Integral Equations and Operator Theory - Volume 69

Integr. Equ. Oper. Theory 69 (2011), 1–27 DOI 10.1007/s00020-010-1841-4 Published online November 23, 2010 c The Author(...

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Integr. Equ. Oper. Theory 69 (2011), 1–27 DOI 10.1007/s00020-010-1841-4 Published online November 23, 2010 c The Author(s) This article is published with open access at Springerlink.com 2010

Integral Equations and Operator Theory

Pathwise Stability of Degenerate Stochastic Evolutions Joris Bierkens Abstract. For linear stochastic evolution equations with linear multiplicative noise, a new method is presented for estimating the pathwise Lyapunov exponent. The method consists of ﬁnding a suitable (quadratic) Lyapunov function by means of solving an operator inequality. One of the appealing features of this approach is the possibility to show stabilizing eﬀects of degenerate noise. The results are illustrated by applying them to the examples of a stochastic partial differential equation and a stochastic differential equation with delay. In the case of a stochastic delay differential equation our results improve upon earlier results. Mathematics Subject Classification (2010). 60Hxx, 93Dxx, 47A62, 34K50. Keywords. Stochastic evolution equation, Lyapunov exponent, pathwise stability, stochastic delay differential equation, operator inequality.

1. Introduction In this paper we discuss a new approach to the estimation of the pathwise Lyapunov exponent of linear stochastic evolution equations with multiplicative noise. From such an estimate pathwise stability may be deduced. Stochastic evolution equations provide a general framework for describing complex systems (high- or inﬁnite-dimensional) under random inﬂuence. There exists an extensive amount of literature on stochastic evolution equations, or stochastic differential equations in inﬁnite dimensions; see for example [15,21,32]. One of the primary ways to characterize dynamical systems qualitatively is to study the question of stability. For stochastic evolutions different notions of stability arise in a natural way: stability in p-th moment, stability in probability and pathwise or almost sure stability. It can be argued that

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from a practical point of view pathwise stability is most relevant, since sample paths represent the observed behaviour of systems modelled by stochastic evolution equations. Important results on moment stability and pathwise stability of linear stochastic evolutions may be found in e.g. [14,19,20]. A survey of results on linear and non-linear stochastic evolution equations may be found in [27]. Stochastic evolutions with delay are our primary inspiration. For this special case results on stability are obtained in e.g. [1]. It is shown there that noise may have a stabilizing eﬀect, a result not obtained in the references mentioned above on stability of general stochastic evolutions. In this paper we wish to obtain this stabilizing eﬀect of noise without using the special structure of delay differential equations. Instead, we employ a general semigroup approach and use only dissipativeness of the semigroup and general estimates on the semigroup and the perturbation by noise. Results of [23] for systems with commuting operators show that in case the noise is non-degenerate it may very well help in stabilizing the stochastic evolution. However in the case of stochastic delay equations, the noise can never inﬂuence the past, resulting in degeneracy of the noise. The result of [23] is improved upon in [11], but here one still depends on commutativity of operators. The generator of the delay semigroup and the noise operator are not commutative, such that results from [11] are still not applicable to stochastic delay differential equations. We propose a different method, in which we use a quadratic Lyapunov function. As is well known to such a quadratic form corresponds a linear operator. We will rephrase the problem of ﬁnding a quadratic form which provides us with the best estimate on the Lyapunov exponent of a given stochastic evolution, as an operator inequality for the mentioned operator. Conditions for the solution of the operator inequality then result in conditions for stability of the stochastic evolution. Our results are applied to stochastic partial differential equations, as a simple illustration, and to stochastic delay differential equations, for which a new stability result is obtained.

2. Preliminary Results Let H be a real Hilbert space, and consider the linear SDE in mild form k t X(t) = S(t)x + S(t − s)Bi X(s) dWi (s), (1) i=1 0

where Wi , i = 1, . . . , k, are independent standard Brownian motions in R, (S(t))t≥0 is a strongly continuous semigroup on H with generator A : D(A) → H, Bi ∈ L(H), i = 1, . . . , k and x ∈ H. A solution to such an equation always exists and is unique (see [15]). Note that (1) is the variation of constants formulation of ⎧ k ⎨ dX(t) = AX(t) dt + Bi X(t) dWi (t), t ≥ 0, i=1 ⎩ X(0) = x.

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Deﬁne the Lyapunov exponent as the random variable 1 λ := lim sup log |X(t; x)|, a.s. (2) t→∞ t with X(·; x) the solution of (1). We say that the stochastic evolution described by (1) is pathwise asymptotically stable if lim |X(t; x)| = 0,

t→∞

almost surely.

We have the following relation between the Lyapunov exponent and the notion of pathwise stability. Proposition 2.1. If λ < 0 almost surely then (1) describes a pathwise asymptotically stable evolution. Proof. Suppose lim supt→∞ |X(t; x)| > ε on some F-measurable E ⊂ Ω. Then 1 0 ≤≤ lim sup log |X(t; x)| ≤ λ < 0 on E, t→∞ t a contradiction. Note that λ ≤ 0 is not sufﬁcient: consider the deterministic evolution described by 0 0 1 x(t) ˙ = x(t), x(0) = , 1 0 0 with solution x(t) =

1 . t

The Lyapunov exponent of (x(t))t≥0 is 0 but the evolution is clearly not asymptotically stable. We recall the following notions concerning linear operators and strongly continuous semigroups. Let A be a closed linear operator and let ρ(A) denote the resolvent set of A:

ρ(A) = λ ∈ C : (λ − A)−1 exists and is bounded . Deﬁne the spectrum σ(A) of A by σ(A) = C\ρ(A). Let s(A) denote the spectral bound of A, and r(B) the spectral radius of B, i.e. s(A) := sup{Re λ : λ ∈ σ(A)}

and r(B) := sup{|λ| : λ ∈ σ(B)}.

Furthermore let ω0 (A) denote the growth bound of A, i.e. ω0 (A) := inf{ω ∈ R : ∃M ≥1 s.t. || exp(At)|| ≤ M eωt for all t ≥ 0}. Proposition 2.2. We have the following relation between s(A), ω0 (A) and r(S(t)): 1 s(A) ≤ ω0 (A) = log r (S(t)) , for all t ≥ 0. (3) t If (S(t))t≥0 is eventually norm continuous then the inequality in (3) becomes an equality: s(A) = ω0 (A).

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See [18], Proposition IV.2.2, and Theorem IV.3.11. Important cases where (S(t)) is eventually norm continuous are the case where H is ﬁnite dimensional, and where the semigroup (S(t)) is (eventually) compact or analytic. Now we consider the particular case of (1) where all operators commute. The following result is slightly sharper and more general than a result on commutative systems found in [23]. A stronger version, allowing for unbounded operators Bi , may be found in [11]. Proposition 2.3. Suppose A and all Bi , i = 1, . . . , k commute. Then

k 1 2 1 Bi , a.s. lim sup log |X(t; x)| ≤ ω0 A − 2 t→∞ t i=1

(4)

Proof. Let (T0 )t≥0 be the strongly continuous semigroup generated by k A− 12 i=1 Bi2 , and let (Ti )t≥0 be the uniformly continuous groups (Ti (t))t≥0 generated by Bi , i = 1, . . . , k. Note that the solution is given by X(t; x) = T0 (t)

k

Ti (Wi (t))x.

t ≥ 0, x ∈ H.

i=1

Suppose ||Ti (t)|| ≤ Mi exp(ω0 (Bi )t), i = 1, . . . , k. Then 1 1 1 1 log |X(t; x)| ≤ log |x| + log ||T0 (t)|| + log ||Ti (Wi (t))||. t t t t i=1 k

(5)

Now using the strong law of large numbers for martingales (see Theorem A.1) lim sup t→∞

ω0 (Bi )|Wi (t)| 1 log Mi log ||Ti (Wi (t))|| ≤ lim sup + = 0 a.s., t t t t→∞

and furthermore 1 lim sup log ||T0 (t)|| = ω0 t→∞ t

A−

1 2

k

Bi2

.

i=1

The stated result is now obtained by combining the above estimates in (5). It may be seen from this proposition that in the commutative case, noise may have a stabilizing eﬀect even when it is degenerate: −1 0 0 0 Example 2.4. Let A = and B = . Then A − 12 B 2 = 0 1 0 2 −1 0 . 0 −1 In the remainder of this paper we will establish a stabilizing eﬀect of degenerate noise in the non-commutative case.

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3. Bounded Case First we consider the case where A ∈ L(H), so that (S(t))t≥0 is a uniformly continuous semigroup, in which case we may apply Itˆ o’s formula. Recall the notion of a coercive operator. Definition 3.1. An operator T ∈ L(H) is called coercive if T x, x ≥ γ|x|2 for all x ∈ H and some γ > 0. Lemma 3.2. Suppose there exists a self-adjoint coercive operator Q ∈ L(H) and λ ∈ R such that, for all x ∈ H, k k QBi x, Bi x − 2λQx, x ≤ 2 Qx, Bi x 2 . Qx, x 2QAx, x + i=1

i=1

Let Y (t) := QX(t), X(t) , with X the solution of (1). Then 1 lim sup log Y (t) ≤ 2λ, a.s. t→∞ t Proof. If x = 0, then P(X(t) = 0) = 1, t ≥ 0, and the required estimate holds trivially. Suppose x = 0. Then by uniqueness of the solution of SDEs and positiveness of Q, P(Y (t) = 0) = 0 for all t ≥ 0. By Itˆo’s formula, k 1 2QAX(t), X(t) + QBi X(t), Bi X(t) d log Y (t) = Y (t) i=1 k 2 − QX(t), Bi X(t) 2 dt Y (t)2 i=1 2 Qx(t), Bi X(t) dWi (t) Y (t) i=1 k

+

2 QX(t), Bi X(t) dWi (t). Y (t) i=1 k

≤ 2λ dt +

i X(t) Now by boundedness of QX(t),B QX(t),X(t) for i = 1, . . . , k and the law of large numbers for martingales (Theorem A.1)

1 t

t 0

2QX(s), Bi X(s) dWi (s) → 0 Y (s)

(t → ∞),

a.s.,

i = 1, . . . , k,

so lim sup t→∞

1 log Y (t) ≤ 2λ t

a.s.

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Proposition 3.3. Suppose there exists a self-adjoint operator Q ∈ L(H), λ ∈ R and bi ∈ R, i = 1, . . . , k such that k

∗

k k k ∗ 2 A+ bi Bi Q + Q A + bi Bi + Bi QBi + 12 bi − 2λ i=1

i=1

i=1

i=1

(6) is negative semidefinite. Then, with Y as in Lemma 3.2, 1 lim sup log Y (t) ≤ 2λ, t→∞ t

a.s.

Proof. Note that (by the abc-formula), for i = 1, . . . , k, Qx, Bi x 2 Qx, Bi x 1 2 + bi ≥ 0, + bi 2 Qx, x Qx, x 4

for all x ∈ H.

(7)

So, by Lemma 3.2, if 2QAx, x +

k

QBi x, Bi x − 2λQx, x

i=1

≤ −2

k

1 2 b Qx, x 2 i=1 i k

bi Qx, Bi x −

i=1

for all x ∈ H,

then the claimed result holds. But this is equivalent to the stated condition.

The following theorem gives a sufﬁcient condition in order for a solution to (6) to exist. Theorem 3.4. Suppose Dj ∈ L(H), j = 1, . . . , k, L is the generator of a strongly continuous semigroup (T (t))t≥0 acting on H such that ||T (t)|| ≤ meωt

for all t ≥ 0,

with m ≥ 1, ω ∈ R, and m2

k

||Dj ||2 + 2ω < 0.

(8)

j=1

Then for any M ∈ L(H) there exists a unique solution Q ∈ L(H) to the equation L∗ Q + QL +

k

Dj∗ QDj = M,

(9)

j=1

which should be interpreted as Qx, Ly + QLx, y +

k

QDj x, Dj y = M x, y

for all x, y ∈ D(L).

j=1

(10)

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This Q also satisfies ⎛ ⎞ ∞ k Q = T ∗ (t) ⎝ Dj∗ QDj − M ⎠ T (t) dt.

7

(11)

j=1

0

We can estimate the norm of Q by ||M ||m2 . k 2ω + m2 j=1 ||Dj ||2

||Q|| ≤ −

(12)

Furthermore, (i) if M = 0 then Q = 0, (ii) if M ≤ 0 then Q ≥ 0, and (iii) if M < 0 then Q > 0. Proof. Deﬁne a recursion by ⎛ ⎞ ∞ k Q0 := 0, Qi+1 := T ∗ (t) ⎝ Dj∗ Qi Dj − M ⎠ T (t) dt. j=1

0

The recursion is actually a contraction, since ∞ k ∗ ∗ ||Qi+1 − Qi || = T (t)Dj (Qi − Qi−1 )Dj T (t) dt j=1 0

2

≤ m

k

2

||Dj ||

j=1

m

2

∞

k j=1

e2ωt dt||Qi − Qi−1 ||

0

||Dj ||2

||Qi − Qi−1 ||. 2ω Note that the recursion is deﬁned such that Qi+1 satisﬁes =−

L∗ Qi+1 + Qi+1 L = M −

k

Dj∗ Qi Dj ,

j=1

a basic result from Lyapunov theory (see [13, Theorem 4.1.23]). Hence there exists a unique ﬁxed point Q ∈ L(H) that satisﬁes both (9) and ⎛ ⎞ ∞ k Dj∗ QDj − M ⎠ T (t) dt. Q = T ∗ (t) ⎝ j=1

0

Hence ∞ ||Q|| ≤ 0

⎛ m2 e2ωt ⎝ ⎛

k j=1

⎞ ||Dj ||2 ||Q|| + ||M ||⎠ dt

⎞ k m2 ⎝ ||Dj ||2 ||Q|| + ||M ||⎠. = −2ω j=1

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By repeating this estimate n times, we obtain

n

i k k n−1 2 2 m2 j=1 ||Dj ||2 m2 ||M || m j=1 ||Dj || ||Q|| + . ||Q|| ≤ −2ω −2ω i=0 −2ω Let n → ∞ to obtain m2 ||M || ||Q|| ≤ −2ω 1 −

1

m

k 2

||Dj ||2 −2ω

j=1

=−

m2 ||M || . k m2 j=1 ||Dj ||2 + 2ω

If M = 0 then Q = 0 by uniqueness of the solution. Now suppose M ≤ 0. Then we can check that the recursion for (Qi ) has the property that Qi ≥ 0 for all i. So Q ≥ 0, and (11) shows that ∞ Q ≥ − T ∗ (t)M T (t) dt. 0

If M < 0, then there exists a unique P ∈ L(H), P > 0 such that M = L∗ P + P L. Then ∞ Q ≥ − T ∗ (t)M T (t) dt = P > 0. 0

So far we only know that if Q ∈ L(H) a solution to (9) with M < 0, then Q > 0. But to obtain equivalence of norms we need Q to be coercive. In the ﬁnite-dimensional case coerciveness of Q is implied by Q > 0 but in inﬁnite dimensions this is not the case. The next proposition shows that we can ﬁnd a coercive solution in case L is dissipative, or equivalently if (T (t))t≥0 is a contraction semigroup. Proposition 3.5. Suppose L, (Dj )j=1,...,k are as in Proposition 3.4 and that (8) holds. Then there exists a Q ∈ L(H) such that (10) holds with M = L + L∗ . Furthermore for this Q we have Q ≥ I and −2ω . ||Q|| ≤ k 2 2 −(2ω + m j=1 ||Dj || ) Note that if L is dissipative, then M x, x = 2Lx, x ≤ 0 for all x ∈ H. Proof. By Proposition 3.4, there exists a unique solution R ∈ L(H), R ≥ 0 k to (10) with M = − j=1 Dj∗ Dj . Furthermore R ≥ 0 and, by (12) k 2 2 j=1 ||Dj || m . ||R|| ≤ k −(2ω + m2 j=1 ||Dj ||2 ) Let Q := I + R. Then Q ≥ I, the claimed estimate for Q holds and L∗ Q + QL +

k

Dj∗ QDj = L∗ + L +

j=1

interpreted in the weak sense of (10).

k j=1

Dj∗ Dj −

k

Dj∗ Dj = L∗ + L,

j=1

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4. Unbounded Case We will now extend the result of the previous section to the case where A is possibly unbounded and does no longer generate a uniformly continuous semigroup. We do this by employing approximations of A by bounded operators (An ). The background for this technique is discussed in Appendix B. Recall that H is a real Hilbert space, and we consider the linear SDE in mild form t X(t) = S(t)x + S(t − s)BX(s) dW (s), (13) 0

where W is a standard Brownian motion in R, (S(t))t≥0 is a strongly continuous semigroup on H with generator A : D(A) → H, B ∈ L(H) and x ∈ H. Remark 4.1. From this point onward we assume for notational convenience k = 1, i.e. the stochastic process X is driven by only one standard Brownian motion. There is however no problem in proving all the results of this section for the case with multiple Brownian motions. Lemma 4.2. Suppose T = (T (t))t≥0 is a strongly continuous semigroup with approximation (Tn )n∈N such that, for some ω < 0 and m ≥ 1, ||T (t)|| ∨ sup ||Tn (t)|| ≤ meωt . n∈N

Let L and Ln denote the generators of (T (t))t≥0 and (Tn (t))t≥0 , n ∈ N, respectively. For n ∈ N let Rn ≥ 0 denote the unique positive semidefinite solution in L(H) to the Lyapunov equation L∗n Rn + Rn Ln = M for some fixed M ∈ L(H), M ≤ 0. Then y, Rn x → y, Rx for any x, y ∈ H, where R ≥ 0 is the unique positive semidefinite solution in L(H) to L∗ R + RL = M. Proof. We have, using the representation (see [13, Theorem 4.1.23]) ∞ ∞ R = − T (t)∗ M T (t) dt, Rn = − Tn (t)∗ M Tn (t) dt, n ∈ N, 0

0

that for any t > 0, |y, Rn x − Rx | ∞ = y, Tn∗ (s)M Tn (s)x − T ∗ (s)M T (s)x ds 0 ∞ = Tn (s)y, M (Tn (s) − T (s))x + (Tn (s) − T (s))y, M T (s)x ds 0

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∞ ||Tn (s)|| ||y|| ||M || |(Tn (s) − T (s))x| ds

≤ 0

∞ |(Tn (s) − T (s))y| ||M || ||T (s)|| ||x|| ds

+ 0

∞

≤ m||M ||

eωs (||y|| |(Tn (s) − T (s))x| + |(Tn (s) − T (s))y| ||x||) ds. 0

∞

For the ﬁrst term we have t ∞ ωs ωs e |(Tn (s) − T (s))x| ≤ e |(Tn (s) − T (s))x| ds + 2m|x| e2ωs ds.

0

0

t

Now pick t large enough such that the second term is smaller than ε/2. Since (Tn ) is an approximation of T , we have uniform convergence in s ∈ [0, t] of |Tn (s)x − T (s)x|. So let N large enough such that |Tn (s)x − T (s)x| ≤ δ for all s ∈ [0, t] and for δ > 0 such that t eωs δ ds < ε/2. 0

Repeating this argument for the second term leads to the stated result. Lemma 4.3. Suppose (T (t))t≥0 is a strongly continuous semigroup with approximation (Tn )n∈N and infinitesemal generators L and (Ln )n∈N , respectively. Suppose that for some m ≥ 1 and ω < 0 we have ||T (t)|| ∨ sup ||Tn (t)|| ≤ meωt , n∈N

and suppose for this m, ω and some D ∈ L(H) condition (8) holds. Let M ∈ L(H) be self-adjoint and negative semidefinite. Let Q and Qn , n ∈ N, denote the unique positive semidefinite solutions to L∗ Q + QL + D∗ QD = M

and

L∗n Qn + Qn Ln + D∗ Qn D = M.

Then for all x, y ∈ H we have that x, Qn y → x, Qy as n → ∞. Proof. For n ∈ N construct a recursion by ∞ n n Q0 := 0 and Qj+1 := Tn∗ (t)(D∗ Qnj D − M )Tn (t) dt,

j ∈ N.

0

Similarly let ∞ Q0 := 0 and Qj+1 := 0

T ∗ (t)(D∗ Qj D − M )T (t) dt,

j ∈ N.

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First we will prove the following Claim. For all x, y ∈ H and j ∈ N we have x, (Qnj − Qj )y → 0 as n → ∞. Proof of claim. For j = 0 the claim holds trivially. Suppose now that the claim holds for value j = k − 1. Let x, y ∈ H. Then for j = k, ∞ n |x, (Qj − Qj )y | = Tn (t)x, (D∗ Qnj−1 D − M )Tn (t)y dt 0 ∞ ∗ − T (t)x, (D Qj−1 D − M )T (t)y dt ∞ ≤

0

|Tn (t)x, D∗ (Qnj−1 − Qj−1 )DTn (t)y | dt

0

∞ ∗ ∗ + x, Tn (t)(D Qj−1 D − M )Tn (t)dt y 0 ∞ − x, T (t)∗ (D∗ Qj−1 D − M )T (t) dt y 0

that

Since by the induction hypothesis Qnj−1 → Qj−1 in weak sense, we have DTn (t)x, (Qnj−1 − Qj−1 )DTn (t)y → 0

as n → ∞

for all t ∈ [0, ∞). By dominated convergence therefore the ﬁrst term proceeds zero as n → ∞. The convergence of the second term is an immediate consequence of Lemma 4.2. So the claim is proven by induction. Now by the proof of Theorem 3.4, Qj → Q and Qnj → Qn in the norm topology of L(H), uniformly in n. Therefore using |x, (Qn − Q)y | ≤ |x, (Qn − Qnj )y | + |x, (Qnj − Qj )y | + |x, (Qj − Q)y | we obtain x, (Qn − Q)y → 0 as n → ∞.

Lemma 4.4. Let B ∈ L(H) and let k ∈ R such that B − kI is stable, i.e. we may estimate ||e−kt eBt || ≤ meλt , t ≥ 0 for some m ≥ 1 and λ < 0. Then for any Q ∈ L(H), Q ≥ 0, and x ∈ H we have QBx, x ≤ k. Qx, x Proof. Let N := (B − kI)∗ Q + Q(B − kI). Then by Lyapunov theory N ≤ 0. Hence 2Q(B − kI)x, x = N x, x ≤ 0

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or equivalently QBx, x ≤ kQx, x . We are now ready to state and prove the main result of this section. Theorem 4.5. Let A be the infinitesemal generator of a strongly continuous semigroup in L(H) and let B ∈ L(H). Suppose there exists b ∈ R such that the semigroup S(t) generated by A + bB satisfies, for some μ ∈ R, ||S(t)|| ≤ eμt ,

t ≥ 0.

Furthermore suppose that for some λ ∈ R, 2(μ + 14 b2 − λ) + ||B||2 < 0. Then lim sup t→∞

1 log |X(t)| ≤ λ, t

where (X(t))t≥0 is the mild solution of dX(t) = AX(t) dt + BX(t) dW (t),

X(0) = x,

with W a one-dimensional standard Brownian motion. Proof. Deﬁne L := A + bB + ( 14 b2 − λ)I, then L is the generator of a semigroup T (t) and we have 1

2

||T (t)|| ≤ e(μ+ 4 b

−λ)t

.

In particular, T is a contraction semigroup and hence also the approximating semigroups (Tn )n∈N (generated by the Yosida approximation (Ln )n∈N ) are contraction semigroups. Furthermore, by (26), for any ε > 0 there exists an N ∈ N such that for all n ≥ N we have 1

2

||Tn (t)|| ≤ e(μ+ 4 b

−λ+ε)t

.

Let ε > 0, small enough such that μ + 14 b2 − λ + 12 ||B||2 + ε < 0 and let N as above. Let ω := μ + 14 b2 − λ + ε. For n ≥ N let Qn be the solution given by Proposition 3.5 to L∗n Qn + Qn Ln + B ∗ Qn B = L∗n + Ln . Similarly let Q be the solution to L∗ Q + QL + B ∗ QB = L∗ + L. Recall that Qn ≥ I, n ≥ N , Q ≥ I and ||Q|| ∨ sup ||Qn || ≤ n≥N

−2ω . −(2ω + ||B||2 )

Let Xn denote the solution to dXn (t) = An Xn (t) dt + BXn (t) dW (t),

Xn (0) = x,

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Pathwise Stability of Deg. Stoc. Evolutions

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where An := Ln − bB − ( 14 b2 − λ)I. Then (An ) is an approximation for A and hence by Proposition B.3, we have almost surely that, for a subsequence (nk ) ⊂ N, sup |Xnk (t) − X(t)| → 0 for all T > 0.

t∈[0,T ]

Since Ln is bounded, we have that, since (An + bB)∗ Qn + Qn (An + bB) + B ∗ Qn B + ( 12 b2 − 2λ)Qn ≤ 0,

n ∈ N,

and by the proofs of Lemma 3.2 and Proposition 3.3 that t logQn Xn (t), Xn (t) ≤ logQn x, x +2λt+2 0

Qn BXn (s), Xn (s) dW (s). Qn Xn (s), Xn (s) (14)

Here we used that Ln + L∗n ≤ 0 by dissipativeness of Ln . In the stochastic integral we have by Lemma 4.3, the almost sure convergence of Xn (s) on [0, t] and the uniform boundedness of ||Qn ||, n ∈ N, that the integrand converges almost surely. By Lemma 4.4 we may apply dominated convergence to have convergence in L2 (Ω) of the stochastic integral t 0

Qn BXn (s), Xn (s) dW (s) → Qn Xn (s), Xn (s)

t 0

QBX(s), X(s) dW (s). QX(s), X(s)

We therefore also have this convergence with probability one for a further subsequence (nkl ) ⊂ N. For this subsequence the left-hand side of (14) converges almost surely to logQX(t), X(t) , and by the uniform estimate on ||Qn ||, the term logQn x, x is bounded by a constant, say M > 0. Hence we have that, with probability one, t logQX(t), X(t) ≤ M + 2λt + 2 0

QBX(s), X(s) dW (s), QX(s), X(s)

for all t ≥ 0.

Dividing by t and by letting t → ∞, using the law of large numbers for martingales (Theorem A.1) we obtain lim sup t→∞

1 logQX(t), X(t) ≤ 2λ t

almost surely.

Using the fact that Q ≥ I, we now have lim sup t→∞

1 log |X(t)| ≤ λ t

almost surely.

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5. Example: Stochastic Heat Equation As a simple illustrative example, let H = L2 (O) where O is some Lipschitz domain in Rn . Consider the stochastic partial differential equation ⎧ ⎨du(t, ξ) = [Δu(t, ξ) + νu(t, ξ)] dt + σu(t, ξ) dW (t), ξ ∈ O, t ≥ 0, u(0, ξ) = v(ξ), ξ ∈ O, (15) ⎩ ∂u(t,ξ) = 0, ξ ∈ ∂O, t ≥ 0. ∂n ∂u the partial derivative in the direcwhere Δ is the Laplace operator on O, ∂n 2 tion of the gradient, ν, σ ∈ R, and v ∈ L (O) is an initial condition. Deﬁne the generator A of the corresponding strongly continuous semigroup (T (t))t≥0 on L2 (O) by A = Δ + νI. Let B = σI. Let b ∈ R and note that the semigroup (S(t)) generated by A + bB = Δ + (ν + σ)I satisﬁes ||S(t))|| ≤ e(ν+bσ)t . By Theorem 4.5, if

λ > ν + bσ + 14 b2 + 12 σ 2 , then the solution of (15) satisﬁes lim sup t→∞

1 log |u(t, ·)|L2 (O) ≤ λ. t

(16)

The optimal lower estimate for λ is obtained by choosing b = −2σ. This results in the conclusion that for λ > ν − 12 σ 2 we have that (16) holds. Note the stabilizing eﬀect of the noise. This result is in agreement with the estimate given by (4) on commuting operators. It could well have been derived with the results of [11] or [23], and is included here mainly to provide an easy example and to show that we are able with our method to obtain this sharpest obtainable bound.

6. Example: Stochastic Delay Differential Equation Consider the stochastic differential equation with delay dY (t) = aY (t) + cY (t − τ ) dt + σY (t) dW (t),

t ≥ 0,

Y (0) = y.

(17)

with a, c ∈ R and τ, σ > 0. Suppose ﬁrst c = 0. Then the solution to the stochastic differential equation is given by Y (t) = exp((a − 12 σ 2 )t + σW (t))y,

t ≥ 0,

and the solution is pathwise asymptotically stable if a < 12 σ 2 . We may now ask ourselves the question: for which c ∈ R do we still have stability? Theorem 6.1. Suppose a < 12 σ 2 and 3

2

|c| < e− 2 σ τ ( 12 σ 2 − a). Then the solution to (17) is pathwise exponentially stable.

(18)

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Proof. For convenience we recast the SDDE (17) in an SDDE with delay time (t) = W (tτ )/√τ , the problem equal to one. By letting Y (t) := Y (τ t) and W may be rewritten as (t), dY (t) = (˜ aY (t) + c˜Y (t − 1)) dt + σ ˜ Y (t) dW √ with a ˜ = τ a, c˜ = τ c and σ ˜ = τ σ. For the remainder of the proof we will omit the tildes. As in Section C, let A be the generator of the delay semigroup in R × L2 ([−1, 0]), with a Φ A := , d 0 dσ where Φu := cu(−1) for u ∈ W 1,2 ([−1, 0]). Furthermore let B ∈ L(R; R × L2 ([−1, 0])) be given by σ 0 B := . 0 0 By Theorem C.1, we have that if, for some b ∈ R, (a + bσ − μ)2 > c2 e−2μ

(19)

μ > a + bσ,

(20)

and then A + bB − μI generates a dissipative semigroup on a renormed space R × L2 ([−1, 0], τ ) (with L2 ([−1, 0], τ ) the Hilbert space consisting of square integrable functions on [0, 1] with inner product 0 f, g τ =

f (s)g(s)τ (s) ds −1

for some suitable weight function τ ∈ L∞ ([−1, 0]). By Theorem 4.5, we have that (17) has a pathwise exponentially stable solution if σ 2 + 12 b2 + 2μ < 0.

(21)

We may reformulate (19) as c2 ≤ (a + bσ − μ)2 e2μ , where b and μ should satisfy (20) and (21). It may be veriﬁed that b = −2σ and μ = −3/2σ 2 − ε, with ε > 0 sufﬁciently small, satisfy these conditions, using that a < 12 σ 2 . Now recall the substitutions a → τ a, c → τ c and √ σ → τ σ and let ε ↓ 0 to obtain the estimate (18). We may rephrase the estimate (18) as a condition on the delay time: assume |c| < 12 σ 2 − a and 1 2 2 2σ − a τ< ln . (22) 3σ 2 |c| Then the solution of (17) is almost surely exponentially stable.

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Remark 6.2. From the proof of Theorem 6.1, we see that the best estimate for |c| in order for the system to remain asymptotically stable (with τ = 1, for convenience) is obtained by solving the nonlinear optimization problem max subject to

(a + bσ − μ)2 e2μ a + bσ − μ < 0 2

and σ +

1 2 2b

(23)

+ 2μ < 0.

over b and μ. The condition a < 12 σ 2 is required for the set of feasible (b, σ) to be non-empty. The above problem may be solved by applying the Karush– Kuhn–Tucker conditions, obtaining μ = − 41 b2 − 12 σ 2 and for b the third-degree equation 1 3 4b

+ b2 σ + b(a + 12 σ 2 − 1) − 2σ = 0.

By solving this equation we obtain an estimate which is sharper than (18), but less readable. Remark 6.3. Note that, for σ = 0, we obtain from (18) the condition |c| < −a, which is the same estimate as that in Corollary C.2. Furthermore we have d −3/2σ2 1 2 1 e ( σ − a) 2 2 = 2 (3a + 1), 2 d(σ ) σ =0

from which we may conclude (see (18)) that adding noise has a stabilizing eﬀect for a > − 13 . Example 6.4. (Population growth under random migration) Consider a population (x(t))t≥0 evolving with constant birth rate β > 0 and constant death rate α > 0. Let r = 1 indicate the development period of an individual. Suppose there is migration with random rate σ which may depend on the size of the population. This leads to the stochastic differential equation with delay dx(t) = [−αx(t) + βx(t − 1)] dt + σx(t) dW (t),

t > 0.

so in (17) we have a = −α and c = β. Then Theorem 6.1 tells us that for 2 β < e−3/2σ ( 12 σ 2 + α) the population will eventually be extinguished with probability one. If, for example α = 0.1, then the graph in Fig. 1 shows upper bounds on values of β for which we know to have pathwise stability. Numerical experiments suggest that for σ not too large the theoretical bounds are quite accurate (see Fig. 2). 6.1. Comparison and Discussion Note that, under the conditions of Theorem 6.1, we do not necessarily have moment stability. Almost all papers on almost sure or pathwise stability of stochastic delay differential equations (e.g. [26,28–30]) ﬁrst establish moment stability and then state sufﬁcient conditions such that moment stability implies pathwise stability. In particular it is usually required that a + |c| < − 21 σ 2 , which should be compared with our ‘basic assumption’ a + |c| < 12 σ 2 . This shows that our result on pathwise stability is much stronger than those in the mentioned papers.

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0.2 exact solution to (23) approximate solution to (23)

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 1. Bound on the birth rate for which the population is eventually extinguished, as function of σ. Here α = 0.1. The solid graph represents the approximation exp(−3/2σ 2 )( 12 σ 2 +α) provided by Theorem 6.1. The dashed graph represent the exact solution to the optimization problem (23), which gives a more relaxed requirement on β An exception is the beautiful result of [1] by Appleby and Mao, which also applies also to nonlinear SDDEs. In the context of our paper, they obtain the following upper bound on c: √ 2 |c| < −a + 14 σ 2 e−σ τ /Φ(σ τ ), where Φ is the CDF of the standard normal distribution. This result may be compared to our bound (18). For some values of parameters a, σ and τ our results are sharper, in particular for larger values of a and σ, see Fig. 3. In the derivation, completely different methods were used in order to obtain these results. The result of [1] is derived explicitly for stochastic delay differential equations. In fact, their result naturally leads to the question: can we obtain a similar result on pathwise stability by taking a more operator theoretic point of view, in the style of Da Prato and Zabczyk [16], to which this paper provides a partial answer. In [11] such operator theoretic results are obtained for stochastic evolutions, but these require a special form of the noise and are therefore not applicable to stochastic delay differential equations. The general result of our analysis, Theorem 4.5, is applicable to general stochastic evolutions.

7. Notes and Remarks The part on stability of degenerate evolutions with bounded generator (applied to ﬁnite dimensional SDEs) will be published separately [8].

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(d)

(c)

Figure 2. The evolution of the size of a population in time with birth rate β = 0.11 and death rate α = 0.1. Without migration or for a small rate of random migration the population increases steadily in size, whereas for a larger rate of migration the population is eventually extinguished

0.2

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(a)

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(b)

Figure 3. Comparison of bound on |c|, depending on σ and a. Here the delay time τ = 1 For more results on ﬁnite dimensional stochastic Lyapunov exponents, see the overview paper [4], the book by Khasminskii [22], and e.g. [2,3,24,31] and [33]. Results on pathwise stability with general decay rate (i.e. not necessarily exponential decay) of stochastic evolutions are given in [9]. In [10] results

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on stabilization by noise of some partial differential equations may be found. In [12] results on stabilization by noise for stochastic reaction-diffusion equations are used to establish existence and uniqueness of invariant measure. In [25] a result on pathwise stability of ﬁnite dimensional stochastic differential equations with jumps is established, using the existence of an invariant measure for the projection of the solution on the unit sphere. An insightful discussion on Itˆ o noise versus Stratonovich noise may be found in [11], together with results on pathwise stability of linear stochastic partial differential equations with linear multiplicative noise. Acknowledgments The author wishes to express his thanks to Onno van Gaans and Sjoerd Verduyn Lunel (Mathematical Institute, Leiden) and also the anonymous reviewer for their insightful remarks which added substantially to the quality of this paper. Open Access. This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Appendix A. Law of Large Numbers for Martingales In this paper we make use of the law of large numbers for martingales. This is not a new result; a more general formulation can be found for example in [29], Theorem 3.4 but without proof. A proof for the formulation below may be found in [6] or [8]. Theorem A.1. (Law of large numbers for martingales) Let (M (t))t≥0 be a continuous local martingale in R with M (0) = 0. If lim sup t→∞

[M ](t) < ∞, t

a.s.,

(24)

then M (t) = 0, t→∞ t lim

a.s.

Appendix B. Approximation of Solutions of Stochastic Differential Equations To be able to extend results from the case of uniformly continuous semigroups to the more general case of strongly continuous semigroups, we will need some results on approximation of solutions of SDEs. The notion of Yosida approximations is well known in semigroup theory, but too restrictive for our purposes. We wish to allow certain perturbations of the Yosida approximation by bounded operators. This leads to our slightly more general notion of approximations which we deﬁne now.

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Definition B.1. Let X be a Banach space. Let A be the inﬁnitesemal generator of a strongly continuous semigroup (S(t))t≥0 in L(X). Let (Tn (t))t≥0,n∈N be a sequence of strongly continuous semigroups in L(X) with generators (An )n∈N . Then (An ) is called an approximation of A if (i) there exists M > 0 and ω ∈ R such that ||T (t)|| ∨ sup ||Tn (t)|| ≤ M eωt n∈N

for all t ≥ 0,

(25)

and (ii) we have that Tn (t)x → T (t)x as n → ∞ for all x ∈ X, uniformly in t on compact sets. An approximation (An ) of A is called a bounded approximation if An ∈ L(X) for all n ∈ N. We will not distinguish between an approximation (An ) of A or an approximation (Tn ) of the corresponding semigroup T . Equivalent conditions for (An ) to be an approximation of A are given by the Trotter–Kato theorem, see [18], Theorem III.4.8. A sufﬁcient condition for a sequence (An ) to be an approximation of A is that (25) holds, and that An x → Ax for all x ∈ D, where D is a core for A. B.1. Yosida Approximation An important example of bounded approximations is the Yosida approximation which we will discuss here. Let X be a Banach space. In this example, A : D(A) → X is the inﬁnitesemal generator of a strongly continuous semigroup (S(t))t≥0 on X satisfying ||S(t)|| ≤ M eωt ,

t ≥ 0,

where M ≥ 1 and ω ∈ R. Recall the notions of the resolvent set of A, ρ(A) := {λ ∈ C : λ − A has a bounded inverse}, and the resolvent of A, R(λ, A) := (λ − A)−1 ,

λ ∈ ρ(A).

Deﬁne the Yosida approximation of A by An := AJn = nAR(n, A) = n2 R(n, A) − nI,

n ∈ N ∩ ρ(A),

where Jn := nR(n, A). These (An )n∈N are bounded operators and therefore generate uniformly continuous semigroups which we denote by (Sn (t))t≥0 , n ∈ N. Furthermore An x → Ax for all x ∈ D(A). By [15], Theorem A.2, ωnt

||Sn (t)|| ≤ M e n−ω , Note that ωn → ω. n−ω

t ≥ 0.

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In particular, for all ω > ω there exists an N ∈ N such that for and all n > N , we have ||Sn (t)|| ≤ M eω t ,

t ≥ 0.

(26)

If we combine the Trotter–Kato approximation theorem [18, Theorem III.4.8] with (26), we obtain the following lemma: Lemma B.2. Sn (t)x → S(t)x for all x ∈ X, uniformly for t ∈ [0, T ] with T > 0. B.2. Convergence of Stochastic Evolutions Proposition B.3. Suppose S is a strongly continuous semigroup with infinitesemal generator A, W is a standard Brownian motion in Rm , p > 2 and X0 ∈ Lp (Ω, F0 ; H). Suppose (An )n∈N are approximations of A. Suppose F : H → H and G : H → L(Rm ; H) are globally Lipschitz. Let X be the unique mild solution to dX(t) = (AX + F (X)) dt + G(X) dW (t),

X(0) = X0 ,

and Xn the unique mild solution to dX(t) = (An Xn + F (Xn )) dt + G(Xn ) dW (t),

X(0) = X0 .

Then for all T > 0, E sup |X(t) − Xn (t)|p → 0

n → ∞.

as

t∈[0,T ]

Moreover, there exists a sequence (nk )k∈N in N such that lim sup |Xnk (t) − X(t)|

k→∞ t∈[0,T ]

Proof. Let ε > 0. E

for all T > 0,

almost surely.

sup |X(t) − Xn (t)|p

t∈[0,T ]

≤3

p−1

E

sup |S(t)X0 − Sn (t)X0 |

p

t∈[0,T ]

p ⎤ t p−1 ⎣ + 3 E sup S(t − s)F (X(s)) − Sn (t − s)F (Xn (s)) ds ⎦ t∈[0,T ] 0 p ⎤ t ⎡ p−1 ⎣ + 3 E sup S(t−s)G(X(s))−Sn (t−s)G(Xn (s), s) dW (s) ⎦. t∈[0,T ] ⎡

0

First term. For the ﬁrst term we have that lim

sup |S(t)X0 − Sn (t)X0 |2 → 0,

n→∞ t∈[0,T ]

almost surely.

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Since (S(t))t∈[0,T ] and (Sn (t))t∈[0,T ] are uniformly bounded in operator norm, by dominated convergence, sup |S(t)X0 − Sn (t)X0 |2 < ε

3p−1 E

t∈[0,T ]

for n large enough. Second term. Note that p s sup S(s − r)F (X(r)) − Sn (s − r)F (Xn (r)) dr s∈[0,t] 0

s

≤ sup s

|S(s − r)F (X(r)) − Sn (s − r)F (Xn (r))|p dr

p−1

s∈[0,t]

0

s

≤ sup (2s)

|Sn (s − r) (F (X(r)) − F (Xn (r))) |p dr

p−1

s∈[0,t]

0

s |Sn (s − r)F (X(r)) − S(s − r)F (X(r))|p dr.

p−1

+ sup (2s) s∈[0,t]

0

with s |Sn (s − r) (F (X(r)) − F (Xn (r))) |p dr

sup s∈[0,t]

0

t ≤ k1

sup |X(r) − Xn (r)|p ds

0

r∈[0,s]

where the constant k1 > 0 may be chosen such that the inequality holds for all t ∈ [0, T ]. Furthermore by dominated convergence, for n large enough, s E sup (2s)

|Sn (s − r)F (X(r)) − S(s − r)F (X(r))|p dr < ε.

p−1

s∈[0,t]

0

Third term. For the third term, estimate the stochastic convolution twice (using [15, Proposition 7.3]), for n large enough, p s E sup S(s − r)G(X(r)) − Sn (s − r)G(Xn (r)) dW (r) s∈[0,t] 0 p s p−1 ≤ 2 E sup S(s − r)G(X(r)) − Sn (s − r)G(X(r)) dW (s) s∈[0,t] 0

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p s + 2p−1 E sup Sn (s − r)(G(X(r)) − G(Xn (r))) dW (s) s∈[0,t] 0

t ≤ ε + k2 E

sup |X(r) − Xn (r)|p dt,

0

r∈[0,s]

where the constant k2 > 0 may be chosen in such a way that the inequality holds for all t ∈ [0, T ]. Combining all the terms, for t ∈ [0, T ] and for n large enough, E sup |X(s) − Xn (s)|p s∈[0,t]

t ≤ 3ε + (k1 + k2 )

E sup |X(r) − Xn (r)|p ds. 0

r∈[0,s]

By Gronwall’s lemma therefore, for t ∈ [0, T ], E sup |X(s) − Xn (s)|p ≤ 3ε exp((k1 + k2 )t), s∈[0,t]

and we may let ε ↓ 0 to obtain the ﬁrst claim. It follows that for every T > 0 there exists a subsequence (n(k))k∈N such that lim sup |X(t) − Xn(k) (t)| = 0,

k→∞ t∈[0,T ]

a.s.

Now deﬁne Ω1 ⊂ Ω, P(Ω1 ) = 1 and a subsequence (n1 (k))k∈N such that lim sup |Xn1 (k) (t) − X(t)| = 0 on Ω1 .

k→∞ t∈[0,1]

Deﬁne recursively, for m ∈ N, m ≥ 2, sets Ωm ⊂ Ω, P(Ωm ) = 1, and further subsequences (nm (k))k∈N of (nm−1 (k))k∈N such that lim

sup |Xnm (k) (t) − X(t)| = 0

k→∞ t∈[0,m]

on Ωm .

˜ := ∩m∈N Ωm (so P(Ω) ˜ = 1), and consider the subsequence (nk (k))k∈N . Let Ω ˜ Let ω ∈ Ω, ε > 0 and M ∈ N. Take K > 0 such that sup |XnM (k)(t) − X(t)|(ω) < ε for all k ≥ K.

t∈[0,M ]

Then for k ≥ K ∨ M , nk (k) ≥ nM (k), and nk (k) ∈ (nM (l))l∈N , and therefore sup |Xnk (k)(t) − X(t)|(ω) < ε.

t∈[0,M ]

˜ with P(Ω) ˜ = 1 and a This proves the second claim: There exists a set Ω ˜ subsequence (nk (k))k∈N of N such that on Ω, we have that for any M ∈ N (and hence any T > 0), lim

sup |Xnk (k) (t) − X(t)| = 0.

k→∞ t∈[0,M ]

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Appendix C. Stochastic Delay Differential Equations In this section we provide the preliminaries for stochastic delay differential equations. Consider ﬁrst a deterministic linear delay differential equation in Rn of the following form ⎧ k ⎨ x(t) ˙ = A0 x(t) + i=1 Ai x(t − θi ) (27) x(0) = c, ⎩ x(t) = f (t), −1 ≤ t < 0, with Ai ∈ L(Rn ), i = 0, . . . , k, and θi ∈ (0, 1], i = 1, . . . , k, c ∈ Rn and f ∈ L2 ([−1, 0]; Rn ). See [5,17] and [13], Section 2.4. This equation has a unique solution (x(t))t≥−1 . Denote the segment process by xt , where xt ∈ L2 ([−1, 0]; Rn ) is deﬁned by xt (s) = x(t + s), t ≥ 0, s ∈ [−1, 0]. The segment process keeps track of the ‘history’ of the delay equation. As discussed there, we may deﬁne a strongly continuous semigroup (T (t)), called the delay semigroup acting on the state space E 2 := Rn × L2 ([−1, 0]; Rn ) with inﬁnitesemal generator A given by k c A0 c + i=1 Ai f (−θi ) A = , d f ds f with domain

# " c 2 1,2 n D(A) = ∈ E : f ∈ W ([−1, 0]; R ), f (0) = c . f c Applying this semigroup to an initial condition ∈ E 2 , corresponds f to solving the delay differential equation (27): x(t) c = T (t) . xt f We may now add nonlinear disturbances and noise. Consider the stochastic differential equation with delay, ⎧ % & k ⎪ dY (t) = A Y (t) + A Y (t − θ ) + ϕ(Y (t), Y ) dt ⎪ 0 i i t i=1 ⎪ ⎨ t≥0 +ψ(Y (t), Yt ) dW (t), (28) ⎪ ⎪ x(0) = c ⎪ ⎩ x(t) = f (t), −1 ≤ t < 0. with (W (t)) an m-dimensional standard Brownian motion, ϕ : E 2 → Rn Lipschitz and ψ : E 2 → L(Rm ; Rn ) Lipschitz. Deﬁne F : E 2 → E 2 and G : E 2 → L(Rm ; E 2 ) by c ϕ(c, f ) c ψ(c, f ) F := , G := . f 0 f 0

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Then F and G are Lipschitz mappings. We may write (28) as a stochastic evolution in E 2 as ⎧ + F (X(t))] dt + G(X(t)) dW (t), t ≥ 0 ⎨ dX(t) =[AX(t) c ⎩ X(0) = . f This equation should always be interpreted in mild form, i.e. t t c X(t) = T (t) + T (t − s)F (X(s)) ds + T (t − s)G(X(s)) dW (s). f 0

0

Y (t) Then X(t) = provides the unique solution of (28). See also [16, Yt Chapter 10] and [6, Chapter 3]. In the example in this paper we are only concerned with the one-dimensional case, n = m = 1, where the linear operators (Ai ) become real constants (ai ), and where ϕ = 0 and ψ(c, f ) = σc, with σ ∈ R, corresponding to the linear equation k ai Y (t − θi ) dt + σY (t) dW (t), dY (t) = a0 Y (t) + i=1

with W a one-dimensional standard Brownian motion. We will now quote a method to transform the delay semigroup into a generalized contraction semigroup by changing the inner product on E 2 . This is a variation on a result in [16, Section 10.3]. Details may be found in [7] or [6, Section 3.6]. Theorem C.1. Suppose ||eA0 t || ≤ eλt for all t ≥ 0. If there exists μ > λ such that k

k

2 −2μθi ||Ai || ||Ai ||e , (λ − μ) > i=1

i=1

then there exists an equivalent inner product on E 2 such that the delay semigroup (T (t)) satisfies ||T (t)|| ≤ eμt ,

t ≥ 0.

By taking μ = 0, this theorem has the following corollary: Corollary C.2. Suppose ||eA0 t || ≤ eλt for all t ≥ 0. If λ<−

k

||Ai ||,

i=1

then there exists an equivalent inner product on E 2 such that the delay semigroup (T (t)) is a contraction semigroup, i.e. ||T (t)|| ≤ 1 for all t ≥ 0.

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References [1] Appleby, J.A.D., Mao, X.: Stochastic stabilisation of functional differential equations. Syst. Control Lett. 54(11), 1069–1081 (2005) [2] Arnold, L.: A formula connecting sample and moment stability of linear stochastic systems. SIAM J. Appl. Math. 44(4), 793–802 (1984) [3] Arnold, L., Oeljeklaus, E., Pardoux, E.: Almost sure and moment stability for linear Itˆ o equations. In: Lyapunov Exponents. Lecture Notes in Mathematics, vol. 1186, pp. 129–159 (1985) [4] Arnold, L., Wihstutz, V.: Lyapunov exponents: a survey. In: Lyapunov Exponents (Bremen, 1984), Lecture Notes in Mathematics, vol. 1186, pp. 1–26. Springer, Berlin (1986) [5] B´ atkai, A., Piazzera, S.: Semigroups for Delay Equations. AK Peters, Ltd. (2005) [6] Bierkens, J.: Long term dynamics of stochastic evolution equations. PhD thesis, Universiteit Leiden (2009) [7] Bierkens, J.: Dissipativity of the delay semigroup. (2010, submitted) [8] Bierkens, J., van Gaans, O., Verduyn Lunel, S.M.: Estimates on the pathwise Lyapunov exponent for linear stochastic differential equations with multiplicative noise. Stoch. Anal. Appl. 28(5), 747–762 (2010) [9] Caraballo, T., Garrido-Atienza, M.J., Real, J.: Asymptotic stability of nonlinear stochastic evolution equations. Stoch. Anal. Appl. 21(2), 301–327 (2003) [10] Caraballo, T., Liu, K., Mao, X.: On stabilization of partial differential equations by noise. Nagoya Math. J. 161, 155–170 (2001) [11] Caraballo, T., Robinson, J.C.: Stabilisation of linear PDEs by Stratonovich noise. Syst. Control Lett. 53(1), 41–50 (2004) [12] Cerrai, S.: Stabilization by noise for a class of stochastic reaction-diffusion equations. Probab. Theory Relat. Fields 133(2), 190–214 (2005) [13] Curtain, R.F., Zwart, H.: An Introduction to Inﬁnite-Dimensional Linear Systems Theory. Springer, New York (1995) [14] Da Prato, G., Gatarek, D., Zabczyk, J.: Invariant measures for semilinear stochastic equations. Stoch. Anal. Appl. 10(4), 387–408 (1992) [15] Da Prato, G., Zabczyk, J.: Stochastic Equations in Inﬁnite Dimensions. Cambridge University Press, Cambridge (1992) [16] Da Prato, G., Zabczyk, J.: Ergodicity for Inﬁnite Dimensional Systems. Cambridge University Press, Cambridge (1996) [17] Delfour, M.C.: The largest class of hereditary systems deﬁning a C0 semigroup on the product space. Can. J. Math. 32(4), 969–978 (1980) [18] Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000) [19] Haussmann, U.G.: Asymptotic stability of the linear Itˆ o equation in inﬁnite dimensions. J. Math. Anal. Appl. 65(1), 219–235 (1978) [20] Ichikawa, A.: Stability of semilinear stochastic evolution equations. J. Math. Anal. Appl. 90(1), 12–44 (1982) [21] Kallianpur, G., Xiong, J.: Stochastic differential equations in inﬁnite dimensional spaces. In: Lecture Notes—Monograph Series. Institute of Mathematical Statistics (1995)

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[22] Khasminskii, R.Z.: Stochastic stability of differential equations. In: Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, vol. 7. Sijthoﬀ & Noordhoﬀ, Alphen aan den Rijn (1980). Translated from the Russian by D. Louvish [23] Kwieci´ nska, A.A.: Stabilization of evolution equations by noise. Proc. Am. Math. Soc. 130(10), 3067–3074 (2002) [24] Leizarowitz, A.: Exact results for the Lyapunov exponents of certain linear Ito systems. SIAM J. Appl. Math. 50(4), 1156–1165 (1990) [25] Li, C.W., Dong, Z., Situ, R.: Almost sure stability of linear stochastic differential equations with jumps. Probab. Theory Relat. Fields 123(1), 121–155 (2002) [26] Liu, K., Truman, A.: Moment and almost sure Lyapunov exponents of mild solutions of stochastic evolution equations with variable delays via approximation approaches. J. Math. Kyoto Univ. 41(4), 749–768 (2001) [27] Liu, K., Truman, A.: Lyapunov function approaches and asymptotic stability of stochastic evolution equations in Hilbert spaces—a survey of recent developments. In: Stochastic Partial Differential Equations and Applications (Trento, 2002). Lecture Notes in Pure and Appl. Math., vol. 227, pp. 337–371. Dekker, New York (2002) [28] Mao, X.: Robustness of exponential stability of stochastic differential delay equations. IEEE Trans. Automat. Control 41(3), 442–447 (1996) [29] Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichester (1997) [30] Mao, X., Shah, A.: Exponential stability of stochastic differential delay equations. Stoch. Stoch. Rep. 60(1–2), 135–153 (1997) ´ Wihstutz, V.: Lyapunov exponent and rotation number of two[31] Pardoux, E., dimensional linear stochastic systems with small diffusion. SIAM J. Appl. Math. 48(2), 442–457 (1988) [32] Peszat, S., Zabczyk, J.: Stochastic partial differential equations with L´evy noise. In: Encyclopedia of Mathematics and its Applications, vol. 113. Cambridge University Press, Cambridge (2007) [33] Scheutzow, M.: Stabilization and destabilization by noise in the plane. Stoch. Anal. Appl. 11(1), 97–113 (1993) Joris Bierkens (B) Centrum Wiskunde & Informatica P. O. Box 94079 1090 GB Amsterdam The Netherlands e-mail: [email protected] Received: February 7, 2010. Revised: October 25, 2010.

Integr. Equ. Oper. Theory 69 (2011), 29–62 DOI 10.1007/s00020-010-1819-2 Published online July 28, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Closable Multipliers V. S. Shulman, I. G. Todorov and L. Turowska Abstract. Let (X, μ) and (Y, ν) be standard measure spaces. A function ϕ ∈ L∞ (X ×Y, μ×ν) is called a (measurable) Schur multiplier if the map Sϕ , defined on the space of Hilbert-Schmidt operators from L2 (X, μ) to L2 (Y, ν) by multiplying their integral kernels by ϕ, is bounded in the operator norm. The paper studies measurable functions ϕ for which Sϕ is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of two special types: if ϕ is of Toeplitz type, that is, if ϕ(x, y) = f (x − y), x, y ∈ G, where G is a locally compact abelian group, then the closability of ϕ is related to the local inclusion of f in the Fourier algebra A(G) of G. If ϕ is a divided difference, that is, a function of the form (f (x)−f (y))/(x−y), then its closability is related to the “operator smoothness” of the function f . A number of examples of non-closable, norm closable and w*-closable multipliers are presented. Mathematics Subject Classification (2000). Primary 47B49; Secondary 47L60, 43A22.

1. Introduction Let (X, μ) and (Y, ν) be standard measure spaces, H1 = L2 (X, μ), H2 = L2 (Y, ν), and let B(H1 , H2 ) be the space of all bounded linear operators from H1 into H2 . There is a method, due mainly to Birman and Solomyak [2–5], to associate to certain bounded measurable functions ϕ on X × Y , linear transformations Sϕ on B(H1 , H2 ); these transformations are called Schur multipliers or, in the more general setting of spectral measures in the place of μ and ν, double operator integrals. Namely, one ﬁrst deﬁnes the map Sϕ on the space of all Hilbert-Schmidt operators by multiplying their integral kernels by ϕ; if Sϕ is bounded in the operator norm, one extends it to the space K(H1 , H2 ) of all compact operators by continuity. The map Sϕ is deﬁned on B(H1 , H2 ) by taking the second dual of the constructed map on K(H1 , H2 ). I. G. Todorov was supported by EPSRC grant D050677/1. L. Turowska was supported by the Swedish Research Council.

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The function ϕ is thus called a Schur multiplier if the multiplication map Sϕ , initially deﬁned on the space of Hilbert-Schmidt operators, is bounded in the operator norm. Equivalently, ϕ is a Schur multiplier if ϕ(x, y)k(x, y) is the integral kernel of a nuclear operator provided k(x, y) is such. The set of all Schur multipliers will be denoted by S(X, Y ). Many years ago, Professor Solomyak informed the ﬁrst author that at the early stages of the development of the theory of double operator integrals, there existed an idea to deﬁne Sϕ for a more general class of functions ϕ (perhaps for all measurable ones) as the closure of a densely deﬁned linear operator in the operator norm, or in the weak operator, topology. However, this approach was not pursued because no information on the closability of the multiplication maps Sϕ was obtained at that time. The aim of this paper is to study the classes of functions ϕ for which Sϕ is closable in the norm topology, or in the weak* topology, of B(H1 , H2 ). By weak* topology we mean the topology σ(B(H1 , H2 ), C1 (H2 , H1 )) induced on B(H1 , H2 ) by its duality with the space C1 (H2 , H1 ) of nuclear operators from H2 into H1 . We denote these classes of functions by Scl (X, Y ) and Sw∗ (X, Y ), respectively. We obtain a satisfactory characterisation of the class Sw∗ (X, Y ). In order to describe it, let us denote by Sloc (X, Y ) the class of functions ϕ with the following property: for each ε > 0 there exist subsets Xε ⊆ X and Yε ⊆ Y of measure not exceeding ε, such that the restriction of ϕ to (X\Xε )×(Y \Yε ) is a Schur multiplier. We prove in Theorem 3.6 that the elements of Sloc (X, Y ) (which we call local Schur multipliers) can be characterised in terms similar to those of Peller’s characterisation of Schur multipliers [26]. Namely, they are precisely the functions of the form (a(x), b(y)), where a(·) (resp. b(·)) is a measurable function from X (resp. Y ) into a separable Hilbert space. Then we show in Theorem 4.4 that the w*-closable multipliers (that is, the elements of Sw∗ (X, Y )) are precisely the functions of the form t(x, y)/s(x, y) where t and s are local Schur multipliers and s(x, y) = 0 for marginally almost all (x, y). In particular, Sw∗ (X, Y ) is an algebra of (equivalent classes of) functions. For any measurable function ϕ on X × Y , there exists a maximal (in a sense that we make precise in Sect. 4) countable family of rectangles on each ∗ of which ϕ is w*-closable; the complement of their union is denoted by κw ϕ . ∗ The “size” of κw ϕ can be considered as a measure of the extent to which ϕ fails to be a w*-closable multiplier. The information we obtain about Scl (X, Y ) is less precise. Roughly speaking, we show that, in order to verify whether ϕ belongs to Scl (X, Y ), ∗ one needs to check whether the set κw ϕ supports a non-zero compact oper∗ ator. More precisely: if κw ϕ does not support a non-zero compact operator then ϕ ∈ Scl (X, Y ) while if there exists a non-zero compact operator in the ∗ / Scl (X, Y ). The difference smallest masa-bimodule with support κw ϕ then ϕ ∈ ∗ between the smallest masa-bimodule with support κw ϕ and the largest one ∗ (the set of all operators supported on κw ϕ ) is subtle; it is the subject of the theory of operator synthesis [1,28], an operator analogue of the theory of

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spectral synthesis [15,20]. We prove that Scl (X, Y ) is an algebra and present various examples of multipliers which are not norm-closable and of normclosable multipliers which are not w*-closable. The product of two measure spaces possesses natural “pseudotopological structures”, namely the ω-topology and the τ -topology, which are related to the problem of closability of multipliers. A set is called τ -open (resp. ω-open) if it is a countable union of measurable rectangles and a null set (resp. a set contained in (X0 × Y ) ∪ (X × Y0 ), where X0 and Y0 are null sets). Denoting by Cτ (X ×Y ) (resp. Cω (X ×Y )) the space of all τ -continuous (resp. ω-continuous) complex valued functions on X × Y , we prove that Scl (X, Y ) ⊆ Cτ (X × Y )

and

Sw∗ (X, Y ) ⊆ Cω (X × Y )

(if one identiﬁes functions equivalent with respect to the product measure). Both inclusions are shown to be strict. We present examples which show that in the chain S(X, Y ) ⊆ Sloc (X, Y ) ⊆ Sw∗ (X, Y ) ⊆ Scl (X, Y ) the ﬁrst and the third inclusions are strict. The question of whether the second inclusion is strict is left open. The paper is organised as follows. In Sect. 2 we state some basic definitions and results about subsets of, and functions on, product measure spaces, bimodules over maximal abelian selfadjoint algebras, Schur multipliers and closable operators. In Sect. 3 we examine local Schur multipliers, while Sects. 4 and 5 are devoted to the study of w*-closable and norm-closable multipliers, respectively. In Sects. 6 and 7 we study multipliers of special types. Given a complex function f deﬁned on a subinterval of the real line, one may consider its divided difference, in other words, the function fˇ on two variables given by fˇ(x, y) = (f (x) − f (y))/(x − y). The corresponding class of multipliers plays an important role in Perturbation Theory and Spectral Theory (see, for example, [26] and the references therein). We show that such multipliers are always norm-closable; in Theorem 6.3 and Corollary 6.4 we formulate necessary and sufﬁcient conditions for fˇ to be a local Schur multiplier. Toeplitz multipliers are (Schur, local, norm-closable or w*-closable) multipliers ϕ of the form ϕ(x, y) = f (x−y), where f is a complex function deﬁned on a locally compact abelian group G (equipped with Haar measure μ). Theorem 7.8 asserts that a function f (x − y) is a w*-closable multiplier if and only if it is a local Schur multiplier, and that this occurs precisely when f is equivalent to a function which belongs locally to the Fourier algebra of G. The closability of Toeplitz multipliers is shown to be related to some questions about sets of multiplicity in harmonic analysis. We describe also those functions of the form f (x − y) which are ((μ × μ)-equivalent to) ω-continuous or τ -continuous functions. En route, we obtain an example of a continuous (hence ω-continuous) function on G × G which is not a norm-closable multiplier.

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2. Preliminary Results 2.1. Pseudo-Topologies on the Product of Measure Spaces In what follows, we write B(Z) = B(Z, γ) for the algebra, with respect to the pointwise product, of all measurable complex valued functions deﬁned on a measure space (Z, γ). Let (X, μ) and (Y, ν) be standard σ-ﬁnite measure spaces, which will be ﬁxed throughout the paper. We will often write B(X × Y ) in the place of B(X × Y, μ × ν). A subset of X×Y is said to be a measurable rectangle (or simply a rectangle) if it is of the form α×β, where α ⊆ X and β ⊆ Y are measurable subsets. A subset E ⊆ X × Y is called marginally null if E ⊆ (X0 × Y ) ∪ (X × Y0 ), where μ(X0 ) = ν(Y0 ) = 0. We call two subsets E, F ⊆ X × Y marginally equivalent (and write E F ) if the symmetric difference of E and F is marginally null. We say that E marginally contains F (or F is marginally contained in E) if F \E is marginally null; E and F are said to be marginally disjoint if E ∩ F is marginally null. A subset E of X × Y is called ω-open if it is marginally equivalent to the union of a countable set of rectangles. The complements of ω-open sets are called ω-closed. It is clear that the class of all ω-open (resp. ω-closed) sets is closed under countable unions (resp. intersections) and ﬁnite intersections (resp. unions); in other words, the ω-open sets form a pseudo-topology. The following lemma shows that in some cases one can form a certain kind of a union of a given, possibly uncountable, family of ω-open subsets. Lemma 2.1. Let E be a family of ω-open subsets of X × Y . Let Eσ be the set of all countable unions of elements of E. Then there exists a (unique up to marginal equivalence) set E ∈ Eσ which marginally contains every set in E. Proof. First assume that the measures μ and ν are ﬁnite. On the set Π of all measurable rectangles we introduce a metric ρ, setting, for R1 = X1 × Y1 , R2 = X2 × Y2 in Π, ρ(R1 , R2 ) = μ(X1 X2 ) + ν(Y1 Y2 ) (here denotes symmetric difference). Then Π is a separable metric space whence the set F of all rectangles that are contained in elements of E is also separable. Let {Rn : n ≥ 1} be a dense sequence in F and E = ∪∞ n=1 Rn . Then it is clear that E marginally contains all R ∈ F and therefore all E ∈ E. ∞ In the general case, let X = ∪∞ n=1 Xn and Y = ∪m=1 Ym with μ(Xn ) < ∞ and ν(Ym ) < ∞, n, m ∈ N. For each pair n, m, let En,m = {E ∩ (Xn × Ym ) : E ∈ E}, let En,m be the ω-union of En,m , and set E = ∪∞ n,m=1 En,m . The uniqueness is obvious. The set whose existence is guaranteed by Lemma 2.1 will be called the ω-union of E. We will say that two functions ϕ, ψ ∈ B(X × Y ) are equivalent, and write ϕ ∼ ψ, if the set D = {(x, y) ∈ X × Y : ϕ(x, y) = ψ(x, y)} is null with respect to the product measure. If D is marginally null then we say that ϕ and ψ coincide marginally everywhere, or that they are marginally equivalent, and write ϕ ψ. By L∞ (X × Y ) we denote as usual the subalgebra of B(X × Y ) of all (equivalence classes of) essentially bounded functions.

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Let E ⊆ X × Y be an ω-open set. A measurable function ϕ : E → C is called ω-continuous if the set ϕ−1 (G) is ω-open for every open subset G ⊆ C. As in [10, Corollary 3.2] one can see that the set Cω (E) of all ωcontinuous functions on E is a subalgebra of B(E). For an arbitrary set M ⊆ Cω (X × Y ) ⊆ B(X × Y ), we let its null set null(M) be the complement of the ω-union of the family E = {h−1 (C\{0}) : h ∈ M}. We will need the following simple result from [21] (see the remark after [21, Proposition 8.1]). Lemma 2.2. Let E ⊆ X × Y be an ω-open set and let f : E → C be an ω-continuous function. If f ∼ 0 then f 0. Thus if two ω-continuous functions are equivalent then they are marginally equivalent, and if a function is equivalent to an ω-continuous function then the latter is deﬁned uniquely up to marginal equivalence. We will also need another pseudo-topology on X × Y . Two subsets E1 , E2 of X × Y will be called μ × ν-equivalent if their symmetric difference is a μ × ν-null set. We will say that a subset E ⊆ X × Y is τ -open if it is μ × ν-equivalent to a countable union of rectangles. It is clear that the pseudo-topology τ is stronger than ω. 2.2. Bimodules If H1 and H2 are Hilbert spaces, we denote by B(H1 , H2 ) the space of all bounded linear operators from H1 into H2 , and by K(H1 , H2 ) (resp. C1 (H1 , H2 ), C2 (H1 , H2 )) the space of compact (resp. nuclear, Hilbert-Schmidt) operators in B(H1 , H2 ). Let T op denote the operator norm of T∈ B(H1 , H2 ). As usual, we write B(H) = B(H, H). The space C1 (H2 , H1 ) (resp. B(H1 , H2 )) can be naturally identiﬁed with the Banach space dual of K(H1 , H2 ) (resp. def

C1 (H2 , H1 )), the duality being given by the map (T, S) → T, S = tr(T S). Here tr A denotes the trace of a nuclear operator A. For a subset W ⊆ C1 (H2 , H1 ), let W ⊥ = {T ∈ B(H1 , H2 ) : T, S = 0, for each S ∈ W}. For the rest of the paper we let H1 = L2 (X, μ) and H2 = L2 (Y, ν). The space L2 (X × Y ) will be identiﬁed with C2 (H1 , H2 ) via the map sending 2 an element k ∈ L (X × Y ) to the integral operator Ik given by Ik ξ(y) = k(x, y)ξ(x)dμ(x), ξ ∈ H1 , y ∈ Y . In a similar fashion, C1 (H2 , H1 ) will be X identiﬁed with the space Γ(X, Y ) of all functions F : X ×Y → C which admit a representation F (x, y) =

∞ i=1

fi (x)gi (y),

∞ ∞ where fi ∈ L2 (X, μ), gi ∈ L2 (Y, ν), i ∈ N, i=1 fi 22 < ∞ and i=1 gi 22 < ∞. Equivalently, Γ(X, Y ) can be deﬁned as the projective tensor product ˆ 2 (Y, ν). It was shown in [10, Theorem 6.5] that Γ(X, Y ) consists of L2 (X, μ)⊗L ω-continuous functions. For brevity, we often identify a function h ∈ Γ(X, Y ) with the corresponding integral operator Ih ∈ C1 (H2 , H1 ). It will be convenient to write Γ(X × Y ) in the place of Γ(X, Y ) (this allows for example to write Γ(κ), where κ is a rectangle).

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If f ∈ L∞ (X, μ), let Mf ∈ B(H1 ) denote the operator of multiplication by f . We will often identify the collection {Mf : f ∈ L∞ (X, μ)} of all such operators with the function space L∞ (X, μ). If α ⊆ X is measurable, we write P (α) = Mχα for the multiplication by the characteristic function of the set α. Similar definitions and identiﬁcations are made for L∞ (Y, ν). A subspace W ⊆ B(H1 , H2 ) will be called a bimodule if Mψ T Mϕ ∈ W for all T ∈ W, ϕ ∈ L∞ (X, μ) and ψ ∈ L∞ (Y, ν). One deﬁnes bimodules in B(H2 , H1 ) in a similar fashion. We say that an ω-closed subset κ ⊆ X × Y supports an operator T ∈ B(H1 , H2 ) (or that T is supported on κ) if P (β)T P (α) = 0 whenever α × β is marginally disjoint from κ. For any subset M ⊆ B(H1 , H2 ), there exists a smallest (up to marginal equivalence) ω-closed set supp M which supports every operator T ∈ M [10]. By [28], for any ω-closed set κ there exists a smallest (resp. largest) w*-closed bimodule Mmin (κ) (resp. Mmax (κ)) with support κ in the sense that if M ⊆ B(H1 , H2 ) is a w*-closed bimodule with supp M = κ then Mmin (κ) ⊆ M ⊆ Mmax (κ). If Mmin (κ) = Mmax (κ), the set κ is called synthetic. By [28, Theorem 4.4], Mmin (κ) = {Ih : h ∈ Γ(X, Y ), hχκ 0}⊥ . Lemma 2.3. Let W ⊆ C1 (H2 , H1 ) be a bimodule and {fn }∞ n=1 ⊆ Γ(X, Y ) be is dense in W. Then null(W) = supp(W ⊥ ) = a sequence such that {Ifn }∞ n=1 ∞ −1 ∩n=1 fn (0). In particular, W is norm dense in C1 (H2 , H1 ) if and only if there exists a sequence {hn }∞ n=1 ⊆ Γ(X, Y ) such that Ihn ∈ W for every n ∈ N and the −1 h (0) is marginally null. set ∩∞ n=1 n Proof. We start by showing the second statement. Since the Hilbert spaces H1 and H2 are separable, the space C1 (H2 , H1 ) is separable and hence there exists a sequence {Kn }∞ n=1 dense in W. Suppose that Kn = Ihn , where hn ∈ Γ(X, Y ) and let E = ∩n h−1 n (0). If E is not marginally null then, by [28, Corollary 4.1], Mmin (E) contains a non-zero operator T . By [28, Theorem 4.4], T, Kn = 0 for all n ∈ N, and hence W is not dense. Conversely, suppose that W is not dense, let hn ∈ Γ, n ∈ N, be such that −1 Ihn ∈ W and set E = ∩∞ n=1 hn (0). The annihilator M of W in B(H1 , H2 ) is a non-zero w*-closed bimodule. By [28], the support F of M is not marginally null, and if an operator Ih belongs to its preannihilator then h vanishes marginally almost everywhere on F . It follows that F is marginally contained in E and hence E is not marginally null. Now we prove the general statement. It was shown in [28] (see the proof of [28, Theorem 2.1]) that null(W) = supp(W ⊥ ) where W is the norm closure of W in C1 (H2 , H1 ). Hence, up to marginally null sets, supp(W ⊥ ) ⊆ −1 ∞ −1 ⊥ null(W) ⊆ ∩∞ n=1 fn (0) and it sufﬁces to show that ∩n=1 fn (0) ⊆ supp(W ). ⊥ Let κ be a rectangle disjoint from supp(W ). By the definition of support, the restriction to κ of the functions corresponding to operators in W form a set dense in Γ(κ). By the ﬁrst part of the proof, the intersection of the null sets of the restrictions of fn , n ∈ N, to κ is marginally null. Hence κ is −1 ⊥ marginally disjoint from ∩∞ n=1 fn (0). Since the complement of supp(W ) is ω-closed, this implies the last remaining inclusion.

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2.3. Schur Multipliers and Peller’s Theorem For a function ϕ ∈ B(X × Y ), set D(ϕ) = {k ∈ L2 (X × Y ) : ϕk ∈ L2 (X × Y )}. We let Sϕ : D(ϕ) → L2 (X × Y ) be the mapping given by Sϕ k = ϕk. We identify Sϕ with a (densely deﬁned linear) map on K(H1 , H2 ) acting by the rule Sϕ (Ik ) = Iϕk . Note that Sϕ depends only on the equivalence class of ϕ. Taking this into account, we will sometimes say that a function ϕ belongs to a certain class of functions, if it is equivalent to a function from this class. When we need to make the distinction, we will write h ∈σ F, if h is equivalent to a function from the class F with respect to the measure σ. Recall that a function ϕ ∈ L∞ (X × Y ) is called a Schur multiplier if the map Sϕ is bounded in the operator norm, that is, if there exists a constant C > 0 such that Sϕ (Ik ) op ≤ C Ik op , for all k ∈ L2 (X × Y ). If ϕ is a Schur multiplier then the mapping Sϕ has a unique weak* continuous extension to B(H1 , H2 ) which will still be denoted by Sϕ . Let S(X, Y ) (or S(X × Y )) be the set of all Schur multipliers; clearly, S(X, Y ) is a subalgebra of B(X, Y ). The following facts follow easily from the definition of a Schur multiplier: Lemma 2.4. (i) If ϕ ∈ S(X × Y ) then ϕ|α×β ∈ S(α × β) for all measurable subsets α ⊆ X and β ⊆ Y . (ii) If X × Y = ∪N p=1 κp , where all κp are rectangles and ϕ|κp ∈ S(κp ) then ϕ ∈ S(X, Y ). Schur multipliers were ﬁrst introduced by Schur in the early 20th century in case of discrete measures μ and ν. A characterisation of this particular class of Schur multipliers was obtained by Grothendieck in [16]. The following generalisation for the class deﬁned above is due to Peller [26]. Theorem 2.5. Let ϕ ∈ L∞ (X × Y ). The following conditions are equivalent: (i) ϕ is a Schur multiplier; (ii) there exist measurable functions a : X → l2 and b : Y → l2 such that ϕ(x, y) = (a(x), b(y))l2 , a.e. on X × Y and sup a(x) 2 sup b(y) 2 < ∞. x∈X

(iii) ϕ(x, y)k(x, y) ∈

μ×ν

y∈Y

Γ(X, Y ) whenever k(x, y) ∈ Γ(X, Y ).

It follows from Peller’s Theorem (and can easily be seen directly) that if the measures μ and ν are ﬁnite then S(X, Y ) ⊆ Γ(X, Y ). Using modern terminology, one can say that Theorem 2.5 identiﬁes the algebra S(X, Y ) with the weak* Haagerup tensor product L∞ (X, μ) ⊗w∗ h L∞ (Y, ν) (see [6] where this tensor product was introduced). 2.4. General Facts on Closable Operators Let X be a Banach space. We denote by X ∗ the dual of X . If S ⊆ X (resp. T ⊆ X ∗ ), we write S ⊥ ⊆ X ∗ (resp. T⊥ ⊆ X ) for the annihilator (resp. preannihilator) of S (resp. T ).

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Let Y be another Banach space. By an operator from X into Y we mean a linear transformation T : D(T ) → Y, where D(T ) is a (not necessarily closed) linear subspace of X called the domain of T . The operator T is called densely deﬁned if D(T ) is norm dense in X . We let Gr T = {(x, T x) : x ∈ D(T )} ⊆ X ⊕ Y be the graph of T . For a subset S ⊆ Y ⊕ X we set S = {(x, y) : (y, x) ∈ S} and let Gr T = (Gr T ) . We recall the definition of the adjoint T ∗ of an operator T : D(T ) → Y. The domain of T ∗ is the subspace D(T ∗ ) = {g ∈ Y ∗ : ∃ f ∈ X ∗ such that g(T x) = f (x), ∀x ∈ D(T )}. For g ∈ D(T ∗ ), one lets T ∗ g equal to f where f ∈ X ∗ is the functional appearing in the definition of D(T ∗ ). Note that g ∈ D(T ∗ ) if and only if the linear map x → g(T x) from D(T ) into C is continuous. By the definition of the operator T ∗ , we have that Gr (−T ∗ ) = (Gr T )⊥ . Recall that an operator T is called closable if the norm closed hull Gr T of its graph is the graph of an operator. Clearly, T is closable if and only if the conditions (xn )∞ n=1 ⊆ X , y ∈ Y, xn → 0 and T xn − y → 0 imply that w∗

⊆ X ∗∗ ⊕ Y ∗∗ of y = 0. We call T w*-closable if the w*-closed hull Gr T ∗∗ ∗∗ its graph is the graph of an operator from X into Y . Here, we identify X and Y with their canonical images in their second duals. We have that T is w*-closable if and only if the conditions (xα )α ⊆ X , G ∈ Y ∗∗ , w-limα xα = 0 and w*-lim T xα = G imply that G = 0. The weak* limit is taken with respect to the weak* topology of Y ∗∗ . In the following proposition the equivalence (iii)⇔(iv) is well-known (see, for example, [19, Chapter III, Section 5]); the other implications can be proved in a similar way. Proposition 2.6. Let T : D(T ) → Y be a densely defined linear operator and set D = D(T ∗ ). Consider the following conditions: (i) T is w*-closable; (ii) D

· w∗

= Y ∗;

(iii) D = Y ∗ ; (iv) T is closable. Then (i)⇐⇒(ii)=⇒(iii)⇐⇒(iv).

3. Local Schur Multipliers We start this section by introducing a class of functions that will play a central role in the paper. For brevity, let us say that a countable family of rectangles covers X × Y , or that it is a covering family, if its union is marginally equivalent to X × Y . Definition 3.1. A function ϕ ∈ B(X × Y ) will be called a local Schur multiplier if there exists a covering family {κm }∞ m=1 of rectangles in X × Y such that ϕ|κm ∈ S(κm ), for each m ∈ N.

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The set of all local Schur multipliers on X × Y will be denoted by Sloc (X, Y ). Proposition 3.2. The set Sloc (X, Y ) is a subalgebra of the algebra Cω (X, Y ) of all ω-continuous functions. Proof. Let ϕ ∈ Sloc (X, Y ). By the σ-ﬁniteness of the measure spaces (X, μ) and (Y, ν), there exists a covering family {κm }∞ m=1 such that ϕ|κm ∈ S(κm ), and (μ × ν)(κm ) < ∞ for each m ∈ N. It follows that ϕ|κm ∈ Γ(κm ). By [10, Theorem 6.5], ϕ|κm is ω-continuous on κm , m ∈ N. Now for an open subset −1 (G)) is ω-open, and hence ϕ G ⊆ C, we have that ϕ−1 (G) = ∪∞ m=1 (κm ∩ ϕ is ω-continuous. It is easy to see that for two functions ϕ, ψ in Sloc (X, Y ), one can ﬁnd a common covering family {κm }∞ m=1 of rectangles on which both ϕ and ψ are Schur multipliers. Since S(κm ) is an algebra, ϕ + ψ and ϕψ belong to Sloc (X, Y ). Let V(X, Y ) be the space of all functions ϕ ∈ B(X × Y ) for which there ∞ exist families {ai }∞ i=1 ⊆ B(X) and {bi }i=1 ⊆ B(Y ) with the properties ∞ i=1

2

|ai (x)| < ∞,

∞

|bi (y)|2 < ∞

i=1

for almost all x ∈ X and y ∈ Y , and ∞ ϕ(x, y) = ai (x)bi (y), almost everywhere on X × Y. i=1

Using a coordinate free language we may say that ϕ ∈ V(X, Y ) if and only if there exists a separable Hilbert space H and measurable functions a : X → H and b : Y → H such that ϕ(x, y) = (a(x), b(y)) for almost all (x, y) ∈ X × Y . We note that Γ(X, Y ) ⊆ V(X, Y ) and S(X, Y ) ⊆ V(X, Y ). Indeed, these function spaces correspond to the cases where the functions a(·) , b(·) are, respectively, square integrable and essentially bounded. ∞ Lemma 3.3. If ϕ ∈ V(X, Y ) then there exist families {Xi }∞ i=1 and {Yj }j=1 of pairwise disjoint subsets of X and Y , respectively, such that ϕ|Xi ×Yj ∈ S(Xi , Yj ), for all i, j ∈ N. We may moreover assume that μ(Xi ) < ∞ and ν(Yj ) < ∞, for all i, j ∈ N.

Proof. Let a : X → 2 and b : Y → 2 be measurable functions such that ϕ(x, y) = (a(x), b(y)), for almost all (x, y). For i, j ∈ N, set Xi = {x ∈ X : i − 1 ≤ a(x) 2 < i} and Yj = {y ∈ Y : j − 1 ≤ b(y) 2 < j}. Then ϕ|Xi ×Yj ∈ S(Xi , Yj ) by Theorem 2.5. Partitioning Xi and Yj into subsets of ﬁnite measure, we obtain the required decompositions. Lemma 3.3 shows, in particular, that V(X, Y ) ⊆ Sloc (X, Y ). Our aim in this section is to show that, in fact, V(X, Y ) = Sloc (X, Y ). Lemma 3.4. Let {κm }∞ m=1 be a covering sequence of ω-open sets. Then there ∞ exist families {Xi }∞ i=1 and {Yj }j=1 of pairwise disjoint measurable subsets of X and Y , respectively, such that

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∞ (i) ∪∞ i=1 Xi and ∪j=1 Yj have full measure, and (ii) each rectangle Xi ×Yj is contained in a finite union of sets from {κm }∞ m=1 .

Proof. Let us say that a subset E ⊆ X × Y is mild if it is contained in a ﬁnite union of sets from the family {κm }∞ m=1 . It sufﬁces to show that there ∞ and {B are increasing sequences {An }∞ n }n=1 of measurable subsets of X n=1 ∞ and Y , respectively, such that ∪n=1 An and ∪∞ n=1 Bn have full measure and all rectangles of the form An × Bn are mild. Indeed, the statement would then follow by setting Xi = Ai \Ai−1 , Yj = Bj \Bj−1 . Since the measure spaces (X, μ), (Y, ν) are standard we may assume that X and Y are equipped with σ-compact topologies with respect to which μ and ν are regular Borel measures. By considering increasing sequences {Un }∞ n=1 and {Vn }∞ n=1 of compact subsets of X and Y , respectively, we reduce the problem to the case where X and Y are compact and μ and ν are ﬁnite. We may clearly assume that each κm is a rectangle. By [10, Lemma 3.4], for any > 0 there exists X ⊆ X, Y ⊆ Y such that μ(X\X ) < , ν(Y \Y ) < and X × Y is contained in a ﬁnite union of rectangles κm . Let n = 2−n , ∞ Ln = ∩∞ k=n X k and Mn = ∩k=n Y k . Then each Ln × Mn is mild since it is contained in X n × Y n . Furthermore, Ln ⊆ Ln+1 , Mn ⊆ Mn+1 , μ(X\Ln ) ≤

∞

μ(X\X k ) < 22−n and ν(Y \Mn ) ≤

k=n

∞

μ(Y \Y k ) < 22−n .

k=n

∞ Thus, μ(X\(∪∞ n=1 Ln )) = 0 and ν(Y \(∪n=1 Mn )) = 0 and the proof is complete.

The following result may be viewed as an analogue of Lemma 2.4 for local multipliers. ∞ Lemma 3.5. Let {Xi }∞ i=1 and {Yj }j=1 be families of pairwise disjoint subsets ∞ of X and Y , respectively, such that X = ∪∞ i=1 Xi and Y = ∪j=1 Yj . Assume that ϕ ∈ B(X × Y ) is such that ϕ|Xi ×Yj ∈ S(Xi , Yj ) for all i, j ∈ N. Then ϕ ∈ V(X, Y ).

Proof. Let ϕi,j (x, y) = ϕ|Xi ×Yj . By our assumption, ϕi,j ∈ S(Xi , Yj ) and hence, by Theorem 2.5, there exist measurable functions ai,j : X → 2 and bi,j : Y → 2 such that ϕ(x, y) = (ai,j (x), bi,j (y)) for almost all (x, y) ∈ Xi × Yj and def

αi,j = sup aij (x) 2 sup bij (y) 2 < ∞. x∈Xi

y∈Yj

We may clearly assume that supx∈Xi aij (x) 2 = supy∈Yj bij (y) 2 . Let H = ⊕i,j Hi,j , where Hi,j = 2 for all i, j ∈ N. Considering ai,j (x) as a vector in Hi,j , we deﬁne a function a : X → H in the following way: if x ∈ Xk then set √ a(x) = ⊕i,j ξi,j (x), where ξk,j (x) = ak,j (x)/j αk,j and ξi,j (x) = 0 for i = k. Similarly, we deﬁne b : Y → H by letting, for y ∈ Yl , b(y) = ⊕i,j ηi,j (y), √ where ηi,l (y) = bi,l (y)/i αi,l and ηi,j (y) = 0 if j = l. Then for each i and

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x ∈ Xi we have a(x) 2H =

∞ ai,j (x) 2 j=1

j 2 αi,j

2

≤ C,

∞ 1 2 where C = j=1 j 2 . Similarly, we see that b(y) H ≤ C. Moreover, for x ∈ Xi and y ∈ Yj , we have that (a(x), b(y))H =

(ai,j (x), bi,j (y))Hi,j ijαi,j

and therefore ϕ(x, y)χXi ×Yj = ijαi,j (a(x), b(y))H , for almost all (x, y) ∈ Xi × Yj . ∞ The next step is to see that there exist families {pi }∞ i=1 and {qj }j=1 of vectors in 2 such that αi,j = (pi , qj )2 , i, j ∈ N. We note ﬁrst that |αi,j | ≤ α = (si , rj )2 , where rj = ej , ci cj , where ci = max{1, |αk,l | : k, l ≤ i} and jci,j i cj αi,j si = j jci cj ej and {ej }∞ is the standard basis of 2 . Observe that since j=1 |αi,j |2 1 2 j j 2 c2 c2 ≤ j j 2 < ∞, we have that si ∈ . Setting pi = ci si and qj = i j

jcj rj , we obtain αi,j = (pi , qj )2 . Now let p(x) = ipi if x ∈ Xi and q(y) = jqj if y ∈ Yj . Then ϕ(x, y) = (p(x), q(y))2 (a(x), b(y))H = (p(x) ⊗ a(x), q(y) ⊗ b(y))2 ⊗H for almost all (x, y) ∈ X × Y .

The following theorem is the main result of the present section. Theorem 3.6. Let ϕ ∈ B(X × Y ). The following are equivalent: (i) ϕ is a local Schur multiplier; (ii) ϕ ∈ V(X, Y ). Proof. (i)⇒(ii) Let {κm }∞ m=1 be a covering family of rectangles from the definition of a local Schur multiplier. By Lemma 3.4, there exist families {Xi }∞ i=1 and {Yj }∞ j=1 of pairwise disjoint measurable sets of X and Y , respectively, whose unions have full measure and each rectangle Xi × Yj is contained in a ﬁnite union of sets of the form κm . Since ϕ|κm ∈ S(κm ) for all m ∈ N, it follows from Lemma 2.4 that ϕ|Xi ×Yj ∈ S(Xi × Yj ). An application of Lemma 3.5 implies (ii). (ii)⇒(i) follows from Lemma 3.3. Let E be the class of all rectangles α × β such that ϕ|α×β is a Schur multiplier. Let κϕ be the complement of the ω-union of E. Then κϕ is the smallest ω-closed set with the property that ϕ is a local Schur multiplier on each rectangle disjoint from it. We call κϕ the set of LM-singularity of ϕ (LM is for “local multiplier”). It may be considered as a measure of how far ϕ is from being a Schur multiplier. In particular, we say that ϕ is extremely non-Schur multiplier if κϕ = X × Y . In Sect. 7 we will give an example of an ω-continuous function which is extremely non-Schur multiplier.

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4. w*-Closable Multipliers We now introduce two classes of functions which, along with local Schur multipliers introduced in the previous section, are the main objects of study in the paper. We recall that (X, μ) and (Y, ν) are ﬁxed standard measure spaces, H1 = L2 (X, μ) and H2 = L2 (Y, ν). Definition 4.1. A function ϕ ∈ B(X × Y ) is called a w*-closable (resp. closable) multiplier if the map Sϕ is w*-closable (resp. closable), when viewed as a densely deﬁned linear operator on K(H1 , H2 ). For the sake of brevity, we will sometimes call a function w*-closable (resp. closable) if it is a w*-closable (resp. closable) multiplier. We recall that we denote by Sw∗ (X, Y ) (resp. Scl (X, Y )) the set of all w*-closable (resp. closable) multipliers. The operator Sϕ∗ acting on C1 (H2 , H1 ) can be easily described. Recall that the map k → Ik establishes an identiﬁcation of Γ(X, Y ) with C1 (H2 , H1 ) and that for f ∈ B(X × Y ), we write f ∈μ×ν Γ(X, Y ) if f is μ × ν-equivalent to a function in Γ(X, Y ). Lemma 4.2.

(i) We have that

D(Sϕ∗ ) = {Ih : h ∈ Γ(X, Y ) and ϕh ∈μ×ν Γ(X, Y )}. In particular, D(Sϕ∗ ) is a bimodule. (ii) For every Ih ∈ D(Sϕ∗ ), we have Sϕ∗ (Ih ) = Iϕh . Proof. (i) For every k ∈ L2 (X × Y ) and h ∈ Γ(X, Y ), we have Ik , Ih = khd(μ × ν). It follows that a function h ∈ B(X × Y ) is equivalent to a function in Γ(X, Y ) if and only if there exists C > 0 such that khd(μ × ν) ≤ C Ik op , for all k ∈ L2 (X × Y ). We now have Ih ∈ D(Sϕ∗ ) ⇔ | Sϕ (Ik ), Ih | ≤ C Ik op for all k ∈ D(Sϕ ) ⇔ ϕkhd(μ × ν) ≤ C Ik op for all k ∈ D(Sϕ ) ⇔ ϕh ∈μ×ν Γ(X, Y ), since D(Sϕ ) is dense in L2 (X × Y ). (ii) is immediate from (i).

Lemma 4.3. Let ϕ ∈ B(X × Y ). The following are equivalent: (i) ϕ is a w*-closable multiplier; (ii) there exists a covering family {κm }m∈N of rectangles and functions sm , tm ∈ Γ(X, Y ) such that sm (x, y) = 0 m.a.e. on κm and ϕ(x, y) =

tm (x, y) , sm (x, y)

a.e. on κm , m ∈ N.

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Proof. (i) ⇒ (ii) If ϕ is a w*-closable multiplier then, by Proposition 2.6, the subspace U = D(Sϕ∗ ) is norm dense in C1 (H2 , H1 ). By Lemma 2.3, there is ∞ ∞ −1 a sequence {hn }∞ n=1 ⊆ Γ(X, Y ) with {Ihn }n=1 ⊆ U such that ∩n=1 hn (0) ∅. ∞ −1 Hence, ∪n=1 hn (C\{0}) X × Y . Since all hn are ω-continuous, the sets h−1 n (C\{0}) are ω-open whence we may assume that they are countable unions of rectangles. We conclude that X × Y is marginally equivalent to a countable union of rectangles κm , m ∈ N, such that for every m ∈ N there exists a function sm ∈ U with sm (x, y) = 0 on κm . By Lemma 4.2, there exists a function tm ∈ Γ(κm ) such that tm ∼ ϕsm . (x,y) almost everywhere on κm . Hence ϕ(x, y) = stm m (x,y) (ii) ⇒ (i) Since ϕsm ∼ tm and tm ∈ Γ(X, Y ), Lemma 4.2 implies that −1 ∗ Ism ∈ D(Sϕ∗ ). Since ∩∞ m=1 sm (0) ∅, the space D(Sϕ ) is norm dense in C1 (H2 , H1 ) by Lemma 2.3. By Proposition 2.6, Sϕ is w*-closable. The following characterisation of w*-closable multipliers is the main result of this section. Theorem 4.4. A function ϕ ∈ B(X × Y ) is a w*-closable multiplier if and only if there exist functions t, s ∈ V(X, Y ) such that s(x, y) = 0 marginally t(x,y) , almost everywhere on almost everywhere on X × Y and ϕ(x, y) = s(x,y) X ×Y. Proof. Let ϕ ∈ B(X × Y ) be a w*-closable multiplier. By Lemma 4.3, there exists a covering family {κm }m∈N of rectangles such that ϕ(x, y) =

tm (x, y) , sm (x, y)

a.e. on κm ,

for some sm , tm ∈ Γ(X, Y ) with sm (x, y) = 0 m.a.e. on κm . Using Lemma 3.3 and the inclusion Γ(X, Y ) ⊆ V(X, Y ) we may, if necessary, partition the sets κm into smaller rectangles and assume that the functions tm and sm belong to S(κm ). ∞ By Lemma 3.4, there exist families {Xk }∞ k=1 and {Yl }l=1 of pairwise disjoint measurable subsets of X and Y , respectively, such that X = ∪∞ k=1 Xk , Y = ∪∞ l=1 Yl and each Xk × Yl is contained in a ﬁnite union of rectangles of the form κm . We show that on each rectangle Xk × Yl the function ϕ can t (x,y) where tk,l , sk,l ∈ Γ(Xk , Yl ) and be written in the form ϕ(x, y) = sk,l k,l (x,y) sk,l (x, y) = 0 marginally almost everywhere on Xk × Yl . Indeed, Xk × Yl is the union of a ﬁnite number of pairwise disjoint rectangles Z × W each of which is the intersections of some rectangles of the form κm and Xk × Yl . Fix (x, y) ∈ Z × W . On Z × W the function ϕ can be written in the form st00 , where t0 , s0 ∈ Γ(X, Y ). We set tk,l (x, y) = t0 (x, y) and sk,l (x, y) = s0 (x, y). Now let us deﬁne functions s and t on X×Y by setting t(x, y) = tk,l (x, y) and s(x, y) = sk,l (x, y) if (x, y) ∈ Xk × Yl . By their definition and Theorem 3.6, s and t belong to V(X, Y ). Conversely, suppose that ϕ ∼ t/s for some functions t, s ∈ V(X, Y ) with s(x, y) = 0 for every (x, y) ∈ X × Y . By Lemma 3.3, X × Y can be decomposed into a countable union of rectangles on each of which t is a

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Schur multiplier. Applying the same lemma to each of these rectangles, we decompose X ×Y into the union of rectangles κm on each of which both t and s are Schur multipliers. By the σ-ﬁniteness of the measure spaces, we may moreover assume that (μ × ν)(κm ) < ∞ for each m ∈ N. It follows that s|κm and t|κm are equivalent to functions from Γ(κm ). An application of Lemma 4.3 now implies that ϕ is a w*-closable multiplier. Corollary 4.5. The set Sw∗ (X, Y ) of all w*-closable multipliers is a subalgebra of B(X × Y ) which contains Sloc (X, Y ). Moreover, every w*-closable multiplier ϕ is equivalent to an ω-continuous function. Proof. The fact that the collection of all w*-closable multipliers is an algebra follows from Theorem 4.4 and Proposition 3.2. Theorems 3.6 and 4.4 imply that every local multiplier is w*-closable. Let ϕ ∈ B(X × Y ) be a w*-closable multiplier. By Theorem 4.3, ϕ = st almost everywhere on X × Y , where s, t ∈ V(X, Y ). By Theorem 3.6, t and s are local multipliers hence they are ω-continuous by Proposition 3.2. It is easy to see that if f is an ω-continuous function and g : C → C is continuous on an open set containing f (X × Y ) then g ◦ f is ω-continuous. Hence 1s is ω-continuous, and since ω-continuous functions form an algebra, t s is ω-continuous. ∗

Let κw ϕ ⊆ X × Y be the complement of the ω-union of the family of all rectangles α × β such that ϕ|α×β ∈ Sw∗ (α, β). The next proposition will be useful for us in the subsequent sections. ∗

∗ Proposition 4.6. Let ϕ ∈ B(X × Y ). Then κw ϕ = null D(Sϕ ).

Proof. It follows from Lemma 2.3 that ϕ is a w*-closable multiplier if and only if null D(Sϕ∗ ) = ∅. Applying this to an arbitrary rectangle α×β ⊆ X ×Y ∗ together with the observation that null D(Sϕ| ) = (α × β) ∩ null D(Sϕ∗ ), we α×β obtain that α × β has a marginally null intersection with null D(Sϕ∗ ) if and only if ϕ|α×β is a w*-closable multiplier. This implies our statement. ∗

It follows from Corollary 4.5 that κw ϕ ⊆ κϕ . It is natural to call the ∗ X × Y extremely non-w∗ -closable functions ϕ ∈ B(X × Y ) for which κw ϕ multipliers. We have that every extremely non-w*-closable multiplier is an extremely non-Schur multiplier.

5. Closable Multipliers In this section we study the class Scl (X, Y ) of closable multipliers. Let ϕ ∈ B(X × Y ). Recall that the transformation Sϕ is deﬁned on the linear manifold D(ϕ) = {h ∈ L2 (X × Y ) : ϕh ∈ L2 (X × Y )} by letting Sϕ h = ϕh and, after identifying L2 (X × Y ) with C2 (H1 , H2 ), is considered as a densely deﬁned operator on the space K(H1 , H2 ) of compact operators from H1 into H2 . The dual space of K(H1 , H2 ) is the space C1 (H2 , H1 ) of nuclear operators; we identify it with Γ(X, Y ), and, by Lemma 4.2, the domain of the adjoint def

operator is D∗ (ϕ) = D(Sϕ∗ ) = {h ∈ Γ(X, Y ) : ϕh ∈μ×ν Γ(X, Y )}. It follows

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from Proposition 2.6 that ϕ ∈ Scl (X, Y ) if and only if D∗ (ϕ) is weak* dense in Γ(X, Y ). Equivalently, ϕ ∈ Scl (X, Y ) if and only if D∗ (ϕ)⊥ = 0, where D∗ (ϕ)⊥ is the set of all compact operators K such that K, h = 0 for all h ∈ D∗ (ϕ). Note that D∗ (ϕ) is a sub-bimodule of the bimodule Γ(X, Y ) over the algebras L∞ (X, μ) and L∞ (Y, ν). Let D ⊆ Γ(X, Y ) be any bimodule. Then, for all measurable sets α ⊆ X, β ⊆ Y and all h ∈ D, the function χα (x)χβ (y)h(x, y) belongs to D. One can choose the sets α and β in such a way that this function is a Schur multiplier. Indeed, if h(x, y) = (a(x), b(y)) for some square integrable Hilbert space valued functions a and b, then it sufﬁces to set α = {x : a(x) ≤ N } and β = {y : b(y) ≤ N }, for some N > 0. Letting N tend to inﬁnity, we moreover see that D ∩ S(X, Y ) is norm dense in D. We will need the following proposition. Proposition 5.1. Let D1 , D2 ⊆ Γ(X, Y ) be weak* dense bimodules, invariant under S(X, Y ). Then D1 ∩ D2 is weak* dense. Proof. We identify the predual of Γ(X, Y ) with K(H1 , H2 ). Let K ∈ (D1 ∩ D2 )⊥ and θi ∈ Di ∩ S(X, Y ), i = 1, 2. By the invariance of D1 and D2 under S(X, Y ), we have that θ1 θ2 ∈ D1 ∩ D2 . Thus, K, θ1 θ2 = 0 and therefore

Sθ1 (K), θ2 = 0 for all θ2 ∈ D2 ∩ S(X, Y ). Since D2 ∩ S(X, Y ) is dense in D2 and Sθ1 (K) is a compact operator, we have that Sθ1 (K) = 0. Thus,

K, θ1 = 0 for all θ1 ∈ D1 ∩ S(X, Y ) and hence K = 0. Theorem 5.2. Scl (X, Y ) is a subalgebra of B(X × Y ). Proof. Let ϕ1 and ϕ2 be closable multipliers. By Theorem 2.5 and Lemma 4.2 (i), the bimodules D∗ (ϕ1 ) and D∗ (ϕ2 ) are invariant under S(X, Y ); moreover, D∗ (ϕ1 ) ∩ D∗ (ϕ2 ) ⊆ D∗ (ϕ1 + ϕ2 ). Propositions 5.1 and 2.6 imply that ϕ1 + ϕ2 is closable. To verify that Scl (X, Y ) is closed under products, it sufﬁces now to show that if ϕ is closable then ϕ2 is closable. Let D = D∗ (ϕ) = {h ∈ Γ(X, Y ) : ϕh ∈μ×ν Γ(X, Y )} and D0 = {h ∈ S(X, Y ) ∩ Γ(X, Y ) : ϕh ∈μ×ν S(X, Y ) ∩ Γ(X, Y )}. Then D0 is dense in D and hence in Γ(X, Y ). The product of a Schur multiplier and a closable multiplier is closable (indeed, if w ∈ S(X, Y ), then D∗ (ϕ) ⊆ D∗ (wϕ) whence D∗ (wϕ) is dense). It def

follows that if h ∈ D0 then ψ = ϕ2 h = ϕ(ϕh) is closable. Fix h ∈ D0 and let k ∈ D∗ (ψ). Then hk ∈ D∗ (ϕ2 ). Hence, if K ∈ D∗ (ϕ2 )⊥ then 0 = K, hk = Sh (K), k. Since D∗ (ψ) is dense, we have that Sh (K) = 0 and K, h = 0. Since D0 is dense, K = 0. Thus D∗ (ϕ2 ) is dense and ϕ2 is closable. Following the analogy with harmonic analysis initiated in [1] and later pursued in [12], let us call an ω-closed set E ⊆ X × Y an operator M -set (respectively, operator M1 -set) if E supports a non-zero compact operator (resp. Mmin (E) contains a non-zero compact operator). Clearly, every operator M1 -set is an operator M -set. We shall show in Sect. 7 that there exist operator M -sets which are not operator M1 -sets. We will shortly see that the property of being or not being an operator M - (resp. M1 -) set is important for

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deciding whether a given function is a closable multiplier. We hence include a consequence of Proposition 5.1 concerning sets which are not operator M or M1 -sets. Proposition 5.3. Let E1 , E2 ⊆ X × Y be ω-closed sets. Suppose that E1 and E2 are not operator M -sets (resp. not operator M1 -sets). Then E1 ∪ E2 is not an operator M -set (resp. not an operator M1 -set). Proof. Suppose that E1 and E2 are not operator M1 -sets. Setting Di = Mmin (Ei )⊥ , we have that Di is a weak* dense sub-bimodule of Γ(X, Y ), i = 1, 2. Note that, by [28], Di = {ψ ∈ Γ(X, Y ) : ψχEi = 0 m.a.e.}, i = 1, 2. It follows that Di is invariant under S(X, Y ), i = 1, 2, and that D1 ∩ D2 = {ψ ∈ Γ(X, Y ) : ψχE1 ∪E2 = 0 m.a.e.}. By [28] again, (D1 ∩D2 )⊥ = Mmin (E1 ∪E2 ). By Proposition 5.1, (D1 ∩D2 )⊥ ∩ K(H1 , H2 ) = {0} and hence E1 ∪ E2 is not an operator M1 -set. Now suppose that E1 and E2 are not operator M -sets. Let Di = {ψ ∈ Γ(X, Y ) : ψ vanishes on an ω-open neighbourhood of Ei }, i = 1, 2. By [28], Di⊥ = Mmax (Ei ), i = 1, 2. It is clear that D1 and D2 are invariant under S(X, Y ) and, since E1 and E2 are not operator M -sets, D1 and D2 are weak* dense in Γ(X, Y ). By Proposition 5.1, D1 ∩ D2 is weak* dense in Γ(X, Y ). However, D1 ∩ D2 equals {ψ ∈ Γ(X, Y ) : ψ vanishes on an ω-open neighbourhood of E1 ∪ E2 } and hence (D1 ∩ D2 )⊥ = Mmax (E1 ∪ E2 ). Thus, E1 ∪ E2 is not an operator M -set. In the next theorem, we relate the notions of operator M - and operator M1 -sets to closability of multipliers. Theorem 5.4. Let ϕ ∈ B(X × Y ). ∗ (i) If κw ϕ is not an operator M -set then ϕ is a closable multiplier. ∗ (ii) If κw ϕ is an operator M1 -set then ϕ is not a closable multiplier. Proof. It follows from Proposition 2.6 that ϕ is closable if and only if D(Sϕ∗ )⊥ ∩ K(H1 , H2 ) = {0}. By [28] and Proposition 4.6 we have ∗

∗

∗ ⊥ w Mmin (κw ϕ ) ⊆ D(Sϕ ) ⊆ Mmax (κϕ )

which clearly implies the statement.

Corollary 5.5. (i) If E is not an operator M -set and if, for every marginally disjoint from E rectangle α × β, the restriction ϕ|α×β is a w*-closable multiplier, then ϕ is a closable multiplier. ∗ (ii) If (μ × ν)(κw ϕ ) = 0 then ϕ is not a closable multiplier. ∗

∗

∗

w w Proof. (i) By the definition of κw ϕ , we have that κϕ ⊆ E, whence κϕ is not an operator M -set. The claim now follows from Theorem 5.4 (i). (ii) Note that any set E of non-zero measure is an operator M1 -set, because it supports a non-trivial Hilbert-Schmidt operator, and all such oper ators belong to Mmin (E) [1]. So it sufﬁces to apply Theorem 5.4 (ii).

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∗

Remark 5.6. (i) If the set κw ϕ is synthetic then ϕ is a closable multiplier if ∗ ) does not contain a non-zero compact operator. and only if Mmax (κw ϕ w∗ (ii) Since κϕ ⊆ κϕ , we obtain that for the closability of ϕ it sufﬁces to show that κϕ does not support non-zero compact operators. ∗ (iii) In Sect. 7 we shall construct a non-closable multiplier ϕ such that κw ϕ is an operator M -set but not an operator M1 -set. Example 5.7. Let E ⊆ X ×Y be ω-closed and let ∂E be its ω-boundary (that is, ∂E = E\E0 , where E0 is the largest, up to marginal equivalence, ω-open set contained in E [10]). If ϕ = χE then for each rectangle α × β marginally contained either in E or in E c , we have that ϕ|α×β is a Schur multiplier and ∗ hence κw ϕ is marginally contained in ∂E. If ∂E is not an operator M -set then, by Theorem 5.4, χE is a closable multiplier. We now present our ﬁrst example of a non-closable multiplier, using a result on spectral (non)-synthesis. Example 5.8. Let U be the bilateral shift acting on the space 2 (Z), that is, the operator given by U en = en+1 , n ∈ Z, where {en }n∈Z is the standard basis of 2 (Z). Fix p > 2. By [29, Proposition 9.9], there exist sequences {an }n∈Z , {bn }n∈Z ∈ 2 (Z) with |an | = |bn |, and an operator X ∈ Cp (2 (Z)) such that (an U n )X(bn U −n ) = 0 and (an U n )∗ X(bn U −n )∗ = 0. n∈Z

n∈Z

Let W : 2 (Z) → L2 (T) be the inverse Fourier transform. Then W U W ∗ is the operator of multiplication by eit and T = W XW ∗ is an operator in Cp (L2 (T)) satisfying Mfn T Mgn = 0 and Mf n T Mgn = 0, n∈Z

n∈Z

where Mfn and Mgn are the multiplication operators by the functions fn and gn given by fn (t) = an eint and gn (t) = bn e−int , respectively. Set dn = an bn and note that {dn } ∈ 1 (Z). Let ψ(t, s) = n∈Z fn (t)gn (s) = n∈Z dn ein(t−s) . As n∈Z |fn (t)|2 = n∈Z |gn (s)|2 = n∈Z |dn | < ∞ for all s, t ∈ T, Theorem 2.5 shows that the function ψ is a Schur multiplier on T × T (equipped with the product Lebesgue measure). Let ψ(t,s) if ψ(t, s) = 0 ϕ(t, s) = ψ(t,s) 0 otherwise. 2 We claim that ϕ is not closable. To see this, assume that {Tn }∞ n=1 ⊆ C2 (L (T)) is a sequence with Tn → T in the operator norm. Then Sψ (Tn ) → Sψ (T ) = Mfn T Mgn = 0. n∈Z

However, Sϕ (Sψ (Tn )) = Sψ (Tn ) → Sψ (T ) =

n∈Z

Mf n T Mgn = 0.

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Example 5.8 will be considerably strengthened later: in Proposition 7.12, we will construct an ω-continuous function which is a non-closable multiplier. On the other hand, the above example has the advantage that it exhibits a multiplier which is not closable in Cp , for each p > 2. Let [0, 1] be the unit interval equipped with the Lebesgue measure, let Δ = {(x, y) ∈ [0, 1]×[0, 1] : x < y} and ϕ = χΔ be the characteristic function of Δ. The multiplier Sϕ is usually called the transformer of triangular truncation (see for example [14]). The following result extends the well-known fact that Sϕ is not a Schur multiplier. Proposition 5.9. The transformer of triangular truncation is closable but not w*-closable. Proof. We ﬁrst show that ϕ is closable. Let Λ = {(x, x) : x ∈ [0, 1]} be the diagonal of the unit square. The set Λ only supports operators of multiplication by functions in L∞ (0, 1); in particular, it is not an operator M -set. Since the function ϕ is constant on each rectangle marginally disjoint from Λ, the claim follows from Corollary 5.5 (i). To show that ϕ is not w*-closable, it sufﬁces, by Corollary 4.5, to show that ϕ is not equivalent to an ω-continuous function. Assume, towards a contradiction, that there exists an ω-continuous function f such that f = ϕ almost everywhere. By Lemma 2.2, f = 0 m.a.e. on Δ and f = 1 m.a.e. on Δ. Note that if a rectangle is marginally disjoint from Δ or Δ then it is marginally disjoint from Λ. It follows that the same is true for any ω-open set. Since f −1 (C\{1}) is marginally disjoint from Δ, we obtain that f = 1 m.a.e on Λ. Similarly f = 0 m.a.e. on Λ. This is a contradiction because Λ is not marginally null. Remark. The proof of Proposition 5.9 implies the following more general statement: Let Δ1 and Δ2 be disjoint ω-open sets and Λ = (Δ1 ∪ Δ2 )c be such that (a) Λ does not support a non-zero compact operator, and (b) for every rectangle κ, κ ∩ Λ ∅ implies that κ ∩ Δi ∅, i = 1, 2. Then χΔ1 is closable but not w*-closable. Example 5.10. Let E ⊆ X ×Y be an ω-closed set such that E\∂E ∅, where ∗ ∂E is the ω-boundary of E, and let ϕ = χE . Then kϕw = null D(Sϕ∗ ) = ∂E. In fact, if a rectangle κ is such that κ ∩ ∂E ∅ then, by the proof of Proposition 5.9, ϕ|κ is not ω-continuous and hence not a w*-closable multi∗ plier, giving that κ is not marginally disjoint from κw ϕ . As ∂E marginally ∗ ∗ w contains κw ϕ (see Example 5.7), we obtain κϕ ∂E. Proposition 5.9 shows that there exist closable multipliers which are not ω-continuous. But they are continuous in the stronger pseudo-topology, τ , introduced in Sect. 2.1. Proposition 5.11. Any closable multiplier is τ -continuous. Proof. Let ϕ ∈ Scl (X, Y ). If U ⊆ C is an open set then ∗

∗

−1 c f −1 (U ) = (f −1 (U ) ∩ κw (U ) ∩ (κw ϕ ) ∪ ((f ϕ ) ).

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∗

w c Since κw ϕ is ω-closed, (κϕ ) is marginally equivalent to a countable ∞ union ∪i=1 αi × βi of rectangles. Moreover, for each i, f |αi ×βi is w*-closable and hence ω-continuous. This implies that f −1 (U ) ∩ (αi × βi ) is marginally equivalent to a countable union of rectangles and hence the same is true for ∗ ∗ c −1 (U ) ∩ κw f −1 (U ) ∩ (κw ϕ ) . It remains to note that (μ × ν)((f ϕ ) = 0 because, ∗ by Corollary 5.5, (μ × ν)(κw ϕ ) = 0.

Remark 5.12. We note that the class of τ -continuous functions is strictly larger than that of closable multipliers; see Proposition 7.12.

6. Divided Differences Let f be a continuous function on a ﬁnite or inﬁnite open subinterval J ⊆ R. The divided difference of f is the function f (x) − f (y) fˇ(x, y) = x−y deﬁned on J × J\Λ, where Λ = {(x, x) : x ∈ R}. Let μ be a regular measure on R whose support contains J. In what follows we will assume that μ is non-atomic and hence f is deﬁned almost everywhere on J × J. The property of fˇ being a Schur multiplier is closely related to a kind of “operator smoothness” of f . Recall that f is called operator Lipschitz (OL) on a subset K ⊆ J if there exists a constant D > 0 such that f (A) − f (B) ≤ D A − B for all selfadjoint operators A, B with spectra in K. The smallest constant D with this property will be denoted by |f |OL . Let O(f ) be the union of all open subintervals I ⊆ J on which f is OL. It is an open subset of J. Its complement will be denoted by E(f ). Lemma 6.1. Let I be a compact subset of J. A function fˇ is a Schur multiplier on I × I if and only if f is OL on I. Proof. If fˇ is a Schur multiplier then, for h1 (x, y) = (x − y)h(x, y), we have Ifˇh1 ≤ C Ih1

(1)

and hence f (A)X − Xf (A) ≤ C AX − XA ,

(2)

2

where A is the operator of multiplication by x on L (I, μ) and X = Ih . By [22, Remark 2.1, Corollary 3.6] and [23, Theorem 3.4], f is OL. Conversely, if f is OL, then (2) holds for each X ∈ B(L2 (I, μ)) by [22, Corollary 3.6]. This implies (1) for L2 -functions of the form h1 (x, y) = (x − y)h(x, y), where h ∈ L2 (I × I, μ × μ). Since functions of this form are dense in L2 (I ×I, μ×μ), and since the L2 -norm majorizes the operator norm, inequality (1) holds for all h1 ∈ L2 (I × I, μ × μ). This means that fˇ is a Schur multiplier. Lemma 6.2. If I1 , I2 are compact intervals and I1 ∩ I2 = ∅ then fˇ|I1 ×I2 ∈ S(I1 , I2 ).

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1 |I1 ×I2 ∈ Proof. Since f (x) − f (y) ∈ S(I1 , I2 ), it sufﬁces to show that x−y S(I1 , I2 ). Without loss of generality we may assume that I1 = [0, a], I2 = [b, c] with b > a. We have ∞ xn 1 =− , (x, y) ∈ I1 × I2 . x−y y n+1 n=0 n an Since yxn+1 ≤ bn+1 , the series converges in S(I1 , I2 ) in norm. S(I1 ,I2 )

The following theorem gives a precise description of the set of LM-singularity for a divided difference. Theorem 6.3. For every continuous function f , we have κfˇ {(x, x) : x ∈ E(f )}. Proof. Write O(f ) as the union of a sequence of disjoint open intervals: O(f ) = ∪∞ n=1 Jn . For each n, Jn × Jn is the union of rectangles Ik × Ik , where Ik are compact subintervals of Jn . Since, by Lemma 6.1, fˇ|Ik ×Ik ∈ S(Ik , Ik ), it follows that fˇ|Jn ×Jn ∈ Sloc (Jn , Jn ). Thus (Jn × Jn ) ∩ κfˇ ∅. Furthermore, κfˇ ⊆ Λ by Lemma 6.2. It follows that, up to a marginally null set, we have κfˇ ⊆ Λ\(∪∞ n=1 Jn × Jn ) = {(x, x) : x ∈ E(f )}. To prove the converse inclusion, it sufﬁces to show by the regularity of μ that if I1 and I2 are compact subsets of J such that fˇ|I1 ×I2 ∈ S(I1 , I2 ) then E(f ) ∩ I1 ∩ I2 = ∅; indeed, we would then have (I1 × I2 ) ∩ {(x, x) : x ∈ E(f )} = {(x, x) : x ∈ E(f ) ∩ I1 ∩ I2 } ∅. Let I = I1 ∩ I2 . By Lemma 2.4 (i), fˇ|I×I ∈ S(I, I), and Lemma 6.1 implies that f is OL on I; therefore I ⊆ O(f ) and hence I ∩ E(f ) = ∅. Corollary 6.4. fˇ is a local Schur multiplier if and only if μ(E(f )) = 0. It is known [21] that the class of all continuous Schur multipliers on X × Y , where X, Y are compact Hausdorﬀ spaces, very weakly depends on the choice of Borel measures on X and Y : it depends only on the support of a measure. The above corollary shows that the class of continuous local Schur multipliers essentially depends on the choice of a measure. Indeed, a change of the measure does not change the set E(f ) while the condition μ(E(f )) = 0 need not be preserved. Corollary 6.5. For each f , the function fˇ is a closable Schur multiplier. Proof. By Theorem 6.3, κfˇ ⊆ {(x, x) : x ∈ J}. Since the diagonal {(x, x) : x ∈ J} does not support a compact operator, it follows from Theorem 5.4 that fˇ is not closable. Proposition 6.6. There exists a function f : [0, 1] → C such that fˇ is a Schur multiplier, fˇ = 0 almost everywhere and 1/fˇ is not a local Schur multiplier. Proof. Let M be a Cantor-like set of non-zero Lebesgue measure (see [18]) and let g be a continuously differentiable function which is equal to zero on M and positive otherwise. Let f be its primitive function: f = g. Then

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f ∈ C 2 ([0, 1]) and hence it is operator Lipschitz [9]; by Lemma 6.1, fˇ is a Schur multiplier. Since f is strictly monotone, fˇ = 0 almost everywhere. The function 1/fˇ, which is deﬁned almost everywhere, is not a local Schur multiplier. In fact, assuming the converse, given > 0, we can ﬁnd subsets X , Y of [0, 1] such that m([0, 1]\X ) < , m([0, 1]\Y ) < and (fˇ)−1 |X ×Y is a Schur multiplier. Hence (fˇ)−1 is equivalent to an essentially bounded function on X × Y . But this is impossible since by construction (fˇ)−1 (x, y) is arbitrary large for (x, y) close to (x, x), x ∈ M and since m(M ) > 0, the set {(x, y) ∈ X ×Y : |(fˇ(x, y))−1 | > C} has positive measure for all C > 0 and sufﬁciently small > 0. The divided difference fˇ can be extended to a continuous function on J × J if and only if f is continuously differentiable. Our next aim is to construct a continuously differentiable function f such that fˇ is not a Schur multiplier on each rectangle with non marginally null intersection with Λ. For this we need an extension of the well-known result of Farforovskaya [11] (see also Peller [26]) which states that a continuously differentiable function on a compact interval need not be OL. Theorem 6.7. There is a function in C 1 ([0, 1]) which is not OL on each subinterval of [0, 1]. Proof. By [11], there exists f ∈ C 1 ([0, 1]) which is not operator Lipschitz on [0, 1]. Such function f can be chosen so that f (0) = f (0) = f (1) = f (1) = 0.

(3)

To see this it sufﬁces to choose a continuously differentiable non OL function g on a subinterval I ⊆ (0, 1) and extend it to a continuously differentiable function f on [0, 1] satisfying (3). Let us denote by C01 the set of all f ∈ C 1 ([0, 1]) satisfying (3). Let Cs = C ∞ ([0, 1]) ∩ C01 . It is well-known that all functions in C ∞ are OL (in fact, it sufﬁces for f to have a continuous second derivative). We claim that for each C > 0, there exists g ∈ Cs , such that g C 1 = 1 and |g|OL > C. C01 , each funcIndeed suppose that this is not true. Since Cs is dense in ∞ ∞ 1 series n=1 gn , where gn ∈ Cs and n=1 gn C 1 < tion f ∈ C0 is the sum of a ∞ ∞. Our assumption gives n=1 |gn |OL < ∞ which easily implies that |f |OL < ∞, and so f ∈ OL. This is a contradiction because, as we know from [11], C01 is not contained in the set of all Operator Lipschitz functions. Now, by [22], we may state that there exist operators A = A∗ and X such that g(A)X − Xg(A) ≥ C/2 AX − XA . Moreover, by [22], A and X can be chosen to have ﬁnite rank. Clearly, the interval [0, 1] can be replaced by an arbitrary closed interval. Let {In } be a sequence of subintervals of [0, 1] such that each subinterval J ⊆ [0, 1] contains at least one (and hence inﬁnitely many) In . We claim that given operators of ﬁnite rank X1 , . . . , Xn−1 , A1 , . . . , An−1, where A∗i = Ai , i = 1, . . . , n − 1, and a number C > 0, there exist ﬁnite rank

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operators A = A∗ and X, and a smooth function g such that supp g ⊆ In , g C 1 ([0,1]) ≤ 1, g(Aj ) = 0, j = 1, . . . , n − 1, and [g(A), X] ≥ C [A, X] . Indeed, since the spectra of all Aj are ﬁnite, one can ﬁnd a subinterval J of In having empty intersection with ∪n−1 j=1 σ(Aj ). Now by the second paragraph, there exists a smooth function g with support in J, such that g OL > C and g C 1 = 1. By the previous arguments this will imply the existence of operators A and X with the required properties. This allows us to construct sequences of operators {Xn }, {An }, of smooth functions {gn } and of positive constants {Cn } such that (1) gn C 1 ≤ 1; (2) supp gn ⊆ In ; (3) each Xn , An are of ﬁnite rank and An = A∗n ; (4) gn (Aj ) = 0 for j < n; (5) [gn (An ), Xn ] ≥ Cn [An , Xn ] ; n−1 (6) Cn ≥ 2n (n + j=1 2−j |gj |OL ). ∞ Let f (t) = j=1 2−j gj (t) so f ∈ C 1 ([0, 1]). Let us prove that f is not OL on any subinterval J ⊆ [0, 1]. Assume the converse; then there exists J ⊂ [0, 1] and C > 0 such that [f (A), X] ≤ C [A, X] for any X and A = A∗ with σ(A) ⊆ J. By the choice of In , given m > 0 there exists n > m such that In ⊆ J. Therefore ∞ n 2−j gj (An ) = 2−j gj (An ). f (An ) = j=1

j=1

Since [f (An ), Xn ] ≤ C [An , Xn ] , we have 2−n [gn (An ), Xn ] ≤ C [An , Xn ] +

n−1 j=1

⎛ ≤ ⎝C +

n−1

2−j [gj (An ), Xn ] ⎞

2−j |gj |OL ⎠ [An , Xn ] .

j=1

On the other hand, [gn (An ), Xn ] ≥ Cn [An , Xn ] . Hence

⎛ Cn ≤ 2n ⎝C +

n−1

⎞ 2−j |gj |OL ⎠ .

j=1

n−1 From condition (6) on the constant Cn we get 2n (n + j=1 2−j gj OL ) ≤ n−1 2n (C + j=1 2−j gj OL ) and hence n ≤ C for every n ∈ N, a contradiction. Corollary 6.8. There exists f ∈ C 1 ([0, 1]) such that kfˇ = {(x, x) : x ∈ [0, 1]}. Proof. Let f be the function constructed in Theorem 6.7. Then O(f ) = ∅ and E(f ) = [0, 1]. The statement now follows from Theorem 6.3.

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7. Multipliers of Toeplitz Type Let G be a locally compact second countable abelian group and therefore metrisable by [17, 8.3]. Let μ = ds be the left invariant Haar measure on G. We write Lp (G) for Lp (G, μ), p = 1, 2, and denote by Cc (G) the space ˆ be the dual of all continuous functions on G with compact support. Let G group of G and A(G) (resp. B(G)) be the Fourier (resp. the Fourier-Stieltˆ under Fourier jes) algebra of G. We recall that A(G) is the image of L1 (G) transform. It is well-known that A(G) coincides with the family of functions t → G f (s − t)g(s)ds = (λ(t)f, g¯), f , g ∈ L2 (G), where λ(t)f (s) = f (s − t). The algebra B(G) is the image under Fourier transform of the convolution algebra M (G) of all bounded Radon measures on G. One has A(G) ⊆ B(G); equality holds if and only if G is compact. It is moreover known that B(G) coincides with the space of all multipliers A(G) (see [31]). For a subset J ⊆ A(G), its null set is deﬁned by null J = {s ∈ G : f (s) = 0 for all f ∈ J}. Conversely, for a closed subset E of G we denote by I(E) (resp. J(E)) the space of all f ∈ A(G) vanishing on E (resp. the closed hull of the space of all f ∈ A(G) vanishing on a neighborhood of E); we have that I(E) is the largest (resp. the smallest) closed ideal of A(G) whose null set is equal to E (see [31]). Let N be the map sending a measurable function f : G → C to the function N f : G × G → C given by N f (s, t) = f (s − t). The functions of the form N f will be called functions of Toeplitz type. It is well-known (see, for example, [7]) that if f ∈ L∞ (G) then N f is a Schur multiplier with respect to Haar measure if and only if f ∈μ B(G). In this section we show that the algebra of w*-closable multipliers of Toeplitz type coincides with that of local Schur multipliers of Toeplitz type; if G is compact then both spaces coincide with the algebra of Schur multipliers of Toeplitz type, that is, with N A(G). We shall start with a result relating the continuity of a function f on G to the ω-continuity of N f . The following lemma is certainly known but, since we were not able to ﬁnd a precise reference, we include its proof for completeness. Let O(X) denote the set of all open subset of a topological space X. Lemma 7.1. Let X be a topological space and ξ : O(C) → O(X) be a union preserving map such that ξ(∅) = ∅, ξ(C) = X and ξ(U ∩ V ) = ∅ whenever U, V ∈ O(C) and U ∩ V = ∅. Then there exists a continuous function g : X → C such that ξ(U ) = g −1 (U ) for all U ∈ O(C). Proof. For t ∈ X, let O(t) denote the union of all U ∈ O(C) with t ∈ / ξ(U ). Since ξ(C) = X, we have that C\O(t) is non-empty. If it contains at least two points, say λ1 and λ2 , then taking disjoint open sets Ui with λi ∈ Ui , i = 1, 2, we obtain that t ∈ ξ(U1 ) ∩ ξ(U2 ). This contradicts the fact that ξ(U1 ) ∩ ξ(U2 ) is empty. We proved that C\O(t) = {λ}, for some λ ∈ C. Setting g(t) = λ, we obtain a function g : X → C. It follows from its definition that g −1 (U ) = ξ(U ), for every U ∈ O(C). Hence, g is continuous.

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For t ∈ G, we denote by Λt the t-shifted diagonal: Λt = {(x, x − t) : x ∈ G}. We say that a subset E of Λt is non-null in Λt , if m({x : (x, x − t) ∈ E}) > 0. For W ⊆ C × G set π(W ) = {t ∈ G : W ∩ Λt is non-null in Λt }. Clearly π(G × G) = G, π(∅) = ∅ and π(W1 ∪ W2 ) = π(W1 ) ∪ π(W2 ). Lemma 7.2. If W is ω-open, then π(W ) is open. Proof. Let s ∈ π(W ). It follows that there exists a rectangle α × β ⊆ W with non-null (in Λs ) intersection with Λs . By the σ-ﬁniteness of the measure spaces, we may moreover assume that α and β have ﬁnite measure. We now have m(α ∩ (β + s)) > 0. Since the function x → m(α ∩ (β + x)) is continuous (being the convolution of the L2 -functions χα and χβ ), m(α ∩ (β + x)) > 0 for all x in a neighborhood V of s. Hence V ⊆ π(W ). Proposition 7.3. Let f : G → C and ϕ = N f . The function ϕ is equivalent to an ω-continuous function if and only if f is equivalent to a continuous function. Moreover, ϕ is ω-continuous if and only if f is continuous. Proof. If f is continuous then N f is continuous and hence ω-continuous. It follows easily that if f is equivalent to a continuous function then N f is equivalent to a continuous function. We hence show the converse assertions. Let ψ : G×G → C be an ω-continuous function equivalent to N f . Thus, def def Z = {(x, y) ∈ G × G : N f (x, y) = ψ(x, y)} is a null set. Then M = π(Z) is a null subset of G. Let us say that a point t ∈ G is good if t ∈ / M. For U ∈ O(C), set ξ(U ) = π(ψ −1 (U )). It follows from Lemma 7.2 that ξ maps O(C) to O(G). The conditions ξ(C) = G, ξ(∅) = ∅ and ξ(U1 ∪ U2 ) = ξ(U1 ) ∪ ξ(U2 ) follow from the corresponding properties of π. We have to show that ξ sends disjoint sets to disjoint sets. Note that if t is good and t ∈ ξ(U ) then f (t) ∈ U . Indeed, ψ(x, x−t) ∈ U for all x belonging to a certain non-null set, by the definition of ξ(U ). Since t is good, for almost all x ∈ G, the pair (x, x − t) does not belong to Z hence there exists x ∈ G such that ψ(x, x − t) = N f (x, x − t) = f (t). Now if U1 ∩ U2 = ∅ and ξ(U1 ) ∩ ξ(U2 ) = ∅ then ξ(U1 ) ∩ ξ(U2 ) is an open set and must contain a good point t ∈ G. But then f (t) ∈ Ui , 1 = 1, 2, a contradiction. Applying Lemma 7.1 we obtain a continuous function g : G → C with ξ(U ) = g −1 (U ), for all U ∈ O(C). By the above argument, g(t) = f (t) for all good t (indeed for each U containing g(t) we have that t ∈ g −1 (U ) = ξ(U ) whence f (t) ∈ U ). Thus f coincides almost everywhere with the continuous function g. If N f is ω-continuous then all points are good and f (t) = g(t), for all t ∈ G. Let us say that a measurable function f : G → C belongs (resp. almost belongs) to A(G) at a point t ∈ G if there exist a neighborhood U of t and

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a function g ∈ A(G) such that f (s) = g(s) everywhere (resp. almost everywhere with respect to Haar measure μ) on U . If f belongs to A(G) at each t ∈ G then we say that f locally belongs to A(G) and write f ∈ A(G)loc . It is obvious that A(G)loc ⊆ C(G) and, using the regularity of A(G), it is easy to show that if G is compact then A(G)loc = A(G). In general we have the inclusions A(G) ⊆ B(G) ⊆ A(G)loc . If f almost belongs to A(G) at each point t ∈ G, it is not difﬁcult to see that f is equivalent to a function in A(G)loc . We recall that in this case we write f ∈μ A(G)loc . For a measurable function f : G → C, let Jf = {h ∈ A(G) : f h ∈μ A(G)} and Ef = null Jf . Clearly, Jf is an ideal of A(G) whence J(Ef ) ⊆ Jf ⊆ I(Ef ). Lemma 7.4. Let f : G → C be measurable. Then Ef = {t ∈ G : f does not almost belong to A(G) at t}.

(4)

Proof. Let E be the set in the right hand side of (4). If t ∈ E c , then f almost belong to A(G) at t and therefore there exists a neighborhood V of t such that f g is equivalent to a function in A(G) for any g ∈ A(G) with V c ⊆ null {g}. Now if g ∈ A(G) takes the values 1 at t and 0 on V c then / null g. Thus, t ∈ Ef and hence Ef ⊆ E. g ∈ Jf and t ∈ To see the reverse inclusion, let t ∈ E and assume that there exists g ∈ A(G) such that f g ∼ h, h ∈ A(G) and g(t) = 0. Then there exists a neighborhood U of t such that |g(s)| > δ > 0 for all s ∈ U . By the regularity of A(G), we can ﬁnd q ∈ A(G) such that q(s)g(s) = 1 for all s ∈ U ; therefore f (s) = f (s)q(s)g(s) for all s ∈ U . Since f gq ∼ hq on U and hq ∈ A(G), the function f almost belongs to A(G) at t. We obtain a contradiction giving E ⊆ Ef . For notational simplicity we let Γ(G) = Γ(G, G). We shall frequently use the map P : Γ(G) → A(G) given by P (f ⊗ g)(t) = (λ(t)g, f¯). Clearly, P is a surjective contraction. For a subset E ⊆ G, let E ∗ = {(x, y) ∈ G × G : x − y ∈ E}. Theorem 7.5. Assume that G is a subgroup of Rn or Tn , n ∈ N. Let f : G → C be a measurable function, ϕ = N f and U, V ⊆ G be measurable sets. The following are equivalent: (i) (U × V ) ∩ Ef∗ ∅; (ii) ϕ|U ×V ∈ Sloc (U, V ); (iii) ϕ|U ×V ∈ Sw∗ (U, V ). Proof. Set E = Ef and note that (U ×V )∩E ∗ ∅ if and only if (U −V )∩E = ∅ for some U ⊆ U and V ⊆ V with μ(U \U ) = μ(V \V ) = 0. We claim that f ψ ∈μ A(G) for every ψ ∈ A(G) ∩ Cc (G) with supp ψ ⊆ E c . Indeed, by Lemma 7.4, for each t ∈ E c there exists a neighborhood Vt of t and a function gt ∈ A(G) such that f ∼ gt on Vt . Since suppψ is compact there exists a ﬁnite set F ⊆ G such that suppψ ⊂ ∪t∈F Vt . It follows from the regularity of A(G)

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that there exist ht ∈ A(G), t ∈ F , such that t∈F ht (x) = 1 if x ∈ suppψ and hs (x) = 0 if x ∈ / Vs for each s ∈ F (see the proof of [17, Theorem 39.21]). Then for every x ∈ G we have f (x)ψ(x) = f (x)ψ(x)ht (x)

t∈F

and hence f ψ ∼ t∈F gt ψht , giving f ψ ∈μ A(G). (i)⇒(ii) Suppose (U × V ) ∩ E ∗ ∅ and let U ⊆ U and V ⊆ V be measurable subsets such that m(U \U ) = m(V \V ) = 0 and (U − V ) ∩ E = ∅. Since G is second countable and U − V ⊆ E c , m(U \U ) = m(V \V ) = 0, ∞ we may choose increasing sequences {Kn }∞ n=1 and {Ln }n=1 of compact sets ∞ ∞ such that, up to a null set, ∪n=1 Kn = U and ∪n=1 Ln = V , and a compact set Mn such that Kn − Ln ⊆ Mn ⊆ E c . Choose, for each n ∈ N, a function ψn ∈ A(G) ∩ Cc (G) supported in E c and taking value 1 on Mn . By the previous paragraph, f ψn ∈μ A(G) and therefore N (f ψn ) is a Schur multiplier. Thus, for each ξ ∈ Γ(G), we have ϕχKn ×Ln ξ = N (f ψn )χKn ×Ln ξ ∈μ×μ Γ(G). It follows that ϕ|Kn ×Ln is a Schur multiplier and hence ϕ ∈ Sloc (U, V ). (ii)⇒(iii) follows from Corollary 4.5. (iii)⇒(i) We will identify Γ(U, V ) with a subset of Γ(G) in a natural way. Let ψ = ϕ|U ×V . By Proposition 2.6, D(Sψ∗ ) is norm dense in Γ(U, V ). Thus, P (D(Sψ∗ )) is norm dense in P (Γ(U, V )). By Lemma 4.2 (i), D(Sψ∗ ) = {h ∈ Γ(U × V ) : ψh ∈μ×μ Γ(U, V )}. Since f P (h) = P (ϕh) ∈ P (Γ(G)) ⊂ A(G) for every h ∈ D(Sψ∗ ), the set {P (h) : h ∈ Γ(U, V ), f P (h) ∈μ A(G)} is dense in P (Γ(U, V )), and hence P (Γ(U, V )) ⊆ Jf . This implies that E = null Jf ⊆ null P (Γ(U, V )). It sufﬁces to show that there exist subsets U ⊆ U , V ⊆ V such that μ(U \U ) = μ(V \V ) = 0 and (U −V )∩null P (Γ(U, V )) = ∅ since this will imply U − V ⊆ null P (Γ(U, V ))c ⊆ E c and hence (U × V ) ∩ E ∗ ∅. Let U , V be the sets of density points of U and V , respectively. Then, by the Lebesgue density theorem (see [25]), μ(U \U ) = μ(V \V ) = 0. To prove the statement, it sufﬁces to show that P (χU ⊗ χV )(s) = μ(U ∩ (V + s)) > 0 if s ∈ U − V . Let s = u − v, u ∈ U , v ∈ V and assume that μ(U ∩ (V + s)) = μ((U − u) ∩ (V − v)) = 0. If B (0) is the closed ball with centre 0 and radius , we then have μ((U − u) ∩ B (0)) + μ((V − v) ∩ B (0)) ≤ μ(B (0)), for each > 0. Applying the Lebesgue density theorem we obtain μ((U − u) ∩ B (0)) μ((V − v) ∩ B (0)) 2 = 1 + 1 = lim + lim ≤ 1,

→0

→0 μ(B (0)) μ(B (0)) a contradiction.

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Remark 7.6. (i) The condition that G be a subgroup of Rn or Tn is only used to prove the implication (iii)⇒(i), where we appeal to the Lebesgue density theorem. The statement remains true for more general groups (in particular, for Lie groups), for which there is an analog of the Lebesgue theorem (see [8,24]). (ii) Taking U = V = G, we see that ϕ ∈ Sw∗ (G, G) implies Ef = ∅ for any group G, since in this case the arguments in the proof of Theorem 7.5 give Jf = P (Γ(G)) = A(G). We note some consequences of Theorem 7.5. In the next corollary, which ∗ gives a precise description of the sets κϕ and κw ϕ , we assume that G satisﬁes the conditions of Theorem 7.5 (see also Remark 7.6 (i)). Corollary 7.7. Let f : G → C be a measurable function and ϕ = N f . Then ∗ ∗ κϕ κ w ϕ (Ef ) . The following theorem shows that the set of local Schur multipliers and that of w*-closable multipliers coincide in the class of Toeplitz functions. Theorem 7.8. Let G be an arbitrary second countable locally compact abelian group. Let f : G → C be a measurable function and ϕ = N f . The following are equivalent: (i) f ∈μ A(G)loc ; (ii) ϕ is a local Schur multiplier; (iii) ϕ is a w*-closable multiplier. If G is compact then the above statements are equivalent to (iv) ϕ is a Schur multiplier. Proof. We note that f ∈μ A(G)loc if and only if Ef = ∅. The equivalence (i)⇔(ii)⇔(iii) follows from Theorem 7.5 and Remark 7.6(ii). Assume that G is compact. Then (i)⇔(iv) follows from the equality A(G) = A(G)loc and the fact that N f is a Schur multiplier if and only if f ∈μ A(G). Our next result shows that the class of ω-continuous functions is strictly larger than the class of w*-closable multipliers. Corollary 7.9. Let G be a compact abelian group, f ∈ C(G)\A(G) and ϕ = N f . Then ϕ is ω-continuous but not w*-closable. Proof. Since ϕ is the composition of the continuous function g, given by g(s, t) = s − t, and the continuous function f , it is ω-continuous. By Theorem 7.8, ϕ is not a w*-closable multiplier. Example 7.10. Let Δ = {(x, y) ∈ R : x ≤ y}. Then χΔ (x, y) = χ(−∞,0] (x − y) is not a w*-closable multiplier, since χ(−∞,0] does not almost belong to A(R) at x = 0. Remark 7.11. It is known that there exists a function f ∈ B(R) such that f > 0 on R, but 1/f ∈ / B(R) ([13, §32]). Since a function of Toeplitz type w(s, t) = f (s − t), s, t ∈ R, f ∈ C(R), is a Schur multiplier if and only if f ∈

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B(R), we obtain a positive Schur multiplier w such that 1/w is not a Schur multiplier. However, 1/w is a local Schur multiplier, since 1/f ∈ A(R)loc . To see this, we note that f belongs locally to A(R) and hence for each s ∈ R there exists a neighbourhood Vs and g ∈ A(R) such that f = g on Vs . Since g(s) = 0 on Vs , using the regularity of A(R) one can ﬁnd h ∈ A(R) such that hg = 1 on Vs . As h = 1/f on Vs and s is arbitrary, we have 1/f ∈ A(G)loc . Note that, by Proposition 6.6, there exists a Schur multiplier w such that w(s, t) = 0 almost everywhere and 1/w is not a local Schur multiplier. Proposition 7.12. There exists an ω-continuous non-closable extremelynon-Schur multiplier. Proof. Since each continuous function on G × G is ω-continuous with respect to the Haar measure, it sufﬁces to exhibit a continuous function f such that N f is non-closable and κf = G × G. Let G = T. By [20, Chapter II, Theorem 3.4], for any set S ⊆ T of Lebesgue measure zero there exists a function h ∈ C(T) whose Fourier series diverges at every point of S. We can choose S so that its closure is T and take the corresponding f ∈ C(T). Let ϕ = N f . By the Riemann Localisation Lemma, any function which belongs to A(T) at x ∈ T has a convergent Fourier series at x. Thus, T ⊆ Ef and hence Ef = T. ∗ 2 By Corollary 7.7, we have κϕ κw ϕ T and therefore ϕ is extremelly-non-Schur multiplier. Moreover, applying now Proposition 4.6, we obtain null D(Sϕ∗ ) = T2 and hence D(Sϕ∗ ) = {0}, showing that ϕ is non-closable. ˆ G is compact, so that G is discrete. Then A(G) = Now assume { χ∈Γ cχ χ : χ∈Γ |cχ | < ∞}. The space of pseudomeasures P M (G) = ˆ via Fourier transform: F → {Fˆ (χ)}χ∈Γ . A(G)∗ can be identiﬁed with ∞ (G) A pseudomeasure F ∈ P M (G) is called a pseudofunction if Fˆ vanishes at inﬁnity. We recall that P M (G) is an A(G)-module with respect to the operation f F (g) = F (f g), for F ∈ P M (G), f, g ∈ A(G), and that the support supp F of a pseudomeasure F is the set {x ∈ G : f F = 0 whenever f (x) = 0, f ∈ A(G)}. If E is a closed subset of G we let P M (E) (resp. N (E)) denote the space of all pesudomeasures supported in E (resp. the weak* closed hull of the space of all measures μ ∈ M (G) supported in E). Here, by the weak* topology we mean the σ(P M (G), A(G))-topology. Clearly, N (E) ⊆ P M (E). Moreover, P M (E) (resp. N (E)) is the largest (resp. smallest) weak* closed subspace the support of whose every element is in E. Moreover, N (E) = I(E)⊥ and P M (E) = J(E)⊥ . Recall that a closed set E ⊆ G is called an M -set (resp. an M1 -set) if P M (E) (resp. N (E)) contains a non-zero pseudofunction. It is known that there exists an M -set which is not an M1 -set, see [15, Section 4.6]. In what follows we shall give some sufﬁcient conditions for a function of Toeplitz type to be a closable or a non-closable multiplier, based on the above notions. In [12], Froelich studied the question of when a given closed set supports a non-zero compact operator and a non-zero pseudo-integral compact operators. The next result uses ideas from [12]. We will use the fact that the

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restriction of the map N (given by N f (x, y) = f (x − y)) to A(G) takes values def ˆ ⊆ Γ(G). in the Varopoulos algebra V (G) = C(G)⊗C(G) We will need a modiﬁcation of the module action of L1 (G) on V (G) described on on page 365 of [30]. For ψ ∈ Γ(G) and r ∈ G, write r · ψ for the function given by r · ψ(s, t) = ψ(s + r, t + r). If f ∈ C(G), let f · ψ = (r · ψ)f (r)dr, where the integral is understood in the Bochner sense. FolG lowing the arguments in [30], one can show that the action of C(G) on Γ(G) extends to an action of L1 (G) on Γ(G) and that if {fα }α is a bounded approximate identity for L1 (G) then fα · ψ → ψ for every ψ ∈ Γ(G). The following lemma establishes that E is an M1 -set if and only if E ∗ is an operator M1 -set. This justiﬁes the terminology introduced in Sect. 5. Lemma 7.13. Let E ⊆ G be a closed set. The space Mmin (E ∗ ) contains a non-zero compact operator if and only if E is an M1 -set. Proof. Let K be a non-zero compact operator supported on E ∗ . Then there ˆ such that cδ,γ def = (Kγ, δ) = 0. Let F be the pseudomeasure exist γ, δ ∈ G given by ˆ Fˆ (χ) = cγ−δ+χ,χ , χ ∈ G. Since K is compact, Fˆ is a pseudofunction. For each v = χ∈Gˆ aχ χ ∈ I(E) (the sum being absolutely convergent), we have aχ Fˆ (χ) = aχ (Kχ, γ − δ + χ) F (v) = ˆ χ∈G

=

K,

χ

aχ χχ(γ − δ)

= K, v˜ ,

ˆ χ∈G

where ·, · is the duality between B(L2 (G)) and Γ(G) and v˜ is the function given by v˜(s, t) = N v(s, t)(γ − δ)(t). Since N v vanishes on E ∗ and K ∈ Mmin (E ∗ ), we have that F (v) = 0, showing that F is a pseudofunction in N (E). Thus E is an M1 -set. Conversely, assume that E is an M1 -set and let F be a non-zero pseudofunction in N (E). We let K be the operator on L2 (G) deﬁned by Kχ = Fˆ (χ)χ ˆ of L2 (G). Then for v ∈ I(E), we have on the orthonormal basis G

K, N v = F (v) = 0. Suppose that ψ ∈ Γ(G) vanishes marginally almost everywhere on E ∗ . ˆ deﬁne the functions ψ χ and ψ˜χ by For χ ∈ G, ψ χ (s, t) = χ · ψ(s, t) and ψ˜χ (s, t) = χ(s)ψ χ (s, t). We have that ψ χ , ψ˜χ ∈ Γ(G), and ψ˜χ (s+r, t+r) = ψ˜χ (s, t) marginally almost everywhere, for each r ∈ G (see [30, Theorems 3.1 and 4.6]). Therefore, by [30, Proposition 4.5], ψ˜χ ∈ N A(G). Since ψ˜χ vanishes on E ∗ , ψ˜χ = N v for some v vanishing on E. By the previous paragraph, K, ψ˜χ = 0. This implies that K, ψ χ = 0. In fact, if χ ≡ 1, this is trivial; if χ ≡ 1 we have ψ χ (s, t) = χ(−s)ψ˜χ (t, s) = χ(−s)v χ (s − t) for some v χ ∈ A(G) and then

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¯ τ ) = 0 for writing v χ (s) = τ ∈Gˆ aτ τ (s) and taking into account that (χτ, ˆ we obtain all τ ∈ G,

K, ψ χ = aτ (K χτ, ¯ τ) = aτ F (χτ ¯ )(χτ, ¯ τ ) = 0. ˆ τ ∈G

ˆ τ ∈G

Finally, we let {uα } be a bounded approximate identity for L1 (G) choˆ For each α, we have that uα ·ψ ∈ span{ψ χ : χ ∈ G} ˆ sen from span{χ : χ ∈ G}. and hence K, uα · ψ = 0 giving K, ψ = 0. Since ψ is an arbitrary element of Γ(G) vanishing on E ∗ , we conclude that K ∈ Mmin (E ∗ ). Proposition 7.14. Let f : G → C be a measurable function and ϕ = N f . Then the following holds: (i) If Ef is not an M -set then ϕ is closable. (ii) If Ef is an M1 -set then ϕ is not closable. Proof. (i) By [12, Theorem 1.2.7, Lemma 1.2.10], Mmax (Ef∗ ) does not contain a non-zero compact operator. The statement now follows from Theorem 5.4 (i) and Corollary 7.7. (ii) follows from Theorem 5.4 (ii), Corollary 7.7 and Lemma 7.13. Remark 7.15. We note that if Ef satisﬁes spectral synthesis then ϕ is closable if and only if Ef is an M -set. We say that a subset E ⊆ G is τ0 -open if E is equivalent, with respect to the Haar measure, to an open subset of G. A function f : G → C is said to be τ0 -continuous if f −1 (U ) is τ0 -open for any open U ⊆ G. Proposition 7.16. Let f : G → C be a measurable function and ϕ = N f . If ϕ is closable then f is τ0 -continuous. Proof. Since f almost belongs to A(G) at each point t ∈ Efc , it is equivalent to a continuous function h on Efc . In fact, for each t ∈ Efc there exists a neighborhood Ut and ht ∈ A(G) such that f = ht almost everywhere on Ut . Since G is second countable there exists a countable number of neighborhoods Uti such that Efc = ∪∞ i=1 Uti . Now set h(t) = hti (t) for t ∈ Uti . Clearly, h is continuous on Efc and f = h a.e. on Efc . Thus, given an open subset U ⊆ G, the set f −1 (U ) ∩ Efc is equivalent to an open set in G. Since f −1 (U ) = (f −1 (U ) ∩ Efc ) ∪ (f −1 (U ) ∩ Ef ), it is enough to show that μ(Ef ) = 0. Assume, by way of contradiction, that μ(Ef ) > 0. Since G is a compact ˆEf vanishes at inﬁnity and hence χEf is a non-zero group, χEf ∈ L1 (G), χ pseudofunction supported in Ef . But since, by Proposition 7.14, Ef is not an M1 -set, we arrive at a contradiction. We will ﬁnish this section by constructing an example of a non-closable ∗ multiplier ϕ for which κw ϕ is an operator M -set but not an operator M1 -set. ∗ The existence of a closable multiplier for which κw ϕ is an operator M -set but not an operator M1 -set remains an open problem.

