1 ∈ K with: Tf Tg Th = T ˜
(f g) ˜ h˜
+ K1 .
Given a set of operator S := {A1 , A2 , . . .} we write A{S} for the algebra generated by S. By the remarks above, A Tf : f ∈ VMO Cn ⊂ Th : h ∈ VO Cn + K and the left-hand side defines a ∗-algebra. We write C ∗ {Tf : f ∈ VMO(Cn )} for the C ∗ -algebra generated by Tf with f ∈ VMO(Cn ). Then we have seen, that there is a dense inclusion of quotient spaces:
* * Th : h ∈ VO Cn K ⊂ C ∗ Tf : f ∈ VMO Cn K.
3.3. Fourier transform and composition formulas $ We derive composition formulas for higher products of Toeplitz operators. Since the spaces t> 2s St for s, t > 0 which appear in Theorem 16 may not be closed under the s -product we did not cover such cases before. However, we only consider bounded symbols contained in the range of the Fourier transform F on a certain subspace of L2 (Cn ) := L2 (Cn , v). In Example 27 we remark a relation between these symbol spaces and the ones considered before. For the rest of this paper we deal with the case t = 1, i.e. operators on H 2 (Cn , μ1 ), but the results extend to t > 0 (1) by appropriate rescalings. We shortly write Tg := Tg for a Toeplitz operator with symbol g. Moreover, in our notations we do not distinguish between the Fourier transform F on L1 (Cn ), the Schwartz space S(Cn ) or its unitary extension to L2 (Cn ). For f ∈ S(Cn ) we define: 1 [F f ](z) = (2π)n
f (u)e−i Re zu dv(u).
Cn
Let Cc (Cn ) be the space of continuous and compactly supported complex-valued functions.
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Lemma 21. The Fourier transform maps Cc (Cn ) into the -algebra F (Cn ) defined above Lemma 6. Moreover, F [Cc (Cn )] generates an associative -sub-algebra in F (Cn ). There is an integral expression for the k-times -product: 1 (F f1 F f2 · · · F fk )(z) = F f1 ⊗ · · · ⊗ fk · e 4 i<j ui uj (z, . . . , z)
(34)
where z ∈ Cn , k ∈ N and f1 , . . . , fk ∈ Cc (Cn ). Proof. Fix f ∈ Cc (Cn ) and α := (α1 , α2 ) ∈ Nn0 × Nn0 . Then it follows by differentiation under the integral and with a suitable compact set K ⊂ Cn : [F f ][α1 ,α2 ] (z) f ∞ (2π)n
∂ |α| −i Re zu e ∂zα1 ∂zα2 dv(u) K
f ∞ 1 = (2π)n 2|α|
α +α u 1 2 dv(u)
K
f ∞ maxu∈K |u| |α| vol(K) . (2π)n 2
Hence, by definition, F f ∈ F (Cn ). Now, let us fix f1 , . . . , fk , g1 , . . . , gl ∈ Cc (Cn ) where k, l ∈ N. We define: 1 F (z) : = F f1 ⊗ · · · ⊗ fk · e 4 i<j ui uj (z, . . . , z),
=:hf (u)
1
G(z) := F g1 ⊗ · · · ⊗ gl · e 4
=:hg (y)
i<j
y i yj
(z, . . . , z)
where the functions hf and hg are compactly supported on Cnk and Cnl , respectively. Now: [F G](z) =
(−1)|γ | F [γ ,0] (z) · G[0,γ ] (z) γ! n
γ ∈N0
=
(−1)|γ | 1 γ! (2π)(l+k)n n γ ∈N0
hf ⊗ hg (u, y)
(−1)|γ | 4|γ |
C(l+k)n
× (u1 + · · · + uk )γ (y1 + · · · + yl )γ e−i Re z(u1 +···+uk +y1 +···+yl ) dv(u, y) = (∗). Since the integrand is compactly supported in Cn(l+k) we can interchange the summation with the integral: 1 (∗) = hf ⊗ hg (u, y) (2π)(l+k)n C(l+k)n
×e
1 4 (u1 +···+uk )(y1 +···+yl )−i Re z(u1 +···+uk +y1 +···+yl )
dv(u, y).
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Defining (w1 , . . . , wk+l ) := (u1 , . . . , uk , y1 , . . . , yl ) ∈ C(l+k)n the integrand has the form: 1
f1 ⊗ · · · ⊗ fk ⊗ g1 ⊗ · · · ⊗ gl (w)e 4
i<j
wi wj −i Re z(w1 +···+wk+l )
.
It follows that 1 [F G](z) = F f1 ⊗ · · · ⊗ fk ⊗ g1 ⊗ · · · ⊗ gl (w)e 4 i<j wi wj (z, . . . , z). The product formula (34) follows inductively from this identity and the result is independent from the order of the multiplication (associativity). 2 In the following we view the integral form of the multiple -product in Lemma 21—i.e. the right-hand side of the identity (34)—as its definition on F [Cc (Cn )]. Since this integral expression makes sense not only for functions fj ∈ Cc (Cn ) but in a wider class of symbols, it can be used to extend the -product. Now, one has to pose suitable growth conditions on the functions fj at infinity that ensure the existence of the Fourier transform on the right-hand side of (34) as a pointwise defined function on Cnk . 1 Next, we estimate the growth of the factor e 4 i<j ui uj . With k ∈ N and a, b ∈ C consider the k × k-matrix ⎛
⎞ 1 ··· 1 ··· 1⎟ ∈ Rk×k . .. ⎟ .⎠
1 ⎜1 where D = ⎜ ⎝ ...
Aa,b := bD + (a − b)Id,
1
1
··· 1
Clearly, Aa,b is symmetric and a − b is an eigenvalue of multiplicity k − 1 with eigenvectors [(1 − k, 1, . . . , 1), (1, 1 − k, 1, . . . , 1), . . . , (1, . . . , 1, k − 1)] of rank k − 1. An additional eigenvector with eigenvalue a + (k − 1)b is (1, . . . , 1). To Aa,b we assign the quadratic form Qa,b (z) := (Aa,b z) · z where z ∈ Ck . In particular, with ε > 0 we are interested in the case 1 Qε,− 1 (z) = ε|z|2 − Re zi zj . 8 4 i<j
Lemma 22. For ε > (k − 1)/8 it holds: e
−Qε,− 1 (z) 8
dv(z)
Ck
πk (ε −
k−1 k 8 )
.
Proof. With our remark above the minimal eigenvalue of the symmetric matrix Aε,− 1 is given by ε − (k − 1) 18 . Hence, one obtains the estimate 1 Aε,− 1 ε − (k − 1) Id. 8 8
8
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It follows, e
−Qε,− 1 (z)
e−(ε−
dv(z)
8
Ck
k−1 2 8 )|z|
dv(z) =
Ck
πk (ε −
k−1 k 8 )
.
2
If we replace z ∈ Ck in Lemma 22 by u = (u1 , . . . , uk ) ∈ Cn × · · · × Cn = Cnk and uj = (u1,j , . . . , un,j )t where j = 1, . . . , k, then we have Qε,− 1 (u) = 8
Therefore it follows with ε > e
−Qε,− 1 (u) 8
n
Qε,− 1 (u ,1 , . . . , u ,k ). 8
=1
k−1 8 :
dv(u) =
n 1
e
−Qε,− 1 (u ,1 ,...,u ,k ) 8
dv(u ,1 , . . . , u ,k )
=1 k C
Cnk
π kn (ε −
k−1 kn 8 )
(35)
.
For k ∈ N and with our notations in (16) consider the following symbol spaces of complexvalued functions on Cn :
D˜ k :=
Dρ
D˜ k . and Sym<0 Cn :=
ρ<(1−k)/8
(36)
k∈N
For symbols f1 , . . . , fk ∈ D˜ k we have 1
f1 ⊗ · · · ⊗ fk (u)e 4
i<j
ui uj
∈ L1 Ckn
and the expression on the right-hand side of (34) exists. By the Riemann–Lebesgue lemma it defines a function in C0 (Cn ). Therefore, we can extend k-times -products to an associative &k ) ⊂ C0 (Cn ). More precisely, with u = (u1 , . . . , uk ) ∈ Cn × · · · × Cn (k times) product on F (D n and z ∈ C we set (F f1 F f2 · · · F fk )(z) 1 1 i<j ui uj −i Re z(u1 +···+uk ) dv(u) 4 := f ⊗ · · · ⊗ f (u)e 1 k (2π)kn Ckn
1 = F f1 ⊗ · · · ⊗ fk · e 4 i<j ui uj (z, . . . , z).
(37)
Here and in the following we will not distinguish in our notation between the product in (37) and the -product defined in (29).
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Definition 23. By AF (Cn ) ⊂ C0 (Cn ) we mean the associative -algebra which is generated by the range F {Sym<0 (Cn )} of the Fourier transform. &k+1 we set f := F f1 F f2 · · · F fk and calculate the Given functions f1 , . . . , fk+1 ∈ D Berezin transform of f F fk+1 : B1 [f F fk+1 ](z) 2 e−|z| = (2π)n(k+1)
Cn
f1 ⊗ · · · ⊗ fk+1 (u) Cn(k+1)
1 × exp ui uj − i Re w(u1 + · · · + uk+1 ) + zw + wz dv(u) dμ1 (w) 4 i<j
−|z|2
e = (2π)n(k+1)
f1 ⊗ · · · ⊗ fk+1 (u) Cn(k+1)
1 i i i i × exp ui uj + z − u1 − · · · − uk+1 z − u1 − · · · − uk+1 dv(u). (38) 4 2 2 2 2 i<j
On the other hand, the Berezin transform of the Toeplitz product Tf TF fk+1 is given by B1 [Tf TF fk+1 ](z) =
f (w) TF fk+1 kz1 (w)kz1 (w) dμ1 (w).
Cn
By a straightforward calculation
|z|2
TF fk+1 kz1
e− 2 (w) = (2π)n
i
i
fk+1 (uk+1 )e(w− 2 uk+1 )(z− 2 uk+1 ) dv(uk+1 ). Cn
After inserting this expression into the integral above one obtains: e−|z| B1 [Tf TF fk+1 ](z) = (2π)n 2
f (w)fk+1 (uk+1 ) C2n
i i × exp w − uk+1 z − uk+1 + wz dv(uk+1 ) dμ1 (w) 2 2 = (∗). Now, we use the integral expression for f (w) which produces the n(k + 2)-times complexdimensional integral:
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e−|z| (∗) = (2π)n(k+1) 2
f1 ⊗ · · · ⊗ fk+1 (u) Cn(k+1)
i i 1 ui uj − uk+1 z − uk+1 × exp 4 2 2 i<j k i × exp −i Re w(u1 + · · · + uk ) + w z − uk+1 + wz dμ1 (w) dv(u). 2 Cn
The inner integral can be calculated explicitly: Cn
i exp −i Re w(u1 + · · · + uk ) + w z − uk+1 + wz dμ1 (w) 2 i i i i = exp z − u1 − · · · − uk z − u1 − · · · − uk+1 , 2 2 2 2
and it follows that
e−|z| (2π)n(k+1) 2
(∗) =
f1 ⊗ · · · ⊗ fk+1 (u) Cn(k+1)
i i 1 ui uj − uk+1 z − uk+1 × exp 4 2 2 i<j k i i i i + z − u1 − · · · − uk z − u1 − · · · − uk+1 dv(u) 2 2 2 2 2 e−|z| = f1 ⊗ · · · ⊗ fk+1 (u) (2π)n(k+1) Cn(k+1)
i i i i 1 ui uj + z − u1 − · · · − uk+1 z − u1 − · · · − uk+1 dv(u) × exp 4 2 2 2 2 i<j k+1 = 2n(k+1) f1 ⊗ · · · ⊗ fk+1 (u) Cn(k+1)
1 × exp − 4
ui uj − i Re z(u1 + · · · + uk+1 ) dμ4 (u).
i<j k+1
A comparison of this expression with the Berezin transform of f F fk+1 shows: B1 [f F fk+1 ] = B1 [Tf TF fk+1 ]. Since B1 is one-to-one on bounded operators we have proved:
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Theorem 24. For {f1 , . . . , fk } ⊂ D˜ k it holds TF f1 TF f2 · · · TF fk = TF f1 F f2 ···F fk . In particular, the quantization map Q : F Sym<0 Cn , f → Tf ∈ L H 2 Cn , μ1 ˜ : AF (Cn ) → L(H 2 (Cn , μ1 )). extends to an algebra homomorphism Q Remark 25. In Theorem 24 one could weaken the assumptions to fj ∈ L1 (Cn ) such that 1
f1 ⊗ · · · ⊗ fk · e 4 i<j ui uj ∈ L1 (Cnk ). Note that the integrals in (38) and (39) exist under this assumption and coincides with the restriction of an entire function on C2n in the variables z and w to the diagonal. The following example indicates that the above composition formulas easily fail in the case of non-real analytic symbols. Example 26. Choose n = 1 and let Cc (C) be the space of compactly supported continuous functions on C. Let fj ∈ Cc (C) for j = 1, 2 be positive-valued. Under the assumption Tf1 Tf2 = 0 it follows for all ∈ N0 :
0 = Tf1 Tf2 z , z = f1 ⊗ f2 (u) exp{u1 u2 }u1 u 2 dμ1 (u1 , u2 ). C2
Multiplying this integral by ( !)−1 and summing over ∈ N0 leads to 2 0 = f1 ⊗ f2 (u)exp{u1 u2 } dμ1 (u1 , u2 ) C2
and since f1 ⊗ f2 is positive-valued and continuous we conclude that either f1 ≡ 0 or f2 ≡ 0. In particular, choose open discs D(xj ) ⊂ C of radius r = 1 centered in xj ∈ C for j = 1, 2, and let 0 = pj ∈ P[z, z] positive-valued and vanishing of arbitrary order at ∂D(xj ). Define fj := pj on D(xj ) and fj ≡ 0 on C \ D(xj ). Then f1 f2 can be defined as in (9) a.e. on C. It vanishes a.e. in the case |x1 − x2 | > 2 such that Tf1 f2 = 0. Since fj = 0 for j = 1, 2, we conclude from our remark above that Tf1 Tf2 = 0 showing that Tf1 f2 = Tf1 Tf2 . In the final part of this section we remark a relation between the Fourier transform and the Berezin transform restricted to the Schwartz space S(Cn ). This can be seen as a link between Theorems 16 and 24. Example 27. Let t > 0, then Bt can be regarded as a convolution operator on S(Cn ). More 2 n precisely, Bt f = f ∗ ht where ht (z) := 2t n exp{− |z|t }. Here f ∗ ht denotes the usual convolution product on the Schwartz space. Defining t 2 gt (z) := F −1 ht (z) = e− 4 |z|
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and choosing t = 1 it follows that B1 f = F [(F −1 f ) · g1 ] and B1 is a pseudo-differential operator on S(Cn ). In particular, one has the inclusion of ranges: 1 2 B1 S Cn ⊂ F e− 4 |·| g: g ∈ S Cn ⊂ F (D˜ 2 ). For symbols f, g ∈ B1 [S(Cn )] the composition formula Tf Tg = Tf g can be obtained from both, Theorems 16 and 24. In addition, Theorem 24 shows that f g ∈ C0 (Cn ). As an application of Theorem 24 we show that the space of Toeplitz operators having symbols in Cc (Cn ) is dense in the ideal of all compact operators. Example 28. For ϕj ∈ Cc (Cn ), j = 1, . . . , k, we can choose sequences (ϕj,l )l∈N ∈ F Sym<0 Cn such that liml→∞ ϕj,l = ϕj uniformly on Cn . It follows from Tϕj ϕj ∞ , Tϕ1 · · · Tϕk = lim Tϕ1,l · · · Tϕk,l = lim Tϕ1,l ···ϕk,l . l→∞
l→∞
Since for each l the product ϕ1,l · · · ϕk,l tends to zero at infinity, the following inclusion is dense (cf. Example 20 for notations): Tϕ : ϕ ∈ Cc Cn ⊂ C ∗ Tϕ : ϕ ∈ Cc Cn = K. 3.4. Fourier–Wiener transform and composition formulas We consider composition formulas for Toeplitz operators arising from the Fourier–Wiener transform. We only deal with the time parameter t := 1. Definition 29. The Fourier–Wiener transform W ∈ L(L2 (Cn , μ1 )) is the Hilbert space isomorphism given as an extension of [Wf ](z) = f (u) exp Q(z, u) dμ2 (u) Cn
where f ∈ Cc (Cn ) and Q(z, u) := 12 {|z|2 − iu¯z − i uz}. ¯ Note, that W can be written as the composition of (unitary) operators Md F Md−1 W : L2 Cn , μ1 −→ L2 Cn −→ L2 Cn −→ L2 Cn , μ1 where d(z) = π −n/2 exp{−| · |2 /2}. In particular, W is an isometry with −1 W f (z) = f (u) exp Q(z, −u) dμ2 (u). Cn
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For α = (α1 , α2 ) ∈ Nn0 × Nn0 and f, g ∈ Cc (Cn ) it follows by a straightforward calculation that [Wf ][α1 ,α2 ] (z) =
f (u)Wα (z, u) exp Q(z, u) dμ2 (u)
Cn
where Wα (z, u) is the polynomial of maximal degree |α| given by
Wα (z, u) :=
γ!
γ α1 ,α2
2|α|−|γ |
α1 γ
α2 (z − iu)α2 −γ (¯z − i u) ¯ α1 −γ . γ
By using W[γ ,0] (z, u) = 2−|γ | (¯z − i u) ¯ γ and W[0,γ ] (z, u) = 2−|γ | (z − iu)γ it can be verified that Wf Wg exists and has the integral expression 3 [Wf Wg](z) = exp |z|2 f ⊗ g(u) 4 C2n
1 i i × exp u1 u2 − z(u1 + 2u2 ) − z(u2 + 2u1 ) dμ2 (u1 , u2 ). 4 4 4
(40)
Our first aim is it to extend this definition from Cc (Cn ) to some larger space. We use the notations introduced in (16): Lemma 30. For c <
3 8
the -product extends to a continuous bilinear form: : W(Dc ) × W(Dc ) → Dκ(c)
where κ(c) = W −1 · Dc .
3 4
+
1 12−32c .
In particular, κ(0) = 10/12 < 1. Here W(Dc ) carries the norm
Proof. By using the integral expression (40) it follows: Wf Wg (z) 1 3 2 f Dc g Dc |z| exp 4 (2π)2n 3 1 1 2 |u| − Re iz(u1 + 2u2 ) + Re iz(u2 + 2u1 ) dv(u) × exp c − 8 4 4 C2n
= (∗). 1
After the transformation u → (3/8 − c)− 2 u one obtains 3
(∗) =
e 4 |z|
2
( 34 − 2c)2n
f Dc g Dc C2n
Re(u1 − u2 )iz dμ1 (u). exp √ 6 − 16c
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Now, the assertion follows from C2n
Re(u1 − u2 )iz 1 dμ1 (u) = exp |z|2 . exp √ 12 − 32c 6 − 16c
2
Example 31. Let 0 < β < 1/2 and hj (z) := exp{β|z|2 }[F −1 gj ](z) where gj ∈ Cc∞ (Cn ) for j = 1, 2, such that hj ∈ Dβ . With the usual convolution product ∗ in S(Cn ) one has 1 | · |2 − β | · |2 F −1 gj exp − Whj = F exp − 2 2 = (2ρ)n exp −ρ| · |2 ∗ gj where ρ := (2 − 4β)−1 > 0. It is easy to check that [Whj ](z) = e
|z|2 2
[B 1 gj ](z) ρ
(41)
and for α < 0 one has [Whj ]
ρn ρα |z|2 2 g exp + |z| . j Dα (ρ − α)n ρ −α 2
Since limα→−∞ ρα(ρ − α)−1 + 1/2 = −ρ + 1/2 = where 0 < ε <
β 1−2β .
β 2β−1
< 0 we conclude that Whj ∈ D−ε
Using (41) we also can give an integral expression for the -product:
1 1 2 [Wh1 Wh2 ](z) = e|z| B 1 g1 ⊗ g2 e−{(ρu1 −(ρ− 2 )z}{ρu2 −(ρ− 2 )z} (z, z). ρ
Let r1 < 1/6 such that κ(r1 ) < 1 (cf. the notation in Lemma 30) and D1/3+r1 ⊂ L2 (Cn , μ1 ). Fix functions h, s ∈ Sym>0 (Cn ) and assume that Wh, Ws ∈ Dr1 and Wh Ws ∈ Dr2 where 0 < r2 < 12 . Using the notations in (18) and X := spanC kz1 : z ∈ Cn ⊂ H 2 Cn , μ1 the following assignments are well defined: MWs MWh P P X −→ Dc1 (1)= 1 −→ Dc2 (1)= 1 −→ D 1 +r1 −→ H 2 Cn , μ1 . 4
3
3
Therefore, by assumption the product TW h TW s and TW hW s can be considered as densely defined operators on H 2 (Cn , μ1 ). In order to prove a composition formula we calculate the Berezin transform of Wh Ws: 2 B1 [Wh Ws (z) = [Wh Ws](x)k 1 (x) dμ1 (x) = (∗). z
Cn
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After inserting the expression (40) for the -product, (∗) equals: 3 1 i h ⊗ s(u) exp |x|2 + u1 u2 − x(u1 + 2u2 + 4iz) (∗) = 4 4 4 C3n
i − x(u2 + 2u1 + 4iz) − |z|2 dμ2 (u) dμ1 (x). 4
It can be checked that the integral exists. Moreover, the integration over the variable x can be carried out explicitly: i i 3 2 e− 4 x(u1 +2u2 +4iz)− 4 x(u2 +2u1 +4iz)+ 4 |x| dμ1 (x) Cn
1 = 4n exp − (u2 + 2u1 + 4iz)(u1 + 2u2 + 4iz) . 4
After inserting this expression above one obtains: 2 h ⊗ s(u)e−u1 u2 +3|z| −iz(u1 +2u2 )−iz(2u1 +u2 ) dμ1 (u). (∗) = C2n
On the other hand, we calculate the Berezin transform of TW h TW s . Note that |z|2 i i 1 n dμ2 (u2 ). TW s kz (v) = 2 s(u2 ) exp 2 v − u2 z − u2 − 2 2 2 Cn
Inserting this integral expression into B1 [TW h TW s ](z) = [Wh](v) TW s kz1 (v)kz1 (v) dμ1 (v) =: (∗∗) Cn
shows that (∗∗) = 2 exp −|z|2
n
|u2 |2 |v|2 − iu2 z + h ⊗ s(u) exp − 2 2
C3n
i i + v 2z − u1 − iu2 + v z − u1 dμ1 (v) dμ2 (u). 2 2 We calculate the integral over the variable v: Cn
|v|2 i i dμ1 (v) exp v 2z − u1 − iu2 + v z − u1 + 2 2 2 i i = 2n exp 2 z − u1 2z − u1 − iu2 , 2 2
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and it follows that (∗∗) =
h ⊗ s(u)e−u1 u2 +3|z|
2 −iz(u
1 +2u2 )−iz(u2 +2u1 )
dμ1 (u).
C2n
Comparing (∗) and (∗∗) above shows that B1 [Wh Ws] = B1 [TW h TW s ] and both are restrictions of an entire functions on the variable z and w to the diagonal. Similar to the proof of Lemma 12 we conclude: Theorem 32. Let h, s ∈ Sym>0 (Cn ) and r1 < 1/6 and r2 < 1/2. In the case Wh, Ws ∈ Dr1 and Wh Ws ∈ Dr2 there is a composition formula TW h TW s = TW hW s as operators on spanC {kz1 : z ∈ Cn }. We show, how the various composition formulas above can be used for explicit symbols spaces. Example 33. For a multi-index γ = (γ1 , γ2 ) ∈ Nn0 × Nn0 consider the monomials mγ (z) := zγ1 zγ2 on Cn . It follows: |z|2 (−i)|γ | − |z|2 ∂γ ∂ |γ | − 2i uz− 2i uz 2 [Wmγ ](z) = e e dμ (u) = e− 2 2 |γ | γ γ γ γ 1 2 1 2 2 ∂z ∂z ∂z ∂z Cn
and the Fourier–Wiener transform of mγ is a polynomial of degree |γ |. Since W is one-to-one and preserves Pn [z, z¯ ] it gives an isomorphism of P[z, z¯ ]. Therefore Corollary 18 also follows from Theorem 32 in the case t = 1. The holomorphic and anti-holomorphic polynomials are W-invariant, as well. For a, b ∈ Cn and c ∈ C we define the complex-valued functions a,b,c (z) := exp az + bz + c|z|2 . Let Re c < explicitly as
1 2
(42)
such that a,b,c ∈ L2 (Cn , μ1 ). The Fourier–Wiener transform can be calculated 1 2ab a0 ,b0 ,c0 (z) [Wa,b,c ](z) = exp (1 − 2c)n 1 − 2c
where a0 := ia/(2c − 1), b0 = ib/(2c − 1) and c0 = c/(2c − 1) with Re c0 < 1/2. Hence the linear hull 1 n S := spanC a,b,c : a, b ∈ C , c ∈ C and Re c < 2 is W-invariant (and by similar reasons W −1 -invariant). In particular, W defines an isomorphism of S . Now, let r, s ∈ Cn , t ∈ C and Re t < 1/2. Then one obtains by a straightforward calculation a,b,c r,s,t = exp{−as}a(1−t)+r,s(1−c)+b,t (1−c)+c .
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3141
If we write t = t1 + it2 and c = c1 + ic2 it can be seen from Re t (1 − c) + c = t1 + c1 − t1 c1 + t2 c2 that the right-hand side of (43) can grow exponentially at infinity even if both factors of the product vanish exponentially at infinity but have “high oscillation” (i.e. t2 c2 is large compared to t1 + c1 − t1 c1 ). In particular, it holds: 0,0,i 0,0,i = 0,0,1+2i .
(44)
The right-hand side does not define a Toeplitz operator on H 2 (Cn , μ1 ) whereas the product exists as a bounded operator. This serves as an example of a case where the product of two Toeplitz operators is not a Toeplitz operator anymore. However, S has a -invariant subspace T20,0,i
S,0 := spanC a,b,t : a, b ∈ Cn and t 0 . The Berezin transform of a function a,b,c ∈ S is given by 1 1 2 [B1 a,b,c ](z) = exp ab + az + bz + c|z| . (1 − c)n 1−c Therefore, B1 [S,0 ] = spanC a,b,t : a, b ∈ Cn , t ∈ (−1, 0] ⊂ S,0 ⊂ Sym>0 Cn . As a corollary to Theorem 16 we obtain: Corollary 34. For f, g ∈ spanC {a,b,t : a, b ∈ Cn , t ∈ (−1, 0]} there is a composition formula of Toeplitz operators: Tf Tg = Tf g . In case of Re c < 0 we also can calculate the Fourier transform of a,b,c : [F a,b,c ](z) =
1 1 2 4ab − 2ibz . exp − − 2iaz − |z| (−2c)n 4c
In particular, spanC {a,b,c : a, b ∈ Cn and c ∈ C s.t. Re c < 0} is invariant under F . For k ∈ N and using (36) we define the spaces 1−k ⊂ D˜ k . D,k := spanC a,b,c : a, b ∈ Cn , Re c < 8 According to Theorem 24 we obtain an identity for multiple products: Corollary 35. Let {f1 , . . . , fk } ⊂ D,k , then it holds: TF f1 TF f2 · · · TF fk = TF f1 F f2 ···F fk .
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Acknowledgments The author wishes to thank Prof. K. Furutani for many hints and helpful discussions as well as for his kind invitation to Tokyo University of Science. Moreover, he gratefully acknowledges many useful hints due to the referee of this paper, who in particular conjectured Theorem 16 under the assumption t > 2s instead of the weaker condition s = t as it was stated originally. References [1] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961) 187–214. [2] W. Bauer, Mean oscillation and Hankel operators on the Segal–Bargmann space, Integral Equations Operator Theory 52 (2005) 1–15. [3] W. Bauer, Toeplitz Operators on Finite and Infinite-Dimensional Spaces and Associated Ψ ∗ -Fréchet Algebras, Shaker Verlag, Aachen, 2005, PhD thesis. [4] W. Bauer, K. Furutani, Compact operators and the pluriharmonic Berezin transform, Int. J. Math. 19 (6) (2008) 1–25. [5] C.A. Berger, L.A. Coburn, Toeplitz operators on the Segal–Bargmann space, Trans. Amer. Math. Soc. 301 (2) (1987) 813–829. [6] F.A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat. 36 (5) (1972) 1134–1167 (in Russian), also in: Math. USSR-Izv. 6 (5) (1972) 1117–1151. [7] F.A. Berezin, Quantization in complex symmetric spaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (2) (1975) 363–402, 472 (in Russian), also in: Math. USSR-Izv. 9 (1975) 341–379. [8] D. Borthwick, A. Lesniewski, H. Upmeier, Non-perturbative deformation quantization on Cartan domains, J. Funct. Anal. 113 (1993) 153–176. [9] L.A. Coburn, Deformation estimates for the Berezin–Toeplitz quantization, Comm. Math. Phys. 149 (1992) 415– 424. [10] L.A. Coburn, The measure algebra of the Heisenberg group, J. Funct. Anal. 161 (1999) 509–525. [11] L.A. Coburn, On the Berezin–Toeplitz calculus, Proc. Amer. Math. Soc. 129 (11) (2001) 3331–3338. [12] L.A. Coburn, Sharp Lipschitz estimates, Proc. Amer. Math. Soc. 135 (4) (2007) 1163–1168. [13] M. Englis, Weighted Bergman kernels and quantization, Comm. Math. Phys. 227 (2002) 211–241. [14] G.B. Folland, Harmonic Analysis in Phase Space, Ann. Math. Stud., vol. 122, Princeton Univ. Press, Princeton, NJ, 1989. [15] V. Guillemin, Toeplitz operators in n dimensions, Integral Equations Operator Theory 7 (1984) 145–205. [16] B. Hall, The range of the heat operator, in: J. Joergensen, L. Walling (Eds.), The Ubiquitous Heat Kernel, Amer. Math. Soc., Providence, RI, 2006, pp. 203–231. [17] E. Hille, A class of reciprocal functions, Ann. of Math. (2) 27 (1926) 427–464.
Journal of Functional Analysis 256 (2009) 3143–3157 www.elsevier.com/locate/jfa
Spectral conditions on Lie and Jordan algebras of compact operators ✩ Matthew Kennedy ∗ , Heydar Radjavi Department of Pure Mathematics, University of Waterloo, 200 University Ave West, Waterloo, Ontario, Canada N2L 3G1 Received 31 March 2008; accepted 19 February 2009 Available online 9 March 2009 Communicated by D. Voiculescu
Abstract We investigate the properties of bounded operators which satisfy a certain spectral additivity condition, and use our results to study Lie and Jordan algebras of compact operators. We prove that these algebras have nontrivial invariant subspaces when their elements have sublinear or submultiplicative spectrum, and when they satisfy simple trace conditions. In certain cases we show that these conditions imply that the algebra is (simultaneously) triangularizable. © 2009 Elsevier Inc. All rights reserved. Keywords: Invariant subspaces; Spectrum; Compact operators; Lie algebras; Jordan algebras
1. Introduction Conditions on the spectrum of an operator have been studied for some time, for instance in the work of Motzkin and Taussky [7], who investigated pairs of n × n matrices A and B with the property that for every scalar λ, the eigenvalues of A + λB are, subject to a slight technical condition, linear functions of the eigenvalues of A and B. Specifically, they required that the eigenvalues {α1 , . . . , αn } and {β1 , . . . , βn }, of A and B respectively, could be expressed as ordered sets {α1 , . . . , αn } and {β1 , . . . , βn }, such that for every scalar λ, the eigenvalues of A+λB are precisely αi + λβi , for i = 1, . . . , n. Such matrices are said to have property L (“L” is for “linear”). ✩
The research was partially supported by NSERC.
* Corresponding author.
E-mail addresses:
[email protected] (M. Kennedy),
[email protected] (H. Radjavi). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.011
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A pair of (simultaneously) triangularizable matrices clearly has property L, and Motzkin and Taussky were interested in conditions under which the converse of this was true. They showed that a finite group of matrices, with every pair of elements in the group having property L, is not just triangularizable, but diagonalizable. Somewhat later, Wales and Zassenhaus [13], Zassenhaus [14], and Guralnick [2] showed that a multiplicative semigroup of n × n matrices is triangularizable under the same circumstances. The following conditions are seemingly much weaker than property L. Definition 1.1. A pair of bounded operators A and B on a Banach space are said to have subadditive spectrum if σ (A + B) ⊆ σ (A) + σ (B), where σ (A) + σ (B) means the set of all α + β, with α and β in σ (A) and σ (B) respectively. Similarly, A and B are said to have sublinear spectrum if σ (A + λB) ⊆ σ (A) + λσ (B) for every complex number λ. A family F of bounded operators on a Banach space is said to have subadditive (resp. sublinear) spectrum if every pair of elements in F has subadditive (resp. sublinear) spectrum. Note that for linear spaces, and in particular for Lie and Jordan algebras, subadditivity of the spectrum clearly implies sublinearity. The conditions of sublinear and subadditive spectrum are, in some sense, even weaker than one might initially suspect. Indeed, as the following example illustrates, there is little hope of obtaining any results about the existence of invariant subspaces for an arbitrary family of operators with subadditive or even sublinear spectrum without imposing a great deal of additional structure. Example 1.2. Consider the matrices ⎛
⎞ 0 1 0 A = ⎝ 0 0 −1 ⎠ , 0 0 0
⎛
0 ⎝ B= 1 0
0 0 1
⎞ 0 0⎠. 0
It is easy to verify that the linear space S spanned by A and B consists entirely of nilpotent matrices, so S has sublinear spectrum, yet S clearly has no nontrivial invariant subspaces. It was shown in [8] that the property of sublinear spectrum implies the triangularizability of a semigroup of compact operators, which is an extension of the classical results mentioned above to the context of an infinite-dimensional Banach space. A natural question is whether this kind of result holds for other types of algebraic structures. In this paper we study Lie and Jordan algebras of compact operators satisfying spectral conditions like sublinearity and submultiplicativity, which have been shown to imply the existence of invariant subspaces for semigroups of compact operators. We will prove that many of the results obtained for semigroups also hold in this nonassociative context.
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Shulman and Turovskii obtained several results of this nature in their development of a radical theory for Lie algebras of compact operators [11]. They showed that a Lie algebra of compact operators with subadditive spectrum has invariant subspaces [11], and it was implicit in their proof that this condition actually implied triangularizability. This gave an extension of Engel’s Theorem for Lie algebras of compact operators, which was also proven recently in [10]. On the other hand, results on the existence of invariant subspaces for Jordan algebras of compact operators have only recently been obtained. In particular, there is currently no Jordan analogue of the theory which was developed in [11]. This is one reason why our results have been developed in a different manner. We begin by analyzing the properties of bounded linear operators which satisfy a certain spectral additivity condition. The results we obtain here, when combined with new “Cartanlike” conditions for the existence of invariant subspaces in Lie and Jordan algebras of compact operators, provides us with a general method of proving the existence of invariant subspaces for these algebras, in particular when their elements have subadditive or submultiplicative spectrum. We will show that this implies a Lie or Jordan algebra of compact operators with subadditive spectrum is triangularizable, obtaining a new proof of Shulman and Turovskii’s result for Lie algebras. Finally, we obtain simple conditions on the trace of the finite-rank elements in the algebra which imply the existence of invariant subspaces. As far as we know, some of our results are new even in finite dimensions. 2. Preliminaries In this paper we confine ourselves to the field C of complex numbers. For a Banach space X , we let B(X ) and K(X ) denote the algebras of all bounded and compact operators on X respectively. For A in B(X ), we let σ (A) and r(A) denote the spectrum and spectral radius of A respectively. If A is of finite rank, then the trace of A is well-defined; we denote it by tr(A). To clarify our exposition, we will sometimes make use of shorthand notation. For example, if S1 and S2 are subsets of B(X ), then we will write S1n for the linear span of {An : A ∈ S1 }, and S1 S2 for the linear span of {AB: A ∈ S1 , B ∈ S2 }. A family F of operators is called reducible if there is a nontrivial closed subspace invariant under every member of F . We say that F is triangularizable if its lattice of invariant subspaces contains a maximal subspace chain C. (If M1 and M2 are two members of C with M1 ⊆ M2 and no other member between M1 and M2 , then dim(M1 M2 ) 1.) To establish the triangularizability of a family of operators, we will require the following lemma from [9, Lemma 7.1.11]. Lemma 2.1 (The Triangularization Lemma). Let P be a property of a family of operators such that (1) every family of operators with property P is reducible, and (2) if F has property P and if M1 and M2 are invariant subspaces of F with M1 ⊆ M2 , then Fˆ has property P, where Fˆ is the set of all quotient operators on M1 /M2 induced by F . Then every family of operators with property P is triangularizable.
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If P is a property of a family of operators which satisfies hypothesis (2) of the Triangularization Lemma, then we say that P is inherited by quotients. The following result from [9, Corollary 8.4.2] will be used in combination with the Triangularization Lemma. Lemma 2.2. The property of sublinear spectrum is inherited by quotients. Definition 2.3. A pair of bounded operators A and B on a Banach space are said to have submultiplicative spectrum if σ (AB) ⊆ σ (A)σ (B), where σ (A)σ (B) means the set of all αβ, with α and β in σ (A) and σ (B) respectively. A family F of bounded operators on a Banach space are said to have submultiplicative spectrum if every pair of elements in F has submultiplicative spectrum. The property of submultiplicative spectrum is not strong enough to imply the reducibility of a semigroup of compact operators; indeed, the existence of finite irreducible matrix groups with sublinear spectrum was shown in [6]. We will show however, that Lie and Jordan algebras of compact operators with submultiplicative spectrum are reducible. 3. Operators with stable spectrum It turns out that the following property, obviously closely related to the property of sublinear spectrum, is of particular importance for obtaining many of our reducibility results. Definition 3.1. Let T be a bounded operator on a Banach space. We say that a bounded operator A has T -stable spectrum if r(A + λT ) r(A) for every complex number λ. A family of bounded operators on a Banach space is said to have T -stable spectrum if each of its elements has T -stable spectrum. It will sometimes be useful to reference a family of bounded operators with T -stable spectrum without making explicit mention of T . Therefore, we will say that a family of bounded operators has stable spectrum if it has T -stable spectrum, for some nonzero T . Example 3.2. Consider the matrices ⎛
⎞ −1 0 0 A = ⎝ 0 0 1⎠, 0 0 0
⎛
⎞ 0 −1 −1 B = ⎝ 0 −1 −1 ⎠ . 1 0 1
Note that A is in Jordan normal form, so σ (A) = {−1, 0}, and B 3 = 0, so σ (B) = {0}. The characteristic polynomial of A + λB is t 3 − t 2 , which implies that σ (A + λB) = σ (A) for all λ in C. This shows that A is B-stable.
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Calculating the inverse of μ − A, we have ⎛ 1 ⎝ (μ − A)−1 B = 2 μ (1 − μ)
0 −1
⎞ μ2 μ2 −μ(1 − μ) −(1 − μ)2 ⎠ ,
μ(1 − μ)
0
μ(1 − μ)
and it is routine to verify that this matrix is nilpotent. Calculating the inverse of 1 − μB, we have ⎛
0 0 −1 (1 − μB) A = ⎝ −1 0 2 0
⎞ 0 −1 ⎠ , 1
and this matrix has characteristic polynomial t 3 − t 2 , showing that it has the same eigenvalues as A. In this section, we will show that the results in Example 3.2 hold in general, which will be important for our main results. An important tool will be the theory of subharmonic functions, based on the result of Vesentini that if f is an analytic function from a domain of the complex numbers into a Banach algebra, then the functions λ → r(f (λ)) and λ → log(r(f (λ))) are subharmonic [12]. We will require the following two fundamental results from the theory of subharmonic functions (see for example [1, Theorem A.1.3] and [1, Theorem A.1.29] resp.). Theorem 3.3 (Maximum Principle for Subharmonic Functions). Let f be a subharmonic function on a domain D of C. If there exists λ0 in D such that f (λ) f (λ0 ) for all λ in D, then f (λ) = f (λ0 ) for all λ in D. We state here only a special case of H. Cartan’s Theorem. Theorem 3.4 (H. Cartan’s Theorem). Let f be a subharmonic function from a domain D of C. If f (λ) = −∞ on a nonempty open ball in D, then f (λ) = −∞ for all λ in D. Remark 3.5. For bounded operators A and T on a Banach space, the function λ → A + λT is analytic, so by Vesentini’s results, the functions λ → r(A + λT ) and λ → log(r(A + λT )) are subharmonic. If A has T -stable spectrum, the Maximum Principle for subharmonic functions immediately implies that r(A + λT ) = r(A) for all complex numbers λ. If A and T have sublinear spectrum and T is quasinilpotent, then A has T -stable spectrum. The following lemma from [11, Lemma 4.2] shows that the quasinilpotence of T is a necessary condition for A to be T -stable. Lemma 3.6. Let A and T be bounded operators on a Banach space. If A has T -stable spectrum, then T is quasinilpotent. Proof. By Remark 3.5, r(A + λT ) = r(A) for all λ in C, so r λ−1 A + T = |λ|−1 r(A)
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for all nonzero λ in C. Thus by the subharmonicity of the function μ → r(μA + T ), r(T ) = lim sup r(μA + T ) = lim sup |μ|r(A) = 0. μ→0
2
μ→0
Lemma 3.7. Let A and T be bounded operators on a Banach space. Then A has T -stable spec/ σ (A). trum if and only if (μ − A)−1 T is quasinilpotent for all μ ∈ Proof. By Remark 3.5, r(A + λT ) = r(A) for all λ in C, so for μ in C with |μ| > r(A), both μ − A and μ − A − λT are invertible. Therefore, λ−1 (μ − A)−1 (μ − A − λT ) = λ−1 − (μ − A)−1 T is invertible for all nonzero λ in C. This means that the values of the operator-valued function / σ (A), are quasinilpotent whenever |μ| > r(A). μ → (μ − A)−1 T , which is analytic for μ ∈ / σ (A). Since Consider the subharmonic function μ → log(r((μ − A)−1 T )) defined for μ ∈ log(r((μ − A)−1 T )) = −∞ whenever |μ| > r(A), by H. Cartan’s Theorem, log(r((μ − / σ (A). In other words, (μ − A)−1 T is quasinilpotent for all A)−1 T )) = −∞ for all μ ∈ μ∈ / σ (A). 2 Lemma 3.8. Let A and T be bounded operators on a Banach space. Then A has T -stable spectrum if and only if σ (A + λT ) ⊆ σ (A) for all λ in C. Proof. Suppose μ ∈ σ (A + λT ), but that μ ∈ / σ (A). Then clearly λ is nonzero, and λ−1 (μ − A)−1 (μ − A − λT ) = λ−1 − (μ − A)−1 T / σ (A), which gives is not invertible. But by Lemma 3.7, (μ − A)−1 T is quasinilpotent for all μ ∈ a contradiction. 2 The next result also follows from [1, Theorem 3.4.14], but it is interesting to see that it can be proved in the following way. Lemma 3.9. Let A and T be bounded operators on a Banach space. If A has T -stable spectrum and σ (A) has no interior points, then σ (A + λT ) = σ (A) for all λ in C. Proof. This follows immediately from Corollary 3.8 and the Spectral Maximum Principle of [1]. 2 Lemma 3.10. Let A and T be bounded operators on a Banach space. If A has T -stable spectrum and σ (A) has no interior points, then σ ((1 − νT )−1 A) = σ (A) for all ν in C. Proof. First suppose λ is nonzero, and that λ ∈ / σ (A). By Lemma 3.9, σ (λ−1 A + νT ) = −1 σ (λ A), and by Lemma 3.6, T is quasinilpotent. These two facts imply that 1 − νT and 1 − λ−1 A − νT are both invertible, and hence that λ(1 − νT )−1 1 − λ−1 A − νT = λ − (1 − νT )−1 A is invertible for all ν in C. Therefore, λ ∈ / σ ((1 − νT )−1 A) for all ν in C.
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Now suppose 0 ∈ / σ (A). Then A is invertible, implying (1 − νT )−1 A is invertible, and hence by the quasinilpotence of T , that 0 ∈ / σ ((1 − νT )−1 A) for all ν in C. We have shown that σ ((1 − νT )−1 A) ⊆ σ (A) for all ν in C. Since σ (A) has no interior points, the result now follows from the Spectral Maximum Principle of [1]. 2 Lemma 3.11. Let f be an entire function from C into B(X ). Suppose that (1) there exists a complex number λ0 such that σ (f (λ0 )) = σ (f (λ)), and (2) there exists N such that rank(f (λ)) N for all complex numbers λ. Then tr(f (λ)) = tr(f (λ0 )) for all λ in C. Proof. Since f takes finite-rank values and is entire, the function λ → tr(f (λ)) is also entire. For each λ, the trace of f (λ) is the sum, with multiplicity, of the eigenvalues of f (λ), so we may write it as a finite sum tr f (λ) = nα (λ)α, α∈σ (f (λ0 ))
where nα (λ) denotes the multiplicity of the eigenvalue α with respect to f (λ). But clearly
nα (λ)|α| rank f (λ) f (λ0 ) N f (λ0 ) ,
α∈σ (f (λ0 ))
which implies that the function λ → tr(f (λ)) is bounded. By Liouville’s Theorem, it now follows that tr(f (λ)) = tr(f (λ0 )) for all λ in C. 2 The next result extends [11, Lemma 4.2], and gives a symmetric trace condition which will be useful for our results. Lemma 3.12. Let A and B be bounded operators on a Banach space. If A is B-stable and one of A or B is of finite rank, then tr(An B) = tr(AB n ) = 0 for all n 1. Proof. First suppose that A is of finite rank. Since B is quasinilpotent by Lemma 3.6, the function ν → (1 − νB)−1 A is entire. Moreover, σ ((1 − νB)−1 A) = σ (A) for all ν in C by Lemma 3.10. Then, taking nth powers, the function v → ((1 − νB)−1 A)n is also entire, and σ (((1 − νB)−1 A)n ) = σ (An ) for all ν in C. Clearly rank(((1 − νB)−1 A)n ) rank(A), so tr(((1 − νB)−1 A)n ) = tr(An ) for all ν in C by Lemma 3.11. For |ν| < B−1 , we may expand (1 − νB)−1 A as a power series in ν, (1 − νB)−1 A =
B k Aν k .
k0
Hence n (1 − νB)−1 A =
k0
n k
B Aν
k
.
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The coefficient of ν k in the above expansion is B k A, and for n 1, the coefficient of ν is BAn + ABAn−1 + · · · + An−1 BA. But we may also expand the constant function tr(((1 − νB)−1 A)n ) as a power series in ν, and the linearity of the trace implies that for n = 1, the coefficient of ν k in this expansion is tr(B k A), and for n 1, that the coefficient of ν is tr BAn + ABAn−1 + · · · + An−1 BA = n tr An B . Comparing the coefficients on the left- and right-hand sides of the equation tr(((1−νB)−1 A)n ) = tr(An ) therefore gives tr(An B) = 0 for all n 1, and AB k = 0 for all k 1. Now suppose that B is of finite rank. The function ν → (1 − νA)−1 B is analytic for −1 / σ (A), with quasinilpotent values by Lemma 3.7. Taking nth powers, the function ν ∈ ν → ((1 − νA)−1 B)n is also analytic with quasinilpotent values for ν −1 ∈ / σ (A). This means tr(((1 − νA)−1 B)n ) = 0 for all ν ∈ / σ (A). As above, for |ν| < A, we may expand ((1 − νA)−1 B)n as a power series in ν, n (1 − νA)−1 B =
n Ak Bν k
.
k0
For n = 1, the coefficient of ν k in the above expansion is Ak B, and for n 1, the coefficient of ν is AB n + BAB n−1 + · · · + B n−1 A. Proceeding as before, we may also expand the constant function tr(((1 − νA)−1 B)n ) as a power series in ν, and the linearity of the trace implies that for n = 1, the coefficient of ν k in this expansion is tr(Ak B), and for n 1, that the coefficient of ν is tr AB n + BAB n−1 + · · · + B n−1 AB = n tr AB n . Comparing the coefficients of the left- and right-hand sides of the equation tr(((1 − νA)−1 B)n ) = 0 therefore gives tr(Ak B) = 0 for all k 1, and tr(AB n ) = 0 for all n 1. 2 4. Spectral conditions on a Lie algebra of compact operators In this section we study Lie algebras of compact operators which satisfy spectral properties like sublinearity and submultiplicativity. Recall that an operator Lie algebra L is a subspace of B(X ) which is closed under the Lie commutator product [A, B] = AB − BA, for A, B ∈ L. A Lie ideal I of L is a Lie subalgebra of L such that [A, B] ∈ I whenever A ∈ L and B ∈ I. For A in L, we define the bounded linear operator adA on L by adA (B) = [A, B]. Definition 4.1. An operator Lie algebra L is said to be an Engel Lie algebra if adA is quasinilpotent for every A in L. An ideal of L is said to be an Engel ideal if it is an Engel Lie algebra.
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There are two particularly important classes of Engel Lie algebras: (1) Commutative operator Lie algebras are Engel Lie algebras, since adA = 0 for every A ∈ L. (2) Operator Lie algebras in which every element is compact and quasinilpotent are Engel Lie algebras. Indeed, for A ∈ L, adA is the restriction of LA − RA to L, where LA and RA denote the (commuting) operators of left and right multiplication by A on B(X ) respectively. We always have σ (LA ) ⊆ σ (A) and σ (RA ) ⊆ σ (A), so LA and RA are quasinilpotent. It follows that LA − RA is quasinilpotent, and hence so is its restriction to L. We require the following two important results of Shulman and Turovskii. The first result from [10, Corollary 11.6] is an extension of Engel’s well-known theorem to Lie algebras of compact operators. The second result from [11, Corollary 4.20] establishes the abundance of finite-rank operators in an irreducible Lie algebra of compact operators. Theorem 4.2. A Lie algebra of compact operators which contains a nonzero Engel Lie ideal is reducible. In particular, an Engel Lie algebra of compact operators is triangularizable. Theorem 4.3. A uniformly closed Lie algebra of compact operators which does not contain any nonzero finite-rank nilpotent elements is triangularizable. For a general Lie algebra of compact operators, it is often more tractable to study its ideal of finite-rank operators due to such niceties as the existence of a trace, so we are especially interested in situations when the reducibility of this ideal implies the reducibility of the entire Lie algebra. Shulman and Turovskii raised the question [11] of whether a Lie algebra of compact operators is reducible whenever its ideal of finite-rank operators is reducible. That such a result holds for an associative algebra of compact operators is a consequence of Lomonosov’s Theorem (see for example [9, Theorem 7.4.7]). The next result, a generalization of [5, Lemma 2.3], provides at least one example of a situation in which this type of result is true in the Lie algebra case. Theorem 4.4. Let L be a uniformly closed Lie algebra of compact operators. If L contains a nonzero element A, with the property that tr(AF ) = 0 for every finite-rank operator F in L, then L is reducible. Proof. Let F denote the ideal of finite-rank operators in L, and let I = {A ∈ L: tr(AF ) = 0 for all F ∈ F }. Using the identity tr [A, B]F = tr A[B, F ] , it is easy to verify that I is a Lie ideal of L. Consider the Lie ideal IF = I ∩ F . If IF = 0, then I is triangularizable by Theorem 4.3, and thus L is reducible by Theorem 4.2. Hence we may suppose that IF = 0. We have tr(AB) = 0 for all A, B ∈ IF , which implies by [4, Theorem 4.5] that [IF , IF ] consists of nilpotent elements. Note that [IF , IF ] is a Lie ideal of L. Indeed, for A ∈ L and F, G ∈ IF ,
A, [F, G] = − F, [G, A] − G, [A, F ]
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by the Jacobi identity. Hence IF is triangularizable by Theorem 4.2. If [IF , IF ] = 0, then IF is a commutative Lie ideal, and hence is triangularizable by Theorem 4.2. Otherwise, the triangularizable ideal [IF , IF ] is nonzero. Either way, L has a nonzero triangularizable ideal, which implies by Theorem 4.2 that it is reducible. 2 Lemma 4.5. Let L be a Lie algebra of compact operators with a nonzero element T . If L has T -stable spectrum, then L is reducible. Proof. By the continuity of the spectrum of compact operators, we may suppose that L is uniformly closed. Let F be the Lie ideal of finite-rank operators in L. By Lemma 3.12, tr(F T ) = 0 for all F in F , so the result follows by Theorem 4.4. 2 Theorem 4.6. A Lie algebra of compact operators with subadditive spectrum is triangularizable. Proof. Let L be a Lie algebra of compact operators with sublinear spectrum. By the Triangularization Lemma and Lemma 2.2, it suffices to show the reducibility of L. As in the proof of Lemma 4.5, we may suppose that L is uniformly closed. Then by Theorem 4.3, if L does not contain a nonzero finite-rank nilpotent element, then it is reducible; hence we may suppose that some nonzero T in L is nilpotent, and of finite rank. Then by the hypothesis of sublinear spectrum, L has T -stable spectrum, so the result follows by Lemma 4.5. 2 Theorem 4.7. A Lie algebra of compact operators with submultiplicative spectrum is reducible. Proof. Let L be a Lie algebra of compact operators with submultiplicative spectrum. As in the proof of Lemma 4.5, we may suppose that L is uniformly closed. Let F be the ideal of finite-rank elements in L. By Theorem 4.3, if L does not contain a nonzero finite-rank nilpotent element, then it is reducible; hence we may suppose that some nonzero T in L is nilpotent, and of finite rank. Then for every F in F , σ (F T ) = 0 by the hypothesis of submultiplicative spectrum, so tr(F T ) = 0. Hence L is reducible by Theorem 4.4. 2 Theorem 4.8. Let L be a uniformly closed Lie algebra of compact operators with the property that tr(F GH ) = tr(F H G) whenever F, G, H ∈ L are of finite rank. Then L is reducible. Proof. Let F be the ideal of finite-rank operators in L. If F is commutative, then it is an Engel ideal of L, and L is reducible by Theorem 4.2. Hence we may suppose that [G, H ] = 0 for some G, H ∈ L. Then for all F in F , tr(F [G, H ]) = 0, so L is reducible by Theorem 4.4. 2 Theorem 4.9. Let L be a uniformly closed Lie algebra of compact operators with the property that tr(F 2 ) = 0 for every finite-rank element F in L. Then L is reducible. Proof. Let F be the ideal of finite-rank operators in L. Then for any F, G ∈ F , 0 = tr (F + G)2 = tr F 2 + tr G2 + tr(F G + GF ) = tr(F G + GF ) = 2 tr(F G),
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so tr(F G) = 0. If F = 0, then L is reducible by Theorem 4.3. Otherwise, L is reducible by Theorem 4.4. 2 5. Spectral conditions on a Jordan algebra of compact operators In this section we turn our attention to Jordan algebras, and show that most of the results we obtained for Lie algebras are also true in this setting. An operator Jordan algebra J is a subspace of B(X ) which is closed under the Jordan anticommutator product {A, B} = AB + BA, for A, B ∈ J . It is easy to verify that this is equivalent to J being closed under taking positive integer powers. A Jordan ideal I of J is a Jordan subalgebra of J such that {A, B} ∈ I whenever A ∈ J and B ∈ I. The methods of this section will differ from those used in the section on Lie algebras. This is mainly because for obtaining reducibility results, the closest Jordan analogue of an Engel Lie ideal will be a Jordan ideal in which every element is quasinilpotent. This is due to the following result, the Jordan analogue of Theorem 4.2, which was recently obtained in [5, Theorem 11.3]. Theorem 5.1. A Jordan algebra of compact operators with a nonzero ideal of quasinilpotent operators is reducible. In particular, a Jordan algebra of compact quasinilpotent operators is triangularizable. Let J be a uniformly closed Jordan algebra of compact operators, and let A be a nonquasinilpotent element of J . For nonzero λ in σ (A), it is well known that the Riesz projection Pλ of A corresponding to λ is of finite rank, and moreover that it may be written as a uniform limit of polynomials in A with zero constant coefficient. It follows that Pλ ∈ J . This fact, combined with Theorem 5.1, implies the following result. Theorem 5.2. A uniformly closed Jordan algebra of compact operators which does not contain any nonzero finite-rank operators is reducible. For Jordan algebras, we are also interested in situations when the reducibility of this ideal implies the reducibility of the entire Jordan algebra. The next result establishes the Jordan analogue of Theorem 4.4. Theorem 5.3. Let J be a uniformly closed Jordan algebra of compact operators. If J contains a nonzero element A with the property that tr(AF ) = 0 for every finite-rank operator F in J , then J is reducible. Proof. Let F denote the Jordan ideal of finite-rank elements in J , and let I = {A ∈ J : tr(AF ) = 0 for all F ∈ F }. Using the identity tr {A, B}F = tr A{B, F } = 0, it is straightforward to verify that I is a Jordan ideal of J . We claim that I consists of quasinilpotent elements. Indeed, suppose otherwise that for some A ∈ I, λ ∈ σ (A) is nonzero, and let Pλ be the Riesz projection of A corresponding to λ. Then Pλ belongs to J , and tr(Pλ APλ ) = nλ, where n is the spectral multiplicity of λ. But
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tr(Pλ APλ ) = tr APλ2 = tr(APλ ) = 0 by hypothesis, which gives a contradiction. The result now follows by Theorem 5.1.
2
Using the structure which is presented in a general Jordan algebra of compact operators, and applying what we know about operators with stable spectrum, we are able to prove the following result. Lemma 5.4. Let J be a Jordan algebra of compact operators with T in J . If J has T -stable spectrum, then AT is quasinilpotent for all A ∈ J . Proof. For A ∈ J , νA + ν 2 A2 + · · · + ν n An + λT ∈ J , for all n 1 and λ, ν ∈ C. Also, σ νA + ν 2 A2 + · · · + ν n An + λT = σ νA + ν 2 A2 + · · · + ν n An , so σ 1 + νA + ν 2 A2 + · · · + ν n An + λT = σ 1 + νA + ν 2 A2 + · · · + ν n An . For sufficiently small ν, taking n → ∞, and using the continuity of the spectrum of compact operators gives σ (1 − νA)−1 + λT = σ (1 − νA)−1 . Hence (1 − νA)−1 has T -stable spectrum, so by Lemma 3.7, (1 − νA)T = T − νAT is quasinilpotent for sufficiently small ν. For such ν, the subharmonic function ν → log(r(T − νAT )) satisfies log(r(T − νAT )) = −∞, so by H. Cartan’s Theorem, log(r(T − νAT )) = −∞, i.e. T − νAT is quasinilpotent for all ν ∈ C. But this means T has AT -stable spectrum, so AT is quasinilpotent by Lemma 3.6. 2 Lemma 5.5. Let J be a Jordan algebra of compact operators, and let T be a nonzero element of J . If J has T -stable spectrum, then J is reducible. Proof. By the continuity of the spectrum of compact operators, we may suppose that J is uniformly closed. Consider the Jordan ideal F of finite-rank operators in J . By Theorem 5.2, if F is zero then J is reducible; hence we may assume that F is nonzero. For A in J and F in F , {A, F } belongs to F , so by Lemma 3.12, tr {A, F }T = tr {F, T }A = 0. If {F, T } = 0 for some F in F , then J is reducible by Lemma 5.3. Otherwise, if {F, T } = 0 for all F in F , then
1 tr(T F ) = tr {F, T } = 0 2 for all F in F , which by Lemma 5.3 again implies the reducibility of J .
2
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Theorem 5.1 roughly says that a Jordan algebra of compact operators with too many quasinilpotent operators is reducible. The next result says that a Jordan algebra of compact operators with too few quasinilpotent operators is also reducible. For its proof, we will require two classical theorems. The Motzkin–Taussky Theorem states that a linear space of finite-rank diagonalizable operators is commutative (see for example [3, Theorem 2.7]). The Kleinecke–Shirokov Theorem states that if A and B are bounded operators on a Banach space, and if [A, [A, B]] = 0, then [A, B] is quasinilpotent (see for example [1, Theorem 5.1.3]). Theorem 5.6. Let J be a uniformly closed Jordan algebra of compact operators. If J does not contain any nonzero finite-rank nilpotent elements, then J is reducible. Proof. Suppose J does not contain any nonzero finite-rank nilpotent elements, and let F be the Jordan ideal of finite-rank elements in J . For F ∈ F , since J contains every polynomial in F with zero constant coefficient, in particular it contains the polynomial (α − F ), p(F ) = F α
where the product is taken over all nonzero α in σ (F ). Since some power of p(F ) is zero, p(F ) is nilpotent, and hence p(F ) = 0 by hypothesis, which implies that F is diagonalizable. Hence every element in F is diagonalizable, so by the Motzkin–Taussky Theorem, the elements of F pairwise commute. Fix some F ∈ F . For A in J , [F, [F, A]] is of finite rank and belongs to J , since
F, [F, A] = F, {F, A} − A, {F, F } . From above, we therefore have that F and [F, [F, A]] commute, i.e. that [F, [F, [F, A]]] = 0, so the Kleinecke–Shirokov Theorem implies that [F, [F, A]] is nilpotent. Since, by hypothesis, J does not contain any nonzero finite-rank nilpotent elements, [F, [F, A]] = 0, and applying the Kleinecke–Shirokov Theorem again implies that [F, A] is nilpotent. But then [F, A]2 is also finite-rank and nilpotent, and moreover, it belongs to J since [F, A]2 = {A, F }2 + 2 A2 , F 2 − A, A, F 2 − F, F, A2 . Hence [F, A]2 = 0, where we again use the hypothesis that J does not contain any nonzero finite-rank nilpotent elements. Let B ∈ J with nonzero β ∈ σ (B), and let P be the Riesz projection of B corresponding to β. Then P is of finite rank, so from above, for A ∈ J , 0 = [P , A]2 = (P A)2 − P A2 P − AP A + (AP )2 . Now consider (1 − P )AP + P A(1 − P ) = {A, P } − 2P AP , which belongs to J . We have 2 (1 − P )AP + P A(1 − P ) = (1 − P )AP A(1 − P ) + P A(1 − P )AP = − (P A)2 − P A2 P − AP A + (AP )2 = −[P , A]2 = 0,
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which implies that the finite-rank element (1 − P )AP + P A(1 − P ) is nilpotent. Since by hypothesis, J does not contain any nonzero finite-rank nilpotent elements, it follows that (1 − P )AP + P A(1 − P ) = 0 for all A ∈ J . We have 0 = (1 − P )AP + P A(1 − P ) = AP + P A − 2P AP , so AP = 2P AP − P A. Multiplying on the left by P then gives P AP = 2P AP − P A = P A, so it follows that the range of P is invariant under A. Since P was chosen to be nontrivial, this shows that J is reducible. 2 Theorem 5.7. A Jordan algebra of compact operators with subadditive spectrum is triangularizable. Proof. Let J be a Jordan algebra of compact operators with sublinear spectrum. By the continuity of the spectrum of compact operators, we may suppose that J is uniformly closed. As in Theorem 4.6, it suffices to show the reducibility of J . By Theorem 5.6, if J does not contain any nonzero quasinilpotent elements, then J is reducible; hence we may suppose that J contains a nonzero quasinilpotent element T . By the hypothesis of sublinear spectrum, J has T -stable spectrum, so the result now follows from Lemma 5.5. 2 Theorem 5.8. A Jordan algebra of compact operators with submultiplicative spectrum is reducible. Proof. Let J be a Jordan algebra of compact operators with submultiplicative spectrum. As in the proof of Theorem 5.7, we may suppose that J is uniformly closed. Let F be the ideal of finite-rank elements in J . By Theorem 5.6, if F does not contain a nonzero nilpotent element, then J is reducible; hence we may suppose that some nonzero T in F is nilpotent. Then for every F in F , σ (F T ) = 0 by the hypothesis of submultiplicative spectrum, so tr(F T ) = 0. Hence J is reducible by Theorem 4.4. 2 Corollary 5.9. Let J be a uniformly closed Jordan algebra of compact operators with the property that tr(F 2 ) = 0 for every finite-rank element F in J . Then J is reducible. Proof. Let F be the ideal of finite-rank elements in J . The hypothesis implies that every F ∈ F is nilpotent; indeed, otherwise some F ∈ F would have nonzero α ∈ σ (F ), and the Riesz projection of F corresponding to α would have tr(P 2 ) = 0, since σ (P ) = {0, 1}. Hence the result follows by Theorem 5.1. 2 Acknowledgments The authors thank Victor Shulman and Yuri Turovskii for their helpful comments and suggestions. References [1] B. Aupetit, A Primer on Spectral Theory, Universitext, Springer-Verlag, New York, 1991.
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[2] R.M. Guralnick, Triangularization of sets of matrices, Linear Multilinear Algebra 9 (1980) 133–140. [3] T. Kato, Perturbation Theory for Linear Operators, second ed., Grundlehren Math. Wiss., vol. 132, Springer-Verlag, Berlin, 1976. [4] M. Kennedy, Triangularization of a Jordan algebra of Schatten operators, Proc. Amer. Math. Soc. 136 (7) (2008) 2521–2527. [5] M. Kennedy, V. Shulman, Yu.V. Turovskii, Invariant subspaces of subgraded Lie algebras of compact operators, Integral Equations Operator Theory 63 (2008) 47–93. [6] M. Lambrou, W. Longstaff, H. Radjavi, Spectral conditions and reducibility of operator semigroups, Indiana Univ. Math. J. 41 (1992) 449–464. [7] T.S. Motzkin, O. Taussky, Pairs of matrices with property L, Trans. Amer. Math. Soc. 73 (1952) 108–114. [8] H. Radjavi, Sublinearity and other spectral conditions on a semigroup, Canad. J. Math. 52 (1) (2000) 197–224. [9] H. Radjavi, P. Rosenthal, Simultaneous Triangularization, Universitext, Springer-Verlag, New York, 2000. [10] V.S. Shulman, Yu.V. Turovskii, Joint spectral radius, operator semigroups, and a problem of W. Wojtynski, J. Funct. Anal. 177 (2000) 383–441. [11] V. Shulman, Yu.V. Turovskii, Invariant subspaces of operator Lie algebras and Lie algebras with compact adjoint action, J. Funct. Anal. 223 (2) (2005) 425–508. [12] E. Vesentini, On the subharmonicity of the spectral radius, Boll. Unione Mat. Ital. 4 (1) (1968) 427–429. [13] D.B. Wales, H.J. Zassenhaus, On L-groups, Math. Ann. 198 (1972) 1–12. [14] H.J. Zassenhaus, On L-semigroups, Math. Ann. 198 (1972) 13–22.
Journal of Functional Analysis 256 (2009) 3158–3191 www.elsevier.com/locate/jfa
Approximate and pseudo-amenability of various classes of Banach algebras Y. Choi 1 , F. Ghahramani ∗,2 , Y. Zhang 3 Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 Received 20 May 2008; accepted 19 February 2009 Available online 10 March 2009 Communicated by N. Kalton
Abstract We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors together with R.J. Loy. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity, and that the Fourier algebra of the free group on two generators is not operator approximately amenable. Further examples are obtained of 1 -semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate contractibility need not imply sequential approximate amenability. Results are also given for Segal algebras on locally compact groups, and algebras of p-pseudo-functions on discrete groups. © 2009 Elsevier Inc. All rights reserved. Keywords: Amenable Banach algebra; Amenable group; Approximately amenable Banach algebra; Approximate diagonal; Approximate identity; Fourier algebra; Segal algebra; Semigroup algebra; Reduced C ∗ -algebra
* Corresponding author.
E-mail addresses:
[email protected] (Y. Choi),
[email protected] (F. Ghahramani),
[email protected] (Y. Zhang). 1 Current address: Département de mathématiques et de statistique, Pavillon Alexandre-Vachon, Université Laval, Québec (Québec), Canada G1V 0A6. 2 Supported by NSERC grant 36640-07. 3 Supported by NSERC grant 238949-05. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.012
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1. Introduction In this article we continue the investigation of various notions of approximate amenability, initiated in [11] and continued in papers by various authors: see, for instance, [5,14,21,25]. Most of this paper is taken up with consideration of certain classes of Banach algebras, such as Fourier algebras, Segal subalgebras of L1 (G), certain 1 -semigroup algebras and the algebras PF p (Γ ) — where Γ is a discrete group — and the problem of determining when such algebras are approximately amenable or pseudo-amenable (the definitions will be given below). The contents of the paper are as follows. After establishing some background definitions and notation in Section 2, we discuss some results for general Banach algebras. If A is a Banach algebra with an approximate diagonal, the forced unitization A# need not possess an approximate diagonal (for example we may take A to be 1 with pointwise multiplication and apply [5, Theorem 4.1]). On the other hand, it follows from [14, Proposition 3.2] that if A has a bounded approximate identity and an approximate diagonal, then A# also has an approximate diagonal. We present a partial extension of this result to the case of multiplier-bounded approximate diagonals: namely, we show that if A has a central b.a.i. and a multiplier-bounded approximate diagonal, then so does A# . One outstanding basic question in this area is the following: does every approximately amenable algebra have a bounded approximate identity? Although we are not able to resolve this here, we obtain some general results (Section 3) showing that slightly stronger notions of approximate amenability guarantee the existence of a bounded approximate identity. As a consequence we are able to show that several classes of Banach algebras, which might plausibly be pseudo-amenable, cannot be boundedly approximately amenable: these include the Schatten classes Sp (H ) for 1 p < ∞ and H a Hilbert space, and all proper Segal algebras on locally compact groups. A related argument shows that for any infinite compact metric space X, the Lipschitz algebra lipα (X), 0 < α 1, is not boundedly approximately contractible. The last four sections are largely independent of each other and can be read interchangeably. Section 4 resolves a question from [13], by showing that the Fourier algebra of A(F2 ) is not (operator) approximately amenable. The proof uses direct manipulation of norm estimates, which rely on the “rapid decay” estimates known for F2 × F2 . It then follows from known restriction theorems that whenever a locally compact group G contains F2 as a closed subgroup, A(G) is not (operator) approximately amenable. Section 5 collects some results on approximate notions of amenability for Segal algebras on locally compact groups. It is observed that Feichtinger’s Segal algebra on an infinite compact abelian group is not approximately amenable. We also show that if S(G) is pseudo-contractible for some Segal subalgebra S(G) ⊆ L1 (G), then G must be compact. It is also shown that whenever G is an SIN-group, every Segal subalgebra S(G) ⊆ L1 (G) is approximately permanently weakly amenable: our proof uses the recent solution by Losert to the derivation problem for group algebras [23]. Section 6 is devoted to the 1 -convolution algebras of totally ordered sets: when the sets are infinite, these algebras are never amenable. We show that these algebras are always boundedly approximately contractible (and in particular are boundedly approximately amenable), but need not be sequentially approximately amenable. This strengthens the observation in [12] that the convolution algebra 1 (Ω∧ ) is boundedly approximately contractible but not sequentially approximately contractible. Finally, in Section 7 we consider the algebras of p-pseudo-functions on discrete groups. As a special case of the results in this section, we show that if Γ is a discrete non-amenable group
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then its reduced C ∗ -algebra is not approximately amenable. This gives some evidence for the tentative conjecture that all approximately amenable C ∗ -algebras are automatically amenable. Further evidence is provided by the fact that not only are the C ∗ -algebras B(H ) and n Mn (C) not amenable, but they are not even approximately amenable — this observation appears to be due to Ozawa, see the remark after Definition 1.2 in [24]. 2. Preliminaries 2.1. Definitions and notation Throughout, if A is a Banach algebra we shall write A# for the forced unitization of A. The adjoined identity element will usually be denoted by 1 unless stated otherwise. A → A that is specified by We will frequently use π to denote the bounded linear map A ⊗ π(a ⊗ b) = ab (a, b ∈ A). Recall that a Banach algebra A is said to be approximately amenable if for every Abimodule X and every bounded derivation D : A → X ∗ there exists a net (Dα ) of inner derivations such that limα Dα (a) = D(a) for all a ∈ A. A is said to be: – boundedly approximately amenable if the net (Dα ) can always be taken to be bounded (in the usual norm of L(A, X ∗ )); – sequentially approximately amenable if the net (Dα ) can always be taken to be a sequence. By the uniform boundedness principle (or a more direct Baire category argument) one sees that sequential approximate amenability implies bounded approximate amenability. The converse is not in general true, as will be shown in Section 6 by combining Theorems 6.1 and 6.4. A Banach algebra A is approximately contractible if for every continuous derivation D : A → X, where X is a Banach A-bimodule, there exists a net (Di ) of inner derivations such that limi Di (a) = D(a) for all a ∈ A. The corresponding variants of bounded and sequential approximate contractibility are defined in analogous fashion to the corresponding notions of approximate amenability. Remark. It is shown in [12, Theorem 2.1] that the concepts of approximate contractibility and approximate amenability are in fact equivalent. However, this is not true for the corresponding sequential variants, and remains unknown (at present) for the bounded variants. It has proved very useful in the classical theory of amenability to have characterizations in terms of virtual diagonals or approximate diagonals. In much of this paper we shall work with approximate diagonals rather than nets of derivations. To fix terminology we recall the following definition. A Definition 2.1. Let A be a Banach algebra. An approximate diagonal for A is a net (Mi ) in A ⊗ such that, for each a ∈ A, aMi − Mi a → 0 and aπ(Mi ) → a.
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We say that the approximate diagonal (Mi ) is multiplier-bounded if there exists a constant K such that for all a ∈ A and all i, each of aMi − Mi a ,
aπ(Mi ) − a and π(Mi )a − a
(2.1)
is bounded by K a . The following equivalence is easily verified. Proposition 2.2. A Banach algebra A is boundedly approximately contractible if and only if A# has a multiplier-bounded approximate diagonal. We shall also make brief use of the notions of pseudo-amenability and pseudo-contractibility. For convenience we recall the relevant definitions from [14]. Definition 2.3. Let A be a Banach algebra. We say that A is pseudo-amenable if it has an approximate diagonal, and pseudo-contractible if it has an approximate diagonal (Mi ) which satisfies aMi = Mi a for all a ∈ A and all i. 2.2. Basics Proposition 2.4. Let S be one of the following classes of Banach algebras: approximately amenable, approximately contractible, sequentially approximately amenable, sequentially approximately contractible, boundedly approximately amenable, boundedly approximately contractible. Let A be a Banach algebra. Then A ∈ S if and only if A# ∈ S. Proof. The case of approximate amenability is given by [11, Proposition 2.4], and in fact the proofs for all the other cases follow the same argument. The key points are that (i) every derivation from A can be extended to a derivation from A# , such that the extended derivation is inner if and only if the original one was; (ii) if D is a derivation from A# to an A-bimodule X, and e denotes the identity of A# , then there is an inner derivation D1 : A# → X such that (D − D1 )(e) = 0. 2 Remark. Note that the proofs of “A approximately contractible ⇔ A# approximately contractible” and “A approximately amenable ⇔ A# approximately amenable” do not rely on the fact that approximate contractibility and approximate amenability are equivalent. Theorem 2.5. Let A be a boundedly approximately contractible Banach algebra. Then there exist A and nets (Fi ), (Gi ) in A such that a constant C > 0 and nets (Mi ) in A ⊗ (i) (ii) (iii) (iv) (v) (vi) (vii)
π(Mi ) = Fi + Gi ; aFi → a for all a ∈ A; aFi C a for all a ∈ A and all i; Gi a → a for all a ∈ A; Gi a C a for all a ∈ A and all i; aMi − Mi a − a ⊗ Gi + Fi ⊗ a → 0 for all a ∈ A; aMi − Mi a − a ⊗ Gi + Fi ⊗ a C a for all a ∈ A and all i.
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For the sake of completeness we give the proof. A# as an A-bimodule in the usual way. Let K be the kernel of the product Proof. Regard A# ⊗ # # map A ⊗ A → A# and let D : A → K be the derivation defined by D(a) = a ⊗ 1 − 1 ⊗ a. Since A is boundedly approximately contractible, there exists a net (ui ) in K such that C := sup sup aui − ui a < ∞ i a 1
and aui − ui a → D(a) for all a ∈ A. A# with the 1 -direct sum A ⊗ A ⊕ A ⊗ 1 ⊕ 1 ⊗ A ⊕ C1 ⊗ 1 , we may Identifying A# ⊗ A and some write each ui in the form ui = (−Mi ) ⊕ (Fi ⊗ 1 ) ⊕ (1 ⊗ Gi ) for some Mi ∈ A ⊗ Fi , Gi ∈ A. We shall show that the nets (Mi ), (Fi ) and (Gi ) have the required properties. First, note that since ui ∈ K for all i we must have 0 = π(ui ) = −π(Mi ) + Fi + Gi
for all i.
Next, since aui − ui a = (−aMi + Mi a + a ⊗ Gi − Fi ⊗ a) ⊕ aFi ⊗ 1 ⊕ (−1 ⊗ Gi a), where the left-hand side is bounded in norm by C a for all a, we have aFi C a , Gi a C a and aMi − Mi a − a ⊗ Gi + Fi ⊗ a C a for all i and all a. Finally, for each a in A we have a ⊗ 1 − 1 ⊗ a = D(a) = lim aui − ui a i = lim (−aMi + Mi a + a ⊗ Gi − Fi ⊗ a) ⊕ aFi ⊗ 1 ⊕ (−1 ⊗ Gi a) i
and matching up terms we conclude that a = lim aFi = lim Gi a i
as required.
i
and 0 = lim aMi − Mi a − a ⊗ Gi + Fi ⊗ a, i
2
Remark. It follows from this that every boundedly approximately contractible Banach algebra has a multiplier-bounded right approximate identity and a multiplier-bounded left approximate identity. We shall use this later, in Section 3. Let κ denote the canonical embedding of A into A∗∗ . We have the following analogue of Theorem 2.5. Theorem 2.6. Let A be a boundedly approximately amenable Banach algebra. Then there exist A)∗∗ and nets (Fi ), (Gi ) in A∗∗ such that a constant C > 0 and nets (Mi ) in (A ⊗
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π(Mi ) = Fi + Gi ; aFi → κ(a) for all a ∈ A; aFi C a for all a ∈ A and all i; Gi a → κ(a) for all a ∈ A; Gi a C a for all a ∈ A and all i; aMi − Mi a − a ⊗ Gi + Fi ⊗ a → 0 for all a ∈ A; aMi − Mi a − a ⊗ Gi + Fi ⊗ a C a for all a ∈ A and all i.
We omit the proof: the argument exactly follows the one for Theorem 2.5. 2.3. Two lemmas using approximate diagonals We record some lemmas here which will be used later. Both are natural adaptations of routine arguments from the setting of amenable Banach algebras. Lemma 2.7. Let B be a unital Banach algebra with identity element 1 , A ⊆ B a closed subalgebra that contains 1 , and suppose that there exists a tracial continuous functional τ on A such that τ (1 ) = 1. If A is pseudo-amenable, then there exists a net (ψα ) in B ∗ such that ψα (1 ) → 1 and ψα (ab − ba)→ 0
sup
b∈B, b 1
for any a ∈ A.
Note that by a trivial rescaling, the net (ψα ) in the conclusion of our lemma can be chosen such that ψα (1 ) = 1 for all α. However, the formally weaker property ψα (1 ) → 1 will suffice for our intended application. Proof. Let (uα ) be an approximate diagonal for A: note that since A has an identity element1 , π(uα ) → 1 . For each α we may write uα = i ciα ⊗ diα , where ciα , diα ∈ A for all i and i ciα diα < ∞. Let τ ∈ B ∗ be any bounded extension of τ to a functional on B, and define
α α ψα (S) = (S ∈ B). τ di Sci i
Then since τ is a trace on A, ψα (1 ) = τ
i
diα ciα
=τ
ciα diα
= τ π(uα ) → τ (1 )
i
and by hypothesis τ (1 ) = 1. A)∗ by For fixed b ∈ B, define a functional φb ∈ (A ⊗ τ (ybx) φb (x ⊗ y) =
(x, y ∈ A).
τ b . By definition of the projective tensor norm, we have φb
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Now for each a ∈ A, the tracial property of τ gives φb (uα a) = φb
ciα ⊗ diα a = τ diα abciα = ψα (ab)
i
i
and φb (auα ) = φb
aciα
⊗ diα
α α = τ di baci = ψα (ba).
i
i
Therefore sup
ψα (ab − ba) =
b∈B, b 1
sup
b∈B, b 1
φb (auα − uα a)
τ auα − uα a → 0 for each a ∈ A. Thus (ψα ) has the required properties.
2
Our second lemma will be needed for the proof of Theorem 6.4. It says, loosely, that the Gelfand transforms of an approximate diagonal must converge pointwise to the indicator function of the diagonal of the character space. Lemma 2.8. Let A be a Banach algebra with non-empty character space ΦA , and suppose A has a (two-sided) bounded approximate identity. If A is approximately amenable, then there exists a A)∗∗ with the following properties: net ( α ) in (A ⊗ (i) limα α , ϕ ⊗ χ = 0 for any ϕ, χ ∈ ΦA with ϕ = χ ; (ii) α , ϕ ⊗ ϕ = 1 for all α and any ϕ ∈ ΦA . Moreover, if A is sequentially approximately amenable, we can take ( α ) to be a sequence. Before giving the proof we note that one could instead appeal to a modification of the proof of [14, Proposition 3.2]. However, since that proposition does not deal explicitly with the sequential case, which will be crucial in our intended application, we have chosen a more direct and only slightly longer argument. Proof of Lemma 2.8. We shall only prove the statement in the case where A is sequentially approximately amenable (the case where we merely assume A to be approximately amenable is completely analogous). Thus, suppose A has a bounded approximate identity and is sequentially approximately amenable. Let E be any weak∗ -limit point in A∗∗ of the bounded approximate identity of A, so that aE = Ea = κ(a) ∈ A∗∗ , κ denoting the canonical embedding of A in its bidual. A → A be the product map and let K = ker π ; this is a sub-A-bimodule of Let π : A ⊗ A. Define a bounded derivation D : A → K ∗∗ by D(a) = a ⊗ E − E ⊗ a (a ∈ A). Since A A⊗ is sequentially approximately amenable there exists a sequence (un ) in K ∗∗ such that a · un − un · a → D(a) for all a ∈ A.
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A)∗∗ . We have Define n = E ⊗ E − un ∈ (A ⊗
n , ϕ ⊗ ϕ = ϕ, E 2 − un , ϕ ⊗ ϕ = 1 − un , π ∗ (ϕ) = 1 − π ∗∗ (un ), ϕ = 1 for all n. Moreover, if ϕ and χ are distinct characters on A, then there exists a ∈ A with ϕ(a) = χ(a). Then
a · un − un · a, ϕ ⊗ χ = un , (ϕ ⊗ χ) · a − un , a · (ϕ ⊗ χ)
= un , ϕ(a)ϕ ⊗ χ − un , χ(a)ϕ ⊗ χ = ϕ(a) − χ(a) un , ϕ ⊗ χ , while
D(a), ϕ ⊗ χ = a ⊗ E − E ⊗ a, ϕ ⊗ χ = ϕ(a) − χ(a)
so that, since a · un − un · a → D(a), ϕ(a) − χ(a) = ϕ(a) − χ(a) lim un , ϕ ⊗ χ . n
Since ϕ(a) − χ(a) = 0, this implies that 1 = limn un , ϕ ⊗ χ , and so lim n , ϕ ⊗ χ = E ⊗ E, ϕ ⊗ χ − lim un , ϕ ⊗ χ = 0, n
as required.
n
2
3. General results Recall that A is approximately contractible if and only if A# has an approximate diagonal (this is [11, Proposition 2.6(a)]). We would like to have a better understanding of just when the presence of an approximate diagonal in A guarantees an approximate diagonal in A# , and to obtain corresponding results for multiplier-bounded approximate diagonals. Note that by combining the proof of (ii) ⇒ (iii) in [14, Proposition 3.2] with (3) ⇒ (1) of [12, Theorem 2.1], we obtain the following result. Proposition 3.1. Let A be a Banach algebra which has a bounded approximate identity and an approximate diagonal. Then A is approximately contractible, and so A# has an approximate diagonal. A natural hope is that the result just stated remains true if we replace ‘approximate diagonal’ with ‘multiplier-bounded approximate diagonal’. We have been unable to verify this: the problem seems to be that while approximate amenability implies approximate contractibility, it is not known if bounded approximate amenability implies bounded approximate contractibility. The following result gives some partial answers.
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Proposition 3.2. Let A be a Banach algebra with a central b.a.i. (eλ )λ∈Λ . Suppose that A has a multiplier-bounded approximate diagonal (Mi ). Then A# has a multiplier-bounded approximate diagonal, and so A is boundedly approximately contractible. Proof. Throughout we denote the adjoined unit of A# by 1 , and the linearized product map A# → A# by π . We shall abuse notation and also use π to denote the restricted map A# ⊗ A → A. A⊗ A# and a constant K > 0 such that: We shall construct a net (nj ) in A# ⊗ π(nj ) − 1 K and b · nj − nj · b K b for all b ∈ A and all j ; (3.1) and lim π(nj ) = 1 j
lim(b · nj − nj · b) = 0
and
j
for all b ∈ A and all j.
(3.2)
If these properties are satisfied, it is then straightforward to show that the net (nj ) has the required properties in Definition 2.1. By hypothesis there exist constants C and K1 such that eλ C for all λ and such that, for all a ∈ A and all i, aπ(Mi ) − a K1 a and a · Mi − Mi · a K1 a . (3.3) Moreover, for any a ∈ A, we have lim aπ(Mi ) = a i
and
lim a · Mi − Mi · a = 0. i
(3.4)
To simplify the ensuing formulas slightly, we let uλ := 2eλ − eλ2 for each λ: note that uλ + (1 − eλ )2 = 1 and uλ 2C + C 2 , for all λ. We now set mλ,i := uλ · Mi + (1 − eλ ) ⊗ (1 − eλ ),
(3.5)
π(mλ,i ) = uλ π(Mi ) + 1 − uλ .
(3.6)
so that
Let I and Λ be the index sets for the nets (Mi ) and (eλ ), respectively. We construct the an iterated limit construction (see [19, p. 26]). Our indexing directed set is required net (nj ) using defined to be J = Λ × λ∈Λ I, equipped with the product ordering, and for each j = (λ, f ) ∈ J we define nj = mλ,f (λ) . Fix λ and i. Using (3.3) and (3.6) gives π(mλ,i ) − 1 = uλ π(Mi ) − uλ K1 uλ K1 2C + C 2 . Also, since each eλ lies in the centre of A, we have for any b ∈ A the identity b · mλ,i − mλ,i · b = uλ b · Mi − uλ · Mi · b + (b − beλ ) ⊗ (1 − eλ ) − (1 − eλ ) ⊗ (b − beλ ).
(3.7)
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Using (3.3) again gives b · mλ,i − mλ,i · b uλ b · Mi − Mi · b + 2 b 1 − eλ 2 CK1 + 2(1 + C)2 b for any b ∈ A. Since λ and i were arbitrary, we have shown that (3.1) holds with, say, K = 2(1 + K1 )(1 + C)2 . It remains to show that (3.2) holds. Using (3.4) and (3.6) we have, for every λ, lim π(mλ,i ) = lim uλ π(Mi ) + 1 − uλ = 1 ; i
i
hence, by [19, Theorem 2.4], lim π(nj ) = lim lim π(mλ,i ) = 1 . j
λ
i
Using (3.4) and (3.7) we have, for every λ, lim b · mλ,i − mλ,i · b = −(b − beλ ) ⊗ (1 − eλ ) + (1 − eλ ) ⊗ (b − beλ ); i
therefore, since (eλ ) is a bounded approximate identity for A, applying [19, Theorem 2.4] we obtain lim(b · nj − nj · bj ) = lim lim(b · mλ,i − mλ,i · b) = 0. j
Thus (3.2) holds and our proof is complete.
λ
i
2
Remark. The result is false if we do not require the central approximate identity in A to be bounded: an example is given by 1 (N) equipped with pointwise multiplication [5, Theorem 4.1]. It is still open whether an approximately amenable Banach algebra must have a bounded approximate identity. If this were the case then one could extend many of the known hereditary properties of amenability to hold for approximate amenability. All presently known examples of approximately amenable Banach algebras have a bounded approximate identity. In addition, all known examples of approximately amenable Banach algebras are in fact boundedly approximately contractible. These last two observations are connected by the following result: every boundedly approximately contractible algebra has a bounded approximate identity (Corollary 3.4 below). We are able to prove a slightly stronger technical result, that allows us to rule out bounded approximate amenability for several classes of Banach algebras. Theorem 3.3. Suppose that the Banach algebra A is boundedly approximately amenable, and has both a multiplier-bounded left approximate identity and a multiplier-bounded right approximate identity. Then A has a bounded approximate identity.
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Proof. Let (eα ) and (fβ ) be, respectively, right and left multiplier-bounded approximate identities for A, so that there exists a constant K > 0 such that aeα K a and fβ a K a for all a ∈ A and all α, β. From this we obtain the following estimates: A and every α, β; (i) fβ · m K m and m · eα K m , for every m ∈ A ⊗ A)∗∗ and every α, β. (ii) fβ · T K T and T · eα K T , for every T ∈ (A ⊗ (The first pair of estimates follows easily from the definition of the projective tensor norm. The second pair follows from the first pair using Goldstine’s theorem and the weak∗ -continuity of the A)∗∗ .) actions of A on (A ⊗ Let (Fi ), (Gi ), (Mi ) and C be the nets (respectively, the constant) satisfying (ii)–(vi) of Theorem 2.6. Suppose that the net (fβ ) is (norm) unbounded. We derive a contradiction as follows. For every i and every β we have fβ · Mi − Mi · fβ − fβ ⊗ Gi + Fi ⊗ fβ C fβ , and so by (ii) above, we have (fβ · Mi − Mi · fβ − fβ ⊗ Gi + Fi ⊗ fβ )eα KC fβ
(3.8)
for every α, β and i. Using the triangle inequality and the left multiplier-boundedness of the set {fβ }, from (3.8) we have fβ Gi · eα KC fβ + fβ · (Mi · eα ) + Mi · (fβ eα ) + Fi fβ eα KC fβ + K Mi · eα + K Mi eα + K Fi eα
(3.9)
for every α, β and i. Hence Gi · eα KC +
K Mi · eα + Mi eα + Fi eα fβ
(3.10)
for every α, β and i. For fixed α and i, combining (3.10) with our assumption that {fβ } is an unbounded set yields Gi · eα KC. Taking limits with respect to i, we then obtain eα KC for each α. But since (eα ) is a right approximate identity and {fγ } is a left multiplier-bounded set, we obtain fγ = lim fγ eα lim sup K eα K 2 C α
α
for all γ . This contradicts our assumption that the net (fβ ) is unbounded. A similar argument, with left and right interchanged, shows that the net (eα ) is also bounded; the existence of a bounded approximate identity is now standard. 2
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Corollary 3.4. Let A be boundedly approximately contractible. Then A has a bounded approximate identity. Proof. This is an immediate consequence of Theorem 3.3, since by Theorem 2.5 every boundedly approximately contractible Banach algebra has a right and a left multiplier-bounded approximate identity. 2 Corollary 3.5. Suppose that A and B are boundedly approximately contractible Banach algebras. Then the direct sum A ⊕ B is boundedly approximately contractible. Proof. This follows from the proof of [11, Proposition 2.7] and Corollary 3.4.
2
The following result is similar to Theorem 3.3, but seems not to imply it nor be implied by it. Proposition 3.6. Suppose that the Banach algebra A is boundedly approximately amenable. Let S be a subset of A which is (left and right) multiplier-bounded, i.e. for some K > 0, we have sa K a and as K a for all a ∈ A, s ∈ S. Then S is norm bounded. Proof sketch. Arguing as at the start of the proof of Theorem 3.3, we note that for every s ∈ S: A; (i) s · m K m and m · s K m , for every m ∈ A ⊗ A)∗∗ . (ii) s · T K m and T · s K s , for every T ∈ (A ⊗ Suppose that S is (norm) unbounded, so that there exists a sequence (sn ) in S with sn → ∞. Then we may argue as in the proof of Theorem 3.3, replacing eα with sm and fβ with sn in Eqs. (3.8)–(3.10), to show that the sequence (sm ) is bounded, giving us a contradiction as before. Hence S is (norm) bounded as claimed. 2 Examples 3.7. The following algebras have multiplier-bounded approximate identities but have no bounded approximate identities. (a) c0 (ω), the space of all sequences such that |an |ωn → 0, equipped with pointwise multiplication, where limn ωn = +∞. (b) 1 (Nmin , ω), the weighted convolution algebra of the semilattice Nmin , where limm ωn = ∞. (c) The Schatten ideals Sp (H ) (H a Hilbert space) where 1 p < ∞. (d) The Fourier algebras of weakly amenable, non-amenable groups (see [6] for the definition and examples). (e) Proper symmetric Segal subalgebras (in the sense of Reiter [27]) of L1 (G). It therefore follows from Theorem 3.3 that none of the above algebras can be boundedly approximately amenable. Remark. It has recently been shown (H.G. Dales and R.J. Loy, private communication) that the algebras of Example 3.7(b) are not even approximately amenable. We can exploit Corollary 3.4 further to show that certain unital Banach algebras are not boundedly approximately contractible.
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Corollary 3.8. Let X be an infinite, compact metric space and let 0 < α 12 . Then the Lipschitz algebra lipα (X) is not boundedly approximately contractible. Proof. Since X is infinite and compact it contains a non-isolated point, x0 say. Let M = {f ∈ lipα (X): f (x0 ) = 0}: then M # ∼ = lipα (X). If lipα (X) were boundedly approximately contractible, then by Proposition 2.4 M would also be boundedly approximately contractible, and hence by Corollary 3.4 would have a bounded approximate identity. By Cohen’s factorization theorem, this would imply that M 2 = M, which is easily seen to be false by considering the function f : x → d(x, x0 )β where α < β < 2α (see also the remarks at the end of [1]). 2 Remark. The above proof works also for 12 < α < 1. However, it is already known that in the latter case lipα (X) is not even approximately amenable since it is abelian and not weakly amenable [1]. F2 ) is not approximately amenable 4. A(F Let F2 denote the free group on two generators. It was observed in [13, Remark 3.4(b)] that A(F2 ) is pseudo-amenable, and the authors asked if it is approximately amenable. In this section we shall answer this question in the negative; indeed, we prove the formally stronger result that A(F2 ) is not even operator approximately amenable. Our techniques are based on direct estimates, exploiting the fact that the norm in A(F2 × F2 ) majorizes a certain weighted 2 -norm. Some consequences for Fourier algebras of more general groups will be given at the end of the section. Background material We state the required definitions and basic properties in the setting of discrete groups, since we will eventually specialize to F2 : some hold in greater generality, but we shall not discuss this here. Let Γ be a discrete group and C00 (Γ ) the space of compactly supported functions on Γ . The Fourier algebra A(Γ ) can be defined as the completion of C00 (Γ ) with respect to the norm f A(Γ ) = inf ξ 2 η 2 : ξ, η ∈ 2 (Γ ); f = ξ ∗ η
f ∈ C00 (Γ ) .
Let λ : 1 (Γ ) → B(2 (Γ )) denote the (faithful) left regular representation of 1 (Γ ) on 2 (Γ ). The WOT-closure of the image of λ is the von Neumann algebra of Γ , and will here be denoted by VN(Γ ). We can identify A(Γ ) with the predual of the group von Neumann algebra VN(Γ ): the pairing between the two satisfies T (g)f (g) λ(T ), f =
f ∈ A(Γ ), T ∈ C00 (Γ ) ,
g∈Γ
from which the following is immediate. Lemma 4.1. For every f ∈ A(Γ ) and every T ∈ C00 (Γ ) we have f (x)T (x) f A(Γ ) λ(T ). x∈Γ
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The norm on A(Γ ) is in general hard to describe, but there are easy upper and lower bounds. Specifically we have 2 (Γ ) ⊆ A(Γ ) and f ∞ f A(Γ ) f 2
for all f ∈ 2 (Γ ).
(4.1)
The following definition first appeared in [13]. Definition 4.2. Let A be a quantized Banach algebra. A is operator approximately amenable if, for each quantized Banach A-bimodule X, every completely bounded derivation A → X ∗ is approximately inner. Clearly, if A is a quantized Banach algebra which happens to be approximately amenable, then it is operator approximately amenable. The following is a ‘quantized’ version of one direction of [5, Proposition 3.3], specialized to the cases of interest. Proposition 4.3. Let Γ be a discrete group and suppose that A(Γ ) is operator approximately amenable. Then for every finite set S ⊂ A(Γ ) and every ε > 0, there exists F ∈ c00 (Γ × Γ ) such that (i) a · F − F · a − a ⊗ π(F ) + π(F ) ⊗ a A(Γ ×Γ ) ε; (ii) a − aπ(F ) A(Γ ) ε for every a ∈ S. For convenience we give a brief outline of how Proposition 4.3 follows from existing results. op to denote the operator projective tensor product of two operator Proof sketch. We use ⊗ op A(Γ )# is a quantized Banach spaces. Since A(Γ ) is a quantized Banach algebra, A(Γ )# ⊗ A(Γ )-bimodule. Let K be the kernel of the (surjective, completely bounded) product map op A(Γ )# → A(Γ )# ; then K and hence K ∗ are also quantized Banach A(Γ )-bimodules. A(Γ )# ⊗ Let D : A(Γ ) → K ∗∗ be the completely bounded derivation defined by D(a) = a ⊗ 1 − 1 ⊗ a (a ∈ A(Γ )). Since K ∗∗ is the dual of a quantized Banach A(Γ )-bimodule, by hypothesis D is approximately inner. Therefore, by combining the proofs of [11, Corollary 2.2] and [5, Proposition 2.1], we obtain the following: for any finite subset S ⊂ A(Γ ) and any ε > 0, there exist F ∈ A(Γ ) ⊗ A(Γ ) and u, v ∈ A(Γ ) such that (1) a · F − F · a + u ⊗ a − a ⊗ v A(Γ )⊗ op A(Γ ) < ε; (2) a − au A(Γ ) < ε and a − va A(Γ ) < ε. By results of Effros and Ruan [8], the operator projective tensor norm on A(Γ ) ⊗ A(Γ ) coincides with its norm as a linear subspace of A(Γ × Γ ). The rest of the proof now follows [5, Propositions 2.3 and 3.3] and we omit the details. 2
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Specializing to F2 Notation. For t ∈ F2 we denote by |t| the word length of t, with the convention that the identity element has length 0. For each n ∈ Z+ let S(n) denote the set {t ∈ F2 : |t| = n}. Elementary calculations show that |S(n)| = 4 · 3n−1 for each n 1. The following Sobolev-type estimate, which we state without proof, is crucial for the argument to follow. It is a special case of [26, Theorem 1.1], and as such really belongs to the province of geometric group theory. Proposition 4.4. Fix m, n ∈ N. Let T ∈ C00 (F2 × F2 ) be supported on S(m) × S(n). Then λ(T ) (m + 1)(n + 1) T 2 . Corollary 4.5. Let F ∈ A(F2 × F2 ). Then F A(F2 ×F2 ) sup
m,n∈Z+
1 (m + 1)(n + 1)
1/2 F (x, y)2 .
(4.2)
x∈S(m) y∈S(n)
Proof. This is a routine deduction from Proposition 4.4, using duality. Let (m, n) ∈ Z2+ and let Tm,n ∈ C00 (F2 × F2 ) be defined by −1 −1 Tm,n (x, y) = (m + 1) (n + 1) F (x, y) 0
if x ∈ S(m) and y ∈ S(n), otherwise.
Then by Proposition 4.4, λ(Tm,n )
F (x, y)2
1/2 ,
(x,y)∈S(m)×S(n)
and since
(x,y)∈F2 ×F2
F (x, y)Tm,n (x, y) =
applying Lemma 4.1 completes the proof.
1 (m + 1)(n + 1)
F (x, y)2 ,
(x,y)∈S(m)×S(n)
2
Proof that A(F2 ) is not operator approximately amenable We start with some notation. In view of the lower bound (4.2), we introduce the following norm on c00 (F2 × F2 ): given H ∈ c00 (F2 × F2 ), let 1 (m + 1)(n + 1) m,n0
H ω×ω = sup
1/2 H (x, y)2 .
x∈S(m) y∈S(n)
For each n ∈ N, we fix a partition of S(n) into two disjoint subsets A(n) and B(n) of equal cardinality, so that |A(n)| = |B(n)| = 12 |S(n)|. We also fix a sequence (γn )n1 of strictly positive
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reals, such that
γn2 S(n) < ∞
(4.3)
n1
(the γn will be chosen later with appropriate hindsight). Now define elements a and b of 2 (F2 ) by a :=
γm 1A(m) ,
m1
b :=
γn 1B(n) .
(4.4)
n1
Finally, let ε > 0. F2 ) is operator approximately amenable. Applying Proposition 4.3 with Suppose that A(F S = {a, b} and using the lower bounds from (4.1) and (4.2), we obtain F ∈ c00 (F2 × F2 ) such that, if we set u = π(F ) ∈ c00 (F2 ): a − ua ∞ ε and b − ub ∞ ε; a · F − F · a − a ⊗ u + u ⊗ a ω×ω ε;
(4.5) (4.6a)
b · F − F · b − b ⊗ u + u ⊗ b ω×ω ε.
(4.6b)
For the moment we shall ignore the relations (4.5), and work exclusively with the information given by (4.6a) and (4.6b). Our task will be simplified by the fact that we have chosen the functions a and b to have ‘large’ yet disjoint supports (this theme, if not the actual calculations, is inspired by the proof of [5, Theorem 4.1]). Remark. Since a and b are fixed in advance of F , both (4.6a) and (4.6b) can always be satisfied by taking F to be of the form c1W ×W for some c ∈ C and some suitably large, finite subset W ⊂ F2 ; hence we will need to use (4.5) at some point if we want to obtain the required contradiction. Our task would be simplified if we furthermore assume that F is constant on sets of the form S(m) × S(n), and indeed the calculations that follow are motivated by this special case. The key step is contained in the following proposition. Proposition 4.6. For each k ∈ N let g(k) :=
1 u(p) |A(k)| p∈A(k)
and h(k) :=
1 u(q). |B(k)| q∈B(k)
Then for every m, n ∈ N we have A(m)1/2 B(n)1/2 g(m) − h(n) (m + 1)(n + 1) γ −1 + γ −1 ε. m
n
(4.7)
For our proof we need a technical lemma, whose essential content is well known, but is stated here for convenience.
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Lemma 4.7. Let I, J be finite index sets and let ci , dj ∈ C for all i ∈ I and all j ∈ J. Let 1 ci |I|
μc :=
and μd :=
i
1 dj . |J| j
Then 1 1 |ci − dj |2 |μc − μd |2 . |I| |J| i∈I j ∈J
Proof sketch. One can prove this by direct calculation. Alternatively, we can use the language of probability theory, as follows. If X and Y are independent complex-valued random variables defined on a common finite probability space (in this case, I × J) then E|X − Y |2 = E(X − Y )(X − Y ) = EXX − (EX)(EY ) − (EX)(EY ) + EY Y = (EX − EY )2 + E|X|2 − |EX|2 + E|Y |2 − |EY |2 (EX − EY )2 . (In the last step we used the fact that |EX|2 E|X|2 and |EY |2 E|Y |2 .) Taking X to be the random variable (i, j ) → ci and Y to be the random variable (i, j ) → dj , the proof is complete. 2 Proof of Proposition 4.6. Eq. (4.6a) implies that a(x) − a(y) F (x, y) − a(x)u(y) + a(y)u(x)2 (m + 1)2 (n + 1)2 ε 2 x∈S(m) y∈S(n)
a(x) − a(y) F (x, y) − a(x)u(y) + a(y)u(x)2
x∈A(m) y∈B(n)
=
a(x)F (x, y) − a(x)u(y)2
x∈A(m) y∈B(n)
= γm2
F (x, y) − u(y)2 .
x∈A(m) y∈B(n)
Therefore (m + 1)(n + 1)ε γm
1/2 F (x, y) − u(y)2 .
(4.8a)
x∈A(m) y∈B(n)
Similarly, using Eq. (4.6b) instead of (4.6a), we have (m + 1)(n + 1)ε
1/2 −b(y)F (x, y) + b(y)u(x)2
x∈A(m) y∈B(n)
= γn
1/2 F (x, y) − u(x)2 .
x∈A(m) y∈B(n)
(4.8b)
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Hence, by using the triangle inequality for the 2-norm on 2 (A(m) × B(n)), we see that (4.8a) and (4.8b) together imply (m + 1)(n + 1) γm−1 + γn−1 ε
1/2 u(x) − u(y)2 .
(4.9)
x∈A(m) y∈B(n)
The desired estimate (4.7) now follows by applying Lemma 4.7 to (4.9).
2
We now show that by fixing our sequence (γn ) appropriately, we can force g to be “slowly varying at infinity”, and play this off against the fact that g has finite support (since u does). For each n, take γn := n−1 |S(n)|−1/2 (this certainly satisfies the condition in (4.3)). If we substitute this into the estimate (4.7) and take m = k, n = k + 1 we get A(k)1/2 B(k + 1)1/2 g(k) − h(k + 1) 1/2 1/2 ε(k + 1)(k + 2) k S(k) + (k + 1)S(k + 1) 1/2 1/2 ε(k + 2)3 S(k) + S(k + 1) ; and since |A(n)| = |B(n)| = 12 |S(n)| = 2 · 3n−1 , we find that (k−1)/2 + 2 · 3k/2 √ g(k) − h(k + 1) ε(k + 2)3 2 · 3 = (1 + 3)ε(k + 2)3 3−k/2 . (k−1)/2 k/2 2·3 ·3
On the other hand, taking m = n = k + 1 in (4.7), an exactly similar argument gives g(k + 1) − h(k + 1) 2ε(k + 2)3 3−k/2 , and we thus obtain the estimate g(k + 1) − g(k) 5ε(k + 2)3 3−k/2 .
(4.10)
By the comparison test, the infinite sum k1 (k + 2)3 3−k/2 converges, with value M say. Moreover, since u has finite support, there exists N 2 such that g(j ) = h(j ) = 0 for all j N . Hence, using (4.10), we get −1 N g(1) = g(N ) − g(1) g(k + 1) − g(k) k=0
5ε
N −1
(k + 2)3 3−k/2 < 5Mε.
k=0
Now observe that, by (4.5), 1 ε a − au ∞ max a(x) − a(x)u(x) = max 1 − u(x). 2 x∈A(1) x∈A(1)
(4.11)
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Let x, y be the two elements of A(1). Then the estimate just given implies that 2ε
1 1 − u(x) + 1 − u(y) 2
1 2 − u(x) − u(y) = 1 − g(1), 2
and combining this with (4.11), we finally arrive at 1 1 − g(1) + g(1) 2ε + 5Mε. As M is, by definition, independent of ε, we obtain a contradiction by taking ε to be sufficiently small. Hence our assumption that A(F2 ) is operator approximately amenable must be false, and the proof is complete. 2 Corollary 4.8. Let G be a locally compact group, into which F2 embeds as a closed subgroup. Then A(G) is not operator approximately amenable. Proof. The hypothesis on G ensures that the restriction homomorphism A(G) → A(F2 ) is completely bounded and a quotient map of Banach spaces [16, Theorem 1]. If A(G) were operator approximately amenable, then A(F2 ) would be also, since operator approximate amenability is inherited by completely bounded quotient algebras. This gives a contradiction. 2 Remark. For a discrete, amenable group G, it was shown in [13] that A(G) is approximately amenable. However, there are discrete groups which are non-amenable yet contain no copy of F2 : Ol’shanskii’s groups, or Burnside groups of sufficiently large rank and exponent. So any attempt to prove that approximate amenability of A(G) implies amenability of G must use different, or additional, methods. 5. Results for Segal algebras Following on from Example 3.7(e) above, we give some results on other notions of approximate amenability in the setting of Segal algebras. Let G be a locally compact group with a left-invariant Haar measure λ. Throughout this section, S(G) denotes a Segal subalgebra of L1 (G) (in the sense of Reiter [27]). We have already seen that a symmetric S(G) is boundedly approximately amenable if and only if it is equal to the whole of L1 (G) and G is amenable. For Feichtinger’s Segal algebra (see [28] for the definition) on a compact abelian group we easily obtain the following: Proposition 5.1. The Feichtinger algebra on an infinite compact abelian group is not approximately amenable. Proof. When G is compact and abelian, the Feichtinger algebra on G is cγ χγ : f = |cγ | < ∞ , S0 (G) = f = γ ∈G
Hence, where χγ is the character of G associated with γ ∈ G. S0 (G) ∼ = 1 (G),
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where the right-hand side is equipped with the pointwise product. But 1 (S) is not approximately amenable if S is an infinite set, due to [5, Theorem 4.1]. So S0 (G) is not approximately amenable. 2 It has been shown in [14] that a Segal algebra on a compact group is pseudo-contractible. The converse is also true and is a consequence of the next proposition. S(G) such that π(N) = 0 and f · N = N · f for f ∈ S(G), Proposition 5.2. If there is N ∈ S(G) ⊗ then G is compact. Proof. Let θ : S(G) → L1 (G) be the inclusion injection. Then the following diagram commutes S(G) S(G) ⊗
θ⊗θ
L1 (G) L1 (G) ⊗
π
π
S(G)
L1 (G).
θ
S(G) be such that π(N) = 0 and f · N = N · f for f ∈ S(G). Let M = θ ⊗ θ (N) ∈ Let N ∈ S(G) ⊗ L1 (G). We have f · M = M · f for all f ∈ L1 (G), and therefore μ · M = M · μ L1 (G) ⊗ (μ ∈ M(G)). In particular, M = δx −1 · M · δx (x ∈ G). Let K be a compact subset of G × G. If we regard M as a function in L1 (G × G), then
M(s, t) ds dt =
K
K
δ
x −1
· M · δx (s, t) ds dt
=
(x)M(s, t) ds dt,
(x,e)K(e,x −1 )
where (x, e)K(e, x −1 ) denotes the set {(xs, tx −1 ): (s, t) ∈ K}, and is the modular function of the group G. Given ε > 0, let R ⊂ G × G be a compact set such that
M(s, t) ds dt < ε.
G×G\R
If G is not compact, then there is x ∈ G such that (x, e)K(e, x −1 ) ⊂ G × G\R and (x) 1, so that (x)M(s, t) ds dt < ε. (x,e)K(e,x −1 )
We then have K |M(s, t)| ds dt < ε. This implies that K |M(s, t)| ds dt = 0, for all compact L1 (G). On the other hand, π(N) = 0 in S(G) and hence K ∈ G × G, and so M = 0 in L1 (G) ⊗ 1 π(M) = θ π(N) = 0 in L (G), a contradiction. Thus, G must be compact. 2
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Remark. Proposition 5.2 holds with S(G) replaced by any Banach algebra B which admits a continuous injective homomorphism B → L1 (G) whose range is dense. Therefore, if such a B exists and is pseudo-contractible, G must be compact. Combining Proposition 5.2 and [14, Theorem 4.5], we then have a characterization of a compact group. Theorem 5.3. The following are equivalent for a locally compact group G. (i) The group G is compact. (ii) There is a Segal algebra on G which is pseudo-contractible. (iii) All Segal algebras on G are pseudo-contractible. (A different proof of the part “(ii) ⇒ (i)” can be seen in [29].) It is natural to ask for an analogous characterization of amenability of G in terms of approximate amenability or pseudo-amenability of Segal algebras on G. First we recall some material from the theory of abstract Segal algebras. A dense left ideal B of a Banach algebra (A, · A ) is called an abstract Segal algebra in A, or simply a Segal algebra in A, with respect to some norm · B if it is a Banach algebra with respect to the norm · B and if b A b B (b ∈ B) [3,22]. It was shown in [22] that if B is a Segal algebra in A, then the mapping J → J A is a bijection from the set of all closed right (two-sided) ideals in B onto the set of all closed right (two-sided) ideals in A and the inverse mapping is I → I ∩ B, where for a set J ⊂ B the notation J A stands for the closure of J in A. The same machinery as in [22, Proposition 2.7] yields the following: Proposition 5.4. Let B be an abstract Segal algebra in a Banach algebra A, let J be a closed two-sided ideal of B and let I = J A . (i) Suppose that A and J both have right approximate identities. Then I has a right approximate identity. (ii) Suppose that B and I both have right approximate identities. Then J has a right approximate identity. Since we only need part (i) of Proposition 5.4, we shall give an independent proof of (i), which is more direct in the sense that it avoids the duality machinery of [22]. Proof of Proposition 5.4. Let F ⊂ I be a finite subset, and let ε > 0. It suffices to find s ∈ I such that maxy∈F ys − y A ε. Since A has a right approximate identity, there exists u ∈ A with yu − y A < ε/2 for all y ∈ F . Therefore, since B is dense in A, there exists u ∈ B such that yu − y A < ε/2 for all y ∈ F. Since I is a right ideal and B is a left ideal in A, yu ∈ I ∩ B = J for every y ∈ F . Therefore, as J has a right approximate identity, there exists w ∈ J such that yu w − yu B ε/2 for all y ∈ F.
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Let s = u w: since w ∈ J ⊆ I and I is also a left ideal, we have s ∈ I ; and for every y ∈ F , ys − y A yu w − yu A + yu − y A yu w − yu B + yu − y A < ε/2 + ε/2 = ε, as required.
2
Remark. Part (ii) of Proposition 5.4 can also be proved by direct ε–δ arguments, similar to the ones just given; again, the duality machinery from [22] can be bypassed. Theorem 5.5. If S(G) is approximately amenable or pseudo-amenable then G is an amenable group. Proof. Let I0 = {f ∈ L1 (G): G f (x) dx = 0} be the augmentation ideal in L1 (G), and let J = I0 ∩ S(G). Then J is a codimension-1 two-sided closed ideal in S(G). If S(G) is approximately amenable or pseudo-amenable, then J has a right approximate identity by [11, Corollary 2.4] or [14, Proposition 2.5] respectively. By Proposition 5.4(i), I0 must also have a right approximate identity. This implies that G is amenable due to [30, Theorem 5.2]. 2 Remark. We do not know whether there is a Segal algebra S(G) that is approximately amenable and that is not identical with L1 (G). It is also an open question whether a Segal algebra on an amenable group is always pseudo-amenable. Partial results can be found in [14]. We now turn to results that do not depend on amenability or compactness of G. While L1 (G) is weakly amenable for every locally compact group G, the same need not be true for Segal algebras: see [9, Remark 3.2] for examples. Following on from results in [9,10] on approximate weak amenability of Segal algebras, we now look at approximate permanent weak amenability. Recall from [4] that a Banach algebra A is said to be n-weakly amenable if every continuous derivation from A into the nth dual space A(n) is inner. A is permanently weakly amenable if it is n-weakly amenable for all n ∈ N. It was shown in [4] that every C ∗ -algebra is permanently weakly amenable, and that every 1 L (G) is n-weakly amenable for all odd, positive integers n. In [18] B.E. Johnson proved that for every free group G, the group algebra 1 (G) is n-weakly amenable for all even, positive n. Combined with [4, Theorem 4.1], this shows that for such groups, 1 (G) is permanently weakly amenable. In an unpublished paper Johnson also showed that for any discrete word-hyperbolic group, the group algebra is permanently weakly amenable. In fact, for any locally compact group G, L1 (G) is permanently weakly amenable. Our proof relies heavily on the following result, proved recently by V. Losert [23, Theorem 1.1]. Theorem 5.6. Let G be a (discrete) group and X a locally compact space on which G has a 2-sided action by homeomorphisms. Then any bounded derivation D : G → M(X) is inner. (The statement in [23] refers only to those X with a left action; however, by standard arguments of Johnson one can reduce the 2-sided case to the 1-sided case, see e.g. [17, §2].) Proof that L1 (G) is permanently weakly amenable. In light of [4] it suffices to show that L1 (G) is 2n-weakly amenable for all n ∈ N.
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Let D : L1 (G) → L1 (G)(2n) be a continuous derivation. By the techniques of [17, §1.d] D extends to a derivation D : M(G) → L1 (G)(2n) , where the measure algebra M(G) acts on L1 (G)(2n) through dualizations of its actions on L1 (G). Now L1 (G)(2n) is isomorphic, as an M(G)-bimodule, to M(X) for some compact space X. The action of point masses on M(X) is equivalent to an action of G on M(X); and g → D(δg ) is a bounded derivation from G into M(X). Hence by Theorem 5.6 this derivation is inner, and this suffices for us to conclude that D : M(G) → L1 (G)(2n) is inner, by w∗ -continuity of D. 2 Theorem 5.7. Let G be a locally compact SIN-group and let S(G) be a Segal algebra on G. Then S(G) is approximately permanently weakly amenable (i.e. for each n ∈ N, every continuous derivation S(G) → S(G)(n) is approximately inner). Note that the case “n = 0” was proved in [9, Theorem 2.1(i)] under the extra hypothesis that our Segal algebra is symmetric. Proof of Theorem 5.7. Since G is SIN, it follows from the results of [20] that S(G) has a central approximate identity (ei ) which is bounded in the L1 -norm. Let n ∈ N and let D : S(G) → S(G)(n) be a continuous derivation. Our approach is to construct from D a net of continuous derivations L1 (G) → L1 (G)(n) , so that we can appeal to Theorem 5.6. The centrality of (ei ), together with the derivation property of D, imply that D S(G) ⊆ Xn := lin a · S(G)(n) · b: a, b ∈ S(G) . Moreover, (ei ) is a two-sided, multiplier-bounded, central approximate identity for Xn . In particular lim ei2 · D(f ) = D(f ) i
f ∈ S(G) ,
(5.1)
where the limit is taken in the norm topology of S(G)(n) . For each i, define a continuous linear mapping τi : L1 (G) → S(G) by τi (f ) = f ∗ ei
f ∈ L1 (G) ,
and let θ : S(G) → L1 (G) denote the (continuous) inclusion map. Both τi and θ are left L1 (G)module morphisms and are also S(G)-bimodule morphisms. Clearly, τi θ (f ) = f ∗ ei = ei ∗ f for f ∈ S(G); so by induction, for each n ∈ N the map (τi θ )(n) : S(G)(n) → S(G)(n) satisfies (τi θ )(n) (F ) = F · ei = ei · F
F ∈ S(G)(n) .
Define i : L1 (G) → L1 (G)(n) by i (f ) =
θ (n) [D(f ∗ ei ) − f · D(ei )] if n is even, (n) τi [D(ei ∗ f ) − D(ei ) · f ] if n is odd
f ∈ L1 (G) .
Then i is a continuous linear map, and for f ∈ S(G) we have, using the derivation property of D,
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i (f ) =
θ (n) (D(f ) · ei ) (n) τi (ei · D(f ))
if n is even, if n is odd.
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(5.2)
Since ei is central, it is straightforward to verify using (5.2) that i (f ∗g) = f · i (g)+ i (f )·g for all f, g ∈ S(G). Therefore, since S(G) is dense in L1 (G), it follows that i is a derivation from L1 (G) to L1 (G)(n) . By Theorem 5.6 there exists ϕi ∈ L1 (G)(n) such that f ∈ L1 (G) .
i (f ) = f · ϕi − ϕi · f
In particular, for f ∈ S(G), Eq. (5.2) implies that for even n we have (n) D(f ) · ei2 = (τi θ )(n) D(f ) · ei = τi i (f ) (n)
(n)
= f · τi (ϕi ) − τi (ϕi ) · f, while for odd n we have ei2 · D(f ) = (τi θ )(n) ei · D(f ) = θ (n) i (f ) = f · θ (n) (ϕi ) − θ (n) (ϕi ) · f. (n)
Take ψi = τi (ϕi ) if n is even, and take ψi = θ (n) (ϕi ) if n is odd. Then ψi ∈ S(G)(n) for all i, and for every f ∈ S(G) we have, by (5.1), D(f ) = lim ei2 · D(f ) = lim f · ψi − ψi · f. i
Thus D is approximately inner, as required.
i
2
Remark. Our construction actually provides a bounded net of inner derivations which approximate D, although the net of implementing elements need not be bounded. 6. 1 -Convolution algebras of totally ordered sets Recall that a semilattice is a commutative semigroup in which every element is idempotent. The 1 -convolution algebras of semilattices provide interesting examples of commutative Banach algebras. However, amenability is too strong a notion for such algebras: if S is a semilattice then the convolution algebra 1 (S) is amenable if and only if S is finite [7, Theorem 10]. It is not clear to the authors exactly which semilattices have approximately amenable 1 -convolution algebras. In the case where the semilattice is totally ordered we can do better. Let Λ be a non-empty, totally ordered set, and regard it as a semigroup by defining the product of two elements to be their maximum. The resulting semigroup, which we denote by Λ∨ , is a semilattice. We may then form the 1 -convolution algebra 1 (Λ∨ ). For every t ∈ Λ∨ we denote the point mass concentrated at t by et . The definition of multiplication in 1 (Λ∨ ) ensures that es et = emax(s,t) for all s and t.
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Remark. One could also turn Λ into a semilattice Λ∧ by defining the product of two elements to be their minimum. This is in some sense more natural, for reasons we shall not discuss here; we have chosen to work with Λ∨ as this fits our main example (in Theorem 6.4) better. Theorem 6.1. Let I be any totally ordered set. Then 1 (I∨ ) is boundedly approximately contractible. Remark. The special case of I = N or Nop was done in [12]. Our arguments are a more abstract version of the ones there. We prove the theorem in several steps. First, by following the proof of [12, Theorem 5.10], # it suffices to prove that 1 (I∨ ) has a multiplier-bounded approximate diagonal, in the sense of # Definition 2.1. Moreover, we can identify 1 (I∨ ) with 1 ( I∨ ), where I denotes the disjoint union of I with an adjoined least element. Clearly I is also a totally ordered set, and so to prove Theorem 6.1 it suffices to prove the following claim: for any totally ordered set I, 1 (I∨ ) has a multiplier-bounded approximate diagonal. It is useful to first consider the case of a finite totally ordered set. More precisely, let F be a finite subset of I, and enumerate its elements in increasing order as min(F ) = c(0) < c(1) < · · · < c(n) = max(F ) 1 (I∨ ) by say. We then define F ∈ 1 (I∨ ) ⊗ F =
n (ec(j −1) − ec(j ) ) ⊗ (ec(j −1) − ec(j ) ) + ec(n) ⊗ ec(n) .
(6.1)
j =1
A small calculation shows that π( F ) = ec(0) , so that eλ π( F ) = eλ
for all λ ∈ F.
(6.2)
It is also easily checked that eλ · F = F · eλ
for all λ ∈ F,
(6.3)
and thus F is a diagonal for the subalgebra 1 (F∨ ) ⊆ 1 (I∨ ). Having seen how to construct a diagonal for the finite case, we now proceed to the general case. Let FIN be the set of all non-empty finite subsets of I, and order FIN with respect to inclusion, so that for any E and F in FIN, E F if and only if E ⊆ F . The following result will, by the remarks above, imply Theorem 6.1. Proposition 6.2. The net ( F )F ∈FIN is a multiplier-bounded approximate diagonal for 1 (I∨ ). We isolate the key technical estimate as a lemma. Lemma 6.3. Let b ∈ 1 (I∨ ), F ∈ FIN. Then b · F − F · b 6 b .
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Proof. By the triangle inequality and the definition of the 1 -norm, we can without loss of generality assume that b = eλ for some λ ∈ I. Thus it suffices to prove that eλ · F − F · eλ 6
for all F ∈ FIN.
(6.4)
This estimate holds trivially if F consists of only one point, so we shall henceforth assume that |F | 2. As before we enumerate the elements of F in increasing order as c(0) < c(1) < · · · < c(n). We consider three possibilities. If λ c(n), then eλ · F = eλ ⊗ ec(n) and F · eλ = ec(n) ⊗ eλ , so that (6.4) certainly holds. At the other extreme, if λ c(0) then eλ · F = F = F · eλ , so that (6.4) once again holds. The third possibility is that c(0) < λ < c(n). Let m = min{k: c(k) > λ} so that 1 m n and c(m − 1) < λ < c(m). When we calculate eλ · F − F · eλ using the formula (6.1), most of the terms cancel and we obtain eλ (ec(m−1) − ec(m) ) ⊗ (ec(m−1) − ec(m) ), eλ · F − F · eλ = −(ec−1(m) − ec(m) ) ⊗ (ec(m−1) − ec(m) )eλ (eλ − ec(m) ) ⊗ (ec(m−1) − ec(m) ), = −(ec(m−1) − ec(m) ) ⊗ (eλ − ec(m) ). Expanding out and using the triangle inequality gives eλ · F − F · eλ 6, as required.
2
Proof of Proposition 6.2. Fix a ∈ 1 (I∨ ). We have already seen in Lemma 6.3 that a · F − F · a 6 a for every F ∈ FIN. Also, since π( F ) = emin(F ) , we have aπ( F ) − a 2 a for every F in FIN.
(6.5)
Thus the ‘multiplier-bounded’ part of the defining condition (2.1) is satisfied. It remains to show that, given ε > 0, there exists F0 ∈ FIN such that aπ( F ) − a < ε
and a · F − F · a < ε
for any F ∈ FIN with F ⊇ F0 . Fix ε > 0 and choose F0 ∈ FIN such that λ∈I \F0 |aλ | ε/6, and let F ∈ FIN with F ⊇ F0 . aλ = 0 Let a denote the obvious truncation of a to the subset F (i.e. aλ = aλ if λ ∈ F and otherwise). Note that a − a ε/6. Since a ∈ 1 (F∨ ), we deduce from Eq. (6.3) and the estimate (6.5) that a ) · F − F · (a − a ε. a ) 6 a − a · F − F · a = (a − Finally, using Eq. (6.2) and Lemma 6.3 we obtain aπ( F ) − a = (a − a )π( F ) − (a − a = ε/3, a ) 2 a − and the proof is complete.
2
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Remark. If the set I is countable, then the net ( F )F ∈FIN has a subnet which is a sequence (take n := {t1 ,...,tn } ). So if I is countable, then 1 (I∨ ) any enumeration of I as {t1 , t2 , . . .} and let is sequentially approximately contractible. A counter-example While sequential approximate amenability implies bounded approximate amenability, the converse is false. This is proved by combining Theorem 6.1 with the following result. Theorem 6.4. Let Λ be an uncountable well-ordered set. Then 1 (Λ∨ ) is not sequentially approximately amenable. (Recall that a totally ordered set is well-ordered if every non-empty subset has a least element: well-ordered sets are precisely those ordered sets which are order-isomorphic to ordinals.) In proving Theorem 6.4 we shall use some basic facts on the character theory of 1 (Λ∨ ). It is clear that the characters on 1 (Λ∨ ) correspond to the non-zero semigroup homomorphisms from Λ∨ to the two-element semigroup {0, 1}; and a little thought gives the following characterization. Proposition 6.5. When regarded as elements of ∞ (Λ), the characters on 1 (Λ∨ ) are all of the form 1Λ\U , where U is a proper (and possibly empty) subset of Λ that is upwards-directed with respect to the given order on Λ. Example 6.6. Take Λ to be the real line with its usual ordering. Then the characters on 1 (Λ∨ ) are either of the form 1(−∞,t) or 1(−∞,t] . If U is a non-empty, upwards-directed subset of a well-ordered set Λ, then U has a least element, u say: hence U = {x ∈ Λ: x u}. Thus the complements of upwards-directed sets are all of the form {y: y < u}. If λ is an element of a well-ordered set and it is not maximal, then there is a unique minimal element greater than λ, which we shall denote by λ + 1. Notation. Let Λ be a well-ordered set and consider the algebra 1 (Λ∨ ). If λ ∈ Λ we denote + 1 is the by λ the character 1<λ . If λ is maximal in Λ then we adopt the convention that λ augmentation character 1Λ . The following is then obvious: we isolate it as a lemma for later reference. Lemma 6.7. Let Λ be a well-ordered set and let λ ∈ Λ. Then δλ = λ + 1 − λ, ∗
where δλ denotes the point mass at λ, regarded as an element of 1 (Λ∨ ) . Our proof of Theorem 6.4 uses our earlier observations on the characters of 1 (Λ∨ ), together with Lemma 2.8. Intuitively, the idea is that the Gelfand transforms of elements in 1 (Λ∨ ) are bad approximations to the indicator function of the set {(λ, λ): λ ∈ Λ}, 1 (Λ∨ ) ⊗ so that if Λ is uncountable then no countable net ( n ) can have the properties described in Lemma 2.8. We make this idea precise as follows.
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Lemma 6.8. Let I be an uncountable index set and let (Fn ) be a countable family in 1 (I × I)∗∗ . Then there exist uncountably many t ∈ I such that Fn , δt ⊗ δt = 0 for all n. Proof. In view of the direct sum decomposition 1 (I × I)∗∗ = 1 (I × I) ⊕ c0 (I × I)⊥ we may write each Fn as κ(fn ) + Gn where Gn ∈ c0 (I × I)⊥ , fn ∈ 1 (I × I) and κ is the natural embedding of 1 (I × I) in its bidual. Let S = n {t ∈ I: fn (t, t) = 0}. Since each fn has countable support, S is countable. In particular I \ S is uncountable, and for any t ∈ I \ S we have Fn , δt ⊗ δt = (fn )t,t + Gn , δt ⊗ δt = 0, as claimed.
2
Proof of Theorem 6.4. Suppose 1 (Λ∨ ) is sequentially approximately amenable. Since Λ is well-ordered it has a least element, and consequently 1 (Λ∨ ) has an identity element. Hence 1 (Λ∨ ))∗∗ such that Lemma 2.8 applies and there is a sequence n ∈ (1 (Λ∨ ) ⊗ n , ϕ ⊗ ϕ = 1 for all n and
lim n , ϕ ⊗ χ = 0 n
(6.6)
for every pair of distinct characters ϕ, χ . By Lemma 6.8 there exists λ ∈ Λ such that n , δλ ⊗ δλ = 0 for all n, and hence by Lemma 6.7, 0=
λ ⊗ λ − n , λ ⊗ λ + 1 , n , − n , λ + 1 ⊗ λ + n , λ + 1 ⊗ λ + 1 .
But by Eq. (6.6) the right-hand side converges to 2 as n → ∞, which is a flagrant contradiction. 2 Remark. The proof just given yields something formally stronger, namely that 1 (Λ∨ ) cannot have an approximate diagonal with countable indexing set. We do not pursue this further in this paper, chiefly because we know of no Banach algebra which has a countably-indexed approximate diagonal and yet has no sequential approximate diagonal. 7. Algebras of pseudo-functions on discrete groups Let Γ be a discrete group, with convolution algebra 1 (Γ ). Given p ∈ (1, ∞) we may consider the left regular representation of Γ on p (Γ ), and this gives an injective continuous algebra homomorphism θp : 1 (Γ ) → B(p (Γ )). We denote by PF p (Γ ) the norm-closure in B(p (Γ )) of the range of θp . Note that PF 2 (Γ ) is nothing but the reduced C ∗ -algebra of Γ .
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If Γ is amenable, then by Johnson’s theorem the convolution algebra 1 (Γ ) is amenable, and since amenability is inherited by closures under Banach algebra norms we deduce that PF p (Γ ) is amenable; in particular the reduced C ∗ -algebra Cr∗ (Γ ) is amenable. The converse result — that amenability of Cr∗ (Γ ) implies amenability of Γ — was proved by Bunce in [2]. With some modifications one can adapt his proof to show that amenability of any one of the algebras PF p (Γ ) is enough to force amenability of Γ . In [11] it was shown that approximate amenability of the group algebra L1 (G) implies amenability of G, by generalizing the well-known argument for amenability of L1 (G). We shall now show that by combining arguments from [2] and [11] we have the following theorem. Theorem 7.1. Let Γ be a discrete group. Then the following are equivalent: (i) (ii) (iii) (iv)
Γ is amenable; PF p (Γ ) is amenable for all p ∈ (1, ∞); PFT p (Γ ) is approximately amenable for some p ∈ (1, ∞); PF p (Γ ) is pseudo-amenable for some p ∈ (1, ∞).
As mentioned above, the implications (i) ⇔ (ii) are already known, while the implication (ii) ⇒ (iii) is trivial; the implication (iii) ⇒ (iv) follows from [14, Proposition 3.2] and only uses the fact that PF p (Γ ) has an identity element. Therefore our contribution is to prove the implication (iv) ⇒ (i). Taking p = 2, our proof will give a slightly streamlined version of Bunce’s arguments, in that we are able to forgo technical arguments with states on C ∗ -algebras in favour of more direct positivity arguments with measures on compact spaces. Our idea is to follow Bunce’s construction up to the point where he produces, from the assumption that Cr∗ (Γ ) is amenable, a non-zero element ρ in ∞ (Γ )∗ which satisfies ρ(g · f ) = ρ(f )
for all f ∈ ∞ (Γ ).
(In [2] ρ is described as being ‘left-invariant’: we adopt the opposite and more usual convention, and say ρ is right-invariant.) In our setting we merely obtain a net (φα ) of functionals on ∞ (Γ ) which satisfies φα (1) → 1 and φα · g − φα → 0 for each g ∈ Γ. We then use this net to obtain a “genuine” invariant mean on ∞ (Γ ), by following the last part of the proof of [11, Theorem 3.2]. For convenience we isolate the relevant argument and state it as the following lemma. Lemma 7.2. Let G be a locally compact group, and let T be a compact G-space on which G acts from the right by homeomorphisms. Equip M(T ) with its usual norm, and regard it as a right Banach G-module. Suppose we have a net (ϕi ) of Radon measures on T which satisfies the following conditions: (i) infi ϕi > 0; (ii) ϕi · g − ϕi → 0 for all g ∈ G. Then there exists a probability measure n on T such that n · g = n.
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We give the proof for sake of completeness (cf. the proof of [11, Theorem 3.2]). Proof. Set ni = ϕi −1 ϕi . The hypothesis that ϕi is bounded below then ensures that ni · g − ni → 0 for all g ∈ G. For any two Radon measures μ, ν on T we have μ − ν |μ| − |ν| , an inequality which can easily be deduced from the definition of the total variation of a measure. Therefore, since |μ · g| = |μ| · g for any μ ∈ M(T ), we have |ni | · g − |ni | = |ni · g| − |ni | ni · g − ni → 0 for every g ∈ G. Take n to be any w∗ -cluster point of the net (|ni |). Since |ni |(1) = 1 for all i, we have n(1) = 1; and for any g ∈ G and f ∈ C(T ), we have (n · g − n)(f ) lim sup |ni | · g − |ni | (f ) = 0, i
so that n · g = n for all g ∈ G.
2
Proof of Theorem 7.1, (iv) ⇒ (i). Our aim is to construct a right-invariant mean on ∞ (Γ ). To fix notation, we recall that the usual left action of Γ on ∞ (Γ ) = 1 (Γ )∗ is defined by (g · f )(x) = f g −1 x
for f ∈ ∞ (Γ ) and g, x ∈ Γ.
For each g ∈ Γ let Lg be the isometric, invertible operator on p (Γ ) given by left translation, i.e. (Lg k)(x) = k(g −1 x) for all k ∈ p (Γ ). We regard PF p (Γ ) as a subalgebra of B(p (Γ )). Take τ to be the functional given by τ (T ) = T δe , δe
T ∈ B p (Γ ) ,
where δe is the basis vector of p (Γ ) that takes the value 1 at e and the value 0 everywhere else. Clearly τ (I ) = 1. A simple calculation shows that for any a, b in the group algebra CΓ , we have τ (θp (a)θp (b)) = τ (θp (b)θp (a)), and so by continuity the restriction of τ to PF p (Γ ) defines a non-zero trace. Suppose that PF p (Γ ) is pseudo-amenable. By Lemma 2.7, there exists a net (ψα ) in B(p (Γ ))∗ such that lim ψα (I ) = 1 α
and lim α
sup
T ∈B(p (Γ )), T 1
ψα (aT − T a) = 0 for all a ∈ PF p (Γ ).
(7.1)
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In particular, for any g ∈ Γ we have sup
M∈B(p (Γ )), M 1
ψα (Lg ML
sup
T ∈B(p (Γ )), T 1
g −1 ) − ψα (M)
ψα (Lg T − T Lg ) → 0.
(7.2)
Regard ∞ (Γ ) as an algebra with pointwise multiplication and supremum norm. There is an embedding of ∞ (Γ ) as a closed unital subalgebra of B(p (Γ )), defined by sending a bounded function f ∈ ∞ (Γ ) to the “diagonal multiplication” operator Mf where (Mf k)(x) = f (x)k(x) for all k ∈ p (Γ ) and x ∈ Γ . Then a direct calculation shows that M(g·f ) = Lg Mf (Lg )−1
for all f ∈ ∞ (Γ ) and g ∈ Γ.
(7.3)
For each α define φα ∈ ∞ (Γ )∗ by φα (f ) = ψα (Mf ), f ∈ ∞ (Γ ). It follows from Eqs. (7.1), (7.2) and (7.3) that limα φα (1) = 1, and that lim φα · g − φα = α
sup
f ∈∞ (Γ ), f 1
φα (g · f ) − φα (f ) = 0 for all g ∈ Γ.
To finish we observe that ∞ (Γ ) may be identified with the space of continuous functions on ˇ a compact Γ -space T (namely, take T to be the Stone–Cech compactification of Γ ), and hence we may identify each φα with a Radon measure on T . By Lemma 7.2, there exists a positive functional n ∈ ∞ (Γ )∗ satisfying n(1) = 1 and n · g = n for all g ∈ Γ , and hence Γ is amenable as claimed. 2 Specializing to the case p = 2 (i.e. the reduced C ∗ -algebra Cr∗ (Γ )), we have the following corollary. Corollary 7.3. The full group C ∗ -algebra C ∗ (Γ ) is approximately amenable if and only if Γ is amenable. Proof. We first recall without proof some basic facts about C ∗ (Γ ): firstly, it is by definition the completion of 1 (Γ ) in a certain C ∗ -norm; and secondly, there is a canonical quotient homomorphism from C ∗ (Γ ) onto Cr∗ (Γ ). Now, suppose that Γ is amenable: then 1 (Γ ) is amenable. As just mentioned, the inclusion homomorphism 1 (Γ ) → C ∗ (Γ ) is continuous with dense range, and therefore C ∗ (Γ ) must also be amenable. Conversely, suppose that C ∗ (Γ ) is approximately amenable. By [11, Proposition 2.2], approximate amenability passes to quotient algebras, and so Cr∗ (Γ ) is approximately amenable. Now apply Theorem 7.1 in the case p = 2. 2 Remark. Using the fact that the canonical tracial state τ on Cr∗ (Γ ) actually extends to a tracial state on the von Neumann algebra VN(Γ ), we can adapt the proof of Theorem 7.1 to show the following result: if A is a closed unital subalgebra of VN(Γ ), with Cr∗ (Γ ) ⊆ A, and furthermore A is approximately amenable, then Γ is amenable.
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Appendix A. Implications In this appendix we give some schematic diagrams. The first illustrates the known implications between various notions of approximate amenability; the second illustrates what is known for commutative Banach algebras; and the third illustrates some partial results concerning approximate identities. We hope that these pictorial representations clarify some of the relationships between the notions considered in this paper. In these diagrams, approximate amenability has been abbreviated to ‘AA’, and approximate contractibility to ‘AC’; similarly for their bounded variants. ‘PsA’ and ‘PsC’ denote pseudoamenability and pseudo-contractibility, respectively. ‘BAI’, ‘CAI’ and ‘MBAI’ respectively denote the presence of a bounded, central and multiplier-bounded approximate identity. In the second diagram, ‘comm.’ is an abbreviation for commutativity. Implications are denoted by solid arrows: dashed arrows with a × in the middle denote the failure of an implication. The label def on an arrow means that the corresponding implication holds “by definition” or a fortiori. Labels in square brackets refer to items in the bibliography. In the third diagram, the labels 3.3 and 3.4 refer, respectively, to Theorem 3.3 and Corollary 3.4 of the present article. A.1. General implications
AA
def
BAA
def
[15] def
[12, Theorem 2.1]
AC
def
contractible
def
PsA def × ()
def
def
BAC
def
amenable
def
PsC
Here the counter-example () to ‘pseudo-amenable implies pseudo-contractible’ follows because unital pseudo-contractible algebras must be contractible [14, Theorem 2.4], while there are unital pseudo-amenable algebras which are not even amenable: perhaps the simplest example is 1 (N, max). A.2. Commutative settings
CAI + AA
[14, Proposition 3.3]
PsA
comm. + PsA def
[11, Lemma 2.2]
comm. + AA
def
×
[5, Theorem 4.1]
comm. + PsC
Here the counterexample to ‘pseudo-contractible implies approximately amenable’ is given by 1 (N) with pointwise multiplication.
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A.3. Approximate identities
PsA + BAI
[14, Proposition 3.2]
AA + BAI
def
BAA + MBAI def
def
BAC 3.4
BAC + BAI
def
3.3
BAA + BAI
References [1] W.G. Bade, P.C. Curtis Jr., H.G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. (3) 55 (2) (1987) 359–377. [2] J.W. Bunce, Finite operators and amenable C ∗ -algebras, Proc. Amer. Math. Soc. 56 (1976) 145–151. [3] J.T. Burnham, Closed ideals in subalgebras of Banach algebras. I, Proc. Amer. Math. Soc. 32 (1972) 551–555. [4] H.G. Dales, F. Ghahramani, N. Grønbæk, Derivations into iterated duals of Banach algebras, Studia Math. 128 (1) (1998) 19–54. [5] H.G. Dales, R.J. Loy, Y. Zhang, Approximate amenability for Banach sequence algebras, Studia Math. 177 (1) (2006) 81–96. [6] J. De Cannière, U. Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (2) (1985) 455–500. [7] J. Duncan, I. Namioka, Amenability of inverse semigroups and their semigroup algebras, Proc. Roy. Soc. Edinburgh Sect. A 80 (3–4) (1978) 309–321. [8] E.G. Effros, Z.-J. Ruan, On approximation properties for operator spaces, Internat. J. Math. 1 (2) (1990) 163–187. [9] F. Ghahramani, A.T.M. Lau, Weak amenability of certain classes of Banach algebras without bounded approximate identities, Math. Proc. Cambridge Philos. Soc. 133 (2) (2002) 357–371. [10] F. Ghahramani, A.T.-M. Lau, Approximate weak amenability, derivations and Arens regularity of Segal algebras, Studia Math. 169 (2) (2005) 189–205. [11] F. Ghahramani, R.J. Loy, Generalized notions of amenability, J. Funct. Anal. 208 (1) (2004) 229–260. [12] F. Ghahramani, R.J. Loy, Y. Zhang, Generalized notions of amenability, II, J. Funct. Anal. 254 (7) (2008) 1776– 1810. [13] F. Ghahramani, R. Stokke, Approximate and pseudo-amenability of the Fourier algebra, Indiana Univ. Math. J. 56 (2) (2007) 909–930. [14] F. Ghahramani, Y. Zhang, Pseudo-amenable and pseudo-contractible Banach algebras, Math. Proc. Cambridge Philos. Soc. 142 (2007) 111–123. [15] F. Gourdeau, Amenability of Lipschitz algebras, Math. Proc. Cambridge Philos. Soc. 112 (3) (1992) 581–588. [16] C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (3) (1973) 91–123. [17] B.E. Johnson, Cohomology in Banach Algebras, Amer. Math. Soc., Providence, RI, 1972. [18] B.E. Johnson, Permanent weak amenability of group algebras of free groups, Bull. London Math. Soc. 31 (5) (1999) 569–573. [19] J.L. Kelley, General Topology, D. Van Nostrand Company, Inc., Toronto, 1955. [20] E. Kotzmann, H. Rindler, Segal algebras on non-abelian groups, Trans. Amer. Math. Soc. 237 (1978) 271–281. [21] M. Lashkarizadeh Bami, H. Samea, Approximate amenability of certain semigroup algebras, Semigroup Forum 71 (2) (2005) 312–322. [22] M. Leinert, A contribution to Segal algebras, Manuscripta Math. 10 (1973) 297–306. [23] V. Losert, The derivation problem for group algebras, Ann. of Math. (2) 168 (1) (2008) 221–246. [24] N. Ozawa, A note on non-amenability of B(p ) for p = 1, 2, Internat. J. Math. 15 (6) (2004) 557–565. [25] A.Yu. Pirkovskii, Approximate characterizations of projectivity and injectivity for Banach modules, Math. Proc. Cambridge Philos. Soc. 143 (2) (2007) 375–385. 1 and A 2 buildings, Geom. Funct. Anal. 8 (4) 1 × A [26] J. Ramagge, G. Robertson, T. Steger, A Haagerup inequality for A (1998) 702–731. [27] H. Reiter, L1 -Algebras and Segal Algebras, Lecture Notes in Math., vol. 231, Springer-Verlag, Berlin, 1971.
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[28] H. Reiter, J.D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups, 2nd ed., London Math. Soc. Monogr., vol. 22, Oxford Univ. Press, New York, 2000. [29] E. Samei, R. Stokke, N. Spronk, Biflatness and pseudo-amenability of Segal algebras, preprint, see arXiv:0801.0731, 2008. [30] G.A. Willis, Approximate units in finite codimensional ideals of group algebras, J. London Math. Soc. (2) 26 (1) (1982) 143–154.
Journal of Functional Analysis 256 (2009) 3192–3235 www.elsevier.com/locate/jfa
Gradient type noises II – Systems of stochastic partial differential equations Michael Hinz ∗ , Martina Zähle Mathematisches Institut, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, 07737 Jena, Germany Received 23 July 2008; accepted 2 February 2009 Available online 6 March 2009 Communicated by Paul Malliavin
Abstract The present paper is the second and main part of a study of partial differential equations under the influence of noisy perturbations. Existence and uniqueness of function solutions in the mild sense are obtained for a class of deterministic linear and semilinear parabolic boundary initial value problems. If the noise data are random, the results may be seen as a pathwise approach to SPDE’s. For typical examples, such as spatially one-dimensional stochastic heat equations with additive or multiplicative perturbations of fractional Brownian type, we recover and extend known results. In addition, we propose to consider partial noises of low order. © 2009 Elsevier Inc. All rights reserved. Keywords: Stochastic partial differential equations; Fractional Brownian sheet; Fractional calculus; Semigroups; Function spaces
1. Introduction We deal with a pathwise approach to systems of stochastic partial differential equations. Its three origins are the classical Brownian sheet approach [39], the study of fractional Brownian sheets [2,20,21] and the theory of Stieltjes type integrals based on fractional calculus and function
* Corresponding author.
E-mail addresses:
[email protected] (M. Hinz),
[email protected] (M. Zähle). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.006
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spaces, [43–45]. The objective of the present paper, which is the second and main part of our study, is to give meaning to parabolic systems of formal type ∂ ∂u (t, x) = −Au(t, x) + F u(t, x) + G u(t, x) , ∇Z(t, x) . (1) ∂t ∂t The operator −A realizes some second order differential operator, F and G are coupling coefficients, linear or sufficiently differentiable. Z is a deterministic non-differentiable Rk -valued vector field on Rn+1 . On bounded smooth domains in Rn , we consider Dirichlet boundary initial value problems associated to (1). We give a meaning to the problem saying that an Rk -valued field u = u(t, x) is a mild solution to (1) with initial condition f if t u(t) = P (t)f +
∂ P (t − s)F u(s) ds + It u, ∇Z , ∂t
t ∈ (0, T ).
(2)
0
Here u(t) = u(t, ·) is understood as Banach space-valued function of t, (P (t))t0 denotes the semigroup associated to −A and u → It (u, ∂t∂ ∇Z) is a suitable integral operator which will be defined in the sequel. Due to the non-differentiability of Z, ∂ ∇Z(t, x) ∂t
(3)
needs a proper interpretation. The gradient will be realized in the sense of Schwartz distributions, the time derivative by means of fractional calculus. This point of view allows to use some semigroup theory to prove existence and uniqueness for mild solutions to the deterministic problem (1). We are particularly interested in cases where Z arises as a sample path of a multiparameter process, such that (1) becomes a stochastic equation and (3) may be interpreted as a random noise. For space dimension n 2, such ‘low order gradient type noises’ as in (3) are partial and directed in space, leading to models different from the classical ones. For n = 1 however, we arrive at usual formulations. There are several well known approaches to stochastic partial differential equations, classical sources are [7,11,15,22,39]. Various formulations of and solutions to parabolic equations under fractional Brownian perturbations have been proposed for instance in [9,12,13,16,17,24,29,36]. Some applications of our deterministic results in the sense of SPDE’s will be described in Section 6. In order to compare our results to some of the mentioned references, let us, apart from some remarks, specify to dimensions k = 1 and n = 1 and consider three cases: In the linear additive case, that is with F ≡ 0 and G constant in (1), and with noises fractional both in time and space, given in terms of Gaussian Fourier series, we a.s. obtain a function solution u if 2H + K > 1. Here 0 < H < 1 denotes the temporal and 0 < K < 1 the spatial Hurst index of the noise. This recovers results familiar from [36], where the fact that the noise itself does not have to be an a.s. locally integrable function was first quantitatively characterized. To express the corresponding conditions of the abstract Hilbert space formulation in terms of Hurst indices H and K, one may follow their Section 3.1. We also refer to [37], where a regularity theory for linear equations on the circle was presented. Earlier references on linear evolution equations in Hilbert spaces under fractional Gaussian noises such as [12] had mostly assumed that H > 1/2 and that the (spatial) covariance operator associated to the noise is nuclear. In this
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case the noise is a locally integrable function in space a.s. From [9] however, it was known that in the case of a noise that is white in space, i.e. if K = 1/2, or in other words, where the covariance operator is the identity, one needs H > n/4 to guarantee the existence of solutions in possibly higher space dimensions n. A pathwise approach to stochastic partial differential equations was proposed in [13], the authors used Young integrals to implement an idea our method is very close to. Their formulation provides a link to the powerful theory of rough paths, [23]. For the specific example of a spatially one-dimensional linear heat equation with additive Gaussian noise, they obtain the a.s. existence of function solutions if, spoken in terms of Hurst indices H and K, the condition 2H + K > 1 is satisfied. The spatial Sobolev regularity δ and temporal Hölder regularity γ of their solutions are such that 2γ + δ < 2H + K − 1. In this case, that is exactly the result we recover under the same hypothesis. We would like to point out that purely pathwise approaches as proposed in the present paper or in [13] can deal also with problems involving non-Gaussian noises, since they rely only on some Hölder type conditions for the driving. For a simple example involving fractional stable noises, [18], see Section 6. Function solutions to semilinear evolution equations with linear multiplicative fractional noise, i.e. with F ≡ 0 and G being linear have been discussed for instance in [16] and [17] in the framework of Itô–Skorohod integration, cf. [27]. There the problems are considered on the whole Euclidean space. In the spatially one-dimensional case, the results of the first paper [16] guarantee the existence and uniqueness of function solutions whenever H > 1/2 and K > 1/2. Actually, it contains results for any space dimension n: For noises originating from an (anisotropic) fractional Brownian sheet with Hurst indices (H, K 1 , . . . , Kn ), H > 1/2, Ki > 1/2, i = 1, . . . , n, function solutions are obtained if 2/(2H − 1) + ni=1 Ki > n. If this condition is violated, the unique solution lies in some Meyer–Watanabe type distribution space. The paper also studies the long-time behaviour of the solutions. Regularity in terms of Hölder or Sobolev exponents is not further discussed. The second reference [17] addresses the case K = 1/2 of white noise in space, where it was observed that using the pointwise product, one may obtain unique function solutions for n = 1 if H > 3/4, and using the Wick product if H > 1/2. Using the Wick formulation, there are short-time function solutions also in space dimension n = 2, for the pointwise product n = 1 is essential. A second purpose of that paper was to relate moments of the solution to weighted intersection local times of Brownian motion. Here we need H > 1/2 and 0 < K < 1 to be such that 2H + K > 2 in order to guarantee the a.s. existence and uniqueness of function solutions if F ≡ 0 and G is linear, note that for K = 1/2 this leads again to the threshold H > 3/4. The regularity of the solution obeys the same parameters as in the linear additive case, but now we measure also the temporal behaviour in some kind of Sobolev norm, similar to [24]. As in the linear additive case, it can happen that for fixed time, the solution is even weakly differentiable. Related semilinear equations with a non-linear noise term, that is with non-linear, but sufficiently nice F and G in (1), seem hard to treat using Itô–Skorohod integration. The papers [24] and [13] studied such models using pathwise techniques. In the specific examples of the first reference, the noise itself is much more regular, namely a locally integrable function, since the square root of the covariance operator is assumed to be nuclear. However, some of our methods are inspired by this paper. The concrete example of the pathwise approach in [13] requires the same hypotheses as we do, namely, apart from some differentiability assumptions on F and G, H > 1/2 and 2H + K > 2. For the regularity of the solution we basically observe the same parameter range as they do. Our regularity results are slightly weaker, in the sense that they consider Hölder continuity in time while we use our Sobolev type norm. On the other hand, they
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obtain a solution only up to some (unknown) explosion time, while in our case a unique solution is seen to exist for an arbitrarily large time interval. Let us emphasize that we restrict ourselves to problems on bounded domains. For linear equations this is not so important, for the semilinear case this is essential, as we use a contraction principle and typical noises exhibit a rather bad behaviour at infinity, cf. [3]. Finally, let us mention some reasons suggesting that systems (1) under noises of form (3) seem worth to be studied: Formal gradients of random fields on Euclidean spaces have already been considered some time ago. They exhibit interesting geometric features, and in the stationary isotropic case they are related to simple models in classical turbulence theory. See [8,19,25,40]. So it seems likely that systems under gradient type noises induced by Rn -valued random fields Z yield interesting models for a number of physical problems. For non-linear equations involving gradients of the solution fields, see for instance [5]. In the classical Brownian sheet approach [39] one cannot expect the solutions to parabolic stochastic differential equations in space dimension n > 1 to be scalar valued processes, at least not if the noise is taken to be white in time and space, formally given by ∂ n+1 Z . ∂t∂x1 · · · ∂xn
(4)
Here Z would be the Brownian sheet on Rn+1 . The roughness of the white noise forces to move on to the study of distribution-valued processes, see [10,39]. Models involving (3) instead lead to a simpler type of calculus which does yield function solutions to (1) for any space dimension under conditions that, apart from additional restrictions caused by non-linearities, actually stem from the usual equations with (4) in space dimension n = 1. As already conjectured by a careful referee and noticed by the authors in the revision process of the present paper, the pathwise approach described here can tell much more. Also for equations involving (4) with suitably chosen Z, there exist function solutions in higher space dimensions n, provided the Hölder respectively Sobolev orders (Hurst parameters) of Z are big enough. In the special case of noises that are white in space one ends up with the conditions familiar from [9] and [17]. It seems reasonable to discuss this matter in a separate follow-up note. Our pathwise method is based on fractional calculus, it has been explained in part I, [14], which had combined [34] and [43]. For related SDE’s see for instance [28] and [44]. Our calculations partly follow [24]. Instead of an abstract general setting, we always measure the spatial regularity in terms of Sobolev spaces. Pointwise products are defined by means of paraproducts, see Lemma C.1 in Appendix C and [33]. This suits the problem surprisingly well and is consistent with the product definitions used in the concrete examples of [24] and [13]. The paper is organized as follows: The next section contains some preliminaries, the main setup and the definition of our integral operator It from (2). In Section 3, a problem under linear multiplicative noise studied, cf. Theorem 3.2. Section 4 generalizes the result to the case of non-linear multiplicative noise, Theorem 4.2. Section 5 points out some refinements related to anisotropic fields and considers a purely linear model, Theorem 5.3. In Section 6 we discuss probabilistic applications. Key results, in particular mapping properties of the pathwise integral operator, are presented in Section 7, they imply the main theorems. Technical proofs are shifted to Appendix A. Although a few facts are used already in these proofs, we have decided to put necessary surveys on semigroup theory, fractional calculus and function spaces into Appendices B and C at the very end, this way the main proofs appear a bit earlier in the text. Note that Lemma B.1 seems to be interesting in its own.
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2. Preliminaries {e1 , . . . , en } denotes the standard basis and | · |n the Euclidean norm in Rn , n is suppressed from notation if n = 1.
Given a normed vector space (E, · E ), the product space kj =1 E is endowed with the k
k l1 -norm j =1 fj E , f = (f1 , . . . , fk ) ∈ j =1 E. j
M(n × k, R) denotes the space of real (n × k)-matrices. For two members B = (bl ) l=1,...,n and C =
j (cl ) l=1,...,n j =1,...,k
of M(n × k, R), with row vectors bl = j
j
j
(bl1 , . . . , blk ),
cl =
j =1,...,k 1 (cl , . . . , clk ),
j
l = 1, . . . , n, and column vectors bj = (b1 , . . . , bn ), cj = (c1 , . . . , cn ), j = 1, . . . , k, we use the notation B, C := b1 , c1 , . . . , bk , ck ,
(5)
where each component of the real k-vector on the right-hand side is given by the standard scalar j j product on Rn , bj , cj = nl=1 bl cl . Obviously B, C = nl=1 bl · cl , where bl · cl = bl1 cl1 , . . . , blk clk ,
(6)
a notation we will prefer at some occasions later on. We do not write the transposition of vectors explicitely, it will always be apparent from the context. a ∧ b and a ∨ b denote the minimum and maximum of two numbers a and b, respectively. Positive constants whose values are not of importance are denoted by c, their values may differ from one occurrence to another. Let k, n ∈ N, k, n 1. Throughout the whole paper, D ⊂ Rn is a bounded C ∞ -domain. 2 (D), obtained as the Let A0 be a self-adjoint operator in L2 (D) with domain dom(A0 ) = H2,0 Friedrichs extension of some second order differential operator AD , (AD f )(x) = −
n
∂ ∂f aik (x) (x) + c(x)f (x) ∂xi ∂xk
i,k=1
f ∈ dom(AD ) = C0∞ (D), satisfying the ellipticity condition i,k aik (x)ξi ξk λ|ξ |2 , x ∈ D, ξ ∈ Rn with some λ > 0, and having real-valued coefficients aik = aki ∈ C ∞ (D), c ∈ C ∞ (D), c(x) 0, x ∈ Rn , which, together with all their derivatives, can be extended continuously to D, see e.g. [1] or [38]. By the choice of the domain, Dirichlet boundary conditions are imposed. The simplest example is the Dirichlet Laplacian − on D ⊂ Rn . Let B be a real (k × k)-matrix, such that all eigenvalues of B are contained in the half plane {z ∈ C: Re z > 0}. We consider A = BA0 , more precisely: Given u = (u1 , . . . , uk ) ∈ C0∞ (D, Ck ), we set Au := B(A0 u1 , . . . , A0 uk ),
(7)
with the usual matrix multiplication. We refer to B as the cross diffusion matrix. From the spectral representation of A0 it can be deduced that A is a sectorial operator, hence −A generates an analytic semigroup (P (t))t0 on L2 (D) of negative type. A proof is carried out in [30] for AD = −, the arguments work for general AD . It is further shown that
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2 (D, Ck ) and complex interpolation shows that for −3/2 α α + κ < 3/2, the dom(A) = H2,0 (D, Ck ) isomorphically onto H α2 (D, Ck ). fractional power Aκ/2 of the operator A maps H α+κ 2 Usually, one refers to such a situation as scale of Hilbert spaces. For the definition and some properties of these spaces we refer to Appendix C. For −1/2 < α < 3/2, the norms · α and f → Aα/2 f 0 are equivalent, for −3/2 α −1/2, · |H2α (D, Ck ) and f → Aα/2 f 0 are equivalent. If (P (t))t0 is the analytic semigroup of negative type on L2 (D, Ck ) generated by −A, these isomorphism properties together with (2.10) permit to consider (P (t))t0 as a strongly continuous and equibounded semigroup on H α2 (D, Ck ) for any fixed −3/2 α < 3/2. Below we will restrict attention to real subspaces, also explained in Appendix C. Now assume Z = (Z 1 , . . . , Z k ), Z j = Z j (t; x1 , . . . , xn ), j = 1, . . . , k, is an Rk -valued vector field on Rn+1 . Below we will consider Z also as Banach space valued function Z(t) of the time j parameter t. In this case we put Zt (s) := 1(0,t) (s)(Z j (s) − Z j (t)) and Zt (s) := 1(0,t) (s)(Z(s) − j Z(t)). The values Z (t) will be assumed to exist for each t > 0 in the pointwise sense. Before we state the definition of the integral operator It from (2), we give a heuristic motivation for it. Assume k = 1, n = 1 and D = (a, b) ⊂ R isa finite interval. Let p(t, x, y) denote the transition densities of the semigroup, i.e. P (t)f (x) = (a,b) p(t, x, y)f (y) dx, and assume for a moment they were regular enough to write
α D0+ p(t − s, x, y)g(s, y)
(−1)α
∂ 1−α D Zt (s, y) dy ds, ∂y t−
(8)
(0,t) (a,b)
where g = g(s, y), denotes a real-valued function and 0 < α < 1. α p(t − s, x, ·)g(s, ·) denotes the left-sided Weyl–Marchaud fractional derivative of s → D0+ order α of the function s → p(t − s, x, ·)g(s, ·), seen as vector-valued function of s. Similarly, 1−α Zt (s, ·) denotes the right-sided Weyl–Marchaud fractional derivative of order 1 − α s → Dt− of s → Zt (s, ·). That means, we integrate p(t − s, x, y)g(s, y) with respect to Zt (s, y) over (0, t) × (a, b) by means of a Stieltjes type integral for two-parameter functions. A similar construction was studied in part I of the present paper, [14], where relations to well-known methods for Stieltjes-type integration via regularization were pointed out, [34,43]. For a survey on fractional integrals and derivatives, we refer to Appendix B. α , carrying out the integration over (a, b) and rearTaking into account the definition of D0+ ranging the terms, (8) is seen to equal
(−1)α Γ (1 − α)
t s
−α
∂ 1−α P (t − s) g(s) Dt− Zt (s) ds ∂y
0
α(−1)α + Γ (1 − α) α(−1)α + Γ (1 − α)
t s
−α−1
(s − σ )
∂ 1−α P (t − s) − P (t − σ ) g(s) Dt Zt (s) dσ ds ∂y
0 0
t s 0 0
∂ 1−α Dt Zt (s) dσ ds. (s − σ )−α−1 P (t − σ ) g(s) − g(σ ) ∂y
(9)
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It seems convenient to express the middle summand of (9) in terms of fractional powers of A. We use the semigroup property together with the analyticity of (P (t))t0 and the fact that s
−α−1
u
Γ (1 − α) α 1 A f − s −α f + I − P (u) f du = α α
∞
u−α−1 P (u)f du,
s
0
for f ∈ dom(Aα ), where I is the identity operator, see Appendix B. Inserting this into (9), the term arising from the summand α −1 s −α f cancels with the first summand in (9), and we arrive at the expression in Definition 2.1 below. Note that in part I we would have corrected the integrand ∂ p(t − s, x, ·)g(s) at s = 0 and added the correction terms P (t)(g(0) ∂y (Z(t) − Z(0))). Here these corrections cancel and may be omitted. In [13], Young integrals were used to realize a similar idea. Young integrals provide a connection to rough paths, while our formulation using fractional calculus is closer to classical PDE theory. The preceding motivates the following rigorous definition. Let k ∈ N \ {0} and suppose j that either g = (gl ) l=1,...,n is a constant real (n × k)-matrix, g ∈ M(n × k, R) or, g is an j =1,...,k
j
M(n × k, R)-valued field on Rn+1 , such that all rows gl = (gl1 , . . . , glk ) with gl = j gl (t; x1 , . . . , xn ), seen as vector valued functions t → gl (t), also admit their values in a Sobolev space contained in that scale. The gradient is taken in distributional sense and always refers to the space variable x = (x1 , . . . , xn ). We use the notation (5). Let 0 < α < 1. For t 0, set
Itα
t ∂ 1−α α g, ∇Z := (−1) Aα P (t − s) g(s), ∇Dt− Zt (s) ds ∂t 0
t s + cα (−1)α
1−α (s − σ )−α−1 P (t − σ ) g(s) − g(σ ) , ∇Dt− Zt (s) dσ ds
0 0
t ∞ + cα (−1)
α 0
1−α σ −α−1 P (σ + t − s) g(s), ∇Dt− Zt (s) dσ ds.
s
The number cα is given by cα = αΓ (1 − α)−1 . Each semigroup operator applies to the entire kvalued term in sharp brackets. The integral terms contain products of functions and distributions. We define them by means of paraproducts as studied in [35] and [33], see Appendix C. This includes the product definitions used in the concrete examples of [24] and [13]. Definition 2.1. For g and Z as above, we define the integral operator It by It
∂ ∂ α g, ∇Z := It g, ∇Z , ∂t ∂t
t 0.
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Remark 2.2. In Section 7 it will be shown that under the respective hypotheses of Theorems 3.2, 4.2 and 5.3 below, It (g, ∂t∂ ∇Z) exists and does not depend on the particular choice of α. 3. Problems with linear multiplicative noise Let G = (G1 , . . . , Gn ) denotes an M(n × k, R)-valued field on Rn , such that each row Gl = (G1l , . . . , Gkl ) : Rk → Rk is a linear mapping. Below we write u → Gu and u → Gl u to emphasize the mappings are linear. Let T > 0 be arbitrary. We study systems of semilinear parabolic equations with linear multiplicative gradient type noise, formally given by ∂u ∂ (t, x) = (−Au)(t, x) + Gu, ∇Z (t, x), ∂t ∂t
(10)
t ∈ (0, T ), x ∈ D, together with the Dirichlet boundary condition u(·, t)∂D = 0, t ∈ (0, T ),
(11)
and with initial condition u(0, x) = f (x),
x ∈ D.
(12)
∂2Z By (5) and (6), we formally have Gu, ∂t∂ ∇Z = nl=1 Gl u · ∂t∂x . l The problem (10)–(12) is made rigorous in the sense of mild solutions: Definition 3.1. A function u is a mild solution to (10)–(12), if it satisfies the integral equation u(t) = P (t)f + It
∂ Gu, ∇Z , ∂t
t ∈ (0, T ).
(13)
Eq. (10) allows to describe diffusion phenomena under couplings caused by the cross diffusion term Au or the noise term G(u), ∂t∂ ∇Z . For k ∈ N, k 1, 0 < γ < 1, 1 < p < ∞ and δ ∈ R, C γ ([0, T ], Hpδ (Rn , Rk )) denotes the space of γ -Hölder continuous Hpδ (Rn , Rk )-valued functions on [0, T ], such that γ u C [0, T ], H δ Rn , Rk := p
sup 0τ
u(t) − u(τ )|Hpδ (Rn , Rk ) (t − τ )γ
< ∞.
(14)
Following essentially [24], we denote by W γ ([0, T ], H˚ 2δ (D, Rk )), 0 < γ < 1, δ ∈ R, the space of H˚ 2δ (D, Rk )-valued functions on [0, T ] such that t u(t) − u(τ ) δ uγ ,δ := sup u(t)δ + dτ < ∞. (t − τ )γ +1 0tT 0
For the definition of the spaces Hpδ (Rn , Rk ) and H˚ 2δ (D, Rk ) we refer to Appendix C.
(15)
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Theorem 3.2. Suppose 0 < α, β, γ , ε < 1 and Z = (Z 1 , . . . , Z k ) is a vector field on Rn+1 such 1−β that Z ∈ C 1−α ([0, T ], Hq (Rn , Rk )), q > 2 ∨ (n/δ). Let G be as specified above, and let f ∈ 2γ +δ+ε H˚ 2 (D, Rk ), 2γ + δ + ε < 3/2. Assume further that β < δ, α < γ < 1 − α and 2γ + δ < 2 − 2α − β. Then problem (10)–(12) has a unique mild solution u in W γ ([0, T ], H˚ 2δ (D, Rk )). The proof relies on the key Proposition 7.2 below. Note that in particular, the temporal regularity 1 − α of the driving field needs to be greater than 1/2. The conditions δ > β and q > 2 ∨ (n/δ) ensure that Lemma C.1 below is applicable in order to evaluate the occurring pointwise product. It is strongly related to usual Sobolev embedding theorems. 4. Problems with non-linear noise term Under familiar dimension conditions, the result can be generalized to systems with coupling and non-linear multiplicative noise term: Let F : Rk → Rk be a C 1 -mapping such that F (0) = 0 and having bounded differential DF . That means, if · L(Rk ,Rk ) denotes a norm in L(Rk , Rk ), we have sup DF (x)L(Rk ,Rk ) < M
(16)
x∈Rk
for some number M > 0. Let G = (G1 , . . . , Gn ) denote an M(n × k, R)-valued field on Rk such that each Gl = 1 (Gl , . . . , Gkl ) : Rk → Rk is a C 2 -mapping, which fulfilles Gl (0) = 0 and has a second order differential D2 Gl , which is bounded and Lipschitz continuous. That means, if · L(Rk ,L(Rk ,Rk )) is a norm in L(Rk , L(Rk , Rk )), we have sup D2 Gl (x)L(Rk ,L(Rk ,Rk )) < M
(17)
x∈Rk
and 2 D Gl (x) − D2 Gl (y)
L(Rk ,L(Rk ,Rk ))
L|x − y|k ,
(18)
x, y ∈ Rk , with some numbers M, L > 0. If, for example, each Gl is a compactly supported C ∞ -mapping, these properties are obvious. We write u → F (u), u → G(u) and u → Gl (u) to point out that F and G are non-linear. We consider semilinear parabolic problems with non-linear multiplicative noise term, given by ∂u ∂ (t, x) = (−Au)(t, x) + F u(t, x) + G(u), ∇Z (t, x), ∂t ∂t t ∈ (0, T ), x ∈ D, together with the former boundary and initial conditions (11) and (12).
(19)
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Definition 4.1. A function u is called a mild solution if it satisfies t u(t) = P (t)f +
∂ P (t − s)F u(s) ds + It G(u), ∇Z , ∂t
t ∈ (0, T ).
0
The composition operators are explained in Appendix C. As usual, their Fréchet derivatives are used to derive key estimates. This involves pointwise multiplication in a single H2δ -space, what in turn forces to restrict to L∞ -functions. δ (D, Rk )), 0 < γ < 1, δ ∈ R, the space of H˚ δ (D, Rk ) = Denote by W γ ([0, T ], H˚ 2,∞ 2,∞ δ H˚ 2 (D, Rk ) ∩ L∞ (Rn , Rk )-valued functions on [0, T ] such that uγ ,δ,∞ := sup 0tT
u(t)
t δ,∞
+ 0
u(t) − u(τ )δ,∞ dτ (t − τ )γ +1
< ∞.
(20)
Here · δ,∞ := · δ + · ∞ , where · δ is the norm in H2δ (Rn , Rk ) and · ∞ that in L∞ (Rn , Rk ). We obtain: 1−β
Theorem 4.2. Suppose 0 < α, β, γ , δ, ε < 1 and Z ∈ C 1−α ([0, T ], Hq (Rn , Rk )), q > 2 ∨ (n/δ). Let F and G be as specified above such that (16)–(18) are satisfied, and 2γ +δ+ε assume f ∈ H˚ 2,∞ (D, Rk ), 2γ + δ + ε < 3/2. Let further β < δ, α < γ < 1 − α and 2γ + δ ∨ (n/2) < 2 − 2α − β. Then problem (19), (11), (12) has a unique mild solution u δ (D, Rk )). in W γ ([0, T ], H˚ 2,∞ This theorem relies on Proposition 7.3. In the hypotheses we have assumed 0 < δ < 1. Though convenient, this is a technical restriction. With refined hypotheses on F and G and some more technical effort, it could be removed. Remark 4.3. Note that Theorem 4.2 forces an additional restriction on the temporal regularity of the driving field Z. Only if n/4 < 1 − α we can find some 0 < β < 1, such that Theorem 4.2 guarantees the existence and uniqueness of a function solution to the non-linear problem (19), (11), (12). This bound had already appeared in [9] and [24]. 5. Complements and refinements First, we refine our hypotheses on Z, and second, refine our results in the case of a related linear problem. Comparing (10) and the proof of Theorem 3.2, respectively Proposition 7.2 below, we j 1−α (Z j )t has no influence on u. This allows lower observe that if Gl ≡ 0, the term ∂x∂ l Dt− (nonnegative) degrees of spatial smoothness of Z j in these directions el . Now let Gj = j j (G1 , . . . , Gn ), j = 1, . . . , k, denote the columns of G. Similar to part I, we define the
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δ n space C γ ([0, T ], Hp,G j (R , R)), 0 < γ < 1, δ ∈ R, 1 < p < ∞, of γ -Hölder continuous δ n Hp,G j (R , R)-valued functions on [0, T ] by
γ u C [0, T ], H δ
n R , R := p,Gj
sup
δ n u(t) − u(τ )|Hp,G j (R , R)
0τ
(t − τ )γ
< ∞,
(21)
δ n where the spaces Hp,G j (R , R) are explained in Appendix C. We immediately obtain:
Corollary 5.1. The assertions of Theorems 3.2 and 4.2 remain valid if the hypotheses on Z there 1−β are replaced by Z j ∈ C 1−α ([0, T ], Hq,Gj (Rn , R)), j = 1, . . . , k, with some q > 2 ∨ (n/δ). j
A particularly simple problem related to (10) arises if G is a constant matrix G = (Gl ) ∈ M(n × k, R), ∂u ∂ (t, x) = (−Au)(t, x) + G, ∇Z (t, x), ∂t ∂t
(22)
t ∈ (0, T ), x ∈ D. Eq. (22) is linear, the noise is additive, and the middle summand in (A.1) vanishes. Definition 5.2. u solves the problem (22), (11), (12) in the mild sense if u(t) = P (t)f + It
∂ G, ∇Z , ∂t
t ∈ (0, T ).
(23)
For 0 < γ < 1 and δ ∈ R, let C γ ([0, T ], H˚ 2δ (D, Rk )) be the space of γ -Hölder continuous H˚ 2δ (D, Rk )-valued functions u on [0, T ], such that u|C γ ([0, T ], H2δ (Rn , Rk )) < ∞. 1−β
Theorem 5.3. Suppose 0 < α, β, γ , ε < 1 and Z ∈ C 1−α ([0, T ], H2 (Rn , Rk )). Let G = 2γ +δ+ε j (Gl ) ∈ M(k × n, R), and assume that f ∈ H˚ 2 (D, Rk ), 2γ + δ + ε < 3/2. Then the mild solution u according to (13) exists and is in C γ ([0, T ], H˚ 2δ (D, Rk )), provided γ + α < 1 and 2γ + δ < 2 − 2α − β. The theorem follows from Proposition 7.1 below. As we do not have to use a contraction principle and the unique solution is already explicitely given by (23), the previous lower bound on γ is no longer necessary. Also, the time regularity here is somewhat stronger than in the previous theorems, note that for any γ > γ , C γ ([0, T ], H˚ 2δ (D, Rk )) is continuously embedded in the space W γ ([0, T ], H˚ 2δ (D, Rk )). Obviously the hypotheses can be refined as before, we omit it. Remark 5.4. In view of the facts listed in Appendix C, we might as well treat boundary initial value problems in general Lp (D, Rk )-spaces, 1 < p < ∞. We refer to [38], in particular to 4.9.1.
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6. Application to random fields We indicate some applications of the discussed models to random fields. Then the equations are to be read in the pathwise sense: There is some Ω1 ⊂ Ω, P(Ω1 ) = 1, such that for any ω ∈ Ω1 , solutions to (10), (19) and (22) are obtained for Z(ω) in place of Z. Note that restricted to Ω1 , the estimates in the corresponding proofs remain valid. Let us survey some possibilities. In most references the noises are chosen to be given in terms of Gaussian Fourier series, bH,K (t, x) =
∞
qj ej (x)βjH (t),
(24)
j =1
where ej are the eigenfunctions of the semigroup (P (t))t0 , the βjH are iid standard fractional Brownian motions with Hurst parameter 0 < H < 1, and the qn are such that j =1 qj2 j −2μ < ∞. −β One can show that (24) is a member of C α ([0, T ], H˚ 2 (D, Rn )) if 0 < α < H and 0 < μ < β < 1. A number 0 < K < 1 just slightly bigger than 1 − μ might be called a Hurst parameter in space, see e.g. [13] or [36]. As mentioned in the introduction, we may for instance consider spatially one-dimensional problems, k = n = 1, with bH,K (t, x) in place of ∂Z ∂x (t, x). For (22) we then need 2H + K > 1, for (10) and (19), H > 1/2 and 2H + K > 2. (If β 1/2 in the multiplicative cases, one actually has to extend bH,K (t, ·) temporarily beyond D to get a distribution on Rn , this is no problem, cf. [38, 4.2.2].) Next, one might like to use globally defined fields. Set u,z X(s, y) := X(s + u, y + z) − X(s, y + z) − X(s + u, y) + X(s, y), s, u ∈ R, y, z ∈ Rn , to denote the rectangular increments u,z X(s, y) of a field X. First, consider centered real-valued Gaussian random fields B H,K on [0, T ] × Rn having stationary increments t−s,x−y B H,K satisfying 2 1/2 c(t − s)H |x − y|K Et−s,x−y B H,K n,
(25)
0 s t T , x, y ∈ Rn , where 0 < H, K < 1, c > 0 is some universal non-random constant, | · |n is the Euclidean norm on Rn and | · | the absolute value on R. A special case is the (spatially isotropic) fractional Brownian sheet with Hurst indices H and K, in this case equality holds in (25). See [2,14,20,21] or [41]. Alternatively, we might want to study centered real-valued Gaussian fields B H,K on [0, T ] × Rn having stationary increments t−s,rel B H,K in each space direction el , l = 1, . . . , n, and such that 2 1/2 Et−s,rel B H,K c(t − s)H |r|Kl ,
(26)
0 s t T , r ∈ R, l = 1, . . . , n, where 0 < H < 1, 0 < Kl < 1, l = 1, . . . , n and c > 0 is nonrandom. As a special case one may consider the (anisotropic) fractional Brownian sheet with Hurst parameters H and K = (K1 , . . . , Kn ), it corresponds to equality in all the conditions (26). In both cases we may choose a version, again denoted by B H,K and B H,K such that a.a. paths of B H,K respectively B H,K are bounded and satisfy certain multiple Hölder conditions
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on any fixed compact set. To see that our results may be applied in these cases, note the following: If a sample of, say B H,K , is multiplied by a compactly supported C ∞ -function that β equals one on a neighbourhood of D, the result is a member of a space Hq (Rn , R) for any fixed t. This neither is true for the uncorrected sample itself nor for a simple non-smooth cut-off using an indicator function. As our method considers only a neighbourhood of D, this makes no difference, hence we assume this has been done and write again B H,K and B H,K . Then β a.s. B H,K ∈ C α ([0, T ], Hq (Rn , R)) if 0 < α < H , 0 < β < K and 1 < q < ∞, and given β
any index vector g = (g1 , . . . , gn ), B H,K ∈ C α ([0, T ], Hq,g (Rn , R)), provided 1 < q < ∞, 0 < α < H and 0 < β < Kl for all l such that gl = 0. 1 1 k k Now assume for instance that Z = (B1H ,K , . . . , BkH ,K ) is an independent vector of fracj
j
tional Brownian sheets BjH ,K of (possibly different) orders 0 < H j , K j < 1. Then (10) and (19) yield systems coupled by cross-diffusion, non-linearity, or by a mixed fractional noise term. If minj =1,...,k H j > 1/2 and 2 minj =1,...,k H j + minj =1,...,k K j > 2, we a.s. obtain function solutions. For linear systems (22), 2 minj =1,...,k H j + minj =1,...,k K j > 1 suffices. The case H j = H , K j = K, j = 1, . . . , k, provides the simplest prototype. 1
k
Or, let Z = (B1H,K , . . . , BkH,K ) be an independent vector consisting of anisotropic fractional j
j
j
j
Brownian sheets BjH,K , 0 < H < 1, K j = (K1 , . . . , Kn ), 0 < Kl < 1. With the refinements described in Corollary 5.1, we may a.s. obtain function solutions to system under anisotropic j noises (10) or (19), as long as for some 1/2 < H < 1, 0 < K < 1, 2H + K > 2 and K < Kl for j all those j, l for which Gl does not vanish identically. As a specific example, Theorem 4.2 yields existence and uniqueness for solutions to onedimensional semilinear heat equations driven by anisotropic fractional Brownian sheets B H,K , ∂ 2 B H,K ∂u (t, x) = u(t, x) + F u(t, x) + G u(t, x) . ∂t ∂t∂x One interesting fact about fields a-priori defined on [0, T ] × Rn is that anisotropic structures as in (26) may be considered. Another interesting fact is that series expansions of type (24) yield noises that do already contain information on the boundary values specified in our problems, while noises obtained from globally defined fields do not. A third motivation to use global fields is that we may easily consider also non-Gaussian β
examples. One particular is the β-fractional α-stable sheet Xα , α ∈ (1, 2), β = (β0 , . . . , βn ), 0 < βi < 1 − 1/α, i = 0, . . . , n. It may be constructed as follows, for details we refer to [18]. The α-stable white noise measure μα on S (Rn+1 ) is given in terms of a Bochner–Minlos formula, S (Rn+1 )
1 ϕ(x)α dx , ei ω,ϕ dμα (ω) = exp − 2 Rn+1
Let I β be an anisotropic Riesz-potential operator of form I β ϕ(x) = cn (β)−1
Rn+1
ϕ(y) |x − y|1−β
dy,
ϕ ∈ S Rn+1 .
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where cn (β) = 2n+1 i Γ (βi ) cos(βi π/2) and |x − y|1−β = i |xi − yi |1−βi . It is known that there is an S (Rn+1 )-valued random variable Tβ such that Tβ ω, ϕ = ω, I β ϕ , for all ϕ ∈ S(Rn+1 ) and a.a. ω ∈ S (Rn+1 ). Under the image (probability) measure μα := μα ◦ Tβ−1 , set β
Xαβ (x0 , x1 , . . . , xd ) := ω, 1[0,x] , where [0, x] = [0, x0 ] × · · · × [0, xn ]. From Proposition 6.3 in [18] it follows easily that (after a β
β
cut-off at infinity) its samples Xα (ω) are P-a.s. members of C α ([0, T ], Hq (Rn , R)), provided 0 < α < β0 and 0 < β < mini=1,...,n βi . For instance if k = n = 1, Theorem 5.3 allows to obtain β
function solutions to linear heat equations driven by β-fractional α-stable noises Xα , β
∂ 2 Xα ∂u (t, x) = u(t, x) + , ∂t ∂t∂x provided 2β0 + β1 > 1. 7. Mapping properties and correctness of the definition The existence and uniqueness statements of Sections 3–5 rely on the mapping properties of the integral operator, which are investigated in this section. As a by-product we prove Remark 2.2. Recall the definitions (14), (15) and (20) of the spaces C γ ([0, T ], Hqδ (Rn , Rk )), δ (D, Rk )). The main steps in proving Theoγ W ([0, T ], H˚ 2δ (D, Rk )) and W γ ([0, T ], H˚ 2,∞ rems 5.3, 3.2 and 4.2 are formulated in the following three propositions, whose proofs are given in Appendix A: j
Proposition 7.1. Given G = (Gl ) ∈ M(n × k, R) and 0 < α, β, γ < 1, the mapping ∂ Z → I(·) G, ∇Z ∂t 1−β
is a continuous linear operator from C 1−α ([0, T ], H2 provided γ + α < 1 and 2γ + δ < 2 − 2α − β.
(27)
(Rn , Rk )) into C γ ([0, T ], H2δ (D, Rk )),
Theorem 5.3 follows from Proposition 7.1 and the mapping properties of the semigroup, (P (t − τ ) − I )P (τ )f δ c(t − τ )ε f 2γ +δ+ε , (t − τ )γ
0 τ < t T.
See in particular formula (B.5) in Appendix B. We introduce the following equivalent norms on W γ ([0, T ], H˚ 2δ (D, Rk )): () uγ ,δ
:= sup e
−t
0tT
where 1 is a parameter, cf. [24].
u(t) + δ
t 0
u(t) − u(τ )δ dτ (t − τ )γ +1
< ∞,
(28)
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Proposition 7.2. Let 0 < α, β, γ < 1, α < γ < 1 − α, β < δ and 2γ + δ < 2 − 2α − β. For j 1−β Z ∈ C 1−α ([0, T ], Hq (Rn , Rk )), q > 2 ∨ (n/δ), and G = (Gl ) such that each Gl is linear, the mapping ∂ u → I(·) Gu, ∇Z ∂t
(29)
is a contraction in W γ ([0, T ], H˚ 2δ (D, Rk )). More precisely, () I(·) Gu, ∂ ∇Z C()u() , γ ,δ ∂t γ ,δ
where C() > 0 tends to zero as goes to infinity. () δ (D, Rk )) be defined as the anaNow let the equivalent norms · γ ,δ,∞ in W γ ([0, T ], H˚ 2,∞ logues of (28), based on (20). Theorem 3.2 now follows from Banach’s fixed point theorem and the mapping properties of the semigroup, (B.5).
Proposition 7.3. Let 0 < α, β, γ , δ < 1. Further assume that α < γ < 1 − α, β < δ and 2γ + 1−β δ ∨ (n/2) < 2 − 2α − β. Let Z ∈ C 1−α ([0, T ], Hq (Rn , Rk )), q > 2 ∨ (n/δ), and let the nonlinear coefficient term G = (G1 , . . . , Gn ), be such that Gl (0) = 0, and each Gl has bounded and Lipschitz second order differential D2 G, i.e. (17) and (18) hold. Then there is a closed ball δ (D, Rk )), such that (29) maps B into itself and for 1 large enough, B0 ⊂ W γ ([0, T ], H˚ 2,∞ 0 () I(·) G(u), ∂ ∇Z − I(·) G(v), ∂ ∇Z ∂t ∂t
γ ,δ,∞
()
C()u − vγ ,δ,∞ ,
(30)
u, v ∈ B0 . Theorem 4.2 now follows similarly as in the previous case, having choosen a common 0 in t Proposition 7.3 and in the following Lemma 7.4. Set J0 (t, u) := 0 P (t − s)F (u(s)) ds. Lemma 7.4. For 0 < γ , δ < 1 such that γ + n/4 < 1 and 0 1 large enough, u → J0 (·, u) maps the closed ball B (0 ) (0, e−0 T ) into itself and for 0 , J0 (·, u) − J0 (·, v)(0 )
γ ,δ,∞
( )
0 C()u − vγ ,δ,∞ ,
u, v ∈ B (0 ) (0, e−0 T ), where C() tends to zero as tends to infinity. As a consequence of Propositions 7.1–7.3 we observe that the integral in the respective sense can be rewritten as forward limit, similar to the forward integral from part I, [14]. For l = 1, . . . , n, set ∂l,r ϕ(x) := ϕ(x + rel ) − ϕ(x),
r > 0,
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to denote the forward difference of a function ϕ in direction el . As in part I, let ∇r+ ϕ(x) = ∂1,r ϕ(x), . . . , ∂n,r ϕ(x) ,
r > 0,
denote the forward pre-gradient of ϕ. Now put Itα
∂ + g, ∇r Z ∂t t
1−α Aα P (t − s) g(s), ∇r+ Dt− Zt (s) ds
:= (−1)
α 0
t s + cα (−1)
α
1−α (s − σ )−α−1 P (t − σ ) g(s) − g(σ ) , ∇r+ Dt− Zt (s) dσ ds
0 0
t ∞ + cα (−1)
α 0
1−α σ −α−1 P (σ + t − s) g(s), ∇r+ Dt− Zt (s) dσ ds,
(31)
s
for t > 0, r > 0 and with some 0 < α < 1. Corollary 7.5. (i) Under the hypotheses of Proposition 7.1, we have Itα
∂ ∂ + α G, ∇Z = lim It G, ∇r Z , r→0 ∂t ∂t
t > 0,
the limit taken in the strong sense in H˚ 2δ (D, Rk ). (ii) Under the hypotheses of Propositions 7.2 and 7.3, we have Itα
∂ ∂ + α G(u), ∇Z = lim It G(u), ∇r Z , r→0 ∂t ∂t
t > 0,
δ δ (D, Rk )) in the strong sense in H˚ 2(,∞) (D, Rk ). for any u ∈ W γ ([0, T ], H˚ 2(,∞)
Here we have written G(u) for linear or non-linear G. A similar assertion is true if the forward differences are replaced by backward differences. The limit representations are helpful in verifying Remark 2.2: Lemma 7.6. Under the hypotheses of Propositions 7.1 and interpreted according to (27), Definition 2.1 is correct, i.e. the existence and the value of the integral do not depend on the particular choice of α: ∂ ∂ Itα G, ∇Z = It G, ∇Z . ∂t ∂t
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If the hypotheses of Proposition 7.2 respectively 7.3 hold, the same is true for the mapping (29) respectively its non-linear version: ∂ ∂ Itα G(u), ∇Z = It G(u), ∇Z . ∂t ∂t The proof can be found in Appendix A. Appendix A. Proofs First notice that in Definition 2.1 and according to (6), we have Itα (g, ∂t∂ ∇Z) := n (l) ∂2Z l=1 It (gl · ∂t∂xl ), t > 0, where (l)
It
∂ 2Z gl · ∂t∂xl ∂ 1−α Aα P (t − s) gl (s) · Dt− Zt (s) ds ∂xl
t := (−1)
α 0
t s + cα (−1)
α
∂ 1−α (s − σ )−α−1 P (t − σ ) gl (s) − gl (σ ) · Dt− Zt (s) dσ ds ∂xl
0 0
t ∞ + cα (−1)
α 0
s
∂ 1−α σ −α−1 P (σ + t − s) gl (s) · Dt− Zt (s) dσ ds. ∂xl
(A.1)
() We further point out that for γ > γ , W γ ([0, T ], H˚ 2δ (D, Rk )) with norm · γ ,δ is continuously () embedded into W γ ([0, T ], H˚ 2δ (D, Rk )) with norm uγ ,δ (with the same parameter ). This will be helpful. Further, we will use of several facts listed in Appendices B and C, in particular formulae (B.3)–(B.5), as well as Lemmas B.1 and C.1.
A.1. A detailed proof for Proposition 7.2 Proof. Step 1: Parameters. We write α to denote the number α as given in Proposition 7.2, i.e. 1−β by hypothesis, Z ∈ C 1−α ([0, T ], Hq (Rn , Rk )), α < γ < 1 − α and 2γ + 2α + δ + β < 2. Consequently there exists some small μ > 0 such that with α := α + μ, we still have α < γ < 1 − α and 2γ + 2α + δ + β < 2.
(A.2)
In Definition 2.1, we use the number α as specified this way. For later use, we record the relation
t e 0
−(t−s) −η
s
−θ
(t − s)
ds
η+θ−1
z sup z>0
e 0
−v
−η −θ
(z − v)
v
dv ,
(A.3)
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0 < η, θ < η + θ < 1, the supremum is bounded by 2 + 4B(1 − η, 1 − θ ), B denoting the Beta function. Step 2: Elementary estimates. Recall that the right-sided Weyl–Marchaud fractional derivative of order 1 − α of Zt is given by 1−α Dt− Zt (s) = 1(0,t) (s)
t Z(s) − Z(t) Z(s) − Z(σ ) (−1)α +α dσ , Γ (1 − α) (t − s)1−α (σ − s)2−α
(A.4)
s
see Appendix B, [14] or [32]. To the first term in brackets on the right-hand side of equality (A.4) we will refer as the boundary correction term, it will be denoted by b(s, t). The second will be called the integral term, it is abbreviated by j (s, t). Recall the definition (14) of the norm in 1−β C 1−α ([0, T ], Hq (Rn , Rk )). Note that for any q > 1 and 0 s < t T , b(s, t) H 1−β Rn , Rk c(t − s)μ Z C 1−α [0, T ], H 1−β Rn , Rk and q q j (s, t) H 1−β Rn , Rk c(t − s)μ Z C 1−α [0, T ], H 1−β Rn , Rk . q q
(A.5)
1−β
1−α In particular, Dt− Zt (s) | Hq (Rn , Rk ) < c. As Z is fixed throughout the whole proof, we 1−β absorb the norm Z | C 1−α ([0, T ], Hq (Rn , Rk )) of Z into the constants c to simplify the notation. For 0 s < τ < t T one deduces
b(s, t) − b(s, τ ) H 1−β Rn , Rk q c(t − s)μ (τ − s)α−1 (t − τ )1−α + c(t − τ )1−α+μ (τ − s)α−1 ,
(A.6)
or, alternatively, b(s, t) − b(s, τ ) H 1−β Rn , Rk q
c(t − τ )
1−α+μ
(τ − s)1−α + c(t − τ )1−α (τ − s)1−α+μ .
(A.7)
The constants c may depend on q. () Step 3: The non-difference part. Recall the definition (28) of the norms · γ ,δ . We begin with an estimate on the first term in brackets there. Fix l = 1, . . . , n and denote by J1 (t), J2 (t) and J3 (t) the summands according to the right-hand side of (A.1) in the order they occur. We consider Gl u(s) in place of gl (s) and write G to abbreviate Gl . By Lemma C.1 below and a simple Fourier multiplier argument, we have Gu(s) · ∂ D 1−α Zt (s) cGu(s) ∂ D 1−α Zt (s) H −β Rn , Rk t− q δ ∂x ∂xl t− l −β cu(s)δ b(s, t) + j (s, t) Hq1−β Rn , Rk for some q > 1. Recall that · δ is our abbreviation for the norm · |H δ (Rn , Rk ).
(A.8)
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Now set δ+β 2
κ :=
and use the mapping property (B.4) of the analytic semigroup (P (t))t0 together with (A.5) and (A.8) to obtain
e
J1 (t)δ ce−t
−t
t
(t − s)−α−κ u(s)δ ds
0 () cuγ ,δ
t
e−(t−s) (t − s)−α−κ ds
0 () cuγ ,δ α+κ−1 .
Similarly, by (A.5),
e
J2 (t)δ ce−t
−t
t s
(t − σ )−κ
0 0
u(s) − u(σ ) dσ ds (s − σ )α+1
() cuγ ,δ κ−1 ,
recall α < γ and the remark preceding this proof. Note also that (t − s) < (t − σ ) and 0 < κ < 1. Finally, due to (A.3),
e
J3 (t)δ ce−t
−t
t
s −α (t − s)−κ u(s)δ ds
0 () cuγ ,δ α+κ−1 .
Consequently, () e−t Ji (t)δ C0 ()uγ ,δ ,
i = 1, 2, 3,
for any 0 t T and with C0 () > 0 tending to zero as goes to infinity. Step 4: The difference part and J1 . Turning to estimates on the difference part of the norms (28), we start with J1 . For 0 τ < t T , c J1 (t) − J1 (τ ) =
t
∂ 1−α Aα P (t − s)G u(s) · D Zt (s) ds ∂yl t−
0
τ − 0
∂ 1−α Aα P (τ − s)G u(s) · D Zτ (s) ds ∂yl τ −
M. Hinz, M. Zähle / Journal of Functional Analysis 256 (2009) 3192–3235
t =
3211
∂ 1−α Aα P (t − s)G u(s) · D Zt (s) ds ∂yl t−
0
τ
∂ 1−α Aα P (t − s)G u(s) · D Zτ (s) ds ∂yl τ −
− 0
τ
∂ 1−α Aα P (t − s)G u(s) · D Zτ (s) ds ∂yl τ −
+ 0
τ
∂ 1−α Aα P (τ − s)G u(s) · D Zτ (s) ds ∂yl τ −
− 0
t = τ
∂ 1−α Aα P (t − s)G u(s) · D Zt (s) ds ∂yl t− τ
∂ 1−α Dt− Zt (s) − Dτ1−α Aα P (t − s)G u(s) − Zτ (s) ds ∂yl
− 0
τ
∂ 1−α Aα P (t − τ ) − I P (τ − s)G u(s) · D Zτ (s) ds, ∂yl τ −
+
(A.9)
0
we have used the semigroup property of (P (t))t0 . By the mapping properties (B.4) and (B.5) of (P (t))t0 , the · δ -norm of the last term on the right-hand side of (A.9) admits the bound τ c(t − τ )
ν
(t − s)−α−κ−ν u(s)δ ds
(A.10)
0
with some γ < ν < 1 being just slightly bigger than γ . Integrating against (t − τ )−γ −1 dτ over (0, t), and multiplying by e−t , we are led to the bound
() cuγ ,δ
t τ
e−(t−s) (t − s)−α−κ−ν ds(t − τ )ν−γ −1 dτ
0 0 () cuγ ,δ
t t
e−(t−s) (t − s)−α−κ−ν ds(t − τ )ν−γ −1 dτ
0 0 ()
cuγ ,δ α+κ+ν−1 .
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For the middle summand on the right-hand side of (A.9), consider 1−α Dt− Zt (s) − Dτ1−α − Zτ (s) = c b(s, t) − b(s, τ ) + c j (s, t) − j (s, τ ) . 1−α Zt (s) − Dτ1−α With c(b(s, t) − b(s, τ )) in place of Dt− − Zτ (s) in that summand, (A.6) yields
τ c
(t − s)−α−κ u(s)δ (τ − s)α−1 (t − τ )1−α (t − s)μ ds
(A.11)
0
plus τ c
(t − s)−α−κ u(s)δ (τ − s)α−1 (t − τ )1−α+μ ds,
(A.12)
0
and after integration, () cuγ ,δ
t τ
e−(t−s) (t − s)μ−α−κ (τ − s)α−1 ds(t − τ )−α−γ dτ
0 0 () cuγ ,δ B(α, 1 − γ
t − α)
e−(t−s) (t − s)μ−γ −α−κ ds
0 () cuγ ,δ α+γ +κ−μ−1 ,
plus ()
t τ
cuγ ,δ
e−(t−s) (t − s)−α−κ (τ − s)α−1 ds(t − τ )μ−α−γ dτ
0 0
cuγ ,δ α+γ +κ−μ−1 , ()
which follows by similar arguments. Recall that γ < 1 − α. For the same summand with t j (s, t) − j (s, τ ) = c τ
Z(s) − Z(σ ) dσ (σ − s)2−α
1−α Zt (s) − Dτ1−α in place of Dt− − Zτ (s), we use Fubini’s theorem to observe that the · δ -norm of
τ c 0
∂ j (s, t) − j (s, τ ) ds Aα P (t − s)G u(s) · ∂yl
M. Hinz, M. Zähle / Journal of Functional Analysis 256 (2009) 3192–3235
3213
is bounded above by τ t
(t − s)−κ−α u(s)δ (σ − s)μ−1 dσ ds
c 0 τ
t τ c τ
(σ − s)μ−α−κ−1 u(s)δ ds dσ
0
() cuγ ,δ
t τ es (σ − s)μ−α−κ−1 ds dσ τ
0
() cuγ ,δ eτ
t
μ−α−κ σ + (σ − τ )μ−α−κ dσ
τ ()
ceτ (t − τ )1−α−κ+μ uγ ,δ ,
(A.13)
note that s < τ < σ < t and 0 < α + κ − μ < 1. Integrating with respect to (t − τ )−γ −1 dτ and taking into account the exponential factors, we obtain the estimate
() cuγ ,δ
t
e−(t−τ ) (t − τ )−α−γ −κ+μ dτ cuγ ,δ α+γ +κ−μ−1 . ()
0
Turn to the first summand on the right-hand side of (A.9). In the · δ -norm it is bounded above by t c(t − τ )
ν
(t − s)−α−κ−ν u(s)δ ds,
τ
ν again just slightly bigger than γ . Integration leads to the bound
() cuγ ,δ
t t
e−(t−s) (t − s)−α−κ−ν ds(t − τ )ν−γ −1 dτ
0 τ () cuγ ,δ
t e
−(t−τ )
0
cuγ ,δ α+γ +κ−1 . ()
t τ
(t − s)−α−κ−ν ds(t − τ )ν−γ −1 dτ
(A.14)
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Clipping the estimates, we see that for 0 t T , e
−t
t 0
J1 (t) − J1 (τ )δ () dτ C1 ()uγ ,δ , (t − τ )γ +1
C1 () tending to zero as goes to infinity. Step 5: The difference part and J2 . For 0 τ < t T , we split the differences of J2 similarly to those of J1 : c J2 (t) − J2 (τ ) t s = 0 0 τ
∂ 1−α (s − σ )−α−1 P (t − σ ) G u(s) − G u(σ ) · D Zt (s) dσ ds ∂yl t− s
−
∂ 1−α (s − σ )−α−1 P (τ − σ ) G u(s) − G u(σ ) · D Zτ (s) dσ ds ∂yl τ −
0 0
t
s
τ
0 τ
=
∂ 1−α (s − σ )−α−1 P (t − σ ) G u(s) − G u(σ ) · D Zt (s) dσ ds ∂yl t− s
−
(s − σ )−α−1 P (t − σ ) G u(s) − G u(σ )
0 0
∂ 1−α · Dt− Zt (s) − Dτ1−α − Zτ (s) dσ ds ∂yl τ s + (s − σ )−α−1 P (t − τ ) − I P (τ − σ ) G u(s) − G u(σ ) 0 0
∂ 1−α · D Zτ (s) dσ ds. ∂yl τ −
(A.15)
Since P (t − σ ) = P (t − s)P (s − σ ), 0 < σ < s < t, the first summand after the last equality sign admits the bound t
(t − s)−κ
τ
s 0
u(s) − u(σ )δ dσ ds, (s − σ )α+1
using t − s < t − τ and integrating, we arrive at () cuγ ,δ
t e 0
−(t−s)
−κ−ν
s
(t − s)
(t − τ )ν−γ −1 dτ ds cuγ ,δ κ+ν−1 ,
0
where again ν is chosen slightly bigger than γ .
()
M. Hinz, M. Zähle / Journal of Functional Analysis 256 (2009) 3192–3235
3215
An estimate of the last summand in (A.15) follows as before combining (B.4) and (B.5), its norm contributes at most τ
−κ−ν
s
(t − τ ) (τ − s) ν
c 0
0
u(s) − u(σ )δ dσ ds, (s − σ )α+1
and after integration, it remains less or equal ()
cuγ ,δ κ+ν−1 . Now consider the middle summand on the right-hand side of the last equality in (A.15) with 1−α Zt (s) − Dτ1−α c(b(s, t) − b(s, τ )) in place of Dt− − Zτ (s). Using (A.7) we observe the bound τ
−κ
s
(τ − s)
c 0
0
u(s) − u(σ )δ dσ (s − σ )α+1
× (t − τ )1−α+μ (τ − s)1−α + (t − τ )1−α (τ − s)1−α+μ ds.
Integration yields () cuγ ,δ
t τ
e−(t−s) (τ − s)1−α−κ (t − τ )−α−γ +μ + (τ − s)1−α−κ+μ (t − τ )−α−γ ds dτ
0 0
cuγ ,δ 2α+γ +κ−μ−2 , ()
note that α + γ < 1. We have used (A.2) to see that t
e−(t−τ ) τ 2−α−κ (t − τ )−α−γ +μ dτ c2α+κ+γ −μ−2 ,
0
and similarly for the other term. For the same summand with j (s, t) − j (s, τ ) inserted, Fubini’s theorem again tells that the norm does not exceed τ t
−κ
s
(t − s) 0 τ
0
() cuγ ,δ eτ
u(s) − u(σ )δ dσ (θ − s)μ−1 dθ ds (s − σ )α+1
t τ (θ − s)μ−κ−1 ds dθ τ
()
t
cuγ ,δ eτ
0
μ−κ θ + (θ − τ )μ−κ dθ
τ () cuγ ,δ eτ (t
− τ )1+μ−κ .
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Performing the integration with respect to τ , we obtain no more than () cuγ ,δ
t
e−(t−τ ) (t − τ )−γ −κ+μ dτ cuγ ,δ γ +κ−μ−1 . ()
0
Combining these estimates, we see that
e
−t
t 0
J2 (t) − J2 (τ )δ () dτ C2 ()uγ ,δ , (t − τ )γ +1
C2 () tending to zero as goes to infinity. Step 6: The difference part and J3 . Splitting the difference as before, c J3 (t) − J3 (τ ) t ∞ = τ
s
∂ 1−α σ −α−1 P (σ )P (t − s)G u(s) · D Zt (s) dσ ds ∂yl t−
τ ∞ + 0
s
τ ∞ − 0
s
∂ 1−α σ −α−1 P (σ ) P (t − τ ) − I P (τ − s)G u(s) · D Zτ (s) dσ ds ∂yl τ − ∂ 1−α Dt− Zt (s) − Dτ1−α σ −α−1 P (σ )P (t − s)G u(s) · − Zτ (s) dσ ds. ∂yl
(A.16)
The norm of the first term on the right-hand side does not exceed t c
s −α (t − s)−κ u(s)δ ds.
(A.17)
τ
Since here t − s < t − τ , integration yields () cuγ ,δ
t t
e−(t−s) s −α (t − s)−κ−ν ds(t − τ )ν−γ −1 dτ cuγ ,δ α+κ+ν−1 ()
0 0
with some ν slightly bigger than γ , in particular α + κ + ν < 1. We have used (A.3). The second summand in (A.16) contributes τ 0
s −α (t − τ )ν (τ − s)−κ−ν u(s)δ ds
(A.18)
M. Hinz, M. Zähle / Journal of Functional Analysis 256 (2009) 3192–3235
3217
with some ν > γ , but close. Integrating and sorting out a Beta function, we arrive at
() cuγ ,δ
t e
−(t−τ )
0
τ
s −α (τ − s)−κ−ν ds(t − τ )ν−γ −1 dτ
0 ()
t
cuγ ,δ
e−(t−τ ) τ 1−α−κ−ν (t − τ )ν−γ −1 dτ
0
cuγ ,δ α+γ +κ−1 ()
by (A.3). The third term in (A.16) with the boundary terms c(b(s, t) − b(s, τ )) in place of 1−α Dt− Zt − Dτ1−α − Zτ (s) contributes τ c
s −α (τ − s)−κ u(s)δ (t − τ )1−α+μ (τ − s)1−α + (τ − s)1−α+μ (t − τ )1−α ds ds,
0
(A.19) here we have used (A.7). For the first summand, integration and evaluation of a Beta function yield
() cuγ ,δ
t τ
e−(t−s) s −α (τ − s)1−α−κ ds(t − τ )−γ −α+μ dτ
0 0 () cuγ ,δ
t
e−(t−s) s −α (t − s)2−2α−γ −κ+μ ds
0 () cuγ ,δ 3α+γ +κ−μ−3 ,
and for the second,
() cuγ ,δ
t τ
e−(t−s) s −α (τ − s)1−α+μ ds(t − τ )−γ −α dτ
0 0
cuγ ,δ 3α+γ +κ−μ−3 . ()
Note that α + κ < 1 and α − ν < 1. Considering the third term with j (s, t) − j (s, τ ) inserted, we proceed as before and use Fubini to get the bound
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τ t
s −α (t − s)−κ u(s)δ (θ − s)μ−1 dθ ds
c 0 τ
τ c
s
−α
u(s)δ
−κ−ν
(τ − s)
t (θ − s)ν+μ−1 dθ ds τ
0 ()
τ
cuγ ,δ
es s −α (τ − s)−κ−ν ds(t − τ )ν+μ ,
(A.20)
0
ν > γ , close to γ . Note that 0 < μ + ν < 1. Integrating, we observe the upper estimate () cuγ ,δ
τ e
−(t−s) −α
s
−κ−ν
(τ − s)
t ds
0
(t − τ )ν−γ +μ−1 dτ
0
() cuγ ,δ α+κ+ν−1 .
This shows that also e
−t
t 0
J3 (t) − J3 (τ )δ () dτ C3 ()uγ ,δ , γ +1 (t − τ )
C3 () tending to zero as goes to infinity, what completes the proof.
2
Next, we comment on the proof of Proposition 7.1. Proof. It is similar, but simpler: Use ∂ 1−α −β n k cD 1−α Zt (s) H 1−β Rn , Rk H R D Z (s) , R 2 t− 2 ∂y t− t l 1−β n c(t − s)μ Z C 1−α [0, T ], H2 R , Rk , and follow the pattern of the preceding proof. Now J2 vanishes, and for J1 and J3 we can modify the former estimates in an obvious way: First split J1 according to (A.9). For the summand corresponding to the last one there, (A.10) yields the bound c(t − τ )ν with some ν > γ . (A.11) and (A.12) yield c(t − τ )1−α for the middle summand with boundary terms b inserted. Recall that γ < 1 − α. With the integral terms j , we can use (A.13), note that (t − τ )1−α−κ+μ cT 1−γ −α−κ+μ (t − τ )γ where κ = (δ + β)/2 and γ + α + κ − μ < 1. The first summand is covered by a bound of type (A.14). Hence J1 (t) − J1 (τ ) c(t − τ )γ , 0 τ < t T . δ
M. Hinz, M. Zähle / Journal of Functional Analysis 256 (2009) 3192–3235
3219
Next, turn to J3 , splitted according to (A.16). For the first and second summand in question, we use (A.17) and (A.18) to observe factors c(t − τ )ν , ν > γ . (A.19) covers the third summand with boundary terms b delivering a factor c(t − τ )1−α . (A.20) contributes a factor (t − τ )ν+μ for the summand with integral terms j . Consequently J3 (t) − J3 (τ ) c(t − τ )γ , δ what finishes the proof.
0 τ < t T,
2
A.2. Proof of Proposition 7.3 Proof. Fix l = 1, . . . , n. Recall that TGl u := Gl (u) = (G1l (u), . . . , Gkl (u)). Step 1: Estimates on the non-linearities. Looking at the equivalent norm (C.1), we deduce that TGl uδ cuδ ,
(A.21)
see also [33, Theorem 5.5.1/1]. For · ∞ in place of · δ a similar assertion is obvious. δ (Rn , Rk ) → Next we consider the Fréchet derivative TG l of the operator TGl : H2,∞ δ (Rn , Rk ). For any u, v ∈ H δ (Rn , Rk ) it is given by H2,∞ 2,∞ TG l (u)v = DGl (u)v.
(A.22)
For fixed x ∈ Rn , DGl (u(x)) ∈ M(k × k, R), v(x) ∈ Rk , and (A.22) is understood in the usual sense of matrix multiplication. For k = 1, the proof of (A.22) is given in [33, Theorem 5.5.3/1]. As we allow k 1, we sketch the arguments for convenience: For fixed x ∈ Rn , Taylor expansion yields Gl (u + v)(x) − Gl (u)(x) − DGl u(x) v(x) 1 =
(1 − θ ) D2 Gl (u(x) − θ v(x))v(x) v(x) dθ.
(A.23)
0
Now given z ∈ Rk , Bz (ξ, η) := D2 Gl (z)ξ − D2 Gl (0)ξ η,
ξ, η ∈ Rk ,
defines some Bz ∈ L(Rk , L(Rk , Rk )), which may be seen as bilinear mapping Bz : δ (Rn , Rk ), Rk × Rk → Rk . We observe that for h, v, w ∈ H2,∞ Bh(x) w(x), v(x) − Bh(y) w(y), v(y) = Bh(x) w(x) − w(y), v(x) + Bh(x) w(y), v(x) − v(y) + Bh(x) − Bh(y) w(y), v(y)
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consequently by the hypotheses on D2 Gl , 2 Bh(x) w(x), v(x) − Bh(y) w(y), v(y) 2 2 c Bh(x) w(x) − w(y), v(x) + c Bh(x) w(y), v(x) − v(y) 2 + c Bh(x) − Bh(y) w(y), v(y) 2 2 cM 2 v2∞ w(x) − w(y) + cM 2 w2∞ v(x) − v(y) 2 + cL2 h(x) − h(y) w2∞ v2∞ . Insert this into the second summand (difference part) of the norm (C.1) and note that the · 0 part of this norm can be estimated in a similar way. We obtain that Bh (w, v) cMwδ,∞ vδ,∞ + cLwδ,∞ vδ,∞ hδ,∞ . δ,∞ This implies that in · δ,∞ , the right-hand side of (A.23) is bounded by c 1 + uδ,∞ + vδ,∞ v2δ,∞ . Hence, Gl (u + v) − Gl (u) − DGl (u)vδ,∞ = 0, vδ,∞ →0 vδ,∞ lim
what proves (A.22). Further, we have T (u)v c uδ,∞ + 1 vδ,∞ . Gl δ,∞
(A.24)
To see this, note first that
j k
∂G l
DGl (u)v =
j =1
∂x1
(u)vj , . . . ,
j k
∂G l
j =1
∂xk
(u)vj .
(A.25)
δ (Rn ) (that is k = 1). For a moment, abuse notation and let · δ,∞ also denote the norm in H2,∞ δ n Since H2,∞ (R ) is a multiplication algebra and due to (C.1), we have for any i, j = 1, . . . , k,
j j j j ∂Gl ∂Gl ∂Gl vj δ,∞ + ∂Gl (0) vj δ,∞ (u)v (u) − (0) j ∂x ∂x ∂xi ∂xi i i δ,∞ δ,∞ δ,∞ M0 uδ,∞ vj δ,∞ + cvj δ,∞ c uδ,∞ + 1 vj δ,∞ , j
∂G
where M0 = supx∈Rk D( ∂xil − implies (A.24).
j
∂Gl ∂xi
(0))(x)L(Rk ,Rk ) . Summing over j according to (A.25) this
M. Hinz, M. Zähle / Journal of Functional Analysis 256 (2009) 3192–3235
3221
By the mean value theorem it now follows that 1 TGl u − TGl vδ
T θ u + (1 − θ )v (u − v) dθ Gl δ
0
cu − vδ,∞ uδ,∞ + vδ,∞ + 1 .
(A.26)
δ (Rn , Rk ), we similarly have Given u1 , u2 , v1 , v2 ∈ H2,∞
TGl u1 − TGl v1 − TGl u2 + TGl v2 1 =
TG l θ u1 + (1 − θ )v1 (u1 − v1 − u2 + v2 ) dθ
0
1 +
TGl θ u1 − (1 − θ )v1 − TG l θ u2 − (1 − θ )v2 (u2 − v2 ) dθ,
0
and taking the · δ -norm, another application of the mean value theorem together with the Lipschitz property of DGl (due to the boundedness of D2 Gl ) yields the bound cu1 − v1 − u2 + v2 δ,∞ u1 δ,∞ + v1 δ,∞ + 1 + cu2 − v2 δ,∞ u1 δ,∞ + v1 δ,∞ + u2 δ,∞ + v2 δ,∞ + 1 , (A.27) cf. [24]. Step 2: An invariant subset. We show that for 0 1 large enough, the integral operator (29) maps the closed ball n k (0 ) δ R , R : uγ ,δ,∞ e−0 T B (0 ) 0, e−0 T := u ∈ W γ [0, T ], H˚ 2,∞ into itself. δ (Rn , Rk )), fix l = 1, . . . , n We follow the proof of Proposition 7.2. Given u ∈ W γ ([0, T ], H˚ 2,∞ and denote by J1 (t, u), J2 (t, u), J3 (t, u) the single summands on the right-hand side of the corresponding special case of representation (A.1). Using (A.21), the estimates involving Ji (t, u)δ and Ji (t, u) − Ji (τ, u)δ , i = 1, 3, 0 τ < t T , carry over from that proof, leading to bounds of type ( )
0 cuγ ,δ,∞ 0α+κ+ν−1 ,
( )
α+κ+γ −1
0 cuγ ,δ,∞ 0
( )
,
3α+γ +κ−μ−3
0 0 cuγ ,δ,∞
( )
α+κ+γ −μ−1
0 cuγ ,δ,∞ 0
,
where κ = (δ + β)/2 and with α, ν slightly bigger than γ as specified there. Now recall that P (t)u ct −n/4 u0 , t > 0, ∞ here · 0 denotes the norm in L2 (D, Rk ), see e.g. [4].
,
or (A.28)
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By isomorphic lifting, this yields estimates on the corresponding terms involving Ji (t, u)∞ and Ji (t, u) − Ji (τ, u)∞ , i = 1, 3, which are analogous to those written in (A.28), but with κ replaced by β/2 + n/4. For the terms with J2 (t, u)δ and J2 (t, u) − J2 (τ, u)δ , bounds of type ( )
0 cuγ ,δ,∞ 0κ+ν−1 ,
( )
2α+γ +κ−μ−2
0 cuγ ,δ,∞ 0
( )
γ +κ−μ−1
0 and cuγ ,δ,∞ 0
(A.29)
follow, ν > γ , but close. Taking the · ∞ -norm instead, the bounds hold with β/2 + n/4 in place of κ. Since uγ ,δ,∞ 1 for u ∈ B (0 ) (0, e−0 T ), it is now sufficient to choose 0 1 large enough to make sure that the images of all u from B (0 ) (0, e−0 T ) have · γ 0,δ,∞ -norm less than e0 T . Step 3: Contractivity. We show that for 1 large enough, (29) is a contraction in B (0 ) (0, e−0 T ). Proceeding as before and using (A.26), we get for instance () e−t J1 (t, u) − J1 (t, v)δ cu − vγ ,δ,∞ uγ ,δ,∞ + vγ ,δ,∞ + 1 α+κ−ν−1 ()
cu − vγ ,δ,∞ α+κ−ν−1 , u, v ∈ B (0 ) (0, e−0 T ), and similary for the other bounds in (A.28). Analogous arguments for J3 (t, u) − J3 (t, v) and Ji (t, u) − Ji (t, v) − Ji (τ, u) + Ji (τ, v), i = 1, 3, yield bounds of type () cu − vγ ,δ,∞ α+κ+ν−1 , similarly for the other versions in (A.28). For · ∞ , κ is to be replaced by β/2 + n/4. On J2 (t, u) − J2 (t, v) and J2 (t, u) − J2 (t, v) − J2 (τ, u) + J2 (τ, v), we use (A.27) to arrive at () the upper bound cu − vγ ,δ,∞ κ+ν−1 , or one of the other bounds from (A.29). κ is to be replaced by β/2 + n/4 if · ∞ is considered. Now choose 0 sufficiently large. 2 A.3. Proof of Corollary 7.5 −β
Proof. Assertion (ii) follows applying Lemma C.1: For v ∈ H2δ (Rn ) and ϕ ∈ Hq (Rn ), 1 + v ∂ ϕ − ∂ ϕ vδ 1 ∂ + ϕ − ∂ ϕ H −β Rn , Rk . (A.30) q r l,r r l,r ∂yl −β ∂yl This tends to zero as r does, since by translation invariance of the Lq (Rn )-norm, −β/2 ∧ ∨ −β/2 ∧ ∨ n Lq R = 0 lim 1 + |ξ |2 ϕ ◦ Ttrel − 1 + |ξ |2 ϕ
r→0
for any t > 0, see part I. Here ψ ◦ Ta (x) = ψ(x + a), a ∈ Rn , denotes the translation, above it is applied in the sense of Schwartz distributions. Assertion (i) follows similarly. 2 A.4. Proof of Lemma 7.6 Proof. It suffices to consider the members of (31) for fixed r > 0. We consider Definition 2.1 interpreted according to (29), the case (27) is similar. Given 0 < α, α < 1, we show that the integral value remains unchanged if α = α + ν, ν > 0 replaces α.
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1−α + In the following, l = 1, . . . , n is fixed, h(s) := Dt− ∂l,r Zt (s) is assumed to exist, and G(s) is written to denote Gl u(s) or Gl (u(s)). Note that we use the definition of the fractional integral ν which includes the factor (−1)−ν , see [14] or [43]. operator It− By semigroup and invertibility properties of fractional integrals and derivatives, the first summand in (A.1) with G in place of g yields
t
ν Aα+ν P (t − s) G(s) · It− h (s) ds
α+ν
(−1)
0
t t Aα+ν P (t − s)(τ − s)ν−1 G(s) · h(τ ) dτ ds.
= (−1)
α s
0
−β
Applying Lemma B.1 to the E = H2 (Rn , Rk )-valued function f := G(·) · h(τ ), we obtain three terms: The first is t α
(−1)
α+ν ν D0+ P (t − ·)G(·) · h(τ ) (τ ) dτ I0+
0
t = (−1)
α
α P (t − ·)G(·) · h(τ ) (τ ) dτ, D0+
0
the second is t −(−1) cα+ν α
ν I0+ Ψt,τ (τ ) dτ, 0
where ∞ Ψt,τ (s) = P (t − s)
u−(α+ν)−1 P (u)G(s) · h(τ ) du,
s
and the third equals t −(−1) cα+ν α
ν I0+ Λt,τ (τ ) dτ, 0
where s Λt,τ (s) = P (t − s) 0
u−(α+ν)−1 G(s) − G(s − u) · h(τ ) du.
(A.31)
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From the second summand in (A.1) we obtain t s α+ν
(−1)
ν (s − σ )−(α+ν)−1 P (t − σ ) G(s) − G(σ ) · It− h(s) dσ ds
cα+ν 0 0
(−1)α+ν cα+ν = Γ (ν)
t t s s
0
(s − σ )−(α+ν)−1
0
× P (t − σ ) G(s) − G(σ ) · h(τ ) dσ (τ − s)ν−1 dτ ds t = (−1) cα+ν α
ν I0+ Λt,τ (τ ) dτ, 0
and from the third summand in (A.1), t ∞ α+ν
(−1)
cα+ν
ν u−(α+ν)−1 P (u)P (t − s) G(s) · It− h(s) du ds
s
0
t = (−1) cα+ν α
ν I0+ Ψt,τ (τ ) dτ. 0
The terms cancel and by (A.31) together with Lemma B.1 we arrive at the integral with α according to Definition 2.1 and interpretation (29). Taking limits as r goes to zero and using Corollary 7.5 (in particular, the estimate (A.30)), the values are seen to agree in H2δ (Rn , Rk ). 2 Finally, we prove Lemma 7.4: Proof. The first assertion is seen as follows. We have
e
J0 (t, u)δ ce−t
−t
t
TF u(s) ds cu() −1 . γ ,δ,∞ δ
0 ()
Replacing · δ by · ∞ , we arrive at cuγ ,δ,∞ n/4−1 . For 0 τ < t T , use t J0 (t, u) − J0 (τ, u) = τ
P (t − s)F u(s) ds +
τ
P (t − τ ) − I P (τ − s)F u(s) ds
0
to arrive at bounds cuγ ,δ γ −1 for the norm · δ , and cuγ ,δ γ +n/4−1 for · ∞ . To deduce the second assertion, use (A.26) to get ()
()
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J0 (t, u) − J0 (τ, v) c δ
t
3225
TF u(s) − TF v(s) ds δ
0
t c
u(s) − v(s)
δ,∞
u(s)
δ,∞
+ v(s)δ,∞ + 1 ds.
0
We arrive at the upper bound cu − vγ ,δ,∞ (uγ ,δ,∞ + vγ ,δ,∞ + 1)−1 , for · ∞ , −1 is to be replaced by n/4−1 . For the term contributed by J0 (t, u) − J0 (t, v) − J0 (τ, u) − J0 (τ, v), () we obtain the estimate cu − vγ ,δ,∞ (uγ ,δ,∞ + vγ ,δ,∞ + 1)γ −1 for the · δ -part, for the · ∞ -part replace γ −1 by γ +n/4−1 . Now note that uγ ,δ,∞ + vγ ,δ,∞ 2, and choose 0 large enough. 2 ()
Appendix B. Fractional calculus and semigroups For general information on fractional calculus we refer to [32] or to part I of the present paper. Here we only sketch some connections to semigroup theory that are used in the main text. Let (E, · E ) be a separable complex Banach space and L(E) the space of bounded linear operators on E, endowed with the operator norm. I denotes the identity operator.
B.1. Semigroups and generators Assume (P (t))t0 ⊂ L(E) is a C0 -semigroup of negative type on E, i.e. P (t) Me−μt , L
t 0,
(B.1)
with some μ, M > 0. Obviously it is equibounded. Let −A denote the infinitesimal generator of (P (t))t0 , a closed linear operator whose domain dom(−A) is dense in E. By (B.1), A is a positive operator. It is most common to express the fractional powers Aα , 0 < α < 1 of A in terms of its resolvent, see [31,42]. The main result of [6] gives a representation for Aα in terms of the semigroup and characterizes its domain: f ∈ E belongs to dom(Aα ), 0 < α < 1, if and only if 1 A f = lim ε→0 Γ (−α)
∞
α
[I − P (u)]f du uα+1
(B.2)
ε
converges strongly in E. From now on, assume in addition that (P (t))t0 is analytic. Then the following useful properties are known: For any f ∈ E, α 0 and t > 0, P (t)f is a member of dom(Aα ), and for any f ∈ dom(Aα ), P (t)Aα f = Aα P (t)f,
t 0.
(B.3)
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Now (B.1) implies the bounds α A P (t) Mα t −α e−μt , L
t > 0,
(B.4)
with some Mα > 0, and P (t)f − f cα t α Aα f , E E
t > 0,
(B.5)
for 0 α < 1, f ∈ dom(Aα ) and with some cα > 0. See [31, Chapter 2.2]. The negative fractional powers A−α , α > 0, admit the representation −α
A
1 f= Γ (α)
∞ uα−1 P (u)f du,
(B.6)
0
strongly convergent for any f ∈ E. For 0 < α1 , α2 < 1, α1 + α2 < 1, we observe Aα1 A−α1 = I and Aα1 Aα2 = Aα1 +α2 , and the mappings Aα1 : dom(Aα1 ) → E as well as Aα1 : dom(Aα1 +α2 ) → dom(Aα2 ) are isomorphisms. In the Hilbert space case, these definitions of fractional powers agree with those deduced from the spectral theorem. We refer to [38]. From the point of view of fractional calculus, (B.2) implies that for any f ∈ E, the (rightα P (·)f of order 0 < α < 1 of the function P (·)f : sided) Weyl–Marchaud fractional derivative D− [0, ∞) → E converges at any t > 0 in the pointwise sense and α D− P (·)f (t) = (−1)α Aα P (t)f.
(B.7)
α , cf. part I, [43] or [32]. Similary, We have used Γ (1 − α) = αΓ (−α) and the definition of D− we observe from (B.6) that the (right-sided) Riemann–Liouville fractional integral I−α P (·)f of order 0 < α < 1 of the function P (·)f : [0, ∞) → E is realized as
I−α P (·)f (t) = (−1)−α A−α P (·)f (t).
(B.8)
B.2. Bounded intervals and scales of Banach spaces Now suppose there is a scale of Banach spaces {(Eδ , · Eδ }δ− <δ<δ+ , δ− < 0 < δ+ , and (P (t))t0 is an analytic semigroup of negative type on E0 with generator A. Assume that for 0 < κ < 1, we have dom(Aκ/2 ) = Eκ , the norms · Eκ and f → Aκ/2 f E0 are equivalent and the fractional powers Aκ/2 : Eκ+δ → Eδ act as isomorphisms. (B.3) then allows to apply the semigroup operators to a member of any Eδ , δ− < δ < δ+ . Let 0 < t < T and f : [0, T ] → E−β , δ− < −β < 0 be a given function. In view of our applications we reverse time and consider the left sided fractional Weyl–Marchaud derivative α P (t − ·)f (·) of order 0 < α < 1 of the function P (t − ·)f (·) : [0, t] → E, formally given by D0+ α P (t − ·)f (·) (s) D0+ s P (t − s)f (s) P (t − s)f (s) − P (t − τ )f (τ ) 1 = 1(0,t) (s) + α dτ (B.9) Γ (1 − α) sα (s − τ )α+1 0
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3227
for 0 < s < t, where cα = αΓ (1 − α)−1 . By α ϕ(s) = I0+
1 Γ (α)
s 0
ϕ(τ ) dτ (s − τ )1−α
we denote the Riemann–Liouville fractional derivative of order 0 < α < 1 of a function ϕ on (0, t). For 0 < α < 1 and δ− < δ < δ+ , let W α ([0, T ], Eδ ) denote the Banach space of functions f : [0, T ] → Eδ such that
α f W [0, T ], Eδ := sup 0tT
f (t)
t Eδ
+ 0
f (t) − f (σ )Eδ dσ (t − σ )α+1
< ∞.
The following seems to be a new result made out of known ingredients. Lemma B.1. Let 0 < α < 1 and f ∈ W α ([0, T ], E−β ), δ− < −β < δ+ . Suppose −β < δ < 2 − 2α − β and δ < δ+ . Then for any 0 < t T , the left sided Weyl–Marchaud fractional derivative α P (t − ·)f (·) of order α of the function P (t − ·)f (·) is given by D0+ α D0+ P (t − ·)f (·) (s) = 1(0,t) (s)ψ(s), where ∞ ψ(s) = A P (t − s)f (s) + cα P (t − s) α
u−α−1 P (u)f (s) du
s
s + cα
u−α−1 P (u)P (t − s) f (s) − f (s − u) du,
(B.10)
0
convergent in L1 ((0, t), Eδ ). Moreover, there exists some ϕ ∈ L1 ((0, t), Eδ ) such that α ϕ, and the identity P (t − ·)f (·) = I0+ α α I0+ D0+ P (t − ·)f (·) = P (t − ·)f (·)
(B.11)
holds in L1 ((0, t), Eδ ). Proof. Put χ(s) := P (t − s)f (s), simplifying the notation. In order to verify the convergence statement, we quote a few facts from [32]. For ε > 0, the truncated Weyl–Marchaud fractional α derivative D0+,ε χ of χ is given by α D0+,ε χ(s) :=
1(0,t) (s) f (s)t −α + αψε (s) , Γ (1 − α)
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where s−ε ψε (s) :=
f (s)−f (τ ) dτ, 0 (s−τ )α+1 f (s) −α − s −α ], α [ε
s > ε,
(B.12)
0 s ε.
Theorem 13.1 in [32] remains valid in our case and asserts that if χ is of form α ϕ χ = I0+
(B.13)
α χ = lim α for some ϕ ∈ L1 ((0, t), Eδ ), then D0+ ε→0 D0+,ε χ = ϕ in L1 ((0, t), Eδ ). On the other hand, Theorem 13.2 in [32] tells that there is some ϕ ∈ L1 ((0, t), Eδ ) such that (B.13) holds if limε→0 ψε exists in L1 ((0, t), Eδ ). The existence of this limit will be checked in the following. Recall (B.10) as well as (B.3) and put N := f |W α ([0, T ], E−β ). Recall that the semigroup operators P (t) are well-defined, bounded and strongly continuous both in E−β and in Eδ , since the fractional powers of A act as isomorphic mappings. We first consider the integral part of (B.12) and observe that for any ε > 0,
s−ε 0
P (t − s)f (s) − P (t − τ )f (τ ) dτ (s − τ )α+1 s
=
P (t − s)[f (s) − P (u)f (s − u)] du uα+1
ε
s = P (t − s)
P (u)[f (s) − f (s − u)] du + uα+1
ε
∞
[I − P (u)]P (t − s)f (s) du uα+1
ε
∞ − P (t − s)
[I − P (u)]f (s) du, uα+1
s
so far with the integrals evaluated in the norm of E−β . Now consider the right-hand side of the last equality. For the last summand there, property (B.4) implies that t ∞ P (t − s) u−α−1 I − P (u) f (s) du ε
s
t
∞ −α−1 I − P (u) f (s) du u
−(δ+β)/2
(t − s)
c
ds Eδ
ε
s
t cN
s −α (t − s)−(δ+β)/2 ds < ∞.
0
Taking into account also (B.5), we obtain
ds E−β
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t s P (t − s) u−α−1 P (u) f (s) − f (s − u) du ε
c
ds
Eδ
ε
t
3229
−(δ+β)/2
s
(t − s)
f (s) − f (s − u)E−β uα+1
ε
du ds
0
t cN
(t − s)−(δ+β)/2 ds < ∞
0
for the first summand. From these bounds we obtain the L1 ((0, t), Eδ )-convergence of the discussed terms. To treat the middle summand, we use the following equality, which was shown in [6, Section 2]:
∞
∞
P (εw)P (t − s)f (s)qα (w) dw =
α
A
[I − P (u)]P (t − s)f (s) du. uα+1
ε
0
The function qα is defined by its Laplace transform, ∞ e
−λu
qα (u) du = λ
−α
0
∞
1 − e−λu du, uα+1
1
Re λ > 0. It is a member of L1 (0, ∞) and satisfies ∞ qα (u) du = Γ (−α), 0
see [6, p. 193]. As P (t − s)f (s) ∈ dom(Aν ) for any ν 0 and 0 s < t, the integral
∞ P (εw)P (t − s)f (s)qα (w) dw 0
is an element of dom(Aα ). If instead it is considered as a member of E−β only, we may pull out P (t − s) from under the integral sign. Hence we may conclude ∞ [I − P (u)]P (t − s)f (s) 1 α du A P (t − s)f (s) − α+1 Γ (−α) u ε
Eδ
∞ 1 α α A = A P (t − s)f (s) − P (εw)P (t − s)f (s)qα (w) dw Γ (−α) 0
Eδ
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∞ 1 α I − P (εw) f (s)qα (w) dw = A P (t − s) Γ (−α)
Eδ
0
c(t − s)−α−(δ+β)/2
∞
I − P (εw) f (s) qα (w) dw. E −β
0
This is uniformly bounded, and the strong continuity of the semigroup yields the desired result. Next, we discuss the remaining part of (B.12) in a similar manner: ε P (t − s)f (s) 1 1 ds − α ε α s α E δ 0
cN α
ε s
−α
−(δ+β)/2
(t − s)
1 ds − α ε
0
cε 1−α−(δ+β)/2 N = α
ε
−(δ+β)/2
(t − s)
ds
0
1 σ 0
−α
t −σ ε
−(δ+β)/2
1 ds −
t −σ ε
−(δ+β)/2
ds ,
0
which for fixed t and small enough ε, is bounded by cε 1−α−(δ+β)/2 , note that 1 − α − (δ + β)/2 > 0. This shows the convergence of the Weyl–Marchaud derivative in L1 ((0, t), Eδ ). The existence of the function ϕ ∈ L1 ((0, t), Eδ ) now follows from [32, Theorem 13.2]. To obtain the identity (B.11), it suffices to take into account [32, Theorem 13.1]. 2 Appendix C. Function spaces We collect basic facts used in the main text. Though the present paper uses an L2 -setting in space, we quote key results for 1 < p < ∞ to indicate that some of our results carry over without effort, cf. Remark 5.4. C.1. Potential spaces Let S(Rn , Ck ) be the space of Ck -valued Schwartz functions and S (Rn , Ck ) the space of tempered distributions on Rn . f → f ∧ and f → f ∨ denote the Fourier transform and its inverse. For 1 < p < ∞ and α ∈ R, the Bessel potential spaces of order α are given by Ck -valued
Hpα Rn , Ck := f ∈ S Rn , Ck : f Hpα Rn , Ck < ∞ , α n k f H R , C := 1 + |ξ |2 α/2 f ∧ ∨ Lp Rn , Ck . p n We suppress Ck from notation if k = 1. Note that Hpα (Rn , Ck ) may be interpreted as the k-fold
k α n product space j =1 Hp (R ). For σ ∈ R, the linear operator f → Iσ f :=
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((1 + |ξ |2 )σ/2 f ∧ )∨ is an isomorphism of Hpα (Rn , Ck ) onto Hpα−σ (Rn , Ck ). If p = 2, we write f α for f |H2α (Rn , Ck ). For 0 < α < 1, f 0 + Rn Rn
|f (x) − f (y)|2k |x − y|2α+n n
1/2 (C.1)
dx dy
determines an equivalent norm in H2α (Rn , Ck ). Here · 0 is the norm in L2 (Rn , Ck ). C.2. Partial potential spaces Let g = (g1 , . . . , gn ) denote an index vector consisting of numbers or mappings gl (we are α (Rn ), only interested in the question whether a particular gl vanishes identically or not). By H2,g α ∈ R, we denote the space n α n α H2,g R := f ∈ S Rn : f H2,g R <∞ , where
α n n f H := 1 + ξ 2 α/2 f ∧ L2 Rn + f L2 R . l 2,g R l:gl =0
l:gl =0
See also part I and [26]. C.3. Spaces on domains Let D be a bounded C ∞ -domain in Rn . By D(D, Ck ) or C0∞ (D, Ck ) we denote the space of smooth compactly supported Ck -valued functions on D, and by D (D, Ck ) its topological dual. As before, Ck is suppressed from the notation if k = 1. For more information on the following see [38]. Given α ∈ R we define the space. H2α D, Ck := f ∈ D D, Ck : ∃g ∈ H2α Rn , Ck such that g|D = f , where g|D denotes the restriction in the sense of distributions. We equip them with the norm α f H D, Ck := inf g H α Rn , Ck : g ∈ H α Rn , Ck such that g|D = f , 2 2 2 the infimum taken over all such g. In particular, f |D H α D, Ck cf H α Rn , Ck 2 2 for f ∈ H2α (Rn , Ck ), cf. [38, 4.2.2]. Keeping in mind this last inequality, we sometimes write f to denote f |D to shorten notation. The space H˚ 2α (D, Ck ) is defined as the completion of C0∞ (D, Ck ) in the norm · |H2α (D, Ck ). One further defines the spaces α D, Ck := f ∈ H α Rn , Ck : supp f ⊂ D , H 2 2
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where supp f denotes the support of f (in distributional sense) and α H2,0 D, Ck := f ∈ H2α D, Ck : f |∂D = 0 , where f |∂D is the restriction (trace) of f to the boundary ∂D of D, see [38, 4.7.1]. α (D, Ck ) = H˚ α (D, Ck ) if −1/2 < α < ∞, α − 1/2 ∈ / Z, and that It is known that H 2 2 α α k ˚ H2 (D, C ) = H2 (D, Ck ) if −∞ < α 1/2, see [38, Section 4.3.2]. We put H α2
D, Ck :=
α (D, Ck ) H 2
if α 0,
H2α (D, Ck )
if α < 0.
C.4. Pointwise multiplication The product of two arbitrary distributions does not make sense. However, in the special case of the spaces we use, one can define products via paraproducts, see [33] or [35]. Choose a function ψ ∈ S(Rn ) with 0 ψ(x) 1 and such that ψ(x) = 1 if |x|n 1 and ψ(x) = 0 if |x|n 3/2. Given f ∈ S (Rn ), consider ∨ S j f (x) := ψ 2−j ξ f ∧ (x), which, according to the Paley–Wiener–Schwartz theorem, is an entire analytic function for any j ∈ N. The product f g of f, g ∈ S (Rn ) is defined as f g := lim S j f S j g, j →∞
whenever the limit exists in S (Rn ). The convergence is part of the assertion below. We refer to [33, Chapter 4], and use a special case of their Theorem 4.4.3/1. To indicate how some results of the present paper can be generalized to an Lp -setting, we state it for arbitrary 1 < p < ∞: Lemma C.1. Let 1 < p, q < ∞ and 0 < β < δ. Assume further that q > p ∨ (n/δ). Then we have −β n f g H R cf Hpδ Rn g Hq−β Rn p −β
for f ∈ Hpδ (Rn ) and g ∈ Hq (Rn ). Now suppose that h is a compactly supported (1 − β )-Hölder continuous function on Rn , 1−β 0 < β < 1. By (C.1) it is seen to be a member of Hq (Rn ) for any 1 < q < ∞, provided −β ∂h ∈ Hq (Rn ) which may be considered in place of g. β < β. If so, it has partial derivatives ∂x l The product preserves locality in the following sense: Lemma C.2. If f, g ∈ S (Rn ) and supp f ∈ D, then also supp f g ∈ D. See [33, Lemma 4.2]. For f, g ∈ S (Rn , Ck ), f = (f 1 , . . . , f k ), g = (g 1 , . . . , g k ), we define the product f · g in the sense of (6), f · g := f 1 g 1 , . . . , f k g k .
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3233
The quoted results carry over. Now consider n k α R , C := Hpα Rn , Ck ∩ L∞ Rn , Ck Hp,∞ and α H˚ p,∞ D, Ck := H˚ pα D, Ck ∩ L∞ Rn , Ck . α (Rn , Ck ) is endowed with the norm · For p = 2, H2,∞ α,∞ := · α + · ∞ , · ∞ denoting the α (Rn , Ck ), α > 0, is a multiplication norm in L∞ (Rn , Ck ). With the entry-wise product (6), Hp,∞ algebra. For p = 2 that means in particular that
w · vα cwα,∞ vα,∞ α (Rn , Ck ). See [33, 4.6.4/2] for the case k = 1. for any v, w ∈ H2,∞
C.5. Real subspaces and composition operators We follow again [33]. Given f ∈ S (Rn , Ck ), the distribution f is defined by requiring f (ϕ) = f (ϕ) for any ϕ ∈ S(Rn , Ck ). The space of Rk -valued Schwartz distributions S (Rn , Rk ) is defined by S Rn , Rk := f ∈ S Rn , Ck : f = f . For 1 < p < ∞, α ∈ R, set Hpα Rn , Rk := Hpα Rn , Ck ∩ S Rn , Rk . This is a closed subspace of Hpα (Rn , Ck ). If α 0, i.e. if f ∈ Hpα (Rn , Ck ) may be seen as locally integrable function, we have f ∈ Hpα (Rn , Rk ) if and only if f is an Rk -valued function in the ordinary sense. In the cases we consider, approximation by smooth functions immediately shows that the product f · g is an Rk -valued distribution, provided f and g are. Given a function G : Rk → R with G(0) = 0 and having bounded differential DG ∈ L∞ (Rk , Rk ), we define the composition operator TG : Hpα (Rn , Rk ) → Hpα (Rn , Rk ), 1 < p < ∞, 0 < α < 1, by TG f := G(f ) = G f 1 , . . . , f k . For p = 2, the written mapping property is guaranteed by (C.1), for general 1 < p < ∞ it follows from well known analogues. References [1] S. Agmon, Elliptic Boundary Value Problems, Van Nostrand, 1965. [2] A. Ayache, S. Leger, M. Pontier, Drap brownien fractionnaire, Potential Anal. 17 (2002) 31–43. [3] A. Ayache, Y. Xiao, Asymptotic properties and Hausdorff dimension of fractional Brownian sheets, J. Fourier Anal. Appl. 11 (2005) 407–439.
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[4] R.F. Bass, Diffusions and Elliptic Operators, Springer, New York, 1997. [5] F.E. Benth, Th. Deck, J. Potthoff, L. Streit, Nonlinear evolution equations with gradient coupled noise, Lett. Math. Phys. 43 (1998) 267–278. [6] H. Berens, P.L. Butzer, U. Westphal, Representation of fractional powers of infinitesimal generators of semigroups, Bull. Amer. Math. Soc. 74 (1968) 191–196. [7] G. DaPrato, J. Zabzcyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. [8] R.M. Dudley, Gaussian processes on several parameters, Ann. Probab. 1 (1973) 66–103. [9] T.E. Duncan, B. Maslowski, B. Pasik-Duncan, Fractional Brownian motion and linear stochastic equations in Hilbert space, Stoch. Dyn. 2 (2002) 225–250. [10] R.C. Dalang, N.E. Frangos, The stochastic wave equation in two spatial dimensions, Ann. Probab. 26 (1998) 187– 212. [11] T. Funaki, Random motion of strings and related stochastic evolution equations, Nagoya Math. J. 89 (1983) 129– 193. [12] W. Grecksch, V.V. Anh, A parabolic stochastic differential equation with fractional Brownian motion input, Statist. Probab. Lett. 41 (1999) 337–345. [13] M. Gubinelli, A. Lejay, S. Tindel, Young integrals and SPDE’s, Potential Anal. 25 (2006) 307–326. [14] M. Hinz, M. Zähle, Gradient type noises I – Partial and hybrid integrals, Complex Var. Elliptic Equ. (2008), in press. [15] H. Holden, J. Ubøe, B. Øksendal, T.S. Zhang, Stochastic Partial Differential Equations – A Modelling White Noise Approach, Birkhäuser, 1996. [16] Y. Hu, Heat equations with fractional white noise potentials, Appl. Math. Optim. 43 (2001) 221–243. [17] Y. Hu, D. Nualart, Stochastic heat equation driven by fractional noise and local time, preprint, 2008. [18] Zh. Huang, Ch. Li, On fractional stable processes and sheets: White noise approach, J. Math. Anal. Appl. 325 (2007) 624–635. [19] K. Itô, Isotropic random currents, in: Proc. Third Berkeley Sympos. Math. Stat. Probab., 1956, pp. 125–132. [20] A. Kamont, On the fractional anisotropic Wiener field, Probab. Math. Statist. 16 (1996) 85–98. [21] T. Lindstrøm, Fractional Brownian fields as integrals of white noise, Bull. London Math. Soc. 25 (1993) 83–88. [22] N.V. Krylov, B.L. Rozovskij, Stochastic evolution equations, Current Problems in Math. Akad. Nauk. SSSR 14 (1979) 71–147 (in Russian). [23] T.J. Lyons, Differential equations driven by rough signals I: An extension of an inequality by L.C. Young, Math. Res. Lett. 1 (1994) 451–464. [24] B. Maslowski, D. Nualart, Evolution equations driven by fractional Brownian motion, J. Funct. Anal. 202 (2003) 277–305. [25] A.S. Monin, A.M. Yaglom, Statistical Fluid Mechanics, MIT Press, Cambridge, MA, 1975. [26] S.M. Nikolskij, Approximation of Functions of Several Variables and Embedding Theorems, Springer, New York, 1975. [27] D. Nualart, The Malliavin Calculus and Related Topics, Springer, New York, 1995. [28] D. Nualart, R. Rascanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (1) (2002) 55–81. [29] B. Øksendal, T.S. Zhang, Multiparameter fractional Brownian motion and quasi-linear stochastic partial differential equations, Stochastics Stochastics Rep. 71 (2000) 141–163. [30] L.A.F. de Oliveira, On reaction-diffusion systems, Electron. J. Differential Equations 1998 (24) (1998) 1–10. [31] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. [32] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, 1993. [33] T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Non-linear Partial Differential Equations, de Gruyter, Berlin–New York, 1996. [34] F. Russo, P. Vallois, Forward, backward and symmetric stochastic integration, Probab. Theory Related Fields 97 (1993) 403–421. [35] M.E. Taylor, Pseudodifferential Operators and Non-linear Partial Differential Equations, Birkhäuser, Boston, 1991. [36] S. Tindel, C.A. Tudor, F. Viens, Stochastic evolution equations with fractional Brownian motion, Probab.Theory Related Fields 127 (2003) 186–204. [37] S. Tindel, C.A. Tudor, F. Viens, Sharp Gaussian regularity on the circle, and applications to the fractional stochastic heat equation, J. Funct. Anal. 217 (2004) 280–313. [38] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, second ed., J.A. Barth, 1995. [39] J.B. Walsh, An introduction to stochastic partial differential equations, in: École d’été de probabilités de Saint-Flour, XIV-1984, in: Lecture Notes in Math., vol. 1180, Springer, 1986.
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Journal of Functional Analysis 256 (2009) 3236–3256 www.elsevier.com/locate/jfa
Rough path limits of the Wong–Zakai type with a modified drift term Peter Friz 1 , Harald Oberhauser 2,∗ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom Received 5 August 2008; accepted 17 February 2009 Available online 9 March 2009 Communicated by Paul Malliavin
Abstract The Wong–Zakai theorem asserts that ODEs driven by “reasonable” (e.g. piecewise linear) approximations of Brownian motion converge to the corresponding Stratonovich stochastic differential equation. With the aid of rough path analysis, we study “non-reasonable” approximations and go beyond a well-known criterion of [Ikeda, Watanabe, North Holland, 1989] in the sense that our result applies to perturbations on all levels, exhibiting additional drift terms involving any iterated Lie brackets of the driving vector fields. In particular, this applies to the approximations by McShane (’72) and Sussmann (’91). Our approach is not restricted to Brownian driving signals. At last, these ideas can be used to prove optimality of certain rough path estimates. © 2009 Elsevier Inc. All rights reserved. Keywords: Iterated Lie brackets in limit processes of differential equations; Rough paths analysis
* Corresponding author.
E-mail addresses:
[email protected] (P. Friz),
[email protected] (H. Oberhauser). 1 Partially supported by a Leverhulme Research Fellowship and EPSRC Grant EP/E048609/1. 2 Partially supported by EPSCR Grant EP/P502365/1 and a DOC-fellowship of the Austrian Academy of Sciences.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.010
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1. Preliminaries 1.1. Rough differential equations Let α ∈ (0, 1]. A weak geometric α-Hölder rough path x over Rd is a continuous path on [0, T ] with values in G[1/α] (Rd ), the step3 -[1/α] nilpotent group over Rd , of finite α-Hölder regularity relative to d, the Carnot–Carathéodory metric on G[p] (Rd ), i.e. xα-Höl;[0,T ] =
sup 0s
d(xs , xt ) < ∞. |t − s|α
For orientation, let us discuss the case α ∈ (1/3, 1/2), which covers Brownian motion (for details see [5,11,17,18]). We realize G2 (Rd ) as the set of all (a, b) ∈ Rd ⊕ Rd×d for which Sym(b) ≡ a ⊗2 /2. (This point of view is natural: a smooth Rd -valued path x = (xti )i=1,...,d , enhanced t s j with its iterated integrals 0 0 dxui dxs , gives canonically rise to a G2 (Rd )-valued path.). Given 2 d (a, b) ∈ G (R ) one gets rid of the redundant Sym(b) by (a, b) → (a, b − a ⊗ /2) ∈ Rd ⊕ so(d). Applied to x enhanced with its iterated integrals over [0, t] this amounts to look at the path x and t i t j j its (signed) areas 0 x0,s dxs − 0 x0,s dxsi , i, j ∈ {1, . . . , d}.4 Without going in too much detail, the group structure on G2 (Rd ) can be identified with the (truncated) tensor multiplication and is relevant as it allows to relate algebraically the path and area increments over adjacent intervals; the mapping (a, b) → (a, b − a ⊗ /2) maps the Lie group G2 (Rd ) to its Lie algebra g2 (Rd ); at last, the Carnot–Carathéodory metric is defined intrinsically as (left-) invariant metric on G2 (Rd ) and satisfies |a| + |b|1/2 d((0, 0), (a, b)) |a| + |b|1/2 . One can then think of a geometric α-Hölder rough path x as a path x : [0, T ] → Rd enhanced with its iterated integrals (equivalently: area integrals) although the later need not make classical sense. For instance, almost every joint realization of Brownian motion and Lévy’s area process is a geometric α-Hölder rough path. Lyons’ theory of rough paths then gives deterministic meaning to the rough differential equation (RDE) dy = V (y) dx,
y(0) = y0
for LipΓ -vector fields (in the sense of Stein5 ), Γ > 1/α 1, and we write yt = π(0, y0 ; x)t for this solution. By considering the space–time rough path x˜ = (t, x) and V˜ = (V0 , V1 , . . . , Vd ) one can consider RDEs with drift. Although well studied [16], with a view towards minimal regularity assumptions on V0 , we shall need certain “Euler” estimates [8] for RDEs with drift which are not available in the current literature. The “Doss–Sussmann method” (implemented for RDEs in Section 3) will provide a quick route to these estimates. 1.2. Perturbed rough paths and iterated Lie brackets Assume we are given a weak geometric α-Hölder rough path x and a path p that takes values in the center of GN (Rd ), N some integer N [1/α] (think of the path p as a perturbation of our 3 [·] gives the integer part of a real number. 4 Given an interval I = [a, b], for brevity we write x ≡ x I a,b ≡ xb − xa . 5 I.e. a function is Lipγ if it is γ -times ( . = [.] − 1 on integers, otherwise equal) differentiable, the γ th derivative
is γ − γ -Hölder continuous and the function and all its derivatives are bounded.
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original path x). Further assume that p is β-Hölder continuous. It is a well-known [17] that x can be lifted uniquely to a α-Hölder path SN (x) with values in GN (Rd ). Then, SN (x)0,· ⊗ p ∈ C min(α,β)-Höl [0, T ], GN Rd . From general facts of such spaces (e.g. [6], relying only on the fact that GN (Rd ) is a geodesic space) we can find a sequence of Lipschitz continuous paths, x n : [0, T ] → Rd , so that d∞ SN x n , SN (x)0,· ⊗ p → 0 as n → ∞
and
supSN x n min(α,β)-Höl < ∞.
n∈N
(The approximations x n are constructed based on geodesics associated to the GN (Rd )-valued increments of SN (x)0,· ⊗ p). By interpolation, it then follows that for all γ < min(α, β), dγ -Höl SN x n , SN (x) ⊗ p → 0 as n → ∞. The interest of such a construction is that the limiting behaviour of ODEs driven by the x n , provided N [1/γ ] and Γ > 1/γ , exhibits additional drift behaviour in terms of the Lie brackets of the driving vector fields and the perturbation p (cf. Section 4.1). We may apply this to Brownian motion and Lévy area, i.e. x = B(ω) = exp(B + A), in which case the approximations x n are constructed in a purely deterministic fashion based on the realization of the Brownian path and its Lévy area. Interesting as it may be, this is not fully satisfactory as it stands in contrast to (probabilistic) non-standard approximation results (McShane [19], Sussmann [22], . . . ) which have the desirable property that the approximations depend only on (finitely many points of) the Brownian path. The first aim of this paper is to give a criterion that covers all these examples in a flexible frame-work of random rough paths. En passant, this allows for a painfree extension to various non-Brownian driving signals (cf. Section 2). We then give a rigorous analysis on how to translate RDEs driven by SN (x) ⊗ p to RDEs driven by x in which one sees the appearance of additional drift vector fields, obtained as contraction of iterated Lie brackets of V = (V1 , . . . , Vd ) and the components of p. (At this stage, we need a good quantitative understanding of RDEs with drift). At last, as a spin-off of these ideas, we show in Section 4.2 optimality of certain rough path estimates, answering a question that was left open in [2]. 2. (Perturbed) approximations 2.1. Criterion for convergence to a “perturbed” limit The discussion in Section 1.2 motivates the following definitions. Definition 1. Let α ∈ (0, 1], N [1/α], x ∈ C α-Höl ([0, T ], G[1/α] (Rd )) and write x = π1 (x) for its projection to a path with values in Rd . (i) Let (Dn ) = (tin : i) be a sequence of dissections of [0, T ] such that supS[1/α] x Dn α-Höl < ∞
n∈N
and d∞ S[1/α] x Dn , x →n→∞ 0
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(x Dn denotes the usual piecewise linear approximation to x based on Dn ). If (x n ) ⊂ C 1-Höl ([0, T ], Rd ) such that −1 pnt := SN x n 0,t ⊗ SN x Dn 0,t takes values in the center of GN (Rd ) for all t ∈ Dn (but not necessarily values in the center for t ∈ / Dn ) then we say that (x n ) is an approximation on (Dn ) with perturbations (pn ) on level N to x. (ii) Let β ∈ (0, 1] such that [1/β] = N . We say that an approximation (x n ) on (Dn ) with perturbations (pn ) on level N to x is min(α, β)-Hölder comparable (with constants c1 , c2 , c3 ) if n ∈D there exists c1 , c2 , c3 ∈ [1, ∞) such that for all neighboring points tin , ti+1 n n x
n ] 1-Höl;[tin ,ti+1
n n β−1 n ] + c 2 t c1 |x Dn |1-Höl;[tin ,ti+1 i+1 − ti
n p c3 |t − s|β s,t
and
for all s, t ∈ Dn .
Although at first sight technical, these definitions are fairly natural: firstly, we restrict our attention to Hölder rough paths x which are the limit of “their (lifted) piecewise linear approximations.” In a probabilistic context (cf. Section 2.2) this covers the bulk of stochastic processes which admit a lift to a rough path including semi-martingales [3,7], fractional Brownian motion with H > 1/4 and many other Gaussian processes [4,10], reversible Markov processes [1] as well as Markov processes with uniformly elliptic generator in divergence form [9,14,21]. Secondly, being min(α, β)-Hölder comparable guarantees that x n remains, at min(α, β)Hölder scale, comparable to the piecewise linear approximations. In particular, the assumption n ] = |x n ] is easy to verify in all examples below. The intuition is that, on |x n |1-Höl;[tin ,ti+1 ˙ n |∞;[tin ,ti+1 n ], D = (t n ), it is n if we assume that x runs at constant speed over any interval I = [tin , ti+1 n i equivalent to saying that length x n I c1 length x Dn I + c2 |I |β
n n β n | + c 2 t = c1 |xtin ,ti+1 . i+1 − ti
Theorem 1. Let α ∈ (0, 1], x ∈ C α-Höl ([0, T ], G[1/α] (Rd )) and let (x n ) be an approximation on some sequence (Dn ) of dissections of [0, T ] with perturbations (pn ) on level N to x. (i) If the approximation is min(α, β)-Hölder comparable (with constants c1 , c2 , c3 ) then there exists a constant c = c(α, β, c1 xα-Höl , c2 , T , N ) such that supSN x n min(α,β)-Höl c supS[1/α] x Dn α-Höl + c3 + 1 < ∞.
n∈N
n∈N
(ii) If pnt → pt for all t ∈ n Dn and n Dn is dense in [0, T ] then p extends to a continuous (in fact, β-Hölder continuous) path with values in the center of GN (Rd ) and for all t ∈ [0, T ], d SN x n 0,t , SN (x)0,t ⊗ p0,t d SN x Dn 0,t , SN (x)0,t + d pn0,t , p0,t and converges to 0 as n → ∞.
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In particular, if the assumptions of both (i) and (ii) are met then d∞ SN x n , SN (x) ⊗ p → n→∞ 0, supSN x n min(α,β)-Höl < ∞
n∈N
and by interpolation, for all γ < min(α, β), dγ -Höl SN x n , SN (x) ⊗ p →n→∞ 0. n ] we have by our assumption on |x n | Proof. (i) Take s < t in [0, T ]. If s, t ∈ [tin , ti+1 1-Höl;[ti ,ti+1 ]
n SN x |t − s|SN x n n ] s,t 1-Höl;[tin ,ti+1 n = |t − s|x 1-Höl; t n ,t n i
i+1
n n
n n β−1 n /t |t − s| c1 xtin ,ti+1 i+1 − ti + c2 ti+1 − ti n α−1 n β−1 − tin + c2 ti+1 − tin |t − s| c1 |x|α-Höl ti+1 c|t ˜ − s|min(α,β) , with c˜ = c(α, ˜ β, c1 xα-Höl , c2 , T ). Otherwise we can find tin tjn so that s tin tjn t and n S N x |t − s|min(α,β) 2c˜ + SN x n n n . s,t t ,t i
j
Using estimates for the Lyons-lift x → SN (x), [17, Theorem 2.2.1], we can further estimate n SN x n n SN x Dn n n + pnn n t ,t t ,t t ,t i
j
i
j
cN,α S[1/α] x Dn
i
j
n t − t n α + c 3 t n − t n β
α-Höl j
i
j
i
α β = cN,α S[1/α] x Dn α-Höl tjn − tin /T T α + c3 tjn − tin /T T β cN,α S[1/α] x Dn α-Höl T α−min(α,β) + c3 T β−min(α,β) |t − s|min(α,β) and, since supn S[1/α] (x Dn )α-Höl < ∞ by assumption, the proof of the uniform Hölder bound is finished. (ii) By assumption, pn is uniformly β-Hölder. By a standard Arzela–Ascoli type argument, it is clear that every pointwise limit (if only on the dense set n Dn ) is a uniform limit. β-Hölder regularity is preserved in this limit and so p is β-Hölder itself. Since t ∈ D , pnt take values n n in the step-N center for all t ∈ Dn , n Dn is dense, and so it is easy to see that p· takes values in the step-N center for all t ∈ [0, T ]. Now take t ∈ Dn . Since elements in the center commute with all elements in GN (Rd ) we have
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d SN x n 0,t , SN (x)0,t ⊗ p0,t −1 −1 −1 = SN x n 0,t ⊗ SN x Dn 0,t ⊗ pn0,t ⊗ SN x Dn 0,t ⊗ SN (x)0,t ⊗ pn0,t ⊗ p0,t −1 −1 = SN x Dn 0,t ⊗ SN (x)0,t ⊗ pn0,t ⊗ p0,t d SN x Dn 0,t , SN (x)0,t + d pn0,t , p0,t . On the other hand, given an arbitrary element t ∈ [0, T ] we can take t n to be the closest neighbour in Dn and so d SN x n 0,t , SN (x)0,t ⊗ p0,t = d SN x Dn 0,t , SN (x)0,t + 2d pn0,t , p0,t n n −1 + d SN (x)−1 0,t ⊗ SN x 0,t , S(x)0,t n ⊗ SN x 0,t n . From the assumptions and Hölder (resp. uniform Hölder) continuity of S(x) (resp. SN (x n )) we see that d(SN (x n )0,t , SN (x)0,t ⊗ p0,t ) → 0, as required. 2 2.2. Convergence to a “perturbed” limit for random rough paths The definitions of Section 2.1 translate in a straightforward way to random rough paths. However, care has to be taken on the type of convergence. Definition 2. Let α ∈ (0, 1], N [1/α], X(ω) ∈ C α-Höl ([0, T ], G[1/α] (Rd )) be a random rough path and write X = π1 (X) for its projection to a random process with values in Rd . (i) Let (Dn ) = (tin : i) be a sequence of dissections of [0, T ] such that supS[1/α] X Dn α-Höl Lq < ∞
∀q ∈ N:
n∈N
d∞ S[1/α] X Dn , X → 0 in probability as n → ∞.
If (X n (ω)) ⊂ C 1-Höl ([0, T ], Rd ) such that, for all ω, −1 Pnt (ω) := SN X n (ω) 0,t ⊗ SN X Dn (ω) 0,t takes values in the center of GN (Rd ) whenever t ∈ Dn (but not necessarily values in the center for t ∈ / Dn ) then we say that (X n ) is an approximation on (Dn ) with perturbations n (P ) on level N to the random rough path X. (ii) Let β ∈ (0, 1] such that [1/β] = N. We say that an approximation (X n ) on (Dn ) with perturbations (Pn ) on level N to the random rough path X ∈ C α-Höl ([0, T ], G[1/α] (Rd )) is min(α, β)-Hölder comparable (with deterministic constants c1 , c2 , c3 ) if there exists n ∈ D , all ω and all q ∈ [1, ∞), c1 , c2 , c3 ∈ [1, ∞) such that for all tin , ti+1 n n X
n ] 1-Höl;[tin ,ti+1
n P s,t
c1 X Dn 1-Höl;[t n ,t n i
c3 |t − s|
β
Lq (P)
i+1 ]
n β−1 + c2 ti+1 − tin ,
for all s, t ∈ [0, T ].
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We now formulate a corollary to Theorem 1 for random rough paths. Corollary 1. Let α ∈ (0, 1], X(ω) ∈ C0α-Höl ([0, T ], G[1/α] (Rd )), write X = π1 (X) for its projection to a process with values in Rd , and let (X n ) be an approximation on some sequence (Dn ) of dissections of [0, T ] with perturbations (Pn ) on level N to X. (i) If the approximation is min(α, β)-Hölder comparable (with constants c1 , c2 , c3 ) then for all γ < min(α, β) ∀q:
supSN x n γ -Höl Lq (P) < ∞.
n∈N
(ii) If Pnt → Pt in probability for all t ∈
n∈N Dn
and
n∈N Dn
is dense in [0, T ] then
d SN X n 0,t , SN (X)0,t ⊗ P0,t → 0 in probability. In particular, if the assumptions of both (i) and (ii) are met then, for all γ < min(α, β), dγ -Höl SN X n , SN (X) ⊗ P → 0 in Lq for all q ∈ [1, ∞). Proof. (i) By a standard Garsia–Rodemich–Rumsey or Kolmogorov argument, the assumption on |||Pns,t |||Lq (P) implies, for any β˜ < β, the existence of C3 ∈ Lq for all q ∈ [1, ∞) so that ∀s < t
˜ in [0, T ] : Pns,t C3 (ω)|t − s|β .
˜ = [1/β] = N and apply Theorem 1 with β˜ instead We pick β˜ close enough to β such that [1/β] of β to see that there exists a deterministic constant c such that c supS[1/α] X Dn α-Höl + 1 + C3 . supSN X n min(α,β)-Höl ˜ n
n
Taking Lq -norms finishes the uniform Lq -bound. (ii) From Theorem 1 d SN X n 0,t , SN (X)0,t ⊗ P0,t d SN X Dn 0,t , SN (X)0,t + d Pn0,t , P0,t which, from the assumptions, obviously converges to 0 (in probability) for every fixed t ∈ n Dn . From general facts, of Lq -convergence of rough paths (cf. [10, Appendix]) this implies the claimed convergence. (Inspection of the proof shows that convergence in probability for all t in a dense set of [0, T ] is enough.) 2 Remark 1. In both examples below we have β = 1/N . However, in order to obtain Corollary 1 as a consequence of Theorem 1, one needs the stated generality for β (i.e. [1/β] = N ) to use a Garsia–Rodemich–Rumsey or Kolmogorov argument.
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2.3. Examples 2.3.1. Sussmann [22] Take any sequence of dissections of [0, T ], say (Dn ) with mesh |Dn | → 0 and think of X(ω) as Brownian motion plus Lévy’s area so that π1 (X) = X is a d-dimensional Brownian motion. The piecewise linear approximation X Dn is nothing but the repeated concatenation of linear chords connecting the points (Xt : t ∈ Dn ). For some fixed v ∈ gN (Rd ) ∩ (Rd )⊗N , N ∈ {2, 3, . . .} we now construct Sussmann’s non-standard approximation X n as (repeated) concatenation of linear chords and “geodesic loops.” First, we require X n (t) = X(t) for all n , t n ) for some i we proceed as follows: t ∈ Dn = (tin : i). For intermediate times, i.e. t ∈ (ti−1 i n n n n ) such as to reach For t ∈ [ti−1 , (ti−1 + ti )/2] we run linearly and at constant speed from X(ti−1 n n n n ) and X(t n ) X(ti ) by time (ti−1 + ti )/2. (This is the usual linear interpolation between X(ti−1 i n n n but run at double speed.) This leaves us with the interval [(ti−1 + ti )/2, ti ] for other purposes n + t n )/2, t n ] → Rd associated and we run, starting at x(tin ) ∈ Rd , through a “geodesic” ξ : [(ti−1 i i n |) ∈ GN (Rd ).6 Since N > 1, π (exp(v/|t n − t n |)) = 0 and so this geodesic to exp(v/|tin − ti−1 1 i i−1 path returns to its starting point in Rd ; in particular n + tin /2 = X n tin = X tin . X n ti−1 It is easy to see (via Chen’s theorem) that this approximation satisfies the assumptions of Corollary 1 with −1 Pns,t := SN x n s,t ⊗ SN x Dn s,t = ev(t−s)
∀s, t ∈ Dn
(so that |Pns,t |Lq = |Pns,t | |t − s|1/N first for all s, t ∈ Dn and then, easy to see, for all s, t) and n , t n] deterministic limit P0,t = evt , β = 1/N . Indeed, the length of x n over any interval I = [ti−1 i is obviously bounded by the length of the corresponding linear chord plus the length of the geodesic associated to exp(v/2n ) = exp(v/|I |), which is precisely equal to exp v/|I | = |I |1/N exp(v) =: c2 |I |1/N . Obviously, nothing here is specific to Brownian motion. An application of Corollary 1 gives the following convergence result which, when applied to Brownian motion and Lévy area and in conjunction with Theorem 2 (of Section 4), implies Sussmann’s non-standard approximation result for stochastic differential equations. Proposition 1 (Rough path convergence of Sussmann’s approximation). Let X(ω) ∈ C0α-Höl ([0, T ], G[1/α] (Rd )) be a (random, α-Hölder rough path) which is the limit of its piecewise linear approximation (as detailed in Definition 2). Then, for any γ < min(α, 1/N ) we have dγ -Höl SN X n , SN (X) ⊗ ev· → 0 in Lq for all q ∈ [1, ∞). 6 A path ξ : [a, b] → Rd is a geodesic associated to g ∈ GN (Rd ) if S (ξ ) N a,b = g and ξ has length equal to the Carnot–Carathéodory norm of g. See [6] and references therein.
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Strictly speaking, Sussmann’s construction did not rely on the concept of geodesics associated to exp(v/2n ) ∈ GN (Rd ). Since exp(v/2n ) is an element of the center of the group he can give a reasonably simple inductive construction of a piecewise linear “approximate geodesic” which is also seen to satisfy the assumptions of our theorem. We also note that he only discusses dyadic approximations and obtains an a.s. convergence result (which can be obtained by a direct application of Theorem 1). In conjunction with Theorem 2 (Section 4), we recover the main result of [22]. 2.3.2. McShane [19] Given x ∈ C([0, T ], R2 ), an interpolation function φ = (φ 1 , φ 2 ) ∈ C 1 ([0, 1], R2 ) with φ(0) = (0, 0) and φ(1) = (1, 1) and a dissection D = Dn = {ti } of [0, T ] we define the McShane interpolation x n ∈ C([0, T ], R2 ) by xtn;i := xtiD + φ (t,i)
t − tD xi D , t D − tD tD ,t
i = 1, 2.
The points tD , t D ∈ D denote the left-, resp. right-, neighbouring points of t in the dissection and
(t, i) :=
i,
if xt1
D
xt2
D
0,
3 − i,
if xt1
D
xt2
D
< 0.
D ,t D ,t
D ,t D ,t
As a simple consequence of this definition, for u < v in [ti , ti+1 ] n S2 x n u,v ≡ exp xu,v + Anu,v 1 2 φ u − ti v − ti = exp xu,v + xti ,ti+1 xti ,ti+1 A , ti+1 − ti ti+1 − ti φ
where Aφ (u, v) ≡ Au,v is the area increment of φ over [u, v] ⊂ [0, 1]. Consider now X(ω) = B(ω) = exp(B(ω) + A(ω)) ∈ C0α-Höl ([0, 1], G[1/α] (R2 )) with α ∈ (1/3, 1/2) and take any (Dn )n∈N with |Dn | → 0. (We know, e.g. from [10], that the lifted piecewise linear approximation S2 (B Dn ) converges to B in 1/α-Hölder rough path topology and in Lq for all q.) It is easy to see (via Chen’s theorem) that McShane’s approximation to twodimensional Brownian motion satisfies the assumptions of Corollary 1 with β = 1/2, N = 2. Indeed, writing S2 (B n ) = exp(B n + An ) it is clear that for any s < t Pns,t = exp xt1
D ,t
2 D x
tD ,t
φ D ×A
s − tD t − tD , t D − tD t D − tD
with D = Dn
and for ti < tj elements of the dissection D = Dn , Pnti ,tj = exp
φ A0,1
j 1 B
tk ,tk+1
tk ,tk+1 .
2 B
k=i+1
It is straight-forward to see that
j
1 2 k=i+1 |Btk ,tk+1 ||Btk ,tk+1 |
converges, in L2 say, to its mean
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j 2 2 (tk+1 − tk ) = |tj − ti | π π k=i+1
j while Ptni ,tj Lq c˜q |tj − ti |1/2 follows directly from Ptni ,tj ∼ ( k=i+1 |Bt1k ,tk+1 ||Bt2k ,tk+1 |)1/2 and j 2 1 B tk ,tk+1 Btk ,tk+1 k=i+1
j 1 2 B tk ,tk+1 Lq Btk ,tk+1 Lq = cq |tj − ti |. Lq
k=i+1
n q c˜ |t − s|1/2 for all s, t since for u < v in [t , t In fact, Ps,t L q i i+1 ]
n P
u,v Lq
1 2 φ v − ti u − ti q/2 1/q = E xti ,ti+1 xti ,ti+1 A , ti+1 − ti ti+1 − ti u−v = cφ,q (ti+1 − ti )1/2 t −t i+1
cφ,q |u − v|
1/2
i
.
At last, we easily see that, for any ti ∈ Dn , n B
1-Höl;[ti ,ti+1 ]
|φ |∞ B Dn 1-Höl;[t ,t
i i+1 ]
.
This shows that all assumptions of Corollary 1 are satisfied and so we have Proposition 2 (Rough path convergence of McShane’s approximation). For all α ∈ [0, 1/2), dα-Höl S2 B n , exp(Bt + At + tΓ ) → 0 in Lq for all q ∈ [1, ∞) where At the usual so(2)-valued Lévy’s area and Γ =
2 φ π A0,1
0 φ
− π2 A0,1
0
∈ so(2).
We note that Corollary 1 also applies to a (minor) variation of the McShane example given in [15]. 3. RDEs with drift: the Doss–Sussmann approach 3.1. Preliminaries on ODEs and flows Consider an ordinary differential equation (ODE), driven by a smooth Rd -valued signal f (t) = (f 1 (t), . . . , f d (t))T on a time interval [t0 , T ] along sufficiently smooth and bounded vector fields V = (V1 , . . . , Vd ) and a drift vector field V0 dy = V0 (y) dt + V (y) df,
y(t0 ) = y0 ∈ Re .
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We also call Ut←t0 (y0 ) ≡ yt the associated flow. Let J denote the Jacobian of U . It satisfies the ODE obtained by formal differentiation w.r.t. y0 . More specifically, d f U (y0 + εa) a → dε t←t0 ε=0
f
is a linear map from Re → Re and we let Jt←t0 (y0 ) denote the corresponding e × e matrix. It is immediate to see that d f d f f f Jt←t0 (y0 ) = M Ut←t0 (y0 ), t · Jt←t0 (y0 ) dt dt
(3.1)
where · denotes matrix multiplication and d f d Vi (y) fti + V0 (y). M (y, t) = dt dt d
i=1
f
f
f
f
f
Note that also Jt2 ←t0 = Jt2 ←t1 · Jt1 ←t0 and that Jt←t0 is invertible with inverse, denoted Jt0 ←t , given as the flow of (3.1) with f replaced by ← − ft
← − f t (.)
= f (t − .), i.e. Jt0 ←t (.) = (Jt←t0 (.))−1 = f
f
Jt←t0 (.). Now for V = (V1 , . . . , Vd ) ∈ Lip2 (Re ) and x ∈ C 1 ([0, T ], Rd ) ODE theory tells us that dy = V (y) dxt has a C 2 -flow. Note that the flow x y0 → π(V ) (0, y0 , x)t ≡ Ut←0 (y0 )
is even globally Lipschitz (thanks to the boundedness which is part of the Lip-definition). The x (.) is itself C 1 and also globally Lipschitz in y as well as its inverse associated Jacobian Jt←0 0 x (.). This is more than enough to see that for V ∈ Lip1 J0←t 0 x J0←t () · V0 (.) ◦ π(V ) (0, , x)t is Lipschitz (in ), uniformly in t in [0, T ]. Obviously, the above expression, as a function of (, t), is also continuous in t. It follows that x z˙ t = J0←t (zt ) · V0 π(V ) (0, zt , xt )t has a unique solution started from z(0) = y0 . An elementary computation shows that y(t) ˜ ≡ π(V ) (0, zt , x)t satisfies dy˜ = V0 (y) ˜ dt + V (y) ˜ dx. This is the Doss–Sussmann method (cf. [20, p.180]) applied in a simple ODE context.
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3.2. Doss–Sussmann method for RDEs We return to the discussion of Section 3.1 and define solutions of RDEs with drift as uniform limits of solutions of ODEs with drift. Definition 3. Let x ∈ C α-Höl ([0, T ], G[1/α] Rd ). If there exists a sequence (x n )n∈N of Lipschitz paths with uniform α-Hölder bounds converging pointwise to x (that is xnt ≡ S[1/α] (x n )t → xt for every t ∈ [0, T ] and supn∈N S[1/α] (x n )α-Höl < ∞) such that for each n ∈ N the RDE (in fact ODE) with drift dy n = V0 y n dt + V y n dxn ,
y n (0) = y0
(3.2)
has a unique solution y n on [0, T ], then we call any limit point in uniform topology of {y n , n ∈ N} a solution of the RDE with drift dy = V0 (y) dt + V (y) dx,
y(0) = y0
and we also write y = π(V ,V0 ) (0, y0 ; (x, t)) for this solution. Proposition 3 (Doss–Sussman for RDE). Assume that (i) x ∈ C0α-Höl ([0, T ], G[1/α] (Rd )), α ∈ (0, 1/2), (ii) V0 ∈ Lip1 (Re ) and V = (V1 , . . . , Vd ) ∈ Lipγ +1 (Re ) for a γ > 1/α, (iii) y0 ∈ Re . Then there exists a unique solution y to the RDE with drift dy = V0 (y) dt + V (y) dx,
y(0) = y0 .
(3.3)
Further, this solution is α-Hölder continuous and given as x y(t) = Ut←0 (zt ),
with z˙ t = W (t, zt ),
(3.4) z(0) = y0 ,
(3.5)
x;. x (.) = π · V0 (π(V ) (0, .; x)t ), Ut←0 where W (t, .) ≡ J0←t (V ) (0, .; x)t is the flow of
dy˜ = V (y) dx,
y(0) ˜ = y0
x,. x . and J0←t = (Dπ(0, .; 0)t )−1 is the inverse of the Jacobian of Ut←0
Proof. We first show that (3.4) has a unique solution. Existence and uniqueness of π(V ) (0, .; x) (and of a C 2 flow) follows from standard rough path theory so this boils down to check that the ODE (3.5) has a unique solution on [0, T ]. However, the vector field W (t, y) is continuous in t and y and W (t, ·) is Lipschitz continuous in · , uniformly in t:
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W (t, y) − W (t, x) = J x;y · V0 π (0, y; x)t − J x;y · V0 π (0, x; x)t (V ) (V ) 0←t 0←t x;y x;x + J0←t · V0 π(V ) (0, x; x)t − J0←t · V0 π(V ) (0, x; x)t x;y x |x − y| + J x;. |x − y||V0 |∞ . J0←t |V0 |Lip U0→t 0←t Lip Lip Thanks to the invariance of the Lipschitz norms |V (. + y)|Lipγ +1 = |V (.)|Lipγ +1 and uniform esx;y
x | , sup |J x timates follow from a routine exercise showing that |Ut←0 Lip y 0←t | and |J0←t |Lip are all bounded by a constant c0 = c0 (α, γ , |V |Lipγ +1 , xα-Höl;[0,T ] ), uniformly in t. The desired Lipschitz regularity of W follows which implies existence of a unique solution z on [0, T ] of (3.5). x (z ) is the unique RDE solution to (3.3) let (x n ) To see that the path yt = Ut←0 t n1 be a sequence of Lipschitz paths with uniform α-Hölder bounds converging pointwise to x. For brevity set xn ≡ S[1/α] (x n ). Following our discussion in Section 3.1 the solutions y n = π(V ,V0 ) (0, y0 ; (xn , t)) are given by solving (3.4) and (3.5) where x is replaced by xn (same reasoning as in the first part of this proof gives existence and uniqueness of solutions zn ). Note that by the universal limit theorem the map (y, x) → π(V ) (0, y, x) is uniformly continuous on bounded sets,7 so if we can show uniform convergence of zn to z the desired conclusion follows. A standard ODE estimate (a simple consequence of Gronwall) is
M n Lt sup ztn − zt e −1 L t∈[0,T ] where M n = sup W n (t, y) − W (t, y) t∈[0,T ] y∈Re
and L =
sup
t∈[0,T ] x=y∈Re
|W (t, x) − W (t, y)| . |x − y| xn ;y
x;y
From the first part of this proof, L < ∞ and to see Mtn → 0 as n → ∞, recall that J0←t → J0←t and V0 (π(V ) (0, y; xn )t ) → V0 (π(V ) (0, y; x)t ) uniformly in y, t as n → ∞ by the universal limit theorem.8 Finally, α-Hölder continuity of the solution follows from the estimate π(V ) (0, zt ; x)t − π(V ) (0, zs ; x)s π(V ) (0, zt ; x)t − π(V ) (0, zt ; x)s + π(V ) (0, zt ; x)s − π(V ) (0, zs ; x)s c1 |t − s|α + |t − s|α c2 |V |Lipγ +1 |zt − zs |e
c2 xaα-Höl |V |α
where c1 = c1 (α, γ , |V |Lipγ +1 xα-Höl;[0,T ] ), c2 = c2 (α, γ ).
Lipγ +1
(3.6)
2
Remark 2. The above proof can be adapted to show uniqueness and (global) existence of RDEs with linear drift term, i.e. 7 (y, x) seen as an element in a product space of two metric spaces, i.e. with metric d((y, x), (y, ˜ x˜ )) = |y − y| ˜ + dα-Höl (x, x˜ ). 8 For the convergence of the Jacobian a localization argument is actually needed.
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dy = Ay dt + V (y) dx,
3249
y(0) = y0 ,
A a (e × e)-matrix, same assumptions as above on V and x. Corollary 2. Let x, V0 and V be as in Proposition 2 and let xn be a sequence of weak geometric α-Hölder paths converging with uniform bounds to x in supremums norm, i.e. supxn α-Höl < ∞, n
d∞ xn , x → 0 as n → ∞.
Denote by y and y n the corresponding solutions of the RDE with drift (3.3). Then supy n α-Höl < ∞, n
sup yt − ytn → 0 as n → ∞,
t∈[0,T ]
and by interpolation for every α < α y − y n
α -Höl
→ 0 as n → ∞.
Proof. The uniform Hölder bounds follow as in (3.6) by noting that the zn are uniformly bounded. For every xn there exists a sequence of Lipschitz paths (x n,m )m with S[1/α] (x n,m ) converging to xn in d∞ with uniform (in m) Hölder bounds and with an associated sequence of RDE solutions y n,m converging to y n . One can choose a subsequence mn such that S[1/α] (x n,mn ) converges to x with uniform Hölder bounds and supt |ytn,mn − ytn | → 0. Hence, supyt − ytn supyt − ytn,mn + supytn,mn − ytn → 0 as n → ∞. t
t
t
2
Remark 3. The auxiliary differential equation for z can be written as x dz = J0←t .W ◦ π(0, zt , x)t dh
(3.7)
where W = V0 and h(t) = t. In fact, one can take W = (W1 , . . . , Wd ), h ∈ C 1-var ([0, T ], Rd ) in which case π(V ) (0, zt , x)t solves dy˜ = V (y) ˜ dx + W (y) ˜ dh. We could make sense of (3.7) as Young-integral equation as long as 1/p + 1/q > 1 and thus obtain RDEs with drift-vector fields driven by h. In this case the pair (x, h) gives rise to a rough path (the cross-integrals of x and h are well-defined Young-integrals); the advantage of the present consideration would be to reduce the regularity assumptions on W .
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3.3. Euler scheme for RDEs with drift We recall the Euler scheme for RDEs, cf. [8]. Definition 4. Let N ∈ N. Given LipN vector fields V = (Vi )i=1,...,d on Re , g ∈ GN (Re ), y ∈ Re . We call N E(V ) (y, g) :=
N
Vi1 · · · Vik I (y)gk;i1 ,...,ik
k=1 i1 ,...,ik ∈{1,...,d}
the step-N Euler scheme (I denotes the identity map). Proposition 4 (Euler-estimate for RDEs with drift). Let N ∈ N, N 1/α. For x ∈ C0α-Höl ([0, T ], GN (Rd )), V0 ∈ Lip1 , V = (Vi )i=1,...,d ∈ Lipγ +1 (Re ), γ > N π(V
0 ,V )
N θ s, ys ; (x, t) s,t − E(V ) (ys , xs,t ) − V0 (ys )|t − s| c|t − s|
where θ 1 + α > 1 and c = c(α, N, |ys |, xα-Höl , |V |LipN , |V0 |Lip1 ). Here ys ∈ Re is a fixed “starting” point. Proof. This is an error estimate for RDEs with drift over the time-interval [s, t]. By shifting time, we may consider w.l.o.g. s = 0, and from [8] we know that π(V ) (0, y0 , x)0,t − E N (y0 , x0,t ) c0 t θ (V ) where c0 = c0 (α, N, y0 , xα-Höl , |V |LipN ), θ = (N + 1)α 1 + α. By the triangle inequality and our definition of “RDE with drift” it then suffices to show that π(V
0 ,V )
0, y0 ; (x, t) 0,t − π(V ) (0, y0 , x)0,t − V0 (y0 )t c1 t θ .
To see this, recall π(V0 ,V ) 0, y0 ; (x, t) 0,t = π(V ) (0, zt , x)t where zt − z0 = zt − y0 = V0 (y0 )t + O(t 2 ). We then have t π(V ) (0, zt , x)t − π(V ) (0, y0 , x)t =
x Jt←0 π
(V ) (0,zs ,x)0,t
dzs
0
t =
t dz +
0
x Jt←0 π
(V ) (0,zs ,x)0,t
0
= V0 (y0 )t + t 1+α and the proof is finished.
2
− I dzs
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4. Applications 4.1. Drift vector fields induced by perturbed driving signals We now show that perturbations of a rough driving signal are picked up by the RDE as a drift term of iterated Lie brackets of the vector fields. Since the RDE solution is a continuous function of the driving signal, we also have continuity under convergence in probability (in suitable Hölder rough path metrics) of random rough driving signals. Combined with the results of Section 2 we arrive at a general criterion for non-standard convergence of stochastic differential equations, more general than [12,13] in the sense that our result applies to perturbations on all levels, exhibiting additional drift terms involving any iterated Lie brackets of the driving vector fields. In particular, the examples given in Section 2 allows us to recover the convergence results of McShane [19] and Sussmann [22]. (In fact, a free benefit of the rough path approach, the respective convergence results will take place at the level of stochastic flows.) Theorem 2. Let x : [0, T ] → G[1/α] (Rd ) be a weak geometric α-Hölder rough path, fix v = (v i1 ,...,iN )i1 ,...,iN =1,...,d ∈ gN (Rd ) ∩ (Rd )⊗N , N [1/α] and set x˜ s,t = exp log SN (x)s,t + vt for s, t ∈ [0, T ]. (This defines a weak geometric min(α, 1/N)-Hölder rough path in GN (Rd ).) Further, assume V0 ∈ Lip1 , V = (Vi )i=1,...,d ∈ Lipγ +1 , γ > max(1/α, N ), vector fields on Re . Then the unique RDE solution of dy = V0 (y) dt + V (y) d˜x,
y(0) = y0
(4.1)
coincides with the unique RDE solution of dz = V0 (z) + W (z) dt + V (z) dx,
z(0) = y0 ,
(4.2)
where
W (z) ≡
i1 ,...,iN ∈{1,...,d}
Vi1 , . . . [ViN−1 , ViN ] . . . z vi1 ,...,iN .
We prepare the proof with Lemma 1. Let k ∈ N. Given a multi-index α = (α1 , . . . , αk ) ∈ {1, . . . , d}k and k, Lipk vector fields V1 , . . . , Vk on Re , define
Vα = Vαk , Vαk−1 , . . . , [Vα2 , Vα1 ] . Further let e1 , . . . , ed denote the canonical basis of Rd . Then gn (Rd ), the step-n free Lie algebra, is generated by elements of form
⊗k , eα = eαk , eαk−1 , . . . , [eα2 , eα1 ] ∈ Rd
kn
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with [ei , ej ] = ei ⊗ ej − ej ⊗ ei and
Vik · · · Vi1 (eα )ik ,...,i1 = Vα .
i1 ,...,ik ∈{1,...,d}
Proof. It is clear that gn (Rd ) is generated by the eα . We prove the second statement by induction: a straightforward calculation shows that it holds for k = 2. Now suppose it holds for k − 1 and denote Vα˜ = [Vαk−1 , . . . , [Vα2 , Vα1 ]]. Then (using summation convention)
i ,...,i Vik . . . Vi1 (eα )ik ,...,i1 = Vik . . . Vi1 eαk ⊗ eαk−1 , . . . , [eα2 , eα1 ] k 1
i ,...,i −Vik . . . Vi1 eαk−1 , . . . , [eα2 , eα1 ] ⊗ eαk k 1
i ,...,i = Vik . . . Vi1 δ αk ,ik ⊗ eαk−1 , . . . , [eα2 , eα1 ] k−1 1
i ,...,i −Vik . . . Vi1 eαk−1 , . . . , [eα2 , eα1 ] k 2 ⊗ δ αk ,i1
i ,...,i = Vαk Vik−1 . . . Vi1 eαk−1 , . . . , [eα2 , eα1 ] k−1 1
i ,...,i −Vik . . . Vi2 eαk−1 , . . . , [eα2 , eα1 ] k 2 Vαk
= Vαk Vα˜ − Vα˜ Vαk = Vαk , Vαk−1 , . . . , [Vα2 , Vα1 ] , where we used the induction hypothesis that
i ,...,i i ,...,i Vik−1 . . . Vi1 eαk−1 , . . . , [eα2 , eα1 ] k−1 1 = Vik . . . Vi2 eαk−1 , . . . , [eα2 , eα1 ] k 2 = Vα˜ .
2
Proof of Theorem 2. By construction W ∈ Lip1 and existence and uniqueness of RDE solutions y, z to (4.2), (4.1) follows from Proposition 3. We have to show that yt = zt for every t ∈ [0, T ]. Therefore fix T˜ ∈ [0, T ], take a dissection D = (ti )i=0,...,|D| of [0, T˜ ] with t0 = 0 and t|D| = T˜ and define zsi = π(V0 +W,V ) ti , yti ; (x, t) s
for s ∈ [ti , T˜ ] ⊂ [0, 1], i = 1, . . . , d.
|D| T
Note that z0˜ = zT˜ and z ˜ = yT˜ , hence T
|D| |D| i 0 z − zi−1 . |zT˜ − yT˜ | = z ˜ − zT˜ ˜ ˜ T T
T
(4.3)
i=1
Using the Lipschitz continuity of RDE flows we get i z − zi−1 T˜ T˜ = π(V0 +W,V ) ti , yti ; (x, t) T˜ − π(V0 +W,V ) ti−1 , yti−1 ; (x, t) T˜ = π(V0 +W,V ) ti , yti ; (x, t) T˜ − π(V0 +W,V ) ti , π(V0 +W,V ) ti−1 , yti−1 ; (x, t) t ; (x, t) T˜ i (x,t) y (4.4) U˜ − π(V0 +W,V ) ti−1 , yti−1 ; (x, t) t ,t . Lip ti−1 ,ti T ←ti
i−1 i
P. Friz, H. Oberhauser / Journal of Functional Analysis 256 (2009) 3236–3256 [1/α ∗ ]
For brevity set α ∗ = min(α, 1/N). By adding/substracting E(V ) V0 (yti−1 )|ti − ti−1 | and splitting up we estimate
|zi˜ T
− zi−1 | (1) + (2) T˜
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(yti−1 , x˜ ti−1 ,ti ) +
with
[1/α ∗ ] (1) = π(V0 ,V ) ti−1 , yti−1 ; (˜x, t) t ,t − E(V ) (yti−1 , x˜ ti−1 ,ti ) − V0 (yti−1 )|ti − ti−1 |, i−1 i [1/α ∗ ] (2) = E(V ) (yti−1 , x˜ ti−1 ,ti ) + V0 (yti−1 )|ti − ti−1 | − π(V0 +W,V ) ti−1 , yti−1 ; (x, t) t ,t . i−1 i
From Proposition 4, (1) c1 |t − s|θ , θ > 1, and by Lemma 1, [1/α ∗ ]
E(V )
[1/α ∗ ]
(yti−1 ; x˜ ti−1 ,ti ) = E(V )
(yti−1 ; xti−1 ,ti ) +
Vi1 · · · ViN I (yti−1 )v i1 ,...,iN |ti − ti−1 |
i1 ,...,iN [1/α ∗ ] = E(V ) (yti−1 ; xti−1 ,ti ) + |ti
− ti−1 |W (yti−1 ).
Again Proposition 4 applies and (2) c2 |t − s|θ . Plugging all this into (4.3) gives |zT˜ − yT˜ | c3
|D|
|ti − ti−1 |θ
i=1
with c3 = c3 (α, N, ˜xα ∗ -Höl , xα-Höl , |V |Lip[1/α∗ ] , |V0 |Lip1 , |W |Lip1 ). Since θ > 1 the sum on the r.h.s. goes to 0 as |D| → 0 and this finishes the proof. 2 4.2. Optimality of RDE estimates At last, we use Theorem 2 to establish optimality of two important RDE estimates (proved in [17] and [11]) for the case of paths with p-variation, p ∈ N. The second part of the following theorem settles a question that was left open in [2]. Theorem 3. Let x : [0, T ] → Gp (Rd ) be a geometric p-rough path, y0 ∈ Re , p ∈ N. If either (1) V = (Vi )di=1 ∈ Lipγ (Re ), γ > p, or (2) V = (Vi )di=1 linear vector fields on Rd , i.e. Vi (y) = Ai · y with Ai a (e × e)-matrix, then there exists a unique RDE solution to dy = V (y) dx,
y(0) = y0 .
Further, in case (1), p |y|∞ C max xp-var;[s,t] , xp-var;[s,t]
(4.5)
with C = C(y0 , p, |V |Lipγ +1 ) and in case (2) p |y|∞ c exp cxp-var
(4.6)
with c = c(y0 , p, (|Vi |)di=1 ) hold and both estimates are optimal (i.e. the bounds are attained).
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We prepare the proof with two lemmas. Lemma 2. For all k ∈ N there exist k smooth, bounded vector fields V1 , . . . , Vk on Re such that
∂ . Vk , . . . , V3 , [V2 , V1 ] . . . = ∂xe k−1 brackets
Proof. Set V = sin xe
∂ , ∂xe
W = − cos xe
∂ , ∂xe
E=
∂ . ∂xe
Note that [V , W ] = E and [V , E] = W . Hence, for k odd: for k even:
V , . . . , V , [V , W ] . . . = E,
V , . . . , V , [V , E] . . . = E.
2
Lemma 3. For all k ∈ N there exist (e × e)-matrices A1 , . . . , Ak (e 2) such that
Ak , . . . , A3 , [A2 , A1 ] . . . i,j =1,...,e = (−δ(i,j )=(1,1) + δ(i,j )=(e,e) )i,j =1,...,e k−1 brackets
where [. , .] denotes the usual matrix commutator. Proof. Let M = (δ(i,j )=(1,e) )i,j =1,...,e , A = (δ(i,j )=(e,1) )i,j =1,...,e ,
N = (−δ(i,j )=(1,1) + δ(i,j )=(e,e) )i,j =1,...,e , 1 δ(i,j )=(1,e) B= . 2 i,j =1,...,e
Note that [A, M] = N and [B, N] = M. Hence for k odd: for k even:
A, B, A, . . . , A, B, [A, M] . . . = N,
A, B, A, . . . , A, [B, N] . . . = N. 2
Proof of Theorem 3. Existence of a unique RDE solution follows from [17]. For (4.6) use [17, p Theorem 2.4.1] with control function ω(s, t) = xp-var;[s,t] to get |yt | K|y0 |xp-var
n0
where K = maxi (|Ai |). However,
a n/p n0 (n/p)!
Kn
xnp-var (n/p)!
,
c0 (p)ec0 (p)a for a 0 and therefore
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p |y|∞,[0,T ] |y0 |c1 exp c1 xp-var . Estimate (4.5) is proved in [11]. To see optimality of both (4.5) and (4.6) define a geometric 1/p-Hölder rough path x˜ on [0, 1] in Gp (Rp ) by x˜ t = exp(λte1,...,p ) where e1,...,p = [ep , . . . , [e3 , [e2 , e1 ]] . . .] ∈ V p (Rp ) for some λ > 0. Note that homogeneity of the p-variation norm implies ˜xp-var = c2 λ1/p .
(4.7)
The corresponding RDE solution dy = V (y) d˜x,
y(0) = y0 .
then coincides with the ODE solution dz = W (z) dt,
z(0) = y0
(4.8)
where W is given by Theorem 2. Case (1): Let V = (Vi )i=1,...,p be the vector fields on Rp from Lemma 2 with k = p. Then the solution (4.8) is easy to write down, yt = (0, . . . , 0, λt)T . p
p
Clearly, |y|∞ = 1/c2 ˜xp-var . To see that the regime of (4.5) where xp-var dominates can be obtained is straightforward by looking at dy = (1, . . . , 1)T dx for any rough path x. Case (2): Let V = (Vi )i=1,...,p be the linear vector fields on Rp given by the matrices (Ai )i=1,...,p of Lemma 3, i.e. Vi (y) = Ai · y. Using Lemma 3 with k = p brackets,9
W (z) = λ Vp , . . . , V3 , [V2 , V1 ] . . . I (z)
= λ Ap , . . . , A3 , [A2 , A1 ] . . . M · z = λ(−δ(i,j )=(1,1) + δ(i,j )=(p,p) )i,j =1,...,e · z. Now choosing y0 = (0, . . . , 0, 1)T , a unique solution to (4.8) is T yt = 0, . . . , 0, eλt . 9 Using the usual identification of linear maps with matrices it is easy check that we have a Lie algebra isomorphism (gl(n, R), [., .]M ) ∼ = (gl(Rn ), [., .]).
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Again by (4.7) p
sup |yt | = eλ = ec2 xp-var t∈[0,1]
and the upper bound of estimate (4.6) is attained.
2
References [1] R.F. Bass, B. Hambly, T. Lyons, Extending the Wong–Zakai theorem to reversible Markov processes, J. Eur. Math. Soc. 4 (3) (2002) 237–269. [2] T. Cass, P. Friz, N. Victoir, Non-degeneracy of Wiener functionals arising from rough differential equations, Trans. Amer. Math. Soc. 361 (2009) 3359–3371. [3] L. Coutin, A. Lejay, Semi-martingales and rough paths theory, Electron. J. Probab. 10 (23) (2005) 761–785 (electronic). [4] L. Coutin, Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields 122 (1) (2002) 108–140. [5] P. Friz, N. Victoir, Approximations of the Brownian rough path with applications to stochastic analysis, Ann. Inst. H. Poincaré Probab. Statist. 41 (4) (2005) 703–724. [6] P. Friz, N. Victoir, A note on the notion of geometric rough paths, Probab. Theory Related Fields 136 (3) (2006) 395–416. [7] P. Friz, N. Victoir, The Burkholder–Davis–Gundy inequality for enhanced martingales, in: C. Donati-Martin, M. Émery, A. Rouault, C. Stricker (Eds.), Séminaire de Probabilités XLI, in: Lecture Notes in Math., Springer, 2008. [8] P. Friz, N. Victoir, Euler estimates for rough differential equations, J. Differential Equations 244 (2) (2008) 388–412. [9] P. Friz, N. Victoir, On uniformly subelliptic operators and stochastic area, Probab. Theory Related Fields 142 (3–4) (2008) 475–523. [10] P. Friz, N. Victoir, Differential equations driven by Gaussian signals I. Ann. Inst. H. Poincaré Probab. Statist. (2009), in press; preprint arXiv:0707.0313v1 [math.PR]. [11] P. Friz, N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications, Cambridge University Press, 2009, forthcoming. [12] I. Gyöngy, G. Michaletzky, On Wong–Zakai approximations with δ-martingales, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2041) (2004) 309–324, stochastic analysis with applications to mathematical finance. [13] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, second edition, North-Holland, Amsterdam, 1989. [14] A. Lejay, Stochastic differential equations driven by a process generated by divergence form operators, ESAIM Probab. Stat. 10 (2006) 356–379. [15] A. Lejay, T. Lyons, On the importance of the Lévy area for studying the limits of functions of converging stochastic processes. Application to homogenization, Current Trends in Potential Theory 7 (2003). [16] A. Lejay, N. Victoir, On (p, q)-rough paths, J. Differential Equations 225 (1) (2006) 103–133. [17] T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (2) (1998) 215–310. [18] T.J. Lyons, M. Caruana, T. Lévy, Differential Equations Driven by Rough Paths, Lecture Notes in Math., vol. 1908, Springer, Berlin, 2007, Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, with an introduction concerning the Summer School by Jean Picard. [19] E.J. McShane, Stochastic differential equations and models of random processes, in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Univ. California, Berkeley, Calif., 1970/1971, in: Probab. Theory, vol. III, Univ. of California Press, Berkeley, 1972, pp. 263–294. [20] L.C.G. Rogers, D. Williams, Diffusions, Markov Processes, and Martingales, vol. 1, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2000, Foundations, reprint of the second edition, 1994. [21] D.W. Stroock, Diffusion semigroups corresponding to uniformly elliptic divergence form operators, in: Séminaire de Probabilités, XXII, in: Lecture Notes in Math., vol. 1321, Springer, Berlin, 1988, pp. 316–347. [22] H.J. Sussmann, Limits of the Wong–Zakai type with a modified drift term, in: Stochastic Analysis, Academic Press, Boston, MA, 1991, pp. 475–493.
Journal of Functional Analysis 256 (2009) 3257–3278 www.elsevier.com/locate/jfa
Interpolation in Bernstein and Paley–Wiener spaces Alexander Olevskii a,1 , Alexander Ulanovskii b,∗ a School of Mathematics, Tel Aviv University, Ramat Aviv, 69978 Israel b Stavanger University, 4036 Stavanger, Norway
Received 21 August 2008; accepted 22 September 2008 Available online 9 October 2008 Communicated by Paul Malliavin
Abstract We consider interpolation of discrete functions by continuous ones with restriction on the size of spectra. We discuss a sharp contrast between the cases of compact and unbounded spectra. In particular we construct ‘universal’ spectra of small measure which deliver positive solution of the interpolation problem in Bernstein spaces for every discrete sequence of knots. © 2008 Elsevier Inc. All rights reserved. Keywords: Paley–Wiener space; Bernstein space; Set of interpolation; Uniform minimality of exponential systems
1. Introduction 1.1. Spaces Throughout this paper Fˆ will denote the Fourier transform of a function F : 1 Fˆ (x) := √ 2π
eitx F (t) dt. R
The same notation will be used for the Fourier transform of a Schwartz distribution F . * Corresponding author. Fax: +47 51831890.
E-mail addresses:
[email protected] (A. Olevskii),
[email protected] (A. Ulanovskii). 1 Partially supported by the Israel Science Foundation.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.09.013
(1)
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Let S be a closed set in R. We say that a function f defined on (another copy of) R belongs to the Bernstein space BS if (i) f is continuous and bounded on R; (ii) f is the Fourier transform of a Schwartz distribution F supported by S. The latter means that
f (x)ϕ(x) ˆ dx = 0, R
for every smooth function ϕ, supported by an interval disjoint from S. The support of F is called the spectrum of f . Endowed with the sup-norm f ∞ := sup f (x), x∈R
BS is a Banach space. If S is a compact, then the elements of BS are entire functions of finite exponential type. We will focus on the situation when S is a closed (not necessary compact) set of finite Lebesgue measure. In this case the elements of BS do not in general admit analytic continuation into the complex plane. We shall also consider the Paley–Wiener spaces PW S := f ∈ L2 (R); f = Fˆ , F = 0 on R \ S , endowed with the L2 -norm · 2 . These spaces can be defined for any measurable S, but again we will be interested in the case when S is a closed set of finite measure. Clearly, there is an embedding with the inequality of norms: mes S 1/2 f 2 . (2) if f ∈ PW S then f ∈ BS and f ∞ 2π 1.2. Interpolation Let Λ = {λj , j ∈ Z} ⊂ R be a uniformly discrete (u.d.) set, that is inf |λj − λk | > 0.
j =k
Definition 1. Λ is called an interpolation set for BS if for every sequence c = {cj , j ∈ Z} ∈ l ∞ (Z), there is a function f ∈ BS such that f (λj ) = cj ,
j ∈ Z.
(3)
The closed graph theorem implies for an interpolation set Λ, that there exists a solution of the interpolation problem (3) satisfying f ∞ C max |cj |, j ∈Z
where the constant C does not depend on c.
(4)
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Definition 2. Λ is called a set of interpolation for PW S if for every c ∈ l 2 (Z) there is a function f ∈ PW S satisfying (3). Again, the solution can be chosen with the additional requirement f 22 C
|cj |2 ,
(5)
j ∈Z
see [11, p. 129] (although for unbounded S the operator f → f |Λ may not act from PW S into l 2 (Z)). 1.3. Density Definition 3. The upper uniform density of a u.d. set Λ is defined as follows: D + (Λ) := lim max l→∞ a∈R
#(Λ ∩ (a, a + l)) . l
A fundamental role of this quantity in the interpolation problem, in the case when S is a single interval, was found by A. Beurling and J.-P. Kahane. Kahane proved [4] that the inequality D + (Λ) <
1 mes S 2π
(6)
D + (Λ)
1 mes S 2π
(7)
is sufficient, while the inequality
is necessary for a u.d. set Λ to be an interpolation set for PW S , where S is an interval. Later, Beurling proved [1] that Λ is an interpolation set for BS if and only if (6) holds. These results are based on the theory of entire functions. 1.4. Disconnected compact spectra The situation becomes much more delicate for disconnected spectra, in particular when S is a union of two intervals. For the sufficiency part, not only the size but also the arithmetic structure of Λ is important. On the other hand, using a new approach Landau [6] succeeded to extend the necessity part to the general case: Theorem A. (See [6].) Let a u.d. set Λ be an interpolation set for PW S , where S is a bounded measurable set. Then (7) holds true. We show that for compact sets S the possibility to interpolate the δ-functions on Λ already implies estimate (7):
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Theorem 1. Let S be a compact set and Λ = {λj , j ∈ Z} be a u.d. set. Suppose that for every j ∈ Z there is a function fj ∈ BS such that fj (λk ) =
1, 0,
k = j, k = j,
k∈Z
(8)
and sup fj ∞ < ∞.
j ∈Z
(9)
Then (7) is true. Due to embedding (2), the same result takes place for PW S . This result admits a geometric interpretation: suppose Λ is such that the exponential system E(Λ) := eiλt λ∈Λ is uniformly minimal in L2 (S). Then (7) holds. 1.5. Unbounded spectra The interpolation results for unbounded spectra are very different from the ones for bounded spectra. The contrast is most striking for Bernstein spaces: not only condition (7) is no longer necessary, but there exist ‘universal’ spectra of arbitrary small measure which deliver positive solution to the interpolation problem for every u.d. Λ: Theorem 2. For every δ > 0 there is a closed set S, mes S < δ, such that every u.d. set Λ is an interpolation set for BS . For the PW S -spaces, the result in such a strong form does not hold. Indeed, no interpolation of a δ-function on Z is possible by a function f ∈ PW S with mes S < 2π , see Proposition 4.1 below. However, for ‘generic’ Λ, we prove that such interpolation is possible, and moreover it can be extended to functions on Λ with a certain decay: Theorem 3. Let a u.d. set Λ be linearly independent over rational numbers (mod π). Then for every δ > 0 there is a set S (a union of some neighborhoods of integers), such that: (i) mes S < δ, (ii) for every sequence c = {cj }, there exists f ∈ PW S satisfying (3).
cj = O |j |−1− , > 0,
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Some results of this paper concerning P W -spaces were preliminary published in our note [8]. We are grateful to the Mathematical Institute in Oberwolfach where this work was started during our 2-weeks stay in 2007. The paper is organized as follows. In Section 2 the compact spectra are considered (including proof of Theorem 1). Theorem 2 is proved in Section 3. Interpolation in P W -spaces with unbounded spectra (including proof of Theorem 3) is discussed in Section 4. Some open problems are presented in Section 5. 2. Compact spectra 2.1. Concentration Definition 4. Given a number c, 0 < c < 1, we say that a linear subspace X of L2 (R) is cconcentrated on a set Q ⊂ R if
f (x)2 dx cf 2 , 2
f ∈ X.
Q
Lemma 2.1. Given sets S, Q ⊂ R and a number 0 < c < 1, let X be a linear subspace of PW S which is c-concentrated on Q. Then dim X
(mes Q)(mes S) . 2πc
This lemma is due to H. Landau [6] (compare (iii) and (iv) on p. 41). For the reader’s convenience, we present the proof. Proof. Let Q, S ⊂ R be two sets of finite positive measure. Let A denote the orthogonal projection of L2 (R) onto the subspace of functions vanishing outside Q, and B denote the orthogonal projection of L2 (R) onto the subspace PW S of functions whose inverse Fourier transform vanishes outside S. Clearly, the operator C := ABA is self-adjoint and positive. It can be written explicitly as 1 Cf (y) = √ 2π
1Q (x)1Q (y)1ˆ S (y − x)f (x) dx,
R
where 1E denotes the characteristic function of E. Since the kernel is square-integrable, C is a compact operator. Denote by lj its eigenvalues arranged in non-increasing order (counting multiplicities). The trace Tr C is equal to the integral of kernel along the ‘diagonal’: Tr C :=
j
lj =
1 (mes Q)(mes S). 2π
(10)
One can easily show that the spectrum of operator D := BAB is identical to the spectrum of operator C.
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Let X be a linear subspace of PW S of dimension k. The quadratic form (Df, f ) on X is given by (Af, f ) = 1Q f 22 . If X is c-concentrated on Q, then inf
f ∈X, f 2 =1
(Df, f ) c.
Due to the minimax principle, among all subspaces of dimension k, the greatest value of inf(Df, f ) on the unit sphere is achieved on a subspace spanned by the first k eigenvectors of D. This means that c lk Tr D/k = Tr C/k. This and (10) prove the lemma. 2 2.2. Interpolation of delta-functions. Proof of Theorem 1 1. Recall that if S is a compact set, then the corresponding Paley–Wiener space PW S consists of entire functions of exponential type. The following lemma is well known (see [11, p. 82]). Lemma 2.2. Suppose Λ = {λj , j ∈ Z} is a u. d. set, and S ⊂ R is a compact set. Then there exists C > 0 such that f 22 C
f (λj )2 ,
for every f ∈ PW S .
j ∈Z
2. We now pass to the proof of Theorem 1. Fix a number δ > 0, and set S(δ) := S + [−δ, δ].
(11)
Take functions fj ∈ BS satisfying (8) and (9), and set sin δ(x − λj ) 2 , gj (x) := fj (x) δ(x − λj )
j ∈ Z.
Clearly, functions gj satisfy (8), (9) and belong to PW S(2δ) . 3. Fix any number R, and denote by #(Λ ∩ (R − r, R + r)) the number of λj ∈ Λ in the interval (R − r, R + r). Since Λ is u.d. we have:
# Λ ∩ (R − r, R + r) Cr,
for all r > 1,
where C > 0 is a constant which does not depend on R. Let us introduce a linear space of functions Wr := g(x) =
k: |λk −R|
Clearly, dim Wr = #(Λ ∩ (R − r, R + r)).
ck gk (x), ck ∈ C .
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4. We shall estimate
the concentration of Wr on the interval (R − r − rδ, R + r + rδ). Choose any function g = j cj gj ∈ Wr . Then g(λj ) = cj when |λj − R| < r, and g(λj ) = 0 when |λj − R| r. By Lemma 2.2, there is a constant C = C(S, Λ) > 0 such that g22 C
g(λj )2 = C λj ∈Λ
|cj |2 .
(13)
|λj −R|
Observe that |x − λj | δr whenever λj ∈ (R − r, R + r) and |x − R| r + δr. Hence, by (9) and (12) we obtain:
g(x)2 dx =
|x−R|r+δr
|x−R|r+δr |λj −R|
Cr
sin δ(x − λj ) 2 2 cj fj (x) dx δ(x − λj )
|cj |2
|λj −R|
|x|>δr
1 δx
4 dx
C δ7 r 2
|cj |2 .
|λj −R|
This and (13) show that for every > 0 there exists r such that g is (1 − )-concentrated on (R − r − δr, R + r + δr) for all r r . 5. It now follows from Lemma 2.1, that
(mes S(2δ))(mes(R − r − δr, R + r + δr)) # Λ ∩ (R − r, R + r) , 2π(1 − ) whenever r r . Since this inequality holds for every R, we obtain: D + (Λ)
1 + δ mes S(2δ) . 1− 2π
Letting δ → 0 and → 0, we conclude that D + (Λ) (2π)−1 mes S. Corollary 2.1. Let S be a compact set. If a uniformly discrete set Λ is an interpolation set for BS , then (7) is true. This is a version of Theorem A for Bernstein spaces. Corollary 2.1 easily follows from Theorem 1. Let Λ be a set of interpolation for BS . Then each interpolation problem (3) can be solved by a function satisfying (4). Hence, there exist functions satisfying (8) and (9), so that the assumptions of Theorem 1 are fullfiled. Due to embedding (2), a similar to Theorem 1 result holds also for PW-spaces: Theorem 2.1. Let S be a compact set and Λ = {λj , j ∈ Z} be a u.d. set. Suppose that for every j ∈ Z there is a function fj ∈ PW S satisfying (8) such that supfj 2 < ∞.
j ∈Z
Then (7) is true.
(14)
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2.3. Uniform minimality of exponentials in L2 (S) Let {ej } be a system of distinct vectors in a Hilbert space H . Set ej r(j ) := dist , span{ek }k=j . ej
Definition 5. A system {ej } is called minimal, if r(j ) > 0 for all j . It is called uniformly minimal, if infj r(j ) > 0. The assumptions of the last theorem admit a geometric interpretation: set E(Λ) := eiλt , λ ∈ Λ . Using Hahn–Banach theorem, one can easily check that conditions (8) and (14) are equivalent to the property of uniform minimality of E(Λ) in L2 (S). Hence, Theorem 2.1 can be reformulated as follows: Theorem 2.2. Let E(Λ) be uniformly minimal in L2 (S), for some compact set S. Then (7) holds. Let us compare two conditions: (i) E(Λ) is uniformly minimal in L2 (S); (ii) L is an interpolation set for PW S . The latter condition is equivalent to the following one (see [11, p. 129]): (iii) There is a constant A > 0 such that the inequality 2 iλj t cj e dt A |cj |2 , S
j
(15)
j
holds for every finite sequence {cj }. Clearly, condition (iii) implies condition (i). A well-known example shows that (iii) is in fact stronger than (i): Example. Set S = [−π, π] and Λ = {λj , j ∈ Z}, where ⎧ ⎨ j + 14 , λj := 0, ⎩ j − 14 ,
j = 1, 2, . . . , j = 0, j = −1, −2, . . . .
The corresponding exponential system E(Λ) is uniformly minimal in L2 (−π, π) (see [10, Theorem 5]). However, Λ is not an interpolation set for L2 (−π, π) (see the remark following Theorem 5 in [10]).
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2.4. Growth of fj 2 and upper density Theorem 2.2 is sharp in the sense that its statement ceases to be true if we allow the r(j ) tend (no matter how slowly) to zero: Theorem 2.3. Given an arbitrarily small number a > 0 and an arbitrarily slowly decreasing sequence 0 < δn < 1, δn → 0, n → ∞, there exists Λ ⊂ R, D + (Λ) 1, such that E(Λ) is minimal in L2 (−a, a) and r(j ) δ|j | , j ∈ Z. Proof. We may assume that a = π/N , where N ∈ N. Clearly, the inequalities r(j ) δ|j | , j ∈ Z, hold if and only if there exists a biorthogonal to {exp(iλj t), j ∈ Z} system in L2 (−π/N, π/N ), such that the norm of j th element is not greater than 1 = δ|j | exp(iλj t)L2 (−π/N,π/N )
N 1 . 2π δ|j |
Hence, applying the Fourier transform (1), to prove the theorem it suffices to find a set Λ = {λj } ⊂ Z, D + (Λ) = 1, and a function f, f |Λ = 0, such that the function gλ (x) := f (x)/f (λ)(x − λ) belongs to PW (−π/N,π/N ) for every λ ∈ Λ and √ N gλj 2 , δ|j |
j ∈ Z.
(16)
Take a rapidly increasing sequence of integers qn ∈ 2N N, and set Qn (z) :=
n z−j , z − Nj
j =1
f (z) := sin
∞ πz Qn (z − qn ). N n=1
Since |Qn (z)| → 1 as |z| → ∞, z ∈ C, we may assume that the qn grow so fast that f is an entire function of exponential type π/N satisfying f (x)/x ∈ L2 (R). Then, by Paley–Wiener theorem, we have f (z)/z ∈ PW (−π/N,π/N ) . Moreover, it is easy to check that f has simple zeros, which we denote by Λ = {· · · < λ−1 < λ0 = 0 < λ1 < · · ·} ⊂ Z, and that D + (Λ) = 1. Fix a constant > 0 and set: qn E := x ∈ R: |x − qn | > , n = 1, 2, . . . . 2 By choosing qn growing fast enough, we may assume that ∞ qn for each n, then 1 − if |x − qn | > Qn (x − qn ) < , 4 n=1
and qj if |x − qj | < Qn (x − qn ) < . for some j, then 1 − 2 n=j
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We need to estimate the norms of gλ . First, let us estimate the derivatives f (λ), λ ∈ Λ. When λ ∈ E, we have: ∞ π π f (λ) = (17) Qn (λ − qn ) > (1 − ). N N n=1
Otherwise, |λ − qk | < qk /2, for some k. In this case we have πx Qk (x − qk ) Qn (λ − qn ). f (λ) = sin N x=λ
n=k
This implies easily that f (λ) > ck > 0,
(18)
where ck depends on k (and N ) only. Now let us estimate f (x)/(x − λ)2 . Set
E :=
∞
3qj 5qj , . Ij := 4 4
Ij ,
j =1
Again, consider two cases. When λ ∈ E, then |x − λ| > qj /4 for x ∈ Ij . Keeping in mind that f (x) < (1 + )sin πx Qj (x − qj ) < C(j ), N
x ∈ Ij ,
we obtain: 2 f (x) 2 dx < 4C(j ) , x − λ qj
j = 1, 2, . . . .
(19)
Ij
On the other hand, 2 f (x) 2 sin πx π2 N 2 dx < (1 + )2 . (1 + ) = x − λ x −λ N 2
(20)
R\E
Second case: |λ − qk | < qk /2, for some k. Then the estimates above also work with one exception: when j = k, instead of (19) we use the following inequality, which is easy to check: f (x) 2 dx < (1 + ) x − λ
Ik
R
2 sin πx N Qk (x − qk ) dx < C (k). x−λ
(21)
√ It follows from (17)–(21) that the norms of gλ can be made arbitrarily close to N when λ ∈ Λ ∩ E, and also these norms can be estimated from above by constants depending only on k
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when λ ∈ Λ, |λ − qk | < qk /2, k = 1, 2, . . . . Hence, condition (16) holds provided the qn grow sufficiently fast. This finishes the proof of Theorem 2.3. 2 However, if the density D + is replaced with the upper density D ∗ , a result similar to Theorem 2.2 holds, at least when the r(j ) do not decay very fast. One says that a positive sequence r(j ) has a ‘non-quasianalytic’ decay if there exists a positive decreasing sequence δj such that r(j ) δ|j | , j = 1, 2, . . . , and we have ∞ log δj
j2
j =1
> −∞.
(22)
In particular, the sequence δj = exp(−|j |α ) has a non-quasianalytic decay if and only if 0 α < 1. We also say that a sequence Rj > 1, j ∈ Z, has a non-quasianalytic growth, if the 1/Rj have a non-quasianalytic decay. Definition 6. Let Λ be a u.d. set. The upper density of Λ is defined as follows: D ∗ (Λ) := lim sup a→∞
#(Λ ∩ (−a, a)) . 2a
Clearly, we have D ∗ (Λ) D + (Λ). Observe that for regularly distributed Λ, for example if Λ = {j + O(1), j ∈ Z}, these two densities are equal. Theorem 2.4. Assume that E(Λ) is minimal in L2 (S) for some compact set S ⊂ R, and that the r(j ) have a non-quasianalytic decay. Then D ∗ (Λ) (2π)−1 mes S. The proof is very similar to the proof of Theorem 1, and so we shall only sketch it briefly. It follows from the assumptions that there exists a biorthogonal to E(Λ) in L2 (S) system whose norms have a non-quasianalytic growth. Hence, applying the Fourier transform, we see that there exist functions fj ∈ PW S satisfying (8) whose norms have a non-quasianalytic growth. Fix a small positive number b. It is well known (see [5]) that assumption (22) implies that for every positive δ there exists ψ ∈ PW (−δ,δ) satisfying |ψ(x)|/δj < 1 for all |x| > bj. Since Λ is uniformly discrete, we may assume that ψ(x) max fj 2 < 1, |x| > br. j : |λj |r
Set ϕ(x) := ψ(x)(sin δx/x)2 . Then ϕ ∈ PW (−3δ,3δ) , and we have ϕ2L2 (|x|br) r max fj 2 → 0, j : |λj |r
r → ∞.
(23)
We may also assume that ϕ(0) = 1. Set gj (x) := fj (x)ϕ(x − λj ) ∈ PW S(3δ) , where S(δ) := S + (−δ, δ). Consider the space of functions cj gj (x) . Wr := g(x) = j : |λj |r
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We wish to show that for every > 0 there exists r > 0 such that Wr is (1 − )-concentrated on (−r(1 + b), r(1 + b)), whenever r r . Firstly, similarly to (13), one can show that the norm of g is ‘large’ compared to the norm of {cj }. On the other hand, by (23), similarly to the second inequality in the step 4 of proof of Theorem 1, one shows that the norm of g outside (−r − br, r + br) is ‘small’ compared to the norm of {cj }. Hence, Lemma 2.1 gives: mes S(3δ) #(Λ ∩ (−r, r)) , mes(−r − rb, r + rb) 2π(1 − )
r r .
Taking the upper limit as r → ∞, and then letting , b, δ → 0, one proves the theorem. 3. Bernstein spaces with unbounded spectra Here we prove Theorem 2 (in a slightly stronger form): Theorem 3.1. For every δ > 0 there is a closed (unbounded) set S, mes S < δ, such that every set Λ which has no finite limit points is an interpolation set for the space BS . In Theorem 3.1 the set S can be chosen of measure zero. We will discuss this improvement in a separate paper. Compared with the statement of Theorem 2, the assumption of uniform discreteness of Λ is relaxed. 3.1. Lemmas Lemma 3.1. For every N 2 there exists a set S(N) ⊂ (−N, N ), mes S(N ) = N sin N x C itx , e dt − 2 Nx N
2 N,
such that
x ∈ R,
(24)
S(N )
where C > 0 is an absolute constant independent on N . Proof. 1. Fix an integer N 2, and let Mj , j = 1, . . . , N, be any even numbers satisfying M1 N 4 ,
Mj +1 N 2 Mj ,
j = 1, . . . , N − 1.
Set Ω(j, k) := j − 1 + S(j, k) := j − 1 +
2k − 1 1 + (−1, 1], Mj Mj
2k − 1 1 + 2 (−1, 1], Mj N Mj
Ω(−j, k) := −Ω(j, k), S(−j, k) := −S(j, k),
(25)
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where j = 1, . . . , N and k = 1, . . . , Mj /2. Set S(N) :=
j /2 N M
S(j, k).
|j |=1 k=1
One can check that Mj /2
Ω(j, k) = (j − 1, j ],
j = 1, . . . , N,
k=1
and that mes S(N) =
2 . N
2. For simplicity, throughout the proof we denote by C different positive constants. Since sin N x 1 = Nx 2N
N eitx dt, −N
to prove (24) it suffices to show that N eitx dt − N 2 eitx dt < C, −N
x ∈ R.
(26)
S(N )
3. Assume first that |x| Ml , for some 1 l N. Using (25), we have for j l, j N , that itx 2 itx e dt − N e dt Ω(j,k)
S(j,k)
sin(x/Mj ) N 2 sin(x/N 2 Mj ) Cx 2 CMl2 = 2 − M3 M3 , x x j j
k = 1, . . . ,
Mj . 2
This and (25) give:
eitx dt − N 2
{l−1|t|N }
{l−1|t|N }∩S(N )
j /2 N M CM 2
l
|j |=l k=1
Clearly, this proves (26) for |x| M1 .
eitx dt
Mj3
C.
(27)
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4. Assume now that |x| > Ml−1 , for some 2 l N. Then, clearly, we have
{|t|l−1}
sin(l − 1)x 2 . eitx dt = 2 M x l−1
(28)
Also, 2 N
e
itx
dt
{l−2|t|l−1}∩S(N )
N 2 mes l − 2 |t| l − 1 ∩ S = 2.
(29)
These estimates and (27) prove (26) for M1 < |x| M2 . Assume that |x| > Ml−1 , for some l 3. Since 2 sin x/(N 2 Mj ) 2N , eitx dt = 2N 2 x Ml−1
2 N
|x| > Ml−1 ,
S(j,k)
by (25), we obtain: 2 N
e
itx
l−2 Mj /2 2N 2 2N 2 Ml−2 dt C. Ml−1 Ml−1 j =1 k=1
{|t|l−2}∩S(N )
From this, (28) and (29) we get
e
itx
{|t|l−1}
dt − N
2
e
itx
dt C,
|x| > Ml−1 , l 3.
{|t|l−1}∩S
Now, this and (27) imply (26) for |x| > Ml−1 , l 3, which completes the proof of Lemma 3.1. 2 Lemma 3.2. For every > 0 there is a compact S = S and a function g = g ∈ BS such that: (i) mes S ; (ii) g∞ = g(0) = 1; (iii) |g(x)| < for |x| > . In addition, S can be chosen disjoint from any given segment. This follows from Lemma 3.1. Indeed, fix an > 0, and choose N in Lemma 3.1 so large that mes S(N ) < , |sin N x/N x| < /2 when |x| > and C/N < /2, where C is the con-
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stant in (24). Given a segment I , set S := R + S(N ), where R is any number such that R + S(N ) ∩ I = ∅. Set N N eitx dt = eiRx eitx dt. g(x) := 2 2 S
S(N )
Conditions (ii) and (iii) follow immediately from Lemma 3.1. 3.2. Proof of Theorem 3.1 Suppose 0 < δ < 1, and take any sequence (j ) > 0, j ∈ Z, such that (j ) < δ.
(30)
j ∈Z
Fix a sequence of disjoint compacts S(j ) , mes S(j ) < (j ), tending to infinity and satisfying the conditions of Lemma 3.2, such that the set S := S(j ) j ∈Z
is closed. Let Λ = {λj , j ∈ Z} be a set without finite limit points. Taking if necessary a subsequence of (j ), we may assume that dj := inf |λk − λj | > (j ), k:k=j
j ∈ Z.
(31)
Conditions (ii) and (iii) of Lemma 3.2 allow one to define a sequence of functions fj ∈ BS such that each function fj has a compact support, 1 = fj ∞ = fj (lj ), and |fj (x)| < (j ),
for |x − lj | dj .
(32)
By (31), we see that fj (λk ) < (j ),
j = k.
Consider a linear operator A : l ∞ → l ∞ defined by the following matrix
j = k, fj (λk ), A := a(j, k) j,k∈Z , a(j, k) = fj (λj ) − 1, j = k. From (30) and (33) one can see that a(j, k) = sup fj (λk ) < 1, A = sup k
j
k j =k
(33)
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so that the operator A + I , I is the identity operator, is invertible in l ∞ . Take an arbitrary “data” c = {cj } ∈ l ∞ , and denote by b = {bj } ∈ l ∞ the solution of the equation (A + I )b = c. Set f (x) :=
bj fj (x).
j ∈Z
By (30)–(32), we see that the series converges uniformly on every finite interval, and sup f (x) (1 + δ)bl ∞ < ∞. x∈R
Also, it is clear that f satisfies the interpolation condition f (lj ) = c(j ), j ∈ Z. Now let ϕ be any smooth test-function supported by a segment disjoint from S. Let (R) → 0 and N(R) → ∞ as R → ∞, be some functions satisfying < (R), |x| R. f (x) − b f (x) j j |j |N (R)
Since each fj has a compact support which lies in S, then (ϕ, ˆ fj ) = 0, j ∈ Z, and so we have (ϕ, ˆ f− ˆ f ) = ϕ, (R)
|j |N (R)
bj fj
ϕ(x) ˆ dx + (1 + δ)bl ∞
ϕ(x) ˆ dx → 0,
R → ∞.
|x|>R
|x|R
Hence, (ϕ, ˆ f ) = 0, which shows that f ∈ BS . 3.3. Remark on sampling sets We finish this section by the following remark. Definition 7. A set Λ is called a sampling set for BS if there is a constant C > 0 such that f ∞ C sup f (λ), λ∈Λ
for every f ∈ BS . Similarly, one may define sampling set for PW S -spaces. When S is a single interval, the sampling sets for BS were completely characterized by Beurling in terms of so-called “lower uniform density” D − (L) (see [2]), by the following condition:
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D − (L) >
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1 mes S. 2π
For the disconnected compacts S no such metrical characterization may exist. However, one can construct a set of critical density which serves as BS -sampling set for every compact of given measure. More precisely, the following is true: Theorem 3.2. There is a set Λ = {j + O(1), j ∈ Z}, which is a sampling set for the Bernstein space BS , for every compact S of measure < 2π . This is a consequence of [7, Theorem 3] (see details in [9]), where such a “universal” sampling set was constructed for Paley–Wiener spaces. The following lemma reduces Theorem 3.2 to Theorem 3 in [7]. Lemma 3.3. If a u.d. set Λ is a sampling set for PW S(a) for some a > 0, then Λ is a sampling set for BS . Recall that S(a) := S + [−a, a]. Assume that Λ is not a sampling set for BS . We have to show that it is not a sampling set for PW S(a) . We have for every > 0, that there exists f ∈ BS such that f ∞ = 1, and |f (λ)| < , λ ∈ Λ. Take a point x0 such that |f (x0 )| 1/2, and set g(x) := f (x)
sin a(x − x0 ) ∈ PW S(a) . a(x − x0 )
Let σ > 0 be so large that S(a) ⊆ [−σ, σ ]. Now we use the Bernstein inequality: g ∞ σ g∞ σ. Hence, |g(x)| 1/2 − σ |x − x0 | when |x − x0 | 1/2σ , and so g2 C, where C depends only on σ . On the other hand, we have sin a(λ − x0 ) 2 2 g(λ)2 2 a(λ − x ) C , 0
λ∈Λ
λ∈Λ
where C depends only on a and the infimum of distances between elements of Λ. Since this construction can be done for every > 0, we see that Λ is not a sampling set for PW S(a) . In a sharp contrast to Theorem 3.2, the following claim is true: Corollary 3.1. Every sampling set for BS , where S is the closed (unbounded) spectrum S constructed in the proof of Theorem 3.1, must be dense in R. Indeed, suppose a set Λ satisfies Λ ∩ (a, b) = ∅, for some a < b. By the construction in the proof of Theorem 3.1, for every > 0 and c ∈ R there exists f ∈ BS such that f (c) = 1 and |f (x)| < for |x − c| > . By choosing c = (a + b)/2 and < (b − a)/2, we see that for every > 0 there exists a function f ∈ BS satisfying f ∞ > (1/)f l ∞ (Λ) , so that Λ is not a sampling set for BS .
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4. Paley–Wiener spaces with unbounded spectra 4.1. In this section we prove Theorem 3, which extends to Paley–Wiener spaces the phenomenon stated in Theorem 2 for Bernstein spaces. However this extension is not literal and looks weaker. The following two propositions give some motivation for the distinctions. Proposition 4.1. No set of measure < 2π admits interpolation of any δ-function on Z by f ∈ PW S . Proof. Suppose some S does, i.e. there exist j ∈ Z and F ∈ L2 (R), supp F ⊆ S, such that Fˆ (j ) = 1,
Fˆ (n) = 0,
n ∈ Z \ {j }.
Consider the “periodization” Fp (x) :=
F (x + 2πj ),
j ∈Z
as a function on the unite circle T. Clearly, Fp is integrable and vanishes on a set of positive measure. On the other hand, the Fourier coefficients of Fp are the same as the Fourier transform Fˆ |Z . Hence, Fp = eij x . A contradiction. 2 This proposition shows that some restriction on Λ are necessary for PW-interpolation with small spectrum. We will consider a “generic” situation, when the elements of Λ are supposed to be rationally independent. However, even under this restriction it is not possible to interpolate the whole l 2 . Proposition 4.2. Consider a random set Λ := {n + ξn , n ∈ Z}, where ξn are independent variables, uniformly distributed on (−a, a), 0 < a < 1/2. Then with probability 1, there is no set S of measure < 2π such that Λ is an interpolation set for PW S . Proof. Clearly, we have with probability one that for every N ∈ N and > 0 there exists k = k(, N) such that |ξk+j − ξk | < ,
for all j, |j | N.
Fix an element of the underlying probability space such that the latter is true, and assume that S ⊂ R is such that Λ is a set of interpolation for PW S . By (15), we have for every c−N , . . . , cN that 2 2 N N N ij t i(k+j +ξk+j )t cj e dt = lim cj e |cj |2 . dt A →0 S
j =−N
S
j =−N
j =−N
Since this is true for every N , we see that Z is also a set of interpolation for PW S . By Proposition 4.1, we conclude that mes S 2π. 2
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4.2. Proof of Theorem 3 Throughout the proof we shall denote by C different positive constants. 1. Without loss of generality we may assume that β < 2. Set S :=
Sj ,
Sj := (−Mj − 4βj , −Mj + 4βj )
(Mj − 4βj , Mj + 4βj ),
j ∈Z
where βj :=
γ , 1 + |j |β
the sequence Mj will be specified in step 4, and γ is any small positive number such that mes S < δ. In what follows we also assume that γ is so small that Sj ∩ Sk = ∅, for j = k. 2. Set Λk := (Λ − λk ) \ {0},
k ∈ Z.
The independence condition on Λ implies, by Kronecker’s theorem, that for every N > 0 the subgroup mλ (mod π), λ ∈ Λk ∩ [−N, N ], m ∈ Z is dense in the l-dimensional torus, l being the number of elements in Λk ∩ [−N, N ]. Hence, the l numbers |cos(Mx)|, x ∈ Λk ∩ [−N, N ], can be made as small as we like by choosing appropriate M ∈ N. 3. Set
sin γj (x − λj ) 4 . gj (x) := cos Mj (x − λj ) γj (x − λj ) Clearly, the spectrum of gj belongs to Sj , and we have gj (λj ) = 1,
(34)
and gj 22
C C 1 + |j |β , γj
j ∈ Z.
4. Since Λ is uniformly discrete, there is a constant C > 0 such that
sin γj (λk − λj ) γj (λk − λj )
4
C , − j )4
γj4 (k
k = j, k, j ∈ Z.
(35)
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Take any small number > 0, and let Nj be so large that we have
sin γj (λk − λj ) γj (λk − λj )
4
(1 + j 2 )(k
− j )2
|λk − λj | Nj , k = j.
,
By step 2, the first factor in the definition of gj can be made arbitrarily small for 0 = |λk − λj | < Nj . We shall choose Mj ∈ N such that cos Mj (λk − λj )
, (1 + j 2 ) max{(k − j )2 , |λk − λj | < Nj }
for all k = j such that |λk − λj | < Nj . This and the previous estimate give gj (λk )
, (1 + j 2 )(j − k)2
k = j, k, j ∈ Z.
One may check that this estimate implies gj (λk )2 j ∈Z, j =k
C 2 . 1 + |k|3
(36)
5. Given a sequence c = {cj , j ∈ Z}, set ∞
c2β :=
|cj |2 1 + |j |β .
j =−∞
Let lβ2 denote the weighted space of all sequences c, cβ < ∞. Define a linear operator R : lβ2 → lβ2 as follows: Rej :=
∞
gj (λk )ek − ej ,
j ∈ Z.
k=−∞
Using (34), we have 2 2 R c e = c g (λ ) j j j j k ek j ∈Z
β
k∈Z
=
j ∈Z,j =k
k∈Z j ∈Z, j =k
c2β
k∈Z
β
2
cj gj (λk ) 1 + |k|β
j ∈Z, j =k
|gj (λk )|2
1 + |k|β . 1 + |j |2β
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Since β < 2, we see from (36) that 2 2 2 R c e j j C cβ , β
j ∈Z
for some constant C > 0. Choose so small that the norm of operator R in lβ2 is less than 1. It follows that the operator T := I + R is invertible in lγ2 , where I is the identity operator. We conclude that for every c ∈ lβ2 the interpolation problem (3) has a solution f given by f (x) :=
bj gj (x),
{bj } = T −1 c ∈ lβ2 .
j ∈Z
Recall, that the spectrum of gj belongs to the set Sj defined in step 1. Since Sj and Sk are disjoint for j = k, it follows that the functions gj and gk are orthogonal in L2 (R). Using this and (35), we see that f 22 = |bj |2 gj 22 Cb2β < ∞. j ∈Z
Recall again that gj ∈ PW Sj ⊂ PW S , j ∈ Z. We conclude that f ∈ PW S . 5. Questions A few problems are stated below, which we leave open: 1. Let Λ be a u.d. sequence, D ∗ (Λ) = 1 (or more concrete: let Λ = n + O(1), n ∈ Z), and S be a finite union of intervals. Suppose that E(Λ) is minimal in L2 (S). Does this imply the estimate mes S 2π ? Theorem 2.4 shows that this is true for “minimality with a slight estimate.” 2. Is it possible to replace D ∗ (Λ) in Theorem 2.4 by the Beurling–Malliavin density (see [3])? 3. Is Landau’s Theorem A still true for unbounded spectra? In particular, does there exist a u.d. set Λ = {n + O(1), n ∈ Z}, which is an interpolation set for PW S , for some (unbounded) closed set S of small measure? Proposition 4.2 shows, that such a Λ, if it exists, must be an exception. 4. Is it possible in Theorem 3 to replace the condition in (ii) by cj = O
1 , |j |
|j | → ∞?
References [1] A. Beurling, Interpolation for an interval in R1 , in: The Collected Works of Arne Beurling, in: Harmonic Analysis, vol. 2, Birkhäuser Boston, Boston, MA, 1989. [2] A. Beurling, Balayage of Fourier–Stieltjes trasnforms, in: The Collected Works of Arne Beurling, in: Harmonic Analysis, vol. 2, Birkhäuser Boston, Boston, MA, 1989.
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[3] A. Beurling, P. Malliavin, On the closure of characters and the zeros of entire functions, Acta Math. 118 (1967) 79–93. [4] J.-P. Kahane, Sur les fonctions moyenne-périodiques bornées, Ann. Inst. Fourier 7 (1957) 293–314. [5] P. Koosis, The Logarithmic Integral, vol. 2, Cambridge Univ. Press, Cambridge, 1992. [6] H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52. [7] A. Olevskii, A. Ulanovskii, Universal sampling of band-limited signals, C. R. Math. Acad. Sci. Paris 342 (12) (2006) 927–931. [8] A. Olevskii, A. Ulanovskii, Interpolation by functions with small spectra, C. R. Math. Acad. Sci. Paris 345 (5) (2007) 261–264. [9] A. Olevskii, A. Ulanovskii, Universal sampling and interpolation of band-limited signals, Geom. Funct. Anal. 18 (3) (2008) 1029–1052. [10] R.M. Redheffer, R.M. Young, Completeness and basis properties of complex exponentials, Trans. Amer. Math. Soc. 277 (1) (1983) 93–111. [11] R.M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 2001.
Journal of Functional Analysis 256 (2009) 3279–3312 www.elsevier.com/locate/jfa
Explicit and almost explicit spectral calculations for diffusion operators Ross G. Pinsky 1 Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel Received 22 August 2008; accepted 25 August 2008 Available online 26 September 2008 Communicated by J. Bourgain
Abstract The diffusion operator HD = −
d 1 d d 1 d d a −b = − exp(−2B) a exp(2B) , 2 dx dx dx 2 dx dx
where B(x) = 0x ab (y) dy, defined either on R+ = (0, ∞) with the Dirichlet boundary condition at x = 0, or on R, can be realized as a self-adjoint operator with respect to the density exp(2Q(x)) dx. The operator 1 b2 d a d + V , where V is unitarily equivalent to the Schrödinger-type operator HS = − 12 dx b,a b,a = 2 ( a + b ). dx We obtain an explicit criterion for the existence of a compact resolvent and explicit formulas up to the multiplicative constant 4 for the infimum of the spectrum and for the infimum of the essential spectrum for these operators. We give some applications which show in particular how inf σ (HD ) scales when a = νa0 and b = γ b0 , where ν and γ are parameters, and a0 and b0 are chosen from certain classes of functions. We also give applications to self-adjoint, multi-dimensional diffusion operators. © 2008 Elsevier Inc. All rights reserved. Keywords: Spectrum; Essential spectrum; Compact resolvent; Diffusion; Schrödinger operator
E-mail address:
[email protected]. URL: http://www.math.technion.ac.il/~pinsky/. 1 The author thanks the hospitality of the Mathematics Department at the Hebrew University, where this work was done during a sabbatical. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.08.012
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1. Introduction and statement of results In this paper, we give an explicit formula up to the multiplicative constant 4 for the bottom of the spectrum and for the bottom of the essential spectrum for diffusion operators on the half-line R+ = (0, ∞) with the Dirichlet boundary condition at 0, and for diffusion operators on the entire line. Assuming a little more regularity, each such operator is unitarily equivalent to a certain Schrödinger-type operator, so we also obtain the same information for these latter operators. Recall that such an operator possesses a compact resolvent if and only if its essential spectrum is empty, or equivalently, if and only if the infimum of its essential spectrum is ∞. Thus, we obtain a completely explicit criterion for the existence of a compact resolvent. A diffusion operator with a compact resolvent is particularly nice because its transition (sub-)probability density p(t, x, y) (with respect to the reversible measure) can be written in the form p(t, x, y) = ∞ ∞ n=0 exp(−λn t)φn (x)φn (y), where {φn }n=0 is a complete, orthonormal set of eigenfunctions ∞ and {λn }n=0 , satisfying 0 λ0 < λ1 λ2 · · · , are the corresponding eigenvalues. We give some applications of the results, which show in particular how inf σ (HD ) scales when a = νa0 and b = γ b0 , where ν and γ are parameters, and a0 and b0 are chosen from certain classes of functions. At the end of the paper, we give applications to self-adjoint, multi-dimensional diffusion operators of the form − 12 ∇ · a∇ − a∇Q · ∇ = − 12 exp(−2Q)∇ · a exp(2Q)∇ on L2 (Rd , exp(2Q) dx). The methods and the statements of the results are analytic, but many of the formulas and results have probabilistic import. We begin with the theory on the half-line, wherein lies the crux of our method. The results for the entire line follow readily the results for the half-line. Let 0 < a ∈ C 1 ([0, ∞)) and xfrom b b ∈ C([0, ∞)). Define B(x) = 0 a (y) dy. Consider the diffusion operator with divergence-form diffusion coefficient a and drift b HD = −
d 1 d d 1 d d a −b = − exp(−2B) a exp(2B) 2 dx dx dx 2 dx dx
on R+ with the Dirichlet boundary condition at x = 0. One can realize HD as a non-negative, self-adjoint operator on L2 (R+ , exp(2B) dx) via the Friedrichs extension of the closure of the non-negative quadratic form 1 QD (f, g) = 2
∞ (f ag ) exp(2B) dx, 0
defined for f, g ∈ C01 (R+ ), the space of continuously differentiable functions with compact support on R+ . Let UB denote the unitary operator from L2 (R+ ) to L2 (R+ , exp(2B) dx) defined by UB f = exp(−B)f. Assuming that b ∈ C 1 (R+ ), define HS = UB−1 HD UB . One can check that HS = − where
1 d d a + Vb,a , 2 dx dx
R.G. Pinsky / Journal of Functional Analysis 256 (2009) 3279–3312
Vb,a =
1 b2 + b . 2 a
3281
(1.1)
2
Assuming that Vb,a = 12 ( ba + b ) is bounded from below, one can realize the Schrödinger-type operator HS as a self-adjoint operator on L2 ((0, ∞)) via the Friedrichs extension of the closure of the semi-bounded quadratic form 1 QS (f, g) = 2
∞ ∞ (f ag ) dx + Vb,a f g dx, 0
0
∞
defined for f, g ∈ C01 (R+ ). Assuming in addition that a(x) dx = ∞, one can prove that HS is in the limit-point case at ∞, which means in particular that HS on C02 (R+ ) is essentially self-adjoint. (A proof in the case a = 1 can be found in [7, Appendix to X.1]. It can easily be extended to a satisfying the above condition.) Thus, the Friedrichs extension is in fact equal to the closure of HS on C02 (R+ ). Note also that UB preserves the Dirichlet boundary condition. From the above considerations, it follows that the spectra and the essential spectra of HD and HS coincide; in particular, HS is also non-negative. Conversely, given a > 0, every potential V 0 can be obtained via some b as in (1.1), and modulo an additive constant, every potential V that is bounded from below can be obtained via − some b as in (1.1). Indeed, let mV = infx∈R+ V (x) and let m− V = mV ∧ 0. Since V − mV 0, it
is easy to show that the Riccati equation 12 b + 12 ba = V − m− V has solutions b which exist for all x 0. d d d d a dx + V ) of Schrödinger-type operators − 12 dx a dx +V The essential spectrum σess (− 12 dx has been well studied. See [8] for the results noted below. For example, if a is bounded and bounded from 0, and if the potential V satisfies limx→∞ V (x) = ∞, then the operator has a compact resolvent. Thus, the spectrum consists of an increasing sequence of eigenvalues acd d a dx + V ) = ∅. On the other hand, if V cumulating only at infinity; in particular, σess (− 12 dx d d a dx , which occurs in particuis a compact (or even relatively compact) perturbation of − 12 dx 1 d d lar if limx→∞ V (x) = 0, then the essential spectrum of − 2 dx a dx + V coincides with that of d d d d a dx ; thus σess (− 12 dx a dx + V ) = [0, ∞). More generally, for arbitrary a > 0, the mini− 12 dx d d a dx + V ), although this max method [7] affords an algorithm for arriving at inf σess (− 12 dx method is mainly of theoretic import and not a practical way of calculating. d d a dx + V is of course given by the well-known variational The bottom of the spectrum of − 12 dx formula: ∞ 1 ( a(f )2 + Vf 2 ) dx 1 d d a + V = inf 0 2 ∞ 2 , inf σ − 2 dx dx 0 f dx 2
where the infimum is over functions 0 = f ∈ C01 (R+ ). The bottom of the spectrum of HD is also given by a variational formula: 1 ∞ 2 2 0 a(f ) exp(2B) dx inf σ (HD ) = inf ∞ 2 , (1.2) 0 f exp(2B) dx where the infimum is over 0 = f ∈ C01 (R+ ).
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The following theorem gives explicit formulas up to the multiplicative constant 4 for the bottom of the spectrum and for the bottom of the essential spectrum of HD . By the spectral invariance, this then extends to the Schrödinger-type operators HS . The formulas take on two ∞ 1 exp(−2B(x)) dx is finite or infinite. In Remark 2 possible forms, depending on whether 0 a(x) after the theorem, it is shown how the proof for the case when the integral is finite can be reduced to the case when the integral is infinite. Remark 4 after the theorem discusses the probabilistic import of the theorem and of the above integral. Theorem 1. Let 0 < a ∈ C 1 ([0, ∞)) and b ∈ C([0, ∞)). Define x B(x) =
b (y) dy. a
0
Consider the self-adjoint diffusion operator HD = −
d 1 d d 1 d d a −b = − exp(−2B) a exp(2B) 2 dx dx dx 2 dx dx
on L2 (R+ , exp(2B) dx) with the Dirichlet boundary condition at 0. ∞ 2 If b ∈ C 1 (R+ ), ba + b is bounded from below and a(x) dx = ∞, consider also the selfadjoint Schrödinger-type operator 1 b2 1 d d a + +b HS = − 2 dx dx 2 a on L2 (R+ ) with the Dirichlet boundary condition at 0. If ∞
1 exp −2B(x) dx = ∞, a(x)
(1.3)
define x
+
1 exp −2B(y) dy a(y)
Ω (b, a) = sup x>0
∞
exp 2B(y) dy
(1.4)
x
0
and x
Ωˆ + (b, a) = lim sup x→∞
0
1 exp −2B(y) dy a(y)
∞
exp 2B(y) dy .
(1.5)
x
If ∞
1 exp −2B(x) dx < ∞, a(x)
(1.6)
R.G. Pinsky / Journal of Functional Analysis 256 (2009) 3279–3312
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let ∞ hb,a (x) =
1 exp −2B(y) dy a(y)
(1.7)
x
and define Ω + (b, a) = sup
x
x>0
h−2 b,a (y)
1 exp −2B(y) dy a(y)
∞
h2b,a (y) exp 2B(y) dy
x
0
−1 = sup h−1 b,a (x) − hb,a (0)
∞
x>0
h2b,a (y) exp
2B(y) dy ,
(1.8)
x
and x
ˆ+
Ω (b, a) = lim sup x→∞
h−2 b,a (y)
1 exp −2B(y) dy a(y)
0
−1 = lim sup h−1 b,a (x) − hb,a (0) x→∞
∞ h2b,a (y) exp
2B(y) dy
x
∞ h2b,a (y) exp
2B(y) dy .
(1.9)
x
Then 1 1 inf σ (HD ) = inf σ (HS ) 8Ω + (b, a) 2Ω + (b, a)
(1.10)
1 1 inf σess (HD ) = inf σess (HS ) . + + ˆ ˆ 8Ω (b, a) 2Ω (b, a)
(1.11)
and
In particular, HD and HS possess compact resolvents if and only if Ωˆ + (b, a) = 0. Remark 1. There does not exist a C for which inf σ (HD ) = Ω +C(b,a) , for all drifts b and all diffusion coefficients a. Indeed, on the one hand, consider the case that b(x) = ±γ , with γ ∈ R, and 2 2 a = 1. Then Vb,a = γ2 and thus by unitary equivalence, inf σ (HD ) = inf σ (HS ) = γ2 . A direct calculation in this case reveals that Ω + (b, a) = 4γ1 2 ; thus, inf σ (HD ) = 8Ω +1(b,a) . On the other hand, consider the case that b(x) = −γ x, with γ > 0, and a = 1. Then limx→∞ Vb,a (x) = ∞, so as noted above, HS and thus also HD have compact resolvents. The unnormalized Hermite function H1 (x) = x is an L2 -eigenfunction of HD corresponding to the eigenvalue γ . Since it is positive, it must in fact be the principal eigenvalue. Thus, the bottom of the spectrum is equal to γ . We have
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R.G. Pinsky / Journal of Functional Analysis 256 (2009) 3279–3312
x
+
Ω (b, a) = sup x>0
exp γ y 2 dy
∞
x
0
1 = sup γ x>0
exp −γ y 2 dy
x
exp y 2 dy
∞
2 .239 . exp −y dy ≈ γ
x
0
Thus, in this case, the bottom of the spectrum is approximately equal to Ω +.239 . In the (b,a) case b(x) = γ x, with γ > 0, and a = 1, one can check that the principal eigenfunction is x exp(−γ x 2 ), with corresponding principal eigenvalue 2. One can calculate that Ω + (a, b) ≈ .097 .194 γ , and thus the bottom of the spectrum is approximately equal to Ω + (b,a) . Writing the bottom of the spectrum in the form sharp; namely, Cb,a
Cb,a , Ω + (b,a)
we do not know whether the upper bound in the theorem is
1 2.
Remark 2. In this remark, we demonstrate how formulas (1.10) and (1.11) in the case (1.6) follow from those formulas in the case (1.3), thereby reducing the proof of the theorem to the case that (1.3) holds. In the case that (1.6) holds, define the h-transform of h 1 HD via the function hb,a in (1.7) by HDb,a u = hb,a HD (hb,a u). When written out, one ob x h h h d d tains HDb,a = − 12 dx a dx − (b + a hb,a ) d . Letting B hb,a (x) = 0 ( ab + hb,a )(y) dy, one has b,a dx b,a ∞ 1 ∞ −2 h b,a (x)) dx = − hb,a hb,a dx = ∞; that is, the diffusion coefficient a with a(x) exp(−2B h
the new drift b + a hb,a satisfies (1.3). The spectrum is invariant under h-transforms [6, Chapter 4, b,a h
h
Sections 3 and 10], so inf σ (HD ) = inf σ (HDb,a ) and inf σess (HD ) = inf σess (HDb,a ). These equalh
) ities along with the fact that (1.3) holds with the diffusion coefficient a and the drift (b + a hb,a b,a h
show that one obtains (1.10) and (1.11) for HD by defining Ω + (b, a) = Ω + (b + a hb,a , a) and b,a
h , a). From (1.4), one has Ωˆ + (b, a) = Ωˆ + (b + a hb,a b,a
∞
x hb,a h 1 hb,a b,a exp −2B (y) dy , a = sup exp 2B (y) dy Ω b+a hb,a a(y) x>0 +
x
0
x = sup x>0
0
h−2 b,a (y)
1 exp −2B(y) dy a(y)
∞ h2b,a (y) exp
2B(y) dy ;
x
(1.12) whence the definition of Ω + (b, a) in (1.8) in the case that (1.6) holds, and likewise for Ωˆ + (b, a). Remark 3. After finishing this paper, the following related result due to Muckenhoupt [3], in the context of weighted Hardy inequalities, was brought to our attention. For 1 p ∞, the inequality
R.G. Pinsky / Journal of Functional Analysis 256 (2009) 3279–3312
3285
p 1 ∞
1 ∞ x p p p V (x)g(x) dx C U (x) g(t) dt dx 0
0
(1.13)
0
holds for all g and some finite C if and only if
1 x
1 ∞ p p p −p U (y) dy V (y) M ≡ sup dy < ∞, x>0
where
1 p
+
1 p
x
0
= 1. And furthermore, if C0 is the least constant C for which the above inequality 1
1
holds, then M C0 p p (p ) p M, for 1 < p < ∞, and C0 = M for p = 1, ∞. (The integrals are interpreted according to the usual convention in the case that p or p is ∞.) Applying this 1 with p = p = 2, U = exp(B) and V = ( a2 ) 2 exp(B), one concludes that inf
1 ∞ 2 2 0 a(f ) exp(2B) dx ∞ 2 0 f exp(2B) dx
lies between
x
8 sup x>0
1 exp −2B(y) dy a(y)
∞
exp 2B(y) dy
−1
x
0
and x
2 sup
x>0
1 exp −2B(y) dy a(y)
∞
exp 2B(y) dy
−1 ,
x
0
where the infimum is over f ∈ C 1 ([0, ∞)) which satisfy f (0) = 0. This is a different variational problem than the one in (1.2) for inf σ (HD ) because the class of admissible functions here is larger than in (1.2). In the case that (1.3) holds, Theorem 1 shows that the same bounds hold for both variational problems, since in this case, x
+
Ω (b, a) = sup x>0
1 exp −2B(y) dy a(y)
∞
exp 2B(y) dy .
x
0
However, when (1.6) holds, Ω + (b, a) is defined differently, and the two variational problems yield different results. Indeed, for example, if b = 1 and a = 1, then one has ∞ x
exp 2B(y) dy = ∞,
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so the infimum in Muckenhoupt’s variational problem is 0; however by (1.8), one calculates that Ω + (b, a) = 1, and it follows from Theorem 1 that the infimum in (1.2) lies between 18 and 12 . (In fact, in this simple case it can be checked directly that inf σ (HD ) = 18 .) Thus, the integral condition (1.6) turns out to be the lower threshold on the size of a exp(2B), the weight that multiplies (f )2 in the variational formulas, so that the two variational formulas, one over f ∈ C01 (R+ ) and one over f ∈ C 1 ([0, ∞)) satisfying f (0) = 0, yield different answers. Muckenhoupt’s proof involves a direct estimation of the integrals in (1.13). We prove Theorem 1 in a completely different way, as will be seen in Sections 3 and 4. Remark 4. Theorem 1 and the reduction noted above in Remark 2 have some probabilistic implications, which we now describe. Let X(t) be generic notation for a Markov diffusion process on the real line. Let Px and Ex denote respectively probabilities and expectations for the process d d d a dx + b dx on (0, ∞), starting at x > 0 and killed at corresponding to the operator −HD = 12 dx time
τ0 ≡ inf t 0: X(t) = 0 ,
(1.14)
the first hitting time of 0. Then Px (τ0 < ∞) = 1, for x > 0, if and only if (1.3) holds [6, Chapter 5]. Consider first the case that (1.3) holds. At the end of Section 3 we show that
inf σ (HD ) = sup λ 0: Ex exp(λτ0 ) < ∞ ,
x > 0.
(1.15)
Thus, in the case that (1.3) holds, (1.10) gives an explicit formula up to the multiplicative constant 4 for sup{λ 0: Ex exp(λτ0 ) < ∞}. h (x) Now consider the case that (1.6) holds. In this case, Px (τ0 < ∞) = hb,a [6, Chapter 5]. The b,a (0) original process, conditioned on {τ0 < ∞}, is itself a Markov diffusion process and it corresponds h h to the h-transformed operator −HDb,a defined in Remark 2 [6, Chapter 7]. Let Ex b,a denote expectations for this conditioned process starting from x > 0. Then it follows from Remark 2 and (1.15) that in the case that (1.6) holds, one has
h inf σ (HD ) = sup λ 0: Ex b,a exp(λτ0 ) < ∞ ,
x > 0.
(1.16)
Note ∞ from (1.4) and (1.10) that when (1.3) holds, a necessary condition for inf σ (HD ) > 0 is exp(2B(y)) dy < ∞. This integral condition is equivalent to Ex τ0 < ∞, for x > 0 [6, that Chapter 5, Section 1]. Thus, when Px (τ0 < ∞) = 1 holds, the finiteness of Ex τ0 is a necessary condition (but not a sufficient one) for inf σ (HD ) > 0. Similarly, when (1.6) holds (in which case h P hb,a (τ0 < ∞) = 1), the finiteness of Ex b,a τ0 is a necessary condition (but not a sufficient one) for inf σ (HD ) > 0. (Of course, this can also be seen from (1.15) and (1.16)—if the first moment does not exist, then a fortiori no exponential moment exists.) An alternative probabilistic representation of inf σ (HD ) is this: inf σ (HD ) = − lim lim
n→∞ t→∞
1 log Px (τ0 ∧ τn > t), t
x > 0,
(1.17)
where τn = inf{t 0: X(t) = n}. (This formula can be found essentially in [6, Chapter 4].)
R.G. Pinsky / Journal of Functional Analysis 256 (2009) 3279–3312
3287
Formulas (1.15) and (1.16) give a probabilistic representation for the bottom of the spectrum of HD . One can also give a similar probabilistic representation for the bottom of the essential spectrum. It follows from (1.15) and (3.9) in Section 3 that if (1.3) holds, then
inf σess (HD ) = lim sup λ 0: Ex exp(λτl ) < ∞, x > l , l→∞
while if (1.6) holds, then
h inf σess (HD ) = lim sup λ 0: Ex b,a exp(λτl ) < ∞, x > l . l→∞
Remark 5. It follows from the theorem that inf σ (HD ) and inf σess (HD ) depend on a and b only through a and B. Remark 6. For the duration of this remark, we consider a to be fixed. By a standard comparison theorem for diffusions, it follows that over the class of drifts b satisfying (1.3), the distribution of τ0 is stochastically increasing with b. Thus, from (1.15), it follows that inf σ (HD ) and inf σess (HD ) are non-increasing over the class of drifts b satisfying (1.3). That is, over drifts satisfying (1.3), the more inward toward 0 the drift, the larger the bottom of the spectrum ∞essential spectrum. (It is not hard to verify that the function x 1and the bottom of the exp(−2B(y)) dy)( x exp(2B(y)) dy) appearing in the definition of Ω + (b, a) is non( 0 a(y) decreasing in b over the class of drifts b satisfying (1.3), but this is not quite enough to arrive at the result in the above sentence.) Despite the above fact and despite Remark 5, it is not true that inf σ (HD ) and inf σess (HD ) are non-increasing as functions of B over the class of b satisfying (1.3). An example will be given at the end of Section 2. We do not know whether inf σ (HD ) and inf σess (HD ) are non-decreasing over the entire class of drifts b satisfying (1.6), so that the more outward toward infinity the drift, the larger the bottom of the spectrum and the bottom of the essential spectrum. To prove that this is true, it would sufh
b,a is non-increasing in b over the class of drifts satisfying (1.6)—that this fice to show that b + a hb,a would suffice follows from (1.16) and the argument above for the class of drifts satisfying (1.3). What is known is this [5]:
For a wide class of a and b which satisfy (1.6) and for which b is on a hb,a a(x) a(x) , one has b + a as x → ∞. = −b + O larger order than x hb,a x
(1.18)
This formula will be useful for one of the calculations in Section 2. Remark 7. For another result in which a two-sided spectral bound is given for elliptic operators, with the upper bound being 4 times the lower one, see [2, Theorem 5.1]. We now line. Let 0 < a ∈ C 1 (R) and b ∈ C(R), and define x bturn to the case of the1 whole d d d and consider the self-adjoint realization on B(x) = 0 a (y) dy. Let HD = − 2 dx a dx − b dx 2 L (R, exp(2B) dx) obtained via the Friedrichs extension of the closure of the quadratic form ∞ 2 QD (f, g) = 12 −∞ (f ag ) exp(2B) dx, for f, g ∈ C01 (R). In the case that b ∈ C 1 (R), ba + b is ∞ d d bounded from below and a(x) dx = −∞ a(x) dx = ∞, define HS = − 12 dx a dx + Vb,a to be
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the self-adjoint operator obtained via the Friedrichs extension of the closure of the quadratic form ∞ ∞ 2 QS (f, g) = 12 −∞ (f ag ) dx + −∞ Vb,a f g dx, where Vb,a = 12 ( ba + b ) and f, g ∈ C01 (R). The first of the two theorems below treats inf σess (HD ) and the second one treats inf σ (HD ). The proofs of these results will be derived in just a few lines from the proof of Theorem 1. Theorem 2. Let a ∈ C 1 (R) and b ∈ C(R). Define x B(x) =
b (y) dy. a
0
Consider the self-adjoint diffusion operator HD = −
d 1 d d 1 d d a −b = − exp(−2B) a exp(2B) 2 dx dx dx 2 dx dx
on L2 (R, exp(2B) dx). ∞ 2 If b ∈ C 1 (R), ba + b is bounded from below and a(x) dx = −∞ a(x) dx = ∞, consider also the self-adjoint Schrödinger-type operator HS = −
1 b2 1 d d a + + b 2 dx dx 2 a
on L2 (R). Let Ω + (b, a) be as in Theorem 1 and define Ω − (b, a) in exactly the same way, using the half-line (−∞, 0) instead of (0, ∞). Let ˆ Ω(b, a) = max Ωˆ + (b, a), Ωˆ − (b, a) . Then 1 1 inf σess (HD ) = inf σess (HS ) . ˆ ˆ 8Ω(b, a) 2Ω(b, a) In particular, HD and HS possess compact resolvents if and only if Ωˆ + (b, a) = Ωˆ − (b, a) = 0. Remark 8. The diffusion is positive recurrent if and only if R exp(2B(x)) dx < ∞ [6, Chapter 5]. It follows from Theorem 2 that inf σess (HD ) = 0 if the diffusion is not positive recurrent. (See also the third to the last paragraph of Remark 4.) Theorem 3. Let a ∈ C 1 (R) and b ∈ C(R). Define x B(x) = 0
Consider the self-adjoint diffusion operator
b (y) dy. a
R.G. Pinsky / Journal of Functional Analysis 256 (2009) 3279–3312
HD = −
3289
d 1 d d 1 d d a −b = − exp(−2B) a exp(2B) 2 dx dx dx 2 dx dx
on L2 (R, exp(2B) dx). ∞ 2 If b ∈ C 1 (R), ba + b is bounded from below and a(x) dx = −∞ a(x) dx = ∞, consider also the self-adjoint Schrödinger-type operator 1 b2 1 d d HS = − a + +b 2 dx dx 2 a on L2 (R). If ∞
1 exp −2B(x) dx = a(x)
−∞
1 exp −2B(x) dx = ∞, a(x)
(1.19)
define Ω(b, a) = ∞.
(1.20)
If ∞
1 exp −2B(x) dx = ∞ a(x)
and −∞
1 exp −2B(x) dx < ∞, a(x)
(1.21)
define x Ω(b, a) = sup x∈R
−∞
1 exp −2B(y) dy a(y)
∞
exp 2B(y) dy .
x
If ∞
1 exp −2B(x) dx < ∞ a(x)
and −∞
1 exp −2B(x) dx = ∞, a(x)
define ∞ Ω(b, a) = sup x∈R
x
1 exp −2B(y) dy a(y)
x −∞
exp 2B(y) dy .
(1.22)
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If ∞
1 exp −2B(x) dx < ∞ a(x)
and −∞
1 exp −2B(x) dx < ∞, a(x)
(1.23)
let ∞ hb,a (x) =
1 exp −2B(y) dy a(y)
x
and define x Ω(b, a) = sup x∈R
−∞
h−2 b,a (y)
1 exp −2B(y) dy a(y)
∞ h2b,a (y) exp
2B(y) dy
x
∞
−1 −1 2 = sup hb,a (x) − hb,a (−∞) hb,a (y) exp 2B(y) dy . x∈R
x
Then 1 1 inf σ (HD ) = inf σ (HS ) . 8Ω(b, a) 2Ω(b, a) Remark 9. The diffusion process X(t) corresponding to −HD is recurrent if (1.19) holds and is transient, otherwise. In the transient case, if (1.21) holds, then Px (limt→∞ X(t) = −∞) = 1; if (1.22) holds, then Px (limt→∞ X(t) = ∞) = 1; if (1.23) holds, then Px (limt→∞ X(t) = −∞) = h (x) 1 − Px (limt→∞ X(t) = ∞) = hb,ab,a(−∞) . (For these results, see [6, Chapter 5].) It follows from Theorem 3 that inf σ (HD ) = 0 if the diffusion is recurrent. Remark 10. Similar to (1.17), one has the following probabilistic representation of inf σ (HD ): inf σ (HD ) = − lim lim
n→∞ t→∞
1 log Px (τ−n ∧ τn > t), t
x ∈ R.
(1.24)
In Section 2 we give some applications of Theorems 1–3. In Section 3 we prove Theorem 1, postponing the proof of a key proposition to Section 4. After the proof of Theorem 1 we give the quick proofs of Theorems 2 and 3. We also prove (1.15) in Section 3. Finally, in Section 5 we show how the one-dimensional result can be used to obtain spectral estimates for self-adjoint, multi-dimensional diffusion operators.
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2. Examples The bottom of the spectrum. One can use Theorem 1 to study the way inf σ (HD ) scales in the parameters γ and ν when b is of the form b = γ b0 and a is of the form a = νa0 . We first consider the effect of the drift alone. Consider for example the following two cases on R+ or on R: l b(x) = −γ 1 + |x| b(x) = −γ |x|
l
and a(x) = 1,
and a(x) = 1,
γ > 0, l ∈ R, γ > 0, l 0.
(2.1) (2.2)
Proposition 1. Consider HD on R+ or on R. 1. Assume that (2.1) holds. (i) If l < 0, then inf σ (HD ) = 0; (ii) If l 0, then there exist constants cl , Cl > 0 such that cl γ 2 inf σ (HD ) Cl γ 2 ,
γ > 1,
and 2
2
cl γ l+1 inf σ (HD ) Cl γ l+1 ,
0 < γ 1.
2. Assume that (2.2) holds. Then 2 2 1 1 γ 1+l inf σ (HD ) γ 1+l , 8Cl 2Cl
γ > 0,
where ⎧ ⎨ sup
x ∞ 2zl+1 2zl+1 x>0 ( 0 exp( l+1 ) dz)( x exp(− l+1 ) dz) Cl = ⎩ C = ( ∞ exp(− 2zl+1 ) dz)2 l 0 l+1
on R+ ; on R.
Remark 11. Note that both on R+ and on R, the rate of growth of inf σ (HD ) for large γ is on a slower order for the drift in (2.2) than for the drift in (2.1). The probabilistic explanation for this follows from the formulas (1.17) and (1.24) and the fact that the latter drifts are small in a (γ -dependent) neighborhood of 0, even as γ becomes large. Note also that for the drift in (2.1), the scaling power is different for γ 1 than for γ 1. The bounds on the infimum of the spectrum in Proposition 1 also hold for the corresponding 1 2 1 d2 2l l−1 in the case Schrödinger operator, HS = − 12 dx 2 + V , where V = 2 γ (1 + x) − 2 γ l(1 + x) 1 1 of (2.1) on R+ and V (x) = 2 γ 2 x 2l − 2 γ x l−1 in the case of (2.2) on R+ , and similarly for R. We now consider simultaneous scaling in a and b. Consider the following case on R+ and on R: l b(x) = −γ 1 + |x|
k and a(x) = ν 1 + |x| ,
where
γ , ν > 0, l, k ∈ R, with l − k > −1 and 2l − k 0.
(2.3)
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(Note that when ν = 1 and k = 0, (2.3) reduces to (2.1) with l 0.) If 2l − k < 0 or if l − k < −1, then one can show that inf σ (HD ) = 0. Proposition 2. Consider HD on R+ or on R. Assume that (2.3) holds. There exist constants cl,k , Cl,k > 0 such that cl,k
γ2 γ2 inf σ (HD ) Cl,k , ν ν
0 < ν < γ,
and cl,k
γ 2−k ν 1−l
1 l−k+1
inf σ (HD ) Cl,k
γ 2−k ν 1−l
1 l−k+1
,
0 < γ ν.
Remark 12. Note that when γ ν, the scaling dependence on the coefficient γ of the drift b has three dramatically different phases, depending on whether the exponent k of the diffusion coefficient satisfies k < 2, k = 2 or k > 2, while the scaling dependence of the coefficient ν of the diffusion coefficient has three dramatically different phases, depending on whether the exponent l of the drift satisfies l < 1, l = 1 or l > 1. However, when γ > ν, there is only one scaling phase, and it is independent of the exponents l and k. The bounds on the infimum of the spectrum in Proposition 2 also hold for the correspond2 d d ing Schrödinger-type operator, HS = − 12 dx (ν(1 + |x|)k ) dx + V , where V = 12 γν (1 + x)2l−k − 1 l−1 in the case of R+ , and similarly for R. The parameter dependence in Proposition 2 2 γ l(1 + x) in the case that γ ν does not seem at all apparent from looking at this operator. We give the proof of Proposition 1; the proof of Proposition 2 is similar. Proof of Proposition 1. We prove the proposition in the case of R+ ; the case of R is handled similarly. To prove part 2, one simply makes an appropriate change of variables in the formula for Ω + (−γ x l , 1) and applies Theorem 1. To get the explicit form of Cl in the case of R, one needs to do a little bit more analysis to show that the supremum over x ∈ R occurs at x = 0. We now prove part 1. If l = −1, then Ω −γ (1 + x)l , 1 = sup +
x>0
x 0
∞
(1 + y)l+1 (1 + y)l+1 dy dy . exp 2γ exp −2γ l+1 l+1 x
(2.4) For l < −1 the right-hand integral is ∞ so Ω + (−γ (1 + x)l , 1) = ∞. Now consider −1 < l < 0. Applying l’Hôspital’s rule to the quotients x 0
l+1
(1 + x)−l exp(2γ (1+x) l+1 shows that
∞
l+1
exp(2γ (1+y) l+1 ) dy
, )
x
l+1
exp(−2γ (1+y) l+1 ) dy l+1
(1 + x)−l exp(−2γ (1+x) l+1
)
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x 0
(1 + y)l+1 (1 + x)l+1 dy ∼ (2γ )−1 (1 + x)−l exp 2γ and exp 2γ l+1 l+1
∞ x
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(1 + y)l+1 (1 + x)l+1 dy ∼ (2γ )−1 (1 + x)−l exp −2γ , exp −2γ l+1 l+1
as x → ∞. This shows that the supremum in (2.4) is ∞; thus Ω + (−γ (1 + x)l , 1) = ∞. One obtains Ω + (−γ (1 + x)l , 1) = ∞ similarly in the case l = −1. Applying Theorem 1 now completes the proof of part 1(i). Consider now part 1(ii); that is, the case l 0. Making the change of variables 1
z = γ l+1 (1 + y), one obtains from (2.4), 2 Ω −γ (1 + x)l , 1 = γ − l+1 +
l+1 ∞ l+1
z z dz dz . exp exp − l+1 l+1
x sup x>γ
1 l+1
γ
(2.5)
x
1 l+1
If l = 0, the integrals on the right-hand side of (2.5) can be calculated explicitly. One finds that the supremum above is equal to 1. Part 1(ii) in the case l = 0 now follows from (2.5) and Theorem 1. From now on, we assume that l > 0. Applying l’Hôspital’s rule in the manner noted above shows that ∞ x
l+1 z x l+1 −l dz ∼ x exp − , exp − l+1 l+1
x 0
l+1 l+1 z x dz ∼ x −l exp , exp − l+1 l+1
as x → ∞;
as x → ∞.
(2.6)
From (2.6) it follows that there exist constants dl , Dl > 0 such that x dl sup x>γ
1 l+1
γ
1 l+1
l+1 ∞ l+1
z z dz dz Dl , exp exp − l+1 l+1
0 < γ 1.
x
Part 1(ii) in the case that 0 < γ 1 now follows from (2.5), (2.7) and Theorem 1. Now consider part 1(ii) in the case that γ > 1. Clearly, 1
2γ l+1
γ
1 l+1
l+1 ∞ l+1
z z dz dz exp exp − l+1 l+1 2γ
1 l+1
(2.7)
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x sup x>γ
1 l+1
γ
1 l+1
x sup x>γ
1 l+1
0
l+1 ∞ l+1
z z dz dz exp exp − l+1 l+1 x
l+1 ∞ l+1
z z exp exp − dz dz . l+1 l+1
(2.8)
x
Using (2.6) to estimate the leftmost and rightmost terms in (2.8), it follows that there exist constants dl , Dl > 0 such that dl γ
2l − l+1
x sup x>γ
1 l+1
γ
1 l+1
l+1 ∞ l+1
2l z z dz dz Dl γ − l+1 , exp exp − l+1 l+1
for γ > 1.
x
(2.9) Part 1(ii) in the case γ > 1 now follows from (2.5), (2.9) and Theorem 1.
2
Theorem 1 allows one to compute the bottom of the spectrum exactly for an ad hoc class of d2 Schrödinger operators, H = − 12 dx 2 + V . Indeed, it follows from the theorem that if a and b ∞ satisfy (1.3) and exp(2B) dx = ∞, then inf σ (HD ) = inf σ (HS ) = 0. Let u = exp(g), where u = 12 ((g )2 + g ), b = uu = g and a = 1. Then b satisfies g is bounded, and define V = 2u 2
2
d 1 d 1 2 the above conditions and HS = − 12 dx 2 + V . Thus, inf σ (− 2 dx 2 + 2 ((g ) + g )) = 0, for all 1 x bounded g. In particular, if g is periodic and not constant, then limx→∞ x 0 V (y) dy > 0 but the bottom of the spectrum is 0. Note that either Ωˆ + (b, a) = Ω + (b, a) = ∞, or Ωˆ + (b, a), Ω + (b, a) < ∞; thus, in R+ either both the bottom of the spectrum and bottom of the essential spectrum equal 0, or else neither of them does. It is not hard to construct examples where the bottom of the spectrum and the bottom of the essential spectrum are both positive and finite but do not coincide. For example, let a = 1 and let b(x) = −1, for x 3. Since Ωˆ + (b, 1) does not depend on {b(x): 0 x 3}, we have Ωˆ + (b, 1) = 14 . Let b(x) = −n, for 1 x 2, and b(x) −1 everywhere. Then the term 3 0 exp(−2B(y)) dy can be made arbitrarily large by choosing n arbitrarily large, and thus for sufficiently large n,
Ω + (b, 1) = sup x>0
x 0
exp −2B(y) dy
∞
1 exp 2B(y) dy > = Ωˆ + (b, 1). 4
x
The bottom of the essential spectrum. We consider operators on R+ . The examples can easily be extended to operators on R by making the analysis on R+ and on R− separately, and applying Theorem 2. Consider first the case that b(x) = −γ (1 + x)l l − k > −1,
and a(x) = ν(1 + x)k , γ , ν > 0, l, k ∈ R, with 2γ , or l − k < −1 and k 1. or l − k = −1 and k 1 + ν
(2.10)
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The set of possible conditions on l, k above are exactly those xfor1 which (1.3) holds. One can ob∞ exp(−2B(y)) dy by applying tain the asymptotic behavior of x exp(2B(y)) dy and of 0 a(y) l’Hôspital’s rule respectively to x 1 ∞ 0 a(y) exp(−2B(y)) dy x exp(2B(y)) dy and . −1 1 b (x) exp(2B(x)) b−1 (x) a(x) exp(−2B(x)) Calculating and applying Theorem 1, one obtains the following result. Proposition 3. Consider HD on R+ . Let a and b satisfy (2.10). 1. Assume that l − k < −1 or that l − k = −1 and 2. Assume that l − k = −1 and γν > 12 . (i) If k > 2, then σess (HD ) = ∅; (ii) If k = 2, then 0 < inf σess (HD ) < ∞; (iii) If k < 2, then inf σess (HD ) = 0. 3. Assume that l − k > −1. (i) If 2l − k > 0, then σess (HD ) = ∅; (ii) If 2l − k = 0, then 0 < inf σess (HD ) < ∞; (iii) If 2l − k < 0, then inf σess (HD ) = 0.
γ ν
12 . Then inf σess (HD ) = 0.
In particular, HD possesses a compact resolvent if and only if 2(i) or 3(i) holds. The bounds on the infimum of the essential spectrum in Proposition 3 also hold for the cor2 d d responding Schrödinger-type operator HS = − 12 dx (ν(1 + x)k ) dx + V , where V = 12 γν (1 + x)2l−k − 12 γ lx l−1 . For certain values of the parameters, the results in Proposition 3 can be deduced directly from looking at HS . For example, if k = 0 and l < 1, then limx→∞ V (x) equals 2 ∞ if l > 0 and is equal to 12 γν if l = 0. It follows from standard perturbations results, mentioned 2
in the first section, that in the former case σess (HS ) = ∅ and in the latter case inf σess (HS ) = 12 γν . However, in fact, Theorem 1 allows one to come to the same type of conclusions as in Proposition 3 in the case that a and b satisfy one of the following general conditions: c1 (1 + x)k a(x) c2 (1 + x)k , x −c2 (1 + x)m 0
b(y) dy −c1 (1 + x)m , (1 + y)k
k ∈ R,
for large x, m > 0, 0 < c1 < c2 ;
(2.11)
or c1 (1 + x)k a(x) c2 (1 + x)k , k 1, x b(y) dy is bounded in x. (1 + y)k 0
It is easy to check that under (2.11) or (2.12), a and b satisfy (1.3).
(2.12)
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Note that now b can be locally erratic, and the bottom of the essential spectrum cannot be deduced directly by looking at HS . Proposition 4. Consider HD on R+ . 1. Assume that a and b satisfy (2.11). (i) If 2m + k − 2 > 0, then σess (HD ) = ∅; (ii) If 2m + k − 2 = 0, then 0 < inf σess (HD ) < ∞; (iii) If 2m + k − 2 < 0, then inf σess (HD ) = 0. 2. Assume that a and b satisfy (2.12). Then inf σess (HD ) = 0. In particular, HD possesses a compact resolvent if and only if 1(i) holds. To prove Proposition 4, one makes the same kind of analysis used for the proof of Proposition 3, along with the following monotonicity property which is easy to verify: for fixed a, if (1.3) holds, then for any x0 > 0, Ωˆ + (b, a) does not depend on {b(x), 0 x x0 } and it is non-decreasing as a function of {b(x), x > x0 }. In Propositions 3 and 4, the coefficients a and b are such that (1.3) holds. When (1.6) holds instead, the analysis is more complicated. We state the following analogous result for the case that (1.6) holds. Consider the following analog of (2.11): c1 (1 + x)k a(x) c2 (1 + x)k , x c1 (1 + x)m 0
b(y) dy c2 (1 + x)m , (1 + y)k
k ∈ R,
for large x, m > 0, 0 < c1 < c2 ,
(2.13)
and the following analog of (2.12): c1 (1 + x)k a(x) c2 (1 + x)k , k > 1, x b(y) dy is bounded in x. (1 + y)k
(2.14)
0
It can be checked that under (2.13) or (2.14), a and b satisfy (1.6). Proposition 5. Consider HD on R+ . Under some mild regularity conditions on a and b one has the following: 1. Assume that a and b satisfy (2.13). (i) If 2m + k − 2 > 0, then σess (HD ) = ∅; (ii) If 2m + k − 2 = 0, then 0 < inf σess (HD ) < ∞; (iii) If 2m + k − 2 < 0, then inf σess (HD ) = 0. 2. Assume that a and b satisfy (2.14). Then inf σess (HD ) = 0. In particular, HD possesses a compact resolvent if and only if 1(i) holds.
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To prove Proposition 5, one uses (1.18). This essentially reduces the problem to the one considered in Proposition 4. We end this section with an example of the phenomenon mentioned in Remark 6. On R+ we give an example with a1 = a2 = 1, and withb1 and b2 chosen appropriately so that (1.3) holds x x for a1 , b1 and a2 , b2 , and such that B1 (x) ≡ 0 b1 (y) dy B2 (x) ≡ 0 b2 (y) dy, but such that d 1 d2 − b inf σ − > 0 and 1 2 dx 2 dx
d 1 d2 − b inf σess − = ∞, 1 2 dx 2 dx
(2.15)
while 1 d2 d d 1 d2 − = inf σ = 0. − b − b inf σ − 2 ess 2 2 dx 2 dx 2 dx 2 dx Let b1 (x) = −x so that B1 (x) =
x 0
(2.16)
2
b1 (y) dy = − x2 . Then
Ω + (b1 , 1) < ∞
and Ωˆ + (b1 , 1) = 0,
so (2.15) holds. It is not hard to construct a b2 so that B2 (x) < B1 (x), but such that for each positive integer n, there exists an interval of length n over which b2 is identically 0. We will now show that Ω + (b2 , 1) = Ωˆ + (b2 , 1) = ∞; thus, (2.16) holds. Using Theorem 1 and the probabilisd d2 tic representation in (1.15), we have for the diffusion corresponding to 12 dx 2 + b2 dx that 1 8Ω + (b
2 , 1)
sup λ 0: Ex exp(λτ0 ) < ∞
1 2Ω + (b
2 , 1)
,
x > 0.
(2.17) 2
d Now for Brownian motion (that is, the driftless diffusion corresponding to the operator 12 dx 2 ) on the interval (0, n), one has Ex exp(λ(τ0 ∧ τn )) < ∞, for x ∈ (0, n), if and only if λ is less than d2 the first eigenvalue for the operator − 12 dx 2 on (0, n) with the Dirichlet boundary condition at 0 2
and n [6, Chapter 3]; that is, if and only if λ < π2n . Since the drift b2 has intervals of length n over which it vanishes, it follows by comparison with the Brownian motion that for the diffusion d2 d corresponding to 12 dx 2 + b2 dx , if xn is chosen along such an interval, then Exn exp(λτ0 ) = ∞, if 2
λ π2n . Since the finiteness or infiniteness of the expectation is independent of the starting point, it follows that in fact this holds for all x > 0, not just for some xn . Since n is arbitrary, it follows that sup{λ 0: Ex exp(λτ0 ) < ∞} = 0. It then follows from (2.17) that Ω + (b2 , 1) = ∞, and then by the definition of Ωˆ + (b2 , 1), also Ωˆ + (b2 , 1) = ∞. 3. Proofs of Theorems 1–3 and of (1.15) Proof of Theorem 1. By Remark 2, it suffices to treat the case in which (1.3) holds. Extend (l,∞) denote the a and b continuously from [0, ∞) to (−1, ∞). For each l ∈ (−1, ∞), let HD corresponding self-adjoint diffusion operator on (l, ∞) with the Dirichlet boundary condition at x = l. Consider the problem
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1 (au ) + bu + λu = 0, 2 u > 0, x ∈ (l, ∞).
x ∈ (l, ∞); (3.1)
Let
λc (l) = sup λ: there is a solution to (3.1) .
(3.2)
By the criticality theory of second-order elliptic operators, there is a positive solution to the (l,∞) above equation for all λ λc (l) [6, Chapter 4, Section 3] and one has inf σ (HD ) = λc (l) [6, Chapter 4, Section 10]. It follows from the criticality theory that λc (l) is right-continuous [6, Chapter 4, Section 4]. However, in what follows we will need left-continuity. We claim that λc (l) is continuous in l.
(3.3)
We postpone the proof of (3.3) until the end of the proof of Theorem 1. Note that any positive solution as above on (l, ∞) with l < 0 is also a positive solution on [0, ∞) and can be normalized by u(0) = 1. Note also that (3.4) below always has a solution if λ = 0. From these facts, it follows that if we consider the problem 1 (au ) + bu + λu = 0, 2 u > 0, x ∈ (0, ∞);
x ∈ (0, ∞);
u(0) = 1,
(3.4)
then
inf σ (HD ) = sup λ 0: there is a solution to (3.4) .
(3.5)
Thus, in order to prove (1.10), it suffices to prove the following proposition. Proposition 6. Assume that (1.3) holds. 1 , there is no solution to (3.4); 2Ω + (b,a) 1 0 < λ < 8Ω + (b,a) , there is a solution to (3.4).
(i) For λ > (ii) For
The proof of part (i) of Proposition 6 is easy, but the proof of part (ii) is non-trivial. The proof of the proposition is given in the next section. Once (1.10) is proved, one proves (1.11) as follows. An old result of Persson [4], slightly modified to accommodate the case of a half-line, states that (l,∞) . inf σess (HD ) = lim inf σ HD l→∞
Letting
(3.6)
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Ωl+ (b, a) = sup x>l
x
1 exp −2B(y) dy a(y)
∞
3299
exp 2B(y) dy ,
(3.7)
x
l (l,∞)
it follows by applying (1.10) to HD 1
8Ωl+ (b, a)
that
(l,∞) inf σ HD
1
(3.8)
. 2Ωl+ (b, a)
We will show that Ωˆ + (b, a) = lim Ωl+ (b, a).
(3.9)
l→∞
Now (1.11) follows from (3.6), (3.8) and (3.9). We now prove (3.9). From the definition of Ωˆ + (b, a), one has for any l > 0, x
Ωl+ (b, a) lim sup x→∞
1 exp −2B(y) dy a(y)
= lim sup x→∞
exp 2B(y) dy
x
l
x
∞
1 exp −2B(y) dy a(y)
∞
exp 2B(y) dy = Ωˆ + (b, a).
(3.10)
x
0
On the other hand, for n = 1, 2, . . . , there exist x0,n and xn with x0,n < xn and limn→∞ xn = ∞, and such that lim sup Ωl+ (b, a) − l→∞
1 n
xn
1 exp −2B(y) dy a(y)
x0,n
xn 0
∞
exp 2B(y) dy
xn
1 exp −2B(y) dy a(y)
∞
exp 2B(y) dy .
(3.11)
xn
Letting n → ∞ in (3.11) and again using the definition of Ωˆ + (b, a), we obtain lim supl→∞ Ωl+ (b, a) Ωˆ + (b, a). Now (3.9) follows from this and (3.10). We now return to prove (3.3). As noted previously, we only need prove left-continuity. Without loss of generality, we prove left-continuity at l = 0. From its definition, λc is non-decreasing. Let λ1 < λ2 < λc (0). It suffices to show that for > 0 sufficiently small, there is a solution to (3.1) with l = − and some λ λ1 . By assumption, there is a solution to (3.1) with l = 0 and λ = λ2 . Let u be such a solution. Then u(0+ ) = limx→0+ u(x) and u (0+ ) = limx→0+ u (x) exist and are finite. This is because any solution to (3.4) must be a linear combination of Φ1 and Φ2 , where Φ1 and Φ2 are two linearly independent solutions to 12 (au ) + bu + λu = 0. If u(0+ ) > 0, then solving the linear equation for x < 0 using the boundary conditions u(0+ ) and u (0+ ) at x = 0, one can extend the solution u a little bit to the left so that it satisfies (3.1) with l = − and λ = λ2 , completing the proof.
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Assume now that u(0+ ) = 0. We will show that there exists a uˆ which is a solution to (3.1) ˆ > 0. Thus, from the previous argument, we can extend with l = 0 and λ = λ1 , and such that u(0) uˆ a little bit to the left so that it satisfies (3.1) with l = − and λ = λ1 , completing the proof. Thus, it remains to show that such a uˆ exists. Let φ be a smooth compactly supported function on R satisfying φ(0) = 1 and ( 12 (aφ ) + bφ + λ1 φ)(0) = 0. Let v = u + δφ, where u is as above and δ > 0. If δ is sufficiently small, then v > 0 on (0, ∞) and 12 (av ) + bv + λ1 v = −(λ2 − λ1 )u + δ( 12 (aφ ) + bφ + λ1 φ) < 0 on (0, ∞). Thus, v is a sub-solution for (3.1) with l = 0 and λ = λ1 , and v(0) = δ > 0. We claim that there is a solution uˆ to (3.1) with l = 0 ˆ = δ. Indeed, let uˆ n solve 12 (a uˆ n ) + buˆ n + λ1 uˆ n = 0 in (0, n), with and λ = λ1 , and with u(0) uˆ n (0) = δ and uˆ n (n) = 0. Then by the maximum principal, uˆ n is increasing in n and uˆ n v; thus uˆ ≡ limn→∞ uˆ n exists and is the desired function. 2 Proof of Theorem 2. For HD on the entire line R, the result of Persson, given in (3.6) for R+ , is (−∞,−l) (l,∞) inf σess (HD ) = lim min inf σ HD , inf σ HD , l→∞
where H (−∞,−l) denotes the corresponding self-adjoint operator on (−∞, −l) with the Dirichlet boundary condition at x = −l. Theorem 2 follows from this and the above proof of Theorem 1. 2 Proof of Theorem 3. By the criticality theory of second-order elliptic operators [6, Chapter 4, Sections 4 and 10], (−l,∞) (−∞,l) inf σ (HD ) = lim inf σ HD = lim inf σ HD . l→∞
l→∞
(3.12)
(−l,∞)
. One simply lets −l play the role played by 0 in Theorem 1. Theorem 1 can be applied to HD x (There is no need to change the definition of B(x) = 0 ab (y) dy, because the lower limit 0 can ∞ 1 be replaced by any x0 without affecting the formulas.) Thus, if a(x) exp(−2B(x)) dx = ∞, then defining + (b, a) = sup Ω−l
x
x>−l
−l
1 exp −2B(y) dy a(y)
∞
exp 2B(y) dy ,
x
we have 1 + 8Ω−l (b, a)
(−l,∞) inf σ HD
1
. + 2Ω−l (b, a)
(3.13)
But lim Ω + (b, a) = l→∞ −l
x sup x∈R
−∞
1 exp −2B(y) dy a(y)
∞ x
exp 2B(y) dy .
(3.14)
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In the case that (1.19) or (1.21) holds, Theorem 3 follows from (3.12)–(3.14). The case that (1.22) holds is obtained from the case that (1.21) holds by interchanging the roles of the positive and negative half-lines. For the case that (1.23) holds, one proceeds as above in the case that (1.21) + (b, a) now defined by holds, but with Ω−l + Ω−l (b, a) =
x sup x>−l
h−2 b,a (y)
−l
1 exp −2B(y) dy a(y)
∞
2B(y) dy .
h2b,a (y) exp
2
x
We end this section by proving that (1.15) holds in the case that (1.3) is in effect, that is, in the case that Px (τ0 < ∞) = 1. From (3.5), it is enough to show that
sup λ 0: there is a solution to (3.4) = sup λ 0: Ex exp(λτ0 ) < ∞ .
(3.15)
Assume first that λ > 0 is such that there exists a solution to (3.4) and let u be a solution. Then u(X(t ∧ τ0 )) exp(λ(t ∧ τ0 )) is a martingale [6, Chapter 2], and thus Ex u X(t ∧ τ0 ) exp λ(t ∧ τ0 ) = u(x).
(3.16)
Letting t → ∞, it follows from Fatou’s lemma that Ex exp(λτ0 ) < ∞. Conversely, assume that λ > 0 is such that Ex exp(λτ0 ) < ∞. Let τn = inf{t 0: X(t) = n}, for n > 0. By the Feynman–Kac formula, un (x) ≡ Ex (exp(λτ0 ); τ0 < τn ) is the solution to the equation 1 (au ) + bu + λu = 0, 2 u(0) = 1, u(n) = 0.
x ∈ (0, n); (3.17)
By the maximum principal, un is increasing in n, and (3.4) will have a solution if and only if limn→∞ un (x) < ∞, in which case u∞ (x) ≡ limn→∞ un (x) is the smallest solution to (3.4). By the monotone convergence theorem and the assumption, we have u∞ (x) = Ex exp(λτ0 ) < ∞. Thus λ is such that there is a solution to (3.4). 4. Proof of Proposition 6 Let λ > 0 and let fn be the unique solution to 1 (af ) + bf + λf = 0 in [0, n]; 2 f (0) = 1, f (n) = 0. Integrating twice and using the boundary conditions gives x fn (x) = 1 + cn 0
1 exp −2B(y) dy a(y)
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x − 2λ
1 dy exp −2B(y) a(y)
0
y
dz fn (z) exp 2B(z) ,
(4.1)
0
where cn =
−1 + 2λ
n 0
dx
x 1 a(x) exp(−2B(x)) 0 dy fn (y) exp(2B(y)) . n 1 0 a(x) exp(−2B(x)) dx
Note that, by the maximum principle, ∞ f1 n 0 and fn is non-decreasing in n. Let f∞ ≡ limn→∞ fn . exp(−2B(x)) dx = ∞. Thus, c∞ ≡ limn→∞ cn = Recall that by assumption, 0 a(x) ∞ 2λ 0 f∞ (x) exp(2B(x)) dx. Letting n → ∞ in (4.1) gives x f∞ (x) = 1 + 2λ
1 dy exp −2B(y) a(y)
∞
dz f∞ (z) exp 2B(z) .
(4.2)
y
0
By the maximum principle and the construction of f∞ , either f∞ is the smallest solution to (3.4) or else f∞ = ∞ and there are no solutions to (3.4). Using this characterization, we now proof the two parts of the proposition. Proof of part (i). We will show that the solution f∞ of (4.2) is equal to ∞ if λ > From (4.2) it follows that f∞ is non-decreasing, and thus also that x f∞ (x) 1 + 2λ
1 exp −2B(y) dy a(y)
∞
exp 2B(y) dy f∞ (x).
1 2Ω(b,a) .
(4.3)
x
0
x 1 ∞ If there exists an x for which 2λ( 0 a(y) exp(−2B(y)) dy)( x exp(2B(y)) dy) 1, then (4.3) cannot hold for such an x unless f∞ (x) = ∞. Recalling the definition of Ω(b, a), we conclude 1 . that there is no finite solution to (4.2) if λ > 2Ω(b,a) 1 Proof of part (ii). We will show that there is a finite solution to (4.2) if 0 < λ < 8Ω(b,a) . We assume that Ω(b, a) < ∞ since otherwise there is nothing to prove. In particular then, we may ∞ exp(2B(z)) dz < ∞. Fix λ > 0 and define the operator assume that
x Tf (x) ≡ 1 + 2λ
1 dy exp −2B(y) a(y)
∞
dz f (z) exp 2B(z) ,
(4.4)
y
0
∞
operating on the domain DT ≡ {f : f 0 and f (z) exp(2B(z)) dz < ∞}. Note that, by assumption, 1 ∈ DT . One can solve (4.2) by iterations. Indeed, it is clear that T n 1 is increasing in n and that f∞ = limn→∞ T n 1, where T n denotes the nth iterate of T . Thus, to prove the existence of a finite solution to (4.2) it is sufficient (and necessary) to show that lim T n 1 < ∞.
n→∞
(4.5)
R.G. Pinsky / Journal of Functional Analysis 256 (2009) 3279–3312
Define a norm by f =
∞
3303
f (x) exp(2B(x)) dx. We will prove (4.5) by showing that
0
lim T n 1 < ∞.
(4.6)
Tf = 1 + 2λS1 f + 2λS2 f,
(4.7)
n→∞
Integrating by parts, we have
where x S1 f (x) =
1 exp −2B(z) dz a(z)
∞
f (z) exp 2B(z) dz ,
x
0
x S2 f (x) =
dz f (z) exp 2B(z)
0
z dt
1 exp −2B(t) . a(t)
(4.8)
0
Thus, T n1 = 1 +
n (2λ)k (S1 + S2 )k 1.
(4.9)
k=1
It is immediate from the definitions of S1 and Ω(b, a) that |S1 1(x)| Ω(b, a), and thus S1 1 Ω(b, a)1.
(4.10)
We will prove the following inequalities: S2 f Ω(b, a)f ; n S S2 f Ω(b, a) S n−1 S2 f + S n f , 1
1
1
(4.11) n 1,
(4.12)
where S 0 is defined to be the identity operator. From (4.10)–(4.12), it follows that k Sδ1 . . . Sδk 1 2Ω(b, a) 1,
(4.13)
where δj = 1 or 2 for each j = 1, . . . , k. From (4.9) and (4.13) it follows that n n n k k T 1 1 + 8λΩ(b, a) . (2λ)k 2k 2Ω(b, a) = 1 + k=1
From (4.14) one concludes that (4.6) holds if λ <
k=1 1 8Ω(b,a) .
(4.14)
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We now prove (4.11) and (4.12). Integrating by parts, we have ∞ S2 f =
dx exp 2B(x)
x
0
dz f (z) exp 2B(z)
z
0
∞ =−
exp 2B(z) dz
dz f (z) exp B(z)
0
∞
∞ dx
∞ Ω(b, a)
z
1 exp −2B(t) dt a(t)
∞ 0
0
x 1 exp −2B(t) exp 2B(z) dz f (x) exp 2B(x) dt a(t)
x
0
1 exp −2B(t) a(t)
0
x
x
+
dt
0
f (x) exp 2B(x) dx = Ω(b, a)f ,
(4.15)
0
proving (4.11). We now turn to (4.12). We will write out the proof for n = 2; the very same technique holds for general n. We have 2 S S2 f = 1
∞
dx exp 2B(x)
x
0
1 exp −2B(z) dz a(z)
0
∞ ×
dt exp 2B(t)
x
t
1 exp −2B(s) ds a(s)
∞
dl exp 2B(l) S2 f (l) .
(4.16)
t
0
Integrating by parts gives ∞
dl exp 2B(l) S2 f (l)
t
∞ =
dl exp 2B(l)
t
l
dr f (r) exp 2B(r)
0
∞ =−
exp 2B(ν) dν
l
r dρ
1 exp −2B(ρ) a(ρ)
1
l
drf (r) exp 2B(r)
0
r
1 exp −2B(ρ) dρ a(ρ)
0
∞ ∞ l + dl dν exp 2B(ν) f (l) exp 2B(l) dρ t
l
0
1 exp −2B(ρ) a(ρ)
∞ t
R.G. Pinsky / Journal of Functional Analysis 256 (2009) 3279–3312
∞
exp 2B(ν) dν
t
t
dr f (r) exp 2B(r)
0
r
1 dρ exp −2B(ρ) a(ρ)
0
∞ ∞ l dν exp 2B(ν) f (l) exp 2B(l) dρ + dl t
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l
1 exp −2B(ρ) . a(ρ)
(4.17)
0
Substituting (4.17) in (4.16) and using the definition of Ω(b, a), we obtain 2 S S 2 f
∞
1
dx exp 2B(x)
0
x
1 exp −2B(z) dz a(z)
0
∞ ×
dt exp 2B(t)
t
x
t ×
+
∞
dr f (r) exp 2B(r)
dx exp 2B(x)
x
0
dν exp 2B(ν)
t
0
0
∞
1 exp −2B(s) ds a(s) r
1 dρ exp −2B(ρ) a(ρ)
0
1 exp −2B(z) dz a(z)
0
∞ ×
dt exp 2B(t)
x
t
1 exp −2B(s) ds a(s)
0
∞ ∞
l 1 × exp −2B(s) dl dν exp 2B(ν) f (l) exp 2B(l) ds a(s) t
l
∞ Ω(b, a)
dx exp 2B(x)
0
∞ ×
dt exp 2B(t)
t
+ Ω(b, a)
dx exp 2B(x)
0
x
1 exp −2B(z) dz a(z)
dr f (r) exp 2B(r)
0
∞
×
0
0
x
∞
x
dt exp 2B(t)
x
r
1 exp −2B(ρ) dρ a(ρ)
0
1 exp −2B(z) dz a(z)
0
t 0
1 exp −2B(s) ds a(s)
= Ω(b, a)S1 S2 f + Ω(b, a)S12 f .
∞ t
dl f (l) exp 2B(l)
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5. Application to multi-dimensional diffusion operators Consider the multi-dimensional diffusion operator 1 1 HD = − ∇ · a∇ − a∇Q · ∇ = − exp(−2Q)∇ · a exp(2Q)∇ 2 2
on Rd , d 2,
(5.1)
where a = {ai,j }ni,j =1 ∈ C 1 (Rd ) is positive definite and Q ∈ C 1 (Rd ). One can realize HD as a non-negative, self-adjoint operator on L2 (Rd , exp(2Q) dx) via the closure of the Friedrichs extension of the non-negative quadratic form QD (f, g) =
1 2
∇f a∇g exp(2Q) dx, Rd
defined for f, g ∈ C01 (Rd ). For l > 0, let Bl (0) ⊂ Rd denote the ball of radius l centered at the origin, and let HDl be the self-adjoint operator on Rd − B¯ l (0) corresponding to HD with the Dirichlet boundary condition at ∂Bl (0). More precisely, HDl is the Friedrichs extension of the closure of the non-negative quadratic form QlD (f, g) =
1 2
∇f a∇g exp(2Q) dx, Rd −B¯ l (0)
defined for f, g ∈ C01 (Rd − B¯ l (0)). The result of Persson [4] noted in Section 3 gives inf σess (HD ) = lim inf σ HDl . l→∞
(5.2)
We will give upper and lower bounds on inf σ (HD ) and inf σ (HDl ) in terms of the corresponding infima for certain one-dimensional operators. From (5.2), this will then also give upper and lower bounds on inf σess (HD ). Applying Theorem 1 to the one-dimensional operators will then yield explicit bounds on inf σ (HD ) and inf σess (HD ). Letting r = |x| and φ ∈ S d−1 denote spherical coordinates, let Arad-har (r, φ) =
x −1 x a (x) |x| |x|
−1 (5.3)
denote the representation in spherical coordinates of the reciprocal of the radially directed x −1 x x quadratic expression ( |x| a (x) |x| ). Let Qr (x) = ∇Q(x) · |x| denote the radial derivative of Q. Write Qr in spherical coordinates as Qr (r, φ). For each φ ∈ S d−1 , define the one-dimensional diffusion operator Hrad-har;φ on R+ by 1 d d d −1 d d Arad-har (r, φ) − Arad-har (r, φ) − Arad-har (r, φ)Qr (r, φ) 2 dr dr 2r dr dr d d 1 1−d Arad-har (r, φ)r d−1 exp 2Q(r, φ) on R+ . (5.4) exp −2Q(r, φ) =− r 2 dr dr
Hrad-har;φ = −
R.G. Pinsky / Journal of Functional Analysis 256 (2009) 3279–3312
3307
x x Let ( |x| a(x) |x| )(r, φ) denote the representation in spherical coordinates of the radially dix x rected quadratic expression ( |x| a(x) |x| ). Let
Arad-avg (r) =
x x S d−1 ( |x| a(x) |x| )(r, φ) exp(2Q(r, φ)) dφ
S d−1
exp(2Q(r, φ)) dφ
(5.5)
and Qr;avg (r) =
S d−1Qr (r, φ) exp(2Q(r, φ)) dφ S d−1
exp(2Q(r, φ)) dφ
(5.6)
.
Define the one-dimensional diffusion operator Hrad-avg on R+ by 1 d d d −1 d d Arad-avg (r) − Arad-avg (r) − Arad-avg (r)Qr;avg (r) 2 dr dr 2r dr dr d d 1 Arad-avg (r) exp 2β(r, φ) on R+ , where = − exp −2β(r) 2 dr dr d −1 1 β(r, φ) = log r + log exp 2Q(r, φ) dφ. 2 2
Hrad-avg = −
(5.7)
S d−1 (l,∞)
(l,∞)
Let Hrad-har,φ and Hrad-avg denote the corresponding operators on (l, ∞) with the Dirichlet boundary condition at r = l, as defined in Section 3. Remark 13. Theorem 1 can be applied to the operators Hrad-har,φ and Hrad-avg even though their (l,∞) (l,∞) drifts are not continuous up to 0. Indeed, the theorem applies directly to Hrad-har,φ and Hrad-avg , for l > 0, and one has (l,∞) inf σ (Hrad-har,φ ) = lim inf σ Hrad-har,φ and l→0
(l,∞) inf σ (Hrad-avg ) = lim inf σ Hrad-avg , l→0
see [6, Chapter 4, Sections x 4 and 10]. The one change that needs to be made is that B should be defined as B(x) = x0 ab (y) dy, for some x0 > 0. (In the proof of Corollary 2 below we use x0 = 1.) We will prove the following theorem. Theorem 4. inf inf σ (Hrad-har;φ ) inf σ (HD ) inf σ (Hrad-avg )
φ∈S d−1
and
(l,∞) (l,∞) (l,∞) inf σ Hrad-avg . inf inf σ Hrad-har;φ inf σ HD
φ∈S d−1
Applying Theorem 1 and (5.2) to Theorem 4, the following corollary is immediate.
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R.G. Pinsky / Journal of Functional Analysis 256 (2009) 3279–3312
Corollary 1. 1 8 supφ∈S d−1
Ω + (β
rad-har;φ , Arad-har (·, φ))
inf σ (HD )
1 2Ω + (β
rad-avg , Arad-avg )
;
1 1 inf σess (HD ) , + + ˆ ˆ 8 supφ∈S d−1 Ω (βrad-har;φ , Arad-har (·, φ)) 2Ω (βrad-avg , Arad-avg ) where d −1 , βrad-har;φ (r) = Arad-har (r, φ) Qr (r, φ) + 2r d −1 , βrad-avg (r) = Arad-avg (r) Qr;avg (r) + 2r and Ω + and Ωˆ + are as in Theorem 1. In particular, Ωˆ + (βrad-avg , Arad-avg ) = 0 is a necessary condition for HD to possess a compact resolvent and Ωˆ + (βrad-har;φ , Arad-har (·, φ)) = 0, for all φ ∈ S d−1 , is a sufficient condition. We give the following application of Corollary 1. Corollary 2. Let HD = − 12 ∇ · a∇ on Rd , d 1. (i) If lim
infφ∈S d−1 Arad-har (r, φ) r2
r→∞
= ∞,
then σess (HD ) = ∅ and HD possesses a compact resolvent; (ii) If lim
r→∞
Arad-avg (r) = 0, r2
then inf σess (HD ) = 0; (iii) If infφ∈S d−1 Arad-har (r, φ) λr 2 , for large r, then inf σess (HD ) (iv) If Arad-avg (r) Λr 2 , for large r, then inf σess (HD )
Λd 2 2 .
λd 2 8 ;
Remark 14. Let Amin (r) =
inf
|v|=1, |x|=r
va(x)v
and Amax (r) =
sup
|v|=1, |x|=r
va(x)v ,
and note that inf Arad-har (r, φ) Amin (r)
φ∈S d−1
and Arad-avg (r) Amax (r).
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3309
Parts (i)–(iii) of the above corollary, with infφ∈S d−1 Arad-har (·, φ) replaced by Amin and Arad-avg replaced by Amax are originally due to Davies [1]. The use of infφ∈S d−1 Arad-har (·, φ) and Arad-avg instead of Amin and Amax is a significant strengthening. For instance, if for |x| 1, the radially x directed vector |x| is an eigenvector for a(x) with eigenvalue γ (|x|) > 1, and all the other eigenvalues of a(x) are equal to 1, then for r 1, one has Arad-har (r, φ) = γ (r) while Amin (r) = 1. A two-dimensional example of such a diffusion matrix is ⎛ a(x) = ⎝
x2
γ (|x|) |x|12 +
x22 |x|2
x1 x2 (γ (|x|) − 1) |x|2 x2 γ (|x|) |x|22
x1 x2 (γ (|x|) − 1) |x|2
+
x12 |x|2
⎞ ⎠.
Switching the roles of the eigenvalues γ (|x|) and 1 above, one has Arad-avg (r) = 1 while Amax (r) = γ (r). A two-dimensional example of such a diffusion matrix is ⎛ a(x) = ⎝
x12 |x|2
x2
x1 x2 (1 − γ (|x|)) |x|2
+ γ (|x|) |x|22
x22 |x|2
x1 x2 (1 − γ (|x|)) |x|2
x2 + γ (|x|) |x|12
⎞ ⎠.
(l,∞)
Proof of Corollary 2. By the standard variational formula for inf σ (HD ), it follows that inf σ (HD(l,∞) ) is non-decreasing in a, and thus by (5.2), inf σess (HD ) is also non-decreasing in a. Also from (5.2), it follows that inf σess (HD ) does not depend on {a(x), 0 < x l}, for any l > 0. In light of these facts, (i) and (ii) follow from (iii) and (iv). Also, by the monotonicity in a, for the proof of (iii) we may assume that Arad-har (r, φ) = λr 2 , for large r, and for the proof of (iv) we may assume that Arad-avg (r) = Λr 2 , for large r. We consider (iii), the proof of (iv) following mutatis mutandi. From Corollary 1, inf σess (HD )
1 8 supφ∈S d−1
Ωˆ + (A
d−1
rad-har (·, φ) 2r
, Arad-har (·, φ))
r For the pair of arguments of Ωˆ + in (5.8), one has exp(2B(r)) = exp( 1 we are assuming that Arad-har (r, φ) = λr 2 for large r, we have ∞
1 Arad-har (r, φ)
d−1 s
.
(5.8)
ds) = r d−1 . Since
r 1−d dr < ∞;
thus, (1.6) holds and ∞ (r) = d−1 rad-har (·,φ) 2r ,Arad-har (·,φ)
hA
r
1 Arad-har (s, φ)
s 1−d ds =
r −d , λd
for large r.
(5.9)
Writing hφ = hA to simplify notation, for any l > 0, one has from the d−1 rad-har (·,φ) 2r ,Arad-har (·,φ) + ˆ definition of Ω ,
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d −1 , Arad-har (·, φ) Ωˆ + Arad-har (·, φ) 2r
r
∞ 1 −2 1−d 2 d−1 hφ (s) hφ (s)s ds = lim sup s Arad-har (s, φ) r→∞ r
l
∞
−1 −1 2 d−1 = lim sup hφ (r) − hφ (l) hφ (s)s ds . r→∞
(5.10)
r
(In the original definition of Ωˆ + , l above is replaced by 0, however using l does not change the −d value of the expression.) Choosing l sufficiently large and substituting hφ (r) = rλd in (5.10), one concludes that d −1 1 , Arad-har (·, φ) = 2 . Ωˆ + Arad-har (·, φ) 2r λd Part (iii) now follows from this and (5.8).
2
Proof of Theorem 4. We will prove the inequalities for HD ; the exactly the same method works (l,∞) for HD . The variational formula for inf σ (HD ) gives
inf σ (HD ) = inf
∇f a∇f exp(2Q) dx , 2 Rd f exp(2Q) dx
1 2 Rd
(5.11)
where the infimum is over f ∈ C01 (Rd ). Using spherical coordinates (r, φ), and letting infradial denote the infimum over radially symmetric functions f ∈ C01 (Rd ), we have from (5.11),
∇f a∇f exp(2Q) dx 2 radial Rd f exp(2Q) dx 1 ∞ x x 2 d−1 dr 2 0 (f (r)) ( S d−1 ( |x| a(x) |x| )(r, φ) exp(2Q(r, φ)) dφ)r ∞ = inf 2 d−1 dr radial 0 f (r)( S d−1 exp(2Q(r, φ)) dφ)r 1 ∞ 2 2 0 (f (r)) Arad-avg (r) exp(2β(r)) dr ∞ = inf , (5.12) 2 radial 0 (f (r) exp(2β(r)) dr
inf σ (HD ) inf
1 2 Rd
1 where β(r) = d−1 2 log r + 2 log S d−1 exp(2Q(r, φ)) dφ. The infimum on the right-hand side of (5.12) is the bottom of the spectrum of the operator Hrad-avg defined in (5.7). This gives the upper bound. We now prove the lower bound. By the Schwarz inequality, x −1 x 2 x fr2 (x) = ∇f (x) · a (x) ∇f (x)a(x)∇f (x) . |x| |x| |x|
(5.13)
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Writing (5.13) in polar coordinates and using the definition of Arad-har , one has (∇f a∇f )(r, φ) Arad-har (r, φ)fr2 (r, φ).
(5.14)
By the variational formula, for any g ∈ C01 (R+ ) and φ ∈ S d−1 , 1 2
∞
2 Arad-har (r, φ) g (r) r d−1 exp 2Q(r, φ) dr
0
∞ inf σ (Hrad-har;φ )
g 2 (r)r d−1 exp 2Q(r, φ) dr.
(5.15)
0
From (5.14) and (5.15) one has for f ∈ C01 (Rd ), 1 2
∞
1 ∇f a∇f exp(2Q) dx 2
Rd
Arad-har (r, φ)fr2 (r, φ)r d−1 exp 2Q(r, φ) dr dφ
S d−1 0
f 2 (r, φ)r d−1 exp 2Q(r, φ) dr dφ
inf σ (Hrad-har;φ )
S d−1
R+
inf inf σ (Hrad-har;φ ) φ∈S d−1
f 2 exp(2Q) dx.
(5.16)
Rd
From (5.16) we conclude that
inf σ (HD ) = inf
1 2 Rd ∇f a∇f exp(2Q(x)) dx 2 Rd f exp(2Q) dx
inf inf σ (Hrad-har;φ ). φ∈S d−1
2
Acknowledgments The author thanks Martin Kolb, a doctoral student at the University of Kaiserslautern, for bringing to his attention the work of Muckenhoupt [3] and for his helpful comments. References [1] E.B. Davies, L1 properties of second order elliptic operators, Bull. London Math. Soc. 17 (1985) 417–436. [2] V. Mazy’a, Analytic criteria in the qualitative spectral analysis of the Schrödinger operator, in: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, Proceedings of Symposia in Pure Mathematics, vol. 76, part 1, Amer. Math. Soc., Providence, RI, 2007, pp. 257–288. [3] B. Muckenhoupt, Hardy’s inequalities with weights, Studia Math. 44 (1972) 31–38. [4] A. Persson, Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator, Math. Scand. 8 (1960) 143–153. [5] R.G. Pinsky, A new approach to the Martin boundary via diffusions conditioned to hit a compact set, Ann. Probab. 21 (1993) 453–481.
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[6] R.G. Pinsky, Positive Harmonic Functions and Diffusion, Cambridge Stud. Adv. Math., vol. 45, Cambridge Univ. Press, Cambridge, 1995. [7] M. Reed, B. Simon, Methods of Modern Mathematical Physics, II, Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. [8] M. Reed, B. Simon, Methods of Modern Mathematical Physics, IV, Analysis of Operators, Academic Press, New York, 1975.
Journal of Functional Analysis 256 (2009) 3313–3341 www.elsevier.com/locate/jfa
Simple compact quantum groups I ✩ Shuzhou Wang 1 Department of Mathematics, University of Georgia, Athens, GA 30602, USA Received 28 August 2008; accepted 22 October 2008 Available online 13 November 2008 Communicated by Alain Connes
Abstract The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups Bu (Q) for Q ∈ GL(n, C) satisfying QQ¯ = ±In , n 2; (b) The quantum automorphism groups Aaut (B, τ ) of finite-dimensional C ∗ -algebras B endowed with the canonical trace τ when dim(B) 4, including the quantum permutation groups Aaut (Xn ) on n points (n 4); (c) The standard deformations Kq of simple compact Lie groups K and their twists Kqu , as well as Rieffel’s deformation KJ . © 2008 Elsevier Inc. All rights reserved. Keywords: Simple quantum groups; Woronowicz C ∗ -algebras; Deformation quantization; Noncommutative geometry; Hopf algebras
1. Introduction The theory of quantum groups saw spectacular breakthroughs in the 1980s when on the one hand Drinfeld [24] and Jimbo [27] discovered the quantized universal enveloping algebras of semisimple Lie algebras based on the work of the Faddeev school on the quantum inverse scattering method, and on the other hand Woronowicz [58–60] independently discovered quantum deformations of compact Lie groups and formulated the axioms for compact quantum groups. Further work of Rosso [38,40], Soibelman and Vaksman, Levendorskii [30,43,44] showed that “compact real forms” Kq of the Drinfeld–Jimbo quantum groups and their twists Kqu are exam✩
Research supported in part by the National Science Foundation grant DMS-0096136. E-mail address:
[email protected]. 1 Fax: +1 706 542 2573.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.020
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ples of compact quantum groups in the sense of Woronowicz. Most notable of these is the work of Soibelman [43] based on his earlier joint work with Vaksman [44], in which a general Kirillov type orbit theory of representations of the quantum function algebras of deformed simple compact Lie groups was developed using the orbits of dressing transformations (i.e. symplectic leaves) in Poisson Lie group theory (see also the monograph [29] for more detailed treatment). Starting in his PhD thesis [48], the author of the present article took a different direction from the above by viewing quantum groups as intrinsic objects and found in a series of papers (including [47] in collaboration with Van Daele) several classes of compact quantum groups that cannot be obtained as deformations of Lie groups. The most important of these are the universal compact quantum groups of Kac type Au (n) and their self-conjugate counterpart Ao (n) [49], the more general universal compact quantum groups Au (Q) and their self-conjugate counterpart Bu (Q) [47,50], where Q ∈ GL(n, C), and the quantum automorphism groups Aaut (B, tr) of finite-dimensional C ∗ -algebras B endowed with a tracial functional tr, including the quantum permutation groups Aaut (Xn ) on the space Xn of n points [53]. Further studies of these quantum groups reveal remarkable properties: (1) According to deep work of Banica [2–4], the representa¯ is a scalar) are tion rings (also called the fusion rings) of the quantum groups Bu (Q) (when QQ all isomorphic to that of SU(2) (see [2, Théorème 1]), and the representation rings of Aaut (B, τ ) (when dim(B) 4, τ being the canonical trace on B) are all isomorphic to that of SO(3) (see [4, Theorem 4.1]), and the representation ring of Au (Q) is almost a free product of two copies of Z (see [3, Théorème 1]); (2) The compact quantum groups Au (Q) admit ergodic actions on both finite and infinite injective von Neumann factors [54]; (3) The special Au (Q)’s for positive Q and Bu (Q)’s for Q satisfying the property QQ¯ = ±In are classified up to isomorphism using respectively the eigenvalues of Q (see [56, Theorem 1.1]) and polar decomposition of Q and eigenvalues of |Q| (see [56, Theorem 2.4]), and the general Au (Q)’s and Bu (Q)’s for arbitrary Q have explicit decompositions as free products of the former special ones (see [56, Theorems 3.1, 3.3 and Corollaries 3.2, 3.4]); (4) Certain quantum symmetry groups in the theory of subfactors were found by Banica [6,7] to fit in the theory of compact quantum groups; (5) The quantum permutation groups Aaut (Xn ) admit interesting quantum subgroups that appear in connection with other areas of mathematics, such as the quantum automorphism groups of finite graphs and the free wreath products discovered by Bichon [15,16]. See also [17] and [8–14] and the references therein for other interesting results related to the quantum permutation groups. The purpose of this article is to initiate a study of simple compact quantum groups. It focuses on the introduction of a notion of simple compact quantum groups and first examples. It is shown that the compact quantum groups mentioned in the last two paragraphs are simple in generic cases. The paper is organized as follows. In Section 2, we recall the notion of a normal quantum subgroup N of a compact quantum group G introduced in [48,49], on which the main notion of a simple compact quantum group in this paper depends. We prove several equivalent conditions for N to be normal, including one that stipulates that the quantum coset spaces G/N and N \G are identical. Further applications of these are contained in [57]. In Section 3 the notion of simple compact quantum groups is introduced. In the classical setting, the notion of a simple compact Lie group can be defined in two ways: one using Lie algebra and the other using the group itself. Though the universal enveloping algebras of simple Lie groups can be deformed into the quantized universal enveloping algebras [24,27], we have no analog of Lie algebras for general quantum groups. Hence we formulate the notion of a simple compact quantum group using group theoretical language so that our notion reduces precisely to the notion of a simple compact Lie group when the quantum group is a compact Lie group:
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Definition 1.1. A compact matrix quantum group is called simple if it is connected and has no non-trivial connected normal quantum subgroups and no non-trivial representations of dimension one. Here a compact quantum group G is called connected if the coefficients of every non-trivial irreducible representation of G generate an infinite-dimensional C ∗ -algebra. In the classical situation, the fact that a simple compact Lie group has no non-trivial representations of dimension one is a consequence of the deep Weyl dimension formula. It is not known if the postulate that a simple compact matrix quantum group has no non-trivial representations of dimension one follows from the other postulates in the definition, for we do not have a dimension formula for irreducible representations of a general simple compact quantum group except the specific examples studied in this paper. After preparatory work in Sections 2 and 3, the main examples of this paper are studied in Sections 4 and 5. Recall [4] that the canonical trace τ on a finite-dimensional C ∗ -algebra B is the restriction of the unique tracial state on the algebra L(B) of operators on B. In Section 4, we prove that Bu (Q) and Aaut (B, τ ) are simple: ¯ = ±In and n 2. Then Theorem 1.2. (See Theorem 4.1.) Let Q ∈ GL(n, C) be such that QQ Bu (Q) is a simple compact quantum group. Theorem 1.3. (See Theorem 4.7.) Let B be a finite-dimensional C ∗ -algebra with dim(B) 4 and τ its canonical trace. Then Aaut (B, τ ) is a simple compact quantum group. The proofs of these two results rely heavily on the fundamental work of Banica [2,4] on the structure of fusion rings (i.e. representative rings) of these quantum groups, as well as the technical results on the correspondence between Hopf ∗-ideals and Woronowicz C ∗ -ideals and the reconstruction of a normal quantum subgroup from the identity in the quotient quantum group, which are developed in Section 4 and are of interest in their own right. It is also shown in Section 4 that the closely related quantum group Au (Q) is not simple for any n and any Q ∈ GL(n, C) (see Proposition 4.5). The last Section 5 is devoted to the standard deformations Kq of simple compact Lie groups, their twists Kqu [30,31,43], and Rieffel’s quantum groups KJ [37], where q ∈ R \ {0}, u ∈ 2 (it) and J is an appropriate skew-symmetric transformation on the direct sum t ⊕ t of Cartan subalgebra t of the Lie algebra of K: Theorem 1.4. (See Theorems 5.1 and 5.6.) Let K be a connected and simply connected simple compact Lie group. Then both Kq and its twists Kqu are simple compact quantum groups. Theorem 1.5. (See Theorem 5.4.) Let K be a simple compact Lie group with a toral subgroup T of rank at least two. Then KJ is a simple compact quantum group. The proofs of Theorems 1.4 and 1.5 make use of the work of Lusztig and Rosso [32,39] on representations of quantized universal enveloping algebras, the work of Soibelman and Levendorskii [30,31,43] on quantum function algebras of Kq and Kqu , and the work of Rieffel [37] and the author [51] on strict deformations of Lie groups and quantum groups, as well as the technical results in Section 4 mentioned earlier.
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Classification of simple compact quantum groups and their irreducible representations up to isomorphism are two of the main goals in the study of compact quantum groups. Namely, one would like to develop a theory of simple compact quantum groups that parallels the Killing– Cartan theory and the Cartan–Weyl theory for simple compact Lie groups. To accomplish the first goal, one must first construct all simple compact quantum groups. Though we have given several infinite classes of examples of these in this article, it should be pointed out that the construction of simple compact quantum groups is far from being complete. In fact it is fair to say that we are only at the beginning stage for this task at the moment. One indication of this is that all the simple compact quantum groups known so far have commutative representation rings, and these rings are order isomorphic to the representation rings of compact Lie groups (we call such quantum groups almost classical ). The universal compact matrix quantum groups Au (Q) have a “very” noncommutative representation ring, being close to the free product of two copies of the ring of integers, according to the fundamental work of Banica [3], where Q ∈ GL(n, C) are positive, n 2. However, Au (Q) are not simple quantum groups (see Section 4). Because of their universal property, Au (Q) should play an important role in the construction and classification of simple compact quantum groups with non-commutative representation rings. A natural and profitable approach seems to be to study quantum automorphism groups of appropriate quantum spaces and their quantum subgroups, such as those in [53–55] and the papers of Banica and Bichon and their collaborators [6]–[17]. In retrospect, both simple Lie groups and finite simple groups are automorphism groups, a similar approach for the theory of simple quantum groups should also play a fundamental role. Convention and notation. We assume that all Woronowicz C ∗ -algebras (also called Woronowicz Hopf C ∗ -algebras) considered in this paper to be full unless otherwise explicitly stated, since morphisms between quantum groups are meaningful only for full Woronowicz C ∗ -algebras (cf. [49,52]). For a compact quantum group G, AG , or C(G), denote the underlying Woronowicz C ∗ -algebra and AG denotes the associated canonical dense Hopf ∗-algebra of quantum representative functions on G. Sometimes we also call AG a compact quantum group, referring to G. See [49,59] for more on other unexplained definitions and notations used in this paper. 2. The notion of normal quantum subgroups Before making the notion of simple quantum groups precise, we recall the notion of normal quantum subgroups (of compact quantum groups) introduced in [48,49] and study their properties further. Let (N, π) be a quantum subgroup of a compact quantum group G with surjections π : AG → AN and πˆ : AG → AN . The quantum group (N, π) should be more precisely called a closed quantum subgroup, but we will omit the word closed in this paper, since we do not consider non-closed quantum subgroups. Define AG/N = a ∈ AG (id ⊗ π)(a) = a ⊗ 1N , AN \G = a ∈ AG (π ⊗ id)(a) = 1N ⊗ a , where is the coproduct on AG , 1N is the unit of the algebra AN . We omit the subscript N in 1N when no confusion arises. Similarly, we define AG/N = AG ∩ AG/N ,
and AN \G = AG ∩ AN \G .
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Note that G/N, N\G shall be denoted more precisely by G/(N, π), (N, π)\G respectively, if there is a possible confusion. Let hN be the Haar measure on N . Let EG/N = (id ⊗ hN π),
EN \G = (hN π ⊗ id).
Then EG/N and EN \G are projections of norm one (completely positive and completely bounded conditional expectations) from AG onto AN \G and AG/N respectively (cf. [34] as well as [54, Proposition 2.3 and Section 6]), and AG/N = EG/N (AG ),
and AN \G = EN \G (AG ).
From this, we see that the ∗-subalgebras AN \G and AG/N are dense in AG/N and AN \G respectively. Assume N is a closed subgroup of an ordinary compact group G. Let π be the restriction morphism from AG := C(G) to AN := C(N ). Let C(G/N ) and C(N \G) be continuous functions on G/N and N\G respectively. Then one can verify that C(G/N ) = AG/N = EG/N (AG ), C(N \G) = AN \G = EN \G (AG ). Therefore we will use the symbols C(G/N) and AG/N (respectively C(N \G) and AN \G ; C(G) and AG ) interchangeably for all quantum groups. Proposition 2.1. Let N be a quantum subgroup of a compact quantum group G. Then the following conditions are equivalent: (1) (2) (3) (4)
AN \G is a Woronowicz C ∗ -subalgebra of AG . AG/N is a Woronowicz C ∗ -subalgebra of AG . AG/N = AN \G . For every irreducible representation uλ of G, either hN π(uλ ) = Idλ or hN π(uλ ) = 0, where hN is the Haar measure on N , dλ is the dimension of uλ and Idλ is the dλ × dλ identity matrix.
Proof. We only need to show that (1) ⇔ (4) ⇔ (3). The proof of the implications (2) ⇔ (4) ⇔ (3) is similar. (3) ⇒ (4). In general one has (AN \G ) ⊆ AN \G ⊗ AG ,
(AG/N ) ⊆ AG ⊗ AG/N .
Letting B = AN \G = AG/N one has (B) ⊆ B ⊗ B. ˆ let nλ be the multiplicity of the trivial representation of N in the representation π(uλ ). For λ ∈ G, We claim that either nλ = dλ or nλ = 0. ˆ such that 1 < nλ < dλ . Note that in general Assume on the contrary that there is a λ ∈ G
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EN \G uλij = (hN π ⊗ id) uλij = hN π uλik uλkj , k
hN π uλkj uλik . EG/N uλij = (id ⊗ hN π) uλij = k
Using unitary equivalence if necessary we choose uλij in such a way that the nλ trivial representations of N appear on the upper left diagonal corner of π(uλ ). Then uλij EN \G uλij = 0 uλij EG/N uλij = 0
if 1 i nλ , 1 j dλ , if nλ < i dλ , 1 j dλ , if 1 i dλ , 1 j nλ , if 1 i dλ , nλ < j dλ .
Since AN \G = AG/N = B and both EN \G and EG/N are projections from AG onto B, we have EN \G = EG/N . Then for nλ < j dλ , 0 = uλij = EN \G uλij = EG/N uλij = 0. This is a contradiction. ˆ consisting of those λ’s such that hN π(uλ ) (4) ⇒ (3). Let S(N ) (or S(N, π)) be the subset of G is Idλ . Then a straightforward calculation using the fact that EN \G and EG/N are projections of AG onto AN \G and AG/N respectively, one gets AN \G = AG/N =
Cuλij λ ∈ S(N ), i, j = 1, . . . , dλ .
(4) ⇒ (1). Let S(N) be defined as in the proof of (4)⇒(3). It is clear that AN \G is a Woronowicz C ∗ -subalgebra of AG and that
uλij λ ∈ S(N ), i, j = 1, . . . , dλ
is a Peter–Weyl basis of the dense ∗-subalgebra AN \G of AN \G . (1) ⇒ (4). Let G1 = N \G. Then by Woronowicz’s Peter–Weyl theorem for compact quantum groups, every irreducible representation uλ of G is either an irreducible representation of G1 or none of the coefficients uλij is in AG1 . That is EN \G uλij =
uλij 0
ˆ 1, if λ ∈ G ˆ \G ˆ 1. if λ ∈ G
By the definition of EN \G and linear independence of the uλij ’s, this implies that
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ˆ 1 , i, k = 1, . . . , dλ , hN π uλik = δik , λ ∈ G λ ˆ \G ˆ 1. hN π uik = 0, λ ∈ G This completes the proof of the proposition.
2
Definition 2.2. A quantum subgroup N of a compact quantum group G is said to be normal if it satisfies the equivalent conditions of Proposition 2.1. Remarks. (a) Condition (4) of Proposition 2.1 plays an important role in this paper. It is a reformulation of the following condition for a normal quantum subgroup N that appears near the end of [49, Section 2]: for every irreducible representation v of G, the multiplicity of the trivial representation of N in the representation π(v) is either zero or the dimension of v. From the proof of the proposition we see that the counit of AG/N is equal to the restriction morphism π|AG/N . (b) Note also that on p. 679 of [49] the following statement is found: “In general, a right quotient quantum group is different from the corresponding left quotient quantum group.” Though in the purely algebraic setting of Hopf algebras, one needs to distinguish between left and right normal quantum subgroups, as indicated in Parshall and Wang [33, 1.5] (see also [1,42,46]), however, in view of Proposition 2.1 above, this cannot happen for normal quantum subgroups of compact quantum groups. Moreover, using Lemmas 4.2–4.4 below, it can be shown that the notion of normality defined in [33] when applied to compact quantum groups is equivalent to our notion of normality. As the main results of this paper do not depend on this equivalence, its proof and other applications are in [57]. (c) The notion of a normal quantum subgroup depends on the morphism π , which gives the “position” of the quantum group N in G. If (N, π1 ) is another quantum subgroup of G with surjection π1 : AG → AN , (N, π1 ) may not be normal even if (N, π) is. This phenomenon already occurs in the group situation. For example a finite group can contain two isomorphic subgroups with one normal but the other not. Examples. We show in (1) and (2) below that the identity group and the full quantum group G are both normal quantum subgroups of G under natural embeddings. These will be called the trivial normal quantum subgroups. See Sections 4, 5 and [57] for examples of non-trivial normal quantum subgroups. (1) Let N = {e} be the one element identity group. Let π = = counit of AG be the morphism from AG to AN . Then by the counital property, one has AG/N = a ∈ AG (id ⊗ )(a) = a ⊗ 1 = AG . That is ({e}, ) is normal and G/({e}, ) = G. (2) Now let N = G and let π : AG → AN be any isomorphism of Woronowicz C ∗ -algebras [49]. Let h be the Haar measure on G and a ∈ AG/N . Since π is an isomorphism and (id ⊗ π)(a) = a ⊗ 1, one has (a) = (id ⊗ π)−1 (a ⊗ 1) = a ⊗ 1. From the invariance of h one has h(a)1 = (1 ⊗ h)(a) = ah(1) = a. Hence AG/N = a ∈ AG (id ⊗ π)(a) = a ⊗ 1 = C1.
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That is (G, π) is normal and G/(G, π) ∼ = {e}. (3) We note that besides the embeddings in (2) it is possible to construct examples of compact quantum groups G with non-normal proper embeddings of G into G. In fact this can happen for compact groups already. The following is a justification of the above notion of normal quantum subgroups. Proposition 2.3. Let A = C(G) be a compact group. Let N be a closed subgroup of G. Let π be the restriction map from A to AN = C(N). Then (N, π) is normal in the sense above if and only if N is a normal subgroup of G in the usual sense. Proof. Under the Gelfand–Naimark correspondence which associates to every commutative C ∗ algebra its spectrum, quotients of G by (ordinary) closed normal subgroups N correspond to Woronowicz C ∗ -subalgebras of C(G), i.e., G/N corresponds to C(G/N ), see [49, 2.6 and 2.12]. Since AG/N = C(G/N) for any closed subgroup N , the proposition follows from Proposition 2.1 above. 2 The following result gives a complete description of quantum normal subgroups of the compact quantum group dual of a discrete group Γ , whose proof is straightforward using e.g. [59] and Proposition 2.1. Proposition 2.4. Let AG = C ∗ (Γ ). Let N be a quantum subgroup of G with surjection π : AG → AN . Then N is normal, L := π(Γ ) is a discrete group and AN = C ∗ (L). Moreover, AG/N = C ∗ (K), where K = ker(π : Γ → L). To distinguish two different quantum subgroups, we include the following result, which should be known to experts in the theory of C ∗ -algebras. Proposition 2.5. Let πk : A → Ak be surjections of unital C ∗ -algebras with kernels Ik (k = 1, 2). Let Pk be the pure state space of Ak . Then the following conditions are equivalent: (1) {φ1 ◦ π1 | φ1 ∈ P1 } = {φ2 ◦ π2 | φ2 ∈ P2 } as subsets of pure states of A. (2) I1 = I2 . (3) There is an isomorphism α : A1 → A2 such that π2 = α ◦ π1 . Proof. (1) ⇒ (2). If I1 = I2 , say, there is an x ∈ I1 \ I2 . Then there is a pure state φ of A/I2 such that φ(π2 (x)) = 0, where we identify A2 with A/I2 . But φπ2 is a pure state of A/I1 ∼ = A1 according to assumption (1). Hence we must have φπ2 (x) = 0. This is a contradiction. (2) ⇒ (3). Let I = I1 = I2 . Let π be the quotient map A → A/I . Let π˜ k be the homomorphism from A/I to Ak such that πk = π˜ k π (k = 1, 2). Then π˜ k are isomorphisms. Put α = π˜ 2 ◦ π˜ 1−1 . Then π2 = α ◦ π1 . (3) ⇒ (1). This follows from P1 = P2 ◦ α. 2 The following proposition is an easy consequence of Proposition 2.1.
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Proposition 2.6. Let (N1 , π1 ) be a normal quantum subgroup of G. Let α : AN1 −→ AN2 be an isomorphism of quantum groups. Then (N2 , απ1 ) is normal. In view of the above discussions, it is reasonable to have the following definition (cf. also remarks after Proposition 2.3). Definition 2.7. Two quantum subgroups (π1 , H1 ) and (π2 , H2 ) of a quantum group G are said to have the same imbedding in G if π1 , π2 satisfy the equivalent conditions of Proposition 2.5. When this happens, we denote (H1 , π1 ) = (H2 , π2 ). Geometrically speaking, two quantum subgroups (H1 , π1 ) and (H2 , π2 ) of a quantum group G are said to have the same imbedding in G if their “images” in G are the same. 3. Simple compact quantum groups To avoid such a difficulty as the classification of finite groups up to isomorphism in developing the theory of simple compact quantum groups, we assume connectivity as a part of the postulates of the latter. We use representation theory to define the notion of connectivity: Definition 3.1. We call a compact quantum group GA connected if for each non-trivial irreˆ A , the C ∗ -algebra C ∗ (uλ ) generated by the coefficients of uλ is of ducible representation uλ ∈ G ij infinite dimension. In virtue of [26, (28.21)], we have Proposition 3.2. Let GA be an ordinary compact group (i.e. AG is commutative). Then GA is connected as a topological space if and only if it is connected in the sense above. Definition 3.3. We call a compact quantum group GA simple if it satisfies the following conditions (1)–(4): (1) (2) (3) (4)
The Woronowicz C ∗ -algebra AG is finitely generated; GA is connected; GA has no non-trivial connected normal quantum subgroups; GA has no non-trivial representations of dimension one.
A (simple) quantum group is called absolutely simple if it has no non-trivial normal quantum subgroups. Similarly a finite quantum group is called simple if it has no non-trivial normal quantum subgroups. Just as the notion of simple compact Lie groups excludes the torus groups, the above notion of simple quantum groups excludes abelian compact quantum groups in the sense of Woronowicz [59], i.e. quantum groups coming from group C ∗ -algebras C ∗ (Γ ) of discrete groups Γ (note that C ∗ (Γ ) is the algebra of continuous functions on the torus Tn when Γ is the discrete
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group Zn ). This is important because it is impossible to classify discrete groups up to isomorphism. However, we do not know if condition (4) in Definition 3.3 (i.e., there is no non-trivial group-like elements) is superfluous, as is the case for simple compact Lie groups because of the Weyl dimension formula. As a justification of this definition, we have the following proposition that shows that our notion of simple compact quantum groups recovers exactly the ordinary notion of simple compact Lie groups. Proposition 3.4. If GA is a simple compact quantum group with A commutative, then the set G := Aˆ of Gelfand characters is a simple compact Lie group in the ordinary sense. Conversely, every simple compact Lie group in the ordinary sense is of this form. The proof Proposition 3.4 follows immediately from [49, Theorem 2.8] and Proposition 3.2 above. We remark that although it is easy as above to prove the characterization of the ordinary simple compact Lie group in terms of our notion of simple compact quantum groups when AG is commutative, it has been highly non-trivial to prove the analogous characterization of ordinary differential manifolds in terms of the axioms of non-commutative manifolds, which is finally achieved in the recent work of Connes [21] (see references therein for earlier, presumably unsuccessful, attempts to such a characterization). Note that a simple compact Lie group is not a direct product of proper connected subgroups. Also, a simple Lie group is not a semi-direct product. Similarly, the following general results are true for quantum groups (for proofs see [57]): Proposition 3.5. If GA is a simple compact quantum group, then AG is neither a tensor product, nor a crossed product by a non-trivial discrete group. To put in perspective the examples of simple compact quantum groups to be studied later, we introduce some properties for compact quantum groups. First we recall that the representation ring (also called the fusion ring) R(G) of a compact quantum group G is an ordered algebra over the integers Z with positive cone (or semiring, which is also a basis) R(G)+ := {χu } consisting w ∈ N ∪ {0} ˆ of G, and structure constants cuv of characters χu of irreducible representations u ∈ G given by the rules χu χv =
w cuv χw ,
ˆ w∈G
where the product χu χv is taken in the algebra AG . Definition 3.6. Let G be a compact quantum group. We say that G has property F if each Woronowicz C ∗ -subalgebra of AG is of the form AG/N for some normal quantum subgroup N of G. We say that G has property FD if each quantum subgroup of G is normal. We say that G is almost classical if its representation ring R(G) is order isomorphic to the representation ring of a compact group. By Proposition 2.3, a compact group trivially has property F . We will give in Sections 4 and 5 non-trivial simple compact quantum groups that are almost classical and have property F . Among compact quantum groups, simple compact quantum groups that are almost classical or
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have property F are closest to ordinary simple compact Lie groups in regard to noncommutative geometry. By Proposition 2.4, as the dual of discrete group Γ , a compact quantum group of the form C ∗ (Γ ) has property FD. When Γ is finite, C ∗ (Γ ) is equal to the dual of the function algebra C(Γ ). This explains the term FD. A compact quantum group G is absolutely simple with property F if and only if every nontrivial representation v of G is faithful, i.e., C ∗ (vij ) = AG , cf. [54]. By a theorem of Handelman [25], the representation ring of a compact connected Lie group is a complete isomorphism invariant. But this fails for compact quantum groups, since the representation rings of a simple compact Lie group K and its standard deformation Kq are order isomorphic. In [5], Banica uses the positive cone R+ (G) of the representation ring R(G) of a compact quantum group G to define what he calls an R+ deformation. This is closely related to almost classical quantum groups. It is clear that a quantum quotient group G/N of an almost classical quantum group G is also almost classical. But a quantum subgroup of an almost classical quantum group need not be almost classical. For example, the quantum permutation groups are almost classical (cf. [4,53] and remarks preceding Theorem 4.7), but according to Bichon [16], their quantum subgroups A2 (Z/mZ) are not almost classical if m 3 (see Corollary 2.7 and the paragraph following Corollary 4.3 of [16]). However, for a compact quantum group with property F , we have the following general result. Theorem 3.7. Let G be a compact quantum group with property F . Then its quantum subgroups and quotient groups G/N (by normal quantum subgroups N ) also have property F . As we will only use the definitions of quantum groups with property F (respectively property FD) but not the assertion in the theorem above, the details for the proof of the theorem is included in a separate paper [57]. The main goals/problems in the theory of simple compact quantum groups are: (1) to construct and classify (up to isomorphism if possible) simple compact quantum groups; (2) to construct and classify irreducible representations of simple compact quantum groups; (3) to analyze the structure of compact quantum groups in terms of simple ones; and (4) to develop applications of simple compact quantum groups in other areas of mathematics and physics. For these purposes, new techniques for compact quantum groups must be developed. The above is a very difficult program at present. Even problem (1) of the program above is daunting. To obtain clues on the general problem (1), it is desirable to find and solve easier parts of it. For this purpose, we propose the following apparently easier problems. Problem 3.8. (1) Construct and classify all simple compact quantum groups with property F (up to isomorphism if possible). (2) Construct and classify all simple compact quantum groups that are almost classical (up to isomorphism if possible). Problem 3.9. Construct simple compact quantum groups with property FD.
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Simple quantum groups in Problems 3.8, 3.9 are most closest to groups known in mathematics. They should be easiest classes to classify. Therefore they should play a fundamental role in the main problems in the theory of simple compact quantum groups. 4. Simplicity of Bu (Q) and Aaut (B, τ ) To prove the main results in this and the next sections, we develop here two technical results, which are of interest in their own right: one on the correspondence between Hopf ∗-ideals and Woronowicz C ∗ -ideals; the other on the reconstruction of a normal quantum group from the identity in the quotient quantum group. We first recall the construction of compact quantum group Bu (Q) associated to a non-singular n × n complex scalar matrix Q (cf. [2,47,49,50]). The (noncommutative) C ∗ -algebra of functions on the quantum group Bu (Q) is generated by noncommutative coordinate functions uij (i, j = 1, . . . , n) that are subject to the following relations: u∗ u = In = uu∗ ,
ut QuQ−1 = In = QuQ−1 ut ,
where u = (uij )ni,j =1 . When QQ¯ is a scalar multiple cIn of the identity matrix In , the quantum group Bu (Q) and the group SU(2) have the same fusion rules for their irreducible representations, as shown by Banica [2], which implies that Bu (Q) is an almost classical quantum group. ¯ = ±In , the isomorphism classification of Bu (Q) is determined by the Under the condition QQ author [56] using polar decomposition of Q and eigenvalues of |Q| (see [56, Theorem 2.4]). For arbitrary Q, Bu (Q) is a free product of its building blocks, involving both Bu (Ql )’s and Au (Pk )’s with Ql Q¯ l being scalar matrices and Pk positive matrices (see [56, Theorem 3.3]). The precise definition of Au (Q) is recalled later in the paragraphs before Proposition 4.5. For positive matrix Q, Au (Q) is classified up to isomorphism in terms of the eigenvalues of Q (see [56, Theorem 1.1]); and for an arbitrary non-singular matrix Q, the general Au (Q) is a free product of Au (Pk )’s with positive matrices Pk (see [56, Theorem 3.1]). In Bichon et al. [18], the same techniques as in [56] were used to classify the unitary fiber functors of the quantum groups Au (Q) and Bu (Q) and their ergodic actions with full multiplicity. Note that for n = 1, Bu (Q) = C(T) is the trivial 1 × 1 unitary group. We will concentrate on the non-trivial case n 2. Note that the isomorphism class of Bu (Q) depends on the normalized Q only if QQ¯ is a scalar matrix [56]. Theorem 4.1. Let Q ∈ GL(n, C) be such that QQ¯ = ±In . Then Bu (Q) is an almost classical simple compact quantum group with property F . In fact it has only one normal subgroup of order 2. Proof. As noted above, the quantum group Bu (Q) is almost classical because its representation ring is order isomorphic to the representation ring of the compact Lie group SU(2) [2]. More precisely, according to [2] irreducible representations of the quantum group Bu (Q) can be parametrized by rk (k = 0, 1, 2, . . .) with r0 trivial and r1 = (uij )nij =1 , so that the fusion rules for their tensor product representations (i.e., decomposition into irreducible representations) read rk ⊗ rl = r|k−l| ⊕ r|k−l|+2 ⊕ · · · ⊕ rk+l−2 ⊕ rk+l ,
k, l 0.
We show that the quantum group Bu (Q) is connected. If k = 2m is even (m > 0), then let rl = rk in the above tensor product decomposition and do the same for the irreducible constituents
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repeatedly, one sees that the algebra C ∗ (r2m ) generated by the coefficients of the representation r2m contains the coefficients of r2s for all s. Hence r2m generates an infinite-dimensional algebra: C ∗ (r2m ) = C ∗ {r2s | s 0} . If k = 2m + 1 is odd (m 0), then let rl = rk in the above tensor product decomposition, one sees that the representation r2 appears therein. Apply the decomposition to r2m+1 ⊗ r2 , one sees that r1 = (uij ) appears therein. Hence the algebra generated by the coefficients of r2m+1 is the same as the algebra generated by those of r1 = (uij ). We conclude from this analysis that there is only one non-trivial Woronowicz C ∗ -subalgebra in Bu (Q), the one C ∗ (r2m ) generated by coefficients of r2m , which is obviously infinite-dimensional as noted above, where m is any nonzero positive number. In particular, the quantum group Bu (Q) is connected. For rest of the proof, we show that the quantum group Bu (Q) has only one normal quantum subgroup, although it has many quantum subgroups. Note that the coordinate functions vij of the matrix group N = {In , −In } satisfy the defining relations of Bu (Q), hence there is a surjection π from the C ∗ -algebra Bu (Q) to the C ∗ -algebra AN of functions on N such that π(uij ) = vij ,
i, j = 1, 2, . . . , n.
It is clear that π is a morphism of quantum groups, hence (N, π) is a quantum subgroup of the quantum group Bu (Q). We show that (N, π) is actually a normal quantum subgroup. To see this, it suffices by Proposition 2.1 to show that π(r2m ) = d2m · v0 ,
π(r2m+1 ) = d2m+1 · v1 ,
where d2m and d2m+1 are dimensions of the representations r2m and r2m+1 respectively, v0 and v1 are the trivial and the non-trivial irreducible representations of N respectively (v1 (±In ) = ±1). By the definition of π and v1 the assertion is clearly true for m = 0. In general, suppose the assertion is true for m. Then π(r2m+1 ) ⊗ π(r1 ) is a multiple of v0 since v12 = v0 . From the decomposition of r2m+1 ⊗ r1 , we get π(r2m+1 ) ⊗ π(r1 ) = π(r2m ) ⊕ π(r2m+2 ). Hence π(r2(m+1) ) = π(r2m+2 ) is a scalar multiple of v0 . Similarly, from π(r2m+2 ) ⊗ π(r1 ) = π(r2m+1 ) ⊕ π(r2m+3 ), we see that π(r2(m+1)+1 ) = π(r2m+3 ) is a multiple of v1 . Since v0 and v1 are one-dimensional representations, the multiples we obtained above must be d2m+2 and d2m+3 respectively. That is (N, π) is normal and AG/N = C ∗ (r2 ) = C ∗ {r2s | s 0} , where for simplicity of notation, the symbol G in G/N refers to the quantum group GBu (Q) . The above also shows that this quantum group has property F .
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We have to show that Bu (Q) has no other normal quantum subgroups, which will imply that it has no connected normal quantum subgroups and is therefore a simple quantum group. Let (N1 , π1 ) be a non-trivial normal quantum subgroup of Bu (Q). We show that (N1 , π1 ) = (N, π) in the sense of Definition 2.7, which will finish the proof of the theorem. Since N1 = 1, by Definition 2.7 and Proposition 2.1 there exists an irreducible representation v of the quantum group Bu (Q) such that π1 (v) is not a scalar and therefore EG/N1 (v) = 0. Hence by the proof of Proposition 2.1 and Woronowicz’s Peter–Weyl theorem [59], AG/N1 = EG/N1 (AG ) = AG . Similarly, we claim that AG/N1 = C1, where 1 is the unit of AG . To prove this, we need three lemmas. It is instructive to compare the second lemma (Lemma 4.3) with the ideal theory for C ∗ -algebras. Lemma 4.2. Let B1 and B2 be Woronowicz C ∗ -algebras with canonical dense Hopf ∗-algebras of “representative functions” B1 and B2 respectively. Assume B2 is full and ψ : B1 → B2 is a morphism of Woronowicz C ∗ -algebras such that the induced morphism ψˆ : B1 → B2 is an isomorphism. Then B1 is full and ψ is also an isomorphism. Remark. The above is false if the roles of B1 and B2 are exchanged, as seen by taking B1 = C ∗ (F2 ) and B2 = Cr∗ (F2 ). ˆ = a for a ∈ B1 . Proof of Lemma 4.2. Since B1 is dense in B1 , it suffices to show that ψ(a) ∗ Since ψ is a morphism of C -algebras, we have ψ(a) a and therefore the first inequality ψ(a) ˆ = ψ(a) a. Since B2 is full, the norm on B2 is the universal C ∗ -norm (see [52]): : π is a ∗-representation of B2 . ψ(a) ˆ ˆ = sup π ψ(a) Taking π = ψˆ −1 in the above, we obtain the second inequality ψ(a) = a. ˆ ψˆ −1 ψ(a) ˆ Combining the first the second inequalities finishes the proof of Lemma 4.2.
2
Lemma 4.3 (Hopf ∗-ideals vs. Woronowicz C ∗ -ideals). (1) Let G be a compact quantum group. Let I be a Hopf ∗-ideal of AG . Then the norm closure I in the C ∗ -algebra AG is a Woronowicz C ∗ -ideal and AG /I is a full Woronowicz C ∗ -algebra. The Hopf ∗-algebra AG /I admits a universal C ∗ -norm and its completion under this norm is a Woronowicz C ∗ -algebra isomorphic to AG /I. (2) The map f (I) = I is a bijection from the set of Hopf ∗-ideals {I} of AG onto the set of Woronowicz C ∗ -ideals {I } of AG such that AG /I is full. The inverse g of f is given by g(I ) = I ∩ AG . Remarks. (a) Note that (2) and the last part of (1) in the lemma above are false if the Woronowicz C ∗ -algebra AG or AG /I is not full, as is shown by the following example. Let AG = C ∗ (F2 )
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be the group C ∗ -algebra of the free group F2 on two generators. Let I be the kernel of the canonical map π : C ∗ (F2 ) → Cr∗ (F2 ) where Cr∗ (F2 ) is the reduced group C ∗ -algebra of F2 . Then I ∩ AG = 0 but I = 0. (b) This lemma strengthens the philosophy in [52] that the “pathology” associated with the ideals between 0 and the kernel of the morphism from the full Woronowicz C ∗ -algebra to reduced one such as π : C ∗ (F2 ) → Cr∗ (F2 ) is not (quantum) group theoretical, but purely functional analytical, and C ∗ (F2 ) and Cr∗ (F2 ) should be viewed as the same quantum group because the same dense Hopf ∗-subalgebra that completely determines the quantum group can be recovered from either the full or the reduced algebra. Similarly, for a general compact quantum group G, the totality of (quantum) group theoretic information is encoded in the purely algebraic object AG , any other (Hopf) algebra should be viewed as defining the same quantum group as AG so long as AG can be recovered from it. The advantage of working with the category of full C ∗ -algebras or the purely algebraic objects AG is that morphisms can be easily defined for them, whereas it is not even possible to define a morphism from the one element group to the quantum group associated with the reduced algebra Cr∗ (F2 ) if is viewed as a different quantum group than the one associated with the full algebra C ∗ (F2 ). Proof of Lemma 4.3. Let I be as in (1). Let π1 : AG → AG /I be the quotient map. Since I is a Hopf ∗-ideal, we have in particular (see Sweedler [45]) (I) ⊂ AG ⊗ I + I ⊗ AG ⊂ ker(π1 ⊗ π1 ). Therefore (I) ⊂ ker(π1 ⊗π1 ). That is, I is a Woronowicz C ∗ -ideal and AG /I is a Woronowicz C ∗ -algebra (see [49, 2.9–2.11]). Denote B1 = AG /I and let πˆ 1 be the induced morphism of the canonical dense Hopf-∗-subalgebras πˆ 1 : AG → B1 . We claim that ker πˆ 1 = I and ψˆ 0 : AG /I → B1 , ψˆ 0 : [a] → π1 (a) is an isomorphism, where [a] ∈ AG /I, a ∈ AG . By [49,59], AG is generated as an algebra by the coefficients uλij of irreducible unitary corepresentations uλ of Hopf ∗-algebra AG . The images [uλij ] of uλij in the quotient Hopf ∗-algebra AG /I give rise to unitary corepresentation of AG /I, and generate it as an algebra (not just as a ∗-algebra). Therefore AG /I is a compact quantum group algebra (CQG algebra) in the sense of Dijkhuizen and Koornwinder [23] (see also [28,52,60])—a more appropriate name for compact quantum group (CQG) algebra might be Woronowicz ∗-algebra (or compact Hopf ∗-algebra), since the quantum group C ∗ -algebra of a compact quantum group G is the C ∗ -algebra C ∗ (G) dual to C(G) according to [35]. Let B2 = AG /I and let B2 be the closure of B2 in the universal C ∗ -norm. Then B2 is a Woronowicz C ∗ -algebra. As the norm on AG is universal, the composition AG −→ AG /I −→ B2 is bounded and extends to a morphism of Woronowicz C ∗ -algebras ρ : AG → B2 . Since I ⊂ ker(ρ), we have I ⊂ ker(ρ) and ρ factors through B1 = AG /I via a C ∗ -algebra morphism ψ: π1
ψ
AG −→ B1 −→ B2 ,
ρ = ψπ1 .
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It is clear that ρ(a) = [a] for a ∈ AG and from this it can be checked that ψˆ and ψˆ 0 are inverse morphisms, where ψˆ : B1 → B2 is the restriction of ψ to the dense Hopf ∗-subalgebra B1 and B2 . Hence ψˆ 0 is an isomorphism as claimed. From ρˆ = ψˆ πˆ 1 (since ρ = ψπ1 ), it is easy to see that ψ is a morphism of Woronowicz C ∗ algebras (see [49, 2.3]). Since ψˆ = ψˆ 0−1 is an isomorphism and B2 is full, by Lemma 4.2, B1 is full and ψ is itself an isomorphism from B1 to B2 . ( We note in passing that since AG / ker(ρ) ∼ = B2 , we have I = ker(ρ).) This proves part (1) of the lemma. To prove part (2) of the lemma, let I be as in (2) and B1 = AG /I. Then by (1) above and [49], B1 is a Woronowicz C ∗ -algebra. Let B1 be the canonical dense Hopf ∗-algebra of B1 and let πˆ 1 : AG → B1 be the morphism associated with the quotient morphism π1 . Then clearly I ⊂ I ∩ AG = gf (I). Conversely if x ∈ I ∩ AG , then x ∈ ker(πˆ 1 ) = I. Hence gf (I) = I. Next let I be as in (2). We show that f g(I ) = I . Let B2 = AG /I —this is not the same B2 as in (1) above. Let π2 be the quotient morphism from AG onto B2 (compare with ρ above). Define I = g(I ) = I ∩ AG . We need to show that I = I . The idea of proof is the same as that of the last part in (1). Using the morphism πˆ 2 : AG → B2 of dense Hopf ∗-algebras associated with π2 , we see that I = ker(πˆ 2 ). Hence I is a Hopf ∗-ideal in AG and AG /I is isomorphic to B2 under the natural map induced from πˆ 2 , and by (1) above, B1 := AG /I is a Woronowicz C ∗ -algebra. Since I ⊂ I , the morphism π2 factors through B1 via a morphism ψ of Woronowicz C ∗ -algebras: π1
ψ
AG −→ B1 −→ B2 ,
π2 = ψπ1 .
Besides being isomorphic to B2 , AG /I is also isomorphic to B1 (under the morphism ψˆ 0 ) according to the proof of (1) earlier. Hence the restriction ψˆ of ψ to the dense Hopf ∗-algebras is an isomorphism from B1 to B2 . Since B2 is full, by Lemma 4.2, ψ itself is an isomorphism, which means that I = I (and B1 = B2 ). This completes the proof of Lemma 4.3. 2 Lemma 4.4 (Reconstruct N from G/N ). Let (N, π) be a normal quantum subgroup of a compact quantum group G. Let πˆ be the associated morphism from AG to AN . Then, + + ker(πˆ ) = A+ G/N AG = AG AG/N = AG AG/N AG ,
where H+ denotes the augmentation ideal (i.e. kernel of the counit) for any Hopf algebra H. Remarks. (a) In the notation of Schneider [42], the result above can be restated as follows: the map Φ is the left inverse of Ψ , where Ψ (ker(πˆ )) := AG/N and Φ(AG/N ) := AG A+ G/N . In the language of Andruskiewitsch and Devoto [1], the result above implies that the sequence 1 −→ N −→ G −→ G/N −→ 1, or the sequence 0 −→ AG/N −→ AG −→ AN −→ 0
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is exact. It is instructive to compare this with the purely algebraic situation in Parshall and Wang [33], where for a given normal quantum subgroup in the sense there, the existence of an exact sequence is not known and the uniqueness does not hold in general (cf. [33, 1.6 and 6.3]). Note that the notion of exact sequence of quantum groups in Schneider [42] is equivalent to that in Andruskiewitsch and Devoto [1] under certain faithful (co)flat conditions. Though a Hopf algebra is not faithfully flat over its Hopf subalgebras if it is not commutative or cocommutative (see Schauenburg [41]), we have the following Conjecture 1. Let G be a compact quantum group. Then the Hopf algebra AG (respectively AG ) is faithfully flat over its Hopf subalgebras. Similarly, AG (respectively AG ) is faithfully coflat over AG /I (respectively AN /I) for every Woronowicz C ∗ -ideal I (respectively Hopf ∗-ideal I). (b) It can be shown using Lemma 4.4 and Schneider [42] that the notion of normal quantum groups in this paper (or in [48,49]) and the one in Parshall and Wang [33] are equivalent for compact quantum groups. For more details, see [57]. Proof of Lemma 4.4. The proof is an adaption of the ones in Sweedler [45, 16.0.2] and of Childs [20, (4.21)] for finite-dimensional Hopf algebras to infinite-dimensional ones considered here. We sketch the main steps here for convenience of the reader. ˆ ) = AG A+ It suffices to prove ker(π) ˆ = A+ G/N AG . The other equality ker(π G/N is proved sim+ ilarly. From these it follows that ker(πˆ ) = AG AG/N AG . Consider the right AN -comodule structures on AN and AG given respectively by N : AN → AN ⊗ AN ,
and (id ⊗ π) ˆ G : AG → AG ⊗ AN ,
where N and G are respectively the coproducts of the Hopf algebras AN and AG . Since AN is cosemisimple by the fundamental work of Woronowicz [59] (see remarks in [49, 2.2]), it follows from of [22, Theorem 3.1.5] that every AN -comodule is projective. Furthermore, one checks that the surjection πˆ : AG → AN is a morphism of AN -comodules. Hence πˆ has a comodule splitting s : AN → AG with πˆ s = idAN . ˆ = 0. Hence A+ ˆ ) and Let x ∈ A+ G/N . By remark (a) following Definition 2.2, π(x) G/N ⊂ ker(π + therefore AG/N AG ⊂ ker(π). ˆ Define a linear map φ on AG by φ = (s πˆ ) ∗ S = m(s πˆ ⊗ S)G , where m and S are respectively the multiplication map and antipode of AG . Then using the coassociativity of G and πˆ s = idAN along with the antipodal property of S, one verifies that φ(AG ) ⊂ AG/N . Since ˆ) ⊂ ker(πˆ ) ⊂ Im(id − s πˆ ), to show ker(πˆ ) ⊂ A+ G/N AG , it suffices to show that Im(id − s π + A+ A . Since ( − id)φ(A ) ⊂ A , the later follows from the identity G G/N G G/N id − s πˆ = ( − id)φ ∗ id = m ( − id)φ ⊗ id G , which one verifies using basic properties of the convolution product along with φ = and the splitting property of s. This proves Lemma 4.4. 2 End of proof of Theorem 4.1. If AG/N1 = C1, we would have AG/N1 = C1 and A+ G/N1 = 0. Let πˆ 1 be the morphism of Hopf algebras from AG to AN1 associated with π1 . Then by
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Lemma 4.4, ker(πˆ 1 ) = A+ G/N1 AG = 0. Since ker(πˆ 1 ) is dense in ker(π1 ) by Lemma 4.3, we would have ker(π1 ) = 0. This contradicts the assumption that N1 is a non-trivial quantum subgroup of G and therefore AG/N1 = C1. Then AG/N1 has to be the only non-trivial Woronowicz C ∗ -subalgebra of Bu (Q), i.e. AG/N1 = C ∗ ({r2m | m 0}) as noted near the beginning of the proof of Theorem 4.1. We infer from Proposition 2.1 that π1 (r2m ) is a multiple of the trivial representation of N1 for any m. From π1 (r1 ) ⊗ π1 (r1 ) = π1 (r0 ) ⊕ π1 (r2 ), we see that π1 (r1 ) ⊗ π1 (r1 ) is a multiple of the trivial representation of N1 . That is
eij ⊗ ekl ⊗ u˜ ij u˜ kl = In ⊗ In ⊗ 1,
ij kl
where u˜ ij are the (i, j )-entries of π1 (r1 ) and eij are matrix units. Hence u˜ ij u˜ kl = 0,
when i = j, or k = l;
u˜ ii u˜ ll = 1,
for all i, l.
Therefore u˜ ij = 0 for i = j and AN1 is commutative. That is, N1 is an ordinary compact group. Now it is clear that u˜ ii = u˜ ll = u˜ −1 ll for all i, l, which we denote by a. Since AN1 is generated by a and N1 is non-trivial, we conclude that N1 is a group of order 2. The map α from AN to AN1 defined by α(vij ) = u˜ ij is clearly an isomorphism such that π1 = απ . Hence (N1 , π1 ) = (N, π) by Definition 2.7. For an example of a non-normal quantum subgroup (H, θ ) of Bu (Q), take a two-elements group H = {In , V }, where
V=
−1 0 , 0 In−1
and θ (uij ) = wij , the coordinate functions on H .
2
Let us also recall the construction of the quantum groups Au (Q) closely related to Bu (Q) [47,49,50]. For every non-singular matrix Q, the quantum group Au (Q) is defined in terms of generators uij (i, j = 1, . . . , n), and relations: u∗ u = In = uu∗ ,
ut QuQ ¯ −1 = In = QuQ ¯ −1 ut .
According to Banica [3], when Q > 0, the irreducible representations of the quantum group Au (Q) are parameterized by the free monoid N ∗ N with generators α and β and antimultiplicative involution α¯ = β (the neutral element is e with e¯ = e). The classes of u and
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u¯ are rα and rβ respectively. Moreover, for each pair of irreducible representations rx and ry (x, y ∈ N ∗ N), one has the following direct sum decomposition (fusion) rules: rx ⊗ ry =
rab .
x=ag, gb=y ¯
In [56], the special Au (Q)’s with Q > 0 are classified up to isomorphism and the general Au (Q)’s with arbitrary Q are shown to be free products of the special Au (Q)’s. The following result was observed by Bichon through private communication (the proof given below was developed by the author): Proposition 4.5. The quantum groups Au (Q) are not simple for any Q ∈ GL(n, C). Proof. To prove this, we first introduce the following notion. A quantum subgroup (N, π) of a compact quantum group G is said to be in the center of G if (π ⊗ id) = (π ⊗ id)σ , where σ (a1 ⊗ a2 ) = a2 ⊗ a1 , a1 , a2 ∈ AG , and is the coproduct of AG . Assume (N, π) is in the center of G. Then using the definitions of AG/N and AN \G in Section 2, it is straightforward to verify that AG/N = AN \G . By Proposition 2.1(3), (N, π) is normal in G. Namely, a quantum subgroup that is in the center of G is always normal, just as in the classical case. Let T be the one-dimensional (connected) torus group and t ∈ C(T) the function such that t ∗ t = 1 = tt ∗ . Then C(T) is generated by t as a C ∗ -algebra: C(T) = C ∗ (t). Define the morphism π : Au (Q) → C(T) by π(uij ) = δij t (note the special case Au (Q) = C(T) when n = 1). Then it is routine to verify that the connected group (T , π) is in the center of the quantum group Au (Q) (not viewed as an algebra) in the sense above and is therefore a normal subgroup therein. Hence Au (Q) is not simple. 2 We remark that although Au (Q) is not simple, for n 2 and Q > 0, it is very close to being normal, satisfying most of the axioms of a simple compact quantum group: its function algebra is finitely generated, it is connected, and its non-trivial irreducible representations are all of dimension greater than one (see Wang [56] for a computation of the dimension of its irreducible representations based on Banica [3]). In particular following problems should be accessible: Problem 4.6. (1) Study further the structure of Au (Q) for positive matrices Q ∈ GL(n, C) and n 2. Determine all of their simple quotient quantum groups. Alternatively, (2) Construct simple compact quantum groups that are not almost classical. A solution of part (1) of the above problem should also give a solution to part (2) and provide the first examples of simple compact quantum groups that are not almost classical because of the highly non-commutative representation ring of Au (Q) (note that all the simple quantum groups known so far are almost classical). It is worth noting that the determination of all simple quotient quantum groups of Au (Q) in the above problem is easier than the determination of all of their
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simple quantum subgroups, the latter being tantamount to finding all simple quantum groups because every compact matrix quantum group is a quantum subgroup of an appropriate Au (Q). These remarks also indicate that Au (Q) should play an important role in the theory of simple compact quantum groups. Next we consider the quantum automorphism group Aaut (B, tr) of a finite-dimensional C ∗ algebra B endowed with a tracial functional tr (cf. [4,53]). This quantum group is defined to be the universal object in the category of compact quantum transformation groups of B that leave tr invariant. Note that the presence of a tracial functional tr is necessary for the existence of the universal object when B is non-commutative (see of [53, Theorem 6.1]). For an arbitrary finitedimensional C ∗ -algebra B, the C ∗ -algebra Aaut (B, tr) is described explicitly in [53] in terms of generators and relations. When B = C(Xn ) is the commutative C ∗ -algebra of functions on the space Xn of n points, the quantum automorphism group Aaut (B) = Aaut (Xn ) (also called the quantum permutation group on n letters) exists without the presence of a (tracial) functional and its description in terms of generators and relations is surprisingly simple. The C ∗ -algebra Aaut (Xn ) is generated by self-adjoint projections aij such that each row and column of the matrix (aij )ni,j =1 add up to 1. That is, aij2 = aij = aij∗ , n
i, j = 1, . . . , n,
aij = 1,
i = 1, . . . , n,
aij = 1,
j = 1, . . . , n.
j =1 n i=1
For more general finite-dimensional C ∗ -algebras B, the description of Aaut (B, tr) in terms of generators and relations is more complicated. We refer the reader to [53] for details. Assume tr is the canonical trace τ on B (see [4, p. 772] or Section 1 for the definition). Then Aaut (B, τ ) is an ordinary permutation group when the dimension of B is less than or equal to 3. However, when the dimension of B is greater than or equal to 4, Aaut (B, τ ) is a non-trivial (noncommutative and noncocommutative) compact quantum group with an infinite-dimensional function algebra [53,54], and as Banica [4] showed, the algebra of symmetries of the fundamental representation of this quantum group is isomorphic to the infinite-dimensional Temply–Lieb algebras TL(n) and the representation ring of Aaut (B, τ ) is isomorphic to that of SO(3). Hence Aaut (B, τ ) is almost classical for all B. It is easy to see that for B = C(Xn ), the canonical trace τ is equal to the unique Sn -invariant state on B, where Sn acts on Xn by permutation. Hence by remark (2) following [53, Theorem 3.1], Aaut (B, τ ) is the same as the quantum permutation group Aaut (Xn ). We refer the reader to [4,53,54] for more on these quantum groups and [15–17] for interesting related results. Note that the description in [4] is not exactly as that in [53] but equivalent to it. We now prove Theorem 4.7. Let B be a finite-dimensional C ∗ -algebra with dim(B) 4. Then Aaut (B, τ ) is an almost classical, absolutely simple compact quantum group with property F . Proof. The argument is similar to the one in Theorem 4.1. By Banica [4], the complete set of mutually inequivalent irreducible representations of the quantum group Aaut (B, τ ) can be
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parametrized by rk (k 0, r0 being the trivial one-dimensional representation). Under this parametrization the fusion rules of its irreducible representations are the same as those of SO(3) and therefore it is almost classical: rk ⊗ rl = r|k−l| ⊕ r|k−l|+1 ⊕ · · · ⊕ rk+l−1 ⊕ rk+l ,
k, l 0.
We claim that there are only two Woronowicz C ∗ -subalgebras in Aaut (B, τ ), namely C1 and Aaut (B, τ ). Let A1 = C1 be a Woronowicz C ∗ -subalgebra of Aaut (B, τ ). Let v be a non-trivial irreducible representation of the compact quantum group of A1 . Then v = rk for some k = 0 and rk ⊗ rk = r0 ⊕ r1 ⊕ r2 ⊕ · · · ⊕ r2k−1 ⊕ r2k ,
k, l 0.
Hence the coefficients of each of the representations r1 , r2 , . . . , r2k are in A1 . Similarly, from the decomposition of r2k ⊗ r2k , we see that the coefficients of each of the representations r1 , r2 , . . . , r4k are in A1 . Inductively, the coefficients of each of the representations r1 , r2 , . . . , r2m k are in A1 (m > 0). Hence A1 = Aaut (B, τ ). Let (π, N) be a normal quantum subgroup of G = Aaut (B, τ ) different from the trivial oneelement subgroup. Then there is a non-trivial irreducible representation uλ = (uλij ) such that π(uλ ) is not a multiple of the trivial representation. Using the same argument as in the proof of Theorem 4.1 we have AG/N = EG/N (AG ) = AG . Therefore we must have AG/N = C1. Then the argument near the end of the proof of Theorem 4.1 (i.e. the paragraph that follows the proof of Lemma 4.4) shows that ker(π) = 0. That is, N is the same quantum group as G. 2 Theorem 4.7 applies in particular to quantum permutation groups Aaut (Xn ) when n 4. As Manin (private communication in July, 2002) pointed out to the author, the reason that these quantum groups are connected could be that there are so many more quantum symmetries that the originally n! isolated permutations are connected together by them. Note however that their function algebras are generated by orthogonal projections aij , so these quantum groups are also disconnected, as observed by Bichon [16]. It would be interesting to find a satisfactory explanation of this paradox. The proofs of the main results of this section do not need explicit description (models) of representations of the quantum groups Bu (Q) and Aaut (B, τ ) and Au (Q). Only the structures of their representation rings (i.e. fusion rules) are used. However, explicit constructions of models of irreducible representations of Lie groups are fundamental and have important applications in other branches of mathematics and physics. Moreover, just as the construction and classification of the representations of simple compact Lie groups is intimately intertwined with the classification of simple compact Lie groups, the same might hold true for simple compact quantum groups. In view of these, we believe an appropriate answer to the following problem should be important in the theory of compact quantum groups in general and the theory of simple compact quantum groups in particular. (Note that the model for the fundamental representation of the quantum group Au (Q) is used in [54] to construct ergodic actions on various von Neumann factors.) Problem 4.8. Construct explicit models of the irreducible representations of the following quantum groups: Au (Q) for Q > 0; Bu (Q) for QQ¯ = ±In ; the quantum automorphism group Aaut (B, τ ) of a finite-dimensional C ∗ -algebra B endowed with the canonical trace τ . Relate the results to the theory simple compact quantum groups if possible.
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5. Simplicity of Kq , Kqu and KJ The compact real forms Kq of Drinfeld–Jimbo quantum groups and their twists Kqu are studied in [43] and [30] respectively. See also [31] for a summary of [30,43] and [29] for more detailed treatment. Motivated by these works, Rieffel constructs in [37] a deformation KJ of compact Lie group K which contains a torus T and raises the question whether Kqu can be obtained as a strict deformation quantization of Kq . This question is answered in the affirmative by the author in [51]. The purpose of this section is to show that the quantum groups Kq , Kqu and KJ are simple in the sense of this paper, provided that the compact Lie group K is simple. We first recall the notation of [30,31,43]. Let G be a connected and simply connected simple complex Lie group with Lie algebra g. Fix a triangular decomposition g = n− ⊕ h ⊕ n+ , together with the corresponding decomposition = + ∪ − of the root system and a fixed basis {αi }ni=1 for + . For each linear functional λ on h, Hλ denotes the element in h corresponding to λ under the isomorphism h ∼ = h∗ determined by the Killing form ( , ) on g. Note that if the reader keeps the context in mind, the symbols α and λ used in this context should not cause confusion with the same symbols used in this paper for other purposes. Let {Xα }α∈ ∪ {Hi }ni=1 be a Weyl basis of g, where Hi = Hαi . This determines a Cartan involution ω0 on g with ω0 (Xα ) = −X−α , ω0 (Hi ) = −Hi . Let k be the compact real form of g defined as the fixed points of ω0 and K the associated compact real form of G. Put hR = ni=1 RHi , t = ihR and T = exp(t), the later being the associated maximal torus of K. Let q = eh/4 (h ∈ R \ {0}). For n, k ∈ N, n k, define [n]q =
q n − q −n , q − q −1
[n]q [n − 1]q . . . [n − k + 1]q n = . k q [k]q [k − 1]q . . . [1]q The quantized universal enveloping algebra Uq (g) [24,27] is the complex associative algebra with generators Xi± , Ki±1 (i = 1, . . . , n) and defining relations Ki Ki−1 = 1 = Ki−1 Ki ,
Ki Kj = Kj Ki ,
Ki Xj± Ki−1 = q ±(αi ,αj ) Xj± , + − K 2 − Ki−2 , Xi , Xj = δij i q − q −1
1−aij
k=0
(−1)k
1 − aij k
qi
k 1−aij −k Xi± Xj± Xi± = 0,
i = j,
where qi = q (αi ,αi ) . On Uq (g) there is a Hopf algebra structure with coproduct Ki±1 = Ki±1 ⊗ Ki±1 ,
Xi± = Xi± ⊗ Ki + Ki−1 ⊗ Xi± ,
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and counit and antipode respectively ε Ki±1 = 1, ε Xi± = 0,
S Xi± = −qi±1 Xi± ,
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S Ki±1 = Ki∓1 .
Under the ∗-structure defined by ± ∗ Xi = Xi∓ ,
Ki∗ = Ki ,
Uq (g) is a Hopf ∗-algebra. Let u = k,l ckl Hk ⊗ Hl ∈ 2 hR . Then it can be shown (cf. [29]) that the following defines a new coproduct on Uq (g), u (ξ ) = exp(−ihu/2)(ξ ) exp(ihu/2), where X ∈ Uq (g) and is the original coproduct on Uq (g). The new Hopf ∗-algebra so obtained is denoted by Uq,u (g). The function algebra AKq of the compact quantum group Kq is defined to be the subalgebra of the dual algebra Uq (g)∗ consisting of matrix elements of finite-dimensional representations ρ of Uq (g) such that eigenvalues of the endomorphisms ρ(Ki ) are positive. The function algebra AKqu of the compact quantum group Kqu is defined to be the subalgebra of the dual algebra Uq,u (g)∗ that has the same elements as AKq , as well as the same ∗-structure, while the product of its elements is defined using u instead of . For each (algebraically) dominant integral weight λ ∈ P+ of (g, h), define matrix elements (i) λ Cμ,i;ν,j of the highest weight Uq (g) module (L(λ), ρλ ) as follows. Let {vν } be an orthonormal λ is defined by weight basis for the unitary Uq (g) module L(λ). Then Cμ,i;ν,j λ Cμ,i;ν,j (X) = ρλ (X)vν(j ) , vμ(i) , λ where X ∈ Uq (g) and , is the inner product on L(λ). The Cμ,i;ν,j ’s is a linear (Peter–Weyl) basis of both AKq and AKqu when λ ranges through the set P+ of dominant integral weights of (g, h).
Theorem 5.1. Let K be a connected and simply connected simple compact Lie group. Then for each q, Kq is an almost classical simple compact quantum group with property F . Proof. First we recall that representations of K and Kq are in one to one correspondence via deformation and the decompositions of tensor products of irreducible representations are not altered under deformation (see Lusztig [32] and Rosso [39] or Chari and Pressley [19]). From this it follows immediately that Kq is almost classical. Let ξ be the map that associates each irreducible representation v of K an irreducible representation ξ(v) of Kq in this correspondence. This map defines an isomorphism of vector spaces from AK to AKq , which we also denote by ξ . It follows from this that Kq is connected and has no non-trivial representations of dimension one. Comparing decompositions of tensor products of representations of K and Kq we see that the ξ maps bijectively the set of Hopf subalgebras of AK onto the set of Hopf subalgebras of AKq . Let ρq be the quotient morphism from AKq to the abelianization Aab Kq , which is by definition the quotient of AKq by the closed two-sided ideal of AKq generated by commutators [a, b],
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a, b ∈ AKq . According to [49], Aab Kq is the algebra of continuous functions on the maximal compact subgroup Aˆ Kq of Kq and ρq gives rise to the embedding of the quantum groups from Aˆ Kq to Kq . It is shown in [43] that the maximal compact subgroup Aˆ Kq is isomorphic to the maximal torus T of K. The associated morphism ρˆq from AKq to AT is given by λ (t) = δij δμν e2πμ(x) , ρˆq Cμ,i;ν,j √ where t = exp(x) ∈ T , x ∈ t = ihR (see [19, p. 438], but −1 should not appear in the formula there). It is clear that one has the same formula as above for restriction morphism ρ from AK to AT : λ ρˆ ξ −1 Cμ,i;ν,j (t) = δij δμν e2πμ(x) ,
i.e.,
ρˆ = ρˆq ◦ ξ.
Let N ⊂ K be a normal subgroup of K with surjections π : AK → AN and πˆ : AK → AN . Then N is a finite subgroup of T and AN = AN is a finite-dimensional Hopf algebra. It is clear that π = ρN ◦ ρ, where ρN is the restriction morphism from AT to AN . Define πq : AKq −→ AN ,
by πq := ρN ◦ ρq .
We claim that (N, πq ) is a normal subgroup of Kq . This follows immediately from the following identities, which one can easily verify using ρˆ = ρˆq ◦ ξ and πˆ = πˆ q ◦ ξ :
a ∈ AKq
AKq /N = ξ(AK/N ), i.e., (id ⊗ πq )(a) = a ⊗ 1 = ξ a ∈ AK (id ⊗ π)(a) = a ⊗ 1 ;
a ∈ AKq
AN \Kq = ξ(AN \K ), i.e., (πq ⊗ id )(a) = 1 ⊗ a = ξ a ∈ AK (π ⊗ id )(a) = 1 ⊗ a .
That is, every normal subgroup N of K gives rise to a normal subgroup (N, πq ) of Kq in the manner above. Conversely, let (N , π ) be a quantum normal subgroup of Kq . Then AKq /N is a Hopf subalgebra of AKq . Since every Hopf subalgebra of AK is of the form AK/N for some normal subgroup N of K (cf. [49]), by the correspondence between Hopf subalgebras of AK and those of AKq noted near the beginning of the proof we have AKq /N = ξ(AK/N ) = AKq /N for some normal subgroup N of K. By Lemma 4.4, we have ker(πˆ q ) = ker(πˆ ). That is (N , π ) and (N, πq ) is the same quantum subgroup of Kq (cf. Definition 2.7 and Lemma 4.3). Since normal subgroups N of K are finite, we conclude from the above that Kq has no non-trivial connected quantum normal subgroups. 2 Examining the proof of Theorem 5.1, we formulate the following general result on the invariance of simplicity of compact quantum groups under deformation, which will be used to prove the simplicity of Kqu and KJ .
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Let G be an almost classical simple compact quantum group with property F and (H, ρ) a quantum subgroup. Assume all normal quantum subgroups of G are quantum subgroups of H . Let Gv be a family of compact quantum groups (“deformation” of G) indexed by a subset {v} of a vector space that includes the origin. Suppose the family Gv satisfies the following conditions: (C1) G0 = G. (C2) There is an isomorphism ξ of vector spaces from AG to AGv . (C3) The coproduct is unchanged under deformation, i.e., v ξ(a) = (ξ ⊗ ξ )(a)
for a ∈ AG .
(C4) For any pair irreducible representations uλ1 and uλ2 of G, if uλ1 ⊗ uλ2 ∼ = uγ1 ⊕ uγ2 ⊕ · · · ⊕ uγl is a decomposition of uλ1 ⊗ uλ2 into direct sum of irreducible subrepresentations uγj (j = 1, 2, . . . , l), then ξ uλ1 ⊗ ξ uλ2 ∼ = ξ uγ1 ⊕ ξ uγ2 ⊕ · · · ⊕ ξ uγl is a decomposition of ξ(uλ1 ) ⊗ ξ(uλ2 ) into direct sum of irreducible representations, where for instance ξ(uλ1 ) denotes the representation of Gv whose coefficients are images of coefficients of uλ1 . (C5) The quantum subgroup H is undeformed. The latter means that there is a morphism ρv of quantum groups from H to Gv such that ρv ξ(a) = ρ(a)
for a ∈ AG .
Under the assumptions above, we have the following result. The proof is the same as that of Theorem 5.1 (H corresponds to T in Theorem 5.1). Theorem 5.2. For each v ∈ {v}, Gv is an almost classical simple quantum group with property F . Remarks. (a) Condition (C4) above is not the same as the requirement that ξ uλ1 ⊗ uλ2 = ξ uλ1 ⊗ ξ uλ2 . The latter requirement together with conditions (C2) and (C3) imply that ξ is an isomorphism of quantum from Gv to G, which is not the case for the quantum groups under consideration here. (b) We believe similar results on invariance of simplicity under deformation hold true without the property F assumption on G. But at the moment we do not know of any simple compact quantum groups that do not satisfy this property, though there are many non-simple quantum groups without this property. Next we recall the construction in [37,51]. Let A = AG be a compact quantum group with coproduct . Suppose that the quantum group G has a toral subgroup (T , ρ)—to obtain nontrivial deformation we assume that T has rank no less than 2. For any element t in T , denote by Et the corresponding evaluation functional on C(T ). Assume that η is a continuous homomorphism
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from a vector space Lie group Rn to T , where n is allowed to be different from the dimension of T . Define an action α of Rd := Rn × Rn on the C ∗ -algebra A as follows: α(s,v) = lη(s) rη(v) , where lη(s) = (Eη(−s) ρ ⊗ id),
rη(v) = (id ⊗ Eη(v) ρ).
For any skew-symmetric operator S on Rn , one may apply Rieffel’s quantization procedure [36] for the action α above to obtain a deformed C ∗ -algebra AJ whose product is denoted ×J , where J = S ⊕ (−S). The family AhJ (h ∈ R) is a strict deformation quantization of A (see [36, Chapter 9]). In [51] the following result is obtained. Theorem 5.3. The deformation AJ is a compact quantum group containing T as a (quantum) subgroup; AJ is a compact matrix quantum group if and only if A is. We denote by GJ the quantum group for AJ . When G is a compact Lie group, the construction GJ above is the same as Rieffel’s construction [37]. By [37, 5.2], GJ is an almost classical compact quantum group if G is a compact Lie group. Combining Theorems 5.3 and 5.2, we obtain Theorem 5.4. Let K be a simple compact Lie group with a toral subgroup T . Then KJ of Rieffel [37] is an almost classical simple compact quantum group with property F . We note that unlike in Theorem 5.1, in the result above we do not need to assume K to be simply connected. This is because AKq is defined using irreducible representations of Uq (g) associated with all dominant integral weights P+ of (g, h), so that AKq becomes the algebra of representative functions on a simply connected K when q → 1. One could also start with a non-simply connected K in Theorem 5.1 too, but then one needs to modify the definition of the quantum algebra AKq by using irreducible representations of Uq (g) associated with analytically dominant integral weights only. This newly defined AKq is a Hopf subalgebra of the Hopf algebra defined originally. It is clear from the proof of Theorem 5.1 that its conclusion remains valid for this newly defined Kq . Finally we consider Kqu . To avoid confusion with the Killing form, we now use s ⊕ v, instead of (s, v) used above, to denote an element of Rd = Rn × Rn . In the present setting, the space Rn is hR , with inner product , = ( , ), where ( , ) is the Killing form of g restricted to hR . We will also use , to denote the inner product on hR ⊕ hR . Noting that the compact abelian group T is also a subgroup of both Kq and Kqu (see [30,43]). The map η there in this case is defined by η(s) = exp(2πis). We can define as above an action of Rd on AKq by αs⊕v = lexp(−2πis) rexp(2πiv) . This action may be viewed as an action of H = T × T in the sense of [36]. For each ν in the weight lattice P of g, the element Hν is in hR . We use the notation Hν ⊕ Hμ to denote Hν + Hμ as an element of hR ⊕ hR . Keep the notation of [36] for the spectral subspaces of the action α (see [36, 2.22]).
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Let uˇ be the map on h* determined by u via the Killing form ( , ) on g. Let p = −(Hν1 ⊕ Hμ1 ),
q = −(Hν2 ⊕ Hμ2 ),
h J= Su ⊕ (−Su ) , 4π where Su is the skew-symmetric operator on hR defined by Su (Hν ) =
ckl ν(Hk )Hl .
k,l
Then one has ih (μ1 , uμ Cμλ11 ,i1 ;ν1 ,j1 ◦ Cμλ22 ,i2 ;ν2 ,j2 = exp ˇ 2 ) − (ν1 , uν ˇ 2 ) Cμλ11 ,i1 ;ν1 ,j1 Cμλ22 ,i2 ;ν2 ,j2 2 = exp −2πip, J q Cμλ11 ,i1 ;ν1 ,j1 Cμλ22 ,i2 ;ν2 ,j2 where ◦ on the left-hand side is the multiplication in AKqu and the right-hand side is the multiplication in AKq . On the other hand one has from [36, 2.22] that Cμλ11 ,i1 ;ν1 ,j1 ×J Cμλ22 ,i2 ;ν2 ,j2 = exp −2πip, J q Cμλ11 ,i1 ;ν1 ,j1 Cμλ22 ,i2 ;ν2 ,j2 . This means that we have the following result [51]. Theorem 5.5. The Hopf ∗-algebras AKqu and (AKq , ×J ) are isomorphic. That is Kqu = (Kq )J in the notation of Theorem 5.3, answering Rieffel’s question [37] in the affirmative. Combining Theorems 5.5, 5.1 and 5.2, we obtain the following Theorem 5.6. Let K be a connected and simply connected simple compact Lie group. Then for each (q, u), Kqu is an almost classical simple compact quantum group with property F . Acknowledgments The research reported here was partially supported by the National Science Foundation grant DMS-0096136. During the writing of this work, the author also received support from the Research Foundation of the University of Georgia in the form of a Faculty Research Grant in the summer of 2000 and the Max Planck Institute of Mathematics at Bonn in the form of a visiting membership for 6 weeks in the summer of 2002. During the summer of 2008, the author also received a very generous research grant from the Mathematics Department of the University of Georgia to enable him to finalize the paper. The author is grateful to these agencies and institutions for generous support.
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[38] M. Rosso, Comparaison des groupes SU(2) quantiques de Drinfeld et Woronowicz, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987) 323–326. [39] M. Rosso, Finite-dimensional representations of the quantum analog of the enveloping algebra of a complex semisimple Lie algebra, Comm. Math. Phys. 117 (1988) 581–593. [40] M. Rosso, Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul differentiel noncommutatif, Duke Math. J. 61 (1990) 11–40. [41] P. Schauenburg, Faithful flatness over subalgebras: Counterexamples, in: Interactions between Ring Theory and Representations of Algebras, Murcia, in: Lect. Notes Pure and Appl. Math., vol. 210, Dekker, New York, 2000, pp. 331–344. [42] H.-J. Schneider, Some remarks on exact sequences of quantum groups, Comm. Algebra 21 (9) (1993) 3337–3357. [43] Y. Soibelman, Algebra of functions on compact quantum group and its representations, Algebra i Analiz 2 (1) (1990) 190–212; Leningrad Math. J. 2 (1) (1991) 161–178. [44] Y. Soibelman, L. Vaksman, The algebra of functions on quantum SU (2), Funct. Anal. i Prilozhen. 22 (3) (1988) 1–14; Funct. Anal. Appl. 22 (3) (1988) 170–181. [45] M.E. Sweedler, Hopf Algebras, Benjamin, New York, 1969. [46] M. Takeuchi, Quotient spaces for Hopf algebras, Comm. Algebra 22 (7) (1994) 2503–2523. [47] A. Van Daele, S.Z. Wang, Universal quantum groups, Internat. J. Math. 7 (1996) 255–264. [48] Wang, S.Z., General constructions of compact quantum groups, PhD thesis, University of California at Berkeley, March, 1993. [49] S.Z. Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (1995) 671–692. [50] S.Z. Wang, New classes of compact quantum groups, Lecture notes for talks at the University of Amsterdam and the University of Warsaw, January and March, 1995. [51] S.Z. Wang, Deformations of compact quantum groups via Rieffel’s quantization, Comm. Math. Phys. 178 (3) (1996) 747–764. [52] S.Z. Wang, Krein duality for compact quantum groups, J. Math. Phys. 38 (1) (1997) 524–534. [53] S.Z. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1) (1998) 195–211; math.OA/ 9807091. [54] S.Z. Wang, Ergodic actions of universal quantum groups on operator algebras, Comm. Math. Phys. 203 (1999) 481–498. [55] S.Z. Wang, Quantum ax + b group as quantum automorphism group of k[x], Comm. Algebra 30 (4) (2002) 1807– 1815. [56] S.Z. Wang, Structure and isomorphic classification of compact quantum groups Au (Q) and Bu (Q), J. Operator Theory 48 (Suppl. 3) (2002) 573–583. [57] Wang, S.Z., Equivalent notions of normal quantum subgroups, compact quantum groups with properties F and FD, and other applications, preprint. [58] S.L. Woronowicz, Twisted SU(2) group. An example of noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987) 117–181. [59] S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987) 613–665. [60] S.L. Woronowicz, Tannaka–Krein duality for compact matrix pseudogroups. Twisted SU(N ) groups, Invent. Math. 93 (1988) 35–76.
Journal of Functional Analysis 256 (2009) 3342–3367 www.elsevier.com/locate/jfa
Semi-classical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space: II. P (φ)2-model on a finite volume ✩ Shigeki Aida 1 Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, 560-8531, Japan Received 2 September 2008; accepted 2 October 2008 Available online 1 November 2008 Presented by the Editors
Abstract We determine the semi-classical limit of the lowest eigenvalue of a P (φ)2 -Hamiltonian on a finite volume interval. © 2008 Elsevier Inc. All rights reserved. Keywords: Semi-classical limit; P (φ)-type Hamiltonian; Quantum field theory
1. Introduction Spatially cut-off P (φ) Hamiltonians play important roles to construct a non-trivial scalar quantum field without cut-off. The Hamiltonian is an infinite-dimensional version of a Schrödinger operator −Hh¯ = −h¯ 2 + U on L2 (RN , dx), where h¯ is the Planck constant. The problem to study the asymptotic behavior of the spectrum of Hh¯ under the limit h¯ → 0 is one of the subject of the semi-classical analysis [10,28]. There are several works on the classical limit of the quantum field, for example [7,8,13]. However, there are not many works concerning the semiclassical analysis of P (φ)-type Hamiltonian (see [1,5]) although there are works on the study of the Schrödinger operators in large dimensions [11,17–20,24,30,31]. In this paper, we deter✩
This research was partially supported by Grant-in-Aid for Scientific Research (C) No. 18540175. E-mail address:
[email protected]. 1 Fax: +81-6-6850-6496.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.001
S. Aida / Journal of Functional Analysis 256 (2009) 3342–3367
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mine the semi-classical limit of the lowest eigenvalue of a P (φ)2 -Hamiltonian on a finite volume interval. The paper is organized as follows. In Section 2, we formulate our problem in the setting of abstract Wiener space and state our main theorem (Theorem 2.9). Also we prepare necessary lemmas. The key of the proof of the main result is a large deviation estimates and Laplace’s type asymptotic formula for Wick polynomials and a lower bound estimate of the Hamiltonian (NGS bound). Hence the main idea of the proof is similar to that of [1,3,4]. In Section 3, we prove the main theorem. In Section 4, we make further remarks. 2. Preliminaries and the main theorem 2
d 2 Let I = [−l/2, l/2]. Let = dx 2 be the Laplace–Beltrami operator on L (I, dx) with periodic boundary condition, where dx denotes the Lebesgue measure. Note that allfunctions and
function spaces in this paper are real-valued ones. Set e0 (x) = 1l and ek (x) = 2l cos( 2πk l x), e−k (x) = 2l sin( 2πk l x) for positive integer k. {en | n = 0, ±1, ±2, . . .} are eigenfunctions of
and constitutes a complete orthonormal system of L2 (I, dx). Since the boundary condition is periodic one, we may consider our function spaces are defined on a circle with the length l. Let us fix a positive number m > 0 and consider a self-adjoint operator A˜ = (m2 − )1/4 . We define the Sobolev spaces: H s (I, dx) = h ∈ D A˜ 2s hH s := A˜ 2s hL2 (I,dx) .
(2.1)
˜ → H and the inverse operator Let H = H 1/2 (I, dx). We have natural one to one map j : D(A) ˜ ⊂ L2 (I, dx). ι is a bounded linear operator into L2 (I, dx) and the adjoint operator ι : H → D(A) ι∗ can be viewed as a bounded linear operator from L2 (I, dx) to H using the Riesz theorem. Let us define a self-adjoint operator (A, D(A)) on H by D(A) = j (D(A˜ 2 )) and Af = j ◦ A˜ ◦ ι. Clearly A and A˜ are unitarily equivalent to each other by the natural unitary transformation Φ : f ∈ L2 (I, dx) → j ◦ A˜ −1 f ∈ H . That is A = Φ ◦ A˜ ◦ Φ −1 holds. Also it is easy to see that A ◦ j = j ◦ A˜ on D(A˜ 2 ) and A˜ ◦ ι = ι ◦ A on D(A). For the separable Hilbert space H , let (W, H, μ) be an abstract Wiener space [15]. μ is the Gaussian measure whose covariance operator is (m2 − )−1/2 on L2 (I, dx). For example, we can take W = H −s0 (I, dx), where s0 is a positive number. That is, the norm is given by w2W := (m2 − )−s0 /2 w2L2 . Let S = A−2γ , where γ = 1 + 2s0 . Then S is a trace class operator on H √ and h2W = Sh2H holds. This triplet fits in the framework in [4]. Let −LA be the generator of the Dirichlet form which is the smallest closed extension of the following closable form on L2 (W, μ): EA (f, f ) =
ADf (w)2 dμ, H
(2.2)
W ∞ where f ∈ FC ∞ A (W ) and Df denotes the H -derivative on W . Here FC A (W ) denotes the set of smooth cylindrical functions f (w) = F (ϕ1 (w), . . . , ϕk (w)) (k ∈ N, F ∈ Cb∞ (Rk ), ϕi ∈ n∈N D(An )). For instance, let us consider the case where ϕi (w) = w, hi . Here hi ∈
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j
˜n n∈N D(A ) and −2 ˜ ◦ A hi . Hence,
the coupling is defined in the sense of distribution. Then D( ϕi , w ) =
ADf (w)2 = H
(∂xi F ) ϕ1 (w), . . . , ϕk (w) (∂xj F ) ϕ1 (w), . . . , ϕk (w) (hi , hj )L2 .
1i,j k
−LA can be identified with the second quantization of (A∗ )2 on H ∗ = H −1/2 (I, dx). The generator of EI which is defined by replacing A in (2.2) by the identity operator is called the Ornstein–Uhlenbeck operator operator. Let us define our Schrödinger operator or the number k be a polynomial function with a a u on L2 (W, dμ). Let P (u) = 2N 2N > 0 and N 2. Let k=0 k g be a periodic positive smooth function on R such that g(x + l) = g(x) for all x. We define the potential function on W by w :, Vλ (w) = λ : V √ λ
w w(x) :V √ := :P √ : g(x) dx, λ λ
(2.3) (2.4)
I
where λ > 0 and : P (w(x)) : is defined by the Wick product with respect to μ. We recall the definition of the Wick product. Note that we do not need the assumption on the periodicity and the smoothness on g to define the Wick product. Let Pn be the projection operator on H onto the linear span of {ek | −n k n} and set wn (x) = (Pn w)(x). We denote cn = Actually cn2 =
Eμ [wn (x)2 ].
n
1 . 2 2 k=−n (ml) + (2πk)
Then for k 2 by the definition of the Wick product,
:
wn (x) √ λ
k
wn (x) :P √ λ
:=
wn (x) √ λ
k +
[k/2]
ck,j
j =1
wn (x) √ λ
k−2j
cn √ λ
2j ,
[k/2]
2N wn (x) wn (x) k−2j cn 2j :=P + ak ck,j √ , √ √ λ λ λ k=2 j =1
k! where ck,l = (− 12 )l l!(k−2l)! . limn→∞
I
(2.5)
n (x) : P ( w√ ) : g(x) dx exists in L2 (μ) and we denote the
λ
2 limit by : V ( √w ) : . The operator (−LA + Vλ , FC ∞ A (W )) is essentially self-adjoint in L (μ) [29] λ and we denote the self-adjoint extension by −LA + Vλ . [26,27,29] are basic references to this operator. −LA + Vλ is a representation of the quantization of the Hamiltonian whose classical field equation is the non-linear Klein–Gordon equation with space-time dimension 2:
1 ∂2 m2 ∂2 w(t, x) + P w(t, x) g(x) = 0, w(t, x) − w(t, x) + 2 ∂x 2 2 ∂t 2
(t, x) ∈ R × I.
(2.6)
S. Aida / Journal of Functional Analysis 256 (2009) 3342–3367
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Physically λ is the inverse of the Planck constant h¯ and our problem is to determine the semiclassical limit of the lowest eigenvalue E0 (λ) of −LA + Vλ as λ → ∞ in terms of the potential function U which is given below. 1 2 Definition / D(A). Here 2.1. Let U (h) = 4 AhH + V (h) for h ∈ D(A) and U (h) = +∞ for h ∈ V (h) = I P (h(x))g(x) dx and h ∈ H .
Note that V can be defined on H by using the following Sobolev embedding theorem. In [4], we assume that V can be extended to a C 3 -function on W . However, this does not hold in the present case and we use this lemma. Lemma 2.2. For any p 1, H ⊂ Lp (I, dx) and the embedding is compact. We can rewrite U in the following form. Lemma 2.3. (1) h ∈ D(A) is equivalent to the following (i) or (ii). (i) h ∈ H 1 (I, dx), (ii) h is an absolutely continuous function with h ∈ L2 (I, dx) and h(−l/2) = h(l/2). (2) It holds that for any h ∈ D(A) U (h) =
1 4
I
h (x)2 dx +
m2 h(x)2 + P h(x) g(x) dx. 4
(2.7)
I
By this lemma, U is a smooth functional on a proper subspace H 1 (I, dx) of H 1/2 (I, dx). The following assumptions are standard in studies of semi-classical limit of Schrödinger operators. The necessary and sufficient condition of the strict positivity of the Hessian of U at hi is given in Lemma 2.6. Assumption 2.4. (A1) U (h) (h ∈ H 1 (I, dx)) is a non-negative function and has finitely many zero point set N = {h1 , . . . , hn }. (A2) Suppose (A1). The Hessian 12 D 2 U (hi ) ∈ L(H 1 (I, dx), H 1 (I, dx)) is a strictly positive operator for all 1 i n. Under (A1) and (A2), we prove that the asymptotic behavior of the bottom of spectrum of −LA + Vλ can be determined by Schrödinger operators with quadratic Wiener functionals in Theorem 2.9. To this end, we prepare a few lemmas. Lemma 2.5. (1) ι∗ = j ◦ A˜ −2 holds. (2) Let v be a bounded measurable function on I . Let Mv be the multiplication operator on L2 (I, dx). Then it holds that ι∗ ◦ Mv ◦ ι = j ◦ A˜ −2 Mv ◦ ι. Moreover ι∗ ◦ Mv ◦ ι is unitar-
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S. Aida / Journal of Functional Analysis 256 (2009) 3342–3367
ily equivalent to A˜ −1 Mv A˜ −1 on L2 (I, dx). In particular ι∗ ◦ Mv ◦ ι is a Hilbert–Schmidt operator on H . Proof. (1) Let ϕ ∈
˜n n∈N D(A )
and h ∈ H .
˜ ι(h), ϕ L2 = Aι(h), A˜ A˜ −2 ϕ L2 = h, j ◦ A˜ −2 ϕ H .
(2.8)
This implies (1). The first and the second statement of (2) follows from (1) and the definition of Φ. The Hilbert–Schmidt property follows from that A˜ −2 is a Hilbert–Schmidt operator. 2 Lemma 2.6. (1) Assume (A1). Then hi can be extended to a C ∞ function on R with period l. Moreover hi satisfies (2.9) −h i (x) + m2 hi (x) + 2P hi (x) g(x) = 0 ( for all x ∈ I ). (2) Let vi (x) = 12 P (hi (x))g(x). Then 12 D 2 V (hi ) = ι∗ Mvi ι. This derivative D is the usual Fréchet derivative on H . (3) The strict positivity of the Hessian of U at hi is equivalent to the positivity of the bottom of the spectrum of the Schrödinger operator − + m2 + 4vi (x) on L2 (I, dx). Proof. (1) We can check that (2.9) holds in the sense of distribution on (−l/2, l/2). By hi ∈ H 1 (I, dx) and the hypoellipticity of the Laplacian, hi is a C ∞ function on (−l/2, l/2). By taking the derivative of U at hi in the direction to the constant function, we have 2 m hi (x) + 2P hi (x) g(x) dx = 0. I
By integrating both the sides of (2.9) on I , we get h i ( 2l −) = h i (− 2l +). This and (2.9) show hi can be extended to a C ∞ function with period l. (2) This is proved by a simple calculation. (3) In the calculation below, the Fréchet derivative is the derivative on the Hilbert space H 1 (I, dx). By the direct calculation, we have for h ∈ H 1 (I, dx), 1 2 1 2 D U (hi )(h, h) = m − 4vi (x) h(x)2 + h (x)2 dx. (2.10) 2 4 I
Now suppose that the Hessian of U at hi is strictly positive. This implies that there exists ε > 0 such that for all h ∈ H 1 (I, dx), RHS of (2.10) εh2H 1 (I,dx) .
(2.11)
Since hH 1 (I,dx) mhL2 (I,dx) , this implies the bottom of spectrum of m2 − + 4vi is positive. We prove the converse. If inf σ (m2 − + 4vi ) > 0, then there exists δ > 0 such that RHS of (2.10) δh2L2 (I,dx) .
(2.12)
S. Aida / Journal of Functional Analysis 256 (2009) 3342–3367
3347
If inf{D 2 U (hi )(f, f ) | f H 1 (I,dx) = 1} = 0, then there exists a sequence fk ∈ H 1 (I, dx) such that fk H 1 (I,dx) = 1 for all k ∈ N and limk→∞ D 2 U (hi )(fk , fk ) = 0. (2.12) implies that limk→∞ fk = 0 in L2 (I, dx). Hence limk→∞ fk H 1 (I,dx) = 0. This is a contradiction. 2 We use the notation Kv = ι∗ Mv ι ∈ L2 (H ). Here we denote the set of all Hilbert–Schmidt operators on a Hilbert space X by L(2) (X) and the Hilbert–Schmidt norm by L(2) (X) . Also we denote the set of all trace class operators on a Hilbert space X by L(1) (X) and the trace norm by L(1) (X) . If there are no confusion, we may omit X. Let K ∈ L(2) (H ). Then we define : (Kw, w)H := lim
n→∞
KPn w, Pn w
H
− trH Pn KPn
in L2 (μ)
(2.13)
for a family of projection operators {Pn } satisfying that Pn converges to IH strongly. Here trH denotes the trace of the operator on H . The definition does not depend on the choice of {Pn }. Hence if K ∈ L(1) (H ), limn→∞ (KPn w, Pn w) converges in L2 (μ) and independent of the choice of {Pn }. We denote the limit by (Kw, w)H . Lemma 2.7. Let v be a bounded measurable function on I . (1) It holds that (Kv Pn w, Pn w)H − trH Pn Kv Pn =
: wn (x)2 : v(x) dx. I
(2) It holds that
I
: w(x)2 : v(x) dx =: (Kv w, w)H :.
Proof. First note that wn (x) = |k|n w, ek ek (x). By the definition of the Wick product, 2 wn (x)2 − cn2 v(x) dx. : wn (x) : v(x) dx = I
I
On the other hand, (Kv Pn w, Pn w)H = trH Pn Kv Pn =
I
wn (x)2 v(x) dx and
Kv j ◦ A˜ −1 ek , j ◦ A˜ −1 ek H
|k|n
1 (Mv ek , ek )L2 2πk |k|n m2 + ( l )2 1 = v(x) dx. 2 + (2πk)2 (ml) |k|n =
(2.14)
I
This proves (1). (2) follows from (1) and the definitions. Below, we use the notation Qv (w) = and Kv,n = Pn Kv Pn .
I
2
: w(x)2 : v(x) dx, Qv,n (w) =
I
: wn (x)2 : v(x) dx
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S. Aida / Journal of Functional Analysis 256 (2009) 3342–3367
Lemma 2.8. Let v be a C 2 -function on R with v(x) = v(x + l) for all x. We assume that m2 − + 4v is a strictly positive operator on L2 (I, dx) and set A˜ v = (m2 − + 4v)1/4 . (1) A˜ 2v − A˜ 2 − 2A˜ −1 Mv A˜ −1 is a trace class operator on L2 (I, dx) and A˜ 2v − A˜ 2 is a Hilbert– Schmidt operator on L2 (I, dx).Moreover A˜ 2v − A˜ 2 − 2A˜ −1 Mv A˜ −1 is unitarily equivalent to thebounded linear operator A4 + 4AKv A − A2 − 2Kv on H . (2) A−1 ( A4 + 4AKv A − A2 )A−1 and A−1 Kv A−1 are trace class operators on H . (3) inf σ (−LA + Qv ) is the eigenvalue of −LA + Qv and 1 ˜2 tr Av − A˜ 2 − 2A˜ −1 Mv A˜ −1 2 2 1 = − A˜ 2v − A˜ 2 A˜ −1 L (L2 (I,dx)) , (2) 4
inf σ (−LA + Qv ) =
tr denotes the trace in L2 (I, dx). (4) Let Ωv be the ground state function of −LA + Qv . Ωv is given by 1 Ωv (w) = det(IH + Tv )1/4 exp − A−1 A4 + 4AKv A − A2 A−1 w, w H , 4
(2.15) (2.16)
(2.17)
where Tv = A−1 ( A4 + 4AKv A − A2 )A−1 . Ωv (w)2 dμ is the Gaussian measure whose covariance operator is (m2 + 4v − )−1/2 on L2 (I, dx). The expression of inf σ (−LA + Qv ) in terms of the Hilbert–Schmidt norm can be found in [12] and [25]. This representation follows from the so-called dressing transformation. Proof. As we noted, the Laplace operator on I with periodic boundary condition can be identified with the Laplace–Beltrami operator on S 1 (l) which is the quotient space of R (1-dimensional Riemannian manifold) by the relation x ∼ y ↔ x − y = l. We identify S 1 (l) [−l/2, l/2). According to this identification, v is also a C 2 -function on S 1 (l). We denote the kernel of Tt = et by pS (t, x, y) (t > 0, x, y ∈ S 1 (l)) with respect to the uniform measure dx on S 1 (l) (Riemannian volume). Then explicitly for any x, y ∈ S 1 (l), pS (t, x, y) =
p(t, x, y + nl),
(2.18)
n∈Z
where p(t, x, y) =
√1 4πt
exp(− |x−y| 4t ). Therefore it holds that 2
C d(x, y)2 pS (t, x, y) √ exp − , 4t t
(2.19)
where d(x, y) denotes the Riemannian distance on S 1 (l), that is, d(x, y) = min{|x − y + nl| | n ∈ Z}. (2.18) can be stated in terms of law of Brownian motion. Let π : R → S 1 (l) be the projection operator. Let γ (t), b(t) be the Brownian motion on S 1 (l) and R respectively whose generators are on S 1 (l) and R and the starting points are π(x) and x ∈ R respectively. Then the laws of γ and π(b) are the same. Also below we use the following relations:
S. Aida / Journal of Functional Analysis 256 (2009) 3342–3367
3349
(i) For a > 0 and γ > 0, it holds that a −γ =
1 (γ )
∞
e−ta dt. t 1−γ
(2.20)
0
(ii) For a > 0 and b > 0 and 1 < γ < 2, it holds that
a
γ −1
−b
γ −1
(2 − γ ) = γ −1
∞
e−tb − e−ta dt. tγ
(2.21)
0
2 First we prove that A˜ 2v − A˜ 2 is a Hilbert–Schmidt operator. Let Tt = e−t (m +4v−) and St = 2 e−t (m −) . For 0 < ε < R < ∞ and f ∈ L2 (S 1 (l), dx), let
1 Tε,R f = √ 2 π
R
S t f − Tt f dt. t 3/2
(2.22)
ε
Then for any f ∈ C ∞ (S 1 (l)), A˜ 2v f − A˜ 2 f =
lim
ε→0,R→∞
Tε,R f
in L2 -sense. By the Feynman–Kac formula, t pS (t, x, y) 1 St (x, y) − Tt (x, y) −m2 t − 0 4v(γ (s)) ds γ (t) = y 1 − e = e E . (2.23) √ x t t 3/2 t Let 0 < ε < a < R < ∞. By the estimate (2.19), a
S (x, y) − T (x, y) a d(x, y)2 t t 4tv∞ C exp − dt dt e t 4t t 3/2 ε
0
e
4av∞
C C1 + log max
1 ,1 d(x, y)2
.
(2.24)
This implies that Tε,R is a Hilbert–Schmidt operator and the strong limit limε→0 Tε,R exists and belongs to Hilbert–Schmidt classes. Noting that max(Tt op , St op ) 1 for all t > 0, I2 = limR→∞ Ta,R converges in norm sense in L2 . It suffices to show that supR Ta,R L(2) (S 1 (l)) < ∞. Actually we can prove that the trace norm is finite. To see it, note that R/n Ta,R = a/n
Snt − Tnt dt. √ 2 nπt 3/2
(2.25)
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S. Aida / Journal of Functional Analysis 256 (2009) 3342–3367
Also by using the identity Tnt − Snt = Tt (T(n−1)t − S(n−1)t ) + (T(n−1)t − S(n−1)t )St − Tt (T(n−2)t − S(n−2)t )St , (2.26) we have Tnt − Snt =
B1 . . . Bk (Tt − St )C1 . . . Cn−k ,
where Bi = ±Tt and Cj = ±St . Let c = min(inf σ (m2 + 4v − ), inf σ (m2 − )) > 0. Then we have B1 . . . Bk op C1 . . . Cn−k op e−nct . Therefore 2 Tnt − Snt L(1) (L2 (S 1 (l),dθ)) Cn e−cnt−m t et − et (−4v) L
2 1 (1) (L (S (l),dθ))
(2.27)
holds. Here we have used that ABCL(1) (X) Aop BL(1) (X) Cop for bounded linear operators on X. By taking n to be large enough, we see that supR Ta,R L(1) (S 1 (l)) < ∞. Now we prove that A˜ 2v − A˜ 2 − 2A˜ −1 Mv A˜ −1 is a trace class operator. To this end, we use Claim A. Let K(x, y) (x, y ∈ I ) be a Borel measurable function and suppose that K ∈ L2 (I × I, dx dy) and for almost all y, x → K(x, y) is absolutely continuous function and ∂x K(·,·) ∈ L2 (I × I, dx dy). Define Tf (x) = I K(x, y)f (y) dy. Then T is a trace class operator on L2 (I, dx). This can be proved by noting that Tf (x) = (J ◦ S)(f )(y) + K(0, ·), f L2 (I,dx) , x ∂ K(x, y)f (y) dy and J g(x) = 0 g(t) dt are Hilbert–Schmidt operators. where Sf (x) = I ∂x Using this we prove the following claim. Claim B. (1) A˜ 2v − A˜ 2 − 2A˜ −2 Mv is a trace class operator. (2) A˜ −1 [A˜ −1 , Mv ] is a trace class operator. We prove (1). We have 2 1 A˜ v − A˜ 2 − 2A˜ −2 Mv (x, y) = √ 2 π
∞
e−m t Kt (x, y) dt, 2
0
where 1 Kt (x, y) = √ Ex t
t 1 1 − e− 0 4v(γ (s))ds − 4v γ (t) δy γ (t) t
(2.28)
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3351
t 1 4 v γ (s) − v γ (t) dsδy γ (t) = √ Ex t t 0
2 1 r t t 1 16 −4τ 0 v γ (s) ds v γ (s) ds e dτ drδy γ (t) − √ Ex t t 0
0
0
=: I1 (t, x, y) + I2 (t, x, y).
(2.29)
∞ 2 Using (2.27), we see that 1 e−m t Kt (x, y) dt is a trace class operator. Hence we need only to 1 −m2 t Kt (x, y) dt. It holds that for 0 < t < 1 consider 0 e C sup Kt (x, y) √ t x,y
(2.30)
and 2
− d(x,y) ∂ 4t e 1 Kt (x, y) C 1 + . + √ ∂x t t
(2.31)
This implies that 1 ∂ −m2 t e Kt (x, y) dt C 1 + log d(x, y) ∂x 0
1 2 which shows that 0 e−m t Kt (x, y) dt is a trace class operator. We prove (2.30). Note that for any 0 < s < t, x, y ∈ R, 2 Ex b(t) − b(s) b(t) = y 2(t − s) + (x − y)2 . Let v(x) ˜ = v(π(x)). We have t |y−x−nl|2 1 1 4 I1 (t, x, y) = √ v˜ x + b(s) − v˜ x + b(t) ds b(t) = y − nl √ Ex e− 4t . t t n∈Z 4πt 0
Hence |y − x − nl| |y−x−nl|2 1 C I1 (t, x, y) C + √ e− 4t √ . t t t n It is easy to check the boundedness of I2 . Therefore (2.30) holds. Next we prove (2.31). Let ηt (s) = st . Then
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t 1 ∂ 4 v˜ x + b(s) − v˜ x + b(t) dsδy π x + b(t) I1 (t, x, y) = √ E ∂x t t 0
t 4 1 +√ E v˜ x + b(s) − v˜ x + b(t) ds Dδy π x + b(t) , ηt t t 0
= I1,1 (t, x, y) + I1,2 (t, x, y),
(2.32)
where D denotes the H -derivative along the direction to ηt . By the same reason as I1 (t, x, y), we have |I1,1 (t, x, y)| √Ct . By the integration by parts formula, t s 1 4 v˜ x + b(t) − v˜ x + b(s) δy π x + b(t) I1,2 (t, x, y) = √ E t t t 0
t b(t) 4 1 δy π x + b(t) . (2.33) +√ E v˜ x + b(s) − v˜ x + b(t) ds t t t 0
By a similar estimate to I1 (t, x, y), we get for 0 < t < 1, |y−x−nl|2 C d(x,y)2 I1,2 (t, x, y) C e− 4t + C e− 4t + C. t n t
(2.34)
Consequently, we obtain (2.31). We prove Claim B (2). By (2.20), we have −1/4 A˜ −1 f (x) = m2 − f (x) =
1 (1/4)
∞ I
e−m t pS (t, x, y)f (y) dt dy. t 3/4 2
(2.35)
0
Let us define a bounded linear operator for 0 < ε < R < ∞, 1 S1,ε,R f (x) = (1/4)
R I
e−m t pS (t, x, y)f (y) dt dy. t 3/4 2
ε
The kernel S1,ε,R (x, y) is a C ∞ -function on I × I and S1,ε,R (x, y)
C , d(x, y)1/2
(2.36)
where C does not depend on ε, R, x, y. By (2.20), 2 −1/4 v(y) Mv m − f (y) = (1/4)
∞ 0
e−m t Tt f (y) dt. t 3/4 2
(2.37)
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Hence −1/4 2 , Mv f = m −
∞
e−m t {Tt (f v) − vTt f } dt. (1/4)t 3/4 2
0
Let 0 < ε < R < ∞. The bounded linear operator R S2,ε,R f =
e−m t {Tt (f v) − vTt f } dt (1/4)t 3/4 2
ε
has the C ∞ -kernel R S2,ε,R (y, z) =
e−m t (v(z) − v(y)) pS (t, y, z) dt. (1/4)t 3/4 2
(2.38)
ε
By a direct calculation, we have S2,ε,R (y, z) C, ∂ C S2,ε,R (y, z) ∂z d(y, z)1/2 ,
(2.39) (2.40)
where C is a constant which is independent of ε, R, y, z. Moreover limε→0,R→∞ S2,ε,R converges to a continuous function on S 1 (l) × S 1 (l) uniformly. Let S3,ε,R (x, z) =
S1,ε,R (x, y)S2,ε,R (y, z) dy.
(2.41)
I
By (2.36), (2.39), (2.40) and
1 1 +1 , dy C log √ d(x, z) d(x, y)d(y, z)
S 1 (l) ∂ supε,R ∂z S3,ε,R L(2) (S 1 (l),dθ) < ∞. This proves that the kernel function of A˜ −1 [A˜ −1 , Mv ] satisfies the properties of Claim A. We prove the unitarily equivalence. Note that A = Φ ◦ A˜ ◦ Φ −1 ,
Φ A˜ −1 Mv A˜ −1 Φ −1 = j ◦ A˜ −1 A˜ −1 Mv A˜ −1 A˜ ◦ ι = Kv , Φ ◦ A˜ 4 + 4Mv ◦ Φ −1 = A4 + 4j ◦ A˜ −1 Mv A˜ ◦ ι and AKv A = A ◦ j ◦ A˜ −2 Mv ◦ ι ◦ A = j ◦ A˜ −1 Mv A˜ ◦ ι which implies the unitary equivalence. For (2), we need only to prove that A−1 Kv A−1 ∈ L(1) (H ). This follows from that −1 Φ A−1 Kv A−1 Φ = A˜ −2 Mv A˜ −2 and A˜ −2 is a Hilbert–Schmidt operator.
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We prove (3) and (4). Let T1 = A˜ 2v − A˜ 2 − 2A˜ −1 Mv A˜ −1 . The square of the Hilbert–Schmidt norm of (A˜ 2v − A˜ 2 )A˜ −1 is the same as the trace of T2 = A˜ −1 (A˜ 2v − A˜ 2 )2 A˜ −1 . The difference of 1 1 2 T1 and − 4 T2 is 1 2 T3 = A˜ −1 2A˜ A˜v A˜ − A˜ 2 A˜ 2v − A˜ 2v A˜ 2 A˜ −1 . 4 Since T3 is a trace class operator, the trace of T3 is 0. So (2.15) and (2.16) are equal. To prove these values are the lowest eigenvalue of −LA + Qv , we introduce a finite-dimensional approximation of −LA + Qv . Let n be a natural number and Pn be the projection operator which we already defined. Let 1/4 Ωv,n (w) = det Pn A−1 A4 + 4AKv AA−1 Pn −1 1 −1 4 2 × exp − A A + 4AKv A − A A Pn w, Pn w H . 4
(2.42)
Then Ωv,n ∈ D(LA ) and (−LA + Q˜ v,n )Ωv,n =
1 trH Pn A4 + 4AKv A − A2 − 2Kv Pn Ωv,n , 2
(2.43)
where ˜ v,n (w) = Qv,n (w) − 1 Pn⊥ A4 + 4AKv A − A2 A−1 Pn w 2 . Q H 4 Taking the limit n → ∞, we complete the proof of (3) and (4).
(2.44)
2
Our main theorem is as follows. Theorem 2.9. Assume that (A1) and (A2) hold. Let E0 (λ) = inf σ (−LA + Vλ ). Then lim E0 (λ) = min Ei ,
λ→∞
(2.45)
1in
where Ei is the lowest eigenvalue of −LA + Qvi (w), where Qvi (w) = vi (x) = 12 P (hi (x))g(x).
I
: w(x)2 : vi (x) dx and
Remark 2.10. In [4], we studied the operator −LA + λV ( √w ) in the case where V is a C 3 λ
function on W . Under the same assumption as in Theorem 2.9, Ki = 12 D 2 V (hi ), AKi A are trace class operators on H and we proved that the limit of the lowest eigenvalue of −LA + λV ( √w ) is λ mini Ei , where Ei =
1 trH A4 + 4AKi A − A2 . 2
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3355
Ei is the lowest eigenvalue of −LA + (Ki w, w). trH in the above is the trace in H . In Theorem 2.9, the potential function of approximate Schrödinger operators is given by Hilbert–Schmidt operators Ki . Ei in Theorem 2.9 can be written as follows: Ei =
1 trH A4 + 4AKi A − A2 − 2Ki , 2
(2.46)
since the operator inside the trace is unitarily equivalent to A˜ 2i − A˜ 2 − 2A˜ −1 Mvi A˜ −1 on L2 (I, dx). The expression of Ei in (2.16) can be found in [12,25]. This implies that Ei is non positive number. It is trivial that Ei 0 because W Qvi (w) dμ(w) = 0. Actually, we see that E0 (λ) 0 for all λ by the identity (3.4). 2
Example 2.11. (1) Assume that g(x) ≡ 1 and set Q(x) = m4 x 2 + P (x). Suppose that Q(x) 0 for all x and let {c1 , . . . , cn } be the zero points. Then the constant functions {c1 , . . . , cn } are minimizers of U and U (ci ) = 0 for all i. We have m2 − + 4vi (x) = − + 2Q (ci ). Thus, (A1) and (A2) are equivalent to that Q (ci ) > 0 for all zero point ci (1 i n). Suppose that g is not a constant function and hi is a constant function for some i. Then P (0) = 0 and hi (x) = 0. Therefore, if g is not a constant function and hi = 0, then hi is not a constant function. 2 2 (2) Let Pa (u) = a(u2 − 1)2 and Qa (u) = m4 u2 + Pa (u). Then for a satisfying a > m8 , Qa 2 achieves the minimum at ±xa , where xa = 1 − m 8a and the Hessian of Qa is strictly positive. Thus for g ≡ 1, Pa − Qa (xa ) satisfies the assumptions of Theorem 2.9. When g is a non-constant function, then we can prove that U has two minimizers {±ha } and satisfies the assumptions in Theorem 2.9 for sufficiently large a as in the same proof in [1]. In this case, hi is not a constant function. We use lower bound estimates of Schrödinger operators (NGS bound [16,29]) and large deviation estimates for Wiener chaos to prove the estimate LHS RHS in Theorem 2.9. Lemma 2.12. In the estimate below, V˜ is a bounded measurable function. (1) It holds that
m 2 (−LA + V˜ )f, f L2 (μ) − log exp − V˜ dμ(w) f 2L2 (μ) . 2 m W
(2) Let T be a trace class self-adjoint operator on H with inf σ (IH + T ) > 0. Then m
(IH + T )Df (w)2 dμ + H
W
V˜ (w)f (w)2 dμ
W
m 2 1 − log exp − V˜ (w) − (T w, w)H − T w2H dμ(w) f 2L2 (μ) 2 m 2
W
m m 2 log det(IH + T ) − tr(T ) − m tr T f 2L2 (μ) . + 2 2
(2.47)
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The following estimate is used for lower bound estimates of −LA + Vλ near zero points {h1 , . . . , hn }. Lemma 2.13. Let V˜ be a bounded measurable function on W . Let v be a C 2 function on R with period√l. We assume that m2 − + 4v is a strictly positive operator on L2 (I, dx). Let cv = inf σ ( m2 − + 4v) and Ev = inf σ (−LA + Qv ). Then it holds that for any f ∈ FC ∞ A (W ),
(−LA + Qv + V˜ − Ev )f, f
cv − log L2 (μ) 2
2 ˜ 2 exp − V (w) Ωv (w) dμ(w) f 2L2 (μ) . cv
W
Proof. We prove this lemma using the finite-dimensional approximation in the proof of Lemma 2.8(4). Let n n0 be natural numbers. Let f (w) = F ( w, e−n0 , . . . , w, en0 ) and set ˜ V˜n = E[V |Fn ]. Here Fn = σ (Pn w). Let An = Pn APn and −LAn be the generator of the Dirichlet form Wn An Df (w)2H dμn (w) on L2 (Wn , dμn ), where μn = (Pn )∗ μ and Wn = Pn W . Let ˜ v,n ). Explicitly, Ev,n = inf σ (−LAn + Q Ev,n =
1 trH Pn A4 + 4AKv A − A2 − 2Kv Pn . 2
−1 , where Ω ˜ v,n and −LA + Q ˜ v,n . Then we have Let f˜ = f Ωv,n v,n is the ground state of −LAn + Q
(−LA + Q˜ v,n + V˜n − Ev,n )f, f L2 (W,dμ) ˜ v,n + V˜n − Ev,n )f, f 2 = (−LAn + Q L (Wn ,dμn ) 2 = An D f˜(w)H Ωv,n (w)2 dμn (w) + V˜n (w)f˜(w)2 Ωv,n (w)2 dμn (w). Wn
(2.48)
Wn
The above Dirichlet form is finite-dimensional one. Let Aand B be strictly positive n × n B n matrices and consider a Gaussian measure dμB (x) = det( 2π ) exp(− (Bx,x) 2 ) dx on R and 2 2 n a Dirichlet form EA,B (f, f ) = Rn |ADf (x)| dμB (x) on L (R , dμB ). Then Γ2 (f, f )(x) inf σ (ABA)ADf (x)2Rn holds, where Γ2 is Bakry–Emery’s Γ2 . In the present case, A = An and B = Pn A−1 A4 + 4AKv AA−1 Pn . We denote Ωv,n (w)2 dμn (wn ) = dμv,n (wn ). Then by the Bakry–Emery criterion, 2 An D f˜(wn )2 dμv,n (w), f˜(wn )2 log f˜(wn )2 /f˜2L2 (μ ) dμv,n (w) Hn v,n cv,n Wn
Wn
where cv,n = inf σ (Pn A4 + 4AKv APn ). Hence by [16], we have cv,n log RHS of (2.48) − 2
2 ˜ 2 exp − Vn (wn ) Ωv,n (wn ) dμn (w) f 2L2 (μ) . cv,n
Wn
By taking the limit n → ∞, we complete the proof.
2
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3357
Now we prepare large deviation estimates. Lemma 2.14. For any δ > 0, lim lim sup
n→∞ λ→∞
1 log μ λ
w wn = −∞. : −V √ > δ w : V √ λ λ
(2.49)
Proof. The proof of this result is essentially found in [6,7,14,21,22] and follows from the hypercontractivity of the semigroup Tt = etL , where −L is the Ornstein–Uhlenbeck operator (number operator) on L2 (W, dμ). We give the sketch of the proof for the sake of completeness. Since
w wn : −V √ > δ w : V √ λ λ
wn w wn wn ⊂ w : V √ :−:V √ : > δ/2 ∪ w : V √ : −V √ > δ/2 . λ λ λ λ It suffices to show
w wn = −∞, :−:V √ : > δ/2 w:V √ λ λ
wn wn 1 = −∞. lim lim sup log μ w : V √ : −V √ > δ/2 n→∞ λ λ λ
1 lim lim sup log μ n→∞ λ→∞ λ
(2.50) (2.51)
λ→∞
wn Note that : V ( √w ) := T(log λ)/2 f (w), : V ( √ ) := T(log λ)/2 fn (w), where f (w) =: V (w) : and λ λ fn (w) =: V (wn ) : . By limn→∞ f − fn L2 (μ) = 0 and the hypercontractivity of Tt , we get (2.50). (2.51) follows from that for any positive δ there exist positive constants Cn , Cn such that
k−2j 2 1+ 2j k/2 wn (x) Cn exp −δ k−2j · Cn λ k−2j . g(x) dx δλ μ w
2
I
Theorem 2.15. Let T be a trace class self-adjoint operator on H . Let χ ∈ Cb∞ (R) be a nonnegative function. Set
w2W w Fλ (w) =: V √ :χ + λ−1 (T w, w)H , λ λ
w ∈ W,
and F (h) = V (h)χ(h2W ) + (T h, h)H for h ∈ H . (1) The image measure of μ by the measurable map Fλ satisfies the large deviation principle with the good rate function: IF (x) =
inf{ 12 h2H | there exists h ∈ H such that F (h) = x}, +∞, there are no h ∈ H such that F (h) = x.
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(2) Assume that I + 2T is a strictly positive operator. Then there exists α0 > 1 such that for any 0 < α < α0 , 1 lim log λ→∞ λ
exp −αλFλ (w) dμ(w)
W
1 = − min h2H + αV (h)χ h2W + α(T h, h)H h ∈ H . 2
(2.52)
Proof. (1) This can be proved by a standard argument [9] by using Lemma 2.14 and the Sobolev embedding theorem (Lemma 2.2). We omit the proof. (2) We prove that for any α > 0
1 lim sup log exp −αVλ (w) dμ(w) < ∞. λ→∞ λ
(2.53)
W
It is well known that W e−αVλ (w) dμ(w) < ∞ for all α > 0. So our task is just to see how the integral on the semi-classical parameter λ carefully. Since there exists β > 1 such that −β(Tdepends w,w)H dμ(w) < ∞, if (2.53) can be proved, we have for α < β, e W
1 exp −αλFλ (w) dμ(w) < ∞. lim sup log λ→∞ λ W
n (x) n (x) By this and a standard argument, we get (2.52). Let I1 = λ : P ( w√ ) : −λP ( w√ ). Let pk,j =
λ
2N k−2j
and qk,j be the positive number such that
1 pk,j
λ
1 + qk,j = 1 and δ be a small positive number.
Then 2N 2j qk,j 2N [k/2] 1 cn (x) w 1 n ck,j δ √ + I1 −C =: I2 , pk,j qk,j δλj −1 λ k=2
(2.54)
j =1
where ck,j and cn are the constants in (2.5). We can find a positive number C , C which depends only on P such that
λP
wn (x) + I2 + λC −C cn2N , √ λ
(2.55)
where N is a natural number which depends on P . Thus
:P
λ I
wn (x) √ λ
: g(x) dx + λC −C cn2N .
(2.56)
Let V¯λ (w) := Vλ (w) + λC .
(2.57)
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By a standard argument using the hypercontractivity of the Ornstein–Uhlenbeck semigroup, for large positive number r, we have C2 (N )|r|1/C3 (N ) , μ {V¯λ −r} e−C1 (N )re
(2.58)
where Ci are positive constants which depend only on P . This completes the proof of (2.53).
2
Lemma 2.16. Let χ be a smooth non-negative function such that {χ = 1} = [−1, 1], {χ = 0} = (−∞, −2] ∪ [2, ∞) and 0 χ 1. Set ρλ,ε (w) = χ( uous functions on I such that infx f2M (x) > 0. Let ϕλ (w) =
2M
:
k=3 I
w(x) √ λ
w2W λε
). Let fk (x) (3 k 2M) be contin-
k : fk (x) dx.
(2.59)
Then for sufficiently small ε, we have lim
λ→∞
e−λϕλ (w)ρλ,ε (w) dμ(w) = 1.
(2.60)
W
1 2 k Proof. Let F (h) = 2M k=3 I h(x) fk (x) dx. Then limhH →∞ ( 2 hH + F (h)) = ∞. Also for any δ > 0 and R > 0, there exists C(δ, R) such that hkLk δ k−2 h2H
for hW C(δ, R),
hH R.
This can be proved by using Lemma 2.2. Therefore, for sufficiently small ε, −1 1 2 2 min hH + 2F (h)χ ε hW h ∈ H 0. 2
Thus, by the Schwarz inequality and the large deviation estimate,
e−λϕλ (w)ρλ,ε (w) dμ(w)
{ρλ,ε (w)=1}
√ 1/2 μ w wW λε
1/2 e−2λϕλ (w)ρλ,ε (w) dμ(w)
W
e
−Cλ
(2.61)
.
Let ψλ (w) =
1/k k
2M : w(x) : f (x) dx √ k λ k=3 I
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and 0 < δ < 1.
e−λϕλ (w)ρλ,ε (w) dμ(w) =
{ρλ,ε (w)=1}
e−λϕλ (w)ρλ,ε (w) dμ(w)
{ρλ,ε (w)=1,ψλ (w)>δ}
e−λϕλ (w) dμ(w)
+ {ρλ,ε (w)=1,ψλ (w)δ}
=: I1 (λ) + I2 (λ).
(2.62)
By the large deviation estimate, limλ→∞ I1 (λ) = 0. We prove that limλ→∞ I2 (λ) = 1. We have
exp −λδψλ (w)2 dμ(w) I2 (λ)
{ρλ,ε (w)=1,ψλ (w)δ}
exp λδψλ (w)2 dμ(w).
{ρλ,ε (w)=1,ψλ (w)δ}
Noting that ψλ (w) =
ψ√ 1 (w) λ
and
√ ρε,λ (w) = 1, ψλ (w) δ = w w2W ελ, ψ1 (w) δ λ , we have
exp −δψ1 (w)2 dμ(w) lim inf I2 (λ) lim sup I2 (λ) λ→∞
W
λ→∞
exp δψ1 (w)2 dμ(w). (2.63)
W
2 Thus, noting that W eδψ1 (w) dμ < ∞ for sufficiently small δ > 0 which follows from Theorem 2.15, we have limλ→∞ I2 (λ) = 1. 2 Remark 2.17. Let G(h) = 12 h2H + F (h) for h ∈ H . Then h = 0 is a zero point of G and is a local minimizer of G. Hence, (2.60) is nothing but the Laplace asymptotic formula. Clearly, if we do not put the cut-off function ρλ,ε on the exponent, the limit may be not 1 if G has other zero points. We need the following lemma to prove the lower bound estimate. The proof of the following lemma is almost similar to Lemma 3.5 in [4]. In [4], we assume that V is a C 1 -function on W . However, we can prove the lemma by modifying the proof noting that the present V is a C ∞ -function on Lp (I, dx) for all p 2N and using the Sobolev embedding theorem.
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We use the following notation. For r > 0 and z ∈ W, k ∈ H , we denote Br (z) = {w ∈ W | w − zW r} and Br,H (k) = {h ∈ H | h − kH r}. Lemma 2.18. √ (1) For any h ∈ D(A) and N ∈ N, Ah2H ( mPN⊥ + APN )h2H . (2) It holds that m 2 hH + V (h) hW R = +∞. lim inf R→∞ 4 (3) For any ε > 0, there exist δ(ε) and N0 ∈ N such that for all N N0 ,
2 1 √ inf mPN⊥ + APN hH + V (h) h ∈ 4
n
c Bε (hi )
∩H
δ(ε).
i=1
Proof. (1) is trivial. (2) follows from lim inf ch2H + V (h) hH = a = +∞
(2.64)
a→∞
for any c > 0. We prove (3). By (2.64) it is enough to show that for any L > 0 such that for any ε > 0, there exist δ(ε) and N0 ∈ N such that for all N N0 ,
2 1 √ inf mPN⊥ + APN hH + V (h) h ∈ 4
n
c Bε (hi )
∩ BL,H (0) δ(ε).
i=1
By a similar proof to [4, Lemma 2.4], there exists ρ(ε) > 0 such that
1 inf Ah2H + V (h) h ∈ 4
n
c
∩ D(A) ρ(ε) > 0.
Bε (hi )
(2.65)
i=1
Since A and PN commute, √ 1 mP ⊥ h + APN h2 + V (h) = m P ⊥ h2 + 1 APN h2 + V (PN h) N N H H H 4 4 4 + V (h) − V (PN h). By Lemma 2.2, we have for h ∈ H with hH L V (h) − V (PN h) CL P ⊥ h N
L2 (I,dx)
(2.66)
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and limN →∞ suphH L PN⊥ hL2 = 0. Therefore there exists N0 ∈ N such that for any N N0 ,
2 ε ε 1 sup V (h) − V (PN h) hH L min ρ , m . 2 2 32
(2.67)
n c Let h ∈ ( ni=1 n Bε (hi )) ∩ cBL,H (0). There are two cases where (i) PN h ∈ i=1 Bεε/2 (hi ), (ii) PN h ∈ ( i=1 Bε/2 (hi )) . Assume that there exists i such that PN h − hi W 2 . Then √ PN⊥ hH PN⊥ hW h − hi W − hi − PN hW ε/2. Hence 14 mPN⊥ h + APN h2H + 2 1 √ 1 ε ⊥ 2 V (h) mε 32 . If (ii) holds, then 4 mPN h + APN hH + V (h) 2 ρ( 2 ). These complete the proof. 2 3. Proof of the main theorem Now we prove Theorem 2.9. Proof of Theorem 2.9. (1) Lower bound estimate. To prove the inequality LHS RHS in (2.45), we need to divide the estimate into two parts: (I) Neighborhood of the zero points of U . (II) Outside neighborhood of the zero points of U . Let χ be a cut-off function as in Lemma 2.16. Let ε > 0 and
χi (w) = χ
(w −
√ n λhi )2W !1 − and χ (w) = χi (w)2 . ∞ ε2 λ i=1
Let f∗ (w) = f (w)χ∗ (w), where ∗ = i, ∞ (1 i n). Then
(−LA + Vλ )f, f =
{∗=1,...,n,∞}
−
(−LA + Vλ )f∗ , f∗ ADχ∗ 2H f (w)2 dμ(w).
(3.1)
{∗=1,...,n,∞} W
Simple calculation shows that there exists a positive constant C such that ADχ∗ (w)2H C μ-a.s. w for all ∗. First, we consider the case where ∗ = 1, . . . , n. ε2 λ (I) Neighborhood of the zero points of U . Let 1 i n. Using the Cameron–Martin formula, (−LA + Vλ )fi , fi
√ √ 2 λ = (ADfi )(w + λhi )H exp − λ(hi , w)H − hi 2H dμ 2 W
√ √ √ λ + Vλ (w + λhi )fi (w + λhi )2 exp − λ(hi , w)H − hi 2H dμ. 2 W
(3.2)
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√ √ Let f¯i (w) = fi (w + λhi ) exp(− 2λ (hi , w)H − λ4 hi 2H ). Note that f¯i L2 (μ) = fi L2 (μ) . Using the integration by parts formula, we have
W
√ √ (ADfi )(w + λhi )2 exp − λ(hi , w)H − λ hi 2 dμ H H 2 √ 2 λ ¯ ¯ = A D fi (w) + hi fi (w) dμ 2 H W
=
√ (AD f¯i )(w)2 dμ + λ H
W
2 f¯i (w)2 λ dμ + A hi , w H 2 4
W
Ahi 2H f¯i (w)2 dμ.
W
Also note that Vλ (w +
√ √ λhi ) = λ P hi (x) dx + λ P hi (x) w(x)g(x) dx + : w(x)2 : vi (x) dx I
+
I 2N
k
: w(x)k :
λ1− 2
k=3
I
P (k) (hi (x)) g(x) dx. k!
(3.3)
I
Lemma 2.6(1) implies 12 (A2 hi , w)H + I P (hi (x))w(x)g(x) dx = 0 μ-a.s. w. By this and U (hi ) = 14 Ahi 2H + I P (hi (x))g(x) dx = 0, we have (−LA + Vλ )fi , fi =
W
AD f¯i (w)2 dμ +
+
Qvi (w)f¯i (w)2 dμ
W
Rλ,i (w)f¯i (w)2 dμ,
(3.4)
W
where Rλ,i (w) =
2N k=3
and gk,i (x) = Rλ,i ,
P (k) (hi (x)) g(x). k!
k
: w(x)k : gk,i (x) dx
λ1− 2 I
By Lemma 2.13, setting ci = inf σ ( m2 + 4vi − ) and V˜ =
(−LA + Vλ − Ei )fi , fi L2 (μ)
ci 2 − log exp − Rλ,i (w)ρε,λ (w) Ωi (w)2 dμ(w) f¯i 2L2 (μ) . 2 ci W
(3.5)
(3.6)
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S. Aida / Journal of Functional Analysis 256 (2009) 3342–3367 w2
Here ρε,λ (w) = χ( 3ε2 λW ). We give a lower bound estimate for the integral on the right-hand side. Note that there exists r0 > 1 such that Ωi ∈ L2r0 (μ). Let 1 < r < r0 and using the Hölder inequality,
2 exp − Rλ,i (w)ρε,λ (w) Ωi (w)2 dμ(w) ci
W
r−1 r exp(−ci,r Rλ,i (w)ρε,λ (w)) dμ(w) Ωi2 Lr (μ) ,
(3.7)
W 2r where we denote ci,r = ci (1−r) . We have limr→1+0 Ωi2 Lr (μ) = 1. By Lemma 2.16, for sufficiently small ε, we have
lim
λ→∞
exp −ci,r Rλ,i (w)ρε,λ (w) dμ(w) = 1
(3.8)
W
which implies lim inf (−LA + Vλ − Ei )fi , fi L2 (μ) 0.
(3.9)
λ→∞
(II) Outside neighborhood of the zero points of U . We estimate ((−LA + Vλ )f∞ , f∞ ). To this end, let √ n 2 3w − λhi W ! χ¯ i (w) = χ χ¯ i (w)2 . and χ¯ ∞ (w) = 1 − ε2 λ
i=1
χ¯ ∞ satisfies that χ¯ ∞ (w) = 1 for w with χ∞ (w) = 0 and
w ∈ W | χ¯ ∞ (w) = 0 ⊂
n i=1
Let ε <
√ε . 3
B
ε
c √ . λ ( λhi ) 3
For this ε , we choose a natural number N0 as in Lemma 2.18 and define T =
( √Am − IH )PN0 ∈ L(1) (H ). We have (−LA + Vλ )f∞ , f∞ 2 1 (IH + T )Df∞ (w) H dμ(w) + Vλ (w) − λδ(ε ) χ¯ ∞ (w)f∞ (w)2 dμ(w) m 2 W
+ W
W
1 λδ(ε )χ¯ ∞ (w)f∞ (w)2 dμ(w). 2
(3.10)
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Note that
1 1 λδ(ε )χ¯ ∞ (w)f∞ (w)2 dμ(w) = λδ(ε )f∞ 2L2 (μ) . 2 2
W
Let V˜λ (w) = (Vλ (w) − 12 λδ(ε ))χ¯ ∞ (w). Applying Lemma 2.12(2), J2 (λ) = m
(IH + T )Df∞ (w)2 dμ(w) + H
W
−
m log 2
V˜λ (w)f∞ (w)2 dμ(w)
W
2 ˜ 1 2 exp − Vλ (w) − (T w, w)H − T wH dμ(w) f∞ 2L2 (μ) m 2
W
m m log det(IH + T ) − tr(T 2 ) − m tr(T ) f∞ 2L2 (μ) . + 2 2
(3.11)
Again by the large deviation estimate, for large λ, we have for any ε > 0, J2 (λ) (−ε λ + Cm )f∞ 2L2 (μ) . Putting the above estimates together, we complete the proof of lower bound estimate. (2) Upper bound estimate. In (3.4), putting f¯i (w) = Ωi (w) and using lim Rλ,i (w)Ωi (w)2 dμ(w) = 0, λ→∞
W
we obtain the upper bound estimate.
2
In this case, we do not need to use Remark 3.1. When g ≡ 1, hi are constant functions. Lemma 2.18. Because inf{ m4 h2H + V (h) | h ∈ ( ni=1 Bε (hi ))c } > 0. This follows from that all hi are constant functions. However, we cannot expect this in general cases. 4. Final remarks It is natural to study the same problem in the case of spatially cut-off P (φ)2 Hamiltonian. In this case, the Cameron–Martin subspace is H 1/2 (R) and we need some modification of the proof. Also the study of the semi-classical asymptotics of the gap of spectrum between the lowest eigenvalue and the second lowest eigenvalue is also basic subject. It is related with the tunneling phenomena and the gap may be exponentially small and the exponent is determined by the Agmon distance [10]. We studied this problem in the case of the perturbation of the number operator by a smooth potential function in [2] and gave a crude upper bound on the gap of the spectrum. We study these problems in forthcoming papers. Besides these problems, we make other remarks below. (1) In the setting of this paper, the semi-group et (LA −Vλ ) is a trace class operator. See [5]. So we may determine the value limλ→∞ tr et (LA −Vλ ) by the sum of the traces of et (LA −Qvi ) . This
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would imply our main results. See [23,32] in finite-dimensional cases. Note that the semigroup of the spatially cut-off P (φ)2 Hamiltonian is not a trace class operator. In [5], the trace formula is studied but the scaling is different. (2) The studies of the Schrödinger operators in large dimension [11,17–20,24,30,31] and some additional works on renormalization may prove our main results. But the relation between them seems not clear at present. References [1] S. Aida, Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space, J. Funct. Anal. 203 (2) (2003) 401–424. [2] S. Aida, Tunneling phenomena of Schrödinger operators in Wiener spaces, preprint, 2004. [3] S. Aida, Semi-classical limit of the bottom of spectrum of a Schrödinger operator on a path space over a compact Riemannian manifold, J. Funct. Anal. 251 (1) (2007) 59–121. [4] S. Aida, Semi-classical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space: I. Unbounded one particle Hamiltonians, 2008, submitted for publication. [5] A. Arai, Trace formula, a Golden–Thompson inequality and classical limit in Boson Fock space, J. Funct. Anal. 136 (1996) 510–547. [6] C. Borell, Tail probabilities in Gauss space, in: Vector Space Measures and Applications. I, in: Lecture Notes in Math., vol. 644, Springer, 1978, pp. 71–82. [7] S. Breen, Leading large order asymptotics for (φ 4 )2 perturbation theory, Comm. Math. Phys. 92 (1983) 179–194. [8] S. Coleman, Fate of the false vacuum: Semiclassical theory, Phys. Rev. D 15 (10) (1977) 2929–2936. [9] J.D. Deuschel, D.W. Stroock, Large Deviations, AMS/Chelsea Ser., vol. 342, Amer. Math. Soc., Providence, RI, 1989. [10] M. Dimassi, J. Sjöstrand, Spectral Asymptotics in the Semi-classical Limit, London Math. Soc. Lecture Note Ser., vol. 268, Cambridge Univ. Press, Cambridge, 1999. [11] S.Y. Dobrokhotov, V.N. Kolokol’tsov, The double-well splitting of low energy levels for the Schrödinger operator of discrete φ 4 -models on tori, J. Math. Phys. 36 (3) (1995) 1038–1053. [12] W.J. Eachus, L. Streit, Exact solution of the quadratic interaction Hamiltonian, Rep. Math. Phys. 4 (3) (1973) 161– 182. [13] J.P. Eckmann, Remarks on the classical limit of quantum field theories, Lett. Math. Phys. 1 (1975/1977) 387–394. [14] S. Fang, J.G. Ren, Sur le squelette et les dérivées de Malliavin des fonctions holomorphes sur un espace de Wiener complexe, J. Math. Kyoto Univ. 33 (3) (1993) 749–764. [15] L. Gross, Abstract Wiener spaces, in: Proc. Fifth Berkeley Sympos. Math. Statist. Probab., Berkeley, CA, 1965/1966, in: Contributions to Probability Theory, part 1, vol. II, Univ. California Press, Berkeley, CA, 1967, pp. 31–42. [16] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975) 1061–1083. [17] B. Helffer, Splitting in large dimension and infrared estimates. II. Moment inequalities, J. Math. Phys. 39 (2) (1998) 760–776. [18] B. Helffer, Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics, Ser. Partial Differential Equations Appl., vol. 1, World Scientific, New York, 2002. [19] B. Helffer, F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker–Planck Operators and Witten Laplacians, Lecture Notes in Math., vol. 1862, Springer, Berlin, 2005. [20] B. Helffer, J. Sjöstrand, Semiclassical expansions of the thermodynamic limit for a Schrödinger equation, I. The one well case, in: Méthodes semi-classiques, vol. 2, Astérisque 210 (1992) 135–181. [21] M. Ledoux, Isoperimetry and Gaussian analysis, in: Lectures on Probability Theory and Statistics, Saint-Flour, 1994, in: Lecture Notes in Math., vol. 1648, Springer, Berlin, 1996, pp. 165–294. [22] M. Ledoux, A note on large deviations for Wiener chaos, in: Séminaire de Probabilités, XXXIV, 1988/1989, in: Lecture Notes in Math., vol. 1426, Springer, Berlin, 1990, pp. 1–14. [23] H. Matsumoto, Semiclassical asymptotics of eigenvalues for Schrödinger operators with magnetic fields, J. Funct. Anal. 129 (1) (1995) 168–190. [24] O. Matte, J.S. Møller, On the spectrum of semi-classical Witten-Laplacians and Schrödinger operators in large dimension, J. Funct. Anal. 220 (2) (2005) 243–264. [25] L. Rosen, Renormalization of the Hilbert space in the mass shift model, J. Math. Phys. 13 (1972) 918–927. [26] B. Simon, Continuum embedded eigenvalues in a spatially cutoff P (φ)2 field theory, Proc. Amer. Math. Soc. 35 (1972) 223–226.
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[27] B. Simon, The P (φ)2 Euclidean (Quantum) Field Theory, Princeton Univ. Press, Princeton, NJ, 1974. [28] B. Simon, Semiclassical analysis of low lying eigenvalues I. Nondegenerate minima: Asymptotic expansions, Ann. Inst. H. Poincaré Sect. A 38 (4) (1983) 295–308. [29] B. Simon, R. Hoegh-Krohn, Hypercontractive semigroups and two-dimensional self-coupled Bose fields, J. Funct. Anal. 9 (1972) 121–180. [30] J. Sjöstrand, Potential wells in high dimensions I, Ann. Inst. H. Poincaré Phys. Théor. 58 (1) (1993) 1–41. [31] J. Sjöstrand, Potential wells in high dimensions II. More about the one well case, Ann. Inst. H. Poincaré Phys. Théor. 58 (1) (1993) 43–53. [32] S. Watanabe, Generalized Wiener functionals and their applications, in: Proc. Fifth Japan–USSR Symposium on Probability Theory and Math. Statistics, in: Lecture Notes in Math., vol. 1299, Springer, Berlin, 1988, pp. 541–548.
Journal of Functional Analysis 256 (2009) 3368–3408 www.elsevier.com/locate/jfa
Compact quantum metric spaces and ergodic actions of compact quantum groups Hanfeng Li 1 Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14260, USA Received 2 September 2008; accepted 9 September 2008 Available online 25 September 2008 Communicated by Alain Connes
Abstract We show that for any co-amenable compact quantum group A = C(G) there exists a unique compact Hausdorff topology on the set EA(G) of isomorphism classes of ergodic actions of G such that the following holds: for any continuous field of ergodic actions of G over a locally compact Hausdorff space T the map T → EA(G) sending each t in T to the isomorphism class of the fibre at t is continuous if and only if the function counting the multiplicity of γ in each fibre is continuous over T for every equivalence class γ of irreducible unitary representations of G. Generalizations for arbitrary compact quantum groups are also obtained. In the case G is a compact group, the restriction of this topology on the subset of isomorphism classes of ergodic actions of full multiplicity coincides with the topology coming from the work of Landstad and Wassermann. Podle´s spheres are shown to be continuous in the natural parameter as ergodic actions of the quantum SU(2) group. We also introduce a notion of regularity for quantum metrics on G, and show how to construct a quantum metric from any ergodic action of G, starting from a regular quantum metric on G. Furthermore, we introduce a quantum Gromov–Hausdorff distance between ergodic actions of G when G is separable and show that it induces the above topology. © 2008 Elsevier Inc. All rights reserved. Keywords: Compact quantum group; Ergodic action; Continuous field; Compact quantum metric space
E-mail address:
[email protected]. 1 Partially supported by NSF Grant DMS-0701414.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.09.009
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1. Introduction An ergodic action of a compact group G on a unital C ∗ -algebra B is a strongly continuous action of G on B such that the fixed point algebra consists only of scalars. For an irreducible representation of G on a Hilbert space H , the conjugate action of G on the algebra B(H ) is ergodic. On the other hand, ergodic actions of G on commutative unital C ∗ -algebras correspond exactly to translations on homogeneous spaces of G. Thus the theory of ergodic actions of G connects both the representation theory and the study of homogeneous spaces. See [14,20,26, 42–44] and references therein. Olesen, Pedersen, and Takesaki classified faithful ergodic actions of an abelian compact group as skew-symmetric bicharacters on the dual group [26]. Landstad and Wassermann generalized their result independently to show that ergodic actions of full multiplicity of an arbitrary compact group G are classified by equivalence classes of dual cocycles [20,43]. However, the general case is quite difficult—so far there is no classification of (faithful) ergodic actions of compact groups, not to mention compact quantum groups. In this paper we are concerned with topological properties of the whole set EA(G) of isomorphism classes of ergodic actions of a compact group G, and more generally, the set EA(G) of isomorphism classes of ergodic actions of a compact quantum group A = C(G). As a consequence of their classification, Olesen, Pedersen, and Takesaki showed that the set of isomorphism classes of faithful ergodic actions of an abelian compact group has a natural abelian compact group structure. From the work of Landstad and Wassermann, the set EA(G)fm of ergodic actions of full multiplicity of an arbitrary compact group G also carries a natural compact Hausdorff topology. There are many ergodic actions not of full multiplicity, such as conjugation actions associated to irreducible representations and actions corresponding to translations on homogeneous spaces (unless G is finite or the homogeneous space is G itself). In the physics literature concerning string theory and quantum field theory, people talk about fuzzy spheres, the matrix algebras Mn (C), converging to the two-sphere S 2 (see the introduction of [36] and references therein). One important feature of this convergence is that each term carries an ergodic action of SU(2), which is used in the construction of this approximation of S 2 by fuzzy spheres. Thus if one wants to give a concrete mathematical foundation for this convergence, it is desirable to include the SU(2) symmetry. However, none of these actions involved are of full multiplicity, and hence the topology of Landstad and Wassermann does not apply here. For compact quantum groups there are even more interesting examples of ergodic actions, 2 , parameterized by a compact subsee [41]. Podle´s introduced a family of quantum spheres Sqt set Tq of the real line, as ergodic actions of the quantum SU(2) group SUq (2) satisfying certain spectral conditions [29]. These quantum spheres carry interesting non-commutative differential geometry [7,8]. One also expects that Podle´s quantum spheres are continuous in the natural parameter t as ergodic actions of SUq (2). Continuous change of C ∗ -algebras is usually described qualitatively as continuous fields of C ∗ -algebras over locally compact Hausdorff spaces [9, Chapter 10]. There is no difficulty to formulate the equivariant version—continuous fields of actions of compact groups [32] or even compact quantum groups (see Section 5 below). Thus if there is any natural topology on EA(G), the relation with continuous fields of ergodic actions should be clarified. One distinct feature of the theory of compact quantum groups is that there is a full compact quantum group and a reduced compact quantum group associated to each compact quantum group G, which may not be the same. A compact quantum group G is called co-amenable if
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the full and reduced compact quantum groups coincide. This is the case for compact groups and SUq (N ) (for 0 < |q| < 1). Our result is simplified in such case. Denote by Gˆ the set of equivalence classes of irreducible unitary representations of G. For each ergodic action of G, one can talk about the multiplicity of each γ ∈ Gˆ in this action [30], which is known to be finite for the compact group case by [14] and, for the compact quantum group case by [5]. Theorem 1.1. Let G be a co-amenable compact quantum group. Then EA(G) has a unique compact Hausdorff topology such that the following holds: for any continuous field of ergodic actions of G over a locally compact Hausdorff space T the map T → EA(G) sending each t ∈ T to the isomorphism class of the fibre at t is continuous if and only if the function counting the multiˆ plicity of γ in each fibre is continuous over T for each γ in G. In particular, fuzzy spheres converge to S 2 as ergodic actions of SU(2) (see [21, Example 10.12]). Podle´s quantum spheres are also continuous as ergodic actions of SUq (2): Theorem 1.2. Let q be a real number with 0 < |q| < 1, and let Tq be the parameter space of 2 Podle´s quantum spheres. The map Tq → EA(SUq (2)) sending t to the isomorphism class of Sqt is continuous. When G is not co-amenable, the more appropriate object to study is a certain quotient space of EA(G). To each ergodic action of G, there is an associated full ergodic action and an associated reduced ergodic action (see Section 3 below), which are always isomorphic when G is co-amenable. Two ergodic actions are said to be equivalent if the associated full (reduced respectively) actions are isomorphic. Denote by EA∼ (G) the quotient space of EA(G) modulo this equivalence relation. We also have to deal with semi-continuous fields of ergodic actions in the general case. Theorem 1.3. Let G be a compact quantum group. Then EA∼ (G) has a unique compact Hausdorff topology such that the following holds: for any semi-continuous field of ergodic actions of G over a locally compact Hausdorff space T the map T → EA∼ (G) sending each t ∈ T to the equivalence class of the fibre at t is continuous if and only if the function counting the multiplicity ˆ of γ in each fibre is continuous over T for each γ in G. Motivated partly by the need to give a mathematical foundation for various approximations in the string theory literature, such as the approximation of S 2 by fuzzy spheres in above, Rieffel initiated the theory of compact quantum metric spaces and quantum Gromov–Hausdorff distances [35,37]. As the information of the metric on a compact metric space X is encoded in the Lipschitz seminorm on C(X), a quantum metric on (the non-commutative space corresponding to) a unital C ∗ -algebra B is a (possibly +∞-valued) seminorm on B satisfying suitable conditions. The seminorm is called a Lip-norm. Given a length function on a compact group G, Rieffel showed how to induce a quantum metric on (the C ∗ -algebra carrying) any ergodic action of G [33]. We find that the right generalizations of length functions for a compact quantum group A = C(G) are Lip-norms on A being finite on the algebra A of regular functions, which we call regular Lip-norms. Every separable co-amenable A has a bi-invariant regular Lip-norm (Corollary 8.10). Then we have the following generalization of Rieffel’s construction (see Section 2 below for more detail on the notation), answering a question Rieffel raised at the end of Section 3 in [37].
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Theorem 1.4. Suppose that G is a co-amenable compact quantum group and LA is a regular Lip-norm on A. Let σ : B → B ⊗ A be an ergodic action of G on a unital C ∗ -algebra B. Define a ( possibly +∞-valued ) seminorm on B via LB (b) = sup LA (b ∗ ϕ)
(1)
ϕ∈S(B)
for all b ∈ B, where S(B) denotes the state space of B and b ∗ ϕ = (ϕ ⊗ id)(σ (b)). Then LB is finite on the algebra B of regular functions and is a Lip-norm on B with rB 2rA , where rB and rA are the radii of B and A respectively. As an important step towards establishing a mathematical foundation for various convergence in the string theory literature, such as the convergence of fuzzy spheres to S 2 , Rieffel introduced a quantum Gromov–Hausdorff distance dist q between compact quantum metric spaces and showed, among many properties of dist q , that the fuzzy spheres converge to S 2 under dist q when they are all endowed with the quantum metrics induced from the ergodic actions of SU(2) for a fixed length function on SU(2) [36]. Two generalizations of dist q are introduced in [15] and [22] in order to distinguish the algebra structures (see also [16]). However, none of these quantum distances distinguishes the group symmetries. That is, there exist non-isomorphic ergodic actions of a compact group such that quantum distances between the compact quantum metric spaces induced by these ergodic actions are zero (see Example 9.1 below). One of the features of our quantum distances in [21,22] is that they can be adapted easily to take care of other algebraic structures. Along the lines in [21,22], we introduce a quantum distance dist e (see Definition 9.3 below) between the compact quantum metric spaces coming from ergodic actions of G as in Theorem 1.4. This distance distinguishes the ergodic actions: Theorem 1.5. Let G be a co-amenable compact quantum group with a fixed left-invariant regular Lip-norm LA on A. Then dist e is a metric on EA(G) inducing the topology in Theorem 1.1. The organization of this paper is as follows. In Section 2 we recall some basic definitions and facts about compact quantum groups, their actions, and compact quantum metric spaces. Associated full and reduced actions are discussed in Section 3. The topologies on EA(G) and EA∼ (G) are introduced in Section 4. We also prove that EA∼ (G) is compact Hausdorff there. In Section 5 we clarify the relation between semi-continuous fields of ergodic actions and the topology introduced in Section 4. This completes the proofs of Theorems 1.1 and 1.3. The continuity of Podle´s quantum spheres is discussed in Section 6. In Section 7 we show that the topology of Landstad and Wassermann on EA(G)fm for a compact group G is simply the relative topology of EA(G)fm in EA(G). Theorems 1.4 and 1.5 are proved in Sections 8 and 9 respectively. 2. Preliminaries In this section we collect some definitions and facts about compact quantum groups and compact quantum metric spaces. Throughout this paper we use ⊗ for the spatial tensor product of C ∗ -algebras, and for the algebraic tensor product of vector spaces. A = C(G) will be a compact quantum group.
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2.1. Compact quantum groups and actions We recall first some definitions and facts about compact quantum groups. See [23,47,48] for more detail. A compact quantum group (A, Φ) is a unital C ∗ -algebra A and a unital ∗-homomorphism Φ : A → A ⊗ A such that (id ⊗ Φ)Φ = (Φ ⊗ id)Φ and that both Φ(A)(1A ⊗ A) and Φ(A)(A ⊗ 1A ) are dense in A ⊗ A, where Φ(A)(1A ⊗ A) (Φ(A)(A ⊗ 1A ) respectively) denotes the linear span of Φ(a)(1A ⊗ a ) (Φ(a)(a ⊗ 1A ) respectively) for a, a ∈ A. We shall write A as C(G) and say that G is the compact quantum group. The Haar measure is the unique state h of A such that (id ⊗ h)Φ = (h ⊗ id)Φ = h. A unitary representation of G on a Hilbert space H is a unitary u ∈ M(K(H ) ⊗ A) such that (id ⊗ Φ)(u) = u12 u13 , where K(H ) is the algebra of compact operators, M(K(H ) ⊗ A) is the multiplier algebra of K(H ) ⊗ A, and we use the leg numbering notation [31, p. 385]. When H is finite-dimensional, uc denotes the contragradient representation acting on the conjugate Hilbert space of H . For unitary representations v and w of G, the tensor product representation v
w is defined as v13 w23 in the leg numbering notation. Denote by Gˆ the set of equivalence classes of irreducible unitary representations of G. For each γ ∈ Gˆ fix uγ ∈ γ acting on Hγ and an orthonormal basis in Hγ . Each Hγ is finite-dimensional. Denote by dγ the dimension of Hγ . Then we may identify B(Hγ ) with Mdγ (C), and hence uγ ∈ Mdγ (A). Denote by Aγ the linear γ span of (uij )ij . Then A∗γ = Aγ c ,
Φ(Aγ ) ⊆ Aγ Aγ ,
(2)
γ and A := γ ∈Gˆ Aγ is the algebra of regular functions in A. For any 1 i, j dγ denote by ρij the unique element in A such that γ β ρij usk = δγβ δis δj k γ
(3) γ
(the existence of such ρij is guaranteed by [47, Theorem 5.7]). Moreover ρij is of the form h(·a) γ for some a ∈ Aγ c . Denote 1idγ ρii by ρ γ . Denote the class of the trivial representations of G by γ0 . There exist a full compact quantum group (Au , Φu ) and a reduced compact quantum group (Ar , Φr ) whose algebras of regular functions and restrictions of comultiplications are the same as (A, Φ|A ). The quantum group G is said to be co-amenable if the canonical surjective homomorphism Au → Ar is an isomorphism [2, Definition 6.1], [3, Theorem 3.6]. There is a unique ∗-homomorphism e : A → C such that (e ⊗ id)Φ = (id ⊗ e)Φ = id on A, which is called the counit. The quantum group G is co-amenable exactly if it has bounded counit and faithful Haar measure [3, Theorem 2.2]. Next we recall some facts about actions of G. See [5,30] for details. Definition 2.1. (See [30, Definition 1.4].) A (left) action of G on a unital C ∗ -algebra B is a unital ∗-homomorphism σ : B → B ⊗ A such that (1) (id ⊗ Φ)σ = (σ ⊗ id)σ , (2) σ (B)(1B ⊗ A) is dense in B ⊗ A.
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The fixed point algebra of σ is B σ = {b ∈ B: σ (b) = b ⊗ 1A }. The action σ is ergodic if B σ = C1B . Remark 2.2. When A has bounded counit, the proof of [12, Lemma 1.4.(a)] shows that (id ⊗ e)σ = id on B and that σ is injective. Let σ : B → B ⊗ A be an action of G on a unital C ∗ -algebra B. For any b ∈ B, ϕ ∈ B and ψ ∈ A set b ∗ ϕ = (ϕ ⊗ id) σ (b) ,
ψ ∗ b = (id ⊗ ψ) σ (b) .
(4)
E γ = id ⊗ ρ γ σ.
(5)
γ
Also set Eij , E γ : B → B by γ γ Eij = id ⊗ ρij σ, Then [5, p. 98] E β E γ = δβγ E γ .
(6)
Set Bγ = E γ (B),
B=
Bγ .
(7)
γ ∈Gˆ
Then B is a dense ∗-subalgebra of B [30, Theorem 1.5], [5, Lemma 11, Proposition 14] (the ergodicity condition in [5] is not used in Lemma 11 and Proposition 14 therein), which we shall call the algebra of regular functions for σ . Moreover, Bγ∗ = Bγ c ,
Bα Bβ ⊆
Bγ ,
σ (Bγ ) ⊆ Bγ Aγ .
(8)
γ α
β
There exist a set Jγ and a linear basis Sγ = {eγ ki : k ∈ Jγ , 1 i dγ } of Bγ [30, Theorem 1.5] such that σ (eγ ki ) =
γ
eγ kj ⊗ uj i .
(9)
1j dγ
The multiplicity mul(B, γ ) is defined as the cardinality of Jγ , which does not depend on the choice of Sγ . Conversely, given a unital ∗-homomorphism σ : B → B ⊗ A for a unital C ∗ algebra B, if there exist a set Jγ and a set of linearly independent elements Sγ = {eγ ki : k ∈ Jγ , 1 i dγ } in B satisfying (9) for each γ ∈ Gˆ such that the linear span of γ ∈Gˆ Sγ is dense in B, then σ is an action of G on B and span Sγ ⊆ Bγ for each γ ∈ Gˆ [30, Corollary 1.6]. In this event, if |Jγ | or mul(B, γ ) is finite, then Bγ = span Sγ . We have that Bγ0 = B σ and that E := E γ0 is a conditional expectation from B onto B σ [5, Lemma 4]. When σ is ergodic, E = ω(·)1B for the unique σ -invariant state ω on B.
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2.2. Compact quantum metric spaces In this subsection we recall some facts about compact quantum metric spaces [33,35,37]. Though Rieffel has set up his theory in the general framework of order-unit spaces, we shall need it only for C ∗ -algebras. See the discussion preceding Definition 2.1 in [35] for the reason of requiring the reality condition (10) below. Definition 2.3. (See [35, Definition 2.1].) By a C ∗ -algebraic compact quantum metric space we mean a pair (B, L) consisting of a unital C ∗ -algebra B and a (possibly +∞-valued) seminorm L on B satisfying the reality condition L(b) = L b∗
(10)
for all b ∈ B, such that L vanishes on C1B and the metric ρL on the state space S(B) defined by ρL (ϕ, ψ) = sup ϕ(b) − ψ(b)
(11)
L(b)1
induces the weak-∗ topology. The radius rB of (B, L) is defined to be the radius of (S(B), ρL ). We say that L is a Lip-norm. Note that L must in fact vanish precisely on C1B and take finite values on a dense subspace of B. Let B be a unital C ∗ -algebra and let L be a (possibly +∞-valued) seminorm on B vanishing on C1B . Then L and · induce (semi)norms L˜ and · ∼ respectively on the quotient space B˜ = B/(C1B ). Notation 2.4. Let
E(B) := b ∈ Bsa : L(b) 1 . For any r 0, let
Dr (B) := b ∈ Bsa : L(b) 1, b r . Note that the definitions of E(B) and Dr (A) use Bsa instead of B. The main criterion for when a seminorm L is a Lip-norm is the following: Proposition 2.5. (See [33, Proposition 1.6, Theorem 1.9].) Let B be a unital C ∗ -algebra and let L be a ( possibly +∞-valued ) seminorm on B satisfying the reality condition (10). Assume that L takes finite values on a dense subspace of B, and that L vanishes on C1B . Then L is a Lip-norm if and only if ˜ and (1) there is a constant K 0 such that · ∼ K L˜ on B; (2) for any r 0, the ball Dr (B) is totally bounded in B for · ; or (2 ) for some r > 0, the ball Dr (B) is totally bounded in B for · . ˜ sa . In this event, rB is exactly the minimal K such that · ∼ K L˜ on (B)
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3. Full and reduced actions In this section we discuss full and reduced actions associated to actions of G. We will use the notation in Section 2.1 freely. Throughout this section, σ will be an action of G on a unital C ∗ -algebra B. Lemma 3.1. The conditional expectation E = (id ⊗ h)σ is faithful on B. If A is co-amenable, then E is faithful on B. Proof. Suppose that E(b) = 0 for some positive b in B. Then for any ϕ ∈ S(B) we have 0 = ϕ(E(b)) = h(b ∗ ϕ). Observe that b ∗ ϕ is in A and is positive. By the faithfulness of h on A [47, Theorem 4.2], b ∗ ϕ = 0. Then (ϕ ⊗ ψ)(σ (b)) = ψ(b ∗ ϕ) = 0 for all ϕ ∈ S(B) and ψ ∈ S(A). Since product states separate points of B ⊗ A [45, Lemma T.5.9 and Proposition T.5.14], σ (b) = 0. From (6) one sees that b ∈ {φ ∗ b: φ ∈ A }. Therefore b = 0. The second assertion is proved similarly, in view of Remark 2.2. 2 For actions σi : Bi → Bi ⊗ A of G on Bi for i = 1, 2, a unital ∗-homomorphism θ : B1 → B2 is said to be equivariant (with respect to σ1 and σ2 ) if σ2 ◦ θ = (θ ⊗ id) ◦ σ1 . Lemma 3.2. Let θ : B1 → B2 be a unital ∗-homomorphism equivariant with respect to actions σ1 , σ2 of G on B1 and B2 . Then Eγ ◦ θ = θ ◦ Eγ , θ (B1 )γ ⊆ (B2 )γ
(12) (13)
ˆ The map θ is surjective if and only if θ (B1 ) = B2 . The map θ is injective on B1 if for all γ ∈ G. and only if θ is injective on B1σ1 . Proof. One has E γ ◦ θ = id ⊗ ρ γ ◦ σ2 ◦ θ = id ⊗ ρ γ ◦ (θ ⊗ id) ◦ σ1 = θ ◦ id ⊗ ρ γ ◦ σ1 = θ ◦ E γ , which proves (12). The formula (13) follows from (12). Since B2 is dense in B2 , if θ (B1 ) = B2 , then θ is surjective. Conversely, suppose that θ is ˆ Thus surjective. Applying both sides of (12) to B1 we get θ ((B1 )γ ) = (B2 )γ for each γ ∈ G. θ (B1 ) = B2 . Since B1σ1 ⊆ B1 , if θ is injective on B1 , then θ is injective on B1σ1 . Conversely, suppose that θ is injective on B1σ1 . Let b ∈ B1 ∩ ker θ . Then θ (E(b∗ b)) = E(θ (b∗ b)) = 0. Thus E(b∗ b) ∈ ker θ . By assumption we have E(b∗ b) = 0. Then b = 0 by Lemma 3.1. 2 Proposition 3.3. The ∗-algebra B has a universal C ∗ -algebra Bu . The canonical ∗-homomorphism B → Bu is injective. Identify B with its canonical image in Bu . The unique ∗σ |B
homomorphism σu : Bu → Bu ⊗ A extending B → B A → Bu ⊗ A is an action of G on Bu . Moreover, the unique ∗-homomorphism πu : Bu → B extending the embedding B → B is equivariant, and the algebra of regular functions for σu is B.
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Proof. Let γ ∈ Gˆ and let Sγ be a linear basis of Bγ satisfying (9). Set bk = Then σ (bk ) =
1i,j,sdγ
γ∗
eγ kj eγ∗ ks ⊗ uj i usi = γ
dγ
∗ i=1 eγ ki eγ ki .
eγ kj eγ∗ ks ⊗ δj s 1A = bk ⊗ 1A .
1j,sdγ
Thus bk ∈ B σ . Note that B σ is a C ∗ -subalgebra of B. So π(·) · on B σ for any ∗representation π of B. Consequently, π(eγ ki )2 bk for any ∗-representation π of B. Thus for any c ∈ B there is some λc ∈ R such that π(c) λc for any ∗-representation π of B. Therefore B has a universal C ∗ -algebra Bu with the canonical ∗-homomorphism φ : B → Bu . Then there is a unique ∗-homomorphism πu : Bu → B such that πu ◦ φ is the canonical embedding ι : B → B. Since ι is injective, so is φ. Thus we may identify B with φ(B). Denote by σu the unique ∗-homomorphism Bu → Bu ⊗ A extending the ∗-homomorphism σ |B
B → B A → Bu ⊗ A. According to the characterization of actions of G in terms of elements satisfying (9) in Section 2.1, σu is an action of G on Bu and B ⊆ Bu . Since σ ◦ πu and (πu ⊗ id) ◦ σu coincide on B, they also coincide on Bu . Thus πu is equivariant. Since B σ is closed and E(B) = B σ , Buσu = E(Bu ) = B σ . Thus πu is injective on Buσu . By Lemma 3.2 the map θ is injective on Bu . Let b ∈ Bu . Then πu (b) ∈ B by Lemma 3.2, and hence πu (b − πu (b)) = 0. Therefore b = πu (b) ∈ B. This proves Bu = B as desired. 2 We refer the reader to [19] for basics on Hilbert C ∗ -modules. Since E : B → B σ is a conditional expectation, B is a right semi-inner-product B σ -module with the inner product ·,·B σ given by x, yB σ = E(x ∗ y) [19, p. 7]. Denote by HB the completion, and by πr the associated representation of B on HB . Denote πr (B) by Br . Proposition 3.4. There exists a unique ∗-homomorphism σr : Br → Br ⊗ A such that σr ◦ πr = (πr ⊗ id) ◦ σ.
(14)
The homomorphism σr is injective and is an action of G on Br . The map πr is equivariant, and is injective on B. The algebra of regular functions for σr is πr (B). Proof. The uniqueness of such σr follows from the surjectivity of πr . Consider the right Hilbert (B σ ⊗ A)-module HB ⊗ A. Denote by B(HB ⊗ A) the C ∗ -algebra of adjointable operators of the Hilbert (B σ ⊗ A)-module HB ⊗ A. Then Br ⊗ A ⊆ B(HB ⊗ A). The argument in the proof of [5, Lemma 3] shows that there is a unitary U ∈ B(HB ⊗ A) satisfying U (b ⊗ a) = ((πr ⊗ id)(σ (b)))(1B ⊗ a) for all a ∈ A, b ∈ B. It follows that U (πr (b) ⊗ 1A ) = ((πr ⊗ id)(σ (b)))U for all b ∈ B. Thus U (Br ⊗ 1A )U −1 ⊆ Br ⊗ A. Define σr : Br → Br ⊗ A by σr (b ) = U (b ⊗ 1A )U −1 . Then (14) follows. Clearly σr is injective. Since σ is an action of G on B, it follows easily that σr is an action of G on Br . The equivariance of πr follows from (14). By Lemma 3.2, πr (B) = Br . It is clear that πr is injective on B σ . Thus by Lemma 3.2 the map πr is injective on B. 2 Definition 3.5. We call the action (Bu , σu ) in Proposition 3.3 the full action associated to (B, σ ), and call the action (Br , σr ) in Proposition 3.4 the reduced action associated to (B, σ ). The action (B, σ ) is said to be full (reduced, co-amenable respectively) if πu (πr , both πu and πr respectively) is an isomorphism.
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Example 3.6. (1) When B is finite-dimensional, the action (B, σ ) is co-amenable. This applies to the actions constructed in [40] and the adjoint action on B(H ) associated to any finitedimensional representation of G on H [41, notation after Theorem 2.5]. n 2, that is, the universal C ∗ -algebra (2) Consider the Cuntz algebra On [6] for an integer n generated by isometries S1 , . . . , Sn satisfying j =1 Sj Sj∗ = 1. Since On is simple, any action of a compact quantum group on On is reduced. Given a compact quantum group A = C(G) and an n-dimensional unitary representation u = (uij )ij of G, one has an action σ of G on On determined by σ (Si ) = nj=1 Sj ⊗ uj i for all 1 i n [18, Theorem 1]. The regular subalgebra B for this action σ contains S1 , . . . , Sn , thus σ is full and hence is co-amenable, because of the universal property of On . This kind of actions has been considered for G being SUq (2) [18,24], SUq (N ) [28], and Au (Q) [41, Section 5]. (3) For the action Φ : A → A ⊗ A of G on A, the C ∗ -algebra for the associated full action is the C ∗ -algebra of the full quantum group [3, Section 3], while the C ∗ -algebra for the associated reduced action is the C ∗ -algebra of the reduced quantum group [3, Section 2]. Thus the action (A, Φ) is full (reduced respectively) exactly if G is a full (reduced respectively) compact quantum group. Remark 3.7. Having isomorphic (B, σ |B ) is an equivalence relation between actions of G on unital C ∗ -algebras. Two actions are equivalent in this sense exactly if they have isomorphic full actions, exactly if they have isomorphic reduced actions. If (A1 , Φ1 ) is another compact quantum group with (A1 , Φ1 |A1 ) isomorphic to (A, Φ|A ), then A1 has also a natural action on Bu . Thus the class of the equivalence classes of actions of G depends only on (A, Φ|A ). Proposition 3.8. The following are equivalent: (1) G is co-amenable, (2) every action of G on a unital C ∗ -algebra is co-amenable, (3) every ergodic action of G on a unital C ∗ -algebra is co-amenable. Proof. (1) ⇒ (2). Let σ be an action of G on a unital C ∗ -algebra B. Then (Br , σr ) is also the reduced action associated to (Bu , σu ). By Lemma 3.1 E is faithful on Bu . Thus the canonical homomorphism Bu → Br is injective, and hence is an isomorphism. Therefore (B, σ ) is coamenable. (2) ⇒ (3). This is trivial. (3) ⇒ (1). This follows from Example 3.6(3). 2 4. Ergodic actions In this section we introduce a topology on the set of isomorphism classes of ergodic actions of G in Definition 4.3 and prove Theorem 4.4. At the end of this section we also discuss the behavior of this topology under taking Cartesian products of compact quantum groups. Notation 4.1. Denote by EA(G) the set of isomorphism classes of ergodic actions of G. Denote by EA∼ (G) the quotient space of EA(G) modulo the equivalence relation in Remark 3.7. What we shall do is to define a topology on EA∼ (G), then pull it back to a topology on EA(G). ˆ let Mγ be the quantum dimension defined after Theorem 5.4 in [47]. One knows For each γ ∈ G,
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that Mγ is a positive number no less than dγ and that Mγ0 = 1. Set Nγ to be the largest integer no bigger than Mγ2 /dγ . Let (B, σ ) be an ergodic action of G. According to [5, Theorem 17], one has mul(B, γ ) Nγ for each γ ∈ Gˆ (the assumption on the injectivity of σ in [5] is not used in the proof of Theorem 17 therein; this can be also seen by passing to the associated reduced action in Proposition 3.4 for which σr is always injective). ˆ The pair (B, σ |B ) consists of the ∗-algebra B and the action σ |B : B → B A. For each γ ∈ G, one has a linear basis Sγ of Bγ satisfying (9), where we take Jγ to be {1, . . . , mul(B, γ )}. If we ˆ then the action σ |B is fixed by (9) and the pair (B, σ |B ) choose such a basis Sγ for each γ ∈ G, is determined by the ∗-algebra structure on B which in turn can be determined by the coefficients appearing in the multiplication and ∗-operation rules on these basis elements. In order to reduce the set of possible coefficients appearing this way, we put one more restriction on Sγ . By the argument on [5, p. 103], one can require Sγ to be an orthonormal basis of Bγ with respect to the inner product x, y = ω(x ∗ y), that is, ω eγ∗ sj eγ ki = δsk δj i .
(15)
We can always choose eγ0 11 = 1B . We shall call a basis Sγ satisfying all these conditions a standard basis of Bγ , and call the union S of a standard basis for each Bγ a standard basis of B. Notation 4.2. Set Gˆ = Gˆ \ {γ0 },
ˆ γ α M = (α, β, γ ) ∈ Gˆ × Gˆ × G:
β .
ˆ set For each γ ∈ G,
Xγ = (γ , k, i): 1 k Nγ , 1 i dγ ,
Xγ = (γ , k, i): 1 k mul(B, γ ), 1 i dγ . Denote by x0 the unique element (γ0 , 1, 1) in Xγ0 . Set Y=
Xα × Xβ × Xγ ,
Xγ × Xγ c ,
γ ∈Gˆ
(α,β,γ )∈M
Y =
Z=
Xα × Xβ × Xγ ,
Z =
Xγ × Xγ c .
γ ∈Gˆ
(α,β,γ )∈M
Fix a standard basis S of B. Since we have chosen ex0 to be 1B , the algebra structure of B is determined by the linear expansion of ex1 ex2 for all x1 ∈ Xα , x2 ∈ Xβ , α, β ∈ Gˆ . By (8) we have ex1 ex2 ∈ γ α Bγ . Thus the coefficients of the expansion of ex1 ex2 under S for all such
β
x1 , x2 determine a scalar function on Y , that is, there exists a unique element f ∈ CY such that for any x1 ∈ Xα , x2 ∈ Xβ , α, β ∈ Gˆ , ex1 ex2 =
(x1 ,x2 ,x3 )∈Y
f (x1 , x2 , x3 )ex3 .
(16)
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Similarly, the ∗-structure of B is determined by the linear expansion of ex1 for all x1 ∈ Xγ , γ ∈ Gˆ . By (8) we have ex∗1 ∈ Bγ c . Thus there exists a unique element g ∈ CZ such that for any x1 ∈ Xγ , γ ∈ Gˆ , ex∗1 =
(17)
g(x1 , x2 )ex2 .
(x1 ,x2 )∈Z
Then (f, g) determines the isomorphism class of (B, σ |B ) and hence determines the equivalence class of (B, σ ) in EA∼ (G). Note that (f, g) does not determine the isomorphism class of (B, σ ) in EA(G) unless (B, σ ) is co-amenable. Since we are going to consider all ergodic actions of G in a uniform way, we extend f and g to functions on Y and Z respectively by f |Y \Y = 0,
g|Z\Z = 0.
(18)
We shall say that (f, g) is the element in CY × CZ associated to S . Denote by P the set of (f, g) in CY × CZ associated to various bases of ergodic actions of G. We say that (f1 , g1 ) and (f2 , g2 ) in P are equivalent if they are associated to standard bases of (B1 , σ1 ) and (B2 , σ2 ) respectively such that (B1 , σ1 |B1 ) and (B2 , σ2 |B2 ) are isomorphic. Then this is an equivalence relation on P and we can identify the quotient space of P modulo this equivalence relation with EA∼ (G) naturally. Definition 4.3. Endow CY × CZ with the product topology. Define the topology on P as the relative topology, and define the topology on EA∼ (G) as the quotient topology from P → EA∼ (G). Also define the topology on EA(G) via setting the open subsets in EA(G) as inverse image of open subsets in EA∼ (G) under the quotient map EA(G) → EA∼ (G). Theorem 4.4. Both P and EA∼ (G) are compact Hausdorff spaces. The space EA(G) is also compact, but it is Hausdorff if and only if G is co-amenable. Both quotient maps P → EA∼ (G) and EA(G) → EA∼ (G) are open. Remark 4.5. Eq. (9) depends on the identification of B(Hγ ) with Mdγ (C), which in turn depends on the choice of an orthonormal basis of Hγ . Then P also depends on such a choice. However, using Lemma 4.11 below one can show directly that the quotient topology on EA∼ (G) does not depend on such a choice. This will also follow from Corollary 5.16 below. In order to prove Theorem 4.4, we need to characterize P and its equivalence relation more explicitly. We start with characterizing P, that is, we consider which elements of CY × CZ come from standard bases of ergodic actions of G. For this purpose, we take f (y) and g(z) for y ∈ Y, z ∈ Z as variables and try to find algebraic conditions they should satisfy in order to construct (B, σ |B ). Set X=
γ ∈Gˆ
Xγ ,
and X0 =
Xγ .
γ ∈Gˆ
Let V be a vector space with basis {vx : x ∈ X0 }. We hope to construct B out of V such that vx becomes ex . Corresponding to (16)–(18) we want to make V into a ∗-algebra with identity vx0 satisfying
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vx1 vx2 =
f (x1 , x2 , x3 )vx3
(19)
(x1 ,x2 ,x3 )∈Y
for any x1 , x2 ∈ X, and
vx∗1 =
g(x1 , x2 )vx2 ,
(20)
(x1 ,x2 )∈Z
for any x1 ∈ X. Corresponding to (9), we also want a unital ∗-homomorphism σV : V → V A satisfying
σV (vγ ki ) =
γ
vγ kj ⊗ uj i
(21)
1j dγ
for (γ , k, i) ∈ X. Thus consider the equations ∗ ∗ vx1 = vx1 , (vx1 vx2 )∗ = vx∗2 vx∗1 , ∗ σV (vx1 ) = σV vx∗1 σV (vx1 vx2 ) = σV (vx1 )σV (vx2 ),
(vx1 vx2 )vx3 = vx1 (vx2 vx3 ),
for all x1 , x2 , x2 ∈ X. Expanding both sides of these equations formally using (19)–(21) and identifying the corresponding coefficients, we get a set E1 of polynomial equations in the variables f (y), g(z) and their conjugates for y ∈ Y, z ∈ Z. For any (f, g) ∈ CY × CZ satisfying E1 , we have a conjugate-linear map ∗ : V → V specified by (20). Set If,g to be the kernel of ∗, and set Vf,g = V /If,g . Denote the quotient map V → Vf,g by φf,g , and denote φf,g (vx ) by νx for x ∈ X0 . Then the formulas νx1 νx2 =
f (x1 , x2 , x3 )νx3 ,
(22)
(x1 ,x2 ,x3 )∈Y
νx∗1 =
g(x1 , x2 )νx2 ,
(23)
(x1 ,x2 )∈Z
σf,g (νγ ki ) =
γ
νγ kj ⊗ uj i
(24)
1j dγ
corresponding to (19)–(21) determine a unital ∗-algebra structure of Vf,g with the identity νx0 and a unital ∗-homomorphism σf,g : Vf,g → Vf,g A. In order to make sure that (f, g) is associated to some standard basis of some ergodic action of G, we need to also take care of (15). Note that ω|B is simply to take the coefficient at 1B . For any γ ∈ Gˆ and any x1 , x2 ∈ Xγ , expand vx∗2 vx1 formally using (20) and (19) and denote by Fx1 ,x2 the coefficient at 1V . Then we want the existence of a non-negative integer mγ ,f,g Nγ for each γ ∈ Gˆ , which one expects to be mul(B, γ ), such that the value of Fγ sj,γ ki at (f, g) is δsk δj i or 0 depending on s, k mγ ,f,g or not. This condition can be expressed as the set E2 of the equations Fx1 ,x2 = 0 for all x1 , x2 ∈ Xγ with x1 = x2 , the equations Fγ si,γ si = Fγ sj,γ sj for all 1 i, j dγ , 1 s Nγ , and the equations Fγ s1,γ s1 Fγ k1,γ k1 = Fγ s1,γ s1 for all 1 k s Nγ (and for all γ ∈ Gˆ ). We also need to take care of (18). Thus denote by E3 the set of equations
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f (x1 , x2 , x3 ) = f (x1 , x2 , x3 )Fx1 ,x1 = f (x1 , x2 , x3 )Fx2 ,x2 = f (x1 , x2 , x3 )Fx3 ,x3 for all (x1 , x2 , x3 ) ∈ Y (the last equation is vacuous when x3 = x0 ), and the equations g(x1 , x2 ) = g(x1 , x2 )Fx1 ,x1 = g(x1 , x2 )Fx2 ,x2 for all (x1 , x2 ) ∈ Z. Notation 4.6. Denote by E the union of E1 , E2 and E3 . Clearly every element in P satisfies E . This proves part of the following characterization of P: Proposition 4.7. P is exactly the set of elements in CY × CZ satisfying E . Let (f, g) ∈ CY × CZ satisfy E . Set
Xf,g = (γ , s, i) ∈ X: 1 s mγ ,f,g , which one expects to parameterize S \ {ex0 }. Since (f, g) satisfies E3 , span{vx : x ∈ X \ Xf,g } ⊆ If,g . Thus νx ’s for x ∈ Xf,g ∪ {x0 } span Vf,g . Clearly Vf,g is the direct sum of Cνx0 and span{νx : x ∈ Xγ } for all γ ∈ Gˆ . Thus it makes sense to talk about the coefficient of ν at νx0 for any ν ∈ Vf,g . This defines a linear functional ϕf,g on Vf,g , which one expects to be ω. Clearly ϕf,g (·)νx0 = (id ⊗ h)σf,g (·)
(25)
on Vf,g . Lemma 4.8. Let (f, g) ∈ CY × CZ satisfy E . Then Vf,g has a universal C ∗ -algebra Bf,g . The canonical ∗-homomorphism Vf,g → Bf,g is injective. Identifying Vf,g with its canonical image in Bf,g one has νx
Fγ Mγ
(26)
for any x = (γ , k, i) ∈ Xf,g , where Fγ denotes the element in Mdγ (C) defined after Theorem 5.4 in [47]. The set S := {νx : x ∈ Xf,g ∪ {x0 }} is a linear basis of Vf,g . Proof. We show first that for each ν ∈ Vf,g there exists some cν ∈ R such that π(ν) cν for any ∗-representation π of Vf,g . Recalling that S spans Vf,g , it suffices to prove the claim for ν = νx for every x ∈ Xf,g . Say x = (γ , k, i). Set −1
Wγ k = Fγ 2 (νγ k1 , . . . , νγ kdγ )T ∈ Mdγ ×1 (Vf,g ), Wγ k = (Wγ k , 0, . . . , 0) ∈ Mdγ ×dγ (Vf,g ). Note that {ν ∈ Vf,g : σf,g (ν) = ν ⊗ 1A } = Cνx0 . The argument in [5, p. 103] shows that
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Wγ∗k Wγ s = δks Mγ νx0
(27)
for all 1 k, s mγ ,f,g . Thus for any ∗-representation π of Vf,g we have π(Wγ k ) and hence 1
π(νx ) Fγ2 Mγ = Fγ Mγ .
Mγ
(28)
Next we show that Vf,g does have a ∗-representation. By [47, Theorem 5.7] one has ˆ Using (25) one sees that h(A∗α Aβ ) = 0 for any α = β ∈ G. ϕf,g νx∗2 νx1 = 0
(29)
for all x1 ∈ Xα , x2 ∈ Xβ , α = β. Using (29) and the assumption that (f, g) satisfies E2 , one observes that ϕf,g ν ∗ ν =
|λx |2 0
(30)
x∈Xf,g ∪{x0 }
for any ν = x∈X0 λx νx ∈ Vf,g . Denote by H the Hilbert space completion of Vf,g with respect to the inner product ν1 , ν2 = ϕf,g (ν1∗ ν2 ), and by H (dγ ) the direct sum of dγ copies of H . By (27) the multiplication by Wγ s extends to a bounded operator on H (dγ ) . Then so does the multiplication by ((νγ k1 , . . . , νγ kdγ )T , 0, . . . , 0) ∈ Mdγ ×dγ (Vf,g ). Consequently, the multiplication by νx for x = (γ , k, i) ∈ Xf,g extends to a bounded operator on H . Since S spans Vf,g , the multiplication of Vf,g extends to a ∗-representation π of Vf,g on H . Now we conclude that Vf,g has a universal C ∗ -algebra Bf,g . It follows from (30) that π ◦ φf,g is injective on span{vx : x ∈ Xf,g ∪ {x0 }}, where φf,g : V → Vf,g is the quotient map. Thus S is a linear basis of Vf,g , and the canonical ∗-homomorphism Vf,g → Bf,g must be injective. The inequality (26) follows from (28). 2 σf,g
For (f, g) as in Lemma 4.8, by the universality of Bf,g , the ∗-homomorphism Vf,g → Vf,g A → Bf,g ⊗ A extends uniquely to a (unital) ∗-homomorphism Bf,g → Bf,g ⊗ A, which we still denote by σf,g . Proposition 4.9. Let (f, g) be as in Lemma 4.8. Then σf,g is an ergodic action of G on Bf,g . The algebra of regular functions for this action is Vf,g . The set S is a standard basis of Vf,g . The element in P associated to this basis is exactly (f, g). Proof. By Lemma 4.8, S is a basis of Vf,g . By (24) and the characterization of actions of G in terms of elements satisfying (9) in Section 2.1, σf,g is an ergodic action of G on Bf,g , and (Bf,g )γ = span{νx : x = (γ , k, i) ∈ Xf,g }, mul(Bf,g , γ ) = mγ ,f,g for all γ ∈ Gˆ . Thus Vf,g is the algebra of regular functions. Denote by ω the unique G-invariant state on Bf,g . By (25) ω extends ϕf,g . Since (f, g) satisfies E2 , we have ω(νx∗ νy ) = δxy for any x = (γ , k, i), y = (γ , s, j ) ∈ Xf,g . Thus S is a standard basis of Bf,g . Clearly the element in P associated to this basis is exactly (f, g). 2 Now Proposition 4.7 follows from Proposition 4.9.
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We are ready to prove the compactness of P. Lemma 4.10. Let (f, g) ∈ P. Then f (x1 , x2 , x3 ) Fα Mα
(31)
for any (x1 , x2 , x3 ) ∈ Y, x1 ∈ Xα and g(x1 , x2 ) Fα Mα
(32)
for any (x1 , x2 ) ∈ Z, x1 ∈ Xα . The space P is compact. Proof. Say, (f, g) is associated to a standard basis S for an ergodic action (B, σ ) of G. Let (HB , πr ) be the GNS representation associated to the unique σ -invariant state ω of B. Then Bα and Bβ are orthogonal to each other in HB for distinct α, β ∈ Gˆ [5, Corollary 12]. In view of (15), S is an orthonormal basis of HB . We may identify B with Vf,g naturally via ex ↔ νx . Then there is a ∗-homomorphism from Bf,g in Lemma 4.8 to B extending this identification. Thus by (26) √ ˆ For any (x1 , x2 , x3 ) ∈ Y , x1 ∈ Xα , by (16), we have ex Fα Mα for any x ∈ Xα , α ∈ G.
f (x1 , x2 , x3 ) = ex , ex ex ex Fα Mα . 3 1 2 1 If y ∈ Y \ Y , then f (y) = 0 by (18). This proves (31). The inequality (32) is proved similarly. By Proposition 4.7 the space P is closed in CY × CZ . It follows from (31) and (32) that P is compact. 2 Next we characterize the equivalence relation on P. For this purpose, we need to consider the relation between two standard bases of B. The argument in the proof of [30, Theorem 1.5] shows the first two assertions of the following lemma: ˆ If bi ∈ B, 1 i dγ satisfy Lemma 4.11. Let γ ∈ G. σ (bi ) =
γ
bj ⊗ uj i
(33)
1j dγ γ
γ
for all 1 i dγ , then bi = Ei1 (b1 ) (see (5)) for all 1 i dγ . Conversely, given b ∈ E11 (B), γ if we set bi = Ei1 (b), then bi ∈ Bγ , 1 i dγ satisfy (33), and b1 = b. For any b1 , . . . , bdγ (b1 , . . . , bd γ respectively) in B satisfying (33) ((33) with bi replaced by bi respectively) we have ω bj∗ bi = δj i ω b1∗ b1
(34)
for all 1 i, j dγ . Proof. We just need to prove (34). By [47, Theorem 5.7] we have γ∗ γ γ 1 h ulj uni = f−1 unl δj i , Mγ
(35)
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where f−1 is the linear functional on A defined in [47, Theorem 5.6]. Thus ∗ γ∗ γ ∗ (33) ∗ bl bn ⊗ ulj uni ω bj bi 1B = (id ⊗ h) σ (bj bi ) = (id ⊗ h) 1l,ndγ (35)
=
bl∗ bn
1l,ndγ
γ 1 f−1 unl δj i . Mγ
Therefore ω bj∗ bi 1B = δj i ω b1∗ b1 1B , which proves (34).
2
By Lemma 4.11, for γ ∈ Gˆ , there is a 1–1 correspondence between standard bases of Bγ and γ orthonormal bases of E11 (B) with respect to the inner product b, b = ω(b∗ b ). It also follows γ from Lemma 4.11 that dim(E11 (B)) = mul(B, γ ). Denote by Un the unitary group of Mn (C). Then γ ∈Gˆ Umul(B,γ ) has a right free transitive action on the set of standard bases of B via γ Un with the acting on the set of orthonormal bases of E11 (B) for each γ ∈ Gˆ . For n m identify subgroup of Um consisting of elements with 1m−n at the lower-right corner. Denote γ ∈Gˆ UNγ by U , equipped with the product topology. Then U has a naturalpartial right (not necessarily free) action τ on P, that is, ξ ∈ U acts at t ∈ P exactly if ξ ∈ γ ∈Gˆ Umγ ,t , where mγ ,t was defined in the paragraph before Notation 4.6, and the image t · ξ is the element in P associated to the standard basis S · ξ of Bt , where S · ξ is the image of the action of ξ at the standard basis S of Bt in Proposition 4.9. Clearly the orbits of this partial action are exactly the fibres of the quotient map P → EA∼ (G), equivalently, exactly the equivalence classes in P introduced before Definition 4.3. Thus we may identify EA∼ (G) with the quotient space P/U . Lemma 4.12. The quotient map P → P/U is open. The quotient topology on P/U is compact Hausdorff. Proof. Denote by π the quotient map P → P/U . To show the openness of π , it suffices to show that π −1 (π(V )) is open for every open subset V of P. Let t ∈ V and ξ ∈ U such that t · ξ is ˆ Replacing ξγ by 1Nγ for γ ∈ Gˆ \ J defined. Say ξ = (ξγ )γ ∈Gˆ . Let J be a finite subset of G.
we get an element ξ ∈ U . Notice that when t ∈ P is close enough to t · ξ , t · (ξ )−1 is defined. Moreover, the restrictions of t · (ξ )−1 on (XJ × XJ × X J ) ∩ Y and (XJ × XJ ) ∩ Z converge to the restrictions of t as t converges to t · ξ , where XJ = γ ∈J Xγ . Clearly we can find a large enough finite subset J of Gˆ such that when t is close enough to t · ξ , the element t · (ξ )−1 is in V . Then t = (t · (ξ )−1 ) · ξ is in π −1 (π(V )). Therefore π −1 (π(V )) is open, and hence π is open. Denote by D the domain of τ , i.e., the subset of P × U consisting of elements (t, ξ ) for which t · ξ is defined. From the equations in E2 it is clear that D is closed in P × U . By Lemma 4.10 the space P is compact. Since U is also compact, so is D. It is also clear that τ is continuous in the sense that the map D → P sending (t, ξ ) to t · ξ is continuous. Thus the set {(t, t ) ∈ P × P: π(t) = π(t )} = {(t, t · ξ ) ∈ P × P: (t, ξ ) ∈ D} is closed in P × P. Since π is open, a standard argument shows that the quotient topology on P/U is compact Hausdorff. 2
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Since EA∼ (G) is Hausdorff by Lemma 4.12, EA(G) is Hausdorff exactly if the quotient map EA(G) → EA∼ (G) is a bijection, exactly if G is co-amenable by Proposition 3.8. Then Theorem 4.4 follows from Lemmas 4.10 and 4.12. Notice that the function t → mγ ,t is continuous on P for each γ ∈ Gˆ . Thus we have Proposition 4.13. The multiplicity function mul(·, γ ) is continuous on both EA(G) and EA∼ (G) ˆ for each γ ∈ G. To end this section, we discuss the behavior of EA(G) when we take Cartesian products of compact quantum groups. Let {Aλ = C(Gλ )}λ∈Λ be a family of compact quantum groups indexed by a set Λ. Then λ Aλ has a unique compact quantum group structure such that the embeddings compact quantum groups Aμ → λ Aλ for μ ∈ Λ are all morphisms between [39, Theorem 1.4, G ). The Haar measure of Proposition 2.6], which we shall denote by C( λ λ λ Aλ is the tensor product λ hλ of the Haar measures hλ of Aλ [39, Proposition 2.7]. ∗ ⊗ Aλ is an action of Gλ on a unital If σλ : Bλ → Bλ C -algebra Bλ for each λ, then the unique seen to be ∗-homomorphism λ σλ : λ Bλ → ( λ Bλ ) ⊗ ( λ Aλ ) extending all σλ ’s is easily an action of λ Gλ . Using the canonical conditional expectation λ Bλ → ( λ Bλ ) λ σλ , one σ checks easily that ( λ Bλ ) λ λ = λ Bλσλ . In particular, λ σλ is ergodic if and only if every σλ is. Proposition 4.14. λ∈Λ be a family of compact quantum groups indexed by a Let {Aλ = C(Gλ )} ) → EA( classes of (Bλ , σλ )’s to the set Λ. The map λ EA(G λ λ Gλ ) sending the isomorphism isomorphism class of ( λ Bλ , λ σλ ) descends to amap λ EA∼ (Gλ ) → EA∼ ( λ Gλ ), that is, there exists a (unique) map λ EA∼ (Gλ ) → EA∼ ( λ Gλ ) such that the diagram
λ EA(Gλ )
EA( λ Gλ ) (36)
∼ λ EA (Gλ )
EA∼ ( λ Gλ )
commutes. Moreover, both of these maps are injective and continuous, where both and λ EA∼ (Gλ ) are endowed with the product topology.
λ EA(Gλ )
Proof. Denote by ∼ λ Gλ the subset of λ Gλ consisting of elements whose all but finitely many components are classes of trivial representations. For any γ ∈ ∼ λ Gλ , say γλ1 , . . . , γλn are the γλ1 γλ2 γλn nontrivial components of γ , the element u1(n+1) u2(n+2) · · · un(2n) (in the leg numbering notation) is an irreducible unitary representation of λ Gλ . Moreover, this map ∼ λ Gλ is bijecλ Gλ → tive [39, Theorem 2.11], and hence we may identify these two sets. Fixing an orthonormal basis of Hγλ we take the tensor products of the bases of Hγλ1 , . . . , Hγλn as an orthonormal basis of Hγ . Let Sλ be a standard basis of Bλ . Say, it consists of a standard λ of (Bλ )αλ for basis Sα ∈ Gλ . Denote by ωλ the σλ -invariant state on Bλ . Then λ ωλ is the λ σλ -invariant each αλ satisfying (9) state of λ Bλ . Using the characterization of ergodic actions in terms of elements σ is B and that the in Section 2.1, one sees that the algebra of regular functions for λ λ λ λ tensor products of Sγλ1 , . . . , Sγλn is a standard basis of ( λ Bλ )γ . This shows the existence of the map λ EA∼ (Gλ ) → EA∼ ( λ Gλ ) making (36) commute. Taking the union of the above
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standard basis of ( λ Bλ )γ , we also get a standard basis of λ Bλ , which we shall denote by ∼ ∼ are trivial at all λ = λ0 λ Sλ . For any fixed λ0 , if we take all γ ∈ λ Gλ whose components and take the sum of the corresponding spectral subspaces of B , we get Bλ0 . Taking norm λ λ . This proves the injectivity of the maps EA(G ) → EA( G ) closure, we get B λ0 λ λ λ λ and ∼ ∼ EA (G ) → EA ( G ). λ λ λ λ → P( λ Gλ ) sending (tλ )λ∈Λ to the element of P( λ Gλ ) asClearly the map λ P(Gλ ) ∼ sociated to the standard basis λ Stλ is continuous, where Stλ is the standard basis of Btλ in Proposition 4.9. Note that the diagram
λ P(Gλ )
P( λ Gλ ) (37)
λ EA
∼
(Gλ )
EA∼ ( λ Gλ )
commutes, where the left vertical map is the productmap. By Theorem 4.4 the map P(Gλ ) → → λ EA∼ (Gλ ) is open. It folEA∼ (Gλ ) is open for each λ. Thus the product map λ P(Gλ ) ∼ ∼ EA (G ) → EA ( G lows from the commutativity of the diagram (37) that the map λ λ λ λ ) is continuous. Then the continuity of the map λ EA(Gλ ) → EA( λ Gλ ) follows from the commutativity of the diagram (36). 2 5. Semi-continuous fields of ergodic actions In this section we prove Theorems 5.11 and 5.12, from which we deduce Theorems 1.1 and 1.3. We start with discussion of semi-continuous fields of C ∗ -algebras. ∗ Notation 5.1. For a field∗ {Ct }t∈T of C -algebras over a locally compact Hausdorff space T , we denote by t Ct the C -algebra of bounded cross-sections (for the supremum norm), and by ∼ ∗ t Ct the C -algebra of bounded cross-sections vanishing at infinity on T .
∗ Note that both t Ct and ∼ t Ct are Banach modules over the C -algebra C∞ (T ) of continuous C-valued functions on T vanishing at infinity. We use Rieffel’s definition of semi-continuous fields of C ∗ -algebras [32, Definition 1.1]. We find that it is convenient to extend the definition slightly. ∗ Definition 5.2. Let {Ct }t∈T be a field ∼ of C -algebras over a locally compact Hausdorff space T∗ , ∗ and let C be a C -subalgebra of t Ct . We say that ({Ct }t∈T , C) is a topological field of C algebras if
(1) the evaluation map πt from Cto Ct is surjective for each t ∈ T , (2) C is a C∞ (T )-submodule of ∼ t Ct . We say that ({Ct }t∈T , C) is upper semi-continuous (lower semi-continuous, continuous respectively) if furthermore for each c ∈ C the function t → πt (c) is upper semi-continuous (lower semi-continuous, continuous respectively). In such case we say that ({Ct }t∈T , C) is semicontinuous.
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Remark 5.3. If we have two upper semi-continuous fields of C ∗ -algebras ({Ct }t∈T , C1 ) and ({Ct }t∈T , C2 ) over T with the same fibres and C1 ⊆ C2 , then C1 = C2 [11, Proposition 2.3]. This is not true for lower semi-continuous fields of C ∗ -algebras. For example, let T be a compact Hausdorff space and let H be a Hilbert space. Take Ct = B(H ) for each t. Set C1 to be the set of all cross-sections c such that t → πt (c) is norm continuous, while set C2 to be the set of all norm-bounded cross-sections c such that both t → πt (c) and t → (πt (c))∗ are continuous with respect to the strong operator topology in B(H ). Then C1 C2 when T is the one-point compactification of N and H is infinite-dimensional. Definition 5.4. By a homomorphism ϕ between two topological fields of C ∗ -algebras ({Ct }t∈T , C) and ({Bt }t∈T , B) over a locally compact Hausdorff space T we mean a ∗-homomorphism ϕt : Ct → Bt for each t ∈ T such that the pointwise ∗-homomorphism t ϕt : t Ct → t Bt sends C into B. ∗ Lemma 5.5. Let {Ct }t∈T be a field of compact Hausdorff space T , C -algebras over a locally and let C be a linear subspace of t Ct . Then a section c ∈ ∼ t Ct is in C := C∞ (T )C if and only if for any t0 ∈ T and ε > 0, there exist a neighborhood U of t0 and c ∈ C such that πt (c − c ) < ε throughout U . If furthermore πt (C ) is dense in Ct for each t and C C , C ∗ ⊆ C, then ({Ct }t∈T , C) is a topological field of C ∗ -algebras over T , which we shall call the topological field generated by C . If furthermore the function t → πt (c) is upper semi-continuous (lower semi-continuous, continuous respectively) for each c ∈ C, then ({Ct }t∈T , C) is upper semi-continuous (lower semi-continuous, continuous respectively).
Proof. The “only if” part is obvious. The “if” part follows from a partition-of-unity argument. The second and the third assertions follow easily. 2 Let ({Ct }t∈T , C) be a topological field of C ∗ -algebras over a locally compact Hausdorff space T . If Θ is another locally compact Hausdorff space and p : Θ → T is a continuous map, then we have thepull-backfield {Cp(θ) }θ∈Θ of C ∗ -algebras over Θ. There is a natural ∗-homomorphism p ∗ : t Ct → θ Cp(θ) sending c to {πp(θ) (c)}θ∈Θ . We will call the topological field generated by p ∗ (C) in Lemma 5.5 the pull-back of ({Ct }t∈T , C) under p. In particular, if Θ is a closed or open subset of T and p is the embedding, we get the restriction of ({Ct }t∈T , C) on Θ. Clearly the pull-back and restriction of homomorphisms between topological fields are also homomorphisms. Lemma 5.6. Let ({Ct }t∈T , C) be a semi-continuous field of unital C ∗ -algebras over a locally compact Hausdorff space T such that the section {f (t)1Ct }t∈T is in C for each f ∈ C∞ (T ). Then for any bounded function g on T vanishing at infinity, the section {g(t)1Ct }t∈T is in C if and only if g ∈ C∞ (T ). Proof. Via restricting to compact subsets of T , we may assume that T is compact. The “if” part is given by assumption. To prove the “only if” part, it suffices to show that when the section {g(t)1Ct }t∈T is in C and g(t0 ) = 0 for some t0 ∈ T , we have g(t) → 0 as t → t0 . Replacing g by g ∗ g, we may assume that g is nonnegative. When the field is upper semi-continuous, the function t → g(t)1Ct = g(t) is upper semi-continuous at t0 and hence g(t) → 0 as t → t0 . When the field is lower semi-continuous, the function t → (g − g(t))1Ct = g − g(t) is lower semi-continuous at t0 and hence g(t) → 0 as t → t0 . 2
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Lemma 5.7. Let ({Ct }t∈T , C) be a topological field of C ∗ -algebras over a locally compact ∗ there is a natural injective ∗-homomorphism Hausdorff space T . Let D be a C -algebra. Then (ϕ(c ⊗ d)) = π (c) ⊗ d for all c ∈ C, d ∈ D, and (C ⊗ D) determined by π ϕ :C⊗D→ ∼ t s s t s ∈ T , where πs and πs denote the coordinate maps t Ct → Cs and t (Ct ⊗ D) → Cs ⊗ D respectively. Identifying C ⊗ D with ϕ(C ⊗ D), the pair ({Ct ⊗ D}t∈T , C ⊗ D) is also a topological field of C ∗ -algebras over T . Proof. For each s ∈ T we have the ∗-homomorphism π s ⊗ id : ( t Ct ) ⊗ D → Cs ⊗ D. Then we have the product ∗-homomorphism ( t Ct ) ⊗ D → t (Ct ⊗ D). Denote by ϕ the restriction of this homomorphism to C ⊗ D. We have πs (ϕ(c ⊗ d)) = πs (c) ⊗ d for all c ∈ C, d ∈ D, and s ∈ T . Clearly this identity also determines ϕ. To show that ϕ is injective, we may assume that Cs is contained in the algebra of bounded linear operators on Hs for some Hilbert space Hs for each s ∈ T , and D is contained in the algebra of bounded linear operators on K for some Hilbert space K. Denote the Hilbert space H by H . Then direct sum T t t t Ct can be identified with the algebra of bounded linear operators c on HT satisfying that c preserves Hs for each s ∈ T and the restriction of c on Hs ∗ is in Cs for each s ∈ T . Now C ⊗ D is naturally a C -algebra of bounded linear operators on the Hilbert space tensor product HT ⊗ K = t∈T (Ht ⊗ K). It is easily checked that for every g ∈ C ⊗ D, g preserves Hs ⊗ K for each s ∈ T , the restriction of c on Hs is equal to πs (ϕ(g)) for each s ∈ T , and the function t → πt (ϕ(g)) on T vanishes at infinity (check this for g ∈ C D first, then approximate g ∈ C ⊗ D by g ∈ C D). It follows that ϕ is injective and maps C ⊗ D into ∼ t (Ct ⊗ D). Clearly the restriction of πs on ϕ(C ⊗ D) is onto Cs ⊗ D for each s ∈ T . Since C is a to be a C∞ (T )-submodule of t (Ct ⊗ D). It follows C∞ (T )-module, ϕ(C D) is easily seen that ϕ(C ⊗ D) is a C∞ (T )-submodule of t (Ct ⊗ D). Thus the pair ({Ct ⊗ D}t∈T , ϕ(C ⊗ D)) is a topological field of C ∗ -algebras over T . 2 From now on, for a topological field ({Ct }t∈T , C) of C ∗ -algebras over a locally compact Hausdorff space T and a C ∗ -algebra D, we shall take ({Ct ⊗ D}t∈T , C ⊗ D) to be the topological field of C ∗ -algebras over T in Lemma 5.7. In general, for a continuous field ({Ct }t∈T , C) of C ∗ -algebras over a compact metrizable space T and a C ∗ -algebra D, the topological field ({Ct ⊗ D}t∈T , C ⊗ D) of C ∗ -algebras may fail to be continuous [17, Theorem A]. The following lemma tells us that if a field ({Ct }t∈T , C) over a locally compact Hausdorff space T can be subtrivialized in the sense that there is a C ∗ -algebra B containing each Ct as a C ∗ -subalgebra so that the elements of C are exactly the continuous maps T → B vanishing at ∞ whose images at each t are in Ct , then the field ({Ct ⊗ D}t∈T , C ⊗ D) can also be subtrivialized and hence is continuous. Lemma 5.8. Let ({Ct }t∈T , C) be a topological field of C ∗ -algebras over a locally compact Hausdorff space T . Suppose that there is a C ∗ -algebra B containing each Ct as a C ∗ -subalgebra so that the elements of C are exactly the continuous maps T → B vanishing at ∞ whose images at each t are in Ct . Let D be a C ∗ -algebra, and identify C ⊗ D with a C ∗ -subalgebra ∼ of t (Ct ⊗ D) as in Lemma 5.7. Then elements of C ⊗ D are exactly the continuous maps T → B ⊗ D vanishing at ∞ whose images at each t are in Ct ⊗ D. Proof. Denote by W the continuous maps T → B ⊗ D vanishing at ∞ whose images at each t are in Ct ⊗ D. This is a C ∗ -subalgebra of ∼ t (Ct ⊗ D).
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Denote by πt the coordinate map C ⊗D → Ct ⊗D for each t ∈ T . Then πt (f ⊗d) = f (t)⊗d for all t ∈ T , f ∈ C, and d ∈ D. It is easy to check that C D ⊆ W . Thus C ⊗ D ⊆ W . ∈ Cs D satisfying Let w ∈ W and let ε > 0. For any s ∈ T , we can find some j bj ⊗ dj w(s) − j bj ⊗ dj < ε. Take fj ∈ C with fj (s) = bj . Then w(t) − ( j fj ⊗ dj )(t) < ε for t = s and hence for all t in some neighborhood of s by continuity. Note that both C and W are Banach modules over C∞ (T ). Now a standard partition of unity argument shows that we can find some g ∈ C D with w − g < ε. Thus C ⊗ D is dense in W and hence C ⊗ D = W . 2 Next we discuss semi-continuous fields of ergodic actions of G. The following definition is a natural generalization of Rieffel’s definition of upper semi-continuous fields of actions of locally compact groups [32, Definition 3.1]. Definition 5.9. By a topological field of actions of G on unital C ∗ -algebras we mean a topological field ({Bt }t∈T , B) of unital C ∗ -algebras over a locally compact Hausdorff space T , and an action σt of G on Bt for each t ∈ T such that the section {f (t)1Bt }t∈T is in B for each f ∈ C∞ (T ) and {σt }t∈T is a homomorphism from ({Bt }t∈T , B) to ({Bt ⊗ A}t∈T , B ⊗ A). If the field ({Bt }t∈T , B) is actually upper semi-continuous (lower semi-continuous, continuous respectively), then we will say that the field of actions is upper semi-continuous (lower semi-continuous, continuous respectively). If each σt is ergodic, we say that this is a field of ergodic actions. Clearly the pull-back of a topological (upper semi-continuous, lower semi-continuous, continuous respectively) field of actions of G on unital C ∗ -algebras is a topological (upper semicontinuous, lower semi-continuous, continuous respectively) field of actions of G. Lemma 5.10. Let ({(Bt , σt )}t∈T , B) be a semi-continuous field of ergodic actions of G over a locally compact Hausdorff space T . Then for any b ∈ B the function t → ωt (πt (b)) is continuous on T , where ωt is the unique σt -invariant state on Bt . Denote by (Bt,r , σt,r ) the reduced action σt ) and byπt,r the canonical ∗-homomorphism Bt → Bt,r . Denote by πr the associated to (Bt , ∗-homomorphism t Bt → t Bt,r given pointwisely by πt,r . Then ({(Bt,r , σt,r )}t∈T , πr (B)) is a lower semi-continuous field of ergodic actions of G over T . Proof. We prove the continuity of the function t → ωt (πt (b)) first. Via taking restrictions to compact subsets of T we may assume that T is compact. The cross-section t → ωt (πt (b))1Bt is simply ((id ⊗ h) ◦ ( t σt ))(b), which is in B. Thus the function t → ωt (πt (b)) is continuous by Lemma 5.6. field of actions. Next we show that ({(Bt,r , σt,r )}t∈T , πr (B)) is a lower semi-continuous ∼ of B , and the evaluationmap Clearly πr (B) is a C ∗ -subalgebra and C∞ (T )-submodule t,r t is surjective for each t. Since σ ◦ π = (π ⊗ id) ◦ t σt , πt : πr (B) → B t,r t,r t,r t,r t t t one sees that t σt,r sends πr (B) into πr (B) ⊗ A. We are left to show that the function t → πt (πr (b)) is lower semi-continuous for each b ∈ B. Note that for any b ∈ B and t ∈ T , 1 1 the norm of πt (πr (b)) is the smallest number K such that ωt (πt (b1∗ b∗ bb1 )) 2 Kωt (πt (b1∗ b1 )) 2 for all b1 ∈ B. It follows easily that the function t → πt (πr (b)) is lower semi-continuous over T for each b ∈ B. This completes the proof of Lemma 5.10. 2 It is well known that there is a continuous field of ergodic actions of the n-dimensional torus Td over the compact space of isomorphism classes of faithful ergodic actions of Td such that the isomorphism class of the fibre at each point is exactly the point (see [1, Theorem 1.1] for a proof
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for the case n = 2; the proof for the higher-dimensional case is similar). We have not been able to extend this to arbitrary compact quantum groups. What we find is that there are two natural semicontinuous fields of ergodic actions of G over P such that the equivalence class of the fibre at each t ∈ P is the image of t under the quotient map P → EA∼ (G) defined before Definition 4.3. By Propositions 4.7 and 4.9, for each t ∈ P, the pair (Vt , σt ) defined after the formula (24) is isomorphic to the regular part of some ergodic action of G. By Propositions 3.3 and 3.4 there exist (unique up to isomorphisms) a full action (Bt,u , σt,u ) and a reduced action (Bt,r , σt,r ) of G whose regular parts are exactly (Vt , σt ). In fact, one can take (Bt , σt ) in Proposition 4.9 as (Bt,u , σt,u ). Recall the quotient map φt : V → Vt defined before (22) for each t ∈ P. Theorem 5.11. The set of cross-sections {φt (v)}t∈P over P for v ∈ V is in t Bt,u ( t Bt,r respectively). It generates an upper (lower respectively) semi-continuous field ({Bt,u }t∈P , Bu ) (({Bt,r }t∈P , Br ) respectively) of C ∗ -algebras over P. Moreover, the field ({(Bt,u , σt,u )}t∈P , Bu ) (({(Bt,r , σt,r )}t∈P , Br ) respectively) is an upper (lower respectively) semi-continuous field of full (reduced respectively) ergodic actions of G. If G is co-amenable, then these two fields coincide and are continuous. Proof. Consider generators wx for x ∈ X0 , θ (y) for y ∈ Y and ζ (z) for z ∈ Z subject to the following relations: (1) wx0 is the identity, (2) Eqs. (22) and (23) with νx , f (y), g(z) replaced by wx , θ (y), ζ (z) respectively, (3) the equations in E with f (y), f (y), g(z), g(z) replaced by θ (y), θ (y)∗ , ζ (z), ζ (z)∗ respectively, (4) θ (y) and ζ (z) are in the center. These relations have ∗-representations since Bf,g,u for any (f, g) ∈ P has generators satisfying these conditions. Consider an irreducible representation π of these relations. Because of (4), π(θ (y)) and π(ζ (z)) have to be scalars. Say π(θ (y)) = f (y) and θ (ζ (z)) = g(z). Then (f, g) ∈ CY × CZ satisfies the equations in E because of (3). Thus the inequalities (31) and (32) hold with |f (y)| and |g(z)| replaced by π(θ (y)) and π(ζ (z)) respectively. Also, there is a ∗homomorphism from Vf,g to the C ∗ -algebra generated by π(wx ), π(θ (y)), π(ζ (z)) sending νx to π(wx ). Thus (26) holds with νx replaced by π(wx ). Consequently, above generators and relations do have a universal C ∗ -algebra Bu . In particular, there is a surjective ∗-homomorphism πf,g : Bu → Bf,g,u for each (f, g) ∈ P sending wx , θ (y), ζ (z) to φf,g (vx ), f (y)φf,g (vx0 ), g(z)φf,g (vx0 ) respectively. These ∗ homomorphisms πt ’s for t ∈ P combine to a ∗-homomorphism π : Bu → t Bt,u . In above we have seen that every irreducible ∗-representation of Bu factors through πt for some t ∈ P. Thus π is faithful and we may identify Bu with π(Bu ). Since Bf,g,u is the universal C ∗ -algebra of Vf,g , one sees easily that ker(πf,g ) is generated by θ (y) − f (y)wx0 and ζ (z) − g(z)wx0 . Since θ (y)−f (y)wx0 → θ (y)−f (y)wx0 and ζ (z)−g (z)wx0 → ζ (z)−g(z)wx0 as (f , g ) → (f, g), the function t → πt (b) is upper semi-continuous on P for each b ∈ Bu . Thanks to the Stone– Weierstrass theorem, the unital C ∗ -subalgebra of Bu generated by θ (y) and ζ (z) is exactly C(P). Thus Bu is a C(P)-submodule of t Bt,u . Therefore ({Bt,u }t∈P , Bu ) is an upper semicontinuous field of C ∗ -algebras over P. Clearly it is generated by the sections {φt (v)}t∈P for v∈V .
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The formula (23) tells us that t σt,u sends the section {φt (vx )}t∈P into Bu ⊗ A for each x ∈ X0 . Since Bu is generated by such sections and C(P), t (σt,u ) sends Bu into Bu ⊗ A. Thus ({(Bt,u , σt,u )}t∈P , Bu ) is an upper semi-continuous field of ergodic actions of G. The assertions about the reduced actions follow from Lemma 5.10. The assertion about the case G is co-amenable follows from Proposition 3.8. 2 Theorem 5.12. Let ({(Bt , σt )}t∈T , B) be a semi-continuous field of ergodic actions of G over a locally compact Hausdorff space T . Let t0 ∈ T . Then the following are equivalent: (1) (2) (3) (4)
the map T → EA(G) sending each t to the isomorphism class of (Bt , σt ) is continuous at t0 , the map T → EA∼ (G) sending each t to the equivalence class of (Bt , σt ) is continuous at t0 , ˆ lim supt→t0 mul(Bt , γ ) mul(Bt0 , γ ) for all γ ∈ G, ˆ limt→t0 mul(Bt , γ ) = mul(Bt0 , γ ) for all γ ∈ G.
ˆ and let cγ si , 1 i Lemma 5.13. Let the notation be as in Theorem 5.12. Let γ ∈ G, mul(Bt0 , γ ), 1 i dγ , be a standard basis of (Bt0 )γ . Then there is a linear map ϕt : (Bt0 )γ → (Bt )γ for all t ∈ T such that the section t → ϕt (c) is in B for every c ∈ (Bt0 )γ , that ϕt0 = id, and that ϕt (cγ si ), 1 s mul(Bt0 , γ ), 1 i dγ , satisfy (9) and (15) (with eγ si and ω replaced by ϕt (cγ si ) and the unique σt -invariant state ωt respectively) throughout a neighborhood of t0 . Proof. We may assume that T is compact. Denote by σ the restriction of t σt on B. Recall γ γ the map Eij defined via (5). Then Eij is also defined on B for the unital ∗-homomorphism σ : B → B ⊗ A. Set m = mul(Bt0 , γ ) and S = {cγ s1 : 1 s m}. For each c ∈ S take b ∈ B γ γ γ with πt0 (b) = c. Then πt (E11 (b)) = E11 (πt (b)) is in E11 (Bt ) for each t ∈ T . By Lemma 4.11 S γ γ γ is a linear basis of E11 (Bt0 ). Set ψt to be the linear map E11 (Bt0 ) → E11 (Bt ) sending each c ∈ S γ to πt (E11 (b)). By Lemma 4.11 we have ψt0 = id. By Lemma 5.10 the function t → ωt (πt (b )) is continuous on T for any b ∈ B, where ωt is the unique σt -invariant state on Bt . Consequently, for any c1 , c2 ∈ S, we have ωt ψt (c1 )∗ ψt (c2 ) → ωt0 ψt0 (c1 )∗ ψt0 (c2 ) = ωt0 (c1 c2 ) = δc1 c2
as t → t0 .
Shrinking T if necessary, we may assume that the matrix Qt = (ωt (ψt (cγ k1 )∗ ψt (cγ s1 )))ks ∈ −1/2 . Then Mm (C) is invertible for all t ∈ T . Set (ct,1 , . . . , ct,m ) = (ψt (cγ 11 ), . . . , ψt (cγ m1 ))Qt γ ∗ ct,k ∈ E11 (Bt ) and ωt (ct,k ct,s ) = δks for all t ∈ T . Note that the section t → ct,s is in B for each γ 1 s m. Thus the section t → Ei1 (ct,s ) is in B for all 1 s m, 1 i dγ . Set ϕt to be the γ linear map (Bt0 )γ → Bt sending cγ si to Ei1 (ct,s ). Then the section t → ϕt (c) is in B for every c ∈ (Bt0 )γ . By Lemma 4.11 these maps have the other desired properties. 2 Remark 5.14. Using Remark 5.3 one can show easily that for an upper semi-continuous field ({(B t , σt )}t∈T , B) of actions of G over a compact Hausdorff space T , the ∗-homomorphism ( t σt )|B : B → B ⊗ A is an action of G on B. Using the well-known fact that upper semicontinuous fields of unital C ∗ -algebras over a compact Hausdorff space T satisfying the hypothesis in Lemma 5.6 correspond exactly to unital C ∗ -algebras containing C(T ) in the centers, one can show further that upper semi-continuous fields of ergodic actions of G over T correspond exactly to actions of G on unital C ∗ -algebras whose fixed point algebras are C(T ) and are in the centers.
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As a corollary of Lemma 5.13 we get Lemma 5.15. Let the notation be as in Theorem 5.12. The function t → mul(Bt , γ ) is lower ˆ semi-continuous on T for each γ ∈ G. We are ready to prove Theorem 5.12. Proof of Theorem 5.12. (1) ⇔ (2) follows from the definition of the topology on EA(G). (2) ⇒ (3) follows from Proposition 4.13. (3) ⇒ (4) follows from Lemma 5.15. We are left to show (4) ⇒ (2). Assume (4). Fix a standard basis S of Bt0 , consisting of a standard basis Sγ ˆ Let J be a finite subset of G. ˆ Then mul(Bt , γ ) = mul(Bt0 , γ ) for each of (Bt0 )γ for each γ ∈ G. γ ∈ J throughout some neighborhood U of t0 . By Lemma 5.13, shrinking U if necessary, we can find a linear map ϕt : (Bt0 )J → (Bt )J for all t ∈ T , where (Bt )J = γ ∈J (Bt )γ , such that the section t → ϕt (c) is in B for every c ∈ (Bt0 )J , that ϕt0 = id, and that ϕt (Sγ ) is a standard basis of (Bt )γ for all γ ∈ J and t ∈ U . For each t ∈ U , extend these bases of (Bt )γ for γ ∈ J to a standard basis St of Bt . Set (ft , gt ) to be the element in P associated to St via (16)–(18). Suppose that α, β ∈ J \ {γ0 }. By Lemma 5.10 the function t → ωt (πt (b)) is continuous for each b ∈ B, where ωt is the unique σt -invariant state on Bt . Then one sees easily that the function t → ft (x1 , x2 , x3 ) is continuous over U for any x1 ∈ Xα , x2 ∈ Xβ , x3 ∈ Xγ , γ ∈ J . Similarly, if α, α¯ ∈ J \ {γ0 }, then the function t → gt (x1 , x2 ) is continuous over U for any x1 ∈ Xα , x2 ∈ Xα¯ . ˆ this means that for any neighborhood W of (ft , gt ) Since J is an arbitrary finite subset of G, 0 0 in P, we can find a neighborhood V of t0 in T and choose a standard basis of Bt for each t ∈ V such that the associated element in P is in W . Therefore (2) holds. 2 Now Theorems 1.1 and 1.3 follow from Theorems 4.4, 5.12 and 5.11. In fact we have a stronger assertion: Corollary 5.16. The topology on EA∼ (G) defined in Definition 4.3 is the unique Hausdorff topology on EA∼ (G) such that the implication (4) ⇒ (2) in Theorem 5.12 holds for all upper semi-continuous (lower semi-continuous respectively) fields of ergodic actions of G over compact Hausdorff spaces. If G is co-amenable, then the topology on EA(G) defined in Definition 4.3 is the unique Hausdorff topology on EA(G) such that the implication (4) ⇒ (2) in Theorem 5.12 holds for all continuous fields of ergodic actions of G over compact Hausdorff spaces. When A is separable and co-amenable, one can describe the topology on EA(G) more explicitly in terms of continuous fields of actions: Theorem 5.17. Suppose that A is separable and co-amenable. Then both P and EA(G) are metrizable. The isomorphism classes of a sequence {(Bn , σn )}n∈N of ergodic actions of G converge to that of (B∞ , σ∞ ) in EA(G) if and only if there exists a continuous field of ergodic actions of G over the one-point compactification N ∪ {∞} of N with fibre (Bn , σn ) at n for 1 n ∞ ˆ and limn→∞ mul(Bn , γ ) = mul(B∞ , γ ) for all γ ∈ G. Proof. Denote by (πA , HA ) the GNS representation of A associated to h. Since A is separable, so is HA . Note that the subspaces Aγ are nonzero and orthogonal to each other in HA ˆ Thus Gˆ is countable. Then Y and Z are both countable. Therefore P and EA(G) for γ ∈ G. are metrizable. The “if” part follows from Theorem 5.12. Suppose that the isomorphism class
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of (Bn , σn ) converges to that of (B∞ , σ∞ ) in EA(G) as n → ∞. By Proposition 4.13 we have ˆ Also the map ξ : N ∪ {∞} → EA(G) sending limn→∞ mul(Bn , γ ) = mul(B∞ , γ ) for all γ ∈ G. 1 n ∞ to the isomorphism class of (Bn , σn ) is continuous. By Theorem 4.4 the quotient map P → EA(G) is open. Thus ξ lifts up to a continuous map η : N ∪ {∞} → P. The pull-back of the continuous field of ergodic actions of G over P in Theorem 5.11 via η is a continuous field of ergodic actions of G over N ∪ {∞} with the desired fibres. This proves the “only if” part. 2 6. Podle´s spheres In this section we prove Theorem 1.2. Fix q ∈ [−1, 1]. The quantum SU(2) group A = C(SUq (2)) [38,46] is defined as the universal C ∗ -algebra generated by α and β subject to the condition that u=
α β
−qβ ∗ α∗
is a unitary in M2 (A). The comultiplication Φ : A → A is defined in such a way that u is a representation of A. Below we assume 0 < |q| < 1. The quantum group SUq (2) is co-amenable [25], [2, Corollary 6.2], [3, Theorem 2.12]. Let
Tq = c(1), c(2), . . . ∪ [0, 1], where 2 c(n) = −q 2n / 1 + q 2n . 2 ) [29] is defined as the universal C ∗ -algebra For t ∈ Tq with t 0, Podle´s quantum sphere C(Sqt generated by at , bt subject to the relations
at∗ = at ,
bt∗ bt = at − at2 + t,
bt at = q 2 at bt ,
bt bt∗ = q 2 at − q 4 at2 + t.
(38)
2 ) is defined as the universal C ∗ -algebra generated by a , b subject For t ∈ Tq with t 0, C(Sqt t t to the relations
at∗ = at , bt at = q 2 at bt ,
bt∗ bt = 1 − t 2 at − at2 + t 2 , bt bt∗ = 1 − t 2 q 2 at − q 4 at2 + t 2 .
(39)
2 ) → C(S 2 ) ⊗ A is determined by The action σt : C(Sqt qt
σt (at ) = at ⊗ 1A + ct ⊗ β ∗ β + bt∗ ⊗ α ∗ β + bt ⊗ β ∗ α, σt (bt ) = −qbt∗ ⊗ β 2 + ct ⊗ αβ + bt ⊗ α 2 ,
(40)
where ct is 1C(S 2 ) − (1 + q 2 )at or (1 − t 2 )1C(S 2 ) − (1 + q 2 )at depending on t 0 or t 0. qt qt As in [13], here we reparametrize the family for 0 c ∞ in [29] for the parameters 0 t 1
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√ √ by t = 2 c/(1 + 1 + 4c) (and c = (t −1 − t)−2 ), and rescale the generators A, B in [29] by at = (1 − t 2 )A, bt = (1 − t 2 )B for 0 t < 1. 2 ) at Proposition 6.1. There is a unique continuous field of C ∗ -algebras over Tq with fibre C(Sqt each t ∈ Tq such that the sections t → at and t → bt are in the algebra B of continuous sections. Moreover, the field {σt }t∈Tq of ergodic actions of SUq (2) is continuous.
Proof. The uniqueness is clear. We start to show that there exists an upper semi-continuous field 2 )} ∗ ({C(Sqt t∈Tq , B) of C -algebras over Tq such that the sections t → at and t → bt are in B. For this purpose, by Lemma 5.5 it suffices to show that the function ηp : t → p(at , bt , at∗ , bt∗ ) is upper semi-continuous over Tq for any noncommutative polynomial p in four variables. Denote by Tq the set of the non-positive numbers in Tq . We prove the upper semi-continuity of ηp over Tq first. We claim that there exists a universal C ∗ -algebra generated by a, b, x subject to the relations (1) the equations in (38) with at , bt , t replaced by a, b, x respectively, (2) the inequality x |c(1)|, (3) x = x ∗ is in the center. 2 ) for t ∈ T has generators satisfying these conditions. Let a, b, x be bounded linear Clearly C(Sqt q operators on a Hilbert space satisfying these relations. We have
1 + q 2 b∗ b + q −2 bb∗ (38) = 1 + q 2 a − a 2 + x + 1 + q 2 a − q 2 a 2 + q −2 x 2 2 = − 1 − 1 + q 2 a + 1 + 1 + q 2 q −2 x .
(41)
Thus 1 − 1 + q 2 a 2 , 1 + q 2 b∗ b 1 + 1 + q 2 2 q −2 x 2 1 + 1 + q 2 q −2 c(1) = 2, and hence −1 1 a 1 + q 2 1 + 22 , − 1 1 b 1 + q 2 2 2 2 . Therefore there does exist a universal C ∗ -algebra C generated by a, b, x subject to these relations. An argument similar to that in the proof of Theorem 5.11 shows that ηp is upper semi-continuous over Tq . The upper semi-continuity of ηp over [0, 1] is proved similarly, replacing (41) by 1 + q 2 b∗ b + q −2 bb∗ 2 2 2 = − 1 − x 2 − 1 + q 2 a + 1 − x 2 + 1 + q 2 q −2 x 2 .
(42)
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This proves the existence of the desired upper semi-continuous field of C ∗ -algebras over Tq . Note that B is generated as a C ∗ -algebra by C(Tq ) and the sections t → at and t → bt . From (40) 2 ), σ )} one sees immediately that ({(C(Sqt t t∈Tq , B) is an upper semi-continuous field of ergodic actions of SUq (2). Since SUq (2) is co-amenable, by Proposition 3.8 and Lemma 5.10 this is actually a continuous field of actions. 2 We are ready to prove Theorem 1.2. 1 1 Proof of Theorem 1.2. It is customary to index SU q (2) by 0, 2 , 1, 1 + 2 , . . . [46, remark after 2 the proof of Theorem 5.8]. Say SU q (2) = {d0 , d1/2 , d1 , d1+1/2 , . . .}. Then mul(C(Sqt ), dk ) = 1, 2 ), d 2 mul(C(Sqt k+ 12 ) = 0 for k = 0, 1, 2, . . . when t 0. And mul(C(Sqt ), dl ) = 1 or 0 depending on l ∈ {0, 1, . . . , n − 1} or not when t = c(n) [30, the note after Proposition 2.5]. Thus the 2 ), γ ) is continuous over T for any γ ∈ SU multiplicity function t → mul(C(Sqt q q (2). Then Theorem 1.2 follows from Proposition 6.1 and Theorem 5.12. 2
7. Ergodic actions of full multiplicity of compact groups In this section we show that the topology of Landstad and Wassermann on the set EA(G)fm of isomorphism classes of ergodic actions of full multiplicity of a compact group G coincides with the relative topology of EAfm in EA(G). Throughout this section we let G = G be a compact Hausdorff group. An ergodic action ˆ Denote by EA(G)fm (B, σ ) of G is said to be of full multiplicity if mul(B, γ ) = dγ for all γ ∈ G. the set of isomorphism classes of ergodic actions of full multiplicity of G. By Proposition 4.13 EA(G)fm is a closed subset of EA(G). Landstad [20] and Wassermann [43] showed independently that EA(G)fm can be identified with the set of equivalence classes of dual cocycles. Let us recall the notation in [20]. Denote by L(G) the von Neumann algebra generated by the left regular representation of G on L2 (G). 2 ∼ One has a natural decomposition L (G) = γ ∈Gˆ Hγ as unitary representations of G. Then L(G) = γ ∈Gˆ B(Hγ ) under this decomposition. Denote by 1γ0 the identity of B(Hγ0 ) for the trivial representation γ0 . One has the normal ∗-homomorphism δ : L(G) → L(G) ⊗ L(G) (tensor product of von Neumann algebras) and the normal ∗-anti-isomorphism ν : L(G) → L(G) determined by δ(x) = x ⊗ x
and ν(x) = x −1
for x ∈ G.
Denote by σ the flip automorphism of L(G) ⊗ L(G) determined by σ (a ⊗ b) = b ⊗ a for all a, b ∈ L(G). One also has Takesaki’s unitary W in B(L2 (G)) ⊗ L(G) defined by (Wf )(x, y) = f (x, xy) for f ∈ C(G × G) and x, y ∈ G. A normalized dual cocycle [20, p. 376] is a unitary w ∈ L(G) ⊗ L(G) satisfying (w ⊗ I ) (δ ⊗ id)(w) = (I ⊗ w) (id ⊗ δ)(w) , (ν ⊗ ν)(w) = σ w ∗ , w(I ⊗ 1γ0 ) = I ⊗ 1γ0 , w(1γ0 ⊗ I ) = 1γ0 ⊗ I, (id ⊗ ν) wσ w ∗ = σ (w)w ∗ ,
wδ(1γ0 ) = δ(1γ0 ), (id ⊗ ν) wW ∗ = W w ∗ .
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Denote by C 2 the set of all normalized dual cocycles. Also denote by H the group of unitaries ξ in L(G) satisfying ξ = ν(ξ ∗ ) and ξ 1γ0 = 1γ0 (on p. 376 of [20] only the condition ξ = ν(ξ ∗ ) is mentioned, but in order for αξ (w) below to satisfy αξ (w)(I ⊗ 1γ0 ) = I ⊗ 1γ0 , one has to require ξ 1γ0 = 1γ0 ; this can be seen using the formula δ(x)(I ⊗ 1γ0 ) = x ⊗ 1γ0 for all x ∈ L(G)). Then H has a left action α on C 2 via αξ (w) = (ξ ⊗ ξ )wδ(ξ ∗ ). The result of Landstad and Wassermann says that EA(G)fm can be identified with C 2 /H [20, Remark 3.13] in a natural way. Note that the unitary groups of L(G) ⊗ L(G) and L(G) are both compact Hausdorff groups with the weak topology. Clearly C 2 and H are closed subsets of the unitary groups of L(G) ⊗ L(G) and L(G) respectively. Thus C 2 is a compact Hausdorff space and H is a compact Hausdorff group, with the relative topologies. It is also clear that the action α is continuous. Therefore C 2 /H equipped with the quotient topology is a compact Hausdorff space. In order to show that the quotient topology on C 2 /H coincides with the relative topology of EA(G)fm in EA(G), we need to recall the map C 2 → EA(G)fm constructed in the proof of ˆ [20, Theorem 3.9]. Let w ∈ C 2 . Set U = wW ∗ ∈ B(L2 (G)) ⊗ L(G). Recall that for each γ ∈ G γ we fixed an orthonormal basis of Hγ and identified B(Hγ ) with Mdγ (C). Let eij , 1 i, j dγ , γ be the matrix units of Mdγ (C) as usual. Then we may write U as γ ∈Gˆ 1i,j dγ bγ ij ⊗ eij for bγ ij ∈ B(L2 (G)). The conjugation of the right regular representation of G on L2 (G) restricts ˆ 1 i, j dγ . on an ergodic action α of G on the C ∗ -algebra B generated by bγ ij for all γ ∈ G, 2 → EA(G) . Furthermore, The isomorphism class of α is the image of w under the map C fm γ each Uγ = 1i,j dγ bγ ij ⊗ eij is a unitary uγ -eigenoperator meaning that Uγ is a unitary in B ⊗ B(Hγ ) satisfying (αx ⊗ id)(Uγ ) = Uγ 1B ⊗ uγ (x)
(43)
for all x ∈ G. If we let σ : B → B ⊗ C(G) = C(G, B) be the ∗-homomorphism associated to α, i.e., (σ (b))(x) = αx (b), then (43) simply means (σ ⊗ id)(Uγ ) = (Uγ )13 (τ (uγ ))23 , where (Uγ )13 and (τ (uγ ))23 are in the leg numbering notation and τ : B(Hγ ) ⊗ C(G) → C(G) ⊗ B(Hγ ) is the flip. It follows that (43) is equivalent to (9) with eγ ki replaced by bγ ki . Then 1j dγ bγ ij bγ∗ kj is easily seen to be σ -invariant and hence is in C1B . One checks easily that Uγ Uγ∗ = 1B ⊗ 1B(Hγ ) means that ω bγ ij bγ∗ kj = δki (44) 1j dγ
for all 1 k, i dγ , where ω is the unique α-invariant state on B. Since G is a compact group, ω is a trace [14, Theorem 4.1]. From Lemma 4.11 one sees that (44) is equivalent to (15) with eγ ki 1/2 replaced by dγ bγ ki . Using W (I ⊗ 1γ0 ) = w(I ⊗ 1γ0 ) = I ⊗ 1γ0 one gets U (I ⊗ 1γ0 ) = I ⊗ 1γ0 . 1/2 ˆ 1 i, j dγ is a standard basis of B. Denote Thus bγ0 11 = 1B . Therefore dγ bγ ij for γ ∈ G, by ψ(w) the associated element in P. Then the diagram C2
C 2 /H
ψ
EA(G)fm
P
EA(G) = EA∼ (G)
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commutes, where we identify EA(G) with EA∼ (G) since G is co-amenable. It was shown in the proof of [20, Theorem 3.9] that one has U12 U13 = (I ⊗ w) (id ⊗ δ)(U )
and (id ⊗ ν)(U ) = U ∗ ,
where U12 and U13 are in the leg numbering notation. It follows that the map ψ is continuous. Consequently, the relative topology on EA(G)fm in EA(G) coincides with the quotient topology coming from C 2 → EA(G)fm . 8. Induced Lip-norm In this section we prove Theorem 1.4. We recall first Rieffel’s construction of Lip-norms from ergodic actions of compact groups. Let G be a compact group. A length function on G is a continuous function l : G → R+ such that l(xy) l(x) + l(y) for all x, y ∈ G, l x −1 = l(x) for all x ∈ G, l(x) = 0 if and only if x = eG . Given an ergodic action α of G on a unital C ∗ -algebra B, Rieffel showed that the seminorm LB on B defined by αx (b) − b : x ∈ G, x = eG LB (b) = sup l(x)
(45)
is a Lip-norm [33, Theorem 2.3]. Note that there is a 1–1 correspondence between length functions on G and left-invariant metrics on G inducing the topology of G, via ρ(x, y) = l(x −1 y) and l(x) = ρ(x, eG ). Since a quantum metric on (the non-commutative space corresponding to) a unital C ∗ -algebra is a Lipnorm on this C ∗ -algebra, a length function for a compact quantum group A = C(G) should be a Lip-norm LA on A satisfying certain compatibility condition with the group structure. The proof of [33, Proposition 2.2] shows that LB in the above is finite on any α-invariant finitedimensional subspace of B, and hence is finite on B. If one applies this observation to the action of G on C(G) corresponding to the right translation of G on itself, then we see that the Lipschitz seminorm LC(G) on C(G) associated to the above metric ρ via LC(G) (a) = sup x=y
|a(x) − a(y)| |a(yx) − a(y)| = sup sup ρ(x, y) l(x) x=eG y
is finite on the algebra of regular functions in C(G). This leads to the following definition: Definition 8.1. We say that a Lip-norm LA on a compact quantum group A = C(G) is regular if LA is finite on the algebra A of regular functions.
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It turns out that a regular Lip-norm is sufficient for us to induce Lip-norms on C ∗ -algebras carrying ergodic actions of co-amenable compact quantum groups. We leave the discussion of the left and right invariance of LA to the end of this section. Remark 8.2. If a unital C ∗ -algebra B has a Lip-norm, then S(B) with the weak-∗ topology is metrizable and hence B is separable. Conversely, if B is a separable unital C ∗ -algebra, then for any countable subset W of B, there exist Lip-norms on B being finite on W [34, Proposition 1.1]. When A = C(G) is separable, A is a countable-dimensional vector space, and hence A has regular Lip-norms. Example 8.3. Let Γ be a discrete group. Then the reduced group C ∗ -algebra Cr∗ (Γ ) is a compact quantum group with Φ(g) = g ⊗ g for g ∈ Γ . Its algebra of regular functions is CΓ . Let l be a length function on Γ . Denote by D the (possibly unbounded) linear operator of pointwise multiplication by l on 2 (Γ ). One may consider the seminorm L defined on CΓ as L(a) = [D, a] and extend it to Cr∗ (Γ ) via setting L = ∞ on Cr∗ (Γ ) \ CΓ . The seminorm L so defined is always finite on CΓ , and hence is regular if it is a Lip-norm. This is the case for Γ = Zd when l is a word-length, or the restriction to Zd of a norm on Rd [34, Theorem 0.1], and for Γ being a hyperbolic group when l is a word-length [27, Corollary 4.4]. Now we try to extend (45) to ergodic actions of compact quantum groups. Let σ : B → B ⊗ C(G) = C(G, B) be the ∗-homomorphism associated to α, i.e., (σ (b))(x) = αx (b) for b ∈ B and x ∈ G. For any b ∈ Bsa , we have LB (b) = sup sup x=eG y
αyx (b) − αy (b) l(x) |ϕ(αyx (b)) − ϕ(αy (b))| l(x) ϕ∈S(B)
= sup sup sup x=eG y
= sup LC(G) (b ∗ ϕ), ϕ∈S(B)
where S(B) denotes the state space of B. Note that for quantum metrics, only the restriction of LB on Bsa is essential. Thus the above formula leads to our definition of the (possibly +∞valued) seminorm LB on B in (1) for any ergodic action σ : B → B ⊗ A of a compact quantum group A = C(G) equipped with a regular Lip-norm LA . Throughout the rest of this section we assume that LA is a regular Lip-norm on A. Lemma 8.4. We have a − h(a)1A 2rA LA (a)
(46)
for all a ∈ Asa . Proof. By Proposition 2.5 we can find a ∈ C1A such that a − a rA LA (a). Then h(a)1A − a = |h(a − a )| a − a rA LA (a). Thus a − h(a)1A a − a + a − h(a)1A 2rA LA (a). 2
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Lemma 8.5. Let LB be the seminorm on a unital C ∗ -algebra B defined via (1) for an action σ : B → B ⊗ A of G on B. Assume that A has bounded counit e. Then for any b ∈ Bsa we have b − E(b) 2rA LB (b), where E : B → B σ is the canonical conditional expectation. Proof. Let ϕ ∈ S(B). Note that h(b ∗ ϕ) = ϕ(E(b)). We have b ∗ ϕ − ϕ E(b) 1A = b ∗ ϕ − h(b ∗ ϕ)1A (46)
(1)
2rA LA (b ∗ ϕ) 2rA LB (b).
Thus sup b − E(b) ∗ ϕ = sup b ∗ ϕ − ϕ E(b) 1A 2rA LB (b). ϕ∈S(B)
ϕ∈S(B)
Therefore by Remark 2.2 we have b − E(b) = e ∗ b − E(b) = sup ϕ e ∗ b − E(b) ϕ∈S(B)
= sup e b − E(b) ∗ ϕ sup b − E(b) ∗ ϕ ϕ∈S(B)
ϕ∈S(B)
2rA LB (b) as desired.
2
For any J ⊆ Gˆ denote
γ ∈J
Aγ and
γ ∈J
Bγ by AJ and BJ respectively.
Lemma 8.6. Assume that A has faithful Haar measure. For any ε > 0 and φ ∈ S(A) there exist ψ ∈ S(A) and a finite subset J ⊆ Gˆ such that ψ vanishes on Aγ for all γ ∈ Gˆ \ J and (φ − ψ)(a) εLA (a)
(47)
for all a ∈ Asa . Proof. Denote by W the set of states of A consisting of convex combinations of states of the form h(a ∗ (·)a) for a ∈ A with h(a ∗ a) = 1. Let ψ ∈ W . Clearly there exists a finite subset F ⊆ Gˆ such that if h(A∗F a AF ) = 0 for some a ∈ A then ψ(a ) = 0. By the faithfulness of h on A and the Peter–Weyl theory [47, Theorems 4.2 and 5.7], for any a ∈ A and any finite subset ˆ h(A∗ a ) = 0 if and only if h(a A∗ ) = 0. Denote by F the set of equivalence classes of J ⊆ G, J J β c irreducible unitary subrepresentations of the tensor products uα
u of all α ∈ F and β ∈ F = c c ˆ {γ : γ ∈ F }. Denote (F ) by J . Suppose that ψ does not vanish on Aγ for some γ ∈ G. Then h(A∗F Aγ AF ) = 0. Thus (8) h(Aγ AF ) ⊇ h(Aγ AF AF c ) = h Aγ AF A∗F {0}.
Since h(Aα Aβ ) = 0 for all α = β c in Gˆ [47, Theorem 5.7], we get γ ∈ J .
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Now we just need to find ψ ∈ W such that (47) holds for all a ∈ A. Since h is faithful, the GNS representation (πA , HA ) of A associated to h is faithful. Thus convex combinations of vector states from (πA , HA ) are weak-∗ dense in S(A) [45, Lemma T.5.9]. Note that A is dense in A. Therefore W is weak-∗ dense in S(A). Take R rA . Since DR (A) is totally bounded by Proposition 2.5, we can find ψ ∈ W such that (φ − ψ)(a) ε
(48)
for all a ∈ DR (A). By Proposition 2.5 we have E(A) = DR (A) + R · 1A . Therefore (48) holds for all a ∈ E(A), from which (47) follows. 2 The next lemma is an analogue of [35, Lemmas 8.3 and 8.4] and [21, Lemma 10.8]. Lemma 8.7. Let B and LB be as in Lemma 8.5. Assume that A is co-amenable. Let ε > 0 and take ψ and J in Lemma 8.6 for φ being the counit e. Denote by Pψ the linear map B → B sending b ∈ B to ψ ∗ b. Then Pψ (B) ⊆ BJ and Pψ (b) b,
and b − Pψ (b) εLB (b)
(49)
for all b ∈ Bsa . ˆ we have Proof. Since ψ vanishes on Aγ for all γ ∈ Gˆ \ J and σ (Bβ ) ⊆ Bβ Aβ for all β ∈ G, ˆ Note that Bβ is finite-dimensional and B = B is dense in B. Pψ (Bβ ) ⊆ BJ for all β ∈ G. β∈Gˆ β Thus Pψ (B) ⊆ BJ . For any b ∈ B clearly Pψ (b) b. If b ∈ Bsa , by Remark 2.2 we have b − Pψ (b) = e ∗ b − Pψ (b) = sup ϕ e ∗ b − Pψ (b) ϕ∈S(B)
(47) = sup e(b ∗ ϕ) − ψ(b ∗ ϕ) sup εLA (b ∗ ϕ) ϕ∈S(B)
ϕ∈S(B)
(1)
= εLB (b).
This finishes the proof of Lemma 8.7.
2
We are ready to prove Theorem 1.4. Proof of Theorem 1.4. We verify the conditions in Proposition 2.5. For any b ∈ B and ϕ in S(B) we have b∗ ∗ ϕ = (b ∗ ϕ)∗ . Since LA satisfies the reality condition (10), so does LB . For any b ∈ B, {b ∗ ϕ: ϕ ∈ S(B)} is bounded and contained in a finite-dimensional subspace of A since σ (B) ⊆ B A. Then LB is finite on B because of the regularity of LA . Clearly LB vanishes ˜ sa . For any ε > 0 let Pψ and J be as on C1B . By Lemma 8.5 we have · ∼ 2rA L˜ B on (B) in Lemma 8.7. Then Pψ (D1 (B)) is a bounded subset of the finite-dimensional space BJ . Thus Pψ (D1 (B)) is totally bounded. Since ε > 0 is arbitrary, D1 (B) is also totally bounded. Therefore Theorem 1.4 follows from Proposition 2.5. 2
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Now we consider the invariance of a (possibly +∞-valued) seminorm on B with respect to an action σ of G. We consider first the case G = G is a compact group. For any action of A = C(G) on B, there is a strongly continuous action α of G on B such that for any b ∈ B, the element σ (b) ∈ B ⊗ A = C(G, B) is given by (σ (b))(x) = αx (b) for all x ∈ G. If a seminorm LB on B is lower semi-continuous, which is the case if LB is defined via (45), and is α-invariant, then for any ψ ∈ S(A) corresponding to a Borel probability measure μ on G, we have LB (ψ ∗ b) = LB
αx (b) dμ(x) LB (b)
G
for all b ∈ B. Conversely, if LB (ψ ∗ b) LB (b) for all b ∈ B and ψ ∈ S(B), taking ψ to be the evaluation at x ∈ G, one sees immediately that LB is α-invariant. Note that the essential information about the quantum metric is the restriction of LB on Bsa . This leads to the following Definition 8.8. Let A = C(G) be a compact quantum group. We say that a (possibly +∞-valued) seminorm LA on A is right-invariant (left-invariant respectively) if LA (ψ ∗ a) LA (a)
LA (a ∗ ψ) LA (a) respectively
for all a ∈ Asa and ψ ∈ S(A). For an action σ : B → B ⊗ A of G on a unital C ∗ -algebra B, we say that a (possibly +∞-valued) seminorm LB on B is invariant if LB (ψ ∗ b) LB (b) for all b ∈ Bsa and ψ ∈ S(A). Proposition 8.9. Let LA be a regular Lip-norm on A. Define ( possibly +∞-valued ) seminorms L A and L A on A by L A (a) = sup LA (ϕ ∗ a), ϕ∈S(A)
and L A (a) = sup LA (a ∗ ϕ) ϕ∈S(A)
for a ∈ A. Assume that A has bounded counit. Then L A (L A respectively) is a right-invariant (left-invariant respectively) regular Lip-norm on A, and L A LA (L A LA respectively). If LA is left-invariant (right-invariant respectively), then so is L A (L A respectively). Proof. An argument similar to that in the proof of Theorem 1.4 shows that L A satisfies the reality condition (10), vanishes on C1A , and is finite on A. Taking ϕ to be the counit we see that L A LA . It follows immediately from Proposition 2.5 that L A is a regular Lip-norm on A. For any a ∈ Asa and ψ ∈ S(A) we have L A (ψ ∗ a) = sup LA ϕ ∗ (ψ ∗ a) = sup LA (ϕ ∗ ψ) ∗ a L A (a), ϕ∈S(A)
ϕ∈S(A)
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where ϕ ∗ ψ is the state on A defined via (ϕ ∗ ψ)(a ) = (ϕ ⊗ ψ)(Φ(a )) for a ∈ A. Therefore L A is right-invariant. Assume that LA is left-invariant. Then for any a ∈ Asa and ψ ∈ S(A) we have L A (a ∗ ψ) = sup LA ϕ ∗ (a ∗ ψ) = sup LA (ϕ ∗ a) ∗ ψ ϕ∈S(A)
sup LA (ϕ ϕ∈S(A)
ϕ∈S(A)
∗ a) = L A (a).
Thus L A is also left-invariant. The assertions about L A are proved similarly.
2
Using Remark 8.2 and applying the construction in Proposition 8.9 twice, we get Corollary 8.10. Every separable compact quantum group with bounded counit has a bi-invariant regular Lip-norm. An argument similar to that in the proof of Proposition 8.9 shows Proposition 8.11. Let σ be an action of G on a unital C ∗ -algebra B. If LA is a right-invariant regular Lip-norm on A, then LB defined via (1) is invariant. 9. Quantum distance In this section we introduce the quantum distance dist e between ergodic actions of G, and prove Theorem 1.5. Throughout this section, A will be a co-amenable compact quantum group with a fixed regular Lip-norm LA . For any ergodic action (B, σ ) of G, we endow B with the Lip-norm LB in Theorem 1.4. In [15,16,21,22,35] several quantum Gromov–Hausdorff distances are introduced, applying to quantum metric spaces in various contexts as order-unit spaces, operator systems, and C ∗ algebras. They are all applicable to C ∗ -algebraic compact quantum metric spaces, which we are dealing with now. Among these distances, the unital version dist nu of the one introduced in [22, Remark 5.5] is the strongest one, which we recall below from [16, Section 5]. To simplify the notation, for fixed unital C ∗ -algebras B1 and B2 , when we take infimum over unital C ∗ algebras C containing both B1 and B2 , we mean to take infimum over all unital injective ∗homomorphisms of B1 and B2 into some unital C ∗ -algebra C. We denote by distCH the Hausdorff distance between subsets of C. Recall that E(B) := {b ∈ Bsa : LB (b) 1}. For any C ∗ -algebraic compact quantum metric spaces (B1 , LB1 ) and (B2 , LB2 ), the distance dist nu (B1 , B2 ) is defined as dist nu (B1 , B2 ) = inf distCH E(B1 ), E(B2 ) , where the infimum is taken over all unital C ∗ -algebras C containing B1 and B2 . Note that dist nu (B1 , B2 ) is always finite since DR (B) is totally bounded and E(B) = DR (B) + R · 1B for any R rB by Proposition 2.5. These distances become zero whenever there is a ∗-isomorphism ϕ : B1 → B2 preserving the Lip-norms on the self-adjoint parts. In particular, as the following example shows, these distances may not distinguish the actions when the Lip-norms LBi come from ergodic actions of G on Bi .