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Example 7.17. Let E ⊆ T be an M -set which is not an M1 -set. Then ∂E = E. We have that Mmin (E ∗ ) does not contain a non-zero compact operator, while Mmax (E ∗ ) contains such an operator, say K. As Mmax (E ∗ ) = Mmin (E ∗ ), we can ﬁnd Ψ ∈ Γ(T) which vanishes on ∗ E such that K,Ψ = 0. 1 Let Ψ1 = n 2n χαn ⊗ χβn , where {αn × βn } is a disjoint family of rectangles such that (E ∗ )c ∪n αn × βn . Then Ψ1 is the limit of elements of Γ(T) which vanish on an ω-open subset of T × T containing E ∗ , and hence

K, Ψ1 = 0 [28]. We note that, moreover, nullΨ1 = E ∗ . Now let Ψ(x,y) (x, y) ∈ (E ∗ )c , ϕ(x, y) = Ψ1 (x,y) 0, (x, y) ∈ E ∗ . As Ψ1 ∈ Γ(T), one can ﬁnd measurable subsets Kn , n ∈ N, with Kn ⊆ Kn+1 , n ∈ N, such that m(Knc ) →n→∞ 0 and Ψ1 χKn ×Kn , ΨχKn ×Kn ∈ S(Kn , Kn ), n ∈ N. Then there exists N such that K, ΨχKN ×KN = 0. On the other hand, we have SΨ1 χKN ×KN (K) = 0. Let Tn ∈ C2 (L2 (T)), Tn → K, def

n → ∞. Then Sn = MχKN Tn MχKN → MχKN KMχKN and SΨ1 (Sn ) = SΨ1 χKN ×KN (Sn ) → SΨ1 χKN ×KN (K) = 0 but Sϕ (SΨ1 (Sn )) = SΨ (Sn ) → SΨ (MχKN KMχKN ) = 0. ∗

∗

w ∗ Thus ϕ is not closable and hence κw ϕ is an operator M -set. As κϕ ⊆ E , ∗ ∗ ∗ w we have Mmin (κw ϕ ) ⊆ Mmin (E ) and hence κϕ is not an operator M1 -set.

8. Open Problems In this section we list some open problems. The most important question which we have left unanswered is the following: Problem 1. Is every w*-closable multiplier a local Schur multiplier? Problem 2. Does Theorem 7.5 hold for all locally compact abelian groups? Problem 3. For which f ∈ C(R) is the divided difference fˇ a w*-closable multiplier? Since all local multipliers are w*-closable, Corollary 6.4 shows that a sufﬁcient condition for this to happen is μ(E(f )) = 0. The last two problems are related to Problem 1. Problem 4. Let f be an Operator Lipschitz function on [a, b], and let f (x) = 0 for all x ∈ [a, b]. Is the inverse function f −1 Operator Lipschitz on [f (a), f (b)]? Problem 5. For which continuous functions f and normal operators A ∈ B(H) is the map on B(H) given by AX − XA → f (A)X − Xf (A) closable in the weak* topology?

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Remark 8.1. (i) It is natural to pose a question, similar to one in Problem 4, in Schatten–von Neumann ideals Cp . As was proved by Potapov and Sukochev [27], a function f is (scalar) Lipschitz if and only if it is operator Lipschitz in Cp , 1 < p < ∞. Therefore, the answer to Problem 4 is negative if the operator Lipschitz condition is just replaced by the operator Lipschitz in Cp , 1 < p < ∞ (one can construct a function f on [a, b] with positive derivative such that the inverse function f −1 is not Lipschitz). However, the answer becomes positive if f is assumed to be continuous. (ii) The map considered in Problem 5 is norm closable. Indeed, if AXn − Xn A →n→∞ 0 and f (A)Xn − Xn f (A) →n→∞ B then [B, A] = lim [[f (A), Xn ], A] = lim [f (A), [Xn , A]] = 0, n→∞

n→∞

that is, B belongs to the commutant {A} of A. If E : B(H) → {A} is a conditional expectation, then B = E(B) = limn→∞ E(f (A)Xn − Xn f (A)) = limn→∞ (f (A)E(Xn ) − E(Xn )f (A)) = 0. (iii) For the case f (z) = z the answer to Problem 5 is negative. More precisely the “Fuglede” map AX −XA → A∗ X −XA∗ is not w*-closable, if σ(A) has non-empty interior and the spectral measure of A is equivalent to the Lebesgue measure on the interior U of σ(A). To see this, we assume for simplicity that A is the operator of multiplication by z on L2 (U, dzdz). Let f be the function on G = R2 given by f (z) = zz and let ϕ = N f be the corresponding Toeplitz multiplier on G × G. It is not difﬁcult to check that the set Ef of all points s ∈ G at which f does not belong to A(G) is the singleton {0}; applying Cor∗ ollary 7.7 we get that κw ϕ = Λ = {(z, z) : z ∈ G}. It follows that the multiplier ϕ is not w*-closable on U × U , so there are Hilbert Schmidt operators Ihn supported in U × U with Ihn → 0 and Iϕhn → B = 0 in the weak* topology. We may assume that hn (z1 , z2 ) vanish on some neighborhoods of the diagonal Λ. Indeed, let Vn be a neighborhood of Λ such that the Hilbert-Schmidt norm hn χVn 2 is less than 1/n. Then ϕhn χVn 2 < 1/n whence ϕhn χVn < 1/n and we may replace hn by hn − hn χVn . Setting pn (z1 , z2 ) = hn (z1 , z2 )/(z1 − z2 ) and Xn = Ipn we get that [A, Xn ] = Ihn → 0 and [A∗ , Xn ] = Iϕhn → B. Acknowledgements The authors are very grateful to M. Roginskaya and T. W. K¨ orner for their friendly and helpful advice, and to the referee for the suggestion to consider a Cp -version of Problem 4 (see Remark 8.1(i)).

References [1] Arveson, W.B.: Operator algebras and invariant subspaces. Ann. Math. (2) 100, 433–532 (1974) [2] Birman, M.S., Solomyak, M.Z.: Stieltjes double-integral operators. II (Russian). Prob. Mat. Fiz. 2, 26–60 (1967)

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[3] Birman, M.S., Solomyak, M.Z.: Stieltjes double-integral operators, III (Passage to the limit under the integral sign) (Russian). Prob. Mat. Fiz. 6, 27–53 (1973) [4] Birman, M.S., Solomyak, M.Z.: Operator integration, perturbations and commutators. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) Issled. Linein. Oper. Teorii Funktsii. 17(170), 34–66 (1989) [5] Birman, M.S., Solomyak, M.Z.: Double operator integrals in a Hilbert space. Integr. Equ. Oper. Theory 47(2), 131–168 (2003) [6] Blecher, D.P., Smith, R.: The dual of the Haagerup tensor product. J. Lond. Math. Soc. (2) 45, 126–144 (1992) [7] Bozejko, M., Fendler, G.: Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group. Colloquium Math. 63, 311–313 (1992) [8] Comfort, W.W., Gordon, H.: Vitali’s theorem for invariant measures. Trans. Am. Math. Soc. 99, 83–90 (1961) [9] Daletskii, J.L., Krein, S.G.: Integration and differentiation of functions of hermitian operators and applications to the theory of perturbations. Am. Math. Soc. Transl. (2) 47, 1–30 (1965) [10] Erdos, J.A., Katavolos, A., Shulman, V.S.: Rank one subspaces of Bimodules over maximal abelian selfadjoint algebras. J. Funct. Anal. 157(2), 554–587 (1998) [11] Farforovskaya, Y.B.: An estimate of the norm ||f (A) − f (B)|| for selfadjoint operators A and B. Zap. Nauchn. Semin. LOMI 56, 143–162 (1976). (English transl. J. Sov. Math. 14, 1133–1149) (1980) [12] Froelich, J.: Compact operators, invariant subspaces and spectral synthesis. J. Funct. Anal. 81, 1–37 (1988) [13] Gelfand, I., Raikov, D., Shilov, G.: Commutative Normed Rings. Translated from the Russian, with a Supplementary Chapter. Chelsea Publishing Co., New York (1964) [14] Gohberg, I.C., Krein, M.G.: Theory and Applications of Volterra Operators in Hilbert Space. Translation of Mathematical Monographs, vol. 24. American Mathematical Society (1970) [15] Graham, C., McGehee, O.C.: Essays in Commutative Harmonic Analysis. Springer-Verlag, New York (1979) [16] Grothendieck, A.: Resume de la theorie metrique des produits tensoriels topologiques. Boll. Soc. Mat. Sao-Paulo 8, 1–79 (1956) [17] Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, vol. I. Structure of Topological Groups, Integration Theory, Group Representations. SpringerVerlag, Berlin (1979) [18] Hewitt, E., Stromberg, K.: Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable. Springer-Verlag, New York (1965) [19] Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1995) [20] Katznelson, Y.: An Introduction to Harmonic Analysis, 3rd edn. Cambridge University Press, Cambridge (2004) [21] Kissin, E., Shulman, V.S.: Operator multipliers. Pac. J. Math. 227(1), 109–141 (2006)

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[22] Kissin, E., Shulman, V.S.: Classes of operator-smooth functions. I. OperatorLipschitz functions. Proc. Edinb. Math. Soc. (2) 48(1), 151–173 (2005) [23] Kissin, E., Shulman, V.S.: On the range inclusion of normal derivations: variations on a theme by Johnson, Williams and Fong. Proc. Lond. Math. Soc. (3) 83(1), 176–198 (2001) [24] Lahiri, B.K.: On translations of sets in topological groups. J. Indian Math. Soc. (N.S.) 39, 173–180 (1975) [25] Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability. Cambridge University Press, Cambridge (1995) [26] Peller, V.: Hankel operators in the theory of perturbations of unitary and selfadjoint operators (Russian). Funktsional. Anal. i Prilozhen. 19(2), 37–51, 96 (1985) [27] Potapov, D., Sukochev, F.: Operator Lipschitz functions in Scahtten-von Neumann classes. Acta Math. (2010, to appear) [28] Shulman, V.S., Turowska, L.: Operator synthesis I: synthetic sets, bilattices and tensor algebras. J. Funct. Anal. 209, 293–331 (2004) [29] Shulman, V.S., Turowska, L.: Operator synthesis II: individual synthesis and linear operator equations. J. Reine Angew. Math. 590, 143–187 (2006) [30] Spronk, N., Turowska, L.: Spectral synthesis and operator synthesis for compact groups. J. Lond. Math. Soc. (2) 66, 361–376 (2002) [31] Rudin, W.: Fourier Analysis on Groups. Wiley, New York (1990) V. S. Shulman Department of Mathematics Vologda State Technical University Vologda, Russia e-mail: shulman [email protected] I. G. Todorov Department of Pure Mathematics Queen’s University Belfast Belfast BT7 1NN, UK e-mail: [email protected] L. Turowska (B) Department of Mathematical Sciences Chalmers University of Technology and the University of Gothenburg 412 96 Gothenburg, Sweden e-mail: [email protected] Received: February 9, 2010. Revised: May 31, 2010.

Integr. Equ. Oper. Theory 69 (2011), 63–71 DOI 10.1007/s00020-010-1842-3 Published online November 27, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Generators for Rings of Compactly Supported Distributions Sara Maad Sasane and Amol Sasane Abstract. Let C denote a closed convex cone in Rd with apex at 0. We denote by E (C) the set of distributions on Rd having compact support contained in C. Then E (C) is a ring with the usual addition and with convolution. We give a necessary and sufficient analytic condition on f1 , . . . , fn for f1 , . . . , fn ∈ E (C) to generate the ring E (C). (Here · denotes Fourier-Laplace transformation.) This result is an application of a general result on rings of analytic functions of several variables by Lars H¨ ormander. En route we answer an open question posed by Yutaka Yamamoto. Mathematics Subject Classification (2010). Primary 46E10; Secondary 93D15, 46F05. Keywords. Rings of distributions, compactly supported distributions, Fourier-Laplace transform, corona type problem.

1. Introduction Let R be a commutative ring with identity. Elements a1 , . . . , an of R are said to generate R if the ideal generated by a1 , . . . , an is equal to R, or equivalently, if there exist b1 , . . . , bn such that a1 b1 + · · · + an bn = 1. For instance, if R = H ∞ (D), the set of all bounded and holomorphic functions on the open unit disc D centered at 0 in C, then the corona theorem says that f1 , . . . , fn ∈ H ∞ (D) generate H ∞ (D) if and only if there exists a C > 0 such that |f1 (z)| + · · · + |fn (z)| > C for all z ∈ D; see [2]. In this note, we address this question when the ring R consists of compactly supported distributions. Let C denote a closed convex cone in Rd with apex at 0. Recall that a convex cone is a subset of Rd with the following properties: 1. If x, y ∈ C, then x + y ∈ C. 2. If x ∈ C and t > 0, then tx ∈ C. Let E (C) be the set consisting of all distributions on Rd having a compact support contained in C. Then E (C) is a commutative ring with the usual

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addition of distributions and the operation of convolution. The Dirac delta distribution δ supported at 0 serves as an identity in the ring E (C). Recall that a distribution f with compact support has a ﬁnite order, and its Fourier-Laplace transform is an entire function given by f(z) = f, e−iz· ,

z ∈ Cd .

We use the notation · for the usual Euclidean 2-norm in Cd . The same notation is also used for the Euclidean norm in Rd . The supporting function of a convex, compact set K (⊂ Rd ) is deﬁned by HK (ξ) = sup x, ξ, x∈K

ξ ∈ Rd .

Our main result is the following: Theorem 1.1. Let C denote a closed convex cone in Rd with apex at 0, and let H denote the supporting function of the compact convex set B := C ∩ {x ∈ Rd : x ≤ 1}, that is, H(ξ) = sup ξ, x. x∈B

Let f1 , . . . , fn ∈ E (C). There exist g1 , . . . , gn ∈ E (C) such that f1 ∗ g1 + · · · + fn ∗ gn = δ if and only if there are positive constants C, N, M such that for all z ∈ Cd , |f1 (z)| + · · · + |fn (z)| ≥ C(1 + z2 )−N e−M H(Im(z)) . (1.1) Theorem 1.1, in the case when d = 1 and C = R was known; see [5].

2. Proof of the Main Result In this section, we will show that our main result follows from a result given in [3]. We recall the Payley–Wiener–Schwartz Theorem below, which will be used in the sequel. Proposition 2.1 (Payley–Wiener–Schwartz). Let K be a convex compact subset of Rd with supporting function H. If u is a distribution with support contained in K, then there exists a positive N such that for all z ∈ Cd , u (z) ≤ C(1 + z2 )N eH(Im(z)) .

(2.1)

Conversely, every entire analytic function in C satisfying an estimate of the form (2.1) is the Fourier-Laplace transform of a distribution with support contained in K. d

Proof. See for instance [4, Theorem 7.3.1]. The only difference is that we have the term (1 + z2 )N instead of (1 + z)N in the estimate (2.1), which follows from the observation that 1 + z2 ≤ (1 + z)2 ≤ 2(1 + z2 ) for every z ∈ Cd (and by replacing N/2 by N ).

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We also recall the main result from H¨ormander [3, Theorem 1, p. 943] for rings of holomorphic functions of several variables, which we will use. For background on complex analysis in several variables, we refer the reader to [6]. Let p be a nonnegative function deﬁned in Cd . Let Ap denote the set of all entire functions F : Cd → C such that there exist positive constants C1 and C2 (which in general depend on F ) such that for all z ∈ Cd , |F (z)| ≤ C1 eC2 p(z) . It is clear that Ap is a ring with the usual pointwise operations. Proposition 2.2 (H¨ ormander). Let p be a non-negative plurisubharmonic function in Cd such that 1. all polynomials belong to Ap , and 2. there exist non-negative K1 , K2 , K3 , K4 such that whenever z, ζ ∈ Cd satisfy z − ζ ≤ e−K1 p(z)−K2 , there holds that p(ζ) ≤ K3 p(z) + K4 . If there exist positive constants C1 , C2 such that for all z ∈ Cd , |F1 (z)| + · · · + |Fn (z)| ≥ C1 e−C2 p(z) ,

(2.2)

then F1 , . . . , Fn ∈ Ap generate Ap . Lemma 2.3. Let C denote a closed convex cone in Rd with apex at 0, and let H denote the supporting function of the compact convex set B := C ∩ {x ∈ Rd : x ≤ 1}, that is, H(ξ) = sup ξ, x. x∈B

Let p be defined by p(z) = log(1 + z2 ) + H(Im(z)) (z ∈ Cd ). Then we have the following: 1. 2. 3. 4.

p is nonnegative and subharmonic. (C). Ap = E Ap contains the polynomials. There exist nonnegative K1 , K2 , K3 , K4 such that whenever z, ζ ∈ Cd satisfy z − ζ ≤ e−K1 p(z)−K2 , there holds that p(ζ) ≤ K3 p(z) + K4 . (That is, condition 2 of Proposition 2.2 is satisfied.)

Proof. 1. Clearly p is non-negative. Also, the complex Hessian at z of the map z → log(1 + z2 ) is easily seen to be 1 1 F (z) := I− zz ∗ . 2 1 + z (1 + z2 )2 So for w ∈ Cd , we have that 1 1 w∗ F (z)w = w2 − |w∗ z|2 2 1 + z (1 + z2 )2 w2 + w2 z2 − |w∗ z|2 = ≥0 (1 + z2 )2 by the Cauchy–Schwarz inequality. So the map z → log(1 + |z|2 ) is plurisubharmonic; see [6, Proposition 4.9, p. 88].

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We will use the fact that a map ϕ : Cd → R that depends only on the imaginary part of the variable is plurisubharmonic if and only if the map is convex; see [6, E.4.8, p.92]. The supporting function HK of any convex compact set K satisﬁes the properties that HK (ξ + η) ≤ HK (ξ) + HK (η),

HK (tξ) = tHK (ξ)

for all ξ, η ∈ R and t ≥ 0. It is then clear that HK is a convex function. In particular our H (the supporting function of B) is convex too. Thus z → H(Im(z)) is plurisubharmonic. Consequently, p, which is the sum of the plurisubharmonic maps z → log(1 + z2 ) and z → H(Im(z)), is plurisubharmonic as well; [6, p.88]. 2. Suppose that f ∈ E (C) has support contained in the compact set K ⊂ C. Then by the Payley–Wiener–Schwartz Theorem, there exist positive C, N , M such that d

for all z ∈ Cd , |f(z)| ≤ C(1 + z2 )N eHK (Im(z)) . Let > 0 be such that K ⊂ B. Then we have for ξ ∈ Rd that HK (ξ) = sup x, ξ ≤ sup x, ξ = −1 sup y, ξ = −1 H(ξ). x∈−1 B

x∈K

y∈B

Thus with M := −1 , we have |f(z)| ≤ C(1 + z2 )N eM H(Im(z)) = CeN log(1+z

2

)+M H(Im(z))

≤ Cemax{N,M }p(z) . So f ∈ Ap . Conversely, if F ∈ Ap , then |F (z)| ≤ C1 eC2 p(z) = C1 (1 + z2 )C2 eC2 H(Im(z)) . But for ξ ∈ Rd we have C2 H(ξ) = C2 sup x, ξ = sup y, ξ = HC2 B (ξ). x∈B

y∈C2 B

So by the Payley–Wiener–Schwartz theorem, there exists an f ∈ E (Rd ) such that f = F and the support of f is contained in C2 B ⊂ C. Thus (C). F ∈ E 3. Let Z+ = {0, 1, 2, 3, . . . }. Let ak z k , Q(z) = k∈Zn + , |k|≤N

where for a multi-index k = (k1 , . . . , kn ) ∈ Zn+ , |k| := k1 + · · · + kn ,

z k = z1k1 . . . znkn ,

and ak ∈ C.

Consider q=

k∈Zn + , |k|≤N

ak

1 i|k|

∂xk11

∂ |k| δ ∈ E (C), . . . ∂xknn

(C) = A , and so Q ∈ A . Then Q = q ∈ E p p

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4. Let K1 and K2 be nonnegative, and let z, ζ satisfy z−ζ ≤ e−K1 p(z)−K2 . Then z − ζ ≤ e−K1 p(z)−K2 = e−K1 p(z) e−K2 ≤ 1 · 1 = 1. In particular, ζ ≤ z + 1. Also, H(Im(ζ − z)) = sup x, Im(ζ − z) ≤ sup xIm(ζ − z) x∈B

x∈B

≤ sup xζ − z ≤ 1 · 1 = 1. x∈B

Thus p(ζ) = log(1 + ζ2 ) + H(Im(ζ)) ≤ 2 log(1 + ζ) + H(Im(z + ζ − z)) ≤ 2 log(2 + z) + H(Im(z)) + H(Im(ζ − z)) ≤ log(8(1 + |z|2 )) + H(Im(z)) + 1 = p(z) + log 8 + 1. This completes the proof. Proof of Theorem 1.1. Necessity of the condition (1.1) is not hard to check. Indeed, if there are g1 , . . . , gn ∈ E (C) such that f1 ∗ g1 + · · · + fn ∗ gn = δ, then upon taking Fourier–Laplace transforms, we obtain g1 (z) + · · · + fn (z) gn (z) = 1 (z ∈ Cd ). f1 (z) By the triangle inequality, g1 (z) + · · · + fn (z) gn (z)| ≤ |f1 (z)|| g1 (z)| + · · · + |fn (z)|| gn (z)|. 1 = |f1 (z) Suppose that gk has support contained in the compact convex set Lk (⊂ C), where k = 1, . . . , n. Then by the Payley–Wiener–Schwartz theorem, we have an estimate | gk (z)| ≤ Ck (1 + z2 )Nk eHLk (Im(z))

(z ∈ Cd )

for each k. Let > 0 be small enough so that Lk ⊂ B for all the k. Then we have for ξ ∈ Rd that HLk (ξ) = sup x, ξ ≤ sup x, ξ = −1 sup y, ξ = −1 H(ξ). x∈−1 B

x∈Lk

y∈B

Thus we have that for all k, | gk (z)| ≤ C(1 + z2 )N eM H(Im(z))

(z ∈ Cd ),

where M := −1 , C := max Ck and N := max Nk . Consequently, k

k

1 ≤ (|f1 (z)| + · · · + |fn (z)|)C(1 + z2 )N eM H(Im(z))

(z ∈ Cd ),

and this yields (1.1), completing the proof of the necessity part. We now show the sufﬁciency of (1.1). Let f1 , . . . , fn ∈ E (C) be such that their Fourier-Laplace transforms satisfy (1.1). Then by Lemma 2.3, f1 , . . . , fn ∈ Ap with p(z) = log(1 + z2 ) + H(Im(z)) (z ∈ Cd ). Moreover,

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this p satisﬁes the conditions 1 and 2 of Proposition 2.2. The condition (1.1) gives |f1 (z)| + · · · + |fn (z)| ≥ C(1 + z2 )−N e−M H(Im(z)) 2

≥ Ce−N log(1+z )−M H(Im(z)) ≥ Ce− max{N,M }p(z) . It then follows from Proposition 2.2 that there are some G1 , . . . , Gn in Ap (C). Hence there exist such that f1 G1 + · · · + fn Gn = 1 on C. But Ap = E g1 , . . . , gn ∈ E (C) such that f1 ∗ g1 + · · · + fn ∗ gn = δ.

3. Special Cases of the Main result 3.1. The Full Space Rd The supporting function H of the unit ball B in Rd is given by H(ξ) = ξ. So we obtain the following consequence of Theorem 1.1. Corollary 3.1. Let f1 , . . . , fn ∈ E (Rd ). There exist g1 , . . . , gn ∈ E (Rd ) such that f1 ∗ g1 + · · · + fn ∗ gn = δ if and only if there are positive constants C, N, M such that for all z ∈ Cd , |f1 (z)| + · · · + |fn (z)| ≥ C(1 + z2 )−N e−M Im(z) .

(3.1)

3.2. The Nonnegative Orthant in Rd Let Rd+ = {x = (x1 , . . . , xd ) ∈ Rd : xk ≥ 0, for all k = 1, . . . , d}. The supporting function H of B = {x ∈ Rd+ : x ≤ 1} in Rd is given by H(ξ) = ξ + , where ξ + := (max{ξ1 , 0), . . . , max{ξd , 0}) for ξ = (ξ1 , . . . , ξd ) ∈ Rd . Theorem 1.1 gives the following. Corollary 3.2. Let f1 , . . . , fn ∈ E (Rd+ ). There exist g1 , . . . , gn ∈ E (Rd+ ) such that f1 ∗ g1 + · · · + fn ∗ gn = δ if and only if there are positive constants C, N, M such that + for all z ∈ Cd , |f1 (z)| + · · · + |fn (z)| ≥ C(1 + z2 )−N e−M (Im(z)) . (3.2)

In particular, in the case when d = 1, we obtain:

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Corollary 3.3. Let f1 , . . . , fn ∈ E (R+ ). There exist g1 , . . . , gn ∈ E (R+ ) such that f1 ∗ g1 + · · · + fn ∗ gn = δ if and only if there are positive constants C, N, M such that for all z ∈ C, |f1 (z)| + · · · + |fn (z)| ≥ C(1 + |z|2 )−N e−M max{Im(z),0} . (3.3) 3.3. The Future Light Cone in Rd+1 Let C be the future light cone, namely, Γ := {(x, t) ∈ Rd × R : x ≤ ct, t ≥ 0}, where c denotes the speed of light. Then the supporting function of the intersection of Γ and the unit ball in Rd+1 is given by ⎧ ⎪ ξ2 + τ 2 if c−1 ξ ≤ τ, ⎪ ⎪ ⎪ ⎨ τ + cξ Φ(ξ, τ ) = √ if − cξ ≤ τ ≤ c−1 ξ, ⎪ 2+1 ⎪ c ⎪ ⎪ ⎩ 0 if τ ≤ −cξ, for (ξ, τ ) ∈ Rd × R. Then we have: Corollary 3.4. Let f1 , . . . , fn ∈ E (Γ). There exist g1 , . . . , gn ∈ E (Γ) such that f1 ∗ g1 + · · · + fn ∗ gn = δ if and only if there are positive constants C, N, M such that for all z ∈ Cd , |f1 (z)| + · · · + |fn (z)| ≥ C(1 + z2 )−N e−M Φ(Im(z)) .

(3.4)

4. Answer to Yamamoto’s Question We remark that Theorem 1.1 answers an open question of Y. Yamamoto; see question number 2 [7, p.282]. There it was asked if for f1 , f2 ∈ E (R), the condition that f1 , f2 have no common zeros in C is enough to guarantee that there are g1 , g2 ∈ E (R) such that f1 ∗ g1 + f2 ∗ g2 = δ. In light of Theorem 1.1 above, the answer is no, since our analytic condition (3.1) (in the case when d = 1) is not equivalent to (and is stronger) than the condition that there is no common zero, as seen in the following example. (The idea behind this example is taken from [5].) Example 4.1. Let c ∈ R+ be the Liouville constant, that is, ∞ 1 c= . n! 10 n=1 (See for example, [1].) Then it can be seen that c is irrational. Also, for K ∈ N, with pK , qK deﬁned by pK = 10K!

K 1 , 10k!

k=1

qK = 10K! ,

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we have that ∞ pK 1 1 1 1 = 0 < c − = (K+1)! + (K+2)! + (K+3)! + · · · k! qK 10 10 10 10 k=K+1 ≤ ≤

1 10(K+1)! 1 (10K! )K

·

∞

10/9 1 1 10 = = (K+1)! · m K! )K 10K! 10 9 (10 10 m=0

=

1 . K qK

(4.1)

Take f1 = δ − δc and f2 = 1[0,1] , where 1[0,1] denotes the indicator function of the interval [0, 1], and δc is the Dirac distribution supported at c. Then f1 , f2 belong to E (R) and we have that ⎧ −iz e −1 ⎪ ⎪ ⎪ if z = 0 ⎨ −iz −icz f1 (z) = 1 − e , f2 (z) = ⎪ de−iz ⎪ ⎪ = 1 if z = 0. ⎩i dz z=0 Moreover, f1 and f2 have no common zeros (otherwise c would be rational!). We now show that (3.1) does not hold. Suppose, on the contrary that there exist C, N, M positive such that (4.2) |f1 (z)| + |f2 (z)| ≥ C(1 + |z|2 )−N e−M |Im(z)| for all z ∈ C. If z = 2πqK , then we have f2 (2πqK ) = 0. On the other hand, |f1 (2πqK )| = |1 − e−ic(2πqK ) | = | sin(πcqK )| = | sin(πcqK − πpK )|. The inequality (4.2) now yields that 2 −N | sin(π(cqK − pK ))| ≥ C(1 + 4π 2 qK ) .

But | sin θ| ≤ |θ| for all real θ, and so we obtain pK 2 −N ≥ C(1 + 4π 2 qK πqK c − ) . qK In light of (4.1), we now obtain 1 2 −N πqK K ≥ C(1 + 4π 2 qK ) , qK and rearranging, we have π

qK (1 + 4π 2 qK )N ≥ C. K qK

Letting K → ∞, we arrive at the contradiction that 0 ≥ C. We remark that in this example f1 , f2 actually belong to E (R+ ), and with the same argument given above, it can be seen that f1 , f2 don’t satisfy (3.3) either. This also gives another example answering question number 1 in [7], namely, for f1 , f2 in E (R+ ), whether the condition that f1 , f2 have no common zeros is enough to guarantee that there are g1 , g2 ∈ E (R+ ) such that f1 ∗ g1 + f2 ∗ g2 = δ.

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References [1] Apostol, T.M.: Modular functions and Dirichlet series in number theory. 2nd edn. Graduate Texts in Mathematics, 41. Springer, New York (1990) [2] Carleson, L.: Interpolations by bounded analytic functions and the corona problem. Ann. Math. Second Series 76, 547–559 (1962) [3] H¨ ormander, L.: Generators for some rings of analytic functions. Bull. Am. Math. Soc. 73, 943–949 (1967) [4] H¨ ormander, L.: The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Grundlehren der Mathematischen Wissenschaften, 256. Springer, Berlin (1983) [5] Petersen, E.K, Meisters, G.H.: Non-Liouville numbers and a theorem of H¨ ormander. J. Funct. Anal. 29(2), 142–150 (1978) [6] Range, M.R.: Holomorphic functions and integral representations in several complex variables. Graduate Texts in Mathematics, 108. Springer, New York (1986) [7] Yamamoto, Y.: Coprimeness of factorizations over a ring of distributions. Problem number 52. In: Blondel, V.D., Sontag, E.D., Vidyasagar, M., Willems, J.C. (eds.) Open Problems in Mathematical Systems and Control Theory. Communications and Control Engineering Series, Springer, London (1999) Sara Maad Sasane Department of Mathematics Stockholm University Stockholm, Sweden e-mail: [email protected] Amol Sasane (B) Department of Mathematics Royal Institute of Technology Stockholm, Sweden e-mail: [email protected]; [email protected] Received: April 6, 2010. Revised: October 24, 2010.

Integr. Equ. Oper. Theory 69 (2011), 73–85 DOI 10.1007/s00020-010-1820-9 Published online July 16, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Interpolating Sequences in Harmonically Weighted Dirichlet Spaces Gerardo R. Chac´on Abstract. In this article we show that interpolating sequences on certain harmonically weighted Dirichlet spaces can be characterized in terms of a separation condition and a Carleson-measure condition. This is the first example of a space with Nevanlinna–Pick kernel with non-radially symmetric weights in which this characterization remains true. Mathematics Subject Classification (2010). Primary 30E05; Secondary 40E20. Keywords. Interpolating sequences, weighted Dirichlet spaces.

1. Introduction Given a positive Borel measure μ defined on the boundary of the unit disc ∂D, define Pμ as the following harmonic function on the unit disc D: 2π Pμ (z) = 0

1 − |z|2 dμ(t) . |eit − z|2 2π

The Dirichlet type space D(μ) is defined as the space of all analytic functions on D such that |f (z)|2 Pμ (z)dA(z) < ∞. D

If μ = 0 then define D(μ) = H 2 , the Hardy space on the unit disc. Notice that if dμ = dm is the arc-length Lebesgue measure on ∂D, then the Dirichlet-type space D(m) coincides with the classical Dirichlet space D. Dirichlet-type spaces were then introduced by Richter in [11] when investigating analytic two-isometries. Richter showed that every analytic twoisometry T such that dim kerT ∗ = 1 can be represented as muliplication by z on a Dirichlet-type space D(μ). These spaces have been studied ever since by several authors, see for example [2,6,7,12–15,17–19].

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It is shown in [11] that the space D(μ) is contained as a set in the space H 2 , consequently a norm on D(μ) can be defined as f 2D(μ) := f 2H 2 + |f (z)|2 Pμ (z)dA(z) D

and it can be shown that evaluation functionals are continuous on D(μ). In this paper, we will investigate interpolating sequences in D(μ) spaces for the case in which μ is a finitely atomic measure on ∂D. Definition 1.1. Let H be a reproducing kernel Hilbert space formed by analytic functions defined on the unit disc D. • A sequence Z = (zj ) of distinct points in D is said to be an interpolating sequence for H if the interpolation problem f (zj ) = aj has a solution f ∈ H whenever (aj /Kzj H ) ∈ l2 where Kzj denotes the reproducing kernel of the space H at the point zj . • A sequence Z = (zj ) of distinct points in D is said to be an interpolating sequence for M(H) (the space of multipliers of H) if the interpolation problem f (zj ) = aj has a solution f ∈ M(H) whenever (aj ) ∈ l∞ . • A sequence Z = (zj ) is said to be H-separated if sup j=l

|kzj (zl )|2 <1 kzj (zj )kzl (zl )

Interpolating sequences in the Dirichlet space were described by Marshall and Sundberg [9] and Bishop [3] simultaneously in the 1994 using different techniques. An important observation proved in [9] is that the interpolating sequences for the Dirichlet space D and the interpolating sequences for its space of multipliers M(D) are the same (just as in the case of H p spaces). Marshall and Sundberg also give a new proof of the characterization of interpolating sequences for H p using Hilbert space techniques. In 2002, Boe [4] characterized the interpolating sequences for the Besov spaces Bp in terms of a separation condition and a Carleson measure condition. He also finds in [5] a necessary and sufficient condition for a sequence to be interpolating in certain Hilbert spaces satisfying the Nevanlinna–Pick property and another technical condition about the Grammian matrix associated with the sequence. Marshall and Sundberg showed in [9] that if a Hilbert space has the Nevanlinna–Pick property, then the interpolating sequences for H and for M(H) are the same. In [19] Shimorin showed that D(μ) spaces have the Nevanlinna–Pick property. Spaces with this property have received a lot of attention lately and several open problems concerning them have been solved for the case of spaces with radially symmetric weights (see for example [1] and the references therein for solutions of the interpolation problem in this setting). The following theorem holds in general. Recall that a finite positive Borel measure ν on the unit disc D is a Carleson measure for the space H if there exists a constant C > 0 such that for every function f ∈ H 2

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the following inequality holds: |f |2 dν ≤ Cf 2H . Throughout this article the notation f g means that there exists a constant K > 0 such that f ≤ Kg and f ∼ g means that f g and g f . In the following proposition, δzj denotes the Dirac measure on zj . Proposition 1.2 (See [16]). Let H be a reproducing kernel Hilbert space of analytic functions on D, and let Z = (zj ) ⊂ D be a sequence of distinct points. Then (a) ⇒ (b) ⇔ (c) ⇒ (d) (a) (b) (c) (d)

Z is an interpolating sequence for M(H). j bj Kzj H j aj Kzj H whenever |bj | ≤ |aj | for every j. j aj Kzj /Kzj H H ∼ (aj )l2 . Z is H-separated and j Kzj −2 δzj is a Carleson measure for H.

For the cases H = H 2 or H = D, the Dirichlet space, these conditions are equivalent (see for example [16]). There is a conjecture that the four conditions are equivalent in every space H with complete Nevanlinna–Pick kernel, in [16] it is also showed that in this setting (b)⇒(a) and the only difficulty is to show that (d) implies any of the previous conditions. In these spaces, Marshall and Sunberg [9] showed that a sequence Z is interpolating for M(H) if and only if it is interpolating for H. In [19], Shimorin shows that the spaces D(μ) have a complete Nevanlinna–Pick kernel. In this article we will use some results due to Serra [17] and some ideas from Sarason [14] and we will show that the four conditions are equivalent in the case in which μ = aδλ , for λ ∈ ∂D and a > 0. Specifically we will show the that (d)⇒(a) which is the only thing that remains to be proven. As far as we know, these are the first examples of spaces with the Nevanlinna–Pick property with weights that are not radially symmetric and where the result remains true. In order to do that, we start by studying Dirichlet type spaces of the form D(aδλ ) and we use the fact that these space can be seen as De Branges–Rovnyak spaces (see [14]). We then use the explicit form of the reproducing kernels to characterize interpolating sequences in terms of a separation condition and a Carleson measure condition. Then, we study the more general case of μ being a finitely atomic measure. In this case, we combine a result from McCullough and Trent [10] with a result from Richter and Sundberg [13] and Aleman [2] to obtain that the quotient of the reproducing kernel of this space and the reproducing kernel of the space D(aδλ ) is positive definite. Then we reduce the problem to the previously proven result.

2. Interpolating Sequences for D(aδλ ) In this section we will show that the four conditions of Theorem 1.2 are equivalent for the space D(aδλ ). First we will show that the space D(aδλ ) can be written as a de Branges–Rovnyak space. We will use a reasoning analogous to that in [14] in which the result is proven for the case a = 1.

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Consider the function bλ (z) :=

(1 − wa )λz 1 − wa λz

where wa is the solution of the equation (x − 1)2 = ax that belongs to the unit disc. That is, let wa :=

2 + a − ((2 + a)2 − 4)1/2 . 2

We will show that the reproducing kernel of the space D(aδλ ) can be written as K λ (z, w) =

1 − bλ (z)bλ (w) , 1 − zw

i.e. D(aδλ ) is the de Branges–Rovnyak space H(bλ ). First, we will use some facts about de Branges–Rovnyak spaces that can be found in [14] and in [8]. If a function γ1 is such that 1 − |γ1 |2 is logintegrable on ∂D and γ1 ∞ = 1 then there is a unique outer function γ2 such that γ2 (0) > 0 and |γ1 |2 + |γ2 |2 = 1 almost everywhere on ∂D. Then for f ∈ H 2 , f belongs to the de Branges–Rovnyak space H(γ1 ) if and only if there is a unique function f + ∈ H 2 such that Tγ1 f = Tγ2 f + , where Tγj is the Toeplitz operator with symbol γj defined on H 2 . In this case, f 2γ1 = f 2H 2 + f + 2H 2 . In our case, let γ1 = bλ and notice that for |z| = 1 (1 − wa )λz 2 1 − |bλ (z)|2 = 1 − 1 − wa λz (1 − wa )2 awa = 1− =1− |1 − wa λz|2 |1 − wa λz|2 =

1 − 2wa Re(λz) + wa2 − awa |1 − wa λz|2

=

wa 2 Re(1 − λz) |1 − wa λz|2

=

a−1 (1 − wa )2 |1 − λz|2 . |1 − wa λz|2

Hence we can define γ2 (z) :=

a−1/2 (1 − wa )(1 − λz) . (1 − wa λz)

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Now, for f, g ∈ H 2 Tγ1 f = Tγ2 g if and only if the function γ1 f − γ2 g is orthogonal to H 2 , but for |z| = 1 a−1/2 (1 − wa )(1 − λz) (1 − wa )λz f (z) − g(z) 1 − wa λz 1 − wa λz 1 − wa λzf (z) − a−1/2 (1 − λz)g(z) = 1 − wa λz 1 − wa λz[f (z) − (z − λ)λa−1/2 g(z)], = 1 − wa λz so Tγ1 f = Tγ2 g if and only if there exists a constant c such that γ1 f − γ2 g =

f (z) = c + a−1/2 λ(z − λ)g(z). 1−wa 2 Here we have used the fact that the function 1−w ¯ is cyclic in H . z λz Finally, since a function f belongs to D(aδλ ) if and only if there exists a function h ∈ H 2 such that f (z) = f (λ) + (z − λ)h(z) and Dλ (f ) = h2H 2 , ¯ −1/2 h(z) belongs to then f ∈ D(aδλ ) if and only if the function g(z) := λa H 2 and g2H 2 = ah2H 2 = aDλ (f ). Thus, the spaces H(bλ ) and D(aδλ ) coincide and

f 2D(aδλ ) = f 2H 2 + aDλ (f ) = f 2H 2 + g2H 2 = f 2H(bλ ) and this completes the proof. We will need the following result by Serra. Proposition 2.1 [17]. A sequence (zj ) ⊆ D is an interpolating sequence for M (D(δ1 ) if and only if (zj ) satisfies the following two conditions: (i) (zj ) is uniformly separated, i.e. zj − zl 1 − zj zl ≥ δ, l = 1, 2, . . . j=l

for some constant δ independent of l. (ii) The sequence 1 − |zj |2 (P1 (zj )) := |1 − zj |2 belongs to l1 . Condition (i) in the previous proposition is also known as the Carleson condition for the sequence {zj }. We will show that condition (d) of Proposition 1.2 implies conditions (i) and (ii) of Proposition 2.1. This will imply that conditions (a) through (d) are equivalent for the space D(δλ ). Actually, for simplifying the notation we will assume λ = 1 but the general result can be proven in a similar way, we will also denote b = b1 . We will use a few lemmas. Lemma 2.2. If the sequence (zj ) is D(δ1 )-separated, then (zj ) is uniformly discrete, i.e. there exist a constant δ > 0 such that zj − z l 1 − zj zl ≥ δ ∀j = l.

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Proof. If (zj ) is D(δ1 )-separated, then there exists a constant 0 < c < 1 such that |Kzj (zl )|2 ≤c Kzj (zj )Kzl (zl ) and since Kzj (zl ) =

1−b(zj )b(zl ) 1−zj zl

we have that

(1 − |zj |2 )(1 − |zl |2 ) (1 − |b(zj )|2 )(1 − |b(zl )|2 ) ≤ c , |1 − zj zl |2 |1 − b(zj )b(zl )|2 but since (1 − |b(zj )|2 )(1 − |b(zl )|2 ) |1 − b(zj )b(zl )|2 we have

b(z ) − b(z ) 2 j l =1− ≤ 1, 1 − b(zj )b(zl )

zj − z l 2 ≤c 1− 1 − zj zj

and consequently (zj ) is uniformly discrete. Lemma 2.3. If a sequence (zj ) ⊂ D is such that (Kzj D(aδ1 ) ) converges to K1 D(aδ1 ) .

2

|1−zj | 1−|zj |2

→ 0, then the sequence

Proof. First note that K1 is well defined since every function in D(aδ1 ) has a nontangential limit at 1, so the evaluation functional f → f (1) is well defined on D(aδ1 ); its kernel is (z − 1)H 2 which is a closed subspace of D(aδ1 ), hence the functional is bounded (see [15]). |1−z |2 Also, note that if 1−|zjj |2 → 0, then zj → 1 and that b(z) := b1 (z) converges to 1 as z converges to 1 because 1−z → 0 as z → 1. |1 − b(z)| = 1 − wa z Consequently for every w ∈ D Kw (z) =

1 − b(w) 1 − b(w)b(z) → 1 − wz 1−w

as z → 1

so K1 (w) =

1 1 − b(w) = , 1−w 1 − wa w

hence K1 2D(aδ1 ) =

1 . 1 − wa

Now, Kzj 2D(aδ1 ) = =

1 − |b(zj )|2 1 − |zj |2 1 2wa (|zj |2 − Re zj ) + 2 |1 − wa zj | (1 − |zj |2 )|1 − wa zj |2

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|zj |2 − Re zj |1 − zj |2 1 = → 0. + 1 − |zj |2 2 2(1 − |zj |2 )

|zj |2 −Re zj 1−|zj |2

→

−1 2 ,

and Kzj 2D(aδ1 ) →

1 1−wa .

Lemma 2.4. Suppose a sequence (zj ) ⊂ D is such that |1−z |

2

j

Kzj −2 D(aδ1 ) δzj

is a D(aδ1 )-Carleson measure, then the sequence ( 1−|zjj |2 ) is bounded away from 0. |1−z

|2

Proof. If there were a subsequence (zjn ) such that the sequence ( 1−|zjjn |2 ) n converges to zero, then by the previous lemma, we have that Kzjn D(aδ1 ) converges to K1 D(aδ1 ) . However, by hypothesis, there exists a constant C > 0 such that for every f ∈ D(aδ1 )

2 2 Kzj −2 D(aδ1 ) |f (zj )| ≤ Cf D(aδ1 ) . j

In particular, taking f ≡ 1 we have that Kzjn −2 D(aδ1 ) → 0 which is a contradiction. Lemma 2.5. Let 0 < ε < 1, and define the set |1 − z|2 ≥ ε , Aε := z ∈ D : 1 − |z|2 then for every z ∈ Aε , wa 1 − |b(z)|2 εwa + 1 − wa ≤ ≤ . |1 − wa z|2 |1 − z|2 ε|1 − wa z|2

(1)

Proof. 1 − |b(z)|2 |1 − wa z|2 − (1 − wa )2 |z|2 = 2 |1 − z| |1 − wa z|2 |1 − z|2 1 − |z|2 − 2wa Re z + 2wa |z|2 = |1 − wa z|2 |1 − z|2 2|z|2 − 2 Re z 1 1 − |z|2 wa = + |1 − z|2 |1 − wa z|2 |1 − wa z|2 |1 − z|2 1 1 − |z|2 = |1 − z|2 |1 − wa z|2 2|z|2 − 2 Re z + 1 − |z|2 wa 1 − |z|2 + − |1 − wa z|2 |1 − z|2 |1 − z|2 2 1 wa 1 − |z| 1 − |z|2 + = 1− |1 − z|2 |1 − wa z|2 |1 − wa z|2 |1 − z|2 2 1 − |z| 1 (1 − wa ) = wa + |1 − wa z|2 |1 − z|2 and since z ∈ Aε inequality (1) follows.

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Now we are ready to prove the main result of this section. Theorem 2.6. Suppose that a sequence (zj ) ⊂ D is D(aδ1 )-separated and that Kzj −2 D(aδ1 ) δzj is a D(aδ1 )-Carleson measure. Then (zj ) is uniformly 1−|z |2

separated and the sequence ( |1−zjj |2 ) belongs to l1 . Proof. Suppose ν is a D(aδ1 )-Carleson measure and let g ∈ H 2 . Define f (z) := (z − 1)g(z), then f ∈ D(aδ1 ) and |z − 1|2 |g(z)|2 dν(z) = |f |2 dν f 2H 2 + a2 g2H 2

g2H 2 .

Hence, |z − 1|2 dν(z) is a Carleson measure for the Hardy space. Conversely, suppose |z − 1|2 dν(z) is a H 2 -Carleson measure and let (1) and f ∈ D(aδ1 ), then g ∈ H 2 , where g(z) := f (z)−f z−1 |f (1) + (z − 1)g(z)|2 dν |f |2 dν = |f (1)|2 ν(D) + |z − 1|2 |g(z)|2 dν(z) f 2D(aδ1 ) + g2H 2 f 2D(aδ1 ) . Thus a measure ν is a D(aδ1 )-Carleson measure if and only if |z − 1|2 dν 2 Kzj −2 is a H -Carleson measure. Hence if D(aδ1 ) δzj is a D(aδ1 )-Carleson measure, then there exists a constant C > 0 such that for every f ∈ H 2

|1 − zj |2 |f (zj )|2 ≤ Cf 2H 2 . (2) 2 K z j D(aδ1 ) j Now, note that by Lemma 2.4 there exists 0 < ε < 1 such that (zj ) ⊂ Aε and consequently by Lemma 2.5 we have that 1 1 1 − |b(zj )|2 ∼ ∼ . 2 2 2 2 (1 − |zj | )|1 − zj | (1 − |zj | )|1 − wa zj | 1 − |zj |2 Thus

|1 − zj |2 1 ∼ , 2 H Kzj D(aδ1 ) Kzj 2 2H 2

(3)

2

where KzHj denotes the reproducing kernel for the space H 2 at zj . Then Eq. (2) can be written as:

|f (zj )|2 f 2H 2 . H 2 2 K 2 z H j j

2 2 Thus j KzHj −2 H 2 δzj is a H -Carleson measure. But it is known (see [16]) that if a sequence (zn ) satisfies this condition and is uniformly discrete

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(which is because of Lemma 2.2), then it is uniformly separated; this proves the first part of the theorem. Finally, for the second part note that by Eq. (3) we have that 1 − |zj |2 1 ∼ Kzj 2D(aδ1 ) |1 − zj |2 and consequently

1 − |zj |2 |1 − zj because result.

|2

∼

1 ≤ C, Kzj 2D(aδ1 )

Kzj −2 D(aδ1 ) δzj is a D(aδ1 )-Carleson measure. This proves the

n 3. Interpolating Sequences for D( k=1 μk δζk )

n In this section we will show that for the case of μ = k=1 μk δζk , μk > 0 for every k = 1, . . . , n, conditions (a) through (d) of Proposition 1.2 are equivalent. For this, we will rely upon the corresponding result for one point mass (Theorem 2.6) and some preliminary results. First, we will need a general result about complete Nevanlinna–Pick reproducing kernels. Recall that a reproducing kernel k on the unit disc is a complete Nevanlinna–Pick kernel (complete NP kernel) if k0 (z) = 1 for all z ∈ D and if there exists a sequence of analytic functions {bn }n≥1 on D such that

1 = bn (z)bn (λ), for all λ, z ∈ D. 1− kλ (z) n≥1

This condition is equivalent to the assumption that 1 − 1/k is positive definite. We mentioned before that Shimorin in [19] showed that the D(μ) spaces have a complete NP kernel. The first result we will need is due to McCullough and Trent [10]. We will say that a subspace M of a Hilbert space H is a multiplier invariant subspace if ϕM ⊂ M for every ϕ ∈ M (H), the space of multipliers of H. Theorem 3.1 [10]. Let k be a complete NP kernel and let M be a multiplier invariant subspace. Then there exists a sequence of multipliers {ϕn } ⊂ M such that

PM = Mϕn Mϕ∗n (SOT ) n≥1

where PM denotes the projection onto M and Mϕn denotes the multiplication operator: f → ϕn f . In particular, notice that if we take the function kz , z ∈ D, we have that

Mϕn Mϕ∗n kz . PM kz = n≥1

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Since Mϕ∗n kz = ϕn (z)kz , then we have that for every w ∈ D,

PM kz (w) = ϕn (w)ϕn (z)kz (w), n≥1

or equivalently,

PM kz (w)

= ϕn (w)ϕn (z), kz (w)

(4)

n≥1

PM kz (w) kz (w)

is positive definite. We will also need the following result which is due to Richter and Sundberg [11,13] and Aleman [2]. i.e.

Theorem 3.2. Let M be a multiplier invariant subspace of D(μ), then dimM zM = 1 and if f ∈ M zM, f D(μ) = 1, then (i) |f (z)| ≤ 1 for all z ∈ D. (ii) f gD(μ) = gD(μf ) , for every g ∈ D(μf ), where dμf = |f |2 dμ. (iii) For every g ∈ M, there exists h ∈ D(μf ) such that g = f h, n Lemma 3.3. Let {ζ1 , . . . , ζn } ⊂ ∂D, and μ1 , . . . , μn > 0. If μ := k=1 μk δζk , and if Kzμ denotes the reproducing kernel of the space D(μ) at z. Then for every j = 1, . . . n, there exists a positive constant aj such that the reproducing Kj kernel Kzj of the space D(aj δζj ) satisfies that K μ is positive definite. Proof. First, notice that the kernel K μ is never zero (see [18]) and consequently the quotient is well defined. Let j ∈ {1, . . . , n} be fixed and define Mj := {f ∈ D(μ) : f (ζk ) = 0 ∀k = j}, then Mj is a multiplier invariant subspace of D(μ). Let φj ∈ Mj zMj , φj D(μ) = 1, then by Theorem 3.2 we have that the multiplication operator Mφj : D(μφj ) → Mj is an onto isometry (and consequently a unitary operator). Here, dμφj

= |φj |2 dμ =

n

μk |φj (ζk )|2 dδζj = μj |φj (ζj )|2 dδζj .

k=1

Define aj := μj |φj (ζj )|2 , then the reproducing kernel for the space Mj is given by KzMj (w) = φj (z)φj (w)Kzj (w). Mj

On the other hand, we also know that Kz

= PMj Kzμ , hence M

Kz j (w) 1 Kzj (w) = μ μ Kz (w) φj (z)φj (w) Kz (w) PMj Kzμ (w) 1 = μ φj (z)φj (w) Kz (w) and since each one of the factors is positive definite, then the result follows.

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From now on, we will use the same notation as in the hypothesis of the previous lemma. A consequence of the lemma is the following: take z = 0, then Kzj (w) = Kzμ (w) = 1 for every w ∈ D and by the positive definiteness Kj of K μ we have that j 2 D(aj δζ ) Kw j . (5) 1≤ μ 2 Kw D(μ) Another consequence of Lemma 3.3 is the following. Lemma 3.4. If a sequence (zj ) ⊂ D is D(μ)-separated, then it is D(ak δζk )separated for every k = 1, . . . , n. k

K Proof. By Lemma 3.3 we have that K μ is positive definite, consequently given z, w ∈ D we have that k k 2 D(ak δζ ) Kzk 2D(ak δζ ) Kw Kz (w) 2 k k μ ≤ μ μ Kz (w) Kz 2D(μ) Kw 2D(μ)

and since the assumption implies that for some C > 0, |Kzμ (w)|2 μ 2 μ 2 Kz D(μ) Kw D(μ)

≤C<1

the result follows.

Notice that we could have proved Lemma 2.2 in a similar fashion: If we consider M := {f ∈ D(δ1 ) : f (1) = 0} and use Theorem 3.2 to identify M with the Hardy space H 2 , then by use of Lemma 3.3 we obtain the result. Lemma 3.5. Let (zj ) ⊂ D be a sequence such that for some m ∈ {1, . . . n},

|ζm −zj |2 1−|zj |2

→ 0, then Kzμj D(μ) → Kζμm D(μ) .

Proof. By Lemma 2.3 we have that Kzmj D(am δζm ) → Kζmm D(am δζm ) and by the reproducing property we have that Kzmj − Kζmm D(am δζm ) → 0. Now, consider the inclusion operator J : D(μ) → D(am δm ), then J is bounded and so is J ∗ . Notice that J ∗ Kzm = Kzμ and by the continuity of J ∗ we have that Kzμj − Kζμm D(μ) → 0 Lemma 3.6. Suppose a sequence (zj ) ⊂ D is such that j Kzμj −2 D(μ) δzj is a D(μ)-Carleson measure, then the sequence ( 0 for every m ∈ {1, . . . , n}.

|ζm −zj |2 1−|zj |2 ),

is bounded away from

Proof. The proof follows using the previous lemma and a reasoning analogous to that on the proof on Lemma 2.4. For each m ∈ {1, . . . , n} and 0 < < 1, define the sets |ζm − z|2 Am := z ∈ D : ≥ ε . ε 1 − |z|2 Notice that by Lemma 3.6, if a sequence (zj ) ⊂ D is such that the mea sure Kzj −2 D(μ) δzj is a D(μ)-Carleson measure, then there exist 0 < ε1 , . . . , εn < 1 such that (zj ) ⊂ A1ε1 ∩ · · · ∩ Anεn .

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Theorem 3.7. Suppose that a sequence (zj ) ⊂ D is D(μ)-separated and that Kzμj −2 D(μ) δzj is a D(μ)-Carleson measure. Then (zj ) is an interpolating sequence for the space of multipliers M (D(μ)). Proof. Fix m ∈ {1, . . . , n} and notice that by the previous lemma, there exists 0 < εm < 1 such that (zj ) ⊂ Am εm so we can use identity (3) to conclude that 1 Kzmj 2D(am δζ

m)

∼

1 − |zj |2 |ζm − zj |2

but since by Eq. (5) for every z ∈ DKzμ D(μ) ≤ Kzm D(am δζm ) , then we have that there exists a constant C > 0 such that

1 − |zj |2

1 ≤ C. μ 2 2 |ζ − z | K zj D(μ) m j j j Thus by Theorem 2.6 (zj ) is an interpolating sequence for M (am D(δζm )). Now we use another result of Serra [17] that says that if (zn ) is interpolating for each M (D(am δζm )), then it is interpolating for M (D(μ)). Acknowledgements This work appeared as part of the author’s doctoral dissertation at the University of Tennessee under the supervision of Dr. Stefan Richter. The author wants to thank Dr. Richter for his help and encouragement. The author would also like to thank the referee for their useful suggestions.

References [1] Agler, J., McCarthy, J.: Pick interpolation and Hilbert function spaces. Graduate Studies in Mathematics, vol. 44. American Mathematical Society, Providence, RI (2002) [2] Aleman, A.: The Multiplication Operators on Hilbert Spaces of Analytic Functions. Habilitationsschrift, Fernuniversit¨ at Hagen (1993) [3] Bishop, C.: Interpolating sequences for the Dirichlet space and its multipliers, Preprint (1994) [4] Boe, B.: Interpolating sequences for Besov spaces. J. Funct. Anal. 192, 319– 341 (2002) [5] Boe, B.: An interpolation Theorem for Hilbert spaces with Nevanlinna–Pick kernel. Proc. Am. Math. Soc. 210 (2003) [6] Chartrand, R.: Toeplitz operators on Dirichlet-type spaces. J. Oper. Theory 48, 3–13 (2002) [7] Chartrand, R.: Multipliers and Carleson measures for D(µ). Integral Equ. Oper. Theory 45, 309–318 (2003) [8] de Branges, L., Rovnyak, J.: Square Summable Power Series. Holt, Rinehart, and Winston, New York (1966) [9] Marshall, D.E., Sundberg, C.: Interpolating sequences for the multipliers of the Dirichlet space, Preprint (1994). Available at http://www.math.washington. edu/∼marshall/preprints/preprints.html

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[10] McCullough, S., Trent, T.: Invariant subspaces and Nevanlinna–Pick kernels. J. Funct. Anal. 178, 226–249 (2000) [11] Richter, S.: A representation theorem for cyclic analytic two-isometries. Trans. Am. Math. Soc. 328, 325–349 (1991) [12] Richter, S., Sundberg, C.: A formula for the local Dirichlet integral. Michigan Math. J. 38, 355–379 (1991) [13] Richter, S., Sundberg, C.: Multipliers and invariant subspaces in the Dirichlet space. J. Oper. Theory 28, 167–187 (1992) [14] Sarason, D.: Local Dirichlet spaces as De Branges–Rovnyak spaces. Proc. Am. Math. Soc. 125, 2133–2139 (1997) [15] Sarason, D.: Harmonically weighted Dirichlet spaces associated with finitely atomic measures. Integral Equ. Oper. Theory 31, 186–213 (1998) [16] Seip, K.: Interpolation and Sampling in Spaces of Analytic Functions. University Lecture Series, vol. 33. American Mathematical Society, Providence, RI (2004) [17] Serra, A.: Interpolating sequences in harmonically weighted Dirichlet spaces. Proc. Am. Math. Soc. 131, 2809–2817 (2003) [18] Shimorin, S.: Reproducing kernels and extremal functions in Dirichlet-type spaces. J. Math. Sci. 107, 4108–4124 (2001) [19] Shimorin, S.: Complete Nevanlinna–Pick property of Dirichlet type spaces. J. Funct. Anal. 191, 276–296 (2002) Gerardo R. Chac´ on (B) Departamento de Medici´ on y Evaluaci´ on Universidad de los Andes M´erida 5101, Venezuela e-mail: [email protected]; [email protected] Received: March 16, 2010. Revised: June 15, 2010.

Integr. Equ. Oper. Theory 69 (2011), 87–99 DOI 10.1007/s00020-010-1827-2 Published online August 11, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Some Closed Range Integral Operators on Spaces of Analytic Functions Austin Anderson To Mary Rose Abstract. Our main result is a characterization of g for which the z operator Sg (f )(z) = 0 f (w)g(w) dw is bounded below on the Bloch space. We point out analogous results for the Hardy space H 2 and the Bergman spaces Ap for 1 ≤ p < ∞. We also show the companion operaz tor Tg (f )(z) = 0 f (w)g (w) dw is never bounded below on H 2 , Bloch, nor BM OA, but may be bounded below on Ap . Mathematics Subject Classification (2010). Primary 47B38; Secondary 47G10, 30D45. Keywords. Volterra operator, Cesaro operator, integral operator, bounded below, closed range, Bloch, Hardy, Bergman, BMOA, multiplication operator.

1. Introduction We examine operators on Banach spaces of analytic functions on the unit disk in the complex plane. The operator Tg , with symbol g(z) an analytic function on the disk, is deﬁned by z Tg f (z) =

f (w)g (w) dw.

0

Tg is a generalization of the standard integral operator, which is Tg when g(z) = z. Letting g(z) = log(1/(1 − z)) gives the Ces´aro operator. Discussion of the operator Tg ﬁrst arose in connection with semigroups of composition operators. (see [11] for background) Characterizing the boundedness and compactness of Tg on certain spaces of analytic functions is of recent A. Anderson was supported by NSF-DGE-0841223.

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interest, as seen in [1,2,5,11], z and open problems remain. Tg and its companion operator Sg f (z) = 0 f (w)g(w) dw are related to the multiplication operator Mg f (z) = g(z)f (z), since integration by parts gives Mg f = f (0)g(0) + Tg f + Sg f. If any two of Mg , Sg , and Tg are bounded, then so is the third. But in some situations one operator is bounded while two are unbounded. Boundedness of Tg on the Hardy and Bergman spaces and BM OA is characterized in [1,2,11]. The pointwise multipliers of these and many other spaces are well known. See [12] for BM OA. In this paper we examine the property of being bounded below for Tg and Sg on spaces of analytic functions. We examine aspects of the problems on Hardy and Bergman spaces, the Bloch space, and BM OA. In doing so we must assume the operators are bounded, and we study characterizations of the symbols for which the operators are bounded. Consideration of Mg is useful as well.

2. Preliminaries The notation f g will mean there exists a universal constant C such that f ≤ Cg. f ≈ g will mean f g f . Let D be the unit disk in the complex plane. Let H(D) denote the set of analytic functions on D. For 1 ≤ p < ∞, the Hardy space H p on D is ⎫ ⎧ 2π ⎬ ⎨ |f (reit )|p dt < ∞ . f ∈ H(D) : f p = sup ⎭ ⎩ 0
The space of bounded analytic functions on D is

H ∞ = f ∈ H(D) : f ∞ = sup |f (z)| < ∞ . z∈D

We deﬁne weighted Bergman spaces, for α > −1, ⎧ ⎫ ⎨ ⎬ Apα = f ∈ H(D) : f Apα = |f (z)|p (1 − |z|2 )α dA(z) < ∞ , ⎩ ⎭ D

where dA(z) refers to Lebesgue area measure on D. The Bloch space is

2 B = f ∈ H(D) : f B = sup |f (z)|(1 − |z| ) < ∞ . z∈D

Note that B is a semi-norm. The true norm accounts for functions differing by an additive constant. A complex measure μ on D is called a (Hardy space) Carleson measure if there exists C > 0 such that μ(S(I)) ≤ C|I| for all arcs I ⊆ ∂D, where S(I) = {reiθ : 1 − |I| < r < 1, eiθ ∈ I} is the Carleson rectangle associated with I, and |I| is the length of I. The smallest such C is called the Carleson constant for the measure μ. Deﬁne, for f ∈ H(D), dμf (z) =

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|f (z)|2 (1 − |z|2 ) dA(z). The space of analytic functions of bounded mean oscillation, BM OA, is the set of f for which μf is Carleson. The BM OA norm f ∗ is comparable to the square root of the Carleson constant for μf . The space of analytic functions of vanishing mean oscillation, V M OA, is the set of f for which lim

|I|→0

μf (S(I)) = 0. |I|

Zhu [14] is a good reference for background on all these spaces. H ∞ is a closed subspace of BM OA, which in turn is a closed subspace 2 of H . H ∞ is also a closed subspace of B. The next lemma will be useful later when studying Tg . Lemma 2.1. Let fn (z) = z n , n = 1, 2, . . .. fn X ≈ 1 for all n and X = H 2 , B, BM OA. Proof. It is well-known that fn H 2 = 1 for all n. Checking the Bloch norm with a calculation, we get fn B ≈ sup0 −1, the differentiation operator and its inverse, the indefinite integral, are isometries between Apα /C and Apα+p , i.e., f Apα ≈ |f (0)| + f Apα+p .

(2.1)

(see [14, 4.28]) Making the natural definition A2−1 = H 2 , the identity holds for p = 2, α = −1as well. This is the well-known Littlewood–Paley identity, f H 2 ≈ |f (0)| + D |f (z)|2 (1 − |z|2 ) dA(z). On all the spaces mentioned, point evaluation is a bounded linear functional. The norm of point evaluation at z in Apα is comparable to 1/(1 − |z|)(2+α)/p . (see [14, Theorem 4.14]) In B and BM OA, the norm of point evaluation is comparable to log(2/(1−|z|)). The following theorem is a generalization of a result on multipliers of Banach spaces in which point evaluation is a bounded. See, for example [6, Lemma 11]. Theorem 2.2. Let X, Y be Banach spaces of analytic functions, and let λ0z and λ1z be linear functionals on X and Y defined by λ0z f = f (z) and λ1z f = f (z). Suppose λ0z and λ1z are bounded. a) Suppose Sg maps X boundedly into Y . Then |g(z)| ≤ Sg

λ1z Y λ1z X

b) Suppose Tg maps X boundedly into Y . Then |g (z)| ≤ Tg

λ1z Y λ0z X

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Proof. Note that, for f ∈ X, |f (z)||g(z)| = |λ1z Sg (f )| ≤ λ1z Y Sg f X Since supf X =1 |f (z)| = λ1z X , taking the sup over f X = 1 of both sides gives us 1 λz |g(z)| ≤ Sg λ1z . X

Y

Hence a). Similarly,

|f (z)||g (z)| = |λ1z Tg (f )| ≤ λ1z Y Tg f X .

Taking the sup over f with norm 1, we get 0 λz |g (z)| ≤ Tg λ1z Y . X

This completes the proof.

Corollary 2.3. If X is a Banach space of analytic functions on which point evaluation of the derivative is a bounded linear functional, and Sg is bounded on X, then g is bounded. In [6], we see a similar result for Mg , i.e., boundedness of Mg on a Banach space in which point evaluation is bounded implies g is bounded. On the Hardy and Bergman spaces on the disk, both Mg and Sg are bounded if and only if g is bounded. That this is necessary for Sg is Corollary 2.3. That it is sufﬁcient follows from integration by parts since Tg and Mg are bounded if g is bounded. (see [1,2] concerning Tg ) A similar situation holds for B. Proposition 2.4. Sg is bounded on B if and only if g ∈ H ∞ . Proof. It is clear g ∈ H ∞ implies Sg is bounded, since Sg f B = sup(|f (z)||g(z)|(1 − |z|2 )) ≤ gH ∞ f B z∈D

The converse follows from Corollary 2.3.

3. The Property of Being Bounded Below An operator T is said to be bounded below if there exists C > 0 such that T f ≥ Cf for all f . It typically is the case for one-to-one operators on Banach spaces that boundedness below is equivalent to having closed range. The analogue of Theorem 3.2 for composition operators is found in Cowen and MacCluer [4]. We include the proof for Tg and Sg , essentially the same, for easy reference. Lemma 3.1. Tg is one-to-one for nonconstant g. Proof. If Tg f1 = Tg f2 , taking derivatives gives f1 (z)g (z) = f2 (z)g (z). Thus f1 (z) = f2 (z) except possibly at the (isolated) points where g vanishes. Since f1 and f2 are analytic, f1 = f2 . When considering the property of being bounded below for Sg , we note that Sg maps any constant function to the 0 function. Thus, it is only useful to consider spaces of analytic functions modulo the constants.

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Theorem 3.2. Let Y be a Banach space of analytic functions on the disk. For nonconstant g, Tg is bounded below on Y if and only if it has closed range. Sg is bounded below on Y /C if and only if it has closed range on Y /C. Proof. Assume Tg is bounded below, i.e., there exists ε > 0 such that Tg f ≥ εf for all f . Suppose {Tg fn } is a Cauchy sequence in the range of Tg . Since fn − fm Tg fn − Tg fm , {fn } is also a Cauchy sequence. Letting f = lim fn , we have Tg fn → Tg f , showing Tg fn converges in the range of Tg . Hence the range is closed. Conversely, assume Tg : Y → Y is closed range. Let {fn } be a sequence in Y such that Tg fn → 0. Tg is one-to-one by Lemma 3.1. Let the closed range of Tg be X. X is a Banach space, and we can deﬁne the inverse Tg−1 : X → Y . Suppose {xn } converges to x = Tg h in X, and Tg−1 xn converges to y in Y . Applying Tg to {Tg−1 xn }, this means xn converges to Tg y. Hence Tg y = Tg h. Since Tg is one-to-one, y = h, and x = Tg−1 y. By the Closed Graph Theorem, Tg−1 is continuous. Thus, fn = Tg−1 (Tg fn ) → 0, implying Tg is bounded below. The same argument holds for Sg as well, but only on spaces modulo constants, since Sg is not one-to-one otherwise. We will show that Tg is never bounded below on H 2 , B, nor BM OA. The sequence {z n } demonstrates the result in each space, since the functions z n have norm comparable to 1, independent of n. (Lemma 2.1) Theorem 3.3. Tg is never bounded below on H 2 , B, nor BM OA. Proof. Let fn (z) = z n . For H 2 ,

lim Tg fn 2 ≈ lim

n→∞

n→∞

|z n |2 |g (z)|2 (1 − |z|2 ) dA(z)

D

We assume Tg is bounded, so g ∈ BM OA by a result of Aleman and Siskakis [2]. Thus μg is a Carleson measure, allowing us to bring the limit inside the integral by the Dominated Convergence Theorem. 2 lim |z n |2 |g (z)|2 (1 − |z|2 ) dA(z) = 0. lim Tg fn ≈ n→∞

n→∞

D

Since fn 2 = 1 for all n, Tg is not bounded below. If Tg is bounded on B, then, by Theorem 2.2, |g (z)|(1 − |z|) = O((1/ log(1/(1 − |z|))) as |z| → 1. Hence Tg fn B = sup |z n ||g (z)|(1 − |z|) sup rn z∈D

0≤r<1

1 . log(2/1 − r)

Given ε > 0, there exists δ < 1 such that 1/ log(2/(1 − r)) < ε for δ < r < 1. For large n, rn < ε for 0 < r < δ. Thus, limn→∞ Tg fn B = 0, and Lemma 2.1 implies Tg is not bounded below on B. On BM OA, Siskakis and Zhao proved Tg being bounded implies g ∈ V M OA [11].

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n→∞

1 ≈ lim sup n→∞ I |I|

Tg fn 2∗

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|z n |2 |g (z)|2 (1 − |z|2 ) dA(z).

S(I)

Let I be an arc in ∂D, and let ε > 0. Since g ∈ V M OA, there exists δ > 0 such that 1 |g (z)|2 (1 − |z|2 ) dA(z) < ε whenever |J| < δ. |J| S(J)

If |I| > δ, divide I into K disjoint intervals of length approximately δ, so I = ∪K i=1 Ji , δ/2 < |Ji | < δ

for all i, and δK ≈ |I|.

Let Sδ (I) = S(I) − ∪i S(Ji ). For large n, (1 − δ/2)2n ≤ ε|I|, and to estimate the integral over Sδ (I) we use the fact that μg is a Carleson measure. 1 |z n |2 |g (z)|2 (1 − |z|2 ) dA(z) |I| S(I)

=

1 |I|

|z n |2 |g (z)|2 (1 − |z|2 ) dA(z)

Sδ (I) K

+

1 |I| i=1

|z n |2 |g (z)|2 (1 − |z|2 ) dA(z)

S(Ji )

≤

1 1 (1 − δ/2)2n Cg2∗ + Kδε ε |I| |I|

for large n. Hence limn→∞ Tg fn ∗ = 0 and Tg is not bounded below on BM OA. In contrast to Theorem 3.3, Tg can be bounded below on weighted Bergman spaces. We state the result here, but the key is Proposition 3.5, proved afterward. Theorem 3.4. Let 1 ≤ p < ∞, α > −1. Tg is bounded below on Apα if and only if there exist c > 0 and δ > 0 such that |{z ∈ D : |g (z)|(1 − |z|2 ) > c} ∩ S(I)| > δ|I|2 . Proof. We must assume Tg is bounded on Apα . By Theorem 2.2, g ∈ B. (That this is also sufﬁcient for Tg to be bounded on Ap0 is in [1].) Tg is bounded below on Apα if and only if p Tg f Apα ≈ |f (z)|p |g (z)|p (1 − |z|2 )α+p dA(z) f pApα . D

By Proposition 3.5, this is true if and only if there exist c > 0 and δ > 0 such that |{z ∈ D : |g (z)|p (1 − |z|2 )p > c} ∩ S(I)| > δ|I|2 for all arcs I ⊆ ∂D. If this holds for some p it holds for all p.

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The proof of [10, Proposition 5.4] shows this result is nonvacuous. Ramey and Ullrich construct a Bloch function g such that |g (z)|(1−|z|) > c0 if 1 − q −(k+1/2) ≤ |z| ≤ 1 − q −(k+1) , for some c0 > 0, q some large positive integer, and k = 1, 2, . . .. Given a Carleson square S(I), let kI be the least positive integer such that q −kI +1/2 ≤ |I|. The annulus E = {z : 1 − q −(kI +1/2) ≤ |z| ≤ 1 − q −(kI +1) } intersects S(I), and |E ∩ S(I)| ≈ |I|((1 − q −(kI +1) ) − (1 − q −(kI +1/2) )) (q 1/2 − 1) 2 q 1/2 − 1 ≥ |I| . q kI +1 q 3/2 Setting c = c0 and δ ≈ 1/q show Theorem 3.4 holds for this example of g, and Tg is bounded below on Apα . We deﬁne H0p = H p /C = {f ∈ H p : f (0) = 0}. The operator Sg can clearly be bounded below, since g(z) = 1 gives the identity operator. A result due to Luecking (see [4, 3.34]) leads to a characterization of functions for which Sg is bounded below on H02 and Apα /C. We state a reformulation useful to our purposes here. = |I|

Proposition 3.5 (Luecking). Let τ be a bounded, nonnegative, measurable function on D. Let Gc = {z ∈ D : τ (z) > c}, 1 ≤ p < ∞, and α > −1. There exists C > 0 such that the inequality p α |f (z)| τ (z)(1 − |z|) dA(z) ≥ C |f (z)|p (1 − |z|)α dA(z) D

D

holds if and only if there exist δ > 0 and c > 0 such that |Gc ∩ S(I)| ≥ δ|I|2 for every interval I ⊂ T . The proof is omitted. Using the Littlewood–Paley identity we get the following: Corollary 3.6. Sg is bounded below on H02 if and only if there exist c > 0 and δ > 0 such that |Gc ∩ S(I)| ≥ δ|I|2 , where Gc = {z ∈ D : |g(z)| > c}. We use Corollary 3.6 to construct a nonexample of boundedness below of Sg on H02 , and compare Mg on H 2 to Sg on H02 . If g(z) is the singular inner z+1 function exp( z−1 ), Sg is not bounded below on H02 . To see this, ﬁx c ∈ (0, 1). Gc is the complement in D of a horodisk, a disk tangent to the unit circle, log c+1 with radius r = 2(log c−1) and center 1 − r. Choosing a sequence of intervals In ⊂ T such that 1 is the center of In and |In | → 0 as n → ∞, we see |Gc ∩ S(In )| →0 |In |2

as n → ∞,

meaning Sg is not bounded below on H02 . Mg is bounded below on H 2 if and only if the radial limit function of g ∈ H ∞ is essentially bounded away from 0 on ∂D. ([8] has this result as a special case of weighted composition operators.) Theorem 3.8 will show this is weaker than the condition for Sg to be bounded below on H02 . The example above of a singular inner function then shows it is strictly weaker. To prove Theorem 3.8 we use a lemma which allows us to estimate an analytic function

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inside the disk by its values on the boundary. Deﬁne the conelike region with aperture α ∈ (0, 1) at eiθ to be

|eiθ − z| <α . Γα (eiθ ) = z ∈ D : 1 − |z| For a function g ∈ H(D), deﬁne the nontangential limit function, for almost all eit , |g ∗ (eit )| =

lim

Γα (eit ) z→eit

|g(z)|.

For any arc I ⊆ ∂D and 0 < r < 2π/|I|, rI will denote the arc with the same center as I and length r|I|. We deﬁne the upper Carleson rectangle Sε (I) = {reit : 1 − |I| < r < (1 − ε|I|), eit ∈ I},

and

S + (I) = S1/2 (I).

Lemma 3.7. Given (1 >) ε > 0 and a point eiθ such that |g ∗ (eiθ )| < ε, there exists an arc I ⊂ ∂D such that |g(z)| < ε for z ∈ Sε (I). Proof. We can choose α close enough to 1 so that Sε (I) ⊂ Γα (eiθ ) for all I centered at eiθ with, say, |I| < 1/4. If |g ∗ (eiθ )| < ε, there exists δ > 0 such that z ∈ Γα (eiθ ),

|z − eiθ | < δ imply |g(z)| < ε.

Choosing I such that S(I) is contained in a δ-neighborhood of eiθ ﬁnishes the proof. Theorem 3.8. If Sg is bounded below on H02 , then Mg is bounded below on H 2 . Proof. Assume Mg is not bounded below on H 2 . Let ε > 0. The radial limit function of g equals g ∗ almost everywhere, so there exists a point eiθ such that |g ∗ (eiθ )| < ε. By Lemma 3.7, there exists S(I) such that |{z : |g(z)| ≥ ε} ∩ S(I)| ≤ ε|I|. Since ε was arbitrary, this violates the condition in Proposition 3.5. We now characterize the symbols g which make Sg bounded below on the Bloch space. It turns out to be a common condition appearing in a few different forms in the literature. The condition appears in characterizing Mg on A20 in McDonald and Sundberg [9]. Our main result is equivalence of (i)-(iii) in Theorem 3.9, and we give references with brief explanations for (iv)–(vi). Theorem 3.9. The following are equivalent for g ∈ H ∞ : (i) g = BF for a finite product B of interpolating Blaschke products and F such that F, 1/F ∈ H ∞ . (ii) Sg is bounded below on B/C. (iii) There exist r < 1 and η > 0 such that for all a ∈ D, sup z∈D(a,r)

|g(z)| > η.

(iv) Sg is bounded below on H02 . (v) Mg is bounded below on Apα for α > −1. (vi) Sg is bounded below on Apα /C for α > −1.

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Proof. (i) ⇒ (ii): Note that Sg1 g2 = Sg1 Sg2 for any g1 , g2 . It follows that if Sg1 and Sg2 are bounded below then Sg1 g2 is also bounded below. We will show that SF and SB are bounded below, implying the result for Sg . It is necessary that g ∈ H ∞ for Sg to be bounded on B. (Corollary 2.3) If F , 1/F ∈ H ∞ , then SF f = sup |F (z)||f (z)|(1 − |z|2 ) ≥ (1/1/F ∞ )f B . z∈D

Hence SF is bounded below. By virtue of the fact beginning this proof, we may assume B is a single interpolating Blaschke product without loss of generality. Let {wn } be the zero sequence of B, so wn − z . B(z) = eiϕ 1 − wn z n Denote the pseudohyperbolic metric ρ(z, w) =

|w − z| , |1 − wz|

for any z, w ∈ D.

For the pseudohyperbolic disk of radius d > 0 and center w ∈ D, we use the notation D(w, d) = {z ∈ D : ρ(z, w) < d}. 1−w z

j Let Bj be B without its jth zero, i.e., Bj (z) = wj −z B(z). Since B is interpolating, there exist δ > 0 and r > 0 such that, for all j, |Bj (z)| > δ whenever z ∈ D(wj , r). In particular, the sequence {wn } is separated, so shrinking r if necessary, we may assume

inf ρ(wk , wj ) > 2r.

j =k

We compare f to SB f = supz∈D |B(z)||f (z)|(1 − |z|2 ). Let a ∈ D be a point where the supremum deﬁning the norm of f is almost achieved, say, |f (a)|(1 − |a|2 ) > f /2. Consider the pseudohyperbolic disk D(a, r). Inside D(a, r) there may be at most one zero of B, say wk . We examine three cases depending on the location and existence of wk . If r/2 ≤ ρ(wk , a) < r, then |B(a)| =

|wk − a| |Bk (a)| > (r/2)δ. |1 − wk a|

Thus we would have SB f ≥ |B(a)||f (a)|(1 − |a|2 ) > (r/2)δf /2, and Sg would be bounded below. On the other hand, suppose ρ(wk , a) < r/2. Consider the disk D(wk , r/2), which is contained in D(a, r). The expression 1−|z|2 is roughly constant on a pseudohyperbolic disk, i.e., sup (1 − |z|2 ) > Cr (1 − |a|2 )

z∈D(a,r)

for some Cr > 0.

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Cr does not depend on a, and is near 1 for small r. By the maximum principle for f , there exists a point za ∈ ∂D(wk , r/2) where |f (za )|(1 − |za |2 ) > |f (a)|Cr (1 − |a|2 ) > Cr f /2. (Since ρ(wk , a) < r/2 and ρ(za , wk ) = r/2, we have ρ(za , a) < r.) This shows that Sg is bounded below, for SB f ≥ |B(za )||f (za )|(1 − |za |2 ) > ρ(wk , za )|Bk (za )|Cr f /2 > (r/2)δCr f /2. Finally, suppose no such wk exists. Then the function ((a − z)/(1 − az))B(z) is also an interpolating Blaschke product, and the previous case applies with wk = a. (ii) ⇒ (iii): Assume (iii) fails. Given ε > 0, choose r near 1 so that 1 − r2 < ε, and choose a ∈ D such that |g(z)| < ε for all z ∈ D(a, r). Consider the test function fa (z) = (a − z)/(1 − az). By a well-known identity, (1 − |z|2 ) |fa (z)| = 1 − (ρ(a, z))2 . Thus fa ∈ B with fa = 1 for all a ∈ D. (The seminorm is 1, but the true norm is between 1 and 2 for all a.) By supposition on g, Sg fa = sup |g(z)||fa (z)|(1 − |z|2 ) z∈D

= max

sup z∈D(a,r)

|g(z)||fa (z)|(1

2

− |z| ),

≤ max

sup z∈D(a,r)

|g(z)|fa ,

sup z∈D\D(a,r)

sup z∈D\D(a,r)

|g(z)||fa (z)|(1

2

− |z| )

2

|g(z)|(1 − r )

< max{ε, g∞ ε} ≤ ε(g∞ + 1). Since fa = 1 and ε was arbitrary, Sg is not bounded below. (iii) ⇒ (i): Assuming (iii) holds, we ﬁrst rule out the possibility that g has a singular inner factor. We factor g = BIg Og where B is a Blaschke product, Ig a singular inner function, and Og an outer function. Let ν be the measure on ∂D determining Ig , so iθ e +z dν(θ) . Ig (z) = exp − eiθ − z Let ε > 0. For any α > 1 and for ν-almost all θ, there exists δ > 0 such that z ∈ Γα (eiθ ),

|z − eiθ | < δ imply |Ig (z)| < ε.

(3.1)

This is [7, Theorem II.6.2]. δ may depend on θ and α, but for nontrivial ν there exists some θ where (3.1) holds. Given r < 1, choose α < 1 such that, for every a near eiθ on the ray from 0 to eiθ , the pseudohyperbolic disk

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D(a, r) is contained in Γα (eiθ ). The disk D(a, r) is a euclidean disk whose euclidean radius is comparable to 1 − a. For a close enough to eiθ , z ∈ D(a, r)

implies |z − eiθ | < δ.

Hence supz∈D(a,r) |g(z)| < εg. This violates (iii), so ν must be trivial, and Ig ≡ 1. A similar argument handles the outer function Og . If for all ε > 0 there exists eit such that |Og∗ (eit )| < ε, we apply Lemma 3.7. The upper Carleson square in Lemma 3.7 contains some pseudohyperbolic disk that violates (iii), so Og∗ is essentially bounded away from 0. There exists η > 0, such that |Og∗ (eit )| ≥ η almost everywhere. Note 1/Og ∈ H ∞ , since for all z ∈ D, 1 log |Og (z)| = 2π

2π

log |Og∗ (eit )|

0

1 − |z|2 dt ≥ log η. |eit − z|2

We have reduced the symbol to a function g = BF , where F, 1/F ∈ H ∞ and B is a Blaschke product, say with zero sequence {wn }. We will show that the measure μB = (1 − |wn |2 )δwn is a Carleson measure, implying B is a ﬁnite product of interpolating Blaschke products. (see, e.g., [9, Lemma 21]) Let r < 1 and η > 0 be as in (iii), so supz∈D(a,r) |B(z)| > η for all a. Given any arc I ⊆ ∂D, we may choose aI and zI such that D(aI , r) ⊆ S(I), zI ∈ D(aI , r), |B(zI )| > η, and (1 − |zI |) ≈ |I| as I varies. μB (S(I)) = (1 − |wnk |2 ) where the subsequence {wnk } = {wn } ∩ S(I). Assume without loss of generality that |I| < 1/2, so |wnk | > 1/2 for all k. This ensures |1 − wnk zI | ≈ |I|. Thus we have

(1 − |zI |2 )(1 − |wn |2 ) 1 k (1 − |wnk |2 ) ≈ |I| |1 − wnk zI |2 k k

= 1 − (ρ(zI , wnk ))2 k

<2

1 − ρ(zI , wn )

n

≤−

n

= − log

log ρ(zI , wn ) |wn − zI | |1 − wn zI | n

= − log |B(z)| ≤ − log η. This shows μB is a Carleson measure. (i) ⇔ (iv) ⇔ (v) ⇔ (vi) Bourdon shows in [3, Theorem 2.3, Corollary 2.5] that (i) is equivalent to the reverse Carleson condition in Corollary 3.6 above, hence (i) ⇔ (iv). This reverse Carleson condition also characterizes boundedness below of Mg on weighted Bergman spaces by Proposition 3.5. Thus (iv) ⇔ (v). A key connection is between Sg and Mg via the differentiation operator and equation

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(2.1), since (Sg f ) = Mg f . The following diagram is commutative: Sg

Apα /C −−−−→ Apα /C ⏐ ⏐ ⏐ ⏐ f →f f →f Mg

Apα+p −−−−→ Apα+p This explains (v) ⇒ (vi). Since A2−1 = H 2 , we can combine (iv) and (vi) to say Sg is bounded below on A2α /C for α ≥ −1.

Concluding Remarks We suspect the results about H 2 can be extended to all H p , 1 ≤ p < ∞, but without the Littlewood–Paley identity the proof is more difﬁcult. Generalizing the results on Bloch to the α-Bloch spaces can be done with adjusted test functions as in [13]. Finally, we have partial results concerning Sg being bounded below on BM OA, but have not completed proving a characterization like the one in Theorem 3.9. Acknowledgements I would like to thank my advisor Dr. Wayne Smith for his invaluable guidance. I also wish to thank the referee for his time and encouraging comments.

References [1] Aleman, A., Cima, J.A.: An integral operator on H p and Hardy’s inequality. J. Anal. Math. 85, 157–176 (2001) [2] Aleman, A., Siskakis, A.G.: Integration operators on Bergman spaces. Indiana Univ. Math. J. 46(2), 337–356 (1997) [3] Bourdon, P.S.: Similarity of parts to the whole for certain multiplication operators. Proc. Am. Math. Soc. 99(3), 563–567 (1987) [4] Cowen, C., MacCluer, B.: Composition Operators on Spaces of Analytic Functions. CRC Press, New York (1995) [5] Dostani´c, M.R.: Integration operators on Bergman spaces with exponential weight. Rev. Mat. Iberoam. 23(2), 421–436 (2007) [6] Duren, P.L., Romberg, B.W., Shields, A.L.: Linear functionals on H p spaces with 0 < p < 1. J. Reine Angew. Math. 238, 32–60 (1969) [7] Garnett, J.B.: Bounded Analytic Functions. Revised First Edition. Springer, New York (2007) [8] Kumar, R., Partington J.R.: Weighted composition operators on Hardy and Bergman spaces. Recent Advances in Operator Theory, Operator Algebras, and Their Applications. Oper. Theory Adv. Appl., vol. 153, pp. 157–167. Birkhuser, Basel (2005) [9] McDonald, G., Sundberg, C.: Toeplitz operators on the disc. Indiana Univ. Math. J. 28(4), 595–611 (1979)

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[10] Ramey, W., Ullrich, D.: Bounded mean oscillation of Bloch pull-backs. Math. Ann. 291(4), 591–606 (1991) [11] Siskakis, A.G., Zhao, R.: A Volterra type operator on spaces of analytic functions. Function spaces (Edwardsville, IL, 1998). Contemp. Math., vol. 232, pp. 299–311. Am. Math. Soc., Providence (1999) [12] Stegenga, D.A.: Bounded Toeplitz operators on H 1 and applications of the duality between H 1 and the functions of bounded mean oscillation. Am. J. Math. 98(3), 573–589 (1976) [13] Zhang, M., Chen, H.: Weighted composition operators of H ∞ into α-Bloch spaces on the unit ball. Acta Math. Sin. (Engl. Ser.) 25(2), 265–278 (2009) [14] Zhu, K.: Operator theory in function spaces. Second edition. Mathematical Surveys and Monographs, vol. 138. Am. Math. Soc., Providence (2007) Austin Anderson (B) Department of Mathematics University of Hawaii Honolulu, HI 96822, USA e-mail: [email protected] Received: March 10, 2010. Revised: July 8, 2010.

Integr. Equ. Oper. Theory 69 (2011), 101–111 DOI 10.1007/s00020-010-1821-8 Published online July 14, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Noncompactness of Integral Operators Modeling Diffuse-Gray Radiation in Polyhedral and Transient Settings ´ Pierre-Etienne Druet and Peter Philip Abstract. While it is well-known that the standard integral operator K of (stationary) diffuse-gray radiation, as it occurs in the radiosity equation, is compact if the domain of radiative interaction is sufficiently regular, we show noncompactness of the operator if the domain is polyhedral. We also show that a stationary operator is never compact when reinterpreted in a transient setting. Moreover, we provide new proofs, which do not use the compactness of K, for 1 being a simple eigenvalue of K for connected enclosures, and for I − (1 − )K being invertible, provided the emissivity does not vanish identically. Mathematics Subject Classification (2010). Primary 45P05; Secondary 45C05. Keywords. Radiosity equation, noncompact, integral operator, diffuse-gray radiation, polyhedral domain, transient setting.

1. Introduction Accounting for diffuse-gray radiative heat transfer is important for the accurate modeling of processes that involve heat transfer at high temperatures through transparent or semi-transparent media with nonspecular surfaces, crystal growth from melt or vapor being two examples [4,11]. The physical modeling of radiative heat transfer is well-understood [14,19], where diffuse-gray radiative heat transfer between points on the surface Σ of a transparent cavity are described by the radiosity equation involving the linear integral operator K (see (1.1) and (1.2) below for details). The same equation also plays an important role in computer graphics applications [3,8]. In [1,2,5,6,12,13], the Lp theory of the radiosity equation is employed to couple the radiosity equation to the heat equation in both stationary and transient This research is supported by the DFG Research Center “Mathematics for Key Technologies” Matheon (FZT 86) in Berlin.

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settings, developing the corresponding existence theory. More regular solutions of the radiosity equation are desirable in the context of optimal control [15,16]. Under suitable hypotheses, one can obtain solutions in Sobolev spaces and in spaces of continuous functions (see [9] and references therein). If Σ is sufﬁciently regular (e.g. at least C 1,α ), then K is known to be compact on Lp (Σ) [5,17,21]. In [12], the compactness of K is used to show that, if Σ is the surface of a connected enclosure, then the eigenvalue 1 of K is simple [12, Lem. 1(iv)] and I − (1 − )K is invertible for a nonvanishing emissivity [12, Lem. 2]. These results are then exploited to couple the radiosity equation to the heat equation and to develop the corresponding existence theory. However, in applications, the involved domains often lack the regularity needed to obtain compactness of K. In Th. 2.1 below, we will show that K is noncompact for polyhedral domains. However, we will also show in Theorems 2.2 and 2.4 how to obtain the abovementioned properties of K without employing compactness. In Sect. 3, we prove that K as well as other stationary bounded linear operators can never be compact if reinterpreted as time-dependent operators in a transient setting, which has sometimes been incorrectly assumed in the literature (cf. remark after the proof of Th. 3.2). For the space domain Ω ⊆ R3 , we assume it consists of two parts Ωs and Ωg , where Ωs represents an opaque solid and Ωg represents a transparent gas. More precisely, we assume (A-1) Ω = Ωs ∪ Ωg , Ωs ∩ Ωg = ∅, and each of the sets Ω, Ωs , Ωg , is a nonvoid, bounded, open subset of R3 such that the interface surface Σ := Ωs ∩ Ωg is Lipschitz and piecewise C 1 , i.e. Σ can be partitioned into ﬁnitely many C 1 -surfaces. (A-2) Ωg is enclosed by Ωs , i.e. Σ = ∂Ωg (see Fig. 1). We will state additional hypotheses on the domains where needed. The above mentioned radiosity equation modeling diffuse-gray radiative heat transfer between points on the surface Σ reads (I − (1 − )K) (R) = σθ4 ,

(1.1)

where I denotes the identity operator, θ represents absolute temperature, represents the emissivity of the solid, σ ∈ R+ represents the Boltzmann radiation constant, R = R(θ) represents radiosity, and K is the linear integral operator deﬁned by (1.2) K(ρ)(x) := V (x, y) ω(x, y) ρ(y) dy for a.e. x ∈ Σ, Σ

ω(x, y) :=

(n(y) · (x − y)) (n(x) · (y − x)) π ((y − x) · (y − x))

0 if Σ ∩ ]x, y[= ∅, V (x, y) := 1 if Σ ∩ ]x, y[= ∅

2

for a.e. (x, y) ∈ Σ × Σ, (1.3)

for (x, y) ∈ Σ × Σ,

(1.4)

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Figure 1. Possible shape of a 2-dimensional section through the 3-dimensional domain Ω = Ωs ∪ Ωg . Here, Ωg has the 3 connected components Ωg,1 , Ωg,2 , Ωg,3 . The picture illustrates that Σ = ∂Ωg can have fewer connected components than Ωg . Note also that, according to (A-2), Ωg is engulfed by Ωs , which can not be seen in the 2-dimensional section ω denoting the view factor, V denoting the visibility factor (being 1 if, and only if, x and y are mutually visible), and n denoting the outer unit normal to the solid domain Ωs , existing almost everywhere on the assumed Lipschitz boundary. Theorem 1.1. Assume (A-1) and (A-2). (a)

The kernel V ω of K is almost everywhere nonnegative (actually positive for V (x, y) = 1), symmetric, and V (z, y)ω(z, y) dy = 1 for a.e. z ∈ Σ. (1.5a) Σ

Moreover, if Ωs and Ωg are polyhedral, then V (z, y)ω(z, y) dy > 0 for every z ∈ Σ,

(1.5b)

Σ

where one can choose each of the finitely many possible values of n(z) if z belongs to more than one face of Σ. (b) For each 1 ≤ p ≤ ∞, the operator K : Lp (Σ) −→ Lp (Σ) given by (1.2) is well-defined, linear, bounded, and positive with K = 1. Proof. See [17, Lem. 3], [20, Lem. 1], and [21, Lem. 2]. For (1.5b), let Σz := {y ∈ Σ : V (z, y) = 1}, and note that, if Σ is a polyhedral enclosure, then meas(Σz ) > 0 for each z ∈ Σ. Since ω(z, y) > 0 for each y ∈ Σz , (1.5b) holds.

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2. Noncompactness of K for Polyhedral Domains It is known that, for polyhedral domains, K is related to the Mellin convolution (see [18]), which is a noncompact operator on all Lp spaces [7]. For conical surfaces Σ, K is known to be noncompact on L2 (Σ) from the spectral analysis in [10]. Here we prove the noncompactness of K for polyhedral domains by providing a simple counterexample. Theorem 2.1. Let p ∈ [1, ∞] and assume (A-1) and (A-2). If Ωs and Ωg are polyhedral, then K : Lp (Σ) −→ Lp (Σ) is not compact. Proof. For the sake of readability and briefness, we present the proof for Ω :=] − 3, 3[3 , Ωg :=]0, 2[×] − 1, 1[×]0, 2[, Ωs := Ω \ Ωg , and leave the adaptation to general polyhedral domains to the reader. For each k ∈ N, we deﬁne − the following subsets A+ k and Ak of Σ = ∂Ωg (see Fig. 2): A+ k := {0} × [−1/(2k), 1/(2k)] × [1/(2k), 1/k], A− k := [1/(2k), 1/k] × [−1/(2k), 1/(2k)] × {0}.

(2.1)

− For each (z, y) ∈ A+ k × Ak , we obtain

n(z) · (y − z) = y1 ≥

1 , 2k

n(y) · (z − y) = z3 ≥

1 . 2k

Figure 2. Illustrating the construction for the proof of Th. 2.1

(2.2)

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Since, on the other hand, z − y2 ≤ ω(z, y) =

√ 3 k ,

105

we estimate

n(z) · (y − z)n(y) · (z − y) k2 ≥ π z − y42 36 π

− for each (z, y) ∈ A+ k × Ak .

(2.3) Fix 1 ≤ p < ∞, and deﬁne, for each k ∈ N, fk (y) := k 2/p χA− (y).

fk : Σ −→ R,

k

(2.4)

Then fk pLp (Σ) = 12 and {fk : k ∈ N} is bounded. Clearly, we also have pointwise convergence fk (y) → 0 for each y ∈ Σ. In particular, if a subsequence of fk were to converge to f in Lp (Σ), then f = 0. However, on the other hand, for each z ∈ A+ k: k 2/p k 2 meas(A− k 2/p k) = . (2.5) (K(fk ))(z) = ω(z, y)fk (y) dy ≥ 36π 72π Σ

In consequence, k2 1 meas(A+ > 0, (2.6) k)= p p 72 π 2 · 72p π p showing that K(fk ) does not converge to 0 in Lp (Σ). For 1 < p < ∞, the pointwise convergence fk → 0 implies weak convergence fk 0 in Lp (Σ) and K(fk ) → 0 shows that K is not compact. For p = 1, consider the sequence gk := f2k . Since, for k = l, int(A− ) and int(A− ) are disjoint, for 2k 2l + + each z ∈ A2k ∪ A2l : (K(gk ))(z) − (K(gl ))(z) = ω(z, y)gk (y) dy + ω(z, y)gl (y) dy K(fk )pLp (Σ) ≥

A+k 2

(2k + 2l )2 ≥ . 72π

A+l 2

(2.7)

Since meas(A+ ∪ A+ ) = 12 (1/22k + 1/22l ), this shows, analogous to (2.6), 2k 2l that K(gk ) − K(gl )L1 (Σ) ≥ 1441 π > 0 for l = k and that K{gk : k ∈ N} is closed, but not compact, hence K is not compact. Finally, for p = ∞, consider the sequence hk := χA− . Then B := {hk : k ∈ N} is bounded in 2k

L∞ (Σ) and the disjointness of the int(A− ) implies that K(B) is closed, but 2k not compact.

The following result was essentially proved as Lemma 1(iv) of [12] for the case where K is compact. We provide a direct proof that does not hinge on the compactness of K. Theorem 2.2. Assume (A-1) and (A-2). If f ∈ L1 (Σ) and K(f ) = f , then f is constant on each Σk , where Σk := ∂Ωg,k is the boundary of a connected component Ωg,k of Ωg (cf. Fig. 1). In particular, if Ωg is connected, then the eigenvalue 1 of K : Lp (Σ) −→ Lp (Σ) is simple (p ∈ [1, ∞], note Lp (Σ) ⊆ L1 (Σ)).

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Proof. Since V (y, z) = 0 if y, z lie in different Σk , we may assume Ωg is connected without loss of generality. We ﬁrst show f takes only one sign on Σ. Introducing the sets Σ+ := {z ∈ Σ : f (z) ≥ 0} and Σ− := {z ∈ Σ : f (z) < 0}, for z ∈ Σ: V (z, y)ω(z, y)f (y) dy + V (z, y)ω(z, y)f (y) dy . f (z) = K(f )(z) = Σ+

Σ−

(2.8) After integrating over Σ+ : ⎛ ⎞ f (z) dz = f (z) ⎝ V (z, y)ω(z, y) dy ⎠ dz Σ+

Σ+

Σ+

+

⎛

f (z) ⎝

Σ−

Σ+

f (z) dz +

≤ Σ+

Σ−

⎞ V (z, y)ω(z, y) dy ⎠ dz ⎛ f (z) ⎝

⎞ V (z, y)ω(z, y) dy ⎠ dz . (2.9)

Σ+

−

From the definition of Σ , we obtain V (z, y)ω(z, y) dy = 0 for a.e. z ∈ Σ− . Σ+

From Th. 1.1(a), we conclude V (z, y) = 0 for almost every z ∈ Σ− and almost every y ∈ Σ+ (Σ− and Σ+ are mutually invisible), implying meas(Σ− ) = 0 or meas(Σ+ ) = 0 due to the assumed connectedness of Ωg . If we now let M := meas(Σ)−1 Σ f denote the mean of f and f˜ := f − M , then (1.5a)

K(f˜) = K(f ) − K(M ) = f − M = f˜,

(2.10)

and we know f˜ takes only one sign on Σ, i.e. f is constant and equal to M . One is usually interested in solving the radiosity equation to obtain R as a function of θ, for example to formulate the coupled conductive-radiative heat ﬂux through Σ, with θ remaining as the only unknown quantity. It is thus desirable to establish the invertibility of I − (1 − )K. It was proved for compact K in [12, Lem. 2]. In the following Th. 2.4, we present a proof that works for polyhedral domains, where we know that compactness is not available. We start by establishing injectivity merely using (A-1) and (A-2). Lemma 2.3. Let p ∈ [1, ∞], and assume (A-1), (A-2). If ∈ L∞ (Σ) with values in [0, 1] is such that, for each connected component Ωg,k of Ωg (cf. Fig. 1), there exists Mk ⊆ Σk := ∂Ωg,k such that Mk has positive surface measure and > 0 on Mk , then the operator (I − (1 − )K) : Lp (Σ) −→ Lp (Σ) p

is injective on L (Σ).

(2.11)

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Proof. As Lp (Σ) ⊆ L1 (Σ), it sufﬁces to consider p = 1. Let f ∈ L1 (Σ). From f = (1 − )K(f ),

(2.12)

we trivially conclude that f vanishes in the set Σ := {z ∈ Σ : (z) = 1}. We extend this conclusion to Σ∗ := {z ∈ Σ : (z) > 0} by noting Th. 1.1(b) f = K(f )L1 (Σ\Σ ) ≤ f L1 (Σ\Σ ) . (2.13) 1 − Σ\Σ

Thus, letting Σ∗k := Σ∗ ∩ Σk , Σ0k := Σk \Σ∗k , we obtain (1 − )(Kk∗ )(f ) = (Kk∗ )(f ) = f , where

Kk∗ ∈ L L1 (Σ0k ), L1 (Σ0k ) , Kk∗ (ρ)(x) := V (x, y)ω(x, y)ρ(y) dy . (2.14) Σ0k

Applying the argument from the proof of Th. 2.2 to Σ+ := {z ∈ Σ0k : f (z) ≥ 0}, Σ− := {z ∈ Σ0k : f (z) < 0}, and Kk∗ instead of K, veriﬁes that f takes only one sign on Σ0k and, hence, on Σk . Thus, on Σk , we have K(f ) ≥ f a.e. if f ≥ 0, K(f ) ≤ f a.e. if f ≤ 0. Since K = 1 by Th. 1.1(b), we obtain K(f ) = f , and Th. 2.2 implies f is constant on each Σk . Finally, according to the hypothesis meas(Mk ) > 0, Mk ⊆ Σ∗k , such that we know f = 0 on Mk , implying f = 0 a.e. on Σk , concluding the proof. For the proof that I − (1 − )K is also surjective, we make use of the following technical condition (A-3). In Lem. 2.5 we will show that Ωs and Ωg being polyhedral is sufﬁcient for (A-3) to hold. (A-3) There exists r0 > 0 such that ess sup V (z, y) ω(z, y) dy < 1, (2.15) z∈Σ

Br0 (z)

where Br0 (z) := {y ∈ Σ : z − y2 < r0 }. Theorem 2.4. Assume the hypotheses of Lem. 2.3 plus (A-3). Then I−(1−)K has an inverse in L(Lp (Σ), Lp (Σ)). Proof. We have to show that, for each g ∈ Lp (Σ), the equation (I − (1 − )K) (f ) = g

(2.16)

has a unique solution f ∈ Lp (Σ). Let r0 > 0 be such that (2.15) holds and deﬁne the auxiliary operators V (z, y) ω(z, y) f (y) dy , (2.17a) (K1 (f ))(z) := Br0 (z)

V (z, y) ω(z, y)f (y) dy .

(K2 (f ))(z) :=

(2.17b)

Σ\Br0 (z)

From K, both K1 and K2 inherit the property of being bounded linear operolder’s inequality and (2.15), we obtain ators from Lp (Σ) into itself. From H¨ K1 L(Lp (Σ),Lp (Σ)) < 1, i.e. I − (1 − )K1 is invertible via the Neumann series

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in L(Lp (Σ), Lp (Σ)). As K = K1 +K2 , applying the inverse (I − (1 − )K1 ) to (2.16) yields the equivalent equation

−1 −1 I − (I − (1 − )K1 ) (1 − )K2 (f ) = (I − (1 − )K1 ) (g). (2.18) −1

With the abbreviation H := (I − (1 − )K1 ) (1 − )K2 , we write the lefthand side of (2.18) as I − H. If we can show that H is compact and I − H is injective, then I − H is invertible by the Riesz–Schauder theorem and we are done. The integral operator K2 is compact, as its kernel k2 (z, y) = χΣ\Br0 (z) (y)V (z, y) ω(z, y)

(2.19)

is uniformly bounded by 1/(πr02 ). In consequence, H is also compact. It remains to prove that I − H is one-to-one, i.e. that 0 is the only solution to the homogeneous version of (2.18). As (2.18) and (2.16) are equivalent, Lem. 2.3 completes the proof. Lemma 2.5. Assuming (A-1), (A-2), and that Ωs , Ωg are polyhedral is sufficient for (A-3) to hold. Proof. Seeking a contradiction, assume there does not exist r0 > 0 such that (2.15) holds. Then there is a sequence Σn ⊆ Σ of measurable sets with meas(Σn ) > 0, such that, for each z ∈ Σn and each n ∈ N: V (z, y)ω(z, y) dy ≥ 1 − 1/n. (2.20a) B1/n (z)

Thus, according to (1.5a), there is a sequence zn in Σ (with zn ∈ Σn ) satisfying V (zn , y)ω(zn , y) dy ≤ 1/n. (2.20b) Σ\B1/n (zn )

Fatou’s lemma now implies lim inf V (zn , y) ω(zn , y) n→∞

= lim inf χΣ\B1/n (zn ) (y)V (zn , y)ω(zn , y) = 0 for a.e. y ∈ Σ. n→∞

(2.21)

As Ωs , Ωg are polyhedral, the outer unit normal n(z) takes only ﬁnitely many values on Σ. As Σ is also compact, there must exist z ∗ ∈ Σ and ξ in the range of n such that, for a subsequence (not relabelled), zn → z ∗ and n(zn ) → ξ. Thus, for almost every y ∈ Σ such that V (z ∗ , y) = 1, we obtain ξ · (y − z ∗ )n(y) · (z ∗ − y) = lim inf n(zn ) · (y − zn )n(y) · (zn − y) = 0. n→∞

(2.22)

However, (2.22) is in contradiction to Σ V (z ∗ , y)ω(z ∗ , y) dy > 0, which must hold according to the polyhedral case of Th. 1.1(a) (z ∗ might lie in the intersection of several faces of Σ, but n(zn ) → ξ guarantees that ξ is the outer unit normal of one of these faces).

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3. Noncompactness of K for Transient Settings Theorem 3.1. For each 1 ≤ p < ∞, the operator ˜ : Lp (0, T, Lp (Σ)) −→ Lp (0, T, Lp (Σ)), (K(ρ))(t) ˜ K := K(ρ(t)), (3.1) is noncompact. Actually, Th. 3.1 is a corollary of the following general result that shows reinterpreting a nontrivial bounded linear operator K : X −→ Y between normed vector spaces X and Y in a transient setting can never result in a ˜ : Lp (0, T, X) −→ Lp (0, T, Y ): compact operator K Theorem 3.2. Let X and Y be normed vector spaces, and let K : X −→ Y be a bounded linear operator. Then, for each 1 ≤ p < ∞, ˜ ˜ : Lp (0, T, X) −→ Lp (0, T, Y ), (K(ρ))(t) := K(ρ(t)), (3.2) K defines a bounded linear operator. If there is x0 ∈ X such that K(x0 ) = 0, ˜ is noncompact. then K ˜ preserves Bochner measurability; the Proof. The continuity of K implies K inequality T p ˜ )(t)p dt ≤ Kp f p p ˜ (3.3) K(f )Lp (0,T,Y ) = K(f Y L (0,T,X) 0

˜ maps Lp (0, T, X) into Lp (0, T, Y ) and is bounded. shows that K Now assume there is x0 ∈ X such that K(x0 ) = 0, let y0 := K(x0 ), δ := x0 X ∈ R+ , and := y0 Y ∈ R+ . Fix p ∈ [1, ∞[. For each n ∈ N, let In := ]T 2−n , T 2−n+1 [

(3.4)

and deﬁne fn ∈ S(0, T, X) as follows: fn : [0, T ] −→ X,

n

fn (t) := 2 p T

−1 p

x0 χIn (t).

(3.5)

Note that (In )n∈N is a sequence of pairwise disjoint measurable subsets of [0, T ] such that meas(In ) = 2−n T . Thus, for each n ∈ N, fn pLp (0,T,X) = fn (t)pX dt = 2n T −1 x0 pX meas(In ) = δ p , (3.6) In

showing that the set B := {fn : n ∈ N} is bounded. Next, it will be shown ˜ that K[B] is closed and noncompact. To that end, for m = n, one computes ˜ n ) − K(f ˜ m )p p K(f

L (0,T,Y )

T =

K (fn (t)) − K (fm (t)) p dt Y

0 p p −1 n mp −1 = 2 T p y0 dt + 2 p T p y0 dt = 2p , Im

Y

In

Y

(3.7)

˜ i.e. the distance between any two distinct elements of K[B] is identical and ˜ ˜ is nonpositive, implying that K[B] is noncompact and closed, showing K compact.

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In [12, Lem. 11], it is incorrectly assumed that stationary compact linear operators are compact when reinterpreted as time-dependent operators in a transient setting. [12, Lem. 11] claims the pseudomonotonicity of a certain transient operator, which is then used to prove existence to a transient heat equation with radiative coupling. For a different existence proof not founding on [12, Lem. 11], see [6]. Acknowledgements We thankfully acknowledge the referee’s helpful suggestions and advise.

References [1] Amosov, A.A.: Nonstationary nonlinear nonlocal problem of radiativeconductive heat transfer in a system of opaque bodies with properties depending on the radiation frequency. J. Math. Sci. 165(3), 1–41 (2010) [2] Amosov, A.A.: Stationary nonlinear nonlocal problem of radiative-conductive heat transfer in a system of opaque bodies with properties depending on the radiation frequency. J. Math. Sci. 164(3), 309–344 (2010) [3] Cohen, M.F., Wallace, J.R.: Radiosity and Realistic Image Synthesis. Academic Press, Cambridge, MA, USA (1993) [4] Dupret, F., Nicod´eme, P., Ryckmans, Y., Wouters, P., Crochet, M.J.: Global modelling of heat transfer in crystal growth furnaces. Int. J. Heat Mass Transf. 33(9), 1849–1871 (1990) ´ Analysis of a coupled system of partial differential equations [5] Druet, P.-E.: modeling the interaction between melt flow, global heat transfer and applied magnetic fields in crystal growth. Ph.D. thesis, Department of Mathematics, Humboldt University of Berlin, Germany, 2008, Available in pdf format at http://edoc.hu-berlin.de/dissertationen/druet-pierre-etienne-2009-02-05/ PDF/druet.pdf ´ Weak solutions to a time-dependent heat equation with nonlocal [6] Druet, P.-E.: radiation boundary condition and arbitrary p-summable right-hand side. Appl. Math. 55(2), 111–149 (2010) [7] Elschner, J.: On spline approximation for a class of non-compact integral equations. Math. Nachr. 146, 271–321 (1990) [8] Gortler, S.J., Schr¨ oder, P., Cohen, M.F., Hanrahan, P.: Wavelet radiosity. Computer Graphics 27, 221–230 (1993) (Proc. SIGGRAPH’93) [9] Hansen, O.: The Radiosity Equation on Polyhedral Domains. Logos Verlag, Berlin (2002) [10] Hansen, O.: On the spectrum of the reflection operator on conical surfaces. Integr. Equ. Oper. Theory 52, 483–503 (2005) [11] Klein, O., Philip, P., Sprekels, J.: Modeling and simulation of sublimation growth of SiC bulk single crystals. Interfaces Free Boundaries 6, 295–314 (2004) [12] Laitinen, M., Tiihonen, T.: Conductive-radiative heat transfer in grey materials. Q. Appl. Math. 59(4), 737–768 (2001) [13] Metzger, M.: Existence for a time-dependent heat equation with non-local radiation terms. Math. Methods Appl. Sci. 22, 1101–1119 (1999)

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[14] Modest, M.F.: Radiative Heat Transfer. McGraw-Hill Series in Mechanical Engineering. McGraw-Hill, New York, USA (1993) [15] Meyer, C., Philip, P., Tr¨ oltzsch, F.: Optimal control of a semilinear PDE with nonlocal radiation interface conditions. SIAM J. Control Optim. 45, 699– 721 (2006) [16] Meyer, C., Yousept, I.: State-constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions. SIAM J. Control Optim. 48, 734–755 (2009) [17] Qatanani, N., Schulz, M.: Analytical and numerical investigation of the Fredholm integral equation for the heat radiation problem. Appl. Math. Comput. 175, 149–170 (2006) [18] Rathsfeld, A.: Edge asymptotics for the radiosity equation over polyhedral boundaries. Math. Methods Appl. Sci. 22, 217–241 (1999) [19] Sparrow, E.M., Cess, R.D.: Radiation Heat Transfer. Hemisphere Publishing Corporation, Washington, D.C., USA (1978) [20] Tiihonen, T.: A nonlocal problem arising from heat radiation on non-convex surfaces. Eur. J. Appl. Math. 8(4), 403–416 (1997) [21] Tiihonen, T.: Stefan–Boltzmann radiation on non-convex surfaces. Math. Methods Appl. Sci. 20(1), 47–57 (1997) ´ Pierre-Etienne Druet Weierstrass Institute for Applied Analysis and Stochastics Mohrenstrasse 39 10117 Berlin Germany e-mail: [email protected] Peter Philip (B) Department of Mathematics Ludwig-Maximilians University (LMU) Munich Theresienstrasse 39 80333 Munich Germany e-mail: [email protected] Received: March 18, 2010. Revised: May 19, 2010.

Integr. Equ. Oper. Theory 69 (2011), 113–132 DOI 10.1007/s00020-010-1825-4 Published online July 16, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Real Linear Operator Theory and its Applications Marko Huhtanen and Santtu Ruotsalainen Abstract. Real linear operators arise in a range of applications of mathematical physics. In this paper, basic properties of real linear operators are studied and their spectral theory is developed. Suitable extensions of classical operator theoretic concepts are introduced. Providing a concrete class, real linear multiplication operators are investigated and, motivated by the Beltrami equation, related problems of unitary approximation are addressed. Mathematics Subject Classification (2010). Primary 47A10; Secondary 47B38. Keywords. Real linear operator on Hilbert space, spectrum, Beurling transform, multiplication operator.

1. Introduction The present paper deals with the theory of bounded real linear operators on a complex separable Hilbert space. The study of complex linear operator theory is classical whereas a more intensive investigation of real linear operators has only fairly recently started, mainly in various mathematical physics applications. Real linearity is central in studies related with problems in planar elasticity [20], in the theory of quasiconformal mappings [15], and in the inverse problem of recovering electrical conductivity distribution in the plane [3]. In this paper basic concepts of bounded real linear operators are developed. The spectrum is studied. The notions of compact and unitary operators are deﬁned. The Beltrami operator is examined from the viewpoint of real linear operator theory. Preceding real linear Toeplitz operator theory, real linear multiplication operators are studied in detail. To elucidate the role of real linear operators, consider the general Beltrami differential equation in the complex plane C ∂f = ν∂f + μ∂f , Supported by the Academy of Finland.

(1.1)

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where ∂ = (∂x + i∂y )/2, ∂ = (∂x − i∂y )/2, z = x + iy, and μ, ν ∈ L∞ (C). It arises in the study of the two dimensional Calder´ on problem and in connection with quasiconformal mappings; see [2,3,14,15]. Under appropriate assumptions, solving (1.1) leads to examining the invertibility of the so-called Beltrami operator I − (μ + ντ )S

(1.2)

on Lp (C)-spaces, where I denotes the identity operator, τ the complex conjugation operator and S is the Beurling transform. In this way, the question concerning the invertibility of the Beltrami operator can be viewed as a real linear spectral problem for the real linear operator (μ + ντ )S. The spectral problem for real linear operators is challenging in general by the fact that the spectrum can be empty. For compact operators, it can contain a continuum; for self-adjoint operators, it is not necessarily real. Thus, the classical complex linear classes of operators need to be carefully inspected in the corresponding real linear case. Aside from these general investigations, we consider a real linear generalization of multiplication operators appearing in (1.2). We study their basic properties and expose their unitary approximation. These investigations pave the way for real linear Toeplitz operator theory. The organization of the paper is as follows. Section 2 is concerned with the general theory of bounded real linear operators and their spectral theory in particular. Recent applications are described. Compact, unitary and ﬁnite rank operators are deﬁned. In Sect. 3, real linear multiplication operators on function spaces are studied. Unitary and scaled unitary approximation of real linear operators of the form (μ + ντ )B, with B complex linear and unitary, is examined.

2. Real Linear Operators and their Spectral Theory 2.1. Basic Definitions Let H be a complex separable Hilbert space. An operator A on H is said to be real linear if A(x + y) = Ax + Ay

and A(λx) = λAx

for all x, y ∈ H and λ ∈ R. The norm of A is deﬁned as A = sup{Ax : x ∈ H, x = 1}. The set of bounded real linear operators is denoted by B(H). Complex linear and antilinear operators represent two extreme cases of real linear operators. An operator A is complex linear, resp. antilinear, if A(λx) = λAx,

resp. A(λx) = λAx,

for all x ∈ H and λ ∈ C. Complex multiplication not being commutative, B(H) is merely a real Banach algebra. Bounded complex linear operators and bounded antilinear operators are real linear subalgebras thereof.

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Of course, the theory of complex linear operators is extensive. On the other hand, real linear operators are abundant in applications, too. Classically, antilinear operators occur in quantum mechanics in the study of time reversal [11, p. 250]. More recent examples are described in what follows. Example 1. The Beltrami equation (1.1) with ν = 0 is central to the two dimensional Calder´ on problem in the inverse problem of ﬁnding L∞ conductivities of a material. There, in the construction of complex geometrical optics solutions, the real linear operator A = P (I − βτ S)−1 ατ

(2.1)

is translated as λI −A which then needs to be inverted. Here P is the Cauchy transform, S the Beurling transform, α ∈ L∞ (C) with support in the unit disk D, and β ∈ L∞ (C) with |β(z)| ≤ kχD (z) for almost all z ∈ C with a constant 0 < k < 1. For more details, see [3, Sect. 4]. Example 2. Purely antilinear operators arise in the study of inverse scattering and nonlinear evolution equations. Namely, the so-called ∂-equation ∂v(k) = −T (k)τ v(k)

(2.2)

2

where v : R → C is assumed to satisfy the condition lim|k|→∞ v(k) = 1 and T : R2 → C to be compactly supported, is then studied. Equation (2.2) is equivalent to 1 MT τ I + C πk v = 1, (2.3) 1 denotes convolution by where 1 is the function having constant value 1, C πk 1 πk and MT the multiplication by T . For further applications and numerical approximation, see [16] and references therein.

2.2. The Complex Linear–Antilinear Representation Every real linear operator A can be represented uniquely as the sum of a complex linear and an antilinear operator. Namely, we have A = A0 + A1

(2.4)

1 2 (A − iAi)

complex linear and A1 = 12 (A + iAi) antilinear. Thus, regarded as having the structure of a Z2 -graded unital algebra

with A0 = B(H) can be over R, i.e. a unital superalgebra over R, where the even elements are complex linear and odd elements are antilinear operators [7, p. 46].

Definition 2.1. An antilinear element κ ∈ B(H) satisfying κ2 = I is said to be an abstract conjugation.1 On any separable Hilbert space H there exists an abstract conjugation by setting κ = ι−1 τ ι, where ι : H → l2 is a Hilbert space isomorphism and τ is the complex conjugation on l2 . Using the representation (2.4) with an abstract conjugation κ, any real linear operator A can be decomposed as A = A0 + B0 κ 1

We do not insist on κ being an isometry which is occasionally required [9].

(2.5)

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with A0 and B0 = A1 κ complex linear. Although appealing, an abstract conjugation may not be natural since it is not unique and depends on the chosen basis of H. Thus, the complex linear–antilinear representation (2.4) can be regarded as canonical in general. In self-conjugate Hilbert spaces, there is a natural decomposition involving the standard complex conjugation. Definition 2.2. A Banach space of complex valued functions is called self-conjugate if it is closed under the complex conjugation τ .2 In self-conjugate spaces, aside from (2.4), a canonical decomposition of a real linear operator A is A = A0 + B0 τ

(2.6)

with unique complex linear operators A0 and B0 . The representation (2.6) is not always possible. Even in realistic applications one must accept dealing with the decomposition (2.4). This is in strong contrast with real linear matrix analysis where the representation (2.6) is always available [13]. Example 3. The so-called Friedrichs operator F of a planar domain D is deﬁned on A2 (D), the Bergman space of square-integrable analytic functions on D [8,20]. Two basic problems in planar elasticity correspond to ﬁnding elements u, v ∈ A2 (D) such that (I + F )u = f,

resp.

(kI − F )v = g,

(2.7)

for some given f, g ∈ A2 (D) and a material constant k ∈ R. The Friedrichs operator on A2 (D) is purely antilinear being deﬁned as F = P τ , where P is the orthogonal projection P : L2 (D) → A2 (D). Because the Bergman space A2 (D) is not self-conjugate, the representation (2.6) does not exist for the Friedrichs operator. It is noteworthy that whether F is of ﬁnite rank or compact, depends on D. For instance, it is known that when D is simply connected and has a C 1,α boundary, F is compact [17]. In view of the preceding example, a Hilbert space H consisting of complex valued functions can be considered as a subspace of the closure of H +τ H denoted here by X. Let us deﬁne F = P τ : H? → H, where P is the orthogonal projection on X onto H. The antilinear operator F can be regarded to measure how self-conjugate H is by the fact that H is self-conjugate if and only if the range of F is H. For example, for A2 (D), where D is the unit disc, F is of rank 1, and for so-called quadrature domains, it is of ﬁnite rank [20]. These Hilbert spaces can be regarded as being very far from self-conjugate. 2.3. The Adjoint of a Real Linear Operator Denote by (·, ·) the inner product in H. The adjoint A∗0 of a complex linear operator A0 is deﬁned as usual by the condition (A0 x, y) = (x, A∗0 y) for all x, y ∈ H. 2 Any Banach space is isometrically isomorphic to a norm closed subspace of C(X), the space of complex valued continuous functions on a compact Hausdorﬀ space, endowed with the supremum norm.

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Deﬁne the adjoint A∗1 of an antilinear operator A1 by (A1 x, y) = (x, A∗1 y) for all x, y ∈ H. Then, for a real linear operator A split according to (2.4), we deﬁne its adjoint by A∗ = A∗0 + A∗1 . This makes B(H) into a real Banach ∗ algebra, i.e. a real Banach algebra with a real linear involution ∗ satisfying A∗∗ = (A∗ )∗ = A

and

(AB)∗ = B ∗ A∗

for all A, B ∈ B(H).3 Definition 2.3. A real linear operator A is called self-adjoint if A = A∗ . For example, the Friedrichs operator in (2.7) is self-adjoint. 2.4. Spectral Theory Although we are dealing with a real linear Banach algebra, complex translates of real linear operators appear regularly in applications. (See the examples above.) In view of this, we set the following definition. Definition 2.4. The spectrum σ(A) of A ∈ B(H) is the set of complex numbers λ, called spectral values, for which λI − A is not boundedly invertible. We call λ ∈ C an eigenvalue of A if there exists a nonzero x ∈ H such that Ax = λx. The point spectrum σp (A) of A consists of the eigenvalues of A. Definition 2.5. [21] The lower bound of an operator A ∈ B(H) is m(A) = inf{Af : f = 1, f ∈ H}. A complex number λ is said to be in the approximate point spectrum σa (A) of A if m(λI − A) = 0. A complex number λ is in the compression spectrum σc (A) of A if the closure of ran(λI − A) differs from H, i.e. λ is an eigenvalue of A∗ . We have σ(A) = σa (A) ∪ σc (A). It is noteworthy that the spectrum can be empty since B(H) is merely 0 1 a real Banach algebra. A ﬁnite dimensional example is given by τ ∈ −1 0 B(C2 ). It is not straightforward to give conditions guaranteeing the existence of the spectrum. It follows from the following proposition that the spectrum of a bounded real linear operator A is compact. Proposition 2.6. Assume A = B + C ∈ B(H) with B invertible and C < m(B). Then A is invertible. Proof. We can factor A = B(I + B −1 C). By using the Neumann series we can C < 1. conclude that the operator I +B −1 C is invertible since B −1 C ≤ m(B) 3

The adjoint can be shown to satisfy Re(Ax, y) = Re(x, A∗ y) for all x, y ∈ H.

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Proposition 2.7. For any μ ∈ C\σ(A) there holds 1 ≤ (μI − A)−1 . dist(μ, σ(A)) Proof. Let α ∈ C be such that |α| < (μI − A)−1 −1 . Then by using the Neumann series, (α + μ)I − A = (μI − A)[(μI − A)−1 α − I] is invertible and α + μ ∈ C\σ(A) from which the result follows. In the complement of σ(A), the resolvent is deﬁned as R(λ; A) = (λI − A)−1 . For |λ| > A, it can be expressed in terms of the Neumann series as ∞ (λ−1 A)k . R(λ; A) = λ−1 k=0

(Hence, maxλ∈σ(A) |λ| ≤ A.) The spectrum of the resolvent can be found by using the following proposition. Proposition 2.8. Suppose A ∈ B(H) be a real linear operator and f (z) = (az + b)(cz + d)−1 , with a, b, c, d ∈ C. If −d/c is not a spectral value of A, then f (σ(A)) = σ(f (A)). Proof. For a polynomial p of degree at most one, we have p(σ(A)) = σ(p(A)). Moreover, from λI − A = λ(A−1 − λ−1 I)A, it follows that if A is invertible, then σ(A−1 ) = 1/σ(A). Combining these observations with −1

b c d c 1 −1 λI − I − − I +A λI − (aA + bI)(cA + dI) = a a a d c c yields the result.

For functions that are not linear fractional transformations, the spectral mapping theorem fails in general. For the adjoint operator there holds σ(A) = σ(A∗ ) by the facts that the operation of taking the adjoint ∗ is additive and that a real linear operator is invertible if and only if its adjoint is. The spectrum is preserved in the orbit of complex linear operators. Proposition 2.9. For an operator A ∈ B(H), set O(A) = {SAS −1 : S complex linear and invertible in B(H)}. Then the spectrum of each element in O(A) coincides with that of A. Proposition 2.10. The spectrum function is upper semi-continuous. This is proved similarly to the complex linear case; see e.g. [4, p. 50]. Since the spectrum may be empty, let us precise: if An → A in operator norm and σ(A) = ∅, then σ(An ) = ∅ for sufﬁciently large n. Theorem 2.11. An operator A ∈ B(H) has a real spectral value r if and only if r is in the spectrum of AC , a canonical complexification of A. Furthermore,

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σp (A) ∩ R = σp (AC ) ∩ R,

119

σa (A) ∩ R = σa (AC ) ∩ R,

σc (A) ∩ R = σc (AC ) ∩ R. Proof. With respect to an abstract conjugation κ, deﬁne the complexiﬁcation of A as4 A B x1 x1 AC : H ⊕ H → H ⊕ H : → , (2.8) x2 x2 B A where A = A + Bκ is split according to the representation (2.5). Let us denote for brevity A = κAκ and x = κx. Clearly, AC is a bounded complex linear operator on H ⊕ H. Note that we have (A∗ )C = (AC )∗ . For a vector of the form (x, x) ∈ H ⊕ H, we have AC (x, x) = (Ax, Ax). Let Ax = rx with x = 0. Then AC (x, x) = (Ax, Ax) = r(x, x), so that σp (A) ∩ R ⊂ σp (AC ) ∩ R. Reasoning similarly, we get the inclusions σa (A) ∩ R ⊂ σa (AC ) ∩ R and σc (A) ∩ R ⊂ σc (AC ) ∩ R. Suppose then that r ∈ σp (AC ) ∩ R, i.e. there is a nonzero (x, y) ∈ H ⊕ H such that Ax + By A B x x = =r . y y B A Bx + Ay Adding the conjugated bottom row to the top row, we get A(x + y) + B(x + y) = r(x + y), i.e. A(x + y) = r(x + y). If x = −y, we have from the top row that Ax − Bx = rx, whence A(ix) = A(ix) + Bκ(ix) = r(ix). Thus r ∈ σp (A) ∩ R. Assume r ∈ σa (AC ) ∩ R, i.e. m(rI − AC ) = 0. This is equivalent to that there is a sequence of vectors (xn , yn ) ∈ H ⊕ H such that (xn , yn )⊕ ≥ ε for some ε > 0 and A − rI B xn = (A − rI)xn + Byn → 0. yn ⊕ B A − rI Bxn + (A − rI)yn ⊕ Since (x, y)2⊕ = x2 + y2 , both (A − rI)xn + Byn and (A − rI)y n + Bxn = Bxn + (A − rI)yn tend to zero. Hence, (A − rI)(xn − y n ) + B(xn − yn ) ≤ (A − rI)xn + Byn +(A − r)y n + Bxn → 0. If the sequence (xn + y n ) is bounded from below by some positive constant, the claim follows. If not, then xn + y n → 0. Since (xn , yn )⊕ is bounded from below, either (xn ) or (yn ) contain a subsequence that is bounded from below by some positive constant. With no loss of generality, assume that (xnk ) is that subsequence. Then (A − rI)xnk − Bxnk ≤ (A − rI)xnk + Bynk + − Bynk − Bxnk → 0, so that (A − rI)ixnk + Bixnk → 0 showing that r ∈ σa (A) ∩ R. 4

The direct sum Hilbert space is equipped as usual with the inner product (x, y)⊕ = (x1 , y1 )H + (x2 , y2 )H .

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Assume r ∈ σc (AC ) ∩ R, i.e. r is an eigenvalue of (AC )∗ Since ∗ ∗ A B ∗ C C ∗ (A ) = (A ) = ∗ , B∗ A reasoning similarly as above with the eigenvalues we can conclude that r ∈ σc (AC ) ∩ R. Although the spectrum of a self-adjoint real linear operator need not lie on the real axis, we do have the following corollary. Corollary 2.12. If A is self-adjoint, then σ(A) ∩ R is nonempty and the spectrum is symmetric with respect to the real axis. Proof. If A is self-adjoint, then also the complex linear AC is self-adjoint. Thus σ(AC ) is real and nonempty and by Theorem 2.11 σ(A) ∩ R is nonempty. Since σ(A) = σ(A∗ ), the spectrum of A is symmetric with respect to the real axis. This combined with Proposition 2.8 yields the following corollary. Corollary 2.13. Assume A = (aB+bI)(cB+dI)−1 for a self-adjoint B ∈ B(H) with a, b, c, d ∈ C. Then the spectrum of A is nonempty. Purely antilinear operators appear often in applications. (See Examples 2 and 3 above.) For them we have the following propositions. Proposition 2.14. Let A be antilinear. Then the point spectrum σp (A), the approximate point spectrum σa (A) and the compression spectrum σc (A) are circularly symmetric with respect to the origin. Proof. Let λ ∈ σa (A), i.e. there is a sequence of unit vectors (fn ) such that Afn − λfn → 0. Then for any θ ∈ R, e2θi λ is an approximate eigenvalue with approximate eigenvectors e−θi fn which can be seen from A(e−θi fn ) − e2θi (e−θi fn ) = eθi (Afn − λfn ) = Afn − λfn . From this it can be seen that the point spectrum is circularly symmetric. Hence the point spectrum of A∗ is also circularly symmetric which implies that the compression spectrum is circularly symmetric. Observe that if A is antilinear, then A2 is complex linear. Proposition 2.15. Let A be antilinear. Then λ ∈ σ(A) if and only if |λ|2 ∈ σ(A2 ). Proof. By Proposition 2.14, we can assume that λ = |λ|. Let λI − A be boundedly invertible where λ ∈ R. Then by symmetry λI + A is also boundedly invertible, and thus (λI − A)(λI + A) = λ2 I − A2 is boundedly invertible. On the other hand, if λ ∈ σa (A), then there is a sequence of unit vectors (fn ) for which (λI − A)fn → 0. Then also (λ2 I − A2 )fn ≤ λI + A(λI − A)fn → 0. If λ ∈ σc (A), then there is a nonzero vector f for which (λI − A∗ )f = 0, and thus (λ2 I − (A∗ )2 )f = 0. Hence λ2 ∈ σc (A).

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Proposition 2.16. If A is antilinear and satisfies A = −A∗ , then σ(A) ⊂ {0}. Proof. By Proposition 2.14, we need to consider only non-negative real numbers. Since A2 = −A∗ A is complex linear self-adjoint and for all x ∈ H we have (A∗ Ax, x) = (Ax, Ax) = (Ax, Ax) ≥ 0, the spectrum of A2 lies in the non-positive real axis. Thus A2 − λ2 I = (A − λI)(A + λI) is invertible for all λ > 0. Thereby only the origin can be in the spectrum of A. For complex linear operators A and B on H, it holds that σ(AB)\{0} = σ(BA)\{0} This stems from the ring-theoretic assertion that, if I − AB is invertible, then I − BA is invertible [10]. For real linear operators an analogy is as follows. Proposition 2.17. Let A be real linear and B complex linear bounded operators. Then σ(AB)\{0} = σ(BA)\{0}. Proof. Let λ = 0. Then AB0 − λI = (ABλ−1 − I)λ = (Aλ−1 B − I)λ is invertible if and only if (BAλ−1 − I)λ is invertible, i.e. (BA − λI) is invertible. These conditions cannot be relaxed in general. Only if B is taken from the other extreme, we have the following identity. Proposition 2.18. Let A be real linear and B antilinear bounded operators. Then σ(AB)\{0} = σ(BA)\{0}. −1

Proof. Let λ = 0. Then AB−λI = (ABλ−1 −I)λ = (Aλ B−I)λ is invertible −1 if and only if (BAλ − I)λ is invertible, i.e. (BA − λI) is invertible. Corollary 2.19. Let A ∈ B(H) be antilinear. Then σ(A2 ) is symmetric with respect to the real axis.5 Proof. Set A = B in Theorem 2.18.

Example 4. Take A = ντ S, where S is the Beurling transform. The norm of A is typically used to establish invertibility of I −ντ S based on the convergence of the Neumann series. For sharper estimates, in view of Proposition 2.15, it is advisable to study the spectrum of the complex linear operator A2 = νS ∗ νS. 2.5. Compact and Unitary Operators In complex linear operator theory, compact operators are regarded as ’small’ and unitary operators as ’large’. Thus, constituting two extremes, compact and unitary operators provide extensively studied classes of operators. For compactness the real linear analogue is straightforward. 5

Propositions 2.14, 2.15, 2.6, 2.17, 2.18 and Corollary 2.19 can be shown to hold as such for bounded real linear operators on a complex Banach space.

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Definition 2.20. A real linear operator A = A0 + A1 split according to (2.4) is said to be compact if A0 and A1 are compact.6 The spectrum of a compact operator can contain a continuum [13]. However, it cannot have interior points by being small in the following sense. Theorem 2.21. Assume A ∈ B(H) is compact and dim H = ∞. Then (i) (ii)

for any line L passing through the origin, σ(A) ∩ L is countable containing the origin with the origin being its only possible limit point, and every spectral value λ = 0 is an eigenvalue.

Proof. We can consider only the real line R, since if the conclusion holds for R, it holds also for any line {reiθ : r ∈ R, with θ ∈ [0, 2π) ﬁxed} by factoring A − reiθ I = eiθ (e−iθ A − rI). If A = A + Bκ is compact, then so is the complexiﬁcation AC given in (2.8) by the following arguments. Let (hn )n∈N ⊂ H ⊕ H be a bounded sequence, where hn = (xn , yn ). Then also (xn ) and (yn ) are bounded sequences. Since A is compact, taking a subsequence four times, we may ﬁnd a subsequence (hn )n∈I , I ⊂ N, such that (Axn ), (Byn ), (Bxn ) and (Ayn ) converge. Thus also (AC hn )n∈I converges, and AC is compact. Since AC is a compact complex linear operator, its spectrum is countable with the origin being its only possible limit point and every nonzero spectral value is an eigenvalue. The result follows from the fact σ(A) ∩ R = σ(AC ) ∩ R. In the inﬁnite dimensional case, the spectrum of A can vanish only if A1 is large. Corollary 2.22. Assume dim H = ∞. For A ∈ B(H) with A1 compact, σ(A) is nonempty. Proof. The result follows from the facts that the essential spectrum of a complex linear operator is always non-empty and compact, and that the essential spectrum is invariant under compact perturbations [18]. Example 5. This is Example 1 continued. It is known that A deﬁned in (2.1) is compact on Lp (C) with 2 < p < k1 and that its spectrum lies in the closed disk centered at origin with radius k (cf. [3] and Proposition 4.1 in particular). That the spectrum is actually nonempty follows from Corollary 2.22. In complex linear operator theory, the way to assess compactness and approximate the spectrum of a compact operator is based on using ﬁnite rank approximants. Analogously, we set the following. 6

This is the same as requiring the image under A of any bounded set in H be relatively compact. Equivalently, for any bounded sequence (fn ) the sequence (Afn ) contains a convergent subsequence. Hence, we have a natural extension of the complex linear notion of compactness.

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Definition 2.23. An operator A ∈ B(H) is said to be of ﬁnite rank, if it can be given as the ﬁnite sum n1 n2 (·, uk )vk + (˜ uk , ·)˜ vk (2.9) A= k=1

k=1

for some uk , vk , u ˜j , v˜j ∈ H, k = 1, . . . , n1 , j = 1, . . . , n2 . The spectrum of ﬁnite rank operators is well understood. Theorem 2.24. Assume dim H = ∞. For an operator A ∈ B(H) given as in (2.9), σ(A) is the union of the origin and a real algebraic plane curve of degree 2(n1 + n2 ) at most. Proof. Since the range of A is contained in a ﬁnite dimensional and hence proper subspace of H, the origin must be in σ(A). The operator A maps H1 = span{uj , u ˜k : j = 1, . . . , n1 , k = 1, . . . , n2 } into H2 = span{vj , v˜k : j = 1, . . . , n1 , k = 1, . . . , n2 }, both subspaces of dimension n1 + n2 at most. Then A = A0 ⊕ 0 with A0 : span{H1 , H2 } → span{H1 , H2 }. Hence, the spectral problem is ﬁnite dimensional. We have A0 x = λx if and only if x ∈ H2 . Thereby we can restrict A0 to H2 . Suppose this restriction is represented as A0 + B0 τ with A0 , B0 ∈ C(n1 +n2 )×(n1 +n2 ) . For a nonzero x ∈ C(n1 +n2 ) it holds A0 x − λx = 0 if and only if A0 − λI A − λI B0 B0 x = 0 ⇐⇒ det 0 = 0. B0 B0 − λI x B0 B0 − λI This determinant is a polynomial in λ and λ and thus its zero set is a real algebraic curve of degree 2(n1 + n2 ) at most. With ﬁnite rank approximations to a compact real linear operator A, Proposition 2.10 and Corollary 2.7 can be used to estimate the spectrum. Although not considered in this paper, there are at least three ways to construct approximations. Certainly, with ﬁnite rank approximations to A0 and A1 , their sum yields a real linear ﬁnite rank approximation to A. However, this is a crude approximation that merely separates the approximation problem in a simple manner. It appears more natural and eﬀective to study approximations in terms of right and left approximation numbers. Definition 2.25. [12] For A ∈ B(H), deﬁne its right approximation numbers as σn (A) = inf{A(I − P ) : P = P ∗ is complex linear, P 2 = I, rank P ≤ n} (2.10) ∗

and its left approximation numbers as σn (A ). For a classical extension of complex linear unitary operators, a real linear operator A is called a symmetry of H if |(Ax, Ay)| = |(x, y)| for every x, y ∈ H [22]. Wigner’s theorem states that every symmetry is either complex linear and unitary or an antilinear bijective isometry [5]. This characterizes symmetries of a Hilbert space thoroughly. This extension is too

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restrictive for our purposes. For an appropriate generalization, we set the following definition. Definition 2.26. A real linear operator A is called unitary if A is a bijective isometry. We denote the group of unitary real linear operators by U(H). Example 6. Assume A = −A∗ . (Then λ ∈ σ(A) ⇐⇒ −λ ∈ σ(A)). Its ∞ k exponential deﬁned as eA = k=0 Ak! is unitary. Proposition 2.27. The spectrum of a unitary operator A ∈ B(H) is a subset of the unit circle. Proof. The origin cannot be in the spectrum, since A is invertible. For a unitary A, we have A = m(A) = 1. If |λ| < 1, then by Proposition 2.6 λI − A is invertible. If |λ| > 1, then it follows that λI − A = λ(I − λ−1 A) is invertible by using the Neumann series. It is noteworthy that the spectrum of a unitary real linear operator can be empty.

3. Real Linear Multiplication Operators on Self-conjugate Hilbert Spaces In this section, motivated by the structure of the Beltrami operator, we study a real linear analogue of multiplication (Laurent) operators and their unitary approximation. 3.1. Basic Properties Recall that the Beurling transform is given as a singular integral operator of Calder´ on–Zygmund type

f (w) 1 dw1 dw2 , w = w1 + iw2 , Sf (z) = − lim π ε→0 (z − w)2 |z−w|>ε

2

for f ∈ L (C). The Beurling transform is unitary which can be seen from the useful identity [1, p. 105] ξ F[f ](ξ), ξ where F denotes the planar Fourier transform

F[f ](z) = e−iπ(zw+zw) f (w) dw1 dw2 . F[Sf ](ξ) =

(3.1)

C

From (3.1) it follows that the spectrum of S is the whole unit circle. Consider the Beltrami operator I − (μ + ντ )S.