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Example 9.1. Let l be a length function on the circle S 1 . Set l to be the length function on the two-torus T2 defined as l(x, y) = l (x) + l (y) for x, y ∈ S 1 . Then l(x, y) = l(x −1 , y) for all (x, y) ∈ T2 . Let θ ∈ R, and let Bθ be the non-commutative two-torus generated by unitaries uθ and vθ satisfying uθ vθ = e2πiθ vθ uθ . Then T2 has a strongly continuous action αθ on Bθ specified by αθ,(x,y) (uθ ) = xuθ and αθ,(x,y) (vθ ) = yvθ . Consider the ∗-isomorphism ψ : Bθ → B−θ determined by ψ(uθ ) = (u−θ )−1 and ψ(vθ ) = v−θ . Then ψ preserves the Lip-norms defined via (45) for the actions αθ and α−θ of T2 , and hence Bθ and B−θ have distances zero under all the quantum distances defined in [15,16,21,22,35]. However, when 0 < θ < 1/2, the actions (Bθ , αθ ) and (B−θ , α−θ ) are not isomorphic, as can be seen from the fact that Cuθ = {b ∈ Bθ : αθ,(x,y) (b) = xb for all (x, y) ∈ T2 } and Cvθ = {b ∈ Bθ : αθ,(x,y) (b) = yb for all (x, y) ∈ T2 }. Notation 9.2. For any C ∗ -algebra C we denote C ⊕ (C ⊗ A) by C . For any action σ : B → B ⊗ A of G on a unital C ∗ -algebra B and any subset X of B we denote by Xσ the graph
b, σ (b) ∈ B : b ∈ X of σ |X . We are going to introduce a quantum distance between ergodic actions of G to distinguish the actions. Modifying the above definition of dist nu , we just need to add one term to take care of the actions: Definition 9.3. Let (B1 , σ1 ) and (B2 , σ2 ) be ergodic actions of A. We set
dist e (B1 , B2 ) = inf distCH
E(B1 ) σ , E(B2 ) σ , 1
2
where the infimum is taken over all unital C ∗ -algebras C containing both B1 and B2 . Clearly dist e dist nu . An argument similar to that in the proof of [22, Theorem 3.15] yields Proposition 9.4. The distance dist e is a metric on EA(G). We relate first continuous fields of ergodic actions of G to the distance dist e . Proposition 9.5. Suppose that LA is left-invariant. Let ({(Bt , σt )}t∈T , B) be a continuous field of ergodic actions of G over a compact metric space T . Let t0 ∈ T . If limt→t0 mul(Bt , γ ) = ˆ then dist e (Bt , Bt ) → 0 as t → t0 . mul(Bt0 , γ ) for all γ ∈ G, 0 To simplify the notation, we shall write Lt for LBt below. ˆ and let b ∈ B Lemma 9.6. Let the notation be as in Proposition 9.5. Let J be a finite subset of G, such that πt (b) ∈ (Bt )J for each t ∈ T . Then the function t → Lt (πt (b)) is continuous on T . Proof. Let s ∈ T . To prove the continuity of t → Lt (πt (b)) at t = s, it suffices to show that for any sequence tn → s one has Ltn (πtn (b)) → Ls (πs (b)). By Remark 8.2 each Bt is separable. Taking restriction to the closure of this sequence, we may assume that B is separable. Since A
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is co-amenable, any unital C ∗ -algebra admitting an ergodic action of A is nuclear [10]. Every separable continuous field of unital nuclear C ∗ -algebras over a compact metric space can be subtrivialized [4, Theorem 3.2]. Thus we can find a unital C ∗ -algebra C and unital embeddings Bt → C for all t ∈ T such that (via identifying each Bt with its image in C) elements in B are exactly those continuous maps T → C whose images at each t are in Bt . Let ϕs ∈ S(Bs ). Extend it to a state of C and let ϕt be the restriction on Bt for each t ∈ T . Then ϕt ∈ S(Bt ) for each t ∈ T and ϕt (πt (c)) → ϕs (πs (c)) as t → s for any c ∈ B. Say, σt πt (b) =
γ
cγ ij (t) ⊗ uij
γ ∈J 1i,j dγ
for all t ∈ T . Then clearly the sections t → cγ ij (t) are in B. Thus πt (b) ∗ ϕt converges to πs (b) ∗ ϕs in AJ as t → s. Since AJ is finite-dimensional, LA is continuous on AJ . Therefore LA (πt (b) ∗ ϕt ) converges to LA (πs (b) ∗ ϕs ) as t → s. Then it follows easily that the function t → Lt (πt (b)) is lower semi-continuous at s. Let ε > 0. Take a sequence t1 , t2 , . . . in T converging to s such that ε + Ltn πtn (b) lim sup Lt πt (b) t→s
for each n 1. Take ϕtn ∈ S(Btn ) for each n 1 such that ε + LA πtn (b) ∗ ϕtn Ltn πtn (b) . Since B is separable, passing to a subsequence if necessary, we may assume that ϕtn ◦ πtn converges to some state ψ of B (in the weak-∗ topology) as n → ∞. Then ψ = ϕs ◦ πs for some ϕs ∈ S(Bs ) by the upper semi-continuity of the field ({Bt }t∈T , B). We have ϕtn (πtn (c)) → ϕs (πs (c)) as n → ∞ for any c ∈ B. As in the second paragraph of the proof, LA (πtn (b) ∗ ϕtn ) converges to LA (πs (b) ∗ ϕs ) as n → ∞. Therefore, 2ε + Ls πs (b) 2ε + LA πs (b) ∗ ϕs lim sup Lt πt (b) . t→t0
Thus the function t → Lt (πt (b)) is upper semi-continuous at s and hence continuous at s.
2
Lemma 9.7. Let V be a finite-dimensional vector space, and let W be a linear subspace of V . Let T be a topological space. Let · t be a norm on V and Lt be a seminorm on V vanishing exactly on W for each t ∈ T such that the functions t → vt and t → Lt (v) are upper semicontinuous and continuous respectively on T for every v ∈ V . Let t0 ∈ T , and let ε > 0. Then ·t
dist H
Et0 (V ), Et (V ) ε
(50)
throughout some neighborhood U of t0 , where Et (V ) = {v ∈ V : Lt (v) 1}. Proof. Via considering V /W we may assume that W = {0}. For any δ > 0, a standard compactness argument shows that
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· t (1 + δ) · t0 , 1 Lt Lt (1 + δ)Lt0 1+δ 0 throughout some neighborhood Uδ of t0 . Then we can find some R > 0 such that · t RLt (·) throughout U1 . Fix δ = R/ε. Let t ∈ U1 ∩ Uδ and v ∈ Et0 (V ). Then v/(1 + δ) ∈ Et (V ), and v − v/(1 + δ) = t
δ vt δvt0 δR = ε. 1+δ
Similarly, for any t ∈ U1 ∩ Uδ and v ∈ Et (V ), we have v/(1 + δ) ∈ Et0 (V ) and v − v/ (1 + δ)t ε. This proves (50). 2 We are ready to prove Proposition 9.5. Proof of Proposition 9.5. As in the first paragraph of the proof of Lemma 9.6 we may assume that there is a unital C ∗ -algebra C containing each Bt as a unital C ∗ -subalgebra and that elements in B are exactly those continuous maps T → C whose images at each t are in Bt . Let ε > 0. Pick ψ ∈ S(A) and J ⊆ Gˆ in Lemma 8.6 for φ being the counit. We may assume that γ0 ∈ J and γ c ∈ J for each γ ∈ J . Then 1Bt ∈ (Bt )J and ((Bt )J )∗ = (Bt )J . By Proposition 8.11 Lt is invariant for all t ∈ T . By Lemma 8.7 we have distCH E(Bt ), E (Bt )J ε
(51)
for all t ∈ T , where E((Bt )J ) := E(Bt ) ∩ (Bt )J . Suppose that limt→t0 mul(Bt , γ ) = mul(Bt0 , γ ) ˆ By Lemma 5.13 there are a neighborhood U of t0 and a linear isomorphism for all γ ∈ G. ϕt : (Bt0 )J → (Bt )J for each t ∈ U such that ϕt0 = id, ϕt ((Bt0 )γ ) = (Bt )γ for each γ ∈ J and t ∈ U , and the map t → ϕt (v) ∈ C is continuous over U for all v ∈ (Bt0 )J . Replacing ϕt by (ϕt + ϕt∗ )(ϕt (1Bt0 ) + ϕt (1Bt0 )∗ )−1 and shrinking U if necessary, we may assume that ϕt is unital and Hermitian throughout U . By Lemma 9.6 we know that { · C ◦ ϕt }t∈U and {Lt ◦ ϕt }t∈U are continuous families of norms and seminorms on (Bt0 )J . By Lemma 9.7, shrinking U if necessary, we have distCH ϕt (X ), E (Bt )J < ε
(52)
throughout U , where X = E((Bt0 )J ). Putting (51) and (52) together, we get distCH E(Bt ), ϕt (X ) < 2ε
(53)
throughout U . Note that
distCH (Y, Z) = distCH (Yσt , Zσt ) for any subsets Y, Z of Bt . Thus
distCH
E(Bt ) σ , ϕt (X ) σ < 2ε t
t
(54)
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throughout U . By Lemma 5.8 we may identify elements of B ⊗ A with the continuous maps T → C ⊗ A whose images at each t are in Bt ⊗ A. Since DR ((Bt0 )J ) is totally bounded and X = DR ((Bt0 )J ) + R · 1Bt0 for any R 2rA by Lemma 8.5, shrinking U if necessary, we may assume that σt (ϕt (x)) − σt0 (x)C⊗A , ϕt (x) − xC < ε for all x ∈ X and t ∈ U . Then
distCH
ϕt (X ) σ , Xσt0 < ε
(55)
t
throughout U . Putting (54) and (55) together, we get
dist e (Bt , Bt0 ) distCH
E(Bt ) σ , E(Bt0 ) σ < 6ε t
throughout U . This finishes the proof of Proposition 9.5.
t0
2
Remark 9.8. Since dist e dist nu and dist nu is the strongest one among the quantum distances defined in [15,16,21,22,35], Proposition 9.5 also holds with dist e replaced by any of them. We are ready to prove Theorem 1.5. Proof of Theorem 1.5. By Proposition 9.4 dist e is a metric on EA(G). By Theorem 5.17 and Proposition 9.5 the topology on EA(G) defined in Definition 4.3 is stronger than that induced by dist e . By Theorem 4.4 the former is compact. Thus these two topologies coincide. 2 Acknowledgments I am grateful to Florin Boca, George Elliott, Marc Rieffel, Shuzhou Wang, and Wei Wu for valuable discussions. I thank Sergey Neshveyev for help on the proof of Lemma 8.6, and thank Magnus Landstad for addressing a question on the topology of EA(G)fm for compact groups. I also would like to thank the referee for several useful comments. References [1] J. Anderson, W. Paschke, The rotation algebra, Houston J. Math. 15 (1) (1989) 1–26. [2] T. Banica, Representations of compact quantum groups and subfactors, J. Reine Angew. Math. 509 (1999) 167–198, math.QA/9804015. [3] E. Bédos, G.J. Murphy, L. Tuset, Co-amenability of compact quantum groups, J. Geom. Phys. 40 (2) (2001) 130– 153, math.OA/0010248. [4] É. Blanchard, Subtriviality of continuous fields of nuclear C ∗ -algebras, J. Reine Angew. Math. 489 (1997) 133–149, math.OA/0012128. [5] F.P. Boca, Ergodic actions of compact matrix pseudogroups on C ∗ -algebras, in: Recent Advances in Operator Algebras, Orléans, 1992, Astérisque 232 (1995) 93–109. [6] J. Cuntz, Simple C ∗ -algebras generated by isometries, Comm. Math. Phys. 57 (2) (1977) 173–185. [7] L. D¸abrowski, G. Landi, M. Paschke, A. Sitarz, The spectral geometry of the equatorial Podle´s sphere, C. R. Math. Acad. Sci. Paris 340 (11) (2005) 819–822, math.QA/0408034. [8] L. D¸abrowski, A. Sitarz, Dirac operator on the standard Podle´s quantum sphere, in: Noncommutative Geometry and Quantum Groups, Warsaw, 2001, in: Banach Center Publ., vol. 61, Polish Acad. Sci., Warsaw, 2003, pp. 49–58. [9] J. Dixmier, C ∗ -Algebras, North-Holland Math. Library, vol. 15, North-Holland, Amsterdam, 1977, translated from the French by Francis Jellett. [10] S. Doplicher, R. Longo, J.E. Roberts, L. Zsidó, A remark on quantum group actions and nuclearity, Rev. Math. Phys. 14 (7–8) (2002) 787–796, math.OA/0204029, dedicated to Professor Huzihiro Araki on the occasion of his 70th birthday.
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Journal of Functional Analysis 256 (2009) 3409–3460 www.elsevier.com/locate/jfa
Harmonic analysis on the Pascal graph J.-F. Quint LAGA, Université Paris 13, 99, avenue Jean-Baptiste Clément, 93430 Villetaneuse, France Received 3 September 2008; accepted 13 January 2009 Available online 29 January 2009 Communicated by P. Delorme
Abstract In this paper, we completely determine the spectral invariants of an auto-similar planar 3-regular graph. Using the same methods, we study the spectral invariants of a natural compactification of this graph. © 2009 Elsevier Inc. All rights reserved. Keywords: Self-similar graphs; Transfer operators; Hyperbolic dynamics
1. Introduction In this whole article, we shall call Pascal graph the infinite, connected and 3-regular graph pictured in Fig. 1. We let Γ denote it. This graph may be constructed in the following way. One writes the Pascal triangle and one erases therein all the even values of binomial coefficients. In this picture, one links each point to its neighbors that have not been erased. One thus obtains a graph in which each point has three neighbors, except the vertex of the triangle which has only two. One then takes two copies of this graph and joins them by their vertices: one thus do get a 3-regular graph. This is the graph Γ . Let ϕ be a complex valued function on Γ . For p in Γ , one sets ϕ(p) = q∼p ϕ(q). The linear operator is self-adjoint with respect to the counting measureon Γ , that is, for any finitely supported functions ϕ and ψ, one has p∈Γ ϕ(p)(ψ(p)) = p∈Γ (ϕ(p))ψ(p). In this article, we will completely determine the spectral invariants of the operator in the space 2 (Γ ) of square-integrable functions on Γ . To set our results, set f : R → R, x → x 2 − x − 3. Let Λ be the Julia set of f , that is, in this case, the set of those x in R for which the sequence (f n (x))n∈N remains bounded. This E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.011
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Fig. 1. The Pascal graph.
is a Cantor set which is √ contained in the interval [−2, 3]. More precisely, if one sets I−2 = √ [−2, 1−2 5 ] and I3 = [ 1+2 5 , 3], for any ε = (εn )n∈N in {−2, 3}N , there exists a unique x in Λ such that, for any n in N, one has f n (x) ∈ Iεn and the thus defined map {−2, 3}N → Λ is a bi-Hölder homeomorphism that conjugates f and the shift mapin {−2, 3}N . For x in Λ, set x ρ(x) = 2x−1 and, if ϕ is a continuous function on Λ, Lρ ϕ(x) = f (y)=x ρ(y)ϕ(y). One easily checks that one has Lρ (1) = 1. Thus, by Ruelle–Perron–Frobenius theorem (see [13, §2.2]), there exists a unique Borel probability measure νρ on Λ such that L∗ρ νρ = νρ . The measure νρ is atom free and f -invariant. Finally, let us note that, if h denotes the function Λ → R, x → 3 − x, one has Lρ h = 2 and, therefore, Λ h dνρ = 2. Let p0 and p0∨ denote the two vertices of the infinite triangles that have been glued to build the graph Γ . Let ϕ0 be the function on Γ with value 1 at p0 , −1 at p0∨ and 0 everywhere else. We have the following Theorem 1.1. The spectrum of in 2 (Γ ) is the union of Λ and the set n∈N f −n (0). The spectral measureof ϕ0 for in 2 (Γ ) is the measure hνρ , the eigenvalues of in 2 (Γ ) are −n the elements of n∈N f (0) and n∈N f −n (−2) and the associate eigenspaces are spanned by finitely supported functions. Finally, the orthogonal complement of the sum of the eigenspaces of in 2 (Γ ) is the cyclic subspace spanned by ϕ0 . The study of the Pascal graph is closely related to the one of the Sierpi´nski graph, pictured in Fig. 2. The spectral theory of the Sierpi´nski graph, and more generally the one of self-similar objects, has been intensively studied, since the original works of Rammal and Toulouse in [14] and Kigami in [10] and [11]. These problems are attacked from a general viewpoint by Sabot in [16], where numerous references may be found; see also Krön [9]. Asymptotics of transition probabilities for the simple random walk on the Sierpi´nski have been computed by Jones in [8] and Grabner and Woess in [3]. The finitely supported eigenfunctions for the Laplace operator on the Sierpi´nski graph and the associate eigenvalues have been determined by Teplyaev in [18]. The Sierpi´nski graph may be seen as the edges graph of the Pascal graph, where two edges are linked when they have a common point. In particular, our description of the eigenvalues associate to finitely supported eigenfunctions on the Pascal graph for the operator is a consequence of the work of Teplyaev. However, the exact determination of the cyclic components of and of its continuous spectrum are new and answer the question asked by Teplyaev in [18, §6.6]. In
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Fig. 2. The Sierpi´nski graph.
Section 14, we shall precisely explain how to connect the study of the Sierpi´nski graph to the one of the Pascal graph. The results cited above rely on the application of the so-called method of Schur complements. This method has recently been succesfully applied to get precise computations for numerous selfsimilar graphs, for example by Grigorchuk and Zuk [7], Bartholdi and Grigorchuk [1], Bartholdi ˘ and Woess [2] and Grigorchuk and Nekrashevych [6]. In [4] and [5], Grigorchuk and Sunik study finite analogues of the Pascal graph. The method we use in this paper is different and rely on the existence of functional equations on the graph. It not only permits very quick computations of spectra, but also the computation of continuous spectral measures of some remarkable vectors, which is the key point in the proof of spectral decomposition theorems. It probably may be applied to self-similar objects satisfying strong homogeneity hypothesis as in [9], but this generalization seems quite difficult to handle. This method allows to describe the spectral theory of other operators connected to the graph Γ . Let Γ0 denote the complete graph with four vertices a, b, c and d. The graph Γ is a covering of Γ0 , as shown by Fig. 3. Let us build, for any integer n, a finite graph in the following way: if the graph Γn has been built, the graph Γn+1 is the graph which is obtained by replacing each point of Γn by a triangle (this process gets more formally detailed in Section 2). We still let denote the operator of summation over the neighbors, acting on functions on Γn . Theorem 1.2. For any nonnegative integers m n, there exist covering maps Γm → Γn and Γ → Γn . The characteristic polynomial of in 2 (Γn ) is (X − 3)(X + 1)3
n−1
p 3 2.3n−1−p p 1+2.3n−1−p f (X) − 2 f p (X) f (X) + 2 .
p=0
Analogous covering maps to those described by this theorem for the Sierpi´nski graph have been recently exhibited by Strichartz in [17]. Close computations of characteristic polynomials ˘ are made by Grigorchuk and Sunik in [4] and [5]. Let us now focus on the initial motivation of this article, which was the study of a phe2 nomenum in dynamical systems. Let X ⊂ (Z/2Z)Z be the three dot system, that is the set of families (pk,l )(k,l)∈Z2 of elements of Z/2Z such that, for any integers k and l, one has pk,l + pk+1,l + pk,l+1 = 0 (in Z/2Z). We equip X with the natural action of Z2 , which is spanned
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Fig. 3. A covering map Γ → Γ0 .
by the maps T : (xk,l ) → (xk+1,l ) and S : (xk,l ) → (xk,l+1 ). This system, which has been introduced by Ledrappier in [12], is an analogue of the natural extension of the angle doubling and tripling on the circle and, as in Furstenberg conjecture, the problem of the classification of the Borel probabilities on X that are Z2 -invariant is open. Let Y denote the set of p in X such that p0,0 = 1. If p is a point of Y , there then exists exactly three elements (k, l) in the set {(1, 0), (0, 1), (−1, 1), (−1, 0), (0, −1), (1, −1)} such that T k S l x belongs to Y . This relation equips the set Y with a 3-regular graph structure (with multiple edges). If p is a point in Y , its connected component Yp for this graph structure is exactly the set of points of the orbit of p under the action of Z2 that belong to Y , that is the equivalence class of p in the equivalence relation induced on Y by the action of Z2 . For any continuous function ϕ on Y , one sets, for any p in Y ,
¯ ϕ(p) =
ϕ T k Sl p .
(k,l)∈{(1,0),(0,1),(−1,1),(−1,0),(0,−1),(1,−1)} T k S l p∈Y
If λ is a Borel probability which is invariant by the action of Z2 on X and such that λ(Y ) > 0 ¯ ∗ μ = 3μ (that is λ is not the Dirac mass at the zero family), the restriction μ of λ to Y satisfies 2 ¯ is a self-adjoint operator in L (Y, μ). and On the origin of this work, we wished to study the homoclinic intersections phenomena in X. Recall that, if φ is a diffeomorphism of a compact manifold M and if p is a hyperbolic fixed point of φ, a homoclinic intersection is an intersection point q of the stable leaf of p and of its unstable leaf. For such a point, one has φ n (q) −−−−−→ p. This notion admits a symbolic n→±∞
dynamics analogue. Let M denote the space (Z/2Z)Z , φ the shift map and p the point of M all of whose components are zero. If q is an element of M all but a finite number of whose components are zero, one has φ n (q) −−−−−→ p. In particular, the point q such that q0 = 1 and all n→±∞
other components are zero possesses this property. In our situation, one checks that there exists a unique element q in X such that one has q0,0 = 1, for any k in Z and l 1, one has qk,l = 0 and, for any k 1 and l −1 − k, one has qk,l = 0. There then exists a graph isomorphism from the graph Γ onto Yq mapping p0 on q: this is the origin of the planar representation of Γ pictured in Fig. 1. We shall now identify Γ with Yq and p0 with q. Note that, if p is the element of X all of whose components are zero, for any integers l > k > 0, one has (T −k S l )n q −−−−−→ p. Let Γ¯ be n→±∞
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the closure of Γ in Y : one can see Γ¯ as a set of pointed planar graphs. Our goal is to determine the structures induced on Γ¯ by the action of Z2 on X. For any p in Γ¯ , let Γp stand for Yp . Note that, if Θp is the edges graph of Γp , the graphs Θp are exactly the graphs that are studied by Teplyaev in [18]. In particular, by [18, §5.4], if Γp does not contain p0 or one of its six images by the natural action of the dihedral group of order 6 on the space X, the spectrum of the operator in 2 (Γp ) is discrete. Every point p in Γ¯ belongs to a unique triangle in Γp . Let a denote the set of elements p such that this triangle is {p, T −1 p, S −1 p}, b the one of points p for which it is of the form {p, Tp, T S −1 p} and c the set of points p for which this triangle is {p, Sp, T −1 Sp}. The set Γ¯ is the disjoint union of a, b and c. Let us denote by θ1 : Γ¯ → {a, b, c} the natural map associate to this partition. We shall say that a function ϕ from Γ¯ into C is 1-triangular if it factors through θ1 . We let E1 denote the space of 1-triangular functions ϕ such that ϕ(a) + ϕ(b) + ϕ(c) = 0: it identifies naturally with C30 = {(s, t, u) ∈ C3 | s + t + u = 0}. We equip C30 with the scalar product which equals one third of the canonical scalar product. and, as above, let Lζ denote the transfer operator associate Let ζ : Λ → R∗+ , x → 13 (x+3)(x−1) 2x−1 with ζ for the dynamics of the polynomial f . As Lζ (1) = 1, there exists a unique Borel probability measure νζ on Λ such that L∗ζ νζ = νζ . Then, if j designs the function Λ → R, x → 13 3−x x+3 , one has Lζ (j ) = 1 and hence Λ j dνζ = 1. Theorem 1.3. For any p in Γ¯ , the set Γp is dense in Γ¯ . There exists a unique Borel probability ¯ ∗ μ = 3μ and the operator ¯ is self-adjoint in L2 (Γ¯ , μ). The spectrum of μ on Γ¯ such that 2 ¯ in L (Γ¯ , μ) is the same as the one of in 2 (Γ ). For any ϕ in E1 , the specthe operator ¯ in L2 (Γ¯ , μ) is ϕ 2 j νζ and the sum of the cyclic spaces spanned by tral measure of ϕ for 2 ¯ in the orthogonal complethe elements of E1 is isometric to L2 (j νζ , C30 ). The spectrum of ment of this subspace is discrete and its eigenvalues are 3, which is simple, and the elements of n∈N f −n (0) ∪ n∈N f −n (−2). The organization of the article is as follows. Sections 2–6 are devoted to the study of the graph Γ . In Section 2, we precisely construct Γ and we establish some elementary properties of its geometry. In Section 3, we determine the spectrum of in 2 (Γ ) and, in Section 4, we prove an essential result towards the computation of the spectral measures of the elements of this space. In Section 5, we describe the structure of the eigenspaces of in 2 (Γ ). Finally, in Section 6, we apply all these preliminary results to the proof of Theorem 1.1. In Section 7, we use the technics developed above to prove Theorem 1.2. In Sections 8–13, we study the space Γ¯ . In Section 8, we precisely describe the geometry of the space Γ¯ and, in Section 9, we introduce some remarkable spaces of locally constant functions ¯ and to the proof of the on this space. Section 10 is devoted to the definition of the operator uniqueness of its harmonic measure. Section 11 extends to Γ¯ the properties proved for Γ in Sec¯ in L2 (Γ¯ , μ). Finally, in Section 13, tions 3 and 4. In Section 12, we study the eigenspaces of we finish the proof of Theorem 1.3. In Section 14, we explain shortly how to transfer our results on the Pascal graph to the Sierpi´nski graph.
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ˆ Fig. 4. Construction of the graph Φ.
2. Geometric preliminaries In all the sequel, we shall call graph a set Φ equipped with a symmetric relation ∼ such that, for any p in Φ, one does not have p ∼ p. For p in Φ, we call neighbors of p the set of elements q in Φ such that p ∼ q. We shall say that Φ is k-regular if all the elements of Φ have the same number k of neighbors. We shall say that Φ is connected if, for any p and q in Φ, there exists a sequence r0 = p, . . . , rn = q of points of Φ such that, for any 1 i n, one has ri−1 ∼ ri . We shall call such a sequence a path from p to q and the integer n the length of this path. If Φ is connected and ϕ is some function on Φ such that, for any points p and q in Φ with p ∼ q, one has ϕ(p) = ϕ(q), then ϕ is constant. We shall say that a subset T of some graph Φ is a triangle if T contains exactly three points p, q and r and one has p ∼ q, q ∼ r and r ∼ p. Let Φ be a 3-regular graph. We let Φˆ denote the set of ordered pairs (p, q) in Φ with p ∼ q equipped with the graph structure for which, if p is a point of Φ, with neighbors q, r and s, the neighbors of (p, q) are (q, p), (p, r) and (p, s). If Φ is connected, Φˆ is connected. Geometrically, Φˆ is the graph one obtains by replacing each point of Φ by a triangle. This process is pictured in Fig. 4. We let Π denote the map Φˆ → Φ, (p, q) → p. Let 2 (Φ) denote the space of functions ϕ : Φ → C such that p∈Φ |ϕ(p)|2 < ∞, equipped with its natural structure √ of a Hilbert space ·,·. If Φ is 3-regular, the map Π induces a bounded ˆ We still let Π denote the adjoint linear map with norm 3, Π ∗ : ϕ → ϕ ◦ Π , 2 (Φ) → 2 (Φ). ∗ 2 2 ˆ ˆ operator of Π : this is the bounded operator (Φ) → (Φ) which, to some function ϕ in 2 (Φ), ∗ associates the function whose value at some point p in Φ is q∼p ϕ(p, q). One has ΠΠ = 3. Let extend the definition of the triangles graph to more general graphs. We shall say that a graph Φ is 3-regular with boundary if every point of Φ has two or three neighbors. In this case, we call the set of points of Φ that have two neighbors the boundary of Φ and we let ∂Φ denote it. If Φ is a 3-regular graph with boundary, we let Φˆ denote the set which is the union of ∂Φ and of the set of ordered pairs (p, q) of elements of Φ with p ∼ q. We equip Φˆ with the graph structure for which, if p is a point of Φ − ∂Φ, with neighbors q, r and s, the neighbors of (p, q) are (q, p), (p, r) and (p, s) and, if p is a point of ∂Φ, with neighbors q and r, the neighbors of p in Φˆ are (p, q) and (p, r) and the neighbors of (p, q) are (q, p), p and (p, r). Thus, Φˆ is itself a 3-regular graph with boundary and there exists a natural bijection between the boundary of Φˆ and the one of Φ (see Fig. 5 for an example when Φ is a triangle). We still let Π denote ˆ and the natural map Φˆ → Φ and Π ∗ and Π the associate bounded operators 2 (Φ) → 2 (Φ) 2 2 ˆ (Φ) → (Φ). Lemma 2.1. Let Φ be a 3-regular graph with boundary. Then, the triangles of Φˆ are exactly the subsets of the form Π −1 (p) where p is some point in Φ. In particular, every point of Φˆ belongs to a unique triangle.
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Fig. 5. The graph Φˆ when Φ is a triangle.
Proof. If p is a point of Φ, the set Π −1 (p) is clearly a triangle. Conversely, pick some point p in Φ − ∂Φ, with neighbors q, r and s. Then, the neighbors of (p, q) are (q, p), (p, r) and (p, s). By definition, as p = q, the point (q, p) can not be a neighbor of (p, r) or of (p, s). Therefore, the only triangle containing (p, q) is Π −1 (p). In the same way, if p belongs to ∂Φ, and if the ˆ p belongs to only one triangle neighbors of p are q and r, as p only has two neighbors in Φ, and, as p = q, no neighbor of (q, p) is also a neighbor of (p, q) and hence (p, q) belongs to only one triangle. 2 If Φ is a graph, we shall say that a bijection σ : Φ → Φ is an automorphism of the graph Φ if, for any p and q in Φ with p ∼ q, one has σ (p) ∼ σ (q). The set of automorphisms of Φ is a subgroup of the permutation group of Φ that is denoted by Aut Φ. If Φ is 3-regular with boundary and if σ is an automorphism of Φ, one has σ (∂Φ) = ∂Φ and there exists a unique automorphism σˆ of Φˆ such that Π σˆ = σ Π . Lemma 2.2. Let Φ be a 3-regular graph with boundary. The map σ → σˆ , Aut Φ → Aut Φˆ is a group isomorphism. Proof. As this map is clearly a monomorphism, it suffices to prove that it is onto. Let thus τ ˆ As τ exchanges triangles of Φ, ˆ by Lemma 2.1, there exists a unique be an automorphism of Φ. bijection σ : Φ → Φ such that Π σˆ = σ Π . Let p and q be points of Φ such that p ∼ q. One then as (p, q) ∼ (q, p), hence τ (p, q) ∼ τ (q, p) and, as these points of Φˆ do not belong to a common triangle, σ (p) = Πτ (p, q) ∼ Πτ (q, p) = σ (q). Therefore, σ is an automorphism of Φ and τ = σˆ , what should be proved. 2 We shall now define an important family of 3-regular graphs with boundary. If a, b and c are three distinct elements, we let T (a, b, c) = T1 (a, b, c) denote the set {a, b, c} equipped with the graph structure for which one has a ∼ b, b ∼ c and c ∼ a and one says that T (a, b, c) is the triangle or the 1-triangle with vertices a, b and c. We consider it as a 3-regular graph with boundary. One then defines by induction a family of 3-regular graphs with boundary by setting, for any n 1, Tn+1 (a, b, c) = Tn (a, b, c). For any n 1, one calls Tn (a, b, c) the n-triangle with vertices a, b and c. One let S(a, b, c) denote the permutation group of the set {a, b, c}. By definition and by Lemma 2.2, one has the following Lemma 2.3. Let a, b and c be three distinct elements. Then, for any n 1, Tn (a, b, c) is a connected 3-regular graph with boundary and ∂Tn (a, b, c) = {a, b, c}. The map that sends an automorphism of Tn (a, b, c) to its restriction to {a, b, c} induces a group isomorphism from Aut Tn (a, b, c) onto S(a, b, c).
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If Φ is a graph and n 1 an integer, we shall say that a subset T of Φ is a n-triangle if there exist points p, q and r in T such that the subset T , endowed with the restriction of the relation ∼, is isomorphic to the graph Tn (p, q, r). By abuse of language, we shall call 0-triangles the points of Φ. Let Φ be a 3-regular graph with boundary. Set Φˆ (0) = Φ, Φˆ (1) = Φˆ and, for any nonnegative ˆ (n) . By induction, for any nonnegative integer n, an automorphism σ of Φ integer n, Φˆ (n+1) = Φ induces a unique automorphism σˆ (n) of Φˆ (n) such that Π n σˆ (n) = σ Π n . By Lemmas 2.1 and 2.2, one immediately deduces, by induction, the following Lemma 2.4. Let Φ be a 3-regular graph with boundary and n be a nonnegative integer. The n-triangles of Φˆ (n) are exactly the subsets of the form Π −n (p) where p is a point of Φ. In particular, every point of Φˆ (n) belongs to a unique n-triangle. The map σ → σ (n) , Aut Φ → Aut Φˆ (n) is a group isomorphism. Corollary 2.5. Let Φ be a 3-regular graph with boundary, n m 1 be integers, p be a point of Φˆ (n) , T be the n-triangle containing p and S be the m-triangle containing p. One has S ⊂ T . If p is a vertex of T and if p does not belong to ∂ Φˆ (n) , the unique neighbor of p in Φˆ (n) that does not belong to the 1-triangle containing p is itself the vertex of some n-triangle of Φˆ (n) . Proof. By Lemma 2.4, one has T = Π −n (Π n p) and S = Π −m (Π m p), hence S ⊂ T . Suppose p belongs to ∂T − ∂ Φˆ (n) . Then, p has a unique neighbor in Φˆ (n) that does not belong to T . Let q be a neighbor of p and R be the n-triangle containing q. If q is not a vertex of R, all the neighbors of q belong to R and hence p ∈ R. As, by Lemma 2.4, p belongs to a unique n-triangle of Φˆ (n) , one has T = R and hence q belongs to T . Therefore, the neighbor of p that does not belong to T is a vertex of the n-triangle which it belongs to. 2 Corollary 2.6. Let n 2 be an integer and a, b and c be three distinct elements. Then, there exist unique elements ab, ba, ac, ca, bc and cb of Tn (a, b, c) such that Tn (a, b, c) is the union of the three (n − 1)-triangles Tn−1 (a, ab, ac), Tn−1 (b, ba, bc) and Tn−1 (c, ca, cb) and that one has ab ∼ ba, ac ∼ ca and bc ∼ cb. Proof. The corollary may be proven directly for n = 2. This case implies the general one by Lemma 2.4. 2 Pick now an element a and two sequences of distinct elements (bn )n1 and (cn )n1 such that, for any n 1, one has bn = a, cn = a and bn = cn . By Corollary 2.6, for any n 1, one can identify Tn (a, bn , cn ) to a subset of Tn+1 (a, bn+1 , cn+1 ) thanks to the unique graph isomorphism sending a to a, bn to abn+1 and cn to acn+1 . One then calls the set n1 Tn (a, bn , cn ), equipped with the graph structure that induces its n-triangle structure on each Tn (a, bn , cn ), n 1, the infinite triangle with vertex a and one let it be denoted by T∞ (a). From the preceding results one deduces the following Lemma 2.7. Let a be an element. The graph T∞ (a) is connected, 3-regular with boundary and ∂T∞ (a) = {a}. If b and c are the two neighbors of a in T∞ (a), there exists a unique isomorphism from T∞ (a) onto T ∞ (a) that sends a to (a, b) and c to (a, c). For any nonnegative integer n, this isomorphism induces a natural bijection between the points of T∞ (a) and the n-triangles of T∞ (a) and every point of T∞ (a) belongs to a unique n-triangle. Finally, T∞ (a) admits a unique nontrivial automorphism; this automorphism is an involution that fixes a and that, for any n 1, exchanges the two vertices of the n-triangle containing a that are different from a.
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In all this article, we fix two distinct elements p0 and p0∨ . One calls Pascal graph the set T∞ (p0 ) ∪ T∞ (p0∨ ) endowed with the graph structure that induces the infinite triangle structure on T∞ (p0 ) and T∞ (p0∨ ) and for which p0 ∼ p0∨ . We let Γ denote the Pascal graph. From Lemma 2.7, one deduces the following Proposition 2.8. The Pascal graph is an infinite, connected and 3-regular graph. If q0 and r0 are the two neighbors of p0 in T∞ (p0 ) and q0∨ and r0∨ the two neighbors of p0∨ in T∞ (p0∨ ), there exists a unique isomorphism from Γ onto Γˆ that sends p0 to (p0 , p0∨ ), p0∨ to (p0∨ , p0 ), q0 to (p0 , q0 ), r0 to (p0 , r0 ), q0∨ to (p0∨ , q0∨ ) and r0∨ to (p0∨ , r0∨ ). For any nonnegative integer n, this isomorphism induces a natural bijection between the points of Γ and the n-triangles of Γ and every point of Γ belongs to a unique n-triangle. A planar representation of the Pascal graph is given in Fig. 1. One shall now identify Γ and Γˆ by the isomorphism described in Proposition 2.8. In particular, from now on, one shall consider Π and Π ∗ as bounded endomorphisms in 2 (Γ ). Let Θ denote the edges graph of Γ . More precisely, Θ is the set of non-ordered pairs {p, q} of elements of Γ with p ∼ q, endowed with the relation for which, if p and q are two neighbor points in Γ , if r and s are the two other neighbors of p and t and u the two other neighbors of q, the neighbors of {p, q} are {p, r}, {p, s}, {q, t} and {q, u}. We call Θ the Sierpi´nski graph. It is an infinite, connected and 4-regular graph. A planar representation of it is given in Fig. 2. If Φ is a k-regular graph, for any function ϕ from Φ into C, one let ϕ denote the function p → q∼p ϕ(q). Then, induces a bounded self-adjoint operator of the space 2 (Φ) with norm k. We call the spectrum of this operator the spectrum of Φ. 3. The spectrum of Γ Let Φ be a 3-regular graph. In this section, we shall study the link between the spectral ˆ Our study is based on the properties of Φ and those of Φ. Lemma 3.1. One has (2 − − 3)Π ∗ = Π ∗ and Π(2 − − 3) = Π . Proof. Let ϕ be a function on Φ, p be a point of Φ and q, r, s be the three neighbors of p. Suppose ϕ(p) = a, ϕ(q) = b, ϕ(r) = c and ϕ(s) = d. Then, one has Π ∗ ϕ(p, q) = a, Π ∗ ϕ(p, q) = 2a + b and 2 Π ∗ ϕ(p, q) = (2b + a) + (2a + c) + (2a + d) = 5a + 2b + c + d. We thus have (2 − − 3)Π ∗ ϕ(p, q) = b + c + d = Π ∗ ϕ(p, q). The second relation is obtained by switching to adjoint operators in the first one. 2 ˆ We shall use elemenWe shall now use Lemma 3.1 to determine the spectrum of in 2 (Φ). tary results from functional analysis. Lemma 3.2. Let E be a Banach space and T be a bounded linear operator in E. Suppose all elements of the spectrum of T have positive real part. Then, if F ⊂ E is a subspace which is stable by T 2 , F is stable by T . Proof. Let 0 < α < β and γ > 0 be such that the spectrum S of T is contained in the interior of the rectangle R = [α, β] + [−γ , γ ]i and U ⊃ R and V be open subsets of C such that the map λ → λ2 induces a biholomorphism from U onto V . There exists a holomorphic function r
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on V such that, for any λ in U , one has r(λ2 ) = λ. As R is simply connected, by Runge theorem, there exists a sequence (rn )n∈N of polynomials in C[X] that converges to r on R 2 . As the spectrum of T 2 is S 2 , which is contained in the interior of R 2 , the sequence rn (T 2 ) then converges to T in the space of endomorphisms of E. For any integer n, rn (T 2 ) stabilizes F , hence T stabilizes F . 2 Lemma 3.3. Let H be a Hilbert space, T be a bounded self-adjoint endomorphism and π be a real polynomial with degree 2. Suppose there exists a closed subspace K of H such that π(T )K ⊂ K and K and T K span H . Then, the image by π of the spectrum of T in H equals the spectrum of π(T ) in K and, if one moreover has T −1 K ∩ K = {0}, the spectrum of T in H is exactly the set of λ in R such that π(λ) belongs to the spectrum of π(T ) in K. Proof. Once π has been written under its canonical form, one can suppose π(X) = X 2 . Let E denote the spectral resolution of T : for any Borel subset B of R, E(B) is a projection of H that commutes with T . Let B be a Borel subset of R such that B = −B. Then, for any Radon measure μ on R, in L2 (μ), the indicator function of B is the limit of a sequence of even polynomials. One hence has E(B)K ⊂ K. The spectrum of T 2 in H is exactly the set of squares of elements of the spectrum of T in H . As T 2 is self-adjoint and K is stable by T 2 , the spectrum of T 2 in K is contained in its spectrum in H , and hence in the set of squares of elements of the spectrum of T . Conversely, suppose there exist elements of the spectrum of T whose square does not belong to the spectrum of T 2 in K. Then, there exists a symmetric open subset V of R such that V contains elements of the spectrum of T in H but that V 2 does not contain elements of the one of T 2 in K. One has E(V )K ⊂ K, but, as V 2 does not contain elements of the spectrum of T 2 in K, E(V )K = 0. Now, as K and T K span H , E(V )K and T E(V )K = E(V )T K span E(V )H . Hence, one has E(V )H = 0, which contradicts the fact that V contains spectral values of T . Therefore, the spectrum of T 2 in K is exactly the set of squares of elements of the spectrum of T in H . Suppose now one has T −1 K ∩ K = {0}. To conclude, it remains to prove that the spectrum of T is symmetric. Suppose this is not the case. Then, one can, after eventually having replaced T by −T , find real numbers 0 < α < β such that U = ]α, β[ contains elements of the spectrum of T but that −U does not contain any. But one then has E(U ) = E(U ∪ (−U )) and hence E(U )K ⊂ K. If L is the image of H by E(U ), one has therefore T 2 (K ∩ L) ⊂ K ∩ L. As the spectrum of the restriction of T to L is contained in R∗+ , one has, by Lemma 3.2, T (K ∩ L) ⊂ K ∩ L and hence, by the hypothesis, K ∩ L = 0. As E(U )K ⊂ K, one thus has E(U ) = 0 on K. As K and T K span H , one has E(U ) = 0, which contradicts the fact that U contains elements of the spectrum of T . Therefore, the spectrum of T is symmetric. The lemma follows. 2 In order to apply these results to spaces of square integrable functions on graphs, we will need results on the geometry of graphs. Let Φ be a connected graph and P and Q be two disjoint subsets of Φ such that Φ = P ∪ Q. We shall say that Φ is split by the partition {P , Q} if any neighbor of an element of P belongs to Q and any neighbor of an element of Q belongs to P . We shall say that Φ is splitable (or bipartite) if there exists a partition of Φ into two subsets that splits it. One easily checks that Φ is splitable if and only if, for any p and q in Φ, the paths joining p to q either all have even length or all have odd length. In particular, if Φ is splitable, the partition {P , Q} that split Φ is unique, two points p and q belonging to the same atom if and only if they may be joined by a path with even length.
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Lemma 3.4. Let Φ be a connected graph and L be the space of functions ϕ on Φ such that, for any p in Φ, ϕ is constant on the neighbors of p. Then, if Φ is not splitable, L equals the space of constant functions. If Φ is split by the partition {P , Q}, L is spanned by constant functions and by the function 1P − 1Q . Proof. Let ϕ be in L, p and q be points of Φ and r0 = p, r1 , . . . , rn = q a path from p to q. For any 1 i n − 1, one has ri−1 ∼ ri ∼ ri+1 , hence ϕ(ri−1 ) = ϕ(ri+1 ) and, if n is even, ϕ(p) = ϕ(q). Therefore, if ϕ(p) = ϕ(q) and if P = {r ∈ Φ | ϕ(r) = ϕ(p)} and Q = {r ∈ Φ | ϕ(r) = ϕ(q)}, the partition {P , Q} splits Φ. The lemma easily follows. 2 We shall use this lemma in the following setting: Lemma 3.5. Let Φ be a connected 3-regular graph. Let ϕ be in 2 (Φ) such that ϕ = 3ϕ. If Φ is infinite, one has ϕ = 0 and, if Φ is finite, ϕ is constant. Let ψ be in 2 (Φ) such that ψ = −3ψ. If Φ is infinite or non-splitable, one has ψ = 0 and, if Φ is finite and split by the partition {P , Q}, ψ is proportional to 1P − 1Q . Proof. As ϕ is in 2 (Φ), the set M = {p ∈ Φ | ϕ(p) = maxΦ ϕ} is not empty. As ϕ = 3ϕ, for any p in M, the neighbors of p all belong to M and, hence, as Φ is connected, M = Φ and ϕ is constant. If Φ is infinite, as ϕ is in 2 (Φ), it is zero. In the same way, suppose ψ = 0 and set P = {p ∈ Φ | ψ(p) = maxΦ ψ} and Q = {q ∈ Φ | ψ(q) = minΦ ψ}. As ψ = −3ψ , one has minΦ ψ = − maxΦ ψ and the neighbors of the points of P belong to Q, whereas the neighbors of the points of Q belong to P . By connectedness, one has P ∪ Q = Φ, the graph Φ is splitable and ψ is proportional to 1P − 1Q . Finally, as ψ is in 2 (Φ), the graph Φ is finite. 2 Recall we let f denote the polynomial x 2 − x − 3. From Lemma 3.1, we deduce the following ˆ Corollary 3.6. Let Φ be a connected 3-regular graph and H be the closed subspace of 2 (Φ) spanned by the image of Π ∗ and by the image of Π ∗ . Then H is stable by and the spectrum of the restriction of to H is, (i) if Φ is infinite, the inverse image by f of the spectrum of in 2 (Φ); (ii) if Φ is finite, but non-splitable, the inverse image by f of the spectrum of in 2 (Φ) deprived from −2; (iii) if Φ is finite and splitable, the inverse image by f of the spectrum of in 2 (Φ) deprived from −2 and 0. Proof. Let K denote the image of Π ∗ . As
√1 Π ∗ 3
induces an isometry from 2 (Φ) onto K,
by Lemma 3.1, the spectrum of f () in K equals the spectrum of in 2 (Φ). We will apply Lemma 3.3 to the space H and the operator . On this purpose, let us study the space −1 K ∩ K. Let L be the space of ϕ in 2 (Φ) such that Π ∗ ϕ belongs to K and let ϕ be in L. If p is a point of ϕ, with neighbors q, r and s, set ϕ(p) = a, ϕ(q) = b, ϕ(r) = c and ϕ(s) = d. Then, one has Π ∗ ϕ(p, q) = 2a + b, Π ∗ ϕ(p, r) = 2a + c and Π ∗ ϕ(p, s) = 2a + d. As Π ∗ ϕ belongs to K, one has b = c = d. Conversely, if ϕ is an element of 2 (Φ) that, for any point p in Φ, is constant on the neighbors of p, ϕ belongs to L. If Φ is infinite, by Lemma 3.4, one has L = {0} and one can apply Lemma 3.3 to H . The spectrum of in H is therefore the inverse image by f of the one of in 2 (Φ). If Φ is finite,
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but non-splitable, by Lemma 3.4, L is the line of constant functions on Φ and we can apply Lemma 3.3 to the orthogonal complement of the space of constant functions in H . We get the result, since f (−2) = f (3) = 3 and, by Lemma 3.5, the constant functions are the only eigenfunctions with eigenvalue 3 in 2 (Φ). Finally, if Φ is finite and split by the partition {P , Q}, by Lemma 3.4, L is spanned by constant functions and by the function 1P − 1Q . Then, Π ∗ (1P − 1Q ) is an eigenvector with eigenvalue 1 in H . One applies Lemma 3.3 to the orthogonal complement of the subspace of H spanned by the constant functions and by Π ∗ (1P − 1Q ). The result follows as f (0) = f (1) = −3 and, still by Lemma 3.5, the eigenfunctions with eigenvalue −3 in 2 (Φ) are the multiples of 1P − 1Q . 2 We now have to determine the spectrum of in the orthogonal complement of H . This is the aim of the following ˆ Lemma 3.7. Let Φ be a connected 3-regular graph and H be the closed subspace of 2 (Φ) ∗ ∗ spanned by the image of Π and by the image of Π . The spectrum of in the orthogonal ˆ is the complement of H is contained in {0, −2}. The eigenspace associate to the value 0 in 2 (Φ) 2 ˆ space of functions ϕ in (Φ) such that Πϕ = 0 and that, for any p and q which are neighbors ˆ is the space in Φ, one has ϕ(p, q) = ϕ(q, p). The eigenspace associate to the value −2 in 2 (Φ) 2 ˆ of functions ϕ in (Φ) such that Πϕ = 0 and that, for any p and q which are neighbors in Φ, one has ϕ(p, q) = −ϕ(q, p). Proof. Let ϕ be orthogonal to H and let p be a point of Φ, with neighbors q, r, s. Set a = ϕ(p, q), b = ϕ(p, r), c = ϕ(q, p) and d = ϕ(r, p). Finally, denote by ψ the indicator function of the set {p} in Φ. As ϕ is orthogonal to Π ∗ ψ and to Π ∗ ψ , one has ϕ(p, s) = −a − b and ϕ(s, p) = −c − d. Thus ϕ(p, q) = c − a and 2 ϕ(p, q) = (a − c) + (d − b) + (−c − d + a + b) = 2a − 2c. Hence, in the orthogonal complement of H , one has 2 + 2 = 0 and, for ϕ in this subspace, one has ϕ = 0 if and only if, for any p an q which are neighbors in Φ, ϕ(p, q) = ϕ(q, p) and ϕ = −2ϕ if and only if, for any p an q which are neighbors in Φ, ϕ(p, q) = −ϕ(q, p). To finish the proof of the lemma, we have to prove that does not have any eigenfunction with eigenvalue 0 or −2 in H . On this purpose, pick some ϕ in H such that ϕ = −2ϕ. By Lemma 3.1, one then has Πϕ = Π(2 − − 3)ϕ = 3Πϕ. If Φ is infinite, by Lemma 3.5, Πϕ is zero. We thus have Πϕ = 0 and Πϕ = −2Πϕ = 0. Therefore, as ϕ belongs to H which is spanned by the images of Π ∗ and of Π ∗ , one has ϕ = 0. If Φ is finite, still by Lemma 3.5, Πϕ is constant. As ϕ is orthogonal to constant functions, one again has Πϕ = 0 and Πϕ = 0, which implies ϕ = 0. If ϕ is now an element of H such that ϕ = 0, one has Πϕ = −3Πϕ. Again, by Lemma 3.5, if Φ is infinite or non-splitable, one has Πϕ = 0 and, hence, ϕ = 0, whereas, if Φ is split by the partition {P , Q}, Πϕ is proportional to 1P − 1Q . But Π ∗ (1P − 1Q ) is an eigenvector with eigenvalue 1 for , hence Πϕ, 1P − 1Q = ϕ, Π ∗ (1P − 1Q ) = 0 and Πϕ = 0, so that ϕ = 0. 2 Recall that, for any nonnegative integer, we let Φˆ (n) denote the graph obtained by replacing each point of Φ by a n-triangle. The space 2 (Φˆ (2) ) contains finitely supported eigenfunctions with eigenvalue −2 and 0, as shown by Fig. 6, where only the nonzero values of the functions have been represented.
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Fig. 6. Eigenfunctions on Φˆ (2) .
We therefore have the following Lemma 3.8. For any n 2, the space 2 (Φˆ (n) ) contains eigenfunctions with eigenvalue −2 and 0. Recall we let Λ denote the Julia set of f . By applying Corollary 3.6 and Lemmas 3.7 and 3.8 to Γ , one gets the following Corollary 3.9. The spectrum of Γ is the union of Λ and of the set
n∈N f
−n (0).
Proof. By Proposition 2.8, Γˆ is isomorphic to Γ . Therefore, by Corollary 3.6 and Lemmas 3.7 and 3.8, the spectrum S of satisfies S = f −1 S ∪ {0, −2}. One easily checks that the set described in the setting of the corollary is the unique compact subset of R verifying this equation. 2 4. The spectral measures of Γ Let still Φ be a connected 3-regular graph. In this section, we will explain how to compute ˆ On this purpose, we use the following the spectral measures of some elements of 2 (Φ). Lemma 4.1. One has ΠΠ ∗ = 6 + and hence, for any ϕ and ψ in 2 (Φ),
Π ∗ ϕ, Π ∗ ψ = 6 ϕ, ψ + ϕ, ψ = 2 Π ∗ ϕ, Π ∗ ψ +
1 2 − − 3 Π ∗ ϕ, Π ∗ ψ . 3
Proof. Let ϕ be in 2 (Φ) and p be a point of Φ, with neighbors q, r, s. Set a = ϕ(p), b = ϕ(q), c = ϕ(r) and d = ϕ(s). Then, one has Π ∗ ϕ(p, q) = 2a + b and hence ΠΠ ∗ ϕ(p) = (2a + b) + (2a + c) + (2a + d) = 6a + (b + c + d), whence the first identity. The second one follows, by applying Lemma 3.1 and the relation ΠΠ ∗ = 3. 2 Let us now study the abstract consequences of this kind of identity. Lemma 4.2. Let H be a Hilbert space, T a bounded self-adjoint endomorphism of H , K a closed subspace of H and π(x) = (x − u)2 + m a real unitary polynomial with degree 2. Suppose one has π(T )K ⊂ K, K and T K span H and there exist real numbers a and b such that, for any v and w in K, one has T v, w = a v, w + b π(T )v, w. Then, for any x = u in the spectrum of T , one has 1+
a − u + bπ(x) 0. x −u
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Proof. Once π has been written under its canonical form, one may suppose one has π(x) = x 2 . Let v be a unitary vector in K. Then, for any real number s, one has 0 T v + sv, T v + sv = T v, T v + 2s T v, v + s 2
= T 2 v, v + 2s a + b T 2 v, v + s 2 . By Lemma 3.3, the squares of the elements of the spectrum of T in H belong to the spectrum of T 2 in K. If x is an element of the spectrum of T , there exists therefore unitary vectors v of K such that T 2 v, v is very close to x 2 . Hence, by the remark above, for any real number s, one has x 2 + 2s(a + bx 2 ) + s 2 0. The discriminant of this polynomial of degree 2 is thus nonpositive, that is one has x 2 − (a + bx 2 )2 0. The lemma follows. 2 Let π(x) = (x − u)2 + m a real unitary polynomial with degree 2. Pick a Borel function θ on R − {u}. Then, if α is a Borel function on R − {u}, one sets, for any y in ]m, ∞[, Lπ,θ α(y) = π(x)=y θ (x)α(x). Let μ be a Borel positive measure on ]m, ∞[. If, for μ-almost all y in ]m, ∞[, θ is nonnegative on the two inverse images of y by π , one let L∗π,θ μ denote the Borel measureν on R − {u} such that, for any nonnegative Borel function α on R − {u}, one has R−{u} α dν = ]m,∞[ Lπ,θ α dμ. We have the following Lemma 4.3. Let H be a Hilbert space, T a bounded self-adjoint endomorphism of H , K a closed subspace of H and π(x) = (x − u)2 + m a real unitary polynomial with degree 2. Suppose one has π(T )K ⊂ K, K and T K span H and there exist real numbers a and b such that, for any v and w in K, one has T v, w = a v, w + b π(T )v, w. Then, for any v in K, if μ is the spectral measure of v for π(T ) and ν its spectral measure for T , if μ(m) = 0, one has ν(u) = 0 and ν = L∗π,θ μ where, for any x = u, one has
a − u + bπ(x) 1 1+ . θ (x) = 2 x −u Proof. Note that, as μ(m) = 0, by Lemma 3.3, the measure μ is concentrated on ]m, ∞[. Moreover, if w is some vector on H with T w = uw, one has π(T )w = mw and, by the hypothesis,
v, w = 0. Thus, one has ν(u) = 0. By Lemma 4.2, the function θ is nonnegative on the spectrum of T , deprived from {0}. Let n be in N. On one hand, one has R−{u}
π(x)n dν(x) = π(T )n v, v =
y n dμ(y).
]m,∞[
On the other hand, for any x = u, one has θ (x) + θ (2u − x) = 1 and hence, for any y in ]m, ∞[, Lπ,θ π n (y) = y n . Thus, one has R−{u} π n dν = ]m,∞[ Lπ,θ π n dμ. In the same way, for any x in R, set α(x) = xπ(x)n . On one hand, one then has
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α(x) dν(x) = T π(T )n v, v = a π(T )n v, v + b π(T )n+1 v, w
R−{u}
=a ]m,∞[
y n dμ(y) + b
y n+1 dμ(y).
]m,∞[
On the other hand, for any x = u, as (2u − x) − u = u − x, one has θ (x)α(x) + θ (2u − x)α(2u − x)
1 a − u + bπ(x) 1 x + (2u − x) + x − (2u − x) π(x)n = 2 2 x−u = aπ(x)n + bπ(x)n+1 and hence, for any y in ]m, ∞[, Lπ,θ α(y) = ay n + by n+1 . Again, one has R−{u} α dν = ]m,∞[ Lπ,θ α dμ. Therefore, for any polynomial α, one has R−{u} α dν = ]m,∞[ Lπ,θ α dμ. In particular, the positive measure L∗π,θ μ is finite and hence, for any compactly supported continu ous function α on R − {u}, one still has R−{u} α dν = ]m,∞[ Lπ,θ α dμ, so that ν = L∗π,θ μ. 2 By applying Lemmas 4.1 and 4.3, one gets the following Corollary 4.4. Let ϕ be in 2 (Φ), μ be the spectral measure of ϕ for in 2 (Φ) and ν be the ˆ Then, one has ν( 1 ) = 0 and, if, for any x = 1 , one sets spectral measure of Π ∗ ϕ for in 2 (Φ). 2 2 ∗ μ. θ (x) = x(x+2) , one has ν = L f,θ 2x−1 Proof. The minimal value on R of the polynomial f is f ( 12 ) = − 13 4 < −3 − 2 . Thus, one has μ(− 13 ) = 0 and the corollary follows from Lemmas 4.1 and 4.3 by an elementary 4 computation. 2 5. Eigenfunctions in 2 (Γ ) In this section, we shall complete the informations given by Lemma 3.7 by describing more eigenvalues −2 and 0. We shall extend these precisely the eigenspaces of in 2 (Γ ) for the results to the eigenvalues in n∈N f −n (−2) and n∈N f −n (0) using the following ˆ spanned by the Lemma 5.1. Let Φ be a 3-regular graph and H be the closed subspace of 2 (Φ) ∗ ∗ image of Π and by the one of Π . Then, for any x in R − {0, −2}, x is an eigenvalue of in H if and only if y = f (x) is an eigenvalue of in 2 (Φ). In this case, the map Rx which sends an eigenfunction ϕ with eigenvalue y in 2 (Φ) to (x − 1)Π ∗ ϕ + Π ∗ ϕ induces an isomorphism between the eigenspace associate to the eigenvalue y in 2 (Φ) and the eigenspace associate to the eigenvalue x in H and, for any ϕ, one has Rx ϕ 22 = x(x + 2)(2x − 1) ϕ 22 . Proof. Let ψ = 0 be in H such that ψ = xψ . As ψ is in H , one has Πψ = 0 or Πψ = 0. As Πψ = xΠψ , one has Πψ = 0. By Lemma 3.1, one has Πψ = yΠψ , hence y is an eigen1 value of in 2 (Φ). In particular, as f ( 12 ) = − 13 4 < −3 − 2 , one has x = 2 . Conversely, 2 if ϕ is an element of (Φ) such that ϕ = yϕ, one has, by Lemma 3.1,
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Rx ϕ = (x − 1)Π ∗ ϕ + Π ∗ ϕ = (x − 1)Π ∗ ϕ + Π ∗ ϕ + ( + 3)Π ∗ ϕ = xΠ ∗ ϕ + x 2 − x Π ∗ ϕ = xRx ϕ. Now, by Lemma 4.1, one has ΠΠ ∗ = 6 + , hence, if ϕ = yϕ, one has, by a direct computation, ΠRx ϕ = x(x + 2)ϕ and, as we have supposed x(x + 2) = 0, Rx is one-to-one and closed. It remains to prove that Rx is onto. In this aim, pick ψ in H such that ψ = xψ but ψ is orthogonal to the image of Rx . For any ϕ in 2 (Φ) such that ϕ = yϕ, one has
ψ, Rx ϕ = (x − 1) Πψ, ϕ + Πψ, ϕ = (2x − 1) Πψ, ϕ and hence, as x = 12 , Πψ, ϕ = 0. As Πψ = yΠψ , one has Πψ = 0. As ψ is in H , one has ψ = 0. The operator Rx is thus an isomorphism. The computation of the norm is then direct, by using Lemmas 3.1 and 4.1. 2 Let us begin by looking at the eigenvalues in n∈N f −n (0). Let n 1 be an integer. Recall that, by corollary 2.5, if T is a n-triangle of Γ and if p is a vertex of T , the neighbor of p that does not belong to T is the vertex of some n-triangle. We shall call the edges linking vertices of n-triangles exterior edges to n-triangles. We let Θn denote the set of edges which are exterior to n-triangles and we endow it with the graph structure for which two edges are neighbors if two of their end points are vertices of the same n-triangle. One easily checks that the graph Θn is naturally isomorphic to the Sierpi´nski graph, introduced in the end of Section 2. We shall from now on identify Θn and Θ. If ϕ is a function on Γ which is constant on edges which are exterior to n-triangles, we let Pn ϕ denote the function on Θ whose value at one point of Θ is the value of ϕ on the associate edge which is exterior to n-triangles. Finally, let us recall that, as Θ is 4-regular, the norm of in 2 (Θ) is 4. By Lemma 3.7, the eigenfunctions with eigenvalue 0 are constant on the edges which are exterior to 1-triangles. We have the following Lemma 5.2. The map P2 induces a Banach spaces isomorphism from the eigenspace of 2 (Γ ) associate to the eigenvalue 0 onto 2 (Θ). Let Q0 denote its inverse. For any ψ in 2 (Θ), one has Q0 ψ 22 (Γ ) = 3 ψ 22 (Θ) − 12 ψ, ψ2 (Θ) . Proof. By using the characterization of Lemma 3.7, one easily checks that, given three values a, b and c on the vertices of some 2-triangle, an eigenfunction with eigenvalue 0 taking these values at the three vertices must take in the interior of the triangle the values that are described in Fig. 7. Recall that one has identified the graphs Θ and Θ2 . For any function ψ on Θ, let Q0 ψ denote the function on Γ that, on each edge which is exterior to 2-triangles, is constant, with value the value of ψ at the point of Θ associate to this edge, and whose values in the interior of the 2triangles are those described by Fig. 7. By an elementary computation, for any real numbers a, b and c, the sum of the squares of the values described in Fig. 7 is 3 2 a + b2 + c2 − ab − ac − bc 2 1 1 1 = 3a 2 − ab − ac + 3b2 − ab − bc + 3c2 − ac − bc , 2 2 2
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Fig. 7. The eigenfunctions with eigenvalue 0.
so that, for any function ψ on Θ, one has Q0 ψ 22 (Γ ) = 3 ψ 22 (Θ) − 12 ψ, ψ2 (Θ) . As −4 ψ 22 (Θ) ψ, ψ2 (Θ) 4 ψ 22 (Θ) , the function ψ belongs to 2 (Θ) if and only if Q0 ψ belongs to 2 (Γ ). The lemma follows.
2
From Lemmas deduce a description of the eigenspaces associate to 5.1 and 5.2, we shall the elements of n∈N f −n (0). For x in n∈N f −n (0), let n(x) denote the integer n such that f n (x) = 0 and κ(x) =
n(x)−1 k=0
f k (x)(2f k (x) − 1) . f k (x) + 2
We have the following Proposition 5.3. Let x be in n∈N f −n (0). The eigenfunctions with eigenvalue x in 2 (Γ ) are constant on edges which are exterior to (n(x) + 1)-triangles in Γ . The map Pn(x)+2 induces a Banach spaces isomorphism from the eigenspace of 2 (Γ ) associate to the eigenvalue x onto 2 (Θ). Let Qx denote its inverse. Then, for any ψ in 2 (Θ), one has Qx ψ 22 (Γ ) = κ(x)(3 ψ 22 (Θ) − 12 ψ, ψ2 (Θ) ). Proof. We shall prove this result by induction on n(x). The case n(x) = 0 has been dealt with in Lemma 5.2. Suppose the lemma has been proved for n(y) with y = f (x). Pick ϕ in 2 (Γ ) such that ϕ = xϕ. Then, as n = n(x) 1, one has x ∈ / {−2, 0} and hence, by Lemma 3.7, ϕ belongs to H . By Lemma 5.1, one thus has ϕ = Rx ψ , for some function ψ such that ψ = yψ . By induction, ψ is constant on edges that exterior to n-triangles. Let p be a vertex of some (n + 1)triangle in Γ and q its exterior neighbor. The points Πp and Πq are vertices of some n-triangle in Γ . One hence has Π ∗ ψ(p) = ψ(Πp) = ψ(Πq) = Π ∗ ψ(q) and Π ∗ ψ(p) = 2ψ(Πp)) + ψ(Πq)) = 3Π ∗ ψ(p). Thus, ϕ(p) = Rx ψ(p) = (x + 2)ψ(p) = ϕ(q): the function ψ is constant on edges that are exterior to (n+1)-triangles and Pn+2 ϕ = (x +2)Pn+1 ψ . As, by induction, Pn+1 induces an isomorphism from the eigenspace associate to the value y onto 2 (Θ), by Lemma 5.1, Pn+2 induces an isomorphism from the eigenspace associate to the value y onto 2 (Θ). The norm computation now follows from the induction and the formula Pn+2 Rx = (x + 2)Pn+1 . 2 Corollary 5.4. For any x in n∈N f −n (0), the eigenspace associate to x in 2 (Γ ) has infinite dimension and is spanned by finitely supported functions.
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Fig. 8. The eigenfunctions with eigenvalue −2.
For the elements of n∈N f −n (−2), there is no analogue of Proposition 5.3. However, we will extend Corollary 5.4. Let us begin by dealing with the eigenvalue −2. Recall we let p0 and p0∨ denote the vertices of the two infinite triangles in Γ . Lemma 5.5. Let ϕ be an eigenfunction with eigenvalue −2 in 2 (Γ ). Then, for any n 1, the sum of the values of ϕ on the vertices of each n-triangle of Γ is zero and ϕ(p0 ) = ϕ(p0∨ ) = 0. The eigenspace associate to the eigenvalue −2 has infinite dimension and is spanned by finitely supported functions. Proof. An immediate computation using Lemma 3.7 shows that the values of ϕ on some 2triangle satisfy the rules described by Fig. 8. In particular, the sum of these values on the vertices of each 2-triangle is zero. By induction, using Corollary 2.6, it follows that, for any n 1, the sum of these values on the vertices of each n-triangle is zero. Let then, for any n 1, pn and qn denote the two other vertices of the n-triangle with vertex p0 . As ϕ is square integrable, one has ϕ(pn ) −→ 0 and ϕ(qn ) −→ 0. Thus ϕ(p0 ) = 0 and, in the same way, ϕ(p0∨ ) = 0. In particular, n→∞
n→∞
for any n 1, ϕ(pn ) + ϕ(qn ) = 0. Let us now prove that ϕ is the limit of some sequence of finitely supported functions. Let still, as in Section 2, T∞ (p0 ) denote the infinite triangle with vertex p0 . Then, as ϕ(p0 ) = 0, ϕ1T∞ (p0 ) is still an eigenvector with eigenvalue −2 and one can suppose ϕ = 0 on T∞ (p0∨ ). For any nonnegative integer n, let ϕn denote the function on Γ that is zero outside the (n + 1)-triangle Tn+1 (p0 , pn+1 , qn+1 ), that equals ϕn on the n-triangle Tn (p0 , pn , qn ) and that is invariant by the action of the elements of signature 1 of the group S(p0 , pn+1 , qn+1 ) on the (n + 1)-triangle Tn+1 (p0 , pn+1 , qn+1 ). In view of Corollary 2.6, the values of ϕn on the vertices of n-triangles are those described by Fig. 9. Then, by Lemma 3.7, for any n 1, ϕn is an eigenvector with eigenvalue −2 and one has ϕn 22 3 ϕ 22 . The sequence (ϕn ) converges weakly to ϕ in 2 (Γ ). The function ϕ belongs to the weak closure of the subspace spanned by finitely supported eigenfunctions with eigenvalue −2 and hence to its strong closure. Finally, the space of eigenfunctions with eigenvalue −2 has infinite dimension since, by Fig. 6, every 2-triangle contains the support of some eigenfunction with eigenvalue −2. 2 For x in n∈N f −n (−2), let n(x) denote the integer n such that f n (x) = −2. By an induction based on Lemma 5.1, one can deduce from Lemma 5.5 the following Corollary 5.6. Let x be in n∈N f −n (−2) and ϕ be an eigenfunction with eigenvalue x in 2 (Γ ). Then, the values of ϕ on the edges that are exterior to (n(x) + 1)-triangles are opposite, for any n n(x) + 1 the sum of the values of ϕ on the vertices of each n-triangle is zero and ϕ(p0 ) = ϕ(p0∨ ) = 0. The eigenspace associate to the eigenvalue x has infinite dimension and is spanned by finitely supported functions.
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Fig. 9. The function ϕn .
6. Spectral decomposition of 2 (Γ ) Let ϕ0 denote the function on Γ that has value 1 at p0 , −1 at p0∨ and 0 everywhere else. 2 (Γ ) is the direct sum of the eigenspaces associate to the In this paragraph, we will prove that −n −n elements of n∈N f (−2) ∪ n∈N f (0) and the cyclic subspace spanned by ϕ0 . Let us begin by studying the latter. By a direct computation, one gets the following Lemma 6.1. One has Π ∗ ϕ0 = ( + 2)ϕ0 . This relation and Corollary 4.4 will allow the determination of the spectral measure of ϕ0 . On this purpose, let us recall the properties of transfer operators we will have to use: they follow from the version of Ruelle–Perron–Frobenius theorem given in [13, §2.2]. If κ is a Borel function on Λ, one let Lκ stand for Lf,κ . Lemma 6.2. Let κ : Λ → R∗+ be a Hölder continuous function. Equip the space C 0 (Λ) with the uniform convergence topology. Then, if λκ > 0 is the spectral radius of the operator Lκ in C 0 (Λ), there exists a unique Borel probability νκ on Λ and a unique continuous positive function lκ on Λ such that one has Lκ lκ = λκ lκ , L∗κ νκ = λκ νκ and Λ lκ dνκ = 1. The spectral radius of Lκ in the space of functions with zero integral with respect to νκ is < λκ and, in particular, for any g in C 0 (Λ), the sequence ( λ1n Lnκ (g))n∈N uniformly converges to Λ g dνκ . The measure νκ is atom κ free and its support is Λ. x For any x in R, set h(x) = 3 − x, k(x) = x + 2 and, for x = 12 , ρ(x) = 2x−1 . One has h ◦ f = hk. From Lemma 6.1, we deduce, thanks to Corollary 4.4, the following
Corollary 6.3. Let νρ be the unique Borel probability on Λ such that L∗ρ νρ = νρ . The spectral measure of ϕ0 is hνρ . Proof. Let μ be the spectral measure of ϕ0 . For x = 12 , set θ (x) = x(x+2) 2x−1 = k(x)ρ(x). The 2 spectral measure of ( + 2)ϕ0 is k μ. Therefore, by Corollary 4.4 and Lemma 6.1, one has μ( 12 ) = 0 and k 2 μ = L∗θ μ. Now, by Lemma 5.5, if ϕ is an eigenfunction with eigenvalue −2 in 2 (Γ ), one has ϕ(p0 ) = ϕ(p0∨ ) = 0 and hence ϕ, ϕ0 = 0, so that μ(−2) = 0. Therefore, one has L∗1 μ = μ. kρ
Besides, by Lemma 3.5, one has μ(3) = 0. Therefore, as h ◦ f = hk, one has L∗ρ ( h1 μ) = 1 ∗ 1 h L 1 (μ) = h μ. kρ
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The Borel measure h1 μ on R is concentrated on the spectrum of . Now, by Proposition 5.3, for any x in n∈N f −n (0) and ϕ in 2 (Γ ) such that ϕ = xϕ, one has ϕ(p0 ) = ϕ(p0∨ ) and hence
ϕ, ϕ0 = 0. Therefore, one has μ( n∈N f −n (0)) = 0 and, by Corollary 3.9, μ is concentrated on Λ. The function ρ is Hölder continuous and positive on Λ and one has Lρ (1) = 1 on Λ. By Lemma 6.2, there exists a unique Borel probability measure νρ on Λ such that L∗ρ (νρ ) = νρ and, for any continuous function g on Λ, the sequence (Lnρ (g))n∈N uniformly converges to the constant function with value Λ g dνρ . Let us prove that the positive Borel measure h1 μ is finite and hence proportional to νρ . Pick some continuous nonnegative function g on Λ, which is zero in the neighborhood of 3 and such that one has 0 < Λ h1 g dμ < ∞. There exists an integer n and a real number ε > 0 such that, for any x in Λ, one has Lnρ (g)(x) ε. As Λ h1 g dμ = 1 n 1 1 of νρ . Now, one Λ h Lρ (g) dμ, one has Λ h dμ < ∞. Therefore, the measure h μ is a multiple 2 has μ(Λ) = ϕ0 2 = 2 and, by a direct computation, Lρ (h) = 2, so that Λ h dνρ = 2. We hence do have μ = hνρ . 2 For any polynomial p in C[X], let pˆ denote the function p()ϕ0 on Γ . By definition, the map g → gˆ extends to an isometry from L2 (hνρ ) onto the cyclic subspace Φ of 2 (Γ ) spanned by ϕ0 . Let l denote the function x → x on Λ. On has the following Proposition 6.4. The subspace Φ is stable by the operators , Π and Π ∗ . For any g in L2 (hνρ ), one has gˆ = lg, Π gˆ = L ρ g, Π ∗ gˆ = k(g ◦ f ). Proof. By definition, Φ is stable by and one has the formula concerning . By a direct computation, one proves that Lρ (l) = 1. Let n be in N. One has Lρ (f n ) = l n Lρ (1) = l n and Lρ (f n l) = l n Lρ (l) = l n . Now, by Lemma 3.1, one has Π(f ()n ϕ0 ) = n Πϕ0 and Π(f ()n ϕ0 ) = n Πϕ0 and hence, as Πϕ0 = Πϕ0 = ϕ0 , the subspace Φ is stable by Π and, for any p in C[X], Π pˆ = L ρ p. Finally, by convexity, for any measurable function g on Λ, one has |Lρ (g)|2 Lρ (|g|2 ), so that, for g in L2 (hνρ ), one has
Lρ (g)2 h dνρ
Λ
Lρ |g|2 h dνρ =
Λ
=
|g|2 kh dνρ 5
Λ
|g|2 (h ◦ f ) dνρ Λ
|g|2 h dνρ , Λ
hence the operator Lρ is continuous in L2 (hνρ ) and, by density, for any g in L2 (hνρ ), Π gˆ = L ρ g. ∗ Finally, by Lemmas 3.1 and 6.1, for any polynomial p in C[X], one has Π (p()ϕ0 ) = p(f ())Π ∗ ϕ0 = p(f ())( + 2)ϕ0 . Therefore, the subspace Φ is stable par Π ∗ and, for any p in C[X], Π ∗ pˆ = k(p ◦ f ). Now, for any g in L2 (hνρ ), one has
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k(g ◦ f )2 h dνρ =
Λ
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k|g ◦ f |2 (h ◦ f ) dνρ Λ
=
Lρ (k)|g|2 h dνρ = 3
Λ
and hence, by density, for any g in L2 (hνρ ), Π ∗ gˆ = k(g ◦ f ).
|g|2 h dνρ Λ
2
In order to determine the complete spectral structure of , we will analyze other remarkable elements of 2 (Γ ). Let us begin by letting ψ0 denote the function on Γ that takes the value 1 at p0 and p0∨ and 0 everywhere else. We have the following Lemma 6.5. One has Π ∗ ψ0 = ψ0 . From this, we deduce the following ψ0 is conCorollary 6.6. The spectral measure of ψ0 is discrete. More precisely, the function tained in the direct sum of the eigenspaces of associate to the elements of n∈N f −n (0). Proof. Let μ be the restriction of the spectral measure of ψ0 to Λ. By Corollary 3.9, it suffices to prove that μ = 0. (x+2) 1 and σ (x) = x(2x−1) . The function σ is Hölder continuous For x ∈ / {0, 12 }, set τ (x) = x(2x−1) and positive on Λ. Proceeding as in the proof of Corollary 6.3, one proves that, as 0 ∈ / Λ, one has, by Lemma 6.5, L∗τ μ = μ. Now, as h ◦ f = hk, for any x in R, for any integer n, one has Lnτ (h) = hLnσ (1). Let λσ denote the spectral radius of Lσ and νσ its equilibrium state, as in 1 Lemma 6.2. By a direct computation one shows that, for any x in Λ, one has Lσ (1)(x) = x+3 . In particular, for x = −2, one has Lσ (1)(x) < 1, so that λσ = Λ Lσ (1) dνσ < 1 and hence n the sequence (Lnσ (1))n∈N uniformly converges to 0 on Λ. Therefore, the sequence (Lτ (h))n∈N uniformly converges to 0 on Λ and one has Λ h dμ = 0, so that μ(Λ − {3}) = 0. By Lemma 3.5, one has μ(3) = 0 and hence μ = 0. 2 that, by using Lemma 6.5, one could establish a formula giving, for any x in Note −n (0), the value of the norm of the projection of ψ0 on the space of eigenfunctions n∈N f with eigenvalue x. Let us now study the spectral invariants of a last element of 2 (Γ ). For this purpose, let q0 and r0 denote the two neighbors of p0 that are different from p0∨ and χ0 the function on Γ that takes the value 1 at q0 , −1 at r0 and 0 everywhere else. In the same way, one let q0∨ and r0∨ denote the two neighbors of p0∨ that are different from p0 and χ0∨ the function on Γ that takes the value 1 at q0 , −1 at r0 and 0 everywhere else. One could again note that one has Π ∗ χ0 = (2 + 2)χ0 and study the spectral measure of χ0 by using the same methods as in Corollaries 6.3 and 6.6. We shall follow another approach, analogous to the one of the proof of Lemma 5.5. By Lemma 2.7, there exists a unique automorphism ι of the graph Γ such that ι(q0 ) = r0 and ι(q0∨ ) = r0∨ and ι is an involution. Let H denote the space of elements ϕ in 2 (Γ ) such that ι(ϕ) = −ϕ and K (resp. K ∨ ) the subspace of H consisting of those elements which are zero on the infinite triangle with vertex p0∨ (resp. p0 ). One has H = K ⊕ K ∨ , χ0 ∈ K, χ0∨ ∈ K ∨ and the subspaces K and K ∨ are stable by the endomorphisms , Π and Π ∗ . For any n 1, let
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Tn denote the n-triangle with vertex p0 and Tn∨ the n-triangle with vertex p0∨ . The permutation groups S(∂Tn ) and S(∂Tn∨ ) act on the triangles Tn and Tn∨ . One let Kn (resp. Kn∨ ) denote the space of functions ϕ on Tn (resp. Tn∨ ) such that, for any s in S(∂Tn ) (resp. in S(∂Tn∨ )), one has sϕ = ε(s)ϕ, where ε is the signature morphism. One identifies Kn and Kn∨ with finite dimensional subspaces of K and K ∨ . One then has Kn ⊂ Kn , Π ∗ Kn ⊂ Kn+1 and, if n 2, ΠKn ⊂ Kn−1 , and the analogous identities in K ∨ . We have the following Lemma 6.7. The spaces K and K ∨ are topologically spanned by the sets ∨ n1 Kn .
n1 Kn
and
Proof. Let ϕ be a function in K. For any integer√n 2, one let ϕn denote the unique element of Kn that equals ϕ on Tn−1 . One has ϕn 2 3 ϕ 2 . Then, for any ϕ, the sequence (ϕn ) weakly converges to ϕ in 2 (Γ ). Hence, the set n1 Kn is weakly dense in H and the vector subspace it spans is therefore strongly dense. The result for K ∨ follows, by symmetry. 2 Corollary 6.8. The spectrum of in H is discrete. Its eigenvalues are exactly the elements of the set n∈N f −n (−2) ∪ n∈N f −n (0). Proof. As, for any n, the subspaces Kn and Kn∨ are stable by and finite dimensional, the fact that the spectrum of in H is discrete immediately follows from Lemma 6.7. The exact determination of the eigenvalues is obtained as in Section 3. A formula for the characteristic polynomial of in Kn is given in Proposition 13.6. 2 We can now finish the proof of Theorem 1.1 with the following Proposition 6.9. Let Φ ⊥ be the orthogonal complement in 2 (Γ ) of the cyclic space Φ spanned the spectrum of in Φ ⊥ is discrete and the set of its eigenvalues is exactly by ϕ0 . Then, −n −n (−2) ∪ n∈N f (0). n∈N f Before to proving this proposition, let us establish a preliminary result. For ϕ and ψ in 2 (Γ ), let μϕ,ψ denote the unique complex Borel measure on R such that, for any polynomial p in C[X], one has R p dμϕ,ψ = p()ϕ, ψ. One has the following Lemma 6.10. For any ϕ and ψ in 2 (Γ ), one has μΠϕ,ψ = f∗ μϕ,Π ∗ ψ . Proof. For any p in C[X], one has, by Lemma 3.1,
∗ p dμΠϕ,ψ = p()Πϕ, ψ = p f () ϕ, Π ψ = (p ◦ f ) dμϕ,Π ∗ ψ . R
2
R
Proof 6.9. By Corollaries 5.4 and 5.6, the eigenspaces associate to the elements of Proposition of n∈N f −n (−2) ∪ n∈N f −n (0) are nonzero. Let P denote the orthogonal projection onto Φ ⊥ in 2 (Γ ). By Proposition 6.4, the operator P commutes with , Π and Π ∗ . To prove the 2 proposition, it suffices to establish that, for any finitely ϕ, for supported any ψ−nin (Γ ), the −n measure μP ϕ,ψ is atomic and concentrated on the set n∈N f (−2) ∪ n∈N f (0).
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Let still q0 , r0 , q0∨ and r0∨ be the neighbors of p0 and p0∨ and, for any integer n, Tn be the n-triangle containing p0 and Tn∨ be the n-triangle containing p0∨ . One let Ln denote the space of functions on Γ whose support is contained in the union of Tn , Tn∨ and the neighbors of the vertices of Tn and Tn∨ . One has, for n 1, ΠLn ⊂ Ln−1 and ΠLn ⊂ Ln−1 . Let us show, by induction on n, that, for any function ϕ in Ln , for any ψ in 2 (Γ ), the measure μP ϕ,ψ is atomic and concentrated on the set n∈N f −n (−2) ∪ n∈N f −n (0). For n = 0, L0 is the space of functions which are zero outside the set {p0 , q0 , r0 , p0∨ , q0∨ , r0∨ }. One easily checks that this space is spanned by the functions ϕ0 , ϕ0 , ψ0 , ψ0 , χ0 and χ0∨ . In this case, the description of the spectral measures follows immediately from Corollaries 6.3, 6.6 and 6.8. If the result is true for some integer n, let us pick some ϕ in Ln+1 . Then, the functions Πϕ and Πϕ are in Ln and, by induction, for any ψ in 2 (Γ ), the measures μΠP ϕ,ψ = and μP Πϕ,ψ −n −n μΠP ϕ,ψ = μP Πϕ,ψ are atomic and concentrated on the set n∈N f (−2) ∪ n∈N f (0). By Lemma 6.10, themeasures μP ϕ,Π ∗ ψ and μP ϕ,Π ∗ ψ = μP ϕ,Π ∗ ψ are thus atomic and con−n centrated on the set n1 f (−2) ∪ n1 f −n (0). Now, by Lemma 3.7, the spectrum of in the orthogonal complement of the subspace of 2 (Γ ) spanned by the image of Π ∗ and by the 2 one of Π ∗ equals {−2, for 0}. Therefore, any ψ−nin (Γ ), the measure μP ϕ,ψ is atomic and −n concentrated on the set n∈N f (−2) ∪ n∈N f (0). The result follows. 2 7. Finite quotients of Γ In this section, we apply the previous methods to the description of the spectrum of certain finite graphs which are strongly related to Γ . Let Φ and Ψ be graphs. We shall say that a map : Φ → Ψ is a covering map if, for any p in Φ, the map induces a bijection from the set of neighbors of p onto the set of neighbors of (p). The composition of two covering maps is a covering map. If Φ and Ψ are 3-regular graphs and if : Φ → Ψ is a covering map, there exists a unique covering map ˆ : Φˆ → Ψˆ such that Π ˆ = Π . Conversely, proceeding as in Lemma 2.2, one proves that every covering map Φˆ → Ψˆ is of this form. Let us fix four distinct elements a, b, c and d. Let Γ0 denote the graph obtained by endowing the set {a, b, c, d} with the relation that links every pair of distinct points: this is a 3-regular graph. Its automorphism group equals the permutation group S(a, b, c, d) of the set {a, b, c, d}. Lemma 7.1. Let Φ be a 3-regular graph and : Φ → Γ0 be a covering map. Then, the map ˜ : Φˆ → Γ0 , (p, q) → (q) is a covering map. The map → ˜ is a S(a, b, c, d)-equivariant bijection from the set of covering maps Φ → Γ0 onto the space of covering maps Φˆ → Γ0 . The construction of the covering ˜ is pictured in Fig. 10. Proof. Let p be a point of Φ and let q, r and s be the neighbors of p. After an eventual permutation of the elements of {a, b, c, d}, suppose one has (p) = a, (q) = b, (r) = c and (s) = d. Then, one has ˜ (p, q) = b, ˜ (q, p) = a, ˜ (p, r) = c and ˜ (p, s) = d and hence ˜ is really a covering map. Conversely, let ω : Φˆ → Γ0 be a covering map. Let still p be a point of Φ, with neighbors q, r and s. Again, after an eventual permutation, suppose one has ω(p, q) = b, ω(p, r) = c and ω(p, s) = d. Then, as ω is a covering map, one necessarily has ω(q, p) = ω(r, p) = ω(s, p) = a.
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Fig. 10. Construction of the covering map ˜.
Thus, there exists a map : Φ → Γ0 such that, for any p and q in Φ with p ∼ q, one has ω(q, p) = (p). By construction, is a covering map and one has ˜ = ω. 2 For any nonnegative integer n, let Γn = Γˆ (n) be the graph obtained by replacing each point of Γ0 by a n-triangle. By Lemma 2.2, the automorphism group of Γn naturally identifies with S(a, b, c, d). From Lemma 7.1, one deduces the following Corollary 7.2. For any nonnegative integers n m, there exist covering maps Γm → Γn . The group S(a, b, c, d) acts simply transitively on the set of these covering maps. Proof. Covering maps Γ0 → Γ0 are simply bijections of Γ0 and the corollary is thus true for m = n = 0. By induction, by Lemma 7.1, the corollary is still true for any nonnegative integer m and n = 0. Finally, as, if Φ and Ψ are 3-regular graphs, there exists a natural bijection between the sets of covering maps Φ → Ψ and Φˆ → Ψˆ , again, by induction, the corollary is true for any nonnegative integers m n. 2 Let us now go back to Γ . Let q0 and r0 be the two neighbors of p0 that are distinct from p0∨ and q0∨ and r0∨ be the two neighbors of p0∨ that are distinct from p0 . We have the following Lemma 7.3. There exists a unique covering map : Γ → Γ0 such that (p0 ) = a, (q0 ) = c, (r0 ) = d, (p0∨ ) = b, (q0∨ ) = c and (r0∨ ) = d. This covering map is pictured in Fig. 3. Proof. Let n 1 be an integer or the infinity and T be a n-triangle. Let : T → Γ0 . Say that is a quasi-covering map if, for any point p in T − ∂T , induces a bijection from the set of neighbors of p onto Γ0 − { (p)} and if, for any point p in ∂T , the values of on the ˜ denote the neighbors of p are distinct elements of Γ0 − { (p)}. In this case, one still let ˜ (p, q) = (q) and that, map Tˆ → Γ0 such that, for any p and q in T with p ∼ q, one has for any p in ∂T = ∂ Tˆ , if the neighbors of p in T are q and r, ˜ (p) is the unique element of Γ0 − { (p), (q), (r)}. Proceeding as in the proof of Lemma 7.1, one easily checks that the map → ˜ is a S(a, b, c, d)-equivariant bijection from the set of quasi-covering maps T → Γ0 onto the set of quasi-covering maps Tˆ → Γ0 . Therefore, for any n 1, if Tn is the n-triangle containing p0 in Γ , there exists a unique quasi-covering map n from Tn into Γ0 such that n (p0 ) = a, n (q0 ) = c and n (r0 ) = d. By uniqueness, n an n+1 coincide on Tn . Hence, there exists a unique quasi-covering map ∞ from the infinite triangle T∞ with vertex p0 into Γ0 such that ∞ (p0 ) = a, ∞ (q0 ) = c ∨ is the infinite triangle with vertex p ∨ , there exists and ∞ (s0 ) = d. In the same way, if T∞ 0
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∨ from T ∨ into Γ such that ∨ (p ∨ ) = b, ∨ (q ∨ ) = c and a unique quasi-covering map ∞ 0 ∞ ∞ 0 ∞ 0 ∨ (s ∨ ) = d. The map : Γ → Γ whose restriction to T is and whose restriction to T ∨ ∞ 0 ∞ ∞ ∞ 0 ∨ is therefore the unique covering map from Γ into Γ enjoying the required properties. 2 is ∞ 0
Again, from Lemmas 7.1 and 7.3, one deduces the following Corollary 7.4. For any nonnegative integer n, there exist covering maps Γ → Γn . The group S(a, b, c, d) acts simply on the set of these covering maps. This action admits two orbits: on one hand, the set of covering maps such that (q0 ) = (q0∨ ), on the other hand, the set of covering maps such that (q0 ) = (r0∨ ). We shall now describe, for any integer n, the spectral theory of the graph Γn . Let still f denote the polynomial X 2 − X − 3. The methods from Sections 3 and 5 allow to prove the following Proposition 7.5. For any nonnegative integer n, the characteristic polynomial of in 2 (Γn ) is
(X − 3)(X + 1)3
n−1
p 3 2.3n−1−p p 1+2.3n−1−p f (X) − 2 f p (X) f (X) + 2 .
p=0
Recall that, in Section 3.9, we have defined splitable graphs. The proof uses the following Lemma 7.6. Let Φ be a 3-regular connected graph. The graph Φˆ is non-splitable. In particular, for any nonnegative integer n, the graph Γn is non-splitable. Proof. As every point of Φˆ is contained in a 1-triangle, every point may be joined to itself by a path with odd length and hence Φˆ is non-splitable. In the same way, every point of Γ0 may be joined to itself by a path with odd length. 2 Proof of Proposition 7.5. We shall prove this result by induction on n. For n = 0, the space 2 (Γ0 ) has dimension 4 and, for the natural action of the group S(a, b, c, d), it is the sum of two irreducible non-isomorphic subspaces, the space of constant functions and the space of functions ϕ such that ϕ(a) + ϕ(b) + ϕ(c) + ϕ(d) = 0. The operator commutes with the action of S(a, b, c, d) and hence stabilizes both these spaces. In the first one, it acts by multiplication by 3 and, in the second one, by multiplication by −1. Its characteristic polynomial is therefore (X − 3)(X + 1)3 . Suppose the result has been proved for n. By Lemma 7.6, Γn is non-splitable. Therefore, if H is the subspace of 2 (Γn ) spanned by the image of Π ∗ and by the one of Π ∗ , by Corollary 3.6 and Lemma 5.1, the characteristic polynomial of in the orthogonal complement in H of the constant functions is 3 2.3n−1−p p+1 1+2.3n−1−p 3 n−1 f p+1 (X) − 2 f p+1 (X) f (X) + 2 f (X) + 1 p=0
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and hence, as f (X) + 1 = (X + 1)(X − 2), the characteristic polynomial of in H may be written (X − 3)(X + 1)3
n n p p 3 2.3n−p p 1+2.3n−p f (X) − 2 f (X) f (X) + 2 . p=0
p=1
It remains to determine the dimensions of the eigenspaces associate to the eigenvalues 0 and −2 in the orthogonal complement of H in 2 (Γn+1 ). They are described by Lemma 3.7. Now, if n 1, the 2-triangles in Γn+1 are the inverse images by Π 2 of the points in Γn−1 and every point in Γn+1 belongs to a unique 2-triangle. Proceeding as in Lemma 5.2, one shows that the eigenspace associate to the eigenvalue 0 in 2 (Γn+1 ) is isomorphic to the space of functions on the edges of Γn−1 . As Γn−1 is a 3-regular graph containing 4.3n−1 points, it has 2.3n edges and the eigenspace associate to the eigenvalue 0 has dimension 2.3n . If n = 0, by using the characterization of Lemma 3.7, one checks through a direct computation the eigenspace associate to the eigenvalue 0 in 2 (Γ1 ) has dimension 2. Then, as, by Corollary 3.6 and Lemma 5.1, H has dimension 2 dim 2 (Γn ) − 1 = 8.3n − 1, the orthogonal complement of the sum of H and of the eigenspace associate to the eigenvalue 0 has dimension 4.3n+1 − (8.3n − 1) − 2.3n = 2.3n + 1. By Lemma 3.7, this space is the eigenspace associate to the eigenvalue −2 of and the characteristic polynomial of in 2 (Γn+1 ) has the form which is given in the setting. 2 8. The planar compactification of Γ We now consider the set X of those elements (pk,l )(k,l)∈Z2 of (Z/2Z)Z such that, for any integers k and l, one has pk,l + pk+1,l + pk,l+1 = 0 in Z/2Z. This is a compact topological space for the topology induced by the product topology. We let T and S denote the two maps from X into X such that, for any p in X, one has Tp = (pk+1,l )(k,l)∈Z2 and Sp = (pk,l+1 )(k,l)∈Z2 . The homeomorphisms T and S span the natural action of Z2 on X. For p in X and k and l in Z, one has pk,l + pk−1,l+1 + pk−1,l = 0 and pk,l + pk,l−1 + pk+1,l−1 = 0. Now, the finite subgroup S of GL2 (Z) which is spanned by the matrices −1 −1 0 1 and −1 exchanges the three pairs of vectors {(1, 0), (0, 1)}, {(−1, 1), (−1, 0)} and −1 0 1 {(0, −1), (1, −1)} of Z2 . In particular, the group S acts on X in a natural way: for any p in X and s in S, for any k and l in Z2 , one has (sp)k,l = ps −1 (k,l) . The action of S on the three pairs of vectors {(1, 0), (0, 1)}, {(−1, 1), (−1, 0)} and {(0, −1), (1, −1)} identifies S with the permutation group of this set with 3 elements. Let Y be the set of points p in X such that p0,0 = 1. The set Y is stable by the action of S. For any p in Y , one let Yp denote the set of points in the orbit of p under the action of Z2 that belong to Y . If p is some point in Y , one has p1,0 + p0,1 = 1 in Z/2Z and hence one and only one of the points Tp and Sp belongs to Y . In the same way, one and only one of the points T −1 p and T −1 Sp belongs to Y and one and only one of the points S −1 p and T S −1 p belongs to Y . For p and q in Y , let us write p ∼ q if q belongs to the set {Tp, Sp, T −1 Sp, T −1 p, S −1 p, T S −1 p}. This relation is symmetric and S-invariant. In the same way, if p belongs to Y , one let Y˜p denote the set of (k, l) in Z2 such that pk,l = 1 and, for any (i, j ) and (k, l) in Y˜p , one writes (i, j ) ∼ (k, l) if (i − k, j − l) belongs to the set {(1, 0), (0, 1), (−1, 1), (−1, 0), (0, −1), (1, −1)}. Then, Y˜p is a 3-regular graph. If the stabilizer of p in Z2 is trivial, Yp is a 3-regular graph and the natural map Y˜p → Yp is a graph isomorphism. 2
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Fig. 11. Connectedness of Y˜p .
Let u denote the unique element of (Z/2Z)Z such that uk,l = 0 if and only if k − l equals 0 modulo 3. The element u is periodic under the action of Z2 and its stabilizer is the set of (k, l) in Z2 such that k − l equals 0 modulo 3. One checks that u belongs to X. Its orbit under the action of Z2 equals {u, T u, Su} and is stable under the action of S: the elements with signature 1 in S fix u, T u and Su and the elements with signature −1 fix u and exchange T u and Su. We have the following 2
Lemma 8.1. Let p be in Y . The graph Y˜p is connected. If p is different from T u and Su, the set Yp , endowed with the relation ∼, is a connected 3-regular graph and the natural map Y˜p → Yp is a covering map. Proof. Let us show that Y˜p is connected. Let (k, l) be in Yp . After an eventual permutation under the group S and an exchange of the roles of (0, 0) and (k, l), one can suppose k and l are nonnegative. In this case, let us prove by induction on k + l that (k, l) belongs to the same connected component of Y˜p as (0, 0). If k + l = 0, this is trivial. Suppose now k + l > 0 and consider ph,k+l−h , for 0 h k + l. Would all these elements of Z/2Z equal 1, one would have, for any nonnegative integers i and j with i + j k + l − 1, pi,j = 0, which is impossible, since p0,0 = 1. After another eventual permutation by S, one can therefore suppose there exists some integer 0 i l − 1 such that, for any 0 j i, one has pk+j,l−j = 1, but pk+i+1,l−i−1 = 0. This situation is pictured in Fig. 11. Then, the points (k + i, l − i) and (k, l) belong to the same connected component in Y˜p and, as pk+i+1,l−i−1 = 0, one has pk+i,l−i−1 = 1 and (k + i, l − i − 1) belongs to Y˜p . As (k + i) + (l − i − 1) = k + l − 1, the result follows by induction. / As the natural map Y˜p → Yp is onto, to conclude, it remains to prove that, for p ∈ {T u, Su}, the points of the set {p, Tp, Sp, T −1 Sp, T −1 p, S −1 p, T S −1 p} are distinct. Set Vp = {Tp, Sp, T −1 Sp, T −1 p, S −1 p, T S −1 p} and let us begin by supposing that p belongs to Vp . Then, after an eventual action of the group S, one can suppose one has p = Tp and p0,−1 = 1 and hence p1,−1 = p0,−1 + p0,0 = 0, which contradicts the fact that Tp = p. One thus has p∈ / Vp . Suppose now two elements of the set Vp equal each other. Again, after an eventual action of the group S, one can suppose one has Tp = Sp, Tp = T −1 p or Tp = T −1 Sp. If Tp = Sp, one has S −1 Tp = p and we just have proved it to be impossible. If Tp = T −1 p, one has T 2 p = p and the family q = (p2k,2l )(k,l)∈Z2 belongs to Y and satisfies T q = q: again, we just have proved it to be impossible. Finally, if T −2 Sp = p, still suppose, after an eventual permutation, one has p0,−1 = 1. Then, one has p1,−1 = 0, and hence, as T −2 Sp = p, p−1,0 = 0. In the same way, one has p−2,0 = p0,−1 = 1 and p−1,−1 = p0,−1 + p−1,0 = 1. Again, this implies
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Fig. 12. The sets Y a , Y b and Y c .
p−3,0 = p−1,−1 = 1, p−2,−1 = p−1,−1 + p−2,0 = 0 and, finally, p−3,−1 = p−2,−1 + p−3,0 = 1, so that the point q = T −3 p again satisfies T −2 Sq = q and q0,0 = q0,−1 = 1. By induction, one deduces that, for any integer k 0, one has pk,0 = 1 if k equals 0 or 1 modulo 3 and that pk,0 = 0 if k equals 2 modulo 3. Proceeding in the same way, one shows that p−1,1 = p−1,0 + p0,0 = 1 and that, as T 2 S −1 p = p, p1,0 = 1. Thus, one has p2,0 = p0,1 = p0,0 + p1,0 = 0, hence p3,0 = p1,1 = p1,0 + p2,0 = 1 and p3,−1 = p1,0 = 1. The point r = T 3 p therefore also satisfies T −2 Sr = r and r0,0 = r0,−1 = 1, so that, for any k in Z, one has pk,0 = 0 if and only if k equals 2 modulo 3. In particular, the sequence (pk,0 )k∈Z is 3-periodic. As T −2 Sp = p, for any l in Z, the sequence (pk,l )k∈Z is 3-periodic and hence T 3 p = p. Therefore, for any k and l in Z, if k − l equals 0 modulo 3, one has T k S l p = p. As one has p0,0 = u1,0 , p−1,0 = u−1,1 and p−1,1 = u−1,2 , one has p = T u. Therefore, if p does not belong to {T u, Su}, the relation ∼ induces a 3-regular graph structure on the set Yp . By definition, the natural map Y˜p → Yp is then a covering map. In particular, Yp is connected. 2 Let ε and η be in {0, 1}. Let X (ε,η) denote the set of elements p in X such that, for any k and l in Z, if (k, l) equals (ε, η) in (Z/2Z)2 , one has pk,l = 0. If p belongs to X (ε,η) , for any k and l in Z, one has p2k+1+ε,2l+1+η = p2k+ε,2l+1+η = p2k+1+ε,2l+η . In particular, for (ε , η ) = (ε, η), one has X (ε,η) ∩ X (ε ,η ) = {0} and, if p is a point in Y (ε,η) = Y ∩ X (ε,η) (one then has (ε, η) = (0, 0)), the point p belongs to a triangle in the graphe Yp . The groupe S acts on (Z/2Z)2 in a natural way and, for any s in S, for any (ε, η) in (Z/2Z)2 , one has X s(ε,η) = sX (ε,η) . From now on, we set a = (1, 1), b = (0, 1), c = (1, 0) and T1 = {a, b, c}. We shall consider T1 as a 1-triangle. The groupe S may be identified with the permutation group S(a, b, c). One sets Yˆ = Y a ∪ Y b ∪ Y c : this is a disjoint union and the set Yˆ is S-invariant. The elements of Y a , Y b and Y c are described by Fig. 12. 2 Let p be a point of Y . We shall let pˆ a , pˆ b and pˆ c denote the elements of (Z/2Z)Z such that, for any k and l in Z, one has a a a a = pˆ 2k−1,2l = pˆ 2k,2l−1 = pk,l and pˆ 2k−1,2l−1 = 0. (i) pˆ 2k,2l b b b b (ii) pˆ 2k,2l = pˆ 2k+1,2l = pˆ 2k+1,2l−1 = pk,l and pˆ 2k+2,2l−1 = 0. c c c c = pˆ 2k,2l+1 = pˆ 2k−1,2l+1 = pk,l and pˆ 2k−1,2l+2 = 0. (iii) pˆ 2k,2l
One checks that, by construction, one has pˆ a = T pˆ b = S pˆ c and that, for any s in S, for any d in sd = s(pˆ d ). T1 , one has sp
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Lemma 8.2. Let d be in T1 . The map p → pˆ d induces a homomorphism from Y onto Y d . Conversely, a point p of Y − {T u, Su} belongs to Yˆ if and only if, for any q in Yp , q belongs to some triangle contained in Yp . In this case, there exists a unique d in T1 and a unique point r of Y such that p = rˆ d and the triangle containing p is {ˆr a , rˆ b , rˆ c }. The proof uses the following Lemma 8.3. Let p be a point of Y − {T u, Su} such that each point of Yp is contained in a triangle. Then the triangle containing p is either {p, T −1 p, S −1 p} or {p, Tp, T S −1 p} or {p, Sp, T −1 Sp}. If it is of the form {p, T −1 p, S −1 p}, the third neighbor q of p is either Tp or Sp. Finally, if q = Tp, the triangle containing q is {q, T q, T S −1 q} and if q = Sp, the triangle containing q is {q, Sq, T −1 Sq}. Proof. Let p be as in the setting. After an eventual action of S, one can suppose that the triangle containing p contains the point T −1 p. Then, by definition, the only possible common neighbors of p and T −1 p are T −1 Sp and S −1 p. Now, as T −1 p belongs to Y , one has p−1,0 = 1, hence p−1,1 = p0,0 + p−1,0 = 0 and T −1 Sp ∈ / Y . Therefore, the triangle containing p is {p, T −1 p, S −1 p}. The other cases follow, by letting S act on the situation. In case the triangle containing p is {p, T −1 p, S −1 p}, the third neighbor q of p is, by construction, necessarily in {Tp, Sp}. Suppose now, still after an eventual permutation under S, one has q = Tp. Then, one has p0,1 = 0 = p1,−1 and hence T −1 Sq and S −1 q do not belong to Y . The triangle containing q is thus {q, T q, T S −1 q}. The other case follows, by symmetry. 2 Proof of Lemma 8.2. One easily checks that, for d in T1 , the point pˆ d belongs to Y d and that the thus defined map induces a homeomorphism from Y onto Y d . Conversely, let p be a point of Y − {T u, Su} such that every element of Yp is contained in a triangle of Yp . Then, by definition and by Lemma 8.1, every point of Y˜p is contained in a triangle of Y˜p . Let (k, l) be a point of Y˜p . By Lemma 8.3, the triangle containing (k, l) is of the form {(k, l), (k − 1, l), (k, l − 1)}, {(k, l), (k + 1, l), (k + 1, l − 1)} or {(k, l), (k, l + 1), (k − 1, l + 1)}. Let (ε(k, l), η(k, l)) denote the unique element of (Z/2Z)2 that does not equal one of the elements of this triangle modulo (2Z)2 . Let us prove that, for any (i, j ) and (k, l) in Y˜p with (i, j ) ∼ (k, l), one has (ε(i, j ), η(i, j )) = (ε(k, l), η(k, l)). If (i, j ) and (k, l) belong to the same triangle, this is clear. Else, after an eventual action of S, by Lemma 8.3, one can suppose that the triangle containing (k, l) is {(k, l), (k − 1, l), (k, l − 1)}, that (i, j ) = (k + 1, l) and hence that the triangle containing (i, j ) is {(k + 1, l), (k + 2, l), (k + 2, l − 1)}. Then, by definition, one has (ε(i, j ), η(i, j )) = (ε(k, l), η(k, l)). As, by Lemma 8.1, the graph Y˜p is connected, the function (ε, η) is constant. By definition, for any integers k and l, one has p2k+ε,2l+η = 0, hence p belongs to Y (ε,η) . The property on triangles immediately follow from the definition of the objects. 2 ¯ and θ1 (p) denote the unique elements of Y and T1 for Let p be a point of Yˆ . One let Πp θ (p) 1 ¯p which one has Π = p. By construction, one has θ1 (p) = a (resp. b, resp. c) if and only if the triangle containing p is {p, T −1 p, S −1 p} (resp. {p, Tp, T S −1 p}, resp. {p, Sp, T −1 Sp}). The maps Π¯ and θ1 are S-equivariant. The map Π¯ is continuous and θ1 is locally constant. For any p in Yˆ , θ1 induces a bijection from the 1-triangle containing p onto T1 .
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Lemma 8.4. Let p be in Y − {T u, Su}. There exists a unique graph isomorphism σ : Yˆp → ¯ = σΠ. Π¯ −1 (Yp ) such that Πσ Proof. By Lemma 2.2, such an isomorphism is necessarily unique. Let us prove that it exists. Let q be the neighbor of p belonging to {Tp, Sp}, r its neighbor in {T −1 p, T −1 Sp} and s its neighbor in {S −1 p, T S −1 p}. One sets σ (p, q) = pˆ a , σ (p, r) = pˆ b and σ (p, s) = pˆ c . Then, the three points σ (p, q), σ (p, r) and σ (p, s) are neighbors in Yp . Let us check, for example, that σ (p, q) is a neighbor of σ (q, p). After an eventual action of S, one can suppose one has q = Tp. Then, one has p = T −1 q and hence σ (q, p) = qˆ b . By construction, one then has T pˆ a = qˆ b , hence σ (q, p) = T σ (p, q), what should be proved. 2 From now on, we shall, for any p in Yˆ , identify the graphs Yˆp and Π¯ −1 (Yp ). We will now construct an element p of Y for which the graph Yp is isomorphic to the Pascal graph. Set, for any k, l 0, p−k,−l = pk+l+1,−l = l+k k in Z/2Z and, for any k, l in Z with either l > 0 or k 1 and k + l 0, pk,l = 0. One easily checks that p belongs to X and hence to Y since p0,0 = 1. We have the following ¯ = p and θ1 (p) = a. There exists an Proposition 8.5. The point p belongs to Yˆ and one has Πp isomorphism from the Pascal graph Γ onto Yp sending p0 to p and p0∨ to Tp. This planar representation of the Pascal graph appears in Fig. 1. The proof uses the following Lemma 8.6. Let 0 k n be integers. Then, the integers other modulo 2.
n 2n 2n+1 and 2n+1 k , 2k , 2k 2k+1 equal each
Proof. Let Aand B be indeterminates. In the characteristic 2 ring Z/2Z[A, B], one has (A + B)n = nk=0 nk Ak B n−k and hence (A + B)2n = nk=0 nk A2k B 2n−2k . Therefore, by 2n 2n n = k and 2k−1 = = 0. By the clasuniqueness, for any 0 k n, one has, in Z/2Z, 2n 2k 2n+1 2n2k+1 2n 2n 2n+1 2n 2n 2n sical identity, one then has 2k = 2k−1 + 2k = 2k and 2k+1 = 2k + 2k+1 = 2k . 2 Proof of Proposition 8.5. By using Lemma 8.6, one checks that one has p = pˆ a . Therefore, p ¯ = p and θ1 (p) = a. By induction, using Lemma 8.4, one deduces that, for any belongs to Yˆ , Πp ¯ = Tp, integer n, p is the vertex of a n-triangle contained in Yp . In the same way, one has Π(Tp) θ1 (Tp) = b and Tp is the vertex of a n-triangle contained in Yp . By Lemma 8.1, the graph Yp is connected and hence it equals the union of both of these infinite triangles. The existence of the isomorphism in question follows. 2 From now on, we shall identify p with p0 , Tp with p0∨ and Γ with Yp . One let Γ¯ denote the closure of Γ in Y and, for any p in Γ¯ , one sets Γp = Yp . One has Π¯ Γ¯ = Γ¯ . We will now describe the set Γ¯ in a more detailed way. For this purpose, let us introduce a partition of Yˆ into six subsets that refines the partition Yˆ = Y a ∪ Y b ∪ Y c . Let b be a point of Yˆ . Then, by Lemma 8.3, the set of neighbors of p is either {Tp, T −1 p, S −1 p} or {Sp, T −1 p, S −1 p} or {T −1 p, Tp, T S −1 p} or {T −1 Sp, Tp, T S −1 p} or {S −1 p, Sp, T −1 Sp} or {T S −1 p, Sp, T −1 Sp}. Let us call the set of q in Yˆ for which one has {(k, l) ∈ Z2 | T k S l p ∼ p} = {(k, l) ∈ Z2 | T k S l q ∼ q} the keel of p. The keels are six closed subsets of Yˆon which the group S act simply 0transi1 −1 tively. We let B0 denote the keel of p0 , i the element −1 of S and r the element . −1 −1 0 1 The element i identifies with the transposition (ab) of {a, b, c} and r with the cycle (cba).
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For any integer n, set Yˆ (n) = Π¯ −n Y . Then, by a direct induction, by Lemmas 8.2 and 8.4, for any integer n, Yˆ (n) is the set of elements p in Y − {T u, Su} for which every point of Yp belongs to a n-triangle in Yp . One hence has Γ ⊂ n∈N Yˆ (n) . Lemma 8.7. Let n be an integer and p and q be in Yˆ (n+1) such that, for any 0 m n, Π¯ m p and Π¯ m q belong to the same keel. Then, for any k and l in Z with k −2n , l −2n and k + l 2n , one has pk,l = qk,l . Proof. Let us prove this result by induction on n. For n = 0, suppose, after an eventual action of S, one has p, q ∈ B0 . Then, one has p−1,0 = p0,0 = p1,0 = p0,−1 = 1, so that p−1,−1 = p−1,0 + p0,−1 = 0 and, in the same way, p1,−1 = p−1,1 = p0,1 = p−1,2 = 0 and p2,−1 = 1. As this is also true for q, the lemma is true for n = 0. Suppose now n 1 and the lemma has been proved for n − 1. Pick p and q as in the setting. Then, as p and q are in the same keel, one has θ1 (p) = θ1 (q). After an eventual action of S, a a ¯ ¯ . The result now follows by one can suppose one has θ1 (p) = a, so that p = Πp and q = Πq induction and by the definition of the map r → rˆ a . 2 Lemma 8.8. Let p be in Yˆ such that the keel of p is B0 . Then, the keel of pˆ a is B0 , the one of pˆ b is iB0 and the one of pˆ c is rB0 and pˆ a is the vertex of a 2-triangle in Γ¯ . If q and r are two points ¯ and Πr ¯ belong to the same keel, there exists r in the 1-triangle containing in Yˆ (2) such that Πq r in Yr such that q and r belong to the same keel. Proof. The first point follows directly from the construction of the objects. Pick q and r as in ¯ and of Πr ¯ is B0 . the setting. After an eventual action of S, one can suppose that the keel of Πq The first part of the lemma now clearly implies the setting. 2 Let Σ be the set of sequences (sn )n∈N of elements of S such that, for any integer n, one has sn ∈ {sn+1 , sn+1 i, sn+1 r}. We equip Σ with the topology induced by the product topology and we let σ : Σ → Σ denote the shift map. One let S act on Σ by left multiplication on all the components. The set Γ¯ is described by the following Proposition 8.9. One has Γ¯ = n∈N Yˆ (n) . For any p in Γ¯ , for any integer n, let sn (p) be the unique element of S such that the keel of Π¯ n p is sn (p)B0 . The thus defined map s induces a S-equivariant homeomorphism from Γ¯ onto Σ and one has σ s = s Π¯ . The image of the point p0 by s is the constant sequence with value e and the fixed points of Π¯ in Γ¯ are exactly the six images of p0 by the action of the group S. Finally, for any p in Γ¯ , the set Γp is dense in Γ¯ . Proof. As the set Γ is included in n∈N Yˆ (n) , so is the set Γ¯ . Conversely, let us note that each of the six points T −1 p0 , T −2 p0 , T−2 S −1 p0 , T −1 S −2 p0 , S −2 p0 and S −1 p0 belong to a different keel. Therefore, if p is a point of n∈N Yˆ (n) , there exists a point q in Γ such that p and q belong to the same keel. By induction, using Lemma 8.8, one deduces that, for any integer n, there exists a point qn of Γ such that, for any 0 m n, Π¯ m p and Π¯ m qn belong to the same keel. By Lemma 8.7, one then has qn −→ p and p belongs to Γ¯ . n→∞ Let p be in Γ¯ . As we have just seen, the point p is completely determined by the sequence s(p) = (sn (p))n∈N . The map s is clearly continuous and S-equivariant and, by definition, one has s Π¯ = σ s. Besides, by Lemma 8.2, if p is a point of Γ¯ , it admits exactly three antecedents
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by Π¯ and, by Lemma 8.8, the keels of these antecedents are s0 (p)B0 , s0 (p)iB0 and s0 (p)rB0 . It follows that the map s takes its values in Σ and that it induces a homeomorphism from Γ¯ onto Σ . ¯ 0 = p0 , for any nonnegBy construction, one has s0 (p0 ) = e and, as, by Proposition 8.5, Πp ative integer n, sn (p0 ) = e. In particular, the other fixed points of Π¯ are the images of p0 by the action of S. Finally, if p is a point of Γ¯ , by Lemmas 2.4 and 8.4, for any nonnegative integer n, the ntriangle containing p in Γp is the set Π¯ −n (Π¯ n p). As r and i span the group S, one easily checks that the subshift of finite type (Σ, σ ) is transitive, so that, for any t in Σ, the set n∈N σ −n (σ n t) is dense in Σ . Therefore, for any p in Γ¯ , Γp is dense in Γ¯ . 2 9. Triangular functions and integration on Γ¯ In this section, we study a particular class of locally constant functions on Γ¯ . We use these functions to determine some properties of a remarkable Radon measure on Γ¯ . For p in Γ¯ , let still, as in Section 8, (sn (p)B0 )n∈N denote the associate keel sequence. Let n 1 be an integer and T be a n-triangle in Γ¯ . Then, by Lemma 8.4, the set Π¯ n−1 T is a 1-triangle of Γ¯ . Therefore, the map θ1 ◦ Π¯ n−1 induces a bijection from the set of vertices of T onto T1 = {a, b, c}. One let an (resp. bn , resp. cn ) denote the set of vertices p of n-triangles of Γ¯ such that θ1 (Π¯ n−1 p) = a (resp. b, resp. c) and θn the map which sends a vertex p of a n-triangle of Γ¯ to the element of {an , bn , cn } to which it belongs. Let Tn be the n-triangle Tn (an , bn , cn ). By Lemma 2.3, the map θn extends in a unique way to a map Γ¯ → Tn , still denoted by θn , that, on each n-triangle T of Γ¯ , induces a graph isomorphism from T onto Tn . This map is locally constant. For n = 1, this definition is coherent with the notations of Section 8, provided one identifies a with a1 , b with b1 and c with c1 . By abuse of language, we will sometimes consider T0 as a set containing only one element and θ0 as the constant map Γ¯ → T0 . For any n 1, the group S act on Tn and identifies with S(an , bn , cn ). We shall identify Tn+1 and Tˆn through the S-equivariant bijection from {an , bn , cn } onto {an+1 , bn+1 , cn+1 } that sends an to an+1 , bn to bn+1 and cn to cn+1 . In particular, one let Π : Tn+1 → Tn denote the triangle contracting map coming from this identification and Π ∗ and Π the associate operators 2 (Tn ) → 2 (Tn+1 ) and 2 (Tn+1 ) → 2 (Tn ). Let, for any n 2, an bn , an cn , bn an , bn cn , cn an and cn bn be the points of Tn defined by Corollary 2.6. The principal properties of the maps θn , n 1, we shall use in the sequel are described by the following Lemma 9.1. Let n 1 be an integer. One has Πθn+1 = θn Π¯ . If p and q are points of Γ¯ such that θn+1 (p) = θn+1 (q), one has θn (p) = θn (q). In particular, one has θn (p) = an if and only if θn+1 (p) is an+1 , bn+1 an+1 or cn+1 an+1 . Let p and q be in Γ¯ , such that θn (p) = θn (q). For any 0 m n, if θn (p) is not contained in a m-triangle which admits one of the vertices of Tn as a vertex, one has sm (p) = sm (q). Proof. Let T be a (n + 1)-triangle of Γ¯ . The map θn+1 induces an isomorphism from T onto ¯ onto Tn . As, by definition, Tn+1 and the map θn induces an isomorphism from the n-triangle ΠT ¯ the maps θn Π and Πθn+1 coincide on the set ∂Tn , one has, by Lemma 2.2, Πθn+1 = θn Π¯ . Let p, q and r be the vertices of T , so that θn+1 (p) = an+1 , θn+1 (q) = bn+1 and θn+1 (r) = cn+1 . By definition, one has θn (p) = an . Let us prove that θn (qp) = θn (rp) = an . This amounts to proving that one has θ1 (Π¯ n−1 qp) = θ1 (Π¯ n−1 rp) = a1 . Now, as above, one has θ2 (Π¯ n−1 qp) =
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Π n−1 θn+1 (qp) = b2 a2 and θ2 (Π¯ n−1 rp) = Π n−1 θn+1 (rp) = c2 a2 , so that we only have to deal with the case where n = 1. Then, with the notations of Section 8, if s = Π¯ 2 p, one checks that a a one has qp = s ˆ b and rp = s ˆ c , whence the result. In particular, if S is some n-triangle in Γ¯ , the restriction of θn to ∂S is completely determined by the restriction of θn+1 to ∂S. By definition, the values of θn are thus determined by those of θn+1 . Finally, let p and q be such that θn (p) = θn (q) and let us show the assumption of the lemma by induction on n 1. For n = 1, this assumption is empty. Suppose n 2 and the assumption has been established for n − 1. Then, one has θn−1 (Π¯ p) = Πθn (p) = Πθn (q) = θn−1 (Π¯ q) and, for any integer m with 1 m n, if θn (p) does not belong to the m-triangle which admits one of the vertices of Tn as a vertex, θn−1 (Π¯ p) does not belong to the (m − 1)triangle which admits one of the vertices of Tn−1 as a vertex and hence, by induction, sm (p) = sm−1 (Π¯ p) = sm−1 (Π¯ q) = sm (q). It remains to handle the case where m = 0. Suppose thus θn (p) is not a vertex of Tn and let us prove that s0 (p) = s0 (q). Note that, by the first part of the proof, one has θ1 (p) = θ1 (q). After an eventual action of S, suppose θ1 (p) = a1 . Then, let T be the n-triangle of Γ¯ containing p and p be the neighbor of p that does not belong to the 1-triangle containing p. As p is not a vertex of T , p belongs to T and, by Lemma 8.3, s0 (p) is either B0 or riB0 , following θ1 (p ) is b1 or c1 . Now, as θn induces an isomorphism from T onto Tn , θn (p ) only depends on θn (p) and hence, still by the first part of the lemma, the value of θ1 at p is completely determined by the value of θn at p. Therefore, the value of s0 at p is determined by the value of θn at p, what should be proved. 2 For any integer n 1, by Proposition 8.5, one has θn (p0 ) = an = θn (rip0 ), so that the coding of Γ¯ by the maps θn , n 1 is ambiguous. This ambiguities are described by the following Corollary 9.2. Let p and q be in Γ¯ such that, for any integer n, one has θn (p) = θn (q). Then, if p = q, there exists s in S such that p belongs to the infinite triangle with vertex sp0 in sΓ and q belongs to the infinite triangle with vertex srip0 in sriΓ . Proof. Suppose one has p = q. Then, by Proposition 8.9, there exists some natural integer m such that sm (p) = sm (q). By Lemma 9.1, for any integer n m, the point θn (p) = θn (q) belongs to the m-triangle which admits one of the vertices of Tn as a vertex. Set p = Π¯ m p and q = Π¯ m q. By Lemma 9.1, for any integer n, one has θn (p ) = Π m θ n+m (p) = Π m θ n+m (q) = θn (q ) and this point is one of the vertices of Tn . As, for any integer n 1, one has θn (an+1 ) = an , θn (bn+1 ) = bn and θn (cn+1 ) = cn , one can suppose, after an eventual action of S, one has, for any integer n 1, θn (p ) = θn (q ) = an . As θ1 (p ) = a1 , the keel of p is B0 or riB0 . After another ¯ is B0 , iB0 or r −1 B0 . action of S, suppose this keel is B0 . Then, by Lemma 8.8, the keel of Πp ¯ is B0 and, by induction, for any integer n, one has As θ1 (Π¯ p ) = Πθ1 (p) = a1 , the keel of Πp sn (p ) = e, so that, by Proposition 8.9, p = p0 and p belongs to the infinite triangle T∞ (p0 ) with vertex p0 in Γ . In the same way, one has q = p0 or q = rip0 and q belongs to the infinite triangle with vertex p0 in Γ or to the infinite triangle with vertex rip0 in riΓ . For any integer n, the map θn induces a bijection from the n-triangle with vertex p0 in Γ onto Tn . Therefore, if p is some point in T∞ (p0 ) such that, for any integer n, one has θn (p ) = θn (p), one has p = p. As we supposed p = q, q belongs to the infinite triangle with vertex rip0 in riΓ , what should be proved. 2 Let n be an integer. We shall say that a function ϕ : Γ¯ → C is n-triangular if it may be written ϕ = ψ ◦ θn , for some function ψ on Tn . When there is no ambiguity, to simplify notations, we
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shall identify ϕ and ψ . By Lemma 9.1, a n-triangular function is (n + 1)-triangular. In particular, triangular functions constitute a subalgebra of the algebra of locally constant functions on Γ¯ . As, for any triangular function ϕ, one has ϕ(p0 ) = ϕ(rip0 ), this subalgebra is not dense in C 0 (Γ¯ ) for the topology of uniform convergence. From now on, we let μ denote the Borel probability measure on Γ¯ whose image under the coding map of Proposition 8.9 is the maximal entropy measure for σ on Σ . In other terms, μ is the unique measure such that, for any sequence t0 , . . . , tn of elements of S, if, for any 0 m n − 1, one has tm ∈ {tm+1 , tm+1 i, tm+1 r}, then μ(t0 B0 ∩ Π¯ −1 t1 B0 ∩ . . . ∩ Π¯ −n tn B0 ) = 6.31 n . By definition, the measure μ is Π¯ -invariant and S-invariant. For any Borel function ϕ on Γ¯ , one sets Π¯ ∗ ϕ = ϕ ◦ Π¯ . For any 1 p ∞, the operator Π¯ ∗ preserves the norm of Lp (Γ¯ , μ). One let Π¯ denote its adjoint, that is, for any Borel function ϕ on 1 ¯ = 1 and, for any 1 p ∞, the operator ¯ ¯ ϕ(q). One has Π1 Γ , one has Πϕ(p) = 3 Π(q)=p ¯ p Π¯ is bounded with norm 1 in L (Γ¯ , μ). Finally, one has Π¯ Π¯ ∗ = 1. The integral of triangular functions with respect to the measure μ may be computed in a natural way: Lemma 9.3. Let n be a nonnegative integer and ϕ be a n-triangular function. One has 1 p∈Tn ϕ(p). 3n
Γ¯
ϕ dμ =
In other terms, the image measure of μ by θn is the normalized counting measure on Tn . Proof. Let us prove the result by induction on n. If n = 0, ϕ is constant and the lemma is evident. ¯ , the function Πϕ ¯ is (n − 1)-triangular and one If n 1, as, by Lemma 9.1, one has Πθn = θn−11Π 1 ¯ has, by induction, Γ¯ ϕ dμ = Γ¯ Π ϕ dμ = 3n−1 p∈Tn−1 3 Π(q)=p ϕ(q) = 31n p∈Tn ϕ(p), whence the result. 2 From now on, for any integer n, one shall identify θn and the associate partition of the measure ¯ space one has (Γ , μ). By Lemma 9.1, this sequence of partitions is increasing. As μ is atom free, μ( s∈S sΓ ) = 0 and, by Corollary 9.2, for any p and q in the total measure set Γ¯ − s∈S sΓ , if, for any integer n, one has θn (p) = θn (q), one has p = q. For any ϕ in L1 (Γ¯ , μ), for any expectation of ϕ knowing θn , that is, for any integer n, one let E(ϕ | θn ) denote the conditional 1 p in Tn , one has E(ϕ | θn )(p) = −1 θ −1 (p) ϕ dμ. μ(θn (p))
n
Lemma 9.4. For any 1 p < ∞, for any ϕ in Lp (Γ¯ , μ), one has E(ϕ | θn ) −→ ϕ in Lp (Γ¯ , μ). n→∞ In particular, the space of triangular functions is dense in Lp (Γ¯ , μ). In the same way the space of triangular functions that take the value zero at the vertices of their definition triangle is dense in Lp (Γ¯ , μ). Finally, for any integer n, for any ϕ in L1 (Γ¯ , μ), one has E(Π¯ ∗ ϕ | θn+1 ) = Π ∗ E(ϕ | θn ), E(Π¯ ϕ | θn ) = 13 ΠE(ϕ | θn+1 ) and, for any p in Tn , E(ϕ | θn )(p) =
1 3
E(ϕ | θn+1 )(q).
q∈Tn+1 θn (q)=p
Proof. The convergence in Lp (Γ¯ , μ) follows from the discussion above and general properties of probability spaces. The density of triangular functions that are zero at the vertices of their definition triangles follows, since, by Lemma 9.4, for any n 1, the measure of the set of elements
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of Γ¯ that are the vertex of some n-triangle is 3n−1 . Finally, the formulae linking conditional expectations knowing θn+1 and θn follow from Lemma 9.1 and the fact that, by Lemma 9.3, the image measure of μ by θn is the normalized counting measure on Tn . 2 Let us finally describe a homeomorphism Γ¯ → Γ¯ that will be useful in the sequel. For any p in Γ¯ , let α(p) denote the unique neighbor of p that does not belong to the triangle containing p. The map α is a fixed point free involution. By Corollary 2.5, for any n 1, α stabilizes the set of points in Γ¯ that are vertex of some n-triangle. For p in Tn − ∂Tn , let still αn (p) denote the unique neighbor of p in Tn that does not belong to the triangle containing p. Lemma 9.5. For any integer n, for any p in Γ¯ , if p is not a vertex of some n-triangle of Γ¯ , one has θn (α(p)) = αn (θn (p)). The map α preserves the measure μ and, for any n 1, ϕ in L1 (Γ¯ , μ) and p in Tn − ∂Tn , one has E(ϕ ◦ α | θn )(p) = E(ϕ | θn )(αn (p)). As triangular functions are not dense in C 0 (Γ¯ ), to check that α preserves the measure μ, we shall use the following Lemma 9.6. Let X be a compact metric space and let A be a complex uniformly closed and conjugation stable subalgebra of C 0 (X). Let Y be the set of elements x in X for which there exists y = x in X such that, for any ϕ in A, ϕ(y) = ϕ(x). The set Y is Borel and, if λ is a Borel complex measure on X such that λ|Y = 0 and that, for any ϕ in A, X ϕ dλ = 0, one has λ = 0. Proof. Let S be the spectrum of the commutative C∗ -algebra A and π : X → S the surjective continuous map which is dual to the natural injection from A into C 0 (X). By the hypothesis, the complex measure π∗ λ is zero on S. Let p denote the projection onto the first component X × X → X and set D = {(x, x) | x ∈ X} ⊂ X × X and E = {(x, y) ∈ X × X | π(x) = π(y)}. One has Y = p(E − D). As E and D are closed subsets of the compact metrizable space X, the space E − D is a countable union of compact sets, hence so are Y and π(Y ). In particular, these subsets are Borel and π induces a Borel isomorphism from X − Y onto S − π(Y ). Therefore, the restriction of λ to X − Y is zero. As its restriction to Y is zero, one has λ = 0. 2 Proof of Lemma 9.5. The first part of the lemma follows from the definition of α and the fact that θn induces a graph isomorphism from the n-triangle containing p onto Tn . Let n 1. As α exchanges the points of Γ¯ that are vertex of some n-triangle, if ϕ is a ntriangular function which is zero at the vertices of Tn , one has, by Lemma 9.3, Γ¯ ϕ ◦ α dμ = ¯ Γ¯ ϕ dμ. As, again by Lemma 9.3, for any integer n 1, the measure of the set of points of Γ n−1 that are vertex of some n-triangle is 3 , one deduces that, for any triangular function ϕ, one has Γ¯ ϕ ◦ α dμ = Γ¯ ϕ dμ. Let A ⊂ C 0 (Γ¯ ) be the uniform closure of the algebra of triangular functions. By Corollary 9.2, ¯ the set of elements p in Γ for which there exists q = p such that, for any ϕ in A, one has ϕ(p) = ϕ(q) is s∈S sΓ . As μ isatom free, this set has zero measure for μ and for α∗ μ. For any ϕ in A, one has Γ¯ ϕ ◦ α dμ = Γ¯ ϕ dμ. By Lemma 9.6, one hence has α∗ μ = μ. 2 Finally, let Θ¯ denote the quotient of Γ¯ by the map α, endowed with the measure λ, which is the image of μ by the natural projection. The space Θ¯ may be seen in a natural way as the set of edges of Γ¯ and may be equipped with a 4-regular graph structure. The image of Γ in Θ¯
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¯ We call then identifies in a natural way with the Sierpi´nski graph Θ and is a dense subset of Θ. triangular functions on Θ¯ the functions coming from triangular functions on Γ¯ that are zero on the vertices of their definition triangle and α-invariant. Then, the results of this section transfer ¯ into analogous results on Θ. ¯ and its harmonic measures 10. The operator ¯ on Γ¯ which is an analogue of the operator on Γ . We shall now study an operator ¯ Let ϕ be a Borel function on Γ¯ . For any p in Γ¯ , one sets ϕ(p) = q∼p ϕ(q). To study the properties of this operator on triangular functions, set moreover, for any nonnegative integer n, for any function ϕon Tn , for any p in Tn − ∂Tn , ϕ(p) = q∼p ϕ(q) and, for any p in ∂Tn , ϕ(p) = ϕ(p) + q∼p ϕ(q). Lemma 10.1. For any integer n 1, the operator is self-adjoint in 2 (Tn ). If ϕ is a n¯ triangular function that is constant on ∂Tn , one has ϕ = ϕ. ¯ is self-adjoint. Besides, as θn induces graph isomorphisms from Proof. One checks easily that ¯ n-triangles of Γ onto Tn , for any p in Tn − ∂Tn , one has ¯ −1 = 1 θ (p) n
q∼p
1θ −1 (q) = 1θ −1 (p) . n
n
In the same way, as, for any point p of Γ¯ that is a vertex of some n-triangle, the unique neighborof p that does not belong to this n-triangle is itself a vertex of some n-triangle, one has ¯ p∈∂ T 1 −1 = p∈∂ T 1 −1 and hence, for any n-triangular function ϕ that is con n θn (p) n θn (p) ¯ = ϕ. 2 stant on ∂Tn , ϕ ¯ in the following We can now set the principal properties of ¯ commutes with the action of S. It is continuous with norm 3 Proposition 10.2. The operator ¯ ∗ μ = 3μ. For any 1 p ∞, the in the space of continuous functions on Γ¯ and one has p ¯ ¯ operator is continuous with norm 3 in L (Γ , μ) and, for p1 + q1 = 1, for any ϕ in Lp (Γ¯ , μ) ¯ ψ = ϕ, ψ. ¯ and ψ in Lq (Γ¯ , μ), one has ϕ, ¯ is positive and 1 ¯ = 3, ¯ is continuous with norm 3 Proof. The first assumption is evident. As in the space of continuous functions. Recall one let α denote the map Γ¯ → Γ¯ that sends some point p to its unique neighbor ¯ = 3Π¯ ∗ Πϕ ¯ + that does not belong to the 1-triangle containing p. For ϕ in C 0 (Γ¯ ), one has ϕ ∗ ϕ ◦ α − ϕ. As the operators Π¯ and Π¯ preserve μ and, by Lemma 9.5, the homeomorphism α ¯ ∗ μ = 3μ. preserves μ, one has ¯ thus acts on Lp (Γ¯ , μ) and it is bounded with For any 1 p ∞, the positive operator norm 3 in this space. Let 1 < p, q < ∞ be such that p1 + q1 = 1. By Lemma 9.4, triangular functions that are zero on the vertices of their definition triangles are dense in Lp (Γ¯ , μ) and ¯ ψ = Lq (Γ¯ , μ). By Lemma 9.4, one hence has, for any ϕ in Lp (Γ¯ , μ) and ψ in Lq (Γ¯ , μ), ϕ, 1 ∞ ¯ ¯
ϕ, ψ. As the operators appearing in this identity are continuous in L (Γ , μ) and L (Γ¯ , μ), it is still true for p = 1 and q = ∞. 2
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We shall now prove that the measure μ is, up to scalar multiplication, the unique Borel ¯ ∗ λ = 3λ. Let us begin by handling the case where λ is complex measure λ on Γ¯ such that S-invariant. ¯ ∗ λ = 3λ. One has Lemma 10.3. Let λ be a S-invariant Borel complex measure on Γ¯ with ¯ λ = λ(Γ )μ. Proof. Let ϕ : p → λ(p), Γ → C. Then, ϕ belongs to 1 (Γ ) and one has ϕ = 3ϕ. By the maximum principle, one hence has ϕ = 0. Therefore, the restriction of ϕ to s∈S sΓ is zero. By Corollary 9.2 and Lemma 9.6, it thus suffices to check that λ is proportional to μ on the space −1 of triangular functions. Let n 1 be an integer and, for any p in Tn , let ϕn (p) = λ(θn (p)) = ¯ 1 dλ. By Lemma 10.1, if p is not a vertex of T , one has 1 = 1 n Γ¯ θn−1 (p) θn−1 (p) θn−1 (p) and ∗ ¯ λ = 3λ, ϕn (p) = 3ϕn (p). Moreover, as λ is S-invariant, ϕn is constant on ∂Tn . hence, as By the maximum principle, ϕn is constant. As p∈Tn ϕn (p) = λ(Γ¯ ), one has, for any p in Tn , ϕn (p) = 31n λ(Γ¯ ), whence the result, by Lemma 9.3. 2 Let us now study the eigenspace associate to the eigenvalue 1 in L1 (Γ¯ , μ). We will need the following Lemma 10.4. Let n 1 be an integer, ϕ be in L1 (Γ¯ , μ) and p be in Tn − ∂Tn . One has ¯ | θn )(p). E(ϕ | θn )(p) = E(ϕ ¯ + ϕ ◦ α − ϕ and hence, ¯ = 3Π¯ ∗ Πϕ Proof. Let still α and αn be as in Lemma 9.5. One has ϕ by Lemmas 9.4 and 9.5, ¯ | θn )(p) = Π ∗ ΠE(ϕ | θn )(p) + E(ϕ | θn ) αn (p) − E(ϕ | θn )(p) E(ϕ = E(ϕ | θn )(p), what should be proved.
2
For any n 1, let Hn denote the space of functions ϕ on Tn such that, for any p in Tn that ¯ = 3ϕ, is not a vertex, one has ϕ(p) = 3ϕ(p). By Lemma 10.4, for any ϕ in L1 (Γ¯ , μ), if ϕ one has E(ϕ | θn ) ∈ Hn . One identifies C3 and the space of complex valued function on T1 by considering (1, 0, 0) (resp. (0, 1, 0), resp. (0, 0, 1)) as the characteristic function of the singleton {a1 } (resp. {b1 }, resp. {c1 }) and one let ηn denote the S-equivariant linear map Hn → C3 , ϕ → (ϕ(an ), ϕ(bn ), ϕ(cn )). Besides, one let (sn )n1 denote the real sequence such that s1 = 1 n . One easily shows that one has sn −→ 0. Let C30 be the set of and, for any n 1, sn+1 = 3s3sn +5 n→∞
elements in C3 the sum of whose coordinates is zero. We have the following Lemma 10.5. Let n 1. For any ϕ in Hn and p in Tn , one has |ϕ(p)| max{|ϕ(an )|, |ϕ(bn )|, |ϕ(cn )|}. In particular, the map ηn is an isomorphism. Suppose n 2. Let ϕ be in Hn such that ηn (ϕ) belongs to C30 and ψ = E(ϕ | θn−1 ). Then ψ belongs to Hn−1 and one has ηn−1 (ψ) = 2 3sn−1 +5 ηn (ϕ).
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Proof. The bound follows from the maximum principle, applied to the operator 13 . It implies that, for any n 1, the operator ηn is injective. Let ϕ be a function on Tn and, for any p in Tn , set δn ϕ(p) = ϕ(p) − 3ϕ(p). If p is not a vertex and δn ϕ(p) = ϕ(p) if p is a vertex. As ηn is injective, so is δn , and hence it is an isomorphism; in particular, ηn is onto, hence it is an isomorphism. 1 n−1 +2 For any n 2, set tn = 3s 3sn−1 +5 and un = 3sn−1 +5 . Recall that, as in Corollary 2.6, if S is a ntriangle and if p and q are two vertices of S, one let pq denote the unique point of S belonging to a (n − 1)-triangle containing p and admitting a neighbor belonging to the (n − 1)-triangle containing q. Let dn and en be the two neighbors of an in Tn . Let us prove by induction on n 2 that, for any ϕ in Hn , one has ϕ(dn ) + ϕ(en ) = sn (ϕ(bn ) + ϕ(cn )) + 2(1 − sn )ϕ(an ) and ϕ(an bn ) = tn ϕ(an ) + un (2ϕ(bn ) + ϕ(cn )). For n = 2, this is an immediate computation. If n 3 and the formula is true for n − 1, pick some function ϕ in Hn . Then, as ϕ(an bn ) = 3ϕ(an bn ), by applying the induction to the restriction of ϕ to the (n − 1)-triangle containing an , one has sn−1 ϕ(an ) + ϕ(an cn ) + 2(1 − sn−1 )ϕ(an bn ) + ϕ(bn an ) = 3ϕ(an bn ). As ηn is an isomorphism, there exists a unique (x, y, z) in C3 such that, for any ϕ in Hn , one has ϕ(an bn ) = xϕ(an ) + yϕ(bn ) + zϕ(cn ). As ηn is S-equivariant, one has, for any ϕ in Hn , ϕ(bn an ) = xϕ(bn ) + yϕ(an ) + zϕ(cn ) and ϕ(an cn ) = xϕ(an ) + yϕ(cn ) + zϕ(bn ). Thus sn−1 (1 + x) + 2(1 − sn−1 )x + y = 3x, sn−1 z + 2(1 − sn−1 )y + x = 3y, sn−1 y + 2(1 − sn−1 )z + z = 3z. By solving this system, one gets x = tn , y = 2un and z = un . Finally, by induction, one has ϕ(dn ) + ϕ(en ) = sn−1 ϕ(an bn ) + ϕ(an cn ) + 2(1 − sn−1 )ϕ(an ) = 3sn−1 un ϕ(bn ) + ϕ(cn ) + 2(1 − sn−1 + sn−1 tn )ϕ(an ), whence the result, since 3sn−1 un = sn = sn−1 (1 − tn ). Then, if ψ = E(ϕ | θn−1 ), one has, by Lemma 9.4, for any p in Tn−1 , ψ(p) = 1 θn−1 (q)=p ϕ(q). As θn−1 induces a graph isomorphism from each of the (n − 1)-triangles 3 of Tn onto Tn−1 , one deduces that ψ belongs to Hn−1 and that, in particular, by Lemma 9.1, if ηn (ϕ) is in C30 , one has 1 ϕ(an ) + ϕ(bn an ) + ϕ(cn an ) 3 1 = (1 + 4un )ϕ(an ) + (tn + un ) ϕ(bn ) + ϕ(cn ) 3 1 + 4un − tn − un 2 ϕ(an ) = ϕ(an ), = 3 3sn−1 + 5
ψ(an ) =
where, for the penultimate equality, one has used the relation ϕ(an ) + ϕ(bn ) + ϕ(cn ) = 0. By Sequivariance, one has the analogous formula at the two other vertices of Tn and hence ηn−1 (ψ) = 2 3sn−1 +5 ηn (ϕ). 2
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¯ = 3ϕ and that Corollary 10.6. Let ϕ be in L∞ (Γ¯ , μ) such that ϕ ϕ = 0.
s∈S ϕ
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◦ s = 0. One has
Proof. For any integer n 1, set ϕn = E(ϕ | θn ). By Lemma 10.4, one has ϕn ∈ Hn . As 3 ϕ s∈S ◦ s = 0, one has ηn (ϕn ) ∈ C0 . Therefore, if n 2, by Lemma 10.5, as ϕn−1 = 2 E(ϕn | θn−1 ), one has ηn−1 (ϕn−1 ) = 3sn−1 +5 ηn (ϕn ). Now, for any n 1, one has ϕn ∞ ϕ ∞ ∞ 2 and, as sn −→ 0, n=1 3sn +5 = 0. Therefore, one necessarily has, for any n 1, ηn (ϕn ) = 0, n→∞
hence, by Lemma 10.5, ϕn = 0, and, by Lemma 9.4, ϕ = 0.
2
We can now describe the eigenvectors with eigenvalue 3 in L1 (Γ¯ , μ): ¯ = 3ϕ. The function ϕ is constant μ-almost Lemma 10.7. Let ϕ be in L1 (Γ¯ , μ) such that ϕ everywhere. Proof. One can suppose that ϕ takes only real values. Let us prove that it suffices to handle the ¯ is positive, one has |ϕ| ¯ ¯ = 3|ϕ| and hence, as case where ϕ is nonnegative. Indeed, as |ϕ| ¯ ¯ has norm 3, |ϕ| = one 3|ϕ|. By studying the functions |ϕ| − ϕ and |ϕ| + ϕ, one can suppose ¯ ∗ λ = 3λ. has ϕ 0. Set ψ = s∈S ϕ ◦ s. The measure λ = ψμ is S-invariant and one has By Lemma 10.3, λ is proportional to μ, that is ψ is constant μ-almost everywhere. In particular, ψ is in L∞ (Γ¯ , μ). As one has 0 ϕ ψ , ϕ is in L∞ (Γ¯ , μ). Then, by Corollary 10.6, on a ϕ − 16 ψ = 0. 2 In order to extend this result to all complex measures on Γ¯ , we shall use a general, surely classical lemma. Let X be a compact metric space. Equip the space C 0 (X) with the uniform convergence topology. If λ is a complex Borel measure on X, recall the total variation |λ| of λ is the finite positive Borel measure on X such that, for any continuous nonnegative function g on X, one has g d|λ| = sup gh dλ X
h∈C 0 (X) h ∞ 1 X
(one may refer to [15, Chapter 6]). In particular, |λ| is the smallest positive Radon measure such that, for any continuous nonnegative function g on X, one has | X g dλ| X g d|λ|. Lemma 10.8. Let X be a compact metric space and P a positive operator with norm 1 on the space of continuous functions on X. For any Borel complex measure λ on X, one has |P ∗ λ| P ∗ |λ|. In particular, if P ∗ λ = λ, one has P ∗ |λ| = |λ|. function g on X, one has P g 0, hence | X P g dλ| Proof. For any nonnegative continuous ∗ ∗ ∗ ∗ X P g d|λ|. As the measure P |λ| is positive, one thus has |P λ| P |λ|. If P λ = λ, one has ∗ |λ| P |λ|, whence the equality, since P has norm 1. 2 We finally deduce the following ¯ ∗ λ = 3λ. One has λ = Proposition 10.9. Let μ be complex Borel measure on Γ¯ such that ¯ λ(Γ )μ.
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¯ one Proof. One can suppose λ takes real values. By applying Lemma 10.8 to the operator 13 , ∗ ¯ |λ| − λ and |λ| + λ, one can suppose λ is has |λ| = 3|λ|, so that, by studying the measures positive. Then, by Lemma 10.3, the measure s λ is proportional to μ. As one has 0 λ ∗ s∈S s∈S s∗ λ, λ is absolutely continuous with respect to μ. By Lemma 10.7, λ is thus proportional to μ. 2 11. Spectrum and spectral measures of Γ¯ ¯ Let us begin by noting that, as in We shall now come to the spectral study of the operator . Lemma 3.1, one has the following ¯2− ¯ and Π( ¯ ¯2− ¯ − 3)Π¯ ∗ = Π¯ ∗ ¯ − 3) = ¯ Π¯ . Lemma 11.1. One has ( Recall we let α denote the map that sends a point p of Γ¯ to the neighbor of p that does not belong to the 1-triangle containing p. As in Section 3, from Lemma 11.1, one deduces the following ¯ is the union of Λ and of the set n∈N f −n (0). The eigenspace Corollary 11.2. The spectrum of ¯ = 0 and associate to the eigenvalue −2 is the space of functions ϕ in L2 (Γ¯ , μ) such that Πϕ ϕ ◦ α = −ϕ. The eigenspace associate to the eigenvalue 0 is the space of functions ϕ in L2 (Γ¯ , μ) ¯ = 0 and ϕ ◦ α = ϕ. such that Πϕ Proof. Let, as in Corollary 3.6, K = Π¯ ∗ L2 (Γ¯ , μ) and H be the closed subspace of L2 (Γ¯ , μ) ¯ ¯ spanned by K and by K. By Lemma 11.1, one has f ()K ⊂ K and, as Π¯ ∗ is an isometry from 2 ¯ ¯ ¯ in L2 (Γ¯ , μ). We will L (Γ , μ) onto K, the spectrum of f () in K equals the spectrum of ¯ ¯ −1 K ∩ K seek to apply Lemma 3.3 to the operator in H . On this purpose, let us prove that 2 ∗ ∗ ¯ ¯ ¯ ¯ only contains constant functions. Let ϕ and ψ be in L (Γ , μ) such that Π ϕ = Π ψ . For any integer n 1, set ϕn = E(ϕ | θn ) and ψn = E(ψ | θn ). By Lemmas 9.4 and 10.4, for any p in Tn+1 − ∂Tn+1 , one has Π ∗ ϕn (p) = Π ∗ ψn (p). By proceeding as in the proof of Corollary 3.6, one deduces that, for any q in Tn , ϕn is constant on the neighbors of q. As any point of Tn is contained in a triangle and Tn is connected, ϕn is constant. As, by Lemma 9.4, one has ϕn −→ ϕ n→∞
¯ −1 K ∩ K equals the line of constant functions. By in L2 (Γ¯ , μ), ϕ is constant. Thus, the space ¯ Lemmas 3.3 and 11.1, the spectrum of in H thus equals the union of {3} and of the inverse ¯ in the space of functions with zero integral in L2 (Γ¯ , μ). image by f of the spectrum of Besides, proceeding as in Lemma 3.7, one sees that the orthogonal complement L of H in ¯ = 0 and L2 (Γ¯ , μ) is the direct sum of the space L−2 of the elements ϕ in L2 (Γ¯ , μ) such that Πϕ 2 ¯ ¯ ϕ ◦ α = −ϕ and of the space L0 of the elements ϕ in L (Γ , μ) such that Πϕ = 0 and ϕ ◦ α = ϕ. ¯ = −2 on L−2 and ¯ = 0 on L0 . Proceeding as in Lemma 3.8 and using Lemma 10.1, One has one sees that these subspaces are not reduced to {0}, since they contain triangular functions. As ¯ in L2 (Γ¯ , μ) equals the union in the proof 3.9, one deduces that the spectrum of of Corollary −n of Λ and n∈N f (0). Finally, as in the proof of Lemma 3.7, it remains to prove that L−2 and L0 are exactly the ¯ associate to the eigenvalues −2 and 0, that is ¯ does not admit the eigenvalue eigenspaces of ¯ ¯ Πϕ ¯ = 3Πϕ ¯ and −2 or 0 in H . Let ϕ be in H such that ϕ = −2ϕ. By Lemma 11.1, one has ¯ ¯ =0 hence by Lemma 10.7, Πϕ is constant. As ϕ is orthogonal to constant functions, one has Πϕ ¯ ¯ ¯ and Π ϕ = −2Πϕ = 0. As ϕ is in H , one thus has ϕ = 0. In the same way, if ϕ is in H and
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¯ = 0, one has ¯ Πϕ ¯ = −3Πϕ. ¯ Now, by an immediate computation, −3 does not belong to if ϕ ¯ Thus Π¯ ϕ = 0 and hence ϕ = 0, what should be proved. 2 the spectrum of . We also have an analogue of Lemma 4.1: ¯ and hence, for any ϕ and ψ in L2 (Γ¯ , μ), ¯ Π¯ ∗ = 2 + 1 Lemma 11.3. One has Π¯ 3 1 ¯ ψ ¯ Π¯ ∗ ϕ, Π¯ ∗ ψ = 2 ϕ, ψ + ϕ,
3 1 2 ¯ − 3 Π¯ ∗ ϕ, Π¯ ∗ ψ . ¯ − = 2 Π¯ ∗ ϕ, Π¯ ∗ ψ + 3 As in Section 4, one deduces the following ¯ in L2 (Γ¯ , μ) and Corollary 11.4. Let ϕ be in L2 (Γ¯ , μ), μ be the spectral measure of ϕ for 1 ∗ 2 ¯ in L (Γ¯ , μ). Then, one has ν( ) = 0 and, if, for any ν be the spectral measure of Π¯ ϕ for 2 x(x+2) 1 x = 2 , one sets τ (x) = 3(2x−1) , one has ν = L∗f,τ μ. 12. Eigenfunctions in L2 (Γ¯ , μ) ¯ In this section, we shall follow the plan of Section 5, in order to describe the eigenspaces of 2 ¯ in L (Γ , μ). As in Section 5, by using Lemmas 11.1 and 11.3, one proves the following analogue of Lemma 5.1: Lemma 12.1. Let H be the closed subspace of L2 (Γ¯ , μ) spanned by the image of Π¯ ∗ and by the ¯ Π¯ ∗ . Then, for any x in R − {0, −2}, x is an eigenvalue of ¯ in H if and only if y = one of ¯ in L2 (Γ¯ , μ). In this case, the map R¯ x which sends an eigenfunction f (x) is an eigenvalue of ¯ Π¯ ∗ ϕ induces an isomorphism between ϕ with eigenvalue y in L2 (Γ¯ , μ) to (x − 1)Π¯ ∗ ϕ + 2 the eigenspace associate to the eigenvalue y in L (Γ¯ , μ) and the eigenspace associate to the eigenvalue x in H and, for any ϕ, one has R¯ x ϕ 22 = 13 x(x + 2)(2x − 1) ϕ 22 . To describe the eigenfunctions with eigenvalue in n∈N f −n (0), we shall proceed as in Section 5. On this purpose, note again that, for any integer n 1, the space of edges that are exterior ¯ If ϕ is a function on Γ¯ which to n-triangles of Γ¯ may be identified in a natural way with Θ. is constant on edges which are exterior to n-triangles, we shall let P¯n ϕ denote the function on Θ¯ whose value at one point of Θ¯ is the value of ϕ on the associate edge which is exterior to ¯ denote the operator that sends a function ψ on Θ¯ to the n-triangles of Γ¯ . Besides, one still let ¯ ∗ λ = 4λ and function whose value at some point p of Θ¯ is q∼p ψ(q). This operator satisfies 2 ¯ λ), where λ is the measure on Θ¯ that has been introduced it is self-adjoint with norm 4 in L (Θ, at the end of Section 9. Lemma 12.2. The map P¯2 induces a Banach spaces isomorphism from the eigenspace ¯ λ). Let Q¯ 0 denote its inverse. For any of L2 (Γ¯ , μ) associate to the eigenvalue 0 onto L2 (Θ, 1 1 2 2 2 ¯ ¯ ψ in L (Θ, λ), one has Q0 ψ L2 (Γ¯ ,μ) = 2 ψ L2 (Θ,λ) − 12
ψ, ψL2 (Θ,λ) . ¯ ¯
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Proof. One proceeds as in Lemma 5.2 by using the characterization of eigenfunctions with eigenvalue 0 given in Corollary 11.2. The formula may be easily checked on triangular functions that are zero at the vertices of their definition triangle and the general case follows by density. 2 Recall that, for x in
n∈N f
−n (0),
κ(x) =
one let n(x) denote the integer n such that f n (x) = 0 and n(x)−1 k=0
f k (x)(2f k (x) − 1) . f k (x) + 2
From Lemmas 12.1 and 12.2, one deduces the following analogue of Proposition 5.3: Proposition 12.3. Let x be in n∈N f −n (0). The eigenfunctions with eigenvalue x in L2 (Γ¯ , μ) are constant on edges which are exterior to (n(x) + 1)-triangles in Γ¯ . The map P¯n(x)+2 induces a Banach spaces isomorphism from the eigenspace of L2 (Γ¯ , μ) associate to the eigenvalue x onto ¯ x denote its inverse. Then, for any ψ in L2 (Θ, ¯ λ). Let Q ¯ λ), one has Qx ψ 2 2 = L2 (Θ, L (Γ¯ ,μ) κ(x)(3 ψ 2L2 (Θ,λ) − 12 ψ, ψL2 (Θ,λ) ). ¯ ¯ Corollary 12.4. For any x in n∈N f −n (0), the eigenspace associate to x in L2 (Γ¯ , μ) has infinite dimension and is spanned by triangular functions that are zero at the vertices of their definition triangle. As in Section 5, the description of the eigenvalues associate to the elements of n∈N f −n (−2) is less precise. Let us begin by the case of the eigenvalue −2. We shall need supplementary informations on triangular functions that are eigenvectors with eigenvalue −2. On this purpose, pick some integer n 1 and some n-triangle S and let ES denote the space of functions ϕ on S such that Πϕ = 0 and, for any point p of S that is not a vertex of S, if q is the neighbor of p that does not belong to the triangle containing p, one has ϕ(q) = −ϕ(p). If S is Tn , one let En stand for ETn . By ¯ = −2ϕ, for Lemmas 9.4 and 9.5 and Corollary 11.2, if ϕ is an element of L2 (Γ¯ , μ) such that ϕ any integer n 1, one has E(ϕ | θn ) ∈ En . Proceeding as in Lemma 5.5, one proves the following Lemma 12.5. Let n 1, S be a n-triangle with vertices p, q and r and ϕ be in ES . One has ϕ(p) + ϕ(q) + ϕ(r) = 0. The space C30 = {(s, t, u) ∈ R3 | s + t + u = 0} is stable under the action of S on C3 . We endow it with the S-invariant hermitian norm . 0 such that, for any (s, t, u) in C30 , one has (s, t, u) 20 = 13 (|s|2 + |t|2 + |u|2 ). For any n 1, one let ρn denote the S-equivariant linear map En → C30 , ϕ → (ϕ(an ), ϕ(bn ), ϕ(cn )), Fn the kernel of ρn and Gn the orthogonal complement of Fn in En for the norm of L2 (Γ¯ , μ). By Lemma 10.1, the elements of Fn are eigenvectors with ¯ eigenvalue −2 of . Lemma 12.6. Let n 1. One has dim Fn = 12 (3n−1 − 1). The map ρn is onto and, for any ϕ in Gn , one has ϕ 2L2 (Γ¯ ,μ) = ( 59 )n−1 ρn (ϕ) 20 . Finally, if n 2 and if ψ = E(ϕ | θn−1 ), ψ belongs to Gn−1 and ρn−1 (ψ) = 23 ρn (ϕ).
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a ,b2
Fig. 13. The functions ϕ2 2
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and ψ2 .
Proof. Let n 1, S be a n-triangle and p and q be distinct vertices of S. Define a function p,q p,q p,q ϕS on S in the following way. If n = 1, one sets ϕS (p) = 1, ϕS (q) = −1 and one says that p,q ϕS is zero at the third point of S. If n 2, let still pq and qp denote the points defined in Corollary 2.6: the point pq belongs to the (n − 1)-triangle P containing p in S, the point qp belongs to the (n − 1)-triangle Q containing q in S and the points pq and qp are neighbors. p,q p,pq qp,q One defines ϕS as the function whose restriction to P is ϕP , whose restriction to Q is ϕQ p,q and whose restriction to the third (n − 1)-triangle of S is zero. One easily checks that ϕS p,q p,q an ,bn belongs to ES . If S = Tn , one let ϕn stand for ϕTn . As one has ρn (ϕn ) = (1, −1, 0) and ρn (ϕnan ,cn ) = (1, 0, −1), the map ρn is onto. For n 2, let ψn denote the function on Tn whose restriction to the (n − 1)-triangle an bn ,an cn bn cn ,bn an An (resp. Bn , resp. Cn ) containing an (resp. bn , resp. cn ) equals ϕA (resp. ϕB , n n
resp. ϕCcnnan ,cn bn ). Then, one easily checks that ψn belongs to Fn . These functions are pictured in Fig. 13. Let us now establish by induction on n 1 the formulae of the lemma on the dimension of Fn and the norm of the elements of Gn . For n = 1, one has F1 = {0} and the map ρ1 is an isomorphism, so that the formula on norms follows from Lemma 9.3. Let thus suppose n 2 and the formulae have been proved for n − 1. We will explicitely construct the inverse map of ρn , depending on the one of ρn−1 . For any triangle S, let FS be the set of elements of ES that are zero at the vertices of S and GS be the orthogonal complement of FS with respect to the natural scalar product on 2 (S). For any (s, t, u) in C30 , let τ (s, t, u) be the unique function on Tn that u−s s−t u−t s−u takes the value s at an , t at bn , u at cn , t−s 3 at an bn , 3 at an cn , 3 at bn an , 3 at bn cn , 3 at cn an and t−u 3 at cn bn and whose restriction to An (resp. Bn , resp. Cn ) belongs to GAn (resp. GBn , resp. GCn ). Then τ (s, t, u) clearly belongs to En and ρn (τ (s, t, u)) = (s, t, u). Besides, one has, by Lemma 9.3 and by induction,
τ (s, t, u), ψn
L2 (Γ¯ ,μ)
=
1
an bn ,an cn τ (s, t, u), ϕA 2 (An ) n n 3
bn cn ,bn an + τ (s, t, u), ϕB 2 (B n ) n
+ τ (s, t, u), ϕCcnnan ,cn bn 2 (C ) n
=
5n−2
1 (t − s) − (u − s) + (u − t) − (s − t) 3 9n−1 + (s − u) − (t − u) = 0.
Conversely, one easily checks, by an analogous scalar product computation, that, if ϕ is an element of En that is orthogonal to ψn and whose restriction to An (resp. Bn , resp. Cn ) is in GAn
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(resp. GBn , resp. GCn ), then ϕ belongs to the image of τ . As both these spaces have dimension 2, they coincide and τ is the inverse map of ρn . In particular, Fn is spanned by ψn and the elements that are zero at the vertices of (n − 1)-triangles, so that dim Fn = 3 dim Fn−1 + 1, whence the dimension computation, by induction. Besides, again by Lemma 9.3 and by induction, for any ϕ in Gn , if ρn (ϕ) = (s, t, u), one has, by the definition of τ , ϕ 2L2 (Γ¯ ,μ) = =
2 2 s − t 2 5n−2 2 2 2 + 2 t − u + 2 s − u |s| + |t| + |u| + 2 n−1 3 3 3 9 1 5n−1 2 |s| + |t|2 + |u|2 n−1 39
(taking in account, for the last equality, that s + t + u = 0). The formula on norms follows, by induction. Finally, for n 2, pick ϕ in Gn and set ψ = E(ϕ | θn−1 ). As in the proof of Lemma 10.5, one deduces from Lemma 9.4 and the fact that θn−1 induces graph isomorphisms between the (n − 1)-triangles of Tn and Tn−1 that, as ϕ belongs to En , ψ belongs to En−1 . As the elements of Fn−1 are zero at the vertices of the (n − 1)-triangles, they belong to Fn too, hence they are orthogonal to ϕ, so that ψ belongs to Gn−1 . By Lemmas 9.1 and 9.4, one has ψ(an ) = 1 3 (ϕ(an ) + ϕ(bn an ) + ϕ(cn an )) and hence, by the formulae above, if ρn (ϕ) = (s, t, u), one has s−u 2 2 ψ(an ) = 13 (s + s−t 3 + 3 ) = 3 s and ρn−1 (ψ) = 3 ρn (ϕ). 2 ¯ associate to the eigenvalue −2: We can now describe the eigenspace of ¯ has infinite dimension and is Lemma 12.7. The eigenspace associate to the eigenvalue −2 of spanned by triangular functions that are zero at the vertices of their definition triangle. Proof. As, by Lemma 12.6, for any n 1, the space Fn has dimension 12 (3n−1 − 1) and, by Lemma 10.1, its elements are eigenfunctions with eigenvalue −2, the eigenspace associate to the eigenvalue −2 has infinite dimension. Let ϕ be an eigenfunction with eigenvalue −2 in L2 (Γ¯ , μ) that is orthogonal to the eigenfunctions that are triangular and zero at the vertices of their definition triangle. Let us prove that ϕ is zero. For any integer n 1, let ϕn = E(ϕ | θn ). By Corollary 11.2, one has Π¯ ϕ = 0 and ϕ ◦ α = −ϕ and hence, by Lemmas 9.4 and 9.5, for any n 1, ϕn belongs to En . As ϕ is orthogonal to the elements of Fn , ϕn belongs to Gn . If n 2, as ϕn−1 = E(ϕn | θn−1 ), by Lemma 12.7, one has ρn−1 (ϕn−1 ) = 23 ρn (ϕn ). Hence, there exists v in C30 such that, for any n 1, one has ρn (ϕn ) = ( 32 )n−1 v, so that, again by Lemma 12.7, ϕn 2L2 (Γ¯ ,μ)
n−1 n−1 2 5 5 ρn (ϕn ) 0 = = v 20 . 9 4
As, by Lemma 9.4, one has ϕn −→ ϕ in L2 (Γ¯ , μ), one thus has necessarily v = 0, hence, for n→∞
any n 1, ϕn = 0 and ϕ = 0, what should be proved.
2
From Lemmas 12.1 and 12.7, one deduces by induction the following
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Corollary 12.8. For any x in n∈N f −n (−2), the eigenspace associate to the eigenvalue x has infinite dimension and is spanned by triangular functions that are zero at the vertices of their definition triangle. 13. Spectral decomposition of L2 (Γ¯ , μ) the space In this section, we shall prove that L2 (Γ¯ , μ) is the orthogonal direct sum of −n of constant functions, the eigenspaces associate to the elements of the set f (−2) ∪ n∈N −n (0) and the cyclic subspaces spanned by 1-triangular functions ϕ such that Πϕ ¯ f = 0. n∈N Let us begin by describing these cyclic subspaces. ¯ ¯ + 2)ϕ = ( ¯ − 1)Π¯ ∗ ϕ and Π¯ ϕ ¯ = (1 + 1 )ϕ. Lemma 13.1. Let ϕ be in E1 . One has ( 3 Proof. Let (s, t, u) = (ϕ(a1 ), ϕ(b1 ), ϕ(c1 )). One has, by definition, s + t + u = 0. Let p be in Γ¯ . After an eventual action of the group S, one can suppose that Π¯ p belongs to the keel ¯ on the 1-triangle containing p and on its B0 from Section 8. Then, the values of ϕ and of ϕ ¯ Π¯ ∗ ϕ on neighbors are those described by Fig. 14. In the same way, the values of Π¯ ∗ ϕ and of the 1-triangle containing p and on its neighbors are those described by Fig. 15. If θ1 (p) = a1 or ¯ + 2)ϕ(p) = 2s + 2t + u = s + t = ( ¯ − 1)Π¯ ∗ ϕ(p); if θ1 (p) = c1 , θ1 (p) = b1 , one hence has ( ∗ ¯ ¯ ¯ ¯ + 2)ϕ = one has ( + 2)ϕ(p) = 2s + t + 2u = s + u = ( − 1)Π ϕ(p). Thus, we do have ( ∗ ¯ ¯ ( − 1)Π ϕ. ¯ = 0, so that, by applying Π¯ to the preceding identity, one gets By definition, one has Πϕ ¯ ¯ ¯ ¯ ¯ ¯ ¯ where, for the last equality, we made use of Π ϕ = Π ( + 2)ϕ = Π Π¯ ∗ ϕ − ϕ = (1 + 13 )ϕ, Lemma 11.3. 2 Thanks to Lemma 13.1, we shall proceed as in Section 6 to determine the spectral measures of the elements of E1 . Let us begin by proving that these measures do not give mass to the points −2 and 0. Lemma 13.2. Let ϕ be in E1 and ψ be an eigenfunction with eigenvalue −2 or 0 in L2 (Γ¯ , μ). One has ϕ, ψ = 0. ¯ =0 Proof. Suppose ψ is an eigenvector with eigenvalue 0. By Corollary 11.2, one has Πψ and ψ ◦ α = ψ and, by Corollary 12.4, one can suppose that, for a certain integer n 2, ψ is n-triangular, with value 0 at the vertices of Tn . Let p, q and r be the vertices of a 2-triangle S of Tn and let pq, qp, pr, rp, qr and rq be the other points of S, with the convention from Corollary 2.6. Then, one has ψ(qp) = ψ(pq) and ψ(rp) = ψ(pr), hence ψ(p) + ψ(qp) + ψ(rp) = 0 and, by using the analogous identities on the other 1-triangles of S, by Lemma 9.1, as ϕ is 1-triangular, one has s∈S
ϕ(s)ψ(s) = ϕ(p) ψ(p) + ψ(qp) + ψ(rp) + ϕ(q) ψ(q) + ψ(pq) + ψ(rq) + ϕ(r) ψ(r) + ψ(pr) + ψ(qr) = 0
and hence, by Lemma 9.3, ϕ, ψ = 0.
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¯ Fig. 14. Values of ϕ and of ϕ.
¯ Π¯ ∗ ϕ. Fig. 15. Values of Π¯ ∗ ϕ and of
Let us now handle the case of the eigenvalue −2. For any n 1, let En and Fn be as in Section 12. Let (s, t, u) = (ϕ(a1 ), ϕ(b1 ), ϕ(c1 )). Let us prove by induction on n that, if ψ belongs to En , one has
ϕ(p)ψ(p) = 2n−1 sψ(an ) + tψ(bn ) + uψ(cn ) .
p∈Tn
For n = 1, the result is trivial. Suppose n 2 and the result has been established for n. Then, by applying the induction to the restriction of ψ to (n − 1)-triangles of Tn , one gets, as ϕ is 1-triangular, by Lemma 9.1, p∈Tn
ϕ(p)ψ(p) = 2n−2 ψ(an )s + ψ(an bn )t + ψ(an cn )u + ψ(bn )t + ψ(bn an )s + ψ(bn cn )u + ψ(cn )u + ψ(cn an )s + ψ(cn bn )t .
Now, one has ψ(an bn ) + ψ(bn an ) = 0 and, by Lemma 12.5, ψ(an ) + ψ(an bn ) + ψ(an cn ) = 0, so that ψ(bn an ) + ψ(cn an ) = ψ(an ). By using this identity and the analogous formulae at the other vertices of Tn , we get
ϕ(p)ψ(p) = 2n−1 sψ(an ) + tψ(bn ) + uψ(cn ) ,
p∈Tn
what should be proved. In particular, for ψ in Fn , one has, by Lemma 9.3, ϕ, ψ = 0 and hence, by Lemma 12.7, this is still true for any eigenvector ψ with eigenvalue −2. 2 Corollary 13.3. Let ϕ be in E1 and ψ be an eigenfunction with eigenvalue in −n (0). One has ϕ, ψ = 0. f n∈N
n∈N f
−n (−2) ∪
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Proof. By Corollary 11.2 and Lemmas 12.1 and 13.2, it suffices to prove that, for x in R, if ψ is ¯ Π¯ ∗ ψ = 0. Now, an eigenvector with eigenvalue x and if ϕ, ψ = 0, one has ϕ, Π¯ ∗ ψ = ϕ, ¯ = 0, hence ϕ, Π¯ ∗ ψ = 0. Besides, by Lemma 13.1, one has by definition, one has Πϕ
1 x ∗ ¯ ¯ ¯ ¯ ¯
ϕ, Π ψ = Π ϕ, ψ = ϕ, 1 + ψ = 1 +
ϕ, ψ = 0, 3 3 what should be proved.
2
1 1 (x+3)(x−1) . As for Corollary 6.3, Set, for any x = −3, j (x) = 13 3−x x+3 and, for x = 2 , ζ (x) = 3 2x−1 we deduce from Lemma 13.1 and Corollary 11.4 the following
Corollary 13.4. Let νζ the unique Borel probability on Λ such that L∗ζ νζ = νζ . For any ϕ in E1 , the spectral measure of ϕ is ϕ 22 j νζ . Proof. As the proof of this result is analogous to the one of Corollary 6.3, we just give its big steps. Let λ be the spectral measure of ϕ. By Lemma 13.2, one has λ(−2) = 0. Set, for j x(x−1)2 x∈ / {−2, 12 }, θ (x) = 3(x+2)(2x−1) . One has θ = j ◦f ζ and, by Corollary 11.4 and Lemma 13.1, λ = L∗θ λ. By Lemma 9.3, ϕ is orthogonal to constant functions. Therefore, by Lemma 10.7, one has λ(3) = 0. Moreover, by Corollaries 11.2 and 13.3, the measure λ is concentrated on Λ. The function ζ is positive on Λ and Lζ (1) = 1. By Lemma 6.2, there exists a unique Borel probability νζ on Λ such that L∗ζ νζ = νζ . Proceeding as in the proof of Corollary 6.3, one proves that the measures λ and j νζ are proportional. As one has Lζ j = 1, one gets λ = ϕ 22 j νζ . 2 1 Let l denote the function x → x on Λ and set, for x = 1, m(x) = x+2 x−1 and, for x = 2 , ξ(x) = 1 x(x−1) 2 ¯ ¯ 3 2x−1 . Let Φ denote the closed subspace of L (Γ , μ) spanned by the elements of E1 and their ¯ and, as in Section 12, let ρ1 design the S-equivariant isomorphism images by the powers of from E1 onto C30 . Still endow C30 with the hermitian norm that equals one third of the canonical norm and denote by ·,·0 the associate scalar product. By Lemma 9.3, the map ρ1 is an isometry. Identify the Hilbert spaces L2 (j νζ , C30 ) and L2 (j νζ ) ⊗ C30 and, for any polynomial p in C[X] ¯ −1 (v). We have an analogue of Proposition 6.4: ⊗ v = p()ρ and for any v in C30 , set p 1
¯ Proposition 13.5. The map g → gˆ induces a S-equivariant isometry from L2 (j νζ , C30 ) onto Φ. ¯ Π¯ and Π¯ ∗ . For any g in L2 (j νζ , C3 ), one has The subspace Φ¯ is stable by the operators , 0 ¯ gˆ = lg, Π¯ gˆ = L ξ g, Π¯ ∗ gˆ = m(g ◦ f ). Proof. Let p be in C[X]. The map C30 × C30 → C,
¯ −1 (v), ρ −1 (w) 2 ¯ (v, w) → p()ρ 1 1 L (Γ ,μ)
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is a S-invariant sesquilinear form. As the representation of S on C30 is irreducible, this sesquilinear form is proportional to the scalar product ·,·0 . By Lemma 9.3 and Corollary 13.4, for v in C30 , one has
¯ −1 (v), ρ −1 (v) 2 ¯ = v 20 p()ρ 1 1 L (Γ ,μ)
pj dνζ , Γ¯
therefore, for any p and q in C[X], for any v and w in C30 , one has
p ⊗ v, q ⊗ wL2 (Γ¯ ,μ) = v, w0 p, qL2 (j νζ ) and hence the map g → gˆ induces an isometry from L2 (j νζ , C30 ) onto a closed subspace of L2 (Γ¯ , μ). As this subspace is spanned by the elements of E1 and their images by the powers ¯ by definition, it equals Φ. ¯ of , The rest of the proof is analogous to the one of Proposition 6.4. ¯ and the formula for ¯ result from the very definition of the objects in The stability of Φ¯ by question. A direct computation shows that Lξ (1) = 0 and that Lξ (l) = 1 + 13 l, so that, for any nonnegative integer n, one has Lξ (f n ) = 0 and Lξ (f n l) = l n (1 + 13 l). Now, by Lemmas 11.1 and 13.1, ¯ () ¯ n ϕ) = 0 and Π¯ (f () ¯ n ϕ) ¯ = ¯ n (1 + 1 )ϕ. ¯ for any ϕ in E1 , one has Π(f The space Φ¯ is 3 ⊗ v = Lξ (p) ⊗ v. As ζ is thus stable by Π¯ and, for any p in C[X] and v in C30 , one has Π¯ p positive on Λ, there exists a real number c > 0 such that, for any x in Λ, one has |ξ(x)| cζ (x), so that, for any Borel function g on Λ, one has |Lξ (g)| cLζ (|g|). Proceeding as in the proof of Proposition 6.4, one proves that Lζ is bounded in L2 (j νζ ). One deduces that Lξ is bounded and the identity concerning Π¯ follows, by density. Finally, by Lemmas 11.1 and 13.1, for any p in C[X] and ϕ in E1 , one has ¯ ¯ ¯ + 2)ϕ. By Corollary 11.2, 1 does not belong to the spec¯ − 1)Π¯ ∗ (p()ϕ) = p(f ())( ( ¯ trum of , so that, by density, for any rational function p whose poles do not belong to the ¯ ¯ and hence the space Φ¯ is stable by Π¯ ∗ . ¯ one has Π¯ ∗ (p()ϕ) = (m(p ◦ f ))()ϕ spectrum of , j (x) x(x+2) 2 Moreover, as, for any x in Λ, one has m(x) j (f (x)) = (x−1)(x+3) , one gets, by an elementary j computation, Lζ (m2 j ◦f ) = 1 and, for any g in L2 (j νζ ),
Λ
m(g ◦ f )2 j dνζ =
j 2 m |g ◦ f | (j ◦ f ) dνζ = |g|2 j dνζ . j ◦f 2
Λ
The formula for Π¯ ∗ follows, by density.
Λ
2
Let us now deal with the other S-isotypic components of the space L2 (Γ¯ , μ). Let ε : S → {−1, 1} denote the signature morphism. We shall say that a function ϕ on Γ¯ is (S, ε)-semiinvariant if, for any s in S, one has ϕ ◦ s = ε(s)ϕ.
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Proposition 13.6. For any integer n 1, the space of S-invariant n-triangular functions on Γ¯ ¯ and the characteristic polynomial of ¯ in this space is is stable by (X − 3)
n−2
p 3n−2−p +2n−2p−1 p 3n−2−p −2n+2p+3 4 4 f (X) f (X) + 2 .
p=0
For any integer n 2, the space of (S, ε)-semi-invariant n-triangular functions on Γ¯ is stable ¯ and the characteristic polynomial of ¯ in this space is by n−2
3n−2−p −2n+2p+3 p 3n−2−p +2n−2p−1 4 4 l ◦ f p (X) f (X) + 2 .
p=0
Proof. These spaces are stable by Lemma 10.1. The computation of the characteristic polynomials is obtained proceeding as in the proof of Proposition 7.5. 2 From this proposition, we deduce, using Lemma 9.4, the following ¯ in the space of S-invariant elements of L2 (Γ¯ , μ) is discrete. Corollary 13.7. The spectrum of ¯ in this space are 3, which is simple, and the elements of n∈N f −n (−2) ∪ The eigenvalues of −n (0). The spectrum of ¯ in the space of (S, ε)-semi-invariant elements of L2 (Γ¯ , μ) is n∈N f ¯ in this space are the elements of n∈N f −n (−2) ∪ n∈N f −n (0). discrete. The eigenvalues of The proof of Theorem 1.3 ends with the following Proposition 13.8. Let Φ¯ ⊥ be the orthogonal complement of Φ¯ in L2 (Γ¯ , μ). The spectrum ¯ in Φ¯ ⊥ is discrete. Its eigenvalues in this space are 3, which is simple, and the elements of of n∈N f −n (−2) ∪ n∈N f −n (0). The proof of this proposition is analogous to the one of Proposition 6.9. It needs us to introduce objects that will play the role of the spaces Ln , n ∈ N, of this proof. Let us use the notations from Section 8 and recall that, by construction, if p is a point of Γ¯ such that θ1 (p) = a1 , the keel of p is B0 or riB0 . For any integer n 1, let Bn be the set which is the union of Tn − ∂Tn and of the set of the six pairs of the form (d, B) where d belongs to ∂Tn and B is one of the two keels for which there exist points p of B with θn (p) = d. Denote by τn the locally constant map Γ¯ → Bn such that, for any p in Γ¯ , if p is not a vertex of some n-triangle, one has τn (p) = θn (p) and, if p is a vertex of some n-triangle, τn (p) is the pair (θn (p), B) where B is the keel containing p. Finally, let say that a function ϕ on Γ¯ is τn -measurable if one has ϕ = ψ ◦ τn , where ψ is some function defined on Bn . The interest of this definition comes from the following Lemma 13.9. Let n 1 be an integer and ϕ be a τn+1 -measurable function on Γ¯ . Then, the ¯ and Π¯ ϕ ¯ are τn -measurable. functions Πϕ Proof. Let p be a point of Γ¯ . If p is not a vertex of some n-triangle, the triangle Π¯ −1 p does not contain a vertex of some (n + 1)-triangle. In the same way, none of the neighbors of Π¯ −1 p
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is a vertex of some (n + 1)-triangle. Thus, for any of the points q appearing in the computa¯ tion of Π¯ ϕ(p) and Π¯ ϕ(p), one has τn (q) = θn (q). Therefore, by the definition of θn and by ¯ ¯ Lemma 9.1, Πϕ(p) and Π¯ ϕ(p) only depend on θn (p). If p is now a vertex of some n-triangle, the neighbor q of p that does not belong to this ntriangle is itself a vertex of some n-triangle and, by Lemma 8.3, the keel of q is determined by the one of p. In particular, τn (q) is determined by τn (p). Only one of the antecedents of the point p by the map Π¯ is a vertex of some (n + 1)-triangle. By Lemma 8.8, it is the one whose keel equals the one of p. In particular, the image by τn+1 of this point r is determined by τn (p). In the same way, the image by τn+1 of the unique antecedent s of q that is a vertex of some (n + 1)-triangle only depends on τn (q), and so on τn (p). The point s is the neighbor of r that does not belong to Π¯ −1 p. Finally, the two other points of Π¯ −1 p and their neighbors that do not belong to Π¯ −1 p are not vertices of some (n + 1)-triangle and hence their image by τn+1 is their image by θn+1 ¯ ¯ that only depend on θn (p). Again, Πϕ(p) and Π¯ ϕ(p) only depend on θn (p). 2 ¯ is simple. By CorollarProof of Proposition 13.8. By Lemma 10.7, the eigenvalue 3 of ies 12.4 and 12.8, the eigenspaces associate to the elements of n∈N f −n (−2) ∪ n∈N f −n (0) are nonzero. Let P¯ denote the orthogonal projector from L2 (Γ¯ , μ) onto Φ¯ ⊥ and, for any ϕ and ψ in L2 (Γ¯ , μ), denote by λϕ,ψ the unique complex Borel measure on R such that, for any ¯ polynomial p in C[X], one has R p dλϕ,ψ = p()ϕ, ψ. By Proposition 13.5, the operator P¯ ∗ ¯ ¯ ¯ commutes with , Π and Π . By Lemma 9.4, to prove the proposition, it suffices to establish that, for any integer n 1, for any τn -measurable function ϕ, for any L2 (Γ¯ , μ), the mea ψ in −n −n sure λP¯ ϕ,ψ is atomic and concentrated on the set n∈N f (3) ∪ n∈N f (0). Let us prove this result by induction on n. For n = 1, the τ1 -measurables functions are those functions that only depend on the keel. One easily checks that this space is spanned by the constant functions, a line of (S, ε)-semi-invariant ¯ In this case, the description of spectral functions, the elements of E1 and their images by . measures immediately follows from Corollaries 13.4 and 13.7. If the result is true for some integer n 1, let us pick some τn+1 -measurable function ϕ. ¯ and Π¯ ϕ ¯ are τn -measurable and, by induction, for any Then, by Lemma 13.9, the functions Πϕ ψ in L2 (Γ¯ , μ), the measures λΠ¯ P¯ ϕ,ψ = λP¯ Πϕ,ψ and λΠ¯ ¯ P¯ ϕ,ψ = λP¯ Π¯ ϕ,ψ are atomic and con¯ ¯ −n −n centrated on the set n∈N f (3) ∪ n∈N f (0). Proceeding as in Lemma 6.10, one deduces that the measures λP¯ ϕ,Π¯ ∗ ψ and λ¯ P¯ ϕ,Π¯ ∗ ψ = λP¯ ϕ,¯ Π¯ ∗ ψ are atomic and concentrated on the set −n (3) ∪ −n (0). Now, by Corollary 11.2, the spectrum of ¯ in the orthogonal n∈N f n1 f 2 ∗ ¯ ¯ ¯ Π¯ ∗ complement of the subspace of L (Γ , μ) spanned by the image of Π and by the one of 2 ¯ equals {−2, 0}. Therefore, for any ψ in L (Γ , μ), the measure λP¯ ϕ,ψ is atomic and concentrated on the set n∈N f −n (3) ∪ n∈N f −n (0). The result follows. 2 ´ 14. The Sierpinski graph In this section, we will quickly explain how the results that have been obtained in this article for the Pascal graph Γ may be transfered to the Sierpi´nski graph Θ pictured in Fig. 2. As explained in Section 2, the graph Θ identifies with the edges graph of Γ . If ϕ is some function on Γ , one let Ξ ∗ ϕ denote the function on Θ such that, for any neighbor points p and q in Γ , the value of Ξ ∗ ϕ on the edge associate to p and q is ϕ(p) + ϕ(q). One let Ξ denote the adjoint of Ξ ∗ and one immediately verifies the following
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Lemma 14.1. One has ( − 1)Ξ ∗ = Ξ ∗ and Ξ Ξ ∗ = 3 + . The restriction of to the orthogonal complement of the image of Ξ ∗ in 2 (Θ) is the operator of multiplication by −2. Through this lemma, all the results of this article transfer from the Pascal graph to the Sierpi´nski graph. They could also be obtained directly in the Sierpi´nski graph, by considering the suitable operators in 2 (Θ). We will only describe the continuous spectrum of Θ and translate Theorem 1.1: this answers the question asked by Teplyaev in [18, §6.6]. For x in R, set k(x) = x + 2 and t (x) = x + 1. From Lemma 14.1, one deduces the following Lemma 14.2. Let ϕ be in 2 (Γ ), μ be the spectral measure of ϕ for in 2 (Γ ) and λ the spectral measure of Ξ ∗ ϕ for in 2 (Θ). Then, one has λ = k(t∗ μ). For any x in R, set g(x) = x 2 − 3x = f (x − 1) + 1. One let Σ = t (Λ) denote the Julia set of g. x−1 For any x in R, let c(x) = (x +2)(4−x) = k(x)h(x −1) and, for x = 32 , γ (x) = 2x−3 = ρ(x −1). One let νγ = t∗ νρ denote the unique Lg,γ -invariant probability measure on Σ. Let still ϕ0 denote the function on Γ that appears in Section 6 and set θ0 = Ξ ∗ ϕ0 (this is the function which is denoted by 1∂∂V in [18, §6]). From Theorem 1.1 and Lemmas 14.1 and 14.2, one deduces the following theorem, that completes the description of the spectrum of Θ by Teplyaev in [18]: Theorem 14.3. The spectrum of in 2 (Θ) is the union of Σ and the set n∈N g −n (−2). The 2 (Θ) is the measure cν , the eigenvalues of in 2 (Θ) are spectral measure for in γ of θ0 −n the elements of n∈N f (−2) ∪ n∈N f −n (−1) and the associate eigenspaces are spanned by finitely supported functions. Finally, the orthogonal complement of the sum of the eigenspaces of in 2 (Θ) is the cyclic subspace spanned by θ0 . References [1] L. Bartholdi, R. Grigorchuk, On the spectrum of Hecke type operators related to some fractal groups, Proc. Steklov Inst. Math. 231 (2000) 1–41. [2] L. Bartholdi, W. Woess, Spectral computations on lamplighter groups and Diestel–Leader graphs, J. Fourier Anal. Appl. 11 (2005) 175–202. [3] P.J. Grabner, W. Woess, Functional iterations and periodic oscillations for simple random walk on the Sierpi´nski graph, Stochatstic Process. Appl. 69 (1997) 127–138. ˘ k, ´ Asymptotic aspects of Schreier graphs and Hanoi Towers groups, C. R. Math. Acad. Sci. [4] R. Grigorchuk, Z. Suni Paris 342 (2006) 545–550. ˘ k, ´ Schreier spectrum of the Hanoi towers group on three pegs, Proc. Sympos. Pure Math., [5] R. Grigorchuk, Z. Suni in press. [6] R. Grigorchuk, V. Nekrashevych, Self-similar groups, operator algebras and Schur complement, J. Mod. Dyn. 1 (2007) 323–370. [7] R. Grigorchuk, A. Zuk, The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps, in: Random Walks and Geometry, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 141–180. [8] O.D. Jones, Transition probabilities for the simple random walk on the Sierpi´nski graph, Stochatstic Process. Appl. 61 (1996) 45–69. [9] B. Krön, Green functions on self-similar graphs and bounds for the spectrum of the Laplacian, Ann. Inst. Fourier 52 (2002) 1875–1900. [10] J. Kigami, A harmonic calculus on the Sierpi´nski spaces, Japan J. Appl. Math. 6 (1989) 259–290. [11] J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc. 335 (1993) 721–755. [12] F. Ledrappier, Un champ markovien peut être d’entropie nulle et mélangeant, C. R. Math. Acad. Sci. Paris 287 (1978) 561–563.
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[13] W. Parry, M. Pollicott, Zeta functions and the periodic orbit structure for hyperbolic dynamics, Astérisque 187–188 (1990), Société Mathématique de France, Paris. [14] R. Rammal, G. Toulouse, Spectrum of the Schrödinger equation on a self-similar structure, Phys. Rev. Lett. 49 (1982) 1194–1197. [15] W. Rudin, Real and Complex Analysis, McGraw–Hill Ser. Higher Math., McGraw–Hill Book Co., New York– Düsseldorf–Johannesburg, 1974. [16] Ch. Sabot, Spectral properties of self-similar lattices and iteration of rational maps, Mem. Soc. Math. Fr. (N.S.) 92 (2003). [17] R. Strichartz, Periodic and almost periodic functions on infinite Sierpi´nski gaskets, Canad. J. Math., in press. [18] A. Teplyaev, Spectral analysis on infinite Sierpi´nski gaskets, J. Funct. Anal. 159 (1998) 537–567.
Journal of Functional Analysis 256 (2009) 3461–3469 www.elsevier.com/locate/jfa
Second fundamental form and gradient of Neumann semigroups ✩ Feng-Yu Wang a,b,∗ a School of Mathematical Sci. and Lab. Math. Com. Sys., Beijing Normal University, Beijing 100875, China b Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, UK
Received 3 September 2008; accepted 11 December 2008 Available online 20 December 2008 Communicated by Paul Malliavin
Abstract The second fundamental form for the boundary of a compact Riemannian manifold is described by a short time behavior for the gradient of Neumann semigroups. As an application, we prove that the manifold is convex if and only if the Neumann heat semigroup Pt satisfies the gradient estimate |∇Pt f | eKt Pt |∇f | for some constant K ∈ R and all t 0, f ∈ C 1 . © 2008 Elsevier Inc. All rights reserved. Keywords: Gradient estimate; Asymptotic formula; Second fundamental form; Neumann semigroup
1. Introduction Let M be a connected compact Riemannian manifold with boundary ∂M. Let N be the inward unit normal vector field of ∂M. Then the second fundamental form of ∂M is defined as I(v1 , v2 ) := −∇v1 N, v2 ,
v1 , v2 ∈ T ∂M.
We call the boundary ∂M (or the manifold M) convex if I(v, v) 0 for all v ∈ T ∂M. ✩
Supported in part by NNSFC (10721091) and the 973-Project.
* Address for correspondence: School of Mathematical Sci. and Lab. Math. Com. Sys., Beijing Normal University,
Beijing 100875, China. E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.12.010
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Let L = + Z for some C 1 vector field Z. Let Pt be the Neumann semigroup generated by L on M. It is well known that if ∂M is convex then the gradient estimate (cf. [5]) |∇Pt f | eKt Pt |∇f |,
f ∈ C 1 (M), t 0
(1.1)
holds provided Ric −∇Z −K,
(1.2)
where K ∈ R is a constant. In general (i.e. ∂M is not necessarily convex), (1.1) implies (1.2) due to a standard argument of Bakry (cf. [2]). So, for convex ∂M these two inequalities are equivalent (see e.g. [2,3,6] for more equivalent statements). On the other hand, for non-convex domains weaker gradient estimates were derived in [7], which implies that for any p > 1 there exists c > 0 such that (see Proposition A.1 below) √ 1/p |∇Pt f | Pt |∇f |p exp[c t],
t ∈ [0, 1], f ∈ C 1 (M)
(1.3)
holds. Obviously, the exponential upper bound in (1.1) for convex M is quite different√from that in (1.3) for non-convex M. Our first result says that (1.3) is sharp in the sense that t cannot be replaced by any smaller function t → S(t) with limt→0 t −1/2 S(t) = 0. Consequently, (1.1) implies the convexity of ∂M. Theorem 1.1. If there exist p 1 and a positive function S with limt→0 t −1/2 S(t) = 0 such that 1/p |∇Pt f | eS(t) Pt |∇f |p ,
t ∈ [0, 1], f ∈ C 1 (M)
(1.4)
holds, then ∂M is convex. Consequently, (1.1) holds if and only if ∂M is convex and (1.2) holds. To prove this result, we study the short time behavior of ∇Pt f. As a result, we have the following asymptotic formula of ∇Pt f at the boundary by using the second fundamental form. Theorem 1.2. For any f ∈ C ∞ (M) with Nf |∂M = 0, |∇f |2 2 |∇Pt f | lim √ log = − √ I(∇f, ∇f ), p 1/p t→0 (P |∇f | ) π t t
p 1.
(1.5)
To prove these results, we shall apply Itô’s formula for the gradient where the local time is involved in. So, in the next section we present a limit theorem for the first moment of the local time, from which we are able to present complete proofs of Theorems 1.1 and 1.2 in Section 3. 2. Short time behavior of the local time The main result of this section is the following limit theorem for the expectation of the local time.
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Theorem 2.1. Let lt be the local time of the reflecting L-diffusion process starting at a point in ∂M. Then 1 lim sup |Elt − 2 t/π | < ∞. t→0 t In particular, Elt 2 lim √ = √ . π t
t→0
To prove this theorem, we need the following upper bound estimate on the second moment of lt (cf. [4] and [7] for the exponential integrability of lt ). Here, we present a sharp upper bound for short time. Lemma 2.2. Let lt be as in Theorem 2.1. There exists a constant c0 > 0 such that Elt2 c0 t for all t ∈ [0, 1]. Proof. Let ρ∂ be the Riemannian distance function to ∂M. We have ∇ρ∂ = N on ∂M, and there exists r0 > 0 such that ρ∂ is C ∞ -smooth on ∂r0 M := x ∈ M: ρ∂ (x) r0 . Let ϕ ∈ C ∞ ([0, ∞)) such that ϕ(r) = r for r ∈ [0, r0 /2] and ϕ(r) = 0 for r r0 . Then ϕ ◦ ρ∂ ∈ C ∞ (M) with ∇ϕ ◦ ρ∂ = N on ∂M. Let xt be the reflecting L-diffusion process with x0 ∈ ∂M. By the Itô’s formula we have √ 2ϕ ◦ ρ∂ (xt ) dbt + Lϕ ◦ ρ∂ (xt ) dt + dlt √ 2ϕ ◦ ρ∂ (xt ) dbt − c1 dt + dlt
dϕ ◦ ρ∂ (xt ) =
for some one-dimensional Brownian motion bt and some constant c1 > 0. This implies √ lt ϕ ◦ ρ∂ (xt ) − 2 ϕ ◦ ρ∂ (xs ) dbs + c1 t. t
0
Therefore,
Elt2
2 3E ϕ ◦ ρ∂ (xt ) + 6E
t
2 ϕ ◦ ρ∂ (xs ) ds + 3c12 t 2
0
2 c2 t + 3E ϕ ◦ ρ∂ (xt ) ,
t ∈ [0, 1]
(2.1)
for some constant c2 > 0. Since by Itô’s formula and noting that ϕ 2 ◦ ρ∂ satisfies the Neumann boundary condition, we have
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√ dϕ 2 ◦ ρ∂ (xt ) = 2 2ϕ ◦ ρ∂ (xt )ϕ ◦ ρ∂ (xt ) dbt + Lϕ 2 ◦ ρ∂ (xt ) dt √ 2 2ϕ ◦ ρ∂ (xt )ϕ ◦ ρ∂ (xt ) dbt + c3 dt for some constant c3 > 0. This implies that Eϕ 2 ◦ ρ∂ (xt ) c3 t and the proof is therefore completed by (2.1). 2 Proof of Theorem 2.1. Let r0 > 0 be such that ρ∂ is C ∞ -smooth on ∂r0 M. Let xt be the reflecting L-diffusion process with x0 ∈ ∂M. Let τ = inf t > 0: ρ∂ (xt ) r0 . By the continuity of the process one has τ > 0. By Itô’s formula we have dρ∂ (xt ) =
√ 2 dbt + Lρ∂ (xt ) dt + dlt ,
t τ
(2.2)
for some one-dimensional Brownian motion bt . Next, let sgn(s) = 1 if s 0; sgn(s) = −1 if s < 0. Let db˜t = sgn(b˜t ) dbt ,
b˜t = 0.
Then b˜t is a Brownian motion satisfying d|b˜t | = dbt + dl˜t ,
(2.3)
where l˜t is the local time of |b˜t | at 0. Since dlt is supported on {ρ∂ (xt ) = 0} while dl˜t on {b˜t = 0}, it follows from (2.2) and (2.3) that √ √ √ √ 2 d ρ∂ (xt ) − 2|b˜t | = 2 ρ∂ (xt ) − 2|b˜t | Lρ∂ (xt ) dt + 2 ρ∂ (xt ) − 2|b˜t | (dlt − 2 dl˜t ) √ √ 2 ρ∂ (xt ) − 2|b˜t | Lρ∂ (xt ) dt c1 ρ∂ (xt ) − 2|b˜t |dt, t τ for some constant c1 > 0. This implies √ 2 c2 E ρ∂ (xt∧τ ) − 2|b˜t∧τ | 1 t 2 . 4
(2.4)
Moreover, (2.2) implies √ Elt∧τ − 2Eρ∂ (xt∧τ ) c2 t for some c2 > 0. Therefore, (2.4) yields Elt∧τ − E|b˜t∧τ | c3 t for some c3 > 0. Combining this with Lemma 2.2 and noting that E|b˜t | = we arrive at
√
2t/π and Eb˜t2 = t,
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√ √ √ Elt − 2√ t = Elt − 2E|b˜t | c3 t + E1{t>τ } lt + 2|b˜t | π c3 t + c4 tP(t > τ )
(2.5)
for some c4 > 0. Noting that √ dρ∂ (xt )2 2 2ρ∂ (xt ) dbt + c dt,
t τ
holds for some c > 0 and a one-dimensional Brownian motion bt , by Proposition A.1 below we have P(t τ ) c1 exp[−c2 /t],
t ∈ (0, 1]
for some constants c1 , c2 > 0. Therefore, the proof is completed by (2.5).
(2.6) 2
3. Proofs of Theorems 1.1 and 1.2 As Theorem 1.1 is a consequence of Theorem 1.2, we first prove the latter. Since on {x ∈ ∂M: ∇f = 0} both sides of (1.5) are zero, we shall only consider points where ∇f = 0. Proof of Theorem 1.2. Since f satisfies the Neumann boundary condition, one has t Pt f = f +
Ps Lf ds. 0
By (1.3) with e.g. p = 2 and noting that Lf ∈ C 1 (M), we obtain t |∇Pt f − ∇f |
t |∇Ps Lf | ds e
0
c
1/2 Ps |Lf |2 ds c(f )t,
t ∈ [0, 1]
0
for some constant c(f ) > 0. This implies |∇f |2 |∇Pt f | =0 lim √ log t→0 |∇f | t and thus,
|∇Pt f | |∇f |2 |∇f |2 |∇Pt f | (Pt |∇f |p )1/p lim √ log log = lim √ − log t→0 |∇f | |∇f | (Pt |∇f |p )1/p t→0 t t |∇f |2 (Pt |∇f |p )1/p . = − lim √ log t→0 |∇f | t
(3.1)
In order to calculate Pt |∇f |p , we shall apply Itô’s formula to |∇f |p (xt ). Thus, below we first consider p 2 for which |∇f |p ∈ C 2 (M).
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(a) Assume that p 2. Let x0 ∈ ∂M with ∇f (x0 ) = 0 and v ∈ Tx0 ∂M. Let f ∈ C ∞ (M) with Nf |∂M = 0. Since for p 2 one has |∇f |p ∈ C 2 (M), the Itô formula yields t |∇f |p (xt ) − |∇f |p (x0 ) =
t L|∇f |p (xs ) ds +
0
N, ∇|∇f |p (xs ) dls .
0
This implies E|∇f |p (xt ) − |∇f |p (x0 ) − E N, ∇|∇f |p (x0 ) lt t E
L|∇f |p (xs )ds + E
0
t
N, ∇|∇f |p (xs ) − N, ∇|∇f |p (x0 ) dls
0
c1 t + c1 E[ηt lt ] for some constant c1 > 0 and ηt := max0st ρ(xs , x0 ), where ρ is the Riemannian distance which is bounded for compact M. By the right continuity of xs and the dominated convergence theorem, we have Eηt2 → 0 as t → 0. Therefore, |E|∇f |p (xt ) − |∇f |p (x0 ) − EN, ∇|∇f |p (x0 )lt | = 0. √ t→0 t lim
(3.2)
Since N, ∇f = 0 on ∂M, we have 0 = vN, ∇f = Hessf (v, N ) − I(v, ∇f ). So,
N, ∇|∇f |p = p|∇f |p−2 Hessf (N, ∇f ) = p|∇f |p−2 I(∇f, ∇f ).
Combining this with (3.2) and Theorem 2.1 we obtain 2 |∇f |2 (x0 ) (Pt |∇f |p (x0 ))1/p = √ I(∇f, ∇f )(x0 ). log √ t→0 |∇f |(x0 ) π t lim
(3.3)
This and (3.1) imply (1.5). (b) By (a) and noting that 1/p 1/2 Pt |∇f | Pt |∇f |p Pt |∇f |2 ,
p ∈ [1, 2],
it suffices to prove 2 |∇f |2 (x0 ) |∇Pt f |(x0 ) = − √ I(∇f, ∇f )(x0 ). log √ t→0 (Pt |∇f |(x0 )) π t lim
Noting that
(3.4)
F.-Y. Wang / Journal of Functional Analysis 256 (2009) 3461–3469
2 2 min Pt−s Ps |∇f |(x0 ) Pt |∇f | (x0 ) → |∇f |(x0 ) > 0
3467
as t → 0,
0st
and by [7, Theorem 1.1] max sup ∇Pt |∇f | < ∞
0t1 M
as ∇f is smooth so that |∇f | is Lipschitz continuous, we have
log
(Pt |∇f |2 )1/2 1 (x0 ) = − Pt |∇f | 2 t = 0
t
2 d log Pt−s Ps |∇f | (x0 ) ds ds
0
Pt−s |∇Ps |∇f ||2 (x0 ) ds ct, Pt−s (Ps |∇f |)2
0t 1
for some constant c > 0 depending on x0 and f. Combining this with (1.5) for p = 2 we prove (3.4). 2 Proof of Theorem 1.1. It suffices to prove the second assertion. By Jensen’s inequality we may assume that p 2. If (1.4) holds and I(v, v) < 0 for some v ∈ Tx0 ∂M for some x0 ∈ ∂M. Taking f ∈ C ∞ (M) such that Nf |∂M = 0 and ∇f (x0 ) = v, we obtain from (1.5) that S(t) |∇Pt f | 2 0 = lim √ lim log (x0 ) = − √ I(v, v) > 0, p 1/p t→0 t→0 (Pt |∇f | ) π t which is impossible.
2
Appendix A The aim of this appendix is to present complete proofs of (1.3) and (2.6) for readers’ reference. Proposition A.1. Let Ric −∇Z −K and I −σ hold for some constant σ , K 0. Then there exists c > 0 such that (1.3) holds. Proof. By [7, Lemma 2.1], for any ϕ ∈ Cb2 ([0, ∞)) with ϕ(0) = 0, ϕ (0) = 1, ϕ 0 and ϕ 0 such that ϕ ◦ ρ∂ ∈ C 2 (M), let c = sup(−Lϕ ◦ ρ∂ ). Then 1/p
|∇Pt f | Pt |∇f |p exp σ ϕ(∞) + t cσ + σ 2 p/(p − 1) + K ,
p > 1, t 0 (A.1)
holds for all f ∈ Cb1 (M). By Corollary 2.3 in [7], there exists r0 > 0 such that ρ∂ is smooth on ∂r0 M := {ρ∂ r0 } and thus, inf Lρ∂ −c
∂r0 M
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holds for some constant c1 > 0. Let h ∈ C ∞ ([0, ∞)) be decreasing such that h(r) = 1 for r ∈ [0, r0 /2] and h(r) = 0 for r r0 . For any t ∈ (0, 1], taking √ ϕ(s) = t
st−1/2
s0
h(r) dr, 0
√ we have ϕ(0) = 0, ϕ (0) = 1, ϕ 0, ϕ 0, ϕ ◦ ρ∂ ∈ C ∞ (M) and ϕ(∞) r0 t, −Lϕ ◦ ρ∂ = h ρ∂ t −1/2 Lρ∂ + t −1/2 h ρ∂ t −1/2 c. Therefore, (1.3) follows from (A.1).
2
The following result is similar to [1, Lemma 2.3] where the hitting time of the boundary was considered. Proposition A.2. Let x0 ∈ ∂M and τ = inf{t 0: ρ∂ (xt ) r0 }, where r0 > 0 is such that ρ∂ is smooth on ∂r0 := {x ∈ M: ρ∂ r0 }. Then there exists a constant C > 0 such that P(τ t) Ce−r0 /(16t) , 2
t > 0.
Proof. Let γt := ρ∂ (xt ), t 0. Since ρ∂2 is smooth on ∂r0 M and satisfies the Neumann boundary condition, by Itô’s formula we obtain √ √ dγt2 = 2 2γt dbt + Lρ∂2 (xt ) dt 2 2γt dbt + c dt,
t τ
for some constant c > 0 and a one-dimensional Brownian motion bt . Thus, for fixed t > 0 and δ > 0,
δ δ2 δ Zs := exp γs2 − cs − 4 2 t t t
s
γu2 du ,
sτ
0
is a supermartingale. Therefore, 2 2 2 P{τ t} = P max γs∧τ r0 P max Zs∧τ eδr0 /t−δc−4δ r0 /t s∈[0,t]
s∈[0,t]
1 exp cδ − δr02 − 4δ 2 r02 . t The proof is completed by taking δ := 1/8.
2
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References [1] M. Arnaudon, A. Thalmaier, F.-Y. Wang, Gradient estimate and Harnack inequality on non-compact Riemannian manifolds, preprint. [2] D. Bakry, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, in: K.D. Elworthy, S. Kusuoka, I. Shigekawa (Eds.), New Trends in Stochastic Analysis, World Scientific, Singapore, 1997. [3] D. Bakry, M. Ledoux, Lévy–Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math. 123 (1996) 259–281. [4] E.P. Hsu, Multiplicative functional for the heat equation on manifolds with boundary, Michigan Math. J. 50 (2002) 351–367. [5] F.-Y. Wang, On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups, Probab. Theory Related Fields 108 (1997) 87–101. [6] F.-Y. Wang, Equivalence of dimension-free Harnack inequality and curvature condition, Integral Equations Operator Theory 48 (2004) 547–552. [7] F.-Y. Wang, Gradient estimates and the first Neumann eigenvalue on manifolds with boundary, Stochastic Process. Appl. 115 (2005) 1475–1486.
Journal of Functional Analysis 256 (2009) 3471–3489 www.elsevier.com/locate/jfa
Radial rapid decay property for cocompact lattices Mattia Perrone Universite d’Aix-Marseille I, LATP, 39 rue F. Joliot-Curie, Marseille, France Received 2 April 2008; accepted 9 February 2009
Communicated by Alain Connes
Abstract We study Haagerup inequality for radial functions on uniform lattices in semisimple Lie groups, with respect to Riemannian metrics and, in some case, to word metrics. In particular we extend the Swiatkowski– Valette results to any lattice acting properly and essentially transitively on classical buildings. © 2009 Elsevier Inc. All rights reserved. Keywords: C ∗ -algebras; Induced representations; Buildings and symmetric spaces; Uniform lattices in Lie groups
1. Introduction 1.1. Property RD Consider a locally compact second countable (lcsc) group G equipped with a proper length function L, i.e. a proper function L : G → R+ with the following property: L(xy) L(x) + L(y) ∀x, y ∈ G, L x −1 = L(x) ∀x ∈ G, L(1G ) = 0.
(1) (2) (3)
Consider also a unitary representation τ : G → U(H) and a subspace E of the space of continuous functions on G with compact support Cc (G), which is stable under the involution f ∗ (x) = f (x −1 ) (in the following we refer to E as a ∗-subspace). Then we say that the triple E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.007
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(τ, L, E) has property RD (rapid decay) if there exist two constants K, C > 0 such that we have the following bound for the operator norm: τ (F )
F (1 + L)K C op 2
∀F ∈ E.
(4)
In particular when τ = λ, the left regular representation and E = Cc (G) we say that G has property RD with respect to L. In this paper we are also interested in the case when τ = λ and E is the space of L-radial functions; in this case we say that G has radial rapid decay property (RRD) with respect to L. Example 1.1 (E. Breuillard). Let G be a finitely generated group, L : Γ → N a word length function with respect to a finite symmetric generating set S = S −1 and let E be the space of L-radial functions with finite support. Then if 1 denotes the trivial representation we have that the triple (1, L, E) has property RD if and only if G is virtually nilpotent. Indeed let sn be the characteristic function of SL (n) := {x ∈ G | L(x) = n} the sphere of radius n. Then 1(sn )
op
= sn 1 = #SL (n) = sn 22 .
(5)
On the other hand we see that any L-radial function F with finite support can be written in the n(F ) form F (x) = k=0 Fk sk (x) and thus n(F ) F (1 + L)k 2 = |Fm |2 (1 + m)2k sm 2 . 2
2
(6)
m=0
Comparing (5) and (6) we obtain that (1, L, E) has property RD if and only if G has polynomial growth with respect to L and hence by Gromov’s famous result (see [10]) if and only if the group G is virtually nilpotent. Using Corollary 2.8 below we also deduce the well-know fact (see [25]) that for amenable finitely generated groups, RRD property is equivalent to virtual nilpotency (just recall that an lcsc group G is amenable if and only if the trivial representation 1 of G is weakly contained in the left regular representation λ of G). Example 1.2 (U. Haagerup). Let G = Fn be the free group on n generators and let LS denote the word length with respect to a free generating set S. In [11] U. Haagerup shows that λ(F )
op
2F (1 + L)2 2
∀F ∈ C[Fn ].
This inequality is actually known in the literature as the Haagerup inequality. In [15] and [14] P. Jolissaint generalizes Haagerup result, defining property RD and showing that a group G has RD with respect to L if and only if the Sobolev space associated to L, HL∞ (G) = F ∈ L2 (G) F (1 + L)k ∈ L2 (G) ∀k 0 ,
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embeds continuously in the reduced C ∗ -algebra of the group G (see also [16]). In this case HL∞ (G) is a dense subalgebra of Cr∗ (G), closed under holomorphic calculus (the locally compact case is due to Schweitzer and Ji, see [13]). In [7] property RD is established for word-hyperbolic groups; A. Connes and H. Moscovici use these results to prove Novikov conjecture for word-hyperbolic groups in [4]. In [15] again it is shown that if Γ < G is a uniform lattice in G and Γ has RD with respect to the restriction L|Γ of a length function L on G, then G has RD with respect to L. The converse has been conjectured by A. Valette in [25] in the case of uniform lattices in semisimple Lie groups. This conjecture is solved only in the rank 1 case (already in [15]) and in the case G = SL3 (F) where F = Qp , F = R and F = C, H, O in [22,18,1] respectively (here O denotes the division algebra of octonions). This question takes particular relevance after V. Lafforgue’s work, that shows in [19] that uniform lattices in semisimple Lie groups with property RD satisfy the Baum–Connes conjecture (see [26]). V. Lafforgue obtained in this way the first examples of discrete groups with Kazhdan property T that satisfy the Baum–Connes conjecture (namely uniform lattices in SL3 (R)). In the recent paper [3] all connected Lie groups with property RD are classified. In particular it is shown that semisimple Lie groups have property RD. This fact is due to C. Herz (see [5] and [6]), as explained in [3]. Let G be a semisimple Lie group, let Γ be a uniform lattice inside G and consider the Riemannian metric d on G. A direct corollary of the Valette conjecture is that the operator norm of the L2 -normalized characteristic function of the set ΓT := {γ ∈ Γ : d(1G , γ ) T } is polynomial in T . In this paper we confirm this fact. The following theorem, that we prove in Section 3, is the central statement of the paper. Theorem 1.3. Let G be an lcsc group, compactly generated, Γ < G a cocompact lattice. Let
1 ∈ Ω = Ω −1 ⊂ G a relatively compact symmetric Borel set such that G = n0 Ω n . Define the length function L(x) := min{n ∈ N | x ∈ Ω n }, and consider EL the space of L-radial functions. Let τ : Γ → U(H) be a positive unitary representation (see Definition 2.4 below) of Γ . Suppose that (indG Γ τ, L, EL ) has RD. Then (τ, L|Γ , EL|Γ ) has RD. Moreover G has RRD property with respect to L if and only if Γ has RRD property with respect to L|Γ . Theorem 0.1 of [3] shows that connected semisimple Lie groups have RD with respect to the length function associated to any relatively compact symmetric generating Borel set, giving the following Corollary 1.4. Let G be a connected semisimple Lie group. Then any cocompact lattice Γ < G has the radial RD property with respect to any G-coherent (see Definition 3.9 below) word length function. Our work was essentially motivated by the work of Swiatkowski and Valette (see [24] and [25]), which shows the connection between radial property RD and the return probability of a simple random walk on the group (see also [2]). In particular Proposition 4 of [25] establishes the equivalence between the radial RD property and the strict N -loop inequality. The following theorem (see Section 3 for the proof) generalizes Theorem 0.6(a) of [24] and Theorem 1 of [25].
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Theorem 1.5. Let Γ be a uniform lattice in a semisimple algebraic group G, defined over a non-archimedean local field F. Consider the Bruhat–Tits building associated to G. Fix a base vertex v0 ∈ (0) and define the length function L (γ ) := d (v0 , γ v0 ), where d is the combinatorial distance on the 1-skeleton of . Then Γ has the radial RD property with respect to L . The proof presented here is quite different from the proof of Swiatkowski and Valette, that uses elaborated arguments involving the geometry of the building together with combinatorial techniques. 2. Operator norm and induced representation 2.1. G-invariant positive cone Positivity plays a crucial role in our approach. We introduce the notion of a G-invariant positive cone, we give various examples, and finally we show that the operator norm preserves positivity (Lemma 2.7) in the case of positive unitary representations. Let H be a Hilbert space. A positive cone H+ is a closed convex subset of H, stable by the R+ -scalar multiplication and such that η, ξ 0 ∀η, ξ ∈ H+ . Observe that in this case there exists an R+ -linear map: 4 p : H+ −→ H
p(ξ1 , . . . , ξ4 ) =
4
(i)j ξj
j =1 4 ) is a vector subspace of H. We say that the positive cone H is such that the image p(H+ + 4 generating if p(H+ ) = H.
Definition 2.1. Let G be an lcsc group, let τ : G → U(H) be a unitary representation. A G-pc is a pair (H+ , s) where H+ is a G-stable generating positive cone of H (i.e. G · H+ ⊂ H+ ) and s is a G-equivariant Lipschitz section: s
4 H+ ←− H s(ξ ) Cs ξ ∀ξ ∈ H 4 → H is the projection on the kth such that p ◦ s = IdH and x4 ◦ s|H+ = IdH+ , where xk : H+ + 4 coordinate of H+ .
For any j ∈ {1, . . . , 4} we will write in what follows for simplicity sj instead of xj ◦ s. This definition is modeled on the following standard example. Example 2.2. Let (X, ν) be a standard Borel G-space with ν a quasi-invariant measure (i.e. [g∗ ν] = [ν] ∀g ∈ G). Then the set L2+ (X, ν) := {F ∈ L2 (X, ν) | F 0} is a G-stable generating
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positive cone for the representation: τ : G −→ U L2 (X, ν) dg −1 ν τ (g)F (x) = (x)F g −1 x . dν Moreover the map: s : L2 (X, ν) −→ L2+ (X, ν)4 ⎛ ⎞ max{(F (x)), 0} ⎜ − min{(F (x)), 0} ⎟ F (x) → ⎝ ⎠ − min{(F (x)), 0} max{(F (x)), 0} is a G-equivariant section and satisfies s(F )2 = F 2 , as a simple verification shows. The following lemma is useful to produce new example of G-pc. Lemma 2.3. Let G and G be two lcsc groups, let (X, ν) be a standard Borel G-space, and let τ : G → U(H) be a unitary representation of G . If (H+ , s) is a G -pc for τ then for any Borel cocycle α : G × X → G the set L2+ (X, H, ν) := F ∈ L2 (X, H, ν) F (x) ∈ H+ , ν-a.e. together with the map σ : L2 (X, H, ν) → L2+ (X, H, ν)4 defined by σ (F )(x) = s(F (x)) ∀x ∈ X, is a G-pc for the representation τα defined by: τα : G −→ U L2 (X, H, ν) dg −1 ν (x)τ α(g, x) F g −1 x . τα (g)F (x) = dν
(7) (8)
Proof. The fact that H+ is G -invariant together with the formula (7) imply that L2+ (X, H, ν) is G-invariant. Moreover we have that for any F, H ∈ L2+ (X, H, ν) we get F, H L2 (X,H,ν) 0. Indeed, recall that this scalar product is defined by the formula F, H L2 (X,H,ν) =
F (x), G(x) H dν(x)
X
and that by construction F (x), G(x) H 0 ν-a.e. On the other hand it is clear that σ is well defined as a section and G-equivariant. Indeed for any j ∈ {1, . . . , 4}:
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dg −1 ν (x)τ α(g, x) sj F g −1 x dν dg −1 ν = sj (x)τ α(g, x) F g −1 x dν = σj τα (g)F (x).
τα (g) σj (F ) (x) =
Finally we have that: 4 σ (F )2 = σj (F )2 2 2 j =1
4 4 2 s F (x) 2 dν(x) σj (F )(x) dν(x) = = j =1 X
X j =1
2 Cs F (x) dν(x) = Cs F 22 .
2
X
Definition 2.4. Let G be an lcsc group and let τ : G → U(H) be a unitary representation. We say that τ is a positive unitary representation if it admits a G-pc (H+ , s). Lemma 2.5. Let G be an lcsc group and let H be a closed subgroup. If τ : H → U(H) is a 2 positive unitary representation then the induced representation indG H τ : G → U(L (G/H , H)) is a positive unitary representation of G. Proof. Given a Borel section σ : G/H → G, we apply Lemma 2.3 for the cocycle α : G × G/H → H , α(g, x) = s(gx)−1 gs(x). Then τα ∼ = indG Hτ. 2 If E is a subspace of Cc (G), denote by E+ the set {|F |: F ∈ E} where |F |(x) := |F (x)|. Proposition 2.6. Let τ : G → U(H) be a positive unitary representation and let H+ be a G-pc. Then the triple (τ, L, E) has property RD if and only if there exist K, C 0 such that: τ (F )ξ, η C F (1 + L)K 2 ξ η ∀η, ξ ∈ H+ ∀F ∈ E+ .
Let us recall that if A ∈ B(H) is a bounded operator on a Hilbert space then Aop =
| Aη, ξ | . η,ξ ∈H−{0} ηξ sup
So the proposition is a straightforward consequence of the following lemma:
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Lemma 2.7. Let G be an lcsc group and let τ : G → U(H) be a positive unitary representation. Then there exists C > 0, which depends on τ such that: τ (F )
τ |F | C op op
∀F ∈ Cc (G).
Moreover if 0 F F ∈ L1 (G) then τ (F )
op
τ (F )op .
(10)
Proof. Fix a G-pc (H+ , s) for the representation τ . Consider the continuous G-equivariant map: 4 q : H+ −→ H
q(ξ1 , . . . , ξ4 ) =
4
ξj
j =1
and put C = supξ 1 q ◦ s(ξ )2 ∞. Let F ∈ Cc (G) be a continuous function, then for any ξ, η ∈ H: 4 4 j k τ (F )ξ, η = F (g) τ (g) (i) sj (ξ ), (i) sk (η) dg j =1
G
k=1
4 j −k = F (g) τ (g)sj (ξ ), sk (η) dg (i) G
j,k=1
4 F (g) (i)j −k τ (g)sj (ξ ), sk (η) dg j,k=1
G
4 F (g) τ (g)sj (ξ ), sk (η) dg
j,k=1
G
= τ |F | q ◦ s(ξ ), q ◦ s(η) . Using identity (9), we have that: τ (F )
=
op
sup
ξ ,η1
sup
ξ ,η1
τ (F )η, ξ
τ |F | q ◦ s(η), q ◦ s(ξ )
C τ |F | op . For the second part we use the fact that for any ξ, η ∈ H+
τ (F − F )ξ, η 0.
2
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A direct consequence of this lemma is the following corollary, that summarizes some wellknow facts (compare to [23]). Recall that given two unitary representations τ, σ of G we say that τ is weakly contained in σ – in symbol τ ≺ σ – if any matrix coefficients of τ can be approximated by a sequence of matrix coefficients of σ uniformly on compact subset of G. Corollary 2.8. Let π : G → U(H) be a positive unitary representation such that π ≺ λ, where λ denotes the left regular representation. Then for every ∗-subspace E ⊆ C0 (G) and for any length function L : G → R+ the triple (λ, L, E) has RD if and only if the triple (π, L, E) has RD. Proof. It is well known that if π ≺ λ then π(F )
op
λ(F )op
∀F ∈ Cc (G).
(11)
On the other hand if π is a positive unitary representation, then Lemma 2.3 of [23] establishes that: π(F )
op
λ(F )op
Lemma 2.7 together with (12) proves the corollary.
∀F 0 ∈ Cc (G).
(12)
2
We recall that a locally compact group G has polynomial growth with respect to a proper length function L if there exist two constants C, K > 0 such that μ(BL (R)) C(1 + R)K ∀R > 0, where μ is any Haar measure on G and BL (R) = {g ∈ G | L(g) R}. Corollary 2.9. Let G be an lcsc amenable group, let L : G → R+ be a proper length function and let E ⊆ C0 (G) be a ∗-subspace containing L-radial functions. Then the following are equivalent: (1) There exists a positive unitary representation π : G → U(H) such that the triple (π, L, E) has RD. (2) For any positive unitary representation τ : G → U(H) the triple (τ, L, E) has RD. (3) The group G has polynomial growth with respect to L. Proof. The implications (3) ⇒ (2) and (2) ⇒ (1) are obvious. So to prove the statement it is enough to show the implication (1) ⇒ (3). Fix R > 0 and consider the characteristic function IBL (R) of the ball of radius R. By amenability of the group G and positivity of π , using inequalities (11) and (12), we have that π(IB (R) ) λ(IB (R) ) 1(IB (R) ) = μ BL (R) . L L L op op op
(13)
On the other hand, as the function IBL (R) is in E and the triple (π, L, E) has RD, there exist constants C, K > 0, not depending on R, such that π(IB
L (R)
Inequalities (13) and (14) imply (3).
1/2 )op C(1 + R)K μ BL (R) . 2
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2.2. Cocompact lattices and induced representations Let G be an lcsc group and let Γ < G be a lattice; let D be a fundamental domain for the right action of Γ on G, i.e. the image of a Borel section: σ : G/Γ → G. Let μ be the Haar measure on G such that μ(D) = 1. In the following, in the case that Γ < G is cocompact (i.e. G/Γ is a compact topological space) we always choose the fundamental domain D to be relatively compact. Proposition 2.10. Let G be an lcsc group, let Γ < G be a cocompact lattice. We define the following linear map Ψ : Cc (G) −→ C[Γ ] Ψ (F )(γ ) = F (g)ϕ(γ , g) dμ(g), G
where ϕ : Γ × G −→ R+ ϕ(γ , g) = μ(Dγ ∩ gD). Then (1) Ψ extends to a continuous map: L1 (G) → l 1 (Γ ), (2) Ψ extends to a continuous map: L2 (G) → l 2 (Γ ), (3) Ψ (F ∗ ) = Ψ (F )∗ for all F in Cc (G). Proof. Remark that
if F ∈ Cc (G), i.e. the support of F is contained in a compact set supp(F ) ⊂ K ⊂ G, then DK ⊂ γ ∈Λ γ D with Λ ⊂ Γ finite and supp(Ψ (F )) ⊂ Λ; this shows that the map Ψ is well defined.
(1) By definition G = γ ∈Γ γ D. Then since the union is disjoint:
ϕ(γ , g) = μ
γ ∈Γ
Dγ ∩ gD
γ ∈Γ
= μ(gD) = 1. Hence F (g)ϕ(γ , g) dμ(g) Ψ (F ) = 1 γ ∈Γ G
F (g)ϕ(γ , g) dμ(g)
γ ∈Γ G
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F (g) ϕ(γ , g) dμ(g) γ ∈Γ
G
F 1 . (2) First we observe that G must be unimodular because it contains a lattice, so μ is also right invariant. Hence according to Fubini
ϕ(γ , g) dμ(g) =
G
χDγ (xg) dμ(x) dμ(g) G D
=
χDγ (g) dμ(g) dμ(x) D G
= μ(D)μ(Dγ ) = 1. And hence, by Cauchy–Schwarz, 2 F (g)ϕ(γ , g) dμ(g) Ψ (F )2 = 2 γ ∈Γ G
F (h)2 ϕ(γ , h) dμ(h) ϕ(γ , g) dμ(g)
γ ∈Γ
G
G
F (h)2 ϕ(γ , h) dμ(h) γ ∈Γ
G
F 22 . (3) Using once more the right invariance of μ: μ(Dγ ∩ gD) = μ D ∩ gDγ −1 = μ g −1 D ∩ Dγ −1 . Hence Ψ (F )∗ (γ ) = Ψ (F ) γ −1 = F (g)ϕ γ −1 , g dμ(g) G
=
F¯ g −1 ϕ(γ , g) dμ(g)
G
= Ψ (F ∗ )(γ ).
2
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Remark 1. We observe that in the proof of Proposition 2.10 we show some useful property of the function ϕ : Γ × G → R+ , namely: ∀g ∈ G γ → ϕ(γ , g)l 1 (Γ ) = 1, ∀γ ∈ Γ g → ϕ(γ , g)L1 (G) = 1, ∀g ∈ G ∀γ ∈ Γ ϕ γ −1 , g = ϕ γ , g −1 . The linear map Ψ plays a crucial role in this paper, as illustrated in the following theorem (compare with Proposition 3.9 of [20]). Theorem 2.11. Let G, Γ be as before and let τ : Γ → U(H) be a unitary representation. Let indG Γ τ denote the induced representation from Γ to G. Then: τ Ψ (F )
op
indG Γ τ (F ) op
∀F ∈ Cc (G).
(15)
Proof. Given a Borel section σ : G/Γ → G define the following cocycle: α : G × G/Γ −→ Γ α(g, x) = σ (gx)−1 gσ (x) and consider the associate representation of G: τα : G −→ L2 (G/Γ, H, μ) ˘ τα (g)X(x) = τ α(g, x) X(xg), where μ˘ is the push-forward of the Haar measure μ under the natural projection π : G → G/Γ . It is well known (see [27]) that τα is isomorphic to indG Γ τ . Remark that the cocycle α depends on the choice of the section σ but if we choose another section σ , the associated cocycle α(σ ) is cohomologous to the first one and then the two representations are isomorphic. For 2 any η ∈ H define the H, μ) ˘ as Xη (x) = η, ∀x ∈ G/Γ . It follows that element2 Xη ∈ L (G/Γ, 2 2 . So we compute: Xη L2 (G/Γ,H,μ) = η d μ(x) ˘ = η G/Γ H H ˘
indG Γ τ (g)Xξ , Xη
= L2 (G/Γ,H,μ) ˘
τ α(g, x) ξ, η H d μ(x) ˘
G/Γ
=
μ˘ x ∈ G/Γ : α(g, x) = γ τ (γ )ξ, η H
γ ∈Γ
=
γ ∈Γ
Finally, if F ∈ Cc (G) we have:
ϕ(γ , g) τ (γ )ξ, η H .
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indG Γ τ (F )Xξ , Xη L2 (G/Γ,H,μ) ˘
=
F (g) indG dμ(g) Γ τ (g)Xξ , Xη L2 (G/Γ,H,μ) ˘
G
=
F (g) G
=
γ ∈Γ
ϕ(γ , g) τ (γ )ξ, η H dμ(g)
γ ∈Γ
Ψ (F )(γ ) τ (γ )ξ, η H
= τ Ψ (F ) ξ, η H .
2
Remark 2. Let ΣG be the set of all classes of unitary representations of G; a subset S of ΣG is closed (in the sense of Fell) if τ ≺ σ and σ ∈ S implies that τ ∈ S. Fell shows that if H is a closed subgroup of G, induction from the representations of H to the representations of G is closed, ∗ ∗ that is indG H S is closed whenever S is closed. Let CS (G) denote the C -algebra associated to the closed subset S of ΣG and let BS (G) denote its dual Banach space (see [8]). If Γ < G is a uniform lattice the theorem above asserts that for all closed subset S ⊂ ΣΓ the linear map Ψ extends to a continuous map ∗ ∗ ΨS : Cind G S (G) −→ CS (Γ ). Γ
Moreover it is known that BS (G) can be realized as a space of bounded functions on G, any element u ∈ BS (G) can be written in the form u(g) = τ (g)ξ, η with uBS (G) = ξ η for some representation τ ∈ S and for some ξ, η ∈ H (see once more [8]). Then the above theorem gives us an explicit formula for the adjoint map: ΨS∗ : BS (Γ ) −→ BindG S (G) Γ γ → τ (γ )ξ, η → g → indG Γ τ (g)Xξ , Xη . 3. Admissible set 3.1. Operator norm for coarse admissible family We introduce in this section some tools from ergodic theory that we need in what follows. We refer to [9] for an exhaustive discussion about coarse admissible families, including several important examples. Definition 3.1. Let G be an lcsc group with left Haar measure μ. An increasing family of bounded Borel subsets {Gt }t∈R+ is said to be coarsely admissible if (1) for every compact subset K ⊂ G, there exists c = c(K) > 0 such that KGt K ⊆ Gt+c
∀t 1,
(2) for every c > 0 there exists D > 0 such that μ(Gt+c ) Dμ(Gt )
∀t 1.
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The following proposition is a special case of Theorem (6.4)(I) of [9, p. 65]. Since we consider only uniform lattices, the proof is elementary and it does not need the tools developed in [9]. Proposition 3.2. Let G be an lcsc group and let Γ ⊆ G be a cocompact lattice in G, let Gt be a coarse admissible family in G and Γt := Γ ∩ Gt . Then there exists C > 0 such that: C −1 μ(Gt ) #|Γt | Cμ(Gt )
∀t 1.
(16)
Proof. For the inequality #|Γt | Cμ(Gt ) see the proof of Lemma (6.5)(I) of [9, p. 66]. Conversely, let B ⊂ G be a Borel relatively compact set such that ∀x ∈ G we have Γ ∩ xB = ∅ (such a set exists because Γ is uniform). Hence for any x ∈ G we can find γ (x) ∈ Γ ∩ xB; defining the Borel retraction: ρ : G −→ Γ x → γ (x). By coarse admissibility there exists c = c(B) such that ρ(Gt ) ⊂ Γ ∩ Gt B ⊂ Γ ∩ Gt+c = Γt+c ,
t 1.
In other words
Gt ⊂
ρ −1 (γ ) ⊂
γ ∈Γt+c
γ B −1 .
γ ∈Γt+c
Always by coarse admissibility we obtain μ(Gt ) Dμ(Gt−c ) Dμ B −1 #|Γt |,
t 1.
2
Theorem 3.3. Let G be an lcsc group, let Γ < G be a cocompact lattice and let τ : Γ → U(H) a positive unitary representation. Suppose that ∃c, k > 0 such that: G ind τ (IG ) c(1 + t)k IG 2 t t Γ op
∀t 1.
Then there exist c , k > 0 such that: τ (IΓ ) c (1 + t)k IΓ 2 t t op
∀t 1.
Proof. We observe that if γ ∈ Γt then the support of the function g → ϕ(γ , g) is contained in DΓt D −1 , where D denotes the fundamental domain as in Section 2. Then there exists C = C(D ∪ D −1 ) > 0 such that supp(g → ϕ(γ , g)) ⊂ Gt+C . It follows that if γ ∈ Γt then: Ψ (IGt+C )(γ ) = Gt+C
ϕ(γ , g) dμ(g) = 1.
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On the other hand, by construction Ψ (IGt+C ) 0, ∀γ ∈ Γ , and thus IΓt Ψ (IGt+C )
∀t 1.
Applying Proposition 3.2, we see that there exists some C1 > 0 such that IGt+C 2 = μ(Gt+C )1/2 C1 #|Γt |1/2 = C1 IΓt 2 . Applying Lemma 2.7 and Theorem 2.11 we have, for t 1 τ (IΓ ) τ Ψ (IG ) t t+C op op G indΓ τ (IGt +C )op c(1 + t + C)k IGt+C 2
c (1 + t)k IΓt 2 .
2
Remark 3. As illustrated in the next section, Theorem 3.3 is useful to deduce upper bounds on operator norm on Γ from upper bounds on operator norm on G. This is why this result looks very close to some result in [9] with a substantial difference: here we are not interested on averages as in ergodic theory but on the existence of some closed ∗-subspace of the reduced C ∗ -algebra of the group Γ . The way from Γ to G seems easier as illustrated in theorem of [15] and the following elementary proposition. Proposition 3.4. Let G be an lcsc group and let Γ < G be a uniform lattice. Let d : G × G → R+ a left-invariant distance on G such that the balls B(t) = {x ∈ G | d(1G , x) t} form a coarse admissible family. Set β(t) = {x ∈ Γ | d(1G , x) t}. Let τ : G → U(H) a positive unitary representation and suppose that ∃c, k > 0 such that: τ|Γ (Iβ(t) )op c(1 + t)k Iβ(t) 2
∀t 1.
Then there exist c , k > 0 such that: τ (IB(t) )op c (1 + t)k IB(t) 2
∀t 1.
Proof. Let d := min{n ∈ N | D ⊂ B(n)}. Then if x ∈ γ D, where D denotes
the fundamental domain, with d(1G , γ ) t we have d(1G , x) t − d. This implies that B(t) ⊂ γ ∈B(t+d+1)∩Γ γ D and hence IB(t) IB(t+d+1)∩Γ ∗ ID and using Lemma 2.7 and Proposition 3.2:
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τ (IB(t) )
op
3485
τ|Γ (IB(t+d+1)∩Γ )op τ (ID )op Cc(2 + t + d)k IB(t+d+1)∩Γ 2
c (1 + t)k IB(t) 2 , where all the inequalities hold for t sufficiently large.
2
Here we are more interested in the case of the left regular representation, so we summarize the previous results in the following corollary. Corollary 3.5. Let G, Γ , B(t) and β(t) be as before. Then there exist two constants C, c > 0 such that for t sufficiently large: C −1
λG (IB(t+c) )op λG (IB(t−c) )op λΓ (Iβ(t) )op C . IB(t−c) 2 Iβ(t) 2 IB(t+c) 2
Proof. This follows from Theorem 3.3 and Proposition 3.4, using the equivalences: ∼ IndG Γ λΓ = λG
and λG|Γ ∼ = [G : Γ ] · λΓ .
2
3.2. Proofs of Theorems 1.3–1.5 and applications Theorem 3.3 establishes property of radial rapid decay for cocompact lattices in large classes of examples. Proof of Theorem 1.3. First observe that {Ω n }n0 is a coarse admissible family; in fact if B and hence BΩ n B ⊂ Ω n+2k . is a bounded domain there exists some k = k(B) > 0 s.t. B ⊂ Ω k
2 compact there exists a finite set F ⊂ G such that Ω ⊂ x∈F xΩ and consequently As Ω is
Ω n+k ⊂ x∈F k xΩ n . This implies μ(Ω n+k ) #|F |k−1 μ(Ω n ). Observe that the same argument is valid for any proper length function with at most exponential growth. So Theorem 3.3 applies. By definition BL|Γ (n) = Ω n ∩ Γ . Hence τ (IBL|Γ (n) )op P1 (n)IBL|Γ (n) 2 . Using Proposition 5 of [25] we conclude the first part of the proof. The second part of the theorem is a direct consequence of Proposition 3.4, together with the following lemma: Lemma 3.6. Let G, Γ, Ω be defined as before. Let τ : G → U(H) be a positive unitary representation of G. Suppose that the triple (τ|Γ , L|Γ , E|Γ ) has RD then the triple (τ, L, E) has RD. Proof. Observe that any L-radial function F ∈ EL with support in a ball of radius n is of the form F = nk=1 Fˆ (k)Sk , where Sn := IΩ n −Ω n−1 . Remark that if G is amenable then the hypothesis implies that G is of polynomial growth with respect to L (see Corollary 2.9), and this is a quasiisometric invariant and there is nothing to show; so we can suppose without restriction that G and Γ are both non-amenable. Because G is a locally compact compactly generated non-amenable group, the Følner condition (see [12]) on G ensures that there exists > 0 such that μ Ω n − Ω n−1 μ Ω n .
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So we have τ (Sn )
op
τ (IΩn )op P2 (n)IΩn 2 P2 (n)Sn 2
and finally τ (F ) P3 (n)F 2 . op This concludes the proof of Theorem 1.3.
2
2
In particular we obtain that if Γ is a uniform lattice in a connected semisimple Lie group G and Ω is any neighborhood of the identity, then Γ has radial rapid decay property with respect to the restriction of the length function LΩ associated to Ω. In fact Theorem 0.1 of [3] shows that G has property RD with respect to the Riemannian metric and hence with respect to LΩ because d(x, 1G ) d(Ω)LΩ (x). Definition 3.7. Let G be a semisimple Lie group and let d : G × G → R+ be a Riemannian metric. For any > 0 and for any n ∈ N consider the set Sn := {x ∈ G: n d(1G , x) < (n + 1)}. Let Γ < G a lattice subgroup, we say that a function f ∈ C[Γ ] is -radial if it is constant on the sets Sn ∩ Γ for all n ∈ N (i.e. f is of the form f = N k=0 fk ISk ∩Γ ). Corollary 3.8. Let G be a semisimple Lie group with finite center, let Γ < G be a irreducible uniform lattice. Then for any > 0 the space of -radial functions has property RD. Proof. We have already seen that the group G has property RD with respect to the Riemannian metric. To prove the corollary is sufficient to observe the growth Riemannian balls. If Γ is an infinite irreducible lattice it is known (see [17] and [21]) that there exist constants α > 1 and β > 0 such that: # B(T ) ∩ Γ ∼ T −β eαT .
(17)
Remark that this growth rate can be used to show directly coarse admissibility of Riemannian balls. On the other hand this implies that ∀ > 0: #(Sn ∩ Γ ) #(B((n + 1)) ∩ Γ ) #(B(n) ∩ Γ ) n→∞ α = − −→ e − 1 > 0. #(B(n) ∩ Γ ) #(B(n) ∩ Γ ) #(B(n) ∩ Γ ) In other words there exists δ = δ() > 0 such that: I(Sn ∩Γ ) 2 δI(B(n)∩Γ ) 2 . See [25] for definitions use in the theorem below.
2
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Proof of Theorem 1.5. The group Γ is a uniform lattice in a simple algebraic group G over a non-archimedean local field F and it follows from Theorem 0.1 of [3] that G has property RD. Proposition 3.13(2) of [9] asserts that the pre-image Gn of the ball Bn of radius n and center v0 forms a coarse admissible family inside G. We can apply Theorem 3.3. 2 Definition 3.9. Let G be an lcsc group and let H < G be a closed subgroup. A length function L : H → R+ is G-coherent if for any compact subset K ⊂ G there exists a constant C = C(K) > 0 such that: sup h∈H ∩Kh0 K −1
L(h) − L(h0 ) C
∀h0 ∈ H.
Example 3.10. Suppose that L = L|H , where L : G → R+ is a proper length function on G. Then L is a proper G-coherent length function on H . Example 3.11. Let G be an lcsc group and let Γ < G be a uniform lattice. Let L : Γ → R+ be a proper length function and let Ω be a compact subset of G that contains the fundamental domain for the right action of Γ on G. Suppose that there exists a constant C > 0 such that: sup γ ∈Γ ∩Ωγ0 Ω −1
L(γ ) − L(γ0 ) C
∀γ0 ∈ Γ.
Then L is G-coherent. Proof of Corollary 1.4. Let L : Γ → N be a G-coherent word length function on Γ . Put LG : G → R+ , LG (g) := γ ∈Γ ϕ(γ , g)L(γ ). The function LG is subadditive: LG (gh) =
ϕ(gh, γ )L(γ )
γ ∈Γ
=
L α(gh, x) d μ(x) ˘
G/Γ
=
L α(g, x)α h, g −1 · x d μ(x) ˘
G/Γ
L α(g, x) d μ(x) ˘ +
G/Γ
L α h, g −1 · x d μ(x) ˘
G/Γ
= LG (g) + LG (h). Let Ω = D ∪ D −1 , we claim that LG is quasi-isometric to LΩ . Indeed, given two elements g ∈ G and γ ∈ Γ such that ϕ(γ , g) 0 one has for obvious reason that LΩ (γ ) LΩ (g) − 2. On the other hand the word length function L dominates the restriction of LΩ to Γ as well as LΩ dominates LG . So there exists a constant a > 0 such that LΩ (γ ) aL(γ ) for any γ in Γ and LG (g) aLΩ (g) for any g ∈ G. We have that:
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aLΩ (g) LG (g) =
ϕ(γ , g)L(γ )
γ ∈Γ
a
γ ∈Γ
a
ϕ(γ , g)LΩ (γ ) ϕ(γ , g) LΩ (g) − 2
γ ∈Γ
= aLΩ (g) − 2a. Hence the function LG is a proper length function on G and defines a metric on G quasi-isometric to the Riemannian one. So G has property RD with respect to LG . On the other hand, as L is G-coherent, there exists a constant c > 0 such that BLG (n − c) ∩ Γ ⊂ BL (n) ⊂ BLG (n + c) ∩ Γ, where BLG (r) = g ∈ G: LG (g) r So we may apply Theorem 3.3.
and BL (n) = γ ∈ Γ : L(γ ) n .
2
Acknowledgments I am very grateful to my advisor Christophe Pittet who introduced me to the theory of groups with RD property and also for the proof of Proposition 3.2. I thank Amos Nevo who explained me some links between ergodic theory and harmonic analysis on Lie groups. I also want to thank François Maucourant who gave me the reference for the asymptotic growth of Riemannian balls for nonpositively curved manifolds, that I use in Corollary 3.8; Claire Anantharaman who gave me the references that I use for Corollary 2.8. I am grateful to Indira Chatterji and Laurent SaloffCoste who invited me to the workshop on property RD at the American institute of Mathematics, Palo Alto, California. This visit was the starting point of my thesis. References [1] I. Chatterji, Property (RD) for cocompact lattices in a finite product of rank one Lie groups with some rank two Lie groups, Geom. Dedicata 96 (2003) 161–177. [2] I. Chatterji, C. Pittet, L. Saloff-Coste, Heat decay and property RD, in preparation. [3] I. Chatterji, C. Pittet, L. Saloff-Coste, Connected Lie groups and property RD, Duke Math. J. 128 (2) (2007). [4] A. Connes, H. Moskovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 (1990) 345–388. [5] M. Cowling, Herz’s ‘principe de majoration’ and the Kunze–Stein phenomenon, in: Harmonic Analysis and Number theory, in: CMS Conf. Proc., vol. 21, Amer. Math. Soc., Providence, 1997, pp. 73–88. [6] M. Cowling, U. Haagerup, R. Howe, Almost L2 matrix coefficients, J. Reine Angew. Math. 387 (1988) 97–110. [7] P. De la Harpe, Groupes hyperboliques, algèbres d’operateurs et un théorème de Jolissaint, C. R. Acad. Sci. Paris 307 (I) (1988) 771–774. [8] P. Eymard, L’ algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964) 181–236. [9] A. Gorodnik, A. Nevo, The Ergodic Theory of Lattice Subgroup, Ann. of Math. Stud., vol. 172, Princenton University Press, in press. [10] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. Inst. Hautes Etudes Sci. 53 (1981) 53–73.
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[11] U. Haagerup, An example of nonnuclear C ∗ -algebra which has the metric approximation property, Invent. Math. 50 (1979) 279–293. [12] J.W. Jenkins, Growth of connected locally compact groups, J. Funct. Anal. 12 (1973) 113–127. [13] R. Ji, L.B. Schweitzer, Spectral invariance of smooth crossed products, and rapid decay for locally compact groups, Topology 10 (1996) 283–305. [14] P. Jolissaint, K-theory of reduced C ∗ -algebras and rapidly deceasing functions on groups, K-Theory 2 (6) (1989) 723–736. [15] P. Jolissaint, Rapidly decreasing functions in reduced C ∗ -algebras of groups, Trans. Amer. Math. Soc. 317 (1990) 167–196. [16] P. Jolissaint, A. Valette, Normes de Sobolev et convoluteurs bornés sur L2 (G), Ann. Inst. Fourier (Grenoble) 41 (1991) 797–822. [17] G. Knieper, On the asymptotic geometry of nonpositively curved manifold, Geom. Funct. Anal. 7 (1997) 755–782. [18] V. Lafforgue, A proof of property RD for discrete cocompact subgroups of SL3 (R), J. Lie Theory 10 (2000) 255– 277. [19] V. Lafforgue, KK-théorie bivariante pour les algèbres de Banach et conjecture de Baum–Connes, Invent. Math. 149 (1) (2002) 1–95. [20] A. Nevo, Spectral transfer and pointwise ergodic theorems for semi-simple Kazhdan groups, Math. Res. Lett. 5 (3) (1998) 305–325. [21] Ch. Pittet, The isoperimetric profile of homogeneous Riemannian manifolds, J. Differential Geom. 54 (2000) 255– 302. [22] J. Ramagge, G. Robertson, T. Steger, A Haagerup inequality for A˜ 1 × A˜ 1 and A˜ 2 buildings, Geom. Funct. Anal. 8 (1988) 702–731. [23] Y. Shalom, Rigidity, unitary representations of semisimple groups and fundamental groups of manifolds with rank one transformation group, Ann. of Math. 152 (2000) 113–182. [24] J. Swiatkowski, On the loop inequality for euclidean buildings, Ann. Inst. Fourier (Grenoble) 47 (4) (1997) 1175– 1194. [25] A. Valette, On the Haagerup inequality and group acting on A˜ n -buildings, Ann. Inst. Fourier (Grenoble) 47 (4) (1997) 1195–1208. [26] A. Valette, Introduction to the Baum–Connes Conjecture, Lectures Math. ETH Zürich, Birkhäuser Verlag, Basel, 2002. [27] R.J. Zimmer, Ergodic Theory and Semi-simple Groups, Monogr. Math., vol. 81, Birkhäuser, Basel, Boston, 1984.
Journal of Functional Analysis 256 (2009) 3490–3509 www.elsevier.com/locate/jfa
Commutators on 1 Detelin T. Dosev 1,2 Department of Mathematics, Texas A&M University, College Station, TX 77843, United States Received 11 April 2008; accepted 10 March 2009 Available online 25 March 2009 Communicated by K. Ball
Abstract The main result is that the commutators on 1 are the operators not of the form λI + K with λ = 0 and K compact. We generalize Apostol’s technique [C. Apostol, Rev. Roumaine Math. Appl. 17 (1972) 1513–1534] to obtain this result and use this generalization to obtain partial results about the commutators X ) for some 1 p < ∞ or p = 0. In particular, on spaces X which can be represented as X ( ∞ p i=0 it is shown that every compact operator on L1 is a commutator. A characterization of the commutators on p1 ⊕ p2 ⊕ · · · ⊕ pn is given. We also show that strictly singular operators on ∞ are commutators. © 2009 Elsevier Inc. All rights reserved. Keywords: Commutators
1. Introduction The commutator of two elements A and B in a Banach algebra is given by [A, B] = AB − BA. A natural problem that arises in the study of derivations on a Banach algebra is to classify the commutators in the algebra. The first major contribution was due to Wintner [14], who proved that the identity in a unital Banach algebra is not a commutator. This immediately implies that E-mail address:
[email protected]. 1 Research supported in part by NSF grant DMS-0503688. 2 This is part of the author’s doctoral dissertation, which is being prepared at Texas A&M University under the direction
of W.B. Johnson. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.03.006
D.T. Dosev / Journal of Functional Analysis 256 (2009) 3490–3509
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no non-zero multiple of the identity is a commutator and with no effort one can also obtain that no operator of the form λI + K, where K is a compact operator and λ = 0, is a commutator in the Banach algebra L(X ) of all bounded linear operators on the Banach space X . The latter fact can be easily seen just by observing that the quotient algebra L(X )/K(X ) (K(X ) is the space of compact operators on X ) also satisfies the conditions of the Wintner’s theorem. Let us note also that instead of considering the ideal of compact operators one can consider any closed proper ideal in L(X ). The observations we made are valid in this case as well. For a Banach space X for which there is a unique maximal proper ideal in L(X ) (which is the case for the spaces Lp and p for 1 p ∞ and c0 ) one can hope to obtain a complete classification of the commutators on the space. The natural conjecture is that the only operators on X that are not commutators are the ones of the form λI + K, where K belongs to the unique maximal ideal in L(X ) and λ = 0. In 1965 Brown and Pearcy [4] made a breakthrough in this direction by proving that this is in fact a classification of the commutators on a Hilbert space. Note that if X = p (1 p < ∞) or X = c0 , the ideal of compact operators K(X ) is the largest proper ideal in L(X ) [13, Theorem 6.2]. Whitley’s proof actually shows that the ideal of strictly singular operators is the largest ideal in the aforementioned spaces, but as he points out, a result of Feldman, Gohberg and Markus [6] shows that the compact operators are in fact the only closed proper ideal in L(X ) for X = p (1 p < ∞) or X = c0 . In 1972, Apostol [1] proved that every non-commutator on the space p for 1 < p < ∞ is of the form λI + K, where K is compact and λ = 0. One year later he proved that the same classification holds in the case of X = c0 [3]. While Apostol’s approach in [1] gave some information about the commutators in L(1 ), he was unable to give a complete characterization. His proof uses the fact that the unit vector basis in p for 1 < p < ∞ is shrinking and this does not hold for 1 . We overcome this obstacle by using the structure of the infinite dimensional subspaces of 1 rather than just the properties of the basis. In Section 2 we study spaces X for which X ( ∞ i=0 X )p , 1 p < ∞ or p = 0, where for p1
∞ i=0
X
:= f = (f1 , f2 , . . .): fi ∈ X , i = 1, 2, . . . , f = p
p
∞
fi < ∞ p
i=0
(for p = 0, the norm is defined by f = sup0i<∞ fi with the additional assumption that limj →∞ fj = 0), and we generally assume that L(X ) has a largest proper ideal. The notion of a decomposition of X will be introduced and it will be shown how it can be used to obtain results about commutators on these spaces. In Section 3 we show that the compact operators on X , where X admits the decomposition X ( ∞ X i=0 )p (in the case p = 1 we will assume that X = L1 or X = 1 ), are commutators and as a corollary we will get that an operator that has a compact restriction to a sufficiently large subspace of X is also a commutator. Section 3 contains our main result—Theorem 18—which shows that the only operators on 1 that are not commutators are the ones of the form λI + K, where K is a compact operator and λ = 0. In the last section we give a characterization of the commutators on p1 ⊕ p2 ⊕ · · · ⊕ pn , where 1 pn < pn−1 < · · · < p1 < ∞ and we also show that the strictly singular operators on ∞ are commutators.
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2. Notation and basic results We say that two Banach spaces X and Y are isomorphic, and denote it by X Y , if there exists an onto, bounded, invertible operator T : X → Y . If in addition T = T −1 = 1, we say that X and Y are isometric and denote it by X ≡ Y . We will follow the ideas in Apostol’s paper [1], which in turn extend those of Brown and Pearcy [4] and earlier work referenced in [4], to develop a base for investigating the commutators on 1 and Lp (1 p < ∞). Definition 1. We say that D = {Xi } is a decomposition of X for some 1 p < ∞ or p = 0 if the following three conditions are satisfied: (1) X ( ∞ i=0 Xi )p , where Xi , i = 0, 1, . . . , are complemented subspaces of X which are also isomorphic to X . −1 (2) There exists a collection of isomorphisms {ψi }∞ i=1 , ψi : Xi → X such that ψi = 1 for i ∈ N and λ = supi∈N ψi < ∞. (3) If Pi is a projection from X onto Xi for i = 0, 1, . . . then for 1 p < ∞ we have nj=1 Pij xp = nj=1 Pij xp for every n > 0 and i1 , i2 , . . . , in > 0 such that il = ik for l = k. For p = 0 we have the corresponding condition nj=1 Pij x = max1j n Pij x for every n > 0 and i1 , i2 , . . . , in > 0 such that il = ik for l = k. Let us also have a collection of uniformly bounded isomorphisms {ϕi }∞ 1 as shown below ϕ
ϕ
ϕ
ϕ
0 1 2 3 X0 −→ X1 −→ X2 −→ X3 −→ ···
which make the following diagram commute ϕi
Xi
Xi+1 ψi+1
ψi
X −1 for every i ∈ N. Clearly ϕi = ψi+1 ◦ ψi and ϕi ψi λ, ϕi−1 λ. Note also, that using condition (3) in the definition of a decomposition we have Pi I − P0 P0 + 1 = C1 which will turn out to be useful in computations. For x = (xi ) ∈ X , xi ∈ Xi define the following two operators:
RD (x) =
∞ i=0
ϕi (xi ),
LD (x) =
∞
ϕi−1 (xi+1 ).
i=0
The operators LD and RD are, respectively, the left and the right shift associated with the decomposition D. As one may observe, the operators LD and RD move the components of x one position to the left/right, respectively, via the isomorphisms ϕi . Throughout the paper we will simply use the letters L and R for the shifts when the decomposition D associated with the shifts is clear from the context. Our first proposition shows some basic properties of the left and the right shift as well as the fact that all the powers of L and R are uniformly bounded, which will play an important role in the sequel.
D.T. Dosev / Journal of Functional Analysis 256 (2009) 3490–3509
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Proposition 2. Let D be a decomposition of X . Then we have
n
L 2λC1 , LR = I,
n
R 2λC1
for every n = 1, 2, . . . ,
RL = I − P0 , RPi = Pi+1 R, Pi L = LPi+1
n
lim L (x) = 0 for all 1 p < ∞ and p = 0.
(1) for i 0,
n→∞
(2) (3)
Proof. The relations LR = I , RL = I − P0 are clear from the definition of the left and the right shift. For x = (xi ) ∈ X , xi ∈ Xi , we have Pi+1 R(x) = Pi+1 Pi L(x) = Pi
∞ i=0
∞
ϕi (xi ) = ϕi (xi ) = RPi (x),
ϕi−1 (xi+1 )
= ϕi−1 (xi+1 ) = LPi+1 (x).
i=0
Now using the fact that −1 ϕk+n ◦ ϕk+n−1 ◦ · · · ◦ ϕk = ψk+n+1 ◦ ψk , −1 −1 ϕk−1 ◦ ϕk+1 ◦ · · · ◦ ϕk+n = ψk−1 ◦ ψk+n+1
we deduce that R n+1 (x) =
∞
ϕi+n ◦ ϕi+n−1 ◦ · · · ◦ ϕi (xi ) =
i=0
Ln+1 (x) =
∞
∞
−1 ψi+n+1 ◦ ψi (xi ),
i=0
−1 −1 ϕi−1 ◦ ϕi+1 ◦ · · · ◦ ϕi+n (xi+n+1 ) =
∞
i=0
ψi−1 ◦ ψi+n+1 (xi+n+1 )
i=0
for n = 0, 1, . . . . Finally we estimate the norms of R n and Ln for p 1:
∞
p ∞
n+1 p
−1
p
−1
R (x) =
ψ ψi+n+1 ◦ ψi (xi ) ◦ ψi xi p
i+n+1
i=0 i=0
∞
p ∞
p p p p p
λ xi = λ P0 x + λ
Pi x
i=0 i=1
p
p = λp P0 xp + λp (I − P0 )x 2p λp P0 + 1 xp ,
p ∞
n+1 p
−1 −1
L (x) = ψ ◦ ψn+1 (xn+1 ) + ψi ◦ ψi+n+1 (xi+n+1 )
0
i=1
∞
p
p
−1
p −1
ψ0 ◦ ψn+1 (xn+1 ) +
2 ψi ◦ ψi+n+1 (xi+n+1 )
i=1
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2
p
λ Pn+1 x + p
p
∞
λ Pi+n+1 x p
p
2p λp
i=1
∞
Pi xp
i=1
p ∞
p
p
= 2λ Pi x = 2p λp (I − P0 )x 2p λp P0 + 1 xp .
i=1
p Note that Lm (x)p 2p λp ∞ i=0 Pi+m x → 0 shows (3) for p 1. In the case p = 0 the computations are somewhat simpler and are shown below:
∞
−1
n+1
−1
R (x) =
ψi+n+1 ◦ ψi (xi ) = max ψi+n+1 ◦ ψi (xi )
0i<∞ i=0
−1
max ψi+n+1 ◦ ψi xi λ max xi = λC1 x, 0i<∞
∞
n+1
L (x) =
ψi−1 ◦ ψi+n+1 (xi+n+1 )
1i<∞
i=0
ψ0−1 ◦ ψn+1 (xn+1 ) + max ψi−1 ◦ ψi+n+1 (xi+n+1 )
1i<∞
2 max ψi−1 ◦ ψi+n+1 xi+n+1 2λ max xi+n+1 0i<∞
0i<∞
2λC1 x. In this case Lm (x) 2λ max0i<∞ xi+m → 0 proves (3) for p = 0.
2
Denote by DS the inner derivation determined by S in L(X ) i.e. DS T = ST − T S. In the notation introduced above, an operator T ∈ L(X ) is a commutator if and only if there exists S ∈ L(X ) such that T ∈ DS L(X ). For a given decomposition D of X denote by A(D) the set A(D) = T ∈ L(X ):
∞
n
n
R T L is strongly convergent .
(4)
n=0
For T ∈ A(D) we define TD =
∞
R n T Ln .
n=0
Our next lemma shows that each operator T ∈ A(D) is a commutator and also gives an explicit expression for T as the commutator of two operators.
D.T. Dosev / Journal of Functional Analysis 256 (2009) 3490–3509
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Lemma 3. Let T ∈ A(D) for some decomposition D = {Xi } of X . Then we have T = DL (RTD ) = −DR (TD L).
(5)
Proof. We will show one of the equalities via direct computation. The proof of the other is similar. DL (RTD ) = LRTD − RTD L = TD − R
∞
n
R TL
n
L
n=0
= TD −
∞
R n T Ln = T .
n=1
In the computation above we used the convention L0 = R 0 = I .
2
Lemma 4. For a decomposition D = {Xi } of X we have the following relations A(D) = DR L(X )RL = DL RLL(X ) . Proof. We will show the first of the relations. The proof of the second, as one may expect, is similar. If T ∈ A(D), then TD L = TD LRL = (TD L)RL ∈ L(X )RL. Then using T = −DR (TD L) from (5) we have T ∈ DR (L(X )RL). Now, to prove the other direction, assume that T ∈ L(X )RL. Then T = SRL for some operator S (hence T R = SR). Then m
R n (DR T )Ln =
n=0
m
R n (RT − T R)Ln =
n=0
=
m n=0
=
m n=0
m
R n+1 T Ln −
n=0
R n+1 SRLLn −
m
m
R n T RLn
n=0
R n SRLRLn
n=0
R n+1 SRLn+1 −
m
R n SRLn
n=0
= R m+1 SRLm+1 − SR = R m+1 T Lm − T R. m m Since lim m→∞ L (x) = 0 for any x ∈ X from (3) and R < 2λC1 for every m > 0, we have m n n limm→∞ n=0 R (DR T )L = −T R. From the last equation we conclude that DR T ∈ A(D) and (DR T )D = −T R. Moreover, from T R = SR we have (DR T )D = −SR and multiplying both sides by L we obtain (DR T )D L = −T . 2
We proved that for a given decomposition D all operators T ∈ A(D) are commutators, but in general the condition in (4) is hard to check for a given operator T . We want to have a condition on T which is easy to check and which ensures the containment T ∈ A(D). To be more precise,
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given an operator T , we want to have a condition on T which will allow us to build a decomposition D for which T ∈ A(D). Our next lemma gives us such a condition (as will become clear later) and it will be our main tool for constructing decompositions in the sequel.
n = Lemma 5. Let T ∈ L(X ) and D = {Xi } be a decomposition of X . Fix ε > 0 and denote P n P , where P is the projection onto X . Let us also assume that i i i=0 i
n )T = lim T (I − P
n ) = 0. (6) lim (I − P n→∞
n→∞
Then there exists an increasing sequence of integers {mj }∞ i=0 such that ∞ ∞ ∞
(I − P
T (I − P
(I − P
mj )T +
mj ) +
mi )T (I − P
mj ) < ε. j =0
j =0
i,j =0
i I − P0 P0 + 1 = C1 for every i ∈ N. This estimate Proof. Note first that I − P follows directly from condition (3) in the definition of a decomposition. Let {nj }∞ j =0 be an increasing sequence of integers such that ∞
T (I − P
nj ) < ε , 3C1 j =0
∞
(I − P
nj )T < ε . 3C1 j =0
Now we can use the inequality ∞ ∞ m
(I − P
(I − P
T (I − P
i )T (I − P
i )T (I − P
nj )
nj )
nj ) + C1 j =0
j =0
j =m+1
to deduce that lim
i→∞
∞
(I − P
i )T (I − P
nj ) = 0. j =0
Using the last equation we can find an increasing sequence of integers {mj }∞ j =0 , mj nj such that ∞ ∞
(I − P
mi )T (I − P
nj ) < ε . 3C1 i=0 j =0
Now it is easy to deduce that the sequence {mj }∞ j =0 satisfies the condition of the lemma. In fact
T (I − P
nj )(I − P
mj ) C1 T (I − P
mj ) = T (I − P
nj ) ,
(I − P
mj )(I − P
nj )T ,
mj )T = (I − P
nj )T C1 (I − P
(I − P
mi )T (I − P
mi )T (I − P
mi )T (I − P
mj ) = (I − P
nj )(I − P
mj ) C1 (I − P
nj ) . This finishes the proof.
2
D.T. Dosev / Journal of Functional Analysis 256 (2009) 3490–3509
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Lemma 6. Let D = {Xi } be a decomposition of X . Then for any T ∈ L(X ) we have Pi T Pj ∈ A(D),
(Pi T Pj )D CPi T Pj
where C depends on D only. Proof. Let us consider the case p 1 first. Note first that Ln 2λC1 and R n 2λC1 . For any x ∈ X we have (see Proposition 2)
m+r
p m+r
p
n n
n n R Pi T Pj L x =
R Pi T Pj L Pj +n x
n=m
n=m
(2λC1 ) Pi T Pj 2p
p
m+r
Pj +n xp
n=m
(2λC1 )2p Pi T Pj p
∞
Pj +n xp
n=m
(2λC1 )
2p
p C1 Pi T Pj p xp .
∞ p n n Since ∞ n=m Pj +n x → 0 as m → ∞ we have that n=0 R Pi T Pj L is strongly convergent and Pi T Pj ∈ A(D). The inequality in the theorem in this case follows from the inequality above with C = 4λ2 C13 . For p = 0 a similar calculation shows
m+r
m+r
R n Pi T Pj Ln x =
R n Pi T Pj Ln Pj +n x
n=m
n=m
= max R n Pi T Pj Ln Pj +n x
mnm+r
4λ2 C12 Pi T Pj
max
mnm+r
Pj +n x.
Since maxmnm+r Pj +n x → 0 as m → ∞ we apply the same argument as in the case p 1 to obtain the conclusion of the theorem. 2 Corollary 7. Let T ∈ L(X ) and D = {Xi } be a decomposition of X . Then we have T P0 = DR LT P0 − (P0 T P0 )D L , P0 T = DL −P0 T R + R(P0 T P0 )D . Proof. We will prove the first equation. Note that from Lemma 3 we have −DR ((P0 T P0 )D L) = P0 T P0 and DL (R(P0 T P0 )D ) = P0 T P0 . Now
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DR LT P0 − (P0 T P0 )D L = RLT P0 − LT P0 R + P0 T P0 = (I − P0 )T P0 + P0 T P0 = T P0 , DL −P0 T R + R(P0 T P0 )D = −LP0 T R + P0 T RL + P0 T P0 = P0 T (I − P0 ) + P0 T P0 = P0 T . Above we used the equality P0 R = 0 = LP0 , which is clear from the definitions of R and L.
2
The following theorem shows the importance of the decompositions in determining whether an operator is a commutator. Theorem 8. Under the hypotheses of Lemma 5, there is a decomposition D of X for which T ∈ A(D),
TD CT + ε
where C depends on D only. In particular, using Lemma 3 we conclude that T is a commutator.
Proof. m0 Using the sequence mi {mj } from Lemma 5, define a decomposition {Xi }, where X0 =
k=0 Xk , and Xi = k=mi−1 +1 Xk for i > 0. Note that the new decomposition also satisfies the conditions in Definition 1 for being a decomposition. Conditions (1) and (3) are clearly satisfied since we are taking only finite direct sums. Hence we will only check condition (2). C λ Let X ( ∞ i=0 X )p and let Xi X for i = 0, 1, . . . (for two Banach spaces X and Y we say λ
that X Y if there exist an onto isomorphism T : X → Y such that T T −1 λ). Then for ∞ λ C C 1 r < s we have sk=r Xk ( sk=r X )p ( sk=r ( ∞ i=0 X )p )p ≡ ( i=0 X )p X , so all 2 the terms of the decomposition after the first one are C λ isomorphic to X . The first term in the new decomposition is also isomorphic to X thus we showed (2) for the new decomposition. For simplicity of notation we will denote the new decomposition by {Xi } and the projections onto Xi by Pi . In the new notation the conclusion from Lemma 5 can be written as ∞ ∞ ∞
(I − P
T (I − P
(I − P
j )T +
j ) +
i )T (I − P
j ) < ε. j =0
j =0
i,j =0
i−1 ) = (I − P
i−1 )Pi = Pi we have Now using Pi (I − P ∞
Pi T Pj P0 T P0 + C1
i,j =0
∞
Pi T + C1
∞
Pi (I − P
i−1 )T
i=1
+ C1
∞
T (I − P
j −1 )Pj
j =1
T Pj +
j =1
i=1
P0 T P0 + C1
∞
∞ i,j =1
Pi T Pj
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+
3499
∞
Pi (I − P
i−1 )T (I − P
j −1 )Pj
i,j =1
P0 T P0 + C12
∞ ∞
(I − P
T (I − P
i−1 )T + C12
j −1 )
j =1
i=1
+ C12
∞
(I − P
i−1 )T (I − P
j −1 )
i,j =1
P0 T P0 + C12 ε. ∞ Since the series ∞ i=0 Pi is strongly convergent to I , we have T = i,j =0 Pi T Pj in the norm topology of L(X ). Using Lemma 6 and the estimate from above, the operator S=
∞
(Pi T Pj )D
i,j =0
is well defined and using Lemma 3 for each term in the sum of the definition of S we have that T = DR (−SL) ∈ A(D). Now DR (TD L − SL) = 0 and by the proof of Lemma 4 we have 0 = − DR (TD L − SL) D L = (TD − S)L. From the equation above we conclude that TD = S and TD CT + ε.
2
3. Compactness and commutators on p and Lp (1 p < ∞) In order to prove the conjecture about the structure of the commutators on a given space we have to show that all the elements in the largest proper ideal are commutators. We prove a lemma that takes care of this in the case X = 1 and also shows that the ideal of compact operators consists of commutators only, provided the space X has some additional structure. Before that we will show a lemma about the operators T on X which do not preserve a copy of X in the cases of X = 1 and X = L1 , which we will use and which is interesting on its own. Lemma 9. Let X = L1 or X = 1 and suppose that T ∈ L(X ) does not preserve a copy of X .
⊂ X, X
≡ X , there exists Y ⊂ X,
such that Y is (1 + δ) Then, for every δ > 0 and for every X isomorphic to X , (1 + δ) complemented in X , and T|Y < δ. Proof. Consider the case X = L1 first. By assumption T does not preserve a copy of L1 which implies that T is not an E-operator (actually this can be taken as an equivalent definition for an operator not to be an E-operator [5, Theorem 4.1]) and hence it is not sign-preserving either
such that Z X
and [11, Theorem 1.5]. Now [11, Lemma 3.1] gives us a subspace Z ⊂ X
T|Z < δ. Using [11, Theorem 1.1] we find Y ⊂ Z, which is (1 + δ) isomorphic to X ≡ L1 ,
and Y clearly satisfies T|Y < δ. If Q is the norm one projection (1 + δ) complemented in X
onto X, and R : X → Y is a projection of norm less than 1 + δ, then P := RQ is a projection from L1 onto Y and P < 1 + δ.
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is isometric to 1 , then X
= span{ψi : For the case X = 1 we use the fact that if X of norm one, such that i = 0, 1, . . .} for some vectors {ψi }∞ i=1 ψj =
λi ei ,
with σj ∩ σk = ∅ for j = k,
i∈σj
where {ei }∞ i=1 is the standard unit vector basis of 1 . This follows trivially from the obser → 1 is an into isometry vation that U ei and U ej must have disjoint supports if U : X (cf. [8, Proposition 2.f.14]). Note also that since every infinite dimensional subspace of 1 contains an isomorphic copy of 1 [8, Proposition 2.a.2], then the operator T is automatically strictly singular and hence compact [6]. Then, {T ψi }∞ i=0 is relatively compact in 1 and hence there exist y ∈ 1 and a subsequence {ψij } such that T ψij → y. Without loss of generality we may assume 2i+1 for i = 0, 1, . . . . Clearly {ϕi }∞ that T ψi → y. Finally, define ϕi = ψ2i −ψ i=0 is a normalized 2
block basis of X such that T ϕi 1 → 0. Assume without loss of generality that T ϕi 1 < ε (this can be easily achieved by passing to a subsequence). Then for Y = span{ϕi : i = 0, 1, . . .}
is 1-complemented in X
as it is the closed span of a we have T|Y < ε. Note also that Y ⊂ X
→ Y be the normalized block basis and clearly is isometric to X [9, Lemma 1]. Finally, let R : X
norm one projection onto Y and Q : 1 → X be the norm one projection onto X. Then clearly P := RQ is a norm one projection onto Y . 2 Lemma 10. Let X be a Banach space for which X ( ∞ i=0 X )p for some 1 p < ∞ or p = 0. In the case p = 1 we will assume that X = L1 or X = 1 . Let T ∈ L(X ) be a compact operator and ε > 0. Then there exists a decomposition D of X such that T ∈ A(D) and TD CT + ε for some constant C depending on D only. Consequently T is a commutator and T = −DR (TD L). Proof. The result is known in the case of X = Lp and X = p for 1 < p < ∞ (cf. [12] and [1]), and for X = c0 and X = C(K) [3]. The proof presented here in these cases follows Apostol’s ideas from [1] and our generalized context gives a shorter proof in the case of Lp for 1 < p < ∞. Partial results were known in the case X = 1 [1, Theorem 2.6]. Case I. p > 1 or p = 0. In this case we proceed as in Theorem 2.4 in [1], but instead of considering a particular type of decomposition as in [1], we consider an arbitrary decomposition D of
n = ni=0 Pi . Now we have X and denote P
n )T = lim T (I − P
n ) = 0. lim (I − P n→∞
n→∞
Choose ϕi , ψi ∈ X such that
(I − P
n )T −
n )T ϕn > (I − P X
T (I − P
n )ψn > T (I − P
n ) − X
1 , n+1
1 , n+1
ϕn = 1,
ψn = 1,
n )ψn = ψn . (I − P
∞ Since the set {T ϕi }∞ i=0 is relatively compact in X and the sequence {(I − Pi )}i=0 converges
strongly to 0 we have limn→∞ (I − Pn )T = 0. On the other hand, the sequence {ψi }∞ i=0 is
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weakly convergent to 0. Now using the fact that T is compact, it follows that the sequence
{T ψi }∞ i=0 converges to 0 in norm and hence limn→∞ T (I − Pn ) = 0. Now Theorem 8 gives the result. Case II. p = 1. Fix ε > 0 and let D = {Xi } be the fixed decomposition ∞ of X defined by Xi = 1 1 L1 [ 2i+1 , 2i ) in the case of X = L1 and by Xi = PNi 1 (where N = i=0 Ni such that card Ni = card N for all i ∈ N and Nj ∩ Nj = ∅ for i = j ) in the case of X = 1 . Using Lemma 9 for each Xi with δ = 2εi will give us 1 + ε complemented subspaces {Yi } of X which are isomorphic to X and T|Yi < 2εi . Set Y0 = (I − ∞ i=1 Pi )X . Note that D = {Yi } is a decomposition for X since all the spaces are complemented and isomorphic to X . This is clear for Yi for i = 1, 2, . . ., and it also holds for Y0 , since X0 ⊂ Y0 is complemented in X , isomorphic to X , and using [5, Corollary 5.3] in the case X = L1 , and [9, Proposition 4] in the case X = 1 , it follows that Y0 is
n = ni=0 Pi , then clearly we have limn→∞ T (I − P
n ) = 0. isomorphic to X as well. Now, if P
Since T is compact operator, then we have limn→∞ (I − Pn )T = 0 as well (the argument provided in Case I above works in this case as well), so using Theorem 8 we conclude that T is a commutator. 2 Remark 11. Using the previous lemma we immediately conclude that [12, Theorem 4.3] holds for p = 1. Namely, a multiplication operator Mφ on L1 is a commutator if and only if the spectrum of Mφ contains more than one limit point or contains zero as the unique limit point. Corollary 12. Let X be a Banach space for which X ( ∞ i=0 X )p for some 1 p < ∞ or p = 0. In the case p = 1 we will assume that X = L1 or X = 1 . Let T ∈ L(X ) and suppose that P is a projection on X such that P X X (I − P )X and that either T P or P T is a compact operator. Then T is a commutator. Proof. First we treat the case when T P is compact operator. Let D = {Xi }∞ i=0 be a decomposition for which T P ∈ A(D) and (T P )D X CT P X + 2ε for a fixed ε > 0 (by Lemma 10). We also want D to be such that (I − P )X = X0 hence we may assume (I − P ) = P0 , where P0 is the projection onto X0 . This can obviously be done for 1 < p < ∞ (since the decomposition used in the proof was arbitrary). In the case of L1 we consider the operator T = GT G−1 where G : P X ⊕ (I − P )X → (I − P0 )X ⊕ X0 is an isomorphism such that GP X = (I − P0 )X , G(I − P )X = X0 . In this case T GP G−1 is compact and clearly we can choose the decomposition as in Lemma 10 and apply the same argument. Now without loss of generality we will assume that T = T . In the case of 1 we can make a similarity as in the previous case and reduce to the case where T PM is a compact operator for some infinite M ⊂ N. Define S = LT (I − P ) − P0 T (I − P )P0 D L − (T P )D L. Use Eq. (5) applied to T P and P0 T (I − P )P0 (recall that P0 T (I − P )P0 ∈ A(D) by Lemma 6) to get −DR (T P )D L = T P , −DR P0 T (I − P )P0 D L = P0 T (I − P )P0 = P0 T (I − P ).
(7) (8)
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Now DR LT (I − P ) = RLT (I − P ) − LT (I − P )R = (I − P0 )T (I − P )
(9)
since (I − P )R = 0. Combining (7)–(9) we conclude that DR S = T . If P T is compact we consider S = −(I − P )T R + R(P0 (I − P )T P0 )D + R(P T )D and a similar calculation shows that T = DL (S). 2 4. Commutators on 1 We already saw in the previous section that the compact operators on 1 are commutators and in order to prove the conjecture in the case of X = 1 we have to show that all operators not of the form λI + K, where K is compact and λ = 0, are commutators. To do that we are going to show that if T is not of the form λI + K, then there exist complemented subspaces X and Y of X which are isomorphic to X , such that X ∩ Y = {0} and T|X : X → Y is an onto isomorphism. As we will see, this last property of T will be enough to show that T is a commutator on any X ) space X for which X ( ∞ p. i=0 Definition 13. The left essential spectrum of T ∈ L(X ) is the set [2, Definition 1.1]
σl.e. (T ) = λ ∈ C: inf (λ − T )x = 0 for all Y ⊂ X s.t. codim(Y ) < ∞ , x∈SY
where SY = {y ∈ Y : y = 1}. For any T ∈ L(X ), σl.e. (T ) is a closed non-void set [2, Theorem 1.4]. The following lemma is an analog of Lemma 4.1 from [1] and the proof follows the steps in the proof there. Lemma 14. Let X = 1 and let T ∈ L(X ), 0 ∈ σl.e. (T ). If T is not compact, then it is similar to an operator T ∈ L(X ) for which there exists a projection PM such that M ⊂ N, card M = card(N − M), and PN−M T PM is not a compact operator. Proof. For simplicity of notation we will denote PM simply by P . By Lemma 3.2 in [1] and using a similarity we can obtain a subset M of N so that n∈N−M T P{n} < ∞, where P{n} is the projection onto the nth coordinate. From this inequality we have that T (I − P ) is compact. If (I − P )T P is not compact we are done, thus we may suppose that (I − P )T P is compact. The equality T = T (I − P ) + (I − P )T P + P T P gives us that P T P is not compact. Using X ≡ P X ⊕1 (I − P )X , let ϕ : P X → (I − P )X be an isometry and define the operators V and V in the following way V (x) = ϕ(P x), V x = ϕ −1 (I − P )x . It is easy to see that P V = V (I − P ) = V P = (I − P )V = V 2 = (V )2 = 0 and V V + V V = I . Define √ 2S = P − (I − P ) + V + V .
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Now a simple check gives us 2S 2 = P − (I − P ) + V + V P − (I − P ) + V + V = P + P V + (I − P ) − (I − P )V + V P + V V − V (I − P ) + V V = 2I + P V − (I − P )V + V P − V (I − P ) = 2I + P V − V + V P − V = 2I, hence S = S −1 . Now consider the operator 2(I − P )S −1 T SP . Again a simple calculation shows that 2(I − P )S −1 T SP = −(I − P ) + (I − P )V T (P + V P ) = −(I − P ) + V T (P + V P ) = −(I − P )T P − (I − P )T V P + V T P + V T V P = −(I − P )T P − (I − P )T (I − P )V P + V P T P + V T (I − P )V P = V P T P + K where K = −(I −P )T P −(I −P )T (I −P )V P +V T (I −P )V P is a compact operator because T (I − P ) and (I − P )T P are compact operators (the first one by construction, the second by assumption). Since V|P X is an isometry, we conclude that V P T P is not compact and hence (I − P )S −1 T SP is not compact either. Taking T = S −1 T S we finish the proof. 2 Proposition 15. Let X = 1 and T ∈ L(X ) be such that there exists a projection P such that P X X , (I − P )X X and the operator (I − P )T P is not compact. Then there exists a complemented subspace Y ⊂ P X such that Y X , (I − P )T P|Y is an isomorphism into and (I − P )T P (Y ) is complemented in X . Proof. Clearly (I − P )T P is not a strictly singular operator since in L(X ) the ideal of compact operators and strictly singular operators coincide [6]. In particular, this implies that there exists an infinite dimensional subspace Z ⊂ P X such that (I − P )T P is an isomorphism on Z. Consider the infinite dimensional subspace (I −P )T P Z. Using [9, Lemma 2] (cf. also [8, Proposition 2.a.2]) we conclude that there exists U ⊂ (I − P )T P Z which is complemented in X and isomorphic to X . Clearly (I − P )T P is an isomorphism on ((I − P )T P )−1 U and since U is complemented in X , we also have that ((I − P )T P )−1 U is complemented in X as well. 2 Theorem 16. Let X ( ∞ i=0 X )p , 1 p < ∞ or p = 0, and let T ∈ L(X ) be an operator for which there exists a projection P such that P X X , (I − P )X X and there exists a complemented subspace Y ⊆ P X such that Y X , (I − P )T P|Y is an isomorphism into and X = (I − P )T P (Y ) is also complemented in X . Then there exists a decomposition D such that T is similar to a matrix operator ∗ L ∗ ∗ on X ⊕ X , where L is the left shift associated with D.
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Proof. Let PX be a projection onto X. Note that X = PX X ⊂ (I − P )X and hence Y ⊂ P X ⊂ (I − PX )X . The previous observation shows that the operator PX T (I − PX )|Y is an isomorphism from Y onto X. Also, X = PX X X by the assumption of the theorem, and (I − PX )X contains Y —a complemented copy of X and hence (I − PX )X X (using a result of Pelczynski [9, Proposition 4]). The observations we made imply that without loss of generality we may assume PX = I − P . Consider two decompositions D1 = {Xi }, D2 = {Yi } of X such that X = Y0 = X1 ⊕ X2 ⊕ · · · , X0 = Y1 ⊕ Y2 ⊕ · · · and Y1 = Y . Define a map S Sϕ = LD1 ϕ ⊕ LD2 ϕ,
ϕ ∈ X,
from X to X ⊕ X . Clearly, S is an isomorphism due to fact that LD1 RD2 = LD2 RD1 = 0 which follows from our choice of the decompositions D1 and D2 . It is also easy to see that the inverse of S is given by S −1 (a, b) = RD1 a + RD2 b. Just using the definition of S and the formula for S −1 we have that ST S −1 (a, b) = ST (RD1 a + RD2 b) = S(T RD1 a + T RD2 b) = (LD1 T RD1 a + LD1 T RD2 b) ⊕ (LD2 T RD1 a + LD2 T RD2 b), hence ST S −1 =
∗ L D 1 T RD 2 ∗ ∗
.
Let A = PY0 T RD2 = (I − P )T RD2
(10)
and note that A|PY0 X ≡ A|(I −P )X : (I − P )X → (I − P )X is onto and invertible since RD2 is an isomorphism on PY0 X and RD2 (PY0 X ) = Y1 = Y . Here we used the fact that PY0 T is an isomorphism on Y (P Y = Y ). Denote by T0 the inverse of A|PY0 X (note that T0 is an automorphism on (I − P )X ) and consider G = I + T0 (I − P ) − T0 A. We will show that G−1 = A + P . In fact, from the definitions of A and T0 it is clear that AT0 (I − P ) = T0 A(I − P ) = I − P ,
P T0 = P A = 0,
(I − P )A = A,
(11)
and since A maps onto (I − P )X and AT0|(I −P )X = I|(I −P )X we also have A − AT0 A = 0. Now using (11) and (12) it is easy to see that
(12)
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(A + P )G = (A + P )(I + T0 (I − P ) − T0 A) = A + AT0 (I − P ) − AT0 A + P = I − P + P = I, G(A + P ) = I + T0 (I − P ) − T0 A (A + P ) = A + P + T0 (I − P )A + T0 (I − P )P − T0 AA − T0 AP = A + P + T0 A − T0 AA − T0 AP = P + (I − T0 A)A + T0 A(I − P ) = P + (I − T0 A)(I − P )A + (I − P ) = I + (I − P ) − T0 A(I − P ) A = I + I − P − (I − P ) A = I. Using a similarity we obtain
I 0
0 G−1
∗ ∗
L D 1 T RD 2 ∗
I 0
0 G
=
∗ L D 1 T RD 2 G . ∗ ∗
It is clear that we will be done if we show that LD1 = LD1 T RD2 G. In order to do this, consider the equation (A + P )G = I ⇔ AG + P G = I . Multiplying both sides of the last equation on the left by LD1 gives us LD1 AG + LD1 P G = LD1 . Using LD1 P ≡ LD1 PX0 = 0 we obtain LD1 AG = LD1 . Finally, substituting A from (10) in the last equation yields LD1 = LD1 AG = LD1 PY0 T RD2 G = LD1 (I − PX0 )T RD2 G = LD1 T RD2 G which finishes the proof.
2
The following theorem was proved in [1] for X = p , 1 < p < ∞, but inessential modifications give the result in the general case. Theorem 17. Let D be a decomposition of X and let L be the left shift associated with it. Then the matrix operator
T1 T2
L T3
acting on X ⊕ X is a commutator. Proof. Let D = {Xi } be the given ∞decomposition. Consider a decomposition D1 = {Yi } such that Y0 = ∞ X and X = 0 i=1 i i=1 Yi . Now there exists an operator G such that DLD G = RD1 LD1 (T1 + T3 ). This can be done using Corollary 7, since RD1 LD1 = I − PY0 = PX0 . By making the similarity T :=
I G
0 I
T1 T2
L T3
I 0 −G I
=
T1 − LG ∗
L T3 + GL
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we have T1 + T3 − LG + GL = T1 + T3 − DL G = T1 + T3 − RD1 LD1 (T1 + T3 ) = PY0 (T1 + T3 ). Using again Corollary 7 we deduce that T1 + T3 − LG + GL is a commutator. Thus by replacing T by T we can assume that T1 + T3 is a commutator, say T1 + T3 = AB − BA and A < 12 (this can be done by scaling). Denote by MT left multiplication by the operator T . Then MR DA < 1 where R is the right shift associated with D. The operator T0 = (MI − MR DA )−1 MR (T3 B − T2 ) is well defined and hence B I B I A 0 T1 L A 0 . − = T0 0 T0 0 T3 A − L T2 T3 T3 A − L In order to verify the computation above, after multiplying and subtracting the corresponding matrices, we have to check that T3 B + (A − L)T0 − T0 A = T2 . This is easy to see using the definition of T0 in the form (MI − MR DA )T0 = MR (T3 B − T2 ) and multiplying both sides of the last equation with L on the left. Then using the fact that LR = I we are done. This finishes the proof. 2 Theorem 18. Let X = 1 . An operator T ∈ L(X ) is a commutator if and only if T − λ is not compact for any λ = 0. Proof. Note first that if T is a commutator, from the remarks we made in the introduction it follows that T − λ cannot be compact for any λ = 0. For proving the other direction we have to consider two cases: Case I. If T is compact operator (λ = 0), the statement of the theorem follows from Lemma 10. Case II. If T − λ is not compact for any λ, then we consider σl.e. (T ). Since σl.e. (T ) is a nonempty set, there exists λ ∈ σl.e. (T ) such that T − λ is not compact and we are in a position to apply Lemma 14 for the operator T − λ. Note that the conclusion of Lemma 14 for T − λ implies that the same claim is true for T as well. Now we are in position to apply Theorem 16 (which . we can because of Proposition 15) and obtain that T is similar to an operator of the form ∗∗ L ∗ Finally, we apply Theorem 17 to complete the proof. 2 5. Commutators on p1 ⊕ p2 ⊕ ··· ⊕ pn and ∞ A B an operator from X ⊕ Y into X ⊕ Y . Lemma 19. Let X and Y be Banach spaces and T = C D If A and D are commutators on the corresponding spaces then T is a commutator on X ⊕ Y . Proof. Let A = [A1 , A2 ] and D = [D1 , D2 ]. Assume without loss of generality that max(A2 , D2 ) < 14 . We need to find operators E1 and E2 such that T=
A1 E2
E1 D1
A2 + I 0
0 D2
−
A2 + I 0
0 D2
A1 E2
E1 D1
,
or equivalently, we have to solve the equations B = E1 D2 − (A2 + I )E1 ,
(13)
C = E2 (A2 + I ) − D2 E2
(14)
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for E1 and E2 . Let G : L(X, Y ) → L(X, Y ) be defined by G(S) = −SA2 + D2 S. It is clear that G < 1 by our choice of A2 and D2 , hence I − G is invertible. Now it is enough to observe that (14) is equivalent to C = (I − G)(E2 ) which will give us E2 = (I − G)−1 C. Analogously we define F : L(Y, X) → L(Y, X) by F (S) = −A2 S + SD2 and then (13) will be equivalent to −B = (I − F )(E1 ). Applying the same argument as above we get that I − F is invertible and hence E1 = (I − F )−1 (−B). 2 Theorem 20. Let X = p ⊕ q where 1 q < p < ∞ and T ∈ L(X ). Let Pp and Pq be the natural projections from X onto p and q respectively. Then T is a commutator if and only if Pp T Pp and Pq T Pq are commutators as operators acting on p and q respectively. Proof. Throughout the proof we will work with the matrix representation of T as an operator A B acting on X . Let T = C where A : p → p , D : q → q , B : q → p , C : p → q . The D well-known fact that the operator C is compact [8, Proposition 2.c.3] will play an important role i Bi in the proof. If T is a commutator, then T = [T1 , T2 ] for some T1 , T2 ∈ L(X ). Write Ti = A Ci Di for i = 1, 2. A simple computation shows that A1 B2 + B1 D2 − A2 B1 − B2 D1 [A1 , A2 ] + B1 C2 − B2 C1 . T= C1 A2 + D1 C2 − C2 A1 − D2 C1 [D1 , D2 ] + C1 B2 − C2 B1 From the classification of the commutators on p for 1 p < ∞ and the fact that the Ci ’s are compact we immediately deduce that the diagonal entries in the last representation of T are commutators. For the other direction we apply Lemma 19 which concludes the proof. 2 The classification given in the theorem can be immediately generalized to a space which is finite sum of p spaces, namely, we have the following Corollary 21. Let X = p1 ⊕ p2 ⊕ · · · ⊕ pn where 1 pn < pn−1 < · · · < p1 < ∞ and T ∈ L(X ). Let Ppi be the natural projections from X onto pi for i = 1, 2, . . . , n. Then T is a commutator if and only if for each 1 i n, Ppi T Ppi is a commutator as an operator acting on pi . Proof. We will proceed by induction on n and clearly Theorem 20 gives us the result for n = 2. If the statement is true for some n, then to show it for n + 1, denote Y = p2 ⊕ p3 ⊕ · · · ⊕ pn . Now X = p1 ⊕ Y and using the same argument as in Theorem 20 we can see that if T is a commutator, then both Pp1 T Pp1 and PY T PY are commutators on p1 and Y respectively. Here we use the induction step to show that compact perturbation of a commutator on Y is still a commutator. The other direction is exactly as in Theorem 20. It is worthwhile noticing that for this direction we do not need any assumption on the spaces in the sum. 2 Our last result shows that every strictly singular operator in L(∞ ) is a commutator. Clearly, this is an essential step in proving the natural conjecture about the classification of the commutators on ∞ , namely, that an operator T ∈ L(∞ ) is not a commutator if and only if T = λI + S for some strictly singular operator S and some λ = 0, but because of the structure of ∞ we cannot apply the method developed in this paper. Note also that the ideal of the strictly singular operators is the largest ideal in L(∞ ) (follows from [13, Theorem 1.2] and [10, Corollary 1.4]), the proof of which we include for completeness. In order to develop (if at all possible) a similar
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approach, one may have to find a suitable substitution for the set A(D) defined in (4) and an analog of the left essential spectrum (Definition 13). Also, a couple of times in this paper we have used the fact that every infinite dimensional subspace of p (1 p < ∞) contains a further subspace isomorphic to p and complemented in p , which does not hold for ∞ . This additional obstacle should be overcome as well. First we will prove Lemma 22. The ideal of strictly singular operators is the largest ideal in L(∞ ). Proof. Assume that T is not a strictly singular operator. Our goal will be to prove that any ideal that contains T must coincide with L(∞ ). Note first that on ∞ the ideals of the weakly compact and the strictly singular operators coincide [13, Theorem 1.2]. Then we use the fact that any nonweakly compact operator is an isomorphism on some subspace Y of ∞ isomorphic to ∞ [10, Corollary 1.4]. The subspaces Y and T Y will be automatically complemented in ∞ because ∞ is an injective space. This automatically yields that I∞ factors through T and hence any ideal containing T coincides with L(∞ ). 2 Theorem 23. Let T ∈ L(∞ ) be a strictly singular operator. Then T is a commutator. Proof. Since T is a strictly singular operator, T is weakly compact [10, Corollary 1.4]. Thus it follows that T ∞ is separable (since any weakly compact subset of the dual to any separable space is metrizable) and let Y = T ∞ . The space ∞ /Y must be non-reflexive since assuming otherwise gives us that Y has a subspace isomorphic to ∞ [7, Theorem 4]. Now consider the quotient map Q : ∞ → ∞ /Y . Q is not weakly compact and hence (using again [10, Corollary 1.4 ]) there exists X ∞ , X ⊂ ∞ such that Q|X is an isomorphism. Let P be a projection onto QX and set P = (Q|X )−1 P Q. P is a projection in ∞ , P Y = {0} and by the construction, P ∞ is isomorphic to ∞ . Thus it follows that P T = 0 and we obtain that T is similar to an operator T for which there exists an infinite M ⊂ N such that PM T = 0. Using [1, Theorem 2.9] we conclude that T is a commutator and hence T is a commutator. 2 Acknowledgment The author thanks W.B. Johnson for useful conversations and suggestions on the subject of this paper. References [1] C. Apostol, Commutators on p spaces, Rev. Roumaine Math. Appl. 17 (1972) 1513–1534. [2] C. Apostol, On the left essential spectrum and non-cyclic operators in Banach spaces, Rev. Roumaine Math. Appl. 17 (1972) 1141–1148. [3] C. Apostol, Commutators on c0 -spaces and on ∞ -spaces, Rev. Roumaine Math. Pures Appl. 18 (1973) 1025–1032. [4] A. Brown, C. Pearcy, Structure of commutators of operators, Ann. of Math. 82 (1965) 112–127. [5] P. Enflo, T.W. Starbird, Subspaces of L1 containing L1 , Studia Math. 65 (1979) 203–225. [6] I.A. Feldman, I.C. Gohberg, A.S. Markus, Normally solvable operators and ideals associated with them, Izv. Moldavsk. Filial Akad. Nauk SSSR 10 (76) (1960) 51–69 (in Russian). [7] J. Lindenstrauss, H.P. Rosenthal, Automorphisms in c0 , 1 and m, Israel J. Math. 7 (1969) 227–239. [8] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces. I., Ergeb. Math. Grenzgeb., vol. 92, Springer-Verlag, Berlin– New York, 1977, xiii+188 pp. [9] A. Pelczynski, Projections in certain Banach spaces, Studia Math. 19 (1960) 209–228. [10] H.P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970) 13–36.
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[11] [12] [13] [14]
H.P. Rosenthal, Embeddings of L1 in L1 , in: Contemp. Math., vol. 26, 1984, pp. 335–349. C. Schneeberger, Commutators on a separable Lp space, Proc. Amer. Math. Soc. 28 (2) (1971) 464–472. R.J. Whitley, Strictly singular operators and their conjugates, Trans. Amer. Math. Soc. 113 (1964) 252–261. A. Wintner, The unboundedness of quantum-mechanical matrices, Phys. Rev. 71 (1947) 738–739.
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Journal of Functional Analysis 256 (2009) 3510–3544 www.elsevier.com/locate/jfa
The convenient setting for non-quasianalytic Denjoy–Carleman differentiable mappings ✩ Andreas Kriegl, Peter W. Michor ∗ , Armin Rainer Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria Received 3 July 2008; accepted 5 March 2009 Available online 21 March 2009 Communicated by J. Bourgain
Abstract For Denjoy–Carleman differentiable function classes C M where the weight sequence M = (Mk ) is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is C M if it maps C M -curves to C M -curves. The category of C M -mappings is cartesian closed in the sense that C M (E, C M (F, G)) ∼ = C M (E × F, G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C M -diffeomorphisms is a C M -Lie group but not better. © 2009 Elsevier Inc. All rights reserved. Keywords: Convenient setting; Denjoy–Carleman classes; Non-quasianalytic of moderate growth
1. Introduction Denjoy–Carleman differentiable functions form spaces of functions between real analytic and C ∞ . They are described by growth conditions on the Taylor expansions, see 2.1. Under appropriate conditions the fundamental results of calculus still hold: Stability under differentiation, composition, solving ODEs, applying the implicit function theorem. See Section 2 for a review of Denjoy–Carleman differentiable functions, which is summarized in Table 1. In [8,16–18,21], see [19] for a comprehensive presentation, convenient calculus was developed ✩
PM was supported by FWF-Project P 21030-N13. AR was supported by FWF-Projects P19392 & J2771.
* Corresponding author.
E-mail addresses:
[email protected] (A. Kriegl),
[email protected] (P.W. Michor),
[email protected] (A. Rainer). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.03.003
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for C ∞ , holomorphic, and real analytic functions: see Appendices A–C for a short overview of the essential results. In this paper we develop the convenient calculus for Denjoy–Carleman classes C M where the weight sequence M = (Mk ) is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth (this holds for all Gevrey differentiable functions G1+δ for δ > 0). By ‘convenient calculus’ we mean that the following theorems are proved: A mapping is C M if it maps C M -curves to C M -curves, see 3.9; this is wrong in the quasianalytic case, see 3.12. The category of C M -mappings is cartesian closed in the sense that C M (E, C M (F, G)) ∼ = C M (E × F, G) for convenient vector spaces, see 5.3; this is wrong for weight sequences of non-moderate growth, see 5.4. The uniform boundedness principle holds for linear mappings into spaces of C M -mappings. For the quasianalytic case we hope for results similar to the real analytic case, but the methods have to be different. This will be taken up in another paper. In chapter 6 some applications to manifolds of mappings are given: The group of C M -diffeomorphisms is a C M -Lie group but not better. 2. Review of Denjoy–Carleman differentiable functions 2.1. Denjoy–Carleman classes C M (Rn , R) of differentiable functions We mainly follow [27] (see also the references therein). We use N = N>0 ∪ {0}. For each multi-index α = (α1 , . . . , αn ) ∈ Nn , we write α! = α1 ! · · · αn !, |α| = α1 + · · · + αn , and ∂ α = ∂ |α| /∂x1α1 · · · ∂xnαn . Let M = (Mk )k∈N be an increasing sequence (Mk+1 Mk ) of positive real numbers with M0 = 1. Let U ⊆ Rn be open. We denote by C M (U ) the set of all f ∈ C ∞ (U ) such that, for all compact K ⊆ U , there exist positive constants C and ρ such that α ∂ f (x) Cρ |α| |α|!M|α| (2.1.1) for all α ∈ Nn and x ∈ K. The set C M (U ) is a Denjoy–Carleman class of functions on U . If Mk = 1, for all k, then C M (U ) coincides with the ring C ω (U ) of real analytic functions on U . In general, C ω (U ) ⊆ C M (U ) ⊆ C ∞ (U ). We assume that M = (Mk ) is logarithmically convex, i.e., Mk2 Mk−1 Mk+1
for all k,
(2.1.2)
or, equivalently, Mk+1 /Mk is increasing. Considering M0 = 1, we obtain that also (Mk )1/k is increasing and Ml Mk Ml+k
for all l, k ∈ N.
(2.1.3)
We also get (see 2.9) M1k Mk Mj Mα1 · · · Mαj
for all αi ∈ N>0 , α1 + · · · + αj = k.
(2.1.4)
Let M = (Mk ) be logarithmically convex. Then Mk = Mk /M0 M1k 1 is increasing by (2.1.4), logarithmically convex, and C M (U ) = C M (U ) for all U open in Rn by (2.1.5). So without loss we assumed at the beginning that M is increasing. Hypothesis (2.1.2) implies that C M (U ) is a ring, for all open subsets U ⊆ Rn , which can easily be derived from (2.1.3) by means of Leibniz’s rule. Note that definition (2.1.1) makes sense
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also for mappings U → Rp . For C M -mappings, (2.1.2) guarantees stability under composition ([23], see also [1, 4.7]; a proof is also contained in the end of the proof of 3.9). A further consequence of (2.1.2) is the inverse function theorem for C M ([14]; for a proof see also [1, 4.10]): Let f : U → V be a C M -mapping between open subsets U, V ⊆ Rn . Let x0 ∈ U . Suppose that the Jacobian matrix (∂f/∂x)(x0 ) is invertible. Then there are neighborhoods U of x0 , V of y0 := f (x0 ) such that f : U → V is a C M -diffeomorphism. Moreover, (2.1.2) implies that C M is closed under solving ODEs (due to [15]): Consider the initial value problem dx = f (t, x), dt
x(0) = y,
where f : (−T , T ) × Ω → Rn , T > 0, and Ω ⊆ Rn is open. Assume that f (t, x) is Lipschitz in x, locally uniformly in t. Then for each relative compact open subset Ω1 ⊆ Ω there exists 0 < T1 T such that for each y ∈ Ω1 there is a unique solution x = x(t, y) on the interval (−T1 , T1 ). If f : (−T , T ) × Ω → Rn is a C M -mapping then the solution x : (−T1 , T1 ) × Ω1 → Rn is a C M -mapping as well. Suppose that M = (Mk ) and N = (Nk ) satisfy Mk C k Nk , for all k and a constant C, or equivalently, sup k∈N>0
Mk Nk
1 k
< ∞.
(2.1.5)
Then, evidently C M (U ) ⊆ C N (U ). The converse is true as well (if (2.1.2) is assumed): One can prove that there exists f ∈ C M (R) such that |f (k) (0)| k!Mk for all k (see [27, Theorem 1]). So the inclusion C M (U ) ⊆ C N (U ) implies (2.1.5). Setting Nk = 1 in (2.1.5) yields that C ω (U ) = C M (U ) if and only if 1
sup (Mk ) k < ∞. k∈N>0
Since (Mk )1/k is increasing (by logarithmic convexity), the strict inclusion C ω (U ) C M (U ) is equivalent to 1
lim (Mk ) k = ∞.
k→∞
We shall also assume that C M is stable under derivation, which is equivalent to the following condition sup k∈N>0
Mk+1 Mk
1 k
< ∞.
(2.1.6) +1
Note that the first order partial derivatives of elements in C M (U ) belong to C M (U ), where M +1 denotes the shifted sequence M +1 = (Mk+1 )k∈N . So the equivalence follows from (2.1.5), by replacing M with M +1 and N with M.
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Definition. By a DC-weight sequence we mean a sequence M = (Mk )k∈N of positive numbers with M0 = 1 which is monotone increasing (Mk+1 Mk ), logarithmically convex (2.1.2), and satisfies (2.1.6). Then C M (U, R) is a differential ring, and the class of C M -functions is stable under compositions. DC stands for Denjoy–Carleman and also for derivation closed. 2.2. Quasianalytic function classes Let Fn denote the ring of formal power series in n variables (with real or complex coefficients). For a sequence M0 = 1, M1 , M2 , . . . > 0, we denote by FnM the set of elements F = α∈Nn Fα x α of Fn for which there exist positive constants C and ρ such that |Fα | Cρ |α| M|α| for all α ∈ Nn . A class C M is called quasianalytic if, for open connected U ⊆ Rn and all a ∈ U , the Taylor series homomorphism 1 ∂ α f (a)x α f → Ta f (x) = Ta : C M (U ) → FnM , α! n α∈N
is injective. By the Denjoy–Carleman theorem [4,5], the following statements are equivalent: (1) C M is quasianalytic. ∞ 1 1/j 1/k . (2) k=1 m = ∞ where mk = inf{(j !Mj ) : j k} is the increasing minorant of (k!Mk ) ∞ k1 1/k = ∞ where Mk∗ = inf{(j !Mj )(l−k)/(l−j ) (l!Ml )(k−j )/(l−j ) : j k l, j < l} (3) k=1 ( Mk∗ ) is the logarithmically convex minorant of k!Mk . ∞ Mk∗ (4) k=0 M ∗ = ∞. k+1
For contemporary proofs see for instance [10, 1.3.8] or [24, 19.11]. Suppose that C ω (U ) and C M (U ) is quasianalytic and logarithmically convex. Then Ta : C M (U ) → FnM is not surjective. This is due to Carleman [4]; an elementary proof can be found in [27, Theorem 3]. C M (U )
2.3. Non-quasianalytic function classes If M is a DC-weight sequence which is not quasianalytic, then there are C M partitions of unity. Namely, there exists a C M function f on R which does not vanish in any neighborhood of 0 but which has vanishing Taylor series at 0. Let g(t) = 0 for t 0 and g(t) = f (t) for t > 0. From g we can construct C M bump functions as usual. 2.4. Strong non-quasianalytic function classes Let M be a DC-weight sequence with C ω (U, R) C M (U, R). Then the mapping Ta : C M (U, R) → FnM is surjective, for all a ∈ U , if and only if there is a constant C such that ∞ k=j
Mj Mk C (k + 1)Mk+1 Mj +1
for any integer j 0.
See [22] and references therein. (2.4.1) is called strong non-quasianalyticity condition.
(2.4.1)
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2.5. Moderate growth A DC-weight sequence M has moderate growth if sup j,k∈N>0
Mj +k Mj Mk
1 j +k
< ∞.
(2.5.1)
Moderate growth implies derivation closed. Moderate growth together with strong nonquasianalyticity (2.4.1) is called strong regularity: Then a version of Whitney’s extension theorem holds for the corresponding function classes (e.g. [3]). 2.6. Gevrey functions Let δ > 0 and put Mk = (k!)δ , for k ∈ N. Then M = (Mk ) is strongly regular. The corresponding class C M of functions is the Gevrey class G1+δ . 2.7. More examples Let δ > 0 and put Mk = (log(k + e))δk , for k ∈ N. Then M = (Mk ) is quasianalytic for 0 < δ 1 and non-quasianalytic (but not strongly) for δ > 1. In any case M is of moderate growth. 2 Let q > 1 and put Mk = q k , for k ∈ N. The corresponding C M -functions are called q-Gevrey regular. Then M = (Mk ) is strongly non-quasianalytic but not of moderate growth, thus not strongly regular. It is derivation closed. 2.8. Spaces of C M -functions Let U ⊆ Rn be open and let M be a DC-weight sequence. For any ρ > 0 and K ⊆ U compact with smooth boundary, define CρM (K) := f ∈ C ∞ (K): f ρ,K < ∞ with
f ρ,K
|∂ α f (x)| n := sup |α| : α∈N , x∈K . ρ |α|!M|α|
It is easy to see that CρM (K) is a Banach space. In the description of CρM (K), instead of compact K with smooth boundary, we may also use open K ⊂ U with K compact in U , like [27]. Or we may work with Whitney jets on compact K, like [13]. The space C M (U ) carries the projective limit topology over compact K ⊆ U of the inductive limit over ρ ∈ N>0 : M lim C (K) . C M (U ) = lim ρ ←− −→ K⊆U ρ∈N>0
One can prove that, for ρ < ρ , the canonical injection CρM (K) → CρM (K) is a compact mapping; M it is even nuclear (see [13], [12, p. 166]). Hence lim −→ρ Cρ (K) is a Silva space, i.e., an inductive
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Table 1 Let M = (Mk ) and N = (Nk ) be increasing () sequences of real numbers with M0 = N0 = 1. By U we denote an open subset of Rn . The mapping Ta : C M (U ) → FnM is the Taylor series homomorphism for a ∈ U (see 2.2). Recall that M is a DC-weight sequence if it is logarithmically convex and stable under derivation. Properties of C M
Properties of M M increasing, M0 = 1, (always assumed below this line)
⇒
C ω (U ) ⊆ C M (U ) ⊆ C ∞ (U )
M is logarithmically convex (always assumed below this line), i.e., Mk2 Mk−1 Mk+1 for all k. Then: (Mk )1/k is increasing, Ml Mk Ml+k for all l, k, and M1k Mk Mj Mα1 · · · Mαj for αi ∈ N>0 , α1 + · · · + αj = k.
⇒
C M (U ) is a ring. C M is closed under composition. C M is closed under applying the inverse function theorem. C M is closed under solving ODEs.
supk∈N>0 (Mk /Nk )1/k < ∞
⇔
C M (U ) ⊆ C N (U )
supk∈N>0 (Mk )1/k < ∞
⇔
C ω (U ) = C M (U )
limk→∞ (Mk )1/k = ∞ supk∈N>0 (Mk+1 /Mk )1/k < ∞
⇔
C ω (U ) C M (U )
⇔
C M is closed under derivation.
(always assumed below this line) ∞ Mk =∞ k=0 (k+1)M
⇔
C M is quasianalytic,
k+1
or, equivalently, ∞ 1 1/k = ∞ k=1 ( k!Mk ) ∞ Mk <∞ k=0 (k+1)M k+1
i.e., Ta : C M (U ) → FnM is injective (not surjective if C ω (U ) C M (U )). ⇔
C M is non-quasianalytic. Then C M partitions of unity exist.
limk→∞ (Mk )1/k = ∞ and ∞ Mj Mk k=j (k+1)Mk+1 C Mj +1
⇔
M has moderate growth, i.e.,
⇒
C ω (U ) C M (U ) and Ta : C M (U ) → FnM is surjective, i.e., C M is strongly non-quasianalytic.
for all j ∈ N and some C
M +k 1/(j +k) ) <∞ supj,k∈N>0 ( M j M j k
C M is cartesian closed will be proved in 5.3
M is strongly regular, i.e., it is strongly non-quasianalytic and has moderate growth.
⇒
Whitney’s extension theorem holds in C M .
δ > 0 and Mk = (k!)δ for k ∈ N. Then M is strongly regular.
⇔
C M is the Gevrey class G1+δ .
limit of Banach spaces such that the canonical mappings are compact; therefore it is complete, webbed, and ultrabornological, see [7], [11, 5.3.3], also [19, 52.37]. We shall use this locally convex topology below only for n = 1 – in general it is stronger than the one which we will define in 3.1, but it has the same system of bounded sets, see 4.6. 2.9. Lemma. For a logarithmically convex sequence Mk with M0 = 1 we have M1k Mk Mj Mα1 · · · Mαj
for all αi ∈ N>0 , α1 + · · · + αj = k.
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Proof. We use induction on k. The assertion is trivial for k = j . Assume that j < k. Then there exists i such that αi 2. Put αi := αi − 1. By induction hypothesis, Mj Mα1 · · · Mαi · · · Mαj M1k−1 Mk−1 . Since Mk+1 /Mk is increasing by (2.1.2), we obtain Mj Mα1 · · · Mαj = Mj Mα1 · · · Mαi · · · Mαj · M1k−1 Mk−1 ·
Mαi Mαi
Mk M1k Mk . Mk−1
2
3. C M -mappings 3.1. Definition: C M -mappings Let M be a DC-weight sequence, and let E be a locally convex vector space. A curve c : R → E is called C M if for each continuous linear functional ∈ E ∗ the curve ◦ c : R → R is of class C M . The curve c is called strongly C M if c is smooth and for all compact K ⊂ R there exists ρ > 0 such that
c(k) (x) : k ∈ N, x ∈ K ρ k k!Mk
is bounded in E.
The curve c is called strongly uniformly C M if c is smooth and there exists ρ > 0 such that
c(k) (x) : k ∈ N, x ∈ R ρ k k!Mk
is bounded in E.
Now let M be a non-quasianalytic DC-weight sequence. Let U be a c∞ -open subset of E, and let F be another locally convex vector space. A mapping f : U → F is called C M if f is smooth in the sense of A.3 and if f ◦ c is a C M -curve in F for every C M -curve c in U . Obviously, the composite of C M -mappings is again a C M -mapping, and the chain rule holds. This notion is equivalent to the expected one on Banach spaces, see 3.9 below. We equip the space C M (U, F ) with the initial locally convex structure with respect to the family of mappings C M (c,)
C M (U, F ) −−−−−→ C M (R, R),
f → ◦ f ◦ c, ∈ E ∗ , c ∈ C M (R, U )
where C M (R, R) carries the locally convex structure described in 2.8 and where E ∗ is the space of all continuous linear functionals on E. For U ⊆ Rn , this locally convex topology differs from the one described in 2.8, but they have the same bounded sets, see 4.6 below. If F
is convenient, then by standard arguments, the space C M (U, F ) is c∞ -closed in the product ,c C M (R, R) and hence is convenient. If F is convenient, then a mapping f : U → F is C M if and only if ◦ f is C M for all ∈ F ∗ .
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3.2. Example: There are weak C M -curves which are not strong By [27, Theorem 1], for each DC-weight sequence M there exists f ∈ C M (R, R) such that k! Mk for all k ∈ N. Then g : R → RN given by g(t)n = f (nt) is C M but not strongly C M . Namely, each bounded linear functional on RN depends only on finitely many coordinates, so we take the maximal ρ for the finitely many coordinates of g being involved. On the other hand, for each ρ and any compact neighborhood L of 0 the set |f (k) (0)|
g (k) (t) : t ∈ L, k ∈ N ρ k k!Mk
has nth coordinate unbounded if n > ρ. 3.3. Lemma. Let E be a convenient vector space such that there exists a Baire vector space topology on the dual E ∗ for which the point evaluations evx are continuous for all x ∈ E. Then a curve c : R → E is C M if and only if c is strongly C M , for any DC-weight sequence M. See 5.2 for a more general version. Proof. Let K be compact in R. We consider the sets
|( ◦ c)(k) (x)| Aρ,C := ∈ E ∗ : C for all k ∈ N, x ∈ K ρ k k!Mk which are closed subsets in E ∗ for the Baire topology. We have ρ,C Aρ,C = E ∗ . By the Baire property there exists ρ and C such that the interior U of Aρ,C is non-empty. If 0 ∈ U then for all ∈ E ∗ there is an > 0 such that ∈ U − 0 and hence for all x ∈ K and all k we have ( ◦ c)(k) (x) 1 ( + 0 ) ◦ c (k) (x) + (0 ◦ c)(k) (x) 2C ρ k k!Mk . So the set
c(k) (x) : k ∈ N, x ∈ K ρ k k!Mk
is weakly bounded in E and hence bounded.
2
3.4. Lemma. Let M be a DC-weight sequence, and let E be a Banach space. For a curve c : R → E the following are equivalent. (1) c is C M . (2) For each sequence (rk ) with rk t k → 0 for all t > 0, and each compact set K in R, the set 1 { k!M c(k) (a)rk : a ∈ K, k ∈ N} is bounded in E. k (3) For each sequence (rk ) satisfying rk > 0, rk r rk+ , and rk t k → 0 for all t > 0, and each 1 c(k) (a)rk k : a ∈ K, k ∈ N} is compact set K in R, there exists an > 0 such that { k!M k bounded in E.
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Proof. (1) ⇒ (2) For K, there exists ρ > 0 such that (k) (k) c (a) k c (a) k!M rk = k!ρ k M · rk ρ k k E E is bounded uniformly in k ∈ N and a ∈ K by 3.3. (2) ⇒ (3) Use = 1. 1 (3) ⇒ (1) Let ak := supa∈K k!M c(k) (a) E . Using [19, 9.2.(4⇒1)] these are the coeffik cients of a power series with positive radius of convergence. Thus ak /ρ k is bounded for some ρ > 0. 2 3.5. Lemma. Let M be a DC-weight sequence. Let E be a convenient vector space, and let S be a family of bounded linear functionals on E which together detect bounded sets (i.e., B ⊆ E is bounded if and only if (B) is bounded for all ∈ S). Then a curve c : R → E is C M if and only if ◦ c : R → R is C M for all ∈ S. Proof. For smooth curves this follows from [19, 2.1 and 2.11]. By 3.4, for any ∈ E , the function ◦ c is C M if and only if: (1) For each sequence (rk ) with rk t k → 0 for all t > 0, and each compact set K in R, the set 1 { k!M ( ◦ c)(k) (a)rk : a ∈ K, k ∈ N} is bounded. k 1 By (1) the curve c is C M if and only if the set { k!M c(k) (a)rk : a ∈ K, k ∈ N} is bounded in E. By k (1) again this is in turn equivalent to ◦ c ∈ C M for all ∈ S, since S detects bounded sets. 2
3.6. C M -curve lemma A sequence xn in a locally convex space E is said to be Mackey convergent to x, if there exists some λn ∞ such that λn (xn − x) is bounded. If we fix λ = (λn ) we say that xn is λ-converging. Lemma. Let M be a non-quasianalytic DC-weight sequence. Then there exist sequences λk → 0, tk → t∞ , sk > 0 in R with the following property: For 1/λ = (1/λn )-converging sequences xn and vn in a convenient vector space E there exists a strongly uniformly C M -curve c : R → E with c(tk + t) = xk + t.vk for |t| sk . Proof. Since C M is not quasianalytic we have k 1/(k!Mk )1/k < ∞. We choose another non¯ quasianalytic DC-weight sequence M¯ = (M¯ k ) with (Mk /M¯ k )1/k → ∞. By 2.3 there is a C M 1 1 function ϕ : R → [0, 1] which is 0 on {t: |t| 2 } and which is 1 on {t: |t| 3 }, i.e. there exist ¯ ρ > 0 such that C, (k) ϕ (t) Cρ ¯ k k!M¯ k
for all t ∈ R and k ∈ N.
For x, v in a absolutely convex bounded set B ⊆ E and 0 < T 1 the curve c : t → ϕ(t/T ) · (x + tv) satisfies (cf. [2, Lemma 2]):
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t t .(x + t.v) + kT 1−k ϕ (k−1) .v T T T −k ¯ k ¯ ¯ k−1 (k − 1)!M¯ k−1 .B ∈ T Cρ k!Mk 1 + .B + kT 1−k Cρ 2 T 1 −k ¯ k ¯ .B + T T −k C¯ ρ k k!M¯ k .B ⊆ T Cρ k!Mk 1 + 2 ρ 3 1 T −k ρ k k!M¯ k .B ⊆ C¯ + 2 ρ
c(k) (t) = T −k ϕ (k)
¯ 3 + 1 ) > 0 which do not depend on x, v and T such that c(k) (t) ∈ So there are ρ, C := C( 2 ρ CT −k ρ k k!M¯ k .B for all k and t. Let 0 < Tj 1 with j Tj < ∞ and tk := 2 j
for |t − tj | Tj .
Then (k) c (t) Cρ k k!M¯ k λj Cρ k k!M¯ k Mk = Cρ k k!Mk B M¯ k Tjk for t = t∞ . Hence c : R → EB (see [19, 2.14.6] or A.1) is smooth at t∞ as well, and is strongly C M by the following lemma. 2 3.7. Lemma. Let c : R \ {0} → E be strongly C M in the sense that c is smooth and for all bounded K ⊂ R \ {0} there exists ρ > 0 such that
c(k) (x) : k ∈ N, x ∈ K ρ k k!Mk
is bounded in E.
Then c has a unique extension to a strongly C M -curve on R. Proof. The curve c has a unique extension to a smooth curve by [19, 2.9]. The strong C M condition extends by continuity. 2 3.8. Corollary. Let M be a non-quasianalytic DC-weight sequence. Then we have: (1) The final topology on E with respect to all strong C M -curves equals the Mackey closure topology. (2) A locally convex space E is convenient A.2 if and only if for any (strongly) C M -curve c : R → E there exists a (strongly) C M -curve c1 : R → E with c1 = c.
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Proof. (1) For any Mackey converging sequence there exists a C M -curve passing through a subsequence in finite time by 3.6. So the final topologies generated by the Mackey converging sequences and by the C M -curves coincide. (2) In order to show that a locally convex space E is convenient, we have to prove that it is c∞ -closed in its completion. So let xn ∈ E converge Mackey to x∞ in the completion.Then by 3.6 there exists a strongly C M -curve c in the completion passing in finite time through a subsequence of the xn with velocity vn = 0. The form of c (in the proof of 3.6) shows that its derivatives c(k) (t) for k > 0 are multiples of the xn and hence have values in E. Then c is a C M -curve and so the antiderivative c of c lies in E by assumption. In particular x∞ ∈ c(R) ⊆ E. Conversely, if E is convenient, then every smooth curve c has a smooth antiderivative c1 in E by [19, 2.14]. Since
ρ k+1 (k
1 1 Mk (k+1) c1 (t) = c(k) (t) k ρ(k + 1)Mk+1 ρ k!Mk + 1)!Mk+1
and since Mk 1 ρ(k + 1)Mk+1 ρM1 by (2.1.2) the antiderivative c1 is (strongly) C M if c is so.
2
3.9. Theorem. Let M = (Mk ) be a non-quasianalytic DC-weight sequence. Let U ⊆ E be c∞ open in a convenient vector space, and let F be a Banach space. For a mapping f : U → F , the following assertions are equivalent. (1) f is C M . (2) f is C M along strongly C M -curves. (3) f is smooth, and for each closed bounded absolutely convex B in E and each x ∈ U ∩ EB there are r > 0, ρ > 0, and C > 0 such that 1 d k (f ◦ iB )(a) k Cρ k L (EB ,F ) k!Mk for all a ∈ U ∩ EB with a − x B r and all k ∈ N. (4) f is smooth, and for each closed bounded absolutely convex B in E and each compact K ⊆ U ∩ EB there are ρ > 0 and C > 0 such that 1 d k (f ◦ iB )(a) k Cρ k L (EB ,F ) k!Mk for all a ∈ K and all k ∈ N. Proof. (1) ⇒ (2) is clear. (2) ⇒ (3) Without loss let E = EB be a Banach space. For each v ∈ E and x ∈ U the iterated directional derivative dvk f (x) exists since f is C M along affine lines. To show that f is smooth it suffices to check that dvkn f (xn ) is bounded for each k ∈ N and each Mackey convergent sequences xn and vn → 0, by [19, 5.20]. For contradiction let us assume that there exist k and sequences
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xn and vn with dvkn f (xn ) → ∞. By passing to a subsequence we may assume that xn and vn are (1/λn )-converging for the λn from 3.6. Hence there exists a strongly C M -curve c in E and with c(t + tn ) = xn + t.vn for t near 0 for each n separately, and for tn from 3.6. But then (f ◦ c)(k) (tn ) = dvkn f (xn ) → ∞, a contradiction. So f is smooth. Assume for contradiction that the boundedness condition in (3) does not hold. Then there exists x ∈ U such that for all r, ρ, C > 0 there is an a = a(r, ρ, C) ∈ U and k = k(r, ρ, C) ∈ N with a − x r but 1 d k f (a) k > Cρ k . L (E,F ) k!Mk By [19, 7.13] we have k d f (a)
Lk (E,F )
(2e)k sup dvk f (a). v 1
1 So for each ρ and n take r = nρ and C = n. Then there are an,ρ ∈ U with an,ρ − x moreover vn,ρ with vn,ρ = 1, and kn,ρ ∈ N such that
1 nρ ,
kn,ρ (2e)kn,ρ dv f (an,ρ ) > n. n,ρ k n,ρ kn,ρ !Mkn,ρ ρ Since K := {an,ρ : n, ρ ∈ N} ∪ {x} is compact, this contradicts the following Claim. For each compact K ⊆ E there are C, ρ 0 such that for all k ∈ N and x ∈ K we have sup v 1 dvk f (x) Cρ k k!Mk . Otherwise, there exists a compact set K ⊆ E such that for each n ∈ N there are kn ∈ N, xn ∈ K, and vn with vn = 1 such that k d n f (xn ) > kn !Mk vn
n
1 λ2n
kn +1 ,
where we used C = ρ := 1/λ2n with the λn from 3.6. By passing to a subsequence (again denoted n) we may assume that the xn are 1/λ-converging, thus there exists a strongly C M -curve c : R → E with c(tn + t) = xn + t.λn .vn for t near 0 by 3.6. Since (f ◦ c)(k) (tn ) = λkn dvkn f (xn ), we get
(f ◦ c)(kn ) (tn ) kn !Mkn
1 kn +1
=
1 kn kn +1 kn dvn f (xn ) λn kn !Mkn
>
1 kn +2 kn +1
→ ∞,
λn
a contradiction to f ◦ c ∈ C M . (3) ⇒ (4) is obvious since the compact set K is covered by finitely many balls. (4) ⇒ (1) We have to show that f ◦ c is C M for each C M -curve c : R → E. By 3.4(2) it suffices to show that for each sequence (rk ) satisfying rk > 0, rk r rk+ , and rk t k → 0 for all
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1 t > 0, and each compact interval I in R, there exists an > 0 such that { k!M (f ◦ c)(k) (a)rk k : k a ∈ I, k ∈ N} is bounded. By 3.4(2) applied to rk 2k instead of rk , for each ∈ E ∗ , each sequence (rk ) with rk t k → 0 1 ( ◦ c)(k) (a)rk 2k : a ∈ I, k ∈ N} is for all t > 0, and each compact interval I in R the set { k!M k
1 bounded in R. Thus { k!M c(k) (a)rk 2k : a ∈ I, k ∈ N} is contained in some closed absolutely k
1 convex B ⊆ E. Consequently, c(k) : I → EB is smooth and hence Kk := { k!M c(k) (a)rk 2k : k a ∈ I } is compact in EB for each k. Then each sequence (xn ) in the set
1 1 (k) K := c (a)rk : a ∈ I, k ∈ N = Kk k!Mk 2k k∈N
has a cluster point in K ∪ {0}: either there is a subsequence in one Kk , or 2kn xkn ∈ Kkn ⊆ B for kn → ∞, hence xkn → 0 in EB . So K ∪ {0} is compact. By Faà di Bruno ([6] for the 1dimensional version) (f ◦ c)(k) (a) = k!
j 0
j
α∈N>0 α1 +···+αj =k
c(α1 ) (a) c(αj ) (a) 1 j d f c(a) ,..., j! α1 ! αj !
and (2.1.4) for a ∈ I and k ∈ N we have 1 (k) k k!M (f ◦ c) (a)rk M1 k
j 0
M1k
j
α∈N>0 α1 +···+αj =k
k − 1 j 0
j −1
j d j f (c(a)) Lj (EB ,F ) c(αi ) (a) B rαi j !Mj αi !Mαi i=1
Cρ j
1 1 = M1k ρ(1 + ρ)k−1 C k . 2k 2
1 2 (f ◦ c)(k) (a)( M1 (1+ρ) )k rk : a ∈ I, k ∈ N} is bounded as required. So { k!M k
2
3.10. Corollary. Let M and N be non-quasianalytic DC-weight sequences with (2.1.5) sup k∈N>0
Mk Nk
1 k
< ∞.
Then C M (U, F ) ⊆ C N (U, F ) for all convenient vector spaces E and F and each c∞ -open U ⊆ E. Moreover C ω (U, F ) ⊆ C M (U, F ) ⊆ C ∞ (U, F ). All these inclusions are bounded. Proof. The inclusions C M ⊆ C N ⊆ C ∞ follow from 3.9 since this is true for condition 3.9(3) applied to ◦ f for ∈ F ∗ . Without loss let F = R. If f is C ω then for each closed absolutely convex bounded B ⊆ E the mapping f ◦ iB : U ∩ EB → R is given by its locally converging Taylor series by [19, 10.1]. So 3.9(3) is satisfied for Mk = 1 and thus for each DC-weight sequence M. So f is C M . All inclusions are bounded by the uniform boundedness principle 4.1 below for C M and [19, 5.26] for C ∞ . 2
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3.11. Corollary. Let M = (Mk ) be a non-quasianalytic DC-weight sequence. Then we have: (1) Multilinear mappings between convenient vector spaces are C M if and only if they are bounded. : (2) If f : E ⊇ U → F is C M , then the derivative df : U → L(E, F ) is C M , and also df U × E → F is C M , where the space L(E, F ) of all bounded linear mappings is considered with the topology of uniform convergence on bounded sets. (3) The chain rule holds. Proof. (1) If f is multilinear and C M then it is smooth by 3.9 and hence bounded by A.3(2). Conversely, if f is multilinear and bounded then it is smooth by A.3(2). Furthermore, f ◦ iB is multilinear and continuous and all derivatives of high order vanish. Thus condition 3.9(3) is satisfied, so f is C M . (2) Since f is smooth, by A.3(3) the map df : U → L(E, F ) exists and is smooth. Let c : R → U be a C M -curve. We have to show that t → df (c(t)) ∈ L(E, F ) is C M . By [19, 5.18] and 3.5 it suffices to show that t → c(t) → (df (c(t)).v) ∈ R is C M for each ∈ F ∗ and v ∈ E. We are reduced to show that x → (df (x).v) satisfies the conditions of 3.9. By 3.9 applied to ◦ f , for each closed bounded absolutely convex B in E and each x ∈ U ∩ EB there are r > 0, ρ > 0, and C > 0 such that 1 d k ( ◦ f ◦ iB )(a) k Cρ k L (EB ,R) k!Mk for all a ∈ U ∩ EB with a − x B r and all k ∈ N. For v ∈ E and those B containing v we then have k d d( ◦ f )( )(v) ◦ iB )(a) k L (EB ,R) k+1 = d ( ◦ f ◦ iB )(a)(v, . . .)Lk (E ,R) B k+1 d ( ◦ f ◦ iB )(a)Lk+1 (E ,R) v EB Cρ k+1 (k + 1)!Mk+1 B M k+1 Cρ k k!Mk (k + 1)ρ Mk Mk+1 1/k k C ρ¯ k!Mk for ρ¯ > ρ sup (k + 1)ρ , Mk k1 is C M . the latter quantity being finite by (2.1.6). By 4.2 below also df (3) This is valid for all smooth f . 2 3.12. Remark. For a quasi analytic DC-weight sequence M Theorem 3.9 is wrong. In fact, take 2 any rational function, e.g. x 2xy+y 2 . Let t → x(t), y(t) be in C M (R, R) with x(0) = 0 = y(0). Then
¯ and y(t) = t r y(t) ¯ for r > 0 and for C M -functions x¯ and y¯ since C M is derivation x(t) = t r x(t) ¯ 2 = 0, since C M is closed. If (x, y) is not constant we may choose r such that x(0) ¯ 2 + y(0) 2 x(t)y(t)2 x(t) ¯ y(t) ¯ quasi-analytic. Then t → x(t)2 +y(t)2 = t r x(t) is C M near 0, but the rational function is not ¯ 2 +y(t) ¯ 2 smooth.
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4. C M -uniform boundedness principles 4.1. Theorem (Uniform boundedness principle). Let M = (Mk ) be a non-quasianalytic DCweight sequence. Let E, F , G be convenient vector spaces and let U ⊆ F be c∞ -open. A linear mapping T : E → C M (U, G) is bounded if and only if evx ◦T : E → G is bounded for every x ∈ U. This is the C M -analogon of A.3(7). Compare with [19, 5.22–5.26] for the principles behind it. They will be used in the following proof and in 4.6 and 4.10. Proof. For x ∈ U and ∈ G∗ the linear mapping ◦ evx = C M (x, ) : C M (U, G) → R is continuous, thus evx is bounded. So if T is bounded then so is evx ◦T . Conversely, suppose that evx ◦T is bounded for all x ∈ U . For each closed absolutely convex bounded B ⊆ E we consider the Banach space EB . For each ∈ G∗ , each C M -curve c : R → U , each t ∈ R, and each compact K ⊂ R the composite given by the following diagram is bounded. E
T
C M (U, G)
evc(t)
G
C M (c,)
EB
C M (R, R)
M lim −→ρ Cρ (K, R)
evt
R
M By [19, 5.24 and 5.25] the map T is bounded. In more detail: Since lim −→ρ Cρ (K, R) is webbed M by 2.8, the closed graph theorem [19, 52.10] yields that the mapping EB → lim −→ρ Cρ (K, R) is continuous. Thus T is bounded. 2
4.2. Corollary. Let M = (Mk ) be a non-quasianalytic DC-weight sequence. (1) For convenient vector spaces E and F , on L(E, F ) the following bornologies coincide which are induced by: • The topology of uniform convergence on bounded subsets of E. • The topology of pointwise convergence. • The embedding L(E, F ) ⊂ C ∞ (E, F ). • The embedding L(E, F ) ⊂ C M (E, F ). (2) Let E, F , G be convenient vector spaces and let U ⊂ E be c∞ -open. A mapping f : U × F → G which is linear in the second variable is C M if and only if f ∨ : U → L(F, G) is well defined and C M . Analogous results hold for spaces of multilinear mappings. Proof. (1) That the first three topologies on L(E, F ) have the same bounded sets has been shown in [19, 5.3 and 5.18]. The inclusion C M (E, F ) → C ∞ (E, F ) is bounded by 3.10 and by the uniform boundedness principle in A.3(7). It remains to show that the inclusion L(E, F ) → C M (E, F ) is bounded, where the former space is considered with the topology of uniform convergence on bounded sets. This follows from the uniform boundedness principle 4.1. (2) The assertion for C ∞ is true by A.3(6).
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If f is C M let c : R → U be a C M -curve. We have to show that t → f ∨ (c(t)) ∈ L(F, G) is C M . By [19, 5.18] and 3.5 it suffices to show that t → (f ∨ (c(t))(v)) = (f (c(t), v)) ∈ R is C M for each ∈ G∗ and v ∈ F ; this is obviously true. Conversely, let f ∨ : U → L(F, G) be C M . We claim that f : U × F → G is C M . By composing with ∈ G∗ we may assume that G = R. By induction we have d k f (x, w0 ) (vk , wk ), . . . , (v1 , w1 ) k d k−1 f ∨ (x) vk , . . . , vˆi , . . . , v1 (wi ) = d k f ∨ (x)(vk , . . . , v1 )(w0 ) + i=1
We check condition 3.9(3) for f : k d f (x, w0 )
Lk (EB ×FB ,R)
d k f ∨ (x)(. . .)(w0 )Lk (E
B
+ ,R)
k k−1 ∨ d f (x) i=1
d k f ∨ (x)Lk (E
B ,L(FB
Cρ k k!Mk w0 B +
k i=1
w0 B + ,R))
Lk−1 (EB ,L(FB ,R))
k k−1 ∨ d f (x) i=1
Lk−1 (EB ,L(FB ,R))
Mk−1 Cρ k−1 (k − 1)!Mk−1 = Cρ k k!Mk w0 B + ρMk
where we used 3.9(3) for L(iB , R) ◦ f ∨ : U → L(FB , R). Thus f is C M .
2
4.3. Proposition. Let M = (Mk ) be a non-quasianalytic DC-weight sequence. Let E and F be convenient vector spaces and let U ⊆ E be c∞ -open. Then we have the bornological identity M C M (U, F ) = lim ←− C (R, F ), s
where s runs through the strongly C M -curves in U and the connecting mappings are given by g ∗ for all reparametrizations g ∈ C M (R, R) of curves s. M M Proof. By 3.9 the linear spaces C M (U, F ), lim ←−s C (R, F ) and lim ←−c C (R, F ) coincide, where c runs through the C M -curves in U : Each element (fc )c determines a unique function f : U → F given by f (x) := (f ◦ constx )(0) with f ◦ c = fc for all such curves c, and f ∈ C M if and only if fc ∈ C M for all such c, by 3.9. Since C M (R, F ) carries the initial structure with respect to ∗ for all ∈ F ∗ we may assume M M F = R. Obviously the identity lim ←−c C (R, R) → lim ←−s C (R, R) is continuous. As projective limit the later space is convenient, so we may apply the uniform boundedness principle 4.1 to conclude that the identity in the converse direction is bounded. 2
4.4. Proposition. Let M = (Mk ) be a non-quasianalytic DC-weight sequence. Let E and F be convenient vector spaces and let U ⊆ E be c∞ -open. Then the bornology of C M (U, F ) is initial with respect to each of the following families of mappings
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(1) iB∗ = C M (iB , F ) : C M (U, F ) → C M (U ∩ EB , F ), (2) C M (iB , πV ) : C M (U, F ) → C M (U ∩ EB , FV ), (3) C M (iB , ) : C M (U, F ) → C M (U ∩ EB , R), where B runs through the closed absolutely convex bounded subsets of E and iB : EB → E denotes the inclusion, and where runs through the continuous linear functionals on F , and where V runs through the absolutely convex 0-neighborhoods of F and FV is obtained by factoring out the kernel of the Minkowsky functional of V and then taking the completion with respect to the induced norm. Warning. The structure in (2) gives a projective limit description of C M (U, F ) if and only if F is complete since then F = lim ←−V FV . Proof. Since iB : EB → E, πV : F → FV and : F → R are bounded linear the mappings iB∗ , C M (iB , πV ) and C M (iB , ) are bounded and linear. The structures given by (1), (2) and (3) are successively weaker. So let, conversely, C M (iB , )(B) be bounded in C M (U ∩ EB , R) for all B and . By 4.3 C M (U, F ) carries the initial structure with respect to all c∗ : C M (U, F ) → C M (R, F ), where c : R → U are the strongly C M -curves and these factor locally as (strongly) C M -curves into some EB . By definition C M (R, F ) carries the initial structure with respect to C M (ιI , ) : C M (R, F ) → C M (I, R) where ιI : I → R are the inclusions of compact intervals into R and ∈ F ∗ . Thus C M (U, F ) carries the initial structure with respect to C M (c|I , ) : C M (U, F ) → C M (I, R), which is coarser than that induced by C M (U, F ) → C M (U ∩ EB , R). 2 4.5. Definition. Let E and F be Banach spaces and A ⊆ E convex. We consider the linear space C ∞ (A, F ) consisting of all sequences (f k )k ∈ k∈N C(A, Lk (E, F )) satisfying 1 f (y)(v) − f (x)(v) = k
k
f k+1 x + t (y − x) (y − x, v) dt
0
for all k ∈ N, x, y ∈ A, and v ∈ E k . If A is open we can identify this space with that of all smooth functions A → F by passing to jets. In addition, let M = (Mk ) be a non-quasianalytic DC-weight sequence and (rk ) a sequence of positive real numbers. Then we consider the normed spaces M C(r (A, F ) := f k k ∈ C ∞ (A, F ): f k (r ) < ∞ k) k
where the norm is given by k f
f k (a)(v1 , . . . , vk ) : k ∈ N, a ∈ A, vi ∈ E . := sup (rk ) k!rk Mk v1 · · · · · vk
If (rk ) = (ρ k ) for some ρ > 0 we just write ρ instead of (rk ) as indices. The spaces M (A, F ) are Banach spaces, since they are closed in ∞ (N, ∞ (A, Lk (E, F ))) via (f k ) → C(r k k) (k → k!rk1Mk f k ).
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4.6. Theorem. Let M = (Mk ) be a non-quasianalytic DC-weight sequence. Let E and F be Banach spaces and let U ⊆ E be open. Then the space C M (U, F ) can be described bornologically in the following equivalent ways, i.e. these constructions give the same vector space and the same bounded sets. M (1) lim ←− lim −→ Cρ (W, F ), K ρ,W
M (2) lim ←− lim −→ Cρ (K, F ), K
(3)
ρ
M lim ←− C(rk ) (K, F ),
K,(rk )
M (4) lim ←− lim −→ Cρ (I, F ). c,I
ρ
Moreover, all involved inductive limits are regular, i.e. the bounded sets of the inductive limits are contained and bounded in some step. Here K runs through all compact convex subsets of U ordered by inclusion, W runs through the open subsets K ⊆ W ⊆ U again ordered by inclusion, ρ runs through the positive real numbers, (rk ) runs through all sequences of positive real numbers for which ρ k /rk → 0 for all ρ > 0, c runs through the C M -curves in U ordered by reparametrization with g ∈ C M (R, R) and I runs through the compact intervals in R. Proof. Note first that all four descriptions describe smooth functions f : U → F , which are given by x → f 0 (x) in (1)–(3) for appropriately chosen K with x ∈ K where f 0 : K → F and by x → fc (t) in (4) for c with x = c(t), t ∈ I and fc : I → F . Smoothness of f follows, since we may test with C M -curves and these factor locally into some K. By 3.9 all four descriptions describe C M (U, F ) as vector space. Obviously the identity is continuous from (1) to (2) and from (2) to (3). The identity from (3) to (1) is continuous, since the space given by (3) is as inverse limit of Banach spaces convenient and the inductive limit in (1) is by construction an (LB)-space, hence webbed, and thus we can apply the uniform S-boundedness principle [19, 5.24], where S = {evx : x ∈ U }. So the descriptions in (1)– (3) describe the same complete bornology on C M (U, F ) and satisfy the uniform S-boundedness principle. Moreover, the inductive limits involved in (1) and (2) are regular: In fact the bounded sets B therein are also bounded in the structure of (3), i.e., for every compact K ⊆ U and sequence (rk ) of positive real numbers for which ρ k /rk → 0 for all ρ > 0:
f k (a)(v1 , . . . , vk ) : k ∈ N, a ∈ A, vi ∈ E, f ∈ B < ∞ sup k!rk Mk v1 · · · · · vk
and so the sequence ak := sup
f k (a)(v1 , . . . , vk ) : a ∈ A, vi ∈ E, f ∈ B < ∞ k!Mk v1 · · · · · vk
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satisfies supk ak /rk < ∞ for all (rk ) as above. By [19, 9.2] these are the coefficients of a power series with positive radius of convergence. Thus ak /ρ k is bounded for some ρ > 0. This means that B is contained and bounded in CρM (K, F ). That also (4) describes the same bornology follows again by the S-uniform boundedness principle, since the inductive limit in (4) is regular by what we said before for the special case E = R and hence the structure of (4) is convenient. 2 4.7. Lemma. Let M be a non-quasianalytic DC-weight sequence. For any convenient vector space E the flip of variables induces an isomorphism L(E, C M (R, R)) ∼ = C M (R, E ) as vector spaces. Proof. For c ∈ C M (R, E ) consider c(x) ˜ := evx ◦c ∈ C M (R, R) for x ∈ E. By the uniform boundedness principle 4.1 the linear mapping c˜ is bounded, since evt ◦c˜ = c(t) ∈ E . ˜ = evt ◦ ∈ E = L(E, R) for t ∈ R. If conversely ∈ L(E, C M (R, R)), we consider (t) Since the bornology of E is generated by S := {evx : x ∈ E}, ˜ : R → E is C M , for evx ◦˜ = (x) ∈ C M (R, R), by 3.5. 2 4.8. Lemma. Let M = (Mk ) be a non-quasianalytic DC-weight sequence. By λM (R) we denote the c∞ -closure of the linear subspace generated by {evt : t ∈ R} in C M (R, R) and let δ : R → λM (R) be given by t → evt . Then λM (R) is the free convenient vector space over C M , i.e. for every convenient vector space G the C M -curve δ induces a bornological isomorphism L λM (R), G ∼ = C M (R, G). We expect λM (R) to be equal to C M (R, R) as it is the case for the analogous situation of smooth mappings, see [19, 23.11], and of holomorphic mappings, see [25] and [26]. Proof. The proof goes along the same lines as in [19, 23.6] and in [8, 5.1.1]. Note first that λM (R) is a convenient vector space since it is c∞ -closed in the convenient vector space C M (R, R) . Moreover, δ is C M by 3.5, since evh ◦δ = h for all h ∈ C M (R, R), so δ ∗ : L(λM (R), G) → C M (R, G) is a well-defined linear mapping. This mapping is injective, since each bounded linear mapping λM (R) → G is uniquely determined on δ(R) = {evt : t ∈ R}. Let M M ∗ M ˜ now
f ∈ C (R, G). Then ◦ f ∈ C (R, R) for every ∈ G and hence f : C (R, R) → ˜ is a well-defined bounded linear map. Since it maps G∗ R given by f (ϕ) = (ϕ( ◦ f ))∈G∗ ˜ evt to f (evt ) = δ(f (t)), where δ : G → G∗ R denotes the bornological embedding given by x → ((x))∈G∗ , it induces a bounded linear mapping f˜ : λM (R) → G satisfying f˜ ◦ δ = f . Thus δ ∗ is a linear bijection. That it is a bornological isomorphism (i.e. δ ∗ and its inverse are both bounded) follows from the uniform boundedness principles 4.1 and 4.2. 2 4.9. Corollary. Let M = (Mk ) and N = (Nk ) be non-quasianalytic DC-weight sequences. We have the following isomorphisms of linear spaces (1) C ∞ (R, C M (R, R)) ∼ = C M (R, C ∞ (R, R)), (2) C ω (R, C M (R, R)) ∼ = C M (R, C ω (R, R)), (3) C N (R, C M (R, R)) ∼ = C M (R, C N (R, R)).
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Proof. For α ∈ {∞, ω, N} we get C M R, C α (R, R) ∼ = L λM (R), C α (R, R) by 4.8, ∼ by 4.7 [19, 3.13.4, 5.3, 11.15], = C α R, L λM (R), R ∼ 2 = C α R, C M (R, R) by 4.8. 4.10. Theorem (Canonical isomorphisms). Let M = (Mk ) and N = (Nk ) be non-quasianalytic DC-weight sequences. Let E, F be convenient vector spaces and let Wi be c∞ -open subsets in such. We have the following natural bornological isomorphisms: (1) (2) (3) (4) (5) (6)
C M (W1 , C N (W2 , F )) ∼ = C N (W2 , C M (W1 , F )). M ∞ C (W1 , C (W2 , F )) ∼ = C ∞ (W2 , C M (W1 , F )). M ω C (W1 , C (W2 , F )) ∼ = C ω (W2 , C M (W1 , F )). M C (W1 , L(E, F )) ∼ = L(E, C M (W1 , F )). M ∞ C (W1 , (X, F )) ∼ = ∞ (X, C M (W1 , F )). k M C (W1 , Lip (X, F )) ∼ = Lipk (X, C M (W1 , F )).
In (5) the space X is an ∞ -space, i.e. a set together with a bornology induced by a family of real valued functions on X, cf. [8, 1.2.4]. In (6) the space X is a Lipk -space, cf. [8, 1.4.1]. The spaces ∞ (X, F ) and Lipk (W, F ) are defined in [8, 3.6.1 and 4.4.1]. Proof. All isomorphisms, as well as their inverse mappings, are given by the flip of coordinates: f → f˜, where f˜(x)(y) := f (y)(x). Furthermore, all occurring function spaces are convenient and satisfy the uniform S-boundedness theorem, where S is the set of point evaluations, by 4.1, [19, 11.11, 11.14, 11.12], and by [8, 3.6.1, 4.4.2, 3.6.6, and 4.4.7]. That f˜ has values in the corresponding spaces follows from the equation f˜(x) = evx ◦ f . One only has to check that f˜ itself is of the corresponding class, since it follows that f → f˜ is bounded. This is a consequence of the uniform boundedness principle, since evx ◦( ˜ ) (f ) = evx (f˜) = f˜(x) = evx ◦f = (evx )∗ (f ). That f˜ is of the appropriate class in (1) and in (2) follows by composing with the appropriate curves c1 : R → W1 , c2 : R → W2 and λ ∈ F ∗ and thereby reducing the statement to the special case in 4.9. That f˜ is of the appropriate class in (3) follows by composing with c1 ∈ C M (R, W1 ) and β C 2 (c2 , λ) : C ω (W2 , F ) → C β2 (R, R) for all λ ∈ F ∗ and c2 ∈ C β2 (R, W2 ), where β2 is in {∞, ω}. Then C β2 (c2 , λ) ◦ f˜ ◦ c1 = (C M (c1 , λ) ◦ f ◦ c2 )∼ : R → C β2 (R, R) is C M by 4.9, since C M (c1 , λ) ◦ f ◦ c2 : R → W2 → C M (W1 , F ) → C M (R, R) is C β2 . That f˜ is of the appropriate class in (4) follows, since L(E, F ) is the c∞ -closed subspace of M C (E, F ) formed by the linear C M -mappings. That f˜ is of the appropriate class in (5) or (6) follows from (4), using the free convenient vector spaces 1 (X) or λk (X) over the ∞ -space X or the Lipk -space X, see [8, 5.1.24 or 5.2.3], satisfying ∞ (X, F ) ∼ = L(1 (X), F ) or satisfying Lipk (X, F ) ∼ = L(λk (X), F ). Existence of these free convenient vector spaces can be proved in a similar way as in 4.8. 2
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5. Exponential law 5.1. Difference quotients For the following see [8, 1.3]. For a subset K ⊆ Rn , α = (α1 , . . . , αn ) ∈ Nn , a linear space E, and f : K → E let: Rk = (x0 , . . . , xk ) ∈ Rk+1 : xi = xj for i = j , K α = x 1 , . . . , x n ∈ Rα1 +1 × · · · × Rαn +1 : xi11 , . . . , xinn ∈ K for 0 ij αj , K α = K α ∩ Rα1 × · · · × Rαn , βi (x) = k!
0j k j =i
1 xi − xj
for x = (x0 , . . . , xk ) ∈ Rk ,
α1 αn δα f x 1, . . . , x n = ... βi1 x 1 . . . βin x n f xi11 , . . . , xinn . i1 =0
in =0
Note that δ 0 f = f and δ α = δnαn ◦ · · · ◦ δ1α1 where δik g x 1 , . . . , x n = δ k g x 1 , . . . , x i−1 ,
, x i+1 , . . . , x n
i x .
Lemma. Let E be a convenient vector space. Let U ⊆ Rn be open. For f : U → E the following conditions are equivalent: (1) f : U → E is C M . (2) For every compact convex set K in U and every ∈ E ∗ there exists ρ > 0 such that
δ α ( ◦ f )(x) : α ∈ Nn , x ∈ K α ρ |α| |α|!M|α|
is bounded in R. Furthermore, the norm on the space CρM (K, R) from 2.8 ( for convex K) is also given by f ρ,K
|δ α f (x)| n α . := sup |α| : α∈N , x∈K ρ |α|!M|α|
Proof. By composing with bounded linear functionals we may assume that E = R. (1) ⇒ (2) If f is C M then for each compact convex set K in U there exists ρ > 0 such that
∂ α f (x) : α ∈ Nn , x ∈ K ρ |α| |α|!M|α|
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is bounded in R. For a differentiable function g : R → R and t0 < · · · < tj there exist si with ti < si < ti+1 such that δ j g(t0 , . . . , tj ) = δ j −1 g (s0 , . . . , sj −1 ). This follows by Rolle’s theorem, see [19, 12.4]. Recursion, for g = ∂ α f , shows that δ α f (x 0 , . . . , x n ) = ∂ α f (s) for some s ∈ K. (2) ⇒ (1) f is C ∞ by [8, 1.3.29] since each difference quotient δ α f is bounded on bounded sets. For g ∈ C ∞ (R, R), using (see [8, 1.3.6]) g(tj ) =
j i−1 1 (tj − tl )δ j g(t0 , . . . , tj ), i! i=0
l=0
induction on j and differentiability of g shows that 1 j +1 δ g(t0 , . . . , tj , ti ), j +1 j
δ j g (t0 , . . . , tj ) =
i=0
where δ j +1 g(t0 , . . . , tj , ti ) := limt→ti δ j +1 g(t0 , . . . , tj , t). If the right hand side divided by ρ |α| |α|!M|α| is bounded, then also δ j g /(ρ |α| |α|!M|α| ) is bounded. By recursion, applied to g = δ β ∂ α−β f , we conclude that f ∈ C M . 2 5.2. Lemma. Let E be a convenient vector space such that there exists a Baire vector space topology on the dual E ∗ for which the point evaluations evx are continuous for all x ∈ E. For a mapping f : Rn → E the following are equivalent: (1) ◦ f is C M for all ∈ E ∗ . (2) For every convex compact K ⊆ Rn there exists ρ > 0 such that
∂ α f (x) n : α∈N , x∈K is bounded in E. ρ |α| |α|!M|α| (3) For every convex compact K ⊆ Rn there exists ρ > 0 such that
δ α f (x) n α is bounded in E. : α ∈ N , x ∈ K ρ |α| |α|!M|α| Proof. (2) ⇒ (1) is obvious. (1) ⇒ (2) Let K be compact convex in Rn . We consider the sets
|∂ α ( ◦ f )(x)| n Aρ,C := ∈ E ∗ : C for all α ∈ N , x ∈ K ρ |α| |α|!M|α| which are closed subsets in E ∗ for the Baire topology. We have ρ,C Aρ,C = E ∗ . By the Baire property there exists ρ and C such that the interior U of Aρ,C is non-empty. If 0 ∈ U then for all ∈ E ∗ there is an > 0 such that ∈ U − 0 and hence for all x ∈ K and all α we have α ∂ ( ◦ f )(x) 1 ∂ α ( + 0 ) ◦ f (x) + ∂ α (0 ◦ f )(x) 2C ρ |α| |α|!M|α| .
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So the set
∂ α f (x) : α ∈ Nn , x ∈ K ρ |α| |α|!M|α|
is weakly bounded in E and hence bounded. (3) ⇒ (1) follows by Lemma 5.1. (1) ⇒ (3) follows as above for the difference quotients instead of the partial differentials. 2 5.3. Theorem (Cartesian closedness). Let M = (Mk ) be a non-quasianalytic DC-weight sequence of moderate growth (2.5.1). Then the category of C M -mappings between convenient real vector spaces is cartesian closed. More precisely, for convenient vector spaces E, F and G and c∞ -open sets U ⊆ E and W ⊆ F a mapping f : U × W → G is C M if and only if f ∨ : U → C M (W, G) is C M . Proof. We first show the result for U = R, W = R, G = R. If f ∈ C M (R2 , R) then clearly for any x ∈ R the function f ∨ (x) = f (x, ) ∈ C M (R, R). To show that f ∨ : R → C M (R, R) is C M it suffices to check 5.1(2) for all ∈ C M (R, R)∗ . Such M M an factors over lim −→ρ Cρ (L) for some compact L ⊂ R. Let K ⊂ R be compact. Since f is C there exists C > 0 and ρ > 0 by Lemma 5.1 such that |δ α f (x, y)| C ρ |α| |α|!M|α|
for α ∈ N2 , (x, y) ∈ (K × L)α .
Since M is of moderate growth (2.5.1) we have Mj +k σ j +k Mj Mk for some σ > 0. Let α = (α1 , α2 ) ∈ N2 . Then: α ∨ δ 1 f (x) α ρ 1 α !M 1
1
= sup
α1 ρ2 ,L
sup
|δ2α2 δ1α1 f (x, y)| : α2 ∈ N, y ∈ Lα2 ρ1α1 α1 !Mα1 ρ2α2 α2 !Mα2
|δ2α2 δ1α1 f (x, y)|
!α2 ! ρ1α1 ρ2α2 (αα11+α (α1 + α2 )!σ −α1 −α2 Mα1 +α2 2 )!
: α2 ∈ N, y ∈ L
α2
|δ α f (x, y)| α2 : α ∈ N, y ∈ L 2 ρ1α1 ρ2α2 σ −|α| 2−|α| |α|!M|α| α
|δ f (x, y)| sup |α| : α2 ∈ N, y ∈ Lα2 C for α1 ∈ N, x ∈ K α1 ρ |α|!M|α|
sup
for ρ1 = ρ2 = 2σρ. So f ∨ : K → CρM2 (L, R) is C M . Thus ◦ f ∨ is C M .
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M M for Conversely, let f ∨ : R → C M (R, R) be C M . Then f ∨ : R → lim −→ρ Cρ2 (L, R) is C 2
M ∗ all compact subsets L ⊂ R. The dual space (lim −→ρ Cρ2 (L, R)) can be equipped with the Baire 2
M ∗ topology of the countable limit lim ←−ρ Cρ2 (L, R) of Banach spaces. 2
f∨
R
M lim −→ρ Cρ2 (L, R)
C M (R, R)
2
f∨
CρM2 (L, R)
K
M M by 5.2. Since the inductive limit Thus the mapping f ∨ : R → lim −→ρ Cρ2 (L, R) is strongly C 2
M lim −→ρ2 Cρ2 (L, R) is countable and regular [7, 7.4 and 7.5] or [19, 52.37], for each compact K ⊂ R there exists ρ1 > 0 such that the bounded set
∂ α1 f ∨ (x) : α1 ∈ N, x ∈ K ρ1α1 α1 !Mα1
is contained and bounded in CρM2 (L, R) for some ρ2 > 0. Thus for α1 ∈ N and x ∈ K we have (using (2.1.3)) α ∨ δ 1 f (y) ∞ > C := sup ρ α1 α !M α1 ∈N y∈K
= sup sup
1
1
α1 ρ2 ,L
α ∨ δ 1 f (x) ρ α1 α !M 1
1
α1 ρ2 ,L
|δ2α2 δ1α1 f (x, y)| : α2 ∈ N, y ∈ Lα2 ρ1α1 α1 !Mα1 ρ2α2 α2 !Mα2 |δ2α2 δ1α1 f (x, y)|
!α2 ! ρ1α1 ρ2α2 (αα11+α (α1 + α2 )!Mα1 +α2 2 )! α
|δ f (x, y)| sup |α| : α2 ∈ N, y ∈ Lα2 ρ |α|!M|α|
: α2 ∈ N, y ∈ Lα2
where ρ = max(ρ1 , ρ2 ). Thus f is C M . Now we consider the general case. Given a C M -mapping f : U × W → G we have to show that f ∨ : U → C M (W, G) is C M . Any continuous linear functional on C M (W, G) factors over some step mapping C M (c2 , ) : C M (W, G) → C M (R, R) of the cone in 3.1 where c2 is a C M curve in W and ∈ G∗ . So we have to check that C M (c2 , ) ◦ f ∨ ◦ c1 : R → C M (R, R) is C M for every C M -curve c1 in U . Since ( ◦ f ◦ (c1 × c2 ))∨ = C M (c2 , ) ◦ f ∨ ◦ c1 this follows from the special case proved above. If f ∨ : U → C M (W, G) is C M then ( ◦ f ◦ (c1 × c2 ))∨ = C M (c2 , ) ◦ f ∨ ◦ c1 is C M for all M C -curves c1 : R → U , c2 : R → W and ∈ G∗ . By the special case, f is then C M . 2
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5.4. Example: Cartesian closedness is wrong in general Let M be a DC-weight sequence which is strongly non-quasianalytic but not of moderate 2 growth. For example, Mk = 2k satisfies this by 2.7. Then by 2.4 there exists f : R2 → R of class C M with ∂ α f (0, 0) = |α|!M|α| . We claim that f ∨ : R → C M (R, R) is not C M . Since M is not of moderate growth there exist jn ∞ and kn > 0 such that
Mkn +jn Mkn Mjn
1 kn +jn
n.
Consider the linear functional : C M (R, R) → R given by (g) =
g (jn ) (0) . jn !Mjn njn n
This functional is continuous since (jn ) g (0) g (jn ) (0) ρ jn C(ρ) g ρ,[−1,1] < ∞ jn !Mjn njn jn !ρ jn Mjn njn n n for suitable ρ where C(ρ) :=
ρ jn
n
1 <∞ n jn
for all ρ. But ◦ f ∨ is not C M since ◦ f ∨
ρ1
sup ,[−1,1] k
sup n
sup n
f (jn ,k) (0, 0) 1 ρ1k k!Mk n jn !Mjn njn 1 ρ1kn kn !Mkn
f (jn ,kn ) (0, 0) jn !Mjn njn
(jn + kn )!Mjn +kn ρ1kn kn !jn !Mkn Mjn njn
sup n
njn +kn ρ1kn njn
=∞
for all ρ1 > 0. 5.5. Theorem. Let M be a non-quasianalytic DC-weight sequence which is of moderate growth. Let E, F , etc., be convenient vector spaces and let U and V be c∞ -open subsets of such. (1) The exponential law holds: C M U, C M (V , G) ∼ = C M (U × V , G) is a linear C M -diffeomorphism of convenient vector spaces.
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The following canonical mappings are C M . ev : C M (U, F ) × U → F , ev(f, x) = f (x), ins : E → C M (F, E × F ), ins(x)(y) = (x, y), ( )∧ : C M (U, C M (V , G)) → C M (U × V , G), ( )∨ : C M (U × V , G) → C M (U, C M (V , G)), comp : C M (F, G) × C M (U, F ) → C M (U, G), C M ( , ) : C M (F, F1 ) × C M (E1 , E) → C M (C M (E, F ), C M (E1 , F1 )) (f, g) → (h → f ◦ h ◦ g),
M
(8) : C (Ei , Fi ) → C M ( Ei , Fi ).
(2) (3) (4) (5) (6) (7)
Proof. (2) The mapping associated to ev via cartesian closedness is the identity on C M (U, F ), which is C M , thus ev is also C M . (3) The mapping associated to ins via cartesian closedness is the identity on E × F , hence ins is C M . (4) The mapping associated to ( )∧ via cartesian closedness is the C M -composition of evaluations ev ◦(ev × Id) : (f ; x, y) → f (x)(y). (5) We apply cartesian closedness twice to get the associated mapping (f ; x; y) → f (x, y), which is just a C M evaluation mapping. (6) The mapping associated to comp via cartesian closedness is (f, g; x) → f (g(x)), which is the C M -mapping ev ◦(Id × ev). (7) The mapping associated to the one in question by applying cartesian closedness twice is (f, g; h, x) → g(h(f (x))), which is the C M -mapping ev ◦(Id × ev) ◦ (Id × Id × ev). (8) Up to a flip of factors the mapping associated via cartesian closedness is the product of the evaluation mappings C M (Ei , Fi ) × Ei → Fi . (1) follows from (4) and (5). 2 6. Manifolds of C M -mappings 6.1. C M -manifolds Let M = (Mk ) be a non-quasianalytic DC-weight sequence of moderate growth. A C M manifold is a smooth manifold such that all chart changings are C M -mappings. Likewise for C M -bundles and C M -Lie groups. Note that any finite dimensional (always assumed paracompact) C ∞ -manifold admits a C ∞ -diffeomorphic real analytic structure thus also a C M -structure. Maybe, any finite dimensional C M -manifold admits a C M -diffeomorphic real analytic structure. 6.2. Spaces of C M -sections Let E → B be a C M vector bundle (possibly infinite dimensional). The space C M (B ← E) of all C M sections is a convenient vector space with the structure induced by C M (B ← E) →
C M uα (Uα ), V ,
α
s → pr2 ◦ψα ◦ s ◦ u−1 α
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α where B ⊇ Uα −→ uα (Uα ) ⊂ W is a C M -atlas for B which we assume to be modelled on a convenient vector space W , and where ψα : E|Uα → Uα × V form a vector bundle atlas over charts Uα of B.
Lemma. For a C M vector bundle E → B a curve c : R → C M (B ← E) is C M if and only if c∧ : R × B → E is C M . Proof. By the description of the structure on C M (B ← E) we may assume that B is c∞ -open in a convenient vector space W and that E = B × V . Then C M (B ← B × V ) ∼ = C M (B, V ). Then the statement follows from the exponential law 5.3. 2 An immediate consequence is the following: If U ⊂ E is an open neighborhood of s(B) for a section s, F → B is another vector bundle and if f : U → F is a fiber respecting C M -mapping, then f∗ : C M (B ← U ) → C M (B ← F ) is C M on the open neighborhood C M (B ← U ) of s in C M (B ← E). We have (d(f∗ )(s)v)x = d(f |U ∩Ex )(s(x))(v(x)). 6.3. Theorem. Let M = (Mk ) be a non-quasianalytic DC-weight sequence of moderate growth. Let A and B be finite dimensional C M manifolds with A compact. Then the space C M (A, B) of all C M -mappings A → B is a C M -manifold modelled on convenient vector spaces C M (A ← f ∗ T B) of C M sections of pullback bundles along f : A → B. Moreover, a curve c : R → C M (A, B) is C M if and only if c∧ : R × A → B is C M . Proof. Choose a C M Riemannian metric on B which exists since we have C M partitions of unity. C M -vector fields have C M -flows by [15]; applying this to the geodesic spray we get the C M exponential mapping exp : T B ⊇ U → B of this Riemannian metric, defined on a suitable open neighborhood of the zero section. We may assume that U is chosen in such a way that (πB , exp) : U → B × B is a C M diffeomorphism onto an open neighborhood V of the diagonal, by the C M inverse function theorem due to [14]. For f ∈ C M (A, B) we consider the pullback vector bundle A ×B T B
f ∗T B
πB∗ f
TB
f ∗ πB
πB f
A
B
Then C M (A ← f ∗ T B) is canonically isomorphic to the space C M (A, T B)f := h. Now let {h ∈ C M (A, T B): πB ◦ h = f } via s → (πB∗ f ) ◦ s and (IdA , h)
→
Uf := g ∈ C M (A, B): f (x), g(x) ∈ V for all x ∈ A , uf : Uf → C M (A ← f ∗ T B), −1 uf (g)(x) = x, exp−1 ◦ (f, g) (x) . f (x) g(x) = x, (πB , exp) Then uf is a bijective mapping from Uf onto the set {s ∈ C M (A ← f ∗ T B): s(A) ⊆ f ∗ U = ∗ (πB∗ f )−1 (U )}, whose inverse is given by u−1 f (s) = exp ◦(πB f ) ◦ s, where we view U → B as
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fiber bundle. The push forward uf is C M since it maps C M -curves to C M -curves by Lemma 6.2. The set uf (Uf ) is open in C M (A ← f ∗ T B) for the topology described above in 6.2. Now we consider the atlas (Uf , uf )f ∈C M (A,B) for C M (A, B). Its chart change mappings are given for s ∈ ug (Uf ∩ Ug ) ⊆ C M (A ← g ∗ T B) by
−1 uf ◦ u−1 ◦ f, exp ◦(πB∗ g) ◦ s g (s) = IdA , (πB , exp) = τf−1 ◦ τg ∗ (s),
where τg (x, Yg(x) ) := (x, expg(x) (Yg(x) )) is a C M diffeomorphism τg : g ∗ T B ⊇ g ∗ U → (g × −1 IdB )−1 (V ) ⊆ A × B which is fiber respecting over A. The chart change uf ◦ u−1 g = (τf ◦ τg )∗ is defined on an open subset and it is also C M since it respects C M -curves. Finally for the topology on C M (A, B) we take the identification topology from this atlas (with the c∞ -topologies on the modeling spaces), which is obviously finer than the compactopen topology and thus Hausdorff. −1 M structure does not depend on the The equation uf ◦ u−1 g = (τf ◦ τg )∗ shows that the C choice of the C M Riemannian metric on B. The statement on C M -curves follows from lemma 6.2. 2 6.4. Corollary. Let A1 , A2 and B be finite dimensional C M manifolds with A1 and A2 compact. Then composition C M (A2 , B) × C M (A1 , A2 ) → C M (A1 , B),
(f, g) → f ◦ g
is C M . However, if N = (Nk ) is another non-quasianalytic DC-weight sequence of moderate growth with (Nk /Mk )1/k 0 then composition is not C N . Proof. Composition maps C M -curves to C M -curves, so it is C M . Let A1 = A2 = S 1 and B = R. Then by (2.1.5) there exists f ∈ C M (S 1 , R) \ C N (S 1 , R). We consider f : R → R periodic. The universal covering space of C M (S 1 , S 1 ) consists of all 2πZ-equivariant mappings in C M (R, R), namely the space of all g + IdR for 2π -periodic g ∈ C M . Thus C M (S 1 , S 1 ) is a real analytic manifold and t → (x → x + t) induces a real analytic curve c in C M (S 1 , S 1 ). But f∗ ◦ c is not C N since: (∂tk |t=0 (f∗ ◦ c)(t))(x) ∂tk |t=0 f (x + t) f (k) (x) = = k!ρ k Nk k!ρ k Nk k!ρ k Nk which is unbounded for x in a suitable compact set and for all ρ > 0 since f ∈ / CN .
2
6.5. Theorem. Let M = (Mk ) be a non-quasianalytic DC-weight sequence of moderate growth. Let A be a compact (⇒ finite dimensional ) C M manifold. Then the group DiffM (A) of all C M diffeomorphisms of A is an open subset of the C M manifold C M (A, A). Moreover, it is a C M regular C M -Lie group: Inversion and composition are C M . Its Lie algebra consists of all C M vector fields on A, with the negative of the usual bracket as Lie bracket. The exponential mapping is C M . It is not surjective onto any neighborhood of IdA .
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Following [20], see also [19, 38.4], a C M -Lie group G with Lie algebra g = Te G is called C M -regular if the following holds: • For each C M -curve X ∈ C M (R, g) there exists a C M -curve g ∈ C M (R, G) whose right logarithmic derivative is X, i.e.,
(0) = e, ∂t g(t) = Te μg(t) X(t) = X(t).g(t).
The curve g is uniquely determined by its initial value g(0), if it exists. • Put evolrG (X) = g(1) where g is the unique solution required above. Then evolrG : C M (R, g) → G is required to be C M also. Proof. The group DiffM (A) is open in C M (A, A) since it is open in the coarser C 1 compact open topology, see [19, 43.1]. So DiffM (A) is a C M -manifold and composition is C M by 6.3 and 6.4. To show that inversion is C M let c be a C M -curve in DiffM (A). By 6.3 the map c∧ : R × A → A is C M and (inv ◦c)∧ : R × A → A satisfies the finite dimensional implicit equation c∧ (t, (inv ◦c)∧ (t, x)) = x for x ∈ A. By the finite dimensional C M implicit function theorem [14] the mapping (inv ◦c)∧ is locally C M and thus C M . By 6.3 again, inv ◦c is a C M curve in DiffM (A). So inv : DiffM (A) → DiffM (A) is C M . The Lie algebra of DiffM (A) is the convenient vector space of all C M -vector fields on A, with the negative of the usual Lie bracket (compare with the proof of [19, 43.1]). To show that DiffM (A) is a C M -regular Lie group, we choose a C M -curve in the space of C M -curves in the Lie algebra of all C M -vector fields on A, c : R → C M (R, C M (A ← T A)). By lemma 6.2 c corresponds to a R2 -time-dependent C M vector field c∧∧ : R2 × A → T A. Since C M -vector fields have C M -flows and since A is compact, c∧ (s) is C M in all variables by [15]. Thus DiffM (A) is a C M -regular C M -Lie evolr (c∧ (s))(t) = Flt group. The exponential mapping is evolr applied to constant curves in the Lie algebra, i.e., it consists of flows of autonomous C M -vector fields. That the exponential map is not surjective onto any C M -neighborhood of the identity follows from [19, 43.5] for A = S 1 . This example can be embedded into any compact manifold, see [9]. 2 Appendix A. Calculus beyond Banach spaces The traditional differential calculus works well for finite dimensional vector spaces and for Banach spaces. For more general locally convex spaces we sketch here the convenient approach as explained in [8] and [19]. The main difficulty is that composition of linear mappings stops to be jointly continuous at the level of Banach spaces, for any compatible topology. We use the notation of [19] and this is the main reference for the whole appendix. We list results in the order in which one can prove them, without proofs for which we refer to [19]. This should explain how to use these results. A.1. The c∞ -topology Let E be a locally convex vector space. A curve c : R → E is called smooth or C ∞ if all derivatives exist and are continuous – this is a concept without problems. Let C ∞ (R, E) be the
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space of smooth functions. It can be shown that the set C ∞ (R, E) does not depend on the locally convex topology of E, only on its associated bornology (system of bounded sets). The final topologies with respect to the following sets of mappings into E coincide: (1) C ∞ (R, E). (2) The set of all Lipschitz curves (so that { c(t)−c(s) t−s : t = s} is bounded in E). (3) The set of injections EB → E where B runs through all bounded absolutely convex subsets in E, and where EB is the linear span of B equipped with the Minkowski functional x B := inf{λ > 0: x ∈ λB}. (4) The set of all Mackey-convergent sequences xn → x (there exists a sequence 0 < λn ∞ with λn (xn − x) bounded). This topology is called the c∞ -topology on E and we write c∞ E for the resulting topological space. In general (on the space D of test functions for example) it is finer than the given locally convex topology, it is not a vector space topology, since scalar multiplication is no longer jointly continuous. The finest among all locally convex topologies on E which are coarser than c∞ E is the bornologification of the given locally convex topology. If E is a Fréchet space, then c∞ E = E. A.2. Convenient vector spaces A locally convex vector space E is said to be a convenient vector space if one of the following equivalent conditions is satisfied (called c∞ -completeness): 1 (1) For any c ∈ C ∞ (R, E) the (Riemann-) integral 0 c(t) dt exists in E. (2) Any Lipschitz curve in E is locally Riemann integrable. (3) A curve c : R → E is smooth if and only if λ ◦ c is smooth for all λ ∈ E ∗ , where E ∗ is the dual consisting of all continuous linear functionals on E. Equivalently, we may use the dual E consisting of all bounded linear functionals. (4) Any Mackey–Cauchy-sequence (i.e., tnm (xn − xm ) → 0 for some tnm → ∞ in R) converges in E. This is visibly a mild completeness requirement. (5) If B is bounded closed absolutely convex, then EB is a Banach space. (6) If f : R → E is scalarwise Lipk , then f is Lipk , for k > 1. (7) If f : R → E is scalarwise C ∞ then f is differentiable at 0. (8) If f : R → E is scalarwise C ∞ then f is C ∞ . Here a mapping f : R → E is called Lipk if all derivatives up to order k exist and are Lipschitz, locally on R. That f is scalarwise C ∞ means λ ◦ f is C ∞ for all continuous linear functionals on E. A.3. Smooth mappings Let E, F , and G be convenient vector spaces, and let U ⊂ E be c∞ -open. A mapping f : U → F is called smooth or C ∞ , if f ◦ c ∈ C ∞ (R, F ) for all c ∈ C ∞ (R, U ). The main properties of smooth calculus are the following.
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(1) For mappings on Fréchet spaces this notion of smoothness coincides with all other reasonable definitions. Even on R2 this is non-trivial. (2) Multilinear mappings are smooth if and only if they are bounded. (3) If f : E ⊇ U → F is smooth then the derivative df : U × E → F is smooth, and also df : U → L(E, F ) is smooth where L(E, F ) denotes the space of all bounded linear mappings with the topology of uniform convergence on bounded subsets. (4) The chain rule holds. (5) The space C ∞ (U, F ) is again a convenient vector space where the structure is given by the obvious injection
C ∞ (c,)
C ∞ (U, F ) −−−−−→
C ∞ (R, R),
f → ( ◦ f ◦ c)c, ,
c∈C ∞ (R,U ),∈F ∗
where C ∞ (R, R) carries the topology of compact convergence in each derivative separately. (6) The exponential law holds: For c∞ -open V ⊂ F , C ∞ U, C ∞ (V , G) ∼ = C ∞ (U × V , G) is a linear diffeomorphism of convenient vector spaces. Note that this is the main assumption of variational calculus. (7) A linear mapping f : E → C ∞ (V , G) is smooth (bounded) if and only if f evv E− → C ∞ (V , G) −−→ G is smooth for each v ∈ V . This is called the smooth uniform boundedness theorem [19, 5.26]. (8) The following canonical mappings are smooth. ev : C ∞ (E, F ) × E → F,
ev(f, x) = f (x),
∞
ins : E → C (F, E × F ), ins(x)(y) = (x, y), ( )∧ : C ∞ E, C ∞ (F, G) → C ∞ (E × F, G), ( )∨ : C ∞ (E × F, G) → C ∞ E, C ∞ (F, G) , comp : C ∞ (F, G) × C ∞ (E, F ) → C ∞ (E, G), C∞(
,
) : C ∞ (F, F1 ) × C ∞ (E1 , E) → C ∞ C ∞ (E, F ), C ∞ (E1 , F1 )
(f, g) → (h → f ◦ h ◦ g), Fi . Ei , : C ∞ (Ei , Fi ) → C ∞ A.4. Remarks. Note that the conclusion of A.3(6) is the starting point of the classical calculus of variations, where a smooth curve in a space of functions was assumed to be just a smooth function in one variable more. It is also the source of the name convenient calculus. This and some other obvious properties already determines the convenient calculus. There are, however, smooth mappings which are not continuous. This is unavoidable and not so horrible as it might appear at first sight. For example the evaluation E × E ∗ → R is jointly continuous if and only if E is normable, but it is always smooth. Clearly smooth mappings are continuous for the c∞ -topology.
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Appendix B. Calculus of holomorphic mappings B.1. Holomorphic curves Let E be a complex locally convex vector space whose underlying real space is convenient – this will be called convenient in the sequel. Let D ⊂ C be the open unit disk and let us denote by H(D, E) the space of all mappings c : D → E such that λ ◦ c : D → C is holomorphic for each continuous complex-linear functional λ on E. Its elements will be called the holomorphic curves. If E and F are convenient complex vector spaces (or c∞ -open sets therein), a mapping f : E → F is called holomorphic if f ◦ c is a holomorphic curve in F for each holomorphic curve c in E. Obviously f is holomorphic if and only if λ ◦ f : E → C is holomorphic for each complex linear continuous (equivalently: bounded) functional λ on F . Let H(E, F ) denote the space of all holomorphic mappings from E to F . B.2. Lemma (Hartog’s theorem). Let Ek for k = 1, 2 and F be complex convenient vector spaces and let Uk ⊂ Ek be c∞ -open. A mapping f : U1 × U2 → F is holomorphic if and only if it is separately holomorphic (i. e. f ( , y) and f (x, ) are holomorphic for all x ∈ U1 and y ∈ U2 ). This implies also that in finite dimensions we have recovered the usual definition. B.3. Lemma. If f : E ⊃ U → F is holomorphic then df : U × E → F exists, is holomorphic and C-linear in the second variable. A multilinear mapping is holomorphic if and only if it is bounded. B.4. Lemma. If E and F are Banach spaces and U is open in E, then for a mapping f : U → F the following conditions are equivalent: (1) f is holomorphic. (2) f is locally a convergent series of homogeneous continuous polynomials. (3) f is C-differentiable in the sense of Fréchet. B.5. Lemma. Let E and F be convenient vector spaces. A mapping f : E → F is holomorphic if and only if it is smooth and its derivative in each point is C-linear. An immediate consequence of this result is that H(E, F ) is a closed linear subspace of C ∞ (ER , FR ) and so it is a convenient vector space if F is one, by A.3(5). The chain rule follows from A.3(4). B.6. Theorem. The category of convenient complex vector spaces and holomorphic mappings between them is cartesian closed, i.e. H(E × F, G) ∼ = H E, H(F, G) . An immediate consequence of this is again that all canonical structural mappings as in A.3(8) are holomorphic.
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Appendix C. Calculus of real analytic mappings C.1. We now sketch the cartesian closed setting to real analytic mappings in infinite dimension following the lines of the Frölicher–Kriegl calculus, as it is presented in [19]. Surprisingly enough one has to deviate from the most obvious notion of real analytic curves in order to get a meaningful theory, but again convenient vector spaces turn out to be the right kind of spaces. C.2. Real analytic curves Let E be a real convenient vector space with continuous dual E ∗ . A curve c : R → E is called real analytic if λ ◦ c : R → R is real analytic for each λ ∈ E ∗ . It turns out that the set of these curves depends only on the bornology of E. Thus we may use the dual E consisting of all bounded linear functionals in the definition. In contrast a curve is called strongly real analytic if it is locally given by power series which converge in the topology of E. They can be extended to germs of holomorphic curves along R in the complexification EC of E. If the dual E ∗ of E admits a Baire topology which is compatible with the duality, then each real analytic curve in E is in fact topologically real analytic for the bornological topology on E. C.3. Real analytic mappings Let E and F be convenient vector spaces. Let U be a c∞ -open set in E. A mapping f : U → F is called real analytic if and only if it is smooth (maps smooth curves to smooth curves) and maps real analytic curves to real analytic curves. Let C ω (U, F ) denote the space of all real analytic mappings. We equip the space C ω (U, R) of all real analytic functions with the initial topology with respect to the families of mappings ∗
c C ω (U, R) − → C ω (R, R), c∗
C ω (U, R) −→ C ∞ (R, R),
for all c ∈ C ω (R, U ), for all c ∈ C ∞ (R, U ),
where C ∞ (R, R) carries the topology of compact convergence in each derivative separately, and where C ω (R, R) is equipped with the final locally convex topology with respect to the embeddings (restriction mappings) of all spaces of holomorphic mappings from a neighborhood V of R in C mapping R to R, and each of these spaces carries the topology of compact convergence. Furthermore we equip the space C ω (U, F ) with the initial topology with respect to the family of mappings λ
∗ C ω (U, F ) −→ C ω (U, R),
for all λ ∈ F ∗ .
It turns out that this is again a convenient space. C.4. Theorem. In the setting of C.3 a mapping f : U → F is real analytic if and only if it is smooth and is real analytic along each affine line in E. C.5. Lemma. The space L(E, F ) of all bounded linear mappings is a closed linear subspace of C ω (E, F ). A mapping f : U → L(E, F ) is real analytic if and only if evx ◦f : U → F is real analytic for each point x ∈ E.
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C.6. Theorem. The category of convenient spaces and real analytic mappings is cartesian closed. So the equation C ω U, C ω (V , F ) ∼ = C ω (U × V , F ) is valid for all c∞ -open sets U in E and V in F , where E, F , and G are convenient vector spaces. This implies again that all structure mappings as in A.3(8) are real analytic. Furthermore the differential operator d : C ω (U, F ) → C ω U, L(E, F ) exists, is unique and real analytic. Multilinear mappings are real analytic if and only if they are bounded. C.7. Theorem (Real analytic uniform boundedness principle). A linear mapping f : E → f evv C ω (V , G) is real analytic (bounded) if and only if E − → C ω (V , G) −−→ G is real analytic (bounded) for each v ∈ V . References [1] E. Bierstone, P.D. Milman, Resolution of singularities in Denjoy–Carleman classes, Selecta Math. (N.S.) 10 (1) (2004) 1–28, MR2061220 (2005c:14074). [2] J. Boman, Differentiability of a function and of its compositions with functions of one variable, Math. Scand. 20 (1967) 249–268, MR0237728 (38 #6009). [3] J. Bonet, R.W. Braun, R. Meise, B.A. Taylor, Whitney’s extension theorem for nonquasianalytic classes of ultradifferentiable functions, Studia Math. 99 (2) (1991) 155–184, MR1120747 (93e:46030). [4] T. Carleman, Les Fonctions Quasi-analytiques, Collection Borel, Gauthier–Villars, Paris, 1926. [5] A. Denjoy, Sur les fonctions quasi-analytiques de variable réelle, C. R. Acad. Sci. Paris 173 (1921) 1320–1322. [6] C.F. Faà di Bruno, Note sur une nouvelle formule du calcul différentielle, Quart. J. Math. 1 (1855) 359–360. [7] K. Floret, Lokalkonvexe Sequenzen mit kompakten Abbildungen, J. Reine Angew. Math. 247 (1971) 155–195, MR0287271 (44 #4478). [8] A. Frölicher, A. Kriegl, Linear Spaces and Differentiation Theory, Pure Appl. Math. (N.Y.), John Wiley & Sons Ltd., A Wiley–Interscience Publication, Chichester, 1988, MR961256 (90h:46076). [9] J. Grabowski, Free subgroups of diffeomorphism groups, Fund. Math. 131 (2) (1988) 103–121, MR974661 (90b:58031). [10] L. Hörmander, The Analysis of Linear Partial Differential Operators. I, Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences), vol. 256, Springer-Verlag, Berlin, 1983, Distribution theory and Fourier analysis, MR717035 (85g:35002a). [11] H. Jarchow, Locally Convex Spaces, Mathematische Leitfäden (Mathematical Textbooks), B.G. Teubner, Stuttgart, 1981, MR632257 (83h:46008). [12] H. Komatsu, Ultradistributions and hyperfunctions, in: Hyperfunctions and Pseudo-Differential Equations, Proc. Conf. on the Theory of Hyperfunctions and Analytic Functionals and Applications, R.I.M.S., Kyoto Univ., Kyoto, 1971, dedicated to the memory of André Martineau, in: Lecture Notes in Math., vol. 287, Springer, Berlin, 1973, pp. 164–179, MR0407596 (53 #11368). [13] H. Komatsu, Ultradistributions. I. Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973) 25–105, MR0320743 (47 #9277). [14] H. Komatsu, The implicit function theorem for ultradifferentiable mappings, Proc. Japan Acad. Ser. A Math. Sci. 55 (3) (1979) 69–72, MR531445 (80e:58007). [15] H. Komatsu, Ultradifferentiability of solutions of ordinary differential equations, Proc. Japan Acad. Ser. A Math. Sci. 56 (4) (1980) 137–142, MR575993 (81j:34066).
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Journal of Functional Analysis 256 (2009) 3545–3567 www.elsevier.com/locate/jfa
A Morita theorem for dual operator algebras ✩ Upasana Kashyap Department of Mathematics, University of Houston, Houston, TX 77204-3008, United States Received 10 July 2008; accepted 24 February 2009 Available online 25 March 2009 Communicated by D. Voiculescu
Abstract We prove that two dual operator algebras are weak∗ Morita equivalent in the sense of [D.P. Blecher, U. Kashyap, Morita equivalence of dual operator algebras, J. Pure Appl. Algebra 212 (2008) 2401–2412] if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak∗ -continuous on appropriate morphism spaces. Moreover, in a fashion similar to the operator algebra case, we characterize such functors as the module normal Haagerup tensor product with an appropriate weak∗ Morita equivalence bimodule. We also develop the theory of the W ∗ -dilation, which connects the non-selfadjoint dual operator algebra with the W ∗ -algebraic framework. In the case of weak∗ Morita equivalence, this W ∗ -dilation is a W ∗ -module over a von Neumann algebra generated by the nonselfadjoint dual operator algebra. The theory of the W ∗ -dilation is a key part of the proof of our main theorem. Published by Elsevier Inc. Keywords: W ∗ -algebra; Operator algebra; Dual operator algebra; Dual operator module; Morita equivalence
1. Introduction and notation An important and well-known perspective of understanding an algebraic object is to study its category of representations. For example, modules correspond to representations of a ring hence rings are commonly studied in terms of their modules. Once we view an algebraic object in terms of its category of representations, it is natural to compare such categories. This leads to the notion of Morita equivalence. The notion of Morita equivalence of rings arose in pure ✩
This work is a part of the authors PhD thesis at the University of Houston. E-mail address:
[email protected].
0022-1236/$ – see front matter Published by Elsevier Inc. doi:10.1016/j.jfa.2009.02.014
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algebra around 1960. Two rings are defined to be Morita equivalent if and only if they have equivalent categories of modules. Morita equivalence is a powerful tool in pure algebra, and it has inspired similar notions in operator algebra theory. In the 1970s Rieffel introduced and developed the notion of Morita equivalence for C ∗ -algebras and W ∗ -algebras. This is a useful and very important tool in modern operator theory. With the advent of operator space theory in the 1990s, Blecher, Muhly and Paulsen generalized Rieffel’s C ∗ -algebraic notion of Morita equivalence to non-selfadjoint operator algebras [10]. Recently we generalized Rieffel’s variant of W ∗ -algebraic Morita equivalence to dual operator algebras. By a dual operator algebra, we mean a unital weak∗ -closed algebra of operators on a Hilbert space which is not necessarily selfadjoint. One can view a dual operator algebra as a nonselfadjoint analogue of a von Neumann algebra. By a non-selfadjoint version of Sakai’s theorem (see e.g. Section 2.7 in [8]), a dual operator algebra is characterized as a unital operator algebra that is also a dual operator space. In [4] we defined two dual operator algebras M and N to be weak∗ Morita equivalent if there exist a dual operator M-N -bimodule X, and a dual operator N -M-bimodule Y , such that M∼ = X ⊗σNh Y as dual operator M-bimodules (that is, via a completely isometric, weak∗ -homeomorphism which is also an M-bimodule map), and N ∼ = Y ⊗σMh X as dual operator N -bimodules. Another notion of Morita equivalence for dual operator algebras introduced by Eleftherakis will be briefly discussed later in the introduction. In the literature of Morita equivalence for rings in pure algebra, there is a popular collection of theorems known as Morita I, II and III. Morita I can be described as the consequences of a pair of bimodules being mutual inverses (X ⊗N Y ∼ = M and Y ⊗M X ∼ = N ). For dual operator algebras, most of the appropriate version of Morita I is proved in [4]. Morita II characterizes module category equivalences as tensoring with an invertible bimodule, and our main theorem here is a Morita II theorem for dual operator algebras. The Morita III theorem states that there is a bijection between the set of isomorphism classes of invertible bimodules and the set of equivalence classes of category equivalences; its appropriate version for dual operator algebras follows as in pure algebra and will be presented in [17]. In [4] we proved that two dual operator algebras that are weak∗ Morita equivalent in our sense have equivalent categories of dual operator modules (one remaining technical detail of this may appear in [6]). In the present work, we prove the converse, a Morita II theorem: if two dual operator algebras have equivalent categories of dual operator modules then they are weak∗ Morita equivalent in the sense of [4]. The functors implementing the categorical equivalences are characterized as the module normal Haagerup tensor product with an appropriate weak∗ Morita equivalence bimodule. We now outline the content of our paper and the strategy of the proof of the main theorem. In Section 2, we explore the connection between the class of operator modules over a dual operator algebra M and those over D, where D is a W ∗ -algebra generated by M. We prove that the class of operator modules over D is a subcategory of the class of operator modules over M. We develop the theory of the W ∗ -dilation, which connects the non-selfadjoint dual operator algebra with the W ∗ -algebraic framework. Every dual operator M-module X dilates to a dual operator module D ⊗σMh X over D which is called the ‘D-dilation’. In particular, when D equals the maximal W ∗ -algebra C generated by a dual operator algebra M, then C ⊗σMh X is called the ‘maximal dilation’. We mostly work with the maximal dilation instead of any arbitrary dilation because every dual operator module is a weak∗ -closed submodule of its maximal dilation. The theory of the W ∗ -dilation is a key part of the proof of our main theorem.
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In Section 3, we present some weak∗ Morita equivalence and W ∗ -dilation results. In the case of weak∗ Morita equivalence the W ∗ -dilation turns out to be a W ∗ -module and hence we are back in the W ∗ -algebraic Morita framework. We consider a notion of Morita equivalence for dual operator algebras which we call ‘weak subequivalence’ which basically dilates to a Morita equivalence of the generated W ∗ -algebras in the sense of Rieffel. In Theorem 5.2 in [4], we proved that weak∗ Morita equivalence implies weak subequivalence. Here, we show that for dual operator algebras weak∗ Morita equivalence is same as weak subequivalence when dilation is over the maximal generated W ∗ -algebra (see Theorem 3.10). This may be viewed as a characterization of our weak∗ Morita equivalence of dual operator algebras. This is also a key point of the proof of our main theorem. In Sections 4 and 5, we prove our main theorem: if two dual operator algebras M and N have equivalent categories of dual operator modules then they are weak∗ Morita equivalent in the sense of [4]. We now briefly discuss the strategy of the proof of the main theorem. It turns out that the functors F and G implementing the categorical equivalence of operator modules over M and N restrict to functors between categories of operator modules and of normal Hilbert modules over the maximal W ∗ -algebras C and D generated by M and N respectively. Thus the W ∗ -algebras C and D are Morita equivalent in the sense of Rieffel (see e.g. Definition 7.4 in [18]). Then following the ideas of [2] and [3], the modules over M and N that we get, dilate to Morita equivalence bimodules between C and D giving weak subequivalence. Since the dilation is over the maximal generated W ∗ -algebra, by the ideas discussed above of Section 3 this weak subequivalence implies weak∗ Morita equivalence between M and N . Many of the techniques and ideas in this paper are taken from [1,3,2,9]. We refer the reader to these papers for earlier ideas, proof techniques, and additional details. In some places we just need to modify the arguments in the present setting of weak∗ -topology, or merely change the tensor product. However, we need to develop new techniques to deal with a number of subtleties that arise in the weak∗ -topology setting. Another notion of Morita equivalence for dual operator algebras was considered in [15] and is called -equivalence. In [16] it was shown that the -equivalence implies weak∗ Morita equivalence in the sense of [4]. That is, any of the equivalences of [15] is one of our weak∗ Morita equivalences. Both the theories have different advantages. For example, the equivalence considered in [15] is equivalent to the very important notion of weak∗ stable isomorphism. On the other hand, our theory contains all examples considered up to this point in the literature of Morita-like equivalence in a dual (weak∗ ) setting. There are certain important examples that do not seem to be contained the other theory but are weak∗ Morita equivalent in our sense. For example, in the selfadjoint setting the second dual of strongly Morita equivalent C ∗ -algebras are Morita equivalent in Rieffel’s W ∗ -algebraic sense. In the non-selfadjoint case, the second dual of strongly Morita equivalent operator algebras in the sense of Blecher, Muhly and Paulsen are weak∗ Morita equivalent in our sense. Also, two ‘similar’ separably acting nest algebras are Morita equivalent in our sense but are not -equivalent. In [4] we showed that weak∗ Morita equivalent dual operator algebra has equivalent categories of normal Hilbert space representations (also known as normal Hilbert modules). However, the converse of this is still an open problem and at present we are working on this aspect. The characterization theorem in [15] is in terms of equivalence of categories of normal Hilbert modules, which intertwines not only the representations of the dual operator algebras, but also their restrictions to the diagonals. We assume that the reader is familiar with the notions from operator space theory. One can refer to [8] and [13] for background and most of the terminology used in this paper. We also
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assume that all dual operator algebras are unital; that is, they each have an identity of norm 1. We will often abbreviate ‘weak∗ ’ to ‘w ∗ ’. We reserve the symbols M and N for dual operator algebras. A normal representation of M is a w ∗ -continuous unital completely contractive homomorphism π : M → B(H ). For a dual space X, we let X∗ denote its predual. We assume that the reader is familiar with the weak∗ -topology and basic duality principles such as the Krein– Smulian theorem (see Theorem A.2.5 in [8]). A concrete dual operator M-N -bimodule is a w ∗ -closed subspace X of B(K, H ) such that θ (M)Xπ(N) ⊂ X, where θ and π are normal representations of M and N on H and K respectively. An abstract dual operator M-N -bimodule is defined to be an operator M-N -bimodule X (by which we mean that X is an operator space and a nondegenerate M-N -bimodule such that the module actions are completely contractive in the sense of 3.1.3 in [8]), which is also a dual operator space, such that the module actions are separately weak∗ -continuous. Such spaces can be represented completely isometrically as concrete dual operator bimodules (see e.g. Theorem 3.8.3 in [8,12]). We shall write M R for the category of left dual operator modules over M. The morphisms in M R are the w ∗ -continuous completely bounded M-module maps. By M H, we mean the category of completely contractive normal Hilbert modules over a dual operator algebra M. That is, elements of M H are pairs (H, π), where H is a (column) Hilbert space (see e.g. 1.2.23 in [8]), and π : M → B(H ) is a normal representation of M. The module action is expressed through the equation m · ζ = π(m)ζ . The morphisms are bounded linear transformations between Hilbert spaces that intertwine the representations; i.e., if (Hi , πi ), i = 1, 2, are objects of the category M H, then the space of morphisms is defined as: BM (H1 , H2 ) = T ∈ B(H1 , H2 ): T π1 (m) = π2 (m)T for all m ∈ M . Any H ∈ M H (with its column Hilbert space structure) is a left dual operator M-module. If E and F are sets, then EF denotes the norm closure of the span of products xy for x ∈ E and y ∈ F. If X and Y are dual operator spaces, we denote by CBσ (X, Y ) the space of completely bounded w ∗ -continuous linear maps from X to Y . Similarly if X and Y are left dual operator M-modules, then CBσM (X, Y ) denotes the space of completely bounded w ∗ -continuous left M-module maps from X to Y . If M is a dual operator algebra, then a W ∗ -cover of M is a pair (A, j ) consisting of a W ∗ algebra A and a completely isometric w ∗ -continuous homomorphism j : M → A, such that j (M) generates A as a W ∗ -algebra. By the Krein–Smulian theorem j (M) is a w ∗ -closed sub∗ (M) is a W ∗ -algebra containing M as a w ∗ -closed algebra of A. The maximal W ∗ -cover Wmax subalgebra that is generated by M as a W ∗ -algebra, and has the following universal property: any normal representation π : M → B(H ) extends uniquely to a (unital) normal ∗-representation ∗ (M) → B(H ) (see [11]). π˜ : Wmax We will refer to Rieffel’s W ∗ -algebraic Morita equivalence (see [18]) as ‘weak Morita equivalence’ for W ∗ -algebras, and the associated equivalence bimodules as ‘W ∗ -equivalencebimodules’ (see e.g. Section 8.5 in [8]). We use the normal module Haagerup tensor product ⊗σMh throughout the paper. We refer to [16] and [4, Section 2] for the universal property and general facts and properties of ⊗σMh . Loosely speaking, the normal module Haagerup tensor product linearizes completely contractive balanced separately weak∗ -continuous bilinear maps.
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2. Dual operator modules over a generated W ∗ -algebra and W ∗ -dilations We begin this section with a weak∗ -topology version of Theorem 3.1 in [1]. Theorem 2.1. Let D be a W ∗ -algebra, B a Banach algebra which is also a dual Banach space, and θ : D → B a unital w ∗ -continuous contractive homomorphism. Then the range of θ is w ∗ closed, and possesses an involution with respect to which θ is a ∗-homomorphism and the range of θ is a W ∗ -algebra. Proof. It is known that (see e.g. Theorem A.5.9 in [8]) the range of a contractive homomorphism between a C ∗ -algebra and a Banach algebra is a C ∗ -algebra and moreover such homomorphisms are ∗-homomorphisms. To see that the range of θ is w ∗ -closed, consider the quotient map D/ ker(θ ) → B which is an isometry, and apply the Krein–Smulian theorem. 2 Thus if X is a left dual operator module over a W ∗ -algebra D, and if we let θ : D → CB(X) be the associated unital w ∗ -continuous contractive (equivalently completely contractive by Proposition 1.2.4 in [8]) homomorphism, then the range of θ is a W ∗ -algebra. Theorem 2.2. Suppose that X is a dual operator module over a dual operator algebra M. Let θ : M → CB(X) be the associated completely contractive homomorphism. Suppose that D is any W ∗ -algebra generated by M. Then the M-action on X can be extended to a D-action with respect to which X is a dual operator D-module if and only if θ is the restriction to M of a w ∗ continuous contractive (equivalently completely contractive) homomorphism φ : D → CB(X). This extended D-action, or equivalently the homomorphism φ, is unique if it exists. Proof. If θ is the restriction to M of a w ∗ -continuous completely contractive homomorphism φ : D → CB(X) then the M-action on X can be extended to a D-action via d · x = φ(d) · x. Note that the D-module action x → dx on X, for x ∈ X and d ∈ D, is a multiplier (see e.g. Theorem 4.6.2 in [8]), hence it is weak∗ -continuous by Theorem 4.1 in [7]. The D-module action on X is separately w ∗ -continuous and completely contractive. Hence X is a dual operator Dmodule. The converse is obvious. To see the uniqueness assertion, suppose that φ1 and φ2 are two w ∗ -continuous contractive homomorphisms D → CB(X), extending θ . By Theorem 2.1, the ranges E1 and E2 , of φ1 and φ2 respectively, are each W ∗ -algebras, but with possibly different involutions and weak∗ -topologies. We will write these involutions as and # respectively. With respect to these involutions φ1 and φ2 are ∗-homomorphisms. Note, CB(X) is a unital Banach algebra and E1 and E2 may be viewed as unital subalgebras of CB(X), with the same unit. Let a ∈ M and f be a state on CB(X) (see e.g. A.4 in [8]). Then f |Ei is a state on Ei for i = 1, 2. Thus f (φ1 (a) ) = f (φ1 (a)) = f (φ2 (a)) = f (φ2 (a)# ). Thus u = φ1 (a) − φ2 (a)# is a Hermitian element in CB(X) with numerical radius 0, hence u = 0. This implies that φ1 (a ∗ ) = φ2 (a ∗ ), since φ1 and φ2 are ∗-homomorphisms. Hence φ1 equals φ2 on the ∗-subalgebra generated by M in D. By weak∗ -density, it follows that φ1 = φ2 on D. 2 This immediately gives the following: Corollary 2.3. Let D be a W ∗ -algebra generated by a dual operator algebra M. If X1 and X2 are two dual operator D-modules, and if T : X1 → X2 is a w ∗ -continuous completely isometric and surjective M-module map, then T is a D-module map.
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Corollary 2.4. Let D be a W ∗ -algebra generated by a dual operator algebra M. Then the category D R of dual operator modules over D is a subcategory of the category M R of dual operator modules over M. Similarly, D H is a subcategory of M H. Next we discuss the W ∗ -dilation which we call the ‘D-dilation’ of a dual operator M-module X, where D is a W ∗ -algebra generated by M. Strictly speaking, it should be called W ∗ -Ddilation, but for brevity we will use the shorter term. Definition 2.5. A pair (E, i) is said to be a D-dilation of a left dual operator M-module X, if the following hold: (1) E is a left dual operator D-module and i : X → E is a w ∗ -continuous completely contractive M-module map. (2) For any left dual operator D-module X , and any w ∗ -continuous completely bounded Mmodule map T : X → X , there exists a unique w ∗ -continuous completely bounded D-module map T˜ : E → X such that T˜ ◦ i = T , and also T cb = T˜ cb . Some authors also use the terminology ‘D-adjunct’ for D-dilation (see [1]). The assertion in (2) above implies that i(X) generates E as a dual operator D-module. To see this, let E = Di(X)w∗ , and consider the quotient map q : E → E/E . Then E/E is a left dual operator D-module such that q ◦ i = 0. Hence the assertion in (2) in the above definition implies that the map q = 0. Thus E = E . Up to a complete isometric module isomorphism there is a unique pair (E, i) satisfying (1) and (2) in the above definition. To see this, let (E , i ) be any other pair satisfying (1) and (2), then there exist the unique w ∗ -continuous completely contractive D-module linear maps ρ : E → E and φ : E → E such that ρ ◦ i = i and φ ◦ i = i. One concludes that ρ ◦ φ is the identity map on i (X) and φ ◦ ρ is the identity map on i(X). Since i(X) and i (X) generate E as a dual operator D-module, and since φ and ρ are w ∗ -continuous complete contractions, this implies that φ and ρ are complete isometries. Remark 2.6. From the above it is clear that the D-dilation (E, i) is the unique pair satisfying (1), and such that for all dual operator D-modules X , the canonical map i ∗ : CBσD (E, X ) → CBσM (X, X ), given by composition with i, is an isometric isomorphism. Note that by using (1.7) and Corollary 1.6.3 in [8], it is easy to see that Mn (CBσ (X, Y )) ∼ = CBσ (X, Mn (Y )) completely isometrically for dual operator spaces X and Y . If X is a left dual operator M-module, then Mn (X) is also a left dual operator M-module via m · [xij ] = [m · xij ] = In ⊗ m · [xij ], where In ⊗m denotes the diagonal matrix in Mn (M) with diagonal entries m. Indeed, if X is a dual operator M-module, the above module action is completely contractive and by Corollary 1.6.3 in [8], this action is separately w ∗ -continuous. This proves that Mn (X) is a dual operator M-module if X is a dual operator M-module. Since i ∗ is an isometry for all dual operator D-modules X , it follows that CBσD (E, Mn (X )) ∼ = CBσM (X, Mn (X )) for all dual operator D-modules X , which ∗ implies that i is a complete isometry. Thus the D-dilation E of X satisfies: CBσD (E, X ) ∼ = CBσM (X, X ) completely isometrically.
(2.1)
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By the dual operator module version of Christensen–Effros–Sinclair theorem (see e.g. Theorem 3.3.1 in [8]), X in Definition 2.5 can be taken to be B(H, K), where K is a normal Hilbert D-module and H is a Hilbert space. In fact, by a modification of Theorem 3.8 in [1], we may take X = K. We are going to prove this important fact in the next theorem but before that we need to recall some tensor products facts. For operator spaces X and Y , we denote the Haagerup tensor product of X and Y by X ⊗h Y . If Z is another operator space, CB(X × Y, Z) denotes the space of completely bounded bilinear maps from X × Y → Z (in the sense of Christensen and Sinclair). It is well known that CB(X × Y, Z) ∼ = CB(X ⊗h Y, Z) completely isometrically (see e.g. 1.5.4 in [8]). If X and Y are two dual operator spaces, we use (X ⊗h Y )∗σ to denote the subspace of (X ⊗h Y )∗ corresponding to the completely bounded bilinear maps from X × Y → C which are separately w ∗ -continuous. Then we define the normal Haagerup tensor product X ⊗σ h Y to be the operator space dual of (X ⊗h Y )∗σ . If Z is another dual operator space, we denote by CBσ (X × Y, Z) the space of completely bounded bilinear maps from X × Y → Z which are separately w ∗ -continuous. By the matrical version of (5.22) in [14], CBσ (X × Y, Z) ∼ = CBσ (X ⊗σ h Y, Z) completely isometrically. That is, the normal Haagerup tensor product linearizes completely contractive, separately weak∗ -continuous bilinear maps. The normal Haagerup tensor product is associative, i.e., if X, Y , Z are dual operator spaces then (X ⊗σ h Y ) ⊗σ h Z ∼ = X ⊗σ h (Y ⊗σ h Z) as dual operator spaces (Lemma 2.2 in [4]). Suppose X is a right dual operator M-module and Y is a left dual operator M-module. A bilinear map u : X × Y → Z is M-balanced if u(xm, y) = u(x, my) for m ∈ A. We let (X ⊗hM Y )∗σ denote the subspace of (X ⊗hM Y )∗ corresponding to the completely bounded balanced bilinear maps from X × Y → C which are separately w ∗ -continuous, where ⊗hM denotes the module Haagerup tensor product (see e.g. 3.4.2, 3.4.3 in [8]). By Proposition 2.1 in [16], the module normal Haagerup tensor product X ⊗σMh Y may be defined to be the operator space dual of (X ⊗hM Y )∗σ . If Z is another dual operator space, we denote by CBMσ (X × Y, Z) the space of completely bounded balanced separately w ∗ -continuous bilinear maps. By Proposition 2.2 in [16], CBMσ (X × Y, Z) ∼ = CBσ (X ⊗σMh Y, Z) completely isometrically.
In order to prove the next lemma, we will introduce some notation. Let CBSσ (X ⊗ Y, Z)
denote the subspace of CB(X ⊗ Y, Z) consisting of completely bounded maps from X ⊗ Y to Z that are induced by the jointly completely bounded bilinear maps from X × Y → Z which
are separately w ∗ -continuous, where ⊗ denotes the operator space projective tensor product (see
e.g. 1.5.11 in [8]). In the case, when Z = C, we denote CBSσ (X ⊗ Y, C) by (X ⊗ Y )∗σ . Lemma 2.7. For any Hilbert spaces H and K and dual operator space X, CBσ (X, B(H, K)) ∼ = CBσ (X ⊗σ h H c , K c ) ∼ = (K r ⊗σ h X ⊗σ h H c )∗ completely isometrically. Proof. Firstly we will prove that for any dual operator space X,
CBSσ X ⊗ H c , K c ∼ = CBσ X, CB H c , K c . θ
From (1.50) in [8], CB(X ⊗ H c , K c ) ∼ = CB(X, CB(H c , K c )) : T → θT where θT (x)(ζ ) = c T (x, ζ ) for x ∈ X and ζ ∈ H . We claim that if T is induced by a separately weak∗ -continuous map from X × H c to K c then θT is a weak∗ -continuous map in CB(X, CB(H c , K c ). To see w∗ this, let xt → x be a bounded net in X. Then, T (xt , ζ ) → T (x, ζ ) for all ζ ∈ H c . By 1.6.1 w∗ in [8], this implies that θT (xt ) → θT (x) in CB(H c , K c ), hence θT ∈ CBσ (X, CB(H c , K c ))
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by Krein–Smulian. Similarly, weak∗ -continuous maps in CB(X, CB(H c , K c )) are mapped into
CBSσ (X ⊗ H c , K c ) which proves the claim. For any dual operator space X, we have the following isometries: CBσ X ⊗σ h H c , K c ∼ = CBσ X × H c , K c
∼ = CBSσ X ⊗ H c , K c ∼ = CBσ X, CB H c , K c ∼ = CBσ X, B(H, K) using Proposition 1.5.14 (1) in [8] and Theorem 3.4.1 in [13]. Consider r CBσ X ⊗σ h H c , K c ∼ = K ∼ = Kr ∼ = Kr ∼ = Kr
∗
⊗ X ⊗σ h H c σ ∗ ⊗h X ⊗σ h H c σ ⊗σ h X ⊗σ h H c ∗ ⊗σ h X ⊗σ h H c ∗ ,
using (1.51) and Proposition 1.5.14(1) in [8], and associativity of the normal Haagerup tensor product. 2 Similarly we have the module version of the above lemma: Lemma 2.8. Let X be a left dual operator M-module and K be a normal Hilbert M-module. Then for any Hilbert space H , CBσM (X, B(H, K)) ∼ = CBσM (X ⊗σ h H c , K c ) ∼ = (K r ⊗σMh X ⊗σ h c H )∗ completely isometrically. Proof. The first isomorphism follows as above with completely bounded maps replaced with module completely bounded maps. Consider r CBσM X ⊗σ h H c , K c ∼ = K ∼ = Kr ∼ = Kr ∼ = Kr
∗
⊗M X ⊗σ h H c σ ∗ ⊗hM X ⊗σ h H c σ ⊗σMh X ⊗σ h H c ∗ ⊗σMh X ⊗σ h H c ∗ ,
using Corollary 3.5.10 in [8], K r ⊗hM − = K r ⊗M − and a variant of Proposition 2.9 in [4].
2
We would like to thank David Blecher for the proof of the following lemma. Lemma 2.9. Let S : X → Y be a w ∗ -continuous linear map between dual operator spaces. The following are equivalent:
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(i) S is a complete isometry and surjective. (ii) For some Hilbert space H , S ⊗ IH : X ⊗σ h H c → Y ⊗σ h H c is a complete isometry and surjective. Proof. Firstly, suppose S is a completely isometric and w ∗ -homeomorphic map. Then, by the functoriality of the normal Haagerup tensor product S ⊗ IH and S −1 ⊗ IH are completely contractive w ∗ -continuous maps, where IH denotes the identity map on H . Also (S −1 ⊗ IH ) ◦ (S ⊗ IH ) = Id on a weak∗ -dense subset X ⊗ H . By w ∗ -density, (S −1 ⊗ IH ) ◦ (S ⊗ IH ) = Id on X ⊗σ h H c . Similarly, (S ⊗ IH ) ◦ (S −1 ⊗ IH ) = Id. Thus S ⊗ IH is a completely isometric and w ∗ -homeomorphic map. Conversely, suppose (ii) holds. Fix an η ∈ H with η = 1. Let v : X → X ⊗σ h Cη : x → x ⊗ η. Since X ⊆ X ⊗h H c completely isometrically via v, and X ⊗h H c ⊆ X ⊗σ h H c completely isometrically, this implies that v is a complete isometry. If S ⊗ IH is a complete isometry, then S ⊗ IH restricted to X ⊗σ h Cη is a complete isometry. Similarly, let u : Y → Y ⊗σ h Cη : y → y ⊗ η. Thus, S = u−1 ◦ (S ⊗ IH ) ◦ v is a complete isometry. To see S is onto, suppose for the sake of contradiction that it is not. Then by the Krein–Smulian theorem G = Ran(S) is a weak∗ -closed proper subspace of Y . Let ϕ ∈ G⊥ and ϕ = 0. Consider a map r : Y ⊗σ h H c → C ⊗σ h H c : y ⊗ ζ → ϕ(y) ⊗ ζ . Then r ◦ (S ⊗ IH ) = 0, since this vanishes on a w ∗ -dense subset Y ⊗ H c . So r = 0. Hence ϕ(y) ⊗ ζ = 0 for all ζ ∈ H and y ∈ Y . This implies ϕ = 0, which is a contradiction. 2 Theorem 2.10. Suppose E is a left dual operator D-module and i : X → E is a w ∗ -continuous completely contractive M-module map. Then (E, i) is the D-dilation of X if and only if the canonical map i ∗ : CBσD (E, K) → CBσM (X, K) as defined above is a complete isometric isomorphism, for all normal Hilbert D-modules K. It is sufficient to take K to be the normal universal representation of D or any normal generator for D H in the sense of [11,18]. Proof. We recall that X in Definition 2.5 can be taken to be B(H, K), where K is a normal Hilbert D-module and H is a Hilbert space. Here we prove that we may take X = K. Consider the following sequence of complete contractions: id⊗i K r ⊗σMh X −→ K r ⊗σMh E ∼ = K r ⊗σDh D ⊗σMh E → K r ⊗σDh E
where the last map in the sequence comes from the multiplication D × E → E. Taking the composition of the above maps, we get a complete contraction S : K r ⊗σMh X → K r ⊗σDh E. Tensoring S with the identity map on H , we get a w ∗ -continuous, completely contractive linear map S1 = S ⊗ idH : K r ⊗σMh X ⊗σ h H c → K r ⊗σDh E ⊗σ h H c by Corollary 2.4 in [4]. From a well-known weak∗ -topology fact, S1 = T ∗ for some T : (K r ⊗σDh E ⊗σ h H c )∗ → (K r ⊗σMh X ⊗σ h H c )∗ . From Lemma 2.8, and standard weak∗ -density arguments, it follows that T equals i ∗ , as defined earlier. Indeed, we use the duality pairing, namely, ψ ⊗ x ⊗ η, T = T (x)(η), ψ, for T ∈ CBσM (X, B(H, K)), x ∈ X, η ∈ H , ψ ∈ K ∗ , to check that (i ∗ )∗ = S1 on the weak∗ -dense subset K r ⊗ X ⊗ H c . Then by weak∗ -density, it follows that (i ∗ )∗ = S1 = T ∗ , so i ∗ = T . Hence, i ∗ is an isometric isomorphism if and only if S1 is an isometric isomorphism if and only if S is an isometric isomorphism by Lemma 2.9. Note that with H = C in Lemma 2.8, CBσM (X, K c ) = (K r ⊗σMh X)∗ . From Lemma 2.8, it is clear that CBσD (E, K c ) ∼ = CBσM (X, K c ) σ σ σ h c c σ h c c ∼ if and only if CBD (E ⊗ H , K ) = CBM (X ⊗ H , K ) for all normal Hilbert D-modules K. For the last assertion, note that every nondegenerate normal Hilbert D-module K is
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a complemented submodule of a direct sum of I copies of the normal universal representation or normal generator, for some cardinal I (see e.g. [11]). Therefore we need to show that if CBσD (E, K) ∼ = CBσM (X, K) completely isometrically then CBσD (E, K I ) ∼ = CBσM (X, K I ) completely isometrically as well, where K I denotes the Hilbert space direct sum of I -copies of K. This follows from the operator space fact that CBσM (X, Y I ) ∼ = MI,1 (CBσM (X, Y )) completely isometrically for any dual operator spaces X and Y which are also M-modules (see page 156 in [12]). Here MI,1 (X) denotes the operator space of columns of length I with entries in X, whose finite subcolumns have uniformly bounded norm. 2 The following lemma shows the existence of the D-dilation. The normal module Haagerup tensor product D ⊗σMh X (which is a dual operator D-module by Lemma 2.3 in [4]) acts as the D-dilation of X. We note that, since by Lemma 2.10 in [4] M ⊗σMh X ∼ = X, there is a canonical w ∗ -continuous completely contractive M-module map i : X → D ⊗σMh X taking x → 1 ⊗M x. Lemma 2.11. For any left dual operator module X over M, the dual operator D-module E = D ⊗σMh X is the D-dilation of X. Proof. If T : X → X is as in Definition 2.5, then by the functoriality of the normal module Haagerup tensor product, ID ⊗ T : D ⊗σMh X → D ⊗σMh X is w ∗ -continuous completely bounded. Composing this with the w ∗ -continuous module action D ⊗σMh X → X gives the required map T˜ . It is routine to check that T˜ has the required properties. 2 Lemma 2.12. If X is a left dual operator M-module, and if D is a W ∗ -algebra generated by M, then the following are equivalent: (1) There exist a dual operator D-module X and a completely isometric w ∗ -continuous Mmodule map j : X → X . (2) The canonical w ∗ -continuous M-module map i : X → D ⊗σMh X, is a complete isometry. Proof. The one direction (2) implies (1) is obvious. For the difficult direction, suppose that m is the module action on X . Then we have the following sequence of canonical w ∗ -continuous completely contractive M-module maps: i
I⊗j
m
X −→ D ⊗σMh X −→ D ⊗σMh X −→ X . The composition of these maps equals j , which is a complete isometry. This forces i to be a complete isometry which proves the assertion. 2 ∗ (M) generated by M defined in the We recall the definition of maximal W ∗ -algebra Wmax ∗ introduction. In the case that D = C = Wmax (M), we call C ⊗σMh X the ‘maximal W ∗ -dilation’ or ‘maximal dilation’. This is a key point in proving our main theorem (Section 4). The reason we work mostly with maximal dilation instead of any arbitrary dilation is the following result.
Corollary 2.13. For any left dual operator M-module X, the canonical M-module map i : X → C ⊗σMh X is a w ∗ -continuous complete isometry.
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Proof. This follows from the previous result, the representation theorem for dual operator modules (see e.g. Theorem 3.8.3 in [8]), and the fact that every normal Hilbert M-module is a normal Hilbert C-module for the maximal W ∗ -algebra generated by M (i.e. the universal property of C). 2 Hence, we may regard X as a w ∗ -closed M-submodule of C ⊗σMh X. There is a similar notion of W ∗ -dilation for right dual operator modules or dual operator bimodules. The results in this section carry through analogously to these cases. 3. Morita equivalence of dual operator algebras In this section, M and N are again dual operator algebras. We reserve the symbols C and D ∗ (M) and W ∗ (N ) generated by M and N respectively. We for the maximal W ∗ -algebras Wmax max refer the reader to [4] if further background for this section is needed. We begin with the following normal Hilbert module characterization of W ∗ -algebras which is proved in Proposition 7.2.12 in [8]. Proposition 3.1. Let M be a dual operator algebra. Then M is a W ∗ -algebra if and only if for every completely contractive normal representation π : M → B(H ), the commutant π(M) is selfadjoint. Corollary 3.2. Suppose M and N are dual operator algebras such that the categories M H and N H are completely isometrically equivalent; i.e., there exist completely contractive functors F : M H → N H and G : N H → M H, such that FG ∼ = Id and GF ∼ = Id completely isometrically, then: (1) If M is a W ∗ -algebra then so is N . (2) Also C H and D H are completely isometrically equivalent. Proof. Suppose F : M H → N H and G : N H → M H, are functors as in the statement of the corollary. If M is a W ∗ -algebra, then for H ∈ M H, BM (H ) is a W ∗ -algebra by Proposition 3.1 (notice BM (H ) = M ). The map T → F (T ) from BM (H ) to BN (F (H )) is a surjective isometric homomorphism (see Lemma 2.2 in [3] or Lemma 4.4 below). Hence by Theorem A.5.9 in [8], this is a ∗-homomorphism if M is a W ∗ -algebra, and consequently its range BN (F (H )) is a W ∗ -algebra. Thus, if M is a W ∗ -algebra, then BN (H ) is a W ∗ -algebra for all normal Hilbert N modules H . From Proposition 3.1, it follows that N is a W ∗ -algebra. For H ∈ M H, we have BC (H ) is a subalgebra of BM (H ). The proof that F restricts to a functor from C H to D H and similar assertion for G, follows identically to the C ∗ -algebra case (see e.g. Proposition 5.1 in [1]). 2 Definition 3.3. (1) Suppose that E and F are weakly Morita equivalent W ∗ -algebras in the sense of Rieffel [18], and that Z is a W ∗ -equivalence F -E-bimodule (see 8.5.12 in [8]), and that W = Z is the conjugate E-F - bimodule of Z. Then we say that (E, F , W, Z) is a W ∗ -Morita context (or W ∗ -context for short).
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(2) Suppose that M and N are dual operator algebras, and suppose that E and F are W ∗ -algebras generated by M and N respectively. Suppose that (E, F , W, Z) is a W ∗ -Morita context, X is a w ∗ -closed M-N -submodule of W , and Y is a w ∗ -closed N -M-submodule of Z. Suppose that the natural pairings Z × W → F and W × Z → E restrict to maps Y × X → N , and X × Y → M respectively, both with w ∗ -dense range. Then we say (M, N, X, Y ) is a subcontext of (E, F , W, Z). If further, E and F are maximal W ∗ -covers (as defined in the introduction) of M and N respectively, then we say that (M, N, X, Y ) is a maximal subcontext. (3) A subcontext (M, N, X, Y ) of a W ∗ -Morita context (E, F , W, Z) is left dilatable if W is the left E-dilation of X, and Z is the left F -dilation of Y . In this case we say that M and N are left weakly subequivalent and (M, N, X, Y ) is a left subequivalence context. There is a similar definition and symmetric theory where we replace the words ‘left’ by ‘right’ or ‘two-sided’. Remark 3.4. Note that (2) in the above definition implies that X and Y are nondegenerate dual operator modules over M and N . Write Lw for the set of 2 × 2 matrices Lw =
a y
x : a ∈ M, b ∈ N, x ∈ X, y ∈ Y . b
Write L for the same set, but with entries from the W ∗ -context (E, F , W, Z). It is well known that L is canonically a W ∗ -algebra, called the ‘linking W ∗ -algebra’ of the W ∗ -context (E, F , W, Z) (see e.g. 8.5.10 in [8]). Saying that (M, N, X, Y ) is a subcontext of (E, F , W, Z) implies that Lw is a w ∗ -closed subalgebra of L . Thus a subcontext gives a linking dual operator algebra Lw . Clearly Lw has a unit. We shall see that Lw generates L as a W ∗ -algebra. The proof of the following theorem is similar to the proof of Theorem 5.2 in [4] with an ∗ (M) and hence we omit it. arbitrary W ∗ -dilation in place of Wmax Theorem 3.5. Suppose that dual operator algebras M and N are linked by a weak∗ Morita context (M, N, X, Y ) in the sense of [4]. Suppose that M is represented normally and completely isometrically as a subalgebra of B(H ) nondegenerately, for some Hilbert space H , and let E be the W ∗ -algebra generated by M in B(H ). Then Y ⊗σMh E is a right W ∗ -module over E. Also (as in the proof of Theorem 5.2 in [4]) Y ⊗σMh E ∼ = Y E w∗ completely isometrically and w ∗ -homeo∗ morphically where w -closure are taken in L , and hence Y ⊗σMh E contains Y as a w ∗ -closed M-submodule completely isometrically. Also, via this module, E is weakly Morita equivalent (in the sense of Rieffel) to the W ∗ -algebra F generated by the completely isometric induced normal representation of N on Y ⊗σMh H . If C is a W ∗ -algebra generated by M, then we shall write F (C) for Y ⊗σMh C ⊗σMh X. From an obvious modification of Theorem 5.2 in [4], we have that F (C) is a W ∗ -algebra containing a copy of N , which is ∗-isomorphic and w ∗ -homeomorphic to (Y CX)−w∗ . The copy of N may be identified with (Y MX)−w∗ . Thus, Theorem 3.5 tells us that C is weakly Morita equivalent to F (C) as W ∗ -algebras.
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Similarly, if D is a W ∗ -algebra generated by N , then we write G(D) for X ⊗σNh D ⊗σNh Y . Again G(D) ∼ = (XDY )−w∗ ∗-isomorphically and w ∗ -homeomorphically. By associativity of the module normal Haagerup tensor product and Lemma 2.10 in [4], G(F (C)) ∼ = C, and F (G(D)) ∼ = D ∗-isomorphically. One can think of F as a mapping between W ∗ -covers of M and N . There is a natural ordering of W ∗ -covers of a dual operator algebra. If (A, j ) and (A , j ) are W ∗ -covers of M, we then define (A, j ) (A , j ) if and only if there is a w ∗ -continuous ∗-homomorphism π : A → A such that π ◦ j = j . It is an easy exercise (using that the range of π is w ∗ -closed) to check that π is surjective. Theorem 3.6. The correspondence C → F (C) is bijective and order preserving. Proof. From the above discussion, the bijectivity is clear. Suppose φ : C1 → C2 is a w ∗ continuous quotient ∗-homomorphism between two W ∗ -algebras generated by M, such that φ|M = IdM . Then by Corollary 2.4 in [4] φ˜ = IdY ⊗ φ ⊗ IdX : Y ⊗σMh C1 ⊗σMh X → Y ⊗σMh C2 ⊗σMh X is a w ∗ -continuous completely contractive map with w ∗ -dense range, which equals the identity when restricted to the copy of N . It is easy to check that φ˜ is a homomorphism on the w ∗ -dense subset Y ⊗ C1 ⊗ X. Therefore by w ∗ -density, φ˜ is a homomorphism. Hence by Proposition A.5.8 in [8], φ˜ is a ∗-homomorphism and is onto. Hence, φ is order preserving. 2 Corollary 3.7. If Lw is the linking dual operator algebra for a weak∗ Morita equivalence of dual operator algebras M and N , and if L is the corresponding linking W ∗ -algebra of the weak ∗ (M) and W ∗ (N ), then W ∗ (Lw ) = L . Morita equivalence of W ∗ -algebras Wmax max max ∗ (M) is normally and faithfully represented on B(H ) for some Hilbert Proof. Suppose Wmax space H . Then, by Lemma 1.1 in [4], H is a normal universal Hilbert M-module. Also M is weak∗ Morita equivalent to Lw , via the dual bimodule M ⊕c Y (see Corollary 4.1 in [4]). By Theorem 3.10 in [4], this induces a normal representation of Lw on the Hilbert space (M ⊕c Y ) ⊗σMh H c . By Proposition 4.2 in [4] we have that
(M ⊕c Y ) ⊗σMh H c ∼ = (H ⊕ K)c unitarily, where K = Y ⊗σMh H c and K is also a normal universal Hilbert N -module (see e.g. ∗ (Lw ) may be taken to be remark on page 6 in [11]). As in the proof of Theorem 5.2 in [4], Wmax ∗ w the W -algebra generated by L in B(H ⊕ K), which is L . 2 The above corollary should have a variant valid for arbitrary W ∗ -covers which we include in [17]. That is, if L is the corresponding linking W ∗ -algebra of the weak Morita equivalence of arbitrary W ∗ -covers then L is a W ∗ -cover of Lw . Proposition 3.8. If (M, N, X, Y ) is a subcontext of a W ∗ -Morita context (E, F , W, Z), then (1) X and Y generate W and Z respectively as left dual operator modules; i.e., W is the smallest w ∗ -closed left E-submodule of W containing X. Similar assertions hold as right dual operator modules, by symmetry. (2) The linking algebra L of (M, N, X, Y ) generates the linking W ∗ -algebra L of (E, F , W, Z). (3) If M or N is a W ∗ -algebra, then (M, N, X, Y ) = (E, F , W, Z).
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Proof. Since the pairing [·,·] : Y × X → N has w ∗ -dense range, we can pick a net et in N which w∗ w∗ is a sum of terms of the form [y, x], for y ∈ Y , x ∈ X, such that et → 1N . Hence wet → w for all w ∈ W . Thus, sums of terms of the form w[y, x], for w ∈ W , x ∈ X, y ∈ Y are w ∗ -dense in W . However, w[y, x] = (w, y)x ∈ EX which shows that EX is w ∗ -dense in W . Thus, X generates W as a left dual operator E -module. Assertions (2) and (3) follow from (1). For example, if M is a W ∗ -algebra, then clearly X = W . Since Y generates Z as a right dual operator module, we have Z = Y E w∗ = Y M w∗ = Y . Since the ranges of the natural pairings Z × W → F and Y × X → N are weak∗ -dense, this implies that F = N . 2 Theorem 3.9. If (M, N, X, Y ) is a weak∗ Morita context which is a subcontext of a W ∗ -Morita context (E, F , W, Z), then it is a dilatable subcontext. Proof. By Proposition 3.8, X and Y generate W and Z, respectively, as left dual operator modules. Hence we have a w ∗ -continuous complete contraction E ⊗σMh X → W with w ∗ -dense range. On the other hand, W∼ = W ⊗σNh N ∼ = W ⊗σNh Y ⊗σMh X ∼ = W ⊗σNh Y ⊗σMh X completely isometrically and w ∗ -homeomorphically. We call the first isomorphism θ1 in the above string, second isomorphism θ2 and third one θ3 . However, the pairing (·,·) : W × Y → E determines a w ∗ -continuous complete contraction W ⊗σNh Y → E, and so we obtain a w ∗ continuous complete
contraction W → E ⊗σMh X. Recall from [4] that N has an ‘approximate t identity’ of the form ni=1 [yit , xit ]. Under the above identifications, θ1
θ2
∗
w −→ w ⊗N 1N −→ w ⊗N w -lim t
−→ w ∗ -lim t
nt
yit
⊗M xit
θ3
−→ w -lim
i=1
nt
nt
i=1
i=1
w, yit xit −→ w ∗ -lim t
∗
t
nt
w ⊗N yit ⊗M xit
i=1
w yit , xit = w.
Hence, the composition of these maps E ⊗σMh X → W → E ⊗σMh X is the identity map, from which it follows that W ∼ = E ⊗σMh X. Similarly Z is the dilation of Y .
2
Theorem 3.10. If (M, N, X, Y ) is a left dilatable maximal subcontext of a W ∗ -context, then M and N are weak∗ Morita equivalent dual operator algebras. Indeed, it also follows that (M, N, X, Y ) is a weak∗ Morita context. Conversely, every weak∗ Morita equivalence of dual operator algebras occurs in this way. That is, every weak∗ Morita context is a left dilatable maximal subcontext of a W ∗ -Morita context. Proof. Every weak∗ Morita context is a left dilatable maximal subcontext of a W ∗ -Morita context is proved in Theorem 5.2 in [4]. For the converse, let C and D be the usual maximal
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W ∗ -algebras of M and N respectively, and let (M, N, X, Y ) be a left dilatable subcontext of (C, D, W, Z). Using Lemmas 2.13 and 2.11, we have Y ⊗σMh X ⊂ D ⊗σNh Y ⊗σMh X ∼ = Z ⊗σMh X ∼ = Z ⊗σC h C ⊗σMh X ∼ = Z ⊗σC h W ∼ = D, complete isometrically and w ∗ -homeomorphically. On the other hand, we have the canonical w ∗ -continuous complete contraction Y ⊗σMh X → N ⊂ D coming from the restricted pairing in Definition 3.3(2). It is easy to check that the composition of maps in these two sequences agree. Thus the canonical map Y ⊗σMh X → N is a w ∗ -continuous completely isometric isomorphism. Similarly, X ⊗σNh Y → M is a w ∗ -continuous completely isometric isomorphism. Hence by the Krein–Smulian theorem, X ⊗σNh Y ∼ = M and Y ⊗σMh X ∼ =N ∗ completely isometrically and w -homeomorphically. Thus M and N are weak∗ Morita equivalent dual operator algebras. 2 4. The main theorem Definition 4.1. Two dual operator algebras M and N are (left) dual operator Morita equivalent if there exist completely contractive functors F : M R → N R and G : N R → M R which are weak∗ continuous on morphism spaces, such that FG ∼ = Id and GF ∼ = Id completely isometrically. Such F and G will be called dual operator equivalence functors. Note that by Corollary 3.5.10 in [8], CBM (V , W ) for V , W ∈ M R is a dual operator space, but CBσM (V , W ) is not a w ∗ -closed subspace of CBM (V , W ). In the above definition, by the funcw∗ tor F being w ∗ -continuous on morphism spaces, we mean that if (ft ) ⊆ CBσM (V , W ), ft → f in w∗
CBM (V , W ), and if f also lies in CBσM (V , W ), then F (ft ) → F (f ) in CBN (F (V ), F (W )). Similarly for the functor G. We also assume that the natural transformations coming from GF ∼ = Id and FG ∼ = Id are weak∗ -continuous in the sense that for all V ∈ M R, the natural transformation wV : GF(V ) → V is a weak∗ -continuous map. Similarly for FG ∼ = Id. There is an obvious analogue to ‘right dual operator Morita equivalence’, where we are con∗ (M) and cerned with right dual operator modules. Throughout, we write C and D for Wmax ∗ Wmax (N) respectively. We now state our main theorem: Theorem 4.2. Two dual operator algebras are weak∗ Morita equivalent if and only if they are left dual operator Morita equivalent, and if and only if they are right dual operator Morita equivalent. Suppose that F and G are the left dual operator equivalence functors, and set Y = F (M) and X = G(N ). Then X is a weak∗ Morita equivalence M-N -bimodule. Similarly Y is a weak∗ Morita equivalence N -M-bimodule; that is, (M, N, X, Y ) is a weak∗ Morita context. Moreover, F (V ) ∼ = Y ⊗σMh V completely isometrically and weak∗ -homeomorphically (as dual operator N -modules) for all V ∈ M R. Thus, F ∼ = Y ⊗σMh − and G ∼ = X ⊗σNh − completely isometrically. Also F and G restrict to equivalences of the subcategory M H with N H, the subcategory C H with D H, and the subcategory C R with D R.
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We will use techniques similar to those of [2] and [3] to prove our main theorem. Mostly this involves the change of tensor product and modification of arguments in the present setting of weak∗ -topology. The following lemmas will be very useful to us. Their proofs are almost identical to analogous results in [2] and therefore are omitted. Lemma 4.3. Let V ∈ M R. Then v → rv is a w ∗ -continuous complete isometry of V onto CBM (M, V ). Here rv ∈ CBM (M, V ) is simply right multiplication by v, i.e., rv (m) = mv. In this case, CBM (M, V ) = CBσM (M, V ) i.e. V ∼ = CBσM (M, V ) completely isometrically and w ∗ homeomorphically. Lemma 4.4. If V , V ∈ M R then the map T → F (T ) gives a completely isometric surjective linear isomorphism CBσM (V , V ) ∼ = CBσN (F (V ), F (V )). If V = V , then this map is a completely isometric surjective homomorphism. Lemma 4.5. For any V ∈ M R, we have F (Rm (V )) ∼ = Rm (F (V )) and F (Cm (V )) ∼ = Cm (F (V )) completely isometrically. Lemma 4.6. The functors F and G restrict to a completely isometric functorial equivalence of the subcategories M H and N H. Proof. Let H ∈ M H. Recall that H with its column Hilbert space structure H c is a left dual operator M-module. We need to show that K = F (H c ) ∈ N H or equivalently F (H c ) is a column Hilbert space. For any dual operator space X and m ∈ N, we have X ⊗h Cm = X ⊗σ h Cm . Hence by Proposition 2.4 in [2], it suffices to show that the identity map K ⊗min Cm → K ⊗σ h Cm is a complete contraction for all m ∈ N. Since all operator space tensor products coincide for Hilbert column spaces, we have Cm (H c ) ∼ = H c ⊗min Cm ∼ = H c ⊗ h Cm ∼ = H c ⊗σ h Cm . Thus K ⊗min Cm ∼ = Cm F H c ∼ = F Cm H c ∼ = F H c ⊗ σ h Cm ∼ = F G(K) ⊗σ h Cm using Lemma 4.5 and G(K) ∼ = H c . Also, using Lemmas 4.3 and 4.4 we have G(K) ∼ = CBM M, G(K) ∼ = CBσN Y, FG(K) ∼ = CBσN (Y, K). By Lemma 2.3 in [4] we get a complete contraction G(K) ⊗σ h Cm → CBσN (Y, K) ⊗σ h Cm . Now CBσN (Y, K) ⊗σ h Cm → CBσN (Y, K ⊗σ h Cm ) : T ⊗ z → y → T (y) ⊗ z for T ∈ CBσN (Y, K) and z ∈ Cm , is a complete contraction. Again using Lemmas 4.3 and 4.4, we have CBσN (Y, K ⊗σ h Cm ) ∼ = CBσM (M, G(K ⊗σ h Cm )) ∼ = G(K ⊗σ h Cm ). Taking the composition of above maps gives a complete contraction G(K) ⊗σ h Cm → G(K ⊗σ h Cm ). Applying F to this map,
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we get a complete contraction F (G(K) ⊗σ h Cm ) → K ⊗σ h Cm . This together with K ⊗min Cm ∼ = F (G(K) ⊗σ h Cm ) gives the required complete contraction K ⊗min Cm → K ⊗σ h Cm . 2 Corollary 4.7. The functors F and G restrict to a completely isometric equivalence of C H and D H. Proof. This is Corollary 3.2 proved earlier.
2
Also, this restricted equivalence is a normal ∗-equivalence in the sense of Rieffel [18], and so C and D are weak Morita equivalent in the sense of Definition 7.4 in [18]. Lemma 4.8. For a dual operator M-module V , the canonical map τV : Y ⊗ V → F (V ) given by y ⊗ v → F (rv )(y) is separately w ∗ -continuous and extends uniquely to a completely contractive map on Y ⊗σMh V . Moreover, this map has w ∗ -dense range. Proof. Since the functor F is w ∗ -continuous on morphism spaces, it is easy to check that τV : Y × V → F (V ) is a separately w ∗ -continuous bilinear map. To see that τV has w ∗ -dense range, suppose the contrary. Let Z = F (V )/N where N = Range(τV )w∗ and let Q : F (V ) → Z be the nonzero w ∗ -continuous quotient map. Then G(Q) : G(F (V )) → G(Z) is nonzero. Thus there exists v ∈ V such that G(Q)wV−1 rv = 0 as a map on M, where wV is the w ∗ continuous completely isometric natural transformation GF(V ) → V coming from GF ∼ = Id. Hence FG(Q)F (wV−1 )F (rv ) = 0, and thus QT F (rv ) = 0 for some w ∗ -continuous module map T : F (V ) → F (V ) since wV−1 is w ∗ -continuous by the Krein–Smulian theorem. By Lemma 4.4, T = F (S) for some w ∗ -continuous module map S : V → V , so that QF (rv ) = 0 for v = S(v) ∈ V . Hence Q ◦ τV = 0, which is a contradiction. Again as in the proof of Lemma 2.6 in [3], τV is a complete contraction. Thus, τV is a separately w ∗ -continuous completely contractive bilinear map. The result follows from the universal property of Y ⊗σMh V . 2 Let (M, N, C, D, F, G, X, Y ) be as above. We let H ∈ M H be the Hilbert space of the normal universal representation of C and let K = F (H ). By Lemma 4.6 and Corollary 4.7, F and G restrict to equivalences of M H with N H, and restrict further to normal ∗-equivalences of C H with D H. By Proposition 1.3 in [18], D acts faithfully on K. Hence, we can regard D as a subalgebra of B(K). Define Z = F (C) and W = G(D). From Lemma 4.8, with V = M, it follows that Y is a right dual operator M-module with module action y · m = F (rm )(y), for y ∈ Y , m ∈ M and rm : M → M : c → cm is simply right multiplication by m. Similarly, X is a right dual operator N -module, and Z and W are dual operator N -C- and M-D-bimodules respectively. The inclusion i of M in C induces a completely contractive w ∗ -continuous inclusion F (i) of Y in Z. One can check that F (i) is a N -M-module map. By Lemma 4.9 below and its proof, it is easy to see that F (i) is a complete isometry. Hence we may regard Y as a w ∗ -closed N -M-submodule of Z and similarly X may be regarded as a w ∗ -closed M-N -submodule of W . With V = X in Lemma 4.8, there is a left N -module map Y ⊗ X → F (X) defined by y ⊗ x → F (rx )(y). Since F (X) = FG(N ) ∼ = N , we get a left N -module map [.] : Y ⊗ X → N . In a similar way we get a module map (.) : X ⊗ Y → M. In what follows we may use the same notation for the unlinearized bilinear maps, so for example we may use the symbol [y, x] for [y ⊗ x]. These maps (.) and [.] have natural extensions to Y ⊗ W → D and X ⊗ Z → C respectively, which
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we denote by the same symbols. Namely, [y, w] is defined via τW for y ∈ Y and w ∈ W . By Lemma 4.8, these maps have weak∗ -dense ranges. Lemma 4.9. The canonical maps X → CBσN (Y, N ) and Y → CBσM (X, M), induced by [.] and (.) respectively, are completely isometrically isomorphic. Similarly, the extended maps W → CBσN (Y, D) and Z → CBσM (X, C) are complete isometries. Proof. By Lemmas 4.3 and 4.4, we have X ∼ = CBσM (M, X) ∼ = CBσN (Y, F (X)) ∼ = CBσN (Y, N ) completely isometrically. Taking the composition of these maps shows that x ∈ X corresponds to the map y → [y, x] in CBσN (Y, N ). Similarly for the other maps. 2 Next consider maps φ : Z → B(H, K), and ρ : W → B(K, H ) defined by φ(z)(ζ ) = F (rζ )(z), and ρ(w)(η) = ωH G(rη )(w), for ζ ∈ H and η ∈ K where ωH : GF(H ) → H is the w ∗ -continuous M-module map coming from the natural transformation GF ∼ = Id. Again rζ : C → H and rη : D → K are the obvious right multiplications. As ωH is an isometric onto map between Hilbert spaces, ωH is unitary and hence also a C-module map by Corollary 2.3. One can check that: ρ(x)φ(z) = (x, z)
and φ(y)ρ(w) = [y, w]V
(4.1)
for all x ∈ X, y ∈ Y , z ∈ Z, w ∈ W and where V ∈ B(K) is a unitary operator in D composed of two natural transformations. A similar calculation as in Lemma 4.3 in [3], shows that the unitary V is in the center of D, hence φ(y)ρ(w) ∈ D for all y ∈ Y and w ∈ W . Lemma 4.10. The map φ (respectively ρ) is a completely isometric w ∗ -continuous N -C-module map (respectively M-D-module map). Moreover, φ(z1 )∗ φ(z2 ) ∈ C for all z1 , z2 ∈ Z, and ρ(w1 )∗ ρ(w2 ) ∈ D, for all w1 , w2 ∈ W . Proof. We will prove that the maps φ and ρ are w ∗ -continuous. The rest of the assertions follow as in Lemma 4.2 in [3] and by von Neumann’s double commutant theorem. To see that φ is w∗ w ∗ -continuous, let (zt ) be a bounded net in Z such that zt → z in Z. For ζ ∈ H , we have F (rζ ) ∈ CBσN (Z, K). Hence F (rζ )(zt ) → F (rζ )(z) weakly. That is, φ(zt ) → φ(z) in the WOT and it follows that φ is weak∗ -continuous. A similar argument works for ρ. 2 We will follow the approach of [2] to prove the selfadjoint analogue of our main theorem, which involves a change of the tensor product. Nonetheless, for completeness we will give the proof. Theorem 4.11. Two W ∗ -algebras A and B are weakly Morita equivalent in the sense of Rieffel if and only if they are dual operator Morita equivalent in the sense of Definition 4.1. Suppose that F and G are the dual operator equivalence functors, and set Z = F (A) and W = G(B). Then, W is a W ∗ -equivalence A-B-bimodule, Z is a W ∗ -equivalence B-A-bimodule, and Z is unitarily and w ∗ -homeomorphically isomorphic to the conjugate W ∗ -bimodule W of W . Moreover, F (V ) ∼ = Z ⊗σAh V completely isometrically and weak∗ -homeomorphically (as dual operator B-modules) for all V ∈ A R. Thus F ∼ = Z ⊗σAh − and G ∼ = W ⊗σBh − completely isometrically. Also F and G restrict to equivalences of the subcategory A H with B H.
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Proof. In [4] we saw that the weakly Morita equivalent W ∗ -algebras (in the sense of Rieffel) are weak∗ Morita equivalent. Hence by Theorem 3.5 in [4], they have equivalent categories of dual operator modules and the assertion about the form of the functors also holds. For the other direction, observe that by Corollary 4.6, the functors F and G restrict to a completely isometric equivalence of A H and B H. Hence, by Definition 7.4 in [18], A and B are weakly Morita equivalent in the sense of Rieffel. We will follow [2] to prove rest of the assertions. By the polarization identity and Lemma 4.10, W is a right C ∗ -module over B with inner product w1 , w2 B = ρ(w1 )∗ ρ(w2 ), for w1 , w2 ∈ W . Similarly, W is a left C ∗ -module over A by setting A w1 , w2 = ρ(w1 )ρ(w2 )∗ . To see that this inner product lies in A, note that, since
n(α) the range of (.) is w ∗ -dense in A, we can choose a net in A of the form eα = k=1 (wk , zk ) = ∗
n(α) w∗ ∗ w k=1 ρ(wk )φ(zk ) where zk ∈ Z and wk ∈ W , such that eα → 1A . Then, eα → 1A . Since ρ is a weak∗ -continuous A-module map, ρ(w)∗ = w ∗ -lim α ρ(eα∗ w)∗ = w ∗ -lim α ρ(w)∗ eα , it follows that ρ(w)ρ(w)∗ is a weak∗ limit of finite sums of terms of the form ρ(w)(ρ(w)∗ ρ(wk ))φ(zk ) = ρ(w)φ(bzk ) = (w, bzk ) ∈ A, where b = ρ(w)∗ ρ(wk ) ∈ B. Thus ρ(w)ρ(w)∗ ∈ A. By the polarization identity ρ(w1 )ρ(w2 )∗ ∈ A. Similarly, Z is both a left and a right C ∗ -module. To see that
n(α) Z is a w ∗ -full right C ∗ -module over A, rechoose a net in A of the form eα = k=1 ρ(wk )φ(zk ) such that eα → IH strongly, so that eα∗ eα → IH weak∗ as done in Theorem 4.4 in [4]. However eα∗ eα = k,l φ(zk )∗ bkl φ(zl ) where bkl = ρ(wk )∗ ρ(wl ) ∈
B. Since P = [bkl ] is a positive
matrix, it has a square root R = [rij ], with rij ∈ B. Thus eα∗ eα = k φ(zkα )∗ φ(zkα ) where zkα = j rkj zj . From this one can easily deduce that the A-valued inner product on Z has w ∗ -dense range. Similarly Z is a weak∗ -full left C ∗ -module over B. Similarly for W . Since ρ and φ are w ∗ -continuous, the inner products are separately w ∗ -continuous. Hence, by Lemma 8.5.4 in [8], W and Z are W ∗ -equivalence bimodules, implementing the weak Morita equivalence of A and B. Note that by Corollary 8.5.8 in [8], CBA (W, A) = CBσA (W, A). Thus by (8.18) in [8] and Lemma 4.9, Z ∼ =W completely isometrically. Let V ∈ A R. By Lemmas 4.3, 4.4 above, Theorem 2.8 in [5], and the fact that Z ∼ = W , we have the following sequence of isomorphisms: F (V ) ∼ = CBσA (W, V ) ∼ = CBσB B, F (V ) ∼ = Z ⊗σAh V as left dual operator B-modules. Thus the conclusions of the theorem all hold.
2
Now we will come back to the setting where M and N are dual operator algebras and C and D are maximal W ∗ -algebras generated by M and N respectively. Theorem 4.12. Suppose M and N are dual operator Morita equivalent dual operator algebras in the sense of Definition 4.1. Suppose that F and G are the dual operator equivalence functors and C and D are maximal W ∗ -algebras generated by M and N respectively. Define Z = F (C) and W = G(D) as above. Then the W ∗ -algebras C and D are weakly Morita equivalent in the sense of Rieffel. In fact Z, which is a dual operator N -C-bimodule, is a W ∗ -equivalence D-C-bimodule. Similarly, W is a W ∗ -equivalence C-D-bimodule, and W is unitarily and w ∗ -homeomorphically isomorphic to the conjugate W ∗ -bimodule Z of Z (and as dual operator bimodules). Proof. By Lemma 4.10, it follows that ρ(W ) is a w ∗ -closed TRO (a closed subspace Z ⊂ B(K, H ) with ZZ ∗ Z ⊂ Z). Hence, by 8.5.11 in [8] and Lemma 4.10, W (or equivalently ρ(W ))
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is a right W ∗ -module over D with inner product w1 , w2 D = ρ(w1 )∗ ρ(w2 ). Since ρ is a complete isometry, the induced norm on W coming from the inner product coincides with the usual norm. Similarly Z is a right W ∗ -module over C. Also, W (or equivalently ρ(W )) is a w ∗ full left W ∗ -module over E = weak∗ closure of ρ(W )ρ(W )∗ , with the obvious inner product ∗ E w1 , w2 = ρ(w1 )ρ(w2 ) . We will show that E = C. Analogous statements hold for D and φ. It will be understood that whatever a property is proved for W , by symmetry, the matching assertions for Z hold. Let Lw be the linking W ∗ -algebra for the right W ∗ -module W , viewed as a weak∗ -closed subalgebra of B(H ⊕ K). We let A = weak∗ closure of ρ(W )φ(Y ). It is easy to check, using the fact that φ(Y )ρ(W ) ∈ D and Lemma 4.10, that A is a dual operator algebra. By the last assertion of Lemma 4.8 and (4.1), M = ρ(X)φ(Y )w∗ ⊆ A and the identity of M is an identity of A. We let U be the weak∗ closure of Dφ(Y ), and we define L to be the following subset of B(H ⊕ K):
A U
ρ(W ) . D
Using (4.1) and Lemma 4.10, it is easy to check that L is a subalgebra of B(H ⊕ K). By explicit computation, Lw L = L and LLw = Lw . To see this, by Lemma 4.10 and the fact that ρ(W ) is a TRO, it follows that Lw L ⊆ L. Again by using (4.1), Lemma 4.10 and the fact that ρ(W )∗ is a left W ∗ -module over D, it follows that LLw ⊆ Lw . Moreover as L and Lw are unital subalgebras of B(H ⊕ K), L ⊆ Lw L and Lw ⊆ LLw . Hence Lw L = L and LLw = Lw . Therefore, we conclude that Lw = L. Comparing corners of these algebras gives E = A and U = ρ(W )∗ . Thus, M ⊆ E , from which it follows that C ⊆ E, since C is the W ∗ -algebra generated by M in B(H ). Thus ρ(W ) is a left C-module, so W can be made into a left C-module in a unique way (by Theorem 2.2). Also by Corollary 2.3, ρ is a left C-module map. By symme∗ try, Z is a left D-module and φ is a D-module map, so that ρ(W )∗ = U = Dφ(Y )w ⊂ φ(Z). ∗ ∗ w∗ By symmetry, φ(Z) ⊂ ρ(W ), so that ρ(W ) = φ(Z). Since, φ(Z) = Dφ(Y ) , by symme∗ try, ρ(W ) = Cρ(X)w∗ . Also, ρ(W )φ(Y ) ⊂ Cρ(X)φ(Y )w ⊂ C and thus E = A ⊂ C. Thus ∗ ∗ w w E = A = C, and that D = φ(Z)φ(Z)∗ = ρ(W )∗ ρ(W ) . This proves the theorem. 2 5. W ∗ -restrictable equivalences Definition 5.1. We say that a dual operator equivalence functor F is W ∗ -restrictable, if F restricts to a functor from C R into D R. We prove our main theorem under the assumption that the functors F and G are W ∗ -restrictable. Later we will prove that this condition is automatic; i.e., the functors F and G are automatically W ∗ -restrictable. Remark 5.2. The canonical equivalence functors coming from a given weak∗ Morita equivalence are W ∗ -restrictable. Suppose that M and N are weak∗ Morita equivalent and let (M, N, X, Y ) be a weak∗ Morita context. Then from Theorem 5.2 in [4] we know that C and D are weakly Morita equivalent W ∗ -algebras, with W ∗ -equivalence D-C-bimodule Z = Y ⊗σMh C. From Theorem 3.5 in [4], F (V ) = Y ⊗σMh V , for V a dual operator M-module. However, if V is a dual operator C-module, Y ⊗σMh V ∼ = Y ⊗σMh C ⊗σC h V ∼ = Z ⊗σC h V . Hence, F restricted to C R is equivalent to σ h ∗ Z ⊗C −, and thus is W -restrictable.
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Theorem 5.3. Suppose that the dual operator equivalence functors F and G are W ∗ -restrictable. Then the conclusions of the Theorem 4.2 all hold. Proof. Clearly, F and G give a dual operator Morita equivalence of C R and D R when restricted to these subcategories. Set Y = F (M), Z = F (C), X = G(N ), and W = G(D) as before. By Theorem 4.11, C and D are weakly Morita equivalent von Neumann algebras with Z and W as W ∗ -equivalence bimodules. From the discussion above Lemma 4.9, Y is a right dual operator M-module and X is a right dual operator N -module. Also Y is a w ∗ -closed N -M-submodule of Z and X is a w ∗ -closed M-N -submodule of W . For any left dual operator C-module X , we have the following sequence of canonical complete isometries by Lemmas 4.3 and 4.4: CBσM (X, X ) ∼ = CBσN N, F (X ) ∼ = F (X ) ∼ CBσ D, F (X ) = D
∼ = CBσC (W, X ). Hence, by the discussion following Definition 2.5, and by Lemma 2.11, we have W ∼ = C ⊗σMh X completely isometrically and as C-modules. It can be checked that this isometry is a right N module map. Similarly, Z ∼ = D ⊗σNh Y . For any dual operator M-module V , we have, Y ⊗σMh V ⊂ (D ⊗σNh Y ) ⊗σMh V ∼ = Z ⊗σMh V completely isometrically, since any dual operator module is contained in its maximal dilation. On the other hand, using Lemmas 4.8, 4.4, and Theorem 4.11, respectively, we have the following sequence of canonical completely contractive N -module maps: Y ⊗σMh V → F (V ) → F C ⊗σMh V ∼ = Z ⊗σC h C ⊗σMh V ∼ = Z ⊗σC h V . The composition of the maps in this sequence coincides with the composition of complete isometries in the last sequence. Hence, the canonical map Y ⊗σMh V → F (V ) is a w ∗ -continuous complete isometry. Since this map has w ∗ -dense range, by the Krein–Smulian theorem, it is a complete isometric isomorphism. Thus F (V ) ∼ = X ⊗σNh U . Fi= Y ⊗σMh V , and similarly G(U ) ∼ σ h σ ∼ ∼ ∼ nally, M = GF(M) = X ⊗N Y , using Lemma 2.10 in [4] and similarly N = Y ⊗Mh X completely isometrically and w ∗ -homeomorphically. 2 Corollary 5.4. Dual operator equivalence functors are automatically W ∗ -restrictable. Proof. Firstly, we will show that W is the maximal dilation of X, and Z is the maximal dilation of Y . In Theorem 4.12, we saw that the set U equals Z. This implies that Y generates Z as a left dual operator D-module. Similarly, X generates W as a left dual operator C-module. By Lemmas 4.3 and 4.4, we have the following sequence of maps CBσM (X, H ) ∼ = CBσN (N, K) ∼ =K ∼ = CBσD (D, K) → CBσM (W, H ). One can check that η ∈ K corresponds under the last two maps in the sequence to the map w → ρ(w)(η), which lies in CBσC (W, H ), since ρ is a left C-module map. Thus, the composition R of the maps in the above sequence has range contained in CBσC (W, H ). Also, R is an
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inverse to the restriction map CBσC (W, H ) → CBσM (X, H ). Thus CBσC (W, H ) ∼ = CBσM (X, H ). Since H is a normal universal representation of C (see the paragraph below Lemma 4.8), it follows from Theorem 2.10, that W is the maximal dilation of X. Similarly Z is the maximal dilation of Y . Let V ∈ C R. By Lemmas 4.3, 4.4, Definition 2.5, Theorem 2.8 in [5], and Theorem 4.12, we have the following sequence of isomorphisms F (V ) ∼ = CBσN N, F (V ) ∼ = CBσM (X, V ) ∼ = CBσC (W, V ) ∼ = Z ⊗σC h V , as left dual operator N -modules. Since Z ⊗σC h V is a left dual operator D-module, we see that F (V ) is a left dual operator D-module and by Theorem 2.2, this D-module action is unique. Also by Corollary 2.3 the map Z ⊗σC h V → F (V ) coming from the composition of the above isomorphisms is a D-module map. This map Z ⊗σC h V → F (V ) is defined analogously to the map τV defined in Lemma 4.8. One can check that if T : V1 → V2 is a morphism in C R, then the following diagram commutes: Z ⊗σC h V1 IZ ⊗T
Z ⊗σC h V2
F (V1 ) F (T )
F (V2 ).
By Corollary 2.4 in [4], IZ ⊗ T is a w ∗ -continuous D-module map and both the horizontal arrows above are w ∗ -continuous D-module maps. Hence, F (T ) is a w ∗ -continuous D-module map; that is, F (T ) is a morphism in D R. Thus F is W ∗ -restrictable. By Theorem 5.3, our main theorem is proved. 2 Acknowledgment The author would like to thank David Blecher for many helpful comments, corrections, and suggestions during the preparation of this work. References [1] D.P. Blecher, Modules over operator algebras and the maximal C ∗ -dilation, J. Funct. Anal. 169 (1999) 251–288. [2] D.P. Blecher, On Morita’s fundamental theorem for C ∗ -algebras, Math. Scand. 88 (2001) 137–153. [3] D.P. Blecher, A Morita theorem for algebras of operators on Hilbert space, J. Pure Appl. Algebra 156 (2001) 153– 169. [4] D.P. Blecher, U. Kashyap, Morita equivalence of dual operator algebras, J. Pure Appl. Algebra 212 (2008) 2401– 2412. [5] D.P. Blecher, U. Kashyap, A characterization and a generalization of W ∗ -modules, preprint, December 2007, arXiv:0712.1236. [6] D.P. Blecher, J. Kraus, Some remarks on w∗ -rigged modules (tentative title), in preparation. [7] D.P. Blecher, B. Magajna, Duality and operator algebras: Automatic weak∗ continuity and applications, J. Funct. Anal. 224 (2005) 386–407. [8] D.P. Blecher, C. Le Merdy, Operator Algebras and Their Modules—An Operator Space Approach, London Math. Soc. Monogr., Oxford University Press, Oxford, 2004. [9] D.P. Blecher, P.S. Muhly, Q. Na, Morita equivalence of operator algebras and their C ∗ -envelopes, Bull. London Math. Soc. 31 (1999) 581–591.
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[10] D.P. Blecher, P.S. Muhly, V.I. Paulsen, Categories of operator modules (Morita equivalence and projective modules), Mem. Amer. Math. Soc. 143 (681) (2000). [11] D.P. Blecher, B. Solel, A double commutant theorem for operator algebras, J. Operator Theory 51 (2004) 435–453. [12] E.G. Effros, Z.-J. Ruan, Representations of operator bimodules and their applications, J. Operator Theory 19 (1988) 137–158. [13] E.G. Effros, Z.-J. Ruan, Operator Spaces, London Math. Soc. Monogr., Oxford University Press, New York, 2000. [14] E.G. Effros, Z.-J. Ruan, Operator space tensor products and Hopf convolution algebras, J. Operator Theory 50 (2003) 131–156. [15] G.K. Eleftherakis, Morita equivalence of dual operator algebras, J. Pure Appl. Algebra 212 (2008) 1060–1071. [16] G.K. Eleftherakis, V.I. Paulsen, Stably isomorphic dual operator algebras, Math. Ann. 341 (2008) 99–112. [17] U. Kashyap, Morita equivalence of dual operator algebras, Ph.D. thesis, University of Houston, 2008. [18] M.A. Rieffel, Morita equivalence for C ∗ -algebras and W ∗ -algebras, J. Pure Appl. Algebra 5 (1974) 51–96.
Journal of Functional Analysis 256 (2009) 3568–3587 www.elsevier.com/locate/jfa
A parametric variational principle and residuality Robert Deville a , Antonín Procházka a,b,∗,1 a Institut de Mathématiques de Bordeaux, Université Bordeaux I, 351 Cours de la Libération, 33405 Talence, France b Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovska 83,
18675 Praha, Czech Republic Received 28 July 2008; accepted 11 March 2009
Communicated by N. Kalton
Abstract We prove a parametric version of a smooth convex variational principle with constraints using a Baire category approach. We examine in depth the necessity of the assumptions of our variational principle by providing counterexamples. © 2009 Elsevier Inc. All rights reserved. Keywords: Smooth variational principle; Perturbed minimization; Continuous dependence of minimizers
1. Introduction A parametric smooth variational principle of Borwein–Preiss kind was introduced by P. Georgiev [3]. Recently L. Veselý [6] modified the proof in order to achieve a parametric smooth variational principle with constraints, i.e. the minimizer after the perturbation is equal to the minimizer before the perturbation for a prescribed set of parameters. In this paper we investigate a possibility of parametrizing (with constraints) the following theorem. Deville–Godefroy–Zizler variational principle. (See [1].) Let (Y, · Y ) be a Banach space. Let (Y, · Y ) be a Banach space of bounded real functions on Y satisfying * Corresponding author.
E-mail address:
[email protected] (A. Procházka). 1 Supported by the grants: Institutional Research Plan AV0Z10190503, A100190801, GA CR ˇ 201/07/0394.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.03.009
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(i) · ∞ · Y , (ii) Y contains a bump, (iii) if g ∈ Y then g(a·) ∈ Y for all a > 0, τy g ∈ Y and τy gY = gY for all y ∈ Y where τy g(x) = g(x − y). Let f : Y → (−∞, +∞] be lower bounded, l.s.c., proper function. Then the set of functions g ∈ Y such that f + g attains its strong minimum is a dense Gδ -subset of Y. The nonparametrized variational principles are usually of interest only in the infinite dimensional Banach spaces (where the original function itself does not have to attain its minimum). We cannot say the same about the parametrized variational principles. Indeed, the existence of a continuous minimizing function for the original function is not granted even in the most simple setting (see Problem 1.1 and Section 5). In order to have a good starting point we consider the following problem. Problem 1.1. Let X be a topological space, Y be a Banach space, and let Y be a fixed space of continuous functions – we call them perturbations – from Y to R. Given a function f : X × Y → (−∞, +∞] which satisfies the following – we call them minimal – conditions: (M1) for every x ∈ X the function f (x, ·) is proper, l.s.c., lower bounded, (M2) for every y ∈ Y the function f (·, y) is a continuous function from X to (−∞, +∞] with its usual topology, is it possible to find : X → Y and v : X → Y continuous such that f x, v(x) + (x) v(x) = inf f (x, y) + (x)(y): y ∈ Y
(1)
for every x ∈ X? First let us comment on the minimal conditions. These requirements are quite natural. Indeed, (M1) is a usual condition of nonparametrized variational principles (Ekeland, Borwein–Preiss, DGZ) and there is no reason why a parametrized version should hold under more general assumptions. In fact, the solution of Problem 1.1 for the case X = singleton is exactly a nonparametrized variational principle. The necessity of (M2) is obvious, consider e.g. X = R = Y , x if x = 0 and sign(0) = 0; δA is the indicator function f (x, y) = δ{sign(x)} (y) where sign(x) = |x| of a set A ⊂ Y , i.e. δA (y) = 0 if y ∈ A otherwise δA (y) = +∞. Nevertheless, Problem 1.1 has, in its general form, a negative solution (see Section 5). Our main theorem (Theorem 4.1) gives a positive answer to the problem when X is a paracompact space, Y is a certain (see Notation 2.3) cone of Lipschitz functions and provided f (apart from obeying (M1), (M2)) is convex in the second variable and satisfies an equi-lower semicontinuity condition (see (A2) in Theorem 4.1) which leads essentially to the lower semicontinuity of inf f (·, Y ) (see Proposition 3.4). In this situation we show that the set of the functions which satisfy (1) is residual in C(X, Y) equipped with the fine topology (see Definition 2.4). This topology enjoys two important properties, it is finer than the uniform topology on C(X, Y) and it is Baire. The latter makes it possible to prove the main theorem in the spirit of the proof of the DGZ variational principle, replacing the points in Y by continuous functions from X to Y . In particular, Theorem 4.1 implies
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Theorem 1.2. Let X be a paracompact Hausdorff topological space, Y be a Banach space with a Fréchet smooth norm, Y be the space of all convex, positive, Lipschitz, Fréchet smooth functions on Y . Let f : X × Y → R satisfy • for all x ∈ X, f (x, ·) is convex, continuous, bounded below, • for all y ∈ Y , f (·, y) is continuous, • for all x0 ∈ X, (f (x0 , ·) − f (x, ·))+ → 0 uniformly on bounded sets of Y as x → x0 . Then for every ε > 0 there exist ∈ C(X, Y) and v ∈ C(X, Y ) such that (x)Y < ε and f (x, ·) + (x) attains its strong minimum at v(x) for all x ∈ X. Moreover x → f (x, v(x)) + (x)(v(x)) is continuous. The additional assumptions appear due to the use of a variant of Michael’s selection theorem in the proof (see Section 3). This includes, apart from the requirements mentioned above, the requirement of X being paracompact. On the other hand, we demonstrate in Section 5 that none of the additional assumptions (convexity, equi-lower semicontinuity) can be dropped without replacement. The organization of the paper is the following. In Section 2 we describe general conditions on the space of perturbations Y, we give some concrete examples of spaces which meet these requirements. We also define and examine the fine topology on the space C(X, Y). In Section 3 we present a version of Michael’s selection theorem (Lemma 3.1) and some lemmata involving the equi-lower semicontinuity (Lemma 3.9 might be of independent interest). In Section 4 we state and prove the main theorem. We state a corollary which can be understood as a localized version of the main theorem and essentially includes as special cases the theorems of Georgiev [3] and Veselý [6]. Section 5 gives some examples to illustrate the limits of the main theorem. Throughout this paper, (Y, · Y ) will be a Banach space. For z ∈ Y and r > 0 we denote BY (z, r) = {y ∈ Y : y − zY r} and SY (z, r) = {y ∈ Y : y − zY = r}. We abbreviate BY = BY (0, 1) and SY = SY (0, 1). For a function g : Y → (−∞, +∞] we denote by dom(g) its effective domain, i.e. dom(g) = {y ∈ Y : g(y) < +∞}. We say that g is proper if dom(g) = ∅. 2. Space of perturbations Definition 2.1. Let Y be a Banach space. A nonnegative convex function b : Y → R is called a convex separating function if for some ε > 0 the set {y ∈ Y : b(y) < ε} is nonempty and bounded. Observe that then {y ∈ Y : b(y) < ε } is nonempty and bounded for all ε > ε. One of the important properties of a convex separating function is described next. Lemma 2.2. Let b : Y → R be a convex separating function. Then for every y0 ∈ Y there exist cy0 > 0 and Cy0 > 0 such that b(y) − b(y0 ) cy0 y − y0 Y for all y ∈ Y \ BY (y0 , Cy0 ). Proof. Since b is a convex separating function it is easily seen that there exist z ∈ Y , c > 0 and C > 0 such that b(y) − b(z) cy − zY for y ∈ Y \ BY (z, C). Now let y0 ∈ Y be given and let us estimate
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b(y) − b(y0 ) = b(y) − b(z) + b(z) − b(y0 ) cy − zY + b(z) − b(y0 ) cy − y0 Y − cy0 − zY + b(z) − b(y0 ) c c y − y0 Y + y − y0 Y − cy0 − zY + b(z) − b(y0 ) 2 2 when y ∈ Y \ BY (z, C). Observe that the term in curly braces becomes positive when y − y0 Y is sufficiently large, say larger than some D > 0. We therefore put cy0 := 2c and Cy0 > D so large that BY (z, C) ⊂ BY (y0 , Cy0 ). 2 Notation 2.3. We will denote by Y some set of convex, Lipschitz functions from Y to [0, +∞) such that (i) Y is a complete positive cone under the norm |g(x) − g(y)| : x, y ∈ Y, x = y , gY = g(0) + sup x − yY (ii) Y contains some convex separating function b, (iii) if g ∈ Y then g(a·) ∈ Y for all a > 0, g − inf g(Y ) ∈ Y, and τy g ∈ Y for all y ∈ Y where τy g(x) = g(x − y). Traditionally, for a norm · on Y with a certain smoothness, 1 + · 2 is a Lipschitz convex separating function with the same smoothness. Thus we may take for Y 1. all convex, Lipschitz functions from Y to [0, +∞), 2. all convex, Lipschitz functions from Y to [0, +∞) which are moreover Gâteaux differentiable, provided · Y is G-differentiable, 3. all convex, Lipschitz functions from Y to [0, +∞) which are moreover Fréchet differentiable, provided · Y is F-differentiable. If X is a Hausdorff topological space, we denote by C(X, Y) the positive cone of all continuous mappings from X to Y together with the fine topology – the definition follows. Definition 2.4. (Cf. [4].) The fine topology on C(X, Y) is the one generated by the neighborhoods of the form Bfine (f, δ) = g ∈ C(X, Y): f (x) − g(x) Y < δ(x) for every x ∈ X where f ∈ C(X, Y), δ ∈ C(X, (0, +∞)). Lemma 2.5. The fine topology on C(X, Y) is Baire. We include the standard proof for the sake of completeness. Proof. In fact, the assertion is true whenever Y is a complete metric space. Let (Gn ) be a sequence of dense open sets in C(X, Y) and let V by any open set in C(X, Y). We claim that there exist sequences (fn ) ⊂ C(X, Y) and (δn ) ⊂ C(X, (0, +∞)) such that
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• Bfine (fn+1 , δn+1 ) ⊂ Bfine (fn , δn ), • sup δn (X) 1/n,
• and Bfine (fn , δn ) ⊂ V ∩ ni=1 Gi . Indeed, let us assume that we have constructed f1 , . . . , fn and δ1 , . . . , δn with the above properties. Since Gn+1 is dense and open we have Bfine (fn , δn ) ∩ Gn+1 ⊃ Bfine (f, 2δ) ⊃ Bfine (f, δ) ⊃ Bfine (f, δ) for some f ∈ C(X, Y), δ ∈ C(X, (0, +∞)). Without loss of generality we may assume that sup δ(X) 1/(n + 1) so clearly the above conditions are satisfied for fn+1 := f and δn+1 := δ. Since Y is complete, this yields that lim fn (x) = f (x) exists for every x ∈ X. Moreover, f is it continuous itself and last but not least a uniform limit of continuous functions fn which makes
f ∈ Bfine (fn , δn ) ⊂ ni=1 Gi ∩ V for every n ∈ N. Thus ∞ i=1 Gi is dense in C(X, Y). 2 3. Existence of approximate minimum A principal step common in the proof of all parametrized variational principles (cf. [3,6]) is the use of some variant of Michael’s selection theorem. Lemma 3.1 (Selection Lemma). Let X be a paracompact Hausdorff topological space and ε ∈ C(X, (0, +∞)). Let f : X × Y → (−∞, +∞] satisfy (a) for every x ∈ X, the function f (x, ·) is proper, lower bounded and convex, (b) for every y ∈ Y , the function f (·, y) is u.s.c. from X to (−∞, +∞], (c) the function inf f (·, Y ) is l.s.c. from X to R. Then there is a continuous function ϕ ∈ C(X, Y ) such that f (x, ϕ(x)) < inf f (x, Y ) + ε(x). Proof. For each y ∈ Y we define Uy = {x ∈ X: f (x, y) < inf f (x, Y ) + ε(x)}. By the assumptions (b) and (c), Uy is open. By the lower boundedness of f (x, ·), the system {Uy }y∈Y covers X. Let {ψs }s∈S be a locally finite partition of unity subordinated to {Uy }y∈Y . For every s ∈ S we find define ys := y. Now ϕ(x) := some Uy such that supp ψs ⊂ Uy and we s∈S ψs (x)ys satisfies the required property. Indeed, f (x, ϕ(x)) s∈S ψs (x)f (x, ys ) < s∈S ψs (x)(inf f (x, Y )+ε(x)) where the first inequality follows from the convexity of f (x, ·) and the second one from the fact that x ∈ Uys if ψs (x) = 0. 2 In order to verify the condition (c) of the previous lemma we look for certain sufficient conditions (see Proposition 3.4). One of them is the equi-lower semicontinuity which we define in such a fashion that allows us to handle the functions with extended values. Definition 3.2. We say that a system {fs : s ∈ S} of functions from a topological space X to (−∞, +∞] is equi-l.s.c. at x0 ∈ X if for every a > 0 and every K > 0 there exists an open neighborhood U of x0 such that for all x ∈ U either fs (x0 ) − a < fs (x), when s ∈ S satisfies fs (x0 ) < +∞, or K < fs (x), when s ∈ S satisfies fs (x0 ) = +∞. A system {fs : s ∈ S} is equil.s.c. if it is equi-l.s.c. at every x0 ∈ X. Observe that when {fs : s ∈ S} is equi-l.s.c. at x0 , {gs : s ∈ S} is equi-l.s.c. at x0 , −∞ < infs∈S fs (x0 ) and −∞ < infs∈S gs (x0 ), then {fs + gs : s ∈ S} is equi-l.s.c. at x0 .
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Observe that, when all fs are real-valued, we may equivalently say that {fs : s ∈ S} are equil.s.c. if for every x0 , (fs (x0 ) − fs (x))+ → 0 uniformly with respect to s ∈ S as x → x0 . Lemma 3.3. Let {fs : s ∈ S} be an equi-l.s.c. system of functions from a topological space X to (−∞, +∞]. Then infs∈S fs is l.s.c. Proof. Let us fix x0 ∈ X. If infs∈S fs (x0 ) = +∞, the conclusion follows immediately from the definition, so we suppose that infs∈S fs (x0 ) < +∞. We choose K > infs∈S fs (x0 ) and ε > 0 arbitrarily. The equi-l.s.c. property provides an open neighborhood U of x0 such that, for all x ∈ U , fs (x0 ) − ε fs (x) if s ∈ S is such that fs (x0 ) < +∞ and inft∈S ft (x0 ) < K < fs (x) if fs (x0 ) = +∞. Consequently, infs∈S fs (x0 ) − ε infs∈S fs (x). 2 Let us abbreviate f (x, y) for f (x, y) + (x)(y), when f : X × Y → (−∞, +∞], ∈ C(X, Y), x ∈ X and y ∈ Y . Proposition 3.4. Let X be a Hausdorff topological space and let f : X ×Y → (−∞, +∞] satisfy (M1), (M2), i.e. (M1) for every x ∈ X the function f (x, ·) is proper, l.s.c., lower bounded, (M2) for every y ∈ Y the function f (·, y) is a continuous function from X to (−∞, +∞] with its usual topology, and moreover (A1) for every x ∈ X, f (x, ·) is convex, (A2) {f (·, y): y ∈ D} is equi-l.s.c. whenever D ⊂ Y is bounded. Let us fix ∈ C(X, Y) and consider the following assertions (i) the set-valued mapping D (x) := {y ∈ Y : f (x, y) < inf(f (x, Y )) + 1} is locally bounded; (ii) the mapping x → inf f (x, Y ) is continuous from X to R. Then (i) implies (ii), and there is a dense set A ⊂ C(X, Y) such that (i) holds for every ∈ A. A key observation permitting to prove the proposition is in the following lemma. Lemma 3.5. Let f : X × Y → (−∞, +∞] satisfy (M1), (M2), (A1) and (A2). Then for all y x0 ∈ X, all y0 ∈ dom(f (x0 , ·)) and all ε > 0 there are an open neighborhood Vε 0 (x0 ) of x0 and y0 rε (x0 ) > 0 with f (x, y) − f (x, y0 ) −εy − y0 Y y
y
for all x ∈ Vε 0 (x0 ) and all y − y0 Y > rε 0 (x0 ). Proof. Let x0 ∈ X, y0 ∈ dom(f (x0 , ·)) and ε > 0 be fixed. By (M2), in particular by the continuity of f (·, y0 ), there is an open neighborhood U of x0 such that p := f (x0 , y0 ) + 1 > f (x, y0 )
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for all x ∈ U . Let us denote q := inf f (x0 , Y ) − 1 and let us choose a positive R > 0 such that (p − q)/R < ε. By the assumption (A2) there exists an open neighborhood V of x0 , V ⊂ U , such that f (x, y) > q for all x ∈ V and all y ∈ BY (y0 , R). Let z ∈ Y , z − y0 Y > R. Set z−y0 y := y0 + R z−y . For x ∈ V we have 0 Y f (x, z) − f (x, y0 ) f (x, y) − f (x, y0 ) q − p −ε z − y0 Y y − y0 Y R y
where the first inequality follows from the convexity of f (x, ·). Thus we set Vε 0 (x0 ) := V and y rε 0 (x0 ) := R. 2 Lemma 3.6. Let ∈ C(X, Y) and D be a bounded subset of Y . Then {(·)(y): y ∈ D} are equi-continuous functions from X to R. Proof. Let us fix x0 ∈ X and ε > 0. Since ∈ C(X, Y) there is an open neighborhood U of x0 such that (x0 ) − (x) < ε Y
for every x ∈ U.
It follows that (x0 )(y) − (x)(y) (x0 )(0) − (x)(0) + εyY 1 + yY ε so the set in question is equi-continuous at x0 .
2
Proof of Proposition 3.4. We first prove that in fact (i) implies (ii). Let x0 ∈ X be fixed. Then there is a neighborhood U of x0 and a bounded set E ⊂ Y such that D (x) ∈ E for all x ∈ U . By the definition of D we have inf f (x, y): y ∈ Y = inf f (x, y): y ∈ E
(2)
for all x ∈ U . By Lemma 3.6 and by the assumption (A2), the functions {f (·, y): y ∈ E} are equi-l.s.c. Using Lemma 3.3 and (2), we conclude that inf f (·, Y ) is l.s.c. Clearly, inf f (·, Y ) is u.s.c. as an infimum of continuous functions. We will now show that there are densely many ∈ C(X, Y) satisfying (i). Let ∈ C(X, Y) and ε ∈ C(X, (0, +∞)) be given. We put h(x)(y) := b(y) · ε(x) and := + h where b ∈ Y is a convex separating function, bY < 1. It follows that h(x)Y = ε(x)bY < ε(x) for all x ∈ X thus ∈ Bfine (, ε). Recall that Lemma 2.2 insures that for each y0 ∈ Y there exist cy0 > 0 and Cy0 > 0 such that b(y) − b(y0 ) cy0 y − y0 Y for all y ∈ Y \ BY (y0 , Cy0 ). We therefore have h(x)(y) − h(x)(y0 ) ε(x)cy0 y − y0 Y
(3)
for all x ∈ X and y − y0 Y > Cy0 . Now D is locally bounded. Indeed, let us fix x0 ∈ X and y0 ∈ dom f (x0 , ·). Then there are an open neighborhood U of x0 and η > 0 such that inf ε(U ) > 2η/cy0 . Since f satisfies (M1), (M2), (A1) and (A2), we
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y
apply Lemma 3.5 for f in order to obtain Vη 0 (x0 ) and rη 0 (x0 ). Without loss of generality y y Vη 0 (x0 ) ⊂ V and rη 0 (x0 ) > Cy0 . This and (3) imply the estimate f+h (x, y) − f+h (x, y0 ) ε(x)cy0 y − y0 Y − ηy − y0 Y ηy − y0 Y y
y
y
for x ∈ Vη 0 (x0 ) and y − y0 Y > rη 0 (x0 ). It follows that D (x) ⊂ BY (y0 , max{rη 0 (x0 ), η1 }) y when x ∈ Vη 0 (x0 ). 2 In the final part of this section we collect some interesting observations which help to understand the effect of the assumptions (M1), (M2), (A1) and (A2). These observations are not used in the proof of the variational principle but we will use them in Section 5. First is a corollary of Proposition 3.4. Corollary 3.7. Let X, Y and f be as in Proposition 3.4 and moreover dim Y < ∞. Let x0 ∈ X be such that A = {y ∈ Y : f (x0 , y) = inf f (x0 , Y )} is bounded. Then the function inf f (·, Y ) is continuous at x0 . The corollary, even if A is just a singleton, does not hold in the case dim Y = ∞. This can be seen in Example 5.4. Proof. We may assume that 0 = inf f (x0 , Y ) and 0 ∈ A ⊂ 12 BY . The lower semicontinuity of f (x0 , ·) and the compactness of SY yield that inf f (x0 , SY ) > a for some a > 0. We may find, using (M2) and (A2), a neighborhood U of x0 such that f (x, 0) < a/3 and inf f (x, SY ) > 2a/3 for all x ∈ U . The assumption (A1) then implies that f (x, y) a3 yY for each x ∈ U and each yY 1. In particular, the set-valued mapping D0 (x) = {y: f (x, y) < inf f (x, Y ) + 1} is bounded at U . Applying Proposition 3.4 we get that inf f (·, Y ) is continuous at x0 . 2 Remark 3.8. Let X be a metrizable space. If we suppose that f maps X × Y into R, i.e. it has no infinite value, and satisfies (M1), (M2) and (A1), then {f (·, y): y ∈ K} is equi-l.s.c. for any compact K ⊂ Y . In particular, f satisfies (A2) automatically provided dim Y < ∞ and X is metrizable. Indeed, for any x ∈ X, the function f (x, ·) is continuous as it is convex, l.s.c. and with finite values (see [5, Proposition 3.3]). Assume that {f (·, y): y ∈ K} is not equi-l.s.c. for some compact K ⊂ Y . Then, since X is metrizable, there are x ∈ X and (xn ) ⊂ X, xn → x, so that f (xn , ·) does not tend uniformly on K to f (x, ·). But f (xn , ·) → f (x, ·) pointwise on Y by (M2) so we get a contradiction with the lemma below. Lemma 3.9. Let f and fn , n ∈ N, be real continuous convex functions defined on an open convex subset V of a Banach space Y such that fn → f pointwise on V . Then fn → f uniformly on compact subsets of V . Proof. Let K ⊂ V be a fixed compact. First we will show that for every ε > 0 there exists n0 ∈ N such that for all n n0 , y ∈ K one has f (y) − ε < fn (y), i.e. (f − fn )+ → 0 uniformly on K, i.e. {f (·, y): y ∈ K} is equi-l.s.c. where f : (N ∪ {∞}) × V → R such that f (n, y) = fn (y) and f (∞, y) = f (y).
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Let us assume that it is not true. Then there exist ε > 0 and a sequence (yn ) ⊂ K such that, without loss of generality, for all n ∈ N, fn (yn ) f (yn ) − ε.
(4)
We may assume, by the compactness of K that yn → y ∈ K. Since fn → f on V , we may use a Baire category argument to get a nonempty open set Up,ε/4 ⊂ V such that for all z ∈ Up,ε/4 and all n > p we have fn (z) − f (z) < ε . 4
(5)
Further, by the compactness of K and the continuity of f , there exist λ ∈ ( 34 , 1) and a nonempty open subset U of Up,ε/4 such that for all a ∈ K, b ∈ U , ε f λa + (1 − λ)b − λf (a) + (1 − λ)f (b) > − . 4
(6)
This requires a proof: fix any b ∈ Up,ε/4 and observe that F (x, y, λ) := f (λx − (1 − λ)y) − (λf (x) + (1 − λ)f (y)) is continuous on K × V × [0, 1]. Further F (a, b , 1) = 0 for all a ∈ K. So for each a ∈ K there are a neighborhood Ua of a, a neighborhood Va of b and an interval Ia = (λa , 1] such that F (x, y, λ) > −ε/4 for all (x, y, λ) ∈ Ua × Va × Ia . The system {Ua : a ∈ K} is an open cover for K. We find its finite open subcover {Ua1 , . . . , Uak } and define U := Va1 ∩ · · · ∩ Vak and λ ∈ ( 34 , 1) such that λ > λa1 , . . . , λak . The set U is nonempty and open as a finite intersection of open neighborhoods of b and obviously satisfies our claim. It is possible to find y˜ ∈ V such that for any n sufficiently large there are zn ∈ U such that y˜ = λyn + (1 − λ)zn . Indeed, choose any z ∈ U and set y˜ := λy + (1 − λ)z, zn := f (y) ˜ + 4ε > λf (yn ) + (1 − λ)f (zn ). It follows
y−λy ˜ n 1−λ .
Hence by (6) we have
fn (y) ˜ λfn (yn ) + (1 − λ)fn (zn ) by the convexity of fn
ε from (4) and (5) λ f (yn ) − ε + (1 − λ) f (zn ) + 4 ε ε f (y) ˜ + − λε + (1 − λ) 4 4 ε f (y) ˜ − 4 which contradicts fn (y) ˜ → f (y). ˜ So we have (f − fn )+ → 0 uniformly on K. On the other hand, if we set Fn (y) := sup{fm (y): m n} for y ∈ V , we have that Fn is a convex, lower semicontinuous function as the supremum of such functions and Fn f pointwise. Hence Fn is real-valued. We may use Proposition 3.3 in [5] to see that Fn is in fact continuous on V . By Dini’s theorem, Fn → f uniformly on K thus (fn − f )+ → 0 uniformly on K. 2
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Remark 3.10. Let us weaken the assumptions of the lemma in the following way. Let K ⊂ Y be a convex compact set. Let f and fn , for n ∈ N be continuous and convex on K such that fn → f pointwise on K. Then these assumptions are not enough to prove that fn → f uniformly on K. Indeed, let Y = 1 , K = co{ enn } and fn (y) = −nyn . Then fn → 0 =: f pointwise, but not even (f − fn )+ tends to 0 uniformly on K. To be sure, let y ∈ K and let us prove that fn (y) → 0. By Choquet’s theorem [2] there existsa probability measure μy on K with μy (Ext K) = 1 such that for all f ∈ ( 1 )∗ one has f (y) = K f (z) dμy (z). Since enn → 0, one may see (with the help of Milman’s theorem [2]) that Ext K = { enn } ∪ {0}. Let us denote zi := eii for i ∈ N and z0 := 0. The ∞ probability measure μy is therefore nothing else than a sequence (λi )i=0 of positive numbers such ∞ that λi = 1 where μy (zi ) = λi for i ∈ N ∪ {0}. We evaluate fn (y) = K fn (z) dμy (z) = i=0 λi fn (zi ) = −λn → 0 as n → ∞. 4. Parametric variational principle Recall that a function h : Y → (−∞, +∞] attains a strong minimum at a point y ∈ Y if it attains minimum at the point y and every minimizing sequence converges to y, i.e. for any sequence (yn ) ⊂ Y one has that h(yn ) → h(y) implies yn → y. Theorem 4.1. Let X be a paracompact Hausdorff topological space. Let f : X × Y → (−∞, +∞] satisfy (M1), (M2), (A1) and (A2), i.e. (M1) for every x ∈ X the function f (x, ·) is proper, l.s.c., lower bounded, (M2) for every y ∈ Y the function f (·, y) is a continuous function from X to (−∞, +∞] with its usual topology, (A1) for every x ∈ X, f (x, ·) is convex, (A2) {f (·, y): y ∈ D} is equi-l.s.c. whenever D ⊂ Y is bounded. Then the set M = ∈ C(X, Y): there is v ∈ C(X, Y ) such that f (x, ·) + (x) attains its strong minimum at v(x) for all x ∈ X
is residual in C(X, Y). Moreover, if ∈ M, then x → inf f (x, Y ) is continuous. In particular, for every ε ∈ C(X, (0, +∞)), there are ∈ C(X, Y) and v ∈ C(X, Y ) such that (x)Y < ε(x) and f (x, ·) + (x) attains its strong minimum at v(x) for all x ∈ X. We remind that we are abbreviating f (x, y) := f (x, y) + (x)(y) whenever ∈ C(X, Y), x ∈ X and y ∈ Y . For the proof we will need one last elementary lemma. Lemma 4.2. Let X be a paracompact topological space, φ : X → R be locally bounded. Then there exists a continuous function ϕ : X → (0, +∞) such that |φ(x)| < ϕ(x) for all x ∈ X. Proof. For every x ∈ X we find an open set Ux x and a constant cx such that φ(Ux ) ⊂ (−cx , cx ). Since X is paracompact we may find a locally finite partition of unity {ψs }s∈S subordinated to the open cover {Ux } of X. For every s ∈ S we define cs := cx for some x ∈ X such that supp ψs ⊂ Ux . The function ϕ(x) := s∈S ψs (x)cs then satisfies the required property. 2
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Proof of Theorem 4.1. Let us consider, for every n ∈ N, the set Un = ∈ C(X, Y): there are vn ∈ C(X, Y ) and δ ∈ C X, (0, +∞) such that 1 for all x ∈ X . f x, vn (x) + δ(x) < inf f (x, y): y − vn (x) Y n Claim. Un is open in C(X, Y). Let x ∈ X be fixed and let us abbreviate f = f (x, ·). Let g1 ∈ Y satisfy 1 f (vn ) + g1 (vn ) + δ < inf f (y) + g1 (y): y − vn Y n for some vn ∈ Y and for some δ > 0. Let g2 ∈ Y such that g1 − g2 Y 1 1 g2 (vn ) − g1 (z) + g2 (z) − δ·n 2 · n for any z ∈ SY (vn , n ). Hence f (z) + g2 (z) f (z) + g1 (z) + g2 (vn ) − g1 (vn ) −
δ·n 2 .
Then g1 (vn ) −
δ 2
f (vn ) + g1 (vn ) + δ + g2 (vn ) − g1 (vn ) −
δ 2
δ f (vn ) + g2 (vn ) + . 2 Since f + g2 is convex, it follows that for any z ∈ Y , z − vn Y
1 n
δ f (z) + g2 (z) f (vn ) + g2 (vn ) + . 2 Now if 1 ∈ Un with vn ∈ C(X, Y ) and δ ∈ C(X, (0, +∞)) and 2 ∈ Bfine (1 , nδ 2 ) then 2 ∈ Un (with the same vn and with δ/2), so Un is open. Claim. The set Un is dense in C(X, Y). Let ∈ C(X, Y) and ε ∈ C(X, (0, +∞)). We need to find ∈ C(X, Y), ∈ Bfine (, ε), δ ∈ C(X, (0, +∞)) and vn ∈ C(X, Y ) such that 1 f x, vn (x) + δ(x) < inf f (x, y): y − vn (x) Y n for every x ∈ X. Thanks to Proposition 3.4 it is enough to consider such that the function x → inf f (x, Y ) is l.s.c. and D (x) = y ∈ Y : f (x, y) < inf f (x, Y ) + 1 is locally bounded. Let b ∈ Y be a convex separating function such that
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(B1) bY < 2 and for all y ∈ Y , τy bY 1, (B2) b(0) + n1 inf{b(y): yY n1 }. Let T : X → (0, +∞) be defined as T (x) := sup{τy bY : y ∈ D (x)}. Then T is locally bounded. Let ϕ be the continuous function that comes from Lemma 4.2 and satisfies T (x) ϕ(x). We may assume that ϕ 1 and use Selection Lemma 3.1 to find vn ∈ C(X, Y ) such that ε(x) f x, vn (x) < inf f (x, y): y ∈ Y + 4nϕ(x) for every x ∈ X. It follows that τvn (x) bY ϕ(x). We define h(x)(y) :=
b(y − vn (x)) · ε(x) . 2τvn (x) bY
It is obvious that h ∈ C(X, Y) and h(x)Y < ε(x) for all x ∈ X, thus ∈ Bfine (, ε) for defined as := + h. It follows that for every x ∈ X f x, vn (x) + (x) vn (x) = f x, vn (x) + h(x) vn (x) ε(x) . < inf f (x, Y ) + h(x) vn (x) + 4nϕ(x)
(7)
If x ∈ X, y ∈ Y and y − vn (x)Y n1 , then by (B2) and the definition of h we get h(x)(y) h(x) vn (x) +
ε(x) 2nτvn (x) bY
which we use immediately in the following estimate f (x, y) + (x)(y) = f (x, y) + h(x)(y) inf f (x, Y ) + h(x)(y) ε(x) + δ(x) inf f (x, Y ) + h(x) vn (x) + 4nϕ(x)
(8)
where δ(x) =
ε(x) 1 1 > 0. − 2n τvn (x) bY 2ϕ(x)
Combining (7) and (8) we conclude that ∈ Un which shows that Un is a dense part of C(X, Y) and the proof of the claim is finished.
Consequently, by Lemma 2.5, n∈N Un is a dense Gδ -subset of C(X, Y).
Claim. Un ⊂ M, i.e. if ∈ Un , then there is v ∈ C(X, Y ) such that f (x, ·) attains its strong minimum at v(x) for every x ∈ X.
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Indeed, for each n ∈ N, let vn ∈ C(X, Y ) be such that 1 . f x, vn (x) < inf f (x, y): y − vn (x) Y n Clearly for every x ∈ X, vp (x) − vn (x)Y < n1 if p n (otherwise, by the choice of vn , we would have f (x, vp (x)) > f (x, vn (x)) as well as, by the choice of vp , we have the opposite strict inequality for every x ∈ X, which is a contradiction). Therefore (vn ) is Cauchy in C(X, Y ) (with the norm · ∞ from Cb (X, Y )) and it converges to some v ∈ C(X, Y ). Let us fix x ∈ X and use the lower semicontinuity of f (x, ·) f x, v(x) lim inf f x, vn (x) n→∞ 1 lim inf inf f (x, y): y − vn (x) Y n→∞ n inf f (x, y): y ∈ Y \ v(x) . So v(x) is a point of minimum for f (x, ·). To see that the minimum attained at v(x) is strong, assume that (zn ) ⊂ Y is a sequence in Y such that f (x, zn ) → f (x, v(x)) but zn v(x). For some subsequence of (zn ) which we will call again (zn ) and for some p ∈ N we have zn − vp (x)Y 1/p for all n ∈ N. Consequently f x, v(x) f x, vp (x) < inf f (x, y): y − vp (x) Y 1/p f (x, zn ) for all n ∈ N which is contradictory to f (x, zn ) → f (x, v(x)). So the proof of the claim is finished. Finally, let ∈ M and x0 ∈ X be fixed. We will show that x → inf f (x, Y ) is continuous at x0 . Indeed, let v ∈ C(X, Y ) such that f (x, v(x)) = inf f (x, Y ). There are an open neighborhood U of x0 and k > 0 such that v(x) ∈ kBY for all x ∈ U . By (A2), {f (·, y): y ∈ kBY } is equi-l.s.c. so by Lemma 3.3 inf f (·, kBY ) = inf f (·, Y ) is l.s.c. at x0 . It is obviously also u.s.c. as the infimum of u.s.c. functions. 2 The following corollary shows that it is possible to localize the points where the minimum is attained. We also include the possibility of not perturbing the function f (x, ·) for x in a certain closed subspace X0 of X. So the corollary actually generalizes a result of Veselý [6, Theorem 4.1] (see also Remark 4.4 below) since it applies in particular when X is metrizable and X0 is its closed subspace. Corollary 4.3. Let X be a paracompact Hausdorff topological space and X0 its closed subspace so that X \ X0 is paracompact. Let f : X × Y → (−∞, +∞] be like in Theorem 4.1. Then for any continuous ε : X → [0, 1) such that ε−1 (0) = X0 and any continuous mapping v0 : X → Y with f x, v0 (x) inf f (x, Y ) + ε(x)2
when x ∈ X
(9)
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there are v ∈ C(X, Y )
and ∈ C(X, Y)
such that (i) f (x, ·) + (x) attains its minimum at v(x) for every x ∈ X, (ii) v(x) − v0 (x)Y ε(x) for every x ∈ X, 2 (iii) (x)Y 2ε(x) c (v0 (x)Y + 1) + ε(x) for some constant c > 0 which depends only on Y. Proof. Let b ∈ Y be a separating convex function which moreover satisfies b(0) = 0, bY = 1 and b c outside BY for some c > 0. The existence of such b is an immediate consequence of conditions posed on Y and Theorem 4.1 with X = {x} and f (x, ·) any convex separating function (possibly without minimum). The assumptions on b imply c 1 and br Y = 1r for r > 0 where br is defined by br (z) := b(z/r). Let us work only on the paracompact space X \ X0 . Observe that bε(x) Y =
1 . ε(x)
We define h(x)(y) :=
2ε(x)2 bε(x) y − v0 (x) c
so h(x)Y 2ε(x) c (v0 (x)Y + 1). By Theorem 4.1, there exist k ∈ C(X \ X0 , Y) and v ∈ C(X \ X0 , Y ) such that k(x)Y < ε(x)2 and f (x, ·) + h(x) + k(x) attains its minimum at v(x). We define = h + k. The condition (i) is satisfied. Further we have (x)Y h(x)Y + 2 k(x)Y 2ε(x) c (v0 (x)Y + 1) + ε(x) . Further, since h(x, v0 (x)) = 0 and from (9), f x, v0 (x) + (x) v0 (x) = f x, v0 (x) + k(x) v0 (x)
inf f (x, Y ) + ε(x)2 + k(x) v0 (x)
while 2ε(x)2 · c + k(x) v0 (x) − ε(x)ε(x)2 c 2 > inf f (x, Y ) + ε(x) + k(x) v0 (x)
f (x, y) + (x)(y) inf f (x, Y ) +
for x ∈ X \ X0 and y ∈ Y such that y − v0 (x)Y = ε(x). From the convexity of f (x, ·) + (x) it follows that v(x) − v0 (x)Y < ε(x). We define |X0 = 0 and v|X0 = v0 . 2
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Remark 4.4. Let us point out the most important differences with respect to the parametrized variational principles [3, Theorem 3.1] and [6, Theorem 1.3]. (a) In both of the cited theorems, the function f has to satisfy the following condition: inf f (·, Y ) is locally lower bounded. We have suppressed completely this assumption in our theorem. Examples 5.3 and 5.4 show functions f which do not meet the above condition, but our Theorem 4.1 still applies for them. We remark that using the respective part of our proof (Lemma 3.5), this condition could be removed also from both cited theorems. (b) If dom(f (x, ·)) = D for all x ∈ X, we are in the setting of Veselý. We show in Example 5.8 that Thoeorem 4.1 goes beyond this setting. functions as perturbations while [3,6] use functions of the form (c) We use Lipschitz νn (x)y − yn (x)2Y , therefore we are “perturbing less” the original function f . Observe that our main theorem stays valid, if we assume the following alternative to Notation 2.3. The space of the above functions from [3,6] already fits in this more general framework. The set Y is a complete (with respect to some norm · Y ) cone of convex continuous functions from Y to [0, +∞) which satisfies: (i) for every bounded subset C of Y there exists a constant MC > 0 such that sup g(C) MC gY for all g ∈ Y; (ii) Y contains some convex separating function b; (iii) whenever g ∈ Y then g(a·) ∈ Y for all a > 0, g − inf g(Y ) ∈ Y, and τy g ∈ Y for all y ∈ Y with y → τy (g)Y continuous. (d) Corollary 4.3 does not cover the situation X = [0, ω1 ], X0 = {ω1 } covered by Theorem 1.3 of [6]. A version of the corollary where we replace the assumption “X \ X0 is paracompact” by the assumption “X0 is discrete” is needed. In order to prove such a version, we can use Lemma 1.2 of [6] in the proof of Theorem 4.1 instead of our Lemma 3.1 and replace the space C(X, Y) by the space CX0 (X, Y) = { ∈ C(X, Y): (x) has a minimum at v0 (x) for every x ∈ X0 }. (e) Let us say that f : X × Y → (−∞, +∞] attains a locally uniformly strong minimum (l.u.s.m.) at v ∈ C(X, Y ) if (a) f (x, v(x)) = inf f (x, Y ) for all x ∈ X, and (b) for each x0 ∈ X and each ε > 0 there are δ > 0 and an open neighborhood U of x0 such that for all x ∈ U and all y ∈ Y the following implication holds true f (x, y) − inf f (x, Y ) < δ
⇒
y − v(x0 ) < ε.
A closer inspection of the proof of Theorem 4.1 reveals that the Gδ dense set { ∈ C(X, Y): there is v ∈ C(X, Y ) such that f attains an l.u.s.m. at v}.
Un equals
5. Examples The functions described in the following proposition will be a prototype for some of our examples.
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Proposition 5.1. Let X = Y ∗ and let us consider f : X × Y → (−∞, +∞] of the form f (x, y) = g(y) − x(y) for y ∈ Y and x ∈ X where g is a proper, l.s.c. and lower bounded function from Y to (−∞, +∞] which satisfies g(y) = +∞. y→∞ yY
(10)
lim
Then f satisfies (M1), (M2) and (A2). If g is convex, then f satisfies also (A1). In contrast with (A2), {f (·, y): y ∈ Y } is equi-l.s.c. if and only if dom(g) is bounded. Of course, if we assume that g is proper, l.s.c., convex and satisfies (10), then g is automatically lower bounded. Proof. The lower boundedness of f (x, ·) is implied by (10). Everything else is trivial.
2
In the first two examples, we will show that the parametric variational principle is still needed even in the spaces which are notorious for having no lower bounded, l.s.c., convex and coercive functions without a minimum, such as the reflexive, Hilbert and finite dimensional spaces. In other words, even if f (x, ·) attains its minimum for every x ∈ X, there does not have to necessarily exist v ∈ C(X, Y ) such that f (x, ·) attains the minimum at v(x). This shows that Y should be reasonably rich. Example 5.2. Let Y be reflexive, X = Y ∗ , and let us define 0, yY 1, g(y) = y2Y − 1, yY > 1 and f (x, y) = g(y) − x(y). Then g is convex and it satisfies (10) so f satisfies (M1), (M2), (A1) and (A2) but every function v : X → Y with f (x, v(x)) = inf f (x, Y ) is discontinuous at 0. Proof. Let x ∈ BX \ {0}. If we denote Mx = {y ∈ Y : f (x, y) = inf f (x, Y )}, then ∅ = Mx is a closed subset of SY and it is not difficult to see that Mx ∩ M−x = ∅ and Mx = Mλx , for 0 < λ < 1/xX . This obviously contradicts continuity of v at 0. 2 In the next example, we examine the continuity of the function x → inf f (x, Y ). Example 5.3. Let X = [0, +∞), Y = R and 0, f (x, y) = |xy − x1 | − x1 ,
x = 0, x = 0.
Then f satisfies (M1), (M2), (A1) and (A2), but 0, x = 0, inf f (x, Y ) = − x1 , x = 0 (attained uniquely at y =
1 ). x2
Obviously x → inf f (x, Y ) is not locally lower bounded at x = 0 (cf. Remark 4.4(a)) so the theorems of Georgiev and Veselý do not apply in this case. Observe, that after application of Theorem 4.1, inf f (·, Y ) is already continuous.
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The previous example may be modified in such a way that f (x, ·) attains a strict minimum for each x ∈ X. Since Corollary 3.7 makes it impossible to construct such an example if dim Y < ∞, we construct it in an infinite dimensional setting. Example 5.4. Let X = [0, 1] and Y = L∞ [0, 1]. Let χA be the characteristic function of a set A, i.e. χA (z) = 1 if z ∈ A, otherwise χA (z) = 0. Let us define functions m : [0, 1] × R → R, m(t, y) := t|y|,
n : [0, 1] × R → R, n(t, y) := t y − t −3 − t −3
and operators M : Y → Y, M(y) := m ·, y(·) ,
N : Y → Y, N (y) := n ·, y(·) .
Further, we define mappings F : X → L1 [0, 1], F (x) := χ[0, x2 ]∪[x,1] ,
G : X → L1 [0, 1], G(x) := χ[ x2 ,x]
and finally f (x, y) = F (x), M(y) + G(x), N (y) . We claim that (a) f satisfies (M1), (M2), (A1) and (A2); (b) for each x ∈ X the function f (x, ·) attains its strict minimum at some v(x) and both v(·) and inf f (·, Y ) are discontinuous at 0; (c) the function inf f (·, Y ) is not locally lower bounded at 0. Proof. Since, for each t ∈ [0, 1], m(t, ·) and n(t, ·) are 1-Lipschitz, we have that M and N are continuous contractions from L∞ [0, 1] to L∞ [0, 1]. On the other hand F and G are continuous. It follows, due to the duality (L1 [0, 1])∗ = L∞ [0, 1], that f (x, ·) is continuous for every x ∈ X and {f (·, y): y ∈ D} is equi-continuous for every bounded D ⊂ Y . Since m(t, ·) and n(t, ·) are convex for each t ∈ [0, 1], the function f (x, ·) is convex for each x ∈ X. This proves the claim (a) with the exception of the lower boundedness of f (x, ·) – it will follow once we prove the claim (b). Now, F (x), M(·) attains a minimum at y if and only if y(t) = 0 for almost all t ∈ [0, x2 ] ∪ [x, 1]. Similarly G(x), N(·) attains its minimum at y if and only if y(t) = −t −3 for almost all t ∈ [ x2 , x]. It follows that f (x, ·) attains its minimum at y if and only if the two above conditions hold simultaneously – this identifies y uniquely, so the minimum is strict. In particular, f (0, ·) attains the strict minimum at v(0) = 0 ∈ Y and the value of the minimum is 0 while f (x, ·) attains its minimum at the uniquely determined v(x) of the norm v(x)Y = 8x −3 and f (x, v(x)) = −x −1 which proves the claims (b) and (c). 2
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Our fourth example shows that Theorem 4.1 need not hold if we drop the convexity assumption (A1) on f (x, ·). Example 5.5. Let Y be a Banach space and we put again X = Y ∗ . Let a ∈ SY be fixed. We put g(y) := inf y − a2 , y + a2 and f (x, y) = g(y) − x(y). We will show that given 0 < ε < 1/16 there are no ∈ C(X, Y), v ∈ C(X, Y ) such that (x)Y < ε and f (x, v(x)) = inf f (x, Y ) for all x ∈ X. Proof. Claim. Let 0 < ε < 1/4. Let φ be any Lipschitz √ function from Y to R√with φY < ε. If g + φ attains its minimum at m ∈ Y , then m − aY < 3ε or m + aY < 3ε. Notice that for every 0 < δ < 1, one has that g(y) < δ
⇒
y − aY <
√ δ
or
y + aY <
√ δ.
(11)
Without loss of generality, let φ(0) = 0. If yY 2 then g(y) (yY − 1)2 . It follows that g(y) + φ(y) g(y) − εyY 2 yY − 1 − εyY 1 − 2ε. Further g(m) + φ(m) = min(g + φ) g(a) + φ(a) 0 + ε, hence mY < 2 thus g(m) − 2ε g(m) + φ(m) ε and finally g(m) 3ε. The claim then follows from (11). To finish the proof of the example fix 0 < ε < 1/16 and suppose there are ∈ C(X, Y), v ∈ C(X, Y ) such that (x)Y < ε and f (x, v(x)) + (x)(v(x)) = inf{f (x, y) + (x)(y): y ∈ Y }. For every x ∈ X such that xX 1/8 we have (x) + xY < 3/16 because xX = xY ∗ = xY . As f (x, ·) + (x)(·) = g(·) − x(·) + (x)(·) attains its minimum at v(x), it follows by the claim that v(x) − aY < 3/4 or v(x) + aY < 3/4. Now let x ∈ 18 SX such that x(a) = 18 , i.e. x is 18 -times tangent to a. It is not difficult to see that then only v(x) − aY < 3/4 holds. Similarly, only v(−x) + aY < 3/4. This shows that the v-image of the connected set 18 BX is contained in the disjoint union BY (a, 34 ) ∪ BY (−a, 34 ) and both v( 18 BX ) ∩ BY (±a, 34 ) are nonempty – so v has to have a point of discontinuity in 18 BX . 2 The next two examples show that the equi-lower semicontinuity of {f (·, D)} for any bounded D ⊂ Y cannot be dropped. In fact, Example 5.7 shows that even if {f (·, K)} is equi-l.s.c. for every compact K ⊂ Y , Theorem 4.1 may still fail.
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Example 5.6. Let X = [0, 1] and Y = R2 with the supremum norm · ∞ . We define g(y) := max{y1 − 3y2 + 2, −y1 − y2 + 2, 2y2 − 2} and g(y , y /x), 1 2 f (x, y) = +∞, g(y1 , 0),
x = 0, x = 0, y2 = 0, x = 0, y2 = 0.
Then f satisfies (M1), (M2), (A1) but not (A2). Given 0 < ε < 15 and ∈ C(X, Y), (x)Y < ε for all x ∈ X, there is no v ∈ C(X, Y ) such that f (x, v(x)) = inf f (x, Y ) for all x ∈ X. Proof. Observe that g enjoys the following properties: a) g(1, 1) = 0 is a strong minimum of g such that g(y) − g(1, 1) 25 y − (1, 1)∞ for all y ∈ Y, b) g(0, 0) = 2 is a strong minimum of g(·, 0) such that g(y1 , 0) − g(0, 0) |y1 | for all y1 ∈ R. Let U = [0, 2x) be a neighborhood of 0 in X. Then for y = (1, x) we have y∞ = 1 and f (x, y) = 0. On the other hand f (0, y) 2 for all y ∈ Y . This shows that {f (·, y): y∞ 1} is not equi-l.s.c. at 0. Further, let (x)Y < 15 for a ∈ C(X, Y). It follows from a) that, for all x ∈ (0, 1], f (x, ·) attains its strong minimum at (1, 1/x) and it follows from b) that f (0, ·) attains its strong minimum at (0, 0). So the uniquely determined v is discontinuous. This shows the breakdown of Theorem 4.1. 2 Example 5.7. Let X = [0, 1] and Y = L2 [0, +∞) with the usual inner product ·,·. Let F : X → Y be defined as F (x) = χ[1/x,1/x+1] , where χA (z) = 1 if z ∈ A, otherwise χA (z) = 0. We define f (x, y) :=
F (x), y + y2Y , x = 0, x = 0. y2Y ,
Then f is real-valued and meets the conditions (M1), (M2), (A1) but not (A2). By Remark 3.8, {f (·, y): y ∈ K} is equi-l.s.c. for any compact subset K of Y . However, Theorem 4.1 fails for f . Proof. It is obvious that f (x, ·) is real-valued, convex, lower bounded and l.s.c. for every x ∈ X. It is also standard that f (·, y) is continuous for every y ∈ Y . An easy computation yields that, F (x) 1 for each x ∈ (0, 1], f (x, ·) attains its minimum at − F (x) 2 , in fact f (x, − 2 ) = − 4 . Moreover, F (x) 2 1 since y + 2 Y = f (x, y) + 4 , one has for every ε > 0, 1 f (x, y) + ε 2 4
⇒
y + F (x) ε. 2 Y
(12)
Let ε > 0 be sufficiently small. And let ∈ C(X, Y) such that (x)Y < ε. Let us assume temporarily that y 2. Then f (x, y) yY . It follows that f (x, y) f (x, y) −
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ε(yY + 1) 2 − 3ε. So if f (x, ·) attains its minimum at v(x), we get
F (x) 1 F (x) ε + (x) − − +ε+ . f x, v(x) = min f (x, Y ) f x, − 2 2 4 2 Thus v(x)Y < 2 and 1 3ε f x, v(x) − 3ε f x, v(x) − ε − ε v(x) Y f x, v(x) − + 4 2 F (x) 9ε whence f (x, v(x)) + 14 9ε 2 . Using (12) we get v(x) + 2 Y 2 and finally v(x)Y 1 9ε 2 − 2 . √ Similarly, if (0)Y < ε and f (0, ·) attains its minimum at v(0), we get that v(0)Y < ε. This contradicts the continuity of v at 0 whenever ε is sufficiently small. 2 Example 5.8. Let X = {0} ∪ { n1 : n ∈ N}, Y = R and let us define f (x, y) =
y − x > 0, +∞, y − x 0. 1 y−x ,
Then f satisfies (M1), (M2), (A1) and (A2). Obviously dom f (x, ·) = dom f (z, ·) if and only if x = z. Our variational principle applies in this situation while the theorems of Georgiev and Veselý do not. References [1] R. Deville, G. Godefroy, V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monogr. Surv. Pure Appl. Math., vol. 64, Longman Scientific & Technical, Harlow, 1993. [2] M. Fabian, P. Habala, P. Hájek, V. Montesinos, J. Pelant, V. Zizler, Functional Analysis and Infinite Dimensional Geometry, CMS Books Math., Springer-Verlag, 2001. [3] P.G. Georgiev, Parametric Borwein–Preiss variational principle and applications, Proc. Amer. Math. Soc. 133 (2005) 3211–3225. [4] J. Munkres, Topology, 2nd edition, Prentice Hall, Upper Saddle River, NJ, 2000. [5] R. Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd edition, Lecture Notes in Math., vol. 1364, Springer, Berlin, 1993. [6] L. Veselý, A parametric smooth variational principle and support properties of convex sets and functions, J. Math. Anal. Appl. 350 (2) (2009) 550–561.
Journal of Functional Analysis 256 (2009) 3588–3642 www.elsevier.com/locate/jfa
Optimal Gaussian Sobolev embeddings ✩ Andrea Cianchi a,∗ , Luboš Pick b a Dipartimento di Matematica e Applicazioni per l’Architettura, Università di Firenze, Piazza Ghiberti 27,
50122 Firenze, Italy b Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83,
186 75 Praha 8, Czech Republic Received 29 August 2008; accepted 4 March 2009 Available online 1 April 2009 Communicated by L. Gross
Abstract A reduction theorem is established, showing that any Sobolev inequality, involving arbitrary rearrangement-invariant norms with respect to the Gauss measure in Rn , is equivalent to a one-dimensional inequality, for a suitable Hardy-type operator, involving the same norms with respect to the standard Lebesgue measure on the unit interval. This result is exploited to provide a general characterization of optimal range and domain norms in Gaussian Sobolev inequalities. Applications to special instances yield optimal Gaussian Sobolev inequalities in Orlicz and Lorentz(–Zygmund) spaces, point out new phenomena, such as the existence of self-optimal spaces, and provide further insight into classical results. © 2009 Elsevier Inc. All rights reserved. Keywords: Logarithmic Sobolev inequalities; Gauss measure; Sobolev embeddings; Rearrangement-invariant spaces; Optimal domain; Optimal range; Orlicz spaces; Lorentz spaces; Hardy operators involving suprema
✩ This research was partly supported by the NATO grant PST.CLG.978798, by the grants 201/01/0333, 201/03/0935, 201/05/2033, 201/07/0388 and 201/08/0383 of the Grant Agency of the Czech Republic, by the grant 0021620839 of the Czech Ministry of Education, by the Neˇcas Center for Mathematical Modeling project No. LC06052 financed by the Czech Ministry of Education, and by the Italian research project “Geometric properties of solutions to variational problems” of GNAMPA (INdAM) 2006. * Corresponding author. E-mail addresses:
[email protected] (A. Cianchi),
[email protected] (L. Pick).
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.03.001
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1. Introduction In connection with the study of quantum fields and hypercontractivity semigroups, extensions of the classical Sobolev inequality in Rn to the setting when the underlying measure space is infinite-dimensional have been investigated. The main motivation for this research is that, in certain circumstances, the study of quantum fields can be reduced to operator or semigroup estimates which are in turn equivalent to inequalities of Sobolev type in infinitely many variables (see [31] and the references therein). The classical Sobolev inequality implies that if u is a weakly differentiable function in Rn , decaying to 0 at infinity, and |∇u|p is integrable on Rn for some p ∈ [1, n), then |u| raised to np the larger power n−p is integrable. When p > n (and the support of u has finite measure), u is in fact essentially bounded. Note, in particular, that the gain in the integrability depends on the dimension n. In attempting to generalize these results to the case where the underlying space is infinitenp dimensional, one immediately meets two problems. First, n−p → p+ as n → ∞, so the gain in integrability is apparently being lost. Second, and more serious, the Lebesgue measure on an infinite-dimensional space is meaningless. These problems were overcome in the fundamental paper by L. Gross [25], where the Lebesgue measure was replaced by the Gauss measure γn , defined on Rn by n
dγn (x) = (2π)− 2 e
−|x|2 2
dx.
(1.1)
Since γn (Rn ) = 1 for every n ∈ N, the extension as n → ∞ is meaningful. The idea was then to seek a version of the Sobolev inequality that would hold on the probability space (Rn , γn ) with a constant independent of n. In [25] an inequality of this kind is proved, which, in particular, entails that u − uγn L2 Log L(Rn ,γn ) C∇uL2 (Rn ,γn )
(1.2)
for some absolute constant C and for every weakly differentiable function u making the right-hand side finite. Here, uγn = Rn u(x) dγn (x), the mean value of u over (Rn , γn ), and L2 Log L(Rn , γn ) is the Orlicz space of those functions u such that |u|2 | log |u|| is integrable in Rn with respect to γn . Observe that (1.2) still provides some slight gain in the integrability from |∇u| to u, even though it is no longer a power-gain. Gross’ result ignited an extensive research on Sobolev inequalities in the Gauss space, including simplified proofs [2], applications [23,35,40], and extensions to the case when |∇u| belongs to a space different from L2 (Rn , γn ) [3,4,6,5,11,10,18,27,33]. For instance, inequalities for functions with |∇u| ∈ Lp (Rn , γn ) for p ∈ [1, ∞) are known [1], and tell us that then p u ∈ Lp Log L 2 (Rn , γn ). Interestingly, in contrast to the Euclidean setting, when |∇u| enjoys a high degree of integrability, stronger than just a power, there is a loss of integrability from |∇u| to u instead of a gain in the Gaussian Sobolev embedding. This happens, in particular, when |∇u| is exponentially integrable [9], or essentially bounded [3]: for instance, in the latter case, one can 2 just infer that u ∈ exp L2 (Rn , γn ), the Orlicz space associated with the Young function et − 1. This phenomenon can be explained by the rapid decay of the Gauss measure at infinity.
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The aim of this paper is to present a comprehensive treatment of optimal Sobolev embeddings in the Gauss space in the general form u − uγn Y (Rn ,γn ) C∇uX(Rn ,γn )
(1.3)
for some constant C and for every u ∈ V 1 X(Rn , γn ), where X(Rn , γn ) and Y (Rn , γn ) are rearrangement-invariant (for short, r.i.) spaces, and V 1 X(Rn , γn ) is the Sobolev-type space built upon X(Rn , γn ), namely V 1 X Rn , γn = u: u is a weakly differentiable function in Rn such that |∇u| ∈ X Rn , γn . Loosely speaking, in an r.i. space the norm of a function depends only on its degree of integrability, namely on the (Gaussian) measure of its level sets. A precise definition is recalled in Section 2, where the necessary prerequisites from the theory of function spaces are collected. Our approach relies on a reduction theorem (Theorem 3.1, Section 3) showing that inequality (1.3) is completely equivalent to a one-dimensional inequality for a suitable Hardy-type operator, involving the same norms as in (1.3), but on the interval (0, 1) endowed with the standard Lebesgue measure. This step requires a symmetrization argument exploiting a general Pólya– Szegö principle on the decrease of r.i. norms of the gradient of Sobolev functions in the Gauss space (Theorem 3.2, Section 3), extending the results of [22] and [38]. Its proof relies upon the Gaussian isoperimetric inequality by Borell [11]. The reduction theorem is a key step in our description of the optimal r.i. spaces X(Rn , γn ) and Y (Rn , γn ) appearing in (1.3). Namely, given X(Rn , γn ), we characterize the optimal, i.e. the smallest, range space Y (Rn , γn ) for which (1.3) holds (Theorem 4.1, Section 4), and, conversely, given Y (Rn , γn ), we characterize the optimal, i.e. the largest, domain space X(Rn , γn ) for which (1.3) holds (Theorem 4.3, Section 4). These results are then employed to establish Sobolev inequalities for concrete spaces. On the one hand, we recover the embeddings mentioned above, corresponding to the choice X(Rn , γn ) = Lp (Rn , γn ), with p ∈ [1, ∞] or X(Rn , γn ) = exp Lβ (Rn , γn ), with β ∈ (0, ∞), and, as a new contribution, we show their sharpness in the framework of all r.i. spaces. On the other hand, and more significantly, we establish new embeddings which involve important customary spaces. Section 5 deals with Gaussian Sobolev inequalities in Orlicz spaces. In Theorem 5.1 of that section we associate with any Young function A another Young function AG such that LAG (Rn , γn ) is the optimal Orlicz space in the inequality u − uγn LAG (Rn ,γn ) C∇uLA (Rn ,γn ) for some absolute constant C and for every u ∈ V 1 LA (Rn , γn ). Sobolev embeddings involving Lorentz spaces are the concern of the subsequent Section 6. In fact, Theorem 6.1 deals with the more general class of Lorentz–Zygmund spaces, which naturally come into play when looking for the optimal range or domain in the Gaussian Sobolev inequality. Finally, in Section 7, a particular feature of Gaussian Sobolev embeddings is pointed out. Indeed, we show that there exist borderline spaces X(Rn , γn ) which are self-optimal in (1.3), in the sense that (1.3) holds with Y (Rn , γn ) = X(Rn , γn ), and the latter is simultaneously the optimal range on the left-hand side and the optimal domain on the right-hand side – see Theorem 7.1. In particular, this is the case when X(Rn , γn ) = LA (Rn , γn ) and A is a Young function given by
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642 1
3591
2
A(t) = e 4 (log t) for large t (Corollary 7.2). In fact, it turns out that LAG (Rn , γn ) = LA (Rn , γn ) for this choice of A. Part of the results of the present paper were announced in the survey [34, Section 9]. 2. Rearrangements and rearrangement-invariant spaces This section contains the basic background from the theory of rearrangements and of r.i. spaces that will be needed in what follows. For an exhaustive treatment of these topics, we refer the reader to [8]. Definitions and basic properties of the spaces which will play a role in our discussion, such as Orlicz, Lorentz and Lorentz–Zygmund spaces, are also recalled below. Let (S, m) be a probability space, namely, a measure space S endowed with a probability measure m. We shall assume throughout that (S, m) is totally σ -finite and that m is non-atomic. In fact, S will either be Rn endowed with the Gaussian measure γn , or (0, 1) endowed with the Lebesgue measure. We shall simply write S instead of (S, m) when no ambiguity can arise. We denote by M(S) the set of real-valued, m-measurable functions on S, and by M+ (S) the set of nonnegative functions in M(S). Let φ ∈ M(S). The decreasing rearrangement φ ∗ : (0, 1) → [0, ∞) of φ is given by φ ∗ (s) = sup t 0: γn x ∈ S: φ(x) > t > s
for s ∈ (0, 1).
Similarly, the signed decreasing rearrangement φ ◦ : (0, 1) → R of φ is defined as φ ◦ (s) = sup t ∈ R: γn x ∈ S: φ(x) > t > s
for s ∈ (0, 1).
We also define φ ∗∗ : (0, 1) → [0, ∞) as 1 φ (s) = s ∗∗
s
φ ∗ (r) dr
for s ∈ (0, 1).
0
Note that φ ∗∗ is also non-increasing, and φ ∗ (s) φ ∗∗ (s) for s ∈ (0, 1). Moreover, (φ + ψ)∗∗ (s) φ ∗∗ (s) + ψ ∗∗ (s)
for s ∈ (0, 1),
(2.1)
for every φ, ψ ∈ M(S). Two measurable functions φ and ψ on S are said to be equimeasurable (or equidistributed) if φ ∗ = ψ ∗ . We shall write φ∼ψ to denote that φ and ψ are equimeasurable. A Banach space X(S) of functions in M(S), equipped with the norm · X(S) , is said to be a rearrangement-invariant space if the following five axioms hold: (P1) 0 ψ φ a.e. implies ψX(S) φX(S) ; (P2) 0 φk φ a.e. implies φk X(S) φX(S) as k → ∞; (P3) 1X(S) < ∞;
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(P4) a constant C exists such that S |φ| dm(x) CφX(S) for every φ ∈ X(S); (P5) φX(S) = ψX(S) whenever φ ∗ = ψ ∗ . A norm · X(S) fulfilling (P1)–(P5) is called an r.i. norm. A consequence of Hardy’s lemma [8, Chapter 2, Proposition 3.6 and Theorem 4.6] entails that if X(S) is any r.i. space and φ, ψ ∈ M(S) are measurable functions in S, then φ ∗∗ (s) ψ ∗∗ (s)
for s ∈ (0, 1)
implies that φX(S) ψX(S) .
(2.2)
We shall also make frequent use of the Hardy–Littlewood inequality [8, Chapter 2, Theorem 2.2], which states that
φ(x)ψ(x) dm(x)
S
1
φ ∗ (s)ψ ∗ (s) ds
(2.3)
0
for every φ, ψ ∈ M(S). Given an r.i. space X(S), the set
X (S) = φ ∈ M(S): φ(x)ψ(x) dm(x) < ∞ for every ψ ∈ X(S) , S
equipped with the norm φX (S) =
sup
ψX(S) 1
φ(x)ψ(x) dm(x),
(2.4)
S
is called the associate space of X(S). It turns out that X (S) is again an r.i. space endowed with the norm given by (2.4), and that X (S) = X(S). Furthermore, the Hölder inequality φ(x)ψ(x) dm(x) φX(S) ψX (S) (2.5) S
holds for every φ ∈ X(S) and ψ ∈ X (S). Let X(S) and Y (S) be r.i. spaces. We write X(S) → Y (S) to denote that X(S) is continuously embedded into Y (S). By [8, Chapter 1, Theorem 1.8], X(S) ⊂ Y (S)
if and only if
X(S) → Y (S).
X(S) → Y (S)
if and only if
Y (S) → X (S),
Moreover, (2.6)
with the same embedding constants. For each r.i. space X(S), there exists a unique r.i. space X(0, 1) on (0, 1) satisfying φX(S) = φ ∗ X(0,1)
for φ ∈ X(S),
(2.7)
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and hence also φX(S) = φ ◦ X(0,1)
for φ ∈ X(S).
(2.8)
Such a space, endowed with the norm defined by 1 f X(0,1) =
sup
ψX (S) 1
f ∗ (s)ψ ∗ (s) ds
0
for f ∈ M(0, 1), is called the representation space of X(S). Let X(S) be an r.i. space. Then, the function ϕX : [0, 1) → [0, ∞) given by ϕX (s) = χ(0,s) X(0,1)
for s ∈ [0, 1),
is called the fundamental function of X(S). The fundamental function ϕX of any r.i. space X(S) is quasiconcave, in the sense that it is non-decreasing on [0, 1), ϕX (0) = 0 and ϕXs(s) is nonincreasing on (0, 1). Moreover, one has that ϕX (s)ϕX (s) = s
for s ∈ [0, 1).
(2.9)
In the remaining part of this section, we recall a few definitions and basic properties of those function spaces that will be involved in our results. The Lebesgue spaces Lp (S), with p ∈ [1, ∞], endowed with the standard norm, are the simplest instance of r.i. spaces. In particular, L1 (S) and L∞ (S) are the largest and the smallest, respectively, r.i. spaces on S, in the sense that if X(S) is any other r.i. space, then L∞ (S) → X(S) → L1 (S). Given any Young function A : [0, ∞) → [0, ∞), namely a convex function vanishing at 0, the Orlicz space LA (S) is the r.i. space of all functions φ ∈ M(S) such that the Luxemburg norm
|φ(x)| φLA (S) = inf λ > 0: dm(x) 1 (2.10) A λ S
: [0, ∞) → [0, ∞), defined by is finite. The function A
= sup st − A(s): s 0 A(t)
for t ∈ [0, ∞),
is also a Young function, called the Young conjugate of A. The Orlicz space LA (S) can be equivalently renormed to become the associate space of LA (S). In particular, one has that φ(x)ψ(x) dm(x) 2φ A ψ A (2.11) L (S) L (S) S
for every φ ∈ LA (S) and ψ ∈ LA (S).
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Since m(S) < ∞, one has that LA (S) = LB (S) (up to equivalent norms) if and only if A and B are Young functions equivalent near infinity, in the sense that A(C1 t) B(t) A(C2 t) for some constants C1 and C2 , and for large t. Classes of Orlicz spaces which will be of particular interest in our applications are the Zygmund spaces of exponential type exp Lβ (S), with β ∈ (0, ∞), and the Zygmund spaces of logarithmic type Lp (log L)α (S), with either p ∈ (1, ∞) and α ∈ R, or p = 1 and α ∈ [0, ∞), β which are generated by Young functions equivalent to et and to t p (log t)α , respectively, near infinity. Let us mention that, in Section 5, Orlicz spaces on (possibly unbounded) intervals different from (0, 1) will also be considered, for technical reasons. The definition of the corresponding Luxemburg norm is then completely analogous. Let p ∈ (0, ∞] and let ω ∈ M+ (0, 1). Then the classical Lorentz spaces Λp (ω)(S) and p Γ (ω)(S) are defined as the sets of those functions φ ∈ M(S) such that the quantities 1 1 ∗ p ω(s) ds) p ( φ (s) if p ∈ (0, ∞), 0 p φΛ (ω)(S) = ∗ ess sup0<s<1 φ (s)ω(s) if p = ∞, and 1 1 ∗∗ p p φΓ p (ω)(S) = ( 0 φ (s) ω(s) ds) ess sup0<s<1 φ ∗∗ (s)ω(s)
if p ∈ (0, ∞), if p = ∞,
respectively, are finite. Clearly, one always has Γ p (ω)(S) ⊂ Λp (ω)(S), and for some p and ω this inclusion may be strict (see [15] and the references therein). In the case when p = ∞, the spaces Λ∞ (ω)(S) and Γ ∞ (ω)(S) are usually called Marcinkiewicz spaces. It should be noted that, for general p and ω, the sets Λp (ω)(S) and Γ p (ω)(S) need not be r.i. spaces (see [19]). In fact, they may even reduce to the trivial space containing only the zero function. The quantity · Λp (ω)(S) is equivalent to an r.i. norm, under which Λp (ω)(S) is an r.i. space if and only if either p ∈ (1, ∞) and 1 sp
r −p ω(r) dr C
s
s ω(r) dr
for s ∈ (0, 1),
0
or p = 1 and 1 s
s
C ω(ρ) dρ r
0
r ω(ρ) dρ
for 0 < r s 1,
0
or p = ∞ and 1 s
s 0
C dr ω(r) ω(s)
for s ∈ (0, 1),
(2.12)
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for some constant C, where we have set ω(s) = ess sup ω(r). 0
Details for the cases where p ∈ (1, ∞) and p = 1 can be found in [36] and [14], respectively. The case where p = ∞ follows quite easily from [37, Theorem 3.1]. Important instances of classical Lorentz spaces are the customary two-parameter Lorentz spaces Lp,q (S) and L(p,q) (S), defined for p, q ∈ (0, ∞] as the sets of those functions φ ∈ M(S) for which the quantities 1 1 − φLp,q (S) = φ ∗ (s)s p q Lq (0,1) and 1 1 − φL(p,q) (S) = φ ∗∗ (s)s p q Lq (0,1) , respectively, are finite. A generalization is provided by the so-called Lorentz–Zygmund spaces Lp,q;α (S) and L(p,q;α) (S), which were introduced in [7], and are defined for p, q ∈ (0, ∞] and α ∈ R as the sets of all functions φ ∈ M(S) such that the quantities 1 1 α − Lp,q;α (S) = φ ∗ (s)s p q 1 + log(1/s) Lq (0,1) and 1 1 α − L(p,q;α) (S) = φ ∗∗ (s)s p q 1 + log(1/s) Lq (0,1) , respectively, are finite. Note that L(p,q;α) (S) = Lp,q;α (S) if and only if p > 1. Furthermore, ∞,∞;− 1
(∞,∞;− 1 )
p,p; α
p (S) β (S) = L β (S) for every β > 0, and Lp (log L)α (S) = L exp Lβ (S) = L if either p > 1 and α ∈ R, or p = 1 and α 0 (up to equivalent norms). We recall that Lp,q;α (S) is an r.i. space (up to equivalent norms) if and only if one of the following conditions is satisfied:
⎧ p = q = 1, α 0, ⎪ ⎪ ⎪ ⎪ ⎨ 1 < p < ∞, 1 q ∞, ⎪ p = ∞, 1 q < ∞, ⎪ ⎪ ⎪ ⎩ p = q = ∞, α 0
α ∈ R, 1 α + < 0, q
(2.13)
(see [7] or [32]). Furthermore, p,q;α (S) = L
⎧ ∞,∞;−α L (S) ⎪ ⎪ ⎨ Lp ,q ;−α (S)
L(1,q ;−α−1) (S) ⎪ ⎪ ⎩ L1,1;−α (S)
if p = q = 1, α 0, if 1 < p < ∞, 1 q ∞, α ∈ R, if p = ∞, 1 q < ∞, α + q1 < 0, if p = q = ∞, α 0,
(2.14)
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up to equivalent norms. Here, and in what follows, we adopt the usual notation p =
∞
p/(p − 1) 1
if p = 1, if 1 < p < ∞, if p = ∞.
We shall also make use of the following characterization of embeddings between Lorentz– Zygmund spaces [7]. Let p, q1 , q2 ∈ [1, ∞] and let α, β ∈ R. Then the embedding Lp,q1 ;α (S) → Lp,q2 ;β (S) holds if and only if one of the following conditions is satisfied: ⎧ 1 1 ⎪ ⎪ 1 q1 q2 ∞, p = ∞, α + β+ , ⎪ ⎪ q1 q2 ⎨ 1 q1 q2 ∞, p < ∞, α β, ⎪ ⎪ 1 1 ⎪ ⎪ ⎩ 1 q2 < q1 ∞, α + >β+ . q1 q2
(2.15)
Let us finally recall that given any quasi-concave, weakly differentiable function ϕ : [0, 1) → [0, ∞) vanishing at 0, and denoting by ϕ its derivative, the spaces Λ1 (ϕ )(S) and Γ ∞ (ϕ)(S) are r.i. spaces (up to equivalent norms), both with fundamental function ϕ. They are the smallest and the largest, respectively, r.i. spaces having this fundamental function. Indeed, if X(S) is any other r.i. space with fundamental function ϕX ≈ ϕ, then Λ1 (ϕ )(S) → X(S) → Γ ∞ (ϕ)(S).
(2.16)
Here and in what follows, the symbol ≈ denotes an equivalence up to multiplicative constants. Because of the first embedding in (2.16), the space Λ1 (ϕ )(S) is usually called the Lorentz endpoint space corresponding to the fundamental function ϕ. If ϕ : (0, 1) → [0, ∞) is the function defined by ϕ(s) =
s ϕ(s)
and lims→0+ ϕ(s) = 0, then 1 Λ (ϕ ) (S) = Γ ∞ (ϕ)(S),
for s ∈ (0, 1),
∞ Γ (ϕ) (S) = Λ1 (ϕ )(S),
(2.17)
up to equivalent norms. 3. Symmetrization and reduction results The reduction theorem for the Sobolev inequality (1.3) reads as follows. Theorem 3.1. Let X(Rn , γn ) and Y (Rn , γn ) be r.i. spaces. (i) If u ∈ V 1 X(Rn , γn ), then u ∈ L1 (Rn , γn ), and, in particular, its mean value uγn is well defined.
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(ii) A constant C1 exists such that u − uγn Y (Rn ,γn ) C1 ∇uX(Rn ,γn )
(3.1)
for every u ∈ V 1 X(Rn , γn ) if and only if a constant C2 exists such that 1 f (r) dr C2 f X(0,1) 1 r 1 + log Y (0,1) s r
(3.2)
for every f ∈ X(0, 1). Moreover, C1 and C2 depend only on each other. The proof of Theorem 3.1 relies upon the Pólya–Szegö principle for the Gaussian symmetrization with arbitrary r.i. norms contained in Theorem 3.2 below. Its statement involves the Gaussian symmetral u• : Rn → R of a measurable function u in Rn defined as u• (x) = u◦ Φ(x1 )
for x = (x1 , . . . , xn ) ∈ Rn ,
(3.3)
where Φ : R → (0, 1) is the function given by 1 Φ(t) = √ 2π
∞
τ2
e− 2 dτ
for t ∈ R.
(3.4)
t
Note that, actually, u ∼ u• , since u ∼ u◦ , and Φ(t) = γn x ∈ Rn : x1 t
for t ∈ R.
(3.5)
An equivalent formulation of the Pólya–Szegö principle can be given in terms of u◦ and of the isoperimetric function of the Gauss space I : (0, 1) → (0, ∞) defined by Φ −1 (s)2 1 I (s) = √ e− 2 2π
for s ∈ (0, 1),
(3.6)
and I (0) = I (1) = 0. The function I owes its name to the fact that the isoperimetric inequality in the Gauss space reads Pγn (E) I γn (E)
(3.7)
for every measurable set E ⊂ Rn [11], with equality if and only if E is (equivalent to) a halfspace [12] (see also [17]). Here, Pγn (E) =
1 (2π)
n 2
∂M E
e−
|x|2 2
dHn−1 (x),
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the Gaussian perimeter of E, where ∂ M E stands for the essential boundary of E (in the sense of geometric measure theory), and Hn−1 denotes the (n − 1)-dimensional Hausdorff measure. Note that the function I is increasing in [0, 12 ], and fulfils I (s) = I (1 − s)
for s ∈ [0, 1].
(3.8)
Moreover, I (s) ≈ s 1 + log
1 s
for s ∈ (0, 1/2],
(3.9)
with absolute equivalence constants. Theorem 3.2. Let u ∈ V 1 L1 (Rn , γn ). Then u◦ is locally absolutely continuous in (0, 1). Moreover, if X(Rn , γn ) is any r.i. space and u ∈ V 1 X(Rn , γn ), then u• ∈ V 1 X(Rn , γn ) and ∇uX(Rn ,γn ) ∇u• X(Rn ,γn ) = I (s) −u◦ (s) X(0,1) . (3.10) Theorem 3.2 is a straightforward consequence of the following lemma and of property (2.2). Lemma 3.3. Let u ∈ V 1 L(Rn , γn ). Then u◦ is locally absolutely continuous in (0, 1), and ∗∗ I (·) −u◦ (·) (s) |∇u|∗∗ (s) for s ∈ (0, 1). (3.11) Proof. The present proof is reminiscent of arguments from [39] and [16, Theorem 6.5 and Lemma 6.6]. Let {(ak , bk )}k∈K be a countable family of disjoint intervals (ak , bk ) ⊂ (0, 1). We have that 2 ∇u(x) dγn (x) = 1 n ∇u(x)e− x2 dx (2π) 2 k∈K {u
◦ (b )
k∈K {u
◦ (b )
◦
=
u (ak )
1 (2π)
n 2
e−
x2 2
dHn−1 (x) dt
k∈Ku◦ (b ) M k ∂ {u>t}
◦
u (ak ) = Pγn {u > t} dt k∈Ku◦ (b ) k ◦
u (ak ) I γn {u > t} dt,
(3.12)
k∈Ku◦ (b ) k
where the second equality holds thanks to the coarea formula and the inequality is a consequence of the isoperimetric inequality (3.7). Now, let 0 < σ < 12 and assume that [ak , bk ] ⊂ [σ, 1 − σ ] for every k ∈ K. Since ak γn {u > t} bk for t ∈ u◦ (bk ), u◦ (ak ) ,
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inequality (3.12) entails that
k∈K {u
∇u(x) dγn (x) I (σ ) u◦ (ak ) − u◦ (bk ) .
(3.13)
k∈K
◦ (b )
On the other hand, γn
◦ u (bk ) < u < u◦ (ak ) = γn u◦ (bk ) < u < u◦ (ak ) (bk − ak ).
k∈K
k∈K
(3.14)
k∈K
Thus, inequality (3.13) yields, via the Hardy–Littlewood inequality (2.3),
u◦ (ak ) − u◦ (bk ) k∈K
k∈K (bk −ak )
1 I (σ )
|∇u|∗ (r) dr.
(3.15)
0
Owing to the arbitrariness of σ , the local absolute continuity of u◦ on (0, 1) follows, since |∇u|∗ ∈ L1 (0, 1). In order to prove (3.11), observe that ◦ (a ) u k
I γn {u > t} dt =
u◦ (bk )
◦ γn ({u>u (bk )})
◦ I γn u > u◦ (r) −u (r) dr
γn ({u>u◦ (ak )})
bk =
I (r) −u◦ (r) dr
for k ∈ K,
(3.16)
ak
where the first equality is a consequence of the (local) absolute continuity of u◦ , and the second one holds since γn ({u > u◦ (r)}) = r if r does not belong to an interval where u◦ is constant and u◦ vanishes in any such interval. From (3.12), (3.16) and the Hardy–Littlewood inequality again, we deduce that, for any family of disjoint intervals {(ak , bk )}k∈K with (ak , bk ) ⊂ (0, 1),
I (r) −u◦ (r) dr
k∈K (bk −ak )
|∇u|∗ (r) dr.
(3.17)
0
k∈K (ak ,bk )
Since each open set in R is a countable union of disjoint open intervals, inequality (3.17) implies that E
I (r) −u◦ (r) dr
|E| |∇u|∗ (r) dr 0
(3.18)
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for every open set E ⊂ (0, 1). In particular, inequality (3.18) tells us that the function I (r)(−u◦ (r)) is integrable on (0, 1). Thanks to the fact that any measurable set can be approximated from outside by open sets, and thanks to the absolute continuity of the Lebesgue integral, inequality (3.18) continues to hold for any measurable set E ⊂ (0, 1). Hence, (3.11) follows, since s
∗ I (·) −u◦ (·) (r) dr = sup
|E|=s
0
for s ∈ (0, 1).
I (r) −u◦ (r) dr
E
2
We are now in a position to prove Theorem 3.1. Proof of Theorem 3.1. (i) By (2.8) and (3.10), it suffices to show that ◦ u (s) − u◦ (1/2) 1 C I (s) −u◦ (s) L1 (0,1) L (0,1)
(3.19)
for some absolute constant C. By Lemma 3.3, u◦ is locally absolutely continuous in (0, 1). Thus, 1
u◦ (s) − u◦ (1/2) =
2
−u◦ (r) dr
for s ∈ (0, 1),
(3.20)
s
and hence 1 1 2 ◦ = −u (r) dr ds L1 (0,1)
◦ u (s) − u◦ (1/2)
s
0
1 2
=
r −u◦ (r) dr +
0
1
(1 − r) −u◦ (r) dr
1 2
12 C 0
1 =C
I (r) −u◦ (r) dr +
1
I (1 − r) −u◦ (r) dr
1 2
I (r) −u◦ (r) dr = C I (r) −u◦ (r) L1 (0,1)
0
for some absolute constant C, where the inequality holds owing to (3.9) and the last but one equality owing to (3.8). Hence, (3.19) follows. (ii) Let us first prove that (3.2) implies (3.1). One has that u − uγn Y (Rn ,γn ) 2u − aY (Rn ,γn )
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for any a ∈ R. Thus, inequality (3.1) will follow if we show that u − u◦ (1/2)
Y (Rn ,γn )
C∇uX(Rn ,γn )
(3.21)
for some constant C = C(C2 ) and for every u ∈ V 1 X(Rn , γn ). By (3.20) and (3.10), inequality (3.21) is in turn reduced to proving that 12 f (r) dr s
C I (s)f (s)X(0,1)
(3.22)
Y (0,1)
for some positive constant C = C(C2 ) and for every f ∈ X(0, 1). By (3.2) applied to f (s) re placed by χ(0, 1 ) (s)s 1 + log 1s f (s), and by (3.9), one has that 2
1 2 χ(0, 1 ) (s) f (r) dr 2
s
Y (0,1)
1 C2 χ(0, 1 ) (s)s 1 + log f (s) 2 s C χ
(0, 12 )
(s)I (s)f (s)
X(0,1)
X(0,1)
(3.23)
for some constant C = C(C2 ) and for every f ∈ X(0, 1). On the other hand, for any such f , 1 2 χ( 1 ,1) (s) f (r) dr 2
s
Y (0,1)
1 2 = χ( 1 ,1) (s) f (1 − r) dr 2
1−s
Y (0,1)
1 2 = χ( 1 ,1) (1 − s) f (1 − r) dr 2
s
1 2 = χ(0, 1 ) (s) f (1 − r) dr 2
s
C χ
since · Y is an r.i. norm Y (0,1)
Y (0,1)
(0, 12 ) (s)I (s)f (1 − s) X(0,1) = C χ 1 (1 − s)I (1 − s)f (s) (0, 2 )
= C χ
( 12 ,1) (s)I (s)f (s) X(0,1)
by (3.23) since · X is an r.i. norm X(0,1) by (3.8) .
Combining (3.23) and (3.24) yields (3.22), and hence (3.2).
(3.24)
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Let us now prove that (3.2) implies (3.1). Given any locally integrable function f : (0, 1) → [0, ∞) such that f (s) = f (1 − s)
for s ∈ (0, 1),
(3.25)
define v : Rn → R by 1
2
f (r) dr I (r)
v(x) =
for x ∈ Rn ,
(3.26)
Φ(x1 )
where Φ : R → [0, 1] is given by (3.4). Owing to (3.8) and (3.25), vγn = 0.
(3.27)
Moreover, by (3.5), we have that 1
v ◦ (s) =
2
f (r) dr I (r)
for s ∈ (0, 1).
(3.28)
s
On the other hand, x2
1 e− 2 ∇v(x) = √1 f Φ(x1 ) = f Φ(x1 ) 2π I (Φ(x1 ))
for a.e. x ∈ Rn ,
(3.29)
where the last equality holds thanks to (3.6). Eq. (3.29) implies that |∇v|∗ (s) = f ∗ (s)
for s ∈ (0, 1).
(3.30)
Owing to (3.27), (3.28) and (3.30), inequality (3.1) applied to u = v implies that 12 f (r) dr I (r) s
C1 f X(0,1)
(3.31)
Y (0,1)
for every f as above. Hence, it is easily seen that 1 2 χ(0, 1 ) (s) f (r) dr 2
s
2C1 χ(0, 1 ) (s)I (s)f (s)X(0,1) 2
Y (0,1)
for every locally integrable function f : (0, 1) → [0, ∞). Now, for any such f ,
(3.32)
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1 f (r) dr s
Y (0,1)
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1 f (r)χ(0, 1 ) (r) dr 2
1 + f (r)χ( 1 ,1) (r) dr 2
1 2 = χ(0, 1 ) (s) f (r) dr 2
1 + f (r)χ( 1 ,1) (r) dr 2
Y (0,1)
s
s
2C1 χ
(0, 12 )
Y (0,1)
(s)I (s)f (s)
s
s
Y (0,1)
Y (0,1)
1 X(0,1)
+
f (r) dr1Y (0,1) ,
(3.33)
1 2
where the last inequality holds thanks to (3.32). Since an absolute constant C exists such that
1 χ(0, 1 ) (s)I (s) Cs 1 + log 2 s
and χ( 1 ,1) (s) Cs 1 + log 2
1 s
for s ∈ (0, 1),
the rightmost side of (3.33) does not exceed 1 s 2C1 C + C1X (0,1) 1Y (0,1) 1 + log . f (s) s X(0,1) Notice that here we have made use of inequality (2.5). Thus, (3.2) follows.
2
4. Optimal range and optimal domain in the Gaussian Sobolev inequality Let X(Rn , γn ) and Y (Rn , γn ) be r.i. spaces. We say that Y (Rn , γn ) is the optimal range for X(Rn , γn ) in the Gaussian Sobolev inequality (1.3) if: i) inequality (1.3) holds; ii) if Z(Rn , γn ) is an r.i. space such that (1.3) holds with Y (Rn , γn ) replaced by Z(Rn , γn ), then Y (Rn , γn ) → Z(Rn , γn ). Analogously, the space X(Rn , γn ) is said to be the optimal domain for Y (Rn , γn ) in the Gaussian Sobolev inequality (1.3) if: i) inequality (1.3) holds; ii) if Z(Rn , γn ) is an r.i. space such that (1.3) holds with X(Rn , γn ) replaced by Z(Rn , γn ), then Z(Rn , γn ) → X(Rn , γn ). Finally, we say that (X(Rn , γn ), Y (Rn , γn )) is an optimal pair in the Gaussian Sobolev inequality (1.3) if Y (Rn , γn ) is the optimal range for X(Rn , γn ) and, simultaneously, X(Rn , γn ) is the optimal domain for Y (Rn , γn ). The optimal range in the Gaussian Sobolev inequality (1.3) for a given domain is characterized in the following theorem.
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Theorem 4.1. Let X(Rn , γn ) be an r.i. space, and let Y (Rn , γn ) be the r.i. space whose associate norm is given by u∗∗ (s) uY (Rn ,γn ) = (4.1) 1 X (0,1) 1 + log s for any u ∈ M(Rn ). Then Y (Rn , γn ) is the optimal range for X(Rn , γn ) in the Gaussian Sobolev inequality (1.3). Our discussion of the optimal domain in (1.3) starts with the following lemma, where a somewhat implicit description is provided for any admissible range Y (Rn , γn ) fulfilling 1 exp L2 Rn , γn → Y Rn , γn → L(log L) 2 Rn , γn .
(4.2)
Note that assumption (4.2) is natural in this setting, since, as anticipated above, exp L2 (Rn , γn ) is the optimal range corresponding to the smallest possible domain L∞ (Rn , γn ), and 1 L(log L) 2 (Rn , γn ) is the optimal range corresponding to the largest possible domain L1 (Rn , γn ) (see Proposition 4.4 below). Lemma 4.2. Let Y (Rn , γn ) be an r.i. space satisfying (4.2). Define 1 h(r) uX(Rn ,γn ) = sup dr 1 0h∼u Y (0,1) s r 1 + log r
(4.3)
for any u ∈ M(Rn ), and let X(Rn , γn ) be the set of all u ∈ M(Rn ) such that uX(Rn ,γn ) < ∞. Then · X(Rn ,γn ) is an r.i. norm, and hence X(Rn , γn ) is an r.i. space equipped with this norm. Moreover, X(Rn , γn ) is the optimal domain for Y (Rn , γn ) in the Gaussian Sobolev embedding (1.3). A more explicit characterization of the optimal domain in (1.3) is given in the next theorem under a slight strengthening of the second embedding in (4.2), which holds in customary situations. It amounts to a boundedness property of the supremum-type Hardy operator T defined for f ∈ M(0, 1) as Tf (s) =
1 + log
f ∗ (r) 1 sup s sr1 1 + log 1
for s ∈ (0, 1).
(4.4)
r
Theorem 4.3. Let Y (Rn , γn ) be an r.i. space such that exp L2 Rn , γn → Y Rn , γn
(4.5)
T is bounded on Y (0, 1).
(4.6)
and
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Then (4.2) holds, and the optimal domain X(Rn , γn ) for Y (Rn , γn ) in the Gaussian Sobolev inequality (1.3) fulfils 1 u∗ (r) dr uX(Rn ,γn ) ≈ 1 Y (0,1) s r 1 + log r
(4.7)
for u ∈ M(Rn ), with absolute equivalence constants. An application of Theorems 4.1 and 4.3, combined with rather standard Hardy-type inequalities, leads to the following result, dealing with some basic examples. Deeper conclusions derived via Theorems 3.1, 4.1 and 4.3, concerning new sharp Sobolev embeddings, are presented in the last three sections. Proposition 4.4. (i) Let p ∈ [1, ∞). Then a constant C = C(p) exists such that u − uγn
p
Lp (log L) 2 (Rn ,γn )
C∇uLp (Rn ,γn )
(4.8)
p
for every u ∈ V 1 Lp (Rn , γn ). Moreover, (Lp (Rn , γn ), Lp (log L) 2 (Rn , γn )) is an optimal pair in (4.8). (ii) An absolute constant C exists such that u − uγn exp L2 (Rn ,γn ) C∇uL∞ (Rn ,γn )
(4.9)
for every u ∈ V 1 L∞ (Rn , γn ). Moreover, (L∞ (Rn , γn ), exp L2 (Rn , γn )) is an optimal pair in (4.9). (iii) Let β ∈ (0, ∞). Then, a constant C = C(β) exists such that u − uγn
2β
exp L 2+β (Rn ,γn )
C∇uexp Lβ (Rn ,γn )
(4.10)
2β
for every u ∈ V 1 exp Lβ (Rn , γn ). Moreover, (exp Lβ (Rn , γn ), exp L 2+β (Rn , γn )) is an optimal pair in (4.10). The examples contained in Proposition 4.4 demonstrate the interesting phenomenon to which we alluded in Section 1: while there is a gain in integrability when the domain is a Lebesgue space, there is actually a loss in integrability when the domain is close to L∞ (Rn , γn ) (observe 2β < β when β > 0). that 2+β We note that the embeddings (4.8)–(4.10) considered in Proposition 4.4 are well known. Our contribution consists in the proof of their optimality. In particular, it follows that in Gross’s original result as well as in its later generalizations, both the range and the domain were already sharp in the broad context of r.i. spaces. Our first concern is to establish Theorem 4.1, Lemma 4.2 and Theorem 4.3; the proof of Proposition 4.4 is postponed to the end of this section.
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Proof of Theorem 4.1. First, we claim that the functional defined by (4.1) actually defines an r.i. norm on (Rn , γn ). To prove this claim, it suffices to show that the functional given by f ∗∗ (s) 1 X (0,1) 1 + log s for any f ∈ M(0, 1) defines an r.i. norm on (0, 1). The positive homogeneity and nontriviality are clear. The triangle inequality follows from the subadditivity of the operation f → f ∗∗ . Properties (P1) and (P2) are satisfied thanks to standard properties of the decreasing rearrangement (see [8]). Property (P3) is a straightforward consequence of the same property for X (0, 1). From property (P4) for X (0, 1), we get that 1 f ∗∗ (s) 1 ds ∗∗ ∗∗ f (1) Cf (1) 1 X (0,1) 1 X (0,1) 1 + log s 1 + log s 1 + log 1s 0 C f L1 (0,1) , for suitable constants C and C depending on X. This proves (P4). Since (P5) is obvious, our claim follows. Owing to Theorem 3.1, the Gaussian Sobolev inequality (1.3) holds with Y (Rn , γn ) as in the statement if (and only if) 1 f (r) dr Cf X(0,1) 1 Y (0,1) s r 1 + log r
(4.11)
for some constant C and every f ∈ X(0, 1). By the very definition of the associate norm and by Fubini’s theorem, we have that 1 1 1 f (r) |f (r)| ∗ dr = sup sup g (s) dr ds sup 1 1 f X(0,1) 1 f X(0,1) 1 gY (0,1) 1 r 1 + log Y (0,1) s r 1 + log r s r 0 1 =
sup
sup
gY (0,1) 1 f X(0,1) 1
0
g ∗∗ (r) = sup gY (0,1) 1 1 + log 1r
∗∗ f (r) g (r) dr 1 + log 1r = 1. X (0,1)
(4.12)
Note that the last equality holds by the definition of the norm in · Y (0,1) . Hence, (4.11) follows. It remains to show that Y (Rn , γn ) is the optimal range for X(Rn , γn ). To this purpose, suppose that Z(Rn , γn ) is another r.i. space such that u − uγn Z(Rn ,γn ) C∇uX(Rn ,γn )
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for some constant C and every u ∈ V 1 X(Rn , γn ). By Theorem 3.1 again, this is equivalent to 1 f (r) dr Cf X(0,1) 1 r 1 + log Z(0,1) s r for some constant C and every f ∈ X(0, 1). Via a chain analogous to (4.12), one can deduce from this inequality that g ∗∗ (r) CgZ (0,1) 1 X (0,1) 1 + log r for every g ∈ Z (0, 1). The last inequality is equivalent to the embedding Z (0, 1) → Y (0, 1), which is in turn equivalent to Y (Rn , γn ) → Z(Rn , γn ). This shows the optimality of Y (Rn , γn ). 2 Let us now come to the proofs concerning the optimal domain in (1.3). We begin with Lemma 4.2. Proof of Lemma 4.2. We shall prove that the functional uX(Rn ,γn ) is an r.i. norm. The fact that X(Rn , γn ) is the optimal domain for Y (Rn , γn ) in the Gaussian Sobolev embedding (1.3) will then immediately follow via Theorem 3.1. It suffices to show that the functional f X(0,1) defined for any function f ∈ M(0, 1) as 1 h(r) f X(0,1) = sup dr , 1 0h∼f r 1 + log Y (0,1) s r is an r.i. norm. We begin by showing that · X(0,1) is actually a norm; we shall then prove that it fulfils properties (P1)–(P5) of r.i. norms. The only nontrivial property of norms to be verified for · X(0,1) is the triangle inequality. To this purpose, let us first observe that, if f, g ∈ M+ (0, 1) and f g a.e. in (0, 1), then f X(0,1) gX(0,1) .
(4.13)
Indeed, by [8, Chapter 2, Corollary 7.6], for any nonnegative function h ∼ f there exists a measure-preserving map H : (0, 1) → (0, 1) such that h = h∗ ◦ H = f ∗ ◦ H . Since f ∗ g ∗ in (0, 1), h g ∗ ◦ H ∼ g, where the equimeasurability of the last two functions holds owing to [8, Chapter 2, Proposition 7.2]. Hence, (4.13) follows. Next, it is not difficult to show that for any simple functions f , g and h in (0, 1) such that h ∼ f + g, there exist (simple) functions hf and hg such that hf ∼ f,
hg ∼ g
and h = hf + hg .
(4.14)
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Now, let f, g ∈ M(0, 1). By a standard result of measure theory there exist sequences of nonnegative simple functions {fk } and {gk } such that fk |f | and gk |g|
as k → ∞.
(4.15)
In particular, ∗ lim (fk + gk )∗ = |f | + |g|
k→∞
in (0, 1).
(4.16)
Given any h ∈ M+ (0, 1) such that h ∼ |f | + |g|, there exists a measure-preserving map H such that ∗ h = h∗ ◦ H = |f | + |g| ◦ H. Define the sequence {hk } by hk = (fk + gk )∗ ◦ H
for k ∈ N.
Thus, hk ∼ fk + gk
for k ∈ N,
and lim hk = h
in (0, 1),
k→∞
by (4.16). Moreover, ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ h∗∗ k (s) = (fk + gk ) (s) fk (s) + gk (s) f (s) + g (s)
for s ∈ (0, 1),
(4.17)
for k ∈ N. By (4.17), the functions hk are equiintegrable in (0, 1), and since the function 1 is bounded in (s, 1) for every s ∈ (0, 1), the functions hk (r) 1 are equiintegrable 1 r 1+log
r 1+log
r
r
in (s, 1) as well. Consequently, 1
lim
k→∞ s
hk (r)
r 1 + log 1r
1
dr = s
h(r)
r 1 + log 1r
dr
for s ∈ (0, 1).
(4.18)
From (4.18), via the Fatou property of r.i. norms [8, Theorem 1.7, Chapter 1], we deduce that 1 1 h(r) hk (r) dr lim inf dr . k→∞ 1 1 r 1 + log r 1 + log Y (0,1) Y (0,1) s s r r
(4.19)
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3609
By property (4.14), there exist two sequences of functions {hfk } and {hgk } such that hfk ∼ fk ,
hgk ∼ gk
and hk = hfk + hgk
for k ∈ N.
On the other hand, there exist two sequences of measure-preserving maps {Hfk } and {Hgk } such that hfk = (hfk )∗ ◦ Hfk = fk∗ ◦ Hfk f ∗ ◦ Hfk ∼ f ∗
for k ∈ N,
and similarly for gk . Therefore, 1 1 1 hfk (r) hgk (r) hk (r) dr dr + dr 1 1 1 Y (0,1) Y (0,1) Y (0,1) s r 1 + log r s r 1 + log r s r 1 + log r 1 1 f ∗ ◦ H (r) g ∗ ◦ H (r) fk gk dr + dr 1 1 Y (0,1) Y (0,1) s r 1 + log r s r 1 + log r f X(0,1) + gX(0,1) ,
(4.20)
for k ∈ N. Since, by (4.13), f + gX(0,1) |f | + |g|X(0,1) , we get from (4.19) and (4.20) that f + gX(0,1) f X(0,1) + gX(0,1) . The triangle inequality for · X(0,1) is thus established. We now pass to the proof of properties (P1)–(P5). The lattice property (P1) is a cosequence of (4.13). As for property (P2), suppose that {fk } is a sequence in M+ (0, 1) such that fk f a.e. in (0, 1). By (4.13), we have that fk X(0,1) fk+1 X(0,1) for k ∈ N. Furthermore, if h is any function such that h ∼ f , then h = f ∗ ◦ H for some measure-preserving transformation H . Consequently, we have that fk ∼ fk∗ ◦ H f ∗ ◦ H = h ∼ f for k ∈ N, whence fk X(0,1)
f X(0,1) . To prove (P3), note that, by (4.2), 1 1X(0,1) C 1 + log s
1 C 1 + log s
Y (0,1)
exp L2 (0,1)
< ∞,
for some absolute constant C and for some constant C = C (Y ). Finally, by (4.2) again, 1 1 |f (r)| |f (r)| f X(0,1) dr C dr C f L1 (0,1) , 1 1 1 Y (0,1) L(log L) 2 (0,1) s r 1 + log r s r 1 + log r for some constants C = C(Y ) and C = C (Y ) and for every f ∈ X(0, 1). This establishes (P4). Since (P5) is obvious, the proof is complete. 2
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Theorem 4.3 will follow from Lemma 4.2, via the next result. Lemma 4.5. Let Y (0, 1) and Z(0, 1) be r.i. spaces. Assume that the operator T satisfies T : Y (0, 1) → Z (0, 1).
(4.21)
Then there exists a constant C = C(Y, Z) such that 1 1 f (r) f ∗ (r) dr C dr 1 1 Y (0,1) Z(0,1) s r 1 + log r s r 1 + log r
(4.22)
for every f ∈ M+ (0, 1). Proof. The conclusion is a consequence of the following chain, which holds for every f ∈ M+ (0, 1): 1 1 1 f (r) f (r) ∗ dr = sup g (s) dr ds 1 1 g (0,1) 1 Y r 1 + log r 1 + log Y (0,1) s s r r 0 1 =
sup
gY (0,1) 1
r 1 + log
0
1 =
f (r)
sup
gY (0,1) 1
0
1
sup
gY (0,1) 1
0
C
C
gY (0,1) 1
0
0
1 sup
hZ (0,1) 1
∗
1 r
f ∗ (r) 1 + log
(T g)∗∗ (r) dr 1 r
1
h (r) 0
(T g)∗∗ (r) dr
1 + log
sup
T gZ (0,1) 1
(T g)∗∗ (r) dr 1 r
f ∗ (r)
1
dr
1 + log 1r
1 + log
sup
g ∗ (s) ds dr
1 r 0
g ∗∗ (r)
f (r)
1 C
r
f (r)
r
f ∗ (s)
s 1 + log 1s
1 f ∗ (s) =C ds , 1 s 1 + log Z(0,1) r s
ds dr
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3611
for some absolute constant C and for some constant C = C (Y, Z). Note that the first inequality in this chain holds since, trivially, g ∗ T g for every g ∈ M(0, 1). The second inequality follows via the Hardy–Littlewood inequality (2.3), since, for any function g ∈ M(0, 1), r (T g)∗∗ (r)
1 + log 1r
0
∗
g (ρ) 1+log ρ1
1 + log 1s supsρ1 r
≈
0
ds
1 + log 1s ds
for r ∈ (0, 1),
with absolute equivalence constants, and the latter isan integral mean of a non-increasing function supsρ1
∗
g (ρ) 1+log ρ1
with respect to the measure
1 + log 1s ds over (0, r), whence it is itself
non-increasing in r (actually, this is the key reason for employing the operator T ). The third inequality follows from (4.21). 2 Proof of Theorem 4.3. We begin by showing that assumption (4.6) implies the second em bedding in (4.2). By property (P3) for Y (0, 1), the constant function f (s) = 1 belongs to
Y (0, 1). By (4.6), Tf (s) =
1 + log 1s ∈ Y (0, 1). This membership is equivalent to the embed-
ding exp L2 (Rn , γn ) → Y (Rn , γn ), and the latter is in turn equivalent to the second embedding in (4.2). As far as equivalence (4.7) is concerned, one trivially has 1 f ∗ (r) dr f X(0,1) 1 r 1 + log Y (0,1) s r for any f ∈ M(0, 1). Conversely, by (4.6) and Lemma 4.5 applied to the case when Y (0, 1) = Y (0, 1) = Z(0, 1) = Z(0, 1), we obtain that 1 f ∗ (r) dr f X(0,1) C 1 Y (0,1) s r 1 + log r for some constant C = C(X, Y ) and for every f ∈ M(0, 1). The proof is complete.
2
In the proof of Proposition 4.4, and in other proofs below, we shall make use of the following characterization of a weighted Hardy inequality established in [30] and [29, Section 1.3.1, Theorem 2]. Proposition 4.6. Let 1 p ∞ and let ν and ω ∈ M+ (0, 1). (i) There exists a constant C such that s ω(s) f (r) dr 0
Cνf Lp (0,1) Lp (0,1)
(4.23)
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for every f ∈ M+ (0, 1) if and only if χ(0,s) < ∞. sup ωχ(s,1) Lp (0,1) ν p 0<s<1 L (0,1)
(4.24)
Moreover, the best constant C in (4.23) is equivalent to the left-hand side of (4.24), up to constants depending on p. (ii) There exists a constant C such that 1 ω(s) f (r) dr s
Cνf Lp (0,1)
(4.25)
Lp (0,1)
for every f ∈ M+ (0, 1) if and only if χ(s,1) < ∞. sup ωχ(0,s) Lp (0,1) ν Lp (0,1) 0<s<1
(4.26)
Moreover, the best constant C in (4.25) is equivalent to the left-hand side of (4.26), up to constants depending on p. Expressions having the form sup sr1
f ∗ (r)
and
1 + log 1r
sup sr1
f ∗∗ (r)
,
1 + log 1r
where f ∈ M(0, 1), will come into play as well. In particular, the following proposition, a special case of a more general result in [24, Theorems 3.2 and 3.5], will be needed. Proposition 4.7. Let p ∈ [1, ∞), and let ν, ω ∈ M+ (0, 1). (i) There exists a constant C such that 1 sup 0
sr1
p
f ∗ (r) 1 + log
1 ω(s) ds C
1 r
f ∗ (s)p ν(s) ds
(4.27)
0
for every f ∈ M(0, 1), if and only if s
ω(r) dr < ∞. 1 p s 0<s<1 (1 + log s ) 2 0 ν(r) dr sup
0
(4.28)
Moreover, the best constant C in (4.27) is equivalent to the left-hand side of (4.28), up to constants depending on p.
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3613
(ii) There exists a constant C such that 1 sup sr1
0
p
f ∗∗ (r) 1 + log
1 ω(s) ds C
1 r
f ∗ (s)p ν(s) ds
(4.29)
0
for every f ∈ M(0, 1), if and only if either p = 1 and s
1 s
sup 0<s<1
ω(r) r 1+log 1r s 0 ν(r) dr
dr < ∞,
(4.30)
or 1 < p < ∞, (4.28) holds, and 1
sup 0<s<1
1 s
p
1
p
ω(r) dr
r 1 + log 1r
s
0
r r ν(ρ) dρ 0
1
p
p
ν(r) dr
< ∞.
(4.31)
Moreover, the best constant C in (4.29) is equivalent to the left-hand side of (4.30) if p = 1, and to the sum of the left-hand sides of (4.28) and (4.31) if 1 < p < ∞, up to constants depending only on p. We next establish some results concerning the operator T to be used in our proofs. Lemma 4.8. There exists an absolute constant C such that (Tf )∗∗ (s) CT (f ∗∗ )(s)
for s ∈ (0, 1),
(4.32)
for every f ∈ M(0, 1). Proof. Let f ∈ M(0, 1). We will show that (Tf )∗∗ (s) C Tf (s) + f ∗∗ (s)
for s ∈ (0, 1),
(4.33)
for some absolute constant C, whence (4.32) obviously follows. To verify (4.33), note that s 0
s f ∗ (ρ) 1 sup (Tf ) (r) dr = 1 + log dr r rρ1 1 + log 1 ∗
ρ
0
s
1 + log 0
f ∗ (ρ) 1 sup dr r rρs 1 + log 1 ρ
s f ∗ (ρ) 1 sup + 1 + log dr r sρ1 1 + log 1 0
ρ
for s ∈ (0, 1).
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We have that s 0
f ∗ (ρ) f ∗ (ρ) 1 1 1 + log dr Cs 1 + log sup sup r sρ1 1 + log 1 s sρ1 1 + log 1 ρ
ρ
= Cs(Tf )(s)
for s ∈ (0, 1),
for some absolute constant C. On the other hand, s 0
s ρ f ∗ (ρ) dτ 1 1 sup sup f ∗ (ρ) 1 + log dr C 1 + log dr r rρs 1 + log 1 r rρs 1 + log τ1 ρ τ ρ 0 2
s 1 + log
C 0
s C
1 sup r rρs
1 1 + log r
0
s C 0
C
s
s r 2
f ∗ (τ ) τ 1 + log τ1
ρ ρ 2
f ∗ (τ ) dτ dr τ 1 + log τ1
f ∗ (τ ) dτ dr τ 1 + log τ1
2τ 1 + log
1 dr dτ r
0
f ∗ (τ ) dτ = C sf ∗∗ (s)
for s ∈ (0, 1),
0
for some absolute constants C and C . Combining these estimates yields (4.33).
2
Corollary 4.9. Let p ∈ (0, ∞) and let ω ∈ M+ (0, 1). If the operator T is bounded on Λp (ω)(0, 1), then it is bounded also on Γ p (ω)(0, 1). Proof. Let f ∈ M(0, 1). By (4.32), 1 Tf Γ p (ω)(0,1) =
1 ∗∗
C
(Tf ) (s) ω(s) ds 0
C
1
p
p
1 T (f
∗∗
p
)(s) ω(s) ds
0
1
1 f ∗∗ (s)p ω(s) ds
p
= C f Γ p (ω)(0,1)
0
for some absolute constant C and for some constant C = C (p, ω).
p
2
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3615
Lemma 4.10. Let p, q and α be such that one of the conditions in (2.13) is satisfied. Then the operator T is bounded on the Lorentz–Zygmund space Lp,q;α (0, 1) if and only if one of the following conditions holds: ⎧ p = q = 1, α 0; ⎪ ⎪ ⎪ ⎪ 1 < p < ∞, 1 q ∞, α ∈ R; ⎪ ⎪ ⎨ 1 p = ∞, 1 q < ∞, α+ q < −1; ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ p = q = ∞, α − 1 . 2
(4.34)
Moreover, if one of the conditions in (4.34) holds, then T is bounded also on L(p,q;α) (0, 1). Proof, sketched. The first assertion follows from Proposition 4.7. The second assertion is a straightforward consequence of Corollary 4.9. 2 Remark 4.11. Note that, by Lemma 4.10, the operator T is bounded on every Lebesgue space Lp (0, 1), with p ∈ [1, ∞), on every Zygmund space Lp (log L)α (0, 1), where either p ∈ (1, ∞) and α ∈ R, or p = 1 and α 0, and on every exponential space exp Lβ (0, 1), with β 2. Instead, it is neither bounded on L∞ (0, 1) nor on any exponential space exp Lβ (0, 1), with β > 2. We are now ready to prove Proposition 4.4. Proof of Proposition 4.4. Throughout the proof, f denotes any function in M(0, 1). (i) From Theorem 4.1 and the weighted Hardy inequality (see Proposition 4.6), we have that the optimal range Y (Rn , γn ) for Lp (Rn , γn ) fulfils f ∗∗ (s) f ∗ (s) ≈ , f Y (0,1) = 1 + log 1s Lp (0,1) 1 + log 1s Lp (0,1) with equivalence constants depending on p. Since the function
1 1+log 1s
is increasing on (0, 1),
an application of [21, Theorem 2.7] tells us that f Y (0,1) ≈ f
p
Lp (log L) 2 (0,1)
, p
with equivalence constants depending on p. This proves the optimality of Lp (log L) 2 (Rn , γn ) as a range for Lp (Rn , γn ). p As for the optimality of the domain, let Y (Rn , γn ) = Lp (log L) 2 (Rn , γn ). Assumption (4.5)
p
is clearly satisfied. Moreover, Y (Rn , γn ) = Lp (log L) 2 (Rn , γn ) when p > 1 and Y (Rn , γn ) = exp L2 (Rn , γn ) when p = 1. Thus, in any case, T is bounded on Y (0, 1), by Lemma 4.10,
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and hence assumption (4.6) is also fulfilled. By Theorem 4.3, the norm in the optimal domain X(Rn , γn ) satisfies 1 1 f ∗ (r) f ∗ (r) 1 f X(0,1) ≈ dr ≈ 1 + log dr , p p p s 1 1 r 1 + log r 1 + log 2 L (log L) (0,1) L (0,1) s s r r with equivalence constants depending on p. By the weighted Hardy inequality (Proposition 4.6), 1 f ∗ (r) 1 dr Cf Lp (0,1) , 1 + log p s 1 r 1 + log L (0,1) s r for some constant C = C(p). Conversely, 1 f ∗ (r) 1 dr 1 + log p s 1 L (0,1) s r 1 + log r 2s f ∗ (r) 1 χ(0, 1 ) (s) 1 + log dr 2 p s 1 L (0,1) s r 1 + log r 2s dr 1 ∗ χ(0, 1 ) (s)f (2s) 1 + log 2 s 1 p r 1 + log s r L (0,1) Cf Lp (0,1) , for some positive constant C = C(p). Altogether, Lp (Rn , γn ) is the optimal domain for p Lp (log L) 2 (Rn , γn ). (ii) First, we show that the domain is optimal in (4.9). Let Y (Rn , γn ) = exp L2 (Rn , γn ). Then 1 Y (Rn , γn ) = L(log L) 2 (Rn , γn ), and hence T is bounded on Y (0, 1), by Lemma 4.10. Thus, we can apply Theorem 4.3. One has that 1 1 f ∗ (r) 1 f ∗ (r) dr ≈ sup dr 1 1 0<s<1 1 + log 1 2 (0,1) r 1 + log r 1 + log exp L s r r s r 1
f L∞ (0,1) sup 0<s<1 1 + log 1 s ≈ f L∞ (0,1) , with absolute equivalence constants. Conversely,
1
s
dr
r 1 + log 1r
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3617
1 1 f ∗ (r) 1 f ∗ (r) dr ≈ sup dr 1 1 0<s<1 1 + log 1 2 (0,1) r 1 + log r 1 + log exp L s s s r r lim s→0+
1
1 1 + log 1s
s
f ∗ (r)
dr
r 1 + log 1r
∗
= 2 lim f (s) = 2f L∞ (0,1) . s→0+
By Theorem 4.3, this shows that L∞ (Rn , γn ) is the optimal domain for exp L2 (Rn , γn ). Assume now that Y (Rn , γn ) is the optimal range for L∞ (Rn , γn ). Then, by Theorem 4.1, 1 f ∗∗ (s) 1 ∗ f Y (0,1) = ≈ f (s) 1 + log ds ≈ f , 1 L(log L) 2 (0,1) s 1 + log 1s L1 (0,1) 0 1
with absolute equivalence constants. Since, by (2.14), (exp L2 ) (Rn , γn ) = L(log L) 2 (Rn , γn ), we deduce that Y (Rn , γn ) = exp L2 (Rn , γn ). (iii) By Theorem 4.1, the optimal range Y (Rn , γn ) for the domain exp Lβ (Rn , γn ) satisfies the chain ∗ 1 1 f ∗∗ (s) f ∗∗ (·) 1 β ≈ (s) 1 + log ds f Y (0,1) ≈ 1 1 s 1 + log (·) 1 + log 1s L(log L) β (0,1) 0 1
1 1 1 1 1 β 1 β +2 ∗ 1 + log sup ds C f (s) 1 + log ds s s sr1 1 + log 1 f ∗∗ (r)
r
0
≈ f
L(log L)
2+β 2β
0
, (0,1)
with the equivalence constants and the constant C depending on β. Note that the last inequality follows from Proposition 4.7(ii). Conversely, by the Hardy–Littlewood inequality (2.3) and by Fubini’s theorem, 1
f Y (0,1) ≈ 0
1 = 0
f ∗∗ (·) 1 1 + log (·)
∗
1 1 1 1 β f ∗∗ (s) 1 β 1 + log (s) 1 + log ds ds s s 1 1 + log s 0
1 1 1 1 β −2 1 1 + log f (r) ds dr ≈ f , 2+β s s L(log L) 2β (0,1) ∗
r
with equivalence constants depending on β. Therefore, Y (Rn , γn ) = L(log L) whence, by (2.14), Y (Rn , γn ) = exp L
2β 2+β
(Rn , γn ).
2+β 2β
(Rn , γn ),
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In order to prove that the domain in (4.10) is optimal, set Y (Rn , γn ) = exp L 2+β (Rn , γn ), and note that T is bounded on Y (0, 1). This follows via Proposition 4.7(i), which entails that 1 Tf
2+β L(log L) 2β
(0,1)
sup
≈ 0
sr1
f ∗ (r) 1 + log 1r
1 1 + log s
1 +1 β
ds
2+β 1 2β f (s) 1 + log ds ≈ f , 2+β s L(log L) 2β (0,1)
1
∗
C 0
with the constant C and the equivalence constants depending on β. Therefore, Theorem 4.3 implies that the optimal domain X(Rn , γn ) for Y (Rn , γn ) fulfils 1 1 f ∗ (r) 1 f ∗ (r) dr ≈ sup dr, f X(0,1) ≈ 2+β 2β 1 1 0<s<1 (1 + log 1 ) 2β exp L 2+β (0,1) s r 1 + log r s r 1 + log r s with equivalence constants depending on β. Since f exp Lβ (0,1) ≈ sup
0<s<1
f ∗ (s) 1
(1 + log 1s ) β
,
it suffices to show that 1
1
sup 0<s<1
(1 + log 1s )
2+β 2β
f ∗ (r)
r 1 + log 1r
s
f ∗ (s)
dr ≈ sup 0<s<1
1
(1 + log 1s ) β
,
(4.35)
with equivalence constants depending on β. We have that 1
1
sup 0<s<1
(1 + log 1s )
sup 0<s<1
2+β 2β
1
(1 + log 1s ) β
sup 0<s<1
f ∗ (s)
0<s<1
dr
r 1 + log 1r
s
f ∗ (s)
C sup
f ∗ (r)
1
(1 + log 1s ) β
1
1 (1 + log 1s )
2+β 2β
s
1
(1 + log 1r ) β dr r 1 + log 1r
,
for some constant C = C(β). To prove the converse estimate, we first observe that there exists an absolute positive constant C such that s
s2
dr
r 1 + log 1r
C 1 + log
1 s
for s ∈ (0, 1/2].
(4.36)
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3619
Next, let s0 ∈ (0, 1] be such that f ∗ (s0 ) 1
(1 + log s10 ) β
f ∗ (s) 1 . sup 2 0<s<1 (1 + log 1 ) β1 s
Clearly,
sup 0<s<1
1
1 (1 + log
2β
1 2+β s s)
f ∗ (r)
r 1 + log
1 r
dr sup 0<s<s0
s0
f ∗ (s0 ) (1 + log
2β
1 2+β s s)
dr
.
r 1 + log 1r
We will be done if we show that there exists an absolute constant δ > 0 such that the right-hand side of the last inequality is not smaller than δf ∗ (s0 ) 1
(1 + log s10 ) β
.
This is easily seen when s0 ∈ [ 12 , 1]: one has just to estimate the supremum from below by the value at, say, s = 14 . In the case when s ∈ (0, 12 ] it suffices to estimate the supremum by the value at s = s02 and make use of (4.36). 2 5. Orlicz spaces Here, we establish an optimal Gaussian Sobolev inequality for the Orlicz–Sobolev space V 1 LA (Rn , γn ) associated with a Young function A. We may assume, without loss of generality, that A(t) dt < ∞. t2
(5.1)
0
Actually, A can be replaced, if necessary, by a Young function equivalent near infinity and fulfilling (5.1), without changing LA (Rn , γn ) (up to equivalent norms). Let E : (0, ∞) → [0, ∞) be the (non-decreasing) function obeying 1 E −1 (t) = r 1 + log+
1 r
LA ( 1t ,∞)
for t > 0,
(5.2)
where log+ t = max{log t, 0}. Note that the right-hand side of (5.2) is actually finite for t > 0, owing to (5.1). Define AG : [0, ∞) → [0, ∞) by t AG (t) = 0
E(s) ds s
for t > 0.
(5.3)
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The main result of this section tells us that LAG (Rn , γn ) is the optimal Orlicz space into which V 1 LA (Rn , γn ) is continuously embedded. Theorem 5.1. Let A be a Young function (modified, if necessary, near 0 in such a way that (5.1) is satisfied). Then, an absolute constant C exists such that u − uγn LAG (Rn ,γn ) C∇uLA (Rn ,γn )
(5.4)
for every u ∈ V 1 LA (Rn , γn ). Moreover, LAG (Rn , γn ) is the optimal Orlicz range space in (5.4). A special Orlicz space LA (Rn , γn ) which is self-optimal in the Sobolev inequality (5.4) will be exhibited in Corollary 7.2, Section 7. We split the proof of Theorem 5.1 in some lemmas. We begin by showing that AG is actually a Young function. Lemma 5.2. Let A be a Young function fulfilling (5.1). Then AG is a Young function. Moreover, E(t/2) AG (t) E(t) for t > 0.
(5.5)
Proof. In order to prove that AG is a Young function, it suffices to show that the function non-decreasing in (0, ∞), or, equivalently, that the function 1 t s 1 + log+
E(t) t
is
1 s
LA (t,∞)
is non-decreasing in (0, ∞). We have that 1 t s 1 + log+
1 s
LA (t,∞)
∞
= inf λ 0: A t
λs 1 + log+
∞
= inf λ 0: A 1
t
1
λs 1 + log+
ds 1
1 s
1 ts
t ds 1
for t > 0.
(5.6)
1 Since the function A is non-decreasing in t for each λ and s, the function 1 λs 1+log+ ts t 1 1 LA (t,∞) is non-decreasing as well. s 1+log+
s
As far as (5.5) is concerned, owing to the monotonicity of the function t E(t/2) t 2
E(s) ds AG (t) = s
t 0
E(s) ds E(t) s
E(t) t ,
for t > 0.
one has that
2
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3621
Define the operator R as 1
Rf (s) = s
f (r)
dr
r 1 + log 1r
for s ∈ (0, 1),
(5.7)
for f ∈ M+ (0, 1). Lemma 5.3. Let A be as in Lemma 5.2. Then, {Rf t}AG (t/2)
1
A f (s) ds
for t 0,
(5.8)
0
for every f ∈ M+ (0, 1) such that 1
A f (s) ds 1.
(5.9)
0
Here, and in what follows, | · | denotes the Lebesgue measure on R. Proof. One has, by (2.11), 1
Rf (s) = s
f (r)
r 1 + log 1r
1 dr 2f LA (0,1) r 1 + log 1r LA (s,1) 2f LA (0,1) E −1 (1/s)
for s ∈ (0, 1).
(5.10)
Thus, −1 {Rf t} s: 2f A L (0,1) E (1/s) t =
1 E( 2f t A
)
for t > 0,
(5.11)
L (0,1)
whence {Rf t}E
t 2f LA (0,1)
1 for t > 0.
Next, given any nonnegative number M, define AM : [0, ∞) → [0, ∞) as AM (t) =
A(t) M
for t 0,
(5.12)
3622
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
and denote by EM the function defined as in (5.2) with A replaced by AM . We claim that, if M 1, then 1 E(t) for t > 0. M
EM (t) ! Indeed, it is easily seen that A M (t) =
−1 (1/t) = EM
1 M A(Mt)
1
s 1 + log+
= inf λ 0:
for t 0. Consequently,
1
!
LAM (t,∞)
s
∞
1 + s
t
∞
= inf λ 0: A t M
∞
inf λ 0: A t M
1 = s 1 + log+ = E −1 (M/t)
1 M A M λs 1 + log
(5.13)
1
λs 1 + log+
1 Ms
ds 1
1
λs 1 + log+
ds 1
1 s
ds 1
1
LA ( Mt ,∞)
s
for t > 0,
(5.14)
whence (5.13) follows. Now, choose 1 M=
A f (s) ds.
0
Assumption (5.9) entails that M 1. The very definition of the Luxemburg norm yields that f LAM (0,1) 1.
(5.15)
On applying (5.12) with A replaced by AM and making use of (5.15) and (5.14), one gets that 1 {Rf t}EM
t 2f LAM (0,1)
E( t ) {Rf t} 2 M Combining (5.16) with (5.5) yields (5.8).
for t > 0. 2
t {Rf t} EM 2 (5.16)
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3623
Lemma 5.4. Let A be a Young function as in Lemma 5.2. Then, Rf LAG (0,1) 8f LA (0,1)
(5.17)
for every f ∈ LA (0, 1). Moreover, LAG (0, 1) is the optimal Orlicz space in (5.17), in the sense that if (5.17) holds with AG replaced by any Young function B, then LAG (0, 1) → LB (0, 1). Proof. Let f be any function in M+ (0, 1) satisfying (5.9). Thus, in particular, f ∈ L1 (0, 1), and hence Rf (s) < ∞ for s ∈ (0, 1). One can easily restrict oneself to the case when lims→0+ Rf (s) = ∞. Let {sk }k∈Z be a sequence in (0, 1) such that Rf (sk ) = 2k
for k ∈ Z.
(5.18)
Notice that sk is non-increasing, since so is Rf . Set fk = f χ[sk ,sk−1 )
for k ∈ Z.
Thus, since Rf (s) Rf (sk ) = 2k
1 AG
if s ∈ (sk , sk−1 ),
sk−1 Rf (s) Rf (s) AG ds = ds 8 8 k∈Z sk
0
sk−1 k 2 ds = AG (sk−1 − sk )AG 2k−3 . 8 k∈Z sk
(5.19)
k∈Z
Now, for every k ∈ Z and s ∈ (sk , sk−1 ), one has that 1 R(fk−1 )(s) sk−1
1
1 fk−1 (r) dr = f (r)χ[sk−1 ,sk−2 ) (r) dr 1 r 1 + log 1r r 1 + log sk−1 r
sk−2
= sk−1
1
f (r)
r 1 + log
1 r
dr = Rf (sk−1 ) − Rf (sk−2 ) = 2k−2 .
Consequently, [sk , sk−1 ) ⊂ Rfk−1 2k−2
for k ∈ Z.
(5.20)
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A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
From inclusion (5.20) and Lemma 5.3 we obtain that (sk−1 − sk )AG 2k−3 Rfk−1 2k−2 AG 2k−3 1
A fk−1 (s) ds
for k ∈ Z.
(5.21)
0
Notice that such an application of Lemma 5.3 with f replaced by fk−1 is possible since, by (5.9), 1
1
A fk−1 (s) ds
0
A f (s) ds 1
for k ∈ Z.
0
Combining (5.19) and (5.21) yields
1 AG
1 1 Rf (s) ds A fk−1 (s) ds = A f (s) ds 1. 8 k∈Z 0
0
(5.22)
0
Thus, we have shown that
1 AG
Rf (s) ds 1 8
0
provided that (5.9) is fulfilled. Hence, (5.17) follows. The proof of the sharpness of the space LAG (0, 1) amounts to showing that if B is any Young function such that Rf LB (0,1) Cf LA (0,1)
(5.23)
for some constant C, and for every f ∈ LA (0, 1), then constants C and t1 exist such that B(t) AG (C t) for t t1 .
(5.24)
By a standard argument in the characterization of Hardy type inequalities (see e.g. [20]), one can show that a necessary condition for (5.23) to hold is the existence of a constant C such that 1 C 1LB (0,t) r 1 + log 1r LA (t,1)
for t ∈ (0, 1).
(5.25)
We have that 1LB (0,t) =
1 B −1 ( 1t )
for t > 0.
(5.26)
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
Moreover, there obviously exists a constant C = C(A) such that 1 1 C
s 1 + log+ 1s LA (t,∞) s 1 + log 1s LA (t,1)
for t ∈ (0, 1/2).
3625
(5.27)
From the first inequality in (5.5), (5.26) and (5.27) we get that 1 −1 A (1/t) E −1 (1/t) CB −1 (1/t) 2 G Hence inequality (5.24) follows. The proof is complete.
for t ∈ (0, 1/2).
2
Proof of Theorem 5.1. The conclusions are straightforward consequences of Theorem 3.2 and Lemma 5.4. 2 6. Lorentz–Zygmund spaces In this section we establish the following sharp Gaussian Sobolev inequality for Lorentz and, more generally, Lorentz–Zygmund spaces. Note that, according to (2.13), the conditions on the parameters p, q and α in the statement are required to ensure that Lp,q;α (Rn , γn ) is actually an r.i. space. Theorem 6.1. (i) Assume that either p = q = 1 and α 0, or p ∈ (1, ∞), q ∈ [1, ∞] and α ∈ R. Then, there exists a constant C = C(p, q, α) such that u − uγn
p,q;α+ 21
L
(Rn ,γn )
C∇uLp,q;α (Rn ,γn )
(6.1) 1
for every u ∈ V 1 Lp,q;α (Rn , γn ). Moreover, (Lp,q;α (Rn , γn ), Lp,q;α+ 2 (Rn , γn )) is an optimal pair in (6.1). (ii) Assume that either q ∈ [1, ∞) and α + q1 < 0, or q = ∞ and α 0. Then, there exists a constant C = C(q, α) such that u − uγn
∞,q;α− 12
L
(Rn ,γn )
C∇uL∞,q;α (Rn ,γn )
(6.2) 1
for every u ∈ V 1 L∞,q;α (Rn , γn ). Moreover, (L∞,q;α (Rn , γn ), L∞,q;α− 2 (Rn , γn )) is an optimal pair in (6.2). Observe that also the new embeddings of Theorem 6.1 exhibit the contrast between the gain of integrability from |∇u| to u in Sobolev embeddings for spaces far from V 1 L∞ (Rn , γn ), and the loss of integrability for spaces near V 1 L∞ (Rn , γn ). Proof. Throughout the proof, we denote by f an arbitrary function from M(0, 1). (i) We shall first prove the optimality of the range. Let p = q = 1 and α 0, and let X(Rn , γn ) = L1,1;α (Rn , γn ). Then X (Rn , γn ) = L∞,∞;−α (Rn , γn ), by (2.14). Let Y (Rn , γn ) be the optimal range for X(Rn , γn ) in (1.3). Then, by Theorem 4.1,
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A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
f ∗∗ (s) 1 −α f ∗∗ (r) 1 + log f Y (0,1) ≈ sup sup s s
= sup 0
Cf
1 + log 1r
∞,∞;−α− 12
L
f ∗∗ (r)
sup 0<s
(0,1)
1 + log
1 s
−α
1 1 −α− 2 = sup f ∗∗ (r) 1 + log r 0
,
with equivalence constants and C depending on α. Note that the last inequality holds by Proposition 4.6. Conversely, f ∗∗ (s) f Y (0,1) ≈ 1 L∞,∞;−α (0,1) 1 + log s s 1 −α 1 f ∗∗ (·) C sup 1 + log s s 0<s<1 1 + log 0
∗ 1 (·)
(r) dr
s 1 −α 1 f ∗∗ (r) dr C sup 1 + log s s 1 0<s<1 1 + log r 0 s dr 1 −α 1 C sup f ∗∗ (s) 1 + log s s 0<s<1 1 + log 1r 0 1 1 −α− 2 C sup f ∗∗ (s) 1 + log C f ∞,∞;−α− 1 , 2 (0,1) L s 0<s<1
for some absolute constants C and C . Here, the first inequality is a consequence of Proposition 4.6, and the second one of the Hardy–Littlewood inequality (2.3). Altogether, Y (Rn , γn ) = 1 1 L∞,∞;−α− 2 (Rn , γn ), or, equivalently, Y (Rn , γn ) = L(log L)α+ 2 (Rn , γn ). Now assume that 1 < p < ∞, 1 q ∞ and α ∈ R, and let X(Rn , γn ) = Lp,q;α (Rn , γn ). Then X (Rn , γn ) = Lp ,q ;−α (Rn , γn ). Let Y (Rn , γn ) be the optimal range for X(Rn , γn ) in (1.3). Then, by Theorem 4.1 and Proposition 4.7(i), f ∗∗ (s) 1 − 1 1 −α f ∗∗ (r) p q f Y (0,1) ≈ 1 + log s sup s 1 Lp ,q ;−α (0,1) 1 Lq (0,1) sr1 1 + log s 1 + log r 1 1 − 1 1 −α− 2 ∗∗ p q 1 + log C s f (s) , q s L (0,1)
with equivalence constants and C depending on p, q and α. Conversely,
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3627
∗ −α f ∗∗ (s) 1 − 1 f ∗∗ (·) s p q 1 + log 1 f Y (0,1) ≈ = (s) q 1 s L (0,1) 1 + log (·) 1 + log 1s Lp ,q ;−α (0,1) ∗ f ∗∗ (·) 1 = 1 + log (s) , p ,q ,−α− 1 1 s 2 (0,1) L 1 + log (·) with equivalence constants depending on p, q and α. Notice that the last equality holds since the function s 0
1 + log 1s is non-increasing in (0, 1). By the Hardy–Littlewood inequality (2.3),
∗ s f ∗∗ (·) 1 1 f ∗∗ (r) 1 + log (r) dr 1 + log dr 1 r r 1 + log 1 1 + log (·) r 0 s =
f
∗∗
s (r) dr
0
f ∗ (r) dr
for s ∈ (0, 1).
0
Thus, by (2.2), ∗ ∗∗ 1 + log 1 f (·) (s) f p ,q ;−α− 1 . p ,q ,−α− 1 2 (0,1) L 1 s 2 (0,1) L 1 + log (·)
1
In conclusion, we have shown that Y (Rn , γn ) = Lp ,q ;−α− 2 (Rn , γn ), whence Y (Rn , γn ) = 1 Lp,q;α+ 2 (Rn , γn ). 1 Next, we shall prove the optimality of the domain. Let Y (Rn , γn ) = Lp,q;α+ 2 (Rn , γn ), where either p = q = 1 and α 0, or 1 < p < ∞, 1 q ∞ and α ∈ R. Let X(Rn , γn ) be the optimal domain for Y (Rn , γn ). Assumption (4.5) is clearly satisfied. Moreover, by Lemma 4.10, the operator T is bounded on Y (0, 1), and hence (4.6) is fulfilled as well. Assume first that p = q = 1 and α 0. Then, by Theorem 4.3, 1 1 1 1 f ∗ (r) f ∗ (r) 1 α+ 2 f X(0,1) ≈ dr = dr 1 + log ds 1,1;α+ 1 s 1 1 2 (0,1) L s r 1 + log r 0 s r 1 + log r 1
= 0
f ∗ (r)
r 1 + log 1r
r
1 1 + log s
α+ 1 2
ds dr ≈ f L1,1;α (0,1) ,
0
with equivalence constants depending on α. Next, assume that 1 < p < ∞, 1 q ∞ and α ∈ R. Then, by Theorem 4.3 and by the weighted Hardy inequality (Proposition 4.6), 1 f ∗ (r) dr f X(0,1) ≈ p,q;α+ 1 1 2 (0,1) L s r 1 + log r
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A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
1 1 1 1 f ∗ (r) 1 α+ 2 p−q = s dr 1 + log q s 1 L (0,1) s r 1 + log r α 1−1 1 p q 1 + log C f ∗ (s) = Cf Lp,q;α (0,1) , s q s L (0,1) with equivalence constants and C depending on p, q and α. Conversely, α+ 1 2s ∗ (r) 2 1 1 1 f − dr f X(0,1) C χ(0, 1 ) (s)s p q 1 + log 2 q s 1 L (0,1) s r 1 + log r α+ 1 2s 2 1 1 1 dr − C χ(0, 1 ) (s)s p q 1 + log f ∗ (2s) 2 s 1 q s r 1 + log r L (0,1) ≈ f Lp,q;α (0,1) , with equivalence constants and C depending on p, q and α. (ii) Note that the case when p = q = ∞ and α 0 has been established in Proposition 4.4, cases (ii) and (iii). We thus have only to deal with the remaining case when p = ∞, 1 q < ∞ and α + q1 < 0. Let us preliminarily observe that, by the Hardy–Littlewood inequality (2.3) and Fubini’s theorem,
∗∗
f ∗∗ (·) 1 1 + log (·)
1 (s) = s
s 0
=
1 s
s 0
f ∗∗ (·) 1 1 + log (·)
f ∗ (ρ)
s
∗
1 (r) dr s
0
dr
r 1 + log 1r
ρ
s
dρ
f ∗∗ (r)
dr
1 + log 1r
for s ∈ (0, 1).
(6.3)
At this stage we have to distinguish the cases where 1 < q < ∞ and q = 1. 1 Assume first that 1 < q < ∞. We begin by proving that L∞,q;α− 2 (Rn , γn ) is the optimal range for L∞,q;α (Rn , γn ). Let X(Rn , γn ) = L∞,q;α (Rn , γn ). Then, by (2.14), X (Rn , γn ) = L(1,q ;−α−1) (Rn , γn ). By Theorem 4.1, the optimal range Y (Rn , γn ) for X(Rn , γn ) fulfils ∗∗ f ∗∗ (s) sup f (r) f Y (0,1) ≈ sr1 1 + log 1 L(1,q ;−α−1) (0,1) 1 + log 1s L(1,q ;−α−1) (0,1) r 1 " s #q 1 q 1 1 (−α−1)q f ∗∗ (ρ) q −1 1 + log = sup dr s ds , s rρ1 1 + log 1 s 0
0
ρ
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3629
with equivalence constants depending on q and α. Since sup rρ1
f ∗∗ (ρ) f ∗∗ (ρ) = max sup , sup rρs 1 + log 1 sρ1 1 + log 1 ρ ρ
f ∗∗ (ρ) 1 + log ρ1
if 0 < r s < 1,
we have that 1 " s #q (−α−1)q q1 1 1 f ∗∗ (ρ) f Y (0,1) C sup dr s q −1 1 + log ds s rρs 1 + log 1 s ρ 0 0 1
q
f ∗∗ (ρ)
sup sρ1 1 + log 1 ρ
+C 0
s
q −1
1 1 + log s
1
(−α−1)q
q
ds
,
for some constant C = C(q, α). Let us call I1 and I2 the first and the second addend, respectively, on the right-hand side of the last inequality. By Proposition 4.7(i), one has that 1 I2 C
f
∗∗
q q −1
(s) s
1 1 + log s
1
(−α− 1 )q
q
2
ds
,
0
for some constant C = C(q, α). As for I1 , note that, by Proposition 4.7(i), s 0
s
f ∗∗ (ρ)
f ∗∗ (r) sup dr C dr 1 rρs 1 + log 1 1 + log ρ r 0
for s ∈ (0, 1),
for some absolute constant C. Thus, 1 s
I1 C 0
C
f ∗∗ (r)
0
1 + log 1r
∗∗
q q −1
1 f
(s) s
q 1 q 1 (−α−1)q 1 1 + log dr ds s s 1 1 + log s
1
(−α− 1 )q
q
2
ds
,
0
for some constants C = C(q, ∞) and C = C (q, α), as a consequence of the weighted Hardy inequality (Proposition 4.6). Thus, we have shown that f Y (0,1) Cf for some constant C = C(q, α).
(1,q ;−α− 21 )
L
(0,1)
(6.4)
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A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
Now, we have to establish a lower bound. On defining ω : (0, 1) → [0, ∞) as 1 (−α−1)q 1 1 + log ω(s) = s s
for s ∈ (0, 1),
and making use of (6.3), we get that q f Y (0,1)
1 s C
s
∗
f (r) 0
r
0
q
dρ
dr ρ 1 + log ρ1
ω(s) ds,
for some constant C = C(q, α). Thus, we only need to show that there exists a positive constant C = C(q, α) such that 1 s
∗
s
f (r) 0
0
r
q
dρ
dr ρ 1 + log ρ1
1 s ω(s) ds C 0
1 f (r) dr 1 + log s ∗
q ω(s) ds.
0
(6.5) In order to prove (6.5), we use a discretization argument making use of (4.36). Define the sequence {sk } as sk = 2−2 , k
for k ∈ N ∪ {0},
(6.6)
so that s0 = 12 , limk→∞ sk = 0 and, for every k ∈ N ∪ {0}, sk+1 = sk2 . Therefore, via (4.36) one can easily verify that 1 s 0
f ∗ (r)
∞ sk s k=0 sk+1
∞
s
∗
f (r) r
0
sk s
∗
0
q
dρ
sk+1
f (r) dr
k=0 sk+1
ω(s) ds
dr ρ 1 + log ρ1
r
0
sk+2 sk
q
dr ρ 1 + log ρ1
sk+1
∗
ω(s) ds
dρ
f (r)
k=0 sk+1 ∞
q
dρ
dr ρ 1 + log ρ1
r
0
s
sk+2
dρ
ω(s) ds q
ρ 1 + log ρ1
ω(s) ds
$ q sk sk+2 ∞ 1 ∗ f (r) dr 1 + log ω(s) ds C sk+1 k=0
0
sk+1
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3631
$ q sk+2 sk+2 ∞ 1 ∗ C f (r) dr 1 + log ω(s) ds sk+3 k=0
C
∞
0 sk+2 s
k=0 sk+3
s 1 16
=C
0
C
f ∗ (r) dr 1 + log
1 s
q
ω(s) ds
0
1 f (r) dr 1 + log s
q
∗
0
1 s 0
sk+3
1 f ∗ (r) dr 1 + log s
ω(s) ds q ω(s) ds,
0
for some constants C, C , C , C depending on q and α. Inequality (6.5) is thus established, and hence also the inequality f Y (0,1) Cf
(1,q ;−α− 21 )
L
(6.7)
(0,1)
1
for some constant C = C(q, α). Combining (6.4) and (6.7) yields Y (0, 1) = L(1,q ;−α− 2 ) (0, 1), 1 whence, by (2.14), Y (Rn , γn ) = L∞,q;α− 2 (Rn , γn ). 1 Now, let us show that L∞,q;α (Rn , γn ) is the optimal domain for L∞,q;α− 2 (Rn , γn ). To this 1 purpose, set Y (Rn , γn ) = L∞,q;α− 2 (Rn , γn ). First, observe that, by (2.15), exp L2 (Rn , γn ) → 1 Y (Rn , γn ), since exp L2 (Rn , γn ) = L∞,∞;− 2 (Rn , γn ). Second, by (2.14), Y (0, 1) = 1 1 L(1,q ;−α− 2 ) (0, 1). By Proposition 4.7(i), T is bounded on L1,q ;−α− 2 (0, 1), and hence, by 1 Corollary 4.9, also on L(1,q ;−α− 2 ) (0, 1). Assumptions (4.5) and (4.6) of Theorem 4.3 are thus fulfilled. According to this theorem, the norm of the optimal domain X(Rn , γn ) for Y (Rn , γn ) satisfies 1 f ∗ (r) f X(0,1) ≈ dr ∞,q;α− 1 1 2 (0,1) L s r 1 + log r 1 1 q 1 1 1 q(α− 2 ) ds q f ∗ (r) 1 + log = dr , s s 1 r 1 + log s r 0 with equivalence constants depending on q and α. We will be done if we show that 1 1
0
s
f ∗ (r)
r 1 + log 1r
q dr
1 1 + log s
q(α− 1 ) 2
ds ≈ s
1
1 qα ds f ∗ (s)q 1 + log , s s
(6.8)
0
with equivalence constants depending on q and α. The upper bound for the left-hand side in terms of the right-hand side in (6.8) follows at once from the weighted Hardy inequality (Proposition 4.6).
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A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
As for the lower bound, we will again use a discretization argument and (4.36). Let sk be as in (6.6) and define ω : (0, 1) → [0, ∞) as 1 1 q(α− 2 ) 1 1 + log ω(s) = s s
for s ∈ (0, 1).
Then, by (4.36), 1 1 0
s
f ∗ (r)
dr r 1 + log 1r
q ω(s) ds
∞ sk 1 k=1 sk+1
sk
∞
s
∞
dr r 1 + log 1r
sk−1
k=1 sk+1
sk
∗
dr r 1 + log 1r
ω(s) ds
sk−1
q sk
q
k=1
C
ω(s) ds q
f ∗ (r)
f (sk−1 )
sk
∞
q
f ∗ (r)
dr
r 1 + log 1r
∗
q
f (sk−1 )
1 + log
k=2
1 sk−1
ω(s) ds sk+1
sk−2 q 2 ω(s) ds sk−1
s q ∞ k−2 1 2 ∗ q C f (s) 1 + log ω(s) ds s
k=2 sk−1 1
=C
2
∗
q
f (s)
1 1 + log s
q
2
ω(s) ds
0
C
1
q 1 2 f ∗ (s)q 1 + log ω(s) ds, s
0
for some constants C, C , C depending on q and α. This chain yields the lower bound in (6.8). The proof in the case where 1 < q < ∞ is complete. Finally, assume that q = 1 (and hence α + 1 < 0). We first prove the optimality of the range in (6.2). Let X(Rn , γn ) = L∞,1;α (Rn , γn ). Then, by (2.14) and Theorem 4.1, the optimal range Y (Rn , γn ) for X(Rn , γn ) fulfils f ∗∗ (s) f Y (0,1) ≈ 1 L(1,∞;−α−1) (0,1) 1 + log s f ∗∗ (·) 1 −α−1 = sup s 1 + log s 0<s<1 1 + log
∗∗ 1 (·)
(s),
(6.9)
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3633
with equivalence constants depending on α. We claim that f ∗∗ (·) 1 −α−1 sup s 1 + log s 0<s<1 1 + log
∗∗ 1 (·)
1 1 −α− 2 (s) ≈ sup f ∗∗ (s)s 1 + log , s 0<s<1
(6.10)
with equivalence constants depending on α. We have that f ∗∗ (·) 1 −α−1 sup s 1 + log s 0<s<1 1 + log
∗∗ (s)
1 (·)
1 1 α 1 −α− 2 1 −α−1 1 sup f (s)s 1 + log 1 + log sup s 1 + log s s s s 0<s<1 0<s<1 1 1 −α− 2 = sup f ∗∗ (s)s 1 + log . s 0<s<1
∗∗
The reverse estimate will follow from (6.3) once we show that s s 1 −α−1 dρ ∗ sup 1 + log f (r) dr 1 s 0<s<1 ρ 1 + log r ρ 0 C sup 0<s<1
1 1 + log s
−α− 1 s 2
f ∗ (r) dr,
(6.11)
0
for some constant C = C(α). On making use of the same discretization sequence {sk } as in (6.6), we have that sup 0<s<1
1 1 + log s
−α−1 s
∗
f (r) 0
sup
sup
k∈N sk+1 ssk
sup
k∈N sk+1 ssk
sup
k∈N sk+1 ssk
C sup
sup
−α−1 s
s
∗
f (r) r
1 1 + log s
−α−1
s
1 1 + log s
k∈N sk+1 ssk
dρ dr ρ 1 + log ρ1
sk+1
∗
f (r) r
0
sup
1 1 + log s
r
dρ dr ρ 1 + log ρ1
0
sup
s
−α−1
sk+2
∗
dρ dr ρ 1 + log ρ1 sk+1
f (r) dr 0
sk+2
dρ
ρ 1 + log ρ1
sk+2 1 2 1 1 −α−1 f ∗ (r) dr 1 + log 1 + log sk+1 sk 0
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A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
C sup
sup
k∈N sk+3 ssk+2
C sup
sk+2 1 1 −α− 2 ∗ f (r) dr 1 + log sk+3 0
s sup
k∈N sk+3 ssk+2
1 1 −α− 2 f ∗ (r) dr 1 + log s
0
1 s 1 −α− 2 C sup 1 + log f ∗ (r) dr, s 0<s<1
0
for some constants C, C and C depending on α. This chain implies (6.11). Eq. (6.10) is thus 1 established. Coupling this equation with (6.9) tells us that Y (0, 1) = L(1,∞;−α− 2 ) (0, 1), whence, 1 by (2.14), Y (Rn , γn ) = L∞,1;α− 2 (Rn , γn ). 1 To prove the optimality of the domain in (6.2), let Y (Rn , γn ) = L∞,1;α− 2 (Rn , γn ). Assumptions (4.5) and (4.6) of Theorem 4.3 are fulfilled by the same reason as in the proof of the case when 1 < q < ∞. Thus, the optimal domain X(Rn , γn ) for Y (Rn , γn ) satisfies 1 1 1 1 f ∗ (r) f ∗ (r) 1 α− 2 ds f X(0,1) ≈ dr = dr 1 + log ∞,1;α− 1 s s 1 1 r 1 + log 2 (0,1) L s r 1 + log r s r 0 1 = 0
r
f ∗ (r)
r 1 + log 1r
1 1 + log s
α− 1 2
0
ds dr ≈ s
1 0
1 1 α+ 2 1 + log dr r r 1 + log 1 f ∗ (r)
r
= f L∞,1;α (0,1) , with equivalence constants depending on α. The proof is complete.
2
7. Self-optimal spaces We conclude by exhibiting special r.i. spaces which are self-optimal in the Gaussian Sobolev inequality (1.3). They are the Lorentz endpoint space and the Marcinkiewicz space whose fundamental function is equivalent to ϕG : (0, 1) → [0, ∞), given by ϕG (s) = e
−2 1+log 1s
for s ∈ (0, 1).
(7.1)
Theorem 7.1. Let ϕG be defined by (7.1). (i) There exists an absolute constant C such that u − uγn Λ1 (ϕ
G )(R
n ,γ ) n
C∇uΛ1 (ϕ
G )(R
n ,γ ) n
(7.2)
)(Rn , γ ). Moreover, (Λ1 (ϕ )(Rn , γ ), Λ1 (ϕ )(Rn , γ )) is an optifor every u ∈ V 1 Λ1 (ϕG n n n G G mal pair in (7.2).
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3635
(ii) There exists an absolute constant C such that u − uγn Γ ∞ (ϕG )(Rn ,γn ) C∇uΓ ∞ (ϕG )(Rn ,γn )
(7.3)
for every u ∈ V 1 Γ ∞ (ϕG )(Rn , γn ). Moreover, (Γ ∞ (ϕG )(Rn , γn ), Γ ∞ (ϕG )(Rn , γn )) is an optimal pair in (7.3). Note that, in fact, Γ ∞ (ϕG ) Rn , γn = Λ∞ (ϕG ) Rn , γn
(7.4)
(up to equivalent norms), as it is easily seen via the weighted Hardy inequality (Proposition 4.6). As a consequence of Theorem 7.1(ii), one has the following corollary. 1
Corollary 7.2. Let A be a Young function fulfilling (5.1) and such that A(t) = e 4 log Then, there exists a constant C = C(A) such that u − uγn LA (Rn ,γn ) C∇uLA (Rn ,γn )
2t
for large t.
(7.5)
for every u ∈ V 1 LA (Rn , γn ). Moreover, (LA (Rn , γn ), LA (Rn , γn )) is an optimal pair in (7.5). Corollary 7.2 follows from Theorem 7.1(ii), via the next proposition. Proposition 7.3. Let A be a Young function and let ωA : (0, 1) → [0, ∞) be the function given by ωA (s) =
1 A−1 ( 1s )
for s ∈ (0, 1).
If there exists δ ∈ (0, 1) such that
1 ds < ∞, A δA−1 s
(7.6)
0
then LA (Rn , γn ) = Γ ∞ (ωA )(Rn , γn ) (up to equivalent norms). Proof. Since ωA is the fundamental function of LA (Rn , γn ), the embedding LA (Rn , γn ) → Γ ∞ (ωA )(Rn , γn ) is a straightforward consequence of (2.16). On the other hand, since any function u such that uΓ ∞ (ωA )(Rn ,γn ) 1 fulfils the inequality u∗ (s) A−1
1 s
for s ∈ (0, 1),
the reverse embedding Γ ∞ (ωA )(Rn , γn ) → LA (Rn , γn ) follows via (7.6). The proof of Theorem 7.1 will make use of the following auxiliary result.
(7.7) 2
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A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
Lemma 7.4. Let ϕ and ψ be non-decreasing, weakly differentiable functions on [0, 1] vanishing at 0, and let T be the operator defined by (4.4). (i) If s 1 + log 1r ψ (r) dr 0 sup < ∞, 0<s<1 ϕ(s) 1 + log 1s
(7.8)
then T : Λ1 (ϕ )(0, 1) → Λ1 (ψ )(0, 1). Moreover, the norm of the operator T does not exceed (an absolute constant times) the left-hand side of (7.8). (ii) If ψ(s) 1 + log 1s sup < ∞, 0<sr<1 ϕ(r) 1 + log 1 r
(7.9)
then T : Γ ∞ (ϕ)(0, 1) → Γ ∞ (ψ)(0, 1). Moreover, the norm of the operator T does not exceed (an absolute constant times) the left-hand side of (7.9). Proof. (i) One has that T : Λ1 (ϕ )(0, 1) → Λ1 (ψ )(0, 1) if and only if 1 sup 0
sr1
f ∗ (r) 1 + log
1 r
1 1 + log ψ (s) ds C s
1
f ∗ (s) ϕ (s) ds
0
for some constant C and for every f ∈ Λ1 (ϕ )(0, 1). Moreover, the optimal constant C in the above inequality equals the norm of T . Thus, the conclusion follows from Proposition 4.7(i). (ii) Let f ∈ M(0, 1). By (4.32), one has that sup (Tf )∗∗ (s)ψ(s) C sup T (f ∗∗ )(s)ψ(s) 0<s<1
0<s<1
ψ(s) 1 + log 1s C sup f ∗∗ (s)ϕ(s) sup , 0<s<1 0<sr<1 ϕ(r) 1 + log 1 r
for some absolute constant C, and the conclusion follows.
2
Now, we are in a position to prove our last main result. Proof of Theorem 7.1. Throughout the proof, we denote by f an arbitrary function in M(0, 1). )(Rn , γ ), then Y (Rn , γ ) is an optimal (i) We begin by showing that if Y (Rn , γn ) = Λ1 (ϕG n n n domain for itself. The space Y (R , γn ) satisfies assumption (4.5) of Theorem 4.3, since
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
1 f Y (Rn ,γn ) = 0
−2 1+log 1
s f ∗ (s) e f ∗ (s) ds sup 0<s<1 1 + log 1 s 1 + log 1s s
1
e
3637
−2 1+log 1s
s
ds
0
≈ f exp L2 (Rn ,γn ) , with absolute equivalence constants. Assumption (4.6) is also fulfilled, as it is easily seen by an )) (0, 1) = Γ ∞ (ϕ )(0, 1), where application of Lemma 7.4(ii), since, by (2.17), (Λ1 (ϕG G ϕ G (s) =
s ϕG (s)
for s ∈ (0, 1).
Let us now note that ϕG (s) ϕG (s) = s 1 + log 1s
for s ∈ (0, 1),
(7.10)
and lim ϕG (s) = 0.
(7.11)
s→0+
Thus, Theorem 4.3 tells us that the optimal domain X(Rn , γn ) for Y (Rn , γn ) obeys 1 1 1 f ∗ (r) f ∗ (r) f X(0,1) ≈ dr = dr ϕG (s) ds 1 1 1 r 1 + log Λ (ϕG )(0,1) s r 1 + log r s r 0 1
= 0
1 =
f (r)
r 1 + log
r 1 r 0
ϕG (s) ds dr
1 = 0
f ∗ (r)ϕG (r) dr = f Λ1 (ϕ
G )(0,1)
ϕG (r) f ∗ (r) dr 1 r 1 + log r
,
0
with absolute equivalence constants, where the third equality holds by (7.11) and the fourth one )(Rn , γ ) is the optimal domain for itself. by (7.10). This shows that Λ1 (ϕG n n 1 Now, let X(R , γn ) = Λ (ϕG )(Rn , γn ). We have to show that X(Rn , γn ) is the optimal range for itself. By Theorem 4.1 and (2.17), the optimal range Y (Rn , γn ) for X(Rn , γn ) satisfies f ∗∗ (s) f Y (0,1) ≈ , 1 Γ ∞ (ϕ G )(0,1) 1 + log s with absolute equivalence constants. Let a ∈ (0, 1/2). Then
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A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
aχ 1 (s) (a, 2 ) ϕY (a) = χ(0,a) Y (0,1) C 1 Γ ∞ (ϕ G )(0,1) s 1 + log s = Ca sup ϕ G (s) 0<s< 12
χ(a, 1 ) (·) ∗∗ 2 (s), 1 (·) 1 + log (·)
for some absolute constant C. Now,
χ(a, 1 ) (·) ∗ 1 2 (s) = χ(0, 1 −a) (s) 2 1 1 (·) 1 + log (·) (a + s) 1 + log a+s
for s ∈ (0, 1),
whence ϕ G (s) ϕY (a) Ca sup s 0<s< 1 2
s 0
ϕ G (s) = Ca sup s 0<s< 1 −a
χ(0, 1 −a) (r) 2 dr 1 (a + r) 1 + log a+r s
dr dr 1 (a + r) 1 + log 2 a+r 0 1 1 2 1+log 1s = 2Ca sup e 1 + log − 1 + log a a+s 0<s< 1 −a 2
2Ca
sup
e
2 1+log 1s
0<s< 12 −a
Let sa be the solution to the equation −(1−2 1+log a1 +log a1 )
1 1 + log − a
1 + log a1 −
1 1 + log . s
(7.12)
1 + log s1a = 1, namely, sa =
. Since lima→0+ sa = 0, the number sa can be used to estimate from bee low the supremum on the rightmost side of (7.12) when a is sufficiently small. Hence, ϕY (a) 2Cae
2 1+log a1 −2
= 2Ce−2 ϕ G (a)
for sufficiently small a.
)(Rn , γ ) → By (2.16), this entails that Y (0, 1) → Γ ∞ (ϕ G )(0, 1), whence, by (2.17), Λ1 (ϕG n n Y (R , γn ). Since we already know that inequality (7.2) holds, we conclude that Y (Rn , γn ) = )(Rn , γ ). Λ1 (ϕG n (ii) Let X(Rn , γn ) = Γ ∞ (ϕG )(Rn , γn ). We begin by showing that X(Rn , γn ) is an optimal range for itself. By Theorem 4.1 and (2.17) such an optimal range Y (Rn , γn ) fulfils
1 f ∗∗ (s) f ∗∗ (r) f Y (0,1) ≈ sup ϕ G (s) ds, 1 Λ1 (ϕ )(0,1) 1 sr1 1 + log s 1 + log r G 0
(7.13)
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3639
with absolute equivalence constants. Note that 1 s s
s ϕ G (r) dr C ϕ G (r) dr 1 r 1 + log r 0
for s ∈ (0, 1),
(7.14)
f ∗ (s)ϕ G (s) ds,
(7.15)
for some absolute constant C, whence by Proposition 4.7(ii), 1
f ∗∗ (r)
sup sr1 1 + log 1 r
0
ϕ G (s) ds
1 C 0
for some absolute constant C. From (7.13) and (7.15) we deduce that Λ1 (ϕ G )(Rn , γn ) → Y (Rn , γn ), and, equivalently, Y (Rn , γn ) → Γ ∞ (ϕG )(Rn , γn ). To prove the converse embedding, observe that, since ϕ G is equivalent to a decreasing function, by the Hardy–Littlewood inequality (2.3) 1
f Y (0,1) ≈ 0
∗
f ∗∗ (·) 1 + log
1 (·)
(s)ϕ G (s) ds
1
C 0
f ∗∗ (s) 1 + log
1 s
ϕ G (s) ds,
with absolute equivalence constants and an absolute constant C. Hence, via Fubini’s theorem and (7.14), 1 f Y (0,1) C 0
1 ≈
ϕ (s) G s 1 + log 1s
s
1
∗
f (r) dr ds = C 0
∗
f (r) 0
f ∗ (r)ϕ G (r) dr = f Λ1 (ϕ
G )(0,1)
1 r
ϕ (s) G ds dr s 1 + log 1s
,
0
with absolute equivalence constants and an absolute constant C. Thus, Y (Rn , γn ) → Λ1 (ϕ G )(Rn , γn ), and hence Γ ∞ (ϕG )(Rn , γn ) → Y (Rn , γn ). This shows that the Gaussian Sobolev embedding (7.3) holds, and that the range is optimal. Now, we shall prove that the domain is optimal as well. To this end, assume that Y (Rn , γn ) = ∞ Γ (ϕG )(Rn , γn ) and note that T is bounded on Y (0, 1) = Λ1 (ϕ G )(0, 1) by Lemma 7.4(i). Thus, assumption (4.6) is fulfilled. Since an absolute constant C exists such that ϕG (s)
C 1 + log 1s
for s ∈ (0, 1),
one has that exp L2 (Rn , γn ) → Λ∞ (ϕG )(Rn , γn ). Hence, by (7.4), assumption (4.5) holds as well. Therefore, by Theorem 4.3, the optimal domain X(Rn , γn ) for Y (Rn , γn ) satisfies
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A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
1 f ∗ (r) f X(0,1) ≈ dr , ∞ 1 r 1 + log Γ (ϕ )(0,1) G s r with absolute equivalence constants. Since we already know that the Gaussian Sobolev embedding (7.3) holds, and since X(Rn , γn ) is the largest possible domain when the range is Γ ∞ (ϕG )(Rn , γn ), the embedding Γ ∞ (ϕG )(Rn , γn ) → X(Rn , γn ) certainly holds. What remains to be proved is the reverse embedding, namely that X(Rn , γn ) → Γ ∞ (ϕG )(Rn , γn ). As a consequence of (2.16), it suffices to show that ϕG (a) CϕX (a)
for a ∈ (0, 1),
(7.16)
for some absolute constant C. Given any a ∈ (0, 1), one has that 1 χ(0,a) (r) ϕX (a) = χ(0,a) X(0,1) ≈ dr ∞ 1 Γ (ϕG )(0,1) s r 1 + log r ϕG (s) = sup s 0<s<1
s 1 0 r
a sup ϕG (s) 0<s
s
χ(0,a) (ρ) dρ dr ρ 1 + log ρ1
dρ ρ 1 + log ρ1
1 1 , = 2 sup ϕG (s) 1 + log − 1 + log s a 0<s
(7.17)
with absolute equivalence constants. Let sa ∈ (0, a) be the solution to the equation 1 + log s1a − −(1+2 1+log a1 +log a1 ) 1 1 + log a = 1; namely, sa = e . Then, on making use of sa to estimate the last supremum in (7.17), we get that ϕX (a) e−2 ϕG (a), whence (7.16) follows. The proof is complete.
(7.18)
2
8. Note added in proof Recently, the manuscript [28], now published, has been brought to our attention by the authors and independently by the Editors of the JFA. It turns out that the reduction Theorem 3.1 of the present paper coincides with [28, Theorem 3, part (i)]. Our Theorem 3.1 was also announced in [34, Theorem 9.1].
A. Cianchi, L. Pick / Journal of Functional Analysis 256 (2009) 3588–3642
3641
Acknowledgment We would like to thank the referee for bringing to our attention the paper [13], in which rearrangement inequalities are used to establish optimizing sequences for functional inequalities, and also the paper [26], dealing with Sobolev inequalities in some Orlicz spaces. The results however do not overlap with ours. References [1] R.A. Adams, General logarithmic Sobolev inequalities and Orlicz imbeddings, J. Funct. Anal. 34 (1979) 292–303. [2] R.A. Adams, F.H. Clarke, Gross’s logarithmic Sobolev inequality: A simple proof, Amer. J. Math. 101 (1979) 1265–1269. [3] S. Aida, T. Masuda, I. Shigekawa, Logarithmic Sobolev inequalities and exponential integrability, J. Funct. Anal. 126 (1994) 83–101. [4] F. Barthe, Log-concave and spherical models in isoperimetry, preprint. [5] F. Barthe, P. Cattiaux, C. Roberto, Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry, Rev. Mat. Iberoamericana 22 (2006) 993–1067. [6] W. Beckner, A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc. 105 (1989) 397–400. [7] C. Bennett, K. Rudnick, On Lorentz–Zygmund spaces, Dissertationes Math. 175 (1980) 1–72. [8] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, New York, 1988. [9] S.G. Bobkov, F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal. 163 (1999) 1–28. [10] S.G. Bobkov, M. Ledoux, On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures, J. Funct. Anal. 156 (1998) 347–365. [11] C. Borell, The Brunn–Minkowski inequality in Gauss space, Invent. Math. 30 (1975) 207–216. [12] E.A. Carlen, C. Kerce, On the cases of equality in Bobkov’s inequality and Gaussian rearrangement, Calc. Var. Partial Differential Equations 13 (2001) 1–18. [13] E.A. Carlen, M. Loss, Extremals of functionals with competing symmetries, J. Funct. Anal. 88 (1990) 437–456. [14] M. Carro, A. García del Amo, J. Soria, Weak-type weights and normable Lorentz spaces, Proc. Amer. Math. Soc. 124 (1996) 849–857. [15] M. Carro, A. Gogatishvili, J. Martín, L. Pick, Functional properties of rearrangement invariant spaces defined in terms of oscillations, J. Funct. Anal. 229 (2005) 375–404. [16] A. Cianchi, D.E. Edmunds, P. Gurka, On weighted Poincaré inequalities, Math. Nachr. 180 (1996) 15–41. [17] A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, On the isoperimetric deficit in the Gauss space, preprint. [18] F. Cipriani, Sobolev–Orlicz imbeddings, weak compactness, and spectrum, J. Funct. Anal. 177 (2000) 89–106. [19] M. Cwikel, A. Kami´nska, L. Maligranda, L. Pick, Are generalized Lorentz “spaces” really spaces? Proc. Amer. Math. Soc. 132 (2004) 3615–3625. [20] D.E. Edmunds, P. Gurka, L. Pick, Compactness of Hardy-type integral operators in weighted Banach function spaces, Studia Math. 109 (1994) 73–90. [21] D.E. Edmunds, R. Kerman, L. Pick, Optimal imbeddings involving rearrangement-invariant quasinorms, J. Funct. Anal. 170 (2000) 307–355. [22] A. Ehrhard, Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes, Ann. École Norm. Sup. 17 (1984) 317–332. [23] G.F. Feissner, Hypercontractive semigroups and Sobolev’s inequality, Trans. Amer. Math. Soc. 210 (1975) 51–62. [24] A. Gogatishvili, B. Opic, L. Pick, Weighted inequalities for Hardy-type operators involving suprema, Collect. Math. 57 (2006) 227–255. [25] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975) 1061–1083. [26] L. Gross, O. Rothaus, Herbst inequalities for supercontractive semigroups, J. Math. Kyoto Univ. 38 (1998) 295–318. [27] M. Ledoux, Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter, J. Math. Kyoto Univ. 35 (1995) 211–220. [28] J. Martín, M. Milman, Isoperimetry and symmetrization for logarithmic Sobolev inequalities, J. Funct. Anal. 256 (2009) 149–178. [29] V.G. Maz’ya, Sobolev Spaces, Springer, Berlin, 1985. [30] B. Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972) 31–38. [31] E. Nelson, The free Markoff field, J. Funct. Anal. 12 (1973) 221–227.
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[32] B. Opic, L. Pick, On generalized Lorentz–Zygmund spaces, Math. Inequal. Appl. 2 (1999) 391–467. [33] E. Pelliccia, G. Talenti, A proof of a logarithmic Sobolev inequality, Calc. Var. Partial Differential Equations 1 (1993) 237–242. [34] L. Pick, Optimality of function spaces in Sobolev embeddings, in: Vladimir Maz’ya (Ed.), Sobolev Spaces in Mathematics I, Sobolev Type Inequalities, Springer, Tamara Rozhkovskaya Publisher, Novosibirsk, ISBN 978-0387-85647-6, 2009, xxix+378 pp., 249–280. [35] O.S. Rothaus, Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities, J. Funct. Anal. 64 (1985) 296–313. [36] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990) 145–158. [37] J. Soria, Lorentz spaces of weak-type, Quart. J. Math. Oxford 49 (1998) 93–103. [38] G. Talenti, An inequality between u∗ and |grad u|∗ , in: General Inequalities, 6, Oberwolfach, 1990, in: Internat. Ser. Numer. Math., vol. 103, Birkhäuser, Basel, 1992, pp. 175–182. [39] G. Talenti, A weighted version of a rearrangement inequality, Ann. Univ. Ferrara 43 (1997) 121–133. [40] F.B. Weissler, Logarithmic Sobolev inequalities and hypercontractive estimates on the circle, J. Funct. Anal. 32 (1979) 102–121.
Journal of Functional Analysis 256 (2009) 3643–3659 www.elsevier.com/locate/jfa
Linking solutions for p-Laplace equations with nonlinearity at critical growth ✩ Marco Degiovanni a,∗ , Sergio Lancelotti b a Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121 Brescia, Italy b Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Received 4 September 2008; accepted 18 January 2009 Available online 11 February 2009 Communicated by J. Coron
Abstract Under a suitable condition on n and p, the quasilinear equation at critical growth −p u = λ|u|p−2 u + ∗
|u|p −2 u is shown to admit a nontrivial weak solution u ∈ W0 structures, for the associated functional, are recognized. © 2009 Elsevier Inc. All rights reserved.
1,p
(Ω) for any λ λ1 . Nonstandard linking
Keywords: p-Laplace equations; Critical growth; Nontrivial solutions; Linking structures
1. Introduction and main results Let Ω be a bounded open subset of Rn , let 1 < p < n and let λ ∈ R. We are interested in the existence of nontrivial solutions u for the quasilinear problem
−p u = λ|u|p−2 u + |u|p u=0
∗ −2
u
in Ω, on ∂Ω,
(1.1)
✩ The research of the authors was partially supported by the MIUR project “Variational and topological methods in the study of nonlinear phenomena” and by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (INdAM). * Corresponding author. E-mail addresses: [email protected] (M. Degiovanni), [email protected] (S. Lancelotti).
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.016
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M. Degiovanni, S. Lancelotti / Journal of Functional Analysis 256 (2009) 3643–3659
where p u := div(|∇u|p−2 ∇u) denotes the p-Laplace operator and p ∗ := np/(n − p) the criti1,p cal Sobolev exponent for the embedding of W0 (Ω) in Lq (Ω). Let us also set p n n |∇u| dx ∞ S = inf R p∗ R \ {0} , : u ∈ C ∗ c ( Rn |u| dx)p/p |∇u|p dx 1,p Ω : u ∈ W0 (Ω) \ {0} λ1 = min p Ω |u| dx 1,p
and denote by ϕ1 ∈ W0 (Ω) a positive solution of −p u = λ1 |u|p−2 u (see Lindqvist [16]). After the seminal paper of Brezis and Nirenberg [4], many works have been devoted to problems at critical growth, mainly when p = 2. In particular, let us recall that, according to the main result of [4], problem (1.1) admits a positive solution u for any λ ∈ ]0, λ1 [ , provided that p = 2 and n 4. The result has been extended by Egnell, García Azorero and Peral Alonso, Guedda and Véron [9,12,14], who have proved that problem (1.1) admits a positive solution u for any λ ∈ ]0, λ1 [ , provided that p > 1 and n p 2 . Such a solution u can be obtained via the Mountain 1,p pass theorem of Ambrosetti and Rabinowitz [1] applied to the C 1 -functional f : W0 (Ω) → R defined as 1 f (u) = p
λ |∇u| dx − p
p
Ω
1 |u| dx − ∗ p
Ω
∗
|u|p dx
p
(1.2)
Ω
and satisfies 1 0 < f (u) < S n/p . n
(1.3)
On the other hand, it is known [4,9,14] that, if Ω is star-shaped and with smooth boundary, then problem (1.1) has no nontrivial solution u for any λ 0. When λ λ1 , it is still meaningful to look for nontrivial solutions u, but the situation is quite different in the two cases p = 2 and p = 2. If p = 2, it has been proved by Capozzi, Fortunato and Palmieri [5] that problem (1.1) has a nontrivial solution u in each of the following cases: (a) λ λ1 and n 5; (b) λ > λ1 , λ ∈ / σ (−2 ) and n 4 (see also Gazzola and Ruf [13, Corollary 1]). Such a solution can be obtained via the Linking theorem of Rabinowitz (see e.g. [19, Theorem 5.3]) applied to the functional f and still satisfies (1.3). 1,p On the other hand, when p = 2 there is in general no direct sum decomposition of W0 (Ω), which allows to recognize a linking structure in a standard way. To our knowledge, the only workable situation amounts to the fact that, if Ω is connected and we set
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λ2 = sup min |∇u|p dx: u ∈ Y, |u|p dx = 1 : Ω
Ω
1,p 1,p W0 (Ω) = (Rϕ1 ) ⊕ Y with Y closed in W0 (Ω) , then λ2 > λ1 and, for every b < λ2 , there exists a decomposition 1,p
W0 (Ω) = (Rϕ1 ) ⊕ Y such that
|∇u| dx = λ1
|u|p dx,
p
Ω
∀u ∈ Rϕ1 ,
Ω
|∇u| dx b
|u|p dx,
p
Ω
∀u ∈ Y.
Ω
Taking advantage of this fact, Arioli and Gazzola [2] have proved that, for any p > 1, problem (1.1) has a nontrivial solution u in each of the following cases: 2
n (a) λ1 λ < λ2 and n+1 > p2 ; (b) λ1 < λ < λ2 and n p 2 .
Such a solution is still obtained via the classical Linking theorem and satisfies (1.3). Our purpose is to provide a complete extension to the case p > 1 of the mentioned result of Capozzi, Fortunato and Palmieri. Because of the lack of decompositions by linear subspaces, we will apply the results of our recent paper [7], which provide an extension of the Linking theorem with linear subspaces substituted by cones. In the line of the case (a), we prove the following: Theorem 1.1. Assume that Ω has C 1,α boundary for some α ∈ ]0, 1[
(1.4)
n3 + p 3 > p2 . n2 + n
(1.5)
and that
Then problem (1.1) has a nontrivial solution u satisfying (1.3) for every λ λ1 . By the way, we also improve the condition on n and p of Arioli and Gazzola, as (1.5) is equivalent to 3
n2 + pn > p2 . n+1
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This is due to a different concentration technique on points “moving to the boundary” of Ω, rather than at a fixed interior point (the key information is contained in Lemma 3.2). Still in the line of (a), we also prove the following results: Theorem 1.2. Assume that n2 > p2 . n+1
(1.6)
Then problem (1.1) has a nontrivial solution u satisfying (1.3) for every λ λ1 . In other words, under the condition of Arioli and Gazzola, the result holds for any λ λ1 , without any smoothness assumption on the boundary of Ω. Theorem 1.3. Assume that Ω is a ball and that (1.6) holds. Then problem (1.1) has a nontrivial radial solution u satisfying (1.3) for every λ λ1 . A comparison between Theorems 1.1 and 1.3 raises an interesting question: if Ω is a ball and n2 n3 + p 3 p2 < 2 , n+1 n +n what about the existence of a nontrivial radial solution u satisfying (1.3), say, when λ = λ1 ? A (negative) answer could come from an extension of the result of Arioli, Gazzola, Grunau and Sassone [3]. In order to state our results in the line of (b), let us set, according to [6,7,17,18], |∇u|p dx 1,p Ω : A ⊆ W0 (Ω) \ {0}, A is symmetric and Index(A) m , λm = inf sup p u∈A Ω |u| dx (1.7) where Index is the Z2 -cohomological index of Fadell and Rabinowitz [10,11]. Then it is well known that (λm ) is a nondecreasing divergent sequence and λ1 is the same as before, while λ2 λ2 . Moreover, in the case p = 2 we have {λm : m 1} = σ (−2 ), but for p = 2 it is only known that the equation −p u = λm |u|p−2 u admits a nontrivial solution u for any m 1. We prove the following result: Theorem 1.4. If n p 2 , then problem (1.1) has a nontrivial solution u satisfying (1.3) for every / {λm : m 1}. λ > λ1 with λ ∈ If Ω is a ball, let |∇u|p dx 1,p (r) Ω : A ⊆ W0,r (Ω) \ {0}, A is symmetric and Index(A) m , λm = inf sup p u∈A Ω |u| dx 1,p
where W0,r (Ω) denotes the corresponding Sobolev space of radial functions. From the results (r)
of [16] it follows that λ1 = λ1 . Then we have
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Theorem 1.5. Assume that Ω is a ball. If n p 2 , then problem (1.1) has a nontrivial radial (r) / {λm : m 1}. solution u satisfying (1.3) for every λ > λ1 with λ ∈ In the next section we recall and prove some preliminary facts, while in Section 3 we prove the results we have stated in the introduction. 2. Linking over cones First of all, let us recall from [7] a generalization of the Linking theorem in which linear subspaces are substituted by symmetric cones. Theorem 2.1. Let X be a Banach space and let f : X → R be a function of class C 1 . Let X− , X+ be two symmetric cones in X such that X+ is closed in X, X− ∩ X+ = {0}, Index X− \ {0} = Index(X \ X+ ) < ∞. Let also e ∈ X \ X− , 0 < r+ < r− , S+ = v ∈ X+ : v = r+ ,
Q = te + w: t 0, w ∈ X− , te + w r− , P = w ∈ X− : w r− ∪ te + w: t 0, w ∈ X− , te + w = r− be such that sup f < inf f, P
S+
sup f < +∞. Q
Define c = inf sup f η Q × {1} , η∈N
where N is the set of deformations η : Q × [0, 1] → X with η(u, t) = u on P × [0, 1]. Then we have inf f c sup f S+
(2.1)
Q
and there exists a sequence (uk ) in X with f (uk ) → 0 and f (uk ) → c. Proof. From [7, Theorem 2.2 and Corollary 2.9] it follows that (2.1) holds. If, by contradiction, there is no sequence (uk ) as required, then there exists σ > 0 such that f (u) σ whenever c − σ f (u) c + σ . In particular, f satisfies (PS)c and from [7, Theorem 2.2 and Corollary 2.9] we deduce that c is a critical value of f , whence a contradiction. 2 Assume now that Ω is a bounded open subset of Rn and that 1 < p < ∞. If we define λm according to (1.7), by [7, Theorem 3.2] the following holds:
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Theorem 2.2. If m 1 is such that λm < λm+1 , then we have
1,p Index u ∈ W0 (Ω) \ {0}: |∇u|p dx λm |u|p dx Ω
Ω
1,p p p |∇u| dx < λm+1 |u| dx = m. = Index u ∈ W0 (Ω): Ω
Ω
In view of the application of Theorem 2.1, the simplest choice is 1,p p p |∇u| dx λm+1 |u| dx , X+ = u ∈ W0 (Ω): Ω
(2.2)
Ω
while X− could be defined as 1,p u ∈ W0 (Ω): |∇u|p dx λm |u|p dx . Ω
(2.3)
Ω
The next result asserts that as X− we can also choose a smaller cone, with better regularity 1,p properties. Let us set u = ( Ω |∇u|p dx)1/p for every u ∈ W0 (Ω) and denote by q the 1,p usual norm in Lq (Ω). We also set M = {u ∈ W0 (Ω): Ω |u|p dx = 1} and denote by B (x) the open ball of center x and radius . Theorem 2.3. Let m 1 be such that λm < λm+1 . Then there exists a symmetric cone X− in 1,p W0 (Ω) such that X− is closed in Lp (Ω) and: (a) we have 1,p 1,α X− ⊆ u ∈ W0 (Ω): |∇u|p dx λm |u|p dx ∩ L∞ (Ω) ∩ Cloc (Ω); Ω
Ω
1,α (Ω); (b) X− ∩ M is bounded in L∞ (Ω) and in Cloc 1,p (c) X− ∩ M is strongly compact in W0 (Ω) and in C 1 (Ω); (d) we have Index(X− \ {0}) = m.
Moreover, if Ω satisfies (1.4), we have that X− ∩ M is bounded in C 1,α (Ω), for some α ∈ ]0, 1[ , and strongly compact in C 1 (Ω). Proof. We only prove the case 1 < p < n. The case p n can be treated with minor modifications. − is a symmetric subset of − be the symmetric cone defined in (2.3). Then M ∩ X Let X 1,p − \ {0}. MoreW0 (Ω) \ {0} with Index(M ∩ X− ) = m, being an odd deformation retract of X p over, M ∩ X− is strongly compact in L (Ω).
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Let us recall that, for every w ∈ Lq (Ω) with q (p ∗ ) , there exists one and only one u ∈ p−1 such that −p u = w. Moreover, if q = n/p, we have u ∈ Lβ(q) (Ω) and u β(q) c(Ω, p, q) w q , where
1,p W0 (Ω)
β(q) =
n(p−1)q
if q < n/p, if q > n/p
n−pq
∞
(see e.g. [14, Propositions 1.2 and 1.3]). In particular, for every w ∈ Lq (Ω) with q/(p − 1) (p ∗ ) , there exists one and only one u ∈ 1,p W0 (Ω) such that −p u = |w|p−2 w. Moreover, if q/(p − 1) = n/p, we have u ∈ Lγ (q) (Ω) and u γ (q) c(Ω, ˜ p, q) w q , where γ (q) =
nq n(p−1)−pq
if q/(p − 1) < n/p, if q/(p − 1) > n/p.
∞
1,p
For every w ∈ M, let J (w) ∈ M be defined as J (w) = u/ u p , where u ∈ W0 (Ω) is the solution of −p u = |w|p−2 w. Then it is easily seen that there exists k 2 such that J k−1 (M) 1,α is bounded in L∞ (Ω). By [8,15,20] it follows that J k (M) is also bounded in Cloc (Ω), or even 1,α in C (Ω) for some α ∈ ]0, 1[ , if Ω satisfies (1.4). Moreover, we have
|w|p−2 w Ω
u p ∇u p
|∇u|p−2 ∇u · ∇
dx = Ω
dx = 1
|w|p dx =
=
u p ∇u p
Ω
p−1
|∇u|p−2 ∇u · ∇w dx ∇u p
∇w p ,
Ω p
which implies, by the convexity of p ,
u p u
w pp + p |w|p−2 w − w dx = 1, p
∇u p ∇u p p
p
Ω
hence
u
∇
u p
p
1 p−1
∇u p
∇w p ,
namely ∇(J (w)) p ∇w p . − ) is a bounded subset of L∞ (Ω) and of C 1,α (Ω) (resp. C 1,α (Ω)) with Therefore J k (M ∩ X loc − ) ⊆ M ∩ X − . Since J is odd and continuous from the topology of Lp (Ω) to that of J k (M ∩ X 1,p W0 (Ω), it follows that − ) = Index(M ∩ X − ) = m Index J k (M ∩ X
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− ) is strongly compact in W (Ω). By the boundedness in C 1,α (Ω), the set and that J k (M ∩ X loc 0 − ) is also strongly compact in C 1 (Ω) (or even in C 1 (Ω), if we have the boundedness J k (M ∩ X in C 1,α (Ω)). Now, if we set − ) , X− = tu: t 0, u ∈ J k (M ∩ X 1,p
1,p − ) is it is clear that X− is a symmetric cone in W0 (Ω) satisfying (a)–(d). Since J k (M ∩ X 1,p k p − ), we also have that X− is closed in L (Ω). 2 compact in W0 (Ω) with 0 ∈ / J (M ∩ X
3. Proof of the main results Let Ω be a bounded open subset of Rn and let p > 1 with p 2 n. For every ε > 0 we set, as in [2], n−p
u∗ε (x) = where c(n, p) > 0 is such that
c(n, p)ε p(p−1) , p n−p p ε p−1 + |x| p−1 p
∗ p ∇u dx =
ε
Rn
∗ p∗ u dx = S n/p . ε
Rn
Up to a different parametrization with respect to ε, the family (u∗ε ) is the same of [9,12,14]. Let also η : R → [0, 1] be a C ∞ -function such that η(s) = 1 for s 1/4 and η(s) = 0 for s 1/2. For every ε, > 0, we set
|x| ∗ uε (x). u ,ε (x) = η
Lemma 3.1. There exist C, σ > 0 such that n−p |∇u ,ε |p dx S n/p + C(ε/ ) p−1 ,
(3.1)
Rn
∗
n
|u ,ε |p dx S n/p − C(ε/ ) p−1 , Rn
|u ,ε | dx p
Rn
(3.2) n−p
σ ε p − C p (ε/ ) p−1 σ ε p log( /ε) − Cε p
if n > p 2 , if n = p 2 ,
(3.3)
for every , ε > 0. Proof. Formulae (3.1) and (3.2) can be found in [2]. Formula (3.3) is similar. Let us prove it for reader’s convenience. Since u ,ε ( x) =
− n−p p
u1,ε/ (x),
M. Degiovanni, S. Lancelotti / Journal of Functional Analysis 256 (2009) 3643–3659
3651
we have
u ,ε (x)p dx = n
Rn
u ,ε ( y)p dy = p
Rn
u1,ε/ (y)p dy.
Rn
On the other hand, it is well known (see e.g. [14]) that
u1,ε (y)p dy
n−p
σ ε p − Cε p−1 σ ε p log(1/ε) − Cε p
Rn
Then formula (3.3) easily follows.
if n > p 2 , if n = p 2 .
2
Now let x ∈ Ω and R > 0 be such that BR (x) ⊆ Ω and ∂ BR (x) ∩ ∂Ω = ∅. If x0 ∈ ∂ BR (x) ∩ ∂Ω and x = x 0 +
x − x0 , |x − x0 |
we have that |x − x0 | = and B (x ) ⊆ Ω for every ∈ ]0, R]. Let ϑ : R → [0, 1] be a C ∞ -function such that ϑ(s) = 0 for s 1/2 and ϑ(s) = 1 for s 1. Let also m 1 with λm < λm+1 , let X+ be as in (2.2), X− be as in Theorem 2.3 and let e ,ε (x) = u ,ε (x − x ),
|x − x | v(x) for every v ∈ X− , v (x) = ϑ
X− = {v : v ∈ X− }.
1,p
Of course, X− also is a symmetric cone in W0 (Ω). Lemma 3.2. Assume that Ω satisfies (1.4). Then there exists C > 0 such that |v | dx Ω
|v| dx − C
p
p
Ω p∗
|v |
dx
Ω
|v|
dx − C
n+p ∗
Ω
for every v ∈ X− and ∈ ]0, R].
,
(3.4)
∗
(3.5)
Ω
|∇v|p dx + C n Ω
dx
|v|p dx,
|∇v |p dx
p/p∗
Ω p∗
Ω
p∗
|v|
n+p
∗
|v|p dx Ω
p/p∗ ,
(3.6)
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Moreover, there exists 0 ∈ ]0, R] such that
e ,ε ∈ / X− and X− is closed in Lp (Ω),
1,p
X− ∩ X+ = {0}, Index X− \ {0} = Index W0 (Ω) \ X+ = m, for every ∈ ]0, 0 ] and ε > 0. Proof. Since Ω is smooth enough, according to Theorem 2.3 there exists C > 0 such that
v(x0 ) = 0, v ∞ + ∇v ∞ C v p
for every v ∈ X− .
(3.7)
For every v ∈ X− and ∈ ]0, R], we have
|v | dx p
Ω
|v|p dx − Ln B (x ) sup |v|p . B (x )
Ω
On the other hand, since v(x0 ) = 0 it holds sup |v| 2 ∇v ∞ .
B (x )
Then (3.4) easily follows. The proof of (3.5) is similar. We also have
|∇v |p dx Ω
|∇v|p dx + CLn B (x ) sup |∇v|p + −p sup |v|p , B (x )
Ω
B (x )
whence assertion (3.6). From (3.4), (3.6) and (3.7) it follows that
1 |∇v | dx (λm + λm+1 ) 2
|v |p dx,
p
Ω
Ω
provided that is small enough. Therefore X− ∩ X+ = {0}. Moreover, for every v ∈ X− we have
1− p∗ p |v| dx Ln B (x )
Ω
Ω
S
−1
p∗
|v|
p
n 1− p∗ p L B (x )
S
λm Ln B (x )
dx
p p∗
+
|v|p dx
Ω\B (x )
|∇v| dx +
|v|p dx
p
Ω −1
1− pp∗
Ω\B (x )
|v| dx + p
Ω
Ω\B (x )
|v|p dx.
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If is small enough, we get
|v|p dx C
Ω
|v|p dx
for every v ∈ X− .
Ω\B (x )
First of all, it follows that e ,ε ∈ / X− and that we have v = 0 only for v = 0. Since {v → v } is
continuous and odd from X− \ {0} to X− \ {0}, it follows
1,p Index X− \ {0} Index X− \ {0} = Index W0 (Ω) \ X+ = m.
1,p
Actually, equality holds, as X− \ {0} ⊆ W0 (Ω) \ X+ . Finally, let (v (k) ) be a sequence in X− (k) with (v ) convergent to some z in Lp (Ω). Then (v (k) ) is bounded in Lp (Ω \ B (x )), hence in 1,p Lp (Ω), hence in W0 (Ω). Up to a subsequence, (v (k) ) is Lp (Ω)-convergent to some element
of X− , whence z ∈ X− . 2 1,p
Now let f : W0 (Ω) → R be the functional defined in (1.2). Lemma 3.3. Assume that Ω satisfies (1.4) and that (1.5) holds. Let m 1 be such that λm < λm+1 , λm λ and let X− be as in Theorem 2.3. Then there exist δ > 0 and two sequences εk → 0+ and k → 0+ with εk / k → 0+ such that 1
p n/p sup f (te k ,εk + w): t 0, w ∈ X−k S n/p 1 − δεk n for every k ∈ N.
Proof. Since X− is a cone, it is easily seen that
sup f (te ,ε + w): t 0, w ∈ X− n/p p p ∇(e ,ε + w) p − λ e ,ε + w p 1
sup = : w ∈ X p − n e ,ε + w p∗ n/p p p p p ( ∇e ,ε p − λ e ,ε p ) + ( ∇w p − λ w p ) 1
sup = : w ∈ X , − p∗ p∗ ∗ n ( e ,ε p∗ + w p∗ )p/p as supt (e ,ε ) ∩ supt (w) is negligible. Writing w = v with v ∈ X− , the assertion we need to prove takes the form sup
p
p
p
p
( ∇e ,ε p − λ e ,ε p ) + ( ∇v p − λ v p ) p∗
p∗
( e ,ε p∗ + v p∗ )p/p
∗
: v ∈ X− S 1 − δε p .
On the other hand, by Lemmas 3.1, 3.2 and the fact that λm λ, we have
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p
p
p
( ∇e ,ε p − λ e ,ε p ) + ( ∇v p − λ v p ) p∗
p∗
( e ,ε p∗ + v p∗ )p/p
∗
n−p n−p n p p S p + C( ε ) p−1 − λσ ε p + λC p ( ε ) p−1 + (C n v p∗ − λC n+p v p∗ ) . n n p∗ p ∗ p/p ∗ ∗ S p − C( ε ) p−1 + v p∗ − C n+p v p∗
p 2 /n2
Now, let εk → 0+ and let k = μεk with μ > 0 small enough, which will be determined later. We need to show that, for every sequence (vk ) in X− , −p εk
n−p n−p n p p p n+p p S p + C( εkk ) p−1 − λσ εk + λC k ( εkk ) p−1 + (C kn vk p∗ − λC k vk p∗ ) −1 n n p∗ n+p ∗ p ∗ p/p ∗ S S p − C( εkk ) p−1 + vk p∗ − C k vk p∗
has strictly negative upper limit as k → ∞. Up to subsequences, it is enough to consider the three cases: (i) vk p∗ → +∞, (ii) vk p∗ → ∈ ]0, +∞[ , (iii) vk p∗ → 0. In case (i) we get n−p n−p n p p p n+p p S p + C( εkk ) p−1 − λσ εk + λC k ( εkk ) p−1 + (C kn vk p∗ − λC k vk p∗ ) → 0, n n p∗ n+p ∗ p ∗ p/p ∗ S S p − C( εkk ) p−1 + vk p∗ − C k vk p∗
while in case (ii) we obtain n−p n−p n p p p n+p p S p + C( εkk ) p−1 − λσ εk + λC k ( εkk ) p−1 + (C kn vk p∗ − λC k vk p∗ ) n n p∗ n+p ∗ p ∗ p/p ∗ S S p − C( εkk ) p−1 + vk p∗ − C k vk p∗ n
→
Sp n
∗
S(S p + p )p/p
∗
< 1.
In both cases, the assertion easily follows. In case (iii), it is equivalent to consider, neglecting higher order terms, the upper limit of −p εk
n−p n p p S p + C( εkk ) p−1 − λσ εk + C kn vk p∗ −1 . n p ∗ p/p ∗ S S p + vk p∗
Since there exists a > 0 such that n n p ∗ p/p ∗ p∗ S p∗ + a vk p∗ , S p + vk p∗ we have
M. Degiovanni, S. Lancelotti / Journal of Functional Analysis 256 (2009) 3643–3659
−p εk
n−p
n
p
p
S p + C( εkk ) p−1 − λσ εk + C kn vk p∗ −1 n p ∗ p/p ∗ S S p + vk p∗
−p εk
n−p
n
p
S p + C( εkk ) p−1 − λσ εk + C kn vk p∗ p∗
n
SS p∗ + aS vk p∗ n−p
−p = εk
p
p
−1 p∗
p
C( εkk ) p−1 − λσ εk + C kn vk p∗ − aS vk p∗ p∗
n
3655
S p + aS vk p∗
.
By Young’s inequality, there exists C1 > 0 such that p C kn vk p∗
np ∗ p ∗ −p
C1 k
p∗ + aS vk p∗
n2 p
p∗
= C1 k + aS vk p∗ .
It follows −p
εk
n n−p n−p n p p p S p + C( εkk ) p−1 − λσ εk + C kn vk p∗ C( εkk ) p−1 − λσ εk + C1 kp −p − 1 εk . n n p ∗ p/p ∗ p∗ S S p + vk p∗ S p + aS vk p∗ 2
If we choose μ > 0 small enough to guarantee that 2
n n2 p 1 p C1 kp = C1 μ p εk λσ εk , 2 n−p
it only remains to control the term ( εkk ) p−1 by requiring n − p p2 n − p − > p. p − 1 n2 p − 1 This is exactly assumption (1.5) and the assertion follows.
2
Proof of Theorem 1.1. Let m 1 be such that λm λ < λm+1 , let X+ be as in (2.2) and X− be as in Theorem 2.3. Since λ < λm+1 , there exist r+ , α > 0 such that f (u) α for every u ∈ X+ with u = r+ . On the other hand, since λ λm , by Lemma 3.2 we also have, for every v ∈ X− , f (v )
C n λ 1 C 1 1 ∗ p p p∗ p∗ p∗
v p∗ − C n+p v p∗ − ∗ v p∗ + ∗ n+p v p∗ α − ∗ v p∗ p p p p 2 2p
if > 0 is small enough. Combining this fact with Lemmas 3.2 and 3.3, we see that there exist
ε, , δ > 0 such that e ,ε ∈ / X− , X− is closed in Lp (Ω) and
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X− ∩ X+ = {0},
1,p Index X− \ {0} = Index W0 (Ω) \ X+ = m,
n/p 1
, sup f (te ,ε + w): t 0, w ∈ X− S n/p 1 − δε p n 1
sup f (w): w ∈ X− α. 2
Since X− is closed in Lp (Ω), we have
te ,ε p∗ + w p∗ b te ,ε + w p∗
for every t ∈ R and w ∈ X−
for some b > 0 (see also [7]). It follows that f (u) → −∞
whenever u → ∞ with u ∈ Re ,ε + X− .
In particular, there exists r− > r+ such that f (u) 0 whenever u ∈ Re ,ε + X− with u = r− . From Theorem 2.1 we deduce that f admits a Palais–Smale sequence at a level c with 0 < c < n1 S n/p . On the other hand, by [14, Theorem 3.4] f satisfies the Palais–Smale condition at such a level. Then f admits a critical point u with 1 0 < f (u) < S n/p . n Of course, u is a nontrivial weak solution of (1.1).
2
Proof of Theorem 1.2. Since the general lines of the argument are the same, we only point out the changes. This time, given B2R (x) ⊆ Ω, we set as in [2] e ,ε (x) = u ,ε (x − x),
|x − x| v(x) for every v ∈ X− . v (x) = ϑ
Without any assumption on ∂Ω, we know from Theorem 2.3 that sup |∇v| + sup |v| C v p BR (x)
for every v ∈ X− .
BR (x)
Then Lemma 3.2 holds with (3.4), (3.5) and (3.6) substituted by Ω
Ω ∗
Ω
|∇v |p dx
(3.8)
,
Ω
∗
∗
|v|p dx, Ω
|∇v|p dx + C n−p Ω
p/p∗
|v|p dx
|v|p dx − C n Ω
∗
|v|p dx − C n
|v |p dx
Ω
|v |p dx
(3.9)
∗
|v|p dx Ω
p/p∗ .
(3.10)
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In the proof of Lemma 3.3, only case (iii) needs some adaptation. Neglecting higher order terms, we have to consider the upper limit of −p εk
n−p
n
p
n−p
S p + C( εkk ) p−1 − λσ εk + C k n p ∗ p/p ∗ S S p + vk p∗
p
vk p∗
−1 ,
which is less than or equal to n−p
−p εk
p
n−p
C( εkk ) p−1 − λσ εk + C k
p∗
p
vk p∗ − aS vk p∗ p∗
n
S p + aS vk p∗
.
By Young’s inequality, there exists C1 > 0 such that n−p
C k
(n−p)p ∗ ∗ −p
p
vk p∗ C1 k p
(n−p)n p
p∗
+ aS vk p∗ = C1 k
p∗
+ aS vk p∗ ,
whence −p
εk
n−p
n
p
n−p
S p + C( εkk ) p−1 − λσ εk + C k n p ∗ p/p ∗ S S p + vk p∗ p 2 /(n−p)n
Here we choose k = μεk
p
vk p∗
n−p
(n−p)n p
p
C( εkk ) p−1 − λσ εk + C1 k −p
− 1 εk
p∗
n
S p + aS vk p∗
.
with μ > 0 small enough to guarantee that (n−p)n p
C1 k
= C1 μ
(n−p)n p
1 p p εk λσ εk . 2
n−p
In the end, to control the term ( εkk ) p−1 , we have to require that p2 n−p n−p − > p. p − 1 (n − p)n p − 1 This is exactly assumption (1.6) and the assertion follows.
2
Proof of Theorem 1.3. We follow step by step the proof of Theorem 1.2, taking as x the center of Ω and working in the space of radial functions. It is easily seen that the proof of Theorem 2.3 and all the other constructions are compatible with radiality. Then the assertion follows in a standard way. 2 Proof of Theorem 1.4. Again the proof is similar to that of Theorem 1.2. We only point out the changes, concerning case (iii) in the proof of Lemma 3.3. First of all, now λm < λ < λm+1 . Since p
p
p
p
p
p
∇v p − λ v p ∇v p − λ v p + C n−p v p∗ + C n v p∗ n−p −Sλ−1 v p∗ + C n v p∗ , m (λ − λm ) v p∗ + C
p
p
p
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up to higher order terms, we have to consider the upper limit of −p εk
n−p
n
S p + C( εkk ) p−1 − λ e k ,εk p − Sλ−1 m (λ − λm ) vk p∗ + C k n p ∗ p/p ∗ S S p + vk p∗ p
p
n−p
p
vk p∗
−1 .
In turn, it is enough to argue on the upper limit of −p εk
n
n−p
p
S p + C( εkk ) p−1 − λ e k ,εk p n
SS p∗
−1
n−p
−p = εk
p
C( εkk ) p−1 − λ e k ,εk p n
.
Sp
Now, in both cases n > p 2 and n = p 2 , it is easily seen that, for every sequence εk → 0+ , there exists some sequence ( k ), going to 0 slowly enough, which guarantees the result. 2 Proof of Theorem 1.5. It is enough to repeat the proof of Theorem 1.4 in the setting of radial functions. 2 References [1] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. [2] G. Arioli, F. Gazzola, Some results on p-Laplace equations with a critical growth term, Differential Integral Equations 11 (1998) 311–326. [3] G. Arioli, F. Gazzola, H.-C. Grunau, E. Sassone, The second bifurcation branch for radial solutions of the Brezis– Nirenberg problem in dimension four, NoDEA Nonlinear Differential Equations Appl. 15 (2008) 69–90. [4] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983) 437–477. [5] A. Capozzi, D. Fortunato, G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985) 463–470. [6] S. Cingolani, M. Degiovanni, Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity, Comm. Partial Differential Equations 30 (2005) 1191–1203. [7] M. Degiovanni, S. Lancelotti, Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) 907–919. [8] E. DiBenedetto, C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983) 827–850. [9] H. Egnell, Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponents, Arch. Ration. Mech. Anal. 104 (1988) 57–77. [10] E.R. Fadell, P.H. Rabinowitz, Bifurcation for odd potential operators and an alternative topological index, J. Funct. Anal. 26 (1977) 48–67. [11] E.R. Fadell, P.H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978) 139–174. [12] J. García Azorero, I. Peral Alonso, Existence and nonuniqueness for the p-Laplacian: Nonlinear eigenvalues, Comm. Partial Differential Equations 12 (1987) 1389–1430. [13] F. Gazzola, B. Ruf, Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations, Adv. Differential Equations 2 (1997) 555–572. [14] M. Guedda, L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (1989) 879–902. [15] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988) 1203–1219. [16] P. Lindqvist, On the equation div(|∇u|p−2 ∇u) + λ|u|p−2 u = 0, Proc. Amer. Math. Soc. 109 (1990) 157–164. [17] K. Perera, Nontrivial critical groups in p-Laplacian problems via the Yang index, Topol. Methods Nonlinear Anal. 21 (2003) 301–309.
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[18] K. Perera, A. Szulkin, p-Laplacian problems where the nonlinearity crosses an eigenvalue, Discrete Contin. Dyn. Syst. 13 (2005) 743–753. [19] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1986. [20] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984) 126–150.
Journal of Functional Analysis 256 (2009) 3660–3687 www.elsevier.com/locate/jfa
Existence of a local smooth solution in probability to the stochastic Euler equations in R 3 Jong Uhn Kim Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA Received 8 September 2008; accepted 19 March 2009 Available online 28 March 2009 Communicated by Paul Malliavin
Abstract We establish the existence of a local smooth solution of the stochastic Euler equations in R 3 . Probabilistic estimate of the random time interval for the existence of a local solution is expressed in terms of expected values of the initial data and the random noise. There are numerous works on the stochastic Euler equations in a two-dimensional domain. Even for the deterministic Euler equations in a three-dimensional domain, the only results are concerned with the local existence of smooth solutions. Our goal is to extend such local existence result to the stochastic equations. © 2009 Elsevier Inc. All rights reserved. Keywords: Stochastic Euler equation; Smooth solutions; Kato’s method
0. Introduction In this paper, we study an initial value problem for the Euler equations with random noise in R 3 . We formulate the problem as follows. ut + (u · ∇)u + ∇p =
∞ j =1
∇ · u = 0,
gj
dBj , dt
(x, t) ∈ R 3 × (0, T ), ω ∈ Ω,
(x, t) ∈ R 3 × (0, T ), ω ∈ Ω,
u(x, 0, ω) = u0 (x, ω),
x ∈ R , ω ∈ Ω, 3
E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.03.012
(0.1) (0.2) (0.3)
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where u = (u1 (x, t, ω), u2 (x, t, ω), u3 (x, t, ω)) is the velocity field, p = p(x, t, ω) is the pressure, gj = gj (x, t, ω), j = 1, 2, . . . , are given random functions, and {Bj }∞ j =1 is a sequence of mutually independent standard Brownian motions over a given sample space Ω. For the deterministic equations ut + (u · ∇)u + ∇p = h,
(x, t) ∈ R 3 × (0, T ),
(0.4)
where h = h(x, t) is a given function, the only known existence results are concerned with the local existence of smooth solutions. See [10–12,17]. Also, see [14] for comments on the known results for the Euler equations, and an extensive list of references. For the stochastic Euler equations in a two-dimensional domain, there are numerous works. We cite only some of them. The existence of martingale solution was proved in [1] and [3], while [2] and [13] obtained strong solutions in the probabilistic sense. In [4], nonstandard analysis was used to obtain solutions on a Loeb space with prescribed Winer process. [6] employed an approach of geodesics on an infinite dimensional Riemannian manifold. But for a three-dimensional domain, there seems to be only one work [15], which discussed ut + (u · ∇)u + ∇p =
dB , dt
(x, t) ∈ R d × (0, T ), ω ∈ Ω, d 2,
(0.5)
where is a constant, and B = (B1 (t), · · · , Bd (t)) is a d-dimensional standard Brownian motion. The solution of the initial value problem for (0.5) can be represented by
t
u(x, t) = uˆ x −
B(η) dη, t + B(t)
(0.6)
0
where uˆ = u(x, ˆ t) is the solution of the deterministic equation with = 0 and the same initial condition. This was derived in [15] and the solution was obtained in the Hölder spaces with weights. Obviously the formula (0.6) is not valid if the random noise depends on the space variables. We note that this formalism is different in spirit from Feyman–Kac representation of solutions of deterministic equations. Recently, [5] discussed representation formula for the deterministic Navier–Stokes equations in terms of a stochastic process. One of the well-known results for the deterministic equations (0.4) is that if u0 ∈ H s (R 3 )3 , for some s > 5/2, ∇ · u0 = 0, and h ∈ L1 (0, T ; H s (R 3 )3 ) ∩ C([0, T ]; H s−1 (R 3 )3 ), there is a unique solution u ∈ C([0, T˜ ]; H s (R 3 )3 ) ∩ C 1 ([0, T˜ ]; H s−1 (R 3 )3 ), for some 0 < T˜ T . Here our goal is to obtain a similar result for the problem (0.1)–(0.3). The main result is Theorem 1.2 below. The method of proof is based on the information from the deterministic case. For the deterministic Euler equations in R 3 , it is known that the existence of a smooth solution is guaranteed on the time interval where the L∞ -norm of the velocity gradient is finite. This information motivates our method to control the nonlinear convection term in (0.1). Our main tool consists of a cut-off function and a stopping time, which also leads to the definition of a local solution of (0.1)–(0.3) in a natural way. See Definition 1.1 below. Our procedure to construct a solution is classical; see the outline given at the end of Section 1. At present, we can handle only additive noise. The case of multiplicative noise is still an open question. In view our assumption on gj ’s in (0.1), our result could be a basis for the standard iteration scheme for the case of multiplicative noise. But it is interesting to note that the cut-off function has both positive and negative effect on the iteration scheme.
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1. Notation and statement of the main result For any nonnegative number s, H s (R 3 ) denotes the usual Sobolev space s/2
f ∈ L2 R 3 1 + |ξ |2 fˆ(ξ ) ∈ L2 R 3 where fˆ(ξ ) is the Fourier transform of f . We denote by ·,·s and · · · s the inner product and the norm of H s (R 3 )3 , respectively. We define, for s 0,
3 Hσs = f ∈ H s R 3 ∇ · f = 0 which is a closed subspace of H s (R 3 )3 . The symbol Π stands for the projection 3 H s R 3 → Hσs . In fact, Π can be explicitly expressed by means of the Fourier transform. For each h = (h1 , h2 , h3 ) ∈ H s (R 3 )3 , (Πh)ˆj = hˆ j −
3 ξj ξk hˆ k , |ξ |2
j = 1, 2, 3.
k=1
For h = (h1 (x), h2 (x), h3 (x)), we write ∇hL∞ (R 3 ) =
3
∂i hj L∞ (R 3 ) .
i,j =1
We also write ∇h ∈ H s R 3 ,
if
∂hi ∈ H s R3 , ∂xj
for all i, j = 1, 2, 3,
and ∇hs =
3 ∂hi ∂x
i,j =1
j
.
H s (R 3 )
Throughout this paper, a stochastic basis (Ω, F , {Ft }, P ) is given and fixed, where {Ft } is a right continuous filtration over the probability space {Ω, F , P } such that F0 contains all P -negligible subsets of Ω. {Bj }∞ j =1 is a sequence of mutually independent standard Brownian motions on (Ω, F , {Ft }, P ). We assume that gj , j 1, is H α (R 3 )3 -valued progressively measurable for some α > 5/2 such that ∞
T
j =1 0
2 E gj (t) α dt < ∞,
for each T > 0.
For the general information on stochastic analysis, see [7] and [9]. We propose the following definition of a local smooth solution of (0.1)–(0.3).