(3.2)

Since the Beurling transform is a Fourier multiplier by (3.1), we have F(μ + ντ )SF ∗ = F(μ + ντ )F ∗ m,

(3.3)

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where m(z) = z/z and m denotes also the multiplication operator mf (ξ) = ξ ∗ ξ f (ξ). Here F, F and m are complex linear and unitary. This is an intriguing structure as the operator given by (3.3) is complex linearly unitarily equivalent to μ + ντ .7 In what follows, assume D ⊂ C is a domain, μ, ν ∈ L∞ (D) and 1 ≤ p ≤ ∞ unless otherwise stated. Denote by χA the characteristic function of a set A. Definition 3.1. A real linear multiplication operator μ + ντ is a mapping on Lp (D) deﬁned by f → (μ + ντ )f = μf + νf . Operators of this type can be viewed as being a real linear generalization of the classical multiplication operators, also called Laurent operators, operating on self-conjugate Hilbert spaces. Being the starting point for Toeplitz operator theory, multiplication operators are of central relevance. (See the highly cited paper [6].) Real linear multiplication operators give also rise to multipliers. Definition 3.2. An operator A on L2 (C) is a real linear Fourier multiplier if it can be given as A = F ∗ (μ + ντ )F. Clearly, a real linear multiplication operator μ + ντ is real linear and bounded on Lp (D). Its norm is given as follows. Theorem 3.3. The norm of μ + τ ν on Lp (D) is |μ| + |ν|∞ . Proof. Let p < ∞ and f ∈ Lp (D). We have

|μ(z)f (z) + ν(z)f (z)|p dA(z) ≤ (|μ(z)| + |ν(z)|)p |f (z)|p dA(z) D

D

≤

|μ| + |ν|p∞ |f (z)|p dA(z)

D

= |μ| + |ν|p∞ f pp and therefore μ + ντ ≤ |μ| + |ν|∞ . Set C = |μ| + |ν|∞ , Uε = (|μ| + |ν|)−1 (B(C, ε)) (the preimage of the ball B(C, ε) under the mapping |μ| + |ν|), and mε = A(Uε ) the measure of Uε . Since C is in the essential range of |μ| + |ν|, mε is positive for all ε > 0. Let us deﬁne fε (z) = ei( Then we have that fε pp

=

|ei(

arg ν(z)−arg μ(z)) ) 2

arg ν(z)−arg μ(z)) ) 2

χUε (z)m−1 ε .

p χUε (z)m−1 ε | dA(z) = 1

D 7

Operators A, B ∈ B(H) are complex linearly unitarily equivalent if there exist complex linear unitary operators U0 and V0 such that A = U0 BV0 .

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and that |μ(z)fε (z) + ν(z)fε (z)| = ||μ(z)|ei arg μ(z) ei(

arg ν(z)−arg μ(z)) ) 2

+|ν(z)|ei arg ν(z) e−i( = ||μ(z)|e

arg ν(z)−arg μ(z)) ) 2

|χUε (z)m−1 ε

i 2 (arg μ(z)+arg ν(z)) i

+|ν(z)|e 2 (arg μ(z)+arg ν(z)) |χUε (z)m−1 ε

= (|μ(z)| + |ν(z)|)χUε (z)m−1 ε ≥ (C − ε)χUε (z)m−1 ε . Thus

p

|μ(z)fε (z) + ν(z)fε (z)| dA(z) ≥ D

(C − ε)p χUε (z)m−1 ε dA(z)

C

= (C − ε)p −→ C p ε→0

and therefore μ + ντ = |μ| + |ν|∞ . For p = ∞, we reason similarly. We have μf + νf ∞ = ess sup |μf + νf | ≤ ess sup(|μ| + |ν|)|f | ≤ |μ| + |ν|∞ f ∞ . D

D

μ(z)) i( arg ν(z)−arg ) 2

Let us deﬁne fε (z) = e the one above, we get

χUε (z). With a calculation similar to

ess sup |μfε + νf ε | ≥ ess sup(C − ε) = C − ε −−−→ C. D

Uε

ε→0

Therefore μ + ντ = |μ| + |ν|∞ on L∞ (D).

A real linear multiplication operator μ + ντ is algebraically invertible if |μ(z)| = |ν(z)| a.e. on D. Its algebraic inverse is the real linear multiplication operator |μ|2

1 (μ − ντ ). − |ν|2

This follows by direct calculation (μ − ντ )(μ + ντ ) = |μ|2 − ντ μ + μντ − ντ ντ = |μ|2 − νμτ + μντ − νντ 2 = |μ|2 − |ν|2 . Whenever |μ(z)| = |ν(z)| a.e., we can divide this equation by |μ|2 − |ν|2 . Proposition 3.4. If zero is not in the essential range of |μ|2 − |ν|2 , then the inverse (μ + ντ )−1 is bounded on Lp (D). Proof. Assume zero is not in the essential range of |μ|2 − |ν|2 . Then there exists ε > 0 such that the measure of the preimage of the ball B(0, ε) is zero. Hence, multiplication by (|μ|2 − |ν|2 )−1 is bounded by 1/ε and the result follows.

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The condition on invertibility can be cast in a different form by using the lower bound of an operator. For the lower bound of a real linear multiplication operator we have the following. Theorem 3.5. The lower bound of a real linear multiplication operator μ + ντ is m(μ + ντ ) = ess inf ||μ| − |ν||. D

Proof. Just as in the proof of Theorem 3.3, set C = ess inf ||μ| − |ν||, Vε = (||μ| − |ν||)−1 (B(C, ε)) (the preimage of the ball B(C, ε) under the mapping ||μ| − |ν||), and mε = A(Vε ) the measure of Vε . Since C is in the essential range of ||μ| − |ν||, mε is positive for all ε > 0. Let us deﬁne fε (z) = ei( We have fε pp

=

|ei(

arg ν(z)−arg μ(z))+π ) 2

arg ν(z)−arg μ(z)+π) ) 2

χVε (z)m−1 ε .

p χVε (z)m−1 ε | dA(z) = 1

D

and |μ(z)fε (z) + ν(z)fε (z)| = ||μ(z)|ei arg μ(z) ei(

arg ν(z)−arg μ(z)+π) ) 2

+|ν(z)|ei arg ν(z) e−i( = ||μ(z)|e

arg ν(z)−arg μ(z)+π) ) 2

|χVε (z)m−1 ε

i 2 (arg μ(z)+arg ν(z)+π) i

+|ν(z)|e 2 (arg μ(z)+arg ν(z)−π) |χVε (z)m−1 ε

= (||μ(z)| − |ν(z)||)χVε (z)m−1 ε ≤ (C + ε)χVε (z)m−1 ε . Hence

p

|μ(z)fε (z) + ν(z)fε (z)| dA(z) ≤ D

(C + ε)p χUε (z)m−1 ε dA(z)

D

= (C + ε)p −→ C p . ε→0

p

On the other hand, let f ∈ L (D) with f p = 1. By the triangle inequality, μf + νf ≥ |μf − νf μf − νf = −μf + νf ess inf |μ| − ess sup |ν| ≥ − ess sup |μ| + ess inf |ν| ess inf |μ| + ess inf −|ν| = ess inf −|μ| + ess inf |ν| = ess inf ||μ| − |ν||

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As a corollary to Proposition 3.4 and to Theorem 3.5, we get a characterization for the resolvent set of a real linear multiplication operator. Corollary 3.6. A complex number λ is not in the spectrum of a real linear multiplication operator μ + ντ if and only if there is a constant C > 0 such that ||μ − λ| − |ν|| > C a.e. on D. Proof. Assume 0 is not in the essential range of |μ|2 − |ν|2 = (|μ| + |ν|)(|μ| − |ν|). Then it cannot be in the essential range of |μ| + |ν| or |μ| − |ν|. The latter implies that there is a constant C > 0 such that ||μ| − |ν|| > C a.e. on D. The greatest constant is of course ess inf D ||μ| − |ν||. Conversely, if 0 < ess inf ||μ| − |ν||, then zero is not in the essential range of |μ| − |ν| nor of |μ| + |ν|. Finally, ﬁrst observe that a real linear multiplication operator μ + ντ is self-adjoint if and only if μ is real valued. Second, there are no non-trivial compact multiplication operators. The structure of unitary real linear multiplication operators is the subject of the next paragraph. 3.2. Unitary Approximation The unitary approximation problem of a complex linear operator is typically connected to energy conservation and probability. (For the unitary approximation, see [21].) Another related problem is that of approximating a complex linear operator with a scalar multiple of a unitary operator. Motivated by the Beltrami equation, we will board these subjects in a special case. The invertibility of the Beltrami operator I − (μ + ντ )S is determined by the spectrum of (μ + ντ )S. By the fact that the Beurling transform S is itself complex linear and unitary, it is natural to view (μ + ντ )S as a perturbation of a real linear unitary operator. Next such a unitary operator is constructed. Proposition 3.7. Unitary real linear multiplication operators are of the form χA (z)eiφ(z) + χD\A (z)eiψ(z) τ, where A ⊂ D is a measurable subset, and φ, ψ are some measurable realvalued functions. Proof. Suppose μ + ντ is unitary and there is a ball B such that |μ| > 0 and |ν| > 0 a.e. on B. We have (μf + νf )(μf + νf ) = (|μ|2 + |ν|2 )|f |2 + 2 Re μνf 2 . Set f (z) =

eiπ/4 a(z)e−i arg μ/2 ei arg ν/2 χB , (|μ||ν|)1/2

where a is real valued. Then

Re μνf 2 = 0

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Real Linear Operator Theory and its Applications

a2 = (μ + ντ )f 2 = |μ||ν|

which leads to

B

2

2

2

(|μ| + |ν| )|f | = B

1 1 − |μ| |ν|

B

2

1 1 + 2 2 |ν| |μ|

129

a2

a2 = 0.

Since a was arbitrary, we see that |μ| = |ν| a.e. on B, but this contradicts invertibility of μ + ντ . Hence we must have μ + ντ = χA μ + χD\A ντ for some A ⊂ D. If 0 < |μ| < 1 a.e. on some B ⊂ D, then taking f = χB g, with an arbitrary g ∈ L2 (D), we have

2 2 2 (μ + ντ )f = |μ| |f | < |f |2 = f 2 B

B

which contradicts the isometricity of μ + ντ . Reasoning similarly in the case |μ| > 1 and for the antilinear part ν, we have the result. Observe that if a unitary real linear multiplication operator is complex linear, i.e. if the set D\A has measure zero, then its spectrum is given by the classical complex linear theory as the essential range of eiφ . On the other hand, if the set D\A has positive measure, then by Corollary 3.6 the spectrum is the whole unit circle irrespective of the essential ranges of φ or ψ. The following theorem by Rogers is needed in what follows. Theorem 3.8. [21,19] Let T be a complex linear operator with ind T = 0.8 Then its distance from the set of complex linear unitary operators is max{T − 1, 1 − m(T )} with the lower bound of T being m(T ) = inf(σ(|T |)). This allows us to approximate (μ + ντ )S with a unitary operator. Theorem 3.9. Assume μ + ντ is an invertible real linear multiplication operator and B is complex linear unitary. Then the best real linear unitary approximant of (μ + ντ )B is U = (χA ei arg μ + χC\A ei arg ν τ )B giving the distance (μ + ντ )B − U = max{ess sup |μ| + |ν| − 1, 1 − ess inf ||μ| − |ν||}. Proof. Let μ, ν ∈ L∞ with μ(z) = |μ(z)|ei arg μ(z) and ν(z) = |ν(z)|ei arg ν(z) . Deﬁne A = {z ∈ C : ∃ε > 0 s.t. |1 − |μ|| + |ν| < |1 − |ν|| + |μ| a.e. on B(z, ε)}. 8

The index of an operator A on a Hilbert space H is deﬁned as ind A = dim ker A − dim coker A, where coker A = H/ ran A.

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A candidate for the unitary approximant of (μ + ντ )B is the operator (χA ei arg μ + χC\A ei arg ν τ )B, since (μ + ντ )B − (χA ei arg μ + χC\A ei arg ν τ )B ≤ μ − χA ei arg μ + (ν − χC\A ei arg ν )τ = |μ − χA ei arg μ | + |ν − χC\A ei arg ν |∞ ess supA |μ − χA ei arg μ | + |ν − χC\A ei arg ν | ≤ max ess supC\A |μ − χA ei arg μ | + |ν − χC\A ei arg ν | = max{ess sup ||μ| − 1| + |ν|, ess sup ||ν| − 1| + |μ|}. A

i arg μ

C\A

i arg ν

Observe that (χA e + χC\A e τ )B is indeed unitary because B and χA ei arg μ + χC\A ei arg ν τ are. As μ + ντ is invertible, the index of (μ + ντ )B is zero and by [21] its distance from the set of complex unitary operators is d = max{(μ + ντ )B − 1, 1 − m((μ + ντ )B)} = max{ess sup |μ| + |ν| − 1, 1 − ess inf ||μ| − |ν||} C

C

= max{ess sup(|μ| + |ν| − 1), ess sup(1 − ||μ| − |ν||)}. C

C

Then we have

ess supA |μ| − 1 + |ν| ess sup ||μ| − 1| + |ν| = ess supA −|μ| + 1 + |ν| A

and

ess supC\A |ν| − 1 + |μ| ess sup ||ν| − 1| + |μ| = ess supC\A −|ν| + 1 + |μ| C\A

≤d

≤ d.

Thus U = (χA ei arg μ + χC\A ei arg ν τ )B is the unitary approximant in the case of an invertible real linear multiplication operator. Finding the spectrum of (μ + ντ )S is challenging in general. However, the problem of ﬁnding the spectrum of its unitary approximant might be easier to approach. It is certainly contained in the unit circle. By the fact that the spectrum is upper semi-continuous, this allows us to make some conclusions about the location of the spectrum of (μ + ντ )S. To end this section, let us note that a mere unitary approximation of (μ + ντ )S is not satisfactory since it does not allow for scaling. It is more natural to approximate (μ + ντ )S with scaled unitary operators. Proposition 3.10. Let μ + ντ be a real linear multiplication operator, μ0 + ν0 τ a unitary real linear multiplication operator and r a non-negative function. Then (μ + ντ ) − r(μ0 + ν0 τ ) = max{ess sup |ν|, ess sup |μ|}, A

C\A

where A = {z ∈ C : ∃ε > 0 s.t. |μ| > |ν| a.e. on B(z, ε)}.

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Proof. Let A ⊂ C be any measurable subset. Then we have M = μ + ντ − r(χA ei arg μ + χC\A ei arg ν τ ) = (|μ| − rχA )ei arg μ + (|ν| − rχC\A )ei arg ν τ = ess sup ||μ| − rχA | + ||ν| − rχC\A | C

= max{ess sup ||μ| − r| + |ν|, ess sup |μ| + ||ν| − r|}. A

(3.4)

C\A

Choosing A = {z ∈ C : ∃r > 0 s.t. |μ| > |ν| a.e. on B(z, r)} and r(z) = χA |μ| + χC\A |ν|, we have M = max{ess sup |ν|, ess sup |μ|}. A

C\A

From (3.4) in the proof of the previous proposition, it is evident how and why the function r and the set A should be chosen to minimize M . When r is constrained to being a constant, choosing it and a suitable set A is more subtle. A crude trial for this is given as follows. Deﬁne A = {z ∈ C : ∃r > 0 s.t. |μ| > |ν| a.e. on B(z, r)}. Then set a = min{ess inf |μ|, ess inf |ν|} and b = max{ess sup |μ|, ess sup |ν|}. A

C\A

A

C\A

Now deﬁne r to be the midpoint and L to be the length of the interval (a, b). With these, we have by invoking (3.4) M = max{ess sup ||μ| − r| + |ν|, ess sup |μ| + ||ν| − r|} A

C\A

≤ max{ess sup ||μ| − r| + ess sup |ν|, ess sup |μ| + ess sup ||ν| − r|} A

A

C\A

C\A

≤ max{ess sup ||μ| − r|, ess sup ||ν| − r|} + max{ess sup |ν|, ess sup |μ|} A

C\A

A

C\A

L ≤ max{ess sup |μ|, ess sup |ν|} + . 2 A C\A

References [1] Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, Princeton (2009) [2] Astala, K., Mueller, J.L., P¨ aiv¨ arinta, L., Siltanen, S.: Numerical computation of complex geometrical optics solutions to the conductivity equation. Appl. Comput. Harmon. Anal. 29, 265–299 (2010) [3] Astala, K., P¨ aiv¨ arinta, L.: Calder´ on’s inverse conductivity problem in the plane. Ann. Math. (2) 163(1), 265–299 (2006) [4] Aupetit, B.: A Primer on Spectral Theory. Springer, New York (1991) [5] Bargmann, V.: Note on Wigner’s theorem on symmetry operations. J. Math. Phys. 5(7), 862–868 (1964) [6] Brown, A., Halmos, P.R.: Algebraic properties of Toeplitz operators. Journal f¨ ur die Reine und Angewandte Mathematik, 213, 89–102 (1963/1964)

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[7] Deligne, P., et al. : Quantum Fields and Strings: A Course for Mathematicians. American Mathematical Society, Providence (1999) [8] Friedrichs, K.: On certain inequalities and characteristic value problems for analytic functions and for functions of two variables. Trans. Am. Math. Soc. 41(3), 321–364 (1937) [9] Garcia, S.R., Putinar, M.: Complex symmetric operators and applications. Trans. Am. Math. Soc. 358, 1285–1315 (2006) [10] Halmos, P.R.: A Hilbert Space Problem Book, 2nd edn. Springer, New York (1982) [11] Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge, reprinted 1991 edition (1985) [12] Huhtanen, M., Nevanlinna, O.: Approximating real linear operators. Stud. Math. 179(1), 7–25 (2007) [13] Huhtanen, M., Nevanlinna, O.: Real linear matrix analysis. In: Perspectives in Operator Theory, Banach Center Publ., vol. 75, pp. 171–189. Polish Acad. Sci., Warsaw (2007) [14] Iwaniec, T., Martin, G.: Geometric Function Theory and Non-linear Analysis. Oxford University Press, New York (2001) [15] Iwaniec, T., Martin, G.: The Beltrami equation. Mem. Am. Math. Soc. 191(893), (2008) [16] Knudsen, K., Mueller, J., Siltanen, S.: Numerical solution method for the dbarequation in the plane. J. Comput. Phys. 198(2), 500–517 (2004) [17] Lin, P., Rochberg, R.: On the Friedrichs operator. Proc. Am. Math. Soc. 123(11), 3335–3342 (1995) [18] M¨ uller, V.: Spectral theory of linear operators. In: Operator Theory: Advances and Applications, vol. 139. Birkh¨ auser Verlag, Basel (2003) [19] Olsen, C.L.: Unitary approximation. J. Funct. Anal. 85, 392–419 (1989) [20] Putinar, M., Shapiro, H.S.: The Friedrichs operator of a planar domain. In: Complex Analysis, Operators, and Related Topics, Oper. Theory Adv. Appl., vol. 113, pp. 303–330. Birkh¨ auser, Basel (2000) [21] Rogers, D.D.: Approximation by unitary and essentially unitary operators. Acta Sci. Math. (Szeged) 39, 141–151 (1977) [22] Wigner, E.P.: Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren. Frederick Vieweg und Sohn, Braunschweig (1931) Marko Huhtanen (B) and Santtu Ruotsalainen Institute of Mathematics Aalto University P. O. Box 11100 00076 Aalto Finland e-mail: [email protected]; [email protected] Received: March 19, 2010. Revised: June 30, 2010.

Integr. Equ. Oper. Theory 69 (2011), 133–148 DOI 10.1007/s00020-010-1810-y Published online June 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Similarity Classification and Properties of Some Extended Holomorphic Curves Kui Ji Abstract. For Ω ⊆ C a connected open set, and U a unital Banach algebra (or a unital C ∗ -algebra), let ξ(U ) and P (U) denote the sets of all idempotents and projections in U, respectively. If e : Ω → ξ(U)(resp. P (U)) is a holomorphic U-valued map, then e is called an extended holomorphic curve on ξ(U)(resp. P (U)). In this article, we focus on discussing the similarity classiﬁcation problem of extended holomorphic curves. First, we introduce the definition of the commutant of extended holomorphic curves. By using K0 -group of the commutant of the extended holomorphic curve, we characterize the curve which has unique ﬁnite (SI) decomposition up to similarity. Subsequently, we also obtain a similarity classiﬁcation theorem. Second, we also discuss the unitary equivalence problem of some curves with respect to inductive limit C ∗ -algebras. Mathematics Subject Classification (2010). Primary 47A13; Secondary 46L80, 47B20, 58B25. Keywords. Ordered K0 -group, similarity classiﬁcation, unitary equivalence, extended holomorphic curve.

1. Introduction Recall that if U is a unital Banach algebra (or a unital C ∗ -algebra), then e ∈ U is called an idempotent in U whenever e2 = e. We will let ξ(U) (or P (U)) denote the set of all idempotents (or projections )in U. Let Ω ⊆ C be a connected open set. If e : Ω → ξ(U)(or P (U)) is a holomorphic U-valued map, then it is called an extended holomorphic curve on ξ(U)(or P (U)). Let e, f : Ω → ξ(U)(or P (U)) be two extended holomorphic curves. We call e and f are similarity equivalent (denoted by e ∼ f ) if there exists X ∈ GL(U)(the set of all invertible elements in U) such that e(λ) = Xf (λ)X −1 , ∀λ ∈ Ω. When U is a unital C ∗ -algebra, we call e and f are This work is supported by Chinese NFSC Grant No. 10901046 and 10731020.

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unitarily equivalent or congruent if there exists a unitary U ∈ U such that u e(λ) = U f (λ)U ∗ , ∀λ ∈ Ω (denoted by e ∼ f )[1]. The similarity classiﬁcation of extended holomorphic curves originates from the systematic research of holomorphic curves initiated by Cowen and Douglas [2]. Let H be a complex separable Hilbert space and Gr(n, H) denote a n-dimensional Grassmann manifold; the set of all n-dimensional subspaces of H. If dim H < +∞, then Gr(n, H) is a complex manifold. A map f : Ω → Gr(n, H) is called a holomorphic curve, if there exists n holomorphic H-valued functions γ1 , γ2 , . . . , γn on Ω such that f (λ) = {γ1 (λ), . . . , γn (λ)} for λ ∈ Ω. In 1978, Cowen and Douglas proved that a kind of curvature function is the unitary invariant of holomorphic curves by means of complex hermitian geometry and operator theory. In 1981, Apostol and Martin [3] introduced some concepts and techniques of C ∗ -algebras to the Cowen–Douglas theory. Martin and Salinas [4] proved that extended ﬂag manifold has a natural intrinsic complex structure and gave a criterion for determining the holomorphic maps from Ω to extended ﬂag manifold. They later proved the congruence theorem for the tuples of elements of a C ∗ -algebra in the Cowen– Douglas class. Since 2002, Chunlan Jiang, Junsheng Fang and many other people obtained a similarity classiﬁcation for Cowen–Douglas operators by using the K0 group of the commutant of single operators as an invariant [5– 7]. In the spirit of the above work, our aim is to characterize the similarity of extended holomorphic curves. Meanwhile, we also discuss the problem of unitary equivalence for extended holomorphic curves with range in inductive limit C ∗ algebras. The paper is organized as follows: In Sect. 1, some notations and well known results will be introduced. In Sect. 2, we will introduce the definition of the commutant of an extended holomorphic curve and discuss its properties. In Sect. 3, we will characterize the extended holomorphic curve with unique ﬁnite (SI) decomposition up to similarity, and we will complete the similarity classiﬁcation of extended holomorphic curves satisfying some properties. In Sect. 4, we will use the spectrum of a homomorphism in an inductive system to characterize the unitary equivalence for some curves. We will ﬁrst introduce some notations and known results. 1.1. [8,9] Let U be a unital Banach algebra, and p, q be idempotents in U. If there exist x, y ∈ U such that p = xy, q = yx, then we call p, q are algebraically equivalent (denoted by p ∼a q). If there exists z ∈ GL(U) with p = zqz −1 , we call e and f are similar equivalent (denoted by p ∼ q). Obviously, e ∼a f and e ∼ f are equivalence relations. 1.2. [9] Let M∞ (U) be the algebraic direct limit of Mn (U) under the embedding a → diag(a, 0), where

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⎧⎛ ⎫ ⎞ ⎨ a11 . . . a1n ⎬ Mn (U) = ⎝ a12 . . . a2n ⎠ : aij ∈ U ⎩ ⎭ a1n . . . ann Proj(U) is the set of algebraic equivalence classes of idempotents in U and (U)). We can deﬁne a binary operation on (A): if [p], [q] ∈ (U) = P roj(M ∞ [p] + [q] = (A), where p ∈ [p], q ∈ [q] with p q = q p = 0, and deﬁne [p + q ]. Therefore, this operation is well deﬁned and makes (A) an abelian semigroup with the identity. Since the two notions coincide on M∞ (U), we can obtain the samesemigroup by using ∼ instead of ∼a . K0 (A) is the Grothendieck group of (A). 1.3. [2] For Ω ⊆ C a connected open set and n a positive integer, let Bn (Ω) denote the set of operators T in L(H) which satisfy: (a) Ω⊂σ(T ) = {λ∈C; T − λ not invertible}; (b) ran(T − λ) := {(T − λ)x; x∈H} = H∀λ ∈ Ω; (c) λ∈Ω ker(T − λ) = H; and (d) dim ker(T − λ) = n, ∀λ ∈ Ω. We call an operator in Bn (Ω) a Cowen-Douglas operator with index n. 1.4. [1] Let n ≥ 1 be a positive integer. (1) Let U be a unital Banach algebra. An n-tuple e = (e1 , e2 , . . . , en ); e2i = ∈ U, e1 ≤ e2 ≤ · · · ≤ en , or {(e1 , e2 , . . . , en )|e2i = ei , ei ej = ej ei = ei 0, i ei = I} is called an extended n-ﬂag in U. (2) Let U be a unital C ∗ -algebra. Then e = (e1 , e2 , . . . , en ); e2i = ei = ei ∗ ∈ U, e1 ≤ e2 ≤ · · · ≤ en , is called a n-ﬂag in U. The sets of all extended n-ﬂags of U and n-ﬂags of U will be denoted by ξn (U) and Pn (U), respectively. We will refer to the spaces ξn (U) and Pn (U) as extended n-ﬂag manifold, and n-ﬂag manifold of U, respectively. When n = 1, we get the spaces ξ(U) and P(U), alternatively called the extended Grassmann manifold, and Grassmann manifold of U. 1.5. [8] Let A = limn→∞ (An , φn,m ) be an inductive limit C ∗ -algebra, that is φ1,2

φ2,3

φ3,4

A1 −→ A2 −→ A3 −→ · · · A where An are all C ∗ -algebras. In the inductive system (An , φn,m ), we understand φn,m = φm−1,m ◦ φm−2,m−1 ◦ · · · φn,n+1 , where φn,m : An → Am , φn,∞ : An → A are all homomorphisms satisfying: ∞ (1) A = n=1 φn,∞ (An ); (2) ||φn,∞ (a) = limm→∞ ||φn,m (a)||, ∀n ∈ N, where a ∈ An ; (3) Ker(φn,∞ ) = {a ∈ An : limm→∞ ||φn,m (a)|| = 0}; (4) If (B, {λn }∞ n=1 ) is another system, and λn = λn+1 ◦ φn,n+1 , then there exists λ : A → B that makes the

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φn,∞

An A AA AA A λn AA

B

/A λ

commutative. Moreover (a) Ker(φn,∞ ) Ker(λn ), ∀n ∈ N ; (b) λ is injective if and only if Ker(λn ) Ker(φn,∞ ), ∀n ∈ N ; ∞ (c) λ is surjective if and only if B = n=1 λn (An ). 1.6. [10] Suppose that X is a path connected compact metric space, T is a closed subset of X, M > 1 is a positive number. Then χT,M , called the test function associated to T, M, is deﬁned as follows: ⎧ x ∈ T; ⎨ 1, 1 χT,M = 1 − M dist(x, T ), dist(x, T ) ≤ M ; ⎩ 1 . 0, dist(x, T ) ≥ M 1.7. In this paper, e ⊕ e ⊕ · · · ⊕ e (n times) is denoted by e(n) . For the convenience of exposition, we will use the symbol • to denote every possible positive integer. For any C ∗ -algebra U, let T U denote the tracial state space.

2. Extended Holomorphic Curve Given a complex manifold Ω, a complex Banach space B, and a C ∞ map f : Ω → B, we denote by ∂f and ∂f the B-valued 1 forms on Ω by 1 1 ∂f = (df − i(df ) ◦ J Ω ), ∂f = (df + i(df ) ◦ J Ω ), 2 2 where df is the differential of f , and J Ω denotes the complex structure of Ω[4]. When U is a unital Banach algebra, we can obtain a homeomorphism n between ξn (U) and Uα = {a ∈ U| i=1 (a − αi ) = 0} , a complex submanifold of U, where α = (α1 , . . . , αn ) ∈ C n [11]. So e : Ω → ξ(U) is holomorphic if and only if ∂e = 0. However, if U is a C ∗ − algebra, we cannot use the condition ∂e = 0 to characterize holomorphic maps. In fact, P (U) is not a complex submanifold of U under these circumstances. Furthermore, if Ω is connected, then ∂e = 0 holds if and only if ∂e = 0, and e is a constant map [1]. Therefore, we need another criterion for determining the holomorphic map from Ω to P (U). Definition 2.1 [1]. A C∞ map e : Ω → P (U) is said to be holomorphic with respect to the almost complex structure J on P (U) if and only if de(λ) ◦ JωΩ = Je(ω) ◦ de(λ), (ω ∈ Ω), or de ◦ J Ω = J ◦ de.

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Lemma 2.2 [1]. Let U be a C ∗ -algebra, and e : Ω → P (U) be a holomorphic map. Then e is holomorphic if and only if ∂e(λ) = e(λ)∂e(λ), ∀λ ∈ Ω.(1.1) We can easily verify that (1.1) is equivalent to [∂e(λ)]e(λ) = 0 ⇐⇒ ∂e(λ) = [∂e(λ)]e(λ) ⇐⇒ e(λ)∂e(λ) = 0, λ ∈ Ω. Example. Let T ∈ L(H) be a Cowen–Douglas operator, and let e(λ) be the projection from H to Ker(T − λ). Then e : Ω → P (L(H)) is a extended holomorphic curve. Example [1]. Suppose a ∈ U. We say a has closed range if 0 is an isolated point in σ(a∗ a) {0}. Let 1 (ζ − a∗ a)−1 dζ, κ(a) = 2πi |ζ|=ε

∗

where ε < inf {σ(a a) \ {0}}. If a : Ω → U is a differentiable map, and a(λ) has closed range, then κ(a(λ)) : Ω → P (U) is an extended holomorphic curve.

∞ Example. Let 0 ∈ Ω ⊂ D, λ ∈ Ω, and R = n=0 |λ|2n , ⎞ ⎛ 2 3 1 λ λ λ R R R R ··· ⎟ ⎜ 2 ⎟ ⎜ λ λ|λ|2 λ |λ|2 |λ|2 ⎟ ⎜ R · · · R R R ⎟ ⎜ 2 3 3 ⎜ 2 ⎟ λ|λ| λ|λ| |λ| P(λ) = ⎜ λ ⎟ · · · R R R ⎟ ⎜ R 4 ⎟ ⎜ λ3 λ2 |λ|2 λ|λ|3 |λ| ⎟ ⎜ R R R ··· ⎠ ⎝ R .. .. .. .. . . . . ∞×∞

then P : Ω → P(L(H)) is an extended holomorphic curve. We can check that ∂P(λ)P(λ) = 0. Without loss of generality, for the two dimensional case, we obtain 1 1 λ λ ∂ 1+λλ 1+λλ 1+λλ 1+λλ λ λ λλ λλ ∂λ 1+λλ 1+λλ 1+λλ 1+λλ ⎛ ⎞ −λ 1 1 λ (1+λλ)2 (1+λλ)2 1+λλ 1+λλ ⎠ =⎝ =0 2 −λ λ (1+λλ)2 (1+λλ)2

λλ λ 1+λλ 1+λλ

Definition 2.3. Let e : Ω → ξ(U) be an extended holomorphic curve. We call {s ∈ U|e(λ)se(λ) = se(λ), ∀λ ∈ Ω} the commutant of e, denoted by A (e). Obviously, A (e) is a Banach subalgebra of U. Furthermore, A (e) is different from the common commutant, {s ∈ U|{se(λ) = e(λ)s, ∀λ ∈ Ω}. Throughout Sects. §1 and §2, we will discuss the extended holomorphic curve e(λ) with the following properties: Property (1) Let s ∈ U. If se(λ) = 0, ∀λ ∈ Ω, then s = 0;

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Property (2) GL(A (e)) ⊆ A (e) A (1 − e), where GL(A (e)) denotes the set of invertible elements in the commutant of e. In fact, let T ∈ L(H) be a Cowen–Douglas operator, and let e(λ) be the projection from H to Ker(T − λ). Then e : Ω → P (U) is an extended holomorphic curve. Meanwhile, for any λ ∈ Ω, we have A (T ) = A (e). In fact, for any S ∈ A (e), (T −λ)Se(λ) = (T −λ)e(λ)Se(λ) = S(T −λ)e(λ) = 0. So T Se(λ) − λSe(λ) = ST e(λ) − λSe(λ) = 0, i.e., T Se(λ) = ST e(λ), ∀λ ∈ Ω. By the properties of a Cowen–Douglas operator, T S = ST . On the other hand, if S ∈ A (e), x ∈ Ker(T − λ), then Sx = Se(λ)x ∈ Ker(T − λ), i.e., Se(λ)y = e(λ)Se(λ)y, ∀y ∈ H. Definition 2.4. Let e : Ω → ξ(U) be an extended holomorphic curve. If there exists a non-zero idempotent pi , i = 1, 2 in A (e) such that e(λ) = p1 e(λ) + p2 e(λ), ∀λ ∈ Ω, and p1 + p2 = I, p1 p2 = p2 p1 = 0, then we say e is decomposable. An extended holomorphic curve is said to be indecomposable if it is not decomposable. Notice that (pe(λ))2 = pepe = p.pe = pe, ∀p2 = p ∈ A (e), pe(λ) is still a extended holomorphic curve on U. Lemma 2.5. An extended holomorphic curve e : Ω → ξ(U) is indecomposable if and only if there exists no nontrivial idempotents in A (e). Proof. Suppose p ∈ A (e), p = 0, I, then we have (1 − p) ∈ A (e). So e(λ) = pe(λ) + (1 − p)e(λ), e(λ) is decomposable. On the other side, it is obvious. Lemma 2.6. Let p ∈ A (e) be an idempotent, then p is minimal if and only if pe(λ) is indecomposable. Proof. “⇒” Let 0 = p1 < p, p1 ∈ A (e), p21 = p1 . Then pe(λ) = p1 pe(λ) + (1 − p1 )pe(λ). (∗) Since pep1 pe = pep1 epe + pep1 (1 − e)pe = pp1 epe + 0 = p1 pe, p1 ∈ A (pe), this is a contradiction as pe is indecomposable. “⇐” If (∗) holds and p1 ∈ A (pe), then p1 p ∈ A (e), p1 p < p. In fact, since pep1 pe = p1 pe, and (1 − p)ep1 pe = (1 − p)epep1 pe = (1 − p)pep1 pe = 0, we know that (p1 p)e = pep1 pe + 0 = (p + (1 − p))ep1 pe = e(p1 p)e. So p1 p ∈ A (e). Moreover, pp1 pe = p(pep1 pe) = p2 ep1 pe = pep1 pe = p1 pe, i.e., pp1 pe(λ) = p1 pe(λ), ∀λ ∈ Ω. By property (1), we can obtain pp1 p = p1 p, p1 p < p. However, p is minimal, which is a contradiction.

3. K0 -Group and Similarity Classification of Extended Holomorphic Curves Definition 3.1. Let e : Ω → ξ(U) be an extended holomorphic curve, a ﬁnite set of idempotents {p1 , p2 , . . . , pn } ⊂ A (e) is called a ﬁnite (SI) decomposition of e if the following are satisﬁed: (1) pi is a minimal idempotent of A (e); (2) pi = I, pi pj = 0, i = j.

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Let {p1 , p2 , . . . , pm } and {q1 , q2 , . . . , qn } are two arbitrary decompositions of e. If the following are satisﬁed: (1) m = n; (2) there exists an invertible element X ∈ A (e) and a permutation π ∈ Sn such that Xqπ(i) X −1 = pi for 1 ≤ i ≤ n, then we say e has a unique (SI) decomposition up to similarity. φ

Lemma 3.2. Let e, f : Ω → ξ(U) be extended holomorphic curves, and A (e) ∼ = A (f ). Then {pi }ni=1 is a decomposition of e if and only if {φ(pi )}ni=1 is a decomposition of f .

Proof. Firstly, pi = I, pi pj = 0(i = j) ⇔ φ(pi ) = I, φ(pi )(pj ) = φ(pi )(pj ) = 0(i = j). Secondly, notice that pi is a minimal idempotent, if there exists a non-zero idempotent q < φ(pi ) of A (f ), i.e., φ(pi )q = qφ(pi ), then φ(pi φ−1 (q)) = φ(φ−1 (q)pi ) = φ(φ−1 (q)). Since φ is a homeomorphism, and pi φ−1 (q) = φ−1 (q)pi = φ−1 (q), we have φ−1 (q) < pi . However, since pi is minimal, this is a contradiction. Lemma 3.3. Let e : Ω → ξ(U) be an extended holomorphic curve, and let p1 , p2 ∈ A (e) be idempotents. If p1 ∼ p2 (inA (e)), then p1 e ∼ p2 e. Proof. By property (2), it is obvious.

Lemma 3.4 [5]. Let {p1 , p2 , . . . , pm , . . . , pmk−1 −1 , . . . , pmk , pmk+1 , . . . , pn } and {q1 , q2 , . . . , qm , . . . , qmk−1 −1 , . . . , qmk , qmk+1 , . . . , qn } be two set of idempotents of A (e). If there exist Xi (i = 1, 2, . . . , k), Y ∈ GL(A (e)) and a permutation π ∈ Sn satisfying (1) Xi pi Xi−1 = qj , mi + 1 ≤ j ≤ mi + 1, i = 0, 1, . . . , k1 , m0 = 0. (2) Y −1 pj Y = qπi , 1 ≤ i ≤ n, then for each r, mk < r < n, there exists Zr such that {Zr qr Zr−1 }nr=mk +1 be a rearrangement of {pr }nr=mk +1 . The following lemma is essentially due to Cao et al. [5]. For the sake of completeness of proof, we need to give the proof in our case. Lemma 3.5. Let {p1 , p2 , . . . , pm , pm+1 , . . . , pn } and {q1 , q2 , . . . , qm , qm+1 , . . . , qn } be two decompositions of e. If the following conditions are satisfied: (1) xi ∈ GL(A (e)), xi pi e(λ)x−1 = qi e, ∀λ ∈ Ω, i = 1, 2, . . . , m, i (2) ∃Y ∈ GL(A (e)), Y pi eY −1 = qi e, then for any qr , r ∈ {m + 1, . . . , n}, there exist r ∈ {m + 1, . . . , n}, and Zr ∈ GL(A (e)) such that Zr (qr e(λ))Zr−1 = pr e(λ). Proof. Firstly, for any r ∈ {m + 1, . . . , n}, there exists pj1 ∈ {pi }ni=1 such that Y qr Y −1 = pj1 by condition (2). If m ≤ j1 ≤ n, then we only need to set Zr = Y . Otherwise, by Y qr e(λ) = pj e(λ) and condition (1), we obtain that Y qr e(λ)Y −1 = pj e(λ) and there exists Xj1 such that Xj1 (pj1 e(λ))Xj1 −1 = qj1 e(λ). Using condition (2) again, there exists j2 = j1 such that Y qj1 Y −1 = pj2 . If j2 ∈ {m + 1, . . . , n}, then we just set Zr = Y Xj1 Y −1 , pr = pj2 . In fact,

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Y −1 = Zr qr (e(λ))Zr−1 = Y Xj1 Y −1 qr (e(λ))Y Xj1 Y −1 = Y Xj1 (pj e(λ))Xj−1 1 −1 = pj2 . Otherwise, after s steps, we can ﬁnd pjs ∈ {pk }nk=m+1 Y qj (e(λ))Y such that Zr qr e(λ)Zr−1 = pr e(λ), where pr = pjs . Then we can set Zr = Y Xjs −1 . . . Y Xj Y . In the main result, we only consider the extended holomorphic curves with following properties: Property (a) Let S = {p1 , p2 , . . . , pN } ⊂ A (e) be ﬁnite idempotents, and pi pj = 0, i = j, then the following statements hold: Let A1 = {p11 , p12 , . . . , p1n1 }, . . . , Ak = {pk1 , pk2 , . . . , pknk } ⊂ S, and pij ∼ i pj , i = 1, 2, . . . , k. Then there exists a common X ∈ GL(A (e)) such that Xpil X −1 = pjl , ∀l = 1, 2, . . . , k. Property (b) V (A (e(n) )) has cancellation, i.e.(p ⊕ 0)e(n) ∼ (q ⊕ 0)e(n) ⇒ pe ∼ qe. Particularly, for two decompositions {p1 , p2 . . . pn }, and {q1 , q2 , . . . , qn }, X

pi ∼i qi , there exists common X such that Xpi X −1 = qi ; And let {p11 , p21 , . . . , pn1 1 , p12 , p22 , . . . , pn2 2 , . . . , p1k , . . . , pnk k } ⊂ A (e) be a X

decomposition of e, and pil ∼l pjl , l = 1, 2, . . . , k. Then, there exists a common invertible element X ∈ MN A (e) such that X(pji ⊕ 0)X −1 = 0 ⊕ · · · pi · · · ⊕ (n ) 0, pi i ∼s p1i + · · · + pni i . In other words, set pi represent similar equivalence i . (∼s denotes stable equivalent). class in {pli }nl=1 Under this situation, we say e∼s (p1 e)(n1 ) ⊕ (p2 e)(n2 ) ⊕ · · · ⊕ (pk e)(nk ) ∈ M∞ (U). Lemma 3.6. Suppose that e(λ) has a unique decomposition up to similarity. Then for any p2 = p ∈ A (e), pe(λ) also has a unique decomposition up to similarity. m Proof. Let {pi }m i=1 , {qi }i=1 be two decompositions of pe(λ). Then

{p1 , p2 , . . . , pm , pm+1 , . . . , pn }, {q1 , q2 , . . . , qm , qm+1 , . . . , qn } are two decompositions of pe(λ). So there exists xi ∈ GL(A (e)) such that = pi , (i = m + 1, . . . , n), (xi = I), Y pi Y −1 = qπi , (i = 1, 2, . . .), Y ∈ xi pi x−1 i GL(A (e)). By using Lemma 3.5, there exists Zi ∈ GL(A (e)) such that Zi qi Zi−1 = pπ(i) , i = 1, 2, . . . , m, where Zi = Y Xjs −1 . . . Y Xj1 Y. Since A (e) satisﬁes property (a), we know there exists Z such that Zqi Z −1 = pπ(i) , i = 1, 2, . . . , m. In the following theorem, we use the K0 -group to characterize some extended holomorphic curves with unique decomposition up to similarity. Theorem 3.7. Let e(n) : Ω → P(Mn (U)), n = 1, 2, . . . be extended holomorphic curves which satisfy above the properties (1),(2),(a) and (b). Then the following statements are equivalent: (1)

e∼s (p1 e)(n1 ) ⊕ (p2 e)(n2 ) ⊕ · · · ⊕ (pk e)(nk ) , pi e are indecomposable, and for any n ∈ N, e(n) has a unique decomposition up to similarity.

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h

(K0 (A (e)), V (A (e)), I) ∼ = (Z k , N k , 1),where h is the isomorphism and h([I]) = (n1 , n2 , . . . , nk ).

Proof. (1)⇒ (2), Let P ∈ A (en ) = Mn (A (e)). By Lemma 3.6, P e(n) has a unique decomposition. Let {Q1 , Q2 , . . . , Qa } be one decomposition of P e(n) , and let {Qa+1 , Qa+2 , . . . , Qb } be a decomposition of (I − P )e(n) , where Qi < P, i = 1, 2, . . . , a, Qj < (I − P ), j = a, . . . , b. Then e(n) = P e(n) +(I −P )e(n) = Q1 e(n) + · · · + Qa e(n) + Qa+1 e(n) + · · · + Qb e(n) , where Qi Qj = 0, i = j. And we have (n1 )

e(n)

p1 ∼

(nk )

(n )

pk ⊕ · · · ⊕ p1 1 (nn ) n e 1 ⊕ ···

(n )

⊕ · · · ⊕ pk k (nn ) n e k .

That means, {pji ⊕ 0 · · · ⊕ 0, · · · , 0 ⊕ 0 ⊕ · · · 0 ⊕ pji , i = 1, 2, . . . , k, j = 1, 2, . . . , ni } is a decomposition of e(n) . By uniqueness of decomposition, and

i,t property (a), we can ﬁnd X ∈ GLA (e(n) ) and mi,t = j lj ≤ ni , t = 1, 2, . . . , n such that ⎛ ⎞ k n i,t l −1 −1 ⎝ pij ⎠ . X P X = X (Q1 + Q2 + · · · + Qa )X = i=1

t=1

j

li,t X

Notice that pij ∼ pi , so we have ⎛ ⎞ k n i,t k n l (m ) ⎝ pij ⎠ ∼s pi it . i=1

Since

k

i=1

n t=1

t=1

(mit )

pi

=

P ∼

i=1 t=1

j

k

k

i=1

pi (Mi ) , where Mi =

t

mit ≤ nni ,

pi (Mi ) (M∞ (A (e))).

i=1

Now we deﬁne h : V (A (e)) → N k as follows: h[P ] = (M1 , M2 , . . . , Mk )

k (Mi ) If [P ] = [P ], then P ∼ P ∼ , so h is well deﬁned; and since i=1 pi

(m ) P can only be similar at most to one projection of the form i pi i , it follows that if h[P ] = h[P ], then [P ] = [P ], i.e., h is one-to-one; And for any (M1 , M2 , . . . , Mk ), there exists a proper integer n such that Mi ≤ nni ,

(m ) so we have h([ pi i ]) = (M1 , M2 , . . . Mk ). Thus h is an isomorphism and h([I]) = (n1 , n2 , . . . , nk ). h

(2)⇒ (1), if V (A (e)) ∼ = N k , then for fi = (0, 0, . . . , 1, . . . , 0) there exist corresponding {Q1 , Q2 , . . . , Qk } ∈ A (e(r) ) such that h([Qi ]) = fi , 1 ≤ i ≤ k.

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Claim. Firstly, we can prove that for any idempotent p ∈ A (e), if P e is indecomposable, then there exists i ∈ N such that h[P ] = ei . In fact, let

k

k

k h[P ] = i=1 λi ei = i=1 h[Qi ], w = r i=1 λi . Then we have k k k (λ ) (λ ) i h[P ] = h λi [Qi ] = h [Qi ] = h Qi i , i=1

where

k

i=1

(λi )

Qi

i=1

i=1

∈ M ri=1 λi (U) = Mw (U). So we can ﬁnd n such that P ⊕ 0n ∼

k i=1

(λi )

Qi

⊕ 0n−w ,

k k (λ ) (λ ) i.e. P e ⊕ 0 ∼ i=1 (Qi e) i ⊕ 0. Since P e is indecomposable, i=1 (Qi e) i is also indecomposable. So k = 1, ∃λi0 = 1, and λi = 0, i = i0 . Secondly, for idempotents P, Q in A (e(n) ), if h[P ] = h[Q], then P e(n) ∼ Qe . Now let {p1 , p2 , . . . , pm } be a decomposition of e, and let h[pi ] =

k j=1 λi,j ei . By pi pj = 0, we have m m k h[I] = h pi pi λi,j ej . =h = (n)

i=1

i=1

i=1 j=1

And by h[I] = (n1 , n2 , . . . , nk ), we also have m k

λi,j =

i=1 j=1

k

ni .

i=1

k That means m ≤ i=1 ni . Thus the decomposition number of e less than

k i=1 ni . Moreover, let (p1 , p2 , . . . , pt ) be a decomposition of e. Then t n h [pi ] = h[I] = fi . i=1

i=1

Since pi e is indecomposable, we can ﬁnd pi1 , pi2 , . . . , pin such that h[pi1 ] = h[pi2 ] = · · · = h[pini ] = fi . And then pij e ∼ pik , 1 ≤ i, j ≤ ni . Let {p1 , . . . ps } be another decomposition of e. Then there exists pj1 , . . . , pjn such that i

h[pj1 ] and h[pik ] =

h[pjl ], 1

≤

exists x ∈ GLA (e) such (n ) pk k e)

3.7.

=

h[pj2 ]

= · · · = h[pjn ] = fi . i

k, l ≤ ni , pik ∼ pjl . Thus, that xpik x−1 = pjl . So e ∼s

pik e ∼ pjl e, and there (n1 )

(p1

(n2 )

e ⊕ p2

e⊕···⊕

and has a unique decomposition. This completes the proof of Theorem

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The following theorem is the main result of this paper: Theorem 3.8 (Main Theorem). Let e(n) , f (n) : Ω → ξ(Mn (U)), n = 1, 2, . . . be extended holomorphic curves in Theorem 2.1. If e(n) , f (n) and e(n) ⊕ f (n) all have unique decompositions up to similarity, and e∼s (p1 e)(n1 ) ⊕ (p2 e)(n2 ) ⊕ · · · ⊕ (pk e)(nk ) , then e ∼s f if and only if the following statements hold: h (1) K0 (A (e ⊕ f ), (A (e ⊕ f )), I) ∼ = (Z (k) , N (k) , 1); (2) h([I]) = (2n1 , 2n2 , . . . , 2nk ). Proof. ⇐ By property (a), we can assume that f ∼s (q1 f )(s1 ) ⊕ (q2 f )(s2 ) ⊕ · · · ⊕ (qm f )(sm ) . For any qi f, i = 1, 2, . . . , m, there exist certain pj e, j = 1, 2, . . . , k such that pj e ∼ qi f , and m = k. Otherwise, we can assume q1 f, q2 f, . . . , ql f, (1 ≤ l ≤ m) are not similar to each pj e, and ql+1 f, . . . , qm f is similar to some pj e, 1 ≤ j ≤ k, then (e⊕f ) ∼s ((p1 e)(t1 ) ⊕(p2 e)(t2 ) ⊕· · ·⊕(pk e)(tk ) ⊕(q1 f )(s1 ) ⊕ · · · ⊕ (ql f )(sl ) ). By assumption and Theorem 3.7, we have (A (e ⊕ f )) ∼ = N (k+l) . This is a contradiction to condition (1). So we have l = m, and m ≤ k. Without loss of generality, we assume q1 f ∼ p1 e, q2 f ∼ p2 e, . . . , qm f ∼ pm e. Then (e ⊕ f ) ∼s ((p1 e)(n1 +s1 ) ⊕ · · · ⊕ (pm e)(nm +sm ) ⊕ (pm+1 e)(nm +1) ⊕ · · · ⊕ (pk e)(nk ) ).

By assumption and Theorem 3.7, we have h[I] = (n1 + s1 , . . . , nm + sm , nm+1 , . . . , nk ). This is a contradiction to condition (2), so m = k. Thus, we can assume pi e ∼ qi f, i = 1, 2 . . . k. That means e⊕f ∼s ((p1 e)(n1 +s1 ) ⊕(p2 e)(n2 +s2 ) ⊕· · ·⊕ (pk e)(nk +sk ) ). By condition (2), ni + si = 2ni , we have si = ni , i = 1, 2 . . . k. So e ∼s f . ⇒ If e ∼s f , then e⊕f ∼s e⊕e ∼s (p1 e)(2n1 ) ⊕(p2 e)(2n2 ) ⊕· · ·⊕(pk e)(2nk ) . By Theorem 3.7, (1) and (2) hold. This completes the proof of Theorem 3.8.

4. Inductive Limits and Extended Holomorphic Curves on C ∗ -algebras We will ﬁrst introduce some notation and well known results from [1]. Let U be a unital C ∗ -algebra, and e : Ω → P (U) be an extended holomorphic curve. Assume X ⊆ U is a ﬁxed subset containing the unit of U. For each λ ∈ Ω and any α ∈ Z+ ∪ {∞}, let J

Bλα (e, X) = {D e(λ)y ∗ xDI e(λ) : I, J ∈ Z+ , I, J ≤ α, x, y ∈ X}, J

where DI = (∂/∂λ)I , D = (∂/∂λ)J . Let Uλα (e, X) be the closure of ∗-subalgebra of U generated by Bλα (e) with property Uλ0 (e, X) ⊆ Uλ1 (e, X) ⊆ · · · ⊆ Uλ∞ (e, X). By the notations mentioned above, M. Martin and N. Salinas introduced a substitute in C ∗ -algebra for Cowen–Douglas class Bn (Ω):

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Definition 4.1 [1]. Let k ≥ 1 be an integer. If the following conditions are satisﬁed, then (e, X) is said to be in the class Ak (Ω, U) (1) Uλ∞ (e, X) is a ﬁnite-dimensional C ∗ -algebra for each λ ∈ Ω. (2) if kλ denotes the cardinal of any maximal collection of mutually orthogonal minimal projections in Uλ∞ (e, X), then kλ ≤ k. In the special case, when X = {I}, (e, X) is equal to e. Definition 4.2 [1]. Let λ ∈ Ω, and α ∈ Z+ be a ﬁxed integer. We say extended holomorphic curves e and f have order of contact α at λ if there exists a unitary ν such that J

J

νD e(λ)DI e(λ)ν ∗ = D f (λ)DI f (λ), I, J ≤ α. Definition 4.3 [1]. We say G ⊂ U is a separating subset of U if {a ∈ U : as = 0, s ∈ G} = {0}. Assume G, T are two separating subsets of U, θ : G → T is a given bijection. We say θ is inner or semi-inner, if there exists unitary u ∈ U such that usu∗ = θ(s), s ∈ G, or if there exists a unitary ν ∈ U such that νt∗ sν ∗ = θ(t)θ(s), s, t ∈ G. U is said to be inner if each semi-inner bijection between two separating subsets of U is inner. Martin and Salinas proved the following congruence theorem for Ak (Ω, U) class on C ∗ -algebra. Lemma 4.4 [1]. Suppose e, f : Ω → P(U ) ∈ Ak (Ω, U). If U is inner, then the following are equivalent: u

(1) e ∼ f ; (2) e and f have order of contact α at each λ ∈ Ω. By using this congruence theorem, we will consider the unitary equivalence problem of some curves associated with inductive limit C ∗ -algebras. Theorem 4.5. Let U = limn→∞ (Un , φn,m ) be a unital C ∗ -algebra, and e, f : Ω → P (U) ∈ Ak (Ω, U) be extended holomorphic curves. If there exists an integer i such that Ui is inner, Uλ∞ ⊆ φi,∞ (Ui ), and φi,∞ is a unit preserving isometric homomorphism, then the following statements are equivalent: (1) e(λ) and f (λ) have order of contact k at each λ ∈ Ω, J

J

uD e(λ)DI e(λ)u∗ = D f (λ)DI f (λ), I, J ≤ k, and where u(λ) = φi,∞ (z(λ)), z(λ) is unitary of Ui . u (2) there exists unitary u such that e(λ) ∼ f (λ), ∀λ ∈ Ω, and where u(λ) = φi,∞ (z(λ)), z(λ) is unitary of Ui .

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Proof. Since Uλ ⊆ φi,∞ (Ui ), then we can assume e(λ) = φi,∞ (e(λ)), DI e(λ) = J

J

φi,∞ (eIi (λ)), D = φi,∞ (eJi (λ)); f (λ) = φi,∞ (λ), DI f (λ) = φi,∞ (fiI (λ), D = J

φi,∞ (f i (λ)). Claim 1. ei (λ) is an extended holomorphic curve, and φi,∞ (DI ei (λ)) = DI φi,∞ (ei (λ)). In fact, for any λ0 ∈ Ω, let λ = x + iy, λ0 = x0 + iy0 . Then lim

x+iy→x0 +iy0

= = =

lim

x+iy→x0 +iy0

lim

x+iy→x0 +iy0

)−ei (x0 +iy0 ) i (x0 +iy0 ) ei (x+iy0x−x − i ei (x0 +iy)−e − e1i (λ0 ) y−y0 0 )−ei (x0 +iy0 ) i (x0 +iy0 ) −φi,∞ (e1i (λ0 )) φi,∞ ei (x+iy0x−x −i ei (x0 +iy)−e y−y 0

0

e(x+iy0 )−e(x0 +iy0 ) x−x0

0 +iy0 ) − i e(x0 +iy)−e(x − ∂e(λ0 ) y−y 0

0

So ∂ei (λ0 ) = e1i (λ0 ). Similarly, we can prove that DI ei (λ) = eIi (λ), D (e(λ)) = φi,∞ (eJi (λ)). Thus ei (λ) and fi (λ) are holomorphic. J

Claim 2. ei , fi ∈ Ak (Ω, Ui ). By e, f ∈ Ak (Ω, U), and J

J

Uλ∞ = Span{φi,∞ (DI ei (λ)D ei (λ))} = φi,∞ (Span(DI ei (λ)D ei (λ)) J ∼ = Span(DI e(λ)D e(λ)) = U ∞ i,λ

we can deduce that ei , fi ∈ Ak (Ω, Ui ). (1) ⇒ (2): By u(λ) = φi,∞ z(λ), we have J

J

z(λ)DI ei (λ)D ei (λ)z(λ)∗ = DI fi (λ)D fi (λ) J

J

⇔ φi,∞ (z(λ)DI ei (λ)D ei (λ))φi,∞ (z(λ)∗ ) = φi,∞ (DI fi (λ)D fi (λ)) J

J

⇔ u(λ)DI e(λ)D e(λ)u(λ)∗ = DI f (λ)D f (λ), ∀I, J ∈ Z+ ∪ {0}. So ei (λ) and fi (λ) have order of contact k at each λ ∈ Ω. Since Ui is z u inner, by Lemma 4.4, we have ei ∼ fi . Then e = φi,∞ (ei ) ∼ φi,∞ (fi ) = f, u = φi,∞ (z). (2) ⇒ (1) is obvious. In the following, we will discuss the unitary equivalence problem of φ1,2

φ2,3

some curves in special inductive limits C ∗ -algebras. Let U1 → U2 → · · · U, = q ∈ Z + , m ≥ n. Assume Un = Mkn (U0 ), n = 1, 2 . . ., where U0 is unital, kkm n homomorphisms φn,m : Un → Um satisfy: ⎛ ⎞ e(x1 (λ)) ⎜ ⎟ e(x2 (λ)) ⎜ ⎟ ∗ φn,m (e(λ)) = un,m ⎜ ⎟ un,m , (1) . .. ⎝ ⎠ e(xq (λ)) for any e(λ) ∈ C ∞ (Ω, Un ). (2) there exists Ω0 ⊂ Ω0 ⊆ Ω such that xi (λ) are holomorphic functions with RanXi ⊆ Ω0 , un,m ∈ Um is unitary.

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{x1 (λ), x2 (λ), . . . , xq (λ)} is called the spectrum of φn,m at λ, denoted by SP (φn,m )λ . And the symbol |SP (φn,m )λ | denotes the number q of the spectrum. SP (φn,m )λ and SP (ψn,m )λ are said to be paired within η, if

SP (φn,m )λ = {x1 , x2 , . . . , xn },

SP (ψn,m )λ = {x1 , x2 , . . . , xn },

and for any i ∈ {1, 2, . . . , n}, dist(xi , xσ(i) ) < η. Inspired by the uniqueness theorem in [10], we also have the following theorem. Theorem 4.6. Let U = limn→∞ (Un , φn,m ), U = limn→∞ (Un , ψn,m ) be C ∗ algebras mentioned above. Let e(λ) : Ω → P(U 0 ) be an extended holomorphic curve. Suppose that for any test function χT,M (see in 1.6), there exists a tracial state s ∈ T U such that s(e(λ)) = χT,M (λ), ∀λ ∈ Ω0 = K. Then for any η, δ > 0 there exists a finite set H ⊂ T (Un ) such that the following statement holds: If φn,m , ψn,m : Un → Um satisfies (1) For any x0 ∈ K, and η-ball Bη (x0 ), |SP (φn,m )λ ∩ Bη (x0 )|/|SP (φn,m )λ | > δ, |SP (ψn,m )λ ∩ Bη (x0 )|/|SP (ψn,m )λ | > δ.

(2) ||T φn,m (! s)(e(λ)) − T ψn,m (! s)(e(λ))|| ≤ δ, ∀! s ∈ T φn,m (H), ∀λ ∈ Ω. Then SP (φn,m )λ and SP (ψn,m )λ can be paired within 3η. Proof. Firstly, divide K into ﬁnite regions with diameter less than η. Denote the set composed of these regions by I. For any subset {Ii1 , Ii2 , · · · , Iik } of I, L ⊂ C(K) denote the set composed of the test functions on Ii1 ∪ Ii2 ∪ · · · ∪ Iik . For any ﬁxed λ ∈ Ω, and W ⊂ (SP φn,m )λ , set

W = {I : W ∩ I = ∅, I ∈ I}, then W ⊆ Bη (W). Let test function ⎧ ⎨ 1, χ(x) = 1 − M dist(x, W), ⎩ 0,

x∈W dist(x, W) ≤ η dist(x, W) ≥ η

and ﬁnite set H = {s : s ∈ T (Un ), ∃χ ∈ L, s(e(λ)) = χ}. Suppose q = rankφn,m = rankψn,m , by T (Un ⊗ Mq (C)) = T (Um ) ∼ = s ∈ T φn,m (H), where T r T (Un ), we have that for any s! = s.T r, ∃s ∈ H, ∀! denotes trace of the matrix. Notice that suppχ ⊆ Bη (W), we have q

q

1 1 T φn,m (W)(! s)(e(λ)) = s!(φn,m (e(λ)) = s(e(xi (λ))) = χ(xi (λ)). q i=1 q i=1

q s)(e(λ))) = i=1 χ(xi (λ)) ≤ q(T φn,m (! s)(e(λ))) + qδ. And |W| ≤ (T φn,m (! In the following ,we will prove that |W| ≤ |(SP ψn,m )λ ∩ B2η (W). " = K, then (SP ψn,m )λ ∩ B2η (W) " = (SP ψn,m )λ , it follows Case 1 If Bη (W) that " = q. |W| ≤ |(SP ψn,m )λ ∩ B2η (W)|

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" = K, then there exists at least one x0 ∈ K, such that Case 2 If Bη (W) " = ∅, Bη (x0 ) ⊂ B2η (W). Thus, by condition (1) Bη (x0 ) ∩ Bη (W) " + qδ ≤ |(SP ψn,m )λ ∩ Bη (W)| " + |(SP ψn,m )λ ∩ Bη (x0 )| |(SP ψn,m )λ ∩ Bη (W)| " ≤ |(SP ψn,m )λ ∩ B2η (W)|

So |W| ≤ q(T φn,m (! s)(e(λ)) + qδ " + qδ ≤ |(SP ψn,m )λ ∩ B2η (W)|. " ≤ |(SP ψn,m )λ ∩ Bη (W) " ⊂ B2η (W), we have B2η (W) " ⊂ B3η (W). So |W| ≤ And because of W |(SP ψn,m )λ ∩ B3η (W)|. Similarly, for any subset W of (SP ψn,m )λ , we can also prove that |W | ≤ |(SP φn,m )λ ∩ B3η (W)|. By Marriage lemma [12], we have that SP (φn,m )λ and SP (ψn,m )λ can be paired within 3η. Corollary 4.7. In Theorem 4.6, if η is small enough such that when u(λ)

|xi (λ) − yσ(i) (λ)| ≤ 3η, ||e(xi (λ)) − e(yσ(i) (λ))|| ≤ 1, then φn,m (e(λ)) ∼ u(λ)

ψn,m (e(λ)); φn,∞ (e(λ)) ∼ ψn,∞ (e(λ)). Acknowledgement The author would like to thank the referee for his helpful comments and suggestions.

References [1] Martin, M., Salinas, N.: Flag Manifolds and the Cowen–Douglas theory. J. Oper. Theory. 38, 329–365 (1997) [2] Cowen, M.J., Douglas, R.G.: Complex geometry and operator theory. Acta Math. 141, 187–261 (1978) [3] Apostol, C., Martin, M.: A C ∗ -algebra approach to the Cowen–Douglas theory. In: Operator Theory: Advances and Applications, vol. 2, pp. 45–51. Birkhauser Verlag, Basel (1981) [4] Martin, M., Salinas, N.: The canonical complex structure of ﬂag manifolds in a C ∗ -algebra. In: Operator Theory: Advances and Applications, vol. 104. Birkhauser Verlag, Basel (1998) [5] Cao, Y., Fang, J.S., Jiang, C.L.: K0 -Group of Banach algebras and decomposition of strongly irreducible operators. J.Oper. Theory 48, 235–253 (2002) [6] Jiang, C.L.: Similarity classiﬁcation of Cowen–Douglas operators. Can. J. Math. 156(4), 742–775 (2004) [7] Jiang, C.L., Guo, X.Z., Ji, K.: K-group and similarity classiﬁcation of operators. J. Funct. Anal. 225(1), 167–192 (2005) [8] Rordam, M., Larsen, F., Laustsen, N.J.: An Introduction to K-Theory for C ∗ Algebras. Cambridge University press (2000) [9] Blackadar, B.: K-Theory for Operator Algebras. Springer, Heidelberg (1986)

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[10] Li, L.: On the classiﬁcation of simple C ∗ -algebras: Inductive limits of matrix algebras over trees. Memoirs of the American Mathematical Society, no. 605, vol. 127 (1997) [11] Corach, G., Porta, H., Recht, L.: Differential geometry of systems of projections in Banach algebras. Pac. J. Math. 143, 209–228 (1990) [12] Halmos, P.R., Vaughan, H.E.: Marriage problem. Am. J. Math. 72, 214– 215 (1950) Kui Ji (B) Department of Mathematics Hebei Normal University Shijiazhuang 050016 People’s Republic of China e-mail: [email protected] Received: March 21, 2010. Revised: April 9, 2010.

Integr. Equ. Oper. Theory 69 (2011), 149–150 DOI 10.1007/s00020-010-1847-y Published online December 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Erratum

Erratum to: Group Representations with Empty Residual Spectrum Yemon Choi

Erratum to: Integr. Equ. Oper. Theory 67 (2010), 95–107 DOI 10.1007/s00020-010-1772-0 We regret to announce that, due to a careless mistake by the author regarding terminology, the article [1] inadvertently claims stronger results than are actually proved. The error lies in mistaking the residual spectrum σr (T ) of an operator T —which is, by definition, the set of λ such that T − λI has non-dense range—for σ(T )\σap (T ), where σap (T ) is the set of approximate eigenvalues of T . See, for instance, [2, §1.2 and §4.6] for the definitions and basic properties. It is the set σ\σap which is studied in the article, rather than σr . The following corrections to pages 95–97 of the article are therefore necessary. (i) A more accurate title for the article would be “Group representations where the spectrum consists of approximate eigenvalues”. Similarly, in lines 3–4 of the abstract, “has empty residual spectrum” should be replaced with “has spectrum consisting of approximate eigenvalues”. (ii) The second paragraph of Section 1, and the question which follows it, should be replaced with the following: In particular, we might consider the approximate point spectrum of such an operator. In many cases this coincides with the whole of the spectrum, motivating the following question: given Γ and X as above, does every a ∈ CΓ, when regarded as an operator on X, have spectrum consisting entirely of approximate eigenvalues? The online version of the original article can be found under doi:10.1007/s00020-010-1772-0.

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(iii) While Theorem 1.1 is correctly quoted, the claim that decomposable operators have empty residual spectrum (which appears at the top of p. 96) is in general false, and should be removed. The correct claim, which as stated in the original can be found as [2, Proposition 1.3.3], is that σ(T ) = σap (T ) for every decomposable operator.1 (iv) On p. 96, the definition of surjunctive pair should be modiﬁed in order to restore the intended meaning, as follows: if X is a Banach space and A ⊆ B(X) is an algebra of operators on it, not necessarily closed, we say (A, X) is a surjunctive pair if σ(a) consists entirely of approximate eigenvalues for each a ∈ A. (This is in fact the definition/characterization used throughout the article, via Lemma 2.1.) (v) The second paragraph of Section 2, which contains an erroneous definition of the residual spectrum, should be removed. With these amendments, all the results claimed in the article are now correct. The author apologizes for any confusion that may have arisen.

References [1] Choi, Y.: Group representations with empty residual spectrum. Integr. Equ. Oper. Theory 67(1), 95–107 (2010) [2] Laursen, K.B., Neumann, M.M.: An Introduction to Local Spectral Theory. London Mathematical Society Monographs. New Series, vol. 20. The Clarendon Press. Oxford University Press, New York (2000) Yemon Choi Department of Mathematics and Statistics McLean Hall University of Saskatchewan 106 Wiggins Road Saskatoon, SK S7N 5E6 Canada e-mail: [email protected]

1 For

example, the multiplication operator M : C[0, 1] → C[0, 1] defined by (M f )(t) = tf (t) is decomposable, and satisfies σ(M ) = σr (M ) = σap (M ): see the discussion on p. 369 of [2].

Integr. Equ. Oper. Theory 69 (2011), 151–170 DOI 10.1007/s00020-010-1856-x Published online January 11, 2011 c The Author(s) This article is published with open access at Springerlink.com 2010

Integral Equations and Operator Theory

Unbounded Jacobi Matrices with a Few Gaps in the Essential Spectrum: Constructive Examples Anne Boutet de Monvel, Jan Janas and Serguei Naboko Abstract. We give explicit examples of unbounded Jacobi operators with a few gaps in their essential spectrum. More precisely a class of Jacobi matrices whose absolutely continuous spectrum ﬁlls any ﬁnite number of bounded intervals is considered. Their point spectrum accumulates to +∞ and −∞. The asymptotics of large eigenvalues is also found. Mathematics Subject Classification (2010). Primary 47B36; Secondary 47A10, 47B25. Keywords. Jacobi matrix, essential spectrum, gaps, asymptotics.

1. Introduction In this paper we look for examples of unbounded Jacobi matrices with several gaps in the essential spectrum. Let 20 = 20 (N) be the space of sequences {fk }∞ 1 with a ﬁnite number of nonzero coordinates. For given real sequences ∞ 2 {ak }∞ 1 and {bk }1 the Jacobi operator J 0 acts in 0 by the formula (J 0 f )k = ak−1 fk−1 + bk fk + ak fk+1

(1.1)

where k = 1, 2, . . . and a0 = f0 = 0. The ak ’s and bk ’s are called the weights and diagonal terms, respectively. In what follows we deal with positive sequences {ak }∞ 1 . By Carleman’s criterion [1] one can extend J 0 to a self-adjoint operator J in 2 = 2 (N) provided that 1 = +∞. ak k

The above mentioned aim of ﬁnding an example of unbounded Jacobi operator J such that R\ σess (J ) consists of the union of several intervals could have been solved by using a general theorem of Stone. This theorem asserts that for a ﬁnite Borel measure μ on R one can ﬁnd sequences {ak }∞ 1 and {bk }∞ 1 (expressed in terms of some moments of μ) such that the operator of multiplication by x in L2 (μ) is unitarily equivalent to the Jacobi operator

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associated to these sequences by (1.1) [23]. Therefore, one can easily prove the existence of unbounded Jacobi operators with arbitrary many gaps. How∞ ever, we want to ﬁnd explicit examples of {ak }∞ 1 and {bk }1 which deﬁne J with a few gaps in its essential spectrum. The construction given below uses the Gilbert–Pearson subordination theory of Jacobi matrices [14] combined with the asymptotic analysis of formal solutions (generalized eigenvectors) of the system an−1 un−1 + bn un + an un+1 = λun

(1.2)

where n = 2, 3, . . . and λ ∈ R. One can rewrite (1.2) in the standard form un+1 = Bn (λ)un where n = 2, 3, . . . and

un := Bn (λ) :=

un−1 , un

0

1

− an−1 an

λ−bn an

(1.3)

.

The matrix Bn (λ) is called the “transfer matrix” and will be used below in our analysis of generalized eigenvectors. It is well known that bounded Jacobi matrices with almost periodic entries have inﬁnitely many gaps in the essential spectrum (in the generic case) [22]. We do not know any paper with similar results in the case of unbounded Jacobi matrices. However, we could expect that these results hold true for them. This is not our aim here because we want to keep control on the form of gaps in the essential spectrum in terms of some parameters which appear in the entries of Jacobi matrices we shall construct below. Moreover, this control might allow to construct examples of unbounded Jacobi matrices with a few so called “mobility edges”. In the case of one gap such a construction has been done in our recent work [13]. Finally, we should mention a recent paper of Christiansen et al. [4] concerning ﬁnite gap bounded Jacobi matrices which describes the so called “isospectral torus”. Recall that to our best knowledge almost all explicit examples of unbounded Jacobi matrices considered in the last 30 years had essential spectra either empty or equal to the whole real line or a half line. Only several years ago appeared the ﬁrst explicit examples of unbounded Jacobi matrices with one gap in the essential spectrum. These examples are given by the following expressions: an = nα + cn , bn ≡ 0, where α ∈ (0, 1] and {cn } is a periodic sequencce of period two with c1 − c2 = 0. It turns out that the Jacobi operator J 1 deﬁned by these sequences has σess (J 1 ) = σac (J 1 ) = (−∞, −|c1 − c2 |] ∪ [|c1 − c2 |, +∞), see [6,8,12,17]. Below we shall try to explain the origin of this phenomena from a general point of view.

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It turns out that a similar strategy (local periodicity) is also crucial for the formation of gaps in the essential spectrum of the unbounded Jacobi matrices considered in the present work. In particular the characteristic polynomial p(λ) [see (3.1)] resemble but is different from the one of bounded Jacobi matrices. By choosing (locally) the entries an as suitable powers of a small parameter ε, it is proved that the absolutely continuous spectrum E of J consists of exactly N disjoint intervals. However, this E is not the same as the corresponding one in the case of bounded periodic Jacobi matrices. In particular, the calculations in the last section show that we can construct examples with an arbitrary number of gaps in σess (J). Note that the essential spectrum of J does not coincide with the absolutely continuous one of the corresponding periodic Jacobi matrix. Hence the results on the existence of gaps do not follow from the classical theory of periodic Jacobi matrices. In this paper we concentrate on explicit examples of unbounded Jacobi matrices with ﬁnitely many (arbitrary many) intervals of absolutely continuous spectrum. However, in our examples the essential spectrum is bounded. The alternate case of ﬁnitely many bounded gaps in the essential spectrum will be considered in a separate paper by using a different technique. The paper is organized as follows. In Sect. 2.1 we present the main idea of the construction in the simplest case of one gap, through analyzed in the above mentioned papers from an essentially different point of view. In Sect. 2.2 we describe the explicit construction of examples of unbounded Jacobi matrices with ﬁnitely many gaps. Its generalization via perturbation theory is considered in Sect. 2.3. The next Sect. 3 deals with the detailed analysis of the case of an arbitrary even “period” N [see (2.1)]. Calculation of the absolutely continuous spectrum of the constructed Jacobi operator (for general even N ) is given in Sect. 5. Moreover, the nondegeneracy of the gaps structure is shown there. Finally, the asymptotics of the discrete spectrum is under consideration in Sect. 4.

2. Construction of Explicit Examples with Several Gaps 2.1. Heuristics The idea of the construction of an unbounded Jacobi matrix J with several gaps in σess (J ) is based on the following heuristic reasoning. Take a family of inﬁnite Jacobi matrices (for simplicity all with zero diagonal, bn ≡ 0) and periodic with respect to some parameters. The essential spectrum of any member of the family consists of a ﬁnite union of intervals which are symmetric with respect to zero (−xM , −yM ) ∪ · · · ∪ (−x1 , −y1 ) ∪ (y1 , x1 ) ∪ · · · ∪ (yM , xM ), for an integer M . We choose some parameters of these Jacobi matrices in such a way that the “internal” intervals (−xM −1 , −yM −1 ), . . . , (yM −1 , xM −1 ) are almost ﬁxed, and try to choose other parameters tending to inﬁnity, with (−xM , −yM ), (yM , xM ), tending to −∞, +∞ respectively. Next we construct

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an unbounded Jacobi matrix J (with bn ≡ 0) which looks similar (locally in index n) to the above ones from the family (but surely not a periodic one). Moreover, each copy of our periodic family appears “approximately” within the matrix J on some (eventually extended) sequence of matrix elements of index n. Additionally we shall try to choose the parameters smoothly or rather adiabatically to be able to track the asymptotics of solutions of (1.2) and therefore to control the a.c. spectrum. These heuristic arguments will be made precise below. The One Gap Case We illustrate the above idea in the one gap case, which was ﬁrst found in [17]. However, the reasoning given in [17] was completely different. Let J p be the periodic Jacobi matrix given by bn ≡ 0 and a2n−1 = w1 ,

a2n = w2 ,

w1 > w2 > 0.

Then the product B2 B1 of the transfer matrices is − w2 λ 0 1 0 1 w1 w1 B2 B1 = w1 = w2 λ λ 1 w2 w2 w1 w1 − wλ1 − w w2 +

λ2 w1 w2

.

Two bands of σac (J p ) are given by {λ | |tr(B2 B1 | ≤ 2}, see [24]. It is easy to check that σac (J p ) is the union of two intervals [−(w1 + w2 ), −(w1 − w2 )] ∪ [w1 − w2 , w1 + w2 ]. We may keep w1 − w2 equal to a ﬁxed positive constant c but with w1 + w2 tending to +∞. For example we can choose w1 = nα + c1 ,

α ∈ (0, 1],

w2 = nα + c2 ,

c = |c1 − c2 |.

In this way we obtain a 2-periodic perturbation of the Jacobi matrix with weights an = nα and zero diagonal which was studied in [5,6,8,12,17,21]. Below we shall exploit a similar idea in the construction of our example. 2.2. Basic Examples According to the above heuristics we deﬁne basic examples of Jacobi matrices as follows. Definition 1 (J (α)). Let z1 , . . . , zN be N positive numbers and α ∈ (0, 1]. The Jacobi matrix J (α) is deﬁned by zero diagonal and weights (l + 1)α , if k = 1, 2, l = 0, 1, . . . al(N +2)+k = (2.1) zk−2 , if k = 3, 4, . . . , N + 2, l = 0, 1, . . . Remark. We must assume that all zi = 0, otherwise J (α) = ⊕Ms , for some ﬁnite Jacobi matrices Ms . One can prove that for N = 4 the spectral picture of J (α) has the following form (in the generic case) (Fig. 1).

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Figure 1. Spectrum of J (α) for N = 4 Here σac (J (α)) = −I2 ∪ −I1 ∪ I1 ∪ I2 with known intervals I1 , I2 and some eigenvalues of J (α) denoted by × which tend to ∓∞. This last assertion is obvious because J (α) is an unbounded Jacobi matrix. The above choice of the position of the weights lα in the l-th block, l = 1, 2, . . .: (lα , lα , z1 , . . . , zN ) could surely be different but deﬁnitively not arbitrary. For another example with the l-th block of weights given by (lα , z1 , . . . , zN , lα ) we obtain a similar spectral picture. What is essential for both examples is that the norm of B(N +2)l+3 (λ)B(N +2)l+2 (λ)B(N +2)l+1 (λ) remains bounded and smooth in l, as l → ∞. However, B(N +2)l+2 (λ)B(N +2)l+1 (λ) → ∞, as l → ∞. In Sect. 5 we shall prove that indeed the spectral picture of J (α) (described above for N = 4) looks similar for general N . Theorem 2. If J (α) is the Jacobi operator defined by (2.1), then ¯ σac (J (α)) ⊃ E where E, defined by (2.5), is a finite collection of intervals filled by purely absolutely continuous spectrum of local multiplicity one a.e. with respect to the Lebesgue measure. Proof. It turns out that the spectral analysis of J (α) can be based on the asymptotic behaviour of generalized eigenvectors corresponding to all λ which do not belong to ∪N i=1 ∂Ii , where ∂Ii is the boundary of Ii . Due to the irregular behaviour (jumps) of products of transfer matrices within blocks we shall study the products over the whole blocks, see [10]. These products behave regularly with respect to the indices s of blocks. Using (1.3) we have u(N +2)s+1 = B(N +2)s (λ) · · · B(N +2)(s−1)+1 (λ)u(N +2)(s−1)+1 .

(2.2)

We want to ﬁnd the asymptotic behaviour of u(N +2)s+1 as s → +∞. By definition of the weights in the s-th block we have B(N +2)s (λ) · · · B(N +2)(s−1)+1 (λ) 0 0 1 0 1 = ... α zN −1 z1 λ λ − zN − z2 z2 − sz1 zN

M (λ)

1 λ z1

0

1

−1

λ sα

C

0 − zsNα

(2.3) 1 . λ sα

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The product of the last three matrices above is ⎛ ⎞ 2 N − λz −1 + sλ2α s2α ⎠. C=⎝ zN λ2 zN 2λ λ3 − − + z1 z1 s2α z1 z1 s2α

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(2.4)

If M (λ) denotes the product of the ﬁrst (N − 1) matrices to the left of C, then we can rewrite the left hand side of (2.3) as A(λ) + s−2α B(λ), where

A(λ) := M (λ) B(λ) := M (λ)

0

−1

zN z1

− 2λ z1

−λzN 2

− λ zz1N

,

λ2 λ3 z1

.

Note that A(λ), B(λ) does not depend on s and B(λ) is not the monodromy matrix generated by the periodic sequence z1 , . . . , zN . Next we observe that zN −1 z1 sα zN ... · · = 1. det[A(λ) + s−2α B(λ)] = zN z2 z1 sα If disc A := (tr A)2 − 4 det A denotes the discriminant of a 2 × 2 matrix A, then 2 disc[A(λ) + s−2α B(λ)] = tr[A(λ) + s−2α B(λ)] − 4.

Notation (elliptic part). We denote E := {λ ∈ R | |tr A(λ)| < 2}.

(2.5)

Below we shall describe some properties of the function λ → tr A(λ). Now we only note that tr A(λ) is a polynomial in λ of degree N . Therefore, each of the equations tr A(λ) = ±2 has at most N solutions. It follows that E consists of at most N intervals. If λ ∈ E then, for some s0 , disc[A(λ) + s−2α B(λ)] < 0 for any s ≥ s0 . Consequently, the product in (2.3) is an elliptic matrix for any s ≥ s0 . Thus, the matrix A(λ) + s−2α B(λ) has two distinct eigenvalues (1)

(2)

μλ (s), μλ (s) for s 1, with

(i) (2) (1) μλ (s) = 1, i = 1, 2 and μλ (s) = μλ (s).

Consider the system of equations ys = A(λ) + s−2α B(λ) ys−1 ,

ys := u(N +2)s+1 ,

s = 2, 3, . . .

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The above properties of the matrices A(λ) + s−2α B(λ) allow to apply [19, Theorem 3.2] and to obtain a basis of solutions of (1.3) with the following asymptotic formula s−1

(k)

u(N +2)s (λ) =

(k)

μλ (r) (fk + o(1)),

r=2

k = 1, 2,

(2.6)

where fk are the eigenvectors of A(λ). By definition of the weights and by direct computation we also obtain the asymptotic behaviour of bases of solutions of (1.3) for the remaining subsequences of indices: (k)

(k)

u(N +2)s+1 (λ) = B(N +2)s (λ)u(N +2)s (λ), zN −1 λ 0 − zN zN (k) −α u(N +2)s+2 (λ) = + (s + 1) 0 0 − zNzN−1

(2.7)

0 −zN +

(k)

(k)

u(N +2)s+3 (λ) = (k) u(N +2)s+4 (λ)

=

× u(N +2)s (λ), 0

(k)

−α

−1 λ(s + 1) 0 −1 zN z1

(2.8)

1

− 2λ z1

u(N +2)s+2 (λ), −2α

+ (s + 1)

(2.9)

−λzN

λ2

−λ2 zN

λ3 z1

(k)

× u(N +2)s+1 (λ), (k)

u(N +2)s+l (λ) =

l−3 i=2

(k)

Ai (λ)u(N +2)s+4 (λ),

where

Ai (λ) =

λ2 zN

(2.10) l = 5, . . . , N + 3,

0

1

− zi−1 zi

λ zi

(2.11)

.

Combining (2.6), (2.7), (2.8), (2.9), (2.10) and (2.11) we see that the system (1.2) has no subordinated solutions for λ ∈ E [see (2.5)]. This completes the proof. Notation (hyperbolic part). Now we turn to the remaining case of S := {λ ∈ R | |tr A(λ)| > 2}.

(2.12)

It happens that in this case we again can ﬁnd the asymptotic behaviour of generalized eigenvectors for λ ∈ S. This hyperbolic asymptotics will allow us to prove that the spectrum of J (α) in S is pure point. In order to prove its discreteness we assume that α ∈ ( 12 , 1). This restriction is not necessary. The results (we prove below) hold true for any α ∈ (0, 1] but their proofs are less elementary. Theorem 3. The spectrum of J (α) in S is discrete.

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Proof. For ys := u(N +2)s+1 we consider the system ys = A(λ) + s−2α B(λ) ys−1 , λ ∈ S, s = 2, 3, . . . By diagonalization of A(λ) we can write A(λ) = V (λ)D(λ)V (λ)−1 with D(λ) diagonal. Hence, for zs := V (λ)−1 ys we have the new system zs = D(λ) + s−2α B1 (λ) zs−1 , s = 2, 3, . . . (2.13) where B1 (λ) = V −1 (λ)B(λ)V (λ). All functions V (λ), V (λ)−1 , D(λ) of λ can be chosen continuous [8]. Fix a bounded interval I ⊂ S. Since α > 12 we can apply a uniform ver (k) sion of the Benzaid–Lutz result [19, Theorem 3.2] and obtain a basis φ n (λ) (k = 1, 2) of solutions of (2.13) such that n−1 −1 (k) (k) (λ) −−−−→ 0, φ μ (λ) − e (2.14) sup k − n i λ∈I n→+∞ i=N0 the spectrum

(2) σ A(λ) + s−2α B(λ) = μ(1) s (λ), μs (λ) −2α and large that B(λ) < −2 for all s ≥ N0 . The case tr A(λ) + s N0 is so−2α B(λ) > 2 is similar. It follows that tr A(λ) + s 1 (2.15) − tr As − disc As < 1, 0 < μ(1) s (λ) := 2 where As := A(λ) + s−2α B(λ). Using (2.14) and (2.15) we have ∞ (1) ∈ 2 . supφn (λ) λ∈I

n=1

This can be easily seen, by repeating the reasoning given in the proofs of (2.7), (2.8), (2.9), (2.10) and (2.11). In particular, the “eigenvalue equation” (1.2) has a nontrivial subordinated solution. ¯ (by Consequently, S ∩ σac (J (α)) = Ø and so E ⊂ σac (J (α)) ⊂ E Theorem 2). The subordination theory shows that σsc (J (α)) is empty [15]. Therefore, S contains only eigenvalues of J (α). Using [19, Lemma 4.2] for a given n ≥ N0 and any ε > 0 one can ﬁnd δ > 0 such that |λ − μ| < δ (1) (λ) − φ (1) (μ) < ε. =⇒ φ (2.16) n n λ, μ ∈ I Now we can repeat the argument of the proof of [19, Theorem 5.3]. Although this argument was given for a particular class of Jacobi matrices it can be applied in general situations. To prove discreteness of σ (J (α)) in S assume that there exists a sequence of distinct eigenvalues {m } of J (α) which accumulates at ∈ S. Altogether (2.14) provides a uniform tails estimate, and (2.16) guarantees the coordinate convergence (1) (). (1) (m ) −−−−→ φ φ n n m→∞

This allows to repeat the reasoning given in [19, p. 184].

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n (s )} of J (α) (for Indeed, by orthogonality of the eigenvectors {φ different s ) we have (1)

(1) (k )}, {φ (1) (m )} 0 = m→∞ lim {φ n n k→∞ k=m

2

=

+∞

(1) ()|2 . |φ s

s=1

The last equality can be easily veriﬁed by using (2.14) and (2.16). Therefore, (1) φ n () = 0, and this contradiction completes the proof of Theorem 3). 2.3. Generalization Slight perturbations of the above examples allow to obtain a class of unbounded Jacobi matrices with the same spectral picture as before. Notation. D1 denotes the space of bounded variation sequences. Definition 4 (J (α)). The Jacobi matrix J (α) is deﬁned by zero diagonal and by the new weights (l + 1)α + ck (l), if k = 1, 2, l = 0, 1, . . . (2.17) a ˜l(N +2)+k = zk−2 + tk (l), if k = 3, . . . , N + 2, l = 0, 1, . . . submitted to the following two assumptions: {tk (l)}l≥0 ∈ 2 ∩ D1 ,

k = 3, 4, . . . , N + 2.

α

α

2

(2.18)

1

{c1 (l)/l }l≥0 & {c2 (l)/l }l≥0 ∈ ∩ D .

(2.19)

Remark. Note that (2.19) allows unbounded sequences {c1 (l)} and {c2 (l)}. By repeating the computations leading to (2.4) we ﬁnd B(N +2)s (λ) · · · B(N +2)(s−1)+1 (λ) 0 1 0 1 = ... +tN −1 (l) λ λ 1 (l) − zz12 +t − zNz−1 +t2 (l) z2 +t2 (l) +t (l) z +t (l) N N N N 0 1 0 1 0 × α lα +c2 (l) l +c (l) +tN (l) − z1 +t1 (l) z1 +tλ1 (l) − lα +c12 (l) lα +cλ2 (l) − zlNα +c 1 (l) Note that due to (2.18) we have the following decomposition 0 1 = Cs + Wsl (λ) + Rsl (λ), +ts−1 (l) λ − zs−1 zs +ts (l) zs +ts (l) where

Cs :=

0

1

− zs−1 zs 0 Wsl (λ) := z

λ zs

, 0

s−1 ts (l)−zs ts−1 (l) zs 2

{Rsl (λ)}l≥0 ∈ 1 ,

− λtzss(l)

s = 2, . . . , N.

,

(2.20)

1 λ lα +c1 (l)

.

(2.21)

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Using (2.21) and distributive rule in (2.20) we can rewrite the product of the ﬁrst N − 1 matrices of the right side of (2.20) as follows: M (λ) +

N −1

CN . . . Cr+1 Wrl (λ)Cr−1 . . . C2 + Rl (1) (λ),

r=2 (1) {Rl (λ)}l≥0

with has the form:

∈ 1 . Since {Wrl (λ)}l≥0 ∈ 2 ∩ D1 the above product (1)

M (λ) + Tl (λ) + Rl (λ), 2

(2.22)

1

where {Tl (λ)}l≥0 ∈ ∩ D . In turn, due to (2.19) the product of the three last matrices of the right side of (2.20) can be written (2)

DN + Vl + Rl (λ) with

DN = ⎛

(2.23)

0

−1

zN z1

−2 zλ1

, (c2 (l) − c1 (l))l−α

0

Vl = ⎝ z 2 t

1 N (l−1)−zN tN (l) z1 2

+

zN (c2 (l)−c1 (l) z1 l−α

0

⎞ ⎠,

(2)

{Rl (λ)}l≥0 ∈ 1 . Combining (2.22) and (2.23) we rewrite the right hand side of (2.20) as (3)

A(λ) + Sl (λ) + Rl (λ),

(2.24)

where A(λ) = M (λ)DN (λ), {Sl (λ)}l≥0 ∈ 2 ∩ D1 , (3)

{Rl (λ)}l≥0 ∈ 1 . By repeating the proofs of Theorems 2 and 3 we obtain ˜ Theorem 5. Let J(α) be the Jacobi operator defined by the weights (2.17) submitted to conditions (2.18) and (2.19) with α > 1/2. Then the statements ˜ of Theorems 2 and 3 remain valid for J(α). Proof (sketch). We ﬁrst note that due to the definitions of Rsl (λ), Tl (λ), M (λ) (1) (2) and (2.24) the error terms Rl (λ) and Rl (λ) are continuous functions of (1) (2) λ. Moreover, both Rl (λ) and Rl (λ) have uniform tail estimates, i.e. for any compact subset X ⊂ R and for any ε > 0 there exists K = K(ε, X) such that ∞ (s) Rl (λ) < ε, s = 1, 2. sup λ∈X

l=K

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(3)

Consequently, Rl (λ) satisﬁes similar tail estimates. Remember that A(λ) and Sl (λ) are also continuous functions of λ (to be more precise they are polynomials in λ). Using a uniform Levinson type theorem (see [19] for the case of uniform 1 perturbations and [20] for the case of uniform D1 ones) one can easily complete the proof of the analogues of Theorems 2 and 3. Remark. By adding a nonzero main diagonal {bk } ∈ 2 ∩ D1 one obtains a still larger class of Jacobi matrices with the same spectral picture as J (α). This surely introduces a compact perturbation of the Jacobi operator and by Weyl’s theorem it preserves the essential spectrum, but our goal is to preserve σac (J (α)).

3. Analysis of Particular Examples One of the aims of particular examples analysis is to verify the nondegeneracy of arbitrary many gaps in the essential spectrum. Properties of the Characteristic Polynomial We shall discuss some properties of the polynomial p(λ) := tr A(λ) which will be useful below. We deﬁne the antidiagonal and diagonal matrices, respectively 0 0 0 1 Ak := , Bk := , k = 1, . . . , N − zk−1 0 0 z1k zk where z0 := zN . In what follows we assume that N is an even number and all zj are positive (without loss of generality). We have p(λ) = tr[(AN + λBN ) · · · (A2 + λB2 )(−A1 − 2λB1 )] =:

N

ck λk . (3.1)

k=0

Lemma 3.1. Let N be an even number. Then the coefficients ck of the characteristic polynomial p(λ) := tr A(λ) satisfy: (i) c2m+1 = 0, for m = 0, 1, 2, . . . N −2m (ii) The sign of c2m coincides with −(−1) 2 . Proof. We pick up the ﬁrst n1 antidiagonal terms Ai (i = N, . . . , N − n1 + 1). After that in the next to the right matrix we take only the term λBN −n1 . The same procedure can be repeated, i.e., we pick up the next n2 antidiagonal matrices As (s = N − n1 − 1, . . . , N − n1 − n2 ), etc. Totally, we get a sequence n1 , n2 , . . . , nk+1 , 0 ≤ ns ≤ N such that !k+1 s=1 ns = N − k. Observe that all the traces tr (AN · · · AN −n1 +1 BN −n1 · · · ) = 0, provided that k is odd. This is clear because then N −k is an odd number and therefore the above product AN · · · AN −n1 +1 BN −n1 · · · has an odd number of anti-diagonal matrices As and a certain number of diagonal matrices Br . Hence the whole product gives an anti-diagonal matrix (therefore with zero

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trace). Moreover, in all nonzero terms tr (AN · · · AN −n1 +1 BN −n1 · · · ) all the numbers n1 , n2 , . . . , nk+1 must be even. Indeed, this fact is a consequence of the following elementary matrix algebra. First note that due to the invariance of the trace with respect to the cyclic permutations one can reduce the proof to the case that n1 should be even. Assume for a moment that n1 is an odd number. Then the ﬁrst to the left product of n1 anti-diagonal matrices AN · · · AN −n1 +1 isagain an anti-diagonal matrix. The next term 0 0 BN −n1 has the form with one nonzero entry in the right lower cor0 ner. The remaining product of matrices contains k − 1 diagonal matrices Bs and N − n1 − k anti-diagonal matrices As . Since the numbers N and k are even and n1 is odd, so N −n1 −k is odd, and therefore the remaining product must also contain an odd number nr of matrices Ai ’s. We can assume that the remaining product begins from the left with an anti-diagonal matrix. In fact, all diagonal factors at the left of the remaining product multiplied by 0 0 . Therefore, the whole BN −n1 would produce a matrix of the form 0 product would have the following structure 0 0 0 0 ∗ ∗ ··· 0 0 0 ∗ ∗ ∗ where all matrices are diagonal. But the product of these diagonal ∗ ∗ matrices contains Bi ’s and so the product of all of them has the ∗ 0 0 form . It follows that the whole product is equal to 0 0 0 0 0 0 0 0 0 = . 0 0 0 0 0 0 This contradiction proves that n1 is even. A more detailed analysis of the product of ni anti-diagonal As shows that it is a positive diagonal matrix ni muliplied by (−1) 2 . This fact can be easily checked using the structure of 0 αs , αs > 0, βs < 0. As = βs 0 Since all Bs have non-negative entries, the sign of ck (for even k) coincides with −(−1)

n1 2

+···+

nk 2

= −(−1)

N −k 2

.

Therefore, the signs of c0 , c2 , . . . , cN form an alternating sequence. In particN ular, the sign of c0 is equal to −(−1) 2 . The proof is complete. Remark 3.2. Note that vanishing of c2m+1 is not surprising, due to the symmetry λ → −λ of σ (J (α)) with vanishing main diagonal of J (α).

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Remark 3.3. Let N = 2M . Denote μ := λ2 . Then √ P (μ) := p( μ) = c2M μM + · · · + c2 μ + c0 , with alternating signs of the coefﬁcients. Observe that c2M = −

2 zN zN −1 . . . z1

and so c2M = 0.

4. Asymptotics of the Discrete Spectrum at Infinity In this section we shall ﬁnd asymptotic formulae for the eigenvalues of J (α) at ±∞. More precisely, we shall compute the main terms of these formulae (modulo bounded corrections). In recent years appeared several papers devoted to the asymptotic behaviour of the eigenvalues of unbounded, selfadjoint Jacobi matrices [3,7,11,16,25]. However, the results found in these works do no apply to J (α). Therefore, we have decided to include asymptotic formulas for the eigenvalues of J (α), although they are not the main concern of this work. The idea of computing approximate values of σp (J (α)) (for sufﬁciently large eigenvalues) is based on the following decomposition J (α) = J 1 + J 2 where J 1 , J 2 are Jacobi matrices and the weights of J 1 are obtained from the weights of J (α) by replacing all zi (i = 1, . . . , N ) by zeros whereas the weights of J 2 are deﬁned by replacing all terms (k α , k α ) (k = 1, 2, . . .) by zeros. The definition of J 1 implies that J1 =

∞ " (Mk ⊕ 0N ),

(4.1)

k=1

where 0N is the (N − 1) × (N − 1) zero matrix and ⎛

0

⎜ α Mk := ⎜ ⎝k 0

kα 0 kα

0

⎞

⎟ kα ⎟ ⎠. 0

By definition of J 2 we have J2 =

∞ " k=1

(02 ⊕ Lk ),

(4.2)

164 where

A. Boutet de Monvel, J. Janas and S. Naboko ⎛

0

⎜ ⎜ z1 ⎜ ⎜ ⎜0 ⎜ ⎜. Lk := L := ⎜ ⎜ .. ⎜ ⎜ ⎜... ⎜ ⎜ ⎝... 0

z1

0

0

...

...

0

z2

0

...

...

z2

0

z3

...

...

..

..

..

..

..

0

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...

...

...

...

...

...

...

...

zN −1

0

⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ .. ⎟ ⎟ . ⎟. ⎟ ⎟ 0 ⎟ ⎟ ⎟ zN ⎠

0

...

...

0

zN

0

.

.

.

.

.

Well known elementary estimates and (4.2) show that π J 2 ≤ 2 max|zi | · cos i N +1 and σess (J 2 ) = {0} ∪ σ(L). Applying [2, Theorem 4, Chap. 9, Sec. 4, p. 219] we have σess (J (α)) ⊂ [−J 2 , J 2 ] = [−L, L].

(4.3)

In what follows we shall consider σp (J (α)) outside the interval [−L, L]. Since the spectrum of J (α) outside this interval is discrete and symmetric with respect to zero (bn ≡ 0) it accumulates at ∓∞. Denote L = a, and take a large number M a. For a self-adjoint operator T , let ΠT (α, β) be the number of eigenvalues of T in the interval (α, β). At this point we use standard arguments of perturbation theory. Namely applying [2, Lemma 3, Chap. 9, Sec. 4, p. 218] twice (with A = J 1 , V = J 2 or A = J (α), V = − J 2 , respectively) we have the inequalities ΠJ 1 (2a, M ) ≤ ΠJ (α) (a, M + a) ≤ ΠJ 1 (0, M + 2a). Note that

(4.4)

√ √ σ(Mk ) = − 2k α , 0, 2k α .

By definition of J 1 and standard arguments we can write α1 M , ΠJ 1 (2a, M ) ∼ √ 2 ΠJ 1 (a, M + a) ∼

M √ 2

α1

+

M √ 2

α1

a αM ,

as M → +∞. Combining (4.4), (4.5) and (4.6) we have 1 M α + rM , ΠJ (α) (a, M + a) = √ 2 1 where rM = O M α −1 , as M → +∞. Let 1 + a < λ1 (J (α)) < λ2 (J (α)) < · · ·

(4.5) (4.6)

(4.7)

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be the sequence of all eigenvalues of J (α) greater than a + 1. The above reasoning and formula (4.7) prove Theorem 6. The large n asymptotic behaviour of λn (J (α)) is given by √ λn (J (α)) = 2 nα (1 + O(1)). (4.8) Open Problems. Two open problems about σp (J (α)). (i) We expect that there is no concentration of σp (J (α)) to the right and left of σess (J (α)). (ii) Is σp (J (α)) in the “interior” gaps finite?

5. Approximate Lengths of Bands (N Even) In this section we consider the Jacobi matrix J (α) associated to a particular sequence z1 , . . . , zN (all are powers εγs of a small positive parameter ε) which will allow to compute the small ε asymptotics of the lengths of the bands of E. This choice of the zs will also allow to give sufﬁcient conditions (in terms of the above exponents γs ) for E being exactly the union of N disjoint intervals. Most of the proofs will be sketchy in order to avoid lengthy but straightforward calculations. Definition 7 (J ε (α)). Let ε be a positive number. We denote J ε (α) the Jacobi matrix J (α) associated to a sequence z1 = εγ1 , . . . , zN = εγN with γ1 < · · · < γN . Lemma 8. Assuming that the sequence {γk } is convex, the coefficients c2k (see (3.1)) admit the following asymptotics, as ε → 0: N

c2k ∼ −(−1) 2 −k (zN zN −1 · · · zN −2k+1 )−1 Φ(k), where Φ(k) :=

zN −2k−1 zN −2k−3 . . . z1 . zN −2k zN −2k−2 . . . z2

Proof. Indeed, the direct computation of ⎛ ⎜ tr ⎜ ⎝

(5.1)

n1 ,...,nk n1 +···+nk =N −k

⎞

⎟ Xn1 . . . Xnk ⎟ ⎠

for a suitable choice of matrices Xi shows that the dominant term among all components of the above traces is given by tr(BN BN −1 · · · BN −2k+1 AN −2k · · · A2 A1 )

(5.2)

for sufﬁciently small ε. This fact can be easily veriﬁed using the convexity of the sequence {γs }. Using our choice of zs = εγs and (5.2) we ﬁnd c2k ∼ ±(−1)k ε−Δk ,

as ε → 0,

(5.3)

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where Δk := γN + γN −1 + · · · + γN −2k+1 +(γN −2k − γN −2k−1 ) + · · · + (γ2 − γ1 ) for k = 1, . . . ,

N , 2

(5.4)

Δ0 := γN − γN −1 + γN −2 − γN −3 + · · · + γ2 − γ1 , with γ0 := 0. Thus, Δs+1 − Δs = 2γN −2s−1 ,

s ≥ 1.

(5.5)

Note that the Δs are positive and increasing. It follows that Δs+1 − Δs = 2γN −2s−1 < 2γN −2s+1 = Δs − Δs−1 ,

(5.6)

i.e., the sequence {Δs } is concave.

Below we shall ﬁnd approximate roots μ−s (respectively μ+s ) of the equation P (μ) = −2 (respectively P (μ) = 2), s = 1, . . . , N = 2M . Since c2M = −

2 < 0, zN zN −1 . . . z1

it is clear that P (μ) → −∞, as μ → +∞, independently on the parity of M . Hence the largest root coincides with μ−1 and corresponds to the equation P (μ) = −2. It turns out that a rough approximation of the roots can be found by the following simple procedure. Although simple this rough approximation dramatically reduces the complexity of ﬁnding the roots and is sufﬁcient for our aim. Surely this procedure works due to the special form of the coefﬁcients of the polynomial P . First Step We look for the solutions of the following simpliﬁed equations (just keeping two leading terms of the polynomial P ): c2k μk + c2k−2 μk−1 = 0,

k = 1, . . . , N2 .

(5.7)

The solutions of (5.7) are: μk ≡ 0, μ ˜k := −

c2k−2 ∼ εΔk −Δk−1 , as ε → 0. c2k

Below (the second step) by using a better approximation of the polynomial P we shall split the zero solution into smaller distinct roots. ˜sk decay, for s = k − 1, k, faster than c2k μ ˜kk Observe that all terms c2s μ k−1 and c2k−2 μ ˜k as ε → 0. Indeed, we have two possibilities: • s < k − 1. By concavity of {Δr } we have the following inequalities: Δk − Δs = Δk − Δk−1 + · · · + Δs+1 − Δs > (k − s)(Δk − Δk−1 ).

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But these inequalities are equivalent to s(Δk − Δk−1 ) − Δs > k(Δk − Δk−1 ) − Δk which by the form of c2s implies faster decay of c2s μ ˜sk . • s > k. Similarly we have Δs−1 − Δk−1 = Δs−1 − Δs−2 + · · · + Δk − Δk−1 < (s − k)(Δk − Δk−1 ). Thus, Δs − Δk < Δs−1 − Δk−1 < (s − k)(Δk − Δk−1 ). Again the last inequality is equivalent to k(Δk − Δk−1 ) − Δk < s(Δk − Δk−1 ) − Δs which implies the desired faster decay of c2s μ ˜sk . Second Step We assume that one can ﬁnd “better” approximate roots by choosing μ ˜±k = μ ˜k (1 + δ± (k)),

k = 1, . . . , N2

(5.8)

as the solutions of the “better” equation c2k μ ˜k±k + c2k−2 μk−1 ±k = ±2

(5.9)

(recall that we look for the solutions of P (μ) = ±2). Combining the asymptotic form for c2k [see (5.3), (5.8) and (5.9)] we obtain δ± (k) ∼ ±2εΔk −k(Δk −Δk−1 )

(5.10)

as ε → 0. Since μ ˜+s ∼ μ ˜−s μ ˜+,s+1 ∼ μ ˜−,s+1 as ε → 0 we have ˜−s = (δ+ (s) − δ− (s)) μ ˜s ∼ 4εΔs −(s−1)(Δs −Δs−1 ) . (5.11) μ ˜+s − μ √ √ √ √ This allows to calculate the lengths of bands ( μ ˜+s , μ ˜−s ) or ( μ ˜−s , μ ˜+s ). In fact, 1 ˜ μ ˜+s − μ √−s ∼ 2εΔs −(s− 2 )(Δs −Δs−1 ) μ ˜+s − μ ˜−s = √ (5.12) μ ˜+s + μ ˜−s as ε → 0. The above arguments lead to Theorem 9. Let J ε (α) be the Jacobi matrix defined by a convex √ √sequence ˜−s , μ ˜+s ) of γ1 < · · · < γN as in Definition 7. The lengths of the bands ( μ the spectrum of J ε (α) lying in R+ (the spectrum is symmetric w.r.t. zero) have the following asymptotics, as ε → 0: 1 μ ˜+s − μ ˜−s ∼ 2εΔs −(s− 2 )(Δs −Δs−1 ) .

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Remark. The above “approximate” lengths of the bands are decreasing as the parameter s increases. Moreover, μ ˜±s have different orders for different s. More precisely ˜+1 μ ˜+2 > μ ˜−2 . . . μ ˜−1 > μ for ε sufﬁciently small. Acknowledgements J. J. is supported in part by “INTAS” and in part by MSHE grant N N201 426533. S. N. is supported in part by “INTAS” and in part by RFBR grant 09-01-00515a. S. N. is also grateful to the Universit´e Paris Diderot (where a part of this work has been done) for its hospitality. Open Access. This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References [1] Berezanski˘ı, J. M.: Expansions in eigenfunctions of selfadjoint operators. In: Bolstein, R., Danskin, J.M., Rovnyak, J., Shulman, L. Translated from the Russian, Translations of Mathematical Monographs, vol. 17. American Mathematical Society, Providence (1968) [2] Birman, M. S., Solomjak, M. Z.: Spectral Theory of Selfadjoint Operators in Hilbert Space. Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht (1987) [3] Boutet de Monvel, A., Naboko, S., Silva, L. O.: The asymptotic behavior of eigenvalues of a modiﬁed Jaynes–Cummings model. Asymptot. Anal. 47(3–4), 291–315 (2006) [4] Christiansen, J. S., Simon, B., Zinchenko, M.: Finite gap Jacobi matrices. I. The isospectral torus. Constr. Approx. 32, 1–65 (2010) [5] Dombrowski, J., Pedersen, S.: Absolute continuity for unbounded Jacobi matrices with constant row sums. J. Math. Anal. Appl. 267(2), 695–713 (2002) [6] Dombrowski, J., Pedersen, S.: Spectral transition parameters for a class of Jacobi matrices. Stud. Math. 152(3), 217–229 (2002) [7] Janas, J., Malejki, M.: Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices. J. Comput. Appl. Math. 200(1), 342–356 (2007) [8] Janas, J., Moszy´ nski, M.: Spectral properties of Jacobi matrices by asymptotic analysis. J. Approx. Theory 120(2), 309–336 (2003) [9] Janas, J., Naboko, S.: Jacobi matrices with power-like weights—grouping in blocks approach. J. Funct. Anal. 166(2), 218–243 (1999) [10] Janas, J., Naboko, S.: Spectral analysis of selfadjoint Jacobi matrices with periodically modulated entries. J. Funct. Anal. 191(2), 318–342 (2002)

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[11] Janas, J., Naboko, S.: Inﬁnite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach. SIAM J. Math. Anal. 36(2), 643–658 (2004) [12] Janas, J., Naboko, S., Stolz, G.: Spectral theory for a class of periodically perturbed unbounded Jacobi matrices: elementary methods. J. Comput. Appl. Math. 171(1-2), 265–276 (2004) [13] Janas, J., Naboko, S., Stolz, G.: Decay bounds on generalized eigenvalues of unbounded jacobi matrices and the singular spectrum. Int. Math. Res. Not. IMRN, pages Art. ID rnn144, 29 (2009) [14] Khan, S., Pearson, D. B.: Subordinacy and spectral theory for inﬁnite matrices. Helv. Phys. Acta 65(4), 505–527 (1992) [15] Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schr¨ odinger operators. Invent. Math. 135(2), 329–367 (1999) [16] Malejki, M.: Approximation of eigenvalues of some unbounded self-adjoint discrete Jacobi matrices by eigenvalues of ﬁnite submatrices. Opuscula Math. 27(1), 37–49 (2007) [17] Moszy´ nski, M.: Spectral properties of some Jacobi matrices with double weights. J. Math. Anal. Appl. 280(2), 400–412 (2003) [18] Pedersen, S.: Absolutely continuous Jacobi operators. Proc. Am. Math. Soc. 130(8), 2369–2376 (2002) [19] Silva, L. O.: Uniform and smooth Benzaid–Lutz type theorems and applications to Jacobi matrices. Operator theory, analysis and mathematical physics. Oper. Theory Adv. Appl., vol. 174, 173–186. Birkh¨ auser, Basel (2007) [20] Silva, L. O.: On the spectral analysis of selfadjoint Jacobi matrices and its applications. PhD thesis. St-Petersburg State University, Russia (2002) [21] Simonov, S.: An example of spectral phase transition phenomenon in a class of Jacobi matrices with periodically modulated weights. Operator theory, analysis and mathematical physics. Oper. Theory Adv. Appl., vol. 174, 187–203. Birkh¨ auser, Basel (2007) [22] Sodin, M., Yuditskii, P.: Almost periodic Jacobi matrices with homogeneous spectrum, inﬁnite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. Geom. Anal. 7(3), 387–435 (1997) [23] Stone, M. H.: Linear transformations in Hilbert space. American Mathematical Society Colloquium Publications, vol. 15. American Mathematical Society, Providence (1990) [24] Teschl, G.: Jacobi Operators and Completely Integrable Nonlinear Lattices. Mathematical Surveys and Monographs, vol. 72. American Mathematical Society, Providence (2000) [25] Zielinski, L.: Eigenvalue asymptotics for a class of Jacobi matrices. In: Hot Topics in Operator Theory: Conference Proceedings, Timi¸soara, June 290–July 4, 2006. Theta Ser. Adv. Math., vol. 9, pp. 217–229. Theta, Bucharest (2008)

170

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Anne Boutet de Monvel Institut de Math´ematiques de Jussieu Universit´e Paris Diderot Paris 7 175 rue du Chevaleret 75013 Paris France e-mail: [email protected] Jan Janas (B) Institute of Mathematics Polish Academy of Sciences ´ Tomasza 30 ul. Sw. 31-027 Krak´ ow Poland e-mail: [email protected] Serguei Naboko Department of Mathematical Physics, Institute of Physics St. Petersburg University Ulianovskaia 1 198904, St. Petergoﬀ, St. Petersburg Russia e-mail: [email protected] Received: June 27, 2009. Revised: October 26, 2010.

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Integr. Equ. Oper. Theory 69 (2011), 171–182 DOI 10.1007/s00020-010-1809-4 Published online June 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Essential Normality of Homogeneous Submodules J¨org Eschmeier Abstract. Let M ⊂ H(B) be a homogeneous submodule of the n-shift Hilbert module on the unit ball in Cn . We show that a modification of an operator inequality used by Guo and Wang in the case of principal submodules is equivalent to the existence of factorizations of the form [Mz∗j , PM ] = (N + 1)−1 Aj , where N is the number operator on H(B). Thus a proof of the inequality would yield positive answers to conjectures of Arveson and Douglas concerning the essential normality of homogeneous submodules of H(B). We show that in all cases in which the conjectures have been established the inequality holds and leads to a unified proof of stronger results. Mathematics Subject Classification (2010). Primary 47A13; Secondary 46H25, 47A53. Keywords. Essential normality, homogeneous submodules.

1. Introduction Let H(B) be the n-shift space on the open unit ball B in Cn , that is, the Hilbert space of analytic functions on B determined by the reproducing kernel 1 . K : B × B → C, K(z, w) = 1 − z, w For an analytic function f : B → C, we denote by f = α∈Nn fα z α and ∞ f = k=0 fk the Taylor expansion and the homogeneous expansion of f . We write Hk for the space of all homogeneous polynomials of degree k in n complex variables. It is well known that the n-shift space H(B), also known as the symmetric Fock space or Drury–Arveson space, consists precisely of all analytic functions f = α∈Nn fα z α with α! |fα |2 < ∞. f 2 = |α|! n α∈N

Here we have used the notation α! = α1 ! · · · αn ! and |α| = α1 + · · · + αn .

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Since the multiplication operators Mzi : H(B) → H(B), f → zi f , with the coordinate functions are bounded, the n-shift space H(B), becomes a Hilbert module over the polynomial ring C[z] in n variables. In Arveson’s language [6] a Hilbert module is essentially normal if all the cross-commutators [Mzi , Mz∗j ] = Mzi Mz∗j − Mz∗j Mzi

(1 ≤ i, j ≤ n)

given by the module structure are compact, and more specifically, q-essentially normal if the cross-commutators belong to the Schatten class C q , where q ∈ [1, ∞). It is well known that the n-shift space H(B), as well as the Hardy space H 2 (B) and the Bergman space L2a (B), are q-essentially normal for q > n. In the case of the n-shift space, Arveson raised the question [3,4] whether every quotient H(B) ⊗ CN /M of the n-shift Hilbert module of finite multiplicity N by a submodule M generated by finitely many CN -valued homogeneous polynomials p1 , . . . , pr is q-essentially normal for q > n. Arveson gave a positive answer for submodules generated by monomials [5]. This result was extended by Douglas [8] to other analytic Hilbert modules. In [6] Arveson reduced the general case to submodules generated by linear CN -valued polynomials. Guo and Wang [13] showed that the conjecture for the n-shift space is true in dimensions ≤ 3 and for all principal homogeneous submodules in arbitrary dimension. In this note we consider the scalar-valued case. By a submodule M ⊂ H(B) we mean a closed subspace M ⊂ H(B) which is invariant under Mz1 , . . . , Mzn . A submodule M ⊂ H(B) is called homogeneous if M = M ∩ Hk . An elementary argument shows that a submodule M ⊂ H(B) k≥0 is homogeneous if and only if it is generated by finitely many homogeneous polynomials, that is, there is a finite tuple p = (p1 , . . . , pr ) of homogeneous polynomials pi such that M = (p) is the closure of the ideal (p) generated by p1 , . . . , pr in C[z]. Let M ⊂ H(B) be a homogeneous submodule. Starting point is the observation that a slight modification of an inequality used by Guo and Wang [13] for principal homogeneous submodules is equivalent to the existence of bounded operators Aj ∈ L(H(B)) satisfying the factorization 1

[Mz∗j , PM ] = (N + 1)− 2 Aj

(j = 1, . . . , n).

Here N is the number operator on H(B) and Mz = (Mz1 , . . . , Mzn ) denotes the tuple of multiplication operators with the coordinate functions on H(B). Let S = PM ⊥ Mz |M ⊥ ∈ L(M ⊥ )n be the quotient tuple induced by Mz on M ⊥ ∼ = H(B)/M . Then any factorization of the above type leads to a corresponding factorization [Sj∗ , Si ] = (N + 1)−1 Bij

(i, j = 1, . . . , n)

of the cross-commutators of S. Using a result from [11] we show that the restriction of (N + 1)−1 to M ⊥ belongs to the Schatten class C q for every q > dim0 Z(M ). Hence it follows that the quotient tuple S is q-essentially normal for every q > dim0 Z(M ) in this case. Here Z(M ) denotes the zero set of the submodule M .

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Thus a proof of the inequality referred to above would give a positive answer to a refinement of the Arveson conjecture due to Douglas [7]. In this note we indicate that, for all classes of homogeneous submodules M ⊂ H(B) which are known to be essentially normal, the above technique applies and gives a direct and very natural proof of the q-essential normality of the quotient tuple S = PM ⊥ Mz |M ⊥ ∈ L(M ⊥ )n for every q > dim0 Z(M ).

2. Main Results Let N be the number operator on H(B), that is, the self-adjoint operator generating the unitary operator group U : R → L (H(B)) defined by (U (t)f ) (z) = f (eit z)

(z ∈ B).

Then N is an unbounded self-adjoint operator on H(B) such that, for all k ∈ N, N h = kh

(h ∈ Hk ).

Here Hk ⊂ C[z] denotes the subspace consisting of all homogeneous polynomials of degree k. For each function f : N → R, wedenote by D(f ) the ∞ linearsubspace of H(B) consisting of all functions h = k=0 hk ∈ H(B) such ∞ that k=0 f (k)hk is the homogeneous expansion of a function in H(B). It is elementary to check that f (N ) : D(f ) → H(B),

h →

∞

f (k)hk

k=0

defines a closed operator whose domain contains at least all polynomials. Let r ∈ Z be an arbitrary integer. By an operator of degree r on H(B) we mean a bounded operator A ∈ L (H(B)) such that AHk ⊂ Hk+r for every non-negative integer k. Here Hk+r = {0} for k + r < 0. For f : N → R, define fr : N → R,

fr (k) = f (k + r)

(= 0 if k + r < 0).

For every operator A ∈ L (H(B)) of degree r ∈ Z and every given function f : N → R, the relations f (N )A = Afr (N ) and Af (N ) = f−r (N )A hold on C[z] = k≥0 Hk . For all non-negative integers m, and every multi-index α ∈ Nn with |α| ≤ m, let Gm,,α : N → R be the function defined by Gm,,α (k) =

k! (k + m − )! (k − ( − |α|))! (k + m)!

(= 0 if k < − |α|).

For p ∈ C[z], we denote by Mp : H(B) → H(B), h → ph, the induced multiplication operator. In [13] it is observed that, for any pair of homogeneous polynomials p ∈ Hm , q ∈ H , the identity Gm,,α (N ) M∂ α q M∂∗α p Mp∗ Mq = α! |α|≤m

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holds on C[z]. Here ∂ α = ∂zα11 . . . ∂zαnn denotes the partial derivatives with respect to the variables z1 , . . . , zn . Let p = (p1 , . . . , pr ) be a tuple of homogeneous polynomials. For k = 1, . . . , n, set ∂k p = (∂k p1 , . . . , ∂k pr ). The submodule M = (p) ⊂ H(B) generated by p is the closure of the range of the operator δp : H(B)r → H(B),

(hi ) →

r

pi h i .

i=1

Lemma 2.1. Let p ∈ Hm be a homogeneous polynomial of degree m and let M ⊂ H(B) be a homogeneous submodule with p ∈ M . Then the identities PM ⊥ Mz∗j Mp = (N + 1)−1 PM ⊥ M∂j p hold for j = 1, . . . , n. Proof. Recall from the section preceding the lemma that the identity G1,m,α (N ) M∂ α p M∂∗α zj Mz∗j Mp = α! |α|≤1

holds on C[z] for j = 1, . . . , n. Since G1,m,ej (k) = (k + 1)−1 for k ≥ m − 1, where ej is the j-th canonical unit vector, we find that PM ⊥ Mz∗j Mp = (N + 1)−1 PM ⊥ M∂j p holds on C[z] for j = 1, . . . , n. Since the operators occurring on both sides are bounded, the same identity holds on all of H(B). If M ⊂ H(B) is a homogeneous submodule generated by a family p = (p1 , . . . , pr ) of homogeneous polynomials, then the preceding lemma shows that the identities PM ⊥ Mz∗j δp = (N + 1)−1 PM ⊥ δ∂j p

(j = 1, . . . , n)

hold on H(B)r . In particular, we find that the operators PM ⊥ δ∂j p vanish on Ker δp . Proposition 2.2. Let M ⊂ H(B) be a submodule generated by a system p = (p1 , . . . , pr ) of homogeneous polynomials. Then for j = 1, . . . , n, the following are equivalent: (i) there is a constant c > 0 such that δ∂∗j p (N + 1)−1 PM ⊥ δ∂j p ≤ c δp∗ δp ; (ii) there is a bounded operator Aj ∈ L(H(B)) such that [Mz∗j , PM ] = 1 (N + 1)− 2 Aj . Proof. Condition (i) is equivalent to the existence of a constant c > 0 such that the inequality 1

(N + 1)− 2 PM ⊥ δ∂j p h2 ≤ cδp h2 holds for all h ∈ H(B)r . This in turn is equivalent to the existence of a bounded operator Aj ∈ L(H(B))with Aj |M ⊥ = 0 and 1

Aj δp = (N + 1)− 2 PM ⊥ δ∂j p .

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Note that [Mz∗j , PM ] = PM ⊥ Mz∗j PM and that Lemma 2.1 yields the identity 1 1 [Mz∗j , PM ]δp = PM ⊥ Mz∗j δp = (N + 1)− 2 (N + 1)− 2 PM ⊥ δ∂j p . Hence the existence of an operator Aj as above is equivalent to the validity of condition (ii). Note that, since the ranges of the operators [Mz∗j , PM ] are contained in M ⊥ , the range of any operator Aj satisfying the identity in condition (ii) of the last proposition is necessarily contained in M ⊥ . Furthermore, any such operator Aj is necessarily a bounded operator of degree −1. Let M ⊂ H(B) be a submodule and let S = Mz /M ∼ = PM ⊥ Mz |M ⊥ denote the quotient tuple induced by Mz . The inessential right spectrum of S

n n ⊥ ⊥ σrf (S) = z ∈ C ; 0 < dim M / (zi − Si )M <∞ i=1

is an analytic subset of the right essential resolvent set of S (cf. [11]). We define d = dim0 (σrf (S)) as the dimension of this analytic set at z = 0. a graded tuple Suppose that M is homogeneous. Then S ∈ L(M ⊥ )n is in the sense of [11] with respect to the decomposition M ⊥ = k≥0 M ⊥ ∩ Hk (see the final remarks following Corollary 2.6 in [11]). Let p = (p1 , . . . , pr ) be a finite tuple of homogeneous polynomials generating M . An elementary exercise shows that (p) = Ann(H(B))/(p)) coincides with the annihilator ideal of H(B)/(p) in C[z]. The polynomial spectral mapping theorem implies that the Taylor spectrum of S is contained in the zero set of (p) = Ann(H(B))/(p)) = {q ∈ C[z]; q(S) = 0}. In particular we obtain that σrf (S) ⊂ Z(p) and hence that d ≤ dim0 Z(p). One can show (Corollary 2.5 in [11]) that there is a one-variable polynomial u ∈ Q[x] with rational coefficients of deg(u) = d − 1 such that dim(M ⊥ ∩ Hk ) = u(k) for sufficiently large natural numbers k. Theorem 2.3. Let M ⊂ H(B) be a submodule generated by a family p = (p1 , . . . , pr ) of homogeneous polynomials. Suppose that there is a constant c > 0 such that δ∂∗j p (N + 1)−1 PM ⊥ δ∂j p ≤ c δp∗ δp

(j = 1, . . . , n).

Then the cross-commutators of the quotient S = PM ⊥ Mz |M ⊥ ∈ L(M ⊥ )n admit factorizations of the form [Sj∗ , Si ] = (N + 1)−1 Bij

(i, j = 1, . . . , n)

with suitable bounded operators Bij ∈ L(M ⊥ ). In particular, the tuple S is q-essentially normal for q > dim0 (σrf (S)).

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Proof. An elementary calculation reveals that [Sj∗ , Si ] ⊕ 0M = PM ⊥ [Mz∗j , Mzi ]PM ⊥ − [Mz∗j , PM ][Mz∗i , PM ]∗ for 1 ≤ i, j ≤ n. In [1] (Proposition 5.3) it was shown that [Mz∗j , Mzi ] = (N + 1)−1 (δij I − Mzi Mz∗j ) for 1 ≤ i, j ≤ n. By the remarks preceding the theorem it follows that tr ((N + 1)−1 |M ⊥ )q =

∞

(k + 1)−q dim(M ⊥ ∩ Hk ) < ∞

k=0

for q > d. By Proposition 2.2 there are bounded operators Aj ∈ L(H(B)) of 1 degree −1 such that the factorizations [Mz∗j , PM ] = (N + 1)− 2 Aj hold. Since [Mz∗j , PM ][Mz∗i , PM ]∗ = (N + 1)−1 Aj A∗i and since the operators complete.

Aj A∗i

(1 ≤ i, j ≤ n)

are reduced by M , the proof of Theorem 2.3 is

One can show that the hypothesis of Theorem 2.3 is satisfied for submodules M ⊂ H(B) generated by a tuple p = (p1 , . . . , pr ) of homogeneous polynomials over the unit ball B in Cn in each of the following cases (i) r = 1 (Guo and Wang [13]), (ii) all pi (1 ≤ i ≤ r) are monomials, (iii) dim0 Z(p) ≤ 1, (iv) (p1 , . . . , pr ) is a stable generating set for M in the sense of Shalit [14], (v) n ≤ 3. For r = 1 and a homogeneous polynomial p of degree m, Guo and Wang [13] gave a direct argument which shows that in this case even the inequality (N + m)−1 δ∂∗j p δ∂j p ≤ m δp∗ δp 1≤j≤n

holds. This inequality obviously implies condition (i) in Proposition 2.2. To treat the remaining cases we first prove an extension of Theorem 2.3. Theorem 2.4. Let M ⊂ H(B) be a submodule generated by a system p = (p1 , . . . , pr ) of homogeneous polynomials. Define L = M ⊥ , Mk = M ∩ Hk and Lk = L ∩ Hk (k ≥ 0). Then, for j = 1, . . . , n, the following conditions are equivalent: 1

(i) PM ⊥ Mz∗j M ⊂ D((N + 1) 2 ); (ii) there is a bounded operator Aj ∈ L(H(B)) such that [Mz∗j , PM ] = 1 (N + 1)− 2 Aj ; (iii) there is a constant c > 0 such that δ∂∗j p (N + 1)−1 PM ⊥ δ∂j p ≤ c δp∗ δp ; (iv) the sequence of operators defined by Ck : Mk → Lk−1 ,

1

f → (N + 1) 2 PM ⊥ Mz∗j f

(k ≥ 0)

is norm-bounded; (v) PMk (Mzj Mz∗j − Mzj PM Mz∗j )PMk = O( k1 ) as k → ∞.

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If these conditions hold for all j = 1, . . . , n, then there are factorizations [Sj∗ , Si ] = (N + 1)−1 Bij

(1 ≤ i, j ≤ n)

with suitable bounded operators Bij ∈ L(M ⊥ ). In particular, the tuple S = PM ⊥ Mz |M ⊥ is q-essentially normal for q > dim0 (σrf (S)). Proof. By Proposition 2.2 conditions (ii) and (iii) are equivalent. We next observe that condition (i) implies conditions (ii) and (iv). If (i) holds, then 1 Aj = (N + 1) 2 PM ⊥ Mz∗j |M ⊕ 0M ⊥ ∈ L(H(B)) is a bounded operator such that, for all h ∈ M , 1

[Mz∗j , PM ]h = (PM ⊥ Mz∗j )h = (N + 1)− 2 Aj h. Hence the operator Aj satisfies condition (ii), and the sequence of operators Ck = Aj |Mk : Mk → Lk−1 is norm-bounded. Obviously condition (ii) implies (i). Indeed, if (ii) holds, then PM ⊥ Mz∗j 1 M = [Mz∗j , PM ]M ⊂ D((N + 1) 2 ). If condition (iv) holds, then the sequence (Ck ) induces a bounded operator C: M= Let h =

k≥0

∞

k=0

Mk → L =

Lk ,

k≥0

∞ k=0

hk →

∞

Ck hk .

k=0

hk ∈ M be given. Then we find that K

1 K ∗ hk −→ Ch. (N + 1) 2 PM ⊥ Mzj k=0

1 2

1

Since (N + 1) is a closed operator, it follows that PM ⊥ Mz∗j h ∈ D((N + 1) 2 ). Thus we have proved the equivalence of the first four conditions. Finally, an elementary computation yields that 1 ∗ C Ck = Ck∗ (N + 1)−1 Ck = (PMk Mzj Mz∗j − Mzj PM Mz∗j )|Mk . k k This observation shows that conditions (iv) and (v) are equivalent. The remaining assertions follow from Theorem 2.3. Let M ⊂ H(B) be a submodule generated by finitely many monomials z αi and let j ∈ {1, . . . , n} be fixed. It was shown in [5] that PM ⊥ Mz∗j |zj M = 0 and that

M zj M ⊂ {z α ; α ∈ Nn with αj ≤ q} with q equal to the maximum of the jth coefficients of the multi-indices αi (i = 1, . . . , r). Let f = α∈Nn fα z α ∈ M zj M be arbitrary. Then fα = 0 for every α ∈ Nn with αj > q. The homogeneous expansion of Mz∗j f is given by ⎛ ⎞ ∞ ∞ αj + 1 ⎝ Mz∗j f = fα+ej z α ⎠ . (Mz∗j f )k = k+1 k=0

k=0

|α|=k

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The estimate ∞ ∞ α! (αj + 1)2 |fα+ej |2 (k + 1)(Mz∗j f )k 2 = k! k + 1 k=0

k=0 |α|=k

=

∞

(αj + 1)

k=0 |α|=k 1

(α + ej )! |fα+ej |2 ≤ qf 2 (k + 1)! 1

1

shows that Mz∗j f ∈ D((N + 1) 2 ) and that (N + 1) 2 Mz∗j f ≤ q 2 f . Since 1 Mz∗j (M zj M ) ⊂ M ⊥ , we find that PM ⊥ Mz∗j M ⊂ D((N + 1) 2 ). Hence condition (i) of Theorem 2.4 holds. Furthermore 1

Aj = (N + 1) 2 PM ⊥ Mz∗j |M ⊕ 0M ⊥ ∈ L(H(B)) 1

is a bounded operator wih Aj ≤ q 2 . Therefore the proof of Proposition 2.2 shows that inequality (i) in Proposition 2.2 holds with c = max1≤i≤r |αi | in this case. Let M ⊂ H(B) be a submodule generated by a tuple p = (p1 , . . . , pr ) of homogeneous polynomials. We use the notations from Theorem 2.4. As in the proof of Theorem 3.1 in [13] it follows that tr PMk (Mzj Mz∗j − Mzj PM Mz∗j )PMk = tr PMk Mzj Mz∗j PMk − tr PMk−1 Mz∗j Mzj PMk−1 . Hence, for sufficiently large k, we obtain that PMk (Mzj Mz∗j − Mzj PM Mz∗j )PMk ≤ tr PMk (Mzj Mz∗j − Mzj PM Mz∗j )PMk ≤ tr PMk

n i=1

(Mzi Mz∗i − Mzi PM Mz∗i )PMk

k+n−1 dim Mk−1 k k+n−1 (dim Hk−1 − dim Lk−1 ) = dim Hk − dim Lk − k k+n−1 u(k − 1) − u(k), = k where u ∈ Q[x] is a polynomial as explained in the section leading to Theorem 2.3. Since the inessential right spectrum σrf (S) of S is contained in the zero set of the annihilator ideal = dim Mk −

(p) = Ann(H(B)/(p)) = {q ∈ C[z]; q(S) = 0}, we find that u = K is a non-negative constant when dim0 Z(p) ≤ 1. Hence in this case we obtain the estimate K(n − 1) PMk (Mzj Mz∗j − Mzj PM Mz∗j )PMk ≤ k for sufficiently large k. Therefore condition (v) of Theorem 2.4 holds.

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Let M ⊂ H(B) be a submodule generated by a tuple p = (p1 , . . . , pr ) of homogeneous polynomials pi of degree mi . Define m = max{m1 , . . . , mr }. Suppose that p = (p1 , . . . , pr ) is a stable generating set for M in the sense of [14]. By definition there is a constant C > 0 such that, for each polynomial g ∈ M , there are polynomials f1 , . . . , fr ∈ C[z] with g = δp (fi ) and

r

pi fi 2 ≤ Cg2 .

i=1

In [14] it is shown that the quotient S = PM ⊥ Mz |M ⊥ is q-essentially normal for q > dim0 Z(p). An elementary argument allows us to deduce that inequality (i) in Proposition 2.2 holds with constant c = mrC. It suffices to show that the inequalities 1

(N + 1)− 2 PM ⊥ δ∂j p (hi )2 ≤ cδp (hi )2

(j = 1, . . . , n)

hold for every tuple (hi ) ∈ C[z]r . A straightforward decomposition into homogeneous parts allows one to assume that hi ∈ Hk−mi (i = 1, . . . , r) for some k ≥ 0. In this case, there are homogeneous polynomials fi ∈ Hk−mi such that δp (hi ) = δp (fi )

and

r

pi fi 2 ≤ cδp (hi )2 .

i=1

As an application of Lemma 2.1 we find that PM ⊥ δ∂j p (hi ) = PM ⊥ δ∂j p (fi ). By the above cited result of Guo and Wang [13], we obtain that 1 (∂j pi )fi 2 = (N + mi )−1 M∂∗j pi M∂j pi fi , fi k ≤ mi Mp∗i Mpi fi , fi ≤ mpi fi 2 for j = 1, . . . , n and i = 1, . . . , r. But this implies the estimate r

2 1 − 12 2 (N + 1) PM ⊥ δ∂j p (hi ) ≤ (∂j pi )fi k i=1 r

2 r

2 ≤m pi fi ≤ mr pi fi i=1

i=1 2

≤ m r C δp (hi ) . Hence also in this case the equivalent conditions of Theorem 2.4 are satisfied for j = 1, . . . , n. Let again M ⊂ H(B) be a submodule generated by a tuple p = (p1 , . . . , pr ) of homogeneous polynomials. Let d be the greatest common divisor of p1 , . . . , pr . It is well known that the polynominal d is homogeneous and that L = {h ∈ C[z]; d h ∈ (p)} is a homogeneous ideal with (p) = d L. Following Guo and Wang [13] (see also [15]) we call this representation the Beurling form of (p). Denote by U = (d) ⊂ H(B) the submodule generated by d. As before we use the notations Mk = M ∩ Hk ,

Uk = U ∩ Hk ,

Lk = L ∩ Hk

(k ≥ 0).

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Let us suppose that dim0 Z(L) ≤ 1. Then there is a natural number such that dim(Hk /Lk ) = for sufficiently large k. Since the map C[z]/L → (d)/(p),

[h] → [dh]

is an isomorphism of graded C[z]-modules (of degree deg(d)), it follows that dim(Uk Mk ) = for sufficiently large k. Theorem 2.5. Let M ⊂ H(B) be a submodule generated by a system p = (p1 , . . . , pr ) of homogeneous polynomials, and let (p) = d L be the Beurling form of (p). Suppose that dim0 Z(L) ≤ 1. Then there is a constant c > 0 such that δ∂∗j p (N + 1)−1 PM ⊥ δ∂j p ≤ c δp∗ δp

(j = 1, . . . , n).

In particular, there are bounded operators Bij ∈ L(M ⊥ ) with [Sj∗ , Si ] = (N + 1)−1 Bij

(i, j = 1, . . . , n)

and S is q-essentially normal for q > dim0 (σrf (S)). Proof. We use the notations explained in the section leading to the theorem. In addition, for a subspace K ⊂ U , we denote by PK ∈ L(H(B)) the orthogonal projection from H(B)) onto K and by QK the orthogonal projection from U onto K. Since everything has been proved already in the case that dim0 Z(p) ≤ 1, we may suppose that d is non-constant. As in [13] the proof is based on the observation that [Mz∗j , PM ] = PU [Mz∗j , PM ] + PU ⊥ Mz∗j PM = PU [Mz∗j , PM ] + [Mz∗j , PU ]PM . By Theorem 2.4 and the cited result from Guo and Wang [13], there are 1 bounded operators Aj ∈ L(H(B)) such that [Mz∗j , PU ]PM = (N + 1)− 2 Aj PM ∗ for j = 1, . . . , n. It is elementary to check that the operators PU [Mzj , PM ] are the trivial extensions of the commutators [(Mzj |U )∗ , QM ] ∈ L(U ) to H(B). Fix j ∈ {1, . . . , n}. To complete the proof it suffices to show that the sequence of operators defined by Cjk : Mk → Uk−1 Mk−1 ,

1

f → (N + 1) 2 QU M (Mzj |U )∗ f

is norm-bounded. Then, exactly as in the proof of Theorem 2.4, it follows 1 that [(Mzj |U )∗ , QM ] = (N + 1)− 2 Bj for some bounded operator Bj ∈ L(U ). Let us denote by Pk ∈ L(H(B)) the orthogonal projections from H(B) onto the subspaces Uk Mk . An elementary calculation shows that ⎛ ⎞ ⎛ ⎞ n n 1 k + n ∗ ⎠ Pk = Cj,k+1 Cj,k+1 Pk − Pk ⎝ Pk ⎝ Mz∗j Pk+1 Mzj ⎠ Pk k + 1 k + 1 j=1 j=1 for k ≥ 1. Hence, with the notations explained in the section preceding the theorem, we obtain that

Vol. 69 (2011)

1 k+1

n

Essential Normality of Homogeneous Submodules ⎛

∗ Cj,k+1 Cj,k+1 ≤ tr ⎝

j=1

≤

1 k+1

⎞

n

∗ ⎠ Cj,k+1 Cj,k+1

j=1

⎛

k+n − tr Pk ⎝ k+1 ⎛

=

181

k+n − tr ⎝ k+1

n

⎞ Mz∗j Pk+1 Mzj ⎠ Pk

j=1

n

⎞

Pk+1 Mzj Pk Mz∗j Pk+1 ⎠

j=1

=

⎛ ⎞ n k+n Pk+1 Mzj PUk⊥ Mz∗j Pk+1 ⎠ − 1 + tr ⎝ k+1 ⎛

=

n−1 + tr ⎝ k+1

j=1

n

⎞

Pk+1 Mzj PU ⊥ Mz∗j Pk+1 ⎠

j=1

for all sufficiently large k. To continue, note that Pk+1 Mzj PU ⊥ Mz∗j Pk+1 = Pk+1 [Mz∗j , PU ]∗ [Mz∗j , PU ]Pk+1 = Pk+1 A∗j (N + 1)−1 Aj Pk+1 =

1 Pk+1 A∗j Aj Pk+1 . k+1

Thus, for sufficiently large k, we obtain the estimates ⎛ ⎛ ⎞ ⎞ n n 1 1 n − 1 ∗ + tr ⎝Pk+1 ⎝ Cj,k+1 Cj,k+1 ≤ A∗j Aj ⎠ Pk+1 ⎠ k + 1 j=1 k+1 k+1 j=1 ⎛ ⎞ n ⎝ ≤ n−1+ A∗j Aj ⎠ . k+1 j=1

This observation completes the proof.

The preceding result implies that in dimension n ≤ 3 condition (i) in Proposition 2.2 is always satisfied. Indeed, in this case, the ideal L occurring in the Beurling form of an ideal (p) generated by a system p = (p1 , . . . , pr ) of homogeneous polynomials has the property that dim0 Z(L) ≤ 1 (see [13]). Corollary 2.6. Let n ≤ 3 and let M ⊂ H(B) be a submodule generated by a system p = (p1 , . . . , pr ) of homogeneous polynomials. Then there is a constant c > 0 such that δ∂∗j p (N + 1)−1 PM ⊥ δ∂j p ≤ c δp∗ δp

(j = 1, . . . , n).

In particular, there are bounded operators Bij ∈ L(M ⊥ ) with [Sj∗ , Si ] = (N + 1)−1 Bij

(i, j = 1, . . . , n)

and S is q-essentially normal for q > dim0 (σrf (S)). Therefore in dimension n ≤ 3 the equivalent conditions of Theorem 2.4 are satisfied for every homogeneous submodule M ⊂ H(B). Whether the same is true in higher dimensions, is an intriguing open question at this moment.

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References [1] Arveson, W.: Subalgebras of C ∗ -algebras. III. Multivariable operator theory. Acta Math. 181, 159–228 (1998) [2] Arveson, W.: The curvature invariant of a Hilbert module over C[z1 , . . . , zn ]. J. Reine Angew. Math. 522, 173–236 (2000) [3] Arveson, W.: The Dirac operator of a commuting tuple. J. Funct. Anal. 189, 53– 79 (2002) [4] Arveson, W.: Several problems in operator theory (2003). http://www.math. berkely.edu/arveson [5] Arveson, W.: p-summable commutators in dimension d. J. Oper. Theory 54, 101–117 (2005) [6] Arveson, W.: Quotients of standard Hilbert modules. Trans. Am. Math. Soc. 359, 6027–6055 (2007) [7] Douglas, R.G.: A new kind of index theorem. In: Analysis, Geometry and Topology of Elliptic Operators (Roskilde, Denmark, 2005). World Scientific Publishing, Singapore (2006) [8] Douglas, R.G.: Essentially reductive Hilbert modules. J. Oper. Theory 55, 117– 133 (2006) [9] Douglas, R.G., Paulsen, V.: Hilbert modules over function algebras. In: Pitman Research Notes in Mathematics, vol. 217 (1989) [10] Douglas, R.G., Sarkar, J.: Essentially reductive weighted shift Hilbert modules. J. Oper. Theory (2010) (to appear) [11] Eschmeier, J.: Grothendieck’s comparison theorem and multivariable Fredholm theory. Arch. Math. 92, 461–475 (2009) [12] Eschmeier, J., Putinar, M.: Spectral decompositions and analytic sheaves. London Mathematical Society, New Series, vol. 10. Clarendon Press, Oxford (1996) [13] Guo, K., Wang, K.: Essentially normal Hilbert modules and K-homology. Math. Ann. 340, 907–934 (2008) [14] Shalit, O.: Stable polynomial division and essential normality of graded Hilbert modules, Preprint, U. Waterloo (2009) [15] Yang, R.W.: The Berger–Shaw theorem in the Hardy module over the bidisc. J. Oper. Theory 42, 379–404 (1999) J¨ org Eschmeier (B) Fachrichtung Mathematik Universit¨ at des Saarlandes Postfach 15 11 50 66041 Saarbr¨ ucken Germany e-mail: [email protected] Received: February 16, 2010. Revised: May 9, 2010.

Integr. Equ. Oper. Theory 69 (2011), 183–201 DOI 10.1007/s00020-010-1822-7 Published online July 13, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Algebraic Properties of Toeplitz Operators on the Harmonic Dirichlet Space Yong Chen, Young Joo Lee and Quang Dieu Nguyen Abstract. We study some algebraic properties of Toeplitz operators on the harmonic Dirichlet space of the unit disk. We first give a characterization for boundedness of Toeplitz operators. Next we characterize commuting Toeplitz operators. Also, we study the product problem of when product of two Toeplitz operators is another Toeplitz operator. The corresponding problems for compactness are also studied. Mathematics Subject Classification (2010). Primary 47B35; Secondary 32A36. Keywords. Toeplitz operator, harmonic Dirichlet space, commutativity, product problem.

1. Introduction Let D be the open unit disk in the complex plane C and dA be the normalized area measure on D. The Sobolev space L 2,1 is the completion of the space of all smooth functions f : D → C such that ⎧ ⎫ 12 2 ⎪ 2 2 ⎪ ⎨ ⎬ ∂f + ∂f dA f = f dA + < ∞. ∂ z¯ ∂z ⎪ ⎪ ⎩ ⎭ D

The space L

D

2,1

is a Hilbert space with the inner product

∂f ∂g ∂f ∂g , , + . f, g = f dA g¯dA + ∂z ∂z L2 ∂ z¯ ∂ z¯ L2 D

D

Here and in what follows, Lp (D, dA) denotes the usual Lebesgue space and the notation ·, ·L2 denotes the standard inner product in L2 (D, dA). The Dirichlet space D is the subspace of L 2,1 consisting of all holomorphic functions Y. Chen is supported by NNSFC (No. 10971195) and ZJNSFC (No. Y6090689). Q. D. Nguyen was supported by the NAFOSTED program.

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on D. Also, the harmonic Dirichlet space Dh is the subspace of L 2,1 consisting of all harmonic functions on D. One can check the relation Dh = D + D where D = {f¯ : f ∈ D}. Also, it is known that both spaces D and Dh are all closed subspaces of L 2,1 . Each point evaluation is easily veriﬁed to be a bounded linear functional on Dh . Hence, for each z ∈ D, there exists a unique function Rz ∈ Dh which has the following reproducing property: u(z) = u, Rz for every u ∈ Dh . Since Dh = D + D, there is a simple relation between Rz and well known reproducing kernel Kz for D: Rz = Kz + Kz − 1.