(1.1)
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Definition 1.1. A pair (X, τ ) is called a local smooth solution of (0.1)–(0.3) if the following conditions are satisfied. (i) X = X(t, ω) is a Hσα (R 3 )-valued right continuous stochastic process adapted to {Ft }, for some α > 5/2. (ii) τ is a stopping time with respect to {Ft } such that τ (ω) = lim τN (ω),
for almost all ω
N →∞
(1.2)
where we define τN (ω) =
inf{0 t < ∞ | ∇X(t, ω)L∞ (R 3 ) N }, ∞, if the above set {· · ·} is empty.
(1.3)
(iii) X(·, ω) ∈ C([0, τ (ω)); Hσα ), for almost all ω, and t X(t ∧ τN ) = u0 −
Π X(s) · ∇ X(s)χ[0,τN ] (s) ds
0
+Π
∞
t
(1.4)
gj (s)χ[0,τN ] (s) dBj (s),
j =1 0
for all 0 t < ∞, and all N 1, for almost all ω, where χ[0,τN ] denotes the characteristic function for the interval [0, τN ]. This is analogous to Definition 5.1 [9, p. 329]. By virtue of (1.2) and (1.3), it holds that τ = τ (ω) > 0, for almost all ω. Our main result is the following. Theorem 1.2. Suppose that u0 is Hσα -valued F0 -measurable for some α > 5/2 such that u0 ∈ L2 (Ω; Hσα ), and that gj , j 1 satisfies (1.1) with the same α. Then, there is a unique local smooth solution of (0.1)–(0.3). Furthermore, we have the following estimate of τ . P {τ > δ} 1 − Cδ
2
∞ E u0 2α +
δ
j =1 0
2 E gj (t) α dt
(1.5)
for all 0 < δ < 1, where C denotes a positive constant independent of u and δ. In the above statement, the uniqueness of a solution holds in the following sense. Suppose ˜ τ˜ ) are solutions of (0.1)–(0.3). Then, for almost all ω, (X, τ ) and (X, τ (ω) = τ˜ (ω),
˜ X(t) = X(t),
We now outline the strategy of proof of Theorem 1.2.
for all t ∈ 0, τ (ω) .
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Step 1. Introduce a cut-off function to control the nonlinear convection term in (0.1). Regularize the initial data and the random noise with respect to the space variables so that the problem can be reduced to essentially a deterministic problem. We then apply Kato’s method [11] to obtain a smooth global solution for each sample point ω. Step 2. Fix T > 0, and obtain energy estimates of approximate solutions on the time interval [0, T ]. By means of these estimates which depend only on the original regularity of the initial data and random noise before approximation, we can construct a pathwise solution of (0.1) modified by a cut-off function. We introduce a stopping time whose ultimate goal is to remove the cut-off function later on. Step 3. Pass T → ∞, and then, pass N → ∞ where N is a parameter in the cut-off function such that N = ∞ makes the cut-off function an identity map. The limit function will be a desired solution. We will present complete details of the proof in the remaining sections. 2. Construction of pathwise approximate solutions We assume the same conditions on u0 and gj ’s as in Theorem 1.2 with fixed α > 5/2. Let ρ = ρ (x), > 0, be the Friedrich mollifier, and define u0, = u0 ∗ ρ , M (t) = Π
∞
(2.1)
t
gj (s) ∗ ρ dBj (s)
(2.2)
j =1 0
where the convolution is taken with respect to the space variables. Then, M is a Hσm -valued continuous square integrable martingale for every m 1. For each integer N 1, define ψN on R such that ⎧ ∀ξ N, ⎨ 1, ψN (ξ ) = 1 + N − ξ, for N ξ N + 1, ⎩ 0, ∀ξ N + 1. Let us fix N 1, > 0, and ω ∈ Ω at which u0, ∈ Hσα+1 and M ∈ C([0, ∞); Hσα+2 ). We then define the nonlinear operator A(t, v) = ψN ∇ v + M (t) L∞ (R 3 ) Π v + M (t) · ∇ and the function f (t, v) = −ψN ∇ v + M (t) L∞ (R 3 ) Π v + M (t) · ∇ M (t). We will apply Kato’s result [11] to the following initial value problem. dv + A(t, v)v = f (t, v), dt v(0) = u0, .
(2.3) (2.4)
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Let Λ = (I − )1/2 where is the Laplacian in R 3 . We now list various properties of Λ, A(t, v), and f (t, v), where β = β(r, T ), λi = λi (r, T ), i = 1, 2, and μi = μi (r, T ), 1 i 4, are certain nonnegative functions defined for r 0 and T > 0 which are nondecreasing in r and T . Here we borrow notation from [11]. (I) Λ is an isomorphism of Hσα+1 onto Hσα . (II) For each v ∈ Hσα+1 with vα+1 r, and each t ∈ [0, T ], A(t, v) + β(r, T ) is m-accretive in Hσα . (III) For each v ∈ Hσα+1 and t 0, there is a bounded linear operator B(t, v) on Hσα such that ΛA(t, v)Λ−1 = A(t, v) + B(t, v), B(t, v) α λ1 (r, T ) L(H ) σ
and B(t, v) − B(t, w)
L(Hσα )
μ1 (r, T )v − wα+1
for all v, w ∈ Hσα+1 with vα+1 , wα+1 r, and all t ∈ [0, T ], where · · · L(Hσα ) denotes the operator norm of bounded linear operators on Hσα . (IV) For each v ∈ Hσα+1 and t 0, A(t, v) is a bounded linear operator from Hσα+1 into Hσα such that A(t, v) − A(t, w)
L(Hσα+1 ,Hσα )
μ2 (r, T )v − wα
for all v, w ∈ Hσα+1 with vα+1 , wα+1 r, and all t ∈ [0, T ], where · · · L(Hσα+1 ,H α ) denotes the operator norm of bounded linear operators from Hσα+1 to Hσα . (V) For each v ∈ Hσα+1 with vα+1 r and t ∈ [0, T ], f (t, v) is Hσα+1 -valued and f (t, v)
α+1
λ2 (r, T ),
and the map t → f (t, v) is continuous from [0, T ] into Hσα . Also, for all v, w with vα+1 r and wα+1 r, f (t, v) − f (t, w) μ3 (r, T )v − wα , α f (t, v) − f (t, w) μ4 (r, T )v − wα+1 . α+1
σ
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Property (I) is obvious by noting that Λ commutes with Π . Property (IV) follows from ψN ∇ v + M (t) ∞ 3 Π v + M (t) · ∇ L (R ) − ψN ∇ w + M (t) L∞ (R 3 ) Π w + M (t) · ∇ L(H α+1 ,H α ) σ σ C ∇(v − w) L∞ (R 3 ) vα + M (t) α + Cv − wα C vα + M (t) α + C v − wα for some positive constants C independent of v, w, and M . Properties (II) and (III) were established in [12] when ψN ≡ 1 and M ≡ 0. But ψN (· · ·) is independent of space variables, and 0 ψN (· · ·) 1. Also, we note that M ∈ C([0, ∞); Hσα+2 ), is a given function, and it plays the same role as v in Π((v + M ) · ∇). Hence, the proof in [12] can be applicable to our case for (II) and (III). Property (V) also follows from the fact M ∈ C([0, ∞); Hσα+2 ). Under the above properties (I) through (V), we can borrow a result from [11] for the following existence result. Proposition 2.1. For fixed N 1, > 0, and ω ∈ Ω at which u0, ∈ Hσα+1 and M ∈ C([0, ∞); Hσα+2 ), there is a unique solution v ∈ C [0, ∞); Hσα+1 ∩ C 1 [0, ∞); Hσα of (2.3)–(2.4). According to [11], we first have a unique local solution v ∈ C [0, T ]; Hσα+1 ∩ C 1 [0, T ]; Hσα for some T > 0. This local solution can be extended to a global solution in time through some estimates. For this, we will need some technical lemmas. The proof of the following version of the Friedrichs lemma can be found in [8]. Lemma 2.2. Let w be a Lipschitz continuous function in R 3 and v ∈ L2 (R 3 ). Then, it holds that (w∂j v) ∗ ρδ − w(∂j v ∗ ρδ )
L2 (R 3 )
C∇wL∞ (R 3 ) vL2 (R 3 )
for some constant C > 0 independent of δ > 0, w and v. For each fixed v and w, the left-hand side tends to zero as δ → 0. The following commutator estimate is a special version of the estimate proved in [12]. Lemma 2.3. Let f, g ∈ H s (R 3 ), for some s > 5/2. It holds that s Λ (f g) − f Λs g
L2 (R 3 )
C ∇f L∞ (R 3 ) gH s−1 (R 3 ) + f H s (R 3 ) gL∞ (R 3 )
for some constant C > 0 independent of f and g.
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It follows that for all v, w ∈ Hσα+1 , Π (w · ∇) Λα+1 v ∗ ρδ , Λα+1 v ∗ ρδ C∇w ∞ 3 v2 , L (R ) α+1 0 α+1 α+1 Π Λ (w · ∇)v ∗ ρδ − Π (w · ∇)Λ v ∗ ρδ 0 C∇wL∞ vα+1 + Cwα+1 ∇vL∞ (R 3 ) , Π (w · ∇)Λα+1 v ∗ ρδ − Π (w · ∇) Λα+1 v ∗ ρδ 0
(2.5)
(2.6)
C∇wL∞ (R 3 ) vα+1 ,
(2.7)
lim Π (w · ∇)Λα+1 v ∗ ρδ − Π (w · ∇) Λα+1 v ∗ ρδ 0 = 0
(2.8)
and δ→0
where C denotes positive constants independent of v, w, and δ > 0. Let v be the solution on the interval [0, T ] in Proposition 2.1. For each δ > 0, it holds that ∂ v ∗ ρδ + ψN ∇(v + M ) L∞ (R 3 ) Π (v + M ) · ∇ v ∗ ρδ ∂t + ψN ∇(v + M ) L∞ (R 3 ) Π (v + M ) · ∇ M ∗ ρδ = 0
(2.9)
for all t ∈ [0, T ]. By virtue of (2.5)–(2.7), we have α+1 Λ (v + M ) · ∇ v ∗ ρδ , Λα+1 v ∗ ρδ 0 C vα+1 + M α+1 ∇vL∞ (R 3 ) + ∇M L∞ (R 3 ) vα+1
(2.10)
for some positive constants C independent of v, M , and δ > 0. On the other hand, we can directly estimate α+1 Λ (v + M ) · ∇ M ∗ ρδ , Λα+1 v ∗ ρδ 0 CM α+2 vα+1 + M α+1 vα+1
(2.11)
for some positive constant C independent of v, M , and δ > 0. By passing δ → 0, it follows from (2.9)–(2.11) that v(t) 2
α+1
2 v(0) α+1 + C + C sup M (s) α+2 0st
t ×
v(s) 2
α+1
0
t ds +
M (s) 2
α+1
ds
(2.12)
0
for all t ∈ [0, T ], for some positive constants C independent of T . This yields that v(t)α+1 is bounded on each bounded time interval. Hence, the solution v can be extended to a global solution. Next we establish the measurability of the above v as a function of ω ∈ Ω.
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Proposition 2.4. For each T > 0, the map ω → v from Ω into C([0, T ]; Hσα+1 ) is FT measurable. Proof. Let us first consider continuous dependence of v on u0, and M . Suppose that as n → ∞, un0 → u0,
in Hσα+1
and M n → M
in C [0, T ]; Hσα+2 ,
for each T > 0.
Let us define the nonlinear operator An (t, v) = ψN ∇ v + M n (t) L∞ (R 3 ) Π v + M n (t) · ∇ and the function fn (t, v) = −ψN ∇ v + M n (t) L∞ (R 3 ) Π v + M n (t) · ∇ M n (t). We also define Bn such that ΛAn (t, v)Λ−1 = An (t, v) + Bn (t, v). Then, An , Bn and fn satisfy all the above properties uniformly in n, and strongly in L Hσα+1 , Hσα , Bn (t, v) → B(t, v) strongly in L Hσα , Hσα ,
An (t, v) → A(t, v)
fn (t, v) → f (t, v)
in Hσα+1
for each v ∈ Hσα+1 and t 0. Let vn be the solution of dvn + An (t, vn )vn = fn (t, vn ), dt vn (0) = un0 . According to Theorem 7 in [11], there is some T > 0 such that vn → v
in C [0, T ]; Hσα+1 .
α+1 ), and {M n }∞ is For any arbitrary T > 0, {vn }∞ n=1 is uniformly bounded in C([0, T ]; Hσ n=1 α+2 uniformly bounded in C([0, T ]; Hσ ). So we can partition [0, T ] into a number of smaller subintervals to establish the continuous dependence of v on u0, and M on the interval [0, T ]. Now the map
ω → (u0, , M )
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from Ω into Hσα+1 × C([0, T ]; Hσα+2 ) is FT -measurable for each T > 0, and the map (u0, , M ) → v is continuous from Hσα+1 × C([0, T ]; Hσα+2 ) into C([0, T ]; Hσα+1 ). Thus, the map ω → v from Ω into C([0, T ]; Hσα+1 ) is FT -measurable for each T > 0.
2
3. Proof of Theorem 1.2 We will construct a solution by means of approximate solutions in the previous section. For clarity, we divide the construction procedure into three steps. 3.1. Fix N 1 and T > 0 We fix any N 1 and any T > 0 throughout this subsection. Recalling (2.1) and (2.2), we can choose a sequence {m } of decreasing positive numbers such that u0,m → u0
in Hσα
and M m → M
in C [0, T ]; Hσα
as m → 0, for almost all ω, where we write M(t) = Π
∞
t
gj (s) dBj (s).
j =1 0
Let um be defined by um = vm + Mm ,
m = 1, 2, . . . ,
where vm is the solution in Propositions 2.1 and 2.4 with = m . Then, um ∈ C [0, ∞); Hσα+1 ,
for almost all ω,
and it is Hσα+1 -valued progressively measurable. It holds that ∂um ∂Mm + ψN ∇um L∞ (R 3 ) Π(um · ∇)um = ∂t ∂t in the sense of distributions over R 3 × (0, ∞), for almost all ω.
(3.1)
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Next we define a stopping time Tm,K for m, K = 1, 2, . . . , by Tm,K =
inf{0 t T | um α K}, T , if the set {· · ·} is empty.
By Ito’s formula, we derive from (3.1) um (t ∧ Tm,K ) 2
α t∧T m,K
2 = um (0) α − 2
ψN ∇um (s) L∞ (R 3 ) um (s) · ∇ um (s), um (s) α ds
0
+2
t∧T m,K ∞
um (s), gj (s) ∗ ρm
j =1
t∧T m,K ∞ Πgj (s) ∗ ρ 2 ds dBj (s) + m α α
j =1
0
(3.2)
0
for all t 0, for almost all ω. It is easy to find that ψN ∇um (s)
L∞ (R 3 )
2 (um (s) · ∇)um (s), um (s) α CN um (s) α
(3.3)
for some positive constant CN independent of m and s. By the Burkholder–Davis–Gundy inequality, ∞ η∧T m,K E sup um (s), gj (s) ∗ ρm α dBj (s) 0ηt
j =1
0
1/2 t∧T m,K ∞ 2 um (s), gj (s) ∗ ρm α ds CE j =1
CE
0
1/2 ∞ t∧T m,K 2 gj (s) ∗ ρ ds sup um (s ∧ Tm,K ) α m α
0st
j =1
(3.4)
0
where C denotes positive constants independent of m, K and t. Combining (3.2)–(3.4), we arrive at 2 E sup um (t ∧ Tm,K ) CN,T 0tT
α
where CN,T is a constant independent of m and K. By passing K → ∞, we obtain E
2 sup um (t) α CN,T .
0tT
(3.5)
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Next we consider the set Ω1 =
∞ ∞ ∞ um C [0,T ];Hσα L . L=1 k=1 m=k
By (3.5), we see that P (Ω1 ) = 1. Let Ω2 be the set of all ω for which, as m → ∞, M m → M
in C [0, T ]; Hσα ,
u0,m → u0
in Hσα ,
and R 3 × (0, ∞),
(3.1) holds in the sense of distributions over
for all m 1.
We then define Ω ∗ = Ω1 ∩ Ω2 . Then, P (Ω ∗ ) = 1. For convenience of notation, we write Mˆ m = Mm and fix any ω∗ ∈ Ω ∗ . Then, there is some L = L(ω∗ ) 1, and a subsequence denoted by {umj } such that umj C([0,T ];Hσα ) L,
for all mj .
(3.6)
The choice of such a subsequence may depend on ω∗ . Since it holds that ∂ (umj − Mˆ mj ) = −ψN ∇umj L∞ (R 3 ) Π(umj · ∇)umj , ∂t in the sense of distributions over R 3 × (0, T ), it follows from (3.6) that ∂ (um − Mˆ m ) j j ∂t
C([0,T ];Hσα−1 )
Cumj 2C([0,T ];H α ) , σ
(3.7)
where C is a positive constant independent of umj and ω∗ . We now fix a positive number β such that 5/2 < β < α.
(3.8)
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By virtue of (3.6)–(3.8), we can further extract a subsequence still denoted by {umj } such that for some function u, umj → u weak star in L∞ 0, T ; Hσα
(3.9)
and umj → u
3 strongly in C [0, T ]; H β (G)
(3.10)
for every bounded open ball G in R 3 , as mj → ∞. Here we have used Corollary 8 in [16]. Next we will show that ∇umj → ∇u
strongly in C [0, T ]; H β−1 R 3 .
(3.11)
For this, choose a function Θ ∈ C ∞ (R 3 ) such that Θ(x) = 1,
for |x| 2,
and Θ(x) = 0,
for |x| 1.
and define ΘR (x) = Θ(x/R),
for R 1.
(3.12)
We will need the following fact. Lemma 3.1. Let s > 0. For all h ∈ H s (R 3 ), it holds that ΘR hs Chs and (∇ΘR )h C hs s R for some positive constants C independent of h and R 1. This can be easily shown when s is an integer. For general s > 0, the above inequalities follow through interpolation. Since the interaction of ΘR with the projection operator Π is not easy to handle for our purpose, we remove Π by considering the vorticity. Let wm = ∇ × um . By (3.1), it holds that ∂ (ΘR wm − ΘR ∇ × Mˆ m ) + ψN ∇um L∞ (R 3 ) (um · ∇)ΘR wm ∂t = ψN ∇um L∞ (R 3 ) (um · ∇ΘR )wm + ψN ∇um L∞ (R 3 ) (ΘR wm · ∇)um .
(3.13)
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By means of (3.6) and Lemma 3.1, we derive from (3.13) ΘR wm (t) − ΘR ∇ × Mˆ m (t) 2 ΘR wm (0) 2 j j j 0 0 t +C
ΘR wm (s) 2 ds + C j 0
0
+
t
ΘR ∇ × Mˆ m (s) ds j 0
0
C +C R2
t
ΘR ∇ × Mˆ m (s) 2 ds j 0
(3.14)
0
for all t ∈ [0, T ], where C denotes positive constants independent of mj , R 1, and t ∈ [0, T ]. Since wmj (0) → ∇ × u0
in Hσα−1
and Mˆ mj → M
in C [0, T ]; Hσα ,
as mj → ∞,
we see that 3 ΘR wmj (0) → 0 strongly in L2 R 3 and ΘR ∇ × Mˆ mj (t) → 0
3 strongly in L2 R 3
as R → ∞ uniformly in mj and t ∈ [0, T ]. It follows that ΘR wm (t) 2 R + C j 0
t
ΘR wm (s) 2 ds, j 0
∀t ∈ [0, T ],
(3.15)
0
where R → 0 as R → ∞ uniformly in mj and t ∈ [0, T ]. By Gronwall’s inequality, it follows from (3.15) that ΘR wm (t) → 0, as R → ∞, uniformly in mj and t ∈ [0, T ]. j 0 On the other hand, by (3.6) and Lemma 3.1, there is a positive constant C independent of mj , R 1, and t ∈ [0, T ] such that ΘR wm (t) C. j α−1 By the interpolation inequality hβ−1 Ch01−λ hλα−1 ,
3 ∀h ∈ H α−1 R 3 ,
for some constant C with λ =
β−1 α−1 ,
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it follows that ΘR wm (t) → 0, j β−1
as R → ∞, uniformly in mj and t ∈ [0, T ].
(3.16)
Again by Lemma 3.1 and the identity ∇ × (ΘR umj ) = ΘR ∇ × umj + ∇ΘR × umj we can infer from (3.16) that ∇ × ΘR um (t) →0 j β−1
(3.17)
as R → ∞ uniformly in mj and t ∈ [0, T ]. We note the identity 3
−1 ∂j ∂i (∂i vk − ∂k vi ) = ∂j vk − −1 ∂j ∂k (∇ · v)
(3.18)
i=1
for j, k = 1, 2, 3, and v = (v1 , v2 , v3 ) ∈ H 1 (R 3 )3 , and that (∇ΘR ) · um j β−1 → 0
(3.19)
as R → ∞, uniformly in mj and t ∈ [0, T ]. Since the Riesz transform is continuous from H s (R 3 ) into itself for any s, it follows from (3.17)–(3.19) that ∇ ΘR umj (t) → 0 in H β−1 R 3 (3.20) as R → ∞ uniformly in mj and t ∈ [0, T ]. Now (3.11) follows from (3.10) and (3.20). Thus, we see that ψN ∇umj L∞ (R 3 ) → ψN ∇uL∞ (R 3 ) strongly in C [0, T ] . This, together with (3.10), yields that ∂u ∂M + ψN ∇uL∞ (R 3 ) Π(u · ∇)u = , ∂t ∂t holds in the sense of distributions over R 3 × (0, T ), at ω∗ . We now show that u ∈ C [0, T ]; Hσβ at ω∗ .
(3.21)
(3.22)
By virtue of (3.9) and (3.21), we have ∂ (u − M) ∈ L∞ 0, T ; Hσα−1 , ∂t which, combined with u0 ∈ Hσα and M ∈ C([0, T ]; Hσα ), yields u ∈ C [0, T ]; Hσα−1 . Since α − 1 > β − 1, (3.11) and (3.23) imply (3.22).
(3.23)
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Next let u1 and u2 be two functions which satisfy (3.21), (3.22) and u1 (0) = u2 (0) = u0 , for the same fixed ω∗ ∈ Ω ∗ . Define τˆN,i =
inf{0 t T | ∇ui (t)L∞ (R 3 ) N }, T , if the set {· · ·} is empty,
i = 1, 2.
Assume τˆN,1 ∧ τˆN,2 > 0. Since ui ∈ C [0, T ]; Hσβ ,
i = 1, 2,
and ∂ (u1 − u2 ) + Π(u1 · ∇)u1 − Π(u2 · ∇)u2 = 0 ∂t in the sense of distributions over R 3 × (0, τˆN,1 ∧ τˆN,2 ), we have u1 (t) − u2 (t) 2 C
t
0
u1 (s) − u2 (s) 2 ds 0
0
for all t ∈ [0, τˆN,1 ∧ τˆN,2 ], for some constant C, which yields u1 (t) = u2 (t),
for all t ∈ [0, τˆN,1 ∧ τˆN,2 ].
Thus, τˆN,1 = τˆN,2 . If τˆN,1 ∧ τˆN,2 = 0, then τˆN,1 = τˆN,2 = 0 is obvious. Hence, for each ω ∈ Ω ∗ , τˆN associated with a limit function u (through the above procedure) of a certain subsequence {umj } is determined uniquely, and also the limit function u itself is unique on the interval [0, τˆN ]. Next we will show that τˆN is a stopping time, and that u(· ∧ τˆN ) is measurable. Lemma 3.2. τˆN is a stopping time. Proof. Choose any 0 t < T . We claim that {τˆN > t} ∩ Ω ∗
∞ ∞ ∞ ∞
1 ∇um C([0,t];L∞ (R 3 )) N − ∩ um C([0,t];Hσα ) L . (3.24) =Ω ∩ ν ∗
L=1 ν=1 k=1 m=k
To see this, let ω belong to the left-hand set. Then, according to the above procedure, there is a certain subsequence {umj } such that (3.6), (3.9)–(3.11) hold for some function u. We know that τˆN can be defined in terms of this limit function u. Since τˆN > t, it must hold ∇uC([0,t];L∞ (R 3 )) N − δ
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for some δ, which implies ∇umj C([0,t];L∞ (R 3 )) N −
δ 2
for all sufficiently large mj . Thus, ω belongs to the right-hand set. Next let ω belong to the righthand set. Since ω ∈ Ω ∗ , there is a subsequence {umj } as above satisfying (3.6), (3.9)–(3.11) for some u, and τˆN can be defined by this u. There must be also another subsequence {u˜ mj } such that for some L 1 and ν 1, u˜ mj C([0,t];Hσα ) L,
for all mj
and 1 ∇ u˜ mj C([0,t];L∞ (R 3 )) N − , ν
(3.25)
for all mj .
By repeating the above procedure on the interval [0, t], we can further extract a subsequence still denoted by {u˜ mj } which satisfies (3.6) and (3.9)–(3.11) for some function u˜ with T replaced by t. By (3.25), it holds that 1 ∇ u ˜ C([0,t];L∞ (R 3 )) N − . ν β
(3.26)
β
Since u˜ ∈ C([0, t]; Hσ ), u ∈ C([0, T ]; Hσ ), and ∂ (u − u) ˜ + Π(u · ∇)u − Π(u˜ · ∇)u˜ = 0 ∂t
(3.27)
holds in the sense of distributions over R 3 × (0, t ∧ τˆN ), we have u ≡ u˜ on [0, t ∧ τˆN ], Hence, it follows from (3.26) that τˆN > t. So ω also belongs to the left-hand set, and (3.24) is valid. Since the right-hand set of (3.24) is Ft -measurable, the set τˆN > t} is Ft -measurable for 0 t < T . For t T , the left-hand set of (3.24) is empty, and it is Ft -measurable. Thus, τˆN is a stopping time. 2 Lemma 3.3. u(· ∧ τˆN ) is Hσα -valued progressively measurable. Proof. For each ω ∈ Ω ∗ , u(· ∧ τˆN ) is well-defined, and belongs to C([0, ∞); Hσ ). Using ΘR , defined by (3.12), we define β
φR (x) = 1 − ΘR (x),
for all x.
Then, for all ω ∈ Ω ∗ , φR u(· ∧ τˆN ) → u(· ∧ τˆN )
3 in C [0, K]; H β R 3 ,
as R → ∞, for every 0 < K < ∞.
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Thus, it is enough to prove measurability of φR u(· ∧ τˆN ) for each R. Fix any 0 < t < ∞, and let Br (p) be the closure of an open ball Br (p) with radius r > 0 and center p in C([0, t]; H β (R 3 )3 ). We will show that ∞ ∞ ∞ ∞
φR u(· ∧ τˆN ) ∈ Br (p) ∩ Ω ∗ = Ω ∗ ∩ φR um (· ∧ τˆN ) ∈ Br+ 1 (p) ν
L=1 ν=1 k=1 m=k
∩ um (· ∧ τˆN ) C([0,t];H α ) L . σ
(3.28)
Suppose ω belongs to the left-hand set. By the same argument as above and by (3.10), ω must belong to the right-hand set. Next let ω belong to the right-hand set. Since ω ∈ Ω ∗ , u(· ∧ τˆN (ω)) is well-defined. At the same time, there is some L 1 such that for each ν 1, there is a certain subsequence {umj } which satisfies (3.6) and (3.9)–(3.11) for some u˜ with T replaced by t ∧ τˆN (ω). Let ˜ inf{0 s t ∧ τˆN (ω) | ∇ u(s) L∞ (R 3 ) N }, τN∗ = t ∧ τˆN (ω), if the set {· · ·} is empty. Suppose τN∗ > 0. Then, u(·) and u(·) ˜ satisfy ∂ (u − u) ˜ + Π(u · ∇)u − Π(u˜ · ∇)u˜ = 0 ∂t in the sense of distributions over R 3 × (0, τN∗ ). It follows that
u ≡ u˜ on 0, τN∗ , and hence, t ∧ τˆN (ω) = τN∗ .
(3.29)
˜ Thus, If τN∗ = 0, (3.29) is still valid because u(0) = u(0).
u ≡ u˜ on 0, t ∧ τˆN (ω) . Since φR umj (·) → φR u(·) ˜ in C([0, t ∧ τˆN (ω)]; H β (R 3 )3 ), we find that φR u˜ · ∧ τˆN (ω) ∈ Br+ 1 (p) ν
and thus, φR u · ∧ τˆN (ω) ∈ Br+ 1 (p). ν
But this is true for all ν 1, and ω belongs to the left-hand set. Thus, (3.28) is valid. Since τˆN is a stopping time, the right-hand set of (3.28) belongs to Ft . Consequently, the map from [0, t] × Ω into H β (R 3 )3 defined by (s, ω) → φR u(s ∧ τˆN )
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is B([0, t]) ⊗ Ft -measurable. So is the map (s, ω) → u(s ∧ τˆN ). For each ω ∈ Ω ∗ , u(· ∧ τˆN ) ∈ C [0, T ]; Hσβ ∩ L∞ 0, T ; Hσα and thus, u(t ∧ τˆN ) ∈ Hσα for all 0 t < ∞, for each ω ∈ Ω ∗ . Meanwhile, every Borel subset of β Hσα is also a Borel subset of Hσ . So u(· ∧ τˆN ) is Hσα -valued progressively measurable. 2 Next we will show that 2 E u(· ∧ τˆN ) L∞ (0,T ;H α ) CN,T
(3.30)
σ
where CN,T is the same constant as in (3.5). Let us choose any constant K > 0. We claim that for each ω ∈ Ω ∗ , um (·) ∞ u · ∧ τˆN (ω) ∞ (3.31) α ∧ K lim α ∧ K. L (0,T ;Hσ )
L (0,T ;Hσ )
m→∞
Fix any ω ∈ Ω ∗ . If the right-hand side is equal to K, then (3.31) holds. Suppose lim um (·) L∞ (0,T ;H α ) ∧ K = γ < K. σ
m→∞
Then, there is a subsequence {umj } such that lim umj (·) L∞ (0,T ;H α ) = γ .
mj →∞
σ
By repetition of the same procedure as above, we can further extract a subsequence still denoted by {umj } such that weak star in L∞ 0, T ; Hσα , u(·) ˜ ∈ C [0, T ]; Hσβ ∩ L∞ 0, T ; Hσα
umj → u˜
(3.32) (3.33)
and u · ∧ τˆN (ω) ≡ u˜ · ∧ τˆN (ω)
(3.34)
for some function u. ˜ Meanwhile, (3.33) implies that u(t) ˜ Hσα is lower semicontinuous in t. Thus, we have u˜ · ∧ τˆN (ω) ∞ ˜ L∞ (0,T ;H α ) u(·) L (0,T ;H α ) σ
σ
which, combined with (3.32) and (3.34), yields u · ∧ τˆN (ω) ∞ γ. L (0,T ;H α ) σ
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Hence, (3.31) is true. Next it follows from Fatou’s lemma and (3.5) that 2 2 E u(· ∧ τˆN ) L∞ (0,T ;H α ) ∧ K limE um (·) L∞ (0,T ;H α ) ∧ K CN,T σ
σ
for all K > 0. By passing K → ∞, we arrive at (3.30). Using this, we will improve the time regularity of u(· ∧ τˆN ). Lemma 3.4. It holds that u(· ∧ τˆN ) ∈ C [0, T ]; Hσα
(3.35)
for almost all ω. Proof. As above, let ρδ be the Friedrichs mollifier in the space variables. It holds that t u(t ∧ τˆN ) ∗ ρδ = u0 ∗ ρδ −
Π u(s) · ∇u(s) ∗ ρδ χ[0,τˆN ] (s) ds
0
+Π
∞
t
gj (s) ∗ ρδ χ[0,τˆN ] (s) dBj (s)
j =1 0
for all 0 t < ∞ and all δ > 0, for each ω ∈ Ω ∗ . Hence, by Ito’s formula, u(t2 ∧ τˆN ) ∗ ρδ 2 − u(t1 ∧ τˆN ) ∗ ρδ 2 α α t2 =−
2 u(s) · ∇u(s) ∗ ρδ , u(s) ∗ ρδ α χ[0,τˆN ] (s) ds
t1 ∞
t2
+
j =1 t1
2 gj (s) ∗ ρδ , u(s) ∗ ρδ α χ[0,τˆN ] (s) dBj (s)
∞ Πgj (s) ∗ ρδ 2 χ[0,τˆ ] (s) ds + N α t2
(3.36)
j =1 t1
˜ where Ω˜ ⊂ Ω ∗ with for all 0 t1 < t2 < ∞, and δ = δn = n1 , n = 1, 2, . . . , for each ω ∈ Ω, ˜ P (Ω \ Ω) = 0.
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By virtue of (2.5)–(2.7) with α + 1 replaced by α, we see t2 u(s) · ∇u(s) ∗ ρδn , u(s) ∗ ρδn α χ[0,τˆN ] (s) ds lim δn →0 t1
t2 CN
u(s) 2 χ[0,τˆ ] (s) ds N α
(3.37)
t1
˜ for some constant CN > 0. for all 0 t1 < t2 < ∞, and all ω ∈ Ω, By the Burkholder–Davis–Gundy inequality and the identity gj ∗ ρδ , u ∗ ρδ α = gj ∗ ρδ ∗ ρδ , uα we have ∞ t 2 gj (s) ∗ ρδ , u(s) ∗ ρδ α χ[0,τˆN ] (s) dBj (s) E sup 0tT
j =1 0
−
∞ j =1 0
CE
t
2 gj (s), u(s) α χ[0,τˆN ] (s) dBj (s)
sup u(t ∧ τˆN ) α
0tT
→ 0,
T
gj (s) ∗ ρδ ∗ ρδ − gj (s) 2 ds
1/2
α
0
as δ → 0, by (1.1) and (3.30).
Therefore, there is a subsequence of {δn } still denoted by {δn } and a subset Ω † ⊂ Ω˜ such that P (Ω \ Ω † ) = 0, and ∞
(·)
j =1 0
2 gj (s) ∗ ρδn , u(s) ∗ ρδn α χ[0,τˆN ] (s) dBj (s)
∞
(·)
→
j =1 0
2 gj (s), u(s) α χ[0,τˆN ] (s) dBj (s)
(3.38)
in C([0, T ]), as δn → 0, for each ω ∈ Ω † . By virtue of (3.37), (3.38) and the fact that u(t ∧ τˆN ) ∈ Hσα , for all 0 t < ∞, for each ω ∈ Ω ∗ , we can pass δ = δn → 0 in (3.36) to arrive at
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u(t2 ∧ τˆN ) 2 − u(t1 ∧ τˆN ) 2 CN α α
t2
3681
u(s) 2 χ[0,τˆ ] (s) ds N α
t1
∞ t2 + 2 gj (s), u(s) α χ[0,τˆN ] (s) dBj (s) j =1 t1
∞ gj (s) 2 χ[0,τˆ ] (s) ds + N α t2
(3.39)
j =1 t1
for all 0 t1 < t2 < ∞, and all ω ∈ Ω † . Since u(· ∧ τˆN ) ∈ C([0, T ]; Hσ ) ∩ L∞ (0, T ; Hσα ) for each ω ∈ Ω ∗ , (3.39) implies (3.35). 2 β
We now summarize what we have achieved so far. With fixed N 1 and T > 0, there is a stopping time τˆN and a function u with the following properties. (i) u(· ∧ τˆN ) ∈ C([0, T ]; Hσα ), for almost all ω, and it is Hσα -valued progressively measurable. (ii) It holds that t u(t ∧ τˆN ) = u0 −
Π u(s) · ∇ u(s)χ[0,τˆN ] (s) ds
0
+Π
∞
t
gj (s)χ[0,τˆN ] (s) dBj (s),
j =1 0
for all 0 t < ∞, for almost all ω. (iii) For almost all ω, inf{0 t T | ∇u(t)L∞ (R 3 ) N }, τˆN = T , if the set {· · ·} is empty. Remark. In the above procedure, the uniqueness of u(·) on the whole interval [0, T ] is not known. We showed the uniqueness on the subinterval [0, τˆN ]. The technical obstruction is the presence of the cut-off function ψN (· · ·). However, if α > 75 , then we can easily show the uniqueness on the whole interval [0, T ], because we can work with the Hσα−1 -norm of u which dominates ∇uL∞ (R 3 ) . 3.2. Fix N 1 and pass T → ∞ Let uκ be the solution obtained above with T = κ, κ = 1, 2, . . . . Let τˆN,κ be a stopping time defined by inf{0 t κ | ∇uκ (t)L∞ (R 3 ) N }, τˆN,κ = κ, if the set {· · ·} is empty.
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There is some subset ΩN such that P (Ω \ ΩN ) = 0, and uκ and τˆN,κ satisfy the above properties (i)–(iii) for all κ = 1, 2, . . . , for each ω ∈ ΩN . It is easy to see that for κ1 < κ2 , uκ1 (t) = uκ2 (t),
for all t ∈ [0, τˆN,κ1 ∧ τˆN,κ2 ],
for each ω ∈ ΩN , and hence, it follows that for all ω ∈ ΩN , τˆN,κ1 τˆN,κ2 , and uκ1 (· ∧ τˆN,κ1 ) ≡ uκ2 (· ∧ τˆN,κ1 ). Thus, we can define, for each ω ∈ ΩN , τ˜N = lim τˆN,κ κ→∞
and u(t ∧ τ˜N ) = lim uκ (t ∧ τˆN,κ ) κ→∞
for all 0 t < ∞. Here we note that if τ˜N < ∞, then there is some κ ∗ such that τˆN,κ ∗ < κ ∗ , and it holds that for all κ κ ∗ , τˆN,κ ∗ = τˆN,κ = τ˜N , and uκ ∗ (t ∧ τˆN,κ ∗ ) = uκ (t ∧ τˆN,κ ) = u(t ∧ τ˜N ),
for all 0 t < ∞.
It follows that ∇u(t)
L∞ (R 3 )
< N,
if t < τ˜N
and ∇u(τ˜N )
L∞ (R 3 )
= N.
On the other hand, if τ˜N = ∞, it holds that for all κ = 1, 2, . . . , τˆN,κ = κ and ∇u(t) ∞ 3 = lim ∇uκ (t ∧ τˆN,κ ) ∞ 3 < N, L (R ) L (R ) κ→∞
for all 0 t < ∞.
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Therefore, we conclude that for each ω ∈ ΩN , τ˜N defined above can be characterized by τ˜N =
inf{0 t < ∞ | ∇u(t)L∞ (R 3 ) N }, ∞, if the set {· · ·} is empty.
(3.40)
We also find that for each ω ∈ ΩN , u(· ∧ τ˜N ) ∈ C [0, ∞); Hσα
(3.41)
and t u(t ∧ τ˜N ) = u0 −
Π u(s) · ∇ u(s)χ[0,τ˜N ] (s) ds
0
+Π
∞
t
gj (s)χ[0,τ˜N ] (s) dBj (s),
(3.42)
j =1 0
holds for all 0 t < ∞. 3.3. Pass N → ∞ For each N = 1, 2, . . . , let uN be the solution obtained in Section 3.2, and let τ˜N be the stopping time associated with uN by (3.40). Let Ω0 =
∞
ΩN
N =1
where ΩN is the same as in Section 3.2 for each N . Then, for any N1 < N2 , it holds that uN1 (t ∧ τ˜N1 ∧ τ˜N2 ) = uN2 (t ∧ τ˜N1 ∧ τ˜N2 ),
for all 0 t < ∞
for each ω ∈ Ω0 . It follows that if N1 < N2 , τ˜N1 τ˜N2 ,
(3.43)
and uN1 (t ∧ τ˜N1 ) = uN2 (t ∧ τ˜N1 ), By (3.43), we can define τ = lim τ˜N N →∞
for all 0 t < ∞.
(3.44)
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and, by (3.44), u(t) = lim uN (t ∧ τ˜N ),
for each 0 t < τ
N →∞
for all ω ∈ Ω0 . In fact, if t < τ˜N ∗ , for some N ∗ 1, at some ω∗ ∈ Ω0 , then it holds that u(t) = uN ∗ (t ∧ τ˜N ∗ ) = uN (t ∧ τ˜N ) for all N N ∗ , and ∇u(t)
L∞ (R 3 )
< N∗
(3.45)
at ω∗ . On the other hand, if τ˜N ∗ < ∞, at ω∗ , u(τ˜N ∗ ) = uN ∗ (τ˜N ∗ ) = uN (τ˜N ∗ )
(3.46)
for all N N ∗ , at ω∗ . It is easy to see that u(·) ∈ C [0, τ ); Hσα ,
for each ω ∈ Ω0 .
We set u(t) = 0,
if t τ.
Then, u(·) is right continuous on [0, ∞), and u(t) is Ft -measurable for each 0 t < ∞. Consequently, it is Hσα -valued progressively measurable. We now define τN =
inf{0 t < ∞ | ∇u(t)L∞ (R 3 ) N }, ∞, if the set {· · ·} is empty.
By (3.45) and (3.46), τN = τ˜N , for all N = 1, 2, . . . , for each ω ∈ Ω0 . Thus, it follows from (3.42) that t u(t ∧ τN ) = u0 −
Π u(s) · ∇ u(s)χ[0,τN ] (s) ds
0
+Π
∞
t
gj (s)χ[0,τN ] (s) dBj (s),
(3.47)
j =1 0
for all 0 t < ∞, and all N 1, for each ω ∈ Ω0 . This (u, τ ) is a local smooth solution of (0.1)–(0.3) according to Definition 1.1. The uniqueness can be easily shown by the same argument as in the previous section. It remains to establish (1.5).
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3.4. Estimate of τ Let (u, τ ) be the solution obtained in Section 3.3, and fix any 0 < δ < 1. Choose an integer N such that 1 1 δ< . N +1 N It holds that
P {τ > δ} P {τN > δ} P P
sup ∇u(t ∧ τN ) L∞ (R 3 ) < N
0tδ
sup u(t ∧ τN ) α < KN
0tδ
where K is a positive constant defined by
K = sup C ∈ R C∇vL∞ (R 3 ) vα , ∀v ∈ Hσα . Since u(· ∧ τN ) ∈ C([0, ∞); Hσα ), for almost all ω, we can define for 0 < L < ∞, TL =
inf{0 t < ∞ | u(t ∧ τN )α L}, τN , if the set {· · ·} is empty.
Then, TL is a stopping time, and τN = lim TL ,
for almost all ω.
L→∞
By means of (2.5)–(2.7) with α + 1 replaced by α, we can derive from (3.47) that u(t ∧ TL ) 2 u0 2 + CN α α
t
u(s ∧ TL ) 2 χ[0,T ] (s) ds L α
0 ∞ t
+2
j =1 0
u(s ∧ TL ), gj (s) α χ[0,TL ] (s) dBj (s)
∞ gj (s) 2 χ[0,T ] (s) ds + L α t
(3.48)
j =1 0
for all 0 t < ∞, for almost all ω, where C is a positive constant independent of u, L, N, t, and ω. By the Burkholder–Davis–Gundy inequality,
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∞ η u(s ∧ TL ), gj (s) α χ[0,TL ] (s) dBj (s) E sup 0ηt
j =1 0
1/2 L ∞ t∧T 2 2 CE u(s ∧ TL ) α gj (s) α ds j =1 0
∞ t∧T L 2 2 1 gj (s) α ds E sup u(s ∧ TL ) α + CE 4 0st
(3.49)
j =1 0
for all 0 t < ∞, where C is a positive constant independent of u, L, N, and t. By the Gronwall inequality, it follows from (3.48) and (3.49) that E
∞ δ 2 2 2 gj (s) ds eCN δ sup u(t ∧ TL ) α C E u0 α + E α
0tδ
j =1 0
where C denotes positive constants independent of u, L, N and δ. By passing L → ∞, we arrive at E
∞ δ 2 2 2 gj (s) ds eCN δ sup u(t ∧ τN ) α C E u0 α + E α
0tδ
j =1 0
where C denotes positive constants independent of u, N and δ. Thus, we have P
sup u(t ∧ τN ) α < KN
0tδ
∞ δ 1 2 gj (s) ds 1 − 2 2 CeCN δ E u0 2α + E α K N j =1 0
1 − Cδ
2
∞ δ 2 2 gj (s) α ds E u0 α + E j =1 0
where C denotes positive constants independent of u and δ. Now the proof of Theorem 1.2 is complete. References [1] H. Bessaih, Martingale solutions for stochastic Euler equations, Stoch. Anal. Appl. 17 (5) (1999) 713–725. [2] H. Bessaih, F. Flandoli, 2-D Euler equation perturbed by noise, NoDEA Nonlinear Differential Equations Appl. 6 (1998) 35–54. [3] Z. Brzezniak, S. Peszat, Stochastic two-dimensional Euler equations, Ann. Probab. 29 (4) (2001) 1796–1832.
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[4] M. Capinski, N. Cutland, Stochastic Euler equations on the torus, Ann. Appl. Probab. 9 (3) (1999) 688–705. [5] P. Constantin, G. Iyer, A stochastic Lagrangian representation of the three-dimensional incompressible Navier– Stokes equations, Comm. Pure Appl. Math. 61 (3) (2008) 330–345. [6] A. Cruzeiro, F. Flandoli, P. Malliavin, Brownian motion on volume preserving diffeomorphisms group and existence of global solutions of 2D stochastic Euler equation, J. Funct. Anal. 242 (1) (2007) 304–326. [7] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. [8] L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. 3, Springer-Verlag, Berlin–Heidelberg– New York, 1985. [9] I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, second ed., Springer, New York–Berlin– Heidelberg, 1997. [10] T. Kato, Nonstationary flows of viscous and ideal fluids in R 3 , J. Funct. Anal. 9 (1972) 296–305. [11] T. Kato, Quasi-linear equations of evolution with applications to partial differential equations, in: Lecture Notes in Math., vol. 448, Springer, 1975, pp. 25–70. [12] T. Kato, G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (7) (1988) 891–907. [13] J.U. Kim, On the stochastic Euler equations in a two-dimensional domain, SIAM J. Math. Anal. 33 (5) (2002) 1211–1227. [14] P.L. Lions, Mathematical Topics in Fluid Mechanics, vol. 1, Clarendon Press, Oxford–New York, 1996. [15] R. Mikulevicius, G. Valiukevicius, On stochastic Euler equation in R d , Electron. J. Probab. 5 (6) (2000) 20. [16] J. Simon, Compact sets in the space Lp (0, T ; B), Ann. Mat. Pura Appl. 146 (1987) 65–96. [17] R. Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal. 20 (1) (1977) 32–43.
Journal of Functional Analysis 256 (2009) 3688–3729 www.elsevier.com/locate/jfa
Universality limits for random matrices and de Branges spaces of entire functions ✩ D.S. Lubinsky School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA Received 8 September 2008; accepted 13 February 2009 Available online 21 March 2009 Communicated by C. Villani
Abstract We prove that de Branges spaces of entire functions describe universality limits in the bulk for random matrices, in the unitary case. In particular, under mild conditions on a measure with compact support, we show that each possible universality limit is the reproducing kernel of a de Branges space of entire functions that equals a classical Paley–Wiener space. We also show that any such reproducing kernel, suitably dilated, may arise as a universality limit for sequences of measures on [−1, 1]. © 2009 Elsevier Inc. All rights reserved. Keywords: Random matrices; Universality limits; de Branges spaces; Orthogonal polynomials
1. Introduction and results Let μ be a finite positive Borel measure on R with all moments x j dμ(x), j 0, finite, and with infinitely many points in its support. Then we may define orthonormal polynomials pn (x) = γn x n + · · · ,
γn > 0,
n = 0, 1, 2, . . . satisfying the orthonormality conditions pn pm dμ = δmn . ✩
Research supported by NSF grant DMS0400446 and US–Israel BSF grant 2004353. E-mail address: [email protected].
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.021
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Throughout we use μ (x) = dμ dx to denote the almost everywhere existing Radon–Nikodym derivative of μ. Orthogonal polynomials play an important role in random matrix theory, especially in the unitary case [2,5,12,27]. One of the key limits there involves the reproducing kernel Kn (x, y) =
n−1
(1.1)
pk (x)pk (y).
k=0
Because of the Christoffel–Darboux formula, it may also be expressed as Kn (x, y) =
γn−1 pn (x)pn−1 (y) − pn−1 (x)pn (y) , γn x−y
x = y.
(1.2)
Define the normalized kernel n (x, y) = μ (x)1/2 μ (y)1/2 Kn (x, y). K
(1.3)
The simplest case of the universality law is the limit n ξ + K lim
n→∞
a ,ξ n (ξ,ξ ) K
+
n (ξ, ξ ) K
b n (ξ,ξ ) K
=
sin π(a − b) , π(a − b)
(1.4)
involving the sinc kernel. It describes the distribution of spacing of eigenvalues of random matrices. Typically this limit holds uniformly for ξ in the interior of the support of μ and a, b in π(a−b) compact subsets of the real line. Of course, when a = b, we interpret sinπ(a−b) as 1. There are a wide variety of methods for establishing universality, and we cannot survey them all here. Perhaps the deepest are Riemann–Hilbert methods, which yield much more than universality, though they require some smoothness properties for the measure [2,5,26]. There are a number of methods that use techniques of mathematical physics [7,31]. Eli Levin observed that first order asymptotics for orthogonal polynomials are sufficient to establish universality [16]. One recently introduced technique [22] (see also [14,18,19]) involves a comparison inequality, and allows one to start with universality for a given measure, and extend it to far more general measures. The disadvantage there is that one needs to start with some measure, with a similar support to the given measure, for which universality is known. However, it has been greatly extended, using devices such as polynomial pullbacks by Totik [38], and his student Findley [6]. Simon [34] obtained equally impressive results by combining this method with Jost functions. In particular, Findley and Totik showed that for regular measures, universality holds a.e. in any interval where log μ is integrable. Here regularity in the sense of Stahl and Totik [36] can be defined as the condition 1/n
lim γn
n→∞
=
1 , cap(supp[μ])
where cap(supp[μ]) is the logarithmic capacity of the support of μ. A perhaps more promising idea was introduced in [21]. It uses classical complex analysis, such as the theory of normal families, entire functions of exponential type, and reproducing kernels for Paley–Wiener spaces. Its advantage is that it does not require a base measure for which
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universality is known, nor regularity. It shows that universality is equivalent to “universality along the diagonal”, or alternatively, ratio asymptotics for Christoffel functions λn (x) = 1/Kn (x, x). Here is a typical result: Theorem 1.1. Let μ be a finite positive Borel measure on the real line with compact support. Let J ⊂ supp[μ] be compact, and such that μ is absolutely continuous in an open set containing J . Assume that μ is positive and continuous at each point of J . The following are equivalent: (I) Uniformly for ξ ∈ J and a in compact subsets of the real line, Kn ξ + lim
a ,ξ n (ξ,ξ ) K
+
a n (ξ,ξ ) K
Kn (ξ, ξ )
n→∞
= 1.
(1.5)
(II) Uniformly for ξ ∈ J and a, b in compact subsets of the complex plane, we have Kn ξ + lim
a ,ξ n (ξ,ξ ) K
+
b n (ξ,ξ ) K
=
Kn (ξ, ξ )
n→∞
sin π(a − b) . π(a − b)
(1.6)
While it is possible that (1.5) always holds under the initial hypotheses of Theorem 1.1, it has been established only when we assume that μ is regular. In [21], it was also shown that instead of continuity of w, we may assume a Lebesgue point type condition. The method may also be applied to varying and exponential weights, and at the “hard” or “soft” edge of the spectrum, where we obtain a Bessel or Airy kernel [15,17,20]. Avila, Last and Simon [1] have shown that this method can be adapted to prove universality for measures whose support is a Cantor set of positive measure, while Simon has extended Theorem 1.1 in a number of other directions [35]. In this paper, we explore the possible limits of subsequences of the sequence {fn }, where fn (a, b) =
Kn ξn +
a ,ξ n (ξn ,ξn ) n K
+
b n (ξn ,ξn ) K
Kn (ξn , ξn )
,
(1.7)
and {ξn } is a sequence of real numbers. Since the {Kn } are reproducing kernels for polynomials, it is scarcely surprising that limits of subsequences of {fn } are reproducing kernels for suitable spaces of entire functions. It turns out that the natural such spaces are de Branges spaces. We can use some of their remarkable theory to characterize universality limits. de Branges spaces [4, p. 50], [25, p. 983 ff], [30, p. 793 ff] are built around the Hermite– Biehler class. An entire function E is said to belong to the Hermite–Biehler class if it has no zeros in the upper half-plane C+ = {z: Im z > 0} and E(z) E(¯z)
for z ∈ C+ .
(1.8)
We write E ∈ HB. Recall that the Hardy space H 2 (C+ ) is the set of all functions g analytic in the upper half-plane, for which ∞ sup y>0 −∞
g(x + iy)2 dx < ∞.
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Given an entire function g, we let g ∗ (z) = g(¯z).
(1.9)
Definition 1.2. The de Branges space H(E) corresponding to the entire function E ∈ HB, is the set of all entire functions g such that both g/E and g ∗ /E belong to H 2 (C+ ), with ∞ 1/2 g 2 g E = < ∞. E
(1.10)
−∞
H(E) is a Hilbert space with inner product ∞ (g, h) = −∞
g h¯ . |E|2
One may construct a reproducing kernel for H(E) from E [25, p. 984], [30, p. 793]. Indeed, if we let K(ζ, z) =
i E(z)E(ζ ) − E ∗ (z)E ∗ (ζ ) , 2π z − ζ¯
(1.11)
then for all ζ , K(ζ, ·) ∈ H(E) and for all complex ζ and all g ∈ H(E), ∞ g(ζ ) = −∞
g(t)K(ζ, t) dt. |E(t)|2
(1.12)
We shall later identify K(ζ¯ , z) with a function f (ζ, z) that arises as a universality limit. We emphasize that the standard reproducing kernel K for a de Branges space involves a conjugate variable, while the standard reproducing kernel Kn for an orthogonal polynomial system does not. The classical de Branges spaces are the Paley–Wiener spaces PW σ , consisting of entire functions of exponential type σ that are square integrable along the real axis. There one may take E(z) = exp(−iσ z), and the norm is just ∞ g L2 (R) =
1/2 |g|
2
−∞
We write H(E) = PW σ
.
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if the two spaces are equal as sets, and have equivalent norms (we do not imply isometric isomorphism). Recall that having equivalent norms means that for some C > 1 independent of g ∈ PW σ , C −1 g L2 (R) g E C g L2 (R) .
(1.13)
The closed graph theorem can be used to show that this norm equivalence follows from mere equality as sets. Our main conclusion is that, under mild conditions, Universality limits in the bulk are reproducing kernels of de Branges spaces that equal classical Paley–Wiener spaces. More precisely: Theorem 1.3. Let μ be a measure with compact support. Let J be a compact set such that μ is absolutely continuous in an open set O containing J , and for some C > 1, C −1 μ C
in O.
Choose {ξn } ⊂ J and define {fn } by (1.7). (a) {fn (·,·)} is a normal family in compact subsets of C2 . (b) Let f (·,·) be the limit of some subsequence {fn (·,·)}n∈S . Then f is an entire function of two variables, that is real-valued in R2 and has f (0, 0) = 1. Moreover, for some σ > 0, f (·,·) is entire of exponential type σ in each variable. (c) Define L(u, v) = (u − v)f (u, v),
u, v ∈ C.
(1.14)
Let a ∈ C have Im a > 0 and let Ea (z) =
√ 2π
L(a, ¯ z) . |L(a, a)| ¯ 1/2
(1.15)
Then f is a reproducing kernel for H(Ea ). In particular, for all z, ζ , f (z, ζ¯ ) =
i Ea (z)Ea (ζ ) − Ea∗ (z)Ea∗ (ζ ) . 2π z − ζ¯
(1.16)
(d) Moreover, H(Ea ) = PW σ
(1.17)
and the norms · Ea of H(Ea ) and · L2 (R) of PW σ are equivalent. We emphasize that there are many de Branges spaces that equal PW σ , but their reproducing kernel is not the sinc kernel sinπtπt . We shall present some examples after Theorem 1.7. A complete description of such spaces is given in [25].
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In the case where there is a little smoothness of w at ξ such as continuity, or a Lebesgue point type condition, and the measure is regular in the sense of Stahl and Totik, indeed f above equals the sinc kernel, as shown in [6,21,22,34,38]. Nor does the above theorem exclude the possibility that f above is always a sinc kernel. We shall show below, however, that for sequences of measures, universality limits can definitely be the reproducing kernel of any de Branges space that equals a classical Paley–Wiener space. More information about f and L are given in the following result: Theorem 1.4. Assume the hypotheses of Theorem 1.3. (a) The function L satisfies the functional equation L(u, v)L(a, b) = L(a, u)L(b, v) − L(b, u)L(a, v)
(1.18)
for all complex a, b, u, v. Moreover, the functions L(·,·) and f (·,·) are uniquely determined by the functional equation (1.18), and the values of the function f (a, ·) for one non-real a. (b) F (z) = zf (0, z)
(1.19)
has countably many real simple zeros {ρj }, and no other zeros. (c) Each g ∈ PW σ admits the expansion g(z) =
∞
g(ρj )
j =−∞
f (ρj , z) , f (ρj , ρj )
(1.20)
which is an orthonormal expansion in H(Ea ), and moreover, ∞ g E
−∞
2 ∞ |g(ρj )|2 = . f (ρj , ρj ) a
(1.21)
j =−∞
Remarks. (a) Note that the right-hand side of (1.21) is independent of a, which is surprising as Ea appears in the left-hand side. This phenomenon is well understood. Indeed, for a non-negative measure ω, we have [30, p. 794] ∞ 2 ∞ 2 g g = dω E E
−∞
−∞
for all g ∈ H(E) iff there is a function A analytic in the interior of C+ , with |A| 1 there, and Im z π
∞ −∞
dω(t) E + E∗A = Re (z), 2 E − E∗A |t − z|
Im z > 0.
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(b) When ξn = ξ , n 1, and ξ is a Lebesgue point of μ , then the exponential type of f in each variable is σ = π sup f (x, x). x∈R
The proof of this given in [21, Lemma 6.4] goes through without change under the above hypotheses. (c) As a consequence of (1.20), we can say a lot about the distribution of the {ρj }, which in the special case of the sinc kernel are just the integers. Define the counting function of {ρj }, ν[a, b] = # j : ρj ∈ [a, b] (1.22) and
ν(t) =
ν([0, t]), ν([t, 0]),
t 0, t 0.
(1.23)
Classical complex analysis [13, p. 126 ff] shows that lim
|t|→∞
ν(t) σ = . |t| π
Much more is true – roughly speaking, for each ε > 0, ν(t) −
1+ε σ t = O log |t| : π
Theorem 1.5. Let p > 0 and τ > 1. Then ∞ −∞
|ν(t) − πσ t|p dt < ∞. (1 + |t|)(log(2 + |t|))p+τ
(1.24)
All of the above results can be proven for a sequence of measures {μn }, rather than a fixed measure μ. The hypotheses (1.26) to (1.28) below in a sense generalize the notion of the bulk of the support to sequences of measures. Theorem1.6. For n 1, let μn be a measure with support on the real line, for which the power moments x j dμn (x), 0 j 2n − 2, are finite. Let Kn denote the nth reproducing kernel for the measure μn , and K˜ n its normalized cousin. Let {ξn } be a sequence of real numbers, and let fn (a, b) =
Kn ξn +
a ,ξ n (ξn ,ξn ) n K
+
Kn (ξn , ξn )
b n (ξn ,ξn ) K
.
(1.25)
Assume that there exists a ∈ C \ R, and C1 , C2 , C3 > 0 with the following property: given A > 0, there exists n0 such that for n n0 and |z| A, fn (a, z) C1 eC2 |Im z| , (1.26)
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and for x ∈ [−A, A], fn (x, x) C3 .
(1.27)
Assume moreover, that for some C0 > 0 and a.e. real t, μn ξn + t Kn (ξn ,ξn ) C0 . lim inf n→∞ μn (ξn )
(1.28)
Then the conclusions of Theorems 1.3, 1.4 and 1.5 hold true for non-constant limits of subsequences of {fn }. Barry Simon has shown [35] that one can weaken the growth assumption (1.26) in a number of ways. We shall also prove a partial converse, showing that any reproducing kernel for a de Branges space that equals a classical Paley–Wiener one, can arise as a multiple of a universality limit: Theorem 1.7. Let H(E) be a de Branges space that equals PW σ for some σ > 0. Let f (ζ¯ , z) be the reproducing kernel for H(E) normalized so that f (0, 0) = 1. Assume also that |E(0)| = 1. Then there exists for n 1, an absolutely continuous measure μn , with support [−1, 1], with μn infinitely differentiable in (−1, 1), with μn (0) = 1, and for which fn (a, b) =
Kn 0 +
a ,0 + b n (0,0) K Kn (0,0)
Kn (0, 0)
satisfies (1.26) and (1.27), while lim fn (a, b) = f (a, b),
n→∞
(1.29)
uniformly for a, b in compact subsets of C. Moreover, given R > 0, (1.28) holds for t ∈ [−R, R]. If in addition, there exists C1 > 1 such that (1.30) C1−1 E(x) C1 , x ∈ R, then (1.28) holds for all t ∈ R. Remarks. (a) The hypothesis f (0, 0) = 1 matches the conclusion in Theorem 1.3(b). It can always be achieved by multiplying E by a suitable constant. However, the hypothesis |E(0)| = 1 is more problematic. Without it, we have to replace (1.29) by 2 2 lim fn (a, b) = f E(0) a, E(0) b .
n→∞
By a dilation of the variable, we can ensure |E(0)| = 1. More precisely, make the substitution t = s|E(0)|2 in the reproducing kernel relation (1.12), and let E1 (z) =
E(|E(0)|2 z) . |E(0)|
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One can check that the reproducing kernel for H(E1 ) is f1 (·,·) = f (|E(0)|2 ·, |E(0)|2 ·). Then E1 (0) = 1 and f1 (0, 0) = 1, and the above result may be applied to H(E1 ). (b) Lyubarskii and Seip [25, p. 1005] presented a range of examples of E(z) other than e−iπz for which H(E) = PW π . For 0 δ < 14 , let ∞ 1− Eδ (z) = (z + i) k=1
z k − δ − ik −4δ
1+
z k − δ + ik −4δ
.
This is an entire function of exponential type π with H(E) = PW π . It satisfies (1.30) only if δ = 0. In fact, if Λδ denotes the zero set of Eδ , then uniformly for all real x, and for some C1 > 1, 2δ C1−1 Eδ (x)/ 1 + |x| dist(x, Λδ ) C1 . Here dist(x, Λδ ) denotes the distance from x to Λδ . For δ > 0, the reproducing kernel fδ of H(Eδ ) is not the sinc kernel. Indeed, using (1.16) for fδ , we see that f (z, −i) =
i Eδ (z)Eδ (i) 2π z+i
and this has a very different zero set, with respect to z, from (c) If we let
sin π(z+i) π(z+i) .
E0 (z) = c sin π(z + i), for some normalizing constant c, a straightforward calculation shows that the reproducing kernel f0 for H(E0 ), given by (1.16), is f (z, ζ ) = c
sinh(2π) sin π(z − ζ ) . 2 π(z − ζ )
If we let E1 (z) =
z + 2i E0 (z), z+i
then on the real line C1 |E1 | C2 , so (1.30) is satisfied. Moreover, it is easily seen that H(E1 ) = H(E0 ) = PW π . However, the reproducing kernel f1 for H(E1 ) is not the sinc kernel. Indeed, (1.16) shows that for some constant C, f1 (z, −2i) = C and this is not a constant multiple of
sin π(z+2i) π(z+2i) .
sin π(z + i) , π(z + i)
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This paper is organized as follows. In Section 2, we present notation and general background, such as on orthogonal polynomials. In Section 3, we present background on entire functions and de Branges spaces. In Section 4, we discuss some polynomial de Branges spaces. In Section 5, we use these to examine de Branges spaces of entire functions associated with general measures μ. In Section 6, we prove Theorems 1.3, 1.4, and 1.5. In Section 7, we prove Theorems 1.6 and 1.7. 2. Notation and background In this section, we record our notation, though some of it has already been introduced earlier. In the sequel C, C1 , C2 , . . . denote constants independent of n, x, y, s, t. The same symbol does not necessarily denote the same constant in different occurrences. We shall write C = C(α) or C = C(α) to respectively denote dependence on, or independence of, the parameter α. We use ∼ in the following sense: given real sequences {cn }, {dn }, we write cn ∼ dn if there exist positive constants C1 , C2 with C1 cn /dn C2 . Similar notation is used for functions and sequences of functions. Throughout, μ denotes a finite positive Borel measure with not necessarily compact support on the real line. Its Radon–Nikodym derivative, which exists a.e., is μ . The corresponding orthonormal polynomials are denoted by {pn }∞ n=0 , so that pn pm dμ = δmn . We denote the zeros of pn by xnn < xn−1,n < · · · < x2n < x1n .
(2.1)
The reproducing kernel Kn (x, t) is defined by (1.1), while the normalized reproducing kernel is defined by (1.3). We let Ln (x, t) = (x − t)Kn (x, t) γn−1 pn (x)pn−1 (t) − pn−1 (x)pn (t) . = γn
(2.2)
The nth Christoffel function is [8, p. 25], [29,32,37], λn (x) = 1/Kn (x, x) =
inf
deg(P )n−1
P 2 dμ . P 2 (x)
(2.3)
When we need to display dependence of pn , Kn or λn on μ (or some other measure), we use pn (μ, ·), Kn (μ, ·, ·), λn (μ, ·), and so on. The Gauss quadrature formula asserts that whenever P
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is a polynomial of degree 2n − 1, n
λn (xj n )P (xj n ) =
(2.4)
P dμ.
j =1
In addition to this, we shall need another Gauss type of quadrature formula [8, p. 19 ff]. Given a real number ξ , there are n or n − 1 points tj n = tj n (ξ ), one of which is ξ , such that
λn (tj n )P (tj n ) =
(2.5)
P dμ,
j
whenever P is a polynomial of degree 2n − 2. The {tj n } are zeros of Ln (ξ, t) =
γn−1 pn (ξ )pn−1 (t) − pn−1 (ξ )pn (t) , γn
regarded as a function of t. Because we consider a sequence {ξn } of points in J , rather than a fixed ξ , we use the quadrature rule that includes ξn , so that tj n = tj n (ξn )
for all j.
Moreover, because we wish to focus on ξn , we shall set t0n = ξn , and order the {tj n } around ξn , treated as the origin: · · · < t−2,n < t−1,n < t0n = ξn < t1n < · · · .
(2.6)
Of course the sequence {tj n } consists of either n − 1 or n points, so terminates, and it is possible that all tj n lie to the left or right of ξn . It is known [8, p. 19] that when (pn pn−1 )(ξn ) = 0, then one zero of Ln (ξn , t) lies in (xj n , xj −1,n ) for each j , and the remaining zero lies outside (xnn , x1n ). For the given sequence {ξn } in J , we shall define for n 1, fn (a, b) =
Kn ξn +
a ,ξ K˜ n (ξn ,ξn ) n
+
b K˜ n (ξn ,ξn )
Kn (ξn , ξn )
(2.7)
and n (a, b) = (a − b)fn (a, b). L
(2.8)
The zeros of fn (0, t) =
Kn ξn , ξn +
t K˜ n (ξn ,ξn )
Kn (ξn , ξn )
will be denoted by {ρj n }j =0 . Since {tj n } = {tj n (ξn )} are the zeros of Ln (ξn , t), we have ρj n = K˜ n (ξn , ξn )(tj n − ξn ).
(2.9)
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We also set ρ0n = 0, corresponding to t0n = ξn . For an appropriate subsequence S of integers, we shall let f (a, b) =
lim
n→∞, n∈S
fn (a, b).
(2.10)
The zeros of f (0, ·) will be denoted by {ρj }j =0 , and we set ρ0 = 0. Our ordering of zeros is · · · ρ−2 ρ−1 < ρ0 = 0 < ρ1 ρ2 · · · .
(2.11)
In Theorem 5.3, and only in that theorem, we shall further restrict the {ρj } to exclude those zeros ρ for which f (ρ, ρ) = 0. This eventuality cannot happen under the hypotheses of Theorems 1.3, 1.6 or 5.4. We shall denote the (exponential) type of f (a, ·) by σ . (We shall show it is independent of a.) We let L(a, b) = (a − b)f (a, b).
(2.12)
3. Background on entire functions We first review some theory that we shall use about entire functions of exponential type. Most of this can be found in the elegant series of lectures of B.Ja. Levin [13]. Recall that if g is entire of order 1, then its exponential type σ is σ = lim sup r→∞
max|z|=r log |g(z)| . r
(3.1)
We say that an entire function g belongs to the Cartwright class and write g ∈ C if it is of exponential type and ∞ −∞
log+ |g(t)| dt < ∞. 1 + t2
(3.2)
Here log+ s = max{0, log s}. We let n(g, r) denote the number of zeros of g in the ball center 0, radius r, counting multiplicity. An important result is that for g ∈ C, that is real-valued on the real axis, lim
r→∞
n(g, r) σ = . 2r π
(3.3)
For this, see [13, Theorem 1, p. 127] or [11, p. 66]. When g is entire of exponential type σ and bounded along the real axis, we have [13, p. 38, Theorem 3] g(z) eσ |Im z| g L (R) , z ∈ C. (3.4) ∞
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When g is entire of exponential type σ and g ∈ L2 (R), we write g ∈ PW σ . (In [13], the notation is g ∈ L2σ .) Here, we have instead of the last inequality [13, p. 149] g(z)
1/2 2 eσ (|Im z|+1) g L2 (R) , π
z ∈ C.
(3.5)
Another useful result is that if g ∈ C has exponential type σ , and has all real zeros, then [13, p. 126, p. 118] log |g(reiθ )| = σ |sin θ |, r→∞ r lim
0 < |θ | < π.
(3.6)
If we do not know that all zeros are real, it is known that [13, p. 118, p. 55, no. 3] lim sup r→∞
log |g(reiθ )| σ |sin θ |, r
0 < |θ | < π.
(3.7)
The Hermite–Biehler class HB was defined in Section 1, as was the de Branges space H(E), for a given entire function E ∈ HB. It is possible to give an abstract definition of a de Branges space [4, pp. 56–57]. de Branges’ original definition involved the notions of mean type and bounded type. One useful alternative involves the reproducing kernel K(ζ, z), defined in terms of E by (1.11). Then [4, p. 53] H(E) is the set of all entire functions g with ∞ 1/2 g 2 <∞ g E = E
(3.8)
−∞
and g(z) K(z, z)1/2 g E
for all z ∈ C.
We emphasize that later on, we shall identify K(ζ, z) with f (ζ¯ , z). For real x, and E as above, we define a phase function ϕ by E(x) = E(x)e−iϕ(x) .
(3.9)
(3.10)
Here ϕ is an increasing continuous function. We have [4, p. 54], [25, p. 984] if E(x) = 0, ϕ (x) =
πK(x, x) . |E(x)|2
(3.11)
There is a sampling series determined by ϕ and a given real number α [4, p. 55], [30, p. 794]. Let {sk } denote the increasing sequence such that ϕ(sk ) = α + kπ,
k = 0, ±1, ±2, . . . .
(3.12)
Assume eiα E − e−iα E ∗ ∈ / H(E).
(3.13)
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Then
K(sk , z) √ K(sk , sk )
(3.14) k
is an orthonormal sequence in H(E), and for all g ∈ H(E), ∞ 2 |g(sk )|2 g π|g(sk )|2 = = , E 2 K(sk , sk ) ϕ (sk )|E(sk )|
−∞
k
(3.15)
k
while for all z, g(z) =
k
K(sk , z) . g(sk ) √ K(sk , sk )
(3.16)
Moreover, there is at most one real α ∈ [0, π) for which (3.13) fails. We shall later show that {ρj } of Theorem 1.4 is a complete interpolating sequence for PW σ . That is, given any sequence {cj } with
|cj |2 < ∞,
j
there exists a unique g ∈ PW σ such that g(ρj ) = cj
for all j.
Such sequences have been characterized in [10,24] using the distribution of {ρj }. In particular, if ν is the counting function defined at (1.22)–(1.23), then h(t) = ν(t) −
σ t π
lies in the class BMO of the real line. That is, 1 sup |h − hI | < ∞, I |I | I
where for any interval I , with length |I |, we let hI =
1 |I |
h. I
It is known that then [9, p. 233, Corollary 2.3], for each p > 0, 1 sup |h − hI |p < ∞. |I | I I
(Garnett considers only p 1, but the case p < 1 follows from Hölder’s inequality.)
(3.17)
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de Branges spaces that equal Paley–Wiener (and more general) spaces have been characterized in [25]. In particular, they showed [25, Theorem 4(ii), p. 982] that if H(E) = PW σ , then uniformly for all real x, 2 ϕ (x)E(x) = πK(x, x) ∼ 1.
(3.18)
4. de Branges spaces of polynomials In this section, n 1 is fixed, and μ is a measure on the real line with 0 j 2n. We assume the notation of Section 2; in particular,
x j dμ(x) finite, for
Ln (u, v) = (u − v)Kn (u, v) γn−1 pn (u)pn−1 (v) − pn−1 (u)pn (v) . = γn
(4.1)
In [23], we used ideas inspired by de Branges spaces to generate formulae for orthogonal polynomials with a weight that is a reciprocal of a positive polynomial. Here, we begin with some simple identities. The first is inspired by the more general theory of de Branges spaces, and the second is well known [28]: Lemma 4.1. (a) For all complex α, β, z, v, Ln (z, v)Ln (α, β) = Ln (α, z)Ln (β, v) − Ln (β, z)Ln (α, v). (b) Ln (z, v) =
γn−1 pn (z)pn (v) Gn (v) − Gn (z) , γn
(4.2)
(4.3)
where Gn (z) =
n 2 (x ) pn−1 (z) γn−1 λn (xj n )pn−1 jn = . pn (z) γn z − xj n
(4.4)
j =1
Proof. (a) Just substitute (4.1) into the right-hand side of (4.2), then multiply out, cancel common factors, and refactorize. (A slightly simpler manipulation is to substitute the formula (4.3) into the right-hand side of (4.2).) (z) (b) Let Gn (z) = ppn−1 . Then (4.3) follows from (4.1). We really only need to prove the n (z) second identity in (4.4). We apply the formula for Lagrange interpolation at the zeros of pn to pn−1 . This gives pn−1 (z) =
n pn−1 (xj n )pn (z) . pn (xj n )(z − xj n ) j =1
(4.5)
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We now use the confluent form of the Christoffel–Darboux formula, λ−1 n (x) = Kn (x, x) =
γn−1 pn (x)pn−1 (x) − pn (x)pn−1 (x) . γn
Setting x = xj n gives λ−1 n (xj n ) =
γn−1 p (xj n )pn−1 (xj n ). γn n
Substituting this into (4.5) gives the second identity in (4.4).
2
Lemma 4.2. (a) If Kn (z, w) = 0, then Im z and Im w have the same sign. In particular, Im z > 0 ⇒ Im w > 0. (b) Let Im a > 0. Then for Im z 0, Kn (a, ¯ z) Kn (a, z); Ln (a, ¯ z) Ln (a, z).
(4.6) (4.7)
In particular, Ln (a, ¯ ·) ∈ HB. Proof. (a) If z is real, then it is known [8, p. 19], that all zeros of Kn (z, ·) are real. Thus in this case Im z = Im w = 0. Now suppose Im z > 0. From (4.3), and the fact that all zeros of pn pn−1 are real, we deduce that Gn (z) = Gn (w). Taking imaginary parts in (4.4), we deduce that (Im z)
n 2 (x ) λn (xj n )pn−1 jn j =1
|z − xj n |2
= (Im w)
n 2 (x ) λn (xj n )pn−1 jn j =1
|w − xj n |2
.
Since both sums are positive, the result follows. (b) The rational function h(z) := Kn (a, z)/Kn (a, ¯ z) is analytic for z in the closed upper half-plane {z: Im z 0}, and for real x, h(x) = 1. Moreover, as a polynomial in z, the coefficients of the Taylor expansion about 0 of Kn (a, ¯ z) are the conjugates of those of Kn (a, z). Then, as z → ∞, |h(z)| → 1. The maximum-modulus principle now shows that h(z) 1 for Im z 0. Since for Im z 0, also |a¯ − z| |a − z|, we obtain (4.7) as well.
2
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From the above, we obtain some de Branges spaces that consist of polynomials. Recall that Kn is the orthogonal polynomial reproducing kernel arising from the measure μ, while K denotes the reproducing kernel for a de Branges space. Recall too, the ∗ notation introduced at (1.9). Theorem 4.3. Fix a with Im a > 0, and let √ 2π
¯ z) Ln (a, . |Ln (a, a)| ¯ 1/2
(4.8)
∗ (z)E ∗ (ζ ) i En,a (z)En,a (ζ ) − En,a n,a . 2π z − ζ¯
(4.9)
En,a (z) = (a) Then Kn (z, ζ¯ ) =
(b) The de Branges space H(En,a ) corresponding to En,a is the space of polynomials of degree n − 1. (c) For all polynomials P of degree n − 1, and all z ∈ C, we have ∞ P (z) =
P (t)
−∞
Kn (t, z) dt. |En,a (t)|2
(4.10)
(d) For all polynomials R of degree 2n − 2, ∞ −∞
R |En,a |2
=
R dμ.
(4.11)
Proof. (a) The identity (4.2), with α = a; β = a; ¯ v = ζ¯ gives Ln (z, ζ¯ )Ln (a, a) ¯ = Ln (a, z)Ln (a, ¯ ζ¯ ) − Ln (a, ¯ z)Ln (a, ζ¯ ).
(4.12)
Since Ln (a, a) ¯ = 2i Im aKn (a, a) ¯ = i Ln (a, a) ¯ , we obtain Kn (z, ζ¯ ) =
Ln (a, ¯ z)Ln (a, ζ¯ ) − Ln (a, z)Ln (a, ¯ ζ¯ ) i , |Ln (a, a)| ¯ z − ζ¯
and (4.9) follows on taking account of (4.8). (b) Note first that En,a ∈ HB by Lemma 4.2(b), so that H(En,a ) is well defined. By definition, it consists of all entire functions g for which both g/En,a and g ∗ /En,a lie in the Hardy class of the upper half-plane, and the norm g En,a is finite. The reproducing kernel K for this space is given by (1.11), with E = En,a : K(ζ, z) =
∗ (z)E ∗ (ζ ) i En,a (z)En,a (ζ ) − En,a n,a . 2π z − ζ¯
(4.13)
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For g ∈ H(En,a ), the reproducing kernel relation (1.12) and Cauchy–Schwarz give, as at (3.9), g(z) K(z, z)1/2 g E , n,a
z ∈ C.
Inasmuch as En,a is a polynomial of degree n − 1, we see that as |z| → ∞, K(z, z) = O |z|2n−1 . Indeed, if we write En,a (t) =
n−1
cj t j ,
j =0
a calculation shows that K(ζ, z) =
i 2π
(cj ck − cj ck )
0j
zj ζ¯ k − zk ζ¯ j z − ζ¯
and then the estimate above follows. Consequently, for g ∈ H(En,a ), as |z| → ∞, g(z) = O |z|n−1/2 , so g is a polynomial of degree n − 1. Conversely, if g is a polynomial of degree n − 1, then g(z)/En,a (z) = O(|z|−1 ) as |z| → ∞, and it follows easily that g/En,a , g ∗ /En,a ∈ H 2 (C+ ), so g ∈ H(En,a ). (c) From (4.9) and (4.13), we see that K(ζ, z) = Kn (z, ζ¯ ). The reproducing kernel relation (1.12) gives, for polynomials P of degree n − 1, ∞ P (ζ ) = −∞
∞ = −∞
∞ = −∞
P (t)K(ζ, t) dt |En,a (t)|2 P (t)Kn (t, ζ¯ ) dt |En,a (t)|2 P (t)Kn (t, ζ ) dt. |En,a (t)|2
(d) We can write R = P S where both P and S are polynomials of degree n − 1. We multiply the identity in (c) by S and then integrate with respect to μ. We obtain
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R dμ =
(P S)(z) dμ(z)
=
∞ Kn (t, z) S(z) P (t) dt dμ(z) |En,a (t)|2 −∞
∞ =
P (t)
1 S(t) dt |En,a (t)|2
∞ −∞
∞ = −∞
S(z)Kn (t, z) dμ(z) dt
1 |En,a (t)|2
−∞
=
P (t)
R . |En,a |2
Here, we have used the reproducing kernel formula for the measure μ. Moreover, the interchange of integrals is justified by absolute convergence of all integrals involved. 2 Remark. The identity in (d) is a real line analogue of a unit circle formula much used in Szeg˝o theory [8, p. 198, Theorem 2.2], but I am not sure it is new. It seems similar to identities in the theory of orthogonal rational functions [3, p. 145], and seems in spirit similar to identities used by Simon [33, p. 456, Theorem 2.1]. 5. de Branges spaces of entire functions Recall the notation fn (a, b) =
Kn ξn +
a ,ξ K˜ n (ξn ,ξn ) n
+
b K˜ n (ξn ,ξn )
Kn (ξn , ξn )
.
We shall prove four general theorems in this section, and we begin by stating them. Throughout this section, we do not assume the hypotheses of Theorem 1.3. Theorem 5.1. Let μ be a measure with support on the real line, with all power moments j x dμ(x), j 0 finite, and with infinitely many points in its support. Let {ξn } be a sequence of real numbers. Assume that there is a non-real complex number a, and an infinite sequence of integers S, for which there exists f (a, z) =
lim
n→∞, n∈S
fn (a, z),
(5.1)
uniformly in compact subsets of C, and that f (a, a) ¯ = 0. Then
(5.2)
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(a) There exists, for all z, v ∈ C, f (z, v) =
lim
n→∞, n∈S
fn (z, v),
and the limit is uniform for z, v in compact subsets of C. (b) Let L(z, v) = (z − v)f (z, v).
(5.3)
L(z, v)L(α, β) = L(α, z)L(β, v) − L(β, z)L(α, v).
(5.4)
For all complex α, β, z, v,
(c) Let Im a > 0. Then for Im z > 0, f (a, ¯ z) f (a, z); L(a, ¯ z) > L(a, z).
(5.5) (5.6)
In particular, for Im z > 0, L(z, z¯ ) > 0
and f (z, z¯ ) > 0.
(5.7)
(d) If f (z, v) = 0, then Im z and Im v have the same sign. In particular, Im z > 0 ⇒ Im v > 0. Consequently, for Im a > 0, L(a, ¯ ·) ∈ HB. The assumption (5.2) is satisfied if a = iy, some y = 0. Indeed, as pn has all real zeros, Kn ξn +
iy K˜ n (ξn , ξn )
, ξn −
iy K˜ n (ξn , ξn )
=
2 n−1 iy pk ξ n + K˜ n (ξn , ξn ) k=0 n−1 pk (ξn )2 = Kn (ξn , ξn ), k=0
so fn (iy, −iy) 1, and also, for all real y, f (iy, −iy) 1.
(5.8)
Of course, it then follows from (5.7) that f (z, z¯ ) > 0 for all non-real z. Theorem 5.2. Assume the hypotheses of Theorem 5.1. Fix a with Im a > 0, and let Ea (z) =
√ 2π
L(a, ¯ z) . |L(a, a)| ¯ 1/2
(5.9)
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(a) Then all zeros of Ea lie in the lower half-plane, and Ea ∈ HB. Moreover, i Ea (z)Ea (ζ ) − Ea∗ (z)Ea∗ (ζ ) . 2π z − ζ¯
f (z, ζ¯ ) =
(5.10)
(b) For all g ∈ H(Ea ), and all z ∈ C, we have ∞ g(z) =
g(t)
−∞
f (z, t) dt. |Ea (t)|2
(5.11)
Moreover, f (z, ·) ∈ H(Ea ) for all z ∈ C. (c) For any a, b, with Im a > 0, Im b > 0, H(Ea ) = H(Eb ) and the norms · Ea and · Eb are equivalent. Theorem 5.3. Assume the hypotheses of Theorem 5.1. Fix a with Im a > 0. (a) Let F (z) = L(z, 0) = zf (0, z),
(5.12)
and let {ρj } be the zeros ρ of F for which f (ρ, ρ) = 0. These are all real and simple. f (ρ ,·) }j is an orthonormal sequence in H(Ea ) and for all g ∈ H(Ea ), (b) The set { √ j f (ρj ,ρj )
|g(ρj )|2 f (ρj , ρj ) j
g E
a
2 ,
(5.13)
while G[g] =
j
g(ρj )
f (ρj , z) ∈ H(Ea ). f (ρj , ρj )
(5.14)
(c) Assume that F ∈ / H(Ea ). Then for all g, h ∈ H(Ea ), we have ∞ −∞
(g h)(ρ ¯ j) g h¯ , = f (ρj , ρj ) |Ea |2
(5.15)
j
and G[g] = g.
(5.16)
Remarks. (a) Note that if ρ is a zero of F , then ρ is necessarily real, but we have not excluded the possibility that f (ρ, ρ) = 0. If this is the case, then g(ρ) = 0 for all g ∈ H(Ea ). This follows easily from the reproducing kernel relation (5.11) and Cauchy–Schwarz. (b) The possibility that f (ρ, ρ) = 0 occurs only in the above theorem. It cannot happen under the hypotheses of Theorems 1.3, 1.6, and 5.4 below.
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Theorem 5.4. Assume, in addition to the hypothesis of Theorem 5.1, that f (a, ·) is an entire function of exponential type σ > 0 and f (t, t) ∼ 1
for t ∈ R.
(5.17)
(a) Then for all complex b, f (b, ·) is an entire function of exponential type σ . (b) For all g ∈ PW σ , g = G[g] ∈ H(Ea ).
(5.18)
In particular, PW σ ⊂ H(Ea ). (c) Assume that there exists C0 > 0 such that for a.e. t ∈ R, μ ξn + lim inf
t K˜ n (ξn ,ξn )
μ (ξn )
n→∞
C0 ,
(5.19)
or, assume that for each r > 0, t r μ ξn + ˜ n (ξn ,ξn ) K dt = 0. − 1 lim n→∞ μ (ξn )
(5.20)
−r
Then PW σ = H(Ea ). We note that we do not assume that μn is absolutely continuous in the above result. Recall from (2.2) and (2.8) our notations Ln (u, v) = (u − v)Kn (u, v) and n (a, b) = (a − b)fn (a, b) L = μn (ξn )Ln ξn +
a
, ξn +
b
.
(5.21)
n (α, β) = L n (α, z)L n (β, v) − L n (β, z)L n (α, v). n (z, v)L L
(5.22)
K˜ n (ξn , ξn )
K˜ n (ξn , ξn )
Lemma 5.5. For all complex α, β, z, v,
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Proof. This is immediate from (5.21) and (4.2).
2
Proof of Theorem 5.1. (a) From Lemma 5.5, we have n (z, v)L n (a, a) n (a, z)L n (a, n (a, n (a, v). L ¯ =L ¯ v) − L ¯ z)L
(5.23)
¯ z) = fn (a, z¯ ) and the symmetry fn (a, b) = Our hypothesis (5.1), the conjugate relation fn (a, fn (b, a) give, uniformly for z in compact subsets of C, lim
n (a, z) = (a − z)f (a, z) = L(a, z); L
lim
n (a, ¯ z) = (a¯ − z)f (a, ¯ z) = L(a, ¯ z); L
lim
n (a, a) ¯ = L(a, a). ¯ L
n→∞, n∈S n→∞, n∈S n→∞, n∈S
By our hypothesis (5.2), and (5.3), L(a, a) ¯ = 2i(Im a)f (a, a) ¯ = 0, so (5.23) gives, uniformly for z, v in compact subsets of C, lim
n→∞, n∈S
n (z, v) = L
1 L(a, z)L(a, ¯ v) − L(a, ¯ z)L(a, v) . L(a, a) ¯
That is, there exists f (z, v) =
lim
n→∞, n∈S
fn (z, v) =
L(a, z)L(a, ¯ v) − L(a, ¯ z)L(a, v) , L(a, a)(z ¯ − v)
(5.24)
and the limit is uniform for z, v in compact sets with z = v. For the case z = v, we can use convergence continuation theorems and the maximum-modulus principle. (b) This follows directly from (5.22), by taking limits. (c), (d) Taking limits in Lemma 4.2(b) gives for Im z 0, f (a, (5.25) ¯ z) L(a, z). ¯ z) f (a, z) and L(a, We must show strict inequality in the second inequality. We first show the assertion on the zeros. Suppose Im v > 0 and f (z, v) = 0. Hurwitz’s theorem and Lemma 4.2(a), show that there exist {zn } with fn (zn , v) = 0 and lim
n→∞, n∈S
zn = z.
By Lemma 4.2(a), Im zn > 0. Then Im z 0. To prove that it is positive, we use our functional relation (5.24). Assume Im z = 0. Then the numerator in (5.24) can be written as 0 = L(a, z)L(a, ¯ v) − L(a, ¯ z)L(a, v) = L(a, z)L(a, ¯ v) − L(a, z)L(a, v).
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Defining h(u) =
L(a, u) L(a, ¯ u)
for Im u 0,
we have that h is meromorphic in the upper half-plane, satisfying there by (5.25), h(u) 1, except perhaps at isolated poles. But these are removable singularities, because of the local boundedness, so we obtain that h is analytic in the upper half-plane. Also, |h(x)| = 1 for real x (again, we can remove isolated singularities), while h(v) = L(a, v) = L(a, z) = 1. L(a, ¯ v) L(a, z) Since Im v > 0, the maximum-modulus principle shows that h = c in the upper half-plane, for some unimodular constant c. Then for all u in the upper half-plane, (5.24) gives f (u, v) =
cL(a, ¯ u)L(a, ¯ v) − L(a, ¯ u)cL(a, ¯ v) = 0. L(a, a)(u ¯ − v)
Hence f (u, v) = 0 for all complex u, and by conjugate symmetry, f (u, v) ¯ = 0 for all complex u. It follows that for each u in the upper half-plane, f (u, ·) has a zero in the upper half-plane. The exact same argument we just used shows that f (u, z) = 0 for all complex z. Hence, f is identically 0 as a function of two complex variables, contradicting that f (0, 0) = 1. So Im z > 0, as desired. It remains to prove strict inequality in (5.6). Suppose we have equality in (5.6) for some z. As above, we form h(u) =
L(a, u) , L(a, ¯ u)
which is analytic for u in the upper half-plane, and has |h| 1 there. We are also assuming |h(z)| = 1, so by the maximum-modulus principle, h = c for some unimodular constant c. As above, we obtain a contradiction. 2 Proof of Theorem 5.2(a), (b). (a) First, Theorem 5.1(d) shows that all zeros of Ea must lie in the open lower half-plane. Moreover, (5.6) shows that |Ea (z)| > |Ea (¯z)| for Im z > 0. So Ea ∈ HB. Next, ¯ , L(a, a) ¯ = 2i(Im a)f (a, a) ¯ = i L(a, a) so the functional equation (5.4) gives ¯ ζ¯ ) − L(a, ¯ z)L(a, ζ¯ ) ¯ = L(a, z)L(a, L(z, ζ¯ )i L(a, a) ⇒ (z − ζ¯ )f (z, ζ¯ )L(a, a) ¯ = i L(a, ¯ z)L(a, ¯ ζ ) − L(a, ¯ z¯ )L(a, ¯ ζ¯ ) . Taking account of the definition (5.9) of Ea , and recalling that Ea∗ (z) = Ea∗ (¯z), gives (5.10).
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(b) By Theorem 5.1, the function Ea ∈ HB, so H(Ea ) is well defined. If K denotes its reproducing kernel, (1.11) and (5.10) show that f (z, ζ¯ ) = K(ζ, z) and (1.12) gives (5.11). By de Branges’ theory, outlined in Section 3, also f (z, ·) ∈ H(Ea ). 2 For the proof of Theorem 5.2(c), we need: Lemma 5.6. (a) For Im a > 0, Im b > 0, and Im z 0, ¯ ¯ L(z, b) 2 |L(a, b)| . L(z, a) ¯ |L(a, a)| ¯
(5.26)
f (u, v)2 f (u, u)f ¯ (v, v). ¯
(5.27)
(b) For all u, v ∈ C,
(c) For all a, b ∈ R, with L(a, b) = 0, and all z ∈ C, f (z, z¯ )
|b − z| |L(a, z)| |Im z| |L(a, b)|
2 f (b, b).
(5.28)
Proof. (a) The functional equation (5.4) gives ¯ ¯ − L(a, ¯ L(z, b)L(a, a) ¯ = L(a, z)L(a, ¯ b) ¯ z)L(a, b). ¯ = If Im z 0, we obtain from Theorem 5.1(c), that |L(a, z)| |L(a, ¯ z)| and |L(a, ¯ b)| ¯ Thus |L(a, b)| |L(a, b)|. L(z, b)L(a, ¯ . ¯ ¯ z)L(a, b) a) ¯ 2L(a, (b) By the Cauchy–Schwarz inequality, Kn (z, w)2 Kn (z, z¯ )Kn (w, w). ¯ After appropriate substitutions in variable, and division by Kn (ξn , ξn ), this leads to fn (u, v)2 fn (u, u)f ¯ n (v, v). ¯ Now let n → ∞ through S. (c) Let a, b ∈ R. The functional equation (5.4) gives L(z, z¯ )L(a, b) = L(a, z)L(b, z¯ ) − L(b, z)L(a, z¯ ). Then
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2|Im z|f (z, z¯ )L(a, b) 2L(a, z)L(b, z) 2L(a, z)|b − z|f (b, b)1/2 f (z, z¯ )1/2 , by (b). Rearranging this gives the result.
2
Proof of Theorem 5.2(c). From (a) of the lemma, we see that for all z in the upper half-plane, Eb (z)/Ea (z) 2
¯ |L(a, b)| . 1/2 ¯ 1/2 |L(a, a)| ¯ |L(b, b)|
Recall that the denominator is positive, in view of (5.7). To show H(Ea ) = H(Eb ), let g ∈ H(Eb ). Then g/Eb , g ∗ /Eb ∈ H 2 (C+ ). The last inequality shows that also g/Ea , g ∗ /Ea ∈ H 2 (C+ ). Thus H(Ea ) ⊇ H(Eb ), and the converse inclusion is then obvious. Finally it follows that for all g, g Eb 2
¯ |L(a, b)| g Ea , ¯ 1/2 |L(a, a)| ¯ 1/2 |L(b, b)|
and the inequality is reversible, and thus the two norms are equivalent.
2
Proof of Theorem 5.3(a). First note that F cannot have any non-real zeros, for it is a uniform limit as n → ∞ through S, of zfn (0, z), which has only real zeros. Define, as at (3.10), the phase function ϕ by Ea (x) = Ea (x)e−iϕ(x) . From (5.10), for real x, F (x) = xf (x, 0) i Ea (x)Ea (0) − Ea∗ (x)Ea∗ (0) = 2π 1 = Ea (x)Ea (0)sin ϕ(x) − ϕ(0) . π
(5.29)
Also, d Ea (x) Ea (0)sin ϕ(x) − ϕ(0) dx 1 + Ea (x)Ea (0)cos ϕ(x) − ϕ(0) ϕ (x). π
1 F (x) = π
(5.30)
It follows from (5.29) and the fact that Ea has non-real zeros, that, F (x) = 0 ⇔ sin ϕ(x) − ϕ(0) = 0. Let α = ϕ(0) and recall that {sj } were defined at (3.12) by ϕ(sj ) = α + j π , j = 0, ±1, ±2, . . . . It follows that after reordering, the {ρj } are just the {sk }. We next show that these zeros with
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f (ρj , ρj ) = 0 are simple. If ρj is not simple, it follows from (5.29) and (5.30) that both ϕ(ρj ) = α + kπ for some k, and ϕ (ρj ) = 0. Then (3.11) with K taken as f shows that 2 1 ϕ (ρj )Ea (ρj ) = 0, π
f (ρj , ρj ) =
a contradiction. Thus, all zeros {ρj } of F with f (ρj , ρj ) = 0 are simple.
2
Proof of Theorem 5.3(b). Recall that α = ϕ(0). The second equation in (5.29) shows that for some constant C, eiα Ea (z) − e−iα Ea∗ (z) = CF (z).
(5.31)
Of course C = 0, as Ea and Ea∗ have zeros in opposite half-planes. If we knew that (3.13) holds, then we could simply apply the de Branges theory, but we do not. So we proceed as follows: we know that f (·,·) is the locally uniform limit of fn (·,·), so the {ρj } are limits of the zeros {ρj n } of fn . Here if j = k , fn (ρj n , ρk n ) =
Kn (tj n , tk n ) = 0. Kn (ξn , ξn )
(Recall (2.2) and (2.9).) Taking appropriate limits with appropriate j = j (n), k = k (n), and using Hurwitz’s theorem, leads to f (ρj , ρk ) = 0,
j = k.
(5.32)
The reproducing kernel relation (5.11) gives ∞ 0= −∞
f (t, ρj )f (t, ρk ) dt. |Ea (t)|2
k ,·) It follows then that { √ff (ρ } is an orthonormal sequence in H(Ea ) and for all g ∈ H(Ea ), we (ρk ,ρk ) k have (in view of (5.11)), the orthonormal expansion
G[g](z) =
j
g(ρj )
f (ρj , z) . f (ρj , ρj ) f (ρj , ρj )
By Bessel’s inequality, ∞ 2 |g(ρj )|2 g 2 . g Ea = E f (ρj , ρj ) a j
−∞
Moreover, every partial sum of G[g] ∈ H(Ea ), and the convergence of the series in the last inequality easily yields that G[g] is the limit of these partial sums in the norm of H(Ea ). As the latter space is a Hilbert space, we obtain (5.14). 2
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Proof of Theorem 5.3(c). Since F ∈ / H(Ea ), then recalling that α = ϕ(0), (5.31) shows that / H(Ea ). eiα Ea − e−iα Ea∗ ∈
(5.33)
This allows one to apply the theory in Section 3. We identified the {ρj } with the {sk }, and can just apply (3.14) to (3.16). 2 Proof of Theorem 5.4(a). We are assuming for a given a, with Im a > 0, that f (a, z) is of exponential type σ . Then the same is true of L(a, z) = (z − a)f (a, z) and L(a, ¯ z). By Lemma 5.6(a), if Im b > 0, Im z 0, ¯ L(z, b) ¯ 2 |L(a, b)| L(z, a) ¯ , |L(a, a)| ¯ and by Theorem 5.1(c), L(¯z, b) ¯ = L(z, b) L(z, b) ¯ . ¯ ·) is no greater than that of L(a, It follows easily that the exponential type of L(b, ¯ ·). The same is ¯ then true for f (b, ·) and f (a, ¯ ·), and hence also f (b, ·) and f (a, ·). The reverse assertion follows by symmetry. By conjugate symmetry, the same is true when Im a < 0 or Im b < 0. Thus when b is non-real, L(b, ·) and f (b, ·) have exponential type σ . It remains to show that if b is real, L(b, z) has type σ . From the functional relation (5.4), if α, β ∈ C with Im α, Im β = 0, and L(α, β) = 0, L(b, z) = L(z, b) =
1 L(α, z)L(β, b) − L(β, z)L(α, b). |L(α, β)|
As both L(α, z) and L(β, z) are of exponential type σ , it follows that L(b, z) is of type at most σ . To show that it is of type σ , we let d be real with L(b, d) = 0, c be non-real, and use Lemma 5.6(b), (c): f (c, z) f (c, c) ¯ 1/2 f (z, z¯ )1/2 f (c, c) ¯ 1/2
|d − z| |L(b, z)| f (d, d)1/2 . |Im z| |L(b, d)|
Thus for |Im z| 1, |f (c, z)| grows no faster than C|z||L(b, z)|. Since both f (c, ·) and L(b, ·) are entire of order σ , the Phragmen–Lindelöf principle allows one to estimate f (c, z) on the strip |Im z| 1. We deduce that the exponential type of f (c, ·) is no smaller than that of L(b, ·). Consequently L(b, ·), and hence f (b, ·), have exponential type σ . Thus they have type σ . 2 For the proof of Theorem 5.4(b), we need: Lemma 5.7. Assume in addition to the hypotheses of Theorem 5.1, that f (a, ·) is of type σ and (5.17) holds. Then
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(a) There exists C > 0 such that the infinitely many zeros {ρj } of L(z, 0) satisfy for all j , ρj +1 − ρj C.
(5.34)
(b) There exists C > 0 such that for all g ∈ PW σ , g(ρj )2 C g 2
L2 (R) .
(5.35)
f (z, z¯ ).
(5.36)
j
(c) For all z ∈ C, ∞ |f (ρj , z)|2 j =1
f (ρj , ρj )
Proof. (a) By (5.17) and (5.27), F (z) = L(z, 0) lies in Cartwright’s class. It also has type σ > 0, so has infinitely many zeros [13, p. 30, Remark 2]. Recall from (5.32) that f (ρj +1 , ρj ) = 0. Next, by hypothesis, f (ρj +1 , ·) is entire of exponential type, and bounded on the real axis. Indeed, our hypothesis (5.17), and (5.27) give f (ρj +1 , x) f (ρj +1 , ρj +1 )1/2 f (x, x)1/2 C1 . Bernstein’s inequality for entire functions of exponential type [13, p. 227] gives for all real t, ∂ f (ρj +1 , t) C1 σ. ∂t Then using our (5.17) again, for some ξ between ρj and ρj +1 , C2 f (ρj +1 , ρj +1 ) = f (ρj +1 , ρj +1 ) − f (ρj +1 , ρj ) ∂ f (ρj +1 , t)|t=ξ (ρj +1 − ρj ) = ∂t C1 σ (ρj +1 − ρj ). (b) This is an immediate consequence of (a) and a well-known estimate [13, p. 150, no. 4]. (c) This follows by applying Bessel’s inequality (5.13) to g(t) = f (t, z), and using the reproducing kernel identity (5.11).
2
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Proof of Theorem 5.4(b). Let g ∈ PW σ and ∞
G(z) = G[g](z) =
g(ρj )
j =−∞
f (ρj , z) . f (ρj , ρj )
We claim that G ∈ H(Ea ). Indeed, by (b) of the previous lemma, and as f (ρj , ρj ) ∼ 1 uniformly in j , |g(ρj )|2 < ∞, f (ρj , ρj ) j
and as in the proof of Theorem 5.3(b), this gives G ∈ H(Ea ). We are going to show that G = g. To this end, let Ψ (z) =
g(z) − G(z) . F (z)
As G(ρj ) = g(ρj ) (recall (5.32)) and F has simple zeros at each ρj (recall Theorem 5.3(a)), so Ψ is entire. As both numerator and denominator are of exponential type, so is Ψ [13, Theorem 5, p. 80]. Next, we claim that also G(z) =
∞
g(ρj )
j =−∞
F (z) . F (ρj )(z − ρj )
Let F (α) = L(α, 0) = 0. Since L(0, ρj ) = 0, the functional equation (5.4) gives L(z, ρj )L(α, 0) = L(α, z)L(0, ρj ) − L(0, z)L(α, ρj ) = F (z)L(α, ρj ) ⇒
f (z, ρj ) =
F (z)L(α, ρj ) . F (α)(z − ρj )
Letting z → ρj , we obtain f (ρj , ρj ) = F (ρj )
L(α, ρj ) . F (α)
Combining these last two identities, we see that f (ρj , z) F (z) = , f (ρj , ρj ) F (ρj )(z − ρj ) and we have (5.37). Next, that identity shows that
1/2 ∞
1/2 ∞ G(z) 2 1 g(ρj ) . F (z) |F (ρj )(z − ρj )|2 j =−∞
j =−∞
(5.37)
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Let ε ∈ (0, π2 ). Here in the cut (double) sector Aε = {z: |z| 1 and ε | arg z| π − ε}, there exists Cε such that for all j , |z − ρj | Cε |i − ρj |. Moreover, ∞ j =−∞
∞ 1 |F (i)|2 1 = |F (ρj )(i − ρj )|2 |F (i)|2 |F (ρj )(i − ρj )|2 j =−∞
=
∞ f (ρj , i) 2 1 | | 2 f (ρj , ρj ) |F (i)| j =−∞
1 |F (i)|2 infx∈R f (x, x)
f (i, ı¯) < ∞,
by (5.36). Then for any n 1, we see that
1/2 1/2 ∞ G(z) 2 1 1 g(ρj ) lim sup . Cε2 |F (ρj )(i − ρj )|2 z→∞, z∈Aε F (z) j =−∞
|j |n
Since this has limit 0 as n → ∞, we have shown that G(z) = 0. lim z→∞, z∈Aε F (z)
(5.38)
Next, F is of exponential type σ , has real zeros, and F (x) = xf (0, x) |x|f (0, 0)1/2 f (x, x)1/2 C|x| by (5.17) and (5.27). Thus it lies in the Cartwright class. From (3.6), for θ ∈ (−π, π) \ {0}, log |F (reiθ )| = σ |sin θ |. r→∞ r lim
Let us now assume g has type τ < σ . Since it is square integrable along the real axis, g also lies in the Cartwright class. By (3.7), for θ ∈ (−π, π) \ {0}, lim sup r→∞
log |g(reiθ )| τ |sin θ |. r
Then for θ ∈ (−π, π) \ {0}, as r → ∞, g iθ re exp (τ − σ )r|sin θ | + o(r) . F
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In particular, for such θ , g lim reiθ = 0. r→∞ F Then for θ ∈ (−π, π) \ {0}, this and (5.38) show that lim |Ψ | reiθ = 0.
r→∞
Inasmuch as Ψ is an entire function of exponential type, the Phragmen–Lindelöf principle (applied on sectors of opening angle less than π ) shows that it is bounded in the plane, and hence constant. As it has limit 0 at ∞, we have Ψ ≡ 0, so g = G ∈ H(Ea ). Finally, if g has type σ , then for ε ∈ (0, 1), the scaled function gε (z) = g(εz) has type εσ < σ , so gε = G[gε ]. It is easily seen that we can let ε → 1− in both sides of this identity.
2
We note that for several of the proofs in this section, one can avoid using de Branges’ machinery, and instead take limits in results that hold for the original reproducing kernels Kn . For the proof of Theorem 5.4(c), we seem to be forced to do the latter. Proof of Theorem 5.4(c). Let g ∈ H(Ea ). Since g/Ea , g/Ea∗ ∈ H 2 (C+ ), while Ea is of exponential type σ , it follows that g has exponential type at most σ . Next, recall that {tj n } = {tj n (ξn )} are the quadrature points for μ including ξn . Fix 1. The Gauss quadrature formula (2.5) and the fact that Kn (tj n , tkn ) = 0 for j = k, gives |g(ρj )|2 Kn (tj n , s) 2 . g(ρ ) dμ(s) = j Kn (tj n , tj n ) Kn (tj n , tj n ) |j |
|j |
Let r > 0 and make the substitution s = ξn +
t t = ξn + ˜ Kn (ξn , ξn )μ (ξn ) Kn (ξn , ξn )
and recall (2.7), (2.9). By dropping the singular part of μ, we obtain, for large enough n, t r |g(ρj )|2 fn (ρj n , t) 2 μ (ξn + K˜ n (ξn ,ξn ) ) g(ρ ) dt . j fn (ρj n , ρj n ) μ (ξn ) fn (ρj n , ρj n )
−r |j |
|j |
(5.39)
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As n → ∞ through S, the right-hand side converges to ∞ |g(ρj )|2 |g(ρj )|2 . f (ρj , ρj ) f (ρj , ρj ) j =−∞
|j |
Recall from Theorem 5.3(b) that this series converges. Next, as n → ∞ through S, uniformly for t in compact sets,
g(ρj )
|j |
fn (ρj n , t) f (ρj , t) → =: G (t) g(ρj ) fn (ρj n , ρj n ) f (ρj , ρj ) |j |
and we have also used the uniform convergence of fn (0, z) to f (0, z), which forces the zeros {ρj n } of fn to converge to those of f . By Fatou’s lemma, t 2 μ ξ + r n f (ρ , t) ˜ n (ξn ,ξn ) n j n K dt g(ρj ) lim inf fn (ρj n , ρj n ) μ (ξn ) n→∞, n∈S −r |j |
r −r
μ ξn + ˜ t 2 Kn (ξn ,ξn ) G (t) lim inf dt μ (ξn ) n→∞, n∈S r
C0
G (t)2 dt,
−r
under our hypothesis (5.19). Alternatively, if we assume our Lebesgue point type condition (5.20), we write the left-hand side of (5.39) as t 2 2 μ ξ + r r n f f (ρ , t) (ρ , t) ˜ n (ξn ,ξn ) n j n n j n K − 1 dt g(ρj ) dt + g(ρj ) fn (ρj n , ρj n ) fn (ρj n , ρj n ) μ (ξn )
−r |j |
r = −r
−r |j |
r t r μ ξn + K˜ n (ξn ,ξn ) 2 2 G (t) dt + o(1) + O − 1 dt = G (t) dt + o(1). μ (ξ ) −r
n
−r
Thus r C0
∞ G (t)2 dt
−r
j =−∞
|g(ρj )|2 , f (ρj , ρj )
C0 = 1 if we have the Lebesgue point type condition. As in the proof of Theorem 5.4(b), where r |g − G |2 → 0 as → ∞, so we obtain −r r C0 −r
∞ g(t)2 dt j =−∞
|g(ρj )|2 . f (ρj , ρj )
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Letting r → ∞ gives ∞ |g|2
C0 −∞
∞ j =−∞
|g(ρj )|2 . f (ρj , ρj )
(5.40)
Thus g ∈ L2 (R), and g is of exponential type at most σ , so also g ∈ PW σ . We have shown that H(Ea ) ⊂ PW σ , and hence H(Ea ) = PW σ . It remains to prove equivalence of the norms. First observe that F ∈ / H(Ea ). Indeed, if F ∈ H(Ea ), as F (ρj ) = 0 for all j , Theorem 5.4(b) shows that identically F = G[F ] = 0, a contradiction. Then (5.15) shows that g 2Ea
∞ g = E
2 ∞ |g(ρj )|2 = C0 g 2L2 (R) f (ρj , ρj ) a
−∞
j =−∞
by (5.40). In the other direction, as f (ρj , ρj ) ∼ 1 uniformly in j , (5.35) shows that g 2Ea =
∞ j =−∞
|g(ρj )|2 C2 g 2L2 (R) . f (ρj , ρj )
2
An alternative proof of the norm equivalence uses the closed graph theorem. Let I denote the identity operator from H(Ea ) to PW σ . Its graph {(f, If ): f ∈ H(Ea )} is all of H(Ea ) × PW σ , so is closed. Then the operator I is a continuous linear operator, and so is bounded. 6. Proof of Theorems 1.3, 1.4, 1.5 Lemma 6.1. Assume the hypotheses of Theorem 1.3. (a) {fn (u, v)}∞ n=1 is uniformly bounded for u, v in compact subsets of the plane. (b) Let f (u, v) denote the locally uniform limit of some subsequence {fn (u, v)}n∈S of {fn (u, v)}∞ n=1 . Then for each fixed u ∈ C, f (u, ·) is an entire function of exponential type. Moreover, for some C1 and C2 independent of u, v, and the subsequence S, f (u, v) C1 eC2 (|Im u|+|Im v|) . (6.1) Proof. This is exactly the same as that of Lemma 5.2 in [21], but we provide some details. We assumed that μ ∼ 1 in some open set O containing compact J . It follows that J is covered by finitely many open intervals in O. By increasing the size of J , we may assume that J consists of finitely many compact intervals. It then suffices to consider the case where J is just one interval, and we now assume this. Since μ is absolutely continuous in the larger open set O, and μ is bounded above and below there, we have the well-known bound [28, Theorem 20, p. 116] 1 Kn (x, x)−1 = λn (x) ∼ , n
(6.2)
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uniformly in n and in each compact subset of O. By reducing O, we can assume this holds in O. By Cauchy–Schwarz, we have 1 Kn (ξ, t) n
1/2 1/2 1 1 Kn (ξ, ξ ) Kn (t, t) C n n
for ξ, t ∈ O. By Bernstein’s growth lemma in the plane [21, Lemma 5.1], applied separately in each variable, we then have for ξ, t ∈ O, |a|, |b| A and n n0 (A), a b 1 K CeC2 (|a|+|b|) . ξ + i , t + i (6.3) n n n n (Strictly speaking, we have to take a slightly smaller set than O, but can relabel.) C and C2 are independent of A, ξ, t, a, b. Of course if u, v lie in a bounded subset of the plane, and ξ ∈ O, Im(u) Re(u) then for n large enough, we may write ξ + un = ξ + Re(u) n + i n , where ξ + n is contained in a slightly larger open set than O. By relabelling, we may assume it lies in O. Then we may recast (6.3) in the form u 1 v Kn ξ + , ξ + CeC2 (|Im u|+|Im v|) . (6.4) n n n Since K˜ n (ξn , ξn ) ∼ n, we see also that for |u|, |v| A and n n0 (A) fn (u, v) C1 eC2 (|Im u|+|Im v|) , where C1 , C2 are independent of n, u, v, A. The stated uniform boundedness follows. (b) Now {fn (u, v)}∞ n=1 is a normal family of two variables u, v. If f (u, v) is the locally uniform limit through the subsequence S of integers, we see that f (u, v) is an entire function in u, v satisfying for all complex u, v, f (u, v) C1 eC2 (|Im u|+|Im v|) .
(6.5)
In particular, f (u, v) is bounded for u, v ∈ R, and is an entire function of exponential type in each variable. 2 Proof of Theorem 1.3. (a) This follows directly from Lemma 6.1. (b) This follows from Lemma 6.1(b) and Theorem 5.4. Note that the hypothesis f (t, t) ∼ 1 there is an easy consequence of (6.2) and the fact the values of f are limits of ratios of Kn taken over smaller and smaller neighborhoods of ξn . Note too that σ > 0. Otherwise (3.4) implies f (0, z) = C for all z, but μ ∼ 1 in O forces f (0, z) to have zeros. (c) This follows from Theorem 5.2. (d) This follows from Theorem 5.4. The hypothesis (5.19) follows easily from our hypothesis μ ∼ 1 in an open set containing J . 2
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Proof of Theorem 1.4. (a) The functional relation (1.18) is (5.4) in Theorem 5.1. Next, once we know f (a, z) for all z, we also know f (a, ¯ z) = f (a, z¯ ) for all z. Moreover, as shown after Theorem 5.1, f (iy, −iy) 1 for all real y, and then (5.7) shows that L(a, a) ¯ = 0. Then for all z, v, L(z, v) =
1 L(a, z)L(a, ¯ v) − L(a, ¯ z)L(a, v) . L(a, a) ¯
So L(z, v) and hence f (z, v) is uniquely determined. (b) This follows from Theorem 5.3. (c) The expansions were established in Theorems 5.3(c) and 5.4(b).
2
Proof of Theorem 1.5. The expansion (1.20) ensures that {ρj } is a complete interpolating sequence for PW σ , as defined in Section 3. Indeed (1.20) shows that each g ∈ PW σ is uniquely determined by its values on {ρj }, and we cannot drop a single ρk , since f (ρk , z) vanishes at all ρj with j = k. By a Theorem of Hruscev, Nikolskii, and Pavlov [10, p. 286], [30, p. 791], the function h(t) = ν(t) − πσ t belongs to BMO. By (3.17), this ensures that for each p > 0, 1 sup |h − hI |p < ∞. (6.6) I |I | I
Next, we apply a well-known inequality [9, p. 223, Lemma 1.1]: if I and J are intervals with |J | > 2|I |, then |hI − hJ | C log |J |/|I | , where C is independent of I and J . This leads easily to the estimate |h[−r,r] | C log r,
r 2.
Together with (6.6), this yields for j 1, 2j +1
|h|(t)p dt C2j j p 2j
and hence 2j +1
2j
|h(t)|p dt Cj −τ . (1 + |t|)(log(2 + |t|))p+τ
Adding over j 1, gives ∞ 2
|h(t)|p dt < ∞. (1 + |t|)(log(2 + |t|))p+τ
The range (−∞, −2) can be treated similarly.
2
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7. Proof of Theorems 1.6 and 1.7 Proof of Theorem 1.6. The assumption (1.26) implies that {fn (a, ·)}∞ n=1 is uniformly bounded in compact subsets of the plane. Using the functional relation (5.23), we deduce that the same is true of {fn (·,·)}∞ n=1 . Note too that if z = x + iy, the fact that each pn has real zeros, ensures that fn (z, z¯ ) fn (x, x). ¯ ∞ Our hypothesis (1.27) ensures that if we fix a non-real a, {fn (a, a)} n=1 is bounded below. Next, let A > 0. The exponential bound (1.26) together with the functional relation (5.22) ensures that for n n0 (A) and |z|, |v| A, fn (z, v) C1 eC1 (|Im z|+|Im v|) . Here C1 and C2 are independent of n, A, z, v. If we take some non-constant subsequential limit f , then it follows that the hypotheses of Theorems 5.1 and 5.4 are fulfilled. Indeed, the hypothesis (5.17) in Theorem 5.4 follows from (1.27), while (1.28) is the requisite modification of (5.19). The proofs of Theorems 5.1 to 5.4 then go through without change. 2 Proof of Theorem 1.7. Step 1. The functions f and E: We assume that f (z, ζ¯ ) = K(ζ, z), the reproducing kernel for H(E) = PW σ and that f (0, 0) = 1. Recall that equality of the spaces implies norm equivalence, and in turn this implies from (3.18), f (x, x) = K(x, x) ∼ 1 in R.
(7.1)
We know from the de Branges theory (cf. (1.11)) that if z = v, i E(z)E ∗ (v) − E ∗ (z)E(v) , 2π z−v
(7.2)
i E (z)E ∗ (z) − E ∗ (z)E(z) . 2π
(7.3)
f (z, v) = while f (z, z) =
By definition of a de Branges space, E has no zeros in {z: Im z > 0}. It follows from (7.1) and (7.3) that it also has no real zeros. For if E(x) = 0, then also E ∗ (x) = 0, and (7.3) gives f (x, x) = 0, contradicting (7.1). Next, as f is a reproducing kernel for H(E), for each fixed u, f (u, ·) ∈ H(E) = PW σ . Thus f (u, ·) is of exponential type at most σ . We use this to show that E is of exponential type at most σ . As usual, define L(z, v) = (z − v)f (z, v) =
i E(z)E ∗ (v) − E ∗ (z)E(v) . 2π
A little manipulation shows that for complex u, v, z, E(z)L(u, v) = L(z, v)E(u) − L(z, u)E(v).
(7.4)
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Choose u, v with L(u, v) = 0. Such a choice is possible, for otherwise f (u, v) = 0 for all u = v, and continuity yields a contradiction to (7.1). Since L(·, u) and L(·, v) are of exponential type at most σ , it follows from (7.4) that E(·) is of exponential type σ . Step 2. The construction of En : Next, as E belongs to the Hermite–Biehler class, for Im z > 0, ∗ E (z) E(z) 1, recall (1.8). This implies that the function E ∗ belongs to the class P , studied in detail in [13, p. 217 ff]. See Corollary 3 in [13, p. 218]. As a consequence [13, Corollary 6, p. 219], there is a sequence of polynomials {Pn } without zeros in the (closed) lower half-plane, that converges to E ∗ , uniformly in compact sets. Define En (z) = Pn∗ (z) = Pn (¯z), a polynomial with zeros only in the open lower half-plane. Then for Im z > 0, |En (z)| |En (¯z)|, so En ∈ HB. We see that uniformly in compact subsets of the plane, lim En (z) = E(z).
n→∞
(7.5)
We may assume that En has degree n. Indeed, it is obvious that we can assume En has degree at most n, and we can multiply by factors 1 − n2z−i to make it up to full degree. Next, for n 1, let Ωn (t) =
1 , |En (t)|2
t ∈ (−∞, ∞).
The measure Ωn (t) dt has the first 2n − 1 finite power moments, and so we can define corresponding orthonormal polynomials {pj (Ωn , ·)}n−1 j =0 . Let Kn (Ωn , ·, ·) denote the nth reproducing kernel formed from these orthogonal polynomials. Then Kn (Ωn , z, v) =
i En (z)En∗ (v) − En∗ (z)En (v) . 2π z−v
(7.6)
This was proved in [23], but also follows easily from the theory of de Branges spaces. Indeed, as En is a polynomial of degree n, so H(En ) is the set of polynomials of degree n − 1. The right-hand side of (7.6) is the reproducing kernel for H(En ) (apart from notational conventions such as conjugate variables). By uniqueness of reproducing kernels, it equals the left-hand side. Next, by (7.5) and (7.2), uniformly for z, v in compact sets, lim Kn (Ωn , z, v) = f (z, v).
(7.7)
lim Kn (Ωn , 0, 0) = f (0, 0) = 1,
(7.8)
n→∞
In particular, n→∞
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and hence Kn (Ωn , 0, 0) 1 = . lim K˜ n (Ωn , 0, 0) = lim 2 n→∞ n→∞ |En (0)| |E(0)|2 Then Kn Ωn , 0 + lim
n→∞
z ,0 + ˜ v K˜ n (Ωn ,0,0) Kn (Ωn ,0,0)
Kn (Ωn , 0, 0)
2 2 = f E(0) z, E(0) v .
(7.9)
Step 3. Truncate the support of Ωn : Choose an > 0 such that |t|an
1 Kn (Ωn , t, t)Ωn (t) dt . n
Let P be a polynomial of degree n − 1, possibly with complex coefficients. Using the Christoffel function inequality, P (t)2 Kn (Ωn , t, t)
∞ |P |2 Ωn ,
t ∈ R,
−∞
we see then that |P |2 Ωn |t|an
1 n
∞ |P |2 Ωn . −∞
Let Jn = [−an , an ]. From this last inequality, and the extremal properties of Christoffel functions, it follows easily that for real x, 1 −1 1 λn (Ωn , x)/λn (Ωn|Jn , x) 1 − . n
(7.10)
More generally, for complex z, the extremal property Kn (Ωn , z, z¯ ) =
|P (z)|2 2 deg(P )n−1 |P | Ωn sup
gives 1 1 Kn (Ωn , z, z¯ )/Kn (Ωn|Jn , z, z¯ ) 1 − . n
(7.11)
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We use this to derive a complex analogue of an inequality that formed the basis of [22]. By the reproducing kernel property of Kn ,
Kn (Ωn|J , z, t) − Kn (Ωn , z, t)2 ωn|J (t) dt n n 2 = Kn (Ωn|Jn , z, z¯ ) − 2Kn (Ωn , z, z¯ ) + Kn (Ωn , z, t) Ωn|Jn (t) dt 2 Kn (Ωn|Jn , z, z¯ ) − 2Kn (Ωn , z, z¯ ) + Kn (Ωn , z, t) Ωn (t) dt = Kn (Ωn|Jn , z, z¯ ) − Kn (Ωn , z, z¯ ).
(7.12)
Using the Christoffel function inequality P (v)2 Kn (Ωn|J , v, v) ¯ n
|P |2 Ωn|Jn
on the polynomial P (t) = Kn (Ωn|Jn , z, t) − Kn (Ωn , z, t), and using (7.12), we obtain for all complex z, v, Kn (Ωn|J , z, v) − Kn (Ωn , z, v)2 Kn (Ωn|J , v, v) ¯ Kn (Ωn|Jn , z, z¯ ) − Kn (Ωn , z, z¯ ) . n n Using (7.11), we continue this as
C Kn (Ωn , v, v)K ¯ n (Ωn , z, z¯ ). n
The constant is independent of z, v, n. From this, (7.9), and (7.11), it follows easily that uniformly for z, v in compact sets, Kn Ωn|Jn , 0 + lim
z v ,0 + ˜ K˜ n (Ωn|Jn ,0,0) Kn (Ωn|Jn ,0,0)
Kn (Ωn|Jn , 0, 0)
n→∞
2 2 = f E(0) z, E(0) v .
(7.13)
Step 4. Scale Ωn|Jn to obtain μn : Define a measure μn on [−1, 1] by μn (x) =
Ωn (an x) , Ωn (0)
x ∈ [−1, 1],
and set μn = 0 outside [−1, 1]. As En has no real zeros, Ωn is infinitely differentiable on the real line, so the same is true of μn on (−1, 1). A substitution in the orthonormality relations shows that pk (μn , x) = pk (Ωn|Jn , an x)[an Ωn (0)]1/2 , and hence Kn (μn , z, v) = Kn (Ωn|Jn , an z, an v)an Ωn (0), and recalling μn (0) = 1, K˜ n (μn , 0, 0) = Kn (Ωn|Jn , 0, 0)an Ωn (0) = an K˜ n (Ωn|Jn , 0, 0).
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Then Kn μn , 0 +
a ,0 + ˜ b K˜ n (μn ,0,0) Kn (μn ,0,0)
Kn (μn , 0, 0)
=
Kn Ωn|Jn , 0 +
a b ,0 + ˜ K˜ n (Ωn|Jn ,0,0) Kn (Ωn|Jn ,0,0)
Kn (Ωn|Jn , 0, 0)
,
so (7.13) gives Kn μn , 0 + lim
a ,0 + ˜ b K˜ n (μn ,0,0) Kn (μn ,0,0)
Kn (μn , 0, 0)
n→∞
2 2 = f E(0) a, E(0) b .
Then (1.29) follows if we assume |E(0)| = 1. Next, the upper bound (1.26) follows easily from the uniform convergence and the fact that f (a, ·) is of exponential type. The lower bound (1.27) follows easily from (7.1) and the uniform convergence. Finally, for each real t, μn 0 +
t K˜ n (μn ,0,0) μn (0)
= Ωn
t K˜ n (Ωn|Jn , 0, 0)
/Ωn (0)
|En (0)| = |En (|E(0)|2 t (1 + o(1)))| 2 |E(0)| → , |E(|E(0)|2 t)|
2
as n → ∞. The condition (1.28) then follows for t in a given finite interval. Of course if |E| is bounded above and below in the real line, it holds throughout the real line. 2 References [1] A. Avila, J. Last, B. Simon, Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with a.c. spectrum, submitted for publication. [2] J. Baik, T. Kriecherbauer, K.T.-R. McLaughlin, P.D. Miller, Uniform Asymptotics for Polynomials Orthogonal with Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles, Princeton Ann. of Math. Stud., 2006. [3] A. Bultheel, P. Gonzalez-Vera, E. Hendriksen, O. Njastad, Orthogonal Rational Functions, Cambridge University Press, Cambridge, 1999. [4] L. de Branges, Hilbert Spaces of Entire Functions, Prentice Hall, New Jersey, 1968. [5] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach, Courant Inst. Lect. Notes, vol. 3, New York University Press, New York, 1999. [6] M. Findley, Universality for regular measures satisfying Szeg˝o’s condition, J. Approx. Theory 155 (2008) 136–154. [7] P.J. Forrester, Log-gases and Random Matrices, online book, http://www.ms.unimelb.edu.au/~matpjf/matpjf.html. [8] G. Freud, Orthogonal Polynomials, Pergamon Press/Akademiai Kiado, Budapest, 1971. [9] J.B. Garnett, Bounded Analytic Functions, Academic Press, Orlando, 1981. [10] S.V. Hruscev, N.K. Nikolskii, B.S. Pavlov, Unconditional bases of exponentials and reproducing kernels, in: V.P. Havin, N.K. Nikoslkii (Eds.), Complex Analysis and Spectral Theory, in: Lecture Notes in Math., vol. 864, Springer, New York, 1981, pp. 214–335. [11] P. Koosis, The Logarithmic Integral I, Cambridge University Press, Cambridge, 1988. [12] A.B. Kuijlaars, M. Vanlessen, Universality for eigenvalue correlations from the modified Jacobi unitary ensemble, Int. Math. Res. Not. 30 (2002) 1575–1600. [13] B.Ya. Levin, Lectures on Entire Functions, in collaboration with Yu. Lyubarskii, M. Sodin, V. Tkachenko, Transl. Math. Monogr., vol. 150, American Mathematical Society, Providence, 1996.
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[14] Eli Levin, D.S. Lubinsky, Universality limits involving orthogonal polynomials on the unit circle, Comput. Methods Funct. Theory 7 (2007) 543–561. [15] Eli Levin, D.S. Lubinsky, Universality limits in the bulk for varying measures, Adv. Math. 219 (2008) 743–779. [16] Eli Levin, D.S. Lubinsky, Universality limits for exponential weights, Constr. Approx. 29 (2009) 247–275. [17] Eli Levin, D.S. Lubinsky, Universality limits at the soft edge of the spectrum via classical complex analysis, manuscript. [18] D.S. Lubinsky, A new approach to universality limits at the edge of the spectrum, in: Contemp. Math. (60th Birthday of Percy Deift), vol. 458, 2008, pp. 281–290. [19] D.S. Lubinsky, Mutually regular measures have similar universality limits, in: M. Neamtu, L. Schumaker (Eds.), Proceedings of Twelfth Texas Conference on Approximation Theory, Nashboro Press, Nashville, 2008, pp. 256– 269. [20] D.S. Lubinsky, Universality limits at the hard edge of the spectrum for measures with compact support, Int. Math. Res. Not. 2008 (2008), Article ID rnn099, 39 pp. [21] D.S. Lubinsky, Universality limits in the bulk for arbitrary measures on compact sets, J. Anal. Math. 106 (2008) 373–394. [22] D.S. Lubinsky, A new approach to universality limits involving orthogonal polynomials, Ann. of Math., in press. [23] D.S. Lubinsky, Explicit orthogonal polynomials for reciprocal polynomial weights on (−∞, ∞), Proc. Amer. Math. Soc., in press. [24] Y.L. Lyubarskii, K. Seip, Complete interpolating sequences for Paley–Wiener spaces and Muckenhoupt’s (Ap ) condition, Rev. Mat. Iberoamericana 13 (1997) 361–376. [25] Y.L. Lyubarskii, K. Seip, Weighted Paley–Wiener spaces, J. Amer. Math. Soc. (2002) 979–1006. [26] K.T.-R. McLaughlin, P. Miller, The ∂¯ steepest descent method for orthogonal polynomials on the real line with varying weights, Int. Math. Res. Not. 2008 (2008), Article ID rnn075, 66 pp. [27] M.L. Mehta, Random Matrices, 2nd ed., Academic Press, Boston, 1991. [28] P. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 213 (1979). [29] P. Nevai, Geza Freud, orthogonal polynomials and Christoffel functions: A case study, J. Approx. Theory 48 (1986) 3–167. [30] J. Ortega-Cerda, K. Seip, Fourier frames, Ann. of Math. 155 (2002) 789–806. [31] L. Pastur, M. Schcherbina, Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles, J. Stat. Phys. 86 (1997) 109–147. [32] B. Simon, Orthogonal Polynomials on the Unit Circle, Parts 1 and 2, American Mathematical Society, Providence, 2005. [33] B. Simon, Orthogonal polynomials with exponentially decaying recursion coefficients, Centre Rech. Math. Proc. Lect. Notes 42 (2007) 453–463. [34] B. Simon, Two extensions of Lubinsky’s universality theorem, J. Anal. Math. 105 (2008) 345–362. [35] B. Simon, The Christoffel–Darboux kernel, in: Perspectives in PDE, Harmonic Analysis and Applications, a Volume in Honor of V.G. Maz’ya’s 70th Birthday, in: Proc. Sympos. Pure Math., vol. 79, 2008, pp. 295–335. [36] H. Stahl, V. Totik, General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992. [37] V. Totik, Asymptotics for Christoffel functions for general measures on the real line, J. Anal. Math. 81 (2000) 283–303. [38] V. Totik, Universality and fine zero spacing on general sets, Ark. Mat., in press.
Journal of Functional Analysis 256 (2009) 3730–3742 www.elsevier.com/locate/jfa
Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains ˘ ckovi´c a , Sönmez Sahuto˘ ˘ Zeljko Cu˘ ¸ glu b,∗,1 a University of Toledo, Department of Mathematics, Toledo, OH 43606-3390, USA b University of Michigan, Department of Mathematics, Ann Arbor, MI 48109-1043, USA
Received 12 September 2008; accepted 25 February 2009 Available online 5 March 2009 Communicated by N. Kalton
Abstract Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let β be a function smooth up to the boundary on a smooth bounded pseudoconvex domain Ω ⊂ Cn . We show that, if Ω is convex or the Levi form of the boundary of Ω is of rank at least n − 2, then compactness of the Hankel operator Hβ implies that β is holomorphic “along” analytic discs in the boundary. Furthermore, when Ω is convex in C2 we show that the condition on β is necessary and sufficient for compactness of Hβ . © 2009 Elsevier Inc. All rights reserved. Keywords: Hankel operators; ∂-Neumann problem; Pseudoconvex; Analytic discs
1. Introduction Hankel operators form an important class of operators on spaces of holomorphic functions. Initially there were two descriptions of Hankel operators, one considered it as an operator on the one-sided sequence space l 2 into itself, and the other as an operator from the Hardy space H 2 of the unit disk into its orthogonal complement in L2 . These operators are closely connected to problems in approximation theory as shown by now the famous work of Nehari [18] on one hand, * Corresponding author.
˘ ckovi´c), [email protected] (S. Sahuto˘ ˘ Cu˘ E-mail addresses: [email protected] (Z. ¸ glu). 1 Supported in part by NSF grant number DMS-0602191.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.018
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and Adamjan, Arov and Krein [1] on the other. These operators also have a close connection to Toeplitz operators, and the commutators of projections and multiplication operators on L2 . More about Hankel operators and related topics can be found [20]. Let Ω be a bounded domain in Cn and dV denote the Lebesgue volume measure. The Bergman space A2 (Ω) is the closed subspace of L2 (Ω) consisting of holomorphic functions on Ω. The Bergman projection P is the orthogonal projection from L2 (Ω) onto A2 (Ω) and can be written explicitly as Pf (z) = Ω K(z, w)f (w) dV (w), where K(z, w) is the Bergman kernel of Ω. For β ∈ L2 (Ω) we can define the Hankel operator Hβ from A2 (Ω) into L2 (Ω) by Hβ (g) = (Id − P )(βg). In general, Hβ is only densely defined on A2 (Ω). When Ω is a bounded pseudoconvex domain, Kohn’s formula P = Id − ∂ ∗ N∂ (N is the (bounded) inverse of complex Laplacian, ∂∂ ∗ + ∂ ∗ ∂, and ∂ ∗ is the Hilbert space adjoint of ∂ on the square integrable (0, 1)-forms on Ω) implies that Hβ (f ) = ∂ ∗ N∂(βf ) = ∂ ∗ N (f ∂β) for f ∈ A2 (Ω) and β ∈ C 1 (Ω). This will be the main tool in this paper as it will allow us to use several complex variables techniques to study Hankel operators. We refer the reader to [7] for more information on the ∂-Neumann operator. The study of the size estimates of Hankel operators on Bergman spaces has inspired a lot of work in the last 20 years. The first result in the study of boundedness and compactness of Hankel operators was done by Axler [2] on the Bergman space of the open unit disk . He showed that, for β holomorphic on , Hβ is bounded if and only if β is in the Bloch space, and Hβ is compact if and only if β is in the little Bloch space. In the case of a general symbol, Zhu [27] showed the connection between size estimates of a Hankel operator and the mean oscillation of the symbol in the Bergman metric. In [4], Bekolle, Berger, Coburn and Zhu studied the same problem in the setting of bounded symmetric domains in Cn with the restriction that Hβ and Hβ are simultaneously bounded and compact with β ∈ L2 (Ω). Stroethoff and Zheng [24,26] independently gave a characterization for compactness of Hankel operators with bounded symbols on . Later Stroethoff [25] generalized these results to the case of the open unit ball and polydisc in Cn . Luecking [16] gave different criteria for boundedness and compactness of Hβ on Ap (Ω) with 1 < p < ∞. Peloso [21] extended Axler’s result to Bergman spaces on smooth bounded strongly pseudoconvex domains. For the same domains, Li [15] characterized bounded and compact Hankel operators Hβ with symbols β ∈ L2 (Ω). Beatrous and Li [3] obtained related results for the commutators of multiplication operators and P on more general domains, that include smooth bounded strongly pseudoconvex domains. The novelty of our approach is that we put an emphasis on the interplay between the geometry of the domain and the symbols of Hankel operators. Although, our symbols are more restricted the domains we consider are much more general and allow rich geometric structures. In several complex variables, compactness of the ∂-Neumann operator has been an active research area for the last couple of decades. We refer the reader to a very nice survey [12] for more information about compactness of the ∂-Neumann operator. Compactness of the canonical solution operators for ∂ on the unit disk has been discussed in [13], where it was in fact shown that this operator restricted to (0, 1)-forms with holomorphic coefficients is a Hilbert–Schmidt operator. Fu and Straube [11] showed that presence of analytic discs in the boundary of a bounded convex domain in Cn is equivalent to the non-compactness of the ∂-Neumann operator. The second author and Straube [23] used their techniques to prove that analytic discs are obstructions for compactness of the ∂-Neumann operator on smooth bounded pseudoconvex domains in Cn whose Levi form has maximal rank. In C2 their result reduces to a folklore result of Catlin [11]. Given Kohn’s formula it is natural to expect a strong relationship between Hankel operators and the ∂-Neumann operator. The following fact confirms this expectation. Compactness of the
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∂-Neumann operator implies compactness of Hankel operators with symbols that are smooth on the closure [12]. Actually, the statement in [12] requires the symbol to have bounded first order derivatives. But any symbol that is continuous on the closure can be approximated uniformly by symbols that are smooth on the closure of the domain. Hence the resulting Hankel operators converge in norm preserving compactness. In this paper we show that the theory for compactness of Hankel operators is somewhat parallel to the theory of compactness of the ∂-Neumann operator in terms of analytic structures in the boundary. Previous work in this direction was done by Knirsch and Schneider [14]. Throughout the paper bΩ denotes the boundary of Ω. Our first result concerns smooth bounded pseudoconvex domains in Cn . Theorem 1. Let Ω be a smooth bounded pseudoconvex domain in Cn for n 2 and β ∈ C ∞ (Ω). Assume that the Levi form of bΩ is of rank at least n − 2. If Hβ is compact on A2 (Ω), then β ◦ f is holomorphic for any holomorphic function f : → bΩ. Remark 1. We note that the statement “β ◦ f is holomorphic” can be interpreted as meaning that β is holomorphic “along” M = f (). However it may not be holomorphic in the transversal directions. Remark 2. One can check that the proof of Theorem 1 shows that compactness of Hβ on A2 (Ω) for β ∈ C ∞ (Ω) still implies that β ◦f is holomorphic for any holomorphic function f : → bΩ when Ω satisfies the following property: If the Levi form of bΩ is of rank k for 0 k n − 1 at p, then there exists an n − k − 1 dimensional complex manifold in bΩ through p. Since in C2 the Levi form has only one eigenvalue the condition on the Levi form in Theorem 1 is always satisfied. Therefore, for n = 2 we have the following corollary. Corollary 1. Let Ω be a smooth bounded pseudoconvex domain in C2 and β ∈ C ∞ (Ω). If Hβ is compact on A2 (Ω) then β ◦ f is holomorphic for any holomorphic function f : → bΩ. For convex domains in Cn we prove the same result without any restriction on the Levi form. Theorem 2. Let Ω be a smooth bounded convex domain in Cn for n 2 and β ∈ C ∞ (Ω). Assume that Hβ is compact on A2 (Ω). Then β ◦ f is holomorphic for any holomorphic function f : → bΩ. In the following theorem we show that, when Ω is convex in C2 , the converse of Theorem 1 is true. Theorem 3. Let Ω be a smooth bounded convex domain in C2 and β ∈ C ∞ (Ω). If β ◦ f is holomorphic for any holomorphic f : → bΩ, then Hβ is compact. Combining Corollary 1 (or Theorem 2) and Theorem 3 we get a necessary and sufficient condition for compactness of Hβ for convex domains in C2 . Corollary 2. Let Ω be a smooth bounded convex domain in C2 and β ∈ C ∞ (Ω). Then Hβ is compact if and only if β ◦ f is holomorphic for any holomorphic f : → bΩ.
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Remark 3. We note that [17] constructed a smooth bounded pseudoconvex complete Hartogs domain Ω in C2 that has no analytic disk in its boundary, yet it does not have a compact ∂Neumann operator. It would be interesting to know whether there exists a symbol β ∈ C ∞ (Ω) such that the Hankel operator HβΩ is not compact on A2 (Ω). Remark 4. We would like to take this opportunity to point out an inaccuracy. Knirsch and Schneider [14, Proposition 1] claim that if there is an affine disk in the boundary of a bounded convex domain in Cn , then the Hankel operator Hzmi is not compact for i = 1, 2, . . . , n and any positive integer m where zi is the ith coordinate function. They correctly prove the result when the disk lies in z1 -coordinate and claim that the proof for i = 2, 3, . . . , n is similar. However, Theorem 3 implies that if Ω is a smooth bounded convex domain in C2 and the set of weakly pseudoconvex points form a disc in z1 -coordinate, then Hz2 is compact. Remark 5. For simplicity we assume that the domains have C ∞ -smooth boundary and the symbols are smooth up to the boundary. However, one can check that the proofs work under weaker but reasonable smoothness assumptions. Remark 6. Recently, Çelik and Straube [6] studied compactness multipliers for the ∂-Neumann problem (we refer the reader to [6] for the definition and some properties of compactness multipliers of the ∂-Neumann problem). This notion is related to that of a symbol of a compact Hankel operator, but there are differences. First of all, the ∂-Neumann operator N is applied to square integrable forms and compactness multipliers are applied after N . In case of Hankel operators, however, one can think of the (0, 1)-form ∂β as acting as a “pre-multiplier” (acting before the canonical solution operator ∂ ∗ N ) on the Bergman space which is more rigid than the space of L2 forms. Secondly, Çelik and Straube proved that on a bounded convex domain, a function that is continuous on the closure is a compactness multiplier if and only if the function vanishes on all the (nontrivial) analytic discs in the boundary. One can show that such symbols produce compact Hankel operators. However, for smooth bounded convex domains in C2 , a symbol smooth on the closure produces a compact Hankel operator if and only if the symbol is holomorphic along (see Remark 1) analytic discs in boundary. (That is, the complex tangential component of the pre-multiplier on any analytic disc in the boundary vanishes.) In general, these connections are not well understood. For example, the following question is still open: Question 1. Assume that Ω is a smooth bounded pseudoconvex domain in Cn and β ∈ C(Ω) is a compactness multiplier for the ∂-Neumann operator on L2(0,1) (Ω). Is Hβ compact on the Bergman space on Ω? 2. Proof of Theorems 1 and 2 Let = 1 denote the unit open disc in C, r denote the disc in C centered at the origin with radius r, and kr denote the polydisc in Ck of multiradius (r, . . . , r). We will be using Hankel operators defined on different domains. So to be more precise, let HφΩ denote the Hankel operator on Ω with symbol φ and RU be the restriction operator onto U . Furthermore, the Bergman projection on U will be denoted by PU . First we will start with a proposition that will allow us to “localize” the proofs.
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In the proofs below we will use generalized constants. That is A B will mean that there exists a constant c > 0 that is independent of the quantities of interest such that A cB. At each step the constant c may change but it will stay independent of the quantities of interest. Proposition 1. Let Ω be a bounded pseudoconvex domain in Cn and φ ∈ L∞ (Ω). Then (i) If HφΩ is compact on A2 (Ω) then for every p ∈ bΩ and U an open neighborhood of p such that U ∩ Ω is a domain, HRUU∩Ω RU ∩Ω is compact on A2 (Ω). ∩Ω (φ) (ii) If for every p ∈ bΩ there exists an open neighborhood U of p such that U ∩ Ω is a domain, and HRUU∩Ω RU ∩Ω is compact on A2 (Ω), then HφΩ is compact on A2 (Ω). ∩Ω (φ) Proof. Let us prove (i) first. For f ∈ A2 (Ω), we have (IdU ∩Ω − PU ∩Ω )RU ∩Ω HφΩ (f ) = (IdU ∩Ω − PU ∩Ω )RU ∩Ω φf − PΩ (φf ) R (f ) + PU ∩Ω RU ∩Ω PΩ (φf ) − RU ∩Ω PΩ (φf ) = HRUU∩Ω ∩Ω (φ) U ∩Ω = HRUU∩Ω R (f ). ∩Ω (φ) U ∩Ω In the last equality we used the fact that PU ∩Ω RU ∩Ω PΩ = RU ∩Ω PΩ on L2 (Ω). Hence R (f ). (IdU ∩Ω − PU ∩Ω )RU ∩Ω HφΩ (f ) = HRUU∩Ω ∩Ω (φ) U ∩Ω RU ∩Ω is also compact. Therefore, if HφΩ is compact, then HRUU∩Ω ∩Ω (φ) To prove (ii) let us choose {p1 , . . . , pm } ⊂ bΩ and open sets U1 , . . . , Um such that (i) Uj is a neighborhood of pj and Uj ∩ Ω is a domain for j = 1, . . . , m, (ii) bΩ ⊂ m j =1 Uj , U ∩Ω
(iii) Sj = HRUj
j ∩Ω
(φ) RUj ∩Ω
is compact for j = 1, . . . , m.
Let U0 = Ω, S0 = HφΩ , and {χj : j = 0, . . . , m} be a C ∞ -smooth partition of unity subject to {Uj : j = 0, . . . , m}. Then for f ∈ A2 (Ω), ∂
m
χj Sj (f ) =
j =0
m m (∂χj )Sj (f ) + χj ∂Sj (f ) j =0
=
m m (∂χj )Sj (f ) + χj (∂φ)f j =0
=
j =0
j =0
m (∂χj )Sj (f ) + (∂φ)f. j =0
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m Hence, since ∂( m j =0 χj Sj (f )) and (∂φ)f are ∂-closed we conclude that j =0 (∂χj )Sj (f ) is ∂-closed. Let S=
m
m χj Sj − ∂ N (∂χj )Sj . ∗
Ω
j =0
(1)
j =0
We write χ0 S0 (f ) as χ0 φf − χ0 PΩ (φf ) and choose a bounded sequence {fj } in A2 (Ω). Let K be a compact set in Ω that contains a neighborhood of the support of χ0 . Cauchy integral formula and Montel’s theorem imply that {fj } and {PΩ (φfj )} have uniformly convergent subsequences on K. Then {χ0 φfj } and {χ0 PΩ (φfj )} have convergent subsequences in L2 (Ω). That is, the operator χ0 S0 is compact on A2 (Ω). Similarly, (∂χ0 )S0 is compact as well. We remind the reader that we assumed that Sj is compact for j = 1, . . . , m and ∂ ∗ N Ω is continuous on bounded pseudoconvex domains. Therefore, (1) implies that S is a compact operator and ∂S(f ) = (∂φ)f . To get the Hankel operator we project onto the complement of A2 (Ω). Hence using HφΩ = (IdΩ − PΩ )S we conclude that HφΩ is compact on A2 (Ω). 2 Lemma 1. Let Ω1 and Ω2 be two bounded pseudoconvex domains in Cn , φ ∈ C ∞ (Ω 2 ), and F : Ω1 → Ω2 be a biholomorphism that has a smooth extension up to the boundary. Assume that Ω1 is compact on A2 (Ω1 ). HφΩ2 is compact on A2 (Ω2 ). Then Hφ◦F Proof. Let g ∈ A2 (Ω1 ), f = g ◦ F −1 , u = ∂ ∗ N Ω2 ∂φf , and w = u ◦ F = F ∗ (u). Then f ∈ A2 (Ω2 ), u = HφΩ2 (f ), and ∂w = ∂F ∗ (u) = F ∗ (∂u) = F ∗ (f ∂φ) = (f ◦ F )∂(φ ◦ F ). So ∂(u ◦ F ) = (f ◦ F )∂(φ ◦ F ) on Ω1 and ∂ ∗ N Ω1 ∂(u ◦ F ) is the canonical solution for ∂w = (f ◦ F )∂(φ ◦ F ) on Ω1 . Then ∗ Ω1 Ω1 Hφ◦F (g) = Hφ◦F (f ◦ F ) = ∂ ∗ N Ω1 ∂(u ◦ F ) = ∂ ∗ N Ω1 ∂ F ∗ HφΩ2 F −1 (g) . Ω1 is a composition of HφΩ2 with continuous operators ∂ ∗ N Ω1 ∂, F ∗ , and (F −1 )∗ . Therefore, Hφ◦F Ω1 is compact on Then since HφΩ2 is assumed to be compact on A2 (Ω2 ) we conclude that Hφ◦F A2 (Ω1 ). 2
Let dbΩ (z) be the function defined on Ω that measures the (minimal) distance from z to bΩ. The Bergman kernel function of Ω satisfies the following relation on the diagonal of Ω × Ω: 2
KΩ (z, z) = sup f (z) : f ∈ A2 (Ω), f L2 (Ω) 1 . The following proposition appeared in [10] for general pseudoconvex domains in Cn and in [22] in the following form. Proposition 2. Let Ω be a bounded pseudoconvex domain in Cn with C 2 -boundary near p ∈ bΩ. If the Levi form is of rank k at p, then there exist a constant C > 0 and a neighborhood U of p such that KΩ (z, z)
C (dbΩ (z))k+2
for z ∈ U ∩ Ω.
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Proof of Theorem 1. We will prove a stronger result. The proof will go along the lines of the proof of Theorem 1 in [23] and the proof of (1) ⇒ (2) in [11] with some additional work. The same strategy has appeared in [5,9,22]. Let us assume that (i) Ω is a smooth bounded pseudoconvex domain in Cn and p ∈ bΩ, (ii) the Levi form of bΩ is of rank k at p through which there exists an n − k − 1 dimensional complex manifold in bΩ, (iii) there exists non-constant holomorphic mapping f : n−k−1 → bΩ and q ∈ such that f (q) = p, Df (q) is full rank (Df is the Jacobian of f ), and ∂(β ◦ f )(q) = 0, (iv) Hβ is compact. Lemma 1 in [23] implies that there exist a neighborhood V of p and a local holomorphic change of coordinates G on V so that G(p) = 0, positive yn -axis is the outward normal direction to the boundary of Ω1 = G(V ∩ Ω) at every point of M = {z ∈ n : zn−k = · · · = zn = 0} ⊂ bΩ1 . Let z = (z , z ) where z = (z1 , . . . , zn−k−1 ) and z = (zn−k , . . . , zn ). We define L to be the k + 1 (complex) dimensional slice of Ω1 that passes through the origin and is orthogonal to M. That is, L = {z ∈ Ck+1 : (0, z ) ∈ Ω1 }. So L is strongly pseudoconvex at the origin when k 1 and is a domain in C when k = 0. Then there exists 0 < λ < 1 such that M1 × L1 ⊂ Ω1 , where L1 is a ball in Ck+1 centered at (0, . . . , 0, −λ) with radius λ and M1 = 12 M. For every j we choose pj = (0, . . . , 0, −1/j ) ∈ M1 × L1 . We take the liberty to abuse the notation and consider pj = (0, . . . , 0, −1/j ) ∈ L1 . Now we define qj = G−1 (pj ) ∈ V ∩ Ω and KΩ (z, qj ) . fj (z) = KΩ (qj , qj ) One can check that {fj } is a bounded sequence of square integrable functions on Ω that converges to zero locally uniformly. Let us define αj = fj ◦ G−1 and β1 = β ◦ G−1 . Then (i) in Proposition 1 implies that HRVV∩Ω RV ∩Ω is compact. In turn, Lemma 1 implies that HβΩ1 1 is ∩Ω (β)
compact. Hence {HβΩ1 1 (αj )} has a convergent subsequence. The strategy for the rest of the proof
will be to prove that {HβΩ1 1 (αj )} has no convergent subsequence. Hence, getting a contradiction.
1 Since ∂(β ◦ f )(q) = 0 without loss of generality we may assume that | ∂β ∂z1 | > 0 at the origin. Then there exist 0 < r < 1 and a smooth function 0 χ 1 on real numbers such that
⊂ M1 , (i) n−k−1 r (ii) χ(t) = 1 for |t| r/2, χ(t) = 0 for |t| 3r/5, n 1 (iii) | ∂β ∂z1 | > 0 on r . Then C =
|z1 |<3r/4 χ(|z1 |) dV (z1 ) > 0.
γ (z)
Let us define γ on Ω1 so that
∂β1 (z) = χ |z1 | · · · χ |zn | ∂z1
and .,. denote the standard pointwise inner product on forms in C. Furthermore, let z = (z1 , w) where w = (z2 , . . . , zn ) and α ∈ A2 (Ω1 ). Then using the mean value property for a holomorphic function α and for fixed w ∈ n−1 3r/4 so that r × {w} ⊂ M1 × L1 we get
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χ |z1 | α(z1 , w) dV (z1 )
Cα(0, w) = |z1 |<3r/4
=
γ (z1 , w)
|z1 |<3r/4
∂β1 (z1 , w) α(z1 , w) dV (z1 ) ∂z1
=
α∂β1 , γ dz1 dV (z1 )
|z1 |<3r/4
∗ Ω ∂∂ N 1 (α∂β1 ), γ dz1 dV (z1 )
= |z1 |<3r/4
∂HβΩ1 1 (α)
=
∂z1
|z1 |<3r/4
γ dV (z1 )
HβΩ1 1 (α)
=− |z1 |<3r/4
∂γ dV (z1 ). ∂z1
Therefore, we have α(0, w)
1/2 Ω1 2 H (α) dV (z1 ) .
β1
|z1 |<3r/4
If we square both sides we get
α(0, w) 2
Ω1 H (α)(z1 , w) 2 dV (z1 ). β1
|z1 |<3r/4
Since |α(0, w)|2 is subharmonic when we integrate over (z2 , . . . , zn−k−1 ) ∈ n−k−2 3r/4 , we get
α(0, z ) 2
Ω1 H (α)(z , z ) 2 dV (z ). β1
(2)
z ∈n−k−1 3r/4
The above inequality applied to αj implies that αj |L1 ∈ L2 (L1 ). Now we use the reproducing property of KL1 on αj |L1 to get αj (pj ) =
KL1 (pj , z)αj |L1 (z) dV (z). L1
Cauchy–Schwartz inequality implies that |αj (pj )| αj |L1 L2 (L1 ) KL1 (pj , .) L2 (L1 ) . On the
other hand KL1 (pj , .) L2 (L1 ) = KL1 (pj , pj ). So we have
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αj |L1 L2 (L1 )
|αj (pj )| KL1 (pj , pj )
=
KΩ (qj , qj ) . KL1 (pj , pj )
Since L1 is a ball in Ck+1 and the rank of the Levi form for Ω (and hence for Ω1 ) is at least k, the asymptotics of the Bergman kernel on balls and Proposition 2 imply the following inequalities: 1 1 KL1 (pj , pj ) , (dbL1 (pj ))k+2 (dbL1 (pj ))k+2 1 KΩ (qj , qj ). (dbΩ (qj ))k+2 We note that pj and qj are related by a diffeomorphism. So for large enough j dbΩ1 (pj ) = ξ > 0 such that ξ < dbL1 (pj ) and they are comparable to dbΩ (qj ). Therefore, there exists αj |L1 L2 (L1 ) for all j . Since {αj } converges to 0 locally uniformly this implies that {αj |L1 } has no convergent subsequence in L2 (L1 ). Also (2) applied to αj − αk implies that αj |L1 − αk |L1 L2 (L1 ) HβΩ1 1 (αj − αk )L2 (Ω ) . 1
Hence {HβΩ1 1 (αj )} has no convergent subsequence in L2 (Ω1 ). Therefore, we have reached a contradiction completing the first proof of Theorem 1. 2 A weaker version of the following lemma appeared in [11]. Lemma 2. Let Ω be a convex domain in Cn and f : → bΩ be a non-constant holomorphic map. Then the convex hull of f () is an affine analytic variety contained in bΩ. Proof. Let K be the convex hull of f () in Cn . First we will show that K is an analytic affine variety. By definition K is an affine set in Cn . Let F (z, w, t) = tf (z) + (1 − t)f (w) for (z, w) ∈ 2 and 0 < t < 1. One can check that
K = F (z, w, t): (z, w, t) ∈ 2 × (0, 1) . If K is open in Cn then we are done. Otherwise, there exists p ∈ K which is a boundary point and, by convexity, there exists (z0 , w0 , t0 ) ∈ 2 × (0, 1) such that after possible rotation and translation p = F (z0 , w0 , t0 ) is the origin and K ⊂ {xn 0}. Let us define g = Re(zn ◦ F ) : 2 × (0, 1) → R. Then g(z0 , w0 , t0 ) = 0 and g(2 × (0, 1)) ⊂ {x ∈ R: x 0}. Maximum principle applied to the harmonic function g implies that g ≡ 0. Hence K ⊂ {xn = 0}. Since f is holomorphic, f must stay in the complex tangent subspace of {xn = 0}. That is, ∂ ∂ f (p) ⊂ span ,..., for every p ∈ . ∂z1 ∂zn−1
(3)
Now it is easy to see that (3) implies that K ⊂ {zn = 0}. So we have demonstrated that if K is not an n dimensional analytic affine variety then it is contained in an n − 1 dimensional analytic affine variety. We use the above argument multiple times if necessary to show that K is open in an analytic affine variety. Hence K is an analytic affine variety.
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Now we will show that K is contained in bΩ. Since K and Ω are convex after some possible rotation and translation, we can assume that f (0) is the origin and f () ⊂ Ω ⊂ {xn 0}. Since ∅ = f () ⊂ K ∩ bΩ the set K is not an open set in Cn . Then, as in the above paragraph, one can show that K ⊂ {xn = 0} ∩ Ω ⊂ bΩ. This completes the proof of the lemma. 2 Proof of Theorem 2. The proof will be very similar to the first part and the proof of (1) ⇒ (2) in [11]. So we will just sketch the proof and point out differences. Let us assume that HβΩ is compact and that there exists a non-constant holomorphic map f : → bΩ. We can choose p ∈ such that |∂(β ◦ f )(p)| > 0. By applying translation and rotation, if necessary, we may assume that f (p) = 0, f (p) = (1, 0, . . . , 0), and positive xn -axis is the outward normal for bΩ at 0. Using Lemma 2 with scaling, if necessary, we may assume that {(z, 0, . . . , 0) ∈ Cn : |z| 1} ⊂ bΩ and | ∂β(0) ∂z1 | > 0. We define
L = (z2 , . . . , zn ) ∈ Cn−1 : (0, z2 , . . . , zn ) ∈ Ω , K (z,q ) pj = (0, . . . , −1/j ) ∈ L, and fj (z) = √ L j . Using the proof of (1) ⇒ (2) in [11] one KL (qj ,qj )
can easily prove that {fj } is a bounded sequence in A2 (L) such that {RλL (fj )}, the restricted sequence of {fj } to λL, has no convergent subsequence in A2 (λL) for any 0 < λ < 1. Then for each j we extend fj to Ω using Ohsawa–Takegoshi theorem [19] to get a bounded sequence {αj } on A2 (Ω). Using similar arguments as in the proof of Theorem 1 and the fact that 1/2 × 12 L ⊂ Ω (this follows from convexity of Ω) one can show that fj − fk L2 ( 1 L) HβΩ (αj − αk )L2 (Ω) . 2
This contradicts the assumption that HβΩ is compact.
2
3. Proof of Theorem 3 We refer the reader to [8, Proposition V.2.3] for a proof of the following standard lemma. Lemma 3. Let T : X → Y be a linear operator between two Hilbert spaces X and Y . Then T is compact if and only if for every > 0 there exist a compact operator K : X → Y and C > 0 so that T (h) h X + C K (h) Y Y
for h ∈ X.
Proof of Theorem 3. Let K denote the closure of the union of all analytic discs in bΩ. Let us choose a defining function ρ for Ω so that ∇ρ = 1. Let β = β1 + iβ2 , ν=
2 ∂ρ ∂ ∂ρ ∂ + ∂xj ∂xj ∂yj ∂yj
and T =
j =1
2 ∂ρ ∂ ∂ρ ∂ − . ∂xj ∂yj ∂yj ∂xj j =1
For sufficiently small and ξ ∈ bΩ, let us define β1 ξ + ν(ξ ) = β1 (ξ ) + T (β2 )(ξ )
and β2 ξ + ν(ξ ) = β2 (ξ ) − T (β1 )(ξ ).
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= β1 + i β2 is a smooth function in a neighborhood of bΩ and it is equal to β on the Then β . One can check as a smooth function on Ω and still call it β boundary of Ω. Let us extend β ) = 0 on bΩ. That is, in some sense β is holomorphic along complex normal that (ν + iT )(β and β are smooth functions = β − β on Ω. Then β direction on the boundary. Let us define β = 0 on bΩ and β is holomorphic on K. Montel’s theorem together with the on Ω such that β can be approximated by smooth functions supported away from the boundary imply fact that β that HβΩ is compact on A2 (Ω). In the rest of the proof we will show that HβΩ is compact on ∞ (Ω) such that ψ = 0 in a neighborhood of K for all A2 (Ω). Let {ψj } be a sequence in C(0,1) j uniformly on Ω. On the boundary, ψj ’s are supported on sets that j and ψj converges to ∂ β satisfy property (P ) (see [11] when Ω is convex). In the following calculation .,.L2 (Ω) denotes the L2 inner product on Ω and N = N Ω . Now we will show that HβΩ is compact. Let g ∈ A2 (Ω). Then we have
∗ ), ∂ ∗ N (g∂ β ) 2 ), g∂ β 2 ∂ N(g∂ β = N (g∂ β L (Ω) L (Ω) ), gψj 2 . = N (g∂ β ), g(∂ β − ψj ) L2 (Ω) + N (g∂ β L (Ω) Let us fix ψj . We choose ψ ∈ C ∞ (Ω) such that 0 ψ 1, ψ ≡ 1 on the support of ψj and ψ is supported away from K. Then for g ∈ A2 (Ω) we have N (g∂ β ), gψj
L2 (Ω)
= ψN (g∂ β ), gψj 2 L (Ω) ) 2 g L2 (Ω) . ψN (g∂ β L (Ω)
(4)
Let us choose finitely many balls B1 , . . . , Bm and φj ∈ C0∞ (Bj ) for j = 0, 1, . . . , m (we take B0 = Ω here) such that m (i) j =0 φj = ψ on Ω, (ii) Ω ∩ Bj is a domain for j = 1, 2, . . . , m, (iii) m j =1 Bj covers the closure of the set {z ∈ bΩ: ψ(z) = 0}, (iv) Ω ∩ Bj has a compact ∂-Neumann operator for j = 1, 2, . . . , m. We note that multiplication with smooth functions preserves the domain of ∂ ∗ and the ∂Neumann operator is compact on Bj ∩ Ω for j = 1, . . . , m. Compactness of N implies the socalled compactness estimates (see for example [12]). Let W −1 (Ω) denote the Sobolev −1 norm for functions and forms. Then for every ε > 0 there exists Cε > 0 such that for h ∈ L2(0,1) (Ω) in the domains of ∂ and ∂ ∗ we have ψh L2 (Ω)
m
φj h L2 (Ω)
j =0
m ε ∂(φj h) j =0
L2 (Ω)
+ ∂ ∗ (φj h)L2 (Ω) + Cε φj h W −1 (Ω)
ε ∂h L2 (Ω) + ∂ ∗ hL2 (Ω) + h L2 (Ω) + Cε h W −1 (Ω) .
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3741
In the calculations above we used interior ellipticity for j = 0 and the fact that multiplication by a smooth function is a continuous operator on Sobolev spaces. Now if we replace h by N h and use the fact that N h L2 (Ω) + ∂N h L2 (Ω) + ∂ ∗ N h L2 (Ω) h L2 (Ω) we get ψN h L2 (Ω) ε h L2 (Ω) + Cε N h W −1 (Ω)
for h ∈ L2(0,1) (Ω).
Then Lemma 3 implies that ψN is compact on L2(0,1) (Ω). Then using the small constant-large constant inequality (2ab a 2 + b2 / ) combined with the inequality above and (4) we get that for any ε > 0 there exists Cε > 0 such that N(g∂ β ), gψj
L2 (Ω)
ε g 2 2
L (Ω)
)2 −1 + Cε N (g∂ β W (Ω)
for g ∈ A2 (Ω).
(5)
uniformly on Ω for every ε > 0 there exists ψj such that Since ψj converges to ∂ β | N(g∂ β ), g(∂ β − ψj )L2 (Ω) | ε g 2L2 (Ω) . Furthermore, the last inequality together with (5) imply that there exists Cε > 0 such that 2 ∗ ∂ N (g∂ β )2 2 = HβΩ (g)L2 (Ω) L (Ω)
)2 −1 g 2L2 (Ω) + C N (g∂ β W (Ω)
for g ∈ A2 (Ω).
The above inequality combined with Lemma 3 and the fact that W −1 (Ω) imbeds compactly into L2 (Ω) imply that HβΩ is compact on A2 (Ω). Therefore, HβΩ is compact. 2 Acknowledgments We would like to thank the referee and Emil Straube for helpful comments. References [1] V.M. Adamjan, D.Z. Arov, M.G. Kre˘ın, Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur–Takagi problem, Mat. Sb. (N.S.) 86 (128) (1971) 34–75. [2] Sheldon Axler, The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math. J. 53 (2) (1986) 315–332. [3] Frank Beatrous, Song-Ying Li, On the boundedness and compactness of operators of Hankel type, J. Funct. Anal. 111 (2) (1993) 350–379. [4] D. Békollé, C.A. Berger, L.A. Coburn, K.H. Zhu, BMO in the Bergman metric on bounded symmetric domains, J. Funct. Anal. 93 (2) (1990) 310–350. [5] David Catlin, Necessary conditions for subellipticity and hypoellipticity for the ∂-Neumann problem on pseudoconvex domains, in: Recent Developments in Several Complex Variables, Proc. Conf., Princeton Univ., Princeton, NJ, 1979, in: Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, NJ, 1981, pp. 93–100. [6] Mehmet Çelik, Emil J. Straube, Observations regarding compactness in the ∂-Neumann problem, Complex Var. Elliptic Equ., in press. [7] So-Chin Chen, Mei-Chi Shaw, Partial Differential Equations in Several Complex Variables, AMS/IP Stud. Adv. Math., vol. 19, American Mathematical Society, Providence, RI, 2001. [8] John P. D’Angelo, Inequalities from Complex Analysis, Carus Math. Monogr., vol. 28, Mathematical Association of America, Washington, DC, 2002. [9] Klas Diederich, Peter Pflug, Necessary conditions for hypoellipticity of the ∂-problem, in: Recent Developments in Several Complex Variables, Proc. Conf., Princeton Univ., Princeton, NJ, 1979, in: Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, NJ, 1981, pp. 151–154.
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[10] Siqi Fu, A sharp estimate on the Bergman kernel of a pseudoconvex domain, Proc. Amer. Math. Soc. 121 (3) (1994) 979–980. [11] Siqi Fu, Emil J. Straube, Compactness of the ∂-Neumann problem on convex domains, J. Funct. Anal. 159 (2) (1998) 629–641. [12] Siqi Fu, Emil J. Straube, Compactness in the ∂-Neumann problem, in: Complex Analysis and Geometry, Columbus, OH, 1999, in: Ohio State Univ. Math. Res. Inst. Publ., vol. 9, de Gruyter, Berlin, 2001, pp. 141–160. [13] Friedrich Haslinger, The canonical solution operator to ∂ restricted to Bergman spaces, Proc. Amer. Math. Soc. 129 (11) (2001) 3321–3329 (electronic). [14] Wolfgang Knirsch, Georg Schneider, Analytic discs in the boundary and compactness of Hankel operators with essentially bounded symbols, J. Math. Anal. Appl. 332 (1) (2007) 570–576. [15] Huiping Li, Hankel operators on the Bergman spaces of strongly pseudoconvex domains, Integral Equations Operator Theory 19 (4) (1994) 458–476. [16] Daniel H. Luecking, Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disk, J. Funct. Anal. 110 (2) (1992) 247–271. [17] P. Matheos, A Hartogs domain with no analytic discs in the boundary for which the ∂-Neumann problem is not compact, PhD thesis, University of California Los Angeles, CA, 1997. [18] Zeev Nehari, On bounded bilinear forms, Ann. of Math. (2) 65 (1957) 153–162. [19] Takeo Ohsawa, Kensh¯o Takegoshi, On the extension of L2 holomorphic functions, Math. Z. 195 (2) (1987) 197– 204. [20] Vladimir V. Peller, Hankel Operators and Their Applications, Springer Monogr. Math., Springer-Verlag, New York, 2003. [21] Marco M. Peloso, Hankel operators on weighted Bergman spaces on strongly pseudoconvex domains, Illinois J. Math. 38 (2) (1994) 223–249. [22] Sönmez Sahuto˘ ¸ glu, Compactness of the ∂-Neumann problem and Stein neighborhood bases, PhD thesis, Texas A&M University, TX, 2006, available at http://handle.tamu.edu/1969.1/3879. [23] Sönmez Sahuto˘ ¸ glu, Emil J. Straube, Analytic discs, plurisubharmonic hulls, and non-compactness of the ∂Neumann operator, Math. Ann. 334 (4) (2006) 809–820. [24] Karel Stroethoff, Compact Hankel operators on the Bergman space, Illinois J. Math. 34 (1) (1990) 159–174. [25] Karel Stroethoff, Compact Hankel operators on the Bergman spaces of the unit ball and polydisk in Cn , J. Operator Theory 23 (1) (1990) 153–170. [26] De Chao Zheng, Toeplitz operators and Hankel operators, Integral Equations Operator Theory 12 (2) (1989) 280– 299. [27] Ke He Zhu, VMO, ESV, and Toeplitz operators on the Bergman space, Trans. Amer. Math. Soc. 302 (2) (1987) 617–646.
Journal of Functional Analysis 256 (2009) 3743–3771 www.elsevier.com/locate/jfa
Concentration–compactness phenomena in the higher order Liouville’s equation ✩ Luca Martinazzi ETH Zurich, Rämistrasse 101, CH-8092, Switzerland Received 12 September 2008; accepted 21 February 2009 Available online 5 March 2009 Communicated by H. Brezis
Abstract We investigate different concentration–compactness and blow-up phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in R2m , then that of a closed manifold and, finally, the particular case of the sphere S 2m . In all cases we allow the sign of the Q-curvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in R2m , blow-up phenomena can occur only at points of positive Q-curvature. As a consequence, on a locally conformally flat manifold of non-positive Euler characteristic we always have compactness. © 2009 Elsevier Inc. All rights reserved. Keywords: Concentration–compactness; Q-curvature; Paneitz operators
1. Introduction and statement of the main results Before stating our results, we recall a few facts concerning the Paneitz operator Pg2m and the Q-curvature Q2m g on a 2m-dimensional smooth Riemannian manifold (M, g). Introduced in [5,25,4,17], the Paneitz operator and the Q-curvature are the higher order equivalents of the Laplace–Beltrami operator and the Gaussian curvature respectively (Pg2 = −g and Q2g = Kg ), and they now play a central role in modern conformal geometry. For their definitions and more related information we refer to [7]. Here we only recall a few properties which shall be used later. First of all we have the Gauss formula, describing how the Q-curvature changes under a conformal change of metric: ✩
This work was supported by ETH Research Grant No. ETH-02 08-2. E-mail address: [email protected].
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.017
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L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771 2m 2mu Pg2m u + Q2m , g = Qgu e
(1)
where gu := e2u g, and u ∈ C ∞ (M) is arbitrary. Then, we have the conformal invariance of the total Q-curvature, when M is closed: 2m Qgu dvolgu = Q2m (2) g dvolg . M
M
Finally, assuming (M, g) closed and locally conformally flat, we have the Gauss–Bonnet–Chern formula (see e.g. [10,7]): Q2m g dvolg =
Λ1 χ(M), 2
(3)
M
where χ(M) is the Euler–Poincaré characteristic of M and Λ1 := QgS 2m dvolgS 2m = (2m − 1)!S 2m
(4)
S 2m
is a constant which we shall meet often in the sequel. In the 4-dimensional case, if (M, g) is not locally conformally flat, we have Q4g +
|Wg |2 dvolg = 8π 2 χ(M), 4
(5)
M
where Wg is the Weyl tensor. Recently S. Alexakis [3] (see also [2]) proved an analogous to (5) for m 3: 2m Λ1 χ(M), (6) Qg + W dvolg = 2 M
where W is a local conformal invariant involving the Weyl tensor and its covariant derivatives. We can now state the main problem treated in this paper. Given a 2m-dimensional Riemannian manifold (M, g), consider a converging sequence of functions Qk → Q0 in C 0 (M), and let gk := e2uk g be conformal metrics satisfying Q2m gk = Qk . In view of (1), the uk ’s satisfy the following elliptic equation of order 2m with critical exponential non-linearity 2muk . Pg2m uk + Q2m g = Qk e
Assume further that there is a constant C > 0 such that vol(gk ) = e2muk dvolg C
for all k.
M
What can be said about the compactness properties of the sequence (uk )?
(7)
(8)
L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771
3745
In general non-compactness has to be expected, at least as a consequence of the noncompactness of the Möbius group on R2m or S 2m . For instance, for every λ > 0 and x0 ∈ R2m , the metric on R2m given by gu := e2u gR2m , u(x) := log 1+λ2 2λ , satisfies Q2m gu ≡ (2m − 1)!. |x−x |2 0
We start by considering the case when (M, g) is an open domain Ω ⊂ R2m with Euclidean metric gR2m . Since PgR2m = (−)m and QgR2m ≡ 0, Eq. (7) reduces to (−)m uk = Qk e2muk . The compactness properties of this equation were studied in dimension 2 by Brézis and Merle [6]. They proved that if Qk 0, Qk L∞ C and e2uk L1 C, then up to selecting a subsequence, one of the following is true:
(i) (uk ) is bounded in L∞ loc (Ω). (ii) uk → −∞ locally uniformly in Ω. (iii) There is a finite set S = {x (i) ; i = 1, . . . , I } ⊂ Ω such that uk → −∞ locally uniformly in Ω \ S. Moreover Qk e2uk Ii=1 βi δx (i) weakly in the sense of measures, where βi 2π for every 1 i I . Subsequently, Li and Shafrir [18] proved that in case (iii) βi ∈ 4πN for every 1 i I . Adimurthi, Robert and Struwe [1] studied the case of dimension 4 (m = 2). As they showed, the situation is more subtle because the blow-up set (the set of points x such that uk (x) → ∞ as k → ∞) can have dimension up to 3 (in contrast to the finite blow-up set S in dimension 2). Moreover, as a consequence of a result of Chang and Chen [8], quantization in the sense of Li–Shafrir does not hold anymore, see also [27,28]. In the following theorem we extend the result of [1] to arbitrary even dimension (see also Proposition 6 below). The function ak in (9) has no geometric meaning, and one can take ak ≡ 1 at first. On the other hand, one can also apply Theorem 1 to non-geometric situations, by allowing ak ≡ 1, see [23]. Theorem 1. Let Ω be a domain in R2m , m > 1, and let (uk )k∈N be a sequence of functions satisfying (−)m uk = Qk e2mak uk ,
(9)
where ak , Q0 ∈ C 0 (Ω), Q0 is bounded, and Qk → Q0 , ak → 1 locally uniformly. Assume that e2mak uk dx C,
(10)
Ω
for all k and define the finite (possibly empty) set S1 := x ∈ Ω: lim lim sup r→0+ k→∞
|Qk |e2mak uk dy
Br (x)
where Λ1 is as in (4). Then one of the following is true.
Λ1 = x (i) : 1 i I , 2
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L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771
2m−1,α (i) For every 0 α < 1, a subsequence converges in Cloc (Ω \ S1 ). (ii) There exist a subsequence, still denoted by (uk ), and a closed nowhere dense set S0 of Hausdorff dimension at most 2m − 1 such that, letting S = S0 ∪ S1 , we have uk → −∞ locally uniformly in Ω \ S as k → ∞. Moreover there is a sequence of numbers βk → ∞ such that
uk →ϕ βk
2m−1,α in Cloc (Ω \ S), 0 α < 1,
where ϕ ∈ C ∞ (Ω \ S1 ), S0 = {x ∈ Ω: ϕ(x) = 0}, and (−)m ϕ ≡ 0,
ϕ 0,
ϕ ≡ 0 in Ω \ S1 .
If S1 = ∅ and Q0 (x (i) ) > 0 for some 1 i I , then case (ii) occurs. We recently proved (see [21]) the existence of solutions to the equation (−)m u = Qe2mu on R2m with Q < 0 constant and e2mu ∈ L1 (R2m ), for m > 1. Scaling any such solution we find a sequence of solutions uk (x) := u(kx) + log k concentrating at a point of negative Q-curvature. For m = 1 that is not possible. On a closed manifold things are different in several respects. Under the assumption (which we always make) that ker Pg2m contains only constant functions, quantization of the total Qcurvature in the sense of Li–Shafrir (see (12) below) holds, as proved in dimension 4 by Druet and Robert [15] and Malchiodi [19], and in arbitrary dimension by Ndiaye [24]. Moreover the concentration set is finite. In [15], however, it is assumed that the Q-curvatures are positive, while in [19,24], a slightly different equation is studied (Pg2m uk + Qk = hk e2muk , with hk constant and Qk prescribed), for which the negative case is simpler. With the help of results from our recent work [21] and a technique of Robert and Struwe [29], we can allow the prescribed Q-curvatures to have varying signs and, contrary to the case of an open domain in R2m , we can rule out concentration at points of negative Q-curvature. Theorem 2. Let (M, g) be a 2m-dimensional closed Riemannian manifold, such that ker Pg = {constants}, and let (uk ) be a sequence of solutions to (7), (8) where the Qk ’s and Q0 are given C 1 functions and Qk → Q0 in C 1 (M). Let Λ1 be as in (4). Then one of the following is true. (i) For every 0 α < 1, a subsequence converges in C 2m−1,α (M). (ii) There exists a finite (possibly empty) set S1 = {x (i) : 1 i I } such that Q0 (x (i) ) > 0 for 1 i I and, up to taking a subsequence, uk → −∞ locally uniformly on (M \ S1 ). Moreover Qk e2muk dvolg
I
Λ1 δx (i)
(11)
i=1
in the sense of measures; then (2) gives Qg dvolg = I Λ1 . M
Finally, S1 = ∅ if and only if vol(gk ) → 0.
(12)
L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771
3747
An immediate consequence of Theorem 2 (identity (12) in particular) and the Gauss–Bonnet– Chern formulas (3) and (5), is the following compactness result: Corollary 3. Under the hypothesis of Theorem 2 assume that either 1. χ(M) 0 and dim M ∈ {2, 4}, or 2. χ(M) 0, dim M 6 and (M, g) is locally conformally flat, and that vol(gk ) 0. Then (i) in Theorem 2 occurs. It is not clear whether the hypothesis that (M, g) be locally conformally flat when dim M 6 is necessary in Corollary 3. For instance, we could drop it if we knew that W 0 in (6), in analogy with (5). Contrary to what happens for the Yamabe equation (see [11–14]), the concentration points of S in Theorem 2 are isolated, as already proved in [15] in dimension 4. In fact, a priori one could expect to have
Qk e2muk dvolg
I
Li Λ1 δx (i)
for some Li ∈ N \ {0},
(13)
i=1
instead of (11). The compactness of M is again a crucial ingredient here; indeed X. Chen [9] showed that on R2 (where quantization holds, as already discussed) one can have (13) with Li > 1. Theorems 1 and 2 will be proven in Sections 2 and 3 respectively. In Section 4 we also consider the special case when M = S 2m . In the proofs of the above theorems we use techniques and ideas from several of the cited papers, particularly from [1,6,15,19,20,29]. In the following, the letter C denotes a generic positive constant, which may change from line to line and even within the same line. 2. The case of an open domain in R 2m An important tool in the proof of Theorem 1 is the following estimate, proved by Brézis and Merle [6] in dimension 2. For the proof in arbitrary dimension see [22]. Notice the role played by the constant γm := Λ21 , which satisfies 1 (−)m − log |x| = δ0 γm
in R2m .
Theorem 4. Let f ∈ L1 (BR (x0 )), BR (x0 ) ⊂ R2m , and let v solve
(−)m v = f in BR (x0 ), m−1 v = v = · · · = v = 0 on ∂BR (x0 ).
(14)
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L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771 γm L1 (BR (x0 ))
Then, for any p ∈ (0, f
), we have e2mp|v| ∈ L1 (BR (x0 )) and e2mp|v| dx C(p)R 2m .
BR (x0 ) p
Lemma 5. Let f ∈ L1 (Ω) ∩ Lloc (Ω \ S1 ) for some p > 1, where Ω ⊂ R2m and S1 ⊂ Ω is a finite set. Assume that
(−)m u = f j u = 0
in Ω, on ∂Ω for 0 j m − 1.
2m,p
Then u is bounded in Wloc (Ω \ S1 ); more precisely, for any B4R (x0 ) ⊂ (Ω \ S1 ), there is a constant C independent of f such that uW 2m,p (BR (x0 )) C f Lp (B4R (x0 )) + f L1 (Ω) .
(15)
The proof of Lemma 5 is given in Appendix A. Proof of Theorem 1. We closely follow [1]. Let S1 be defined as in the statement of the theorem. Clearly (10) implies that S1 = {x (i) ∈ Ω: 1 i I } is finite. Given x0 ∈ Ω \ S1 , we have, for some 0 < R < dist(x0 , ∂Ω), |Qk |e2mak uk dx < γm .
α := lim sup k→∞
BR (x0 )
For such x0 and R write uk = vk + hk in BR (x0 ), where
(−)m vk = Qk e2mak uk vk = vk = · · · =
m−1
in BR (x0 ), vk = 0 on ∂BR (x0 ),
− + + + and (−)m hk = 0. Set h+ k := χ{hk 0} hk , hk := hk − hk . Since hk uk + |vk |, we have
+ h k
L1 (BR (x0 ))
u+ k L1 (B
R (x0 ))
+ vk L1 (BR (x0 )) .
+ 2mak uk on B (x ), hence (10) implies Observe that, for k large enough mu+ R 0 k 2mak uk e
u+ k dx
C
BR (x0 )
As for vk , observe that 1 <
γm α ,
e2mak uk dx C.
BR (x0 )
hence by Theorem 4
2m|vk | dx
BR (x0 )
BR (x0 )
e2m|vk | dx CR 2m ,
(16)
L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771
3749
with C depending on α and not on k. Hence + h k
L1 (BR (x0 ))
C.
(17)
We distinguish 2 cases. Case 1. Suppose that hk L1 (BR/2 (x0 )) C uniformly in k. Then by Proposition 11 we have that hk is equibounded in C (BR/8 (x0 )) for every 0. Moreover, by Pizzetti’s formula (identity (79) in Appendix A) and (17), − hk (x) dx = 2 − h+ (x) dx − − hk (x) dx k BR (x0 )
BR (x0 )
C − − hk (x) dx
BR (x0 )
BR (x0 )
=C−
m−1
ci R 2i i hk (x0 ) C.
i=0
Hence we can apply Proposition 11 locally on all of BR (x0 ) and obtain bounds for (hk ) in (B (x )) for any 0. Cloc R 0 Fix p ∈ (1, γm /α). By Theorem 4 e2m|vk | Lp (BR (x0 )) C(p), hence, using that ak → 1 uniformly on BR (x0 ), we infer (−)m vk
Lp (B)
= Qk e2mak hk e2mak vk Lp (B) C(B, p)
(18)
for every ball B BR (x0 ) and for k large enough. In addition vk L1 (BR (x0 )) C, hence by elliptic estimates, vk W 2m,p (B) C(B, p)
for every ball B BR (x0 ).
0,α By the immersion W 2m,p → C 0,α , (vk ), is bounded in Cloc (BR (x0 )), for some α > 0. Going m back to (18), we now see that vk is uniformly bounded in L∞ loc (BR (x0 )), hence
vk W 2m,p (B) C(B, p) for every p > 1, B BR (x0 ), and by the immersion W 2m,p → C 2m−1,α we obtain that (vk ), 2m−1,α (BR (x0 )). hence (uk ), is bounded in Cloc Case 2. Assume that hk L1 (BR/2 (x0 )) =: βk → ∞ as k → ∞. Set ϕk := 1. m ϕk = 0, 2. ϕk L1 (BR/2 (x0 )) = 1, 3. ϕk+ L1 (BR (x0 )) → 0 by (17).
hk βk ,
so that
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(B (x )) for every 0, hence a subsequence As above we have that ϕk is bounded in Cloc R 0 2m (B (x )) to a function ϕ, with converges in Cloc R 0
1. m ϕ = 0, 2. ϕL1 (BR/2 (x0 )) = 1, 3. ϕ + L1 (BR (x0 )) = 0, hence ϕ 0. Let us define S0 = {x ∈ BR (x0 ): ϕ(x) = 0}. Take x ∈ S0 ; then by (79), ϕ(x), . . . , m−1 ϕ(x) cannot all vanish, unless ϕ ≡ 0 on Bρ (x) ⊂ BR (x0 ) for some ρ > 0, but then by analyticity, we would have ϕ ≡ 0, a contradiction. Hence there exists j with 1 j 2m − 3 such that D j ϕ(x) = 0,
D j +1 ϕ(x) = 0,
i.e. S0 ⊂
2m−3
x ∈ BR (x0 ): D j ϕ(x) = 0, D j +1 ϕ(x) = 0 .
j =1
Therefore S0 is (2m − 1)-rectifiable. Since ϕ < 0 on BR (x0 ) \ S0 , we infer hk = βk ϕk → −∞,
e2mak hk → 0
locally uniformly on BR (x0 ) \ S0 . Then, as before, from (−)m vk = Qk e2mak hk e2mak vk , 2m−1,α we have that vk is bounded in Cloc (Ω \ S0 ). Then uk = hk + vk → −∞ uniformly locally away from S0 .
Since Cases 1 and 2 are mutually exclusive, covering Ω \ S1 with balls, we obtain that either a 2m−1,α (Ω \ S1 ), or a subsequence uk → −∞ locally uniformly subsequence uk is bounded in Cloc on Ω \ (S0 ∪ S1 ). In this latter case, the behavior described in case (ii) of the theorem occurs. Indeed fix any BR (x0 ) ⊂ Ω \ S1 and take βk as above. Then, on a ball Bρ (y0 ) ⊂ Ω \ S1 , we can write uk = v˜k + h˜ k as above, where h˜ k → −∞ locally uniformly away from a rectifiable ˜ set S0 of dimension at most (2m − 1), h˜k → ϕ, ˜ where β˜k = h˜ k L1 (Bρ/2 (y)) , and v˜k is bounded 2m−1,α in Cloc (Bρ (y0 )). Then
(a) (b)
h˜ k βk h˜ k βk
and and
uk βk uk βk
v˜k βk
βk
2m−1,α → 0 in Cloc (Bρ (y0 )), and we have that either
2m−1,α are bounded in Cloc (Bρ (y0 )), or
go to −∞ locally uniformly away from S0 .
Since the 2 cases are mutually exclusive, and on BR (x0 ) case (a) occurs, upon covering Ω \ S1 with a sequence of balls, we obtain the desired behavior for βukk . We now show that if I 1 and Q0 (x (i) ) > 0 for some 1 i I , then Case 2 occurs. Assume by contradiction that Q0 (x0 ) > 0 for some x0 ∈ S1 and Case 1 occurs, i.e. (uk ) is bounded in 2m−1,α (Ω \ S1 ), so that fk := Qk e2mak uk is bounded in L∞ Cloc loc (Ω \ S1 ). Then there exists a finite (Ω \ S ) such that signed measure μ on Ω, with μ ∈ L∞ 1 loc
L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771
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fk μ as measures, p
fk μ in Lloc (Ω \ S1 ) for 1 p < ∞. Let us take R > 0 such that BR (x0 ) ⊂ Ω, BR (x0 ) ∩ S1 = {x0 } and Q0 > 0 on BR (x0 ). By our assumption, (−)j uk −C
on ∂BR (x0 ) for 0 j m − 1.
(19)
Let zk be the solution to
(−)m zk = Qk e2mak uk in BR (x0 ), m−1 zk = zk = · · · = zk = 0 on ∂BR (x0 ).
By Proposition 13 and (19) uk zk − C.
(20)
2m−1,α (BR (x0 ) \ {x0 }), where By Lemma 5, up to a subsequence, zk → z in Cloc
(−)m z = μ
in BR (x0 ),
z = z = · · · = m−1 z = 0 on ∂BR (x0 ). 1 Since Q0 (x0 ) > 0, we have μ γm δx0 = (−)m ln |x−x , and Proposition 13 applied to the 0| 1 implies function z(x) − ln |x−x 0|
z(x) ln
1 − C, |x − x0 |
hence e2mz dx
1 C
BR (x0 )
BR (x0 )
1 dx = +∞. |x − x0 |2m
Then (20) and Fatou’s lemma imply
lim inf k→∞
BR (x0 )
e
2mak uk
dx
lim inf e2mak uk dx k→∞
BR (x0 )
1 C
lim inf e2mak zk dx k→∞
BR (x0 )
1 C
e2mz dx = +∞, BR (x0 )
contradicting (10).
2
(21)
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The following proposition gives a general procedure to rescale at points where uk goes to infinity. Proposition 6. In the hypothesis of Theorem 1, assume that ak ≡ 1 for every k and that case (ii) occurs. Then, for every x0 ∈ S such that supBR (x0 ) uk → ∞ for every 0 < R < dist(x0 , ∂Ω) as k → ∞, there exist points xk → x0 and positive numbers rk → 0 such that vk (x) := uk (xk + rk x) + ln rk 0 ln 2 + vk (0),
(22)
2m−1,α (R2m ), where and as k → ∞ either a subsequence vk → v in Cloc
(−)m v = Q0 (x0 )e2mv , or vk → −∞ almost everywhere and there are positive numbers γk → +∞ such that 2m−1,α 2m R , in Cloc
vk →p γk
where p is a polynomial on even degree at most 2m − 2. Proof. Following [1], take x0 such that supBR (x0 ) uk → +∞ for every R and select, for R < dist(x0 , ∂Ω), 0 rk < R and xk ∈ Brk (x0 ) such that (R − rk )euk (xk ) = (R − rk ) sup euk = max (R − r) sup euk =: Lk . 0r
Brk (x0 )
Then Lk → +∞ and sk :=
R−rk 2Lk
Br (x0 )
→ 0 as k → ∞, and
vk (x) := uk (xk + sk x) + ln sk 0 in BLk (0) satisfies ˜ k e2mvk , (−)m vk = Q
˜ k (x) := Qk (xk + sk x), Q
and BLk (0)
˜ k e2mvk dx = Q
Qk e2muk dx C.
B 1 (R−r ) (xk ) 2
k
We can now apply the first part of the theorem to the functions vk , observing that there are no concentration points (S1 = ∅), since vk 0, and using Theorem 12 to characterize the function p. 2
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3753
3. The case of a closed manifold To prove Theorem 2 we assume that supM uk → ∞ and we blow up at I suitably chosen sequences of points xi,k → x (i) with uk (xi,k ) → ∞ as k → ∞, 1 i I . We call the x (i) ’s blow-up (or concentration) points. Then we show the following: (i) If x (i) is a blow-up point, then Q0 (x (i) ) > 0. (ii) The profile of the uk ’s at any blow-up point is the function η0 defined in (27) below, hence it carries the fixed amount of energy Λ1 , see (29). (iii) uk → −∞ locally uniformly in M \ {x (i) : 1 i I }. (iv) The neck energy vanishes in the sense of (47) below, hence in the limit only the energy of the profiles at the blow-up points appears. Parts (i) and (ii) (Proposition 8) follow from Lemma 7 below and the classification results of [22] (or [32]) and [21]. For parts (iii) and (iv) we adapt a technique of [15], see also [19,24] for a different approach. The following lemma (compare [19, Lemma 2.3]) is important, because its failure in the noncompact case is responsible for the rich concentration–compactness behavior in Theorem 1. Its proof relies on the existence and on basic properties of the Green function for the Paneitz operator Pg2m , as proven in [24, Lemma 2.1] (here we need the hypothesis ker Pg2m = {constants}). Lemma 7. Let (uk ) be a sequence of functions on (M, g) satisfying (7) and (8). Then for = 1, . . . , 2m − 1, we have
p ∇ uk dvolg C(p)r 2m−p ,
1p<
2m ,
Br (x)
for every x ∈ M, 0 < r < rinj and for every k, where rinj is the injectivity radius of (M, g). 1 Proof. Set fk := Qk e2muk − Q2m g , which is bounded in L (M) thanks to (8). Let Gξ be the Green’s function for Pg2m on (M, g) such that
uk (ξ ) = − uk dvolg + Gξ (y)fk (y) dvolg (y). M
(23)
M
For x, ξ ∈ M, x = ξ , [24, Lemma 2.1] implies ∇ Gξ (x) ξ
C , dist(x, ξ )
1 2m − 1.
Then, differentiating (23) and using (24) and Jensen’s inequality, we get ∇ uk (ξ )p C
M
p 1 fk (y) dvolg (y) dist(ξ, y)
(24)
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C
fk L1 (M)
p
dist(ξ, y) M
|fk (y)| dvolg (y). fk L1 (M)
From Fubini’s theorem we then conclude ∇ uk (ξ )p dvolg (ξ ) Cfk p 1 sup L (M)
y∈M Br (x)
Br (x)
1 dvolg (ξ ) dist(ξ, y)p
2
Cr 2m−p .
Let expx : Tx M ∼ = R2m → M denote the exponential map at x. Proposition 8. Let (uk ) be a sequence of solutions to (7), (8) with max uk → ∞ as k → ∞. Choose points xk → x0 ∈ M (up to a subsequence) such that uk (xk ) = maxM uk . Then Q0 (x0 ) > 0 and, setting μk := 2
(2m − 1)! Q0 (x0 )
1 2m
e−uk (xk ) ,
(25)
we find that the functions ηk : B rinj ⊂ R2m → R, given by μk
(2m − 1)! 1 log , ηk (y) := uk expxk (μk y) + log μk − 2m Q0 (x0 ) 2m−1,α 2 converge up to a subsequence to η0 (y) = ln 1+|y| (R2m ). Moreover 2 in Cloc
lim
lim
R→+∞ k→∞ BRμk (xk )
Qk e2muk dvolg = Λ1 .
(26)
Remark. The function η0 (x) := log
2 1 + |x|2
(27)
satisfies (−)m η0 = (2m − 1)!e2mη0 , which is (9) with Qk ≡ (2m − 1)! and ak ≡ 1. In fact η0 has a remarkable geometric interpretation: If π : S 2m → R2m is the stereographic projection, then ∗ e2η0 gR2m = π −1 gS 2m ,
(28)
where gS 2m is the round metric on S 2m . Then (28) implies (2m − 1)! R2m
e2mη0 dx = S 2m
QS 2m dvolgS 2m = (2m − 1)!S 2m = Λ1 .
(29)
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Proof of Proposition 8. Step 1. Set σk = e−uk (xk ) , and consider on B rinj ⊂ R2m the functions σk
zk (y) := uk expxk (σk y) + log(σk ) 0,
(30)
and the metrics g˜ k := (expxk ◦Tk )∗ g, ˆ k (y) := Qk (expx (σk y)), and pulling where Tk : R2m → R2m , Tk y = σk y. Then, setting Q k back (7) via expxk ◦Tk , we get −2m ˆ 2mzk Pg˜2m zk + Q2m . Qk e g˜ k = σk k
(31)
, Q2m = σk2m Q2m Setting now gˆ k := σk−2 g˜ k , we have Pgˆ2m = σk2m Pg˜2m g˜ k , and from (31) we infer gˆ k k
k
ˆ 2mzk . Pgˆ2m zk + Q2m gˆ k = Qk e k
(32)
Then, since the principal part of the Paneitz operator is (−g )m , we can write Pgˆk = (−gˆk )m + Ak , where Ak is a linear differential operator of order at most 2m − 1; moreover the coefficients k (R2m ) for all k 0, since gˆ → g k 2m of Ak are going to 0 in Cloc k R2m in Cloc (R ) for all k 0, and m PgR2m = (−) . Then (32) can be written as ˆ 2mzk . (−gˆk )m zk + Ak zk + Q2m gˆ k = Qk e
(33)
2m−1,α Step 2. We now claim that zk → z0 in Cloc (R2m ), where
(−)m z0 = Q0 (x0 )e2mz0 ,
e2mz0 dx < ∞.
(34)
R2m
h =0 We first assume m > 1. Fix R > 0 and write zk = hk + wk on BR = BR (0), where m gˆ k k and (−gˆk )m wk = (−gˆk )m zk in BR , (35) m−1 wk = 0 on ∂BR . wk = wk = · · · = ˆ k e2mzk L∞ (BR ) C, and clearly Q2m = σ 2m Q2m → 0 in L∞ (R2m ). From zk 0 we infer Q k loc g˜ k gˆ k
2m Lemma 7 implies that (Ak zk ) is bounded in Lp (BR ), 1 p < 2m−1 , hence from (35) and elliptic 2m 2m,p (BR ), 1 p < 2m−1 , hence in C 0 (BR ). estimates we get uniform bounds for (wk ) in W Again using Lemma 7, we get gˆk hk L1 (BR ) C zk W 2,1 (BR ) + wk W 2,1 (BR ) C.
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Since m−1 (gˆk hk ) = 0, elliptic estimates (compare Proposition 11) give gˆ k
gˆk hk C (BR/2 ) C()
for every ∈ N.
(36)
This, together with |hk (0)| = |wk (0)| C, and hk −wk C and elliptic estimates (e.g. [16, Theorem 8.18]), implies that hk L1 (BR/2 ) C, hence, again using elliptic estimates, hk C (BR/4 ) C()
for every ∈ N.
(37)
2m Therefore (zk ) is bounded in W 2m,p (BR/4 ), 1 p < 2m−1 . We now go back to (35), replacing R with R/4 and redefining hk and wk accordingly on BR/4 . We now have that (Ak zk ) is bounded 2m in Lp (BR/4 ) for 1 p < 2m−2 by Sobolev’s embedding, and we infer as above that (wk ) is 2m 2m,p (BR/4 ), 1 p < 2m−2 , and hk is bounded in C (BR/16 ), 0. Iterating, we bounded in W 2m,p find that (zk ) is bounded in W (BR/42m ) for every p ∈ [1, ∞[. By letting R → ∞ and extract2m−1,α ing a diagonal subsequence, we infer that (zk ) converges in Cloc (R2m ). Then (34) follows from Fatou’s lemma, letting R → ∞, and the claim is proven. When m = 1, since Pg2 = −g , (32) implies at once that (gˆk zk ) is locally bounded in L∞ . Then, since zk 0 and zk (0) = 0, the claim follows from elliptic estimates (e.g. [16, Theorem 8.18]).
Step 3. We shall now rule out the possibility that Q0 (x0 ) 0. Case Q0 (x0 ) = 0. By the maximum principle one sees that, for m = 1, (34) has no solution (see e.g. [21, Theorem 3]), a contradiction. If m 2, still by [21, Theorem 3], any solution z0 to (34) is a non-constant polynomial of degree at most 2m − 2, and there are 1 j m − 1 and a < 0 such that j z0 ≡ a. Following an argument of [29], see also [19], we shall find a contradiction. Indeed we have lim
k→∞ BR
j zk dx =
j z0 dx = |a|ω2m R 2m + o R 2m 2m
as R → +∞.
BR
Scaling back to uk , we find lim
k→∞
2j −2m σk
2j ∇ uk dvolg C −1 R 2m + o R 2m
as R → +∞,
BRσk (xk )
while, from Lemma 7,
2j ∇ uk dvolg C(Rσk )2m−2j .
BRσk (xk )
This yields the desired contradiction as k, R → +∞.
(38)
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3757
Case Q0 (x0 ) < 0. By [21, Theorem 1] there exists no solution to (34) for m = 1, a contradiction. If m 2, from [21, Theorem 2] we infer that there are a constant a = 0 and 1 j m − 1 such that j z0 (x) = a,
lim
|x|→+∞ x∈C
where C := {tξ ∈ R2m : t 0, ξ ∈ K} and K ⊂ S 2m−1 is a compact set with H2m−1 (K) > 0. Then, as above,
2j 2j −2m ∇ uk dvolg lim σk k→∞ BRσk (xk )
C
−1
j z0 dx
BR ∩C
C −1 R 2m + o R 2m ,
again contradicting (38). Then we have shown that Q0 (x0 ) > 0. Step 4. Since Qk (x0 ) > 0, μk and ηk are well defined. Repeating the procedure of Step 2, we 2m−1,α 2m−1,α (R2m ) such that ηk → η in Cloc (R2m ), where (compare (34)) find a function η ∈ Cloc (−) η = (2m − 1)!e m
2mη
e2mη dx < +∞.
, R2m
By [22, Theorem 2], either η is a standard solution, i.e. there are x0 ∈ R2m , λ > 0 such that η(y) = log
2λ , 1 + λ2 |y − y0 |2
(39)
or j η(x) → a as |x| → ∞ for some constant a < 0 and for some 1 j m − 1. In the latter case, as in Step 3, we reach a contradiction. Hence (39) is satisfied. Since maxM ηk = ηk (0) = log 2 for every k, we have y0 = 0, λ = 1, i.e. η = η0 . Since, by Fatou’s lemma
lim lim
R→∞ k→∞ Rμk (xk )
(26) follows from (29).
Qk e
2muk
dvolg = (2m − 1)!
e2mη0 dx,
R2m
2
Proof of Theorem 2. Assume first that uk C. Then Pg2m uk is bounded in L∞ (M) and Lemma 7 and by elliptic estimates uk − uk is bounded in W 2m,p (M) for every 1 p < ∞, hence in C 2m−1,α (M) for every α ∈ [0, 1[, where uk := − M uk dvolg . Observe that by Jensen’s inequality and (8), uk C. If uk remains bounded (up to a subsequence), then by Ascoli–Arzelà’s theorem, for every α ∈ [0, 1[, uk is convergent (up to a subsequence) in C 2m−1,α (M), and we are in case (i) of Theorem 2.
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If uk → −∞, we have that uk → −∞ uniformly on M and we are in case (ii) of the theorem, with S1 = ∅. From now on we shall assume that maxM uk → ∞ as k → ∞, and closely follow the argument of [15]. Step 1. There are I > 0 converging sequences xi,k → x (i) ∈ M with uk (xi,k ) → ∞ as k → ∞, such that: (A1 ) Q0 (x (i) ) > 0, 1 i I . (A2 )
dist(xi,k ,xj,k ) μi,k
→ +∞ as k → +∞ for all 1 i, j I , i = j , where
(2m − 1)! μi,k := 2 Q0 (x (i) )
1 2m
e−uk (xi,k ) .
(A3 ) Set ηi,k (y) := uk (expxi,k (μi,k y)) − uk (xi,k ). Then for 1 i I ηi,k (y) → η0 (y) = log
2 1 + |y|2
2m 2m R (k → ∞). in Cloc
(40)
(A4 ) For 1 i I lim
lim
R→+∞ k→+∞ BRμi,k (xi,k )
(A5 ) There exists C > 0 such that for all k sup euk (x) Rk (x) C,
Qk e2muk dx → Λ1 .
(41)
Rk (x) := min dist(x, xi,k ). 1iI
x∈M
Step 1 follows from Proposition 8 and induction as follows. Define x1,k = xk as in Proposition 8. Then (A1 ), (A3 ) and (A4 ) are satisfied with i = 1. If supx∈M [euk (x) dist(x1,k , x)] C, then I = 1 and also (A5 ) is satisfied, so we are done. Otherwise we choose x2,k such that R1,k (x2,k )euk (x2,k ) = max R1,k (x)euk (x) → ∞, x∈M
R1,k (x) := dist(x, x1,k ).
(42)
Then (A2 ) with i = 2, j = 1 follows at once from (42), while (A2 ) with i = 1, j = 2 follows from (A3 ), as in [15]. A slight modification of Proposition 8 shows that (x2,k , μ2,k ) satisfies (A1 ), (A3 ) and (A4 ), and we continue so, until also property (A5 ) is satisfied. The procedure stops after finitely many steps, thanks to (A2 ), (A4 ) and (8). Step 2. With the same proof as in Step 2 of [15, Theorem 1]: sup Rk (x) ∇ uk (x) C,
= 1, 2, . . . , 2m − 1.
(43)
x∈M
Step 3. uk → −∞ locally uniformly in M \ S1 , S1 := {x (i) : 1 i I }. This follows easily from (43) above and (46) below (which implies that uk → −∞ locally uniformly in
L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771
3759
Bδν (x (i) ) \ {x (i) } for any 1 i I , ν ∈ [1, 2[ and δν as in Step 4), but we also sketch an instructive alternative proof, which does not make use of (46). Our Theorem 1 can be reproduced on a closed manifold, with a similar proof and using Proposition 3.1 from [19] instead of Theorem 4 above. Then either 2m−1,α (M \ S1 ), or (a) uk is bounded in Cloc (b) uk → −∞ locally uniformly in M \ S1 , or (c) there exists a closed set S0 ⊂ M \ S1 of Hausdorff dimension at most 2m − 1 and numbers βk → +∞ such that
2m−1,α M \ (S0 ∪ S) , in Cloc
uk →ϕ βk
(44)
where m g ϕ ≡ 0,
ϕ 0,
ϕ ≡ 0 on M \ S1 ,
ϕ ≡ 0 on S0 .
(45)
Case (a) can be ruled out using (8) as in (21) at the end of the proof of Theorem 1. Case (c) contradicts Lemma 7, by considering any ball BR (x0 ) Ω \ S1 with BR (x0 ) |∇ϕ| dvolg > 0 and using (44). Hence case (b) occurs, as claimed. Step 4. We claim that for every 1 ν < 2, there exist δν > 0 and Cν > 0 such that for 1 i I 2m(ν−1)
dist(x, xi,k )2mν e2muk (x) Cν μi,k
for x ∈ Bδν (xi,k ).
(46)
Then on the necks Σi,k := Bδν (xi,k ) \ BRμi,k (xi,k ) we have e
2muk
2m(ν−1) dvolg Cν μi,k
Σi,k
dist(x, xi,k )−2mν dvolg (x)
Σi,k 2m(ν−1)
δν
Cν μi,k
r 2m−1−2mν dr
Rμi,k 2m(ν−1) 2m(1−ν) δν ,
= Cν R 2m(1−ν) − Cν μi,k whence lim
lim
R→+∞ k→+∞ Σi,k
Qk e2muk dvolg = 0.
(47)
This, together with (26) and Step 3 implies (11), assuming that x (i) = x (j ) for i = j . This we be shown in Step 4c below. Then (12) follows at once from (2). Let us prove (46). Fix 1 ν < 2 and set for 1 i I R˜ i,k := min dist(xi,k , xj,k ). j =i
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Step 4a. Let i ∈ {1, . . . , I } be such that for some θ > 0 we have R˜ i,k θ R˜ j,k
for 1 j I, k 1.
(48)
Set ϕi,k (r) := r
2mν
exp
−
(49)
2muk dσg
∂Br (xi,k )
for 0 < r < rinj , where dσg is the measure on ∂Br (xi,k ) induced by g. Observe that (rμi,k ) < 0 ϕi,k
−
rμi,k < −ν
if and only if
∂uk dσg ∂n
−1 .
(50)
∂Brμi,k (xi,k )
From (40) we infer ∂uk μi,k ∂n ∂Bμ
→ i,k r (xi,k )
2 ∂ −2r log = , ∂r 1 + r2 1 + r2
hence −
μi,k
∂Bμi,k r (xi,k )
∂uk 2r dσg → − ∂n 1 + r2
and (50) implies that for any R 2Rν := 2 (rμi,k ) < 0 ϕi,k
ν 2−ν ,
for r > 0 as k → ∞,
there exists k0 (R) such that
for k k0 (R), r ∈ [2Rν , R].
(51)
Define ri,k := sup r ∈ [2Rν μi,k , R˜ i,k /2]: ϕi,k (ρ) < 0 for ρ ∈ [2Rν μi,k , r) .
(52)
From (51) we infer that ri,k = +∞. k→+∞ μi,k
(53)
lim ri,k = 0.
(54)
lim
Let us assume that k→∞
Consider vi,k (y) := uk expxi,k (ri,k y) − Ci,k ,
Ci,k :=
− ∂Bri,k (xi,k )
uk dσg ,
(55)
L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771
3761
and let ˆ i,k (y) := Qk expx (ri,k y) , Q i,k
−2 gˆ i,k := ri,k (expxi,k ◦Ti,k )∗ g,
where Ti,k (y) := ri,k y
for y ∈ R2m .
Then 2m 2m ˆ Pgˆ2m vi,k + ri,k Qgˆi,k = ri,k Qi,k e2m(vi,k +Ci,k ) i,k 2m(1−ν) = ri,k ϕi,k (ri,k )Qˆ i,k e2mvi,k .
(56)
We also set Ji = j = i: dist(xi,k , xj,k ) = O(ri,k ) as k → ∞
(57)
and (i)
x˜j,k :=
1 exp−1 xi,k (xj,k ), ri,k
(i)
x˜j = lim x˜j,k ,
(58)
k→∞
(i)
after passing to a subsequence, if necessary. Thanks to (48) and (52), we have that |x˜j | 2 for all j ∈ Ji and that (i) x˜ − x˜ (i) 2 j θ
for all j, ∈ Ji , j = .
By (43) and the choice of Ci,k in (55), vi,k is uniformly bounded in (i) 2m−1 2m R \ 0, x˜j : j ∈ Ji . Cloc Thanks to (52) and (53), given R > 2Rν , there exists k0 (R) such that ϕi,k (ri,k ) < ϕi,k (Rμi,k ) for all k k0 . From (40), we infer μ2m i,k exp
−
2muk dσ
= exp
∂BRμi,k (xi,k )
−
2m(uk + log μi,k ) dσ
∂BRμi,k (xi,k )
= C(R) + o(1)
as k → ∞,
(59)
where C(R) → 0 as R → ∞. Then, together with (53), letting k → +∞ we get
(60)
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L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771 2m(1−ν)
ri,k
2m(1−ν)
ϕi,k (ri,k ) ri,k
ϕi,k (Rμi,k ) 2m(ν−1) 2mν μi,k = μ2m exp − 2mu dσ R k i,k ri,k ∂BRμi,k (xi,k )
→ 0.
(61)
Therefore the right-hand side of (56) goes to 0 locally uniformly in (i) R2m \ 0, x˜j : j ∈ Ji ; moreover gˆ i,k → gR2m
2m k R for every k 0, in Cloc
2m ˆ ri,k Qi,k → 0
2m 1 R . in Cloc
(62)
It follows that, up to a subsequence, vi,k → hi
(i) 2m−1,α 2m R \ 0, x˜j : j ∈ Ji , in Cloc
(63)
where, taking (43) into account, m hi (x) = 0,
(i) x ∈ R2m \ 0, x˜j : j ∈ Ji ,
and ˜ ∇ hi (x) C R(x)
(i) for = 1, . . . , 2m − 1, x ∈ R2m \ 0, x˜j : j ∈ Ji ,
˜ with R(x) := min{|x|, |x − x˜j(i) |: j ∈ Ji }. Then Proposition 15 from Appendix A implies that
hi (x) = −λ log |x| −
(i) λj logx − x˜j + β
(64)
j ∈ Ji
for some λ, β, λj ∈ R. We now recall that the Paneitz operator is in divergence form, hence we can write vi,k = divgˆi,k (Agˆi,k vi,k ) Pgˆ2m i,k
(65)
for some differential operator Agˆi,k of order 2m − 1, with coefficients converging to the coefficient of (−1)m ∇m−1 uniformly in B1 , thanks to (62). Then integrating (56), using (62), (63) and (65), we get 2m(1−ν) Qk e2muk dvolg = lim ϕi,k (ri,k )ri,k Qˆ i,k e2mvi,k dvolgˆi,k lim k→∞ Bri,k (xi,k )
k→∞
= lim
k→∞ B1
B1
2m divgˆi,k (Agˆi,k vi,k ) + ri,k Qgˆi,k dvolgˆi,k
L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771
3763
= lim
k→∞ ∂B1
n · (Agˆi,k vi,k ) dσgˆi,k
Λ1 ∂m−1 hi dσ = λ , ∂n 2
= (−1)m
(66)
∂B1
where n denotes the exterior unit normal to ∂B1 and the last identity can be inferred using (14) and the following:
∂m−1 hi dσ = λ ∂n
∂B1
∂m−1 log 1 |x| ∂n
∂B1
+
j ∈ Ji
dσ
m log
λj B1
1 |x − x˜j(i) |
dx.
≡0 on B1
From (43) with = 1, we get uk exp
xi,k (ri,k y1 )
− uk expxi,k (ri,k y2 ) Cri,k r
sup
|∇uk | C
(67)
∂Bri,k r (xi,k )
for 0 r 32 , |y1 | = |y2 | = r. For 2Rν μi,k Rμi,k r ri,k , we infer from (59) 2m(ν−1)
ϕi,k (r) ϕi,k (Rμi,k ) C(R)μi,k
2m(ν−1) . + o μi,k
This, (49), (59), (60) and (67) imply that for any η > 0 there exist Rη 2Rν and kη ∈ N such that 2m(ν−1)
dist(x, xi,k )2mν e2muk ημi,k
for x ∈ Bri,k (xi,k ) \ BRη μi,k (xi,k ), k kη .
It now follows easily that lim
lim
R→+∞ k→∞ Bri,k (xi,k )\BRμi,k (xi,k )
Qk e2muk dx = 0
and from (41) lim
k→+∞ Bri,k (xi,k )
Qk e2muk dx = Λ1 .
(68)
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L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771 (i)
This implies that λ = 2. With a similar computation, integrating on Bδ (x˜j ) for δ small instead of B1 (0), one proves that λj 2 for all j ∈ Ji . Now set hi (r) := − hi dσ. ∂Br (0)
Then λj d 2mν 2mhi (r) r 2 r 2mν−1 e2mhi (r) r = 2m ν − 2 − e (i) 2 dr 2| x ˜ | j j ∈ Ji for 0 < r < 32 . In particular d 2mν 2mhi (r) r e <0 dr r=1 (r ) < 0. This implies that hence, for k large enough, ϕi,k i,k
ri,k =
R˜ i,k 2
for k large.
(69)
This in turn implies limk→∞ R˜ i,k = 0, when i satisfies (48) and limk→∞ ri,k = 0. For i satisfying (48) and lim supk→∞ R˜ i,k > 0, we infer, instead, that lim supk→∞ ri,k > 0. In both cases (68) holds. Step 4b. Now assume that lim sup R˜ i,k > 0
for every 1 i I.
(70)
k→∞
Then (48) is satisfied for every 1 i I , hence lim supk→∞ ri,k > 0, 1 i I . Up to selecting a subsequence, we can set δν := inf
1iI
1 lim ri,k > 0. 2 k→∞
Take now η = 1 in (68), and let R1 be the corresponding Rη . Then (46) is true for x ∈ Bδν (xi,k ) \ BR1 μi,k (xi,k ). On the other hand, thanks to (A3 ), we have uk (x) uk (xi,k ) + C on BR1 μi,k (x). Then, using (25), we get dist(x, xi,k )2mν e2muk (x) C(R1 μi,k )2mν e2muk (xi,k ) 2m(ν−1)
CR12mν μi,k
for x ∈ BR1 μi,k (xi,k ).
This completes the proof of (46), under the assumption that (70) holds. Step 4c. We now prove that in fact (70) holds true. Choose 1 i0 I so that, up to a subsequence, R˜ i0 ,k = min R˜ i,k 1iI
for every k ∈ N,
L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771
3765
and assume by contradiction that limk→∞ R˜ i0 ,k = 0. Clearly (48) holds for i = i0 , hence also (69) holds for i = i0 , by Step 4a. Then, setting Ji0 as is (57), we claim that, for any i ∈ Ji0 , there exists θ (i) > 0 such that R˜ i,k θ (i)R˜ j,k
for 1 j I.
Indeed R˜ i,k = O(ri0 ,k ) = O(R˜ i0 ,k )
as k → ∞.
It then follows that (48) holds for all i ∈ Ji0 , and that Step 4a applies to them. Observing that Ji0 = ∅ thanks to Step 4a (identity (69) with i0 instead of i), we can pick i ∈ Ji0 such that, up to a subsequence, dist(xi,k , xi0 ,k ) dist(xj,k , xi0 ,k ) (i)
for all j ∈ Ji0 , k > 0. (i)
(i)
(i)
Recalling the definition of x˜j for j ∈ Ji , we get |x˜i0 | |x˜j − x˜i0 | for all j ∈ Ji . A consequence of this inequality is that the scalar product (i)
(i)
x˜i0 · x˜j > 0
(71)
for all j ∈ Ji . In other words all the x˜j(i) ’s with j ∈ Ji lie in the same half space orthogonal (i)
(i)
to x˜i0 and whose boundary contains 0 = x˜i . Multiplying (56) by ∇vi,k and integrating over Bδ = Bδ (0) (δ > 0 small), we get 2m ˆ Pgˆ2m v ∇v dvol = − ri,k Qi,k ∇vi,k dvolgˆi,k i,k i,k gˆ i,k i,k Bδ
Bδ
2m(1−ν)
+
ri,k
ϕi,k (ri,k )
2m
Qˆ i,k ∇e2mvi,k dvolgˆi,k
Bδ (0)
=: (I )k + (II)k .
(72)
Recalling (62) and (63), we see at once that limk→∞ (I )k = 0. Integrating by parts, we also see that
2m(1−ν) r (II)k C i,k ϕi,k (ri,k ) 2m
Bδ (0)
+
2m(1−ν) ri,k
2m
ˆ i,k ∇Q ˆ i,k e2mvi,k dvolgˆ Q i,k ˆ i,k Q
ϕi,k (ri,k )
O(1) dσgˆi,k
∂Bδ (0)
→ 0 as k → ∞, where the last term vanishes thanks to (61), and the first term on the right of (II)k vanishes thanks to (66) and the remark that
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L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771
ˆ i,k ∇Q → 0 in L∞ (Bδ ). ˆ i,k Q
(73)
Recalling (63), using (43) and (62), we arrive at ∇hi (−)m hi dx = 0.
(74)
Bδ
Let us assume m even. Then, integrating by parts, we get
1 0= 2
2 m (−) 2 hi n dσ
∂Bδ
∂(−)m−1−j hi dσ ∇(−)j hi ∂n
m 2 −1
−
j =0 ∂B
δ
+
∂(−)j hi (−)m−1−j hi dσ. ∇ ∂n
m 2 −1
j =0 ∂B
(75)
δ
Then, taking the limit as δ → 0, and writing hi (x) = 2 log
1 + Gi (x) |x|
we see that all terms in (75) vanish (Gi is regular in a neighborhood of 0 and the vector function 1 ∇ log |x| is anti-symmetric), up to at most lim
δ→0 ∂Bδ
1 dσ = 2γm ∇Gi (0), (−∇Gi )∂ν (−)m−1 2 log |x|
see (14). But then (75) gives 2γm ∇Gi (0). Also when m is odd, in a completely analogous way, we get ∇Gi (0) = 0, a contradiction with (64) and (71). This ends the proof of Step 4. Step 5. Finally, if case (ii) occurs and S = ∅, then (41) implies −1 lim sup vol(gk ) Q0 x (1) Λ1 > 0. k→∞
This justifies the last claim of the theorem.
2
L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771
3767
4. The case M = S 2m In the case of the 2m-dimensional sphere, the concentration–compactness of Theorem 2 becomes quite explicit: only one concentration point can appear and, by composing with suitable Möbius transformations, we have a global understanding of the concentration behavior. This was already noticed in [31,20], in dimension 2 and 4 under the assumption, which we now drop, that the Q-curvatures are positive. Theorem 9. Let (S 2m , g) be the 2m-dimensional round sphere, and let uk : M → R be a sequence of solutions of Pg uk + (2m − 1)! = Qk e2muk ,
(76)
where Qk → Q0 in C 0 for a given continuous function Q0 . Assume also that vol(gk ) =
e2muk dvolg = S 2m ,
(77)
S 2m
where gk := e2muk g. Then one of the following is true. (i) For every 0 α < 1, a subsequence converges in C 2m−1,α (S 2m ). (ii) There is a point x0 ∈ S 2m such that up to a subsequence uk → −∞ locally uniformly in S 2m \ {x0 }. Moreover Q0 (x0 ) > 0, Qk e2muk dvolg Λ1 δx0 and there exist Möbius diffeomorphisms Φk such that the metrics hk := Φk∗ gk satisfy hk → g in H 2m S 2m ,
Qhk → (2m − 1)!
in L2 S 2m .
(78)
Proof. On the round sphere Pg = m−1 i=0 (−g + i(2m − i − 1)); moreover ker g = {constants} and the non-zero eigenvalues of −g are all positive. That easily implies that ker Pg2m = {constants}. From Theorem 2, and the Gauss–Bonnet–Chern theorem, we infer that in case (ii) we have Λ1 =
Qg dvolg = I Λ1 , M
hence I = 1, and Qk e2muk dvolg Λ1 δx0 . In fact, in order to apply Theorem 2, we would need Qk → Q0 in C 1 (M), but this hypothesis is only used in (73) in the last part of the proof of Theorem 2, in order to show that the concentration points are isolated. Since in the case of the sphere only one concentration point appears, that part of the proof is superfluous, and the assumption Qk → Q0 in C 0 (M) suffices.
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L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771
To prove the second part of the theorem, for every k we define a Möbius transformation Φk : S 2m → S 2m such that the normalized metric hk := Φk∗ gk satisfies x dvolhk = 0. S 2m
Then (78) follows by reasoning as in [20, bottom of p. 16].
2
Acknowledgment I am grateful to Professor Michael Struwe for many stimulating discussions. Appendix A. A few useful results Here we collect a few results which have been used above. For the proofs of Lemma 10, Propositions 11 and 13, and Theorem 12, see e.g. [22]. The following lemma can be considered a generalized mean value identity for polyharmonic function. Lemma 10. (See Pizzetti [26].) Let m h = 0, in BR (x0 ) ⊂ Rn , for some m, n positive integers. Then there are positive constants ci = ci (n) such that m−1
− h(z) dz = ci R 2i i h(x0 ).
(79)
i=0
BR (x0 )
Proposition 11. Let m h = 0 in B2 ⊂ Rn . For every 0 α < 1, p ∈ [1, ∞) and 0 there are constants C(, p) and C(, α) independent of h such that hW ,p (B1 ) C(, p)hL1 (B2 ) , hC ,α (B1 ) C(, α)hL1 (B2 ) . A simple consequence of Lemma 10 and Proposition 11 is the following Liouville-type theorem. Theorem 12. Consider h : Rn → R with m h = 0 and h(x) C(1 + |x| ) for some integer 0. Then h is a polynomial of degree at most max{, 2m − 2}. Proposition 13. Let u ∈ C 2m−1 (B 1 ) such that
(−)m u C (−)j u C
in B1 , on ∂B1 for 0 j < m.
Then there exists a constant C independent of u such that u C in B1 .
(80)
L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771
3769
Lemma 14. Let u ∈ L1 (Ω) and u = 0 on ∂Ω, where Ω ⊂ Rn is a bounded domain. Then for n we have every 1 p < n−1 uW 1,p (Ω) C(p)uL1 (Ω) . n Proof. Let u ∈ C ∞ (Ω) and u|∂Ω = 0. If 1 p < n−1 , then q := e.g. [30, p. 91]) and the imbedding W 1,q → L∞ we infer
p p−1
> n. From Lp -theory (see
∇uLp (Ω) C
∇u · ∇ϕ dx = C
sup 1,q
sup
ϕ∈L∞ (Ω) ϕL∞ (Ω) 1 Ω
−uϕ dx
ϕ∈W0 (Ω) Ω ∇ϕLq (Ω) 1
C
sup 1,q
ϕ∈W0 (Ω) Ω ∇ϕLq (Ω) 1
−uϕ dx CuL1 .
To estimate uLp (Ω) we use Poincaré’s inequality. For the general case one can use a standard mollifying procedure. 2 Proof of Lemma 5. By Lemma 14, m−1 uW 1,r (Ω) C(r)f L1 (Ω) for 1 r < by Lp -theory, uW 2m−1,r (Ω) C(r)f L1 (Ω) , and by Sobolev’s embedding, uLs (Ω) C(s)f L1 (Ω)
for all 1 s < ∞.
2m 2m−1 . Then,
(81)
Now fix B = B4R (x0 ) (Ω \ S1 ) and write u = u1 + u2 , where
(−)m u2 = f j u2 = 0
in B4R (x0 ), on ∂B4R (x0 ) for 0 j m − 1.
By Lp -theory u2 W 2m,p (B4R (x0 )) C(p, B)f Lp (B4R (x0 )) ,
(82)
with C(p, B) depending on p and the chosen ball B. Together with (81), we find u1 L1 (B4R (x0 )) C(p, B) f Lp (B4R (x0 )) + f L1 (Ω) . By Proposition 11 u1 W 2m,p (BR (x0 )) C(p, B) f Lp (B4R (x0 )) + f L1 (Ω) , and (15) follows.
2
Proposition 15. Let S = {x1 , . . . , xI } ⊂ R2m be a finite set and let h ∈ C ∞ (R2m \ S) satisfy m h = 0 and dist(x, S)∇h(x) C
for x ∈ R2m \ S.
(83)
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L. Martinazzi / Journal of Functional Analysis 256 (2009) 3743–3771
Then there are constants β and λi , 1 i I , such that h(x) =
I
λi log
i=1
1 + β. |x − xi |
(84)
Proof. Thanks to (83), h ∈ L1loc (R2m ), so that m h is well defined in the sense of distributions and it is supported in S. Therefore m h =
I
βi δxi
i=1
for some constants βi . Then, recalling (14), if we set v(x) := h(x) −
I
i=1
λi log
1 , |x − xi |
λi := (−1)m
βi , γm
we get m v ≡ 0 in R2m in the sense of distributions (hence v is smooth) and ∇v(x)|x| C
in R2m .
(85)
Then |v(x)| C(log(1 + |x|) + 1). By Theorem 12 v is a polynomial, which (85) forces to be constant, say v ≡ −β. Now (84) follows at once. 2 References [1] Adimurthi, F. Robert, M. Struwe, Concentration phenomena for Liouville’s equation in dimension 4, J. Eur. Math. Soc. 8 (2006) 171–180. [2] S. Alexakis, On the decomposition of global conformal invariants II, Adv. Math. 206 (2006) 466–502. [3] S. Alexakis, The decomposition of Global Conformal Invariants: On a conjecture of Deser and Schwimmer, preprint, 2007. [4] T. Branson, The Functional Determinant, Global Anal. Res. Center Lect. Notes Ser., vol. 4, Seoul National Univ., 1993. [5] T. Branson, B. Oersted, Explicit functional determinants in four dimensions, Comm. Partial Differential Equations 16 (1991) 1223–1253. [6] H. Brézis, F. Merle, Uniform estimates and blow-up behaviour for solutions of −u = V (x)eu in two dimensions, Comm. Partial Differential Equations 16 (1991) 1223–1253. [7] S.-Y.A. Chang, Non-linear Elliptic Equations in Conformal Geometry, Zur. Lect. Adv. Math., EMS, 2004. [8] S.-Y.A. Chang, W. Chen, A note on a class of higher order conformally covariant equations, Discrete Contin. Dyn. Syst. 63 (2001) 275–281. [9] X. Chen, Remarks on the existence of branch bubbles on the blowup analysis on equation u = e2u in dimension 2, Comm. Anal. Geom. 7 (1999) 295–302. [10] S.-S. Chern, A simple intrinsic proof of the Gauss–Bonnet theorem for closed Riemannian manifolds, Ann. of Math. 45 (1944) 747–752. [11] O. Druet, From one bubble to several bubbles: The low dimensional case, J. Differential Geom. 63 (2003) 399–473. [12] O. Druet, Compactness for the Yamabe equation in low dimensions, Int. Math. Res. Not. IMRN 23 (2004) 1143– 1191. [13] O. Druet, E. Hebey, Blow-up examples for second order elliptic PDEs of critical Sobolev growth, Trans. Amer. Math. Soc. 357 (2005) 1915–1929.
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[14] O. Druet, E. Hebey, F. Robert, Blow-up Theory for Elliptic PDEs in Riemannian Geometry, Math. Notes, vol. 45, Princeton Univ. Press, 2004. [15] O. Druet, F. Robert, Bubbling phenomena for fourth-order four-dimensional PDEs with exponential growth, Proc. Amer. Math. Soc. 3 (2006) 897–908. [16] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1977. [17] C.R. Graham, R. Jenne, L. Mason, G. Sparling, Conformally invariant powers of the Laplacian, I: Existence, J. Lond. Math. Soc. 46 (2) (1992) 557–565. [18] Y. Li, I. Shafrir, Blow-up analysis for solutions of −u = V eu in dimension 2, Indiana Univ. Math. J. 43 (1994) 1255–1270. [19] A. Malchiodi, Compactness of solutions to some geometric fourth-order equations, J. Reine Angew. Math. 594 (2006) 137–174. [20] A. Malchiodi, M. Struwe, Q-curvature flow on S 4 , J. Differential Geom. 73 (2006) 1–44. [21] L. Martinazzi, Conformal metrics on R2m with non-positive constant Q-curvature, Rend. Lincei Mat. Appl. 19 (2008) 279–292. [22] L. Martinazzi, Classification of the entire solutions to the higher order Liouville’s equation on R2m , Math. Z. (2009), doi:10.1007/s00209-008-0419-1. [23] L. Martinazzi, A threshold phenomenon for embeddings of H0m into Orlicz spaces, preprint, 2009, Calc. Var. Partial Differential Equations, submitted for publication. [24] C.B. Ndiaye, Constant Q-curvature metrics in arbitrary dimension, J. Funct. Anal. 251 (2007) 1–58. [25] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008), Paper 036, preprint, 1983. [26] P. Pizzetti, Sulla media dei valori che una funzione dei punti dello spazio assume alla superficie di una sfera, Rend. Lincei 18 (1909) 182–185. [27] F. Robert, Concentration phenomena for a fourth order equation with exponential growth: The radial case, J. Differential Equations 231 (2006) 135–164. [28] F. Robert, Quantization effects for a fourth order equation of exponential growth in dimension four, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 531–553. [29] F. Robert, M. Struwe, Asymptotic profile for a fourth order PDE with critical exponential growth in dimension four, Adv. Nonlinear Stud. 4 (2004) 397–415. [30] C.G. Simader, On Dirichlet’s Boundary Value Problem, Lecture Notes in Math., vol. 268, Springer, 1972. [31] M. Struwe, A flow approach to Nirenberg’s problem, Duke Math. J. 128 (1) (2005) 19–64. [32] X. Xu, Uniqueness and non-existence theorems for conformally invariant equations, J. Funct. Anal. 222 (2005) 1–28.
Journal of Functional Analysis 256 (2009) 3772–3805 www.elsevier.com/locate/jfa
Compactness properties of operator multipliers K. Juschenko a , R.H. Levene b,∗ , I.G. Todorov c , L. Turowska a a Department of Mathematical Sciences, Chalmers & Göteborg University, SE-412 96 Göteborg, Sweden b School of Mathematics, Trinity College, Dublin 2, Ireland c School of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, Northern Ireland
Received 12 September 2008; accepted 10 December 2008 Available online 22 January 2009 Communicated by N. Kalton
Abstract We continue the study of multidimensional operator multipliers initiated in [K. Juschenko, I.G. Todorov, L. Turowska, Multidimensional operator multipliers, Trans. Amer. Math. Soc., in press]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C ∗ -algebras is automatically compact. We give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the C ∗ -algebra of compact operators in terms of tensor products, generalising results of Saar [H. Saar, Kompakte, vollständig beschränkte Abbildungen mit Werten in einer nuklearen C ∗ -Algebra, Diplomarbeit, Universität des Saarlandes, Saarbrücken, 1982]. © 2008 Published by Elsevier Inc. Keywords: Operator multiplier; Complete compactness; Schur multiplier; Haagerup tensor product
1. Introduction A bounded function ϕ : N×N → C is called a Schur multiplier if (ϕ(i, j )aij ) is the matrix of a bounded linear operator on 2 whenever (aij ) is such. The study of Schur multipliers was initiated by Schur in the early 20th century and since then has attracted considerable attention, much of * Corresponding author.
E-mail addresses: [email protected] (K. Juschenko), [email protected] (R.H. Levene), [email protected] (I.G. Todorov), [email protected] (L. Turowska). 0022-1236/$ – see front matter © 2008 Published by Elsevier Inc. doi:10.1016/j.jfa.2008.12.018
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which was inspired by A. Grothendieck’s characterisation of these objects in his Résumé [9]. Grothendieck ∞it has the2 form showed that a function ϕ is a Schur multiplier precisely when a (i)b (j ), where a , b : N → C satisfy the conditions sup ϕ(i, j ) = ∞ k k k i k=1 k k=1 |ak (i)| < ∞ ∞ 2 and supj k=1 |bk (j )| < ∞. In modern terminology, this characterisation can be expressed by saying that ϕ is a Schur multiplier precisely when it belongs to the extended Haagerup tensor product ∞ ⊗eh ∞ of two copies of ∞ . Special classes of Schur multipliers, e.g. Toeplitz and Hankel Schur multipliers, have played an important role in analysis and have been studied extensively (see [19]). Compact Schur multipliers, that is, the functions ϕ for which the mapping (aij ) → (ϕ(i, j )aij ) on B(2 ) is compact, were characterised by Hladnik [11], who identified them with the elements of the Haagerup tensor product c0 ⊗h c0 . A non-commutative version of Schur multipliers was introduced by Kissin and Shulman [14] as follows. Let A and B be C ∗ -algebras and let π and ρ be representations of A and B on Hilbert spaces H and K, respectively. Identifying H ⊗ K with the Hilbert space C2 (H d , K) of all Hilbert–Schmidt operators from the dual space H d of H into K, we obtain a representation σπ,ρ of the minimal tensor product A ⊗ B acting on C2 (H d , K). An element ϕ ∈ A ⊗ B is called a π ,ρ-multiplier if σπ,ρ (ϕ) is bounded in the operator norm of C2 (H d , K). If ϕ is a π ,ρ-multiplier for any pair of representations (π, ρ) then ϕ is called a universal (operator) multiplier. Multidimensional Schur multipliers and their non-commutative counterparts were introduced and studied in [12], where the authors gave, in particular, a characterisation of universal multipliers as certain weak limits of elements of the algebraic tensor product of the corresponding C ∗ -algebras, generalising the corresponding results of Grothendieck and Peller [9,18] as previously conjectured by Kissin and Shulman in [14]. Let A1 , . . . , An be C ∗ -algebras. Like Schur multipliers, elements of the set M(A1 , . . . , An ) of (multidimensional) universal multipliers give rise to completely bounded (multilinear) maps. Requiring these maps to be compact or completely compact, we define the sets of compact and completely compact operator multipliers denoted by Mc (A1 , . . . , An ) and Mcc (A1 , . . . , An ), respectively. The notion of complete compactness we use is an operator space version of compactness which was introduced by Saar [21] and subsequently studied by Oikhberg [15] and Webster [27]. Our results on operator multipliers rely on the main result of Section 3 where we prove a representation theorem for completely compact completely bounded multilinear maps. In [3] Christensen and Sinclair established a representation result for completely bounded multilinear maps which implies that every such map Φ : K(H2 , H1 ) ⊗h · · · ⊗h K(Hn , Hn−1 ) → K(Hn , H1 ) (where, for Hilbert spaces H and H , we denote by K(H , H ) the space of all compact operators from H into H ) has the form Φ(x1 ⊗ · · · ⊗ xn−1 ) = A1 (x1 ⊗ I )A2 . . . (xn−1 ⊗ I )An ,
(1)
for some index set J and bounded block operator matrices A1 ∈ M1,J (B(H1 )), A2 ∈ MJ (B(H2 )), . . . , An ∈ MJ,1 (B(Hn )). In other words, Φ arises from an element u = A1 · · · An ∈ B(H1 ) ⊗eh · · · ⊗eh B(Hn ) of the extended Haagerup tensor product of B(H1 ), . . . , B(Hn ). Moreover, if Φ is A1 , . . . , An modular for some von Neumann algebras A1 , . . . , An , then the entries of Ai can be chosen
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from Ai . We show in Section 3 that a map Φ as above is completely compact precisely when it has a representation of the form (1) where u = A1 · · · An ∈ K(H1 ) ⊗h B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 ) ⊗h K(Hn ). This extends a result of Saar [21] in the two-dimensional case. If, additionally, A1 , . . . , An are von Neumann algebras and Φ is A1 , . . . , An -modular then u can be chosen from K (A1 ) ⊗h (A2 ⊗eh · · · ⊗eh An−1 ) ⊗h K (An ), where K (A) denotes the ideal of compact operators contained in A in its identity representation. As a consequence of this and a result of Effros and Kishimoto [4] we point out the completely isometric identifications ∗∗ ∗∗ CC K(H2 , H1 ) K(H1 ) ⊗h K(H2 ) CB B(H2 , H1 ) , where CC(X ) and CB(X ) are the spaces of completely compact and completely bounded maps on an operator space X , respectively. In Section 4 we pinpoint the connection between universal operator multipliers and completely bounded maps. This technical result is used in Section 5 to define the symbol uϕ of an operator multiplier ϕ ∈ M(A1 , . . . , An ) which, in the case n is even (resp. odd) is an element of An ⊗eh Aon−1 ⊗eh · · · ⊗eh A2 ⊗eh Ao1 (resp. An ⊗eh Aon−1 ⊗eh · · · ⊗eh Ao2 ⊗eh A1 ). Here Ao is the opposite C ∗ -algebra of a C ∗ -algebra A. This notion extends a similar notion that was given in the case of completely bounded masa-bimodule maps by Katavolos and Paulsen in [13]. We give a symbolic calculus for universal multipliers which is used to establish a universal property of the symbol related to the representation theory of the C ∗ -algebras under consideration. The symbol of a universal multiplier is used in Section 6 to single out the completely compact multipliers within the set of all operator multipliers. In fact, we show that ϕ ∈ Mcc (A1 , . . . , An ) if and only if uϕ ∈
K(An ) ⊗h (Aon−1 ⊗eh · · · ⊗eh Ao3 ⊗eh A2 ) ⊗h K(Ao1 ) K(An ) ⊗h (Aon−1 ⊗eh · · · ⊗eh A3 ⊗eh Ao2 ) ⊗h K(A1 )
if n is even, if n is odd,
which is equivalent to the approximability of ϕ in the multiplier norm by operator multipliers of finite rank whose range consists of finite rank operators. It follows that a multidimensional Schur multiplier ϕ ∈ ∞ (Nn ) is compact if and only if ϕ ∈ c0 ⊗h (∞ ⊗eh · · · ⊗eh ∞ ) ⊗h c0 . In Section 7 we use Saar’s construction [21] of a completely bounded compact mapping which is not completely compact to show that the inclusion Mcc (A1 , . . . , An ) ⊆ Mc (A1 , . . . , An ) is proper if both K(A1 ) and K(An ) contain full matrix algebras of arbitrarily large sizes. However, if both K(A1 ) and K(An ) are isomorphic to a c0 -sum of matrix algebras of uniformly bounded sizes then the sets of compact and completely compact multipliers coincide. The case when only one of K(A1 ) and K(An ) contains matrix algebras of arbitrary large size remains, however, unsettled. Finally, for n = 2, we characterise the cases where every universal multiplier is automatically compact: this happens precisely when one of the algebras A1 and A2 is finite dimensional and the other one coincides with its algebra of compact elements. 2. Preliminaries We start by recalling standard notation and notions from operator space theory. We refer the reader to [1,6,16,20] for more details.
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If H and K are Hilbert spaces we let B(H, K) (resp. K(H, K)) denote the set of all bounded linear (resp. compact) operators from H into K. If I is a set we let H I be the direct sum of |I | copies of H and set H ∞ = H N . Writing H ⊗ K for the Hilbertian tensor product of two Hilbert spaces, we observe that H I = H ⊗ 2 (I ) as Hilbert spaces. An operator space E is a closed subspace of B(H, K), for some Hilbert spaces H and K. The opposite operator space E o associated with E is the space E o = {x d : x ∈ E} ⊆ B(K d , H d ). Here, and in the sequel, H d = {ξ d : ξ ∈ H } denotes the dual of the Hilbert space H , where ξ d (η) = (η, ξ ) for η ∈ H . Note that H d is canonically conjugate-linearly isometric to H . We also adopt the notation x d ∈ B(K d , H d ) for the Banach space adjoint of x ∈ B(H, K), so that x d ξ d = (x ∗ ξ )d for ξ ∈ K. As usual, E ∗ will denote the operator space dual of E . If n, m ∈ N, by Mn,m (E) we denote the space of all n by m matrices with entries in E and let Mn (E) = Mn,n (E). The space Mn,m (E) carries a natural norm arising from the embedding Mn,m (E) ⊆ B(H m , K n ). Let I and J be arbitrary index sets. If v is a matrix with entries in E and indexed by I × J , and I0 ⊆ I and J0 ⊆ J are finite sets, we let vI0 ,J0 ∈ MI0 ,J0 (E) be the matrix obtained by restricting v to the indices from I0 × J0 . We define MI,J (E) to be the space of all such v for which def v = sup vI0 ,J0 : I0 ⊆ I , J0 ⊆ J finite < ∞. Then MI,J (E) is an operator space [6, §10.1]. Note that MI,J (B(H, K)) can be naturally identified with B(H J , K I ) and every v ∈ MI,J (B(H, K)) is the weak limit of {vI0 ,J0 } along the net {(I0 , J0 ): I0 ⊆ I, J0 ⊆ J finite}. We set MI (E) = MI,I (E). For A = (aij ) ∈ MI (E), we write Ad = (aijd ) ∈ MI (E o ). 2.1. Completely bounded maps and Haagerup tensor products If E and F are operator spaces, a linear map Φ : E → F is called completely bounded if the maps Φ (k) : Mk (E) → Mk (F ) given by Φ (k) ((aij )) = (Φ(aij )) are bounded for every k ∈ N and def
Φ cb = supk Φ (k) < ∞. Given linear spaces E1 , . . . , En , we denote by E1 · · · En their algebraic tensor product. If E1 , . . . , En are operator spaces and a k = (aijk ) ∈ Mmk ,mk+1 (Ek ), mk ∈ N, k = 1, . . . , n, we define the multiplicative product a 1 · · · a n ∈ Mm1 ,mn+1 (E1 · · · En ) n 1 ⊗ a2 by letting its (i, j )-entry (a 1 · · · a n )ij be i2 ,...,in ai,i i2 ,i3 ⊗ · · · ⊗ ain ,j . If E is another 2 operator space and Φ : E1 × · · · × En → E is a multilinear map we let Φ (m) : Mm (E1 ) × · · · × Mm (En ) → Mm (E) be the map given by (m) 1 1 Φ a , . . . , a n ij = Φ ai,i , ai22 ,i3 , . . . , ainn ,j , 2 i2 ,...,in k ) ∈ M (E ), k = 1, . . . , n. The multilinear map Φ is called completely bounded where a k = (as,t m k if there exists a constant C > 0 such that, for all m ∈ N, (m) 1 Φ a , . . . , a n C a 1 . . . a n , a k ∈ Mm (Ek ), k = 1, . . . , n.
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Set Φ cb = sup{ Φ (m) (a 1 , . . . , a n ) : m ∈ N, a 1 , . . . , a n 1}. It is well known (see [6,17]) that a completely bounded multilinear map Φ gives rise to a completely bounded map on the Haagerup tensor product E1 ⊗h · · · ⊗h En (see [6] and [20] for its definition and basic properties). The set of all completely bounded multilinear maps from E1 × · · · × En into E will be denoted by CB(E1 × · · · × En , E). If E1 , . . . , En and E are dual operator spaces we say that a map Φ ∈ CB(E1 × · · · × En , E) is normal [3] if it is weak∗ continuous in each variable. We write CBσ (E1 × · · · × En , E) for the space of all normal maps in CB(E1 × · · · × En , E). The extended Haagerup tensor product E1 ⊗eh · · · ⊗eh En is defined [5] as the space of all normal completely bounded maps u : E1∗ × · · · × En∗ → C. It was shown in [5] that if u ∈ E1 ⊗eh 1 )∈M · · · ⊗eh En then there exist index sets J1 , J2 , . . . , Jn−1 and matrices a 1 = (a1,s 1,J1 (E1 ), n ∗ 2 2 n a = (as,t ) ∈ MJ1 ,J2 (E2 ), . . . , a = (at,1 ) ∈ MJn−1 ,1 (En ) such that if fi ∈ Ei , i = 1, . . . , n, then
def u, f1 ⊗ · · · ⊗ fn = u(f1 , . . . , fn ) = a 1 , f1 . . . a n , fn ,
(2)
k )) and the product of the (possibly infinite) matrices in (2) is defined to where a k , fk = (fk (as,t be the limit of the sums
i1 ∈F1 ,...,in−1 ∈Fn−1
1 2 f2 ai1 ,i2 . . . fn ainn−1 ,1 f1 a1,i 1
along the net {(F1 × · · · × Fn−1 ): Fj ⊆ Jj finite, 1 j n − 1}. We may thus identify u with the matrix product a 1 · · · a n ; two elements a 1 · · · a n and a˜ 1 · · · a˜ n coincide if a 1 , f1 . . . a n , fn = a˜ 1 , f1 . . . a˜ n , fn for all fi ∈ Ei∗ . Moreover, u eh = inf a 1 . . . a n : u = a 1 · · · a n . The space E1 ⊗eh · · · ⊗eh En has a natural operator space structure [5]. If E1 , . . . , En are dual operator spaces then by [5, Theorem 5.3] E1 ⊗eh · · · ⊗eh En coincides with the weak∗ Haagerup tensor product E1 ⊗w∗h · · · ⊗w∗h En of Blecher and Smith [2]. Given operator spaces Fi and completely bounded maps gi : Ei → Fi , i = 1, . . . , n, Effros and Ruan [5] define a completely bounded map g = g1 ⊗eh · · · ⊗eh gn : E1 ⊗eh · · · ⊗eh En → F1 ⊗eh · · · ⊗eh Fn ,
a 1 · · · a n → a 1 , g1 · · · a n , gn where a k , gk = (gk (aijk )). Thus
g(u), f1 ⊗ · · · ⊗ fn = u, (f1 ◦ g1 ) ⊗ · · · ⊗ (fn ◦ gn )
(3)
for u ∈ E1 ⊗eh · · · ⊗eh En and fi ∈ Fi∗ , i = 1, . . . , n. The following fact is a straightforward consequence of a well-known theorem due to Christensen and Sinclair [3], and it will be used throughout the exposition.
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Theorem 2.1. Let Hi be a Hilbert space and Ri ⊆ B(Hi ) be a von Neumann algebra, i = 1, . . . , n. There exists an isometry γ from R1 ⊗eh · · · ⊗eh Rn onto the space of all R1 , . . . , Rn -modular maps in CBσ (B(H2 , H1 ) × · · · × B(Hn , Hn−1 ), B(Hn , H1 )), given as follows: if u ∈ R1 ⊗eh · · · ⊗eh Rn has a representation u = A1 · · · An where Ai ∈ MJ (Ri ) ⊆ B(Hi ⊗ 2 (J )) for some index set J , then γ (u)(T1 , . . . , Tn−1 ) = A1 (T1 ⊗ I )A2 . . . An−1 (Tn−1 ⊗ I )An , for all Ti ∈ B(Hi+1 , Hi ), i = 1, . . . , n − 1, where I is the identity operator on 2 (J ). We now turn to the definition of slice maps which will play an important role in our proofs. Given ω1 ∈ B(H1 )∗ we set Lω1 = ω1 ⊗eh idB(H2 ) . After identifying C ⊗ B(H2 ) with B(H2 ) we obtain a mapping Lω1 : B(H1 ) ⊗eh B(H2 ) → B(H2 ) called a left slice map. Similarly, for ω2 ∈ B(H2 )∗ we obtain a right slice map Rω2 : B(H1 ) ⊗eh B(H2 ) → B(H1 ). If u = i∈I vi ⊗ wi ∈ B(H1 ) ⊗eh B(H2 ) where v = (vi )i∈I ∈ M1,I (B(H1 )) and w = (wi )i∈I ∈ MI,1 (B(H2 )), then Lω1 (u) =
i∈I
ω1 (vi )wi
and Rω2 (u) =
ω2 (wi )vi .
i∈I
Moreover,
Rω2 (u), ω1 = u, ω1 ⊗ ω2 = Lω1 (u), ω2 = ω1 (vi )ω2 (wi ).
(4)
i∈I
It was shown in [24] that if E ⊆ B(H1 ) and F ⊆ B(H2 ) are closed subspaces then, up to a complete isometry, E ⊗eh F = u ∈ B(H1 ) ⊗eh B(H2 ): Lω1 (u) ∈ F and Rω2 (u) ∈ E for all ω1 ∈ B(H1 )∗ and ω2 ∈ B(H2 )∗ = u ∈ B(H1 ) ⊗eh B(H2 ): Lω1 (u) ∈ F and Rω2 (u) ∈ E for all ω1 ∈ B(H1 )∗ and ω2 ∈ B(H2 )∗ .
(5)
E ⊗h F = u ∈ B(H1 ) ⊗h B(H2 ): Lω1 (u) ∈ F and Rω2 (u) ∈ E for all ω1 ∈ B(H1 )∗ and ω2 ∈ B(H2 )∗ .
(6)
Moreover [23],
Thus, E ⊗h F can be canonically identified with a subspace of B(H1 ) ⊗h B(H2 ) which, on the other hand, sits completely isometrically in B(H1 ) ⊗eh B(H2 ). These identifications are made in the statement of the following lemma which will be useful for us later.
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Lemma 2.2. If H1 , H2 , H3 are Hilbert spaces and E1 , E2 ⊆ B(H1 ), F1 , F2 ⊆ B(H2 ) and G1 , G2 ⊆ B(H3 ) are operator spaces, then (E1 ⊗eh F1 ) ∩ (E2 ⊗h F2 ) = (E1 ∩ E2 ) ⊗h (F1 ∩ F2 )
and
(E1 ⊗eh F1 ⊗eh G1 ) ∩ (E2 ⊗h F2 ⊗h G2 ) = (E1 ∩ E2 ) ⊗h (F1 ∩ F2 ) ⊗h (G1 ∩ G2 ). Proof. Since ⊗eh and ⊗h are both associative, the second equation follows from the first. If u ∈ (E1 ⊗eh F1 ) ∩ (E2 ⊗h F2 ) ⊆ B(H1 ) ⊗h B(H2 ) then Lϕ (u) ∈ F1 ∩ F2 and Rψ (u) ∈ E1 ∩ E2 whenever ϕ ∈ B(H1 )∗ and ψ ∈ B(H2 )∗ . By (6), u ∈ (E1 ∩ E2 ) ⊗h (F1 ∩ F2 ). The converse inclusion follows immediately in light of the injectivity of the Haagerup tensor product. 2 2.2. Operator multipliers We now recall some definitions and results from [14] and [12] that will be needed later. Let H1 , . . . , Hn be Hilbert spaces and H = H1 ⊗ · · · ⊗ Hn be their Hilbertian tensor product. Set HS(H1 , H2 ) = C2 (H1d , H2 ) and let θH1 ,H2 : H1 ⊗ H2 → HS(H1 , H2 ) be the canonical isometry given by θ (ξ1 ⊗ ξ2 )(ηd ) = (ξ1 , η)ξ2 for ξ1 , η ∈ H1 and ξ2 ∈ H2 . When n is even, we inductively define def HS(H1 , . . . , Hn ) = C2 HS(H2 , H3 )d , HS(H1 , H4 , . . . , Hn ) , and let θH1 ,...,Hn : H → HS(H1 , . . . , Hn ) be given by θH1 ,...,Hn (ξ2,3 ⊗ ξ ) = θHS(H2 ,H3 ),HS(H1 ,H4 ,...,Hn ) θH2 ,H3 (ξ2,3 ) ⊗ θH1 ,H4 ,...,Hn (ξ ) , where ξ2,3 ∈ H2 ⊗ H3 and ξ ∈ H1 ⊗ H4 ⊗ · · · ⊗ Hn . When n is odd, we let def
HS(H1 , . . . , Hn ) = HS(C, H1 , . . . , Hn ). If K is a Hilbert space, we will identify C2 (Cd , K) with K via the map S → S(1d ). The isomorphism θH1 ,...,Hn in the odd case is given by θH1 ,...,Hn (ξ ) = θC,H1 ,...,Hn (1 ⊗ ξ ). We will omit the subscripts when they are clear from the context and simply write θ . If ξ ∈ H1 ⊗ H2 we let ξ op denote the operator norm of θ (ξ ). By · 2 we will denote the Hilbert–Schmidt norm. Let (H1 ⊗ H2 ) (H2 ⊗ H3 )d · · · (Hn−1 ⊗ Hn ) if n is even, Γ (H1 , . . . , Hn ) = (H1 ⊗ H2 )d (H2 ⊗ H3 ) · · · (Hn−1 ⊗ Hn ) if n is odd. We equip Γ (H1 , . . . , Hn ) with the Haagerup norm · h where each of the terms of the algebraic tensor product is given the opposite operator space structure to the one arising from the embedding H ⊗K → (C2 (H d , K), · op ). We denote by · 2,∧ the projective norm on Γ (H1 , . . . , Hn ) where each of the terms is given its Hilbert space norm.
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Suppose n is even. For each ϕ ∈ B(H ) we let Sϕ : Γ (H1 , . . . , Hn ) → B(H1d , Hn ) be the map given by d d d Sϕ (ζ ) = θ ϕ(ξ1,2 ⊗ ξ3,4 ⊗ · · · ⊗ ξn−1,n ) θ η2,3 θ η4,5 . . . θ ηn−2,n−1 d ··· ξ where ζ = ξ1,2 η2,3 n−1,n ∈ Γ (H1 , . . . , Hn ) is an elementary tensor. In particular, if Ai ∈ B(Hi ), i = 1, . . . , n, and ϕ = A1 ⊗ · · · ⊗ An then
d A2 θ ξ1,2 Ad1 . Sϕ (ζ ) = An θ (ξn−1,n ) . . . Ad3 θ η2,3 Now suppose that n is odd and let ζ ∈ Γ (H1 , . . . , Hn ) and ξ1 ∈ H1 . Then ξ1 ⊗ ζ ∈ H1 Γ (H1 , . . . , Hn ) = Γ (C, H1 , . . . , Hn ). For ϕ ∈ B(H ) we let Sϕ (ζ ) be the operator defined on H1 by Sϕ (ζ )(ξ1 ) = S1 ⊗ ϕ (ξ1 ⊗ ζ ). Note that S1 ⊗ ϕ (ξ1 ⊗ ζ ) ∈ C2 (Cd , Hn ); thus, Sϕ (ζ )(ξ1 ) can be viewed as an element of Hn . It was d ⊗ξ shown in [12] that Sϕ (ζ ) ∈ B(H1 , Hn ). If ζ = η1,2 2,3 ⊗ · · · ⊗ ξn−1,n and ϕ = A1 ⊗ · · · ⊗ An for Ai ∈ B(Hi ), i = 1, . . . , n, then d A1 . Sϕ (ζ ) = An θ (ξn−1,n ) . . . A3 θ (ξ2,3 )Ad2 θ η1,2 As observed in [12, Remark 4.3], for any ϕ ∈ B(H ) and ζ ∈ Γ (H1 , . . . , Hn ), Sϕ (ζ )
op
ϕ ζ 2,∧ .
(7)
On the other hand, an element ϕ ∈ B(H ) is called a concrete operator multiplier if there exists C > 0 such that Sϕ (ζ ) op C ζ h for each ζ ∈ Γ (H1 , . . . , Hn ). When n = 2, this is equivalent to Sϕ (ζ ) op C θ (ζ ) op for each ζ ∈ H1 ⊗ H2 . We call the smallest constant C with this property the concrete multiplier norm of ϕ. Now let Ai be a C ∗ -algebra and πi : Ai → B(Hi ) be a representation, i = 1, . . . , n. Set π = π1 ⊗ · · · ⊗ πn : A1 ⊗ · · · ⊗ An → B(H1 ⊗ · · · ⊗ Hn ) (here, and in the sequel, by A ⊗ B we will denote the minimal tensor product of the C ∗ -algebras A and B). An element ϕ ∈ A1 ⊗ · · · ⊗ An is called a π1 , . . . , πn -multiplier if π(ϕ) is a concrete operator multiplier. We denote by ϕ π1 ,...,πn the concrete multiplier norm of π(ϕ). We call ϕ a universal multiplier if it is a π1 , . . . , πn multiplier for all representations πi of Ai , i = 1, . . . , n. We denote the collection of all universal multipliers by M(A1 , . . . , An ); from this definition, it immediately follows that A1 · · · An ⊆ M(A1 , . . . , An ) ⊆ A1 ⊗ · · · ⊗ An . It was observed in [12] that if ϕ ∈ M(A1 , . . . , An ) then def ϕ m = sup ϕ π1 ,...,πn : πi is a representation of Ai , i = 1, . . . , n < ∞.
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It is obvious that if Ai and Bi are C ∗ -algebras and ρi : Ai → Bi is a ∗-isomorphism, i = 1, . . . , n, then (ρ1 ⊗ · · · ⊗ ρn ) M(A1 , . . . , An ) = M(B1 , . . . , Bn ). If ϕ is an operator, and {ϕν } a net of operators, acting on H1 ⊗ · · · ⊗ Hn we say that {ϕν } converges semi-weakly to ϕ if (ϕν ξ, η) →ν (ϕξ, η) for all ξ, η ∈ H1 · · · Hn . The following characterisation of universal multipliers was established in [12] (see Theorem 6.5, the subsequent remark and the proof of Proposition 6.2) and will be used extensively in the sequel. Theorem 2.3. Let Ai ⊆ B(Hi ) be a C ∗ -algebra, i = 1, . . . , n, and ϕ ∈ A1 ⊗ · · · ⊗ An . Suppose that n is even. The following are equivalent: (i) ϕ ∈ M(A1 , . . . , An ); (ii) there exists a net {ϕν } where ϕν = Aν1 Aν2 · · · Aνn and Aνi is a finite block operator ma trix with entries in Ai such that ϕν → ϕ semi-weakly, ϕν m ni=1 Aν2i ni=1 Aνd 2i−1 ν ν d and the operator norms Ai for i even and Ai for i odd, are bounded by a constant depending only on n. For every net {ϕν } satisfying (ii) we have that Sϕν (ζ ) → Sϕ (ζ ) weakly for all ζ = ξ1,2 ⊗ · · · ⊗ ξn−1,n ∈ Γ (H1 , . . . , Hn ) and that supν ϕν m is finite. Moreover, the net ϕν can be chosen in (ii) so that Aνi → Ai (resp. Aνi d → Adi ) strongly for i even (resp. for i odd) for some bounded block operator matrix Ai with entries in Ai (resp. (Adi ) ) such that Sid⊗···⊗id(ϕ) (ζ ) = An θ (ξn−1,n ) ⊗ I . . . θ (ξ1,2 ) ⊗ I Ad1 , for all ζ = ξ1,2 ⊗ · · · ⊗ ξn−1,n ∈ Γ (H1 , . . . , Hn ). A similar statement holds if n is odd. Finally, recall that an element a of a C ∗ -algebra A is called compact if the operator x → axa on A is compact. Let K(A) be the collection of all compact elements of A. It is well known [7,29] that a ∈ K(A) if and only if there exists a faithful representation π of A such that π(a) is a compact operator. Moreover, π can be taken to be the reduced atomic representation of A. The notion of a compact element of a C ∗ -algebra will play a central role in Sections 6 and 7 of the paper. 3. Completely compact maps We start by recalling the notion of a completely compact map introduced in [21] and studied further in [27] and [15]. By way of motivation, recall that if X and Y are Banach spaces then a bounded linear map Φ : X → Y is compact if and only if for every ε > 0, there exists a finite dimensional subspace F ⊆ Y such that dist(Φ(x), F ) < ε for every x in the unit ball of X . Now let X and Y be operator spaces. A completely bounded map Φ : X → Y is called completely compact if for each ε > 0 there exists a finite dimensional subspace F ⊆ Y such that dist Φ (m) (x), Mm (F ) < ε,
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for every x ∈ Mm (X ) with x 1 and every m ∈ N. We extend this definition to multilinear maps: if Y, X1 , . . . , Xn are operator spaces and Φ : X1 × · · · × Xn → Y is a completely bounded multilinear map, we call Φ completely compact if for each ε > 0 there exists a finite dimensional subspace F ⊆ Y such that dist Φ (m) (x1 , . . . , xn ), Mm (F ) < ε, for all xi ∈ Mm (Xi ), xi 1, i = 1, . . . , n, and all m ∈ N. We denote by CC(X1 × · · · × Xn , Y) the space of all completely bounded completely compact multilinear maps from X1 × · · · × Xn into Y. A straightforward verification shows the following: Remark 3.1. A completely bounded map Φ : X1 × · · · × Xn → Y is completely compact if and only if its linearisation Φ˜ : X1 ⊗h · · · ⊗h Xn → Y is completely compact. In view of this remark, we frequently identify the spaces CC(X1 × · · · × Xn , Y) and CC(X1 ⊗h · · · ⊗h Xn , Y). The next result is essentially due to Saar (see Lemmas 1 and 2 of [21]). Proposition 3.2. (i) CC(X1 × · · · × Xn , Y) is closed in CB(X1 × · · · × Xn , Y). (ii) Let E, F and G be operator spaces. If Φ ∈ CC(E, F ) and Ψ ∈ CB(F , G) then Ψ ◦ Φ ∈ CC(E, G). If Φ ∈ CC(F , G) and Ψ ∈ CB(E, F ) then Φ ◦ Ψ ∈ CC(E, G). Let H1 , . . . , Hn be Hilbert spaces. Recall the isometry γ : B(H1 ) ⊗eh · · · ⊗eh B(Hn ) → CBσ B(H2 , H1 ) × · · · × B(Hn , Hn−1 ), B(Hn , H1 ) from Theorem 2.1. Let us identify a completely bounded map defined on B(H2 , H1 ) × · · · × B(Hn , Hn−1 ) with the corresponding completely bounded map defined on def
Bh = B(H2 , H1 ) ⊗h · · · ⊗h B(Hn , Hn−1 ). For u ∈ B(H1 ) ⊗eh · · · ⊗eh B(Hn ) we let γ0 (u) be the restriction of γ (u) to def
Kh = K(H2 , H1 ) ⊗h · · · ⊗h K(Hn , Hn−1 ). Proposition 3.3. The map γ0 is an isometry from B(H1 ) ⊗eh · · · ⊗eh B(Hn ) onto CB(Kh , B(Hn , H1 )). Proof. Let Φ ∈ CB(Kh , B(Hn , H1 )). Since Φ is completely bounded, its second dual Φ ∗∗ : B(H2 , H1 ) ⊗σ h · · · ⊗σ h B(Hn , Hn−1 ) → B(Hn , H1 )∗∗ is completely bounded (here ⊗σ h denotes the normal Haagerup tensor product [5]). Let Q : B(Hn , H1 )∗∗ → B(Hn , H1 ) be the canonical projection. The multilinear map Φ˜ : B(H2 , H1 ) × · · · × B(Hn , Hn−1 ) → B(Hn , H1 )
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corresponding to Q ◦ Φ ∗∗ is completely bounded and, by (5.22) of [5], weak∗ continuous in each variable. By Theorem 2.1, there exists an element u ∈ B(H1 ) ⊗eh · · · ⊗eh B(Hn ) such that ˜ K = Φ. Thus γ0 is surjective. Φ˜ = γ (u). Hence γ0 (u) = γ (u)|Kh = Φ| h Fix u ∈ B(H1 ) ⊗eh · · · ⊗eh B(Hn ). From the definition of γ0 we have γ0 (u) cb γ (u) cb = u eh . On the other hand, the restrictions of the maps Q ◦ γ0 (u)∗∗ and γ (u) to Kh coincide, and since both maps are weak∗ continuous, γ (u) = Q ◦ γ0 (u)∗∗ |Bh . Hence, u eh Q ◦ γ0 (u)∗∗ cb γ0 (u)∗∗ cb = γ0 (u) cb . Thus, γ0 is an isometry.
2
Theorem 3.4. Let H1 , . . . , Hn be Hilbert spaces. The image under γ0 of the operator space def
def
E = K(H1 ) ⊗h (B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 )) ⊗h K(Hn ) is F = CC(Kh , K(Hn , H1 )). Proof. We first establish the inclusion γ0 (E) ⊆ F . If Φ = γ0 (u) where u ∈ E then, by Proposition 3.3, Φ is the limit in the cb norm of maps of the form γ0 (v), where v = a B b ∈ K(H1 ) B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 ) K(Hn ), a and b have finite rank and B is a finite matrix with entries in B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 ). But each such map has finite rank and hence is completely compact. Moreover, every operator in the image of γ0 (v) has range contained in the range of a, which is finite dimensional. It follows that Φ takes compact values; it is completely compact by Proposition 3.2. To see that F ⊆ γ0 (E), let Φ ∈ F . We will assume for technical simplicity that H1 , . . . , Hn are separable. Let {pk }k (resp. {qk }k ) be a sequence of projections of finite rank on H1 (resp. Hn ) such that pk → I (resp. qk → I ) in the strong operator topology. Let Ψk : K(Hn , H1 ) → K(Hn , H1 ) be the complete contraction given by Ψk (x) = pk xqk . Let ε > 0. Since Φ is completely compact there exists a subspace F ⊆ K(Hn , H1 ) of dimension < ∞ such that dist(Φ (m) (x), Mm (F )) < ε whenever x ∈ Mm (Kh ) has norm at most one. Denote the restriction of Ψk to F by Ψk,F and let ι be the inclusion map ι : F → K(Hn , H1 ). By [6, Corollary 2.2.4], Ψk,F − ι cb Ψk,F − ι . Since F ⊆ K(Hn , H1 ), we have that Ψk,F (x) → x in norm for each x ∈ F . It follows easily that there exists k0 such that Ψk,F − ι cb < ε whenever k k0 . Let x ∈ Mm (Kh ) be of norm at most one. Then there exists y ∈ Mm (F ) such that Φ (m) (x) − y < ε. Note that y Φ (m) (x) − y + Φ (m) (x) ε + Φ cb . Let Φk = Ψk ◦ Φ. If k k0 then (m) Φ − Φ (m) (x) Φ (m) (x) − Ψ (m) (y) + Ψ (m) (y) − y + y − Φ (m) (x) k k k k (m) = Ψk Φ (m) (x) − y + (Ψk,F − ι)(m) (y) + y − Φ (m) (x) 2ε + ε ε + Φ cb . This shows that Φk − Φ cb → 0.
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By Proposition 3.2, it only remains to prove that each Φk lies in γ0 (E). By Proposition 3.3, there exists an element u = A1 A2 · · · An−1 An ∈ B(H1 ) ⊗eh · · · ⊗eh B(Hn ) where A1 : H1∞ → H1 , Ai : Hi∞ → Hi∞ , i = 2, . . . , n − 1 and An : Hn → Hn∞ are bounded operators, such that Φ = γ0 (u). Observe that Φk = γ0 (uk ) where uk = (pk A1 ) A2 · · · An−1 (An qk ). It therefore suffices to show that uk ∈ E for each k. Fix k and let p = pk , q = qk . The operator pA1 : H1∞ → H1 has finite dimensional range and is hence compact. For i = 1, . . . , n, let Qi,r : Hi∞ → Hi∞ be a projection with block matrix whose first r diagonal entries are equal to the identity operator while the rest are zero. Then by compactness, (pA1 )Q1,r → pA1 and Qn,r (An q) → An q in norm as r → ∞. Let B = A2 · · · An−1 , Cr = (pA1 )Q1,r B Qn,r (An q), r ∈ N, and C = (pA1 ) B (An q). Then Cr − C eh Cr − (pA1 )Q1,r B (An q) eh + (pA1 )Q1,r B (An q) − C eh (pA1 )Q1,r B Qn,r (An q) − An q + (pA1 )Q1,r − pA1 B An q . It follows that Cr − C eh → 0 as r → ∞. Our claim will follow if we show that Cr ∈ E. To this end, it suffices to show that if A1 = [a1 , . . . , ar , 0, . . .] and An = [b1 , . . . , br , 0, . . .]t , where ai , bi are operators of finite rank, then A1 B An ∈ E. Let A1 and An be as stated and let B = (Q2,r A2 ) A3 · · · An−2 (An−1 Qn,r ). Then A1 B An = A1 B An+1 belongs to the algebraic tensor product K(H1 ) (B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 )) K(Hn ) and hence to E = K(H1 ) ⊗h (B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 )) ⊗h K(Hn ). 2 Remarks 3.5. (i) It follows from Theorem 3.4 that if Φ : Kh → K(Hn , H1 ) is a mapping of finite rank whose image consists of finite rank operators then there exist finite rank projections p and q on H1 and Hn , respectively, and u ∈ (pK(H1 )) ⊗h (B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 )) ⊗h (K(Hn )q) such that Φ = γ0 (u). (ii) The identity E1 ⊗h (E2 ⊗eh E3 ) = (E1 ⊗h E2 ) ⊗eh E3 does not hold in general; for an example, take E1 = E3 = B(H ) and E2 = C. J (iii) For every Φ ∈ CC(Kh , K(Hn , H1 )) there exist A1 ∈ K(H1J1 , H1 ), Ai ∈ B(HiJi , Hi i−1 ), i = J 2, . . . , n − 1 and An ∈ K(Hn , Hn n−1 ) such that Φ(x1 ⊗ · · · ⊗ xn−1 ) = A1 (x1 ⊗ I )A2 . . . (xn−1 ⊗ I )An , whenever xi ∈ K(Hi+1 , Hi ), i = 1, . . . , n − 1. Indeed, by Proposition 3.4 we have Φ(x1 ⊗ · · · ⊗ xn−1 ) = A1 (x1 ⊗ I )A2 . . . (xn−1 ⊗ I )An for some A1 A2 · · · An ∈ K(H1 ) ⊗h (B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 )) ⊗h K(Hn ). Using an idea of Blecher and Smith [2, Theorem 3.1], we can choose A1 = [tj ]j ∈J1 ∈ MJ1 ,1 (K(H1 )) ⊆ B(H1J1 , H1 ) and An = J [si ]i∈Jn−1 ∈ M1,Jn−1 (K(Hn )) ⊆ B(Hn , Hn n−1 ) such that the sums i si si∗ and j tj∗ tj conF verge uniformly. Then A1 is the norm limit of AF 1 = [tj ]i∈J1 , where F is a finite subset of J1 and tjF = tj if j ∈ F and tjF = 0 otherwise. Therefore A1 ∈ K(H1J1 , H ). Similarly, An ∈ K(Hn , HnJn −1 ).
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In the case n = 2, Theorem 3.4 reduces to the following result which was established by Saar (Satz 6 of [21]) using the fact that every completely compact completely bounded map on K(H1 , H2 ) is a linear combination of completely compact completely positive maps. Corollary 3.6. A completely bounded map Φ : K(H1 , H2 ) → K(H1 , H2 ) is completely compact if and only if there exist an index set J and families {ai }i∈J ⊆ K(H1 ) and {bi }i∈J ⊆ K(H2 ) such that the series i∈J bi bi∗ and i∈J ai∗ ai converge uniformly and Φ(x) =
bi xai ,
x ∈ K(H1 , H2 ).
i∈J
We note in passing that Theorem 3.4 together with a result of Effros and Kishimoto [4] yields the following completely isometric identification: Corollary 3.7. CC(K(H2 , H1 ))∗∗ (K(H1 ) ⊗h K(H2 ))∗∗ CB(B(H2 , H1 )). Saar [21] constructed an example of a compact map Φ : K(H ) → K(H ) which is not completely compact (see Section 7 where we give a detailed account of this construction). We note that a compact completely positive map Φ : K(H ) → K(H ) is automatically completely compact. Indeed, the Stinespring Theorem implies that there exist an index set J and a row operator A = [ai ]i∈J ∈ B(H J , H ) such that Φ(x) = i∈J ai xai∗ , x ∈ K(H ). The second dual Φ ∗∗ : B(H ) → B(H ) of Φ is a compact map given by the same formula. A standard Banach space argument shows that Φ ∗∗ takes values in K(H ), and hence Φ ∗∗ (I ) ∈ K(H ). This means that AA∗ ∈ K(H ) and so A ∈ K(H J , H ) which easily implies that Φ is completely compact. The previous paragraph shows that there exists a compact completely bounded map on K(H ) which cannot be written as a linear combination of compact completely positive maps. We finish this section with a modular version of Theorem 3.4. Given von Neumann algebras Ai ⊆ B(Hi ), i = 1, . . . , n, we let CCA1 ,...,An (Kh , K(Hn , H1 )) denote the space of all completely compact multilinear maps from Kh into K(Hn , H1 ) such that the corresponding multilinear map from K(H2 , H1 ) × · · · × K(Hn , Hn−1 ) into K(Hn , H1 ) is A1 , . . . , An -modular. Corollary 3.8. Let Ai ⊆ B(Hi ), i = 1, . . . , n, be von Neumann algebras. Set K (Ai ) = K(Hi ) ∩ Ai , for i = 1 and i = n. Then γ0 K (A1 ) ⊗h (A2 ⊗eh · · · ⊗eh An−1 ) ⊗h K (An ) = CCA1 ,...,An Kh , K(Hn , H1 ) . Proof. By Theorems 2.1 and 3.4, the image of K (A1 ) ⊗h (A2 ⊗eh · · · ⊗eh An−1 ) ⊗h K (An ) under γ0 is contained in CCA1 ,...,An (Kh , K(Hn , H1 )). For the converse, fix an element Φ ∈ CCA1 ,...,An (Kh , K(Hn , H1 )). By Theorem 3.4, there exists a unique u ∈ K(H1 ) ⊗h (B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 )) ⊗h K(Hn ) such that γ0 (u) = Φ. By Theorem 2.1, u ∈ A1 ⊗eh · · · ⊗eh An . Lemma 2.2 now shows that u ∈ K (A1 ) ⊗h (A2 ⊗eh · · · ⊗eh An−1 ) ⊗h K (An ). 2 4. Complete boundedness of multipliers Our aim in this section is to clarify the relationship between universal operator multipliers and completely bounded maps, extending results of [12]. We begin with an observation which will
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allow us to deal with the cases of even and odd numbers of variables in the same manner. We use the notation established in Section 2. Proposition 4.1. Let A1 , . . . , An be C ∗ -algebras and ϕ ∈ M(A1 , . . . , An ). Let πi be a representation of Ai on a Hilbert space Hi , i = 1, . . . , n, and π = π1 ⊗ · · · ⊗ πn . The map Sπ(ϕ) takes values in K(H1 , Hn ) if n is odd, and in K(H1d , Hn ) if n is even. Proof. For even n, this is immediate as observed in [12]. Let n be odd. Assume without loss of generality that Ai = B(Hi ) and πi is the identity representation. We call an element ζ ∈ Γ (H1 , . . . , Hn ) thoroughly elementary if d ⊗ ξ2,3 ⊗ · · · ⊗ ξn−1,n ζ = η1,2 d d d where all ηj,j +1 = ηj ⊗ ηj +1 and ξj −1,j = ξj −1 ⊗ ξj are elementary tensors. The linear span of the thoroughly elementary tensors is dense in the completion of Γ (H1 , . . . , Hn ) in · 2,∧ . Moreover, the linear span of the elementary tensors ϕ = ϕ1 ⊗ · · · ⊗ ϕn is dense in B(H1 ) ⊗ · · · ⊗ B(Hn ). By (7) and since Sϕ (ζ ) is linear in both ϕ and ζ , it suffices to show that Sϕ (ζ ) is compact when ϕ is an elementary tensor and ζ is a thoroughly elementary tensor. However, in this case Sϕ (ζ ) has rank at most 1, since for every ξ1 ∈ H1 ,
Sϕ (ζ )ξ1 = ϕn θ (ξn−1,n ) . . . ϕ2d θ
d η1,2 ϕ1 ξ1 =
n−1
(ϕj ξj , ηj ) ϕn ξn .
2
j =1
We now establish some notation. Let A1 , . . . , An be C ∗ -algebras and ϕ ∈ A1 ⊗ · · · ⊗ An . Assume that n is even and let π1 , . . . , πn be representations of A1 , . . . , An on H1 , . . . , Hn , respectively. Set π = π1 ⊗ · · · ⊗ πn . Using the natural identifications, we consider the map Sπ(ϕ) : Γ (H1 , . . . , Hn ) → H1 ⊗ Hn as a map (denoted in the same way) d Sπ(ϕ) : C2 H1d , H2 · · · C2 Hn−1 , Hn → C2 H1d , Hn . We let d , Hn · · · C2 H1d , H2 → C2 H1d , Hn Φπ(ϕ) : C2 Hn−1 be the map given on elementary tensors by Φπ(ϕ) (Tn−1 ⊗ · · · ⊗ T1 ) = Sπ(ϕ) (T1 ⊗ · · · ⊗ Tn−1 ). Note that if ϕ ∈ M(A1 , . . . , An ) then Φπ(ϕ) is bounded when the domain is equipped with the Haagerup norm and the range with the operator norm. In this case, Φπ(ϕ) has a unique extension (which will be denoted in the same way) d , Hn ⊗h · · · ⊗h K H1d , H2 , · h → K H1d , Hn , · op . Φπ(ϕ) : K Hn−1 If n is odd then the map Φπ(ϕ) is defined in a similar way. The map Φπ(ϕ) will be used extensively hereafter.
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The main result of this section is Theorem 4.3, where we explain how the complete boundedness of the mappings Φπ(ϕ) relates to the property of ϕ being a multiplier. We will need the following lemma. Lemma 4.2. Let Ai ⊆ B(Hi ) be a C ∗ -algebra, i = 1, . . . , n, and let k ∈ N. Let ϕ ∈ A1 ⊗ · · · ⊗ An and write ψ = (id(k) ⊗· · ·⊗id(k) )(ϕ). Suppose that n is even. If Ti ∈ Mk (C2 (Hid , Hi+1 )) for odd i d )) for even i then and Ti ∈ Mk (C2 (Hi , Hi+1 Φϕ(k) (Tn−1 · · · T1 ) = Φψ (Tn−1 ⊗ · · · ⊗ T1 ), (k)
where we identify the operator spaces Mk (C2 (Hid , Hi+1 )) and C2 ((Hid )(k) , Hi+1 ) for odd i, and (k)
d )) and C (H , (H d )(k) ) for even i. A similar statement holds for odd n. Mk (C2 (Hi , Hi+1 2 i i+1
Proof. To simplify notation, we give the proof for n = 2; the proof of the general case is similar. If ϕ = a1 ⊗ a2 is an elementary tensor then Φϕ (T ) = a2 T a1d for T ∈ C2 (H1d , H2 ) and it is easily checked that the statement holds. By linearity, it holds for each ϕ ∈ A1 A2 . Suppose now that ϕ ∈ A1 ⊗ A2 is arbitrary. Let {ϕm } ⊆ A1 A2 be a sequence converging in the operator norm to ϕ and ψm = (id(k) ⊗ id(k) )(ϕm ). By (7), Φϕm (T ) → Φϕ (T ) in the operator norm, for all T ∈ (k) (k) C2 (H1d , H2 ). This implies that if S ∈ Mk (C2 (H1d , H2 )), then Φϕm (S) → Φϕ (S) in the operator norm of Mk (C2 (H1d , H2 )). Since ψm → ψ in the operator norm, we conclude that Φψm (S) → (k) (k) Φψ (S) in the operator norm of C2 ((H1d )(k) , H2 ). It follows that Φψ (S) = Φϕ (S). 2 Theorem 4.3. Let A1 , . . . , An be C ∗ -algebras and ϕ ∈ A1 ⊗ · · · ⊗ An . The following are equivalent: (i) ϕ ∈ M(A1 , . . . , An ); (ii) if πi is a representation of Ai , i = 1, . . . , n, and π = π1 ⊗ · · · ⊗ πn then the map Φπ(ϕ) is completely bounded; (iii) there exist faithful representations πi of Ai , i = 1, . . . , n, such that if π = π1 ⊗ · · · ⊗ πn then the map Φπ(ϕ) is completely bounded. Moreover, if the above conditions hold and π is as in (iii) then ϕ m = Φπ(ϕ) cb . Proof. For technical simplicity we only consider the case n = 3. (i) ⇒ (ii) Let ϕ ∈ M(A1 , A2 , A3 ) and πi : Ai → B(Hi ) be a representation, i = 1, 2, 3. Then π(ϕ) ∈ M(π1 (A1 ), π(A2 ), π3 (A3 )); thus, it suffices to assume that Ai ⊆ B(Hi ) are concrete C ∗ -algebras and that πi is the identity representation, i = 1, 2, 3. Fix k ∈ N and let ψ = (id(k) ⊗ id(k) ⊗ id(k) )(ϕ). Since ϕ ∈ M(A1 , A2 , A3 ), the map Φψ : K H2d(k) , H3(k) K H1(k) , H2d(k) → K H1(k) , H3(k) is bounded with norm not exceeding ϕ m . By Lemma 4.2, Φϕ(k) ϕ m . Since this inequality holds for every k ∈ N, the map Φϕ is completely bounded. (ii) ⇒ (iii) is trivial.
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(iii) ⇒ (i) We may assume that Ai ⊆ B(Hi ) and that πi is the identity representation, i = 1, 2, 3. Let λ be a cardinal number, ρi = id(λ) be the ampliation of the identity representation of multiplicity λ, ψ = (ρ1 ⊗ ρ2 ⊗ ρ3 )(ϕ), and H˜ i = Hiλ , i = 1, 2, 3. Fix ε > 0 and ζ ∈ Γ (H˜ 1 , H˜ 2 , H˜ 3 ). Let T˜ = T˜2 T˜1 ∈ C2 H˜ 2d , H˜ 3 C2 H˜ 1 , H˜ 2d be the element canonically corresponding to ζ . Then there exist k ∈ N and canonical projections Pi from H˜ i onto the direct sum of k copies of Hi such that if T0 = (P3 T˜2 (P2d ⊗ I )) (k) (k) (k) ((P2d ⊗ I )T˜1 P1 ) and if ζ0 is the element of Γ (H1 , H2 , H3 ) corresponding to T0 then ζ − ζ0 2,∧ ε. Set ψ0 = (id(k) ⊗ id(k) ⊗ id(k) )(ϕ). Arguing as in Lemma 4.2, we see that Φψ0 (T0 ) op = Φψ (T0 ) op . Using (7) and Lemma 4.2 we obtain Sψ (ζ )
op
Sψ (ζ − ζ0 ) op + Sψ (ζ0 ) op = Sψ (ζ − ζ0 ) op + Φψ (T0 ) op ψ ζ − ζ0 2,∧ + Φψ (T0 ) ε ϕ + Φ (k) (T0 ) 0
op
ϕ
op
ε ϕ + Φϕ cb T0 h ε ϕ + Φϕ cb P3 T˜2 P2d ⊗ I op P2d ⊗ I T˜1 P1 op ε ϕ + Φϕ cb T˜2 op T˜1 op . It follows that ϕ id(λ) ,id(λ) ,id(λ) Φϕ cb . Now let ρ1 , ρ2 , ρ3 be arbitrary representations of A1 , A2 , A3 , respectively. Then there exists a cardinal number λ such that each of the representations ρi is approximately subordinate to the representation id(λ) (see [26] and [10, Theorem 5.1]). By Theorem 5.1 of [12], ϕ ρ1 ,ρ2 ,ρ3 ϕ id(λ) ,id(λ) ,id(λ) ; now the previous paragraph implies that ϕ ρ1 ,ρ2 ,ρ3 Φϕ cb . It follows that ϕ ∈ M(A1 , A2 , A3 ) and ϕ m Φϕ cb . As the reversed inequality was already established, we conclude that ϕ m = Φϕ cb . 2 5. The symbol of a universal multiplier Our aim in this section is to generalise the natural correspondence between a function ϕ ∈ ∞ ⊗eh ∞ and the Schur multiplier Sϕ on B(2 (N)) given by Sϕ ((aij )) = (ϕ(i, j )aij ). To each universal operator multiplier we will associate an element of an extended Haagerup tensor product which we call its symbol. This will be used in the subsequent sections to identify certain classes of operator multipliers. Recall that if A is a C ∗ -algebra, its opposite C ∗ -algebra Ao is defined to be the C ∗ -algebra whose underlying set, norm, involution and linear structure coincide with those of A and whose multiplication · is given by a · b = ba. If a ∈ A we denote by a o the element of Ao corresponding to a. If π : A → B(H ) is a representation of A then the map π d : a o → π(a)d from Ao into B(H d ) is a representation of Ao . Clearly, π is faithful if and only if π d is faithful. If πi : Ai → B(Hi ) are faithful representations, i = 1, . . . , n (n even), then by [5, Lemma 5.4] there exists a d ⊗ · · · ⊗ π d from A ⊗ Ao o complete isometry πn ⊗eh πn−1 eh eh 1 n eh n−1 ⊗eh · · · ⊗eh A1 into B(Hn ) ⊗eh o d ) ⊗ · · · ⊗ B(H d ) which sends a ⊗ a o d B(Hn−1 eh eh n 1 n−1 ⊗ · · · ⊗ a1 to πn (an ) ⊗ πn−1 (an−1 ) ⊗ · · · d ⊗ π1 (a1 ) .
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Henceforth, we will consistently write π = π1 ⊗ · · · ⊗ πn and d πn ⊗eh πn−1 ⊗eh · · · ⊗eh π2 ⊗eh π1d if n is even, π = d πn ⊗eh πn−1 ⊗eh · · · ⊗eh π2d ⊗eh π1 if n is odd. Let n ∈ N, A1 , . . . , An be C ∗ -algebras, πi be a representation of Ai , i = 1, . . . , n, and ϕ ∈ M(A1 , . . . , An ). Assume that n is even. By Theorem 4.3, the map d , Hn ⊗h · · · ⊗h K H1d , H2 → K H1d , Hn Φπ(ϕ) : K Hn−1 is completely bounded. By Proposition 3.3, there exists a unique element uπϕ ∈ B(Hn ) ⊗eh d ) ⊗ · · · ⊗ B(H d ) such that γ (uπ ) = Φ B(Hn−1 eh eh 0 ϕ π(ϕ) . For example, if each Ai is a concrete 1 ∗ C -algebra and ai ∈ Ai , i = 1, . . . , n, then d d uid a1 ⊗a2 ⊗···⊗an−1 ⊗an = an ⊗ an−1 ⊗ · · · ⊗ a2 ⊗ a1 .
If n is odd then we define uπϕ similarly. The main result of this section is the following. Theorem 5.1. Let A1 , . . . , An be C ∗ -algebras and ϕ ∈ M(A1 , . . . , An ). There exists a unique element An ⊗eh Aon−1 ⊗eh · · · ⊗eh A2 ⊗eh Ao1 if n is even, uϕ ∈ An ⊗eh Aon−1 ⊗eh · · · ⊗eh Ao2 ⊗eh A1 if n is odd with the property that if πi is a representation of Ai for i = 1, . . . , n then uπϕ = π (uϕ ).
(8)
The map ϕ → uϕ is linear and if ai ∈ Ai , i = 1, . . . , n, then ua1 ⊗···⊗an =
o ⊗ · · · ⊗ a2 ⊗ a1o an ⊗ an−1 o an ⊗ an−1 ⊗ · · · ⊗ a2o ⊗ a1
if n is even, if n is odd.
Moreover, ϕ m = uϕ eh . Definition 5.2. The element uϕ defined in Theorem 5.1 will be called the symbol of the universal multiplier ϕ. In order to prove Theorem 5.1 we have to establish a number of auxiliary results. If ω ∈ B(H )∗ we let ω˜ ∈ B(H d )∗ be the functional given by ω(a ˜ d ) = ω(a). Note that if ω = ωξ,η is the vector functional a → (aξ, η) then ω˜ = ωηd ,ξ d . Lemma 5.3. Let Ai ⊆ B(Hi ) be a C ∗ -algebra, ξi , ηi ∈ Hi and ωi = ωξi ,ηi , i = 1, . . . , n. Suppose that ϕ ∈ M(A1 , . . . , An ). Then uid ˜ n−1 ⊗ · · · ⊗ ω˜ 1 if n is even, ϕ , ωn ⊗ ω ϕ(ξ1 ⊗ · · · ⊗ ξn ), η1 ⊗ · · · ⊗ ηn = (9) id uϕ , ωn ⊗ ω˜ n−1 ⊗ · · · ⊗ ω1 if n is odd.
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Proof. We only consider the case n is even; the proof for odd n is similar. Suppose that ϕ is an d d elementary tensor, say ϕ = a1 ⊗ · · · ⊗ an . Then uid ϕ = an ⊗ an−1 ⊗ · · · ⊗ a1 and thus n
˜ n−1 ⊗ · · · ⊗ ω˜ 1 . ϕ(ξ1 ⊗ · · · ⊗ ξn ), η1 ⊗ · · · ⊗ ηn = (ai ξi , ηi ) = uid ϕ , ωn ⊗ ω i=1
By linearity, (9) holds for each ϕ ∈ A1 · · · An . Now let ϕ be an arbitrary element of M(A1 , . . . , An ). By Theorem 2.3, there exists a net ν id ν {ϕν } ⊆ A1 · · · An and representations uid ϕ = An · · · A1 and uϕν = An · · · A1 , where d ν Ai are finite matrices with entries in Ai if i is even and in Ai if i is odd, such that ϕν → ϕ semi-weakly, Aνi → Ai strongly and all norms Ai , Aνi are bounded by a constant depending only on n. As in (2), we have
uid ˜ n−1 ⊗ · · · ⊗ ω˜ 1 = An , ωn An−1 , ω˜ n−1 . . . A1 , ω˜ 1 . ϕ , ωn ⊗ ω
(10)
Moreover, all norms Aνi , ωi (for even i) and Aνi , ω˜ i (for odd i) are bounded by a constant depending only on n, and the strong convergence of Aνi to Ai implies that Aνi , ωi converges strongly to Ai , ωi . Indeed, it is easy to check that if ξ, η ∈ H , A ∈ MI (B(H )) = B(H ⊗ 2 (I )) and ζ ∈ 2 (I ) for some index set I then A, ωξ,η ζ 2 = A(ξ ⊗ ζ ), η ⊗ A, ωξ,η ζ . This implies that Ai − Aνi , ωi η C (Ai − Aνi )(ξi ⊗ η) for some constant C > 0, and the strong convergence follows. Since operator multiplication is jointly strongly continuous on bounded sets, it now follows from (10) that
uid ˜ n−1 ⊗ · · · ⊗ ω˜ 1 → uid ˜ n−1 ⊗ · · · ⊗ ω˜ 1 . ϕν , ωn ⊗ ω ϕ , ωn ⊗ ω
On the other hand, since ϕν → ϕ semi-weakly, ϕν (ξ1 ⊗ · · · ⊗ ξn ), η1 ⊗ · · · ⊗ ηn → ϕ(ξ1 ⊗ · · · ⊗ ξn ), η1 ⊗ · · · ⊗ ηn . The proof is complete.
2
Lemma 5.4. Let Hi be a Hilbert space and Ei ⊆ B(Hi ) be an operator space, i = 1, . . . , n. Suppose that X and Y are closed subspaces of E1 and En , respectively and let u, v ∈ E1 ⊗eh · · · ⊗eh En . If Rω (u) ∈ X
and Lω (v) ∈ Y
whenever ω = ω2 ⊗ · · · ⊗ ωn and ω = ω1 ⊗ · · · ⊗ ωn−1 where every ωi , ωi ∈ B(Hi )∗ is a vector functional, then
u ∈ X ⊗eh E2 ⊗eh · · · ⊗eh En
and v ∈ E1 ⊗eh · · · ⊗eh En−1 ⊗eh Y.
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Proof. Let Fi be the span of the vector functionals on B(Hi ). By linearity, Rω (u) ∈ X for each ω ∈ F2 · · · Fn . Now suppose that ω ∈ B(H2 ) ⊗eh · · · ⊗eh B(Hn ) ∗ = C1 (H2 ) ⊗h · · · ⊗h C1 (Hn ). There exists a sequence (ωm ) ⊆ F2 · · · Fn such that ωm → ω in norm. Hence Rω (u) − Rω (u) m B(H1 ) ω − ωm u eh → 0, whence Rω (u) = limm Rωm (u) ∈ X . Spronk’s formula (5) now implies that u ∈ X ⊗eh E2 ⊗eh · · · ⊗eh En . The assertion concerning v has a similar proof. 2 We will use slice maps defined on the minimal tensor product of several C ∗ -algebras as follows. Assume that Ai ⊆ B(Hi ) and ωi ∈ B(Hi )∗ , i = 1, . . . , n, and let ϕ ∈ A1 ⊗ · · · ⊗ An . If 1 i1 < · · · < ik n and {1 < 2 < · · · < n−k } is the complement of {i1 , . . . , ik } in {1, . . . , n}, let Λωi1 ,...,ωik : A1 ⊗ · · · ⊗ An → A1 ⊗ · · · ⊗ An−k be the unique norm continuous linear mapping given on elementary tensors by Λωi1 ,...,ωik (a1 ⊗ · · · ⊗ an ) = ωi1 (ai1 ) . . . ωik (aik ) a1 ⊗ · · · ⊗ an−k . Proposition 5.5. Let Ai ⊆ B(Hi ), i = 1, . . . , n, be C ∗ -algebras and let ϕ ∈ M(A1 , . . . , An ). Then An ⊗eh Adn−1 ⊗eh · · · ⊗eh A2 ⊗eh Ad1 if n is even, id uϕ ∈ An ⊗eh Adn−1 ⊗eh · · · ⊗eh Ad2 ⊗eh A1 if n is odd. d Proof. We only consider the case n = 3. Let u = uid ϕ ; by definition, u ∈ B(H3 ) ⊗eh B(H2 ) ⊗eh B(H1 ). Let ξi , ηi ∈ Hi and ωi = ωξi ,ηi , i = 1, 2, 3. Then by (4) and Lemma 5.3,
Rω˜ 2 ⊗ω1 (u)ξ3 , η3 = Rω˜ 2 ⊗ω1 (u), ω3 = u, ω3 ⊗ ω˜ 2 ⊗ ω1 = ϕ(ξ1 ⊗ ξ2 ⊗ ξ3 ), η1 ⊗ η2 ⊗ η3 = Λω1 ,ω2 (ϕ)ξ3 , η3 . Thus Rω˜ 2 ⊗ω1 (u) = Λω1 ,ω2 (ϕ) ∈ A3 . Lemma 5.4 now implies that u ∈ A3 ⊗eh B(H2d ) ⊗eh B(H1 ). Let w = Rω1 (u). By the previous paragraph, w ∈ A3 ⊗eh B(H2d ). By (4) and Lemma 5.3,
Lω3 (w)η2d , ξ2d = Lω3 (w), ω˜ 2 = Rω1 (u), ω3 ⊗ ω˜ 2 = u, ω3 ⊗ ω˜ 2 ⊗ ω1 = Λω1 ,ω3 (ϕ)ξ2 , η2 = Λω1 ,ω3 (ϕ)d η2d , ξ2d .
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Hence Lω3 (w) = Λω1 ,ω3 (ϕ)d ∈ Ad2 and, by Lemma 5.4, w ∈ A3 ⊗eh Ad2 . Applying this lemma again shows that u ∈ A3 ⊗eh Ad2 ⊗eh B(H1 ). Continuing in this fashion we see that u ∈ A3 ⊗eh Ad2 ⊗eh A1 . 2 Lemma 5.6. Let A1 , . . . , An be C ∗ -algebras and let ρi : Ai → B(Ki ),
θi : ρi (Ai ) → B(Hi )
be representations, i = 1, . . . , n. Suppose that (κ)
(i) for any cardinal number κ, the representations θi : ρi (Ai ) → B(Hiκ ) are strongly continuous, and (ii) whenever ϕ ∈ M(A1 , . . . , An ) and {ϕν } is a net in A1 · · · An such that ρ(ϕν ) → ρ(ϕ) semi-weakly and supν ϕν m < ∞ then Φθ◦ρ(ϕν ) → Φθ◦ρ(ϕ) pointwise weakly. θ◦ρ
Then uϕ
= θ (uϕ ), for each ϕ ∈ M(A1 , . . . , An ). ρ
Proof. We suppose that n is even, the proof for odd n being similar. If ϕ = a1 ⊗ · · · ⊗ an is an ρ o ⊗ · · · ⊗ a1o ), so elementary tensor, then uϕ = ρ (an ⊗ an−1 o uϕθ◦ρ = (θ ◦ ρ) an ⊗ an−1 ⊗ · · · ⊗ a1o = θ uρϕ . By linearity, the claim also holds for ϕ ∈ A1 · · · An . If ϕ ∈ M(A1 , . . . , An ) is arbitrary then ρ(ϕ) ∈ M(ρ1 (A1 ), . . . , ρn (An )) and by Theorem 2.3 and Proposition 5.5, there exist a net {ϕν } ⊆ A1 · · · An such that ρ(ϕν ) → ρ(ϕ) semi-weakly, ρ a representation uϕ = An · · · A1 , where Ai ∈ Mκ (ρi (Ai )) ⊆ B(Kiκ ) if i is even and Ai ∈ Mκ (ρid (Aoi )) ⊆ B(Kiκ )d if i is odd (κ being a suitable index set), whose operator matrix entries ρ belong to ρi (Ai ) if i is even and to ρid (Aoi ) if i is odd, and representations uϕν = Aνn · · · Aν1 ν where the Ai are finite matrices with operator entries in ρi (Ai ) if i is even and ρid (Aoi ) if i is odd such that Aνi → Ai strongly and all norms Aνi , Ai are bounded. ρ ρ Now θ (uϕ ) = A˜ n · · · A˜ 1 and θ (uϕν ) = A˜ νn · · · A˜ ν1 where A˜ i and A˜ νi are the images (κ) of Ai and Aνi under θi or (θid )(κ) according to whether i is even or odd. By assumption (i), γ0 θ uρϕν (Tn−1 ⊗ · · · ⊗ T1 ) → γ0 θ uρϕ (Tn−1 ⊗ · · · ⊗ T1 ) (11) d , H ),. . .,T ∈ C (H d , H ). On the other hand, assumption (ii) and weakly for all Tn−1 ∈ C2 (Hn−1 n 1 2 2 1 the first paragraph of the proof show that = Φθ◦ρ(ϕν ) → Φθ◦ρ(ϕ) = γ0 uϕθ◦ρ γ0 θ uρϕν = γ0 uϕθ◦ρ ν
pointwise weakly. Using (11) we conclude that γ0 (uϕ ) = γ0 (θ (uϕ )); since γ0 is injective we θ◦ρ ρ have that uϕ = θ (uϕ ). 2 θ◦ρ
ρ
Proof of Theorem 5.1. We will only consider the case n is even. Let ρi : Ai → B(Ki ) be the d ⊗ · · · ⊗ ρd. universal representation of Ai , i = 1, . . . , n. Set ρ = ρ1 ⊗ · · · ⊗ ρn and ρ = ρn ⊗ ρn−1 1 ρ ρ −1 By Proposition 5.5, uϕ lies in the image of ρ ; we define uϕ = (ρ ) (uϕ ).
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Let κ be a nonzero cardinal number and let σi = ρi . If θi = idρi (Ai ) = σi ◦ ρi−1 then it follows from the proof of Proposition 6.2 of [12] that the hypotheses of Lemma 5.6 are satisfied, so writing θ = θ1 ⊗ · · · ⊗ θn , we have uσϕ = uϕθ◦ρ = θ uρϕ = (θ ◦ ρ )(uϕ ) = σ (uϕ ). (κ)
(κ)
Now let πi be an arbitrary representation of Ai . It is well known (see e.g. [25]) that πi is unitarily (κ) equivalent to a subrepresentation of σi = ρi for some κ. Hence there exist unitary operators vi , i = 1, . . . , n (acting between appropriate Hilbert spaces) and subspaces Hi of Kiκ , such that if τi (x) = vi xvi∗ |Hi then πi = τi ◦ σi . Examining the proof of Proposition 6.2 of [12], we see that τ = τ1 ⊗ · · · ⊗ τn satisfies the hypotheses of Lemma 5.6, so uπϕ = uτϕ◦σ = τ uσϕ = (τ ◦ σ ) (uϕ ) = π (uϕ ). The uniqueness of uϕ follows from the injectivity of γ0 . The linearity of the map ϕ → uϕ and its values on elementary tensors are straightforward. The fact that ϕ m = uϕ eh follows from Proposition 3.3 and Theorem 4.3. 2 Remarks. (i) Let Ai ⊆ B(Hi ), i = 1, . . . , n be concrete C ∗ -algebras of operators. Taking πi to be the identity representation for i = 1, . . . , n and writing id = π1 ⊗ · · · ⊗ πn gives uϕ = uid ϕ if we identify Aoi with Adi . (ii) Theorem 5.1 implies that if Ai , i = 1, . . . , n, are concrete C ∗ -algebras then the entries of the block operator matrices Ai appearing in the representation of ϕ in Theorem 2.3 can be chosen from Ai , i = 1, . . . , n. 6. Completely compact multipliers In this section we introduce the class of completely compact multipliers and characterise them within the class of all universal multipliers using the notion of the symbol introduced in Section 5. We will need the following lemma. Lemma 6.1. Let Ai ⊆ B(Hi ) be a C ∗ -algebra, i = 1, . . . , n, a ∈ A1 , b ∈ An and ϕ ∈ M(A1 , . . . , An ). Let ψ ∈ A1 ⊗ · · · ⊗ An be given by (a ⊗ I ⊗ · · · ⊗ I ⊗ b)ϕ if n is even, ψ= (I ⊗ · · · ⊗ I ⊗ b)ϕ(a ⊗ I ⊗ · · · ⊗ I ) if n is odd. Then ψ ∈ M(A1 , . . . , An ) and Φψ (x) =
bΦϕ (x)a d bΦϕ (x)a
if n is even, if n is odd.
(12)
Proof. For technical simplicity, we will only consider the case n = 2. Let ai ∈ Ai , i = 1, 2, and ϕ = a1 ⊗ a2 . In this case ψ = (aa1 ) ⊗ (ba2 ) so Φψ (T ) = ba2 T (aa1 )d = ba2 T a1d a d = bΦϕ (T )a d . By linearity, (12) holds whenever ϕ ∈ A1 A2 .
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Assume that ϕ ∈ M(A1 , A2 ) is arbitrary. Fix an operator T ∈ C2 (H1d , H2 ). By Theorem 2.3, there exists a net {ϕν } ⊆ A1 A2 such that ϕν → ϕ semi-weakly, supν ϕν m < ∞ and Φϕν (T ) → Φϕ (T ) weakly. Let ψν = (a ⊗ b)ϕν ; then ψν → ψ semi-weakly. Clearly, ψν ∈ A1 A2 ; in particular ψν ∈ M(A1 , A2 ). By the previous paragraph, Φψν (·) = bΦϕν (·)a d and hence Φψν (T ) → bΦϕ (T )a d weakly. If ϕν = B1ν B2ν then ψν = (aB1ν ) ((b ⊗ I )B2ν ). It follows from Theorem 2.3 that ψ ∈ M(A1 , A2 ) and that Φψν (T ) → Φψ (T ) weakly. Thus Φψ (T ) = bΦϕ (T )a d . 2 Given faithful representations π1 , . . . , πn of the C ∗ -algebras A1 , . . . , An , respectively, we define π (A1 , . . . , An ) = ϕ ∈ M(A1 , . . . , An ): Φπ(ϕ) is completely compact Mcc Mffπ (A1 , . . . , An ) = ϕ ∈ M(A1 , . . . , An ): the range of Φπ(ϕ) is a finite dimensional space of finite-rank operators . Theorem 6.2. Let Ai ⊆ B(Hi ) be a C ∗ -algebra, i = 1, . . . , n, and ϕ ∈ M(A1 , . . . , An ). The following are equivalent: id (A , . . . , A ); (i) ϕ ∈ Mcc 1 n (K(Hn ) ∩ An ) ⊗h (Adn−1 ⊗eh · · · ⊗eh A2 ) ⊗h (K(H1d ) ∩ Ad1 ) (ii) uid ∈ ϕ (K(Hn ) ∩ An ) ⊗h (Adn−1 ⊗eh · · · ⊗eh Ad2 ) ⊗h (K(H1 ) ∩ A1 )
if n is even, if n is odd;
(iii) there exists a net {ϕα } ⊆ Mffid (A1 , . . . , An ) such that ϕα − ϕ m → 0. Proof. We will only consider the case n is even. (i) ⇒ (ii) Theorem 3.4 implies that d d uid ϕ ∈ K(Hn ) ⊗h B Hn−1 ⊗eh · · · ⊗eh B(H2 ) ⊗h K H1 while, by Proposition 5.5, d d uid ϕ ∈ An ⊗eh An−1 ⊗eh · · · ⊗eh A2 ⊗eh A1 .
The conclusion now follows from Lemma 2.2. (ii) ⇒ (i) By Theorem 3.4, Φϕ = γ0 (uid ϕ ) is completely compact. (ii) ⇒ (iii) Let p ∈ B(H1 ) (resp. q ∈ B(Hn )) be the projection onto the span of all ranges of operators in K(H1 ) ∩ A1 (resp. K(Hn ) ∩ An ), and let {pα } ⊆ K(H1 ) ∩ A1 (resp. {qα } ⊆ K(Hn ) ∩ An ) be a net of finite rank projections which tends strongly to p (resp. q). It is easy to see that Φϕ (Tn−1 ⊗ · · · ⊗ T1 ) = qΦϕ (Tn−1 ⊗ · · · ⊗ T1 )p d , for all T1 ∈ K(H1d , H2 ), . . . , Tn−1 ∈ d , H ). Let ϕ = (p ⊗ I ⊗ · · · ⊗ I ⊗ q )ϕ. By Lemma 6.1, ϕ ∈ M(A , . . . , A ) and K(Hn−1 n α α α α 1 n Φϕα (·) = qα Φϕ (·)pαd ; hence ϕα ∈ Mffid (A1 , . . . , An ). We have already seen that Φϕ is completely compact, and it follows from the proof of Theorem 3.4 that Φϕα → Φϕ in the cb norm. By Theorem 4.3, ϕ − ϕα m → 0. (iii) ⇒ (i) is immediate from Proposition 3.2 and Theorem 4.3 and the fact that finite rank maps are completely compact. 2
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Now consider the sets Mcc (A1 , . . . , An ) =
π Mcc (A1 , . . . , An ),
π
Mff (A1 , . . . , An ) =
Mffπ (A1 , . . . , An )
π
where the unions are taken over all π = π1 ⊗ · · · ⊗ πn , each πi being a faithful representation of Ai . We refer to the first of these as the set of completely compact multipliers. Lemma 6.3. If ρi is the reduced atomic representation of Ai , i = 1, . . . , n, and ρ = ρ1 ⊗ · · · ⊗ ρn ρ then Mff (A1 , . . . , An ) = Mff (A1 , . . . , An ). Proof. Again, we give the proof for the even case only. We must show that Mffπ (A1 , . . . , An ) ⊆ ρ Mff (A1 , . . . , An ) whenever π = π1 ⊗ · · · ⊗ πn where each πi is a faithful representation of Ai . Without loss of generality, we may assume that each πi is the identity representation of Ai ⊆ B(Hi ). Let ϕ ∈ Mffπ (A1 , . . . , An ) so that the range of Φϕ is finite dimensional and consists of finite rank operators. By Remark 3.5 (i) there exist finite rank projections p and q on H1d and Hn , respectively, such that uid ϕ lies in the intersection of d ⊗eh · · · ⊗eh B(H2 ) ⊗h K H1d p qK(Hn ) ⊗h B Hn−1 and An ⊗eh · · · ⊗eh Ad1 . By Lemma 2.2, uid ϕ lies in d ⊗eh · · · ⊗eh B(H2 ) ⊗h K H1d p ∩ Ad1 . qK(Hn ) ∩ An ⊗h B Hn−1 id Hence there exists a representation uid ϕ = An · · · A1 of uϕ such that An = qAn and A1 = A1 p. Suppose that An = [b1 , b2 , . . .], where bj∈ An for each j , and let qj be the orthogsee that {Qm } is an increasing onal projection onto the range of bj . Setting Qm = m j =1 qj we sequence of projections in An dominated by q. It follows that ∞ m=1 Qm ∈ An . We may thus assume that q ∈ An . Similarly, we may assume that p ∈ Ad1 . Now ρ (uϕ ) = ρn (q)ρn (An ) · · · ρ1 (A1 )ρ1 (p) . ρ
By [29], ρn (q) and ρ1 (p) have finite rank. By Lemma 6.1, ϕ ∈ Mff (A1 , . . . , An ).
2
We are now ready to prove the main result of this section. Theorem 6.4. Let A1 , . . . , An be C ∗ -algebras and ϕ ∈ M(A1 , . . . , An ). The following are equivalent: (i) ϕ ∈ Mcc (A1 , . . . , An ); K(An ) ⊗h (Aon−1 ⊗eh · · · ⊗eh A2 ) ⊗h K(Ao1 ) (ii) uϕ ∈ K(An ) ⊗h (Aon−1 ⊗eh · · · ⊗eh Ao2 ) ⊗h K(A1 )
if n is even, if n is odd;
(iii) there exists a net {ϕα } ⊆ Mff (A1 , . . . , An ) such that ϕα − ϕ m → 0.
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Proof. We will only consider the case n is even. π (A , . . . , A ); after identifying A (i) ⇒ (ii) Choose π = π1 ⊗ · · · ⊗ πn such that ϕ ∈ Mcc 1 n i with its image under πi , we may assume that each πi is the identity representation of a concrete C ∗ -algebra Ai ⊆ B(Hi ). By Theorem 6.2, uid ϕ lies in K(Hn ) ∩ An ⊗h Aon−1 ⊗eh · · · ⊗eh A2 ⊗h K H1d ∩ Ao1 . The conclusion follows from the fact that K(Hi ) ∩ Ai ⊆ K(Ai ) for i = 1, n. (ii) ⇒ (i) Let ρi be the reduced atomic representation Ai → B(Hi ) for i = 1, . . . , n. Since ρ ρ is an isometry, uϕ = ρ (uϕ ) lies in d o ρn K(An ) ⊗h ρn−1 An−1 ⊗eh · · · ⊗eh ρ2 (A2 ) ⊗h ρ1d K Ao1 . ρ
By Theorem 7.5 of [28], K(Hi ) ∩ ρi (Ai ) = ρi (K(Ai )). By Theorem 6.2, ϕ ∈ Mcc (A1 , . . . , An ). (i) ⇒ (iii) is immediate from Theorem 6.2. (iii) ⇒ (i) Suppose that {ϕα } ⊆ Mff (A1 , . . . , An ) is a net such that ϕα − ϕ m → 0. By ρ Lemma 6.3, {ϕα } ⊆ Mff (A1 , . . . , An ), where ρ is the tensor product of the reduced atomic repρ resentations of A1 , . . . , An . By Theorem 6.2, ϕ ∈ Mcc (A1 , . . . , An ) ⊆ Mcc (A1 , . . . , An ). 2 In the next theorem we show that in the case n = 2 one more equivalent condition can be added to those of Theorem 6.4. Theorem 6.5. Let A and B be C ∗ -algebras and ϕ ∈ M(A, B). The following are equivalent: (i) ϕ ∈ Mcc (A, B); (ii) there exists a sequence {ϕk }∞ k=1 ⊆ K(A) K(B) such that ϕk − ϕ m → 0 as k → ∞. o ⊗h K(Ao ); thus uϕ = ∞ Proof. (i) ⇒ (ii) By Theorem 6.4, uϕ ∈ K(B) i=1 bi ⊗ ai where ∞ ∞ o o∗ o ∗ o ai ∈ K(A ), bi ∈ K(B), i ∈ N, and the series i=1 bi bi and i=1 ai ai converge in norm. Let ϕk = ki=1 ai ⊗ bi ∈ A B. By Theorem 5.1, uϕk = ki=1 bi ⊗ aio and ϕ − ϕk m = uϕ − uϕk eh → 0 as k → ∞. (ii) ⇒ (i) Assume that A and B are represented concretely. It is clear that ϕk ∈ Mcc (A, B). By Theorem 4.3, Φid(ϕ) − Φid(ϕk ) cb = ϕ − ϕk m . Proposition 3.2 now implies that Φid(ϕ) is completely compact, in other words, ϕ ∈ Mcc (A, B). 2 7. Compact multipliers In this section we compare the set of completely compact multipliers with that of compact multipliers. We exhibit sufficient conditions for these two sets of multipliers to coincide, and show that in general they are distinct. Finally, we address the question of when any universal multiplier in the minimal tensor product of two C ∗ -algebras is automatically compact. We show that this happens precisely when one of the C ∗ -algebras is finite dimensional while the other coincides with the set of its compact elements. 7.1. Automatic complete compactness We will need the following result complementing Theorem 3.4. Notation is as in Section 3.
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Proposition 7.1. If Φ : Kh → K(Hn , H1 ) is a compact completely bounded map then γ0−1 (Φ) ∈ K(H1 ) ⊗eh B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 ) ⊗eh K(Hn ). Proof. Fix ε > 0. By compactness, there exist y1 , . . . , y ∈ K(Hn , H1 ) such that min1i Φ(x) − yi < ε for each x ∈ Kh with x 1. Let {pα } (resp. {qα }) be a net of finite rank projections in K(H1 ) (resp. K(Hn )) such that pα → I (resp. qα → I ) strongly and let Φα : Kh → K(Hn , H1 ) be the map given by Φα (x) = pα Φ(x)qα . Let u = γ0−1 (Φ) and uα = γ0−1 (Φα ). Since each yi is compact there exists α0 such that pα yi qα − yi < ε for i = 1, . . . , and α α0 . Moreover, for any x ∈ Kh , x 1 and α α0 , we have Φα (x) − Φ(x) min Φα (x) − pα yi qα + pα yi qα − yi + yi − Φ(x) 1i
min 2 Φ(x) − yi + pα yi qα − yi 3ε, 1i
so Φα − Φ → 0. Remark 3.5 (i) shows that uα ∈ K(H1 ) ⊗h (B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 )) ⊗h K(Hn ); it follows that for every ω ∈ (B(H2 )⊗eh · · ·⊗eh B(Hn−1 )⊗eh B(Hn ))∗ we have Rω (uα ) ∈ K(H1 ). Suppose that ξi , ηi ∈ Hi and let ωi = ωξi ,ηi be the corresponding vector functional. Lemma 5.3 and a straightforward verification shows that if v ∈ B(H1 ) ⊗eh · · · ⊗eh B(Hn ) has a representation of the form v = A1 · · · An and ω = ω2 ⊗ · · · ⊗ ωn then Rω (v)ξ1 , η1 = v, ω1 ⊗ · · · ⊗ ωn = γ0 (v)(ζ )ξn , η1 ,
(13)
where ζ=
∗ ∗ ⊗ ξn−2 ⊗ ηn∗ ⊗ ξn−1 ∈ Kh η2 ⊗ ξ1 ⊗ η3∗ ⊗ ξ2 ⊗ · · · ⊗ ηn−1
is an elementary tensor whose components are rank one operators. Since γ0 (uα ) → γ0 (u) in norm, (13) implies that Rω (uα ) → Rω (u) in the operator norm of K(H1 ). Since Rω (uα ) ∈ K(H1 ), we obtain Rω (u) ∈ K(H1 ). By Lemma 5.4, u ∈ K(H1 ) ⊗eh B(H2 ) ⊗eh · · · ⊗eh B(Hn ). Similarly we see that u ∈ B(H1 ) ⊗eh B(H2 ) ⊗eh · · · ⊗eh K(Hn ); the conclusion now follows. 2 Remark. The converse of Proposition 7.1 does not hold, even for n = 2. Indeed, let {pi }∞ i=1 be a family of pairwise orthogonal rank one projections on a Hilbert space H and let u = ∞ p ⊗ p . Then u ∈ K(H ) ⊗ K(H ) and the range of γ (u) consists of compact operai eh 0 i=1 i tors, but γ0 (u)(pi ) = pi for each i, so γ0 (u) is not compact. Given C ∗ -algebras A1 , . . . , An , we let Mc (A1 , . . . , An ) be the collection of all ϕ ∈ M(A1 , . . . , An ) for which there exist faithful representations π1 , . . . , πn of A1 , . . . , An , respectively, such that if π = π1 ⊗ · · · ⊗ πn then the map Φπ(ϕ) is compact. We call the elements of Mc (A1 , . . . , An ) compact multipliers. As a consequence of the previous result we obtain the following fact.
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Proposition 7.2. Let A1 , . . . , An be C ∗ -algebras and let ϕ ∈ Mc (A1 , . . . , An ). Then uϕ ∈
K(An ) ⊗eh Aon−1 ⊗eh · · · ⊗eh A2 ⊗eh K(Ao1 ) K(An ) ⊗eh Aon−1 ⊗eh · · · ⊗eh Ao2 ⊗eh K(A1 )
if n is even, if n is odd.
Proof. We only consider the case n is even. We may assume that Ai ⊆ B(Hi ) is a concrete nondegenerate C ∗ -algebra, i = 1, . . . , n, and that Φϕ is compact. By Propositions 5.5 and 7.1, uid ϕ belongs to d ⊗eh · · · ⊗eh B(H2 ) ⊗eh K H1d ∩ An ⊗eh · · · ⊗eh Ad1 . K(Hn ) ⊗eh B Hn−1 Since K(Hn ) ∩ An ⊆ K(An ) and K(H1d ) ∩ Ad1 ⊆ K(Ad1 ), an application of (5) shows that uid ϕ ∈ d d K(An ) ⊗eh An−1 ⊗eh · · · ⊗eh A2 ⊗eh K(A1 ). 2 If {Aj }j ∈J is a family of C ∗ -algebras, we will denote by and ∞ -direct sums, respectively.
c0
j ∈J
Aj and
∞
j ∈J
Aj their c0 -
and suppose that K(A1 ) is isomorphic to Theorem 7.3. Let A1 , . . . , An be C ∗ -algebras, c0 c0 M and K(A ) is isomorphic to M m n n j j where J is some index set and supj ∈J mj j ∈J j ∈J and supj ∈J nj are finite. Then Mc (A1 , . . . , An ) = Mcc (A1 , . . . , An ). Proof. We give the proof for n = 3; the case of a general n is similar. Let m = sup{mj , nj : def
j ∈ J }. By hypothesis, K(A1 ) and K(A3 ) may both be embedded in the C ∗ -algebra C = c0 j ∈J Mm for some m ∈ N; without loss of generality, we may assume that this embedding is an inclusion and that Ai is represented faithfully on some Hilbert space Hi such that H1 and H3 both contain the Hilbert space H = j ∈J Cm . Given ϕ ∈ Mc (A1 , A2 , A3 ), Proposition 7.2 implies that the symbol uϕ of ϕ can be written in the form uϕ = A3 A2 A1 , where the entries of A3 and A1 belong to C. Let us write {eij : i, j = 1, . . . , m} for the canonical matrix unit ∞ system of Mm and let Pk = j ∈J ekk ∈ j ∈J Mm , k = 1, . . . , m. For k, , s, t = 1, . . . , m, we s,t set Ak, 3 = Pk A3 (P ⊗ I ) and A1 = (Ps ⊗ I )A1 Pt and define s,t uk,,s,t = Ak, 3 A2 A1
and Φk,,s,t = γ0 (uk,,s,t ).
Then γ0 (uϕ ) = Φ = k,,s,t Φk,,s,t so it suffices to show that each of the maps Φk,,s,t is completely compact. Now s,t Φk,,s,t (T2 ⊗ T1 ) = Pk Φ(P T2 ⊗ T1 Ps )Pt = Ak, 3 (P T2 ) ⊗ I A2 (T1 Ps ) ⊗ I A1 . Thus, Φk,,s,t can be considered as a completely bounded multilinear map from K(H2d , P H ) × K(Ps H, H2d ) into K(Pt H, Pk H ). Since Φ is compact, it follows that Φk,,s,t is compact. j Take a basis {ei : i = 1, . . . , m, j ∈ J } of H = j ∈J Cm , where for each j ∈ J , the standard j
basis of the j th copy of Cm is {ei : i = 1, . . . , m}. Let Uk : Pk H → P1 H be the unitary operator
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j
defined by Uk ek = e1 . Consider the mapping Ψ : K(H2d , P1 H )×K(P1 H, H2d ) → K(P1 H, P1 H ) given by Ψ (T2 ⊗ T1 ) = Uk Φk,,s,t (U T2 ⊗ T1 Us )Ut . To show that Φk,,s,t is completely compact it suffices to show that Ψ is. Let C0 = P1 CP1 ; then C0 is isomorphic to c0 and its commutant C0 has a cyclic vector. Moreover, Ψ is a C0 modular multilinear map. Let {pα } be a net of finite rank projections belonging to C0 , such that s-lim pα = IP1 H . Consider the completely bounded multilinear maps Ψα (x) = pα Ψ (x)pα . Since the range of each pα is finite dimensional, Ψα has finite rank, so is completely compact. Since Ψ is compact, we may argue as in the proof of Proposition 7.1 to show that Ψα − Ψ → 0. Now the maps Ψ and Ψα are C0 -modular and C0 has a cyclic vector, so by the generalisation [12, Lemma 3.3] of a result of Smith [23, Theorem 2.1], Ψα − Ψ cb = Ψα − Ψ → 0. Proposition 3.2 now implies that Ψ is completely compact.
2
The following corollary extends Proposition 5 of [11] to the case of multidimensional Schur multipliers. Let n 2 be an integer. We recall from [12] that with every ϕ ∈ ∞ (Nn ) we associate a mapping Sϕ : 2 (N2 ) · · · 2 (N2 ) → 2 (N2 ) which extends the usual Schur multiplication in the case n = 2. We equip the domain of Sϕ with the Haagerup norm where each of the terms is given its operator space structure arising from its embedding into the corresponding space of Hilbert–Schmidt operators endowed with the operator norm. Corollary 7.4. Let n > 2 and ϕ ∈ ∞ (Nn ). The following are equivalent: (i) Sϕ is compact; (ii) ϕ ∈ c0 ⊗h (∞ ⊗eh · · · ⊗eh ∞ ) ⊗h c0 . n−2
Proof. Assume first that Sϕ is compact. It follows from [12, Section 3] that the map Sϕ induces a completely bounded compact map Sˆϕ : C2 × · · · × C2 → C2 defined by Sˆϕ (Tf1 , . . . , Tfn ) = TSϕ (f1 ,...,fn ) , where Tf is the Hilbert–Schmidt operator with kernel f . By Proposition 7.1, ϕ = γ0−1 (Sˆϕ ) ∈ K(2 ) ⊗eh B(2 ) ⊗eh . . . ⊗eh B(2 ) ⊗eh K(2 ). Since Sϕ is bounded, ϕ is a Schur multiplier and by [12, Theorem 3.4], ϕ ∈ ∞ ⊗eh . . . ⊗eh ∞ . Hence ϕ ∈ c0 ⊗eh ∞ ⊗eh . . . ⊗eh ∞ ⊗eh c0 . We may now argue as in the last paragraph of the preceding proof to show that ϕ ∈ c0 ⊗h (∞ ⊗eh · · · ⊗eh ∞ ) ⊗h c0 . 2 Our next aim is to show that if both K(A1 ) and K(An ) contain full matrix algebras of arbitrarily large sizes then the completely compact multipliers form a proper subset of the compact multipliers. Saar [21] has provided an example of a compact completely bounded map on K(H ) (where H is a separable Hilbert space) which is not completely compact. It turns out that Saar’s
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example also shows that the sets of compact and completely compact multipliers are distinct, in the case under consideration. We will need some preliminary results. Let A and B be C ∗ -algebras. Recall that a linear map Φ : A → B is called symmetric (or hermitian) if Φ = Φ ∗ where Φ ∗ : A → B is the map given h = {a ∈ S : a = a ∗ }. The by Φ ∗ (a) = (Φ(a ∗ ))∗ . By SA we denote the unit ball of A and set SA A following lemma is a special case of Satz 6 of [21]. We include a direct proof for the convenience of the reader. Lemma 7.5. Let H be a Hilbert space. If Φ : A → K(H ) is a symmetric, completely compact linear map with Φ cb 1, then there exists a positive operator c ∈ K(H ) such that Φ (n) (a) h c ⊗ 1n for all a ∈ SM and all n ∈ N. Moreover, c can be chosen to have norm arbitrarily n ( A) close to one. Proof. We first show that for a given ε > 0 there exists a finite rank projection p on H such that (n) Φ (a) − (p ⊗ 1n )Φ (n) (a)(p ⊗ 1n ) ε
for any a ∈ SMn (A) .
(14)
Since Φ is completely compact, there exists a finite dimensional subspace F ⊂ K(H ) such that dist(Φ (n) (a), Mn (F )) ε/3 for any a ∈ Mn (A), a 1 and any n ∈ N. Let SF,1+ε = {x ∈ F : x 1 + ε} and let k = dim F . Choose a finite rank projection p ∈ K(H ) such that x − pxp <
ε k(3 + ε)
for all x ∈ SF,1+ε
and let Ψ : F → K(H ) be defined by Ψ (x) = x − pxp. By [6, Corollary 2.2.4], Ψ is completely bounded and Ψ cb k Ψ . This implies that (n) Ψ (y) k Ψ y
ε ε y 3+ε 3
for all y ∈ Mn (F ) with y 1 + ε/3. h Now for a ∈ SM let y ∈ Mn (F ) be such that Φ (n) (a) − y ε/3. Then y n ( A) Φ (n) (a) + ε/3 1 + ε/3. Hence (n) Φ (a) − (p ⊗ 1n )Φ (n) (a)(p ⊗ 1n ) Φ (n) (a) − y + Ψ (n) (y) + (p ⊗ 1n ) y − Φ (n) (a) (p ⊗ 1n ) ε/3 + ε/3 + ε/3 = ε, proving (14). Next we fix ε > 0 and choose a finite rank projection q1 on H such that (n) Φ (a) − (q1 ⊗ 1n )Φ (n) (a)(q1 ⊗ 1n ) ε , 2
a ∈ Mn (A), a 1, n ∈ N.
Let r1 : A → K(H ) be the mapping given by r1 (a) = Φ(a) − q1 Φ(a)q1 , a ∈ A. Then r1 = Ψ ◦ Φ, where Ψ : K(H ) → K(H ) is the completely bounded map given by Ψ (x) = x − q1 xq1 . By Proposition 3.2, r1 is completely compact. Moreover, r1 cb ε/2 and Φ(a) =
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q1 Φ(a)q1 + r1 (a), a ∈ A. Proceeding by induction, we can find sequences of finite rank projections qi and completely compact symmetric mappings ri such that ri cb ε/2i and Φ(a) = q1 Φ(a)q1 +
∞
qi+1 ri (a)qi+1 ,
a ∈ A.
i=1
Let c = q1 +
∞
ε i=1 2i qi+1 .
(n)
We have that Φ (n) and ri
Φ (n) (a) = (q1 ⊗ 1n )Φ (n) (a)(q1 ⊗ 1n ) +
are symmetric and
∞ (n) (qi+1 ⊗ 1n )ri (a)(qi+1 ⊗ 1n ), i=1
for each a ∈ A. Now Φ
(n)
(a) (q1 ⊗ 1n ) Φ cb +
∞ i=1
∞ ε (qi+1 ⊗ 1n ) ri cb q1 + qi+1 ⊗ 1n = c ⊗ 1n 2i i=1
h for all a ∈ SM . By construction, c is compact and c 1 + ε. n ( A)
2
Let H be an infinite dimensional separable Hilbert space and {qk }k∈N be a family n of pairwise q = I . Set p = orthogonal projections in B(H ) with rank qk = k and ∞ k n k=1 k=1 qk , n ∈ N. Let Φk : B(qk H ) → B(qk H ), k ∈ N, be symmetric linear maps such that Φk cb = 1,
Φk → 0
as k → ∞,
and
∞
Φk 22 < ∞,
(15)
k=1
where Φk 2 denotes the norm of the mapping Φk when B(qk H ) C2 (qk H ) is equipped with the Hilbert–Schmidt norm. Identifying B(qk H ) with qk B(H )qk , let Φ : K(H ) → B(H ) be the map given by the norm-convergent sum Φ(x) =
∞
⊕
Φk (qk xqk ),
x ∈ K(H ).
(16)
k=1
An example of such a map is obtained by taking Φk = k −1 τk where τk is the transposition map B(qk H ) Mk → Mk B(qk H ), which is symmetric and an isometry for both the operator and the Hilbert–Schmidt norm. It is well known [20, p. 419] that τk cb = k and hence conditions (15) are satisfied. The next lemma is a straightforward extension of [21, pp. 32–34]. Lemma 7.6. If Φ is a map satisfying (15) and (16) then the range of Φ consists of compact operators. Moreover, Φ is completely contractive and compact but not completely compact. Proof. Fix x ∈ K(H ). Since Φk → 0 as k → ∞, we have pn Φ(x)pn → Φ(x) in norm, so Φ(x) ∈ K(H ). Each of the maps x → Φk (qk xqk ) is completely contractive, so Φ is completely contractive.
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def
Next, note that Φ maps the unit ball of K(H ) into U = U1 ⊕ U2 ⊕ · · ·, where Uk is the ball of radius Φk in qk B(H )qk . Since U is compact, the map Φ is compact. If Φ were completely compact then by Lemma 7.5, there would exist a positive compact operator c on H such that Φ (k) (x) c ⊗ 1k
h for all x ∈ SM and all k ∈ N. k (K(H ))
h , Hence for every k ∈ N and x ∈ SM k (K(H )) (k)
Φk
(qk ⊗ 1k )x(qk ⊗ 1k ) = (qk ⊗ 1k )Φ (k) (x)(qk ⊗ 1k ) qk cqk ⊗ 1k .
(k)
However, Φk = Φk cb = 1 by [22], so 1 (k) h qk cqk = qk cqk ⊗ 1k sup Φk (x) : x ∈ SM , k (qk K(H )qk ) 2 which is impossible since c is compact.
2
Lemma 7.7. Given a map Φ be as above, let C = sal multiplier ϕ ∈ M(C d , C) with Φ = Φid(ϕ) .
c0
k∈N B(qk H ) ⊆ K(H ).
There exists a univer-
Proof. Let ϕk ∈ B(qk H )d ⊗ B(qk H ) be such that Φid(ϕk ) = Φk , k ∈ N, where the family {Φk }∞ k=1 n satisfies (15). Then ϕ = Φ . Let ψ = ϕ . If n < m then ψ − ψ = k min k 2 n k m n min k=1 m Φ so k 2 k=n+1
ψm − ψn min
1/2
m
Φk 22
.
k=n+1
By (15), the sequence {ψn } converges to an element ϕ ∈ C d ⊗ C. Moreover, for every x ∈ C2 (H ) we have Φid(ϕ) (x) = lim pn Φid(ϕ) (x)pn = lim Φid(ψn ) (x) = Φ(x), n→∞
n→∞
where the limits are in the operator norm. So Φid(ϕ) = Φ which is completely contractive by Lemma 7.6, so ϕ ∈ M(C d , C) by Theorem 4.3. 2 Given C ∗ -algebras Ai ⊆ B(Hi ), i = 1, . . . , n, and ψ = c2 ⊗ · · · ⊗ cn−1 ∈ A2 · · · An−1 , we may define a bounded linear map A1 ⊗ An → B1 ⊗ A2 ⊗ · · · ⊗ An , where B1 = A1 if n is even and B1 = Ad1 if n is odd, by a ⊗ b →
a⊗ψ ⊗b ad ⊗ ψ ⊗ b
if n is even, if n is odd.