(1.1)

We let P : L 2,1 → D and Q : L 2,1 → Dh be the Hilbert space orthogonal projections respectively. Since P ϕ(z) = u, Kz for ϕ ∈ L 2,1 and z ∈ D, the formula (1.1) leads us to the following representation of the projection Q: Q(ϕ) = P (ϕ) + P (ϕ) − P (ϕ)(0)

(1.2)

for functions ϕ ∈ L 2,1 . Given a function ϕ ∈ L 2,1 , the Toeplitz operator Tϕ : Dh → Dh with symbol ϕ is densely deﬁned by Tϕ f = Q(ϕf ) whenever f ϕ ∈ L 2,1 . In this paper, we study some algebraic properties of Toeplitz operators on the harmonic Dirichlet space Dh . In recent papers [14,15], Zhao considered certain harmonic symbols inducing bounded Toeplitz operators, and then studied (semi-)commuting problem and product problem for Toeplitz operators. In the present paper, we continue to study the same characterizing problems for general symbols. In order to handle general symbols, we ﬁrst show that boundary vanishing property of a symbol induces a Toeplitz operator which is a simple bounded linear functional on Dh ; see Proposition 2.1. We then use this fact to obtain a useful connection between Toeplitz operators with general symbol and the harmonic symbol which is the Poisson projection of the boundary value of given general symbol. For the existence of the boundary values of functions in L 2,1 , see Section 2. With these observations, we ﬁrst characterize bounded and compact Toeplitz operators in terms of certain Carleson measures and boundary vanishing property of symbol respectively; see Theorems 2.2 and 2.3. We then characterize commuting Toeplitz operators and three general symbols U, V, H for which the product TU TV − TH is equal to 0; see Theorems 3.4 and 4.3 respectively. As consequences, we characterize semi-commuting Toeplitz operators and zero Toeplitz operators. As special cases of when the symbols are harmonic, our results recover several known results in [14,15] mentioned above. As a corresponding problem, we also study the compact product problem of when TU TV − TH is compact on Dh ; see Theorem 4.10.

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185

The corresponding problems for Toeplitz operators acting on the Hardy space, (harmonic) Bergman space or Dirichlet space D have been well studied as in [1–4,7] or [8] and references therein.

2. The Boundedness and Compactness In this section, we characterize boundedness and compactness for Toeplitz operators. For ψ ∈ L 2,1 , it turns out that ψ(reiθ ) is absolutely continuous on r ∈ [0, 1) for almost every θ ∈ [0, 2π] and absolutely continuous on θ ∈ [0, 2π] for almost every r ∈ [0, 1). In particular, the radial limit ψ|∂D := limr→1 ψ(reiθ ) exists for almost every θ ∈ [0, 2π]. Moreover, we have ψ|∂D ∈ L1 (∂D) and the Poisson extension of ψ|∂D belongs to L 2,1 . See [5] and [6] for details and related facts. We let Δ0 = {ψ ∈ L 2,1 : ψ|∂D = 0}. We start with the following proposition showing that the boundary vanishing property of a symbol gives a simple behavior of the corresponding Toeplitz operator. Proposition 2.1. Let u ∈ Δ0 . Then we have Tu ϕ = uϕdA D

for every polynomials ϕ ∈ Dh . In particular, Tu can be extended to a bounded linear functional on Dh . Proof. Fix a polynomial ϕ in Dh . To prove the ﬁrst part, it sufﬁces to show Tu ϕ, ψ = uϕdA ψdA D

D

for every polynomials ψ in Dh . Write ψ = ψ1 + ψ2 where ψ1 , ψ2 ∈ D are polynomials. Since Tu ϕ, ψ = uϕ, ψ, it is enough to show ∂(uϕ) ∂(uϕ) ψ2 dA = 0. ψ1 + ∂z ∂ z¯ D

First, we claim that

D

∂(uϕ) ψ1 dA = 0. ∂z

(2.1)

We can further assume ψ1 = z n , n ≥ 1 and put v = uϕ. Note v ∈ Δ0 . Using the polar coordinates integration and Fubini’s theorem, we see (2.1) is equivalent to ⎞ ⎞ ⎛ 1 ⎛ 2π 2π 1 ∂v ∂v e−inθ ⎝ rn dr⎠ dθ − i rn−1 ⎝ e−inθ dθ⎠ dr = 0. (2.2) ∂r ∂θ 0

0

0

0

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On the other hand, since v is absolutely continuous in r and θ separately, we obtain by integration by parts for almost all r and θ, 1 1 2π 2π n ∂v n−1 −inθ ∂v dr = −n r dθ = in e−inθ v dθ r v dr, e ∂r ∂θ 0

0

0

0

where we use the fact v|∂D = 0. Combining the above with (2.2), we have (2.1). Also by an analogous argument we see ∂(uϕ) ψ dA = 0. ∂ z¯ 2 D

This completes the proof of the ﬁrst part of the proposition. To prove the second part, we note that there exists a constant C such that |F |2 dA ≤ C||F ||2 (2.3) D

for every F ∈ L

2,1

; see [9, Proposition 1] for example. It follows that ||Tu ϕ|| = | uϕ dA| ≤ C||u||||ϕ|| D

for every polynomials ϕ ∈ Dh . Thus Tu is bounded on a dense subset of Dh and hence can be extended to a bounded linear functional on Dh . The proof is complete. The boundedness of Toeplitz operators will be in terms of certain Carleson measures. A positive Borel measure μ on D is called an D-Carleson measure if there exists a constant C > 0 for which ⎛ ⎞ 12 ⎝ |f |2 dμ⎠ ≤ C||f || D

for every f ∈ D. See [10] or [11] for characterizations for D-Carleson mea2 sures. We let M be the space of all U ∈ L 2,1 for which u ∈ L∞ (D), | ∂u ∂z | dA 2 and | ∂u ∂ z¯ | dA are D-Carleson measures where u is the Poisson extension of U |∂D . Now, we characterize bounded Toeplitz operators in terms of the space M . Theorem 2.2. Let U ∈ L 2,1 . Then TU is bounded on Dh if and only if U ∈ M . In the proof, we will use some auxiliary functions. For a ∈ D, we let z 1 , Sa (z) = Ea (z) = , z ∈ D. 1−a ¯z (1 − a ¯z)2 Put ea = (1 − |a|2 )Ea and sa = (1 − |a|2 )Sa . Then we check ea = sa and ||ea || = 1. Also, we can easily see that ea converges weakly to 0 in Dh as |a| → 1. It is well known that f (a) = f, Sa L2 for holomorphic f ∈ L1 (D, dA) and ψ(a) = ψsa , sa L2 for harmonic ψ ∈ L1 (D, dA); see [16, Section 4] for details.

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Proof. Suppose TU is bounded on Dh . First assume U is harmonic and write U = f + g¯ for some f, g ∈ D. Fix a ∈ D. Since f ea is holomorphic in L2 (D, dA), we have f ea , Sa L2 = af (a). It follows that TU ea , ea = U ea , ea = f ea , sa L2 + U sa , sa L2 = (1 − |a|2 )f ea , Sa L2 + U (a) = (1 − |a|2 )f (a)a + U (a).

(2.4)

Since ||ea || = 1, it follows that |U (a)| ≤ |TU ea , ea | + |a(1 − |a|2 )f (a)| ≤ ||TU || + (1 − |a|2 )|f (a)| for each a ∈ D. Recall that D ⊂ B0 , the little Bloch space consisting of all holomorphic functions ψ on D for which (1 − |a|2 )|ψ (a)| → 0 as |a| → 1; see [16, Chapter 5]. Thus, the above observation shows U ∈ L∞ (D). Next we show that |f |2 dA is D-Carleson on D. Given k ∈ D, we let z ψ(z) := ψf [k](z) =

f (ζ)k(ζ) dζ,

z ∈ D.

0

Then we have TU k, ψ = U k, ψ = U k , f kL2 + f k, f kL2 and hence |f k, f kL2 | ≤ |TU k, ψ| + |U k , f kL2 | ≤ ||TU ||||k||||ψ|| + ||U k ||L2 ||f k||L2 ≤ ||TU ||||k||||f k||L2 + ||U ||∞ ||k||||f k||L2 where || ||L2 is the usual norm in L2 (D, dA). Thus ||f k||L2 ≤ (||TU || + ||U ||∞ )||k|| and |f |2 dA is an D-Carleson measure. Now, to prove that |g |2 dA is D-Carleson, let k ∈ D. Using ψ = ψg [k], we have ¯ = U k¯ , g kL2 + g k, g kL2 TU k, ψ and by the similar argument as above |g k, g kL2 | = |g k, g kL2 | ≤ ||TU ||||k||||g k||L2 + ||U ||∞ ||k||||g k||L2 , which shows that |g |2 dA is an D-Carleson measure and hence U ∈ M . Conversely, one can easily check that each harmonic U ∈ M induces a bounded Toeplitz operator on Dh . Now suppose general U ∈ L 2,1 and let u be the Poisson extension of U |∂D . Since u − U ∈ Δ0 , we have by Proposition 2.1 (2.5) TU ϕ − Tu ϕ = (U − u)ϕdA D

for every polynomials ϕ in Dh . Thus, the above together with Proposition 2.1 shows that the boundedness of TU and Tu are equivalent. Now, by the previous harmonic case, we have the desired result. The proof is complete.

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We remark in passing that if U, V ∈ M and U V ∈ L 2,1 , then U V ∈ M . Indeed, for U, V ∈ M , let u, v be the Poisson extensions of U |∂D , V |∂D respectively. Since U V − uv ∈ Δ0 , we have by Proposition 2.1 TU V ϕ − Tuv ϕ = (U V − uv)ϕ dA (2.6) D

for every polynomials ϕ in Dh . Thus boundedness of TU V and Tuv are equivalent by Proposition 2.1 again. But, since u, v ∈ M , using a routine argument using (2.3), we can check the boundedness of Tuv . We now characterize compact Toeplitz operators. For compactness of a bit general operators which have the form of TU TV − TH , see Theorem 4.10 in Section 4. Theorem 2.3. Let U ∈ L 2,1 . Then TU is compact on Dh if and only if U ∈ Δ0 . Proof. First suppose TU is compact on Dh . By (2.5) and Proposition 2.1, Tu is also compact on Dh where u is the Poisson extension of U |∂D . Recall ea introduced in the proof of Theorem 2.2 converges weakly to 0 in Dh as |a| → 1. Since D ⊂ B0 , (2.4) (for Tu ) shows that u vanishes on ∂D and hence U ∈ Δ0 . Conversely, if U ∈ Δ0 , then TU is a continuous linear functional on Dh by Proposition 2.1 again. Thus TU is compact, as desired. The proof is complete.

3. Commuting Toeplitz Operators In this section, we characterize commuting Toeplitz operators. Given two Toeplitz operators TU and TV , we let [TU , TV ] := TU TV − TV TU ,

[TU , TV ) := TU TV − TU V

be the commutator and semi-commutator respectively. Let U, V ∈ M and U V ∈ L 2,1 . Fix a polynomial ϕ ∈ Dh . By Proposition 2.1, we have TV ϕ = Tv ϕ + (V − v)ϕdA, D

(U − u)dA = u +

Q(U ) = TU 1 = Tu 1 + D

(U − u)dA. D

It follows from Proposition 2.1 again that TU TV ϕ = TU Tv ϕ + Q(U ) (V − v)ϕdA

D

(U − u)Tv ϕdA + Q(U )

= Tu Tv ϕ + D

(V − v)ϕdA D

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(U − u)Tv ϕdA + u

= Tu Tv ϕ + D

D

(V − v)ϕdA D

(U − u)dA

+

(V − v)ϕdA

(3.1)

D

and hence by (2.6)

(U − u)Tv ϕ dA + u

[ TU , TV )ϕ = [Tu , Tv )ϕ + D

D

(V − v)ϕdA D

(U − u) dA

+

189

(V − v)ϕ dA −

D

D

= [Tu , Tv )ϕ + [u − u(0)]

(U V − uv)ϕ dA

(V − v)ϕdA D

+ [TU , TV )ϕ(0) − [Tu , Tv )ϕ(0).

(3.2)

By interchanging the roles of U, V in (3.2) and subtracting one from the other, one can see [ TU , TV ]ϕ = [Tu , Tv ]ϕ + [TU , TV ]ϕ(0) − [Tu , Tv ]ϕ(0) + [u − u(0)] (V − v)ϕdA − [v − v(0)] (U − u)ϕdA D

(3.3)

D

for every ϕ ∈ Dh . The following technical lemma will be useful in the sequel. In the following, we will use the notation z to denote not only the identity function but also points in D. Lemma 3.1. Let u, v ∈ M be harmonic and write u(z) =

∞ k=0

ak z k +

∞

a−k z¯k ,

k=1

v(z) =

∞ k=0

bk z k +

∞

b−k z¯k

(3.4)

k=1

for the power series expansions of u, v respectively. Then, we have Tu Tv [z n ](z) − Tu Tv [z n ](0) ⎛ ⎞ ∞ ∞ ∞ nb−n [u − u(0)] ⎝ at−n−j bj z t + a−t−n−j bj z t ⎠ − = n+1 t=1 j=−∞ j=−∞ and Tu Tv [¯ z n ](z) − Tu Tv [¯ z n ](0) ⎛ ⎞ ∞ ∞ ∞ nbn [u − u(0)] t ⎝ at+n−j bj z + a−t+n−j bj z t ⎠ − = n+1 t=1 j=−∞ j=−∞ for every integers n ≥ 0 and z ∈ D.

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Proof. It is well known that the projection P : L 2,1 → D can be represented as ∂ψ z (w) dA(w), z ∈ D P ψ(z) = ψ dA + 1 − zw ¯ ∂w D

for functions ψ ∈ L that

D

2,1

; see [12] or [13] for details. Using this, one can check

⎧ n−m ⎪ , if n > m, ⎨z m n 1 P [¯ z z ](z) = n+1 , if n = m, ⎪ ⎩ 0, if n ≤ m.

Fix an integer n ≥ 0 and z ∈ D. By (1.2) and direct computations using the above formula, we can see ∞

Tv z n (z) =

k=1 ∞

Tv z¯n (z) =

bk−n z k +

∞

b−k−n z¯k +

b−n , n+1

b−k+n z¯k +

bn . n+1

k=1

bk+n z k +

k=1

∞ k=1

Now applying Tu and using the corresponding formulas above for u, we can get Tu Tv z n (z) =

∞ ∞

[bk−n at−k + b−k−n at+k ]z t

t=1 k=1 ∞ ∞

+

[bk−n a−t−k + b−k−n a−t+k ]¯ zt

t=1 k=1 ∞ a−k bk−n + ak b−k−n

b−n u(z) k+1 n+1 k=1 ⎛ ⎞ ⎛ ⎞ ∞ ∞ ∞ ∞ ⎝ ⎝ = at−n−j bj ⎠ z t + a−t−n−j bj ⎠ z¯t +

t=1

−

t=1

j=−∞

nb−n [u − u(0)] + n+1

+

∞ k=1

j=−∞

b−n a−k bk−n + ak b−k−n + u(0), k+1 n+1

which gives the ﬁrst relation. By the similar argument, we can obtain the second one. This completes the proof. We let P denote the set of all U ∈ L 2,1 such that for all integers n ≥ 0 Uw ¯ n dA = (n + 1) uw ¯ n dA and U wn dA = (n + 1) uwn dA D

D

D

D

where u is the Poisson extension of U |∂D . Note that P = ∅; see the remark just next to Theorem 3.4. Also, for harmonic U ∈ L 2,1 , we can check that U ∈ P if and only if U is constant.

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We ﬁrst have the following necessary condition for commuting Toeplitz operators. Proposition 3.2. Let U, V ∈ M and u, v be the Poisson extensions of U |∂D , V |∂D respectively. Suppose [TU , TV ] = 0. Then the following statements hold; (a) If U ∈ / P, then v − v(0) = α[u − u(0)] for some constant α. (b) If V ∈ / P, then u − u(0) = β[v − v(0)] for some constant β. Proof. Considering power series expansions of u, v as in (3.4) and using an application of Lemma 3.1, we obtain n [a−n (v − v(0)) − b−n (u − u(0))] [Tu , Tv ](z n ) − [Tu , Tv ](z n )(0) = n+1 for each integers n ≥ 0. Combining the above with (3.3), one can see [TU , TV ](z n ) − [TU , TV ](z n )(0) ⎤ ⎡ nb −n ⎦ = [u − u(0)] ⎣ (V − v)wn dA − n+1 D ⎤ ⎡ na −n ⎦ −[v − v(0)] ⎣ (U − u)wn dA − n+1 D ⎡ ⎤ = [ u − u(0)] ⎣ V wn dA − (n + 1) vwn dA⎦ D

⎡ −[v − v(0)] ⎣

D

U wn dA − (n + 1)

D

⎤ uwn dA⎦

D

for every integers n ≥ 0. Here we use the following identities a−n b−n , uwn dA = vwn dA = n+1 n+1 D

(3.5)

(3.6)

D

for each n. Also, using the same argument, we have [ TU , TV ](z n ) − [TU , TV ](z n )(0) ⎤ ⎡ = [ u − u(0)] ⎣ V wn dA − (n + 1) vwn dA⎦ D

⎡ −[v − v(0)] ⎣

D

U wn dA − (n + 1) D

D

⎤ uwn dA⎦

(3.7)

U wn dA = (n + 1) for every integers n ≥ 0. So, if U ∈ / P, we have either D m n uw dA for some integer n ≥ 0 or D U w dA = (m + 1) D uwm dA for D some integer m ≥ 0. Since [TU , TV ] = 0, in both cases, (3.5) and (3.7) show that (a) holds. Also, (3.5) and (3.7) show that (b) holds. The proof is complete.

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As an immediate consequence, we recover Theorem 1.1 in [14] where harmonic symbols have been considered. Corollary 3.3. Let u, v ∈ M be harmonic symbols. Then Tu and Tv commute if and only if a nontrivial linear combination of u and v is constant on D. Proof. First suppose that Tu and Tv commute each other. If u, v ∈ P, then the result clearly holds because u, v are constants. If one of u, v is not contained in P, Proposition 3.2 shows that a nontrivial linear combination of u and v is constant. The converse implication is clear. The proof is complete. Now, we characterize commuting Toeplitz operators with general symbols. Theorem 3.4. Let U, V ∈ M and u, v be the Poisson extensions of U |∂D , V |∂D respectively. Then the following statements holds; (a) If U, V ∈ P, then TU and TV commute if and only if [U Q(V ϕ) − V Q(U ϕ)] dA = 0 D

for every ϕ ∈ Dh . (b) If U ∈ / P, then TU and TV commute if and only if v = αu + δ and [U Q(V ϕ) − V Q(U ϕ)] dA + [u − u(0)] [V − αU − δ]ϕ dA = 0 (3.8) D

D

for every ϕ ∈ Dh and some constants α, δ. (c) If V ∈ / P, then TU and TV commute if and only if u = αv + δ and [U Q(V ϕ) − V Q(U ϕ)] dA + [v − v(0)] [αV − U + δ]ϕ dA = 0 D

D

for every ϕ ∈ Dh and some constants α, δ. Proof. First we prove (a). Since U, V ∈ P, we have using (3.5) and (3.7) [TU , TV ]ϕ = [TU , TV ]ϕ(0) for every ϕ ∈ Dh , which gives (a) because [TU , TV ]ϕ(0) = [U Q(V ϕ) − V Q(U ϕ)] dA. (3.9) D

To prove (b), suppose U ∈ / P and TU TV = TV TU . By Proposition 3.2, v−v(0) = α[u−u(0)] for some constant α and then Tu Tv = Tv Tu by Corollary 3.3. It follows from (3.3) that [u − u(0)] [V − αU − v(0) + αu(0)]ϕ dA = 0 D

for every ϕ ∈ Dh . Thus, by (3.9), (3.8) holds with δ = v(0) − αu(0). Conversely, if v = αu + δ and (3.8) holds, then [Tu , Tv ] = 0 by Corollary 3.3

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again. By (3.3), (3.8) and (3.9), we have

193

[TU , TV ]ϕ = [TU , TV ][ϕ](0) + [u − u(0)]

[V − αU − δ]ϕ dA = 0, D

so TU and TV commute. The proof of (c) is similar. The proof is complete.

We remark in passing that case (a) in Theorem 3.4 can indeed happen. To be more precise, consider functions U, V given by U (reiθ ) = (6r − 5r2 )(eiθ + e−iθ ),

V (reiθ ) = 1 + (10r − 9r2 )e2iθ . Simple calculations show that U, V ∈ P and D [U Q(V ϕ) − V Q(U ϕ)] dA = 0 for every ϕ ∈ Dh .

4. Products of Toeplitz Operators In this section, we study the product problem characterizing three symbols U, V, H for which product TU TV − TH is equal to 0. Also, we give the corresponding characterization for compactness. Let U, V, H ∈ M and U V ∈ L 2,1 . Also, let u, v and h be the Poisson extensions of U |∂D , V |∂D and H|∂D respectively. By Proposition 2.1, we notice TH ϕ = Th ϕ + (H − h)ϕdA D

for every polynomials ϕ ∈ Dh . It follows from (3.1) that TU TV = TH if and only if (Tu Tv − Th )ϕ = −u (V − v)ϕdA + Ψ(ϕ) D

(4.1) where

(H − h)ϕdA −

Ψ(ϕ) := D

(U − u)dA

D

(V − v)ϕdA −

D

(U − u)Tv ϕdA D

is a linear functional on Dh . We ﬁrst obtain a necessary condition for product problem in the special case V ∈ / P. Lemma 4.1. Let U, V, H ∈ M and suppose V ∈ / P. If TU TV = TH , then u is constant where u is the Poisson extension of U |∂D . Proof. Let v, h be the Poisson extensions of V |∂D , H|∂D respectively. Consider the power series expansions of u, v as in (3.4) and also write h(z) =

∞ k=0

ck z k +

∞ k=1

c−k z¯k

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for the power series expansion of h. Let n ≥ 0 and m ≥ 1 be integers. Using Lemma 3.1 and direct calculations, we can obtain ∞ 1 n m − 1 am bn , am+n−j bj + m Tu Tv z¯ , z = m n+1 j=−∞ ∞ 1 n m − 1 a−m bn , a−m+n−j bj + m Tu Tv z¯ , z¯ = m n+1 j=−∞ ∞ 1 − 1 am b−n am−n−j bj + m Tu Tv z n , z m = m n+1 j=−∞ ∞ 1 n m − 1 a−m b−n . Tu Tv z , z¯ = m a−n−m−j bj + m n+1 j=−∞ Also, we note Th z¯n , z m = mcm+n ,

Th z¯n , z¯m = mcn−m ,

Th z n , z m = mcm−n ,

Th z n , z¯m = mc−m−n .

Note Ψ(z n ), z m = 0. Thus, in view of (4.1), we obtain a bunch of identities; ⎛ ⎞ ∞ (4.2) am+n−j bj − cm+n = −am ⎝ V w ¯ n dA − bn ⎠ , j=−∞ ∞

D

an−m−j bj − cn−m

⎛ ⎞ = −a−m ⎝ V w ¯ n dA − bn ⎠ ,

j=−∞ ∞

⎛ am−n−j bj − cm−n = −am ⎝

j=−∞ ∞

D

(4.3)

⎞ V wn dA − b−n ⎠ ,

(4.4)

⎛ ⎞ = −a−m ⎝ V wn dA − b−n ⎠

(4.5)

D

a−m−n−j bj − c−m−n

j=−∞

D

for all integers n ≥ 0 and m ≥ 1. Consider the following two conditions; (c1) D V dA − b0 = 0. (c2) There exists an integer N ≥ 1 such that D V wN dA − bN = 0. First, we claim that if (c1) and (c2) hold, then a1 a−1 = 0. To prove the claim, suppose (c1) and (c2) hold. First, replace m with m + and insert n = 0 in (4.2). Then, compare the series obtained just before with (4.2) with n = . The result is ⎛ ⎞ ⎛ ⎞ (4.6) ¯ dA − b ⎠ = am+ ⎝ V dA − b0 ⎠ am ⎝ V w D

D

for every integers m ≥ 1 and ≥ 0. It follows that am = α am+N ,

m = 1, 2, . . .

(4.7)

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where

⎛ α(= αN ) := ⎝

⎞ ⎛ V dA − b0 ⎠

⎝

D

195

⎞ Vw ¯ N dA − bN ⎠ .

D

Similarly, for m > N , replace m with m − N and insert n = 0 in (4.3). Then, comparing the series obtained just before with (4.3) with n = N , we have a−m = α aN −m ,

m > N.

(4.8)

It follows from (4.7) and (4.8) that akN +1 = α−k a1 and a−kN −1 = αk a−1 for all k = 1, 2, . . .. Therefore akN +1 a−kN −1 = a1 a−1 for every k ≥ 1. Since both ak and a−k go to 0 as k → ∞, we have a1 a−1 = 0 and the claim follows. Recall that V ∈ P if and only if D V dA − b0 = 0 and Vw ¯ n dA − bn = V wn dA − b−n = 0 D

D

for all integers n ≥ 1. There are two cases to consider; (d) V dA − b0 = 0; (e) V dA − b0 = 0. D

D

First assume (d). Since V ∈ / P, there are three cases (d1), (d2) and (d3) below to consider; ¯ n dA − bn = 0 for every n ≥ 1. Then (4.6) shows (d1) Suppose D V w am = 0 for m > 1. Also, let n = 2, m = 1 in (4.3) and m = 1, n = 0 in (4.2). By comparing these two equations, we see a1 = 0. Thus am = 0 for all m ≥ 1. On the other hand, by comparing (4.3) and (4.5) with n = 0, we can also see a−m = 0 for all m ≥ 1. Hence u is constant. (d2) Suppose Vw ¯ N dA − bN = 0, Vw ¯ M dA − bM = 0 D

D

for some N, M ≥ 1. Then, by (4.6) we have am+N = 0 for every m ∈ Z+ . Also, (4.6) shows am = αM am+M for m ≥ 1. These facts imply am = 0 for 1 ≤ m ≤ N by induction. Therefore, am = 0 for all m ≥ 1. By the similar argument, (4.3) with n = N and (4.5) with n = 0 imply a−j = 0 for all j ≥ N + 1. Also, (4.8) shows a−m = αM aM −m for m ≥ M + 1. These facts imply a−j = 0 for 1 ≤ m ≤ N . Therefore, a−m = 0 for all m ≥ 1 and hence u is constant. ¯ n dA−bn = 0 for every n ≥ 1. Then (4.7) shows (d3) Now suppose D V w am = α1 am+1 for all m ≥ 1 in particular. It follows that aj+1 = α1−j a1 for all j ≥ 1. On the other hand, if we consider n = 2, m = 1 in (4.3) and m = 1, n = 0 in (4.2), we see that a1 = 0 if and only if a−1 = 0. But, since (c1) and (c2) hold, we have a1 a−1 = 0 by the claim and then a1 = 0. So aj = 0 for all j ≥ 1. Also, (4.8) shows a−m = α1 a1−m for all m ≥ 2. So a−j = α1j a−1 for all j ≥ 1. Now, by the observation above, we have a−j = 0 for all j ≥ 1 also and hence u is constant.

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Now assume (e). Since V ∈ / P, there are two cases (e1) and (e2) below to consider; ¯ N dA − bN = 0 for some N ≥ 1. Then (4.6) shows (e1) Suppose D V w aj = 0 for all j ≥ 1. Also, considering (4.3) with n = N and (4.5) with n = 0 replacing m with m − N , we see that a−j = 0 for all j ≥ 1 and hence u is constant. ¯ n dA − bn = 0 for every n but D V wN − b−N = 0 (e2) Suppose D V w for some N ≥ 1. For m > N , replacing m by m − N in (4.2) with n = 0 and combining with (4.4) with n = N , we get aj = 0 for j > N . Also, letting n = m = N in (4.3) and (4.4), we have aN = 0. Also, taking n = N in (4.4) and replacing m by m − N in (4.3) with n = 0, we have aj = 0 for 1 ≤ j < N . Thus aj = 0 for all integers m ≥ 1. On the other hand, taking n = N in (4.5) and replacing m by m + N in (4.3) with n = 0, we have a−j = 0 for all j ≥ 1 also and hence u is constant. The proof is complete. Before studying the product problem, we need the following lemma. Lemma 4.2. Let u, v ∈ M be harmonic and consider the power series expansions of u, v as in (3.4). Then, we have ⎛ ⎞ ∞ ∞ ∞ ⎝ Tuv [z n ](z) − Tuv [z n ](0) = at−n−j bj z t + a−t−n−j bj z t ⎠ t=1

and Tuv [z n ](z) − Tuv [z n ](0) =

∞ t=1

j=−∞

⎛ ⎝

∞

j=−∞

at+n−j bj z t +

j=−∞

∞

⎞ a−t+n−j bj z t ⎠ .

j=−∞

for every integers n ≥ 0 and z ∈ D. Proof. We only prove the ﬁrst part because the second one is similar. By the similar arguments as in Lemma 3.1, we can see P [uvz n ](z) = uvwn dA + ai bj z i+j+n + ai b−j z i−j+n D

+ =

i≥0,j≥0

i≥0,j>0,i>j−n

uvwn dA +

= P [uvz n ](0) +

a−i b−j z n−i−j

i>0,j>0,i+j
j≥0,i>0,j>i−n

D

a−i bj z j−i+n + ∞ ∞

at−n−j bj z t

t=1 j=−∞ ∞ ∞

at−n−j bj z t .

t=1 j=−∞

Also, by the similar argument, we obtain ∞ ∞ P [uvz n ](z) = P [uvz n ](0) + a−t−n−j bj z t . t=1 j=−∞

Combining the above with (1.2), we have the ﬁrst relation. This completes the proof.

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Now, we give a characterization on when the product TU TV − TH is equal to 0. Theorem 4.3. Let U, V, H ∈ M and U V ∈ L 2,1 . Suppose u is the Poisson extension of U |∂D . Then TU TV = TH if and only if the following conditions hold. (a) U V − H ∈ Δ0 . (b) D [U Q(V ϕ) − Hϕ] dA = 0 for every ϕ ∈ Dh . (c) Either V ∈ P or u is constant. Proof. First assume that TU TV = TH . By Theorem 2.3, we have (a). Since (TU TV − TH )ϕ(0) = [U Q(V ϕ) − Hϕ] dA D

for each ϕ ∈ Dh , we have (b). Also, (c) is a consequence of Lemma 4.1. To prove the converse implication, let v be the Poisson extension of V |∂D and consider the power series expansions of u, v as in (3.4). By Lemmas 3.1 and 4.2, we see nb−n [u − u(0)] , n+1 (4.9) nbn [u − u(0)] n n [Tu , Tv ) z − [Tu , Tv )z (0) = − . n+1 It follows from (3.2) and (3.6) that ⎛ ⎞ [TU , TV )z n = [u − u(0)] ⎝ V wn dA − (n + 1) vwn dA⎠ + [TU , TV )z n (0) [Tu , Tv ) z n − [Tu , Tv )z n (0) = −

⎛ [TV , TV )z n = [u − u(0)] ⎝

D

D

V wn dA − (n + 1)

D

⎞ vwn dA⎠ + [TU , TV )z n (0)

D

for every n ≥ 0. Thus, conditions (c) yields [TU , TV )ϕ = [TU , TV )ϕ(0) for every ϕ ∈ Dh . On the other hand, by (a) and Proposition 2.1, we note TU V −H (ϕ) = TU V −H (ϕ)(0) for every polynomials ϕ ∈ Dh . It follows from (b) that (TU TV − TH )ϕ = [TU , TV )(ϕ) + TU V −H (ϕ) = [TU , TV )ϕ(0) + TU V −H (ϕ)(0) = (TU TV − TH )(ϕ)(0) = [U Q(V ϕ) − Hϕ] dA D

= 0 for every polynomials ϕ ∈ Dh . Thus we have TU TV = TH as desired. The proof is complete. As immediate consequences, we obtain several results. First, taking U = V = 0 in Theorem 4.3, we characterize the zero Toeplitz operator.

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Corollary 4.4. Let H ∈ M . Then TH = 0 if and only if the following conditions hold. (a) H ∈ Δ0 . (b) D Hϕ dA = 0 for every ϕ ∈ Dh . In addition, if H is harmonic, then TH = 0 if and only if H = 0. In view of harmonic case above, one might ask whether condition (b) is essential in Corollary 4.4 or not. Let’s consider the function H given by H(reiθ ) = (1 − r)eiθ . Then clearly H ∈ Δ0 and by Proposition 2.1 z ) = (1 − r)r dA = 0. TH (¯ D

Thus, (b) in Corollary 4.4 cannot be relaxed. As a special case of Theorem 4.3 when the symbols are all harmnoic, we have the following corollary which extends Theorem 1.1 of [15]. Corollary 4.5. Let u, v, h ∈ M be harmonic symbols. Then Tu Tv = Th if and only if (a) uv − h = 0 on D. (b) Either u or v is constant. Proof. Since harmonic functions in P are only constant functions, we have (b) by Theorem 4.3. Then the harmonic function uv − h must be 0 on ∂D by Theorem 4.3 again and hence on D. Now, suppose (a) and (b). Since uv − h is harmonic by (b), Tuv = Th by (a) and Corollary 4.4. It follows from (b) that Tu Tv = Tuv = Th . Thus Tu Tv = Th as desired. The proof is complete. In a special case h = 0 in Corollary 4.5, we show that the zero product has only a trivial solution as shown in the following which recovers Corollary 1.2 of [15]. Corollary 4.6. Let u, v ∈ M be harmonic symbols. Then Tu Tv = 0 if and only if either u = 0 or v = 0. As another consequence of Theorem 4.3, we characterize semi-commuting Toeplitz operators as shown in the following. Corollary 4.7. Let U, V ∈ M and U V ∈ L 2,1 . Suppose u is the Poisson extension of U |∂D . Then TU TV = TU V if and only if the following conditions hold. (a) D [U Q(V ϕ) − U V ϕ] dA = 0 for every ϕ ∈ Dh . (b) Either V ∈ P or u is constant. Specially, if we take U, V as harmonic functions in Corollary 4.7, we have the following simple corollary which recovers Theorem 1.2 of [14]. Corollary 4.8. Let u, v ∈ M be harmonic. Then Tu Tv = Tuv if and only if either u or v is constant.

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We remark that there are nontrivial examples U, V ∈ M for which V ∈ P and U, V satisfy (a) in Corollary 4.7. For example, the functions U, V given by U (reiθ ) = (20r2 − 12r)e−iθ and V (reiθ ) = 1 + (3 − 2r)eiθ will work. In the rest of the paper, we study the corresponding compact product problem of when the product TU TV − TH is compact on Dh . We ﬁrst prove that (semi-)commutators always have ﬁnite ranks. Proposition 4.9. Let U, V ∈ M and U V ∈ L 2,1 . Then [TU , TV ) and [TU , TV ] are finite ranks operators on Dh . Proof. Let u, v be the Poisson extensions of U |∂D , V |∂D respectively. Suppose u, v have the power series expansions as in (3.4). Using (3.2) and (4.9), one can see ⎛ ⎞ −nb−n + (V − v)wn dA⎠ + [TU , TV )z n (0), [TU , TV )z n = [u − u(0)] ⎝ n+1 D ⎛ ⎞ −nb n + (V − v)z n dA⎠ + [TU , TV )z n (0) [TU , TV )z n = [u − u(0)] ⎝ n+1 D

for each integer n ≥ 0. Thus, for any polynomial ϕ(z) =

N n=0

cn z n +

M

dm z¯m ∈ Dh ,

m=0

where N, M ≥ 0 are any integers, we have [TU , TV )ϕ

⎡

= [u−u(0)] ⎣

D

⎤ N M nb−n cn mbm dm⎦ − +[TU , TV )ϕ(0), (V −v)ϕdA− n + 1 m+1 n=0 m=0

which implies that the rank of [TU , TV ) is at most two on a dense subspace of Dh . So it can be extended to a continuous functional of ﬁnite rank on Dh . Finally, noting [TU , TV ] = [TU , TV ) − [TV , TU ), we see [TU , TV ] also has ﬁnite rank. The proof is complete. As an easy application of the above proposition, we have the following theorem. Theorem 4.10. Let U, V, H ∈ M and U V ∈ L 2,1 . Then the following statements are equivalent. (a) TU TV − TH is compact on Dh . (b) TU TV − TH has finite rank on Dh . (c) U V − H ∈ Δ0 . Proof. Note that TU TV − TH = [TU , TV ) − TH−U V .

(4.10)

Thus by Proposition 4.9, TU TV − TH is compact if and only if TH−U V is compact, which is in turn equivalent to U V − H ∈ Δ0 by Theorem 2.3. So

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we have (a) ⇔ (c). Also, (b) ⇒ (a) is clear. Finally, if we assume (c), then TH−U V has rank one by Proposition 2.1 and then (b) follows from (4.10). The proof is complete. Acknowledgements This work was partly done during the third named author’s visits to Department of Mathematics, Chonnam National University, Korea in the academic year 2008-2009 and to School of Mathematical Sciences, Fudan University, China in November of 2009. He wishes to express his gratitude to these institutions for hospitality and ﬁnancial support.

References ˘ ckovi´c, Z.: ˘ A theorem of Brown–Halmos type for Bergman space [1] Ahern, P., Cu˘ Toeplitz operators. J. Funct. Anal. 187, 200–210 (2001) ˘ ckovi´c, Z.: ˘ Commuting Toeplitz operators with harmonic sym[2] Axler, S., Cu˘ bols. Integral Equ. Oper. Theory 14, 1–11 (1991) [3] Axler, S., Zheng, D.: Compact operators via the Berezin transform. Indiana Univ. Math. J. 47, 387–400 (1998) [4] Brown, L., Halmos, P.R.: Algebraic properties of Toeplitz operators. J. Reine Angew. Math. 213, 89–102 (1964) [5] Chen, Y.: Commuting Toeplitz operators on the Dirichlet space. J. Math. Anal. Appl. 357, 214–224 (2009) [6] Chen, Y., Nguyen, Q.D.: Toeplitz and Hankel operators with L∞,1 symbols on Dirichlet space. J. Math. Anal. Appl. 369, 368–376 (2010) [7] Choe, B.R., Lee, Y.J.: Commuting Toeplitz operators on the harmonic Bergman spaces. Michigan Math. J. 46, 163–174 (1999) [8] Lee, Y.J.: Algebraic properties of Toeplitz operators on the Dirichlet space. J. Math. Anal. Appl. 329, 1316–1329 (2007) [9] Lee, Y.J., Nguyen, Q.D.: Toeplitz operators on the Dirichlet spaces of planar domains. Proc. Am. Math. Soc. (to appear) [10] Stegenga, D.: Multipliers of Dirichlet space. Illinois J. Math. 24, 113–139 (1980) [11] Wu, Z.: Carleson measures and multipliers for Dirichlet spaces. J. Funct. Anal. 169, 148–163 (1999) [12] Wu, Z.: Hankel and Toeplitz operators on the Dirichlet spaces. Integral Equ. Oper. Theory 15, 503–525 (1992) [13] Wu, Z.: Operator theory and function theory on the Dirichlet space. In: Axler, S., McCarthy, J., Sarason, D. (eds.) Holomorphic Spaces. MSRI, pp. 179–199 (1998) [14] Zhao, L.: Commutativity of Toeplitz operators on the harmonic Dirichlet space. J. Math. Anal. Appl. 339, 1148–1160 (2008) [15] Zhao, L.: Products of Toeplitz operators on the harmonic Dirichlet space, preprint [16] Zhu, K.: Operator Theory in Function Spaces, 2nd edn. American Mathematical Society, Providence (2007)

Vol. 69 (2011)

Toeplitz Operators on Harmonic Dirichlet Space

Yong Chen College of Mathematics, Physics and Information Engineering Zhejiang Normal University 321004 Jinhua, People’s Republic of China and School of Mathematical Sciences Fudan University 200433 Shanghai, People’s Republic of China e-mail: [email protected] Young Joo Lee (B) Department of Mathematics Chonnam National University Gwangju 500-757, Korea e-mail: [email protected] Quang Dieu Nguyen Department of Mathematics Ha Noi National University of Education 136 Xuan Thuy, Ha Noi, Vietnam e-mail: dieu [email protected] Received: March 30, 2010. Revised: May 7, 2010.

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Integr. Equ. Oper. Theory 69 (2011), 203–215 DOI 10.1007/s00020-010-1826-3 Published online July 30, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

On Zeros of Certain Analytic Functions Vladimir Bolotnikov Abstract. Given a function s which is analytic and bounded by one in modulus in the open unit disk D and given a ﬁnite Blaschke product ϑ of degree k, we relate the number of zeros of the function s − ϑ inside D to the number of boundary zeros of special type of the same function. Mathematics Subject Classification (2010). 30C15, 30D50. Keywords. Angular derivatives, boundary zeros, ﬁxed points.

1. Introduction Let S denote the Schur class of analytic functions mapping the open unit disk D into the closed unit disk D and let Bk stand for the set of all Blaschke products of degree k. Let f be a nonzero function of the form f = s − ϑ ≡ 0,

where

s ∈ S, ϑ ∈ Bk

(1.1)

and let ND (f ) denote the number of zeros of f in D counted with multiplicities. If f ∞ := supz∈D |f (z) < 1, it follows from the Rouche’s theorem that ND (f ) = k. On the other hand, the Schwarz–Pick lemma implies that ND (f ) cannot exceed k, and simple examples show that ND (f ) indeed may be equal to any nonnegative integer n ≤ k. The objective of this paper is to compute ND (f ) in terms of the number of boundary zeros of f of certain type. To be more speciﬁc, let us say that t0 ∈ T is a boundary zero of an analytic function f : D → C of multiplicity mf (t0 ) = m if f (z) = O(1) (z − t0 )m

and

lim

z→ t0

f (z) =∞ (z − t0 )m+1

as z →t0 .

(1.2)

In what follows, we will use notation f (t0 ) := lim f (z) and f (j) (t0 ) := lim f (j) (z) (j ≥ 1) z→ t0

z→ t0

provided the latter limits exist. The symbol z →t0 means that a point z ∈ D tends to a boundary point t0 ∈ T nontangentially, i.e., so that |z − t0 | < V. Bolotnikov was partially supported by National Science Foundation Grant DMS 0901124.

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α(1 − |z|) for some α > 1. We will write z → t0 if z tends to t0 unrestrictedly (in D or in C which will be always clear from the context). Observe that in general, the ﬁrst condition in (1.2) means that the limits f (j) (t0 ) exist and equal zero for j = 0, . . . , m − 1 and that f (m) (z) stays bounded as z →t0 . The second condition says that in case the limit f (m) (t0 ) exists and equals zero, the nontangential convergence of f (m) (z) to zero is not too fast. The following proposition (the proof will be presented in Sect. 2) shows that conditions (1.2) imply some more if f is of the form (1.1). Proposition 1.1. Let us assume that a function f of the form (1.1) has zero at t0 ∈ T of multiplicity mf (t0 ) = 2n − 1. Then the limit f (2n−1) (t0 ) exists and the following number is real: Δf,n (t0 ) := (−1)n−1 t02n−1 ϑ(t0 )f (2n−1) (t0 ) ∈ R.

(1.3)

For boundary zeros of functions f of the form (1.1), we introduce the additional characteristic τf (t0 ) which we will refer to as to the nonpositive degree of t0 . This definition is well justiﬁed by Proposition 1.1. Definition 1.2. Let f be of the form (1.1) and let t0 ∈ T be its boundary zero of multiplicity mf (t0 ). Then the nonpositive degree τf (t0 ) of t0 is deﬁned as follows: ⎧ if mf (t0 ) = 2n, ⎨ n, if mf (t0 ) = 2n − 1 and Δf,n (t0 ) ≤ 0, (1.4) τf (t0 ) = n, ⎩ n − 1, if mf (t0 ) = 2n − 1 and Δf,n (t0 ) > 0. We will say that t0 ∈ T is a boundary zero of f of nonpositive type if τf (t0 ) ≥ 1. The following result was proved in [5]. Theorem 1.3. Let s ∈ S and ϑ ∈ Bk . Then ND (s − ϑ) < k if and only if the function f = s − ϑ has a boundary zero of nonpositive type. To make the latter statement more precise, let us introduce the total nonpositive degree τ (f ) for f of the form (1.1) as τ (f ) := i τf (ti ) where the sum is taken over all boundary zeros of f . The next theorem is the main result of the paper. Theorem 1.4. Let f be as in (1.1). Then ND (f ) + τ (f ) = k. The paper is organized as follows. Section 2 contains the proof of Proposition 1.1 and some needed background on Nevanlinna–Pick type interpolation for Schur-class functions. The proof of Theorem 1.4 is given in Sect. 3. The last section discusses an extension of the classical Carath´eodory–Julia– Wolﬀ theorem on ﬁxed points to the classes of generalized Schur functions.

2. Preliminaries In this section we justify Proposition 1.1 and collect some results needed for the proof of Theorem 1.4. We ﬁrst recall one result from [10].

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Proposition 2.1. Let s ∈ S, t0 ∈ T, and let us assume that s(z) − ϑ(z) = O(|z − t0 |2n−1 )

(2.1)

for some rational function ϑ analytic and unimodular on T. Then the nontangential limits s(j) (t0 ) exist for j = 0, . . . , 2n − 1 as well as the limit ds,n (t0 ) := lim

z→ t0

1 ∂ 2n−2 1 − |s(z)|2 < ∞. ((n − 1)!)2 ∂z n−1 ∂ z¯n−1 1 − |z|2

(2.2)

Furthermore, these limits are related as follows: s(2n−1) (t0 )s(t0 ) (2n − 1)! n−2 n−1 j i+j+1 s(n+i+1) (t0 )s(n−j−1) (t0 ) j . (2.3) (−1) t + i 0 (n + i + 1)!(n − j − 1)! i=0 j=0

ds,n (t0 ) = (−1)n−1 t02n−1

Proof of Proposition 1.1. The assumption in this proposition means that condition (2.1) holds and thus Proposition 2.1 applies to s. On the other hand, Proposition 2.1 applies to the function ϑ (instead of s) giving the equality ϑ(2n−1) (t0 )ϑ(t0 ) (2n − 1)! n−2 n−1 j i+j+1 ϑ(n+i+1) (t0 )ϑ(n−j−1) (t0 ) (2.4) (−1)j t + i 0 (n + i + 1)!(n − j − 1)! i=0 j=0

dϑ,n (t0 ) = (−1)n−1 t02n−1

where dϑ,n (t0 ) is deﬁned via formula (2.2). Since ϑ is analytic at t0 , it follows from (2.1) that s(j) (t0 ) = ϑ(j) (t0 ) for j = 0, . . . , 2n − 2. (2n−1)

(2.5)

(2n−1)

Since the limit s (t0 ) exists, the limit f (t0 ) exists as well. Upon subtracting (2.4) from (2.3) and making use of (2.5) we get (−1)n−1 t02n−1 b(t0 ) (2n−1) s (t0 ) − ϑ(2n−1) (t0 ) ds,n (t0 ) − dϑ,n (t0 ) = (2n − 1)! =

(−1)n−1 t02n−1 ϑ(t0 ) (2n−1) f (t0 ). (2n − 1)!

(2.6)

is positive on D×D and For every Schur-class function s, the kernel 1−s(z)s(ζ) 1−zζ therefore, the number ds,n (t0 ) deﬁned in (2.2) is nonnegative. Since ϑ ∈ S, the number dϑ,n (t0 ) is nonnegative as well. Now it is clear that the expression on the left hand side of (2.6) is equal to a real number, which completes the proof. We next recall some needed interpolation results for Schur-class functions. Let us denote by IP the following interpolation problem. IP: Given a finite Blaschke product ϑ ∈ Bk and distinct points z1 , . . . , zr ∈ D and zr+1 , . . . , z ∈ T given along with multiplicities n1 , . . . , n , find all functions g ∈ S such that the function g − ϑ

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has zero at zi of multiplicity at least ni for i = 1, . . . , r; has boundary zero at zi of nonpositive degree at least ni for i = r + 1, . . . , .

For the given tuple n = (n1 , . . . , n ) of multiplicities, we let |n| := n1 + · · · + n so that for every solution g of the problem IP, |n| ≤ ND (g − ϑ) + τ (g − ϑ). Recall that ϑ is analytic on D and introduce the numbers cij :=

ϑ(j) (zi ) j!

j = 0, . . . , ni − 1, if i ∈ {1, . . . , r}, j = 0, . . . , 2ni − 1, if i ∈ {r + 1, . . . , }.

for

(2.7)

The problem IP can be reformulated in terms of these numbers as follows: find all g ∈ S such that g (j) (zi ) = j! · cij

j = 0, . . . , ni − 1; i = 1, . . . , r,

for

g (j) (zi ) := lim g (j) (z) = j! · cij z→ zi

for j = 0, . . . , 2ni − 2; i = r + 1, . . . , , and

(−1)ni −1 zi2ni −1 ci,0 g (2ni −1) (zi ) − (2ni − 1)! · ci,2ni −1 ≤ 0

(2.8)

for i = r +1, . . . , . This problem is well known (see e.g., [2–4,7–9,12]) and its solution is recalled below. Let us consider the |n| × |n| matrix P = [Pij ]i,j=1 with the block entries Pij ∈ Cni ×nj given by ⎤β=0,...,nj −1 ⎡ ∂ α+β 1 − ϑ(z)ϑ(ζ) ⎥ 1 ⎦ α!β! ∂z α ∂ ζ¯β 1 − z ζ¯

⎢ Pij = ⎣ lim

z → zi ζ → zj

.

(2.9)

α=0,...,ni −1

Straightforward differentiation produces explicit entry-wise formulas in terms of the Taylor coefﬁcients (2.7): min{α,β}

[Pij ]α,β =

s=0

−

ziβ−s z¯jα−s (α + β − s)! (α − s)!s!(β − s)! (1 − zi z¯j )α+β−s+1

β min{γ,δ} α γ=0 δ=0

s=0

δ−s γ−s (γ + δ − s)! zi z¯j ci,α−γ cj,β−δ (γ − s)!s!(δ − s)! (1 − zi z¯j )γ+δ−s+1

if i = j or if i = j ∈ {1, . . . , r}, and [Pii ]α,β =

γ β

(−1)δ

γ δ

ziγ+δ+1 ci,α+δ+1 ci,j−γ

γ=0 δ=0

for i = r+1, . . . , . The matrix P constructed above is called the Schwarz–Pick matrix of ϑ bases on points z1 , . . . , z taken with the respective multiplicities n1 , . . . , n . Since ϑ ∈ Bk , it follows (see e.g., [6]) that P is positive semidefinite and that moreover, rank P = min{|n|, k}.

(2.10)

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The problem IP has a unique solution if and only if the matrix P is singular (see e.g., [9]), that is, if and only if |n| > k. It is obvious that this unique solution is ϑ. In other words, if g ∈ S is such that ND (g − ϑ) + τ (g − ϑ) ≥ |n| > k, then g ≡ ϑ. We thus arrive at the following half of Theorem 1.4. Remark 2.2. Let s ∈ S and ϑ ∈ Bk and let us assume that s ≡ ϑ. Then ND (s − ϑ) + τ (s − ϑ) ≤ k. If |n| ≤ k, the matrix P is invertible by (2.10) and the solution set of the problem IP can be parametrized by a linear fractional formula. To recall this formula, let T be the block diagonal matrix with the i-th diagonal block equal the upper triangular ni × ni Jordan block with the number z i on the main diagonal: ⎤ ⎡ z¯i 1 ... 0 ⎡ ⎤ 0 T1 ⎢ .. ⎥ ⎢0 . ⎥ z¯i ⎢ ⎥ .. ⎥ , (2.11) ⎢ T =⎣ = , where T ⎦ i . ⎥ ⎢ . .. .. ⎣ .. . . 1⎦ 0 T 0 ... 0 z¯i let E be the row-vector E = E1 . . . E ,

where

Ei = 1 0 . . . 0 ∈ C1×ni

and let C ∈ Cn be deﬁned from the numbers cij as follows: C = C1 . . . C , where Ci = ci,0 . . . ci,ni −1 ∈ C1×ni .

(2.12)

(2.13)

Remark 2.3. The matrix P deﬁned in (2.9) satisﬁes the equality P + C ∗C = T ∗P T + E∗E

(2.14)

where T , E and C are deﬁned as in (2.11)–(2.13). Indeed, upon applying the operator identity

∂ α+β 1 α!β! ∂z α ∂ ζ¯β

to both parts of the

1 − ϑ(z)ϑ(ζ) 1 − ϑ(z)ϑ(ζ) + ϑ(z)ϑ(ζ) = z ζ¯ · +1 ¯ 1 − zζ 1 − z ζ¯ and then passing to the limits as z → zi and ζ → zj we verify the equality between the (α, β)-entries in the matrix equality Pij + Ci∗ Cj = Ti∗ Pij Tj + Ei∗ Ej . Appropriately varying α, β, i, j, we then get the whole equality (2.14). A simple consequence of equality (2.14) is that the matrix 1 1 u11 u12 P 2T (2.15) = (P + C ∗ C)−1 P 2 C ∗ u21 u22 E is contractive and admits ⎡ u11 U = ⎣u21 u31

a unitary extension of the form ⎤ ⎡ |n| ⎤ ⎡ |n| ⎤ u12 u13 C C u22 u23 ⎦ : ⎣ C ⎦ → ⎣ C ⎦ . u32 0 C C

(2.16)

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Then the 2 × 2 matrix function Σ deﬁned via the unitary realization u a(z) b(z) u22 u23 −1 u12 u13 (2.17) + z 21 (I − zu11 ) Σ(z) = = u32 0 u31 c(z) d(z) is rational of McMillan degree deg Σ = |n| (by the McMillan degree of a rational matrix function F we mean the dimension of the state space in the minimal realization of F or equivalently, the sum of pole multiplicities at all poles of F in the extended complex plane). Also the function Σ is inner in D which means that the matrix Σ(z) is contractive if z ∈ D and unitary if z ∈ T. Therefore, the functions a, b, c, d are rational Schur-class functions. Their Taylor coefﬁcients at the points z1 , . . . , z are discussed in the next lemma. Recall that z1 , . . . , zr ∈ D and zr+1 , . . . , z ∈ T. Lemma 2.4. Let a, b, c, d be the functions constructed from the matrices P , T , E, C via formulas (2.15)–(2.17). Then 1. 2. 3. 4. 5.

j = 0, . . . , ni − 1, if i ∈ {1, . . . , r}, j = 0, . . . , 2ni − 1, if i ∈ {r + 1, . . . , }. |d(zi )| = 1 for i = r + 1, . . . , . b(j) (zi ) = 0 for j = 0, . . . , ni − 1 and i = 1, . . . , c(j) (zi ) = 0 for j = 0, . . . , ni − 1 and i = r + 1, . . . , . b(ni ) (zi ) = 0, c(ni ) (zi ) = 0 for i = r + 1, . . . , and moreover, a(j) (zi ) = j! · cij

for

i (ni ) (zi )d(zi )a(zi ). b(ni ) (zi ) = (−1)ni −1 z 2n i c

(2.18)

Proof. For the proof of all statements concerning the boundary points zr+1 , . . . , z we refer to Theorem 6.4 and Lemma 6.5 in [9]. Parts of statements (1) and (3) concerning the interior points z1 , . . . , zr follow from the general fact established in [13] and reproduced later in [9, Section 6]: If P is a positive definite operator satisfying identity (2.14) and if Σ is the function constructed via formulas (2.15)–(2.17), then the entries of the row-vector functions F (z) = (zI − T ∗ )−1 (E ∗ a(z) − C ∗ )

and

G(z) = (zI − T ∗ )−1 E ∗ b(z)

belong to the Hardy space H 2 of the unit disk. In particular, F and G are analytic at z1 , . . . , zr and therefore, Resz=zi F (z) = 0 and

Resz=zi G(z) = 0 for i = 1, . . . , r.

(2.19)

It is not hard to conclude from the structure (2.11)–(2.13) of the matrices T , E, C (see e.g., [3, Section 16.2]) that ⎡ ⎡ ⎤ ⎤ a(zi ) − ci,0 b(zi ) ⎢ ⎢ b (zi ) ⎥ ⎥ a (zi ) − ci,1 ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥, .. .. , Resz=zi G(z) = ⎢ Resz=zi F (z) = ⎢ ⎢ ⎥ ⎥ . . ⎣ (n −1) ⎣ (n −1) ⎦ ⎦ a i (zi ) b i (zi ) (ni −1)! − ci,ni −1 (ni −1)! which together with (2.19) imply all the desired equalities for the Taylor coefﬁcients of a and b at z1 , . . . , zr .

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The next theorem shows that the entries of the Σ serve as the coefﬁcients in the Redheffer type linear fractional parametrization of the solution set of the problem IP. Theorem 2.5. Let us assume that |n| < k so that the matrix P defined in (2.9) is positive definite and let Σ be given by (2.17). Then: bc g ( g ∈ S), establishes a one-to-one g ] := a + 1. The formula g = RΣ [ 1 − d g correspondence between S and the set of all solutions g of the problem IP. g ] satisfies the strict inequality 2. The function g = RΣ [

(−1)ni −1 zi2ni −1 ci,0 g (2ni −1) (zi ) − (2ni − 1)! · ci,2ni −1 < 0 at zi ∈ T (see (2.8)) if and only if the parameter g is subject to g(zi ) = d(zi )

and

| g (zi )| < ∞.

The ﬁrst statement of the theorem was proved in [12] in a very general operator-valued setting of the Abstract Interpolation Problem. The ﬁnite dimensional adaptation of this approach used here can be found in [7–9]. The second statement was proved in [9, Section 7]. It shows that for most of g ] solves the problem IP with equaliparameters g ∈ S, the function g = RΣ [ ties in (2.8). In conclusion we remark that the function Σ is determined from the matrices (2.11)–(2.13) and P (that is, from the data set of the problem IP) essentially uniquely: a different choice of a unitary extension U in (2.15) changes the entries b, c and d in (2.17) respectively to βb, γc and βγd for some unimodular numbers β and γ; the latter aﬀects neither the parametrization formula from Theorem 2.5 nor the properties of a, b, c, d mentioned in Lemma 2.4.

3. Proof of Theorem 1.4 Given s ∈ S and ϑ ∈ Bk (s ≡ ϑ), let {(zi , ni )}ri=1 be the set of all zeros of the function f = s − ϑ in D given along with their multiplicities and let {(zi , ni = τf (zi ))}i=r+1 be the set of all boundary zeros accompanied with their respective nonpositive degrees. Let us assume that |n| := n1 + · · · + n = ND (s − ϑ) + τ (s − ϑ) < k

(3.1)

and let us consider the interpolation problem IP associated with ϑ, the points zi and multiplicities ni for i = 1, . . . , . This problem is nondegenerate (in the sense that the matrix P deﬁned in (2.9) is positive definite) due to assumption (3.1). Furthermore, both ϑ and s are solutions of this problem. By Theorem 2.5, they are of the form s=a+

bc s 1 − d s

and ϑ = a +

bcϑ 1 − dϑ

(3.2)

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for some s, ϑ ∈ S. As a consequence of (3.2), we have s−ϑ=

bc( s − ϑ) . − d (1 − dϑ)(1 s)

(3.3)

We next observe that ϑ is a ﬁnite Blaschke product of degree at least one. Indeed, since ϑ and Σ are rational, then ϑ is rational. A well known property of the Redheffer transformation is that McMillan degrees of ϑ, ϑ and Σ are Since deg ϑ = k and deg Σ = |n| we conrelated by deg ϑ ≤ deg Σ + deg ϑ. clude from (3.1) that deg ϑ ≥ k − |n| > 0. Since Σ and ϑ are inner in D, the identity 1 2) |b|2 (1 − |ϑ| bϑ 2 ∗ 1 − |ϑ| = + 1, (I − ΣΣ ) bϑ 2 |1 − ϑd| 1 − dϑ 1−dϑ

(this identity follows directly from (2.17) and the second equality in (3.2)) shows that ϑ is also inner. Since ϑ is rational, it is a ﬁnite Blaschke product. By construction, ϑ solves the problem IP with equalities in (2.8). Since |d(zi )| = 1 for i = r + 1, . . . , (see statement (2) in Lemma 2.4), it follows from statement (2) in Theorem 2.5 that i )d(zi ) = 0 for 1 − ϑ(z

i = r + 1, . . . , .

(3.4)

= 0. To this end, let us observe that Now we will show that ND ( s − ϑ) since d, ϑ and s are Schur-class functions, the denominator in (3.3) does not vanish on D (if it did, we would have by the maximum modulus principle, s ≡ ϑ ≡ d = const and then s ≡ ϑ, which is not the case). Assuming that s(ζ) = ϑ(ζ) for some ζ ∈ D \ {z1 , . . . , zr } we conclude from (3.3) that s(ζ) = ϑ(ζ) which is impossible since z1 , . . . , zr are all zeros of f = s − ϑ i ) for some i ∈ {1, . . . , r}, then we in D. On the other hand, if s(zi ) = ϑ(z conclude from (3.3) and the third statement in Lemma 2.4, that the function f = s − ϑ has zero at zi of multiplicity at least ni + 1 which contradicts the =0 s − ϑ) assumption that s − ϑ has zero of multiplicity ni at zi . Thus ND ( and by Theorem 1.3, the function f = s − ϑ has a boundary zero of nonpositive type at some point t0 ∈ T. The latter means that the limits s(t0 ) and s (t0 ) exist and satisfy

0 ) and t0 ϑ(t 0 ) s (t0 ) − ϑ (t0 ) ≤ 0. (3.5) s(t0 ) = ϑ(t The point t0 belongs either to the set {zr+1 , . . . , z } or to its complement in T. We will show that either case cannot occur which will imply that (1.2) cannot hold and thus will complete the proof of the theorem. Case 1: Let t0 = zα for some α ∈ {r + 1, . . . , }. We will show that the following limit exists (ﬁnitely) and is nonpositive: L := lim

z→ zα

(−1)nα zα2nα +1 ϑ(zα )(s(z) − ϑ(z)) ≤ 0. (z − zα )2nα +1

(3.6)

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To this end, we ﬁrst plug in (3.3) into (3.6) to get L = lim

z→ zα

= lim

z→ zα

(−1)nα zα2nα +1 ϑ(zα )b(z)c(z)(s(z) − ϑ(z)) (1 − d(z)ϑ(z))(1 − d(z) s(z))(z − zα )2nα +1 (−1)nα zα2nα +1 ϑ(zα ) b(z)c(z) s(z) − ϑ(z) · lim · lim . z−zα zα (1 − d(z)ϑ(z))(1 − d(z) s(z)) z→zα (z − zα )2nα z→ (3.7)

α ) = 1/d(zα ) by (3.4), we have Since s(zα ) = ϑ(z (−1)nα zα2nα +1 ϑ(zα ) (−1)nα zα2nα +1 ϑ(zα ) = . 2 (1 − d(z)ϑ(z))(1 − d(z) s(z)) (1 − d(zα )ϑ(z))

lim

z→ zα

(3.8)

By parts (3) and (4) in Lemma 2.4, lim

z→zα

b(z) b(nα ) (zα ) = n α (z − zα ) nα !

and

lim

z→zα

c(z) c(nα ) (zα ) = n α (z − zα ) nα !

which together with (2.18) gives lim

z→zα

b(z)c(z) b(nα ) (zα )c(nα ) (zα ) = (z − zα )2nα (nα !)2 =

(nα ) α (zα )|2 d(zα )a(zα ) (−1)nα −1 z 2n α |c . (nα !)2

(3.9)

α ), it follows that Finally, since the limit s (zα ) exists and since s(zα ) = ϑ(z lim

z→ zα

s(z) − ϑ(z) = s (zα ) − ϑ (zα ). z − zα

(3.10)

Substituting (3.8)–(3.10) into (3.7) and taking into account that zα ∈ T and a(zα ) = ϑ(zα ) ∈ T (by part (1) in Lemma 2.4 and since ϑ is a ﬁnite Blaschke product), we arrive at L=−

s (zα ) − ϑ (zα )) zα |c(nα ) (zα )|2 d(zα )( . α ))2 (nα !)2 (1 − d(zα )ϑ(z

(3.11)

α )( Since zα ϑ(z s (zα ) − ϑ (zα )) ≤ 0 (by the second relation in (3.5)) and since α )d(zα ) = eiφ and α )d(zα )| = 1 so that ϑ(z |ϑ(z α )d(zα ϑ(z 1 ≤ 0, = 2 2 cos φ − 2 (1 − d(zα )ϑ(zα )) it follows that the expression on the right hand side of (3.11) is nonpositive which proves (3.6). It follows from (3.11) that the function f = s − ϑ has boundary zero at zα of multiplicity mf (zα ) ≥ 2nα +1 and that the limit (3.6) is equal to L = lim

z→ zα

(−1)nα zα2nα +1 ϑ(zα )f (2nα +1) (zα )(zα )) = Δf,nα +1 (zα ) ≤ 0. (2nα + 1)!

Therefore τf (zα ) ≥ nα + 1 (see the definition (1.4)) which contradicts to the assumption that τf (zα ) = nα .

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Case 2: Let t0 ∈ T \ {zr+1 , . . . , z }. The existence of boundary limits s(t0 ) and s (t0 ) implies the existence of s(t0 ) and s (t0 ). The ﬁrst relation in (3.5) guarantees that s(t0 ) = ϑ(t0 ). Then it follows from (3.3) that s (t0 ) − ϑ (t0 ) =

s (t0 ) − ϑ (t0 )) b(t0 )c(t0 )( . (1 − d(t0 )b(t0 ))2

(3.12)

For the rest of the case all the relevant functions will be evaluated at t0 and we will write g rather than g(t0 ). Since the matrix Σ(t0 ) = ac db is unitary, we have 1 − |a|2 = |b|2 and ac = −bd which together with the second equality in (3.2) gives |b|2 bcϑ |b|2 dϑ = |b|2 + = . 1 − aϑ = 1 − a a + 1 − dϑ 1 − dϑ 1 − dϑ Therefore, ϑbcϑ ϑ ϑ (ϑ − a) bcϑ 1 − ϑa |b|2 · = = = = 2 2 (1 − dϑ) 1 − dϑ 1 − dϑ 1 − dϑ 1 − dϑ |1 − dϑ| which being combined with (3.12) implies

t0 ϑ s − ϑ |b|2 ϑbc( s − ϑ ) ϑbc ϑ t 0 t0 ϑ (s − ϑ ) = = t0 ϑ s − ϑ · = 2 2 2 (1 − dϑ) (1 − dϑ) |1 − dϑ| which is nonpositive, due to the second relation in (3.5). Thus, s(t0 ) = ϑ(t0 ) and t0 ϑ(t0 ) (s (t0 ) − ϑ (t0 )) ≤ 0 which means that the function f = s − ϑ has boundary zero at t0 of nonpositive type which is impossible since zr+1 , . . . , z are all boundary zeros of f of nonpositive type. This completes the proof of Theorem 1.4.

4. Fixed Points of Generalized Schur Functions Given a function g analytic or meromorphic on D, a point z0 ∈ D is called a ﬁxed point of g if g(z0 ) = z0 (if z0 ∈ T, then g(z0 ) is understood as the nontangential boundary limit). The set of all ﬁxed points of g will be denoted by Fix(g). To each ﬁxed point z0 ∈ Fix(g) we assign the ﬁxed point index ig (z0 ) to be the multiplicity of the zero of the function g − id at z0 where id is the identity map. In the boundary case where z0 ∈ T, we let ig (z0 ) := mg−id (z0 ) where mg (z0 ) is deﬁned according to (1.2). If g = id is a Schur-class function then it may have at most one ﬁxed point z0 in D and moreover, ig (z0 ) = 1 and |g (z0 )| ≤ 1, which follows from the Schwarz–Pick inequality. Another classical result, the Carath´eodory–Julia– Wolﬀ theorem, states that if g ∈ S has no ﬁxed points in D, then there exists a unique point t0 ∈ T such that g(t0 ) = t0 and 0 ≤ g (t0 ) ≤ 1. Furthermore, if g ≡ const, then ig (t0 ) = 1 by the Julia lemma [11]. On the other hand, it is well known that g ∈ S may have other ﬁxed boundary points ti (each one of multiplicity one) so that g (t0 ) > 1. Summarizing we conclude that besides trivial exceptional cases, every function g ∈ S has exactly one fixed point in z0 ∈ D such that |g (z0 )| ≤ 1.

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In [5] we raised the question to what extent this conclusion holds for meromorphic functions g of the form g = s/b where s ∈ S and b is a ﬁnite Blaschke product. Via nontangential boundary limits, these functions (commonly known as generalized Schur functions) can be identiﬁed with the functions from the unit ball of L∞ (T) which admit meromorphic continuation inside the unit disk with the ﬁnite total pole multiplicity. Originally they appeared in [1,15] in an interpolation context and have been studied later in [14]. We denote by Sk the class of generalized Schur functions of the form s (4.1) g = , where s ∈ S and b ∈ Bk b do not have common zeros in D. It follows from the Carath´eodory-Julia theorem that if a generalized Schur function g admits a unimodular boundary limit g(t0 ), then the limit g (t0 ) exists and equals either a real number or +∞ (see e.g., [10]). Definition 4.1. Let g be a generalized Schur function. We will say that a point t0 ∈ T is a boundary fixed point of nonpositive type for g if g(t0 ) = t0

and

g (t0 ) ≤ 1.

(4.2)

Note that if ig (t0 ) ≥ 2, then g(t0 ) = t0 , g (t0 ) = 1, so that t0 is a ﬁxed point of g of nonpositive type. Also observe that if g is of the form (4.1), then g(z) − z =

s(z) − ϑ(z) , b(z)

where

ϑ(z) := zb(z) ∈ Bk+1

(4.3)

and thus z0 is a ﬁxed point of g if and only if its is a zero of the function s − ϑ in which case ig (z0 ) = ms−ϑ (z0 ). It is not hard to check that t0 ∈ T is a boundary ﬁxed point of g of nonpositive type if and only if t0 is a boundary zero of s − ϑ of nonpositive type. It then makes sense to assign to each ﬁxed point t0 ∈ T of nonpositive type its nonpositive degree τg (t0 ) by letting τg (t0 ) := τs−ϑ (t0 )

(4.4)

where τs−ϑ (t0 ) is determined via formula (1.4). To deﬁne τg (t0 ) in terms of g rather than in terms of representation (4.1), we can proceed as follows. If t0 ∈ T is a ﬁxed boundary point of g ∈ Sκ of multiplicity ig (t0 ) = mg−id (t0 ) = 1, we let τg (t0 ) = 1 if g (t0 ) ≤ 1 and τg (t0 ) = 0 if g (t0 ) > 1. If the multiplicity ig (t0 ) is greater than one, then we let ⎧ if ig (t0 ) = 2n, ⎨ n, if ig (t0 ) = 2n − 1 and (−1)n−1 t02n−2 g (2n−1) (t0 ) ≤ 0, τg (t0 ) = n, ⎩ n − 1, if ig (t0 ) = 2n − 1 and (−1)n−1 t02n−2 g (2n−1) (t0 ) > 0. (4.5) To justify this definition we have to show that if ig (t0 ) = 2n − 1, then the angular boundary limit g (2n−1) (t0 ) exists ﬁnitely and that furthermore, the number t02n−2 g (2n−1) (t0 ) is real. Indeed, if ig (t0 ) = 2n−1 and g is of the form (4.1), then we have from (4.3), that ms−ϑ = 2n − 1 and therefore, s(2n−1) (t0 )

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exists by Proposition 1.1. Therefore, g (2n−1) (t0 ) exists as well. Furthermore, since ig (t0 ) = mg−id (t0 ) = 2n − 1, we have g(t0 ) = t0 ,

g (t0 ) = 1,

g (j) (t0 ) = 0 for j = 2, . . . , 2n − 2.

Using the latter equalities, it is not hard to show by a straightforward computation that in case n > 1, (−1)n−1 t02n−2 g (2n−1) (t0 ) = (−1)n−1 t02n−1 ϑ(t0 )(s(2n−1) (t0 ) − ϑ(2n−1) (t0 )) =: Δs−ϑ,n (t0 ). The latter equality shows not only that the requisite number is indeed real (by Proposition 1.1), but also that formula (4.5) is compatible with (4.4). Theorem 4.2. Let g ∈ Sk . Then ig (z) + z∈Fix(g)∩D

τg (t0 ) = k + 1.

z∈Fix(g)∩T

Theorem 4.2 is a consequence of Theorem 1.4 (specialized to the case ϑ(z) = zb(z)) and definitions introduced just above. It makes more precise Theorem 3.1 from [5] stating that if a function g ∈ Sk has at most k ﬁxed points (counted with multiplicities) in D, then it has at least one boundary ﬁxed point of nonpositive type.

References [1] Akhiezer, N.I.: On a minimum problem in the theory of functions, and on the number of roots of an algebraic equation which lie inside the unit circle. Izv. Akad. Nauk 9, 1169–1189 (1930) [2] Ball, J.A.: Interpolation problems of Pick-Nevanlinna and Loewner type for meromorphic matrix functions. Integr. Equ. Oper. Theory 6, 804–840 (1983) [3] Ball, J.A., Gohberg, I., Rodman, L.: Interpolation of Rational Matrix Functions. OT 45. Birkh¨ auser (1990) [4] Ball, J.A., Helton, J.W.: Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: parameterization of the set of all solutions. Integr. Equ. Oper. Theory 9, 155–203 (1986) [5] Bolotnikov, V.: On a certain generalization of the Carath´eodory-Julia-Wolﬀ theorem. Bull. Belg. Math. Soc. Simon Stevin (to appear) [6] Bolotnikov, V.: A uniqueness result on boundary interpolation. Proc. Am. Math. Soc. 136(5), 1705–1715 (2008) [7] Bolotnikov, V., Dym, H.: On degenerate interpolation maximum entropy and extremal problems for matrix Schur functions. Integr. Equ. Oper. Theory 32(4), 367–435 (1998) [8] Bolotnikov, V., Dym, H.: On boundary interpolation for matrix Schur functions. Mem. Am. Math. Soc. 181, no 856 (2006) [9] Bolotnikov, V., Kheifets, A.: The higher order Carath´eodory–Julia theorem and related boundary interpolation problems. Oper. Theory Adv. Appl. OT 179, 63–102. Birkh¨ auser, Basel (2007)

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[10] Bolotnikov, V., Kheifets, A.: Carath´eodory-Julia type conditions and symmetries of boundary asymptotics for analytic functions on the unit disk. Math. Nachr. 282(11), 1513–1536 (2009) [11] Julia, G.: Extension d’un lemma de Schwartz. Acta Math. 42, 349–355 (1920) [12] Katsnelson, V., Kheifets, A., Yuditskii, P.: An abstract interpolation problem and extension theory of isometric operators. Oper. Theory Adv. Appl. OT 95, 283–298. Birkh¨ auser Verlag, Basel (1997) [13] Kheifets, A.: Scattering matrices and Parseval equality in abstract interpolation problem. Ph.D. Thesis, Khrakov State University (1990) ¨ [14] Kre˘ın, M.G., Langer, H.: Uber einige Fortsetzungsprobleme, die eng mit der angen. I. Einige Theorie hermitescher Operatoren im Raume Πκ zusammenh¨ Funktionenklassen und ihre Darstellungen. Math. Nachr. 77, 187–236 (1977) [15] Takagi, T.: On an algebraic problem related to an analytic theorem of Carath´eodory and Fej´er. Jpn. J. Math. 1, 83–93 (1924) Vladimir Bolotnikov Department of Mathematics The College of William and Mary Williamsburg, VA 23187-8795, USA e-mail: [email protected] Received: April 17, 2010. Revised: July 2, 2010.

Integr. Equ. Oper. Theory 69 (2011), 217–232 DOI 10.1007/s00020-010-1831-6 Published online September 30, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Lp-Approximation of the Integrated Density of States for Schr¨ odinger Operators with Finite Local Complexity Michael J. Gruber, Daniel H. Lenz and Ivan Veseli´c Abstract. We study spectral properties of Schr¨ odinger operators on Rd . The electromagnetic potential is assumed to be determined locally by a colouring of the lattice points in Zd , with the property that frequencies of finite patterns are well defined. We prove that the integrated density of states (spectral distribution function) is approximated by its finite volume analogues, i.e. the normalised eigenvalue counting functions. The convergence holds in the space Lp (I) where I is any finite energy interval and 1 ≤ p < ∞ is arbitrary. Mathematics Subject Classification (2010). 35J10, 81Q10. Keywords. Integrated density of states, random Schr¨ odinger operators, finite local complexity.

1. Introduction Spectral properties play a key role in the analysis of selfadjoint operators. This is in particular the case for Hamiltonians describing the time evolution of quantum mechanical systems. In the context of mathematical physics one often studies the integrated density of states, in the following abbreviated by IDS. It is very natural to think of the IDS as the normalized eigenvalue counting function of the restriction of the Hamiltonian to a large but ﬁnite volume system. This leads to the question in what sense and how well one can approximate the IDS of the full Hamilton operator by the spectral distribution functions of appropriately chosen ﬁnite-volume analogues. This question has been pursued for various types of selfadjoint operators, resp. Hamiltonians, in particular in the mathematical physics and geometry literature. Let us mention the seminal papers [12,14] and the recent reviews [8,16]. There one can ﬁnd also an overview of the literature up to 2007. c 2010 by the authors. Faithful reproduction of this article, in its entirety, by any means is permitted for non-commercial purposes.

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It turns out that in the discrete and in the one-dimensional setting one can control the convergence of ﬁnite volume approximants to the IDS very well. Let us state this more precisely: • •

•

For difference operators (with ﬁnite range) on combinatorial graphs the eigenvalue counting functions are bounded. This leads to uniform convergence (in supremum norm) for the IDS [9,11]. For Schr¨ odinger operators on metric graphs (so called quantum graphs) with constant edge lengths one can achieve uniform convergence for the IDS as well [3]. Here one has to assume that the randomness satisﬁes a ﬁnite local complexity condition. Note that for quantum graphs the eigenvalue counting functions are unbounded: However, the technically relevant objects are the spectral shift functions, which are still bounded. Metric graphs with non-constant edge lengths lead to unbounded shift functions; in this case, convergence holds locally uniformly, as well as globally uniformly with respect to a weighted supremum norm [4].

See also [4] for an overview. For electromagnetic Schr¨odinger operators on Rd even the perturbation by a compactly supported potential may lead to a locally unbounded spectral shift function. Thus the shift function diverges not only at inﬁnity but also on compact energy intervals. This is in particular the case for Landautype Hamiltonians, see, e.g., [5,13] and references therein. One may expect that the situation will be better for certain random perturbations of the Landau Hamiltonian. In order to obtain continuity of the IDS of Landau-type Hamiltonians plus a random, ergodic potential one has to pose appropriate conditions on the randomness. They amount to regularity conditions on the random distribution (see [1,2,7,17] and references therein). The results of the present paper apply to highly “singular” distributions, though: those of Bernoulli type; at each lattice site in Zd , local electric and magnetic potentials are chosen randomly from a ﬁnite set of prototypes. For such models there are no results on the continuity of the IDS. Thus this property cannot help us proving uniform convergence of the distribution functions. Our main result is that one can still achieve a strong form of convergence. More precisely, we show that convergence holds in the space Lp (I) for any ﬁnite interval I ⊂ R and any ﬁnite p. In the following section we describe our model and assumptions and state the main theorems. Section 3 provides bounds on the spectral shift function. They are applied in Sect. 4 which establishes certain almost additivity properties and thus concludes the proof of the main theorem. The latter is applied to certain types of alloy type random Schr¨ odinger operators in the ﬁnal section.

2. Model and Results Throughout the paper we will consider electromagnetic Schr¨ odinger operators which satisfy the following regularity

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Assumption 1. Let U be an open set in Rd , A : U → Rd a magnetic vector potential, each component of which is locally square integrable, V = V+ − V− : U → R a scalar electric potential, such that its positive part V+ ≥ 0 is locally integrable and its negative part V− ≥ 0 is in the Kato class. This implies that V− is relatively form bounded with respect to −ΔU , the Dirichlet Laplacian on U, with relative bound δ strictly smaller than one. Under these conditions the magnetic Schr¨ odinger operator H U = (−i∇ − A)2 + V

(1)

is well deﬁned via the corresponding lower semi-bounded quadratic form with core Cc∞ (U) [15]. Due to this choice of core, we say that H U has Dirichlet boundary conditions. Let us mention locally uniform Lp -integrability conditions which are sufﬁcient for V− to be in the Kato-class. More precisely, if V− satisﬁes ⎛ ⎞1/p ⎜ ⎟ V− Lploc,unif (Rd ) = sup ⎝ |V− (y)|p dy ⎠ <∞ x∈Rd

|x−y|≤1

for p = 1 if d = 1 and p > d/2 if d ≥ 2, then it belongs to the Kato-class. Next we want to introduce the notions of a colouring and a pattern. For this purpose we denote the set of all ﬁnite subsets of Zd by Fﬁn (Zd ) and an arbitrary ﬁnite set by A. A colouring is a map C : Zd → A and a pattern is a map P : D(P ) → A, where D(P ) ∈ Fﬁn (Zd ) is called the domain of P . We denote the set of all patterns by P. For a ﬁxed Q ∈ Fﬁn (Zd ) we denote the subset of P which contains only the patterns with domain Q by P(Q). Given a set Q ⊂ D(P ) and an element x ∈ Zd we deﬁne a restriction of a pattern P |Q and the translate of a pattern P + x by the vector x in the following way P |Q : Q → A,

g → P |Q (g) = P (g),

P + x : D(P ) + x → A,

y + x → P (y)

Two patterns P1 , P2 are equivalent if there exists an x ∈ Zd such that P2 = P1 + x. The equivalence class of a pattern P in P is denoted by P˜ . ˜ For two patterns P and P This induces on P a set of equivalence classes P. the number of occurrences of the pattern P in P is denoted by P (P ) := {x ∈ Zd | D(P ) + x ⊂ D(P ),

P |D(P )+x = P + x}.

Here, denotes the cardinality of a ﬁnite set. Next we deﬁne the notion of a van Hove sequence and of the frequency of a pattern along a given van Hove sequence. A sequence (Uj )j∈N of ﬁnite, non-empty subsets of Zd is called a van Hove sequence if for all M ∈ N :

lim

j→∞

∂ M Uj = 0. Uj

(2)

Here ∂ M U = {x ∈ U | dist(x, Zd \ U ) ≤ M } ∪ {x ∈ Zd \ U | dist(x, U ) ≤ M }.

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It is sufﬁcient to check the relation (2) for M = 1, it then follows for all M ∈ N, cf. for instance Lemma 2.1 in [10]. If for a pattern P and a van Hove sequence (Uj )j∈N the limit νP := lim

j→∞

P (C|Uj ) . Uj

exists, we call νP the frequency of P along (Uj )j∈N in the colouring C. In our setting A will be a ﬁnite collection of pairs (a, v) where a is a function Rd → Rd such that all its components are in L2 and v is a function Rd → R such that its positive part is in L1 and its negative part is in the Kato class, cf. Assumption 1. Moreover, both the support of a and of v are contained in W0 := [0, 1]d . Given a colouring C : Zd → A we denote by Ca its ﬁrst and by Cv its second component. To each colouring C we associate an electromagnetic potential (AC , VC ) : Rd → Rd × R AC (x) := Ca (k)(x − k), VC (x) := Cv (k)(x − k) k∈Zd

k∈Zd

and a Schr¨ odinger operator HC := (−i∇ − AC )2 + VC . Note that for any open subset U of Rd the restriction HCU of HC to U with Dirichlet boundary conditions satisﬁes Assumption 1. Now we want to deﬁne the IDS of HC and of ﬁnite restrictions thereof. For this purpose we need some more notation. In order to associate ﬁnite subsets of Zd with bounded subsets of Rd , we deﬁne

W : Fﬁn (Zd ) → B(Rd ), Q → WQ := (W0 + t) t∈Q

where we use the natural embedding Z ⊂ R and denote the Borel-σodinger algebra by B. Furthermore, if U is an open set in Rd , H U a Schr¨ d

d

◦

operator deﬁned on U and Q ∈ Fﬁn (Zd ) such that WQ ⊂ U we deﬁne, by a slight but natural abuse of notation, ◦

H Q := H WQ , i.e. the restriction of H U to the interior of WQ in the sense of quadratic forms as above. Let us denote by χ(−∞,λ] (HC ) and χ(−∞,λ] HCQ the spectral families of HC and HCQ , respectively. The IDS of HCQ is the distribution function

N (λ, Q) := Tr χ(−∞,λ] HCQ , divided by the volume volWQ = Q. Since Q is ﬁnite, HCQ is an elliptic operator on a bounded domain and χ(−∞,λ] HCQ is trace-class. However,

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χ(−∞,λ] (HC ) is not, which is the reason why we need the existence theorem below, and why HC may display any interesting spectral features at all. For M ∈ N we denote by CM ⊂ Zd the cube at the origin with side length M − 1, i.e. CM := {x ∈ Zd : 0 ≤ xj ≤ M − 1, j = 1, . . . , d}. Now we are prepared to state our main theorem: Theorem 2. Let C : Zd →A be a colouring, (Uj )j∈N a van Hove sequence such that for all patterns P ∈ M ∈N P(CM ) the frequencies νP exist. Let I ⊂ R be a finite interval. Then there exists a function N belonging to all Lp (I) with 1 ≤ p < ∞ and independent of the van Hove sequence such that for j → ∞ we have p N λ) − 1 N (λ, Uj ) dλ → 0 (3) Uj I

for any any p ∈ [1, ∞). More precisely, for any M ∈ N the above integral is bounded by ∂ M U 1 d C j + C(T + C) 2 + cp,d C p M Uj P (C|Uj ) d +C(T + C) 2 Uj − νP P ∈P(CM )

where T = sup I. The dependencies of the constants appearing in the estimate are as follows: cp,d depends only on the dimension d and the exponent p, and C depends only on d and on (the Kato norm of ) VC,− . Remark 3. • Obviously, the theorem yields a function N ∈ Lp (R) such that on each ﬁnite interval I the above holds, and we may extend this to (upper) semibounded intervals since our operators are uniformly semibounded below. • In particular one may choose the van Hove sequence Uj := Cj or a similar family of expanding cubes (if the frequencies exist) so that one gets an explicit 1j -decay for the second term in the error estimate. A sequence of cubes is a common choice for deﬁning the IDS. • In the random setting, introduced in Sect. 5, there is an alternative definition of the IDS (Pastur–Shubin formula) which coincides with N , as we show there.

3. Bounds on the Spectral Shift Function of Facets In this section we prove certain bounds on the spectral shift function (SSF) which are needed in Sect. 4. They concern the SSFs of two electromagnetic Schr¨ odinger operators which differ by an (additional) Dirichlet boundary condition on a facet. This difference can be understood as a generalized (positive) compactly supported potential. Hence we can use results established in [5], either verbatim or with slight modiﬁcations.

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Recall that if V− is in the Kato class, then there exists some number δ smaller than one such that V− is relatively Δ-bounded with relative bound δ. We quote Lemma 5 from [5]: Lemma 4. Let H U be a Schr¨ odinger operator defined on the open set U ⊂ Rd of finite volume which satisfies Assumption 1. Denote by En the nth eigenvalue counted from below including multiplicity of H U . Then there exists a constant C1 such that 2/d n 2π(1 − δ)d − C1 for all n ∈ N. (4) En ≥ e |U| The constant C1 depends on the Kato norm of V− only. Lemma 4 will be used in the proof of the next result, which is a slight modiﬁcation of Theorem 1 in [5]. We will frequently use certain subsets of d − 1 dimensional hyperplanes in Rd . They are unit squares in the hyperplanes. More formally we deﬁne: Definition 5. A set S ⊂ Rd is called canonical facet if there exists a j ∈ {1, . . . , d} such that S = {(x1 , . . . , xd ) | xj = 0

and xi ∈ [0, 1] for i = j}. A set S is called a facet if there exists a canonical facet S˜ and a vector x ∈ Zd such that ˜ S = x + S˜ := {y ∈ Rd | y − x ∈ S}. Let U be an open subset of Rd , S a facet as deﬁned in Definition 5, and U˜ := U \ S. Let H1 be a Schr¨ odinger operator on U, satisfying Assumption 1. ˜ Using quadratic forms we deﬁne H2 as the Dirichlet restriction of H1 to U. −H1 −H2 −H −H 2 1 ,e , and Veﬀ := e −e are well deﬁned by Then the operators e the spectral calculus. Theorem 6. (a) The operator Veﬀ is compact. (b) Denote by μn the nth singular value of Veﬀ counted from above including multiplicity. Then there are finite positive constants c and C2 such that the singular values of the operator Veﬀ obey 1/d

μn ≤ C2 e−cn

.

(5)

The constant c may be chosen depending only on d, while C2 depends only on the Kato-class norm of V− . Proof. To prove part (a), i.e. that the operator Veﬀ is compact, it is sufﬁcient to ﬁnd a family AR , R > 0 of compact operators such that the operator norm of the difference DR := Veﬀ − AR tends to zero as R → ∞. Indeed, such an operator family will be constructed in the proof of the quantitative statement (b). The operators in the family will be even trace-class. The proof of (b) is almost the same as the one of Theorem 1 in [5]. We use the same notation as there and explain only the step in the proof which is slightly different in the present setting. Let B := BR ⊂ Rd be an open ball of radius R > 0 containing the facet S in its interior, and S c the complement of S.

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Denote by H1B the Dirichlet restriction of H1 to the set U ∩ B and by the Dirichlet restriction of H2 to the set U˜ ∩ B. Set B B D = Veﬀ − e−H2 − e−H1 .

Let Ex and Px denote expectation and probability for Brownian motion, bt , starting at x. For any open set U ⊂ Rd denote by τU = inf{t > 0 | bt ∈ U } the ﬁrst exit time from U . We use the Feynman-Kac-Itˆo formula to express e−H1 f for f ∈ Cc (U) as

1 1 e−H1 f (x) = Ex e−iSA (b) e− 0 V (bs )ds χ{τU >1} (b)f (b1 ) t where SA is a real valued stochastic process (Itˆo integral) corresponding to the magnetic vector potential A of the Schr¨ odinger operator; this representation holds for more general f [15] but a dense set sufﬁces for our purposes. B B Analogous representations hold for the operators e−H2 , e−H1 , and e−H2 if one replaces the condition χ{τU >1} by

χ{τU˜ >1} ,

χ{τU ∩B >1} , and χ{τU˜∩B >1} , respectively.

Since the operator D can be expressed in terms of four different exponentials it follows that

1 1 (Df )(x) = Ex ρ(b) e−iSA (b) e− 0 V (bs )ds f (b1 ) where ρ = χ{τU >1} − χ{τU˜ >1} − χ{τU ∩B >1} + χ{τU˜∩B >1} . A simple transformation (using χ{τU >1} − χ{τU˜ >1} = χ{τU >1} (1 − χ{τU˜ >1} ) = χ{τU >1} χ{τU˜ ≤1} = χ{τU >1} χ{τSc ≤1} etc.) shows that ρ = χ{τU >1} χ{τSc ≤1} χ{τB ≤1} . older inequality implies We abbreviate B := {τS c ≤ 1} ∩ {τB ≤ 1}. The H¨ that 1/4 1 1/2 1/4 Ex |f (b1 )|2 (Ex [χB (b)]) . |Df |(x) ≤ Ex χ{τU >1} e−4 0 V (bs )ds At this point we make the dependence of the operator D on the radius of the ball B = BR explicit and denote it consequently by DR . From this point on we can follow exactly the proof of Theorem 1 in [5] to conclude that −R2 DR ≤ const exp . 32 With the choice R = Rn = n1/2d the desired estimate (5) follows.

The next Lemma is an abstraction of the proof of Theorem 2 in [5], which in turn relies on [6]. The abstract formulation may be of use also in other contexts. Let A, B be two selfadjoint operators such that Veﬀ = e−B − e−A is trace class. This implies that the sequence μ = {μn }n∈N of singular values of Veﬀ (enumerated in decreasing order including multiplicity) converges to zero. Let F : [0, ∞) → [0, ∞) be a convex function with F (0) = 0. In

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particular, F is isotone because for x ∈ [0, ∞), α ∈ [0, 1] convexity gives F (αx) = F (αx + (1 − α)0) ≤ αF (x) + (1 − α)F (0) = αF (x) which, since F (x) ≥ 0, implies F (αx) ≤ F (x). Set φ(n) = F (n)−F (n−1) for n ∈ N, n ≥ 2. Lemma 7. (a)

Let F be as above. Then T F (|ξ(λ, B, A)|) dλ ≤ eT φ, μ2 (N)

(6)

−∞

(b) Let h : R → R be a bounded measurable function with support in (−∞, T ]. Then

T h(λ) ξ(λ, B, A) dλ ≤ e φ, μ2 (N) + T

G (|h(λ)|) dλ

(7)

−∞

R

where G denotes the Legendre transform of F , i.e. G(y) = sup{xy − F (x) | x ≥ 0} for y ≥ 0. Of course the usefulness of the Lemma depends heavily on a priori information about the decay of the sequence μ. However, for the choice of operators A = H1 and B = H2 we do know that the singular values decay at an almost exponential rate. Remark 8. Depending on how many additional properties we assume for the function F , we obtain correspondingly more information about its Legendre transform G. This will be discussed next. 1.

2. 3.

The following properties hold under no additional assumptions on F : G(0) = 0, G is convex (because xy−F (x) is convex in y) and G(y) ≥ 0 for all y. If lim F (x) x = ∞ then G takes on ﬁnite values only (and vice versa). x→∞

If F is twice differentiable, f := F > 0 and F > 0 on [0, ∞), then 0 if y ≤ f (0), G(y) = −1 −1 yf (y) − F (f (y)) if y > f (0).

Here f −1 denotes the inverse of the function f . Consequently, T G (|h(λ)|) dλ −∞

=

|h(λ)|f −1 (|h(λ)|) − F (f −1 (|h(λ)|)) dλ.

{λ≤T : |h(λ)|>f (0)}

4.

If there exist a positive constant C and an exponent p > 1 such that μn ≤ Cn−p , then one can choose the function F as F (x) = xq+1 , where q is any number smaller than p − 1. Indeed, for this choice of F we have φ(n) ≤ (q + 1)nq . Thus

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Lp -approximation of the integrated density of states

μn φ(n) ≤ (q + 1)C

n

n−p nq < ∞.

n

y q+1

Note that in this case G(y) = q 5.

225

q+1 q

.

If there exist positive constants c, C and p such that μn ≤ Ce−cn for all n ∈ N, then for each value of t < c, the choice x p ety − 1 dy F (x) = p

0

gives a ﬁnite right hand side in (6). Indeed, φ(n) = p etn and thus p p φ, μ2 (N) ≤ C e−cn etn < ∞.

n n−1

p ety − 1 dy ≤

n

The Legendre transform for such a choice of F satisﬁes 1/p log(1 + y) −1 G(y) ≤ yf (y) = for all y ≥ 0. t Thus in this speciﬁc case inequality (7) reads p h(λ) ξ(λ, B, A) dλ ≤ eT C e−(c−t)n R

n∈N

|h(λ)|

+

log(1 + |h(λ)|) t

1/p dλ

(8)

which recovers the result of [5]. Now we prove Lemma 7. Proof. Since Veﬀ is trace class but not necessarily the operator difference A − B, the SSF is deﬁned via the invariance principle T

T F (|ξ(λ, B, A)|) dλ =

−∞

F (|ξ(e−λ , e−B , e−A )|) dλ.

−∞

Now a change of variables gives us T

−λ

F (|ξ(e −∞

−B

,e

−A

,e

∞ )|) dλ ≤ e

T

F (|ξ(s, e−B , e−A )|) ds.

e−T

To the last expression we can apply the estimate of [6] and bound it above by ∞ F (|ξ(s, e−B , e−A )|) ds ≤ μn (Veﬀ )φ(n). e−T

n∈N

This establishes claim (a). To prove (b), we note that by the very definition of the Legendre transform, the Young inequality

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|h · ξ| ≤ F (|ξ|) + G(|h|) holds. Integrating over λ we obtain

T

h(λ)ξ(λ) dλ ≤

F (|ξ(λ, B, A)|) dλ +

G(|h(λ)|) dλ.

−∞

Together with (a) this completes the proof.

4. Almost Additivity for the Eigenvalue Counting Functions In this section we prove Theorem 2. For this aim we will apply a Banach space-valued ergodic theorem obtained in [9]. Actually, for our purposes it will be convenient to quote a slightly streamlined version of this result from [10]. To spell it out we need to introduce the notion of a boundary term and the properties of almost additivity and invariance. Definition 9. A function b : Fﬁn (Zd ) → [0, ∞) is called a boundary term if the following three properties hold: (i) b(Q) = b(Q + x) for all x ∈ Zd and all Q ∈ Fﬁn (Zd ), b(U ) (ii) lim Ujj = 0 for any van Hove sequence (Uj )j∈N and j→∞

(iii)

there exists a constant D ∈ (0, ∞) such that b(Q) ≤ D Q for all Q ∈ Fﬁn (Zd )

Definition 10. Let (X, ·) be a Banach space and F a function Fﬁn (Zd ) → X. (a)

The function F is said to be almost-additive if there exists a boundary term b such that m m m F (Qk ) ≤ b(Qk ) F (∪k=1 Qk ) − k=1

k=1

for all m ∈ N and all pairwise disjoint sets Qk ∈ Fﬁn (Zd ), k = 1, . . . , m. (b) Let C : Zd −→ A be a colouring. The function F is said to be C-invariant if F (Q) = F (Q + x) whenever x ∈ Zd and Q ∈ Fﬁn (Zd ) obey C|Q + x = C|Q+x . In this case there exists a function F deﬁned on the (classes of) patterns such that F (P ) = F (Q) if C|Q = P . If F : Fﬁn (Zd ) → X is almost additive and invariant there exists a K ∈ (0, ∞) such that F (Q) ≤ K Q for all Q ∈ Fﬁn (Zd ).

(9)

Now we are in the position to quote the Banach space valued ergodic theorem from [9], see [10, Sect. 5.1] as well.

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Theorem 11. Let A be a finite set of colours, C : Zd → A a colouring and (Uj ) a van Hove sequence along which the frequencies of all patterns P ∈ d M ∈N P(CM ) exist. Let F : F(Z ) → X be a C-invariant and almost-additive function. Then the limit F (Uj ) = lim j→∞ Uj M →∞

F := lim

νP

P ∈P(CM )

F(P ) CM

exists in X. Furthermore, for j, M ∈ N the bound M F − F (Uj ) ≤ 2 b(CM ) +(K + D) ∂ Uj +K d Uj M Uj

P ∈P(CM )

P (C|Uj ) Uj −νP (10)

holds. We want to apply the ergodic theorem to the eigenvalue counting functions of Schr¨ odinger operators, considered as elements of X := Lp (I) for a ﬁxed ﬁnite interval I ⊂ R and p ∈ [1, ∞). More precisely, we study the function F : Fﬁn (Zd ) → Lp (I), Q → N ·, H Q with N λ, H Q := Trχ(−∞,λ] H Q for λ ∈ R. Note that this notation is slightly different from the one used in Theorem 2. The reason is that in the proofs we use a more general class of operators than which was necessary to formulate the main result and thus need a bit more ﬂexibility. To conclude Theorem 2 we need to show that F fulﬁls the hypotheses of Theorem 11. This is done in the following Lemma 12. The function F : Ffin (Zd ) → Lp (I) is invariant and almostadditive. Proof. Note that H is a local operator, thus H U depends only on A|U and V |U , for any open U ⊂ Rd . The translation operator Ty f (x) = f (x − y) is unitary for any y ∈ Rd . Thus the spectrum of H, resp. H U , is invariant under conjugation by Ty . It follows that the function m F is C-invariant. To prove almost-additivity, let Q = k=1 Qk with disjoint Qk ∈ pairwise m ), k = 1, . . . , m. We need to compare F (Q) to k=1 F (Qk ). Note that Fﬁn (Zd m WQ = k=1 WQk but the WQk need not be pairwise disjoint because their boundaries can touch. There are two extreme cases: ◦ ◦ m are pairwise disjoint. Then W = W 1. All WQ Q Qk , and consequently k k=1 m m H Q = k=1 H Qk and F (Q) = k=1 F (Qk ). 2. No WQk is disjoint from all others. Since ∂WQk consists of at most 2d ∂Qk facets (where ∂Qk denotes the combinatorial boundary of ◦ ◦ m m Qk ⊂ Zd ) the sets WQ and k=1 WQk differ by at most 2d k=1 ∂Qk facets. In fact, the latter case gives an upper bound for the general case (where the “isolated” Qk simply do not contribute), and we can drop the factor 2

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because touching facets need to be counted once only (they are counted twice in the sum). ◦ m So, let M ≤ d k=1 ∂Qk be the number of facets by which WQ and ◦ m k=1 WQk differ and enumerate them arbitrarily as S1 , . . . , SM . Set H0 := H Q and where Uj := WQ \

Hj := H Uj

j

Si

i=1

m m for j = 1, . . . , M . Clearly, HM = k=1 H Qk and N (·, HM ) = k=1 F (Qk ). This relation allows us to write the difference that we want to estimate as a sum of SSFs: F (Q) −

m

F (Qk ) =

M

(N (·, Hj−1 ) − N (·, Hj )) =

j=1

k=1

M

ξ(·, Hj−1 , Hj )

j=1

Hj−1 and Hj differ exactly by a Dirichlet condition at one facet Sj so that 1/d Theorem 6 applies and gives the estimate μn ≤ Ce−cn for the singular values. Now we can apply Lemma 7 with T := sup I, A := Hj , B := Hj−1 and F (x) := xp . Then inequality (6) reads 1/d |ξ(·, Hj−1 , Hj )|p ≤ eT C2 pnp−1 e−cn =: C˜ p (11) n

I

with a constant C˜ which is independent of j. The triangle inequality thus gives m m m F (Qk ) ≤ M C˜ ≤ d C˜ ∂Qk =: b(Qk ). F (Q) − k=1

Lp (I)

k=1

k=1

The function b : Fﬁn (Zd ) → R, Q → d C˜ ∂Q satisﬁes the three conditions required for a boundary term. Proof of Theorem 2. Given the previous Lemma, we can apply the abstract ergodic theorem to our counting functions associated to ﬁnite subsets Q. In order to check the form of the error estimate, we note that the constant C˜ in Eq. (11) satisﬁes C˜ p = C2 eT const(p, d) with C2 being the constant from Theorem 6 depending on the Kato-norm of VC,− . Also, b(Q) = dC˜ ∂Q so that the bound D on b in the general abstract ergodic theorem can be ˜ ˜ It follows that 2b(CdM ) ≤ 4d2 C/M . taken to be dC. M Finally, the uniform lower estimate on the eigenvalues established in Lemma 4 gives a uniform upper estimate on the number of eigenvalues in (−∞, T ], namely

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Lp -approximation of the integrated density of states d/2

(T + C1 )

229

d/2 e d) |U|. 2π(1 − δ)

Hence we can take d

K = C3 (T + C1 ) 2

as the uniform bound on the function F in the abstract ergodic theorem. Here the constants C1 , C2 , C3 depend only on d and the Kato norm of VC,− .

5. Application to Random Operators In order to apply our results to random operators we quote the necessary random versions of the definition and main theorem from [4]: Let (Ω, P) be a probability space such that Zd acts ergodically on (Ω, P). We denote the Zd -action on Ω by x : ω → ω − x. A random A-colouring is a map A with C(ω − y)x−y = C(ω)x C : Ω −→ Zd

for all x, y ∈ Zd . Note that for each ﬁxed ω we obtain a (usual) Acolouring. By the (usual) ergodic theorem for scalar functions the frequencies of patterns exist almost surely. Thus we can apply our abstract Banach space valued ergodic theorem: Theorem 13. Let A be a finite set, C be a random A-colouring and (X, · ) a Banach space. Let (Uj )j∈N be a van Hove sequence. For each fixed ω ∈ Ω let Fω : Ffin (Zd ) −→ X be a C(ω)-invariant, almost-additive bounded function. Assume that the family (Fω )ω∈Ω is Zd -homogeneous, i.e. Fω+x (Q + x) = Fω (Q) for all x ∈ Zd , Q ∈ Ffin (Zd ). Then, for almost every ω ∈ Ω the limits Fω (Uj ) = lim j→∞ |Uj | M →∞

F ω := lim

P ∈P(CM )

νP

F(P ) |CM |

exist in the topology of (X, · ) and are equal. In particular, F ω is almost surely independent of ω. Concretely, we apply this to colourings given by local models for the potentials like in Sect. 2, but now with a randomly chosen colouring so that all operators and counting functions additionally depend on the random variable ω. For the formulation of the theorem we introduce a distribution function N : R → R deﬁned by a trace per unit volume formula (sometimes called Pastur–Shubin formula) (12) N (λ) := Tr χW0 χ(−∞,λ] (Hω ) dP(ω). Ω

By applying the random version of the ergodic theorem we get the random version of Theorem 2: Theorem 14. Let C : Ω −→ Zd A be a random colouring, (Uj )j∈N a van Hove sequence. Let I ⊂ R be a finite interval and p ∈ [1, ∞). Then for j → ∞ we have for almost all ω ∈ Ω

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I

Remark 15. Of course there are similar error estimates as in Theorem 2. Proof. The existence of the limit in Eq. 13 follows directly from our abstract theorem since we checked all requirements in the proof of Theorem 2 already. For the proof of the Shubin–Pastur formula (12) we use a variation of the proof of Theorem 3 in [3]: First notice that 1 Tr χW Uj χ(−∞,λ] (Hω ) dP(ω) N (λ) = Uj Ω

independently of Uj due to additivity and invariance. Now it sufﬁces to show that Uj 1 Tr χW Uj e−tHω − e−tHω → 0, j → ∞ (*) Uj U

Zd \U

for all t. We use the abbreviation ⊕Hω = Hω j ⊕ Hω j and the linearity of the trace to conclude Uj Tr χW Uj e−tHω − e−tHω Uj −tHω −t⊕Hω = Tr χW Uj (e −e ) + Tr χW Uj e−t⊕Hω − e−tHω . The second term actually vanishes. Thus it sufﬁces to estimate the ﬁrst term, which we do next. Let us note that for a compact operator K and a bounded operator B on the same Hilbert space the corresponding singular values obey the relation μn (BK) ≤ Bμn (K). In particular this holds for K = Veﬀ as in Theorem 6 and B = χW Uj . Now we can proceed exactly as in Lemma 12, using that the addition of the boundary conditions by which ⊕Hω differs from Hω gives a boundary term in Lp (I) for the corresponding SSF. Now the claim (*) follows from the van Hove property.

References [1] Combes, J.-M., Hislop, P.D.: Landau Hamiltonians with random potentials: localization and the density of states. Comm. Math. Phys. 177(3), 603–629 (1996) [2] Combes, J.-M., Hislop, P.D., Klopp, F.: An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schr¨ odinger operators. Duke Math. J. 140(3), 469–498 (2007) [3] Gruber, M.J., Lenz, D.H., Veseli´c, I.: Uniform existence of the integrated density of states for random Schr¨ odinger operators on metric graphs over Zd . J. Funct. Anal., 253(2), 515–533, 2007. arXiv:math.SP/0612743

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[4] Gruber, M.J., Lenz, D.H., Veseli´c, I.: Uniform existence of the integrated density of states for combinatorial and metric graphs over Zd . In: Exner, P., Keating, J., Kuchment, P., Sunada, T., Teplyaev, A. (eds.), Analysis on Graphs and its Applications. Proceedings of Symposia in Pure Mathematics, vol. 77, pp. 87–108. American Mathematical Society, Providence, RI (2008). arXiv:0712.1740 [5] Hundertmark, D., Killip, R., Nakamura, S., Stollmann, P., Veseli´c, I.: Bounds on the spectral shift function and the density of states. Comm. Math. Phys. 262(2), 489–503 (2006). arXiv:math-ph/0412078 [6] Hundertmark, D., Simon, B.: An optimal Lp -bound on the Krein spectral shift function. J. Anal. Math. 87, 199–208 (2002) [7] Hupfer, T., Leschke, H., M¨ uller, P., Warzel, S.: The absolute continuity of the integrated density of states for magnetic Schr¨ odinger operators with certain unbounded random potentials. Comm. Math. Phys. 221(2), 229–254 (2001) [8] Kirsch, W., Metzger, B.: The integrated density of states for random Schr¨ odinger operators. In: Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon’s 60th Birthday. Proceedings of Symposia in Pure Mathematics, vol. 76, pp. 649–696. American Mathematical Society, Providence, RI, (2007) [9] Lenz, D., M¨ uller, P., Veseli´c, I.: Uniform existence of the integrated density of states for models on Zd . Positivity 12(4), 571–589 (2008). arXiv:mathph/0607063 [10] Lenz, D., Schwarzenberger, F., Veseli´c, I.: A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states. Geom. Dedicata (2010). arXiv:1003.3620 [11] Lenz, D., Veseli´c, I.: Hamiltonians on discrete structures: jumps of the integrated density of states and uniform convergence. Math. Z. 263(4):813–835 (2009). arXiv:0709.2836 [12] Pastur, L.A.: Selfaverageability of the number of states of the Schr¨ odinger equation with a random potential. Mat. Fiz. i Funkcional. Anal. (Vyp. 2):111–116, 238 (1971) [13] Raikov, G.D., Warzel, S.: Spectral asymptotics for magnetic Schr¨ odinger operators with rapidly decreasing electric potentials. C. R. Math. Acad. Sci. Paris 335(8), 683–688 (2002) [14] Shubin, M.A.: Spectral theory and the index of elliptic operators with almostperiodic coefficients. Uspekhi Mat. Nauk, 34(2(206)), 95–135 (1979) [15] Simon, B.: Functional integration and quantum physics, volume 86 of Pure and Applied Mathematics. Academic Press Inc./Harcourt Brace Jovanovich Publishers, New York (1979) [16] Veseli´c, I.: Existence and regularity properties of the integrated density of states of random Schr¨ odinger operators. Lecture Notes in Mathematics, vol. 1917. Springer, Berlin (2008) [17] Wang, W.-M.: Microlocalization, percolation, and Anderson localization for the magnetic Schr¨ odinger operator with a random potential. J. Funct. Anal. 146(1), 1–26 (1997)

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Michael J. Gruber (B) Institut f¨ ur Mathematik TU Clausthal 38678 Clausthal-Zellerfeld Germany e-mail: [email protected] URL: http://www.math.tu-clausthal.de/∼mjg/ Daniel H. Lenz Mathematisches Institut Friedrich-Schiller-Universit¨ at Jena 07743 Jena Germany URL: http://www.analysis-lenz.uni-jena.de/ Ivan Veseli´c Emmy-Noether-Projekt ‘Schr¨ odingeroperatoren’ Fakult¨ at f¨ ur Mathematik TU Chemnitz 09107 Chemnitz Germany URL: http://www.tu-chemnitz.de/mathematik/enp Received: April 21, 2010. Revised: September 13, 2010.

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Integr. Equ. Oper. Theory 69 (2011), 233–246 DOI 10.1007/s00020-010-1836-1 Published online November 5, 2010 c The Author(s) This article is published with open access at Springerlink.com 2010

Integral Equations and Operator Theory

A Local Lifting Theorem for Jointly Subnormal Families of Unbounded Operators Witold Majdak and Jan Stochel Abstract. A local lifting theorem for bounded operators that intertwine a pair of jointly subnormal families of unbounded operators is proved. Each family in question is assumed to be composed of operators deﬁned on a common invariant domain consisting of “joint” analytic vectors. This result can be viewed as a generalization of the local lifting theorem proved by Sebesty´en, Thomson and the present authors for pairs of bounded subnormal operators. Mathematics Subject Classification (2010). Primary 47B20; Secondary 47A20. Keywords. Jointly subnormal family of operators, minimal normal extension, lift of intertwining operators, commutant lifting theorem.

1. Introduction A local lifting theorem, originally formulated for pairs of bounded subnormal operators (cf. [6, Theorem 4.2]), states that an intertwining operator between two subnormal operators lifts to an intertwining operator between their minimal normal extensions if and only if (1) the restriction of the intertwining operator to each cyclic invariant subspace lifts, and (2) the supremum of the norms of the cyclic lifts is finite (recall that the first commutant lifting theorem for bounded subnormal operators was proved by Bram in [2]). Our aim in this paper is to generalize the local lifting theorem of [6] to the case of pairs of jointly subnormal families of unbounded operators (see [12–14,17] for basic results on unbounded subnormal operators). The first problem to be overcome here concerns the question of which kind of minimality for normal extensions of unbounded operators should be chosen. A careful analysis W. Majdak was supported by the AGH local grant 10.420.03. J. Stochel was supported by the MNiSzW grant N201 026 32/1350.

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of the results on lifting strong commutants contained in [11] reveals that the notion of minimality of cyclic type is appropriate for this purpose. The other difficulty is to guarantee the existence of minimal normal extensions of cyclic type for cyclic parts of a jointly subnormal family of operators. This can be achieved by assuming that each family in question have a dense set of “joint” analytic vectors (recall that the notion of an analytic vector was introduced by Nelson in [8]). The latter enables us to use a generalization of the Maserick theorem [7, Theorem 3.2], which was proved in [12], to establish the main result of the paper (cf. Theorem 5.3). In Sect. 6 we dwell upon a special kind of local intertwining referred to as local commutativity.

2. Preliminaries As usual, C, R and N stand for the sets of complex numbers, real numbers and nonnegative integers, respectively. All linear spaces in this paper are assumed to be complex and all operators under consideration are assumed to be linear. Given two Hilbert spaces H and K, we denote by B(H, K) the set of all bounded operators from H into K. To simplify the writing, we put B(H) := B(H, H). The identity operator on H is denoted by IH . If X is a subset of H, then linX stands for the linear span of X. A family {fγ }γ∈Λ of vectors in a linear space X is said to be finite if fγ = 0 for all but a finite number of γ’s. Let A be an operator in H. Denote by D(A), A∗ and A¯ the domain, the adjoint and the closure of A (in case they exist). We write A ⊆ B if the operator B is an extension of A (A and B may act in distinct Hilbert spaces). We say that a closed linear subspace M of H reduces A if P A ⊆ AP , where P is the orthogonal projection of H onto M; if this is the case, then A|M stands for the restriction of A to M, i.e., D(A|M ) = D(A) ∩ M and A|M f = Af for f ∈ D(A|M ). If A is closable and E is a linear subspace of D(A) such that A ⊆ A|E (or equivalently A¯ = A|E ), then E is called a core of A (here A|E is the usual restriction of the mapping A to the set E). ∞ For an operator A in H, we put D∞ (A) = n=1 D(An ). Elements of D∞ (A) are referred to as C∞ -vectors. In the present paper we explore a special class of C∞ -vectors called analytic ones. Following [8], we say that f ∈ D∞ (A) is an analytic vector for A if there exists a positive real number t such that ∞ n=0

An f

tn < ∞. n!

The set of all analytic vectors for A, denoted by A (A), is a linear subspace of H which is invariant for A, i.e. A(A (A)) ⊆ A (A). Now, we recall some definitions from [12, Sect. 4]. Suppose that (Ω, +, ∗) is a commutative ∗-semigroup with the zero element 0 and E is a linear space. By a form over (Ω, E) we mean a mapping ϕ : Ω × E × E → C such that ϕ(υ; ·, -) : E × E → C is a sesquilinear form for all υ ∈ Ω. A form ϕ over

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(Ω, E) is called positive definite if k

ϕ(υn∗ + υm ; fm , fn ) ≥ 0,

{υm }km=0 ⊆ Ω, {fm }km=0 ⊆ E, k ∈ N.

m,n=0

We say that a form ϕ over (Ω, E) is weakly positive definite if for every f ∈ E, ϕ(·; f, f ) is positive definite as a form over (Ω, C). Given a weakly positive definite form ϕ over (Ω, E) and υ ∈ Ω, we set ∞ tn ∗ <∞ . Aϕ (υ) = f ∈ E ∃ t > 0 : ϕ(n · (υ + υ); f, f ) n! n=0 the set of all characters of Ω (by a character of Ω we mean Denote by Ω a nonzero additive involution preserving complex mapping on Ω). Let G be a set of ∗-generators of a ∗-semigroup Ω. As in [12, p. 40], we write CG for the Cartesian product υ∈G Cυ , where Cυ := C for every υ ∈ G. In → CG given by the formula what follows, we consider the mapping jG : Ω jG (χ) = {χ(υ)}υ∈G for χ ∈ Ω. The following lemma is a version of [12, Corollary 3] (see also [12, Remarks 3 and 5]). = CG . If ϕ is a Lemma 2.1. Let Ω, E and jG be as above. Assume that jG (Ω) weakly positive definite form over (Ω, E) such that Aϕ (υ) = E for all υ ∈ G, then the form ϕ is positive definite. Let Σ be a nonempty set. Denote by NΣ the set of all mappings α : Σ → N such that α(σ) = 0 for all but a finite number of elements σ ∈ Σ. The set ΩΣ := NΣ × NΣ becomes a commutative ∗-semigroup when equipped with the standard coordinate-wise addition and involution (α, β)∗ = (β, α) for α, β ∈ NΣ . In this particular case, G := {(δσ , 0) : σ ∈ Σ} is a set of ∗-generators of ΩΣ (δσ is the characteristic function of the singleton set {σ}). As observed in [12, p. 50],

jG (Ω Σ ) = CG .

(2.1)

3. Normal Extensions Let E be an inner product space and let H be its Hilbert space completion. Denote by L(E) the algebra of all operators A : E → E (with composition as multiplication), and by L# (E) the subalgebra of L(E) consisting of all operators A ∈ L(E) for which there exists an operator A# ∈ L(E) such that Af, g = f, A# g for all f, g ∈ E. Such an operator A# is uniquely determined. It is plain that L# (E) is a ∗-algebra with the involution A → A# . Note that if E is a linear subspace of a Hilbert space K and A is a densely defined operator in K such that E ⊆ D(A), then A|E belongs to L# (E) if and only if E ⊆ D(A∗ ), A(E) ⊆ E and A∗ (E) ⊆ E; if this is the case, then A# = A∗ |E . Given a family A = {Aσ }σ∈Σ ⊆ L(E) of commuting operators (i.e. α(σ) Aσ Aτ = Aτ Aσ for all σ, τ ∈ Σ) and α ∈ NΣ , we set Aα = σ∈Σ Aσ . A family A = {Aσ }σ∈Σ ⊆ L(E) is said to be jointly subnormal if there exist

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a Hilbert space K and a family M = {Mσ }σ∈Σ of spectrally commuting normal operators1 acting in K such that E ⊆ K and Aσ ⊆ Mσ for all σ ∈ Σ; M is then called a normal extension of A (cf. [12, p. 49]). We say that a family A = {Aσ }σ∈Σ of operators acts in H if each operator Aσ acts in H. Given a family A = {Aσ }σ∈Σ of operators in H, we set D∞ (A) = { D(Aσ0 · · · Aσn ) : σ0 , . . . , σn ∈ Σ, n ∈ N}. Clearly, D∞ (A) is the greatest linear subspace of σ∈Σ D(Aσ ) that is invariant (in the usual set-theoretical sense) for every Aσ . If each operator Aσ is densely defined, then we write A∗ := {A∗σ }σ∈Σ . Now we move on to the concept of minimality of a normal extension. By [12, Remark 8] and the results contained in [18, p. 423], we may formulate the ensuing lemma. Lemma 3.1. Let M = {Mσ }σ∈Σ be a family of spectrally commuting normal operators in K. Then (i) D∞ (M ) = D∞ (M ∗ ) = D∞ (M ∪ M ∗ ), (ii) Mσ Mτ f = Mτ Mσ f , Mσ Mτ∗ f = Mτ∗ Mσ f and Mσ∗ Mτ∗ f = Mτ∗ Mσ∗ f for all f ∈ D∞ (M ) and σ, τ ∈ Σ, (iii) {Mσ |D∞ (M ) }σ∈Σ ⊆ L# (D∞ (M )), ∗ (iv) (Mσ |D∞ (M ) )# = M σ |D∞ (M ) for all σ ∈ Σ, (v) if E is a subset of σ∈Σ D(Mσ ) such that Mσ (E) ⊆ E for all σ ∈ Σ, then E ⊆ D∞ (M ∪ M ∗ ). Sketch of the proof. Note first that if N1 , . . . , Nn are spectrally commuting normal operators in K and E is the joint spectral measure2 of (N1 , . . . , Nn ), then D(Nn · · · N1 ) = D(Nn∗ · · · N1∗ ) ⎫ ⎧ n k ⎬ ⎨ |zj |2 E(dz)f, f < ∞ , = f ∈ K: ⎭ ⎩ Cn k=1 j=1 ⎫ ⎪ Ni Nj , Nj Ni ⊆ zi zj E(dz),⎪ ⎪ ⎪ ⎪ ⎪ n ⎪ C ⎪ ⎪ ⎪ ⎬ ∗ ∗ Ni Nj , Nj Ni ⊆ zi z¯j E(dz), ⎪ ⎪ Cn ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ∗ ∗ ∗ Ni Nj , Nj Ni ⊆ z¯i z¯j E(dz),⎪ ⎪ ⎪ ⎭

(3.1)

(3.2)

Cn

for all i, j ∈ {1, . . . , n}. It follows from (3.1) that D∞ (M ) = D∞ (M ∗ ), which immediately implies that (i) holds. This combined with (3.2) gives (ii). The conditions (iii)–(v) follow from (i) and (ii). 1 2

We say that normal operators spectrally commute if their spectral measures commute. We refer the reader to [1] for more information on joint spectral measures.

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It is now clear from Lemma 3.1 that M α f , M ∗α f and M ∗α M β f make #α β M f for all f ∈ D∞ (M ) and α, β ∈ NΣ , sense and that M ∗α M β f = M #

:= {(Mσ |D∞ (M ) )# }σ∈Σ . In view of := {Mσ |D∞ (M ) }σ∈Σ and M where M Lemma 3.1, any jointly subnormal family {Aσ }σ∈Σ ⊆ L(E) is always commutative. We say that a normal extension M = {Mσ }σ∈Σ of a jointly subnormal family A = {Aσ }σ∈Σ ⊆ L(E) is minimal of cyclic type if FM (E) := lin{M ∗α f : f ∈ E, α ∈ NΣ }

(3.3)

is a core of Mσ for every σ ∈ Σ (according to Lemma 3.1, this definition is correct). We now prove that each jointly subnormal family possessing a rich set of analytic vectors has a minimal normal extension of cyclic type which can be built up from a normal extension. Theorem 3.2. Let A = {Aσ }σ∈Σ ⊆ L(E) be a jointly subnormal family of operators such that E = A (Aσ ) for all σ ∈ Σ and let M = {Mσ }σ∈Σ be a normal extension of A. Then for every σ ∈ Σ, the closed linear space FM (E) reduces Mσ to Nσ := Mσ |FM (E) , i.e., Nσ = Mσ |FM (E) . Moreover, N = {Nσ }σ∈Σ is a minimal normal extension of A of cyclic type. σ }σ∈Σ ⊆ Proof. Owing to Lemma 3.1, FM (E) ⊆ D∞ (M ∪ M ∗ ) and {M # σ ) for L (FM (E)), where Mσ := Mσ |FM (E) . By our assumption, E ⊆ A (M #α (E), all σ ∈ Σ. Fix σ ∈ Σ. Since FM (E) is the linear span of α∈NΣ M #α σ and M σ# (use again Lemma 3.1), and the operators M commute with M we deduce from [12, Theorem 1] that Nσ is a normal operator. Hence, by [15, Corollary 1], the closed linear space FM (E) reduces Mσ to Nσ . This implies that Nσ∗ ⊆ Mσ∗ . As σ is arbitrary, we deduce that FM (E) = FN (E). As a consequence, we see that FN (E) is a core of Nσ for every σ ∈ Σ. The normal operators Nσ , σ ∈ Σ, spectrally commute as restrictions of spectrally commuting normal operators Mσ , σ ∈ Σ. This completes the proof. In our paper we frequently consider unbounded subnormal operators A in a Hilbert space H such that D(A) = A (A). The reader should be aware of the fact that such operators can never be closed (cf. [16, Theorem 7]). Let A = {Aσ }σ∈Σ ⊆ L(E) be a jointly subnormal family, and let M = {Mσ }σ∈Σ and N = {Nσ }σ∈Σ be its normal extensions acting in Hilbert spaces K and L respectively. We say that M and N are H-unitarily equivalent if there exists a unitary operator U ∈ B(K, L) such that U |H = IH and U Mσ = Nσ U for all σ ∈ Σ. We show that minimal normal extensions of cyclic type of a jointly subnormal family are determined up to H-unitary equivalence (this was suggested in [12, Remark 8]). Proposition 3.3. Let A = {Aσ }σ∈Σ ⊆ L(E) be a jointly subnormal family of operators, and let M and N be minimal normal extensions of cyclic type of A. Then M and N are H-unitarily equivalent.

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Proof. Let K and L be Hilbert spaces in which M = {Mσ }σ∈Σ and N = {Nσ }σ∈Σ act. In view of Lemma 3.1, for all α, β ∈ NΣ and f, g ∈ E we have M ∗α f, M ∗β g = M β M ∗α f, g = M ∗α M β f, g = M β f, M α g = Aβ f, Aα g = N β f, N α g = · · · = N ∗α f, N ∗β g . Thus, by the cyclic minimality of M and N , we deduce that there exists a unitary operator U ∈ B(K, L) such that U (FM (E)) = FN (E) and U (M ∗α f ) = N ∗α f for all α ∈ NΣ and f ∈ E. Substituting α = 0, we deduce that U |H = IH . By Lemma 3.1, we have U Mσ (M ∗α f ) = U (M ∗α Mσ f ) = U (M ∗α Aσ f ) = N ∗α Aσ f = N ∗α Nσ f = Nσ N ∗α f = Nσ U (M ∗α f ), α ∈ NΣ , σ ∈ Σ, f ∈ E. This, when combined with U (FM (E)) = FN (E), implies that U (Mσ |FM (E) ) = (Nσ |FN (E) )U . Taking closures and using cyclic minimality, we deduce that U Mσ = Nσ U for all σ ∈ Σ. This completes the proof.

4. Lifting Criteria Let A = {Aσ }σ∈Σ and B = {Bσ }σ∈Σ be families of operators acting in Hilbert spaces H1 and H2 , respectively. If T ∈ B(H1 , H2 ) intertwines the families A and B, i.e. T Aσ ⊆ Bσ T for all σ ∈ Σ, then we write T A ⊆ BT . The class of all operators T ∈ B(H1 , H2 ) intertwining A and B will be denoted by I(A, B). Consider the following general situation. H1 and H2 are closed linear subspaces of Hilbert spaces K1 and K2 , respectively; E1 and E2 are dense linear subspaces of H1 and H2 , respectively; A = {Aσ }σ∈Σ ⊆ L(E1 ) and B = {Bσ }σ∈Σ ⊆ L(E2 ) are jointly subnormal families of operators; M = {Mσ }σ∈Σ and

(4.1)

N = {Nσ }σ∈Σ are their minimal normal extensions of cyclic type acting in K1 and K2 , respectively. Definition 4.1. We say that an operator T ∈ B(H1 , H2 ) lifts to I(M , N ) if there exists an operator R ∈ I(M , N ) such that T ⊆ R; R is called a lift of T to I(M , N ). We first state a Bram type criterion for the existence of a lift, which is a multioperator version of [11, Proposition 4.1] for intertwining operators. Proposition 4.2. Suppose that (4.1) holds and T ∈ B(H1 , H2 ) is an operator such that T E1 ⊆ E2 . Then the following conditions are equivalent: (i) there exists a (unique) lift T of T to I(M , N ), (ii) there exists a real number c ≥ 0 such that B α T fβ , B β T fα ≤ c Aα fβ , Aβ fα (4.2) α,β

α,β

for all finite families {fα }α∈NΣ ⊆ E1 .

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Moreover, if (i) holds, then (iii) T A ⊆ BT , (iv) T2 = min{c ≥ 0 : c satisfies (ii)}. Proposition 4.2 can be proved in a similar manner as [11, Proposition 4.1]: first, we may formulate a multioperator version of [11, Proposition 3.1] for intertwining operators, and then, with its help, adapt the proof of [11, Proposition 4.1] to the present context (use Lemma 3.1 to make the reasoning applicable). Throughout what follows, a unique lift of T is denoted by T. Now we are ready to formulate the main result of this section. Its proof resembles to some extent the proof of [6, Theorem 3.2]. To justify the implication (ivM )⇒(iiiM ) below, the pivotal part of the proof, we have to adapt the original idea from [6] to the context of unbounded operators, which seems to be the main difficulty in the whole procedure. Theorem 4.3. Suppose that (4.1) holds. Assume that E1 = A (Aσ ) and E2 = A (Bσ ) for all σ ∈ Σ. Let M be a linear subspace of E1 such that E1 = lin{Aα f : f ∈ M, α ∈ NΣ },

(4.3)

and let T ∈ B(H1 , H2 ) be an operator such that T E1 ⊆ E2 . Then the following conditions are equivalent: (i) T lifts to I(M , N ), (ii) there exists a real number c ≥ 0 such that (4.2) holds for all finite families {fα }α∈NΣ ⊆ E1 , (iii) there exists a real number c ≥ 0 such that B β+γ T fα,β , B α+δ T fγ,δ ≤ c Aβ+γ fα,β , Aα+δ fγ,δ (4.4) α,β,γ,δ

α,β,γ,δ

for all finite families {fα,β }α,β∈NΣ ⊆ E1 , (iiiM ) T A ⊆ BT and there exists a real number c ≥ 0 such that (4.4) holds for all finite families {fα,β }α,β∈NΣ ⊆ M, (iv) there exists a real number c ≥ 0 such that B β+γ T f, B α+δ T f λα,β λγ,δ ≤ c Aβ+γ f, Aα+δ f λα,β λγ,δ α,β,γ,δ

α,β,γ,δ

(4.5) for every f ∈ E1 and for all finite families {λα,β }α,β∈NΣ ⊆ C, (ivM ) T A ⊆ BT and there exists a real number c ≥ 0 such that (4.5) holds for every f ∈ M and for all finite families {λα,β }α,β∈NΣ ⊆ C. Moreover, if (i) holds, then (v) T A ⊆ BT , (vi) the smallest real number c ≥ 0 satisfying the condition (ii) (resp. (iii), (iiiM ), (iv), (ivM )), is equal to T2 . Proof. The equivalence of conditions (i) and (ii) as well as the implication (i)⇒(v) follow from Proposition 4.2.

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(ii)⇒(iii) From the implications (ii)⇒(i) and (i)⇒(v), we infer that T A ⊆ BT . Let c ≥ 0 be as in (ii). Take a finite family {fα,β }α,β∈NΣ ⊆ E1 . We define a finite family {gα }α∈NΣ ⊆ E1 by gα = Aβ fα,β , α ∈ NΣ . (4.6) β

Then T A ⊆ BT implies that

BαT B T gβ , B T gα = α

β

α,β

β

A fβ,γ

,B T

γ

α,β

=

γ

B

γ+α

δ

A fα,δ

δ

T fβ,γ , B

β+δ

T fα,δ .

(4.7)

α,β,γ,δ

Clearly, we have Aα gβ , Aβ gα = Aγ+α fβ,γ , Aβ+δ fα,δ . α,β

(4.8)

α,β,γ,δ

That (4.4) is satisfied with the constant c follows from (4.2), (4.7) and (4.8). (iii)⇒(ii) Let c ≥ 0 be as in (iii). Take a finite family {fα }α∈NΣ ⊆ E1 . For α, β ∈ NΣ , set gα,β = fα if β = 0 and gα,β = 0 otherwise. Then, we have B β+γ T gα,β , B α+δ T gγ,δ = B γ T fα , B α T fγ , α,γ

α,β,γ,δ

A

β+γ

gα,β , A

α+δ

gγ,δ =

Aγ fα , Aα fγ .

α,γ

α,β,γ,δ

This and (4.4) imply (4.2) with the same constant c. Consider the commutative ∗-semigroup ΩΣ = NΣ × NΣ (see Section 2). Given a real number c ≥ 0, we define the form ϕc over (ΩΣ , M) by ϕc ((α, β); f, g) = cAα f, Aβ g − B α T f, B β T g ,

f, g ∈ M, (α, β) ∈ ΩΣ .

(ivM )⇒(iiiM ) Let c ≥ 0 be as in (ivM ). It follows from (ivM ) that the form ϕc is weakly positive definite. Consider the set G = {(δσ , 0) : σ ∈ Σ} of ∗-generators of the ∗-semigroup ΩΣ . We show that Aϕc (δσ , 0) = M

for every σ ∈ Σ.

(4.9)

Indeed, if σ ∈ Σ and f ∈ M ⊆ A (Aσ ), then there exists a real number t = t(f ) > 0 such that ∞

Anσ f

n=0

tn < ∞. n!

(4.10)

Since ϕc is weakly positive definite, we get 0 ≤ ϕc (n · ((δσ , 0)∗ + (δσ , 0)); f, f ) = ϕc (n · (δσ , δσ ); f, f ) = cAnσ f 2 − Bσn T f 2 ≤ cAnσ f 2 ,

n ∈ N, σ ∈ Σ.

This together with (4.10) yields ∞ n=0

ϕc (n · ((δσ , 0)∗ + (δσ , 0)); f, f )

tn < ∞. n!

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Hence, f ∈ Aϕc (δσ , 0) for every σ ∈ Σ, which leads to (4.9). A combination of (2.1), (4.9) and Lemma 2.1 implies that the form ϕc is positive definite. As a consequence, (iiiM ) holds. The implication (iiiM )⇒(ivM ) follows directly from the fact that every positive definite form is weakly positive definite. Arguing as above with M = E1 , we see that (iii) and (iv) are equivalent (in the proofs of (iiiM )⇔(ivM ) and (iii)⇔(iv), we do not use the inclusion T A ⊆ BT ). (iiiM )⇒(ii) Let c ≥ 0 be as in (iiiM ). By (4.3), for a finite family {gα }α∈NΣ ⊆ E1 , there exists a finite family {fα,β }α,β∈NΣ ⊆ M satisfying (4.6). Arguing as in the proof of the implication (ii)⇒(iii), we see that (4.2) holds with {gα }α∈NΣ in place of {fα }α∈NΣ (here we make of use of the assumption that T A ⊆ BT ). (iiiM ) follows from (ii), because, as shown above, (ii) implies (iii) and (v). The reader can easily convince himself that all of conditions (ii), (iii), (iiiM ), (iv) and (ivM ) are equivalent to each other with the same constants c. Thus, (vi) follows from the implication (ii)⇒(iv) of Proposition 4.2. This completes the proof.

5. A Local Lifting Theorem for Unbounded Operators Let E be a dense linear subspace of a Hilbert space H. Consider a family of commuting operators A = {Aσ }σ∈Σ ⊆ L(E). For f ∈ E and σ ∈ Σ, we define EA,f = lin{Aα f : α ∈ NΣ },

QA,f = EA,f

and

Aσ,f = Aσ |EA,f .

The family {Aσ,f }σ∈Σ will be denoted briefly by Af . Clearly, Af ⊆ L(EA,f ). Since A is a family of commuting operators, so also is Af . If A is jointly subnormal and M = {Mσ }σ∈Σ is a normal extension of A, then for every f ∈ E, we have Aσ,f ⊆ Aσ ⊆ Mσ ,

σ ∈ Σ,

so the family Af is jointly subnormal. For g ∈ D∞ (M ) and σ ∈ Σ, we put FM ,g = lin{M ∗α M β g : α, β ∈ NΣ },

QM ,g = FM ,g ,

Mσg = Mσ |FM ,g .

The definition of Mσg makes sense due to Lemma 3.1. Let us denote by M g the family {Mσg }σ∈Σ . If f ∈ E, then EA,f ⊆ FM ,f and clearly QA,f ⊆ QM ,f . Therefore, Aσ,f ⊆ Mσf ⊆ Mσ ,

σ ∈ Σ.

A relationship between families Af and M f is elucidated below. Theorem 5.1. Let M = {Mσ }σ∈Σ be a normal extension of a jointly subnormal family A = {Aσ }σ∈Σ ⊆ L(E) such that E = A (Aσ ) for all σ ∈ Σ. If f ∈ E, then (i) QM ,f reduces Mσ to Mσf for every σ ∈ Σ, (ii) M f is a minimal normal extension of Af of cyclic type.

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Proof. Since E = A (Aσ ) for every σ ∈ Σ, we see that EA,f = A (Aσ,f ) for all σ ∈ Σ. Noticing that FM (EA,f ) = FM ,f (see (3.3) for the notation), we may apply Theorem 3.2 to the jointly subnormal family Af and its normal extension M . This completes the proof. Let E1 and E2 be dense linear subspaces of Hilbert spaces H1 and H2 , respectively. Consider families of operators A = {Aσ }σ∈Σ ⊆ L(E1 ) and B = {Bσ }σ∈Σ ⊆ L(E2 ). Set S = (A, B). Take an operator T ∈ B(H1 , H2 ) such that T A ⊆ BT . Then T E1 ⊆ E2 and T (EA,f ) ⊆ EB,T f for all f ∈ E1 . Owing to the boundedness of T , this yields T (QA,f ) ⊆ QB,T f for all f ∈ E1 . Define the operator TS,f via TS,f = T |QA,f ∈ B(QA,f , QB,T f ). It follows from T A ⊆ BT that TS,f Af ⊆ B T f TS,f ,

f ∈ E1 .

(5.1)

Now we formulate a version of [6, Lemma 4.1] for families of operators. Lemma 5.2. Suppose that (4.1) holds. Assume that E1 = A (Aσ ) and E2 = A (Bσ ) for all σ ∈ Σ. Take an operator T ∈ B(H1 , H2 ) such that T A ⊆ BT and fix a vector f ∈ E1 . Then the following conditions are equivalent (with S := (A, B)): (i) TS,f lifts to I(M f , N T f ), (ii) there exists a real number c ≥ 0 such that (4.5) holds for all finite families {λα,β }α,β∈NΣ ⊆ C. 2 Moreover, if (i) holds, then T

S,f = min{c ≥ 0 : c satisfies (ii)}. Proof. In view of (5.1) and Theorem 5.1, we can apply Theorem 4.3 to M = Cf , the intertwining operator TS,f , jointly subnormal families Af and B T f , and their minimal normal extensions of cyclic type M f and N T f , respectively. The following theorem, which is the main result of the paper, generalizes a local lifting theorem (cf. [6, Theorem 4.2]) to the case of pairs of jointly subnormal families of unbounded operators. Theorem 5.3. Suppose that (4.1) holds. Assume that E1 = A (Aσ ) and E2 = A (Bσ ) for all σ ∈ Σ. Let M be a linear subspace of E1 such that E1 = lin{Aα f : f ∈ M, α ∈ NΣ } and let T ∈ B(H1 , H2 ) be an operator such that T E1 ⊆ E2 . Then the following conditions are equivalent (with S := (A, B)): (i) (ii)

T lifts to I(M , N ), T A ⊆ BT , the operator TS,f lifts to I(M f , N T f ) for every f ∈ M, and sup T

S,f < ∞.

f ∈M

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Moreover, if (i) holds, then T = sup T

S,f .

(5.2)

f ∈M

Proof. The proof of the implication (i)⇒(ii) requires more care than in the case of bounded operators. Let T be the lift of T to I(M , N ). Fix f ∈ M. By the Putnam-Fuglede theorem (cf. [10]), TM ⊆ N T implies TM ∗ ⊆ N ∗ T. These two inclusions and T ⊆ T guarantee that T|QM ,f ∈ B(QM ,f , QN ,T f ) and TS,f ⊆ T|QM ,f . Next, we show that Th ∈ D(NσT f ) for all h ∈ D(Mσf ) and σ ∈ Σ.

(5.3)

Indeed, since h ⊆ D(Mσ ) and TM ⊆ N T, we deduce that Th ∈ , we have Th = T|QM ,f h ∈ QN ,T f . This, combined with D(Nσ ). As h ∈ Q N ,T f reduces the operator Nσ to NσT f (cf. Theorem 5.1), leads the fact that Q to Th ∈ D(Nσ ) ∩ QN ,T f = D(N T f ), ∈ D(Mσf ) M ,f

σ

which proves (5.3). The condition (5.3) together with TM ⊆ N T implies that T|QM ,f M f ⊆ N T f T|QM ,f . Hence, by the uniqueness of T

S,f , we have T

= T | , which also shows that the right-hand side of (5.2) is less than M ,f S,f Q or equal to its left-hand side. The implication (ii)⇒(i) can be deduced from Lemma 5.2 and Theorem 4.3. On the way we also verify that the left-hand side of (5.2) is less than or equal to its right-hand side. This completes the proof.

6. Local Commutativity Let us concentrate on a single subnormal operator A ∈ B(H). For simplicity, we write “minimal normal extension” in place of “minimal normal extension of cyclic type”. Suppose for a moment that T ∈ B(H) is an operator that commutes with A (this requirement is necessary for T to lift to the commutant of a minimal normal extension of A). Then T QA,f ⊆ QA,T f and consequently T |QA,f A|QA,f = A|QA,T f T |QA,f for all f ∈ H. The question arises under what circumstances T QA,f ⊆ QA,f for all f ∈ H. Note that if this is the case, then T |QA,f A|QA,f = A|QA,f T |QA,f for all f ∈ H, which is referred to as the local commutativity of A and T . This question can be extended to the case in which A and T are not assumed to commute. In Proposition 6.1 we provide a complete answer to the extended question. It is worth pointing out that the operators A and T must commute if all cyclic invariant subspaces QA,f of A are invariant for T . Given an operator A ∈ B(H) and a set A of closed linear subspaces of H, we write Lat(A) for the set of all closed linear subspaces of H invariant for A and Alg(A) for the set of all operators T ∈ B(H) such that A ⊆ Lat(T ). Denote by W(A) the closure in the weak operator topology (equivalently: in the strong operator topology) of the algebra generated by {A, IH }. It is clear that W(A) ⊆ Alg(Lat(A)). In general, the reverse implication is not true

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(cf. [5], [19, Example 7]; for more information on the subject see also [3,4]). The Olin-Thomson theorem states that W(A) = Alg(Lat(A)) for any subnormal operator A ∈ B(H) (cf. [9, Theorem 3]). Proposition 6.1. Let A ∈ B(H) be a subnormal operator. If T ∈ B(H), then the following conditions are equivalent: (i) (ii)

T QA,f ⊆ QA,f for all f ∈ H, where QA,f is the smallest closed linear subspace of H containing f and invariant for A, T ∈ W(A).

Proof. (i)⇒(ii) In view of the Olin-Thomson theorem, it is enough to show that T ∈ Alg(Lat(A)). Take M ∈ Lat(A). If f ∈ M, then f ∈ QA,f ⊆ M. Hence, by (i), we have T f ∈ QA,f ⊆ M. This means that M ∈ Lat(T ). (ii)⇒(i) Take f ∈ H. Since QA,f ∈ Lat(A), we see that QA,f ∈ Lat(p(A)) for every complex polynomial p in one indeterminate, which implies that QA,f ∈ Lat(R) for every R ∈ W(A). This, together with (ii), gives (i). We conclude the paper with a simple example of an operator T which lifts to the commutant of a minimal normal extension of a subnormal operator A, and which has the property that not all cyclic invariant subspaces of A are invariant for T . This shows that a (global) lift of a member T of the commutant of a subnormal operator A may exist though the local commutativity of A and T does not hold. In fact, it may even happen that for some vector f the linear spaces QA,f and T QA,f are orthogonal. Example 6.2. Given a bounded Borel function ψ on T := {z ∈ C : |z| = 1}, we denote by Mψ the operator of multiplication by ψ on L2 (T). Let ξ be the identity function on T. Set T = Mξ |H 2 and A = T 2 , where H 2 stands for the Hardy space regarded as a closed linear subspace of L2 (T). The operator A is subnormal (as an isometric operator) and T A = AT . Observe that if f = 1, then An f = ξ 2n for all n ∈ N, which implies that QA,f = n∈N C · ξ 2n . Since T f = ξ, we deduce that T QA,f = n∈N C·ξ 2n+1 . Therefore, the linear spaces 2 QA,f and T QA,f are orthogonal. Note that the unitary operator M2ξ is a 2minimal normal extension of the isometry A (because n∈N Mξ∗n 2 (H ) = L (T), ¯ 2n and M ∗n ¯ 2n−1 , n ≥ 1). which follows from the equalities Mξ∗n 2 f = ξ ξ2 ξ = ξ Clearly, Mξ is the lift of T to the commutant of Mξ2 . Acknowledgements The authors would like to thank the referee for a careful reading of the manuscript and for many suggestions that helped to improve the final version of the paper. Open Access. This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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References [1] Birman, M.Sh., Solomjak, M.Z.: Spectral Theory of Selfadjoint Operators in Hilbert Space. D. Reidel Publishing Co., Dordrecht (1987) [2] Bram, J.: Subnormal operators. Duke Math. J. 22, 75–94 (1955) [3] Conway, J.B.: The theory of subnormal operators, Mathematical Surveys and Monographs, Providence, Rhode Island (1991) [4] Conway, J.B.: A course in operator theory, Graduate Studies in Mathematics 21, American Mathematical Society, Providence, Rhode Island (2000) [5] Deddens, J.A., Fillmore, P.A.: Reﬂexive linear transformations. Linear Algebra Appl. 10, 89–93 (1975) [6] Majdak, W., Sebesty´en, Z., Stochel, J., Thomson, J.E.: A local lifting theorem for subnormal operators. Proc. Amer. Math. Soc. 134, 1687–1699 (2006) [7] Maserick, P.H.: Spectral theory of operator-valued transformations. J. Math. Anal. Appl. 41, 497–507 (1973) [8] Nelson, E.: Analytic vectors. Ann. Math. 70, 572–615 (1959) [9] Olin, R.F., Thomson, J.E.: Algebras of subnormal operators. J. Funct. Anal. 37, 271–301 (1980) [10] Putnam, C.R.: On normal operators in Hilbert space. Amer. J. Math. 73, 357– 362 (1951) [11] Stochel, J.: Lifting strong commutants of unbounded subnormal operators. Integr. Equ. Oper. Theory 43, 189–214 (2002) [12] Stochel, J., Szafraniec, F.H.: On normal extensions of unbounded operators. I. J. Oper. Theory 14, 31–55 (1985) [13] Stochel, J., Szafraniec, F.H.: On normal extensions of unbounded operators. II. Acta Sci. Math. (Szeged) 53, 153–177 (1989) [14] Stochel, J., Szafraniec, F.H.: On normal extensions of unbounded operators. III. Spectral properties. Publ. Res. Inst. Math. Sci. 25, 105–139 (1989) [15] Stochel, J., Szafraniec, F.H.: The normal part of an unbounded operator. Nederl. Akad. Wetensch. Indag. Math. 51, 495–503 (1989) [16] Stochel, J., Szafraniec, F.H.: C∞ -vectors and boundedness. Ann. Polon. Math. 66, 223–238 (1997) [17] Stochel, J., Szafraniec, F.H.: The complex moment problem and subnormality: a polar decomposition approach. J. Funct. Anal. 159, 432–491 (1998) [18] Stochel, J., Szafraniec, F.H.: Domination of unbounded operators and commutativity. J. Math. Soc. Jpn. 55(2), 405–437 (2003) [19] Wogen, W.R.: Some counterexamples in nonselfadjoint algebras. Ann. Math. 126, 415–427 (1987) Witold Majdak Faculty of Applied Mathematics AGH Science and Technology University Al. Mickiewicza 30 30059 Krak´ ow, Poland e-mail: [email protected]

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Jan Stochel (B) Instytut Matematyki Uniwersytet Jagiello´ nski ul. L ojasiewicza 6 30348 Krak´ ow, Poland e-mail: [email protected] Received: May 27, 2010. Revised: October 8, 2010.

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Integr. Equ. Oper. Theory 69 (2011), 247–268 DOI 10.1007/s00020-010-1840-5 Published online November 11, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Compact Differences of Composition Operators in Several Variables Katherine Heller, Barbara D. MacCluer and Rachel J. Weir Abstract. When ϕ and ψ are linear–fractional self-maps of the unit ball BN in CN , N ≥ 1, we show that the difference Cϕ − Cψ cannot be nontrivially compact on either the Hardy space H 2 (BN ) or any weighted Bergman space A2α (BN ). Our arguments emphasize geometrical properties of the inducing maps ϕ and ψ.

1. Introduction For a domain Ω in CN , where N ≥ 1, and an analytic map ϕ : Ω → Ω, we deﬁne the composition operator Cϕ by Cϕ (f ) = f ◦ ϕ, where f is analytic in Ω. In the case that Ω = D, the unit disk in C, every composition operator acts boundedly on the Hardy space ⎫ ⎧ 2π ⎬ ⎨ dθ <∞ |f (reiθ )|2 H 2 (D) = f analytic in D : f 2H 2 ≡ lim ⎭ ⎩ 2π r→1− 0

and weighted Bergman spaces ⎧ ⎨ A2α (D) = f analytic in D : ⎩ f 2α ≡ (α + 1)

D

⎫ ⎬ |f (z)|2 (1 − |z|2 )α dA(z) < ∞ , ⎭

where α > −1 and dA is normalized area measure. If D is replaced by the unit ball BN in CN , N > 1, it is no longer the case that every composition operator is bounded on the Hardy or weighted Bergman space of the ball (these spaces are deﬁned in Sect. 3). However for a large class of maps ϕ, R. J. Weir would like to thank the Allegheny College Academic Support Committee for funding provided during the development of this paper.

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including the rich class of linear–fractional maps, boundedness does continue to hold. Many properties of composition operators have been studied over the past four decades; the monographs [4] and [14] give an overview of the work before the mid-1990s. Recently there has been considerable interest in studying algebras of composition operators, often modulo the ideal of compact operators (see, for example, [6–10]). In this direction, the question of when a difference Cϕ − Cψ is compact naturally arises. The main result of this paper shows that if ϕ and ψ are linear– fractional self-maps of BN , then Cϕ − Cψ cannot be non-trivially compact; i.e. if the difference is compact, either Cϕ and Cψ are individually compact (this happens precisely when ϕ∞ < 1 and ψ∞ < 1), or ϕ = ψ. While our focus is on the several variable case, we begin with a simpliﬁed proof of this result in one variable. The fact that a difference of linear–fractional composition operators cannot be non-trivially compact on H 2 (D) or A2α (D) was ﬁrst obtained by Bourdon [2] and Moorhouse [12] as a consequence of results on the compactness of a difference of more general composition operators in one variable. Our approach here is self-contained, and takes a geometric perspective, which will allow us to generalize our arguments to several variables. The analogy between the one and several variable arguments is not perfect, owing to a number of phenomena that are present when N > 1 but do not occur when N = 1. Nevertheless, our geometric approach when N = 1 leads us to a tractable way to proceed when N > 1, and highlights the new phenomena which must be addressed. Since our arguments are essentially the same for either the Hardy or weighted Bergman spaces, in what follows we will let H denote any of these spaces. Our starting point, in either the disk or the ball, will be the following necessary condition for compactness of Cϕ − Cψ . Theorem 1 [11,12]. Suppose ϕ, ψ are holomorphic self-maps of D (respectively, BN ) and suppose that there exists a sequence of points zn tending to the boundary of D (BN ) along which

1 − |zn |2 1 − |zn |2 + ρ(ϕ(zn ), ψ(zn )) (1) 1 − |ϕ(zn )|2 1 − |ψ(zn )|2 does not converge to zero, where ρ(ϕ(zn ), ψ(zn )) is deﬁned by 2

1 − (ρ(ϕ(zn ), ψ(zn ))) =

(1 − |ϕ(zn )|2 )(1 − |ψ(zn )|2 ) . |1 − ϕ(zn ), ψ(zn ) |2

(2)

Then Cϕ − Cψ is not compact on H. For z and w in D or BN , the quantity ρ(z, w) will be referred to as pseudohyperbolic distance between z and w, so that the ﬁrst factor in is the pseudohyperbolic distance between ϕ(zn ) and ψ(zn ). In the disk, pseudohyperbolic distance has the simpler expression z−w . ρD (z, w) = 1 − zw

the (1) the

(3)

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2. Results in One Variable Throughout this section H denotes either the Hardy space H 2 (D) or a weighted Bergman space A2α (D), as deﬁned in the previous section. Theorem 2. Suppose that ϕ and ψ are linear–fractional self-maps of D. If the difference Cϕ − Cψ is compact on H, then either ϕ = ψ, or both ϕ∞ and ψ∞ are strictly less than 1, so that Cϕ and Cψ are individually compact. The key step in our proof of Theorem 2 is contained in the following result. Theorem 3. Suppose ϕ and ψ are non-automorphism linear–fractional selfmaps of D with ϕ(ζ) = ψ(ζ) ∈ ∂D and ϕ (ζ) = ψ (ζ) for some ζ ∈ ∂D. If ϕ = ψ then Cϕ − Cψ is not compact on H. Proof. By pre- and post-composing with rotations, we may assume without loss of generality that ζ = 1 and ϕ(ζ) = ψ(ζ) = 1. Since ϕ and ψ are linear– fractional, we may also assume without loss of generality that ϕ(D) ⊆ ψ(D), so that τ1 ≡ ψ −1 ◦ ϕ is a well-deﬁned linear–fractional self-map of D. Note that τ1 (1) = 1 and τ1 (1) = 1. Since ϕ = ψ, τ1 is not the identity. Thus τ1 is conjugate via the Cayley transform 1+z C(z) = i 1−z to a translation w → w + b of the upper half-plane H = {w : Im w > 0} for some b = 0 with Im b ≥ 0. Moreover, it is easy to see that ψ ◦ τ 1 = τ2 ◦ ψ for some linear–fractional τ2 which is also conjugate to a translation in the upper half-plane. Specifically, if τ1 is conjugate to translation by b, then τ2 is conjugate to translation by c = b/|ψ (1)|; see Lemma 5 in [9]. Since b = 0, so also c = 0. For any positive number k, the line {Im w = k} corresponds under the Cayley transform to Ek ≡ {z : |1 − z|2 = k1 (1 − |z|2 )}, which is an internally tangent circle in D passing through 1. The radius of this circle is equal to (k + 1)−1 . By choosing k sufﬁciently large, this circle will be contained in ψ(D) ∪ {1}. Fix such a k and choose points wn on {Im w = k} with wn → ∞. The corresponding points vn = C −1 (wn ) in the disk tend to 1 along the circle Ek , and each vn is the image under ψ of some zn belonging to the internally tangent circle ψ −1 (Ek ) = Ek . Notice that zn → 1 as n → ∞. Next we compute the pseudohyperbolic distance between ϕ(zn ) and ψ(zn ). To simplify the computations, we deﬁne the pseudohyperbolic distance in the upper half-plane H by ρH (u, v) = ρD (C −1 u, C −1 v) for u and v in H. Using this definition and (3) it is straightforward to see that u − v . ρH (u, v) = u − v

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Since ϕ = ψ ◦ τ1 = τ2 ◦ ψ, ρD (ϕ(zn ), ψ(zn )) = ρD (τ2 (ψ(zn )), ψ(zn )) = ρD (τ2 (vn ), vn ) = ρH (C(τ2 (vn )), C(vn )) = ρH (Cτ2 C −1 (wn ), CC −1 (wn )) = ρH (wn + c, wn ) c c . = = 2iIm wn + c 2ik + c Thus for all n, the pseudohyperbolic distance between ϕ(zn ) and ψ(zn ) is a positive constant. Turning to the second factor in Eq. (1), we have k (|1 − zn |2 ) 1 − |zn |2 = 2 1 − |ψ(zn )| k(|1 − ψ(zn )|2 ) by the geometry of the sequence {zn } already noted. Thus since ψ is differentiable at 1 with ψ (1) = 0 and ψ(1) = 1, 1 − |zn |2 k = = 0. n→∞ 1 − |ψ(zn )|2 k|ψ (1)|2 lim

Thus we have shown that ρ(ϕ(zn ), ψ(zn ))

1 − |zn |2 1 − |ψ(zn )|2

has a positive limit as n → 1 (where zn → 1). Theorem 1 guarantees that Cϕ − Cψ is not compact on H. Proof of Theorem 2. If either ϕ∞ < 1 or ψ∞ < 1, then the compactness of the difference Cϕ − Cψ implies the compactness of each operator individually. Thus we may assume ϕ∞ = ψ∞ = 1. Suppose ϕ(ζ) = η where ζ, η are in ∂D. Since ϕ and ψ are linear fractional, both ϕ (ζ) and ψ (ζ) exist and are non-zero. If either ϕ(ζ) = ψ(ζ) or ϕ (ζ) = ψ (ζ), then by Theorem 9.16 of [4] the essential norm of Cϕ − Cψ satisﬁes Cϕ − Cψ 2e ≥ |ϕ (ζ)|β for some positive number β depending on the particular choice of H in question; when H = H 2 (D) we may take β = 1, and when H = A2α (D), β = α + 2. This gives a positive lower bound on the essential norm of the difference, so that if the difference is compact we must have ϕ(ζ) = ψ(ζ) and ϕ (ζ) = ψ (ζ). Note that this argument also shows that if Cϕ − Cψ is compact but non-zero, neither ϕ nor ψ can be an automorphism. An appeal to Theorem 3 ﬁnishes the proof.

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3. Results in Several Variables In this section H will denote either the Hardy space H 2 (BN ) or a weighted Bergman space A2α (BN ), where BN is the ball ⎫ ⎧ N ⎬ ⎨ |zj |2 < 1 (z1 , z2 , . . . , zN ) ∈ CN : ⎭ ⎩ j=1

in CN , N > 1. These Hilbert spaces are deﬁned by ⎧ ⎫ ⎨ ⎬ H 2 (BN ) = f analytic on BN : f 2 ≡ sup |f (rζ)|2 dσ(ζ) < ∞ , ⎩ ⎭ 0
where σ is normalized Lebesgue surface area measure on ∂BN , and for α > −1, ⎧ ⎫ ⎨ ⎬ A2α (BN ) = f analytic on BN : f 2α ≡ |f (z)|2 wα (z)dν(z) < ∞ , ⎩ ⎭ BN

where wα (z) =

Γ(N + α + 1) (1 − |z|2 )α , Γ(N + 1)Γ(α + 1)

and ν is normalized Lebesgue volume measure on BN . In H 2 (BN ) the reproducing kernel for the bounded linear functional of evaluation at w ∈ BN is 1 (4) Kw (z) = (1 − z, w )N which has norm (1 − |w|2 )(−N/2) . The reproducing kernel for evaluation at w in A2α (BN ) is 1 Kw (z) = . (5) (1 − z, w )N +1+α By a linear–fractional map of BN we mean a map that is analytic in BN and of the form Az + B , (6) ϕ(z) = z, C + d where A is an N × N matrix, B and C are N × 1 column vectors, d is a complex scalar, and ·, · is the usual inner product on CN . If ϕ maps BN into itself, then necessarily |d| > |C|, so that in particular d = 0, and ϕ is analytic in a neighborhood of BN . When ϕ is a linear–fractional self-map of BN , Cϕ is bounded on H 2 (BN ) and A2α (BN ) for α > −1 [3]. Moreover, every automorphism of BN is linear–fractional ([13], Theorem 2.2.5). An important geometric property of linear–fractional maps is that they take aﬃne sets into aﬃne sets ([3], Theorem 7). By an aﬃne set in CN we mean the translate of a complex subspace; the dimension of the aﬃne set is the dimension of the subspace. An “aﬃne subset of BN ” is the intersection of BN with an aﬃne set in CN .

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The goal of this section is to obtain the following result. Theorem 4. If ϕ and ψ are linear–fractional self maps of BN with Cϕ − Cψ compact on H, then either ϕ = ψ, or ϕ∞ and ψ∞ are both strictly less than 1, so that Cϕ and Cψ are compact on H. A special case of Theorem 4 appears in [11], and the work in [11] has been extended in [5]. Our techiques here are different, and rely on an extension to several variables of the key geometric arguments of the last section, which we turn to next. The Cayley upper half space HN is deﬁned by

HN = (w1 , w ) : Im w1 > |w |2 . where w = (w2 , . . . , wN ) and |w |2 = |w2 |2 + · · · + |wN |2 . Its boundary is of course

(w1 , w ) : Im w1 = |w |2 . Let e1 = (1, 0, . . . , 0) = (1, 0 ). The (linear–fractional) Cayley transform e1 + z C(z) = i 1 − z1 is a biholomorphic map of the ball BN onto HN ; its inverse is

w1 − i 2w , C −1 (w) = . w1 + i w1 + i If b = (b1 , b ) ∈ CN , an H-translation is a map of the form hb (w1 , w ) = (w1 + 2iw , b + b1 , w + b ).

(7)

If Im b1 ≥ |b |2 it maps HN into itself. It is an automorphism of HN if Im b1 = |b |2 . The following two facts, which generalize results we used in the previous section, are easily checked: • An H-translation hb maps the set Γk ≡ {(w1 , w ) : Im w1 − |w |2 = k} ˜ where k˜ = into the corresponding set {(w1 , w ) : Im w1 − |w |2 = k} k + Im b1 − |b |2 . • For any k > 0, 1 −1 2 2 1 − |z| C (Γk ) = E(k, e1 ) ≡ z ∈ BN : |1 − z1 | = . k The set E(k, e1 ) is an internally tangent ellipsoid at e1 = (1, 0 ); a computation shows that E(k, e1 ) consists of the points (z1 , z ) satisfying 2

2 1 z1 − k + 1 |z |2 = . 1 + k 1+k 1+k In particular, for t real, points of the form (t + i(k + |w |2 ), w ) in Γk pull back under C −1 to points on the ellipsoid E(k, e1 ). For ﬁxed w , these pull-back points tend to e1 as t → ∞, and for ﬁxed t, they tend to e1 as |w | → ∞. Recall that the pseudohyperbolic metric ρBN (·, ·) on BN is deﬁned by 1 − ρBN (z, w)2 =

(1 − |z|2 )(1 − |w|2 ) . |1 − z, w |2

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For points v, u in HN , write ρH (v, u) for ρBN (C −1 v, C −1 u); we will call this the pseudohyperbolic metric on HN . Since the pseudohyperbolic metric ρBN is easily seen to be automorphism invariant, it follows that for any automorphism Λ of HN , ρH (Λv, Λu) = ρH (v, u). In the next theorem, we will use this observation with Λ an automorphic H-translation. By a parabolic linear–fractional map in BN ﬁxing e1 we mean a linear–fractional map τ of BN into BN with τ (e1 ) = e1 and D1 τ1 (e1 ) = 1, but ﬁxing no other point in BN . By [1] any parabolic linear–fractional self-map ϕ of BN that ﬁxes e1 is conjugate to a self-map of HN of the form Φ(w1 , w ) = (w1 + 2iw , δ + b, Aw + γ) (where δ and γ are in CN −1 , b ∈ C, and certain conditions hold, including |A| ≤ 1). Note that the H-translations of Eq. (7) are a special case of this family of maps. Conjugating Φ by the Cayley transform C we see that the ﬁrst coordinate function of C −1 ΦC is (2i − b)z1 − 2z , δ + b . (8) −bz1 − 2z , δ + 2i + b We will need this explicit formula in the proof of the next result. Theorem 5. Suppose ϕ and ψ are parabolic linear–fractional self-maps of the ball ﬁxing e1 , so that D1 ϕ1 (e1 ) = 1 = D1 ψ1 (e1 ). If ϕ = ψ, then Cϕ − Cψ is not compact on H. Proof. We will show that for distinct maps ϕ and ψ as in the hypothesis, there exists a sequence of points {z (n) } in BN such that (a) z (n) → e1 as n → ∞. (b) For all n, ρBN (ϕ(z (n) ), ψ(z (n) )) has a strictly positive constant value. (c) (1 − |ϕ(z (n) )|2 )/(1 − |z (n) |2 ) has ﬁnite positive limit as n → ∞. An appeal to Theorem 1 will then complete the proof. We will use the corresponding upper case letters for a self-map of BN conjugated to HN , so that Φ = CϕC −1 and Ψ = CψC −1 . These maps have the forms Φ(w1 , w ) = (w1 + 2iw , δ1 + b1 , A1 w + γ1 ) and Ψ(w1 , w ) = (w1 + 2iw , δ2 + b2 , A2 w + γ2 ) for some δi and γi in CN −1 , scalars bi and (N − 1) × (N − 1) matrices Ai , i = 1, 2. (n) Fix a sequence of points {w(n) } = {(w1 , c)} in HN , where c is a conN −1 stant in C to be determined, satisfying (n)

(i) Im w1 − |c|2 = k (n) (ii) C −1 (w1 , c) → e1

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where k > 0 is ﬁxed but arbitrary. By (i), the points C −1 (w(n) ) lie on the ellipsoid E(k, e1 ). We can ensure that condition (ii) holds by requiring that (n) Rew1 → ∞. Let (n) (n) P1 = Φ w(n) = w1 + 2ic, δ1 + b1 , A1 c + γ1 and (n)

P2

(n) = Ψ w(n) = w1 + 2ic, δ2 + b2 , A2 c + γ2 .

Since the pseudohyperbolic metric is automorphism invariant, we have (n) (n) (n) (n) ρH P1 , P2 = ρH h P1 , h P2 where h is the automorphic H-translation given by (n) h(v1 , v ) = v1 − Re w1 , v . (n)

(n)

Thus for any positive integer n, the distance ρH (P1 , P2 ) is equal to ρH ((i(k+|c|2 )+2ic, δ1 +b1 , A1 c+γ1 ), (i(k+|c|2 )+2ic, δ2 +b2 , A2 c+γ2 )). Note that this quantity is independent of the particular point w(n) in our sequence chosen to satisfy (i) and (ii), and that if ϕ = ψ (so that not all of b1 = b2 , δ1 = δ2 , A1 = A2 and γ1 = γ2 hold) we may certainly choose c so that this quantity is not 0. Thus for such a choice, condition (ii) gives the existence of a sequence of points {z (n) } in BN tending to e1 along E(k, e1 ) for which ρBN (ϕ(z (n) ), ψ(z (n) )) is a positive constant value; the z (n) ’s being just the inverse images under the Cayley transform C of our chosen points w(n) in HN . Hence conditions (a) and (b) hold. For property (c), ﬁrst note that the images under Φ of points of the form (w1 , c) with Im w1 − |c|2 = k look like (w1 + 2ic, δ1 + b1 , A1 c + γ1 ) , and for these points we see that Im (w1 +2ic, δ1 +b1 )−|A1 c+γ1 |2 = k+|c|2 +Im(2ic, δ1 +b1 ) − |A1 c+γ1 |2 , which is constant, say k . In other words, the image under ϕ of our points z (n) lie on some ellipsoid E(k , e1 ) and 2 2

1 (n) (n) 1 − ϕ1 (z ) = 1 − ϕ z . k Moreover, since the points z (n) lie on E(k, e1 ), we have 2 2

1 (n) 1 − z (n) . 1 − z1 = k

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2 2 1 − ϕ z (n) k 1 − ϕ1 z (n) 2 = 2 . k (n) 1 − z (n) 1 − z1

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Since ϕ1 is differentiable at e1 , we have a Taylor series expansion of ϕ1 in a neighborhood of e1 : ϕ1 (z) = ϕ1 (e1 ) + D1 ϕ1 (e1 )(z1 − 1) +

N

Dj ϕ1 (e1 )zj +

j=2

+

N

1 2 D ϕ1 (e1 )(z1 − 1)2 2! 1

D1 Dj ϕ1 (e1 )(z1 − 1)zj

j=2

+

N

N

Dk Dj ϕ1 (e1 )zk zj +

k,j=2;k=j

1 2 D ϕ1 (e1 )zj2 + · · · 2! j=2 j

where Dj = ∂z∂ j . Recall that by hypothesis D1 ϕ1 (e1 ) = 1. Direct computation using Eq. (8) shows that Dj ϕ1 (e1 ) = 0, for j = 2, . . . , N (this also follows more generally from the fact that e1 is a ﬁxed point of ϕ; see Lemma 6.6 in [4]) and Dk Dj ϕ1 (e1 ) = 0 for k, j = 2, . . . , N . Thus ⎡ 1 ⎣ 2 D1 ϕ1 (e1 )(z1 − 1)2 ϕ1 (z) − 1 = (z1 − 1) + 2! ⎤ N +2 D1 Dj ϕ1 (e1 )(z1 − 1)zj ⎦ + · · · , j=2

where the + · · · indicates higher order terms of the form Dν ϕ1 (e1 )(z − e1 )ν , ν! jN with ν = (j1 , j2 , . . . , jN ) a multi-index of order at least 3, Dν = D1j1 D2j2 · · · DN , jN ν j1 j2 ν! = j1 !j2 ! · · · jN !, and (z − e1 ) = (z1 − 1) z2 · · · zN . Since for z ∈ BN we have 1 − |z1 |2 1 − |z1 | |z2 |2 + · · · + |zN |2 ≤ ≤2 ≤2 |1 − z1 | |1 − z1 | |1 − z1 |

we see that 1 − ϕ1 (z) → 1 as z → e1 in BN . 1 − z1 By (9) this implies that 1 − |ϕ(z (n) )|2 1 − |z (n) |2 has a positive ﬁnite limit as n → ∞, and property (c) holds as desired.

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To prove Theorem 4 we will use the preceding result and the following qualitative generalization of Theorem 9.16 in [4], specialized to linear–fractional maps. In the statement we use the notation ψζ for the coordinate of ψ in the ζ−direction, that is ψζ (z) = ψ(z), ζ . Moreover the derivative of ψζ in the η direction, denoted Dη ψζ , is deﬁned by Dη ψζ (z) ≡ ψ (z)η, ζ . Note that when ζ = η = e1 this is just D1 ψ1 (z). For η ∈ ∂BN , write [η] for the complex line containing η and 0; that is, the one-dimensional subspace of BN consisting of all points of the form {αη : α ∈ C}. In particular, the complex line [e1 ] intersected with BN consists of all points in the ball whose last N − 1 coordinates are 0. Theorem 6. Suppose ϕ and ψ are linear–fractional self-maps of BN , and suppose ϕ(ζ) = ζ for some ζ ∈ ∂ BN . If either ψ(ζ) = ζ, or ψ(ζ) = ζ and Dζ ϕζ (ζ) = Dζ ψζ (ζ), then Cϕ − Cψ is not compact. Proof. The argument follows that of Theorem 9.16 in [4]. Without loss of generality we may assume ζ = e1 . First suppose ψ(e1 ) = e1 . If we can ﬁnd a sequence of points z (n) tending to ∂BN , so that

Kz(n) (Cϕ − Cψ )∗ Kz(n) is bounded away from 0, where Kz(n) denotes the kernel function in H at z (n) (see Eqs. (4) and (5)), then Cϕ − Cψ is not compact, since the normalized kernel functions Kz(n) /Kz(n) tend weakly to 0 as z (n) → ∂BN . Using the fact that for any bounded composition operator Cτ we have Cτ∗ (Kz ) = Kτ (z) , we see that 2

(Cϕ − Cψ )∗ (Kz ) = Kϕ(z) 2 + Kψ(z) 2 − 2Re Kϕ(z) (ψ(z)), and thus (Cϕ − Cψ )∗ (Kz /Kz )2 ≥

1 − |z|2 1 − |ϕ(z)|2

β − 2Re

Kϕ(z) (ψ(z)) Kz 2

(10)

where β = N in H 2 (BN ) and β = N +1+α in A2α (BN ). With our assumption that ψ(e1 ) = e1 , it is easy to see that as z → e1 radially, the second term on the right hand side of Eq. (10) tends to 0. By Julia–Caratheodory theory (see for example, [13], Section 8.5), the ﬁrst term tends to the positive value (D1 ϕ1 (e1 ))−β . This shows that Cϕ − Cψ is not compact if ψ(e1 ) = ϕ(e1 ). Now suppose ψ(e1 ) = ϕ(e1 ) = e1 but D1 ψ1 (e1 ) = D1 ϕ1 (e1 ). As before, if we can ﬁnd a sequence of points z (n) in BN tending to e1 along which (Cϕ − Cψ )∗ (Kz(n) /Kz(n) ) is bounded away from 0, then we can conclude that Cϕ − Cψ is not compact. (n) The sequence {z (n) } will be chosen so that z (n) = (z1 , 0 ) where (n)

(n)

|1 − z1 | |1 − z1 | = =M (n) 1 − |z (n) |2 1 − |z1 |2

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for a ﬁxed and suitably large value of M ; that is, the points z (n) will approach e1 along the boundary of a non-tangential approach region, of large aperture, in the complex line [e1 ]. To analyze the second term on the right hand side of Eq. (10), we ﬁrst consider 1 − ϕ(z), ψ(z) 1 − |ϕ(z)|2 ϕ(z) − e1 , ϕ(z) − ψ(z) = + 2 1 − |z| 1 − |z|2 1 − |z|2 e1 , ϕ(z) − ψ(z) + . 1 − |z|2

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The third term on the right hand side of Eq. (11) has modulus |ϕ1 (z) − ψ1 (z)| 1 − ψ1 (z) 1 − ϕ1 (z) |1 − z1 | = − , 1 − z1 1 − |z|2 1 − z1 1 − |z|2 and if z is chosen to approach e1 in [e1 ] along the curve |1−z1 |/(1−|z1 |2 ) = M this will tend to |D1 ψ1 (e1 ) − D1 ϕ1 (e1 )|M . Since D1 ϕ1 (e1 ) = D1 ψ1 (e1 ) by assumption, this can be made as large as desired by choosing M large. The ﬁrst term on the right hand side of Eq. (11) tends to |D1 ϕ1 (e1 )| along any sequence of points approaching e1 non-tangentially in [e1 ]. We claim that the second term in (11) tends to 0 along any such sequence. To see this, it’s enough to show that |ϕ(z) − e1 ||ϕ(z) − ψ(z)| 1 − |z|2 tends to 0 as z approaches e1 non-tangentially in [e1 ]. Since |ϕ(z) − ψ(z)| ≤ |ϕ(z) − e1 | + |e1 − ψ(z)| it sufﬁces to show that |ϕ(z) − e1 |2 1 − |z|2 and |ϕ(z) − e1 ||ψ(z) − e1 | 1 − |z|2 both tend to 0. We have 2 2 |ϕ(z) − e1 |2 |ϕ1 (z) − 1| + |ϕ (z)| |1 − z1 | = 1 − |z|2 |1 − z1 | 1 − |z|2 where ϕ (z) denotes the (N − 1)-tuple (ϕ2 (z), . . . , ϕN (z)). We’re consider(n) (n) (n) ing points z (n) = (z1 , 0 ) tending to e1 for which |1 − z1 |/(1 − |z1 |2 ) is some constant value M . Along such a sequence, |1 − ϕ1 (z)|/|1 − z1 | tends to |D1 ϕ1 (e1 )|, so that |ϕ1 (z) − 1|2 |1 − z1 | → 0. |1 − z1 | 1 − |z|2 By the Julia–Caratheodory theorem ([13], Theorem 8.5.6), we also have for 2 ≤ k ≤ N, |ϕk (z)|2 →0 |1 − z1 |

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along any non-tangential sequence approaching e1 in [e1 ]. Since a similar analysis applies to show that |ψ(z) − e1 |2 →0 1 − |z|2 along the sequences under consideration, it follows that |ϕ(z) − e1 ||ψ(z) − e1 | →0 1 − |z|2 as desired. Thus we have shown the following: If ϕ(e1 ) = ψ(e1 ) = e1 but D1 ϕ1 (e1 ) = D1 ψ1 (e1 ), then given > 0 there exists M < ∞ so that if (n) z (n) = (z1 , 0 ) approaches e1 as n → ∞, where (n)

|1 − z1 | (n)

1 − |z1 |2

= M,

then lim sup n→∞

|Kϕ(z(n) ) (ψ(z (n) ))| < . Kz(n) 2

By Eq. (10) this says Cϕ − Cψ is not compact.

Remark. It is clear that a version of Theorem 6 holds, with essentially the same proof, when ϕ and ψ are more general analytic self-maps of BN , if in the statement of the theorem the values of ϕ, ψ, Dζ ϕζ and Dζ ψζ at ζ are replaced by their radial limits there. Also a version of the result, with the hypothesis ϕ(ζ) = ζ replaced by ϕ(ζ) = η for ζ, η ∈ ∂BN can be formulated. Since we do not need these more general versions, we leave the precise statements to the interested reader; see also Theorem 2.1 in [5]. Proposition 1. Suppose that ϕ is a linear–fractional self-map of BN such that the restriction of ϕ to the the complex line [e1 ] in BN is the identity on [e1 ] ∩ ∂BN . Then ϕ(z1 , z2 , . . . , zN ) = (z1 , A z ), where z denotes (z2 , . . . , zN ) and A is an (N − 1) × (N − 1) matrix. Proof. Since ϕ is linear–fractional we have Az + B ϕ(z) = c1 z1 + c2 z2 + · · · cN zN + 1 for some N × N matrix A = (ajk ), N × 1 matrix B, and constants ck . By hypothesis we must have ak1 λ + bk = 0 for 2 ≤ k ≤ N and all λ ∈ C with |λ| = 1. Thus ak1 = bk = 0 for 2 ≤ k ≤ N . From ϕ(e1 ) = e1 and ϕ(−e1 ) = −e1 we see that b1 = c1 and a11 = 1. Using this, and the requirement that a11 λ + b1 = λ(c1 λ + 1) for |λ| = 1, we must have b1 = 0 and thus also c1 = 0. Moreover, since ϕ ﬁxes e1 , we have by Lemma 6.6 of [4] that Dk ϕ1 (e1 ) = 0

for k = 2, 3, . . . , N.

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Since ϕ(−e1 ) = −e1 , we may apply the same lemma to −ϕ(−z) to see that Dk ϕ1 (−e1 ) = 0

for k = 2, 3, . . . , N.

(13)

Eqs. (12) and (13) together say a1k = ck = 0

for k = 2, . . . , N.

This completes the proof.

To move from Theorem 5, which deals with parabolic maps, to the full result of Theorem 4, we will need the notion of the Krein adjoint of the linear–fractional map ϕ. If ϕ is as given in Eq. (6), its Krein adjoint is deﬁned to be the linear fractional map A∗ z − C σϕ (z) = . (14) z, −B + d This will be a self-map of BN whenever ϕ is, and when ϕ is an automorphism, its Krein adjoint is equal to ϕ−1 . Moreover, ϕ and σϕ have the same ﬁxed points on ∂BN ; for these and other basic facts, see [3] and [11]. Properties of the map ϕ ◦ σϕ appear in the next result, which uses some ideas from Section 3 of [11]. Theorem 7. Suppose ϕ and ψ are linear fractional maps with ϕ∞ = ψ∞ = 1. Assume further that at least one of the maps ϕ, ψ is univalent. If Cϕ − Cψ is compact on H, then ϕ = ψ. Proof. By the symmetric roles of ϕ and ψ we may assume that ϕ is univalent. The hypothesis ϕ∞ = 1 implies that there exists ζ in ∂BN with |ϕ(ζ)| = 1. Composing on the left and right by unitaries, there is no loss of generality in assuming ζ = e1 and ϕ(e1 ) = e1 . By Theorem 6, we must have ψ(e1 ) = e1 as well. Let σϕ be the Krein adjoint of ϕ as deﬁned in Eq. (14). Since Cϕ − Cψ is compact and Cσϕ is bounded, Cσϕ (Cϕ − Cψ ) = Cϕ◦σϕ − Cψ◦σϕ is also compact. Set τ = ϕ ◦ σϕ and ξ = ψ ◦ σϕ . We have τ (e1 ) = ϕ ◦ σϕ (e1 ) = e1 and thus by Theorem 6, ξ(e1 ) = e1 and D1 τ1 (e1 ) = D1 ξ1 (e1 ). A computation shows that D1 τ1 (e1 ) = 1, (details of this computation can be found in the proof of Theorem 2 in [11]) so that D1 ξ1 (e1 ) = 1. We claim that τ = ξ. To see this, note that it is immediate by Theorem 5 if neither τ nor ξ have any ﬁxed point in the open ball BN . Suppose next that τ has a ﬁxed point in the open ball and lying in the complex line [e1 ]. Restricting τ to the intersection of [e1 ] and the ball, we see that τ must be the identity on [e1 ] ∩ BN , since D1 τ1 (e1 ) = 1 (see, for example, Problem 2.38 in [4], p. 60). By Proposition 1 we see that τ has the form τ (z1 , z ) = (z1 , Az ) for some (N − 1) × (N − 1) matrix A. We label the entries of A as ajk for j, k = 2, . . . , N . Since Cτ − Cξ is compact, we appeal to Theorem 6 to see

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that, since τ is the identity at each point of [e1 ] ∩ ∂BN , so is ξ. Applying Proposition 1 again, we see that ξ(z1 , z ) = (z1 , M z ) for an (N − 1) × (N − 1) matrix M = (mjk ), j, k = 2, . . . , n. Our goal is to show that A = M . Fix j with 2 ≤ j ≤ N , and consider the pseudohyperbolic distance ρ(τ (ωt ), ξ(ωt )) at points of the form √ √ (15) ωt = (t, 0, . . . , 1 − t, 0 . . . , 0) = (t, 0 , 1 − t, 0 ), √ for 0 < t < 1, where the 1 − t appears in the j th component. These points lie in the ball BN . A computation shows that 1 − ρ2 (τ (ωt ), ξ(ωt )) is equal to [1 − t2 − (1 − t)

N

|1 −

2 2 k=2 |akj | ][1 − t − (1 − t) N t2 − (1 − t) k=2 akj mkj |2

N

k=2

|mkj |2 ]

,

and further computation shows that as t ↑ 1 this has limit equal to (2 −

(2 − λ2 )(2 − γ 2 ) k akj mkj )(2 − k akj mkj )

where

λ=

N

(16)

1/2 2

|akj |

k=2

and

γ=

N

1/2 |mkj |2

.

k=2

Observe that λ and γ are at most 1, since τ and ξ map the ball into itself. Write Z = (a2j , a3j , . . . , aN j ) and W = (m2j , m3j , . . . , mN j ), so that λ = Z and γ = W . Moreover, the denominator in (16) is |2 − Z, W |2 , and by the Cauchy-Schwarz inequality, |Z, W | ≤ λγ with equality only if either Z = cW for some c ∈ C or one of Z, W is 0. We investigate the condition under which the expression in (16) is equal to 1. We have (2 − λ2 )(2 − γ 2 ) (2 − λ2 )(2 − γ 2 ) (2 − λ2 )(2 − γ 2 ) ≤ ≤ ≤ 1, |2 − Z, W |2 (2 − |Z, W |)2 (2 − λγ)2 with the last inequality following from its equivalence to (λ − γ)2 ≥ 0. Thus if the expression in (16) is equal to 1, we must have λ = γ and |2 − Z, W | = 2 − |Z, W | = 2 − ZW . Together these force Z = W , which says that the (j − 1)st column of A is the same as the (j − 1)st column of M . Thus if A = M , ρ(τ, ξ) has a strictly positive limit along some path as in Eq. (15). The ratio 1 − |z|2 1 − |τ (z)|2

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has the positive limit (2 − λ2 )−1 along the same path. Applying Theorem 1 we have a contradiction to the hypothesis that Cτ − Cξ is compact. Thus A = M , verifying the claim under the assumption that τ has a ﬁxed point in [e1 ] ∩ BN . Finally, suppose τ has a ﬁxed point in the intersection of the open ball and the complex line through η and e1 for some η ∈ ∂BN , but not in [e1 ]. Since the automorphisms act doubly transitively on ∂BN , we may ﬁnd an automorphism Φ of the ball, ﬁxing e1 so that τ ≡ Φ−1 τ Φ ﬁxes e1 and a point of [e1 ] ∩ BN . A computation shows that D1 τ1 (e1 ) = 1; this computation is aided by the fact that D1 τ1 (e1 ) = τ (e1 )e1 , e1 and the observation that since τ, Φ and Φ−1 all ﬁx e1 , we have Dk τ1 (e1 ) = 0, Dk Φ1 (e1 ) = 0, Dk Φ−1 1 (e1 ) = 0

for all k = 2, 3, . . . , N

([4], Lemma 6.6). Conjugating ξ by Φ as well to get ξ ≡ Φ−1 ξΦ, we apply the and hence τ = ξ, in this case as well. previous argument to see that τ = ξ, Thus compactness of Cϕ − Cψ implies that τ = ξ, or equivalently ϕ ◦ σϕ = ψ ◦ σϕ

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on BN , where σϕ is the Krein adjoint of ϕ. From this we see that ϕ and ψ agree on the range of σϕ . Since we are assuming that ϕ is univalent, so is σϕ [3] and it follows that ϕ = ψ, since the range of σϕ is an open set in BN . The ﬁnal step is to remove the hypothesis of univalence in the last result to obtain the full proof of Theorem 4, which we turn to next. It will be helpful to recast our Hilbert space H as weighted Hardy spaces, deﬁned below, and consider restriction and extension operators on these weighted Hardy spaces. If f is analytic in BN , then f has a homogeneous expansion fs , f= s

where, for each z ∈ BN , we have fs (z) =

cα z α .

(18)

|α|=s

Here, α = (α1 , . . . , αN ) and |α| = α1 + · · · + αN . The Hardy space H 2 (BN ) is equivalently deﬁned as ∞ 2 fs 2 < ∞ , (19) f analytic in BN : s=0 2

where · 2 is the norm in L (σ). The sum in (19) is f 2H 2 (BN ) . More generally, given a suitable sequence of positive numbers {β(s)}, the weighted Hardy space H 2 (β, BN ) is the set of functions f which are analytic in BN and which satisfy ∞ f 2H 2 (β,BN ) ≡ fs 22 β(s)2 < ∞. s=0

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Note that, since the monomials z α are orthogonal on L2 (σ) ([13], Section 1.4), fs 22 =

|cα |2 z α 22 =

|α|=s

|α|=s

|cα |2

(N − 1)!α! . (N − 1 + s)!

The next result realizes the weighted Bergman spaces A2γ (BN ) as weighted Hardy spaces. Lemma 1. (a) Let γ > −1 and set β(s)2 = (s + 1)−(γ+1) . We have H 2 (β, BN ) = A2γ (BN ), with equivalent norms. (b) Let K be an integer with 1 ≤ K < N , and let β(s)2 =

(N − 1)!(K − 1 + s)! (s + 1)−(γ+1) (K − 1)!(N − 1 + s)!

where γ ≥ −1. Then H 2 (β, BK ) = A2N −K+γ (BK ), with equivalent norms. Proof. We prove (a) ﬁrst. Let f be analytic in BK with homogeneous expansion f = fs , where fs is as in Eq. (18). If β(s)2 = (s + 1)−(γ+1) , we have ∞ 1 α!(N − 1)! · |cα |2 f 2H 2 (β,BN ) = Γ(N + s) (s + 1)γ+1 s=0 |α|=s

and f 2A2γ (BN ) =

∞ s=0 |α|=s

|cα |2

α!Γ(N + γ + 1) . Γ(N + s + γ + 1)

This last formula follows from the fact that z α 2A2γ (BK ) =

α!Γ(N + γ + 1) Γ(N + |α| + γ + 1)

(see Lemma 1.11 in [15].) The result in (a) will follow if we can show that

Γ(N + s + γ + 1) α!(N − 1)! 1 · · Γ(N + s) (s + 1)γ+1 α!Γ(N + γ + 1) is bounded above and below by positive constants, depending only on N and γ, for all s ≥ 0. This follows easily from the fact that, by Stirling’s formula, lim

s→∞

Γ(N + s + γ + 1) = 1. (s + 1)γ+1 Γ(N + s)

BK ) where β(s) 2 = From (a) we know that A2N −K+γ (BK ) = H 2 (β, −(N −K+γ+1) (s + 1) . Thus it sufﬁces to show that ! " # (N − 1)!(K − 1 + s)! (s + 1)−(γ+1) · (s + 1)(N −K+γ+1) (K − 1)!(N − 1 + s)!

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is bounded above and below by positive constants (depending on N and K) for all s ≥ 0. Straightforward estimates show that

N −K (N − 1)! (N − 1)!(K − 1 + s)! 1 (N − 1)! N −K ≥ · (s + 1) ≥ (K − 1)! (K − 1)!(N − 1 + s)! (K − 1)! N + 1 for all s ≥ 0, and the desired result follows.

Since the proof of Theorem 4 ultimately relies on an inductive argument, we will work with certain restriction and extension operators on weighted Hardy spaces. These are deﬁned next. Let K, N ∈ N with 1 ≤ K < N . Given a sequence {β(s)} of positive numbers, we deﬁne the associated sequence {β(s)} by 2 = (N − 1)!(K − 1 + s)! β(s)2 . β(s) (K − 1)!(N − 1 + s)! BK ) → H 2 (β, BN ) by We can then deﬁne the extension operator E : H 2 (β, (Ef )(z1 , . . . , zN ) = f (z1 , . . . , zK ), BK ) by and the restriction operator R : H 2 (β, BN ) → H 2 (β, (Rf )(z1 , . . . , zK ) = f (z1 , . . . , zK , 0 ). The next result establishes properties of these operators; it is an extension of Proposition 2.21 in [3] which applies to the case K = 1. BK ) into Lemma 2. (a) The extension operator E is an isometry of H 2 (β, 2 H (β, BN ). (b) The restriction operator R is a norm-decreasing map of H 2 (β, BN ) onto BK ). H 2 (β, BK ) with homogeneous expansion f = fs Proof. For (a), let f ∈ H 2 (β, with fs as in Eq. (18) for z ∈ CK . Then Ef = fs also, and writing fs 2,K for the norm of fs in L2 (∂BK , σ) we have 2 = fs 22,K β(s) f 2H 2 (β,B ) K

s

=

|cα |2

(K − 1)!α! 2 β(s) (K − 1 + s)!

|cα |2

(K − 1)!α! (N − 1)!(K − 1 + s)! · β(s)2 (K − 1 + s)! (K − 1)!(N − 1 + s)!

|cα |2

(N − 1)!α! β(s)2 (N − 1 + s)!

s |α|=s

=

s |α|=s

=

s |α|=s

=

s

fs 22,N β(s)2

= Ef 2H 2 (β,BN ) . Therefore, E is an isometry.

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For (b) let f ∈ H 2 (β, BN ) have homogeneous expansion f = fs with fs as in Eq. (18) for z ∈ BN . For each nonnegative integer s, let As = {α : α = (α , 0 )}, where α is a multi-index with K entries and 0 denotes the zero vector in CN −K , and let Bs consist of all other multi-indices α with N entries satisfying |α| = s. Writing fs (z) = cα z α + cα z α , As

Bs

it follows that (Rf )(z1 , . . . , zK ) =

s

As

αK cα z1α1 . . . zK ,

and Rf 2H 2 (β,B

K)

=

s

= ≤

(K − 1)!α! 2 β(s) (K − 1 + s)!

|cα |2

(N − 1)!α! β(s)2 (N − 1 + s)!

As

s

|cα |2

As

s

|cα |2

|α|=s

(N − 1)!α! β(s)2 (N − 1 + s)!

= f 2H 2 (β,BN ) . Therefore, R is norm-decreasing. To see that R is surjective, let f ∈ BK ) with H 2 (β, α α f (z1 , . . . , zK ) = cα z1 1 . . . zKK , s

|α |=s

where α is a multi-index with K entries. Then f = RF , where F (z) = cα z α . s

As

and cα ≡ cα for α = (α , 0 ) as before. Also, (N − 1)!α! F 2H 2 (β,BN ) = β(s)2 |cα |2 (N − 1 + s)! s As

=

s

=

so F ∈ H 2 (β, BN ).

As

s

=

|cα |2

(K − 1)!α! 2 β(s) (K − 1 + s)! α

α

|cα |2 z1 1 . . . zKK 22,K

|α |=s

f 2H 2 (β,B K)

< ∞,

Recall that by an aﬃne subset of dimension k in BN , we mean the intersection of BN with a translate of a k-dimensional subspace of CN .

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Proof of Theorem 4. Suppose that ϕ and ψ are linear–fractional maps with ϕ∞ = ψ∞ = 1. We will show that if Cϕ − Cψ is compact on H, then ϕ = ψ. The hypothesis ϕ∞ = 1 implies that there exists a point ζ in ∂BN with |ϕ(ζ)| = 1. Composing on the left and right by unitaries, there is no loss of generality in assuming ζ = e1 and ϕ(e1 ) = e1 . By Theorem 6, we must have ψ(e1 ) = e1 as well. The argument proceeds exactly as in the proof of Theorem 7 up to the point where the relationship ϕ ◦ σϕ = ψ ◦ σϕ ,

(20)

is obtained. Since Theorem 7 covers the case that at least one of ϕ and ψ is one-to-one, we now only consider the case that neither is univalent. This implies that there is a smallest k1 with 1 ≤ k1 < N so that ϕ(BN ) is contained in an aﬃne set of dimension k1 , and there is a smallest k2 with 1 ≤ k2 < N so that ψ(BN ) is contained in an aﬃne set of dimension k2 . Since the roles of ϕ and ψ can be reversed, there is no loss of generality in assuming k1 ≥ k2 . Our ﬁrst goal is to show that equality k1 = k2 holds as a consequence of Eq. (20). By Proposition 13 in [3], σϕ (BN ) is also contained in a k1 -dimensional aﬃne set. Set σϕ (0) = p, and let φp be an automorphism of BN sending p to 0 and satisfying φp = φ−1 p . Since φp ◦ σϕ ﬁxes the origin, and maps the ball into a k1 -dimensional aﬃne set, we may write, as in [3], φp ◦ σ ϕ = L ◦ τ where L is linear of rank k1 and τ is a one-to-one linear fractional map. Specifically, if φp ◦ σ ϕ =

Az z, C + 1

we can choose L(z) = Az and τ (z) =

z . z, C + 1

(Note that L and τ need not be self-maps of BN , though their composition is, and both are deﬁned and analytic on a neighborhood of the closed ball). From this it follows that σϕ = φp ◦ L ◦ τ . Taking Krein adjoints on both sides we have ϕ = στ ◦ L∗ ◦ σφp , where στ (z) = z − C. We have σφp = φ−1 p = φp , so that ϕ ◦ σϕ = στ ◦ L∗ ◦ σφp ◦ φp ◦ L ◦ τ = στ ◦ L∗ L ◦ τ,

(21)

where L∗ L is linear with rank k1 , τ is univalent, and στ is a translation. Thus the image of the ball BN under ϕ◦σϕ cannot be contained in a k dimensional aﬃne set for any k < k1 , and the relation ϕ(σϕ (BN )) = ψ(σϕ (BN )) ⊆ ψ(BN ) says that strict inequality k1 > k2 is impossible, and therefore k1 = k2 as desired. We denote the common value of k1 and k2 by K. Thus ϕ(BN ) is contained in a K-dimensional aﬃne set A1 and is not contained in any aﬃne set of smaller dimension, and ψ(BN ) is contained in a K-dimensional aﬃne set A2 , and is not contained in any aﬃne set of smaller

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dimension. We have A2 ⊇ ψ(BN ) ⊇ ψ(σϕ (BN )) = ϕ(σϕ (BN )) where A1 ⊇ ϕ(σϕ (BN )) = στ L∗ Lτ (BN ) for linear L∗ L of rank K and univalent linear fractional τ and στ . This forces A1 = A2 ; that is, the range of ϕ and the range of ψ are contained in the same K-dimensional aﬃne set, which we will simply denote A. Note that e1 ∈ A. Our goal is to show that ϕ = ψ. Let ζ ∈ ∂BN with ζ = e1 . Let Λζ be a K-dimensional aﬃne subset of BN , containing e1 and ζ in its bound$ ary, whose intersection with BN is a K-dimensional ball. We will write B K N −K for {(z1 , z2 , . . . , zK , 0 ) ≡ (z , 0 ) ∈ BN }, where 0 denotes the 0 in C . $ Let ρ1 be an automorphism of BN ﬁxing e1 with ρ1 (B K ) = Λζ and let ρ2 $ be an automorphism of BN ﬁxing e1 with ρ2 (A) = B K ; such automorphisms exist because of the two-fold transitivity of the automorphisms on ∂BN . Note that ρ2 ◦ ϕ ◦ ρ1 and ρ2 ◦ ψ ◦ ρ1 are linear–fractional self-maps of BN with $ $ $ $ ρ2 ◦ ϕ ◦ ρ1 (B K ) ⊆ BK and ρ2 ◦ ψ ◦ ρ1 (BK ) ⊆ BK . Let π be the projection of N K C onto C deﬁned by π(z , z ) = z and deﬁne maps μ and ν on BK by μ(z ) = π ◦ ρ2 ◦ ϕ ◦ ρ1 (z , 0 ) and ν(z ) = π ◦ ρ2 ◦ ψ ◦ ρ1 (z , 0 ). These are linear–fractional self-maps of BK ﬁxing (1, 0, . . . , 0) ∈ ∂BK . Write H as H 2 (β, BN ) with β(s)2 = (s + 1)−(γ+1) , where γ = −1 if H = H 2 (BN ) and γ = α if H = A2α (BN ), up to an equivalent norm. We BK ) claim that Cμ − Cν is compact on the weighted Hardy space H 2 (β, where 2 = (N − 1)!(K − 1 + s)! β(s)2 . β(s) (K − 1)!(N − 1 + s)! BK ) = A2 Since, by Lemma 1, H 2 (β, N −K+γ (BK ) with equivalent norms, it will follow that Cμ − Cν is compact on A2N −K+γ (BK ). To prove the claim it sufﬁces to show that if {fn } is a bounded sequence BK ) and fn → 0 almost uniformly on BK , then on H 2 (β, (Cμ − Cν )fn H 2 (β,B K ) → 0. BK ). Deﬁne functions Fn on BN by Let fn be such a sequence in H 2 (β, Fn = Efn , BK ) → H (β, BN ) is the extension operator deﬁned by where E : H (β, 2

2

(Ef )(z1 , . . . , zN ) = f (z1 , . . . , zK ). By Lemma 2, E is an isometry, so the functions Fn form a bounded sequence in H and Fn → 0 uniformly on compact subsets of BN . Since Cρ2 ◦ϕ◦ρ1 − Cρ2 ◦ψ◦ρ1 = Cρ1 (Cϕ − Cψ )Cρ2 is compact on H, we have Fn ◦ ρ2 ◦ ϕ ◦ ρ1 − Fn ◦ ρ2 ◦ ψ ◦ ρ1 H 2 (β,BN ) → 0

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BK ) by Deﬁne the restriction operator R : H 2 (β, BN ) → H 2 (β, (Rf )(z1 , . . . , zK ) = f (z1 , . . . , zK , 0 ). BK ) By Lemma 2, R is a norm-decreasing map of H 2 (β, BN ) onto H 2 (β, and so R(Fn ◦ ρ2 ◦ ϕ ◦ ρ1 ) − R(Fn ◦ ρ2 ◦ ψ ◦ ρ1 )H 2 (β,B K) → 0 But R(Fn ◦ρ2 ◦ϕ◦ρ1 ) = fn ◦μ on BK and R(Fn ◦ρ2 ◦ψ ◦ρ1 ) = fn ◦ν on BK , so the claim is veriﬁed, and Cμ − Cν is compact on A2N −K+γ (BK ). Since K < N and μ and ν are linear–fractional self-maps of BK with μ∞ = ν∞ = 1, by induction this forces μ = ν, which in turn says that ϕ = ψ on the aﬃne set Λζ containing ζ and e1 . Since ζ is an arbitrary point in ∂BN , this says ϕ = ψ in BN .

References [1] Bracci, F., Contreras, M., Diaz-Madrigal, S.: Classification of semigroups of linear fractional maps in the unit ball. Adv. Math. 208, 318–350 (2007) [2] Bourdon, P.: Components of linear–fractional composition operators. J. Math. Anal. Appl. 279, 228–245 (2003) [3] Cowen, C., MacCluer, B.: Linear fractional maps of the ball and their composition operators. Acta. Sci. Math. (Szeged) 66, 351–376 (2000) [4] Cowen, C., MacCluer, B.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995) [5] Jiang, L., Ouyang, C.: Compact differences of composition operators on holomorphic function spaces in the unit ball (preprint) [6] Jury, M.: C ∗ -algebras generated by groups of composition operators. Indiana Univ. Math. J 56, 3171–3192 (2007) [7] Jury, M.: The Fredholm index for elements of Toeplitz-composition C ∗ -algebras. Integral Equ. Operator Theory 58, 341–362 (2007) [8] Kriete, T., MacCluer, B., Moorhouse, J.: Toeplitz-composition C ∗ -algebras. J. Operator Theory 58, 135–156 (2007) [9] Kriete, T., MacCluer, B., Moorhouse, J.: Composition operators within singly generated composition C ∗ -algebras, Israel J. Math. arXiv:math/0610077 (to appear) [10] Kriete, T., MacCluer, B., Moorhouse, J.: Spectral theory for algebraic combinations of Toeplitz and composition operators. J. Funct. Anal. 257, 2378–2409 (2009) [11] MacCluer, B., Weir, R.: Linear–fractional composition operators in several variables. Integral Equ. Operator Theory 53, 373–402 (2005) [12] Moorhouse, J.: Compact differences of composition operators. J. Funct. Anal. 219, 70–92 (2005) [13] Rudin, W.: Function Theory in the Unit Ball of CN . Springer, New York (1980) [14] Shapiro, J.: Composition Operators and Classical Function Theory. Springer, New York (1993)

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[15] Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Springer, New York (2005) Katherine Heller and Barbara D. MacCluer (B) Department of Mathematics University of Virginia P. O. Box 400137 Charlottesville VA 22904, USA e-mail: [email protected]; [email protected] Rachel J. Weir Department of Mathematics Allegheny College Meadville PA 16335, USA e-mail: [email protected] Received: June 15, 2010. Revised: October 13, 2010.

Integr. Equ. Oper. Theory 69 (2011), 269–300 DOI 10.1007/s00020-010-1832-5 Published online October 7, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Second Order Elliptic Differential-Operator Equations with Unbounded Operator Boundary Conditions in UMD Banach Spaces B. A. Aliev and Ya. Yakubov Abstract. In a UMD Banach space E, we consider a boundary value problem for a second order elliptic differential-operator equation with a spectral parameter when one of the boundary conditions, in the principal part, contains a linear unbounded operator in E. A theorem on an isomorphism is proved and an appropriate estimate of the solution with respect to the space variable and the spectral parameter is obtained. In this way, Fredholm property of the problem is shown. Moreover, discreteness of the spectrum and completeness of a system of root functions corresponding to the homogeneous problem are established. Finally, applications of obtained abstract results to nonlocal boundary value problems for elliptic differential equations with a parameter in nonsmooth domains are given. Mathematics Subject Classification (2010). Primary 47E05; Secondary 47A75, 34L10, 34B05, 35P10, 35J25. Keywords. Differential-operator equations, elliptic equations, isomorphism, completeness, root functions, spectral parameter.

1. Introduction Boundary value problems for elliptic differential-operator equations were considered in monographs by Krein [16], Dezin [10], Gorbachuk and Gorbachuk [15], Yakubov [23,24], Yakubov and Yakubov [27], Shklyar [20] and in many papers, for example, by Amann [5], deLaubenfels [8], Aibeche [1,2], Yakubov and Aliev [26], Dore and Yakubov [11], Yakubov [25], Aliev and Yakubov [4] and others. All these studies, usually, treat problems in Hilbert spaces. The development of the theory of general boundary value problems for elliptic differential-operator equations in Banach spaces became possible due to a Ya. Yakubov was supported by the Israel Ministry of Absorption.

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paper by Weis [22] in 2001. Let us mention here very recent studies by Favini et al. [13] and Favini and Yakubov [14], where regular boundary value problems for second and fourth orders, respectively, elliptic differential-operator equations have been studied. In particular, an isomorphism, i.e., also maximal Lp -regularity of the problems have been established. We refer the reader to these papers for some relevant references on the subject. In the present paper, we essentially use the ideas and methods of paper [25] and study a boundary value problem for a second order elliptic differential-operator equation with a parameter when one of the boundary conditions, in the principal part, contains a linear unbounded operator (the principal part of boundary conditions in [25] is pure differential), an operator B in (1.2) below. Moreover, using very recent studies, we prove our results in UMD Banach spaces in contrast to Hilbert spaces framework in [25]. Thus, we treat, in a UMD Banach space E, the following boundary value problem: L(λ)u := λu(x) − u (x) + Au(x) + (A1 u)(x) = f (x), L1 u := αu (1) + Bu(0) +

N1

L2 u := βu (0) +

(1.1)

γ1j u (x1j ) + T1 u = f1 ,

j=1 N2

x ∈ (0, 1),

(1.2)

γ2j u (x2j ) + T2 u = f2 ,

j=1

where λ is a spectral parameter; A is a linear closed, densely deﬁned operator in E; A1 is a linear operator in Lp ((0, 1); E), p ∈ (1, ∞); B is a linear (unbounded) closed operator in E, which is bounded in appropriate spaces; Tk , k = 1, 2, are linear operators from Lp ((0, 1); E) into E; α, β, γkj are complex numbers, α = 0, β = 0; xkj ∈ (0, 1) ; Nk , k = 1, 2, are some natural numbers. Let us emphasize that the main operator of the equation is A and of the boundary conditions is B. The operators A1 , T1 , and T2 are perturbation operators and can be treated by standard perturbation arguments. Similar to paper [25], in this paper, for sufﬁciently large |λ| from some sector containing a positive semi-axis, we prove a theorem on an isomorphism for problem (1.1)–(1.2) (in particular, it implies maximal Lp -regularity), establish some estimates for a solution of (1.1)–(1.2) (with respect to x-variable and λ) in Lp ((0, 1); E). Fredholm property of problem (1.1)–(1.2), for λ = 0, is also proved. Further, discreteness of the spectrum and completeness of a system of root functions (eigenfunctions and associated functions) corresponding to homogeneous problem of (1.1)–(1.2) in Lp ((0, 1); E) are proved. Note that the presence of the unbounded operator B in the boundary condition inﬂuence on the estimates with respect to λ for the solution of problem (1.1)–(1.2). In other words, we prove that in spite of the fact that an operator L(λ), generated by the boundary value problem (1.1)–(1.2), is coercive on the space variable, it is not coercive on the spectral parameter. In particular, the resolvent of the operator corresponding to the problem with homogeneous boundary conditions, for sufﬁciently large |λ| from some sector, containing − 1+ 1 the positive semi-axis, decreases as |λ| ( 2 2p ) and not as |λ|−1 .

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 271 We would like to mention here two more recent papers which concern to our present paper. The ﬁrst one is by Cheggag et al. [7]. The authors also treat abstract boundary value problems with an unbounded operator in the main part of boundary conditions, but they put some more restrictive assumptions on the main operators of the problem: the operators are commutative in the resolvent sense and they have bounded imaginary powers (BIP). Moreover, by our method it is possible to add perturbation terms into both the equation and boundary conditions: the operator A1 into the equation [see (1.1)] and values of the derivative of the unknown function at intermediate points of the interval and the operators Tk into the boundary conditions [see (1.2)]. The second paper is by Aibeche and Laidoune [3]. They consider other boundary conditions with other perturbation operators. Our perturbation operators act on vector-valued functions and they treat operators which act on elements of the underlying Banach space. We also present all rigorous proofs of the calculations, while they do not present some necessary calculations which are new in the framework of UMD Banach spaces in contrast to Hilbert spaces. Another thing is that our completeness theorem covers problems in Banach spaces, while their proof is true only in the framework of Hilbert spaces. It is well-known that usually boundary value problems for elliptic equations in non-smooth domains do not have the maximal regularity for a solution. We have found, in an application, a class of elliptic problems in cylindrical domains (i.e., in non-smooth domains) which has a maximal regularity for a solution, i.e., the solution belongs to Wp2 -Sobolev spaces for any 1 < p < ∞. Note, that our considered equations [see (7.1) and (7.7)] do not contain mixed derivatives between x-variable and y-variable. We give now some necessary definitions. Let E1 and E2 be Banach ˙ 2 of all the vectors of the form (u, υ), where u ∈ E1 , υ ∈ spaces. The set E1 +E E2 , with ordinary coordinatewise linear operations and the norm (u, υ)E1 +E ˙ 2 := uE1 + υE2 is a Banach space and is said to be a direct sum of Banach spaces E1 and E2 . By (E1 , E2 )θ,p , 0 < θ < 1, 1 ≤ p ≤ ∞, we denote the standard (real) interpolation space (see, e.g., [21] for definitions and properties). Let A be a linear closed operator in a Banach space E with the domain D (A). The domain D(A) is turned into a Banach space E (A) with respect to the norm 1 2 2 2 uE(A) := uE + AuE . By B(E1 , E) we denote a Banach space of all bounded operators from E1 into E with ordinary operator norm; B(E) := B(E, E). By Lp ((0, 1); E), 1 < p < ∞, we denote a Banach space of functions x → u (x) : [0, 1] → E, strongly measurable and summable in the p-th power, with the norm ⎛ 1 ⎞ p1 p uLp ((0,1);E) := ⎝ u(x)E dx⎠ < ∞. 0

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Wp ((0, 1); E), 1 < p < ∞, 0 ≤ is an integer, is a Banach space of functions u(x) with values in E, which have generalized derivatives up to the -th order inclusive, on (0, 1) with the norm uW ((0,1);E) := p

⎛

⎞ p1 1

p

⎝

u(k) (x) dx⎠ < ∞.

k=0

E

0

By Wp2 ((0, 1); E(A), E) we denote Wp2 ((0, 1); E(A), E) := {u : u ∈ Lp ((0, 1); E(A)) , u ∈ Lp ((0, 1); E)} with the ﬁnite norm uWp2 ((0,1);E(A),E) := uLp ((0,1);E(A)) + u Lp ((0,1);E) . It is known that this space is a Banach space (see for more general spaces [21, Lemma 1.8.1]; see also [27, Sect. 1.7.7]). The following notions are rather known but we would like to bring them here for the reader convenience. A Banach space E is said to be of class HT, if the Hilbert transform is bounded on Lp (R; E) for some (and then all) p > 1. Here the Hilbert transform H of a function f ∈ S(R; E), the Schwartz space of rapidly decreasing E-valued functions, is deﬁned by 1 1 Hf := P V ∗ f, π t

) dτ . These spaces are often also called UMD i.e., (Hf )(t) := π1 lim |τ |>ε f (t−τ τ ε→0

Banach spaces, where the UMD stands for the property of unconditional martingale differences. Definition 1.1. Let E be a complex Banach space, and A is a closed linear operator in E. The operator A is called sectorial if the following conditions are satisﬁed: 1. D(A) = E, R(A) = E, (−∞, 0) ⊂ ρ(A); 2. λ(λ + A)−1 ≤ M for all λ > 0, and some M < ∞. Definition 1.2. Let E and F be Banach spaces. A family of operators T ⊂ B(E, F ) is called R-bounded, if there is a constant C > 0 and p ≥ 1 such that for each natural number n, Tj ∈ T , uj ∈ E and for all independent, symmetric, {−1, 1}-valued random variables εj on [0, 1] (e.g., the Rademacher functions εj (t) = sign sin(2j πt) ) the inequality

n

n

εj T j u j

≤C

εj u j

j=1

j=1

Lp ((0,1);F )

Lp ((0,1);E)

is valid. The smallest such C is called R-bound of T and is denoted by R{T }E→F . If E = F , the R-bound will be denoted by R{T }E or simply R{T }.

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 273 Remark 1.3. From the definition of R-boundedness it follows that every R-bounded family of operators is (uniformly) bounded (it is enough to take n = 1). On the other hand, in a Hilbert space H every bounded set is R-bounded (see, e.g., Kunstmann and Weis [18, p. 75]). Therefore, in a Hilbert space, the notion of R-boundedness is equivalent to boundedness of a family of operators (see also Denk et al. [9, p. 26]). Definition 1.4. A sectorial operator A is called R-sectorial if RA (0) := R{λ(λ + A)−1 : λ > 0} < ∞. The number φR A := inf{θ ∈ (0, π) : RA (π − θ) < ∞}, where RA (θ) := R{λ(λ + A)−1 : | arg λ| ≤ θ}, is called an R-angle of the operator A.

2. Homogeneous Equations First, we consider the following boundary value problem in a Banach space E L0 (λ)u := λu(x) − u (x) + Au(x) = 0, L10 u := αu (1) + Bu(0) +

N1

x ∈ (0, 1),

γ1j u (x1j ) = f1 ,

j=1

L20 u := βu (0) +

N2

(2.1)

(2.2)

γ2j u (x2j ) = f2 .

j=1

Theorem 2.1. Let the following conditions be fulfilled: 1. α, β, γkj are complex numbers; α = 0, β = 0; xkj ∈ (0, 1); 2. An operator A is closed, densely defined and invertible in a UMD Banach space E, R{λR(λ, A) : arg λ = π} < ∞, and R(λ, A) ≤ C(1 + |λ|)−1 for | arg λ| ≥ π − ϕ, for some 0 ≤ ϕ < π;1 1 3. A closed operator B is bounded from E(A 2 ) into E and from E (A) into 1 E(A 2 ). Then, for fk ∈ (E(A), E) 1 + 1 ,p , p ∈ (1, ∞), and for sufficiently large |λ| 2 2p from the sector |arg λ| ≤ ϕ, problem (2.1)–(2.2) has a unique solution u ∈ Wp2 ((0, 1); E(A), E), such that u(0) ∈ D(B), and for the solution it holds the estimate |λ|uLp ((0,1);E) + u Lp ((0,1);E) + AuLp ((0,1);E) 2 1 1 − 2p 2 ≤C fk E . fk (E(A),E) 1 + 1 ,p + |λ| k=1

1

2

2p

The operator A is, in particular, R-sectorial in E with the R-angle φR A < π. Therefore, by, e.g., [9, Theorem 2.3], there exist fractional powers of A. If ϕ = 0 then, by Remark 1.3, it is enough to claim only R{λR(λ, A) : arg λ = π} < ∞.

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Proof. From condition 2, by virtue of [27, Lemma 5.4.2/6], for | arg λ| ≤ ϕ < π, there exists the holomorphic, for x > 0, and strongly continuous, for 1

x ≥ 0, semigroup e−x(A+λI) 2 . By the proof of [13, Theorem 2], any solution of equation (2.1), belonging to Wp2 ((0, 1); E(A), E), for |arg λ| ≤ ϕ, has the representation 1

1

u (x) = e−x(A+λI) 2 g1 + e−(1−x)(A+λI) 2 g2 ,

(2.3)

with g1 , g2 ∈ (E(A), E) 1 ,p . By [21, Theorem 1.8.2] (see also [27, Theorem 2p 1 1 1.7.7/1]), u(0) ∈ (E(A), E) 2p ,p . It is proved below that (E(A), E) 2p ,p = 1

1

(E(A 2 ), E(A))1− p1 ,p and that, for each 0 < θ < 1, (E(A 2 ), E(A))θ,p ⊂ D(B). Therefore, u(0) ∈ D(B). The function u (x) of the form (2.3) satisﬁes (2.2), if 1 1 1 α − (A + λI) 2 e−(A+λI) 2 g1 + (A + λI) 2 g2 N1 1 1 1 + B g1 + e−(A+λI) 2 g2 + γ1j − (A + λI) 2 e−x1j (A+λI) 2 g1 j=1

1 + (A + λI) e−(1−x1j )(A+λI) 2 g2 = f1 , 1 1 1 β − (A + λI) 2 g1 + (A + λI) 2 e−(A+λI) 2 g2 1 2

+

N2

(2.4)

1 1 γ2j − (A + λI) 2 e−x2j (A+λI) 2 g1

j=1

1 1 + (A + λI) 2 e−(1−x2j )(A+λI) 2 g2 = f2 . Rewrite system (2.4) in the form ⎡ 1 2

⎣−α (A + λI) e

1 −(A+λI) 2

−

N1

⎤ 1 2

γ1j (A + λI) e

1 −x1j (A+λI) 2

+ B ⎦ g1

j=1

⎡

1 −(A+λI) 2

1 2

+ ⎣α (A+λI) +Be

+

N1

⎤ 1 2

γ1j (A+λI) e−(1−x1j )(A+λI)

j=1

⎡ 1 2

⎣−β (A + λI) −

N2

1 2

γ2j (A + λI) e

1 2

+ ⎣β (A+λI) e

1

−(A+λI) 2

+

N2 j=1

⎦g2 = f1 ,

⎤ 1

−x2j (A+λI) 2

(2.5)

⎦ g1

j=1

⎡

1/2

⎤ 1 2

γ2j (A+λI) e

1

−(1−x2j )(A+λI) 2

⎦ g2 = f2 .

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 275 ˙ (E(A), E) 1 + 1 ,p Let us write this system in the space E := (E(A), E) 1 + 1 ,p + 2 2p 2 2p in the form of the operator equation g1 f1 (A(λ) + R(λ)) = , (2.6) g2 f2 where A(λ) and R(λ) are operator-matrices of dimension 2 × 2: ⎞ ⎛ 1 B α (A + λI) 2 ⎟ ⎜ A(λ) := ⎝ ⎠, 1 2 −β (A + λI) 0 ˙ D(A(λ)) := (E(A), E) 1 ,p + (E(A), E) 1 ,p 2p

and

⎛ R(λ) := ⎝

2p

K11 (λ)

K12 (λ)

K21 (λ)

K22 (λ)

⎞ ⎠,

D(R(λ)) := E,

with 1

1

K11 (λ) = −α (A + λI) 2 e−(A+λI) 2 −

N1

1

1

γ1j (A + λI) 2 e−x1j (A+λI) 2 ,

j=1 1

K12 (λ) = Be−(A+λI) 2 +

N1

1

1

γ1j (A + λI) 2 e−(1−x1j )(A+λI) 2 ,

j=1

K21 (λ) = −

N2

1

1

γ2j (A + λI) 2 e−x2j (A+λI) 2 ,

j=1 1

1

K22 (λ) = β (A + λI) 2 e−(A+λI) 2 +

N2

1

1

γ2j (A + λI) 2 e−(1−x2j )(A+λI) 2 .

j=1

Obviously, the operator A(λ) in the space E, for |arg λ| ≤ ϕ < π, has an inverse. Moreover, ⎞ ⎛ −1 2 1 0 − β (A + λI) ⎟ ⎜ −1 ⎟. A (λ) = ⎜ ⎠ ⎝ −1 −1 1 2 2 −2 1 1 (A + λI) (A + λI) B (A + λI) α αβ g1 On the other hand, for any ∈ D(A(λ)), it holds the identity g2 g1 g1 −1 A(λ) A(λ) = . g2 g2 Then, from equation (2.6) we have: g f1 −1 −1 1 I + A(λ) R(λ) = A(λ) . g2 f2

(2.7)

276

B. A. Aliev and Ya. Yakubov

IEOT

Show now that, for |arg λ| ≤ ϕ < π, the operator A(λ)−1 is bounded ˙ (E(A), E) 1 ,p and it holds the estimate from E into (E(A), E) 1 ,p + 2p 2p

−1 ≤ C,

A(λ)

(2.8) B E, (E(A),E)

˙ 1 ,p +(E(A),E) 1 ,p 2p 2p

where C > 0 is some constant independent on λ. To this end, it is sufﬁcient to show: −1 (a) the operator (A + λI) 2 , for |arg λ| ≤ ϕ < π, is bounded from (E(A), E) 1 + 1 ,p into (E(A), E) 1 ,p and it holds the estimate 2 2p 2p

1

−

≤ C, (2.9)

(A + λI) 2 B (E(A),E) 1 + 2

1 ,p , 2p

(E(A),E)

1 ,p 2p

where C > 0 is a constant independent on λ; −1 −1 (b) the operator (A + λI) 2 B (A + λI) 2 , for |arg λ| ≤ ϕ < π, is 1 bounded from (E(A), E) 12 + 2p 1 ,p into (E(A), E) 2p ,p and it holds the estimate

−1 −1

≤ C, (2.10)

(A + λI) 2 B (A + λI) 2 B (E(A),E) 1 + 2

1 ,p ,(E(A),E) 1 ,p 2p 2p

where C > 0 is some constant independent on λ. −1 Let us prove (a). Since the operator (A + λI) 2 , for |arg λ| ≤ ϕ, is 1 1 bounded from E into E(A 2 ) and from E(A 2 ) into E (A) then, by the interpolation theorem [21, Theorem 1.3.3/(a)] and [27, Lemma 5.4.2/6], the 1 −1 operator (A + λI) 2 , for |arg λ| ≤ ϕ, is bounded from (E, E(A 2 ))θ,p into 1 (E(A 2 ), E(A))θ,p , for any θ ∈ (0, 1), and it holds the estimate

1

(A + λI)− 2 1 1 B (E,E(A 2 ))θ,p , (E(A 2 ),E(A))θ,p

1−θ

−1

≤ C (A + λI) 2 B

1 E,E(A 2

)

θ

−1

(A + λI) 2

1

B E(A 2 ),E(A)

1−θ

1 1 θ

−1

−1 ≤ C A 2 (A + λI) 2

A (A + λI) 2 A− 2

B(E)

B(E)

≤ C.

Further, by [21, formula 1.15.4/(2) and formula 1.15.2/(4)] and [21, Theorem 1.3.3/(b)], we have 1

(E(A 2 ), E(A))θ,p = (E, E(A)) 1 (1−θ)+θ,p = (E, E(A)) 1 + θ ,p 2

2

2

= (E(A), E) 1 − θ ,p , 2

2

1 2

(E, E(A ))θ,p = (E, E(A)) 1 θ,p = (E(A), E)1− θ ,p . 2

Take θ = 1 − 1

1 p.

2

Then, substituting 1

2 1 1 (E(A 2 ), E(A))θ,p = (E(A), E) 2p ,p and (E, E(A ))θ,p = (E(A), E) 12 + 2p ,p

into the last inequality, we prove (2.9). This proves (a). Let us now prove (b). By condition 3 and the interpolation theorem 1 [21, Theorem 1.3.3/(a)], the operator B is bounded from (E(A 2 ), E(A))θ,p

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 277 1

into (E, E(A 2 ))θ,p for any θ ∈ (0, 1). So, using the last inequality, the opera1 1 1 tor (A + λI)− 2 B(A + λI)− 2 , for |arg λ| ≤ ϕ, is bounded from (E, E(A 2 ))θ,p 1 into (E(A 2 ), E(A))θ,p for any θ ∈ (0, 1). Again, by the interpolation theorem [21, Theorem 1.3.3/(a)], [27, Lemma 5.4.2/6], and condition 3, for |arg λ| ≤ ϕ, we have

1 1

(A + λI)− 2 B(A + λI)− 2 1 1 B (E,E(A 2 ))θ,p , (E(A 2 ),E(A))θ,p

1 1 1−θ

≤ C (A + λI)− 2 B(A + λI)− 2

1

B E,E(A 2 )

1 1 θ

× (A + λI)− 2 B(A + λI)− 2

1

B E(A 2 ),E(A)

1 1 1 1−θ

≤ C A 2 (A + λI)− 2 B(A + λI)− 2

B(E)

1 1 1 θ

× A(A + λI)− 2 B(A + λI)− 2 A− 2

B(E)

1

1 1 1−θ

1 1−θ

1 1−θ

2 ≤ C A (A + λI)− 2

BA− 2

A 2 (A + λI)− 2

B(E) B(E) B(E)

θ

1

θ

1

θ

1

2

2

2 − 12

−1

− 12

× A (A + λI)

A BA

A (A + λI)

B(E)

B(E)

B(E)

≤ C.

Taking now again θ = 1 − p1 , by the above considerations, we prove (2.10), i.e., (b) has been proved too. Therefore, as it was mentioned above, estimate (2.8) follows from (2.9) and (2.10). −1 Further, from the formulas of A (λ) and R (λ), we get that the prod−1 uct A (λ) R (λ) is an operator-matrix the elements of which are a linear combination of the operators 1

1

− 12

e−(A+λI) 2 , (A + λI) − 12

(A + λI)

1

1

Be−(A+λI) 2 , e−xkj (A+λI) 2 , e−(1−xkj )(A+λI) 2 , 1

− 12

Be−x2j (A+λI) 2 , (A + λI)

1

Be−(1−x2j )(A+λI) 2 . −1

Let us show that all the above operators in the operator-matrix A (λ) R (λ), 1 1 for |arg λ| ≤ ϕ, are bounded from (E(A), E) 2p ,p into (E(A), E) 2p ,p . 1

−1

For example, we show this for the operator (A + λI) 2 Be−(A+λI) 2 . By [27, Lemma 5.4.2/6] and condition 3, for |arg λ| ≤ ϕ, we have

1

1

(A + λI)− 2 Be−(A+λI) 2

B(E)

1

− 12 − 12 12 −(A+λI) 2

= (A + λI) BA A e

−1

≤ (A + λI) 2

B(E)

− 12

≤ C(1 + |λ|)

1

BA− 2

1 −ω|λ| 2

e

B(E)

1 −ω|λ| 2

≤ Ce

B(E)

1 −(A+λI) 12

A 2 e

B(E)

,

∃ω > 0;

(2.11)

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B. A. Aliev and Ya. Yakubov

IEOT

1

1

(A + λI)− 2 Be−(A+λI) 2

B(E(A))

1

− 12 −(A+λI) 2 −1

A (A + λI) =

Be A

B(E)

1

1 1 − 12 −1 −(A+λI) 2

2 2 = A (A + λI) A BA e

1

−1

≤ A 2 (A + λI) 2

B(E)

1 −ω|λ| 2

≤ Ce

,

B(E)

1

2

A BA−1

B(E)

−(A+λI) 12

e

B(E)

∃ω > 0.

(2.12)

Then, from the estimates (2.11) and (2.12), by the interpolation theorem −1

1

[21, Theorem 1.3.3/(a)], for |arg λ| ≤ ϕ, the operator (A + λI) 2 Be−(A+λI) 2 is bounded from (E(A), E)θ,p into (E(A), E)θ,p , for any θ ∈ (0, 1), in particular, from (E(A), E) 1 ,p into (E(A), E) 1 ,p , and it holds the estimate 2p 2p

1

1

(A + λI)− 2 Be−(A+λI) 2

B (E(A),E)

1 ,p 2p

1

1− 2p

1

− 12 −(A+λI) 2

≤ C (A + λI) Be

B(E(A))

1

1 2p

− 12 −(A+λI) 2

(A + λI) ×

Be

B(E)

1

≤ Ce−ω|λ| 2 ,

∃ω > 0.

(2.13)

In a similar way, we prove that the rest members of the operator-matrix A(λ)−1 R(λ) are bounded in (E(A), E) 1 ,p and for the norms of these mem2p bers, for |arg λ| ≤ ϕ, the estimate of the form (2.13) holds. Therefore, for sufﬁciently large |λ| from the sector |arg λ| ≤ ϕ, the operator A(λ)−1 R(λ) is ˙ (E(A), E) 1 ,p and bounded in (E(A), E) 1 ,p + 2p

2p

A(λ)−1 R(λ)

B (E(A),E)

1 2p

˙ +(E(A),E) ,p

1 ,p 2p

1

≤ Ce−ω|λ| 2 < 1.

Hence, by the Neumann identity, for | arg λ| ≤ ϕ and |λ| → ∞, ∞ −1 k −1 −1 I + A(λ) R(λ) =I+ (−1)k A(λ) R(λ) ,

(2.14)

(2.15)

k=1

where the series converges in the norm of the space of bounded operators in ˙ (E(A), E) 1 ,p . (E(A), E) 1 ,p + 2p 2p By (2.15), from (2.7), provided that |arg λ| ≤ ϕ and |λ| → ∞, we have −1 f1 g1 −1 −1 A (λ) = I + A(λ) R(λ) . g2 f2 Consequently, for sufﬁciently large |λ| from the sector |arg λ| ≤ ϕ, the elements g1 and g2 can be represented in the form: gk = (Ck1 (λ) + Rk1 (λ)) f1 + (Ck2 (λ) + Rk2 (λ)) f2 ,

k = 1, 2,

(2.16)

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 279 − 12

with the operators C12 (λ) = − β1 (A + λI)

, C21 (λ) = −1 2

− 12

1 α

− 12

(A + λI)

,

1 C11 (λ) = 0, C22 (λ) = αβ (A + λI) B (A + λI) ; Rkj (λ), by (2.8) and 1 (2.14), are some bounded operators from (E(A), E) 12 + 2p 1 ,p into (E(A), E) 2p ,p provided that |arg λ| ≤ ϕ and |λ| → ∞. Moreover, using the estimates (2.8) and (2.14), one can show that, for |arg λ| ≤ ϕ and |λ| → ∞, B (E(A),E) 1 +

Rkj (λ)

2

1 ,p ,(E(A),E) 1 ,p 2p 2p

1

≤ Ce−ω|λ| 2 ,

∃C, ω > 0.

(2.17)

From the representation of A(λ)−1 and A(λ)−1 R(λ) it also follows that, for sufﬁciently large |λ| from the sector |arg λ| ≤ ϕ, for the operators Rkj (λ) the estimate 1

Rkj (λ)B(E) ≤ Ce−ω|λ| 2 ,

∃C, ω > 0,

holds. Substituting (2.16) into (2.3) we get 2 1 u (x) = e−x(A+λI) 2 (C1k (λ) + R1k (λ)) k=1

1 + e−(1−x)(A+λI) 2 (C2k (λ) + R2k (λ)) fk .

Then, for sufﬁciently large |λ| from the sector |arg λ| ≤ ϕ, we have |λ| uLp ((0,1);E) + u Lp ((0,1);E) + AuLp ((0,1);E) ⎧ ⎡⎛ ⎞ p1

p 1

2 ⎪ ⎨ 1

⎢ −x(A+λI) 2 ⎠ ≤C C1k (λ) fk

|λ| ⎣⎝

e

dx ⎪ E k=1 ⎩ 0

⎛

⎞ p1

p 1

−x(A+λI) 12

⎠ +⎝

R1k (λ) fk

e

dx ⎛

E

0

⎞ p1

p 1

−(1−x)(A+λI) 12

⎠ +⎝

C2k (λ) fk

e

dx E

0

⎞ p1 ⎤

p 1

−(1−x)(A+λI) 12

⎠ ⎥ +⎝

R2k (λ) fk

⎦

e

dx ⎛

E

0

−1

+ 1 + A (A + λI)

B(E) ⎡⎛ ⎞ p1

p 1

1

⎢ −x(A+λI) 2 ⎠ × ⎣⎝

C1k (λ) fk

(A + λI) e

dx 0

E

(2.18)

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B. A. Aliev and Ya. Yakubov

IEOT

⎛

⎞ p1

p 1

1

−x(A+λI) 2 ⎠ +⎝

R1k (λ) fk

(A + λI) e

dx ⎛

E

0

⎞ p1

p 1

1

−(1−x)(A+λI) 2 ⎠ +⎝

C2k (λ)fk

(A + λI)e

dx E

0

⎞ p1 ⎤⎫

p ⎪ 1

⎬ 1

⎥ −(1−x)(A+λI) 2

⎠ (A + λI) e +⎝

R (λ)f dx ⎦ . 2k k

⎪ ⎭ E ⎛

(2.19)

0

By [27, Theorem 5.4.2/1] and the form of the operators C1k , for the ﬁrst term of the right hand side of inequality (2.19) we have, for | arg λ| ≤ ϕ and |λ| sufﬁciently large, ⎛

⎞ p1

p 1

−x(A+λI) 12

⎠ |λ| ⎝

C1k (λ) fk

e

dx E

0

⎛

⎞ p1

p 1

−x(A+λI) 12

1 − ⎠ ≤ C |λ| ⎝

(A + λI) 2 fk

e

dx E

0

−1

≤ C |λ| (A + λI)

⎛ ×⎝

B(E)

1

⎞ p1

p 1

1

(A + λI) 2 e−x(A+λI) 2 fk dx⎠

E

0

≤C

2 k=1

fk (E(A),E) 1 2

+ 1 ,p 2p

+ |λ|

1 1 2 − 2p

fk E .

By the same [27, Theorem 5.4.2/1] and estimates (2.17) and (2.18), for sufﬁciently large |λ| from the sector |arg λ| ≤ ϕ, for the second term of the right hand side of inequality (2.19), we have ⎞ p1

p 1

1

−x(A+λI) 2

⎠ R1k (λ) fk

|λ| ⎝

e

dx ⎛

0

−1

≤ C |λ| (A + λI)

E

B(E)

⎛

⎞ p1

p 1

1

−x(A+λI) 2 ⎠ ×⎝

R1k (λ) fk

(A + λI) e

dx E

0

≤C

2 k=1

R1k (λ) fk (E(A),E)

1 ,p 2p

1 1− 2p

+ |λ|

R1k (λ) fk E

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 281 1

−ω|λ| 2

≤ Ce ≤C

2 k=1

2 fk (E(A),E) 1 2

k=1

fk (E(A),E) 1 2

+ 1 ,p 2p 1

+ 1 ,p 2p

+ |λ| 2

+ |λ|

1 − 2p

1 1− 2p

fk E

fk E .

In the third term of the right hand side of inequality (2.19), the integral with C21 (λ) is estimated as the integral with C12 (λ) in the ﬁrst term β C12 (λ). Let us of the right hand side of inequality (2.19) since C21 (λ) = − α estimate the integral with C22 (λ) in the third term of the right hand side of inequality (2.19). Again, by [27, Theorem 5.4.2/1], estimate (2.10), [27, Lemma 5.4.2/6], and condition 3 we have ⎞ p1 ⎛ 1

p

−(1−x)(A+λI) 12

⎠ C22 (λ) f2

|λ| ⎝

e

dx 0

E

⎛

−1

≤ C |λ| (A + λI)

B(E)

1

1

⎝ (A + λI) e−(1−x)(A+λI) 2

0

− 12

× (A + λI) ! ≤C

− 12

B (A + λI)

⎞ p1

p

⎠ f2

dx E

−1 −1

(A + λI) 2 B (A + λI) 2 f2

(E(A),E)

1 1− 2p

+ |λ|

−1 −1

(A + λI) 2 B (A + λI) 2 f2

E

≤ C f2 (E(A),E) 1 2

1 1− 2p

+ 1 ,p 2p

≤C

k=1

− 12 − 12 12

BA A (A + λI) f2

B(E) E 1 1 1− − + |λ| 2p |λ| 2 f2 E 1 1 2 − 2p + |λ| f2 E

−1

(A + λI) 2

+ |λ| ≤ C f2 (E(A),E) 1 1 + ,p 2 2p ≤ C f2 (E(A),E) 1 1 2

1 ,p 2p

"

2

+

2p

,p

fk (E(A),E) 1 2

+ 1 ,p 2p

+ |λ|

1 1 2 − 2p

fk E .

Similarly, we can estimate all other terms of the right hand side of inequality (2.19) and to get the estimate of the theorem. Remark 2.2. In the framework of Hilbert spaces, by Remark 1.3, one has to remove R-boundedness restriction from condition 2—it is enough the normboundedness.

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#N1 γ1j u (x1j ) = Remark 2.3. When considering L10 := αu (1) + Bu(1) + j=1 # N 1 f1 , instead of αu (1) + Bu(0) + j=1 γ1j u (x1j ) = f1 , in problem (2.1)–(2.2), one can prove the same theorem as Theorem 2.1 but with the following addi1 1 −1 2 B(E) < |α| tional assumptions: BA− 2 B(E) < |α| c0 , A BA c0 , where the 1 1 −2 2 constant c0 > 0 is such that A (A + λI) B(E) ≤ c0 uniformly in the sector | arg λ| ≤ ϕ < π (the existence of such c0 follows, e.g., from [27, Lemma 5.4.2/6]).

3. Nonhomogeneous Equations Let us now consider a boundary value problem for a nonhomogeneous equation with a parameter L0 (λ)u := λu(x) − u (x) + Au(x) = f (x), L10 u := αu (1) + Bu(0) +

N1

x ∈ (0, 1),

γ1j u (x1j ) = f1 ,

j=1

L20 u := βu (0) +

N2

(3.1)

(3.2)

γ2j u (x2j ) = f2 .

j=1

Theorem 3.1. Let the following conditions be fulfilled: 1. α, β, γkj are complex numbers; α = 0, β = 0; xkj ∈ (0, 1); 2. An operator A is closed, densely defined and invertible in a UMD Banach space E and R{λR(λ, A) : | arg λ| ≥ π − ϕ} < ∞ for some 0 ≤ ϕ < π;2 1 3. A closed operator B is bounded from E(A 2 ) into E and from E (A) into 1 E(A 2 ). Then, the operator L0 (λ) : u → L0 (λ)u := (L0 (λ)u, L10 u, L20 u), for sufficiently large |λ| from the sector | arg λ| ≤ ϕ, is an isomorphism from Wp2 ((0, 1); E(A), E) onto ˙ (E(A), E) 1 + Lp ((0, 1); E) + 2

1 2p ,p

˙ (E(A), E) 1 + + 2

1 2p ,p

and for these λ the following estimate is true for the solution of problem (3.1)–(3.2) (also u(0) ∈ D(B)) |λ| uLp ((0,1);E) + u Lp ((0,1);E) + AuLp ((0,1);E) $ 2 1 − 1 ≤ C |λ| 2 2p f Lp ((0,1);E) + fk (E(A),E) 1 k=1

2

+|λ|

+ 1 ,p 2p

1 1 2 − 2p

% fk E . (3.3)

2

In fact, condition 2 is equivalent to that A is an invertible R-sectorial operator in E with the R-angle φR A < π − ϕ.

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 283 Proof. Injectivity of the mapping L0 (λ) follows from Theorem 2.1 since a homogeneous boundary value problem corresponding to boundary value problem (3.1)–(3.2) for sufﬁciently large |λ| from the sector | arg λ| ≤ ϕ < π has only a trivial solution. Thus, it is sufﬁcient to show that L0 (λ) is surjec1 tive, i.e., for any f ∈ Lp ((0, 1); E) and f1 , f2 ∈ (E(A), E) 12 + 2p ,p , there exists a solution of problem (3.1)–(3.2) belonging to Wp2 ((0, 1); E(A), E). Deﬁne f˜ (x) := f (x) if x ∈ [0, 1] and f˜ (x) = 0 if x ∈ [0, 1]. The solution of problem (3.1)–(3.2) is represented in the form of the sum u (x) = u1 (x) + u2 (x), where u1 (x) is the restriction on [0, 1] of the solution of the equation L0 (λ) u ˜1 (x) = f˜ (x), x ∈ R = (−∞, +∞), (3.4) and u2 (x) is a solution of the problem Lk0 u2 = fk − Lk0 u1 ,

L0 (λ) u2 = 0,

k = 1, 2.

(3.5)

It was shown in the proof of [13, Theorem of equation

iμx4] that a solution −1 1 ˜(μ)dμ, where (3.4) is given by the formula u ˜1 (x) = 2π e L (λ, iμ) F f 0 R F f˜ is the Fourier transform of the function f˜(x) and L0 (λ, σ) is a characteristic operator pencil of equation (3.4), i.e., L0 (λ, σ) = −σ 2 + A + λI. Moreover, the solution belongs to Wp2 (R; E(A), E) and for the solution it holds the estimate (see [13, formula (4.11)]) |λ| ˜ u1 Lp (R;E) + ˜ u1 Wp2 (R;E(A),E) ≤ Cf˜Lp (R;E) ,

|arg λ| ≤ ϕ.

(3.6)

Therefore, u1 ∈ Wp2 ((0, 1); E(A), E) and from (3.6), for |arg λ| ≤ ϕ, it follows that |λ| u1 Lp ((0,1);E) + u1 Wp2 ((0,1);E(A),E) ≤ C f Lp ((0,1);E) .

(3.7)

By [27, Theorem 1.7.7/1] (see also [21, Theorem 1.8.2]) and inequal(s) 1 ity (3.7), we have u1 (x0 ) ∈ (E(A), E) s2 + 2p ,p , ∀x0 ∈ [0, 1] , s = 0, 1. On 1

the other hand, since the operator B is bounded from (E(A 2 ), E(A))θ,p into 1 (E, E(A 2 ))θ,p , 0 < θ < 1 (see the proof of Theorem 2.1), then, using [21, formulas 1.15.2/(4) and 1.15.4/(2)], the operator B is also bounded from 1 (E(A), E) 1 ,p into (E(A), E) 12 + 2p 1 ,p . Then, Lk0 u1 ∈ (E(A), E) 12 + 2p ,p . Thus, 2p by Theorem 2.1, for sufﬁciently large |λ| from the sector |arg λ| ≤ ϕ, problem (3.5) has a unique solution u2 (x) that belongs to Wp2 ((0, 1); E(A), E). Moreover, for the solution of problem (3.5), for |arg λ| ≤ ϕ, |λ| → ∞, we have |λ| u2 Lp ((0,1);E) + u2 Lp ((0,1);E) + Au2 Lp ((0,1);E) 2 1 1 2 − 2p ≤C fk − Lk0 u1 E fk − Lk0 u1 (E(A),E) 1 1 + |λ| 2

k=1

≤C

2

k=1

fk (E(A),E) 1

+ Lk0 u1 (E(A),E) 1 2

2

+

2p

,p

1

+ 1 ,p 2p

+ |λ| 2 1

+ 1 ,p 2p

+ |λ| 2

1 − 2p

1 − 2p

fk E

Lk0 u1 E

284

B. A. Aliev and Ya. Yakubov $

2

IEOT

1 1 2 − 2p

fk E fk (E(A),E) 1 1 + |λ| + ,p 2 2p k=1 + u1 (1)(E(A),E) 1 1 + Bu1 (0)(E(A),E) 1 + ,p +

≤C

2

2p

+ u1 (x1j )(E(A),E) 1 2

2

+ 1 ,p 2p

+

u1 (0)(E(A),E) 1 + 2

1

+ u1 (x2j )(E(A),E) 1

1 ,p 2p 1 ,p 2p

1 − 2p

(u1 (1)E + Bu1 (0)E % & (3.8) + u1 (x1j )E + u1 (0)E + u1 (x2j )E . + 1 ,p 2 2p

+ |λ| 2

Hence, by [27, Theorem 1.7.7/1] (see also [21, Theorem 1.8.2]) and (3.7), for any x0 ∈ [0, 1], we have, for s = 0, 1, (s)

u1 (x0 )(E(A),E) s + 2

1 ,p 2p

≤ C u1 Wp2 ((0,1);E(A),E) ≤ C f Lp ((0,1);E) .

(3.9)

By [27, Theorem 1.7.7/2], for any complex number μ ∈ C and any u1 ∈ Wp2 ((0, 1); E), for s = 0, 1, 1 (s) 2−s 2+ 1 u1 (x0 )E ≤ C |μ| p u1 Wp2 ((0,1);E) + |μ| p u1 Lp ((0,1);E) . |μ| (3.10) 1 p

Dividing (3.10) by |μ| and substituting λ = μ2 , for λ ∈ C (and u1 ∈ Wp2 ((0, 1); E)), we have, for s = 0, 1, (s) 1− s − 1 (3.11) |λ| 2 2p u1 (x0 )E ≤ C u1 W 2 ((0,1);E) + |λ| u1 Lp ((0,1);E) . p

Then, from (3.7) and (3.11), for | arg λ| ≤ ϕ, we have s

1

|λ|1− 2 − 2p u1 (x0 )E ≤ Cf Lp ((0,1);E) , (s)

s = 0, 1.

(3.12)

Taking into account (3.8), (3.9), (3.12), condition 3, and that the operator B is bounded from (E(A), E) 1 ,p into (E(A), E) 1 + 1 ,p we get, for | arg λ| ≤ ϕ, 2p

2

2p

|λ| u2 Lp ((0,1);E) + u2 Lp ((0,1);E) + Au2 Lp ((0,1);E) $ 2 1 1 2 − 2p ≤C fk E + f Lp ((0,1);E) fk (E(A),E) 1 1 + |λ| 2

k=1 1

+ |λ| 2

1 − 2p

,p

1

1

the operator A

2p

%

A 2 u1 (0)E .

Estimate now |λ| 2 1 2

+

1 − 2p

(3.13)

1

A 2 u1 (0)E . Since u1 (0) ∈ (E(A), E)

is bounded from (E(A), E) 1 2

1 2p ,p

into (E(A), E) 1 + 2

1 2p ,p 1 2p ,p

and (see

[21, Theorem 1.15.2/(e)]) then, obviously, A u1 (0) ∈ (E(A), E) 1 + 1 ,p ⊂ E. 2 2p Then, by (3.9),

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 285 1

1

A 2 u1 (0)E ≤ CA 2 u1 (0)(E(A),E) 1 + 2

1 ,p 2p

≤ C u1 (0)(E(A),E)

1 ,p 2p

≤ C f Lp ((0,1);E) . So, for |arg λ| ≤ ϕ, 1

|λ| 2

1 − 2p

1

1

A 2 u1 (0)E ≤ C |λ| 2

1 − 2p

f Lp ((0,1);E) .

(3.14)

Taking into account (3.14), from (3.13) we have |λ| u2 Lp ((0,1);E) + u2 Lp ((0,1);E) + Au2 Lp ((0,1);E) $ 2 1 − 1 ≤ C |λ| 2 2p f Lp ((0,1);E) + fk (E(A),E) 1 1 2

k=1

+

2p

,p

+|λ|

1 1 2 − 2p

% fk E

.

(3.15) Hence, (3.3) follows from (3.7) and (3.15) since u = u1 + u2 .

Remark 3.2. In the framework of Hilbert spaces, by Remark 1.3, condition 2 has to be changed to the condition: 2 . A is a closed, densely deﬁned operator in a Hilbert space H and R(λ, A) ≤ C(1 + |λ|)−1 ,

| arg λ| ≥ π − ϕ,

with some 0 ≤ ϕ < π, and A is invertible.

4. Fredholm Property Consider now a boundary value problem without the spectral parameter λ in the equation but with perturbed both the equation and the boundary conditions Lu := −u (x) + Au(x) + (A1 u)(x) = f (x), L1 u := αu (1) + Bu (0) +

N1

L2 u := βu (0) +

(4.1)

γ1j u (x1j ) + T1 u = f1 ,

j=1 N2

x ∈ (0, 1),

(4.2)

γ2j u (x2j ) + T2 u = f2 .

j=1

Let us recall that the main operator of the equation is A and of the boundary conditions is B. The operators A1 , T1 , and T2 are treated below by standard perturbation arguments. Theorem 4.1. Let the following conditions be fulfilled: 1. α, β, γkj are complex numbers; α = 0, β = 0; xkj ∈ (0, 1); 2. An operator A is closed, densely defined in a UMD Banach space E and R{λR(λ, A) : λ ≤ −M } < ∞ for some M ≥ 0;3 3. The embedding E (A) ⊂ E is compact; 1 4. The operator B is bounded from E(A 2 ) into E and from E(A) into 1 E(A 2 ); 3

For example, any R-sectorial operator in E with the R-angle φR A < π satisfies condition 2.

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5. For any ε > 0 and u ∈ Wp2 ((0, 1);E(A), E), p ∈ (1, ∞), A1 uLp ((0,1);E) ≤ ε uW 2 ((0,1);E(A),E) + C (ε) uLp ((0,1);E) ; p

6. For any ε > 0 and u ∈ Tk u(E(A),E) 1 2

+ 1 ,p 2p

Wp2 ((0, 1); E(A), E),

k = 1, 2,

≤ ε uWp2 ((0,1);E(A),E) + C (ε) uLp ((0,1);E) .

Then, the operator L : u → Lu := (Lu, L1 u, L2 u) from Wp2 ((0, 1); E(A), E) into ˙ (E(A), E) 1 + Lp ((0, 1); E) + 2

1 2p ,p

˙ (E(A), E) 1 + + 2

1 2p ,p

is bounded and Fredholm, and u(0) ∈ D(B). Proof. Without loss of generality, we can assume that condition 2 is satisﬁed for all λ ≤ 0 (λ = 0 means that A is invertible). Our general case is reduced to the latter if the operator A + M I is considered instead of the operator A, and the operator A1 − M I is considered instead of the operator A1 . We can represent the operator L in the form L = L0 (λ0 ) + L1 ,

(4.3)

where L0 (λ) u := (L0 (λ) u, L10 u, L20 u), with L0 (λ) := −u (x) + (A + λI) u (x) , L10 u := αu (1) + Bu(0) +

N1

γ1j u (x1j ),

j=1

L20 u := βu (0) +

N2

γ2j u (x2j ),

j=1

and L1 u := (−λ0 u (x) + (A1 u)(x), T1 u, T2 u). By Theorem 3.1, for sufﬁciently large λ0 > 0, the operator L0 (λ0 ) from ˙ (E(A), E) 1 + 1 ,p + ˙ (E(A), E) 1 + 1 ,p Wp2 ((0, 1); E(A), E) onto Lp ((0, 1); E) + 2 2p 2 2p is invertible. From condition 3, by [27, Theorem 5.2.1/1], it follows that the embedding Wp2 ((0, 1); E(A), E) ⊂ Lp ((0, 1); E) is compact. By conditions 5 and 6, for any u ∈ Wp2 ((0, 1); E(A), E), we have L1 uLp ((0,1);E)+(E(A),E) ˙ 1 2

˙ +(E(A),E) 1 + 1 ,p + 1 ,p 2p 2 2p

≤ λ0 uLp ((0,1);E) +A1 uLp ((0,1);E) + T1 u(E(A),E) 1 1 + T2 u(E(A),E) 1 1 + ,p + ,p 2 2p 2 2p ≤ C uLp ((0,1);E) + ε uW 2 ((0,1);E(A),E) + C (ε) uLp ((0,1);E) p

≤ ε1 uWp2 ((0,1);E(A),E) + C (ε1 ) uLp ((0,1);E) . Hence, by [27, Lemma 1.2.7/2], the operator L1 from Wp2 ((0, 1); E(A), E) ˙ (E(A), E) 1 + 1 ,p + ˙ (E(A), E) 1 + 1 ,p is compact. Applying into Lp ((0, 1); E)+ 2 2p 2 2p [17, Sect. 14, Theorem 14.1] to the operator L, we complete the proof.

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 287 Remark 4.2. In the framework of Hilbert spaces, by Remark 1.3, condition 2 has to be changed to the condition: 2 . A is a closed, densely deﬁned operator in a Hilbert space H and R(λ, A) ≤ C(1 + |λ|)−1 ,

arg λ = π, |λ| → ∞.

Remark 4.3. By [21, Theorem 1.8.2] (see also [27, Theorem 1.7.7/1]), the 1 operator u(x) → u(0) : Wp2 ((0, 1); E(A), E) → (E(A), E) 2p ,p is bounded. It is 1

2 1 1 shown in the proof of Theorem 2.1 that (E(A), E) 2p ,p = (E(A ), E(A))1− p ,p , 1

1 (E, E(A 2 ))1− p1 ,p = (E(A), E) 12 + 2p ,p , and that the operator B is bounded 1

1

from (E(A 2 ), E(A))θ,p into (E, E(A 2 ))θ,p for any 0 < θ < 1. Therefore, the operator u(x) → Bu(0) is bounded from Wp2 ((0, 1); E(A), E) into 1 (E(A), E) 12 + 2p ,p . By [27, Lemma 1.2.7/2 and Theorem 5.2.1/1], from condition 6, it follows that the operator T1 is compact from Wp2 ((0, 1); E(A), E) into (E(A), 1 E) 12 + 2p ,p , i.e., the term Bu(0) in the ﬁrst boundary condition of (4.2) is, in general, in the principal part of the boundary condition and cannot be combined with the perturbation term T1 u.

5. Isomorphism of the Whole Problem: With a Linear Parameter in the Equation and Perturbation Operators in Both the Equation and the Boundary Conditions Let s > 0 and let E, E1 be Banach spaces. Denote, for 0 < s ≤ 1, p ∈ (1, ∞), a Banach space ' & ' & Bps (0, 1); (E1 , E)1− 2s ,2 , E := Wp2 ((0, 1); E1 , E), Lp ((0, 1); E) 1− s ,2 2

(5.1)

and, for 1 < s < 2, a Banach space ' & Bps (0, 1); (E1 , E)1− 2s ,2 , E := Wp2 ((0, 1); E1 , E), Bp1 ((0, 1); (E1 , E) 12 ,2 , E)

2−s,2

.

(5.2)

Consider now our full boundary value problem in E mentioned in Sect. 1, i.e., L (λ) u := λu (x) − u (x) + Au (x) + (A1 u)(x) = f (x) , L1 u := αu (1) + Bu(0) +

N1

L2 u := βu (0) +

j=1

(5.3)

γ1j u (x1j ) + T1 u = f1 ,

j=1 N2

x ∈ (0, 1),

γ2j u (x2j ) + T2 u = f2 .

(5.4)

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Theorem 5.1. Let the following conditions be fulfilled: 1. α, β, γkj are complex numbers; α = 0, β = 0; xkj ∈ (0, 1); 2. An operator A is closed, densely defined in a UMD Banach space E and R{λR(λ, A) : | arg λ| ≥ π − ϕ, |λ| ≥ M } < ∞ for some 0 ≤ ϕ < π and some M ≥ 0;4 3. The embedding E(A) ⊂ E is compact; 1 4. The operator B is bounded from E(A 2 ) into E and from E(A) into 1 E(A 2 ); 5. The operator A1 is bounded in Lp ((0, 1); E), p ∈ (1, ∞); 6. For any ε > 0 and u ∈ Wp2 ((0, 1); E(A), E), k = 1, 2, Tk u(E(A),E) 1 2

+ 1 ,p 2p

≤ ε uW 2 ((0,1);E(A),E) + C(ε) uLp ((0,1);E) , p

Tk uE ≤ ε u

1+ 1 p

Bp

(0,1);(E(A),E) 1 − 2

+C(ε) u Lp ((0,1);E) .

1 ,2 ,E 2p

Then, the operator L (λ) : u → L (λ) u := (L(λ)u, L1 u, L2 u), for sufficiently large |λ| from the sector |arg λ| ≤ ϕ, is an isomorphism from Wp2 ((0, 1); E(A), E) onto ˙ (E(A), E) 1 + Lp ((0, 1); E) + 2

1 2p ,p

˙ (E(A), E) 1 + + 2

1 2p ,p

,

the estimate |λ| uLp ((0,1);E) + u Lp ((0,1);E) + AuLp ((0,1);E) $ 2 1 1 2 − 2p ≤ C |λ| f Lp ((0,1);E) + fk (E(A),E) 1 k=1

2

+ 1 ,p 2p

+ |λ|

1 1 2 − 2p

% fk E (5.5)

holds for the solution of problem (5.3)–(5.4), and u(0) ∈ D(B). Proof. As in beginning of the proof of Theorem 4.1, we can assume, without loss of generality, that condition 2 is satisﬁed for the whole sector | arg λ| ≥ π − ϕ (including λ = 0, which means that A is invertible). Let u ∈ Wp2 ((0, 1); E(A), E) be a solution of problem (5.3)–(5.4). Then, u(x) is a solution of the problem L0 (λ) u = f − A1 u, Lk0 u = fk − Tk u, k = 1, 2,

4

For example, any R-sectorial operator in E with the R-angle φR A < π − ϕ satisfies condition 2.

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 289 where Lk0 u, k = 1, 2, are determined by equalities (3.2). Then, by Theorem 3.1, for sufﬁciently large |λ| from the sector |arg λ| ≤ ϕ, we have |λ| uLp ((0,1);E) + u Lp ((0,1);E) + AuLp ((0,1);E) $ 1

≤ C |λ| 2 +

2

1 − 2p

f − A1 uLp ((0,1);E)

fk − Tk u(E(A),E) 1 2

k=1

+ 1 ,p 2p

+ |λ|

1 1 2 − 2p

% fk − Tk uE

.

(5.6)

By condition 5, 1

1

|λ| 2 − 2p f − A1 uLp ((0,1);E) 1

1

1

1

1

1

≤ |λ| 2 − 2p f Lp ((0,1);E) + |λ| 2 − 2p A1 uLp ((0,1);E) 1

1

≤ |λ| 2 − 2p f Lp ((0,1);E) + C|λ| 2 − 2p uLp ((0,1);E) .

(5.7)

By condition 6, fk − Tk u(E(A),E) 1 2

+ 1 ,p 2p

≤ fk (E(A),E) 1 2

+ C(ε) uLp ((0,1);E) , ε > 0, u ∈

+ 1 ,p 2p

Wp2

+ ε uW 2 ((0,1);E(A),E) p

((0, 1); E(A), E).

(5.8)

Obviously, 1

|λ| 2 1 p

1 − 2p

1

fk − Tk uE ≤ |λ| 2

1 − 2p

(fk E + Tk uE ) .

(5.9)

By virtue of (5.2) and [27, Lemma 1.7.3/8] (take E1 = E(A) and s = 1+ 2 1 in (5.2) and E0 := Wp ((0, 1); E(A), E), E1 := Bp (0, 1); (E(A), E) 12 ,2 , E ,

and θ = 1 − |μ|

1 1− p

1 p

u

in [27, Lemma 1.7.3/8]), we have ! 1+ 1 p

Bp

(0,1);(E(A),E) 1 − 2

"

+uWp2 ((0,1);E(A),E)

,

1 ,2 ,E 2p

≤C

Bp1 (0,1);(E(A),E) 1 ,2 ,E

|μ| u

2

μ ∈ C, u ∈ Wp2 ((0, 1); E(A), E).

Replacing λ = μ2 , as a result, we have, for λ ∈ C, ! 1

|λ| 2

1 − 2p

u

1+ 1 Bp p

(0,1);(E(A),E) 1 −

+ uW 2 ((0,1);E(A),E) p

2

"

,

1 ,2 ,E 2p

≤C

1

Bp1 (0,1);(E(A),E) 1 ,2 ,E

|λ| 2 u

2

u ∈ Wp2 ((0, 1); E(A), E) .

(5.10)

Further, in (5.1) we take E1 = E(A), s = 1, and θ = 12 . Then, ' & Bp1 (0, 1); (E(A), E) 12 ,2 , E := Wp2 ((0, 1); E(A), E) , Lp ((0, 1); E) 1 ,2 . 2

290

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By [27, Lemma 1.7.3/8], for λ ∈ C, u ∈ Wp2 ((0, 1); E(A), E), 1 ≤ C u + |λ| u |λ| 2 u 1 2 W ((0,1);E(A),E) Lp ((0,1);E) . Bp (0,1);(E(A),E) 1 ,2 ,E

p

2

(5.11) Then, taking into account (5.11) into (5.10), we get 1

|λ| 2

1 − 2p

u

1+ 1 p

Bp

(0,1);(E(A),E) 1 − 2

1 ,2 ,E 2p

≤ C uW 2 ((0,1);E(A),E) + |λ| uLp ((0,1);E) .

p

(5.12)

From condition 6 and inequality (5.12) we have 1 − 1 |λ| 2 2p Tk uE ≤ Cε uWp2 ((0,1);E(A),E) + |λ|uLp ((0,1);E) 1

1

+ C(ε)|λ| 2 − 2p uLp ((0,1);E) .

(5.13)

Then, taking into account (5.13) into (5.9), we have 1 1 1 − 1 |λ| 2 − 2p fk − Tk uE ≤ |λ| 2 2p fk E + Cε uWp2 ((0,1);E(A),E) 1 − 1 + |λ| uLp ((0,1);E) + C(ε) |λ| 2 2p uLp ((0,1);E) . (5.14) Hence, by (5.7), (5.8), and (5.14), from (5.6) we have − 1+ 1 − 1+ 1 1 − Cε − C |λ| ( 2 2p ) − C(ε) |λ| ( 2 2p ) |λ| uLp ((0,1);E) $ 1 − 1 + u Lp ((0,1);E) + AuLp ((0,1);E) ≤ C |λ| 2 2p f Lp ((0,1);E) 2 + fk (E(A),E) 1 k=1

2

+ 1 ,p 2p

+ |λ|

1 1 2 − 2p

% fk E

.

(5.15)

First, choose ε0 such that Cε0 < 1. Furthermore, choose |λ| so that Cε0 + − 1+ 1 − 1+ 1 C |λ| ( 2 2p ) + C(ε0 ) |λ| ( 2 2p ) < 1. Then, from (5.15) we get (5.5). Consequently, for sufﬁciently large |λ| from the sector |arg λ| ≤ ϕ, the solution of problem (5.3)–(5.4) in Wp2 ((0, 1); E(A), E) is unique. By Theorem 4.1, for ˙ each such λ, the operator L (λ) from Wp2 ((0, 1); E(A), E) into Lp ((0, 1); E) + ˙ (E(A), E) 1 + 1 ,p is a Fredholm operator. Then, an isomor(E(A), E) 1 + 1 ,p + 2 2p 2 2p phism follows from the uniqueness and the Fredholm property. Remark 5.2. In the framework of Hilbert spaces, by Remark 1.3, condition 2 has to be changed to the condition: 2 . A is a closed, densely deﬁned operator in a Hilbert space H and R(λ, A) ≤ C(1 + |λ|)−1 ,

| arg λ| ≥ π − ϕ, |λ| → ∞,

with some 0 ≤ ϕ < π. Remark 5.3. A similar remark, as Remark 2.3, is true for Theorems 3.1, 4.1, and 5.1.

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 291

6. Completeness of a System of Root Functions Let us, ﬁrst, formulate and prove a completeness theorem in the framework of Hilbert spaces. Consider a homogeneous problem for (5.3)–(5.4) in a Hilbert space H, i.e., L(λ)u := λu (x) − u (x) + Au (x) + (A1 u)(x) = 0, L1 u := αu (1) + Bu(0) +

N1

x ∈ (0, 1),

γ1j u (x1j ) + T1 u = 0,

j=1

L2 u := βu (0) +

N2

(6.1)

(6.2)

γ2j u (x2j ) + T2 u = 0.

j=1

A number λ0 is called an eigenvalue of problem (6.1)–(6.2) if the problem L(λ0 )u = 0 , Lk u = 0, k = 1, 2, has a non-trivial solution u0 (x) that belongs to W22 ((0, 1); H(A), H) and u0 (x) is called an eigenfunction of problem (6.1)–(6.2) corresponding to the eigenvalue λ0 . A solution um (x), for a natural number m ≥ 1, of the problem L (λ0 ) um + um−1 = 0 , Lk um = 0, k = 1, 2, belonging to W22 ((0, 1); H(A), H), is called an m-th associated function to the eigenfunction u0 (x) of problem (6.1)–(6.2). We combine the eigenfunctions and associated functions of problem (6.1)–(6.2) under the general name of root functions of problem (6.1)–(6.2). Let an operator C from a Hilbert space H into a Hilbert space H1 be bounded. Then its adjoint operator C ∗ from H1 into H is bounded and, for u ∈ H, u1 ∈ H1 , we have (Cu, u1 )H1 = (u, C ∗ u1 )H . Since (C ∗ C)∗ = C ∗ C ∗∗ = C ∗ C, the operator C ∗ C in H is selfadjoint. From (C ∗ Cu, u)H = (Cu, Cu)H ≥ 0 it follows that the operator C ∗ C in H is nonnegative. In turn, it implies that there exists a unique non-negative selfadjoint 1 operator T := (C ∗ C) 2 in H. If C from a Hilbert space H into a Hilbert space 1 H1 is compact, then, in addition to the above, the operator T = (C ∗ C) 2 in H is compact. The eigenvalues of the operator T are called singular numbers of the compact operator C and are denoted by sj (C; H, H1 ). Enumerate the singular numbers in decreasing order, taking into account their multiplicities, so that sj (C; H, H1 ) := λj (T ),

j = 1, . . . , ∞.

Theorem 6.1. Let the following conditions be fulfilled: 1. α, β, γkj are complex numbers; α = 0, β = 0; xkj ∈ (0, 1); 2. The embedding H(A) ⊂ H is compact and for some t > 0, for the embedding operator J, it holds that sj (J; H(A), H) ≤ Cj −t , j = 1, 2, . . .; 3. The operator A is closed, densely defined in a Hilbert space H and, for 2π < ϕ < π, some 2+t R(λ, A) ≤ C(1 + |λ|)−1 ,

| arg λ| ≥ π − ϕ, |λ| → ∞;

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1

4. The operator B is bounded from H(A 2 ) into H and from H(A) into 1 H(A 2 ); 5. The operator A1 is bounded in L2 ((0, 1); H); 6. For any ε > 0 and u ∈ W22 ((0, 1); H(A), H), k = 1, 2, Tk u(H(A),H) 3 4

,2

≤ ε uW22 ((0,1);H(A),H) + C(ε) uL2 ((0,1);H) , 3 B22 (0,1);(H(A),H) 1 ,2 ,H

Tk uH ≤ ε u

+ C(ε) uL2 ((0,1);H) .

4

Then, the spectrum of problem (6.1)–(6.2) is discrete and a system of root functions of problem (6.1)–(6.2) is complete in the space L2 ((0, 1); H). Proof. In the space H := L2 ((0, 1); H), consider an operator A which is deﬁned by the equalities D(A) := W22 ((0, 1); H(A), H; Lk u = 0, k = 1, 2) , Au = −u (x) + Au(x) + (A1 u)(x).

(6.3)

Apply [27, Theorem 2.2.2/1] to the operator A in H. Using the same technique as in the proof of [25, Theorem 4], one can show that W22 ((0, 1); H(A), H; Lk u = 0, k = 1, 2) is dense in L2 ((0, 1); H), i.e., the ﬁrst condition of [27, Theorem 2.2.2/1] is fulﬁlled. By [27, Theorem 5.2.1/1], the embedding W22 ((0, 1); H(A), H) ⊂ L2 ((0, 1); H) = H is compact, i.e., the embedding H(A) ⊂ H is also compact. By [27, Lemmas 1.2.10/3 and 1.7.8/6], ' & 2t sj (J; H(A), H) ≤ C sj J; W22 ((0, 1); H(A), H) , L2 ((0, 1); H) ≤ Cj − 2+t , i.e., the second condition of [27, Theorem 2.2.2/1] is also fulﬁlled (with p = 2t 2+t in [27, Theorem 2.2.2/1]). By Theorem 5.1 (in the framework of Hilbert spaces and for p = 2), in view of Remark 1.3, for sufﬁciently large |λ| from the angle |arg λ| ≤ ϕ < π, for a solution of the equation λu + Au = f it holds the estimate 1

|λ| uL2 ((0,1);H) + u L2 ((0,1);H) + AuL2 ((0,1);H) ≤ C |λ| 4 f L2 ((0,1);H) . From the last estimate, for sufﬁciently large |λ| from the angle |arg λ| ≤ ϕ < π, we have R (λ, −A) ≤ C |λ|

− 34

.

Consequently, R (−λ, A) ≤ C |λ|

− 34

,

|arg(−λ)| ≥ π − ϕ, |λ| → ∞.

2π πt Since ϕ > 2+t then π − ϕ < 2+t . Therefore, condition (3) of [27, Theorem 2t 2.2.2/1] is also satisﬁed (with η = 34 and, previously chosen, p = 2+t in [27, Theorem 2.2.2/1]).

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 293 In order to formulate a completeness theorem in the framework of Banach spaces, we need a definition of approximation numbers (of a compact operator) which coincide with (the operator’s) singular numbers in the framework of Hilbert spaces (see, e.g., [27, Theorem 1.2.10/2]). Let C be a compact operator from a Banach space E into a Banach space E1 . Then, s˜j (C; E, E1 ) :=

inf

dim R(K)<j

C − KB(E,E1 )

K∈B(E,E1 )

are said to be the approximation numbers of C, where R(K) denotes the image of the operator K. Consider now problem (6.1)–(6.2) in a UMD Banach space E and in the space E := Lp ((0, 1); E) introduce an operator A which is deﬁned by the equalities D(A) := Wp2 ((0, 1); E(A), E; Lk u = 0, k = 1, 2) , Au = −u (x) + Au(x) + (A1 u)(x). Theorem 6.2. Let the spectrum of the operator A be non empty; for some 0 < s < 2, s˜j (J; Wp2 ((0, 1); E(A), E), Lp ((0, 1); E)) ≤ Cj −s ; and let all conditions of Theorem 5.1 be satisfied. Moreover, condition 2 of Theorem 5.1 is satisfied for some 2−s 2 π < ϕ < π. Then, the spectrum of problem (6.1)–(6.2) in E is discrete and a system of root functions of problem (6.1)–(6.2) is complete in the space Lp ((0, 1); E). Proof. The proof repeats the steps of the proof of Theorem 6.1, taking Remark 1.3 into account, but one has to use, instead of [27, Theorem 2.2.2/1], the Burgoyne’s theorem [6, Theorem 4.5], which is also presented in a more convenient form for using in application in [28, Theorem 1]. As well, one should to use [28, Lemma 2] instead of [27, Lemma 1.2.10/3]. Remark 6.3. Due to Remark 2.3 (see also Remark 5.3), one can consider the #N1 γ1j u (x1j ) + T1 u = 0 following boundary condition αu (1) + Bu(1) + j=1 # N1 γ1j u (x1j )+T1 u = 0, in boundary conditions instead of αu (1)+Bu(0)+ j=1 (6.2), with the same additional assumptions as in Remark 2.3.

7. Application of Abstract Results to Elliptic Partial Differential Equations We will consider a very particular model nonlocal boundary value problem for a second order elliptic differential equation with a parameter in the square Ω = [0, 1] × [0, 1], namely, when an underlying space E is a Hilbert space, a principal operator of the equation is selfadjoint and positive-definite, and boundary conditions on two sides of the square are Dirichlet boundary conditions. For ideas of possible general boundary value problems, namely, multidimensional (and not two-dimensional) case, a completely Banach space framework, a nonselfadjoint main operator of the equation, and not necessarily Dirichlet boundary conditions on all sides of the square, we refer the

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reader to [13]. Thus, consider, for (x, y) ∈ Ω, L(λ)u := λu(x, y) − Dx2 u(x, y) − Dy (a(y)Dy u(x, y)) +a1 (x, y)u(x, y) = f (x, y),

(7.1)

L1 u := αDx u (1, y) + b (y) Dy u (0, y) + c(y)u(0, g(y)) 1 (b(t, y)Dt u(0, t) + c(t, y)u(0, h(t))) dt + 0

+

N1

y ∈ [0, 1] ,

γ1j Dx u (x1j , y) + (T1 u) (y) = f1 (y) ,

(7.2)

j=1

L2 u := βDx u (0, y) +

N2

γ2j Dx u (x2j , y) + (T2 u) (y)

j=1

= f2 (y) ,

y ∈ [0, 1] , L3 u := u (x, 0) = 0,

x ∈ [0, 1] ,

L4 u := u (x, 1) = 0,

x ∈ [0, 1] ,

(7.3)

where λ is a complex parameter; α, β, γkj are complex numbers; xkj ∈ (0, 1); a (y), a1 (x, y), b (y), c(y), g(y), b(x, y), c(x, y), and h(y) are some functions, ∂ ∂ g(y) and h(y) map [0, 1] into itself; Dx = ∂x , Dy = ∂y ; Tk are operators from Lp ((0, 1); L2 (0, 1)) into L2 (0, 1). Denote an interpolation space of Sobolev spaces by ' & s Bq,p (0, 1) := Wqs0 (0, 1), Wqs1 (0, 1) θ,p , where 0 ≤ s0 , s1 are integers, 0 < θ < 1, 1 < q < ∞, 1 ≤ p ≤ ∞, and s = (1 − θ) s0 + θs1 . Set ' & s Wps (0, 1) := Bp,p (0, 1) := Wps0 (0, 1) , Wps1 (0, 1) θ,p 1 1+ p

if 0 < s = integer. For the definition of Bp

in Theorem 7.1 see (5.2).

Theorem 7.1. Let the following conditions be fulfilled: 1. α, β, γkj are complex numbers; α = 0, β = 0; xkj ∈ (0, 1); a(y) ∈ C 1 [0, 1] , a(y) > 0 for y ∈ [0, 1]; a1 (x, y) ∈ C([0, 1]2 ); b(y) ∈ C 1 [0, 1], b(0) = b(1) = 0; c(y), g(y), h(y) ∈ C 1 [0, 1], g(y) and h(y) map [0, 1] into itself, c(0) = c(1) = 0; b(t, y), c(t, y) ∈ C 1 ([0, 1]2 ), b(t, 0) = b(t, 1) = c(t, 0) = c(t, 1) = 0 for t ∈ [0, 1];5 2. For any ε > 0 and u ∈ Wp2 ((0, 1);W22 (0, 1), L2 (0, 1)), p ∈ (1, ∞), Tk u

1− 1 p

B2,p

(0,1)

≤ ε uW 2 ((0,1);W 2 (0,1),L2 (0,1)) + C (ε) uLp ((0,1);L2 (0,1)) , p

Tk uL2 (0,1) ≤ ε u

1+ 1 p

Bp

5

2

!

1+ 1 p

(0,1);W2

" +C (0,1),L2 (0,1)

(ε) uLp ((0,1);L2 (0,1)) .

The restriction c(0) = c(1) = 0 can be replaced by g(0) ∈ {0, 1}, g(1) ∈ {0, 1} or g(0) ∈ {0, 1}, c(1) = 0 or g(1) ∈ {0, 1}, c(0) = 0.

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 295 Then, the operator L (λ) : u → L (λ) u := (L(λ)u, L1 u, L2 u), for sufficiently large |λ| from the sector |arg λ| ≤ ϕ < π, is an isomorphism from Wp2 ((0, 1); W22 (0, 1), L2 (0, 1)) onto 1− 1

1− 1

p p ˙ 2,p,∗ ˙ 2,p,∗ (0, 1)+B (0, 1), Lp ((0, 1); L2 (0, 1)) +B

where

⎧ 1− 1 ⎪ B2,p p (0, 1), 1 2,

and for the solution u(x, y) of problem (7.1)–(7.3) it holds that u(0, y) ∈ W21 (0, 1) and the following estimate is true |λ|u(x, y)Lp ((0,1);L2 (0,1)) + Dx2 u(x, y)Lp ((0,1);L2 (0,1)) + Dy (a(y)Dy u(x, y))Lp ((0,1);L2 (0,1)) $ ≤ C |λ| + |λ|

1 1 2 − 2p

1 1 2 − 2p

f (x, y)Lp ((0,1);L2 (0,1)) + %

fk (y)L2 (0,1)

2

fk (y)

k=1

1− 1

B2,p p (0,1)

.

Proof. In the space E := L2 (0, 1), consider operators A and B which are deﬁned by the following equalities D(A) := W22 ((0, 1); u(0) = u(1) = 0), D(B) :=

W21 (0, 1),

Bu := b(y)u (y) + c(y)u(g(y)) +

1

Au := −(a(y)u (y)) ,

(7.4) (7.5)

(b(t, y)u (t) + c(t, y)u(h(t))) dt

0

and, in the space Lp ((0, 1); E), consider an operator A1 which is deﬁned by the equalities D(A1 ) := Lp ((0, 1); E),

(A1 u)(x) := a1 (x, y)u(x, y).

(7.6)

Then, problem (7.1)–(7.3) can be rewritten in the operator form (5.3)–(5.4) and one can apply Theorem 5.1. Let us check all conditions of Theorem 5.1 for all our above operators. From condition 1, it follows that the operator A is selfadjoint and positive-definite in E = L2 (0, 1). Therefore, condition 2 of Theorem 5.1 is fulﬁlled with any 0 ≤ ϕ < π, taking Remark 1.3 into account. The embedding W22 (0, 1) ⊂ L2 (0, 1) is compact (see, e.g., [21, Theorem 3.2.5]), i.e., condition 3 of Theorem 5.1 is fulﬁlled too. By [12] (see also [21, Theorem 4.3.3]), 1

D(A 2 ) = W21 ((0, 1); u(0) = u(1) = 0).

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1

The operator B is bounded from E(A 2 ) into E = L2 (0, 1). Indeed, for any 1 u ∈ D(A 2 ), since W21 (0, 1) ⊂ C[0, 1], BuL2 (0,1) ≤ b(y)u (y)L2 (0,1) + c(y)u(g(y))L2 (0,1)

1

1

c(t, y)u(h(t))dt

+

b(t, y)u (t)dt +

L2 (0,1)

0

0

L2 (0,1)

≤ C(u (y)L2 (0,1) + uC[0,1] ) ≤ C uW21 (0,1) = C u

1

E(A 2 )

.

On the other hand, for any u ∈ D(A), since W22 (0, 1) ⊂ C 1 [0, 1], we have Bu

1 E(A 2

)

= BuL2 (0,1) + (Bu) L2 (0,1) ≤ C uW21 (0,1) + uC 1 [0,1] + (b(y)u (y)) L2 (0,1) ≤ C uW22 (0,1) = C uE(A) . 1

Thus, the operator B is also bounded from E(A) into E(A 2 ) (note that (Bu)(0) = (Bu)(1) = 0) and, therefore, condition 4 of Theorem 5.1 is fulﬁlled. Obviously, the operator A1 is bounded in Lp ((0, 1); L2 (0, 1)), i.e., condition 5 of Theorem 5.1 is also satisﬁed. Further, Wp2 ((0, 1); E(A), E) ⊂ Wp2 ((0, 1); W22 (0, 1), L2 (0, 1)) and, by [21, Theorem 4.3.3], ' 2 & 1 (E(A), E) 12 + 2p ,p = W2 ((0, 1); u(0) = u(1) = 0), L2 (0, 1) 1 + 1 ,p 2 2p ⎧ 1− 1 p ⎪ B (0, 1), 1 2 1− 1

p (0, 1) = B2,p,∗

and ' 2 & 1 (E(A), E) 12 − 2p ,2 = W2 ((0, 1); u(0) = u(1) = 0), L2 (0, 1) 1 − 2

1 1+ p

= W2

1 2p ,2

((0, 1); u(0) = u(1) = 0).

Therefore, condition 6 of Theorem 5.1 is also fulﬁlled.

Let us now consider a homogeneous boundary value problem in the square Ω = [0, 1] × [0, 1], corresponding to problem (7.1)–(7.3), namely L (λ) u := λu (x, y) − Dx2 u (x, y) − Dy (a (y) Dy u (x, y)) + a1 (x, y)u(x, y) = 0,

(7.7)

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 297 L1 u := αDx u (1, y) + b (y) Dy u (0, y) + c(y)u(0, g(y)) 1 + (b(t, y)Dt u(0, t) + c(t, y)u(0, h(t))) dt 0

+

N1

γ1j Dx u (x1j , y) + (T1 u) (y) = 0,

y ∈ [0, 1] ,

(7.8)

j=1

L2 u := βDx u (0, y) +

N2

γ2j Dx u (x2j , y) + (T2 u) (y)

j=1

= 0,

y ∈ [0, 1] , L3 u := u (x, 0) = 0, L4 u := u (x, 1) = 0,

x ∈ [0, 1] , x ∈ [0, 1] .

(7.9)

Theorem 7.2. Let the following conditions be fulfilled: 1. α, β, γkj are complex numbers; α = 0, β = 0; xkj ∈ (0, 1); a(y) ∈ C 1 [0, 1] , a(y) > 0 for y ∈ [0, 1]; a1 (x, y) ∈ C([0, 1]2 ); b(y) ∈ C 1 [0, 1], b(0) = b(1) = 0; c(y), g(y), h(y) ∈ C 1 [0, 1], g(y) and h(y) map [0, 1] into itself, c(0) = c(1) = 0; b(t, y), c(t, y) ∈ C 1 ([0, 1]2 ), b(t, 0) = b(t, 1) = c(t, 0) = c(t, 1) = 0 for t ∈ [0, 1];6 2. For any ε > 0 and u ∈ W22 (Ω), Tk u

1

W22 (0,1)

≤ ε uW22 (Ω) + C (ε) uL2 (Ω) ,

Tk uL2 (0,1) ≤ ε u

3

W22 (Ω)

+ C (ε) uL2 (Ω) ,

k = 1, 2.

Then, the spectrum of problem (7.7)–(7.9) is discrete and a system of root functions of problem (7.7)–(7.9) is complete in the space L2 (Ω). Proof. In the space E := L2 (0, 1), consider operators A, A1 , and B which are deﬁned by equalities (7.4)–(7.6). Then, we can rewrite problem (7.7)–(7.9) in the operator form (6.1)–(6.2) and apply Theorem 6.1. Conditions 3–6 of Theorem 6.1 can be checked in the same way as in the proof of Theorem 7.1. Note only that L2 ((0, 1); L2 (0, 1)) = L2 (Ω) and W22 ((0, 1); W22 ((0, 1); u(0) = u(1) = 0), L2 (0, 1)) ⊂ W22 (Ω), moreover, ' & (E(A), E) 34 ,2 = W22 ((0, 1); u(0) = u(1) = 0), L2 (0, 1) 3 ,2 4 ⎛ ⎞ 1 1 2 ⎝ (0, 1); (min{x, 1 − x})−1 |u(x)|2 dx < ∞⎠ = B2,2 ⎛ 1 2

= W2 ⎝(0, 1);

0

1

⎞ (min{x, 1 − x})−1 |u(x)|2 dx < ∞⎠ ,

0

6

The restriction c(0) = c(1) = 0 can be replaced by g(0) ∈ {0, 1}, g(1) ∈ {0, 1} or g(0) ∈ {0, 1}, c(1) = 0 or g(1) ∈ {0, 1}, c(0) = 0.

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' & (E(A), E) 14 ,2 = W22 ((0, 1); u(0) = u(1) = 0), L2 (0, 1) 1 ,2 4

3 2

= B2,2 ((0, 1); u(0) = u(1) = 0) 3

= W22 ((0, 1); u(0) = u(1) = 0), and, by (5.1)–(5.2), 3 B22 (0, 1); (E(A), E) 14 ,2 , E 3 3 = B22 (0, 1); W22 ((0, 1); u(0) = u(1) = 0), L2 (0, 1) 3 3 ⊂ B22 (0, 1); W22 (0, 1), L2 (0, 1) 3 ' & = B22 (0, 1); W22 (0, 1), L2 (0, 1) 1 ,2 , L2 (0, 1) 4 2 1 2 = W2 (Ω), B2 (0, 1); (W2 (0, 1), L2 (0, 1)) 12 ,2 , L2 (0, 1) 1 2 ,2 3 ' 2 & ' 2 & 2 1 = W2 (Ω), W2 (Ω), L2 (Ω) 1 ,2 1 = W2 (Ω), W2 (Ω) 1 ,2 = W22 (Ω). 2

2 ,2

2

It follows from [19, ch. 2, Sect. 4, Theorem 2] that for eigenvalues of the operator A, λj (A) ∼ j 2 , j = 1, 2, . . .. On the other hand, it is easy to see that sj (J; E(A), E) = (λj (A))−1 (see, e.g., [4, p. 1099]). Then, condition 2 of Theorem 6.1 is fulﬁlled with t = 2. Remark 7.3. Due to Remarks 5.3 and 6.3, one can take the term 1 b(y)Dy u(1, y) + c(y)u(1, g(y)) +

(b(t, y)Dt u(1, t) + c(t, y)u(1, h(t))) dt, 0

instead of the term 1 b(y)Dy u(0, y) + c(y)u(0, g(y)) +

(b(t, y)Dt u(0, t) + c(t, y)u(0, h(t))) dt, 0

in boundary conditions (7.2) and (7.8) with the additional restriction that max{b(·)C 1 [0,1] , c(·)C 1 [0,1] , b(·, ·)C 1 ([0,1]2 ) , c(·, ·)C 1 ([0,1]2 ) } is sufﬁciently small. Finally, let us give some examples of the operators Tk which satisfy condition 2 of Theorem 7.2 and, therefore, condition 2 of Theorem 7.1 (at least for p = 2): #M k 1) Tk u = j=1 δjk u(xjk , y), where δjk ∈ C, xjk ∈ [0, 1], k = 1, 2 (and y ∈ [0, 1]);

1 1# m+n 2) Tk u = 0 0 m+n≤1 Tkmn (x, y, z) ∂ ∂xmu(x,z) ∂z n dxdz, where all functions 3 Tkmn (x, y, z) ∈ L2 ([0, 1] ), Tkmn (x, y, z) are continuously differentiable with respect to y and all the mentioned derivatives also belong to L2 ([0, 1]3 ).

Vol. 69 (2011) Second Order Elliptic Differential-Operator Equations 299 It was shown in [4] that the above operators Tk satisfy condition 2 of Theorem 7.2.

References [1] Aibeche, A.: Coerciveness estimates for a class of nonlocal elliptic problems. Differ. Equ. Dyn. Syst. 1(4), 341–351 (1993) [2] Aibeche, A.: Fold-completeness of generalized eigenvectors of a class of elliptic problems. Result. Math. 33, 1–8 (1998) [3] Aibeche, A., Laidoune, K.: Some properties of the solution of a second order elliptic abstract differential equation. Aust. J. Math. Anal. Appl. 5(2), 1–15 (2008) [4] Aliev, B.A., Yakubov, Ya.: Elliptic differential-operator problems with a spectral parameter in both the equation and boundary-operator conditions. Adv. Differ. Equ. 11(10), 1081–1110 (2006). [Erratum in 12(9), 1079 (2007)] [5] Amann, H.: Dual semigroups and second order linear elliptic boundary value problems. Israel J. Math. 45, 225–254 (1983) [6] Burgoyne, J.: Denseness of the generalized eigenvectors of a discrete operator in a Banach space. J. Oper. Theory 33, 279–297 (1995) [7] Cheggag, M., Favini, A., Labbas, R., Maingot, S., Medeghri, A.: Sturm-Liouville problems for an abstract differential equation of elliptic type in UMD spaces. Diﬀ. Int. Equ., 21, 9–10 (2008), 981–1000 [8] deLaubenfels, R.: Incomplete iterated Cauchy problems. J. Math. Anal. Appl. 168(2), 552–579 (1992) [9] Denk, R., Hieber, M., Pr¨ uss, J.: R-boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type. Mem. Am. Math. Soc., Providence (2003) [10] Dezin, A.A.: General Questions of the Boundary Value Problem Theory (Russian). Nauka, Moscow (1980) [11] Dore, G., Yakubov, S.: Semigroup estimates and noncoervice boundary value problems. Semigroup Forum 60, 93–121 (2000) [12] Evzerov, I.D., Sobolevskii, P.E.: Fractional powers of ordinary differential operators (Russian). Diﬀerents. Uravneniya, 9(2), 228–240 (1973) [13] Favini, A., Shakhmurov, V., Yakubov, Ya.: Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. Semigroup Forum 79(1), 22–54 (2009) [14] Favini, A., Yakubov, Ya.: Regular boundary value problems for elliptic differential-operator equations of the fourth order in UMD Banach spaces. Scientiae Mathematicae Japonicae 70(2), 183–204 (2009) [15] Gorbachuk, V.I., Gorbachuk, M.L.: Boundary Value Problems for DiﬀerentialOperator Equations (Russian). Naukova Dumka, Kiev (1984) [16] Krein, S.G.: Linear Differential Equations in Banach Space. Providence (1971) [17] Krein, S.G.: Linear Equations in Banach Space. Birkh¨ auser (1982) [18] Kunstmann, P.C., Weis, L.: Maximal Lp -regularity for Parabolic Equations, Fourier Multiplier Theorems and H ∞ -Functional Calculus in Functional analytic methods for evolution equations. Lect. Notes Math. 1855, 65–311 (2004) [19] Naimark, M.A.: Linear Differential Operators. Ungar, New York (1967)

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[20] Shklyar, A.Ya.: Complete Second Order Linear Differential Equations in Hilbert Spaces. Birkh¨ auser Verlag, Basel (1997) [21] Tribel, H.: Interpolation Theory. Functional Spaces. Differential Operators. North-Holland, Amsterdam (1978) [22] Weis, L.: Operator-valued Fourier multiplier theorems and maximal Lp -regularity. Math. Ann. 319, 735–758 (2001) [23] Yakubov, S.Ya.: Linear Differential-Operator Equations and their Applications (Russian). Elm, Baku (1985) [24] Yakubov, S.: Completeness of Root Functions of Regular Differential Operators. Longman, New-York (1994) [25] Yakubov, S.: Problems for elliptic equations with operator-boundary conditions. Integr. Equat. Oper. Theory 43, 215–236 (2002) [26] Yakubov, S.Ya., Aliev, B.A.: A boundary value problem with an operator in boundary conditions for a second order elliptic diﬀerental-operator equation (Russian). Sibir. Math. Zhurnal., 26(4), 176–188 (1985) [English translation: Sibirian Math. J., 26(4), 618-628 (1985)] [27] Yakubov, S., Yakubov, Ya.: Differential-Operator Equations. Ordinary and Partial Differential Equations. Chapman and Hall/CRC, Boca Raton (2000) [28] Yakubov, Ya.: Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. I. Abstract theory. J. Math. Pures Appl. 92(3), 263–275 (2009) B. A. Aliev National Academy of Sciences of Azerbaijan Institute of Mathematics and Mechanics 9, F. Agayev str. Baku 1141 Azerbaijan Republic e-mail: [email protected] Ya. Yakubov (B) Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences Tel-Aviv University Ramat-Aviv 69978 Israel e-mail: [email protected] Received: July 21, 2009. Revised: June 13, 2010.

Integr. Equ. Oper. Theory 69 (2011), 301–316 DOI 10.1007/s00020-010-1857-9 Published online January 12, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

An Index Theorem for Band-Dominated Operators with Slowly Oscillating Coefficients (after Deundyak and Shteinberg) Rufus Willett Abstract. We provide a proof of an index theorem for band-dominated operators with slowly oscillating coefficients. The statement is essentially the same as the main result of the announcement of Deundyak and Shteinberg (Funct Anal Appl 19(4):321–323, 1985), but our methods are very different from those hinted at there. The index theorem we prove can also be seen as a partial generalization to higher dimensions of the main result of the article of Rabinovich et al. (Integr Equ Oper Theory 49:221–238, 2004). Mathematics Subject Classification (2010). Primary 47A53; Secondary 19K56. Keywords. Fredholm operator, stable Higson corona.

1. Introduction In this piece we prove an index theorem for band-dominated operators (BDOs) with slowly oscillating coefﬁcients on l2 (ZN ); this is a special case of the results of Deundyak and Shteinberg announced in [5]. It also partially generalizes results of Rabinovich et al. [14] (see [16] for a different proof), who give an index theorem for BDOs on l2 (Z). The main ingredient in our argument is K-theoretic, making use of the stable Higson corona of Emerson and Meyer [8], and asymptotic morphisms in the sense of Connes and Higson [10] to facilitate certain computations. We have given relatively elementary demonstrations of some properties of the stable Higson corona elsewhere [23]. The other important ingredient is an index theorem of Semenjuta and Simonenko [20] for so-called generalized discrete convolution operators (these

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are called BDOs with continuous coefficients in [15]).1 Indeed, our approach is to use the K-theory computations alluded to above to reduce the general case to the index theorem of [20]; note, however, that the ﬁnal statement makes no use of K-theory—see Theorem 6.1 below. We have not been able to prove an index theorem for general BDOs (i.e. with coefﬁcients that are not necessarily slowly oscillating) on l2 (ZN ); indeed Corollary 6.2 and Remark 6.3 below imply the existence of a ‘dimension obstruction’ to straightforwardly extending the current result to the general case (see also [15, pp. 151–152]). The result below is also restricted to the case of BDOs on Hilbert space, due to our reliance on C ∗ -algebraic methods; note, however, the results of Roch [17] imply that the main theorems also apply to band operators with slowly oscillating coefﬁcients on lp -spaces. Outline of the Piece Section 2 elaborates on [19, Section 4] to give a picture of the symbol calculus for BDOs in terms of C ∗ -algebra crossed products. This is used extensively in the following computations, and may be of some interest in its own right—see [24, Section 2] for some developments along these lines. Section 3 introduces BDOs with slowly oscillating coefﬁcients, following [15, Section 2.4]. Section 4 introduces BDOs with continuous coefﬁcients, and states the index theorem of Semenjuta and Simonenko [20] that applies in this case. Section 5 introduces the stable Higson corona and gives a K-theoretic statement and proof of the main result. Finally, Sect. 6 gives a non-K-theoretic restatement of the main theorem, as well as pointing out the ‘dimension obstruction’ alluded to above, and sketching an index theorem for locally compact operators along the lines of that of Rabinovich and Roch in [13]. Notation Throughout the piece, m = (m1 , . . . , mN ) is used to denote an N -tuple of integers in ZN . l2 (ZN ) denotes the Hilbert space of complex-valued square summable functions on ZN ; we will denote its usual basis by {δm : m ∈ ZN }. L(E) denotes the algebra of bounded operators on a Banach space E, and K(E) the compact operators. For k a positive integer, Mk (C) denotes the algebra of k × k matrices over C. If X is a compact Hausdorﬀ topological space, Ck (X) denotes the C ∗ -algebra of continuous functions from X to Mk (C) (equipped with pointwise operations and supremum norm); if X is just assumed locally compact then lk∞ (X) denotes the C ∗ -algebra of all bounded functions from X to Mk (C) and C0,k (X) denotes the C ∗ -algebra of continuous functions from X to Mk (C) that vanish at inﬁnity. The symbol ‘· ⊗ ·’ always denotes a completed tensor product: either of Hilbert spaces, or the spatial tensor product of C ∗ -algebras. Finally, K∗ (A) := K0 (A) ⊕ K1 (A) denotes the topological K-theory group of a C ∗ -algebra A, and K ∗ (X) the topological K-theory group of a space X. 1 An earlier version of this paper used the Atiyah–Singer index theorem for pseudodifferential operators on the n-torus [2] instead of [20]; we would like to thank the referee for pointing out the current more direct approach.