We write ιψ for the restriction of this map to M(A1 , An ).
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Lemma 7.8. (i) The range of ιψ is contained in M(B1 , A2 , . . . , An ). (ii) ιψ (Mcid (A1 , An )) ⊆ Mcid (B1 , A2 , . . . , An ). d ) ⊗ · · · ⊗ B(H )) . Writing (iii) Suppose that n is even and ω ∈ (B(Hn−1 eh eh 2 ∗ d ⊗eh · · · ⊗eh B(H2 ) ⊗eh B H1d → B(Hn ) ⊗eh B H1d Mω : B(Hn ) ⊗eh B Hn−1 for the “middle slice map” Mω = Rω ⊗eh idB(H d ) , we have 1
˜ ϕ Mω (uιψ (ϕ) ) = ω(ψ)u d where ψ˜ = cn−1 ⊗ · · · ⊗ c2 . The same is true, mutatis mutandis, if n is odd.
Proof. Let ϕ ∈ M(A1 , An ). By Theorem 2.3, there exist a net {ϕν } ⊆ A1 An and representad ν ν ν id tions uid ϕν = A2 A1 and uϕ = A2 A1 , where Ai are finite matrices with entries in A1 if i = 1 ν ν and in An if i = 2, such that ϕν → ϕ semi-weakly, Ai → Ai strongly and supi,ν Ai < ∞. (i) It is easy to see that ιψ (ϕν ) satisfies the boundedness conditions of Theorem 2.3 and converges semi-weakly to ιψ (ϕ), which is therefore a universal multiplier. (ii) Suppose that n is even and let ι = ιψ . It is immediate to check that if ϕ ∈ A1 An and d , H ) then T1 ∈ K(H1d , H2 ), . . ., Tn−1 ∈ K(Hn−1 n d . . . c 2 T1 . Φι(ϕ) (Tn−1 ⊗ · · · ⊗ T1 ) = Φϕ Tn−1 cn−1 Note that this equation holds for any ϕ ∈ M(A1 , An ) since Φϕν (T ) → Φϕ (T ) and Φι(ϕν ) (Tn−1 ⊗ · · · ⊗ T1 ) → Φι(ϕ) (Tn−1 ⊗ · · · ⊗ T1 ) weakly for any T , T1 , . . . , Tn−1 . Since Φι(ϕ) is the compod sition of the bounded mapping Xn−1 ⊗ · · · ⊗ X1 → Xn−1 cn−1 . . . c2 X1 with Φϕ , it follows that if ϕ is a compact operator multiplier then so is ι(ϕ). (iii) We have that d Φι(ϕν ) (Tn−1 ⊗ · · · ⊗ T1 ) = Aν2 (Tn−1 ⊗ I ) cn−1 ⊗ I . . . (c2 ⊗ I )(T1 ⊗ I )Aν1 d → A2 (Tn−1 ⊗ I ) cn−1 ⊗ I . . . (c2 ⊗ I )(T1 ⊗ I )A1 weakly. On the other hand, Φι(ϕν ) (Tn−1 ⊗ · · · ⊗ T1 ) → Φι(ϕ) (Tn−1 ⊗ · · · ⊗ T1 ) which implies that d ˜ ϕ. 2 uι(ϕ) = A2 (cn−1 ⊗ I ) · · · (c2 ⊗ I ) A1 . It follows that Mω (uι(ϕ) ) = ω(ψ)u Theorem 7.9. Let A1 , . . . , An be C ∗ -algebras with the property that both K(A1 ) and K(An ) contain full matrix algebras of arbitrarily large sizes. Then the inclusion Mcc (A1 , . . . , An ) ⊆ Mc (A1 , . . . , An ) is proper. Proof. We may assume that Ai ⊆ B(Hi ), i = 1, . . . , n for some Hilbert spaces H1 , . . . , Hn . First suppose that n = 2. By hypothesis, we may assume that there is an infinite separable cdimensional 0 Mk ⊆ K(H ) as in Hilbert space H with H d ⊆ H1 and H ⊆ H2 , and a C ∗ -algebra C = k∈N Lemma 7.7 with C d ⊆ A1 and C ⊆ A2 . By the injectivity of the minimal tensor product of C ∗ algebras, C d ⊗ C ⊆ A1 ⊗ A2 .
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Let ϕ ∈ C d ⊗ C be given by Lemma 7.7. It follows from Lemma 7.6 that ϕ ∈ Mc (A1 , A2 ) \ id (A , A ). Since faithful representations of A and A restrict to representations of C conMcc 1 2 1 2 taining the identity subrepresentation up to unitary equivalence, we have that ϕ ∈ Mc (A1 , A2 ) \ Mcc (A1 , A2 ). Suppose now that n is even. Let ϕ ∈ Mc (A1 , An ) \ Mcc (A1 , An ), fix any non-zero ψ = c2 ⊗ · · · ⊗ cn−1 ∈ A2 · · · An−1 and let us write ι = ιψ . Suppose that ι(ϕ) is a completely compact multiplier. By Theorem 6.4, uι(ϕ) ∈ K(An ) ⊗h (Aon−1 ⊗eh · · · ⊗eh A2 ) ⊗h K(Ao1 ). d d ) ⊗ · · · ⊗ B(H )) ⊗ · · · ⊗ c2 ∈ Adn−1 ⊗eh · · · ⊗eh A2 and fix ω ∈ (B(Hn−1 Let ψ˜ = cn−1 eh eh 2 ∗ ˜ = 0. By Lemma 7.8 (iii), Mω (uι(ϕ) ) = ω(ψ)u ˜ ϕ and hence uϕ ∈ K(An ) ⊗h K(Ao ) such that ω(ψ) 1 which by Theorem 6.4 contradicts the assumption that ϕ is not a completely compact multiplier. If n is odd then the same proof works with minor modifications. 2 Remark 7.10. We do not know whether the sets Mcc (A, B) and Mc (A, B) are distinct if K(A) contains matrix algebras of arbitrarily large sizes, while K(B) does not (and vice versa). Let C be the C ∗ -algebra defined in Lemma 7.7. To show that the inclusion Mcc (C, c0 ) ⊆ Mc (C, c0 ) is proper it would suffice to exhibit mappings Φk : Mk → Mk which satisfy (15) and are left Dk modular (where Dk is the subalgebra of all diagonal matrices of Mk ). This modularity condition would enable us to find ϕk ∈ Mkd ⊗ Dk such that Φk = Φid(ϕk ) using the method of Lemma 7.7 and we could then conclude from Lemma 7.6 that Mcc (C, c0 ) Mc (C, c0 ). However, we do not know if such mappings Φk exist. This prompts the following question: if D is a masa in B(H ), does there exist a constant C such that whenever Φ : K(H ) → K(H ) is a bounded and left D-modular map then Φ cb C Φ ? If such a version of Smith’s automatic complete boundedness result holds then it would follow that Mcc (C, c0 ) = Mc (C, c0 ). 7.2. Automatic compactness We now turn to the question of when every universal multiplier is automatically compact. We will restrict to the case n = 2 for the rest of the paper. We will first establish an auxiliary result in a different but related setting. Suppose that A and B are commutative C ∗ -algebras and assume that A = C0 (X) and B = C0 (Y ) for some locally compact Hausdorff spaces X and Y . The C ∗ -algebra C0 (X) ⊗ C0 (Y ) will be identified with C0 (X × Y ) and M(A, B) with a subset of C0 (X × Y ). Elements of the Haagerup tensor product C0 (X) ⊗h C0 (Y ), as well as of ˆ 0 (Y ), will be identified with functions in C0 (X × Y ) in the projective tensor product C0 (X)⊗C ˆ 0 (Y ) the natural way. Note that, by Grothendieck’s inequality, C0 (X) ⊗h C0 (Y ) and C0 (X)⊗C coincide as sets of functions. Proposition 7.11. Let X and Y be infinite, locally compact Hausdorff spaces. Then C0 (X) ⊗h C0 (Y ) ⊆ M(C0 (X), C0 (Y )) and this inclusion is proper. Proof. The inclusion C0 (X) ⊗h C0 (Y ) ⊆ M(C0 (X), C0 (Y )) follows from Corollary 6.7 of [14]. To show that this inclusion is proper, suppose first that X and Y are compact. By Theorem 11.9.1 of [8], there exists a sequence (fi )∞ i=1 ⊆ C(X) ⊗h C(Y ) such that supi∈N fi h < ∞, converging uniformly to a function f ∈ C(X × Y ) \ C(X) ⊗h C(Y ). By Corollary 6.7 of [14], f ∈ M(C(X), C(Y )). The conclusion now follows. Now assume that both X and Y are locally compact but not compact (the case where one of the spaces is compact while the other is not is similar). Let X˜ = X ∪ {∞} and Y˜ = Y ∪ {∞} be the
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˜ = C0 (X) + C1 and C(Y˜ ) = C0 (Y ) + C1, one point compactifications of X and Y . Then C(X) where 1 denotes the constant function taking the value one. Moreover, it is easy to see that ˜ ⊗ C(Y˜ ) = C0 (X × Y ) + C0 (X) + C0 (Y ) + C1 C(X) and ˜ ⊗C( ˆ Y˜ ) = C0 (X)⊗C ˆ 0 (Y ) + C0 (X) + C0 (Y ) + C1. C(X)
(17)
˜ C(Y˜ )) \ C(X) ˜ ⊗h C(Y˜ ). Write ϕ = By the first part of the proof, there exists ϕ ∈ M(C(X), ϕ1 + ϕ2 + ϕ3 + ϕ4 where ϕ1 ∈ C0 (X × Y ), ϕ2 ∈ C0 (X), ϕ3 ∈ C0 (Y ) and ϕ4 ∈ C1. Suppose that ˜ ⊗C( ˆ Y˜ ), a contradiction. 2 ϕ1 ∈ C0 (X) ⊗h C0 (Y ). By (17), ϕ ∈ C(X) Theorem 7.12. Let A and B be C ∗ -algebras. The following are equivalent: (i) either A is finite dimensional and K(B) = B, or B is finite dimensional and K(A) = A; (ii) Mc (A, B) = M(A, B); (iii) Mcc (A, B) = M(A, B). Proof. (i) ⇒ (iii) Suppose that A is finite dimensional and K(B) = B, and that A ⊆ B(H1 ) and B ⊆ B(H2 ) for some Hilbert spaces H1 and H2 where H1 is finite dimensional. Fix ϕ ∈ M(A, B). Then ϕ is the sum of finitely many elements of the form a ⊗ b where a has finite rank and b ∈ K(H2 ); such elements are completely compact multipliers by Theorem 6.4. (iii) ⇒ (ii) is trivial. (ii) ⇒ (i) Assume that both A and B are infinite dimensional and are identified with their image under the reduced atomic representation. If either K(A) or K(B) is finite dimensional then there exists an elementary tensor a ⊗ b ∈ (A B) \ (K(A) K(B)). By Proposition 7.2, a ⊗ b ∈ Mc (A, B). We can therefore assume that both K(A) and K(B) are infinite dimensional. Then, up to a ∗-isomorphism, c0 is contained in both K(A) and K(B). By Proposition 7.11, there exists ϕ ∈ M(c0 , c0 ) \ (c0 ⊗h c0 ). Then ϕ ∈ M(A, B) and Φid(ϕ) is not compact by Hladnik’s characterisation [11]. Since the restrictions to c0 of any faithful representations of A, B contain representations unitarily equivalent to the identity representations, we see that ϕ is not a compact multiplier. Thus at least one of the C ∗ -algebras A and B is finite dimensional; assume without loss of generality that this is A. Suppose that B = K(B) and fix an element b ∈ B \ K(B). Let a ∈ A be a non-zero element. By Proposition 7.2, the elementary tensor a ⊗ b is not a compact multiplier. 2 Acknowledgments We are grateful to V.S. Shulman for stimulating results, questions and discussions. We would like to thank M. Neufang for pointing out to us Corollary 3.7 and R. Smith for a discussion concerning Remark 7.10. The first named author is grateful to G. Pisier for the support of the one semester visit to the University of Paris 6 and the warm atmosphere at the department, where one of the last drafts of the paper was finished. The first named author was supported by The Royal Swedish Academy of Sciences, Knut och Alice Wallenbergs Stiftelse and Jubileumsfonden of the University of Gothenburg’s Research
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Foundation. The second and the third named authors were supported by Engineering and Physical Sciences Research Council grant EP/D050677/1. The last named author was supported by the Swedish Research Council. References [1] D.P. Blecher, C. Le Merdy, Operator Algebras and Their Modules—An Operator Apace Approach, Oxford University Press, 2004. [2] D.P. Blecher, R. Smith, The dual of the Haagerup tensor product, J. London Math. Soc. (2) 45 (1992) 126–144. [3] E. Christensen, A.M. Sinclair, Representations of completely bounded multilinear operators, J. Funct. Anal. 72 (1987) 151–181. [4] E.G. Effros, A. Kishimoto, Module maps and Hochschild–Johnson cohomology, Indiana Math. J. 36 (1987) 257– 276. [5] E.G. Effros, Z.J. Ruan, Operator spaces tensor products and Hopf convolution algebras, J. Operator Theory 50 (2003) 131–156. [6] E.G. Effros, Z.J. Ruan, Operator Spaces, London Math. Soc. Monogr. Ser., vol. 23, Oxford University Press, New York, 2000. [7] J.A. Erdos, On a certain elements of C ∗ -algebras, Illinois J. Math. 15 (1971) 682–693. [8] C.C. Graham, O.C. McGehee, Essays in Commutative Harmonic Analysis, Springer, 1979. [9] A. Grothendieck, Resume de la theorie metrique des produits tensoriels topologiques, Boll. Soc. Mat. Sao-Paulo 8 (1956) 1–79. [10] D.W. Hadwin, Nonseparable approximate equivalence, Trans. Amer. Math. Soc. 266 (1) (1981) 203–231. [11] M. Hladnik, Compact Schur multipliers, Proc. Amer. Math. Soc. 128 (2000) 2585–2591. [12] K. Juschenko, I.G. Todorov, L. Turowska, Multidimensional operator multipliers, Trans. Amer. Math. Soc., in press. [13] A. Katavolos, V. Paulsen, On the ranges of bimodule projections, Canad. Math. Bull. 48 (2005) 97–111. [14] E. Kissin, V.S. Shulman, Operator multipliers, Pacific J. Math. 227 (2006) 109–141. [15] T. Oikhberg, Direct sums of operator spaces, J. London Math. Soc. (2) 64 (1) (2001) 144–160. [16] V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2002. [17] V. Paulsen, R.R. Smith, Multilinear maps and tensor norms on operator systems, J. Funct. Anal. 73 (1987) 258–276. [18] V.V. Peller, Hankel operators in the perturbation theory of unitary and selfadjoint operators, Funktsional. Anal. i Prilozhen. 19 (2) (1985) 37–51, 96. [19] G. Pisier, Similarity Problems and Completely Bounded Maps, Springer-Verlag, Berlin–New York, 2001. [20] G. Pisier, Introduction to Operator Space Theory, Cambridge University Press, 2003. [21] H. Saar, Kompakte, vollständig beschränkte Abbildungen mit Werten in einer nuklearen C ∗ -Algebra, Diplomarbeit, Universität des Saarlandes, Saarbrücken, 1982. [22] R.R. Smith, Completely bounded maps between C ∗ -algebras, J. London Math. Soc. (2) 27 (1983) 157–166. [23] R.R. Smith, Completely bounded module maps and the Haagerup tensor product, J. Funct. Anal. 102 (1991) 156– 175. [24] N. Spronk, Measurable Schur multipliers and completely bounded multipliers of Fourier algebras, Proc. London Math. Soc. (3) 89 (2004) 161–192. [25] M. Takesaki, Theory of Operator Algebras I, Springer, 2001. [26] D. Voiculescu, A non-commutative Weyl–von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976) 97–113. [27] C. Webster, Matrix compact sets and operator approximation properties, arXiv: math/9804093, 1998. [28] K. Ylinen, Compact and finite-dimensional elements of normed algebras, Ann. Acad. Sci. Fenn. Ser. AI 428 (1968) 1–38. [29] K. Ylinen, A note on the compact elements of C ∗ -algebras, Proc. Amer. Math. Soc. 35 (1972) 305–306.
Journal of Functional Analysis 256 (2009) 3806–3829 www.elsevier.com/locate/jfa
Spectral inclusion for unbounded block operator matrices ✩ Christiane Tretter Mathematisches Institut, Universität Bern, Sidlerstr. 5, 3012 Bern, Switzerland Received 15 September 2008; accepted 23 December 2008 Available online 23 January 2009 Communicated by N. Kalton
Abstract In this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W 2 (A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W 2 (A). This provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non-convex set W 2 (A). Several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given. © 2009 Elsevier Inc. All rights reserved. Keywords: Spectrum; Unbounded linear operator; Block operator matrix; Numerical range; Quadratic numerical range
1. Introduction The spectra of linear operators play a crucial role in many branches of mathematics and in numerous applications. Analytic information on the spectrum is, in general, hard to obtain and numerical approximations may not be reliable, in particular, if the operator is not self-adjoint or normal. In this paper, we suggest a new analytic enclosure for the spectrum of unbounded linear operators that admit a so-called block operator matrix representation. ✩ The support for this work of Deutsche Forschungsgemeinschaft DFG, grant No. TR368/6-1, and of Schweizerischer Nationalfonds, SNF, grant No. 200021-119826/1, is greatly appreciated. E-mail address: [email protected].
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.12.024
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The classical tool to enclose the spectrum of a linear operator A in a Hilbert space H is the numerical range W (A) = {(Ax, x): x ∈ D(A), x = 1}. It is easy to see that the point spectrum σp (A) of A is contained in W (A) and that the approximate point spectrum σapp (A) of A is contained in the closure of W (A), σp (A) ⊂ W (A),
σapp (A) ⊂ W (A);
(1.1)
the inclusion σ (A) ⊂ W (A) holds if A is closed and every component of the complement C \ W (A) contains at least one point of the resolvent set ρ(A) of A (see [10, Theorem V.3.2]). By the Toeplitz–Hausdorff theorem, the numerical range is a convex subset of C (see [10, Theorem V.3.1]). At first sight, the convexity seems to be a useful property, e.g. to show that the spectrum lies in a half-plane. However, the numerical range often gives a poor localization of the spectrum and it cannot capture finer structures of the spectrum like being separated in two parts. In view of these shortcomings, the concept of quadratic numerical range was introduced in [15]. It is defined if a decomposition H = H1 ⊕ H2 of the underlying Hilbert space H is given and A is a linear operator that admits a corresponding block operator matrix representation A=
A C
B , D
D(A) = D(A) ∩ D(C) ⊕ D(B) ∩ D(D) ,
(1.2)
with densely defined operator entries A, B, C, and D (note that this holds automatically if A is bounded, but not for unbounded A). The case that A has bounded off-diagonal entries B and C was considered in [15]; the case that A itself is bounded was studied in detail in the subsequent papers [13,14]. The case that all entries are unbounded is of particular interest in view of applications, e.g. to systems of (partial) differential equations of mixed order (and type). For a block operator matrix A as in (1.2), the quadratic numerical range W 2 (A) is defined as the set of all eigenvalues of all 2 × 2 matrices Af,g :=
(Af, f ) (Cf, g)
(Bg, f ) (Dg, g)
∈ M2 (C)
with elements f ∈ D(A) ∩ D(C), g ∈ D(B) ∩ D(D) having norm 1, that is, W 2 (A) :=
σp (Af,g ).
(f,g)t ∈D (A) f =g=1
Unlike the numerical range, the quadratic numerical range is no longer convex; it consists of at most two components which need not be convex either (even in the finite-dimensional case). For block operator matrices with bounded off-diagonal corners, it was shown in [15] that the quadratic numerical range shares the spectral inclusion property (1.1) with the numerical range. This result was used to prove that certain block operator matrices are exponentially dichotomous, i.e. their spectrum lies in two sectors in the left and the right half-plane separated by a strip around the imaginary axis. For bounded block operator matrices, it was proved in [13] that the quadratic numerical range is always contained in the numerical range W 2 (A) ⊂ W (A). For bounded self-adjoint block operator matrices, the quadratic numerical range was used successfully by V. Kostrykin, K.A. Makarov, and A.K. Motovilov in [12] to prove sharp results on the
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perturbation of the spectrum and the spectral subspaces of the block diagonal part. Other methods to prove such perturbation results for unbounded block operator matrices include an analysis of the Schur complements associated with a block operator matrix (1.2) (see [11]) and norm estimates for solutions of corresponding algebraic Riccati equations (see [2]). In the present paper, we introduce and study the quadratic numerical range for block operator matrices all entries of which are unbounded. To this end, we distinguish diagonally dominant and off-diagonally dominant block operator matrices, depending on the position and strength of the dominanting entries in each column. The main result of this paper is that, for these two classes of block operator matrices, the quadratic numerical range has the spectral inclusion property. We show that this new method to enclose the spectrum of unbounded linear operators yields tighter analytic enclosures than the usual numerical range or classical perturbation theory, both in the non-self-adjoint case and in the self-adjoint case. The paper is organized as follows. In Section 2 we define diagonally dominant and offdiagonally dominant block operator matrices and use these notions to establish criteria for a block operator matrix to be closed. In Section 3 we introduce the quadratic numerical range of an unbounded block operator matrix and prove some elementary properties; in particular, we show that the quadratic numerical range is always contained in the usual numerical range. In Section 4 we prove that the quadratic numerical range has the spectral inclusion property if the block operator matrix is diagonally dominant or off-diagonally dominant of order 0; here the order is the maximum of the relative bounds in each column with respect to the dominating entries. In Section 5 we consider block operator matrices with separated diagonal entries and symmetric corners (Re W (D) < Re W (A) and C = B ∗ ); block operator matrices with self-adjoint diagonal entries and anti-symmetric corners (A = A∗ , D = D ∗ , and C = −B ∗ ); and self-adjoint block operator matrices with semi-bounded diagonal entries. By means of these three classes, we show that the spectral enclosures in terms of the quadratic numerical range cannot be obtained by previously available methods. Applications to the Dirac operators and to two-channel Hamiltonians illustrate our results. 2. Diagonally dominant and off-diagonally dominant block operator matrices Let H1 and H2 be Banach spaces. In the Banach space H = H1 ⊕ H2 we consider the linear operator A given by the block operator matrix A=
A C
B ; D
(2.1)
here A, D are densely defined closed linear operators in H1 and H2 , respectively, and B, C are densely defined closed linear operators from H2 to H1 and from H1 to H2 , respectively. Then A with its natural domain D(A) := D(A) ∩ D(C) ⊕ D(B) ∩ D(D) is also densely defined, but not necessarily closed. In the following we distinguish two classes of block operator matrices, diagonally dominant and off-diagonally dominant. To this end, we recall the notion of relative boundedness (see [10, Section IV.1.1]).
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If E, F , and G are Banach spaces and T , S are linear operators from E to F and from E to G, respectively, then S is called T -bounded (or relatively bounded with respect to T ) if D(T ) ⊂ D(S) and there exist constants aS , bS 0 such that Sx aS x + bS T x,
x ∈ D(T ).
(2.2)
The infimum δS of all bS so that (2.2) holds for some aS 0 is called T -bound of S (or relative bound of S with respect to T ). The operator S is called T -compact (or relatively compact with respect to T ) if D(T ) ⊂ D(S) ∞ and, for every bounded sequence (xn )∞ 1 ⊂ D(T ) such that (T xn )1 ⊂ F is bounded, the sequence ∞ (Sxn )1 ⊂ G contains a convergent subsequence (see [10, Section IV.1.3]). Note that if T is closed and S is closable, then D(T ) ⊂ D(S) already implies that S is T bounded (see [10, Remark IV.1.5]). From the inequality Sx aS x + bS T x aS x + bS (T + S)x + bS Sx,
x ∈ D(T ),
(2.3)
it is easy to see that if S is T -bounded with T -bound δS < 1, then S is (T + S)-bounded with (T + S)-bound δS /(1 − δS ) [10, Chapter IV, §1, Problem 1.2]. It is also not difficult to prove (see [10, Section V.4.1, (4.1), (4.2)]) that (2.2) is equivalent to Sx2 aS 2 x2 + bS 2 T x2 ,
x ∈ D(T ),
(2.4)
with constants aS , bS 0; moreover, (2.2) holds with bS < δ for some δ > 0 if and only if (2.4) holds with bS < δ. Hence the T -bound of S can also be defined as the infimum of all bS so that (2.4) holds for some aS 0. If T is closed, then DT = (D(T ), ·T ) with the graph norm xT := x+T x, x ∈ D(T ), is a Banach space. Obviously, S is T -bounded if and only if S is a bounded operator from DT to G, and S is T -compact if and only if S is compact from DT to G. Hence a T -compact operator is always T -bounded. Definition 2.1. The block operator matrix A in (2.1) is called (i) diagonally dominant if C is A-bounded and B is D-bounded, (ii) off-diagonally dominant if A is C-bounded and D is B-bounded. Remark. Since the entries of A are assumed to be closed, the type of dominance may be read off from domain inclusions (see [10, Remark IV.1.5]): A is diagonally dominant
⇐⇒
D(A) ⊂ D(C),
D(D) ⊂ D(B),
A is off-diagonally dominant
⇐⇒
D(C) ⊂ D(A),
D(B) ⊂ D(D);
in both cases, the block operator matrix A is densely defined with domain D(A) = D1 ⊕ D2 :=
D(A) ⊕ D(D)
if A is diagonally dominant,
D(C) ⊕ D(B)
if A is off-diagonally dominant.
(2.5)
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To ensure that the block operator matrix A is closed, we need more refined assumptions on the strength of the entries with respect to each other. Definition 2.2. Let δ 0. The block operator matrix A is called (i) diagonally dominant of order δ if C is A-bounded with A-bound δC , B is D-bounded with D-bound δB , and δ = max {δB , δC }, (ii) off-diagonally dominant of order δ if A is C-bounded with C-bound δA , D is B-bounded with B-bound δD , and δ = max{δA , δD }. Remark. If H1 = H2 , then A is off-diagonally dominant (of order δ) if and only if GA :=
0 I
I 0
A C
B D
=
C A
D B
(2.6)
is diagonally dominant (of order δ); in this particular situation, the following statements for the off-diagonally dominant case may be deduced from the corresponding statements for the diagonally dominant case. Proposition 2.3. Define the block operator matrices T :=
A 0 , 0 D
S :=
0 C
B 0
.
(2.7)
Then the following implications hold: (i) If A is diagonally dominant of order δ, then S is T -bounded with T -bound δ. (ii) If A is off-diagonally dominant of order δ, then T is S-bounded with S-bound δ. Proof. We prove (i); the proof of (ii) is similar. Let ε > 0 be arbitrary. By the assumptions 0 such that and by the introductory remarks (see (2.4)), there exist constants aB , aC , bB , bC δB bB < δB + ε, δC bC < δC + ε and Bg2 aB 2 g2 + bB 2 Dg2 ,
g ∈ D(D),
(2.8)
2 Af 2 , Cf 2 aC 2 f 2 + bC
f ∈ D(A).
(2.9)
Hence we obtain, for (f g)t ∈ D(A) ⊕ D(D), 0 C
B 0
2 f = Bg2 + Cf 2 a 2 g2 + b 2 Dg2 + a 2 f 2 + b 2 Af 2 B B C C g 2 2 f 2 2 A 0 f . max aB , aC g + max bB , bC 0 D g
} < max{δ + ε, δ + ε} = δ + ε, this shows that S is T -bounded with T Since max{bB , bC B C bound < δ. That the T -bound indeed equals δ follows by contradiction if we first set f = 0 and then g = 0 in the above inequality. 2
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An immediate consequence of Proposition 2.3 and the introductory remarks (see inequality (2.3)) is the following. Corollary 2.4. If, in Proposition 2.3(i) or (ii), the order of dominance δ is < 1, then S or T , respectively, is A-bounded with A-bound δ/(1 − δ). Theorem 2.5. The block operator matrix A is closed if one of the following holds: (i) A is diagonally dominant of order δ < 1; (ii) A is off-diagonally dominant of order δ < 1. Proof. The assertions follow from Proposition 2.3 and from classical perturbation results on the stability of closedness under relatively bounded perturbations with relative bound < 1 (see [10, Theorem IV.1.1]). 2 In the above theorem the assumptions are symmetric for the two columns of the block operator matrix. In fact, a stronger dominance in one column may compensate for a weaker one in the other column. Theorem 2.6. The block operator matrix A is closed if one of the following holds: (i) A is diagonally dominant and the relative bounds δC of C and δB of B satisfy δC2 1 + δB2 < 1 or
δB2 1 + δC2 < 1;
(ii) A is off-diagonally dominant and the relative bounds δA of A and δD of D satisfy 2 2 δA 1 + δD < 1 or
2 2 δD 1 + δA < 1.
Proof. We prove (i); the proof of (ii) is analogous. Let e.g. δC2 (1 + δB2 ) < 1. Then δC < 1 and δB δC < 1. We consider the block operator matrices T :=
A B , 0 D
S :=
0 C
0 . 0
First we prove that T is closed. Suppose that ((xn yn )t )∞ 1 ⊂ D(A) ⊕ D(D) is a sequence such that (xn yn )t → (x y)t , n → ∞, and T
Axn + Byn v xn = → , w yn Dyn
n → ∞,
with some v ∈ H1 , w ∈ H2 . Since D is assumed to be closed, this shows that y ∈ D(D) ⊂ D(B) and Dy = w. The assumption that B is D-bounded implies that there exist aB , bB 0 with B(yn − y) aB yn − y + bB D(yn − y) → 0,
n → ∞.
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Hence Byn → By and Axn → v − By, n → ∞. Since A is assumed to be closed, we obtain x ∈ D(A) and Ax + By = v. Thus T is closed. To prove the claim, it now suffices to show that S is T -bounded with T -bound < 1. Choose ε > 0 so that (δC + ε)2 (1 + (δB + ε)2 ) < 1 and let 0 with δ b < δ + ε, δ b < δ + ε be as in (2.8), (2.9); the constants aB , aC , bB , bC B B C C B C in particular, we have 2 bC 1 + bB 2 < 1.
(2.10)
For (f g)t ∈ D(A) ⊕ D(D) and arbitrary γ > 0, we obtain, using the elementary inequality (ξ1 + ξ2 )2 (1 + γ −1 )ξ12 + (1 + γ )ξ22 for arbitrary ξ1 , ξ2 ∈ R, 0 C
2 f = Cf 2 g 2 2 Af + Bg + Bg aC 2 f 2 + bC 2 2 aC 2 f 2 + 1 + γ −1 bC Af + Bg2 + (1 + γ )bC Bg2 2 2 2 aC 2 f 2 + (1 + γ )aB 2 bC g2 + 1 + γ −1 bC Af + Bg2 + (1 + γ )bB 2 bC Dg2 2 2
2 f 2 2 f 2 −1 2 A B . max aC , (1 + γ )aB bC + bC max 1 + γ , (1 + γ )bB 0 D g g 0 0
2 max 1 + γ −1 , (1 + γ )b 2 < 1. The Hence S has T -bound < 1 if there is a γ > 0 such that bC B latter holds if and only if 2 bC
2 1 − bC
<
2 1 − bB 2 bC 2 bB 2 bC
,
. which is equivalent to the inequality (2.10) satisfied by bB and bC
2
Corollary 2.7. The block operator matrix A is closed if one of the following holds: (i) C is A-bounded, B is D-bounded and at least one relative bound is 0; (ii) A is C-bounded, D is B-bounded and at least one relative bound is 0. In the next three corollaries we consider particular cases of operators with relative bound 0. As in Corollary 2.7, we distinguish the diagonally dominant case (i) and the off-diagonally dominant case (ii). Clearly, if E, F , and G are Banach spaces and S is bounded from E to G, then S is T -bounded with T -bound 0 for every operator T from E to F . Thus the following corollary is obvious. Corollary 2.8. The block operator matrix A is closed if one of the following holds: (i) (a) D(A) ⊂ D(C) and B is bounded, (b) D(D) ⊂ D(B) and C is bounded, or if one of the following holds:
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(ii) (c) D(C) ⊂ D(A) and D is bounded, (d) D(B) ⊂ D(D) and A is bounded. Suppose that E, F , and G are Banach spaces and that T and S are linear operators acting from E to F and from F to G, respectively. If E and F are reflexive, at least one of the operators T and S is closable, and S is T -compact, then S is T -bounded with T -bound 0 (see [8, Corollary III.7.7]). This immediately implies the next corollary. Corollary 2.9. The block operator matrix A is closed if H1 , H2 are reflexive Banach spaces and one of the following holds: (i) (a) D(A) ⊂ D(C) and B is D-compact, (b) D(D) ⊂ D(B) and C is A-compact, or if one of the following holds: (ii) (c) D(C) ⊂ D(A) and D is B-compact, (d) D(B) ⊂ D(D) and A is C-compact. A more subtle criterion for an operator to have relative bound 0 is a domain inclusion for some fractional power. More exactly, suppose that E, F are Banach spaces, S is closable from E to F , and T is sectorial in E (i.e. (−∞, 0) ⊂ ρ(T ) and (T −λ)−1 < M/|λ|, λ ∈ (−∞, 0), with some M 0). If there exists a γ ∈ (0, 1) with D(T γ ) ⊂ D(S), then S is T -bounded with T -bound 0 (see [17, Corollary 2.6.11]). If T is not sectorial, but E is a Hilbert space, this implication still applies to the sectorial operator |T |. Corollary 2.10. If H1 , H2 are Hilbert spaces, then the block operator matrix A is closed if one of the following holds: (i) (a) D(A) ⊂ D(C) and D(|D|γ ) ⊂ D(B) for some γ ∈ (0, 1), (b) D(D) ⊂ D(B) and D(|A|γ ) ⊂ D(C) for some γ ∈ (0, 1), or if one of the following holds: (ii) (c) D(C) ⊂ D(A) and D(|B|γ ) ⊂ D(D) for some γ ∈ (0, 1), (d) D(B) ⊂ D(D) and D(|C|γ ) ⊂ D(A) for some γ ∈ (0, 1). 3. The quadratic numerical range The quadratic numerical range was first introduced in [15] to enclose the spectrum of block operator matrices with bounded off-diagonal corners B, C; for bounded block operator matrices it was studied in greater detail in [13,14]. Here we generalize it to diagonally dominant and offdiagonally dominant block operator matrices. From now on, we always assume that H1 and H2 are Hilbert spaces; for convenience, we denote both scalar products with (·,·). Definition 3.1. For f ∈ D1 = D(A) ∩ D(C), g ∈ D2 = D(B) ∩ D(D), f = g = 1, let Af,g :=
(Af, f ) (Cf, g)
(Bg, f ) (Dg, g)
∈ M2 (C).
(3.1)
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Then the set of all eigenvalues of all matrices Af,g ,
W 2 (A) :=
(3.2)
σp (Af,g ),
f ∈D1 , g∈D2 , f =g=1
is called the quadratic numerical range of the unbounded block operator matrix A (with respect to the block operator representation (2.1)). Sometimes it is more convenient to have the following, clearly equivalent, description of the quadratic numerical range, which uses non-zero elements f , g that need not have norm one. Remark. For f ∈ D1 , g ∈ D2 , f, g = 0, we set Af,g :=
(Af,f ) f 2 (Cf,g) f g
(Bg,f ) f g (Dg,g) g2
∈ M2 (C)
(3.3)
and (f, g; λ) := det
(Af, f ) − λ(f, f ) (Cf, g)
(Bg, f ) (Dg, g) − λ(g, g)
= f 2 g2 det(Af,g − λ).
Then W 2 (A) =
σp (Af,g )
f ∈D1 , g∈D2 f,g =0
= λ ∈ C: ∃f ∈ D1 , g ∈ D2 , f, g = 0, det(Af,g − λ) = 0
= λ ∈ C: ∃f ∈ D1 , g ∈ D2 , f, g = 0, (f, g; λ) = 0 . In the particular case when A = A∗ , D = D ∗ , and either C ⊂ B ∗ or C ⊂ −B ∗ , the two solutions λ± (f, g) of the quadratic equation det(Af,g − λ) = 0 are given by f 1 (Af, f ) (Dg, g) = λ+ + ± g 2 f 2 g2
(Af, f ) (Dg, g) − f 2 g2
2 +4
|(Bg, f )||(Cf, g)| . f 2 g2
Here the expression under the square root is always real. If C ⊂ B ∗ , it is non-negative; if √ √ C ⊂ −B ∗ , we choose a branch of the square root such that z 0 if z 0 and Im z > 0 if z < 0. The numerical range W (T ) of a closed linear operator T with domain D(T ) in a Hilbert space H is defined as
W (T ) := (T x, x): x ∈ D(T ), x = 1 .
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Whereas the numerical range is always a convex subset of C (see [10, Theorem V.3.1]), the quadratic numerical range of a block operator matrix consists of at most two components which need not be convex either, even in the finite-dimensional case (see [13, p. 110, Fig. 3]). The following proposition shows that the quadratic numerical range is contained in the usual numerical range. Proposition 3.2. W 2 (A) ⊂ W (A). Proof. Let λ0 ∈ W 2 (A). Then there are f ∈ D1 , g ∈ D2 , f = g = 1, and c ∈ C2 , c = 1, with Af,g c = λ0 c. Taking the scalar product with c =: (c1 c2 )t , we find c1 c1 c1 f c1 f , = A , . λ0 = Af,g c2 c2 c2 g c2 g Since c1 f 2 + c2 g2 = |c1 |2 + |c2 |2 = c2 = 1, this implies that λ0 ∈ W (A).
2
Proposition 3.3. The numerical ranges of the diagonal elements A and D satisfy (i) dim H2 2 ⇒ W (A) ⊂ W 2 (A), (ii) dim H1 2 ⇒ W (D) ⊂ W 2 (A). Proof. We prove (i); the proof of (ii) is analogous. Let dim H2 2. Since D2 is dense in H2 , for each f ∈ D1 , f = 1, there exists an element g ∈ D2 , g = 1, with (Cf, g) = 0. To see this, let e1 , e2 ∈ D2 , e1 = e2 = 1, be linearly independent. If (Cf, e1 ) = 0 or (Cf, e2 ) = 0, we can take g = e1 or g = e2 , respectively. If (Cf, e1 ) = 0 and (Cf, e2 ) = 0, there are non-zero constants α1 , α2 ∈ C such that for g = α1 e1 + α2 e2 we have (Cf, g ) = α1 (Cf, e1 ) + α2 (Cf, e2 ) = 0; in this case we choose g := g / g . With this choice of g, we obtain Af,g =
(Af, f ) (Bg, f ) ; 0 (Dg, g)
hence (Af, f ) ∈ σp (Af,g ) ⊂ W 2 (A), and W (A) ⊂ W 2 (A) follows.
2
Note that in the case dim H1 = 1 or dim H2 = 1 the above inclusions do not hold in general, even in the finite-dimensional case (see [13, p. 109, Figs. 1, 2]). 4. Spectral inclusion One of the most important features of the numerical range of a linear operator T is the spectral inclusion property σp (T ) ⊂ W (T ),
σapp (T ) ⊂ W (T )
for the point spectrum and the approximate point spectrum, which is defined as
σapp (T ) := λ ∈ C: ∃(xn )∞ 1 ⊂ D(T ), xn = 1, (T − λ)xn → 0, n → ∞ .
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In the following we prove analogues of these spectral inclusions for the quadratic numerical range of unbounded block operator matrices. Theorem 4.1. σp (A) ⊂ W 2 (A). Proof. If λ0 ∈ σp (A), then there is a non-zero vector (f g)t ∈ D(A) = D1 ⊕ D2 with Af + Bg = λ0 f, Cf + Dg = λ0 g. g ∈ D2 , f = g = 1, such that f = f f, g = g g , and take the If we choose f∈ D1 , scalar product with f and g , respectively, in the above equations, we find (Af, f) + (Bg, f) = λ0 (f, f), (Cf, g ) + (Dg, g ) = λ0 (g, g ). These equations can be written equivalently as f f = λ Af, 0 g g g 2 and hence λ0 ∈ σp (Af, g ) ⊂ W (A).
2
Theorem 4.2. If A is diagonally dominant of order 0, then σapp (A) ⊂ W 2 (A). t ∞ Proof. Let λ0 ∈ σapp (A). Then there exists a sequence (fn )∞ 1 = ((fn gn ) )1 ⊂ D(A) = D(A) ⊕ D(D), fn 2 + gn 2 = 1, such that (A − λ0 )fn → 0, n → ∞, i.e.
(A − λ0 )fn + Bgn =: hn → 0, Cfn + (D − λ0 )gn =: kn → 0,
n → ∞.
(4.1)
Since the dominance order of A is 0 and hence < 1, the operator S=
0 C
B 0
is A-bounded by Corollary 2.3. Thus (A − λ0 )fn → 0, n → ∞, implies that (Sfn )∞ 1 and hence ∞ are bounded. Then, by (4.1), ((A − λ )f )∞ and ((D − λ )g )∞ are and (Cf ) also (Bgn )∞ n 0 n 0 n 1 1 1 1 bounded as well. gn ∈ D(D) so that fn = gn = 1, fn = fn fn , gn = gn gn Now choose fn ∈ D(A), for n ∈ N, and consider (Afn , fn ) − λ (B gn , fn ) (fn , gn ; λ) = det , λ ∈ C. (C fn , gn ) (D gn , gn ) − λ
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Assume first that lim infn→∞ fn > 0 and lim infn→∞ gn > 0, without loss of generality fn , gn > 0, n ∈ N. By (4.1), we have 1 (hn , fn ) − gn (B gn , fn ) , fn 1 (kn , (C fn , gn ) = gn ) − gn (D gn , gn ) − λ 0 , fn
(Afn , fn ) − λ0 =
and thus (fn , gn ; λ0 ) = det
1 fn (hn , fn ) 1 gn ) fn (kn ,
(B gn , fn )
(D gn , gn ) − λ 0
(4.2)
.
The elements of the first column tend to 0 and the sequences with elements B gn = Bgn /gn gn = (D − λ0 )gn /gn are bounded. Hence and (D − λ0 ) (fn , gn ; λ0 ) → 0,
n → ∞.
gn ; ·) is a monic quadratic polynomial, we can write As (fn , (fn , gn ; λ) = λ − λ1n λ − λ2n ,
n ∈ N,
(4.3)
where λ1n , λ2n are the solutions of the quadratic equation (fn , gn ; λ) = 0 and so λ1n , λ2n ∈ W 2 (A). 1 2 From (4.2) and (4.3) it follows that λn → λ0 or λn → λ0 , n → ∞, and thus λ0 ∈ W 2 (A). Next we consider the case when lim infn→∞ gn = 0. Then fn 2 + gn 2 = 1 implies that lim infn→∞ fn > 0. Without loss of generality, we may assume that limn→∞ gn = 0 and fn > γ , n ∈ N, for some γ ∈ (0, 1]. First suppose that dim H2 2. We show that Bgn → 0, n → ∞. For this let ε > 0 be arbitrary. Since ((D − λ0 )gn )∞ 1 is bounded, there exists an M > 0 such that (D − λ0 )gn < M, n ∈ N. Because A is diagonally dominant of order 0, the operator B is D-bounded with D-bound 0 and hence there exist constants aB , bB 0 such that bB < ε/(2M) and Bgn aB gn + bB Dgn aB + bB |λ0 | gn + bB (D − λ0 )gn ,
n ∈ N.
If we choose N ∈ N such that gn < ε/(2(aB + bB |λ0 |)), n N , it follows that Bgn < ε for n N . Hence Bgn → 0, n → ∞. This and the first relation in (4.1) show that (A − λ0 )fn → 0, n → ∞, and so λ0 ∈ σapp (A). Together with Proposition 3.3, we obtain λ0 ∈ σapp (A) ⊂ W (A) ⊂ W 2 (A). If dim H2 = 1, then B and D are bounded operators. Then gn → 0 implies that Bgn → 0, (D − gn chosen as above, (A − λ0 )fn → 0, C fn → λ0 )gn → 0, n → ∞, and hence by (4.1), with fn , 0, n → ∞ (note that fn −1 γ −1 , n ∈ N). The boundedness of B and D also implies that gn )∞ (B gn )∞ 1 , ((D − λ0 ) 1 are bounded. Therefore (fn , gn ; λ0 ) = det
(Afn , fn ) − λ0 (C fn , gn )
(B gn , fn ) (D gn , gn ) − λ 0
→ 0,
n → ∞.
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Now, in the same way as above, it follows that λ0 ∈ W 2 (A). The case lim infn→∞ fn = 0 is treated analogously if we use that C is A-bounded with A-bound 0. 2 The next corollary is an immediate consequence of Theorem 4.2 if we observe that boundedness and relative compactness both imply relative boundedness with relative bound 0 (see [8, Corollary III.7.7] and the introductory remarks in Section 2). Corollary 4.3. If B is bounded or D-compact and C is bounded or A-compact, then σapp (A) ⊂ W 2 (A). Corollary 4.3 shows that Theorem 4.2 generalizes [15, Theorem 2.1], where it was assumed that B and C are bounded; note that, by the closed graph theorem, the approximate point spectrum σapp (A) is the complement of the set r(A) of points of regular type of A, σapp (A) = C \ r(A), where
r(A) := λ ∈ C: ∃Cλ > 0, (A − λ)x Cλ x, x ∈ D(T ) . The inclusion of the approximate point spectrum for the off-diagonally dominant case has not been considered so far. Theorem 4.4. If A is off-diagonally dominant of order 0 and B, C are boundedly invertible, then σapp (A) ⊂ W 2 (A). Proof. The first part of the proof is analogous to the proof of Theorem 4.2, now with fn , fn ∈ gn ∈ D(B); we continue to use the same notation. In fact, by Corollary 2.3, the D(C) and gn , operator T =
A 0 0 D
is A-bounded. Thus, in the same way as in the proof of Theorem 4.2, we can show that ∞ ∞ ∞ λ0 ∈ σapp (A) implies that all sequences ((A − λ0 )fn )∞ 1 , (Bgn )1 , (Cfn )1 , ((D − λ0 )gn )1 are gn ; λ0 ) → 0, n → ∞, if lim infn→∞ fn > 0 and lim infn→∞ gn > 0. bounded, and (fn , Hence λ0 ∈ W 2 (A) in this case. It remains to consider the case lim infn→∞ gn = 0; the case lim infn→∞ fn = 0 is analogous. Again, without loss of generality, we may suppose that limn→∞ gn = 0 and fn > γ , n ∈ N, with some γ ∈ (0, 1]. If dim H2 = 1, the proof is the same as the respective part of the proof of Theorem 4.2. If ∞ dim H2 2, we prove that (Dgn )∞ 1 tends to 0. Let ε > 0 be arbitrary. Since (Bgn )1 is bounded, there exists an M > 0 such that Bgn < M, n ∈ N. Because A is off-diagonally dominant of order 0, the operator D is B-bounded with B-bound 0 and hence there exist constants aD , bD 0 such that bD < ε/(2M) and Dgn aD gn + bD Bgn ,
n ∈ N.
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If we choose N ∈ N such that gn < ε/(2aD ), n N , it follows that Dgn < ε for n N . ∞ Since C is boundedly invertible, the fact that (gn )∞ 1 , (Dgn )1 tend to 0 and the second relation in (4.1) yield fn = −C −1 (D − λ0 )gn + C −1 kn → 0, a contradiction to fn > γ > 0, n ∈ N.
n → ∞,
2
Corollary 4.5. If A is bounded or C-compact, D is bounded or B-compact, and if B, C are boundedly invertible, then σapp (A) ⊂ W 2 (A). The following example shows that Theorem 4.4 and Corollary 4.5 do not hold without the assumption that B, C are boundedly invertible. Example 4.6. Let A = C = D = 0 and let B be a bijective closed linear operator from H2 to H1 with dense domain D(B) H2 . Then the block operator matrix A=
0 B 0 0
,
D(A) = H1 ⊕ D(B),
is closed and off-diagonally dominant with W 2 (A) = {0}. If λ ∈ C \ {0}, then A − λ is injective, the range R(A − λ) = H1 ⊕ D(B) H1 ⊕ H2 is dense and hence λ0 ∈ σc (A) ⊂ σapp (A); if λ = 0, then A − λ is not injective and hence λ0 ∈ σp (A) ⊂ σapp (A). Thus σapp (A) = C is not contained in W 2 (A) = {0}. In contrast to the bounded case, for an unbounded linear operator the inclusion σ (T ) ⊂ W (T ) of the spectrum only holds if every component of C \ W (T ) contains a point μ ∈ ρ(T ), or, equivalently, a point μ with R(T − μ) = H (see [10, Theorem V.3.2]). The analogue for the quadratic numerical range is as follows. Theorem 4.7. Let A be closed and either diagonally dominant of order 0 or off-diagonally dominant of order 0 with B, C boundedly invertible in the latter case. If Ω is a component of C \ W 2 (A) that contains a point μ ∈ ρ(A), then Ω ⊂ ρ(A); in particular, if every component of C \ W 2 (A) contains a point μ ∈ ρ(A), then σ (A) ⊂ W 2 (A). Proof. For every point λ ∈ r(A), the range R(A − λ) is closed and the mapping λ → dim R(A − λ)⊥ is constant on every component of r(A) (see e.g. [10, Theorem V.3.2]). Thus, by Theorem 4.2 or 4.4, respectively, the same is true on each component of C \ W 2 (A) ⊂ C \ σapp (A) = r(A). By assumption, it follows that R(A − λ) = H for all λ ∈ C \ W 2 (A), that is, C \ W 2 (A) ⊂ ρ(A). 2
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Remark. Since the numerical range is convex, the complement C \ W (A) has at most two components. So far there is no information about the number of components of the complement C \ W 2 (A) of the closure of the quadratic numerical range. The following lemma may be used to verify the assumptions of Theorem 4.7. Lemma 4.8. Let E, F be Banach spaces and let T , S be linear operators from E to F . Suppose that there exists a ray Θρ,ϕ := {reiϕ : r ρ} with ρ 0, ϕ ∈ (−π, π] and a constant M 0 such that Θρ,ϕ ⊂ ρ(T ) and (T − λ)−1 M , |λ|
λ ∈ Θρ,ϕ .
(4.4)
If S is T -bounded with T -bound < 1/(M + 1), then there is an R ρ with ΘR,ϕ ⊂ ρ(T + S). Proof. Let λ ∈ Θρ,ϕ . By the assumption on S, there exist aS , bS 0 with bS < 1/(M + 1) and Sx aS x + bS T x aS + bS |λ| x + bS (T − λ)x ,
x ∈ D(T ).
The theorem on the stability of bounded invertibility (see [10, Theorem IV.1.16]), applied to T − λ and S, shows that λ ∈ ρ(T + S) if aS + bS |λ| (T − λ)−1 + bS < 1. Due to assumption (4.4), this inequality is satisfied if aS
M + (1 + M)bS < 1. |λ|
The latter holds if |λ| R with R ρ such that R>
aS M . (1 − (1 + M)bS )
2
Remark. If we weaken the assumptions of this paper and allow the dominanting entries A, D of the diagonally dominant block operator matrix A in Theorem 4.2 (and hence A itself) to be closable, then the quadratic numerical range of A still provides spectral enclosures for the closure A by means of the inclusions σp (A) ⊂ σapp (A),
σapp (A) = σapp (A).
(4.5)
Note that, in the off-diagonally dominant case in Theorem 4.4, the dominating entries B, C are always closed since they are assumed to be boundedly invertible.
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5. Applications In this section we use the quadratic numerical range to enclose the spectrum of three different classes of unbounded block operator matrices: block operator matrices with separated diagonal entries and symmetric corners (Re W (D) < Re W (A), and C = B ∗ ); block operator matrices with self-adjoint diagonal entries and anti-symmetric corners (A = A∗ , D = D ∗ , and C = −B ∗ ); and self-adjoint block operator matrices with semi-bounded diagonal entries. In all three cases, the quadratic numerical range yields tighter spectral enclosures than classical perturbation theorems or the usual numerical range. In order to estimate the quadratic numerical range for the above classes of block operator matrices we use the following elementary lemma for 2 × 2 matrices. Lemma 5.1. Let a, b, c, d ∈ C be complex numbers and A :=
a c
b d
∈ M2 (C).
Then the eigenvalues λ1 , λ2 of A have the following properties: (i) If Re d < 0 < Re a and bc 0, then (a) Re λ2 Re d < 0 < Re a Re λ1 , (b) min{Im a, Im d} Im λ1 , Im λ2 max{Im a, Im d}, (c) λ1 , −λ2 ∈ {z ∈ C: | arg z| max{| arg a|, π − | arg d|}}. (ii) If Re d < Re a and bc 0, then (a) Re d Re λ2 Re √λ1 Re a, √ √ (b) Re λ2 Re d + |bc| < Re a − |bc| Re λ1 if |bc| < (Re a − Re d)/2, and λ1 , λ2 ∈ R if, in addition, a, d ∈ R, (c) √ Re λ1 = Re λ2 = (a + d)/2, | Im λ1 | = | Im λ2 | = |bc| − (a − d)2 /4 if a, d ∈ R and |bc| (a − d)/2. (iii) If a, d ∈ R and c = b (the complex conjugate of b), then 2|b| 1 arctan , λ1 = max{a, d} + |b| tan 2 |a − d| 2|b| 1 λ2 = min{a, d} − |b| tan arctan . 2 |a − d| Proof. (i) All statements were proved in [15, Lemma 3.1]. (ii)(a) If Re λ < Re d (< Re a) or Re λ > Re a (> Re d), then the eigenvalue equation (a − λ)(d − λ) = bc 0 cannot hold. In fact, decomposing all numbers therein into real and imaginary parts, one can show that Im a − Im λ and Im d − Im λ have different signs and Re((a − λ)(d − √ λ)) > 0. √ (b) If Re d + |bc| < Re λ < Re a − |bc|, then det(A − λ) |a − λ||d − λ| − |bc| | Re a − Re λ|| Re d − Re λ| − |bc| > 0, hence λ is not an eigenvalue of A. The √relation Re λ1 + Re λ2 = Re a + Re d excludes the possibility that e.g. Re λ1 , Re λ2 Re d + |bc|.
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Fig. 1. Assumptions on A and D in Theorem 5.2.
The assertions in (b) for a, d ∈ R and in (c) are immediate from the formula λ1/2 =
a+d ± 2
(a − d)2 + bc. 4
(5.1)
(iii) The claim can be easily deduced from formula (5.1) with c = b for the solutions of the quadratic equation (a − λ)(d − λ) − |b|2 = 0. 2 The next theorem shows that if Re W (D) < Re W (A) and C = B ∗ , then the gap between the diagonal entries A and D is retained as a spectral gap for the whole block operator matrix A; it is remarkable that this holds even in the off-diagonally dominant case where B ∗ and B are stronger than A and D, respectively. Theorem 5.2. Let the block operator matrix A be of the form A=
A B∗
B D
and define the sector Σω := {reiφ : r 0, |φ| ω} for ω ∈ [0, π). If there exist α, δ > 0 and angles ϕ, ϑ ∈ [0, π/2] such that W (D) ⊂ {z ∈ −Σϕ : Re z −δ},
W (A) ⊂ {z ∈ Σϑ : Re z α}
(see Fig. 1), then, with θ := max{ϕ, ϑ}, σp (A) ⊂ {z ∈ −Σθ : Re z −δ} ∪˙ {z ∈ Σθ : Re z α}. Suppose, in addition, that A is either diagonally dominant or off-diagonally dominant of order 0 and, in the latter case, B is boundedly invertible. If there exists a point μ ∈ ρ(A) ∩ ρ(D) ∩ {z ∈ C: −δ < Re z < α}, then σ (A) ⊂ {z ∈ −Σθ : Re z −δ} ∪˙ {z ∈ Σθ : Re z α}.
Proof. The above elementary Lemma 5.1(i) (see also [15, Lemma 3.1]) yields ˙ ∈ Σθ : Re z α} =: Ξ σp (Af,g ) ⊂ {z ∈ −Σθ : Re z −δ}∪{z
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for all f ∈ D1 , g ∈ D2 , f = g = 1, and thus W 2 (A) ⊂ W 2 (A) ⊂ Ξ . By Theorem 4.2, we then have σp (A) ⊂ W 2 (A) ⊂ Ξ . For the inclusion of the spectrum, we first note that in the off-diagonally dominant case, B ∗ is boundedly invertible since so is B. The inclusion σ (A) ⊂ W 2 (A) follows from Theorems 4.4 and 4.7 if we show that C \ Ξ contains a point μ ∈ ρ(A). To prove this, we consider the block operator matrices T :=
A 0 , 0 D
S :=
0 B∗
B 0
.
Due to the assumptions on A, D, we have σ (A) ⊂ W (A), σ (D) ⊂ W (D) and −1 (T − iμ)−1 (cos θ ) , |μ|
μ ∈ R \ {0}.
If B is boundedly invertible, it is closed and hence S is self-adjoint so that (S − iμ)−1 1 , |μ|
μ ∈ R \ {0}.
If A is diagonally dominant of order 0, then S is T -bounded with T -bound 0; if A is offdiagonally dominant of order 0, then T is S-bounded with S-bound 0 (see Proposition 2.3). In both cases, the assumptions of Lemma 4.8 are satisfied and so {iμ: |μ| R} ⊂ ρ(T + S) = ρ(A) for some R > 0. 2 Corollary 5.3. If A is self-adjoint, then, clearly, there exists a point μ ∈ ρ(A) ∩ ρ(D) ∩ {z ∈ C: −δ < Re z < α}; in this case, ˙ [α, ∞). σ (A) ⊂ (−∞, −δ] ∪ Remark. For bounded B, the spectral inclusion in Corollary 5.3 seems to have been proved first by C. Davis and W.M. Kahan using some geometric considerations on the rotation of spectral subspaces (see [7, Theorem 8.1]); another proof was given by V. Adamjan and H. Langer who showed that the inverse of the Schur complement S1 (λ) := A − λ − B(D − λ)−1 B ∗ , and thus the resolvent (A − λ)−1 , exist for λ ∈ (−δ, α) (see [1, Theorem 2.1]). For unbounded B, J. Weidmann gave a prove using the spectral theorem (see [19, Theorem 7.25]); in addition, A.K. Motovilov and A.V. Selin established sharp bounds on the change of the spectral subspaces of A corresponding to the intervals (−∞, −δ] and [α, ∞) (see [16, Theorem 1]). Examples for off-diagonally dominant self-adjoint block operator matrices with separated diagonal elements are furnished by Dirac operators in R3 , which describe the behaviour of a quantum mechanical particle of spin 1/2 (see [18]). Example 5.4. Denote by m and e the mass and the charge, respectively, of a relativistic spin 1/2 particle, by c the velocity of light, by h¯ the Planck constant, and by σ = (σ 1 , σ 2 , σ 3 ) the vector of the Pauli spin matrices σ 1 :=
0 1 , 1 0
σ 2 :=
0 i
−i , 0
σ 3 :=
1 0 . 0 −1
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Let φ : R3 → R be a scalar potential and A = (a1 , a2 , a3 ) : R3 → R3 a vector potential generating respectively. Then the Dirac operator an electric field E = ∇φ and a magnetic field B = rot A, in R3 with electromagnetic potential is the block operator matrix (see [18, (4.14)]) HΦ =
mc2 + eφ cσ · (−ih¯ ∇ − ec A)
cσ · (−ih¯ ∇ − ec A) −mc2 + eφ
(5.2)
in L2 (R3 )4 = L2 (R3 )2 ⊕ L2 (R3 )2 . If we suppose that the vector potential satisfies 3 A ∈ L2,loc R3 ,
∈ L∞ R 3 , A
∈ L3/2 R3 B
(5.3)
and set := y ∈ L2 R3 2 : ih¯ ∂ν + e aν y ∈ L2 R3 2 , ν = 1, 2, 3 , H1 (A) c is self-adjoint (see [9, Theorem 2.2]; then the operator H0 with domain D(H0 ) = H1 (A)⊕H 1 (A) 3 note that the condition A ∈ L∞ (R ) in (5.3) may be replaced by A being locally uniformly Hölder continuous with exponent η ∈ [0, 1]). For bounded scalar potential φ, the Dirac operator ⊕ H1 (A). If eφ < mc2 , then the assumptions HΦ is self-adjoint on D(HΦ ) = D(H0 ) = H1 (A) 2 of Theorem 5.2 are satisfied (with α = mc − eφ, δ = −mc2 + eφ, ϕ = ϑ = 0) and we immediately obtain σ (HΦ ) ∩ −mc2 + eφ, mc2 − eφ = ∅, that is, the gap between the diagonal entries of HΦ is retained as a spectral gap for the Dirac operator HΦ . Next we consider the case A = A∗ , D = D ∗ , and C = −B ∗ ; note that the corresponding block operator matrix A is symmetric with respect to the indefinite inner product [·,·] := (J ·, ·) on H1 ⊕ H2 with J :=
I 0 0 −I
,
that is, J A is symmetric in H. Theorem 5.5. Let the block operator matrix A of the form A=
A −B ∗
B D
with A = A∗ , D = D ∗ be either diagonally dominant or off-diagonally dominant of order 0 and, in the latter case, let B be boundedly invertible.
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(i) If W 2 (A) ⊂ R, then W 2 (A) consists of one or two intervals and σ (A) ⊂ W 2 (A) ⊂ W (A). (ii) If A and D are bounded from below, a− := min σ (A),
d− := min σ (D),
then σ (A) ∩ R ⊂ min{a− , d− }, ∞ ,
a − + d− . σ (A) \ R ⊂ z ∈ C: Re z 2
analogous statements hold for A and D bounded from above. (iii) If B is bounded, then σ (A) \ R ⊂ {z ∈ C: | Im z| B}; if, in addition, we have δ := dist(W (A), W (D)) > 0, then B δ/2
⇒
B > δ/2
⇒
σ (A) ⊂ R,
σ (A) \ R ⊂ z ∈ C: | Im z| B2 − δ 2 /4 .
Proof of Theorem 5.1. In a similar way as in the proof of Theorem 5.2, we consider the block operator matrices T :=
A 0
0 , D
S :=
0 −B ∗
B 0
;
T is self-adjoint and, if B is boundedly invertible and thus closed, so is iS. (i) If W 2 (A) ⊂ R, its at most two components must be intervals. By Theorem 4.7, for the inclusion σ (A) ⊂ W 2 (A), it suffices to prove that there exist points μ+ , μ− ∈ ρ(A) in the upper and lower half-plane, respectively. To this end, we note that (T − iμ)−1 1 , μ ∈ R \ {0}, |μ| −1 S − (ν ± iμ0 ) 1 , ν ∈ R \ {0}, |ν| with an arbitrary fixed μ0 ∈ (0, ∞). As in the proof of Theorem 5.2, we find that the assumptions of Lemma 4.8 are satisfied for T and S ∓ iμ0 , respectively. As a consequence, there exists an R > 0 such that {±iμ: |μ| R} ⊂ ρ(T + S) = ρ(A) in the diagonally dominant case and {ν ± iμ0 : |ν| R} ⊂ ρ(S + T ) = ρ(A) in the off-diagonally dominant case. (ii) Lemma 5.1(ii), applied to the 2 × 2 matrices Af,g , yields that the asserted inclusions hold with W 2 (A) instead of σ (A). By Theorem 4.7, it suffices to prove that there is a μ ∈ ρ(A) in the half-plane {z ∈ C: Re z < (a− + d− )/2}. Note that
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T − (ν0 + iμ) −1 1 , |μ| (S − μ)−1 1 , |μ|
μ ∈ R \ {0}, μ ∈ R \ {0},
with an arbitrary fixed ν0 ∈ (−∞, (a− + d− )/2). Hence, by Lemma 4.8 applied to T − ν0 and S, respectively, there is an R > 0 with {ν0 + iμ: |μ| R} ⊂ ρ(T + S) = ρ(A) in the diagonally dominant case and {±μ: |μ| R} ⊂ ρ(S + T ) = ρ(A) in the off-diagonally dominant case. (iii) If B is bounded, then A is a bounded perturbation of the self-adjoint operator T and the first claim follows from standard perturbation theorems (see e.g. [10, Problem V.4.8]); in particular, there are μ+ , μ− ∈ ρ(A) in the upper and lower half-plane, respectively. As in the proof of (ii), the estimates in the case δ > 0 follow from Lemma 5.1(ii), applied to the 2 × 2 matrices Af,g . 2 Remark. Under some additional assumptions on the positions of σ (A) and σ (D), which guarantee that the spectrum of A is real, S. Albeverio, A.K. Motovilov, and A.A. Shkalikov proved spectral enclosures finer than those in Theorem 5.5(iii) (see [2, Theorem 8.1]). Their method relies on sharp norm bounds for the solution K of the algebraic Riccati equation KA − DK = C − KBK associated with a block operator matrix (1.2) and even applies in more general situations, e.g. when C = −B ∗ . Finally, we consider the case A = A∗ , D = D ∗ , C = B ∗ and bounded B, where the corresponding block operator matrix A is self-adjoint. It is a classical problem dating back to C. Davis and W.M. Kahan (see [5–7]) to study the perturbation of the spectrum of the block diagonal operator diag(A, D) in terms of the operator B. If A and D (and hence A) are bounded, then V. Kostrykin, K.A. Makarov, and A.K. Motovilov recently obtained sharp estimates for the spectrum of A by means of the quadratic numerical range (see [12]). If A and/or D are unbounded, but both bounded from below or both bounded from above, then Theorem 4.2 allows us to generalize [12, Lemma 1.1] and obtain lower and upper bounds, respectively, for the spectrum of A. Theorem 5.6. Let the block operator matrix A be of the form A=
A B∗
B D
with A = A∗ , D = D ∗ either both semi-bounded from below or from above and bounded B. Then the spectrum of A satisfies the following estimates: (i) If A and D are bounded from below and δB−
2B 1 arctan , := B tan 2 | min σ (A) − min σ (D)|
then A is bounded from below with
min min σ (A), min σ (D) − δB− min σ (A) min min σ (A), min σ (D) .
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(ii) If A and D are bounded from above and 1 2B , δB+ := B tan arctan 2 | max σ (A) − max σ (D)| then A is bounded from above with
max max σ (A), max σ (D) max σ (A) max max σ (A), max σ (D) + δB+ . Proof. The proof of the fact that the quadratic numerical range satisfies the stated estimates is analogous to the proof of [12, Lemma 1.1]. In order to obtain the estimates of the spectrum, we use Corollary 4.3, Theorem 4.7, and the fact that C \ R ⊂ ρ(A) since A is self-adjoint. 2 In the special case when the spectra of the diagonal entries A, D have positive distance, a combination of Theorems 5.2 and 5.6 yields the following generalization of [12, Theorem 1.3]. Theorem 5.7. Let the block operator matrix A be of the form A=
A B∗
B D
with A = A∗ , D = D ∗ , δA,D := dist(σ (A), σ (D)) > 0, and bounded B. Define 2B 1 . arctan δB := B tan 2 δA,D (i) Then
σ (A) ⊂ λ ∈ C: dist λ, σ (A) ∪˙ σ (D) δB . (ii) If B <
√ ˙ σ2 with σ1 , σ2 = ∅ and 3δA,D /2, then δB < δA,D /2 and σ (A) = σ1 ∪
σ1 ⊂ λ ∈ C: dist λ, σ (A) δB ⊂ λ ∈ C: dist λ, σ (A) < δA,D /2 ,
σ2 ⊂ λ ∈ C: dist λ, σ (D) δB ⊂ λ ∈ C: dist λ, σ (D) < δA,D /2 .
(iii) If (conv σ (A)) ∩ σ (D) = ∅ and B < σ1 , σ2 = ∅ and
√
2δA,D , then δB < δA,D and σ (A) = σ1 ∪˙ σ2 with
σ1 ⊂ λ ∈ C: dist λ, σ (A) δB ⊂ λ ∈ C: dist λ, σ (A) < δA,D ,
σ2 ⊂ λ ∈ C: dist λ, σ (A) δA,D . Proof. Since A is self-adjoint and hence C \ R ⊂ ρ(A), the inclusion σ (A) ⊂ W 2 (A) holds by Corollary 4.3 and Theorem 4.7. That W 2 (A) satisfies the estimates claimed for the spectrum follows in the same way as [12, Theorem 1.3] if we use Theorem 5.6 and Corollary 5.3. 2
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Remark. Under the assumptions of Theorem 5.7(iii), V. Kostrykin, K.A. Makarov, and A.K. Motovilov used the Schur complement S1 (λ) := A − λ − B(D − λ)−1 B ∗ to establish an even finer estimate σ (A) taking into account the exact position of σ (A) with respect to σ (D) (see [11, Theorem 3.2]). As an example, we apply Theorem 5.6 to two-channel Hamiltonians arising in non-relativistic quantum mechanics (see [3,4]). Example 5.8. A simple model of interaction between a confined channel (e.g. a quark/anti-quark system) and a scattering channel (e.g. a two-hadron system) leads to a two-channel Hamiltonian of the form Hc V V −∇ 2 /2 + Uc =: HV := V −∇ 2 /2 V Hs in the Hilbert space L2 (R3 ) ⊕ L2 (R3 ); here h, ¯ the quark mass and the hadronic ground state mass have been normalized to unity (see [3]). For the Hamiltonian Hc = −∇ 2 /2 + Uc in the confined channel, we assume that the potential Uc is such that Hc is self-adjoint in L2 (R3 ) and bounded from below, Hc −ω0 where ω0 > 0, with discrete spectrum σ (Hc ) = σp (Hc ) = {μn : n ∈ N0 } ⊂ [−ω0 , ∞) accumulating only at ∞; this includes e.g. the 3-dimensional harmonic oscillator. The unperturbed Hamiltonian Hs = −∇ 2 /2 in the scattering channel, which is the kinetic energy operator for relative motion between the two hadrons, is self-adjoint and non-negative in L2 (R3 ) with continuous spectrum σ (Hs ) = σc (Hs ) = [0, ∞). The off-diagonal terms V and V represent the coupling between the channels. If the potential V is essentially bounded, V ∈ L∞ (R3 ), then HV is self-adjoint on D(HV ) = D(Hc ) ⊕ D(Hs ) and semi-bounded. Whereas classical perturbation theory yields σ (HV ) ⊂ [−ω0 − V , ∞), Theorem 5.6 provides the sharper lower bound σ (HV ) ⊂ −ω0 − δV− , ∞ ,
δV−
2V 1 < V , arctan := V tan 2 ω0
since | min σ (Hc ) − min σ (Hs )| = ω0 > 0. Summary The spectral enclosures established in the above theorems can neither be obtained by perturbation theory nor by means of the usual numerical range; in fact, the spectral shift is either independent of the perturbation or, in the bounded case, strictly less than the norm of the perturbation, more precisely: (i) In Theorem 5.2, the strip {z ∈ C: −δ < Re z < α} that remains free of spectrum is not affected by the size of the operator B at all; moreover, it is impossible to show that this strip separates the spectrum into two parts by means of the numerical range because of its convexity. (ii) In Theorem 5.5(ii), the lower (upper, respectively) bounds for the real part of real and the non-real spectrum do not depend on the size of the operator B at all; in Theorem 5.5(iii), the estimate for the modulus of the imaginary part of the spectrum is less than B.
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(iii) In Theorem 5.6, if min σ (A) = min σ (D) (max σ (A) = max σ (D), respectively), then the spectrum shifts at most by δB− < B to the left (by δB+ < B to the right, respectively). (iv) In Theorem 5.7, the spectrum expands at most by δB < B. Whereas classical perturbation theory only yields that the spectrum of A remains separated in two parts as long as √ /2, we can improve this bound by a factor of 3 in general and by a factor of B < δ A,D √ 2 2 if, in addition, (conv σ (A)) ∩ σ (D) = ∅. References [1] V.M. Adamjan, H. Langer, Spectral properties of a class of rational operator valued functions, J. Operator Theory 33 (2) (1995) 259–277. [2] S. Albeverio, A.K. Motovilov, A.A. Shkalikov, Bounds on variation of spectral subspaces under J-self-adjoint perturbations, arXiv: 0808.2783, 2008. [3] R.F. Dashen, J.B. Healy, I.J. Muzinich, Potential scattering with confined channels, Ann. Physics 102 (1) (1976) 1–70. [4] R.F. Dashen, J.B. Healy, I.J. Muzinich, Theory of multichannel potential scattering with permanently confined channels, Phys. Rev. D (3) 14 (10) (1976) 2773–2789. [5] C. Davis, The rotation of eigenvectors by a perturbation, J. Math. Anal. Appl. 6 (1963) 159–173. [6] C. Davis, The rotation of eigenvectors by a perturbation. II, J. Math. Anal. Appl. 11 (1965) 20–27. [7] C. Davis, W.M. Kahan, The rotation of eigenvectors by a perturbation. III, SIAM J. Numer. Anal. 7 (1970) 1–46. [8] D.E. Edmunds, W.D. Evans, Spectral Theory and Differential Operators, Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York, 1987, Oxford Science Publications. [9] W.D. Evans, R.T. Lewis, Eigenvalue estimates in the semi-classical limit for Pauli and Dirac operators with a magnetic field, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 455 (1981) (1999) 183–217. [10] T. Kato, Perturbation Theory for Linear Operators, Classics Math., Springer, Berlin, 1995, reprint of the 1980 edition. [11] V. Kostrykin, K.A. Makarov, A.K. Motovilov, On the existence of solutions to the operator Riccati equation and the tan Θ theorem, Integral Equations Operator Theory 51 (1) (2005) 121–140. [12] V. Kostrykin, K.A. Makarov, A.K. Motovilov, Perturbation of spectra and spectral subspaces, Trans. Amer. Math. Soc. 359 (1) (2007) 77–89. [13] H. Langer, A.S. Markus, V.I. Matsaev, C. Tretter, A new concept for block operator matrices: The quadratic numerical range, Linear Algebra Appl. 330 (1–3) (2001) 89–112. [14] H. Langer, A.S. Markus, C. Tretter, Corners of numerical ranges, in: Recent Advances in Operator Theory, Groningen, 1998, in: Oper. Theory Adv. Appl., vol. 124, Birkhäuser, Basel, 2001, pp. 385–400. [15] H. Langer, C. Tretter, Spectral decomposition of some nonselfadjoint block operator matrices, J. Operator Theory 39 (2) (1998) 339–359. [16] A.K. Motovilov, A.V. Selin, Some sharp norm estimates in the subspace perturbation problem, Integral Equations Operator Theory 56 (4) (2006) 511–542. [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., vol. 44, Springer, New York, 1983. [18] B. Thaller, The Dirac Equation, Texts Monogr. Phys., Springer, Berlin, 1992. [19] J. Weidmann, Linear Operators in Hilbert Spaces, Grad. Texts in Math., vol. 68, Springer, New York, 1980, translated from the 1976 German original.
Journal of Functional Analysis 256 (2009) 3830–3840 www.elsevier.com/locate/jfa
New examples of c0-saturated Banach spaces II I. Gasparis Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece Received 16 September 2008; accepted 26 November 2008 Available online 5 December 2008 Communicated by N. Kalton
Abstract For every Banach space Z with a shrinking unconditional basis satisfying an upper p-estimate for some p > 1, an isomorphically polyhedral Banach space is constructed which has an unconditional basis and admits a quotient isomorphic to Z. It follows that reflexive Banach spaces with an unconditional basis and non-trivial type, Tsirelson’s original space and ( c0 )p for p ∈ (1, ∞), are isomorphic to quotients of isomorphically polyhedral Banach spaces with unconditional bases. © 2008 Elsevier Inc. All rights reserved. Keywords: c0 -saturated space; Polyhedral space; Upper p estimates; Quotient map
1. Introduction An infinite-dimensional Banach space is c0 -saturated if every closed, linear, infinite-dimensional subspace contains a closed, linear subspace isomorphic to c0 . It is classical result [16] that every C(K) space with K being a countable infinite, compact metric space, is c0 -saturated. This result was generalised in [8] to the class of the so-called Lindenstrauss–Phelps spaces, i.e., spaces whose dual closed unit ball has but countably many extreme points. These spaces in turn (see [9]), belong to the class of the isomorphically polyhedral spaces. We recall that a Banach space is polyhedral if the closed unit ball of each of its finite-dimensional subspaces has finitely many extreme points. It is isomorphically polyhedral if it is polyhedral under an equivalent norm. It was proved in [9] that separable isomorphically polyhedral spaces are c0 -saturated. Not much is known about the behavior of isomorphically polyhedral, or more generally c0 saturated spaces, under quotient maps. It was asked in [17] if the dual of a separable isomorphiE-mail address: [email protected]. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.021
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cally polyhedral space is 1 -saturated that is, every closed, linear, infinite-dimensional subspace of the dual contains a further subspace isomorphic to 1 . It is an open problem [15,17,18] if every quotient of a C(K) space with K a countable and compact metric space, is c0 -saturated. It was shown in [10] that for every p ∈ (1, ∞), p is isomorphic to a quotient of an isomorphically polyhedral space with an unconditional basis. The purpose of this article is to extend this result by showing the following. Theorem 1.1. Let Z be a Banach space with a shrinking, unconditional basis satisfying an upper p-estimate for some p > 1. Then, there exists an isomorphically polyhedral space with an unconditional basis which admits a quotient isomorphic to Z. We obtain in particular, that reflexive spaces with unconditional bases and non-trivial type, are isomorphic to quotients of isomorphically polyhedral spaces with unconditional bases. The same property holds for ( c0 )p , for all p ∈ (1, ∞). Using the fact that Tsirelson’s space T [7] is isomorphic to its modified version [5,12], we also obtain that Tsirelson’s original space T ∗ [19] is isomorphic to a quotient of an isomorphically polyhedral space with an unconditional basis. We note that in [10] it is shown that for every p ∈ (1, ∞), there exists an isomorphically polyhedral space Ep , with an unconditional basis, which admits a quotient isomorphic to p . Moreover, Ep is not isomorphic to a subspace of a C(K) space for any countable and compact metric space K. In the present construction, starting with a reflexive space with an unconditional basis and non-trivial type, we do not know if the resulting isomorphically polyhedral space embeds isomorphically into some C(K) space with K countable and compact. A consequence of Theorem 1.1 is the existence of new examples of c0 -saturated spaces admitting reflexive quotients. The first example of such a space was given in [4], where a certain Orlicz function space was shown to admit 2 as a quotient. It was proved in [13] that 2 is a quotient of a c0 -saturated space with an unconditional basis. More general results were obtained in [2] with the use of interpolation methods. They showed that every reflexive space with an unconditional basis has a block subspace which is isomorphic to a quotient of a c0 -saturated space. This result has been recently extended to cover all separable reflexive spaces. It is shown in [3] that every such space is a quotient of a c0 -saturated space with a basis. 2. Preliminaries Our notation is standard as may be found in [14]. We shall consider Banach spaces over the real field. If X is a Banach space then BX stands for its closed unit ball. By a subspace of a Banach space we shall always mean a closed, linear subspace. A bounded subset B of the dual X ∗ of X is norming, if there exists ρ > 0 such that supx ∗ ∈B |x ∗ (x)| ρx, for all x ∈ X. In case B ⊂ BX∗ and ρ = 1, B is said to isometrically norm X. X is said to contain an isomorph of the Banach space Y (or, equivalently, that X contains Y isomorphically), if there exists a bounded linear injection from Y into X having closed range. A sequence (xn ) in a Banach space is semi-normalized if infn xn > 0 and supn xn < ∞. It is called a basic sequence provided it is a Schauder basis for its closed linear span in X. A Schauder basis (xn ) for the space X is shrinking, if the sequence of functionals (xn∗ ), biorthogonal to (xn ), is a Schauder basis for X ∗ . and (yn ) are basic If (xn ) sequences, then (xn ) dominates (yn ) if there is a constant C > 0 so that ni=1 ai yi C ni=1 ai xi , for every choice of scalars (ai )ni=1 and all n ∈ N. The
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basic sequences (xn ) and (yn ) are equivalent, if each one of them dominatesthe other. A basic sequence (xn ) is called suppression 1-unconditional, if i∈F ai xi ni=1 ai xi , for all ⊂ {1, . . . , n}. Evidently, such a basic sequence n ∈ N, all choices of scalars (ai )ni=1 , and every F is unconditional, that is every series of the form n an xn converges unconditionally, whenever it converges. If (xn ) is a basic sequence in some Banach space X, then a sequence (un ) of non-zero vectors (Fn ) in X, is a block basis of (xn ) if there exist a sequence of non-zero scalars (an ) and a sequence of successive finite subsets of N (i.e., max Fn < min Fn+1 for all n ∈ N), so that un = i∈Fn ai xi , for all n ∈ N. We then call Fn the support of un for all n ∈ N. Any member of a block basis of (xn ) will be called a block of (xn ). A Banach space with an unconditional basis (en ) satisfiesan upper p-estimate, for some p > 1, if there exists a constant C > 0 so that ni=1 ui C( ni=1 ui p )1/p , for every choice (ui )ni=1 of disjointly supported blocks of (en ). Given finite subsets E, F of N, then the notation E < F indicates that max E < min F . If μ, ν are finitely supported signed measures on N, then we write μ < ν if supp μ < supp ν. A family F of finite subsets of N is said to be compact if it is compact in the topology of pointwise convergence in 2N . We next recall the Schreier family S1 = F ⊂ N: |F | min F ∪ {∅}. The higher ordinal Schreier families {Sα : α < ω1 }, were introduced in [1] where it is shown that α Sα is homeomorphic to the ordinal interval [1, ωω ], for all α < ω1 . 3. The main construction Let Z be a Banach space with a normalized, shrinking, unconditional basis (zn ). By renorming if necessary, we may assume that n an zn = n |an |zn , for every (an ) ∈ c00 . Define φZ : N → R, by k ui : (ui )ki=1 are finitely and disjointly supported blocks of (zn ), φZ (k) = sup i=1 ui 1, ∀i k ,
∀k ∈ N.
It is easy to see that φZ is a submultiplicative function, that is φZ (mn) φZ (m)φZ (n) for all integers m, n. It follows from the results of [11] (cf. also Theorem 1.f.12 in [14]), that Z satisfies an upper p-estimate for some p > 1 if and only if φZ (k) < k for some k 2. To define the desired isomorphically polyhedral space, we need to introduce some notations and definitions. Let Z satisfy an upper p-estimate for some p > 1. For simplicity, we shall write φ instead of φZ . Notation. Fix some k0 2 with φ(k0 ) < k0 and choose λ ∈ (φ(k0 )/k0 , 1). Set n = (1/k0 )n and δn = λn , for all n ∈ N ∪ {0}. We next choose a sequence (Fn ) of successive finite subsets of N (i.e., max Fn < min Fn+1 for all n ∈ N) so that n (1/|Fn |) < 1 and 1 + (1/δn ) < n+1 min Fn+1 , for all n ∈ N ∪ {0}.
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Notation. Given n ∈ N, let en∗ denote the point mass measure at n. Let P denote the set of nonnegative, finitely supported measures μ on N of the form μ = i λi ej∗i , where λi ∈ [0, 1] and ji ∈ Fi for all i ∈ N. Let P1 = {μ ∈ P:
i
μ(Fi ) 1}.
Definition 3.1. A measure μ ∈ P is said to be Z-bounded, provided that i μ(Gi ) 1 for every sequence (G i ) of subsets of N with Gi an initial segment of Fi (we allow Gi = ∅) for all i ∈ N, such that i (|Gi |/|Fi |)zi Z 1. Remark 3.2. A typical example of a Z-bounded measure is as follows: Let (ρi ) be a sequence of non-negative scalars such that i ρi zi∗ Z ∗ 1. For every i ∈ N, choose an∗ initial segment Gi of Fi . Let I be a finite subset of N and define μ = i∈I ρi (|Gi |/|Fi |)emax Gi . Then μ is Z-bounded. Indeed, let (Hi ) bea sequence of finite subsets of N with each Hi being an initial segment of Fi , such that i (|Hi |/|Fi |)zi Z 1. Let i ∈ I . Then, either Gi is an initial segment of Hi , or, max Gi > max Hi . If the former, then μ(Hi ) = ρi (|Gi |/|Fi |) and |Gi |/|Fi | |Hi |/|Fi |. If the latter, then μ(Hi ) = 0. Let I1 be the subset of I consisting of those elements of I for which the first alternative occurs. Then i∈I1 (|Gi |/|Fi |)zi Z 1 by our initial assumptions on (zi ), and so
μ(Hi ) =
i
i∈I1
μ(Hi ) =
ρi |Gi |/|Fi | 1.
i∈I1
Definition 3.3. A finite sequence (μi )ki=1 of non-zero, disjointly supported members of P1 is called admissible, if for every n ∈ N we have that Fn ∩ supp μi = ∅ for at most one i k, and, moreover, if Fn ∩ supp μi = ∅ for some n ∈ N and i k, then k min Fn . Note in particular that {min supp μi : i k} ∈ S1 if (μi )ki=1 is admissible. We can now describe a norming subset of the space we wish to construct. M = μ ∈ P, μ is Z-bounded, μ =
k i=1
(μi )ki=1
μi , k ∈ N and
is an admissible sequence in P1 ∪ en∗ : n ∈ N ∪ {0}.
It is easy to see that μ|I ∈ M for every μ ∈ M and all I ⊂ N. We can now define a norm · M on c00 by
μ {i} x(i) : μ ∈ M , xM = max
∀x ∈ c00 .
i
Let XM be the completion of (c00 , · M ). Since M is closed under restrictions to subsets of N, we obtain that the natural basis (en ) of c00 becomes a normalized, suppression 1-unconditional ∗ . Our objective is basis for XM . Note also that M is an isometrically norming subset of BXM
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to show that XM is isomorphically polyhedral and admits a quotient isomorphic to Z. The first task is accomplished through the next result, proved in [10] via Elton’s theorem [6]. Proposition 3.4. Let X be a Banach space with a normalized Schauder basis (en ). Let (en∗ ) denote the sequence of functionals biorthogonal to (en ). Assume there is a bounded norming subset B of X ∗ with the following property: There exists a compact family F of finite subsets of N such that for every b∗ ∈ B there exist F ∈ F and a finite sequence (bk∗ )k∈F of finitely supported ∗ ∗ ∗ absolutely sub-convex combinations of (en ) so that b = k∈F bk and min supp bk∗ k for all ∗ k ∈ F . Then, n |b∗ (en )| < ∞, for all b∗ ∈ B w and X is isomorphically polyhedral. Corollary 3.5. XM is isomorphically polyhedral and (en ) is an unconditional, shrinking normalized basis for XM . Proof. The fact that (en ) is normalized and unconditional follows directly from the definition of M. Next, let μ ∈ M. We verify that the conditions given in Proposition 3.4 are fulfilled by μ with F = S1 . Indeed, this is obvious when μ = en∗ for some n ∈ N. Otherwise, μ = ki=1 μi for some k ∈ N and an admissible family (μi )ki=1 of members of P1 . Since every μi is a finitely supported sub-convex combination of (en∗ ), and {min supp μi : i k} ∈ S1 , the first assertion of the corollary follows from Proposition 3.4. Since XM is c0 -saturated, it cannot contain any isomorph of 1 and thus a classical result due to James yields that (en ) is shrinking. 2 ∗ . In the sequel, we shall write · , resp. · ∗ , instead of · M , resp. · XM The next lemma describes a simple method for selecting subsequences of (en ), equivalent to the c0 -basis.
sequence of finite subsets of N Lemma 3.6. Let I be a finite subset of N and (Gn )n∈I a with G an initial segment of F for all n ∈ I , such that n n∈I (|Gn |/|Fn |)zn Z 1. Then n n∈I k∈Gn ek 1. Proof. Set u = n∈I k∈Gn ek . We show that μ(u) 1 for all μ ∈ M. In case μ = en∗ for some n ∈ trivially holds. Every other element μ ∈ M is Z-bounded and so N, then the assertion μ(u) = n∈I μ(Gn ) 1, as n∈I (|Gn |/|Fn |)zn Z 1. 2 4. Z is isomorphic to a quotient of XM The main result of this section is the following Theorem 4.1. Let u∗n = i∈Fn (1/|Fi |)ei∗ , for all n ∈ N. Then (u∗n ) is equivalent to (zn∗ ), the sequence of functionals biorthogonal to (zn ). The proof of this result will follow after a series of lemmas, where we first show that (u∗n ) dominates (zn∗ ) and then that it is dominated by (zn∗ ). Note that our initial assumptions on (zn ) yield that (zn∗ ) is a normalized, suppression 1-unconditional basis for Z ∗ . Lemma 4.2. (u∗n ) dominates (zn∗ ).
I. Gasparis / Journal of Functional Analysis 256 (2009) 3830–3840
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∗ ∗ ∗ Proof. Note first that (un ) is a normalized (convex) block basis of (en ) in XM . This is so since i∈Fn ei = 1, for all n ∈ N (observe that the support of every member of M meets each Fn in at most one point). We thus obtain that (u∗n ) is normalized 1-unconditional. n and suppression ∗ ∗ 1. We next find scalars )ni=1 be scalars in [0, 1] with a z Now let n ∈ N and (ai i i Z i=1 (bi )ni=1 in [0, 1] so that ni=1 bi zi Z 1 and ni=1 ai bi = ni=1 ai zi∗ Z ∗ . For each i n choose an initial segment Gi of Fi so that
|Gi |/|Fi | bi < |Gi |/|Fi | + 1/|Fi | . This choice ensures, thanks to our initial assumptions on (zn ), that n n
|Gi |/|Fi | zi bi zi 1. i=1
Let u =
n i=1
k∈Gi ek .
Z
i=1
Z
We deduce from Lemma 3.6, that u 1. It follows now that
n n n n
∗
∗ ∗ 1/|F a u a u (u) = a | e (u) = ai |Gi |/|Fi | i i i i i i k k∈Fi i=1 i=1 i=1 i=1 ∗ n n n
∗
1/|Fi | . ai bi − 1/|Fi | ai zi − ∗ i=1
i=1
Z
i=1
Next suppose that n ∈ N and (ai )ni=1 is a scalar sequence satisfying ni=1 ai zi Z ∗ = 1. Let I + = {i n: ai 0} and I − = I \ I + . Our preceding work yields that ∗
∗ 1/|Fi | , ∀j ∈ {+, −}. ai ui ai zi − i∈I j
∗
Z∗
i∈I j
i∈I j
We deduce now from the above and the fact that (u∗n ) is suppression 1-unconditional, that n ∞
∗ 1/|Fi | > 0. ai ui 1 − 2 ∗
i=1
Therefore, letting A = (1/2)(1 −
i=1
∞
i=1 (1/|Fi |)) > 0,
we obtain that
n n ∗ ∗ ai ui A ai zi i=1
∗
i=1
Z∗
for every n ∈ N and all choices of scalars (ai )ni=1 ⊂ R. The proof of the lemma is now complete. 2 vector in XM , with u 1 and ai 0, Lemma 4.3. Let u = i ai ei be a finitely supported for all i ∈ N. Let μ ∈ P be Z-bounded and write μ = i∈I λi ej∗i , where I is a finite subset of N, ji ∈ Fi and λi ∈ (0, 1] for all i ∈ I . Suppose that there exists n ∈ N ∪ {0} with n < min I such that aji n+1 , for all i ∈ I . Then, μ∗ 2.
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Proof. Set I1 = {i ∈ I : λi 1/2} and I2 = I \ I1 . The assertion of the lemma will follow once we show that μ|Is ∈ M for every s 2. To this end, we first claim that |I1 | 2/ n+1 . Indeed, if the claim were false, let d = 2/ n+1 and choose d + 1 elements i1 < · · · < id+1 in I1 . Since d < min Fi , for all i ∈ I (by the initial choice of the sequence (Fi )) we have that is an admissible sequence of members of P1 . Moreover, since μ is Z-bounded, so is (λs ej∗i )d+1 s s=1 ∗ μ|J , for every J ⊂ N. Therefore, d+1 s=1 λs eji ∈ M and so our assumptions on u yield that s
1 u
d+1
λs ej∗is (u) =
s=1
d+1
λs ajis (d + 1)( n+1 /2) > 1,
s=1
a contradiction that proves our claim. It follows now that (λi ej∗i )i∈I1 is an admissible family and hence μ|I1 ∈ M. We next show that μ|I2 ∈ M. We first choose a non-empty initial segment J1 of I2 which is maximal with respect to the condition i∈J1 λi 1. In case J1 = I2 , the assertion follows as P1 ⊂ M. If J1 is a proper initial segment of I2 , then, by maximality, we must have 1/2 <
λi 1,
i∈J1
as λi < 1/2, for every i ∈ I2 . We now set μ1 = i∈J1 λi ej∗i . This measure belongs to P1 and satisfies μ1 (u) > n+1 /2 because aji n+1 , for all i ∈ I . We repeat the same process to I2 \ J1 and obtain a non-empty initial segment J2 of I2 \ J1 , and a measure μ2 = i∈J2 λi ej∗i in P1 so that either J1 ∪ J2 = I2 , or J2 is a proper initial segment of I2 \ J1 satisfying μ2 (u) > n+1 /2. If the former, the process stops. If the latter, the process steps, say k. continues. Because I2 is finite, this process will terminate after a finite number of k < · · · < J of I with I = We shall then have produced successive subintervals J 1 k 2 2 r=1 Jr , and ∗ measures μ1 < · · · < μk in P1 with μr = i∈Jr λi eji , for all r k. Moreover, μr (u) > n+1 /2, for all r < k. We claim that k d = 2/ n+1 . Indeed, assuming the contrary, we have by the choice of k, that μr (u) n+1 /2, for all r d. Butalso, 2/ n+1 < min Fi , as i n + 1 for all i ∈ I , and thus, (μr )dr=1 is admissible. Since ν = dr=1 μr = μ| dr=1 Jr , it is Z-bounded and so ν ∈ M. Therefore, 1 u
d
μr (u) > d n+1 /2 = 1,
r=1
which is a contradiction. Hence, k 2/ n+1 min Fi , for all i ∈ I . This implies that (μr )kr=1 is admissible and so μ|I2 = kr=1 μr ∈ M, completing the proof of the lemma. 2 Lemma 4.4. Let u = i ai ei be a finitely supported vector in XM , with u 1 and ai 0, for all i ∈ N. Let I be a finite subset of N and n ∈ N ∪ {0} with n < min I . Suppose that for each i ∈ I there exists an initial segment Gi of Fi such that amax Gi n+1 . Let (ρi ) be a sequence of non-negative scalars such that i ρi zi∗ Z ∗ 1. Assume further that there exists a family J of pairwise disjoint subsets of I such that every member J of J satisfies the following conditions:
I. Gasparis / Journal of Functional Analysis 256 (2009) 3830–3840
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(1) i∈J (|Gi |/|Fi |)zi Z 1, and (2) i∈J ρi (|Gi |/|Fi |) δn . Then |J | < (1/ n+1 )(1 + [1/δn ]), and
ρi |Gi |/|Fi | (1 + 1/δn )φ(k0 )n+1 .
J ∈J i∈J
Proof. We set d = (1/ n+1 )(1 + [1/δn ]) and assume, to the contrary,that |J | d. We can now ∗ select pairwise disjoint members J1 , . . . , Jd of J and define μr = i∈Jr ρi (|Gi |/|Fi |)emax Gi , ∗ for all r d. Since i ρi zi Z ∗ 1, (1) implies that μr ∈ P1 for all r d. But also, n + 1 i, because of our initial assumptions on (Fi ). It folfor all i ∈ I and thus d < min Fi for all i ∈ I , lows now that (μr )dr=1 is admissible. Let μ = dr=1 μr . Since μ is Z-bounded (see Remark 3.2), we infer from the above that μ ∈ M and therefore, 1 μ(u) =
d
d
ρi |Gi |/|Fi | amax Gi ρi |Gi |/|Fi | n+1 n+1 δn d,
r=1 i∈Jr
by (2).
r=1 i∈Jr
Hence, d 1/( n+1 δn ) < (1/ n+1 )(1 + [1/δn ]) = d. This contradiction shows that |J | < d, as required. We next verify the second assertion of the lemma. To this end, J ∈J i∈J
ρi |Gi |/|Fi | =
ρi zi∗
i
(|Gi |/|Fi |)zi J ∈J i∈J
|G |/|F | z i i i J ∈J i∈J
Z
φ |J | , by (1),
φ (1/ n+1 ) 1 + [1/δn ]
φ(1/ n+1 )φ 1 + [1/δn ] ,
(1 + 1/δn )φ k0n+1
by the submultiplicativity of φ,
(1 + 1/δn )φ(k0 )n+1 , using once again the fact that φ is submultiplicative. This concludes the proof of the lemma.
2
Lemma 4.5. There exists a constant C > 0 such that for every I ⊂ N, finite, and every collection of scalars (ρi )i∈I with i ρi zi∗ Z ∗ 1, we have that i∈I ρi u∗i ∗ C. Consequently, (zi∗ ) dominates (u∗i ). Proof. The unconditionality of (zi∗ ) clearly allows us establish the assertion of the lemma under the additional assumption that ρi 0, for all i ∈ I . Given n ∈ N ∪ {0}, let In = {i ∈ I : i > n}. Let u = i ai ei be a finitely supported vector in XM , with u 1 and ai 0, for all i ∈ N. We have the following initial estimate
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I. Gasparis / Journal of Functional Analysis 256 (2009) 3830–3840
ρi u∗i (u) =
i∈I
ρi
aj /|Fi | =
j ∈Fi
i∈I ∞
n n +
ρi
∞
ρi
aj /|Fi |
j ∈Fi : aj ∈( n+1 , n ]
n=0 i∈I
aj /|Fi |.
(4.1)
j ∈Fi : aj ∈( n+1 , n ]
n=0 i∈In
n=0
∞
Fix some n ∈ N ∪ {0}. For each i ∈ In , let jin denote the largest j ∈ Fi so that aj ∈ ( n+1 , n ]. In case jin fails to exist, then the corresponding summand in (4.1) equals 0 and thus it has no effect into our estimates. Let Gni be the initial segment of Fi with max Gni = jin . It is clear that j ∈Fi : aj ∈( n+1 , n ]
aj /|Fi | ( n / n+1 )ajin |Gni |/|Fi | ,
Call a subset J of In bad, if n
|G |/|F | z i i 1 and i Z
i∈J
∀i ∈ In .
(4.2)
ρi |Gni |/|Fi | δn .
i∈J
It is clear that we can extract a maximal, under inclusion, family Jn consisting of pair. We now observe the following: If J ⊂ I \ Jn and wise disjoint, bad subsets of I n n i∈J (|Gni |/|Fi |)zi Z 1, then i∈J ρi (|Gni |/|Fi |) < δn . Indeed, if that were not so, then J would be bad, contradicting the maximality of Jn . Letting μn = i∈In \ Jn ρi (|Gni |/|Fi |)ej∗n , we infer from the preceding observation, that i (1/δn )μn is a Z-bounded measure. On the other hand, ajin n+1 , for each i ∈ In for which jin exists, and therefore Lemma 4.3 yields that μn ∗ 2δn . Taking in account (4.2), we obtain the estimate
i∈In \
ρi Jn
aj /|Fi | ( n / n+1 )
j ∈Fi : aj ∈( n+1 , n ]
i∈In \
Jn
ρi ajin |Gni |/|Fi |
= ( n / n+1 )μn (u) 2k0 δn .
(4.3)
We next employ Lemma 4.4 to obtain the estimate
ρi |Gni |/|Fi | (1 + 1/δn )φ(k0 )n+1 ,
J ∈Jn i∈J
and thus, taking (4.2) into account, we reach the estimate i∈
Jn
ρi
aj /|Fi | ( n / n+1 )
j ∈Fi : aj ∈( n+1 , n ]
i∈
= k0
J ∈Jn i∈J
k0 n
Jn
ρi ajin |Gni |/|Fi |
ρi ajin |Gni |/|Fi |
J ∈Jn i∈J
ρi |Gni |/|Fi |
(4.4)
I. Gasparis / Journal of Functional Analysis 256 (2009) 3830–3840
k0 n (1 + 1/δn )φ(k0 )n+1 , by (4.4)
n = k0 φ(k0 )(1 + 1/δn ) φ(k0 )/k0
n
n = k0 φ(k0 ) φ(k0 )/k0 + φ(k0 )/λk0 .
3839
(4.5)
Let Bn = k0 φ(k0 )[(φ(k0 )/k0 )n + (φ(k0 )/λk0 )n ]. Eqs. (4.3) and (4.5) now yield that i∈In
ρi
aj /|Fi | 2k0 δn + Bn ,
∀n 0,
j ∈Fi : aj ∈( n+1 , n ]
and so, finally, (4.1) gives us the estimate
ρi u∗i (u)
i∈I
∞ (n n + 2k0 δn + Bn ) < ∞, n=0
as 0 < φ(k0 )/k0 < λ < 1. The assertion of the lemma now follows as (en ) is an unconditional basis for XM . 2 Proof of Theorem 4.1. It follows directly from Lemmas 4.2 and 4.5 that (u∗n ) and (zn∗ ) are equivalent. 2 Corollary 4.6. Z is isomorphic to a quotient of XM and the map Q : XM → Z given by Q an en = ak /|Fn | zn n
n
k∈Fn
is a well-defined, bounded, linear surjection. Proof. Since (zn∗ ) dominates (u∗n ), we have that Q is a well-defined, bounded, linear operator. It is easy to see now, that Q∗ (zn∗ ) = u∗n , for all n ∈ N, and thus Q∗ is an isomorphic embedding ∗ , by Theorem 4.1. It follows now that Q is a surjection. 2 of Z ∗ into XM References [1] D. Alspach, S.A. Argyros, Complexity of weakly null sequences, Dissertationes Math. 321 (1992) 1–44. [2] S.A. Argyros, V. Felouzis, Interpolating hereditarily indecomposable Banach spaces, J. Amer. Math. Soc. 13 (2000) 243–294. [3] S.A. Argyros, T. Raikoftsalis, The cofinal property of indecomposable reflexive Banach spaces, preprint. [4] P.G. Casazza, N.J. Kalton, L. Tzafriri, Decompositions of Banach lattices into direct sums, Trans. Amer. Math. Soc. 304 (1987) 771–800. [5] P.G. Casazza, E. Odell, Tsirelson’s space and minimal subspaces, in: Texas Functional Analysis Seminar 1982– 1983, Austin, TX, in: Longhorn Notes, University Texas Press, Austin, TX, 1983, pp. 61–72. [6] J. Elton, Extremely weakly unconditionally convergent series, Israel J. Math. 40 (1981) 255–258. [7] T. Figiel, W.B. Johnson, A uniformly convex Banach space which contains no p , Compos. Math. 29 (1974) 179– 190. [8] V. Fonf, A property of Lindenstrauss–Phelps spaces, Funct. Anal. Appl. 13 (1) (1979) 66–67. [9] V. Fonf, Polyhedral Banach spaces, Mat. Zametki 30 (4) (1981) 627–634. [10] I. Gasparis, New examples of c0 -saturated Banach spaces, preprint available from http://xxx.lanl.gov/PS_cache/ arxiv/pdf/0809/0809.1808v1.pdf.
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[11] W.B. Johnson, On finite dimensional subspaces of Banach spaces with local unconditional structure, Studia Math. 51 (1974) 223–238. [12] W.B. Johnson, A reflexive Banach space which is not sufficiently Euclidean, Studia Math. 55 (1976) 201–205. [13] D.H. Leung, On c0 -saturated Banach spaces, Illinois J. Math. 39 (1995) 15–29. [14] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I, II, Springer-Verlag, New York, 1977. [15] E. Odell, On quotients of Banach spaces having shrinking unconditional bases, Illinois J. Math. 36 (1992) 681–695. [16] A. Pelczynski, Z. Semadeni, Spaces of continuous functions III. Spaces C(Ω) for Ω without perfect subsets, Studia Math. 18 (1959) 211–222. [17] H.P. Rosenthal, Some aspects of the subspace structure of infinite-dimensional Banach spaces, in: Approximation Theory and Functional Analysis, College Station, 1990, Academic Press, 1991, pp. 151–176. [18] H.P. Rosenthal, The Banach spaces C(K), in: W.B. Johnson, J. Lindenstrauss (Eds.), Handbook of the Geometry of Banach Spaces, vol. 2, North-Holland, 2003, pp. 1547–1602. [19] B.S. Tsirelson, Not every Banach spaces contains p or c0 , Funct. Anal. Appl. 8 (1974) 138–141.
Journal of Functional Analysis 256 (2009) 3841–3846 www.elsevier.com/locate/jfa
The relative commutant of separable C∗-algebras of real rank zero ✩ Ilijas Farah Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada M3J 1P3 Received 18 September 2008; accepted 7 October 2008 Available online 5 November 2008 Communicated by Alain Connes
Abstract We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C∗ -algebra in its ultrapower depend on the choice of the ultrafilter? © 2008 Elsevier Inc. All rights reserved. Keywords: C∗ -algebras; Ultrapowers; Continuum Hypothesis
All algebras and all subalgebras in this note are C∗ -algebras and C∗ -subalgebras, respectively, and all ultrafilters are nonprincipal ultrafilters on N. Our C∗ -terminology is standard (see e.g. [2]). In the following U ranges over nonprincipal ultrafilters on N. With AU denoting the (norm, also called C∗ -) ultrapower of a C∗ -algebra A associated with U we have FU (A) = A ∩ AU , the relative commutant of A in its ultrapower. This invariant plays an important role in [8] and [7]. Theorem 1. For every separable infinite-dimensional C ∗ -algebra A of real rank zero the following are equivalent. ✩ Partially supported by NSERC. I would like to thank N. Christopher Phillips for many useful comments on the first draft of this paper. In this version Theorem 1 was proved only for UHF algebras, and Chris’s suggestion to use instead of helped me extend the result to its present form. E-mail address: [email protected]. URL: http://www.math.yorku.ca/~ifarah.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.007
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(1) FU (A) ∼ = FV (A) for any two nonprincipal ultrafilters U and V on N. A (2) AU ∼ = V for any two nonprincipal ultrafilters U and V on N. (3) The Continuum Hypothesis. The equivalence of (3) and (2) in Theorem 1 for every infinite-dimensional C∗ -algebra A of cardinality 2ℵ0 that has arbitrarily long finite chains in the Murray–von Neumann ordering of projections was proved in [6, Corollary 3.8], using the same Dow’s result from [4] used here. We shall prove (1) implies (3) and (2) implies (3) in Corollary 10 below. The reverse implications are well-known consequences of countable saturatedness of ultrapowers associated with nonprincipal ultrafilters on N (see [1, Proposition 7.6]). The implication from (3) to (1) holds for every separable C∗ -algebra A and the implication from (3) to (2) holds for every C∗ -algebra A of size 2ℵ0 . The point is that if A is separable then the isomorphism between diagonal copies of A extends to an isomorphism between the ultrapowers. Countable saturation of AU can be proved directly from its analogue, due to Keisler, in classical model theory. This also follows from the argument in [6, Theorem 3.2 and Remark 3.3]. While the Continuum Hypothesis implies that any two ultrapowers of B(H ) associated with nonprincipal ultrafilters on N are isomorphic, it does not imply that the relative commutants of B(H ) in those ultrapowers are isomorphic. As a matter of fact, it implies the opposite (see [5]). For a C∗ -algebra A let P(A) = {p: p ∈ A is a projection} ordered by p q if and only if pq = p. Our proof depends on the analysis of types of gaps in P(A ∩ AU ) (see Definition 4). Gaps in P(N)/ Fin and related quotient structures are well studied; for example, analysis of such gaps is very important in the consistency proof of the statement ‘all Banach algebra automorphisms of C(X) into some Banach algebra are continuous’ (see [3]). It was recently discovered that the gap-spectrum of P(C(H )) (where C(H ) is the Calkin algebra, B(H )/K(H )) is much richer than the gap-spectrum of P(N)/ Fin [12]. Notational convention. We denote elements of ultraproducts by boldface Roman letters such as p and their representing sequences by p(n), for n ∈ N. We shall follow von Neumann’s convention and identify a natural number n with the set {0, . . . , n − 1}. The symbol ω is used for ultrafilters in the operator algebra literature and it is reserved for the least infinite ordinal in the set-theoretic literature. I will avoid using it in this note. By σ (a) we denote the spectrum of a normal operator a. Lemma 2 below is well known. A sharper result can be found e.g., in [9, Lemma 2.5.4] but we include a proof for reader’s convenience. Lemma a√ and a projection r, if a − r < ε < 1 then σ (a) ⊆ √ For a self-adjoint √ √ 2. + 2 ε). If in addition ε < 1/16 then there is a projection r in (−2 ε, 2 ε) ∪ (1 − 2 ε, 1√ C ∗ (a) such that r − a < 2 ε. Proof. Since a < 1 + ε < 2, we have a 2 − a a(a r) + a − r < 4ε. √− r) + r(a −√ Thus |x(1 − x)| < 4ε for all x ∈ σ (a) and in turn |x| < 2 ε or |1 − x| < 2 ε. Now assume ε < 1/16. In this case 1/2 ∈ / σ (a). Define a continuous function f with domain σ (a) as √ follows. Let f (t) = 0 for −∞ < t < 1/2 and f (t) = 1 for 1/2 t < ∞. Since |f (t) − t| < 2 ε for all t ∈ σ (a), f (a) is a projection in C ∗ (a) as required. 2 A representing sequence p(n) of a projection p in an ultrapower can be chosen so that each p(n) is a projection (see [6, Proposition 2.5(1)], this also follows immediately from [10, Lemma 4.2.2] or [9, Lemma 2.5.5]).
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Lemma 3. For projections p, q in AU the following are equivalent. (1) p q. (2) There is a representing sequence p (i), for i ∈ N, of p such that p (i) q(i) for all i. (3) There is a representing sequence q (i), for i ∈ N, of q such that p(i) q (i) for all i. Proof. Both (3) implies (1) and (2) implies (1) are trivial. We shall prove (1) implies (2). Assume p q. For every n 1 the set Xn = j : q(j )p(j )q(j ) − p(j ) < 1/(4n) belongs to U . We may assume n Xn = ∅. Let p (j ) = 0 if j ∈ / X0 . If j ∈ Xn \ Xn+1 then C ∗ (a(j )) such that Lemma 2, with a(j ) = √ q(j )p(j )q(j ), implies there is a projection p (j ) ∈ √ p (j ) − a(j ) < 1/(2 n). Then p (j ) q(j ) and p (j ) − p(j ) < 1/ n for all j ∈ Xn . Therefore p (j ), for j ∈ N, is a representing sequence of p as required. In order to prove (1) implies (3) apply the above to 1 − p 1 − q in the ultrapower of the unitization of A to find an appropriate representing sequence for 1 − q. 2 By NN we denote the set of all nondecreasing functions f from N to N such that limn f (n) = ∞, ordered pointwise. Write f U g if {n: f (n) g(n)} ∈ U and denote the quotient linear ordering by NN / U . Following [4], for an ultrafilter U we write κ(U) for the coinitiality of NN / U , i.e., the minimal cardinality of X ⊆ NN such that for every g ∈ NN there is f ∈ X such that f U g. (It is not difficult to see that this is equal to κ(U) as defined in [4, Definition 1.3].) Recall that ℵ0 is the cardinality of N. Definition 4. Let λ be a cardinal. An (ℵ0 , λ)-gap in a partially ordered set P is a pair consisting of a P -increasing family am , for m ∈ N, and a P -decreasing family bγ , for γ < λ, such that am P bγ for all m and γ but there is no c ∈ P such that am P c for all m and c P bγ for all γ . Assume r 0 (n) r 1 (n) · · · r l(n)−1 (n) are projections in A and limn→∞ l(n) = ∞. For h : N → N define r h via its representing sequence (let r i (n) = r l(n)−1 (n) for i l(n)) r h (n) = r h(n) (n). Let pm = r m¯ , where m(j ¯ ) = m for all j . Lemma 5. With notation from the previous paragraph, for every projection s in AU such that pm s for all m there is h : N → N such that pm r h for all m and r h s. Proof. Since pm s, for each m ∈ N the set Xm = i: r m (i)s(i) − r m (i) < 1/m belongs to U . Since the value of r m (i)s(i) − r m (i) is increasing in m we have Xm ⊇ Xm+1 . We / X0 and for i ∈ Xm \ Xm+1 may assume m Xm = ∅. Define h : N → N by letting h(i) = 0 for i ∈ let h(i) = m.
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For each m and i ∈ Xm we have h(i) m and therefore r h pm . Also, i ∈ Xm implies r h (i)s(i) − r h (i) < 1/m hence r h s. 2 The proof of Proposition 6 was inspired by Alan Dow’s [4, Proposition 1.4]. Dow’s result was independently proved by Saharon Shelah ([11, Theorem VI.3.14] with λ = ℵ0 ). By A1 we denote the unit ball of a C∗ -algebra A. Proposition 6. Assume A is a separable C ∗ -algebra and there are finite self-adjoint sets F0 ⊆ F1 ⊆ F2 ⊆ · · · ⊆ A1 whose union is dense in A1 and such that for each n there is a -increasing chain Cn of projections in Bn = Fn ∩ A of length at least n. Then for every nonprincipal ultrafilter U on N and every cardinal λ there is an (ℵ0 , λ)-gap in P(A ∩ AU ) if and only if κ(U) = λ. Proof. First we prove the converse implication. Assume gγ , for γ < λ = κ(U), is a U -decreasing and U -unbounded below chain of functions in NN . Let 0 = r 0 (n) r 1 (n) · · · r n−1 (n) be an enumeration of Cn . Claim 7. For all f, g in NN the following are equivalent. (1) f U g. (2) rf rg . Proof. Assume f U g. Then X = {j : f (j ) g(j )} ∈ U and rf (j ) rg (j ) for all j ∈ X hence (2) follows. If f U g then X = {j : f (j ) > g(j )} ∈ U and for all j ∈ X we have r f (i)r g (i) − r g (i) = 1, hence rf rg . 2 Let qγ = rgγ , for γ < λ. By Claim 7 we have pm pm+1 qδ qγ for all m and all γ < δ < λ. All of pm and qγ belong to A ∩ AU . We shall show that this family forms a gap in P(AU ) (and therefore it forms a gap in P(A ∩ AU )). Assume s ∈ AU is such that s qγ for all γ . By Lemma 5 there is h such that pm r h s for all m. By Claim 7 we have h U gγ for all γ and m ¯ U h for all m, a contradiction. In order to prove the direct implication, assume that pm , qγ form an (ℵ0 , λ)-gap in P(A ∩ AU ). By successively using Lemma 3 for m = 1, 2, . . . find representing sequences pm (i)i∈N , for pm such that pm (i) pm+1 (i) for all i. Choose an increasing sequence 0 = m0 < m1 < m2 < · · · such that the following holds for all k. (∗) for all j < mk and all a ∈ Fmk , if l mk+1 then [pj (l), a] < 1/k. For n ∈ N and i such that for some k we have i < mk and mk+1 n let r i (n) = pi (n). Thus we have projections r 0 (n) r 1 (n) · · · r mk (n)
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whenever n mk+1 . For h : N → N define r h as in the paragraph before Lemma 5, by its representing sequence (let r i (n) = r mk (n) if i mk ) r h (n) = r h(n) (n). Claim 8. If h : N → N then r h ∈ A ∩ AU . Proof. Fix any b in the unit ball of A and ε > 0. If k > 1/ε and there is b ∈ F2k satisfying b − b < ε/2 then for i > n2k in Y we have that [pj (i), b ] < ε/2 and therefore [r h (i), b] < ε for U -many i. 2 Using Lemma 5 for each qγ find hγ such that rγ = rhγ satisfies pi rγ qγ for all i. Since is a linear ordering and λ is a regular cardinal, we can find a cofinal subset Z of λ such that for γ < δ in Z we have rδ rγ . By reenumerating we may assume Z = λ and then rγ , for γ ∈ Z, together with pi , for i ∈ N, form an (ℵ0 , λ)-gap. However, rδ rγ is equivalent to hδ U hγ , and therefore hγ , for γ < λ, form a U -decreasing and U -unbounded below sequence in NN / U , and therefore λ = κ(U). 2 NN / U
The proof of Proposition 6 can be modified (by removing some of its parts) to a proof of the following. Proposition 9. Assume A is a separable C ∗ -algebra and P(A) has arbitrarily long finite chains. Then for every nonprincipal ultrafilter U on N and every cardinal λ there is an (ℵ0 , λ)-gap in P(AU ) if and only if κ(U) = λ. Corollary 10. Assume the Continuum Hypothesis fails. If A is an infinite-dimensional separable C∗ -algebra of real rank zero then there are nonprincipal ultrafilters U and V on N such that FU (A) ∼ = FV (A) and AU ∼ = AV . Proof. By [4, Theorem 2.2] we can find U and V so that κ(U) = ℵ1 and κ(V) = ℵ2 (here ℵ1 and ℵ2 are the least two uncountable cardinals; all that matters for us is that they are both less or equal than 2ℵ0 and different). Therefore P(A ∩ AU ) has an (ℵ0 , ℵ1 )-gap while P(A ∩ AV ) does not, and A ∩ AU and A ∩ AV cannot be isomorphic. It remains to prove that if A is an infinite-dimensional C∗ -algebra of real rank zero then P(A) has an infinite chain of projections. We may assume A is unital. Recursively find a decreasing sequence rn for n ∈ N in P(A) so that rn Arn is infinite-dimensional for all n. Assume rn has / {0, rn }. If qAn q is been chosen. Since A has real rank zero, in rn Arn we can fix a projection q ∈ infinite-dimensional then let rn+1 = q. Otherwise, let rn+1 = rn − q and note that rn+1 Arn+1 is infinite-dimensional. 2 It is likely that Theorem 1 and Corollary 10 can be extended to all infinite-dimensional separable C∗ -algebras (possibly by considering the Cuntz ordering of positive elements instead of P(A)).
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References [1] I. Ben Yaacov, A. Berenstein, C.W. Henson, A. Usvyatsov, Model theory for metric structures, in: Z. Chatzidakis, et al. (Eds.), Model Theory with Applications to Algebra and Analysis, vol. II, in: London Math. Soc. Lecture Note Ser., vol. 350, Cambridge Univ. Press, 2008, pp. 315–427. [2] B. Blackadar, Operator Algebras. Theory of C ∗ -Algebras and von Neumann Algebras, Operator Algebras and Noncommutative Geometry, III, Encyclopaedia Math. Sci., vol. 122, Springer, Berlin, 2006. [3] H.G. Dales, W.H. Woodin, An Introduction to Independence for Analysts, London Math. Soc. Lecture Note Ser., vol. 115, Cambridge Univ. Press, Cambridge, 1987. [4] A. Dow, On ultrapowers of Boolean algebras, Topology Proc. 9 (2) (1984) 269–291. [5] I. Farah, N.C. Phillips, J. Stepr¯ans, The commutant of L(H ) in its ultrapower may or may not be trivial, preprint, http://arxiv.org/abs/0808.3763v1. [6] L. Ge, D. Hadwin, Ultraproducts of C ∗ -algebras, in: Recent Advances in Operator Theory and Related Topics, Szeged, 1999, in: Oper. Theory Adv. Appl., vol. 127, Birkhäuser, Basel, 2001, pp. 305–326. [7] E. Kirchberg, Central sequences in C ∗ -algebras and strongly purely infinite algebras, in: Operator Algebras: The Abel Symposium 2004, in: Abel Symp., vol. 1, Springer, Berlin, 2006, pp. 175–231. [8] E. Kirchberg, N.C. Phillips, Embedding of exact C ∗ -algebras in the Cuntz algebra O2 , J. Reine Angew. Math. 525 (2000) 17–53. [9] H. Lin, An Introduction to the Classification of Amenable C ∗ -algebras, World Scientific, River Edge, NJ, 2001. [10] T.A. Loring, Lifting Solutions to Perturbing Problems in C ∗ -Algebras, Fields Inst. Monogr., vol. 8, Amer. Math. Soc., Providence, RI, 1997. [11] S. Shelah, Classification Theory and the Number of Nonisomorphic Models, second ed., Stud. Logic Found. Math., vol. 92, North-Holland, Amsterdam, 1990. [12] B. Zamora-Aviles, There is an analytic gap of projections in the Calkin algebra, preprint, 2008, York University.
Journal of Functional Analysis 256 (2009) 3847–3859 www.elsevier.com/locate/jfa
Global Div-Curl lemma on bounded domains in R3 Hideo Kozono a,∗ , Taku Yanagisawa b a Mathematical Institute, Tohoku University, Sendai 980-8578, Japan b Department of Mathematics, Nara Women’s University, Nara 630-8506, Japan
Received 13 October 2008; accepted 12 January 2009 Available online 4 February 2009 Communicated by H. Brezis Dedicated to Professor Tetsuro Miyakawa on the occasion of his 60th birthday
Abstract We consider a global version of the Div-Curl lemma for vector fields in a bounded domain Ω ⊂ R3 with ∞ r the smooth boundary ∂Ω. Suppose that {uj }∞ j =1 and {vj }j =1 converge to u and v weakly in L (Ω) and
Lr (Ω), respectively, where 1 < r < ∞ with 1/r + 1/r = 1. Assume also that {div uj }∞ j =1 is bounded in s (Ω) for s > max{1, 3r /(3 + r )}, Lq (Ω) for q > max{1, 3r/(3 + r)} and that {rot vj }∞ is bounded in L j =1
∞ 1−1/q,q (∂Ω), or {v × ν| respectively. If either {uj · ν|∂Ω }∞ j ∂Ω }j =1 is bounded in j =1 is bounded in W W 1−1/s,s (∂Ω) (ν: unit outward normal to ∂Ω), then it holds that Ω uj · vj dx → Ω u · v dx. In particular, if either uj · ν = 0 or vj × ν = 0 on ∂Ω for all j = 1, 2, . . . is satisfied, then we have that Ω uj · vj dx → Ω u · v dx. As an immediate consequence, we prove the well-known Div-Curl lemma for any open set in R3 . The Helmholtz–Weyl decomposition for Lr (Ω) plays an essential role for the proof. © 2009 Elsevier Inc. All rights reserved.
Keywords: Div-Curl lemma; Helmholtz–Weyl decomposition; Elliptic system of boundary value problem; Compact imbedding
1. Introduction Let Ω be an open set in R3 . It is well known that if uj u, vj v weakly in L2 (Ω) and if ∞ 2 {div uj }∞ j =1 and {rot vj }j =1 are bounded in L (Ω), then it holds that uj · vj u · v in the sense of * Corresponding author.
E-mail addresses: [email protected] (H. Kozono), [email protected] (T. Yanagisawa). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.010
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distributions in Ω. This is the original Div-Curl lemma. For instance, we refer to Tartar [9]. The purpose of this paper is to deal with a similar lemma to bounded domains where the convergence uj · vj → u · v holds in the sense that
uj · vj dx →
Ω
u · v dx
as j → ∞.
(1.1)
Ω
Our result may be regarded as a global version of the Div-Curl lemma, which includes the previous one. To obtain such a global version, we need to pay an attention to the behaviour of {uj }∞ j =1 ∞ , or that of and {vj }∞ on the boundary ∂Ω of Ω. Indeed, an additional bound of {u · ν| } j ∂Ω j =1 j =1 1/2 (∂Ω) on the boundary ∂Ω plays an essential role for our convergence, {vj × ν|∂Ω }∞ j =1 in H where ν denotes the unit outward normal to ∂Ω. In what follows, we impose the following assumption on the domain Ω: Assumption. Ω is a bounded domain in R3 with C ∞ -boundary ∂Ω. Our method is based on our previous paper [5] which shows the Helmholtz–Weyl decomposition for vector fields in general Lr -spaces over Ω. In fact, it turns out in [5] that every u ∈ Lr (Ω) can be expressed uniquely as u = h + rot w + ∇p,
(1.2)
¯ with div h = 0, rot h = 0 and w, p ∈ W 1,r (Ω). We call h, w and p the harwhere h ∈ C ∞ (Ω) monic part, the vector and the scalar potentials of u, respectively. In the case r = 2, this exhibits an orthogonal decomposition in L2 (Ω). Hence, the convergence (1.1) may be reduced to that of each part. Since the space of harmonic vector fields is of finite dimension, the convergence of the harmonic part is easily handled. So, we need only to deal with the convergence in two parts of vector and scalar potentials. Roughly speaking, we have higher regularity for w and p such as w ∈ W 2,s (Ω) and p ∈ W 2,q (Ω) under the additional assumptions that rot u ∈ Ls (Ω) and that div u ∈ Lq (Ω), respectively. If we take q and s so that 1/r > 1/q − 1/3 and 1/r > 1/s − 1/3, re spectively, then we have compact embeddings W 2,q (Ω) ⊂ W 1,r (Ω) and W 2,s (Ω) ⊂ W 1,r (Ω). ∞ q s Hence, assuming that {div uj }∞ j =1 is bounded in L (Ω) and that {rot vj }j =1 is bounded in L (Ω), we show existence of strongly convergent subsequences of the gradient of the scalar potential part ∞ r r of {uj }∞ j =1 in L (Ω) and of the rotational of the vector potential part of {vj }j =1 in L (Ω). As a conclusion, we obtain (1.1). However, this argument is so formal that we need to justify it more carefully. Indeed, the gain of one more regularity for w and p in (1.2) follows from the a priori estimates in W 2,s (Ω) and W 2,q (Ω) for the elliptic boundary value problems. This is the reason ∞ 1−1/q,q (∂Ω), or {v × ν| why we assume boundedness of either {uj · ν|∂Ω }∞ j ∂Ω }j =1 in j =1 in W W 1−1/s,s (∂Ω). Before stating our result, we first recall the generalized trace theorem for u · ν and u × ν on q q ∂Ω defined on the Banach spaces Ediv (Ω) and Erot (Ω) for 1 < q < ∞, where q Ediv (Ω) ≡ u ∈ Lq (Ω); div u ∈ Lq (Ω) with the norm u E q = u q + div u q , div q q q Erot (Ω) ≡ u ∈ L (Ω); rot u ∈ L (Ω) with the norm u E q = u q + rot u q . rot
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Here and in what follows, · q denotes the usual Lq -norm over Ω. It is known that there are q q bounded operators γν and τν on Ediv (Ω) and Erot (Ω) with properties that
¯ γν u = u · ν|∂Ω if u ∈ C 1 (Ω),
γν : u ∈ Ediv (Ω) → γν u ∈ W 1−1/q ,q (∂Ω)∗ , q
¯ τν u = u × ν|∂Ω if u ∈ C 1 (Ω),
τν : u ∈ Erot (Ω) → τν u ∈ W 1−1/q ,q (∂Ω)∗ , q
respectively, where 1/q + 1/q = 1. The range W 1−1/q ,q (∂Ω)∗ of γν and τν is the dual space of W 1−1/q ,q (∂Ω) which is the image of the trace on ∂Ω of functions in W 1,q (Ω). Indeed, the following generalized Stokes formula holds
q
(u, ∇p) + (div u, p) = γν u, γ0 p ∂Ω
for all u ∈ Ediv (Ω) and all p ∈ W 1,q (Ω), q
(u, rot φ) = (rot u, φ) + τν u, γ0 φ ∂Ω
for all u ∈ Erot (Ω) and all φ ∈ W
1,q
(Ω),
(1.3) (1.4)
where γ0 denotes the usual trace operator from W 1,q (Ω) onto W 1−1/q ,q (∂Ω), and ·,· ∂Ω is the duality pairing between W 1−1/q ,q (∂Ω)∗ and W 1−1/q ,q (∂Ω). Here and in what follows, q q (·,·) denotes the duality pairing between L (Ω) and L (Ω). For a detail of (1.3) and (1.4), we refer to Borchers and Sohr [2], [5], Simader and Sohr [7] and Temam [10]. Our result now reads: Theorem 1. Let Ω be as in the Assumption. Let 1 < r < ∞ with 1/r + 1/r = 1. Suppose that ∞ r r {uj }∞ j =1 ⊂ L (Ω) and {vj }j =1 ⊂ L (Ω) satisfy uj u weakly in Lr (Ω),
vj v
weakly in Lr (Ω)
(1.5)
for some u ∈ Lr (Ω) and v ∈ Lr (Ω), respectively. Assume also that q {div uj }∞ j =1 is bounded in L (Ω) for some q > max 1, 3r/(3 + r)
(1.6)
s {rot vj }∞ j =1 is bounded in L (Ω) for some s > max 1, 3r /(3 + r ) ,
(1.7)
and that
respectively. If either 1−1/q,q (∂Ω), or (i) {γν uj }∞ j =1 is bounded in W ∞ 1−1/s,s (∂Ω), (ii) {τν vj }j =1 is bounded in W
then it holds that
uj · vj dx → Ω
u · v dx
as j → ∞.
(1.8)
Ω
In particular, if either γν uj = 0, or τν vj = 0 for all j = 1, 2, . . . is satisfied, then we have also (1.8).
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As an immediate consequence of our theorem, we have the following Div-Curl lemma in an arbitrary open set in R3 . Corollary 1.1. (See Tartar [9].) Let D be an arbitrary open set in R3 . Let 1 < r < ∞. Suppose ∞ r r that {uj }∞ j =1 ⊂ L (D) and {vj }j =1 ⊂ L (D) satisfy uj u weakly in Lr (D),
vj v
weakly in Lr (D)
(1.9)
for some u ∈ Lr (D) and v ∈ Lr (D), respectively. Assume also that
∞ r r {div uj }∞ j =1 and {rot vj }j =1 are bounded in L (D) and L (D),
(1.10)
respectively. Then it holds that uj · vj u · v
in the sense of distributions in D.
(1.11)
Remarks. (i) Since Ω is a bounded domain, we may assume that 3r/(3 + r) < q r and q ∞ s 3r /(3 + r ) < s r , and hence it holds that {uj }∞ j =1 ⊂ Ediv (Ω) and that {vj }j =1 ⊂ Erot (Ω).
1−1/q ,q (∂Ω)∗ and {τ v }∞ ⊂ W 1−1/s ,s (∂Ω)∗ . Then we have that {γν uj }∞ ν j j =1 j =1 ⊂ W 1−1/r,r (∂Ω) (ii) In Theorem 1, it is unnecessary to assume both bounds of {γν uj }∞ j =1 in W
1−1/r ,r (∂Ω). Indeed, what we need is only one of these bounds. and {τν vj }∞ j =1 in W (iii) Robbin, Rogers and Temple [6] treated the Div-Curl lemma by means of the de Rham– Hodge decomposition for the general k-form on a domain in Rn . Since they did not take any care of the boundary condition of k-forms, the global convergence such as (1.8) is excluded in their argument. On the other hand, our Helmholtz–Weyl decomposition allows us to handle the boundary conditions u · ν|∂Ω and v × ν|∂Ω , which yields the convergence of uj · vj on the whole domain Ω. (iv) The corresponding result in higher dimensions should be considered. Since our proof depends on the Helmholtz–Weyl decomposition in Lr of vector fields on Ω in Section 2 below, we need to extend it to the general differential forms in terms of the exterior differential operator d and its formal adjoint δ ≡ d ∗ . Another aspect on the Div-Curl lemma stands on an application of the theory of the Hardy space. See e.g., Coifman, Lions, Meyer and Semmes [3]. In this direction, Dafni [4] introduced the local Hardy space h1r (Ω) and showed that uj · vj → u · v weakly-∗ in h1r (Ω). It seems an interesting problem to investigate the relation between our Lr decomposition and the structure of u · v in h1r (Ω), which will be discussed in the forthcoming paper.
2. Lr -Helmholtz–Weyl decomposition In this section, we recall the Helmholtz–Weyl decomposition for vector fields in Lr (Ω). For a detail, we refer [5]. According to the two types u · ν = 0 and u × ν = 0 of boundary conditions on ∂Ω, we first define harmonic vector spaces Xhar (Ω) and Vhar (Ω) as ¯ div h = 0, rot h = 0 in Ω with h · ν = 0 on ∂Ω , Xhar (Ω) = h ∈ C ∞ (Ω); ¯ div h = 0, rot h = 0 in Ω with h × ν = 0 on ∂Ω . Vhar (Ω) = h ∈ C ∞ (Ω);
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Moreover, for 1 < r < ∞ let us define divergence-free vector fields Xσr (Ω) and Vσr (Ω) by Xσr (Ω) ≡ u ∈ W 1,r (Ω); div u = 0, γν u = 0 , Vσr (Ω) ≡ u ∈ W 1,r (Ω); div u = 0, τν u = 0 . Then we have the following decomposition theorem. For a detail, we refer Kozono and Yanagisawa [5]. Proposition 2.1. (See [5].) Let Ω be as in the Assumption. Let 1 < r < ∞. (1) Both Xhar (Ω) and Vhar (Ω) are finite dimensional vector spaces. (2) For every u ∈ Lr (Ω), there are p ∈ W 1,r (Ω), w ∈ Vσr (Ω) and h ∈ Xhar (Ω) such that u can be represented as u = h + rot w + ∇p.
(2.1)
Such a triplet {p, w, h} is subordinate to the estimate
p W 1,r + w W 1,r + h r C u r
(2.2)
with the constant C = C(Ω, r) independent of u. The above decomposition (2.1) is unique. In fact, if u has another expression u = h˜ + rot w˜ + ∇ p˜ for p˜ ∈ W 1,r (Ω), w˜ ∈ Vσr (Ω) and h˜ ∈ Xhar (Ω), then we have ˜ h = h,
rot w = rot w, ˜
∇p = ∇ p. ˜
(2.3)
(3) For every u ∈ Lr (Ω), there are p ∈ W01,r (Ω), w ∈ Xσr (Ω) and h ∈ Vhar (Ω) such that u can be represented as u = h + rot w + ∇p.
(2.4)
Such a triplet {p, w, h} is subordinate to the estimate
p W 1,r + w W 1,r + h r C u r
(2.5)
with the constant C = C(Ω, r) independent of u. The above decomposition (2.4) is unique. In fact, if u has another expression u = h˜ + rot w˜ + ∇ p˜ for p˜ ∈ W01,r (Ω), w˜ ∈ Xσr (Ω) and h˜ ∈ Vhar (Ω), then we have ˜ h = h,
rot w = rot w, ˜
p = p. ˜
(2.6)
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An immediate consequence of the above theorem is Corollary 2.1. Let Ω be as in the Assumption. (1) By the unique decompositions (2.1) and (2.4) we have two kinds of direct sums in algebraic and topological sense Lr (Ω) = Xhar (Ω) ⊕ rot Vσr (Ω) ⊕ ∇ W 1,r (Ω),
(2.7)
Lr (Ω) = Vhar (Ω) ⊕ rot Xσr (Ω) ⊕ ∇ W01,r (Ω)
(2.8)
for 1 < r < ∞. (2) Let Sr , Rr and Qr be projection operators associated with both (2.1) and (2.4) from Lr (Ω) onto Xhar (Ω), rot Vσr (Ω) and ∇ W 1,r (Ω), and from Lr (Ω) onto Vhar (Ω), rot Xσr (Ω) and ∇ W01,r (Ω), respectively, i.e., Sr u ≡ h,
Rr u ≡ rot w,
Qr u ≡ ∇p.
(2.9)
Then we have
Sr u r C u r ,
Rr u r C u r ,
Qr u r C u r
(2.10)
for all u ∈ Lr (Ω), where C = C(r) is the constant depending only on 1 < r < ∞. Moreover, there holds ⎧ 2 Sr∗ = Sr , ⎨ Sr = Sr , (2.11) R 2 = Rr , Rr∗ = Rr , ⎩ r2 ∗ Qr = Qr , Qr = Qr ,
where Sr∗ , Rr∗ and Q∗r denote the adjoint operators on Lr (Ω) of Sr , Rr and Qr , respectively. Remark 1. (1) The scalar potential p ∈ W 1,r (Ω) and the vector potential w ∈ Vσr (Ω) of u ∈ Lr (Ω) in (2.1) are formally obtained by solving the following boundary value problems p = div u in Ω,
∂p =u·ν ∂ν
on ∂Ω,
(2.12)
and ⎧ ⎨ rot(rot w) = rot u div w = 0 ⎩ w×ν =0
in Ω, in Ω, on ∂Ω,
(2.13)
respectively. Since u does not have any regularity, we need to solve (2.12) and (2.13) in a general ized sense for the given div u ∈ W01,r (Ω)∗ and for the given rot u ∈ Vσr (Ω)∗ with r = r/(r − 1), respectively. Indeed, the solvability in such a generalized sense follows from the variational inequalities of the quadratic forms
H. Kozono, T. Yanagisawa / Journal of Functional Analysis 256 (2009) 3847–3859
∇p r C
w W 1,r
|(∇p, ∇ψ)|
∇ψ r ψ∈W 1,r (Ω) sup
for all p ∈ W 1,r (Ω),
L
|(rot w, rot Ψ )|
(w, ψj )
C sup +
Ψ
W 1,r Ψ ∈V r (Ω)
for all w ∈ Vσr (Ω),
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(2.14)
(2.15)
j =1
σ
with the constant C = C(Ω, r), respectively. In (2.15), we set L = dim Vhar (Ω) with {ψ1 , . . . , ψL } an orthonormal basis in L2 (Ω) of Vhar (Ω). Then the harmonic part h of u in (2.1) is defined by h ≡ u − ∇p − rot w. For the proof of (2.14) and (2.15), see Simader and Sohr [7, Theorem 1.3] and [5, Lemma 4.1(2)], respectively. (2) Similarly, the scalar potential p ∈ W01,r (Ω) and the vector potential w ∈ Xσr (Ω) of u ∈ Lr (Ω) in (2.4) are formally given by the following boundary value problems p = div u
in Ω,
p = 0 on ∂Ω,
(2.16)
and ⎧ rot(rot w) = rot u ⎪ ⎪ ⎨ div w = 0 ⎪ rot w × ν = u × ν ⎪ ⎩ w·ν =0
in Ω, in Ω, on ∂Ω, on ∂Ω,
(2.17)
respectively. In the same way as in (1), the solvability of (2.16) and (2.17) for general u ∈ Lr (Ω) follows from the variational inequalities
∇p r C
w W 1,r C
sup
ϕ∈W01,r (Ω)
|(∇p, ∇ϕ)|
∇ϕ r
for all p ∈ W01,r (Ω),
N
|(rot w, rot Φ)|
(w, ϕj )
+
Φ W 1,r Φ∈X r (Ω)
sup
for all w ∈ Xσr (Ω),
(2.18)
(2.19)
j =1
σ
with the constant C = C(Ω, r), respectively. In (2.19), we set N = dim Xhar (Ω) with {ϕ1 , . . . , ϕN } an orthonormal basis in L2 (Ω) of Xhar (Ω). Then the harmonic part h of u in (2.4) is defined by h ≡ u − ∇p − rot w. For the proof of (2.18) and (2.19), see Simader and Sohr [8] and [5, Lemma 4.1(1)]. (3) In (2.1), we may take canonical p ∈ W 1,r (Ω) and w ∈ Vσr (Ω) in such a way that p(x) dx = 0,
(w, ψj ) = 0,
j = 1, . . . , L.
Ω
Hence by (2.14) and (2.15), such p and w are subject to the estimates
p W 1,r C u r ,
w W 1,r C u r ,
which yields (2.2). Similarly, we may choose w ∈ Xσr (Ω) in (2.4) in such a way that
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(w, ϕj ) = 0,
j = 1, . . . , N,
which, by means of (2.19), yields (2.5). If u has an additional regularity such as div u ∈ Lq (Ω) and rot u ∈ Lq (Ω) for some 1 < q r, then we may choose the scalar and the vector potentials p and w in (2.1) and (2.4) in the class W 2,q (Ω). More precisely, we have Proposition 2.2. Let Ω be as in the Assumption and let 1 < r < ∞. Suppose that u ∈ Lr (Ω). (1) Let us consider the decomposition (2.1). (i) If, in addition, rot u ∈ Lq (Ω) for some 1 < q r, then the vector potential w of u in (2.1) can be chosen as w ∈ W 2,q (Ω) ∩ Vσr (Ω) with the estimate
(2.20)
w W 2,q C rot u q + u r . (ii) If, in addition, div u ∈ Lq (Ω) with γν u ∈ W 1−1/q,q (∂Ω) for some 1 < q r, then the scalar potential p of u in (2.1) can be chosen as p ∈ W 2,q (Ω) ∩ W 1,r (Ω) with the estimate
(2.21)
p W 2,q C div u q + u r + γν u W 1−1/q,q (∂Ω) . (2) Let us consider the decomposition (2.4). (i) If, in addition, div u ∈ Lq (Ω) for some 1 < q r, then the scalar potential p of u in (2.4) can be chosen as p ∈ W 2,q (Ω) ∩ W01,r (Ω) with the estimate
p W 2,q C div u q .
(2.22)
(ii) If, in addition, rot u ∈ Lq (Ω) with τν u ∈ W 1−1/q,q (∂Ω) for some 1 < q r, then the vector potential w of u in (2.4) can be chosen as w ∈ W 2,q (Ω) ∩ Xσr (Ω) with the estimate
(2.23)
w W 2,q C rot u q + u r + τν u W 1−1/q,q (∂Ω) . Here C = C(Ω, r, q) is the constant depending only on Ω, r and q. Proof. (1) (i) In the decomposition (2.1), the vector potential w is a solution of the boundary value problem (2.13). More precisely, based on the variational inequality (2.15), we can choose a canonical w ∈ Vσr (Ω) in such a way that (rot w, rot Ψ ) = (u, rot Ψ )
for all Ψ ∈ Vσr (Ω)
(2.24)
with the estimate
w W 1,r C u r ,
(2.25)
where C = C(Ω, r) is a constant depending only on Ω and r. See [5, Lemma 4.2(2)]. Since div w = 0 in Ω and since rot u ∈ Lq (Ω), it follows from (2.24) that − w = rot u in the sense of
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distributions in Ω, and we may regard w as a weak solution of the boundary value problem ⎧ ⎨ − w = rot u in Ω, (2.26) div w = 0 on ∂Ω, ⎩ w×ν =0 on ∂Ω. Hence it follows from [5, Lemma 4.3(1)] and the classical theory of Agmon, Douglis and Nirenberg [1] that the solution w of the homogeneous boundary value problem (2.26) belongs to W 2,q (Ω) and that the estimate
(2.27)
w W 2,q C rot u q + w q 1
−1
holds with a constant C depending only on Ω and q. Since w q |Ω| q r w r , the desired estimate (2.20) follows from (2.25) and (2.27). (ii) The scalar potential p in (2.1) is a solution of (2.12). More precisely, based on the variational inequality (2.14), we may choose a canonical p ∈ W 1,r (Ω) in such a way that (∇p, ∇ψ) = (u, ∇ψ)
for all ψ ∈ W 1,r (Ω)
(2.28)
with the estimate
p W 1,r C u r .
(2.29)
See Simader and Sohr [7, Theorems 1.3, 1.4]. Since div u ∈ Lq (Ω) and since γν (∇p − u) = 0, we may regard p as a weak solution of (2.12). Since γν u ∈ W 1−1/q,q (∂Ω), the well-known a priori estimate for the inhomogeneous Neumann problem of the Poisson equation states that p ∈ W 2,q (Ω) with the estimate
p W 2,q C div u q + p q + γν u W 1−1/q,q (∂Ω) . 1
−1
Since p q |Ω| q r p r , from (2.29) and the above estimate we obtain (2.21). (2) (i) In the decomposition of (2.4), the scalar potential p is a solution of (2.16). More precisely, based on the variational inequality (2.18), we may choose a canonical p ∈ W01,r (Ω) in such a way that (∇p, ∇ϕ) = (u, ∇ϕ)
for all ϕ ∈ W01,r (Ω)
(2.30)
with the estimate
p W 1,r C u r . See Simader and Sohr [8]. Since div u ∈ Lq (Ω), by (2.30) we may regard p as a weak solution of (2.16). Hence the regularity theorem of the Dirichlet problem of the Poisson equation with the homogeneous boundary condition states that p ∈ W 2,q (Ω) ∩ W01,r (Ω) with the estimate
p W 2,q C div u q , which yields (2.22).
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(ii) The vector potential w ∈ Xσr (Ω) in (2.4) is a solution of (2.17). More precisely, we can choose a canonical w ∈ Xσr (Ω) in such a way that
(rot w, rot Φ) = (u, rot Φ)
for all Φ ∈ Xσr (Ω)
(2.31)
with the estimate
w W 1,r C u r .
(2.32)
See [5, Lemma 4.3(1)]. Since div w = 0 in Ω and since rot u ∈ Lq (Ω), it follows from (2.31) that − w = rot u in the sense of distributions in Ω, and we may regard w as a weak solution of the boundary value problem − w = rot u
rot w × ν = u × ν w·ν =0
in Ω, on ∂Ω, on ∂Ω.
(2.33)
Since τν u ∈ W 1−1/q,q (∂Ω), it follows from [5, Lemma 4.3(2)] and Agmon, Douglis and Nirenberg [1] that the solution w of the inhomogeneous boundary value problem (2.33) belongs to W 2,q (Ω) and that the estimate
w W 2,q C rot u q + w q + τν u W 1−1/q,q (∂Ω) 1
−1
holds with a constant C = C(Ω, q) depending only on Ω and q. Since w q |Ω| q r w r , from (2.32) and the above estimates we obtain the desired estimate (2.23). This completes the proof of Proposition 2.2. 2 3. Proof of Theorem 1 1−1/q,q (∂Ω). In such a case, (i) Let us first consider the case when {γν uj }∞ j =1 is bounded in W we make use of the decomposition (2.1). Let Sr , Rr and Qr be the projection operators from Lr (Ω) onto Xhar (Ω), rot Vσr (Ω) and ∇W 1,r (Ω) defined by (2.9), respectively. Notice that the identity
(u, v) = (Sr u, Sr v) + (Rr u, Rr v) + (Qr u, Qr v)
(3.1)
holds for all u ∈ Lr (Ω) and all v ∈ Lr (Ω). Indeed, by the generalized Stokes formula (1.3) and (1.4), we have (∇p, h) = −(p, div h) + γν h, γ0 p ∂Ω = 0, (rot w, h) = (w, rot h) + τν w, γ0 h ∂Ω = 0 for all p ∈ W 1,r (Ω), w ∈ Vσr (Ω) and h ∈ Xhar (Ω). Similarly, we have (rot w, ∇p) = γν (rot w), γ0 p ∂Ω = 0 Thus we obtain (3.1).
for all w ∈ Vσr (Ω), p ∈ W 1,r (Ω).
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Now, by (3.1), we see that the convergence (1.8) can be reduced to (Sr uj , Sr vj ) → (Sr u, Sr v),
(3.2)
(Rr uj , Rr vj ) → (Rr u, Rr v),
(3.3)
(Qr uj , Qr vj ) → (Qr u, Qr v).
(3.4)
By Proposition 2.1(1), the ranges of Sr and Sr are of finite dimension, which means that both Sr and Sr are finite rank operators, therefore compact. Hence, we have by (1.5) that Sr uj → Sr u
Sr v j → Sr v
strongly in Lr (Ω),
strongly in Lr (Ω),
from which it follows (3.2). Next, we apply Proposition 2.2(1) to (3.3) and (3.4). Since Ω is bounded, we may assume that 3r 3r < s r . < q r, max 1, max 1, 3+r 3 + r By (1.7) and (2.20) with q and r replaced by s and r , respectively, we see that Rr vj ≡ rot w˜ j with w˜ j ∈ Vσr (Ω) satisfies w˜ j ∈ W 2,s (Ω) ∩ Vσr (Ω) with the estimate
w˜ j W 2,s C rot vj s + vj r M
for all j = 1, 2, . . .
with a constant M independent of j . Since 1/r > 1/s − 1/3, the embedding W 2,s (Ω) ⊂ W 1,r (Ω) is compact, and hence we see that {w˜ j }∞ j =1 has a strongly convergent subsequence
r in W 1,r (Ω), and hence {Rr vj }∞ j =1 has a strongly convergent subsequence in L (Ω). Since
(1.5) yields rot w˜ j = Rr vj Rr v weakly in Lr (Ω), it holds, in fact, that R r v j → Rr v
strongly in Lr (Ω).
(3.5)
Obviously by (1.5), Rr uj Rr u weakly in Lr (Ω), and hence (3.3) follows. 1−1/q,q (∂Ω), we see from (1.6) and (2.21) that Q u = ∇p Since {γν uj }∞ r j j j =1 is bounded in W 2,q satisfies that pj ∈ W (Ω) with the estimate
pj W 2,q C div uj q + uj r + γν uj W 1−1/q,q (∂Ω) M
for all j = 1, 2, . . .
with a constant M independent of j . Since 1/r > 1/q − 1/3, again by the compact embedding W 2,q (Ω) ⊂ W 1,r (Ω) and by the weak convergence ∇pj = Qr uj Qr u in Lr (Ω), implied by (1.5), it holds that Qr uj → Qr u strongly in Lr (Ω).
(3.6)
Since (1.5) yields Qr vj Qr v weakly in Lr (Ω), we see that (3.4) follows. 1−1/s,s (∂Ω). In this case, (ii) We next consider the case when {τν vj }∞ j =1 is bounded in W we make use of the decomposition (2.4). Then the argument is quite similar to the former
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case (i) above. However, we give the proof for the sake of completeness. By the same notations Sr , Rr and Qr , we denote the projection operators from Lr (Ω) onto Vhar (Ω), rot Xσr (Ω) and ∇W01,r (Ω) defined by (2.9), respectively. By the generalized Stokes formula, it is easy to see that the identity (3.1) holds, and hence we may prove (3.2), (3.3) and (3.4). Since the range of Sr is Vhar (Ω), it follows from Proposition 2.1(1) that the convergence (3.2) holds. 1−1/s,s (∂Ω), by (1.7) and (2.23) with q and r replaced by Since {τν vj }∞ j =1 is bounded in W
s and r , we find that Rr vj = rot w˜ j with w˜ j ∈ Xσr (Ω) satisfies, in fact, that w˜ j ∈ W 2,s (Ω) ∩ Xσr (Ω) with the estimate
w˜ j W 2,s C rot vj s + vj r + τν vj W 1−1/s,s (∂Ω) M
for all j = 1, 2, . . .
with a constant M independent of j . By the compact embedding W 2,s (Ω) ⊂ W 1,r (Ω) and by the weak convergence rot w˜ j = Rr vj Rr v in Lr (Ω), implied by (1.5), it holds that R r v j → Rr v
strongly in Lr (Ω).
(3.7)
Since (1.5) yields Rr uj Rr u weakly in Lr (Ω), it follows (3.3). From (1.6) and (2.22) we see that Qr uj = ∇pj with pj ∈ W01,r (Ω) satisfies, in fact, that pj ∈ W 2,q (Ω) ∩ W01,r (Ω) with the estimate
pj W 2,q C div uj q M
for all j = 1, 2, . . .
with a constant M independent of j . Hence again by the compact embedding W 2,q (Ω) ⊂ W 1,r (Ω) and by the weak convergence ∇pj = Qr uj Qr u in Lr (Ω), implied by (1.5), it holds that Qr uj → Qr u strongly in Lr (Ω).
(3.8)
Since (1.5) yields Qr vj Qr v weakly in Lr (Ω), we see that (3.4) follows. This proves Theorem 1. Proof of Corollary 1.1. We may prove that for every ϕ ∈ C0∞ (D)
ϕuj · vj dx → D
ϕu · v dx. D
Let us take a bounded domain Ω ⊂ R3 with the smooth boundary ∂Ω so that supp ϕ ⊂ Ω ⊂ D. Then it suffices to prove that ϕuj · vj dx → ϕu · v dx. (3.9) Ω
Ω
Obviously by (1.9), it holds that ϕuj ϕu
weakly Lr (Ω),
vj v
weakly Lr (Ω).
(3.10)
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Since div(ϕuj ) = ϕ div uj + uj · ∇ϕ, we see by (1.9) and (1.10) that {div(ϕuj )}∞ j =1 is bounded in Lr (Ω) with γν (ϕuj ) = 0,
j = 1, 2, . . . .
(3.11)
r Since (1.10) states that {rot vj }∞ j =1 is also bounded in L (Ω), by taking q = r and s = r in (1.6) and (1.7), respectively, we see that the convergence (3.9) follows from (3.10), (3.11) and Theorem 1(i). This proves Corollary 1.1. 2
Acknowledgment The authors would like to express their thanks to the referee for her/his valuable comments. References [1] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math. 17 (1964) 35–92. [2] W. Borchers, H. Sohr, On the equations rot v = g and div u = f with zero boundary conditions, Hokkaido Math. J. 19 (1990) 67–87. [3] R. Coifman, P.-L. Lions, Y. Meyer, S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. 72 (1993) 247–286. [4] G. Dafni, Nonhomogeneous div-curl lemmas and local Hardy spaces, Adv. Differential Equations 10 (2005) 505– 526. [5] H. Kozono, T. Yanagisawa, Lr -variational inequality for vector fields and the Helmholtz–Weyl decomposition in bounded domains, Indiana Univ. Math. J., in press. [6] J.W. Robbin, R. Rogers, B. Temple, On weak continuity and Hodge decomposition, Trans. Amer. Math. Soc. 303 (1987) 609–618. [7] C.G. Simader, H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in Lq -spaces for bounded and exterior domains, in: G.P. Galdi (Ed.), Mathematical Problems Relating to the Navier–Stokes Equations, in: Ser. Adv. Math. Appl. Sci., World Scientific, Singapore, New Jersey, London, Hong Kong, 1992, pp. 1–35. [8] C.G. Simader, H. Sohr, The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains, Pitman Res. Notes Math. Ser., vol. 360, Longman, 1996. [9] L. Tartar, Compensated compactness and applications to partial differential equations, in: R.J. Knops (Ed.), Nonlinear Analysis and Mechanics: Heriot–Watt Symposium, vol. 4, in: Res. Notes Math., Pitman, 1979. [10] R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, New York, Oxford, 1979.
Journal of Functional Analysis 256 (2009) 3861–3891 www.elsevier.com/locate/jfa
The classification of certain non-simple C ∗ -algebras of tracial rank zero Xiaochun Fang 1 Department of Mathematics, Tongji University, Shanghai 200092, China Received 21 September 2008; accepted 5 March 2009 Available online 21 March 2009 Communicated by D. Voiculescu
Abstract It is proved that the Z2 -graded ordered K-theory group with order unit (K∗ (A), K∗ (A)+ , [1A ]) is a complete invariant for the class of unital C ∗ -algebras which are inductive limits of finite direct sums of unital amenable separable simple C ∗ -algebras with tracial rank zero which satisfy the UCT. © 2009 Elsevier Inc. All rights reserved. Keywords: C ∗ -algebra; Tracial rank zero; Classification
1. Introduction The Elliott conjecture asserts that all nuclear separable simple C ∗ -algebras are classified up to isomorphism by an invariant, called the Elliott invariant. In the last twenty years important progress has been made in the classification of amenable C ∗ -algebras by many authors (see [13,51]), for example by D. Dadalart (see [3–8]), G. Elliott (see [10–20]), G. Gong (see [23–27]), H. Lin (see [29–45]), and so on. In order to search for more classifiable C ∗ -algebras, tracial rank for C ∗ -algebras was introduced by H. Lin in [35], and it was proved that simple C ∗ -algebras with tracial rank zero have real rank zero, stable rank one, weakly unperforated K0 groups, and are quasidiagonal. A classification theorem for unital amenable separable simple C ∗ -algebras with tracial rank zero which satisfy the UCT was given in [40]. In this paper, we prove that the ordered K-theory group (K∗ (A), K∗ (A)+ , [1A ]) is a complete invariant for the class of unital C ∗ -algebras which are inductive limits of finite direct sums of E-mail address: [email protected]. 1 The research reported in this article was supported by the National Natural Science Foundation of China (10771161).
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.03.005
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unital amenable simple C ∗ -algebras with tracial rank zero which satisfy the UCT. A C ∗ -algebra in this class will be called a unital ATAF algebra, and by the example in [5] it is known that there exists a unital ATAF algebra which is not an AH algebra. The paper is organized as follows: In Section 2, we list some preliminary results which include the formal definition of tracial rank zero. In Section 3, we obtain the uniqueness theorem for unital ATAF algebras: Let A be a unital ATAF algebra. Then for any ε > 0, and any finite subset F ⊆ A, there exists a K-triple (P , G, δ), where δ is a positive number, P is a finite subset of P (A), and G is a finite subset of A, which satisfies that for any unital K-injective C ∗ -algebra B of real rank zero and stable rank one and with K0 (B) weakly unperforated (in particular B is a unital ATAF algebra), and any two unital G-δ-multiplicative completely positive linear maps L1 , L2 : A → B, if L1# (p) = L2# (p) for any p ∈ P , then there exists a unitary element U ∈ B such that Ad(U ) ◦ L1 ≈ε L2 on F . In Section 4 we obtain the existence theorem for unital ATAF algebras: Let A and B be unital ATAF algebras. Then for any α ∈ KK(A, B) with the induced map α∗ : K(A) → K(B) satisfying that α∗ (K∗ (A)+ ) ⊆ K∗ (B)+ and α∗ ([1A ]) = [1B ], and for any K-triple (P , G, δ), where δ is a positive number, P is a finite subset of P (A), and G is a finite subset of A, there is a unital G-δ-multiplicative completely positive linear map L : A → B such that L# (p) = α∗ ([p]) for any p ∈ P . In Section 5 we obtain the classification theorem for ATAF algebras: Suppose that A and B are two unital ATAF algebras. If K∗ (A), K∗ (A)+ , [1A ] ∼ = K∗ (B), K∗ (B)+ , [1B ] , then A ∼ = B. Moreover the range of the invariant for ATAF algebras is also discussed. 2. Preliminaries and definitions Let a and b be two positive elements in a C ∗ -algebra A. We write [a] [b], if there exists a partial isometry v ∈ A∗∗ such that, for every c ∈ Her(a), v ∗ c, cv ∈ A, vv ∗ = Pa , where Pa is the range projection of a in A∗∗ , and v ∗ cv ∈ Her(b). We write [a] = [b] if v ∗ Her(a)v = Her(b). Let n be a positive integer. We write n[a] [b], if there are mutually orthogonal positive elements b1 , b2 , . . . , bn ∈ Her(b) such that [a] [bi ], i = 1, 2, . . . , n. Let A and B be C ∗ -algebras, let L : A → B be a map, let ε > 0, and let F be a finite subset of A. L is called F -ε-multiplicative if L(xy) − L(x)L(y) < ε for all x, y ∈ F . Let L : A → B be another map, then we write L ≈ ε L
on F,
if L(x) − L (x) < ε
for all x ∈ F.
Definition 2.1. (See [36, Definition 3.6.2].) A simple unital C ∗ -algebra A is said to have tracial rank zero, if for any ε > 0, any finite subset F , and any nonzero positive element a, there exist a nonzero projection p ∈ A and a finite dimensional C ∗ -subalgebra B of A with 1B = p, such that (1) xp − px < ε for all x ∈ F , (2) pxp ∈ε B for all x ∈ F , and (3) [1 − p] [a].
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If A has tracial rank zero, then we will write T R(A) = 0. A C ∗ -algebra with tracial rank zero is also called a TAF algebra. Moreover if in the above definition we replace the finite dimensional C ∗ -subalgebra B of A by a C ∗ -subalgebra B of A which is a unital hereditary C ∗ -subalgebra of a C ∗ -algebra with the form C(X) ⊗ F , where X is a one dimensional finite CW complex and F is a finite dimensional C ∗ -algebra, then A is called to have tracial rank no more than one and denoted by T R(A) 1. Definition 2.2. A C ∗ -algebra A will be called an ATAF algebra (approximate TAF algebra), if A= ∞ n=1 An , where An ⊆ An+1 , and each An is a finite direct sum of unital amenable separable simple C ∗ -algebras with tracial rank zero which satisfy the UCT. Clearly, an ATAF algebra is still a (non-simple generally) C ∗ -algebra with tracial rank zero. Moreover an AF algebra is an ATAF algebra, and by the example in [5] it is known that there exists a unital ATAF algebra which is not an AH algebra. Definition 2.3. (See [1, Definition 6.7.1].) Let (G, G+ ) be an ordered group. We write x 0 if x ∈ G+ ; we write x > 0 if x ∈ G+ \ {0}. An ordered group (G, G+ ) is called unperforated if nx 0 for some integer n > 0 implies that x 0; an ordered group (G, G+ ) is called weakly unperforated if nx > 0 for some integer n > 0 implies x > 0. A unital C ∗ -algebra A is said to have stable rank one, and written as tsr(A) = 1, if GL(A) is dense in A, i.e., the set of invertible elements is dense in A. A unital C ∗ -algebra A is said to have real rank zero, and written as RR(A) = 0, if the set of invertible self-adjoint elements is dense in Asa (see [2,9,21,22,46–49]). Theorem 2.4. (See [36, Theorem 3.6.11].) Every unital simple C ∗ -algebra A with tracial rank zero has stable rank one and real rank zero; and (K0 (A), K0 (A)+ , [1A ]) is a weakly unperforated ordered group with the ordered unit [1A ]. 3. Uniqueness theorem for ATAF algebras Let A and B be C ∗ -algebras, let L : A → B be a contractive completely positive linear map, and let Xn denote the Moore space with a fixed base point obtained by attaching the disk to the circle by a degree n 2 map, which satisfies K0 (C0 (Xn )) = Z/nZ and K1 (C0 (Xn )) = 0. Then L ⊗ idC(Xn ) : A ⊗ C(Xn ) → B ⊗ C(Xn ), L ⊗ idSC(Xn ) : A ⊗ SC(Xn ) → B ⊗ SC(Xn ), and L ⊗ idC(T)⊗C(Xn ) : A ⊗ C(T) ⊗ C(Xn ) → B ⊗ C(T) ⊗ C(Xn ) are contractive completely positive linear maps. For any integer n > 1, positive number ε > 0, and finite subsets F1 ⊆ A ⊗ C(Xn ), F2 ⊆ A ⊗ SC(Xn ), and F3 ⊆ A ⊗ C(T) ⊗ C(Xn ), there exist η > 0 and finite subset G ⊆ A, which are independent of the C ∗ -algebra B, such that if L is G-η-multiplicative, then L ⊗ idC(Xn ) is F1 -ε-multiplicative, L ⊗ idSC(Xn ) is F2 -ε-multiplicative, and L ⊗ idC(T)⊗C(Xn ) is F3 -ε-multiplicative (see Lemma 2.2 in [4]). If A is a C ∗ -algebra, we denote by Proj(A) the set of projections in ∞ k=1 Mk (A) ⊆ A ⊗ K, and set M M Proj A ⊗ C(T) ⊗ C(Xn ) , Proj (A) = Proj A ⊗ C(T) ∪ n=2
P (A) = Proj (A) = Proj A ⊗ C(T) ∪ ∞
∞ n=2
Proj A ⊗ C(T) ⊗ C(Xn ) .
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One defines Ki (A, Z/nZ) = Ki A ⊗ C0 (Xn ) (i = 0, 1), K∗ (A) = K0 (A) ⊕ K1 (A) ∼ = K0 A ⊗ C(T) . Then K∗ (A, Z/nZ) = K0 (A, Z/nZ) ⊕ K1 (A, Z/nZ) ∼ = K∗ A ⊗ C(T) ⊗ C0 (Xn ) , K∗ (A) ⊕ K∗ (A, Z/nZ) ∼ = K∗ A ⊗ C(T) ⊗ C(Xn ) , and we will identify the isomorphic groups above. Set ∞ C(T) ⊗ C(Xn ) , C = C(T) ⊕ C0 = C(T) ⊕
C = C ⊗ K,
n=2
∞ C(T) ⊗ C0 (Xn ) ,
C 0 = C0 ⊗ K,
n=2
then we define K(A) = K0 (A ⊗ C0 ) = K0 (A ⊗ C 0 ) = K∗ (A) ⊕
∞
K∗ (A, Z/nZ) .
n=2
⊆ K(A). By the group identification above, we have K0 (A ⊗ C) = K0 (A ⊗ C) For any contractive completely positive linear map ϕ : A → B, we associate a map ϕ# : Proj(A) → K0 (B), which is defined as follows: For p ∈ B ⊗ K, ϕ# (p) = [χ(ϕ ⊗ idK (p))], if ϕ ⊗ idK (p)2 − ϕ ⊗ idK (p) < 1/4; otherwise ϕ# (p) = 0, where χ is the characteristic function of the interval (1/2, 1]. Set
ϕ = ϕ ⊗ idC : A ⊗ C → B ⊗ C, then by abusing the notation we also associate a map
= K0 (B ⊗ C) ⊆ K(B), ϕ# : ProjM (A) → K0 (B ⊗ C) ϕ (p))] if
ϕ (p)2 −
ϕ (p) < which is defined as follows: For any p ∈ A ⊗ C ⊗ K, ϕ# (p) = [χ(
1/4, otherwise ϕ# (p) = 0, where χ is the characteristic function of the interval (1/2, 1]. Two sequences (an ) and (bn ) are called congruent if there is n0 such that an = bn for n n0 . Congruence is denoted by (an ) ≡ (bn ), or even an ≡ bn by abusing the notation. We say (ϕn ) is a (unital) contractive completely positive asymptotic morphism, if {ϕn } is a sequence of (unital) contractive completely positive linear maps from A to Bn , which satisfies that for any a, a ∈ A, lim ϕn (aa ) − ϕn (a)ϕn (a ) = 0. n→∞
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A contractive completely positive asymptotic morphism (ϕn ) induces a sequence of maps ϕn# : ProjM (A) → K(Bn ). Note that if p, q ∈ ProjM (A) have the same class in K(A), then pi , qi ∈ ϕn# (p) ≡ ϕn# (q). Assume that A is unital. For any x ∈ K(A), we fixed the projections N ([p ] − [q ]) and set ϕ (x) = (ϕ ProjM (A) (1 i N ) such that x = N i i n# i=1 i=1 n# (pi ) − ϕn# (qi )) ∈ K(A). The sequence (ϕn# (x)) depends on the particular projections that we use to represent x, but only up to congruence. While in general the map ϕn# : K(A) → K(Bn ) are not group morphisms, the sequence {ϕn# (x)} satisfies ϕn# (x + y) ≡ ϕn# (x) + ϕn# (y) for all x, y ∈ K(A). Let P ⊆ Proj(A) ∩ Mk (A) and G ⊆ A be finite sets, and let δ > 0. We say that (P , G, δ) is a K0 -triple if for any contractive completely positive G-δ-multiplicative linear map ϕ : A → B, the ϕ (p)2 −
ϕ (p) < 1/4 map
ϕ = ϕ ⊗ idMk (C) : Mk (A) → Mk (B) is P - 14 -multiplicative, and so
for any p ∈ P . Let P ⊆ Proj(M) (A) ∩ Mk (A ⊗ C) (where C = C(T) ⊕ ( ∞ n=2 (C(T) ⊗ C(Xn )))) and G ⊆ A be finite sets, and let δ > 0. We say that (P , G, δ) is a K-triple if for any contractive completely positive G-δ-multiplicative linear map ϕ : A → B, the map
ϕ = ϕ ⊗ idC : A ⊗ C → B ⊗ C is 1
P - 4 -multiplicative, where C = C ⊗ K. In this case ϕ# (p) = [χ(
ϕ (p))] for all p ∈ P , where χ is the characteristic function of the interval (1/2, 1]. Theorem 3.1. (See [36, Theorem 6.3.3].) Let A be a unital amenable separable simple C ∗ algebra with tracial rank zero which satisfies the UCT. Then for any ε > 0, and any finite subset F ⊆ A, there exists a K-triple (P , G, δ), where δ is a positive number, P is a finite subset of P (A), and G is a finite subset of A, which satisfies that for any unital C ∗ - algebra B of real rank zero and stable rank one and with weakly unperforated K0 (B), and any two unital G-δmultiplicative contractive completely positive linear maps L1 , L2 : A → B, if for any p ∈ P , L1# (p) = L2# (p), then there exists a unitary element U ∈ B such that Ad(U ) ◦ L1 ≈ε L2 on F . Lemma 3.2. Let A be a unital C ∗ -algebra with the cancellation property. Then (1) For any projection p ∈ A, the canonical morphism i∗ : K0 (pAp) → K0 (A) is injective. (2) If (K0 (A), K0 (A)+ ) is a weakly unperforated ordered group, then for any projection p ∈ A, (K0 (pAp), K0 (pAp)+ ) is a weakly unperforated ordered group. Proof. (1) For any x ∈ K0 (pAp), if i∗ (x) = 0, then we need to show that x = 0 in K0 (pAp). Let x ∈ K0 (pAp), without loss of generality we may assume that x = [r] − [s], where s, r ∈ Ml (pAp). If i∗ (x) = 0, then we have [r] = [s] in K0 (A). Since A has the cancellation property, r ∼ s, i.e., there exists a partial isometry v ∈ Ml (pAp) such that v ∗ v = r, vv ∗ = s. Take w = diag(p, p, . . . , p)v diag(p, p, . . . , p) (where p repeats l times), then w ∈ Ml (pAp), w ∗ w = r, ww ∗ = s, so we have [r] = [s] in K0 (pAp), i.e., x = 0 in K0 (pAp). So the canonical morphism i∗ : K0 (pAp) → K0 (A) is injective. (2) For any x ∈ K0 (pAp), if there exists an integer k such that kx > 0, then we need to show that x 0. Without loss of generality we may assume that x = [r] − [s], where s, r ∈ Ml (pAp). Since kx > 0, and by (1) the canonical morphism i∗ : K0 (pAp) → K0 (A) is injective, we have kx > 0 in K0 (A). Since (K0 (A), K0 (A)+ ) is a weakly unperforated ordered group, we have x > 0 in K0 (A), i.e., [r] − [s] > 0 in K0 (A). Therefore, there exists a nonzero projection
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q ∈ Mi (A) (suppose that i > l), such that [r] − [s] = [q ] in K0 (A), i.e., [r] = [s] + [q ] in K0 (A). Regard r, s as projections in Mi (pAp). Since A has the cancellation property, there exists a unitary element u ∈ M2i (A) such that u∗ diag(s, q )u = diag(r, 0). Set q = u∗ diag(0, q )u, then q diag(r, 0) diag(p, p), and so we have q ∈ M2i (pAp), and [q] = [q ] in K0 (A). So we have [r] = [s] + [q] in K0 (A). Since r, s are projections in M2i (pAp), and A has the cancellation property, we have [r] = [s] + [q] in K0 (pAp), i.e., [r] − [s] = [q]. Therefore we have [r] − [s] 0, i.e., x 0. 2 Definition 3.3. A C ∗ -algebra B will be called K-injective, if for any nonzero projection p ∈ B, the canonical map from K(pBp) = K 0 ((p ⊗ 1M(C0 ) )(B ⊗ C0 )(p ⊗ 1M(C0 ) )) to K(B) = K0 (B ⊗ C0 ) is injective, where C0 = C(T) ⊕ ( ∞ n=2 (C(T) ⊗ C0 (Xn ))). Lemma 3.4. Suppose that A = rn=1 An , where each An is a unital amenable separable simple C ∗ -algebra with tracial rank zero which satisfies the UCT. Then for any ε > 0, and any finite subset F ⊆ A, there exists a K-triple (P , G, δ), where δ is a positive number, P is a finite subset of P (A), and G is a finite subset of A, which satisfies that for any unital K-injective C ∗ - algebra B of real rank zero and stable rank one and with K0 (B) weakly unperforated, and any two unital G-δ-multiplicative contractive completely positive linear maps L1 , L2 : A → B, if for any p ∈ P, L1# (p) = L2# (p), then there exists a unitary element U ∈ B such that Ad(U ) ◦ L1 ≈ε L2 on F . Proof. We prove this theorem by two steps. (I) We show the assertion that for any ε > 0, and any finite subset F ⊆ A, there exists a Ktriple (P , G, δ), where δ is a positive number, P is a finite subset of P (A), and G is a finite subset of A, which satisfies the condition C: For any unital K-injective C ∗ -algebra B of real rank zero and stable rank one and with K0 (B) weakly unperforated, and any two unital G-δ-multiplicative contractive completely positive linear maps L1 , L2 : A → B, which satisfy that for any p ∈ P , L1# (p) = L2# (p), and L1 (1An ) = pn , L2 (1An ) = qn , where pn and qn are projections in B such that pn pm = 0 (1 n < m r) and qn qm = 0 (1 n < m r), there exists a unitary element U ∈ B such that Ad(U ) ◦ L1 ≈ε L2 on F . Let in : An → A and πn : A → An be the canonical inclusion and projection homomorphisms respectively. For ε > 0 and F ⊆ A above, we have πn (F ) ⊆ An . We will find P , G, δ satisfying the condition C above. For ε > 0 and πn (F ) ⊆ An above, by Theorem 3.1, there exists a K-triple (Pn , Gn , δn ), where δn is a positive number, Pn is a finite subset of P (An ), and Gn is a finite subset of An , which satisfies the following condition: For any unital C ∗ -algebra C of real rank zero and stable rank one and with K0 (C) weakly unperforated, and any two unital G-δ-multiplicative completely positive linear maps L1 , L2 : A → C which satisfy that for any p ∈ Pn Ln1# (p) = Ln2# (p), there exists a unitary element Un ∈ C such that Ad(Un ) ◦ Ln1 ≈ε/2 Ln2 on πn (F ).
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Moreover without loss of generality, we may assume that 1An ∈ Gn ∩ Pn , and that Gn is sufficiently large and δn is sufficiently small such that for any two unital Gn -δn -multiplicative
, H : A → C, if H ≈ H on G , then for any p ∈ P , completely positive linear maps Hn1 n n n n2 n1 δn n2
Hn1# (p) = Hn2# (p).
Take G=
r
Gn ∪ F,
n=1
P=
r
Pn ,
δ = min(1, δ1 , δ2 , . . . , δr , ε)/K,
n=1
where K is a sufficiently large integer which will be determined later. We will prove that this K-triple (P , G, δ) satisfies the condition C above. First of all, for convenience, we may assume that G is in the unit ball of A. Let B be a unital K-injective C ∗ -algebra of real rank zero and stable rank one and with K0 (B) weakly unperforated, and let L1 , L2 : A → B be two unital G-δmultiplicative contractive completely positive linear maps such that L1 (1An ) = pn , L2 (1An ) = qn , where pn and qn are projections in B such that pn pm = 0 (1 n < m r) and qn qm = 0 (1 n < m r), and L1# (p) = L2# (p) for any p ∈ P . Since L1# (1An ) = L2# (1An ) in K0 (B), i.e., [pn ] = [qn ] in K0 (B), and tsr(B) = 1, there exist partial isometries vn (n = 1, 2, . . . r) such that vn vn ∗ = qn , vn ∗ vn = pn . Set Ln1 = pn (L1 ◦ in )pn , Ln2 = vn ∗ qn (L2 ◦ in )qn vn and C = pn Bpn . Since L1 , L2 are two unital G-δmultiplicative completely contractive positive linear maps, and L1 ◦ in ≈2δ pn (L1 ◦ in )pn , L2 ◦ in ≈2δ qn (L1 ◦ in )qn on Gn , we may choose the integer K above sufficiently large such that Ln1 = pn (L1 ◦ in )pn ,
Ln2 = vn ∗ qn (L2 ◦ in )qn vn : An → pn Bpn
are Gn -δn -multiplicative. Since tsr(pn Bpn ) = 1, and RR(pn Bpn ) = 0, by Lemma 3.2 we have (K0 (pn Bpn ), K0 (pn Bpn )+ ) is a weakly unperforated ordered group. Since L1# (p) = L2# (p) for all p ∈ P , we have (L1 ◦ in )# (p) = (L2 ◦ in )# (p),
for all p ∈ Pn .
Since L1 ◦ in ≈2δ pn (L1 ◦ in )pn on Gn , we may choose the integer K above sufficiently large such that (L1 ◦ in )# (p) = pn (L1 ◦ in )pn # (p),
for all p ∈ Pn ,
in K(B). Similarly L2 ◦ in ≈2δ qn (L2 ◦ in )qn on Gn , and so we may choose the integer K above sufficiently large such that (L2 ◦ in )# (p) = qn (L2 ◦ in )qn # (p),
for all p ∈ Pn ,
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in K(B). Then we have (Ln1 )# (p) = pn (L1 ◦ in )pn # (p) = qn (L2 ◦ in )qn # (p),
for all p ∈ Pn ,
in K(B). Since qn (L2 ◦ in )qn # (p) = vn ∗ qn (L2 ◦ in )qn vn # (p) = (Ln2 )# (p),
for all p ∈ Pn ,
in K(B), (Ln1 )# (p) = (Ln2 )# (p) in K(B) for all p ∈ Pn . Since both Ln1 and Ln2 are maps from An to pn Bpn , and B is K-injective, we have (Ln1 )# (p) = (Ln2 )# (p),
for all p ∈ Pn ,
in K(pn Bpn ). So there exist unitary elements Un ∈ pn Bpn (n = 1, 2, . . . , r) such that Ad(Un ) ◦ pn (L1 ◦ in )pn = Ad(Un ) ◦ Ln1 ≈ε/2 Ln2 = vn ∗ qn (L2 ◦ in )qn vn in πn (F ). Take u = U1 + U2 + U3 + · · · + Ur . Since L1 is unital, we have p1 + p2 + p3 + · · · + pr = 1B . Then u ∈ B is a unitary element and Ad(U1 ) ◦ p1 (L1 ◦ i1 )p1 + Ad(U2 ) ◦ p2 (L1 ◦ i2 )p2 + · · · + Ad(Ur ) ◦ pr (L1 ◦ ir )pr ≈ε/2 v1 ∗ q1 (L2 ◦ i1 )q1 v1 + v2 ∗ q2 (L2 ◦ i2 )q2 v2 + · · · + vr ∗ qr (L2 ◦ ir )qr vr on F . Since p1 (L1 ◦ i1 )p1 + p2 (L1 ◦ i2 )p2 + · · · + pr (L1 ◦ ir )pr ≈2δr L1 on G, set v = v1 + v2 + v3 + · · · + vr , we have v ∈ B is a unitary element and v1 ∗ q1 (L2 ◦ i1 )q1 v1 + v2 ∗ q2 (L2 ◦ i2 )q2 v2 + · · · + vr ∗ qr (L2 ◦ ir )qr vr ≈2δr Ad(v) ◦ L2 on G. Set U = uv ∗ . We may choose the integer K above sufficiently large such that 8δr < ε, and then we have Ad(U ) ◦ L1 ≈ε L2 on F . (II) For generality, we have to prove that the assertion in (I) holds still without the assumption in the condition C that L1 and L2 satisfy that L1 (1An ) = pn and L2 (1An ) = qn , where {pn }, {qn } are two sequences of orthogonal projections in B. For the finite subset F ⊆ A and ε/2, by the
δ1 ), where δ1 is a positive number less than ε/4, P discussion above there exists a K-triple (P , G,
is a finite subset of A, which satisfies that for any unital K-injective is a finite subset of P (A), G C ∗ -algebra B of real rank zero and stable rank one and with K0 (B) weakly unperforated, and
1 -multiplicative completely positive linear maps H1 , H2 : A → B such that any two unital G-δ for any p ∈ P , H1# (p) = H2# (p),
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and H1 (1An ) = pn , H2 (1An ) = qn , where pn and qn are projections in B such that pn pm = 0 and qn qm = 0 (1 n < m r), there exists a unitary element U ∈ B such that Ad(U ) ◦ H1 ≈ε/2 H2 .
and Without loss of generality, we may assume that for any n (with 1 n r), 1An ∈ G
and for any two unital G-δ
1 -multiplicative contractive completely positive linear maps F ⊆ G, H1 , H2 : A → B, if H1 ≈δ1 H2
then for any p ∈ P , on G, H1# (p) = H2# (p). Take a positive number δ < δ1 and a finite subset G of A such that rn=1 Gn ⊆ G, where
∪G
2 ), and πn is the canonical projection map from A onto An , then G
∪G
2 ⊆ G. Gn = πn (G Let L1 , L2 : A → B be two unital G-δ-multiplicative contractive completely positive linear maps such that L1# (p) = L2# (p),
for all p ∈ P .
Since 1An ∈ G for any n (with 1 n r), we have L1 (1A ) − L1 (1A )L1 (1A ) < δ. n n n Then with δ sufficiently small, there exist projections pn , qn ∈ B such that √ L1 (1A ) − pn < δ, n √ L2 (1A ) − qn < δ. n For any n = m with 1 n, m r, since L1 (1A )L1 (1A ) = L1 (1A )L1 (1A ) − L1 (1A 1A ) < δ, n m n m n m we have √ pn pm < 2 δ + δ and √ qn qm < 2 δ + δ. By Lemma 2.5.6 in [36], there exist mutually orthogonal projections p1 , p2 , . . . , pr and mutually orthogonal projections q1 , q2 , . . . , qr such that for any n (with 1 n r), we have pn − p < h(δ) n
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and qn − q < h(δ), n
where h(δ) is independent of the choice of L1 , L2 and lim h(δ) = 0.
δ→0
We have then
√ p L1 (1A )p − p < δ + h(δ), n n n n √
q L2 (1A )q − q < δ + h(δ). n n n n With δ sufficiently small, we have that pn L1 (1An )pn is invertible in pn Bpn and qn L2 (1An )qn is invertible in qn Bqn . Then we have
√ √ p L1 (1A )p −1/2 − p < δ + h(δ) / 1 − δ + h(δ) , n n n n
√ p L1 (1A )p −1/2 < 1/ 1 − δ + h(δ) , n n n and
√ √ q L1 (1A )q −1/2 − q < δ + h(δ) 1 − δ + h(δ) , n n n n
√ q L1 (1A )q −1/2 − q < 1 1 − δ + h(δ) . n n n n For any n (with 1 n r), we define L n1 , L n2 : An → B as follows: −1/2
−1/2 L n1 (x) = pn L1 (1An )pn
pn L1 (x)pn pn L1 (1An )pn
, and −1/2
−1/2 L n2 (x) = qn L1 (1An )qn
qn L2 (x)qn qn L1 (1An )qn
, for any x ∈ An . For convenience, we may assume that G is in the unit ball of A. Then we have L ni ≈σ (δ) Li ,
i = 1, 2,
on Gn , where σ (δ) =
√ √ √ √ δ + h(δ) 1 − δ + h(δ) 1/ 1 − δ + h(δ) + 1 + 2 δ + h(δ) + δ ,
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which is independent of the choice of L1 , L2 and limδ→0 σ (δ) = 0. Define L 1 , L 2 : A = r A n=1 n → B as follows: L 1 (x) =
r
L n1 (xn ),
L 2 (x) =
n=1
for any x =
r
n=1 xn
r
L n2 (xn ),
n=1
∈ A. Then
L 1 (1An ) = L n1 (1An ) = pn ,
L 2 (1An ) = L n2 (1An ) = qn ,
and L i ≈rσ (δ) Li ,
i = 1, 2,
∪G
2 . Since L1 , L2 are two unital G-δ-multiplicative contractive completely positive on G ⊇ G
we have linear maps, for any x, y ∈ G,
L (xy) − L (x)L (y) 3rσ (δ) + δ. i i i Take δ sufficiently small such that 3rσ (δ) + δ < δ1 . Then L 1 , L 2 are two contractive completely
1 -multiplicative linear maps such that positive G-δ L i ≈δ1 Li ,
i = 1, 2,
Then we have on G. L i # (p) = Li # (p),
i = 1, 2,
for any p ∈ P . Since L1# (p) = L2# (p),
for all p ∈ P ,
L 1# (p) = L 2# (p),
for all p ∈ P .
we have
By (I) there exists a unitary element U ∈ B such that Ad(U ) ◦ L 1 ≈ε/2 L 2
we have on F . Since F ⊆ G, L i ≈δ1 Li ,
i = 1, 2,
on F . Therefore we have Ad(U ) ◦ L1 ≈ε/2+2δ1 L 2 on F . Since δ1 < δ < ε/4, we have Ad(U ) ◦ L1 ≈ε L2 on F .
2
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Theorem 3.5. Let A be a unital C ∗ -algebra and suppose that A = ∞ n=1 An , where An ⊆ An+1 , and each An is a finite direct sum of unital simple C ∗ -algebras. Then A is K-injective. ∞ Proof. Set C0 = C(T) ⊕ ( ∞ n=2 (C(T) ⊗ C0 (Xn ))) and C = C(T) ⊕ ( n=2 (C(T) ⊗ C(Xn ))). We will prove that for any projection p ∈ A the canonical map from K0 ((p ⊗ 1M(C0 ) )(A ⊗ C0 )(p ⊗ 1M(C0 ) )) = K0 (pAp ⊗ C0 )) to K0 (A ⊗ C0 ) is injective. Let π : C(Xn ) → C be the value map at the base point of Xn . We have the split exact sequence: idB⊗C(T) ⊗i
idB⊗C(T) ⊗π
0 −→ B ⊗ C(T) ⊗ C0 (Xn ) −−−−−−−→ B ⊗ C(T) ⊗ C(Xn ) −−−−−−−→ B ⊗ C(T) −→ 0 for any unital C ∗ -algebra B. Then we have that K0 (A⊗C(T)⊗C0 (Xn )) = Ker((idA⊗C(T) ⊗π)∗ ) and K0 (pAp ⊗ C(T) ⊗ C0 (Xn )) = Ker((idpAp⊗C(T) ⊗π)∗ ). Therefore, it is enough to prove that for any projection p ∈ A the canonical map from K0 (pAp ⊗ C)) to K0 (A ⊗ C) is injective. We will prove it by three steps. Let C be a unital C ∗ -algebra (in particular the C ∗ -algebra C). (I) Suppose that A is a unital simple C ∗ -algebra. We will show that for any nonzero projection p ∈ A the canonical map ι : K0 ((p ⊗ 1C )(A ⊗ C)(p ⊗ 1C )) → K0 (A ⊗ C) is injective. Suppose that ι(x) = 0 in K0 (A ⊗ C) for some x ∈ K0 ((p ⊗ 1C )(A ⊗ C)(p ⊗ 1C )). We need to show that x = 0 in K0 ((p ⊗ 1C )(A ⊗ C)(p ⊗ 1C )). Without loss of generality we may assume that x = [r] − [s], where r, s ∈ Mk ((p ⊗ 1C )(A ⊗ C)(p ⊗ 1C )). Since ι(x) = 0, there exists an integer m such that diag(r, 1A⊗C , . . . , 1A⊗C ) ∼ diag(s, 1A⊗C , . . . , 1A⊗C ) in Mk+m (A ⊗ C), where 1A⊗C repeats m times. Since A is a unital C ∗ -algebra, there exist an integer n and elements xi ∈ A (1 i n) n simple ∗ such that 1A = i=1 xi pxi , and so 1A diag(p, p, . . . , p), where p repeats n times. Then we have 1A⊗C diag(p ⊗ 1C , p ⊗ 1C , . . . , p ⊗ 1C ), where p ⊗ 1C repeats n times. Since diag(r, 1A⊗C , . . . , 1A⊗C ) ∼ diag(s, 1A⊗C , . . . , 1A⊗C ) in Mk+m (A ⊗ C), where 1A⊗C repeats m times, we have diag(r, p ⊗ 1C , . . . , p ⊗ 1C ) ∼ diag(s, p ⊗ 1C , . . . , p ⊗ 1C ) in Mk+mn (A ⊗ C), where p ⊗ 1C repeats mn times. Since diag(r, p ⊗ 1C , . . . , p ⊗ 1C ) ∈ Mk+mn (p ⊗ 1C )(A ⊗ C)(p ⊗ 1C ) and diag(s, p ⊗ 1C , . . . , p ⊗ 1C ) ∈ Mk+mn (p ⊗ 1C )(A ⊗ C)(p ⊗ 1C ) ,
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by the same discussion as the proof of Theorem 3.2(1), we have diag(r, p ⊗ 1C , . . . , p ⊗ 1C ) ∼ diag(s, p ⊗ 1C , . . . , p ⊗ 1C ) in Mk+mn ((p ⊗ 1C )(A ⊗ C)(p ⊗ 1C )). So [r] − [s] = 0 in K0 ((p ⊗ 1C )(A ⊗ C)(p ⊗ 1C )), i.e., x = 0 in K0 ((p ⊗ 1C )(A ⊗ C)(p ⊗ 1C )). (II) Suppose that A = rn=1 An where each An is a unital simple C ∗ -algebra. We will show that for any nonzero projection p ∈ A the canonical map ι from K0 ((p ⊗ 1C )(A ⊗ C)(p ⊗ 1C )) to K0 (A ⊗ C) is injective. Since p ∈ A = rn=1 An , we have p = rn=1 pn , where each pn ∈ An is a projection. Then the following diagram r
K0 ((p ⊗ 1C )(A ⊗ C)(p ⊗ 1C ))
n=1 K0 ((pn
⊗ 1C )(An ⊗ C)(pn ⊗ 1C ))
r
ι
n=1 ιn
r
K0 (A ⊗ C)
n=1 K0 (An
⊗ C)
commutes, where all the maps are canonical. By (I), for each n, the canonical map ιn : K0 ((pn ⊗ 1C )(An ⊗ C)(pn ⊗ 1C )) → K0 (An ⊗ C) is injective, so the canonical map ι from K0 ((p ⊗ 1C )(A ⊗ C)(p ⊗ 1C )) to K0 (A ⊗ C) is injective. (III) Suppose that A = ∞ n=1 An , where An ⊆ An+1 , and each An is a finite direct sum of unital simple C ∗ -algebras. We will show that for any nonzero projection p ∈ A the canonical map ι : K0 ((p ⊗ 1C )A ⊗ C(p ⊗ 1C )) → K0 (A ⊗ C) is injective. Let ε be a positive number such that ε < 1/7. Since A = ∞ n=1 An and p ∈ A is a projection, for convenience we may assume that for any integer n, there exists a projection pn ∈ An such that pn − p < ε. Then there exists a unitary element un ∈ A such that u∗n pn un = p
and un − 1 < 2ε.
Suppose that ι(x) = 0, where x ∈ K0 ((p ⊗ 1C )(A ⊗ C)(p ⊗ 1C )), we need to show that x = 0 in K0 ((p ⊗ 1C )(A ⊗ C)(p ⊗ 1C )). We assume that x = [r] − [s], where r, s ∈ Mk ((p ⊗ 1C ) × (A ⊗ C)(p ⊗ 1C )) for some integer k. Since A = ∞ n=1 An , we have A⊗C =
∞
(An ⊗ C).
n=1
Then there exist a large integer n and projections rn , sn ∈ Mk ((pn ⊗ 1C )(An ⊗ C)(pn ⊗ 1C )) such that rn − r < 3ε
and sn − s < 3ε,
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and so we have [r] = [rn ]
and [s] = [sn ]
in K0 (A ⊗ C). Let ιn be the canonical map from K0 ((pn ⊗ 1C )(An ⊗ C)(pn ⊗ 1C )) to K0 (An ⊗ C), and let γn be the canonical map from K0 (An ⊗ C) to K0 (A ⊗ C), then γn ◦ ιn [rn ] − [sn ] = ι [r] − [s] = 0. By Theorem 6.3.2 in [50] we may assume that ιn ([rn ] − [sn ]) = 0 in K0 (An ⊗ C). By (II) the canonical map ιn is injective, so we have [rn ] − [sn ] = 0 in K0 ((pn ⊗ 1C )(An ⊗ C)(pn ⊗ 1C )), i.e., there exists an integer m such that diag(rn , pn ⊗ 1C , . . . , pn ⊗ 1C ) ∼ diag(sn , pn ⊗ 1C , . . . , pn ⊗ 1C ) in Mk+m ((pn ⊗ 1C )(An ⊗ C)(pn ⊗ 1C )), where pn ⊗ 1C repeats m times. Set (un ⊗ 1C )k = diag(un ⊗ 1C , . . . , un ⊗ 1C ), (pn ⊗ 1C )k = diag(pn ⊗ 1C , . . . , pn ⊗ 1C ), where un ⊗ 1C and pn ⊗ 1C repeat k times respectively. Then (un ⊗ 1C )∗ rn (un ⊗ 1C )k − r < 7ε, k (un ⊗ 1C )∗ sn (un ⊗ 1C )k − s < 7ε. k So we have (un ⊗ 1C )∗k rn (un ⊗ 1C )k = [r], (un ⊗ 1C )∗k sn (un ⊗ 1C )k = [s], in K0 ((p ⊗ 1C )(A ⊗ C)(p ⊗ 1C )). Since diag(rn , pn ⊗ 1C , . . . pn ⊗ 1C ) ∼ diag(sn , pn ⊗ 1C , . . . pn ⊗ 1C ) in Mk+m ((pn ⊗ 1C )(An ⊗ C)(pn ⊗ 1C )), we have diag (un ⊗ 1C )∗k rn (un ⊗ 1C )k , (un ⊗ 1C )∗ (pn ⊗ 1C )(un ⊗ 1C ), . . . , (un ⊗ 1C )∗ (pn ⊗ 1C )(un ⊗ 1C ) ∼ diag (un ⊗ 1C )∗k sn (un ⊗ 1C )k , (un ⊗ 1C )∗ (pn ⊗ 1C )(un ⊗ 1C ), . . . , (un ⊗ 1C )∗ (pn ⊗ 1C )(un ⊗ 1C ) in Mk+m (A ⊗ C). Since both the projections above are in Mk+m ((p ⊗ 1C )(A ⊗ C)(p ⊗ 1C )), we have (un ⊗ 1C )∗k rn (un ⊗ 1C )k = (un ⊗ 1C )∗k sn (un ⊗ 1C )k , in K0 ((p ⊗1C )(A⊗C)(p ⊗1C )), and so x = [r]−[s] = 0 in K0 ((p ⊗1C )(A⊗C)(p ⊗1C )).
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Lemma 3.6. Let A be a unital C ∗ -algebra with A = ∞ n=1 An , where An ⊆ An+1 , An has the cancellation property, and K0 (An ) is weakly unperforated (in particular if An is a finite direct sum of unital amenable separable simple C ∗ -algebras with tracial rank zero). Then for any projection p ∈ A, (K0 (pAp), K0 (pAp)+ ) is a weakly unperforated ordered group. Proof. By Theorem 3.2(2), we only have to show that (K0 (A), K0 (A)+ ) is a weakly unperforated ordered group. For any x ∈ K0 (A) and integer k such that kx > 0, we will show that x 0. Without loss of generality we may assume that x = [r] − [s], where r, s ∈ Ml (A). Since A= ∞ n=1 An , there exist a sufficiently large integer n1 and projections rn1 , sn1 ∈ Ml (An1 ) such that [r] = [rn1 ], [s] = [sn1 ] in K0 (A). Since k([r]−[s]) > 0 in K0 (A), we have k([rn1 ]−[sn1 ]) > 0 in K0 (A). Therefore, there exists a nonzero projection t ∈ Mj (A) such that k([rn1 ] − [sn1 ]) = [t] in K0 (A). Since A = ∞ n=1 An , there exist an integer n2 (with n2 > n1 ), and a projection tn2 ∈ Mj (An2 ) such that [tn2 ] = [t] in K0 (A). We have k([rn1 ] − [sn1 ]) = [tn2 ] in K0 (A). Let i = max{l, j }, we may regard r, s as projections in Mi (A), and regard rn1 , sn1 and tn2 as projections in Mi (An2 ). Take ⎛r ⎜ ⎜ e=⎜ ⎜ ⎝
⎞
n1
⎟ ⎟ ⎟, ⎟ ⎠
rn1 ..
. rn1
⎛s ⎜ ⎜ f =⎜ ⎜ ⎝
⎞
n1
⎟ ⎟ ⎟ ⎟ ⎠
sn1 ..
. sn1
0
tn2
where rn1 and sn1 repeat k times, then we have [e] = [f ] in K0 (A), where e, f ∈ Mi(k+1) (An2 ). Since A has the cancellation property, e ∼ f in Mi(k+1) (A). Since A = ∞ n=1 An , we have a n3 (with n3 > n2 ) such that e ∼ f in Mi(k+1) (An3 ), and so k([rn1 ] − [sn1 ]) = [tn2 ] in K0 (An3 ). Since tn2 = 0 and An3 has the cancellation property, we have k([(rn1 )] − [(sn1 )]) > 0 in K0 (An3 ). Since (K0 (An3 ), K0 (An3 )+ ) is a weakly unperforated ordered group, we have ([rn1 ] − [sn1 ]) > 0 in K0 (An3 ). So ([rn1 ] − [sn1 ]) 0 in K0 (A). Therefore ([r] − [s]) 0 in K0 (A). 2 Theorem 3.7. Let A be a unital ATAF algebra. Then for any ε > 0, and any finite subset F ⊆ A, there exists a K-triple (P , G, δ), where δ is a positive number, P is a finite subset of P (A), and G is a finite subset of A, which satisfies that for any unital K-injective C ∗ -algebra B of real rank zero and stable rank one and with K0 (B) weakly unperforated (in particular if B is a unital ATAF algebra by Theorem 3.5 and Lemma 3.6), and any two unital G-δ-multiplicative completely positive linear maps L1 , L2 : A → B, if for any p ∈ P , L1# (p) = L2# (p), then there exists a unitary element U ∈ B such that Ad(U ) ◦ L1 ≈ε L2 on F . Proof. Suppose A = ∞ n=1 An , where An ⊆ An+1 , 1A = 1An , and each An is a finite direct sum of unital amenable separable simple C ∗ -algebras with tracial rank zero which satisfy the UCT. Let Jn : An → A be the canonical inclusion morphism. For ε > 0, and a finite subset F ⊆ A,
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there exist a sufficiently large number n and a finite subset F ⊆ An , such that for any x ∈ F , x ≈ε/3 Jn (F ), i.e., for any x ∈ F , there exists x ∈ F such that x − Jn (x ) < ε/3. For ε/3 > 0, and the finite subset F ⊆ An , by Lemma 3.4, there exists a K-triple (Pn , Gn , δn ), where δn is a positive number, Pn is a finite subset of P (An ), and Gn is a finite subset of An , which satisfies the following condition: For any unital K-injective C ∗ - algebra B of real rank zero stable rank one and with K0 (B) weakly unperforated, and any two unital Gn -δn -multiplicative contractive completely positive linear maps Hn1 , Hn2 : An → B, if for any p ∈ Pn , Hn1 # (p) = Hn2 # (p), then there exists a unitary element U ∈ B such that Ad(U ) ◦ (Hn1 ) ≈ε/3 Hn2 on F . Take G = Jn (Gn ), P = Jn (Pn ), and δ = δn , then (P , G, δ) is a K-triple. Let B be a unital Kinjective C ∗ -algebra of real rank zero and stable rank one and with K0 (B) weakly unperforated, and let L1 , L2 : A → B be two unital G-δ-multiplicative completely positive linear maps such that L1# (p) = L2# (p),
for all p ∈ P .
Then L1 ◦ Jn , L2 ◦ Jn : An → B are two unital Gn -δn -multiplicative completely positive linear maps such that (L1 ◦ Jn )# (p) = (L2 ◦ Jn )# (p),
for all p ∈ Pn .
From above we can find a unitary element U ∈ B such that Ad(U ) ◦ (L1 ◦ Jn ) ≈ε/3 L2 ◦ Jn on F , i.e., for any x ∈ F , ∗ U (L1 ◦ Jn )(x )U − (L2 ◦ Jn )(x ) < ε/3. By the relation of F and F , for any x ∈ F , there exists x ∈ F such that x − Jn (x ) < ε/3. Since L1 , L2 are two unital completely positive linear maps, we have L2 (x) − (L2 ◦ Jn )(x ) < ε/3. L1 (x) − (L1 ◦ Jn )(x ) < ε/3, Therefore, for any x ∈ F , L2 (x) − U ∗ L1 (x)U L2 (x) − (L2 ◦ Jn )(x ) + (L2 ◦ Jn )(x ) − U ∗ (L1 ◦ Jn )(x )U + U ∗ (L1 ◦ Jn )(x )U − U ∗ L1 (x)U < ε, i.e., Ad(U ) ◦ L1 ≈ε L2 on F .
2
4. Existence theorem for ATAF algebras In this section, we are going to keep the notations K(A) = K0 (A) ⊕ K1 (A)
∞ 1 i=0
n=2
Ki (A, Z/nZ) = K∗ (A)
∞ n=2
K∗ (A, Z/nZ).
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Here Ki (A, Z/nZ)) = Ki (A ⊗ C0 (Xn )), Xn is the Moore space with a fixed base point obtained by attaching the disk to the circle by a degree n 2 map, and we identify K∗ (A) = K0 (A) ⊕ K1 (A) with K0 (A ⊗ C(T)) and identify K∗ (A, Z/n) = K0 (A, Z/n) ⊕ K1 (A, Z/n) with K0 (A ⊗ C0 (Xn ) ⊗ C(T)). Definition 4.1. Let A be a unital C ∗ -algebra, and let K∗ (A) = K0 (A) ⊕ K1 (A). We define K∗ (A)+ to be the collection of all the elements of the form {[p], [u ⊕ (1Mk(A) − p)]} ∈ K0 (A)⊕K1 (A), where p ∈ Mk (A) is a projection, u ∈ pMk (A)p is a unitary element and 1Mk (A) is the unit of Mk (A). It is well known that if A is of real rank zero and stable rank one, or of the form m i=1 Pi Mni (C(Xi ))Pi , where Xi are finite CW complexes, and Pi ∈ Mni (C(Xi )) are projections, then (K∗ (A), K∗ (A)+ ) is an ordered group with the order unit [1A ]. Lemma 4.2. (See [28, Theorem 2.3] and [41, Lemma 5.1].) Let X be a finite CW complex, C = QMk (C(X))Q, where Q ∈ Mk (C(X)) is a projection, and let B be a unital simple C ∗ -algebra with tracial rank zero. Then for any α ∈ KK(C, B) with the induced map α∗ : K(C) → K(B) satisfying that α∗ (K0 (C)+ \ {0}) ⊆ K0 (B)+ \ {0} and α∗ ([1C ]) = [qB ] for a projection qB in B, there is a monomorphism φ : C → B such that φ∗ = α∗ : K(C) → K(B) and φ(1C ) = qB . Lemma 4.3. Let A be a unital amenable separable simple C ∗ -algebra with tracial rank zero which satisfies the UCT, and let B be a unital ATAF algebra. Then for any α ∈ KK(A, B) with the induced map α∗ : K(A) → K(B) satisfying that α∗ (K∗ (A)+ ) ⊆ K∗ (B)+ and α∗ ([1A ]) = [qB ] for a projection qB in B, and for any K-triple (P , G, δ), where δ is a positive number, P is a finite subset of P (A), and G is a finite subset of A, there is a G-δ-multiplicative contractive completely positive linear map L : A → B such that L(1A ) = qB and L# (p) = α∗ ([p]) for any p ∈P. Proof. Let B = ∞ m=1 Bm , where Bm ⊆ Bm+1 , and each Bm is a finite direct sum of unital separable amenable simple C ∗ -algebras with tracial rank zero which satisfy the UCT. Since A is a unital amenable separable simple C ∗ -algebra with tracial rank zero which satisfies the UCT, A is a unital simple AH algebra. By [18, 2.16] we may assume that A = ∞ n=1 An , where An ⊆ An+1 , and each An is isomorphic to a C ∗ -algebra of the form Qn Mkn (C(Xn ))Qn with Xn being a finite CW complex and Qn being a projection in Mkn (C(Xn )). For the K-triple (P , G, δ), where δ is positive number, P is a finite subset of P (A), and G is a finite subset of A, without loss of generality, we may assume that 1A ∈ P , 1A ∈ G. Moreover we may assume that G is so large and δ is so small that ϕ(p) − ϕ(p)2 < 1/32 for any p ∈ P and any G-δ-multiplicative contractive completely positive linear map ϕ : A → B. Set M = max{ x + 2: x ∈ G ∪ P }, and f : [0, 1] → [0, 1] be a continuous nonnegative function such that f (x) = 0 (for x ∈ [0, 1/4]) and f (x) = 1 (for x ∈ [3/4, 1]). Take a positive number ε such that ε < min(δ/8M, 1) which will be determined later. Then there exist finite subsets G1 ⊆ An and P1 ⊆ P (An ) (for some sufficiently large integer n) which satisfy that for any x ∈ G and p ∈ P there exist x ∈ G1 and p1 ∈ P1 such that x − x < ε, Then we have [p] = [p1 ] in K(A).
p − p1 < ε.
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Let ϕ : A → B be a G1 -δ/2-multiplicative completely positive linear map. Since for any x, y ∈ G, there exist x1 , y1 ∈ G1 such that x − x1 < ε and y − y1 < ε. Therefore xy − x1 y1 < 2Mε. Then ϕ(xy) − ϕ(x)ϕ(y) ϕ(xy) − ϕ(x1 y1 ) + ϕ(x1 y1 ) − ϕ(x1 )ϕ(y1 ) + ϕ(x1 )ϕ(y1 ) − ϕ(x)ϕ(y) < 2Mε + δ/2 + 2Mε < δ, and so ϕ is a G-δ-multiplicative completely positive linear map. By discussion above we know any G1 -ε-multiplicative contractive completely positive linear map ϕ : A → B is also G-δ-multiplicative, and so ϕ(p) − ϕ(p)2 < 1/32 (for all p ∈ P ). Moreover we may choose a large G1 and a small ε so that ϕ(p1 ) − ϕ(p1 )2 < 1/16 (for all p1 ∈ P1 ) and f (ϕ(p)) − f (ϕ(p1 )) < 1 for any G1 -ε-multiplicative contractive completely positive linear map ϕ : A → B. Then (G1 , P1 , ε) is also a K-triple, and for any p ∈ P , ϕ# (p) = f ϕ(p) = f ϕ(p1 ) = ϕ# (p1 ). Let Jn : An → A,
jm : Bm → B
be the canonical inclusion morphisms, and let [Jn ] ∈ KK(An , A) and [jm ] ∈ KK(Bm , B) be induced by Jn and jm respectively. Set αn = [Jn ] ◦ α ∈ KK(An , B), then (αn )∗ (K∗ (An )+ ) ⊆ K∗ (B)+ and (αn )∗ ([1An ]) = [qB ]. Since An is isomorphic to Qn Mkn (C(Xn ))Qn , where Xn is a finite CW complex and Qn is a projection in Mkn (C(Xn )), K∗ (An )+ , and so K∗ (An ), is finitely generated. Then we have KK(An , B) = limm→∞ KK(An , Bm ), and so there is β ∈ KK(An , Bm ) (for a sufficiently large m) such that [Jn ] ◦ α = αn = β ◦ [jm ]. Moreover we may have β∗ (K∗ (An )+ ) ⊆ K∗ (Bm )+ and β∗ ([1An ]) = [qBm ] for a projection qBm in Bm . Let An = tl=1 An,l and 1An = tl=1 1An,l , where each An,l is isomorphic to a C ∗ -algebra of a connected finite CW complex and Ql = 1An,l being the form Ql Mnl (C(Xl ))Ql with Xl being a projection in Mnl (C(Xl )), and let Bm = sk=1 Bm,k and qBm = sk=1 qk for some projections qk ∈ Bm,l , where each Bm,k is a unital separable amenable simple C ∗ -algebras with tracial rank zero which satisfy the UCT. Since KK(An , Bm ) ∼ =
s
KK(An , Bm,k ),
k=1
s we denote by γ the isomorphism from k=1 KK(An , Bm,k ) onto KK(An , Bm ). Since β ∈ KK(An , Bm ), there exist βk ∈ KK(An , Bm,k ) (k = 1, 2, . . . , s) such that γ (β1 ⊕ β2 ⊕ · · · ⊕ βs ) = β. Let ik : Bm,k → Bm ,
πk : Bm → Bm,k
X. Fang / Journal of Functional Analysis 256 (2009) 3861–3891
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be the canonical inclusion morphism and the canonical projection morphism respectively. Denote by [ik ] the element in KK(Bm,k , Bm ) induced by ik , and by [πk ] the element in KK(Bm , Bm,k ) induced by πk . Then by definition, we have βk = β ◦ [πk ],
(βk )∗ K∗ (An )+ ⊆ K∗ (Bm,k )+ ,
(βk )∗ [1An ] = [qk ],
and β = γ (β1 ⊕ β2 ⊕ · · · ⊕ βs ) = β1 ◦ [i1 ] + β2 ◦ [i2 ] + · · · + βs ◦ [is ]. Since KK(An , Bm,k ) = KK
t
An,l , Bm,k ∼ =
l=1
t
KK(An,l , Bm,k ),
l=1
we denote this isomorphism by
τk :
t
KK(An,l , Bm,k ) → KK(An , Bm,k ).
l=1
Then for any βk ∈ KK(An , Bm,k ) there exist βlk ∈ KK(An,l , Bm,k ) (1 l t) such that τk (β1k ⊕ β2k ⊕ · · · ⊕ βtk ) = βk . Let il : An,l → An ,
πl : An → An,l
be the canonical inclusion morphism and the canonical projection morphism respectively. Denote by [il ] the element in KK(An,l , An ) induced by il , and by [πl ] the element in KK(An , An,l ) induced by πl . Then βlk = [il ] ◦ βk , and βk = τk (β1k ⊕ β2k ⊕ · · · ⊕ βtk ) = π1 ◦ β1k + π2 ◦ β2k + · · · + πr ◦ βtk . Then for each βlk ∈ KK(An,l , Bm,k ) above, (βlk )∗ : K(An,l ) → K(Bm,k ) satisfies that (βlk )∗ (K∗ (An,l )+ ) ⊆ K∗ (Bm,k )+ and (βlk )∗ ([1An,l ]) = [qlk ], where qlk (1 l t) are the pro jections in Bm,k such that tl=1 [qlk ] = [qk ]. We claim that for the above βlk ∈ KK(An,l , Bm,k ), there exists a homomorphism φlk : An,l → Bm,k such that (φlk )∗ = (βlk )∗ on K(An,l ), and φlk (1An,l ) = qlk .
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Case (I): qlk = 0. Since (βk )∗ (K∗ (An,l )+ ) ⊆ K∗ (Bm,k )+ and (βk )∗ ([1An,l ]) = [qlk ], we have (βk )∗ = 0 on K∗ (An,l ). Since the following diagrams (0)
(0)
K0 (An,l )
ρn
K0 (An,l , Z/nZ)
(βlk )∗
βn
(βlk )∗
K0 (Bn,k )
(0) ρn
K0 (Bn,k , Z/nZ)
K1 (An,l ) (βlk )∗
(0) βn
K1 (Bn,k )
and (0)
(0)
K1 (An,l )
ρn
K1 (An,l , Z/nZ)
(βlk )∗
βn
(βlk )∗
K1 (Bn,k )
(0) ρn
K1 (Bn,k , Z/nZ)
K0 (An,l ) (βlk )∗
(0) βn
K0 (Bn,k )
commute (see [36, 5.8.10]), we have (βlk )∗ = 0 on K(An,l ). Then we let φlk = 0, and this completes the proof of the claim in this case. Case (II): qlk = 0. Let p ∈ Proj(An,l ) \ {0}. By definition An,l is isomorphic to a C ∗ -algebra of the form Ql Mnl (C(Xl ))Ql with Xl being a connected finite CW complex and Ql being a projection in Mnl (C(Xl )), and so there is an integer M such that [Ql ] = [1An,l ] M[p] in K0 (An,l ). Then 0 = [qlk ] = (βlk )∗ ([1An,l ]) M(βlk )∗ ([p]), and so (βlk )∗ ([p]) ∈ K0 (Bm,k )+ \ {0}. By Lemma 4.2, there exists a monomorphism φlk : An,l → Bm,k such that (φlk )∗ = (βlk )∗ on K(An,l ), and φlk (1An,l ) = qlk , and this completes the proof of the claim. Let Ik : Bm,k ⊕ Bm,k ⊕ · · · ⊕ Bm,k → Bm,k ⊗ K, where Bm,k repeats t times, be the canonical inclusion morphism, and for each k, set ψk = Ik ◦
t φlk ◦ πl : An → Bm,k ⊗ K, l=1
then ψk : An → Bm,k ⊗ K is a homomorphism and ψk (1An ) is a projection in Bm,k ⊗ K. For any p ∈ P (An ),
t
ψk ∗ [p] = Ik ◦ φlk ◦ πl [p] l=1
∗
X. Fang / Journal of Functional Analysis 256 (2009) 3861–3891
= (Ik )∗
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t t (φlk )∗ πl (p) = (Ik )∗ (βlk )∗ πl (p) l=1
l=1
= (β1k )∗ π1 (p) + (β2k )∗ π2 (p) + · · · + (βtk )∗ πt (p) = (βk )∗ [p] , where we identify K(Bm,k ) with K(Bm,k ⊗ K), i.e., (ψk )∗ = (βk )∗ on K(An ). Set ψ=
s (ik ⊗ idK ) ◦ ψk , k=1
then ψ is a homomorphism from An to Bm ⊗ K. For any p ∈ P (An ), we have ψ∗ [p] =
s [p] (ik ⊗ idK ) ◦ ψk k=1
=
∗
s s (ik ⊗ idK )∗ (ψk )∗ [p] = (ik ⊗ idK )∗ (βk )∗ [p] k=1
k=1
= [i1 ] ◦ β1 + [i2 ] ◦ β2 + · · · + [is ] ◦ βs ∗ [p] = β∗ [p] , i.e., ψ∗ = β∗ on K(An ). Set Q = ψ(1An ). Since ψ∗ (1An ) = β∗ ([1An ]) = [qBm ], [Q] = [qBm ] in K0 (Bm ). Since Bm has the cancellation property, there exists v ∈ Bm ⊗ K such that v ∗ v = qBm , vv ∗ = Q. Replacing ψ with v ∗ ψv, we may assume that ψ is a homomorphism from An to Bm such that ψ∗ = β∗ on K(An ), and ψ(1An ) = qBm . Since [jm (qBm )] = [qB ], there exists u ∈ B such that u∗ u = jm (qBm ), uu∗ = qB . Set φ = u(jm ◦ ψ)u∗ , then φ is a homomorphism from An to B such that φ∗ = (αn )∗
on K(An ),
and φ(1An ) = qB .
Since qB BqB is amenable and An is a subalgebra of A , for the K-triple (G1 , P1 , ε) above, we have a G1 -ε-multiplicative contractive completely positive linear map L : A → B such that L(1A ) = qB , and L# (p1 ) = φ∗ ([p1 ]) (for all p1 ∈ P1 ). Then, by the discussion in the beginning, L : A → B is a G-δ-multiplicative contractive completely positive linear map, and for any p ∈ P , there exists p1 ∈ P1 such that p − p1 < ε < 1 and L# (p) = L# (p1 ). Therefore, for any p ∈ P , L# (p) = L# (p1 ) = φ∗ [p1 ] = (αn )∗ [p1 ] = α∗ Jn (p1 ) = α∗ [p1 ] = α∗ [p] .
2
Lemma 4.4. Let A = rn=1 An , where each An is a unital amenable separable simple C ∗ algebra with tracial rank zero which satisfies the UCT, and let B be a unital ATAF algebra. Then for any α ∈ KK(A, B) with the induced map α∗ : K(A) → K(B) satisfying that α∗ (K∗ (A)+ ) ⊆ K∗ (B)+ and α∗ ([1A ]) = [1B ], and for any K-triple (P , G, δ), where δ is a positive number, P is a finite subset of P (A), and G is a finite subset of A, there is a unital G-δ-multiplicative completely positive linear map L : A → B such that L# (p) = α∗ ([p]) for any p ∈ P .
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Proof. Since KK(A, B) = KK
r
An , B ∼ =
n=1
r
KK(An , B),
n=1
we may denote this isomorphism by τ:
r
KK(An , B) → KK(A, B).
n=1
Then for any α ∈ KK(A, B) there exist αn ∈ KK(An , B) (1 n r) such that τ (α1 ⊕ α2 ⊕ · · · ⊕ αr ) = α. Let in : An → A,
πn : A → An
be the canonical inclusion morphism and the canonical projection morphism respectively. Denote by [in ] the element in KK(An , A) induced by in , and by [πn ] the element in KK(A, An ) induced by πn . Then αn = α ◦ [in ],
and
α = τ (α1 ⊕ α2 ⊕ · · · ⊕ αr ) = α1 ◦ [π1 ] + α2 ◦ [π2 ] + · · · + αr ◦ [πr ]. Then for each αn ∈ KK(An , B) above, (αn )∗ : K(An ) → K(B) satisfies that α∗ (K∗ (A)+ ) ⊆ K∗ (B)+ and (αn )∗ ([1An ]) = [pn ], where pn (1 n r) are the projections in B such that r n=1 [pn ] = [1B ]. For the K-triple (P , G, δ), where δ is positive number, P is a finite subset of P (A), and G is a finite subset of A, without loss of generality, we may assume that 1A ∈ P and 1A ∈ G. Then 1An ∈ πn (G) ⊆ An , 1An ∈ πn (P ) ⊆ P (An ), and (πn (P ), πn (G), δ) is also a K-triple. For the K-triple (πn (P ), πn (G), δ), by Lemma 4.3, there exists a contractive completely positive πn (G)-δ-multiplicative linear map φn : An → B such that for any p ∈ P , φn# (πn (p)) = (αn )∗ ([πn (p)]) in K(B), and φn (1An ) = pn . Let I : B ⊕ B ⊕ · · · ⊕ B → B ⊗ K, where B repeats r times, be the canonical inclusion morphism, and set r (φn ◦ πn ), ϕ=I ◦ n=1
then ϕ(1A ) is a projection, and ϕ : A → B ⊗ K is a contractive completely positive G-δmultiplicative linear map.
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For any p ∈ P , ϕ# (p) = I ◦
r
(φn ◦ πn )
n=1
= (I )∗
(p) #
r r φn# πn (p) = (I )∗ (αn )∗ πn (p) n=1
n=1
= (α1 )∗ π1 (p) + (α2 )∗ π2 (p) + · · · + (αr )∗ πr (p) = α∗ [p] , where we identify K(B) with K(B ⊗ K). Set Q = ϕ(1A ). Since ϕ# (1A ) = α∗ ([1A ]) = [1B ], [Q] = [1B ] in K0 (B). Since B has the cancellation property, there exists v ∈ B ⊗ K such that v ∗ v = 1B and vv ∗ = Q. Set L = v ∗ ϕv, then L(1A ) = 1B , and L is contractive completely positive G-δ -multiplicative linear map, where δ is a positive number depending on δ such that δ → 0 when δ → 0. Then without loss of generality we may assume that L is G-δ-multiplicative, and this completes the proof. 2 Theorem 4.5. Let A and B be unital ATAF algebras. Then for any α ∈ KK(A, B) with the induced map α∗ : K(A) → K(B) satisfying that α∗ (K∗ (A)+ ) ⊆ K∗ (B)+ and α∗ ([1A ]) = [1B ], and for any K-triple (P , G, δ), where δ is a positive number, P is a finite subset of P (A), and G is a finite subset of A, there is a unital G-δ-multiplicative completely positive linear map L : A → B such that L# (p) = α∗ ([p]) for any p ∈ P . Proof. First we may assume that G is so large and δ is so small that ϕ(p) − ϕ(p)2 < 1/32 for any p ∈ P and any G-δ-multiplicative contractive completely positive linear map ϕ : A → B. Set M = max{ x + 2: x ∈ G ∪ P }, and f : [0, 1] → [0, 1] be a continuous nonnegative function such that f (x) = 0 (for x ∈ [0, 1/4]) and f (x) = 1 (for x ∈ [3/4, 1]). Suppose A = ∞ n=1 An , where An ⊆ An+1 , 1A = 1An , and each An is a finite direct sum of unital amenable separable simple C ∗ -algebras with tracial rank zero which satisfy the UCT. Take a positive number ε such that ε < min(δ/8M, 1) which will be determined later. Then there exist finite subsets G1 ⊆ An and P1 ⊆ P (An ) (for some sufficiently large integer n) which satisfy that for any x ∈ G and p ∈ P there exist x ∈ G1 and p1 ∈ P1 such that x − x < ε,
p − p1 < ε.
Then we have [p] = [p1 ] in K(A). Let ϕ : A → B be a G1 -δ/2-multiplicative completely positive linear map. Since for any x, y ∈ G, there exist x1 , y1 ∈ G1 such that x − x1 < ε and y − y1 < ε. Therefore xy − x1 y1 < 2Mε. Then ϕ(xy) − ϕ(x)ϕ(y) ϕ(xy) − ϕ(x1 y1 ) + ϕ(x1 y1 ) − ϕ(x1 )ϕ(y1 ) + ϕ(x1 )ϕ(y1 ) − ϕ(x)ϕ(y) < 2Mε + δ/2 + 2Mε < δ, and so ϕ is a G-δ-multiplicative completely positive linear map.
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By discussion above we know any G1 -ε-multiplicative contractive completely positive linear map ϕ : A → B is also G-δ-multiplicative, and so ϕ(p) − ϕ(p)2 < 1/32 (for all p ∈ P ). Moreover we may choose a large G1 and a small ε so that ϕ(p1 ) − ϕ(p1 )2 < 1/16 (for all p1 ∈ P1 ) and f (ϕ(p)) − f (ϕ(p1 )) < 1 for any G1 -ε-multiplicative contractive completely positive linear map ϕ : A → B. Then (G1 , P1 , ε) is also a K-triple, and for any p ∈ P ϕ# (p) = f ϕ(p) = f ϕ(p1 ) = ϕ# (p1 ). Let Jn : An → A be the canonical inclusion morphism, and denote by [Jn ] the element in KK(An , A) induced by Jn , then α ◦ [Jn ] ∈ KK(An , B). Furthermore, the induced map α∗ ◦ Jn∗ : K(An ) → K(B) is order preserving, and (α∗ ◦ Jn∗ )([1An ]) = [1B ]. For α ◦ [Jn ] and K-triple (P1 , G1 , ε), by Lemma 4.4, there exists a unital G1 -ε-multiplicative completely positive linear map φ : An → B such that for any p1 ∈ P1 , φ# (p1 ) = (α∗ ◦ Jn∗ )([p1 ]). Since A is amenable C ∗ -algebra, there exists a unital completely positive linear map L:A→B such that L# (p1 ) = φ# (p1 ) (for all p1 ∈ P1 ), and L is G1 -ε-multiplicative. By discussion above L : A → B is a unital G-δ-multiplicative completely positive linear map. Furthermore, for any p ∈ P (A), L# (p) = L# (p1 ) = φ# (p1 ) = (α∗ ◦ Jn ∗ ) [p1 ] = α∗ [p1 ] = α∗ [p] .
2
5. Classification theorem and the range of the invariant for ATAF algebras Theorem 5.1. Let A and B be two unital ATAF algebras. If K∗ (A), K∗ (A)+ , [1A ] ∼ = K∗ (B), K∗ (B)+ , [1B ] , then A ∼ = B. Moreover the isomorphism between the invariances could be induced by the isomorphism between the algebras. Proof. Since (K∗ (A), K∗ (A)+ , [1A ]) ∼ = (K∗ (B), K∗ (B)+ , [1B ]) and both A and B satisfy the UCT, by the Theorem 2.4.6 in [51], there exist α ∈ KK(A, B), β ∈ KK(B, A) such that α ◦ β = [idA ], β ◦ α = [idB ], and α∗ , β∗ realize the isomorphisms above in both directions. Let {Fn } and {Fn } be two increasing of finite subsets of the unit ball of A and B ∞ sequences
respectively such that ∞ n=1 Fn and n=1 Fn are two dense subsets of the unit ball of A and B respectively. Let {Pn } and {Pn } be two increasing sequences of finite subsets of P (A) and P (B) ∞
K(A) and K(B) respectively such that ∞ n=1 {[p] : p ∈ Pn } and n=1 {[p] : p ∈ Pn } generate respectively. Let {εn } be a decreasing sequence of positive numbers such that ∞ n=1 εn < ∞. Set C = C(T) ⊕ ( ∞ (C(T) ⊗ C(X ))), where X is the Moore space obtained by attaching n n n=2 the disk to the circle by a degree n 2 map. For any completely positive contractive linear map φ from a C ∗ -algebra D1 to a C ∗ -algebra D2 , set φ∞ = φ ⊗ idC ⊗K : D1 ⊗ C ⊗ K → D2 ⊗ C ⊗ K.
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If moreover φ∞ is P -δ-multiplicative for some finite subset P of P (D1 ), where 0 < δ < 14 , then √ √ the spectrum of φ∞ (p) is a subset of [0, δ] ∪ [1 − δ, 1] for any p ∈ P . We define the map φ∼ : P ⊆ P (D1 ) → P (D2 ),
by φ∼ (p) = χ[1/2,1] φ∞ (p)
(for all p ∈ P ),
then φ# (p) = φ∼ (p)
√ and φ∼ (p) − φ∞ (p) < δ.
By [36, Lemma 2.5.11] there exists ε0 < 18 such that for any C ∗ -algebra E and a, b ∈ E+ with the spectra in [0, 1/4] ∪ [3/4, 1], if a − b < ε0 , then χ[1/2,1] (a) − χ[1/2,1] (b) < 1. For ε = ε1 and F = F1 , we have a K-triple (P1 , G1 , δ1 ) given by Theorem 3.7, where δ1 > 0, P1 is a finite subset of P (A), and G1 is a finite subset of A. Moreover we may assume that δ1 < ε1 , F1 ⊆ G1 , P1 ⊆ P1 , and for any contractive completely positive G1 -δ1 -multiplicative linear map ϕ : A → D with D being a C ∗ -algebra, the map ϕ∞ : A ⊗ C ⊗ K → D ⊗ C ⊗ K is P1 -ε02 -multiplicative. For α ∈ KK(A, B) and K-triple (P1 , G1 , δ1 /2), by Theorem 4.5 there is a G1 -δ1 /2multiplicative unital completely positive linear map L1 : A → B such that for any p ∈ P1 , L1# (p) = α∗ ([p]). Then L1∞ : A ⊗ C ⊗ K → B ⊗ C ⊗ K is P1 -ε02 -multiplicative, therefore
L1∼ (p) − L1∞ (p) < ε02 = ε0 (for all p ∈ P1 ). For ε = ε1 and F = F1 ∪ L1 (G1 ), we have a K-triple (P1 , G 1 , η1 ) given by Theorem 3.7, where η1 > 0, P1 is a finite subset of P (B), and G 1 is a finite subset of B. Moreover we may assume that P1 ∪ L1∼ (P1 ) ⊆ P1 , η1 < δ1 , F1 ∪ L1 (G1 ) ⊆ G 1 , and for any contractive completely
: positive G 1 -η1 -multiplicative linear map ϕ : B → D with D being a C ∗ -algebra, the map ϕ∞ B ⊗ C ⊗ K → D ⊗ C ⊗ K is P1 -ε02 -multiplicative. For β ∈ KK(B, A) and K-triple (P1 , G 1 , η1 /2), by Theorem 4.5 there is a G 1 -η1 /2multiplicative unital completely positive linear map λ 1 : B → A such that for any p ∈ P1 , λ 1# (p) = β∗ ([p]). Similarly we have that the map λ 1∞ : B ⊗ C ⊗ K → A ⊗ C ⊗ K is P1 -ε02 multiplicative, and λ 1∼ (p) − λ 1∞ (p) < ε02 = ε0 (for all p ∈ P1 ). Since L1 (G1 ) ⊆ G 1 , λ 1 ◦ L1 : A → A is a unital completely positive G1 -δ1 -multiplicative linear map. In fact for any x, y ∈ G1 , λ 1 ◦ L1 (xy) =δ1 /2 λ 1 L1 (x)L1 (y) =η1 /2 λ 1 ◦ L1 (x)λ 1 ◦ L1 (y). Therefore (λ 1 ◦ L1 )∞ is P1 -ε02 -multiplicative, and so for any p ∈ P1 , the spectrum of (λ 1 ◦ L1 )∞ (p) is in [0, 1/8] ∪ [7/8, 1] by ε0 < 18 . In addition, since L1∼ (p) − L1∞ (p) < ε0 (for all p ∈ P1 ),
λ
1∞
◦ L1∼ (p) − λ 1 ◦ L1 ∞ (p) = λ 1∞ ◦ L1∼ (p) − λ 1∞ ◦ L1∞ (p) < ε0 <
1 8
(for all p ∈ P1 ).
Then for any p ∈ P1 , the spectrum of λ 1∞ ◦ L1∼ (p) is in [0, 1/4] ∪ [3/4, 1], and
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χ[1/2,1] λ ◦ L1∼ (p) − λ ◦ L1 (p) 1∞ 1 ∼ = χ[1/2,1] λ 1∞ ◦ L1∼ (p) − χ[1/2,1] λ 1 ◦ L1 ∞ (p) < 1. Therefore,
λ1 ◦ L1 # (p) = λ 1 ◦ L1 ∼ (p) = χ[1/2,1] λ 1∞ ◦ L1∼ (p) = λ 1# L1∼ (p) . On the other hand, since L1∼ (P1 ) ⊆ P1 , for any p ∈ P1 , λ 1# L1∼ (p) = β∗ L1∼ (p) = β∗ L1# (p) = β∗ α∗ [p] = idA# (p). Therefore, (λ 1 ◦ L1 )# (p) = idA# (p) for any p ∈ P1 . By Theorem 3.7 there exists a unitary element u1 ∈ A such that idA ≈ε1 Ad(u1 ) ◦ (λ 1 ◦ L1 ) on F1 . Set λ1 = Ad(u1 ) ◦ λ 1 , then idA ≈ε1 λ1 ◦ L1 on F1 . Moreover we still have that λ1 : B → A is a G 1 -η1 /2-multiplicative unital completely positive linear map such that λ1# (p) = β∗ ([p]) for any p ∈ P1 . Therefore the map λ1∞ : B ⊗ C ⊗ K → A ⊗ C ⊗ K is P1 -ε02 -multiplicative, and
λ1∼ (p) − λ1∞ (p) < ε02 = ε0 (for all p ∈ P1 ). For ε = ε2 and F = F2 , we have a K-triple (P2 , G2 , δ2 ) given by Theorem 3.7, where δ2 > 0, P2 is a finite subset of P (A), and G2 is a finite subset of A. Moreover we may assume that P2 ∪ λ1∼ (P1 ) ⊆ P2 , δ2 < min{η1 , ε2 }, F2 ∪ λ1 (G 1 ) ⊆ G2 , and for any contractive completely positive G2 -δ2 -multiplicative linear map ϕ : A → D with D being a C ∗ -algebra, the map ϕ∞ : A ⊗ C ⊗ K → D ⊗ C ⊗ K is P2 -ε02 -multiplicative. For α ∈ KK(A, B) and K-triple (P2 , G2 , δ2 /2), by Theorem 4.5, there is a unital completely positive G2 -δ2 /2-multiplicative linear map L 2 : A → B such that for any p ∈ P2 , L 2# (p) = α∗ ([p]). Similarly we have that L 2 ◦ λ1 : B → B is a unital completely positive G 1 -η1 -multiplicative linear map, and (L 2 ◦ λ1 )# (p) = idB # (p) for any p ∈ P1 . By Theorem 3.7 there exists a unitary element v2 ∈ B, such that idB ≈δ1 /2 Ad(v2 ) ◦ (L 2 ◦ λ1 ) on F1 . Set L2 = Ad(v2 ) ◦ L 2 , then idB ≈ε1 L2 ◦ λ1 on F1 . Moreover we still have that L2 : A → B is a unital completely positive G2 -δ2 /2-multiplicative linear map such that L2# (p) = α∗ ([p]) for any p ∈ P2 , and L2∞ is P2 -ε02 -multiplicative with L2∼ (p) − L2∞ (p) < ε0 (for all p ∈ P2 ). Therefore, we obtain the following diagram id → A A − L1 ↓ λ1 ↓ L2
B
id − →
B,
where the upper triangle is F1 -ε1 -commutative and the lower triangle is F1 -ε1 -commutative. Continuing in this fashion, we obtain the following approximately intertwining diagram id → A −
A
id − → A −→ · · · −→ A
L1 ↓ λ1 ↓ L2 λ2 id B − →
B
id − → B −→ · · · −→ B,
where the upper triangle is Fn -εn -commutative and the lower triangle is Fn -εn -commutative.
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Since Ln is Fn -εn -multiplicative and λn is Fn -εn -multiplicative, by Theorem 1.10.16 in [36], there exist isomorphisms h1 : A → B and h2 : B → A, such that h−1 2 = h1 . Moreover since ∞ ∞
} generate K(A) and K(B) respectively, we have {[p]: p ∈ P } and {[p]: p ∈ P n n m=1 m=1 that h1∗ = α∗ . 2 ∗ Lemma 5.2. Let A = ∞ n=1 An , where An ⊆ An+1 , be a unital C -algebra, and let in,m : An → Am and in : An → A be two canonical inclusion maps. + (in )∗ (K∗ (An )), and K∗ (A)+ = ∞ (1) K∗ (A) = ∞ n=1 n=1 (in )∗ (K∗ (An ) ); ∞ (2) Ker((in )∗ ) = m=1 Ker((in,m )∗ ). ∞ Proof. It is known that K∗ (A) = ∞ n=1 (in )∗ (K∗ (An )) and Ker((in )∗ ) = m=1 Ker((in,m )∗ ). It is also easy to see that (in )∗ (K∗ (An )+ ) ⊆ K∗ (A)+ . Let x = ([p], [u ⊕ 1A − p]) ∈ K∗ (A)+ , where p ∈ A is a projection and u ∈ pAp is a unitary element, and let 0 < ε < 1. We have a sufficiently large n, a projection pn ∈ An , and an element vn ∈ pn An pn such that p − pn < ε, vn vn∗ − pn < ε, vn∗ vn − pn < ε, and u − vn < ε. Then |vn | is invertible, and let vn = un |vn | be the polar decomposition. By the standard discussion, without loss of generality, we may assume that un ∈ pn An pn is a unitary element with u − un < ε. Then [p] = [pn ] in K0 (A), [u ⊕ 1A − p] = [un ⊕ 1An − pn ] in K1 (A). Moreover set xn = ([pn ], [un ⊕ 1An − pn ]), then xn ∈ K∗ (An )+ with (in )∗ (xn ) = x. ∞ Since Mk (A) = n=1 Mk (An ) for any integer k, we have completed the proof. 2 Definition 5.3. An ordered group (G, G+ ) is called weakly unperforated with torsion if the torsion-free ordered group G/Gtor , where Gtor is the torsion subgroup of G, is unperforated, and G has the following property: If g ∈ G+ , t ∈ Gtor , and ng + mt ∈ G+ with n, m ∈ Z, and n 1, then t = t + t
with t , t
∈ Gtor , mt = 0, and g + t
∈ G+ . An ordered group (G, G+ ) is called to have the Riesz interpolation property if for any x1 , x2 , y1 , y2 ∈ G such that xi yj for i, j ∈ {1, 2}, then there is a z ∈ G such that xi z yj for i, j ∈ {1, 2}. It is clear that if (G, G+ ) is weakly unperforated (see Definition 2.3), then it is weakly unperforated with torsion; and if (G, G+ ) is simple, then it is weakly unperforated with torsion if and only if it is weakly unperforated. Theorem 5.4. Let A be an ATAF algebra. Then (K∗ (A), K∗ (A)+ ) is weakly unperforated with torsion, and has the Riesz interpolation property. mn Proof. Let A = ∞ n=1 An , where An ⊆ An+1 , An = j =1 Anj , and each Anj is a unital amenable separable simple C ∗ -algebra with tracial rank zero which satisfies the UCT. Since A has stable rank one and real rank zero, by Theorem 3.2 of [12], the ordered group + ) has the Riesz interpolation property. (K∗ (A), K∗ (A) mn mn n + + Since An = m j =1 Anj , K∗ (An ) = j =1 K∗ (Anj ) and K∗ (An ) = j =1 K∗ (Anj ) . Since each Anj is a simple AH-algebra with slow dimension growth, by Theorem 4.12 of [17], the ordered group (K∗ (Anj ), K∗ (Anj )+ ) is weakly unperforated with torsion. By simple computation we have (K∗ (An ), K∗ (An )+ ) is also weakly unperforated with torsion. Let x ∈ K∗ (A) such that k x¯ 0, where x¯ ∈ K∗ (A)/K∗ (A)tor is the image of x under the quotient map and k is a positive integer, i.e., there is a positive integer m and y ∈ K∗ (A) such
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that my = 0 and kx + y ∈ K∗ (A)+ . By Lemma 5.2, there is a sufficiently large (positive) integer n and xn , yn ∈ K∗ (An ) such that myn = 0, kxn + yn ∈ K∗ (An )+ , and (in )∗ (xn ) = x, (in )∗ (yn ) = y. Since (K∗ (An ), K∗ (An )+ ) is weakly unperforated with torsion, there is a zn ∈ K∗ (An )tor such that xn + zn ∈ K∗ (An )+ , and so x + (in )∗ (zn ) ∈ K∗ (A)+ and (in )∗ (zn ) ∈ K∗ (A)tor . Therefore K∗ (A)/K∗ (A)tor is unperforated. Let g ∈ K∗ (A)+ , t ∈ K∗ (A)tor such that kg + mt ∈ K∗ (A)+ for integer m and positive integer k. By Lemma 5.2, there is a sufficiently large (positive) integer n, and gn ∈ K∗ (An )+ , tn ∈ K∗ (An )tor such that (in )∗ (gn ) = g, (in )∗ (tn ) = t, and kgn + mtn ∈ K∗ (An )+ . Since (K∗ (An ), K∗ (An )+ ) is weakly unperforated with torsion, tn = tn + tn
with tn , tn
∈ K∗ (An )tor , mtn = 0, and gn + tn
∈ K∗ (An )+ . Set t = (in )∗ (tn ) and t
= (in )∗ (tn
), then t = t + t
, mt = 0, and g + t
∈ K∗ (A)+ , and this completes the proof. 2 To end this paper, we present the example of ATAF algebra in [5] which is not an AH algebra. Denote by D the two by two matrix algebra over the 3∞ UHF algebra and by C the unique simple unital AD algebra of real rank zero with K0 (C) = Z[ 13 ], K1 (C) = Z/2Z ordered by K∗ (C)+ = (x, y): x > 0 or (x = 0, y = 0) . Recall that an AD algebra is a C ∗ -algebra isomorphic to a countable inductive limit of C ∗ algebras of the form ri=1 Mk(i) (I˜n(i) ), where In(i) = a ∈ C [0, 1], Mn : a(0) = 0, a(1) ∈ C1n , and I˜n(i) is the unitalization of In(i) . In the case that tor K0 (−) = 0 and m tor K1 (−) = 0, the invariant ρ0
β0
m m K(−; m) : K0 (−) −−→ K0 (−; Z/mZ) −−→ K1 (−)
is complete for AD algebras with real rank zero (and stable rank one). Therefore there is a unique unital ∗-homomorphism ϕ : C → D, unique up to approximately inner equivalence, having K(ϕ; 2) given by K0 (C; 2) K(ϕ;2)
K0 (D; 2)
Z[ 13 ]
ρ
Z/2Z ⊕ Z/2Z [01]
2
Z[ 13 ]
β
ρ
Z/2Z
Z/2Z 0
β
0
Let E be the inductive limit of C ∗ -algebras of the form Ek = C ⊕ k1 D with the connecting map ϕk : Ek → Ek+1 defined by ϕk (c, (d1 , d2 , . . . , dk )) = (c, (d1 , d2 , . . . , dk , ϕ(c))) for any (c, (d1 , d2 , . . . , dk )) ∈ Ek . Then ∞ E = c, (dn ) ∈ C ⊕ D : ϕ(c) − dn → 0 . n=1
Therefore K1 (E) = Z/2Z, and
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! " ∞ ! " 1 1 ⊕ : yn → 2x , K0 (E) = x, (yn ) ∈ Z Z 3 3 n=1 K0 (E)+ = x, (yn ) : x 0, yn 0 . It follows that Inf K0 (E) = 0. Choosing generators approximately, we get ∞ Z/2Z : wn → z K0 (E; Z/2Z) = v, z, (wn ) ∈ Z/2Z ⊕ Z/2Z ⊕ n=1
with ρ20 x/3r , yn /3rn = x + 2Z, 0, (yn + 2Z) ,
β20 (v, z, wn ) = z.
Moreover
! " ∞ ! " 1 1 K∗ (E) = x, z, (yn ) ∈ Z ⊕ Z/2Z ⊕ : yn → 2x Z 3 3 n=1
ordered by K∗ (E)+ =
x, z, (yn ) : yn → 2x, and (x > 0, yn 0) or (x = z = 0, yn 0) .
It is proved in [5] that C is isomorphic to an AH algebra C of real rank zero, and one may use only 3-dimensional finite CW-complexes in the construction of C . Therefore E is an ATAF algebra. Moreover in [5] it is proved that E is not an AH algebra and fits into a split and qua∞ sidiagonal extension of C by D, which splits by c → (c, ϕ(c), ϕ(c), . . .) and in which the 1
projections en = (1, . . . , 1, 0, . . .) (1 repeats n times) are central in E and form an approximate unit for ∞ 1 D. Concerning the range of the invariant, the point is whether all the possibilities allowed by Theorem 5.4 actually occur. Moreover it is interesting to see a description of exactly which values of the invariant of ATAF algebras arise from an AH algebras. A full discussion of the range of the invariant of ATAF algebras will follow. Acknowledgment The author would like to thank Huaxin Lin for his interest in the classification for the class of C ∗ -algebras considered in this paper and many helpful discussions with him. References [1] B. Blackadar, K-theory for operator algebras, Springer-Verlag, New York, 1986. [2] L.G. Brown, G.K. Pedersen, C ∗ -algebras of real rank zero, J. Funct. Anal. 99 (1991) 131–149. [3] M. Dadarlat, Approximately unitarily equivalent morphisms and inductive limit C ∗ -algebras, K-Theory 9 (1995) 117–137. [4] M. Dadarlat, Morphisms of simple tracially AF algebras, Internat. J. Math. 15 (2004) 919–957. [5] M. Dadarlat, S. Eilers, Approximate homogeneity is not a local property, J. Reine Angew. Math. 507 (1999) 1–13.
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[6] M. Dadarlat, S. Eilers, On the classification of nuclear C ∗ -algebras, Proc. London Math. Soc. 85 (2002) 168–210. [7] M. Dadarlat, G. Gong, A classification result for approximately homogeneous C ∗ -algebras of real rank zero, Geom. Funct. Anal. 7 (1997) 646–711. [8] M. Dadarlat, T.A. Loring, A universal multicoefficient theorem for the Kasparov groups, Duke Math. J. 84 (1996) 355–377. [9] K.R. Davidson, C ∗ -Algebras by Example, Fields Inst. Monogr., Amer. Math. Soc., Providence, RI, 1996. [10] G.A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976) 29–44. [11] G.A. Elliott, Dimension groups with torsion, Internat. J. Math. 1 (1990) 361–380. [12] G.A. Elliott, On the classification of C ∗ -algebras of real rank zero, J. Reine Angew. Math. 443 (1993) 179–219. [13] G.A. Elliott, The classification problem for amenable C ∗ -algebras, in: Proceedings of the International Congress of Mathematicians, vols. 1, 2, Zurich, 1994, Birkhäuser, Basel, 1995, pp. 922–932. [14] G.A. Elliott, A classification of certain simple C ∗ -algebras II, J. Ramanujan Math. Soc. 12 (1997) 97–134. [15] G.A. Elliott, D.E. Evans, H. Su, Classification of inductive limits of the Toeplitz algebra tensored with K, in: Operator Algebras and Quantum Fields Theory, Rome, 1996, Int. Press, Cambridge, MA, 1997, pp. 36–50. [16] G.A. Elliott, X. Fang, Simple inductive limits of C ∗ -algebras with building blocks from spheres of odd dimension, in: Operator Algebra and Operator Theory, in: Contemp. Math., vol. 228, 1998, pp. 79–86. [17] G.A. Elliott, G. Gong, On the classification of C ∗ -algebras of real rank zero II, Ann. of Math. 144 (1996) 497–610. [18] G.A. Elliott, G. Gong, L. Li, Injectivity of the connecting maps in AH inductive limit systems, Canad. Math. Bull. 48 (2005) 50–68. [19] G.A. Elliott, G. Gong, L. Li, On the classification of simple inductive limit C ∗ -algebras II: The isomorphism theorem, Invent. Math. 168 (2007) 249–320. [20] G.A. Elliott, G. Gong, H. Lin, C. Pasnicu, Abelian C ∗ -subalgebras of C ∗ -algebras of real rank zero and inductive limit C ∗ -algebras, Duke Math. J. 85 (1996) 511–554. [21] X. Fang, Graph C ∗ -Algebras and their ideals defined by Cuntz–Krieger family of possibly row-infinite directed graphs, Integral Equations Operator Theory 54 (2006) 301–316. [22] X. Fang, The real rank zero property of crossed product, Proc. Amer. Math. Soc. 134 (2006) 3015–3024. [23] G. Gong, Classification of C ∗ -algebras of real rank zero and unsuspended E-equivalence types, J. Funct. Anal. 152 (1998) 281–329. [24] G. Gong, On the classification of simple inductive limit C ∗ -algebras I: The reduction theorem, Doc. Math. 7 (2002) 255–461. [25] G. Gong, H. Lin, Almost multiplicative morphisms and almost commuting matrices, J. Operator Theory 40 (1998) 217–275. [26] G. Gong, H. Lin, Classification of homomorphisms from C(X) to simple C ∗ -algebras of real rank zero, Acta Math. Sin. 16 (2000) 181–206. [27] G. Gong, H. Lin, Almost multiplicative morphisms and K-theory, Internat. J. Math. 11 (2000) 983–1000. [28] L. Li, C ∗ -algebra homomorphisms and KK-theory, K-Theory 18 (1999) 161–172. [29] H. Lin, Approximation by normal elements with finite spectra in C ∗ -algebras of real rank zero, Pacific J. Math. 173 (1996) 443–489. [30] H. Lin, On the classification of C ∗ -algebras of real rank zero with zero K1 , J. Operator Theory 35 (1996) 147–178. [31] H. Lin, Almost multiplicative morphisms and some applications, J. Operator Theory 37 (1997) 121–154. [32] H. Lin, Classification of simple C ∗ -algebras with unique traces, Amer. J. Math. 120 (1998) 1289–1315. [33] H. Lin, Lifting automorphisms, K-Theory 16 (1999) 105–127. [34] H. Lin, Homomorphisms from C ∗ -algebras of continuous trace, Math. Scand. 86 (2000) 249–272. [35] H. Lin, The tracial topological rank of C ∗ -algebras, Proc. London Math. Soc. 83 (2001) 199–234. [36] H. Lin, An Introduction to the Classification of Amenable C ∗ -Algebras, World Scientific, New Jersey/London/ Singapore/Hong Kong, 2001. [37] H. Lin, Classification of simple tracially AF C ∗ -algebras, Canad. J. Math. 53 (2001) 161–194. [38] H. Lin, Stable approximate unitary equivalence of homomorphisms, J. Operator Theory 47 (2002) 343–378. [39] H. Lin, Classification of simple C ∗ -algebras and higher dimensional non-commutative tori, Ann. of Math. 157 (2003) 521–544. [40] H. Lin, Classification of simple C ∗ -algebras with tracial topological rank zero, Duck Math. J. 125 (2004) 91–119. [41] H. Lin, The range of approximate unitary equivalence classes of homomorphisms from AH-algebras, preprint arXiv:0801.3858v1, 2008. [42] H. Lin, N.C. Phillips, Classification of direct limits of even Cuntz-circle algebras, Mem. Amer. Math. Soc. 118 (565) (1995).
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[43] H. Lin, N.C. Phillips, Approximate unitary equivalence of homomorphisms from O∞ , J. Reine Angew. Math. 464 (1995) 173–186. [44] H. Lin, M. Rordam, Extensions of inductive limits of circle algebras, J. London Math. Soc. 51 (1995) 603–613. [45] H. Lin, H. Su, Classification of direct limits of generalized Toeplitz algebras, Pacific J. Math. 181 (1997) 89–140. [46] S. Liu, X. Fang, Extension algebras of Cuntz algebra, J. Math. Anal. Appl. 329 (2007) 655–663. [47] T.A. Loring, C ∗ -algebras generated by stable relations, J. Funct. Anal. 112 (1993) 159–203. [48] G.J. Murphy, C ∗ -Algebras and Operator Theory, Academic Press, London, 1990. [49] G.K. Pedersen, C ∗ -Algebras and Their Automorphism Groups, Academic Press, London/New York/San Francisco, 1979. [50] M. Rordam, An Introduction to K-Theory for C ∗ -Algebras, London Math. Soc. Stud. Texts, vol. 49, Cambridge, 2000. [51] M. Rordam, Classification of Nuclear Simple C ∗ -Algebras, Encyclopaedia Math. Sci., Springer-Verlag, Berlin/ Heidelberg/New York, 2002.
Journal of Functional Analysis 256 (2009) 3892–3915 www.elsevier.com/locate/jfa
A Kre˘ın space coordinate free version of the de Branges complementary space Damir Z. Arov a , Olof J. Staffans b,∗ a Division of Mathematical Analysis, Institute of Physics and Mathematics, South-Ukrainian Pedagogical University,
65020 Odessa, Ukraine b Åbo Akademi University, Department of Mathematics, FIN-20500 Åbo, Finland
Received 22 September 2008; accepted 23 October 2008 Available online 14 November 2008 Communicated by J. Coron
Abstract Let Z be a maximal nonnegative subspace of a Kre˘ın space X , and let X /Z be the quotient of X modulo Z. Define H(Z) = h ∈ X /Z sup −[x, x]X x ∈ h < ∞ . It is proved that sup{−[x, x]X | x ∈ h} 0 for h ∈ H(Z), and that H(Z) is a Hilbert space with norm 1/2 , hH(Z ) = sup −[x, x]X x ∈ h which is continuously contained in X /Z, and the properties of this space are studied. Given any fundamental decomposition X = −Y [] U of X , the subspace Z can be written as the graph of a contraction A : U → Y. There is a natural isomorphism between X /Z and Y, and under this isomorphism the space H(Z) is mapped isometrically onto the complementary space H(A) of the range space of A studied by de Branges and Rovnyak. The space H(Z) is used as state space in a construction of a canonical passive state/signal shift realization of a linear observable and backward conservative discrete time invariant state/signal system with a given passive future behavior, equal to a given maximal nonnegative right-shift 2 (W) of all 2 -sequences on Z+ with values in the Kre˘ın invariant subspace Z of the Kre˘ın space X = k+ * Corresponding author.
E-mail address: [email protected] (O.J. Staffans). URL: http://web.abo.fi/~staffans/ (O.J. Staffans). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.10.019
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signal space W. This state/signal realization is related to the de Branges–Rovnyak model of a linear observable and backward conservative scattering input/state/output system whose scattering matrix is a given Schur class function in the same way as H(Z) is related to H(A). © 2008 Elsevier Inc. All rights reserved. Keywords: Kre˘ın space; Hilbert space; Quotient space; Maximal nonnegative subspace; Contractive operator; Passive system; State space; Signal space; Behavior
1. Introduction The main result of this article concerns the geometry of Kre˘ın spaces, and it describes the relationship between the orthogonal companion Z [⊥] of a maximal nonnegative subspace Z of a Kre˘ın space X and a certain Hilbert space H(Z) that is continuously (but not necessarily densely) contained in the quotient space X /Z. This result was discovered as a byproduct of our continuing research on passive linear discrete time invariant s/s (state/signal) systems, which so far has resulted in the publications [2–5]. The subspace H(Z) can be interpreted as a coordinate free version of the de Branges complement of the range space of a contractive operator A between two Hilbert spaces U and Y, as will be explained in more detail in Section 3. Let Z be a maximal nonnegative subspace of a Kre˘ın space X , and let X /Z be the quotient of X modulo Z. Define H(Z) = h ∈ X /Z sup −[x, x]X x ∈ h < ∞ .
(1.1)
As we shall prove in Theorem 2.3, sup{−[x, x]X | x ∈ h} 0 for h ∈ H(Z), and H(Z) is a Hilbert space with norm 1/2 , hH(Z ) = sup −[x, x]X x ∈ h
h ∈ H(Z),
(1.2)
which is continuously contained in X /Z. In Lemma 2.4 we prove that the following “Schwarz type” inequality [x, z]X 2 [z, z]X [x, x]X + h2 H (Z ) ,
h ∈ H(Z), x ∈ h, z ∈ Z,
(1.3)
holds, and that it collapses to an equality if and only if either [z, z]X = 0 or [z, z]X = 0 and x = ([x, z]X /[z, z]X )z + z† for some z† ∈ Z [⊥] , where Z [⊥] is the orthogonal companion of Z in X , i.e., Z [⊥] = x ∈ X [x, z]X = 0 for all z ∈ Z .
(1.4)
In Lemma 2.4 and Theorem 2.5 we prove a number of additional results about the space H(Z), such as the following. Define the subspace H0 (Z) of X /Z by H0 (Z) := z† + Z z† ∈ Z [⊥] ,
(1.5)
where z† + Z stands for the equivalence class in X /Z which contains z† . Then H0 (Z) ⊂ H(Z), and the supremum in (1.1) is achieved if and only if h ∈ H0 (Z). The space H0 (Z) has a natural positive inner product induced by X , namely
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† z1 + Z, z2† + Z H0 (Z ) = − z1† , z2† X ,
z1† , z1† ∈ Z [⊥] ,
(1.6)
and this inner product coincides with the inner product inherited from H(Z). Moreover, H0 (Z) is dense in H(Z). This means that the Hilbert space H(Z) is the completion of H0 (Z). Furthermore, if we define Z0 = Z ∩ Z [⊥] , then Z [⊥] /Z0 can be identified in a natural way with H0 (Z) with the positive inner product inherited from −X , so that we may also regard H(Z) as the completion of Z [⊥] /Z0 with respect to this inner product. Since Z [⊥] can be interpreted as a maximal nonnegative subspace of the anti-space −X of the Kre˘ın space X , it follows that there is a dual version of the space H(Z) that we denote by H(Z [⊥] ). Connections between the spaces H(Z) and H(Z [⊥] ) are studied in Theorem 2.12. All our results on the Hilbert spaces H(Z) and H(Z [⊥] ), including those mentioned above, are formulated and proved in Section 2. As we show in Section 3, there is a close connection between the Hilbert space H(Z) and the de Branges complementary space H(A) induced by the contraction A that appear in the graph representation of Z with respect to some fundamental decomposition of the Kre˘ın space X . All the proofs that we give in Section 2 are “coordinate free” in the sense that they make no use of such a graph representation. They are also selfcontained in the sense that they require no a priori knowledge whatsoever of the de Branges complementary space H(A). The connection between H(Z) and the space H(A) is described in detail in Section 3. In this section we have also included some alternative proofs, which are based on the above graph representation, of some of the results in Section 2. The main reason for including these proofs is that they illustrate the connection between the space H(Z) and the space H(A). Our coordinate free proofs in Section 2 are written in the spirit of the original proof by de Branges and Rovnyak of Theorem 7 in [10], given on pp. 24–26 of that book, whereas the alternative proofs in Section 3 are written in the spirit of more recent proofs of the same result. More precisely, to each space H(Z) there corresponds not only one space H(A), but a whole family of spaces H(A). As is well known, if X = −Y [] U is a fundamental decomposition of the Kre˘ın space X , then each maximal nonnegative subspace Z of X is the graph of a linear contraction A : U → Y, and conversely, the graph of every linear contraction A : U → Y between the Hilbert spaces U and Y is a maximal nonnegative subspace of the Kre˘ın space X = −Y []U . However, the correspondence between Z and the contraction A is far from one-to-one, since A obviously does not depend only on Z but also on the choice of the fundamental decomposition X = −Y [] U . It is easy to see that X is the direct sum of Z and −Y, and this implies that there is a natural isomorphism T : X /Z → Y. It turns out that the restriction of T to H(Z) is a unitary map of H(Z) onto the de Branges complement H(A) of the range space M(A) of the operator A. As we mentioned above, this makes it is possible to prove some of the results in Section 2 by appealing to known results about H(A) and M(A) due to de Branges and Rovnyak. However, it is also possible to proceed in the opposite direction, and to prove results about the spaces H(A) by appealing to the results about the Hilbert space H(Z) given in Section 2. As we mentioned at the beginning, the present article is an outgrowth of our research on passive linear discrete time invariant s/s (state/signal) systems. In Section 4 we describe how the 2 (W), the set of all 2 space H(Z) is used in passive s/s systems theory. There we take X = k+ + sequences on Z with values in the Kre˘ın signal space W, with the indefinite inner product in 2 (W) induced by the inner product in W. The subspace Z is a maximal nonnegative right-shift k+ invariant subspace of X , or in the state/system terminology, Z is a passive future behavior. The space H(Z) is used as the state space of the canonical shift realization of a linear observable and backward conservative discrete time invariant s/s system with the given passive future behavior Z that we construct in Section 4. This s/s realization is related to the de Branges–Rovnyak model of
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a linear observable and backward conservative scattering input/state/output system whose scattering matrix is a given Schur class function in the same way as H(Z) is related to H(A). The latter model has been studied in [9,10], and the more recently in [1], with a different terminology: there the given Schur class function is realized as the “characteristic function” of a “coisometric and closely outer connected colligation.” The idea of using a quotient of two vector valued sequence spaces as the state space of a (not necessarily passive) finite-dimensional input/state/output realization of a given rational transfer function with finite-dimensional state, input, and output spaces goes back to Kalman; see, e.g., the last part of [11]. Notations and conventions. The space of bounded linear operators from one Kre˘ın space X to another Kre˘ın space Y is denoted by B(X ; Y). The domain, range, and kernel of a linear operator A are denoted by D(A), R(A), and N (A), respectively. The restriction of A to some subspace Z ⊂ D(A) is denoted by A|Z . The identity operator on X is denoted by 1X . The projection onto a closed subspace Y of a space X along some complementary subspace U is denoted by PYU , or by PY if Y and U are orthogonal with respect to a Hilbert or Kre˘ın space inner product in X . We locally convex topological vector spaces Y and U denote the ordered product of the two by UY , and sometimes write Y U for UY (interpreting Y U as an ordered sum), identifying y vectors 0 and u0 with y and u for y ∈ Y and u ∈ U . The inner product in a Hilbert space X is denoted by (·,·)X , and by [·,·]X in the case of a Kre˘ın space X . The orthogonal sum of two Hilbert spaces Y and U is denoted by Y ⊕ U , and the orthogonal sum of two Kre˘ın spaces Y and U is denoted by Y [] U . We identify U and Y with the appropriate subspaces of these sums. A Hilbert space Y is continuously contained in a topological vector space X if Y is a subspace of X , and the inclusion map of Y → X is continuous. If X is a Kre˘ın space with inner product [·,·]X , then the Kre˘ın space −X is the same vector space with the inner product −[·,·]X . We call −X the anti-space of X . 2. The Hilbert space H(Z) 2.1. Preliminaries on Kre˘ın spaces We assume the reader to be familiar with basic notions and results in Kre˘ın space theory. A short introduction to this theory can be found in, e.g. [1], and more detailed treatments in [6] or [7]. Nevertheless, we include here a short summary of Kre˘ın space theory in order to establish the notations. Let X be a Kre˘ın space. This means that X is a vector space with an indefinite inner product [·,·]X , and that X has a fundamental decomposition X = −Y [] U , where Y and U are Hilbert spaces. The topology of X is the one induced by the Hilbert space norm x2 = −[y, y]X + [u, u]X , where x = u + y with y ∈ Y and u ∈ U (different fundamental decompositions give different but equivalent norms). Such a norm is called an admissible norm. A subspace Z of X is nonnegative if [x, x]X 0 for every x ∈ Z. It is maximal nonnegative if it is not properly contained in any other nonnegative subspace of X . Nonpositive and maximal nonpositive subspaces are defined analogously. The orthogonal companion Z [⊥] of a subspace Z is defined by (1.4). It is well known that (Z [⊥] )[⊥] = Z if and only if Z is closed. The subspace Z is neutral if [z, z]X = 0 for all z ∈ Z, or equivalently, if Z ⊂ Z [⊥] . It is called Lagrangian if Z [⊥] = Z.
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A nonnegative subspace Z is uniformly positive if it is a Hilbert space with respect to the inner product inherited from X , and it is uniformly negative if it is a Hilbert space with respect to the inner product norm inherited from −X . If Z is both uniformly positive and maximal nonnegative, and only in this case, Z induces a fundamental decomposition X = Z [⊥] [] Z,
(2.1)
and Z [⊥] uniformly negative and maximal nonpositive. In general the intersection Z0 := Z ∩ Z [⊥]
(2.2)
can be different from {0}. It is the maximal neutral subspace contained in Z, and at the same time the maximal neutral subspace contained in Z [⊥] . In the proof of Theorem 2.3 below we shall need the following lemma. Lemma 2.1. If Z is a maximal nonnegative subspace of a Kre˘ın space X , then X =Z Y
(2.3)
for every uniformly negative and maximal nonpositive subspace Y of X . Proof. If x ∈ Z ∩ Y, then, on one hand, [x, x]X 0 since x ∈ Z, and on the other hand [x, x]X 0 since x ∈ Y. The uniform negativity of Y implies that x = 0. Thus, Z ∩ Y = {0}. We next show that Z + Y is dense in X , or equivalently, that x = 0 whenever x ∈ (Z + Y)[⊥] . The condition x ∈ (Z + Y)[⊥] is equivalent to x ∈ Z [⊥] ∩ Y [⊥] . Since Z is nonpositive and Y [⊥] is uniformly positive, this implies that x = 0 (by an argument analogous to the one above). Finally, we show that Z + Y is closed in X . Let xn = zn + yn → x in X , with zn ∈ Z and yn ∈ Y. Let PY and PY [⊥] be the complementary orthogonal projections onto Y and Y [⊥] , respectively. Then, for each n and m, 0 [zn − zm , zn − zm ]X = PY (zn − zm ), PY (zn − zm ) X + PY [⊥] (zn − zm ), PY [⊥] (zn − zm ) X , and hence 0 − PY (zn − zm ), PY (zn − zm ) X PY [⊥] (zn − zm ), PY [⊥] (zn − zm ) X = PY [⊥] (xn − xm ), PY [⊥] (xn − xm ) X . Here the final expression tends to zero as n, m → ∞, hence so do the other two. As both −Y and Y [⊥] are uniformly positive, this implies that both PY (zn − zm ) and PY [⊥] (zn − zm ) tend to zero in X as n, m → ∞, and consequently zn − zm → 0 in X as n, m → ∞. By the completeness of X , the limit limn→∞ zn := z exists, and hence also limn→∞ yn := y = x − z exists. Both Z and Y are closed, so z ∈ Z, y ∈ Y, and x = z + y ∈ Z + Y. 2
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The partial converse of Lemma 2.1 is also true: if Z is nonnegative and (2.3) holds for some uniformly negative and maximal nonpositive subspace, then Z is maximal nonnegative. 2.2. Preliminaries on quotient spaces Let X be a topological vector space, and let Z be a closed subspace of X . The quotient space of X modulo Z (or over Z) is denoted by X /Z. Each element in X /Z is an equivalence class of vectors in X , where x1 and x2 ∈ X are considered to be equivalent if x1 − x2 ∈ Z. Thus, each equivalence class is a closed affine subset of X . The equivalence class in X /Z which contains a particular x ∈ X is denoted by x + Z. The quotient X /Z is a vector space with addition and scalar multiplication defined by (x1 + Z) + (x2 + Z) = (x1 + x2 ) + Z and λ(x + Z) = (λx) + Z. The quotient map x → x + Z is denoted by πZ , or shortly by π . The quotient topology in X /Z is the one inherited from X through the quotient map π , i.e., Ω ⊂ X /Z is open in X /Z if and only if its inverse image π −1 (Ω) of Ω is open in X . The quotient map π is obviously linear, and it is both continuous and open with respect to the quotient topology in X /Z. If the topology in X is induced by a Hilbert space norm · X , then the topology in X /Z is induced by the Hilbert space quotient norm hX /Z = min xX x ∈ h .
(2.4)
In particular, this is true if X is a closed subspace of a Kre˘ın space (since the topology of a Kre˘ın space is induced by a Hilbert space norm). In both these cases the quotient map π has a bounded right-inverse, since π is surjective, and since the topologies in X /Z and X are induced by Hilbert space norms. If Z is a maximal nonnegative subspace of a Kre˘ın space X , then we can say more: Lemma 2.2. If Z is a maximal nonnegative subspace of a Kre˘ın space X and Y is an arbitrary uniformly negative and maximal nonpositive subspace of X , then the quotient map π : X → X /Z has a unique bounded right-inverse T with range Y. Proof. By Lemma 2.1, W = Z Y. This implies that the restriction of the quotient map πZ to Y is a continuous linear bijection from Y to X /Z. Since the topology in X /Z is induced by a Hilbert space norm, this implies that the inverse T of this map is continuous. This map T is the unique right-inverse of πZ with R(T ) = Y. 2 2.3. The space H(Z) As an introduction to our first main result, presented in Theorem 2.3, we first consider the case where Z is a uniformly positive maximal nonnegative subspace of X . Then (2.1) is a fundamental decomposition of X , and every x ∈ X has a unique decomposition x = zx† + zx
with zx† ∈ Z [⊥] and zx ∈ Z;
here zx† = PZ [⊥] x and zx = PZ x. When x is decomposed in this way we get, for every z ∈ Z, [x − z, x − z]X = zx† + (zx − z), zx† + (zx − z) X = zx† , zx† X + [zx − z, zx − z]X ,
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where zx − z ∈ Z. Hence, since Z is nonnegative, sup −[x − z, x − z]X = − zx† , zx† X ,
z∈Z
(2.5)
and zx is the unique vector in Z for which the supremum is achieved. The right-hand side is the square of the norm of zx† in the Hilbert space −Z [⊥] , and the left-hand side can be interpreted as the square of a Hilbert norm in the quotient space X /Z. We denote X /Z equipped with the above norm by H(Z), and denote the norm of h ∈ H(Z) by (1.2). With this notation (2.5) becomes x + Z2H(Z ) = − zx† , zx† X ,
(2.6)
where x + Z stands for the equivalence class in X /Z to which x belongs. The mapping x → zx† := PZ [⊥] x is a unitary map of H(Z) onto −Z [⊥] , whose inverse z† → z† + Z is the restriction of the quotient map π to Z [⊥] . We now proceed to discuss the general case where Z is maximal nonnegative but not necessarily uniformly positive. In this case the supremum in (1.1) can be infinite for some equivalence classes h ∈ X /Z, and, if finite, it need not always be achieved for some z ∈ Z. Nonetheless, we define H(Z) to be the subset of X /Z for which the supremum in (1.2) is finite. As we show in the following theorem, it is still true that H(Z) is a Hilbert space which is continuously contained in X /Z. Theorem 2.3. Let Z be a maximal nonnegative subspace of a Kre˘ın space X . Define H(Z) by (1.1), and define · H(Z ) by (1.2). Then H(Z) is a Hilbert space with the norm · H(Z ) which is continuously contained in X /Z. Proof. Step 1 (The supremum in (1.1) in nonnegative). If h = Z, then the supremum in (1.1) is zero since Z is nonnegative. We claim that the supremum is strictly positive if it is finite and h = Z. Suppose that x0 ∈ h but x0 ∈ / Z, and that sup{−[x, x]X | x ∈ h} 0, i.e., that [x0 + z, x0 + z]X 0,
z ∈ Z.
(2.7)
Define Z = {λx0 + z | λ ∈ C, z ∈ Z}. Then Z is a subspace which strictly contains Z. We claim that Z is nonnegative. If λ = 0 then [λx0 + z, λx0 + z]X 0 because of the nonnegativity of Z, whereas if λ = 0, then
1 1 0 [λx0 + z, λx0 + z]X = |λ|2 x0 + z, x0 + z λ λ X
(2.8)
because of (2.7). This proves that Z is nonnegative. However, Z was assumed to be maximal nonnegative, so it cannot have a nontrivial nonnegative extension. This shows that (2.7) cannot hold, and consequently the supremum in (1.1) is strictly positive.
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Step 2 (λhH(Z ) = λhH(Z ) for all h ∈ H(Z)). If λ = 0, then both sides are equal to zero, and if λ = 0 this follows from the same identity that was used in (2.8). Step 3 ( · H(Z ) satisfies the parallelogram law). It is easy to verify that for all x1 , x2 ∈ X and z1 , z2 ∈ Z we have [x1 + x2 + z1 , x1 + x2 + z1 ]X + [x1 − x2 + z2 , x1 − x2 + z2 ]X
1 1 = 2 x1 + (z1 + z2 ), x1 + (z1 + z2 ) 2 2 X
1 1 + 2 x2 + (z1 − z2 ), x2 + (z1 − z2 ) , 2 2 X and hence x1 + x2 + Z2H(Z ) + x1 − x2 + Z2H(Z ) = sup −[x1 + x2 + z1 , x1 + x2 + z1 ]X z1 ,z2 ∈Z
− [x1 − x2 + z2 , x1 − x2 + z2 ]X
1 1 = 2 sup − x1 + (z1 + z2 ), x1 + (z1 + z2 ) 2 2 z1 ,z2 ∈Z X
1 1 − x2 + (z1 − z2 ), x2 + (z1 − z2 ) 2 2 X
= 2 sup − x1 + z1 , x1 + z1 X − x2 + z2 , x2 + z2 X z1 ,z2 ∈Z
= 2x1 + Z2H(Z ) + 2x2 + Z2H(Z ) . This shows that · H(Z ) satisfies the parallelogram law. Step 4 (H(Z) is a subspace and · H(Z ) is a norm in H(Z) induced by an inner product). It follows from the homogeneity property proved in Step 2 and the parallelogram law proved in Step 3 that if both sup{−[x, x]X | x ∈ h1 } < ∞ and sup{−[x, x]X | x ∈ h2 } < ∞, then sup{−[x, x]X | x ∈ λ1 h1 + λ2 h2 } < ∞ for all λ1 , λ2 ∈ C. Thus H(Z) is a subspace of X /Z. Since · H(Z ) is a strictly positive homogeneous function on H(Z) satisfying the parallelogram law, it is a norm on H(Z) induced by an inner product in H(Z), which can be defined in terms of · H(Z ) via the standard polarisation identity 4(x1 + Z, x2 + Z)H(Z ) = x1 + x2 + Z2H(Z ) − x1 − x2 + Z2H(Z ) + ix1 + ix2 + Z2H(Z ) − ix1 − ix2 + Z2H(Z ) . Step 5 (The inclusion map H(Z) → X /Z is continuous). Let hn → 0 in H(Z), and let R be a bounded right-inverse to the quotient map π with a uniformly negative range (such a right-inverse exists by Lemma 2.2). Then Rhn ∈ hn , and hence −Rhn , Rhn X hn 2H(Z ) . This implies that
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lim infn→∞ Rhn , Rhn X 0. Together with the fact that the range of R is uniformly negative, this implies that Rhn → 0 in X as n → ∞. Consequently, hn = πRhn → 0 in X /Z. Step 6 (The space H(Z) with the norm · H(Z ) is complete). Let {hn }∞ n=1 be a Cauchy sequence in H(Z). Since the inclusion map H(Z) → X /Z is continuous, {hn }∞ n=1 is also a Cauchy sequence in X /Z, and since X /Z is complete, hn converges to some limit h in X /Z. We claim that h ∈ H(Z), and that hn converges to h in H(Z). Let > 0, and choose N so large that hn − hm H(Z ) whenever both n N and m N . Let R be a continuous right-inverse to the quotient map π (such a right-inverse exists by Lemma 2.2), and define xn = Rhn and x = Rh. Then hn = xn + Z, h = x + Z, xm → x in X as m → ∞, and for every z ∈ Z, −[x − xn + z, x − xn + z]X = lim −[xm − xn + z, xm − xn + z]X , m→∞
where −[xm − xn + z, xm − xn + z]X xm − xn + Z2H(Z ) 2 whenever both n N and m N . Hence −[x − xn + z, x − xn + z]X 2 for all z ∈ Z when n N . By the definition of · H(Z ) , x − xn + Z ∈ H(Z) and x − xn + ZH(Z ) . Since xn + Z ∈ H(Z) also x ∈ H(Z), and hn = xn + Z → x + Z = h in H(Z) as n → ∞. 2 2.4. Properties of the space H(Z) We continue to study some properties of the Hilbert space H(Z). In particular we will show that the subspace H0 (Z) defined in (1.5) is a dense subspace of H(Z). We begin with a preliminary lemma. Since Z is an nonnegative subspace of X , the Schwarz inequality says that [x, z]X 2 [z, z]X [x, x]X ,
x, z ∈ Z.
A generalisation of this inequality is presented in part (1) of the following lemma. Lemma 2.4. Let Z be a maximal nonnegative subspace of a Kre˘ın space X . (1) The inequality (1.3) holds. (2) The inequality (1.3) collapses to an equality if and only if either [z, z]X = 0 or [z, z]X = 0 and x = ([x, z]X /[z, z]X )z + z† where z† ∈ Z [⊥] . (3) The supremum in (1.1) is achieved if and only if h = z† + Z for some z† ∈ Z [⊥] , and in this case the supremum is equal to max − z† + z, z† + z X z ∈ Z = − z† , z† X .
(2.9)
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(4) If z† ∈ Z [⊥] , x ∈ X , and x + Z ∈ H(Z), then z† + Z ∈ H(Z) and † z + Z, x + Z H(Z ) = − z† , x X .
(2.10)
Proof. Proof of claim (1). Let h ∈ H(Z), x ∈ h, z ∈ Z, and λ ∈ C. Then −[x − λz, x − λz]X h2H(Z ) , or equivalently, |λ|2 [z, z]X − 2[x, λz]X + [x, x]X + h2H(Z ) 0.
(2.11)
If [z, z]X = 0, then this implies that [x, z]X = 0 (since the inequality is true for all λ ∈ C), so (1.3) holds in the trivial form 0 = 0. If [z, z]X = 0, then we can take λ = [x, z]X /[z, z]X in (2.11) and multiply the resulting formula by [z, z]X to get (1.3). Proof of claim (3). It h = z† + Z for some z† ∈ Z [⊥] , then − z† + z, z† + z X = − z† , z† X − [z, z]X − z† , z† X . The supremum in (1.1) is achieved by taking z = 0, and it is equal to −[z† , z† ]X . Conversely, if the supremum in (1.1) is achieved at some point x0 ∈ h, then it follows from (1.3) with x replaced by x0 that [x0 , z]X = 0 for all z ∈ Z. Consequently, x0 ∈ Z [⊥] . Proof of claim (2). If the inequality (1.3) holds in the form of an equality and [z, z]X = 0, then we get equality in (2.11) by taking λ = [x, z]X /[z, z]X . This implies that the supremum in (1.1) is achieved for the vector x − ([x, z]X /[z, z]X ). By claim (3), x − ([x, z]X /[z, z]X ) ∈ Z [⊥] . Proof of claim (4). It follows from part (3) that z† + Z ∈ H(Z). In order to prove (2.10) is suffices to prove that, for all z† ∈ Z [⊥] and x ∈ X with x + Z ∈ H(Z) we have
x + z† + Z 2
H (Z )
2 = x + Z2H(Z ) − z† , x X − x, z† X + z† + Z H(Z ) ,
(2.12)
because (2.10) then follows from the polarisation identity. However, for all z† ∈ Z [⊥] , x ∈ X with x + Z ∈ H(Z), and z ∈ Z, − x + z† + z, x + z† + z X = −[x + z, x + z]X − x, z† X − z† , x X − z† , z† X .
(2.13)
After taking the supremum over all z ∈ Z and using the fact that z† + Z2H(Z ) = −[z† , z† ]X we get (2.12). 2 The following theorem contains a geometrical interpretation of the Hilbert space H(Z) as the completion of the pre-Hilbert space H0 (Z) with the inner product given in (1.6).
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Theorem 2.5. Let Z be a maximal nonnegative subspace of a Kre˘ın space X . Then the subspace H0 (Z) of X /Z defined in (1.5) is a dense subspace of the Hilbert space H(Z) defined in Theorem 2.3, and the inner product in H0 (Z) inherited from H(Z) is given by (1.6). Thus, H(Z) is a Hilbert space completion of the space H0 (Z) with the inner product (1.6). Proof. It follows from part (3) of Lemma 2.4 that H0 (Z) ⊂ H(Z). That the inner product in H0 (Z) inherited from H(Z) is given by (1.6) follows from (2.10). To show that the H0 (Z) is dense in H(Z) it suffices to show that (H0 (Z))⊥ = {0}. Let x ∈ X and x + Z ∈ H(Z), and suppose that (x + Z, z† + Z)H(Z ) = 0 for all z† ∈ Z [⊥] . Then by (2.10), [x, z† ]X = 0 for all z† ∈ Z [⊥] , and hence x ∈ (Z [⊥] )[⊥] = Z and x + Z = Z is the zero vector in X /Z. 2 2.5. The spaces Z [⊥] /Z0 and U(Z) Up to now we have concentrated our attention on the two subspaces H(Z) and H0 (Z) of X /Z. It turns out that the latter of these spaces is closely related to the space Z [⊥] /Z0 , where as before Z0 = Z ∩ Z [⊥] . The space Z [⊥] /Z0 is defined in the standard way as the quotient of Z [⊥] modulo its closed subspace Z0 . Since the topology in Z [⊥] inherited from −X is induced by a Hilbert space norm, it follows that also the standard quotient topology in Z [⊥] /Z0 is induced by a Hilbert space norm. In particular, Z [⊥] /Z0 is complete with respect to the quotient topology. We define the space U(Z) to be the same vector space as Z [⊥] /Z0 , but with a different topology induced by the positive inner product inherited from −X , i.e., † z1 + Z0 , z2† + Z0 U (Z ) = − z1† , z2† X ,
z1† , z2† ∈ Z [⊥] .
(2.14)
That this is, indeed, a positive inner product on the vector space Z [⊥] /Z0 follows from the fact that Z [⊥] is a nonnegative subspace of −X , and that Z0 is the maximal neutral subspace in Z [⊥] . The topology induced by this inner product is weaker than the standard quotient topology of Z [⊥] /Z0 , so that the embedding of Z [⊥] /Z0 in U(Z) is continuous. However, the inverse of this embedding map need not be continuous, and U(Z) need not be complete. Thus, U(Z) is a unitary space (a pre-Hilbert space), but U(Z) need not be a Hilbert space. It is a Hilbert space if and only if Z [⊥] is the direct sum of Z0 and a uniformly negative subspace in X , or equivalently, if and only if Z is the direct sum of Z0 and a uniformly positive subspace in X . In this case (and only in this case) the topologies of Z [⊥] /Z0 and U(Z) coincide. Theorem 2.6. Let Z be a maximal nonnegative subspace of a Kre˘ın space X . Define Z0 = Z ∩ Z [⊥] , and let U(Z), H0 (Z), and H(Z) be the spaces defined earlier in this section. Then the formula S z† + Z0 = z† + Z,
x † ∈ Z [⊥] ,
(2.15)
defines an linear isometric map S from the unitary space U(Z) into the Hilbert space H(Z) with R(S) = H0 (Z). In particular R(S) is dense in H(Z). Proof. That (2.15) defines a linear isometric operator S from U(Z) onto H0 (Z) follows from the formulas (2.14) and (1.6) for the inner products in U(Z) and H0 (Z), respectively. That H0 (Z) is dense in H(Z) is part of the conclusion of Theorem 2.5. 2
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Remark 2.7. The linear bijection S defined in (2.15) is an isomorphism with respect to the two inner products in U(Z) and H0 (Z), and it is still continuous if we replace the topology in U(Z) by the quotient topology of Z [⊥] /Z0 or if we replace the topology in H0 (Z) by the quotient topology inherited from X /Z. This follows from the fact that the quotient topology in Z [⊥] /Z0 in stronger than the inner product topology of U(Z), and that the inner product topology of H0 (Z) is stronger than the quotient topology inherited from X /Z. However, the inverse S −1 need not be continuous with respect to the quotient topologies. It is continuous if and only if Z is the direct sum of Z0 and a uniformly positive subspace. In this case the spaces U(Z) and H0 (Z) are complete, and hence H0 (Z) = H(Z) and S is a unitary map of U(Z) onto H(Z). Theorems 2.14 and 3.7 list a number of other equivalent conditions for this case to occur. 2.6. Further properties of the space H(Z) Convergence of a sequence in H(Z) is related to convergence of the corresponding representatives in X as follows. Lemma 2.8. Let Z be a maximal nonnegative subspace of a Kre˘ın space X , and let H(Z) be the Hilbert space defined in Theorem 2.3. (1) If yn + Z → x + Z in H(Z) as n → ∞, then there exists a sequence xn ∈ X such that xn → x in X and xn + Z = yn + Z → x + Z in H(Z) as n → ∞. The same claim remains true if we throughout replace the strong convergence by weak convergence. (2) Given any x + Z ∈ H(Z) there exists a sequence xn ∈ Z + Z [⊥] such that xn → x in X and xn + Z → x + Z in H(Z) as n → ∞. (3) If zn† ∈ Z [⊥] and supn0 (−[zn† , zn† ]X ) < ∞, then there exist a vector x ∈ X , a subsequence zn†j , and a sequence znj ∈ Z such that zn†j + znj → x weakly in X and zn†j + Z → x + Z weakly in H(Z) as j → ∞. Proof. Proof of claim (1). Since H(Z) is continuously contained in X /Z, the sequence yn + Z converges to x + Z also in the topology of X /Z. The quotient map πZ has a bounded right-inverse, and this implies that there exists a sequence xn ∈ X which tends to a limit x ∈ X such that xn + Z = yn + Z for all n and x + Z = x + Z. In particular, x − x ∈ Z. Define xn = xn + x − x . Then xn + Z = yn + Z for all n, xn + Z → x + Z in H(Z), and xn → x in X as n → ∞. The version where the strong convergence has been replaced by weak convergence is proved in the same way. Proof of claim (2). Since H0 (Z) is dense in H(Z), for each x +Z ∈ H(Z) there exists a sequence yn ∈ Z [⊥] such that yn + Z → x + Z in H(Z). By applying claim (1) to this sequence we can find a sequence xn satisfying the conclusion of claim (2), since the condition xn + Z = yn + Z implies that xn ∈ Z + Z [⊥] . Proof of claim (3). For each zn† we have zn† + Z2H(Z ) = −[zn† , zn† ]X , so the given condition im-
plies that the sequence zn† + Z is bounded in H(Z). The unit ball in H(Z) is weakly sequentially compact, and hence some subsequence znj + Z converges weakly to a limit x + Z in H(Z). The conclusion of claim (3) now follows from claim (1). 2
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Proposition 2.9. Let Z be a maximal nonnegative subspace of a Kre˘ın space, and define H(Z) as in Theorem 2.3. Then x + ZH(Z ) defined in (1.2) (finite or infinite) is equal to x + ZH(Z ) = sup z† , x X z† ∈ Z [⊥] and − z† , z† X 1 .
(2.16)
Proof. If x ∈ X and x + Z ∈ H(Z), then (2.16) follows from (2.10) and density of H0 (Z) in H(Z). Conversely, suppose that the supremum in (2.16) is finite. Then the linear functional F (z† ) := [z† , x]X : Z [⊥] → C is bounded on Z [⊥] (with respect to the semi-norm inherited from −X ). However, every such functional can be interpreted as a bounded linear functional on H0 (Z) with respect to the norm inherited from H(Z). Since H0 (Z) is dense in H(Z) and H(Z) is a Hilbert space, there is some y + Z ∈ H(Z) such that F (z† ) = [z† + Z, y + Z]H(Z ) for every z† ∈ Z [⊥] . By (2.10), [z† + Z, y + Z]H(Z ) = [z† , y]X . Thus, F (z† ) = [z† , x]X = [z† , y]X for all z† ∈ Z [⊥] . This implies that x − y ∈ Z, and so x + Z = y + Z ∈ H(Z). As we observed above, this implies (2.16). 2 Given a maximal nonnegative subspace Z of a Kre˘ın space X we define L(Z) by L(Z) = x + Z x ∈ Z + Z [⊥] .
(2.17)
Lemma 2.10. The set L(Z) defined above is a closed subspace of X /Z. Proof. It is easy to see that L(Z) is a subspace. To see that it is closed we argue as follows. Let hn ∈ L(Z), and let hn → h in X /Z as n → ∞. Let R be a bounded right-inverse of the quotient map πZ , and define xn = Rhn and x = Rh. Then xn → x in X as n → ∞, and xn + Z = hn ∈ L(Z) for all n. This implies that xn − yn ∈ Z for some yn ∈ Z + Z [⊥] , and consequently xn ∈ Z + Z [⊥] . Therefore also x = limn→∞ xn ∈ Z + Z [⊥] . Thus h = x + Z ∈ L(Z). 2 Proposition 2.11. Let Z be a maximal nonnegative subspace of a Kre˘ın space, and let H(Z) and H0 (Z) be the spaces defined earlier in this section. Then the closure in X /Z of each of the spaces H0 (Z) and H(Z) in X /Z is equal to L(Z) defined in (2.17). Proof. In view of claim (2) of Lemma 2.8 and the continuous inclusion of H(Z) in X /Z, H(Z) ⊂ L(Z). Consequently, H0 (Z) ⊂ H(Z) ⊂ L(Z) = L(Z). To complete the proof we still have to show that L(Z) ⊂ H0 (Z). Take any h ∈ L(Z), and choose some x ∈ Z + Z [⊥] such that h = x + Z. Then there exists a sequence xn ∈ Z + Z [⊥] such that xn → x in X as n → ∞. By the definition of H0 (Z), xn + Z ∈ H0 (Z) for all n. By the continuity of the quotient map πZ , xn + Z → x + Z in X /Z, and so h = x + Z ∈ H0 (Z). 2 2.7. Relation between H(Z) and H(Z [⊥] ) If Z is a maximal nonnegative subspace of a Kre˘ın space X , then Z [⊥] can be interpreted as a maximal nonnegative subspace of the anti-space −X of X . We can therefore repeat the same construction presented above with X replaced by −X , and with Z replaced by Z [⊥] to get the
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Hilbert space H(Z [⊥] ) which is a Hilbert completion of the inner product space H0 (Z [⊥] ). The unitary space U(Z [⊥] ) is obtained in an analogous way, and it can be canonically embedded in the Hilbert space H(Z [⊥] ). As the following theorem shows, there are certain connections between the two spaces H(Z) and H(Z [⊥] ). To explore this connection we investigate the following two subspaces of X : X (Z) = x ∈ X x + Z ∈ H(Z) , X Z [⊥] = x ∈ X x + Z [⊥] ∈ H Z [⊥] .
(2.18) (2.19)
Indeed, they are subspaces, since X (Z) is the inverse image of H(Z) under the quotient map πZ , and X (Z [⊥] ) is the inverse image of H(Z [⊥] ) under the quotient map πZ [⊥] . Theorem 2.12. Let Z be a maximal nonnegative subspace of a Kre˘ın space Z, let Z0 = Z ∩Z [⊥] , and let X (Z), X (Z [⊥] ), H(Z), H0 (Z), H(Z [⊥] ), and H0 (Z [⊥] ) be the spaces defined above. Then the following claims are true. (1) X (Z) = X (Z [⊥] ). (2) Z + Z [⊥] ⊂ X (Z) ⊂ Z + Z [⊥] . (3) Z0 is the maximal subspace of Z + Z [⊥] which is orthogonal to Z + Z [⊥] , and the same statement remains true if we replace Z + Z [⊥] by X (Z) or by Z + Z [⊥] . (4) Let F be one of the spaces listed in part (2). Then the formula [y1 + Z0 , y2 + Z0 ]F /Z0 = [y1 , y2 ]X ,
y1 , y2 ∈ F ,
(2.20)
defines a nondegenerate indefinite inner product in the quotient space F /Z0 . (5) The formula Q(x + Z0 ) =
x +Z , x + Z [⊥]
x ∈ X (Z),
(2.21)
defines a linear isometric map from the space X (Z)/Z0 with the inner product defined in (2.20) with F = X (Z) into the Kre˘ın space −H(Z) [] H(Z [⊥] ). The image of (Z + Z ⊥ )/Z0 under Q is −H0 (Z)[] H0 (Z [⊥] ). In particular, both R(Q|(Z +Z ⊥ )/Z0 ) and R(Q) are dense in −H(Z) [] H(Z [⊥] ). Proof. Proof of claim (1). Let x ∈ X and x + Z ∈ H(Z). It follows from (1.6) and (2.12) that x + z† , x + z† X = [x, x]X + z† , x X + x, z† X + z† , z† X
2 = x + Z2H(Z ) + [x, x]X − x + z† + Z H(Z ) . Taking the supremum over all z† ∈ Z [⊥] we find that x + Z [⊥] ∈ H(Z [⊥] ), and that x + Z [⊥] 2H(Z [⊥] ) = x + Z2H(Z ) + [x, x]X −
2
inf x + z† + Z H(Z ) .
z† ∈Z [⊥]
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Since H0 (Z) is dense in H(Z), the infimum is zero, and hence
2 [x, x]X = −x + Z2H(Z ) + x + Z [⊥] H(Z [⊥] ) .
(2.22)
That x + Z ∈ H(Z) whenever x + Z [⊥] ∈ H(Z [⊥] ) can be proved in the same way (or by replacing X by −X and Z by Z [⊥] ). Proof of claim (2). That Z + Z [⊥] ⊂ X (Z) follows from part (3) of Lemma 2.4, and that X (Z) ⊂ Z + Z [⊥] follows from Proposition 2.11. Proof of claims (3), (4). The straightforward proofs of claims (3) and (4) are left to the reader. Proof of claim (5). That Q is well defined it follows from the fact that N (πZ ) ∩ N (πZ [⊥] ) = Z0 . That Q is an isometry follows from (2.20) and (2.22). The remaining claims are obvious. 2 Remark 2.13. The operator Q in (2.21) is also a well-defined continuous linear map from X /Z Z + Z [⊥] /Z0 into X /Z [⊥] , but the range of this extended map is not necessarily contained in −H(Z) [] H(Z [⊥] ), neither does it necessarily contain −H(Z) [] H(Z [⊥] ). In fact, the space X (Z) is the maximal subspace of X whose image under this extended map Q is contained in −H(Z) [] H(Z [⊥] ). However, the image of X (Z) need not be all of −H(Z) [] H(Z [⊥] ). To see this it suffices to observe that the intersection of the image with the closed subspace −H(Z) of −H(Z) [] H(Z [⊥] ) is equal to −H0 (Z). It is not difficult to show that the image is all of −H(Z) [] H(Z [⊥] ) if and only if H0 (Z) = H(Z).
Theorem 2.14. Let Z be a maximal nonnegative subspace of a Kre˘ın space X , let Z0 = Z ∩Z [⊥] , and let H(Z), H(Z [⊥] ), H0 (Z), H0 (Z [⊥] ), U(Z), U(Z [⊥] ), and X (Z) be the spaces defined above. Then the following conditions are equivalent. (1) Z = Z0 [] Z+ where Z+ is a uniformly positive subspace of Z. (2) Z [⊥] = Z− [] Z0 where Z− is a uniformly negative subspace of Z. (3) Z + Z [⊥] = Z− [] Z0 [] Z+ , where Z− and Z+ are uniformly negative and positive subspaces, respectively, of X . (4) Z + Z [⊥] is closed in X . (5) U(Z) is a Hilbert space. (6) U(Z [⊥] ) is a Hilbert space. (7) H0 (Z) = H(Z). (8) H0 (Z [⊥] ) = H(Z [⊥] ). (9) (Z + Z [⊥] )/Z0 is a Kre˘ın space with the inner product defined in (2.20). (10) X (Z)/Z0 is a Kre˘ın space with the inner product defined in (2.20). (11) X (Z) = Z + Z [⊥] . Proof. Proof of the equivalence (2) ⇔ (5). Let Z− be an arbitrary direct complement to Z0 in Z ⊥ . Then by, e.g. [7, Lemmas 5.1 and 5.2, p. 11], Z− with the inner product inherited from −X is isometrically isomorphic to U(Z). Thus, U(Z) is a complete if and only if Z− is complete, and this is true if and only if Z− is uniformly negative.
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Proof of the equivalence (5) ⇔ (7). The operator S defined in (2.15) is an isometric map of U(Z) onto H0 (Z), so U(Z) is complete if and only H0 (Z) is complete, and this is true if and only if H0 (Z) = H(Z) (since H(Z) is a completion of H0 (Z). Proof of the equivalence (7) ⇔ (9). The operator Q defined in (2.21) is an isometric map of (Z + Z [⊥] )/Z0 onto −H0 (Z) [] H0 (Z [⊥] ). Thus, (Z + Z [⊥] )/Z0 is complete if and only if −H0 (Z) [] H0 (Z [⊥] ) is complete, which is true if and only if H0 (Z) = H(Z). Proof of the equivalence (7) ⇔ (11). This follows from the definitions of H0 (Z) and X (Z). Proof of the implication (9) ⇒ (10). The operator Q defined in (2.21) is an isometric map of X (Z)/Z0 into −H(Z) [] H(Z [⊥] ) which contains the image of (Z + Z [⊥] )/Z0 . If (Z + Z [⊥] )/Z0 is complete, then this image coincides with −H(Z) [] H(Z [⊥] ) (being dense in −H(Z) [] H(Z [⊥] )), and hence also the image of X (Z)/Z0 must coincide with −H(Z) [] H(Z [⊥] ). This means that X (Z)/Z0 is complete. Proof of the implication (10) ⇒ (7). If (10) holds, then the image of X (Z)/Z0 under the isometric operator Q in (2.21) coincides with −H(Z) [] H(Z [⊥] ) (being dense in −H(Z) [] H(Z [⊥] )). In particular, the range must contain every vector of the form 0h , where h is an arbitrary vector in H(Z) (and the H(Z [⊥] )-component is zero). However, it is easy to see that the intersection of the image of X (Z)/Z0 under Q with the subspace where the H(Z [⊥] )-component is zero is equal to H0 (Z). This implies that H(Z) ⊂ H0 (Z), and so H0 (Z) = H(Z). Proof of the equivalence (1) ⇔ (2), (5) ⇔ (6), and (7) ⇔ (8). These equivalences follow from the equivalence of (1), (5), (7), and (9) and the fact that (9) is invariant under the interchange of Z and Z [⊥] . Proof of the implication (1) & (2) ⇒ (3). This implication is trivial. [⊥] [⊥] Proof of the implication (3) ⇒ (4). If (3) holds, then X = Z− []Z− and also X = Z+ []Z+ (see, e.g., [7, Theorem 3.4, p. 104]). This means that there exist bounded orthogonal projections P∓ of X onto Z∓ . Out of these P− vanishes on Z0 [] Z+ and P+ vanishes on Z− [] Z0 . For each x ∈ X , define P0 x = x − P− x − P+ x. Then also P0 is bounded, and for every x ∈ Z + Z [⊥] we have x = P− x + P0 x + P+ x, where P− x ∈ Z− , P0 x ∈ Z0 , and P+ x ∈ Z+ . Let xn ∈ Z + Z ⊥ , and let xn → x ∈ X . Then xn = P− xn + P0 xn + P+ xn → P− x + P0 x + P+ x = x, and hence x ∈ Z− [] Z0 [] Z+ = Z + Z [⊥] . Thus Z + Z [⊥] is closed.
Proof of the implication (4) ⇒ (11). This follows from part (2) of Theorem 2.12.
2
3. Connection with the complementary space H(A) In this section we shall discuss the connection between the space H(Z) for some maximal nonnegative subspace Z of a Kre˘ın space X , and the de Branges complementary space H(A) induced by the contraction A that appears in the graph representation of Z with respect to some fundamental decomposition of X . We shall also give alternative proofs of some of the results in Section 2 that depend on the standard graph representation of a maximal nonnegative subspace of a Kre˘ın space.
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3.1. The graph representation of a maximal nonnegative subspace Lemma 3.1. Let A be a linear contraction from a Hilbert spaces U to a Hilbert space Y, and let
Au Z = z= u∈U u
(3.1)
be the graph of A. Then Z is a maximal nonnegative subspace of the Kre˘ın space −Y [] U . Conversely, let Z be a maximal nonnegative subspace of a Kre˘ın space X , and let X = −Y [] U be a fundamental decomposition of X . Then Z has the graph representation (3.1) for a unique linear contraction A : U → Y. Proof. See, e.g., [7, Theorem 1.7, p. 54 and Theorem 4.2, pp. 105, 106].
2
Lemma 3.2. The orthogonal companion Z [⊥] of the maximal nonnegative subspace Z in Lemma 3.1 has the graph representation
y Z [⊥] = z = y ∈ Y . A∗ y Proof. This is well known, and it is a simple corollary of Lemma 3.1.
(3.2) 2
Alternative proof of Lemma 2.1. Let U = Y [⊥] . Then X = Y [] U is a fundamental decomposition of X , and hence Z has the graph representation (3.1) for some A ∈ B(U; −Y). This implies that every x ∈ X has the unique decomposition
y Au y − Au x= = + , u u 0 where
Au u
∈ Z and
y−Au 0
∈ Y.
(3.3)
2
The following lemma is a slight extension of Lemma 2.2. Lemma 3.3. Let Z be a maximal nonnegative subspace of a Kre˘ın space X , and let X = −Y [] U be a fundamental decomposition of X . Then the operator T in Lemma 2.2 is a bounded linear operator X /Z → Y with a bounded inverse, and T (x + Z) = y − Au,
z=
y Y ∈ . u U
(3.4)
Proof. Most of this follows from Lemma 2.2. The explicit formula (3.4) for T (x + Z) follows from (3.3). 2
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3.2. The de Branges complement of a range space Let A ∈ B(U; Y). By definition, the range space M(A) of A is the range of A endowed with the range norm which makes A|[N (A)]⊥ a unitary operator [N (A)]⊥ → M(A). In other words, to each y ∈ R(A) we can find a unique vector u ∈ [N (A)]⊥ such that y = Au, and define yM(A) = uU . The basic properties of the space M(A) can be found in many books, including [12, pp. 2, 3]. We next assume that A is a contraction, and proceed to define the de Branges complement H(A) of M(A). One starts by defining H(A) to be the following subset of Y: H(A) = y ∈ Y yH(A) < ∞ ,
(3.5)
y2H(A) = sup y − Au2 − u2U .
(3.6)
where
u∈U
It is known from the work of de Branges and Rovnyak [9,10] that H(A) is a linear subspace of Y, that · H(A) defined in (3.6) is a norm in H(A) induced by a Hilbert space inner product, and that H(A) with this norm is continuously (but not necessarily densely) contained in Y. The following well-known facts explain in which sense H(A) can be interpreted as a complement of M(A) (see, e.g., [12, Chapter 1] for the proofs): (1) Y = M(A)+H(A), i.e., every y ∈ Y can be written as a sum y = y1 +y2 , where y1 ∈ M(A) and y2 ∈ H(A). The sum is direct (i.e., M(A) and H(A) are closed and M(A) ∩ H(A) = 0) if and only if it is orthogonal, i.e., H(A) = M(A)⊥ , and this is true if and only if A is a partial isometry (i.e., A is an isometry on [N (A)]⊥ ). (2) If y = y1 + y2 with y1 ∈ M(A) and y2 ∈ H(A), then y2Y y1 2M(A) + y2 2H(A) . Moreover, for each y ∈ Y there exist unique vectors y1 ∈ M(A) and y2 ∈ H(A) such that y = y1 + y2 and y2Y = y1 2M(A) + y2 2H(A) , namely y1 = AA∗ y and y2 = (1Y − AA∗ )y. The above definition of H(A) follows the original approach taken by de Branges and Rovnyak in [9,10]. It was later realized that H(A) also can be characterised in a different way, namely (3) H(A) = M((1Y − AA∗ )1/2 ). A proof of this fact can be found in, e.g., [12, Note (NI-6), pp. 7, 8]. 3.3. The connection between H(Z) and H(A) We proceed to investigate the connection between H(Z) and H(A). Lemma 3.4. The bounded linear operator T : X /Z → Y defined in Lemma 2.2 maps the subset H(Z) ⊂ X /Z defined in (1.1) one-to-one onto the subspace H(A) ⊂ Y defined in (3.5), and x + ZH(Z ) = T (x + Z)H(A) for all x + Z ∈ H(Z).
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Proof. Because of (1.1) and (3.5), it suffices to show that x + ZH(Z ) = T (x + Z)H(A) for all x ∈ X with x + Z ∈ H(Z) (finite or infinite), or equivalently, that T −1 y2H(Z ) = y2H(A) for all y ∈ H(A). However, this follows from the fact that the right-hand side is given by (3.6), whereas the left-hand side is given by
−1 2
T y
H (Z )
= sup −[y − z, y − z]X z∈Z
y Au y Au − , − = sup − 0 u 0 u u∈U X 2 2 2 = sup y − AuY − uU = yH(A) . 2 u∈U
Alternative proof of Theorem 2.3. Theorem 2.3 follows from Lemma 3.4 and the fact that H(A) is a Hilbert space which is continuously (but not necessarily densely) contained in Y. 2 Corollary 3.5. The restriction of the operator T in Lemma 3.4 to H(Z) is a unitary map from H(Z) to H(A). Proof. This, too, follows from Lemma 3.4.
2
Lemma 3.6. The bounded linear operator T : X /Z → Y defined in Lemma 2.2 maps the subspace H0 (Z) of X /Z defined in (1.5) one-to-one onto the range of the operator (1Y − AA∗ ). y Proof. Take any z† ∈ Z [⊥] . Then z† = A∗ y for some y ∈ Y. Consequently, T (z† + Z) = y − A(A∗ y) = (1Y − AA∗ )y. Thus, R(T|H0 (Z ) ) ⊂ R(1Y − AA∗ ). Conversely, suppose that x ∈ X , and that T (x + Z) = (1Y − AA∗ )y for some y ∈ Y. Then ∗ x + Z = (1Y − AA∗ )y + Z = { (1Y −AAu )y+Au | u ∈ U}. In particular, by replacing u by A∗ y + u y y we find that x + Z = A∗ y + Z. Thus, x + Z = z† + Z, where z† = A∗ y ∈ Z [⊥] . Thus, R(1Y − AA∗ ) ⊂ R(T|H0 (Z ) ). 2 y Alternative proof of part (4) of Lemma 2.4. Write z† = A∗ y , and denote T (x + Z) by y . Then y = (1Y − AA∗ )1/2 y1 for some y1 ∈ [N ((1Y − AA∗ )1/2 )]⊥ , and † z + Z, x + Z H(Z ) = T z† + Z , T x + Z H(A) = (1Y − AA∗ )y, y H(A) = (1Y − AA∗ )y, (1Y − AA∗ )1/2 y1 M((1 −AA∗ )1/2 ) Y ∗ 1/2 = (1Y − AA ) y, y1 Y = y, (1Y − AA∗ )1/2 y1 Y = (y, y )Y = − z† , y X = − z† , x X . 2 Alternative proof of Theorem 2.5. That H0 (Z) is a dense subset of H(Z) follows from Corollary 3.5 and the fact that T (H0 (Z)) = R(1Y − AA∗ ) is a dense subspace of T (H(Z)) = H(A) = M((1Y − AA∗ )1/2 ). That the inner product in H0 (Z) inherited from H(Z) is given by (1.6) follows from part (iv) of Lemma 2.4. 2
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Alternative proof of part (3) of Lemma 2.4. If h ∈ H0 (Z), then h = z† + Z for some z† ∈ Z [⊥] , and by Theorem 2.5, sup −[x, x]X = − z† , z† X . x∈h
Thus, the supremum is achieved for x = z† . Conversely, suppose that the supremum in (1.1) is achieved for some x0 ∈ h. Let y = T h, where T is the operator in Lemma 3.3. Then x0 = y + z0 for some z0 ∈ Z, and max −[y + z, y + z]X z ∈ Z = −[x + z0 , x + z0 ]X . By using the graph representation (3.1) we can write z = to get
Au u
and z0 =
Au0 u0
for some u, u0 ∈ U
max y + Au2Y − u2U u ∈ U = y − Au0 2Y − u0 2U .
(3.7)
We claim that u0 ∈ (N (A))⊥ , and prove this as follows. It is always possible to write u0 = u1 + u2 where u1 ∈ (N (A))⊥ and u2 ∈ N (A). If u2 = 0, then y − Au0 2Y − u0 2U = y − Au1 2Y − u1 2U − u2 2U < y − Au1 2Y − u1 2U , contradicting (3.7). Thus, u0 ∈ (N (A))⊥ . Define y0 = Au0 and y = y − y0 . Then y = y + (−y0 ), y ∈ H(A), −y0 ∈ M(A), and y 2Y = y2H(A) + −y0 2M(A) . Consequently, by property (2) of the complementary spaces H(A) and M(A) listed earlier in this section, y = (1Y − AA∗ )y , and hence y ∈ R(1Y − AA∗ ). By Lemma 3.6, h = T −1 y ∈ H0 (Z). 2 We end this section with the following addition to Theorem 2.14. Theorem 3.7. Let Z be a maximal nonnegative subspace of a Kre˘ın space X , and let H(Z), H(Z [⊥] ), and X (Z) be the spaces defined in Section 2. Then the following conditions are equivalent to each other, and they are also equivalent to conditions (1)–(11) in Theorem 2.14, (12) H(Z) is closed in X /Z. (13) H(Z [⊥] ) is closed in −X /Z [⊥] . (14) X (Z) is closed in X . Proof. When we in this proof refer to conditions (1)–(11) we mean the corresponding conditions in Theorem 2.14. Proof of the equivalence (7) ⇔ (12). Choose some fundamental decomposition W = −Y [] U of W, let T be the operator defined in Lemma 2.2, and let A be the contraction in the graph representation (3.1). Then T is an isomorphism X /Z → Y which maps H(Z) onto
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H(A) = R((1 − AA∗ )1/2 ) and H0 (Z) onto R(1 − AA∗ ). Thus, the condition H0 (Z) = H(Z) is equivalent to the condition R(1 − AA∗ ) = R((1 − AA∗ )1/2 ), whereas the condition that H(A) is closed in X /Z is equivalent to the condition that R((1 − AA∗ )1/2 ) is closed in Y. However, both of these conditions are equivalent to the condition that ⊥ R (1 − AA∗ )1/2 = N (1 − AA∗ ) = R (1 − AA∗ ) . Thus (7) and (12) are equivalent. Proof of the implication (4) ⇒ (14). This follows from part (2) of Theorem 2.12. Proof of the implication (14) ⇒ (12). If X (Z) is closed in X , then it follows from part (2) of Theorem 2.12 that X (Z) = Z + Z [⊥] . By (2.17), (2.18), and Lemma 2.10, H(Z) is closed in X /Z. Proof of the equivalence (12) ⇔ (13). Both of these are equivalent to (14) (since X (Z) = X (Z [⊥] )), and hence equivalent to each other. 2 Above we have given alternative proofs of some of the coordinate free results in Section 2 by appealing to known results about the de Branges complementary spaces H(A). It is also possible to proceed in the opposite direction and to re-derive results about the spaces H(A) from the results in Section 2 using the isomorphism T in Corollary 3.5. We leave this to the reader. 4. Application to passive state/signal systems theory As was mentioned in the introduction, the results presented in this article were obtained as byproducts of our study of the realization problem in passive state/signal systems theory [2–5]. Here we shall only give a short outline of one of the motivating applications. A passive linear discrete time invariant s/s (state/signal) system Σ = (V ; X , W) has a Kre˘ın signal space W (enabling connections to the external environment), a Hilbert state space X (representing an internal memory), a generating subspace V of the Kre˘ın space K = −X [] X [] W (defining the dynamics) with the properties (i)–(iv) listed in [2], and the set of trajectories, which + consists of sequences (x(·), w(·)) ∈ (X × W)Z satisfying x(n + 1) ∈ V , n ∈ Z+ , x(n) w(n) where Z+ = 0, 1, 2, . . . . This system is passive if V is a maximal nonnegative subspace of K. The nonnegativity of V equivalent to the requirement that the trajectories of Σ satisfy
x(n + 1) 2 − x(n) 2 w(n), w(n) , n ∈ Z+ , (4.1) X X W and the maximal nonnegativity of V is equivalent to the requirement that, in addition, the adjoint system Σ∗ = (V∗ ; X , −W) defined in a natural way has the same property. See [3] for details. Trajectories (x(·), w(·)) with x(0) = 0 (i.e., trajectories whose internal memory is zero at the starting time zero) are called externally generated. The future behavior Wfut of a passive s/s system Σ consists of all sequences w(·) of signals in 2+ (W) = 2 (Z+ ; W) that are obtained from the externally generated trajectories (x(·), w(·)) of Σ by ignoring the state component x(·),
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Wfut = W ∩ 2+ (W). 2 (W) It is not difficult to show that Wfut is a maximal nonnegative subspace of the Kre˘ın space k+ of sequences w(·) ∈ 2+ (W) with indefinite inner product ∞ w1 (k), w2 (k) W . w1 (·), w2 (·) k 2 (W ) = +
k=0
2 (W). Moreover, Wfut is S+ -shift invariant, where S+ is the right shift in k+ The inverse problem to the one described above is the following: is it true that every maximal 2 (W) can be realized as a future behavior of some passive nonnegative S+ -invariant subspace in k+ s/s system? This inverse problem is more difficult to solve, but it turns out that it has a positive answer (given in Theorem 4.1 below), even if we impose some additional constraints on the system Σ, which will be discussed below. A s/s system Σ is forward conservative if (4.1) holds in the form of an equality for all trajectories of Σ, and it is backward conservative if the adjoint system Σ∗ is forward conservative. Thus, Σ = (V ; X , W) is passive and forward conservative if and only if V is maximal nonnegative and V ⊂ V [⊥] (this inclusion means that V is neutral), and Σ is passive and backward conservative if and only if V is maximal nonnegative and V [⊥] ⊂ V . Both of these conditions hold if and only if V is a Lagrangian subspace of K, in which case Σ is called conservative. The subspace of X that we get by taking the closure in X of all states x(n) that appear in externally generated trajectories (x(·), w(·)) of Σ is called the (approximately) reachable subspace, and we denote it by RΣ . If RΣ = X , then Σ is called controllable. The subspace of all x0 ∈ X with the property that (x(·), w(·)) with x(0) = x0 and w(n) = 0 for all n ∈ Z+ is a trajectory of Σ is called the unobservable subspace, and it is denoted by UΣ . If UΣ = {0}, then Σ is called (approximately) observable. A s/s system is called simple if X = RΣ + U⊥ Σ , or = {0}. equivalently, if UΣ ∩ R⊥ Σ The following solution to the inverse problem is given in [3] (see [3, Theorem 3.8] and its proof).
Theorem 4.1. Let W be a Kre˘ın space, and let Z be an arbitrary maximal nonnegative S+ 2 (W). Then there exists a passive s/s system Σ with invariant subspace of the Kre˘ın space k+ future behavior Z satisfying one of the following sets of additional conditions: (1) Σ is observable and backward conservative. (2) Σ is controllable and forward conservative. (3) Σ is simple conservative s/s system. Each of the above three s/s systems are defined by Z up to unitary similarity. The notion of unitary similarity of s/s systems used above is defined in a natural way; see [3]. The idea behind the proof of Theorem 4.1 given in [3] is the following. First one chooses a fundamental decomposition W = −Y [] U of W, which induces the fundamental decom2 (W) = −2 (Y) [] 2 (U) of k 2 (W). A maximal nonnegative right-shift invariant position k+ + + + 2 (W) has the graph representation (3.1) with respect to this fundamental desubspace Z of k+ 2 (W), where the operator A is a contractive linear block Toeplitz operator composition of k+
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from 2+ (U) to 2+ (Y). The symbol of this operator is a B(U; Y)-valued Schur function (i.e., an analytic and contractive-valued function) D(z) in the unit disk. There exist three different de Branges–Rovnyak i/s/o (input/state/output) models with the same scattering matrix (characteristic function) equal to the given Schur function D(z). All of these three models are passive discrete-time invariant i/s/o scattering systems, with one of the following sets of additional properties: (1) the first one is observable and backward conservative, (2) the second one is controllable and forward conservative, and (3) the third one is simple and conservative. In operator theory one calls systems with the above properties “operator colligations” (nodes) that are (1) “co-isometric and closely outer connected,” or (2) “isometric and closely inner connected,” or (3) “unitary and closely connected”; see, e.g., [1, Chapter 2]. The state space of the observable and backward conservative de Branges–Rovnyak model is the de Branges–Rovnyak space H(A), where A is the contractive shift-invariant operator of multiplication by D(z), acting from the Hardy space H+2 (U) to the Hardy space H+2 (Y), and the main operator in this model is the incoming shift operator y(z) → [y(z) − y(0]/z. The three passive s/s systems constructed in the proofs of parts (1)–(3) of Theorem 4.1 are the unique passive s/s systems whose i/s/o representations corresponding to the fundamental decomposition W = −Y [] U are the time domain versions of the three de Branges–Rovnyak models (1)–(3) described above. By the time domain versions of these models we mean the models that one gets by mapping the Hardy spaces H+2 (U) and H+2 (Y) isometrically onto the sequence spaces 2+ (U) and 2+ (Y) by means of the inverse Fourier transform. In the time domain the inverse i/s/o problem becomes the problem of realizing a contractive right-shift invariant map from 2+ (U) to 2+ (Y) as the i/o (input/output) map of a scattering passive systems. The main disadvantage with the proofs outlined above is that they do not in each case produce just one single s/s realization but infinitely many, all of which are unitarily equivalent to each other. In all cases the realizations that we obtain depend on the fundamental decomposition W = −Y [] U that we start with. This is obvious from the fact that, for example, in case (1) the state space is a subspace of 2+ (Y), so different choices of Y result in different (albeit unitarily similar) s/s realizations. The results presented in Section 2 were obtained in our search for a one and only canonical (or coordinate free) s/s realization in each of cases (1)–(3). By “canonical” we mean that this realization should be uniquely determined by the given data, i.e., by the original maximal non2 (W) that we want to realize. In particular, it must not negative shift-invariant subspace Z of k+ depend on some arbitrary choice of a fundamental decomposition of the signal space W. Indeed, our search was successful, and in case (1) it led to the following result. Theorem 4.2. Let W be a Kre˘ın space, and let Z be a maximal nonnegative S+ -invariant sub2 (W). Let X space of k+ obc = H(Z), and let Vobc =
∗w+Z S+ w+Z w(0)
H(Z) ∈ H(Z) w ∈ X (Z) , W
2 (W). Then Σ where X (Z) is the space defined in (2.18) with X = k+ obc = (Vobc ; Xobc , W) is a passive observable backward conservative s/s system with future behavior Wfut = Z.
As a part of the proof of this theorem one shows thatVobc is well defined, i.e., that ∗ w + Z ∈ H(Z) whenever w ∈ X (Z). S+
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Analogous canonical shift realization models can also be obtained for cases (2) and (3) based on the results presented in Section 2. The proof of Theorem 4.2 and the corresponding passive s/s realizations of the types (2) and (3) will be given elsewhere. Remark 4.3. As we have seen above, our construction in Theorem 2.3 of the Hilbert space H(Z) (contained in the quotient of a Kre˘ın space X over the maximally nonnegative subspace Z of X ) is related to the corresponding construction in [10] of the Hilbert space H(A), where A is a contraction between two Hilbert spaces. That construction was extended by Louise de Branges in [8] to the case where A is a contraction between two Kre˘ın spaces, in which case the resulting space H(A) is a Kre˘ın space. The primary motivation for our interest in H(Z) was explained above: we need the space H(Z) in our construction of a canonical model of an observable and backward conservative passive state/signal system with a Hilbert state space. However, it seems plausible that there also exists a coordinate free version of the construction in [8] that would lead to canonical models of state/signal systems with a Kre˘ın state space. We leave this as an open question. Acknowledgments Damir Z. Arov thanks Åbo Akademi for its hospitality and the Academy of Finland and the Magnus Ehrnrooth Foundation for their financial support during his visits to Åbo in 2003–2008. References [1] Daniel Alpay, Aad Dijksma, James Rovnyak, Henrik de Snoo, Schur Functions, Operator Colligations, and Reproducing Kernel Hilbert Spaces, Operator Theory Adv. Appl., vol. 96, Birkhäuser, Basel, 1997. [2] Damir Z. Arov, Olof J. Staffans, State/signal linear time-invariant systems theory. Part I: Discrete time systems, in: The State Space Method, Generalizations and Applications, in: Operator Theory Adv. Appl., vol. 161, Birkhäuser, Basel, 2005, pp. 115–177. [3] Damir Z. Arov, Olof J. Staffans, State/signal linear time-invariant systems theory. Passive discrete time systems, Internat. J. Robust Nonlinear Control 17 (2007) 497–548. [4] Damir Z. Arov, Olof J. Staffans, State/signal linear time-invariant systems theory. Part III: Transmission and impedance representations of discrete time systems, in: Operator Theory, Structured Matrices, and Dilations, Tiberiu Constantinescu Memorial Volume, Bucharest, Romania, in: Theta Foundation, Amer. Math. Soc., Providence, RI, 2007, pp. 101–140. [5] Damir Z. Arov, Olof J. Staffans, State/signal linear time-invariant systems theory. Part IV: Affine representations of discrete time systems, Complex Anal. Oper. Theory 1 (2007) 457–521. [6] Tomas Ya. Azizov, Iosif S. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric, Wiley, New York, 1989. [7] János Bognár, Indefinite Inner Product Spaces, Ergeb. Math. Grenzgeb., vol. 78, Springer-Verlag, Berlin, 1974. [8] Louis de Branges, Complementation in Kre˘ın spaces, Trans. Amer. Math. Soc. 305 (1) (1988) 277–291. [9] Louis de Branges, James Rovnyak, Canonical models in quantum scattering theory, in: Perturbation Theory and Its Applications in Quantum Mechanics, Proc. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Inst., Univ. of Wisconsin, Madison, WI, 1965, Wiley, New York, 1966, pp. 295–392. [10] Louis de Branges, James Rovnyak, Square Summable Power Series, Holt, Rinehart & Winston, New York, 1966. [11] Rudolf E. Kalman, Peter L. Falb, Michael A. Arbib, Topics in Mathematical System Theory, McGraw–Hill, New York, 1969. [12] Donald Sarason, Sub-Hardy Hilbert Spaces in the Unit Disk, Univ. Arkansas Lecture Notes in Math. Sci., vol. 10, Wiley–Interscience, New York, 1994.
Journal of Functional Analysis 256 (2009) 3916–3976 www.elsevier.com/locate/jfa
Spectral controllability for 2D and 3D linear Schrödinger equations K. Beauchard a,1 , Y. Chitour b,∗,2 , D. Kateb c , R. Long d,3 a CMLA, ENS Cachan, CNRS, Universud, 61 avenue du Président Wilson, F-94230 Cachan, France b L2S, Université Paris-Sud XI, CNRS, Supélec, 3 Rue Joliot-Curie, 91192 Gif-sur-Yvette, France c Centre de Recherche de Royallieu, LMAC, 60020 Compiègne, France d CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
Received 23 September 2008; accepted 5 February 2009 Available online 10 March 2009 Communicated by J. Coron
Abstract We consider a quantum particle in an infinite square potential well of Rn , n = 2, 3, subjected to a control which is a uniform (in space) electric field. Under the dipolar moment approximation, the wave function solves a PDE of Schrödinger type. We study the spectral controllability in finite time of the linearized system around the ground state. We characterize one necessary condition for spectral controllability in finite time: (Kal) if Ω is the bottom of the well, then for every eigenvalue λ of −D Ω , the projections of the dipolar moment onto every (normalized) eigenvector associated to λ are linearly independent in Rn . In 3D, our main result states that spectral controllability in finite time never holds for one-directional dipolar moment. The proof uses classical results from trigonometric moment theory and properties about the set of zeros of entire functions. In 2D, we first prove the existence of a minimal time Tmin (Ω) > 0 for spectral controllability, i.e., if T > Tmin (Ω), one has spectral controllability in time T if condition (Kal) holds true for (Ω) and, if T < Tmin (Ω) and the dipolar moment is one-directional, then one does not have spectral controllability in time T . We next characterize a necessary and sufficient condition on the dipolar moment
* Corresponding author.
E-mail addresses: [email protected] (K. Beauchard), [email protected] (Y. Chitour), [email protected] (D. Kateb), [email protected] (R. Long). 1 The work of the first author has been supported ANR C-QUID. 2 Member of Digiteo http://www.digiteo.fr. The work was in part carried out while the second author was working as Marie Curie Fellow at the Department of Mathematics and Statistics, University of Kuopio, Finland, supported by the European Commission 6th framework program “Transfer of Knowledge” through the project “Parametrization in the Control of Dynamic Systems” (PARAMCOSYS, MTKD-CT-2004-509223). 3 Member of Digiteo http://www.digiteo.fr. 0022-1236/$ – see front matter © 2009 Published by Elsevier Inc. doi:10.1016/j.jfa.2009.02.009
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insuring that spectral controllability in time T > Tmin (Ω) holds generically with respect to the domain. The proof relies on shape differentiation and a careful study of Dirichlet-to-Neumann operators associated to certain Helmholtz equations. We also show that one can recover exact controllability in abstract spaces from this 2D spectral controllability, by adapting a classical variational argument from control theory. © 2009 Published by Elsevier Inc. Keywords: Schrödinger equation; Spectral controllability; Minimality of trigonometric families; Generic controllability; Shape differentiation; Helmholtz equation; Layer potentials
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of the control problem, notations and statement of the results . 2.1. Definition of the control problem . . . . . . . . . . . . . . . . . . . . . . 2.2. Previous 1D results, difficulties of the 2D and 3D generalizations 2.2.1. 1D controllability of (3) . . . . . . . . . . . . . . . . . . . . . . 2.2.2. 1D controllability of (4) . . . . . . . . . . . . . . . . . . . . . . 2.3. Statement of the main results . . . . . . . . . . . . . . . . . . . . . . . . . 3. Spectral controllability in 2D and 3D . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Haraux and Jaffard’s result . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Proof of Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Proof of Theorem 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 2D exact controllability in abstract spaces . . . . . . . . . . . . . . . . . . . . . 5. Generic spectral controllability for the quantum box . . . . . . . . . . . . . . 5.1. Reduction of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Proof strategy for the genericity of (Bk ) . . . . . . . . . . . . . . . . . . 5.3. Proof strategy for Proposition 5.4 . . . . . . . . . . . . . . . . . . . . . . 5.4. Evaluations of the singular parts of Mb (uq∗ ) and Md (uq∗ ) . . . . . 5.4.1. Expression of Mb (uq∗ ) . . . . . . . . . . . . . . . . . . . . . . . 5.4.2. Contribution of Md (uq∗ ) . . . . . . . . . . . . . . . . . . . . . . 5.5. Proof of Proposition 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusion, conjectures, perspectives . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Shape differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Main definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. Regularity of the eigenvalues and eigenfunctions . . . . . . . . . . . . A.3. Local variations of the eigenvalues and eigenfunctions . . . . . . . . Appendix B. The Dirichlet-to-Neumann map for the Helmholtz equation . . . B.1. Preliminary results on Helmholtz equation . . . . . . . . . . . . . . . . B.1.1. Fundamental solution . . . . . . . . . . . . . . . . . . . . . . . . B.1.2. Jump relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.3. Mapping properties in Sobolev spaces . . . . . . . . . . . . . B.2. Dirichlet-to-Neumann map . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.1. Singular perturbation problem and reduced resolvent . . . B.2.2. Normal derivative of the double-layer potential . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Let us consider a quantum particle in an infinite square potential well of Rn , n ∈ {1, 2, 3} subjected to a uniform (in space) time dependent electric field u : t → u(t) ∈ Rn . Let Ω be the domain of Rn corresponding to the bottom of the well. This physical system is modeled by a wave function ψ:
R+ × Ω → C, (t, q) → ψ(t, q),
such that |ψ(t, q)|2 dq represents the probability of the particle to be in the volume dq surrounding the point q at time t. Thus, the wave function ψ lives on the L2 (Ω, C)-sphere S as it is well known that the L2 (Ω, C)-norm of the wave function ψ is preserved over time. Under the dipolar moment approximation, this wave function solves the following Schrödinger equation ∂ψ (t, q) = −ψ(t, q) − u(t), μ(q) ψ(t, q), (t, q) ∈ R+ × Ω, i ∂t ψ(t, q) = 0, (t, q) ∈ R+ × ∂Ω,
(1)
where μ ∈ C 0 (Ω, Rn ) is the dipolar moment and . , . denotes the usual scalar product on Rn . The system (1) is a nonlinear control system in which • the state is the wave function ψ with ψ(t) ∈ S, for every t 0, • the control is the electric field u : t ∈ R+ → u(t) ∈ Rn . Studying controllability properties of the control system (1) reveals interesting features. For instance, Turinici proved in [45] that, the system (1) is not controllable in H 2 ∩ H01 (Ω, C) with controls u in Lrloc (R+ , Rn ), r ∈ (1, +∞). This result is a corollary of a more general result about the controllability of bilinear control systems, due to Ball, Marsden and Slemrod in [7]. However, it has been proved in [8] that the system (1) in 1D, with Ω = (−1/2, 1/2) and μ(q) = q is locally controllable around the ground state in H 7 ((−1/2, 1/2), C) with H01 ((0, T ), R) controls, when T is large enough. This system is even controllable between eigenstates, as proved in [9]. Therefore the noncontrollability result emphasized in [45] is essentially due to a choice of functional spaces that do not allow the controllability, but this controllability holds in other satisfying functional spaces. At the moment, in 2D or 3D, no positive exact controllability result is known for (1). We can also consider a similar nonlinear system. The quantum particle is now placed in a moving infinite square potential well of Rn , n ∈ {1, 2, 3}. Let Ω be the domain of Rn corresponding to the bottom of the well. It is proved by Rouchon in [38] that this physical system is represented by the following Schrödinger equation ⎧ ∂ψ ⎪ ⎪ i (t, q) = −ψ(t, q) − u(t), μ(q) ψ(t, q), ⎪ ⎪ ∂t ⎨ ψ(t, q) = 0, ⎪ ⎪ ˙ = s(t), ⎪ d(t) ⎪ ⎩ s˙ (t) = u(t),
(t, q) ∈ R+ × Ω, (t, q) ∈ R+ × ∂Ω,
(2)
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where ψ is the wave function of the particle in the moving frame, u := d¨ is the acceleration of the well, s is the speed of the well, d is the position of the well and μ(q) = q (but in this article, we will study this system for more general functions μ). The system (2) is a nonlinear control system with state, the triple (ψ, s, d) with ψ(t) ∈ S, for every t 0, and control, the acceleration of the well u : t ∈ R+ → Rn . In 1D, with Ω = (−1/2, 1/2), the local controllability around the eigenstates and the controllability between eigenstates of (2) is proved in [9]. A classical approach to prove the local controllability of nonlinear systems such as (1) and (2) around a reference trajectory consists in proving first, the controllability of the linearized system around the reference trajectory and second, the local controllability of the nonlinear system around the reference trajectory, with the help of an inverse mapping theorem. If the linearized system around the reference trajectory is not controllable, one may use the return method advocated by Coron (cf. [14,15] and references therein, and [8,9] for applications to 1D Schrödinger equations). This method relies on the study of another reference trajectory of the nonlinear system admitting a controllable linearized system. Therefore, it is natural to linearize (1) and (2) along “simple” trajectories, for instance, along the one corresponding to the zero control, u ≡ 0 and to study the controllability of the resulting linear system. For k ∈ N∗ , the eigenstate ψk (t, q) := φk (q)e−iλk t defines such a trajectory ((ψ = ψk , u ≡ 0) for (1) and (ψ = ψk , s ≡ 0, d ≡ 0, u ≡ 0) for (2)), where (φk )k∈N∗ is a complete orthonormal system of eigenfunctions for −D Ω , the Laplacian operator on Ω with Dirichlet boundary condition, and (λk )k∈N∗ are the corresponding nondecreasing sequence of eigenvalues counted with their multiplicity. In the particular case k = 1, ψ1 is called the ground state and the following systems are the linearized systems respectively of (1) around the ground state, ⎧ ⎨ ∂Ψ (t, q) = −Ψ (t, q) − v(t), μ(q) ψ1 (t, q), i ∂t ⎩ Ψ (t, q) = 0,
(t, q) ∈ R+ × Ω,
(3)
(t, q) ∈ R+ × ∂Ω,
and of (2) around the trajectory ((ψ = ψ1 , s ≡ 0, d ≡ 0), u ≡ 0), ⎧ ∂Ψ ⎪ (t, q) = −Ψ (t, q) − v(t), μ(q) ψ1 (t, q), ⎪i ⎪ ⎪ ⎨ ∂t Ψ (t, q) = 0, ⎪ ⎪ ˙ = s(t), ⎪ d(t) ⎪ ⎩ s˙ (t) = v(t).