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2. Crossed Products of C ∗ -Algebras and Fredholm Theory for BDOs Definition 2.1. Let lk∞ (ZN ) act on l2 (ZN ) ⊗ Ck by pointwise matrix multiplication; throughout the piece, we will abuse notation by writing f for both a function f : ZN → Mk (C) in lk∞ (ZN ) and the corresponding operator in L(l2 (ZN ) ⊗ Ck ). For m ∈ ZN , let Vm ∈ L(l2 (ZN ) ⊗ Ck ) be the left unitary shift operator deﬁned by Vm : δn ⊗ v → δm+n ⊗ v for any n ∈ ZN and v ∈ Ck . A band-dominated operator (BDO) is an element of the C ∗ -subalgebra of L(l2 (ZN ) ⊗ Ck ) generated by lk∞ (ZN ) and the shifts {Vm : m ∈ ZN }; denote this C ∗ -algebra by Ak . Any BDO2 T ∈ Ak has a unique formal representation as a (possibly inﬁnite) sum fm Vm T = m∈ZN

where fm ∈ lk∞ (ZN ) for all m ∈ ZN (the sum need not converge, even in the weak operator topology, however). We call the elements fm appearing in such a representation the coefficients of T . The following proposition is essentially due to Higson and Yu. It generalizes [14, Proposition 2.1]; the proof is a slight adaptation of that of [4, Proposition 5.1.3] and is thus omitted. Proposition 2.2. Let α be the natural left shift action of ZN on lk∞ (ZN ) (which is spatially implemented by the unitaries Vm ). Let B be a C ∗ -subalgebra of l∞ (ZN ) that is preserved by α and let Bk := B ⊗ Mk (C) be concretely represented on l2 (ZN ) ⊗ Ck by pointwise matrix multiplication. Then the C ∗ -subalgebra of L(l2 (ZN ) ⊗ Ck ) generated by elements of the form bVm , b ∈ Bk , m ∈ ZN , is canonically isomorphic to the reduced crossed product Bk r ZN of Bk by ZN with respect to the action α. Assume moreover that Bk contains C0 (ZN ) ⊗ Mk (C) ∼ = C0,k (ZN ). Then N the isomorphism above takes the subalgebra C0,k (Z ) r ZN of Bk r ZN to K(l2 (ZN ) ⊗ Ck ). Corollary 2.3. Ak is canonically isomorphic to lk∞ (ZN ) r ZN . Proof. Set B in the above to be all of l∞ (ZN ).

Let B be a unital C ∗ -subalgebra of l∞ (ZN ) which contains C0 (ZN ) and is preserved under the shift action α of ZN . The Gelfand–Naimark theorem implies that B is canonically isomorphic to C(ZN ) for some compact Hausdorﬀ space ZN which is an equivariant compactification of ZN : ZN contains ZN as an open dense subset in such a way that the (left) shift action of ZN on itself extends to a continuous action of ZN on ZN by homeomorphisms. 2

Indeed, any bounded operator on l2 (ZN ) ⊗ Ck .

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Corollary 2.4. Let ZN be an equivariant compactification of ZN and ∂ZN := ZN \ZN the associated corona space. Then there exists a short exact sequence of C ∗ -algebras 0 → K(l2 (ZN ) ⊗ Ck ) → Ck (ZN ) r ZN → Ck (∂ZN ) r ZN → 0. σ

Proof. There is a short exact sequence of C ∗ -algebras 0 → C0,k (ZN ) → Ck (ZN ) → Ck (∂ZN ) → 0, where all the ∗-homomorphisms are equivariant for the natural (left) ZN actions on each algebra. As ZN is an exact group in the sense of C ∗ -algebra theory (see for example [4, Chapter 5]), this gives rise to a short exact sequence of crossed product algebras 0 → C0,k (ZN ) r ZN → Ck (ZN ) r ZN → Ck (∂ZN ) r ZN → 0. Proposition 2.2 implies that Ck (ZN ) r ZN is naturally represented on l2 (ZN ) ⊗ Ck as the C ∗ -algebra generated by Ck (ZN ) and the shifts Vm , and moreover that this representation takes C0,k (ZN )r ZN to the compact operators on l2 (ZN ) ⊗ Ck , so we are done. As a corollary, note that an operator F ∈ Ck (ZN ) r ZN ⊆ L(l2 (ZN ) ⊗ Ck ) is Fredholm if and only if its ‘symbol’ σ(F ) ∈ Ck (∂ZN ) r ZN is invertible. This is closely related to the results on Fredholmness of BDOs in [15], in particular the symbol calculus from [15, Section 2.2.4]; we will not need this in this piece, but see [24, Section 2] for a study of the precise relationship and some corollaries.

3. The Higson Compactification and BDOs with Slowly Oscillating Coefficients In this section we introduce BDOs with slowly oscillating coefﬁcients. Definition 3.1. Let X be a locally compact metric space, and let f : X → E be a continuous bounded function from X to a normed space E. f is called slowly oscillating if for all R > 0 the function ∇R f deﬁned by (∇R f )(x) = sup{f (x) − f (y) : d(x, y) ≤ R} tends to zero at inﬁnity in X. The previous definition restricts to the class of functions studied in [15, Section 2.4.1] in the case X = ZN (equipped with the restriction of the Euclidean metric from RN ), but we need it in slightly more generality. Definition 3.2. The slowly oscillating functions from X to C form a commutative C ∗ -algebra when equipped with the supremum norm, which we denote SO(X); we write SO for SO(ZN ) when there is no risk of confusion. The C ∗ algebra of slowly oscillating functions on X with values in Mk (C) is denoted SOk (X) or just SOk when X = ZN .

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The space of multiplicative linear functionals on SO(X) is denoted X , h and the associated corona ∂h X := X \X (here X is identiﬁed with the space h of point evaluations on SO(X)). We call X the Higson compactification of X and ∂h X its Higson corona; see Remark 3.3 below for a justiﬁcation of the name. Finally, the C ∗ -subalgebra of L(l2 (ZN ) ⊗ Ck ) generated by SOk (acting by pointwise matrix multiplication) and the shifts Vm will be denoted A(SOk ) or A(SOk (ZN )) if we need to be more speciﬁc. Operators in this algebra are called BDOs with slowly oscillating coefficients. h

Remark 3.3. X has been extensively studied in coarse geometry, index theory and K-homology; the name ‘Higson compactification’ is after Nigel Higson, who introduced it in these areas [11]. The only examples we will use in this piece are the Higson compactifications of X = ZN and X = RN . Note that in [15, Section 2.4], the authors use the notation M (SO) and M ∞ (SO) h

for what we have called ZN and ∂h ZN respectively. Now, SO(ZN ) ⊆ l∞ (ZN ) is unital, contains C0 (ZN ), and is preserved h

by the shift action α of ZN on l∞ (ZN ). Hence ZN is an equivariant compactification of ZN in the sense of the previous section, whence Corollary 2.4 gives a short exact sequence 0 → K(l2 (ZN ) ⊗ Ck ) → Ak (SO) → Ck (∂h ZN ) r ZN → 0. σ

(1)

Lemma 3.4. The action of ZN on ∂h ZN is trivial. Proof. It sufﬁces to prove that if f is an element of SOk , η is a functional in ∂h ZN and m ∈ ZN , then η(Vm∗ f Vm − f ) = 0. Let (mi )i∈I be any net in ZN converging to η in the Gelfand topology on M (SOk ), so in particular (mi ) tends to inﬁnity in ZN . It follows that η(Vm∗ f Vm − f ) = lim (Vm∗ f Vm )(mi ) − f (mi ) i∈I = lim f (mi − m) − f (mi ) = 0 i∈I

by definition of slowly oscillating functions.

Hence one has the identiﬁcations Ck (∂h ZN ) r ZN ∼ = Ck (∂h ZN ) ⊗ Cr∗ (ZN ) ∼ = Ck (∂h ZN ) ⊗ C(TN ) ∼ = Ck (∂h ZN × TN ), where the second isomorphism uses the Fourier transform to identify the reduced group C ∗ -algebra of ZN and the continuous functions on the N torus TN . The short exact sequence in line (1) above can thus be rewritten 0 → K(l2 (ZN ) ⊗ Ck ) → Ak (SO) → Ck (∂h ZN × TN ) → 0. σ

(2)

We will not need this fact, but it is not hard to check that σ as in the above line is the same as the map smb from [15, p. 102].

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4. The Spherical Compactification and BDOs with Continuous Coefficients In this section we introduce the spherical compactification of ZN . The spherical compactification is used to deﬁne BDOs with continuous coefficients in [15, Section 2.3.6]; we repeat the definition below for the reader’s convenience. We then state an index theorem of Semenjuta and Simonenko [20] for BDOs with continuous coefﬁcients; this is an important ingredient in our main result. Definition 4.1. Let S N −1 be the N − 1 dimensional unit sphere considered s as a subset of N -dimensional Euclidean space RN . Let ZN be equal as a set to the disjoint union ZN S N −1 , and topologize it by stipulating that: • the induced subspace topologies on S N −1 and ZN are the usual ones; N N −1 if and only if • a sequence (mk )∞ k=0 in Z \{0} converges to x ∈ S k the sequence of norms (m RN ) converges to inﬁnity, and the sequence (mk /mk RN ) converges to x ∈ S N −1 . s

Equipped with this (metrizable) topology, ZN will be called the spherical compactification of ZN . The natural action of ZN on itself extends to an action on the spherical s compactification; in the language of Sect. 2, this says that ZN is an equivariant compactification of ZN . In particular, Corollary 2.4 above implies that there is a short exact sequence of C ∗ -algebras s

0 → K(l2 (ZN ) ⊗ Ck ) → Ck (ZN ) r ZN → Ck (S N −1 ) r ZN → 0.

(3)

However, it follows from the argument of Lemma 3.4 that the action of ZN is trivial on the sphere at inﬁnity, whence ∼ Ck (S N −1 ) ⊗ C ∗ (ZN ) ∼ Ck (S N −1 ) r ZN = = Ck (S N −1 × TN ) r

∼ = Ck (S N −1 × TN ),

just as in the case of C(∂h ZN ) r ZN looked at earlier. The short exact sequence in line (3) above thus becomes s

S Ck (S N −1 × TN ) → 0. 0 → K(l2 (ZN ) ⊗ Ck ) → Ck (ZN ) r ZN →

σ

(4)

The map σS is called the Semenjuta–Simonenko symbol map in what follows. s In [15, Section 2.3.6], BDOs in the subalgebra C(ZN ) r ZN of Ak are called BDOs with continuous coefficients. In [21]3 Semenjuta and Simonenko prove an index formula for operators s in Ck (ZN ) r ZN , which we now describe. s Assume that k = N in the discussion above, and that F ∈ CN (ZN ) r ZN is Fredholm, so that σS (F ) is a continuous map σS (F ) : S N −1 × TN → GLN (C) from S N −1 × TN into the N -dimensional complex general linear group. Let now GLN −1 (C) be embedded in GLN (C) as the stabilizer of the point 3

See also [20] for the definitions used in [21].

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(1, 0, . . . , 0) ∈ CN , so that GLN (C)/GLN −1 (C) identiﬁes with CN \{0}; we may deform retract this latter space onto the sphere of radius 1 in CN , a copy of S 2N −1 . This process gives rise to a map N −1 σ × TN → GLN (C) → CN \{0} → S 2N −1 . S (F ) : S

(5)

Fix orientations on the domain and codomain of σ S (F ) as follows: that on S 2N −1 is the orientation it inherits from the complex structure on CN ; that on S N −1 ×TN is the product orientation where S N −1 (respectively, TN ) inherits its orientation as a subspace (respectively, quotient) of the same copy of RN (how the original RN is orientated makes no difference). Now, as σ S (F ) is a continuous map between oriented manifolds of the same dimension, it has an integer degree Degree(σ S (F )). The following theorem is a restatement of [20, formula (2), p. 135], the main theorem of that paper. This theorem can also be seen as a special case of index formulas of Fedosov [9] and Atiyah–Singer [2].4 Theorem 4.2. Say F, σ S (F ) are as above. Then Index(F ) = (−1)

N (N +1) −1 2

Degree(σ S (F )) . (N − 1)!

Note that a map S N −1 × TN → GLk (C) for any k ∈ N can be considered as a map S N −1 × TN → GLk (C) where k = max{k, N } and GLk (C) is either identiﬁed with GLk (C), or embedded in it as a subgroup in the natural way. From here, the map can be homotoped (through maps with range in GLk (C)) to one of the form S N −1 × TN → GLN (C) ⊕ {1k −N } ⊆ GLk (C), where k = max{k, N } and 1k −N is the identity in k − N dimensions (see [1, p. 239]). Thus one may deﬁne a map IndS : K 1 (S N −1 × TN ) → Z

(6)

by ﬁrst homotoping a class [x] so that it has image in GLN (C) ⊕ {1k −N }, and then applying the formula from Theorem 4.2 to the part of the image in GLN (C); it is easy to see that this is well-deﬁned on the level of K-theory. Note that if [σS (F )] ∈ K 1 (S N −1 × TN ) is the class deﬁned by the symbol of a Fredholm operator F , then IndS ([σS (F )]) = Index(F ).

(7)

We will use this later. The homomorphism in line (6) will be called the Semenjuta–Simonenko index map; note, of course, that it is just a concrete instantiation of the usual index map in K-theory arising from the short exact sequence in line (4)—indeed, this follows from the formula in line (7). 4

We thank the referee for pointing out the references [20] and [9].

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5. The Stable Higson Compactification and Proof of the Main Result In this section we give a proof of the main index theorem (Theorem 5.4 below, restated later as Theorem 6.1). The main ingredient in the proof is the stable Higson corona of Emerson and Meyer [8], which is used to ‘organize’ certain K-theoretic computations. The stable Higson corona is not a topological corona associated to some compactification of ZN , but rather a noncommutative C ∗ -algebra that plays a similar role: the point is that the Higson corona ∂h ZN is known to have very bad topological properties [6,7,12]; we use the stable Higson corona as a manageable substitute for it on a K-theoretical level. The following definition comes from [8]. Definition 5.1. Let K be an abstract copy of the C ∗ -algebra of compact operators on a separable inﬁnite dimensional Hilbert space and X be a locally compact metric space. The stable Higson compactification of X, denoted ¯c(X), is the C ∗ -algebra of (continuous, bounded) slowly oscillating functions from X to K. Note that it contains the C ∗ -algebra of continuous functions from X to K that vanish at inﬁnity, denoted C0 (X, K), as an ideal. The stable Higson corona of X, denoted c(X), is the quotient C ∗ -algebra c(X) :=

¯c(X) . C0 (X, K)

The following proposition collects together some basic facts from coarse geometry that will be of use to us here; all are simple special cases of more general facts. s

h

Proposition 5.2. 1. There is a natural inclusion Ck (ZN ) → Ck (ZN ), which fixes C0,k (ZN ) and thus passes to an inclusion of quotients Ck (S N −1 ) ∼ = 2.

s

h

Ck (ZN ) ∼ Ck (ZN ) → = Ck (∂h ZN ). N C0,k (Z ) C0,k (ZN ) h

There is an inclusion Ck (ZN ) → ¯c(ZN ), which maps C0,k (ZN ) into C0 (ZN , K) and thus passes to an inclusion of quotients h

¯c(ZN ) ∼ Ck (ZN ) Ck (∂h Z ) ∼ → = c(ZN ). = C0,k (ZN ) C0 (ZN , K) N

3.

The inclusion of ZN into RN induces ∗-homomorphisms h

h

Ck (RN ) → Ck (ZN ) and ¯c(RN ) → ¯c(ZN ); these induce isomorphisms on the ‘corona algebras’ Ck (∂h RN ) ∼ = Ck (∂h ZN ) and c(RN ) ∼ = c(ZN ) respectively.

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Sketch of proofs. For part (1), note that the restriction of any function in s C(ZN ) to ZN is slowly oscillating when restricted to ZN , hence extends h

(uniquely) to ZN ; this deﬁnes the inclusions in the statement. For part (2), identify Mk (C) with a sub-C ∗ -algebra of K via any ∗-homomorphism taking rank one projections to rank one projections. The h

restriction of a function in Ck (ZN ) to ZN is then a slowly oscillating function from ZN to K; this deﬁnes the inclusions in the statement. Note that while this map involves a choice of embedding Mk (C) → K, the choice does not matter on the level of K-theory. For part (3), the ∗-homomorphisms are both deﬁned by restriction of functions from RN to ZN . The isomorphisms in the statement come from the fact that the inclusion ZN → RN is a coarse equivalence, and that the Higson corona and stable Higson coronas deﬁne functors on the coarse category; for details, see [18, Proposition 2.41] for the case of the Higson corona and [8, Proposition 13] for the stable Higson corona. h

Consider now the map C(RN ) → Cb ([1, ∞), C(S N −1 )) deﬁned by sending a function f on the Higson compactification of RN to the map sending t ∈ [1, ∞) to the restriction of f to the sphere of radius t about zero in RN . Passing to quotients by functions vanishing at inﬁnity and using the second part of Proposition 5.2 above, this gives rise to a map α : C(∂h ZN ) ∼ =

h

Cb ([1, ∞), C(S N −1 )) C(RN ) → . N C0 (R ) C0 ([1, ∞), C(S N −1 ))

In particular, then, α is an asymptotic morphism from C(∂h ZN ) to C(S N −1 ) in the sense of [10], so induces a map on K-theory α∗ : K ∗ (∂h ZN ) → K ∗ (S N −1 ).

(8)

Analogously, there is a ∗-homomorphism αc : c(ZN ) ∼ =

¯c(ZN ) Cb ([1, ∞), C(S N −1 ) ⊗ K) → C0 (ZN , K) C0 ([1, ∞), C(S N −1 ) ⊗ K)

i.e. an asymptotic morphism from c(ZN ) to C(S N −1 ) ⊗ K. It thus induces a map on K-theory α∗c : K∗ (c(ZN )) → K∗ (C(S N −1 ) ⊗ K).

(9)

Remarks 5.3. • The asymptotic morphism α acts on K-theory roughly as follows; see for example [3, Chapter 25] for details. Assume that [p] ∈ K 0 (∂h ZN ), where p is a projection in Mk (C(∂h ZN )). Let p˜ be any h

lift of p to Mk (C(RN )); this exists because of the canonical identiﬁcation C(∂h ZN ) ∼ = C(∂h RN ). Let αt (p) be the restriction of p˜ to the sphere of radius t in RN for all t > 0. One then has that αt (p)2 − αt (p) and αt (p)∗ − αt (p) both tend to zero as t → ∞, i.e. αt (p) gets ‘close’ to being a projection as t → ∞. It thus (via the functional calculus) deﬁnes a class in K 0 (S N −1 ) for all t suitably large; we let α∗ [p] be the class of any αt (p) for t large. The action of α on K 1 , and that of αc on K-theory, are described similarly.

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• In the notation of Deundyak and Shteinberg [5], α∗ is equal to the composition i∗1 ◦ (i∗∞ )−1 as a map on K-theory; in this sense, one can think of α∗ as a concrete way of describing the map i∗1 ◦ (i∗∞ )−1 . We do not need this fact here, so will not prove it, but see [22, Chapters 5, 6]. We may now state the main result of this section. Theorem 5.4. Let F ∈ A(SOk (ZN )) be a Fredholm BDO on ZN with slowly oscillating coefficients. Its symbol σ(F ) ∈ Ck (∂h ZN ×TN ) as in line (2) above is thus invertible and so defines a class [σ(F )] ∈ K 1 (∂h ZN × TN ). Let IndS : K 1 (S N −1 × TN ) → Z be the Semenjuta–Simonenko index map of line (6) above. Then Index(F ) = IndS ◦ (α ⊗ 1)∗ [σ(F )]. Theorem 5.4 is proved by studying the commutative diagram introduced in line (10) below. The vertical arrows between the bottom and middle rows in this diagram are the inclusions from the ﬁrst part of Proposition 5.2, while those between the second and third rows are as in the second part of that proposition. 0

/ C0 (ZN , K) O

/ ¯c(ZN ) O

/ c(ZN ) O

/ 0.

jh

0

/ C0,k (ZN )

/ C (ZN h ) k O

/ Ck (∂h ZN ) O

/0

(10)

js

0

/ C0,k (ZN )

/ C (ZN s ) k

/ Ck (S N −1 )

/0

The maps in this diagram are all ZN -equivariant. We may thus take crossed products by ZN everywhere, getting a new commutative diagram. 0

/ (C0 (ZN ) r ZN ) ⊗ K O

/ ¯c(ZN ) r ZN O

/ c(ZN ) r ZN O

/ 0.

j h r 1

0

/ C0,k (ZN ) r ZN

/ C (ZN h ) ZN k O r

/ Ck (∂h ZN ) r ZN O

/0

j s r 1

0

/ C0,k (ZN ) r ZN

/ C (ZN s ) ZN k r

/ Ck (S N −1 ) r ZN

/0 (11)

Note that, as Z is an exact group, the rows are still exact. We make the following identiﬁcations in the above: • all the ZN actions in the right-hand column are trivial (by the argument of Lemma 3.4), whence ‘· r ZN ’ has the same eﬀect as ‘· ⊗ C(TN )’; h • by Proposition 2.2, Ck (ZN ) r ZN ∼ = A(SOk (ZN )) and C0,k (ZN ) r N ∼ 2 N k ∼ 2 N Z = K(l (Z ) ⊗ C ) = K(l (Z )) ⊗ Mk (C). N

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Diagram (11) above thus becomes 0

/ c(ZN ) ⊗ C(TN ) O

/ ¯c(ZN ) r ZN O

/ K(l2 (ZN )) ⊗ K O

/ 0,

j h ⊗1

0

/ K(l2 (ZN )) ⊗ Mk (C)

/ A(SOk (ZN )) O

/ Ck (∂h ZN × TN ) O

/0

(12)

j s ⊗1

0

/ K(l2 (ZN )) ⊗ Mk (C)

/ C (ZN s ) ZN k r

/ Ck (S N −1 × TN )

/0

which in turn gives rise to a commutative diagram of six-term exact sequences in K-theory. We will be interested only in the ‘index map’ portion of these exact sequences, as studied below. Note that the map K(l2 (ZN )) ⊗ Mk (C) → K(l2 (ZN )) ⊗ K between the middle and top rows on the left-hand-side of (12) above induces an isomorphism on K-theory. Hence the portion of the commutative diagram of six term exact sequences containing the index maps looks like K1 (c(ZN ) ⊗ C(TN )) O

Indc

/Z.

Ind

/Z

IndS

/Z

(13)

(j h ⊗1)∗

K 1 (∂h ZN × TN ) O (j s ⊗1)∗

K 1 (S N −1 × TN )

Here the horizontal maps are all connecting maps (‘index maps’) in the Ktheory six term exact sequences associated to (12); the bottom one is of course simply the Semenjuta–Simonenko index map from line (6) in Sect. 4 above. To complete the proof of Theorem 5.4, we need the following two lemmas. Lemma 5.5. Let α∗ , α∗c and j h be as in lines (8), (9) and (10) above respectively. They are related by the formula α∗ = α∗c ◦ j∗h : K ∗ (∂h ZN ) → K ∗ (S N −1 ), where we have identified Ck (∂h ZN ) and C(∂h ZN , K) on the level of K-theory in order to make sense of this. Proof. Both maps are given by the same formulas up to the natural isomorphisms on K-theory induced by the inclusions Ck (∂h ZN ) → C(∂h ZN , K) and Ck (S N −1 ) → C(S N −1 , K) Lemma 5.6. j h ◦ j s : Ck (S N −1 ) → c(ZN ) induces an isomorphism on Ktheory, with inverse α∗c .

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Moreover, there are maps between K∗ (Ck (S N −1 ×TN )) and K∗ (c(ZN )⊗ C(T )) functorially induced by j h ◦ j s and α∗S which also induce mutually inverse isomorphisms on K-theory. N

Proof. As RN is a CAT (0) space, up to the isomorphisms in K-theory induced by the inclusion Mk (C) → K and the canonical identiﬁcation of c(ZN ) and c(RN ), the map j h ◦ j s is the same as the map i from [23, Proposition 4.2]5 , which is an isomorphism. Consider the map induced on K-theory by αc ◦ j h ◦ j s . Pre-composing with the stabilization isomorphism again, this takes a class [f ] from K∗ (C(S N −1 , K))6 to the restriction of any lift f˜ of f to the sphere of radius t for t suitably large (cf. Remark 5.3). However, as f is just a function on S N −1 , we may assume that f˜ simply is f on all spheres of radius t for t ≥ 1. Hence αc ◦ j h ◦ j s induces the identity on K-theory. This says that α∗c is a one-sided inverse to j∗h ◦ j∗s ; as the latter map is an isomorphism, however, α∗c must be a two-sided inverse as required. To complete the proof, it sufﬁces to show that the ∗-homomorphism (j h ◦ j s ) ⊗ 1 : C(S N −1 ) ⊗ K ⊗ C(TN ) → c(ZN ) ⊗ C(TN ) and asymptotic morphism αc ⊗ 1 : c(ZN ) ⊗ C(TN ) →

Cb ([1, ∞), C(S N −1 ) ⊗ K ⊗ C(TN )) C0 ([1, ∞), C(S N −1 ) ⊗ K ⊗ C(TN ))

induce mutually inverse isomorphisms at the level of K-theory. A slight adaptation of [23, Proposition 4.3] again shows that ((j h ◦ j s ) ⊗ 1)∗ is an isomorphism, however, and then the same argument as above shows that (αc ⊗ 1)∗ is a one-sided, whence also two-sided, inverse. We are now ﬁnally ready to prove Theorem 5.4. Proof of Theorem 5.4. Let F be as in Theorem 5.4. It sufﬁces to compute Ind[σ(F )], which is equal to Indc ◦ (j h ⊗ 1)∗ [σ(F )] = Indc ◦ (j h ⊗ 1)∗ ◦ (j s ⊗ 1)∗ ◦ (αc ⊗ 1)∗ ◦(j h ⊗ 1)∗ [σ(F )] using naturality of the index map in K-theory and Lemma 5.6. Using naturality of the index map again, this is in turn equal to IndS ◦ (αc ⊗ 1)∗ ◦ (j h ⊗ 1)∗ [σ(F )]. Finally, applying Lemma 5.5, this is equal to IndS ◦ (α ⊗ 1)∗ [σ(F )], which proves Theorem 5.4. 5

This also follows from results in [8]. Stability of C(S N −1 , K) implies that classes of this form for f ∈ C(S N −1 , K) generate this K-theory group.

6

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6. Restatement of the Main Result In this section we give a more concrete restatement of Theorem 5.4, which is much closer to the statement of [5]. We also discuss some ‘dimension corollaries’, and sketch an extension of Theorem 5.4 along the lines of [13]. Theorem 6.1. Let F ∈ A(SON (ZN )) be a Fredholm BDO on ZN with slowly oscillating coefficients and values in MN (C). Then its index may be computed by the following ‘recipe’: •

Using the fact that ∂h ZN = ∂h RN , extend all the matrix entries of (fij ) := σ(F ) : ∂h ZN × TN → C

•

•

to functions on RN ×TN .7 Denote by f ij the extension of fij so obtained. For some t0 ∈ [1, ∞) and all t ≥ t0 , the matrix (f ij ) will be invertible when restricted to the sphere of radius t. It thus defines a function, which we denote σt (F ), from S N −1 × TN to GLN (C). N −1 For any t ≥ t0 , there is thus a map σ ×TN → S 2N −1 defined t (F ) : S as in line (5) above. Just as in Theorem 4.2, then Index(F ) = (−1)

N (N +1) −1 2

Degree(σ t (F )) . (N − 1)!

for any t ≥ t0 . Just as in Sect. 4, if F ∈ A(SOk (ZN )) for some k ∈ N, then for t suitably large σt (F ) : S N −1 × TN → GLk (C) can be homotoped (through maps with image in GLk (C)) to a map σt (F ) : S N −1 × TN → GLN (C) ⊕ {1k −N } ⊆ GLk (C), where k = max{k, N }, and 1k −N is the identity in k −N dimensions (see [1, p. 239]). Having performed this homotopy, Index(F ) can be computed using Theorem 6.1 above above, whatever the original k is. Moreover, as the degree of the map σ t (F ) is easily seen to be zero whenever k < N (essentially as the image ends up being in a lower dimensional sphere), one has the following corollary. Corollary 6.2. There exist Fredholm operators in A(SOk (ZN )) of non-zero index if and only if k ≥ N . Proof. One half of this has already been proved. For the existence of Fredholm operators of index one for all k ≥ N , it is sufﬁcient to note that there are maps S N −1 × TN → S 2N −1 of degree (N − 1)! coming from maps S N −1 × TN → GLN (C) (cf. [1, p. 239]—this statement follows easily from results stated there), and use Theorem 6.1. 7 This sounds difficult. In concrete situations, however, the coefficients of the original F will be given as restrictions of slowly oscillating functions on RN , so it is not really a problem.

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Remark 6.3. On the other hand, there are Fredholm BDOs with non-trivial index, non-slowly-oscillating coefﬁcients, and with scalar values on l2 (ZN ) for all N . Indeed, let P be the orthogonal projection onto the closed subspace of l2 (ZN ) spanned by {δm=(m1 ,...,mN ) : m1 ≤ 0, m2 = m3 = · · · = mN = 0}, and let U be the unitary shift deﬁned on basis elements by U : δm1 ,...,mN → δm1 +1,m2 ,...,mN . Then P U P + (1 − P ) is a Fredholm band operator of index one. The above corollary is thus of interest as it suggests that any index theorem applying to all BDOs on ZN would have to be significantly different from Theorems 5.4 and 6.1. Note that this ‘dimension problem’ does not apply in the case N = 1 studied in [14,16]. Recall ﬁnally that Rabinovich and Roch [13] have used the results of [14] to prove an index theorem for locally compact operators on RN . We can partially extend their theorem using our machinery; a brief sketch is as follows. There is a natural definition of locally compact BDO with slowly oscillating coefﬁcients on RN ; one can moreover show that the C ∗ -algebra of such operators is isomorphic to ¯c(ZN ) r ZN . Such operators are rich in the sense of [15] if and only if they are actually in the (much smaller) subalgeh

bra C(ZN , K) r ZN . Fredholm operators in this algebra can be handled ∼ C(ZN h , Mk (C)) r ZN , in exactly the same way as those in A(SOk (ZN )) = thus proving an index theorem precisely analogous to Theorems 5.4 and 6.1 above. Acknowledgments This piece is adapted from part of the author’s Ph.D. thesis [22]; I would like to thank my advisor John Roe for his support, encouragement and many useful comments during this work. I would also like to thank Steffen Roch for interesting discussions, and showing me the paper [5]. Finally, I would like to thank the referee for pointing out the references [20] and [9], and some other helpful comments.

References [1] Atiyah, M.: Algebraic topology and elliptic operators. Comm. Pure Appl. Math. 20, 237–249 (1967) [2] Atiyah, M., Singer, I.: The index of elliptic operators I. Ann. Math. 87(3), 484– 530 (1968) [3] Blackadar, B.: K-Theory for Operator Algebras, 2nd edn. Cambridge University Press, Cambridge (1998) [4] Brown, N., Ozawa, N.: C ∗ -Algebras and Finite-Dimensional Approximations. Graduate Studies in Mathematics, vol. 88. American Mathematical Society (2008)

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[5] Deundyak, V.M., Shteinberg, B.Y.: Index of convolution operators with slowly varying coefficients on abelian groups. Funct. Anal. Appl. 19(4), 321–323 (1985) [6] Dranishnikov, A., Ferry, S.: On the Higson-Roe corona. Russ. Math. Surv. 52(5), 1017–1028 (1997) [7] Dranishnikov, A., Keesling, J., Uspenskii, V.V.: On the Higson corona of uniformly contractible spaces. Topology 37(4), 791–803 (1998) [8] Emerson, H., Meyer, R.: Dualizing the coarse assembly map. J. Inst. Math. Jussieu 5, 161–186 (2006) [9] Fedosov, B.: Analytic formulae for the index of elliptic operators. Trans. Moscow Math. Soc. 30, 159–240 (1974) [10] Guentner, E., Higson, N., Trout, J.: Equivariant E-theory. Mem. Am. Math. Soc. 148(703), (2000) [11] Higson, N.: On the relative K-homology theory of Baum and Douglas. Preprint (1989) [12] Keesling, J.: The one dimensional Cˇech cohomology of the Higson compactification and its corona. Topol. Proc. 19, 129–148 (1994) [13] Rabinovich, V.S., Roch, S.: The Fredholm index of locally compact banddominated operators on Lp (R). Integr. Equ. Oper. Theory 57(2), 263– 281 (2007) [14] Rabinovich, V.S., Roch, S., Roe, J.: Fredholm indices of band-dominated operators on discrete groups. Integr. Equ. Oper. Theory 49, 221–238 (2004) [15] Rabinovich, V.S., Roch, S., Silbermann, B.: Limit Operators and Their Applications in Operator Theory. Operator Theory: Advances and Applications, vol. 150. Birkh¨ auser, Boston (2004) [16] Rabinovich, V.S., Roch, S., Silbermann, B.: The finite sections approach to the index formula for band-dominated operators. In: Recent Advances in Operator Theory and Applications. Operator Theory: Advances and Applications, vol. 187, pp. 185–193. Birkh¨ auser, Boston (2009) [17] Roch, S.: Band-dominated operators on lp -spaces: Fredholm indices and finite sections. Acta Sci. Math. (Szeged) 70, 783–797 (2004) [18] Roe, J.: Lectures on Coarse Geometry. University Lecture Series, vol. 31. American Mathematical Society (2003) [19] Roe, J.: Band-dominated Fredholm operators on discrete groups. Integr. Equ. Oper. Theory 51, 411–416 (2005) [20] Semenjuta, V.N., Simonenko, I.B.: Computation of the index of multidimensional discrete convolutions. Mat. Issled. 4(14), 134–141 (1969, in Russian) [21] Semenjuta, V.N., Simonenko, I.B.: The index of multidimensional discrete convolutions. Mat. Issled. 4(2), 88–94 (1969, in Russian) [22] Willett, R.: Band-dominated operators and the stable Higson corona. PhD thesis, The Pennsylvania State University (2009) [23] Willett, R.: Some ‘homological’ properties of the stable Higson corona. J. Noncommut. Geom. (2009, accepted) [24] Willett, R.: Crossed products, fixed point algebras, and band-dominated Fredholm operators. (2010, submitted)

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Rufus Willett (B) Vanderbilt University 1326 Stevenson Center Nashville TN 37240 USA e-mail: [email protected] Received: October 2, 2009. Revised: December 6, 2010.

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Integr. Equ. Oper. Theory 69 (2011), 317–346 DOI 10.1007/s00020-010-1846-z Published online November 25, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Toeplitz Operators with Special Symbols on Segal–Bargmann Spaces K. Jotsaroop and S. Thangavelu Abstract. We study the boundedness of Toeplitz operators on Segal– Bargmann spaces in various contexts. Using Gutzmer’s formula as the main tool we identify symbols for which the Toeplitz operators correspond to Fourier multipliers on the underlying groups. The spaces considered include Fock spaces, Hermite and twisted Bergman spaces and Segal–Bargmann spaces associated to Riemannian symmetric spaces of compact type. Mathematics Subject Classification (2010). 47B35, 43A85, 22E30. Keywords. Segal–Bargmann transform, weighted Bergman spaces, Toeplitz operators, Fourier multipliers, Gutzmer’s formula, Hermite and Laguerre functions, symmetric spaces.

1. Introduction Given a domain Ω in Cn let H(Ω, dμ) stand for a weighted Bergman space of holomorphic functions contained in L2 (Ω, dμ). Let g be a Lebesgue measurable function on Ω such that gF ∈ L2 (Ω, dμ) for all F from a dense subspace of H(Ω, dμ). We can then deﬁne the Toeplitz operator Tg by Tg F = P (gF ) where P : L2 (Ω, dμ) → H(Ω, dμ) is the natural orthogonal projection. Such Toeplitz operators have been studied extensively in the literature. Suppose now that Ω is invariant under the action of a Lie group G. The group g has a natural action on H(Ω, dμ). Let us further assume that there is an isometric isomorphism B between L2 (G) and H(Ω, dμ). Using this, we can transfer the Toeplitz operator Tg into the operator B −1 Tg B acting on L2 (G). Then the boundedness of Tg becomes equivalent to that of this transferred operator which might turn out to be easier to study using harmonic analysis on the group G. The simplest bounded operators on L2 (G) are given by Fourier multilpiers and hence it is natural to ask which Toeplitz operators give rise to such multiplier transformations.

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In this article we are interested in the case where B is the Segal –Bargmann transform on the group G and H(Ω, dμ) is the image of L2 (G) under B. It turns out that we can identify a large class of symbols g for which B −1 Tg B reduces to Fourier multipliers. The groups for which such results can be proved include Rn , Heisenberg groups and compact Lie groups. An important role is played by the so called Gutzmer’s formula. Thus for Fock spaces, Hermite and twisted Bergman spaces and Segal–Bargmann spaces associated to compact Lie groups and symmetric spaces we have identiﬁed special classes of symbols for which Tg ’s correspond to multiplier transforms. The plan of the paper is as follows. In the next section we look at Toeplitz operators on the classical Fock spaces. In Sect. 3 we study Toeplitz operators on Hermite-Bergman spaces which give rise to Hermite multipliers when conjugated with the Hermite semigroup. In Sect. 4 we characterise all Toeplitz operators on the twisted Bergman spaces that correspond to Weyl multipliers. Finally, in the last section we consider Toeplitz operators on Segal–Bargmann spaces associated to compact Lie groups and symmetric spaces. For results closely related to the theme of this paper we refer to [2,10–12].

2. Toeplitz Operators on Fock Spaces In this section we look at Toeplitz operators on Fock spaces which have been studied by several authors, see [2] and the references there. First we consider Toeplitz operators with radial symbols and obtain a necessary and sufﬁcient condition for Tg to be bounded. Toeplitz operators with radial symbols on F(C) with a different assumption have been studied by Grudsky and Vasilevski [9]. The condition involves the heat ﬂow g ∗ q1/4 and under a mild decay assumption we prove boundedness when g is radial. Later we consider symbols g(x + iy) which depend only on y and show that they correspond to Fourier multipliers. For such symbols we show that the conjecture of Berger and Coburn [2] is true. 2.1. Radial Symbols In this subsection we consider Toeplitz operators associated to radial symbols on the Fock space F(Cn ). F(Cn ) := {f ∈ O(Cn ) : |f (z)|2 dμ(z) < ∞}, Cn 1

2

where dμ(z) = (2π)−n e− 2 |z| dz. It is known that F(Cn ) is a Hilbert space z·w ¯ with the reproducing kernel explicitly given by K(z, w) = e 2 . Recall that the Toeplitz operator with symbol g is given by ¯ dμ(w). Tg f (z) = P (f g)(z) = g(w)f (w)ez·w/2 Cn

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Toeplitz Operators

319

An orthonormal basis for the Fock space F(Cn ) is given by ζα (z) =

zα 2

|α| 2

(α!)1/2

.

As Tg ζα , ζβ = gζα , ζβ we can easily check that Tg ζα , ζβ = δαβ Tg ζα , ζα whenever g is radial. This leads to the following result for Toeplitz operators with radial symbols. In what follows we let 1

qt (z) = (4πt)−n e− 4t |z|

2

stand for the heat kernel on Cn associated to the standard Laplacian and 1 2 − 1 |z|2 n−1 |z| (z) = L ϕn−1 e 4 k k 2 for the Laguerre functions of type (n − 1). In general, Laguerre functions of − 1 |z|2 α 1 2 e 4 , where Lα type α > −1 are deﬁned as ϕα k (z) = Lk 2 |z| k are the Laguerre polynomials of type α > −1 given by Lα k (x) =

k j=0

(−x)j Γ(k + α + 1) . Γ(k − j + 1)Γ(j + α + 1) j!

For further details refer to [20]. In the rest of the paper ϕk (z) stands for (z) unless otherwise speciﬁed. ϕn−1 k Theorem 2.1. Let g be a radial measurable function on Cn such that gζα ∈ L2 (Cn , dμ) for all α ∈ Nn . Then the Toeplitz operator Tg is bounded on F(Cn ) if and only if the sequence k!(n − 1)! g ∗ q1/4 (w)ϕk (2w)dw (k + n − 1)! Cn

is bounded. Proof. Using the result (see Lemma 3.2.6 in [26]) (n − 1)! 2−|α| ζα (ω)ζβ (ω)dσ(ω) = δαβ (|α| + n − 1)! S 2n−1

we easily calculate that whenever g is radial k!(n − 1)! Tg ζα , ζβ = δαβ (k + n − 1)! where k = |α|. Therefore, Tg F (z) =

α∈Nn

where

∞ g(r) 0

r2k+2n−1 − 1 r2 e 2 dr 2k k!

R|α| (g)F, ζα ζα (z)

320

K. Jotsaroop and S. Thangavelu k!(n − 1)! Rk (g) = (k + n − 1)!

∞ g(r) 0

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r2k+2n−1 − 1 r2 e 2 dr. 2k k!

The theorem now follows from the following lemma.

Lemma 2.2. Let g be a radial function as in the theorem. Then for any k ∈ N, k!(n − 1)! g ∗ q1/4 (w)ϕk (2w)dw. Rk (g) = cn (k + n − 1)! Cn

Proof. We make use of the following formula satisﬁed by Laguerre functions (see Szego [20]) −x2

e

2 Lα k (x )

1 = k!

∞

2

e−t t2k+α

0

Jα (2tx) α t dt tα xα

which can be rewritten as −2x2

e

2 Lα k (2x )

1 = k!

∞

1 2

e− 8 t

0

t2 8

k

Jα (tx) 2α+1 t dt. tα xα

Inverting the Hankel transform and making a change of variables we get 1 − 1 t2 e 2 k!

t2 2

k

∞ =

2

2 e−2x Lα k (2x )

0

Jα (2tx) 2α+1 x dx. (2tx)α

(2.1.1)

As both sides are holomorphic in t the above equation remains true when t is replaced by it. Under the assumption that g is radial we observe that 2 g ∗ q1/4 (z) = π −n g(w)e−|z−w| dw Cn

reduces to a constant multiple of −|z|2

∞

e

g(r)e−r

0

2

Jn−1 (2irs) 2n−1 r dr. (2irs)n−1

Therefore, g ∗ q1/4 (w)ϕk (2w)dw Cn

∞ ∞ = cn 0

0

2

e−r g(r)e−s

2

Jn−1 (2irs) n−1 2 −s2 2n−1 2n−1 L (2s )e r s drds. (2irs)n−1 k

Using Fubini, which is justiﬁed by our assumptions on g, and making use of the above identity (2.1.1) satisﬁed by Laguerre functions, we obtain

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Toeplitz Operators

∞ g ∗ q1/4 (w)ϕk (2w)dw = cn

g(r) 0

Cn

321

r2k+2n−1 − 1 r2 e 2 dr. 2k k!

This completes the proof of the lemma.

Corollary 2.3. Let g be a radial function as in the previous theorem. Further assume that |g ∗ q1/4 (z)| ≤ C|z|−1 , for all z = 0. Then Tg is bounded on F(Cn ). Proof. In view of the theorem we only need to check that the sequence k!(n − 1)! (k + n − 1)!

∞

2

h(r)Ln−1 (2r2 )e−r r2n−1 dr k

0

is bounded where h(z) = g ∗ q1/4 (z). Under the assumption on g ∗ q1/4 this can be easily veriﬁed using the following estimates on integrals of Laguerre functions. ∞ k!(n − 1)! 2 n−1 2 −r 2n−1 dr (k + n − 1)! h(r)Lk (2r )e r 0

≤ cn

k!(n − 1)! (k + n − 1)!

∞

r−1 |Ln−1 (r2 )|e−r k

2

/2 2n−1

r

dr.

0

(r2 ), r ∈ R by We deﬁne Ln−1 k 1/2 2 k!(n − 1)! n−1 2 Ln−1 (r2 )rn−1 e−r /2 . Lk (r ) = k (k + n − 1)! It follows from Lemma 1.5.4 in [26] that ∞

|Ln−1 (r2 )|r−β rdr ∼ k 1/2−β/2 k

0 k!(n−1)! when k is large. By Stirling’s formula for large k, (k+n−1)! ∼ k −(n−1) . By using the estimates above after putting β = −(n − 2) we have ∞ k!(n − 1)! 2 n−1 2 −r 2n−1 dr (k + n − 1)! h(r)Lk (2r )e r

≤

0

k!(n − 1)! (k + n − 1)!

1/2 ∞

|Ln−1 (r2 )|rn−2 rdr k

0

∼ k −(n−1)/2 k 1/2+(n−2)/2 = 1. This proves the corollary.

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2.2. Toeplitz Operators and Fourier Multipliers A conjecture of Berger and Coburn [2] says that Tg is bounded on F(Cn ) if and only if g ∗ q1/4 is bounded. In this subsection we verify this conjecture when the symbol g(x + iy) depends only on y. In such a case the problem reduces to checking if a certain Fourier multiplier is bounded on L2 (Rn ). As the Fock space is closely related to the weighted Bergman space associated to the Segal–Bargmann/heat kernel transform we consider Toeplitz operators on the space Bt (Cn ) consisting of entire functions that are square integrable with respect to qt/2 (y)dxdy where qt is the standard heat kernel on Rn . By the results of Segal and Bargmann [1] we know that F ∈ Bt (Cn ) if and only if F = f ∗ qt for some f ∈ L2 (Rn ) and |F (x + iy)|2 qt/2 (y)dxdy = cn |f (x)|2 dx. Rn

R2n

Let g be a measurable function on C such that gF belongs to L2 (Cn , qt/2 (y)dz) whenever F ∈ Bt (Cn ) and let Tg be the associated Toeplitz operator. n

Theorem 2.4. Let g(x + iy) = g0 (y) be as above. Then Tg is bounded on Bt (Cn ) if and only if g0 ∗ qt/2 is bounded where the convolution is on Rn . Proof. When Fj = fj ∗ qt ∈ Bt (Cn ), j = 1, 2 Plancherel’s theorem leads to 2 F1 (x + iy)F2 (x + iy)dx = cn fˆ1 (ξ)fˆ2 (ξ)e−2t|ξ| e−2y·ξ dξ. Rn

Rn

Integrating the above with respect to g(y)qt/2 (y)dy we see that Tg F1 (x + iy)F2 (x + iy)qt/2 (y)dxdy = cn mt (ξ)fˆ1 (ξ)fˆ2 (ξ)dξ Cn

Rn

where mt (ξ) = e−2t|ξ|

2

e−2y·ξ g0 (y)qt/2 (y)dy.

Rn

From this it is clear that Tg is bounded if and only if mt deﬁnes a bounded Fourier multiplier on L2 (Rn ) which happens precisely when mt is a bounded function. An easy calculation shows that mt (ξ) = g0 ∗ qt/2 (ξ) which proves the theorem. Remark 2.1. We can read out properties of Fourier multipliers mt (ξ) that correspond to Toeplitz operators from the work of Hille [14]. Indeed, when t = 1/2 which corresponds to the Fock space, the multiplier m and the symbol g are related via 2 −n/2 g0 (y)e−|ξ−y| dy. m(ξ) = (2π) Rn

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Assuming n = 1, let ∞

|g0 (y)|e−y

2

+α|y|

dy < ∞

−∞

∞ for some α > 0. Then if g0 (y) = k=0 ak Hk (y) is the expansion of g0 in terms ∞ of thek Hermite polynomials Hk , Hille [14] has proved that m(z) = k=0 ak (2z) for all z ∈ C with |z| < α.

3. Toeplitz Operators on Hermite–Bergman Spaces In this section we study Toeplitz operators on Hermite-Bergman spaces which are Segal–Bargmann spaces associated to the Hermite semigroup e−tH . As in the case of Fock spaces we show that the transferred operator etH Tg e−tH is a pseudo-differential operator whose Weyl symbol is related to the heat ﬂow of g. This leads to a conjecture similar to that of Berger and Coburn. By making use of Gutzmer’s formula for Hermite expansions we identify certain special symbols g which lead to Hermite multipliers. 3.1. Hermite–Bergman Spaces On R2n consider the weight function Ut given by Ut (x, y) = 4n (sinh(4t))−n/2 etanh(2t)|x|

2

−coth(2t)|y|2

.

The Hermite Bergman space Ht (Cn ) is the space of all entire functions F which are square integrable with respect to Ut (x, y)dxdy. It is known that F ∈ Ht (Cn ) if and only if F = e−tH f for some f ∈ L2 (Rn ) where e−tH is the Hermite semigroup, see [3]. Moreover, |F (x + iy)|2 Ut (x, y)dxdy = cn |f (x)|2 dx Rn

R2n

whenever F = e−tH f. In the above the Hermite semigroup is deﬁned by e−tH f = e−(2|α|+n)t f, Φα Φα α∈Nn

where Φα are the normalised Hermite functions which are eigenfunctions of the Hermite operator H = −Δ + |x|2 with eigenvalues (2|α| + n). See [25] for more about Hermite functions. An important tool in studying the above space is an analogue of Gutzmer’s formula for Hermite expansions which we now proceed to state. Let π(x, u) be the family of unitary operators deﬁned on L2 (Rn ) by 1

π(x, u)ϕ(ξ) = ei(x·ξ+ 2 x·y) ϕ(ξ + y). These are related to the Schr¨odinger representation of the Heisenberg group, see [7,21]. It is clear π(z, w)F (ξ) makes sense even for (z, w) ∈ Cn × Cn whenever F is holomorphic. However, the resulting function need not be in L2 (Rn ) unless further assumptions are made on F. When F = Φα (or any ﬁnite linear combination of the Hermite functions) π(z, w)F (ξ) is indeed in

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L2 (Rn ) and using Mehler’s formula for the Hermite functions we can prove that n |π(z, w)Φα (ξ)|2 dξ = (2π) 2 e(u·y−v·x) Φα,α (2iy, 2iv) Rn

where Φα,α are the special Hermite functions which are expressible in terms of Laguerre functions. Gutzmer’s formula says that a similar result is true for π(z, w)F (ξ) under some assumptions on F. In order to state Gutzmer’s formula we need to introduce one more notation. Let Sp(n, R) stand for the symplectic group consisting of 2n × 2n real matrices that preserve the symplectic form [(x, u), (y, v)] = (u · y − v · x) on R2n and have determinant one. Let O(2n, R) be the orthogonal group and we deﬁne K = Sp(n, R)∩O(2n, R). Then there is a one to one correspondence between K and the unitary group U (n). Let σ = a + ib be an n × n complex matrix with real and imaginary parts a and b. Then σ is unitary if and only a −b if the matrix A = is in K. For these facts we refer to Folland [7]. b a By σ.(x, u) we denote the action of the correspoding matrix A on (x, u). This action has a natural extension to Cn × Cn denoted by σ.(z, w) and is given by σ.(z, w) = (a.z − b.w, a.w + b.z) where σ = a + ib. Theorem 3.1. For a holomorphic function F we have the following formula for any z = x + iy, w = u + iv ∈ Cn : |π(σ.(z, w))F (ξ)|2 dσdξ Rn K (u·y−v·x)

=e

∞ k!(n − 1)! ϕk (2iy, 2iv) Pk f 22 (k + n − 1)!

k=0

where f stands for the restriction of F to Rn . In the above formula Pk are the spectral projections of the Hermite operator deﬁned by Pk f (x) = f, Φα Φα (x) |α|=k

and

ϕk (z, w) = Ln−1 k

2 2 1 1 2 (z + w2 ) e− 4 (z +w ) 2

are the holomorphically extended Laguerre functions of type (n − 1). The above formula means that if either the integral or the sum is ﬁnite then they are equal. Note that the sum is clearly ﬁnite when f = e−tH g for some g ∈ L2 (Rn ). We refer to [24] for a proof of the above formula. The characterisation of Ht (Cn ) as the image of L2 (Rn ) under the Hermite semigroup e−tH can be proved using Gutzmer’s formula, see [24]. The only other ingredient

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needed is the formula k!(n − 1)! p2t (2y, 2v)ϕk (2iy, 2iv)dydv = e2(2k+n)t (k + n − 1)! R2n

where pt (y, v) stands for the heat kernel associated to the special Hermite operator, see Sect. 4. 3.2. Toeplitz Operators on Ht (Cn ) Let P : L2 (Cn ) → Ht (Cn ) be the orthogonal projection which is explicitly given by P F (z) = F (u, v)Kt (z, u + iv)Ut (u, v)dudv. R2n

Here Kt (z, w) is the reproducing kernel of Ht (Cn ) deﬁned by e−2(2|α|+n)t Φα (z)Φα (w). ¯ Kt (z, w) = α∈Nn

Using Mehler’s formula we can show that n

1

Kt (z, w) = (sinh(4t))− 2 e− 2 coth(4t)(z

2

+w2 )

1

e sinh(4t) z,w ,

where z, w is the standard Hermitian inner product on Cn and z 2 = z12 + ... + zn2 etc. For a measurable function g on Cn such that gKt (., w) belongs to L2 (Cn , dμt ) for all w (we will refer to this condition as ∗), we deﬁne the Toeplitz operator Tg on Ht (Cn ) by Tg f (z) = g(w)f (w)Ks (z, w)dμs (w). Cn

By the condition (∗), it is easy to see that Tg is a densely deﬁned operator on Ht (Cn ). Another important consequence of (∗) is that g ∗ qs is well 2 n 1 deﬁned for 0 < s < 12 sinh 4t, where qs (x) = (4πs)− 2 e− 4s |x| is the heat kernel corresponding to the standard Laplacian on Rn . In fact, it is a C ∞ function on Cn . By using the semigroup property we get g ∗ qs+r = (g ∗ qr ) ∗ qs , 1 2

when 0 < s + r < sinh 4t. Now we ﬁnd some necessary and sufﬁcient conditions on g such that Tg is a bounded operator. These conditions are given in terms of g ∗ qs for 0 < s < 12 sinh 4t. In order to do this we transfer Tg to L2 (Rn ) and ﬁnd the corresponding Weyl symbol of the resulting operator. Following Folland [7] we deﬁne the Weyl pseudo-differential operator on L2 (Rn ) with symbol σ ∈ S (R2n ) by 1 (x + y), ξ e−i(x−y).ξ f (y)dydξ. (3.2.1) σ σ(D, X)f (x) = (2π)−n 2 Rn Rn

We recall that (see [7]) σ(D, X) = W (ˆ σ ), where W is the Weyl transform and σ ˆ is the Fourier transform of a tempered distribution. We deﬁne for σ ∈ S (R2n )

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σt (x, ξ) = σ(cosh(2t)x, − sinh(2t)ξ).

(3.2.2)

2n

Note that σ → σt is an isomorphism on S (R ). Theorem 3.2. Let Tg , defined as above, be bounded. Then we have g ∗qs ∞ ≤ 1 1 c(s) Tg for all s ∈ 8 sinh 4t, 2 sinh 4t . Conversely, if we assume that g ∗ qs ∞ < ∞ for some 0 < s < 18 sinh 4t, then Tg is bounded. Moreover, we have Tg ≤ c(s) g ∗ qs ∞ . Proof. First let us assume that Tg is bounded. For 14 sinh 4t ≤ s < 12 sinh 4t the proof is trivial. We look at the Berezin Transform of Tg deﬁned by (see [7]) T˜g (z) = Tg kz , kz Ht .

(3.2.3)

It is easy to check that T˜g (z) = g ∗ q 14 sinh4t (z). Here kz (w) =

Kt (w,z) √ Kt (z,z)

is the

normalized reproducing kernel. In fact, even if Tg is not bounded the Berezin transform is well deﬁned because of the condition (∗) and it is the same as above. By applying Cauchy-Schwarz inequality to (3.2.3) we get |g ∗ q 14 sinh 4t (z)| ≤ Tg , So, by the semigroup property, when 0 < s < = (g ∗ q 14 sinh 4t ) ∗ qs (z) and

z ∈ Cn . 1 2

(3.2.4)

sinh 4t we get g∗qs+ 14 sinh 4t (z)

g ∗ qs+ 14 sinh 4t ∞ ≤ c(s) Tg ,

(3.2.5)

where c(s) is independent of g. For proving the estimate for the other half of the interval in the statement of the theorem, we make use of the boundedness of the operator etH Tg e−tH on L2 (Rn ). Let etH Tg e−tH = W (σˆt ) for some σ ∈ S (R2n ). In order to ﬁnd the explicit form of σt we calculate the Berezin transform of Tg in terms of σ. By using (3.2.1) an easy computation shows that Tg kz , kz Ht = e−tH σt (D, X)etH kz , kz Ht , σt (D, X)etH kz , etH kz L2 (Rn ) = σ ∗ q 18 sinh 4t (z).

(3.2.6)

By equating (3.2.3) and (3.2.6) we get g ∗ q 14 sinh 4t (z) = σ ∗ q 18 sinh 4t (z),

z ∈ Cn .

(3.2.7)

Given that g satisﬁes (∗) and σ ∈ S (R2n ), for a ﬁxed z ∈ Cn it is easy to check the following two facts : (i) s −→ g ∗ qs (z) extends as a holomorphic function to the domain

1 1 D1 = ζ ∈ C : |ζ − sinh 4t| < sinh 4t 4 4 and (ii) s −→ σ ∗ qs (z) extends as a holomorphic function to D2 = {ζ ∈ C : ζ > 0}. By using the above two facts we get that g ∗ q 18 sinh 4t ∗ qs ≡ σ ∗ qs for all 0 < s < 38 sinh 4t. Now taking the limit s −→ 0 we get σt (x, y) = g ∗ q 18 sinh 4t (cosh(2t)x, − sinh(2t)y).

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Using the fact that B(L2 (Rn )) (Banach space of bounded linear operators on L2 (Rn ) with . := . op ) is the dual of the space of all trace class operators on L2 (Rn ), we get the following: |tr(W (f )W (σˆt ))| ≤ Tg W (f ) tr

(3.2.8)

for all f ∈ L2 (Cn ) such that W (f ) is trace class. In particular, (3.2.8) holds for all f in Schwartz class. It is easy to compute that tr(W (¯ z )W (f )W (¯ z )∗ W (σˆt )) = fˆ ∗ σt (z)

(3.2.9)

for all z when f ∈ S(R2n ) and σ ∈ S (R2n ). If we choose f in (3.2.9) such that fˆ(w) = qt1 (u)qt2 (v), w = u + iv where t1 = s cosh 2t, t2 = s sinh 2t and z = (cosh 2t)−1 x + i(sinh 2t)−1 y we get tr(W (¯ z )W (f )W (¯ z )∗ W (σˆt )) = σ ∗ qs (x + iy) for all s > 0. By (3.2.8) |σ ∗ qs (x + iy)| ≤ c(s) Tg , where σ = g ∗ q 18 sinh 4t and this implies |(g ∗ q 18 sinh 4t ) ∗ qs (x + iy)| ≤ c(s) Tg

(3.2.10)

for all z ∈ Cn . Finally, the boundedness of Tg implies that g ∗ qs ∞ ≤ c(s) Tg whenever s > 18 sinh 4t. Conversely, let g ∗ qs ∞ < ∞ for some 0 < s < ing as in Berger and Coburn [2]

1 8

sinh 4t then proceed-

σt ∗ ≡ Σ|μ|+|β|≤2n+1 Dξμ Dxβ σt ∞ < ∞, where σ = g ∗ q 18 sinh 4t . Now we can appeal to Theorem 2.73 in [7] by which σt (D, X) is bounded with σt (D, X) ≤ σt ∗ . The Berezin symbol of e−tH σt (D, X)etH [see (3.2.6) and (3.2.3)] is given by (e−tH σt (D, X)etH )˜(z) = σ ∗ q 18 sinh 4t (z) = g ∗ q 14 sinh 4t (z) which implies that T˜g ≡ (e−tH σt (D, X)etH )˜. Hence by the uniqueness of the Berezin transform Tg = e−tH σt (D, X)etH . Therefore, the boundedness of σt (D, X) implies that Tg ≤ σt ∗ . As shown in [2] we have σt ∗ ≤ c(n, s) g ∗ qs ∞ . Hence the theorem is proved. Remark 3.1. The above theorem is the analogue of Theorems 11 and 12 in [2]. As in [2] we conjecture that Tg is bounded if and only if g ∗ q 18 sinh 4t is bounded. We have a class of symbols supporting this conjecture, see Sect. 3.3

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3.3. Hermite Multipliers and Toeplitz Operators In this subsection we are interested in ﬁnding a necessary and sufﬁcient condition on the symbol g so that etH Tg e−tH is a Hermite multiplier. Using the fact that W (σ) is a function of the Hermite operator if and only if the symbol σ is a radial distribution we get the following result. Theorem 3.3. Given Tg on Ht (Cn ) the operator etH Tg e−tH is a Hermite multiplier if and only if g ∗ q 18 sinh(4t) (cosh(2t)y, − sinh(2t)v) is a radial function on R2n . Corollary 3.4. Let g be as in the theorem. Then Tg is bounded on Ht (Cn ) if and only if the sequence k!(n − 1)! g ∗ q 18 sinh(4t) (cosh(2t)y, − sinh(2t)v)ϕk (2y, 2v)dydv (k + n − 1)! R2n

is bounded. Example 3.1. An example of symbol satisfying the condition given in Theo2 2 rem 3.3 is provided by g(y, v) = eα|y| +β|v| under suitable conditions on α and β. A simple calculation shows that g ∗ q 18 sinh(4t) (cosh(2t)y, − sinh(2t)v) = e

α coth(2t) sinh(4t) |y|2 2−α sinh(4t)

e

β tanh(2t) sinh(4t) |v|2 2−β sinh(4t)

and hence g ∗ q 18 sinh(4t) (cosh(2t)y, − sinh(2t)v) is radial if and only if α coth(2t) β tanh(2t) = . 2 − α sinh(4t) 2 − β sinh(4t) After simpliﬁcation we get the condition α coth(2t) − β tanh(2t) = αβ which is necessary and sufﬁcient for the radiality of the function g ∗ q 18 sinh(4t) (cosh(2t)y, − sinh(2t)v),

g(y, v) = eα|y|

2

+β|v|2

.

When the above condition is veriﬁed, by Corollary 3.4 the operator Tg is bounded on Ht (Cn ) if and only if the sequence α coth(2t) sinh(4t) 2 2 k!(n − 1)! e 2−α sinh(4t) (|y| +|v| ) ϕk (2y, 2v)dydv (k + n − 1)! R2n

is bounded. Again, by repeating the method in the Theorem 2.1 this is equivalent to the boundedness of the sequence 2 k!(n − 1)! eλ|z| ϕk (2z)dz (k + n − 1)! = cn Cn

Cn λ

e 1+λ |z|

2

|z|2k − 1 |z|2 e 2 dz = cn k!2k

α coth(2t) sinh(4t) . Thus the condition for 2−α sinh(4t) 1+λ | 1−λ | ≤ 1 or λ ≤ 0. In terms of α

1+λ 1−λ

k

where λ = the boundedness of Tg reduces to the condition reads as 2 |α| sinh(4t) − 2α ≥ 0. So, the necessary and sufﬁcient condition for Tg

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to be bounded is the boundedness of g ∗ q 18 sinh(4t) . It is worth comparing this example with a similar example given in [2]. The condition in Corollary 3.4 on g is not easy to check. However, using Gutzmer’s formula we can get a sufﬁcient condition in a more convenient form for certain special class of symbols. Consider radial functions h(y, v) on R2n for which 2 2 h(y, v)e|y| +|v| (|y|2 + |v|2 )k/2 < ∞ (3.3.11) R2n

for all k ∈ N. Deﬁne a function g by the equation g(ξ, v)Ut (ξ, v) = e−2y·ξ h(y, v)dy.

(3.3.12)

Rn

Theorem 3.5. Suppose g is given by (3.3.12) where h satisfies (3.3.11). Then we have etH Tg e−tH = mt (H) where k!(n − 1)! mt (2k + n) = e−2(2k+n)t h(y, v)ϕk (2iy, 2iv)dydv. (k + n − 1)! R2n

Consequently, Tg is bounded on Ht (Cn ) if and only if |mt (2k + n)| ≤ C for all k ∈ N. Proof. As we mentioned we prove this theorem by using Gutzmer’s formula. Indeed, polarising Gutzmer’s formula we obtain π(k.(iy, iv))F1 (ξ)π(k.(iy, iv))F2 (ξ)dξdk K Rn

=

∞ k!(n − 1)! −2t(2k+n) e Pk f1 , f2 ϕk (2iy, 2iv) (k + n − 1)!

k=0 −tH

fj , j = 1, 2 are from Ht (Cn ). Integrating the above identity where Fj = e with respect to h(y, v)dydv we obtain π(k.(iy, iv))F1 (ξ)π(k.(iy, iv))F2 (ξ)h(y, v)dξdkdydv R2n K Rn ∞

=

k=0

k!(n − 1)! −2t(2k+n) e Pk f1 , f2 (k + n − 1)!

h(y, v)ϕk (2iy, 2iv)dydv.

R2n

When the function h is K invariant, π(k.(iy, iv))F1 (ξ)π(k.(iy, iv))F2 (ξ)h(y, v)dξdkdydv R2n K Rn

=

π(iy, iv)F1 (ξ)π(iy, iv)F2 (ξ)h(y, v)dξdydv. R2n Rn

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Recalling the definition of π(iy, iv) the above integral can be rewritten as F1 (ξ + iv)F2 (ξ + iv)e−2y·ξ h(y, v)dξdydv. R2n Rn

Suppose now g(ξ, v) satisﬁes the equation g(ξ, v)Ut (ξ, v) = e−2y·ξ h(y, v)dy Rn

and mt (2k + n) is deﬁned by mt (2k + n) = e−2t(2k+n)

k!(n − 1)! (k + n − 1)!

h(y, v)ϕk (2iy, 2iv)dydv. R2n

Then it is clear that we have obtained F1 (ξ + iv)F2 (ξ + iv)g(ξ, v)Ut (ξ, v)dξdv R2n

=

∞

mt (2k + n)Pk f1 , f2

k=0

which simply means that (Tg F1 , F2 )Ht = mt (H)f1 , f2 L2 where mt (H)f =

∞

mt (2k + n)Pk f

k=0

is the Hermite multiplier. Thus the boundedness of the Toeplitz operator Tg on Ht (Cn ) is equivalent to the boundedness of mt (H) on L2 (Rn ). Remark 3.2. In the above proof of sufﬁciency we have not used Theorem 3.3 but the condition stated in that theorem can be veriﬁed. Indeed, when g satisﬁes the equation (3.3.12) a simple calculation shows that 2

2

g ∗ q 18 sinh(4t) (cosh(2t)x, − sinh(2t)y)etanh(2t)(|x| +|y| ) 2 2 2 = h(ξ, v)etanh(2t)(|ξ| +|v| ) e− cosh(2t) (x·ξ+y·v) dξdv R2n

from which it is clear that g ∗q 18 sinh(4t) (cosh(2t)x, − sinh(2t)y) is radial whenever h(ξ, v) is radial. The above equation also suggests a relation between g and h. Remark 3.3. The radiality of the function g ∗ q 18 sinh(4t) (cosh(2t)x, − sinh(2t)y) is not equivalent to the factorisation given in (3.3.12). Indeed, consider the 2 2 symbol g(x, y) = eα|x| +β|y| considered earlier with the conditions α coth (2t) − β tanh(2t) = αβ and (α) < 12 (sinh(2t))−1 . If there exists a function h such that g(ξ, v)Ut (ξ, v) = e−2ξ·y h(y, v)dy, Rn

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then we have the relation i i ξ, v Ut ξ, v = e−iξ·y h(y, v)dy. g 2 2 Rn

This leads to the equation − 14 (tanh(2t)+α)|y|2 −(coth(2t)−β)|v|2

e

e

=

e−iξ·y h(y, v) dy.

Rn

By Fourier inversion we see that 1

2

h(y, v) = c e− tanh(2t)+α |y| e−(coth(2t)−β)|v|

2

which is not radial in general. Remark 3.4. Since

Ut (ξ, v) = ct

p2t (2y, 2v)e−2y·ξ dy

Rn

the equation (3.3.12) is equivalent to g(ξ, v) = g1 (y, v)e−2y·ξ dy. Rn

Indeed, if g satisﬁes the above equation, then the function h(y, v) = g1 (y − u, v)p2t (2u, 2v)du Rn

satisﬁes

h(y, v)e−2y·ξ dy = g(ξ, v)Ut (ξ, v)

Rn

as can be easily veriﬁed. Thus for such symbols Theorem 3.5 is valid.

4. Toeplitz Operators on Twisted Bergman Spaces In this section we take up the study of Toeplitz operators on twisted Bergman spaces which are Segal–Bargmann spaces associated to the special Hermite semigroup e−tL . These spaces arise naturally in the study of Segal–Bargmann transform on the Heisenberg group, see [15]. We show that etL Tg e−tL is a Weyl multiplier if and only if the symbol g(x + iy, u + iv) depends only on (y, v). By means of Gutzmer’s formula we study boundedness of Tg which correspond to multipliers for special Hermite operators.

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4.1. Twisted Bergman Spaces By the term twisted Bergman spaces we mean the Hilbert space of entire functions F (z, w) on C2n which are square integrable with respect to the weight function Wt (z, w) = e(u·y−v·x) p2t (2y, 2v) where 1

p2t (y, v) = cn (sinh(2t))−n e− 4 coth(2t)(|y|

2

+|v|2 )

is the heat kernel associated to the special Hermite operator L, see [23]. Thus the special Hermite semigroup e−tL is given by e−tL f = f × pt , the twisted convolution of f with pt . These spaces, denoted by Bt∗ (C2n ), arise naturally in the study of Segal–Bargmann transform on the Heisenberg group [15]. The following result proved in [15] characterises Bt∗ (C2n ). Theorem 4.1. An entire function F on C2n belongs to Bt∗ (C2n ) if and only if its restriction to R2n is of the form e−tL f (x, u) for some f ∈ L2 (R2n ). Moreover, the norm of F in Bt∗ (C2n ) is the same as the norm of f in L2 (R2n ). Another proof of this was found in [22] which is based on the following Gutzmer’s formula for the special Hermite expansion. Recall that ϕk (x, u) = ϕk (x + iu) are the Laguerre functions of type (n − 1) introduced earlier. They extend to entire functions on C2n and are denoted by ϕk (z, w). The twisted convolutions f × ϕk are the projections onto the k-th eigenspace of L and the special Hermite expansion of an f ∈ L2 (R2n ) is written as f = (2π)−n

∞

f × ϕk

k=0

where the series converges in L2 . The heat kernel pt associated to the special Hermite operator can also be written as pt (x, u) = (2π)−n

∞

e−(2k+n)t ϕk (x, u).

k=0 2n

Theorem 4.2. For any F ∈ O(C ) we have e(u.y−v.x) |F (σ(x + iy, u + iv))|2 dxdudσ R2n K

=

∞ k!(n − 1)! f × ϕk 22 ϕk (2iy, 2iv), (k + n − 1)!

k=0

where f is the restriction of F to R2n . Clearly when F = e−tL f the above formula holds. So, Theorem 4.1 easily follows from Gutzmer’s formula once we have the identity (k + n − 1)! (2k+n)2t e p2t (2y, 2v)ϕk (2iy, 2iv)dydv = . (4.1.1) k!(n − 1)! R2n

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This has been proved in [22], see Lemma 6.3. The following extension of this result is needed for the study of Toeplitz operators. Lemma 4.3. i(u·y−v·x) 2 e pt (x − y, u − v)ϕk (iy, iv)dydv = ϕk (ix, iu)e(2k+n)t . (4.1.2) R2n

˜ Proof. Recall that the symplectic Fourier transform f of a function f is 1 deﬁned by f˜(x, y) = fˆ 2 (−y, x) . We know that ϕk ’s are eigenfunctions of the symplectic Fourier transform with eigenvalues (−1)k , i.e. i(η.y−ξ.v) −n 2 ϕk (ξ, η)e dξdη = (−1)k ϕk (y, v). (2π) R2n

The above equation remains true even if we replace (y, v) by (iy, iv). So we get (η.y−ξ.v) (2π)−n ϕk (ξ, η)e− 2 dξdη = (−1)k ϕk (iy, iv). (4.1.3) R2n

Now putting (4.1.3) in (4.1.2) and by using Fubini’s theorem we get i(u·y−v·x) (η.y−ξ.v) 2 2 coth t k 2 (−1) e e− 4 ((x−y) +(u−v) ) e− 2 ϕk (ξ, η)dydvdξdη R2n R2n

= (tanh t)n (−1)k

e

η.x−ξ.u 2

e−

tanh t ((u−iη)2 +(x−iξ)2 ) 4

ϕk (ξ, η)dξdη.

R2n

Now look at the function η.x−ξ.u 2 2 tanh t F (t) = (cosh t)−n e 2 e− 4 ((u−iη) +(x−iξ) ) ϕk (ξ, η)dξdη. R2n

If we replace t by z with |z| < π/2, it is easy to see that the integral converges absolutely. In fact, F can be extended as a holomorphic function to the strip |z| < π/2 containing the real line. Consider η.x−ξ.u 2 2 tanh t −n F (−t) = (cosh t) e 2 e 4 ((u−iη) +(x−iξ) ) ϕk (ξ, η)dξdη R2n

which after using e

tanh t ((u−iη)2 +(x−iξ)2 ) 4

reads as F (−t) = (cosh t)−n

e

η.x−ξ.u 2

= e−

e−

tanh t ((η+iu)2 +(ξ+ix)2 ) 4

tanh t ((η+iu)2 +(ξ+ix)2 ) 4

ϕk (ξ, η)dξdη.

R2n

This is nothing but the twisted convolution of ϕk with p t at (iu, ix). It is easy to calculate F (−t) by recalling ∞ e−(2k+n)t (−1)k ϕk (ξ, η). p t (ξ, η) = k=0

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Using the above along with the fact that ϕk × ϕj = cn δj,k ϕk we get F (−t) = (−1)k e−(2k+n)t ϕk (ix, iu). The right hand side of the above equation is also a holomorphic function of t and both sides agree on the negative real axis. Therefore, they agree everywhere and changing t into −t we get the lemma. Note that when x = u = 0 this lemma reduces to (4.1.1) as ϕk (0, 0) =

(k+n−1)! k!(n−1)! .

4.2. Toeplitz Operators and Special Hermite Multipliers In this subsection we get some necessary and sufﬁcient conditions for the boundedness of Tg on Bt∗ (C2n ) for a special class of symbols, by making use of Gutzmer’s Formula for the special Hermite expansion. First note that by Theorem 4.1 e−tL Φα,β (z, w) = e−(2|β|+n)t Φα,β (z, w) form an orthonormal basis for Bt∗ (C2n ). We denote e−tL Φα,β by φα,β in this section. Consider a measurable function g on C2n for which |g(z, w)φα,β (z, w)φμ,ν (z, w)|Wt (z, w)dzdw < ∞. (4.2.4) C2n

Now we can deﬁne a densely deﬁned bilinear form on Bt∗ (C2n ) by Tg φα,β , φμ,ν Bt∗ (C2n ) := g(z, w)φα,β (z, w)φμ,ν (z, w)Wt (z, w)dzdw. C2n

These are nothing but the matrix entries of Tg . We consider special symbols g for which the above densely deﬁned sesquilinear form becomes a diagonal form. Such symbols are provided by functions of the form g(x + iy, u + iv) = g0 (y, v) where g0 is a radial function on R2n . Theorem 4.4. Let g be as above and satisfy (4.2.4). Then Tg is bounded if and only if the sequence −(2k+n)2t k!(n − 1)! g0 (y, v)p2t (2y, 2v)ϕk (2iy, 2iv)dydv e (k + n − 1)! R2n

is bounded. Proof. Clearly Tg φα,β , φμ,ν Bt∗ (C2n ) is well deﬁned for all (α, β) and (μ, ν). As done in Section 3.3 we can polarize Gutzmer’s formula to obtain e(u.y−v.x) e−tL f1 (σ(x + iy, u + iv))e−tL f2 (σ(x + iy, u + iv))dxdudσ R2n K

=

∞ k!(n − 1)! −(2k+n)2t e f1 × ϕk , f2 × ϕk L2 (R2n ) ϕk (2iy, 2iv). (k + n − 1)!

k=0

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When f1 = Φα,β and f2 = Φμ,ν the above identity reduces to

R2n

e(u.y−v.x) φα,β (σ(z, w))φμ,ν (σ(z, w))dxdudσ

K

=

∞ j!(n − 1)! −(2j+n)2t Φα,β × ϕj , Φμ,ν × ϕj L2 (R2n ) ϕj (2iy, 2iv) (4.2.5) e (j + n − 1)! j=0

= δα,μ δβ,ν

k!(n − 1)! −(2k+n)2t ϕk (2iy, 2iv), e (k + n − 1)!

where |β| = k. Writing the matrix coefﬁcients explicitly g0 (y, v)φα,β (z, w)φμ,ν (z, w)Wt (z, w)dzdw. Tg φα,β , φμ,ν Bt∗ (C2n ) = C2n

The above integral converges absolutely. Now, replace (z, w) by σ(z, w) where σ ∈ K. Since g0 (y, v)Wt (z, w)dzdw is invariant under the action of K we get Tg φα,β , φμ,ν Bt∗ (C2n ) = g0 (y, v)φα,β (σ(z, w))φμ,ν (σ(z, w))Wt (z, w)dzdwdσ. K C2n

The integral converges absolutely and hence by applying Fubini’s theorem and using (4.2.5) we get Tg Φα,β , Φμ,ν Bt∗ (C2n ) k!(n − 1)! −(2k+n)2t e = δα,μ δβ,ν (k + n − 1)!

g0 (y, v)p2t (2y, 2v)ϕk (2iy, 2iv)dydv,

R2n

where |β| = k. Thus the operator Tg is diagonal in the basis {φα,β : α, β ∈ Nn }

and the theorem follows.

Let h be a radial measurable function on R2n and assume that 2 2 |h(y, v)|es(|y| +|v| ) dydv < ∞ (4.2.6) R2n

for all s > 0. Consider the symbol deﬁned by g(x + iy, u + iv)p2t (2y, 2v) = h × p2t (2y, 2v). Corollary 4.5. In the above theorem let g be as above with h satisfying (4.2.6). Then Tg is bounded if and only if the sequence k!(n − 1)! h(y, v)ϕk (iy, iv)dydv (k + n − 1)! R2n

is bounded.

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Proof. As h and pt are both radial, so is g0 (y, v) = g(iy, iv). Hence by Theorem 4.4 we know that Tg is bounded if and only if k!(n − 1)! −(2k+n)2t e g0 (y, v)p2t (2y, 2v)ϕk (2iy, 2iv)dydv. (k + n − 1)! R2n

As g(iy, iv)p2t (2y, 2v) = h × p2t (2y, 2v) the above simpliﬁes to k!(n − 1)! −(2k+n)2t e h × p2t (2y, 2v)ϕk (2iy, 2iv)dydv. (k + n − 1)!

(4.2.7)

R2n

Because of (4.2.6) we can use Fubini’s theorem to change the order of integration. By Lemma 4.3 we have p2t (x − 2y, u − 2v)ei(u.y−v.x) ϕk (2iy, 2iv)dydv = e(2k+n)2t ϕk (ix, iu). R2n

Using this in (4.2.7) we obtain the corollary.

From now on let us assume that g is a measurable function on C2n such

that C2n |g(z, w)φα,β (z, w)|2 Wt (z, w)dzdw < ∞ for all α, β. We will refer to this condition as (∗∗). Note that the condition (∗∗) on g implies that it belongs to L2 (C2n , e(u.y−v.x) e−

(|x|2 +|u|2 ) 2

1

e(− coth 2t+ 2 )(|y|

2

+|v|2 )

dzdw).

Bt∗ (C2n ) 2

For such symbols the Toeplitz operator on is deﬁned by Tg (f ) := P (gf ) where P is the orthogonal projection from L (C2n , Wt ) onto Bt∗ (C2n ). We study the class of symbols g for which Tg is bounded and etL Tg e−tL is a right Weyl multiplier, i.e. W etL Tg e−tL f = W (f )Mt for some Mt ∈ B(L2 (R2n )). Before proving the next theorem we prove a lemma which will be used. Let Vt (z, w) = e(u.y−v.x) e−

(|x|2 +|u|2 ) 2

1

e(− coth 2t+ 2 )(|y|

2

+|v|2 )

and consider the measure dτ (z, w) = Vt (z, w)dzdw, where dzdw is the Lebesgue measure on R4n . Let P(R4n ) be the set of all polynomials on R4n . Note that P(R4n ) ⊂ L2 (C2n , dτ (z, w)). Lemma 4.6. P(R4n ) is dense in L2 (C2n , dτ (z, w)). Proof. By abuse of notation let us denote any polynomial p ∈ P(R4n ) by p(z, w). As the weight function Vt (z, w) corresponding to dτ is nowhere van1/2 ishing, it is enough to show that the linear span of p(z, w) (Vt (z, w)) is 2 2n 2 2n dense in L (C ). More precisely, if there exists g ∈ L (C ) such that 1/2 g(z, w)p(z, w) (Vt (z, w)) dzdw = 0 (4.2.8) C2n

for all p ∈ P(R4n ) then we need to show g = 0. Now suppose that there exists g satisfying (4.2.8). It is easy to see that by completing the square in

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Vt (z, w) (4.2.8) can be rewritten as 2 2 2 2 coth 2t−1 1 g(z − v, w + y)p(z − v, w + y)e− 4 (|x| +|u| ) e− 2 (|y| +|v| ) dzdw = 0 C2n

(4.2.9) 4n

for all p ∈ P(R ). If we let g˜(z, w) = g(z + v, w − y) then it is clear that g L2 (C2n ) = g L2 (C2n ) . So, it is g˜ ∈ L2 (C2n ) whenever g ∈ L2 (C2n ) and ˜ enough to show that g˜ = 0. The equation (4.2.9) means that 2 2 2 2 coth 2t−1 1 g˜(z, w)q(z, w)e− 4 (|x| +|u| ) e− 2 (|y| +|v| ) dzdw = 0 C2n

for all q ∈ P(R4n ). As the linear span of functions of the form 1

q(z, w)e− 4 (|x| 2

2

+|u|2 ) − coth22t−1 (|y|2 +|v|2 )

e

2n

is dense in L (C ) the last equation implies g˜ = 0 proving the lemma.

2n

Theorem 4.7. Let a Lebesgue measurable function g on C satisfy (∗∗) and let Tg be the corresponding Toeplitz operator on Bt∗ (C2n ). Then Tg = 0 if and only if g = 0 a.e. Proof. When g = 0 a.e. clearly Tg = 0. Conversely, let Tg = 0. We need to prove that g = 0 a.e. By using the explicit form of the functions φα,β namely, z 2 +w2

φα,β (z, w) = Pα,β (z, w)e− 4 , where Pα,β are holomorphic polynomials on C2n of degree |α| + |β| the condition (∗∗) takes the form |g(z, w)Pα,β (z, w)|2 Vt (z, w)dzdw < ∞ C2n

for all α, β. The above also implies that g ∈ L2 (C2n , Vt (z, w)dzdw) in particular. In view of the previous lemma, proving g = 0 a.e. is equivalent to proving that g(z, w)p(z, w)Vt (z, w)dzdw = 0 (4.2.10) C2n 4n

for all p ∈ P(R ). The assumption Tg = 0 gives us for all α, β, μ, ν g(y, v)φα,β (z, w)φμ,ν (z, w)Wt (z, w)dzdw = 0. Tg φα,β , φμ,ν Bt∗ (C2n ) = C2n

Again by using the explicit form of φα,β we get g(y, v)Pα,β (z, w)Pμ,ν (z, w)Vt (z, w)dzdw = 0. C2n

We claim that for every α, β, the monomial z α wβ belongs to the linear span of {Pμ,ν (z, w) : |μ|+|ν| = |α|+|β|}. This claim would then prove (4.2.10) which in turn would prove g = 0 a.e. In fact, once we have the claim (4.2.10) ¯ ν which in turn will be true for all polynomials of the form p(z, w) = z α wβ z¯μ w α β μ ν will prove (4.2.10) for all monomials x y u v and hence for all polynomials.

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Returning to the claim it is sufﬁcient to prove it for z, w purely real. We know that the special Hermite functions Φμ,ν (x, u) give all the eigenfunctions of the dilated Hermite operator H(1/2) = − + 14 (|x|2 + |u|2 ) on R2n . (see [25]) More precisely, H(1/2)Φμ,ν = (|μ| + |ν| + n)Φμ,ν . If Hα,β (x, u) stand for the (ordinary) Hermite polynomials on R2n (adapted to H(1/2)) then it can be written as a linear combination of Pμ,ν (x, u) with |μ| + |ν| = |α| + |β|. It is well known that xα uβ can be written as a linear combination of Hμ,ν and hence as a linear combination of Pμ,ν as well. Thus g is orthogonal to all polynomials in L2 (C2n , Vt ) and this proves the result. We now characterise all the symbols g for which etL Tg e−tL reduces to a Weyl multiplier. For this characterisation we need to consider symbols g so that ga,b (z, w) := g(z + a, w + b) satisﬁes condition (∗∗) for all (a, b) ∈ R2n . Theorem 4.8. Let ga,b satisfy (∗∗) for all (a, b) ∈ R2n and let the corresponding Tg be a bounded operator on Bt∗ (C2n ). Then etL Tg e−tL is a right Weyl multiplier if and only if g(z, w) = g(iy, iv). Proof. Let us ﬁrst assume that Tg is bounded and corresponds to a right Weyl multiplier Mt . As proved in [4] we know that Mt = W (σ), for some σ ∈ S (R2n ). Therefore, we have etL Tg e−tL f = f × σ,

(4.2.11)

for all f ∈ L2 (R2n ). Recall that the twisted translations of functions on R2n are deﬁned by τ (a, b)f (x, u) := e−i/2(a.u−b.x) f (x − a, u − b), (a, b) ∈ R2n . Clearly, τ (a, b) is a unitary map on L2 (R2n ). It is easy to check that τ (a, b)f × g = τ (a, b)(f × g) when f, g ∈ L2 (R2n ). This implies that etL Tg e−tL commutes with twisted translations (see (4.2.11)). As e−tL f = f × pt , e−tL f is equivariant under twisted translations. By using the fact that e−tL is a unitary map from L2 (R2n ) onto Bt∗ (C2n ) and its equivariance under twisted translations, we get that τ (a, b)F, τ (a, b)GBt∗ (C2n ) = F, GBt∗ (C2n ) for all (a, b) ∈ R2n and F, G ∈ Bt∗ (C2n ). (Here, τ (a, b) on Bt∗ (C2n ) is the natural extension to holomorphic functions.) We will now show that g(z, w) = g(iy, iv) a.e. In view of Theorem 4.7 and the hypothesis on ga,b it is enough to show that Tg φα,β , φμ,ν Bt∗ (C2n ) = Tga,b φα,β , φμ,ν Bt∗ (C2n ) for all (a, b) and α, β, μ, ν. But this can be easily shown to be true. Indeed, by making the change of variable (z, w) −→ (z + a, w + b) in the equation Tg τ (a, b)φα,β , τ (a, b)φμ,ν Bt∗ (C2n ) = g(z, w)τ (a, b)φα,β (z, w)τ (a, b)φμ,ν (z, w)Wt (z, w)dzdw C2n

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it is easy to see that Tg τ (a, b)φα,β , τ (a, b)φμ,ν Bt∗ (C2n ) = Tga,b φα,β , φμ,ν Bt∗ (C2n ) . Therefore, it is enough to show that Tg φα,β , φμ,ν Bt∗ (C2n ) = Tg τ (a, b)φα,β , τ (a, b)φμ,ν Bt∗ (C2n )

(4.2.12)

2n

for all (a, b) ∈ R and multi-indices (α, β) and (μ, ν). In other words, we need to show that Tg commutes with twisted translations, which is immediate as etL Tg e−tL commutes with twisted translations and e−tL is equivariant under them. This proves the ﬁrst part of the theorem. Conversely, assume that g(z, w) = g(iy, iv) and Tg is bounded. Clearly, Tg φα,β , φμ,ν Bt∗ (C2n ) = Tga,b φα,β , φμ,ν Bt∗ (C2n ) for all (a, b) ∈ R2n . As shown earlier, this implies that Tg φα,β , φμ,ν Bt∗ (C2n ) = Tg τ (a, b)φα,β , τ (a, b)φμ,ν Bt∗ (C2n ) . So, Tg commutes with twisted translations. Again, as shown before etL Tg e−tL commutes with twisted translations as well. This means that there exists σ ∈ S (R2n ) such that etL Tg e−tL f = f × σ for all f ∈ L2 (R2n ). When we take the Weyl transform on both sides we get W (etL Tg e−tL f ) = W (f )W (σ). As Tg is bounded, this proves that etL Tg e−tL is a right Weyl multiplier.

5. Toeplitz Operators Associated to Symmetric Spaces Segal–Bargmann spaces associated to Riemannian symmetric spaces have been studied by Hall [10], Stenzel [19] and Kr¨ otz et al. [16]. The situation of non-compact symmetric spaces is much more complicated whereas the compact case is well understood as a weighted Bergman spaces. In both cases we have Gutzmer’s formula using which we can study Toeplitz operators that correspond to Fourier multilpiers on the underlying group. In this section we study such operators in the case of compact symmetric spaces, extending some results of Hall [12]. The case of noncompact Riemannian symmetric spaces will be taken up elsewhere. 5.1. Lassalle–Gutzmer Formula Consider a compact symmetric space X = U/K where (U, K) is a compact symmetric pair. We may assume that K is connected and U is semisimple. We let u = k + p stand for the Cartan decomposition of u and let a be a Cartan subspace of p. Functions f on X can be viewed as right K−invariˆ then it can be shown that fˆ(π) = 0 unless π ant functions on U. If π ∈ U is K−spherical, i.e., the representation space V of π has a unique K−ﬁxed vector u. It then follows that fˆ(π)v = (v, u)fˆ(π)u for any v ∈ V which ˆK stand for the equivalence classes of means that fˆ(π) is of rank one. Let U K−spherical representations of U. Then there is a one to one correspondence

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ˆK and a certain discrete subset P of a∗ called the set between elements of U of restricted dominant weights. For each λ ∈ P let (πλ , Vλ ) be a spherical representation of U of dimension dλ . Let {vjλ , 1 ≤ j ≤ dλ } be an orthonormal basis for Vλ with v1λ being the unique K-ﬁxed vector. Then the functions ϕλj (g) = πλ (g)v1λ , vjλ form an orthogonal family of right K−invariant analytic functions on U and we can consider them as functions of the symmetric space. When x = g.o ∈ X, we simply denote by ϕλj (x) the function ϕλj (g.o). The function ϕλ1 (g) is K biinvariant, called an elementary spherical function. It is usually denoted by ϕλ . The Fourier coefﬁcients of f ∈ L2 (X), are deﬁned by ˆ fj (λ) = f (x)ϕλj (x)dm0 (x) X

and the Fourier series is written as f (x) =

dλ

dλ

fˆj (λ)ϕλj (x) .

j=1

λ∈P

Then the Plancherel theorem reads as ⎛ ⎞ dλ |f (x)|2 dm0 (x) = dλ ⎝ |fˆj (λ)|2 ⎠ . j=1

λ∈P

X

Let UC (resp. KC ) be the universal complexiﬁcation of U (resp. K). The group KC sits inside UC as a closed subgroup. We may then consider the complex homogeneous space XC = UC /KC , which is a complex variety and gives the complexiﬁcation of the symmetric space X = U/K. The Lie algebra uC of UC is the complexiﬁed Lie algebra uC = u + iu. For every g ∈ UC there exists u ∈ U and X ∈ u such that g = u exp iX. Let Ω be any U invariant domain in XC and let O(Ω) stand for the space of holomorphic functions on Ω. The group U acts on O(Ω) by T (g)f (z) = f (g −1 z). For each λ ∈ P the matrix coefﬁcients ϕλj extend to XC as holomorphic functions. When f ∈ O(Ω), it can be shown that the series f (z) =

dλ

λ∈P

dλ

fˆj (λ)ϕλj (z)

j=1

converges uniformly over compact subsets of Ω. The above series is called the Laurent expansion of f and we have the following formula known as Gutzmer’s formula for X. Theorem 5.1. For every f ∈ O(XC ) and H ∈ ia, we have ⎛ ⎞ dλ |f (g. exp(H).o)|2 dg = dλ ⎝ |fˆj (λ)|2 ⎠ ϕλ (exp(2H).o). U

λ∈P

j=1

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This theorem is due to Lassalle, see [17,18] for a proof. This formula has been used by Faraut [5,6] to give an elegant proof of a theorem of Stenzel [19] on the Segal–Bargmann transform for the compact symmetric space X. The second author has used the same to study holomorphic Sobolev spaces in [27]. 5.2. Segal–Bargmann Transform on X Let Δ stand for the Laplace-Beltrami operator on X suitably shifted so that its spectrum consists of |λ + ρ|2 where λ ∈ P and ρ is the half sum of positive roots (see Faraut [5]). The solution of the heat equation associated to Δ with initial condition f ∈ L2 (X) is given by the expansion ⎛ ⎞ dλ 2 u(x, t) = dλ e−t|λ+ρ| ⎝ fˆj (λ)ϕλj (x)⎠ . j=1

λ∈P

By deﬁning the heat kernel γt by γt (x) =

λ∈P

dλ e−t|λ+ρ|

2

⎛ ⎞ dλ ⎝ ϕλj (x)⎠ j=1

the solution can be written as u(g, t)=f ∗γt (g) where the convolution is taken on U. For f ∈ L2 (X) it can be shown that the solution u extends to XC as a holomorphic function. The map taking f into u(z, t)=f ∗γt (g), z=g.o is called the Segal–Bargmann transform and has been studied by Hall [10], Stenzel [19] and others. The image of L2 (X) under the Segal–Bargmann transform has been characterised by Stenzel [19] as a weighted Bergman space. The weight function wt is given in terms of the heat kernel on the noncompact dual Y of X. Consider the group G=K exp(ip) whose Lie algebra is k + ip. Under the assumption that U is semisimple, G turns out to be a real semisimple group and K a maximal compact subgroup. The noncompact dual is then deﬁned as Y =G/K. Let ΔG be the Laplace-Beltrami operator on Y with heat kernel deﬁned by 2 2 γt1 (g) = e−t(|λ| +|ρ| ) ψλ (g)|c(λ)|−2 dλ. (ia)∗

Here ψλ are the spherical functions on Y. Deﬁne a weight function wt (z) on 1 (exp(2H)), z=u exp(H), u ∈ U, H ∈ ia. Then we have the XC by wt (z)=γ2t following result. Theorem 5.2. The Segal–Bargmann transform is an isometric isomorphism between L2 (X) and the space of all holomorphic functions on XC that are square integrable with respect to wt (z)dm where dm is the invariant measure on XC .

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This theorem is due to Stenzel [19]; for an elegant proof using Gutzmer’s formula see Faraut [5]. The key ingredient in Faraut’s proof is Lassalle’s formula and the following relation between ϕλ and ψλ , namely ϕλ (exp(H)) = ψ−i(λ+ρ) (exp(H)),

H ∈ ia.

To conclude this subsection let us recall the following integration formulas on XC and Y : f (x)dm(x) = c f (u exp(H).o)J(H)dudH, U ia

XC

f (x)dm1 (x) = c

f (u exp(H).o)J1 (H)dudH. K ia

Y

Here the Jacobians J and J1 are deﬁned in terms of the roots, see Faraut [5]. We need the following fact that J(H) = J1 (2H). 5.3. Toeplitz Operators and Fourier Multipliers Given a symbol g(z) deﬁned on XC we consider the Toeplitz operator Tg on the Segal–Bargmann space HL2 (XC , wt ). In this subsection we are interested in ﬁnding symbols g so that etΔ Tg e−tΔ is a Fourier multiplier on L2 (X, dm0 ). Given a bounded function a(λ) the Fourier multiplier a(D) is deﬁned by ⎛ ⎞ dλ dλ a(λ) ⎝ fˆj (λ)ϕλj (x)⎠ a(D)f (x) = j=1

λ∈P

for all f ∈ L2 (X, dm0 ). It is clear that a(D) is bounded if and only if a is bounded. Using Gutzmer’s formula we can easily prove the following result. 1 Theorem 5.3. Suppose h is a K−biinvariant distribution on g so that h∗γ2t is 1 well defined. Let g(z)wt (z) = h∗γ2t (exp(2H)) whenever z = u exp(H), u ∈ U, H ∈ ia. Then etΔ Tg e−tΔ = a(D) where a(λ) = h(exp(H))ψ−i(λ+ρ) (exp(H))J1 (H)dH. ia

Proof. When F, F ∈ HL2 (XC , wt ) we can use the polarised form of Gutzmer’s formula to get F (u exp(H).o)F (u exp(H).o)du U

=

λ∈P

⎛ dλ e−2(|λ+ρ|

2

)t

⎝

dλ j=1

⎞ fˆj (λ)fˆ j (λ)⎠ ϕλ (exp(2H))

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where F = f ∗ γt and F = f ∗ γt . Integrating the above with respect to 1 (exp(2H))J(H)dH and recalling the definition of g(z) we obtain h ∗ γ2t 2 F (z)F (z)g(z)wt (z)dm(z) = dλ e−2(|λ+ρ| )t XC

⎛ ×⎝

dλ

⎞ fˆj (λ)fˆ j (λ)⎠

j=1

λ∈P

1 ϕλ (exp(2H))h ∗ γ2t (exp(2H))J(H)dH.

ia

As J(H) = J1 (2H) and ϕλ (exp(H)) = ψ−i(λ+ρ) (exp(H)) the integral on the right hand side reduces to 2 1 ˜ h ∗ γ2t (exp(H))ψ−i(λ+ρ) (exp(H))J1 (H)dH = e2(|λ+ρ| )t h(−i(λ + ρ)). ia

Thus we have Tg F (z)F (z)wt (z)dm(z) = a(D)f (x)f (x)dm0 (x) XC

which proves the theorem.

X

Remark 5.1. When we take h to be the distribution p(Δ)δe where p is a polynomial it follows that a(λ)=p(−|λ + ρ|2 ) so that a(D)=p(iΔ). Hence the differential operator p(iΔ) corresponds to the Toeplitz operator with Tg with symbol g(z)=γ2t (exp(2H))−1 p(Δ)γ2t (exp(2H)), z=u exp(H).o. In the context of compact Lie groups U , Hall [12] has considered more general differential operators on U and studied the symbols of Toeplitz operators corresponding to them using a different method. 5.4. Some Remarks on Compact Lie Groups Let us rewrite our theorem in the previous section as follows. Given a K−biinvariant function g0 on Y = G/K deﬁne g(z) = g0 (exp(2H)), z = u exp(H).o, H ∈ ia. Then we have Corollary 5.4. Let g be as above. Then the Toeplitz operator Tg is bounded on HL2 (XC , wt dm) if and only if 2 1 g0 (exp(H))γ2t (exp(H))ψ−i(λ+ρ) (exp(H))J1 (H)dH ≤ Ce2t|λ+ρ| ia

for all λ ∈ P. Let U be a compact semisimple Lie group which can be treated as a compact symmetric space. In this case the group G turns out to be a complex Lie group and hence the heat kernel γt1 is explicitly known, see Gangolli [8]. We also have explicit expressions for the spherical functions ϕλ (Weyl character formula) and ψλ . More precisely, isλ(H) s∈W c(sλ)e ψλ (exp(H)) = Πα∈Q (eα(H) − e−α(H) )

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where W is the Weyl group, c is the Harish-Chandra c−function and Q is the set of positive roots. The heat kernel is given by 2 1 α(H) e− 4t |H| . −α(H) −e )

2

γt1 (exp(H)) = Ct e−t|ρ| Πα∈Q

(eα(H)

These two results are proved in Gangolli [8]; see also Helgason [13]. Deﬁning π(λ) = Πα∈Q α(Hλ ) where Hλ ∈ ia corresponds to λ we have the simple formula c(λ) = π(ρ)/π(iλ). The Jacobian factor J1 (H) appearing in the integration formula for Y = G/K is also expressible in terms of the roots α ∈ Q. Thus it can be checked that 1 g0 (exp(H))γ2t (exp(H))ψ−i(λ+ρ) (exp(H))J1 (H)dH ia −t|ρ|2

= Ct e

c(−is(λ + ρ))

s∈W

1

2

g0 (exp(H))es(λ+ρ)(H) π(H)e− 8t |H| dH.

ia

Note that when g0 = 1 the integral 1 γ2t (exp(H))ψλ (exp(H))J1 (H)dH ia

reduces to e−2t|ρ|

2

s∈W

= e−2t|ρ|

2

c(sλ)

ia

2

1

π(H)eisλ(H) e− 8t |H| dH 2

c(sλ)π(isλ) e−2t|λ| = Ct e−2t(|λ|

2

+|ρ|2 )

s∈W

which is the deﬁning relation for the heat kernel. Theorem 5.5. Let Tg be a Toeplitz operator on the Segal–Bargmann space associated to a compact Lie group U where g(u exp(H).o) = g0 (exp(H)), u ∈ U, H ∈ ia. Then Tg is bounded if and only if 2 1 g0 (exp(H))e(λ+ρ)(H) π(H)e− 8t |H| dH ≤ Ct |π(i(λ + ρ))|e2t|λ+ρ|2 . ia

Deﬁning g1 (H) = g0 (exp(H))π(H) the above condition can be put in the form g1 (H)e− 8t1 |H−4t(λ+ρ)|2 dH ≤ Ct |π(i(λ + ρ))| ia

for all λ ∈ P. This has an obvious resemblance with the sufﬁcient condition we obtained for the Fock spaces.

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Acknowlegments The second author is supported by J. C. Bose Fellowship from the Department of Science and Technology (DST) and also by a grant from UGC via DSA-SAP.

References [1] Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. Comm. Pure Appl. Math. 14, 187–214 (1961) [2] Berger, C.A., Coburn, L.A.: Heat ﬂow and Berezin–Toeplitz estimates. Am. J. Math. 3, 563–590 (1994) [3] Byun, D.-W.: Inversions of Hermite semigroup. Proc. Am. Math. Soc. 118, 437–445 (1993) [4] Daubechies, I.: On the distributions corresponding to bounded operators in the Weyl quantisation. Commun. Math. Phys. 75, 229–238 (1980) [5] Faraut, J.: Espaces Hilbertiens invariant de fonctions holomorphes, Seminaires et Congres 7(2003), Societe Math. France, 101–167 [6] Faraut, J.: Analysis on the crown of a Riemannian symmetric space, pp. 99–110 in Lie groups and symmetric spaces. In: Amer. Math. Soc. Transl. Ser.2, vol. 210 , American Mathematical Society, Providence (2003) [7] Folland, G.B.: Harmonic Analysis on Phase space. Princeton University Press, Princeton (1989) [8] Gangolli, R.: Asymptotic behaviour of spectra of compact quotients of certain symmetric spaces. Acta Math. 121, 151–192 (1968) [9] Grudsky, S.M., Vasilevski, N.L.: Toeplitz operators on the Fock space: radial component eﬀects. Integr. Equ. Oper. Theory 44, 10–37 (2002) [10] Hall, B.: The Segal–Bargmann “coherent state” transform for compact Lie groups. J. Funct. Anal. 122(1), 103–151 (1994) [11] Hall, B., Lewkeeratiyutkul, W.: Holomorphic Sobolev spaces and the generalised Segal–Bargmann transform. J. Funct. Anal. 217, 192–220 (2004) [12] Hall, B.: Berezin-Toeplitz quantization on Lie groups. J. Funct. Anal. 255, 2488–2506 (2008) [13] Helgason, S.: Groups and Geometric Analysis, vol. 83. American Mathematical Society, Providence (2002) [14] Hille, E.: A class of reciprocal functions. Ann. Math. 27, 427–464 (1926) [15] Kr¨ otz, B., Thangavelu, S., Xu, Y.: The heat kernel transform for the Heisenberg group. J. Funct. Anal. 225(2), 301–336 (2005) [16] Kr¨ otz, B., Olafsson, G., Stanton, R.: The image of the heat kernel transform on Riemannian symmetric spaces of noncompact type. Int. Math. Res. Notes, no. 22, 1307–1329 (2005) [17] Lassalle, M.: S´eries de Laurent des fonctions holomorphes dans la complexiﬁcation d’un espace sym´etrique compact. Ann. Sci. ’Ecole Norm. Sup. (4) 11(2), 167–210 (1978) [18] Lassalle, M.: L’espace de Hardy d’un domaine de Reinhardt g´en´eralis´e. J. Funct. Anal. 60(3), 309–340 (1985)

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[19] Stenzel, M.: The Segal–Bargmann transform on a symmetric space of compact type. J. Funct. Anal. 165, 44–58 (1999) [20] Szeg¨ o, G.: Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Providence (1967) [21] Thangavelu, S.: Harmonic analysis on the Heisenberg group. In: Programme in Mathematics, vol. 159, Birkh¨ auser, Boston (1998) [22] Thangavelu, S.: Gutzmer’s formula and Poisson integrals on the Heisenberg group. Paciﬁc J. Math. 231, 217–238 (2007) [23] Thangavelu, S.: Hermite and Laguerre semigroups: some recent developments. Seminaires et Congres (to appear) [24] Thangavelu, S.: An analogue of Gutzmer’s formula for Hermite expansions. Studia Math. 185(3), 279–290 (2008) [25] Thangavelu, S.: Lectures on Hermite and Laguerre expansions. Princeton University Press, Princeton (1993) [26] Thangavelu, S.: An introduction to the uncertainty principle: Hardy’s theorem on Lie groups. In: Progress in Mathematics, vol. 217, Birkh¨ auser, Boston (1998) [27] Thangavelu, S.: Holomorphic Sobolev spaces associated to compact symmetric spaces. J. Funct. Anal. 251, 438–462 (2007) K. Jotsaroop and S. Thangavelu (B) Department of Mathematics Indian Institute of Science Bangalore 560 012, India e-mail: [email protected]; [email protected] Received: April 30, 2010 Revised: October 30, 2010.

Integr. Equ. Oper. Theory 69 (2011), 347–363 DOI 10.1007/s00020-010-1839-y Published online November 5, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Left–Right Browder and Left–Right Fredholm Operators ˇ Zivkovi´ ˇ Sneˇzana C. c-Zlatanovi´c, Dragan S. Djordjevi´c and Robin E. Harte Abstract. We consider left and right Browder operators, left and right Fredholm operators, spectra related with these operators, and various operator quantities. Mathematics Subject Classification (2010). 47A53. Keywords. Left (right) Fredholm, left (right) Browder.

1. Introduction Let X denote an inﬁnite dimensional Banach space. We use B(X) to denote the set of all linear bounded operators on X. Also, K(X) and F (X), respectively, denote the set of all compact and ﬁnite rank operators on X. For A ∈ B(X) we use N (A) and R(A), respectively, to denote the null-space and the range of A. We use Gl (X) and Gr (X), respectively, to denote the set of all left and right invertible operators on X. It is well-known that A ∈ Gl (X) if and only if A is injective and R(A) is a closed and complemented subspace of X. Also, A ∈ Gr (X) if and only if A is onto and N (A) is a complemented subspace of X. The set of all invertible operators on X is denoted by G(X). Let α(A) = dim N (A) if N (A) is ﬁnite dimensional, and let α(A) = ∞ if N (A) is inﬁnite dimensional. Similarly, let β(A) = dim X/R(A) = codim R(A) if X/R(A) is ﬁnite dimensional, and let β(A) = ∞ if X/R(A) is inﬁnite dimensional. Sets of upper and lower Fredholm operators, respectively, are deﬁned as Φ+ (X) = {A ∈ B(X) : α(A) < ∞ and R(A) is closed}, and Φ− (X) = {A ∈ B(X) : β(A) < ∞}. This work is supported by the Ministry of Science and Technological Development, Republic of Serbia, grant no. 144003.

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Operators in Φ± (X) = Φ+ (X) ∪ Φ− (X) are called semi-Fredholm operators. For such operators the index is deﬁned by i(A) = α(A) − β(A). Let + Φ− + (X) = {A ∈ Φ+ (X) : i(A) ≤ 0} and Φ− (X) = {A ∈ Φ− (X) : i(A) ≥ 0}. The set of Fredholm operators is deﬁned as Φ(X) = Φ+ (X) ∩ Φ− (X). The set of Weyl operators is deﬁned as Φ0 (X) = {A ∈ Φ(X) : i(A) = 0}. Let S be a subset of a Banach space A. The perturbation class of S, denoted by P (S), is the set P (S) = {a ∈ R : a + s ∈ S for every s ∈ S}. The Calkin algebra over X is the quotient algebra C(X) = B(X)/K(X), and π : B(X) → C(X) denotes the natural homomorphism. Let re (A) denote spectral radius of the element π(A) in C(X), A ∈ B(X), i.e. re (A) = 1 limn→∞ ( π(An ) ) n and it is called essential spectral radius of A. An operator A ∈ B(X) is Riesz if {λ ∈ C : A − λ ∈ Φ(X)} = C\{0}, i.e. re (A) = 0. For A ∈ B(X) set A P Φ = inf{ A − P : P ∈ P (Φ(X))}. 1

It is known that re (A) = limn→∞ ( An P Φ ) n . An operator A ∈ B(X) is relatively regular (or g-invertible) if there exists B ∈ B(X) such that ABA = A. It is well-known that A is relatively regular if and only if R(A) and N (A) are closed and complemented subspaces of X. Sets of left and right Fredholm operators, respectively, are deﬁned as Φl (X) = {A ∈ B(X) : R(A) is a closed and complemented subspace of X and α(A) < ∞}, and Φr (X) = {A ∈ B(X) : N (A) is a complemented subspace of X and β(A) < ∞}. It is known that the sets Φl (X) and Φr (X) are open [1] (Chap. 5.2, Theorem 6), and P (Φl (X)) = P (Φ(X)) = P (Φr (X)) [1] (Chap. 5.2, Corollary 3). An operator A ∈ B(X) is left (right) Weyl if A is left (right) Fredholm operator and i(A) ≤ 0 (i(A) ≥ 0). We use Wl (X) (Wr (X)) to denote the set of all left (right) Weyl operators. The ascent of A ∈ B(X), denoted by asc(A), is the smallest n ∈ N such that N (An ) = N (An+1 ). If such n does not exist, then asc(A) = ∞. The descent of A, denoted by dsc(A), is the smallest n ∈ N such that R(An ) = R(An+1 ). If such n does not exist, then dsc(A) = ∞. An operator A ∈ B(X) is upper semi-Browder if it is upper semiFredholm of ﬁnite ascent, and A is lower semi-Browder if it is lower semiFredholm of ﬁnite descent. Let B+ (X) (B− (X)) denote the set of all upper (lower) semi-Browder operators. The set of Browder operators is deﬁned as B(X) = B+ (X) ∩ B− (X).

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The operator A ∈ B(X) is left Browder if it is left Fredholm of ﬁnite ascent, and A is right Browder if it is right Fredholm of ﬁnite descent. Let Bl (X) (Br (X)) denote the set of all left (right) Browder operators. From [4] (Theorem 7.9.2) and [1] (Chap. 5.2, Theorem 7), for A ∈ B(X) and K ∈ K(X) which commutes with A, it follows that A is left Browder ⇐⇒ A + K is left Browder, A is right Browder ⇐⇒ A + K is right Browder.

(1.1) (1.2)

The following assertions [13] (Theorems 7 and 8) tell us that it holds more generally. If A ∈ B(X), and if E Riesz which commutes with A, then A is left Browder ⇐⇒ A + E is left Browder , A is right Browder ⇐⇒ A + E is right Browder. Moreover, the following hold. Theorem 1.1. If A ∈ B(X) and E ∈ B(X) is Riesz, then lef t lef t AE − EA ∈ P (Φ(X)) =⇒ σw (A) = σw (A + E),

AE − EA ∈ P (Φ(X)) =⇒ AE = EA =⇒ AE = EA =⇒

right right (A) = σw (A + E), σw lef t lef t σb (A) = σb (A + E), σbright (A) = σbright (A + E).

(1.3) (1.4) (1.5) (1.6)

The following theorem gives a characterization of left and right Browder operators [13] (Theorems 5 and 6). Theorem 1.2. Let A ∈ B(X). Then A is left (right) Browder iﬀ there exist closed subspaces X1 and X2 invariant with respect to A such that X = X1 ⊕ X2 , dim X1 < ∞, the reduction A1 = A|X1 : X1 → X1 is nilpotent and the reduction A2 = A|X2 : X2 → X2 is left (right) invertible. Corresponding spectra of A ∈ B(X) are deﬁned as: σl (A) = {λ ∈ C : A − λ ∈ / Gl (X)}-the left spectrum, / Gr (X)}-the right spectrum, σr (A) = {λ ∈ C : A − λ ∈ σa (A) = {λ ∈ C : A − λ is not bounded below}-the approximate point spectrum, σd (A) = {λ ∈ C : A − λ is not onto}-the defect spectrum, / B(X)}-the Browder spectrum, σb (A) = {λ ∈ C : A − λ ∈ σb+ (A) = {λ ∈ C : A − λ ∈ / B+ (X)}-the Browder essential approximate point spectrum, / B− (X)}-the Browder essential defect specσb+ (A) = {λ ∈ C : A − λ ∈ trum, σblef t (A) = {λ ∈ C : A − λ ∈ / Bl (X)}-the left Browder spectrum, right σb (A) = {λ ∈ C : A − λ ∈ / Br (X)}-the right Browder, / Φ0 (X)}-the Weyl spectrum, σw (A) = {λ ∈ C : A − λ ∈ lef t (A) = {λ ∈ C : A − λ ∈ / Wl (X)}-the left Weyl spectrum, σw right (A) = {λ ∈ C : A − λ ∈ / Wr (X)}-the right Weyl spectrum, σw + σw (A) = {λ ∈ C : A − λ ∈ / Φ− + (X)}-the essential approximate point spectrum,

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− (A) = {λ ∈ C : A − λ ∈ / Φ+ σw − (X)}-the essential defect spectrum, / Φ(X)}-the Fredholm spectrum, σe (A) = {λ ∈ C : A − λ ∈ / Φl (X)}-the left Fredholm spectrum, σelef t (A) = {λ ∈ C : A − λ ∈ / Φr (X)}-the right Fredholm spectrum, σeright (A) = {λ ∈ C : A − λ ∈ / Φ+ (X)}-the upper semi-Fredholm spectrum, σe+ (A) = {λ ∈ C : A − λ ∈ / Φ− (X)}-the lower semi-Fredholm spectrum. σe− (A) = {λ ∈ C : A − λ ∈

2. Properties of Corresponding Spectra We prove the following auxiliary assertion. Lemma 2.1. Let A ∈ B(X) and let X be a direct sum of closed subspaces X1 and X2 which are A-invariant. If A1 = A|X1 : X1 → X1 and A2 = A|X2 : X2 → X2 , then the following statements hold: (2.1.1) The operator A is g-invertible if and only if A1 and A2 are g-invertible. (2.1.2) The operator A ∈ Φl (X) if and only if A1 ∈ Φl (X1 ) and A2 ∈ Φl (X2 ), and in that case i(A) = i(A1 ) + i(A2 ). (2.1.3) The operator A ∈ Φr (X) if and only if A1 ∈ Φr (X1 ) and A2 ∈ Φr (X2 ), and in that case i(A) = i(A1 ) + i(A2 ). (2.1.4) The operator A ∈ Bl (X) if and only if A1 ∈ Bl (X1 ) and A2 ∈ Bl (X2 ), and in that case i(A) = i(A1 ) + i(A2 ). (2.1.5) The operator A ∈ Br (X) if and only if A1 ∈ Br (X1 ) and A2 ∈ Br (X2 ), and in that case i(A) = i(A1 ) + i(A2 ). Proof. (2.1.1): The operator A has the following matrix form with respect to the decomposition X = X1 ⊕ X2 : A1 0 X1 X1 A= : → X2 X2 0 A2 Suppose that A is g-invertible. Then there exists B ∈ B(X) such that ABA = A, and B has the following matrix form: B11 B12 X1 X1 B= : → B21 B22 X2 X2 Therefore

and we get

A1 0

0 A2

B11 B21

A1 B11 A1 A2 B21 A1

B12 B22

A1 0

0 A1 = A2 0

A1 B12 A2 A1 = 0 A2 B22 A2

0 , A2

0 , A2

which implies A1 B11 A1 = A1 and A2 B22 A2 = A2 . Thus, A1 and A2 are g-invertible operators. Conversely, suppose that A1 ∈ B(X1 ) and A2 ∈ B(X2 ) are g-invertible operators. Then there exist B1 ∈ B(X1 ) and B2 ∈ B(X2 ) such that A1 B1 A1 =

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Left–Right Browder and Left–Right Fredholm

A1 and A2 B2 A2 = A2 . Let

B1 B= 0

351

0 . B2

Then we have B ∈ B(X) and ABA = A, so A is a g-invertible operator. (2.1.2): Since N (A) = N (A1 ) ⊕ N (A2 ) and R(A) = R(A1 ) ⊕ R(A2 ), it follows that α(A) = α(A1 ) + α(A2 ) and β(A) = β(A1 ) + β(A2 ). Therefore, α(A) < ∞ if and only if α(A1 ) < ∞ and α(A2 ) < ∞. Hence, according to (2.1.1), A ∈ Φl (X) if and only if A1 ∈ Φl (X1 ) and A2 ∈ Φl (X2 ), and in that case i(A) = α(A)−β(A) = (α(A1 )+α(A2 ))−(β(A1 )+β(A2 )) = i(A1 )+i(A2 ). (2.1.3): Similarly to (2.1.2). (2.1.4): Since N (An ) = N (An1 ) ⊕ N (An2 ) for n ∈ N, we conclude that asc(A) < ∞ if and only if asc(A1 ) < ∞ and asc(A2 ) < ∞. Now the statements follows from (2.1.2). (2.1.5): From R(An ) = R(An1 ) ⊕ R(An2 ) for n ∈ N, we see that dsc(A) < ∞ if and only if dsc(A1 ) < ∞ and dsc(A2 ) < ∞. Then the conclusion follows from (2.1.3). Let P(X) denote the set of all projections P ∈ B(X) such that codim R(P ) < ∞. For A ∈ B(X) and P ∈ P(X), the compression AP : R(P ) → R(P ) is deﬁned by AP y = P Ay, y ∈ R(P ), i.e. AP = P A|R(P ) , where A|R(P ) : R(P ) → X is the restriction of A. Clearly, R(P ) is a Banach space and AP ∈ B(R(P )). Zem´anek [12] gave the proof of the fact that if P ∈ P(X), then A is semi-Fredholm if and only if AP is semi-Fredholm and i(A) = i(AP ). We prove the following result in that case. Theorem 2.2. Let A ∈ B(X), P ∈ P(X). Then (2.2.1) A ∈ Φl (X) if and only if AP ∈ Φl (R(P )), and in that case i(AP ) = i(A). (2.2.2) A ∈ Φr (X) if and only if AP ∈ Φr (R(P )), and in that case i(AP ) = i(A). (2.2.3) If AP = P A, then A ∈ Bl (X) if and only if AP ∈ Bl (R(P )), and in that case i(AP ) = i(A). (2.2.4) If AP = P A, then A ∈ Br (X) if and only if AP ∈ Br (R(P )), and in that case i(AP ) = i(A). Proof. (2.2.3), (2.2.4): Suppose that P ∈ P(X), A ∈ B(X) and AP = P A. Then X = R(P ) ⊕ N (P ) and subspaces R(P ) and N (P ) are invariant for P AP ∈ B(X). The operator P AP has the following matrix form: R(P ) R(P ) AP 0 : → . P AP = 0 0 N (P ) N (P ) Since dim N (P ) < ∞, from (2.1.4) and (2.1.5) it follows that P AP is left (right) Browder if and only if AP is left (right) Browder and i(P AP ) = i(AP ) + i(0) = i(AP ). Since A = P A + (I − P )A = P AP + P A(I − P ) + (I − P )A,

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and since P A(I −P )+(I −P )A is a ﬁnite rank operator, which commutes with P AP , by (1.1) and (1.2) it follows that P AP is a left (right) Browder operator if and only if A is left (right) Browder, and in that case i(P AP ) = i(A). (2.2.1) and (2.2.2) can be proved similarly, using (2.1.2) and (2.1.3). It is known [12] that + (A) = σw

σa (AP ),

− σw (A) =

P ∈P(X)

σb+ (A) =

σd (AP ),

P ∈P(X)

σa (AP ),

σb− (A) =

P ∈P(X), AP =P A

σd (AP ).

P ∈P(X), AP =P A

We prove analogous assertion for the left and right Browder and Weyl spectra. Theorem 2.3. Let A ∈ B(X). Then σblef t (A) =

σl (AP ),

(2.1)

σr (AP ).

(2.2)

P ∈P(X), AP =P A

σbright (A) =

P ∈P(X), AP =P A

Proof. To prove the inclusion “⊂” in (2.1) (or (2.2)), suppose that λ ∈ / σl (AP ) (λ ∈ / σr (AP )) for some P ∈ P(X) such that AP = P A. Then AP − λIP = (A − λ)P is left (right) invertible, and so (A − λ)P is left (right) Browder. By Theorem 2.2 it follows that A − λ is left (right) Browder, i.e. λ ∈ / σblef t (A) (λ ∈ / σbright (A)). / To prove the converse inclusion, suppose that λ ∈ / σblef t (A) (λ ∈ right σb (A)). Then A − λ ∈ Bl (X) (A − λ ∈ Br (X)). By Theorem 1.2, X is a direct sum of closed subspaces X1 and X2 , which are A − λ-invariant. Consequently, they are A-invariant, and they have the following properties: dim X1 < ∞ and A1 − λ is nilpotent on X1 , where A1 = A|X1 : X1 → X1 and if A2 = A|X2 : X2 → X2 , then A2 − λ is left (right) invertible. Let P be the projection of X onto X2 along X1 . Clearly, P ∈ P(X). Since the subspaces X1 and X2 are invariant for A, we see that AP = P A and (A − λ)P = A2 − λ. / σl (AP ) (λ ∈ / σr (AP )). Thus AP − λIP is left (right) invertible and so λ ∈ Combining (2.2.1) and (2.2.2) with the proof of Theorem 2 in [12] we get the following theorem. Theorem 2.4. Let A ∈ B(X). Then lef t (A) = σw

σl (AP ),

(2.3)

σr (AP ).

(2.4)

P ∈P(X) right (A) = σw

P ∈P(X)

Proof. To prove the inclusion “⊂” in (2.3) (or (2.4)), suppose that λ ∈ / σl (AP ) (λ ∈ / σr (AP )) for some P ∈ P(X), then AP − λIP = (A − λ)P is left (right) invertible, and so (A − λ)P is left (right) Weyl. By (2.2.1) (or (2.2.2)) it lef t right (A) (λ ∈ / σw (A)). follows that A − λ is left (right) Weyl, i.e. λ ∈ / σw

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lef t (A). Then A − λ ∈ To prove the converse in (2.3), suppose that λ ∈ / σw Φl (X) and i(A−λ) ≤ 0. Since α(A−λ) ≤ β(A−λ), there exists a subspace V such that dim V = dim N (A−λ) < ∞ and V ∩R(A−λ) = {0}. There exists a joint closed complement W of V and N (A − λ), that is X = V ⊕ W = N (A − λ) ⊕ W . Let P be the projection such that R(P ) = W and N (P ) = V . Then P ∈ P(X) and we show that (A−λ)P is left invertible. From A−λ ∈ Φl (X) it follows that (A−λ)P is a left Fredholm operator on R(P ), by (2.2.1). To prove that (A − λ)P is injective, suppose that w ∈ W and (A − λ)P w = 0. Then P (A−λ)w = 0 and hence (A−λ)w ∈ N (P )∩R(A−λ) = V ∩R(A−λ) = {0}, which implies w ∈ W ∩ N (A − λ) = {0}. Therefore, (A − λ)P is injective. / σl (AP ). This proves that (A − λ)P is left invertible, and hence λ ∈ right To prove the converse in (2.4), suppose that λ ∈ / σw (A). Then A−λ ∈ Φr (X) and i(A − λ) ≥ 0. Hence α(A − λ) ≥ β(A − λ) and β(A − λ) < ∞. Let M be a subspace of N (A−λ) such that dim M = codim R(A−λ) < ∞. There exists a closed subspace V of X such that X = M ⊕ V . Since codim V = codim R(A − λ) < ∞, there exists a joint complement W of V and R(A − λ), that is

X = W ⊕ V = W ⊕ R(A − λ).

(2.5)

Let P be the projection such that R(P ) = V and N (P ) = W , clearly, P ∈ P(X). Since X = M ⊕ V and M ⊂ N (A − λ), we see that (A − λ)V = (A − λ)X, so R((A − λ)P ) = P ((A − λ)V ) = P (R(A − λ)). From (2.5) we get P (R(A − λ)) = V . Therefore, R((A − λ)P ) = V , i.e. (A − λ)P is onto. By (2.2.2) it follows that (A − λ)P is right Fredholm, and so (A − λ)P is right invertible. Hence λ ∈ / σr (AP ). lef t The following example shows that in general σw (A) = σblef t (A) and right = σb (A). This example was used in [9] and [7].

right σw (A)

Example. Let H be a separable Hilbert space, let V be the right shift on H and let N ∈ B(H) be quasinilpotent. If A = V ⊕ V ∗ ⊕ N , then σblef t (A) = lef t right σbright (A) = D and σelef t (A) = σeright (A) = σw (A) = σw (A) = ∂D ∪ {0}, where D is the closed unit ball. Proof. Since σb (A) = D [9] and σb+ (A) = σb− (A) = D [7], from σb+ (A) ⊂ σblef t (A) ⊂ σb (A) and σb− (A) ⊂ σbright (A) ⊂ σb (A) we get σblef t (A) = σbright (A) = D. From σw (A) = ∂D ∪ {0} [9], ∂σw (A) ⊂ σe+ (A) ⊂ σe (A) ⊂ σw (A) and ∂σw (A) ⊂ σe− (A) ⊂ σw (A) we obtain σe+ (A) = σe− (A) = σe (A) = ∂D ∪ {0}. Since σe+ (A) ⊂ σelef t (A) ⊂ σe (A) and σe− (A) ⊂ σeright (A) ⊂ σe (A), it follows lef t (A) ⊂ σw (A) and that σelef t (A) = σeright (A) = ∂D ∪ {0}. As σelef t (A) ⊂ σw right right lef t right (A) ⊂ σw (A) ⊂ σw (A), we get σw (A) = σw (A) = ∂D ∪ {0}. σe Recall that for A, B ∈ B(X) the following hold: If A, B ∈ Φl (X) (Φr (X)), then BA ∈ Φl (X) (Φr (X)); If BA ∈ Φl (X), then A ∈ Φl (X); If BA ∈ Φr (X), then B ∈ Φr (X). Also recall that for A, B ∈ B(X) it holds [4] (Theorem 7.9.2):

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If AB = BA, then A, B ∈ B+ (X) (B− (X)) if and only if AB ∈ B+ (X) (B− (X)). Now it is easy to see that the next statements hold. Lemma 2.5. Let A, B ∈ B(X) and AB = BA. Then (2.5.1) A, B ∈ Bl (X) (2.5.2) A, B ∈ Br (X)

⇐⇒ ⇐⇒

AB ∈ Bl (X), AB ∈ Br (X).

Theorem 2.6. Let A ∈ B(X) and let f be an analytic function deﬁned in a neighborhood of σ(A). Then f (σblef t (A)) = σblef t (f (A)), f (σbright (A)) = σbright (f (A)). Proof. Follows from Lemma 2.5, [6] (Chap. I, Theorems 6.4 and 6.8), (2.1.4), (2.1.5) and the fact that the left (right) Browder spectrum of any operator is non-empty set. Let us remark that previous theorem can be proved also in the following way: for A ∈ B(X), σblef t (A) = σelef t (A) ∪ σb+ (A) and it is well-known that f (σelef t (A)) = σelef t (f (A)) [3] and f (σb+ (A)) = σb+ (f (A)) [7] (Theorem 3.4) for every analytic function f deﬁned in a neighborhood of σ(A). Thus, f (σblef t (A)) = f (σelef t (A) ∪ σb+ (A)) = f (σelef t (A)) ∪ f (σb+ (A)) = σelef t (f (A)) ∪ σb+ (f (A)) = σblef t (f (A)). Similarly for the right Browder spectrum. Let (Gn ) be a sequence of compact subsets of C. The limit superior, lim sup Gn , is the set of all λ in C such that every neighborhood of λ intersects inﬁnitely many Gn . It is known that B+ (X) and B− (X) are open subsets in B(X) [5] (Satz 4). Since the sets Φl (X) and Φr (X) are open, we conclude that Bl (X) and Br (X) are open subsets in B(X) and consequently, for A ∈ B(X) the mapping A → σblef t (A) is upper semi-continuous, i.e. if An ∈ B(X) and An → A, then lim sup σblef t (An ) ⊂ σblef t (A). Analogously, the mapping A → σbright (A) is upper semi-continuous. If X and Y are inﬁnite dimensional Banach spaces, A ∈ B(X), B ∈ B(Y ) and C ∈ B(Y, X), we denote A C MC = ∈ B(X ⊕ Y ). 0 B Theorem 2.7. For each j ∈ {e, w, b} and ∗ ∈ {+, −, lef t, right} there is inclusion σj∗ (MC ) ⊂ σj∗ (A) ∪ σj∗ (B). Particulary, if A and B are left (resp. right, upper, lower) Browder (Weyl), then MC is left (resp. right, upper, lower) Browder (Weyl).

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A 0 Proof. Let M = . By Lemma 2.1 it follows that σj∗ (M ) = σj∗ (A) ∪ 0 B σj∗ (B). Observe that 0 I 0 A C I A k1 C = = MCk → M as k → ∞. 0 kI 0 B 0 k1 I 0 B Since MCk and MC are similar, it follows that σj∗ (MCk ) = σj∗ (MC ). By openess of all the relevant semigroups the mappings σj∗ are each upper semicontinuous: thus indeed σj∗ (MC ) = lim sup σj∗ (MCk ) ⊂ σj∗ (A) ∪ σj∗ (B).

3. Geometric Characteristics For A ∈ B(X), the injectivity radius of A, denoted by sinj (A), is deﬁned as follows: sinj (A) = inf{|λ| : λ ∈ σa (A)} = max{ ≥ 0 : |λ| < =⇒ A − λ is bounded below}. The surjectivity radius of the operator A, denoted by ssur (A), is deﬁned as follows: ssur (A) = inf{|λ| : λ ∈ σd (A)} = max{ ≥ 0 : |λ| < =⇒ A − λ is onto}. The semi-Fredholm radius of A is s(A) = inf{|λ| : A − λ is not semi − Fredholm} = max{ ≥ 0 : |λ| < =⇒ A − λ is semi − Fredholm}. Zem´anek [11] proved the following results: If A ∈ B(X) is bounded below, then s(A) =

sup sinj (A + F ).

F ∈F (X)

If A ∈ B(X) is surjective, then s(A) =

sup ssur (A + F ).

F ∈F (X)

For A ∈ B(X) we deﬁne the Gl -radius sl (A) and Gr -radius sr (A): sl (A) = inf{|λ| : λ ∈ σl (A)} = max{ ≥ 0 : |λ| < =⇒ A − λ ∈ Gl (X)}, sr (A) = inf{|λ| : λ ∈ σr (A)} = max{ ≥ 0 : |λ| < =⇒ A − λ ∈ Gr (X)}. Analogously, we deﬁne left and right Fredholm, Weyl and Browder radius of A: s∗ω (A) = dist(0, σω∗ (A)),

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where ω = e, w, b, and ∗ = lef t, right, and also upper and lower semi-Browder radius of A: + s+ b (A) = dist(0, σb (A)), − s− b (A) = dist(0, σb (A)).

Using Zem´anek’s method of removing jumping points, we prove the following result. Theorem 3.1. (3.1.1) Let A ∈ B(X) be a left invertible operator. Then t (A) = slef b

(3.1.2)

sup AF =F A,F ∈F (X)

sl (A + F ) =

sl (A + E).

sup AE=EA,E∈R(X)

Let A ∈ B(X) be a right invertible operator. Then (A) = sright b

sup AF =F A,F ∈F (X)

sr (A + F ) =

sr (A + E).

sup AE=EA,E∈R(X)

t Proof. (3.1.1): Let A ∈ Gl (X), and let D = {λ ∈ C : |λ| < slef (A)}. Then b A − λ ∈ Φ+ (X) for every λ ∈ D. According [2] (Theorem 3.2.20), α(A − λ) is equal to 0 everywhere in the disk D, except possibly in the set which is at most countable, and all points of this set are isolated. These points are called jumping points. The set of all accumulation points of the set of all jumping points can only be a subset of the boundary of D. From (1.5) we obtain t (A) = dist(0, σblef t (A)) = dist(0, σblef t (A + E)) ≥ sl (A + E) slef b

for every E ∈ R(X) which commute with A. Hence, t slef (A) ≥ b

sup

sl (A + E) ≥

AE=EA,E∈R(X)

sup AF =F A,F ∈F (X)

sl (A + F ). (3.1)

If A does not have any jumping point in D, then t slef (A) = sl (A) ≤ b

sup AF =F A,F ∈F (X)

sl (A + F ).

(3.2)

From (3.1) and (3.2) we get (3.1.1). Suppose that A has the jumping points in D. Denote the jumping points such that t |λ1 | ≤ |λ2 | ≤ · · · |λn | ≤ · · · < slef (A). b

Therefore, sl (A) = |λ1 |. Since A − λ1 ∈ Bl (X), from [6] (Theorem 20.10) it follows that X is a direct sum of closed subspaces X1 and X2 in X, which are invariant for A − λ1 , i.e. they are invariant for A, dim X1 < ∞, A − λ1 is nilpotent on X1 , and for the reduction A2 = A|X2 : X2 → X2 we have A2 − λ1 is injective. t (A) and F = μP , where P is Let μ ∈ C such that |μ| > A + slef b the projection from X onto X1 along X2 . Let λ ∈ D. Then A − λ ≤ t A + slef (A) < |μ|, so A − λ + μ is invertible. Hence the reduction (A + μ − b λ)|X1 = (A + F − λ)|X1 : X1 → X1 is invertible on X1 and N (A + F − λ) = N ((A + F − λ)|X1 ) ⊕ N ((A + F − λ)|X2 ) = {0} ⊕ N ((A2 − λ)|X2 ) = {0}, for all λ ∈ D\{λ2 , . . . , λn , . . . }. For all λ ∈ D it holds A − λ ∈ Bl (X). Since

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F ∈ F (X) and AF = F A, by (1.1) it follows that A + F − λ ∈ Bl (X). Therefore, A + F − λ is left invertible for all λ ∈ D\{λ2 , . . . , λn , . . . }. Let > 0. Then there exist only ﬁnitely many jumping points λi such t that |λi | < slef (A) − . Therefore, applying the previous method ﬁnitely b many times, we obtain the operator F1 ∈ F (X) such that A + F1 − λ is left t invertible for |λ| < slef (A) − , i.e. b t (A) − . sl (A + F1 ) ≥ slef b

(3.3)

From (3.3) and (3.1) we get (3.1.1). The statement (3.1.2) can be proved similarly.

Theorem 3.2. Let A ∈ B(X). (3.2.1) If A is bounded below, then s+ b (A) =

sup AF =F A,F ∈F (X)

sinj (A + F ) =

sup

sinj (A + E).

AE=EA,E∈R(X)

(3.2.2) If A is surjective, then s− b (A) =

sup AF =F A,F ∈F (X)

ssur (A + F ) =

sup

ssur (A + E).

AE=EA,E∈R(X)

Proof. Analogously to Theorem 3.1, using [8] (Theorem 7).

Theorem 3.3. Let J(X) be any non zero ideal of Riesz operators. (3.3.1) If A ∈ B(X) is left invertible, then t lef t slef (A) = w (A) = se

=

sup sl (A + F ) =

F ∈F (X)

sup E∈R(X),AE−EA∈P (Φ(X))

sup sl (A + E) E∈J(X)

sl (A + E).

(3.3.2) If A ∈ B(X) is right invertible, then (A) = sright (A) = sright w e =

sup sr (A + F ) =

F ∈F (X)

sup E∈R(X),AE−EA∈P (Φ(X))

sup sr (A + E) E∈J(X)

sr (A + E).

Proof. The ﬁrst equality in (3.3.1) and (3.3.2) follows from the continuity of the index. The other equalities in (3.3.1) and (3.3.2) follow from (1.3), (1.4), [6] (Theorem 16.21 and Corollary 12.4), and [1] (Chap. 5.2, Theorem 7), analogously to the proof of Theorem 3.1. For A ∈ B(X), set t mlef (A) = dist(A, B(X)\Φl (X)), e

(A) = dist(A, B(X)\Φr (X)). mright e We extend some results from [14] to left and right Fredholm operators. Theorem 3.4. Let A, B ∈ B(X). Then: t (3.4.1) mlef (A) > 0 ⇐⇒ A ∈ Φl (X), e lef t t (A) for each A ∈ B(X) ⇐⇒ B ∈ P (Φ(X)), (3.4.2) me (A + B) = mlef e lef t lef t (3.4.3) me (A + B) ≤ me (A) + B ,

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t t (A + B) ≤ mlef (A) + B P Φ . mlef e e lef t If B P Φ < me (A), then A, A + B ∈ Φl (X) and i(A) = i(A + B). t (I), then I − A ∈ Φ(X) and i(I − A) = 0. If A P Φ < mlef e n t (I) for some n > 1, then I − A ∈ Φ(X) and If A P Φ < mlef e i(I − A) = 0. (3.4.8) If AB − BA ∈ P (Φ(X)) and

(3.4.4) (3.4.5) (3.4.6) (3.4.7)

1

t (An )) n , re (B) < lim (mlef e n→∞

then A, A + B ∈ Φl (X) and i(A + B) = i(A). 1 t t (A) ≥ limn→∞ (mlef (An )) n . (3.4.9) slef e e Proof. (3.4.1): Clearly, since Φl (X) is open. t t (A+B) = mlef (A) for each A ∈ B(X). (3.4.2): (=⇒) Suppose that mlef e e t t (A) > 0 by (3.4.1), and so mlef (A + B) > 0. It If A ∈ Φl (X), then mlef e e follows that A + B ∈ Φl (X). Therefore, B ∈ P (Φl (X)) = P (Φ(X)). (⇐=) Let B ∈ P (Φ(X)) = P (Φl (X)). Then −B ∈ P (Φl (X)) and we have that C ∈ Φl (X) if and only if C + B ∈ Φl (X). Thus, C ∈ B(X)\Φl (X) if and only if C ∈ −B + B(X)\Φl (X). Consequently, t mlef (A) = inf{ A − C : C ∈ B(X)\Φl (X)} e

= inf{ A − (−B + C1 : C1 ∈ B(X)\Φl (X)} = inf{ (A + B) − C1 : C1 ∈ B(X)\Φl (X)} t (A + B). = mlef e

(3.4.3): Clearly. (3.4.4): Let P ∈ P (Φ(X)). According to (3.4.2) and (3.4.3) we have t t t (A + B) = mlef (A + B + P ) ≤ mlef (A) + B + P , mlef e e e t t which implies mlef (A + B) ≤ mlef (A) + inf{ B + P : P ∈ P (Φ(X))} = e e lef t me (A) + B P Φ . t (A) and let λ ∈ [0, 1]. From (3.4.4) (3.4.5): Suppose that B P Φ < mlef e it follows that t t t mlef (A) = mlef (A + λB + (−λB)) ≤ mlef (A + λB) + − λB P Φ e e e t t t = mlef (A + λB) + λ B P Φ < mlef (A + λB) + mlef (A), e e e t and so mlef (A + λB) > 0. It follows that A + λB ∈ Φl (X), and hence e A, A + B ∈ Φl (X). Since the index is locally constant, we obtain i(A + B) = i(A). t (I). From (3.4.5) we get I − A ∈ Φl (X) and (3.4.6): Let A P Φ < mlef e i(I − A) = i(I) = 0. Thus, I − A ∈ Φ(X). t (I) for some n > 1 and let λ ∈ [0, 1]. Then (3.4.7): Let An P Φ < mlef e n n n n t (I) and from (3.4.6) it follows (λA) P Φ = λ A P Φ ≤ A P Φ < mlef e n that I − (λA) ∈ Φ(X). Since

I − (λA)n = (I − λA)(I + λA + · · · + λn−1 An−1 ) = (I + λA + · · · + λn−1 An−1 )(I − λA),

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we conclude I − λA ∈ Φ(X). Consequently, I − A ∈ Φ(X) and i(I − A) = i(I) = 0. (3.4.8): Suppose that AB − BA ∈ P (Φ(X)) and 1

t (An )) n . re (B) < lim (mlef e n→∞

1

t (An )) n . Since Let be such that re (B) < < limn→∞ (mlef e 1

re (B) = lim ( B n P Φ ) n n→∞

1

1

t it follows that limn→∞ ( B n P Φ ) n < < limn→∞ (mlef (An )) n . Thus there e 1 1 n lef t n is an odd n ∈ N such that ( B P Φ ) n < < (me (A )) n , and so B n P Φ < t (An ). By (3.4.5) we get An +B n ∈ Φl (X). Since P (Φ(X)) two-sided ideal mlef e of B(X), from AB − BA ∈ P (Φ(X)) it follows that An + B n = C(A + B) + P where C = An−1 −BAn−2 +· · · +B n−1 and P ∈ P (Φ(X)). Thus, C(A+B) ∈ Φl (X) and so A + B ∈ Φl (X). Let us remark that the proof above shows that A + λB ∈ Φl (X) for 0 ≤ λ ≤ 1, which implies that i(A + B) = i(A). 1 t (3.4.9): Suppose that limn→∞ (mlef (An )) n > 0. For λ ∈ C, |λ| < e 1 t limn→∞ (mlef (An )) n and B = λI it follows that e 1

t (An )) n . re (B) = |λ| < lim (mlef e n→∞

t (A) ≥ Since AB = BA, from (3.4.8) we get λI − A ∈ Φl (X). Hence slef e 1 lef t n n limn→∞ (me (A )) .

The next theorem is a dual part of Theorem 3.4. Theorem 3.5. Let A, B ∈ B(X). Then (A) > 0 ⇐⇒ A ∈ Φr (X), mright e (A + B) = mright (A) for each A ∈ B(X) ⇐⇒ B ∈ P (Φ(X)), mright e e right (A + B) ≤ mright (A) + B , me e right (A + B) ≤ m (A) + B P Φ . mright e e (A), then A, A + B ∈ Φr (X) and i(A) = i(A + B). If B P Φ < mright e (I), then I − A ∈ Φ(X) and i(I − A) = 0. If A P Φ < mright e (I) for some n > 1, then I − A ∈ Φ(X) and If An P Φ < mright e i(I − A) = 0. (3.5.8) If AB − BA ∈ P (Φ(X)) and

(3.5.1) (3.5.2) (3.5.3) (3.5.4) (3.5.5) (3.5.6) (3.5.7)

1

(An )) n , re (B) < lim (mright e n→∞

then A, A + B ∈ Φr (X) and i(A + B) = i(A). 1 (A) ≥ limn→∞ (mright (An )) n . (3.5.9) sright e e Theorem 3.6. Let A, B ∈ B(X). (3.6.1) If t t (A) + mlef (B), A − B P Φ < mlef e e

then A, B ∈ Φl (X) and i(A) = i(B).

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(3.6.2) If (A) + mright (B), A − B P Φ < mright e e then A, B ∈ Φr (X) and i(A) = i(B). t t (A) + mlef (B). Then Proof. (3.6.1): Suppose that A − B P Φ < mlef e e t t (A) + mlef (B) > 0, mlef e e t t and so mlef (A) and mlef (B) can not be at the same time equal to zero. If e e lef t t (A) > 0. Now from one of them, say me (B), is equal to zero, then mlef e lef t t (B) > A−B P Φ < me (A), by (3.4.5) we conclude B ∈ Φl (X), that is mlef e lef t lef t 0, which is a contradiction. Therefore, me (A) > 0 and me (B) > 0, and so A, B ∈ Φl (X). There exists P ∈ P (Φ(X))) such that t t A − B − P < mlef (A) + mlef (B). e e t t Let C = B + P . From (3.4.2) it follows that mlef (C) = mlef (B), and so we e e get t t A − C < mlef (A) + mlef (C). e e t Therefore, the open ball centered at A with radii mlef (A) and the open e lef t ball centered at C with radii me (C) have a non-empty intersection. Hence their union is linearly connected set contained in Φl (X) ⊂ Φ+ (X). Since the index is locally constant, it follows that i(A) = i(C). For λ ∈ [0, 1] we have λP ∈ P (Φ(X)), which implies B + λP ∈ Φl (X). Again from local constancy of the index we conclude i(B) = i(B + P ). Therefore i(A) = i(B).

Let us remark that P (Wl (X)) = P (Φ(X)) and P (Wr (X)) = P (Φ(X)) and analogous assertions can be formulated for left and right Weyl operators and the quantities: t mlef w (A) = dist(A, B(X)\Wl (X)),

(A) = dist(A, B(X)\Wr (X)), mright w

A ∈ B(X).

t right Notice that if mlef (A) > 0), i.e. if A ∈ Wl (X) (A ∈ Wr (X)), w (A) > 0 (mw t lef t (A) then because of local constancy of the index it holds mlef w (A) = me right (A) = m (A)). (mright w e Moreover, the following more general assertions can be proved analogously.

Theorem 3.7. Let U be an open subset of Φ± (X) such that μU ⊂ U for every μ = 0. For A ∈ B(X), set mU (A) = dist(A, B(X)\U), A P (U ) = inf{ A + P : P ∈ P (U)}, where P (U) is the perturbation class of U. Then, for A, B ∈ B(X), the following hold: (3.7.1) mU (A) > 0 ⇐⇒ A ∈ U; (3.7.2) mU (A + B) = mU (A) for every A ∈ B(X) ⇐⇒ B ∈ P (U); (3.7.3) mU (A + B) ≤ mU (A) + B ; (3.7.4) mU (A + B) ≤ mU (A) + B P (U ) ;

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If B P (U ) < mU (A), then A, A + B ∈ U and i(A) = i(A + B). If A P (U ) < mU (I), then I − A ∈ Φ(X) and i(I − A) = 0. If An P (U ) < mU (I) for n > 1, then I − A ∈ Φ(X) and i(I − A) = 0. If A − B P (U ) < mU (A) + mU (B), then A, B ∈ U and i(A) = i(B).

Theorem 3.8. Let U be an open subset of Φ± (X) such that (i) μU ⊂ U for every μ = 0, (ii) I ∈ U, (iii) K(X) ⊂ P (U). Then 1

re (A) = lim ( An P (U ) ) n .