(t, q) ∈ R+ × Ω, (t, q) ∈ R+ × ∂Ω,
(4)
In this paper, we only study controllability properties of systems (3) and (4). Let us recall classical results about the controllability of these two systems in 1D, results being the starting point of the strategies developed in [8] and [9] for the nonlinear systems (1) and (2). Their proof will be sketched in Section 2 in order to explain the difficulties arising in their generalization to the 2D and 3D cases. For system (3), Ω = (0, 1) and, if s is a nonnegative s ((0, 1), C) be equal to D(As/2 ) where D(A) := H 2 ∩ H 1 ((0, 1), C) and real number, let H(0) 0 Aϕ := −ϕ . Then, up to a condition satisfied by the dipolar moment μ (see Proposition 2.2 for 3 ((0, 1), C) with control functions in a detailed statement), the system (3) is controllable in H(0) L2 ((0, T ), R) for every T > 0. As regards controllability for system (4), we show that it is not exact controllable in finite time for the 1D problem and we describe the reachable set. The crucial technical reason for that lies in the fact that the eigenvalues of D Ω verify a uniform gap condition,
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i.e., there exists ρ > 0 such that, for every positive integer, we have λk+1 − λk ρ. However, in 2D, the existence of a regular domain Ω of R2 such that the eigenvalues of D Ω present a uniform gap is still an open problem and in 3D, no uniform gap is possible because of the Weyl formula. Therefore, exact controllability of (3) and (4) in 2D and 3D is not a trivial question and it is thus natural to study a weaker controllability property for this system. This is why we investigate, in this article, the spectral controllability of systems (3) and (4). To define that concept of controllability, let us denote D, the linear span of the eigenvectors φk , k ∈ N∗ , and TS ϕ, the tangent space to the sphere S at the point ϕ ∈ S. We say that system (3) is spectral controllable in time T if, for every Ψ0 ∈ D ∩ TS ψ1 (0), Ψf ∈ D ∩ TS ψ1 (T ), there exists v ∈ L2 ((0, T ), Rn ) such that the trajectory Ψ (·) of (3) starting at Ψ0 satisfies Ψ (T ) = Ψf . For system (4), that definition must be adapted as follows. Let . , .L2 denote the L2 (Ω, C)-scalar product. Then, system (4) is spectral controllable in time T if, for every Ψ0 ∈ D ∩ TS ψ1 (0), Ψf ∈ D ∩ TS ψ1 (T ) with Ψf , ψ1 (T ) = Ψ0 , ψ1 (0) and for every d0 ∈ Rn , there exists v ∈ L2 ((0, T ), Rn ) such that the trajectory (Ψ, s, d)(·) of (4) starting at (Ψ0 , 0, d0 ) satisfies (Ψ, s, d)(T ) = (Ψf , 0, 0). Our main results deal with the spectral controllability of (3) and (4). Before describing them, let us make a general remark. Since we are dealing with controls only depending on time, the control systems under consideration can be put into the general form x˙ = Ax + B(x)u where the state belongs to some C-valued functional space X, the control u is Rn -valued, the drift A is an (unbounded) linear operator admitting a complete orthonormal system of eigenfunctions and the controlled vector field B(·) has rank one. Using the classical moment theory, it is easy to characterize two necessary conditions for spectral controllability in some finite time T > 0. The first one corresponds to the Kalman condition for controllability in finite dimension. In our context, it means that (Kal) for every eigenvalue λ of A, the projections bkj := μ(q)φ1 , φkj , 1 j m(λ), of the controlled vector field B(·) on each eigenvector associated to λ are linearly independent in Rn . The above condition implies that the multiplicity of every eigenvalue λ of A is less than or equal to n. Note also that if A has simple spectrum (this will be referred as condition (Simp)), then condition (Kal) simply reads: the projections bk := μ(q)φ1 , φk of the controlled vector field B(·) on each (normalized) eigenvector is nonzero. We refer to the latter condition as (NonZ). The second condition is specific to the infinite dimension (for the state space) and it is related to the minimality of the family (e±i(λk −λ1 )t )k∈N in L2 ((0, T ), C) (see Definition 3.1). By applying a result of Haraux and Jaffard [19], we show that minimality never occurs in 3D for system (4) and also for system (3) if, in addition, the dipolar moment has a constant direction. In 2D, we show that minimality holds for both systems (3) and (4) if T is larger than a minimal time Tmin (Ω). In turn, if the dipolar moment has a constant direction, spectral controllability in time T > 0 for system (4) enables one to define a Hilbert subspace H of L2 (Ω, C) in which (4) is controllable, with L2 ((0, T ), R)-controls, when T > Tmin (Ω). In order to get spectral controllability in time T > Tmin (Ω), it therefore amounts, for a 2D domain Ω and a dipolar moment function μ, to check the validity of (Kal). Since the latter is difficult to verify for a given 2D domain Ω, we rather investigate conditions on the dipolar moment μ to insure that (Kal) holds true generically with respect to domains Ω with C 3 boundary. There is a trivial necessary condition on μ for (Kal) to hold true generically with respect to the domain: μ must be nowhere locally constant (NLC), i.e., its level sets are all of empty interior. (Indeed, simply consider a 2D domain where μ is constant. Then (Kal) does not hold, because
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of the L2 (Ω, C)-orthogonality of the eigenvectors φk .) One of our main results says that condition (NLC) for a C 1 dipolar moment μ is also sufficient to prove that condition (Kal) holds true, generically with respect to domains Ω with C 3 boundary. To do so, we start from the well-known fact that the spectrum of the Laplacian operator on a domain Ω ⊂ R2 with Dirichlet boundary conditions is generically simple. Therefore, it amounts to prove that condition (NLC) for a C 1 dipolar moment μ is also sufficient for condition (NonZ) to hold true, generically with respect to domains Ω with C 3 boundary. In summary, we can finally show that, in 2D, spectral controllability in finite time, for both systems (4) and (3) holds true, generically with respect to domains with C 3 boundary, if and only if the C 1 dipolar moment μ is nowhere locally constant. Before giving the plan of the paper, let us sketch the argument showing that (NLC) implies (NonZ), generically with respect to the domain. First of all, we must consider a topology for domains with C 3 boundary. Following [42], the latter is defined by taking as base of neighborhoods the sets V (Ω, ε) defined, for Ω any domain with C 3 boundary and ε > 0 small enough, as the images of Ω by Id2 + u, u ∈ W 4,∞ (Ω, R2 ) and uW 4,∞ < ε. We use D3 to denote the Banach space of domains with C 3 boundary equipped with the topology defined previously. A property is said to be generic in D3 if the subset of domains in D3 verifying that property is everywhere dense in D3 . We now fix a domain Ω with C 3 boundary and a C 1 dipolar moment μ verifying (NLC). Without loss of generality we assume that (Simp) is verified by Ω and we first reduce the argument to showing, for every positive integer k 2, the existence of a sequence (Ωn ) of domains with C 3 boundary converging to Ω such that (NonZ)k (i.e., bk = 0 along the sequence (Ωn )) holds true along the sequence. We proceed with a contradiction argument and we thus assume that there exists ε > 0 such that, for every u ∈ W 4,∞ (Ω, R2 ) with uW 4,∞ < ε, the corresponding bk is equal to zero. We compute the shape derivative of the relation bk = 0 at u ≡ 0 and we can express it as an integral along the boundary of Ω, i.e.,
u(q), ν(q) M(q) dσ (q) = 0,
∂Ω
where ν denotes the outer unit normal vector field and M(·) is a R2 -valued function defined on ∂Ω. As we will see below, in order to define M, one must introduce ξ1 and ξk , solutions of inhomogeneous Helmholtz equations (see (31) below). We at once deduce that M(·) ≡ 0 on ∂Ω. Reaching a contradiction in our argument amounts to show that the functions ξ1 , ξk introduced above actually do not exist. Unfortunately, we are not able to do that. By pushing further the contradiction argument, we compute the shape derivative of bk = 0 at every u ∈ W 4,∞ (Ω, R2 ) with uW 4,∞ < ε. That translates into the following relation: for ε > 0 small enough and for every u, v ∈ W 4,∞ (Ω, R2 ) with uW 4,∞ < ε and vW 4,∞ < ε, one has
v(q), ν(u)(q) M(u)(q) dσ (q) = 0,
∂(Id2 +u)Ω
where ν(u) denotes the outer unit normal vector field defined on ∂(Id2 + u)(Ω) and M(u)(·) is an R2 -valued function defined on ∂(Id2 + u)(Ω). The expression of M(u)(·) requires to define ξ1 (u), ξk (u), solutions of inhomogeneous Helmholtz equations. Of course, M(0), ξ1 (0) and ξk (0) are equal to M, ξ1 and ξk defined previously and we have that M(u)(·) ≡ 0 on ∂(Id2 + u)(Ω) for uW 4,∞ < ε.
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At this stage, we are again not able to derive a contradiction. So we again take the shape derivative of M(u)(·) ≡ 0 on ∂Ω and end up with the relation M (u)(q) = −(u.ν)(q)
∂M(0) (q), ∂ν
q ∈ ∂Ω,
(5)
for uW 4,∞ < ε. We now start a strategy first introduced in [11], which consists in defining M (u) for functions u defined on ∂Ω which are continuous except at some point q¯ of ∂Ω. For instance, we will take u = uq¯ as a Heaviside function H0 (q) ¯ admitting a single jump of discontinuity at an arbitrary point q ∈ ∂Ω. The key remark is the following: if (u.ν) belongs to the Sobolev space H s (∂Ω) for some s > 0 then, by standard elliptic theory arguments, M (u) belongs to H s−1 (∂Ω). In order to take advantage of the gap of regularity between the two sides of Eq. (5), we embark in the computation of the singular part of M (uq¯ )(·) at q¯ (in the distributional sense) and eventually come up with the following expression,
M (uq¯ )(σ ) = M0
1 + R(σ ), p.v. σ
where σ denotes the arc-length (with σ = 0 corresponding to q) ¯ and R(·) belongs to H 1/2−ε (∂Ω) for every ε > 0. Plugging back the above expression into Eq. (5), one must necessarily have M0 = 0. Recalling that q¯ ∈ ∂Ω is arbitrary, we end up with M0 (·) ≡ 0 on ∂Ω. In [11], the previous relation on M0 was providing additional information with respect to the relation M(u) ≡ 0 on ∂Ω, which allowed to conclude the contradiction argument. However, in the present situation, it turns out that M0 (·) is proportional to M(0)(·) and hence is trivially equal to zero. One must therefore compute the first nontrivial term in the “singular” expansion of M (uq¯ ) + (uq¯ .ν) ∂M(0) ∂ν at q, ¯ in the distributional sense. That procedure requires a detailed study of Dirichlet-to-Neumann operators associated to several Helmholtz equations. Once the nontrivial term is characterized, we easily conclude. The rest of the paper is organized as follows. In Section 2, we provide the main notations and precise definitions of the control systems (3) and (4), complete 1D results with their proofs and the statements of the main theorems of this article. Then, in Section 3, we give the proofs for the spectral controllability results in 2D and 3D. As for Section 4, the construction of some abstract spaces where we have 2D exact controllability is described. Section 5 contains the proof of the sufficiency of condition (NLC) to get generic controllability in 2D for the quantum box and Section 6 presents some conjectures. Finally, we gather in Appendix A the main results on shape differentiation used in the paper and Appendix B contains material on the Dirichlet-to-Neumann map for the Helmholtz equation with the proof of several technical lemmas which are needed in Section 5. 2. Definition of the control problem, notations and statement of the results 2.1. Definition of the control problem Let Ω be a domain of Rn (i.e., a bounded nonempty open subset of Rn ), n ∈ {1, 2, 3}, with a C 1 boundary. We use −D Ω to denote the Laplacian operator on Ω with Dirichlet boundary conditions, i.e.,
K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
2 1 D D Ω = H ∩ H0 (Ω, C),
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−D Ω φ = −φ.
The space L2 (Ω, C) has a complete orthonormal system (φk )k∈N∗ of eigenfunctions for −D Ω, φk ∈ H 2 ∩ H01 (Ω, C),
−D Ω φk = λk φk ,
where (λk )k∈N∗ is a nondecreasing sequence of positive real numbers. With this notation, the eigenvalues λk are counted as many times as their multiplicity. For t ∈ R and q ∈ Ω, we define the function ψ1 by ψ1 (t, q) := φ1 (q)e−iλ1 t . 0 2 We recall that −iD Ω generates a C -group of isometries of L (Ω, C) defined by
e−it ϕ :=
ϕ, φk e−iλk t φk ,
∀ϕ ∈ L2 (Ω, C).
k∈N∗
In this paper, we study controllability properties of the linear systems (3) and (4). In order to consider them as control systems, we first need a concept of trajectories associated to these systems. For that purpose, recall that the unit sphere S of L2 (Ω, C) is defined as follows, S := ϕ ∈ L2 (Ω, C); ϕL2 (Ω) = 1 , and, for ϕ ∈ S, the tangent space to the sphere S at the point ϕ is given by
2 TS ϕ := ξ ∈ L (Ω, C); ξ(q)ϕ(q) dq = 0 . Ω
Definition 2.1 (Weak solutions). Let T > 0, μ ∈ C 0 (Ω, R2 ), Ψ0 ∈ TS φ1 and v ∈ L1 ((0, T ), Rn ). A weak solution to the Cauchy problem ⎧ ∂Ψ ⎪ (t, q) = −Ψ (t, q) − v(t), μ(q) ψ1 (t, q), ⎨i ∂t Ψ ⎪ ⎩ (t, q) = 0, Ψ (0) = Ψ0 ,
(t, q) ∈ R+ × Ω, (t, q) ∈ R+ × ∂Ω,
(6)
is a function Ψ ∈ C 0 ([0, T ], L2 (Ω, C)) such that for every t ∈ [0, T ], t Ψ (t) = e
it
Ψ0 + i
ei(t−s) v(s), μ ψ1 (s) ds
0
Then (Ψ, v) is a trajectory of the control system (3) on [0, T ]. Let s0 , d0 ∈ Rn . A weak solution to the Cauchy problem
in L2 (Ω, C).
(7)
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⎧ ∂Ψ ⎪ ⎪ i (t, q) = −Ψ (t, q) − v(t), μ(q) ψ1 (t, q), ⎪ ⎪ ∂t ⎪ ⎪ ⎨ Ψ (t, q) = 0, Ψ (0) = Ψ0 , ⎪ ⎪ ⎪ ⎪ ⎪ s ⎪ ˙ (t) = v(t), ⎩ ˙ = s(t), d(t)
(t, q) ∈ R+ × Ω, (t, q) ∈ R+ × ∂Ω,
(8)
s(0) = s0 , d(0) = d0 ,
is a function (Ψ, s, d) with s ∈ W 1,1 ((0, T ), Rn ), d ∈ W 2,1 ((0, T ), Rn ) solutions of s˙ (t) = v(t) ˙ = s(t) d(t)
in L1 (0, T ), Rn ,
in L1 (0, T ), Rn ,
s(0) = s0 , d(0) = d0 ,
and Ψ ∈ C 0 ([0, T ], L2 (Ω, C)) such that for every t ∈ [0, T ], (7) holds. Then ((Ψ, s, d), v) is a trajectory of the control system (4) on [0, T ]. The following proposition recalls a classical existence and uniqueness result for the solutions of (6), from which one can deduce the similar result for (8). Theorem 2.1. For every T > 0, Ψ0 ∈ TS φ1 , v ∈ L1 ((0, T ), Rn ), there exists a unique weak solution to the Cauchy problem (6) and Ψ (t) ∈ TS ψ1 (t) for every t 0. Then, the system (3) is a control system where • the state is the function Ψ , with Ψ (t) ∈ TS ψ1 (t) for every t ∈ R+ , • the control is v : t ∈ R+ → v(t) ∈ Rn , L1loc (R+ , Rn ) is the set of admissible controls and the system (4) is a control system where • the state is the triple (Ψ, s, d), with Ψ (t) ∈ TS ψ1 (t) for every t ∈ R+ , • the control is v : t ∈ R+ → v(t) ∈ Rn and L1loc (R+ , Rn ) is the set of admissible controls. More precisely, in this paper, we investigate the following controllability property for (3). Definition 2.2 (Spectral controllability for (3)). The system (3) is spectral controllable in time T if, for every Ψ0 ∈ D ∩ TS ψ1 (0), Ψf ∈ D ∩ TS ψ1 (T ), there exists v ∈ L2 ((0, T ), Rn ) such that the solution of (6) satisfies Ψ (T ) = Ψf , where D := Span φk ; k ∈ N∗ . For the system (4), this definition needs to be adapted because of the presence of s and d in the state variable and because the directions Ψ (t), ψ1 (t) and s(t) are linked. Indeed, any solution of (8) satisfies n (j ) Ψ (t), ψ1 (t) = Ψ0 , ψ1 (0) + μ φ1 , φ1 s (j ) (t) − s (j ) (0) , j =1
∀t,
(9)
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where, for x ∈ Rn , x (j ) denotes its components, x = (x (1) , . . . , x (n) ) and . , . denotes the L2 (Ω, C)-scalar product. Therefore, we study the following controllability property for (4). Definition 2.3 (Spectral controllability for (4)). The system (4) is spectral controllable in time T if for every Ψ0 ∈ D ∩ TS ψ1 (0), Ψf ∈ D ∩ TS ψ1 (T ) with Ψf , ψ1 (T ) = Ψ0 , ψ1 (0), for every d0 ∈ Rn , there exists v ∈ L2 ((0, T ), Rn ) such that the solution of (8) with s0 = 0 satisfies (Ψ, s, d)(T ) = (Ψf , 0, 0). The notations Ω, n ∈ {1, 2, 3}, φk , ψ1 , . , ., S, TS , D, x = (x (1) , . . . , x (n) ) ∈ Rn introduced in this section are used all along this article. We also denote (ej )1j n the canonical basis of Rn and ωk := λk − λ1 , for every k ∈ N∗ . We use the same notation for the Rn -scalar product and the L2 (Ω)-scalar product but if a confusion is possible we precise the space in subscript . , .L2 (Ω) or . , .Rn . When some confusion is possible, we also precise the domain on the eigenvalues and Ω eigenfunctions of the Laplacian: λΩ k , φk . 2.2. Previous 1D results, difficulties of the 2D and 3D generalizations In this section, we recall classical results about the controllability of the systems (3) and (4) in 1D, that are the starting point of the strategies developed in [8] and [9] for the nonlinear systems (1) and (2). We also give their proof in order to explain the difficulties arising in their generalization to the 2D and 3D cases. We take Ω = (0, 1), so φk (q) =
√ 2 sin(kπq),
λk = (kπ)2
and we use the following notations
s (0, 1), C := D As/2 H(0)
where D(A) := H 2 ∩ H01 (0, 1), C , Aϕ := −ϕ .
2.2.1. 1D controllability of (3) For the control system (3), we have the following result. Proposition 2.2. Let Ω = (0, 1) and μ ∈ W 3,∞ ((0, 1), R). (1) We assume that ∃c1 , c2 > 0,
c2 c1 μφ1 , φk 3 , 3 k k
∀k ∈ N∗ .
(10)
3 ((0, 1), C) with control functions in Then, for every T > 0, the system (3) is controllable in H(0) 3 ((0, 1), C) with Ψ ∈ T ψ (0) and Ψ ∈ T ψ (T ), L2 ((0, T ), R): for every T > 0, Ψ0 , Ψf ∈ H(0) 0 S 1 f S 1 2 there exists v ∈ L ((0, T ), R) such that the solution of (6) satisfies Ψ (T ) = Ψf . (2) We assume that there exists m ∈ N∗ such that μφ1 , φm = 0 and
∃c1 , c2 > 0,
c2 c1 μφ1 , φk 3 , 3 k k
∀k ∈ N∗ such that μφ1 , φk = 0.
(11)
Then, the system (3) is not controllable: for every T > 0, Ψ0 ∈ L2 ((0, 1), C) and v ∈ L1 ((0, T ), R) the solution of (6) satisfies
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Ψ (T ), φk = Ψ0 , φk e−iλk T ,
∀k ∈ N∗ such that μφ1 , φk = 0.
3 ((0, 1), C) with Ψ ∈ But one can characterize the reachable set: for every T > 0, Ψ0 , Ψf ∈ H(0) 0 −iλ T ∗ TS ψ1 (0), Ψf ∈ TS ψ1 (T ), Ψf , φk = Ψ0 , φk e k for every k ∈ N such that μφ1 , φk = 0, there exists v ∈ L2 ((0, T ), R) such that the solution of (6) satisfies Ψ (T ) = Ψf .
Remark 2.1. Let us emphasize that the assumption (10) is generic with respect to μ ∈ W 3,∞ ((0, 1), R). Indeed, thanks to Baire’s Lemma, it is easy to prove that the property “μφ1 , φk = 0, ∀k ∈ N∗ ” holds generically with respect to μ ∈ W 3,∞ ((0, 1), R). Moreover, for such a function μ, integrations by parts lead to 1 μφ1 , φk = 2 0
4k[(−1)k+1 μ (1) − μ (0)] 1 μ(q) sin(πq) sin(kπq) dq = +o 3 . (k 2 − 1)2 k
Thus, the asymptotic behavior in 1/k 3 of these coefficients is equivalent to the property “μ (1) ± μ (0) = 0,” that is also generic in W 3,∞ ((0, 1), R). The key ingredient for the proof of Proposition 2.2 is the following theorem due to Kahane [26, Theorem III.6.1, p. 114]. Theorem 2.3. Let (μk )k∈N∗ ⊂ R such that μ1 = 0 and μk+1 − μk ρ > 0,
∀k ∈ N∗ .
(12)
Let T > 0 be such that lim
x→+∞
T N (x) < , x 2π
where, for x > 0, N(x) is the largest number of μk ’s contained in an interval of length x. Then, there exists C > 0 such that, for every c = (ck )k∈N∗ ∈ l 2 (N∗ , C) with c1 ∈ R, there exists w ∈ L2 ((0, T ), R) such that wL2 ((0,T ),R) Ccl 2 (N∗ ,C) and T w(t)eiμk t dt = ck ,
∀k ∈ N∗ .
0
Remark 2.2. The proof of Theorem 2.3 relies on an Ingham inequality for the family iμ t −iμ t 1, e k , e k ; k ∈ N∗ , k 2 , which corresponds to the Riesz basis property of this family in L2 ((0, T ), C). For the proof of Theorem 2.3, see, for example Krabs [30, Section 1.2.2], Komornik and Loreti [29, Chapter 9], or Avdonin and Ivanov [5, Chapter II, Section 4]. For the proof of similar results, we also refer to the prior works by Ingham [22], and to Beurling [10, pp. 341–365], Haraux [18], Redheffer [37],
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Russel [39, Section 3], Schwartz [40]. Improvements of Theorem 2.3 have been obtained by Jaffard, Tucsnak and Zuazua [24,25], Jaffard and Micu [23], Baiocchi, Komornik and Loreti [6], Komornik and Loreti [28], [29, Theorem 9.4, p. 177]. Proof of Proposition 2.2. We assume (10). Let T > 0 and Ψ0 ∈ TS ψ1 (0). By definition, the weak solution of (6) with some control v ∈ L2 ((0, T ), R) is Ψ (t, q) =
∞
xk (t)φk (q)
where xk (t)
k=1
t
= Ψ0 , φk + iμφ1 , φk
v(τ )e
iωk τ
dτ e−iλk t ,
∀k ∈ N∗ ,
0
with convergence in L2 ((0, 1), C) for every t ∈ [0, T ], where ωk := λk − λ1 , for every k ∈ N∗ . Since μφ1 , φk = 0, for every k ∈ N∗ , the equality Ψ (T ) = Ψf in L2 ((0, 1), C) is equivalent to the following trigonometric moment problem on the control v, T v(t)eiωk t dt = dk ,
∀k ∈ N∗ ,
(13)
0
where dk :=
Ψf , φk eiλk T − Ψ0 , φk , iμφ1 , φk
∀k ∈ N∗ .
(14)
Thanks to (10), the right-hand side (dk )k∈N∗ belongs to l 2 (N∗ , C) if and only if Ψf − e−iAT Ψ0 ∈ 3 ((0, 1), C), and in that case, (13) has a solution v ∈ L2 ((0, T ), R) for every T > 0, thanks to H(0) Theorem 2.3. The proof of the statement (2) is similar. 2 Now, let us discuss the generalization of Proposition 2.2 to the 2D and 3D cases. In 2D and 3D, the equality Ψ (T ) = Ψf for a solution of (6) is equivalent to
T
i μφ1 , φk L2 (Ω) ,
v(t)e 0
iωk t
= Ψf , φk eiλk T − Ψ0 , φk ,
dt
∀k ∈ N∗ .
(15)
Rn
Thus, the property μφ1 , φk = 0,
∀k ∈ N∗
is still a necessary condition for the controllability of (3). Let us assume that this property holds, then (15) is satisfied in particular when
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T v(t)eiωk t dt = −i 0
μφ1 , φk Ψf , φk eiλk T − Ψ0 , φk , 2 |μφ1 , φk |
∀k ∈ N∗ .
Thus, the controllability of (3) can be reduced to the solvability of n trigonometric moment problems on the real valued functions v (1) , . . . , v (n) . In 2D, the existence of a regular domain Ω of R2 such that the eigenvalues of D Ω present a uniform gap (which corresponds to the assumption (12)) is an open problem. For general 2D regular domains, we only have Weyl’s formula ∃c = c(Ω) > 0, ∃α = α(Ω) ∈ (0, 1),
k ∈ N∗ ; λk ∈ [0, t] = ct + O t α
when t → +∞.
This formula is not sufficient to ensure the existence of a uniform gap between the frequencies ωk . Therefore the classical result given in Theorem 2.3 cannot be applied: the controllability of (3) is a more difficult problem in 2D than in 1D. In 3D, with Weyl’s formula, ∃c = c(Ω) > 0, ∃α = α(Ω) ∈ (0, 3/2),
k ∈ N∗ ; λk ∈ [0, t] = ct 3/2 + O t α
when t → +∞, no uniform gap is possible. Thus, the noncontrollability of (3) is expected. The exact controllability of (3) in 2D and 3D being a difficult problem, it is natural to study a weaker controllability property for this system. This is why we investigate its spectral controllability in this article. Notice that the spectral controllability in time T of (3) is equivalent to the existence of a solution v ∈ L2 ((0, T ), Rn ) of (15) for any right-hand side with finite support. This remark will be used in the study of the spectral controllability of (3) (see Section 3.2). 2.2.2. 1D controllability of (4) For the control system (4), we have the following result. Proposition 2.4. Let Ω = (0, 1) and μ ∈ W 3,∞ ((0, 1), R). (1) The system (4) is not controllable: for every Ψ0 ∈ TS ψ1 (0), s0 , d0 ∈ R, v ∈ L1loc (R+ , R), the solution of (8) satisfies (9). (2) If (11) holds, then, one can characterize the reachable set for (4): for every T > 0, Ψ0 , Ψf ∈ 3 ((0, 1), C), s , s , d , d ∈ R with Ψ , ψ (T ) = Ψ , ψ (0) + iμφ , φ (s − s ) H(0) 0 f 0 f f 1 0 1 1 1 f 0 and
Ψ (T ), φk = Ψ0 , φk e−iλk T ,
∀k 2 such that μφ1 , φk = 0,
(16)
there exists v ∈ L2 ((0, T ), R) such that the solution of (8) satisfies (Ψ, s, d)(T ) = (Ψf , sf , df ). The proof of this proposition is similar to the one of Proposition 2.2. Notice that, in 2D and 3D, the equality (Ψ, s, d)(T ) = (Ψf , 0, 0) for the solution of (8) with s0 = 0, Ψ0 ∈ TS ψ1 (0), Ψf ∈ TS ψ1 (T ) such that Ψ0 , ψ1 (0) = Ψf , ψ1 (T ) is equivalent to
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⎧ T ⎪ ⎪ ⎪ iωk t ⎪ i μφ1 , φk L2 (Ω) , v(t)e dt = Ψf , φk eiλk T − Ψ0 , φk , ⎪ ⎪ ⎪ ⎪ Rn ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ T v(t) dt = 0, ⎪ ⎪ ⎪0 ⎪ ⎪ ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ ⎪ ⎪ tv(t) dt = d0 . ⎪ ⎩
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∀k 2,
(17)
0
Thus, the spectral controllability in time T of (4) is equivalent to the existence of a solution v ∈ L2 ((0, T ), Rn ) of (17), for any right-hand side with finite support. This remark will be used in the study of the spectral controllability of (4) (see Section 3.3). 2.3. Statement of the main results In order to state our results, we first give several definitions relative to the domain and the dipolar moment. Definition 2.4 (Kalman condition (Kal)). Let Ω be a domain of Rn , n = 2, 3 with C 1 boundary. Then Ω verifies property (Kal) if (Kal) any eigenvalue λ of −D Ω has a multiplicity m n and the vectors μφ1 , φk1 , . . . , μφ1 , φkm are linearly independent in Rn , where k1 < · · · < km and φk1 , . . . , φkm are the eigenvectors associated to λ. Definition 2.5 (Simplicity of the spectrum (Simp)). Let Ω be a domain of Rn , n = 2, 3 with C 1 boundary. Then Ω verifies property (Simp) if (Simp)
the eigenvalues of −D Ω are simple.
Definition 2.6 (Nonzero projection (NonZ)). Consider μ ∈ C 0 (Ω, Rn ), n = 2, 3 and (φk )k∈N∗ the complete orthonormal system of eigenvectors of −D Ω . Then μφ1 has a nonzero projection on (φk )k∈N∗ if, for every integer k 2, we have (NonZ)k
μφ1 , φk = 0.
In that case, we say that μ verifies property (NonZ). Remark that if a domain Ω satisfies (Simp), then condition (Kal) reduces to condition (NonZ). The next theorem gathers our result regarding the spectral controllability properties for system (3). Theorem 2.5. (1) Let Ω be a domain of R2 with C 1 boundary and μ ∈ C 0 (Ω, R2 ) verifying (Kal). Then, there exists Tmin = Tmin (Ω) > 0 such that
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(1.a) for every T > Tmin , system (3) is spectral controllable in time T ; (1.b) for every T < Tmin , system (3) is not spectral controllable in time T , under the additional assumption μ(x) = μ(x)e ˜ 1
where μ˜ ∈ C 0 (Ω, R).
(18)
(2) Let Ω be a domain of Rn , n = 2, 3, with C 1 boundary and μ ∈ C 0 (Ω, Rn ) such that (Kal) is not verified. Then, system (3) is not spectral controllable. (3) Let Ω be a domain of R3 with C 1 boundary and μ ∈ C 0 (Ω, R3 ) of the form (18). Then, system (3) is not spectral controllable. Remark 2.3. Let us emphasize that (Kal) holds true generically with respect to the pair (Ω, μ) because conditions (Simp) and (NonZ) hold true simultaneously generically with respect to the pair (Ω, μ), where Ω is a domain of R2 with C 1 boundary and μ ∈ C 0 (Ω, R2 ). Indeed, the genericity of (Simp) with respect to the domain Ω is a classical result (see, for instance, [21]). Moreover, for a domain Ω of R2 with C 1 boundary verifying (Simp), the set
μ ∈ C 0 Ω, R2 ; μφ1 , φk = 0, ∀k ∈ N∗ is dense in C 0 (Ω, R2 ) (it can be proved thanks to Baire’s Lemma). As for system (4), we prove the following result. Theorem 2.6. (1) Let Ω be a domain of R2 with C 1 boundary and μ ∈ C 0 (Ω, R2 ) verifying (Kal). Let Tmin = Tmin (Ω) be as in Theorem 2.5. Then, system (4) is spectral controllable in time T > Tmin . (2) Let Ω be a domain of Rn , n = 2, 3, with C 1 boundary and μ ∈ C 0 (Ω, Rn ) such that (Kal) is not verified. Then, system (4) is not spectral controllable. (3) Let Ω be a domain of R3 with C 1 boundary and μ ∈ C 0 (Ω, R3 ). Then system (4) is not spectral controllable: for every T > 0 and m ∈ N∗ , there exists d0 ∈ R3 such that (iφm , 0, d0 ) is not zero controllable in time T . Remark 2.4. Notice that in item (3) of Theorem 2.6, the dipolar moment μ is not necessarily one-dimensional. Thus, we prove a stronger noncontrollability result for this 3D system, than the one given in Theorem 2.5(3). This improvement is due to the presence of s and d in the state variable. The proofs of Theorems 2.5 and 2.6 are given in Section 3. In Section 4, we prove that, one can recover the exact controllability, in some abstract spaces, for the system (3) in 2D with μ of the form (18) thanks to the previous spectral controllability result. Such abstract spaces may be used for the study of the nonlinear system. This is an open problem. According to Theorem 2.6, one knows that, in 2D, property (Kal) is a necessary and sufficient condition for the spectral controllability of (3) and (4) in time T > Tmin (Ω). We next use that characterization to prove that spectral controllability of (3) and (4) in large time holds true generically with respect to the 2D domain Ω. For that purpose, let us first precise the topology
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on domains we are using, then define genericity and finally state the condition on the dipolar moment μ that ensures the genericity. For l 1, the set Dl of domains Ω of R2 with C l boundary. Following [42], we define next a topology on Dl . Consider the Banach space W l+1,∞ (Ω, R2 ) equipped with its standard norm. For Ω ∈ Dl , u ∈ W l+1,∞ (Ω, R2 ), let Ω + u := (Id+ u)(Ω) be the subset of points y ∈ R2 such that y = x + u(x) for some x ∈ Ω and ∂Ω + u := (Id+ u)(∂Ω) its boundary. For ε > 0, let V (Ω, ) be the set of all Ω + u with u ∈ W l+1,∞ (Ω, R2 ) and uW l+1,∞ ε. The topology of Dl is defined by taking the sets V (Ω, ε) with ε small enough as a base of neighborhoods of Ω. Then, Dl is a Banach space. Definition 2.7. We say that a property (P ) is generic in Dl if the set of domains of Dl on which this property holds true is dense in Dl : for every Ω ∈ Dl , there exists ρ > 0 such that the set {u ∈ Eρ (Ω); Ω + u satisfies (P )} is dense in Eρ (Ω), where Eρ (Ω) := {u ∈ W l+1,∞ (Ω, R2 ); uW l+1,∞ < ρ}. Definition 2.8 (Nonlocally constant (NLC)). A map μ ∈ C 0 (R2 , R2 ) is said to be nowhere locally constant if, for every μ0 ∈ R2 , the level set {q ∈ R2 μ(q) = μ0 } has an empty interior. Note that if μ is (NLC) and continuously differentiable, then the subset of Rn , n = 2, 3, where the differential of μ is not zero, must be open and dense. We now state one of the main results of the paper. Theorem 2.7. Let μ ∈ C 1 (R2 , R2 ). The spectral controllability in large time for system (4) is generic in D3 if and only if μ is nowhere locally constant. According to item (2) of Theorem 2.6, the proof of the previous theorem reduces to establishing the next proposition, since (Simp) and (NonZ) both verified imply that (Kal) holds true. Proposition 2.8. Let μ ∈ C 1 (R2 , R2 ). If Ω ∈ D1 , we say that Ω has property (A) if (Simp) and (NonZ) hold true for Ω. Then, property (A) is generic in D3 if and only if μ is nowhere locally constant. Section 5 is devoted to the proof of the above proposition. 3. Spectral controllability in 2D and 3D The goal of this Section is the proof of Theorems 2.5 and 2.6. This section is organized as follows. In Section 3.1, we state a sufficient condition for the minimality in L2 ((0, T ), C) of a family of complex exponentials. This condition, due to Haraux and Jaffard [19], involves Weyl’s formula. In Section 3.2, we prove Theorem 2.5, thanks to Haraux and Jaffard’s result. In Section 3.3, we prove Theorem 2.6. The proofs of the two first statements also rely on Haraux and Jaffard’s result. The proof of the third statement involves different ideas, about the set of zeros of holomorphic functions. 3.1. Haraux and Jaffard’s result First, let us recall the definition of the minimality of a family of vectors.
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Definition 3.1. Let X be a Banach space over K = R or C. A family (zk )k∈Z of vectors of X is minimal in X if, for every m ∈ Z, zm does not belong to the closure in X of the vector space generated by {zk ; k ∈ Z − {m}}, zm ∈ / ClX Span zk ; k ∈ Z − {m} ,
∀m ∈ Z.
When X is an Hilbert space, we have the following classical equivalent definitions. Proposition 3.1. Let (X, . , .X ) be a Hilbert space and (zk )k∈Z be a family of vectors of X. The following statements are equivalent. (1) (zk )k∈Z is minimal in X. (2) For every m ∈ Z, there exists Cm > 0 such that, for every f ∈ X of the form f = k∈K fk zk where K ⊂ Z is finite, Cm |fm | f X . (3) There exists a family (Zk )k∈Z of vectors of X bi-orthogonal to (zk )k∈Z , i.e., zm , Zk X = δm,k ,
∀k, m ∈ Z.
(4) For every (dk )k∈Z ⊂ K with finite support, there exists v ∈ X solution of the moment problem v, zk X = dk ,
∀k ∈ Z.
(19)
Proof of Proposition 3.1. For (1) ⇒ (2), the largest value for the constant Cm is Cm := dist zm , Span zk ; k ∈ Z − {m} . The implications (2) ⇒ (1), no (1) ⇒ no (3) and (3) ⇔ (4) are easy. For (1) ⇒ (3), one can take
Zm := Pm
zm zm 2X
where Pm is the orthogonal projection from X to Vm⊥ , the orthogonal supplementary of Vm := ClX (Span{zk ; k ∈ Z − {m}}) in X, which is a closed vector subspace of X. 2 Remark 3.1. The statement (4) is particularly important in this article. Indeed, as seen in Section 2.2, the spectral controllability in time T of (3) is equivalent to the solvability of a moment problem of the form (19) with X = L2 ((0, T ), Rn ), z0 := μφ1 , φ1 , zk := μφ1 , φk+1 cos(ωk+1 t), z−k := μφ1 , φk+1 sin(ωk+1 t), ∀k ∈ N∗ . Thus, the spectral controllability in time T of (3) is equivalent to the minimality of the family (zk )k∈Z in L2 ((0, T ), Rn ). The following theorem is the key point of Section 3. It has been proved by Haraux and Jaffard in [19, Corollary 2.3.5], as a consequence of the Beurling–Malliavin theorem, thanks to the computation of the Beurling–Malliavin density of a sequence that satisfies Weyl’s formula.
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Theorem 3.2. Let (μk )k∈Z be a sequence of real numbers such that {k ∈ Z; 0 μk t} = dt + O t α ,
{k ∈ Z; −t μk 0} = dt + O t α ,
for some d 0 and α ∈ (0, 1). Then, (1) for every T > 2πd, the family {eiμk t ; k ∈ Z} is minimal in L2 ((0, T ), C), (2) for every T < 2πd, the family {eiμk t ; k ∈ Z} is not minimal in L2 ((0, T ), C). Remark 3.2. Notice that, when μ0 = 0 and μk = −μ−k > 0, for every k ∈ N∗ , then the minimality of the family {eiμk t ; k ∈ Z} in L2 ((0, T ), C) is equivalent to the minimality of the family {1, cos(μk t), sin(μk t); k 0} in L2 ((0, T ), R). 3.2. Proof of Theorem 2.5 The goal of this section is the proof of Theorem 2.5 thanks to Theorem 3.2. Proof of Theorem 2.5. (1) Let Ω be a domain of R2 with C 1 boundary and μ ∈ C 0 (Ω, R2 ) be such that (Kal) holds. Thanks to Weyl’s formula, there exists d = d(Ω) ∈ (0, +∞) and α = α(Ω) ∈ (0, 1) such that k ∈ N∗ ; ωk ∈ [0, t] = dt + O t α when t → +∞.
(20)
Let Tmin = Tmin (Ω) := 2πd. (1.a) Let T > Tmin , Ψ0 ∈ D ∩ TS ψ1 (0), Ψf ∈ D ∩ TS ψ1 (T ) and let us prove that there exists v ∈ L2 ((0, T ), R2 ) solution of (15). We introduce Λ1 := k ∈ N∗ ; λk is a simple eigenvalue of D Ω ,
Λ2 := k ∈ N∗ ; λk = λk+1 .
For every k ∈ Λ2 , the vectors μφ1 , φk and μφ1 , φk+1 are linearly independent in R2 , thus there exists a unique Zk ∈ C2 such that
μφ1 , φk L2 (Ω) , Zk
R2
= −idk ,
μφ1 , φk+1 L2 (Ω) , Zk
R2
= −idk+1 ,
where dj := Ψf , φj eiλj T − Ψ0 , φj , for every j ∈ N∗ . For a function v ∈ L2 ((0, T ), R2 ), (15) is satisfied in particular when T v(t)eiωk t dt = −idk 0
μφ1 , φk , |μφ1 , φk |2
∀k ∈ Λ1 ,
T v(t)eiωk t dt = Zk ,
∀k ∈ Λ2 ,
(21)
0
i.e., when v (1) and v (2) solve a trigonometric moment problem with a finite supported right-hand side. The solvability of (21) is equivalent to the minimality of the family
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1, cos(ωk t), sin(ωk t); k 2 in L2 ((0, T ), R) (see Proposition 3.1), which holds true thanks to Theorem 3.2. For the proof of (1.b) and (3), let us first emphasize that, when (18) and (Kal) hold, then the spectral controllability in time T of (3) is equivalent to (and not only implied by) the minimality of the family {1, cos(ωk t), sin(ωk t); k 2} in L2 ((0, T ), R). (1.b) Let T < Tmin and let us assume (18). Theorem 3.2 ensures that the family {1, cos(ωk t), sin(ωk t); k 2} is not minimal in L2 ((0, T ), R), thus (4) is not spectral controllable in time T . (2) Let Ω be a domain of Rn with C 1 boundary, n = 2, 3, and μ ∈ C 0 (Ω, Rn ). We assume that (Kal) does not hold. There exists k ∈ N∗ such that λk has multiplicity m and there exists (α1 , . . . , αm ) ∈ Rm − {0} such that α1 μφ1 , φk1 + · · · + αm μφ1 , φkm = 0, where k1 , . . . , km are all the integers such that λk = λk1 = · · · = λkm . Let Ψ0 ∈ D ∩ TS ψ1 (T ) of the form Ψ0 = β1 φk1 + · · · + βm φkm where β1 , . . . , βm ∈ C and α1 β1 + · · · + αm βm = 0. Any solution of (6) satisfies, for j ∈ {1, . . . , m},
T
Ψ (T ), φkj = Ψ0 , φkj + i μφ1 , φkj ,
e−iλk T .
v(t)eiωk t dt Rn
0
We then have α1 Ψ (T ), φk1 + · · · + αm Ψ (T ), φkm = (α1 β1 + · · · + αm βm )e−iλk T = 0, implying that Ψ0 is not zero controllable in time T . (3) Let Ω be a domain of R3 with C 1 boundary and μ ∈ C 0 (Ω, R3 ) of the form (18) be such that (Kal) holds true (otherwise, we already know that (3) is not spectral controllable thanks to (2)). Let T > 0. Thanks to Weyl’s formula, we have ωk ∈ [0, t] = dt 3/2 + O t α ,
when t → +∞,
where d ∈ (0, +∞) and α ∈ (0, 3/2). Thus, there exists a subsequence (ωσ (k) )k∈N∗ of (ωk )k∈N∗ such that k ∈ N∗ ; ωσ (k) ∈ [0, t] = d t + O t α when t → +∞, for some d > T /2π and some α ∈ (0, 1). Theorem 3.2 ensures that the family iω t −iω t e σ (k) , e σ (k) ; k ∈ N∗ is not minimal in L2 ((0, T ), C). Thus, the family {1, eiωk t , e−iωk t ; k 2} is not minimal in L2 ((0, T ), C). Therefore, (3) is not spectral controllable. 2 Remark 3.3. When a domain Ω of R2 with C 1 boundary and μ ∈ C 0 (Ω, R2 ) are such that (Kal) holds but (18) does not hold, then Tmin (Ω) := 2πd(Ω) may not be the minimal time for the spectral controllability of (3). Indeed, let us consider μ = (μ(1) , μ(2) ) such that
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μ(1) φ1 , φk = 0 if and only if k ∈ N∗ is odd and (2) μ φ1 , φk = 0 if and only if k ∈ N∗ is even.
Then, the minimal time for the spectral controllability of (3) is Tmin (Ω, μ) = πd(Ω). Remark 3.4. In order to remove the assumption (18), one could try to adapt Haraux and Jaffard’s result to families of vector exponentials of the form iω t bk e k ; k ∈ Z where bk ∈ Rn − {0}. Indeed, the spectral controllability of (3) is equivalent to the minimality in L2 ((0, T ), Cn ) of this family with bk = μφ1 , φk . This generalization is an open problem. 3.3. Proof of Theorem 2.6 The goal of this section is the proof of Theorem 2.6. The proof of the statement (1) can be deduced from the following lemma in the same way as the proof of Theorem 2.5(1.a) was deduced from Theorem 3.2(1). Lemma 3.3. Let (μk )k∈Z be a sequence of real numbers such that μ0 = 0 and {k ∈ Z; 0 μk t} = dt + O t α , {k ∈ Z; −t μk 0} = dt + O t α , for some d > 0 and α ∈ (0, 1). Then, for every T > 2πd, the family {t, eiμk t ; k ∈ Z} is minimal in L2 ((0, T ), C). Proof of Lemma 3.3. Let T > 2πd and let us assume that the family {t, eiμk t ; k ∈ Z} is not minimal in L2 ((0, T ), C). Thanks to Theorem 3.2, the family {eiμk t ; k ∈ Z} is minimal in L2 ((0, T ), C) thus, necessarily, t ∈ ClL2 ((0,T ),C) Span eiμk t ; k ∈ Z .
(22)
With successive integrations, we see that t k ∈ ClC 0 ([0,T ],C) Span t, eiμk t ; k ∈ Z ,
∀k ∈ N, k 2.
The Stone Weirstrass theorem ensures that Span{1, t k ; k ∈ N, k 2} is dense in C 0 ([0, T ], C), thus it is also dense in L2 ((0, T ), C). From (22), we deduce that the vector space Span{eiμk t ; k ∈ Z} is dense in L2 ((0, T ), C). This is a contradiction, because, thanks to Theorem 3.2, for every α ∈ R − {μk ; k ∈ Z}, the family {eiαt , eiμk t ; k ∈ Z} is minimal in L2 ((0, T ), C) i.e., / ClL2 ((0,T ),C) Span eiμk t ; k ∈ Z . eiαt ∈
2
Item (2) of Theorem 2.6 is a direct consequence of Theorem 2.5(2). The proof of the statement (3) of Theorem 2.6 involves different ideas. A useful preliminary result is stated in the next
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lemma (see [31, Lecture 2, Section 2.3, pp. 10–11]) that has already been used in [11, Lemma 16] for similar purposes. Lemma 3.4. Let f : C → C be a holomorphic function such that f (s) C0 eC0 |s| .
∃C0 > 0, such that ∀s ∈ C,
Assume that f = 0. Let n : [0, +∞) → N be defined by n(R) := s ∈ C; f (s) = 0 and |s| R . Then, R ∃C1 > 0, ∀R ∈ (1, +∞),
n(t) dt C1 R. t
1
Proof of item (3) of Theorem 2.6. Let Ω be a regular domain of R3 such that (Kal) holds (otherwise, the system (4) is already known to be nonspectral controllable thanks to (2)). Let T > 0 and m ∈ N∗ . We assume (iφm , 0, el ) is zero controllable in time T for l = 1, 2, 3: there exists vl ∈ L2 ((0, T ), R3 ) such that ⎧ T ⎪ ⎪ ⎪ iω t k ⎪ μφ1 , φk L2 (Ω) , vl (t)e dt = −δk,m , ⎪ ⎪ ⎪ ⎪ 3 R ⎪ 0 ⎪ ⎪ ⎪ T ⎪ ⎪ ⎨ vl (t) dt = 0, ⎪ ⎪ ⎪ ⎪0 ⎪ ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ ⎪ tvl (t) dt = el , ⎪ ⎪ ⎩
∀k 2,
(23)
0
for l = 1, 2, 3. In particular, for every k ∈ N∗ − {1, m}, the vector μφ1 , φk L2 (Ω) ∈ R3 − {0} belongs to the kernel of the matrix C(iωk ), where ⎛ T
0 ⎜ T C(λ) := ⎝ 0 T 0
(1)
v1 (t)eλt dt v2(1) (t)eλt dt (1)
v3 (t)eλt dt
T 0 T 0 T 0
(2)
v1 (t)eλt dt v2(2) (t)eλt dt (2)
v3 (t)eλt dt
T 0 T 0 T 0
(3)
v1 (t)eλt dt
⎞
⎟ v2(3) (t)eλt dt ⎠ . (3)
v3 (t)eλt dt
Thus G(λ) := det[C(λ)] satisfies G(iωk ) = 0, for every k ∈ N∗ − {1, m}. It is easy to see that G is a holomorphic function verifying the growth condition of Lemma 3.4. Then using Weyl’s formula and Lemma 3.4, we deduce that G ≡ 0. However, thanks to the last two equalities in (23), we have
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⎛
λ + o(λ) ⎝ C(λ) = o(λ) o(λ)
⎞ o(λ) o(λ) λ + o(λ) o(λ) ⎠ o(λ) λ + o(λ)
3937
when λ → 0,
so G(λ) = λ3 + o(λ3 ) = 0 when λ → 0, which is a contradiction.
2
4. 2D exact controllability in abstract spaces The goal of this section is the proof of the following result. Theorem 4.1. Let Ω be a domain of R2 with C 1 boundary and μ ∈ C 0 (Ω, R2 ) be of the form (18) such that condition (Kal) holds true. Let d ∈ (0, +∞) and α ∈ (0, 1) be such that (20) holds, T > 2πd and (xm )m∈N∗ ⊂ R∗+ be such that ∞ m=1 xm = 1. ∗ For every m ∈ N , there exists Cm > 0 such that, for every ϕT ∈ TS ψ1 (T ), the solution of ⎧ ∂ϕ ⎪ ⎪ = −ϕ, (t, q) ∈ R+ × Ω, ⎨i ∂t ϕ(t, q) = 0, (t, q) ∈ R × ∂Ω, ⎪ ⎪ ⎩ ϕ(T ) = ϕT ,
(24)
satisfies 2 Cm ϕT , φm
T
2 μψ ˜ 1 (t), ϕ(t) dt,
∀m ∈ N∗ .
(25)
0
We introduce the Hilbert spaces
" ∞ 2 H := ϕ: Ω → C; ϕ, ψ1 (T ) = 0 and Cm xm ϕ, φm < +∞ , ∗
m=1
∞ H := ϕ : Ω → C; ϕ, ψ1 (T ) = 0 and m=1
" 2 1 ϕ, φm < +∞ . Cm x m
Then, for every Ψf ∈ H , there exists v˜ ∈ L2 ((0, T ), R) such that the solution of (6) with Ψ0 = 0 and v(t) = v(t)e ˜ 1 satisfies Ψ (T ) = Ψf . Remark 4.1. Notice that TS ψ1 (T ) ⊂ H ∗ and H ⊂ TS ψ1 (T ) because Cm xm → 0 when m → +∞. The space H is a regular space, its regularity depends on the asymptotic behavior of the sequence (Cm xm )m∈N∗ . Remark 4.2. The spaces H and H ∗ are defined in order to have an observability inequality in H ∗ . Indeed, considering the product of the inequality (25) with xm and summing over m ∈ N∗ , we get
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T ϕT 2H ∗
2 μψ ˜ 1 (t), ϕ(t) dt,
∀ϕT ∈ H ∗ .
(26)
0
Remark 4.3. Trying to apply the classical approach in order to get the controllability thanks to (26), we introduce the functional J:
H ∗ → R, 1 ϕT → 2
T
2 μψ ˜ 1 (t), ϕ(t) dt + ϕT , Ψf .
0
In the classical situation, J is continuous, convex and coercive on H ∗ , thus inf{J (ϕT ); ϕT ∈ H ∗ } ˜ := μψ ˜ 1 (t), ϕ(t) that is achieved at some point ϕT . Writing dJ (ϕT ) = 0, we get a control v(t) steers (3) from Ψ (0) = 0 to Ψ (T ) = Ψf . In our situation, this classical approach does not work because the functional J may not be well defined on H ∗ . Thus, an adaptation of this approach is needed. Proof of Theorem 4.1. First, let us prove (25). For ϕT ∈ TS ψ1 (T ), the solution of (24) is ϕ(t) =
∞ ϕT , φk e−iλk (t−T ) φk k=1
so ∞
μφ ˜ 1 , φk ϕT , φk eiλk (t−T ) − ϕT , φk e−iλk (t−T ) . μψ ˜ 1 (t), ϕ(t) = 2i k=2
Applying Theorem 3.2 and Proposition 3.1, there exists a constant C˜ m > 0 such that, for every ϕT ∈ TS ψ1 (T ), 2 2 ˜ 1 , φm ϕT , φm C˜ m μφ
T
2 μψ ˜ 1 (t), ϕ(t) dt.
0
We get (25) with Cm := C˜ m |μφ ˜ 1 , φm |2 . Now, let us prove the controllability result. Let Ψf ∈ H . For > 0 we introduce the functional J : TS ψ1 (T ) → R, 1 J (ϕT ) := 2
T
2 μψ ˜ 1 (t), ϕ(t) dt + Ψf , ϕT + ϕT 2 2
L (Ω)
,
0
where ϕ is the solution of (24). The functional J is convex, continuous and coercive because J (ϕT ) ϕT 2L2 − Ψf L2 ϕT L2 .
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Thus, there exists ϕT ∈ TS ψ1 (T ) such that J ϕT = min J (ϕT ); ϕT ∈ TS ψ1 (T ) . Then, ϕT solves the Euler equation associated to this optimization problem, T
v˜ (t) μψ ˜ 1 (t), ξ(t) dt + Ψf , ξT + 2 ϕT , ξT = 0,
∀ξT ∈ TS ψ1 (T ),
(27)
0
where v˜ (t) := μψ ˜ 1 (t), ϕ (t) , ϕ (resp. ξ ) is the solution of (24) with ϕT = ϕT (resp. ϕT = ξT ). For 0 < 1 < 2 , we have J1 J2 thus the sequence (J (ϕT ))>0 decreases when decreases to zero. Thus, J ϕT M1 := J1 ϕT1 ,
∀ ∈ (0, 1).
There exists M2 > 0 such that, # # #ϕ #
T H∗
M2 ,
∀ ∈ (0, 1).
Indeed, thanks to (26), we have, # # 1 # #2 M1 J ϕT #ϕT #H ∗ − Ψf H #ϕT #H ∗ . 2 The sequence (v˜ )∈(0,1) is bounded in L2 ((0, T ), R). Indeed, we have # # 1 M1 J ϕT v˜ 2L2 − Ψf H #ϕT #H ∗ , 2 thus
v˜ 2L2 (0,T ) 2 M1 + M2 Ψf H . Therefore, there exists v˜ ∈ L2 ((0, T ), R) such that v˜ → v˜ weakly in L2 ((0, T ), R). Passing to the limit → 0 in (27) with ξT ∈ H , we get T
v(t) ˜ μψ ˜ 1 (t), ξ(t) dt + Ψf , ξT = 0,
∀ξT ∈ H,
0
because # # 2 ϕ , ξT 2 #ϕ # T
T H ∗ ξT H
2M2 ξT H .
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Since H is dense in TS ψ1 (T ), we have T
v(t) ˜ μψ ˜ 1 (t), ξ(t) dt + Ψf , ξT = 0,
∀ξT ∈ TS ψ1 (T ).
(28)
0
Let Ψ be the solution of (6) with Ψ0 = 0. Using the fact that ξ solves (24) and Ψ solves (6) with Ψ0 = 0, we deduce from (28) that Ψ (T ), ξT = Ψf , ξT , Thus Ψ (T ) = Ψf .
∀ξT ∈ TS ψ1 (T ).
2
Remark 4.4. A uniform gap condition for the eigenvalues of −D Ω , cf. (12), would imply that the constants Cm , m ∈ N∗ admit a uniform positive lower bound and, in that case, H can be taken as the subset of TS ψ1 (T ) made of the functions φ with H 1+ finite norm. As we mentioned before, the existence of a planar domain verifying (12) is not even known. One could maybe define weaker gaps conditions in order to relate H to some Sobolev spaces. 5. Generic spectral controllability for the quantum box The goal of this Section is the proof of Proposition 2.8. Consider μ ∈ C 1 (R2 , R2 ). If μ is not nowhere constant, then there exists an open ball B where μ is constant. Taking an open neighborhood of domains of D3 included in B, condition (NonZ) will never be satisfied for those domains, thus property (A) is not generic in D3 . For the rest of the section, we fix μ ∈ C 1 (R2 , R2 ) which is nowhere constant. In Section 5.1, we reduce the proof of the genericity of property (A) (Proposition 2.8) to the proof of the genericity of a weaker property (Bk ). In Section 5.2, we present the strategy for the proof of the genericity of property (Bk ): it is sufficient to prove a weaker result, stated in Proposition 5.4. In Section 5.3, we present the strategy for the proof of Proposition 5.4. In Section 5.4, we perform some preliminary results for the proof of Proposition 5.4, which is achieved in Section 5.5. 5.1. Reduction of the problem The goal of this section is to reduce the proof of the genericity of the property (A), (Proposition 2.8) to the proof of the genericity of a weaker property (Bk ). For that purpose, we introduce the properties (Ak ) and (Bk ). Ω Ω For the rest of the paper, the notations λj 0 and φj 0 are used to denote respectively the j th eigenvalue and one corresponding normalized eigenvector associated to −D Ω0 . If, in the course of a definition or an argument, one domain under consideration is denoted Ω, then we simply Ω use λj and φj instead of λΩ j and φj . Definition 5.1. Let k ∈ N∗ , k 2 and Ω ∈ D3 . We say that Ω satisfies property (Ak ) if μ(q)φ1 (q)φk (q) dq = 0. Ω
K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
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Definition 5.2. Let k ∈ N∗ , k 2 and Ω ∈ D3 . We say that Ω satisfies property (Bk ) if μ(q)φ1 (q)φk (q) dq = 0,
either Ω
μ(q)φ1 (q)φk (q) dq = 0 and M(·) is not identically equal to zero,
or
(29)
Ω
where M : ∂Ω → R2 is given by M(q) :=
∂ξk ∂φk ∂ξ1 ∂φ1 (q) (q) + (q) (q), ∂ν ∂ν ∂ν ∂ν
(30)
ν is the unit outward normal to ∂Ω and ξ1 , ξk are the solutions of the following systems, ⎧ −( + λ1 )ξk = μφk , in Ω, ⎪ ⎪ ⎪ ⎨ ξk = 0, on ∂Ω, ⎪ ξk φ1 = 0, ⎪ ⎪ ⎩
⎧ −( + λk )ξ1 = μφ1 , in Ω, ⎪ ⎪ ⎪ ⎨ ξ1 = 0, on ∂Ω, ⎪ ξ1 φk = 0. ⎪ ⎪ ⎩
Ω
(31)
Ω
A first reduction is given in the next proposition. Its proof is standard and relies on Baire Lemma, we write it for sake of completeness. Proposition 5.1. If (Ak ) is generic in D3 for every k 2, then (A) is generic in D3 . Proof of Proposition 5.1. Let Ω ∈ D3 . We want to prove that the set
G := u ∈ W 4,∞ Ω, R2 ; Ω + u satisfies (A) is dense in W 4,∞ (Ω, R2 ). For k ∈ N∗ , we introduce the set Gk of functions u ∈ W 4,∞ (Ω, R2 ) such that λΩ+u < · · · < λΩ+u λΩ+u k 1 k+1 · · ·
μ(q)φ1Ω+u (q)φjΩ+u (q) dq = 0,
and
∀j ∈ {2, . . . , k}.
Ω+u
Then, G1 = W 4,∞ (Ω, R2 ), Gk+1 is an open subset of Gk for every k$∈ N∗ (thanks to the continuity of u → λΩ+u and u → φjΩ+u for j = 2, . . . , k + 1) and G = k∈N∗ Gk . Thanks to Baire j Lemma, it is sufficient to prove that, for every k ∈ N∗ , Gk+1 is a dense in Gk . Let k ∈ N∗ , u0 ∈ Gk − Gk+1 and > 0. We have Ω
Ω
Ω
0 λ1 0 < · · · < λk 0 λk+1 ···,
Ω
Ω
μ(q)φ1 0 (q)φj 0 (q) dq = 0, Ω0
∀j ∈ {2, . . . , k},
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K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976 Ω
Ω
0 and λk 0 = λk+1 or
Ω
Ω
0 μ(q)φ1 0 (q)φk+1 (q) dq = 0,
Ω0
where Ω0 := Ω + u0 . Thanks to the generic simplicity of the eigenvalues of the Laplacian and Ω +u the continuity of u → φj 0 for 2 j k (see [21]), there exists u1 ∈ W 4,∞ (Ω0 , R2 ) with u1 W 4,∞ < such that Ω1 Ω1 1 λΩ 1 < · · · < λk < λk+1 · · ·
and
μ(q)φ1Ω1 (q)φjΩ1 (q) dq = 0,
∀j ∈ {2, . . . , k},
Ω1
where Ω1 := Ω0 + u1 . Thanks to the genericity of (Ak+1 ) and the continuity of u → λjΩ1 +u for
2 j k + 1, u → φjΩ1 +u for 2 j k there exists u2 ∈ W 4,∞ (Ω1 , R2 ) with u2 W 4,∞ < , such that Ω2 Ω2 Ω2 λ1 < · · · < λk < λk+1 · · · and μ(q)φ1Ω2 (q)φjΩ2 (q) dq = 0, ∀j ∈ {2, . . . , k + 1}. Ω2
Then, u := (I + u2 ) ◦ (I + u1 ) ◦ (I + u0 ) − I is arbitrarily close to u0 in W 4,∞ (Ω, R2 ) and u ∈ Gk+1 . 2 A second reduction is given in the next proposition. Its proof is also standard. The argument goes by contradiction and relies on shape differentiation with respect to the domain Ω. It has been introduced by Albert [3] and recently used in [11]. We gathered in Appendix A well-known facts about shape differentiation which will be used in the proof. Proposition 5.2. Let k 2. If (Bk ) is generic in D3 , then (Ak ) is generic in D3 . Proof of Proposition 5.2. Let Ω0 ∈ D3 , k ∈ N∗ , k 2. We want to prove that the set
G := u ∈ W 4,∞ Ω0 , R2 ; Ω0 + u satisfies (Ak ) is dense in W 4,∞ (Ω0 , R2 ). We argue by contradiction. Let us assume the existence of u0 ∈ W 4,∞ (Ω0 , R2 ) and ρ0 > 0 such that, for every u ∈ W 4,∞ (Ω0 , R2 ) with u0 − uW 4,∞ < ρ0 , we have u ∈ / G. Thanks to the genericity of (Bk ), we can assume that Ω := Ω0 + u0 satisfies (Bk ). Then, there exists ρ > 0 such that, for every u ∈ Eρ (Ω) := {v ∈ W 4,∞ (Ω, R2 ); vW 4,∞ < ρ}, we have μ(q)φ1Ω+u (q)φkΩ+u (q) dq = 0, ∀u ∈ Eρ (Ω). (32) Ω+u
Thus, the directional derivative of the integral appearing in (32) in the direction u is equal to zero, for every u ∈ Eρ (Ω). By classical results on shape differentiation (cf. [42] or Appendix A below), we get
K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
μ φ1 (u)φk + φ1 φk (u) dq = 0,
∀u ∈ Eρ (Ω),
3943
(33)
Ω
where φ1 (u) et φk (u) are solutions of ⎧ −( + λ1 )φ1 (u) = λ1 (u)φ1 , ⎪ ⎪ ⎪ ⎪ ⎨ φ1 (u) = −u, ∇φ1 , ⎪ ⎪ φ1 φ1 (u) = 0, ⎪ ⎪ ⎩
⎧ −( + λk )φk (u) = λk (u)φk , in Ω, ⎪ ⎪ ⎪ ⎨ φk (u) = −u, ∇φk , on ∂Ω, ⎪ ⎪ ⎪ φk φk (u) = 0. ⎩
in Ω, on ∂Ω,
(34)
Ω
Ω
In order to transform (33) in a linear form in u, we introduce the dual systems (31). Note that these systems have unique solutions, thanks to (32). Using Green’s second formula and systems (34), we have
μ φ1 (u)φk + φ1 φk (u) dq
− Ω
φ1 (u)( + λ1 )ξk dq
= Ω
Ω
∂φ1 (u) ∂ξk φ1 (u) − ξk dσ (q) ( + λ1 )φ1 (u)ξk dq + ∂ν ∂ν
= Ω
+
∂Ω
( + λk )φk (u)ξ1 dq +
Ω
=
φk (u)( + λk )ξ1 dq
+
∂φ (u) ∂ξ1 φk (u) − ξ1 k dσ (q) ∂ν ∂ν
∂Ω
φ1 (u)
∂ξk ∂ξ1 + φk (u) dσ (q). ∂ν ∂ν
∂Ω
Then, (33) is equivalent to
∂φ1 ∂ξk ∂φk ∂ξ1 + dσ (q) = 0, u, ν ∂ν ∂ν ∂ν ∂ν
∀u ∈ Eρ (Ω).
(35)
∂Ω
This implies that M ≡ 0 which is a contradiction because Ω satisfies (Bk ).
2
5.2. Proof strategy for the genericity of (Bk ) According to Propositions 5.1 and 5.2, it remains to show that the property (Bk ) is generic in D3 for every k 2. To proceed in that direction, fix k 2 and Ω ∈ D3 . Without loss of generality, we assume from now that
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1. the spectrum of −D is simple on Ω; 2. there exists q¯ ∈ ∂Ω such that dμ(q) ¯ · τq¯ = 0,
(36)
where τq¯ is the unit tangent vector on ∂Ω at the point q. ¯ Indeed, the second condition is generic and open. Therefore, for a given domain Ω ∈ D3 , one can choose an arbitrarily close domain Ω ∈ D3 verifying condition 2. The latter holding in an open neighborhood of Ω , one can pick a domain Ω ∈ D3 arbitrarily close to Ω verifying both conditions 1 and 2. Arguing by contradiction, we assume there exists ρ > 0 such that μ(q)φ1Ω+u (q)φkΩ+u (q) dq = 0, ∀u ∈ Eρ (Ω), (37) Ω+u
and M(u) ≡ 0 on ∂Ω + u, ∀u ∈ Eρ (Ω),
(38)
where Eρ (Ω) := {v ∈ W 4,∞ (Ω, R2 ); vW 4,∞ < ρ} and M(u) : ∂(Ω + u) → R2 is defined by M(u)(q) =
∂ξ1 (u) ∂φ1 (u) ∂ξk (u) ∂φk (u) (q) (q) + (q) (q), ∂ν ∂ν ∂ν ∂ν
(39)
where φ1 (u), and φk (u) are normalized eigenvectors of D Ω+u associated to λ1 (u) and λk (u) respectively and ξ1 (u) and ξk (u) are the solutions of (31) associated to Ω + u. (Such systems have solutions since (37) holds true.) In the sequel, we (sometimes) drop the variable (u) when it corresponds to u = 0. The next step consists in shape differentiating the condition M(u) ≡ 0 for u ∈ Eρ (Ω). Applying the classical shape differentiation formula regarding Dirichlet boundary condition (see Theorem A.2), we get M (u) = −u, ν
∂M(0) ∂ν
on ∂Ω.
(40)
Remark 5.1. For technical details on regular extension of outward normal vector, we refer to [42, Théorème 4.1, Chapitre IV, p. 69]. After computations, we get
∂ξ1 ∂φk ∂ξ1 ∂φ1 ∂ξk ∂φ1 ∂ξk + + (u) + (u) (u) ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν
∂ ∂φ1 ∂ξk ∂ ∂φk ∂ξ1 ∂φk ∂ ∂ξ1 + + = −u, ν ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν
∂φ1 ∂ ∂ξk + on ∂Ω. ∂ν ∂ν ∂ν
∂φk ∂ν
(u)
(41)
K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
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The relation between the first shape derivative of a normal derivative ( ∂φ ∂ν ) (u) and the normal
derivative of a first shape derivative and reads as follows.
∂φ ∂ν
is given in [20, Théorème 5.5.2, formula (5.74), p. 205]
Lemma 5.3. With the notations above, We have
∂φ ∂ν
=
∂ ∂φ ∂ 2φ ∂φ − u, ν − 2 − ∇φ, ∇Γ u, ν on ∂Ω, ∂ν ∂ν ∂ν ∂ν
where ∇Γ is the tangential gradient and of φ (a bilinear form) applied to (ν, ν).
∂2φ ∂ν 2
(42)
is understood as the image of the second derivative
Using the above lemma and the fact that the involved functions vanish on ∂Ω, (41) is rewritten as follows ∂φk (u) ∂ξ1 ∂φk ∂ξ1 (u) ∂φ1 (u) ∂ξk ∂φ1 ∂ξk (u) + + + ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν
2 2 2 ∂ φk ∂ξ1 ∂φk ∂ ξ1 ∂ φ1 ∂ξk ∂φ1 ∂ 2 ξk + = −u, ν + + ∂ν ∂ν 2 ∂ν ∂ν 2 ∂ν 2 ∂ν ∂ν 2 ∂ν
on ∂Ω,
(43)
where φ1 (u) and φk (u) are defined in (34) and ξ1 (u) and ξk (u) are solutions of ⎧ −( + λ1 )ξk (u) = λ1 (u)ξk + μφk (u), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ξ (u) = −u, ν ∂ξk , k ∂ν ⎪
⎪ ⎪ φ1 ξk (u) + φ1 (u)ξk dq = 0, ⎪ ⎪ ⎩
in Ω, on ∂Ω,
Ω
and ⎧ −( + λk )ξ1 (u) = λk (u)ξ1 + μφ1 (u), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ξ (u) = −u, ν ∂ξ1 , 1 ∂ν ⎪
⎪ ⎪ φk ξ1 (u) + φk (u)ξ1 dq = 0. ⎪ ⎪ ⎩
in Ω, on ∂Ω,
(44)
Ω
As a consequence of the previous computations, the genericity of (Bk ) in D3 results from the next proposition. Proposition 5.4. Let k 2 and Ω ∈ D3 . Assume that (37) and (38) hold true. Then, there does not exist ρ > 0 such that (34) and (44) admit solutions satisfying (43) for every u ∈ Eρ (Ω). Remark 5.2. Let J (Ω) be a smooth functional depending on the domain Ω and u a variation belonging to W k,∞ (Ω, R2 ). As pointed out in [42], we have
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K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
J (Ω)(u, u) = (J ) (Ω)(u, u) − J (Ω)(u · ∇u).
(45)
This equation says that J (Ω), the second derivative with respect to the domain at the point Ω, applied to the function u is not in general equal to the first derivative of the function J (Ω)(u) at the point Ω applied to u. The difference between them is equal to the first shape derivative of the function J (Ω) applied to u · ∇u. However, in our case, they are equal because the first shape derivative is equal to zero by assumption. Thus, (43) exactly corresponds to the second shape derivative of (37). 5.3. Proof strategy for Proposition 5.4 To prove Proposition 5.4, our strategy is similar to that developed in [11] and, in order to describe it, we first need information on the regularity of the solutions of (34) and (44). For that purpose, we consider the following standard definitions of Sobolev spaces and distributions on Ω (cf. [32]). If m is a positive integer, we use H m (Ω) to denote the Sobolev space of order m on Ω defined by H m (Ω) := Ψ D α Ψ ∈ L2 (Ω), |α| m , where D α =
∂ α1 +α2 α α , ∂x1 1 ∂x2 2
and |α| = α1 + α2 . Here the differential operators D α are defined in the
distributional sense on Ω, with D (Ω) the space of distributions on Ω being dual to D(Ω), the set of smooth functions with compact support in Ω (cf. [32]). Let ρ : Ω → R+ be a function of ¯ equal to the distance function to ∂Ω (ρ(x) = d(x, ∂Ω)) for d(x, ∂Ω) small enough. class C 2 (Ω) Such a function exists as noted in [32, Chapter 1, §11.2, p. 62]. According to [32], for s ∈ N, we set Ξ s (Ω) := Ψ ρ |α| D α Ψ ∈ L2 (Ω), |α| s , equipped with the norm Ψ Ξ s (Ω) =
|α|s
# |α| α #2 #ρ D Ψ # 2
1/2
L (Ω)
.
Then Ξ s (Ω) is a Hilbert space so that H s (Ω) ⊂ Ξ s (Ω) ⊂ Ξ 0 (Ω) = L2 (Ω) with a continuous embedding. Let Ξ −s (Ω) := (Ξ s (Ω)) be the dual space of Ξ s (Ω) for the L2 (Ω) scalar-product. Then, Ξ −s (Ω) is a distribution space as proved in [32]. Remark 5.3. By interpolations techniques, we can also define the spaces Ξ s (Ω) for all real positive number s. Then, we have H s (Ω) ⊂ Ξ s (Ω) ⊂ Ξ s (Ω) ⊂ L2 (Ω) if 0 < s < s (see [32, p. 184] for more details). We can now apply the general theorems stated in [32] to the present situation. Let A := + λ and B0 be the Dirichlet trace operator. We set s DA (Ω) = Ψ Ψ ∈ H s (Ω), AΨ ∈ Ξ s−2m (Ω) ,
0 < s < 2m,
K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
3947
s (Ω) is a Hilbert space. with the norm defined by Ψ = (Ψ 2H s (Ω) + AΨ 2Ξ s−2m )1/2 . Then, DA We write system (34) with new notations
AΨ = f
in Ω
and B0 Ψ = g
on ∂Ω,
(46)
with j = 1, k.
(47)
where f = −λ (u)φj
and g = −u, ν
∂φj , ∂ν
We apply [32, Theorem 7.4, p. 202] for m = 1 (one boundary condition) and m0 = 0 (there is not derivation in the trace operator). As φ1 is an eigenfunction, f is in every distribution space, in particular it is an element of every Ξ s (Ω) for s < 0. Then, if 0 < s < 2, we have f ∈ Ξ s−2 (Ω). s (Ω). We now apply [32, Theorem 7.3, s−1/2 If g ∈ H (Ω), by [32, Theorem 7.4, p. 202], Ψ ∈ DA ∂ s−3/2 (Ω). We summarize these results p. 201] with B1 = ∂ν and m1 = 1. Then, we have ∂Ψ ∂ν ∈ H in the following lemma. Lemma 5.5. Let s ∈ (0, 2) and j ∈ {1, k}. With the notations above, if the Dirichlet boundary condition g = u, ν
∂φj ∂ν
∈ H s−1/2 (∂Ω), then we have φj (u) ∈ H s (Ω) and
∂φj (u) ∂ν
∈ H s−3/2 (∂Ω).
As already mentioned in the introduction, the starting remark for the argument of Proposition 5.4 goes as follows. By taking into account Lemma 5.5, the right-hand side of (43) is in H s−1/2 (∂Ω) and, at the same time, the left-hand side in H s−3/2 (∂Ω), for s ∈ (0, 2). To take advantage of that gap of regularity between the two sides of (43), we first consider variations exhibiting just one jump of discontinuity on Ω, let say at some point q∗ ∈ ∂Ω, so that, for all the quantities involved in (43), an irregular part only occurs at the point q∗ . If we are able to compute exactly this irregular part, we would infer that it has to be equal to zero by using (43). It would provide some extra information at the point q∗ , of the type F (q∗ ) = 0 where F is an R2 -valued map defined on ∂Ω. Since the point q∗ is arbitrary, we would end with the relation F ≡ 0 on ∂Ω, similar to (38). Using this new information together with M(0) ≡ 0, one hopes to derive a contradiction. Let us provide more details. Fix q∗ ∈ ∂Ω and a parametrization σ of C1 , the connected component of ∂Ω containing q∗ , so that σ ∈ [−L, L) and q∗ corresponds to σ = 0. Fix an open neighborhood Vα of q∗ in C1 parameterized by (−α, α) with α < L. We consider an admissible variation uq∗ (see Definition 5.3 below) defined as follows: on (−α, 0), uq∗ , ν = 0, on (0, α), uq∗ , ν = 1 and uq∗ , ν is smooth in C1 except at σ = 0. According to Remark 5.4 below, we can extend the definition of M (u) to functions u which are not regular enough to perform shape differentiation (such as uq∗ ). We then show that M (uq∗ ) admits, in the distributional sense, the following Taylor expansion valid in (−α, α),
M (uq∗ )(σ ) = M0
1 + M1 ln |σ | + M3 σ ln |σ | + R(σ ), p.v. σ
and we also have, according to (40), M (uq∗ )(σ ) = M2 H0 (σ ) + R(σ ).
(48)
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K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
In the above equations, the coefficients Mi , 0 i 3, are R2 -valued, R denotes a (generic) C 1 function over (−α, α) and H0 belongs to H 1/2− (∂Ω) for every > 0. We will then prove that Mi , 0 i 2, are always equal to zero and, from the relation M ≡ 0 on ∂Ω, we will therefore be left with the relation M3 σ ln |σ | + R(σ ) = 0 on (−α, α).
(49)
It would immediately yield M3 = 0. Moreover, we will compute M3 as a function of the values of φ1 , φk , ξ1 , ξk and their normal derivatives at σ = 0 (i.e., at q∗ ). Therefore, M3 can be seen as a function defined on ∂Ω and, since q∗ is arbitrary, we will get from (49) that M3 (·) ≡ 0 on ∂Ω. It will provide us with a new nontrivial relationship between φ1 , φk , ξ1 , ξk and their normal derivatives and we will reach shortly after a contradiction, hence concluding the proof of Proposition 5.4. In order now to access to (48) and get a hold on the Mi ’s, we split M (uq∗ ) as follows, M (uq∗ ) = Mb (uq∗ ) + Md (uq∗ ),
(50)
where Mb (uq∗ ) =
(u ) (u ) ∂φk (uq∗ ) ∂ξ1 ∂φk ∂ξ1,b ∂φ (uq ) ∂ξk ∂φ1 ∂ξk,b q∗ q∗ + + 1 ∗ + , ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν
(51)
Md (uq∗ ) =
∂φk ∂ξ1,d (uq∗ ) ∂φ1 ∂ξk,d (uq∗ ) + , ∂ν ∂ν ∂ν ∂ν
(52)
where Mb (uq∗ ) and Md (uq∗ ) are the contributions of respectively the boundary ∂Ω and the domain Ω to M (uq∗ ). In (51) and (52), we choose the variation uq∗ (see Definition 5.3) such that φ1 (uq∗ ) and φk (uq∗ ) are solutions of ⎧ −( + λ1 )φ1 (uq∗ ) = λ1 (uq∗ )φ1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ φ (uq ) = −uq , ν ∂φ1 , ∗ ∗ 1 ∂ν ⎪ ⎪ ⎪ ⎪ φ1 φ1 (uq∗ ) = 0, ⎪ ⎩
in Ω, on ∂Ω,
Ω
⎧ −( + λk )φk (uq∗ ) = λk (uq∗ )φk , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ φ (uq ) = −uq , ν ∂φk , ∗ ∗ k ∂ν ⎪ ⎪ ⎪ ⎪ φk φk (uq∗ ) = 0, ⎪ ⎩
in Ω, on ∂Ω,
(53)
Ω (u ), ξ (u ), ξ (u ) and ξ and the ξ1,b q∗ k,b q∗ k,d are defined as the solutions of the following 1,d q∗ Helmholtz equations,
K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
⎧ −( + λ1 )ξk,b (uq∗ ) = 0, in Ω, ⎪ ⎪ ⎪ ⎪ ⎪ ∂ξ ⎪ ⎨ ξ (uq ) = −uq , ν k , on ∂Ω, ∗ ∗ k,b ∂ν ⎪ ⎪ ⎪ ⎪ φ1 ξk,b (uq∗ ) = 0, ⎪ ⎪ ⎩ Ω
⎧ −( + λk )ξ1,b (uq∗ ) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ξ (uq ) = −uq , ν ∂ξ1 , ∗ ∗ 1,b ∂ν ⎪ ⎪ ⎪ ⎪ φk ξ1,b (uq∗ ) = 0, ⎪ ⎪ ⎩
3949
in Ω, on ∂Ω,
(54)
Ω
⎧ −( + λ1 )ξk,d (uq∗ ) = λ1 (u)ξk + μφk (uq∗ ), ⎪ ⎪ ⎪ ⎪ ⎨ ξk,d (uq∗ ) = 0, ⎪ ⎪ φ1 ξk,d (uq∗ ) = 0, ⎪ ⎪ ⎩
in Ω, on ∂Ω, (55)
Ω
⎧ −( + λk )ξ1,d (uq∗ ) = λk (u)ξ1 + μφ1 (uq∗ ), ⎪ ⎪ ⎪ ⎪ ⎨ ξ1,d (uq∗ ) = 0, ⎪ ⎪ (uq∗ ) = 0. ⎪ φk ξ1,d ⎪ ⎩
in Ω, on ∂Ω, (56)
Ω (u )+ξ (u )+c φ and ξ (u ) = ξ (u )+ξ (u )+c φ , By linearity, ξ1 (uq∗ ) = ξ1,b q∗ 1 k 2 1 k q∗ k,b q∗ k,d q∗ 1,d q ∗ where c1 = − Ω φk (u)ξ1 dq and c2 = − Ω φ1 (u)ξk dq. We simply intend here to compute ξj (uq∗ ), j = 1, k, as the sum of two terms, one coming from the boundary condition and the second from the inhomogeneous part of the PDE. Each of these terms requires the study of a Dirichlet-to-Neumann operator associated to a Helmholtz equation. In the next section, we develop in details these computations.
5.4. Evaluations of the singular parts of Mb (uq∗ ) and Md (uq∗ ) In what follows, p and q denote points of R2 and x, y denotes respectively the first and second coordinates of a point in R2 . For the rest of the paper, we fix a point q∗ ∈ ∂Ω and, with no loss of generality, we assume that ∂Ω has only one connected component. We next choose a parametrization of ∂Ω by arc-length σ ∈ [−L, L) so that q∗ corresponds to (x(0), y(0)). The initial control problem (4) is clearly invariant by rotation and thus we can assume that the tangent vector at σ = 0 is equal to (−1, 0)T . We finally proceed to a translation of vector q∗ which implies that (x(0), y(0)) = (0, 0). That transformation only modifies the PDEs and ξ , j = 1, k, replacing q by q + q in (32), (31), (54) and (55). governing ξj , ξj,d ∗ j,b Since Ω is of class C 3 , there exists a neighborhood N0 of 0 ∈ R such that for every σ ∈ N0 , we have x(σ ) = −σ + O σ 3 , y(σ ) =
κ(0) 2 σ + O σ3 , 2
(57) (58)
where κ is the curvature function of ∂Ω. Let Na be the subset of ∂Ω made of points q(σ ) = (x(σ ), y(σ )) with σ ∈ N0 and ν(·) be the unit outward normal along ∂Ω, which is of class C 2 , and has direction (y (·), −x (·)).
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We now consider a variation uq∗ which exhibits a unique jump of discontinuity at q∗ , i.e., uq∗ is only defined through its normal part uq∗ , ν given next ⎧ ⎨ 0, uq∗ , ν(σ ) = 1, ⎩ η(σ ),
σ ∈ [−α, 0), σ ∈ [0, α), σ ∈ [−L, −α) ∪ [α, L),
(59)
where 0 < α is small enough so that [−α, α] ⊂ N0 and η is smooth so that uq∗ , ν is 2L-periodic and smooth except at σ = 0. We sometimes refer to uq∗ , ν as the Heaviside function on ∂Ω and use H0 to denote it. Remark 5.4. Strictly speaking, uq∗ cannot be considered as a variation of domain since it is not in W 4,∞ (Ω, R2 ). However, it is rather easy to see that solutions of the differential systems obtained after shape differentiation can be defined by standard density arguments for function spaces containing W 4,∞ (Ω, R2 ). For instance, M (u) is first defined by shape differentiation for u ∈ Eρ (Ω), and that requires to consider the functions φj (u) and ξj (u), j = 1, k verifying (34) and (44). On the other hand, these functions only need u, ν, the normal component of the variation, to be defined. Thus, for u, ν ∈ H s (∂Ω), s 1, one still can define by density (unique) solutions of (34) and (44) associated to u and thus traces on ∂Ω of these elements. Finally, using Lemma 5.5, the function defined in the left-hand side of (43) is well defined and, by an obvious abuse of notation, we use M (u) to denote it. We now have defined M (uq∗ ) and we refer to it as the shape differential of M for the variation uq∗ . Remark 5.5. For presentation ease, we use the arc-length σ for parameterizing all points q in a neighborhood of the fixed point q∗ ∈ ∂Ω. Definition 5.3. Let Ω be a domain of D3 not verifying condition (Bk ). A variation u (defined with u, ν ∈ H s (∂Ω), s ∈ (0, 2)) is said to be admissible if u, ν
∂φ1 ∂ξk dσ (q) = 0. ∂ν ∂ν
(60)
∂Ω
By applying Green’s second formula and using (33) and (38), one sees that condition (60) is necessary (and sufficient) for the existence of solutions of the PDEs given in (53), (54) and then k (55) after an appropriate choice of c1 and c2 . Moreover, remark that if ∂ξ ∂ν ≡ 0 on ∂Ω (and thus ∂ξ1 ∂ν ≡ 0), then every variation is admissible. Lemma 5.6. For every q∗ ∈ ∂Ω, one can choose the smooth function η and the parameter α introduced in (59) such that uq∗ is an admissible variation. ∂ξ1 k Proof of Lemma 5.6. We may assume that ∂ξ ∂ν (and thus ∂ν ) is not identically equal to 0 on k ∂Ω. Assume first that ∂ξ ∂ν (q∗ ) = 0. Eq. (60) can clearly be stated as an affine relation L(η) = l, k where L is a linear form and l ∈ R. Notice that L is not null. Indeed, ∂ξ ∂ν (q) is not equal to zero k in an open neighborhood of q∗ . Then, by choosing α small enough, ∂ξ ∂ν (q(σ )) is not equal to zero for some σ in (−L, −α) ∪ (α, L). It is therefore always possible to select η so that uq∗
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is an admissible variation. It is immediate to extend the above construction to the case where ∂ξk ∂ξk ∂ν (q∗ ) = 0 and there exists a sequence of points q ∈ ∂Ω converging to q∗ such that ∂ν (q) = 0. k It remains to treat the case where ∂ξ ∂ν ≡ 0 on an open neighborhood N of q∗ ∈ ∂Ω. It is then possible to choose α > 0 small enough so that q(σ ) ∈ N for σ ∈ (−2α, 2α) and η ≡ 0 on (2α, L) ∪ (−L, −α). Then, the corresponding uq∗ is admissible. 2 Definition 5.4. We say that a function g defined on ∂Ω is 2-regular if there exists two smooth ˜ ) for σ in an (i.e., C ∞ ) functions h, h˜ defined on ∂Ω such that g(σ ) = σ 2 ln(|σ |)h(σ ) + h(σ open neighborhood of zero. We will use sometimes the symbol R2 to denote an arbitrary 2regular function. In addition, we use the symbol R1 to denote an arbitrary C 1 function in an open neighborhood of zero. Note that a 2-regular function is necessarily of class C 1 . Finally, we use the notation O(σ ) to denote an arbitrary C 1 function equal to zero at σ = 0 and with uniformly bounded derivative over some open neighborhood of zero. ∂ξ1,b ∂φk ∂φ1 ∂ν (uq∗ ), ∂ν (uq∗ ), ∂ν (uq∗ ), ∂ξk,d ∂ν (uq∗ ) involved in M (uq∗ ) = Mb (uq∗ ) + Md (uq∗ ) only occur at the
In the next paragraph, we will prove that the irregular parts of
∂ξ1,d ∂ξk,b ∂ν (uq∗ ), ∂ν (uq∗ ) and point q∗ and we intend to
calculate them exactly.
5.4.1. Expression of Mb (uq∗ ) The main result of this section is the following theorem. Theorem 5.7. There exists an open neighborhood of zero N1 ⊂ N0 such that, if σ ∈ N1 , one has Mb (uq∗ )(σ ) =
∂φ1 (0) ∂ξk (0) ∂φk (0) ∂ξ1 (0) 1 + λk λ1 σ ln |σ | + R1 . π ∂ν ∂ν ∂ν ∂ν
(61)
For the rest of the paper, we set a1 := −
1 , 2π
a2 :=
1 . 8π
(62)
Note that the constant 1/π appearing in the right-hand side of (61) is equal to −4(a1 + 2a2 ). The proof of this theorem is based on the following proposition. Proposition 5.8. We have
∂φ1 (uq∗ ) 1 ∂φ1 ∂ ∂φ1 (σ ) = −2 a1 (0) p.v. + a1 (0) ln |σ | ∂ν ∂ν σ ∂τ ∂ν
∂ 2 ∂φ1 ∂φ1 + a1 2 + (a1 + 2a2 )λ1 (0)σ ln |σ | ∂ν ∂ν ∂τ
∂φ1 ∂ ∂φ1 − a1 (0)L1 (σ ) − a1 (0)L2 (σ ) + R1 , ∂ν ∂τ ∂ν
(63)
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∂φk (uq∗ ) ∂φk ∂ ∂φk 1 + a1 (0) ln |σ | (σ ) = −2 a1 (0) p.v. ∂ν ∂ν σ ∂τ ∂ν
∂ 2 ∂φk ∂φk + a1 2 + (a1 + 2a2 )λk (0)σ ln |σ | ∂ν ∂ν ∂τ
∂φk ∂ ∂φk (0)L1 (σ ) − a1 (0)L2 (σ ) + R1 , − a1 ∂ν ∂τ ∂ν (u ) ∂ξ1,b q∗
∂ν
(u ) ∂ξk,b q∗
∂ν
1 ∂ξ1 ∂ ∂ξ1 (σ ) = −2 a1 (0) p.v. + a1 (0) ln |σ | ∂ν σ ∂τ ∂ν
∂ 2 ∂ξ1 ∂ξ1 + a1 2 + (a1 + 2a2 )λk (0)σ ln |σ | ∂ν ∂ν ∂τ
∂ξ1 ∂ ∂ξ1 (0)L1 (σ ) − a1 (0)L2 (σ ) + R1 , − a1 ∂ν ∂τ ∂ν
1 ∂ξk ∂ ∂ξk (σ ) = −2 a1 (0) p.v. + a1 (0) ln |σ | ∂ν σ ∂τ ∂ν
∂ 2 ∂ξk ∂ξk + a1 2 + (a1 + 2a2 )λ1 (0)σ ln |σ | ∂ν ∂ν ∂τ
∂ξk ∂ ∂ξk (0)L2 (σ ) + R1 , − a1 (0)L1 (σ ) − a1 ∂ν ∂τ ∂ν
(64)
(65)
(66)
where L1 (σ ) := T0 (p.v.( σ1 )), L2 (σ ) := T0 (ln |σ |), with T0 the linear operator defined in (111). Recall that R1 is used to denote an arbitrary C 1 function of ∂Ω. Proof of Proposition 5.8. Explicit computation is only provided for (63) since expressions for (u ) ∂φk (uq∗ ) ∂ξ1,b q∗ , ∂ν ∂ν
and
(u ) ∂ξk,b q∗ ∂ν
are derived in a similar way. From (107), we first easily get that
∂φ the contribution of λ1 (uq∗ )φ1 to ∂ν1 (q∗ ) is a term of class C 2 1 Proposition B.8 with g = ∂φ ∂ν . It yields
E1
and thus of type R2 . We next apply
∂φ1 (0) ∂ ∂φ1 ∂φ1 1 H0 (σ ) = a1 p.v. + (0)a1 ln |σ | ∂ν ∂ν σ ∂τ ∂ν
∂ 2 ∂φ1 ∂φ1 + a1 2 + (a1 + 2a2 )λ1 (0)σ ln |σ | + R2 . ∂ν ∂ν ∂τ
According to Theorem B.4, we get
∂φ1 ∂φ1 (0) ∂ ∂φ1 1 (σ ) = −2 a1 p.v. + (0)a1 ln |σ | ∂ν ∂ν σ ∂τ ∂ν
∂ 2 ∂φ1 ∂φ1 + a1 2 + (a1 + 2a2 )λ1 (0)σ ln |σ | ∂ν ∂ν ∂τ
(67)
K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
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∂φ1 (0) ∂ ∂φ1 a1 L1 (σ ) − (0)a1 L2 (σ ) ∂ν ∂τ ∂ν
∂ 2 ∂φ1 ∂φ1 + (a1 + 2a2 )λ1 (0)L3 (σ ) + R1 , − a1 2 ∂ν ∂ν ∂τ
−
where L3 (σ ) := T0 (σ ln |σ |). Recalling that ln |σ | belongs to H 1/2−ε (∂Ω) for every ε > 0 and the regularizing effect of the operator T0 , one immediately gets that σ ln |σ | ∈ H 3/2−ε (∂Ω) and L3 (σ ) ∈ H 5/2−ε (∂Ω) for every ε > 0. It implies that L3 (·) is a C 1 function of ∂Ω. 2 Remark 5.6. For the rest of the paper, we will need information about the regularity of Lj (σ ), j = 1, 2. As done in the above argument, we have that p.v.( σ1 ) ∈ H −1/2−ε (∂Ω) for every ε > 0 and, thanks to the regularizing effect of the operator T0 , we get that L1 (·) ∈ H 1/2−ε (∂Ω) for every ε > 0. Similarly, we get that L2 (·) ∈ H 3/2−ε (∂Ω) and T0 (H 3/2−ε (∂Ω)) ⊂ R1 for every ε > 0. We are now able to prove Theorem 5.7. Proof of Theorem 5.7. Let σ ∈ N0 and we eventually reduce the size of the neighborhood later on. Our first goal consists in computing explicitly the coefficient associated to p.v.( σ1 ) in Mb (uq∗ ). Using Proposition 5.8 and Remark 5.6, we have (u ) ∂ξ1,b ∂φk (uq∗ ) ∂ξ1 ∂φk q∗ (σ ) (σ ) + (σ ) (σ ) ∂ν ∂ν ∂ν ∂ν (u ) ∂ξk,b ∂φ (uq ) ∂ξk ∂φ1 q∗ (σ ) + (σ ) (σ ) + 1 ∗ (σ ) ∂ν ∂ν ∂ν ∂ν
∂φk 1 ∂ξ1 (0)a1 p.v. + P0 (σ ) (0) + O(σ ) = −2 ∂ν σ ∂ν
∂ξ1 1 ∂φk −2 (0)a1 p.v. + P0 (σ ) (0) + O(σ ) ∂ν σ ∂ν
1 ∂ξk ∂φ1 (0)a1 p.v. (0) + O(σ ) −2 + P0 (σ ) ∂ν σ ∂ν
∂ξk 1 ∂φ1 −2 (0)a1 p.v. + P0 (σ ) (0) + O(σ ) ∂ν σ ∂ν
∂φk ∂ξ1 ∂φ1 ∂ξk 1 = −4a1 (0) (0) + (0) (0) p.v. + P0 (σ ). ∂ν ∂ν ∂ν ∂ν σ
Mb (uq∗ )(σ ) =
where P0 (σ ) denotes any function belonging to H 1/2−ε (∂Ω) for every ε > 0 in some open neighborhood of zero. Then, we have Mb (uq∗ )(σ ) = −4a1 M(0) p.v
1 + P0 (σ ). σ
(68)
Since M ≡ 0 on ∂Ω, we have in particular M(0) = 0. In consequence, there is not any term in p.v( σ1 ) in Mb (uq∗ ).
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The next step consists in identifying the least regular term of P0 . We begin by writing that
∂ ∂φk ∂ 2 ∂φk ∂φk Mb (uq∗ )(σ ) = − 2a1 (0) ln |σ | + 2 a1 2 + (a1 + 2a2 )λk (0)σ ln |σ | ∂τ ∂ν ∂ν ∂ν ∂τ
∂φk ∂ ∂φk (0)L1 (σ ) − a1 (0)L2 (σ ) + R1 (σ ) − a1 ∂ν ∂τ ∂ν
∂ξ1 ∂ ∂ξ1 × (0) + (0)σ + O σ 2 ∂ν ∂τ ∂ν
∂ ∂ξ1 ∂ 2 ∂ξ1 ∂ξ1 (0) ln |σ | + 2 a1 2 + (a1 + 2a2 )λk (0)σ ln |σ | − 2a1 ∂τ ∂ν ∂ν ∂ν ∂τ
∂ξ1 ∂ ∂ξ1 (0)L2 (σ ) + R1 (σ ) − a1 (0)L1 (σ ) − a1 ∂ν ∂τ ∂ν
∂ ∂φk ∂φk (0) + (0)σ + O σ 2 × ∂ν ∂τ ∂ν
∂ ∂φ1 ∂ 2 ∂φ1 ∂φ1 − 2a1 (0) ln |σ | + 2 a1 2 + (a1 + 2a2 )λ1 (0)σ ln |σ | ∂τ ∂ν ∂ν ∂ν ∂τ
∂φ1 ∂ ∂φ1 (0)L1 (σ ) − a1 (0)L2 (σ ) + R1 (σ ) − a1 ∂ν ∂τ ∂ν
∂ ∂ξk ∂ξk (0) + (0)σ + O σ 2 × ∂ν ∂τ ∂ν
∂ ∂ξk ∂ 2 ∂ξk ∂ξk (0) ln |σ | + 2 a1 2 + (a1 + 2a2 )λ1 (0)σ ln |σ | − 2a1 ∂τ ∂ν ∂ν ∂ν ∂τ
∂ξk ∂ ∂ξk − a1 (0)L1 (σ ) − a1 (0)L2 (σ ) + R1 (σ ) ∂ν ∂τ ∂ν
∂φ1 ∂ ∂φ1 × (0) + (0)σ + O σ 2 . ∂ν ∂τ ∂ν Rearranging the terms and using Remark 5.6, we get
∂M ∂ 2M (0) 2 ln |σ | + σ L1 (σ ) + L2 (σ ) + M(0)L1 (σ ) + 2 2 (0)σ ln |σ | ∂τ ∂τ ∂φ1 (0) ∂ξk (0) ∂φk (0) ∂ξ1 (0) + λk σ ln |σ | + R1 (σ ). − (4a1 + 8a2 ) λ1 ∂ν ∂ν ∂ν ∂ν
Mb (uq∗ )(σ ) = −a1
Since M(0) ≡ 0 on ∂Ω, we have ∂M ∂τ (0) ≡ 0 and above equation reduces Eq. (61). 2
∂2M (0) ∂τ 2
= 0 on ∂Ω. As a consequence, the
5.4.2. Contribution of Md (uq∗ ) We prove in this section the following theorem regarding the Taylor expansion of Md (uq∗ ) in an open neighborhood of zero.
K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
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Theorem 5.9. There exists an open neighborhood of zero N2 ⊂ N0 such that, if σ ∈ N2 , one has Md (uq∗ )(σ ) =
1 ∂φ1 ∂φk μ(q∗ ) (0) (0)σ ln |σ | + R1 . π ∂ν ∂ν
(69)
The proof of this theorem is based on the following proposition. Proposition 5.10. We keep the notations above, then we have (u ) ∂ξ1,d q∗
∂ν ∂ξk,d (uq∗ ) ∂ν
(σ ) = −(2a1 + 4a2 )μ(q∗ )
∂φ1 (0)σ ln |σ | + R1 , ∂ν
(70)
(σ ) = −(2a1 + 4a2 )μ(q∗ )
∂φk (0)σ ln |σ | + R1 . ∂ν
(71)
1 Proof of Theorem 5.9. We note that 2a1 + 4a2 = − 2π . Assuming Proposition 5.10. We have ∂φk ∂ξ1,d (uq∗ ) ∂φ1 ∂ξk,d (uq∗ ) + ∂ν ∂ν ∂ν ∂ν
∂φk ∂φ1 1 = (0) + O(σ ) μ(q∗ ) (0)σ ln |σ | + R1 ∂ν 2π ∂ν
1 ∂φ1 ∂φk (0) + O(σ ) μ(q∗ ) (0)σ ln |σ | + R1 + ∂ν 2π ∂ν
Md (uq∗ )(σ ) =
=
1 ∂φk ∂φ1 μ(q∗ ) (0) (0)σ ln |σ | + R1 . π ∂ν ∂ν
2
. Recall now the system verified by ξ1,d
⎧ −( + λk )ξ1,d (uq∗ ) = λk (uq∗ )ξ1 + μ(q + q∗ )φ1 (uq∗ ) in Ω, ⎪ ⎪ ⎪ ⎪ ⎨ ξ1,d (uq∗ ) = 0 on ∂Ω, ⎪ ⎪ φk ξ1,d (uq∗ ) = 0. ⎪ ⎪ ⎩
(72)
Ω verifies a similar system by exchanging the indices 1 and k and we will omit The function ξk,d the corresponding argument.
Remark 5.7. By classical elliptic regularity theory presented in [32], we know that φ1 ∈ ∂ξ
∈ H 3− (Ω). By taking the trace, we have ∂ν1,d ∈ H 3/2− (∂Ω). H 1− (Ω), and then ξ1,d A straightforward computation shows that the last term σ ln |σ | in our expansion of Mb (uq ∗ )
is in H 3/2− (∂Ω). Hence, it is necessary to compute exactly the first singular term of
∂ξ1,d ∂ν .
Remark 5.8. The term Md (uq ∗ ) cannot be treated by a direct functional analysis argument: if there were a family of functional spaces X s with a well established theory of elliptic equations such as that for Sobolev spaces and if there would exist s1 = s2 such that H0 ∈ X s1 and
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K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976 ∂ξ
ln |σ | ∈ X s2 , then ∂ν1,d and σ ln |σ | would not be in the same X s . However, we cannot distinguish these two functions even in the family of Besov spaces. We can also note that H0 is a bounded variation function and ln |σ | is not, but elliptic theory in the space of bounded variation functions is not easy. For these reasons, it seems that an exact computation of the first term of ∂ξ1,d ∂ν
is necessary.
Let us first prove the following technical lemma, which expresses integrals over Ω by means of boundary integrals over ∂Ω. Lemma 5.11. Let k, m be two distinct positive integers. Assume that a function h verifies ( + λk )h = 0 in Ω. Then we have ∂ ∂νp
Ω
∂h ∗ −1 ∗ (Em − Ek )(h|∂Ω ) + Kk − Km , h(q)Gm (p, q) dq = λm − λk ∂ν
(73)
where Gm (·,·) is the fundamental solution of the Helmholtz equation corresponding to λm and ∗ are defined in Secverifying the Sommerfeld condition and the operators Ek , Em , Kk∗ , Km tion B.1. Proof of Lemma 5.11. Green’s second formula says that
g1 (g2 ) − g2 (g1 ) =
Ω
∂g2 ∂g1 g1 − g2 dσ (q), ∂ν ∂ν
∂Ω
where g1 , g2 are arbitrary functions such that the above integrals exist. Choose g1 = h and g2 = cGk where c is a real number to be determined later. We have c
h(q) ( + λk )Gm (p, q) dq = c
Ω
∂Gm ∂h h(q) (p, q) − Gm (p, q) ∂νq ∂νq
dσ (q). (74)
∂Ω
Since ( + λm )Gm (p, q) = δp , we then get ch(p) + c(λk − λm )
h(q)Gm (p, q) dq Ω
∂Gm ∂h h(q) dσ (q). =c (p, q) − Gm (p, q) ∂νq ∂νq
(75)
∂Ω
Setting c :=
1 λm −λk ,
we then get, for p ∈ Ω,
h(q)Gm (p, q) dq = Ω
∂h 1 −Dm (h)(p) + Sm (p) + h(p) , λm − λk ∂ν
(76)
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where Sm , Dm are respectively the single-layer and double-layer potentials associated to Gm (cf. Section B.1). By applying the normal derivative operator to the two side terms of Eq. (76) and then taking ∂ into account the jump relations (103) and ( 12 I + Kk∗ ) ∂ν h = Ek (h), we get that ∂ ∂ν
Ω
∂h ∂ 1 −Dm (h)(p) + Sm (p) + h(p) h(q)Gm (p, q) dq = λm − λk ∂ν ∂ν
1 ∂h ∂h −1 ∗ = −Em (h|∂Ω ) + + Km + (p) λm − λk 2 ∂ν ∂ν
1 ∂h 1 ∗ ∂h −Em (h|∂Ω ) + Km + (p) = λm − λk ∂ν 2 ∂ν
% & 1 ∂h ∗ ∂h = −Em (h|∂Ω ) + Km + Ek (h|∂Ω ) − Kk∗ (p) λm − λk ∂ν ∂ν
∂h ∗ 1 ∗ (Ek − Em )(h|∂Ω ) + Km − Kk (p). 2 = λm − λk ∂ν
We are now able to provide an argument for Proposition 5.10. Proof of Proposition 5.10. According to (107), we first easily get that the contribution of λk (uq∗ )ξ1 to
∂ξ1,d ∂ν (q∗ )
is a term of class C 2 and that
∂ξ1,d ∂ 1 ∗ I + Kk (p) = μ(q + q∗ )φ1 (q)Gk (p, q) dq + R1 . 2 ∂ν ∂νp
(77)
Ω
We need the Taylor expansion of the right-hand side of (77) when a boundary point p belongs to an open neighborhood (in ∂Ω) of q∗ (i.e., (0, 0)). For that purpose, we perform the following decomposition. ∂ ∂νp
μ(q
+ q∗ )φ1 (q)Gk (p, q) dq
Ω
=
μ(q + q∗ )φ1 (q)
Ω
∂Gk (p, q) dq ∂νp
= μ(p + q∗ )I1 (p) + I2 (p), where I1 (p) = Ω
φ1 (q)
∂Gk ∂ (p, q) dq = ∂νp ∂νp
φ1 (q)Gk (p, q) dq,
Ω
and I2 = Ω
∂Gk (p, q) dq. μ(q + q∗ ) − μ(p + q∗ ) φ1 (q) ∂νp
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We first treat I1 (p). Since φ1 verifies ( + λ1 )φ1 = 0, we can apply Lemma 5.11 and we get I1 (p) =
∂φ1 −1 (Ek − E1 ) φ1 ∂Ω + K1∗ − Kk∗ . λk − λ1 ∂ν
(78)
Using the arc-length σ , recall that p = O(σ ). Thus, we have p + q∗ = q∗ + O(σ ) and we write I1 (σ ) for I1 (p(σ )). According to (116), we first deduce that 1 1 ∂φ1 (0)σ ln |σ | + R2 (Ek − E1 ) φ1 = (a1 + 2a2 )(λk − λ1 ) λk − λ1 λk − λ1 ∂ν = (a1 + 2a2 )
∂φ1 (0)σ ln |σ | + R2 . ∂ν
(79)
As Kk∗ and K1∗ have the same principal part, by Lemma B.5, we know that (Kk∗ − K1∗ ) 2-regular term. Then, I1 (σ ) = −(a1 + 2a2 )
∂φ1 (0)σ ln |σ | + R2 . ∂ν
∂φ1 ∂ν
is a
(80)
We now treat I2 (p). Taking into account the Taylor expansion of μ at p + q∗, we can rewrite I2 (p) = dμ(p + q∗ )J2 (p) + R(p) where J2 (p) =
(q − p) Ω
∂Gk (p, q)φ1 (q) dq, ∂νp
and R(p) = Ω
∂Gk O q − p2 (p, q)φ1 (q) dq. ∂νp
Since R(·) is a more than J2 (·), it is enough to prove that J2 is of class C 1 . regular term Note that J2 = Ω H (p, q)φ (q) dq with H (·,·) the convolution kernel given by H (p, q) := k (q − p) ∂G ∂νp (p, q), p = q. The kernel H is no longer singular (it is actually uniformly bounded on its domain of definition) and straightforward computations yield that H defines a pseudodifferential operator of class −3/2. Recall that H0 ∈ H 1/2−ε (∂Ω) for every ε > 0, we deduce that φ1 ∈ H 1−ε (Ω) for every ε > 0, then I2 ∈ H 5/2−ε (∂Ω) for every ε > 0. Then σ → J2 (p(σ )) admits a continuous derivative in an open neighborhood of zero. We conclude that the contribution ∂ξ
of J2 to ∂ν1,d (σ ) yields an R1 term. By Theorem B.4, we finally get ∂ξ1,d
∂ν
(σ ) = −(2a1 + 4a2 )μ(q∗ )
∂φ1 (0)σ ln |σ | + R1 . ∂ν
2
(81)
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5.5. Proof of Proposition 5.4 Collecting the results of Theorems 5.7 and 5.9 in (43), we get that, for σ in some open neighborhood of zero, one has M (uq∗ )(σ ) =
∂φ1 (0) ∂ξk (0) ∂φk (0) ∂ξ1 (0) 1 ∂φ1 (0) ∂φk (0) λ1 + λk + μ(q∗ ) σ ln |σ | + R1 π ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν
= −H0
∂M (0). ∂ν
(82)
The left-hand side of (82) is continuous at σ = 0, which implies that ∂M ∂ν (0) = 0. Then the left1 hand side of (82) must be of class C at σ = 0, implying that the coefficient of σ ln |σ | must also be equal to zero. We finally get that λ1
∂φ1 (0) ∂ξk (0) ∂φk (0) ∂ξ1 (0) ∂φ1 (0) ∂φk (0) + λk + μ(q∗ ) = 0, ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν
(83)
and, since q∗ is an arbitrary point of ∂Ω, we get λ1
∂ξk ∂ξ1 ∂φ1 ∂φk ∂φ1 ∂φk (q) (q) + λk (q) (q) + μ(q) (q) (q) = 0 on ∂Ω. ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν
1 Consider now equations (38) and (84) as a linear system with ∂ξ ∂ν (q) and After an elementary algebraic manipulation, we have, for every q ∈ ∂Ω,
∂ξ1 1 ∂φk ∂φ1 (q) (q) − (q) = 0, μ(q) ∂ν ∂ν λ1 − λk ∂ν ∂φ1 ∂ξk 1 ∂φk (q) − (q) = 0. μ(q) ∂ν ∂ν λk − λ1 ∂ν
(84)
∂ξk ∂ν (q) as unknowns.
(85) (86)
As φ1 and φk are eigenfunctions of −D Ω , by Holmgren uniqueness theorem (see [43, Proposition 4.3, p. 433]), their normal derivatives cannot be equal to zero on a subset of ∂Ω with nonnull measure. Then, by a simple density argument, we get ∂ξ1 1 ∂φ1 (q) − (q) = 0 on ∂Ω, μ(q) ∂ν λ1 − λk ∂ν 1 ∂ξk ∂φk (q) − (q) = 0 on ∂Ω. μ(q) ∂ν λk − λ1 ∂ν
(87) (88)
What we have proved so far is that, if property (Bk ), k > 1, is not generic then a certain property (Ck ) is not as well, where the latter property is defined exactly as in Definition 5.2 except that the function M defined in (30) is replaced by the function S : ∂Ω → R2 defined by S(q) :=
∂φ1 ∂ξ1 1 μ(q) (q) − (q) ∂ν λ1 − λk ∂ν
for q ∈ ∂Ω.
(89)
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As in Proposition 5.4, it now amounts to prove that the function S defined in (89) cannot be identically equal to zero on any Eρ (Ω) with ρ > 0. We can follow the same strategy developed in Section 5.2 and use the computations made in Section 5.4. Reasoning by contradiction, we assume that S ≡ 0 on ∂Ω. Taking the shape differentiation of that equation and using a variation uq∗ for an arbitrary q∗ ∈ ∂Ω, we get ∂ξ1 (uq∗ ) ∂φ (uq ) 1 1 ∂ ∂ξ1 ∂φ1 (q) − (q) − (q) . μ(q) 1 ∗ (q) = −H0 μ(q) ∂ν λ1 − λk ∂ν ∂ν ∂ν λ1 − λk ∂ν
(90)
Using Propositions 5.8 and 5.10, we have at σ = 0
∂ξ1 1 ∂φ1 1 (0) − (0)μ(q∗ ) a1 p.v. ∂ν λ1 − λk ∂ν σ
1 ∂ ∂φ1 ∂ ∂ξ1 (0) − (0)μ(q∗ ) a1 ln |σ | + O σ ln |σ | +2 ∂τ ∂ν λ1 − λk ∂τ ∂ν 1 ∂ ∂ξ1 ∂φ1 − μ(q) (0). = −H0 ∂ν ∂ν λ1 − λk ∂ν
−2
By (87), we simplify the previous equation and get
∂ξ1 ∂ ∂φ1 1 (0) − (0)μ(q∗ ) a1 ln |σ | + O σ ln |σ | ∂ν λ1 − λk ∂τ ∂ν
1 ∂ ∂ξ1 ∂φ1 − μ q(σ ) = H0 ∂ν ∂ν λ1 − λk ∂ν
∂ 2 ∂τ
Since the right-hand side remains bounded in neighborhood of σ = 0, it is necessary that ∂ ∂τ
∂ξ1 1 ∂ ∂φ1 (0) − (0)μ(q∗ ) = 0. ∂ν λ1 − λk ∂τ ∂ν
(91)
On the other hand, by taking the tangent derivative of (87) at q = q∗ , we have
∂S ∂ q(τ ) = ∂τ ∂τ
∂ ∂φ1 ∂ξ1 1 ∂φ1 (0) − (0)μ(q∗ ) + (0)dμ(q∗ ) · τ0 = 0, (92) ∂ν λ1 − λk ∂τ ∂ν ∂ν
where τ0 is the unit tangent vector on ∂Ω at the point q∗ . From (91) and (92), we end up with ∂φ1 (0) dμ(q∗ ) · τ0 = 0. ∂ν
(93)
As the previous reasoning is valid almost everywhere on ∂Ω, we have ∂φ1 (q) dμ(q) · τq = 0, ∂ν
for q ∈ ∂Ω.
(94)
1 By condition (36) and by continuity of the map q → dμ(q) · τq for q ∈ ∂Ω, we get that ∂φ ∂ν is equal to zero on an open neighborhood of q¯ on ∂Ω (defined in (36)). This is not possible by
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Holmgren uniqueness theorem. We finally derived a contradiction and Proposition 5.4 is now proved. 6. Conclusion, conjectures, perspectives We recapitulate all the controllability results known for (3) in the following array.
1D Ω = (0, 1)
2D
Spectral controllability in time T of (3) yes under (H1) ∀T > 0
Exact controllability in time T of (3) yes under (H2) (i.e., generically with respect to μ) 3 ((0, 1), C) with L2 ((0, T ), R)-controls in H(0) ∀T > 0
no under no (H1) yes under (H3) (i.e., generically with respect to (Ω, μ)) with T > Tmin (Ω)
no under no (H1) yes under (H3) and (H4) in abstract spaces with T > Tmin (Ω)
no under (H3) and (H4) with T < Tmin (Ω) no under no (H3) no under (H4)
3D
no under no (H3) no under (H4)
In this array, we have used the notation Tmin (Ω) := 2πd(Ω) where d(Ω) > 0 is such that k ∈ N∗ ; λk − λ1 ∈ [0, t] ∼ d(Ω)t,
when t → +∞.
and the assumptions (H1) μϕ1 , ϕk L2 (Ω) = 0, for every k ∈ N∗ , (H2) there exists c1 , c2 > 0 such that, c2 c1 , μϕ , ϕ 1 k k3 k3
∀k ∈ N∗ ,
(H3) any eigenvalue λ of −D Ω has multiplicity m n (n = 2, 3 is the space dimension: Ω ⊂ Rn ) and the vectors μφ1 , φk1 , . . . , μφ1 , φkm are linearly independent in Rn , where k1 < · · · < km and φk1 , . . . , φkm are the eigenvectors associated to λ. (H4) there exists μ˜ ∈ C 0 (Ω, R) such that μ(q) = μ(q)e ˜ 1. The assumption (H4) is not necessary for the nonspectral controllability of (3) in small time in 2D (see Remark 3.3). We conjecture that, in 2D, the system (3) is not spectral controllable in small time under (H3). This is an open problem.
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Similarly, the assumption (H4) is not necessary for the nonspectral controllability of (3) in any time T > 0 in 3D. We conjecture that, in 3D, and with any time T > 0, the system (3) is not spectral controllable in time T under (H3). This is an open problem. A strategy to prove these conjectures could be the adaptation of Haraux and Jaffard’s result (Theorem 3.2) to vector exponential families. Acknowledgments The authors would like to thank Jean-Michel Coron and Enrique Zuazua for helpful comments. This work was supported by Digiteo and Region Ile-de-France. Appendix A. Shape differentiation The material presented here is borrowed from [42] and [35]. A.1. Main definitions Let Ω be a domain in D3 . For a positive integer l, we consider perturbations u in the space W l,∞ (Ω, R2 ) with norm ul,∞ := sup ess D α u(x); 0 α l, x ∈ Ω . Then, the domain Ω + u is defined by Ω + u := (Id+ u)(Ω) = x + u(x), x ∈ Ω . Lemma A.1. (Cf. [42].) Let l ∈ N∗ and u ∈ W l,∞ (Ω, R2 ) be such that ul,∞ 1/2. Then, the map Id+ u is invertible. Furthermore, there exits w ∈ W l,∞ (Ω + u, R2 ) such that (Id+ u)−1 = Id+ w and wl,∞ Cl ul,∞ where Cl is a constant independent on u. Remark A.1. According to this result, if Ω is of class C j , we can choose l = j + 1 so that the new domain Ω + u is also of class C j . In particular, if we need domains of class C 3 , we can choose W 4,∞ (Ω, R2 ) as the perturbation space. We now consider a function
v : u ∈ W l,∞ Ω, R2 → v(u) ∈ W m,r (Ω + u) where 1 r < ∞ and m l are integers. In practice, v(u) is solution of a suitable problem, which depends on the perturbation function u. We are interested in the study of the regularity of the function v(u) with respect to the perturbation function u. Definition A.1 (First order local variation). Let k m 1 and 1 r < ∞. We say that the function v(u) has a first order local variation at u = 0 on W m, r (Ω + u) for all u ∈ W l,∞ (Ω, R2 ) m−1,r if there exists a linear map v (u) from u ∈ W l,∞ (Ω, R2 ) to v (u) ∈ Wloc (Ω) such that, for every open set ω Ω,
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v(u) = v(0) + v (u) + θ (u)
3963
in ω,
when ul,∞ is small enough and θ (u)m−1,r → 0 as ul,∞ → 0. ul,∞ Remark A.2. The first order local variation can also be defined as v(tu)|ω − v(0)|ω t→0 t
v (u) = lim
in W m−1,r (ω),
where ω Ω. The following theorem provides sufficient conditions for the existence of the first order local variation. Theorem A.2. (Cf. [42].) Let Ω be a C 0,1 domain. Consider the map u → v(u) ∈ W m,r (Ω + u) defined on a neighborhood of u = 0 in W k,∞ (Ω, R2 ), with k m 1 and 1 r < ∞. Assume ˙ ∈ W m, r (Ω) such that that there exists a linear continuous map u ∈ W k,∞ (Ω) → v(u) ˙ + θ (u) v(u) ◦ (Id+ u) = v(0) + v(u)
in W m, r (Ω),
for all u ∈ W k,∞ (Ω, R2 ) small enough, where θ (u)m−1,r → 0 as uk,∞ → 0. uk,∞ Furthermore, we assume that for every u ∈ W k,∞ (Ω, R2 ) small enough, v(u) = 0 on ∂Ω + u. Then, for each ω Ω, the function u → v(u)|ω ∈ W m−1,r (ω) defined on a neighborhood of u = 0 in W k,∞ (Ω, R2 ) is differentiable at u = 0. Moreover, the map u → v(u)|ω has a first order local variation and this variation at u = 0 in the direction u1 denoted by v (u1 ) verifies v (u1 ) ∈ W m−1,r (Ω) and v (u1 ) = −u1 , ν
∂v(0) ∂ν
on ∂Ω.
(95)
A.2. Regularity of the eigenvalues and eigenfunctions By applying [36, Theorem 3] in the same way as in [35], we get the following result. Theorem A.3. Let Ω ⊂ R3 be an open bounded domain of class C 1 . Let λ be an eigenvalue of multiplicity h of −D Ω , with associated orthonormal eigenfunctions y1 , . . . , yh . Then, there exist , and h continuous functions with values in H 2 ∩ h real-valued continuous functions, u → λΩ+u i 1 H0 (Ω, R), u → yi (u), for i = 1, . . . , h, defined in a neighborhood U of u = 0 in W 4,∞ (Ω, R3 ) such that the following properties hold,
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• λΩ i = λ for i = 1, . . . , h, • for every u ∈ U , ϕiΩ+u := yi (u) ◦ (I + u)−1 is an eigenfunction of −D Ω+u associated to the , eigenvalue λΩ+u i • for every u ∈ U , the family (ϕ1Ω+u . . . , ϕhΩ+u ) is orthonormal in L2 (Ω + u, R), • for each open interval I ⊂ R, such that the intersection of I with the set of eigenvalues of −D Ω contains only λ, there exists a neighborhood UI ⊂ U such that, for every u ∈ UI , there exist exactly h eigenvalues (counting the multiplicity), λΩ+u , 1 i h, of −D Ω+u i contained in I , • for each u ∈ W 2,∞ (Ω, C) and 1 i h, the map R → R × H 2 ∩ H01 (Ω, R),
, yi (tu) t → λΩ+tu i is analytic in a neighborhood of t = 0. A.3. Local variations of the eigenvalues and eigenfunctions Let Ω ⊂ R2 be an open bounded domain of class C 1 . Let λ be an eigenvalue of multiplicity h of −D Ω , with associated orthonormal eigenfunctions y1 , . . . , yh . Let ϕi (u) ∈ H 2 ∩ H01 (Ω + u, R), i = 1, . . . , h be the eigenfunctions of −D Ω+u associated to the eigenvalues λi (u), i = 1, . . . , h, where λi (0) = λ for i = 1, . . . , h. According to the result of the previous section, the functions t → λi (tu) and t → ϕi (tu) are analytic in a neighborhood of 0. Let us denote by &
dϕi λi (u0 ) resp. du u0 the value of the directional derivative of λi (resp. ϕi ) at u = 0 in the direction u0 , λi (tu0 ) − λi (0) . t→0 t
λi (u0 ) := lim For i = 1, . . . , h,
dϕi du ]u0
∈ H 2 (Ω, R) and, for every open subset ω Ω,
dϕi du
& := lim u0
t→0
ϕi (tu0 )|ω − ϕi (0)|ω t
in H 2 ω, R3 .
We have, for every t ∈ R, ⎧ −ϕi (tu0 ) = λi (tu0 )ϕi (tu0 ) in Ω + tu0 , ⎪ ⎪ ⎪ ⎨ ϕi (tu0 ) = 0 on ∂(Ω + tu0 ), ϕi (tu0 )(q)2 dq = 1. ⎪ ⎪ ⎪ ⎩ Ω+tu0
Thus, using classical results on shape differentiation (see [42]), we get
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& ⎧ dϕi ⎪ ⎪ −( + λ ) = λi (u0 )ϕi i ⎪ ⎪ du ⎪ u0 ⎪ & ⎪ ⎪ ⎨ dϕi = −u0 .∇ϕi on ∂Ω, du u0 ⎪ ⎪ & ⎪ ⎪ dϕi ⎪ ⎪ ⎪ ϕ (q) (q) dq = 0. i ⎪ ⎩ du u0
3965
in Ω, (96)
Ω
Remark A.3. We note that all results stated above can be easily extended for C 3 domains and variations u ∈ W 4,∞ (Ω, R2 ). Appendix B. The Dirichlet-to-Neumann map for the Helmholtz equation Let Ω ⊂ R2 be a bounded domain with a connected boundary ∂Ω of class C 3 and outward unit normal ν. For k > 0, we consider the following problem
+ k 2 u = F, in Ω, u = f,
(97)
on ∂Ω.
The goal of this section is to study the Dirichlet-to-Neumann map associated to (97) when −k 2 is an eigenvalue of the interior Dirichlet problem. In Section B.1, we recall some useful background results (see, for instance, [4,13,34,44]). In Section B.2, we study precisely the Dirichlet-to-Neumann map associated to (97). B.1. Preliminary results on Helmholtz equation A standard approach for studying the Helmholtz equations consists in the representation of the solution using the single and double layer potentials respectively defined by Sk (f )(p) :=
Gk (p, q)f (q) dσ (q),
∀p ∈ R2 \∂Ω,
(98)
∂Ω
and Dk (f )(p) :=
∂Gk (p, q) f (q) dσ (q), ∂νq
∀p ∈ R2 \∂Ω,
(99)
∂Ω
where Gk (. , .) is the fundamental solution of the Helmholtz equation that satisfies the Sommerfeld condition and f ∈ L2 (∂Ω). Here the notation ∂ν∂ q stands for the outward unit normal to ∂Ω at the point q. Then the solution of (97) is given by the third Green formula,
u = −Sk where
∂u + Dk (f ) + F ∗ Gk , ∂ν
(100)
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F ∗ Gk (p) :=
F (q)Gk (p, q) dq,
∀p ∈ Ω.
Ω
For the reader’s convenience, we recall the following useful standard result which highlights the difference between the fundamental solution G0 of the Laplace equation and Gk (see [2] and [33]). B.1.1. Fundamental solution Proposition B.1. Let k > 0. The fundamental solution for the Helmholtz equation is
i Gk (p, q) = − H01 k|p − q| 4
(101)
where H01 denotes the Hankel function of the first kind of order 0. If G0 (p, q) := the fundamental solution of the Laplace equation, then we have
1 2π
ln |p − q| is
Gk (p, q) = G0 (p, q) + gk (p, q), (1)
(102)
(2)
where gk = gk + gk with gk(1) (p, q) := −
∞ k k|p − q| (−1)j k|p − q| 2j 1 1 , ln + ln 2π 2 2π 2 2 (j !)2 j =1
and
∞
(−1)j i k|p − q| 2j (2) gk (p, q) := − J0 k|p − q| + ψ(j + 1) , 4 2 (j !)2 j =1
with ψ, the digamma function and J0 , the Bessel function of first kind. The interested reader can find details in [1] and [12]. B.1.2. Jump relations Now, let us state jump relations satisfied by the layer potentials and their normal derivative. We recall the standard notations f |± (p) = lim f (x ± tνp ), t→0+
p ∈ ∂Ω,
and ∂ f |± (p) = lim ∇f (p ± tνp ), νp , νp t→0+ We quote from [4, Lemma 11.1, p. 186] the following result.
p ∈ ∂Ω.
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Theorem B.2. Let Ω be a C 3 domain in R2 and let f ∈ L2 (∂Ω). We have
Sk (f ) + (p) = Sk (f ) − (p) = Sk (f )(p), a.e. p ∈ ∂Ω,
1 ∂ Sk (f ) (p) = ± I + Kk∗ f (p), a.e. p ∈ ∂Ω, ∂νp 2 ±
1 Dk (f )|± (p) = ∓ I + Kk f (p), a.e. p ∈ ∂Ω, 2
(103)
where Kk is the operator defined by Kk φ(p) := p.v.
∂Gk (p, q) φ(q) dσ (q), ∂νq
p ∈ ∂Ω,
(104)
∂Ω
and where Kk∗ is its L2 (∂Ω)-adjoint. An other operator will be of interest and will play a major role in our computations. It is the normal derivative of Dk (f ), Ek (f )(p) :=
∂ ∂νp
∂Gk (p, q)f (q) dσ (q) , ∂νq
p ∈ ∂Ω.
(105)
∂Ω
Remark B.1. There is not a jump relation for the normal derivative of the double-layer potential across the boundary ∂Ω. B.1.3. Mapping properties in Sobolev spaces The following results are also needed (see [44, Chapter 7] and [13, Chapter 3]). Theorem B.3. Let Ω be a C 3 domain and s ∈ R. Then, (i) (ii) (iii) (iv) (v)
the operator Sk is bounded from H s (∂Ω) into H s+1 (∂Ω), the operators Kk and its adjoint Kk∗ are bounded from H s (∂Ω) into H s+1 (∂Ω), the operators I2 ± Kk∗ and I2 ± Kk are bounded from: H s (∂Ω) into H s (∂Ω), the operator Kk∗ − K0∗ is continuous from H s (∂Ω) into H s+3 (∂Ω). the operator Ek is bounded from H s (∂Ω) into H s−1 (∂Ω).
Proof. The results concerning the single and double layer potential are developed in (cf. [34, Chapter 4, paragraph 2])) where are studied the boundedness properties of singular integral operators whose kernels are the restriction to ∂Ω of kernels defined in R2 . (1) In R2 , the layer potential kernel associated to Helmholtz equation is K(x) = H0 (k|x|) where (1) H0 is the Hankel function of order 0. A Taylor expansion shows that the kernel is pseudohomogeneous of class −1. Thanks to [34, Theorem 4.3.1], we conclude that Sk is bounded from H s (∂Ω) into H s+1 (∂Ω), Concerning the double layer potential, its regularity property is due to the fact that its kernel is pseudo-homogeneous of class −1. From [34, Theorem 4.3.1], Kk and Kk∗ are bounded from
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H s (∂Ω) into H s+1 (∂Ω) for every real s. We point out that one can find the detailed computations in [34, Example 4.5, Section 4.3.3 ]. A Taylor expansion shows that the kernel of the operator Kk∗ − K0∗ has the same property as E(x, y), the kernel of the single layer potential corresponding to the biharmonic equation (cf. [17]). We recall that
∂E(x, y) 1 νy , x − y 2 ln |x − y| + 1 . = ∂νy 8π The factor νy , x − y is regular on ∂Ω × ∂Ω and furthermore for small |x − y| it satisfies
νy , x − y = O |x − y|2 .
(106)
Thus, for an element (x, y) living near the diagonal ∂Ω × ∂Ω, we have
∂E(x, y) = O |x − y|2 ln |x − y| . ∂νy It follows that E(x, y) and the kernel of Kk∗ − K0∗ have the same smoothing effects. Furthermore, from [34, Example 4.3, p. 216], we get that the kernel of Kk∗ − K0∗ is pseudo-homogeneous of class −3. Thanks to [34, Theorem 4.3.1], it comes that Kk∗ − K0∗ is continuous from H s (∂Ω) into H s+3 (∂Ω), for every real s. To finish, we see Ek as a pseudodifferential operator on ∂Ω whose leading symbol is of the form p(ξ ) = − 12 |ξ |. Consequently, the operator Ek is continuous from H s (∂Ω) into H s−1 (∂Ω). 2 B.2. Dirichlet-to-Neumann map The goal of this section is the study of the singularities of the normal derivative of the solution of (97). From (100), (103) and (105), we deduce
1 ∂ ∗ ∂u I + Kk = Ek (f ) + (F ∗ Gk ). 2 ∂ν ∂ν
(107)
In Section B.2.1, we study the inverse of the operator ( 12 I + Kk∗ ), thanks to the reduced resolvent theory. In Section B.2.2, we study the normal derivative of the double-layer potential Ek (f ). B.2.1. Singular perturbation problem and reduced resolvent Notice that, when −k 2 is an eigenvalue of the interior Dirichlet problem for the Laplacian, the integral equation (107) is not invertible. The associated operator ( 12 I + Kk∗ ) is in fact invertible except for these critical values. In this subsection, we show how to solve (107) in an efficient manner. More precisely, we consider a general right-hand side v, which is assumed to belong to the range of 12 I + Kk∗ and whose Taylor expansion in an open neighborhood of zero takes the following form,
1 v(σ ) = α1 p.v. + α2 ln |σ | + α3 σ ln |σ | + R2 , σ
(108)
K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
3969
where α1 , α2 and α3 are arbitrary real numbers and where σ denotes the oriented counterclockwise arc-length of the boundary ∂Ω and R2 is an error term defined in Definition 5.4. The main idea is to break up the explicit formula of ∂u ∂ν into two parts. The first part reflects ∂u the singular behavior of ∂ν and it will not depend on the eigenvalue −k 2 of the Laplacian. The second part is a regular remainder of the type R2 . Precisely, the goal of this subsection is the proof of the following result. Theorem B.4. Assume that
∂u ∂ν
satisfies the equation
1 ∗ ∂u I + Kk = v, 2 ∂ν
(109)
where v is given by (108). Then, we have ∂u = 2v + T0 v + R2 , ∂ν
(110)
where the linear operator T0 given by
T0 := −2
1 I + K0∗ 2
−1
K0∗ ,
(111)
defines a bounded operator from H s (∂Ω) into H s+1 (∂Ω), for every s ∈ R. For the proof of the result, precise information on (Kk∗ − K0∗ ) p.v.( σ1 ) is needed. Although we know the higher smoothing effect of Kk∗ − K0∗ the operator, we have to show the following result. Lemma B.5. Let k > 0. The distribution (Kk∗ − K0∗ ) p.v.( σ1 ) is of the type R2 . Note that the above distribution makes sense thanks to Remark 5.5. Proof. We are led to study the Taylor expansion of L I (σ0 ) = −L
1 dσ, (σ − σ0 )2 ln |σ − σ0 | p.v. σ
(112)
for σ0 in an open neighborhood of zero. We may assume σ0 > 0 and we fix α > 0 small enough. We have to evaluate the following limit − lim
→0
1 (σ − σ0 ) ln |σ − σ0 | dσ + σ
α
2
−α
1 (σ − σ0 ) ln |σ − σ0 | dσ σ 2
α = lim
→0
dσ (σ − σ0 )2 ln |σ − σ0 | − (σ + σ0 )2 ln |σ + σ0 | σ
= I1 (σ0 ) + I2 (σ0 ) + I3 (σ0 ),
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K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
where we set α I1 (σ0 ) := lim
σ ln
→0
|σ − σ0 | dσ, σ0 + σ
α I2 (σ0 ) := −2σ0 lim
→0
ln |σ − σ0 | + ln |σ + σ0 | dσ,
α I3 (σ0 ) := σ02 lim
ln
→0
|σ − σ0 | dσ . σ0 + σ σ
estimate I1 (σ0 ). The function in the integral is integrable at σ = 0, then I1 (σ0 ) = α We first |σ −σ0 | σ ln 0 σ0 +σ dσ . We first make the change of variable t = σ/σ0 . We get I1 (σ0 ) = 1 X |t−1| 2 σ0 (C0 + J1 (α/σ0 )), where C0 = 0 t ln |t−1| t+1 dt and J1 (X) = 1 t ln t+1 dt for X 1. By integrating by part J1 , we obtain I1 (σ0 ) = C0 σ02 +
α 2 − σ02 α − σ0 − σ0 (α − σ0 ). ln 2 α + σ0
Then I1 is of class C 2 in a neighborhood of zero. We next consider I2 (σ0 ). We have α−σ 0 I2 (σ0 ) = −2σ0
"
α+σ 0
ln |s| ds + −σ0
ln |s| ds σ0
= −2σ0 (α − σ0 ) ln |α − σ0 | + (α + σ0 ) ln |α + σ0 | − 2α , which show that I2 is real analytic in an open neighborhood of zero. Finally, we estimate I3 (σ0 ). We have α I3 (σ0 ) = σ02
lim
→0
= σ02 C1
v=
t−1 t+1
α/σ 0
/σ0
ln
|1 − t| dt 1+t t
+ σ02 H1 (α/σ0 ),
X dt and H1 (X) = 1 ln t−1 for X 1. Making the change of variable β ln v t+1 t 0 in H1 , we have H1 (α/σ0 ) = 2 0 1−v 2 dv, where β = α−σ α+σ0 . We note that β < 1. Then,
where C1 =
1
|σ − σ0 | dσ = σ02 lim ln →0 σ0 + σ σ
0
ln 1−t 1+t
β 0
dt t
ln v dv = 1 − v2 =
β ln v 0
v
dv =
n0
β 2n+1 ln β n0
β
2n
2n + 1
(ln v)v 2n dv
n0 0
−
n0
β 2n+1 = S1 + S2 . (2n + 1)2
K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
3971
For S1 , we have
n β (−β)n 1 − S1 = ln(β) 2 n n n1
n1
1 = ln(β) − ln(1 − β) + ln(1 + β) 2 1 1+β = ln ln β 2 1−β
1 α 2σ0 = ln 1 − ln . 2 α + σ0 σ0 dS2 dσ0 .
For S2 , we begin by computing
β 2n dβ dS2 2α 1 β 2n+1 =− = dσ0 2n + 1 dσ0 β 2n + 1 (α + σ0 )2 n0
=
α2
n0
α α 1 ln = 2 σ − σ0 0 α(1 −
σ02 ) α2
ln
α . σ0
Recall that I3 (σ0 ) = σ02 C1 + σ02 (S1 + S2 ), the computations above show that I3 is a 2-regular term. 2 We are now ready to prove Theorem B.4. Proof of Theorem B.4. We subdivide the proof in several steps. Step 1: We begin to recall some results on the reduced resolvent theory (cf. [27, Chapter I, paragraph 5]). Since λ = 0 is an eigenvalue of ( 12 I + Kk∗ ), the resolvent
R(λ) =
−1 1 − λ I + Kk∗ 2
has a singularity at λ = 0. Since the dimension of the eigenspace associated to λ = 0 is equal to one, the resolvent is expanded as a Laurent series
−1 ∞ 1 A−1,k n n+1 ∗ − λ I + Kk + = λ A0,k 2 λ n=0
in a neighbourhood of λ = 0. The notations A−1,k and A0,k stand for A−1,k :=
1 2iπ
Γ
and
−1 1 − λ I + Kk∗ dλ, 2
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K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
1 A0,k := 2iπ
1 λ
−1 1 ∗ − λ I + Kk dλ, 2
Γ
where Γ is a small positively oriented circle enclosing 0 in C. According to [27], the operator P0 := −A−1,k is a projector on the null space associated to λ = 0 and moreover
A0,k P0 = P0 A0,k = 0,
1 1 ∗ ∗ I + Kk A0,k = A0,k I + Kk = I − P0 . 2 2
The last equalities show that A0,k is the “inverse” of ( 12 I + Kk∗ ) restrained to the complementary subspace to the null space associated to λ = 0. Step 2: Using the reduced resolvent method, one gets ∂u = A0,k v + U(v), ∂ν
(113)
where U() is an arbitrary element belonging to Ker( 12 I + Kk∗ ). Recall that Ker( 12 I + Kk∗ ) coincides with the span of the traces of all normal derivatives on ∂Ω of Dirichlet eigenfunctions of the Laplacian with eigenvalue −k 2 (see [16, p. 684]). We then deduce that U() is of type R2 . We can now rewrite Eq. (113) as follows −1 & %
1 v + U(v) I + K0∗ v + A0,k − 2
−1 −1
−1 & %
1 1 dλ 1 1 = v + U(v) I + K0∗ − λ I + Kk∗ I + K0∗ v+ − 2 2iπ λ 2 2
∂u = ∂ν
1 I + K0∗ 2
1 I + K0∗ v 2 −1 −1 & %
∗
1 1 1 dλ v + U(v) − λ I + Kk∗ I + K0∗ + K0 − Kk∗ + λI 2iπ λ 2 2
=
Γ
−1
=
−1
Γ
1 I + K0∗ 2
−1
v + A0,k
% −1 & %
−1 &
∗
1 1 K0 − Kk∗ v + A−1,k I + K0∗ I + K0∗ v 2 2
+ U(v).
(114)
Writing
it follows that
1 I + K0∗ 2
−1
1 = 2I − 2 I + K0∗ 2
−1
K0∗ ,
K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
3973
−1
1 ∂u ∗ = 2v − 2 I + K0 K0∗ v + V(v) + W(v), ∂ν 2 where % V(v) := A0,k
−1 &
∗
1 ∗ ∗ K0 − Kk v, I + K0 2
and %
W(v) := A−1,k
1 I + K0∗ 2
−1 &
v + U(v).
Since −A−1,k is a projector on the null eigenspace associated to the zero eigenvalue, the remainder W(v) belongs to R2 . The smoothing effects of Kk∗ − K0∗ described in Lemma B.5 and Theorem B.3(iv) show that V(v) belongs also to R2 . Concerning the term T0 v, its regularity is deduced from the fact that 1 I + K0∗ : H s (∂Ω) → H s (∂Ω) 2 is an isomorphism and that K0∗ is a bounded operator from H s (∂Ω) → H s+1 (∂Ω) for every real s. 2 B.2.2. Normal derivative of the double-layer potential In [16], the normal derivative of a double-layer potential is investigated in dimension three. For our purpose, we adapt their computations in dimension two. Lemma B.6. Let k ∈ C with Im k 0 and f ∈ D (∂Ω). We have ( ' ∂f ∂ψ 2 , +k ψ(p) f (q)Gk (p, q)νq , νp dσ (q) dσ (p), Ek (f ), ψ = − Gk ∗ ∂τ ∂τ
∂Ω
∂Ω
∀ψ ∈ D(∂Ω), where . , . refers to the D (∂Ω)/D(∂Ω)-duality, and ∗ is the convolution product on ∂Ω. Remark B.2. For details about the convolution product defined on ∂Ω, we can refer to [41, Chapitre IV, pp. 166–168]. Lemma B.7. Let f := H0 g where H0 is the Heaviside function with a jump at zero and g : ∂Ω → R is smooth. We have
2
∂g
∂ g ∂Gk 2 p(σ ) + (0)Gk + Gk ∗ H0 + k g + O σ2 , Ek (f ) p(σ ) = g(0) 2 ∂τ ∂τ ∂τ in the space of distributions D (∂Ω).
(115)
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K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
Proof of Lemma B.7. We apply Lemma B.6 to f = H0 g. On the one hand, we have ( ' (
' ∂g ∂ψ ∂f ∂ψ , = − Gk ∗ δ0 g(0) + H0 , − Gk ∗ ∂τ ∂τ ∂τ ∂τ ( ( ' '
∂g ∂ 2g ∂Gk ,ψ + (0)Gk , ψ + Gk ∗ H0 2 , ψ . = g(0) ∂τ ∂τ ∂τ On the other hand, using νp(0) , νq(σ ) = 1 + O(σ 3 ) in a neighborhood of σ = 0, we get
k
2
ψ(p)
∂Ω
=k
2
f (q)Gk (p, q)νq , νp dσ (q) dσ (p)
∂Ω
ψ(p)
∂Ω
f (q)Gk (p, q) 1 + O σ 2 dσ (q) dσ (p)
∂Ω
= k Gk ∗ f, ψ + O σ 2 , ψ . 2
2
Then, we have the following result. Proposition B.8. Let f = H0 g, with H0 the Heaviside function with jump at zero and g : ∂Ω → R a smooth function. We have
∂g ∂ 2g 1 + (0)a1 ln |σ | + a1 2 + (a1 + 2a2 )k 2 g (0)σ ln |σ | Ek (f ) p(σ ) = g(0)a1 p.v. σ ∂τ ∂τ + R2 ,
(116)
where a1 and a2 are defined in (62). Proof. According to Proposition B.1, we get
Gk (p, 0) = a1 ln |p| + ck + a2 k 2 |p|2 ln |p| + O |p|2 when p → 0,
(117)
where ck is a constant depending on k 2 . Thus, using (57) and (58) we get
Gk p(σ ), 0 = a1 ln |σ | + ck + a2 k 2 σ 2 ln |σ | + O σ 2 .
(118)
Similar computations show that
∂Gk 1 p(σ ), 0 = a1 p.v. + 2a2 k 2 σ ln |σ | + O(σ ). ∂τ σ 2
(119)
˜ Since Gk is a compactly supported Consider g˜ = ∂∂τg2 + k 2 g and calculate now Gk ∗ H0 g. distribution, then Gk ∗ H0 is a primitive of Gk (see for example [41, Chapitre IV, p. 168]). Thanks to the fact that g˜ is a smooth function, we conclude that
K. Beauchard et al. / Journal of Functional Analysis 256 (2009) 3916–3976
3975
Gk ∗ H0 g˜ = g(0)a ˜ 1 σ ln |σ | + αk + O(σ ), where αk is a constant of integration. This ends the proof of the proposition.
(120) 2
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Journal of Functional Analysis 256 (2009) 3977–3995 www.elsevier.com/locate/jfa
Eigenvalue inequalities for Klein–Gordon operators Evans M. Harrell II, Selma Yıldırım Yolcu School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA Received 1 October 2008; accepted 10 December 2008 Available online 7 January 2009 Communicated by J. Coron
Abstract We consider the pseudodifferential operators Hm,Ω associated by the prescriptions of quantum mechan ics to the Klein–Gordon Hamiltonian |P|2 + m2 when restricted to a bounded, open domain Ω ∈ Rd . When the mass m is 0 the operator H0,Ω coincides with the generator of the Cauchy stochastic process with a killing condition on ∂Ω. (The operator H0,Ω is sometimes called the fractional Laplacian with power 12 , cf. [R. Bañuelos, T. Kulczycki, Eigenvalue gaps for the Cauchy process and a Poincaré inequality, J. Funct. Anal. 211 (2) (2004) 355–423; R. Bañuelos, T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal. 234 (2006) 199–225; E. Giere, The fractional Laplacian in applications, http://www.eckhard-giere.de/math/publications/review.pdf].) We prove several universal inequalities for the eigenvalues 0 < β1 < β2 · · · of Hm,Ω and their means βk := k1 k=1 β . Among the inequalities proved are: βk cst.
k 1/d |Ω|
for an explicit, optimal “semiclassical” constant depending only on the dimension d. For any dimension d 2 and any k, βk+1
d +1 βk . d −1
Furthermore, when d 2 and k 2j , 1/d k d 1/d . 2 (d − 1) j βj βk
E-mail addresses: [email protected] (E.M. Harrell II), [email protected] (S. Yıldırım Yolcu). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.12.008
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E.M. Harrell II, S. Yıldırım Yolcu / Journal of Functional Analysis 256 (2009) 3977–3995
Finally, we present some analogous estimates allowing for an operator including an external potential energy field, i.e., Hm,Ω + V (x), for V (x) in certain function classes. © 2009 Elsevier Inc. All rights reserved. Keywords: Fractional Laplacian; Weyl law; Dirichlet problem; Riesz means; Universal bounds; Cauchy process; Dirac equation; Klein–Gordon equation; Semiclassical; Relativistic particle
1. Introduction The quantum-mechanical operator corresponding to the Klein–Gordon Hamiltonian is a firstorder pseudodifferential operator used to model relativistic particles in quantum mechanics. On unrestricted space the part representing kinetic energy |P|2 + m2 can be defined as the square root of −Δ + m2 , where m is a nonnegative constant corresponding to the mass, in units where the speed of light is set to 1. We restrict it to functions supported within bounded, open domain Ω ∈ Rd and designate the quantum version of |P|2 + m2 |Ω as Hm,Ω . (A full definition of Hm,Ω is provided below.) The operator Hm,Ω is positive definite with purely discrete spectrum consisting of positive eigenvalues 0 < β1 < β2 · · · . When m = 0 the operator H0,Ω reduces to the generator of the Cauchy stochastic process [5,35], and because H0,Ω Hm,Ω H0,Ω + m,
(1.1)
we shall sometimes be able to confine ourselves to this case without loss of generality. Our aim is to find analogues for Hm,Ω of some familiar inequalities of a general nature that apply to the eigenvalues 0 < λ1 < λ2 · · · of the Dirichlet problem for the Laplacian on a bounded, open domain. In some of these the spectrum is constrained by the shape and size of the domain Ω; for example the volume of Ω appears in both the Faber–Krahn lower bound for λ1 and in the Weyl estimate of λk as k → ∞. In addition, there are universal bounds, whereby either λk individually, or else some quantity involving many eigenvalues such as an average, a gap or a ratio, is controlled by a different spectral quantity, independently of the geometry of Ω. Various aspects of the well-developed subjects of geometric and universal bounds are treated, for instance, in [1,3,6,8,23]. One way to generate geometric and universal bounds for the Laplacian is based on identities for traces of commutators of operators [2,16,19–21,28], and with the benefit of hindsight these algebraic methods can be perceived implicitly in most of the classic universal spectral bounds for Laplacians [24,33,40]. Moreover, comparable universal bounds have been obtained with the same strategy for Schrödinger operators on Euclidean spaces [21], and both Laplacians and Schrödinger operators on embedded manifolds [10,11,14,16,17,19,29,32,41]. In many cases examples can be identified in which the inequalities are saturated. The plan of attack is to use trace identities to derive universal spectral bounds and geometric spectral bounds for Hm,Ω . The generator of the Cauchy process, corresponding to the case m = √ 0, is often referred to as the fractional Laplacian and designated −Δ [4,5,15]. The √ latter is, unfortunately, ambiguous notation, since this operator is distinct from the operator −ΔΩ as defined by the functional calculus for the Dirichlet–Laplacian −ΔΩ , except when Ω is all of Rd . For this reason we shall avoid the ambiguous notation when Ω is a proper subset of Rd . (For the spectral theorem and the functional calculus, see, e.g., [34].) Whereas several universal eigenvalue bounds, mostly of unknown or indifferent sharpness, have been obtained for higherorder partial differential operators such as the bilaplacian (e.g., [12,20,27,38,39]), and for some
E.M. Harrell II, S. Yıldırım Yolcu / Journal of Functional Analysis 256 (2009) 3977–3995
3979
first-order Dirac operators [9], universal bounds for pseudodifferential operators appear not to have been studied before. In a final section we study interacting Klein–Gordon operators of the form H = Hm,Ω + V (x),
(1.2)
which is used in a semi-relativistic approximation to model the quantum dynamics of a fastmoving spinless particle in an external field. Klein–Gordon operators can be conveniently defined using the Fourier transform on the dense subspace of test functions Cc∞ (Rd ). With the normalization 1 ϕ (ξ ) = F [ϕ] := (2π)d/2
exp (−iξ · x)ϕ(x) dx, Rd
the Laplacian is given by −Δϕ := F −1 |ξ |2 ϕ (ξ ), and therefore
−Δ + m2 ϕ := F −1 |ξ |2 + m2 ϕ (ξ ).
(1.3)
The semigroup generated on L2 (Rd ) is known explicitly, so that, for instance with m = 0, √ exp (− −Δt)[ϕ](x) = p0 (t, ·) ∗ ϕ,
(1.4)
where for t > 0 the transition density (= convolution kernel) is p0 (t, x) :=
cd t (t 2 + |x|2 )
d+1 2
,
(1.5)
with cd := (4π)d/2 Γd!(1+d/2) . (Cf. [5]. We note that cd is the same “semiclassical” constant that appears in the Weyl estimate for the eigenvalues of the Laplacian. It is given in [5] and some d+1 other sources as π − 2 Γ ( d+1 2 ), which is equal to cd by an application of the duplication formula of the gamma function.) If Ω is a nonempty, bounded, open subset of Rd , then we define Hm,Ω as follows. Consider the quadratic form on Cc∞ (Ω) given by ϕ→
ϕ −Δ + m2 ϕ.
Ω
√ (Here −Δ + m2 is calculated for Rd .) Since this quadratic form is positive and defined on a dense subset of L2 (Ω), it extends to a unique minimal positive operator (the Friedrichs extension) on L2 (Ω), which we designate Hm,Ω . The semigroup e−tHm,Ω has an integral kernel pm,Ω (t, x, y), the form √ of which is typically not known explicitly, but is bounded by comparison
with the operator e−t −Δ+m on L2 (Rd ), which is known explicitly [31, p. 183], and is bounded for t > 0. Consequently, e−tHm,Ω is Hilbert–Schmidt and Hm,Ω has purely discrete spectrum. 2
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We remark that the Fourier transform can be more directly applied to Hm,Ω than to the square root of the Dirichlet–Laplacian according to the functional calculus, which dominates it in the following sense. Suppose that ϕ ∈ Cc∞ (Ω) ⊂ Cc∞ (Rd ). Then
2 ϕ, Hm,Ω ϕ
= Hm,Ω ϕ = 2
Ω
= Rd
−1
2 F |ξ |2 + m2 ϕˆ
χΩ F −1 |ξ |2 + m2 ϕˆ 2 −1
2 F |ξ |2 + m2 ϕˆ
Rd
=
ϕ −Δ + m2 ϕ
Rd
=
ϕ −Δ + m2 ϕ,
Ω
because supp(ϕ) ∈ Ω and −Δ is a local operator. Therefore, if βk denotes the kth eigenvalue of Hm,Ω , and λk is the kth eigenvalue of −Δ, βk
λk + m2 .
(1.6)
2. Trace formulae and inequalities for spectra of Hm,Ω In [18] universal bounds for spectra of Laplacians were found as consequences of differential inequalities for Riesz means defined on the sequence of eigenvalues. The strategy here is the same, as adapted to the eigenvalues βj , j = 1, . . . of the first-order pseudodifferential operator Hm,Ω . However, as the earlier article made heavy use of the fact that the Laplacian is of second order and acts locally, neither of which circumstance applies here, the results we obtain here and the details of the argument are quite different. An essential lemma is an adaptation of a result of [21,22]. Lemma 2.1 (Harrell–Stubbe). Let H be a self-adjoint operator on L2 (Ω), Ω ∈ Rd , with discrete spectrum β1 β2 · · · < inf σess (H ), interpreted as +∞ when σess (H ) is empty. Denoting the corresponding normalized eigenfunctions {uj }, assume that for a Cartesian coordinate xα , the functions xα uj and xα2 uj are in the domain of definition of H . Then for any z < inf σess (H ), j : βj z
and
2
(z − βj ) uj , xα , [H, xα ] uj − 2[H, xα ]uj 0,
(2.1)
E.M. Harrell II, S. Yıldırım Yolcu / Journal of Functional Analysis 256 (2009) 3977–3995
2
(z − βj )2 uj , xα , [H, xα ] uj − 2(z − βj )[H, xα ]uj 0.
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(2.2)
j : βj z
So that this article is self-contained, we provide a proof of the lemma under the simplifying assumption that the spectrum is purely discrete. Proof. Elementary calculations show that, subject to the domain assumptions made in the statement of the theorem, [H, xα ]uj = (H − βj )xα uj , and
uj , xα , [H, xα ] uj = 2 xα uj , (H − βj )xα uj .
These two identities can be combined and slightly rearranged to yield: 2
(z − βj ) uj , xα , [H, xα ] uj − 2[H, xα ]uj
= 2 (z − βj ) − (H − βj ) xα uj , (H − βj )xα uj
= 2 (z − H )xα uj , (H − βj )xα uj .
(2.3)
Using the completeness of the eigenfunctions of H , (H − βj )xα uj =
(βk − βj ) xα uj , uk uk ,
k
so the right side of (2.3) can be rewritten as 2
(z − βk ) uk , xα uj (βk − βj ) xα uj , uk
k
=2
2 (z − βk )(βk − βj ) uk , xα uj
k
2
2 (z − βk )(βk − βj ) uk , xα uj ,
(2.4)
k: βk
provided that βj z. If we now sum (2.3) over j with βj z, i.e., the same values of j as for k in (2.4), then after symmetrizing in j, k, j : βj z
2
(z − βj ) uj , xα , [H, xα ] uj − 2[H, xα ]uj
j,k: βk ,βj
2
(z − βk ) − (z − βj ) (βk − βj ) uk , xα uj ,
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which simplifies to
−
2 (βk − βj )2 uk , xα uj 0,
j,k: βk ,βj
as claimed in (2.1). In order to establish (2.2), multiply (2.4) by (z − βj ) and then sum on j for βj < z. The summand on the right side is odd in the exchange of j and k, and thus the right side equates to 0. 2 Some consequences of more general forms of the lemma are worked out in [22]. Before deriving a differential inequality that will be useful to control the spectrum, we first follow the strategy of [21] to obtain a universal bound on βn+1 in terms of the statistical distribution of the lower eigenvalues. For this purpose we introduce notation for the normalized moments of the eigenvalues. Definition. Given a real number r and an integer n > 0, βnr := simply write
1 n
n
r j =1 βj .
When r = 1 we
βn = βn1 .
Theorem 2.1. If d 2, then for each positive integer n, the eigenvalues βn of Hm,Ω satisfy 1
βn+1 (d
− 1)βn−1
d+
d 2 − d 2 − 1 βn βn−1 .
(2.5)
Before giving the proof we note two slightly weaker but more appealing variants of (2.5) using the Cauchy–Schwarz inequality, 1 βn βn−1 , with the aid of which the universal bound (2.5) simplifies to d +1
βn+1 (d
− 1)βn−1
d +1 βn . d −1
(2.6)
In particular, β2 d + 1 , β1 d − 1
(2.7)
regardless of any property of the domain other than boundedness. In this connection, recall that R. Bañuelos and T. Kulczycki have proved in [4] that the fundamental gap of the Cauchy process is controlled by the inradius in the case of a bounded convex domain Ω of inradius Inr(Ω), viz., for m = 0, √ √ λ2 − (1/2) λ1 (Ω), β2 − β1 Inr where λ1 and λ2 are the first and second eigenvalues for the Dirichlet–Laplacian for the unit ball, B1 in Rd . (Recall that the inradius Inr(Ω) of a region Ω is defined by Inr(Ω) = sup d(x): x ∈ Ω ,
E.M. Harrell II, S. Yıldırım Yolcu / Journal of Functional Analysis 256 (2009) 3977–3995
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where d(x) = min{|x − y|: y ∈ / Ω} [13].) Since a ratio bound like (2.7) is algebraically equivalent to a gap bound, (2.7) provides an independent upper bound on the gap β2 − β1 . Continuing to set m = 0, (1.6) and (2.7) in the 2 β1 imply: form β2 − β1 d−1 Corollary 2.2. If β1∗ and λ∗1 denote the fundamental eigenvalues of H0,Ω and −Δ, respectively, on the unit ball of Rd , then β2 − β1
∗ λ1 β1∗ 2 2 . d − 1 Inr(Ω) d − 1 Inr(Ω)
(2.8)
Proof. Since H0,Ω is defined by closure from a core of functions in Cc∞ , its fundamental eigenvalue satisfies the principle of domain monotonicity. That is, if Ω1 ⊃ Ω2 , then β1 (Ω1 ) β1 (Ω2 ). β∗ In particular, if Ω is a ball of radius r, then β1 (Ω) r1 , which is the fundamental eigenvalue of the unit ball B1 by scaling. The first inequality follows from (2.7), and the second one by (1.6) 2 Proof of Theorem 2.1. We make the special choice H = Hm,Ω and calculate the first and second commutators with the aid of the Fourier transform: Writing Hm,Ω = χΩ F −1 |ξ |2 + m2 F , [Hm,Ω , xα ]ϕ = (Hm,Ω xα − xα Hm,Ω )ϕ = χΩ F −1 |ξ |2 + m2 F [xα ϕ] − χΩ xα F −1 |ξ |2 + m2 ϕˆ
∂ ϕˆ ∂ = χΩ F −1 |ξ |2 + m2 − |ξ |2 + m2 ϕˆ ∂ξα ∂ξα ξα ϕ. ˆ = −iχΩ F −1 |ξ |2 + m2
(2.9)
Similarly, xα , [Hm,Ω , xα ] ϕ = χΩ F −1
ξα2 ϕˆ . − |ξ |2 + m2 (|ξ |2 + m2 )3/2 1
Due to (2.9) and (2.10), there are simplifications when we sum over α: d [Hm,Ω , xα ]ϕ 2 ϕ, ˆ α=1
|ξ |2 ϕ ˆ 1 |ξ |2 + m2
and d α=1
ξα2 − |ξ |2 + m2 (|ξ |2 + m2 )3/2 1
In consequence, (2.2) implies that
=
(d − 1)|ξ |2 + d m2 d −1 . 2 2 3/2 (|ξ | + m ) |ξ |2 + m2
(2.10)
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(d − 1)
n n
−1 (z − βj )2 uj , Hm,Ω uj − 2 (z − βj ) 0, j =1
(2.11)
j =1
provided z ∈ [βn , βn+1 ]. Because −1 Hm,Ω uj =
1 uj βj
and (z − βj ) = −
(z − βj )(z − βj − z) , βj
inequality (2.11) can be rewritten as (d + 1)
n (z − βj )2 j =1
βj
− 2z
n (z − βj )
βj
j =1
0,
(2.12)
or, equivalently, (d − 1)βn−1 z2 − 2dz + (d + 1)βn 0.
(2.13)
Setting z = βn+1 , we see that βn+1 must be less than the larger root of (2.13), which is the conclusion of the theorem. 2 For future purposes we note that this theorem extends with small modifications to semirelativistic Hamiltonians of the form Hm,Ω + V (x). More specifically, (2.11) is valid when {uk } and {βk } are the eigenfunctions and eigenvalues of Hm,Ω + V (x). We next apply similar reasoning to a function related to Riesz means. With a+ := max(0, a), let U (z) :=
(z − βk )2+ k
βk
,
(2.14)
where z is a real variable. Note that if z ∈ [βj , βj +1 ], then U (z) = βj−1 z2 − 2z + βj . j
(2.15)
Theorem 2.3. The function z−(d+1) U (z) is nondecreasing in the variable z. Moreover, for d 2 and any j 1, the “Riesz mean” R1 (z) := k (z − βk )+ satisfies R1 (z) for all z ( d+1 d−1 )βj .
2j (d − 1)d (d + 1)d+1 βj d
zd+1
(2.16)
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Proof. In notation that suppresses n, Eq. (2.12) can be written (d + 1)
(z − βk )2+ βk
k
− 2z
(z − βk )+ βk
k
0,
(2.17)
which for the function U reads (d + 1)U (z) − zU (z) 0, or, equivalently, d U (z) 0, dz zd+1
(2.18)
proving the claim about U . Eq. (2.11) tells us that R1 (z) Since
U (z) zd+1
d −1 U (z). 2
(2.19)
is nondecreasing, when z zj ∗ βj , U (z)
z zj ∗
d+1 U (zj ∗ ).
(2.20)
From (2.15) with the Cauchy–Schwarz inequality we get 1 U (z) (z − βj )2 , j βj
(2.21)
so that with (2.19) and (2.20) we obtain R1 (z)
(d − 1)j 2βj
z zj ∗
d+1 (zj ∗ − βj )2 .
We now choose an optimized value of zj ∗ to maximize the coefficient of zd+1 , viz., zj ∗ = Substituting this into (2.22), we get (2.16), as claimed. 2
(2.22) d+1 d−1 βj .
The Legendre transform of R1 (z) is a straightforward calculation, to be found explicitly for example in [18,26]. The result for k − 1 < w < k is
R1∗ (w) = w − [w] β[w]+1 + [w]β[w] ,
(2.23)
where [w] denotes the greatest integer w. When w approaches an integer value k from below, R1∗ (k) = kβ k . With the Legendre transform of the right side of (2.16), we get
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kβ k
d+1 d βj k d . 21/d j 1/d (d − 1)
(2.24)
This leads us to the following upper bound for ratios of averages of eigenvalues of Hm,Ω : Corollary 2.4. For k > 2j , Eq. (2.24) implies βk βj
1 k d d . 21/d (d − 1) j
(2.25)
Remark 2.5. The reason for the restriction on k, j is that in Theorem 2.3, we assumed that ∗ z ( d+1 d−1 )βj . Since the maximizing value of zj in the calculation of the Legendre transform of the right side of (2.16) depends monotonically on w, we get w = 2j
(d − 1)zj ∗ (d + 1)βj
d .
(2.26)
Thus the inequality is valid under the assumption that k > w 2j . 3. Weyl asymptotics and semiclassical bounds for Hm,Ω In this section we consider the eigenvalues√βk of Hm,Ω as k → ∞. In view of the elementary |ξ |2 +m2
inequalities (1.1), and the fact that lim|ξ |→∞ = 1, it suffices to consider the case m = 0. |ξ | We begin with the analogue of the Weyl formula for the Laplacian, adapting one of the standard proofs of the latter, which relies on an estimate of the partition function Z(t) := e−βj t for t > 0. Recall that the function Z(t) can be written as Z(t) = where N(β) := tion function is
βj β
e−βt dN (β),
(3.1)
1 is the usual counting function. Another standard formula for the parti Z(t) =
pΩ (x, x, t) dx.
(3.2)
Ω
√ If we accept that Hm,Ω is well approximated by −ΔΩ in the “semiclassical limit,” then the analogue for N(β) of the Weyl asymptotic formula for √ the Laplacian should be identical to the usual Weyl formula, with the identification of βk with λk . This intuition is confirmed by the following: Proposition 3.1. As β → ∞, N (β) ∼
|Ω| βd . (4π)d/2 Γ (1 + d/2)
(3.3)
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Equivalently, as k → ∞, √ Γ (1 + d/2)k 1/d βk ∼ 4π . |Ω|
(3.4)
Moreover, the function U of (2.14) satisfies U (z) ∼
(4π)d/2 (d 2
2|Ω| zd+1 . − 1)Γ (1 + d/2)
Proof. By Karamata’s Tauberian theorem [36], if we can show that for t → 0, t d Z(t) → cd |Ω|, then the first claim follows from (3.1). The further claims for βk and U (z) are easy consequences of (3.3). By a standard comparison, pΩ (x, y, t) < p0 (x − y, t)
(3.5)
on Ω, where pΩ is the integral kernel of the semigroup e−tH0,Ω . Define rΩ (x, y, t) := p0 (x − y, t) − pΩ (x, y, t), and let δΩ (x) := dist(x, ∂Ω). According to [5], 0 rΩ (x, y, t)
t
cd P d+1 δΩ (x)
y
(τΩ < t),
where P y (τΩ < t) is the probability that a path originating at y exits Ω before time t. From (3.5),
pΩ (x, x, t) dx Ω
p0 (0, t) dx = cd
|Ω| , td
Ω
and we proceed to calculate:
pΩ (x, x, t) dx = Ω
pΩ (x, x, t) dx +
√ {x: δΩ (x)< t }
p0 (0, t) − rΩ (x, x, t) dx
√ {x: δΩ (x)> t }
√ pΩ (x, x, t) dx + x: δΩ (x) > t cd t −d
= √ {x: δΩ (x)< t }
−
rΩ (x, x, t) dx.
√ {x: δΩ (x)> t }
The first integral on the right side of (3.6) becomes
(3.6)
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0
pΩ (x, x, t) dx
√ {x: δΩ (x)< t }
p0 (0, t) dx
√ {x: δΩ (x)< t }
√ cd t −d x: δΩ (x) < t = o t −d
as t → 0. As for the final integral of (3.6), 0 rΩ (x, x, t) dx √ {x: δΩ (x)> t }
√ {x: δΩ (x)> t }
t d+1 δΩ (x)
(3.7)
dx
t
|Ω| (1−d)/2 = o t −d . =0 t t (d+1)/2
(3.8)
With (3.7) and (3.8) we thus validate the condition allowing the application of Karamata’s Tauberian theorem. 2 An easy corollary of Theorem 2.3 is a counterpart for H0,Ω to the Li–Yau inequality for the Laplacian [30]. (As noted in [26], the Li–Yau inequality is equivalent to an earlier inequality by Berezin [7] through the Legendre transform. See also [31].) 2cd |Ω| Since we know that z−(d+1) U (z)↑ d!(d 2 −1) , and that because of (2.6) a choice of z safely guaranteed to exceed βk is z =
d+1 d−1 βk ,
with the aid of (2.21) we obtain
2 −(d+1) 2 d +1 k 2cd |Ω| βk βk . d −1 d!(d 2 − 1) βk d − 1 This leads directly to the semiclassical estimate: √ (d − 1)21/d 4π Γ (1 + d/2)k 1/d βk . d +1 |Ω|
(3.9)
However, a better estimate, improving (d − 1)21/d to d, can be derived by following the argument of Li and Yau [30] more closely. As a first step we slightly generalize the lemma attributed in [30] to Hörmander: Lemma 3.1. Let f : Rd → R satisfy 0 f (ξ ) M1 and f (ξ )w |ξ | dξ M2 ,
(3.10)
Rd
where the weight function w is nonnegative and nondecreasing. Define R = R(M1 , M2 ) by the condition that BR
w |ξ | dξ = ωd−1
R w(r)r d−1 dr = 0
M2 , M1
(3.11)
E.M. Harrell II, S. Yıldırım Yolcu / Journal of Functional Analysis 256 (2009) 3977–3995
where ωd−1 := |Sd−1 | =
2π d/2 Γ (d/2) .
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Then f (ξ ) dξ
Rd
π d/2 M1 Rd . Γ (1 + d/2)
(3.12)
1
2 (d+p) d+p As a special case, if w(ξ ) = |ξ |p , then R = [ M , and so M1 wd−1 ]
−p
d 1 (d + p)M2 d+p (wd−1 M1 ) d+p d
f (ξ ) dξ Rd
=
d +p M2 d
d d+p
π d/2 M1 Γ (1 + d/2)
p d+p
.
Proof. Let g(ξ ) := M1 χ{|ξ |R} and note that according to the definition of R, w(|ξ |)g(ξ ) dξ = M2 . We observe that (w(|ξ |) − w(R))(f (ξ ) − g(ξ )) 0 for all ξ . (Check |ξ | R and |ξ | > R separately.) Hence w(R)
f (ξ ) − g(ξ ) dξ
w |ξ | f (ξ ) − g(ξ ) = 0,
(3.13)
π d/2 M1 Rd . Γ (1 + d/2)
(3.14)
and, consequently,
f (ξ ) dξ
g(ξ ) dξ = |BR |M1 =
2
For the application to H0,Ω , note that β = u , H0,Ω u = Choosing w(|ξ |) = |ξ | in the lemma, with f (ξ ) = k=
2 |ξ | uˆ (ξ ) dξ .
k
π d/2 f (ξ ) dξ f ∞ Γ (1 + d/2)
ˆ (ξ )|2 , =1 |u
1 d+1
k
(3.15)
we find
β
=1
d +1 d
d d+1
(3.16)
or k =1
β
Γ (1 + d/2) 1/d 1+ 1 d k d. d + 1 π d/2 f ∞
(3.17)
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As for f ∞ , k k uˆ (ξ ) 2 = =1
=1
=
2 1 ix·ξ e u (x) dx (2π)d Ω
k
2 1 ix·ξ |Ω| e , uk d (2π) (2π)d =1
by Bessel’s inequality, as eix·ξ 22 = |Ω|. In conclusion, we have an analogue of the Li–Yau inequality [30]. Theorem 3.2. For all k = 1, . . . , the eigenvalues βk of H0,Ω satisfy √ βk
4πd Γ (1 + d/2)k 1/d . d +1 |Ω|
(3.18)
We observe that, just like the Li–Yau inequality for the Laplacian, (3.18) has the best possible coefficient consistent with the Weyl-type law of Proposition 3.1. Moreover, in view of (1.6), Theorem 3.2 has a corollary for the Dirichlet–Laplacian: √ k 4π d Γ (1 + d/2)k 1/d 1 λ , k d +1 |Ω|
(3.19)
=1
which is comparable to the Li–Yau inequality, but neither implies it nor is directly implied by it. (For an alternative route to (3.19) see Theorem 5.1 of [22].) 4. Universal bounds for Hm,Ω + V (x) We turn now to the Klein–Gordon Hamiltonian with an external interaction, H = Hm,Ω + V (x).
(4.1)
As mentioned in the introduction, these operators are used in semi-relativistic quantum mechanics. In addition, Hamiltonian operators similar to (4.1) have been of interest as models of nonrelativistic charge carriers traveling in a two-dimensional hexagonal structure like carbon graphene, a novel material with remarkable electronic properties [25,37]. (What distinguishes graphene from the common material graphite is that graphene sheets are only one atom thick.) Our point of departure to derive useful spectral bounds for (4.1) is (2.11), which remains valid for interacting operators H . We shall impose conditions on V that guarantee that operators of the form (4.1) are self-adjoint with the same domain of definition as for Hm,Ω . Theorem 4.1. Let βk denote the eigenvalues of (4.1), where V is a real-valued locally L1 func (z−β )2 tion, and, as in (2.14), set U (z) := k βkk + .
E.M. Harrell II, S. Yıldırım Yolcu / Journal of Functional Analysis 256 (2009) 3977–3995
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(a) If V 0, then for each k, (2.6) holds. That is d +1
βk+1 (d
− 1)βk−1
d +1 βk . d −1
(4.2)
Moreover, for k > 2j , (2.25) holds. That is, 1 k d d 1/d . 2 (d − 1) j βj βk
(4.3)
(b) If V ∈ Ls for some 2 d < s < ∞, and α :=
V s (d − 2)!(s − 1)
s−1 s
√ (d−1)2 1−2d d−s s−1 π2 d Γ ( d2 ) d (d|Ω|) sd (s − d) s
< 1,
then for each k, the eigenvalues βk satisfy βk+1 βk Moreover,
U (z) z((d+1)−α(d−1))
βk−1 βk+1 1 +
2 . (d − 1)(1 − α)
(4.4)
is a nondecreasing function of z ∈ R, and for k > 2j ,
1/(d−α(d−1)) k d − α(d − 1) . 1/(d−α(d−1)) j (d − 1)(1 − α)2 βj βk
(4.5)
Proof. From (2.11), we obtain (d − 1)
n
−1 (z − βj )2 uj , Hm,Ω uj − 2 (z − βj ) 0. j =1
j
(a) Observe that V 0 implies −1 (Hm,Ω + V )−1 Hm,Ω
and
1 −1 uj , Hm,Ω uj . βj Thus, (4.2) and (4.3) follow in strict analogy with (2.6) and (2.25). (b) By using the resolvent formula we get
1 −1 −1 = uj , (Hm,Ω + V )−1 uj = uj , Hm,Ω uj − uj , (Hm,Ω + V )−1 V Hm,Ω uj βj
1 −1 −1 uj , V Hm,Ω uj − uj , = uj , Hm,Ω βj
(4.6)
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which implies
1 −1 −1 1 + uj , V Hm,Ω uj = uj , Hm,Ω uj βj
(4.7)
1 −1 −1 1 − V Hm,Ω uj uj , Hm,Ω uj . βj
(4.8)
and
If 2 d s < ∞, we now claim that V H −1 ϕ αϕ2 m,Ω 2
(4.9)
for any ϕ ∈ L2 . Granting the claim, with ϕ = uj in (4.8), we get
1−α −1 uj , Hm,Ω uj . βj
(4.10)
To establish (4.9) begin by noting that by Hölder’s inequality, V H −1 ϕ V s H −1 ϕ m,Ω m,Ω 2
2s s−2
.
(4.11)
Because Hm,Ω H0,Ω > 0, −1 H m,Ω ϕ
2s s−2
−1 H0,Ω ϕ
2s s−2
(4.12)
.
Inequality (3.5) for the transition density implies e−tH0,Ω (x, y, t) p0 (x − y, t) =
−( d−1 ) −cd ∂ 2 2 . t + |x − y|2 d − 1 ∂t
−1 Applying the Laplace transform, the kernel of H0,Ω is less than
∞ 0
−( d−1 ) −cd ∂ 2 cd 2 dt = t + |x − y|2 |x − y|−(d−1) . d − 1 ∂t d −1
Together with (4.11) and (4.12) we get V H −1 ϕ cd V s |x|−(d−1) ∗ ϕ 2s . m,Ω 2 s−2 d −1 According to Young’s convolution inequality, −(d−1) |x| ∗ ϕ 2s |x|−(d−1) s−2
so
s s−1
ϕ2 ,
E.M. Harrell II, S. Yıldırım Yolcu / Journal of Functional Analysis 256 (2009) 3977–3995
V H −1 ϕ m,Ω 2
Γ ( d+1 2 ) V s |x|−(d−1) s ϕ2 . s−1 π (d+1)/2 (d − 1)
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(4.13)
s , choose R ∗ as the radius of the ball BR ∗ centered at the For an upper bound to |x|−(d−1) s−1 origin having the same volume as Ω. Since by rearrangement,
−(d−1) |x|
s
L s−1 (Ω)
s−d s−1 (R ∗ ) s−1 (s − 1) s s = ωd−1 , L s−1 (BR ∗ ) s −d
|x|−(d−1)
we get the estimate −(d−1) |x|
s s−1
<2
d−1 d
π
d−1 2
1−d s−1 d
s−d s − 1 s d sd Γ d|Ω| . 2 s −d
(4.14)
With (4.13) this implies (4.9) and consequently (4.10). Because α < 1 by assumption, (4.6) together with (4.10) yield (d − 1)
n 1−α j =1
βj
(z − βj )2 − 2
n (z − βj ) 0,
(4.15)
j =1
or, equivalently, (d − 1)(1 − α)βk−1 z2 − 2 d − α(d − 1) z + d + 1 − α(d − 1) βk 0.
(4.16)
By setting z = βk+1 , we see that βk+1 must be smaller than the larger root of (4.16), i.e., after some algebra, βk+1
(d − 1)(1 − α) + 1 +
1 − ((d + α − αd)2 − 1)(βk βk−1 − 1)
(d − 1)(1 − α)βk−1
.
(4.17)
As was the case for (2.6), with the Cauchy–Schwarz inequality in the form 1 βk βk−1 , (4.17) implies the simpler but slightly weaker inequalities (4.4). Now observe that (4.15) differs from (2.12) only in the extra factor 1 − α > 0, and therefore all of the consequences of that inequality can be recovered with suitable changes of some constants. U (z) is nondecreasing and, therefore, In particular, the function z(d+1)−α(d−1) U (z)
z zj ∗
(d+1)−α(d−1) U (zj ∗ )
(4.18)
when z zj ∗ βj . At the same time, by (4.15) we have (d − 1)(1 − α) U (z) R1 (z). 2
(4.19)
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By (4.19) and the fact that
R1 (z)
U (z) j
1 βj
(z − β j )2 , we obtain
(d − 1)(1 − α)j 2β j
z zj ∗
(d+1)−α(d−1)
2 zj ∗ − βj .
(4.20)
To maximize the coefficient of zd+1−α(d−1) we optimize zj ∗ and get zj ∗ =
(d + 1) − α(d − 1) βj . (d − 1)(1 − α)
Substituting this into (4.20) gives R1 (z)
2j [(d − 1)(1 − α)]d−α(d−1) [(d + 1) − α(d − 1)](d+1)−α(d−1) βj
d−α(d−1)
z(d+1)−α(d−1)
(4.21)
for all z (d+1)−α(d−1) (d−1)(1−α) βj . With the Legendre transform of the right side of (4.21), we obtain kβk
[d − α(d − 1)]βj k 1+1/(d−α(d−1)) . [(d − 1)(1 − α)]21/(d−α(d−1)) j 1/(d−α(d−1))
(4.22)
1/(d−α(d−1)) k d − α(d − 1) 1/(d−α(d−1)) j [(d − 1)(1 − α)]2
(4.23)
Therefore, βk βj as claimed.
2
Acknowledgments The authors are grateful to Mark Ashbaugh, Lotfi Hermi, and Joachim Stubbe for conversations and references. References [1] M.S. Ashbaugh, The universal eigenvalue bounds of Payne–Pólya–Weinberger, Hile–Protter, and H.C. Yang, in: Spectral and Inverse Spectral Theory, Goa, 2000, in: Proc. Indian Acad. Sci. Math. Sci., vol. 112, 2002, pp. 3–30. [2] M.S. Ashbaugh, L. Hermi, A unified approach to universal inequalities for eigenvalues of elliptic operators, Pacific J. Math. 217 (2004) 201–220. [3] C. Bandle, Isoperimetric Inequalities and Applications, Pitman Monogr. Stud. Math., vol. 7, Pitman, Boston, 1980. [4] R. Bañuelos, T. Kulczycki, Eigenvalue gaps for the Cauchy process and a Poincaré inequality, J. Funct. Anal. 211 (2) (2004) 355–423. [5] R. Bañuelos, T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal. 234 (2006) 199–225. [6] P.H. Bérard, Spectral Geometry: Direct and Inverse Problems, Lecture Notes in Math., vol. 1207, Springer-Verlag, Berlin, 1986. [7] F. Berezin, Convariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR 37 (1972) 1134–1167 (in Russian); English transl. in Math. USSR Izv. 6 (1972) 1117–1151. [8] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984.
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[9] D. Chen, Extrinsic eigenvalue estimates of the Dirac operator, 2007, preprint, arXiv:math/0701847v1. [10] Q.M. Cheng, H.C. Yang, Estimates on eigenvalues of Laplacian, Math. Ann. 331 (2005) 445–460. [11] Q.M. Cheng, H.C. Yang, Inequalities for eigenvalues of Laplacian on domains and complex hypersurfaces in complex projective spaces, J. Math. Soc. Japan 58 (2006) 545–561. [12] Q.M. Cheng, H.C. Yang, Inequalities for eigenvalues of a clamped plate problem, Trans. Amer. Math. Soc. 358 (2006) 2625–2635. [13] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. [14] A. El Soufi, E.M. Harrell II, S. Ilias, Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds, Trans. Amer. Math. Soc., in press. [15] E. Giere, The fractional Laplacian in applications, http://www.eckhard-giere.de/math/publications/review.pdf. [16] E.M. Harrell II, Some geometric bounds on eigenvalue gaps, Comm. Partial Differential Equations 18 (1993) 179– 198. [17] E.M. Harrell II, Commutators, eigenvalue gaps, and mean curvature in the theory of Schrödinger operators, Comm. Partial Differential Equations 32 (2007) 401–413. [18] E.M. Harrell II, L. Hermi, Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues, J. Funct. Anal. 254 (2008) 3173–3191. [19] E.M. Harrell II, P.L. Michel, Commutator bounds for eigenvalues, with applications to spectral geometry, Comm. Partial Differential Equations 19 (1994) 2037–2055; Comm. Partial Differential Equations 20 (1995) 1453, Erratum. [20] E.M. Harrell II, P.L. Michel, Commutator bounds for eigenvalues of some differential operators, in: G. Ferreyra, G. Goldstein, F. Neubrander (Eds.), Evolution Equations, Dekker, New York, 1994, pp. 235–244. [21] E.M. Harrell II, J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997) 1797–1809. [22] E.M. Harrell II, J. Stubbe, Universal bounds and semiclassical estimates for eigenvalues of abstract Schrödinger operators, 2008, preprint, arXiv: 0808.1133. [23] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Front. Math., Birkhäuser, 2006. [24] G.N. Hile, M.H. Protter, Inequalities for eigenvalues of the Laplacian, Indiana Univ. Math. J. 29 (1980) 523–538. [25] R. Jackiw, S.-Y. Pi, Chiral Gauge theory for graphene, Phys. Rev. Lett. 98 (2007) 266402. [26] A. Laptev, T. Weidl, Recent Results on Lieb–Thirring inequalities, Journées “Équations aux Dérivées Partielles,” La Chapelle sur Erdre, 2000, XX (2002), Univ. Nantes, Nantes, 2000, 14 pp. [27] H.A. Levine, M.H. Protter, Unrestricted lower bounds for eigenvalues for classes of elliptic equations and systems of equations with applications to problems in elasticity, Math. Methods Appl. Sci. 7 (1985) 210–222. [28] M. Levitin, L. Parnovski, Commutators, spectral trace identities, and universal estimates for eigenvalues, J. Funct. Anal. 192 (2002) 425–445. [29] P. Li, Eigenvalue estimates on homogeneous manifolds, Comment. Math. Helv. 55 (1980) 347–363. [30] P. Li, S.-T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983) 309–318. [31] E.H. Lieb, M. Loss, Analysis, second ed., Grad. Stud. Math., vol. 14, Amer. Math. Soc., Providence, RI, 2001. [32] P.L. Michel, Eigenvalue gaps for self-adjoint operators, Georgia Institute of Technology PhD Dissertation, 1994. [33] L.E. Payne, G. Pólya, H.F. Weinberger, On the ratio of consecutive eigenvalues, J. Math. Phys. 35 (1956) 289–298. [34] M. Reed, B. Simon, Methods of Modern Mathematical Physics, I, Functional Analysis, Academic Press, New York, 1972. [35] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Stud. Adv. Math., vol. 68, Cambridge University Press, Cambridge, 1999. [36] B. Simon, Functional Integration and Quantum Physics, Academic Press, New York, 1979. [37] P.R. Wallace, The band theory of graphite, Phys. Rev. 71 (1947) 622–634. [38] Q. Wang, C. Xia, Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds, J. Funct. Anal. 245 (2007) 334–352. [39] F.E. Wu, L.F. Cao, Estimate for eigenvalues of Laplacian operator with any order, Sci. China Ser. A 50 (2007) 1078–1086. [40] H.C. Yang, Estimates of the difference between consecutive eigenvalues, 1995, preprint, revision of International Centre for Theoretical Physics preprint IC/91/60, Trieste, Italy, April 1991. [41] P.C. Yang, S.-T. Yau, Eigenvalues of the Laplacian of a compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 4 (1980) 55–63.
Journal of Functional Analysis 256 (2009) 3996–4029 www.elsevier.com/locate/jfa
Separation and duality in locally L0-convex modules Damir Filipovi´c ∗ , Michael Kupper, Nicolas Vogelpoth 1 Vienna Institute of Finance, 2 University of Vienna and Vienna University of Economics and Business Administration, Heiligenstaedter Strasse 46-48, A-1190 Vienna, Austria Received 1 October 2008; accepted 24 November 2008 Available online 30 December 2008 Communicated by Paul Malliavin
Abstract Motivated by financial applications, we study convex analysis for modules over the ordered ring L0 of random variables. We establish a module analogue of locally convex vector spaces, namely locally L0 -convex modules. In this context, we prove hyperplane separation theorems. We investigate continuity, subdifferentiability and dual representations of Fenchel–Moreau type for L0 -convex functions from L0 -modules into L0 . Several examples and applications are given. © 2008 Elsevier Inc. All rights reserved. Keywords: L0 -modules; Hahn–Banach extension; Hyperplane separation; Locally L0 -convex modules; L0 -convex functions; Lower semi continuity; Subdifferentiability; Fenchel–Moreau duality
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I. Separation in locally L0 -convex modules . . . . 2.1. Main results . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Hahn–Banach extension theorem . . . . . . . . . 2.3. Locally L0 -convex modules . . . . . . . . . . . . . 2.3.1. The countable concatenation property 2.3.2. The index set of nets . . . . . . . . . . . .
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* Corresponding author.
E-mail address: [email protected] (D. Filipovi´c). 1 Financial support from Munich Re Grant for doctoral students is gratefully acknowledged. 2 Supported by WWTF (Vienna Science and Technology Fund).
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.015
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2.4. The gauge function . . . . . . . . . . . . . . . . . . . . . . 2.5. Hyperplane separation . . . . . . . . . . . . . . . . . . . . 3. Part II. Duality in locally L0 -convex modules . . . . . . . . . 3.1. Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Financial applications . . . . . . . . . . . . . . . . . . . . 3.3. Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . 3.4. Lower semi continuous functions . . . . . . . . . . . . 3.5. Lower semi continuous L0 -convex functions . . . . 3.6. Subdifferentiability . . . . . . . . . . . . . . . . . . . . . . 3.7. Proof of the Fenchel–Moreau duality Theorem 3.8 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Various fundamental results in mathematical finance draw from convex analysis. For instance, arbitrage theory or duality of risk and utility functions are concepts built on the Hahn–Banach extension theorem and its consequences for hyperplane separation in locally convex vector spaces, cf. [6,10]. The simplest situation is a one period setup: π,ρ,u
R
p
0
E
(1.1)
T.
Random future (date T ) payments are modeled as elements of a locally convex vector space E endowed with semi norms p. Price, risk or utility assessments π , ρ, or u, map E linearly, convexly, or concavely, into the real line R, respectively. However, the idea of hedging random future payments develops its power in a multi period setting. We therefore randomize the initial data, and let π = π(ω, ·), ρ = ρ(ω, ·), or u = u(ω, ·), be ω dependent, where ω ∈ Ω denotes the initial states modeled by a probability space (Ω, F , P ). Here F is understood as the information available at some future initial date t < T . While classical convex analysis perfectly applies in the one period model (1.1), its application in a multi period framework is rather delicate. Take, for instance, the convexity properties of the risk measure ρ. These properties have to be extended to ω wise convexity properties of ρ(ω, ·) for almost all ω ∈ Ω. But ω wise convex duality correspondences for ρ(ω, ·) have to be made measurable in ω to assert intertemporal consistency in a recursive multi period setup. This would require heavy measurable selection criteria. We propose instead to consider π = π(ω, ·), ρ = ρ(ω, ·), or u = u(ω, ·), as maps into L0 = L0 (Ω, F , P ), the ordered ring of (equivalence classes of) random variables: R
L0
π,ρ,u
E
p
0
t
The space E, in turn, is considered as module over L0 .
T.
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This requires hyperplane separation and convex duality results on topological modules, which seem to be new in the literature. In this paper, we provide a comprehensive treatment of convex analysis for topological L0 -modules. While our emphasis is on financial applications as outlined above, the results in this paper are of theoretical nature. We illustrate the scope of applications that can be covered by our results in Section 3.2 below. The paper is divided into two parts. The first part covers Hahn–Banach extension and hyperplane separation theorems. In the second part, as an application of the first, duality results are established. The related literature is discussed in the course of the text. The remainder of the paper is as follows: Part I. In Section 2.1 we state the main results on locally L0 -convex topologies and hyperplane separation in locally L0 -convex modules. For the sake of readability, all proofs are postponed to the subsequent respective sections. In Section 2.2 we prove a Hahn–Banach type extension theorem in the context of L0 -modules. Instead of sublinear and linear functions on a vector space we study L0 -sublinear and L0 -linear functions on an L0 -module. In Section 2.3 we characterize a class of topological L0 -modules, namely locally L0 -convex modules. An important feature of a locally L0 -convex module E is that the neighborhoods of 0 absorb E over L0 . This is the key difference to the notion of a locally convex module which is merely absorbent over the real line, cf. [13,20,23]. The neighborhood base of a locally L0 -convex module is constructed by means of L0 -semi norms. Such vector valued, or vectorial, norms go back to [14]. In Section 2.4 we establish some preliminary results for L0 -valued gauge functions. In Section 2.5 we prove the hyperplane separation theorems in locally L0 -convex modules. We separate a non-empty open L0 -convex set from an L0 -convex set and we strictly separate a point from a non-empty closed L0 -convex set by means of continuous L0 -linear functions. Part II. In Section 3.1 we state the main Fenchel–Moreau type duality results in locally L0 -convex modules. Section 3.2 illustrates the scope of financial applications. As in part one, all proofs are postponed to the subsequent respective sections. In Section 3.3 we prove that L0 -convex functions share a certain local property. In Section 3.4 we characterize lower semi continuous functions. In Section 3.5 we establish continuity results for L0 -convex functions. For instance, under topological assumptions on E, proper L0 -convex functions are automatically continuous on the interior of their effective domain. In Section 3.6 we prove that proper lower semi continuous L0 -convex functions are subdifferentiable on the interior of their effective domain. In Section 3.7 we prove our Fenchel–Moreau type dual representation for proper lower semi continuous L0 -convex functions. 2. Part I. Separation in locally L0 -convex modules 2.1. Main results Let (Ω, F , P ) be a probability space. Denote by L0 the ring of real valued F -measurable random variables. Random variables and sets which coincide almost surely are identified. Recall that L0 equipped with the order of almost sure dominance is a lattice ordered ring. Throughout, the strict inequality X > Y between two random variables is to be understood as point-wise almost surely (in other texts, “X > Y ” is sometimes interpreted as “X Y and X = Y ”). Define L0+ := {Y ∈ L0 | Y 0} and L0++ := {Y ∈ L0 | Y > 0}. By L¯ 0 we denote the space ¯ := R ∪ {±∞} and we define of all F -measurable random variables which take values in R 0 0 ¯ ¯ L+ := {Y ∈ L | Y 0}. Throughout, we follow the convention 0 · (+∞) := 0.
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The order of almost sure dominance allows to define the following topology on L0 . We let Bε := Y ∈ L0 |Y | ε denote the ball of radius ε ∈ L0++ centered at 0 ∈ L0 . A set V ⊂ L0 is a neighborhood of Y ∈ L0 if there is ε ∈ L0++ such that Y + Bε ⊂ V . A set V ⊂ L0 is open if it is a neighborhood of all Y ∈ V . Inspection shows that the collection of all open sets is a topology on L0 , which is referred to as topology induced by | · |. By construction, U := {Bε | ε ∈ L0++ } is a neighborhood base of 0 ∈ L0 . Throughout, we make the convention that L0 = (L0 , | · |) is endowed with this topology. Notice that (L0 , | · |) is not a real topological vector space, in general. Indeed, suppose (Ω, F , P ) is atom-less. Then the scalar multiplication R → L0 , α → α · 1 is not continuous at α = 0. The topology on L0 induced by | · | is finer than the topology of convergence in probability, which is often used in convex analysis on L0 , such as in [3]. For example, L0++ is open in (L0 , | · |) but not in the topology of convergence in probability. However, it follows from Theorem 2.4 below that (L0 , | · |) is a topological ring or, equivalently, a topological L0 -module in the following sense: Definition 2.1. A topological L0 -module (E, T ) is an L0 -module E endowed with a topology T such that the module operations (i) (E, T ) × (E, T ) → (E, T ), (X1 , X2 ) → X1 + X2 and (ii) (L0 , | · |) × (E, T ) → (E, T ), (Y, X) → Y X are continuous w.r.t. the corresponding product topologies. Locally L0 -convex topologies in our framework are defined as follows: Definition 2.2. A topology T on E is locally L0 -convex if (E, T ) is a topological L0 -module and there is a neighborhood base U of 0 ∈ E for which each U ∈ U is (i) L0 -convex: Y X1 + (1 − Y )X2 ∈ U for all X1 , X2 ∈ U and Y ∈ L0 with 0 Y 1, (ii) L0 -absorbent: for all X ∈ E there is Y ∈ L0++ such that X ∈ Y U , (iii) L0 -balanced: Y X ∈ U for all X ∈ U and Y ∈ L0 with |Y | 1. In this case, (E, T ) is a locally L0 -convex module. Note that an L0 -convex set K ⊂ E with 0 ∈ K satisfies Y K ⊂ K for all Y ∈ L0 with 0 Y 1; in particular, 1A K ⊂ K for all A ∈ F . Next we show how to construct, and actually characterize all, locally L0 -convex modules. Let E be an L0 -module. Definition 2.3. A function · : E → L0+ is an L0 -semi norm on E if: (i) Y X = |Y | X for all Y ∈ L0 and X ∈ E, (ii) X1 + X2 X1 + X2 for all X1 , X2 ∈ E.
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If, moreover, (iii) X = 0 implies X = 0, then · is an L0 -norm on E. Any family P of L0 -semi norms on E induces a topology in the following way. For finite Q ⊂ P and ε ∈ L0++ we define UQ,ε := X ∈ E sup X ε
· ∈Q
and U := UQ,ε Q ⊂ P finite and ε ∈ L0++ .
(2.2)
We then proceed as for (L0 , | · |) above and define a topology, referred to as topology induced by P, on E with neighborhood base U of 0. We thus obtain a locally L0 -convex module, as the following theorem states: Theorem 2.4. A topological L0 -module (E, T ) is locally L0 -convex if and only if T is induced by a family of L0 -semi norms. Proof. This follows from Lemma 2.16 and Corollary 2.24.
2
By convention, an L0 -normed module (E, · ) is always endowed with the locally L0 -convex topology induced by · . Notice that any L0 -norm · on E = L0 satisfies 1 > 0 and · =
1 | · |. An important L0 -normed module is given in the following example. Recall that a function μ : E → L0 is L0 -linear if μ(Y1 X1 + Y2 X2 ) = Y1 μ(X1 ) + Y2 μ(X2 ) for all X1 , X2 ∈ E and Y1 , Y2 ∈ L0 . Example 2.5. Let (Ω, E, P ) be a probability space with F ⊂ E , and let p ∈ [1, +∞]. We define the function · p : L¯ 0 (E) → L¯ 0+ (F ) by
X p :=
limn→∞ E[|X|p ∧ n | F ]1/p if p < +∞, ess.inf{Y ∈ L¯ 0 (F ) | Y |X|} if p = +∞,
(2.3)
and denote p LF (E) := X ∈ L0 (E) X p ∈ L0 (F ) . p
In [15], it is shown that (LF (E), · p ) is an L0 -normed module, which is complete in the p p sense that any Cauchy net in LF (E) has a limit in LF (E). Moreover, for p < ∞, the L0 -module p q of all continuous L0 -linear functions μ : LF (E) → L0 can be identified with LF (E), where q := p/(p − 1) (:= +∞ if p = 1).
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p
Since X/ X p ∈ Lp (E) for X ∈ LF (E), we conclude that LF (E) = L0 (F ) · Lp (E) as sets. In particular, for F = {∅, Ω} the function · p can be identified with the classical Lp -norm. p In turn L{∅,Ω} (E) can be identified with the classical Lp space Lp (E). In fact, whenever F = p σ (A1 , . . . , An ) is finitely generated, we can identify Lσ (A1 ,...,An ) (E) with Lp (E). Hahn–Banach type extension theorems for modules appear already in the fifties. This started with [11], where modules over totally ordered rings were considered. Modules over rings which are algebraically and topologically isomorphic to the space of essentially bounded measurable functions on a finite measure space were considered in [12,21,19]. Nowadays, it is well known, cf. [4,22], that a Hahn–Banach type extension theorem for modules over more general ordered rings can be established. In particular, this is the case for L0 -modules. However, to our knowledge, the following hyperplane separation theorems for L0 -modules are new in the literature. The proofs are given in Section 2.5 below. Theorem 2.6 (Hyperplane separation I). Let E be a locally L0 -convex module and let K, M ⊂ E be L0 -convex, K open and non-empty. If 1A M ∩1A K = ∅ for all A ∈ F with P [A] > 0 then there is a continuous L0 -linear function μ : E → L0 such that μY < μZ
for all Y ∈ K and Z ∈ M.
For the second hyperplane separation theorem we need to impose some technical assumption on the topology. Definition 2.7. A topological L0 -module has the countable concatenation property if for every of 0 ∈ E and for every countable partition (An ) ⊂ F countable collection (Un ) of neighborhoods (An ∩ Am = ∅ for n = m and n∈N An = Ω) the set
1An Un
n∈N
again is a neighborhood of 0 ∈ E. Notice that any L0 -normed module has the countable concatenation property. Theorem 2.8 (Hyperplane separation II). Let E be a locally L0 -convex module that has the countable concatenation property and let K ⊂ E be closed L0 -convex and non-empty. If X ∈ E satisfies 1A {X} ∩ 1A K = ∅ for all A ∈ F with P [A] > 0 then there is ε ∈ L0++ and a continuous L0 -linear function μ : E → L0 such that μY + ε < μX
for all Y ∈ K.
2.2. Hahn–Banach extension theorem In this section, we establish a Hahn–Banach type extension theorem. We recall that the main result of this section, Theorem 2.14, is already contained in [4,22]. Nevertheless, for the sake of completeness, we provide a self contained proof which is tailored to our setup. The fact that not all elements in L0 possess a multiplicative inverse leads to difficulties in showing that the
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“one step extension” from the proof of the classical Hahn–Banach theorem is well defined in our framework. For this reason, we derive some preliminary results first. The following lemma recalls that F is a complete lattice w.r.t. the partial order of almost sure set inclusion. Lemma 2.9. Every non-empty collection D ⊂ F has a supremum denoted by ess.sup D and called essential supremum of D. Further, if D is directed upwards(A ∪ B ∈ D for all A, B ∈ D) there is an increasing sequence (An ) in D such that ess.sup D = n∈N An . If D ⊂ F is empty we set ess.sup D := ∅. Proof. For a countable set C ⊂ D define AC :=
A∈C
A. Then AC ∈ F and the upper bound
c := sup P [AC ] C ⊂ D countable is attained by some Csup ; indeed, take a sequence (Cn ) in D with P [ACn ] → c and Csup := n∈N Cn . Then, Csup ∈ F and P [ACsup ] = c. We conclude that ess.sup D := ACsup is as required. Indeed, ess.sup D is an upper bound of D, otherwise there would be A ∈ D with P [A \ ess.sup D] > 0 and in turn P [ACsup ∪{A} ] > P [ACsup ] = c. To see that ess.sup D is a least upper bound, observe ess.sup D ⊂ A whenever A ∈ F with A ⊂ A for all A ∈ D. By construction, there is an increasing sequence approximating ess.sup D if D is directed upwards. 2 Let E be an L0 -module. For a set C ⊂ E, we define the map M(· | C) : E → F , M(Z | C) := ess.sup{A ∈ F | 1A Z ∈ C}.
(2.4)
If C is an L0 -submodule of E the collection {A ∈ F | 1A Z ∈ C} is directed upwards for all Z ∈ E and hence there exists an increasing sequence (Mn ) ⊂ F such that M(Z | C) =
Mn .
(2.5)
n∈N
Definition 2.10. A set C ⊂ E has the closure property if 1M(Z|C) Z ∈ C
for all Z ∈ E.
(2.6)
By Cˆ we denote the smallest subset of E that has the closure property and contains C. Note that Cˆ is given by Cˆ = {1M(Z|C) Z | Z ∈ E} and therefore Cˆ always exists and is well defined. By definition, the closure property is a property in reference to E. In particular, E has the closure property. The next example illustrates the situation where an L0 -submodule C of E does not have the closure property. Lemma 2.11. Let C ⊂ E be an L0 -submodule. Then Cˆ is again an L0 -submodule.
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Proof. Let X ∈ Cˆ and Y ∈ L0 . Denote Z = Y X. By definition, there exists some X ∈ E with X = 1M(X |C) X . Since C is an L0 -submodule of E there exist an increasing sequence (Mn ) ⊂ F with Mn M(X | C) such that 1Mn X ∈ C. Hence 1Mn Z = Y 1Mn X ∈ C, and thus Mn ⊂ M(Z | C), for all n ∈ N. We conclude that M(X | C) ⊂ M(Z | C) and thus ˆ Y X = Y 1M(X |C) X = 1M(Z|C) Z ∈ C. Now let X = 1A X , Y = 1B Y ∈ Cˆ where A := M(X | C) and B := M(Y | C), for some ∈ E. Denote
X , Y
Z = X + Y = 1A\B X + 1A∩B (X + Y ) + 1B\A Y. As above there exist increasing sequences (An ), (Bn ) ⊂ F with An A and Bn B such that 1An X , 1Bn Y ∈ C and thus 1An \B X = 1A\B 1An X ∈ C, 1An ∩Bn (X + Y ) = 1Bn 1An X + 1An 1Bn Y ∈ C, 1Bn \A Y = 1B\A 1Bn Y ∈ C. Define the disjoint union Mn = (An \ B) ∪ (An ∩ Bn ) ∪ (Bn \ A). We obtain 1Mn Z = 1An \B X + 1An ∩Bn (X + Y ) + 1Bn \A Y ∈ C, and thus Mn ⊂ M(Z | C), for all n ∈ N. Since Mn A ∪ B, we conclude that A ∪ B ⊂ M(Z | C) and thus ˆ X + Y = 1M(Z|C) Z ∈ C. Hence the lemma is proved.
2
For a set C ⊂ E we denote by spanL0 (C) :=
n
Yi Xi Xi ∈ C, Yi ∈ L0 , 0 ≤ i ≤ n, n ∈ N
i=1
the L0 -submodule of E generated by C. The next example illustrates the situation where an L0 -submodule C of E does not have the closure property. Example 2.12. Consider the probability space Ω = [0, 1], F = B[0, 1] the Borel σ -algebra and P the Lebesgue measure on [0, 1]. Let E = L0 , and define C := spanL0 {1[1−2−(n−1) ,1−2−n ] | n ∈ N}. ˆ Then, 1 ∈ / C but 1 ∈ C.
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Proposition 2.13. Let C ⊂ E be an L0 -submodule of E, Z ∈ E and Z := 1M(Z |C)c Z . Then (i) M(Z | C) = M(Z | C), (ii) X = X and Y = Y on M(Z | C)c whenever X + Y Z = X + Y Z for X, X ∈ C and Y, Y ∈ L0 , and (iii) for W ∈ 1M(Z|C)c L0 and an L0 -linear function μ : C → L0 μ(X ¯ + Y Z) := μX + Y W
for all X ∈ C and Y ∈ L0 ,
(2.7)
defines the unique L0 -linear extension of μ to spanL0 (C, Z) which satisfies μZ ¯ = W. If in addition to this C has the closure property, (iv) spanL0 (C, Z ) = spanL0 (C, Z). Proof. (i) By definition of Z, M(Z | C) ⊂ M(Z | C), and since P [M(Z | C) \ M(Z | C)] > 0 would contradict the definition of M(Z | C) we have M(Z | C) = M(Z | C). (ii) X + Y Z = X + Y Z is equivalent to X − X = (Y − Y )Z. If B := {Y − Y = 0} ∩ M(Z | C)c had positive measure then on B, Z = (X − X )/(Y − Y ) ∈ C in contradiction to the definition of M(Z | C). Hence Y = Y and in turn X = X on M(Z | C)c . (iii) This is an immediate consequence of (ii). (iv) By definition of Z, spanL0 (C, Z) ⊂ spanL0 (C, Z ). Since C has the closure property, 1M(Z |C) Z ∈ C and hence spanL0 (C, Z) = spanL0 (C, Z ). 2 We can now state and prove the main result of this section. Theorem 2.14 (Hahn–Banach). Consider an L0 -sublinear function p : E → L0 , an L0 submodule C of E and an L0 -linear function μ : C → L0 such that μX p(X) for all X ∈ C. Then μ extends to an L0 -linear function μ¯ : E → L0 such that μX ¯ p(X) for all X ∈ E. Proof. Step 1: In view of Lemma 2.15 below we can assume that C has the closure property and that there exists Z ∈ E \ C. Then Z := 1M(Z |C)c Z ∈ / C and Z = 0. We will show that μ extends L0 -linearly to μ¯ : spanL0 (C, Z) → C, such that μX ¯ p(X) for all X ∈ spanL0 (C, Z).
(2.8)
More precisely, we claim that
W := 1M(Z|C)c ess.sup μX − p(X − Z) X∈C
and μ¯ defined as in (2.7) satisfies μX + Y W p(X + Y Z)
for all X ∈ C and Y ∈ L0 ,
(2.9)
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which, apparently, is equivalent to (2.8). To verify this claim, let X, X ∈ C and observe μX + μX = μ(X + X ) p(X + X ) = p(X + Z + X − Z) p(X + Z) + p(X − Z). Hence, μX − p(X − Z) p(X + Z) − μX
for all X, X ∈ C.
(2.10)
Since Z = 0 on M(Z | C) we have μX − p(X − Z) 0 on M(Z | C) as well as p(X + Z) − μX 0 on M(Z | C) for all X, X ∈ C. Hence, (2.10) implies μX − p(X − Z) W p(X + Z) − μX
for all X, X ∈ C,
(2.11)
and in turn μX ± W p(X ± Z)
for all X ∈ C.
From this we derive 1A (μX + W ) 1A p(X + Z) = 1A p(X + 1A Z),
(2.12)
1Ac (μX − W ) 1Ac p(X − Z) = 1Ac p(X − 1Ac Z)
(2.13)
for all A ∈ F . Adding up the inequalities in (2.12) and (2.13) yields
μX + (1A − 1Ac )W p X + (1A − 1Ac )Z for all X ∈ C and A ∈ F .
(2.14)
Further, for all Y ∈ L0 with P [Y = 0] = 1 we have Y/|Y | = 1A − 1Ac , where A := {Y > 0} ∈ F . Thus, (2.14) implies Y X Y X + W |Y |p + Z |Y | μ |Y | |Y | |Y | |Y | for all X ∈ C and Y ∈ L0 with P [Y = 0] = 1. From this we derive μX + Y W p(X + Y Z)
for all X ∈ C and Y ∈ L0 with P [Y = 0] = 1.
(2.15)
But this already implies the required inequality in (2.9). Indeed, for X ∈ C and arbitrary Y ∈ L0 we define Y := Y 1A + 1Ac , where A := {Y = 0}, and derive from (2.15) 1A (μX + Y W ) = 1A (μX + Y W ) 1A p(X + Y Z) = 1A p(X + Y Z), 1Ac (μX + Y W ) = 1Ac (μX) 1Ac p(X) = 1Ac p(X + Y Z). Adding up (2.16) and (2.17), we see that (2.15) implies (2.9) and complete this step.
(2.16) (2.17)
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Step 2: The set
0
C ⊂ D L -linear ⊂ E, D has the closure property I := (D, μ) ¯ L0 -linear 0 μ¯ : D → L , μ| ¯ C = μ and μX ¯ p(X) for all X ∈ D
is partially ordered by (D, μ) ¯ (D , μ¯ )
if and only if
D ⊂ D
and μ¯ |D = μ. ¯
We will show that a totally ordered subset {(Di , μ¯ i ), i ∈ I } of I (that is, for all i, j either (Di , μ¯ i ) (Dj , μ¯ j ) or (Di , μ¯ i ) (Dj , μ¯ j )) has an upper bound and then we will apply Zorn’s lemma. To this end, observe that D given by C ⊂ D :=
Di ⊂ E
i∈I
¯ Di := μ¯ i is an L0 -module since {(Di , μ¯ i ), i ∈ I } is totally ordered. μ¯ : D → L0 given by μ| is L0 -linear, dominated by p on all of D and μ| ¯ C = μ. Further, in view of Lemma 2.15 below, we can assume that D has the closure property. Hence, (D, μ) ¯ ∈ I is an upper bound for {(Di , μ¯ i ), i ∈ I } and Zorn’s lemma yields the existence of a maximal element (Dmax , μ¯ max ) ∈ I, i.e. (Dmax , μ¯ max ) (D, μ) ¯ ∈I
implies (Dmax , μ¯ max ) = (D, μ). ¯
Assume that Dmax = E. Then, by the first step of this proof, μ¯ max extends to μ¯ max : spanL0 (Dmax , Z) → L0 , where Z ∈ E \ Dmax , which contradicts the maximality of (Dmax , μ¯ max ). Hence, Dmax = E and μ¯ max is as desired. 2 Lemma 2.15. Let C, μ, p be as in Theorem 2.14. Then μ extends uniquely to an L0 -linear ˆ function μˆ : Cˆ → L0 such that μX ˆ p(X) for all X ∈ C. Proof. For Z ∈ E, let μ(1 ˆ M(Z|C) Z) := lim μ(1Mn Z), n→∞
where M(Z | C) =
n∈N Mn
(2.18)
as in (2.5). Since for all n m μ(1Mn Z) = μ(1Mm Z)
on Mn ,
ˆ (2.18) uniquely and unambiguously defines the L0 -linear extension μˆ : Cˆ → L0 of μ to C. ˆ 2 Further, (2.18) guarantees that μX ˆ p(X) for all X ∈ C.
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2.3. Locally L0 -convex modules In this section we establish some facts about locally L0 -convex modules. For more background on general topological spaces we refer to the comprehensive Chapter 2 of [1]. Let us first recall some basic definitions. Let T be a topology on some set E. Then K ⊂ ˚ ∂K, K, ¯ E is closed if K c ∈ T . The interior, boundary and closure of K are denoted by K, ˚ and K closed if and only respectively. Moreover, K˚ ∩ ∂K = ∅, K is open if and only if K = K, ˚ ∂K, K¯ is an interior, boundary, closure point of K, respectively. ¯ An element X ∈ K, if K = K. Now let E be an L0 -module and T the topology induced by some family P of L0 -semi norms on E, see Definition 2.3 and below. The following result gives one direction in the proof of Theorem 2.4. The converse direction is proved in Corollary 2.24 below. Lemma 2.16. (E, T ) is a locally L0 -convex module. Proof. Let U denote the neighborhood base given in (2.2). It follows by inspection that each U ∈ U is L0 -convex, L0 -absorbent and L0 -balanced as in Definition 2.2. To establish (i) and (ii) of Definition 2.1, let O ∈ T . (i) We show that O˜ := {(X, Y ) | (X, Y ) ∈ E × E, X + Y ∈ O} is open. Let (X, Y ) ∈ O˜ and U = UQ,ε ∈ U such that X + Y + U ⊂ O. Then V = UQ,ε/2 satisfies V + V ⊂ U and hence ˜ This means that (X, Y ) is an interior point of O˜ and (i) follows. (X + V ) × (Y + V ) ⊂ O. (ii) We show that O˜ := {(X, Y ) | (X, Y ) ∈ E × L0 , XY ∈ O} is open. Consider (X, Y ) ∈ O˜ and U = UQ,ε ∈ U such that XY + U ⊂ O. We find ε ∈ L0++ and W ∈ U such that W × Z ∈ L0 |Z − Y | ε ⊂ O˜ as follows. As in the proof of (i) let V ∈ U be such that V + V ⊂ U and let ε ∈ L0++ be such that εX ∈ V , which is possible since V is L0 -absorbing. Further, since V is L0 -balanced, (Z − Y )X ∈ V
if |Z − Y | ε.
V is of the form V = UQ,δ , hence W := UQ,δ/(ε+|Y |) satisfies (ε + |Y |)W ⊂ V and since W is L0 -balanced ZW ⊂ V
for all Z ∈ L0 with |Z| ε + |Y |.
Finally, for |Z − Y | ε and X ∈ W we derive Z(X + X ) − Y X = (Z − Y )X + ZX ∈ V + V ⊂ U and the assertion is proved.
2
Here is a trivial example. Example 2.17 (Chaos topology). The locally L0 -convex topology T induced by the trivial L0 semi norm · ≡ 0 on L0 consists of the sets ∅ and L0 . T is called chaos topology and it is an example for a locally L0 -convex topology which is not Hausdorff. Note that T is locally convex and locally L0 -convex at the same time.
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2.3.1. The countable concatenation property A technicality we encounter is a certain concatenation property. This concatenation property is crucial in the context of hyperplane separation, cf. Lemma 2.28, Theorem 2.8 and Examples 2.29 and 2.30 in Section 2.5 below. The following result motivates the subsequent definition. Lemma 2.18. Let P be a family of L0 -semi norms inducing a locally L0 -convex topology T on E. (i) For A ∈ F and · ∈ P, 1A · is an L0 -semi norm. (ii) For a finite collection · 1 , . . . , · n ∈ P, supi=1,...,n · i is an L0 -semi norm. (iii) Define P := P ∪ 1A · A ∈ F , · ∈ P , P := P ∪ sup · Q ⊂ P finite
· ∈Q
and denote T and T the induced locally L0 -convex topologies, respectively. Then T = T = T ; in other words, we may always assume that, with every · ∈ P, P contains 1A · for all A ∈ F and that P is closed under finite suprema. Proof. (i) and (ii) follow from the properties of L0 -semi norms. (iii) Since P ⊂ P ⊂ P we have T ⊂ T ⊂ T . The inclusion T ⊂ T follows from the fact that for all ε ∈ L0++ , U{ · },ε ⊂ U{1A · },ε
for all · ∈ P and A ∈ F
U{ · 1 ,..., · n },ε = U{supi=1,...,n · i },ε
and
for all · 1 , . . . , · n ∈ P.
2
For a finite collection UQ1 ,ε1 , . . . , UQn ,εn and a finite collection of pairwise disjoint sets A1 , . . . , An ∈ F (Ai ∩ Aj = ∅ for i = j ), the preceding lemma shows that ni=1 1Ai UQi ,εi is a neighborhood of 0 ∈ E. Indeed, let
· :=
n
1Ai sup · = sup 1Ai sup ·
i=1
· ∈Qi
i=1,...,n
· ∈Qi
and ε := ni=1 1Ai εi . Then, ni=1 1Ai UQi ,εi = U{ · },ε . In the case of a countably infinite sequence (UQn ,εn ) and a pairwise disjoint sequence (An ) ⊂ F (Ai ∩ Aj = ∅ for i = j ) the next example illustrates that the above reasoning does not apply, as the L0 -semi norm given by
· :=
n∈N
1An sup · = sup 1An sup ·
· ∈Qn
cannot be assumed to belong to P in general.
n∈N
· ∈Qn
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Example 2.19. Consider the probability space Ω = [0, 1], F = σ (An | n ∈ N) the σ -algebra generated by the sets An := [1 − 2−(n−1) , 1 − 2−n ], and P the Lebesgue measure. Define Bn := 0 0 A mn m , and let E := L . For the family P of L -semi norms | · |n := 1An | · |, n ∈ N, we subsequently derive the following: (i) | · | = n∈N | · |n ∈ / P. (ii) For all ε ∈ L0++ , U{|·|},ε = n∈N 1An U{|·|n },ε is not a neighborhood of the origin in the locally L0 -convex topology induced by P. (iii) The sequence (1Bn n1 + 1Bnc )n∈N converges to 0 w.r.t. the locally L0 -convex topology induced by P but it does not converge to 0 in the locally L0 -convex topology induced by P ∪ {| · |}. This leads us to the following definition. Definition 2.20. A family P of L0 -semi norms has the countable concatenation property if 1An · n ∈ P n∈N
for every pairwise disjoint sequence (An ) ⊂ F and for every sequence of L0 -semi norms ( · n ) in P. If P is a family of L0 -semi norms which has the countable concatenation property then (E, T ) has the countable concatenation property in the sense of Definition 2.7. Conversely, if (E, T ) is a topological L0 -module which has the countable concatenation property, where T is induced by a family P of L0 -semi norms, we can always assume that P has the countable concatenation property. Indeed, inspection shows that
1An · n (An ) ⊂ F pairwise disjoint, · n ⊂ P n∈N
also induces T . In view of Lemma 2.18 we can always assume that a finite family of L0 -semi norms has the countable concatenation property. 2.3.2. The index set of nets The neighborhood base U of 0 ∈ E given in (2.2) is indexed with the collection of all finite subsets of P and L0++ . We introduce a direction “” on this index set as follows: (R2 , α2 ) (R1 , α1 )
if and only if
R2 ⊂ R1
and α1 α2 ,
(2.19)
for all finite R1 , R2 ⊂ P and α1 , α2 ∈ L0++ . We denote nets w.r.t. this index set by (XR,α ). If E is a topological L0 -module, not necessarily locally L0 -convex, nets are denoted by (Xα )α∈D or (Xα ) for corresponding index set D. 2.4. The gauge function Let E be an L0 -module.
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Definition 2.21. The gauge function pK : E → L¯ 0+ of a set K ⊂ E is defined by pK (X) := ess.inf Y ∈ L0+ X ∈ Y K .
(2.20)
The gauge function pK of an L0 -absorbent set K ⊂ E maps E into L0+ . Moreover: Proposition 2.22. The gauge function pK of an L0 -absorbent set K ⊂ E satisfies: (i) pK (X) 1 for all X ∈ K. (ii) 1A pK (1A X) 1A pK (X) for all X ∈ E and A ∈ F . (iii) YpK (1{Y >0} X) = pK (Y X) for all X ∈ E and Y ∈ L0+ ; in particular, YpK (X) = pK (Y X) if Y ∈ L0++ . Proof. (i) This assertion follows immediately from the definition of pK . (ii) Let X ∈ E and A ∈ F . We have 1A ess.inf Z = 1A ess.inf 1A Z 1A ess.inf 1A Z X∈ZK
X∈ZK
1A X∈1A ZK
= 1A ess.inf 1A Z = 1A ess.inf Z, 1A X∈ZK
1A X∈ZK
(2.21)
where the inequality in (2.21) follows since X ∈ ZK implies 1A X ∈ 1A ZK. Hence, 1A pK (X) 1A pK (1A X). (iii) Let X ∈ E, Y ∈ L0+ and define A := {Y > 0}. We have Y ess.inf Z = ess.inf Y Z 1A X∈ZK
1A X∈ZK
Z :=Y Z
=
ess.inf
1A XY ∈1A Z K
Z
= ess.inf Z = ess.inf Z, 1A XY ∈ZK
and hence YpK (1A X) = pK (Y X).
XY ∈ZK
2
A non-empty L0 -absorbent L0 -convex set K ⊂ E always contains the origin; indeed, let X ∈ E and Y1 , Y2 ∈ L0++ be such that X/Y1 , −X/Y2 ∈ K. Then, since K is L0 -convex, Y2 −X X−X Y1 X + = = 0 ∈ K. Y1 + Y2 Y1 Y1 + Y2 Y2 Y1 + Y2
(2.22)
Depending on the choice of K ⊂ E, the gauge function pK can be L0 -sublinear or an L0 -semi norm. Proposition 2.23. The gauge function pK of an L0 -absorbent L0 -convex set K ⊂ E satisfies: (i) pK (X) = ess.inf{Y ∈ L0++ | X ∈ Y K} for all X ∈ E. (ii) YpK (X) = pK (Y X) for all Y ∈ L0+ and X ∈ E. (iii) pK (X + Y ) pK (X) + pK (Y ) for all X, Y ∈ E.
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(iv) For all X ∈ E there exists a sequence (Zn ) in L0 such that Zn pK (X) a.s.
(2.23)
In particular, since 0 ∈ K (cf. (2.22)), pK is L0 -sublinear. If in addition to this K is L0 -balanced then pK satisfies: (v) pK (Y X) = |Y |pK (X) for all Y ∈ L0 and for all X ∈ E. In particular, pK is an L0 -semi norm. Proof. (i) As “” follows from the definition of pK we only prove the reverse inequality. To this end, let Y ∈ L0+ with X = Y Z for some Z ∈ K. Then {Y = 0} ⊂ {X = 0} and in turn A := {Y > 0} ⊃ {X = 0}. Thus, with Yε := 1A Y + 1Ac ε for ε ∈ L0++ we have X = 1A X = Y 1A Z = Yε 1A Z ∈ Yε 1A K ⊂ Yε K. The claim now follows since ess.infε∈L0 Yε = Y . ++ (ii) To prove this assertion we first show that 1A pK (1A X) = 1A pK (X)
for all X ∈ E and A ∈ F .
(2.24)
(ii) then follows from (iii) of Proposition 2.22 together with (2.24). To establish (2.24), we only have to prove the reverse inequality in (2.21). To this end, let Y1 , Y2 ∈ L0+ with 1A X = 1A Y1 Z1 , X = Y2 Z2 for Z1 , Z2 ∈ K and A ∈ F . In particular, 1Ac X = 1Ac Y2 Z2 . We have X = 1A Y1 Z1 + 1Ac Y2 Z2 = (1A Y1 + 1Ac Y2 )(1A Z1 + 1Ac Z2 ) and since L0 -convexity of K implies that 1A Z1 + 1Ac Z2 = 1A Z1 + (1 − 1A )Z2 ∈ K the required inequality follows. (iii) Let X1 , X2 ∈ E and Y1 , Y2 ∈ L0++ such that X1 /Y1 , X2 /Y2 ∈ K. Since K is L0 -convex Y1 X1 Y2 X2 X1 + X2 + = ∈ K. Y1 + Y2 Y1 Y1 + Y2 Y2 Y1 + Y2 2 Thus, pK ( XY11 +X +Y2 ) 1, and hence pK (X1 + X2 ) Y1 + Y2 . Since Y1 and Y2 are arbitrary, we may take the essential infimum over all such pairs Y1 , Y2 and — in view of (i) — we derive
pK (X1 + X2 ) pK (X1 ) + pK (X2 ). (iv) As in the proof of (2.24), L0 -convexity of K implies that the set Y ∈ L0+ X ∈ Y K is directed downwards (and upwards) for all X ∈ E.
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(v) Let X ∈ E, Y ∈ L0 and A := {Y 0}. Then (2.24) and (ii) imply pK (Y X) = 1A |Y |pK (X) + 1Ac |Y |pK (−X), and hence it remains to prove that pK (−X) = pK (X). But since K is L0 -balanced we have −K = K and hence pK (−X) = p−K (−X) = pK (X).
2
As a consequence of Proposition 2.23, we can now complete the proof of Theorem 2.4: Corollary 2.24. Any locally L0 -convex topology T on E is induced by a family of L0 -semi norms. Proof. Let U be a neighborhood base of 0 ∈ E such that every U ∈ U is L0 -absorbent, L0 convex and L0 -balanced. Then, the family of gauge functions P := {pU | U ∈ U}, by Proposition 2.23, is a family of L0 -semi norms and the topology induced by P coincides with T . 2 Proposition 2.25. The gauge function pK of an L0 -absorbent L0 -convex set K ⊂ E (recall that 0 ∈ K, cf. (2.22)) satisfies: / 1A K for all A ∈ F with P [A] > 0. (i) pK (X) 1 for all X ∈ E with 1A X ∈ If in addition to this, E is a locally L0 -convex module, then pK satisfies: ˚ (ii) pK (X) < 1 for all X ∈ K. Proof. To prove (i) let us assume that {pK (X) < 1} has positive P -measure for some X ∈ E / K for all A ∈ F with P [A] > 0. With (iv) of Proposition 2.23 we know that there is with X1A ∈ Y ∈ L0+ such that B := {Y < 1} has positive P -measure and X ∈ Y K. But this is a contradiction as we derive X1B ∈ Y 1B K ⊂ 1B K, where the last inclusion follows from the L0 -convexity of 1B K. (Note that 0 ∈ K.) ˚ Then there exists a neighborhood UQ,ε (Q ⊂ P finite and ε ∈ L0++ ) of 0 ∈ E (ii) Let X ∈ K. such that X + UQ,ε ⊂ K. In view of Proposition 2.18 we can assume that P is closed under finite suprema and that UQ,ε = U{ · sup },ε , where · sup := sup · ∈Q · . Then, for all δ ∈ L0++ , X − X(1 + δ)
sup
= δ X sup .
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Thus, choosing δ such that δ X sup ε, we derive X(1 + δ) ∈ K and hence pK (X) 1/(1 + δ) < 1. 2 2.5. Hyperplane separation Let E be a locally L0 -convex module. Let X ∈ E be such that there is an L0 -linear bijection μ : spanL0 (X) → L0 . Then, necessarily μ(Y X) = Y μX
for all Y ∈ L0 ,
(2.25)
and μ−1 : L0 → spanL0 (X) is L0 -linear as well. Since μ is a surjection we derive from (2.25) that P [μX = 0] = 1. Further,
Y = μ μ−1 (Y ) = μ(Y¯ X) = Y¯ μX for all Y ∈ L0 . Hence, Y¯ = Y/μX and in turn μ−1 (Y ) = Y X/μX. On replacing μ by μ/(μX), we can always assume that μX = 1. In this case, μ(Y X) = Y and μ−1 Y = Y X for all Y ∈ L0 . Lemma 2.26. Let K, M ⊂ E be L0 -convex, K open and non-empty. If 1A M ∩ 1A K = ∅ for all A ∈ F with P [A] > 0, then there is an L0 -linear function μ : E → L0 such that μY < μZ
for all Y ∈ K and Z ∈ M.
(2.26)
Proof. We can assume that M is non-empty. Step 1: Suppose first that M = {X} is a singleton. Without loss of generality, we may assume that 0 ∈ K. Indeed, if 0 ∈ / K replace X by X − Y and K by K −Y for some Y ∈ K which is possible since K = ∅. Note that {X −Y }, K −Y remain L0 -convex, that K − Y remains open non-empty and that an L0 -linear function μ : E → L0 separates {X} from K — in the sense of (2.26) — if and only if it separates {X − Y } from K −Y. Thus, let K be L0 -convex open non-empty and 0 ∈ K. (Note that K is L0 -absorbent.) By / K for all A ∈ F with P [A] > 0. In particular, 1A X = 0 for all A ∈ F with assumption, 1A X ∈ P [A] > 0. Hence, Y X = Y X implies Y = Y for all Y, Y ∈ L0 and μ : spanL0 (X) → L0 , μ(Y X) := Y
for all Y ∈ L0 ,
(2.27)
is a well-defined L0 -linear bijection of spanL0 (X) into L0 . By Proposition 2.23, the gauge function pK : E → L0 is L0 -sublinear. We show pK (Z) μZ for all Z ∈ spanL0 (X). For Z ∈ spanL0 (X) let Y ∈ L0 be the unique element with Z = Y X. From (2.24) in the proof of Proposition 2.23 we derive pK (Y X) = 1A pK (1A Y X) + 1Ac pK (1Ac Y X)
(2.28)
for A := {Y 0}. Further, with (ii) of Proposition 2.23 and (i) of Proposition 2.25 we know that 1A pK (1A Y X) = 1A YpK (X) 1A Y = 1A μ(Y X)
(2.29)
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and since pK 0 1Ac pK (1Ac Y X) 1Ac Y = 1Ac μ(Y X).
(2.30)
Adding up (2.29) and (2.30), together with (2.28), yield pK (Y X) μ(Y X). Hence, pK (Z) μZ for all Z ∈ spanL0 (X) and therefore μ extends by the Hahn–Banach Theorem 2.14 to μ : E → L0 such that μY pK (Y )
for all Y ∈ E.
In particular, for all Y ∈ K μY pK (Y ) < 1 = μX, where the strict inequality follows from (ii) of Proposition 2.25 and the equality follows from (2.27). Step 2: Now let M be as in the lemma. Then, K − M is L0 -convex open non-empty and 1A {0} ∩ 1A (K − M) = ∅ for all A ∈ F with P [A] > 0. Thus, from the first step of this proof, there is an L0 -linear function μ : E → L0 with μ(Y − Z) < 0 for all Y ∈ K and Z ∈ M, and the assertion is proved.
2
Lemma 2.27. Let K ⊂ E be open L0 -convex with 0 ∈ K. If μ : E → L0 is L0 -linear such that μ(X) pK (X)
for all X ∈ E,
then μ is continuous. Proof. It suffices to show that μ−1 Bε is a neighborhood of 0 ∈ E for each ball Bε centered at 0 ∈ L0 . Thus, let ε ∈ L0++ . The set U := εK ∩ −εK is a neighborhood of 0 ∈ E. (Indeed, let V := UQ,δ ⊂ K, be a neighborhood of 0 ∈ E, which exists since K is open and 0 ∈ K. Then, εV = UQ,εδ is an L0 -balanced neighborhood of 0 ∈ E. Further, εV ⊂ εK, −εV ⊂ −εK and since εV is L0 -balanced εV = −εV and in turn εV ⊂ εK ∩ −εK.) Further, for all X ∈ U we have μ(X) pK (X) ε
and
−μ(X) = μ(−X) pK (−X) ε. Thus, |μ(X)| ε and hence U ⊂ μ−1 Bε . We can now prove Theorem 2.6.
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Proof of Theorem 2.6. We can assume that M is non-empty. Define L := K − M. For X ∈ L, the set L − X is L0 -convex open and 0 ∈ L − X. By assumption, 0 ∈ / 1A L for all A ∈ F with / 1A (L − X). From the first step of the proof of Lemma 2.26 we know P [A] > 0 and so 1A (−X) ∈ that there is an L0 -linear function μ : E → L0 such that μY pL−X (Y )
for all Y ∈ E.
By Lemma 2.27, μ is continuous. Further, μY < μ(−X) for all Y ∈ L − X, 2
and Theorem 2.6 is proved.
Lemma 2.28. Let P be a family of F -semi norms inducing a locally L0 -convex topology on E and let K ⊂ E be closed with 1A X + 1Ac X ∈ K for all A ∈ F and X, X ∈ K. If P has the countable concatenation property and X ∈ E satisfies 1A {X} ∩ 1A K = ∅ for all A ∈ F with P [A] > 0, then there is an L0 -convex, L0 -absorbent and L0 -balanced neighborhood U of 0 ∈ E such that 1A (X + U ) ∩ 1A (K + U ) = ∅ for all A ∈ F with P [A] > 0. Proof. We can assume that K = ∅. Via translation by X, it suffices to construct an L0 -convex, L0 -absorbent and L0 -balanced neighborhood U of 0 ∈ E such that 1A U ∩ 1A (K + U ) = ∅ for all A ∈ F with P [A] > 0. Step 1: In this step we construct an L0 -convex, L0 -absorbent, L0 -balanced neighborhood U of 0 ∈ E such that 1A U ∩ 1A K = ∅ for all A ∈ F with P [A] > 0. To this end, define ε ∗ := 1 ∧ ess.sup ess.inf ε ∈ L0++ UQ,ε ∩ K = ∅ . Q⊂P finite
(Note that for all Q ⊂ P finite there is ε ∈ L0++ such that UQ,ε ∩ K = ∅ since all neighborhoods of 0 ∈ E are L0 -absorbent.) Successively we show that ε ∗ satisfies: (i) ε ∗ ∈ L0++ . (ii) There is an L0 -semi norm · ∗ ∈ P such that ε∗ < ess.inf ε ∈ L0++ U{ · ∗ },ε ∩ K = ∅ . 2 (iii) 1A U{ · ∗ },ε∗ /2 ∩ 1A K = ∅ for all A ∈ F . (Note that U{ · ∗ },ε∗ /2 is L0 -convex, L0 -absorbent, L0 -balanced and closed.)
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(i) Suppose P [A] > 0, A := {ε ∗ = 0}. Then for all Q ⊂ P finite and for all α ∈ L0++ there is XQ,α ∈ K such that 1A XQ,α ∈ UQ,1/α ∩ 1A K. Hence, for X ∈ K the net (1A XQ,α + 1Ac X) converges to 1Ac X and 1A XQ,α + 1Ac X ∈ K for all Q ⊂ P finite and for all α ∈ L0++ . Since K is closed, we derive 1Ac X ∈ K, which is impossible as it would imply 0 = 1A 1Ac X ∈ 1A K. (ii) For all finite Q ⊂ P, let εQ := ess.inf ε ∈ L0++ UQ,ε ∩ K = ∅ . For finite Q, Q ⊂ P, UQ∪Q ,ε ⊂ UQ,ε , UQ ,ε . Thus, the collection {εQ | Q ⊂ P finite} is directed upwards and hence there is an increasing sequence (εQn ) with 1 ∧ εQn ε ∗ a.s. Let A1 := εQ1 > ε ∗ /2 , An := εQn > ε ∗ /2 \ An−1 for all n 2. Then,
n∈N An
Ω since ε ∗ > ε ∗ /2. Further, the L0 -semi norm
· ∗ :=
n∈N
1An sup ·
· ∈Qn
is an element of P since P has the countable concatenation property and · ∗ is as required. (iii) Finally, assume there is A ∈ F , P [A] > 0, and X ∈ K such that 1A X ∈ 1A U{ · ∗ },ε∗ /2 . Then ε∗ 1A ess.inf ε ∈ L0++ U{ · ∗ },ε ∩ K = ∅ 1A , 2 in contradiction to the statement in (ii). Step 2: From the first step we have · ∈ P and ε ∈ L0++ such that 1A U{ · },ε ∩ 1A K = ∅ for all A ∈ F with P [A] > 0. This implies 1A U{ · },ε/2 ∩ 1A (K + U{ · },ε/2 ) = ∅ for all A ∈ F with P [A] > 0 and the assertion follows. 2 The next example illustrates, that the countable concatenation property, as an assumption on P in Lemma 2.28, cannot be omitted. Example 2.29. Let (Ω, F , P ), An , and the family P of L0 -semi norms on E = L0 be as in Example 2.19. From Example 2.19 we know that P does not have the countable concatenation property. We now further derive the following: (i) The set K := {X ∈ E | X 1} is closed with respect to the locally L0 -convex topology on E induced by P. Indeed, if X ∈ / K then there is n ∈ N such that 0 < 1 − X =: c ∈ R on An . But then X + U{1An |·|},c/2 defines a neighborhood of X which is disjoint of K. Hence K c is open.
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(ii) 1A K ∩ {0} = ∅ for all A ∈ F with P [A] > 0. This follows as 1An K ∩ {0} = ∅, for all atoms An , n ∈ N. (iii) For every neighborhood U of 0 ∈ E there exists A ∈ F with P [A] > 0 such that 1A K ∩ U = ∅. Indeed, for every neighborhood U of 0 ∈ E there are n ∈ N and ε ∈ L0++ such that U{1An |·|},ε ⊂ U . Note that P [An ] < 1. But now, 1Acn K ⊂ 1Acn E = 1Acn U{1An |·|},ε ⊂ U . We can now prove Theorem 2.8: Proof of Theorem 2.8. Recall we can assume a family P of L0 -semi norms induces the locally L0 -convex topology on E and that P inherits the countable concatenation property from E. By Lemma 2.28, there is an L0 -convex, L0 -absorbent and L0 -balanced neighborhood U of 0 ∈ E such that 1A (X + U ) ∩ 1A (K + U ) = ∅ for all A ∈ F with P [A] > 0. Since K + U˚ , X + U˚ are L0 -convex open and K + U˚ is non-empty Theorem 2.6 yields a continuous L0 -linear function μ : E → L0 such that for all Y ∈ K + U˚ and Z ∈ X + U˚ .
μY < μZ
Further, from the first step of the proof of Lemma 2.26 we know that there is X0 ∈ E such that μ(Y X0 ) = Y
for all Y ∈ L0 .
Since U˚ is L0 -absorbent and L0 -balanced there is ε ∈ L0++ such that −εX0 ∈ U˚ . Thus, μY < μ(X − εX0 ) = μX − ε
for all Y ∈ K + U˚ .
In particular, μY + ε < μX whence Theorem 2.8 is proved.
for all Y ∈ K,
2
We provide an example which illustrates that the countable concatenation property, as an assumption on P in Theorem 2.8, cannot be omitted. Example 2.30. Let (Ω, F , P ), An , and the family P of L0 -semi norms on E = L0 be as in Example 2.29. Then the closed subset K := {X ∈ E | X 1} of E cannot be separated from 0 by a continuous L0 -linear function. Indeed, as every L0 -linear function μ : E → L0 is of the form μX =
n∈N
1An an X
for all X ∈ E,
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for some sequence (an ) ⊂ R, we conclude that an > 0 for all n ∈ N if μ separates 0 from K. Such μ, however, is not continuous at 0. To see this, let Z := n∈N 1An an , ε ∈ L0++ and observe that μ−1 Y ∈ L0 |Y | ε = X ∈ E |X/Z| ε is not a neighborhood of 0 ∈ E. 3. Part II. Duality in locally L0 -convex modules 3.1. Main results We first recall and introduce some terminology. Let E be an L0 -module. The effective domain of a function f : E → L¯ 0 is denoted by dom f := {X ∈ E | f (X) ∈ L0 }. The epigraph of f is denoted by epi f := {(X, Y ) ∈ E × L0 | f (X) Y }. The function f is proper if f (X) > −∞ for all X ∈ E and dom f = ∅. Definition 3.1. Let E be an L0 -module and f : E → L¯ 0 a proper function. (i) f is L0 -convex if f (Y X1 + (1 − Y )X2 ) Yf (X1 ) + (1 − Y )f (X2 ) for all X1 , X2 ∈ E and Y ∈ L0 with 0 Y 1. (ii) f has the local property if 1A f (X) = 1A f (1A X) for all X ∈ E and A ∈ F . As a first result in this part, we obtain that L0 -convexity enforces the local property. The proof is given in Section 3.3 below. Theorem 3.2. Let E be an L0 -module. A proper function f : E → L¯ 0 is L0 -convex if and only if f has the local property and epi f is L0 -convex. We now address some topological properties of L0 -convex functions. Definition 3.3. Let E be a topological L0 -module. A function f : E → L¯ 0 is lower semi continuous if for all Y ∈ L0 the level set {X ∈ E | f (X) Y } is closed. As one expects from the real case, lower semi continuity of an L0 -convex function can also be characterized in terms of its epigraph. In fact, the following result is proved in Section 3.4. Proposition 3.4. Let E be a locally L0 -convex module that has the countable concatenation property. A proper function f : E → L¯ 0 that has the local property is lower semi continuous if and only if epi f is closed. A subset B of a topological L0 -module E is an L0 -barrel if it is L0 -convex, L0 -absorbent, L0 balanced and closed. A locally L0 -convex module E is an L0 -barreled module if every L0 -barrel is a neighborhood of 0 ∈ E. It follows by inspection that L0 -normed modules are L0 -barreled. The following result is proved in Section 3.5.
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Proposition 3.5. Let E be an L0 -barreled module. A proper lower semi continuous L0 -convex ˚ f. function f : E → L¯ 0 is continuous on dom We now turn to our main, Fenchel–Moreau type, duality results. Let E be a topological L0 module, and denote by L(E, L0 ) the L0 -module of continuous L0 -linear functions μ : E → L0 . The conjugate f ∗ : L(E, L0 ) → L¯ 0 of a function f : E → L¯ 0 is defined by
f ∗ (μ) := ess.sup μX − f (X) .
(3.31)
X∈E
Further, the conjugate f ∗∗ : E → L¯ 0 of f ∗ is defined by
f ∗∗ (X) := ess.sup μX − f ∗ (μ) .
(3.32)
μ∈L(E,L0 )
Definition 3.6. Let E be a topological L0 -module. An element μ of L(E, L0 ) is a subgradient of a function f : E → L¯ 0 at X0 ∈ dom f if μ(X − X0 ) f (X) − f (X0 )
for all X ∈ E.
The set of all subgradients of f at X0 is denoted by ∂f (X0 ). A pre stage of Theorem 3.7 below, which we will prove in Section 3.6, is given in Kutateladze [18,16,17]. However, Kutateladze entirely remains within an algebraic scope as he does not address topological aspects such as continuity. More precisely, he provides necessary and sufficient conditions for the existence of algebraic subgradients of L0 -sublinear functions in terms of the underlying ring. Further, Kutateladze only covers the case of L0 -sublinear functions which take values in L0 adjoint +∞, that is, L0 ∪ {+∞} rather than functions which take values in L¯ 0 . Theorem 3.7. Let E be an L0 -barreled module that has the countable concatenation property. Let f : E → L¯ 0 be a proper lower semi continuous L0 -convex function. Then, ˚ f. ∂f (X) = ∅ for all X ∈ dom Here is the generalized Fenchel–Moreau duality theorem, the proof of which is given in Section 3.7. Theorem 3.8. Let E be a locally L0 -convex module that has the countable concatenation property. Let f : E → L¯ 0 be a proper lower semi continuous L0 -convex function. Then, f = f ∗∗ . 3.2. Financial applications In this section we illustrate the scope of applications that can be covered by our results. The entropic risk measure ρ0 : L¯ 0 → [−∞, +∞] is defined as
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ρ0 (X) := log E exp(−X) . Its restriction to the locally convex vector space Lp , p ∈ [1, +∞], is proper, convex, lower semi continuous. Classical convex analysis yields the dual representation
ρ0 (X) = sup E[ZX] − ρ0∗ (Z) Z∈Lq
with conjugate function
ρ0∗ (Z) = sup E[ZX] − ρ0 (X) X∈Lp
= E −Z log(−Z) if defined and = +∞ otherwise
where q := p/(p − 1) (:= +∞ if p = 1), cf. [8]. For p = +∞, in particular, ρ0 is continuous and subdifferentiable on dom˚ ρ0 = L∞ with unique subgradient − exp(−X)/E[exp(−X)] at X ∈ L∞ . Market models in stochastic finance involve filtrations which represent the flow of information provided by the market. Let (Ω, F , (Ft )t∈N , P ) be a filtered probability space. We shall write L0 (F ), L0 (Ft ), etc. to express the respective reference σ -algebra. The [−∞, +∞]-valued entropic risk measure ρ0 can be made contingent on the information available at t by modifying it to ρt : L¯ 0 (F ) → L¯ 0 (Ft ), ρt (X) := log E exp(−X) Ft . As in the deterministic case, subdifferentiability and dual representation of ρt are important aspects in risk management applications. For this reason, ρt must be restricted to a space which allows for convex analysis. The restriction ρt to bounded risks, that is L∞ (F ), has been analyzed in [2,5,7,9]. It turns out that ρt maps L∞ (F ) into L∞ (Ft ). Convex analysis of ρt can then be carried out by means of scalarization, an idea which goes back to [12,19,21]. However, L∞ (F ) is a too narrow model space for financial risks. For instance, it does not contain normal distributed random variables. The space Lp (F ), for p ∈ [1, +∞), is larger and already sufficient for many applications. But ρt restricted to Lp (F ) takes values in L¯ 0 (Ft ) and the scalarization method used in the previous literature can no longer be applied. Exploiting our results, we thus propose to view ρt as a function on the L0 (Ft )-module p LFt (F ), defined in Example 2.5, which in fact is much larger than Lp (F ) and thus even better apt p for applications. The function ρt : LFt (F ) → L¯ 0 (Ft ) is proper L0 -convex. Fatou’s generalized lemma and Lemma 3.10 show that ρt is lower semi continuous. Moreover, from Theorem 3.8 we know that the following dual representation applies
ρt (X) = ess.sup E[ZX | Ft ] − ρt∗ (Z) q
Z∈LF (F ) t
=
ess.sup Y ∈L0 (Ft ), Z ∈Lq (F )
Y E[Z X | Ft ] − ρt∗ (Y Z ) .
For time-consistent dynamic risk assessment, compositions of the form ρt ◦ (−ρt+1 ) are another important aspect, cf. [5,9]. For the entropic risk measure we derive in an ad hoc manner
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that ρt ◦ (−ρt+1 ) = ρt on L¯ 0 (F ). Hence, our results immediately apply to the dynamic risk assessment by means of the entropic risk measure. An extension to more general dynamic risk measures and lower semi continuity as well as subdifferentiability aspects of compositions of lower semi continuous functions is subject to future research. 3.3. Proof of Theorem 3.2 To prove the if statement, let X1 , X2 ∈ E and Y ∈ L0 , 0 Y 1. The inequality
f Y X1 + (1 − Y )X2 Yf (X1 ) + (1 − Y )f (X2 )
(3.33)
is trivially valid on {f (X1 ) = +∞} ∪ {f (X2 ) = +∞}. Since f is proper there is X ∈ dom f . Since f has the local property X1 := 1{f (X1 )<+∞} X1 + 1{f (X1 )=+∞} X ∈ dom f, X2 := 1{f (X2 )<+∞} X2 + 1{f (X2 )=+∞} X ∈ dom f. From L0 -convexity of epi f we derive
f Y X1 + (1 − Y )X2 Yf X1 + (1 − Y )f X2 .
(3.34)
The local property of f together with (3.33) and (3.34) yields
f Y X1 + (1 − Y )X2 Yf (X1 ) + (1 − Y )f (X2 ), that is, f is L0 -convex. To establish the only if statement, observe that epi f is L0 -convex if f is L0 -convex. Thus, it suffices to prove that f has the local property. This, however, follows from the inequalities f (1A X) = f (1A X + 1Ac 0) ≤ 1A f (X) + 1Ac f (0) = 1A f (1A (1A X) + 1Ac X) + 1Ac f (0) ≤ 1A f (1A X) + 1Ac f (0) which become equalities if multiplied with 1A . 3.4. Lower semi continuous functions Lemma 3.9. Let E be a topological L0 -module. The essential supremum of a family of lower semi continuous functions fi : E → L¯ 0 , i ∈ I , I an arbitrary index set, is lower semi continuous. Proof. The assertion follows from the identity X X ∈ E and fi (X) Y X X ∈ E and ess.sup fi (X) Y = i∈I
for all Y ∈ L0 .
2
i∈I
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The essential limit inferior ess.lim infα Xα of a net (Xα ) ⊂ L0 is defined by ess.lim inf Xα := ess.sup ess.inf Xβ . α
α
βα
Lemma 3.10. Let E be a locally L0 -convex module that has the countable concatenation property. A proper function f : E → L¯ 0 that has the local property is lower semi continuous if and only if ess.lim inf f (Xα ) f (X)
(3.35)
α
for all nets (Xα ) ⊂ E with Xα → X for some X ∈ E. Proof. Assume that f has the local property, is lower semi continuous and let (Xα ) ⊂ E be such that Xα → X for some X ∈ E. Let Y ∈ L0 be such that Y < f (X) which is possible since f is proper. By lower semi continuity of f , the set V := {Z ∈ E | f (Z) Y } is closed and by the local property we have 1A X + 1Ac X ∈ V for all A ∈ F and X , X ∈ V . Further, / 1A V 1A X ∈ for all A ∈ F with P [A] > 0. By Lemma 2.28 there is a neighborhood U of 0 ∈ E such that 1A (X + U ) ∩ 1A V = ∅ for all A ∈ F with P [A] > 0. Since Xα → X there is α0 such that / 1A V for all β α0 and A ∈ F Xβ ∈ X + U for all β α0 . Due to the local property, 1A Xβ ∈ with P [A] > 0. Hence, f (Xβ ) > Y for all β α0 and in turn ess.lim inf f (Xα ) = ess.sup ess.inf f (Xβ ) α
α
βα
ess.inf f (Xβ ) Y. βα0
Since Y was arbitrary, we deduce (3.35). Now assume (3.35) and let Y ∈ L0 . We have to show that the set V := Z ∈ E f (Z) Y is closed. To this end, let (Xα ) ⊂ V and X ∈ E with Xα → X for some X ∈ E. Then, from the inequality f (Xα ) Y for each α, we obtain f (X) ess.lim inf f (Xα ) Y, α
so X ∈ V . That is, V is closed, and hence f is lower semi continuous. Next, we prove Proposition 3.4. Proof of Proposition 3.4. Define φ : E × L0 → L¯ 0 by φ(X, Y ) := f (X) − Y.
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From Lemma 3.10 and the definition of the product topology we derive that lower semi continuity of f on E is equivalent to lower semi continuity of φ on E × L0 . For all Z ∈ L0 we have
(X, Y ) ∈ E × L0 φ(X, Y ) Z = epi f − (0, Z).
Since E × L0 is a topological L0 -module we derive that {(X, Y ) ∈ E × L0 | φ(X, Y ) Z} is closed if and only if epi f is closed. This proves Proposition 3.4. 2 3.5. Lower semi continuous L0 -convex functions Lemma 3.11. Let E be a topological L0 -module. If in the neighborhood of X0 ∈ E a proper L0 -convex function f : E → L¯ 0 is bounded above by Y0 ∈ L0 then f is continuous at X0 . Proof. On replacing f by f (· + X0 ) − f (X0 ), we assume that X0 = f (X0 ) = 0. Let δ ∈ L0++ and f (X) Y0 for all X in a neighborhood V of 0 ∈ E. We have to show that there is a neighborhood Wδ of 0 ∈ E such that |f (X)| δ for all X ∈ Wδ . Without loss of generality we can assume that Y0 is such that ε := δ/Y0 > 0 is well defined and ε < 1. Since E is a topological L0 -module W := V ∩ −V is a symmetric (W = −W ) neighborhood of 0 ∈ E. We will show that the neighborhood Wδ := εW is as required. Indeed, for all X ∈ εW we have ±X/ε ∈ V and hence L0 -convexity of f implies f (X) (1 − ε)f (0) + εf (X/ε) εY0 = δ
and
f (X) (1 + ε)f (0) − εf (−X/ε) −εY0 = −δ. Thus, |f (X)| δ for all X ∈ Wδ , whence the required continuity follows.
2
Proposition 3.12. Let E be a topological L0 -module. Let f : E → L¯ 0 be a proper L0 -convex function. The following statements are equivalent: (i) There is a non-empty open set O ⊂ E on which f is bounded above by Y0 ∈ L0 . ˚ f and dom ˚ f = ∅. (ii) f is continuous on dom ˚ f and for every δ ∈ L0++ (F ) there is a neighProof. (ii) implies (i) since for every X0 ∈ dom borhood V of X0 such that f (X0 ) − δ f (X) f (X0 ) + δ for all X ∈ V . O := V˚ and Y0 := f (X0 ) + δ are then as required. ˚ f , whence dom ˚ f = Conversely, let O and Y0 be as in (i) and take X0 ∈ O. Then, X0 ∈ dom ˚ f , let X1 ∈ dom ˚ f . Observe that there is Y1 ∈ L0++ , ∅. To see that f is continuous on dom ˚ f . Since E is a topological L0 -module the Y1 > 1, such that X2 := X0 + Y1 (X1 − X0 ) ∈ dom map H : E → E given by H (X) := X2 −
Y1 − 1 (X2 − X) for all X ∈ E, Y1
is continuous and has continuous inverse H −1 . As H transforms X0 into X1 , it transforms O into an open set H (O) containing X1 . By L0 -convexity of f , we have for all X ∈ H (O)
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Y1 − 1 −1 1 H (X) + X2 Y1 Y1
Y1 − 1 1 f H −1 (X) + f (X2 ) Y1 Y1 Y1 − 1 1 Y0 + f (X2 ). Y1 Y1
f (X) = f
˚ f there is a neighborhood of X1 on which f is bounded In other words, for every X1 ∈ dom 0 above by an element of L . By Lemma 3.11, f is continuous at X1 . 2 Corollary 3.13. Let E be a topological L0 -module and X ∈ E. Every proper L0 -convex function ˚ f. f : spanL0 (X) → L¯ 0 is continuous (with respect to the trace topology) on dom ˚ f , else translate. Then there is a Proof. Without loss of generality we assume that 0 ∈ dom ˚ f . From L0 neighborhood U of 0 ∈ spanL0 (X) and Y ∈ L0++ such that X˜ := Y X ∈ U ⊂ dom ˜ on the open set convexity it follows that f is bounded above by sup(f (0), f (X)) λX˜ 0 < λ < 1, λ ∈ L0 ˚ f. and hence, by Proposition 3.12, f is continuous on dom
2
We can now prove Proposition 3.5. ˚ f . By translation, we may assume Proof of Proposition 3.5. Assume that there is X0 ∈ dom X0 = 0. Take Y0 ∈ L0 such that f (0) < Y0 . By assumption, the level set C := {X ∈ E | f (X) Y0 } is closed. Further, for all X ∈ E the net (X/Y )Y ∈L0 converges to 0 ∈ E. By Corol++ lary 3.13, the restriction of f to spanL0 (X) is continuous at 0, hence f (X/Y ) < Y0 for large Y which implies that C is L0 -absorbent. Hence, C ∩ −C is an L0 -barrel and in turn a neighborhood of 0 ∈ E. Thus, C is a neighborhood of 0 ∈ E and since f is bounded above by Y0 on all of C it is continuous at 0. This proves Proposition 3.5. 2 3.6. Subdifferentiability Let E be a topological L0 -module. Recall the definitions (3.31) and (3.32) of the conjugates and f ∗∗ of a function f : E → L¯ 0 and f ∗ , respectively. The effective domain of f ∗ is given by the set
f∗
μ ∈ L E, L0 ∃Y ∈ L0 : ess.sup μX − f (X) Y . X∈E
If f is proper, then f ∗ maps its effective domain into L0 and f ∗ is L0 -convex if f is so. The effective domain of f ∗∗ is given by the set
X ∈ E ∃Y ∈ L0 : ess.sup μX − f ∗ (μ) Y . μ∈L(E,L0 )
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Again, if f ∗ is proper f ∗∗ maps its effective domain into L0 and f ∗∗ is L0 -convex if f ∗ is so. Since for all X ∈ E and μ ∈ L(E, L0 ), f ∗ (μ) μX − f (X)
(3.36)
f (X) f ∗∗ (X).
(3.37)
we have for all X ∈ E
For μ ∈ L(E, L0 ) and X0 ∈ dom f we have μ ∈ ∂f (X0 )
if and only if
f (X0 ) = μX0 − f ∗ (μ).
(3.38)
Indeed, μ ∈ ∂f (X0 ) by definition means
f (X0 ) μX0 − μX − f (X) for all X ∈ E. This is equivalent to
f (X0 ) μX0 − ess.sup μX − f (X) = μX0 − f ∗ (μ) X∈E
which, by (3.36), is equivalent to f (X0 ) = μX0 − f ∗ (μ). With (3.37) and (3.38) we know that μ ∈ ∂f (X0 ) maximizes (3.32) at X0 , i.e. f ∗∗ (X0 ) = μX0 − f ∗ (μ). Lemma 3.14. Let E be an L0 -barreled module that has the countable concatenation property. Let f : E → L¯ 0 be a proper lower semi continuous function that has the local property. Equivalent are: ˚ f = ∅. (i) dom ˚ (ii) epi f = ∅. / 1A epi˚ f for all A ∈ F with Further, for all X ∈ dom f , (X, f (X)) ∈ ∂ epi f and 1A (X, f (X)) ∈ P [A] > 0. ˚ f . We claim Proof. To prove that (i) implies (ii), let ε ∈ L0++ and X ∈ dom
X, f (X) + ε ∈ epi˚ f .
(3.39)
To verify this, we show that there is a neighborhood U of (X, f (X) + ε) such that U ⊂ epi f . By Proposition 3.5, f is continuous at X. Hence, there is a neighborhood UE of X such that f (X) + ε/3 f (X ) This implies
for all X ∈ UE .
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X, f (X) + ε ∈ UE × UL0 ⊂ epi f, where UL0 := Y ∈ L0 f (X) + ε − Y ε/3 . U := UE × UL0 is as required and (3.39) is proved. Conversely, to prove that (ii) implies (i), let (X, Y ) ∈ epi˚ f . Then there are neighborhoods UE and UL0 of X and Y respectively such that U := UE × UL0 ⊂ epi f . In particular, f (X ) < +∞ ˚ f. for all X ∈ UE and hence X ∈ dom Next, let X ∈ dom f . To prove (X, f (X)) ∈ ∂ epi f we show that every U ⊂ E × L0 of the form U := UE × Y ∈ L0 f (X) − Y ε , UE ⊂ E a neighborhood of X, satisfies U ∩ epi f = ∅ = U ∩ epi f c . Observe (X, f (X) − ε/2), (X, f (X) + ε/2) ∈ U and (X, f (X) − ε/2) ∈ / epi f and (X, f (X) + ε/2) ∈ epi f , which proves (X, f (X)) ∈ ∂ epi f . For fixed A ∈ F with P [A] > 0, we show in a similar way that 1A (X, f (X)) ∈ / 1A epi˚ f . Observe that every U ⊂ E × L0 of the form U := UE × Y ∈ L0 1A f (X) − Y ε , UE ⊂ E a neighborhood of 1A X, satisfies U ∩ epi f c = ∅. Indeed, 1A (X, f (X) − ε/2) ∈ U and yet 1A (X, f (X) − ε/2) ∈ / 1A epi f by the local property / 1A epi˚ f . 2 of f . This proves 1A (X, f (X)) ∈ Next, we prove Theorem 3.7. ˚ f . We separate (X0 , f (X0 )) from epi˚ f by means of Proof of Theorem 3.7. Let X0 ∈ dom ˚ Theorem 2.6. By Lemma 3.14, epi f is non-empty, (X0 , f (X0 )) ∈ ∂ epi f and 1A X0 , f (X0 ) ∩ 1A epi˚ f = ∅ for all A ∈ F with P [A] > 0. Hence, there are continuous L0 -linear functions μ1 : E → L0 and μ2 : L0 → L0 such that μ1 X + μ2 Y < μ1 X0 + μ2 f (X0 )
for all (X, Y ) ∈ epi˚ f .
(3.40)
From (3.40) together with the fact that μ2 Y = Y μ2 1 for all Y ∈ L0 we derive that μ2 1 < 0. We will show that −μ1 /μ2 1 ∈ ∂f (X0 ). To this end, let X ∈ E, A := {f (X) = +∞} and ˜ f (X)) ˜ ∈ ∂ epi f . Thus, there is a net X˜ := 1A X0 + 1Ac X. Then, X˜ ∈ dom f and in turn (X, ˚ ˜ ˜ (XR,α , YR,α ) ⊂ epi f which converges to (X, f (X)) and for which
D. Filipovi´c et al. / Journal of Functional Analysis 256 (2009) 3996–4029
μ1 XR,α + YR,α μ2 1 < μ1 X0 + μ2 f (X0 )
for all R, α.
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(3.41)
Since μ1 is continuous we may pass to limits in (3.41) yielding −μ1 (X˜ − X0 ) ˜ − f (X0 ). f (X) μ2 1 Finally, from the local property of f and μ1 we derive μ1 (X − X0 ) f (X) − f (X0 ) μ2 1 and since X ∈ E was arbitrary we conclude that −μ1 /μ2 1 indeed is a subgradient of f at X0 . This proves Theorem 3.7. 2 3.7. Proof of the Fenchel–Moreau duality Theorem 3.8 In this section, we prove Theorem 3.8. The proof follows a known pattern, cf. Proposition A.6 in [10]; however, it contains certain subtleties due to our L0 -convex framework. We fix X0 ∈ E, and proceed in two steps. Step 1: Let β ∈ L0 with β < f (X0 ). In this step, we show there is a continuous function h : E → L0 of the form h(X) = μX + Z,
(3.42)
where μ : E → L0 is continuous L0 -linear and Z ∈ L0 , such that h(X0 ) = β and h(X) f (X) for all X ∈ E. To this end, we separate (X0 , β) from epi f by means of Theorem 2.8. It applies since β < f (X0 ) and the local property of f imply 1A (X0 , β) ∩ 1A epi f = ∅ for all A ∈ F with P [A] > 0. (Note, epi f is closed by Proposition 3.4.) Hence, there are continuous L0 -linear functions μ1 : E → L0 and μ2 : L0 → L0 such that δ := ess.sup μ1 X + μ2 Y < μ1 X0 + μ2 β.
(3.43)
(X,Y )∈epi f
This has two consequences: (i) μ2 1 0. Indeed, μ2 Y = Y μ2 1 for all Y ∈ L0 . Further, (X, Y ) ∈ epi f for arbitrarily large Y as long as f (X) Y . Hence, for large Y ∈ L0 , μ1 X + μ2 Y is large on {μ2 1 > 0} and yet bounded above by μ1 X0 + μ2 β. This implies P [μ2 1 > 0] = 0. (ii) {f (X0 ) < +∞} ⊂ {μ2 1 < 0}. Indeed, define X˜ 0 := 1{f (X0 )<+∞} X0 + 1{f (X0 )=+∞} X for some X ∈ dom f . (f is proper by assumption.) By L0 -convexity of f , X˜ 0 ∈ dom f . Local property of f and (3.43) imply on {f (X0 ) < +∞}
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μ1 X0 + μ2 f (X0 ) = μ1 X˜ 0 + μ2 f (X˜ 0 ) < μ1 X0 + μ2 β. Hence, f (X0 )μ2 1 = μ2 f (X0 ) < μ2 β = βμ2 1 on {f (X0 ) < +∞} and so μ2 1 < 0 on {f (X0 ) < +∞}. / dom f . We distinguish the cases X0 ∈ dom f and X0 ∈ Case 1. Assume X0 ∈ dom f . By (ii), μ2 1 < 0. Thus, define h by h(X) := −
μ1 (X − X0 ) +β μ2 1
for all X ∈ E,
which is as required. Indeed, h(X) f (X) for all X ∈ dom f as a consequence of (3.43). If X∈ / dom f we have 1B h(X) = 1B h(X ) 1B f (X ) = 1B f (X),
(3.44)
where X = 1B X + 1B c X for some X ∈ dom f and B = {f (X) < +∞}. Hence, h(X) f (X) for all X ∈ E. Case 2. Assume X0 ∈ / dom f . Then chose any X0 ∈ dom f and let h be the correspond0 ing L -affine minorant as constructed in case 1 above. Define A1 := {μ2 1 < 0}, A2 := Ac1 and h1 , h2 : E → L0 , h1 (X) := 1A1 h2 (X) :=
μ1 (X − X0 ) − +β , μ2 1
1A2 (h (X) + β − h (X0 )) 1A2 (h (X) +
β−h (X ˜ 0) h(X
0)
˜ h(X))
on {h (X0 ) β}, on {h (X0 ) < β},
with the convention 0/0 := 0, where h˜ : E → L0 , ˜ h(X) := δ − μ1 X. ˜ 0 ) < 0 and h(X) ˜ 0 for all X ∈ dom f . It follows that Note that on {μ2 1 = 0}, h(X h := h1 + h2 is as required. (As in (3.44) we see h(X) f (X) for all X ∈ E.) Step 2: Recall f f ∗∗ , cf. (3.37). By way of contradiction, assume f (X0 ) > f ∗∗ (X0 ) on a set of positive measure. Then there is β ∈ L0 with β > f ∗∗ (X0 ) on a set of positive measure and β < f (X0 ) (a.s.). The first step of this proof yields h : E → L0 , h(X) = μX + Z
for all X ∈ E,
for continuous L0 -linear μ : E → L0 and Z ∈ L0 , such that h(X0 ) = β and h(X) f (X) for all X ∈ E. We derive a contradiction as
D. Filipovi´c et al. / Journal of Functional Analysis 256 (2009) 3996–4029
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f ∗∗ (X0 ) μX0 − f ∗ (μ)
= μX0 − ess.sup μX − f (X) X∈E
μX0 − ess.sup μX − h(X) = β X∈E
negates β > f ∗∗ (X0 ) on a set of positive measure. This finishes the proof of Theorem 3.8. Acknowledgments We thank Norbert Brunner, Freddy Delbaen and Eberhard Mayerhofer for helpful comments. References [1] C.D. Aliprantis, K.C. Border, Infinite Dimensional Analysis. A Hitchhiker’s Guide, third ed., Springer, 2006. [2] J. Bion-Nadal, Conditional risk measures and robust representation of convex conditional risk measures, CMAP preprint No. 557, 2004. [3] W. Brannath, W. Schachermayer, A bipolar theorem for subsets of L0+ (Ω, F , P ), in: Séminaire de Probabilités XXXIII, in: Lecture Notes in Math., vol. 1709, Springer, 1999, pp. 349–354. [4] W.W. Breckner, E. Scheiber, A Hahn–Banach type extension theorem for linear mappings into ordered modules, Mathematica 19 (42) (1977) 13–27. [5] P. Cheridito, F. Delbaen, M. Kupper, Dynamic monetary risk measures for bounded discrete-time processes, Electron. J. Probab. 11 (2006). [6] F. Delbaen, W. Schachermayer, The Mathematics of Arbitrage, first ed., second printing, Springer Finance, vol. 16, Springer, 2008. [7] K. Detlefsen, G. Scandolo, Conditional and dynamic convex risk measures, Finance Stoch. 9 (4) (2005) 539–561. [8] D. Filiovi´c, G. Svindland, Convex risk measures beyond bounded risks, or the canonical model space for lawinvariant convex risk measures is L1 , Vienna Institute of Finance Working Paper No. 2, 2008. [9] H. Föllmer, I. Penner, Convex risk measures and the dynamics of their penalty functions, Statist. Decisions 24 (1) (2006) 61–96. [10] H. Föllmer, A. Schied, Stochastic Finance, an Introduction in Discrete Time, second ed., de Gruyter Stud. Math., vol. 27, 2002. [11] A.L. Ghika, The extension of general linear functionals in semi-normed modules, Acad. Rep. Pop. Romane Bul. Sti. Ser. Mat. Fiz. Chim. 2 (1950) 399–405. [12] R.E. Harte, A generalization of the Hahn–Banach theorem, J. London Math. Soc. 40 (1965) 283–287. [13] R.E. Harte, Modules over a Banach algebra, PhD thesis, Cambridge, 1964. [14] L.V. Kantorovic, The method of successive approximations for functional equations, Acta Math. 71 (1939) 63–97. [15] M. Kupper, N. Vogelpoth, Complete L0 -normed modules and automatic continuity of monotone L0 -convex functions, working paper, 2008. [16] S.S. Kutateladze, Convex operators, Uspekhi Mat. Nauk 1 (34) (1979) 167–196. [17] S.S. Kutateladze, Modules admitting convex analysis, Soviet Math. Dokl. 3 (21) (1980). [18] S.S. Kutateladze, Convex analysis in modules, Siberian Math. J. 4 (22) (1981) 118–128. [19] M. Orhon, On the Hahn–Banach theorem for modules over C(S), J. London Math. Soc. (2) 1 (1969) 363–368. [20] M. Orhon, T. Terzioglu, Diagonal operators on spaces of measurable functions, Mem. Soc. Math. Fr. 31–32 (1972) 265–270. [21] G. Vincent-Smith, The Hahn–Banach theorem for modules, Proc. London Math. Soc. (3) 17 (1967) 72–90. [22] D. Vuza, The Hahn–Banach extension theorem for modules over ordered rings, Rev. Roumaine Math. Pures Appl. 9 (27) (1982) 989–995. [23] R. Welland, On Köthe spaces, Trans. Amer. Math. Soc. 112 (1964) 267–277.
Journal of Functional Analysis 256 (2009) 4030–4070 www.elsevier.com/locate/jfa
Noncommutative hyperbolic geometry on the unit ball of B(H)n ✩ Gelu Popescu Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA Received 3 October 2008; accepted 3 February 2009 Available online 20 February 2009 Communicated by N. Kalton
Abstract In this paper we introduce a hyperbolic (Poincaré–Bergman type) distance δ on the noncommutative open ball 1/2 <1 , B(H)n 1 := (X1 , . . . , Xn ) ∈ B(H)n : X1 X1∗ + · · · + Xn Xn∗ where B(H) is the algebra of all bounded linear operators on a Hilbert space H. It is proved that δ is invariant under the action of the free holomorphic automorphism group of [B(H)n ]1 , i.e., δ Ψ (X), Ψ (Y ) = δ(X, Y ),
X, Y ∈ B(H)n 1 ,
for all Ψ ∈ Aut([B(H)n ]1 ). Moreover, we show that the δ-topology and the usual operator norm topology coincide on [B(H)n ]1 . While the open ball [B(H)n ]1 is not a complete metric space with respect to the operator norm topology, we prove that [B(H)n ]1 is a complete metric space with respect to the hyperbolic metric δ. We obtain an explicit formula for δ in terms of the reconstruction operator RX := X1∗ ⊗ R1 + · · · + Xn∗ ⊗ Rn ,
X := (X1 , . . . , Xn ) ∈ B(H)n 1 ,
associated with the right creation operators R1 , . . . , Rn on the full Fock space with n generators. In the particular case when H = C, we show that the hyperbolic distance δ coincides with the Poincaré–Bergman ✩
Research supported in part by an NSF grant. E-mail address: [email protected].
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.003
G. Popescu / Journal of Functional Analysis 256 (2009) 4030–4070
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distance on the open unit ball Bn := z = (z1 , . . . , zn ) ∈ Cn : z2 < 1 . We obtain a Schwarz–Pick lemma for free holomorphic functions on [B(H)n ]1 with respect to the hyperbolic metric, i.e., if F := (F1 , . . . , Fm ) is a contractive (F ∞ 1) free holomorphic function, then δ F (X), F (Y ) δ(X, Y ),
X, Y ∈ B(H)n 1 .
As consequences, we show that the Carathéodory and the Kobayashi distances, with respect to δ, coincide with δ on [B(H)n ]1 . The results of this paper are presented in the more general context of Harnack parts of the closed ball [B(H)n ]− 1 , which are noncommutative analogues of the Gleason parts of the Gelfand spectrum of a function algebra. © 2009 Elsevier Inc. All rights reserved. Keywords: Noncommutative hyperbolic geometry; Noncommutative function theory; Poincaré–Bergman metric; Harnack part; Hyperbolic distance; Free holomorphic function; Free pluriharmonic function; Schwarz–Pick lemma
0. Introduction Poincaré’s discovery of a conformally invariant metric on the open unit disc D := {z ∈ C: |z| < 1} of the complex plane was a cornerstone in the development of complex function theory. The hyperbolic (Poincaré) distance is defined on D by z−w , δP (z, w) := tanh−1 1 − z¯ w
z, w ∈ D.
Some of the basic and most important properties of the Poincaré distance are the following: (1) the Poincaré distance is invariant under the conformal automorphisms of D, i.e., δP ϕ(z), ϕ(w) = δP (z, w),
z, w ∈ D,
for all ϕ ∈ Aut(D); (2) the δP -topology induced on the open disc is the usual planar topology; (3) (D, δP ) is a complete metric space; (4) any analytic function f : D → D is distance-decreasing, i.e., satisfies δP f (z), f (w) δP (z, w),
z, w ∈ D.
Bergman (see [2]) introduced an analogue of the Poincaré distance for the open unit ball of Cn , Bn := z = (z1 , . . . , zn ) ∈ Cn : z2 < 1 ,
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which is defined by βn (z, w) =
1 1 + ψz (w)2 ln , 2 1 − ψz (w)2
z, w ∈ Bn ,
where ψz is the involutive automorphism of Bn that interchanges 0 and z. The Poincaré–Bergman distance has properties similar to those of δP (see (1)–(4)). There is a large literature concerning invariant metrics, hyperbolic manifolds, and the geometric viewpoint of complex function theory (see [15,16,45,17] and the references therein). There are several extensions of the Poincaré–Bergman distance and related topics to more general domains. We mention the work of Suciu [38,40,39], Foia¸s [8], and Andô, Suciu and Timotin [1] on Harnack parts of contractions and Harnack type distances between two contractions on Hilbert spaces. Some of their results will be recover (with a different proof) in the present paper, in the particular case when n = 1. In this paper, we continue our program to develop a noncommutative function theory on the unit ball of B(H)n (see [30,31,33,32,34]). The main goal is to introduce a hyperbolic metric δ on the noncommutative ball 1/2 B(H)n 1 := (X1 , . . . , Xn ) ∈ B(H)n : X1 X1∗ + · · · + Xn Xn∗ < 1 , where B(H) denotes the algebra of all bounded linear operators on a Hilbert space H, which satisfy properties similar to those of the Poincaré metric δP (see (1)–(3)), and which is a noncommutative extension of the Poincaré–Bergman metric βn on the open unit ball of Cn . The secondary goal is to obtain a Schwarz–Pick lemma for free holomorphic functions on [B(H)n ]1 with respect to the hyperbolic metric. We should mention that the noncommutative ball [B(H)n ]1 can be identified with the open unit ball of B(Hn , H), which is one of the infinite-dimensional Cartan domains studied by L.A. Harris [11–13]. He has obtained several results, related to our topic, in the more general setting of J B ∗ -algebras (see also the book by H. Upmeier [42]). We also remark that the group of all free holomorphic automorphisms of [B(H)n ]1 [34], can be identified with a subgroup of the group of automorphisms of [B(Hn , H)]1 considered by R.S. Phillips [19] (see also [44]). However, the hyperbolic distance δ that we introduce in this paper is different from the Kobayashi distance on [B(H)n ]1 , with respect to the Poincaré distance on D, and also different from the one considered, for example, in [11]. In [30,31,33,32,34], we obtained several results concerning the theory of free holomorphic (resp. pluriharmonic) functions on [B(H)n ]1 and provided a framework for the study of arbitrary n-tuples of operators on a Hilbert space H. Several classical results from complex analysis [3, 9,14,37] have free analogues in the noncommutative multivariable setting. To put our work in perspective, we need to set up some notation and recall some definitions. Let F+ n be the unital free semigroup on n generators g1 , . . . , gn and the identity g0 . The length of α ∈ F+ n is defined by |α| := 0 if α = g0 and |α| := k if α = gi1 · · · gik , where i1 , . . . , ik ∈ {1, . . . , n}. If (X1 , . . . , Xn ) ∈ B(H)n , we set Xα := Xi1 · · · Xik and Xg0 := IH . Throughout this paper, we assume that E is a separable Hilbert space. A map F : [B(H)n ]1 → B(H) ⊗min B(E) is called free holomorphic function on
[B(H)n ]1 with coefficients in B(E) if there exist A(α) ∈ + B(E), α ∈ Fn , such that lim supk→∞ |α|=k A∗(α) A(α) 1/2k 1 and F (X1 , . . . , Xn ) =
∞ k=0 |α|=k
Xα ⊗ A(α) ,
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where the series converges in the operator norm topology for any (X1 , . . . , Xn ) ∈ [B(H)n ]1 . The set of all free holomorphic functions on [B(H)n ]1 with coefficients in B(E) is denoted by Hol(B(H)n1 ). Let H ∞ (B(H)n1 ) denote the set of all elements F in Hol(B(H)n1 ) such that F ∞ := supF (X1 , . . . , Xn ) < ∞, where the supremum is taken over all n-tuples of operators (X1 , . . . , Xn ) ∈ [B(H)n ]1 and any Hilbert space H. According to [30] and [33], H ∞ (B(H)n1 ) can be identified to the operator ¯ algebra Fn∞ ⊗B(E) (the weakly closed algebra generated by the spatial tensor product), where Fn∞ is the noncommutative analytic Toeplitz algebra (see [23,21,25]). We say that a map u : [B(H)n ]1 → B(H) ⊗min B(E) is a self-adjoint free pluriharmonic function on [B(K)n ]1 if u = Re f := 12 (f ∗ + f ) for some free holomorphic function f . We also recall that u is called positive if u(X1 , . . . , Xn ) 0 for any (X1 , . . . , Xn ) ∈ [B(K)n ]1 , where K is an infinite-dimensional Hilbert space. H
In Section 1, we introduce an equivalence relation ∼ on the closed ball [B(H)n ]− 1 , and study H
the equivalence classes (called Harnack parts) with respect to ∼. Two n-tuples of operators A := (A1 , . . . , An ) and B := (B1 , . . . , Bn ) in [B(H)n ]− 1 are called Harnack equivalent (and denote H
A ∼ B) if and only if there exists a constant c 1 such that 1 Re p(B1 , . . . , Bn ) Re p(A1 , . . . , An ) c2 Re p(B1 , . . . , Bn ) c2 for any noncommutative polynomial p ∈ C[X1 , . . . , Xn ] ⊗ Mm , m ∈ N, with matrix-valued coefficients such that Re p 0. Here Mm denotes the algebra of all m × m matrices with entries H
in C. We also use the notation A ∼ B to emphasize the constant c in the inequalities above. The c
Harnack parts of [B(H)n ]− 1 are noncommutative analogues of the Gleason parts of the Gelfand spectrum of a function algebra (see [10]). In Section 1, we use several results (see [27–29,36]) concerning the theory of noncommutative Poisson transforms on Cuntz–Toeplitz C ∗ -algebras (see [4]) and free pluriharmonic functions (see [33,32]) to obtain useful characterizations for the Harnack equivalence on the closed ball [B(H)n ]− 1 . On the other hand, a characterization of positive free pluriharmonic functions (see [33]) and dilation theory (see [41]) are used to obtain a Harnack type inequality (see [3]) for positive free pluriharmonic function on [B(H)n ]1 . More precisely, we show that if u is a positive free pluriharmonic function on [B(H)n ]1 with operator-valued coefficients in B(E) and 0 < r < 1, then u(0)
1+r 1−r u(X1 , . . . , Xn ) u(0) 1+r 1−r
for any (X1 , . . . , Xn ) ∈ [B(H)n ]− r . This result is crucial in order to prove that the open unit ball [B(H)n ]1 is a distinguished Harnack part of [B(H)n ]− 1 , namely, the Harnack part of 0. In Section 2, we introduce a hyperbolic (Poincaré–Bergman type) metric on the Harnack parts n − + of [B(H)n ]− 1 . More precisely, given a Harnack part of [B(H) ]1 we define δ : × → R by setting δ(A, B) := ln ω(A, B),
A, B ∈ ,
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where
H ω(A, B) := inf c > 1: A ∼ B . c
We prove that δ is a metric on . Consider the particular case when = [B(H)n ]1 and let δ : [B(H)n ]1 × [B(H)n ]1 → [0, ∞) be the hyperbolic metric defined above. We prove, in Section 2, that δ is invariant under the action of the group Aut([B(H)n ]1 ) of all the free holomorphic automorphisms of the noncommutative ball [B(H)n ]1 , i.e., δ Ψ (A), Ψ (B) = δ(A, B),
A, B ∈ B(H)n 1 ,
for all Ψ ∈ Aut([B(H)n ]1 ). We mention that the group Aut([B(H)n ]1 ) was determined in [34]. Using a characterization of the Harnack equivalence on [B(H)n ]− 1 in terms of free pluriharmonic kernels, we obtain an explicit formula for the hyperbolic distance in terms of the reconstruction operator. More precisely, we show that δ(A, B) = ln max CA CB−1 , CB CA−1 ,
A, B ∈ B(H)n 1 ,
where CX := (X ⊗ I )(I − RX )−1 and RX := X1∗ ⊗ R1 + · · · + Xn∗ ⊗ Rn is the reconstruction operator associated with the n-tuple X := (X1 , . . . , Xn ) ∈ [B(H)n ]1 and with the right creation operators R1 , . . . , Rn on the full Fock space with n generators. In particular, we show that δ|Bn ×Bn coincides with the Poincaré–Bergman distance on Bn , i.e., δ(z, w) =
1 1 + ψz (w)2 ln , 2 1 − ψz (w)2
z, w ∈ Bn ,
where ψz is the involutive automorphism of Bn that interchanges 0 and z. We mention that similar results concerning the invariance under the automorphism group Aut([B(H)n ]1 ) as well as an explicit formula for the hyperbolic metric hold on any Harnack part of [B(H)n ]− 1. In Section 3, we study the relations between the δ-topology, the dH -topology (which will be introduced), and the operator norm topology on Harnack parts of [B(H)n ]− 1 . We prove that the hyperbolic metric δ is a complete metric on any Harnack part of − B0 (H)n 1 := (X1 , . . . , Xn ) ∈ B(H)n 1 : r(X1 , . . . , Xn ) < 1 , and that all the topologies above coincide on the open ball [B(H)n ]1 . In particular, we deduce that [B(H)n ]1 is a complete metric space with respect to the hyperbolic metric δ and that the δ-topology and the usual operator norm topology coincide on [B(H)n ]1 . A very important property of the Poincaré–Bergman distance βm : Bm × Bm → R+ is that any holomorphic function f : Bn → Bm is distance-decreasing, i.e., βm f (z), f (w) βn (z, w),
z, w ∈ Bn .
In Section 4, we extend this result and prove a Schwarz–Pick lemma for free holomorphic functions on [B(H)n ]1 with operator-valued coefficients, with respect to the hyperbolic metric on the noncommutative ball [B(H)n ]1 .
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More precisely, let Fj : [B(H)n ]1 → B(H) ⊗min B(E), j = 1, . . . , m, be free holomorphic functions with coefficients in B(E), and assume that F := (F1 , . . . , Fm ) is a contractive H
(F ∞ 1) free holomorphic function. If X, Y ∈ [B(H)n ]1 , then we prove that F (X) ∼ F (Y ) and δ F (X), F (Y ) δ(X, Y ), where δ is the hyperbolic metric defined on the Harnack parts of the noncommutative ball [B(H)n ]− 1 . This result is used to show that the Carathéodory and the Kobayashi distances, with respect to δ, coincide with δ on [B(H)n ]1 . The present paper makes connections between noncommutative function theory (see [27,30, 33,34]) and classical results in hyperbolic complex analysis and geometry (see [15–17,3,9,14, 37]). In particular, we obtain a new formula of the Poincaré–Bergman metric on Bn using Harnack inequalities for positive free pluriharmonic functions on [B(H)n ]1 , as well as a formula in terms of the left creation operators on the full Fock space with n generators. It would be interesting to see if the results of this paper can be extended to more general infinite-dimensional bounded domains such as the J B ∗ -algebras of Harris [11], the domains considered by Phillips [19], or the noncommutative domains from [35]. Since our results are based on noncommutative function theory, dilation and model theory for row contractions, we are inclined to believe in a positive answer at least for the latter domains. 1. Harnack equivalence on the closed unit ball [B(H)n ]− 1 H
In this section, we introduce a preorder relation ≺ on the closed ball [B(H)n ]− 1 and provide H
several characterizations. This preorder induces an equivalence relation ∼ on [B(H)n ]− 1 , whose equivalence classes are called Harnack parts. Several characterizations for the Harnack parts are provided. We obtain a Harnack type inequality for positive free pluriharmonic functions and use it to prove that the open unit ball [B(H)n ]1 is a distinguished Harnack part of [B(H)n ]− 1 , namely, the Harnack part of 0. Let Hn be an n-dimensional complex Hilbert space with orthonormal basis e1 , e2 , . . . , en , where n = 1, 2, . . . , or n = ∞. We consider the full Fock space of Hn defined by F 2 (Hn ) := C1 ⊕
Hn⊗k ,
k1
where Hn⊗k is the (Hilbert) tensor product of k copies of Hn . Define the left (resp. right) creation operators Si (resp. Ri ), i = 1, . . . , n, acting on F 2 (Hn ) by setting Si ϕ := ei ⊗ ϕ,
ϕ ∈ F 2 (Hn ),
(resp. Ri ϕ := ϕ ⊗ ei , ϕ ∈ F 2 (Hn )). The noncommutative disc algebra An (resp. Rn ) is the norm closed algebra generated by the left (resp. right) creation operators and the identity. The noncommutative analytic Toeplitz algebra Fn∞ (resp. R∞ n ) is the weakly closed version of An (resp. Rn ). These algebras were introduced in [23] in connection with a noncommutative von Neumann type inequality [43], and have been studied in several papers (see [21,24–26,5,6], and the references therein).
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We need to recall from [27] a few facts about noncommutative Poisson transforms associated with row contractions T := (T1 , . . . , Tn ) ∈ [B(H)n ]− 1. Let F+ be the unital free semigroup on n generators g1 , . . . , gn , and the identity g0 . We denote n eα := ei1 ⊗ · · · ⊗ eik and eg0 := 1. Note that {eα }α∈F+n is an orthonormal basis for F 2 (Hn ). For each 0 < r 1, define the defect operator T ,r := (IH − r 2 T1 T1∗ − · · · − r 2 Tn Tn∗ )1/2 . The noncommutative Poisson kernel associated with T is the family of operators KT ,r : H → T ,r H ⊗ F 2 (Hn ),
0 < r 1,
defined by KT ,r h :=
∞
r |α| T ,r Tα∗ h ⊗ eα ,
h ∈ H.
k=0 |α|=k
When r = 1, we denote T := T ,1 and KT := KT ,1 . The operators KT ,r are isometries if 0 < r < 1, and KT∗ KT = IH − SOT- lim
k→∞
Tα Tα∗ .
|α|=k
Thus KT is an isometry if and only if T is a pure row contraction, i.e., SOT- lim
k→∞
Tα Tα∗ = 0.
|α|=k
We denote by C ∗ (S1 , . . . , Sn ) the Cuntz–Toeplitz C ∗ -algebra generated by the left creation operators. The noncommutative Poisson transform at T := (T1 , . . . , Tn ) ∈ [B(H)n ]− 1 is the unital completely contractive linear map PT : C ∗ (S1 , . . . , Sn ) → B(H) defined by PT [f ] := lim KT∗ ,r (IH ⊗ f )KT ,r , r→1
f ∈ C ∗ (S1 , . . . , Sn ),
where the limit exists in the norm topology of B(H). Moreover, we have PT Sα Sβ∗ = Tα Tβ∗ ,
α, β ∈ F+ n.
When T := (T1 , . . . , Tn ) is a pure row contraction, we have PT [f ] = KT∗ (IDT ⊗ f )KT , where DT = T H. We refer to [27,28,36] for more on noncommutative Poisson transforms on C ∗ -algebras generated by isometries. For basic results concerning completely bounded maps and operator spaces we refer to [18,20,7]. When T ∈ [B(H)n ]− 1 is a completely noncoisometric (c.n.c.) row contraction, i.e., there is no h ∈ H, h = 0, such that
G. Popescu / Journal of Functional Analysis 256 (2009) 4030–4070
T ∗ h2 = h2 α
4037
for any k = 1, 2, . . . ,
|α|=k
an Fn∞ -functional calculus was developed in [24]. We showed that if f = then ΓT (f ) = f (T1 , . . . , Tn ) := SOT- lim
∞
r→1
a S α∈F+ n α α
is in Fn∞ ,
r |α| aα Tα
k=0 |α|=k
exists and ΓT : Fn∞ → B(H) is a completely contractive homomorphism and WOT-continuous (resp. SOT-continuous) on bounded sets. Moreover, we showed (see [36]) that ΓT (f ) = PT [f ], f ∈ Fn∞ , where PT [f ] := SOT- lim KT ,r (IH ⊗ f )KT ,r , r→1
f ∈ Fn∞ ,
(1.1)
is the extension of the noncommutative Poisson transform to the noncommutative analytic Toeplitz algebra Fn∞ . We introduced in [33] the noncommutative Poisson transform Pμ of a completely bounded linear map μ : R∗n + Rn → B(E) by setting (Pμ)(X1 , . . . , Xn ) := (id ⊗ μ) P (X, R) ,
X := (X1 , . . . , Xn ) ∈ B(H)n 1 ,
where the free pluriharmonic Poisson kernel P (X, R) is given by P (X, R) :=
∞ k=1 |α|=k
Xα∗ ⊗ R α +I +
∞ k=1 |α|=k
∗ Xα ⊗ R α,
X ∈ B(H)n 1 ,
and the series are convergent in the operator norm topology. We recall that the joint spectral radius associated with an n-tuple of operators (X1 , . . . , Xn ) ∈ B(H)n is given by 1/2k ∗ Xα Xα . r(X1 , . . . , Xn ) := lim k→∞ |α|=k
We remark that the free pluriharmonic Poisson kernel P (X, R) makes sense for any n-tuple of operators X := (X1 , . . . , Xn ) ∈ B(H)n with r(X1 , . . . , Xn ) < 1. According to [36], X ∈ [B(H)n ]− 1 if and only if r(X1 , . . . , Xn ) 1 and P (rX, R) 0,
r ∈ [0, 1).
We say that a free pluriharmonic function u is positive if u(X1 , . . . , Xn ) 0 for any (X1 , . . . , Xn ) ∈ [B(K)n ]γ and any Hilbert space K. We recall [33] that u 0 if and only if u(rS1 , . . . , rSn ) 0 for any r ∈ [0, 1). In particular, if p is a noncommutative polynomial with operator-valued coefficients, then Re p 0 if and only if Re p(S1 , . . . , Sn ) 0.
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Now, we introduce a preorder relation ≺ on the closed ball [B(H)n ]− 1 and provide several characterizations. Let A := (A1 , . . . , An ) and B := (B1 , . . . , Bn ) be in [B(H)n ]− 1 . We say that H
A is Harnack dominated by B, and denote A ≺ B, if there exists c > 0 such that Re p(A1 , . . . , An ) c2 Re p(B1 , . . . , Bn ) for any noncommutative polynomial with matrix-valued coefficients p ∈ C[X1 , . . . , Xn ] ⊗ Mm , H
m ∈ N, such that Re p 0. When we want to emphasize the constant c, we write A ≺ B. c
Theorem 1.1. Let A := (A1 , . . . , An ) and B the following statements are equivalent:
:= (B1 , . . . , Bn ) be in [B(H)n ]− 1
and let c > 0. Then
H
(i) A ≺ B;
c
(ii) (PA ⊗min id)[q ∗ q] c2 (PB ⊗min id)[q ∗ q] for any polynomial q = |α|k Sα ⊗ C(α) with matrix-valued coefficients C(α) ∈ Mm , and k, m ∈ N, where PX is the noncommutative Poisson transform at X ∈ [B(H)n ]− 1; (iii) P (rA, R) c2 P (rB, R) for any r ∈ [0, 1), where P (X, R) is the free pluriharmonic Poisson kernel associated with X ∈ [B(H)n ]1 ; (iv) u(rA1 , . . . , rAn ) c2 u(rB1 , . . . , rBn ) for any positive free pluriharmonic function u with operator-valued coefficients and any r ∈ [0, 1); (v) c2 PB − PA is a completely positive linear map on the operator space A∗n + An · .
Proof. First we prove the equivalence (i) ↔ (ii). Assume that (i) holds, and let q = |α|k Sα ⊗ C(α) be an arbitrary polynomial in An ⊗ Mm . Since the left creation operators are isometries with orthogonal ranges, q ∗ q has form Re p(S1 , . . . , Sn ) for some noncommutative polynomial p ∈ C[X1 , . . . , Xn ] ⊗ Mm , m ∈ N, such that Re p 0. Using the properties of the noncommutative Poisson transform, one can see that (i) ⇒ (ii). Conversely, assume that (ii) holds. Let p ∈ C[X1 , . . . , Xn ] ⊗ Mm , m ∈ N, be a polynomial of degree k such that Re p 0. Then Re p(S1 , . . . , Sn ) is a positive multi-Toeplitz operator with respect to R1 , . . . , Rn acting on the Hilbert space F 2 (Hn ) ⊗ Cm , i.e., ∗ Ri ⊗ ICm Re p(S1 , . . . , Sn ) (Rj ⊗ ICm ) = δij G,
i, j = 1, . . . , n.
According to the Fejér type factorization theorem of [25], there exists a multi-analytic operator q in B(F 2 (Hn ) ⊗ Cm ) such that Re p(S1 , . . . , Sn ) = q ∗ q and ∗ Sα ⊗ ICm q(1 ⊗ h) = 0 for any h ∈ Cm and |α| > k. Therefore, q is a polynomial, i.e., q = operators C(α) ∈ B(Cm ). Note that
|α|k Sα
⊗ C(α) for some
Re p(A1 , . . . , An ) = (PA ⊗min id) q ∗ q c2 (PB ⊗min id) q ∗ q = c2 Re p(B1 , . . . , Bn ), which proves (i).
G. Popescu / Journal of Functional Analysis 256 (2009) 4030–4070 (m)
4039 (m)
(m)
Let us prove that (i) ⇒ (iii). For each m ∈ N, consider R (m) := (R1 , . . . , Rn ), where Ri , i = 1, . . . , n, is the compression of the right creation operator Ri to Pm := span{eα : α ∈ F+ n, (m) + |α| m}. Note that Rα = 0 for any α ∈ Fn with |α| m + 1 and, consequently, we have
P rX, R (m) =
r |α| Xα∗ ⊗ R α +I + (m)
1|α|m
(m) ∗
r |α| Xα ⊗ R α
.
1|α|m
Note that R (m) is a pure row contraction and the noncommutative Poisson transform id ⊗ PR (m) is a completely positive map. We recall that X → P (X, R) is a positive free pluriharmonic function with coefficients in B(F 2 (Hn )). Hence P (rX, R) 0 for any X ∈ [B(H)n ]− 1 and r ∈ [0, 1). Applying id ⊗ PR (m) , we obtain P rX, R (m) = (id ⊗ PR (m) ) P (rX, S) 0. Now, applying (i), we obtain P rA, R (m) c2 P rB, R (m) for any m ∈ N and r ∈ [0, 1). Using Lemma 8.1 from [33], we deduce that P (rA, R) c2 P (rB, R) for any r ∈ [0, 1). Therefore (iii) holds. To prove the implication (iii) ⇒ (iv), assume that condition (iii) holds and let u be a positive free pluriharmonic function with coefficients in B(E). According to Corollary 5.5 from [33], there exists a completely positive linear map μ : R∗n + Rn → B(E) such that u(Y ) = (Pμ)(Y ) := (id ⊗ μ) P (Y, R) for any Y ∈ [B(H)n ]1 . Hence and using the fact that c2 P (rB, R) − P (rA, R) 0, we deduce that c2 u(rB1 , . . . , rBn ) − u(rA1 , . . . , rAn ) = (id ⊗ μ) c2 P (rB, R) − P (rA, R) 0, which proves (iv). Now, we prove the implication (iv) ⇒ (v). Let g ∈ A∗n + An · ⊗ Mm be positive. Then, according to Theorem 4.1 from [33], the map defined by u(X) := (PX ⊗ id)[g],
X ∈ B(H)n 1 ,
(1.2)
is a positive free pluriharmonic function. Condition (iv) implies u(rA1 , . . . , rAn ) c2 u(rB1 , . . . , rBn ) for any r ∈ [0, 1). On the other hand, by relation (1.2), we have c2 (PrB ⊗ id)[g] − (PrA ⊗ id)[g] = c2 u(rB1 , . . . , rBn ) − u(rA1 , . . . , rAn ) 0 for any r ∈ [0, 1). Since g ∈ A∗n + An · ⊗ Mm , we have (PA ⊗ id)[g] = lim (PrA ⊗ id)[g] r→1
and (PB ⊗ id)[g] = lim (PrB ⊗ id)[g], r→1
(1.3)
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where the convergence is in the operator norm topology. Taking r → 1 in (1.3), we deduce item (v). To prove the implication (v) ⇒ (i), let p ∈ C[X1 , . . . , Xn ] ⊗ Mm , m ∈ N, be a noncommutative polynomial with matrix coefficients such that Re p 0. Due to the proprieties of the noncommutative Poisson transform, we have c2 Re p(B1 , . . . , Bn ) − Re p(A1 , . . . , An ) = c2 (PB ⊗ id) Re p(S1 , . . . , Sn ) − (PA ⊗ id) Re p(S1 , . . . , Sn ) . Since Re p(S1 , . . . , Sn ) 0 and c2 PB − PA is a completely positive linear map on the operator space A∗n + An · , we deduce item (i). This completes the proof. 2 We remark that each item in Theorem 1.1 is equivalent to the following: P rA, R (m) c2 P rB, R (m) for any m ∈ N, r ∈ [0, 1), where R (m) is defined in the proof of Theorem 1.1. In what follows, we characterize the elements of the closed ball [B(H)n ]− 1 which are Harnack dominated by 0. H
Theorem 1.2. Let A := (A1 , . . . , An ) be in [B(H)n ]− 1 . Then A ≺ 0 if and only if the joint spectral radius r(A1 , . . . , An ) < 1. Proof. Note that the map X → P (X, R) is a positive free pluriharmonic function on [B(H)n ]1 with coefficients in B(F 2 (Hn )) and has the factorization ∗ −1 P (X, R) = (I − RX )−1 − I + I − RX ∗ −1 ∗ ∗ I − RX − I − R X (I − RX ) + I − RX (I − RX )−1 = I − RX ∗ −1 I − X1 X1∗ − · · · − Xn Xn∗ ⊗ I (I − RX )−1 , = I − RX where RX := X1∗ ⊗ R1 + · · · + Xn∗ ⊗ Rn is the reconstruction operator associated with the ntuple X := (X1 , . . . , Xn ) ∈ [B(H)n ]1 . We remark that, due to the fact that the spectral radius of RX is equal to r(X1 , . . . , Xn ), the factorization above holds for any X ∈ [B(H)n ]− 1 with r(X1 , . . . , Xn ) < 1. H
Now, using Theorem 1.1, part (iii) and the factorization obtained above, we deduce that A ≺ 0 if and only if there exists c 1 such that ∗ −1 I − rRA I − r 2 A1 A∗1 − · · · − r 2 An A∗n ⊗ I (I − rRA )−1 c2 I for any r ∈ [0, 1). Similar inequality holds if we replace the right creation operators by the left creation operators. Then, applying the noncommutative Poisson transform id ⊗ Peiθ R we obtain ∗ I − r 2 A1 A∗1 − · · · − r 2 An A∗n ⊗ I c2 I − re−iθ RA I − reiθ RA for any r ∈ [0, 1) and θ ∈ R.
(1.4)
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4041
H
Assume now that A := (A1 , . . . , An ) ∈ [B(H)n ]− 1 is such that A ≺ 0. Then r(A1 , . . . , An ) 1. Suppose that r(A1 , . . . , An ) = 1. Since r(RA ) = r(A1 , . . . , An ), there exists λ0 ∈ T in the approximative spectrum of RA . Consequently, there is a sequence {hm } in H ⊗ F 2 (Hn ) such that hm = 1 and λ0 hm − RA hm m1 for m = 1, 2, . . . . Hence and taking r = 1 − m1 in relation (1.4), we deduce that
2 1 2 ∗ 1 2 ¯ I − 1− RA RA hm , hm c hm − 1 − λ0 RA hm m m 2 1 4c2 2 c λ0 hm − RA hm + RA hm 2 . m m
(1.5)
Combining this result with the fact that RA hm 1, we deduce that
1 1− 1− m
2
1 2 4c2 1− 1− RA hm 2 2 . m m
Hence, we obtain 2m 4c2 + 1 for any m = 1, 2, . . . , which is a contradiction. Therefore, we have r(A1 , . . . , An ) < 1. Conversely, assume that A := (A1 , . . . , An ) ∈ [B(H)n ]− 1 has the joint spectral radius −1 r(A1 , . . . , An ) < 1. Note that M := supr∈(0,1) (I − rRA ) exists and, therefore, ∗ −1 I − r 2 A1 A∗1 − · · · − r 2 An A∗n ⊗ I (I − rRA )−1 M 2 I I − rRA H
for any r ∈ (0, 1), which, due to Theorem 1.1, shows that A ≺ 0. The proof is complete.
2
We mention that in the particular case when n = 1 we can recover a result obtained in [1]. H
H
n − Since ≺ is a preorder relation on [B(H)n ]− 1 , it induces an equivalent relation ∼ on [B(H) ]1 , H
which we call Harnack equivalence. The equivalence classes with respect to ∼ are called Harnack n − parts of [B(H)n ]− 1 . Let A := (A1 , . . . , An ) and B := (B1 , . . . , Bn ) be in [B(H) ]1 . It is easy to H
see that A and B are Harnack equivalent (we denote A ∼ B) if and only if there exists c 1 such that 1 Re p(B1 , . . . , Bn ) Re p(A1 , . . . , An ) c2 Re p(B1 , . . . , Bn ) c2
(1.6)
for any noncommutative polynomial with matrix-valued coefficients p ∈ C[X1 , . . . , Xn ] ⊗ Mm , H
H
H
c
c
c
m ∈ N, such that Re p 0. We also use the notation A ∼ B if A ≺ B and B ≺ A. A completely positive (c.p.) linear map μX : C ∗ (S1 , . . . , Sn ) → B(H) is called representing c.p. map for the point X := (X1 , . . . , Xn ) ∈ [B(H)n ]− 1 if μ(Sα ) = Xα
for any α ∈ F+ n.
Next, we obtain several characterizations for the Harnack equivalence on the closed ball [B(H)n ]− 1 . The result will play a crucial role in this paper.
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Theorem 1.3. Let A := (A1 , . . . , An ) and B := (B1 , . . . , Bn ) be in [B(H)n ]− 1 and let c > 1. Then the following statements are equivalent: H
(i) A ∼ B; c
(ii) the noncommutative Poisson transform satisfies the inequalities 1 (PB ⊗min id) q ∗ q (PA ⊗min id) q ∗ q c2 (PB ⊗min id) q ∗ q 2 c
for any polynomial q = |α|k Sα ⊗ A(α) with matrix-valued coefficients A(α) ∈ Mm , and k, m ∈ N; (iii) the free pluriharmonic kernel satisfies the inequalities 1 P (rB, R) P (rA, R) c2 P (rB, R) c2 for any r ∈ [0, 1); (iv) for any positive free pluriharmonic function u with operator-valued coefficients and any r ∈ [0, 1), 1 u(rB1 , . . . , rBn ) u(rA1 , . . . , rAn ) c2 u(rB1 , . . . , rBn ); c2 (v) c2 PB − PA and c2 PA − PB is a completely positive linear map on the operator space A∗n + An · , where PX is the noncommutative Poisson transform at X ∈ [B(H)n ]− 1; (vi) there are representing c.p. maps μA and μB for A and B, respectively, such that 1 μB μA c 2 μB . c2 Proof. The first five equivalences follow using Theorem 1.1. It remains to show that (i) ↔ (vi). H
Assume that A ∼ B. Then according to item (v), PA − c12 PB and PB − c12 PA are completely posc
itive linear maps on the operator space A∗n + An · . Using Arveson’s extension theorem, we find some completely positive linear maps ϕ and ψ on the Cuntz–Toeplitz algebra C ∗ (S1 , . . . , Sn ) which are extensions of PA − c12 PB and PB − c12 PA , respectively. Define the c.p. maps μA , μB : C ∗ (S1 , . . . , Sn ) → B(H) by setting μA :=
c2 2 c2 2 c c ψ +ϕ . ϕ + ψ and μ := B 4 4 c −1 c −1
(1.7)
Note that for any f ∈ A∗n + An · , we have 1 1 1 PA [f ] = ϕ(f ) + 2 PB [f ] = ϕ(f ) + 2 ψ(f ) + 2 PA . c c c Solving for PA [f ], we obtain PA [f ] = μA (f ). Similarly, we obtain PB [f ] = μB (f ). Since PA [Sα ] = Aα and PB [Sα ] = Bα for any α ∈ F+ n , it is clear that μA , μB are representing c.p.
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maps for A := (A1 , . . . , An ) and B := (B1 , . . . , Bn ), respectively. Now, since c > 1 and using relation (1.7), it is a routine to show that 1 μB μA c 2 μB . c2
(1.8)
Conversely, assume that (vi) holds for some c > 1, and let p(S1 , . . . , Sn ) := |α|q Sα ⊗ M(α) be a polynomial such that Re p 0. Since μA , μB are representing c.p. maps for A := (A1 , . . . , An ) and B := (B1 , . . . , Bn ), respectively, relation (1.8) implies 1 Re p(B1 , . . . , Bn ) Re p(A1 , . . . , An ) c2 Re p(B1 , . . . , Bn ). c2 H
This shows that A ∼ B and completes the proof. c
2
We remark that the first five equivalences in Theorem 1.3 remain true even when c 1. The next result is a Harnack type inequality for positive free pluriharmonic functions on [B(H)n ]1 . Theorem 1.4. If u is a positive free pluriharmonic function on [B(H)n ]1 with operator-valued coefficients in B(E) and 0 r < 1, then u(0)
1−r 1+r u(X1 , . . . , Xn ) u(0) 1+r 1−r
for any (X1 , . . . , Xn ) ∈ [B(H)n ]− r . Proof. Notice that the free pluriharmonic Poisson kernel satisfies the relation P (rX, R) =
∞
k r k RX +I +
k=1
∞
∗ k r k RX ,
0 r < 1,
k=1
for any X := (X1 , . . . , Xn ) ∈ [B(H)n ]1 , where RX := X1∗ ⊗ R1 + · · · + Xn∗ ⊗ Rn is the recon∗R = struction operator. Since R1 , . . . , Rn are isometries with orthogonal ranges, we have RX X
n n ∗ ∗ 1/2 < 1. Let U be the minimal unitary i=1 Xi Xi ⊗ I and therefore RX = i=1 Xi Xi dilation of RX on a Hilbert space M ⊃ H ⊗ F 2 (Hn ), in the spirit of Sz.-Nagy–Foia¸s. Then k =P k we have RX H⊗F 2 (Hn ) U |H⊗F 2 (Hn ) for any k = 1, 2, . . . . Since RX is a strict contraction, U is a bilateral shift which can be identified with Meiθ ⊗ IG for some Hilbert space G, where Meiθ is the multiplication operator by eiθ on L2 (T). Consequently, there is a unitary operator W : L2 (T) ⊗ G → M such that P (rX, R) = PH⊗F 2 (Hn ) W M(r) ⊗ IG W ∗ H⊗F 2 (H ) , n
where M(r) :=
∞ k=1
r k Mekiθ + I +
∞ k=1
r k Mek−iθ .
(1.9)
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For each f ∈ L2 (T), we have k=∞ M(r)f, f = f 22 r |k| eiθk = f 22
k=−∞
1 − r2 . 1 − 2r cos θ + r 2
Since 1−r 1+r 1 − r2 , 1+r 1 − 2r cos θ + r 2 1 − r we have
1−r 1+r I
M(r)
1+r 1−r I .
Hence and due to (1.9), we deduce that 1−r 1+r I P (rX, R) I 1+r 1−r
(1.10)
for any X ∈ [B(H)n ]1 and 0 r < 1. Since Y → P (Y, R) is a pluriharmonic function on [B(H)n ]1 , hence continuous, we deduce that (1.10) holds for any X ∈ [B(H)n ]− 1. On the other hand, since u is a positive pluriharmonic function on [B(H)n ]1 , one can use [33] (see Corollary 5.5) to find a completely positive linear map μ : R∗n + Rn → B(E) such that u is the noncommutative Poisson transform of μ, i.e., u(Y1 , . . . , Yn ) = (id ⊗ μ) P (Y, R) ,
Y := (Y1 , . . . , Yn ) ∈ B(H)n 1 .
Hence, u(rX1 , . . . , rXn ) = (id ⊗ μ)[P (rX, R)] for any X := (X1 , . . . , Xn ) ∈ [B(H)n ]− 1 . Now, using inequalities (1.10) and the fact that μ is a completely positive linear map, we obtain u(0)
1−r 1+r u(rX1 , . . . , rXn ) u(0) , 1+r 1−r
This completes the proof.
− X ∈ B(H)n 1 .
2
We recall that if f ∈ An ⊗ Mm and (A1 , . . . , An ) ∈ [B(H)n ]− 1 , then, due to the noncommutative von Neumann inequality [23] (see also [24]), it makes sense to define f (A1 , . . . , An ) ∈ B(H) ⊗min Mm . In this case we have f (A1 , . . . , An ) f . Our next task is to determine the Harnack part of 0. First, we need the following technical result. H
Lemma 1.5. Let A := (A1 , . . . , An ) and B := (B1 , . . . , Bn ) be in [B(H)n ]− 1 such that A ∼ B, and let {fk }∞ be a sequence of elements in A ⊗ M , m ∈ N, such that f n m k 1 for any k ∈ N. k=1 Then lim fk (A1 , . . . , An ) = 1 if and only if
k→∞
lim fk (B1 , . . . , Bn ) = 1.
k→∞
Proof. Assume that limk→∞ fk (A1 , . . . , An ) = 1. Then there is a sequence of vectors m {hk }∞ k=1 ⊂ H ⊗ C such that
G. Popescu / Journal of Functional Analysis 256 (2009) 4030–4070
hk = 1 and
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lim fk (A1 , . . . , An )hk = 1.
k→∞
Let {αk }∞ k=1 ⊂ D be such that |αk | → 1 as k → ∞. According to Theorem 1.5 from [31], the inverse noncommutative Cayley transform Γ −1 of αk fr is in An ⊗ Mm and Re Γ −1 [αk fk ] 0. Therefore, gk := Γ −1 [αk fk ] := (I − αk fk )−1 (I + αk fk ) ∈ An ⊗ Mm
(1.11)
H
and Re gk 0 for all k ∈ N. Since A ∼ B, Theorem 1.1 implies the existence of a constant c 1 such that 1 Re gk (A1 , . . . , An ) Re gk (B1 , . . . , Bn ) c2 Re gk (A1 , . . . , An ). c2
(1.12)
For each k ∈ N, we define the vectors yk := I − αk fk (A1 , . . . , An ) hk
and xk := I − αk fk (B1 , . . . , Bn ) yk .
(1.13)
Note that due to relations (1.11) and (1.13), we have
gk (B1 , . . . , Bn )yk , yk = I + αk fk (B1 , . . . , Bn ) xk , I − αk fk (B1 , . . . , Bn ) xk 2 = xk 2 − |αk |2 fk (B1 , . . . , Bn )xk + i Im αk fk (B1 , . . . , Bn )xk , xk .
Consequently, we deduce that 2 Re gk (B1 , . . . , Bn )yk , yk = xk 2 − |αk |2 fk (B1 , . . . , Bn )xk . Similarly, we obtain 2 Re gk (A1 , . . . , An )yk , yk = hk 2 − |αk |2 fk (A1 , . . . , An )hk . Hence and using the second inequality in (1.12), we deduce that 2 2 xk 2 − |αk |2 fk (B1 , . . . , Bn )xk c2 hk 2 − |αk |2 fk (A1 , . . . , An )hk . Consequently, since hk = 1, limk→∞ fk (A1 , . . . , An )hk = 1 and |αk | → 1 as k → ∞, we deduce that 2 xk 2 − |αk |2 fk (B1 , . . . , Bn )xk → 0,
as k → ∞.
(1.14)
Now, suppose that limk→∞ fk (B1 , . . . , Bn ) = 1. Due to the noncommutative von Neumann inequality we have fk (B1 , . . . , Bn ) fk 1. Passing to a subsequence, we can assume that there is t ∈ (0, 1) such that fk (B1 , . . . , Bn ) t < 1 for all k ∈ N. Due to relation (1.14) and the fact that 2 0 1 − t 2 xk 2 xk 2 − |αk |2 fk (B1 , . . . , Bn )xk ,
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we deduce that xk → 0 as k → ∞. Using relation (1.13), we obtain yk = [I − αk fk (B1 , . . . , Bn )]−1 xk . Consequently, and using again relation (1.14), we have I − αk fk (A1 , . . . , An ) hk = yk → 0,
as k → ∞,
(1.15)
∞ for any sequence {αk }∞ k=1 ⊂ D with the property that |αk | → 1 as k → ∞. Let {βk }k=1 ⊂ D be another sequence with the same property and such that αk + βk → 0 as k → ∞. Then, due to (1.15), we have
lim 2hk − (αk + βk )fk (A1 , . . . , An )hk = 0.
k→∞
Taking into account that fk (A1 , . . . , An )hk 1 for all k ∈ N, we deduce that hk → 0 as k → ∞, which contradicts the fact that hk = 1 for all k ∈ N. Therefore, we must have limk→∞ fk (B1 , . . . , Bn ) = 1. The converse follows in a similar manner if one uses the first inequality in (1.12). The proof is complete. 2 Now, we have all the ingredients to determine the Harnack part of 0 and obtain a characterization in terms of the free pluriharmonic Poisson kernel. Theorem 1.6. Let A := (A1 , . . . , An ) be in [B(H)n ]− 1 . Then the following statements are equivalent: H
(i) A ∼ 0; (ii) A ∈ [B(H)n ]1 ; (iii) r(A1 , . . . , An ) < 1 and P (A, R) aI for some constant a > 0, where P (A, R) is the free pluriharmonic Poisson kernel at A. H
Proof. First, we prove the implication (i) ⇒ (ii). Assume that A ∼ 0 and A = 1. For each k ∈ N define ⎡S
1
⎢0 ⎢ fk := ⎢ . ⎣ .. 0
· · · Sn ⎤ ··· 0 ⎥ ⎥ ⎥ ∈ A n ⊗ Mn . ... ⎦ ···
0
Then fk (A1 , . . . , An ) = A = 1. Applying Lemma 1.5, we deduce that 0 = fk (0) → 1, as k → ∞, which is a contradiction. Therefore A < 1. Now, we prove that (ii) ⇒ (i). Let A := (A1 , . . . , An ) be in [B(H)n ]1 and let r := A < 1. According to Theorem 1.4, we have u(0)
1−r 1+r u(X1 , . . . , Xn ) u(0) 1+r 1−r
for any positive free pluriharmonic function u on [B(H)n ]1 with operator-valued coefficients H
in B(E). Applying now Theorem 1.3, we deduce that A ∼ 0.
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To prove that (ii) ⇒ (iii), let A ∈ [B(H)n ]1 . Note that r(A1 , . . . , An ) A < 1, which implies the operators I − RA and I − A1 A∗1 − · · · − An A∗n 0 are invertible. Hence, and using the fact that ∗ −1 P (A, R) = I − RA I − A1 A∗1 − · · · − An A∗n ⊗ I (I − RA )−1 , one can easily deduce that there exists a > 0 such that P (A, R) aI . Therefore (iii) holds. It H
remains to show that (iii) ⇒ (i). Assume that (iii) holds. Due to Theorem 1.2, we have A ≺ 0 and P (X, S) =
∞
Xα∗ ⊗ S α +I +
k=1 |α|=k
∞ k=1 |α|=k
∗ Xα ⊗ S α,
X ∈ B(H)n 1 ,
where the series are convergent in the operator norm topology. On the other hand, since P (A, S) aI , applying the noncommutative Poisson transform id ⊗ PrR , we obtain P (rA, R) = (id ⊗ PrR ) P (A, S) aI = aP (0, R) H
for any r ∈ [0, 1). Due to Theorem 1.1, equivalence (i) ↔ (iii), we have 0 ≺ A. Therefore, item (i) holds. The proof is complete. 2 We remark that, when n = 1, we recover a result obtained by Foia¸s [8]. 2. Hyperbolic metric on the Harnack parts of the closed ball [B(H)n ]− 1 In this section we introduce a hyperbolic (Poincaré–Bergman type) metric δ on the Harnack n parts of [B(H)n ]− 1 , and prove that it is invariant under the action of the group Aut([B(H) ]1 ) n of all the free holomorphic automorphisms of the noncommutative ball [B(H) ]1 . We obtain an explicit formula for the hyperbolic distance in terms of the reconstruction operator and show that δ|Bn ×Bn coincides with the Poincaré–Bergman distance on Bn , the open unit ball of Cn . H
Given A, B ∈ [B(H)n ]− 1 in the same Harnack part, i.e., A ∼ B, we introduce
H ω(A, B) := inf c > 1: A ∼ B . c
(2.1)
Lemma 2.1. Let be a Harnack part of [B(H)n ]− 1 and let A, B, C ∈ . Then the following properties hold: (i) (ii) (iii) (iv)
ω(A, B) 1; ω(A, B) = 1 if and only if A = B; ω(A, B) = ω(B, A); ω(A, C) ω(A, B)ω(B, C).
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Proof. The items (i) and (iii) are obvious due to relations (1.6) and (2.1). If ω(A, B) = 1, then H
A ∼ B and, due to Theorem 1.1, part (iii), we have P (rA, R) = P (rB, R) for any r ∈ [0, 1). 1
Applying this equality to vectors of the form h ⊗ 1, h ∈ H, we obtain
r |α| A∗α h ⊗ eα =
|α|1
r |α| Bα∗ h ⊗ eα .
|α|1
Hence, we deduce that Ai = Bi for any i = 1, . . . , n. Therefore, (ii) holds. Note that, due to Theorem 1.3, we have 1 P (rB, R) P (rA, R) ω(A, B)2 P (rB, R) ω(A, B)2 and 1 P (rC, R) P (rB, R) ω(B, C)2 P (rC, R) ω(B, C)2 for any r ∈ [0, 1). Consequently, we deduce that 1 ω(A, B)2 ω(B, C)2
P (rC, R) P (rA, R) ω(A, B)2 ω(B, C)2 P (rC, R)
for any r ∈ [0, 1). Applying again Theorem 1.3, we have ω(A, C) ω(A, B)ω(B, C). This completes the proof. 2 Now, we can introduce a hyperbolic (Poincaré–Bergman type) metric on the Harnack parts of [B(H)n ]− 1. + Proposition 2.2. Let be a Harnack part of [B(H)n ]− 1 and define δ : × → R by setting
δ(A, B) := ln ω(A, B),
A, B ∈ .
(2.2)
Then δ is a metric on . Proof. The result follows from Lemma 2.1.
2
In [34], we showed that any free holomorphic automorphism Ψ of the unit ball [B(H)n ]1 which fixes the origin is implemented by a unitary operator on Cn , i.e., there is a unitary operator U on Cn such that Ψ (X1 , . . . , Xn ) = ΨU (X1 , . . . , Xn ) := [X1 · · · Xn ]U,
(X1 , . . . , Xn ) ∈ B(H)n 1 .
The theory of noncommutative characteristic functions for row contractions (see [22,29]) was used to find all the involutive free holomorphic automorphisms of [B(H)n ]1 . They turned out to be of the form
G. Popescu / Journal of Functional Analysis 256 (2009) 4030–4070
Ψλ = −Θλ (X1 , . . . , Xn ) := λ − λ IK −
n
4049
−1 λ¯ i Xi
[X1 · · · Xn ]λ∗ ,
i=1
for some λ = (λ1 , . . . , λn ) ∈ Bn , where Θλ is the characteristic function of the row contraction λ, and λ , λ∗ are certain defect operators. Moreover, we determined the group Aut([B(H)n ]1 ) of all the free holomorphic automorphisms of the noncommutative ball [B(H)n ]1 and showed that if Ψ ∈ Aut([B(H)n ]1 ) and λ := Ψ −1 (0), then there is a unitary operator U on Cn such that Ψ = ΨU ◦ Ψλ . The following result is essential for the proof of the main result of this section. Lemma 2.3. Let A := (A1 , . . . , An ) and B := (B1 , . . . , Bn ) be in [B(H)n ]− 1 and let Ψ ∈ H
H
c
c
Aut([B(H)n ]1 ). Then A ≺ B if and only if Ψ (A1 , . . . , An ) ≺ Ψ (B1 , . . . , Bn ). Proof. Let p(S1 , . . . , Sn ) :=
Sα ⊗ M(α) ,
M(α) ∈ Mm ,
|α|k
be a polynomial in An ⊗ Mm such that Re p(S1 , . . . , Sn ) 0. Due to the results from [34], if Ψ ∈ Aut([B(H)n ]1 ), then Ψ (S1 , . . . , Sn ) = (Ψ1 (S1 , . . . , Sn ), . . . , Ψn (S1 , . . . , Sn )) is a row contraction with entries Ψi (S1 , . . . , Sn ), i = 1, . . . , n, in the noncommutative disc algebra An . Since the noncommutative Poisson transform at Ψ (S1 , . . . , Sn ) is a c.p. linear map PΨ (S1 ,...,Sn ) : C ∗ (S1 , . . . , Sn ) → B(H), we deduce that Q(S1 , . . . , Sn ) := Re
Ψα (S1 , . . . , Sn ) ⊗ M(α)
|α|k
= PΨ (S1 ,...,Sn ) ⊗ id Re p(S1 , . . . , Sn ) 0. H
If we assume that A ≺ B, then Theorem 1.1, part (v) implies c
Q(A1 , . . . , An ) = (PA ⊗ id) Q(S1 , . . . , Sn ) c2 (PB ⊗ id) Q(S1 , . . . , Sn ) = c2 Q(B1 , . . . , Bn ), where PA and PB are the noncommutative Poisson transforms at A and B, respectively. Therefore, we have Re p Ψ1 (A1 , . . . , An ), . . . , Ψn (A1 , . . . , An ) c2 Re p Ψ1 (B1 , . . . , Bn ), . . . , Ψn (B1 , . . . , Bn ) , H
which shows that Ψ (A1 , . . . , An ) ≺ Ψ (B1 , . . . , Bn ). c
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G. Popescu / Journal of Functional Analysis 256 (2009) 4030–4070 H
Conversely, assume that Ψ (A1 , . . . , An ) ≺ Ψ (B1 , . . . , Bn ). Applying the first part of the proof, c
we deduce that H Ψ −1 Ψ (A1 , . . . , An ) ≺ Ψ −1 Ψ (B1 , . . . , Bn ) . c
H
Since Ψ −1 ◦ Ψ = id on the closed ball [B(H)n ]− B. The proof is com1 , we deduce that A ≺ c plete. 2 Here is the main result of this section. Theorem 2.4. Let δ : [B(H)n ]1 ⊗ [B(H)n ]1 → [0, ∞) be the hyperbolic metric. Then the following statements hold. (i) If A, B ∈ [B(H)n ]1 , then δ(A, B) = ln max CA CB−1 , CB CA−1 , where CX := (X ⊗ I )(I − RX )−1 and RX := X1∗ ⊗ R1 + · · · + Xn∗ ⊗ Rn is the reconstruction operator associated with the right creation operators R1 , . . . , Rn and X := (X1 , . . . , Xn ) ∈ [B(H)n ]1 . (ii) For any free holomorphic automorphism Ψ of the noncommutative unit ball [B(H)n ]1 , δ(A, B) = δ Ψ (A), Ψ (B) ,
A, B ∈ B(H)n 1 .
(iii) δ|Bn ×Bn coincides with the Poincaré–Bergman distance on Bn , i.e., δ(z, w) =
1 1 + ψz (w)2 ln , 2 1 − ψz (w)2
z, w ∈ Bn ,
where ψz is the involutive automorphism of Bn that interchanges 0 and z. H
Proof. Let A, B ∈ [B(H)n ]1 . Due to Theorem 1.6, we have A ∼ B. In order to determine H
ω(A, B), assume that A ∼ B for some c 1. According to Theorem 1.3, we have c
1 P (rB, R) P (rA, R) c2 P (rB, R) c2 for any r ∈ [0, 1). Since A < 1 and A < 1, we can take the limit, as r → 1, in the operator norm topology, and obtain 1 P (B, R) P (A, R) c2 P (B, R). c2
(2.3)
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We recall that the free pluriharmonic kernel P (X, R), X := (X1 , . . . , Xn ) ∈ [B(H)n ]1 , has the ∗ C , where C := ( ⊗ I )(I − R )−1 . Note also that C is an factorization P (X, R) = CX X X X X X invertible operator. It is easy to see that relation (2.3) implies CA∗ −1 CB∗ CB CA−1 c2 I
and CB∗ −1 CA∗ CA CB−1 c2 I.
Therefore, d := max CA CB−1 , CB CA−1 c, which implies d ω(A, B). On the other hand, since CB CA−1 d and CA CB−1 d, we have CA∗ −1 CB∗ CB CA−1 d 2 I
and CB∗ −1 CA∗ CA CB−1 d 2 I.
Hence, we deduce that 1 ∗ C CB CA∗ CA d 2 CB∗ CB , d2 B which is equivalent to 1 P (B, S) P (A, S) d 2 P (B, S), d2 where S := (S1 , . . . , Sn ) is the n-tuple of left creation operators. Applying the noncommutative Poisson transform id ⊗ PrR , r ∈ [0, 1), and taking into account that it is a positive map, we deduce that 1 P (rB, R) P (rA, R) d 2 P (rB, R) d2 H
for any r ∈ [0, 1). Due to Theorem 1.3, we deduce that A ∼ B and, consequently, ω(A, B) d. d
Since the reverse inequality was already proved, we have ω(A, B) = d, which together with (2.2) prove part (i). To prove (ii), let Ψ ∈ Aut([B(H)n ]1 ). If A, B ∈ [B(H)n ]1 , then, due to Theorem 1.6, we have H
A ∼ B. Applying Lemma 2.3, the result follows. Now, let us prove item (iii). Let z := (z1 , . . . , zn ) ∈ Bn . Due to part (i) of this theorem, we have δ(z, 0) = ln max Cz , Cz−1 , where Cz := (1 − z2 )1/2 (I −
n
−1 i=1 z¯ i Ri ) .
First, we show that
n z¯ i Ri = 1 + z2 . I − i=1
(2.4)
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Indeed, since R1 , . . . , Rn are isometries with orthogonal ranges, we have n n 1/2 2 z¯ i Ri = |zi | = z2 . i=1
i=1
Consequently, n z¯ i Ri 1 + z2 . I −
(2.5)
i=1
Note that, due to Riesz representation theorem, we have sup w=(w1 ,...,wn )∈Bn
z¯ i wi = 1 + z2 . 1 +
(2.6)
i=1
On the other hand, due to the noncommutative von Neumann inequality [23], we have n 1 + z¯ i wi I − z¯ i Ri i=1
(2.7)
i=1
for any (w1 , . . . , wn ) ∈ Bn . Combining relations (2.5), (2.6), and (2.7), we deduce (2.4). Consequently, we have −1 C =
z
1 + z2 1 − z2
1/2 (2.8)
.
Note also that n n 2 −1 n z¯ i Ri z¯ i Ri + z¯ i Ri + · · · I− 1+ i=1
i=1
i=1
= 1 + z2 + z22 =
+ ···
1 . 1 − z2
Consequently, we have Cz
1 + z2 1 − z2
1/2 (2.9)
.
Due to relations (2.8) and (2.9), we have ω(z, 0) =
1 + z2 1 − z2
1/2 .
(2.10)
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Now, we consider the general case. For each w ∈ Bn , let Ψz be the corresponding involutive automorphism of [B(H)n ]1 . We recall (see [34]) that Ψw (0) = w and Ψw (w) = 0. Due to part (ii) of this theorem and relation (2.10), we have δ(z, w) = δ Ψw (z), Ψw (w) = δ Ψw (z), 0 = ln ω Ψw (z), 0 =
1 1 + Ψz (w)2 ln . 2 1 − Ψz (w)2
Since, according to [34], Ψw is a noncommutative extension of the involutive automorphism of Bn that interchanges 0 and z, i.e., Ψw (z) = ψw (z) for z ∈ Bn , item (iii) follows. The proof is complete. 2 Corollary 2.5. For any X, Y ∈ [B(H)n ]1 , δ(X, Y ) ln
X Y . (1 − X)(1 − Y )
Proof. According to Theorem 1.6 and Proposition 2.2, [B(H)n ]1 is the Harnack part of 0 and δ is a metric on the open ball [B(H)n ]1 . Therefore δ(X, Y ) δ(X, 0) + δ(0, Y ). Theorem 2.4, part (i) implies −1 , δ(X, 0) = ln max CX , CX where CX := (I ⊗ X )(I − RX )−1 and RX := X1∗ ⊗ R1 + · · · + Xn∗ ⊗ Rn . Since R1 , . . . , Rn are isometries with orthogonal ranges, we have I − RX 1 + RX = 1 + X and (I − RX )−1 1 + RX + RX 2 + · · · = 1 + X + X2 + · · · =
1 . 1 − X
On the other hand, since X < 1, we have −1 2 1 + XX ∗ + XX ∗ 2 + · · · X
=
1 . 1 − X2
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Now, one can easily see that −1 C = (I − RX ) I ⊗ −1 X X 1 + X −1 X 1 + X 1/2 1 − X and CX
X . 1 − X
Note also that, due to the fact that I − XX ∗ 1 − XX ∗ , we have 1 + X 1/2 X . 1 − X 1 − X Therefore −1 ln X . δ(X, 0) = ln max CX , CX 1 − X Taking into account that δ(X, Y ) δ(X, 0) + δ(0, Y ), the result follows. The proof is complete. 2 In what follows we prove that the hyperbolic metric δ, on the Harnack parts of [B(H)n ]− 1, is invariant under the automorphism group Aut([B(H)n ]1 ), and can be written in terms of the reconstruction operator. First, we need the following result. Lemma 2.6. Let A := (A1 , . . . , An ) and B := (B1 , . . . , Bn ) be in [B(H)n ]− 1 . Then the following properties hold. H
H
(i) A ∼ B if and only if rA ∼ rB for any r ∈ [0, 1) and supr∈[0,1) ω(rA, rB) < ∞. In this case, ω(A, B) = sup ω(rA, rB) r∈[0,1)
and δ(A, B) = sup δ(rA, rB). r∈[0,1)
H
(ii) If A ∼ B, then the functions r → ω(rA, rB) and r → δ(rA, rB) are increasing on [0, 1). H
H
Proof. Assume that A ∼ B and let a := ω(A, B). Due to relation (1.6), it is clear that A ∼ B. By a
Theorem 1.3, we deduce that 1 P (rB, R) P (rA, R) a 2 P (rB, R) a2 H
for any r ∈ [0, 1), which shows that rA ∼ rB for any r ∈ [0, 1), and supr∈[0,1) ω(rA, rB) a. a
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Conversely, assume that supr∈[0,1) ω(rA, rB) < ∞. Denote cr := ω(rA, rB) for r ∈ [0, 1), H
and let c1 := supr∈[0,1) cr < ∞. Since rA ∼ rB and 1 cr c1 , we have cr
1 1 P (trB, R) 2 P (trB, R) P (trA, R) cr2 P (trB, R) c12 P (trB, R) 2 c c1 r H
for any t, r ∈ [0, 1). Due to Theorem 1.3 we deduce that A ∼ B. Thus ω(A, B) c1 . Since the c1
reverse inequality was already proved above, we have a = c1 . The second part of item (i) is now obvious. To prove (ii), let s, t ∈ [0, 1) be such that s < t. Applying part (i), we have ω(sA, sB) sup ω(rtA, rtB) ω(tA, tB). r∈[0,1)
Hence, we deduce item (ii). The proof is complete.
2 H
Theorem 2.7. Let A := (A1 , . . . , An ) and B := (B1 , . . . , Bn ) be in [B(H)n ]− 1 such that A ∼ B. Then (i) δ(A, B) = δ(Ψ (A), Ψ (B)) for any Ψ ∈ Aut([B(H)n ]1 ); (ii) the metric δ satisfies the relation −1 −1 , sup CrB CrA , δ(A, B) = ln max sup CrA CrB r∈[0,1)
r∈[0,1)
where CX := (X ⊗ I )(I − RX )−1 and RX := X1∗ ⊗ R1 + · · ·+ Xn∗ ⊗ Rn is the reconstruction operator associated with the right creation operators R1 , . . . , Rn and X := (X1 , . . . , Xn ) ∈ [B(H)n ]1 . Proof. Due to Lemma 2.6 and Theorem 2.4, the result follows.
2
3. Metric topologies on Harnack parts of [B(H)n ]− 1 In this section we study the relations between the δ-topology, the dH -topology (which will be introduced), and the operator norm topology on Harnack parts of [B(H)n ]− 1 . We prove that the hyperbolic metric δ is a complete metric on certain Harnack parts, and that all the topologies above coincide on the open ball [B(H)n ]1 . First, we need some notation. Denote − − B0 (H)n 1 := (X1 , . . . , Xn ) ∈ B(H)n 1 : r(X1 , . . . , Xn ) < 1 , where r(X1 , . . . , Xn ) is the joint spectral radius of (X1 , . . . , Xn ). Note that, due to Theorems 1.2 and 1.6, we have
− − H B(H)n 1 ⊂ B0 (H)n 1 = X ∈ B(H)n 1 : X ≺ 0 .
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H
If A, B are in [B0 (H)n ]− 1 , then A ≺ 0 and B ≺ 0. Consequently, there exists c 1 such that, for any f ∈ An ⊗ B(E) with Re f 0, where E is a separable infinite-dimensional Hilbert space, we have Re f (A) c2 Re f (0)
and
Re f (B) c2 Re f (0).
Hence, we deduce that Re f (A) − Re f (B) 2c2 Re f (0). n − Therefore, it makes sense to define the map dH : [B0 (H)n ]− 1 × [B0 (H) ]1 → [0, ∞) by setting
dH (A, B) := sup u(A) − u(B): u ∈ Re An ⊗ B(E) , u(0) = I, u 0 . Proposition 3.1. For any A, B ∈ [B0 (H)n ]− 1, dH (A, B) = supRe p(A) − Re p(B), where the supremum is taken over all polynomials p ∈ C[X1 , . . . , Xn ] ⊗ Mm , m ∈ N, with Re p(0) = I and Re p 0. Proof. Let f ∈ An ⊗ B(E) be such that Re f 0 and (Re f )(0) = I . According to [25], f has a unique formal Fourier representation f=
Sα ⊗ C(α) ,
C(α) ∈ B(E).
α∈F+ n
|α| Moreover, limr→1 fr = f in the operator norm topology, where fr = ∞ |α|=k r Sα ⊗ C(α) k=1 is in An ⊗ B(E) and the series is convergent in the operator norm. Consequently, for any > 0, there exist r ∈ [0, 1) and N ∈ N such that pr ,N − f < , 2
(3.1)
|α| 1 where pr ,N := N |α|=k r Sα ⊗ C(α) . Define the polynomial q,r ,N := 1+ (pr ,N + I ) k=0 and note that (Re q,r ,N )(0) = I . On the other hand, due to (3.1), we have Re pr ,N − Re f < 2 , which, due to the fact that Re f 0, implies Re q,r ,N 0. Now, notice that q,r ,N − f
1 pr ,N − f + I + f , 1+ 1+
which together with
relation (3.1) show that f can be approximated, in the operator norm, with polynomials q = N k=0 Sα ⊗ D(α) , D(α) ∈ B(E), such that Re q 0 and (Re q)(0) = I . Consider now an orthonormal basis {ξ1 , ξ2 , . . .} of E and let Em := span{ξ1 , . . . , ξm }. Setting qm := PF 2 (Hn )⊗Em q|F 2 (Hn )⊗Em =
N k=0
Sα ⊗ PEm D(α) |Em ,
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it is easy to see that Re qm 0, (Re qm )(0) = I , and q(A) − q(B) = lim qm (A) − qm (B). m→∞
This can be used to complete the proof.