(3.4)

n→∞

1

1

Proof. Let λ ∈ C and |λ| > (mU (I))− n ( An P (U ) ) n for some n ∈ N. Then mU (I) > (A/λ)n P (U ) and by (3.7.7) it follows that λI − A ∈ Φ(X). There1 1 fore re (A) ≤ (mU (I))− n ( An P (U ) ) n for all n ∈ N. Notice that from I ∈ U it follows that mU (I) > 0. Therefore, 1

1

1

re (A) ≤ lim (mU (I))− n lim ( An P (U ) )) n = lim ( An P (U ) ) n . n→∞

n→∞

n→∞

Since K(X) ⊂ P (U), it follows that A P (U ) ≤ π(An ) for every n ∈ N. Thus n

1

1

1

re (A) ≤ lim ( An P (U ) ) n ≤ lim ( An P (U ) ) n ≤ lim ( π(An ) ) n = re (A), n→∞

n→∞

n→∞

which implies (3.4). Theorem 3.9. Let U be an open subset of Φ± (X) such that (i) (ii) (iii) (iv) (v)

μU ⊂ U for every μ = 0, I ∈ U, K(X) ⊂ P (U), GU ⊂ U and UG ⊂ U, (∀A, B ∈ B(X))(AB ∈ U =⇒ A ∈ U) or (∀A, B ∈ B(X))(AB ∈ U =⇒ B ∈ U). Then, for A, B ∈ B(X), the following hold:

(3.9.1) If AB − BA ∈ P (U) and 1

re (B) < lim (mU (An )) n , n→∞

then A, A + B ∈ U and i(A + B) = i(A). 1 (3.9.2) sU (A) ≥ limn→∞ (mU (An )) n , where sU (A) = max{ ≥ 0 : |λ| < ⇒ A − λ ∈ U}. Proof. Follows from Theorem 3.8 and (3.7.5).

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For A ∈ B(X), let A P Φ+ = inf{ A + P : P ∈ P (Φ+ (X))}, A P Φ− = inf{ A + P : P ∈ P (Φ− (X))}, and m+ e (A) = dist(A, B(X)\Φ+ (X)), m− e (A) = dist(A, B(X)\Φ− (X)). If we take Φ+ (X) or Φ− (X) for U in (3.7.8), we get: Corollary 3.10. Let A, B ∈ B(X). If + A − B P Φ+ < m+ e (A) + me (B),

or − A − B P Φ− < m− e (A) + me (B),

then i(A) = i(B). From Corollary 3.10 we obtain Theorem 4 in [10]: + Let T, S ∈ B(X). If π(T − S) < m+ e (T ) + me (S) or π(T − S) < − − me (T ) + me (S), then i(T ) = i(S).

References [1] Caradus, S.R.: Generalized Inverses and Operator Theory, Queen’s Papers in Pure and Applied Mathematics, No. 38. Queen’s University, Kingston (1978) [2] Caradus, S.R., Pfaffenberger, W.E., Yood, B.: Calkin Algebras and Algebras of Operators on Banach Spaces. Marcel Dekker, New York (1974) [3] Gramsch, B., Lay, D.: Spectral mapping theorems for essential spectra. Math. Ann. 192, 17–32 (1971) [4] Harte, R.: Invertibility and Singularity for Bounded Linear Operators. Marcel Dekker, New York (1988) [5] Kroh, H., Volkman, P.: St¨ orungss¨ atze f¨ ur Semifredholmoperatoren. Math. Z. 148, 295–297 (1976) [6] M¨ uler, V.: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Birkh¨ auser Verlag, Basel (2001) [7] Rakoˇcevi´c, V.: Approximate point spectrum and commuting comact perturbations. Glasgow Math. J. 28, 193–198 (1986) [8] Rakoˇcevi´c, V.: Semi-Browder operators and perturbations. Studia Math. 122, 131–137 (1996) [9] Salinas, N.: Operators with essentially disconnected spectrum. Acta Sci. Math. (Szeged) 33, 193–205 (1972) [10] Zem´ anek, J.: The semi-fredholm radius of a linear operator. Bull. Pol. Acad. Sci. Math. 32, 67–76 (1984) [11] Zem´ anek, J.: Geometric characteristics of semi-Fredholm operators and their asymptotic behaviour. Studia Math. 80, 219–234 (1984) [12] Zem´ anek, J.: Compressions and the Weyl-Browder spectra. Proc. R. Irish Acad. A 86, 57–62 (1986)

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ˇ [13] Zivkovi´ c-Zlatanovi´c, S., Djordjevi´c, D.S., Harte, R.E.: On left and right Browder operators (submitted) ˇ [14] Zivkovi´ c, S.: Semi-Fredholm operators and perturbations. Publ. Inst. Math. Beograd 61, 73–89 (1997) ˇ Zivkovi´ ˇ Sneˇzana C. c-Zlatanovi´c and Dragan S. Djordjevi´c (B) Faculty of Sciences and Mathematics University of Niˇs P. O. Box 224, 18000 Nis, Serbia e-mail: [email protected]; [email protected] Robin E. Harte Trinity College Dublin 2, Ireland e-mail: [email protected] Received: May 13, 2010. Revised: October 18, 2010.

Integr. Equ. Oper. Theory 69 (2011), 365–372 DOI 10.1007/s00020-010-1835-2 Published online October 30, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

The Maximum Principle for Holomorphic Operator Functions Andrzej Daniluk Abstract. We show that if an operator-valued analytic function f of a complex variable attains its maximum modulus at z0 , then the coefficients of the nonconstant terms in the power series expansion about z0 cannot be invertible, provided a complex uniform convexity condition holds. One application is that the norm of the resolvent of an operator on a complex uniformly convex space cannot have a local maximum. Mathematics Subject Classification (2010). Primary 32A10; Secondary 30C80, 46E40. Keywords. Maximum principle, holomorphic operator function, Banach space, uniformly convex space.

0. Introduction It is well known that the maximum modulus of a non-constant analytic function with values in a (complex) Banach space L can have a local maximum unless all points of the the unit sphere of L are complex extreme [10]. Here we consider analytic functions f : Ω → B(L) (for Ω ⊂ C a domain), noting that the unit sphere of B(L) may have points that are not complex extreme even for L a Hilbert space of dimension at least 2. If z0 ∈ Ω satisﬁes f (z0 ) ≥ f (z) for z ∈ Ω, we show that f (z) fails to be invertible for z near z0 , provided that L is complex uniformly convex. The same result also applies to higher derivatives f (k) (z) for k ≥ 1. An earlier result due to the author, dealing with the resolvent of a Hilbert space operator, appears in B¨ ottcher [1, Prop. 6.1], and an extension to Lp spaces L (1 < p < ∞) is given in [2, Th. 5.1]. In recent work Shargorodsky [8] and Shargorodsky and Shkarin [9] have given examples of operators on Banach spaces (which can be separable, reﬂexive and strictly convex) where the norm of the resolvent does have a local maximum.

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1. Basic Definitions and Preliminary Results Let Ω ⊂ C denote a domain. We will denote by L a complex Banach space and by B(L) the set of all bounded linear operators on L. We will also denote S 1 = {x ∈ L : x = 1}. Here and subsequently Mf = sup {f (z) : z ∈ Ω}. For r > 0 and a ∈ Ω we set D(a, r) = {z ∈ Ω : |z − a| < r}. It is a well-known fact [see 6, Theorem 3.13.1 or 3, Chap. III, Sect. 14] that if L is a complex Banach space and f : Ω → B(L) is a holomorphic operator function and f (z0 ) = Mf for some z0 ∈ Ω then ∀z ∈ Ω f (z) = Mf , i.e. the function f (·) satisﬁes the maximum principle. Remark 1.1. Obviously the above statement holds not only for holomorphic operator functions, but in general for holomorphic functions into any Banach space. However, this is not a close analogue of classical maximum principle for holomorphic (complex) functions, since there |f (z)| = sup |f | implies not only |f | = const, but f = const. For operator holomorphic functions such a conclusion however would be false. As a counterexample it sufﬁces to take L = C2 , Ω = D(0, 1) and f (z) = ( 10 z0 ). There is another analogy yet. Note that for complex holomorphic functions the condition f = const is equivalent to the fact that in expansion of the form f (ζ) = a0 + (ζ − z)a1 + (ζ − z)2 a2 + · · · , valid in D(z, r), all coefﬁcients except the ﬁrst one disappear. One can equivalently say that f (ζ) − f (z) = 0 for all ζ ∈ D(z, r). We shall prove that this zeroing in C of the right-hand side of the expansion corresponds to noninvertibility in B(L) of the coefﬁcients in the operator expansion. Definition 1.2. A Banach space L is called uniformly convex if for every η > 0 there exists δ > 0 such that x + y ∀x, y ∈ S 1 x − y ≥ η =⇒ 2 < 1 − δ. Definition 1.3. A Banach space L is called complex uniformly convex if for every ε > 0 there exists δ > 0 such that x, y ∈ L, y ≥ ε and x + ζy ≤ 1, ∀ζ ∈ D(0, 1) =⇒ x ≤ 1 − δ. Example 1.4. For 1 < p < ∞, measure spaces Lp (X, dμ) are uniformly convex (and hence also complex uniformly convex). Nevertheless, the measure space L1 (X, dμ) is a complex uniformly convex space, but not uniformly convex [4]. Lemma 1.5. Let M > 0 and L be a uniformly convex Banach space. Then ∀ξ > 0 ∃ε > 0 ∀y1 , y2 ∈ L ∀ϕ ∈ (S 1 )∗ (y1 ≤ M, y2 ≤ M, |ϕy1 − M | < ε, |ϕy2 − M | < ε =⇒ y1 − y2 < ξ) .

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The Maximum Principle for Holomorphic Operator

Proof. Take any ε > 0 such that ε < min the antecedent we have

1

1 2 M, 2 ξ

367

. Then for y1 , y2 satisfying

M ≥ yi ≥ |ϕyi | = |ϕyi − M + M | ≥ M − ε,

i = 1, 2,

hence |y1 − y2 | ≤ ε. Set yi = yyii for i = 1, 2 (obviously yi ∈ S 1 ). Then we have 1 1 |ϕyi − yi | ≤ |ϕyi − yi | yi M −ε 2 2 ≤ |ϕyi − yi | = |ϕyi − M + M − yi | M M 4ε 2 2 (|ϕyi − M | + M − yi ) ≤ (ε + ε) = . ≤ M M M

|ϕyi − 1| =

Also y1 + y2 1 1 1 1 4ε ϕ − 1 = (ϕy1 − 1)+ (ϕy2 − 1) ≤ |ϕy1 − 1|+ |ϕy2 − 1| ≤ , 2 2 2 2 2 M thus

Take η = such that

y1 + y2 ≥ ϕ y1 + y2 ≥ 1 − 4ε . 2 2 M 1 2M ξ.

By the definition of the uniform convexity there exists δ > 0 y1 + y2 y1 − y2 < η 2 > 1 − δ =⇒

for any y1 , y2 ∈ S 1 . Hence, if we claim additionally that ε ≤ 14 M δ (which can be done as this neither contradicts the initial assumptions about ε, nor δ depends on its initial choice), we will have y1 − y2 <

1 ξ. 2M

Then y1 − y2 = y1 y1 − y2 y2 = y1 (y1 − y2 ) + (y1 − y2 )y2 1 1 1 ξ +ε ≤ ξ + ξ = ξ, ≤ y1 y1 −y2 +|y1 − y2 | y2 ≤ M · 2M 2 2 which is our assertion. We will also need another lemma concerning holomorphic complex functions: Lemma 1.6 (Approximative version of the maximum principle for holomorphic complex functions). Let q be a complex function which is analytic in D(0, r) and let for some c > 0 |q(z)| < |q(0)| + c,

|z| < r.

|q(z) − q(0)| < c ,

|z| < r ,

Then

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where

2 1/2

c = 2 |q(0)| c + c

√ ,

2 r. 2

r =

Proof. Let us expand q into the series q(z) = q0 + zq1 + z 2 q2 + · · · ,

|z| < r.

Take ρ < r. Then we have, as a particular form of Parseval’s identity [7, Theorem 10.22] ∞ 1 2 2 |q(z)| |dz| = |qn | ρ2n . 2πρ n=0 |z|=ρ

On the other hand, by assumption we have 1 1 2 2 2 2 |q(z)| |dz| ≤ (|q(0)| + c) |dz| = (|q(0)|+c) = |q0 | +c2 . 2πρ 2πρ |z|=ρ

|z|=ρ

Thus 2

2

ρ2 |q1 | + ρ4 |q2 | + · · · ≤ c2 . Taking z such that |z| <

√

2 2 ρ

we get, using the Schwarz inequality: z z2 |q(z) − q(0)| = q1 z + q2 z 2 + · · · = q1 ρ · + q2 ρ2 · 2 + · · · ρ ρ 1/2 2 4

1/2 2 2 |z| |z| 2 ≤ |q1 ρ| + q2 ρ + · · · · + 4 + ··· ρ2 ρ 1/2 1/2 1 1 + + ··· ≤ c2 · = c . 2 4

As ρ was arbitrary, the same is true for |z| < of the lemma.

√ 2 2 r

= r . This is the assertion

2. The Maximum Principle For Holomorphic Operator Functions In this section we present a generalization of the maximum principle for holomorphic operator functions. We broaden the notion of being zero in C of the coefﬁcients in the complex expansion to the idea of non-invertibility in B(L) of the coefﬁcients in the operator expansion and then we will be in a position to prove the desired generalization. We will give two proofs of the result: the ﬁrst one for uniformly convex spaces, and the second one for complex uniformly convex spaces. The second proof follows the lines of [5] and it was suggested to the author by one of the referees. It should be noted that an extra proof for uniformly convex spaces would not be necessary, because uniformly convex spaces are complex uniformly convex, but both of the proofs are presented, because each is based on another idea and therefore each has the potential to be useful when tackling spaces that cannot be included yet.

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Theorem 2.1. Let L be a uniformly convex complex Banach space, Ω ⊂ C be a domain, f : Ω → B(L) be a holomorphic operator function and Mf = sup {f (z) : z ∈ Ω}. Assume that f (z0 ) = Mf for some z0 ∈ Ω. Then ∃r > 0 ∀z ∈ D(z0 , r ) f (z) − f (z0 ) is not invertible. Furthermore, there exists a sequence {xn }, xn ∈ S 1 , such that (f (z) − f (z0 )) xn −−−−→ 0 n→∞

(2.1)

uniformly in z ∈ D(z0 , r ). Proof. (Valid for uniformly convex spaces only.) Let {xn } be a sequence such that xn ∈ S 1 and f (z0 )xn −−−−→ f (z0 ) = Mf n→∞

(2.2)

(such a sequence exists by the definition of the norm of the operator f (z0 )) and let ϕn ∈ (S 1 )∗ be such that ϕn f (z0 )xn = f (z0 )xn (such a sequence exists by the Hahn-Banach Theorem). Let ε be chosen to ξ as in Lemma 1.5 (with M = Mf ). Due to the convergence (2.2) there is some N ∈ N such that for all n > N ε ε2 , Mf − f (z0 )xn ≤ min . 3 16Mf Consider (for a ﬁxed n > N ) the complex holomorphic function q deﬁned on D(0, r) by q(z − z0 ) = ϕn f (z)xn for z ∈ D(z0 , r). Setting c = Mf − f (z0 )xn we have, according to the above inequality,

ε 2 ε2 ε2 + < , 2Mf c + c2 ≤ 2Mf · 16Mf 3 4 hence c = (2Mf c + c2 )1/2 < 2ε . Moreover |q(z − z0 )| ≤ f (z) ≤ Mf = f (z0 )xn + c = q(0) + c, hence from Lemma 1.6 it follows that |q(z − z0 ) − q(0)| < c <

ε 2

for

|z − z0 | < r

and thus ε ε ε + c ≤ + < ε. 2 2 3 Setting y1 = f (z0 )xn , y2 = f (z)xn we see that they satisfy the assumptions of Lemma 1.5 (with ϕ = ϕn ), hence |q(z − z0 ) − Mf | ≤ |q(z − z0 ) − q(0)| + |q(0) − Mf | ≤

y1 − y2 < ξ, which is our assertion.

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Obviously, uniform convergence implies that for each z ∈ D(z0 , r ) (f (z) − f (z0 )) xn −−−−→ 0, n→∞

which proves that f (z) − f (z0 ) is not invertible for z ∈ D(z0 , r ).

Corollary 2.2. Under the assumptions of Theorem 2.1, if the expansion of a holomorphic operator function f : Ω → B(L) into a power series around z0 ∈ Ω is of the form f (z) =

∞

(z − z0 )j Aj ,

j=0

then all A1 , A2 , . . . are not invertible. Moreover, there exists {xn }, xn ∈ S 1 , such that Ak xn −−−−→ 0.

∀k ∈ N

n→∞

Proof. Theorem 2.1 shows that ∀ξ > 0 ∃N ∈ N ∀n > N ∀z ∈ D(z0 , r ) (f (z) − A0 ) xn < ξ. For each r < r and k ≥ 1 we have f (z) 1 1 Ak = dz = 2πi (z − z0 )k+1 2πi |z−z0 |=r

and accordingly Ak xn = hence 1 Ak xn ≤ 2π

|z−z0 |=r

1 2πi

|z−z0 |=r

|z−z0 |=r

f (z) − A0 dz (z − z0 )k+1

(f (z) − A0 ) xn dz, (z − z0 )k+1

(f (z) − A0 ) xn |z − z0 |

k+1

|dz| ≤ |z−z0 |=r

ξ ξ |dz| = k . 2πrk+1 r

Since ξ may be chosen arbitrarily, we conclude that for each k ∈ N Ak xn −−−−→ 0, n→∞

which is the desired conclusion.

As we mentioned at the beginning of this section, we present here another proof of Theorem 2.1, valid for complex uniformly convex spaces. To this end we need one more lemma. Lemma 2.3. Let L be a complex Banach space and f be an L-valued function analytic in a neighbourhood of 0 ∈ C. If f (z) = f (0) for every z in a neighbourhood of 0, then there exists r > 0 such that f (0) + ζ (f (z) − f (0)) ≤ f (0) , where D(a, ρ) := {w ∈ C : |w − a| ≤ ρ}.

∀ζ ∈ D(0, 1), ∀z ∈ D(0, r),

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Proof. Let f (z) = f (0), ∀z ∈ D(0, R), R > 0. The argument in the ﬁrst four lines of the proof of Lemma 1.1, [5] implies that | f (0), u| + | f (z) − f (0), u| ≤ f (0) ,

∀z ∈ D(0, r), r := R/3,

for any u ∈ L∗ with u = 1. Take ∀ζ ∈ D(0, 1) and ∀z ∈ D(0, r). There exists u ∈ L∗ with u = 1 such that

f (0) + ζ (f (z) − f (0)) , u = f (0) + ζ (f (z) − f (0)) . Hence

f (0) + ζ (f (z) − f (0)) ≤ | f (0), u + | f (z) − f (0), u| ≤ f (0) , ∀z ∈ D(0, r).

Proof. (Of Theorem 2.1 for complex uniformly convex spaces.) It follows from the maximum principle [see, e.g., 6, Theorem 3.13.1 or 3, Chap. III, Sect. 14] that f (z) = Mf , ∀z ∈ Ω. Hence Lemma 2.3 applies to the function f (·+z0 ). Let r be the number from that lemma. Let xn ∈ L, xn = 1 be a sequence such that f (z0 )xn → Mf = f (z0 ) ,

n → ∞.

(2.3)

Then it has to satisfy (2.1). Suppose the contrary. Then there exist ε0 > 0, a subsequence of (xn )n∈N which we denote by (yn )n∈N and zn ∈ D(0, r), n ∈ N such that (f (zn ) − f (z0 )) yn ≥ ε0 ,

∀n ∈ N.

Take the δ from Definition 1.3 which corresponds to ε = ε0 /Mf . According to (2.3), there exists N > 0 such that f (z0 )yn > Mf (1 − δ), ∀n ≥ N . Then x := M1f f (z0 )yN ∈ L and y := M1f (f (zN ) − f (z0 )) yN ∈ L satisfy the following conditions: x > 1 − δ, y ≥ M1f ε0 = ε, and it follows from Lemma 2.3 that 1 x + ζy = (f (z )y + ζ (f (z ) − f (z )) y ) 0 N N 0 N Mf ≤

1 1 f (z0 ) + ζ (f (zN ) − f (z0 )) ≤ Mf = 1, Mf Mf

∀ζ ∈ D(0, 1).

This contradicts the complex uniform convexity of L and proves (2.1).

Remark 2.4. Note that the assumption of the uniform convexity of the space L cannot be dropped. Namely Shargorodsky [8] gives an example of a bounded linear operator on a Banach space whose resolvent norm is constant in a neighbourhood of zero. The operator in that example is a weighted bilateral shift on the space l∞ (Z) equipped with the equivalent norm x = supk =0 |xk | + |x0 |. Such a space is obviously not uniformly convex, and the resolvent of the operator is the holomorphic function which contradicts the assertion of the Theorem 2.1.

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Acknowledgements The author wishes to express his thanks to Prof. Jan Stochel, Ph.D., from Jagiellonian University and Prof. Jaroslav Zem´ anek, Ph.D., from IMPAN, for fruitful discussions at the Operator Theory Seminar. The author is also grateful to an anonymous referee who suggested the proof of the main theorem for complex uniformly convex spaces.

References [1] B¨ ottcher, A.: Pseudospectra and singular values of large convolution operators. J. Integral Equ. Appl. 6(3), 267–301 (1994) [2] B¨ ottcher, A., Grudsky, S.M., Silbermann, B.: Norms of inverses, spectra, and pseudospectra of large truncated Wiener-Hopf operators and Toeplitz matrices. New York J. Math. 3, 1–31 (1997) [3] Dunford, N., Schwartz, J.T.: Linear Operators. I. General Theory. Interscience Publishers, New York, London (1958) [4] Globevnik, J.: On complex strict and uniform convexity. Proc. Am. Math. Soc. 47(1), 175–178 (1975) [5] Globevnik, J., Vidav, I.: On operator-valued analytic functions with constant norm. J. Funct. Anal. 15, 394–403 (1974) [6] Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups. American Mathematical Society, Providence (1957) [7] Rudin, W.: Real and Complex Analysis. Mc Graw-Hill, Singapore (1986) [8] Shargorodsky, E.: On the level sets of the resolvent norm of a linear operator. Bull. Lond. Math. Soc. 40, 493–504 (2008) [9] Shargorodsky, E., Shkarin, S.: The level sets of the resolvent norm and convexity properties of Banach spaces. Arch. Math. 93, 59–66 (2009) [10] Thorp, E., Whitley, R.: The strong maximum modulus theorem for analytic functions into a Banach space. Proc. Am. Math. Soc. 18(4), 640–646 (1967) Andrzej Daniluk (B) Faculty of Mathematics Jagiellonian University Krak´ ow, Poland e-mail: [email protected] Received: May 17, 2010. Revised: October 6, 2010.

Integr. Equ. Oper. Theory 69 (2011), 373–391 DOI 10.1007/s00020-011-1866-3 Published online February 1, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

On Integral Equations Related to Weighted Toeplitz Operators Carme Cascante, Joan F`abrega and Daniel Pascuas Abstract. For weighted Toeplitz operators TϕN defined on spaces of holomorphic functions in the unit ball, we derive regularity properties of the solutions f to the equation TϕN (f ) = h in terms of the regularity of the symbol ϕ and the data h. As an application, we deduce that if f ≡ 0 is a function in the Hardy space H 1 such that its argument f¯/f is in a Lipschitz space on the unit sphere S, then f is also in the same Lipschitz space, extending a result of Dyakonov to several complex variables. Mathematics Subject Classification (2010) . Primary 45E05; Secondary 47B35, 32A35. Keywords. Toeplitz operators, Lipschitz symbols.

1. Introduction The goal of this paper is to study the regularity of solutions to certain equations related to weighted Toeplitz operators in several complex variables. We will start by stating some particular cases of the main results in this paper, which involve classical spaces and integral operators and illustrate the object of this paper, although they can be applied in a more general setting. Let B denote the open unit ball in Cn and S its boundary, and we denote by H = H(B) the space of holomorphic functions on B. If 1 ≤ p < +∞, H p (B) is the classical Hardy space on the unit ball. The Cauchy transform P : L1 (S) → H is deﬁned as follows: ψ(ζ) P(ψ)(z) := ¯ n dσ(ζ) (z ∈ B). (1 − ζz) S

Here dσ denotes the normalized Lebesgue measure on S and if z = (z1 , . . . , zn ) ¯ = n ζ zi . ∈ Cn , ζ = (ζ1 , . . . , ζn ) ∈ Cn , ζz i i=1 Partially supported by DGICYT Grant MTM2008-05561-C02-01 and DURSI Grant 2009SGR 1303.

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It is well known that if 1 < p < +∞, the Cauchy transform maps Lp (S) to the Hardy space H p (B) boundedly [12, § 6.3] and that this fact fails for p = 1. For ϕ ∈ L∞ (S), and f ∈ H 1 (S), the corresponding Toeplitz operator with symbol ϕ is deﬁned by Tϕ (f ) := P(ϕf ). Then, it is clear that Tϕ is bounded from Lp (S) to H p (B), for 1 0, then Tϕ maps H 1 (S) to H 1 (B) boundedly. As it is usual, if 1 ≤ p < +∞, we will write H p to denote both H p (B) and the space of its admissible boundary values H p (S), and therefore we write Tϕ : H p → H p (it will be clear from the context which is the kind of Hardy space considered). Then we have: Theorem 1.1. Let τ > 0 and ϕ ∈ Λτ be a non-vanishing function on S. If f ∈ H 1 and Tϕ (f ) ∈ Λτ , then f ∈ Λτ . This result extends [7, Theorem 3.1], which deals with the case n = 1 and the regularity of the solutions to the equation Tϕ (f ) = 0. We remark that if we drop the condition 0 ∈ / ϕ(S), then Theorem 1.1 is not true in general. Indeed, we only need to consider the symbol ϕ(ζ) = (1 − ζ1 )τ and the function f (ζ) = (1 − ζ1 )−τ with 0 < τ < n (to ensure that f ∈ H 1 ). As in the one variable case (see [7]), the above theorem implies some interesting properties of the holomorphic Lipschitz functions. For instance, / ϕ(S), and ϕf ∈ Λτ + ker P, Corollary 1.2. If f ∈ H 1 , ϕ ∈ Λτ , so that 0 ∈ then f ∈ Λτ . In particular, we have: Corollary 1.3. If f ∈ H 1 \ {0} and its argument function ϕ = f /f is in Λτ , then f ∈ Λτ . The preceding corollary is proved in [7] for n = 1. 0 = We consider symbols in some spaces of bounded functions on B, Gττ,k τ0 k Gτ,k (B) ⊂ Λτ0 (B) ∩ C (B), whose intersection with H(B) coincide with the space Λτ (B) ∩ H(B). These spaces are deﬁned in terms of the growth of the derivatives of the function, in a way similar to the well-known characterization of the harmonic and invariant harmonic Lipschitz functions on B. In 0 0 0 · Gττ,k = Gττ,k , we will show that these spaces fact, since they satisfy that Gττ,k of symbols contain sums of products of harmonic and invariant harmonic extensions of Lipschitz spaces with respect to different metrics (see Sect. 2.3 for a precise definition of these spaces and their properties). For N > 0, let dνN (z) := cN (1−|z|2 )N −1 dν(z), where ν is the Lebesgue ) measure on B and cN = Γ(n+N n!Γ(N ) , so that νN (B) = 1. Let L1N := L1 (B, dνN ) and consider the weighted Bergman transform N P : L1N → H deﬁned by ψ(w) N P (ψ)(z) := dνN (w). (1 − wz)n+N B

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1 := L1N ∩ H and for ϕ ∈ L∞ (B), deﬁne the weighted Toeplitz operaLet B−N N 1 → H by TϕN (f ) := P N (ϕf ). Since the operator P N is bounded tor Tϕ : B−N p p on LM := L (B, dνM p ), 1 ≤ p < ∞, M < N (see, for instance, [14, Theorem p 2.11]), the operator TϕN is also bounded on B−M := H ∩ LpM . If ϕ ∈ Λτ0 , 1 then the boundedness of TϕN also holds on B−N (see Proposition 2.25). In order to unify the statement of some results, we denote dσ by dν0 . This usual notation (see, for instance, [3, § 0.3]) is motivated by the fact that dνN converges to dσ in the weak* sense as N → 0+ , that is, ψ(ζ) dσ(ζ) = lim+ ψ(z) dνN (z), for every ψ ∈ C(B). N →0

S

B

The proof of this fact can be easily veriﬁed by integrating in polar coordinates. It is also easy to show that, if ϕ ∈ Λτ , for τ > 0, and f ∈ H 1 (B), then lim TϕN (f )(z) = Tϕ (f ∗ )(z),

N →0+

for every z ∈ B.

Here f ∗ ∈ H 1 (S) denotes the boundary values of f ∈ H 1 (B). From now on, we will write f ∗ = f . Now we state the two main results of this paper. 1 0 such that 0 ∈ / ϕ(S). If f ∈ B−N satisfies TϕN (f ) = Theorem 1.4. Let ϕ ∈ Gττ,k ∞ ∞ 1 h ∈ Bτ , then f ∈ Bτ and f Bτ∞ ≤ C(f B−N + hBτ∞ ), where C > 0 is a finite constant only depending on ϕ, N > 0 and n. In particular, f Bτ∞ ≤ 1 Cf B−N , for any f ∈ ker TϕN . 0 Theorem 1.5. Let ϕ be the restriction to S of a function in Gττ,k and let 1 ∞ ∞ f ∈ H . If 0 ∈ / ϕ(S) and Tϕ (f ) = h ∈ Bτ , then f ∈ Bτ and f Bτ∞ ≤ C(f H 1 + hBτ∞ ), where C > 0 is a finite constant only depending on ϕ and n. In particular, f Bτ∞ ≤ Cf H 1 , for any f ∈ ker Tϕ .

Note that the inequalities in the above theorems are in fact equivalences due to the continuity of both the Toeplitz operator and the embeddings 1 . Bτ∞ ⊂ H 1 ⊂ B−N 0 For τ > 1/2, the restriction to S of Gττ,k contains the space Λτ , and so Theorem 1.5 includes the result of Theorem 1.1 for these cases. However, the same techniques used to prove the above theorems allow us to extend this result to the whole scale of spaces Λτ . As a consequence of the above two theorems, we have: 1 , for N > 0, or f ∈ H 1 and N = 0, ϕ ∈ Corollary 1.6. If either f ∈ B−N τ0 τ0 / ϕ(S) and ϕf ∈ Gτ,k + ker P N , then f ∈ Bτ∞ . Gτ,k , 0 ∈

Since g − g(0) ∈ ker P, for every g ∈ H 1 , in particular, we have: Corollary 1.7. If f, g ∈ H 1 satisfy ϕ = g/f ∈ Γτ and 0 ∈ / ϕ(S), then f, g ∈ Bτ∞ .

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The preceding result generalizes Corollary 1.3, and extends [7, Corollary 3.2] to dimension n > 1. The paper is organized as follows. In Sect. 2 we state some properties of the spaces considered in this work and we also recall some integral representation formulas that will be used in the proof of the main theorems. In Sect. 3 we state our main technical theorem (Theorem 3.1) from which we deduce Theorems 1.4 and 1.5 and its corollaries. We also construct some counterexamples. Finally, Theorem 3.1 is proved in Sect. 4.

2. Preliminaries 2.1. Notations Throughout the paper, the letter C will denote a positive constant, which may vary from place to place. The notation f (z) g(z) means that there exists C > 0, which does not depend on z, f and g, such that f (z) ≤ Cg(z). We write f (z) ≈ g(z) when f (z) g(z) and g(z) f (z). ∂ , for j = 1, . . . , n. For any multiindex α = (α1 , . . . , αn ) ∈ Let ∂j := ∂z j n Nn , where N is the set of non-negative integers, let |α| := j=1 αj and α1 ∂ |α| k ∂α := ∂zα = ∂1 · · · ∂nαn . We write |∂ k ϕ| := |α|=k |∂α ϕ| and |d ϕ| := complex tangential |α|+|β|=k |∂α ∂ β ϕ|. When n > 1, we also consider the differential operators Di,j := z i ∂j − z j ∂i and |∂T ϕ| := 1≤i<j≤n |Di,j ϕ|. For n = 1, we write |∂T ϕ| := 0. We also introduce the following two functions on B which will be used 0 . in the definition of the spaces Gττ,k 1 For ϕ ∈ C (B), let ϕ(z) := (1 − |z|2 )|∂ϕ(z)| + (1 − |z|2 )1/2 |∂ T ϕ(z)|. For t ∈ R, let

ωt (z) :=

(1 − |z|2 )min(t,0) , if t = 0, e log 1−|z| if t = 0. 2,

(2.1)

(2.2)

2.2. Holomorphic Besov Spaces In this section we recall some well-known properties of the holomorphic Besov spaces. The proofs of them can be found for instance in [3,14]. Let 1 ≤ p < ∞, k ∈ N and δ > 0. The weighted Sobolev space Lpk,δ is the completion of the space C ∞ (B), endowed with the norm ⎧ ⎫1/p k ⎨ ⎬ ψLpk,δ := |dj ψ(z)|p (1 − |z|2 )δp−1 dν(z) . ⎩ ⎭ j=0 B

When k = 0, we will just write Lpδ = Lp0,δ . We extend this definition to the case p = ∞, so that L∞ k,δ is the subspace of functions ψ in the Sobolev space L1k,δ+1 satisfying

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ess sup |dj ψ(z)|(1 − |z|2 )δ < ∞.

j=0

z∈B

If 1 ≤ p ≤ ∞ and s ∈ R, the holomorphic Besov space Bsp is deﬁned to be Bsp := H ∩ Lpk,k−s , for some k ∈ N, k > s. It is well known that · Lpm,m−s and · Lpk,k−s are equivalent norms on Bsp , for any m, k ∈ N, m, k > s (see [3, Theorem 5.13] and [14, Chapter 6] for p < +∞, and [14, Chapter 7] for p = +∞). We can found other equivalent norms on Bsp in [3, § 5] and [14]. It is also well-known the following result: Proposition 2.1. If f ∈ H satisfies either p Ck,s (f ) =

⎧ ⎨ ⎩

B

⎫1/p ⎬ |∂ k f (z)|p (1 − |z|2 )(k−s)p−1 dν(z) < ∞, ⎭

when 1 ≤ p < ∞,

or p Ck,s (f ) = sup |∂ k f (z)|(1 − |z|2 )k−s < ∞, z∈B

when p = ∞,

for some nonnegative integer k > s, then f ∈ Bsp . p (f ) the sum of the modulus of the coefficients Moreover, adding to Ck,s of the Taylor polynomial of f of order k − 1 at the origin, we also obtain an equivalent norm on Bsp . See, for instance, [14, Prop. 6.2]. Note that if s = 0, then B0∞ is the Bloch space. If s > 0 then Bs∞ coincides with the space of holomorphic functions on B whose boundary values are in the corresponding Lipschitz–Zygmund space Λs (see the next subsection for more details). Proposition 2.2. [3, Theorems 5.13,14] Let 1 ≤ p ≤ q ≤ ∞ and let s, t ∈ R. Then: (i). If s > t, then Bsp ⊂ Btp . 1 . (ii). For any ε > 0, B01 ⊂ H 1 ⊂ B−ε 1 ∞ . (iii). If s − n/p = t − n/q, then Bs+n/p ⊂ Bsp ⊂ Btq ⊂ Bs−n/p 0 2.3. The Space Gττ,k

0 In this section we deﬁne the spaces Gττ,k and we state some of their main properties.

Definition 2.3. Let 0 < τ0 ≤ τ, τ0 < 1/2, and let k > τ be an integer. The 0 space Gττ,k consists of all the functions ϕ ∈ C k (B) ∩ C(B) satisfying ϕGττ,k 0 :=

k

sup

j=0 z∈B

|∂ j ϕ(z)| ϕ(z) + sup < ∞. ωτ −j (z) z∈B (1 − |z|2 )τ0

The next lemma states some properties of these spaces. 0 Lemma 2.4. The spaces Gττ,k satisfy:

(i).

0 Gττ,k · Lpδ ⊂ Lpδ .

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0 0 ⊂ Gϑϑ,m , provided that ϑ0 ≤ τ0 , ϑ ≤ τ and m ≤ k. Gττ,k τ0 H ∩ Gτ,k = Bτ∞ .

0 Proof. (i) follows from the inclusion Gττ,k ⊂ L∞ (B). The embedding (ii) is a τ0 consequence of the definition of Gτ,k and the fact that (1 − |z|2 )s ≤ (1 − |z|2 )t 0 and ωs (z) ωt (z), if s > t. Note that if ϕ ∈ H ∩ Gττ,k , then ϕ˜ = 0 and therefore (iii) follows easily from Proposition 2.1. 0 In order to obtain multiplicative properties of the spaces Gττ,k , we ﬁrst state some properties of the function ωt .

Lemma 2.5. Let a, b ∈ R. Then (1 − |z|2 )a ωb (z) (1 − |z|2 )c , for every c ∈ R such that c < a and c ≤ a + b. Proof. Just note that 2 a

(1 − |z| ) ωb (z) =

(1 − |z|2 )min(a+b,a) , e (1 − |z|2 )a log 1−|z| 2,

if b = 0 if b = 0

(1 − |z|2 )c ,

for every c ∈ R such that c < a and c ≤ a + b.

Lemma 2.6. Let ϑ, τ > 0, k ∈ R and m ∈ Z such that m ≥ 0. Then ϑ,τ Sm,k :=

m

ωϑ−i ωτ +i−k ωϑ−m + ωτ −k .

i=0

Proof. We estimate the different products ωϑ−i ωτ +i−k as follows: • If i > k − τ , then ωϑ−i ωτ +i−k = ωϑ−i ωϑ−m , since ϑ − m ≤ ϑ − i. • If i < ϑ, then ωϑ−i ωτ +i−k = ωτ +i−k ωτ −k , since τ − k ≤ τ + i − k. • If i > ϑ and i ≤ k − τ , then ωϑ−i (z)ωτ +i−k (z) = (1 − |z|)ϑ−i ωτ +i−k (z) (1 − |z|)τ −k = ωτ −k (z), by Lemma 2.5, since τ − k < τ − k + ϑ = (ϑ − i) + (τ + i − k) and τ − k ≤ −i < −ϑ < 0. • If i = ϑ and i < k − τ , then ωϑ−i (z)ωτ +i−k (z) = (1 − |z|)τ +i−k ω0 (z) (1 − |z|)τ −k = ωτ −k (z), by Lemma 2.5, since τ − k < τ − k + ϑ = τ + i − k < 0. • If ϑ = i = k − τ , then ωϑ−i (z)ωτ +i−k (z) = ω0 (z)2 (1 − |z|)τ −k = ωτ −k (z), since τ − k = −ϑ < 0. Proposition 2.7. ϕψGϑ0 ϕGττ,k 0 ψ ϑ0 G ϑ,m

ϑ,m

provided that ϑ0 ≤ τ0 , ϑ ≤ τ and m ≤ k.

0 0 (ϕ ∈ Gττ,k , ψ ∈ Gτϑ,m ),

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Proof. If α ∈ Nn , |α| = l ≤ m, then ϑ,τ |∂α (ϕψ)| |∂β ϕ||∂γ ψ| ϕGττ,k 0 ψ ϑ0 S l,l G ϑ,m

β+γ=α

ϕGττ,k 0 ψ ϑ0 ωϑ−l , G ϑ,m

ϑ,τ Sl,l

since, by Lemma 2.6, ωϑ−l + ωτ −l ωϑ−l . On the other hand, + ϕ(z)|ψ(z)| ϕψ(z) ≤ |ϕ(z)|ψ(z) ϕ τ0 ψ Gτ,k

ϑ

0 Gϑ,m

(1 − |z|2 )ϑ0 ,

and the proof is complete.

0 Our next goal is to show the relationship between the spaces Gττ,k and both the classical Lipschitz–Zygmund spaces Λτ and the non-isotropic Lipschitz–Zygmund spaces Γτ .

Definition 2.8. If 0 < τ < 1, the classical Lipschitz–Zygmund space on S with respect to the Euclidean metric, Λτ = Λτ (S), consists of all the functions ϕ ∈ C(S) such that ϕΛτ := ϕ∞ + sup

ζ,η∈S ζ=η

|ϕ(ζ) − ϕ(η)| < ∞. |ζ − η|τ

If k is a positive integer and k < τ < k + 1, then Λτ = Λτ (S) consists of all the functions ϕ ∈ C k (S) such that ϕΛτ := ϕC k + ∂α ∂ β ϕΛτ −k (S) < ∞. |α|+|β|=k

When τ is a positive integer, Λτ is defined analogously by using second order differences, or by interpolation between Λk−1/2 and Λk+1/2 . The spaces Λτ (B) are deﬁned in a similar way. The main properties of the spaces Λτ can be found, for instance, in the expository paper [8]. In order to show the relation between these Lipschitz spaces and the 0 , we need the following result. spaces Gττ,k Theorem 2.9. [8, § 15] A continuous function ϕ on S is in Λτ if and only if, for some (every) integer k > τ , its harmonic extension Φ on B satisfies sup(1 − |z|2 )k−τ |dk Φ(z)| < ∞. z∈B

(2.3)

We recall that if (2.3) holds for some function ϕ ∈ C k (B), then ϕ ∈ Λτ (B) (see [8, Theorem 15.7]). 2 τ0 −1 give This fact and the estimate |dϕ(z)| ϕGττ,k 0 (1 − |z| ) 0 0 ⊂ Λτ0 (B), and, in particular, the restriction of Gττ,k Proposition 2.10. Gττ,k to S is contained in Λτ0 (S).

Moreover, we have: Proposition 2.11. Let ϕ ∈ Λτ (S), and let Φ be its harmonic extension on B.

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0 . If n = 1, then Φ ∈ Gττ,k If n > 1 and τ > 1/2, then τ −1/2 • Φ ∈ Gτ,k , when 1/2 < τ < 1. 0 • Φ ∈ Gττ,k , for any 0 < τ0 < 1/2, when τ ≥ 1.

Corollary 2.12. If either n = 1 or n > 1 and τ > 1/2, then every function 0 , for some τ0 > 0 and for ϕ ∈ Λτ is the restriction of a function Φ ∈ Gττ,k any integer k > τ . If n > 1, then we also consider the Lipschitz–Zygmund space on S with respect to the pseudodistance d(ζ, η) = |1 − ζη|, which is denoted by Γτ = Γτ (S). If 0 < τ < 1/2, this space is deﬁned similarly to Λτ , but replacing the Euclidean distance |ζ − η| by d(ζ, η). For values τ ≥ 1/2 the definition is given in terms of Lipschitz conditions of certain complex tangential derivatives (see [4, pp. 670-1] and the references therein for the precise definitions and main properties). We recall that if f ∈ H(B) has boundary values f ∗ , then f ∗ ∈ Λτ if and only if f ∗ ∈ Γτ (see [13; 12, §6.4; 10, §8.8] and the references therein; see also [4, pp. 670-1]). Similarly to what happens in the holomorphic case, the complex tangential derivatives of the functions in the space Γτ are more regular, in the sense that Dij ϕ ∈ Γτ −1/2 for i, j = 1, . . . , n. 0 , we will use the In order to state the relation between Γτ and Gττ,k following characterization of Γτ , for 0 < τ < n, in terms of its invariant harmonic extensions. Theorem 2.13. If 0 < τ < n, a continuous function ϕ on S is in Γτ if and only if its invariant harmonic extension Φ on B satisfies (2.3), for some (every) integer k > τ . It is also well-known the estimate |∂α ∂ β Di0 j0 · · · Dim jm Di0 j0 · · · Dil jl Φ(z)| (1 − |z|2 )τ −|α|−|β|−m/2−l/2 , which holds for any α, β ∈ Nn and m, l ∈ N so that τ −|α|−|β|−m/2−l/2 < 0. See [4, pp. 670-1] for a more complete list of characterizations of these spaces. The characterization given in Theorem 2.13 fails to be true when τ ≥ n (see [9, Chapter 6] for more details). As a consequence of Theorem 2.13, we have: Proposition 2.14. If n > 1, 0 < τ < n and ϕ ∈ Γτ , then, for any integer k > τ , its invariant harmonic extension Φ satisfies that: • •

Φ ∈ Gττ,k , when 0 < τ < 1/2. 0 Φ ∈ Gττ,k , for any 0 < τ0 < 1/2, when τ ≥ 1/2. Since Γτ is a subspace of Λτ , Proposition 2.14 and Corollary 2.12 give:

Corollary 2.15. Every function ϕ ∈ Γτ , τ > 0, is the restriction of a function 0 , for some τ0 > 0 and for any integer k > τ . Φ ∈ Gττ,k

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2.4. Representation Formulas and Estimates In this subsection we recall some well-known results on the integral representation formulas obtained in [6]. We begin by introducing the following nonnegative integral kernels and their corresponding integral operators. Definition 2.16. Let N, M, L ∈ R such that N > 0 and L < n. Then N (w, z) := KM,L

(1 − |w|2 )N −1 |1 − wz|M D(w, z)L

(z, w ∈ B, z = w),

where D(w, z) := |1 − wz|2 − (1 − |w|2 )(1 − |z|2 ). The associated integral N : operator is also denoted by KM,L N N (ψ)(z) := KM,L (w, z)ψ(w) dν(w). KM,L B

Note that D(w, z) = |(w − z)z|2 + (1 − |z|2 )|w − z|2 , so, for every z ∈ B such that 1 − |z|2 ≥ δ > 0, we have that

|w − z|−2L , if |w − z| < (1 − |z|)/2, N (w, z) KM,L (2.4) (1 − |w|2 )N −1 , if |w − z| ≥ (1 − |z|)/2. N This estimate ensures that KM,L (w, z) ∈ L1 (dν(w)), for every z ∈ B.

Theorem 2.17. [6] Let N > 0. Then every function ψ ∈ C 1 (B) decomposes as ψ = P N (ψ) + KN (∂ψ), where KN (∂ψ)(z) :=

(2.5)

KN (w, z) ∧ ∂ψ(w)

B N

and K (w, z) is an (n, n − 1)-form (on w) of class C ∞ on B × B outside its diagonal. In particular, if ψ is holomorphic on B then ψ = P N (ψ). Moreover, KN (w, z) satisfies the estimate N |KN (w, z) ∧ ∂ψ(w)| KN −n+1,n−1/2 (w, z)ψ(w),

(2.6)

for any ψ ∈ C 1 (B), where ψ is defined as in (2.1). Then, it is clear that N |P N (ψ)| Kn+N,0 (|ψ|)

N and |KN (∂ψ)| KN −n+1,n−1/2 (ψ).

(2.7)

Remark 2.18. The representation formula (2.5) will be applied in a more 0 general setting: namely, to functions ψ = ϕf , where ϕ ∈ Gττ,k and either 1 1 f ∈ B−N , for N > 0, or f ∈ H , for N = 0. The validity of the formula for this class of functions is obtained by applying the dominated convergence theorem and Theorem 2.17 to the functions ψr (z) = ψ(rz).

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Lemma 2.19. [6, Lemma I.1] N KM,L (w, z) dν(w) ωt (z), B N where t := n + N − M − 2L is the so-called type of the kernel KM,L . N Observe that from the above estimate we deduce that if KM,L is a ker2 −δ N nel of type 0, 0 < δ < N and ψ(z) = (1 − |z| ) , then KM,L (ψ) ψ. As a consequence of that result and Schur’s lemma we have: N N Lemma 2.20. If KM,L is a kernel of type 0 and 0 < δ < N , then KM,L maps p boundedly Lδ to itself.

By applying H¨ older’s inequality we deduce the following pointwise estiN , which will be often used in the forthcoming mate of the operators KM,L sections. Lemma 2.21. Let N ≥ 0, τ > 0, p ≥ 1 and 0 < ε < N + τ . Then (N +τ −ε)p N +τ p p (KN −n+1,n−1/2 (|ψ|)) KN p−n+1,n−1/2 (|ψ| ). In the next lemma we state some well-known differentiation formulas for both operators P N and KN (see, for instance, [5, § 5]). Lemma 2.22. Let N ≥ 0, α ∈ Nn and k = |α|. (i) If ψ ∈ C k (B), then ∂α P N (ψ) = P N +k (∂α ψ). (ii) If ψ ∈ C k+1 (B), then ∂α KN (∂ψ) = KN +k (∂∂α ψ). Proof. For the sake of completeness, we give a brief sketch of the proof. For ∂ P N +1 (w, z) and N > 0, (i) follows from the equation ∂z∂ j P N (w, z) = − ∂w j integration by parts, while (ii) is just a direct consequence of the representation formula (2.5) and (i). The case N = 0 is deduced from the corresponding formulas for N > 0 by taking N → 0+ . It is also possible to prove this result by differentiation under the integral, using the identity n n ζj dσ(ζ) = (−1)j−1 cn dζi dζ l i=1 i=j

and Stokes theorem.

l=1

Remark 2.23. The above differentiation formulas will be applied to functions 0 ψ = ϕf , where ϕ ∈ Gττ,k and f ∈ Bs∞ , for s > 0. The validity of the formulas in this more general setting can be shown by applying Lemma 2.22 to ψr (z) = ψ(rz) and the dominated convergence theorem. Now we state some continuity properties of the integral operator P N . Proposition 2.24. (i). If 0 < δ < N and 1 ≤ p < ∞, then P N maps conp tinuously Lpδ to B−δ . N 0 to Bτ∞ . (ii). If N ≥ 0, then P maps continuously Gττ,k (iii). If N = 0, then P maps continuously Λτ to Bτ∞ .

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Proof. The proof of (i) can be found in [14, Theorem 2.10]. The proof of (ii) 0 satisﬁes reduces to show that every ψ ∈ Gττ,k N +τ 2 τ −k |∂ k P N (ψ)(z)| = |P N +k (∂ k ψ)(z)| Kn+N , +k,0 (1)(z) (1 − |z| )

which follows from Lemmas 2.22 and 2.19. Assertion (iii) for 0 < τ < 1 can be found in [12, § 6.4]. The case τ ≥ 1 follows from (ii) and Corollary 2.12. 0 Proposition 2.25. If ϕ ∈ Gττ,k , then:

(i). (ii). (iii).

1 . For N > 0, TϕN is bounded on B−N For N = 0, Tϕ is bounded on H 1 . For N ≥ 0, TϕN is bounded on Bτ∞ .

Proof. By Proposition 2.10, ϕ ∈ Λτ0 (B), so |ϕ(w) − ϕ(z)| |w − z|τ0 N |1 − wz|τ0 /2 . Then, since TϕN (f )(z) = Tϕ−ϕ(z) (f )(z) + ϕ(z)f (z), we have that N |TϕN (f )| Kn+N −τ0 /2,0 (|f |) + |f |. N By Fubini’s Theorem and Lemma 2.19, Kn+N −τ0 /2,0 (|f |)L1N f L1N . N Therefore Tϕ (f )L1N f L1N , and the proof of (i) is complete. The proof of (ii) follows as a consequence of Proposition 2.10 and [12, Theorem 6.5.4], and also arguing as in case (i). Finally, (iii) follows from Lemma 2.4 (iii), Proposition 2.7 and Proposition 2.24 (ii). 0 3. Toeplitz Operators with Symbols in Gττ,k

In this section we state a general theorem from which we will deduce the results stated in the introduction. The proof of this general theorem will be postponed to the next section. 1 0 and f ∈ B−N , N ≥ 0, satisfy Observe that if the functions ϕ ∈ Gττ,k the equation TϕN (f ) = h ∈ Bτ∞ , then, taking into account Remark 2.18, formula (2.5) gives that ϕf = KN (f ∂ϕ) + h.

(3.1)

Note that, by (2.6), N +τ0 N |KN (f ∂ϕ)| KN ˜ KN −n+1,n−1/2 (|f |ϕ) −n+1,n−1/2 (|f |) 1 , for and therefore by (2.4), KN (f ∂ϕ) is pointwise deﬁned even if f ∈ B−N 0 1 1 some N < N0 < N + τ0 . This fact and the inclusion H ⊂ B−N0 for any N0 > 0, allow us to unify the proofs of Theorems 1.4 and 1.5, using the following result: 0 be such that 0 ∈ / ϕ(S). If 0 < Theorem 3.1. Let N ≥ 0 and let ϕ ∈ Gττ,k 1 ∞ N0 < N + τ0 , f ∈ B−N0 and h ∈ Bτ satisfy (3.1), then f ∈ Bτ∞ and 1 + hBτ∞ . f Bτ∞ f B−N 0

Now we easily deduce Theorems 1.4 and 1.5 all at once: 0 Theorem 3.2. Let ϕ ∈ Gττ,k be such that 0 ∈ / ϕ(S).

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1 and TϕN (f ) ∈ Bτ∞ , for some N > 0, then f ∈ Bτ∞ and If f ∈ B−N 1 f Bτ∞ f B−N + hBτ∞ . If f ∈ H 1 and Tϕ (f ) ∈ Bτ∞ , then f ∈ Bτ∞ and f Bτ∞ f H 1 + hBτ∞ .

Proof. As we pointed out at the beginning of the section, if TϕN (f ) = h ∈ Bτ∞ , N ≥ 0, then ϕ and f satisfy (3.1). Therefore (i) directly follows from 1 , for every t > 0, Theorem 3.1 (case N > 0). By Proposition 2.2, H 1 ⊂ B−t 1 1 and, in particular, H ⊂ B−N0 , for every 0 < N0 < τ0 , so (ii) also follows from Theorem 3.1 (case N = 0). As an immediate consequence of Theorem 3.2 we obtain the following corollaries. Corollary 3.3. Let τ > 0 and assume that ϕ satisfy that 0 ∈ / ϕ(S), and one of the following conditions: (i). (ii). (iii).

n = 1 and ϕ ∈ Λτ . n > 1, τ > 12 and ϕ ∈ Λτ . n > 1, τ ≤ 12 and ϕ ∈ Γτ .

If f ∈ H 1 and Tϕ (f ) ∈ Bτ∞ , then f ∈ Bτ∞ . Proof. This is a consequence of Theorem 3.2 and Corollaries 2.12 and 2.15. 0 be such that 0 ∈ / ϕ(S). Corollary 3.4. Let ϕ ∈ Gττ,k

(i). (ii).

1 0 If N > 0 and f ∈ B−N satisfies that ϕf ∈ Gττ,k +ker P N , then f ∈ Bτ∞ . τ 0 If f ∈ H 1 satisfies that ϕf ∈ Gτ,k + ker P, then f ∈ Bτ∞ .

Proof. This is a consequence of Theorem 3.2 and Proposition 2.24(ii).

0 , then ker(TϕN − λI) ⊂ Bτ∞ , for any λ ∈ C \ ϕ(S) Corollary 3.5. If ϕ ∈ Gττ,k / ϕ(S). and N ≥ 0. In particular, ker TϕN ⊂ Bτ∞ , whenever 0 ∈

N Proof. Since TϕN − λI = Tϕ−λ , it directly follows from Theorem 3.2.

Remark 3.6. If the condition 0 ∈ / ϕ(S) is omitted, then ker TϕN is not neces∞ sarily contained in Bτ . For n > 1, this result follows by taking ϕ(z) = z 1 , 1 , if N > 0, (H 1 , if and observing that ker TϕN contains any function in B−N N = 0), which does not depend on the first variable. For n = 1 we may consider the symbol ϕ(z) = z m+1 (1 − z)m+α and the function f (z) = (1 − z)−α , 0 where 0 < α < 1 and m ∈ N such that m + α ≥ τ , which satisfy ϕ ∈ Gττ,k and f ∈ ker TϕN \ Bτ∞ . Now we extend Corollary 3.3. Theorem 3.7. Let τ > 0 and let ϕ ∈ Λτ be a non-vanishing function on S. If f ∈ H 1 and Tϕ (f ) ∈ Bτ∞ , then f ∈ Bτ∞ .

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Proof. If τ > 1/2, the result is just a consequence of Corollary 2.12 and part (ii) of Theorem 3.2. Now assume that τ ≤ 1/2. Since |w − z|2 ≤ 2|1 − wz|, we have that Λτ ⊂ Γτ /2 . And then Corollary 2.15 and part (ii) of Theorem 3.2 show that f ∈ Bτ∞/2 . Thus |∂f (z)| (1 − |z|2 )τ /2−1 , but we want to prove that |∂f (z)| (1 − |z|2 )τ −1 , or equivalently |Φ(z)∂f (z)| (1 − |z|2 )τ −1 , Φ being the harmonic extension of ϕ to B. (Recall that, since 0 ∈ ϕ(S), there is 0 < r < 1 so that |Φ(z)| 1 for r ≤ |z| ≤ 1.) In order to show the estimate note that |dΦ(z)| (1 − |z|2 )τ −1 , which implies that |Φ(z) − Φ(w)| |z − w|τ |1 − wz|τ /2 , for z, w ∈ B. On the 1 so ∂j f = P 1 (∂j f ) and therefore other hand, since f ∈ Bτ∞/2 , ∂j f ∈ B−1 Φ(z)∂j f (z) = P 1 ((Φ(z) − Φ)∂j f )(z) + P 1 (∂j (Φf ))(z) − P 1 (f ∂j Φ)(z). By Lemma 2.22, P 1 (∂j (Φf )) = ∂j Tϕ f . Hence τ /2

τ (1)(z), |Φ(z)∂j f (z)| Kn+1−τ /2,0 (1)(z) + (1 − |z|2 )τ −1 + Kn+1,0

and then Lemma 2.19 shows that |Φ(z)∂j f (z)| (1 − |z|2 )τ −1 .

Since P maps Λτ to Bτ∞ , we deduce Corollary 3.8. If f ∈ H 1 and ϕ ∈ Λτ satisfy 0 ∈ / ϕ(S) and ϕf ∈ Λτ + ker P, then f ∈ Bτ∞ . Now we obtain Corollary 1.3: Corollary 3.9. If f, g ∈ H 1 \ {0} satisfy ϕ = g/f ∈ Λτ and 0 ∈ / ϕ(S), then f, g ∈ Bτ∞ . In particular, if f ∈ H 1 \ {0} and its argument function f /f is in Λτ , then f ∈ Bτ∞ . Proof. Since Tϕ (f ) = P(g) = g(0) ∈ Bτ∞ , Theorem 3.7 shows that f ∈ Bτ∞ . Therefore g = f ϕ ∈ Λτ and hence g ∈ Bτ∞ .

4. Proof of Theorem 3.1 This section is devoted to the proof of Theorem 3.1. It is splitted into three steps composed of several lemmas that will give succesive improvements on the regularity of the solutions to the equation TϕN (f ) = h. First we will show 1 1 that any solution f to (3.1) which is in B−N is in fact in any B−t , t > 0. 0 ∞ Then we will obtain that the solution is in B−t for any t > 0, and ﬁnally we will deduce that it is in Bτ∞ . Throughout this section we will assume that ϕ and h satisfy the hypotheses of Theorem 3.1. Since |ϕ(ζ)| ≥ ρ > 0 on S, we can choose r0 such that |ϕ(z)| ≥ ρ/2 > 0 on the corona C = { z ∈ B : r0 ≤ |z| ≤ 1 }. Let χ be a real C ∞ -function on Cn supported on the corona C0 = { z ∈ B : r0 ≤ |z| ≤ 1 + r0 }, such that 0 ≤ χ ≤ 1 and χ ≡ 1 on a neighborhood of S. Then (3.1) shows that χ χ (4.1) f = KN (f ∂ϕ) + h + (1 − χ)f. ϕ ϕ

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The function (1 − χ)f is a C ∞ function with compact support on B. χ χ 0 0 It is easy to prove that ∈ Gττ,k , and so h ∈ Gττ,k , by Proposition 2.7. ϕ ϕ χ 0 Therefore (1 − χ)f + h ∈ Gττ,k and ϕ χ 1 f B−N (4.2) + hBτ∞ (1 − χ)f + hGττ,k 0 ≤ Cϕ 0 ϕ Hence, in order to prove that f ∈ Bτ∞ , we have just to show that χ N 0 K (f ∂ϕ) ∈ Gττ,k . ϕ 1 , for any t > 0. Step 1. The ﬁrst couple of lemmas will show that f ∈ B−t 1 Lemma 4.1. Let f ∈ B−s , for some 0 < s < N + τ0 , and assume it satisfies (3.1).

(i). (ii).

1 If s ≤ τ0 then f ∈ B−t , for every t > 0. 1 If s > τ0 then f ∈ B−(s−τ . 0)

Proof. First note that (2.7) shows that N |KN (f ∂ϕ)| = |KN (∂(f ϕ))| KN −n+1,n− 1 (|f |ϕ), 2

and so N +τ0 |KN (f ∂ϕ)| ϕGττ,k 0 K N −n+1,n−1/2 (|f |).

(4.3)

By integrating and using Fubini’s Theorem, for any t > 0 we have that KN (f ∂ϕ)L1t ϕGττ,k |f (w)|gt (w) dν(w), 0 B

where gt (w) = (1 − |w|2 )N +τ0 −1

t KN −n+1,n−1/2 (z, w) dν(z).

B

Now Lemmas 2.19 and 2.5 show that gt (w) (1 − |w|2 )N +τ0 −1 ωt−N (w) (1 − |w|2 )s−1 ,

(4.4)

provided that s ≤ t + τ0 . (recall that s < N + τ0 ). Therefore, if s ≤ t + τ0 , KN (f ∂ϕ)L1t ϕGττ,k 0 f L1 , s

(4.5)

1 . We conclude that: so, by (4.1) and (4.2), f ∈ B−t

(i) (ii)

If s ≤ τ0 then s ≤ t + τ0 and (4.4) holds for every t > 0, and hence 1 , for every t > 0. f ∈ B−t 1 . If s > τ0 then (4.4) holds for t = s − τ0 , and consequently f ∈ B−(s−τ 0)

1 1 Lemma 4.2. If f ∈ B−N satisfies (3.1) then f ∈ B−t , for every t > 0. 0

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Proof. For N0 ≤ τ0 the result follows directly from Lemma 4.1, (i). So assume that N0 > τ0 . Let k be the greatest positive integer such that kτ0 < N0 . Then 1 , Lemma 4.1 (ii) implies that f ∈ kτ0 < N0 ≤ (k + 1)τ0 . Now, since f ∈ B−N 0 1 1 1 B−(N0 −τ0 ) , so f ∈ B−(N0 −2τ0 ) , . . . , so f ∈ B−(N . But 0 < N0 −kτ0 ≤ τ0 0 −kτ0 ) 1 and therefore Lemma 4.1 (i) shows that f ∈ B−t , for every t > 0. Remark 4.3. Observe that the above arguments, (4.1) and (4.5) give in par1 1 ticular the estimate f B−t f B−N + hBτ∞ . 0

∞ Step 2. The next couple of lemmas will show that the function f is in B−t , for any t > 0. We follow the ideas in [7]. p , for some 1 ≤ p < ∞ and for every s > 0. If Lemma 4.4. Let f ∈ B−s q f satisfies (3.1) then f ∈ B−s , for every s > 0 and for every q such that τ0 1 1 p < q < ∞ and p − n < q . p p ⊂ B−t . Consequently, we only Proof. If −t < −s < 0, then the space B−s have to prove the lemma, for s sufﬁciently small. Let p < q < ∞ and 0 < ε < N + τ0 . Assume f satisﬁes (3.1). Then, as we have shown in the proof of Lemma 4.1, (4.3) holds, and so Lemma 2.21 gives (N +τ −ε)q

0 q |KN (f ∂ϕ)|q ϕqGτ0 KN q−n+1,n− 1 (|f | ). τ,k

By Proposition 2.2(iii), which implies that

p B−s

2

n

−s− p ∞ p (1 − |w|2 ) ⊂ B−s−n/p and |f (w)| f B−s , n

|f (w)|q = |f (w)|q−p |f (w)|p f q−p (1 − |w|2 )(p−q)(s+ p ) |f (w)|p , Bp −s

and, by integrating, we get (N +τ −ε)q

N (ε,s)

q−p 0 q (|f |p ), KN q−n+1,n− 1 (|f | ) f B p KM,L −s

2

where N (ε, s) = (N + τ0 − ε)q + (p − q)(s + np ) = sp + (N − s)q + nq( τ0n−ε − 1 1 p + q ), M = N q − n + 1 and L = n − 1/2. Therefore 1− p

q KN (f ∂ϕ)Lqs ϕGττ,k 0 f p Iε,s , B −s

N (ε,s)

where Iε,s = KM,L (|f |p )L1sq .

q Thus we only have to prove that Iε,s f pB p , for ε, s > 0 small enough and −s

− τn0 < 1q , because then the previous estimate shows that KN (f ∂ϕ)Lqs q and hence, by (4.1) and (4.2), we conclude that f ∈ B−s . In ϕGττ,k 0 f B p −s q order to estimate Iε,s , ﬁrst apply Fubini’s Theorem to get ⎛ ⎞ sq q Iε,s = |f (w)|p (1 − |w|2 )N (ε,s)−1 ⎝ KM,L (z, w) dν(z)⎠ dν(w), 1 p

B

B

and since n + sq − M − 2L = (s − N )q, then apply Lemma 2.19 to obtain q Iε,s |f (w)|p (1 − |w|2 )N (ε,s)−1 ω(s−N )q (w) dν(w). B

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Now we consider two cases: Case N = 0. Then (s − N )q = sq > 0 so q |f (w)|p (1 − |w|2 )N (ε,s)−1 dν(w) ≤ f pB p , Iε,s −s

B

provided that N (ε, s) > sp, which holds for ε, s > 0 small enough and p1 − 1q < τ0 n , since lim (N (ε, s) − sp) = nq{ τn0 − ( p1 − 1q )} > 0.

ε,s0

Case N > 0. Let 0 < s < N . Then (s − N )q < 0 and so q Iε,s |f (w)|p (1 − |w|2 )N (ε,s)+(s−N )q−1 dν(w) ≤ f pB p , −s

B

provided that N (ε, s) + (s − N )q > sp, which holds for ε > 0 small enough and p1 − 1q < τn0 , since N (ε, s) + (s − N )q − sp = nq{ τ0n−ε − ( p1 − 1q )} > 0.

And the proof is complete. Lemma 4.5. Let f ∈ for every t > 0.

1 B−s ,

for every s > 0. If f satisfies (3.1) then f ∈

∞ B−t ,

τ0 τ0 Proof. Let k be the greatest positive integer such that k 2n < 1. Then k 2n < τ0 1 ≤ (k + 1) 2n . Let

pj =

1 τ0 1 − j 2n

(j = 0, . . . , k).

τ0 1 1 − 2n > pj−1 − τn0 , for j = 1, . . . , k. Then pj ≥ 1, for j = 0, . . . , k, and p1j = pj−1 p0 Now, since f ∈ B−s , for every s > 0, and f satisﬁes (3.1), Lemma 4.4 p1 p2 pk shows that f ∈ B−s so f ∈ B−s , . . . , so f ∈ B−s , for every s > 0. But τ0 τ0 1 pk − 2n = 1 − (k + 1) 2n ≤ 0 and therefore Lemma 4.4 once again shows q q ∞ , for every q > pk and every s > 0. Since B−s ⊂ B−s− that f ∈ B−s n , by q ∞ Proposition 2.2 (iii), we conclude that f ∈ B−t , for every t > 0.

Remark 4.6. Observe that the above arguments and (4.1) give the estimate ∞ f 1 f B−t B−t + hBτ∞ . Step 3. In what follows we will ﬁnally deduce that f ∈ Bτ∞ . ∞ Lemma 4.7. Let f ∈ B−t , for every t > 0. If f satisfies (3.1) then f ∈ H ∞ .

Proof. Since f satisﬁes (3.1), (4.3) holds, as we have shown in the proof of Lemma 4.1. But N +τ0 N +τ0 −t ∞ K (|f |) f B−t (1) KN −n+1,n− 1 N −n+1,n− 1 2

and, by Lemma 2.19, N

2

N +τ0 −t KN (1) −n+1,n− 12

fore K (f ∂ϕ)∞ ϕ

τ0 Gτ,k

f

∞ B−t

ωτ0 −t 1, for any 0 < t < τ0 . There-

, and, by (3.1) and (4.2) f ∈ H ∞ .

In order to prove that f ∈ Bτ∞ , we will use the following formula.

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0 and f ∈ H ∞ . Then Lemma 4.8. Let N ≥ 0, ϕ ∈ Gττ,k ϕ∂α f = ∂α P N (ϕf ) − cα,β P N +k (∂γ ϕ∂β f ) + KN +k (∂ϕ∂α f ), (4.6)

β+γ=α |β|
for every α ∈ Nn , where k = |α| and cα,β = α!/(β!γ!). Proof. First assume that ϕ ∈ C k (B) and f ∈ H(B). By Theorem 2.17 we have that ϕ∂α f = P N +k (ϕ∂α f ) + KN +k (∂ϕ∂α f ). Moreover, ϕ∂α f = ∂α (ϕf ) − cα,β ∂γ ϕ∂β f, β+γ=α |β|
and Lemma 2.22 shows that P N +k (∂α (ϕf )) = ∂α P N (ϕf ). Hence we obtain (4.6). By a standard approximation argument (based on the dominated convergence theorem), we deduce the general case from the regular case just proved above. Lemma 4.9. (i). If f ∈ H ∞ satisfies (3.1), then for every 0 < t ≤ τ0 , f ∈ Bt∞ . (ii). Let f ∈ Bs∞ , for some 0 < s < τ . If f satisfies (3.1), then for every 0 < t ≤ min(τ, s + τ0 ), f ∈ Bt∞ . Proof. Let f and t be as in either (i) or (ii), and assume f satisﬁes (3.1). Let k ∈ Z and α ∈ Nn such that |α| = k > τ . We want to prove that |∂α f (z)| (1 − |z|2 )t−k . Since |∂α f | ≤ χ/ϕ∞ |ϕ∂α f | + (1 − χ)|∂α f |,

(4.7)

we only need to estimate |ϕ∂α f |. Lemma 4.8, (3.1) and (2.7) show that N +k N +k |ϕ∂α f | |∂α h| + Kn+N +k,0 (Fα ) + KN +k−n+1,n− 12 (|∂α f |ϕ), where Fα = |∂β f | |∂γ ϕ|. Note that we only need to prove that

(4.8)

β+γ=α |β|
|∂α f (w)|ϕ(w) (1 − |w|2 )t−k

and Fα (w) (1 − |w|2 )t−k ,

because then (4.8) and Lemma 2.19 show that N +t N +t |(ϕ∂α f )(z)| |∂α h(z)| + KN (1)(z) + Kn+N +k,0 (1)(z) +k−n+1,n− 1 2

ωt−k (z) + ωτ −k (z) = (1 − |z|2 )t−k , and therefore, by (4.7), we conclude that |∂α f (z)| (1 − |z|2 )t−k . (i) Let f ∈ H ∞ . Then |∂β f (w)| f ∞ (1 − |w|2 )−|β| , for every multiindex β. So 2 τ0 −k 2 t−k ϕGττ,k ϕGττ,k , |∂α f (w)|ϕ(w) 0 f ∞ (1 − |w| ) 0 f ∞ (1 − |w| )

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for every t ∈ R such that t ≤ τ0 . Next Lemma 2.5 shows that Fα (w) ϕGττ,k 0 f ∞

k−1

(1 − |w|2 )−i ωτ −k+i (z)

i=0

2 t−k ϕGττ,k , 0 f ∞ (1 − |w| )

for every t ∈ R such that t < 1 and t ≤ τ . (ii) Let f ∈ Bs∞ , for some 0 < s < τ . Then |∂β f (z)| f Bs∞ ωs−|β| (z), for every multiindex β, so Lemma 2.5 gives that 2 s+τ0 −k f Bs∞ ϕGττ,k |∂α f (w)|ϕ(w) 0 (1 − |w| ) 2 t−k ≤ f Bs∞ ϕGττ,k , 0 (1 − |w| )

for every t ∈ R such that t ≤ s + τ0 < s + 1, and Lemma 2.6 shows that s,τ Fα (w) ϕGττ,k 0 f B ∞ S k−1,k (w) s 2 τ −k ϕGττ,k } 0 f B ∞ {ωs+1−k (w) + (1 − |w| ) s 2 t−k ϕGττ,k , 0 f B ∞ (1 − |w| ) s

for every t ∈ R such that t ≤ s + 1 and t ≤ τ .

Lemma 4.10. If f ∈ H ∞ satisfies (3.1) then f ∈ Bτ∞ . Proof. Let f ∈ H ∞ . If τ = τ0 there is nothing to prove by Lemma 4.9 (i). So assume that τ0 < τ , and let k ≥ 1 be the greatest integer such that kτ0 < τ . Since f ∈ H ∞ , Lemma 4.9 (i) shows that f ∈ Bτ∞0 , and then Lemma 4.9 (ii) ∞ ∞ ∞ , so f ∈ B3τ , . . ., so f ∈ Bkτ , and hence f ∈ Bτ∞ . implies that f ∈ B2τ 0 0 0 Remark 4.11. Observe that the above arguments and (4.1) give the estimate ∞ + hB ∞ . f Bτ∞ f B−t τ Remark 4.12. The Remarks 4.3, 4.6 and 4.11 show that 1 + hBτ∞ . f Bτ∞ f B−N 0

Note that the opposite estimate is always fulﬁlled. This follows from the con1 tinuous embedding Bτ∞ ⊂ B−N , and the estimate hBτ∞ ≤ TϕN f Bτ∞ . 0 Therefore 1 + hBτ∞ . f Bτ∞ ≈ f B−N

Acknowledgments The authors would like to thank the referee for his careful reading of the ﬁrst version of this paper and his valuable suggestions and comments to improve the ﬁnal version.

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References [1] Ahern, P., Bruna, J.: Maximal and area integral characterizations of Hardy– Sobolev spaces in the unit ball of Cn . Rev. Mat. Iberoamericana 4(1), 123–153 (1988) [2] Beatrous, F.: Estimates for derivatives of holmorphic functions in pseudoconvex domains. Math. Z 191, 91–116 (1986) [3] Beatrous, F., Burbea, J.: Holomorphic Sobolev spaces on the ball. Dissertationes Math. (Rozprawy Mat.) 276, 60 (1989) [4] Bonami, A., Bruna, J., Grelier, S.: On Hardy, BMO and Lipschitz spaces of invariantly harmonic functions in the unit ball. Proc. Lond. Math. Soc. (3) 77(3), 665–696 (1998) [5] Bruna, J., Ortega, J.M.: Interpolation by holomorphic functions smooth to the boundary in the unit ball of Cn . Math. Ann. 274(4), 527–575 (1986) [6] Charpentier, P.H.: Formules explicites pour les solutions minimales de l’equa¯ = f dans la boule et dans le polydisque de Cn . Ann. Inst. Fourier tion ∂u Grenoble 30, 121–154 (1980) [7] Dyakonov, K.M.: Toeplitz operators and arguments of analytic functions. Math. Ann. 344(2), 353–380 (2009) [8] Krantz, S.G.: Lipschitz spaces, smoothness of functions, and approximation theory. Exposition. Math. 1(3), 193–260 (1983) [9] Krantz, S.G.: Partial differential equations and complex analysis. In: Studies in Advanced Mathematics. CRC Press, Boca Raton (1992) [10] Krantz, S.G.: Function theory of several complex variables. Reprint of the 1992 edition. AMS Chelsea Publishing, Providence (2001) [11] Ortega, J.M., F` abrega, J.: Pointwise multipliers and decomposition theorems in analytic Besov spaces. Math. Z 235, 53–81 (2000) [12] Rudin, W.: Function Theory in the Unit Ball of Cn . Springer Verlag, New York (1980) [13] Stein, E.M.: Singular integrals and estimates for the Cauchy–Riemann equations. Bull. AMS 79, 440–445 (1973) [14] Zhu, K.: Spaces of holomorphic functions in the unit ball. In: Graduate Texts in Mathematics, vol. 226. Springer, New York (2005) abrega and Daniel Pascuas Carme Cascante (B), Joan F` Dept. Matem` atica Aplicada i An` alisi Universitat de Barcelona Gran Via 585, 08071 Barcelona, Spain e-mail: [email protected]; joan− [email protected]; daniel− [email protected] Received: June 23, 2010. Revised: January 10, 2011.

Integr. Equ. Oper. Theory 69 (2011), 393–404 DOI 10.1007/s00020-010-1845-0 Published online November 23, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

A Classification of Positive Solutions of Some Integral Systems Feiyao Ma, Xiaotao Huang and Lihe Wang Abstract. In this paper, we investigate the symmetry of some nonlinear integral systems with Riesz potentials. With the method of moving planes, we study the symmetry of positive solutions in two cases, on Rn and on bounded domains. These results can be extended to integral equations with Bessel potentials. Mathematics Subject Classification (2010). Primary 45K05, 45P05; Secondary 35J67. Keywords. Riesz potentials and Bessel potentials, symmetry and monotonicity, integral equation systems, moving planes.

1. Introduction In this paper, we study the symmetry of the following integral system: u(x) = Rn |x − y|α−n ev(y) dy, (1.1) v(x) = Rn |x − y|β−n eu(y) dy, where 0 < α, β < n. Obviously, the single integral equation |x − y|α−n eu(y) dy u(x) =

(1.2)

Rn

implies the following differential equation (−Δ)α/2 u = eu(y) . It is well-known that the critical case (α = n) (−Δ)n/2 u = (n − 1)!enu(y) ,

x ∈ Rn ,

is important in many ﬁelds, e.g., the prescribing scalar curvature problem, the bacterial aggregation modeling and others. This kind of critical case was This work was completed with the support NSFC: 10771166.

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studied by Lin [13] and Li [15]. In [13], Lin proved in the special cases n = 2, 4 that if u ∈ L1 (Rn ) then u is radially symmetric about some point x0 in Rn . In this paper, we investigate system (1.1) for all 0 < α, β < n. Our main results are as follows. Theorem 1.1. Assume {u, v} is a pair of positive solutions of (1.1). If eu(y) ∈ Lr (Rn ),

ev(y) ∈ Ls (Rn ),

where r, s > 1, n/s + n/r = α + β. Then u and v are radially symmetric and monotone decreasing about some point x0 in Rn . Recently, Li et al. [14] studied the symmetry of domains and solutions of integral equations. In this paper, we also consider the symmetry of both the domain and solutions of our system. Theorem 1.2. Suppose Ω ⊂ Rn is a bounded C 1 domain, {u, v} is a pair of positive solutions of ⎧ ⎨ u(x) = Ω |x − y|α−n ev(y) dy, (1.3) v(x) = Ω |x − y|β−n eu(y) dy, ⎩ u = A, v = D on ∂Ω, where α, β, A, D are constants satisfying 0 < α, β < n, A, D > 0. If eu(y) ∈ Lr (Ω),

ev(y) ∈ Ls (Ω),

where r, s > 1, n/s + n/r = α + β, then Ω is a ball, u and v must be radially symmetric and monotone decreasing with respect to the radius. The results for single equation are also correct. It is easy to verify that Corollary 1.3. (i) Assume u is a positive solution of (1.2). If eu ∈ Ln/α (Rn ), then u is radially symmetric and monotone decreasing about some point in Rn . (ii) Assume Ω ⊂ Rn is a bounded C 1 domain, u is a positive solution of u(x) = Ω |x − y|α−n eu(y) dy, u = D, on ∂Ω, where α, D are constants satisfying 0 < α < n, D > 0. If eu(y) ∈ Ln/α (Ω), then Ω is a ball, u is radially symmetric and monotone decreasing with respect to the radius. Remark 1.4. From [13] we know that if u ∈ Ln/α (Rn ) then positive solution u(x) must be radially symmetric in the special cases α = n = 2, 4. For the subcritical cases 0 < α < n, we show in Corollary 1.3 (i) that if u ∈ Ln/α (Rn ), then u is radially symmetric and monotone decreasing about some point in Rn . This adds some results to the well-known conformally invariant equations. The tool used in this paper is the method of moving planes, which was introduced by Alexandroﬀ [1] in the early 1950s. Later, this method was reﬁned by Serrin [16], and further developed by Gidas et al. [10], Caffarelli et al. [5] and others. It was ﬁrst observed by Chen et al.[7,8] that instead of

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the maximum principle to partial differential equations, Hardy–Littlewood– Sobolev type inequality could be used to stduy integral equations. The existence, the regularity and the symmetry of integral systems with Riesz potentials were investigate in [6,9,11,12]. In Sect. 2 we show the symmetry of positive solutions of system (1.1) in Rn , in Sect. 3 we prove the symmetry of both the bounded domain and positive solutions of system (1.3). Finally, in Sect. 4 we investigate the symmetry of integral equation system with Bessel potential.

2. Proof of Theorem 1.1 In this section, we study the symmetry and monotonicity of positive solutions of (1.1). First, we give a regularity lemma. Lemma 2.1. For any given r ≥ 1, if eu ∈ Lr (Rn ), then u ∈ Lp (Rn ) for all p ≥ 1. Proof. The condition eu ∈ Lr (Rn ) implies that C . λr Then for any p ≥ 1, there exits a λ0 large enough such that for all λ > λ0 , |{x ∈ Rn : |eu | > λ}| ≤

|{x ∈ Rn : |u| > λ}| ≤

C C ≤ p+1 . λ r (e ) λ

Therefore, ∞ ||u||Lp (Rn ) = p

|{x ∈ Rn : |u| > λ}|λp−1 dλ

0

λ0 ≤ p 0

λp−1 dλ + C (eλ )r

∞

λ−2 dλ ≤ C.

λ0

For all λ ∈ R, we deﬁne Tλ = {(x1 , . . . , xn ) ∈ Rn , x1 = λ} as the moving plane. Let xλ = (2λ−x1 , . . . , xn ), Σλ = {(x1 , . . . , xn ) ∈ Rn , x1 ≤ λ}, uλ (y) = u(y λ ) and vλ (y) = v(y λ ). A direct computation yields that Lemma 2.2. If (u, v) is a pair of positive solutions of (1.1), then λ λ u(y ) − u(y) = (|y − z|α−n − |y λ − z|α−n )[ev(z ) − ev(z) ]dz, Σλ

v(y λ ) − v(y) =

Σλ

(|y − z|β−n − |y λ − z|β−n )[eu(z

λ

)

− eu(z) ]dz.

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To prove Theorem 1.1, we compare u(x) with u(xλ ) and v(x) with v(xλ ) on Σλ while we move the plane Tλ from λ = −∞ towards λ = +∞. The proof consists of two steps. Step 1. We show that there exists a constant N > 0 such that for all λ < −N , we have u(xλ ) ≥ u(x),

v(xλ ) ≥ v(x),

∀x ∈ Σλ .

(2.1)

Thus we can start moving the plane Tλ continuously from λ ≤ −N towards the right as long as (2.1) holds. Lemma 2.3. There exists a constant N > 0, such that for all λ < −N , u(xλ ) ≥ u(x),

v(xλ ) ≥ v(x),

∀x ∈ Σλ .

Proof. Deﬁne Σuλ = {y ∈ Σλ : u(y) > u(y λ )}, Σvλ = {y ∈ Σλ : v(y) > v(y λ )}. From Lemma 2.1, we have λ λ u(y) − u(y ) ≤ (|y − z|α−n − |y λ − z|α−n )[ev(z) − ev(z ) ]dz, Σv λ

≤

(|y − z|α−n − |y λ − z|α−n )ev(y) [v(z) − v(z λ )]dz,

Σv λ

≤ Σv λ

1 ev(y) [v(z) − v(z λ )]dz. |y − z|n−α

It follows from the Hardy–Littlewood–Sobolev inequality that ||u − uλ ||Lp (Σuλ ) ≤ ||u − uλ ||Lp (Σλ ) ≤ C||ev ||Ls (Σvλ ) ||v − vλ ||Lq (Σvλ ) , (2.2) where 1/p + α/n = 1/s + 1/q. Similarly, we have ||v − vλ ||Lq (Σvλ ) ≤ ||v − vλ ||Lq (Σλ ) ≤ C||eu ||Lr (Σuλ ) ||u − uλ ||Lp (Σuλ ) , (2.3) where 1/q + β/n = 1/r + 1/p. Combining (2.2) and (2.3), ||u − uλ ||Lp (Σuλ ) ≤ C||ev ||Ls (Σvλ ) ||eu ||Lr (Σuλ ) ||u − uλ ||Lp (Σuλ ) ,

(2.4)

where 1/p − 1/q = 1/s − α/n = 1/r − β/n. The integrability conditions eu ∈ Lr (Rn ) and ev ∈ Ls (Rn ) ensure that there exists a constant N > 0 sufﬁciently large such that for all λ < −N , C||ev ||Ls (Σvλ ) ||eu ||Lr (Σuλ ) ≤ C||ev ||Ls (Σλ ) ||eu ||Lr (Σλ ) ≤

1 . 2

Then (2.4) implies that ||u − uλ ||Lp (Σuλ ) = 0 and Σuλ must be measure zero and hence empty for all λ < −N . Similarly, Σvλ must also be empty for any λ < −N . This completes the proof of the lemma.

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Step 2. Lemma 2.3 shows that the plane Tλ can be moved starting from ¯ < ∞, u(x) and v(x) λ < −N . Now we prove that if the plane stop at x1 = λ must be symmetric and monotone decreasing about Tλ¯ . Since the direction of x1 can be chosen arbitrarily, we deduce that u(x) and v(x) must be radially symmetric and monotone decreasing about some points in Rn . This completes the proof of Theorem 1.1. Lemma 2.4. Define ¯ = sup{μ : u(y λ ) ≥ u(y), v(y λ ) ≥ v(y), ∀λ < μ, y ∈ Σλ }. λ ¯ < ∞, then for all y ∈ Σλ , If λ u(y λ ) ≡ u(y), v(y λ ) ≡ v(y). ¯ must be less then +∞. This is a direct result Proof. First, we point out that λ from Lemma 2.3 with the method of moving planes from +∞ towards −∞. We prove this lemma by contradiction. If it is not true, Lemma 2.2 shows that ¯

u(y λ ) > u(y),

¯

v(y λ ) > v(y),

∀y ∈ Σλ¯ .

Now we show that Tλ can be moved further, more precisely, there exists an small enough such that ¯<λ<λ ¯ + , (2.5) u(y λ ) ≥ u(y), v(y λ ) ≥ v(y), ∀y ∈ Σλ , λ ¯ which is a contradiction with the definition of λ. Deﬁne λ V λ ΣU λ = {u(y) > u(y ), y ∈ Σλ }, Σλ = {v(y) > v(y ), y ∈ Σλ }, V u U v V then |ΣU ¯ Σλ ⊂ Σλ ¯ Σλ ⊂ Σλ ¯ | = |Σλ ¯ | = 0 and limλ→λ ¯ , limλ→λ ¯. λ From (2.4), we can choose an > 0 sufﬁciently small such that for all ¯ λ ¯ + ), λ ∈ (λ, 1 C||ev ||Ls (Σvλ ) ||eu ||Lr (Σuλ ) ≤ . 2 Then we deduce that ||uλ − u||Lp (Σuλ ) = 0 and therefore Σuλ must be empty ¯ λ ¯ + ). ¯ λ ¯ + ). Similarly, Σv must also be empty for any λ ∈ (λ, for any λ ∈ (λ, λ This veriﬁes (2.5) and therefore completes the proof.

3. Proof of Theorem 1.2 We also use the method of moving planes to prove Theorem 1.2. For any λ ∈ R, let Tλ = {(x1 , x2 , . . . , xn ) ∈ Ω : x1 = λ} as the hyperplane vertical to x1 axis, deﬁne Σλ = {x ∈ Ω : x1 > λ}, xλ = (2λ − x1 , . . . , xn ), Σλ = {x ∈ Ω : xλ ∈ Σλ } and Ωλ := Ω\(Σλ ∪ Σλ ). Since Ω is bounded and ∂Ω ∈ C 1 , while moving the plane Tλ from ¯ = ∅} and at x1 = +∞ to x1 = −∞, there exists a λ0 = max{λ : Tλ ∩ Ω the beginning Σλ will remain in Ω. Then we move the plane continuously towards x1 = −∞ until one of the following occurs: ¯ Tλ ; (i) Σλ is internally tangent to ∂Ω at some point ˆz ∈ (ii) Tλ is orthogonal to the boundary of Ω at some ˆz ∈ ∂Ω.

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ˆ as the ﬁrst value of λ such that Tλ reaches one of the above posiDenote λ tions. First, we give a lemma whose proof is a direct computation. ˆ λ0 ), Lemma 3.1. For any x ∈ Σλ , λ ∈ [λ,

λ Kα ev(y ) − eu(y) dy + (−Kα )ev(y) dy, u(xλ ) − u(x) = Σλ

v(xλ ) − v(x) =

Ωλ

u(y λ ) u(y) Kβ e −e dy + (−Kβ )eu(y) dy,

Σλ

Ωλ

where Ki (x, y) =

1 1 − λ , |x − y|n−i |x − y|n−i

i = α, β.

Moreover, we have Kα (x, y) > 0, Kα (x, y) < 0,

Kβ (x, y) > 0, Kβ (x, y) < 0,

∀ x, y ∈ Σλ , ∀ x ∈ Σλ , y ∈ Ωλ .

To prove Theorem 1.2, we compare u(x) with u(xλ ) and v(x) with v(xλ ) on Σλ . The proof consists of three steps. ˆ λ0 ) such that for Step 1. We show that there exist a constant λ1 ∈ [λ, all λ ∈ [λ1 , λ0 ), u(xλ ) ≥ u(x),

v(xλ ) ≥ v(x),

∀x ∈ Σλ .

(3.1)

Thus we can start moving the plane Tλ continuously from λ ∈ [λ1 , λ0 ) to the left as long as (3.1) holds. ˆ λ0 ) such that for all λ ∈ [λ1 , λ0 ), Lemma 3.2. There exists a λ1 ∈ [λ, u(xλ ) ≥ u(x), Σuλ

v(xλ ) ≥ v(x),

∀x ∈ Σλ .

λ

:= {x ∈ Σλ : u(x) > u(x )}, Σvλ := {x ∈ Σλ : v(x) > Proof. Deﬁne λ v(x )}. From Lemma 3.1, we have λ λ u(y) − u(y ) ≤ (|y − z|α−n − |y λ − z|α−n )[ev(z) − ev(z ) ]dz, Σv λ

≤ Σv λ λ

v(y) − v(y ) ≤ Σu λ

1 ev(y) [v(z) − v(z λ )]dz. |y − z|n−α 1 eu(y) [u(z) − u(z λ )]dz. |y − z|n−β

It follows from the Hardy–Littlewood–Sobolev inequality that ||u − uλ ||Lp (Σuλ ) ≤ ||u − uλ ||Lp (Σλ ) ≤ C||ev ||Ls (Σvλ ) ||v − vλ ||Lq (Σvλ ) ,

||v − vλ ||Lq (Σvλ ) ≤ ||v − vλ ||Lq (Σλ ) ≤ C||eu ||Lr (Σuλ ) ||u − uλ ||Lp (Σuλ ) ,

where 1/p + α/n = 1/s + 1/q and 1/q + β/n = 1/r + 1/p.

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Combining the above two inequalities, we have ||u − uλ ||Lp (Σuλ ) ≤ C||ev ||Ls (Σvλ ) ||eu ||Lr (Σuλ ) ||u − uλ ||Lp (Σuλ ) . u

r

v

(3.2)

s

The integrability conditions e ∈ L (Ω), e ∈ L (Ω) ensure that there exists a constant N > 0 sufﬁciently large such that for all λ < −N , 1 C||ev ||Ls (Σvλ ) ||eu ||Lr (Σuλ ) ≤ C||ev ||Ls (Σλ ) ||eu ||Lr (Σλ ) ≤ . 2 Then (3.2) implies that ||u − uλ ||Lp (Σuλ ) = 0 and Σuλ must be measure zero and hence empty for all λ < −N . Similarly, Σvλ must also be empty for any λ < −N . This completes the proof of the lemma. ˆ Step 2. We show that the plane can be moved continuously until λ = λ with (3.1) holding. Lemma 3.3. Define ¯ := inf{λ ∈ [λ, ˆ λ0 ), u(xλ ) ≥ u(x), v(xλ ) ≥ v(x), ∀ x ∈ Σλ }. λ ¯ = λ. ˆ Then λ ˆ < λ, ¯ it will be proved that Proof. We prove the lemma by contradiction. If λ the plane can be moved further, i.e., there exists an such that ˆ<λ ¯ − ≤ λ < λ, ¯ (3.3) u(xλ ) ≥ u(x), v(xλ ) ≥ v(x)∀x ∈ Σλ , λ ¯ which is a contradiction with the definition of λ. ˆ ¯ Suppose λ < λ, then Ωλ¯ = ∅. Lemma 3.1 shows that ¯

u(xλ ) > u(x), v(xλ ) > v(x), ¯ implies that The definition of λ

x ∈ Σλ¯ .

lim |Σuλ | = lim |Σvλ | = 0.

¯ λ→λ

(3.4)

¯ λ→λ

We can choose an > 0 sufﬁciently small such that 1 C||ev ||Ls (Σvλ ) ||eu ||Lr (Σuλ ) ≤ . 2 u Following from (3.2) we have ||u − uλ ||Lp (uλ ) = 0, therefore λ must be ¯ − , λ). ¯ Similarly, Σv must also be empty for all λ ∈ empty for all λ ∈ [λ λ ¯ ¯ [λ − , λ). This veriﬁes (3.3) and completes the proof of the lemma. ¯ = λ, ˆ Ω must be symmetric about the Step 3. Finally, we show that at λ plane Tλˆ , u(x) and v(x) must be symmetric and monotone decreasing about the plane Tλˆ . Since the direction of x1 can be chosen arbitrarily, we deduce that Ω must be a ball, u(x) and v(x) must be radially symmetric and monotone decreasing with respect to the radius. This complete our proof of Theorem 1.2. Lemma 3.4. Σλˆ ∪ (Ω ∩ Tλˆ ) ∪ Σλˆ = Ω,

ˆ

u(xλ ) ≡ u(x),

∀ x ∈ Σλˆ .

ˆ we prove the lemma in two cases: Proof. From the definition of λ, ¯ Tλ . Case 1. Σλ is internally tangent to ∂Ω at some zˆ∈

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Suppose Σλˆ ∪ (Ω ∩ Tλˆ ) ∪ Σλˆ = Ω, which implies that Ωλˆ = ∅. Then Lemmas 3.1 and 3.3 show that

1 1 ˆ v(y) λ u(ˆ z ) − u(ˆ z) ≥ e − dy > 0, |ˆ z − y|n−α |ˆ z λˆ − y|n−α Ωλ ˆ

1 1 ˆ u(y) λ v(ˆ z ) − v(ˆ z) ≥ e − dy > 0. |ˆ z − y|n−β |ˆ z λˆ − y|n−β Ωλ ˆ

ˆ

But it is a contradiction with the fact that zˆ, zˆλ ∈ ∂Ω and the boundary conditions u ≡ A,

v ≡ D, on ∂Ω.

Case 2. Tλ is orthogonal to the boundary of Ω at some point zˆ ∈ ∂Ω. From the assumption, it is easy to see that ∂x1 u(ˆ z ) = 0,

∂x1 v(ˆ z ) = 0.

(3.5)

If Σλˆ ∪ (Ω ∩ Tλˆ ) ∪ Σλˆ = Ω which means that Ωλˆ = ∅, then there exists a ˆ

m → zˆ and (xm )λ → zˆ. ball B ⊂ Ωλˆ . Let {xm }∞ ˆ \Tλ ˆ such that x m=1 ⊂ ∂Σλ ˆ Without loss of generality, we assume B lies on the left of {(xm )λ }∞ m=1 , more ˆ m λ precisely, there exists a constant δ > 0 such that (x )1 − y1 ≥ δ for any ˆ ˆ ˆ (xm )λ = ((xm )λ1 , . . . , (xm )λn ) and y = (y1 , . . . , yn ) ∈ B.

It is easy to verify that ˆ m λ m v(y) u((x ) ) − u(x ) ≥ e Ωλ ˆ

1 |(xm )λˆ − y|n−α

≥ (n − α)δ

eu(y) (n − α)

B

≥ (n − α)δ

eu(y)

B

1 − m |x − y|n−α

dy ˆ

(¯ xm − y)(xm − (xm )λ ) dy |¯ xm − y|n−α+2

ˆ

|(xm )λ − xm | dy. |¯ xm − y|n−α+2

That is, ˆ

lim inf

u((xm )λ ) − u(xm )

m→∞

|(xm )λˆ − xm |

> 0.

And this contradicts with (3.5). From above two cases, we know that Ω is symmetric about the plane Tλˆ ˆ (3.1) holds. From the opposite direction we use the method and when λ = λ of moving planes again, the results in Lemmas 3.2–3.4 are also valid. So we deduce that ˆ

u(xλ ) ≡ u(x),

ˆ

v(xλ ) ≡ v(x),

∀x ∈ Σλˆ ; and Σλˆ ∪ (Ω ∩ Tλˆ ) ∪ Σλˆ = Ω.

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4. The Symmetry of Integral Systems with Bessel Potentials In this section, we extend our results to the following integral equation system u(x) = Rn Gα (x − y)ev(y) dy, x ∈ Rn , (4.1) v(x) = Rn Gβ (x − y)eu(y) dy, x ∈ Rn , where 0 < α, β < n are constants and Gα is the Bessel kernel deﬁned as ∞ α−n dδ 1 π|x|2 δ Gα (x) = exp − exp − δ 2 γ(α) δ 4π δ (4π) α2 Γ

α

0

with γ(α) = 2 . Our symmetry result of the above system is Theorem 4.1. Assume {u, v} is a pair of positive solutions of (4.1). If eu(y) ∈ Lr (Rn ),

ev(y) ∈ Ls (Rn ),

where r, s > 1, n/s + n/r = α + β, then u and v are radially symmetric and monotone decreasing about some point x0 in Rn . We also consider the following Dirichlet problem ⎧ v(y) dy, x ∈ Ω, ⎪ ⎨ u(x) = Ω Gα (x − y)e u(y) dy, x ∈ Ω, v(x) = Ω Gβ (x − y)e ⎪ ⎩ u = C1 , v = C2 on ∂Ω.

(4.2)

We have Theorem 4.2. Suppose Ω ⊂ Rn is a bounded C 1 domain, {u, v} is a pair of positive solutions of system (4.2). If eu(y) ∈ Lr (Ω),

ev(y) ∈ Ls (Ω),

where r, s > 1, n/s + n/r = α + β, then Ω is a ball, u and v must be radially symmetric and monotone decreasing with respect to the radius. The proof of the above two Theorems are similar to the proof of systems with Riesz potential. We just show some key points. Lemma 4.3. (Hardy–Littlewood–Sobolev inequality for Bessel potential) For any α > 0, let u = Gα ∗ f . If f ∈ Lp (Rn ) with 1 < p < ∞, then u ∈ Lr (Rn ) where −n/r ≤ α − n/p and p < r < ∞. Moreover, we have ||u||Lr (Rn ) ≤ C||f ||Lp (Rn ) where C = C(α, n, p). Proof. The result is a direct corollary from the Sobolev Embedding Theorem and properties of Bessel potentials in [2]. Ma and Chen have proved a similar result in [3] with which they studied the radial symmetry of some (systems of) integral equations with Bessel potentials in [3,4].

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Lemma 4.4. Assume (u, v) is a pair of positive solutions of (4.1). Then λ λ gα (x, y)[ev(z ) − ev(z) ]dy, u(x ) − u(x) = Σλ

v(xλ ) − v(x) =

gβ (x, y)[eu(z

λ

)

− eu(z) ]dy,

Σλ

where Σλ = {(x1 , . . . , xn ) ∈ Rn , x1 ≤ λ} and for all x, y ∈ Σλ , gi (x, y) := Gi (x − y) − Gi (xλ − y) ≥ 0, i = α, β. From the estimate in Lemma 4.3 and the identities in Lemma 4.4, we could prove Theorem 4.1 similarly as the proof of Theorem 1.1. To prove Theorem 4.2, we need the following two lemmas which are respectively similar to Lemma 3.1 and the second case in Lemma 3.4. The others are similar to the ones of Theorem 1.2. ˆ ≤ λ < λ0 , Lemma 4.5. For any x ∈ Σλ , λ λ u(xλ ) − u(x) = gα (x, y)[ev(z ) − ev(z) ]dy + (−gα (x, y))ev(z) dy, Σλ λ

Ωλ

v(x ) − v(x) =

λ

u(z )

gβ (x, y)[e

u(z)

−e

Σλ

]dy +

(−gβ (x, y))eu(z) dy,

Ωλ

where Σλ , Ωλ are defined in Sect. 3 and gi (x, y) := Gi (x − y) − Gi (xλ − y) ≥ 0,

i = α, β.

Furthermore, we have gα (x, y) > 0,

gβ (x, y) > 0,

∀x, y ∈ Σλ ,

gα (x, y) < 0,

gβ (x, y) < 0,

∀x ∈ Σλ ,

y ∈ Ωλ .

Lemma 4.6. Assume Tλˆ is orthogonal to ∂Ω at some point zˆ ∈ Tλ ∩∂Ω where ˆ is defined in Sect. 3. Then Ω is symmetric about the plane T ˆ . λ λ Proof. We prove it by contradiction. Assume that if Σλˆ ∪ (Ω ∩ Tλˆ ) ∪ Σλˆ = Ω. Then Ωλˆ = ∅. From the boundary condition, it is easy to see that ∂x1 u(ˆ z ) = 0 and ∂x1 v(ˆ z ) = 0.

(4.3) ˆ

m → zˆ, it follows that (xm )λ → zˆ. If Let {xm }∞ ˆ \Tλ ˆ such that x m=1 ⊂ ∂Σλ Ωλˆ = ∅, there exists a ball B ⊂ Ωλˆ . Without loss of generality, we may ˆ assume B lies on the left of {(xm )λ }∞ m=1 , and there exists > 0 such that ˆ ˆ ˆ ˆ m λ m λ m λ (x )1 −y1 ≥ for any (x ) = ((x )1 , . . . , (xm )λn ) and y = (y1 , . . . , yn ) ∈ B.

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For any y ∈ B, let x ¯m be on the segment from (xm )λ to xm such that ˆ

Gα ((xm )λ − y) − Gα (xm − y) ∞ ˆ π|(xm )λ − y|2 1 π|xm − y|2 exp − = − exp − γ(α) δ δ 0 α−n dδ δ × exp − δ 2 4π δ ∞ ˆ xm − y)((xm )λ − xm ) 1 π|¯ xm − y|2 −2π(¯ = exp − γ(α) δ δ 0 α−n dδ δ × exp − . δ 2 4π δ ˆ

m Since (xm )λ1 ≤ x ¯m 1 ≤ x1 , we have ˆ

ˆ

ˆ

m m λ m m λ (¯ xm − y)(xm − (xm )λ ) = (¯ xm 1 − y1 )(x1 − (x )1 ) ≥ (x1 − (x )1 ) ˆ

= |xm − (xm )λ |. It is easy to verify that ˆ

u((xm )λ ) − u(xm ) ∞ −π|¯ xm − y|2 2π ˆ v(y) e exp ≥ |xm − (xm )λ | γ(α) δ Ωλ ˆ

× exp

−δ 4π

0

δ

α−n 2

dδ dy, δ2

that is, ˆ

lim inf m→∞

u((xm )λ ) − u(xm ) |(xm )λˆ − xm |

which is in contradiction to (4.3).

>0

Acknowledgments We wish to thank the referees for their thoughtful comments and helpful suggestions.

References [1] Alexandroﬀ, A.D.: A characteristic property of the spheres. Ann. Math. Pura. Appl. 58, 303–354 (1962) [2] Adams, R.: Sobolev Spaces, Pure and Applied Mathematics, vol. 65. Academic Press, New York (1975) [3] Chen, D., Ma, L.: Radial symmetry and monotonicity for an integral equation. J. Math. Anal. Appl. 342, 943–949 (2008)

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[4] Chen, D., Ma, L.: Radial symmetry and uniqueness for positive solutions of a Schr¨ oinger type system. Math. Comput. Model. 49, 379–385 (2009) [5] Caffarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42, 271–297 (1989) [6] Chen, W., Li, C.: Regularity of solutions for a system of integral equations. Commun. Pure Appl. Anal. 4, 1–8 (2005) [7] Chen, W., Li, C., Ou, B.: Classiﬁcation of solutions for an integral equation. Commun. Pure Appl. Math. 59, 330–343 (2006) [8] Chen, W., Li, C., Ou, B.: Classiﬁcation of solutions for a system of integral equations. Commun. Partial Differ. Equ. 30, 59C65 (2005) [9] Chen, W., Li, C., Ou, B.: Qualitative properties of solutions for an integral equation. Disc. Cont. Dyn. Syst. 12, 347–354 (2005) [10] Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979) [11] Jin, C., Li, C.: Symmetry of solutions to some systems of integral equations. Proc. Am. Math. Sci. 134, 1661–1670 (2005) [12] Li, C., Lim, J.: The singularity analysis of solutions to some integral equations. Commun. Pure Appl. Anal. 6, 453–464 (2007) [13] Lin, C.: A classiﬁcation of solutions of a conformally invariant fourth order equation in Rn . Comment. Math. Helv. 73, 206–231 (1998) [14] Li, D., Str¨ ohmer, G., Wang, L.: Symmetry of integral equations on bounded domains. Proc. Am. Math. Soc. 137, 3695–3702 (2009) [15] Li, Y.: Prescribing scalar curvature on S n and related problems, part 1. J. Differ. Equ. 120, 319–410 (1995) [16] Serrin, J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304–318 (1971) Feiyao Ma and Xiaotao Huang (B) Department of Mathematics Xi’an Jiaotong University Xi’an 710049 People’s Republic of China e-mail: [email protected]; xiaotao [email protected] Lihe Wang The University of Iowa Iowa City IA 52242-1419, USA e-mail: [email protected] Received: July 24, 2010 Revised: October 28, 2010.

Integr. Equ. Oper. Theory 69 (2011), 405–444 DOI 10.1007/s00020-010-1848-x Published online December 4, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves Vladimir Rabinovich and Stefan Samko Dedicated to the memory of Professor I. Simonenko Abstract. The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces Lp(·) (Γ, w) on a class of composed Carleson curves Γ where the weights w have a ﬁnite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces Lp(·) (R+ , dμ) where dμ is an invariant measure on multiplicative group R+ = {r ∈ R : r > 0}. (2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefﬁcients acting on Lp(·) (Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on R+ and local invertibility of singular integral operators on R. (3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces Lp(·) (Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a ﬁnite set of slowly oscillating singularities. Mathematics Subject Classification (2000). Primary 47G30. Keywords. Pseudodifferential operators, H¨ ormander class, Singular operators, Variable exponent, Generalized Lebesgue space, Fredholmness.

1. Introduction Last decade there arose a big interest to investigation of the classical operators of Analysis, i.e. singular and maximal operators, Hardy operators, pseudodifferential operators, in the Lp(·) -spaces with variable exponents p(·). Many papers have been devoted to the extension of various results on the boundedness of operators, well known for the constant p, to the case of variable p(·).

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This extension is essentially nontrivial and demands new ideas and methods, see for instance [7–11,28,44] and references therein. Similar to the case of the constant p, the Fredholm theory of the mentioned operators in spaces related to Lp(·) has also a big interest. With respect to one-dimensional singular integral operators in variable exponent Lebesgue spaces we refer, for instance, to [17–26,41]. In our paper [41] we proved the boundedness of pseudodifferential oper0 acting in the variable exponent Lebesgue spaces ators of the class OP S1,0 p(·) n L (R ) and obtained the necessary and sufﬁcient conditions of the Fredhom 0 with symbols slowly oscillating at property of operators of the class OP S1,0 inﬁnity, in the spaces Lp(·) (Rn ). The proof of the sufﬁciency of conditions of the Fredholmness is more or less standard being based on the calculus of pseudodifferential operators, the boundedness theorems and the interpolation in the spaces Lp(·) (Rn ), while the proof of the necessity of those conditions meet big difﬁculties. (In particular, they are connected with the fact that the shift and dilation operators are unbounded in Lp(·) ). The main aim of the paper is the Fredholm theory of singular integral operators (SIOs) on composed curves Γ with whirling points and coefﬁcients having slowly oscillating discontinuities acting in the weighted spaces Lp(·) (Γ, w). Applying results from [41] we prove that singular integral operators are bounded in Lp(·) (Γ, w) and they are the local type operators in the Simonenko sense [46–48]. Consequently, for the investigation of the Fredholm property we can apply the Simonenko local principle. This principle reduces the investigation of the Fredholm property of local type operators to the investigation of the local invertibility of their local representatives which are simpler operators than the original one. For instance, the investigation of the Fredholm property of the SIO A = aI + bSΓ , with continuous coefﬁcients a and b and a Lyapunov curve Γ, in the space Lp (Γ), 1 < p < ∞, is known to be reduced to investigation of local representatives at every point t0 ∈ Γ which are operators of the type At0 = a(t0 )I + b(t0 )SR . Their local invertibility in Lp (R) coincides with the invertibility which is equivalent to the condition a(t0 )±b(t0 ) = 0. The investigation of the Fredholm property of the operator A = aI + bSΓ with piece-wise continuous coefﬁcients on a simple Lyapunov curve Γ in the space Lp (Γ, w) with power weight w, is reduced to the investigation of the local invertibility of the homogeneous operators of the form aI + bSR acting in Lp (R), where a, b are piecewise constant functions with the only discontinuity at the origin and inﬁnity. These operators are realized as Mellin convolutions and conditions of their invertibility are given in the terms of the Mellin transform of the kernel. In [2–5,33–36,38], the Simonenko local method was applied to SIO on some composed Carleson curves with discontinuous coefﬁcients acting on weighted Lp -spaces, and in the paper [39] for SIO acting on weighted H¨ older spaces. In this case the local representatives are Mellin pseudodifferential operators with variable symbols. The symbols of local representatives

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(the local symbols) explain the appearance of the logarithmic double spirales and spiral horns in the local spectrums of SIO. Note that in the theory of Gohberg et al. [14] and Spitkovsky [49] for SIO on Lyapunov curves in Lp spaces with Muckenhoupt weights, the typical local spectra are circular arcs and circular horns. We extend here the results of the mentioned papers to the case of variable exponent p(·). The local representatives of the SIO at the singular points t ∈ Γ appear as Mellin pseudodifferential operators with a symbol depending on the curve, weight and coefﬁcients and also on the values of p(·) at singular points t. Making use of the results on local invertibility of Mellin pseudodifferential operators, we obtain necessary and sufﬁcient conditions of the local invertibility of SIOs at singular points of the curves, weights and coefﬁcients. Finally, the application of the Simonenko local principle allows to obtain the necessary and sufﬁcient conditions of Fredhomness in Lp (Γ, w). The methods of localization developed in the paper can be applied to the study of the Fredholm property of multidimensional SIOs and pseudodifferential operators on compact and noncompact manifolds, boundary value problems in Sobolev and Besov spaces connected with Lp(·) . We hope to do this in forthcoming papers. Another approach to the investigation of the algebra of operators generated by the operator SΓ of singular integration along a general composed Carleson curve Γ and operators of multiplication by piece-wise continuous functions, acting in Lp (Γ, w), where 1 < p < ∞, and w is a Muckenhoupt weight, based on the Wiener–Hopf factorization and theory of submultiplicative functions was given by B¨ ottcher and Karlovich (see book [1] and references therein). In [18–20], some results of the book [1] were transferred to algebras of SIO acting in the Lebesgue spaces with variable exponents. The paper is organized as follows. In Sect. 2 we consider pseudodifferential operators on R acting in the variable exponent Lebesgue spaces Lp(·) (R). The main result of this section is a criterion of local invertibility, at the point +∞, of pseudodifferential operators with slowly oscillating symbols, and a criterion of local invertibility of pseudodifferential operators and singular integral operators at the point x0 ∈ R. In Sect. 3 the results of Sect. 2 are reformulated for the Mellin pseudodifferential operators acting on Lp(·) (R+ , dμ) with the invariant measure dμ = dr r on the multiplicative group R+ . In Sect. 4 we apply the results of Sects. 2 and 3 to the investigation of boundedness, local invertibility and Fredholmness of singular integral operators on composed Carleson curves acting on the Lebesgue spaces Lp(·) (Γ, w) with weights having a ﬁnite set of oscillating singularities. We obtain here the following results: (1)

Theorem on the boundedness of SIO on composed Carleson curves Γ acting on the Lebesgue spaces Lp (Γ, w) with weights having a ﬁnite set of oscillating singularities. The proof of this theorem is based on the local boundedness of Mellin pseudodifferential operators on the spaces Lp(·) (R+ , dμ) and an admissible partition of unity on the curve

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Γ. The pseudodifferential operators approach demands that the curve near every node is inﬁnitely smooth. But in fact we use the existence of only a ﬁnite number of derivatives. (2) Criterion of the local invertibility and Fredholmness of SIOs on slowly oscillating composed curves with piecewise slowly oscillating coefﬁcients, in the spaces Lp(·) (Γ, w) with the weight w slowly oscillating at the nodes. The main tools of this section is the local principle of Simonenko and necessary and sufﬁcient conditions of local invertibility of Mellin at the point 0, and pseudodifferential operators acting in Lp(·) R+ , dr r pseudodifferential and singular integral operators acting in Lp(·) (R) at the point x0 ∈ R. Section 5 is devoted to a comparison of the used class of oscillating weights with the Bary-Stechkin type weights. In particular, we show in Lemma 56 that our assumption on the differentiability of weights near the nodes is inessential in the sense that any function in the Bary-Stechkin class is equivalent to N times differentiable function in this class, for any given ﬁnite N , the Matuszewska-Orlicz indices coinciding under the equivalence, as is known. However, the conditions on the weights in terms of the Simonenko indices are somewhat stricter than in terms of the Matuszewska-Orlicz indices, see Remark 57. We will use the following notations: • for a Banach space X, B(X) stands for the space of all bounded operators in X, • C ∞ (R) is the linear space of inﬁnitely differentiable functions on R, • C0∞ (R) is a subspace of C ∞ (R) of functions with compact support, • Cb∞ (R) is a subspace of C ∞ (R) of functions bounded on R with all their derivatives, • S(R) is the L. Schwartz space of functions in C ∞ (R) decreasing at −n inﬁnity with all their derivatives faster than every power |x| , n ∈ N. • If a is a function or matrix, by aI we denote the operator of multiplication by a.

2. Pseudodifferential Operators on R 2.1. Some Properties In this section we give an auxiliary material on pseudodifferential operators (more information may be found for instance in [38, Chapter 4], or [37]). Definition 1. (i) We say that a function a ∈ C ∞ (R × R) is a symbol of the m if class S1,0 |a|l1 ,l2 =

−m+α sup ∂xβ ∂ξα a(x, ξ) ξ < ∞,

2 α≤l1 ,β≤l2 (x,ξ)∈R

(2.1)

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1/2 2 . To a symbol for every l1 , l2 ∈ N0 = {0} ∪ N, where ξ = 1 + |ξ| a we relate the pseudodifferential operator (ψdo) 1 Op(a)u(x) = dξ a(x, ξ)u(y)ei(x−y)·ξ dy, 2π R

(2.2)

R

0 where u ∈ C0∞ (R); by OP S1,0 we denote the class of ψdo s with symbols 0 in S1,0 . (ii) We say that a function a ∈ C ∞ (R × R × R) is a double symbol of the m if class S1,0,0 α−m |a|l1 ,l2 ,l3 = sup ∂xβ ∂yγ ∂ξα a(x, y, ξ) ξ < ∞ (2.3) 3 α≤l1 ,β≤l2 ,γ≤l3 (x,y,ξ)∈R

for every l1 , l2 , l3 ∈ N0 . To a symbol a we relate the pseudodifferential operator with double symbol 1 dξ a(x, y, ξ)u(y)ei(x−y)·ξ dy, (2.4) Opd (a)u(x) = 2π R

where u ∈ C0∞ (R), m m by OP S1,0,0 . S1,0,0

R

and we denote the class of ψdo s with symbols in

Proposition 2. (Calderon–Vaillancourt, see for instance [38, Theorem 0 . Then the operator Op(a) is bounded in L2 (R) 4.1.12]). Let Op(a) ∈ OP S1,0 and Op(a)B(L2 (R)) ≤ C |a|2,2 ,

(2.5)

where C does not depend on a. Proposition 3. (see [38, Chapter 4]) (i)

m

Let aj ∈ S1,0j , j = 1, 2 and C = Op(a1 )Op(a2 ). Then C ∈ m1 +m2 , C = Op(c) where OP S1,0 1 a(x, ξ + η)b(x + y, ξ)e−iy·η dydη. (2.6) c(x, ξ) = 2π R2

Moreover, c(x, ξ) = a(x, ξ)b(x, ξ) + t(x, ξ), (ii)

where t Let a ∈

m1 +m2 −1 . ∈ S1,0 m S1,0,0 . Then Opd (a)

a# (x, ξ) =

1 2π

(2.7)

m ∈ OP S1,0 , Opd (a) = OP (a# ) where

a(x, x + y, ξ + η)e−iy·η dydη.

R2

m−1 Moreover, a# (x, ξ) = a(x, x, ξ) + t(x, ξ), where t ∈ S1,0 .

(2.8)

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m are bounded in S(R) (see for instance Note that the operators in OP S1,0 [38, Proposition 4.1.5]). m We say that an operator Aτ is formally adjoint to A = Op(a) ∈ OP S1,0 if

(Aτ u, v) = (u, Av) for all u, v ∈ S(Rn ), where (·, ·) is the standard scalar product corresponding to L2 (R). m Proposition 4. Let a ∈ S1,0 Then the operator Aτ formally adjoint to A = m Op(a) belongs to OP S1,0 and Aτ = Op(aτ ) with 1 a ¯(x + y, ξ + η)e−iy·η dydη. (2.9) aτ (x, ξ) = 2π R2

m−1 ¯(x, ξ) + t(x, ξ), where t ∈ S1,0 . aτ (x, ξ) = a

The integrals in (2.6), (2.8), (2.9) are understood as oscillatory (see [38, Chap. 4.1.2], [37, Chap. 2]). 0 Definition 5. (i) We say that a symbol a ∈ S1,0 is slowly oscillating at the point +∞, if β α ∂x ∂ξ a(x, ξ) ≤ Cαβ (x) ξ−α , (2.10)

and limx→+∞ Cαβ (x) = 0 for all α ∈ N0 and β ∈ N. We denote this class by SO+∞ and the corresponding class of ψdo s by OP SO+∞ . 0 is slowly oscillating at the (ii) We say that a double symbol a ∈ S1,0,0 point +∞, if β γ α ∂x ∂y ∂ξ a(x, y, ξ) ≤ Cαβγ (x, y) ξ−α where limx→+∞ Cαβγ (x, y) = 0 uniformly with respect y for all α, γ ∈ N0 and β ∈ N, and limy→+∞ Cαβγ (x, y) = 0 uniformly with respect x for all α, β ∈ N0 and γ ∈ N. We denote this class by SO+∞,d and the corresponding class of ψdo s by OP SO+∞,d . ˚+∞ , if the coefﬁcient Cαβ (x) in estimate (2.10) (iii) We say that a ∈ S satisﬁes the condition limx→+∞ Cαβ (x) = 0 for all α, β ∈ N0 . The ˚+∞ . corresponding class of ψdo s is denoted by OP S Proposition 6. ( [38, Chap. 4]) (i) Let Op(aj ) ∈ OP SO+∞ , j = 1, 2 and B = Op(a1 )Op(a2 ). Then B ∈ OP SO+∞ and B = Op(b) with b(x, ξ) = a1 (x, ξ)a2 (x, ξ) + q(x, ξ), (ii)

˚+∞ . where q ∈ S Let Opd (a) ∈ OP SO+∞,d . Then Opd (a) = Op(a# ) ∈ OP SO+∞ , where a# (x, ξ) = a(x, x, ξ) + q(x, ξ), ˚+∞ . and q ∈ S

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Let Op(a) ∈ OP SO+∞ . Then the formal adjoint operator (Op(a))τ = Op(aτ ) is in OP SO+∞ with aτ (x, ξ) = a ¯(x, x, ξ) + q(x, ξ), m ˚+∞ . and q ∈ S

2.2. Pseudodifferential Operators on Lebesgue Space with Variable Exponent We give the definition of variable exponent Lebesgue spaces for the general case where the underlying space is an arbitrary quasimetric measure space, because such spaces will be used in various settings in this paper. Let (X, d, μ) be a quasimetric measure space, i.e. a topological space endowed with the quasimetric d : X × X → R1+ and nonnegative Borel measure μ (we refer to [6,12,15] for quasimetric measure spaces). Let p : X → (1, ∞) be a measurable function on X. Definition 7. The variable exponent Lebesgue space Lp(·) (X) is introduced via the modular p(·) p(x) dμ(x) < ∞ (2.11) IX (f ) = |f (x)| X

by the norm

f Lp(·) (X) = inf λ > 0 :

p(·) IX

f ≤1 . λ

p(·)

We also use a similar space Ln (X) of vector-functions on X with values in Cn , deﬁned via norm

f p(·) f Lp(·) = inf λ > 0 : In ≤1 , (X) n λ

p(·) p(x) where In (f ) := X f (x)Cn dμ(x) < ∞. (i)

Everywhere in the sequel we assume that p(·) satisﬁes the conditions: there exists numbers p− , p+ ∈ (1, ∞) such that 1 < p− ≤ p(x) ≤ p+ < ∞.

(ii)

(2.12)

there holds the log-condition |p(x) − p(y)| ≤

A log

1 d(x,y)

,

x, y ∈ R,

d(x, y) ≤

1 , 2

(2.13)

∗ Under condition (2.12) the space Lp(·) (X) is reﬂexive and Lp(·) (X) = 1 1 Lq(·) (X) where p(x) + q(x) = 1, x ∈ X. n The case X = R will be the main one in this paper and in this case we also suppose that (iii) there exists the limit lim p(x) = p(∞) and |x|→∞

|p(x) − p(∞)| ≤

A , log (2 + |x|)

x ∈ Rn .

(2.14)

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Note that under condition (2.12) for a function a ∈ L∞ (X) we have aIB(Lp(·) (X)) ≤ aL∞ (X)

(2.15)

which follows from the definition of the norm in Lp(·) (X), and that the modular convergence is equivalent to the norm convergence. The latter follows from the properties: c1 ≤ f Lp(·) (X) ≤ c2

=⇒ c3 ≤ I p(·) (f ) ≤ c4

(2.16)

and C1 ≤ I p(·) (f ) ≤ C2 =⇒ C3 ≤ f Lp(·) (R) ≤ C4 (2.17) p− p+ p− p+ 1/p 1/p with c3 = min c1 , c1 , c4 = max c2 , c2 , C3 = min C2 − , C2 + , 1/p 1/p C4 = max C2 − , C2 + . In the case X = Rn the imbeddings C0∞ (Rn ) ⊂ S(Rn ) ⊂ Lp(·) (Rn ) hold; they are dense under under assumptions (2.12), (2.13), (2.14) (see, for instance, Theorem 2.11 in [29]). Proposition 8. ( [10]) Let pj : Rn → [1, ∞), j = 1, 2, be bounded measurable functions, A be a linear operator defined on Lp1 (·) (Rn ) ∩ Lp2 (·) (Rn ) and AuLpj (·) (Rn ) ≤ Cj uLpj (·) (Rn ) ,

j = 1, 2.

(2.18)

Then A is also bounded on the intermediate space Lpθ (·) (Rn ), where 1 θ 1−θ = + , θ ∈ [0, 1] , pθ (x) p1 (x) p2 (x) and θ

1−θ

AB(Lpθ (·) ) ≤ AB(Lp1 (·) ) AB(Lp2 (·) ) . The following proposition is an extension of the well-known theorem of Krasnosel skii [30] on the interpolation of the compactness property in Lp -spaces with a constant p. Proposition 9. ( [41, Proposition 2.2]) Let pj : Rn → [1, ∞), j = 1, 2, be bounded measurable functions satisfying assumptions (2.12)–(2.14) and let a linear operator A defined on Lp1 (·) (Rn ) ∩ Lp2 (·) (Rn ) satisfy the boundedness assumptions in (2.18). If A : Lp1 (·) (Rn ) → Lp1 (·) (Rn ) is a compact operator, then A : Lpθ (·) (Rn ) → Lpθ (·) (Rn ) is a compact operator in every intermediate space Lpθ (·) (Rn ), θ ∈ (0, 1]. 0 is bounded Theorem 10. ([41, Theorem 5.1]) An operator Op(a) ∈ OP S1,0 p(·) in L (R) and there exists M > 0 and C > 0 not depending on a such that

Op(a)B(Lp(·) (R)) ≤ C |a|M,M

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Proposition 3 and Theorem 10 imply the following 0 Corollary 11. An operator Op(a) ∈ OP S1,0,0 is bounded in Lp(·) (R) and there exists M > 0 and C > 0 not depending on A such that

Op(a)B(Lp(·) (R)) ≤ C |a|M,M,M , where C > 0 and M > 0 do not depend on a. Note that, because S(R) is dense in Lp(·) (R), the formal adjoint Aτ to 0 coincides with the operator A∗ adjoint to the operator A = Op(a) ∈ OP S1,0 p(·) 0 the operator A acting in L (R). Hence A∗ = Op(aτ ) ∈ OP S1,0 , where aτ is deﬁned by (2.9). Proposition 12. Let χR be the characteristic function of the segment [R, +∞), ˚+∞ . Then Q = Op(q) ∈ OP S lim χR QB(Lp(·) (R)) = lim QχR IB(Lp(·) (R)) = 0.

R→+∞

R→+∞

(2.19)

Proof. Let ϕ ∈ C ∞ (R) be a real-valued function such that 1, x ≥ 1 ϕ(x) = , 0, x ≤ 1/2 x ˚+∞ , we and ϕR (x) = ϕ( R ), R > 0. We have ϕR Q = Op(ϕR q). Since q ∈ S have

lim |ϕR q|l1 ,l2 = 0

R→∞

for every l1 , l2 ∈ N0 . Applying Theorem 10, we obtain that lim ϕR R→∞

QB(Lp(·) (R)) = 0. Now we will prove that

lim QϕR IB(Lp(·) (R)) = 0.

R→+∞

˚+∞ by We have QϕR IB(Lp(·) (R)) = ϕR Q∗ B(Lq(·) (R)) , where Q∗ ∈ OP S statement (iii) of Proposition 6. Hence lim QϕR IB(Lp(·) (R)) = lim ϕR Q∗ B(Lq(·) (R)) = 0.

R→∞

R→∞

Since ϕR χR = χR , equality (2.20) implies (2.19).

(2.20)

2.3. Local Invertibility at +∞ Definition 13. We say that an operator A ∈ B(Lp(·) (R)) is locally invertible at the point +∞, if there exist operators LR and RR such that LR AχR I = χR I, χR ARR = χR I.

(2.21)

We also need the following propositions, where by Vh u(x) = u(x − h) we denote the translation operator. Proposition 14. ([41, Proposition 6.3]) Let a sequence (R ) hm → +∞, and wm (∈ C (R)) be a sequence converging in the sup-norm on R to a function w ∈ C(R). Suppose also that there exists a constant C > 0 such that C C for every m ∈ N and |w(x)| ≤ |x| . Then |wm (x) | ≤ |x| lim Vhm wm Lp(·) (R) = wLp(+∞) (R) .

m→∞

(2.22)

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Proposition 15. ([41, Proposition 6.4]) Let Op(a) ∈ OP SO+∞ , and a sequence hm → +∞. Then there exists a subsequence hmk of hm and a symbol 0 not depending on x, such that for every function u ∈ C0∞ (R) a(h) ∈ OP S1,0 lim V−hmk Op(a)Vhmk u = Op(a(h) )u

k→∞

in the topology of S(R). In what follows, if a is a symbol and h ∈ R, then ah denotes the symbol shifted in x, that is, ah (x, ξ) = a(x + h, ξ). Note that V−h Op(a)Vh = Op(ah ). ˚+∞ , and a sequence hm → +∞. Then Proposition 16. Let Op(a) ∈ OP S O for every function u ∈ C0∞ (R) lim V−hm Op(a)Vhm uLp(·) (R) = 0.

m→∞

(2.23)

Proof. We have V−hm Op(a)Vhm = Op(ahm ). Let ϕ ∈ C0∞ (R) such that ϕu = u. Hence V−hm Op(a)Vhm u = Opd (ahm ϕ)u. Applying formula (2.8) we obtain that Opd (ahm ϕ) = Op(bm ) where 1 bm (x, ξ) = a(x + hm , ξ + η)ϕ(x + y)e−iy·η dydη. 2π

(2.24)

R2

Then applying the definition of the oscillatory integral in (2.24) we obtain that lim sup ∂xβ ∂ξα bm (x, ξ) = 0 m→∞ (x,ξ)∈R2

for all α, β ∈ N0 . Theorem 10 implies that lim Op(bm )B(Lp(·) (R)) = lim Opd (ahm ϕ)B(Lp(·) (R)) = 0. (2.25)

m→∞

m→∞

Hence the statement of the proposition follows from formula (2.25).

Theorem 17. Let Op(a) ∈ OP SO+∞ . Then the operator Op(a) : Lp(·) (R) →Lp(·) (R) is locally invertible at the point +∞ if and only if lim inf inf |a(x, ξ)| > 0. x→+∞ ξ∈R

(2.26)

Proof. (a) First we prove that condition (2.26) is sufﬁcient for the local invertibility of Op(a) at the point +∞. Let ϕR be the function from the proof of Proposition 16. Condition (2.26) implies that there exists an R0 > 0 such that bR0 = ϕR0 a−1 ∈ SO+∞ . Hence by Proposition 6 Op(bR0 )Op(a) = ϕR0 I + QR0 ,

(2.27)

˚+∞ . Equality (2.27) implies that where QR0 ∈ OP S Op(bR0 )Op(a)χR I = (I + QR0 χR I)χR I,

(2.28)

where R is such that ϕR0 χR = χR . By Proposition 12 we can choose an R such that QχR IB(Lp(·) (R)) < 1. Hence (I + QχR I)−1 Op(bR )Op(a)χR I = χR I.

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Thus, the operator Op(a) is locally left invertible at the point +∞. In the same way we prove that Op(a) is locally right invertible at the point +∞. (b) Now we prove the necessity of condition (2.26) for the local invertibility of Op(a) at the point +∞. Let Op(a) : Lp(·) (R) → Lp(·) (R) be a locally invertible operator. Then there exists C > 0 and R > 0 such that Op(a)χR uLp(·) (R) ≥ C χR uLp(·) (R)

(2.29)

for every u ∈ C0∞ (R). Let a sequence hm ∈ R tend to +∞, and u ∈ C0∞ (R). Then for a ﬁxed R > 0 there exists m0 > 0 such that χR Vhm u = Vhm u for m ≥ m0 . Hence for such m Vhm (V−hm Op(a)Vhm u)Lp(·) (R) = Op(a)χR Vhm uLp(·) (R) ≥ C Vhm uLp(·) (R) .

(2.30)

Let hmk be a subsequence of hm deﬁned in Proposition 15 and let wk = V−hmk Op(a)Vhmk u = Op ahmk u. Applying Proposition 15, we obtain that wk → w = Op(a(h) )u in the space S(R). Then we can use Proposition 14 to pass to the limit in the inequality ≥ C Vhmk u p(·) , Vhmk wk p(·) (R)

L

and obtain that

Op(a(h) )u

Lp(+∞) (R)

L

(R)

≥ C uLp(+∞) (R) ,

(2.31)

where the symbol a(h) depends only on ξ. Estimate (2.31) implies the condition (2.32) inf a(h) (ξ) > 0. ξ∈R

Thus, we proved that for every sequence hm → +∞ there exists a subse0 such that the sequence a(hmk , ξ) quence hmk and a limit symbol a(h) ∈ S1,0 converges uniformly on R to the limit function a(h) (ξ) for which condition (2.32) holds. Suppose now that condition (2.26) is not satisﬁed. Then there exists a sequence (hm , ξm ), hm → +∞ such that lim a(hm , ξm ) = 0.

m→∞

(2.33)

Choose a subsequence hmk of the sequence hm such that a(hmk , ξ) converges uniformly with respect to ξ ∈ R to the limit function ah (ξ) for which condition (2.32) holds. Then lim a(hmk , ξmk ) = 0

(2.34)

lim a(hmk , ξmk ) − a(h) (ξmk ) = 0.

(2.35)

k→∞

and

k→∞

Hence (2.34) and (2.35) contradict to (2.32).

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0 (n) (OP SO+∞ (n)) we denote the class of ψdo s Op(a), where By OP S1,0 0 a is a matrix with entries aij ∈ S1,0 (SO+∞ ). Theorem 17 is reformulated for the matrix case in the following form. p(·)

p(·)

Theorem 18. Let Op(a) ∈ OP SO+∞ (n). Then Op(a) : Ln (R) →Ln (R) is locally invertible at the point +∞ if and only if lim inf inf |det(a(x, ξ))| > 0. x→+∞ ξ∈R

(2.36)

2.4. Local Invertibility at the Point x0 ∈ R Definition 19. We say that A ∈ B(Lp(·) (R)) is locally invertible at the point x0 ∈ R, if there exist an interval Iε (x0 ) = (x0 − ε, x0 + ε ) and operators Lx0 ,ε , Rx0 ,ε ∈ B(Lp(·) (R)) such that Lx0 ,ε Aχxε 0 I = χxε 0 I, χxε 0 ARx0 ,ε = χxε 0 I, where χxε 0 = χIε (x0 ) is the characteristic function of Iε (x0 ). The operators Lx0 ,ε (Rx0 ,ε ) are called left (right) locally inverse operators. 0 0 of symbols in S1,0 for which there exist We consider a subclass S˜1,0 ± ∞ functions a ∈ Cb (R) such that (2.37) lim sup a(x, ξ) − a± (x) = 0. ξ→±∞ x∈R

0 Let Op(a) ∈ OP S˜1,0 . Then we set σx0 (A) = a+ (x0 ), a− (x0 )

and say that σx0 (Op(a)) is the local symbol of the operator Op(a) at the 0 point x0 ∈ R. Note that if Op(aj ) ∈ OP S˜1,0 , j = 1, 2, then σx0 (Op(a1 )Op(a2 )) = σx0 (Op(a1 ))σx0 (Op(a2 ) + − − := a+ 1 (x0 )a2 (x0 ), a1 (x0 )a1 (x0 ) . 0 The ψdo Op(a) ∈ OP S˜1,0 is called elliptic at the point x0 , if the local symbol σx0 (Op(a)) is invertible, that is, a± (x0 ) = 0. In this section we also need the following propositions. 0 and Proposition 20. Let t ∈ S1,0

lim t(x, ξ) = 0.

(x,ξ)→0

(2.38)

Then Op(t) is a compact operator in Lp(·) (R), where p(·) satisfies conditions (2.12)–(2.14). Proof. Condition (2.38) implies that Op(t) is compact in L2 (R) (see [37, Theorem 5.8.3]). We can ﬁnd a function r : R → (1, ∞) satisfying (2.12)–(2.14) such that Lp(·) (R) is an intermediate space between L2 (R) and Lr(·) (R). Hence Op(t) is a compact operator in Lp(·) (R) by Proposition 9. Let ϕ ∈ C0∞ (R), ϕ(x) = 1 if |x| ≤ 12 , supp ϕ = [−1, 1] , and 0 ≤ ϕ(x) ≤ 1. 0 We set ϕxε 0 (x) = ϕ( x−x ε ).

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0 and Proposition 21. Let t ∈ S1,0

lim sup |t(x, ξ)| = 0.

(2.39)

lim Op(t)χxε 0 IB(Lp(·) (R)) = lim χxε 0 Op(t)B(Lp(·) (R)) = 0

(2.40)

lim Op(t)ϕxε 0 IB(Lp(·) (R)) = lim ϕxε 0 Op(t)B(Lp(·) (R)) = 0.

(2.41)

ξ→∞ x∈Rn

Then ε→0

ε→0

and ε→0

ε→0

Proof. Fix ε0 > 0 and let 0 < ε < ε0 . Then Op(t)χxε 0 I = Op(t)χxε00 χxε 0 I. The operator Op(t)χxε00 I is compact by Proposition 20, and χxε 0 I → 0 if ε → 0 strongly in Lp(·) (R). Hence limε→0 Op(t)χxε 0 IB(Lp(·) (R)) = 0. Passing to the adjoint operators and taking into account that (2.39) implies the convergence lim sup |tτ (x, ξ)| = 0,

ξ→∞ x∈Rn

we obtain that

∗ lim χxε 0 Op(t)B(Lp(·) (R)) = (Op(t)) χxε 0 I B(Lq(·) (R)) = 0.

ε→0

Formula (2.41) follows from (2.40). Proposition 22. Let (τx0 ,δ u)(x) = δ

1 − p(x)

u

x−x δ

0

, δ > 0. Then

lim τx0 ,δ uLp(·) (R) = uLp(x0 ) (R)

δ→0

for every function u ∈ C0∞ (R). Proof. Fix a function u ∈ C0∞ (R) and set x−x0 p(x) u p(·) δ F (λ, δ) = Iλ (τx0 ,δ u) = δ −1 dx, λ

λ>0

R

0 After the change of the variables x−x = y we get δ u (y) p(x0 +δy) dy. F (λ, δ) = λ

(2.42)

R

Passing to the limit in (2.42) as δ → 0, we obtain u (y) p(x0 ) lim F (λ, δ) = dx := F (λ, 0) δ→0 λ

(2.43)

R

where the convergence is uniform with respect to λ > 0 on every segment [a, b] ⊂ R. Note that F : (0, +∞) × [0, 1] → R+ is a continuous function. Moreover, there exists a partial derivative Fλ (λ, δ) < 0 for every (λ, δ) ∈ (0, +∞) × [0, 1]. Hence for every ﬁx δ ∈ [0, 1], F (·, δ) is a monotonically decreasing function of λ on (0, ∞). It implies that τx0 ,δ uLp(·) (R) = inf {λ > 0 : F (λ, δ) ≤ 1} = λ(δ)

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where λ(δ) is a solution of the equation F (λ, δ) = 1. One can see that for δ = 0 the equation F (λ, 0) = 1 has an unique solution λ(0) = uLp(x0 ) (R) . Moreover, Fλ uLp(x0 ) (R) , 0 = 0. Hence by the Implicit Function Theorem we obtain that there exists a unique solution λ(δ) of the equation F (λ, δ) = 1 for small δ and λ(δ) is a continuous function in a neighborhood of the point 0. Hence uLp(x0 ) (Rn ) = λ(0) := lim λ(δ) = lim τx0 ,δ uLp(·) (Rn ) δ→0

δ→0

for every function u ∈ C0∞ (R).

Let φ ∈ C0∞ (R) be a real-valued function such that φ(ξ) = 1 if |ξ| ≤ 1, φ(ξ) = 0 if |ξ| ≥ 2, and 0 ≤ φ(ξ) ≤ 1. Let also φR (ξ) = φ(ξ/R) and ψR = 1− φR . 0 Theorem 23. Let a ∈ S˜1,0 . Then Op(a) : Lp(·) (R) → Lp(·) (R) is locally invertible at a point x0 ∈ R if and only if Op(a) is an elliptic operator at the point x0 .

Proof. (i) First we prove that the local ellipticity of Op(a) at the point x0 implies the local invertibility at this point. Let a0 (x, ξ) = a+ (x)θ(ξ) + 0 a− (ξ)(1 − θ(ξ)), where θ is the characteristic function of R+ . Since a ∈ S˜1,0 , we then obtain (2.44) sup (a(x, ξ) − a0 (x, ξ))ψR (ξ) = 0. lim R→+∞ (x,ξ)∈R2

and

lim sup a± (x) − a± (x0 ) ϕxε 0 (x) = 0.

ε→0 x

Hence lim

sup (a(x, ξ) − a0 (x0 , ξ))ϕxε 0 (x)ψR (ξ) = 0.

ε→0,R→+∞ (x,ξ)∈R2

(2.45)

(2.46)

In view of the ellipticity of Op(a) at the point x0 and relation (2.46) we obtain that there exist ε0 and R0 such that the symbol b(x, ξ) = 0 . Then a−1 (x, ξ)ϕxε00 (x)ψR0 (ξ) is in S1,0 Op(b)Op(a) = Op(ϕxε00 ψR0 ) + Op(tε0 ,R0 ),

(2.47)

−1 where tε0 ,R0 ∈ S1,0 by formula (2.7). Formula (2.47) implies that

Op(b)Op(a) = ϕxε00 I + ϕxε00 Op(φR0 ) + Op(tε0 ,R0 ).

(2.48)

Choose ε > 0 such that χxε 0 ϕxε00 = χxε 0 . Then from (2.48) we get Op(b)Op(a)χxε 0 I = χxε 0 I + Qε , where Qε = ϕxε00 Op(φR0 )χxε 0 I + Op(tε0 ,R0 )χxε 0 I

(2.49)

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is a compact operator in Lp(·) (R) by Proposition 20. Since we have the strong convergence χxε 0 I → 0 in Lp(·) (R), we can choose ε > 0 small enough such that Qε χxε0 I < 1. Hence (I + Qε χxε0 I)−1 Op(b)Op(a)χxε0 I = χxε0 I. Hence L = (I + Qε χxε0 I)−1 Op(b) is the left locally inverse operator at the point x0 ∈ R. In the same way we prove that there exists a right locally inverse operator at the point x0 . (ii) Now we prove that the local invertibility of A = Op(a) at the point x0 implies the local ellipticity of Op(a) at this point. We denote A0 = a+ P+ + a− P− , Ax0 = a+ (x0 )P+ + a− (x0 )P− , where P± =

1 (I ± SR ) 2 0

and

(SR u)(x) =

x0

1 πi

R

u(y)dy . y−x

p(·)

Note that the SIOs A and A are bounded in L (R) (see for instance [41]). By the multiplicative inequality (see for instance [50, p. 22], or [37, Proposition 5.8.1]) formula (2.44) implies that lim sup ∂xβ ∂ξα ((a(x, ξ) − a0 (x, ξ))ψR (ξ)) = 0, R→+∞ (x,ξ)∈R2

By Theorem 10, for each η > 0 we can ﬁnd an R0 > 0 such that lim (A − A0 )Op(ψR0 )B(Lp(·) (R)) < η. R→∞

(2.50)

Continuity of the coefﬁcients a± at the point x0 implies that for every η > 0 there exists an ε0 > 0 such that for all ε ∈ (0, ε0 ) 0 (A − Ax0 )ϕxε 0 I p(·) < η. (2.51) B(L (R)) Furthermore, (A − A0 )ϕxε 0 I p(·) ≤ (A − A0 )Op(ψR0 )ϕxε 0 I B(Lp(·) (R)) B(L (R)) + (A − A0 )Op(φR0 )ϕxε 0 I B(Lp(·) (R)) , (2.52) and

(A − A0 )Op(φR0 )ϕxε 0 I p(·) B(L (R)) ≤ (A − A0 )B(Lp(·) (R)) Op(φR0 )ϕxε 0 IB(Lp(·) (R)) . By Proposition 21, for small ε > 0 we have Op(φR0 )ϕxε 0 IB(Lp(·) (R)) <

Hence

η . (A − A0 )B(Lp(·) (R))

(A − A0 )Op(φR )ϕxε 0 I p(·) < η. 0 B(L (R))

(2.53)

Thus, estimates (2.50), (2.52) and (2.53) yield that (A − Ax0 )ϕxε 0 IB(Lp(·) (R)) < 3η

(2.54)

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for small ε > 0. Let A be locally invertible at x0 . Then there exist ε > 0 and C > 0 such that the following estimate holds Aχxε0 uLp(·) (R) ≥ C χxε0 uLp(·) (R) for every u ∈ (2.55) implies

C0∞ (R).

Note that for ε > 0 small enough

Aϕxε 0 uLp(·) (R) ≥ C ϕxε 0 uLp(·) (R) , Let η =

C 6.

(2.55) χxε0 ϕxε 0

u ∈ C0∞ (R).

=

ϕxε 0 .

Then (2.56)

Then (2.54) and (2.56) yield that

C ϕxε 0 uLp(·) (R) , u ∈ C0∞ (R). (2.57) 2 We replace u in (2.57) by τx0 ,δ u where δ > 0. Then for δ > 0 small enough ϕxε 0 (τx0 ,δ u) = τx0 ,δ u. Since Ax0 commutes with the operator τx0 ,δ , from (2.57) we obtain C (2.58) τx0 ,δ Ax0 uLp(·) (R) ≥ τx0 ,δ uLp(·) (R) . 2 Passing to the limit as δ → 0 in (2.58) and applying Proposition 22, we obtain the estimate C Ax0 uLp(x0 ) (R) ≥ uLp(x0 ) (R) (2.59) 2 for every u ∈ C0∞ (R). In the same way, from the estimate Ax0 ϕxε 0 uLp(·) (R) ≥

A∗ χxε 0 uLq(·) (R) ≥ C χxε 0 uLnq(·) (R) ,

u ∈ C0∞ (R)

(2.60)

we obtain that C vLq(x0 ) (R) , v ∈ C0∞ (R). (2.61) 2 Since C0∞ (R) is dense in Lp(x0 ) (R), estimates (2.60) and (2.61) imply the invertibility of Ax0 in Lp(x0 ) (R). It remains to note that the invertibility of the SIO Ax0 in the space Lp (R) with constant p ∈ (1, ∞) implies, as is well known, the condition a± (x0 ) = 0 (see for instance [47]). (Ax0 )∗ vLq(x0 ) (R) ≥

Theorem 24. Let A0 = a+ P+ + a− P− be a SIO with coefficients a± ∈ L∞ (R) continuous at a point x0 ∈ R. Then A : Lp(·) (R) → Lp(·) (R) is locally invertible at the point x0 , if and only if a± (x0 ) = 0. Proof. (i) Let the condition a± (x0 ) = 0 hold. By the continuity of a± at the point x0 , for every η > 0 we can ﬁnd an ε > 0 such that 0 A − Ax0 ϕxε 0 I p(·) < η. (2.62) B(L (R)) Hence A0 ϕxε 0 I = Ax0 ϕxε 0 I + Tε ,

(2.63)

where Tε < η. The condition a± (x0 ) = 0 implies that there exists the −1 −1 inverse operator (Ax0 ) = a+ (x0 )−1 P+ + a− (x0 )−1 P− . Let η < (Ax0 ) . Then there exists an ε such that ϕxε 0 χxε0 = χxε0 . From formula (2.63) we get (I + Tε χxε0 I)−1 (Ax0 )

−1

A0 χxε0 I = χxε0 I.

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Hence there exists a left locally inverse operator for A0 at the point x0 . In the same way we prove that there exists a right locally inverse operator. (ii) Let A0 be a locally invertible operator at the point x0 . Then (2.62) implies that Ax0 is also locally invertible at the point x0 . Hence for every u ∈ C0∞ (R) estimate (2.59) holds. As in the part (ii) of the proof of Theorem 23 we obtain that a± (x0 ) = 0.