2
Due to the next result, we call dH the kernel metric on [B0 (H)n ]− 1. Proposition 3.2. dH is a metric on [B0 (H)n ]− 1 satisfying relation dH (A, B) = P (A, R) − P (B, R) for any A, B ∈ [B0 (H)n ]− 1 , where P (X, R) is the free pluriharmonic Poisson kernel associated − n with X ∈ [B0 (H) ]1 . Moreover, the map [0, 1) r → dH (rA, rB) ∈ R+ is increasing and dH (A, B) = sup dH (rA, rB). r∈[0,1)
Proof. It is easy to see that dH is a metric. Since r(A) < 1, the map v : [B(K)n ]1 → B(K) ⊗min B(H) defined by vA (Y ) :=
∞
Yα˜ ⊗ A∗α + I +
k=1 |α|=k
∞ k=1 |α|=k
Yα˜∗ ⊗ Aα
is a free pluriharmonic function on the open ball [B(K)n ]γ for some γ > 1, where K is an infinite-dimensional Hilbert space. Therefore vA is continuous on [B(K)n ]γ and vA (S) :=
∞
Sα˜ ⊗ A∗α + I +
k=1 |α|=k
∞ k=1 |α|=k
Sα∗˜ ⊗ Aα
is in Re(An ⊗ B(H)). Consequently, vA (S) = limr→1 vA (rS) in the norm topology. Since a similar result holds for vB , and R is unitarily equivalent to S, we have P (A, R) − P (B, R) = vA (S) − vB (S) = lim vA (rS) − vB (rS) r→1 = lim P (rA, R) − P (rB, R). r→1
Due to the noncommutative von Neumann inequality the map [0, 1) r → P (rA, R) − P (rB, R) ∈ R+ is increasing.
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For each r ∈ [0, 1), the map ur (Y ) := P (Y, rR) is a free pluriharmonic function on [B(H)n ]γ for some γ > 1, with coefficients in B(F 2 (Hn )). Moreover, since ur is positive on [B(H)n ]1 and ur (0) = I , the definition of dH implies ur (A) − ur (B) = P (A, rR) − P (B, rR) dH (A, B)
(3.2)
for any r ∈ [0, 1). Taking r → 1 and using the first part of the proof, we deduce that P (A, R) − P (B, R) dH (A, B).
(3.3)
Now, let G ∈ Re(An ⊗ B(E)) be with G(0) = I and G 0. Since Y → G(Y ) is a positive free pluriharmonic function of [B(H)n ]1 , we can apply Corollary 5.5 from [33] to find a completely positive linear map μ : R∗n + Rn → B(E) with μ(I ) = I and G(Y ) = (id ⊗ μ) P (Y, R) ,
Y ∈ B(H)n 1 .
Consequently, for each r ∈ [0, 1), we have G(rA) − G(rB) μP (A, R) − P (B, R). Since G ∈ Re(An ⊗ B(E)), Theorem 4.1 from [33] shows that G(A) = limr→1 G(rA) and G(B) = limr→1 G(rB) in the operator norm topology. Due to the fact that μ = 1, we have G(A) − G(B) P (A, R) − P (B, R). Therefore, dH (A, B) P (A, R) − P (B, R), which together with (3.3) prove the equality. The last part of the proposition can be easily deduced from the considerations above. The proof is complete. 2 Theorem 3.3. Let dH be the kernel metric on [B0 (H)n ]− 1 . Then the following statements hold: (i) the metric dH is complete on [B0 (H)n ]− 1; (ii) the dH -topology is stronger than the norm topology on [B0 (H)n ]− 1; (iii) the dH -topology coincides with the norm topology on the open unit ball [B(H)n ]1 . Proof. First we prove that A − B P (A, R) − P (B, R),
− A, B ∈ B0 (H)n 1 .
(3.4)
Indeed, for each r ∈ [0, 1), we have 2π
1 rRA = 2π
eit P A, reit R dt,
0
where RA := A∗1 ⊗ R1 + · · · + A∗n ⊗ Rn is the reconstruction operator. Using the noncommutative von Neumann inequality, we obtain
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rA − rB = rRA − rRB 1 2π it it it = e P A, re R − P B, re R dt 2π 0
sup P A, reit R − P B, reit R t∈[0,2π]
P (A, rR) − P (B, rR) for any r ∈ [0, 1). Since r(A) < 1 and r(B) < 1, we have lim P (A, rR) − P (B, rR) = P (A, R) − P (B, R),
r→1
which proves our assertion. (k) (k) n − Now, to prove (i), let {A(k) := (A1 , . . . , An )}∞ k=1 be a dH -Cauchy sequence in [B0 (H) ]1 . Due to relation (3.4), we have (k) A − A(p) P A(k) , R − P A(p) , R = dH A(k) , A(p) n − for any k, p ∈ N. Hence, {A(k) }∞ k=1 is a Cauchy sequence in the norm topology of [B(H) ]1 . − n (k) Therefore, there exists T := (T1 , . . . , Tn ) in [B(H) ]1 such that T − A → 0, as k → ∞. Now let us prove that the joint spectral radius r(T ) < 1. Since {A(k) }∞ k=1 is a dH -Cauchy (k) (k ) 0 sequence, there exists k0 ∈ N such that dH (A , A ) 1 for any k k0 . On the other hand, H
(k0 ) ≺ 0. Applying Theorem 1.1, we find since A(k0 ) ∈ [B0 (H)n ]− 1 , Theorem 1.2 shows that A (k ) 2 0 c 1 such that P (rA , R) c for any r ∈ [0, 1). Hence and using inequality (3.2), we have
P rA(k) , R P rA(k) , R − P rA(k0 ) , R + P rA(k0 ) , R I dH A(k) , A(k0 ) + P rA(k0 ) , R I 1 + c2 I for any k k0 and r ∈ [0, 1). Taking k → ∞ and using the continuity of the free pluriharmonic functions in the operator norm topology, we obtain P (rT , R) (1+c2 )I for r ∈ [0, 1). Applying H
again Theorem 1.1, we deduce that T ≺ 0. Now, Theorem 1.2 implies r(T ) < 1, which shows that T ∈ [B0 (H)n ]− 1 and proves part (i). Note that part (ii) follows from Proposition 3.2 and inequality (3.4). To prove part (iii), we assume that A, B ∈ [B(H)n ]1 . First, recall from the proof of Corollary 2.5 that (I − RX )−1 for any X ∈ [B(H)n ]1 . Consequently, we have
1 1 − X
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P (A, R) − P (B, R) 2(I − RA )−1 − (I − RB )−1 2(I − RA )−1 (I − RB )−1 RA − RB
2A − B . (1 − A)(1 − B)
Hence, using part (ii) and the fact that dH (A, B) = P (A, R) − P (B, R), we deduce that the dH -topology coincides with the norm topology on the open unit ball [B(H)n ]1 . This completes the proof. 2 Corollary 3.4. If A, B ∈ [B0 (H)n ]− 1 , then A − B dH (A, B). Moreover, if A, B ∈ [B(H)n ]1 , then dH (A, B)
2A − B . (1 − A)(1 − B)
In what follows we obtain another formula for the hyperbolic distance that will be used to prove the main result of this section. We mention that if f ∈ An ⊗ Mm , m ∈ N, then we call Re f strictly positive and denote Re f > 0 if there exists a constant a > 0 such that Re f aI . H
Proposition 3.5. Let A := (A1 , . . . , An ) and B := (B1 , . . . , Bn ) be in [B(H)n ]− 1 such that A ∼ B. Then Re f (A1 , . . . , An )x, x 1 , (3.5) δ(A, B) = sup ln 2 Re f (B1 , . . . , Bn )x, x where the supremum is taken over all f ∈ An ⊗ Mm , m ∈ N, with Re f > 0 and x ∈ H ⊗ Cm with x = 0. Proof. Denote the right-hand side of (3.5) by δ (A, B). If f ∈ An ⊗Mm , m ∈ N, with Re f aI , then applying the noncommutative Poison transform, we have Re f (Y1 , . . . , Yn ) aI for any H
(Y1 , . . . , Yn ) ∈ [B(H)n ]− 1 . Assume that A ∼ B with c 1. Due to Theorem 1.1, we deduce that c
1 Re f (A1 , . . . , An )x, x c2 2 Re f (B1 , . . . , Bn )x, x c for any x ∈ H ⊗ Cm with x = 0. Hence, we have δ (A, B) ln c, which implies δ (A, B) δ(A, B). To prove the reverse inequality, note that if g ∈ An ⊗ Mm with Re g 0, then f := g + I has the property that Re f I for any > 0. Consequently, Re f (A1 , . . . , An )x, x 1 δ (A, B) sup ln 2 Re f (B1 , . . . , Bn )x, x
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for any x ∈ H ⊗ Cm with x = 0. Note that the latter inequality is equivalent to 1 Re g(B1 , . . . , Bn )x, x + 2δ (A,B) x2 e2δ (A,B) e Re g(B1 , . . . , Bn )x, x + x2 e2δ (A,B) Re g(B1 , . . . , Bn )x, x + e2δ (A,B) x2 .
Taking → 0, we deduce that ω(A, B) eδ (A,B) , which implies δ(A, B) δ (A, B) and completes the proof. 2 We remark that, under the conditions of Proposition 3.5, one can also prove that Re p(A1 , . . . , An )x, x 1 , δ(A, B) = sup ln 2 Re p(B1 , . . . , Bn )x, x where the supremum is taken over all noncommutative polynomials p ∈ C[X1 , . . . , Xn ] ⊗ Mm , m ∈ N, with Re p > 0, and x ∈ H ⊗ Cm with x = 0. Indeed, if f ∈ An ⊗ Mm with Re f aI,
a > 0, and 0 < < a, then, as in the proof of Proposition 3.1, we can find a polynomial p = |α|N Sα ⊗ C(α) , C(α) ∈ Mm , such that f − p < . Hence, Re f − Re p < and, consequently, Re p > 0. Now, our assertion follows. Here is the main result of this section. Theorem 3.6. Let δ be the Poincaré–Bergman type metric on a Harnack part of [B0 (H)n ]− 1. Then the following properties hold: (i) δ is complete on ; (ii) the δ-topology is stronger then the dH -topology on ; (iii) the δ-topology, the dH -topology, and the operator norm topology coincide on the open ball [B(H)n ]1 . Proof. Let A := (A1 , . . . , An ) and B := (B1 , . . . , Bn ) be in a Harnack part of [B0 (H)n ]− 1. H
Then A ∼ B and Re f (A1 , . . . , An ) ω(A, B)2 Re f (B1 , . . . , Bn ) for any f ∈ An ⊗ Mm with Re f 0. Hence, we have Re f (A1 , . . . , An ) − Re f (B1 , . . . , Bn ) ω(A, B)2 − 1 Re f (B1 , . . . , Bn ). H
(3.6)
On the other hand, since B ≺ 0, we have r(B) < 1 so P (B, R) makes sense. Also, due to the fact that the noncommutative Poisson transform id ⊗ PrR is completely positive, and P (B, S) P (B, R)I , we deduce that P (rB, R) = (id ⊗ PrR ) P (B, S) P (B, R)I = P (B, R)P (0, R)
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for any r ∈ [0, 1). Applying now the equivalence (iii) ↔ (iv) of Theorem 1.1, when c2 = P (B, R)), we obtain Re f (rB1 , . . . , rBn ) P (B, R) Re f (0) for any r ∈ [0, 1). Taking the limit, as r → 1, in the operator norm topology, we get Re f (B1 , . . . , Bn ) P (B, R) Re f (0). Combining this inequality with (3.6), we have Re f (A1 , . . . , An ) − Re f (B1 , . . . , Bn ) ω(A, B)2 − 1 P (B, R) Re f (0). A similar inequality holds if one interchange A with B. If, in addition, we assume that Re f (0) = I , then we can deduce that −sI Re f (A1 , . . . , An ) − Re f (B1 , . . . , Bn ) sI, where s := [ω(A, B)2 − 1] max{P (A, R), P (B, R)}. Since Re f (A1 , . . . , An ) − Re f (B1 , . . . , Bn ) is a self-adjoint operator, we have Re f (A1 , . . . , An ) − Re f (B1 , . . . , Bn ) s. Due to the definition of the metric dH , we deduce that dH (A, B) s. Consequently, we obtain dH (A, B) max P (A, R), P (B, R) e2δ(A,B) − 1 .
(3.7)
(k) ∞ Now, we prove that δ is a complete metric on . Let {A(k) := (A(k) 1 , . . . , An )}k=1 ⊂ be ∞ (k) a δ-Cauchy sequence. First, we prove that the sequence {P (A , R)}k=1 is bounded. For any > 0 there exists k0 ∈ N such that
δ A(k) , A(p) < H
for any k, p k0 .
(3.8)
H
Since A(k) ∼ A(k0 ) and A(k0 ) ≺ 0, for any f ∈ An ⊗ Mm with Re f 0, we have 2 Re f A(k) ω A(k) , A(k0 ) Re f A(k0 ) c2 Re f (0),
(3.9)
where c := P (A(k0 ) , R)1/2 ω(A(k) , A(k0 ) ). Consequently, due to Theorem 1.1, we have H
A(k) ≺ 0 and P (A(k) , R) c2 for any k k0 . Combining this with relation (3.8), we obtain (k) (k ) 2 P A , R P A 0 , R e for any k k0 . This shows that the sequence {P (A(k) , R)}∞ k=1 is bounded. Consequently, due to inequality (3.7), we deduce that {A(k) } is a dH -Cauchy sequence. According to Theorem 3.3, there exists A := (A1 , . . . , An ) ∈ [B0 (H)n ]− 1 such that dH A(k) , A → 0 as k → ∞.
(3.10)
Now, let f ∈ An ⊗ Mm with Re f 0 and Re f (0) = I . Due to relations (3.9) and (3.8), we have 2 Re f A(k) ω A(k) , A(k0 ) Re f A(k0 ) e2 Re f A(k0 )
(3.11)
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for k k0 . According to (3.10), Re f (A(k) ) → Re f (A), as k → ∞, in the operator norm topology. Consequently, (3.11) implies Re f (A) e2 Re f A(k0 ) .
(3.12)
A similar inequality can be deduced in the more general case when f ∈ An ⊗ Mm with Re f 0. Indeed, for each > 0 let g := I + f , Y = Re g(0), and ϕ := Y −1/2 gY −1/2 . Since Re ϕ 0 and Re ϕ(0) = I , we can apply inequality (3.12) to ϕ and deduce that I + Re f (A) e2 I + Re f A(k0 ) for any > 0. Taking → 0, we get Re f (A) e2 Re f A(k0 )
(3.13)
for any f ∈ An ⊗ Mm with Re f 0. This shows that H
A ≺ A(k0 ) .
(3.14)
H
On the other hand, since A(k0 ) ≺ A(k) for any k k0 , we deduce that 2 Re f A(k0 ) ω A(k0 ) , A(k) Re f A(k) e2 Re f A(k) for k k0 . According to Theorem 3.3, the dH -topology is stronger than the norm topology (k) → A ∈ [B (H)n ]− in the operator norm on [B0 (H)n ]− 0 1 . Therefore, relation (3.10) implies A 1 topology. Passing to the limit in the inequality above, we deduce that Re f A(k0 ) e2 Re f (A)
(3.15) H
for any f ∈ An ⊗ Mm with Re f 0. Consequently, we have A(k0 ) ≺ A. Hence and using (3.14), H
we obtain A ∼ A(k0 ) , which proves that A ∈ . From the inequalities (3.13) and (3.15), we have ω(A(k0 ) , A) e2 . Hence, δ(A(k0 ) , A) < , which together with (3.8) imply δ(A(k) , A) < 2 for any k k0 . Consequently, δ(A(k) , A) → 0 as k → ∞, which proves that δ is complete on . Note that we have also proved part (ii) of this theorem. Now, we prove part (iii). To this end, assume that A, B ∈ [B(H)n ]1 . Due to the fact that B < 1, P (B, R) is a positive invertible operator. Since P (B, R)−1 P (B, R)−1 , we have I P (B, R)−1 P (B, R), which implies H
I P (B, R)−1 P (rB, R) for any r ∈ [0, 1). According to Theorem 1.1, we deduce that 0 ≺ B and Re f (0) P (B, R)−1 Re f (B) for any f ∈ An ⊗ Mm with Re f 0. If, in addition, we assume that Re f (0) = I , then the latter inequality implies
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Re f (A)x, x P (B, R)−1 Re f (A) − Re f (B) x, x −1 Re f (B)x, x x P (B, R)−1 dH (A, B) for any x ∈ H ⊗ Cm , x = 0. Hence, we deduce that ln
Re f (A)x, x ln 1 + P (B, R)−1 dH (A, B) . Re f (B)x, x
One can obtain a similar inequality interchanging A with B. Combining these two inequalities, we obtain Re f (A)x, x −1 ln , P (A, R)−1 dH (A, B) . (3.16) Re f (B)x, x ln 1 + max P (B, R) Consider now the general case when g ∈ An ⊗ Mm with Re g > 0. Then Y := Re g(0) is a positive invertible operator on H ⊗ Cm and f := Y −1/2 gY −1/2 has the properties Re f 0 and Re f (0) = I . Applying (3.16) to f when x := Y −1/2 y, y ∈ H ⊗ Cm , and y = 0, we deduce that 2δ(A, B) ln 1 + max P (B, R)−1 , P (A, R)−1 dH (A, B) .
(3.17)
n n Now, let {A(k) }∞ k=1 be a sequence of elements in [B(H) ]1 and let A ∈ [B(H) ]1 be such that dH (A(k) , A) → 0 as k → ∞. Due to Proposition 3.2, we deduce that P (A(k) , R) → P (A, R) in the operator norm topology, as k → ∞. On the other hand, the operators P (A(k) , R) and P (A, R) are invertible due to the fact that A(k) < 1 and A < 1. Consequently, and using the well-known fact that the map Z → Z −1 is continuous on the open set of all invertible operators, we deduce that P (A(k) , R)−1 → P (A, R)−1 in the operator norm topology. Hence, the sequence (k) −1 {P (A(k) , R)−1 }∞ k=1 is bounded. Therefore, there exists M > 0 with P (A , R) M for any k ∈ N. Applying now inequality (3.17), we deduce that
2δ A(k) , A ln 1 + MdH A(k) , A
for any k ∈ N.
Since dH (A(k) , A) → 0 as k → ∞, the latter inequality implies that δ(A(k) , A) → 0 as k → ∞. Therefore the dH -topology on [B(H)n ]1 is stronger than the δ-topology. Due to the first part of this theorem, the two topologies coincide on [B(H)n ]1 . Applying now Theorem 3.3, we complete the proof. 2 Corollary 3.7. Let be a Harnack part of [B0 (H)n ]− 1 . Then δ(A, B)
dH (A, B) 1 ln 1 + , 2 max{P (A, R), P (B, R)}
A, B ∈ .
Moreover, if A, B ∈ := [B(H)n ]1 , then δ(A, B)
1 ln 1 + dH (A, B) max P (A, R)−1 , P (B, R)−1 . 2
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Combining Corollary 3.4 with Corollary 3.7, one can obtain inequalities involving the hyperbolic metric δ and the metric induced by the operator norm on [B0 (H)n ]− 1 . In particular, if A, B ∈ [B(H)n ]1 , then we have 1 A − B ln 1 + δ(A, B) 2 max{P (A, R), P (B, R)} and δ(A, B)
2A − B 1 ln 1 + max P (A, R)−1 , P (B, R)−1 . 2 (1 − A)(1 − B)
4. Schwarz–Pick lemma with respect to the hyperbolic metric on [B(H)n ]1 A very important property of the Poincaré–Bergman distance βm : Bm × Bm → R+ is that βm f (z), f (w) βn (z, w),
z, w ∈ Bn ,
for any holomorphic function f : Bn → Bm . In this section we extend this result and obtain a Schwarz–Pick lemma for free holomorphic functions on [B(H)n ]1 with operator-valued coefficients, with respect to the hyperbolic metric on the noncommutative ball [B(H)n ]1 . Lemma 4.1. Let A := (A1 , . . . , An ) and B := (B1 , . . . , Bn ) be pure row contractions. Then H
A ≺ B if and only if c
Re f (A1 , . . . , An ) c2 Re f (B1 , . . . , Bn ) ¯ for any f ∈ Fn∞ ⊗B(E) with Re f 0. H
¯ Proof. Assume that A, B are pure row contractions with A ≺ B and let f ∈ Fn∞ ⊗B(E) be such c that Re f 0. Then f has a unique representation of the form f (S1 , . . . , Sn ) =
∞
Sα ⊗ A(α) ,
A(α) ∈ B(E).
k=0 |α|=k
Due to the results from [30], for each r ∈ [0, 1), fr (S1 , . . . , Sn ) := f (rS1 , . . . , rSn ) is in An ⊗ B(E). Moreover, applying the noncommutative Poisson transform P[rS1 ,...,rSn ] ⊗ id to H
the inequality Re f (S1 , . . . , Sn ) 0, we deduce that Re fr (S1 , . . . , Sn ) 0. Since A ≺ B, Theoc rem 1.1 shows that there exists c 1 such that Re fr (A1 , . . . , An ) c2 Re fr (B1 , . . . , Bn ). Due to the Fn∞ -functional calculus for pure row contractions (see [24]),
(4.1)
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f (A1 , . . . , An ) := SOT- lim fr (A1 , . . . , An ) r→1
and
f (B1 , . . . , Bn ) := SOT- lim fr (B1 , . . . , Bn ) r→1
exist. Consequently, taking the limit, as r → 1, in inequality (4.1), we get Re f (A1 , . . . , An ) c2 Re f (B1 , . . . , Bn ). Since the converse is obvious, the proof is complete.
2
Now we prove a Schwarz–Pick lemma for free holomorphic functions on [B(H)n ]1 with operator-valued coefficients, with respect to the hyperbolic metric. Theorem 4.2. Let Fj : [B(H)n ]1 → B(H) ⊗min B(E), j = 1, . . . , m, be free holomorphic functions with coefficients in B(E), and assume that F := (F1 , . . . , Fm ) is a contractive free holoH
morphic function. If X, Y ∈ [B(H)n ]1 , then F (X) ∼ F (Y ) and δ F (X), F (Y ) δ(X, Y ), where δ is the hyperbolic metric defined on the Harnack parts of the noncommutative ball [B(H)n ]− 1. Proof. Assume that each Fj has a representation of the form Fj (X1 , . . . , Xn ) =
∞
Xα ⊗ A(α,j ) ,
A(α,j ) ∈ B(E).
k=0 |α|=k
Since Fj is a bounded free holomorphic function, due to [30] (see also [33]), there exists ¯ fj (S1 , . . . , Sn ) ∈ Fn∞ ⊗B(E) such that Fj (X1 , . . . , Xn ) = (PX ⊗ id) fj (S1 , . . . , Sn ) ,
X := (X1 , . . . , Xn ) ∈ B(E)n 1 .
Moreover, fj (S1 , . . . , Sn ) = SOT- lim Fj (rS1 , . . . , rSn ) r→1
and Fj (rS1 , . . . , rSn ) ∈ An ⊗ B(E). Now, let L1 , . . . , Lm be the left creation operators on the full Fock space F 2 (Hm ) with m generators, and consider p(L1 , . . . , Lm ) =
Lα ⊗ M(α) ,
M(α) ∈ B Ck
|α|q
to be an arbitrary polynomial with Re p(L1 , . . . , Lm ) 0. Since F = (F1 , . . . , Fm ) is a contractive free holomorphic function, we deduce that (see [30]) f1 (S1 , . . . , Sn ), . . . , fm (S1 , . . . , Sn ) = F ∞ 1.
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Applying now the noncommutative Poisson transform P(f1 (S1 ,...,Sn ),...,fm (S1 ,...,Sn )) ⊗ id to the inequality Re p(L1 , . . . , Lm ) 0, we obtain Re p f1 (S1 , . . . , Sn ), . . . , fm (S1 , . . . , Sn ) 0.
(4.2)
¯ Note that p(f (S)) := p(f1 (S1 , . . . , Sn ), . . . , fm (S1 , . . . , Sn )) in is Fn∞ ⊗B(E) ⊗min B(Cm ). n Let X := (X1 , . . . , Xn ) and Y := (Y1 , . . . , Yn ) be in the open ball [B(H) ]1 . Due to TheoH
H
rem 1.6, we have X ∼ Y . Assume that X ∼ Y for some c 1. Due to (4.2) and Lemma 4.1, we c deduce that 1 Re p F (Y ) Re p F (X) c2 Re p F (Y ) . 2 c H
Consequently, F (X) ∼ F (Y ) and ω(F (X), F (Y )) c, where ω is defined by relation (2.1). This c
implies that ω(F (X), F (Y )) ω(X, Y ), which completes the proof.
2
We remark that the hyperbolic metric δ coincides with the Carathéodory type metric defined by cball (X, Y ) := sup δ F (X), F (Y ) , F
X, Y ∈ B(H)n 1 ,
where the supremum is taken over all free holomorphic functions F : [B(H)n ]1 → [B(H)n ]1 . Indeed, due to Theorem 4.2, we have cball (X, Y ) δ(X, Y ). Taking F = id, we also deduce that cball (X, Y ) δ(X, Y ), which proves our assertion. Corollary 4.3. Let F := (F1 , . . . , Fm ) be a contractive free holomorphic function with coeffiH
cients in B(E). If z, w ∈ Bn , then F (z) ∼ F (w) and δ F (z), F (w) δ(z, w). We remark that if m = n = 1 in Corollary 4.3, we obtain a very simple proof of Suciu’s result [38]. Note also that, in particular (if E = C), for any free holomorphic function F : [B(H)n ]1 → [B(H)m ]1 , we have δ F (X), F (Y ) δ(X, Y ), which extends the result mentioned at the beginning of this section. Corollary 4.4. If f ∈ H ∞ (D) is a contractive analytic function on the open unit disc and A, B ∈ H
B(H) are strict contractions, then f (A) ∼ f (B) and δ f (A), f (B) δ(A, B).
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A few remarks are necessary. Our hyperbolic metric δ is different from the Kobayashi distance δK on unit ball [B(H)n ]1 , with respect to the Poincaré distance on D. Indeed, when n = 1, one can show that δ coincides with the Harnack distance introduced by Suciu. In this case, according to [39] (and due to a result from [42]), we have δ(0, A) < δK (0, A) =
1 1 + A ln 2 1 − A
for certain strict contractions A ∈ B(H) with dim H 2. This also shows that δ is different from the metric for the ball [B(H)n ]1 , as defined in [11]. We define now a Kobayashi type pseudo-distance on domains M ⊂ B(H)n , n ∈ N, with respect to the hyperbolic metric δ of the ball [B(H)n ]1 , as follows. Given two points X, Y ∈ M, we consider a chain of free holomorphic balls from X to Y . That is, a chain of elements X = X0 , X1 , . . . , Xk = Y in M, pairs of elements A1 , B1 , . . . , Ak , Bk in [B(H)n ]1 , and free holomorphic functions F1 , . . . , Fk on [B(H)n ]1 with values in M such that Fj (Aj ) = Xj −1
and Fj (Bj ) = Xj
for j = 1, . . . , k.
Denote this chain by γ and define its length by (γ ) := δ(A1 , B1 ) + · · · + δ(Ak , Bk ), where δ is the hyperbolic metric on [B(H)n ]1 . We define M (X, Y ) := inf (γ ), δball
where the infimum is taken over all chains γ of free holomorphic balls from X to Y . If there is M (X, Y ) = ∞. In general, δ M is not a true distance on M. However, it no such chain, we set δball ball becomes a true distance in some special cases. It is well known that the Kobayashi distance on the open unit disc D coincides with the Poincaré metric. A similar result holds in our noncommutative setting. M is a true distance and δ M = δ. Proposition 4.5. If M = [B(H)n ]1 , then δball ball
Proof. If γ is a chain, as defined above, we use Theorem 4.2 and the fact that δ is a metric to deduce that δ(X, Y ) δ(X0 , X1 ) + δ(X1 , X2 ) + · · · + δ(Xk−1 , Xk ) = δ F1 (A1 ), F1 (B1 ) + δ F2 (A2 ), F2 (B2 ) + · · · + δ Fk (Ak ), Fk (Bk ) δ(A1 , B1 ) + δ(A2 , B2 ) + · · · + δ(Ak , Bk ) = (γ ). Taking the infimum over all chains γ of free holomorphic balls from X to Y , we deduce that M (X, Y ). Taking F the identity on [B(H)n ] , we obtain δ M (X, Y ) δ(X, Y ). 2 δ(X, Y ) δball 1 ball It would be interesting to find, as in the classical case, classes of noncommutative domains M M is a true distance. in B(H)n so that δball
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Acknowledgment We would like to thank the referee for useful comments on the results of the paper and for bringing to our attention several references. References [1] T. Ando, I. Suciu, D. Timotin, Characterization of some Harnack parts of contractions, J. Operator Theory 2 (2) (1979) 233–245. [2] S. Bergman, The Kernel Function and Conformal Mapping, Math. Surveys, vol. V, Amer. Math. Soc., Providence, RI, 1970, x+257 pp. [3] J.B. Conway, Functions of One Complex Variable. I, second ed., Grad. Texts in Math., vol. 159, Springer-Verlag, New York, 1995. [4] J. Cuntz, Simple C ∗ -algebras generated by isometries, Comm. Math. Phys. 57 (1977) 173–185. [5] K.R. Davidson, D. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras, Math. Ann. 311 (1998) 275–303. [6] K.R. Davidson, D. Pitts, Invariant subspaces and hyper-reflexivity for free semigroup algebras, Proc. Lond. Math. Soc. 78 (1999) 401–430. [7] E.G. Effros, Z.J. Ruan, Operator Spaces, London Math. Soc. Monogr. New Ser., vol. 23, Clarendon Press, Oxford Univ. Press, New York, 2000. [8] C. Foia¸s, On Harnack parts of contractions, Rev. Roumaine Math. Pures Appl. 19 (1974) 315–318. [9] J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. [10] A.M. Gleason, Function Algebras, Seminars on Analytic Functions, vol. 2, Institute for Advanced Study, Princeton, NJ, 1957. [11] L.A. Harris, Bounded symmetric homogeneous domains in infinite dimensional spaces, in: Proceedings on Infinite Dimensional Holomorphy, Internat. Conf., Univ. Kentucky, Lexington, KY, 1973, in: Lecture Notes in Math., vol. 364, Springer-Verlag, Berlin, 1974, pp. 13–40. [12] L.A. Harris, Schwarz–Pick systems of pseudometrics for domains in normed linear spaces, in: Advances in Holomorphy, Proc. Sem. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, in: North-Holland Math. Stud., vol. 34, North-Holland, Amsterdam, 1979, pp. 345–406. [13] L.A. Harris, Analytic invariants and the Schwarz–Pick inequality, Israel J. Math. 34 (3) (1979) 177–197. [14] K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, 1962. [15] S. Kobayashi, Hyperbolic Complex Spaces, Grundlehren Math. Wiss., vol. 318, Springer-Verlag, Berlin, 1998, xiv+471 pp. [16] S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, World Scientific Publishing Co., Hackensack, NJ, 2005, xii+148 pp. [17] S.G. Krantz, Geometric Function Theory, Explorations in Complex Analysis. Cornerstones, Birkhäuser Boston Inc., Boston, MA, 2006, xiv+314 pp. [18] V.I. Paulsen, Completely Bounded Maps and Dilations, Res. Notes Math., vol. 146, Pitman, New York, 1986. [19] R.S. Phillips, On symplectic mappings of contraction operators, Studia Math. 31 (1968) 15–27. [20] G. Pisier, Similarity Problems and Completely Bounded Maps, Lecture Notes in Math., vol. 1618, Springer-Verlag, New York, 1995. [21] G. Popescu, Multi-analytic operators and some factorization theorems, Indiana Univ. Math. J. 38 (1989) 693–710. [22] G. Popescu, Characteristic functions for infinite sequences of noncommuting operators, J. Operator Theory 22 (1989) 51–71. [23] G. Popescu, Von Neumann inequality for (B(H )n )1 , Math. Scand. 68 (1991) 292–304. [24] G. Popescu, Functional calculus for noncommuting operators, Michigan Math. J. 42 (1995) 345–356. [25] G. Popescu, Multi-analytic operators on Fock spaces, Math. Ann. 303 (1995) 31–46. [26] G. Popescu, Noncommutative disc algebras and their representations, Proc. Amer. Math. Soc. 124 (1996) 2137– 2148. [27] G. Popescu, Poisson transforms on some C ∗ -algebras generated by isometries, J. Funct. Anal. 161 (1999) 27–61. [28] G. Popescu, Curvature invariant for Hilbert modules over free semigroup algebras, Adv. Math. 158 (2001) 264–309. [29] G. Popescu, Operator theory on noncommutative varieties, Indiana Univ. Math. J. 55 (2) (2006) 389–442. [30] G. Popescu, Free holomorphic functions on the unit ball of B(H)n , J. Funct. Anal. 241 (2006) 268–333. [31] G. Popescu, Free holomorphic functions and interpolation, Math. Ann. 342 (2008) 1–30.
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Journal of Functional Analysis 256 (2009) 4071–4094 www.elsevier.com/locate/jfa
Rearrangement invariance of Rademacher multiplicator spaces Serguei V. Astashkin a , Guillermo P. Curbera b,∗,1 a Department of Mathematics, Samara State University, ul. Akad. Pavlova 1, 443011 Samara, Russia b Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, Sevilla 41080, Spain
Received 6 October 2008; accepted 23 December 2008 Available online 19 January 2009 Communicated by N. Kalton
Abstract Let X be a rearrangement invariant function space on [0, 1]. We consider the Rademacher multiplicator space Λ(R, X) of all measurable functions x such that x · h ∈ X for every a.e. converging series h = an rn ∈ X, where (rn ) are the Rademacher functions. We study the situation when Λ(R, X) is a rearrangement invariant space different from L∞ . Particular attention is given to the case when X is an interpolation space between the Lorentz space Λ(ϕ) and the Marcinkiewicz space M(ϕ). Consequences are derived regarding the behaviour of partial sums and tails of Rademacher series in function spaces. © 2009 Elsevier Inc. All rights reserved. Keywords: Rademacher functions; Rearrangement invariant spaces
Introduction In this paper we study the behaviour of the Rademacher functions (rn ) in function spaces. Let R denote the set of all functions of the form an rn , where the series converges a.e. For a rearrangement invariant (r.i.) space X on [0, 1], let R(X) be the closed linear subspace of X given by R ∩ X. The Rademacher multiplicator space Λ(R, X) of all space of X is the measurable functions x : [0, 1] → R such that x an rn ∈ X, for every an rn ∈ R(X). It is a * Corresponding author.
E-mail addresses: [email protected] (S.V. Astashkin), [email protected] (G.P. Curbera). 1 Partially supported D.G.I. #BFM2003-06335-C03-01 (Spain).
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.12.021
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Banach function space on [0, 1] when endowed with the norm an rn xΛ(R,X) = sup x an rn ∈ X, : X
an rn 1 . X
The space Λ(R, X) can be viewed as the space of operators from R(X) into the whole space X given by multiplication by a measurable function. The Rademacher multiplicator space Λ(R, X) was firstly considered in [8] where it was shown that for a broad class of classical r.i. spaces X the space Λ(R, X) is not r.i. This result was extended in [2] to include all r.i. spaces such that the lower dilation index γϕX of their fundamental function ϕX satisfies γϕX > 0. This result motivated the study the symmetric kernel Sym(R, X) of the space Λ(R, X), that is, the largest r.i. space embedded into Λ(R, X). The space Sym(R, X) was studied in [2], where it was shown that, if X is an r.i. space satisfying the Fatou property and X ⊃ LN , where LN is the Orlicz space with N (t) = exp(t 2 ) − 1, then Sym(R, X) is the r.i. space with the norm x := x ∗ (t) log1/2 (2/t)X . It was also shown that any space X which has the Fatou property and is an interpolation space for the couple (L log1/2 L, L∞ ) can be realized as the symmetric kernel of a certain r.i. space. The opposite situation is when the Rademacher multiplicator space Λ(R, X) is r.i. The simplest case of this situation is when Λ(R, X) = L∞ . In [1] it was shown that Λ(R, X) = L∞ holds for all r.i. spaces X which are interpolation spaces for the couple (L∞ , LN ). It was shown in [3] that Λ(R, X) = L∞ if and only if the function log1/2 (2/t) does not belong to the closure of L∞ in X. In this paper we investigate the case when the Rademacher multiplicator space Λ(R, X) is an r.i. space different from L∞ . Examples of this situation were considered in [2,8,9]. In all cases they were spaces X consisting of functions with exponential growth. The paper is organized as follows. Section 1 is devoted to the preliminaries. In Section 2 we study technical conditions on an r.i. space X and its fundamental function ϕ. In Section 3 we present a sufficient condition for Λ(R, X) being r.i. (Theorem 3.4). For this, two results are needed. Firstly, that the symmetric kernel Sym(R, X) is a maximal space (Proposition 3.1), and secondly, a condition, of independent interest, on the behaviour of logarithmic functions on an r.i. space (Proposition 3.3). Section 4 is devoted to the study of necessary conditions for Λ(R, X) being an r.i. space. This is done by separately studying conditions on partial sums and tails of Rademacher series (Propositions 4.1 and 4.2). Theorem 4.4 addresses the case when X in an interpolation space for the couple (Λ(ϕ), M(ϕ)), where Λ(ϕ) and M(ϕ) are, respectively, the Lorentz and Marcinkiewicz spaces with the fundamental function ϕ. Theorem 4.5 specializes the previous result for the case of X = M(ϕ). We end presenting, in Section 5, examples which highlight certain features of the previous results. 1. Preliminaries Throughout the paper a rearrangement invariant (r.i.) space X is a Banach space of classes of measurable functions on [0, 1] such that if y ∗ x ∗ and x ∈ X then y ∈ X and yX xX . Here x ∗ is the decreasing rearrangement of x, that is, the right continuous inverse of its distribution function: nx (τ ) = λ{t ∈ [0, 1]: |x(t)| > τ }, where λ is the Lebesgue measure on [0, 1]. Functions x and y are said to be equimeasurable if nx (τ ) = ny (τ ), for all τ > 0. The associated 1 space (or Köthe dual) of X is the space X of all functions y such that 0 |x(t)y(t)| dt < ∞, for every x ∈ X. It is an r.i. space. The space X is a subspace of the topological dual X ∗ . If X is a
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norming subspace of X ∗ , then X is isometric to a subspace of the space X = (X ) . The space X is maximal when X = X . We denote by X0 the closure of L∞ in X. If X is not L∞ , then X0 coincides with the absolutely continuous part of X, that is, the set of all functions x ∈ X such that limλ(A)→0 xχA X = 0. Here and next, χA is the characteristic function of the set A ⊂ [0, 1]. The fundamental function of X is the function ϕX (t) := χ[0,t] X . Important examples of r.i. spaces are Marcinkiewicz, Lorentz and Orlicz spaces. Let ϕ : [0, 1] → [0, +∞) be a quasi-concave function, that is, ϕ increases, ϕ(t)/t decreases and ϕ(0) = 0. The Marcinkiewicz space M(ϕ) is the space of all measurable functions x on [0, 1] for which the norm ϕ(t) 0
t
xM(ϕ) = sup
x ∗ (s) ds < ∞.
0
If ϕ : [0, 1] → [0, +∞) is an increasing concave function, ϕ(0) = 0, then the Lorentz space Λ(ϕ) consists of all measurable functions x on [0, 1] such that 1 xΛ(ϕ) =
x ∗ (s) dϕ(s) < ∞.
0
Let M be an Orlicz function, that is, an increasing convex function on [0, ∞) with M(0) = 0. The norm of the Orlicz space LM is defined as follows
1
xLM = inf λ > 0:
|x(s)| M ds 1 . λ
0
The fundamental functions of these spaces are ϕM(ϕ) (t) = ϕΛ(ϕ) (t) = ϕ(t), and ϕLM (t) = 1/M −1 (1/t), respectively. The Marcinkiewicz M(ϕ) and Lorentz Λ(ϕ) spaces are, respectively, the largest and the smallest r.i. spaces with fundamental function ϕ, that is, if the fundamental function of an r.i. space X is equal to ϕ, then Λ(ϕ) ⊂ X ⊂ M(ϕ). If ψ is a positive function defined on [0, 1], then its lower dilation index is γψ := lim
log(sup 0<s1
t→0+
ψ(st) ψ(s) )
log t
,
and its upper dilation index is δψ := lim
log(sup0<s1/t
ψ(st) ψ(s) )
log t
t→+∞
.
If a quasi-concave function ϕ satisfies δϕ < 1, then we have the following equivalence for the norm in the Marcinkiewicz space M(ϕ) xM(ϕ) sup ϕ(t)x ∗ (t), 0
(1.1)
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[11, Theorem II.5.3]. Here, and throughout the paper, A B means that there exist constants C > 0 and c > 0 such that c · A B C · A. For each t > 0, the dilation operator σt x(s) := x(s/t)χ[0,1] (s/t), s ∈ [0, 1], is bounded on any r.i. space X. Given Banach spaces X0 and X1 continuously embedded in a common Hausdorff topological vector space, a Banach space X is an interpolation space with respect to the couple (X0 , X1 ) if X0 ∩ X1 ⊂ X ⊂ X0 + X1 and for every linear operator T with T : Xi → Xi (i = 0, 1) continuously, we have T : X → X. 1. We have already deThe Rademacher functions are rn (t) := sign sin(2n πt), t ∈ [0, 1], n fined R(X) := R ∩ X where R is the set of all a.e. converging series an rn , that is, (an ) ∈ 2 [15, Theorem V.8.2]. For X = Lp , 1 p < ∞, Khintchin inequality shows that R(X) is isomorphic to 2 . If X = L∞ , then R(X) = 1 . The Orlicz space LN , for N (t) = exp(t 2 ) − 1, will be of major importance in our study. A result of Rodin and Semenov shows that R(X) ≈ 2 if and only if (LN )0 ⊂ X, [14]. Hence, for spaces X satisfying this condition we have an rn (1.2) (an ) 2 . X
The fundamental function of LN is (equivalent to) ϕ(t) = log−1/2 (2/t). Since N (t) increases very rapidly, LN coincides with the Marcinkiewicz space with fundamental function ϕ, [13]. This, together with δϕ = 0 < 1, gives xLN sup x ∗ (t) log−1/2 (2/t). 0
In particular, for every 0 < t 1 we have x ∗ (t) CxLN log1/2 (2/t).
(1.3)
Hence, for an r.i. space X, LN ⊂ X is equivalent to log1/2 (2/t) ∈ X. k n We denote the dyadic intervals of [0, 1] by Δkn := [ k−1 2n , 2n ) for n ∈ N and k = 1, . . . , 2 . The set of all dyadic step functions is D = n Dn , where, for n ∈ N, Dn is the set of all dyadic step functions of order n: n
f (t) =
2
ck χΔkn (t),
ck ∈ R.
k=1
For convenience of computations, all logarithms will be considered with base 2. For any undefined notion regarding function spaces, r.i. spaces, and interpolation of linear operators, we refer the reader to the monographs [6,7,11,12]. 2. Conditions on r.i. spaces In this section we collect together the conditions and results of technical nature on an r.i. space X and its fundamental function ϕ that will be needed in the following sections. Definition 2.1. Let X be an r.i. space on [0, 1] and ϕ be a function defined on [0, 1].
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(a) The operator Q is defined, over adequate functions x(t) on [0, 1], as follows Qx(t) := x t 2 ,
0 t 1.
(b) ϕ satisfies the Δ2 -condition if it is nonnegative, increasing, concave, and there exists C > 0 such that ϕ(t) C · ϕ t 2 ,
0 < t 1.
(2.1)
(c) X satisfies the log-condition if there exists C > 0 such that, for all u ∈ (0, 1),
1/2 2 1/2 2u log log χ χ C (0,u] (0,u] . t t X X
(2.2)
(d) If ϕ is increasing and quasi-concave, we define the function ϕ(t) ¯ := t −1/2 ϕ 21−1/t , (e) ϕ(t) satisfies the that
0 < t 1.
√ 2-condition if it is increasing, quasi-concave, and there exists n0 ∈ N such
cϕ := sup nn0
1 ϕ(2−2n ) √ . ϕ(2−n ) 2
(2.3)
The following proposition is rather elementary. However, since we will refer next to it many times, we include the proofs of the less trivial implications (3) and (4). Proposition 2.2. Let X be an r.i. space on [0, 1] with fundamental function ϕ(t). (1) If Q is bounded on X, then ϕ ∈ Δ2 . (2) If ϕ ∈ Δ2 and X is an interpolation space for the couple (Λ(ϕ), M(ϕ)), then Q is bounded on X. (3) If log1/2 (2/t) ∈ X and ϕ ∈ Δ2 , then X satisfies the log-condition. (4) If the lower dilation index of ϕ¯ satisfies γϕ¯ > 0, then ∞
C0 := sup n=1,2,...
√
ϕ(2−k ) 1 < ∞. √ −n nϕ(2 ) k k=n
1/2 (5) If the function √ log (2/t)ϕ(t) is increasing on some interval (0, t0 ), for t0 > 0, then ϕ satisfies the 2-condition.
Proof. (3) For 0 < u 1, we have
1/2 2 1/2 2 1/2 2 log χ(0,u] log χ(0,u2 ] + log χ(u2 ,u] . t t t X
X
X
(2.4)
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√ If 0 < t u2 , then t t/u. Therefore,
√ 1/2 2 √ 1/2 2u 1/2 2 log χ(0,u2 ] 2log χ(0,u] √ χ(0,u2 ] 2log . t t t X X X By (2.1),
√ √ 1/2 2 log 2 log1/2 2 χ 2 X 2C log1/2 2 ϕ u2 χ 2 (u ,u] (u ,u] t u u X
√ 2 √ 1/2 2u 1/2 2u ϕ u 2C log χ(0,u] = 2C log . 2 t u X The last two formulas and (2.4) imply the log-condition (2.2). (4) If γϕ¯ > 0, then there are δ > 0 and C > 0 such that for all 0 < u 1 and t 1 we have ¯ that is, ϕ(tu) ¯ Ct δ ϕ(u), ϕ 21−1/(tu) Ct δ+1/2 ϕ 21−1/u . Fix n ∈ N, set u = 1/n, and apply the previous inequality with t = 2−j , where j = 0, 1, 2, . . . . This, together with the quasi-concavity of ϕ, implies that j ϕ 2−2 n C2−(δ+1/2)j ϕ 2−n ,
j = 0, 1, 2, . . . .
Therefore, j +1
∞ ∞ 2 n−1 ∞ ϕ(2−k ) ϕ(2−k ) j/2 √ −2j n 2 nϕ 2 = √ √ k k j k=n j =0 j =0 k=2 n
∞ 2δ √ √ . 2−(δ+1/2)j 2j/2 = C nϕ 2−n δ C nϕ 2−n 2 −1
2
j =0
Remark 2.3. (a) Definition 2.1(b) was introduced in [4] in connection with extrapolation of operators in r.i. spaces. (b) Due to ϕ being concave and increasing, the Δ2 -condition (2.1) is equivalent to its discrete analog: there exist β > 1 and C > 0 such that ϕ 2−n Cϕ 2−βn ,
n ∈ N.
(2.5)
(c) The log-condition (2.2) is equivalent to its discrete analog: there exists C > 0 such that, for all n ∈ N,
1/2 2 1/2 2 log χ(0,2−n ] C log χ(0,2−n ] (2.6) . nt t 2 X X (d) The same proof as of Proposition 2.2(3) shows that if inequality (2.1) holds for u ∈ E, for some E ⊂ [0, 1], then inequality (2.2) also holds for u ∈ E.
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(e) Note that the condition γϕ¯ > 0 implies that ϕ(t) C log−1/2−δ (2/t) (0 < t 1) with some C > 0 and δ > 0. Therefore, log1/2 (2/t) ∈ Λ(ϕ). The condition γϕ¯ > 0 means, roughly speaking, that the distance from ϕ to the fundamental function log−1/2 (2/t) of the space LN is sufficiently large. 3. Sufficient conditions for Λ(R, X) being an r.i. space We begin with a sharpening of results of §2 of [2], that will be needed for the main result of this section. Proposition 3.1. Let X be an r.i. space on [0, 1]. Then Sym(R, X) = Sym(R, X ). In particular, Sym(R, X) is a maximal r.i. space. Proof. First, suppose log1/2 (2/t) ∈ / X0 . Since (X )0 = X0 , then log1/2 (2/t) ∈ / (X )0 . From [2, Theorem 3.2], Sym(R, X) = Sym(R, X ) = L∞ . Suppose now that log1/2 (2/t) ∈ X0 . Since X ⊂ X , then Sym(R, X) ⊂ Sym(R, X ), [2, Corollary 3.4]. Thus, we only have to prove that Sym(X ) ⊂ Sym(R, X). By [2, Corollary 2.11], we have xSym(R,X ) x ∗ (t) log1/2 (2/t)X . 1/2 ∗ Therefore, if x ∈ Sym(R, X ) then x (t) log (2/t) ∈ X . Let a = (ak ) ∈ 2 . It is well known that the function xa := ak rk ∈ (LN )0 , which implies, by [11, Lemma II.5.4], that
−1/2
lim log
t→0+
t 2 1 xa∗ (s) ds = 0, t t 0
whence lim xa∗ (t) log−1/2 t→0+
2 = 0. t
For 0 < h 1 and 0 < t 1, we have x ∗ (t)xa∗ (t)χ[0,h] (t) sup xa∗ (s) log−1/2 (2/s) · x ∗ (t) log1/2 (2/t), 0<sh
whence ∗ x (t)x ∗ (t)χ[0,h] (t) a
X
sup xa∗ (s) log−1/2 (2/s) · x ∗ (t) log1/2 (2/t)X . 0<sh
Since x ∗ (t) log1/2 (2/t) ∈ X , by (3.1), ∗ x (t)x ∗ (t)χ[0,h] (t) a
X
→0
as h → 0+ .
(3.1)
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This means that x ∗ (t)xa∗ (t) ∈ (X )0 = X0 . Since this holds for every a ∈ 2 , then [2, Corollary 2.5] implies that x ∈ Sym(R, X0 ) ⊂ Sym(R, X). The maximality of Sym(R, X) follows from [2, Proposition 2.6]. 2 The following corollary is a straightforward consequence of Proposition 3.1 and [2, Corollary 2.11]. Corollary 3.2. For every r.i. space X on [0, 1] such that log1/2 (2/t) ∈ X xSym(R,X) x ∗ (t) log1/2 (2/t)X . The next result gives a useful sufficient condition for the rearrangement invariance of Λ(R, X). Proposition 3.3. Let X be an r.i. space on [0, 1] such that log1/2 (2/t) ∈ X0 . Suppose there exists A > 0 such that for n ∈ N 2n 2n
2 2 ck χΔkn · log1/2 ck χΔkn · log1/2 n A , t 2 t +1−k k=1
k=1
X
(3.2)
X
for every c1 c2 · · · c2n 0. Then Λ(R, X) is an r.i. space. Proof. Since Sym(R, X) ⊆ Λ(R, X), we just have to show the opposite inclusion. n Let f ∈ Dn , that is, f = 2k=1 ck χΔkn , with ck ∈ R. Since Sym(R, X) is r.i., we can assume n that the sequence (ck )2k=1 is nonnegative and decreasing. In this case, by Corollary 3.2, 2n
1/2 2 f Sym(R,X) ck χΔkn · log t
.
(3.3)
X
k=1
On the other hand, by the definition of the norm in Λ(R, X), ∞ ∞ f Λ(R,X) sup f · ak rk : ak rk 1 . k=n+1
X
k=n+1
X
Since log1/2 (2/t) ∈ X0 , we have R(X) ≈ 2 so, by (1.2), for all n, m ∈ N, n+m 1 √ rk C1 . m k=n+1
X
Hence, f Λ(R,X) C1−1
n+m 1 sup f · √ rk . m m∈N k=n+1
X
(3.4)
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Note that, for n ∈ N and k = 1, . . . , 2n , we have m ∗ ∗ n+m n 2 t , ri (t) = ri χΔkn · i=n+1
4079
0 t 2−n .
i=1
Therefore, since X is r.i., we get 2n n+m 1 ri = ck χΔkn · f · √ m i=n+1 k=1 X 2n = ck χΔkn · k=1
n+m 1 ri √ m i=n+1
1 √ m
m
X
∗
ri
i=1
n 2 t +1−k . X
Using the central limit theorem (see [14] or [12, Theorem 2.b.4]), we have ∗
m 1 2 2 1/2 , 0
(3.5)
i=1
This and inequality (3.4) imply that 2n
2 ck χΔkn · log1/2 n f Λ(R,X) M 2 t +1−k k=1
. X
By assumption, log1/2 (2/t) ∈ X0 . Therefore, since (X )0 = X0 isometrically, we can change the norm of X by the norm of X, i.e., we have 2n
2 1/2 f Λ(R,X) M ck χΔkn · log . 2n t + 1 − k k=1
X
This inequality, (3.3), and (3.2) imply that for f ∈ D we have f Sym(R,X) Cf Λ(R,X) .
(3.6)
Next, we extend inequality (3.6) to L∞ . Since log1/2 (2/t) ∈ X0 , we have Sym(R, X) = L∞ [2, Theorem 3.2]. In particular, the fundamental function of Sym(R, X) tends to 0 as t → 0+ . By Lusin’s theorem, we deduce that for every g ∈ L∞ there is a sequence {fn } ⊂ D such that fn − gSym(R,X) → 0. The embedding Sym(R, X) ⊂ Λ(R, X) implies that fn − gΛ(R,X) → 0. By (3.6), we have that fn Sym(R,X) Cfn Λ(R,X) . Consequently gSym(R,X) CgΛ(R,X) ,
g ∈ L∞ .
(3.7)
Let h ∈ Λ(R, X) be a nonnegative function. There exists a sequence {gn } ⊂ L∞ such that 0 gn ↑ h a.e. on [0, 1]. Then, (3.7) implies that gn Sym(R,X) Cgn Λ(R,X) ChΛ(R,X) .
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Since, by Proposition 3.1, Sym(R, X) is maximal, we deduce that h ∈ Sym(R, X) and hSym(R,X) ChΛ(R,X) . 2 We can now prove the main result of this section. Theorem 3.4. If X is an r.i. space on [0, 1] such that the operator Qx(s) = x(s 2 ) is bounded in X, then Λ(R, X) = Sym(R, X). / X0 then, by [3], Λ(R, X) = Sym(R, X) = L∞ . Thus, Proof. First we note that if log1/2 (2/t) ∈ we may assume that log1/2 (2/t) ∈ X0 . In view of Proposition 3.3, we just have to establish (3.2). Denote n
g(t) =
2
ck χΔkn · log
1/2
k=1
2n 2 2 ck χΔkn · log1/2 n and h(t) = , t 2 t +1−k k=1
where c1 c2 · · · c2n 0. We write g in the form g = g1 + g2 , g1 (t) = c1 χΔ1n (t) · log1/2 (2/t)
and g2 (t) = g(t) − g1 (t).
(3.8)
Since Q is bounded in X, by Proposition 2.2(1), (3), we have
2 1/2 ChX . χ · log g1 X Cc1 Δ1n 2 n t X
(3.9)
For estimating g2 X , we consider the function g¯ 2 (t) :=
n
j
2
j =1 i=2j −1 +1
ci χΔin (t) log1/2
. j −n 2
2
It is easily seen that g2 (t) g¯ 2 (t),
0 < t 1.
(3.10)
√ Let Q−1 be the inverse operator to Q, i.e., Q−1 x(t) := x( t). A straightforward calculation shows that −1
Q
g¯ 2 (t) =
n
j
2
j =1 i=2j −1 +1
ci χ
2
[ (i−1) 2n , 2
i2 ] 22n
(t) log
. j −n 2
1/2
2
Thus, in order to prove the inequality −1 ∗ Q g¯ 2 (t) h∗ (t),
0 < t 1,
(3.11)
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it suffices to show that, for every j = 1, 2, . . . , n and i satisfying 2j −1 < i 2j , we have
i 2 2k − 1 i2 λ t ∈ [0, 1]: h(t) ci log1/2 j −n = . 2 22n 22n k=1
Indeed, let k = 1, 2, . . . , i. Since c1 c2 · · · c2n 0, we have
2 λ t ∈ Δkn : h(t) ci log1/2 j −n 2
−n 2 2 1/2 1/2 log λ t ∈ 0, 2 : log 2n t 2j −n = 2j −2n . Hence,
2 i · 2j −2n i 2 · 2−2n . λ t ∈ [0, 1]: h(t) ci log1/2 j −n 2 So, inequality (3.11) is proved. By (3.10) and (3.11) we get g2 X Q Q−1 g¯ 2 X QQ−1 g¯ 2 X QhX . This inequality together with (3.9) yields (3.2) so, the result is proved.
2
Combining Theorem 3.4 and Proposition 2.2(2) we have the following. Corollary 3.5. If X is an interpolation space for the couple (Λ(ϕ), M(ϕ)) and ϕ ∈ Δ2 , then Λ(R, X) = Sym(R, X). In particular, X may be the Lorentz space Λ(ϕ) or the Marcinkiewicz space M(ϕ). Remark 3.6. Boundedness of the operator Q in X is not a necessary condition for the equality Λ(R, X) = Sym(R, X). Indeed, for every increasing concave function ϕ ∈ Δ2 there is an r.i. space X such that Q is not bounded on X [5, Example 2.12]. Actually, X is a subspace of the Marcinkiewicz space M(ϕ) and therefore X = M(ϕ). Then, Proposition 3.1, Corollary 3.5, and [3, Corollary 3] imply that Sym(R, X) = Sym(R, X ) = Sym R, M(ϕ) = Λ R, M(ϕ) ⊃ Λ(R, X). This, together with Λ(R, X) ⊃ Sym(R, X), gives Λ(R, X) = Sym(R, X). Remark 3.7. The proofs of Proposition 3.3 and Theorem 3.4, and [3, Remark 5] show that the following tail-assertion holds: if X is an r.i. space on [0, 1] such that the operator Q is bounded on X, then there exists A > 0 such that for every f ∈ Dn we have ∞ ∞ ci ri : ci ri 1 . f Sym(R,X) A sup f · i=n+1
X
i=n+1
X
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4. Necessary conditions for Λ(R, X) being an r.i. space Now we pass to the consideration of the opposite problem: finding necessary conditions for an r.i. space X satisfying Λ(R, X) = Sym(R, X). We require some specific sets formed by unions of dyadic intervals. Consider the matrix A = (θi,j ), where θi,j is the value of the function rj on the interval Δi2n , where n ∈ N, j = 1, 2, . . . , 2n, and i = 1, 2, . . . , 22n . Let Ωn ⊂ {1, 2, . . . , 22n } be the set of all rearrangements of signs on {1, 2, . . . , 2n } such that for every i ∈ Ωn we have θi,j +n = θi,j ,
j = 1, 2, . . . , n.
Denote by A(Ωn ) the submatrix of A corresponding to the set Ωn . We will consider A(Ωn ) as 2n an operator acting from 2n 2 into 2 . Define Un =
Δi2n .
i∈Ωn
Since card Ωn = 2n , then λ(Un ) = 2−n . Now, we prove a first result assuming certain tail-estimates. Proposition 4.1. Suppose that X is an r.i. space on [0, 1] with fundamental function ϕ such that γϕ¯ > 0. Let I ⊂ N be such that for every n ∈ I we have ∞ ∞ χ[0,2−n ] Sym(R,X) A sup χ[0,2−n ] · ci ri : ci ri 1 , i=n+1
X
i=n+1
X
where A > 0 does not depend on n. Then, there exist β > 1 and C > 0 such that, for all n ∈ I , we have (2.5), that is, ϕ 2−n Cϕ 2−βn , n ∈ I. Proof. Note that, by Remark 2.3(e), log1/2 (2/t) ∈ X0 . We only have to consider the case when I is infinite. In view of the hypothesis and arguing as in the proof of Proposition 3.3, we get
1/2 2 1/2 2 log χ[0,2−n ] C1 log χ[0,2−n ] , n ∈ I. n t 2 t X X Since Λ(ϕ) ⊂ X ⊂ M(ϕ) [11, Theorems 2.5.5, 2.5.7], then
1/2 2 1/2 2 log −n −n log χ χ C , 1 [0,2 ] [0,2 ] n t 2 t M(ϕ) Λ(ϕ)
n ∈ I.
The right-hand side of (4.1) is equal to 2−k ∞ k=n
2−k−1
1/2
log
2 2n t
dϕ(t) 2
∞ k=n
log
−k ϕ 2 − ϕ 2−k−1 . n−k 2
1/2
2
(4.1)
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For j > n, by applying the Abel transformation, we have j
log1/2
k=n
=
−k ϕ 2 − ϕ 2−k−1 n−k 2
2
j −1 √ √ ϕ 2−k−1 ( k + 2 − n − k + 1 − n ) + ϕ 2−n − ϕ 2−j −1 j + 1 − n k=n
j ϕ(2−k ) + ϕ 2−n . √ k−n k=n+1
Therefore,
1/2 2 log χ[0,2−n ] n 2 t
2C1 ·
Λ(ϕ)
∞ −n ϕ(2−k ) . +ϕ 2 √ k−n k=n+1
On the other hand,
1/2 2 log −n χ [0,2 ] t
√ −n nϕ 2 .
M(ϕ)
Thus (4.1) implies that √
nϕ 2−n 2C1
∞ −n ϕ(2−k ) . +ϕ 2 √ k−n k=n+1
Hence, if n ∈ I is large enough, we have ∞ √ −n ϕ(2−k ) 3C1 nϕ 2 . √ k−n k=n+1
(4.2)
Let ε > 0 to be chosen later. Then [(1+ε)n] k=n+1
ϕ(2−k ) ϕ 2−n √ k−n
[(1+ε)n]−n k=1
1 √ k
(1 + ε)n − n 2ϕ 2−n √ √ 2 εϕ 2−n n.
(4.3)
Since γϕ¯ > 0, by Proposition 2.2(4), we have ∞ ∞ √ ϕ(2−k ) √ ϕ(2−k ) 2 2C0 ϕ 2−2n n. √ √ k−n k k=2n k=2n
(4.4)
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We choose ε so that 0 < ε (12C )−2 . Then inequalities (4.2)–(4.4) imply that √ −n 6C1 nϕ 2
−2n √ ϕ(2−k ) + 2C0 ϕ 2 n . √ k−n k=[(1+ε)n]+1 2n−1
Since 2n−1
1 √ ϕ(2−k ) ϕ 2−[(1+ε)n] √ √ 2ϕ 2−[(1+ε)n] n, k−n k k=1 k=[(1+ε)n]+1 n
we then get √ −n √ 12C1 max(1, C0 ) nϕ 2−[(1+ε)n] nϕ 2 or ϕ 2−n Cϕ 2−[(1+ε)n] . Choosing β ∈ (1, 1 + ε), we have that [(1 + ε)n] > βn for large enough n ∈ N. Therefore, inequality (2.5) holds for large enough n ∈ I for such β. Changing C, if necessary, we obtain (2.5) for all n ∈ I . 2 We now assume certain head-estimates. This requires the intervals [0, 2−n ] to be replaced with the sets Un defined previously (see Example 5.1 below). Proposition 4.2. Suppose that X is an r.i. space on [0, 1] such that log1/2 (2/t) ∈ X0 . Let I ⊂ N be such that, for every n ∈ I , 2n 2n χUn Sym(R,X) A sup χUn · ci ri : ci ri 1 , j =1
X
j =1
X
where A > 0 does not depend on n. Then, there exist β > 1 and C > 0 such that, for all n ∈ I , we have (2.5), that is, ϕ 2−n Cϕ 2−βn , n ∈ I. Proof. Since log1/2 (2/t) ∈ X0 , we have R(X) ≈ 2 . Thus 2n 2n 2n ci ri : ci ri 1 sup χUn · ci ri : (cj ) 2 1 . sup χUn · j =1
X
j =1
j =1
X
X
i For c = (cj )2n j =1 and θi,j the value of rj on Δ2n , we have
χUn ·
2n j =1
cj rj =
i∈Ωn
2n j =1
cj θi,j
· χΔin =
i∈Ωn
bi χΔin ,
(4.5)
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4085
where bi := (A(Ωn )c)i for i ∈ Ωn . Then, if (cj )2 1, 2n √ √ |bi | = cj θi,j (cj )2 2n 2n,
i ∈ Ωn .
(4.6)
j =1
Moreover, the choice of Ωn implies that bi =
2n
cj θi,j =
j =1
n (cj + cj +n )εji ,
i ∈ Ωn ,
j =1
for some rearrangement of signs (εji )nj=1 . Note that, since (cj )2 1,
n
1/2 (cj + cj +n )
2
√ 2.
(4.7)
j =1
Given ε > 0, to be chosen later, consider the set √ Bn := k ∈ Ωn : |bk | ε n . We estimate the size of Bn . For this, we use the exponential estimate of Rademacher sums, [10, Theorem II.2.5], and (4.7): n √ card Bn = card εk = ±1: (ck + ck+n )εk ε n k=1 n √ n = 2 λ t ∈ [0, 1]: (ck + ck+n )rk (t) ε n k=1
n 1/2 n ε √n n 2 2 λ t ∈ [0, 1]: (ck + ck+n )rk (t) √ (cj + cj +n ) 2 j =1 k=1
2 · 2n · e−ε
2 n/8
,
whence 2
card Bn 2 · 2n(1−αε ) ,
for α = (log2 e)/8.
Using this estimate, equality (4.5), and (4.6), we have, for the fundamental function ϕ of X, 2n cj rj bi χΔin + bi χΔin χUn · X X j =1
X
i∈Bn
i∈Ωn \Bn
√ √ 2n · χ[0,card Bn ·2−2n ] X + ε nχUn X √ √ 2 2n · ϕ 2 · 2−n(1+αε ) + ε nϕ 2−n .
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Consequently, 2n 2n √ 2 sup χUn · ci ri : ci ri 1 C1 n ϕ 2 · 2−n(1+αε ) + εϕ 2−n . j =1
j =1
X
X
On the other hand, since log1/2 (2/t) ∈ X0 , then by [2, Theorem 2.8] √ χUn Sym(R,X) χ[0,2−n ] log1/2 (2/t)X nϕ 2−n ,
n ∈ N.
From the last two inequalities and the hypothesis it follows that, for every n ∈ I , 2 ϕ 2−n C2 ϕ 2 · 2−n(1+αε ) + εϕ 2−n . For ε > 0 small enough, we get ϕ 2−n C3 ϕ 2−βn , where C3 > 0 does not depend on n ∈ I and β = 1 + αε 2 > 1.
2
Combining Propositions 4.1 and 4.2 we obtain the following result. Theorem 4.3. Let X be an r.i. space on [0, 1] such that log1/2 (2/t) ∈ X0 . Suppose that there exists A > 0 such that, for every n ∈ N, χUn Sym(R,X) AχUn Λ(R,X) .
(4.8)
Then X satisfies the discrete log-condition (2.6), that is, there exists C > 0 such that for all n ∈ N
1/2 2 1/2 2 log χ(0,2−n ] C log χ(0,2−n ] . n t 2 t X X If, in addition, γϕ¯ > 0, where ϕ is the fundamental function of X, then ϕ ∈ Δ2 . Proof. Denote by I1 the set of all n ∈ N such that ∞ ∞ χUn Sym(R,X) 2A sup χUn · ci ri : ci ri 1 . i=2n+1
X
i=2n+1
X
The definition of Λ(R, X) and the properties of Rademacher functions yield 2n 2n χUn Λ(R,X) sup χUn · ci ri : ci ri 1 i=1 i=1 X X ∞ ∞ + sup χUn · ci ri : ci ri 1 . i=2n+1
X
i=2n+1
X
(4.9)
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Set I2 := N \ I1 . By (4.8) and (4.9) we have, for every n ∈ I2 , 2n 2n χUn Sym(R,X) 2A sup χUn · ci ri : ci ri 1 . i=1
i=1
X
4087
(4.10)
X
It is clear that χUn Sym(R,X) = χ[0,2−n ] Sym(R,X) and ∞ ∞ ci ri : ci ri 1 sup χUn · i=2n+1 i=2n+1 X X ∞ ∞ = sup χ[0,2−n ] · ci ri : ci ri 1 , i=n+1
since the functions χUn · (4.9) is equivalent to
∞
i=2n+1 ci ri
X
and χ[0,2−n ] ·
i=n+1
X
∞
i=n+1 ci ri
are equimeasurable. Therefore,
∞ ∞ χ[0,2−n ] Sym(R,X) 2A sup χ[0,2−n ] · ci ri : ci ri 1 , i=n+1
X
i=n+1
that is (as in the proof of Proposition 3.3), to the inequality
1/2 2 1/2 2 log −n −n log χ χ C (0,2 ] (0,2 ] , n t 2 t X X
(4.11)
X
n ∈ I1 .
By Proposition 4.2, inequality (4.10) implies that there exist β1 > 1 and C1 > 0 such that inequality (2.5) holds for n ∈ I2 . This, together with Proposition 2.2(3) and Remark 2.3(d), give (2.6) for n ∈ I2 . Consequently, (2.6) holds for all n ∈ N. If γϕ¯ > 0 then, by (4.11) and Proposition 4.1, we have (2.5) for some β2 > 1 and C2 > 0 and all n ∈ I1 . Combining this with Proposition 4.2 and (4.10) we obtain (2.5) for all n ∈ N. From Remark 2.3(b), we conclude that ϕ ∈ Δ2 . 2 Theorem 4.3, Corollary 3.5, Proposition 4.1, and Remark 3.7 yield the following result. Theorem 4.4. Let X be an interpolation space for the couple (Λ(ϕ), M(ϕ)), where the function ϕ satisfies the condition: γϕ¯ > 0. The following conditions are equivalent: (i) Λ(R, X) = Sym(R, X). (ii) There exists A > 0 such that χUn Sym(R,X) AχUn Λ(R,X) ,
n ∈ N.
(iii) ϕ ∈ Δ2 . (iv) There exists A > 0 such that for all n ∈ N and for every f ∈ Dn we have ∞ ∞ ak rk : ak rk 1 . f Sym(R,X) A sup f · k=n+1
X
k=n+1
X
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(v) There exists A > 0 such that for every n ∈ N ∞ ∞ χ[0,2−n ] Sym(R,X) A sup χ[0,2−n ] · ak rk : ak rk 1 . k=n+1
k=n+1
X
X
The previous result can be improved for Marcinkiewicz spaces. Theorem 4.5. Suppose that X is a Marcinkiewicz space M(ϕ) with log1/2 (2/t) ∈ X0 and δϕ < 1. The following conditions are equivalent: (i) Λ(R, X) = Sym(R, X). (ii) X satisfies the discrete log-condition (2.6), that is, there exists C > 0 such that
1/2 2 1/2 2 log χ(0,2−n ] C log χ(0,2−n ] , n t 2 t X
n ∈ N.
X
(iii) There exists C > 0 such that √
√ n · ϕ 2−n C sup k − n · ϕ 2−k ,
n ∈ N.
kn
Moreover, if ϕ satisfies the
√ 2-condition (2.3), then conditions (i)–(iii) are equivalent to:
(iv) ϕ ∈ Δ2 . Proof. We start proving the equivalence between (ii) and (iii). Since u log1/2
2 2 dt u log1/2 , t u
0 < u 1,
0
we have, by the quasi-concavity of ϕ(t),
t 1/2 2 2 ϕ(t) 1/2 log −n χ ds = sup log [0,2 ] 2n t 2n s 0
=
sup
0
sup
0
sup kn
ϕ(t) 2n t
2n t 1/2 2 ds log s 0
ϕ(t) log1/2
2 2n t
√ k − n · ϕ 2−k ,
S.V. Astashkin, G.P. Curbera / Journal of Functional Analysis 256 (2009) 4071–4094
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for every n ∈ N. Similarly,
√ 1/2 2 log χ[0,2−n ] sup k · ϕ 2−k . t X
kn
Therefore, inequality (2.6) is equivalent to sup
√ √ k · ϕ 2−k C sup k − n · ϕ 2−k ,
kn
kn
which, in turn, is equivalent to (iii). The implication (i) ⇒ (ii) is a straightforward consequence of Theorem 4.3. Let us prove the opposite statement. Assume that (ii) is fulfilled. From Proposition 3.3, it suffices to prove (3.2), that is, xX CyX , where n
x(t) :=
2
ak χΔkn (t) log
k=1
1/2
2 , t
n
y(t) :=
2
ak χΔkn (t) log1/2
k=1
2 , 2n t + 1 − k
the constant C > 0 does not depend on n ∈ N, and a1 a2 · · · a2n 0. Since x(t) is a decreasing function and δϕ < 1, by (1.1) we have xX sup x(t)ϕ(t). 0
Hence, using the quasi-concavity of ϕ(t), we have
xX sup ak log1/2 1k2n
2 k ϕ n sup a2j n − j + 1ϕ 2j −n . −n k2 2 0j n
(4.12)
Since a2j
j −n 1/2 2 χ[0,2j −n ] a2j log n − j + 1ϕ 2 , t X
by (ii), we have a2j
j −n 1/2 a C n − j + 1ϕ 2 log j 2
χ[0,2j −n ] . n−j 2 t X 2
Note that, for σa the dilation operator by a > 0,
log1/2
2 1/2 −n (t) = σ (t) . log χ χ j −n j [0,2 ] 2 2n−j t [0,2 ] 2n t 2
(4.13)
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This implies that the functions
1/2
log
2 2n−j t
j
χ[0,2j −n ] (t)
and
2
χΔin (t) log
1/2
i=1
2 n 2 t +1−i
have the same distribution function. Therefore, since a1 · · · a2j 0, we get 2j
2 2 1/2 a j log1/2 χ a χ yX . j −n ] i (t) log i [0,2 Δ 2 nt + 1 − i n 2n−j t 2 X i=1
X
The result follows from the previous inequality together with (4.12) and (4.13). √ We prove that (iii), together with the 2-condition, implies (iv). By Remark 2.3(b), we only have to show that there exist β > 1 and C > 0 such that (2.5) holds, that is, ϕ 2−n Cϕ 2−βn , n ∈ N. By assumption, for every n ∈ N there exists kn n such that √ −n C1 kn − nϕ 2−kn . nϕ 2
(4.14)
Let us show that we can assume, if n is large enough, that kn can be chosen satisfying C12 + 1 C12
n kn 2n.
(4.15)
The first inequality follows directly from (4.14) since, as kn n, we have √ n C1 kn − n. For the second inequality in (4.15), suppose that kn 2n. Then, for some m ∈ N, we have 2m n √ m+1 n. This, together with the 2-condition (2.3), implies that, for large enough n, we kn < 2 have
m m+1 √ kn − nϕ 2−kn 2 2 nϕ 2−2 n √ √ cϕ−1 ( 2cϕ )m 2nϕ 2−2n √ cϕ−1 2nϕ 2−2n .
√ But, then (4.14) holds also for kn = 2n (with the constant cϕ−1 2C1 instead of C1 ). Thus, (4.15) holds. Combining (4.14) and (4.15), we conclude that ϕ 2−n Cϕ 2−βn , where n ∈ N is large enough, β := (C12 + 1)/C12 > 1, and the constant C > 0 does not depend on n. By changing, if necessary, the constant C, we have the above inequality for all n ∈ N. Since, by Proposition 2.2(3), (iv) implies (ii) the proof is completed. 2
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Remark 4.6. The proof of Theorem 4.5 and [3, Remark 5] show that the following tail-assertion holds: if X is a Marcinkiewicz space M(ϕ) such that δϕ < 1, then Λ(R, X) is r.i. if and only if there exists A > 0 such that for every f ∈ Dn we have ∞ ∞ f Sym(R,X) A sup f · ci ri : ci ri 1 . i=n+1
X
i=n+1
X
5. Examples We end with two examples which highlight certain features of the previous results. Example 5.1. The sets Un appearing in Proposition 4.2 are, in some sense, necessary since they cannot be replaced by the simpler sets, of equal measure, [0, 2−n ]. This is so even in the case when the r.i. space X is “very close” to L∞ . To see this, define ψ(t) := 21−log
1/2 (2/t)
0 < t 1.
,
It can be easily checked that, for sufficiently small t > 0, ψ (t) > 0 and ψ (t) < 0. Therefore, there exists a nonnegative increasing concave function ϕ(t) on [0, 1] such that ϕ(t) = ψ(t) for small enough t > 0. Let X be the Marcinkiewicz space M(ϕ). Since the upper dilation index of ϕ satisfies δϕ = 0, we have X ⊂ Lp for all p < ∞. Since limt→0+ ψ(t) log1/2 (2/t) = 0, we have log1/2 (2/t) ∈ X0 . Therefore, n n sup χ[0,2−n ] · ci ri : ci ri 1 i=1 i=1 X X n sup χ[0,2−n ] · ci ri : (ai )2 1 i=1 X n √ 1 √ χ[0,2−n ] · n2− n . √ ri n i=1
X
From [2, Corollary 2.11] and [11, Theorem 2.5.3] it follows that χ[0,2−n ] Sym(R,X)
sup log1/2 (2/t) · 21−log
1/2 (2/t)
0
√ √ C n2− n .
Thus, for all n ∈ N, we have n n χ[0,2−n ] Sym(R,X) C sup χ[0,2−n ] · ci ri : ci ri 1 . i=1
X
i=1
X
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However, since √ ϕ(2−2n ) 2− n → 0 if n → ∞, −n ϕ(2 )
it follows that ϕ ∈ / Δ2 . This example also allows to see that the discrete log-condition in Theorem 4.5(ii) cannot be replaced with the weaker condition χ[0,2−n ) Sym(M(ψ)) χ[0,2−n ) Λ(R,M(ψ)) ,
n ∈ N.
√ To see this, note that ψ(t) satisfies the 2-condition (due to Proposition 2.2(5)) and δψ = 0. However, since ψ ∈ / Δ2 , we conclude that Λ(R, M(ψ)) = Sym(R, M(ψ)). Example 5.2. The implication (i) ⇒ (iv) in Theorem 4.5 is not valid in general, that is, the √ 2-condition is essential. k k To see this, let nk = 22 , tk = 2−nk , and γk = 2k−1−2 for k ∈ N. Note that nk+1 = n2k and nk γk = 2k−1 . Define a continuous function ϕ(t) on the interval (0, 2−4 ] as follows: ϕ(t) = k1 t γk if tk2 < t tk and ϕ(t) is linear on the interval (tk+1 , tk2 ], k ∈ N. Direct calculation shows that ϕ(tk ) =
1 −1/2 and ϕ tk2 = nk+1 , k
1 −1/2 n k k
k ∈ N.
(5.1)
The function ϕ(t) is quasi-concave and limt→0+ ϕ(t) = 0. Let us check that X = M(ϕ) and ϕ satisfy the conditions of Theorem 4.5. Condition log1/2 (2/t) ∈ (M(ϕ))0 is equivalent to lim
√
i→∞
i + 1ϕ 2−i = 0.
(5.2)
To see this, let nk i < 2nk , then, by (5.1), √
1 −1/2 2 i + 1ϕ 2−i 2nk + 1ϕ(tk ) = 2nk + 1 nk < . k k
If 2nk i < nk+1 , again by (5.1), we have √ 1 −1/2 2 i + 1ϕ 2−i nk+1 + 1ϕ tk2 = nk+1 + 1 nk+1 < . k k This gives (5.2). In order to calculate the upper dilation index δϕ , we consider the dilation function Mϕ of ϕ. Since ϕ(2−n ) = 2nk γk , −n−nk ) n=0,1,... ϕ(2
Mϕ 2nk = sup
S.V. Astashkin, G.P. Curbera / Journal of Functional Analysis 256 (2009) 4071–4094
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then δϕ γk → 0, whence δϕ = 0. Moreover, ϕ ∈ / Δ2 since, by (5.1), ϕ(2−2nk ) ϕ(tk2 ) k−1 = = 2−2 → 0 as k → ∞. ϕ(2−nk ) ϕ(tk ) Lastly, we show that Λ(R, M(ϕ)) = Sym(M(ϕ)). In view of the proof of Theorem 4.5, it suffices to prove that there exists C > 0 such that √ √ n · ϕ 2−n C sup k − n · ϕ 2−k ,
n ∈ N.
kn
For this, it suffices to find a constant C > 0 such that for all n ∈ N there exists m ∈ N such that m ϕ 2−n C2m/2 ϕ 2−2 n .
(5.3)
Suppose first that nk n < 2nk . Set m := [log(nk+1 /n)]. Since nk+1 nk+1 1 nk+1 2m = nk , n 2n 4nk 4 by using (5.1), we get m ϕ 2−2 n ϕ 2−nk+1 =
1 1 −1/2 nk+1 = n−1 k+1 k+1 k 1 k −1/2 nk ϕ 2−nk 2−m/2 ϕ 2−n . k+1 4
If 2nk n < nk+1 , set m := [log(nk+2 /n)]. Then, nk+2 nk+2 1 nk+2 2m = nk+1 . n 2n 2nk+1 2 Therefore, again by (5.1), m ϕ 2−2 n ϕ 2−nk+2 =
1 1 −1/2 nk+2 = n−1 k+2 k + 2 k+1 1 k −1/2 nk+1 ϕ 2−2nk √ 2−m/2 ϕ 2−n . k+2 3 2
Thus (5.3) is proved and so, Λ(R, M(ϕ)) = Sym(M(ϕ)). References [1] S.V. Astashkin, On multiplicator space generated by the Rademacher system, Math. Notes 75 (2004) 158–165. [2] S.V. Astashkin, G.P. Curbera, Symmetric kernel of Rademacher multiplicator spaces, J. Funct. Anal. 226 (2005) 173–192. [3] S.V. Astashkin, G.P. Curbera, Rademacher multiplicator spaces equal to L∞ , Proc. Amer. Math. Soc. 136 (2008) 3493–3501.
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[4] S.V. Astashkin, K.V. Lykov, Extrapolatory description for the Lorentz and Marcinkiewicz spaces “close” to L∞ , Siberian Math. J. 47 (2006) 797–812. [5] S.V. Astashkin, K.V. Lykov, Strongly extrapolation spaces and interpolation, Siberian Math. J., in press. [6] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988. [7] Yu.A. Brudny˘ı, N.Ya. Krugljak, Interpolation Functors and Interpolation Spaces, North-Holland, Amsterdam, 1991. [8] G.P. Curbera, A note on function spaces generated by Rademacher series, Proc. Edinb. Math. Soc. 40 (1997) 119– 126. [9] G.P. Curbera, V.A. Rodin, Multiplication operators on the space of Rademacher series in rearrangement invariant spaces, Math. Proc. Cambridge Philos. Soc. 134 (2003) 153–162. [10] B.S. Kashin, A.A. Saakyan, Orthogonal Series, Amer. Math. Soc., Providence, RI, 1989. [11] S.G. Krein, Ju.I. Petunin, E.M. Semenov, Interpolation of Linear Operators, Amer. Math. Soc., Providence, RI, 1982. [12] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces, vol. II, Springer-Verlag, Berlin, 1979. [13] G.G. Lorentz, Relations between function spaces, Proc. Amer. Math. Soc. 12 (1961) 127–132. [14] V.A. Rodin, E.M. Semenov, Rademacher series in symmetric spaces, Anal. Math. 1 (1975) 207–222. [15] A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, 1977.
Journal of Functional Analysis 256 (2009) 4095–4127 www.elsevier.com/locate/jfa
Intrinsic ultracontractivity of a Schrödinger semigroup in RN ✩ Bénédicte Alziary a , Peter Takáˇc b,∗,1 a Université Toulouse 1 (Sciences Sociales), CEREMATH – UMR MIP, 21 Allées de Brienne,
F-31000 Toulouse cedex, France b Universität Rostock, Fachbereich Mathematik, Universitätsplatz 1, D-18055 Rostock, Germany
Received 7 October 2008; accepted 23 February 2009 Available online 4 March 2009 Communicated by C. Kenig
Abstract We give a (possibly sharp) sufficient condition on the electric potential q : RN → [0, ∞) in the Schrödinger operator A = − + q(x)• on L2 (RN ) that guarantees that the Schrödinger heat semigroup {e−At : t 0} on L2 (RN ) generated by −A is intrinsically ultracontractive. Moreover, if q(x) ≡ q(|x|) is radially symmetric, we show that our condition on q is also necessary (i.e., truly sharp); it reads ∞ q(r)−1/2 dr < ∞
for some r0 ∈ (0, ∞).
r0
Our proofs make essential use of techniques based on a logarithmic Sobolev inequality, Rosen’s inequality (proved via a new Fenchel–Young inequality), and a very precise asymptotic formula due to H ARTMAN and W INTNER. © 2009 Elsevier Inc. All rights reserved. Keywords: Schrödinger operator and heat semigroup; Ground state; Intrinsic ultracontractivity; Logarithmic Sobolev and Rosen’s inequalities; WKB-type asymptotic formula; Semigroup and resolvent compactness
✩
The work was supported in part by le Ministère des Affaires Étrangères (France) and the German Academic Exchange Service (DAAD, Germany) within the exchange program “PROCOPE”. * Corresponding author. E-mail addresses: [email protected] (B. Alziary), [email protected] (P. Takáˇc). 1 A part of this research was performed when the author was a visiting professor at CEREMATH – UMR MIP, Université des Sciences Sociales, Toulouse, France. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.013
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1. Introduction The question of intrinsic ultracontractivity of the Schrödinger heat semigroup {e−At : t 0} on L2 (RN ) generated by the self-adjoint linear operator −A = − q(x)• has been an interesting open problem since the late 1970s. It has been studied or discussed in a number of articles, including R. Bañuelos [8], E.B. Davies [10,11], E.B. Davies and B. Simon [12], L. Gross [17], M. Murata [23], and M. Reed and B. Simon [26]. As usual, def
A ≡ Aq = − + q(x) •
on L2 RN
(1)
denotes the Schrödinger operator with a given (electric) potential q, where q : RN → R is assumed to be a continuous function that satisfies the following standard hypothesis: def
q0 = inf q > 0 and q(x) → +∞ as |x| → ∞. RN
(2)
It is well known ([11,14] or [27]) that, under this hypothesis, the Schrödinger operator A on L2 (RN ), defined to be the Friedrichs extension of A|Cc2 (RN ) , is self-adjoint and positive definite, and its inverse A−1 on L2 (RN ) is compact. The principal eigenvalue Λ ≡ Λq of the operator A is simple with the associated eigenfunction ϕ ≡ ϕq normalized by ϕ > 0 throughout RN and ϕL2 (RN ) = 1. In the physics literature, Λ and ϕ, respectively, are called the ground state energy and the ground state of the Schrödinger operator A. Let {e−At : t 0} denote the strongly continuous semigroup on L2 (RN ) generated by −A. This semigroup is called intrinsically ultracontractive if the linear operator T (t) = e−At is bounded from L2 (RN ) to X for each t > 0, where def X ≡ Xq = f ∈ L2 RN : f/ϕ ∈ L∞ RN
(3)
is a linear space endowed with the natural norm def f X = ess sup |f |/ϕ . RN
Hence, X is a Banach space continuously embedded into L2 (RN ). The main goal of this article is to investigate a natural sufficient condition for the intrinsic ultracontractivity of the semigroup {e−At : t 0} that looks like or is only slightly stronger than ∞ r0
max|x|=r q(x)1/2 dr < ∞ min|x|r q(x)
for some 0 < r0 < ∞,
(4)
see Y. Pinchover [25, p. 587]. Indeed, our pair of hypotheses (12) and (25) is very close to (4). If an additional hypothesis on the variation and the growth of q(x) as |x| → ∞ is imposed (which is typically “strong” with respect to the angular variable x = x/|x| and “weak” with respect to the radial variable r = |x|, cf. (13) below), then condition (4) becomes equivalent with
B. Alziary, P. Takáˇc / Journal of Functional Analysis 256 (2009) 4095–4127
∞ r0
−1/2 min q(x) dr < ∞ for some 0 < r0 < ∞,
|x|=r
4097
(5)
cf. M. Murata [23, Ineq. (6.5), p. 376] and Y. Pinchover [25, p. 587]. We are not able to verify whether (5) alone implies the intrinsic ultracontractivity; we will replace it by a slightly stronger sufficient condition — the pair of inequalities (12) and (25) below. This condition still covers all standard cases treated in [10–12,17,23,26]. Closely related conditions on q are imposed in B. Alziary, J. Fleckinger, and P. Takáˇc [2, Theorem 2.1, p. 128] (for N = 2), [3, Theorem 2.1, p. 365] and [4, Eq. (19), p. 219] (for N 2), and in B. Alziary and P. Takáˇc [5, Theorem 2.1, p. 284] and [6, Eq. (2.6), p. 39]. More precisely, we will show that if the potential q(x) is “squeezed” between two radially symmetric potentials that are not too far apart from each other (Hypothesis (Hq )), then indeed condition (5) implies that {e−At : t 0} is intrinsically ultracontractive, see Theorem 3.1 below. However, to better understand intrinsic ultracontractivity, one needs to address also the question of possible equivalence of the following four statements: (i) The semigroup {e−At : t 0} is intrinsically ultracontractive on L2 (RN ). (ii) The operator T (t) = e−At is compact from X into itself for each t > 0. (iii) The resolvent −1
(A − λI )
∞ =
e−At eλt dt
(6)
0
of A = − + q(x)• is compact from X into itself for each λ < Λ. (iv) Every eigenfunction v ∈ L2 (RN ) of A belongs to X, that is, if Av = λv for some λ ∈ R and 0 = v ∈ L2 (RN ) then v/ϕ ∈ L∞ (RN ). In addition to our proof of (i), standard arguments from the general theory of semigroups of positive operators on Banach lattices show that the following implications hold: (i) ⇒ (ii) ⇒ (iii) ⇒ (iv), by Proposition 3.3 (proved in Section 6.3). For an arbitrary potential q(x) satisfying only conditions (2), the validity of any of the reversed implications (iv) ⇒ (iii) ⇒ (ii) ⇒ (i) ⇒ (5) is still an open question. Only if q(x) is a bounded perturbation of a radially symmetric potential with a power growth near infinity, the implication (iv) ⇒ (5) has been obtained in M. Murata [23, Corollary 6.4, p. 376]. We establish this implication in Proposition 4.7 for q(x) ≡ q(|x|) radially symmetric without such a severe growth restriction on q(r) as r → ∞. There are several other problems closely related to (5), (iii), and (iv), such as the ground-state positivity of the resolvent (A − λI )−1 for each λ < Λ investigated in [4, Theorem 3.1, p. 221], [5, Theorem 2.1, p. 284], and [6, Theorem 3.1, p. 41], and the anti-maximum principle (i.e., the ground-state negativity) for the solution u of the Schrödinger equation −u + q(x)u = λu + f (x)
in L2 RN ,
(7)
for Λ < λ < Λ + δ (δ > 0 — small enough, depending on f ∈ X \ {0}, f 0) established in [2, Theorem 2.1, p. 128] for N = 2, and in [3, Theorem 2.1, p. 365], [4, Theorem 3.4, p. 222], [6, Theorem 3.4, p. 42], and [25, Theorem 5.3, p. 575] for any N 2 (under various hypotheses on q). The reader is referred to [4] for the corresponding definitions.
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Next, let us focus on statements (iii) and (iv) with a radially symmetric potential q(x) ≡ q(r) where r = |x| 0, because they involve only the stationary Schrödinger operator A. For technical reasons, let us assume that q(r) does not oscillate “too fast” as r → ∞; cf. (13). We are able to verify the implication (iv) ⇒ (5) under quite different hypotheses on q(r) than in [23, Theorem 4.3, p. 352, Proposition 6.1, p. 376]. The implication (5) ⇒ (iii) has been verified in [4, Theorem 3.2, Part (a), p. 221]. Hence, in this case all three statements (5), (iii), and (iv) are equivalent. To interpret this result, first, consider the harmonic oscillator, that is, q(r) = r 2 for r 0. One finds immediately that, except for the ground state ϕ itself, no other eigenfunction v of A (associated with an eigenvalue λ = Λ) can satisfy v ∈ X. We refer to E.B. Davies [11, Section 4.3, pp. 113–117] for greater details when N = 1. On the other hand, if q(r) = r 2+ε for r 0 (ε > 0 — a constant), then v ∈ X holds for every eigenfunction v of A, again by results from [11, Corollary 4.5.5, p. 122], combined with [11, Lemma 4.2.2, p. 110, Theorem 4.2.3, p. 111].The reader is referred to E.B. Davies and B. Simon [12, Theorem 6.3, p. 359] and to M. Hoffmann-Ostenhof [20, Theorem 1.4(i), p. 67] for the same result under much weaker restrictions on q(x). From these simple examples it is clear that, if (iv) is to be satisfied, then the potential q(x) has to grow fast enough as |x| → ∞. We will see in this article that for q(x) ≡ q(r) radially symmetric, precisely (5) is the necessary and sufficient condition for the validity of (iv); see Proposition 3.4. There are essentially two main directions in proving the intrinsic ultracontractivity (i) of the Schrödinger heat semigroup {e−At : t 0}, or either of the results (iii) or (iv) implied by (i): Let each qj : R+ → R (j = 1, 2) be a continuous function, such that the auxiliary potential qj (x) ≡ qj (|x|) (x ∈ RN ) satisfies (2). Assume that q1 (|x|) q(x) q2 (|x|) holds for all x ∈ RN . The first main direction allows for a relatively large gap q2 (r) − q1 (r) as r → ∞, but (except for our present work) it limits the growth of qj (r) to power growth as r → ∞; see [10–12]. The logarithmic Sobolev and Rosen’s inequalities are important tools in this approach. The second main direction allows only for a relatively small gap q2 (r) − q1 (r) as r → ∞, sometimes even bounded or q(x) ≡ qj (|x|) radially symmetric, but it does not limit the growth of qj (r) to power growth as r → ∞; see [2–6,20,25]. This approach takes advantage of comparison methods with sub- and supersolutions and rather tight asymptotic estimates with radially symmetric potentials. It has been used with success to establish directly (iii) (in [4,6]), (iv) (in [20]), the ground-state positivity (in [4–6]), and the anti-maximum principle (in [2,3,25]). Last but not least, assuming both restrictions, i.e., q2 (r)−q1 (r) uniformly bounded for all r 0 and power growth of qj (r) as r → ∞, M URATA established the equivalence of (i) and (5) in [23, Theorem 6.3, Corollary 6.4, p. 376]. We should also mention that intrinsic ultracontractivity for the corresponding Dirichlet problem in certain domains Ω ⊂ RN (in place of RN ) has been investigated in a number of articles, [7,8,25] among them, by methods of potential and probability theories, which are much different from our methods. Similarly as in [4] also in our present work we treat potentials q(x) that are not necessarily radially symmetric. We impose quite general hypotheses on the potential q that guarantee the validity of statement (i); hence, that of (ii), (iii), and (iv), as well. As in [4], our method makes use of rather precise estimates of the asymptotic behavior at infinity of the ground state ϕ for the Schrödinger operator A (Lemma 4.4). The two main ingredients in our proof of statement (i) (Theorem 3.1) are the logarithmic Sobolev and Rosen’s inequalities, (63) and (64), respectively. The former is taken from DAVIES ’ monograph [11, Ineq. (3.2.1), p. 83], whereas the latter is proved in our Proposition 5.1. Our version of Rosen’s inequality is sharper and more concrete than that used in the works of Davies and Simon [12, Theorem 5.2, p. 357, Theorem 6.1, p. 358]
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and Murata [23, Ineq. (6.17), p. 378]. It provides a more concrete estimate for proving intrinsic ultracontractivity; see Ineq. (66). We give a simple proof of this inequality which combines an asymptotic formula (near infinity) for the ground state ϕ, due to Ph. Hartman and A. Wintner [19, Eq. (xxv), p. 49 (or Eq. (158), p. 80)], stated in our Corollary 4.6, with the Fenchel–Young inequality (Lemma 5.2). This article is organized as follows. In the next section (Section 2) we describe the type of potentials q(x) we are concerned with, together with some basic notations. Section 3 contains our main new results, Theorem 3.1 and Proposition 3.4. These results are proved in Sections 4 through 6. 2. Hypotheses and notations Let us consider the inhomogeneous stationary Schrödinger equation (7), i.e., −u + q(x)u = λu + f (x)
in L2 RN .
Here, f ∈ L2 (RN ) is a given function, λ ∈ C is a complex parameter, and the potential q : RN → R is a continuous function; we always assume that q satisfies (2), i.e., def
q0 = inf q > 0
and q(x) → +∞ as |x| → ∞.
RN
We interpret Eq. (7) as the operator equation Au = λu + f in L2 (RN ), where the Schrödinger operator (1), def
A ≡ Aq = − + q(x) •
on L2 RN ,
is defined formally as follows: We first define the (Hermitian) quadratic form
∇v · ∇ w¯ + q(x)v w¯ dx
(8)
def Vq = f ∈ L2 RN : Qq (f, f ) < ∞ .
(9)
def
Qq (v, w) =
RN
for every pair v, w ∈ Vq where
Then A is defined to be the Friedrichs representation of the quadratic form Qq in L2 (Ω); L2 (Ω) is endowed with the natural inner product def
(v, w)L2 (RN ) =
v w¯ dx,
v, w ∈ L2 (Ω).
RN
This means that A is a positive definite, self-adjoint linear operator on L2 (Ω) with domain dom(A) dense in Vq and
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(Av)w¯ dx = Qq (v, w) for all v, w ∈ dom(A); RN
see T. Kato [21, Theorem VI.2.1, p. 322]. Notice that Vq is a Hilbert space with the inner product (v, w)q = Qq (v, w) and the norm vVq = ((v, v)q )1/2 . The embedding Vq → L2 (RN ) is compact, by (2). The principal eigenvalue Λ ≡ Λq of the operator A ≡ Aq can be obtained from the Rayleigh quotient Λ ≡ Λq = inf Qq (f, f ): f ∈ Vq with f L2 (RN ) = 1 ,
Λ > 0.
(10)
This eigenvalue is simple with the associated eigenfunction ϕ ≡ ϕq normalized by ϕ > 0 throughout RN and ϕL2 (RN ) = 1; ϕ is a minimizer for the Rayleigh quotient above. The reader is referred to D.E. Edmunds and W.D. Evans [14] or M. Reed and B. Simon [27, Chapter XIII] for these and other basic facts about Schrödinger operators. def
We set r = |x| for x ∈ RN , so r ∈ R+ , where R+ = [0, ∞). If q is a radially symmetric potential, q(x) = q(r) for x ∈ RN , then also the eigenfunction ϕ must be radially symmetric. This follows directly from Λ being a simple eigenvalue. Our technique includes a comparison argument with radially symmetric potentials, which are assumed to satisfy certain differentiability and growth conditions in the radial variable r = |x|, r ∈ R+ . More precisely, we bound the potential q : RN → R by such radially symmetric potentials from below and above. In order to formulate our hypotheses on the potential q(x), x ∈ RN , we first introduce the following class (Q) of auxiliary functions Q(r) of r = |x| 0: (Q) Q : R+ → (0, ∞) is a locally absolutely continuous function that satisfies the following conditions, for some 0 < r0 < ∞: r Q(r)
1/2
Q(t)
r
dt · P log
r0
1/2
Q(t)
dt
for all r > r0 ,
(11)
r0
where P : R → (0, ∞) is a strictly monotone increasing, continuous function that satisfies also P (R) = (0, ∞) and ∞
P (ξ )−1 dξ < ∞,
(12)
0
and there is a constant γ , 1 < γ 2, such that
∞
d γ −1/2 1/2
dr Q(r)
Q(r) dr < ∞. r0
(13)
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Condition (13) is taken from P H . H ARTMAN’s monograph [18, Exercise 17.5, Part (b), p. 320]. Originally, it appeared in the work of Ph. Hartman and A. Wintner [19, Eq. (xxiv), p. 49, Eq. (157), p. 80], in an equivalent form ∞
Q (r)/Q(r) γ Q(r)1/2 1−γ dr < ∞ r0
where we have corrected the exponent γ − 1 to 1 − γ . Condition (13) replaces another condition, d Q(r)−1/2 → 0 dr
as r → ∞,
from [18, Exercise 17.5, Part (a), p. 320]. Also this condition appeared originally in [19], on p. 49, as the last condition in Eq. (xxii), and on p. 79, Eq. (153). Remark 2.1. We note that (11) combined with the properties of P force Q(r)/r → ∞ as r → ∞ (cf. [4, Remark 2.1, pp. 218–219]). Indeed, first we get def
r1
Q(r) Q1 =
1/2
Q(t)
r1
dt · P log
r0
1/2
Q(t)
dt
>0
r0
for all r r1 = r0 + 1, which is then inserted into Ineq. (11) to get r1 1/2 1/2 Q1 · P log (r − r1 )Q1 → +∞ as r → ∞. Q(r)/r 1 − r
Furthermore, there is no potential Q(r) of class (Q) that would satisfy conditions (11) and (13) with γ = 1 simultaneously, by [4, Remark 2.2, p. 219]. Several examples of radially symmetric potentials q(x) = q(r) that do or do not belong to class (Q) have been given in Alziary, Fleckinger, and Takáˇc [4, Section 3.2, pp. 223–226]. These examples illustrate how “large” class (Q) actually is. We remark that the class (Q) used in [4, Section 2, p. 218] is somewhat larger than in our present article (see Proposition 2.2 below), but the examples from [4] apply to the present class (Q), as well, without any change. In [4], the pair of conditions (11) and (12) is replaced by the weaker condition (14). In our present work we impose conditions (11) and (12) for technical reasons only; the following weaker condition, ∞
Q(r)−1/2 dr < ∞
(14)
r0
(cf. (5)), may still be sufficient, cf. Proposition 2.2, Parts (c) and (d). Somewhat simpler (and, perhaps, also more useful) sufficient conditions for a function Q to be in class (Q) are the following ones, where Ineq. (11) is replaced by a stronger, but simpler inequality (15) below:
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(Q ) Q : R+ → (0, ∞) is locally absolutely continuous, monotone increasing on some interval [r0 , ∞), 0 < r0 < ∞, and satisfies the following conditions: Q(r)1/2 r · P log Q(r)
for all r > r0 ,
(15)
instead of Ineq. (11), with the remaining conditions on P and Q unchanged, see (12) and (13). The following proposition clarifies the equivalence relation between conditions (12) for P and (14) for Q. Its proof shows how to find the function P provided Q satisfies certain (simple) growth and monotonicity conditions, in addition to (14). We remark that condition (13) on Q enters only Part (b) of this proposition. Proposition 2.2. Assume that Q : R+ → (0, ∞) is a locally absolutely continuous function. Then the following statements are valid: (a) Ineqs. (11) and (12) imply (14). (Hence, Q is in class (Q) ⇒ Q verifies (14).) (b) Q is in class (Q ) ⇒ Q is in class (Q). (Hence, Q is in class (Q ) ⇒ Q verifies (14).) (c) If the function
−1
r r → Q(r)
1/2
Q(t)
: (r1 , ∞) → (0, ∞)
dt
r0
is strictly monotone increasing, for some 0 < r0 < r1 < ∞, with the limit
−1
r
= ∞,
Q(t)1/2 dt
lim Q(r)
r→∞
(16)
r0
then condition (14) is equivalent with (12) for P determined by
r 1/2
P log
Q(t)
dt
r = cQ(r)
r0
−1 1/2
Q(t)
dt
for all r > r1 ,
(17)
r0
where c ∈ (0, 1) is an arbitrary constant. In particular, P can be extended to a strictly monotone increasing, continuous function P : R → (0, ∞) that satisfies also P (R) = (0, ∞). (d) If the function r → r −2 Q(r) : (r0 , ∞) → (0, ∞) is strictly monotone increasing with the limit lim r −2 Q(r) = ∞,
r→∞
then condition (14) is equivalent with (12) for P determined by
(18)
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P log Q(r) = cr −1 Q(r)1/2
for all r > r0 ,
4103
(19)
where c ∈ (0, 1) is an arbitrary constant. In particular, P can be extended to a strictly monotone increasing, continuous function P : R → (0, ∞) that satisfies also P (R) = (0, ∞). r Proof. To prove Part (a), we make the substitution ξ = log( r0 Q(t)1/2 dt) for all r r1 > r0 , r r where r1 is sufficiently large so that r01 Q(t)1/2 dt 1. Denote ξ1 = log( r01 Q(t)1/2 dt) 0. We compute, using (11), ∞
−1/2
Q(r)
∞ dr
r1
r1
Q(r)1/2 dr r r ( r0 Q(t)1/2 dt) · P [log( r0 Q(t)1/2 dt)]
r
−1
∞ r d 1/2 1/2 log Q(t) dt · Q(t) dt = P log dr r1
∞ =
r0
r0
P (ξ )−1 dξ.
ξ1
Proof of Part (b). If the function Q belongs to class (Q ) then Q(r) is monotone increasing for r r0 . Hence, we have r (r − r0 ) · Q(r)
1/2
Q(t)1/2 dt.
(20)
r0
Condition (11) in class (Q) is now readily obtained from condition (15) in class (Q ), with a possibly different function P . r Proof of Part (c). Let P be determined by Eq. (17) and set ξ1 = log( r01 Q(t)1/2 dt). We continue P from the interval (ξ1 , ∞) to its complement (−∞, ξ1 ] = R \ (ξ1 , ∞) simply by setting P (ξ ) = P (ξ1 ) exp(ξ − ξ1 ) for all ξ ξ1 . Hence, P is strictly monotone increasing and continuous on R and satisfies also P (R) = (0, ∞) together with (11) (where r0 needs to be replaced by r1 ). Finally, an analogous calculation as in the proof of Part (a) shows that conditions (12) and (14) are indeed equivalent. Proof of Part (d). Now let P be determined by Eq. (19) and set ξ0 = log Q(r0 ). We continue P from the interval (ξ0 , ∞) to its complement (−∞, ξ0 ] = R \ (ξ0 , ∞) again by setting P (ξ ) = P (ξ0 ) exp(ξ − ξ0 ) for all ξ ξ0 . Hence, P is strictly monotone increasing and continuous on R and satisfies also P (R) = (0, ∞) together with (15). In analogy with the proof of Part (a), we first substitute ξ = log Q(r) for all r r0 and then, using (19), we calculate ∞
−1
P (ξ ) ξ0
∞ dξ = r0
−1 P log Q(r) d log Q(r)
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1 = c
∞
1/2
r/Q(r)
1 Q (r)/Q(r) dr = c
r0
2 =− c
∞
Q(r)−3/2 Q (r)r dr
r0
∞
∞ d 2 Q(r)−1/2 r dr = r1 Q(r1 )−1/2 + Q(r)−1/2 dr , dr c
r0
r0
by integration by parts combined with (18). Thus, conditions (12) and (14) are equivalent. The proposition is proved. 2 Remark 2.3. In what follows we will define the function P always only on a half-line [ξ0 , ∞) for some ξ0 ∈ R. The continuation of P to the whole of R described above (proof of Proposition 2.2) is then used without further explicit notice. The definition of P on the whole of R is needed for some technical reasons which will become apparent later in Proposition 5.1 (Rosen’s inequality (47)). Finally, in order to have simple auxiliary functions Q(r) of r 0, let us introduce the following class of such functions: (Q ) Q : R+ → (0, ∞) is locally absolutely continuous, such that the function r → r −2 Q(r) : (r0 , ∞) → (0, ∞) is strictly monotone increasing, for some 0 < r0 < ∞, and unbounded at infinity, i.e., limr→∞ (r −2 Q(r)) = ∞ as in (18), and both conditions (13) and (14) hold. Clearly, Parts (b) and (d) of Proposition 2.2 guarantee: Q is in class (Q ) ⇒ Q is in class (Q ) ⇒ Q is in class (Q). Remark 2.4. In order that the function r → r −2 Q(r) is strictly monotone increasing on some interval (r1 , ∞), 0 < r0 < r1 < ∞, and unbounded at infinity, it suffices to assume that (cf. Part (c) of Proposition 2.2) r r → Q(r)
−1 1/2
Q(t)
: (r0 , ∞) → (0, ∞)
dt
r0
is monotone increasing (not necessarily strictly), for some 0 < r0 < ∞, and unbounded at infinity, i.e.,
1/2
lim Q(r)
Q(t)
r→∞
r0
as in (16).
−1
r dt
=∞
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To verify this claim, let us first note that the first derivative of the function def
r
(r) =
1/2 1/2
Q(t)
of r ∈ (r0 , ∞)
dt
r0
is given by 1 (r) = Q(r)1/2 2
−1/2
r 1/2
Q(t)
dt
,
r0 < r < ∞.
r0
r Consequently, r → Q(r)( r0 Q(t)1/2 dt)−1 is monotone increasing if and only if is convex on (r0 , ∞). Furthermore, by l’Hôspital’s rule combined with (16), we have lim (r)/r = lim (r) = ∞.
r→∞
r→∞
Next, let r and s be arbitrary numbers, such that r0 < r < s < ∞, and find θ ∈ (0, 1) such that r = θ r0 + (1 − θ )s. The following inequality is a special case of the definition of convexity of the function on (r0 , ∞): (s) (r) θ r0 (s) (r0 ) − − . r s r0 s r Now take r1 ∈ (r0 , ∞) sufficiently large, such that (s)s −1 − (r0 )r0−1 > 0 holds for all s r1 . We observe that the function r → (r)/r is strictly monotone increasing for r r1 . Finally, taking advantage of the factorization Q(r)1/2 /r = 2 (r)( (r)/r), we conclude that also the function r → r −2 Q(r) must be strictly monotone increasing on (r1 , ∞) and unbounded at infinity, as claimed. Another sufficient condition for r → r −2 Q(r) to be strictly monotone increasing on some interval (r0 , ∞), 0 < r0 < ∞, is the inequality rQ (r) > 2Q(r) owing to
for all r > r0 ,
(21)
d −2 −3 dr (r Q(r)) = r (rQ (r) − 2Q(r)).
Example 2.5. Typical examples of (positive) auxiliary functions Q(r) (r 0) belonging to class (Q ) and, hence, to classes (Q) and (Q ) as well, are given by Q(r)1/2 = r 1+δ , r(log r)1+δ , r log r(log log r)1+δ , . . .
for r r0 ,
(22)
with r0 > 0 large enough, where δ > 0 is a constant. Thus, we can write them as Q(r)1/2 = r 1+δ or 1+δ Q(r)1/2 = r · log r · log log r · · · · · (log)m (r) · (log)m+1 (r)
(23)
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for r r0 , using the abbreviation (log)m (r) = log log · · · log(r), m times def
where m 0 is an integer. We have defined (log)0 (r) = r for r 0. These auxiliary functions obey all three conditions, (13) with any choice of γ > 1, (14), and (21). To see this, first, we employ (23) to calculate d 1 Q (r)/Q(r) = log Q(r)1/2 2 dr d log r + log log r + · · · + (log)m+1 (r) + (1 + δ)(log)m+2 (r) = dr −1 = r −1 + (r · log r)−1 + · · · + r · log r · log log r · · · · · (log)m (r) −1 + (1 + δ) r · log r · log log r · · · · · (log)m+1 (r) . Consequently, we get 12 r(Q (r)/Q(r)) > 1 for every r r0 , i.e. (21), provided r0 > 0 has been chosen large enough. Moreover, either r · drd log(Q(r)1/2 ) = 1 + δ (for Q(r)1/2 = r 1+δ ) or else r · drd log(Q(r)1/2 ) → 1 as r → ∞ (for m 0). Applying the last fact to the following formula, d d Q(r)−1/2 = −Q(r)−1/2 · log Q(r)1/2 , dr dr
(24)
we arrive at (13) for any choice of γ > 1. Ineq. (14) is verified directly by a chain of logarithmic substitutions. Last but not least, we remark that the cases Q(r)1/2 = r 1+δ and Q(r)1/2 = r · (log r)1+δ (m = 0), respectively, with δ > 0, have been treated in [12, Theorem 6.1, Parts (a) and (b), p. 358]. It is mentioned there [12, Remarks, p. 359] that also the case Q(r)1/2 = r · log r · (log log r)1+δ (m = 1) can be treated by the same methods. In the present article we are able to treat Q given by formula (23) for any integer m 0. Denoting by r = |x| the radial variable (x ∈ RN ), we impose the following restrictions on the variation and the growth of q(x): Hypothesis. We assume that q : RN → (0, ∞) is a continuous function satisfying (2) together with (Hq ) there exists a function Q : R+ → (0, ∞) of class (Q) and a constant r0 > 0 such that the inequalities
r 1/2
Q(t)
r
dt · P log
r0
hold for all x ∈ RN with r = |x| > r0 .
1/2
Q(t) r0
dt
q(x) Q(r)
(25)
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We remark that this hypothesis is not weaker than the corresponding hypothesis in [4, pp. 219– 220], due to the strict monotonicity hypothesis on P here. The following hypothesis is stronger (but simpler) than Hypothesis (Hq ) above: Hypothesis. q : RN → (0, ∞) is a continuous function satisfying (2) together with (Hq ) there exists a function Q : R+ → (0, ∞) of class (Q ) and a constant r0 > 0 such that the inequalities r · Q(r)1/2 · P log Q(r) q(x) Q(r)
(26)
hold for all x ∈ RN with r = |x| > r0 . Finally, recalling the choice of function P in Eq. (19) with a constant c ∈ (0, 1), we observe that (Hq ) ⇒ (Hq ) ⇒ (Hq ), where (Hq ) reads as follows: Hypothesis. q : RN → (0, ∞) is a continuous function satisfying (2) together with (Hq ) there exists a function Q : R+ → (0, ∞) of class (Q ) and constants 0 < c < 1 and r0 > 0 such that the following inequalities hold, cQ(r) q(x) Q(r)
for all x ∈ RN , r = |x| > r0 .
(27)
Remark 2.6. We remark that for auxiliary functions Q considered in Example 2.5, Eq. (22), condition (27) is more restrictive than condition (26). Hence, it is better to impose the latter, condition (26), by making a better choice of function P than that from Eq. (19). Indeed, we can choose P as follows: If Q(r)1/2 = r 1+δ , we take P to be P (ξ ) = cξ 1+δ
for all ξ ξ0 ,
where c > 0 and δ > 0 are arbitrary constants and ξ0 = log r0 > 0 is large enough. If Q is given by formula (23), we set 1+δ P (ξ ) = cξ · log ξ · · · · · (log)m−1 (ξ ) · (log)m (ξ )
for all ξ ξ0 ,
where c ∈ (0, ∞) and δ ∈ (0, δ) are arbitrary constants and ξ0 = log r0 > 0 is large enough. The function P is strictly monotone increasing on the interval [ξ0 , ∞) where it is also continuous. It satisfies hypothesis (12) provided it is properly extended to the entire real line R; cf. Remark 2.3 above. To verify also (15), we first notice that the case Q(r)1/2 = r 1+δ is trivial. So let Q be given by formula (23). Then, if r0 > 0 is sufficiently large, we have for all r r0 ,
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δ−δ Q(r)1/2 = (r/c)P (log r) · (log)m+1 (r) δ−δ 1 (r/c)P log Q(r) · (log)m+1 (r) 4 δ−δ (8c)−1 r · P log Q(r) · (log)m+1 (r) r · P log Q(r) . 3. Main results We consider the strongly continuous semigroup T (t) = e−At , for t 0, generated by the self-adjoint linear operator −A = − q(x)• on L2 (RN ). The following theorem is our main result. (Recall (Hq ) ⇒ (Hq ) ⇒ (Hq ).) Theorem 3.1. Let Hypothesis (Hq ) be satisfied. Then the Schrödinger heat semigroup {e−At : t 0} on L2 (RN ) is intrinsically ultracontractive, that is, the linear operator e−At is bounded from L2 (RN ) to X for each t > 0. There are several related versions of this theorem in the literature: E.B. Davies and B. Simon [12, Theorem 6.1, Parts (a) and (b), p. 358], respectively, have proved this result for a special case of the potentials q(x) = |x|2(1+δ) and q(x) = |x|2 (log |x|)2(1+δ) in one space dimension (N = 1), where δ > 0 and |x| r0 > 0. They mention [12, Remarks, p. 359] that also the case q(x) = |x|2 (log |x|)2 (log log |x|)2(1+δ) can be treated by the same methods. Another important result also covered by our Theorem 3.1 has been obtained in [12, Theorem 6.3, p. 359] and later generalized in E.B. Davies [10, Lemma 2, p. 183, Lemma 4, p. 185], and [11, Section 4.5, pp. 119–125], especially Theorem 4.5.4 on p. 121. There it is assumed that c1 |x|a q(x) c2 |x|b
holds for all |x| r0 > 0,
(28)
where a, b, c1 , c2 , and r0 are some positive constants, such that 2 < a b < 2(a − 1). In the previous articles [2–6], electric potentials with such a large variation were not allowed; cf. [4, condition (20), p. 220]. Unfortunately, our Theorem 3.1 does not cover potentials of the kind considered in [10, Lemma 3, p. 183]; a different approach to Rosen’s inequality is required for such potentials. Remark 3.2. Indeed, Ineqs. (28) guarantee the validity of Hypothesis (Hq ) and, hence, (Hq ) as well. To see this, take Q(r) = c2 r b for all r r0 > 0 and let −1/2
P (ξ ) = c1 c2
a − 1 − (b/2) ξ for all ξ ∈ R. exp b+1
Only Ineq. (15) remains to be verified; it holds if, for instance, r0 > 0 is chosen large enough, such that 1/2 P (b + 1) · log r c2 r (b/2)−1
for all r > r0 .
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However, fixing arbitrary constants δ > 0 and c3 > 0 and taking P such that (cf. (22)) c3−1 P (ξ ) = ξ 1+δ , ξ(log ξ )1+δ , ξ log ξ(log log ξ )1+δ , . . .
for ξ ξ0 ,
with ξ0 > 0 large enough, we can obtain Ineqs. (26) which are strictly weaker than (28), thanks to (b/2) + 1 < a b. Our strategy for the proof of Theorem 3.1 follows familiar steps from the aforementioned work in [10–12,23]. A crucial difference in our approach is a better, more general version of Rosen’s inequality (see Section 1) than the one used in [10, Lemma 2, p. 183, Lemma 4, p. 185], [11, Section 4.5, pp. 119–125], [12, Theorem 5.1, p. 355, Theorem 6.3, p. 359], and [23, Ineq. (6.17), p. 378]. We have already mentioned in the Introduction (Section 1) that statements (ii), (iii), and (iv) formulated there contain important consequences of Theorem 3.1 (which claims statement (i) — intrinsic ultracontractivity). A rigorous formulation of these statements needs the following preliminaries. Recall our hypothesis that q : RN → R is a continuous function that satisfies (2). By the weak maximum principle for parabolic Cauchy problems, the operator e−At is positive for each t 0, that is, for f ∈ L2 (RN ) and u(t) = e−At f we have ⇒
f 0 a.e. in RN
u(t) 0 a.e. in RN .
(29)
Consequently, given any constant C > 0, we have also |f | Cϕ
in RN
⇒
u(t) Ce−Λt ϕ
in RN ,
(30)
by linearity. Hence, the operator e−At maps X into itself with the operator norm e−Λt , by (30). For any complex number λ ∈ C that is not an eigenvalue of the operator A = − + q(x)• on L2 (RN ), we denote by −1 (A − λI )−1 = − + q(x) • −λI the resolvent of A on L2 (RN ) given by u(x) = (A − λI )−1 f (x),
x ∈ RN .
Given any λ < Λ, we combine formula (6) (where the integral on the right-hand side converges in the strong operator topology on L2 (RN )) with (29) and (30) to conclude that f 0
a.e. in RN
⇒
u 0 a.e. in RN
and |f | Cϕ
in RN
⇒
|u| C(Λ − λ)−1 ϕ
in RN .
Furthermore, the following implications are well known; cf. [11, Section 2.1, pp. 59–63] or [12, Section 3, pp. 343–349]:
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Proposition 3.3. Assume that q : RN → R is a continuous function that obeys (2) and let −∞ < λ < Λ. Then the following implications hold for the semigroup {e−At : t 0} and the resolvent (A − λI )−1 : (i) ⇒ (ii) ⇒ (iii) ⇒ (iv). For reader’s convenience, we verify this proposition in Section 6.3. Finally, for q(x) ≡ q(|x|) radially symmetric, we show the equivalence of statements (5), (iii), and (iv). Notice that in this case (5) reads ∞
q(r)−1/2 dr < ∞.
(31)
0
More precisely, we have Proposition 3.4. Assume that q : R+ → (0, ∞) is a locally absolutely continuous function that satisfies (2) and (13) with q(r) in place of Q(r), for some 0 < r0 < ∞ and 1 < γ 2. Then we have the equivalence relations (31) ⇔ (iii) ⇔ (iv). The implication (31) ⇒ (iii) has been obtained in [4, Theorem 3.2, Part (a), p. 221], even for q somewhat more general (within the class of radially symmetric potentials) than that allowed here, and also for q not necessarily radially symmetric. Furthermore, also (iii) ⇒ (iv) has been established in [4, Theorem 3.2], the proof of Part (a) ⇒ Part (b) on p. 247. Only the implication (iv) ⇒ (31) is new under our hypotheses on q(r); it follows immediately from Proposition 4.7 in our present work. Its proof is based on an asymptotic formula due to Ph. Hartman and A. Wintner [19, Eq. (xxv), p. 49] which we state in Lemma 4.4, Eq. (38). 4. Preliminary results In this section we first prove a ramification of B. S IMON’s comparison result [29, Theorem 8, p. 324], for the ground states corresponding to two different electric potentials (see Lemma 4.1 and Corollary 4.2 below). Then we state an asymptotic formula (Lemma 4.4) for the ground state ϕ ≡ ϕQ associated with a potential Q(r) of class (Q). 4.1. Simon’s comparison result The following comparison lemma is an important tool in our approach. In a closely related form it has already appeared in B. Simon [29, Theorem 8, p. 324]. Lemma 4.1. Let q1 , q2 : RN → R be two continuous functions, such that q1 (x) q2 (x) holds for all x ∈ RN satisfying |x| R, with some R 0, and q2 (x) → +∞ as |x| → ∞. Assume that the ground state energy Λq2 of the Schrödinger operator Aq2 = − + q2 (x)• on L2 (RN ) vanishes, Λq2 = 0, that is, the corresponding ground state ϕq2 of Aq2 satisfies the equation Aq2 ϕq2 = 0 in L2 (RN ). Furthermore, let uj : ΩR → (0, ∞); j = 1, 2, be two positive continuous functions on ΩR = {x ∈ RN : |x| R}, such that uj ∈ W 1,2 (RN ), qj u2j ∈ L1 (RN ), and both inequalities
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−u1 + q1 (x)u1 0 in ΩR ,
(32)
−u2 + q2 (x)u2 0 in ΩR
(33)
hold in the sense of distributions supported in the exterior domain ΩR = {x ∈ RN : |x| > R}. Then we have u2 (x) u2 (x) = sup < ∞. u (x) |x|=R u1 (x) |x|R 1 sup
A simple standard proof is given below. We will apply this lemma in the following special form due to [12, Lemma 6.2, p. 359]. Corollary 4.2. Assume that q1 , q2 : RN → R are two strictly positive, continuous functions, qj (x) q0 > 0 for every x ∈ RN ; j = 1, 2, such that q1 (x) → +∞ and q2 (x) − q1 (x) → +∞ as |x| → ∞. Then the ground states ϕq1 and ϕq2 corresponding to the potentials q1 , and q2 , respectively, satisfy ϕq2 γ ϕq1 in RN , with some constant 0 < γ < ∞. The conclusion of this corollary follows from Lemma 4.1 by setting uj = ϕqj ; j = 1, 2, and taking R 0 sufficiently large, such that q2 (x) − Λq2 − q1 (x) − Λq1 = q2 (x) − q1 (x) − (Λq2 − Λq1 ) 0 holds for all |x| R. The ground states ϕq1 and ϕq2 are continuous, by the following remark. Remark 4.3. Let u ∈ dom(A) be a solution of Eq. (7). Standard local Lp -regularity theory app 2,p plied to this equation with f ∈ Lloc (RN ) for some p with 2 p < ∞ guarantees u ∈ Wloc (RN ); see Gilbarg and Trudinger [16, Theorem 9.15, p. 241]. In particular, if p > N then u ∈ C 1 (RN ), by the Sobolev imbedding theorem [16, Theorem 7.10, p. 155]. Now it follows that the ground state ϕ ≡ ϕq satisfies ϕ ∈ C 1 (RN ). Finally, we apply the strong maximum and boundary point principles, which are due to J.-M. Bony [9] for weak solutions (see also P.-L. Lions [22]), in order to conclude that ϕ > 0 everywhere in RN . Proof of Lemma 4.1. Take γ = sup|x|=R (u2 (x)/u1 (x)); hence, 0 < γ < ∞, by the positivity and continuity of u1 and u2 on the sphere ∂ΩR = {x ∈ RN : |x| = R}. Consequently, the function v = γ u1 − u2 defined in ΩR satisfies v 0 on the sphere ∂ΩR . Moreover, in the sense of distributions supported in the exterior domain ΩR we have −v + q2 (x)v = γ −u1 + q2 (x)u1 − −u2 + q2 (x)u2 γ −u1 + q1 (x)u1 − −u2 + q2 (x)u2 0,
(34)
by q1 q2 in ΩR followed by Ineqs. (32) and (33). Next, notice that the negative part v − of v, def
the function v − = max{−v, 0}, vanishes identically on the sphere ∂ΩR ; we extend it from ΩR to the entire space RN by setting v − = 0 in BR (0) = x ∈ RN : |x| R = RN \ ΩR .
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Then v − satisfies also v − ∈ W 1,2 (RN ) and q2 (v − )2 ∈ L1 (RN ), by 0 v − u2 in RN . In particular, we have v − ∈ Vq2 . (Recall that the Friedrichs energy space Vq has been introduced in (9) with the help of the quadratic form Qq defined in (8).) Now we take advantage of Ineq. (34) to find 0 Qq2 v, v − = −Qq2 v − , v − 0 which forces Qq2 (v − , v − ) = 0. The ground state ϕq2 is the unique minimizer, up to a scalar multiple, of the quadratic form Qq2 . But the ground state energy Λq2 = 0, by our hypothesis, and therefore we must have v − = cϕq2 in RN , where c 0 is a constant. From v − = 0 in BR (0) we deduce c = 0. We have verified v = γ u1 − u2 0 in ΩR as claimed. 2 4.2. The Hartman–Wintner formula To state the Hartman–Wintner asymptotic formula [19], let us consider a more general setting for the eigenvalue problem Aϕ = Λϕ for the ground state ϕ corresponding to a potential q(x) = Q(|x|) (x ∈ RN ) of class (Q), namely, −u + Q |x| u = λu in ΩR = x ∈ RN : |x| > R
(35)
for some 0 < R < ∞. Here, λ ∈ R is arbitrary and a weak solution u is any function u ∈ 1,2 (RN ) satisfying Eq. (35) in the sense of distributions on ΩR . If u(x) ≡ ψ(|x|) is radially Wloc symmetric, then ψ : (R, ∞) → R satisfies the radial Schrödinger equation −ψ (r) −
N −1 ψ (r) + Q(r)ψ(r) = λψ(r), r
r > R.
(36)
Consequently, ψ is a C 2 function on (R, ∞). The following asymptotic formula for a positive solution ψ of (36), with ψ(r) → 0 as r → ∞, plays an essential role in our present work. Lemma 4.4. Let Q(r) be of class (Q) and λ ∈ R. Assume that, for some 0 < R < ∞, ψ : (R, ∞) → (0, ∞) is a C 2 function that satisfies Eq. (36), such that ψ(r) → 0 as r → ∞. Denote def
V (r) = Q(r) − λ +
(N − 1)(N − 3) 4r 2
for r > r0 ,
(37)
with r0 R large enough, so that V (r) > 0 for all r > r0 . Then we have r
(N −1)/2
−1/4
ψ(r) = cV (r)
exp η(r) −
1/2
V (t) r0
where c > 0 is a constant and η(r) → 0 as r → ∞.
r
dt ,
r > r0 ,
(38)
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This lemma is proved in Ph. Hartman and A. Wintner [19, pp. 81–82] for the unknown function v(r) = r (N −1)/2 ψ(r) > 0 satisfying the equation −v (r) + V (r)v(r) = 0,
r > R,
and the “boundary condition” r −(N −1)/2 v(r) = ψ(r) → 0 as r → ∞. It suffices to realize that this equation is equivalent with the radial Schrödinger equation (36) for ψ above. Formula (38) above corresponds to Eq. (xxv) on p. 49 and to Eq. (158) on p. 80 in [19]. Remark 4.5. Notice that also the potential V (r) defined in (37) belongs to class (Q) provided r0 > 0 is chosen large enough, so that V (r) > 0, by Q(r) → ∞ as r → ∞. Formula (38) still −3) remains valid if the potential V is replaced by Q. Here, the term −λ + (N −1)(N has been 4r 2 added for convenience only (easy comparison with the setting in [19]); it may be left out by taking r0 > 0 large enough. Corollary 4.6. Under the hypotheses of Lemma 4.4 above, there exist constants c1 , c2 > 0 such that r −log ψ(r) c1
Q(t)1/2 dt + c2 ,
r > r0 .
(39)
V (t)1/2 dt + c2 ,
r > r0 ,
(40)
r0
Proof. First, we prove the estimate r −log ψ(r) c1 r0
where c1 , c2 0 are some constants independent from r (r > r0 ). We rewrite (38) as N −1 1 −log ψ(r) = log r − log c + log V (r) − η(r) + 2 4
r V (t)1/2 dt,
r > r0 .
(41)
r0
Owing to Remark 2.1 about potentials of class (Q), we have V (r)/r → ∞ as r → ∞. Hence, it suffices to verify
log V (r) c1
r
V (t)1/2 dt + c2 ,
r > r0 ,
(42)
r0
where c1 , c2 > 0 are some constants. Owing to 1 < γ 2, the conjugate exponent γ = γ /(γ − 1) satisfies 2 γ < ∞. Hence, for r0 < r s < ∞ we have
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r log V (r) − log V (r0 ) =
V (t)V (t)−1 dt
r0
r
1/γ 1/γ d V (t)−1/2 V (t)1/2 V (t)1/2 dt. dt
= −2 r0
We apply Hölder’s inequality to estimate
log V (r) − log V (r0 ) 2
r
1/γ r
1/γ
d γ −1/2 1/2 1/2
V (t) dt V (t) dt
dt V (t)
r0
c
r0
1/γ
r V (t)1/2 dt
,
r0
where ∞
1/γ
d γ −1/2 1/2
V (t) dt c =2 < ∞,
dt V (t)
r0
by condition (13). These inequalities yield (42), as claimed. Thus, (40) follows by applying (42) to (41). Now we derive (39) from (40). The following inequalities hold for all t r0 , provided r0 > 0 is large enough: V (t)1/2 − Q(t)1/2 =
V (t) − Q(t) V (t)1/2 + Q(t)1/2
=
(N −1)(N −3) 4t 2 1/2 V (t) + Q(t)1/2
−λ +
|(N − 1)(N − 3)| |λ| + Q(t)−1/2 . 4t 2
The last inequality implies r
r 1/2
V (t) r0
dt
Q(t)1/2 dt + c ,
r > r0 ,
(43)
r0
where
|(N − 1)(N − 3)| c = |λ| + 4r02
∞
Q(t)−1/2 dt < ∞,
r0
by (14), provided r0 > 0 is large enough, such that V (r) > 0 for all r > r0 . Finally, we apply (43) to the estimate (40) to conclude that (39) holds with c2 = c1 c + c2 . 2
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If the Schrödinger heat semigroup {e−t Aq : t 0} on L2 (RN ) happens to be intrinsically ultracontractive, then every eigenfunction v ∈ L2 (RN ) of Aq , that is, Aq v = λv for some λ ∈ R, v = 0, must satisfy v ∈ Xq or, equivalently, supRN (|v|/ϕq ) < ∞, by Proposition 3.3. As a consequence, we obtain the following necessary condition for the intrinsic ultracontractivity, provided q(x) = Q(|x|) is a radially symmetric potential of class (Q). We verify the necessity of this condition below. Proposition 4.7. Let Q(r) be of class (Q) and −∞ < λ1 < λ2 < ∞. Assume that, for some 0 < R < ∞ and each j = 1, 2, ψj : (R, ∞) → R is a C 2 function that satisfies the radial Schrödinger equation (36) with λ = λj , such that ψj (r) → 0 as r → ∞. Furthermore, assume that there are constants C 0 and r0 R large enough, such that
0 < ψ2 (r) Cψ1 (r) Then condition (14) must hold, i.e.,
∞ r0
holds for all r > r0 .
(44)
Q(t)−1/2 dt < ∞.
This result is proved in Murata [23, Corollary 6.4, p. 376], for a special subclass of our class (Q ); he assumes that Q(r) satisfies the growth hypotheses (21) and rQ (r) kQ(r)
for all r > r0 ,
(45)
where k ∈ (2, ∞) is a constant; see [23, (6.3), p. 376, (1.7), p. 345], respectively. (Notice that Ineqs. (21) and (45) combined entail (13) for any γ > 1.) We do not need such a severe restriction on the growth of Q(r) as in (45) which is equivalent to r → r −k Q(r) being monotone decreasing for r > r0 . It is noteworthy that Murata [23] proves this result also for the nonradial Schrödinger equation with a potential q(x) in place of Q(|x|) in Eq. (36), such that |q(x) − Q(|x|)| C ≡ const < ∞ for all x ∈ RN . Proof of Proposition 4.7. We employ the asymptotic formula (38) from Lemma 4.4 above, with λ = λj and V (r) = Vj (r) = Q(r) − λj +
(N − 1)(N − 3) 4r 2
for r > r0 ,
so that Vj (r) > 0 for all r > r0 (j = 1, 2), to compute the expression 1 V2 (r) |ψ2 (r)| = c − · log + η(r) + log ψ1 (r) 4 V1 (r)
r
V1 (t)1/2 − V2 (t)1/2 dt,
(46)
r0
r > r0 , where c ∈ R is a constant, r0 R is large enough, and η(r) → 0 as r → ∞. From (37) and Q(r) → +∞ as r → ∞ we deduce also V2 (r)/V1 (r) → 1 as r → ∞. Consequently, we may apply (44) to (46) to conclude that the integral above converges, ∞ V1 (t)1/2 − V2 (t)1/2 dt < ∞. r0
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Notice that V1 (t) > V2 (t) holds by λ1 < λ2 . Now the conclusion of our proposition follows from these facts combined with the identity ∞ ∞ V1 (t)1/2 − V2 (t)1/2 dt = (λ2 − λ1 ) r0
r0
dt . (Q(t) − λ1 )1/2 + (Q(t) − λ2 )1/2
2
5. Rosen’s inequality J. ROSEN’s inequality [28, Eq. (5), p. 369], if valid, is an important “sufficient” tool for proving intrinsic ultracontractivity; cf. [10–12,23]. Below we give a version of this inequality, (47), that improves some earlier versions from [10, Lemma 2, p. 183, Lemma 4, p. 185], [11, Section 4.5, pp. 119–125], [12, Theorem 5.1, p. 355, Theorem 6.3, p. 359], and [23, Ineq. (6.17), p. 378]. Proposition 5.1. Let Hypothesis (Hq ) be satisfied. Recall that, in Ineq. (25), P : R → (0, ∞) is a strictly monotone increasing, continuous function that satisfies also P (R) = (0, ∞) together with (12). We denote by P˜ the inverse of the composition P ◦ log, that is, P˜ : (0, ∞) → (0, ∞) is the strictly monotone increasing, continuous function that satisfies P˜ ◦ P = exp on R, where exp stands for the (natural) exponential function. Then there exist some constants C1 > 0 and C2 > 0 (both large enough), such that the inequality −log ϕq (x) εq(x) + C1 · P˜ (C1 /ε) + C2
(47)
holds for all x ∈ RN and for all ε > 0. We prove Proposition 5.1 (Rosen’s inequality) in several steps. We begin with the following direct consequence of the classical Fenchel–Young inequality (see, e.g., R.A. Adams and J.J.F. Fournier [1, Ineq. (2), p. 264]): Lemma 5.2. Assume that f : R → (0, ∞) and f˜ : (0, ∞) → (0, ∞) are two strictly monotone increasing, continuous functions, such that f (R) = (0, ∞), f˜((0, ∞)) = (0, ∞), and f˜ ◦ f = exp. Then we have ab a · f (log a) + b · f˜(b)
for all a, b ∈ (0, ∞).
(48)
Proof. The functions f ◦ log and f˜ map (0, ∞) onto itself and are inverse to one another, by f˜ ◦ f = exp . Moreover, both of them tend to 0 (+∞, respectively) as their arguments go to 0 (+∞). Consequently, we may apply the classical Fenchel–Young inequality to conclude that a
b f (log s) ds +
ab 0
f˜(t) dt
for all a, b ∈ R+ .
0
Since both integrands f ◦ log and f˜ on the right-hand side are monotone increasing functions, Ineq. (48) follows. 2
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Proof of Proposition 5.1. In Lemma 5.2 above we make the following substitutions: First, we ˜ ˜ take the functions r f = P and f = P as defined in the statement of this proposition. Then we substitute a = r0 Q(t)1/2 dt for r > r0 , and replace b by b/ε for b > 0 and ε > 0, thus obtaining r
r 1/2
Q(t)
b
dt ε
1/2
Q(t)
r0
r
dt · P log
r0
1/2
Q(t)
dt
+ b · P˜ (b/ε)
r0
for all r > r0 , b > 0, and ε > 0, by (48) multiplied by ε. Consequently, applying the lower bound on q(x) from (25) to the right-hand side, we arrive at |x| b Q(t)1/2 dt εq(x) + b · P˜ (b/ε)
(49)
r0
for all x ∈ RN with |x| > r0 , b > 0, and ε > 0, provided r0 > 0 is large enough. Next, we apply (49) to the right-hand side of (39) in Corollary 4.6, used with the potential 2Q in place of Q, to get −log ϕ2Q |x| c1
|x| √ ε 1/2 2Q(t) dt + c2 c1 2 q(x) + P˜ (b/ε) + c2 b
(50)
r0
for all x ∈ RN with |x| > r0 , b > 0, and ε > 0, where c1 , c2 > 0 are some constants independent def √ from x, b, and ε. Now we make the special choice b = C1 = c1 2 in (50) which then becomes −log ϕ2Q |x| εq(x) + C1 · P˜ (C1 /ε) + c2
(51)
for all x ∈ RN with |x| > r0 , and for all ε > 0. Finally, we employ Corollary 4.2 (Simon’s comparison result) with q1 = q and q2 = 2Q, hence, q2 (x) − q1 (x) Q(|x|) for |x| r0 (r0 > 0 — large enough), to conclude that there exists a constant γ > 0 such that ϕ2Q γ ϕq in RN . We apply this inequality to the left-hand side of (51) to conclude that −log γ ϕq (x) −log ϕ2Q |x| εq(x) + C1 · P˜ (C1 /ε) + c2
(52)
for all x ∈ RN with |x| > r0 , and for all ε > 0. Consequently, choosing a constant C2 > 0 sufficiently large, such that C2 max −log ϕq (x) |x|r0
we derive the desired inequality (47) from (52).
and C2 c2 + log γ , 2
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6. Proofs of the main results In this section we give the proofs of our main theorem, Theorem 3.1, and Proposition 3.3 stated in Section 3. We follow similar procedures as in [11, Section 2.2, pp. 63–71], [12, Section 4, pp. 349–355], or [23, Section 6, pp. 375–385]. Let u ∈ C(R+ → L2 (RN )) be a mild solution of the following heat equation: ∂u = −Au + Λu in L2 RN , for t > 0; ∂t u(·, 0) = u0 in L2 RN ,
(53)
where the Schrödinger operator A = − + q(x)• on L2 (RN ), together with its ground state energy Λ ∈ R and its ground state ϕ ∈ L2 (RN ), have been specified in Section 2. Recall that Λ is a simple eigenvalue of A with the associated eigenfunction ϕ normalized by ϕ > 0 throughout RN and ϕL2 (RN ) = 1. The initial values at t = 0 satisfy u0 ∈ L2 (RN ). Hence, u(·, t) = e−At u0 in L2 (RN ) for t 0, which implies u(·, t) ∈ dom(A) for every t > 0, as the semigroup {e−At : t 0} on L2 (RN ) is holomorphic. We want to show that the function v(x, t) = u(x, t)/ϕ(x) of x ∈ RN is (essentially) bounded for every t > 0.
6.1. Preliminary calculations for
|u/ϕ|p(t) ϕ 2
Given any monotone increasing C 1 -function p : [0, T ) → [2, ∞), 0 < T ∞, we compute for t ∈ (0, T ), d dt
v(x, t) p(t) ϕ(x)2 dx
RN
d = dt
u(x, t) p(t) 2
ϕ(x) ϕ(x) dx RN
p−2
u dp
u
p
u
2 u ∂u
ϕ dx + =p
ϕ log ϕ ϕ dx ϕ ϕ ∂t dt RN
= −p
RN
p−2
u dp
u
p
u
2
u −u + q(x)u − Λu dx + log
ϕ
ϕ
ϕ ϕ dx dt
RN
RN
= −p
|v|
p−2
dp vϕ − + q(x) − Λ (vϕ) dx + dt
RN
|v|p log |v|ϕ 2 dx,
RN
by (53). We apply integration by parts to calculate the first integral above:
|v|p−2 vϕ(−)(vϕ) dx =
RN
RN
∇ |v|p−2 vϕ · ∇(vϕ) dx
(54)
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4119
(p − 1)|v|p−2 ϕ∇v + |v|p−2 v∇ϕ · (ϕ∇v + v∇ϕ) dx
= RN
= (p − 1)
|v|p−2 |∇v|2 ϕ 2 dx
RN
+p
|v|p−2 vϕ(∇v · ∇ϕ) dx +
RN
RN
= (p − 1)
|v|p−2 |∇v|2 ϕ 2 dx +
RN
∇ |v|p ϕ · ∇ϕ dx
RN
= (p − 1)
|v|p |∇ϕ|2 dx
|v|p−2 |∇v|2 ϕ 2 dx +
RN
|v|p ϕ(−ϕ) dx,
RN
which renders
|v|p−2 vϕ − + q(x) − Λ (vϕ) dx
RN
= (p − 1)
|v|
p−2
|v|p ϕ − + q(x) − Λ ϕ dx.
|∇v| ϕ dx + 2 2
RN
RN
We insert this integral into the time derivative (54) to get d dt
v(x, t) p(t) ϕ(x)2 dx
RN
= −p(p − 1)
+
dp dt
|v|
p−2
|∇v| ϕ dx − p 2 2
RN
|v|p ϕ − + q(x) − Λ ϕ dx
RN
|v|p log |v|ϕ 2 dx RN
2 2 2 2 = −p(p − 1) |∇g| ϕ dx − p |v|p ϕ − + q(x) − Λ ϕ dx p +
2 dp · p dt
RN
RN
|g|2 log |g|ϕ 2 dx,
(55)
RN
where we have substituted g = |v|(p/2)−1 v; hence ∇g = p(t) 2 for 0 t < T .)
p (p/2)−1 ∇v. 2 |v|
(Recall that p =
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6.2. Proof of Theorem 3.1 First, let us note that in our definition of class (Q) of “auxiliary” potentials Q(r) introduced in Section 2 we may assume that, in addition to the pair of inequalities (11) and (12), the function P (ξ ) satisfies also the following limited growth condition, P (ξ ) P (0) exp eξ − 1 for all ξ ∈ R.
(56)
Indeed, we may simply replace P (ξ ) by for every ξ ∈ R. Pˆ (ξ ) = min P (ξ ), P (0) exp eξ − 1 Clearly, if Ineqs. (11) and (12) are satisfied with a function P then so they are with Pˆ . Analogous arguments apply to the smaller class (Q ) with the pair of inequalities (15) and (12) (in place of (11) and (12), respectively). As for (12), it follows from for all ξ ∈ R. Pˆ (ξ )−1 P (ξ )−1 + P (0)−1 exp 1 − eξ Second, recall that C1 > 0 is one of the two constants from Rosen’s inequality (47) (see Proposition 5.1). Set ∞ T1 = 2C1 1 2
P (ξ )−1 dξ ;
(57)
log 2
hence 0 < T1 < ∞, thanks to (12). Let T ∈ (0, T1 ] be arbitrary, but fixed, and let the number ξ0 ≡ ξ0 (T ) ∈ R be determined by ∞ 1 2
P (ξ )−1 dξ = T /(2C1 );
(58)
log 2+ξ0
hence ξ0 0. Notice that ξ0 (T ) ∞ as T 0, owing to T1 < ∞. Next, we define an auxiliary function : [2, ∞) → (0, ∞) by 1 log p + ξ0 (p) = C1 P 2
for every p 2.
(59)
It satisfies (p) 0 as p ∞ and, by (58), also ∞ 2
∞
(p) dp = 2C1 p 1 2
P (ξ )−1 dξ = T .
(60)
log 2+ξ0
Now we are ready to specify the function p : [0, T ) → [2, ∞) used in Section 6.1 above. It is determined by the differential equation
B. Alziary, P. Takáˇc / Journal of Functional Analysis 256 (2009) 4095–4127
(p) dp · = 1 at all times t ∈ [0, T ), p dt
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(61)
subject to the initial condition p(0) = 2. (This initial value problem has a unique solution, by the separation of variables combined with (p) > 0 and dp/dt > 0.) Moreover, we have p(t) ∞ as t T , thanks to (60). Third, p is a monotone increasing C 1 -function which satisfies also the following differential inequality, owing to 2(p − 1) p for p 2, 2 d log(p − 1) dt (p)
for all t ∈ [0, T ).
Since also 2(p − 1) p 2 /2 for p 2, this differential inequality entails also p d log p dt 2(p)
for all t ∈ [0, T ).
We will take advantage of the following equivalent forms of these two inequalities later, −2(p − 1) +
dp 2 dp 0 and −p + · 0 for all t ∈ [0, T ), dt p dt
(62)
respectively. The main difficulty inthe remaining part of this proof is to find a suitable upper bound for the time-dependent integral RN |g|2 log |g|ϕ 2 dx in (55) above. To do this, we employ the expres sions RN |∇f |2 dx and RN |f |2 (−log ϕ) dx, where we have set f = gϕ. Ineqs. (62) will play an important role here. We begin with the well-known logarithmic Sobolev inequality |f |2 log |f | dx f 2L2 (RN ) log f L2 (RN ) RN
+ ε∇f 2L2 (RN ) +
γN N log f 2L2 (RN ) 4 ε
(63)
which holds for every ε > 0, where γN > 0 is some constant depending only on the dimension N . This version of the logarithmic Sobolev inequality can be found in Davies [11, Ineq. (3.2.1), p. 83]; it follows from Theorem 2.2.3 (p. 64) combined with Theorem 2.3.6 (p. 73) in [11]. We substitute f = gϕ in Ineq. (63) above to get
|g| log |g|ϕ dx = 2
RN
2
|f | log |f | dx + 2
RN
|g|2 (−log ϕ)ϕ 2 dx
RN
N γN f 2L2 (RN ) log f L2 (RN ) + ε∇f 2L2 (RN ) + log f 2L2 (RN ) 4 ε + |f |2 (−log ϕ) dx. RN
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In order to estimate the last integral from above, we need Rosen’s inequality (47) (see Proposition 5.1) −log ϕ(x) ε q(x) − Λ + Θ(ε),
x ∈ RN ,
(64)
ε > 0.
(65)
which holds for every ε > 0, where we have abbreviated def Θ(ε) = C1 · P˜ (C1 /ε) + C2 + εΛ,
Recall that C1 > 0 and C2 > 0 are some constants. Thus, we arrive at |g|2 log |g|ϕ 2 dx RN
γN N f 2L2 (RN ) f 2L2 (RN ) log f L2 (RN ) + ε∇f 2L2 (RN ) + log 4 ε q(x) − Λ |f |2 dx + Θf 2L2 (RN ) +ε RN
= f 2L2 (RN ) log f L2 (RN ) + ε +
|∇f |2 + q(x) − Λ |f |2 dx
RN
N γN log + Θ f 2L2 (RN ) 4 ε
= f 2L2 (RN ) log f L2 (RN ) +ε |∇g|2 ϕ 2 dx + |g|2 ϕ − + q(x) − Λ ϕ dx +
RN
RN
γN N log + Θ f 2L2 (RN ) 4 ε
(66)
where we have used |∇f |2 = |∇g|2 ϕ 2 + ∇(g 2 ϕ) · ∇ϕ. We apply the last inequality to estimate the last integral in (55): d f (·, t)2 2 N = d L (R ) dt dt
u(x, t) p(t) 2
ϕ(x) ϕ(x) dx RN
d = dt
g(x, t) 2 ϕ(x)2 dx
RN
2 2 |∇g|2 ϕ 2 dx − p |v|p ϕ − + q(x) − Λ ϕ dx −p(p − 1) p RN
RN
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2 dp · f 2L2 (RN ) log f L2 (RN ) p dt 2 dp 2 2 2 ε + · |∇g| ϕ dx + |g| ϕ − + q(x) − Λ ϕ dx p dt +
RN
RN
γN 2 dp N · log + Θ f 2L2 (RN ) p dt 4 ε dp 2 −2(p − 1) + = |∇g|2 ϕ 2 dx ε p dt +
RN
2 dp ε |g|2 ϕ − + q(x) − Λ ϕ dx + −p + · p dt RN
2 dp · f 2L2 (RN ) log f L2 (RN ) p dt γN 2 dp N log + Θ f 2L2 (RN ) . + · p dt 4 ε +
(67)
At this point we take advantage of our special choice of the function p(t) for 0 t < T (by Eq. (61) with p(0) = 2) and, in addition, set ε = (p(t)) defined in (59). We apply Ineqs. (62) together with (cf. (10)) ϕ − + q(x) − Λ ϕ 0
in RN
(68)
to (67) to estimate the time derivative (for 0 < t < T ) d f (·, t)2 2 N 2 · dp f 2 2 N log f 2 N L (R ) L (R ) L (R ) dt p dt γN 2 dp N log + Θ (p) f 2L2 (RN ) . + · p dt 4 (p) Next, we make use of the identity d 1 2 dp 2/p −2(p−1)/p d 2/p f L2 (RN ) = f L2 (RN ) f L2 (RN ) log f L2 (RN ) · f 2L2 (RN ) − 2 · dt p dt dt p to get d 2 dp 2/p f (·, t)2/p f L2 (RN ) log f L2 (RN ) 2· L2 (RN ) dt dt p γN 2 dp N 2/p log + Θ (p) f L2 (RN ) + 2· dt 4 (p) p
(69)
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2 dp 2/p f L2 (RN ) log f L2 (RN ) · p 2 dt 2 dp N γN 2/p = 2· log + Θ (p) f L2 (RN ) . dt 4 (p) p −
Finally, this implies (for 0 < t < T ) d log dt
1/p(t)
u(x, t) p(t)
ϕ(x)2 dx
ϕ(x)
RN
2/p γN d 2 dp N log + Θ (p) . = log f (·, t) L2 (RN ) 2 · dt dt 4 (p) p
(70)
In order to guarantee that the Lp(t) (RN )-norm of u(·, t)/ϕ remains uniformly bounded for all times t ∈ [0, T ), we will verify that γN 1 dp N log + Θ (p) ∈ L1 (0, T ) · (p) p 2 dt 4
as a function of t ∈ (0, T ).
This condition is equivalent with 1 N γN log + Θ (p) ∈ L1 (2, ∞) (p) p2 4
as a function of p ∈ (2, ∞),
(71)
which is verified as follows. Making use of (56) and (59) we obtain 1 log 1/(2) log 1/(p) = −log C1 + log P log p + ξ0 2 1 log p + ξ0 − 1 = eξ0 p 1/2 + K log P (0)/C1 + exp 2 for all p 2, where K = log(P (0)/C1 ) − 1 is a constant. This estimate guarantees p −2 log 1/(p) ∈ L1 (2, ∞)
as a function of p ∈ (2, ∞).
(72)
Next, recall that Θ = Θ((p)) has been defined in (65). Using (59) again we get also ∞
P˜ C1 /(p) p −2 dp =
2
∞ dp 1 log p + ξ0 P˜ P 2 p2 2
∞ =
∞ √ dp 1 ξ0 log p + ξ0 exp =e p −3/2 dp = eξ0 2 < ∞. 2 2 p
2
Finally, we combine (72) and (73) to get condition (71).
2
(73)
B. Alziary, P. Takáˇc / Journal of Functional Analysis 256 (2009) 4095–4127
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To complete the proof of Theorem 3.1, we let t T which entails p(t) ∞. Thus, with a help from RN ϕ(x)2 dx = 1, integrating Ineq. (70) with respect to time t ∈ (0, T ) we derive u(·, T ) = u(·, T )/ϕ ∞ N X L (R )
1/p(t)
u(x, t) p(t) 2
lim inf Cu0 L2 (RN ) ,
ϕ(x) ϕ(x) dx tT RN
where C ≡ C(T ) ∈ (0, ∞) is a constant depending only on the L1 (2, ∞)-norm of the function in (71). Since T ∈ (0, T1 ] is arbitrary, where T1 is independent from the initial data u0 ∈ L2 (RN ), by (57), we conclude that u(·, t)/ϕ ∈ L∞ (RN ) for every t > 0 and e−At (t > 0) is a bounded linear operator from L2 (RN ) to X, as desired. 6.3. Proof of Proposition 3.3 To obtain the first implication, (i) ⇒ (ii), we follow [4, Section 8, pp. 243–245] and [11, Section 2.1, pp. 59–63]. Let X = L1 (RN ; ϕ dx) denote the weighted Lebesgue space of all (equivalence classes of) Lebesgue-measurable functions f : RN → C with the norm def
f X =
|f |ϕ dx < ∞.
RN
Clearly, the Banach space X defined in (3) is the dual space of X with respect to the duality induced by the natural inner product on L2 (RN ). The embeddings X → L2 RN → X are dense and continuous. Now let us fix any real number t > 0 and consider the operator T (t) = e−At on L2 (RN ). We denote by e−At |X the restriction of e−At to X. Hence, e−At |X is a bounded linear operator on X with the operator norm e−Λt , by (30). Furthermore, e−At possesses a unique extension e−At |X to a bounded linear operator on X (by interpolation, see e.g. [4, Lemma 4.3, p. 227] for details on extensions of symmetric operators). Finally, it is obvious that e−At |X : X → X is the adjoint of e−At |X : X → X . By the definition of intrinsic ultracontractivity, the linear operator T (t) = e−At is bounded from L2 (RN ) to X for each t > 0. Hence, its extension e−At |X is a bounded linear operator from X to L2 (RN ). This implies that also the product e−2At = e−At e−At is a bounded linear operator from X to X. It follows that e−2At is a weakly compact operator from X to X , by the Dunford–Pettis criterion for weak compactness in the Lebesgue space X (see Dunford and Schwartz [13, Theorem IV.8.9, p. 292] or Edwards [15, Theorem 4.21.2, p. 274]). Finally, a corollary of the Dunford–Pettis theorem [13, Theorem VI.8.12, p. 508, Corollary VI.8.13, p. 510] guarantees that the square e−4At = e−2At e−2At of e−2At is a (strongly) compact operator from X to X . As t > 0 is arbitrary, each operator e−At on X is compact. The compactness of e−At on X now follows by Schauder’s theorem ([13, Theorem VI.5.2, p. 485] or [15, Corollary 9.2.3, p. 621]).
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The implication (ii) ⇒ (iii) is obtained directly from the proofs of Theorems 3.2 and 3.3 in A. Pazy [24, Section 2.3, pp. 48–49]. One calculates the resolvent using the Laplace transform (6) of the semigroup, i.e., −1
(A − λI )
∞ −At λt e f= f e dt
for all f ∈ L2 RN and λ < Λ,
(74)
0
where the Bochner integral converges absolutely in L2 (RN ). By implication (30) we know that for each t > 0 the operator T (t) = e−At maps the Banach space X into itself. Moreover, assuming (ii) we know also that the restriction T (t)|X of T (t) to X is compact from X into itself. We combine the compactness with implication (30) to conclude that the Laplace transform (74) exists as a Bochner integral that converges absolutely in X for all f ∈ X and λ < Λ. The compactness of the restriction (A − λI )−1 |X of (A − λI )−1 to X now follows from that of each T (t)|X (t > 0) exactly as in the proofs of Theorems 3.2 and 3.3 in [24, Section 2.3, pp. 48–49]. Here, one has to employ the fact that the set of all compact linear operators on X is a closed linear subspace in the Banach space of all bounded linear operators on X endowed with the uniform operator norm. The remaining implication (iii) ⇒ (iv) has been established in [4, Theorem 3.2], the proof of Part (a) ⇒ Part (b) on p. 247. The proposition is proved. Acknowledgments This work was supported in part by le Ministère des Affaires Étrangères (France) and the German Academic Exchange Service (DAAD, Germany) within the exchange program “PROCOPE” between France and Germany. References [1] R.A. Adams, J.J.F. Fournier, Sobolev Spaces, second ed., Academic Press, New York, 2003. [2] B. Alziary, J. Fleckinger, P. Takáˇc, An extension of maximum and anti-maximum principles to a Schrödinger equation in R2 , J. Differential Equations 156 (1999) 122–152. [3] B. Alziary, J. Fleckinger, P. Takáˇc, Positivity and negativity of solutions to a Schrödinger equation in RN , Positivity 5 (4) (2001) 359–382. [4] B. Alziary, J. Fleckinger, P. Takáˇc, Ground-state positivity, negativity, and compactness for a Schrödinger operator in RN , J. Funct. Anal. 245 (2007) 213–248, doi:10.1016/j.jfa.2006.12.007. [5] B. Alziary, P. Takáˇc, A pointwise lower bound for positive solutions of a Schrödinger equation in RN , J. Differential Equations 133 (2) (1997) 280–295. [6] B. Alziary, P. Takáˇc, Compactness for a Schrödinger operator in the ground-state space over RN , in: Proceedings of the 2006 International Conference on Partial Differential Equations and Applications, in Honor of Jacqueline Fleckinger, Toulouse, France, June 30–July 1, 2006, Electron. J. Differ. Equ. Conf. 16 (2007) 35–58. [7] A. Ancona, First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains, J. Anal. Math. 72 (1997) 45–92. [8] R. Bañuelos, Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators, J. Funct. Anal. 100 (1991) 181–206. [9] J.-M. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris Sér. A 265 (1967) 333–336. [10] E.B. Davies, Criteria for ultracontractivity, Ann. Inst. H. Poincaré 43 (2) (1985) 181–194. [11] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge, 1989. [12] E.B. Davies, B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal. 59 (2) (1984) 335–395.
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Journal of Functional Analysis 256 (2009) 4128–4161 www.elsevier.com/locate/jfa
Remarks on the non-commutative Khintchine inequalities for 0 < p < 2 Gilles Pisier a,b,∗,1 a Texas A&M University, College Station, TX 77843, USA b Université Paris 6 (UPMC), Institut Math. Jussieu (Analyse Fonctionnelle), Case 186, 75252 Paris cedex 05, France
Received 9 October 2008; accepted 18 November 2008 Available online 28 November 2008 Communicated by N. Kalton
Abstract We show that the validity of the non-commutative Khintchine inequality for some q with 1 < q < 2 implies its validity (with another constant) for all 1 p < q. We prove this for the inequality involving the Rademacher functions, but also for more general “lacunary” sequences, or even non-commutative analogues of the Rademacher functions. For instance, we may apply it to the “Z(2)-sequences” previously considered by Harcharras. The result appears to be new in that case. It implies that the space n1 contains (as an operator space) a large subspace uniformly isomorphic (as an operator space) to Rk + Ck with k ∼ n1/2 . This naturally raises several interesting questions concerning the best possible such k. Unfortunately we cannot settle the validity of the non-commutative Khintchine inequality for 0 < p < 1 but we can prove several would be corollaries. For instance, given an infinite scalar matrix [xij ], we give a necessary and sufficient condition for [±xij ] to be in the Schatten class Sp for almost all (independent) choices of signs ±1. We also characterize the bounded Schur multipliers from S2 to Sp . The latter two characterizations extend to 0 < p < 1 results already known for 1 p 2. In addition, we observe that the hypercontractive inequalities, proved by Carlen and Lieb for the Fermionic case, remain valid for operator space valued functions, and hence the Kahane inequalities are valid in this setting. Published by Elsevier Inc. Keywords: Khintchine inequalities; Non-commutative Lp space
* Address for correspondence: Université Paris 6 (UPMC), Institut Math. Jussieu (Analyse Fonctionnelle), Case 186,
75252 Paris cedex 05, France. E-mail address: [email protected]. 1 Partially supported by NSF grant 0503688 and ANR-06-BLAN-0015. 0022-1236/$ – see front matter Published by Elsevier Inc. doi:10.1016/j.jfa.2008.11.014
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The non-commutative Khintchine inequalities play a very important rôle in the recent developments in non-commutative Functional Analysis, and in particular in Operator Space Theory, see [28,27]. Just like their commutative counterpart for ordinary Lp -spaces, they are a central tool to understand all sorts of questions involving series of random variables, or random vectors, in relation with unconditional or almost unconditional convergence in non-commutative Lp [30]. The commutative version is also crucial in the factorization theory for linear maps between Lp spaces [21,22] in connection with Grothendieck’s theorem. The non-commutative analogues of Grothendieck’s theorem reflect the same close connection with the Khintchine inequalities, see e.g. the recent paper [36]. Moreover, in the non-commutative case, further motivation for their study comes from Random Matrix Theory and Free Probability. For instance one finds that the Rademacher functions (i.e. i.i.d. ±1-valued) independent random variables satisfy the same inequalities as the freely independent ones in non-commutative Lp for p < ∞. For reasons that hopefully will appear below, the case p < 2 is more delicate, and actually the case p < 1 is still open. When p < 2, let us say for convenience that a sequence (fk ) in classical Lp satisfies the classical Khintchine inequality KI p if there is a constant cp such that for all finite scalar sequences (aj ) we have
|aj |2
1/2
aj fj . cp p
in L2 . Then it is easy to see that if p < q < 2, KI q imNow assume that (fk ) is orthonormal plies KI p . Indeed, let S = aj fj . Let θ be such that 1/q = (1 − θ )/p + θ/2. We have
|aj |2
1/2
cq Sq cq Sp1−θ Sθ2 = cq Sp1−θ
|aj |2
θ/2 ,
and hence after a suitable division we obtain KI p with cp = (cq )1/(1−θ) . The heart of this simple argument is that the span of the sequence (fk ) is the same in Lp and in Lq or in L2 . In sharp contrast, the analogue of this fails for operator spaces. The span of the Rademacher functions in Lp is not isomorphic as operator space to its span in Lq , although they have the same underlying Banach space. This is reflected in the form of the non-commutative version of the Khintchine inequalities first proved by Lust-Piquard in [16] and labelled as (Khq ) below for the case of noncommutative Lq . Nevertheless, it turns out that the above simple minded extrapolation argument can still be made to work, this is our main result but this requires a more sophisticated version of Hölder’s inequality, that (apparently) forces us to restrict ourselves to p 1. Let 1 q 2. Let (rk ) be the Rademacher functions on Ω = [0, 1]. Let (xk ) be a finite sequence in a non-commutative Lq -space. The non-commutative Khintchine inequalities say (when 1 q 2) that there is a constant βq independent of x = (xk ) such that q 1/q |||x|||q βq rk (t)xk dt
(0.1)
q
where def
|||x|||q =
inf
xk =ak +bk
1/2 1/2 ak∗ ak bk bk∗ + . q
q
(0.2)
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This was first proved in [16] for 1 < q < 2 and in [18] for q = 1 (the converse inequality is easy and holds with constant 1). One of the two proofs in [18] derives this from the non-commutative (little) Grothendieck inequality proved in [25]. In this paper, we follow an approach very similar to the original one in [25] to show that the validity of (0.1) for some q with 1 < q < 2 implies its validity (with another constant) for all 1 p < q; we also make crucial use of more recent ideas from [13]. For that deduction the only assumption needed on (rk ) is its orthonormality in L2 ([0, 1]). Thus our approach yields (0.1) also for more general “lacunary” sequences than the Rademacher functions. For instance, we may apply it to the “Z(2)-sequences” considered in [9] (see also [10,2]). The result appears to be new in that case. Our argument can be viewed as an operator space analogue of the classical fact, in Rudin’s style [31], that if a sequence of characters Λ spans a Hilbert space in Lq (G) (G compact Abelian group, e.g. G = T) for some q < 2 then it also does for all p < q. It implies that the space n1 contains (as an operator space) a large subspace uniformly isomorphic (as an 1
operator space) to Rk + Ck with k ∼ n 2 . Another corollary (see Theorem 4.2) is that there is a constant c such that, for any n, the usual “basis” of S1n contains a c-unconditional subset of size n3/2 . This opens the door to various questions concerning the best possible size of such subspaces and subsets. See the end of Section 1 for some speculation on this. Unfortunately we cannot prove our result (at the time of this writing) for 0 < p < 1, for lack of a proof of Step 3 below. Thus we leave open the validity of (0.1) for 0 < q < 1. Nevertheless we will be able to prove several partial results in that direction. In particular (see Sections 3 and 5), if 0 < p 2, given arbitrary scalar coefficients [xij ], we give a necessary and sufficient condition for the random matrix [±xij ] to be in the Schatten class Sp for almost all choices of signs. This happens iff [xij ] admits a decomposition of the form xij = aij + bij with i
p/2 |aij |2
<∞
and
j
j
p/2 |bij |2
< ∞.
i
We also show that [xij ] defines a bounded Schur multiplier from S2 to Sp iff it admits a decomposition of the form xij = ψij + χij with i
sup |ψij |2p/(2−p) < ∞ and j
j
sup |χij |2p/(2−p) < ∞. i
In those two results, only the case 0 < p < 1 is new. In passing we remind the reader that when 0 < p < 1, Lp -spaces (commutative or not), in particular the Schatten class Sp , are not normed spaces. They are only p-normed, i.e. for any pair x, y in the space we have x + yp xp + yp . In the final section, we turn to the Kahane inequalities. Recall that the latter are a vector valued version of the Khintchine inequalities valid for functions with values in an arbitrary Banach space. It is natural to wonder whether there are non-commutative analogues when one uses the operator space valued non-commutative Lp -spaces introduced in [27]. We observe that the hypercontractive inequalities, proved by Carlen and Lieb [7] for the Fermionic case, remain
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valid for operator space valued functions, and hence the Kahane inequalities are valid in this Fermionic setting. The point of this simple remark is that Kahane’s inequality now appears as the Bosonic case. The same remark is valid in any setting for which hypercontractivity has been established. This applies in particular to Biane’s free hypercontractive inequalities [5]. 1. The case 1 p < 2 Actually, L2 ([0, 1]) or L2 (G) can be replaced here by any non-commutative L2 -space L2 (ϕ) associated to a semi-finite generalized (i.e. “non-commutative”) measure space, and (rk ) is then replaced by an orthonormal sequence (ξk ) in L2 (ϕ). Then the right-hand side of (0.1) is replaced by ξk ⊗ xk Lq (ϕ×τ ) . More precisely, by a (semi-finite) generalized measure space (N, ϕ) we mean a von Neumann algebra N equipped with a faithful, normal, semi-finite trace ϕ. Without loss of generality, we may always reduce consideration to the σ -finite case. Throughout this paper, we will use freely the basics of non-commutative integration as described in [23] or [33, Chapter IX]. Let us fix another generalized measure space (M, τ ). The inequality we are interested in now takes the following form:
(Kq )
⎧ ∃βq such that for any finite sequence ⎪ ⎪ ⎪ ⎪ ⎨ x = (xk ) in Lq (τ ) we have ξ |||x||| β ⊗ x ⎪ q q k k ⎪ ⎪ Lq (ϕ×τ ) ⎪ ⎩ where ||| · |||q is defined as in (0.2).
In the Rademacher case, i.e. when (ξk ) = (rk ), we denote (Khq ) instead of (Kq ), and we refer to these as the non-commutative Khintchine inequalities. We can now state our main result for the case q 1. Theorem 1.1. Let 1 < q < 2. Recall that (ξk ) is assumed orthonormal in L2 (ϕ). Then (Kq ) ⇒ (Kp ) for all 1 p < q. Here is a sketch of the argument. We denote S=
ξk ⊗ xk .
Let D be the collection of all “densities,” i.e. all f in L1 (τ )+ with τ (f ) = 1. Fix p with 0 < p q. Then we denote for x = (xk )
ξk ⊗ yk Cq (x) = inf q
where · q is the norm in Lq (ϕ ⊗ τ ) and the infimum runs over all sequences y = (yk ) in Lq (τ ) for which there is f in D such that 1 1 1−1 − xk = f p q yk + yk f p q /2.
Note that Cp (x) = Sp .
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Remark 1.2. Assume xk = xk∗ for all k. Then 1/2
|||x|||q = inf αk∗ αk
(1.1)
q
where the infimum runs over all decompositions xk = Re(αk ) = αk + αk∗ /2. Indeed, xk = ak + bk implies xk = Re(ak + bk∗ ). Let αk = ak + bk∗ . We have (assuming q 1) 1/2 1/2 1/2 ak∗ ak bk bk∗ αk∗ αk + . q
q
q
Therefore inf ( αk∗ αk )1/2 q |||x|||q . Since the converse inequality is obvious, this proves (1.1). The proof of Theorem 1.1 is based on a variant of “Maurey’s extrapolation principle” (see [21, 22]). This combines three steps: (here C , C , C , . . . are constants independent of x = (xk ) and we wish to emphasize that here p remains fixed while the index q in Cq (x) is such that p < q 2). Step 1. Assuming (Kq ) we have |||x|||p C Cq (x). Step 2. C2 (x) C |||x|||p . Actually we will prove also the converse inequality (up to a constant). Step 3. Cq (x) C Cp (x)1−θ C2 (x)θ θ where q1 = 1−θ p + 2 . (Recall p < q < 2 so that 0 < θ < 1.) The three steps put all together yield
θ |||x|||p C C Cp (x)1−θ C |||x|||p and hence |||x|||p C Cp (x) = C Sp .
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Proof of Step 1. This is easy: We simply apply (Kq ) to y = (yk ). More precisely, fix ε > 0. Let 1
−1
1
−1
y = (yk ) and f in D such that xk = (f p q yk + yk f p q )/2 and ξk ⊗ yk < Cq (x)(1 + ε). q
By (Kq ) we have |||y|||q < βq Cq (x)(1 + ε). Let ak , bk be such that yk = ak + bk with 1/q 1/2 bk bk∗ ak∗ ak + βq Cq (x)(1 + ε) q
we have 1
2xk = f p
− q1
1
− q1
(ak + bk ) + (ak + bk )f p
.
But it is easy to check that for some g, h ∈ D there are αk , βk such that 1
ak = αk g q
− 12
1
bk = h q
,
− 12
βk
with
αk 22
1/2
1/2 ak∗ ak
q
and
βk 22
1/2
1/2 bk bk∗ . q
Thus we find 1
2xk = f p Let
1 r
=
1 p
− q1
1
αk g q
− 12
1
+f p
− q1
1
hq
− 12
1
βk + αk g q
− 12 . Note that by Hölder’s inequality (since
1 p
− 12
fp
1
−
1 2
− q1
1
+ hq
− 12
1
− q1
1
hq
− 12
1
βk + αk g q
|||x |||p
− 12
1
fp
αk 22
− q1
1/2
. Then, again by Hölder, we have +
βk 22
1/2
βq Cq (x)(1 + ε).
Similarly, let 1
xk = f p
− q1
1
αk g q
− 12
1
+ hq
− 12
1
βk f p
We claim that |||x |||p βq Cq (x)(1 + ε).
− q1
.
= ( p1 − q1 ) + ( q1 − 12 ))
1 1 1 1 1 −1 1−1 f p q h q 2 1 and g q − 2 f p − q 1. r r Let xk = f p
1
βk f p
− q1
.
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Thus we obtain, since 2x = x + x 2|||x|||p |||x |||p + |||x |||p 2βq Cq (x)(1 + ε), and hence Step 1 holds with C = βq . We now check the above claim. Define θ by 1
xk (z) = f p
1 q
=
1 − q(z)
+
1−θ p
1
αk g q(z)
θ 2
and let
− 12
def
1 z where q(z) = 1−z p + 2 . We will use the probability measure μθ on the boundary of the complex strip S = {0 < (z) < 1} that is the Jensen (i.e. harmonic) measure for the point θ . This gives mass θ (resp. 1 − θ ) to the vertical line { (z) = 1} (resp. { (z) = 0}). By perturbation, we may assume that f and g are suitably bounded below so that xk (.) is a “nice” Lp (τ )-valued analytic ¯ Then, since q(θ ) = q, we have by Cauchy’s function on S, i.e. bounded and continuous on S. formula 1 1 1 1 − − f p q αk g q 2 = xk (θ ) = xk (z) dμθ (z), (z)∈{0,1} 1
but ∀t ∈ R xk (it) = U (it)αk V (it)g p U (it) = U (1 + it) = f
it ( p1 − 12 ) 1
fp
− q1
− 12
1
and xk (1 + it) = f p
and V (it) = V (1 + it) = g 1
αk g q
− 12
(0)
1
= (1 − θ )αk g p
− 12
− 12
U (1 + it)αk V (1 + it), where
it ( 12 − p1 ) 1
+ θf p
are unitary. This yields
− 12
(1)
αk
where α (1) (resp. α (0) ) are the corresponding averages over { (z) = 1} (resp. { (z) = 0}) satis (0) (1) fying ( αk 22 )1/2 ( αk 22 )1/2 and ( αk 22 )1/2 ( αk 22 )1/2 , and hence we find 1 1 1 1 1 1 1 1 − − − − |||(f p q αk g q 2 )|||p ( αk 22 )1/2 . Similarly, we find |||(h q 2 βk f p q )|||p ( βk 22 )1/2 . Thus we obtain as claimed 1/2 1/2 + βq Cq (x)(1 + ε). 2 |||x |||p αk 22 βk 22 Proof of Step 2. Assume |||x|||p < 1, i.e. xk = ak + bk with 1/2 1/2 bk bk∗ ak∗ ak + < 1. p
p
By semi-finiteness of τ , we may assume there f0 > 0 in D. (In the finite case we can exists simply take f0 = 1.) Let f = (ε(f0 )2/p + ak∗ ak + bk bk∗ )1/2 . We can choose ε > 0 small enough so that f p < 2 p
(using the fact that Lp/2 (τ ) is p/2-normed). We can then write xk = ak f + f bk
G. Pisier / Journal of Functional Analysis 256 (2009) 4128–4161
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where ak = ak (f )−1 and bk = (f )−1 bk . Let f = (f )p (τ (f p ))−1 . Note that f ∈ D and we have 1
xk = αk f p
− 12
1
+f p
− 12
(1.2)
βk
where 1
1
αk = f p ak f 2 , Note that
ak ∗ ak = (f )−1
αk 22
βk = f p f 2 bk .
ak∗ ak (f )−1 1 and similarly
1/2
bk bk ∗ 1. Therefore
1/2 1 1 ak ∗ ak = f p f 2 f p 2 p , 2
and similarly
βk 22
1/2
1
2p .
We will now modify this to obtain αk = βk . More precisely we claim there are yk in L2 (τ ) such 1
that xk = (f p
− 12
1
yk + yk f p
Let
1 r
=
1 p
− 12
yk 22
)/2 and
1/2
2
αk 22
1/2
+
βk 22
1/2
(1.3)
.
− 12 . To prove this claim, let E be the dense subspace of L2 (τ ) ⊕ · · · ⊕ L2 (τ ) formed 1
1
of families h = (hk ) such that f r hk + hk f r ∈ L2 (τ ) for all k. Then for all h in E we have 1 1 xk , hk = αk f r + f r βk , hk and hence 1/2 1/2 1 1/2 1/2 hk f 1r 2 f r hk 2 αk 22 xk , hk + βk 22 . 2 2 1
1
1
By an elementary calculation one verifies easily that f r hk 22 f r hk + hk f r 22 and similarly 1
1
1
hk f r 22 f r hk + hk f r 22 . Therefore we find 1/2 1/2 1 1/2 f r hk + hk f 1r 2 αk 22 + βk 22 . xk , hk 2 From this our claim that there are (yk ) in L2 (τ ) satisfying (1.3) follows immediately by duality. Then (1.3) implies 1/2 1 ξk ⊗ y k = yk 22 4 · 2p , 2
and we obtain C2 (x) 2
2+ p1
|||x|||p , completing Step 2.
2
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Remark 1.3. Conversely we have |||x|||p C2 (x). 1
Indeed, if C2 (x) < 1 then xk = (f p
− 12
1
yk + yk f p
− 12
(1.4)
)/2 with
1/2 ξk ⊗ y k = yk 22 < 1. 2
Let now 1 1 1 −1 − ak = yk f p 2 /2 and bk = f p 2 yk /2.
We have (recall the notation |T | = 2 and hence (setting
1 r
=
1 p
√
T ∗T )
ak∗ ak
1/2
1/2 1 1 − = yk∗ yk f p 2
− 12 ) by Hölder
1/2 1/2 1 1 p−2 yk∗ yk ak∗ ak < 1. 2 f r 2
p
Similarly ( bk bk∗ )1/2 p < 1/2. Thus we obtain |||x|||p < 1. By homogeneity this proves (1.4). 1
1
Proof of Step 3. Fix ε > 0. Let yk be such that xk = (f r yk + yk f r )/2 with ξk ⊗ yk < C2 (x)(1 + ε). 2
Let us assume that (M, τ ) is Mn equipped with usual trace. We will use the orthonormal basis for which f is diagonal with coefficients denoted by (fi ). We have then 1 1 −1 (yk )ij = 2 fi r + fjr (xk )ij .
We define yk (θ ) by setting θ θ −1 yk (θ )ij = 2 fi r + fjr (xk )ij .
Note that yk (0) = xk while yk (1) = yk . Let T (θ ) = ξk ⊗ yk (θ ). 1 1−θ θ We claim that if q = p + 2 and 1 p < q 2 T (θ ) cT (0)1−θ T (1)θ p 2 q
(1.5)
for some constant c depending only on p and q. As observed in [13], when p > 1, this is easy to prove using the boundedness of the triangular projection on Sp . The case p = 1 is a consequence
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of Theorem 1.1a in [13] (the latter uses [26, Theorem 4.5]). See Appendix A for a detailed justification. Therefore we obtain θ Cq (x) T (θ )q 21−θ cCp (x)1−θ C2 (x)(1 + ε) , i.e. we obtain Step 3 in the matricial case. Note that the argument works assuming merely that the density f has finite spectrum. 2 Proof of Theorem 1.1. Combining the 3 steps, we have already indicated the proof in the case M = Mn or assuming merely that the density f has finite spectrum. We will now prove the general semi-finite case. We return to Step 2. We claim that for any δ > 0 we can find (xk ) such that |||(xk ) − (xk )|||p < δ|||x|||p and such that 1
C2 (x ) 2 · 2 p (1 + δ)|||x |||p , where the definition of C2 (x ) is now restricted to densities with finite spectrum. Indeed, one may assume by homogeneity that 0 < |||x|||p < 1. Let r be defined by 1r = p1 − 12 . Let δ = (δ/n)|||x|||p . Then let f, yk , . . . be as in the above proof of Step 2 and let g ∈ D be an 1
1
1
element with finite spectrum such that f r − g r r < (2 · 2 p )−1 δ . Note that g exists by the semi-finiteness of τ . Then let 1 1 xk = g r yk + yk g r /2. Note that (by Hölder) xk − xk p < δ and hence (assuming p 1) |||x − x |||p < δ|||x|||p . We now observe that the proof of Step 3 applies if we replace (x, f ) by (x , g). Thus if we apply the three steps to x we obtain for some constant C4 ξk ⊗ xk . |||x |||p C4 Cp (x ) = C4 p
But since (xk ) is an arbitrary close perturbation of (xk ) in Lp -norm, we conclude that (Kp ) holds. 2 Remark 1.4. In Theorem 1.1, the assumption that (ξk ) is orthonormal in L2 (ϕ) (that is only used in Step 2) can be replaced by the following one: for any finite sequence y = (yk ) in L2 (M, τ ) we have ξk ⊗ y k
L2 (ϕ×τ )
yk 22
1/2
.
(1.6)
The proof (of Step 2) for that case is identical. Assume for simplicity that (M, τ ) is Mn equipped with its usual trace. Let S = ξk ⊗ xk , xk ∈ Mn . Equivalently S = [Sij ] with Sij ∈ L2 (ϕ). Consider f ∈ D. The proof of Step 3 becomes straightforward if (1.5) holds. In the case p 1, we invoked [13] to claim that (1.5) is indeed true, but we do not know whether it still holds when 0 < p < 1.
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Nevertheless, there is a situation when (1.5) is easy to check, when the following condition (γ , γ ) holds: Condition (γ , γ ). Let γ , γ be positive numbers. We say that S = ξk ⊗ xk satisfies the 1 1 r r condition (γ , γ ) if we can find f in D and yk such that xk = (f yk + yk f )/2 and such that T = ξk ⊗ yk satisfies simultaneously the following two bounds: T 2 γ C2 (x), 1 ⊗ f 1r T γ Sp . p
(1.7) (1.8)
If we set F = 1 ⊗ f , we can rewrite (1.8) as 1 F r T γ F 1r T + T F 1r /2, p p
(1.9) 1
1
and hence by the triangle inequality (or its analogue for p < 1), since S = (F r T + T F r )/2, we have automatically for a suitable γ (depending only on γ and p) T F 1r γ Sp . p
(1.10)
Remark 1.5. The reason why condition (γ , γ ) resolves our problem is that the one-sided version of (1.5) is quite easy: we have −θ F r S S1−θ F − 1r S θ . p 2 q
(1.11)
1
Indeed, if we let T = F − r S then (1.11) becomes 1−θ F r T F 1r T 1−θ T θ 2 p q
(1.12)
and the latter holds by Lemma 1.8 below. Theorem 1.6. Let (ξk ) be a sequence in L2 (ϕ) orthonormal or merely satisfying (1.6). Let 0 < p < q < 2. Then, if we assume the condition (γ , γ ) (as above but for any S), the implication (Kq ) ⇒ (Kp ) holds, where the resulting constant βp depends on p, q, βq and also on γ , γ . For simplicity we will prove this again assuming that (M, τ ) is Mn equipped with its usual trace. See the above proof of Theorem 1.1 for indications on how to check the general case. Remark 1.7. If (Kp ) holds, then there are constants (γ , γ ) depending only on p such that the condition (γ , γ ) holds. Indeed by the above proof of Step 2 we have 1
1
2xk = f r yk + yk f r with
yk 22
1/2
C |||x|||p
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and hence by Hölder and (1.6) 1/2 1 2 F r T T 2 y C |||x|||p . k 2 p Now if (Kp ) holds we have 1 1 |||x|||p βp Sp = βp /2F r T + T F r p 1
1
1
therefore we find F r T p C βp /2F r T + T F r p i.e. (1.9) holds. The following two lemmas will be used. Lemma 1.8. Let (M, τ ) be a generalized measure space. Consider F 0 in L1 (τ ). Assume θ 0 < p < q < 2. Let 1r = p1 − 12 and let θ be such that q1 = 1−θ p + 2 . Then for any V in L2 (τ ) we have 1−θ F r V F 1r V 1−θ V θ 2 p q and 1 1−θ V F 1−θ r V F r V θ2 . p q Proof. It suffices to show 1 1 1 1 V F (1−θ)( p − 2 ) V F p − 2 1−θ V θ 2 p q
(1.13)
since we obtain the other inequality by replacing V by V ∗ . Since the complex interpolation of non-commutative Lp -spaces is valid in the whole range 0 < p < ∞ [35], this can be deduced from the 3 line lemma. Alternatively, this also follows from Hölder’s inequality, together with [15]. Indeed, 1 1 1 1 1 1 V F (1−θ)( p − 2 ) = |V |F (1−θ)( p − 2 ) = |V |θ |V |1−θ F (1−θ)( p − 2 ) q q q and hence by Hölder (recall
1 q
=
1−θ p
+ θ2 )
( 1 − 1 )(1−θ) V θ2 |V |1−θ F p 2
p 1−θ
.
But by [15] (see also [1]) we have 1−θ ( 1 − 1 )(1−θ) |V | F p 2 and hence we obtain (1.13).
2
p 1−θ
1 1 1−θ − |V |F p 2 p
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Lemma 1.9. (See [13].) Let Qj (j = 1, . . . , n) be mutually orthogonal projections in M and let λj (j = 1, . . . , n) be non-negative numbers. (i) For any 1 q ∞ and any x in Lq (τ ) n λ ∨λ i j Qi xQj λi + λj
3 xLq (τ ) 2
n λ ∧λ i j Qi xQj λi + λj
1 xLq (τ ) . 2
i,j =1
Lq (τ )
and
i,j =1
Lq (τ )
(ii) For any 1 < q < ∞ there is a constant t (q), depending only on q, such that for any x in Lq (τ ) n λi Qi xQj λi + λj i,j =1
t (q)xLq (τ )
Lq (τ )
and n λj Qi xQj λi + λj i,j =1
t (q)xLq (τ ) .
Lq (τ )
(iii) For any s with 1 < q < s ∞, any density f ∈ D and any x ∈ Ls (τ ), we have 1−1 1 1 1 − 1 1 − 1 − max f q s x q , xf q s q t (q)f q s x + xf q s L
q (τ )
.
Proof. This was used in [13] (see also [11,12] for related facts). For the convenience of the reader we sketch the argument. We may easily reduce to the case Qj = 1. (i) expresses the fact that λi ∨ λj λi + λj
and
λi ∧ λj λi + λj
are (completely) contractive Schur multipliers on Lq (Mn ) for any 1 q ∞ (see [13]). (ii) Using a permutation of the (Qj ) we may assume 0 λ1 λ2 · · · λn . By the boundedness of the triangular projection when 1 < q < ∞ (see the seminal paper [20] and [30, §8] for more references to the literature), it suffices to check (ii) when x is either upper or lower triangular with respect to the decomposition (Qj ). More precisely it suffices to check this when either x = x + or x = x − where x + = ij Qi xQj and x − = i>j Qi xQj . But since λi ∨ λj = λj and λi ∧ λj = λi if i j , the case when x is upper triangular (i.e. x = x + ) follows from the first part. The lower triangular case (i.e. x = x − ) is similar.
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(iii) By density we may reduce this to the case when f has finite spectrum, so that f = λj Qj . Then (iii) essentially reduces to (ii). 2 Proof of Theorem 1.6. We choose q > 1 with 0 < p < q < 2. We use the same notation as for Theorem 1.1. By the observations made before Theorem 1.6, it suffices to verify (1.5). By Lemma 1.8 applied with V = T and V = T ∗ we have 1−θ F r T F 1r T 1−θ T θ , 2 p q 1−θ 1 1−θ T F r T F r T θ . 2 p q θ
Let λi = fi r . By Lemma 1.9 we have θ 1 θr f + f r −1 f r Tij t (q)F 1−θ r T i j i q q
(1.14)
and similarly θ 1 θr f + f r −1 Tij f r t (q)T F 1−θ r . i j j q q
(1.15)
Note that we have θ 1 1 θ −1 r fi + fjr Tij . T (θ ) = fi r + fjr
Therefore by the triangle inequality and (1.14), (1.15) T (θ ) t (q) F 1r T 1−θ + T F 1r 1−θ T θ 2 p p q and hence by condition (γ , γ ) T (θ ) t (q) (γ )1−θ + (γ )1−θ F 1r T + T F 1r /21−θ T θ 2 p q 1−θ 1−θ 1−θ θ Sp T 2 . t (q) (γ ) + (γ ) Therefore we obtain (1.5). By condition (γ , γ ) we have T (1)2 = T 2 γ C2 (x), and also Cq (x) T (θ )q so we conclude that Step 3 holds. 2 Remark 1.10. Theorem 1.1 implies as a special case the following fact possibly of independent interest: if for some 0 < q < 2 we have ⎧ ⎨ ∃C ∀ak ∈ Lp (τ ), 1/2 (Cq ) ⎩ ξk ⊗ ak ak∗ ak C , Lq (τ )
Lq (ϕ×τ )
then (Cp ) holds (for a different constant C) for all p with 0 < p < q. In this case Step 3 is easy to verify (only right multiplication appears in this case).
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Remark 1.11. If 0 < p 2, the converse inequality to (Kp ) is valid assuming that ϕ(1) = 1 and ξk ∈ L2 (N, ϕ) is orthonormal or satisfies (1.6). Indeed, for any t 0 in N ⊗ M since p/2 1 and ϕ(1) = 1, by the operator concavity of t → t p/2 (see [4, pp. 115–120]), we have tp/2 EM (t)p/2 and hence, if S =
ξk ⊗ xk , we have
1/2 1/2 1/2 ∗ Sp = S ∗ S p/2 EM S ∗ S p/2 xk xk , p
and similarly 1/2 xk xk∗ Sp . p
From this we easily deduce Sp c(p)|||x|||p 1
−1
where c(p) = 1 if 1 p 2 and c(p) = 2 p if 0 < p 1. The preceding remark shows that the assumption that ϕ is finite cannot be removed. Remark 1.12. To extend Theorem 1.1 to the case 0 < p < 1 the difficulty lies in Step 3, or in proving a certain form of Hölder inequality such as (1.5). Note that a much weaker estimate allows to conclude: It suffices to show that there is a function ε → δ(ε) tending to zero with ε > 0 such that when f ∈ D we have (α = p1 − 12 = 1r ) (1 < q < 2): x2 1, f α x + xf α p ε
⇒
α(1−θ) f x + xf α(1−θ) q δ(ε).
This might hold even if Step 3 poses a problem. In the case 2 q < ∞, the formulation of (Kq ) must be changed. When 2 < q < ∞, and x = (xk ) is a finite sequence in Lq (τ ), we set 1/2 1/2
def |||x|||q = max xk∗ xk xk xk∗ , . q
q
We will then say (when 2 < q < ∞) that (ξk ) satisfies (Kq ) if there is a constant βq such that for any such x = (xk ) we have ξk ⊗ xk βq |||x|||q . q
By [16], this holds when (ξk ) are the Rademacher functions on [0, 1].
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In operator space theory, all Lq -spaces, in particular the Schatten class Sq , can be equipped with a “natural” operator space structure (see [28, §9.5 and 9.8]). Let Cq (resp. Rq ) denote the closed span of {ei1 | i 1} (resp. {e1j | j 1}) in Sq . We denote by Cq,n and Rq,n the corresponding n-dimensional subspaces. By definition, the “sum space” Rq + Cq is the quotient space (Rq ⊕ Cq )/N where N = {(x, −t x) | x ∈ Rq }. We define similarly Rq,n + Cq,n . The intersection Rq ∩ Cq is defined as the subspace {x, t x} in Rq ⊕ Cq . Here the direct sums are meant (say) in the operator space sense, i.e. in the ∞ -sense. Let us denote by Rad(n, q) (resp. Rad(q)) the linear span of the first n (resp. the sequence of all the) Rademacher functions in Lq ([0, 1]). The operator space structure induced on the space Rad(q) is entirely described by the non-commutative Khintchine inequalities (see [28, §9.8]): the space Rad(q) is completely isomorphic to Rq + Cq when 1 q 2 and to Rq ∩ Cq when 2 q < ∞. The case 0 < q < 1 is open. n Note that Rad(q, n) is an n-dimensional subspace of 2q , so (Khq ) implies that Rq,n + Cq,n n uniformly embeds into 2q for all 1 q < 2. The next result improves significantly the dimension of the embedding. First recall that two operator spaces E, F are called completely c-isomorphic if there is an isomorphism u : E → F such that ucb u−1 cb c. Theorem 1.13. Let 1 q < 2. For any n, there is a subspace of nq with dimension k = [n1/2 ] that is completely c-isomorphic to Rq,k + Cq,k where c is a constant depending only on q. Proof. By [9], we know that there is a subset of [1, eit , . . . , eint ] with cardinality k = [n1/2 ] such that the corresponding set {ξ1 , . . . , ξk } satisfies (Kq ) for all q such that 2 q 4 and hence by duality for all q such that 4/3 q 2. By Theorem 1.1, the same set satisfies (Kp ) for all 1 p 2 (with a constant βp independent of n). 2 Problem. What is the correct estimate of k in Theorem 1.13? In particular is it true for k ∼ [nα ] with α any number in (0, 1) instead of α = 12 ? Is it true for k ∼ [δn] (0 < δ < 1)? The case q = 1 is particularly interesting. It is natural to expect a positive answer with k proportional to n by analogy with the Banach space case (see [8]). One could even dream of an operator space version of the Kashin decomposition (cf. [32])! Another bold conjecture would be the operator space generalization of Schechtman’s and Bourgain, Lindenstrauss and Milman’s results, as refined by Talagrand in [34]: Problem. Let E be an n-dimensional operator subspace of S1 . Assume that for some p > 1 E embeds c-completely isomorphically into Sp . Is it true that E can then be embedded c completely isomorphically (the constant c being a function of p and c) into S12n ? 2. Conditional expectation variant Again we consider (ξk ) in L2 (N, ϕ) and coefficients (xk ) in L2 (M, τ ), but, in addition, we give ourselves a von Neumann subalgebra M ⊂ M such that ϕ|M is semi-finite and we denote by E : M → M the conditional expectation with respect to M. Recall that E extends to a contractive projection (still denoted abusively by E) from Lq (M, τ ) onto Lq (M, τ ) for all 1 q ∞.
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Consider x = (xk ) with xk ∈ Lq (τ ). In this section, we define |||x|||q,M =
inf
xk =ak +bk
1/2 1/2 ak∗ ak ak ak∗ E + E . q
q
We then define again
Cq,M (x) = inf ξk ⊗ yk q
where the infimum runs over all yk in Lq (τ ) such that there is f in D ∩ L1 (M) such that 1
1
xk = (f r yk + yk f r )/2. Then the proof described in Section 1 extends with no change to this situation and shows that if there is βq (M) such that for all finite sequences x = (xk ) in Lq (τ ) we have ξk ⊗ xk |||x|||q,M βq (M)
(2.1)
q
then for any p with 1 p < q there is a constant βp (M) such that (2.1) holds for any x = (xk ) in Lp (τ ) when q is replaced by p. The main new case we have in mind is the case when Lq (M, τ ) = Sq (Schatten class) and M is the subalgebra of diagonal operators (so that Lq (M) q ) on 2 . Thus we state for future reference: Theorem 2.1. Both Theorems 1.1 and 1.6 remain valid when ||| · |||q is replaced by ||| · |||q,M . Proof. The verification of this assertion is straightforward. Note that the conditional expectation E satisfies E(axb) = aE(x)b whenever a, b are in M and x in Lq (M, τ ). This is used to verify Step 2. The proof of the other steps require no significant change. 2 Let Lq (M, τ ) = Sq (Schatten q-class) and let M ⊂ B(2 ) be the subalgebra of diagonal operators with conditional expectation denoted by E. Consider a family x = (xij ) with xij ∈ Sq . We will denote by (xij )k the entries of each matrix xij ∈ Sq , and we set xˆij = (xij )ij eij
and xˆ = (xˆij ).
Let us denote
def
xRq =
xij xij∗
ij
1/2
q
def
and xCq =
ij
xij∗ xij
1/2 . q
Lemma 2.2. For any q 1 we have x ˆ Rq xRq
and x ˆ Cq xCq ,
(2.2)
and consequently |||x||| ˆ q |||x|||q .
(2.3)
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Proof. Indeed, this follows from the convexity of the norms involved and the identity xˆij =
zi zj D(z )xij D(z ) dm(z ) dm(z )
where z = (zi ), z = (zj ) denote elements of TN equipped with its normalized Haar measure m. 2 Now consider λij ∈ C. We define p/2 p/2 1/p 2 2 |aij | + |bij | [λ]p = inf i
j
j
(2.4)
i
where the inf runs over all possible decompositions λij = aij + bij . Lemma 2.3. Let xˆij = λij eij (i.e. λij = (xij )ij ). We have x ˆ Cq =
j
q/2 1/q |λij |
and x ˆ Rq =
2
i
i
q/2 1/q |λij |
2
.
j
Moreover, for any 0 < q < ∞ |||x||| ˆ q,M = [λ]q |||x|||q,M where M is the subalgebra of diagonal operators on 2 . Proof. The first assertion is an immediate calculation. We now show that for any 0 < q < ∞ [λ]q |||x|||q,M .
(2.5)
Indeed, let us denote 1/2 ∗ xij xij xCq ,M = E
q
ij
1/2 ∗ . xij xij and xRq ,M = E q
Then a simple verification shows that xCq ,M =
q/2 1/q q/2 1/q (xij )k 2 (xk )k 2 k
k
ij
and similarly we find xRq ,M
k
(xk )k 2
q/2 1/q .
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Then (2.5) follows immediately. In particular [λ]q |||x||| ˆ q,M , and since the converse is immediate we obtain |||x||| ˆ q,M = [λ]q . 2 Remark 2.4. It seems worthwhile to point out that (2.2) is no longer valid when 0 < q < 1 (even with a constant). Indeed, restricting to the case when xij = 0 ∀i = j , these inequalities imply 1/2 1/q ∗ . (xjj )jj q xjj xjj j
(2.6)
q
j
Now let us consider the case xjj =
n
ej k .
k=1 ∗ x = nP where P is the rank one orthogonal projection onto On one hand we have xjj jj ∗ ∗ x )1/2 = nP and hence ( xjj xjj )1/2 q = n. But on the other hand n−1/2 ek , so that ( xjj jj (xjj )jj = 1 and hence
1/q (xjj )jj q = n1/q . This shows that (2.6) and hence (2.2) fails for q < 1. The same example shows (a fortiori) that (2.3) also fails for q < 1. Remark 2.5. Let j : Lp (M, τ ) → Lp (M , τ ) be an isometric embedding. Let yk = j (xk ). Then clearly p p rk (t)xk dt. rk (t)yk dt = p
p
However, when 0 < p < 1, we do not see how to prove that there is a constant C such that (xk ) C (yk ) , p p although when p 1 this holds with C = 1 using a conditional expectation. This may be an indication that (Khp ) does not hold for 0 < p < 1, at least in the same form as for p 1. 3. The case 0 < p < 1 In (Kq ), we may consider the case when Lq (τ ) = Sq (Schatten q-class) and the sequence x = (xk ) is of the form xij = λij eij with λij ∈ C. For this special case, the approach used in the preceding section works for all 0 < p < q. Thus, we obtain
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Theorem 3.1. Let (εij ) be an i.i.d sequence of {+1, −1}-valued random variables on a probability space with P(εij = ±1) = 1/2. Then for any 0 < p < 1 there is a constant βp such that p 1/p [λ]p βp εij λij eij dP Sp
where [λ]p is defined in (2.4). Remark. Of course, by [16,18], the case 1 p 2 is already known. Remark 3.2. Since Sp is p-normed when 0 < p < 1, the converse inequality is obvious: we have p 1/p p/2 1/2 2 ap aij eij = |aij | i
p
j
i
j
and similarly bp ( j ( i |bij |2 )p/2 )1/2 . Therefore εij λij eij [λ]p . sup p
εij =±1
By well-known general results (cf. [14]) this allows us to formulate Corollary 3.3. Let λij ∈ C be arbitrary complex scalars. The following are equivalent. (i) The matrix [εij λij ] belongs to Sp for almost all choices of signs εij = ±1. (ii) Same as (i) for all choices of signs. (iii) There is a decomposition λij = aij + bij with i
p/2 |aij |2
j
<∞
and
j
p/2 |bij |2
<∞
i
i.e. in short [λ]p < ∞. Remark 3.4. Note that when 0 < p < 1, the spaces Sp or Lp (τ ) are p-normed, i.e. their norm satisfies for any pair of elements x, y x + yp xp + yp .
(3.1)
Remark 3.5. Assume here that 0 < p 2. Note that [λ]p < 1 implies that there is a sequence fi > 0 with fi 1 such that, if we set 1r = p1 − 12 we have 1 1r 1/2 f + f r −1 λij 2 2. i j ij
(3.2)
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Indeed, we have λij = aij + bij with ( i ( j |aij |2 )p/2 )1/p + ( j ( i |bij |2 )p/2 )1/p < 1 we then set ai = ( j |aij |2 )p/2 and bj = ( i |bij |2 )p/2 so that ( ai ) + ( bj ) < 1. Note that: 1 1 1r a + b r −1 (λij ) a − r |aij | + |bij |b−1/r i j i j
and hence by Hölder 1 1 r 1/2 −1/r 1/p 2 1/2 −1/r 1/p 2 1/2 a + b r −1 λij 2 a b ai + bj i j i j ij
= Let fi = ai + bi . Then for all i.
1/2 1/2 + 2. ai bj
fi < 1, (3.2) holds and, if we perturb fi slightly, we may assume fi > 0
We will use the following well-known elementary fact. Proposition 3.6. Let X be a p-normed space (0 < p 1), i.e. we assume ∀x, y ∈ X
x + yp xp + yp .
Then there is a constant χp such that for any finite sequence (xk ) in X and any sequence of real numbers (αk ) we have αk rk xk εk xk χp sup |αk | . (3.3) Lp (X)
Lp (X)
k
Here (rk ) denote the Rademacher functions on [0, 1) and Lp (X) = Lp ([0, 1]; X). Proof. If αk ∈ {−1, 1}, we have equality in (3.3) with χp = 1. If αk ∈ {−1, 0, 1} we can write αk = (βk + γk )/2 with βk ∈ {−1, 1}, γk ∈ {−1, 1} and then we obtain (3.2) (using the p-triangle 1
−1
p inequality . For the general case, we can write any αk in [−1, 1] as a series ∞ (3.1)) with χp = 2 αk = 1 αk (m)ξk (m) with αk (m) ∈ {−1, 0, 1} and |ξk (m)| 2−m . We then obtain (3.3) with
χp = 2
1 p −1
∞
1/p 2−mp
1
= 2p
−1 p
2 −1
−1/p
.
2
1
Proof of Theorem 3.1. Let S = εij λij eij and let xij = eij λij . We already know by [16,18] the case 1 p < 2. We will show that the condition (γ , γ ) holds, and hence that Theorem 3.1 follows from Theorem 1.6. Again we assume that M = Mn . Let M ⊂ M be the subalgebra of diagonal matrices with associated conditional expectation denoted by E. Let x = (xij ). Then by Lemma 2.3 for any 0 < q < 2 we have [λ]q = |||x|||q,M .
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So the inequality in Theorem 3.1 boils down to |||x|||p,M βp εij xij p . We need to observe that when we run the proof of Theorem 1.6 with |||x|||p,M in place of |||x|||p we only need to know (Kq ) for a family (yk ) such that each yk lies in the closure in Lq (τ ) of elements in Mxk M. When M is the algebra of diagonal operators, that means that yk is obtained from xk by a Schur multiplier, so that in any case when (xk ) is the family (xij ) given as above by xij = eij λij , then all the families (yij ) are also of the same form i.e. we have yij = eij μij for some scalars μij , and for the latter we know by [16,18] that the M-version of (Kq ) holds for 1 q 2. So we will be able to conclude if we can verify the condition (γ , γ ). We claim that for some constant C 1 1 1 −1 1 1 f p 2 T C f p − 2 T + Tf p − 2 p p where f is any positive diagonal matrix and T = above. Indeed, we have 1
fi p
− 12
1
fi p
− 12
(3.4)
εij yij , with yij of the form yij = eij μij as 1
+ fjp
− 12
and hence, by Proposition 3.6, (3.4) holds with C = χp . Thus, we have condition (γ , γ ) with γ = χp and by Remark 3.5 we can arrange to have, say, γ = 4. Thus, modulo the above observation, we may view Theorem 3.1 as a corollary to Theorem 2.1. 2 Remark 3.7. Assume λij ∈ Lp (M, τ ) (or simply λij ∈ Sp ) and let xij = eij ⊗ λij ∈ Lp (B(2 ) ⊗ M). Then, atthe time of this writing, we do not know whether Theorem 3.1 remains valid for the series εij eij ⊗ λij , with [λ]p replaced by 1/2 p 1/p 1/2 p 1/p ∗ ∗ [[λ]]p = inf aij aij + bij bij i
p
j
j
p
i
where the infimum runs over all decomposition, λij = aij + bij in Lp (τ ). By (Khp ) this clearly holds when p 1. 4. Remarks on σ (q)-sets and σ (q)cb -sets In [9] (see also [10]) the following notion is introduced: Definition 4.1. A subset E ⊂ N × N is called a σ (q)-set (0 < q ∞) if the system {eij | (i, j ) ∈ E} is an unconditional basis of its closed linear span in Sq . Equivalently, there is a constant C such that for any finitely supported family of scalars {λij | (i, j ) ∈ E} and any bounded family of scalars (αij ) with sup |αij | 1 we have α λ e ij ij ij (i,j )∈E
Sq
C λ e ij ij .
The smallest such constant C is denoted by σq (E).
(i,j )∈E
Sq
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The “operator space” version of this notion is as follows: E is called a σ (q)cb -set if there is a C such that for any finitely supported family {λij | (i, j ) ∈ E} in Sq and any (αij ) as before we have αij eij ⊗ λij C eij ⊗ λij . (i,j )∈E
Sq (2 ⊗2 )
Sq (2 ⊗2 )
(i,j )∈E
We then denote by σqcb (E) the smallest such constant C. It is not known whether σ (q)-sets are automatically σ (q)cb -sets when q = 2. (The case q = 2 is trivial: every subset E is σ (2)cb .) By the non-commutative Khintchine inequalities [16,18], if 1 q < 2, E ⊂ N × N is a σ (q)-set (resp. σ (q)cb -set) iff there is a constant C such that for all families {λij | (i, j ) ∈ E} with λij scalar (resp. λij ∈ Sq ) we have λ e [λ]q C ij ij (i,j )∈E
resp.
Sq
[[λ]]q C e ⊗ λ ij ij (i,j )∈E
.
Sq (2 ⊗2 )
The proof of Theorem 1.1, modified as in Theorem 2.1, yields the following complement to [9]: Theorem 4.2. Assume 1 p < q < 2. Any σ (q)-set (resp. σ (q)cb -set) E ⊂ N × N is a σ (p)-set (resp. σ (p)cb -set). Corollary 4.3. There is a constant c 1 such that, for any n, the usual “basis” {eij } of S1n contains a c-unconditional subset of size n3/2 . Proof. By [9, Theorem 4.8] there is a constant c 1 such that, for any n, the set [n] × [n] contains a further (“Hankelian”) subset that is a σ (4)cb -set (and hence by duality also σ (4/3)cb ) with constant c and cardinal n3/2 . 2 Problem. What is the “right” order of growth in the preceding statement? Can 3/2 be replaced by any number < 2? Remark 4.4. As observed in [9], if 2 < p < q, it is easy to show by interpolation that any σ (q)set (resp. σ (q)cb -set) is a σ (p)-set (resp. σ (p)cb -set). Moreover, any such set is a σ (q )-set (resp. σ (q )cb -set) where q −1 = 1 − q −1 . However, the fact that e.g. σ (q ) ⇒ q(1) is new as far as we know. 5. Grothendieck–Maurey factorization for Schur multipliers (0 < p < 1) Consider a bounded linear map u : H → Lp (τ ) on a Hilbert space H with 0 < p 2. To avoid technicalities, we assume that the range of u lies in a finite dimensional von Neumann subalgebra of M on which τ is finite. When p 1, it is known that there is f in L1 (τ )+ with τ (f ) = 1 and a bounded linear map u˜ : H → L2 (τ ) such that
G. Pisier / Journal of Functional Analysis 256 (2009) 4128–4161
∀x ∈ H
1
u(x) = f p
− 12
1
p u(x) ˜ + u(x)f ˜
4151
− 12
and u ˜ Kp u where Kp is a constant independent of u. In the case p = 1, this fact is easy to deduce from the dual form proved in [25] for maps from M to H ; the latter is often designated as the non-commutative “little GT” (here GT stands for Grothendieck’s theorem). It is easy to deduce this statement from (Khp ) (see [18] for more details) in the case 1 p < 2 (note that p = 2 is trivial). See [17] for a proof that the best constant Kp remains bounded when p runs over [1, 2]. We refer the reader to [17,19,13] for various generalizations. It seems natural to conjecture that the preceding factorization of u remains valid for any p with 0 < p < 1. Unfortunately, we leave this open. Nevertheless, in analogy with Section 3, we are able to prove the preceding factorization in the special case of Schur multipliers as follows. Theorem 5.1. Let 0 < p < 1. Let r be such that
1 r
=
1 p
− 12 . Consider a Schur multiplier
uϕ : [xij ] → [xij ϕij ] where ϕij ∈ C. The following are equivalent: (i) uϕ is bounded from S2 to Sp . (ii) ϕ admits a decomposition as ϕ = ψ + χ with i supj |ψij |r < ∞ and j supi |χij |r < ∞. 1 1 (iii) There is a sequence fi 0 with fi < ∞ such that |ϕij | fi r + fjr . Proof. (Sketch) (ii) ⇔ (iii) is elementary, and (ii) ⇒ (i) is easy. The main point is (i) ⇒ (ii). To prove this, the scheme is the same as in Section 3. We again use extrapolation starting from the knowledge that Theorem 5.1 holds when p = q for some q with 1 q < 2. Let us fix p with 0 < p < 1. For any q with p q 2, we denote Cq (ϕ) = inf uy : S2 → Sq where the infimum runs over all y = (yij ) for which there is fi 0 with ϕij = (fi
1 1 p−q
1 1 p−q
yij + yij fj
fi 1 such that
)/2. We also denote ]ϕ[p = inf ψr (∞ ) + t χ
r (∞ )
where the infimum runs over all decompositions ϕ = ψ + χ . Note that Cp (ϕ) = uϕ : S2 → Sp . Let 1 q < 2 and q1 = same arguments as in Section 1: Step 1 . ]ϕ[p C Cq (λ). Step 2 . C2 (ϕ) C ]ϕ[p . Step 3 . Cq (ϕ) C Cp (ϕ)1−θ C2 (ϕ)θ .
1−θ p
+ θ2 . We have then by the
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Note that obviously uy : S2 → S2 = sup |yij | so that we have again equivalence in Step 2 . To verify Step 3 we argue exactly as for Theorem 3.1. 2 Corollary 5.2. Let 0 < p 2 q ∞. Let 1r = p1 − q1 . With this value of r, the properties (i) and (ii) in the preceding theorem are equivalent to: (i) uϕ is bounded from Sq to Sp . Proof. Assume (i) . Since Sp has cotype 2 [35], uϕ factors through a Hilbert space by [25]. By an elementary averaging argument (see e.g. [29]), the factorization can be achieved using only Schur multipliers. Thus we must have ϕ = ϕ1 ϕ2 with ϕ1 (resp. ϕ2 ) bounded from S2 to Sp (resp. Sq to S2 ). If we now apply Theorem 5.1 (resp. the results of [29,36]) to ϕ1 (resp. ϕ2 ), and 1
−1
1
use an arithmetic/geometric type inequality of the form f p 2 g 2 f, g 0, we obtain (iii). The other implications are easy. 2
− q1
1
c(f p
− q1
1
+gp
− q1
) for all
Problem. Characterize the bounded Schur multipliers from Sq to Sp when p < q < 2 or when 2 < p < q ∞. Some useful information on this problem can be derived from [13]. The difficulty is due to the fact that, except when q = 1, 2, ∞, we have no characterization of the bounded Schur multipliers on Sq . Remark. By general results, actually Theorem 5.1 implies Theorem 3.1. Indeed, the same idea as in [18] can be used to see this. Moreover, as pointed out by Q. Xu, the converse implication is also easy: just observe that, by Theorem 3.1, any Schur multiplier bounded from S2 to Sp must be a bounded “multiplier” from 2 (N × N) to p (2 ) + t p (2 ). Then a well-known variant of Maurey’s classical factorization yields (ii) or (iii) in Theorem 5.1. Although the recent paper [13] established several important factorization theorems for maps between non-commutative Lp -spaces, there seems to be some extra difficulty to extend the Maurey factorization theorem when 0 < p < 1. The next result points to the obstacle. To avoid technicalities we again restrict to the finite dimensional case, so we assume (M, τ ) as before but with M finite dimensional. For any ε > 0, we denote Dε = {f ∈ D | f ε1}. For any x in M, we let T (x)y = xy + yx. Note that if x > 0 then T (x) is an isomorphism on M so that T (x)−1 makes sense. Let B be any Banach space. Given a linear map u : B → Lp (τ ), we denote by Mp (u) the smallest constant C such that for any finite sequence (xj ) in B 1/2 2 (uxj ) C x . j p
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We denote by Mp (u) the smallest constant C such that there is ε > 0 and a probability λ on Dε such that ∀x ∈ B where (as before) We have then
1 r
=
1 p
1 −1 2 T f r ux dλ(f ) C 2 x2 2
(5.1)
− 12 .
Proposition 5.3. There is a constant β > 0 such that for any u as above we have 1 Mp (u) Mp (u) Mp (u). β Proof. The main point is to observe that if ε is chosen small enough (compared to dim(M)) we have for any finite sequence y = (yj ) in Lp (τ ) 1 1/2 (yj ) inf T f r −1 yj 2 β (yj )p 2 p f ∈Dε
(5.2)
where β is a fixed constant, independent of the dimension of M. Then Mp (u) Mp (u) follows immediately. To prove the converse, assume Mp (u) 1. Then by (5.2) we have inf
y∈Dε
1 −1 2 T f r xj 2 . u(xj ) β 2 2
By a well-known Hahn–Banach type argument (see e.g. [28, Exercise 2.2.1]), there is a net (λi ) on Dε such that ∀x ∈ B
lim i
2 1 −1 T f r u(x) dλi (f ) β 2 x2 . 2
We may as well assume that the net corresponds to an ultrafilter. Setting λ = lim λi , we obtain (5.1) and hence Mp (u) β. 2 Remark 5.4. Now assume 1 p < 2. Note then that
1 r
=
1 p
− 12 satisfies −1 − 2r = 1 − p2 < 0.
2
Therefore the function t → t − r is operator convex (see e.g. [4, p. 123]). Using this and assuming Mp (u) 1, we claim that there is, for some ε > 0, a density F in Dε such that 1/r −1 T F ux 2 β x,
∀x ∈ H.
(5.3)
Indeed we first observe that 1 −1 2 1 2 2 T f r y T (f )− r y = T (f )− r y, y 2
2
(5.4)
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where means that the (squared) norms are equivalent with equivalence constants depending only on r, and hence if we set F=
f dλ(f )
we deduce from (5.1) that, for some constant c, we have
T (F )−2/r ux, ux cx2
and hence using (5.4) again we obtain (5.3). Note that if B is Hilbertian any bounded linear u from B to Lp (τ ) satisfies the factorization of the form (5.3) if 1 p 2. This follows immediately by duality from either [17] or [19]. However, what happens for 0 < p < 1 is unclear: Can we still get rid of λ as in the preceding remark? 6. A non-commutative Kahane inequality In vector-valued probability theory, the following inequalities due to Kahane (see [14]) play an important role. For any 0 < p < q < ∞, there is a constant K(p, q) such that for any Banach space X and any finite sequence (xk ) of elements of X we have rk xk
Lq (X)
rk xk K(p, q)
Lp (X)
(6.1)
where (rk ) denotes as before the Rademacher functions. As observed by C. Borell (see [6]) Kahane’s result can be deduced from the hypercontractive inequality for the semi-group T (t) defined on L2 ([0, 1]) by T (t) k∈A rk = e−t|A| k∈A rk for any finite set A ⊂ N. The hypercontractivity says that if 1 < p < q < ∞ and if e−2t (p − 1)(q − 1)−1 then T (t) : Lp → Lq = 1. Since T (t) 0 for all t 0, this implies that for any Banach space X, we also have T (t) : Lp (X) → Lq (X) = 1. In particular, if S =
rk xk then T (t)S = e−t S and hence we find SLq (X) (q − 1)1/2 (p − 1)−1/2 SLp (X) ,
which yields (6.1) for p > 1 (and the case 0 < p 1 can be easily deduced from this using Hölder’s inequality). The goal of this section is to remark that this approach is valid mutatis mutandis in the “anti-symmetric” or Fermionic setting considered in [7]. Let (M, τ ) be a von Neumann algebra equipped with a faithful normal trace τ such that τ (1) = 1. Let {Qk | k 0} be a spin system in M. By this we mean that Qk are self-adjoint unitary operators such that ∀k =
Qk Q = −Q Qk .
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For any finite set A ⊂ N, ordered so that A = {k1 , . . . , km } with k1 < k2 < · · · < km , we set QA = Qk1 Qk2 . . . Qkm , with the convention Qφ = 1. We will assume that M is generated by {Qk }. In that case, M is the so-called hyperfinite factor of type II 1 , i.e. the non-commutative analogue of the Lebesgue interval [0, 1]. Let V (t) : L2 (τ ) → L2 (τ ) be the semi-group defined for all A ⊂ N (|A| < ∞) by V (t)QA = e−t|A| QA . Carlen and Lieb [7] observed that the semi-group V (t) is completely positive (see [7, (4.2), p. 36]) and proved that if e−2t (p − 1)(q − 1)−1 V (t) : Lp (τ ) → Lq (τ ) = 1. We take the occasion of this paper to point out that the Kahane inequality remains valid in this setting provided one works with the “vector-valued non-commutative Lp -spaces” Lp (τ ; E) introduced in [27]. Here E is an operator space, i.e. E ⊂ B(H ) for some Hilbert space H , and Lp (τ ; E) is defined as the completion if Lp (τ ) ⊗ E for the norm denoted by · Lp (τ ;E) defined as follows. For any f in the algebraic tensor product Lp (τ ) ⊗ E f Lp (τ ;E) = inf aL2p (τ ) bL2p (τ )
(6.2)
where the infimum runs over all possible factorizations of f of the form f =a·g·b
(6.3)
with g ∈ M ⊗ E such that gM⊗min E 1. In (6.3), the map (a, g, b) → a · g · b is obtained by linear extension from a, (m ⊗ e), b → amb ⊗ e. Our observation boils down to the following. Lemma 6.1. If T : Lp (τ ) → Lq (τ ) (1 p, q ∞) is completely positive and bounded, then for any operator space E = {0} the operator T ⊗ idE extends to a bounded operator from Lp (τ ; E) to Lq (τ ; E) such that T ⊗ idE : Lp (τ ; E) → Lq (τ ; E) = T : Lp (τ ) → Lq (τ ).
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Proof. By density, it suffices to prove this when M is generated by a finite subset {Q0 , Q1 , . . . , Qn }, say with even cardinality (i.e. n odd). Then M = M2k with k = (n + 1)/2 and it is well known (see e.g. [24]) that any c.p. map T : M → M is of the form T (x) =
aj∗ xaj
(6.4)
for some finite set aj in M. Now assume f ∈ Lp (τ ) ⊗ E with f Lp (τ ;E) < 1. We can write f = a ∗ · g · b with a M⊗min E < 1. Assume T : Lp (τ ) → Lq (τ ) = 1. Let α = 2p , b2p < 1 and g ( aj∗ a ∗ aaj )1/2 and β = ( aj∗ b∗ baj )1/2 . Since α 2 = T (a ∗ a) and β 2 = T (b∗ b) we have α2q < 1 and β2q < 1. Fix ε > 0. We have 1/2 aaj = αj ε1 + α 2 , 1/2 baj = βj ε1 + β 2 where αj = aaj (ε1 + α 2 )−1/2 and βj = baj (ε1 + β 2 )−1/2 satisfy and similarly
−1/2 2 1/2 αj∗ αj = ε1 + α 2 α ε1 + α 2 1
βj∗ βj 1. This implies clearly (by the defining property of an operator space!) αj∗ · g · βj
M⊗min E
< 1.
We have 1/2 1/2 gˆ ε1 + β 2 (T ⊗ idE )(f ) = ε1 + α 2 where gˆ =
αj∗ · g · βj , and hence we conclude by (6.2) (T ⊗ idE )(f ) L
q (τ ;E)
1/2 ε1 + β 2 1/2 ε1 + α 2 2q 2q 2 1/2 2 1/2 ε + β ε + α 1 + ε, q
q
and since ε > 0 is arbitrary, we obtain the announced result by homogeneity.
2
Remark 6.2. Q. Xu pointed out to me that Lemma 6.1 remains valid in the non-hyperfinite case.One can check this using the following fact: consider y in Lp (τ ) ⊗ Mn , then y ∈ BLp (τ ;Mn ) λ y iff there are λ, μ in BLp (τ ) such that y ∗ μ 0 where 0 is meant in Lp (τ × tr) (see e.g. [28, Exercise 11.5] for the result at the root of this fact). A similar statement is valid with B(H ) in place of Mn . Theorem 6.3. Let 1 < p < q < ∞. Assume e−2t (p − 1)(q − 1)−1 , then for any operator space E V (t) : Lp (τ ; E) → Lq (τ ; E) 1.
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Consequently, for any 1 p < q < ∞ there is a constant K (p, q) such that for any E and any finite sequence xk in E we have Qk ⊗ xk
Lq (τ ;E)
Qk ⊗ xk K (p, q)
Lp (τ ;E)
Proof. The first part follows from the preceding lemma by [7]. Let f = if 1 < p < q < ∞ we have
.
Qk ⊗ xk . In particular,
f Lq (τ ;E) (q − 1)1/2 (p − 1)−1/2 f Lp (τ ;E) .
(6.5)
Let 0 < θ < 1 be defined by 1 1−θ θ = + . p 1 q By [27, p. 40] we have isometrically Lp (τ ; E) = L1 (τ ; E), Lq (τ ; E) θ and hence 1−θ f θLq (τ ;E) , f Lp (τ ;E) f L 1 (τ ;E)
which when combined with (6.5) yields 1 f Lq (τ ;E) (q − 1)1/2 (p − 1)−1/2 1−θ f L1 (τ ;E) .
2
Remark. Obviously Theorem 6.3 is also valid for other hypercontractive semi-groups, as the ones in [5]. Acknowledgments I am very grateful to Quanhua Xu for many stimulating suggestions and improvements. I also thank the referee for his/her very careful reading and the resulting corrections. Appendix A The main technical difficulty in our proof of Step 3 above is (1.5). We will first show how this follows from Theorem 1.1 in [13]. We will then also outline a direct more self-contained argument. Let (M, τ ) be a generalized (possibly non-commutative) measure space, with associated space Lp (τ ). Since it is easy to pass from the finite to the semifinite case, we assume τ finite. Consider a density f > 0 in M with τ (f ) = 1, with finite spectrum, i.e. we assume that f = N 1 fj Qj N where 0 < f1 f2 · · · fN , 1 = 1 Qj and Qj are mutually orthogonal projections in M. We now introduce for any x in Lp (τ ) (1 p ∞)
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1 xLp (f ) = f p x L
p (τ )
1 + xf p L
p (τ )
(A.1)
.
We will denote by Lp (f ) the space Lp (τ ) equipped with the norm · Lp (f ) . Then [13, Theorem 1.1] implies in particular that for any 0 < θ < 1 and any 1 < p < ∞ we have Lp(θ) (f ) M, Lp (f ) θ
(A.2)
θ where p(θ )−1 = 1−θ ∞ + p = θ/p, and where means that the norms on both sides are equivalent with equivalence constants depending only on p and θ . Note that by the triangle inequality and by Lemma 1.9(ii), we have
1 1 f p x + xf p L
p (τ )
1 1 xLp (f ) 2t (p)f p x + xf p L
p (τ )
(A.3)
.
Let us denote T (f )x = f x + xf. With this notation, the dual norms xLp (f )∗ = sup τ (xy) yLp (f ) 1 satisfy for any x in Lp (τ ) the following dual version to (A.3) −1 1 −1 x L 2t (p) T f p
p (τ )
1 −1 xLp (f )∗ T f p x L
p (τ )
.
(A.4)
Note that with our simplifying assumptions on f , T (f ) is an isomorphism on Lp (τ ). Here and in the sequel we will denote by c1 , c2 , . . . constants depending only on p and θ . Recall (see e.g. [3]) that we have isometrically for any 0 < θ < 1 ∗ M, Lp (f ) θ = L1 (τ ), Lp (f )∗ θ . Therefore (A.2) implies in particular that for any x in Lp (τ ) 1−θ xθLp (f )∗ . xLp(θ) (f )∗ c1 xL 1 (τ )
(A.5)
Using (A.4), (A.5) implies θ −1 T f p (x)
Lp(θ) (τ )
1 1−θ p −1 θ T f c2 xL x L 1 (τ )
p (τ )
.
(A.6)
In Step 3 of the present paper, we used the special case p = 2. If we denote q = p(θ ) we have 1 1−θ θ q = 1 + 2 so that (A.6) becomes 1 −1 θ θ −1 T f 2 (x)q c2 x11−θ T f 2 x 2 ,
(A.7)
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and we obtain (1.5) for p = 1. The case 1 < p < 2 can be derived by the same argument, but this is anyway much easier because of the simultaneous boundedness on Lp and L2 of the triangular projection. For the convenience of the reader, we now give a direct argument, based on the same ideas 1 as [13]. We want to show (A.7). Note that it is equivalent to (change x to T (f 2 y)): for all y in M θ −1 1 1 1−θ T f 2 T f 2 y q c4 T f 2 y 1 yθ2 .
(A.8)
By the triangle inequality and by Lemma 1.9(ii) we have θ −1 1 θ −1 θ 1−θ θ −1 1−θ θ T f 2 T f 2 y q T f 2 f 2 f 2 y q + T f 2 yf 2 f 2 q 1−θ 1−θ t (q) f 2 y q + yf 2 q . Therefore to show (A.7) (or (A.8)) it suffices to show 1 1−θ 1 1−θ f 2 y + yf 1−θ 2 c6 f 2 y + yf 2 yθ2 . 1 q q Recall that f =
N 1
(A.9)
fj Qj . We denote y+ =
y− =
Qi yQj ,
Qi yQj .
i>j
ij
Note that y + (resp. y − ) is the upper (resp. lower) triangular part of y (with respect to the decomposition I = Qj ). We recall that, whenever 1 < q < ∞, y → y + and y → y − are bounded linear maps on Lq (τ ) with bounds independent of N , but this fails in case q = 1 or q = ∞ (see [20] and [30, §8] for references on this). By the triangle inequality, since y = y + + y − , to prove (A.9) it suffices to show ∀y ∈ M 1 1−θ 1−θ 1 1−θ max f 2 y + q , y + f 2 q c7 f 2 y + yf 2 1 yθ2
(A.10)
+ and similarly with y − in place of y + . Let L− p (τ ) = {x ∈ Lp (τ ) | x = 0}. Let Λp = − + − − Lp (τ )/Lp (τ ). Note that x + Lp (τ ) = x + Lp (τ ). We will denote abusively by x + Λp the + norm in Λp of the equivalence class of x + modulo L− p (τ ). Note that x Λ1 x1 for all x 1
1
1
1
in L1 (τ ) and hence f 2 y + + y + f 2 Λ1 f 2 y + yf 2 1 for all y in L2 (τ ). Moreover we have y + Λ2 = y + 2 . Therefore to show (A.10) it suffices to show 1−θ + f 2 y c7 f 12 y + + y + f 12 1−θ y + θ Λ Λ q 1
(A.11)
2
1−θ
and similarly for y + f 2 . We now observe that, by Lemma 1.9(i), the maps T1 : x →
λi ∧ λj λi + λj
Qi xQj
and T2 : x →
λi ∨ λ j λi + λj
Qi xxj
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have norm 3/2 on Lq (τ ) for all 1 q ∞, in particular on L1 (τ ). Since these maps preserve 1
2 L− 1 (τ ), the “same” maps are contractive on Λ1 . Applying this with λi = fi and assuming as before that f1 · · · fN , we have fi ∧ fj = fi and fi ∨ fj = fj for all i j and hence 1
1
1
1
1
T1 (f 2 y + + y + f 2 ) = f 1/2 y + and T2 (f 2 y + + y + f 2 ) = y + f 2 . This gives us 1 1 1 1 max f 2 y + Λ , y + f 2 Λ (3/2)f 2 y + + y + f 2 Λ . 1
1
1
Thus to show (A.11) it suffices to show 1−θ + f 2 y c7 f 12 y + 1−θ y + θ , Λ Λ q 1
2
1−θ
and similarly for y + f 2 . Now by [26, Theorem 4.5] and by duality we have (Λ1 , Λ2 )θ Λq with equivalent norms (and equivalence constants independent of N ). Using the analytic function z z → f 2 and a by now routine application of the 3 line lemma (this is essentially the “Stein interpolation principle”) this gives us (recall y + Λ2 = y + 2 ) 1−θ + f 2 y
Λq
1 1−θ θ c8 f 2 y + Λ y + 2 . 1
But now since the “triangular projection” y → y + is bounded on Lq (τ ) when 1 < q < ∞ (and since (f
1−θ 2
y)+ = f
1−θ 2
y + ) we obtain finally 1−θ + f 2 y c9 f 12 y + 1−θ y + θ . Λ 2 q 1
By the preceding successive reductions, this completes the proof of (A.7) and hence also of (1.5) for p = 1. References [1] H. Araki, Golden–Thompson and Peierls–Bogolubov inequalities for a general von Neumann algebra, Comm. Math. Phys. 34 (1973) 167–178. [2] W. Banks, A. Harcharras, New examples of noncommutative Λ(p) sets, Illinois J. Math. 47 (4) (2003) 1063–1078. [3] J. Bergh, J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, New York, 1976. [4] R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997. [5] P. Biane, Free hypercontractivity, Comm. Math. Phys. 184 (2) (1997) 457–474. [6] C. Borell, On the integrability of Banach space valued Walsh polynomials, in: Séminaire de Probabilités, XIII, Univ. Strasbourg, Strasbourg, 1977/1978, in: Lecture Notes in Math., vol. 721, Springer-Verlag, Berlin, 1979, pp. 1–3. (Reviewer: D.J. Eustice) 42C10 (30B20 30C85 60B11). [7] E. Carlen, E.H. Lieb, Optimal hypercontractivity for Fermi fields and related noncommutative integration inequalities, Comm. Math. Phys. 155 (1) (1993) 27–46. [8] T. Figiel, J. Lindenstrauss, V.D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1–2) (1977) 53–94. [9] A. Harcharras, Fourier analysis, Schur multipliers on S p and non-commutative Λ(p)-sets, Studia Math. 137 (3) (1999) 203–260. [10] A. Harcharras, S. Neuwirth, K. Oleszkiewicz, Lacunary matrices, Indiana Univ. Math. J. 50 (4) (2001) 1675–1689. [11] F. Hiai, H. Kosaki, Means for matrices and comparison of their norms, Indiana Univ. Math. J. 48 (3) (1999) 899–936. [12] F. Hiai, H. Kosaki, Means of Hilbert Space Operators, Lecture Notes in Math., vol. 1820, Springer-Verlag, Berlin, 2003. [13] M. Junge, J. Parcet, Rosenthal’s theorem for subspaces of noncommutative Lp , Duke Math. J. 141 (1) (2008) 75–122.
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Journal of Functional Analysis 256 (2009) 4162–4186 www.elsevier.com/locate/jfa
Adjoints of composition operators on Hardy spaces of the half-plane Sam Elliott Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK Received 10 October 2008; accepted 29 November 2008 Available online 16 December 2008 Communicated by N. Kalton
Abstract Building on techniques used in the case of the disc, we use a variety of methods to develop formulae for the adjoints of composition operators on Hardy spaces of the upper half-plane. In doing so, we prove a slight extension of a known necessary condition for the boundedness of such operators, and use it to provide a complete classification of the bounded composition operators with rational symbol. We then consider some specific examples, comparing our formulae with each other, and with other easily deduced formulae for simple cases. © 2008 Elsevier Inc. All rights reserved. Keywords: Composition operator; Adjoint; Hardy space; Aleksandrov–Clark measure
0. Introduction A great deal of work has already taken place in studying the properties of analytic composition operators on Hardy spaces on the unit disc D of the complex plane. It has long been known that all such operators are bounded on all the Hardy spaces (and indeed on a great many other spaces too), and a number of characterisations of compactness and weak compactness have also been produced, including those of Cima and Matheson [4], Sarason [18] and Shapiro [19]. In contrast, relatively little is known about composition operators acting on Hardy spaces of a half-plane. Although corresponding Hardy spaces of the disc and half-plane are isomorphic, composition operators act very differently in the two cases. It is known, for example, that not all E-mail address: [email protected]. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.025
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analytic composition operators are bounded, though no satisfactory characterisation of boundedness has yet been found; moreover, Valentin Matache showed in [12] that there are in fact no compact composition operators in the half-plane case. The question of when an operator is isometric has now been dealt with in both cases, however: in the disc by Nordgren [14], and more recently in the half-plane by Chalendar and Partington [3]. Lately, a good deal of research has concentrated on describing the adjoints of analytic composition operators on the disc. Much of this work has been concerned with the Hardy space H 2 (D) which, being a subspace of L2 (T), is a Hilbert space and hence self-dual, meaning adjoints play a particularly important rôle in its structure. Here T denotes the unit circle in the complex plane. In [7] Carl Cowen produced the first adjoint formulae for the case where the composing map is fractional linear. It has since been shown that for all the Hardy spaces on the disc, an important generalisation of Aleksandrov’s disintegration theorem [1], gives rise to a formula for the preadjoint of a composition operator in terms of what are now called Aleksandrov–Clark (AC) measures: these measures were initially studied by Douglas Clark [6] in relation to perturbations of unitary operators, but have since found applications in a number of apparently unrelated areas. The same method has been shown to produce pre-adjoint formulae for the Lp spaces on T, and even the space of Borel measures on T. In particular, since H 2 and L2 are both Hilbert spaces, in these cases this formula gives a description of the adjoint of the composition operator as well. More recently, John McDonald [13] produced an explicit adjoint formula for operators induced by finite Blaschke products. In the last few years, Cowen together with Eva Gallardo– Gutiérrez [8] developed a method, later corrected by Hammond, Moorehouse and Robbins [10], which gave a characterisation in the more general case of an operator on H 2 with rational symbol. The formula shows that the adjoint of each such operator is a so-called ‘multiple-valued weighted composition operator’, plus an additional term. A simplified proof of the formula has since been given by Paul Bourdon and Joel Shapiro [2]. We begin by generalising the notion of Aleksandrov–Clark measures to the half-plane (we choose the upper half-plane C+ , as its boundary is the most natural to work with for our purposes). A similar generalisation has already been made by Alexei Poltoratski in [15] with the intention of studying perturbations of self-adjoint operators, much in the same way as Clark did with unitary operators. We show that, subject to a certain condition necessary for a composition operator to be bounded, a characterisation of the pre-adjoint of a composition operator can also be made on the half-plane using AC measures; again this will give an adjoint formula for the case where p = 2. The middle sections of this paper will then be devoted to the study of composition operators with rational symbol. We prove a complete characterisation of the boundedness of such operators, as well as a number of other results along the same lines. Having made this characterisation, we use integral methods in the vein of [10] to find an explicit formula for the adjoint of a composition operator on H 2 (C+ ) with rational symbol, which will turn out to be a multiple-valued weighted composition operator, but this time without any additional terms. Finally, we present some examples including the simplest case (an operator with linear symbol), and a slightly more complicated function known to be an isometry by the results of [3].
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1. Preliminaries For 1 p < ∞, the Hardy space H p (C+ ) is the Banach space of analytic functions f : C+ → C such that the norm f p = sup y>0
f (x + iy)p dx
1
p
< ∞.
R
The space H ∞ (C+ ) is the space of all bounded analytic functions on C+ together with the supremum norm. It can easily be shown that each H p -space is a subspace of the corresponding Lp (R)-space by equating each Hardy space function with its boundary function, reached via non-tangential limits; equivalently, for p < ∞ it is possible to extend any Lp function to the half-plane by integrating with respect to the Poisson kernels. As such, we see that H 2 is in fact a Hilbert space, being a subspace of L2 . An analagous construction may be made for the disc, and a natural identification of the disc with half-plane induces an isomorphism between each H p (C+ ) and the equivalent Hardy space of the disc. We will explore this identification further in Section 3. For an analytic map ϕ : C+ → C+ , we may define the composition operator with symbol ϕ, which can be considered to act on any of the spaces H p (C+ ) or Lp (R). Given such a mapping ϕ, this operator, written Cϕ is defined by the formula Cϕ f = f ◦ ϕ. For f ∈ Lp , we may either extend f to the half-plane and compose it with ϕ, or extend ϕ to R and use this for composition, the two methods are entirely equivalent. In the case of disc, the Aleksandrov–Clark (AC) measures of an analytic function, ψ : D → D, were constructed via the collection of functions given by β + ψ(z) , uβ (z) = β − ψ(z) for β ∈ T. Each uβ can be shown to be positive and harmonic on the disc, and so, via the Riesz–Herglotz representation theorem, each may be written as the Poisson integral of a (finite) positive measure on the unit circle. This collection, indexed by T, is known as the collection of Aleksandrov–Clark (AC) measures associated with ψ, and denoted Aψ . For a full description of the construction, see for example [5,11,17]. A number of results are well known about AC measures in the disc case, most particularly the following theorem from [1], reproduced in a number of other works, including for example [5, p. 216]. Theorem 1 (Aleksandrov’s disintegration theorem). Let ψ be an analytic self-map of the disc, and Aψ = {μβ : β ∈ T} be the collection of AC measures associated with ψ. Then for each function f ∈ L1 (T), f (ζ ) dμβ (ζ ) dm(β) = f (ζ ) dm(ζ ), T
T
where m denotes normalised Lebesgue measure on T.
T
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In the upper half-plane case, the equivalent construction is as follows: given an analytic selfmap of the upper half-plane, ϕ, we note that the function
i(1 + αϕ(z)) uα (z) = ϕ(z) − α
is positive, and harmonic for each α ∈ R. In fact, this is precisely the function we get by transforming the plane to the disc via the standard Möbius identification: J :D →
1−z , z → i 1+z
C+ ,
J −1 : C+ → D,
s →
i −s , i +s
taking the function uα from the disc case, and transforming back to the plane. Since α is simply a constant with respect to z, the functions given by i(1 + αϕ(z)) 1 vα (z) = ϕ(z) − α π(1 + α 2 ) are also all positive and harmonic, and we will see later that it will be more convenient to use this system for our purposes. We continue with the following theorem from [9]. Theorem 2 (The upper half-plane Herglotz theorem). We denote by Py (x − t) the upper halfplane Poisson kernel, namely Py (x − t) =
y 1 . π (x − t)2 + y 2
If v : C+ → R is a positive, harmonic function, then v may be written as v(x + iy) = cy +
Py (x − t) dμ(t), R
where c 0 and μ is a positive measure such that
dμ(t) < ∞. 1 + t2
We notice that, unlike the disc case, in the half-plane we lose the finiteness of our measures, and there is an additional term of c (z). This additional term corresponds to a point mass existing at a notional point ‘∞,’ or equivalently to a point mass at −1 on the boundary of the disc. Using Theorem 2, we see that each vα may be written vα (x + iy) = cα y +
Py (x − t) dμα (t). R
(1)
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We will call the collection of pairs (μα , cα ) the Aleksandrov–Clark (AC) measures associated with ϕ, and much as in the disc, denote this collection Aϕ . 2. The half-plane Aleksandrov operator We begin by noting the following result, which will simplify our future calculations. Lemma 3. For any function ϕ : C+ → C+ , the constant cα in (1) takes the value zero for malmost every α. Proof. We take the function ϕ, and construct the collection of functions vα as above. For α ∈ R, we denote by αˆ the corresponding point on the circle T, via the standard identification. We may also translate the functions ϕ and vα to equivalent functions on the disc: we denote ϕ˜ : D → D, v˜α : D → R+ ,
ϕ˜ = J −1 ◦ ϕ ◦ J, v˜α = vα ◦ J.
We observe that i(1 + αϕ(J (z))) 1 v˜α (z) = ϕ(J (z)) − α π(1 + α 2 ) αˆ + ϕ(z) ˜ 1 , = αˆ − ϕ(z) ˜ π(1 + α 2 ) by construction. So the functions v˜α are positive multiples of the functions uαˆ , and the measures they define via Herglotz’ theorem will have point masses in the same places. By Garnett [9, p. 19], the value of cα corresponds to the point mass of the measure given by v˜α at −1. As such, cα is zero if and only if that measure has no point mass at −1, or equivalently by the above, the AC measure associated with ϕ˜ and αˆ has no point mass at −1. Let us suppose that the AC measure associated with ϕ˜ and αˆ had a non-zero point mass at −1 for a set of αˆ of positive Lebesgue measure. We denote by {μα } the collection of all AC measures associated with ϕ. ˜ Let f be an L1 function on T, then by the standard Aleksandrov disintegration theorem (Theorem 1 above) we have f (ζ ) dm(ζ ) = f (ζ ) dμα (ζ ) dm(α) T
T T
f (ζ ) dμα (ζ ) dm(α) +
= T T\{−1}
f (ζ ) dμα (ζ ) dm(α) T {−1}
f (ζ ) dμα (ζ ) dm(α) + f (−1)
= T T\{−1}
kα dm(α), T
where kα is the value of the point mass of μα at −1. If we change the value of f at the point −1 (a set of Lebesgue measure zero), the left-hand side of this equality will remain unchanged, but
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the right-hand side will change, since kα is non-zero on a set of positive Lebesgue measure. This is a contradiction, hence μα ({−1}) cannot be non-zero on a set of positive Lebesgue measure, and thus cα = 0 for m-almost every α. 2 We now define the Aleksandrov operator, Aϕ , of symbol ϕ, to be the operator Aϕ f (α) = f (t) dμα (t). R
This operator may be allowed to act on any number of function spaces on the upper-half plane, but for the moment, we will simply consider this definition to be true ‘whenever the integral makes sense.’ It is clear that this is a linear operator. 2.1. Functions which map ∞ to itself We begin by looking at how the Aleksandrov operator acts on a Poisson kernel. Taking z = x + iy, we let fz (t) = Py (x − t). By definition, we have Aϕ fz (α) = Py (x − t) dμα (t) m-almost everywhere, by Lemma 3
1 + αϕ(z) 1 i ϕ(z) − α π(1 + α 2 ) imaginary 2 − α 2 ϕ(z) ϕ(z) − α+ i α|ϕ(z)| = π(1 + α 2 ) ((ϕ(z)) − α)2 + (ϕ(z))2 R
=
Rearranging, we get 1 (1 + α 2 ) (ϕ(z)) 2 ) ((ϕ(z)) − α)2 + (ϕ(z))2 α π( 1+ = P (ϕ(z)) ((ϕ(z)) − α) =
= fϕ(z) (α). We aim to show a level of duality between the Aleksandrov operator, and the composition operator Cϕ . Let us for the moment assume that ϕ(∞) = lim ϕ(z) = ∞. |z|→∞
Then for each M ∈ N, there is some N ∈ N such that |ϕ(z)| > M whenever |z| > N . In particular, if g has compact support in R, then supp(g) ⊆ z: |z| < M0 for some M0 ∈ N, and so supp Cϕ (g) = z: ϕ(z) ∈ supp(g) ⊆ z: |z| < N0 for some N0 ∈ N. In other words, Cϕ g has compact support.
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We begin by taking fz as above, which is a continuous Lp -function on R, for each 1 p ∞. We also take g to be a continuous function on R with compact support. By the above, we have
Aϕ fz (α)g(α) dm(α) =
R
P ϕ(z) ϕ(z) − α g(α) dm(α)
R
since Poisson kernels are reproducing kernels for Lp
= g ϕ(z) = Cϕ g(z) = Py (x − t)Cϕ g(t) dm(t),
(2)
R
since Cϕ g has compact support. In order to continue, we will need the following. Theorem 4. Let ϕ be an analytic self map of C+ , which maps ∞ to itself. Then the operator, Aϕ is bounded on Lp (R) if and only if Cϕ is bounded on Lq (R), where 1/p + 1/q = 1. Proof. Suppose Cϕ is bounded on Lq (R). We begin by taking fz as above. Since Cϕ is bounded on Lq , it must map Lq into itself. Moreover, fz ∈ Lp for each p, and so, by taking Lq limits of the compact support function g in (2), we have that Aϕ fz (α)g(α) dm(α) = fz (t)Cϕ g(t) dm(t), R
R
for all g ∈ Lq (R), since functions of compact support are dense in each Lq . Taking suprema over all possible g of norm 1, we get (by the duality of Lp and Lq ) Aϕ fz p = sup
g=1
fz (t)Cϕ g(t) dm(t) R
fz p Cϕ Lq →Lq , and so Aϕ is bounded on Poisson kernels, and similarly, on finite linear combinations of Poisson kernels. We know, however, that the linear span of Poisson kernels is dense in each Lp , and hence by the Hahn–Banach theorem, Aϕ must be bounded on the whole of Lp . Suppose now that Cϕ is not bounded on Lq . Then given M ∈ N, there is a g ∈ Lq with g = 1 such that Cϕ g > M. As such, by the density of linear combinations of Poisson kernels in Lp , there must be some finite linear combination of Poisson kernels, f , with f = 1 and f (t)Cϕ g(t) dm(t) > M. R
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Hence Aϕ f p > M, giving Aϕ > M, and so Aϕ is not bounded.
2
All this leads us to our first important result. Theorem 5. Assume ϕ : C+ → C+ is analytic, with ϕ(∞) = ∞, and 1 < p, q < ∞ with 1/p + 1/q = 1. Whenever Cϕ : Lq → Lq is bounded, it is the adjoint of Aϕ : Lp → Lp . Proof. We return to the identity (2). Provided we ensure the integral remains finite, we may take linear combinations, and then limits of Poisson kernels, and similarly for continuous functions of compact support, and the same identity will clearly hold. Therefore, whenever both Aϕ and Cϕ are bounded, taking Lp and Lq limits respectively, since the Poisson kernels are dense in each Lp , and the continuous functions of compact support are dense in each Lq , the identity (2) remains true. In particular, Cϕ : Lq → Lq is the adjoint of Aϕ : Lp → Lp for each such p and q. 2 2.2. More general analytic functions We will now remove the assumption that ϕ must map ∞ to itself. We first note the following, which is a slight extension of Corollary 2.2 from [12]: Proposition 6. If ϕ : C+ → C+ is bounded on some set of infinite measure on R, then Cϕ is not a bounded operator on Lp (R), or H p (C+ ) for any 1 p < ∞. It is also not bounded on C0 (R). Proof. For 1 p < ∞, the function fp : R → R given by fp (z) =
1 1 + |z|2/p
is in Lp (R). Moreover, each such function is in C0 (R). If ϕ is bounded on some set of infinite measure, say Σ , then we have ϕ(z) < K on Σ, for some K ∈ N. Now
Cϕ fp (z) =
1 1 > 2/p 1 + |ϕ(z)| 1 + K 2/p
on Σ. Since Σ is of infinite measure, we have, setting 1/(1 + K 2/p ) = εp m z: Cϕ f (z) > εp = ∞,
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so Cϕ fp ∈ / Lp (R), and hence Cϕ is not bounded on Lp (R). Since Cϕ fp (z) > εp for arbitrarily / C0 (R), and hence Cϕ is not large z, it is also clear that Cϕ fp (z) 0 as z → ∞, so Cϕ fp ∈ bounded on C0 (R). For the case of H p (C+ ), we take gp to be the function gp (z) =
1 , (i + z)2/p
/ H p (C+ ), indeed it will not even which is in H p (C+ ). The same argument will give that Cϕ gp ∈ p p + be in L (R). As such, Cϕ is not bounded on H (C ). 2 We know now that no function which is bounded on some set of infinite measure can give rise to a bounded composition operator. Let us suppose, therefore, that ϕ is unbounded on every set of infinite measure, then for all M ∈ N, m z: |ϕ(z)| < M < ∞. Indeed, for each M ∈ N, given δ > 0, there exists an N ∈ N such that m(K) < δ, where K = z: ϕ(z) < M, |z| > N . So, for all M ∈ N, given ε > 0, there is an Nε ∈ N such that m ϕ(Kε ) < ε, where Kε = z: ϕ(z) < M, |z| > Nε . Now, let g have compact support, then there is some M ∈ N with supp(g) ⊆ z: |z| < M . Given ε > 0, we can find an Nε ∈ N such that m ϕ(K ) < . We now set gε = g · χR\ϕ(Kε ) . If |z| > Nε , then either ϕ(z) M, in which case g(ϕ(z)) = 0, or z ∈ Kε , in which case χR\ϕ(Kε ) (ϕ(z)) = 0. Either way, Cϕ gε (z) = gε ◦ ϕ(z) = 0 for |z| > Nε , so Cϕ gε has compact support. This motivates our next main result, which is a more general version of Theorem 5.
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Theorem 7. Let ϕ : C+ → C+ be analytic, and let 1 < p, q < ∞ with 1/p + 1/q = 1. Whenever Cϕ : Lq → Lq is bounded, it is the adjoint of Aϕ : Lp → Lp . Proof. If Cϕ is bounded, then ϕ must not be bounded on any set of infinite measure. We recall Eq. (2), taking now g to be continuous with compact support, and gε as above, we have
Aϕ fz (α)gε (α) dm(α) =
R
since Poisson kernels are reproducing kernels for LP
P (ϕ(z)) (ϕ(z)) − α gε (α) dm(α)
R
= gε ϕ(z) = Cϕ gε (z) = Py (x − t)Cϕ gε (t) dm(t),
(2 )
R
which remains valid since Cϕ gε has compact support. We note that lim gε = g
ε→0
in each Lp -norm, so functions of this form are dense in the continuous functions of compact support, which are in turn dense in each Lp . Taking linear spans and closures, therefore, we have that Aϕ : Lp → Lp is bounded if and only if Cϕ : Lq → Lq is, and if both are bounded, then Cϕ is the adjoint of Aϕ . 2 Given that L2 is a Hilbert space, we may then deduce the following corollary. Corollary 8. If Cϕ : L2 (C+ ) → L2 (C+ ) is bounded, then Aϕ is its adjoint. If we replace the use of Poisson kernels in the preceding results with the reproducing kernels for the H p spaces, namely the functions kz (t) =
1 , z−t
(3)
we obtain precisely the same results for the H p spaces. In particular, we have: Theorem 9. Let 1 p < ∞. If Cϕ : H p (C+ ) → H p (C+ ) is bounded, then it is the adjoint of Aϕ : H q → H q , where 1/p + 1/q = 1. Corollary 10. If Cϕ : H 2 (C+ ) → H 2 (C+ ) is bounded, then Aϕ : H 2 (C+ ) → H 2 (C+ ) is its adjoint. 3. Rational self-maps of the upper half-plane We use the mapping given in [3], which identifies the Hardy space on the right half-plane, H p (C+ ), with the equivalent Hardy space on the disc, and the space Lp (T) with Lp (iR). We will
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need a slight alteration to work with the upper half-plane, C+ , but this change is essentially trivial. We begin by identifying the disc with the upper half-plane, via the mapping we have already mentioned, namely 1−z + z → i J :D → C , , 1+z J −1 : C+ → D,
s →
i −s . i +s
This natural mapping then gives rise to a unitary equivalence between H p (D), and H p (C+ ) (1 p < ∞), given by V : H p (D) → H p C+ , (V g)(s) =
π 1/p (i
1 g J −1 (s) , 2/p + s)
−1 (2i)2/p π 1/p V G (z) = G J (z) , (1 + z)2/p the same mapping also identifies Lp (T) with Lp (R). Lemma 11. If ϕ : C+ → C+ is an analytic self-map of the upper half-plane, then the composition operator Cϕ : H p (C+ ) → H p (C+ ) (similarly Lp (R) → Lp (R)) is unitarily equivalent to the weighted composition operator LΦ : H p (D) → H p (D) (similarly Lp (T) → Lp (T)), given by 1 + Φ(z) 2/p CΦ f (z), (LΦ f )(z) = 1+z where Φ = J −1 ◦ ϕ ◦ J . Proof. Let f ∈ H p (C+ ) (or f ∈ Lp (R)), then −1 (2i)2/p π 1/p V ◦ Cϕ ◦ Vf = (Cϕ ◦ Vf )(J (z)) (1 + z)2/p (2i)2/p π 1/p 1 = · f J −1 ◦ ϕ ◦ J (z) . 2/p (1 + z) π 1/p (i + ϕ(J (z)))2/p Combining factors, we get 1 i − ϕ(J (z)) 2/p 1 i + ϕ(J (z)) = f Φ(z) · + ϕ(J (z)) i + ϕ(J (z)) 1+z i+ 1 + Φ(z) 2/p = CΦ f (z), 1+z
as required.
2
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We now recall Proposition 6 above. For what follows, we will need the following corollary: Corollary 12. If r : C+ → C+ is a rational map such that r(∞) = ∞, then Cr is not bounded on Lp (R), or H p (C+ ) for any 1 p < ∞. Proof. If r(∞) = ∞, then r must tend to some finite limit as z → ±∞ (being rational). As such, there must be some n ∈ N such that r is bounded on {z: |z| > n}, which has infinite measure, so by Proposition 6, Cr is not bounded on any of the spaces mentioned. 2 We now aim to prove that each rational map which does map ∞ to itself must give rise to a bounded operator on all the appropriate spaces. Proposition 13. Let r = a/b : C+ → C+ be a rational map written in its lowest terms, and let r(∞) = ∞. Then Cr is bounded on each of the spaces, H p (C+ ), Lp (R) for 1 p < ∞. Proof. We recall that 1 + Φr 2/p CΦr is bounded on H p (D), Lp (T) 1+z
where Φr = J −1 ◦ r ◦ J 1 + Φr (z) 2/p ⇔ sup CΦr f ∞. 1+z f =1 p
Cr is bounded on H C+ , Lp (R) ⇔ p
Now, 1 + Φr (z) 2/p 1 + Φr (z) 2/p · CΦ f p sup CΦr f sup r 1+z 1+z f =1 f =1 p ∞ 1 + Φr (z) 2/p · CΦ . = r 1+z ∞ However CΦr < ∞ since all composition operators on the disc are bounded on the relevant spaces, so Cr will be bounded on H p (C+ ) and Lp (R), provided 1 + J −1 ◦ r ◦ J (z) 2/p < ∞. 1+z ∞ We note, however, that 1 + J −1 ◦ r ◦ J (z) 2/p 1+z
1 + J −1 ◦ r ◦ J (z) 2/p = 1+z ∞ ∞ −1 1 + J ◦ r ◦ J (z) 2/p = sup , 1+z z∈D
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so Cr will be bounded on all the spaces simultaneously, provided 1 + J −1 ◦ r ◦ J (z) < ∞. sup 1+z z∈D
Since J −1 ◦ r ◦ J : D → D, we have |J −1 ◦ r ◦ J (z)| < 1, so our inequality is clearly satisfied for z away from −1. Hence 1 + J −1 ◦ r ◦ J (z) <∞ sup 1+z z∈D
1 + J −1 ◦ r ◦ J (z) < ∞. lim z→−1 1+z
⇔
Now, making the substitution z = −k,
1 + i−r i 1+k 1−k 1+k 1 + J −1 ◦ r ◦ J (z) i+r i 1 1−k = lim . lim = 2 lim 1+k z→−1 k→1 k→1 (1 − k) i + r i 1+z 1−k 1−k We recall that r = a/b, where a and b are polynomials with no common factors, so b i 1+k 1 1−k 2 lim 1+k . 1+k = 2 lim 1+k k→1 (1 − k) i + r i k→1 (1 − k) ib i +a i 1−k
Making the change of variables t =
1 1−k ,
1−k
1−k
that is k = 1 − 1t , we get
b i 1+k b(−i(2t − 1)) 1−k . t = 2 lim 2 lim 1+k 1+k t→∞ (ib(i(2t − 1)) + a(i(2t − 1))) k→1 (1 − k) ib i +a i 1−k
1−k
If we let deg(b) = m, then the degree of the numerator of the fraction is m + 1. Since r(∞) = ∞, we must have deg(a) > deg(b), so the degree of the denominator of the fraction is greater than or equal to m + 1, so b(−i(2t − 1)) < ∞, lim t t→∞ (ib(i(2t − 1)) + a(i(2t − 1)) and hence Cr is bounded on each Lp (R), and each H p (C+ ).
2
Corollary 14. For a rational map r : C+ → C+ , Cr is bounded on each Lp (R), and each H p (C+ ) if and only if r(∞) = ∞. 4. Further observations about rational maps Proposition 15. Let r be a rational map such that r(∞) = ∞ and r(C+ ) ⊆ C+ . Then both r −1 (C+ ) and r −1 (C− ) contain an unbounded component.
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Proof. The fact that r −1 (C+ ) contains such a component is trivial, since C+ ⊆ r −1 (C+ ). For r −1 (C− ), we observe the following: Let h(z) = 1z = h−1 (z). Consider the mapping hrh, it is easy to see that r(∞) = ∞ if and only if hrh(0) = 0. Let AK be the region {z: |z| > K}. Since r(∞) = ∞, r(AK ) ⊆ AK for sufficiently large K. Similarly, if BK = {z: h(z) ∈ AK } then hrh(BK ) ⊆ BK for sufficiently large K. Moreover, r(C+ ) ⊆ C+ , so hrh(C− ) ⊆ C− .
Now BK is an open neighbourhood of 0, and hrh is an open mapping, with hrh(0) = 0, so hrh(BK ) is an open neighbourhood of 0. As such, hrh(BK ) C− , and there is at least one point in BK (indeed, an open subset of BK ) which is mapped to C+ by hrh. Thus, there is an open subset of AK mapped to C− by r, but this is true for all sufficiently large K, so there are points of arbitrarily large modulus sent to C− . Since r is rational, r −1 (C− ) has at most finitely many components, so r −1 (C− ) must have an unbounded component. 2 Proposition 16. Let r be a rational map such that r(∞) = ∞ and r(C+ ) ⊆ C+ . If r is of the form r(z) =
an z n + · · · + a1 z + a0 bm z m + · · · + b1 z + b0
with an , bm = 0, then (i) n = m + 1, (ii) bamn ∈ R, and in particular, (iii)
an bm
> 0,
( ab00 ) 0.
Proof. (i) For |z| large enough, r(z) ≈
an n−m . bn z
θ=
Taking
an bm
= ceiγ , we set
−γ . n−m
3π 2
If n − m 2, then keiθ ∈ C+ , but for sufficiently large k, 3πi r keiθ ≈ ck n−m e 2 ∈ C− , so r(C+ ) C+ , which is a contradiction.
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(ii) Since n = m + 1, we have r(z) ≈
an bm z,
for sufficiently large z. Suppose
an bm
∈ / R+ . Then
an = ceiγ , bm where γ = 0 (mod 2π), that is, γ ∈ (0, 2π). We have that r(keiθ ) ≈ ckei(θ+γ ) , so setting θ = π − γ2 , we get γ r keiθ ≈ ckei(π+ 2 ) ∈ C− , γ
but kei(π− 2 ) ∈ C+ , so r(C + ) C+ , which is a contradiction. (iii) For z sufficiently small, we have r(z) ≈
a0 . b0
If ( ab00 ) < 0, then r ke for k sufficiently small, but ke
iπ 2
iπ 2
≈
a0 ∈ C− , b0
∈ C+ , so r(C+ ) C+ , which is a contradiction.
2
Altogether, this gives us a refinement of Corollary 14, namely: Corollary 17. For a rational map r : C+ → C+ , Cr is bounded on each Lp (R), and each H p (C+ ) if and only if the degree of the numerator of r is precisely 1 larger than the degree of the denominator of r. 5. A note on maps which are quotients of linear combinations of powers of z A slightly larger class of function which are of interest is the following: we denote by QLP(A) the collection of maps from A to A which are quotients of linear combinations of powers of z. That is, all those maps of the form ϕ(z) =
λ1 za1 + λ2 za2 + · · · + λm zam , μ1 zb1 + μ2 zb2 + · · · + μn zbn
where each ai and each bj is a non-negative real number. We assume without loss of generality that the powers ai , and bi are written in descending order. A number of the methods we have used so far to work with rational maps will also work for these functions, and we present the results for completeness. We note that each map ϕ ∈ QLP(C+ ) has a well-defined (possibly infinite) limit as |z| → ∞, so by the same argument used in Corollary 12, for such a Cϕ to be bounded, we must have lim ϕ(z) = ∞,
|z|→∞
that is to say we must have a1 > b1 . Indeed more than this, we have the following:
S. Elliott / Journal of Functional Analysis 256 (2009) 4162–4186
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Proposition 18. If ϕ ∈ QLP(C+ ), given by ϕ(z) =
λ1 za1 + λ2 za2 + · · · + λm zam μ1 zb1 + μ2 zb2 + · · · + μn zbn
is such that a1 − b1 < 1, then ϕ does not give rise to a bounded compostion operator on any Lp (R), or H p (C+ ) for any 1 p < ∞. Proof. Let 1 p < ∞, and let ε > 0. Then the function fp,ε given by fp,ε (z) =
1 1 + |z|
1+ε p
is in Lp (R). Let us suppose that ϕ ∈ QLP(C+ ), with a1 − b1 < 1. Then in particular, 1 for some ε > 0. Now a1 − b1 < 1+ε Cϕ fp,ε (z) =
1
a1 a 1+ε mz m p 1 + λμ1 zzb1+···+λ +···+μ zbn
n
1
1 1+
|λ1 |
|μ1 | |z|
1
1+ε
1+ε , p
for sufficiently large z. This is clearly not an Lp (R) function, and so Cϕ is not bounded on Lp (R). We note again, that much as with rational functions, the map f (z) = will do for the H p case.
1 (i + z)
1+ε p
2
So if a1 − b1 < 1, Cϕ cannot be bounded. It remains only to show that if a1 − b1 1, then Cϕ must be bounded. Proposition 19. Let ϕ ∈ QLP(C+ ), with representation ϕ(z) =
λ1 za1 + λ2 za2 + · · · + λm zam , μ1 zb1 + μ2 zb2 + · · · + μn zbn
(4)
moreover, let a1 − b1 1. Then ϕ gives rise to a bounded composition operator on each of the spaces H p (C+ ), Lp (R) for 1 p < ∞. Proof. We begin by writing σ (z) = λ1 za1 + λ2 za2 + · · · + λm zam , τ (z) = μ1 zb1 + μ2 zb2 + · · · + μn zbn , then ϕ = σ/τ . Using the same argument as in Proposition 13, we get that Cϕ is bounded, provided τ (−i(2t − 1)) < ∞. lim t t→∞ iσ (−i(2t − 1)) + τ (−i(2t − 1))
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The leading power in the numerator of the fraction is b1 + 1, and in the denominator, it is a1 , but a1 − b1 1, so this limit is indeed finite. 2 Corollary 20. A map of the form (4) induces a bounded composition operator if and only if a1 − b1 1. 6. An adjoint formula for rational-symbol composition operators We begin by making some elementary calculations concerning Cϕ∗ . Let ϕ be a rational selfmap of C+ with ϕ(∞) = ∞, and let f ∈ H 2 (C+ ). If we denote by kz the reproducing kernel for H 2 at z as defined in (3), then
Cϕ∗ f (z) = Cϕ∗ f, kz = f, Cϕ kz 1 1 · dt = f (t) · 2πi z − ϕ(t) R
1 = 2πi
R
f (t) ϕ(t) − z
(5)
dt.
Now let us consider the closed curve γε in C+ shown below:
We note that 1 2πi
γε
f (t) ϕ(t) − z
dt =
1 2πi
(− 1ε , 1ε )
f (t + εi) ϕ(t + εi) − z
dt +
1 2πi
κ
f (t) ϕ(t) − z
dt,
where κ denotes the semicircular section of γε . Taking limits as ε → 0, we get 1 ε→0 2πi
lim
γε
f (t) ϕ(t) − z
1 ε→0 2πi
dt = lim
(− 1ε , 1ε )
f (t + εi) ϕ(t + εi) − z
dt +
1 lim 2πi ε→0
κ
f (t + εi) ϕ(t + εi) − z
dt.
S. Elliott / Journal of Functional Analysis 256 (2009) 4162–4186
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Let us now consider the collection of functions f in H 2 , such that f = O(z−1 ) near ∞. For f in this collection, f (t) dt = 0 lim ε→0 ϕ(t) − z κ
since ϕ(∞) = ∞. As such, f (t) lim dt = lim ε→0 ε→0 ϕ(t) − z γε
f (t + εi) ϕ(t + εi) − z
(− 1ε , 1ε )
dt = R
f (t) ϕ(t) − z
dt,
(6)
since t = t for t ∈ R. Combining (5) and (6), we get that ∗ 1 f (t) f (s) dt = ,s = t , Res Cϕ f (z) = lim ε→0 2πi ϕ(s) − z ϕ(t) − z t∈C+ γε
ϕ(t)=z
by the residue theorem. Since the collection of functions which are O(z−1 ) near ∞ are dense in H 2 , we can write any function f in H 2 as f = lim fn , n→∞
where the fn are O(z−1 ) near ∞. As such, 1 lim ε→0 2πi
γε
1 dt = lim ε→0 2πi ϕ(t) − z f (t)
(− 1ε , 1ε )
f (t + εi) ϕ(t + εi) − z
dt + lim lim
n→∞ ε→0
κ
0 fn (t) dt, ϕ(t) − z
and the same result carries through. This gives us a formula for Cϕ∗ , namely
∗ Cϕ f (z) =
t∈C+ ∩ϕ −1 (z)
We note further that, if we assume that ∗ Cϕ f (z) =
t∈C+ ∩ϕ −1 (z)
=
t∈C+ ∩ϕ −1 (z)
Res lim
s→t
1 ϕ(s)−z
f (s) ,s = t . Res ϕ(s) − z
has only simple poles, then
f (s) ϕ(s) − z
,s = t =
s −t ϕ(s) − z
t∈C+ ∩ϕ −1 (z)
lim
s→t
(s − t)f (s) ϕ(s) − z
lim f (s) ,
s→t
the last line being possible because t is only a simple pole, and f has no poles, being analytic. Overall, this gives us
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∗ Cϕ f (z) =
lim
t∈C+ ∩ϕ −1 (z)
s→t
s −t ϕ(s) − z
f (t),
(7)
which means that Cϕ∗ is in fact a so-called ‘multiple-valued weighted composition operator.’ We note finally that, since ϕ is rational, it will have only simple poles for all but at most finitely many z, hence the above formula is valid except for possibly finitely many z, that is to say, it is true on a dense subset of C+ . 7. Some examples Using the formulae we have derived, we will calculate the adjoints of a number of composition operators. First though, in order to use our Aleksandrov operator characterisation, we will need to work out how to calculate the AC measures associated with an analytic function ϕ. The following are the equivalent of a number of useful results on the disc from Saksman’s excellent introduction to AC measures [17]. Let μ = μa dm + dσ be a measure on R, and let us also denote its Poisson extension by μ, that is to say μ(z) =
Pz (ζ ) dμ(ζ ). R
From Theorem 11.24, and a simple extension of Exercise 19, Section 11 in [16], we have lim μ(b + ir) =
r→0+
μa ∞
for m-almost every b ∈ R, for σ -almost every b ∈ R.
(8)
Given an analytic self-map, ϕ of C+ , we recall that the Aleksandrov–Clark (AC) measures, (μα , cα ), of ϕ are defined by the formula 1 i(1 + αϕ(z)) = Py (x − t) dμα (t) + cα y, ϕ(z) − α π(1 + α 2 ) R
where z = x + iy. Proposition 21. If ϕ : C+ → C+ is an analytic function, and {μα } is its collection of AC measures, then
)) 1 i(1+αϕ(ζ ϕ(ζ )−α π(1+α 2 )
if ϕ(ζ ) ∈ C \ R,
0
if ϕ(ζ ) ∈ R.
Py (x − t) dμα (t) + cα y =
i(1 + αϕ(z)) 1 , ϕ(z) − α π(1 + α 2 )
μaα (ζ ) = Proof. R
S. Elliott / Journal of Functional Analysis 256 (2009) 4162–4186
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where Py (x − t) =
y 1 . π (x − t)2 + y
Hence, 1 π
R
i(1 + αϕ(z)) y 1 − cα y, dμα (t) = ϕ(z) − α (x − t)2 + y 2 π(1 + α 2 )
(9)
= μα (x+iy)
recalling that μα denotes both a measure and its Poisson extension. We now take limits as y → 0. The left-hand side of (9) is μaα (x) (m-a.e.) by (8), and the right-hand side is 1 1 i(1 + αϕ(x)) if ϕ(x) ∈ C \ R, i(1+αϕ(x)) ϕ(x)−α π(1+α 2 ) = ϕ(x) − α π(1 + α 2 ) 0 if ϕ(x) ∈ R. We note that m-almost everywhere equality is the best we could hope for, given that μaα is an L1 function. 2 Proposition 22. supp(σα ) ⊆ x ∈ R: ϕ(x) = α . Proof. Suppose x ∈ R, such that either lim ϕ(x + iy) = α,
y→0+
or this limit does not exist. Then there exists some ε > 0 and some sequence yn 0 with ϕ(x + iyn ) − α ε, for each n ∈ N. But
i(1 + αϕ(x + iyn )) 1 μα (x + iyn ) = − cα yn . ϕ(x + iyn ) − α π(1 + α 2 )
Since ϕ(x) = α, we have lim inf μα (x + iy) < ∞, y→0+
so by (8), σ (x) = 0. As such, supp(σα ) ⊆ x ∈ R: ϕ(x) = α .
2
We now move on to our first example, which is the simplest possible case of a composition operator with linear symbol. It is worth noting that by results from Section 3, strictly linear maps
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are the only fractional linear maps which yield bounded composition operators. Moreover, in order to map C+ into itself, we must have positive real co-efficient of z, and a constant term with non-negative imaginary part, by Proposition 16. Example 1. We begin by noting that, if ϕ(z) = az + b, where a ∈ R+ , and (b) 0, then Cϕ f, g =
Cϕ f (z)g(z) dz =
R
f (az + b)g(z) dz, R
setting x = az + b,
=
f (x)g R
x −b 1 x − b dz dx = f (x) g dx, a dx a a R
since the analytic extension of g to the lower half-plane is g(z). So the adjoint of Cϕ is the weighted composition operator given by z−b 1 Cϕ∗ f (z) = f . a a
(10)
The calculation of the same adjoint using AC measures is as follows. We must split the example into two cases: (i) The case where (b) = 0. (ii) The case where (b) > 0. (i) Since (b) = 0, we have that ϕ(x) ∈ R for all x ∈ R, so by Proposition 21, the absolutely continuous part of each AC measure associated with ϕ is identically 0, or in other words, each measure is entirely singular. By Proposition 22, the singular part of each μα lives on the preimages of α under ϕ, so the support of each μα is just the single point α−b a . α−b In order to determine the value of the point mass at a , we use the defining equation for the AC measures of ϕ, namely cα y + R
i(1 + αϕ(x + iy)) 1 , Py (x − t) dμα (t) = ϕ(x + iy) − α π(1 + α 2 )
where Py (x − t) = π1 (x−t)y2 +y 2 . Since ∞ has no preimages under ϕ, cα = 0 for each α. Setting x = 0 and y = 1 in the above, we get 1 π
R
1 1 i(1 + α(ai + b)) 1 · . dμ (t) = α (ai + b) − α 1 + t2 π 1 + α 2
S. Elliott / Journal of Functional Analysis 256 (2009) 4162–4186
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As such,
1+
1 α−b 2 μα a
α−b a
1 a i(1 + α(ai + b)) = = , 2 ai + b − α 1+α (b − α)2 + a 2
so μα
α−b a
1 = . a
Now Cϕ∗ f (α) = Aϕ f (α) =
R
α−b 1 f (t) dμα (t) = f , a a
which is precisely the same as (10), since b ∈ R. (ii) Since (b) > 0, Proposition 22 tells us that each μα is absolutely continuous and Proposition 21 that i(1 + αϕ(t)) 1 ϕ(t) − α π(1 + α 2 ) i(1 + α(at + b) 1 = . at + b − α π(1 + α 2 )
μaα (t) =
So ∗ Cϕ f (α) =
R
i(1 + α(at + b) 1 dt f (t) at + b − α π(1 + α 2 )
=
f (t) R
(b) 1 dt. π (at − α + (b))2 + (b)2 =(α−(b)−at)2
Since α ∈ R, (α) = α and (α) = 0, so α−b 11 a f (t) dt π a α−b − t 2 + α−b 2 a a R 1 = f (t)P( α−b ) (t) dt a a
∗ Cϕ f (α) =
R
α−b 1 , = f a a just as in (10).
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Our final method of calculation is the residue formula from Section 6. We note that since ϕ is linear, it has a well defined inverse, and no repeated roots, so in fact formula (7) describes the adjoint of ϕ everywhere. As such, we have
∗ Cϕ f (z) =
lim
t∈C+ ∩ϕ −1 (z)
=
s→t
s −t ϕ(s) − z
f (t)
z−b f . lim a s→ z−b as + b − z s−
z−b a
a
So a simple application of L’Hôpital’s theorem gives us z−b 1 (Cϕ∗ f )(z) = f , a a just as in (10). Example 2. Let us consider the map 1 ϕ(z) = z − , z which we know to give rise to an isometric composition operator on H 2 (C+ ) by Proposition 2.1 of [3]. We observe that √ z ± z2 + 4 −1 ϕ (z) = . 2 For calculating the AC measures of ϕ, we note that, if dμα = μaα dm + dσα , then: (a) z −
1 z
∈ R for all z = 0, so μaα (x) = 0 for all x,
(b) σα lives on {x: ϕ(x) = α} = {x: x −
1 x
√ 2 = α} = { α± 2α +4 }.
Moreover, μ0 ≡ 0. Setting x = 0 and y = 1, we get 1 π
R
i(1 + αϕ(i)) 1 1 − cα , dμα (t) = ϕ(i) − α 1 + t2 π(1 + α 2 )
(11)
and setting x = 0 and y = 2, we get 1 π
R
i(1 + αϕ(2i)) 1 1 − 2cα . dμ (t) = α ϕ(2i) − α 4 + t2 π(1 + α 2 )
(12)
It is easy to show that cα = 0 for all α, so solving (11) and (12) as simultaneous equations gives us
S. Elliott / Journal of Functional Analysis 256 (2009) 4162–4186
α+
√ √ 2 α2 + 4 α +4+α = √ 2 2 α2 + 4
α−
√ √ 2 α2 + 4 α +4−α = √ . 2 2 α2 + 4
σα
4185
and σα So ∗ Cϕ f (α) =
√ √ √ 2 α + α2 + 4 α − α2 + 4 α2 + 4 + α α +4−α + √ . f f √ 2 2 2 α2 + 4 2 α2 + 4
√
We note as an aside that μα = σα+ + σα− =
√ √ α 2 + 4 + α + α 2 + 4 − α = 1, √ 2 2 α +4
for each α. The residue method calculation proceeds as follows: let us suppose that z = 2i, that is to say that the two values of ϕ −1 (z) are distinct. Then ∗ s−t lim f (t). Cϕ f (z) = s→t ϕ(s) − z + −1 t∈C ∩ϕ
(z)
So, using L’Hôpital’s theorem to evaluate the limit, we get ∗ Cϕ f (z) =
√ √ √ √ 2 z + z2 + 4 z − z2 + 4 z2 + 4 + z z +4−z + √ , f f √ 2 2 2 z2 + 4 2 z2 + 4
for z = 2i. Moreover, we observe that for z = 2i, we have only one solution to ϕ(t) = z, namely t = i. In this case, f (s) Res , s = i = f (i), ϕ(s) − 2i so since √ z2 + 4 + z z2 + 4 − z + √ = 1, √ 2 z2 + 4 2 z2 + 4
√
the formula is still valid for z = 2i. By observing the above examples, we have reason to hope that some other well known results from the disc may have natural analogues in the half-plane. In Example 1 above, Cϕ is an isometry when b = 0, and |a| = 1, and moreover, we have already observed that Example 2 gives an isometry. In both these cases, ϕ is inner (it maps the boundary of C+ to itself), and the AC
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measures associated with ϕ are of constant modulus equal to 1. We see in [17], for example, that this is precisely the condition for a composition operator on the disc to give rise to an isometry. We note finally, that by virtue of the mapping at the beginning of Section 3, every composition operator on C+ is equivalent to a weighted composition operator on the disc. As such, our observations here may be used to study a certain class of weighted composition operators, which are also of interest. Acknowledgments The author would like to gratefully acknowledge the financial support of the UK Engineering and Physical Sciences Research Council (EPSRC), and the School of Mathematics at the University of Leeds. He is also most deeply indebted to Prof. Jonathan Partington for his ongoing support and guidance. References [1] A.B. Aleksandrov, The multiplicity of boundary values of inner functions, Invent. Akad. Nauk ArmSSR, Mat. 22 (1987) 490–503. [2] P.S. Bourdon, J.H. Shapiro, Adjoints of rationally induced composition operators, J. Funct. Anal. 255 (8) (2008) 1995–2012. [3] I. Chalendar, J.R. Partington, On the structure of invariant subspaces for isometric composition operators on H 2 (D) and H 2 (C+ ), Arch. Math. 81 (2003) 193–207. [4] J.A. Cima, A.L. Matheson, Essential norms of composition operators and Aleksandrov measures, Pacific J. Math. 179 (1997) 59–63. [5] J.A. Cima, A.L. Matheson, W.T. Ross, The Cauchy Transform, American Mathematical Society, 2006. [6] D.N. Clark, One-dimensional perturbations of restricted shifts, J. Anal. Math. 25 (1972) 169–191. [7] C.C. Cowen, Linear fractional composition operators on H 2 , Integral Equations Operator Theory 11 (1988) 151– 160. [8] C.C. Cowen, E.A. Gallardo-Gutiérrez, A new class of operators and a description of adjoints of composition operators, J. Funct. Anal. 238 (2006) 447–462. [9] J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. [10] C. Hammond, J. Moorehouse, M.E. Robbins, Adjoints of composition operators with rational symbol, J. Math. Anal. Appl. 341 (2008) 626–639. [11] V. Kapustin, A. Poltoratski, Boundary convergence of vector-valued pseudocontinuable functions, J. Funct. Anal. 238 (2006) 313–326. [12] V. Matache, Composition operators on Hardy spaces of a half-plane, Proc. Amer. Math. Soc. 127 (1999) 1483–1491. [13] J.N. McDonald, Adjoints of a class of composition operators, Proc. Amer. Math. Soc. 131 (2003) 601–606. [14] E.A. Nordgren, Composition operators, Canad. J. Math. 20 (1968) 442–449. [15] A. Poltoratski˘ı, Kre˘ın’s spectral shift and perturbations of spectra of rank one, Algebra Anal. 10 (1998) 143–183. [16] W. Rudin, Real and Complex Analysis, McGraw–Hill Book Co., New York, 1987. [17] E. Saksman, An elementary introduction to Clark measures, in: D. Girela, C. González (Eds.), Topics in Complex Analysis and Operator Theory, Univ. Málaga, 2007, pp. 85–136. [18] D. Sarason, Composition Operators as Integral Operators, Dekker, New York, 1990. [19] J.H. Shapiro, The essential norm of a composition operator, Ann. of Math. 125 (1987) 375–404.
Journal of Functional Analysis 256 (2009) 4187–4196 www.elsevier.com/locate/jfa
Invariant subspaces for operator semigroups with commutators of rank at most one ✩ Roman Drnovšek Department of Mathematics, Faculty of Mathematics and Physics, Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia Received 16 October 2008; accepted 11 March 2009 Available online 31 March 2009 Communicated by D. Voiculescu
Abstract Let X be a complex Banach space of dimension at least 2, and let S be a multiplicative semigroup of operators on X such that the rank of ST − T S is at most 1 for all {S, T } ⊂ S. We prove that S has a non-trivial invariant subspace provided it is not commutative. As a consequence we show that S is triangularizable if it consists of polynomially compact operators. This generalizes results from [H. Radjavi, P. Rosenthal, From local to global triangularization, J. Funct. Anal. 147 (1997) 443–456] and [G. Cigler, R. Drnovšek, D. Kokol-Bukovšek, T. Laffey, M. Omladiˇc, H. Radjavi, P. Rosenthal, Invariant subspaces for semigroups of algebraic operators, J. Funct. Anal. 160 (1998) 452–465]. © 2009 Elsevier Inc. All rights reserved. Keywords: Invariant subspaces; Triangularizability; Semigroups
1. Introduction Throughout the paper, let X be a complex Banach space of dimension at least 2. A subspace of X means a closed linear manifold of X. Trivial subspaces of X are {0} and X. The dual space of X is denoted by X ∗ . By an operator on X we mean a bounded linear transformation from X into itself. By I we denote the identity operator. The Banach algebra of all operators on X is denoted by B(X). We denote by T ∗ the adjoint operator of T ∈ B(X). The notation [S, T ] ✩
This work was supported by the Slovenian Research Agency. E-mail address: [email protected].
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.03.010
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is used as an abbreviation for the commutator ST − T S, where S, T ∈ B(X). Given y ∈ X and φ ∈ X ∗ , the rank-one operator y ⊗ φ on X is defined by (y ⊗ φ)x = φ(x)y. A subspace M of X is invariant under an operator T ∈ B(X) whenever T (M) ⊆ M. Let C be a collection of operators in B(X). A subspace M of X is invariant under C if M is invariant under every T ∈ C. If, in addition, the subspace M is invariant under every S ∈ B(X) that commutes with all operators of C, M is said to be hyperinvariant under C. A collection C is triangularizable if there is a chain of invariant subspaces for C which is maximal as a subspace chain. In many situations the existence of a non-trivial invariant subspace already implies triangularizability, as the Triangularization Lemma shows (see [6] or [7]). In order to recall it, some definitions are needed. Let C be a collection of operators in B(X). If M and N are invariant subspaces under C with N ⊂ M, then C induces a collection Cˆ of quotients as follows: for T ∈ C, the operator Tˆ ∈ Cˆ is defined on M/N by Tˆ (x + N ) = T x + N . Any such Cˆ is called a set of quotients of the collection C. A property of collections of operators is said to be inherited by quotients if every set of quotients of a collection having the property also has the same property. Lemma 1.1 (Triangularization Lemma). Let P be a property of collections of operators that is inherited by quotients. If every collection of operators (on a space of dimension greater than one) which satisfies P has a non-trivial invariant subspace, then every collection satisfying P is triangularizable. A collection of operators is called an (operator) semigroup if it is closed under multiplication. It is a well-known fact that every commutative semigroup of matrices is triangularizable. In 1978 Laffey extended it as follows (see [4]). Theorem 1.2. If S is a semigroup of matrices such that the rank of [S, T ] is at most 1 for all {S, T } ⊂ S, then S is triangularizable. As a generalization of the preceding theorem, Radjavi and Rosenthal proved the following theorem (see [6, Corollary 2]). Theorem 1.3. If S is a semigroup contained in the Schatten class Cp such that the rank of [S, T ] is at most 1 for all {S, T } ⊂ S, then S is triangularizable. Applying the remarkable result of Turovskii [8] the last theorem can be easily extended to compact operators on X (see [7, Theorem 9.2.10]). Another infinite-dimensional extension of Laffey’s result was shown by the group of authors in [1]. Recall that an operator T on X is said to be algebraic if there exists a non-zero complex polynomial p such that p(T ) = 0. Theorem 1.4. Let S be a semigroup of algebraic operators on X such that the rank of [S, T ] is at most 1 for all {S, T } ⊂ S. Then S is triangularizable.
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In [2] we showed the following theorem related to the subject. Theorem 1.5. Let S be a non-commutative semigroup in B(X) generated by two elements. If the rank of [S, T ] is at most 1 for all {S, T } ⊂ S, then S has a non-trivial hyperinvariant subspace. The preceding theorem is an easy consequence of the following theorem [2, Theorem 2.2]. Theorem 1.6. Let A and B be operators on X such that the rank of [A, B] is 1 and the ranks of each of [A2 , B], [A, B 2 ], [A2 , B 2 ], [AB, BA] are at most 1. Then {A, B} has a non-trivial hyperinvariant subspace. It is clear from the proof of Theorem 1.6 that the following is also true. Theorem 1.7. Under the assumptions of Theorem 1.6, let [A, B] = y ⊗ φ for a non-zero vector y ∈ X and a non-zero functional φ ∈ X ∗ . Then φ(y) = 0, and so [A, B]2 = 0. The proof of Theorem 1.6 is essentially based upon the following (easily proved) observation [2, Lemma 2.1]. Lemma 1.8. Let y ∈ X and φ ∈ X ∗ be non-zero vectors. Assume that z ∈ X and ψ ∈ X ∗ are such that the rank of the sum y ⊗ φ + z ⊗ ψ is at most 1. Then either z is a multiple of y or ψ is a multiple of φ. 2. Results The key result of this paper is the following theorem. Theorem 2.1. Let S be a semigroup in B(X) such that the rank of [S, T ] is at most 1 for all {S, T } ⊂ S. Suppose that [A, B] = y ⊗ φ for some {A, B} ⊂ S and for some non-zero vector y ∈ X and non-zero functional φ ∈ X ∗ . Then φ(Cy) = 0 for all C ∈ S ∪ {I }. Proof. With no loss of generality we can assume that CS = S, that is, S is closed for scalar multiples of its members. Since φ(y) = 0 by Theorem 1.7, we must prove that φ(Cy) = 0 for all C ∈ S. Assume on the contrary that φ(Cy) = 0 for some C ∈ S. As CS = S, we can assume that φ(Cy) = 1. Since φ(y) = 0 and (C ∗ φ)(y) = φ(Cy) = 1, neither y is an eigenvector of C nor φ is an eigenvector of C ∗ . Therefore, it follows from Lemma 1.8 that the rank of [AB, C] − [BA, C] = [A, B], C = (y ⊗ φ)C − C(y ⊗ φ) = y ⊗ (C ∗ φ) − (Cy) ⊗ φ is equal to 2. Since the ranks of [AB, C] and [BA, C] are at most 1, they are both equal to 1. In fact, we must have [AB, C] = (αy + βCy) ⊗ (γ φ + δC ∗ φ) for some scalars α, β, γ and δ. We claim that δ = 0. Assume on the contrary that δ = 0. Clearly, we can assume that γ = 1. An application of Theorem 1.7 for the pair {AB, C} yields φ(αy + βCy) = 0, and so β = 0. Then the rank of [BA, C] = [AB, C] − [A, B], C = y ⊗ (αφ − C ∗ φ) + (Cy) ⊗ φ
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is equal to 2. This contradiction proves the claim. Now, with no loss of generality we can assume that δ = 1, and so [AB, C] = (αy + βCy) ⊗ (γ φ + C ∗ φ) and [BA, C] = [AB, C] − [A, B], C = y ⊗ αγ φ + (α − 1)C ∗ φ + Cy ⊗ (1 + βγ )φ + βC ∗ φ . Since rank[BA, C] 1 and Cy is not a multiple of y, by Lemma 1.8 there exists a scalar λ such that αγ φ + (α − 1)C ∗ φ = λ (1 + βγ )φ + βC ∗ φ . Since C ∗ φ and φ are linearly independent, we have λ(1 + βγ ) = αγ
and λβ = α − 1.
Eliminating α we obtain that λ = γ , and so α = 1 + βγ . An application of Theorem 1.7 for the pair {AB, C} yields (γ φ + C ∗ φ)(αy + βCy) = 0, which implies that 1 + 2βγ + βφ C 2 y = 0. It follows that β = 0, and we may define k = 1/β + γ . Therefore, we have [AB, C] = β(ky + Cy) ⊗ (γ φ + C ∗ φ), [BA, C] = β(γ y + Cy) ⊗ (kφ + C ∗ φ) and φ C 2 y = −k − γ . Now, the rank of AB, C 2 = [AB, C]C + C[AB, C] = β(ky + Cy) ⊗ γ C ∗ φ + (C ∗ )2 φ + β kCy + C 2 y ⊗ (γ φ + C ∗ φ) is at most 1, and so, by Lemma 1.8 again, one of the following two cases must occur. Case (I): kCy + C 2 y = λ(ky + Cy) for some scalar λ. Applying the functional φ to this equation we obtain that λ = k + φ(C 2 y). Since φ(C 2 y) = −k − γ , we have λ = −γ , so that C 2 y + (k + γ )Cy + kγ y = 0 or
(C + k)(C + γ )y = 0.
Now, decompose X as X = lin{y} ⊕ lin{Cy} ⊕ ker(φ) ∩ ker(C ∗ φ). Note that ker(φ) = lin{y} ⊕ ker(φ) ∩ ker(C ∗ φ). With respect to this decomposition the operator C has the matrix C=
0 1 0
−kγ −k − γ 0
c13 c23 c33
.
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If x ∈ ker(φ) ∩ ker(C ∗ φ), then φ(Cx) = 0, so that Cx ∈ ker(φ) = lin{y} ⊕ ker(φ) ∩ ker(C ∗ φ). This implies that c23 = 0. Since [A, B]x = 0 for all x ∈ ker(φ) and [A, B]Cy = y, we have [A, B] =
0 0 0
0 0 . 0
1 0 0
Since [AB, C]y = β(ky + Cy) and [AB, C]Cy = −βk(ky + Cy), we have [AB, C] =
k 1 0
−k 2 −k 0
γ 1 0
−γ 2 −γ 0
0 0 0
k = β 1 ( 1 −k 0
(1)
0 ).
Similarly, we obtain that [BA, C] = β
0 0 0
γ = β 1 ( 1 −γ 0
0 ).
Denoting D = AB = (dij )3i,j =1 , we have [AB, BA] = [A, B]D − D[A, B] =
d21 0 0
d22 − d11 −d21 −d31
d23 0 0
.
Since rank[AB, BA] 1, we conclude that d21 = 0, and either d23 = 0 or d31 = 0. Therefore, we must consider two subcases. Subcase (Ia): d23 = 0. Then [AB, C] = [AB − d11 I, C] d12 − c13 d31 −(k + γ )d12 + kγ (d22 − d11 ) − c13 d32 = −d12 d22 − d11 d32 − c33 d31 −kγ d31 − (k + γ )d32 − c33 d32
d13 c33 − c13 (d33 − d11 ) −d13 d31 c13 + d33 c33 − c33 d33
.
Comparing with (1) and simplifying we get d13 = 0, c13 d31 = 0,
d12 = βk, c13 d32 = 0,
d22 = d11 + β, d32 = c33 d31 .
(2)
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We thus have
βk 0 D = AB = d11 + β 0 d32 d33 d11 βγ BA = AB − [A, B] = 0 d11 + β d32 d31 d11 0 d31
and 0 0 d33
.
Now compute the commutator [ABBA, C] = [AB, C]BA + AB[BA, C] k d11 βγ 0 = β 1 ( 1 −k 0 ) 0 d11 + β 0 0 d32 d33 d31 d11 γ βk 0 + 0 d11 + β 0 β 1 ( 1 −γ 0 ) d32 d33 d31 0 k d11 γ + βk = β 1 ( d11 −kd11 − 1 0 ) + β ( 1 −γ d11 + β 0 d31 γ + d32
0 ).
Since its rank is at most 1, two subsubcases are possible by Lemma 1.8. Subsubcase (Ia1): (d11 γ + βk, d11 + β, d31 γ + d32 ) = μ(k, 1, 0) for some scalar μ. Eliminating μ we obtain that d11 = 0 and d31 γ + d32 = 0.
(3)
We now compute the commutator [BAAB, C] = [BA, C]AB + BA[AB, C] 0 βk 0 γ = β 1 ( 1 −γ 0 ) 0 β 0 0 d31 d32 d33 0 βγ 0 k + 0 β 0 β 1 ( 1 −k 0 ) d31 d32 d33 0 γ βγ = β 1 (0 1 0) + β β ( 1 −k 0 d31 k + d32
0 ).
Since its rank is at most 1, we conclude that d31 k + d32 = 0. Using (3) we obtain that d31 = 0 and d32 = 0 as k = γ . Therefore, the two-dimensional subspace lin{y, Cy} is invariant under ˜ D˜ and E˜ denote the restrictions to this subspace of C, D = AB and {C, AB, BA}. Let C, E = BA, respectively. Then
R. Drnovšek / Journal of Functional Analysis 256 (2009) 4187–4196
C˜ = D˜ = β
k (0 1
1)
0 −kγ 1 −k − γ
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,
and E˜ = β
γ ( 0 1 ). 1
˜ Now, we could apply Theorem 1.2 to obtain triangularizability of the semigroup generated by C, D˜ and E˜ which would contradict the fact that the commutator ˜ C] ˜ = 1 −k − γ [D˜ − E, 0 −1 is not nilpotent. However, there exists a quick way to get a contradiction directly. Since k γ D˜ C˜ = β ( 1 −k − γ ) and E˜ C˜ = β ( 1 −k − γ ), 1 1 we have
k γ + and ( 0 −k ) (1 k + γ ) 1 1
γ k 2 ˜ ˜ ˜ [E C, D] = β ( 0 −γ ) + (1 k + γ ) . 1 1 ˜ E] ˜ = β2 [D˜ C,
Since the ranks of both commutators are at most 1, we conclude first that k = 0 and then γ = 0. This is not possible, and so Subsubcase (Ia1) is finished. Subsubcase (Ia2): (d11 , −kd11 − 1, 0) = μ(1, −γ , 0) for some scalar μ. It follows that d11 = μ = −β. In this case we compute [CAB, BA] = C[AB, BA] − [BA, C]AB 0 −kγ c13 β = 1 −k − γ 0 0 (0 1 0 0 c33 −d31 γ −β βk − β 1 ( 1 −γ 0 ) 0 0 0 d31 d32 0 γ = β ( 0 1 0 ) − β 1 ( −β −d32 0
0) 0 0 d33 βk
0 ),
where we have used two equalities from (2). Since the rank of this commutator is at most 1, we obtain that γ = 0 and d32 = 0. Similarly, we obtain that [CBA, AB] = C[BA, AB] − [AB, C]BA 0 0 c13 −β = 1 −k 0 0 (0 1 0) 0 0 c33 d31
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k −β 0 0 − β 1 ( 1 −k 0 ) 0 0 0 0 d31 0 d33 0 k = −β ( 0 1 0 ) − β 1 ( β 0 0 ). 0 0 It follows that k = 0, so that γ = k = 0. This contradiction completes the proof in this subsubcase, and so Subcase (Ia) as well. Subcase (Ib): d31 = 0. Then
d11 d12 d13 0 −kγ c13 [AB, C] = 0 d22 d23 1 −k − γ 0 0 d32 d33 0 0 c33 0 −kγ c13 d11 d12 d13 d12 − 1 −k − γ 0 0 d22 d23 = d22 − d11 0 d32 d33 0 0 c33 d32
∗ ∗ ∗
∗ ∗ . ∗
Comparing with (1) we obtain d12 = βk, d22 = d11 + β and d32 = 0. Therefore, the twodimensional subspace lin{y, Cy} is invariant under {C, AB, BA}, and so we can proceed as in ˜ D˜ and E˜ the restrictions to this invariant Subsubcase (Ia1) to get a contradiction. Denote by C, subspace of C, D = AB and E = BA, respectively. Then C˜ = D˜ =
d11 0
βk d11 + β
0 −kγ 1 −k − γ
and E˜ =
,
d11 0
βγ d11 + β
.
˜ D, ˜ E} ˜ is triangularizable. However, this implies By Theorem 1.2, the semigroup generated by {C, a contradiction, as the commutator ˜ C] ˜ = [D˜ − E,
1 −k − γ 0 −1
is not nilpotent. This concludes this subcase, and so Case (I) as well. Case (II): γ C ∗ φ + (C ∗ )2 φ = λ(γ φ + C ∗ φ) for some scalar λ. Computing both sides of this equality at the vector y yields λ = γ + φ(C 2 y). Since φ(C 2 y) = −k − γ , we obtain that λ = −k. Therefore, we have (C ∗ )2 φ + (k + γ )C ∗ φ + kγ φ = 0. Thus, the semigroup S ∗ = {S ∗ : S ∈ S} satisfies the same conditions as the semigroup S in Case (I). Indeed, if x → Fx denotes the isometric embedding of X to X ∗∗ , then we have [B ∗ , A∗ ] = φ ⊗ Fy , Fy (φ) = φ(y) = 0 and Fy (C ∗ φ) = (C ∗ φ)(y) = φ(Cy) = 0. So, we can use Case (I) to obtain a contradiction. This completes the proof of the theorem. 2
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Corollary 2.2. If S is a non-commutative semigroup in B(X) such that the rank of [S, T ] is at most 1 for all {S, T } ⊂ S, then S has a non-trivial invariant subspace. Proof. Since S is not commutative, we have [A, B] = y ⊗ φ for some {A, B} ⊂ S and for some non-zero vector y ∈ X and non-zero functional φ ∈ X ∗ . Then the closed linear span of the set {Sy: S ∈ S ∪ {I }} is a non-zero subspace invariant under S. Since φ(Sy) = 0 for all S ∈ S ∪ {I } by Theorem 2.1, it is contained in ker(φ), and so it is non-trivial. 2 The preceding corollary does not hold for commutative semigroups, since S may be generated by an operator without non-trivial invariant subspaces. In order to cover the commutative case as well, the following property of operators was introduced in [2]. Let R be the property of operators on Banach spaces of dimension at least 2 such that: (a) R is inherited by quotients, (b) each commutative semigroup of operators with the property R has a non-trivial invariant subspace. Since every non-zero compact operator on an infinite-dimensional Banach space has a nontrivial hyperinvariant subspace by the famous Lomonosov’s result [5], the property of being a compact operator is an example of such property R. Another example is the property of being an algebraic operator, as eigenspaces of an algebraic operator are hyperinvariant subspaces. The main result of the paper is the following theorem. Theorem 2.3. Let S be a semigroup of operators on X with the property R. If the rank of [S, T ] is at most 1 for all {S, T } ⊂ S, then S is triangularizable. Proof. By the Triangularization Lemma, it suffices to show that S has a non-trivial invariant subspace. This is true by Corollary 2.2 if S is not commutative. Otherwise, this holds by the condition (b) of property R. 2 Compact and algebraic operators are special cases of polynomially compact operators. Recall that an operator T on X is said to be polynomially compact if there exists a non-zero complex polynomial p such that the operator p(T ) is compact. Triangularizability of collections of polynomially compact operators was studied by Konvalinka in [3]. Since the Lomonosov’s result is strong enough to give non-trivial hyperinvariant subspaces of polynomially compact operators (that are not algebraic), the property of being a polynomially compact operator is also an example of property R. So, we have the following corollary of Theorem 2.3 that extends both Theorems 1.3 and 1.4. Corollary 2.4. Let S be a semigroup of polynomially compact operators on X. If the rank of [S, T ] is at most 1 for all {S, T } ⊂ S, then S is triangularizable. References [1] G. Cigler, R. Drnovšek, D. Kokol-Bukovšek, T. Laffey, M. Omladiˇc, H. Radjavi, P. Rosenthal, Invariant subspaces for semigroups of algebraic operators, J. Funct. Anal. 160 (1998) 452–465. [2] R. Drnovšek, Hyperinvariant subspaces for operator semigroups with commutators of rank at most one, Houston J. Math. 26 (3) (2000) 543–548.
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[3] M. Konvalinka, Triangularizability of polynomially compact operators, Integral Equations Operator Theory 52 (2) (2005) 271–284. [4] T.J. Laffey, Simultaneous triangularization of matrices – low rank cases and the nonderogatory case, Linear Multilinear Algebra 6 (1978) 269–305. [5] V.I. Lomonosov, Invariant subspaces of the family of operators that commute with a completely continuous operator, Funktsional. Anal. i Prilozhen. 7 (3) (1973) 55–56. [6] H. Radjavi, P. Rosenthal, From local to global triangularization, J. Funct. Anal. 147 (1997) 443–456. [7] H. Radjavi, P. Rosenthal, Simultaneous Triangularization, Universitext, Springer-Verlag, New York, 2000. [8] Yu.V. Turovskii, Volterra semigroups have invariant subspaces, J. Funct. Anal. 162 (1999) 313–322.
Journal of Functional Analysis 256 (2009) 4197–4214 www.elsevier.com/locate/jfa
Density minoration of a strongly non-degenerated random variable Paul Malliavin a , Eulalia Nualart b,c,∗ a 10 rue Saint-Louis-en-l’Ile, 75004 Paris, France b Institut Galilée, Université Paris 13, 93430 Villetaneuse, France c Departamento de Estadística e Investigación Operativa, Universidad Pública de Navarra, 31006 Pamplona, Spain
Received 4 November 2008; accepted 24 November 2008 Available online 10 December 2008 Communicated by Paul Malliavin
Abstract We obtain a lower bound for the density of a d-dimensional random variable on the Wiener space under exponential moment condition of the divergence of covering vector fields. © 2008 Elsevier Inc. All rights reserved. Keywords: Malliavin calculus; Lower bounds for densities; Divergence; Riesz transforms
1. Introduction Finding lower bounds for densities of random variables on the Wiener space has been a current subject of research in probability theory for the last twenty years. As is well known, the use of stochastic calculus of variations (Malliavin calculus) is a tool for studying existence, smoothness of densities of random variables on the Wiener space, as well as finding explicit lower bounds. The work by Kusuoka and Stroock became the starting point of the use of Malliavin calculus to obtain lower bounds for densities. In [9], they obtained a lower bound of Gaussian type for the density of a uniformly hypoelliptic diffusion whose drift is a smooth combination of its diffusion coefficient. Their results are the first known extensions of the analytical results obtained in [5,15]. Fifteen years later, Kohatsu-Higa [7,8] extended the result by Kusuoka and Stroock by giving * Corresponding author.
E-mail addresses: [email protected] (P. Malliavin), [email protected] (E. Nualart). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.11.016
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a general definition of uniformly elliptic random variables on the Wiener space and obtained Gaussian type lower bounds for those random variables. He applied his results to the solution of the non-linear heat equation and to non-homogeneous uniformly elliptic diffusions. The result by Kohatsu-Higa was applied in [4] to solutions of non-linear hyperbolic SPDEs to derive results on potential theory. Later on, Bally [2] relaxed the uniformly elliptic condition of Kohatsu-Higa. The ideas of Bally were used in [6] to obtain a lower bound for the density of a non-linear Landau process with a degenerate diffusion coefficient. In this paper we are interested in finding lower bounds for densities of abstract random variables on the Wiener space; we extend the one-dimensional result obtained in [13], which is a lower bound for the density of a real random variable F on the Wiener space, under an exponential moment on the divergence of a covering vector field of F ; the methodology used in [13] was the resolution of a one-dimensional variational problem. The present work will use the following tricks to solve the case of R d valued random variables: radial averaging combined with Riesz transform estimates on the sphere and Riesz transform estimates on balls. 2. Notations and main theorem We first introduce some elements of the differential calculus on Gaussian probability spaces (see for instance [11,14]). Let W = {W (h), h ∈ H } be an isonormal Gaussian process associated with a Hilbert space H . Let S denote the class of smooth random variables of the form F = f (W (h1 ), . . . , W (hn )), where h1 , . . . , hn are in H , n 1, and f belongs to CP∞ (R n ), the set of functions f such that f and all its partial derivatives have at most polynomial growth. Given F in S, its derivative is the H -valued random variable given by DF =
n
∂i f W (h1 ), . . . , W (hn ) hi .
i=1
For h ∈ H fixed, we define the operator Dh on the set S by Dh F = DF, hH . More generally, the kth order derivative of F ∈ S is obtained by iterating the derivative operator k times and is denoted D k F . Then for every p 1 and any natural number k, we denote by p Dk the clousure of S with respect to the norm · k,p defined by k p p F k,p = E |F |p + E D j F H ⊗j . j =1
The derivative operator D is a closed and unbounded operator with values in L2 (Ω; H ); it is defined on the dense subset D12 of L2 (Ω). We denote by δ the adjoint of the operator D, which is an unbounded operator on L2 (Ω, H ) taking values in L2 (Ω). In particular, if u belongs to Dom δ, then δ(u) is the element of L2 (Ω) characterized by the duality relation: (2.1)a E F δ(u) = E DF, uH , for any F ∈ D12 . The operator δ is called the divergence operator. An important application of Malliavin calculus is the following criterion for existence and smoothness of densities:
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We say that an R d -valued random variable F is non-degenerated if F ∈ D2∞ and (det γF )−1 ∈ Lp (Ω) for all p > 1, where γF denotes the Malliavin matrix of F , that is, (γF )ij = DFi , DFj H ,
1 i, j d.
(2.1)b
Then the law of a non-degenerated R d -valued random variable F has a probability density function which is Hölderian (see for instance [11, p. 72, Theorem 4.1], [14]). Consider a non-degenerated R d -valued random variable F . Let us recall the definition of system of covering vector fields of F (see for instance [12]): An H ⊗d -valued random variable (A1 , . . . , Ad ) is a system of covering vector fields of F if Ai ∈ Dom δ, and if (∂i φ) ◦ F = DAi (φ ◦ F )
(2.1)c
for all φ smooth and i = 1, . . . , d. i,j For instance, denoting γF the inverse of the Malliavin matrix γF , then Ai :=
i,j
γF DFj
(2.1)d
j
is system of covering vector fields of F (there exist many other possible choices of covering vector fields of F ). Then, for any system of covering vector fields, we have (see for instance [11,12,14]) E ∂i φ(F ) = E φ(F )δ(Ai ) ,
for all i = 1, . . . , d.
(2.1)e
The main theorem of this paper is the following (see [13] for the one-dimensional case). Theorem. Assume that there exists a system of covering vector fields Ai of F , γ > 1 and c > 0 such that,
E exp c
d
|δ(Ai )|
γ
< ∞.
(2.2)a
i=1
Then the law of F has a probability density function p in R d such that there exists cγ > 0 satisfying γ p(x) cγ exp −cγ x γ −1 ,
for all x ∈ R d .
Reduction to an inequality on Rd Let φ : R d → R be smooth. By the integration by parts formula and (2.1)e ,
φ(x)∂i p(x) dx =
Rd
Rd
φ(x)E δ(Ai ) F = x p(x) dx.
(2.2)b
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Therefore,
∂i p (x) = E δ(Ai ) F = x , p
for all i = 1, . . . , d.
(2.3)a
Moreover, writing qi := E[δ(Ai ) | F ], for any γ > 1, it holds that ∇pγ (F ) = qγ |qi |γ , γ p d
i=1
and, thus, for any γ > 1 and c > 0, using Jensen’s inequality, d
d
γ ∇pγ γ
δ(Ai ) E exp c (F ) E exp c |qi | E exp c . (2.3)b pγ i=1
i=1
Hence, hypothesis (2.2)a , implies that there exists γ > 1 and c > 0 such that
γ exp c∇ log p(x) p(x) dx < +∞.
Rd
That is, we have that
γ exp c∇f (x) − f (x) dx < +∞,
where f = − log p.
(2.3)c
x 1.
(2.3)d
Rd
We want to prove that (2.3)c implies that γ
f (x) cγ x γ −1 ,
for all x ∈ R d ,
This would prove (2.2)b . We have reduced our problem to an implication on R d : does (2.3)c → (2.3)d ? The methodology used to prove this implication will depend on three steps: a radial averaging method, an estimation of Riesz transform on the unit sphere of R d and, finally, an estimation of Riesz transform on the unit ball of R d . 3. Radial averaging For any C 1 positive function f on R d , γ > 1, and c > 0, we define the functional Iγ (f ) := Rd
γ exp c∇f (x) exp −f (x) dx.
(3.1)a
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The average of a function over the orthogonal group G with respect its Haar measure σ is defined as ∗ (3.1)b f (x) := f (gx)σ (dg), for all x ∈ R d . G
Theorem. For any C 1 positive function f on R d and for all γ > 1, Iγ (f ∗ ) Iγ (f ).
(3.1)c
Proof. In two dimensions, the average of f can be written as 1 f (x) := 2π ∗
2π f (Rθ x) dθ,
for all x ∈ R 2 ,
0
where Rθ is the rotation matrix Rθ =
cos θ sin θ
− sin θ cos θ
.
Let γ > 1 be fixed and let f1 , f2 be C 1 functions on R d . Then f1 + f2 γ 1 ∇f1 γ + 1 ∇f2 γ , ∇ 2 2 2
(3.2)a
as f1 + f2 1 ∇f1 + ∇f2 ∇ 2 2 and the function ξ → ξ γ is convex. Hence, Iγ
f1 + f2 2
1 1 c c γ γ exp ∇f1 exp ∇f2 exp − f1 exp − f2 dx 2 2 2 2
Rd
Iγ (f1 ) × Iγ (f2 ),
(3.2)b
the last inequality obtained using Cauchy–Schwarz inequality. We set Φx (θ ) = f (Rθ x) and kπ 1 , n 0. fn (x) = n Φx 2 2n n 0k<2
Iterating the last inequality in R 2 it holds that for all n 0, Iγ (fn (x)) Iγ (f ). In particular, lim inf Iγ fn (x) Iγ (f ). (3.2)c n→∞
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Moreover, for all x ∈ R 2 , limn→+∞ fn (x) = f ∗ (x). Hence, γ γ lim exp c∇fn (x) exp −fn (x) = exp c(∇f )∗ (x) exp −f ∗ (x) .
n→+∞
By Fatou’s lemma, ∗
Iγ (f )
γ lim exp c∇fn (x) exp −f (x) dx lim inf Iγ (fn ). n→∞
n→+∞
(3.2)d
R2
This proves the result in dimension 2. In dimension d, we pick an orthonormal basis of the Lie algebra of G and we average on the rotations corresponding to each of these elements. 2 Corollary. Let f be a C 1 positive function on R d such that for some γ > 1 and c > 0, Iγ (f ) < +∞. Then, there exists a constant cγ > 0 such that for all x ∈ R d sufficiently large, γ
f ∗ (x) cγ x γ −1 .
(3.2)e
Proof. By the last Theorem, Iγ (f ∗ ) < +∞. Because f ∗ is a radial, changing variables x = r, +∞ γ Iγ (f ) = ωd exp c f (r) exp −f (r) r d−1 dr. ∗
0
Because the last integral can be bounded below by +∞ γ exp −f (r) dr, exp c f (r) 1
Appendix A implies the desired result.
2
4. Riesz transform on the sphere Set G the orthogonal group of R d ; set G its Lie algebra of infinitesimal derivations on the left; then G is isomorphic to d × d antisymmetric matrices a, a T + a = 0. The unit sphere S of R d is an homogeneous space under the action of G. Choosing a point s0 ∈ S we get a map Ψ : G → S defined by Ψ (g) = g −1 (s0 ); functional spaces on S are lifted by Ψ ∗ to functional spaces on G. Instead of establishing Riesz transform on S it will be at the same time easier and more general to establish Riesz transform on G. Denote by ad the adjoint action of G on itself: ad(a)(X) = [a, X] = aX − Xa,
a, X ∈ G.
For i < j set ai,j ∈ G the matrix having all its coefficients equal to zero at the exception of the i ( or j ) line and of the j (or i) column, the absolute value of the non-vanishing coefficients
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being equal to 1. Then {ai,j }i<j constitutes a basis of G; we define an euclidean metric on G by imposing that the basis {ai,j }i<j will be orthonormal. The euclidean structure so choosen on G defines on G a structure of Riemannian manifold. Using [3] formula (2.1)d we obtain that the Ricci tensor of G is equal to Ricci = −
2 d − 1 1 ad(ai,j ) = × Identity. 4 4
(4.1)a
i<j
Set 0 the Laplace Beltrami operator of G operating on functions; it generates a diffusion process on G; it can be proved (see [3]) that this diffusion can be defined by solving the following Stratonovitch stochastic differential equation dgb (t) =
ai,j
◦ dbi,j (t) gb (t),
gb (0) = Identity,
(4.1)b
i<j
where bi,j are independent scalar valued Brownian motions. The law of gb (t) converges when t → ∞ towards the Haar measure σ of G. Using the right invariant parallelism on G it is possible to construct a 1-differential form on G to a map G → G. Set 1 the Hodge–Laplace–Beltrami operator operating on 1-differential forms, set d the exterior differential sending p-differential forms into p + 1-differential forms and set d ∗ its adjoint. We then have 0 = d ∗ d,
1 = dd ∗ + d ∗ d,
together with the basic Hodge commutations: 0 d ∗ = d ∗ 1 .
d0 = 1 d,
(4.1)c
Theorem (Riesz transform). Let f be a C 1 function on G with vanishing mean value. Then there exists a kernel K ∈ L1σ (G; G) such that f (g0 ) = ∇f ∗ K(g0 ) :=
∇f g −1 g0 K(g) σ (dg),
G
where
K(g)p σ (dg) =: Kp p G
Lσ (G;G )
G
for all p <
d +2 d +1 ,
where d =
d(d−1) 2
is the dimension of G.
Proof. Set λ = ∇f = df . Then, 0 f = d ∗ λ.
< ∞,
(4.2)a
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Therefore, ∗ f = −1 0 d λ.
(4.2)b
∗ ∗ −1 Using the commutation −1 0 d = d 1 , we get
f = d ∗ −1 1 λ.
(4.2)c
Using [10], we must lift the situation to the orthonormal frame bundle O(G) of G. Set H the orthogonal group of the euclidean space G, and set H its Lie algebra then O(M) = H × G. The explicit expression of the Christoffel symbol characterizing the Levi–Civita connection has been computed in [3] formula (2.1)a , as ∇a X =
1 ad(a)(X) := aX − Xa, 2
then ad(a) ∈ H,
(4.2)d
and the lift a˜ i,j of the vector field ai,j to H × G is defined as a˜ i,j (γ , g) =
1 ad(ai,j )γ , ai,j g . 2
˜ 0 of 0 defines the process (γ ˜ (t), gb (t)), where γ ˜ (t) satisfies the system The lifted Laplacian b b of equations 1 dγb˜ (t) = 2
˜ ad aij ◦ d bi,j (t) γb˜ (t),
γb˜ (0) = Identity,
i<j
˜ db(t) = γb˜ (t) d b(t) ,
(4.2)e
and gb (t) is given in (4.1)b (see [1, Section 3]). According to [10, Proposition 2.4.1], ˜ 0 − Ricci. 1 = Given a differential form ω ∈ C 0 (G; G), we have d −1 t × γb˜ (t) ω gb (t)g0 , exp(t1 )ω (g0 ) = E exp − 4 and −1 1 ω (g0 ) =
∞ exp(t1 )ω (g0 ) dt. 0
(4.2)f
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In order to realize (4.2)c we have to apply d ∗ to (4.2)f , that is to differentiate one-time relatively to the initial condition g0 ; it is well known that such derivation can be obtained by a Girsanov transformation (see [11, p. 245]); bi,j (t) ∗ i,j d −1 . (4.2)g × exp − t × γb˜ (t) ω(gb (t)g0 ) d exp(t1 )ω (g0 ) = E t 4 i<j
Set πt (g) the probability density of the law of gb (t) relatively to the measure σ . We deduce from (4.2)g that p KLp (G;G ) σ
∞
πt (g) p d −1 t dt exp −p σ (dg); √ 4 t G
0
finally
πt (g) √ t
p σ (dg)
G
1
p
t2
R
d
|ξ |2 dξ − p2 −(p−1) d2 exp −p t .
2t t pd2
2
Theorem. Let ϕ be a C 1 function on the d-dimensional unit sphere S. Then
exp(ϕ) dσ S
exp KL1σ (G;G ) ∇ϕG dσ × exp ϕ dσ .
S
(4.3)a
S
Moreover, there exist two constants c1 , c2 such that ϕ dσ . exp ϕL∞ (S) < c1 + 2 exp c2 ∇ϕG dσ × exp S
(4.3)b
S
Proof. By substracting a constant to ϕ we reduce to the case where the mean value vanishes: then, expanding in Taylor serie the exponential, we get that 1 exp(ϕ) dσ = |ϕ|p dσ. p! S
S
By convexity of the norm Lp , we have ∇ϕ ∗ KLp ∇ϕLp × KL1 . Then, using (4.2)a we obtain (4.3)a . Let us proceed to the proof of (4.3)b ; set q > d + 1 the conjugate exponent of p < Then, using (4.2)a together with Hölder’s inequality, ϕL∞ KLp ∇ϕLq .
d +2 d +1 .
(4.3)c
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On the other hand, 1 1 ϕnL∞ + ϕnL∞ . exp ϕL∞ = n! n! n
Using (4.3)c , 1 1 ϕnL∞ KnLp ∇ϕnLq , n! n!
nq
nq
where ∇ϕnLq
n q
=
∇ϕG dσ
q
,
S
and, using the fact that n q, it yields that
n q
∇ϕG dσ S
q
∇ϕn dσ. S
Hence, 1 n ϕL∞ exp KLp × ∇ϕG dσ. n!
nq
S
In (4.3)b , we choose, c2 := KLp > KL1 . Consider the polynomial P (ξ ) =
1 ξ n, n! n
ξ > 0.
Then, lim exp(−ξ )P (ξ ) = 0.
ξ →+∞
Set R such that P (ξ )
1 exp(ξ ), 2
ξ > R.
Then, P (ξ ) exp(ξ ) − P (ξ ),
ξ > R.
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Finally, we take c1 := max P (ξ ), ξ ∈[0,R]
which concludes the proof of (4.3)b .
2
4.1. Exceptional set of radius Set ϕr (σ ) = f (rσ ),
1 and note that ∇ϕr G = ∇f G . r
Then, by (2.3)c ,
∞ r d−1 dr
exp cr −γ ∇ϕr γ − ϕr dσ
S
0
γ exp c∇f (x) − f (x) dx < +∞.
Rd
Set −γ γ Θ = r ∈ [1, +∞[; exp cr ∇ϕr − ϕr dσ < 1 .
(4.4)a
S
Then, r d−1 dr < ∞.
(4.4)b
Θc
Using Cauchy–Schwarz inequality,
2 −γ c −γ γ γ exp r ∇ϕr dσ exp cr ∇ϕr − ϕr dσ × exp(ϕr ) dσ, 2
S
S
S
and, in particular,
2 c exp r −γ ∇ϕr γ dσ exp(ϕr ) dσ, 2
S
∀r ∈ Θ.
(4.4)c
S
Using (4.3)a , setting c˜ := KL1σ (G;G ) , we have S
2 c exp r −γ ∇ϕr γ dσ exp c∇ϕ ˜ dσ × exp ϕ dσ , r r 2 S
S
∀r ∈ Θ.
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Using (3.2)e ,
2 γ c exp r −γ ∇ϕr γ dσ exp cγ r 1−γ × exp c∇ϕ ˜ r dσ, 2
S
∀r ∈ Θ.
S
Finally,
c −γ γ exp r ∇ϕr dσ sup(A, B) 2
γ where A := exp cγ r 1−γ ,
(4.4)d
S
and B=
exp c2 ∇ϕr dσ.
S
Theorem. We have γ
ϕr L∞ (S) cγ
r 1−γ ,
∀r ∈ Θ.
(4.5)a
Proof. Looking to the inequality (4.4)d , we have two cases to consider, either A B or A < B. In the first case we have γ 1−γ exp c2 ∇ϕr dσ ; exp cγ r S
we conclude by using (4.3)b and (3.2)e . In the second case, we have c −γ γ exp r ∇ϕr dσ exp c2 ∇ϕr dσ. 2 S
(4.5)b
S
Consider the set γ c −γ γ E := s ∈ S; r ∇ϕr 2c2 ∇ϕr = s ∈ S; ∇ϕr c3 r 1−γ , 2
(4.5)c
and set ζ=
exp c2 ∇ϕr dσ,
E
then S
exp c2 ∇ϕr dσ =
+
E
Ec
γ ζ + exp c3 r 1−γ
(4.5)d
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and E
c −γ γ exp r ∇ϕr dσ exp 2c2 ∇ϕr dσ. 2
(4.5)e
E
By Cauchy–Schwarz inequality,
exp 2c2 ∇ϕr dσ
E
exp c2 ∇ϕr dσ
2 = ζ 2.
E
Using (4.5)b together with (4.5)e and (4.5)d , we get γ ζ 2 ζ + exp c3 r 1−γ ,
which implies that γ 1 1−γ . ζ < exp c3 r 2 Going back to (4.5)d we conclude (4.5)a by using (4.3)b .
2
5. Riesz transform on the unit ball In the last section we have given the wanted estimate on all R d at the exception of a very thin set of spheres corresponding to the spheres which have their radius in Θ; we fill these missing places by using now Riesz transform on balls of R d . Let P (x, y) and G(x, y) denote, respectively, the Poisson kernel and the Green function of the d-dimensional ball B of radius 1, that is, P (x, y) =
1 − x2 , ωd x − yd
y ∈ ∂B = S,
(5.1)a
and, for d 3, G(x, y) = where ωd =
2π d/2 Γ (d/2) ;
2−d y 1 x − y2−d − y × , − x 2 d(d − 2)ωd y
(5.1)b
remark that
G(x, y) = 0,
∀x ∈ ∂B
or
∀y ∈ ∂B
and that x G(x, y) = δy ,
where δy denotes the Dirac mass at y. For d = 2 the Green function has the following expression: 1 1 log x − y − log y − yx G(x, y) = ; 2π y we shall not discuss later in detail the case d = 2, it is left to the reader.
(5.1)c
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Lemma. We have y y2 − x y − x,
∀x, y ∈ B.
(5.1)d
Proof. Considering the plane containing x and y the statement is a statement in dimension 2; use the formalism of complex number set y = ρ real and x = λ = ξ + iη complex of modulus smaller than 1. We have to check that (λ − ρ)(λ¯ − ρ) λ − ρ −1 λ¯ − ρ −1
− 2ρξ + ρ 2 −2ρ −1 ξ + ρ −2 ,
or
the inequality is satisfied for ξ = 0; it stays valid until the solution ξ0 of the equation −2ρξ0 + ρ 2 = −2ρ −1 ξ0 + ρ −2 −→ ξ0 =
ρ + ρ −1 > 1. 2
2
Lemma. We have ∀x, y ∈ B,
∇y G(x, y) h(x − y)
where h(z) := cz1−d × 1z<2 .
(5.1)e
Proof. For the terms where do not appear the differential of y−2 the estimate results from (5.1)c and (5.1)d ; the differential of y−2 contributes as 1−d y 1 . × − x y y2 As the singularity is in y = 0, we can assume that y < 12 . Then for all x ∈ B, we have 1−d y y 1−d 1 2 2 × × × yd−1 → 0 − x = 2 2 y y y y y
when y → 0.
Remark. The indicator function 1z<2 is needed to realize that hLp < ∞ for all p <
2 d d−1 .
Theorem (Riesz transform). Let f be a C 1 positive function on the d-dimensional ball B of radius 1. Then, for all x ∈ B, f (x) =
f (y)P (x, y) σ (dy) −
∂B
(∇y G) · ∇f (y) dy.
(5.2)a
B
Proof. By the Green representation formula, for any x ∈ B, and for any f of class C 2 , we have f (x) = ∂B
f (y), P (x, y) σ (dy) +
G(x, y)f (y) dy. B
P. Malliavin, E. Nualart / Journal of Functional Analysis 256 (2009) 4197–4214
4211
Decomposing the laplacian in a sum of second derivatives and making an integration by parts, taking into account the vanishing of the Green function at the boundary, it yields f (x) = f (y)P (x, y) σ (dy) − (∇y G) · ∇f (y) dy, ∂B
B
approximating a C 1 function by a sequence of C 2 functions and passing to the limit we prove (5.2)a . 2 Theorem. There exist two constants c1 , c2 such that for any C 1 function ϕ on the unit ball B, we have (5.3)a exp ϕL∞ (B) < c1 + 2 exp c2 ∇ϕR d dx × exp ϕL∞ (∂B) . S
Proof. We use the decomposition (5.2)a ; the norm L∞ (B) of the Poisson integral is equal to the norm in L∞ (∂B) as for all x ∈ B, P (x, y) σ (dy) = 1. ∂B
It remains to evaluate the Green kernel integral; set ψ(x) = ∇ϕR d × 1B (x). Then, using (5.1)e , (∇y G) · ∇ϕ(y) dy
L∞ (B)
B
As h ∈ Lp (R d ) for all p <
d d−1 ,
ψ ∗ hL∞ (R d ) .
we proceed as for the proof of (4.3)b .
(5.3)b
2
Theorem. Set ϕr (x) = f (rx),
x ∈ B, r ∈ Θ.
(5.4)a
Then, γ
ϕr L∞ (B) cγ
r 1−γ . Proof. We proceed as for the proof of (4.5)a .
(5.4)b
2
Proof of main theorem. Then (2.3)b results from (5.4)b combined with the fact that (4.4)b implies that Θ ∩ [r, 2r]
is always non-empty for r > r0 .
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That is, we conclude that for x sufficiently large γ
f (x)
sup y∈B(0,x)
which proves (2.2)b .
f (y) cγ
x 1−γ ,
2
Acknowledgments E.N. would like to thank Jonathan Mattingly, Scott McKinley and Frederi Viens for stimulating discussions on the subject. Appendix A The one-dimensional version of theorem was proved in [13]. For the sake of completeness of this paper, we supply a sketch of the original proof. Theorem. Let f be a C 1 real positive function such that there exists γ > 1 and c > 0 such that ∞
γ exp c f (x) exp −f (x) dx < +∞.
0
Then, there exists c˜ > 0 such that for all x > 0 sufficiently large, γ
f (x) < cx ˜ γ −1 f. Proof. For all n ∈ N , define αn = inf x 0: f (x) n . Write ∞
∞ γ exp c f (x) exp −f (x) dx =
α n+1
γ exp c f (x) exp −f (x) dx
n=0 αn
0
∞ n=0
α n+1
γ exp c f (x) dx.
exp −(n + 1)
αn
Consider the functional α n+1
γ exp c f (x) dx.
Jn (f ) := αn
P. Malliavin, E. Nualart / Journal of Functional Analysis 256 (2009) 4197–4214
4213
We want to find the minimum of Jn (f ) over all the C 1 positive decreasing functions f : [αn , αn+1 ] → R such that α n+1
f (x) dx = 1.
αn
By the Lagrange multipliers method it suffices to find the minimum of the functional α n+1
f (x) dx,
Jn (f ) − λ αn
where λ is a constant. The corresponding Euler–Lagrange equation is γ γ cf (x)γ −1 exp c f (x) = const, α this implies that f (x) is a constant determined by the condition αnn+1 f (x) dx = 1. If we denote ln = αn+1 − αn we obtain that the minimum of Jn (f ) is reached when f (x) = ln−1 . Thus, ∞
∞ γ −γ exp c f (x) exp −f (x) dx ln exp −(n + 1) + cln . n=0
0
−γ
By hypothesis, this serie is convergent. Hence, ln exp(−(n+1)+cln ) converges to 0 as n → ∞. This implies that there exists n0 > 0 such that for all n n0 , −γ
(n + 1) − cln > 0. In particular, for any q n0 , we have that q
ln = αq+1 − αn0 >
n=n0
This implies that αq+1 > cγ (q + 1) the desired result follows. 2
q n=n0
γ −1 γ
c1/γ > c1/γ (n + 1)1/γ
q+1 n0 +1
1 x 1/γ
dx.
. But if x is such that x < αq+1 , then f (x) < q + 1. Hence,
References [1] H. Airault, P. Malliavin, Quasi-invariance of Brownian measures on the group of circle homemorphisms and infinitedimensional Riemannian geometry, J. Funct. Anal. 241 (2006) 99–142. [2] V. Bally, Lower bounds for the density of locally elliptic Itô processes, Ann. Probab. 34 (2006) 2406–2440. [3] A.B. Cruzeiro, P. Malliavin, Non-existence of infinitesimally invariant measures on loop groups, J. Funct. Anal. 254 (2008) 1974–1987. [4] R.C. Dalang, E. Nualart, Potential theory for hyperbolic SPDEs, Ann. Probab. 32 (2004) 2099–2148.
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[5] C.L. Fefferman, A. Sanchez-Calle, Fundamental solutions of second order subelliptic operators, Ann. of Math. 124 (1986) 247–272. [6] H. Guérin, S. Méléard, E. Nualart, Estimates for the density of a nonlinear Landau process, J. Funct. Anal. 238 (2006) 649–677. [7] A. Kohatsu-Higa, Lower bounds for densities of uniformly elliptic random variables on Wiener space, Probab. Theory Related Fields 126 (2003) 421–457. [8] A. Kohatsu-Higa, Lower bounds for densities of uniformly elliptic non-homogeneous diffusions, in: Proceedings of the Stochastic Inequalities Conference in Barcelona, in: Progr. Probab., vol. 56, 2003, pp. 323–338. [9] S. Kusuoka, D. Stroock, Applications of the Malliavin calculus III, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987) 391–442. [10] P. Malliavin, Formules de la moyenne, calcul de pertubations et théorèmes d’annulation pour les formes harmoniques, J. Funct. Anal. 17 (1974) 274–291. [11] P. Malliavin, Stochastic Analysis, Springer-Verlag, 1997. [12] P. Malliavin, A. Thalmaier, Stochastic Calculus of Variations in Mathematical Finance, Springer-Verlag, 2006. [13] E. Nualart, Exponential divergence estimates and heat kernel tail, C. R. Acad. Sci. Paris, Ser. I 338 (2004) 77–80. [14] D. Nualart, The Malliavin Calculus and Related Topics, second ed., Springer-Verlag, 2006. [15] A. Sanchez-Calle, Fundamental solutions and geometry of the sum of square of vector fields, Invent. Math. 78 (1986) 143–160.
Journal of Functional Analysis 256 (2009) 4215 www.elsevier.com/locate/jfa
Corrigendum
Corrigendum to “Nonergodicity of Euler fluid dynamics on tori versus positivity of the Arnold-Ricci tensor” [J. Funct. Anal. 254 (7) (2008) 1903–1925] A.B. Cruzeiro a,∗ , Paul Malliavin b a Dep. Matemática I.S.T., T.U.L., Lisbon and Grupo de Física-Matemática U.L., Av. Rovisco Pais,
1049-001 Lisboa, Portugal b 10 rue S. Louis-en-l’Ile, 75004 Paris, France
Available online 9 March 2009
As one can check from the proof of Theorem 6.2, the following definition of the Ricci curvature tensor, was used Ω(ei , u, ei ) Ricci(u) = i
where Ω denotes the Riemannian curvature and ei an orthonormal basis of the tangent space. This corresponds to a change of sign is the standard definition of the Ricci, which explains the difference with the signs of the curvatures appearing in Arnold’s approach to Hydrodynamics (cf. [1, p. 200]). Although it was mentioned in the introduction that a different sign convention was used, the choice was clearly bad and may lead to misunderstandings. One misunderstanding is already in the paper, in the comments regarding the non-validity of Myers type theorems in an infinite dimensional setting (cf. introduction, end of paragraph 3). We apologize for the mistake and thank Christian Loeschcke for pointing it out to us. Reference [1] V.I. Arnold, B.A. Khesin, Topological Methods in Hydrodynamics, Appl. Math. Sci., vol. 125, Springer, 1988.
DOI of original article: 10.1016/j.jfa.2007.08.002. * Corresponding author.
E-mail addresses: [email protected] (A.B. Cruzeiro), [email protected] (P. Malliavin). 0022-1236/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.022