3. Mellin Pseudodifferential Operators 3.1. Main Property In this section we reformulate the results of Sect. 2 for the Mellin pseudodifferential operators. (See, for instance [38, Chapter 4.5]). Definition 25. (i) We say that a matrix-function a = (aij )ni,j=1 belongs to E(n), if aij ∈ C ∞ (R+ ×R) and (r∂r )β ∂ξα aij (r, ξ) ξβ < ∞, sup |a|l1 ,l2 = max 1≤i,j≤n (r,ξ)∈R+ ×R α≤l1 ,β≤l2 (3.1) 2 1/2 ξ = (1 + ξ ) (ii)

for all l1 , l2 ∈ N0 . We say that a matrix-function a = (aij )ni,j=1 belongs to Ed (n), if aij ∈ C ∞ (R+ × R+ ×R) and

|a|l1 ,l2 ,l3 = max

1≤i,j≤n (r,ρ,ξ)∈R+ ×R+ ×R

β (r∂r )α (ρ∂ρ )γ ∂ξ aij (r, ρ, ξ) ξβ < ∞,

sup

α≤l1 ,β≤l2 ,γ≤l3

(3.2) (iii)

for all l1 , l2 , l3 ∈ N0 . Let a ∈ E(n). The operator −1 dξ a(r, ξ) (rρ−1 )iξ u(ρ)ρ−1 dρ, (Op(a)u)(r) = (2π) R

(iv)

(3.3)

R+

where u ∈ C0∞ (R+ , Cn ), is called the Mellin pseudodifferential operator (M ψdo) with symbol a ∈ E(n). We denote by OP E(n) the class of all such operators and by OP Ed (n) the class of the double M ψdo s Opd (a) with symbols a ∈ Ed (n) which are deﬁned by formula (3.3) with the symbol a of two variables replaced by the double symbol a of three variables. We say that a matrix-function a (∈ E(n)) is slowly oscillating at the point r = 0 and belongs to Esl (n), if lim sup |(r∂r )β ∂ξα aij (r, ξ)|ξα = 0,

r→+0 ξ∈R

(3.4)

for all α ∈ N0 and β ∈ N. By E0 (n) we denote the set of matrixfunctions satisfying condition (3.4) for all α, β ∈ N0 .

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We say that the matrix-function a = (aij )ni,j=1 ∈ Ed (n) is slowly oscillating at the point 0 and belongs to Esl,d (n) if lim

sup

r→+0 (ρ,ξ)∈R ×R +

|(r∂r )β (ρ∂ρ )γ ∂ξα aij (r, ρ, ξ)|ξα = 0

for all β ∈ N and every γ, α ∈ N0 , and lim

sup

ρ→+0 (r,ξ)∈R ×R +

|(r∂r )β (ρ∂ρ )γ ∂ξα aij (r, ρ, ξ)|ξα = 0

for all γ ∈ N and every β, α ∈ N0 . The corresponding classes of Mellin ψdo s are denoted by OP Esl (n), OP Esl,d (n), OP E0 (n). Remark 26. Note that the Mellin ψdo s are ψdo s on the multiplicative group R+ with the invariant measure dμ = dr r . The M ψdo s are obtained from ψdo s −x on R by means of the change of the variables : R+ r = e , x ∈ R which maps the point +∞ to the point 0. The main properties of M ψdo s easily follow from the corresponding properties of ψdo s on R (see [38, Chap. 4.5]). By L2n (R+ , dμ) we denote the space of measurable Cn -valued functions u on R+ with the norm ⎛ ⎞1/2 ⎜ ⎟ 2 uL2 (R+ ,dμ) = ⎝ u(r)Cn dμ⎠ . n

R+

Proposition 27. ([38, Chap. 4]) Let A = Op(a) ∈ OP E(n). Then the operator A is bounded in L2n (R+ , dμ) and there exists C > 0 not depending on A such that AB(L2 (R+ ,dμ)) ≤ C |a|2,2 . n

(3.5)

Proposition 28. ([38, Chap. 4]) (i)

Let Op(a), Op(b) ∈ OP E(n). Then C = Op(a)Op(b) ∈ OP E(n), and C = Op(c) with 1 c(r, ξ) = a(r, ξ + η)b(rρ, ξ)ρ−iη dρdη. (3.6) 2π R+ R

(ii)

Let Opd (a) ∈ OP Ed (n). Then Opd (a) ∈ OP E(n), Opd (a) = Op(a# ) and 1 a(r, rρ, ξ + η)ρ−iη dρdη. (3.7) a# (r, ξ) = 2π R2

(iii)

Let A = Op(a) ∈ OP E(n) and acting in L2n (R+ , dμ). Then the adjoint operator A∗ ∈ OP E(n), and A∗ = Op(b) 1 b(r, ξ) = a∗ (rρ, ξ + η)ρ−iη dρdη, (3.8) 2π R2

∗

where a (r, ξ) is the Hermite adjoint matrix to a(r, ξ).

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The integrals in (3.6), (3.7), (3.8) are understood as oscillatory (see [38, Chap. 4]). Proposition 29. (i) Let Op(a), Op(b) ∈ OP Esl (n). Then Op(a)Op(b) = Op(c) ∈ OP Esl (n), where c(r, ξ) = a(r, ξ)b(r, ξ) + q(r, ξ), and q(r, ξ) ∈ E0 (n). (ii) Let Opd (a) ∈ OP Ed,sl (n). Then Opd (a) = Op(a# ) ∈ OP Esl (n), where a# (r, ξ) = a(r, r, ξ) + q(r, ξ) (ii)

and q(r, ξ) ∈ E0 (n). Let Op(a) ∈ OP Esl (n) and act in L2 (R+ , dμ, Cn ). Then the adjoint operator Op(a)∗ = Op(b) ∈ OP Esl (n) and b(r, ξ) = a∗ (r, ξ) + q(r, ξ), where a∗ (r, ξ) is the Hermite adjoint matrix to a∗ (r, ξ), and q ∈ E0 (n). Let w=exp v, where v ∈ C ∞ (R+ ) is a real valued function such that

k d (3.9) sup r v(r) < ∞ dr r∈R+

for every k ∈ N. Moreover, we assume that there exists an interval (c, d) 0 such that the function κv = rv satisﬁes the condition c < inf κv (r) ≤ sup κv (r) < d. r∈R+

(3.10)

r∈R+

We say that w = ev is the weight of the class R(c, d), if conditions (3.9) and (3.10) hold, and of the class Rsl (c, d), if w ∈ R(c, d) and lim rκv (r) = 0.

r→0

(3.11)

The weights in Rsl (c, d) are called slowly oscillating at the point 0. Definition 30. We say that a symbol a deﬁned on R+ × R belongs to E(n, (c, d)), if a is analytically extended with respect to the second variable ξ into the strip Π = {ξ ∈ C : I(ξ) ∈ (c, d)} and (r∂r )β ∂ α aij (r, ξ + iη) < ∞ sup (r,ξ+iη)∈R+ ×Π

for all α, β ∈ N0 . By OP E(n, (c, d)) we denote the corresponding class of M ψdo s with analytical symbols. The class OP Ed (n, (c, d)) of M ψdo s with double symbols deﬁned on R+ × R+ × R and analitically extended, with respect to the third variable, into the strip Π is introduced in the obvious way. Proposition 31. ([38, Chap. 4]) . (i) Let a ∈ E(n, (c, d)) and w = ev ∈ R(c, d). Then wOp(a)w−1 = Opd (aw ),

(3.12)

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where aw (r, ρ, ξ) = a(r, ρ, ξ + iϑv (r, ρ)) and 1 ϑv (r, ρ) =

κv (r1−τ ρτ )dτ.

0

(Note that condition (3.10) yields that ϑv (r, ρ) ∈ (c, d) for all r, ρ ∈ R+ ). (ii) Let A = Op(a) ∈ OP Esl (n, (c, d)) , w ∈ Rsl (c, d). Then wOp(a)w−1 ∈ OP Esl (n) and wOp(a)w−1 = Op(˜ aw ) + Op(q)

(3.13)

where a ˜w (r, ξ) = a(r, ξ + iκv (r)) and q ∈ E0 (n). 3.2. Mellin ψdo in the Spaces Lp(·) n (R+ , dμ) p(·)

In this subsection we deal with the space Ln (R+ , dμ) with variable exponent, deﬁned in a general form by Definition 7; now we take X = R+

and

dμ(r) =

dr . r

Let p : R+ → (1, ∞) be a measurable function satisfying condition (2.12) on X = R1+ . We suppose that the function p satisﬁes the log-condition of the form |p(r) − p(ρ)| ≤

A log

1

(3.14)

|log ρr |

√ e. Note that condition (3.14) is nothing else but condition (2.13) with the metric d(r, ρ) = log ρr . Correspondingly to (2.14) we also suppose that there exist the coinciding limits for all r, ρ ∈ R+ such that

√1 e

≤

r ρ

≤

p(0) := lim p(r) = p(∞) := lim p(r) r→+0

r→+∞

and |p(r) − p(0)| ≤

C , log (2 + | log r|)

C > 0, r ∈ R+ .

(3.15)

Note that the mapping R x → exp x ∈ R+ generates an isomorphism p(·) p(·) ˜ of the spaces Ln (R+ , dμ) and Ln (R), where p(r) = p˜(log r), so that conditions (3.14) and (3.15) have their obvious origin in conditions (2.13) and (2.14) with d(x, y) = |x − y| on R1 . Theorem 32. Let p satisfy assumption (2.13) and conditions (3.14) and (3.15). Then every operator Op(a) ∈ OP E(n) (Opd (a) ∈ OP Ed (n)) is p(·) bounded in Ln (R+ , dμ) and there exist M > 0 such that ≤ C |a|M,M , Op(a)B(Lp(·) (R+ ,dμ)) n

(3.16)

≤ C |a|M,M,M ). (Opd (a)B(Lp(·) (R+ ,dμ)) n

(3.17)

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Proof. Let u be a measurable function on R with values in Cn . We set (Ψu)(r) = u(− log r), r ∈ R+ . It is evident that the mapping ˜ Ψ : Lnp(·) (R) →Lnp(·) (R+ , dμ)

where p(r) = p˜(− log r), r ∈ R+ , is an isomorphism between the Banach spaces. This isomorphism generates the isomorphism of the spaces of operators ˜ ˜ : B(Lp(·) (R+ , dμ)) → B(Lp(·) Ψ n n (R)) p(·) ˜ ˜ by the formula Ψ(A) = Ψ−1 AΨ,A ∈ B(Ln (R+ , dμ)). Moreover, Ψ 0 (OP E(n)) = OP S1,0 (n). Hence Theorem 32 follows from Theorem 10 and Corollary 11.

Let w be a weight, that is, a.e. positive measurable function on R+ . We p(·) introduce the weighted space Ln (R+ , w, dμ) by the norm uLp(·) = wuLp(·) < ∞. (R+ ,w,dμ) (R+ ,dμ) n n Theorem 33. Let Op(a) ∈ OP E(n, (c, d)), w = ev ∈ R(c, d). Then Op(a) is p(·) bounded in Ln (R+ , w, dμ) and there exist constants M > 0, C > 0, not depending of a such that ≤ C |a|M,M |v|M , Op(a)B(Lp(·) (R+ ,w,dμ)) n

(3.18)

where |v|M =

M k=1

sup v (k) (r) .

r∈R+

p(·)

Proof. The boundedness of A in Ln (R+ , w, dμ) is equivalent to the boundp(·) edness of wAw−1 in Ln (R+ , dμ). Applying formula (3.12) and Theorem 32 we obtain estimate (3.18). 3.2.1. Local Invertibility of Mellin Pseudodifferential Operators. Let A ∈ p(·) B(Ln (R+ , dμ)). We say that A is locally invertible at the point 0, if there p(·) exists an R > 0 and operators LR , RR ∈ B(Ln (R+ , dμ)) such that LR Aχ[0,R] I = χ[0,R] I,

χ[0,R] ARR = χ[0,R] I. p(·)

Theorem 34. Let Op(a) ∈ OP Esl (n) and act in Ln (R+ , μ). Then Op(a) is locally invertible at the point 0 , if and only if lim inf inf |det a(r, ξ)| > 0. r→+0 ξ∈R

p(·)

Proof. Note that the operator A ∈ B(Ln (R+ , μ)) is locally invertible at p(·) ˜ the point 0, if and only if the operator ΨAΨ−1 ∈ B(Ln (R)) is locally ˜ invertible at the point +∞. Moreover, Ψ(OP Esl (n)) = OP SO+∞ (n). Hence Theorem 34 follows from Theorem 18.

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4. Singular Integral Operators on Some Carleson Curves 4.1. Curves, Weights, Coefficients We say that a complex-valued function a ∈ C m (0, ε), ε > 0, if a ∈ C m (0, ε) and

j d a(r) < ∞ sup r dr r∈(0,ε) for every j = 0, 1, . . . , m, and a ∈ C ∞ (0, ε) if m = ∞. We say that a ∈ m ˜∞ C˜m (0, ε) if κa = r da dr ∈ C (0, ε), and a ∈ C (0, ε) if m = ∞. A function m a ∈ C (0, ε), m ≥ 1 is said to be slowly oscillating at the point 0 and belong m (0, ε) if to the class Csl lim κa (r) = 0.

r→0

∞ m We write Csl (0, ε) if m = ∞. We denote by C˜sl (0, ε), m ≥ 1 the class of m m ˜ functions a ∈ C (0, ε) such that κa ∈ Csl (0, ε). We write C˜∞ (0, ε) if m = ∞. If a ∈ C˜m (0, ε), m ≥ 1 we set

1 ϑa (r, ρ) =

κa (r1−τ ρτ )dτ.

0

A set γ ⊂ C is called a simple locally Lyapunov arc, if there exists a homeomorphism ϕ : [0, 1] → γ such that ϕ ∈ C 1 ((0, 1)), ϕ (r) = 0 for all r ∈ (0, 1), and for every segment [a, b] ⊂ (0, 1) there exist C > 0 and α ∈ (0, 1] such that |ϕ (r) − ϕ (ρ)| ≤ C |r − ρ|

α

r, ρ ∈ [a, b].

for all

The points ϕ(0) and ϕ(1) are called the endpoints of γ. We refer to a set Γ(⊂ C) as a composed curve if Γ = ∪K k=1 Γk , where Γ1 , . . . , ΓK are oriented and rectiﬁable simple locally Lyapunov arc, each pair of which has at most endpoints in common. A node of Γ is a point which is endpoint of at least one of the arcs Γ1 , . . . , ΓK . The set of all the nodes is denoted by F. Let t0 ∈ F. We suppose that there exists an ε > 0 such that the portion Γ(t0 , ε) = {t ∈ Γ : |t0 − t| < ε} is of the form n(t0 )

Γ(t0 , ε) = {t0 } ∪ Γ1t0 ∪ · · · ∪ Γt0 where

Γjt0 = z ∈ C : z = t0 + reiϕt0 ,j (r) : r ∈ (0, ε), (j = 1, . . . , n(t0 )) ,

and ϕt0 ,j (r) = ψt0 (r) + ψt0 ,j (r),

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where ψt0 , ψt0 ,1 , . . . , ψt0 ,n(t0 ) are real-valued functions such that: ψt0 ∈ C˜∞ (0, ε), ψt0 ,j ∈ C ∞ (0, ε), and 0 ≤ m1 < ψt0 ,1 (r) < M1 < m2 < ψt0 ,2 (r) < M2 < · · · < mnt0 < ψt0 ,nt0 (r) < Mnt0 < 2π for all r ∈ (0, ε) with certain constants mj , Mj . Note that the function ψt0 deﬁnes the rotation, and the functions ψt0 ,j deﬁne the oscillations of the curves Γjt0 near the node t0 . We suppose that these conditions hold for every node, and we denote such class of curves by L. ∞ ∞ (0, ε), ψt0 ,j ∈ Csl (0, ε) in the above conditions for every node If ψt0 ∈ C˜sl t0 ∈ F, then we say that the curve Γ is slowly oscillating at every node t0 . We denote such class of curves by Lsl . For example, if ϕt0 ,,j (r) = δt0 log r + μt0 ,j ,

j = 1, . . . , n(t0 ),

r ∈ (0, ε)

with 0 ≤ μt0 1 < μt0 2 < · · · < μt0 nt0 < 2π, then the above conditions on the node 0 are fulﬁlled. Let Γ be a locally rectiﬁable composed curve with Lebesgue length measure. The curve Γ is said to be a Carleson curve ( an Ahlfors–David regular curve) (see for instance [1, p. 2]) if CΓ = sup sup t∈Γ ε>0

|Γ(t, ε)| < ∞, ε

where |Γ(t, ε)| is the length of the portion Γ(t, ε). Taking into account that d(reiϕ(r) ) = (1 + (rϕ (r))2 dr, it easy to see that Γ ∈ L is a Carleson curve. Throughout what follows we assume that Γ ∈ L and we suppose for simplicity that Γ is a compact curve. Let p : Γ → (1, ∞) be a measurable function satisfying assumption (2.13) on X = Γ\F. For t0 ∈ F we suppose that there exist an ε > 0 such that the functions pt0 j (r) := p(t0 + reiϕt0 ,j (r) ) = pt0 (r),

r ∈ (0, ε)

(4.1)

∞

do not depend on j and belong to C (0, ε) and satisfy assumption (2.12) and conditions (3.14) and (3.15). It follows from condition (3.15) that pt0 is a continuous function at the origin and lim pt0 (r) = pt0 (0) = p(t0 ).

r→0

Let w : Γ → [0, ∞] be a measurable function referred in the sequel as a weight. The weighted variable exponent Lebesgue space Lp(·) (Γ, w) is deﬁned 1 as the space of functions f such that [w(x)] p(x) ∈ Lp(·) (Γ), the latter being introduced by Definition 7. We write Lp(·) (Γ) if w ≡ 1. We consider weights on R+ of the form w = exp v,

(4.2)

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where a real-valued function v ∈ C˜∞ (0, ε). We denote such a class of weights by R0 . Let κv = rv , and + κw = lim sup κv (r) = lim sup r→0

r→0

rw (r) , w(r)

(4.3)

rw (r) . w(r)

(4.4)

and − κw = lim inf κv (r) = lim inf r→0

r→0

∞ By Rsl we denote the class of weights w = exp v with v ∈ C˜sl (0, ε). For 0 instance, if

r ∈ (0, ε)

v(r) = f (log(− log r)) log r, and f ∈

Cb∞ (R),

then w ∈ R0 . For f = sin x sl

κv (r) = cos(log(− log r)) + sin(log(− log r)) √ π = 2 cos log(− log r) − 2 √ √ + − and κw = 2, κw = − 2. Proposition 35. Let w = ev ∈ R0 . Then for every δ > 0 there exists an ε ∈ (0, ε) such that +

−

w(ρ)rκw +δ ≤ w(r) ≤ w(ρ)rκw −δ for ρ, r ∈ (0, ε ).

(4.5)

Proof. Let 1

κv (ρ

ϑv (r, ρ) := 0

Then

1 r )dτ = ln ρr

1−τ τ

r ρ

v(r) − v(ρ) κv (t) dt = . t ln ρr

ϑv (r,ρ) w(r)w−1 (ρ) = ev(r)−v(ρ) = eϑv (r,ρ)(log r−log ρ) = rρ−1 .

(4.6)

For every δ > 0 we can ﬁnd ε ∈ (0, ε) such that − + − δ < ϑv (r, ρ) < κw +δ κw

(4.7)

for all r, ρ ∈ (0, ε ). Estimate (4.5) follows then from (4.6) and (4.7).

Let w be a weight on the curve Γ. We suppose that for every point tj ∈ F there exists a neighborhood Uj such that w and w−1 belong L∞ (Γ\ ∪tj ∈F (Γ ∩ Uj )). We say that w ∈ RΓ , if for every point t0 ∈ F and for every j ∈ {1, . . . , n(t0 )} the function wt0 (r) = w(t0 + reiϕt0 ,j (r) ) = evt0 (r) ,

r ∈ (0, ε)

(4.8)

does not depend on j and wt0 = e ∈ R0 . By Ap(·) (Γ) we denote the class of weights in RΓ such that 1 1 − < lim inf κvt0 (r) ≤ lim sup κvt0 (r) < 1 − , (4.9) r→0 p(t0 ) p(t0 ) r→0 vt0

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for every node t0 ∈ F, and by Asl p(·) (Γ) the class of weights in Ap(·) (Γ) such that wt0 ∈ Rsl for every node t ∈ F. 0 0 Proposition 36. If w ∈ Ap(·) (Γ), then w ∈ Lp(·) (Γ) and w−1 ∈ Lq(·) (Γ). Proof. First we prove that if t0 ∈ F, then there exists an ε > 0 such that w ∈ Lp(·) (Γ(t0 , ε)). We will prove that p(·) IΓ(t0 ,ε) (w) = w(t)p(t) |dt| < ∞. (4.10) Γ(t0 ,ε)

Applying expressions (4.1) and (4.8) for the weight w and exponent p on the portion Γ(t0 , ε), we obtain that nt0 p(·) w(t)p(t) |dt| IΓ(t0 ,ε) (w) = j=1 j Γt0

nt0

ε

=

p

wt0t0

(r)

2 (r) 1 + (rϕt0 ,j (r)) dr.

j=1 0

Applying Proposition 35 we obtain that for every δ > 0 there exist an ε ∈ (0, 1) such that wt0 (r) ≤ Cr

− κw −δ t 0

,

r ∈ (0, δ),

where − κw = lim inf κvt0 (r) > − t0 r→0

1 . p(t0 )

(4.11)

Note that pt0 is a continuous function and pt0 (0) = p(t0 ). Then applying estimate (4.5) we can ﬁnd ﬁrst a δ > 0 and then an ε > 0 such that − − δ) > −1. γt0 = inf pt0 (r)(κw t0 r∈(0,ε)

(4.12)

Estimate (4.12) yields that p (r) wt0t0 (r)

≤C

pt0 (r)

r

− −δ pt0 (r) κw t 0

= C1 rγt0 .

(4.13)

p(·)

Hence IΓ(t0 ,ε) (w) < ∞, because γt0 > −1. In the same way applying the

right hand side inequality from (4.9), we obtain that w−1 ∈ Lq(·) (Γ(t0 , ε)) for sufﬁciently small ε > 0. Since w and w−1 are L∞ −functions outside the union of small neighborhoods of the nodes tk ∈ F , and Lp(·) (K) ⊃ L∞ (K) for every compact set K, we obtain that w ∈ Lp(·) (Γ) and w−1 ∈ Lq(·) (Γ). A function a : Γ → C is said to be piecewise slowly oscillating on Γ, if a ∈ C(Γ\F) and for each node t0 ∈ F we have a(t0 + reiϕt0 ,j (r) ) = at0 ,j (r),

r ∈ (0, ε),

j ∈ {1, . . . , n(t0 )} ,

∞ and at0 ,j ∈ Csl (0, ε). We denote the class of piecewise slowly oscillating functions by P SO(Γ).

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4.2. Representation of a Singular Integral Operator at the Node as a Mellin Pseudodifferential Operators We suppose that Γ is a compact Carleson curve of the class L and w ∈ Ap(·) (Γ) is a weight satisfying conditions given in Sect. 4.1. We consider the Cauchy SIO deﬁned on Γ as f (τ )dτ , t ∈ Γ. (4.14) (SΓ f ) (t) = lim ε→0 τ −t Γ\Γ(t0 ,ε)

For the point t0 ∈ F we introduce the mapping (·)

p

t0 ((0, ε), dμ), Φt0 : Lp(·) (Γ(t0 , ε), w) → Ln(t 0)

where

⎛

⎞

1

r pt0 (r) wt0 (r)f (t0 + reiϕt0 ,1 (r) ) · · ·

⎜ ⎜ ⎜ (Φt0 f ) (r) = ⎜ ⎜ ⎝

(4.15)

⎟ ⎟ ⎟ ⎟ = f˜(r), ⎟ ⎠

1

r ∈ (0, ε).

r pt0 (r) wt0 (r)f (t0 + reiϕt0 ,n(t0 ) (r) ) ˜ ˜ ˜ The inverse mapping Φ−1 t0 transforms the vector-function f = (f1 , . . . , fn(t0 ) ) ∈ p

(·)

n(t )

t0 ((0, ε), dμ) to the function f on the curve Γ(t0 , ε) = ∪j=10 Γt0 j by the Ln(t 0) rule

f |Γt0 j (t0 + reiϕt0 ,j (r) ) = r

1 t0 (r)

−p

wt−1 (r)f˜j (r). 0

Proposition 37. The mapping Φt0 is an isomorphism between the corresponding Banach spaces. Proof. We have

p(·) IΓ(t0 ,ε) (f, w) =

p(τ )

|w(τ )f (τ )|

n(t0 )

|dτ | =

|w(τ )f (τ )|p(τ ) |dτ | .

j=1 Γ

Γ(t0 ,ε)

t0 j

(4.16) After the change of variables τ = t0 + reiϕt0 ,j (r) we obtain |w(τ )f (τ )|p(τ ) |dτ |

(4.17)

Γ(t0 ,ε) n(t0 ) ε

=

|w(t0 + reiϕt0 ,j (r) )f (t0 + reiϕt0 ,j (r) )|pt0 (r)

2 1 + rϕt0 j (r) dr

j=1 0 n(t0 ) ε

=

1

|r pt0 (r) wt0 (r)f (t0 + reiϕt0 ,j (r) )|pt0 (r)

2 1 + rϕt0 j (r) dμ(r).

j=1 0

Since

0 < inf

(0,ε)

1+

2 rϕt0 j (r)

≤ sup

(0,ε)

2 1 + rϕt0 j (r) < ∞,

(4.18)

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estimate (4.18) implies that the modular Γ(t0 ,ε) |w(τ )f (τ )|p(τ ) |dτ | is bounded if and only the modulars ε 1 2 |r pt0 (r) wt0 (r)f (t0 + reiϕt0 ,j (r) )|pt0 (r) 1 + rϕt0 j (r) dμ(r) 0

are bounded for every j = 1, 2, . . . , n(t0 ). Hence the mapping p

(·)

Φt0 : Lp(·) (Γ(t0 , ε), w) → Lnt(t00 ) ((0, ε), dμ) is bounded. In the same way we show that (·)

t0 p(·) Φ−1 (Γ(t0 , ε), w) t0 : Ln(t0 ) ((0, ε), dμ) → L

p

is bounded. Hence Φt0 is an isomorphism between the corresponding Banach spaces. To formulate the main results, we need the following notation. Put εk = 1, if t0 is the starting point of an oriented arc Γt0 k and εk = −1, if t0 is its ending point. Deﬁne ν : [0, 2π) × (C\iZ) → C by

ν(δ, z) =

coth(πz), e(π−δ)z sinh(πz) ,

δ=0 δ ∈ (0, 2π).

(4.19)

Let φt0 ∈ C0∞ (Γ(t0 , ε)) and equal to 1 in a smaller neighborhood of t0 . Proposition 38. Let Γ be a composed compact curve of the class L, the exponent p(·) satisfy the above conditions on Γ and w ∈ Ap(·) (Γ). Then for every point t0 ∈ F the operator t0 S t0 := Φt0 φt0 SΓ φt0 Φ−1 t0 = Op(s ), n(t )

0 is a Mellin ψdo in the class OP Ed (n) with the double symbol st0 = (stjk0 )j,k=1 where

t

0 sjk (r, ρ, ξ)

⎧ 1 ξ+i +ϑv (r,ρ) ⎪ pt (r) t0 ⎪ 0 ⎪εk φ ˜k,t (ρ) 1+iρϕt0 ,k (ρ) ν(2π + ψt ,j (r) − ψt ,k (ρ), ˜j,t (r)φ ), ⎪ 0 0 0 0 ⎪ 1+iϑψ (r,ρ) 1+iϑψt (r,ρ) ⎪ t0 0 ⎨ ξ+i( 1 +ϑvt (r,ρ)) 1+iρϕt ,k (ρ) pt (r) 0 = φ 0 0 ˜j,t (ρ)εk ˜j,t (r)φ ), 0 0 ⎪ 1+iϑϕt ,k (r,ρ) ν(0, 1+iϑψ (r,ρ) ⎪ t0 0 ⎪ ⎪ ξ+i( 1 +ϑvt (r,ρ)) ⎪ 1+iρϕt ,k (ρ) ⎪ pt (r) 0 ⎩εk φ 0 0 ˜kt (ρ) ˜jt (r)φ ν(ψt ,j (r) − ψt ,k (ρ), ), 0

0

1+iϑψt

0

,k (r,ρ)

0

0

1+iϑψt (r,ρ) 0

j < k, j = k,, j > k, (4.20)

and φ˜j,t0 (r) = φt0 (t0 + reiϕt0 ,j (r) ). Remark 39. Proposition 38 has been proved ﬁrst in [35] for the constant p : 1 < p < ∞ (see the proof of Proposition 3.4 in [35]). The detailed proof is contained in the book [38, Chapter 4.6]. The proof for the variable exponents uses Propositions 28, 29 , 31 and repeats, word for word, the proof for the constant p.

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4.3. Boundedness of the Singular Integral Operator in Lp(·) (Γ, w) Note that in the case of simple Carleson curves and the weights of the form N

ωj (|t − tj |)

w(t) = j=1

where ωj may grow and have oscillations at the point 0, the boundedness of SΓ : Lp(·) (Γ, w) → Lp(·) (Γ, w) has been established in [23], but we may not use this result for composed Carleson curves. In the following theorem we give some conditions of the boundedness of SΓ : Lp(·) (Γ, w) → Lp(·) (Γ, w) which are based on the boundedness of the Mellin pseudodifferential operators. We say that a nonnegative function φt0 ∈ C0∞ (Γ(t0 , ε)) is a smooth cut-oﬀ function of a neighborhood Γ(t0 , ε) of the point t0 , if there exists an ε < ε such that φt0 (t) = 1 for all t ∈ Γ(t0 , ε ). Theorem 40. Let Γ be a composed compact curve of the class L, and let p(·) satisfy the above conditions on Γ and w ∈ Ap(·) (Γ). Then SΓ : Lp(·) (Γ, w) → Lp(·) (Γ, w) is a bounded operator. Proof. Let N

φk (t) = 1,

t∈Γ

(4.21)

k=0

be a partition of unity on Γ, where N is a number of nodes on Γ, φ0 ∈ C0 (Γ) (the class of continuous functions with a compact support), φj , j = 1, . . . , N, be smooth cut-oﬀ functions such that supp φj contains only one node tj , and let mapping (4.15) be deﬁned on supp φj , j = 1, . . . , N . It is clear that Γ∩ supp φ0 is a Lyapunov curve, and w and w−1 belong L∞ (supp φ0 ). Let ψj be another smooth cut-oﬀ function of a neighborhood of the point tj with supp ψj in a small neighborhood of supp ϕj and ψj (t) = 1 for t ∈ supp ϕj . Then SΓ =

N

ψ j S Γ ϕj I +

j=0

N

(1 − ψj )SΓ ϕj I.

(4.22)

j=0

The boundedness ϕ0 SΓ ψ0 I : Lp(·) (Γ, w) → Lp(·) (Γ, w) follows from [23] because ϕ0 SΓ ψ0 I is deﬁned on a simple Lyapunov portion of Γ, and w and w−1 belong L∞ on this portion. It follows from Proposition 38 that for every j = 1, . . . , N the operator S tj := Φtj ψj SΓ ϕj Φ−1 tj I is the Mellin pseudodifferential operator in OP Ed (n(tj )) with a double symbol deﬁned by formulas (4.19) and (4.20). By pt (·)

Theorem 32 S tj is bounded on Ln(tj j ) (R+ , dμ). Hence ψj SΓ ϕj I is a bounded operator in Lp(·) (Γ, w). Let us consider the operator Kij = (1 − ψj )SΓ ϕj I. Since supp(1 − ψj ) ∩ suppϕj = ∅, the operator Kij has a smooth kernel, and Kij is bounded from L1 (Γ) in L∞ (Γ). By Proposition 36, w ∈ Lp(·) (Γ) and

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older inequality for the space Lp(·) with variw−1 ∈ Lq(·) (Γ). Applying the H¨ able exponents p(·), we obtain that the operator u → w−1 u is a bounded operator from Lp(·) (Γ) into L1 (Γ). Since the operator Kij is bounded from L1 (Γ) into L∞ (Γ) and the operator v → wv is bounded from L∞ (Γ) to Lp(·) (Γ), we obtain that wKij w−1 I is a bounded operator in the space Lp(·) (Γ). This concludes the proof. 4.4. The Fredholm Property of Singular Integral Operators in Lp(·) (Γ, w) 4.5. Local Invertibility Definition 41. We say that an operator A ∈ B(Lp(·) (Γ, w)) is locally invertible at the point t0 ∈ Γ, if there exist a neighborhood Ut0 (⊂ Γ) of the point t0 , and operators RUt0 , LUt0 ∈ B(Lp(·) (Γ, w)) such that RUt0 AχUt0 I = χUt0 I

and ALUt0 χUt0 I = χUt0 I,

where χUt0 is a characteristic function of Ut0 . We set stjk0 )m σ ˜ t0 (SΓ ) = (˜ j,k , where s˜tjk0 (r, ξ) = stjk0 (r, r, ξ) ⎧ ξ+i p(t1 ) +rvt (r) ⎪ ⎪ 0 0 ⎪ εk ν(2π + ψt0 ,j (r) − ψt0 ,k (r), ), ⎪ 1+irψt (r) ⎪ 0 ! ⎪ " ⎪ ⎨ ξ+i p(t1 ) +rvt (r) 0 0 , = εk ν 0, 1+irψt (r) ⎪ 0 ⎪ ⎪

⎪ ⎪ ξ+i p(t1 ) +rvt 0 (r) ⎪ 0 ⎪ ⎩ ν ψt0,j (r) − ψt0 ,k (r), , 1+irψ (r) t 0

If a ∈ P SO(Γ) and t0 ∈ F, then we set ⎛ at0 ,1 (r) ⎜ at0 ,2 (r) ⎜ · σ ˜ t0 (aI)(r) = ⎜ ⎜ ⎝ ·

j < k, j = k, . j > k,

⎞ ⎟ ⎟ ⎟. ⎟ ⎠ at0 ,n(t0 ) (r)

Let AΓ = aI + bSΓ ,

a, b ∈ P SO(Γ).

(4.23)

Then we deﬁne σ ˜ t0 (AΓ )(r, ξ) = σ ˜ t0 (aI)(r)+ σ ˜ t0 (bI)(r)˜ σ t0 (SΓ )(r, ξ),

r ∈ (0, ε),

ξ ∈ R, (4.24)

and σ ˜ t0 (AΓ ) = {a(t0 ) + b(t0 ), a(t0 ) − b(t0 )}

(4.25)

if t0 ∈ Γ\F. In the following theorem we deal with the class Lsl of slowly oscillating curves and the class Asl p(·) (Γ) of weights slowly oscillating at every node of Γ.

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Theorem 42. Let Γ ∈ Lsl , w = exp v ∈ Asl p(·) (Γ) and AΓ be an operator of form (4.23) which acts in Lp(·) (Γ, w). Then: (i) AΓ is locally invertible at the point t0 ∈ F, if and only if (4.26) ˜ t0 (AΓ )(r, ξ) > 0. lim inf inf det σ r→0

(ii)

ξ∈R

AΓ is locally invertible at the point t0 ∈ Γ\F, if and only if σ ˜ t0 (AΓ ) is invertible, that is a(t0 ) ± b(t0 ) = 0.

(4.27)

Proof. (i) Note that AΓ : Lp(·) (Γ, w) → Lp(·) (Γ, w) is locally invertible at the point t0 ∈ Γ, if and only if the operator

dr dr pt0 (·) pt0 (·) : L AtΓ0 = Φt0 φt0 Aφt0 Φ−1 (0, ε), 0, ε), → L n n t0 t0 t0 r r is locally invertible at the point 0, where the operator AtΓ0 is a Mellin ψdo with double symbol in the class OP Ed (n(t0 )) given by formulas (4.19) and (4.20). The conditions Γ ∈ Lsl , w ∈ Asl p(·) (Γ) and a, b ∈ P SO(Γ) and Proposition 38 t0 imply that AΓ ∈ OP Ed,sl (n(t0 )) (see for instance [38, Chapter 4.6.5]). It follows from statement (ii) of Proposition 27 that the Mellin symbol σ(AtΓ0 ) of AtΓ0 is of the form σ(AtΓ0 )(r, ξ) = σ ˜ t0 (AΓ )(r, ξ) + qt0 (r, ξ), n(t )

0 where qt0 = (qtij0 )i,j=1 and

lim sup ∂ξα (r∂r )β qtij0 (r, ξ) = 0

r→0 ξ∈R

for all α, β ∈ N0 . By Theorem 42 condition (4.26) is necessary and sufﬁcient for the local invertibility of the Mellin ψdo AtΓ0 at the point 0. Hence condition (4.26) is necessary and sufﬁcient for the local invertibility of AΓ : Lp(·) (Γ, w) → Lp(·) (Γ, w) at the point t0 ∈ F. Note that the condition of the local invertibility in the spaces Lp(·) (Γ, w) depends on the value p(·) only at the point t0 . (ii) Let t0 ∈ Γ\F. Then there exist a simple locally Lyapunov curve Γj ⊂ Γ such that t0 ∈ int Γj , where ϕj : (0, 1) → int Γj is the parametrization of the curve int Γj . Let ϕj (r0 ) = t0 , and ϕj (r0 ) = 1. Let ε > 0 be sufﬁciently small and Γtj0 ,ε = ϕj (It0 ,ε ), It0 ,ε = (r0 − ε, r0 + ε). The restriction ϕtj0 ,ε of the mapping ϕj on It0 ,ε is the homeomorphism It0 ,ε on Γtj0 ,ε . Let ˜ (It0 ,ε ), Φtj0 ,ε : Lp(·) (Γtj0 ,ε ) → Lp(·)

with p˜(x) = p(ϕj (x)) be the isomorphism deﬁned as (Φtj0 ,ε u)(x) = u(ϕjt0 ,ε (x)),

−1 ˜ : Lp(·) (It0 ,ε ) → Lp(·) (Γtj0 ,ε ) be the inverse mapping. and Φtj0 ,ε It is well known (see for instance [2]) that −1 Φtj0 ,ε χε SΓ χε Φtj0 ,ε =χ ˜ ε SR χ ˜ ε I + Tε ,

(4.28)

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˜ε are the characteristic functions of Γtj0 ,ε and It0 ,ε , respecwhere χε and χ tively, Tε is a compact operator in Lp (It0 ,ε ) for every constant p ∈ (1, ∞). −1 Moreover, it follows from (4.28) and boundedness of Φtj0 ,ε χε SΓ χε Φtj0 ,ε ˜ ˜ and χ ˜ ε SR χ ˜ε I in Lp(·) (It0 ,ε ) that Tε is also a bounded operator in Lp(·) (It0 ,ε ) if p(·) satisﬁes conditions (2.12), (2.13). By Proposition 9 we obtain that Tε ˜ is a compact operator in Lp(·) (It0 ,ε ). Let φ ∈ C0 ((−1, 1)) and φ(0) = 1. We set

x − x0 φδ (x) = φ , φ˜δ (t) = φδ (ϕ−1 j (t)). δ Then φδ χε = φδ for sufﬁciently small δ > 0. Hence we obtain from (4.28) that −1 Φtj0 ,ε φδ SΓ φδ Φtj0 ,ε = φ˜δ SR φ˜δ I + φ˜δ Tε φ˜δ I, (4.29) φ˜δ (t) = φδ (ϕ−1 (t)). j

The sequence φ˜δ I strongly converges to 0 in Lp(·) (It0 ,ε ) as δ → 0. Hence lim φ˜δ Tε φ˜δ I p(·) = 0. (4.30) B(L

δ→0

(It0 ,ε ))

It yields that AΓ : L (Γ, w) → L (Γ, w) is locally invertible at the point t0 , if and only if the operator ˜ ˜ (It ,ε ) → Lp(·) (It ,ε ), φ˜δ At0 φ˜δ I : Lp(·) p(·)

p(·)

R

ϕtj0 ,ε )I

0

0

ϕtj0 ,ε )SR ,

= (a ◦ + (b ◦ is locally invertible at the point x0 = with Φjt0 ,ε (t0 ) ∈ R. Applying Theorem 24 we obtain that φ˜δ AtR0 φ˜δ I is locally invertible at the point x0 ∈ R, if and only if AtR0

(a ◦ ϕjt0 ,ε )(r0 ) ± (b ◦ ϕtj0 ,ε )(r0 ) = a(t0 ) ± b(t0 ) = 0. p(·)

4.6. Simonenko’s Local Principle in L (X) We prove here Simonenko’s local principle in variable exponent Lebesgue spaces Lp(·) (X) in the general setting where the underlying space X is a quasimetric measure space, as introduced by Definition 7. In this subsection we assume that X is a Hausdorﬀ compact space. Definition 43. An operator A ∈ B(Lp(·) (X)) is called an operator of local type, if for every two closed set F1 and F2 such that F1 ∩ F2 = ∅, the operator χF1 AχF2 I is compact. Definition 44. An operator A ∈ B(Lp(·) (X)) is called locally Fredholm at the point x0 ∈ X, if there exist a neighborhood U of the point x0 and operators Lx0 , Rx0 ∈ B(Lp(·) (X)) such that Lx0 AχU I = χU I + T1

and χU ARx0 = χU I + T2 , p(·)

(4.31)

where T1 , T2 are compact operators in L (X). If T1 = 0 and T2 = 0, A is called a locally invertible operator at the point x0 .

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Remark 45. We say that the space X does not have discrete components, if for every point x0 ∈ X there exists a sequence U1 ⊃ U2 ⊃ · · · ⊃ Uj ⊃ · · ·. of neighborhoods of the point x0 such that lim μ(Uj ) = 0.

j→∞

(4.32)

If X does not have discrete components, the local Fredholmness coincides with the local invertibility. Indeed, let Lx0 AχU I = χU I + T1 , and U ⊃ U1 ⊃ U2 ⊃ · · · ⊃ Uj ⊃ · · · , then we obtain Lx0 AχUj I = (I + T1 χUj I)χUj I.

(4.33)

Condition (4.32) implies that the sequence χUj I strongly tends to 0 in Lp(·) (X). Hence lim T1 χUj I B(Lp(·) (X)) = 0. j→∞

It implies that the operators I + T1 χUj I are invertible for sufﬁciently large j. Then (I + T1 χUjj I)−1 Lx0 is a left local inverse operator at x0 . In the same way one can prove the existence of a right local inverse operator. Theorem 46. (Simonenko’s local principle [46–48]) Let A ∈ B(Lp(·) (X, μ)) be an operator of local type. Then A is a Fredholm operator if and only if A is a locally Fredholm operator at every point x ∈ X. If the space X does not have discrete components, we can replace the local Fredholmness by the local invertibility. The proof of Theorem 46 for variable p(·) repeats word by word the Simonenko’s proof for a constant p (See for instance [48, pp 21–24]). 4.7. Fredholmness of SIO Theorem 47. Let Γ be a composed compact curve of the class L, let p(·) satisfy the above conditions on Γ and w ∈ Ap(·) (Γ). Then SΓ : Lp(·) (Γ, w) → Lp(·) (Γ, w) is a local type operator in the sense of Simonenko, that is, for every closed set F1 , F2 ⊂ Γ such that F1 ∩ F2 = ∅ the operator χF1 SΓ χF2 I is a compact operator in Lp(·) (Γ, w). Proof. The operator χF1 SΓ χF2 I has a kernel k ∈ C ∞ (Γ × Γ). Hence χF1 SΓ χF2 I : L1 (Γ) → L∞ (Γ) is a compact operator. Because u → w−1 u is a bounded operator from Lp(·) (Γ, w) in L1 (Γ) and v → wv is a bounded operator from L∞ (Γ) to Lp(·) (Γ, w), the operator χF1 SΓ χF2 I is compact in Lp(·) (Γ, w). Theorem 48. Let AΓ be an operator of form (4.23) and Γ and w satisfy the assumptions of Theorem 42. Then AΓ : Lp(·) (Γ, w) → Lp(·) (Γ, w) is a Fredholm operator, if and only if there hold condition (4.26) for every point t0 ∈ F and condition (4.27) for every point t0 ∈ Γ\F. Proof. Make use of Theorems 42, 46 and 47.

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Remark 49. If we freeze the variable exponent p(·) at the point t0 , condition (4.26) coincides with the Fredholmness condition obtained in paper [3] for the case of the constant Lebesgue exponent p ∈ (1, ∞), while condition (4.27) is classical and does not depend on p(·). Let N = AM Γ

N

M

Ajk Γ,

(4.34)

j=1 k=1

where Ajk Γ = ajk I + bjk SΓ , and ajk , bjk ∈ P SO(Γ). N at the point t ∈ Γ by the formula We deﬁne the local symbol of AM Γ N )= σ ˜ t (AM Γ

N

M

σ ˜ t (Ajk Γ ),

j=1 k=1

where σ ˜

t

(Ajk Γ )

are deﬁned by formulas (4.24) and (4.25). Note that t MN N t N σ ˜ t (AM )= σ ˜+ (AΓ ), σ ˜− (AM ) , Γ Γ

in the case t ∈ Γ\F, where N

t N (AM )= σ ˜± Γ

M

(ajk (t) ± bjk (t)).

j=1 k=1 N We say that the symbol σ ˜ t (AM ) is invertible if Γ N ˜ (AM )(r, ξ) > 0, lim inf inf det σ Γ r→0

ξ∈R

for t ∈ F, and σ ˜ = 0 for t ∈ Γ\F. Theorem 48 and the Simonenko local principle imply the following result. t MN ) + (AΓ

N , where Γ and w satisfy the assumptions of Theorem 50. The operator AM Γ Theorem 42, is a Fredholm operator in Lp(·) (Γ, w), if and only if the local N ) is invertible for every point t ∈ Γ. symbol σ ˜ t (AM Γ

Remark 51. The statement of Theorem 50 can be extended on operaN in tors in the Banach algebra obtained by the closure of operators AM Γ B(Lp(·) (Γ, w)). We are going to do it in a forthcoming paper. 4.7.1. Index Formula. Let A = aI + bSΓ , where a, b ∈ P SO(Γ) and Γ ∈ Lsl . Let A be a Fredholm operator in Lp(·) (Γ, w), where w ∈ Asl p(·) (Γ). Then the p(·) p(·) Fredholm index of A : L (Γ, w) → L (Γ, w) is given by the formula # $ K a(t) + b(t) −1 (2π) index A = − arg a(t) − b(t) t∈Γj j=1 −

L j=1

−1

(2π)

% &∞ ˜ (Atj )(r, ξ) ξ=−∞ . lim arg det σ

r→0

(4.35)

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In this formula, K is the number of the oriented and rectiﬁable simple smooth arcs generating the composed curve Γ, and L is the number of nodes of the curve Γ. The index formula (4.35) is proved by the method of separation of singularities, and this proof is similar to that for the constant p (see for instance [3,4,36]). Remark 52. All the results of the paper remain valid if we replace the clas∞ ∞ (0, ε), C˜sl (0, ε) in the assumptions on the curve Γ ses C ∞ (0, ε), C˜∞ (0, ε), Csl m m (0, ε), C˜sl (0, ε) and the weights near nodes by the classes C m (0, ε), C˜m (0, ε), Csl where m is sufﬁciently large. In relation to Remark 52, see also Definition 55 and Lemma 56 in the next section.

5. On Comparison of the Used Class of Oscillating Weights with the Bary–Stechkin Type Weights We wish to compare the class of weights w used in this paper with the class of oscillating weights known as Bary–Stechkin class which was used in various papers, see for instance [23,27]. In the proofs in this section we follow some ideas of paper [43]. We call two non-negative functions f and g equivalent, if c1 f (x) ≤ g(x) ≤ c2 f (x),

c1 > 0, c2 > 0.

Note that the weighted variable exponent spaces obviously does not change if we replace the weight by an equivalent weight; for us it is also important to observe that the Bary-Stechkin class, deﬁned below, is also closed with respect to the equivalence of functions. We need some definitions. Recall that a non-negative function f on [0, ], 0 < < ∞, is called almost increasing (almost decreasing), if there exists a constant C(≥ 1) such that f (x) ≤ Cf (y) for all x ≤ y (x ≥ y, respectively). Equivalently, a function f is almost increasing (almost decreasing), if it is equivalent to an increasing (decreasing, resp.) function g. 5.1. Bary–Stechkin class Φ Definition 53. Let 0 < < ∞. 1) By W = W ([0, ]) we denote the class of functions ϕ continuous and positive on (0, ] such that there exists the ﬁnite limit limx→0 ϕ(x); 2) by W0 = W0 ([0, ]) we denote the class of functions '=W ' ([0, ]) we denote the class ϕ ∈ W almost increasing on (0, ); 3) by W of functions w ∈ W such that xa w(x) ∈ W0 for some a = a(w) ∈ R1 ; 4) by W = W ([0, ]) we denote the class of functions w ∈ W such that there exists is almost decreasing. a number b ∈ R1 such that ft(t) b '0 , W are known to be characterized in terms of the The classes W 0 Matuszewska-Orlicz indices m(w) and M (w) of w: '0 ⇐⇒ −∞ < m(w) ≤ ∞, w∈W w ∈ W 0 ⇐⇒ −∞ ≤ m(w) < ∞;

(5.1) (5.2)

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We refer to [31,32] for the Matuszewska-Orlicz indices and to [16] and [42] for statements (5.1) and (5.2). Definition 54. We introduce the Bary–Stechkin class Φ as the class of functions in W with ﬁnite Matuszewska-Orlicz indices, that is, ' ∩ W. Φ=W

(5.3)

Note that the Bary–Stechkin class is usually introduced as the two-parameter ' class Φα β of functions w ∈ W satisfying the conditions x 0

w(t) w(x) dt ≤ C , t1+α xα

w(t) w(x) dt ≤ C , t1+β xβ

(5.4)

x

non-empty if and only if α < β; we have ( Φ=

Φα β,

−∞<α<β<+∞

which follows from the fact that w ∈ Φα β , if and only if α < m(w) ≤ M (w) < β, see [16,42]. 5.2. Simonenko Type Class S2 Let 0 < < ∞. The indices xw (x) p(w) = inf , 0<x≤ w(x)

xw (x) 0<x≤ w(x)

q(w) = sup

(5.5)

which appeared in (4.3) and (4.4) are known as Simonenko indices, see [45], and it is known that p(w) ≤ m(w) ≤ M (w) ≤ q(w),

(5.6)

see [32, Theorem 11.11]. The class of functions on (0, ) with ﬁnite Simonenko indices may be called Simonenko class. We introduce a slight generalization of this notion as inspired by conditions (3.9) and (3.11). Definition 55. We say that a weight function w = ev(x) is in the Simonenko type class SN , N = 1, 2, 3, . . . , if

k d (5.7) sup x v(x) < ∞, k = 1, 2, . . . , N dx x∈(0,) and

2 d v(x) = 0. x x→0 dx lim

(5.8)

Obviously, SN +1 ⊂ SN . We are mainly interested in the case N = 2. A connection of this class with Simonenko indices becomes clear, if we observe that conditions (5.7) and (5.8) with N = 2 in terms of the weight w itself have the form xw (x) d xw (x) <∞ < ∞, sup x (5.9) sup w(x) dx w(x) x∈(0,)

x∈(0,)

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x→0

d dx

xw (x) w(x)

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= 0.

'0 ∩ W , for every N = 1, 2, 3, . . . . there Lemma 56. Given a function w ∈ W 0 exists a function '0 ∩ W wN ∈ C N ([0, ]) ∩ W 0

equivalent to w, and such that v(x) = log wN (x) satisfies conditions (5.7). It may be chosen as N −1 x w(t) ln xt α dt (5.10) wN (x) = x t1+α 0

with any α such that α < m(w). Proof. Let ﬁrst N = 1. The proof of the equivalence w1 (x) ∼ w(x) is direct, via the usage of the properties w(x) is almost increasing , (5.11) m(w) = sup μ > 0 : xμ w(x) is almost decreasing , (5.12) M (w) = inf ν > 0 : xν see [16, Theorem 3.6], for the proof of (5.11) and (5.12). By direct differentiation of w1 (x) we obtain d w1 (x) = αw1 (x) + w(x). (5.13) dx Then the ﬁrst inequality in (5.9), corresponding to the case N = 1, holds '0 ∩ W ' because w1 ∼ w. Note that w ∈ W 0 =⇒ w1 ∈ W0 ∩ W 0 and w and w1 as equivalent functions have equal Matuszewska-Orlich indices, see [32, Theorem 11.4]. For N > 1 the statement is obtained by iteration of the procedure. Indeed, by the already proved equivalence w1 ∼ w, we have x

x w(x) ∼ w1 (x) ∼ x

α 0

w1 (t) dt = xα t1+α

x 0

w(s)ds s1+α

x

dt = w2 (x). t

s

By direct differentiation we obtain x

d w2 (x) = αw2 (x) + w1 (x). dx

(5.14)

w1 (x) xw2 (x) =α+ , w2 (x) w2 (x)

(5.15)

Consequently

whence the ﬁrst inequality in (5.9) follows in view of the equivalence w1 ∼ w2 .

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Furthermore, differentiating (5.15), by (5.14) and (5.15) we obtain

2 w1 w d xw2 (x) = − x dx w2 (x) w2 w2 whence the second inequality in (5.9) follows in view of the equivalence w ∼ w1 ∼ w2 . For N > 2 the statement is obtained by induction following (5.13) and (5.15). Remark 57. It is known that the interval deﬁned by Matuszewska-Orlicz indices in general is narrower than that deﬁned by Simonenko indices, namely [m(w), M (w)] ⊆ [p(w), q(w)],

(5.16)

see [32, Theorem 11.11]. Therefore, any function having ﬁnite Simonenko indices, also has ﬁnite Matuszewska-Orlicz indices and consequently belongs to the Bary–Steckin class Φ. Acknowledgements In the case of the ﬁrst author the work was supported by the CONACYT project 81615. The authors are thankful to the anonymous referee for the very careful reading of the manuscript and the comments which essentially improved the presentation in the paper.

References [1] B¨ ottcher, A., Karlovich, Yu.I.: Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Birkh¨ auser, Basel (1997) [2] B¨ ottcher, A., Karlovich, Yu.I., Rabinovich, V.S.: Emergence, persistence, and disapperance of logarithmic spirals in the spectra of singular integral operators. Integr. Equat. Oper. Theory 25, 405–444 (1996) [3] B¨ ottcher, A., Karlovich, Yu.I., Rabinovich, V.S.: Mellin pseudodifferential operators with slowly varying symbols and singular integrals on Carleson curves with Muckenhoupt weights. Manuscripta Math. 95, 363–376 (1998) [4] B¨ ottcher, A., Karlovich, Yu.I., Rabinovich, V.S.: The method of limit operators for one-dimensional singular integrals with oscillating data. J. Oper. Theory 43, 171–198 (2000) [5] B¨ ottcher, A., Karlovich, Y.I., Rabinovich, V.S.: Singular integral operators with complex conjugation from the viewpoint of pseudodifferential operators. In: Elschner J., Gohberg I., Silbermann B. (eds.) The Book: Problems and Methods in Mathematical Physics, Operator Theory: Advances and Applications, vol. 121, pp. 36–59. Birkh¨ auser (2001) [6] Coifman, R.R., Weiss, G.: Analyse harmonique non-commutative sur certaines espaces homegenes. Lecture Notes Math. 242 (1971) [7] Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.J.: The maximal function on variable Lp -spaces. Ann. Acad. Sci. Fenn. Math. 28, 223–238 (2003) [8] Diening, L.: Maximal function on generalized Lebesgue spaces Lp (·). Math. Inequal. Appl. 7(2), 245–253 (2004)

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[9] Diening, L.: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129(8), 657–700 (2005) [10] Diening, L., H¨ ast¨ o, P., Nekvinda, A.: Open problems in variable exponent Lebesgue and Sobolev spaces, “Function Spaces, Differential Operators and Nonlinear Analysis”. In: Proceedings of the Conference held in Milovy, Bohemian-Moravian Uplands, May 28–June 2, 2004, Math. Inst. Acad. Sci. Czech Republick, Praha, pp. 38–58 (2005) [11] Diening, L., Ruˇ z iˇ cka, M.: Calderon-Zygmund operators on generalized Lebesgue spaces Lp(x) and problems related to ﬂuid dynamics J. Reine Angew. Math. 563, 197–220 (2003) [12] Genebashvili, I., Gogatishvili, A., Kokilashvili, V., Krbec, M.: Weight theory for integral transforms on spaces of homogeneous type. Longman Scientiﬁc and Technical (1998) [13] Gohberg, I., Krupnik, N.: One-Dimensional Linear Singular Integral Equations. Vols. I and II. Birkh¨ auser, Basel (1992) [14] Gohberg, I.C., Krupnik, N.Ya., Spitkovsky, I.M.: Banach algebra of singular integral operators with piecewise continuous coefﬁcients, general contours and weight. Integr. Equ. Oper. Theory 17, 322–327 (1993) [15] Heinonen, J.: Lectures on analysis on metric spaces. Springer-Verlag, Universitext, New York (2001) [16] Karapetiants, N., Samko, N.: Weighted theorems on fractional integrals in the generalized H¨ older spaces H0ω (ρ) via the indices mω and Mω . Fract. Calc. Appl. Anal. 7(4), 437–458 (2004) [17] Karlovich, A.Yu.: Fredholmness of singular integral operators with piecewise continuous coefﬁcients on weighted Banach function spaces. J. Integr. Eq. Appl. 15(3), 263–320 (2003) [18] Karlovich, A.Yu.: Singular integral operators on variable Lebesgue spaces over arbitrary Carleson curves. Operator Theory: Advances and Applications, Vol. 202, pp. 321–336. Birkh¨ auser Verlag, Basel/Switzerland (2009) [19] Karlovich, A.Yu.: Singular integral operators on variable Lebesgue spaces with radial oscillating weights. arXiv:0708.0778v3 [math.FA] (2009) [20] Karlovich, A.Yu.: Singular integral operators on Nakano spaces with weights having ﬁnite sets of discontinuites. arXiv:1002.4813v1 [math.FA] (2010) [21] Kokilashvili, V., Paatashvili, V., Samko, S.: Boundary value problems for analytic functions in the class of Cauchy-type integrals with density in Lp(·) (Γ). Bound. Value Probl. 1, 43–71 (2005) [22] Kokilashvili, V., Paatashvili, V., Samko, S.: Boundedness in Lebesgue spaces with variable exponent of the Cauchy singular operators on Carleson curves. In: Erusalimsky Yu., Gohberg I., Grudsky S., Rabinovich V., Vasilevski, N. (eds.) “Operator Theory: Advances and Applications”, v. 170 (dedicated to 70th birthday of Prof. I.B.Simonenko), pp. 167–186. Birkh¨ auser (2006) [23] Kokilashvili, V., Samko, N., Samko, S.: Singular operators in variable spaces Lp(·) with oscillating weights. Math. Nachr. 280, 1145–1156 (2007) [24] Kokilashvili, V., Samko, S.: Singular Integrals in Weighted Lebesgue Spaces with Variable Exponent. Georgian Math. J. 10(1), 145–156 (2003) [25] Kokilashvili, V., Samko, S.: Singular integral equations in the Lebesgue spaces with variable exponent. Proc. A. Razmadze Math. Inst. 131, 61–78 (2003)

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[26] Kokilashvili, V., Samko, S.: Weighted boundedness in Lebesgue spaces with variable exponents of classical operators on Carleson curves. Proc. A. Razmadze Math. Inst. 138, 106–110 (2005) [27] Kokilashvili, V., Samko, S.: The maximal operator in weighted variable exponent spaces on metric spaces. Georgian Math. J. 15(4), 683–712 (2008) [28] Kokilashvili, V., Samko, S.: Weighted boundedness of the maximal, singular and potential operators in variable exponent spaces, In: Kilbas A.A., Rogosin S.V. (eds.) Analytic Methods of Analysis and Differential Equations, Cambr. Sci. Publ. 139–164 (2008) [29] Kov´ ac˘ık, O., R´ akosn˘ık, J.: On spaces Lp(x) and Wk;p(x) . Czechoslovak Math. J. 41(116), 592–618 (1991) [30] Krasnosel’skii, M.A.: On a theorem of M. Riesz. Sov. Math. Dokl. 1, 229– 231 (1960) [31] Maligranda, L.: Indices and interpolation, Dissertationes Math. (Rozprawy Mat.), 234, 49 (1985) [32] Maligranda, L.: Orlicz spaces and interpolation. Departamento de Matem´ atica, Universidade Estadual de Campinas, Campinas SP Brazil (1989) [33] Rabinovich, V.S.: Singular integral operators on a composed contour with oscillating tangent and pseudodifferential Mellin operators. Sov. Math. Dokl. 44, 791–796 (1992) [34] Rabinovich, V.S.: Singular integral operators on composed contours and pseudodifferential operators. Math. Notes 58, 722–734 (1995) [35] Rabinovich, V.S.: Algebras of singular integral operators on compound contours with nodes that are logarithmic whirl points (In Russian). Izvestia AN Rossii, ser. mathem., 60(6), 169–200, 1996; Engl. transl.: Izvestia: Mathematics 60(6), 1261–1292 (1996) [36] Rabinovich, V.: Mellin pseudodifferential operators technique in the theory of singular integral operators on some Carleson curves. Operator Theory: Advances and Applications, Vol. 102, p. 201–218. Birkh¨ auser, Basel (1998) [37] Rabinovich, V.: An Introductionary Course on Pseudodifferential Operators, Textos de Matem´ atica, Centro de Matem´ atica Aplicada, Instituto Superior T´ecnico, Portugal (1998) [38] Rabinovich, V., Roch, S., Silbermann, B.: Limit Operators and their Applications in Operator Theory, Operator Theory: Advances and Applications, vol 150. Birkh¨ auser, Basel (2004) [39] Rabinovich, V., Samko, N., Samko, S.: Local Fredholm Properties of Singular Integral Operators on Carleson Curves Acting on Weighted Holder Spaces. Integr. Equ. Oper. Theory 56(2), 257–283 (2006) [40] Rabinovich, V., Samko, S.G.: Essential Spectra of Pseudodifferential Operators in Sobolev Spaces with Variable Smoothness and Variable Lebesgue Indices, Doklady Mathematics, 76(3), 835–838, 2007,(Original Russian Text in Dokl). AN Rossii 417(2), 167–170 (2007) [41] Rabinovich, V., Samko, S.: Boundedness and Fredholmness of pseudodifferential operators in variable exponent spaces. Integr. Equ. Oper. Theory 60(4), 507–537 (2008) [42] Samko, N.: On non-equilibrated almost monotonic functions of the ZygmundBary-Stechkin class, Real Anal. Exch. 30(2), 727–745 (2004/2005)

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[43] Samko, N.: Note on Matuzsewska-Orlich indices and Zygmund inequalities. Armenian J. Math. 3(1), 22–31 (2010) [44] Samko, S.: On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integr. Transf. and Spec. Funct 16(5–6), 461–482 (2005) [45] Simonenko, I.B.: Interpolation and extrapolation of linear operators in Orlicz spaces (in Russian). Mat. Sb. (N.S.), Ser. Mat. 63(4), 536–553 (1964) [46] Simonenko, I.B.: A new general method of investigatigating linear operator equations of singular integral equations type, I, II, Izv. Acad. Nauk SSSR, Ser. Mat. 3, 567–586, 4, 757–782 (1965) [47] Simonenko, I.B., Min, C.N.: The local principle in the theory of onedimensional singular integral operators with piece-wise continuous coefﬁcients, Rostov on Don, Rostov State University (1986) [48] Simonenko, I.B.: The local method in the theory of operators invariant with respect to shifts and their envelopes. Rostov State University, Rostov-na-Donu (2007). ISBN 978-5-9153-161-5 [49] Spitkovsky, I.M.: Singular integral operators with P C symbols on spaces with general weights. J. Funct. Anal. 105, 129–149 (1992) [50] Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Second Edition. Springer, Berlin (2001) [51] Taylor, M.E.: Pseudodifferential Operators. Princeton Univ. Press, Princeton (1981) Vladimir Rabinovich Instituto Politecnico Nacional Esime Zacatenco Av. IPN 07738 Mexico DF, Mexico e-mail: [email protected] Stefan Samko (B) Universidade do Algarve FCT Campus de Gambelas 8005-139 Faro Portugal e-mail: [email protected] Received: July 26, 2010. Revised: November 1, 2010.

Integr. Equ. Oper. Theory 69 (2011), 445–446 DOI 10.1007/s00020-010-1854-z Published online January 19, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Erratum

Erratum to: Joint Hyponormality of Rational Toeplitz Pairs In Sung Hwang and Woo Young Lee

Erratum to: Integr. Equ. Oper. Theory (2009) 65:387–403 DOI 10.1007/s00020-009-1729-3 In Theorem 1 of the article “Joint Hyponormality of Rational Toeplitz Pairs” [1], the condition ‘φ and ψ have a common pole’ has to be assumed. Due to a technical problem, we omitted an assumption in Theorem 1 of [1]. For the sake of completeness, we restate Theorem 1, in its correct form. Theorem 1. Let φ and ψ be rational functions in L∞ which have a common pole. If T = (Tφ , Tψ ) is hyponormal then φ − βψ ∈ H 2 for some constant β. The proof of [1, Theorem 1] is correct and complete under the additional assumption ‘φ and ψ have a common pole.’ In turn, Corollary 2 of [1] should also have an additional assumption ‘θ0 and θ2 are not coprime,’ which is equivalent to the condition ‘φ and ψ have a common pole.’ We would remark that Corollary 2 of [1] is still a generalization of the case of trigonometric polynomial symbols because if φ and ψ are trigonometric (non-analytic) polynomials then they have a common pole at z = 0.

Reference [1] Hwang, I.S., Lee, W.Y.: Joint hyponormality of rational Toeplitz pairs. Integr. Equ. Oper. Theory 65, 387–403 (2009)

The online version of the original article can be found under doi:10.1007/s00020-009-1729-3.

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In Sung Hwang Department of Mathematics Sungkyunkwan University Suwon 440-746 Korea e-mail: [email protected] Woo Young Lee (B) Department of Mathematics Seoul National University Seoul 151-742 Korea e-mail: [email protected]

IEOT

Integr. Equ. Oper. Theory 69 (2011), 447–449 DOI 10.1007/s00020-010-1855-y Published online January 15, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Erratum

Erratum to: Finite Section Method for a Banach Algebra of Convolution Type Operators on Lp(R) with Symbols Generated by P C and SO Alexei Yu. Karlovich, Helena Mascarenhas and Pedro A. Santos

Erratum to: Integr. Equ. Oper. Theory (2010) 67:559–600 DOI 10.1007/s00020-010-1784-9 We correct Theorem 3.2 and Corollary 3.3 from [2]. This correction amounts to the observation that the proof of the main result in [2] contains a gap in Lemma 10.6 for p = 2. The results of [2] are true for p = 2.

1. Spectra of the Operators P(α,β) and Q(α,β) Let the notation be as in [2]. Theorem 3.1 in [2] was misstated (see [1, Chap. 9, Theorem 4.1]), it is valid only for α, β ∈ R. This led to the mistake in the calculation of the spectra of P(α,β) and Q(α,β) if either α = −∞ or β = +∞ in Theorem 3.2 of [2]. This theorem must be replaced by the following. Theorem 1.1. (a) If −∞ < α < β < +∞, then spB(Lp (α,β)) (P(α,β) ) = spB(Lp (α,β)) (Q(α,β) ) = Lp . (b) If −∞ < β < +∞, then spB(Lp (−∞,β)) (P(−∞,β) ) = Ap (0, 1),

spB(Lp (−∞,β)) (Q(−∞,β) ) = Aq (0, 1).

The online version of the original article can be found under doi:10.1007/s00020-010-1784-9.

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(c) If −∞ < α < +∞, then spB(Lp (α,+∞)) (P(α,+∞) ) = Aq (0, 1),

spB(Lp (α,+∞)) (Q(α,+∞) ) = Ap (0, 1).

Proof. The proof of part (a) given in [2, Theorem 3.2] remains unchanged. (b) It is clear that λ belongs to the spectrum of P(−∞,β) (resp. Q(−∞,β) ) if and only if 2λ − 1 (resp. 1 − 2λ) is in the spectrum of S(−∞,β) . By [3, Theorem IV(ii)], spB(Lp (−∞,β)) (S(−∞,β) ) = Ap (−1, 1). Hence spB(Lp (−∞,β)) (P(−∞,β) ) = {λ ∈ C : 2λ − 1 ∈ Ap (−1, 1)} = Ap (0, 1), spB(Lp (−∞,β)) (Q(−∞,β) ) = {λ ∈ C : 1 − 2λ ∈ Ap (−1, 1)} = Ap (1, 0). It remains to observe that Ap (1, 0) = Aq (0, 1). (c) The proof is analogous to that of part (b) because in view of [3, Theorem IV(ii)], spB(Lp (α,+∞))) (S(α,+∞) ) = Aq (−1, 1). Since spB(Lp (−∞,−1)) (Q(−∞,−1) ) and spB(Lp (1,+∞)) (P(1,+∞) ) are contained in Lp , the result stated in Corollary 3.3(a),(b) of [2] remains valid and the proof is the same. However, Corollary 3.3(c) in [2] is wrong and must be replaced by the following. Corollary 1.2. Let χ+ and χ− be the characteristic functions of (0, +∞) and (−∞, 0), respectively, and B := B(Lp (R)). Then spB (χ+ PR χ+ I + χ− QR χ− I) = Aq (0, 1). Proof. Let λ ∈ C. The operator χ+ PR χ+ I + χ− QR χ− I − λI is represented ˙ p (0, +∞) by the matrix with respect to the direct sum Lp (R) = Lp (−∞, 0)+L Q(−∞,0) − λI 0 . 0 P(0,+∞) − λI Hence, by Theorem 1.1(b)–(c), spB (χ+ PR χ+ I + χ− QR χ− I) = spB(Lp (−∞,0)) (Q(−∞,0) ) ∪ spB(Lp (0,+∞)) (P(0,+∞) ) = Aq (0, 1) ∪ Aq (0, 1) = Aq (0, 1), which ﬁnishes the proof.

2. Gap in the Proof of Lemma 10.6 in [2] Since the proof of [2, Lemma 10.6] uses the wrong [2, Corollary 3.3(c)], it contains a gap. Replacing [2, Corollary 3.3(c)] by Corollary 1.2 in the proof of [2, Lemma 10.6], we arrive at the following.

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Lemma 2.1. Let η ∈ M∞ (SO). Then Aq (0, 1) ⊂ spLJ∞,η ΦJ ∞,η (PX− W− X− P + QX− W+ X− Q) ⊂ Lp , Aq (0, 1) ⊂ spLJ∞,η ΦJ ∞,η (PX+ W+ X+ P + QX+ W− X+ Q) ⊂ Lp . We believe that both spectra coincide with Lp , however we were so far unable to prove this. If p = 2 then both A2 (0, 1) and L2 collapse to the segment [0, 1]. Hence for p = 2 the previous result reads as follows. Lemma 2.2. Let p = 2 and η ∈ M∞ (SO). Then spLJ∞,η ΦJ ∞,η (PX− W− X− P + QX− W+ X− Q) = [0, 1], spLJ∞,η ΦJ ∞,η (PX+ W+ X+ P + QX+ W− X+ Q) = [0, 1]. Thus, the main result of [2] remains true (and new) at least for p = 2.

References [1] Gohberg, I., Krupnik, N.: One-Dimensional Linear Singular Integral Equations, vols. 1 and 2. Operator Theory: Advances and Applications, vol. 53–54. Birkh¨ auser, Basel (1992) [2] Karlovich, A.Yu., Mascarenhas, H., Santos, P.A.: Finite section method for a Banach algebra of convolution type operators on Lp (R) with symbols generated by P C and SO. Integr. Equ. Oper. Theory 67, 559–600 (2010) [3] Widom, H.: Singular integral equations on Lp . Trans. Am. Math. Soc. 97, 131– 160 (1960) Alexei Yu. Karlovich (B) Departamento de Matem´ atica Faculdade de Ciˆencias e Tecnologia Universidade Nova de Lisboa Quinta da Torre 2829-516 Caparica Portugal e-mail: [email protected] Helena Mascarenhas and Pedro A. Santos Departamento de Mathem´ atica Instituto Superior T´ecnico Universidade T´ecnica de Lisboa Av. Rovisco Pais 1049-001 Lisboa Portugal e-mail: [email protected] [email protected]

Integr. Equ. Oper. Theory 69 (2011), 451–478 DOI 10.1007/s00020-011-1867-2 Published online February 16, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Remarks on the Trotter–Kato Product Formula for Unitary Groups Pavel Exner, Hagen Neidhardt and Valentin A. Zagrebnov Dedicated to our friend Takashi Ichinose in occasion of his 70th birthday Abstract. Let A and B be non-negative self-adjoint operators in a separable Hilbert space such that their form sum C is densely deﬁned. It is shown that the Trotter product formula holds for imaginary parameter values in the L2 -norm, that is, one has lim

T 2 −itA/n −itB/n n e h − e−itC h dt = 0 e

n→+∞ −T

for each element h of the Hilbert space and any T > 0. This result is extended to the class of holomorphic Kato functions, to which the exponential function belongs. Moreover, for a class of admissible functions: φ(·), ψ(·) : R+ −→ C, where R+ := [0, ∞), satisfying in addition e (φ(y)) ≥ 0, m (φ(y) ≤ 0 and m (ψ(y)) ≤ 0 for y ∈ R+ , we prove that s- lim (φ(tA/n)ψ(tB/n))n = e−itC n→∞

holds true uniformly on [0, T ] t for any T > 0. Mathematics Subject Classification (2010). Primary 47A55, 47D03, 81Q30; Secondary 47B25. Keywords. Trotter product formula, Trotter–Kato product formula, unitary groups, Feynman path integrals, holomorphic Kato functions, admissible functions.

1. Introduction In the resent paper we ﬁrst give a short survey of the main results on the Trotter–Kato product formula for imaginary parameters, and reformulate some of them in a form suitable for further generalizations. This allows us to

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extend the L2 -convergence of the imaginary parameter Trotter product formula to holomorphic Kato functions. Using the concept of admissible functions introduced in [6] we prove this result also for the Trotter–Kato product formula. It is a longtime open problem to prove that for non-negative self-adjoint operators A and B in a separable Hilbert space H the strongly convergent Trotter product formula n (1.1) s- lim e−itA/n e−itB/n = e−itC n→∞

holds uniformly in t ∈ [0, T ] for any T > 0, where C is the form sum of A and B, cf. [13, Problem 11.3.9]. Apart from a pure mathematical interest such a product formula is tightly related to certain physical problems. In particular, the Trotter formula provides a natural way to deﬁne Feynman path integrals [21], cf. [4,13]. Note that extensions of such a definition beyond the essentially self-adjoint case allows one to treat in this way Schr¨ odinger operators for a much wider class of potentials. In order to put our message into a proper context we recall ﬁrst some known results relevant for our presentation. Let −A and −B be two generators of contraction semigroups in the Banach space X. In the seminal paper [23] Trotter proved that if the operator −C, where C := A + B, is a generator of a contraction semigroup on X, then the formula n (1.2) e−tC = s- lim e−tA/n e−tB/n , n→∞

holds for all t ∈ [0, T ] and any T > 0. The formula is usually called Trotter, or Lie–Trotter product formula. The result was generalized by Chernoﬀ in [2] to Banach spaces X in the following form: Let F (·) : R+ −→ B(X) be a strongly continuous operator-valued family of contractions such that F (0) = I and the strong derivative F (+0) exists being a densely deﬁned operator in X. If −C, C := −F (+0), is a generator of a C0 -contraction semigroup, then the generalized Lie–Trotter product formula e−tC = s- lim F (t/n)n , n→∞

(1.3)

holds for t ≥ 0. In [3, Theorem 3.1] it was shown that the strong convergence in the last formula is in fact uniform in t ∈ [0, T ] for any T > 0. Moreover, in [3, Theorem 1.1] this result was generalized as follows: Let F (·) : R+ −→ B(X), where R+ = [0, ∞), be a family of linear contractions on a Banach space X. Then the generalized Lie–Trotter product formula (1.3) holds uniformly in t ∈ [0, T ] for any T > 0 if and only if there is a λ > 0 such that (λ + C)−1 = s- lim (λ + Sτ )−1 , τ →+0

where I − F (τ ) , τ > 0. τ Using these results, Kato [15] was able to prove the following claim: let A and B be two non-negative self-adjoint operators in a separable Hilbert Sτ :=

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space H. Assume that the intersection dom(A1/2 ) ∩ dom(B 1/2 ) is dense in H. . If C := A + B is the form sum of the operators A and B, then the Lie–Trotter product formula, n (1.4) e−tC = s- lim e−tA/n e−tB/n , n→∞

holds uniformly in t ∈ [0, T ] for any T > 0. In fact, the Lie–Trotter formula was extended by Kato to more general products of the form (f (tA/n) n g(tB/n)) , where f (and similarly g) is a real valued Borel measurable function f (·) : R+ −→ R+ obeying 0 ≤ f (t) ≤ 1, f (0) = 1 and f (+0) = −1, which we call Kato functions in the following. Usually the product formulæ of that type are known under the name Lie–Trotter–Kato or Trotter–Kato. It is a longstanding open question whether the Lie–Trotter product formula (1.4) remains valid for imaginary parameters t under the same assumptions which justify the formula (1.2), see [3, Remark p. 91; 9; 10; 21]. Note that if A and B are non-negative self-adjoint operators in H and the limit in the left-hand side of (1.1) exists for all t ∈ R, then dom(A1/2 ) ∩ dom(B 1/2 ) is dense in H, see [13, Proposition 11.7.3]. Hence, we assume in the following that dom(A1/2 ) ∩ dom(B 1/2 ) is dense in H. Furthermore, applying Trotter’s result [23] one immediately gets that formula (1.1) is valid if the operator C := A + B is self-adjoint. However, if A + B is not essentially self-adjoint, then all attempts to verify the Lie–Trotter product formula (1.1) for imaginary parameters have failed so far. A somewhat weaker result is proved in [18, Proposition 3.2], see also [13, Proposition 11.7.4]. It was shown there that n . −itA/n −itB/n ϕ(t) e e dt = ϕ(t)e−itC dt, C := A + B, (1.5) s- lim n→∞

R

R

holds for all ϕ ∈ L1 (R). In [10] Ichinose proposed a modified Trotter-type product formula. He proved in that paper that e−itA/n (EA ([0, nδ/t]) + e−ita/n EA ((nδ/t, ∞) e−itC = s- lim n→∞ n , t ≥ 0, × e−itB/n (EB ([0, nδ/t]) + e−itb/n EB ((nδ/t, ∞) (1.6) where EA (·) and EB (·) denote the spectral measures of the operators A and B, respectively, and a ≥ 0, b ≥ 0, 0 < δ < π/2. If one introduces the functions f (λ) := e−iλ χ[0,δ] (λ) + χ(δ,∞) (λ),

λ ≥ 0,

(1.7)

then the result of [10] for a = b = 0 acquires the form s- lim (f (tA/n)f (tB/n)) = e−itC n

n→∞

(1.8)

for any t ≥ 0. Notice that the above function f (λ) is admissible in the sense of [6], i.e. |f (x)| ≤ 1,

x ∈ [0, ∞),

f (0) = 1,

and f (+0) = −i,

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and satisﬁes in addition the conditions e (f (x)) ≥ 0 and m (f (x)) ≤ 0, x ∈ R+ . In [10, Section 3] this result was generalized to functions ζ(t, λ) from a class denoted by Fν,μ (τ, γ, ε), 0 < τ ≤ ∞, 0 < μ < ν ≤ 1, γ ∈ R and ε = ±1, deﬁned in a slightly cumbersome way. Consider a particular case. Let f be an admissible function. Choosing γ = 0 and ε = −1 one can verify that ζ(t, λ) := f (tλ) ∈ Fν,μ (τ, 0, −1) if and only if f (·) is continuous, 1 m (f (x)) ≤ 0 and |1 − f (x)| ≤ min 2μ, | m (f (x))| , x ∈ R+ . ν (1.9) In particular, there exists a δ > 0 such that the conditions (1.9) are satisﬁed for 1 χ[0,δ] (x) + χ(δ,∞) (x) , 1 + ix

x ≥ 0,

e−itC = s- lim (f (tA/n)f (tB/n)) ,

t ≥ 0.

f (x) = which yields

n

n→∞

In [16], see also [19] or [13, Corollary 11.3.5], Lapidus showed a slightly stronger result, namely that

n (1.10) e−itC = s- lim (I + itA/n)−1 (I + itB/n)−1 n→∞

holds uniformly in t ∈ [0, T ], T > 0. Averaging formulas were proposed in [17] for real parameters for the cases of linear and non-linear semigroups. It was Cachia who for the ﬁrst time linked the imaginary parameter averaging formulas to the L2 convergence. In [1] he proved that 2 T e−2itA/n + e−2itB/n n −itC lim h−e h dt = 0 n→∞ 2 0

holds for any h ∈ H and T > 0. In fact, the notion of holomorphic Kato functions also appeared for the ﬁrst time in [1]. A Kato function f (·) is called holomorphic, if it admits a holomorphic extension to the right complex halfplane, Cright := {z ∈ C : e (z) > 0}, such that |f (z)| ≤ 1, z ∈ Cright . For holomorphic Kato functions the limit f (iy) := lim→+0 f ( + iy) exists for a.e. y ∈ R. In the following we are going to show that there is a Borel measurable function f(·) : iR −→ C satisfying |f(iy)| ≤ 1, y ∈ R, such that f (iy) = f(iy) for a.e. y ∈ R, cf. Lemma 3.2. Since the f(·) is Borel measurable the expression f(isA) is well deﬁned by the functional calculus for any s ∈ R. Moreover, one has f(isA) ≤ 1 for s ∈ R. It was shown in [1] that if f and g are holomorphic Kato functions, then

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2 T f(2itA/n) + g(2itB/n) n lim h − e−itC h dt = 0 n→∞ 2 0

holds for any h ∈ H and T > 0. Before we close this introductory survey, let us mention another family of related results. Note that the paper [1] was inspired, in fact, by results obtained by Ichinose and by one of us in [5]. This article was devoted to the so-called Zeno product formula, which can be regarded as a kind of degenerated Lie–Trotter product formula. In this formula one replaces the unitary factor e−itA by an orthogonal projection onto some closed subspace h ⊆ H and deﬁnes C as the self-adjoint operator associated with the quadratic form √ √ √ Bh, Bk , h, k ∈ dom(k) := dom( B) ∩ h, where it is assumed k(h, k) := that dom(k) is dense in h. It was proved in [5] that T n lim P e−itB/n P h − e−itC h dt = 0

n→∞

0

holds for any h ∈ h and T > 0, where P is the orthogonal projection from H onto h. Subsequently, an attempt was made in [6] to replace the strong L2 topology of [5] by the usual strong topology of H. For admissible functions φ satisfying m (φ(x)) ≤ 0, x ∈ R+ , it was shown in [6] that e−itC = s- lim (P φ(tB/n)P ) , n

n→∞

holds uniformly in t ∈ [0, T ] for any T > 0. We would like to stress that the function φ(x) = e−ix , x ∈ R+ , is admissible but does not satisfy the condition m (e−ix ) ≤ 0 for x ∈ R+ , thus the question about convergence of the Zeno product formula in the strong topology of H remains open. Our present paper is organized as follows. In Sect. 2 we show that the Trotter product formula makes sense in L2 -topology, that is, it holds T 2 −itA/n −itB/n n e h − e−itC h dt = 0 lim e

n→∞ −T

(1.11)

for h ∈ H and any T > 0 without any additional assumptions, cf. Theorem 2.2. This observation follows directly from the Lapidus result (1.5). Of course, it does not solve under our hypotheses the strong convergence problem of (1.1). Nevertheless, (1.11) implies the existence of a subsequence nk such that one has pointwise (i.e., the strong) convergence along it for a.e. t ∈ [−T, T ]. From the physical point of view our result seems to be quite satisfactory, see a discussion on that point in [8, Section 11]. Using the concept of the holomorphic Kato functions we prove the Trotter–Kato product formula in the L2 -topology in Sect. 3, that is, T 2 n g (itB/n) h − e−itC h dt = 0 lim f (itA/n)

n→∞ −T

(1.12)

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for h ∈ H and any T > 0, where f, g are holomorphic Kato functions and f, g are Borel measurable extensions of f and g on the imaginary axis, see Lemma 3.2 and Theorem 3.3. Moreover, we propose a characterization of the class of holomorphic Kato functions. Finally, in Sect. 4 we give a generalization of the results due to Ichinose [10], to the class of admissible functions deﬁned above. We show that s- lim (φ(tA/n)ψ(tB/n)) = e−itC , n

(1.13)

n→∞

where φ and ψ are admissible functions such that e (φ(y)) ≥ 0, m (φ(y) ≤ 0 and m (ψ(y)) ≤ 0 for y ∈ R+ , cf. Theorem 4.7. Choosing φ(y) = ψ(y) = (1+ iy)−1 , y ∈ R, one recovers Lapidus’ result (1.10), see [19] and [13, Corollary 11.3.5]. Moreover, it turns out that admissible functions can be always slightly modiﬁed so that the Trotter–Kato product formula is valid, see Corollary 4.9. In particular, it follows from Corollary 4.9 that the modiﬁed Trotter product formula, n s- lim e−itA/n EA ([0, πn/2t])e−itB/n EB ([0, πn/2t]) = e−itC , (1.14) n→∞

holds uniformly in t ∈ [0, T ], T > 0, cf. (1.7) and (1.8). Notice that (1.14) is similar to (1.6).

2. Lapidus’ Results Revisited We start by proving the following important technical lemma. Lemma 2.1. Let {Fn (·)}n∈N be a family of measurable operator-valued functions Fn (·) : iR −→ B(H) such that Fn (it) ≤ 1 holds for a.e. t ∈ R. Furthermore, let C be a densely defined self-adjoint operator. Then the following assertions are equivalent: (i) For each ϕ ∈ L1 (R) one has w- lim ϕ(t)Fn (it)dt = ϕ(t)e−itC dt. n→∞

R

(2.1)

R

(ii) For each h ∈ H and T > 0 it holds T lim

n→∞ −T

Fn (it)h − e−itC h 2 dt = 0.

(2.2)

(iii) For each ϕ ∈ L1 (R) one has s- lim ϕ(t)Fn (it)dt = ϕ(t)e−itC dt. n→∞

R

(2.3)

R

Proof. (i) =⇒ (ii) Since

Fn (it)h − e−itC h 2 ≤ 2 h 2 − 2e (Fn (it)h, e−itC h),

t ∈ R,

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and e−itC h =

∞ (−it)k

k!

k=0

C k h,

t ∈ R,

for h ∈ EC ([a, b])H, −∞ < a < b < ∞, we ﬁnd ∞ ik tk −itC 2 2 k (Fn (it)h, C h) h ≤ 2 h − 2e

Fn (it)h − e k! k=0

for a.e. t ∈ R, which leads to T

Fn (it)h − e−itC h 2 dt

−T

⎛ ≤ 4T h 2 − 2e ⎝

∞ k T i

k=0

k!

⎞ tk (Fn (it)n h, C k h)dt⎠

−T

or T

Fn (it)h − e−itC h 2 dt

−T

⎛ ≤ 4T h 2 − 2e ⎝

∞ k i k=0

k!

⎛ ⎝

T

⎞⎞ tk Fn (it)h dt, C k h⎠⎠

−T

for t ≥ 0. From (2.1) we get ⎛ T ⎞ ⎛ T ⎞ lim ⎝ tk Fn (it)h, C k h⎠ = ⎝ tk e−itC h dt, C k h⎠ . n→∞

−T

−T

Hence T

Fn (it)h − e−itC h 2 dt

−T

⎛ ≤ 4T g 2 − 2e ⎝

∞ k i k=0

k!

⎛ ⎝

T

⎞⎞ tk e−itC h dt, C k h⎠⎠ .

−T

Therefore T lim sup n→∞

Fn (it)h − e−itC h 2 dt

−T 2

T

≤ 4T h − 2e −T

which proves (2.2).

(e−itC h, e−itC h)dt = 0

(2.4)

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(ii) =⇒ (iii) The following estimate holds: T T

−itC h dt ϕ(t) Fn (it) − e |ϕ(t)| Fn (it)h − e−itC h dt. ≤ −T

−T

From (1.11) we obtain the convergence in measure, that is, for each ε > 0 one has lim {t ∈ [−T, T ] : Fn (it)h − e−itC h ≥ ε} = 0. n→∞

Setting Δε,n := {t ∈ [−T, T ] : Fn (it)h − e−itC h ≥ ε} we ﬁnd the estimate T

−itC h dt ϕ(t) Fn (it) − e −T ≤ ε |ϕ(t)| dt + 2 |ϕ(t)|dt, n ∈ N. Δε,n

[−T,T ]\Δε,n

In view of (2.2) we obtain in the limit n → ∞ the inequality T

−itC lim sup h dt ≤ ε |ϕ(t)|dt ϕ(t) Fn (it) − e n→∞ −T

R

for any ε > 0. Hence for any ε small enough we have

−itC ≤ ε lim sup ϕ(t) Fn (it) − e h dt |ϕ(t)|dt + 2 n→∞ R

R

|ϕ(t)|dt.

R\[−T,T ]

Since T can be chosen sufﬁciently large and ε was arbitrary we get

−itC h dt lim sup ϕ(t) Fn (it) − e = 0, n→∞ R

which yields

s- lim

n→∞

ϕ(t)Fn (it)h = R

ϕ(t)e−itC h dt,

h ∈ H.

R

(iii) =⇒ (i) Obviously (2.3) implies (2.1).

Lemma 2.1 allows us to reformulate the Lapidus result of [18, Proposition 3.2], mentioned as (1.5) above, in the following form: Theorem 2.2. Let A and B two non-negative self-adjoint operators on the . Hilbert space H. If the form sum C := A + B is densely defined, then (1.11) holds for any h ∈ H and T > 0.

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Proof. We set

n Fn (it) := e−itA/n e−itB/n ,

n ∈ N,

t ∈ R.

From [18, Proposition 3.2] we get (1.5), which yields (2.1). Applying now Lemma 2.1 we obtain (1.11). We note that Theorem 2.2 partially solves the question posed in [13, Problem 11.3.9] by a slight change of topology. Indeed, from Theorem 2.2 we get that (1.1) holds in measure, that is, for any real number η > 0, h ∈ H and T > 0 one has n lim t ∈ [−T, T ] : e−itA/n e−itB/n h − e−itC h ≥ η = 0, (2.5) n→∞

where | · | denotes the Lebesgue measure, while [13, Problem 11.3.9] requires a uniform convergence of t ∈ [−T, T ], i.e. for any η > 0, h ∈ H and T > 0 one has n lim sup e−tA/n e−tB/n h − e−tC h = 0. n→∞ t∈[−T,T ]

Notice that convergence in measure (2.5) takes place if and only if any

−itA/n −itB/n n

subsequence of e e contains a subsequence e−itA/nk n∈N nk e−itB/nk which converges strongly almost everywhere to e−itC , i.e k∈N

s- lim

k→∞

e−itA/nk e−itB/nk

nk

= e−itC

holds for a.e. t ∈ [−T, T ]. Remark 2.3. From the viewpoint of physical applications, the formula (1.11) allows us to extend the Trotter-type definition of Feynman integrals for Schr¨ odinger operators to a wider class of potentials. Following [21], see also t [13, Deﬁnition 11.2.21], the Feynman integral FTP (V ) associated with the potential V is the strong operator limit n t FTP (V ) := s- lim e−itH0 /n e−itV /n n→∞

− 12 Δ

where H0 := and −Δ is the Laplacian operator in L2 (Rd ) deﬁned in the usual way. From [14], cf. [13, Corollary 11.2.22], one gets that the Feynman integral exists if V : Rd −→ R is Lebesgue measurable, non-negative, and satisﬁes V ∈ L2loc (Rd ). With Theorem 2.2 in mind it is possible to extend the Trotter-type definition of Feynman integrals if one replaces the L2 (Rd )-topology by the L2 ([−T, T ] × Rd )-topology. Indeed, let us deﬁne the generalized Feynman t (V ) by integral FgTP T 2 −itH0 /n −itV /n n t lim e h − FgTP (V )h dt = 0 e

n→∞ −T

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t (V ) yields the for h ∈ L2 (Rd ) and T > 0. Obviously, the existence of FTP t existence of FgTP (V ) while the converse is in general not true. By Theorem 2.2 one can immediately conclude that the generalized Feynman integral exists if V : Rd −→ R is Lebesgue measurable, non-negative, and satisﬁes V ∈ L1loc (Rd ). This substantially extends the class of admissible potentials. The same class of potentials is covered by the so-called modiﬁed Feynman t (V ) deﬁned by integral FM

n t FM (V ) := s- lim [I + i(t/n)H0 ]−1 [I + i(t/n)V ]−1 , n→∞

see [13, Deﬁnition 11.4.4] and [13, Corollary 11.4.5]. However, in this case the exponents are replaced by resolvents which leads to loss of the typical structure of Feynman integrals and the related physical insights.

3. Lapidus’ Result Generalized The Lapidus result (1.5) relies on the so-called Vitali’s classical theorem and the Vitali extended theorem, cf. [13, Theorem 11.7.1]. We reformulate them in application to our situation as follows: Let Φn (z), n ∈ N be a sequence of contractive holomorphic function in Cright which for x ∈ R+ converges to a function Φ(x), that is, limn→∞ Φn (x) = Φ(x) for x ∈ R+ . Then Φ(x) admits a contractive holomorphic continuation Φ(z) to Cright such that Φ(z) = limn→∞ Φn (z). Since Φn (z) and Φ(z) are contractive holomorphic functions the limits Φn (iy) := lim→+0 Φn ( + iy), n ∈ N, and Φ(iy) := lim→+0 Φ( + iy) exist for a.e. y ∈ R. The Vitali extended theorem now yields that ϕ(y)Φn (iy)dy = ϕ(y)Φ(iy)dy lim n→∞

R

R

for any ϕ ∈ L1 (R). Notice that this conclusion cannot be deduced from Theorem 11.7.1 of [13], since it is required that the functions Φn (z) and Φ(z) must admit continuous extension to Cright . However, applying Lemma 2 of [1], which is a slight generalization of Theorem 11.7.1 from [13], one gets that the conclusion holds. Let us make precise the notion of holomorphic Kato functions (cf. Sect. 1) in the following way: Definition 3.1. Let f (·) : R+ −→ R+ be a Kato function. The function is called a holomorphic Kato function if f (·) admits a holomorphic continuation to Cright such that |f (z)| ≤ 1,

z ∈ Cright .

Standard holomorphic Kato functions are fk (x) := (1+x/k)−k , x ∈ R+ , and, of course, f (x) = e−x , x ∈ R+ . At the end of this section we give a description of holomorphic Kato functions and indicate some non-standard examples of holomorphic Kato functions. It turns out that for standard holomorphic functions the limit to the imaginary axis exists everywhere. This yields f (tA) = s-lim→+0 f (( + it)A) for any t ≥ 0. However, if

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f is a holomorphic Kato function, then in general the relation f (itA) = s-lim→+0 f (( + it)A) cannot be expected. Indeed, this is due the fact that the limit f (iy) = lim→+0 f ( + iy) exists only for a.e. y ∈ R. Hence the limit function f (iy) is not in general Borel measurable which makes it impossible to apply the functional calculus for self-adjoint operators. However, the limit function f (iy), deﬁned for all those y ∈ R for which the limit f (iy) exists, admits an extension to the whole real axis which is Borel measurable. Lemma 3.2. Let f (·) : R+ −→ R+ be a holomorphic Kato function. Then there is a Borel measurable function f(·) : iR −→ C satisfying |f(iy)| ≤ 1, y ∈ R, such that f(iy) = lim→+0 f ( + iy) for a.e. y ∈ R. Proof. We set fR (z) := e (f (z)) and fI (z) = m (f (z)), z ∈ Cright . Since |f (z)| ≤ 1, z ∈ Cright , we ﬁnd |fR (z)|2 + |fI (z)|2 ≤ 1,

z ∈ Cright .

Further let fR± (z) := max{0, ±fR (z)} ≥ 0 and fI± (z) := max{0, ±fI (z)} ≥ 0, z ∈ Cright . Since the function f (·) is holomorphic the functions fR± (·) and fI± (·) are Borel measurable. Obviously, we have f (z) = fR+ (z) − fR− (z) + i(fI+ (z) − fI− (z)),

z ∈ Cright ,

and |fR+ (z)|2 + |fR− (z)|2 + |fI+ (z)|2 + |fI− (z)|2 ≤ 1,

z ∈ Cright .

We set fR± (iy) := lim inf →+0 fR± ( + iy) and fI± (iy) := lim inf →+0 fI± ( + iy), y ∈ R. Since fR± (z) and fI± (z), z ∈ Cright , are Borel measurable functions, the functions fR± (iy) and fI± (iy) are also Borel measurable. From inf fR± (η + iy) ≤ fR± ( + iy) and

0<η≤

inf fI± (η + iy) ≤ fI± ( + iy),

y ∈ R,

0<η≤

we ﬁnd 2 2 j j inf f (η + iy) + inf f (η + iy) 0<η≤ I 0<η≤ R j=±

≤ |fR+ ( + iy)|2 + |fR− ( + iy)|2 + |fI+ ( + iy)|2 + |fI− ( + iy)|2 ≤ 1, y ∈ R, which yields |fR+ (iy)|2 + |fR− (iy)|2 + |fI+ (iy)|2 + |fI− (iy)|2 ≤ 1, Setting f(iy) := f+ (iy) − f− (iy) + i(f+ (iy) − f− (iy)), R

R

I

I

y ∈ R. y ∈ R,

we deﬁne a Borel measurable function. From |f(iy)|2 = |f+ (iy) − f− (iy)|2 + |f+ (iy) − f− (iy)|2 R

R

I

I

≤ |fR+ (iy)|2 + |fR− (iy)|2 + |fI+ (iy)|2 + |fI− (iy)|2 ≤ 1,

y ∈ R.

The relation f(iy) = lim→+0 f ( + iy) for a.e. y ∈ R is obvious.

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Since |f(iy)| ≤ 1, y ∈ R, the expression f(iτ A), τ ∈ R, is well deﬁned for any self-adjoint operator A. Setting F (z) := f (zA)g(zB), z ∈ Cright , it turns out that the operatorvalued family F (z) is contractive and holomorphic, if f and g are holomorphic Kato functions. Theorem 3.3. Let A and B two non-negative self-adjoint operators on the . Hilbert space H. Assume that C := A + B is densely defined. If f and g are holomorphic Kato functions, then (1.12) holds for any h ∈ H and T > 0. Proof. We set Fn (z) := F (z/n)n , z ∈ Cright , n ∈ N. We note that if z = t ∈ R+ , then n n Fn (t) := (f (tA/n)g(tB/n)) = e−tA/n e−tB/n ,

t ∈ R+ ,

n ∈ N.

From [15] we ﬁnd s- lim Fn (t) = e−tC n→∞

for t ∈ R+ which yields lim (Fn (t)h, k) = (e−tC h, k)

n→∞

for t ∈ R+ and h, k ∈ H. Setting Φn (t) := (Fn (t)h, k), n ∈ N, and Φ(t) = (e−tC h, k), t ∈ R+ , by Vitali’s classical theorem, see [13, Theorem 11.7.1.(i)], we ﬁnd limn→∞ Φn (z) = Φ(z), z ∈ Cright . Moreover, taking into account the extended Vitali theorem, see Lemma 2.1 of [1], we get

ϕ(t)Φn (it) dt =

lim

n→∞ R

ϕ(t)Φ(it) dt R

for all ϕ ∈ L1 (R). The last relation yields

ϕ(t)Fn (it) dt =

w- lim

n→∞ R

ϕ(t)e−itC dt

R

for all ϕ ∈ L1 (R) where Fn (it) := s-lim→+0 Fn ( + it) for a.e. t ∈ R, cf. [22, Section V.2]. Applying Lemma 2.1 we obtain T lim

n→∞ −T

for T > 0.

Fn (it)h − e−itC h 2 dt = 0

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n It remains to show that Fn (it) can be replaced by f(itA/n) g (itB/n) for each n ∈ N. We ﬁnd T

2 dt f (( + it)A/n)h − f(itA/n)h

−T

T =

d(EA (λ)h, h) |f (( + it)λ/n) − f(itλ/n)|2

dt −T

[0,∞)

T

=

d(EA (λ)h, h)

dt |f (( + it)λ/n) − f(itλ/n)|2

−T

[0,∞)

for each n ∈ N and h ∈ H. For any λ ∈ [0, ∞) we have T lim

→+0 −T

dt |f (( + it)λ/n) − f(itλ/n)|2 = 0

which yields T

2 dt f (( + it)A/n)h − f(itA/n)h = 0

lim

→+0 −T

for each n ∈ N and h ∈ H. Since also T

2

dt g(( + it)B/n)h − g(itB/n)h = 0

lim

→+0 −T

holds for each n ∈ N and h ∈ H we immediately ﬁnd that T lim

n

→+0 −T

dt (f (( + it)A/n)g(( + it)B/n)) h

n 2 − f(itA/n) g (itB/n) h = 0

for each n ∈ N and h ∈ H. Hence T 0 = lim

→+0 −T

T =

n 2 dt Fn ( + it)h − f(itA/n) g (itB/n) h

n 2 dt Fn (it)h − f(itA/n) g (itB/n) h

−T

for each n ∈ N and h ∈ H which yields Fn (it) = (f(itA/n) g (itB/n))n for a.e. t ∈ R and n ∈ N.

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Using the standard Kato function fk (x), see above, we get T n

lim (I + itA/kn)−k (I + itB/kn)−k h − e−itC h dt = 0

n→+∞ −T

for any h ∈ H and T > 0. We note that for the particular case k = 1 Lapidus demonstrated in [16] that (1.10) holds uniformly in t ∈ [0, T ] for any T > 0. By Theorem 3.3 one gets that formula (1.10) is valid in a weaker topology than in [16]. This discrepancy will be clariﬁed in the next section. The set of holomorphic Kato functions was characterized in [7]. For the sake of completeness we recall these results here: Theorem 3.4 ([7, Theorem 5.1]). If f is a holomorphic Kato function, then (i) there is an at most countable set of complex numbers {ξk }k , ξk ∈ Cright with m (ξk ) ≥ 0 satisfying the condition κ := 4

e (ξk ) k

|ξk |2

≤ 1;

(3.1)

(ii) there is a Borel measure ν defined on R+ = [0, ∞) obeying ν({0}) = 0 and 1 dν(t) < ∞ 1 + t2 R+

1 such that the limit β := limx→+0 π2 R+ x2 +t 2 dν(t) exists and satisfies the condition β ≤ 1 − κ; (iii) the Kato function f admits the representation ⎧ ⎫ ⎪ ⎪ ⎨ 2x ⎬ 1 f (x) = D(x) exp − dν(t) e−αx , x ∈ R+ , (3.2) ⎪ ⎪ x2 + t2 ⎩ π ⎭ R+

where α := 1 − κ − β and D(x) is a Blaschke-type product given by D(x) := k

x2 − 2xe (ξk ) + |ξk |2 , x2 + 2xe (ξk ) + |ξk |2

x ∈ R+ .

(3.3)

The factor D(x) is absent if the set {ξk }k is empty; in that case we set κ := 0. Conversely, if a real function f admits the representation (3.2) such that the assumptions (i) and (ii) are satisfied and the condition α + κ + β = 1 holds, then f is a holomorphic Kato function and its holomorphic extension to Cright is given by ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ 2z 1 dν(t) e−αz , z ∈ Cright . f (z) = D(z) exp − ⎪ ⎪ z 2 + t2 ⎭ ⎩ π R+

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Above we have indicated several standard holomorphic Kato functions such as f (z) = e−z , fk (z) = (1 + iz/k)−k , k ∈ N, z ∈ Cright . The last theorem allows us to give examples of some non-standard holomorphic Kato functions. 1. If a holomorphic Kato function f (·) has no zeros in Cright and ν ≡ 0, then f (z) = e−z , z ∈ Cright , where α = 1 follows from condition α = 1 − κ − β where κ = β = 0. Obviously, we have f(iy) = lim→+0 f ( + iy) = e−iy for y ∈ R. 2. If a holomorphic Kato function f (·) has zeros and the measure ν ≡ 0, then f (·) is of the form f (z) = D(z)e−αz , where the Blaschke-type product D(z) is given by (3.3). In particular, if n = 1 we ﬁnd the representation f (z) =

z 2 − 2ze (ξ) + |ξ|2 −αz e , z 2 + 2ze (ξ) + |ξ|2

z ∈ Cright ,

where ξ ∈ Cright is such that α+4

e (ξ) = 1. |ξ|2

This gives the representation 2 z 2 − 2η z − 1−α e−αz , f (z) = 2 2 z + 2η z + 1−α 0<η≤ ! τ=

z ∈ Cright ,

4 1−α , 0

4 (1−α)2

≤ α ≤ 1, where we have denoted ξ = η + iτ, η > 0, and 2 2 − η − 1−α . We have

f(iy) = lim f ( + iy) = →+0

1 y 2 + 4η 1−α + 2iηy 1 y 2 − 4η 1−α + 2iηy

e−iαy ,

y ∈ R.

3. If a holomorphic Kato function f (z) has no zeros and the measure ν is atomic, then f (z) admits the representation # " 1 2z ν({sl }) e−αz , z ∈ Cright , f (z) = exp − π z 2 + s2l l

where {sl }l is the point where ν({sl }) = 0. In the particular case when dν(t) = cδ(t − s)dt, s > 0, we have 1 2zc e−αz , f (z) = exp − π z 2 + s2 2c 1 π s2

= 1, which yields c = 12 (1 − α)πs2 and s2 f (z) := exp −z(1 − α) 2 e−αz , z ∈ Cright . z + s2

and α +

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One gets " 2 lim→+0 f ( + iy) = exp iy(1 − α) y2s−s2 e−iαy f (iy) = 0

y= ±s y = ±s

where y ∈ R. 4. If a holomorphic Kato function f (z) has no zeros and the measure ν is absolutely continuous, that is, dν(t) = h(t)dt, h(t)(1 + t2 )−1 ∈ L1 (R+ ), then f (z) admits the representation ⎫ ⎧ ⎬ ⎨ 2z ∞ h(t) dt e−αz , z ∈ Cright , f (z) = exp − ⎩ π z 2 + t2 ⎭ 0

such that 2 α + lim x→+0 π

∞ 0

h(t) dt = 1. x2 + t2

∞ If h(·) is H¨older continuous, then lim→+0 2(+iy) π 0 ists for each y ∈ R and one gets ⎧ ⎫ ∞ ⎨ ⎬ 2( + iy) h(t) f(iy) = exp − lim e−iy , ⎩ →+0 π ( + iy)2 + t2 ⎭

h(t) (+iy)2 +t2 dt

ex-

y ∈ R.

0

In particular, if f (x) = (1 + xk )−k , x ∈ R+ , k ∈ N, then ⎧ ⎫ ⎪ ⎪

⎨ kz ⎬ 1 t2 f (z) = exp − ln 1 + dt ⎪ ⎪ z 2 + t2 k2 ⎩ π ⎭ R+

for z ∈ Cright and f(iy) = (1 + iy/k)−k , y ∈ R, k ∈ N.

4. Ichinose’s Result Revisited Recall that the notion of admissible functions was introduced in [6, Deﬁnition 1]. Definition 4.1 A Borel measurable function φ : R+ −→ C is called admissible if the conditions |φ(y)| ≤ 1,

y ∈ [0, ∞),

φ(0) = 1,

φ (+0) = −i

are satisﬁed. We set φR (y) := e (φ(y)) and φI (y) := m (φ(y)), y ∈ R+ . Obviously we have |φR (y)| ≤ 1,

y ∈ R+ ,

φR (0) = 1 and φR (+0) = 0

as well as |φI (y)| ≤ 1,

y ∈ R+ ,

φI (0) = 0 and φI (+0) = −1.

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Let Σ := {y ∈ R+ : φ(y) = 0} and Ω := R \ Σ. We set " 1 , y∈Ω ϕ(y) := φ(y) 1, y∈Σ. Notice that χΩ (y) = ϕ(y)φ(y), y ∈ Ω, where χΩ (·) is the characteristic function of Ω. The function ϕ obeys |ϕ(y)| ≥ 1,

y ∈ R+ ,

ϕ(0) = 1,

Moreover, we ﬁnd

"

ϕR (y) := e (ϕ(y)) = and

" ϕI (y) := m (ϕ(y)) =

ϕ (0) = i.

φR (y) |φ(y)|2 ,

y∈Ω

1,

y∈Σ

φI (y) − |φ(y)| 2, 0,

y∈Ω y ∈ Σ.

as well as ϕR (0) = 1,

ϕR (+0) = 0 and ϕI (0) = 0,

ϕI (+0) = 1.

Let Eτ := χΩ (τ A), τ > 0. Obviously, Eτ is an orthogonal projection. We consider the operator-valued function K(τ ) :=

I − ψ(τ B) ϕ(τ A) − I + Eτ Eτ , τ τ

τ > 0.

We note that KR (τ ) := e (K(τ )) =

I − ψR (τ B) ϕR (τ A) − I + Eτ Eτ τ τ

and ϕI (τ A) − Eτ ψI (τ B)Eτ . (4.1) τ If φI (y) ≤ 0 and ψI (y) ≤ 0, y ∈ R+ , then KI (τ ) ≥ 0. Furthermore, we KI (τ ) := m (K(τ )) =

set Lγ (τ ) := γKR (τ ) + KI (τ ),

γ ∈ [0, 1].

Let us introduce the functions fγ (y) := γ(ϕR (y) − 1) + ϕI (y),

y ∈ R+ ,

(4.2)

gγ (y) := γ(1 − ψR (y)) − ψI (y),

y ∈ R+ ,

(4.3)

and for γ ∈ [0, 1]. If φ(·) and ψ(·) are admissible functions, then fγ (0) = 0,

fγ (+0) = 1 and gγ (0) = 0,

gγ (+0) = 1.

(4.4)

Using the functions fγ (·) and gγ (·) one gets the representation Lγ (τ ) =

gγ (τ B) fγ (τ A) + Eτ Eτ τ τ

(4.5)

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If φR (y) ≥ 0 and φI (y) ≤ 0 for y ∈ R+ , then " fγ (y) =

γφR (y)(I−φR (y))−φI (y)(1+γφI (y)) , |φ(y)|2

y∈Ω y∈Σ

0,

yields fγ (y) ≥ 0 for y ∈ R+ and γ ∈ [0, 1]. Hence fγ (τ A) ≥ 0 for τ > 0 and γ ∈ [0, 1]. Similarly, if ψI (y) ≤ 0 for y ∈ R+ , then gγ (y) ≥ 0 for y ∈ R+ which implies gγ (τ B) ≥ 0 for τ > 0 and γ ∈ [0, 1]. Hence one has Lγ (τ ) ≥ 0 and τ > 0 and γ ∈ [0, 1], which shows that (μI + Lγ (τ ))−1 exists and is bounded for μ > 0, τ > 0 and γ ∈ [0, 1]. Lemma 4.2 Let A and B be non-negative self-adjoint operators such that the intersection dom(A1/2 ) ∩ dom(B 1/2 ) is dense in H. If f (·) : R+ −→ R and g(·) : R+ −→ R are finite-valued non-negative Borel measurable functions satisfying f (0) = 0,

f (+0) = 1

and

g(0) = 0,

g (+0) = 1,

(4.6)

and 0 ≤ g(y) ≤ 1,

y ∈ R+ ,

(4.7)

then s- lim (μI + L(τ ))−1 = (μI + C)−1

(4.8)

τ →+0

for μ > 0 where 1 1 f (τ A) + Eτ g(τ B)Eτ , τ > 0. (4.9) τ τ and Eτ := χΩ (τ A), Ω ⊇ supp(f ) ∪ {0}, supp(f ) := {y ∈ R+ : f (y) > 0}. L(τ ) :=

Proof. Since f (·) takes only ﬁnite values the operator f (τ A), τ > 0, is densely deﬁned. Moreover, the operator g(τ B) is bounded. Hence the operator L(τ ) is well-deﬁned. We set 1 and q(y) := 1 − g(y), y ∈ R+ . p(y) := (4.10) 1 + f (y) We note that 0 ≤ p(y) ≤ 1,

y ∈ R+ ,

p(0) = 1,

and p (0) = −1

0 ≤ q(y) ≤ 1,

y ∈ R+ ,

q(0) = 1,

and q (0) = −1.

as well as Hence p(·) and q(·) are Kato functions. We have

I − p(τ A) $ I − q(τ B) $ L(τ ) = p(τ A)−1/2 + p(τ A)Eτ Eτ p(τ A) p(τ A)−1/2 . τ τ Let F% (τ ) := p(τ A)1/2 Eτ q(τ B)Eτ p(τ A)1/2 ,

τ ≥ 0.

Since p(y) = 1 for y ∈ Σ := R \ Ω ⊆ ker(f ), ker(f ) := {y ∈ R+ : f (y) = 0}, we ﬁnd L(τ ) = p(τ A)−1/2 Eτ S% τ Eτ p(τ A)−1/2 ,

(4.11)

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where I − F% (τ ) , τ > 0. S% τ := τ Hence the representation −1 $ $ (μI + L(τ ))−1 = p(τ A) μp(τ A) + Eτ S% τ Eτ p(τ A) holds. Since 0 ≤ p(τ A) ≤ I we obtain −1 $ $ p(τ A) μI + Eτ S% τ Eτ p(τ A) ≤ (μI + L(τ ))−1

(4.12)

for μ > 0. By the formula 1 (μI + Eτ S% τ Eτ ))−1 = Eτ⊥ + Eτ (μEτ + Eτ S% τ Eτ )−1 Eτ μ we get

(4.13)

−1 $ $ p(τ A)Eτ μEτ + Eτ S% τ Eτ Eτ p(τ A) ≤ (μI + L(τ ))−1

for τ > 0. $ Setting p% (y)$:= p(y)χΩ (y), y ∈ R+ , we ﬁnd the representation % F (τ ) = p% (τ A)q(τ B) p% (τ A), τ > 0. Since p% (·) and q(·) are Kato functions we obtain s- lim (μI + S% τ (τ ))−1 = (μI + C)−1 τ →+0

for μ > 0 using [3] and [15]. Taking into account formula (4.13) we ﬁnd s- lim Eτ (μEτ + Eτ S% τ Eτ )−1 Eτ = (μI + C)−1 τ →+0

(4.14)

for μ > 0. From (4.12) and (4.14) we ﬁnally get

−1 ((μI + C)−1 h, h) ≤ lim inf (μI + L(τ ))−1 h, h τ →+0

for h ∈ H, μ > 0. Moreover, from (4.5) we ﬁnd g(τ B) f (τ A) EA ([0, a)) + Eτ EB ([0, b))Eτ , a, b ∈ (0, ∞), τ τ which gives the estimate

−1 g(τ B) f (τ A) EA ([0, a)) + Eτ EB ([0, b))Eτ (μI + L(τ ))−1 ≤ μI + τ τ L(τ ) ≥

for μ > 0 and a, b ∈ (0, ∞). Using s-limτ →+0 Eτ = I we obtain

−1 lim sup (μI + L(τ ))−1 h, h ≤ (μI + AEA ([0, a)) + BEB ([0, b))) h, h . τ →+0

for h ∈ H, μ > 0 and a, b ∈ (0, ∞). Since a, b ∈ (0, ∞) are arbitrary we obtain

lim sup (μI + L(τ ))−1 h, h ≤ (μI + C)−1 h, h τ →+0

for h ∈ H, μ > 0. Hence w- lim (μI + L(τ ))−1 = (μI + C)−1 τ →+0

(4.15)

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for μ > 0, and consequently, w- lim (μI + L(τ ))−1/2 = (μI + C)−1/2 τ →+0

(4.16)

for μ > 0. From (4.15) and (4.16) we immediately get (4.18).

Lemma 4.3 Let A and B be non-negative self-adjoint operators such that the intersection dom(A1/2 ) ∩ dom(B 1/2 ) is dense in H. If φ and ψ are admissible functions such that φR (y) ≥ 0,

φI (y) ≤ 0

and

ψI (y) ≤ 0,

y ∈ R+ ,

(4.17)

then the self-adjoint operators Lγ (τ ), τ > 0 are well-defined and non-negative, and it holds s- lim (μI + Lγ (τ ))−1 = (μI + C)−1 τ →+0

(4.18)

for μ > 0 and γ ∈ [0, 1]. Proof. One easily veriﬁes that the functions fγ (·) and gγ (·) deﬁned by (4.2) and (4.3) satisfy the assumptions (4.6) and (4.7) for each γ ∈ [0, 1]. Setting Ω := supp(φ) := {y ∈ R+ : φ(y) = 0} we ﬁnd Ω ⊇ supp(fγ ) ∪ {0} for γ = [0, 1]. Moreover, the definition of Lγ (τ ) given by (4.5) coincides with that one of L(τ ) for each τ > 0 and γ ∈ [0, 1], see (4.9). Applying Lemma 4.2 we arrive at the sought conclusion. For purposes of the next statement we introduce the operators Mγ (τ ) := Lγ (τ ) + (1 + γ)ϕR (τ A) + (1 − γ)ϕI (τ A)

(4.19)

with τ > 0 and γ ∈ [0, 1]. Since Lγ (τ ) ≥ 0, ϕR (τ A) ≥ 0 and ϕI (τ A) ≥ 0 we get Mγ (τ ) ≥ 0 for γ ∈ [0, 1]. Lemma 4.4 Let A and B be non-negative self-adjoint operators such that the intersection dom(A1/2 ) ∩ dom(B 1/2 ) is dense in H. If φ and ψ are admissible functions such that the conditions (4.17) are satisﬁed, then Mγ (τ ) ≥ 0 and s- lim (μI + Mγ (τ ))−1 = ((1 + μ + γ)I + C)−1 τ →+0

(4.20)

holds for μ > 0 and γ ∈ [0, 1]. Proof. We note that Mγ (τ ) ≥ Lγ (τ ) + (1 + γ)ϕR (τ A) ≥ 0 for τ > 0 and (n) γ ∈ [0, 1]. Let ΩR := {y ∈ R : ϕR (y) ≤ n}. We set " (n) ϕR (y), y ∈ ΩR (n) ϕR (y) := (n) n, y ∈ R+ \ ΩR (n)

(n)

for any n ∈ N. Obviously we have 0 ≤ ϕR (y) ≤ n, y ∈ R+ , and 0 ≤ ϕR (y) ≤ ϕR (y), y ∈ R+ . Therefore one obtains (n)

Mγ (τ ) ≥ Lγ (τ ) + (1 + γ)ϕR (τ A) ≥ 0 for τ > 0 and γ ∈ [0, 1] which yields (n)

(μI + Mγ (τ ))−1 ≤ (μI + Lγ (τ ) + (1 + γ)ϕR (τ A))−1 ,

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(n)

for μ > 0 and γ ∈ [0, 1]. Since s-limτ →+0 ϕR (τ A) = I we obtain from Lemma 4.3 that (n)

s- lim (μI + Lγ (τ ) + (1 + γ)ϕR (τ A))−1 = ((1 + μ + γ)I + C)−1 . τ →+0

Hence

lim sup (μI + Mγ (τ ))−1 h, h ≤ ((1 + μ + γ)I + C)−1 h, h τ →+0

(4.21)

for μ > 0 and γ ∈ [0, 1]. Furthermore, we note that Mγ (τ ) ≤ (1 + γ)I + Lγ (τ ) + (1 + γ)ρ(τ A), where ρ(y) := ϕR (y) + ϕI (y) − 1, y ∈ R+ . One has ρ(0) = 0 and ρ (0) = 1. Hence we ﬁnd ρ(τ A) Mγ (τ ) ≤ (1 + γ)I + Lγ (τ ) + τ0 (1 + γ) τ for 0 < τ ≤ τ0 . By 1 + ρ(y) =

φR (y) − φI (y) φ2 (y) + φ2I (y) ≥ R ≥ 1, 2 |φ(y)| |φ(y)|2

y ∈ R+ ,

we ﬁnd ρ(y) ≥ 0, y ∈ R+ . We set fγ (y) :=

1 (fγ (y) + τ0 (1 + γ)ρ(y)) , (1 + τ0 + τ0 γ)

y ∈ R+ ,

τ0 > 0,

where fγ (y) is given by (4.2). It holds fγ (0) = 0 and fγ (0) = 1 as well as fγ (y) = 0 for y ∈ Σ. One gets gγ (τ B) ρ(τ A) fγ (τ A) Lγ (τ ) + τ0 (1 + γ) ≤ (1 + τ0 + γτ0 ) + Eτ Eτ τ τ τ for γ ∈ [0, 1] and τ ∈ (0, τ0 ] where gγ (y) is given by (4.3). Setting γ (τ ) := fγ (τ A) + Eτ gγ (τ B) Eτ , L τ τ

τ > 0,

γ ∈ [0, 1],

we obtain γ (τ ), Mγ (τ ) ≤ (1 + γ)I + (1 + τ0 + γτ0 )L

τ > 0,

γ ∈ [0, 1],

which yields γ (τ ))−1 ≤ (μI + Mγ (τ ))−1 , ((1 + μ + γ)I + (1 + τ0 + γτ0 )L μ > 0, 0 < τ ≤ τ0 , and γ ∈ [0, 1]. Let λ :=

1+μ+γ 1+τ0 +γτ0 ,

we ﬁnd

γ (τ ))−1 ≤ (1 + τ0 + γτ0 )(μI + Mγ (τ ))−1 (λI + L for μ > 0, 0 < τ ≤ τ0 and γ ∈ [0, 1]. Applying Lemma 4.2 we immediately get that ((λI + C)−1 h, h) ≤ (1 + τ0 + γτ0 ) lim inf ((μ + Mγ (τ ))−1 h, h) τ →+0

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for μ > 0, τ0 > 0, γ ∈ [0, 1] and h ∈ H. Since τ0 > 0 is arbitrary we ﬁnally obtain (((1 + μ + γ)I + C)−1 h, h) ≤ lim inf ((μI + Mγ (τ ))−1 h, h)

(4.22)

τ →+0

for μ > 0, γ ∈ [0, 1] and h ∈ H. From (4.21) and (4.22) we deduce that w- lim (μI + Mα (τ ))−1 = ((1 + μ + γ)I + C)−1 τ →+0

(4.23)

holds for μ > 0 and γ ∈ [0, 1]. Since the relation (4.23) is valid for every μ > 0 we get w- lim (μI + Mα (τ ))−1/2 = ((1 + μ + γ)I + C)−1/2 τ →+0

for μ > 0 which yields s- lim (μI + Mα (τ ))−1/2 = ((1 + μ + γ)I + C)−1/2 τ →+0

for μ > 0. The last relation proves (4.20). Let us introduce the operator-valued function 1 1 T (τ ) := $ (KR (τ ) + ϕR (τ A) − ϕI (τ A)) $ , I + M0 (τ ) I + M0 (τ )

τ > 0, (4.24)

where M0 (τ ) = KI (τ ) + ϕR (τ A) + ϕI (τ A) ≥ 0, see (4.1), (4.5) and (4.19). Lemma 4.5 Let A and B be non-negative self-adjoint operators in a separable Hilbert space H such that dom(A1/2 ) ∩ dom(B 1/2 ) is dense in H. If φ and ψ are admissible functions such that the conditions (4.17) are satisﬁed, then T (τ ) ≥ −I, τ > 0, and s- lim (iI + T (τ ))−1 = (iI + (2I + C)−1 )−1 , τ →+0

(4.25)

where T (τ ) is deﬁned by (4.24). Proof. Since Mγ (τ ) ≥ 0 for γ ∈ [0, 1] and τ > 0 we ﬁnd I + T (τ ) ≥ 0, which yields T (τ ) ≥ −I. Hence γT (τ ) ≥ −γI holds for γ ∈ [0, 1], and therefore the operator I + γT (τ ) is boundedly invertible for γ ∈ [0, 1) and we have the representation 1 1 (I + γT (τ ))−1 $ (I + Mγ (τ ))−1 = $ I + M0 (τ ) I + M0 (τ ) for γ ∈ [0, 1). Setting γ = 0 we ﬁnd from Lemma 4.4 that s- lim $ τ →+0

1 I + M0 (τ )

= (2I + C)−1/2 .

Since s-limτ →+0 (I + Mγ (τ ))−1 = ((2 + γ)I + C)−1 for γ ∈ [0, 1], by Lemma 4.4 we get that w-limτ →+0 (I + γT (τ ))−1 exists for γ ∈ [0, 1) and is given by 2I + C , γ ∈ [0, 1). w- lim (I + γT (τ ))−1 = τ →+0 (2 + γ)I + C

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Hence we get w- lim (νI + T (τ ))−1 = τ →+0

2I + C = (νI +(2I +C)−1 )−1 , I + ν(2I + C)

ν ∈ (1, ∞).

However, in standard manner we obtain from this relation s- lim (νI + T (τ ))−1 = (νI + (2I + C)−1 )−1 , τ →+0

ν ∈ (1, ∞),

which immediately implies (4.25).

Finally, for technical reasons we need the following lemma. First we recall that a bounded operator X is called accretive, if e (Xh, h) ≥ 0 for any h ∈ H. Lemma 4.6 Let {X(τ )}τ >0 be a sequence of bounded accretive operators on H. If there is self-adjoint operator Y such that w- lim (X(τ ) − ξ)−1 = (iY − ξ)−1 τ →+0

for some e (ξ) ≤ 0, then s- lim (X(τ ) − ξ)−1 = (iY − ξ)−1 . τ →+0

Proof. We set W (τ ) := (X(τ ) + ξ)(X(τ ) − ξ)−1

and W := (iY + ξ)(iY − ξ)−1 ,

τ > 0. One easily veriﬁes that {W (τ )}τ >0 is a family of contractions. Obviously, we have w-limτ →+0 W (τ ) = W . By

W (τ )h − W h 2 = W (τ )h 2 + h 2 − 2e ((W (τ )h, W h)),

τ > 0,

we ﬁnd lim sup W (τ )h − W h 2 ≤ 2 h 2 − 2 lim (W (τ )h, W h) = 0, τ →+0

τ →+0

which completes the proof.

Theorem 4.7 Let A and B be non-negative self-adjoint operators such that the intersection dom(A1/2 ) ∩ dom(B 1/2 ) is dense in H. If φ and ψ admissible functions such that the conditions (4.17) are satisfied, then (1.13) holds uniformly for t ∈ [0, T ] and T > 0. Proof. Taking into account the representation 1 1 $ (iI + T (τ ))−1 $ = (iI + (1 + i)ϕ(τ A) + K(τ ))−1 I + M0 (τ ) I + M0 (τ ) we ﬁnd & & 1 1 A)(Z(τ ) − ξ0 )−1 φ(τ A) $ (iI + T (τ ))−1 $ = φ(τ I + M0 (τ ) I + M0 (τ ) (4.26) for τ > 0, where φ(y) :=

1 ϕ(y) , y

∈ R+ , ξ0 = −(1 + i),

A) + Sτ , Z(τ ) := iφ(τ

τ > 0,

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and Sτ :=

&

& A), φ(τ A)K(τ ) φ(τ

IEOT

τ > 0.

A straightforward computation shows that

I − φ(τ A) $ I − ψ(τ B) Sτ = Eτ + φ(τ A) Eτ = Eτ Sτ Eτ τ > 0, τ τ (4.27) where I − F (τ ) , τ > 0, Sτ := τ $ $ and F (τ ) := φ(τ A)ψ(τ B) φ(τ A), τ > 0. Since for each τ > 0 the opera A)) ≥ 0, τ ≥ 0, the operator Z(τ ) is accretive tors Sτ are accretive and e (φ(τ and the inverse operator (Z(τ ) − ξ0 )−1 exists and its norm is bounded by one for τ > 0. From the representation (4.26), Lemma 4.4 and Lemma 4.5 we get w- lim (Z(τ ) − ξ0 )−1 = (iC − ξ)−1 , ξ = −1 − 2i, (4.28) τ →+0 & A) = I. Since Sτ is accretive we ﬁnd where we have used s-limτ →+0 φ(τ (iI + Sτ − ξ0 )−1 − (Z(τ ) − ξ0 )−1 = i(iI + Sτ − ξ0 )−1 A) − I)(Z(τ ) − ξ0 )−1 × (φ(τ

(4.29)

for τ > 0. From (4.28) and (4.29) we get w- lim (Sτ − ξ)−1 = (iC − ξ)−1 . τ →+0

Applying Lemma 4.6 we ﬁnd s- lim (Sτ − ξ)−1 = (iC − ξ)−1 τ →+0

which yields s- lim (μI + Sτ )−1 = (μI + iC)−1 , τ →+0

μ > 0.

(4.30)

Since 1 Sτ = − (I − Eτ ) + Sτ , τ

τ > 0,

we have (μI + Sτ )−1 − (μI + Sτ )−1 = (μI + Sτ )−1 (Sτ − Sτ )(μI + Sτ )−1 =

1 (μI + Sτ )−1 (I − Eτ ), μτ

τ > 0.

Let Δ = [0, d), d > 0, then we have (μI + Sτ )−1 EA (Δ)h − (μI + Sτ )−1 EA (Δ)h =

1 (μI + Sτ )−1 (I − Eτ )EA (Δ)h μτ

for τ > 0. Since (I − Eτ )EA (Δ)h = 0 if τ is sufﬁciently small we ﬁnd from (4.30) that s- lim (μI + Sτ )−1 = (μI + iC)−1 , τ →+0

μ > 0.

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From [2] we get s-limn→∞ F (t/n)n = e−itC uniformly in t ∈ [0, T ], T > 0, which completes the proof. Corollary 4.8 ([19]). Let A and B be non-negative self-adjoint operators such that dom(A1/2 ) ∩ dom(B 1/2 ) is dense in H. If φ(y) = ψ(y) = (1 + iy)−1 , y ∈ R+ , then

n s- lim (I + itA/n)−1 (I + itB/n)−1 = e−itC n→∞

uniformly in t ∈ [0, T ], T > 0. Proof. One easily veriﬁes that φ(·) and ψ(·) are admissible functions. Moreover, one has 1 y ≥ 0 and φI (y) = ψI (y) = − ≤0 φR (y) = ψR (y) = 1 + y2 1 + y2 which shows that the assumptions (4.17) are satisﬁed. Applying Theorem 4.7 we arrive at the sought conclusion. The function φ(y) = e−iy , y ∈ R+ , does not satisfy the conditions e (φ(y)) ≥ 0 and m (φ(y)) ≤ 0. However, its modiﬁcation φ(y) := e−iy χ[0,π/2] (y), y ∈ R+ , obeys e (φ(y)) ≥ 0 and m (φ(y)) ≤ 0, y ∈ R+ . In particular, the function (1.7) satisﬁes the conditions e (f (y)) ≥ 0 and m (f (y)) ≤ 0. The last observation leads to the following claim. Corollary 4.9 Let A and B non-negative self-adjoint operators in a separable Hilbert space H such that dom(A1/2 ) ∩ dom(B 1/2 ) is dense in H. Let φ and ψ be admissible functions. Then there are real numbers δφ > 0 and δψ > 0 such that s- lim (φ(tA/n)EA (([0, nδφ /t])ψ(tB/n)EB (([0, nδψ /t])) = e−itC n

n→+∞

holds uniformly in t ∈ [0, T ], T > 0. Proof. If the function φ is admissible, then φ(0) := limy→+0 φ(y) = 1 and φ (+0) = limy→+0 φ(y)−1 = −i. In particular, this yields e (φ(0)) = y limy→+0 φR (y) = 1 and m (φ(0)) = limy→+0 φI (y) = 0 as well as e (φ (+0)) = limy→+0 φR (y)−1 = 0 and m (φ (+0)) = limy→+0 φIy(y) = −1 y where φR (y) := e (φ(y)) and φI (y) := m (φ(y)), y ∈ R+ . Hence there is a δφ > 0 such that φR (y) ≥ 0 and φI (y) ≤ 0 for y ∈ [0, δφ ], and consequently, the function φ(y)χ[0,δφ ] (y) satisﬁes the assumptions of Theorem 4.7. Similar considerations are valid for ψ. Corollary 4.9 shows that the modiﬁed Trotter product formula (1.14) mentioned in the introduction is valid.

5. Concluding Remarks To conclude the paper let us list some open problems related to the Trotter–Kato product formula:

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(i) The relation between holomorphic Kato and admissible functions is an open question. Of course, the class of admissible functions is (in some sense) larger than the class of holomorphic Kato functions, even if the conditions (4.17) are satisﬁed. This follows from the fact that far from zero an admissible function can be chosen arbitrarily, in particular, one can extend it by zero. However, a holomorphic Kato function, which is zero on a set of positive Lebesgue measure, equals zero identically. On the other hand, it is not clear whether a holomorphic Kato function satisﬁes the conditions of admissible functions at zero, cf. Definition 4.1. (ii) Is it possible to verify the Trotter–Kato product formula (1.13) for admissible functions if one strengthens slightly the hypotheses made in Sect. 4, for instance, supposing that dom(B 1/2 ) ⊆ dom(A1/2 )? (iii) Are there non-negative self-adjoint operators A and B such that the Trotter–Kato product formula (1.13) does not hold for a pair of holomorphic Kato functions φ and ψ? (iv) What can be said about the operator norm convergence of the Trotter–Kato product formula (1.13)? It is known that for the real parameter there are several conditions, which guarantee the operator norm convergence, see [12,20] and references therein. For imaginary parameter, however, the available results are rather restricted, see [11]. Acknowledgements The authors are grateful for the hospitality they enjoyed, P.E. in WIAS and H.N. in Doppler Institute, during the time when the work was done. The research was supported by the Czech Ministry of Education, Youth and Sports within the project LC06002. Further we wish to thank Takashi Ichinose who draw our attention to the paper [10]. Finally, we would like to thank the referees for various comments, hints and valuable suggestions which helped us to improve our ﬁrst version of this paper essentially.

References [1] Cachia, V.: On a product formula for unitary groups. Bull. London Math. Soc. 37(4), 621–626 (2005) [2] Chernoﬀ, P.R.: Note on product formulas for operator semigroups. J. Funct. Anal. 2, 238–242 (1968) [3] Chernoﬀ, P.R.: Product formulas, nonlinear semigroups, and addition of unbounded operators. American Mathematical Society, Providence, R. I., 1974 (Memoirs of the American Mathematical Society, No. 140) [4] Exner, P.: Open quantum systems and Feynman integrals. Fundamental Theories of Physics. D. Reidel Publishing Co., Dordrecht (1985) [5] Exner, P., Ichinose, T.: A product formula related to quantum Zeno dynamics. Ann. Henri Poincar´e 6(2), 195–215 (2005) [6] Exner, P., Ichinose, T., Neidhardt, H., Zagrebnov, V.A.: Zeno product formula revisited. Integral Equ. Oper. Theory 57(1), 67–81 (2007)

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[7] Exner, P., Neidhardt, H.: Trotter–Kato product formula for unitary groups (2009) arXiv:0907.1199v1 [math-ph] [8] Facchi, P., Pascazio, S.: Quantum Zeno dynamics: mathematical and physical aspects. J. Phys. A 41(49), 493001 (2008) [9] Faris, W.G.: The product formula for semigroups deﬁned by Friedrichs extensions. Pac. J. Math. 22, 47–70 (1967) [10] Ichinose, T.: A product formula and its application to the Schr¨ odinger equation. Publ. Res. Inst. Math. Sci. 16(2), 585–600 (1980) [11] Ichinose, T., Tamura, H.: Note on the norm convergence of the unitary Trotter product formula. Lett. Math. Phys. 70(1), 65–81 (2004) [12] Ichinose, T., Tamura, H., Tamura, Hiroshi, Zagrebnov, Valentin A.: Note on the paper: “The norm convergence of the Trotter-Kato product formula with error bound” by T. Ichinose and H. Tamura [Comm. Math. Phys. 217 (2001), no. 3, 489–502; MR1822104 (2002e:47048a)]. Comm. Math. Phys. 221(3), 499– 510 (2001) [13] Johnson, G.W., Lapidus, M.L.: The Feynman integral and Feynman’s operational calculus. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford Science Publications, New York 2000 [14] Kato, T.: Schr¨ odinger operators with singular potentials. Israel J. Math. 13(1972), 135–148 (1973) [15] Kato, T.: Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups. In: Topics in functional analysis (essays dedicated to M. G. Kre˘ın on the occasion of his 70th birthday), Adv. in Math. Suppl. Stud., vol. 3, pp. 185–195. Academic Press, New York (1978) [16] Lapidus, M.: Formules de moyenne et de produit pour les r´esolvantes imaginaires d’op´erateurs auto-adjoints. C. R. Acad. Sci. Paris S´er. A-B 291(7), A451–A454 (1980) [17] Lapidus, M.L.: Generalization of the Trotter-Lie formula. Integral Equ. Oper. Theory 4(3), 366–415 (1981) [18] Lapidus, M.L.: The problem of the Trotter-Lie formula for unitary groups of operators. S´eminaire Choquet: Initiation ` a l’Analyse. Publ. Math. Univ. Pierre et Marie Curie (Paris IV). 20 `eme ann´ee, 1980/81, 46:1701–1746, (1982) [19] Lapidus, M.L.: Product formula for imaginary resolvents with application to a modiﬁed Feynman integral. J. Funct. Anal. 63(3), 261–275 (1985) [20] Neidhardt, H., Zagrebnov, V.A.: Trotter–Kato product formula and operator-norm convergence. Comm. Math. Phys 205(1), 129–159 (1999) [21] Nelson, E.: Feynman integrals and the Schr¨ odinger equation. J. Math. Phys. 5, 332–343 (1964) [22] B´ela Sz.-Nagy and Ciprian Foia¸s. Harmonic analysis of operators on Hilbert space. North-Holland Publishing Co., Amsterdam (1970) (Translated from the French and revised) [23] Trotter, H.F.: On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545–551 (1959)

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Pavel Exner Department of Theoretical Physics, NPI Academy of Sciences ˇ z 25068 Reˇ Czech Republic and Doppler Institute, Czech Technical University Bˇrehov´ a 7, 11519 Prague Czech Republic e-mail: [email protected] Hagen Neidhardt (B) Weierstrass Institute for Applied Analysis and Stochastics Mohrenstrasse 39, 10117 Berlin, Germany e-mail: [email protected] Valentin A. Zagrebnov Centre de Physique Th´eorique, Universit´e de la M´editerran´ee (Aix-Marseille II) Luminy-Case 907, 13288 Marseille Cedex 9, France e-mail: [email protected] Received: July 13, 2009. Revised: January 4, 2011.

Integr. Equ. Oper. Theory 69 (2011), 479–508 DOI 10.1007/s00020-010-1859-7 Published online February 9, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Scattering Theory for CMV Matrices: Uniqueness, Helson–Szeg˝ o and Strong Szeg˝ o Theorems L. Golinskii, A. Kheifets, F. Peherstorfer and P. Yuditskii Abstract. We develop a scattering theory for CMV matrices, similar to the Faddeev–Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for uniqueness, which are connected with the Helson–Szeg˝ o and the strong Szeg˝ o theorems. The first condition is given in terms of the boundedness of a transformation operator associated with the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblunsky parameters of the CMV matrices, corresponding spectral measures and scattering functions. Mathematics Subject Classification (2010). 30H15, 30H10, 47B36. Keywords. Spectral measure, scattering function, γ-generating pair, Arov regularity, Schur algorithm, Faddeev–Marchenko space, transformation operator, Gelfand–Levitan–Marchenko equation.

1. Introduction To a given collection of numbers {αn }n≥0 in the open unit disk D, called the Verblunsky coefficients, and α−1 in the unit circle T, we deﬁne the CMV matrix A = Aod Ae , where ⎡ ⎤ ⎤ ⎡ −α−1 A0 ⎢ ⎥ A 1 ⎢ ⎥ ⎥ ⎢ A2 Aod = ⎢ ⎥ , Ae = ⎣ ⎦, A3 ⎣ ⎦ .. . .. . F. Peherstorfer: Deceased. The work of A. Kheifets was partially supported by the University of Massachusetts Lowell Research and Scholarship Grant, project number: H50090000000010. The work of F. Peherstorfer and P. Yuditskii was partially supported by the Austrian Science Found FWF, project number: P20413-N18.

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and the Ak ’s are the 2 × 2 unitary matrices

αk ρk , ρk = 1 − |αk |2 . Ak = ρk −αk Unlike the standard convention [29, p. 265], we do not ﬁx the value α−1 = −1. Our reasons will become clear later on. Note that A is a unitary operator on l2 (Z+ ). The initial vector e0 of the standard basis is cyclic for A. Indeed, by the definition for n = 0, 1, . . . A{e2n ρ2n − e2n+1 α2n } = e2n+1 α2n+1 + e2n+2 ρ2n+1 −1

A

{e2n+1 ρ2n+1 − e2n+2 α2n+1 } = e2n+2 α2n+2 + e2n+3 ρ2n+2

(1.1)

A−1 e0 = −α−1 (e0 α0 + e1 ρ0 ). That is, acting in turn by A−1 and A on e0 and taking the linear combinations, we can get any vector of the standard basis. CMV matrices were introduced in [10]. More recent surveys on this topic are [21,29,30]. 2 are the standard Hardy spaces We will use the following notations: H± of analytic/conjugate-analytic functions of the unit disk, H ∞ is the space of bounded analytic functions in the unit disk; the Schur class is the unit ball of H ∞ . 1.1. Spectral Characteristics Since A is a unitary operator, the following function A+z t+z e0 , e0 = σ(dt) R(z) := A−z t−z

(1.2)

T

has a nonnegative real part in the unit disk, which yields the integral formula in (1.2). The measure σ = σ(A) is called the spectral measure of A with respect to the cyclic vector e0 . The standard Lebesgue decomposition is σ(dt) = w(t)m(dt) + σs (dt)

(1.3)

where m(dt) is the normalized Lebesgue measure, and σs is the singular component. We will say that A is absolutely continuous if σs = 0. Note that

R(0) = e0 , e0 = σ(dt) = 1, T

so σ is a probability measure. We deﬁne the function φ by the equation 1 − α−1 φ(z) 1 − R(z) , R(z) = . φ(z) = α−1 1 + R(z) 1 + α−1 φ(z)

(1.4)

Then |φ| ≤ 1, φ(0) = 0. An important relation is w(t) = R(t) =

1 − |φ(t)|2 |1 + α−1 φ(t)|2

(1.5)

a.e. on T. Given a CMV matrix A, the spectral measure σ is uniquely deﬁned by (1.2). Conversely, by the ﬁrst formula in (1.4), the measure σ uniquely deﬁnes α−1 φ. Hence, to recover φ (and by that αn ), we need to know α−1 . Therefore,

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the pair {σ, α−1 }, not just σ, determines uniquely the CMV matrix A. That is why we consider the pair {σ, α−1 } as the spectral data. The one-to-one correspondences A ←→ {σ, α−1 } ←→ {R, α−1 } ←→ {φ, α−1 } are studied in the theory of orthogonal polynomials on the unit circle (OPUC) [29] and in the Schur analysis [28]. 1.2. Direct Scattering By definition, the matrix A is in the Szeg˝ o class, A ∈ Sz, if |αk |2 < ∞. By the Szeg˝o–Kolmogorov–Krein theorem (see, e.g., [29, Theorem 2.3.1]), A ∈ Sz if and only if in representation (1.3) of the spectral measure σ we have log w ∈ L1 .

(1.6)

There are no constraints on the singular part σs of the spectral measure. We o class and σs = 0. say that A ∈ Szac if A is in the Szeg˝ A standard fact from the theory of Hardy classes reads that property (1.6) yields w(t) = |D(t)|2

(1.7)

2 function, D(0) > 0. a.e. on T, where D is a boundary value of an outer H+ D is known as the Szeg˝ o function of the measure σ. By the same theorem ([29, Theorem 2.3.1]), ∞ D(0) = ρk . (1.8) k=0

It follows from (1.5) that A ∈ Sz ⇔ log(1 − |φ|2 ) ∈ L1 , so there exists a unique outer function ψ such that |ψ(t)|2 + |φ(t)|2 = 1,

ψ(0) > 0.

(1.9)

In this case we will also say that the function φ is of the Szeg˝ o class. By (1.5) 2 ψ(t) (1.10) w(t) = 1 + α−1 φ(t) a.e. Hence D is of the form ψ(z) , D(z) = 1 + α−1 φ(z)

ψ(0) = D(0) =

∞

ρk .

(1.11)

k=0

Definition 1.1. The scattering function of A ∈ Sz is deﬁned as s(t) = −α−1

D(t) D(t)

=

ψ(t) (−α−1 ) − φ(t) , ψ(t) 1 − (−α−1 )φ(t)

t ∈ T.

(1.12)

Note that |s(t)| = 1 a.e. on T. In the Faddeev–Marchenko theory the scattering function appears as a coefﬁcient in the leading term of certain asymptotics. In our context we have

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Theorem 1.2. Let A ∈ Sz. Then there exists a unique generalized eigenvector Ψ(t) = {Ψn (t)}∞ n=0 such that Ψ0 (t) Ψ1 (t) . . . A = t Ψ0 (t) Ψ1 (t) . . . , t ∈ T, (1.13) and the following asymptotics holds in L2 -norm Ψ2n (t) = tn + o(1),

Ψ2n+1 (t) = s(t)t−n−1 + o(1),

n → ∞.

(1.14)

Theorem 1.2 is a restatement of the classical Szeg˝o theorem on the asymptotic behavior of OPUC [29, Theorem 2.4.1], since we can choose Ψ2n (t) = D(t)t−n p2n (t),

Ψ2n+1 (t) = −α−1 D(t)tn p2n+1 (t)

as a solution of (1.13), where pn are orthonormal polynomials with respect to σ (cf. [29, Lemma 4.3.14]). 1.3. Main Objectives and Results The main objective of this paper is to solve the inverse scattering problem (the heart of the Faddeev–Marchenko theory [12,22,23]), i.e., to reconstruct the CMV matrix A from its scattering function s. In general, the solution of this inverse problem is not unique. In particular, s does not contain any information about the (possible) singular measure. Even in the class of absolutely continuous measures the correspondence A → s is not one to one (see Examples 3.4 and 7.13). In this paper we show that the uniqueness in the inverse scattering is equivalent to the Arov regularity (Definition 2.4) of the function φ (see Theorem 3.1 below). We also consider two interesting subclasses of the uniqueness class, namely, the Helson–Szeg˝o class and the strong Szeg˝ o class. The ﬁrst class is exactly the one for which a certain transformation operator1 is invertible. We obtain a complete description of the corresponding spectral measures and the scattering functions in Sect. 6. The second class is the one for which the transformation operators have a determinant. For this class a complete description is given to the Verblunsky coefﬁcients, the spectral measures and the scattering functions in Sect. 7. This paper is the result of a substantial revision of the manuscript [16]. The authors are thankful to Peller for useful discussions concerning the Besov spaces. They are also thankful to the reviewer for a number of comments that improved the write up of the paper.

2. Adamyan–Arov–Krein Theory We begin with the following Definition 2.1. Pairs (φ, ψ) with properties 1. φ, ψ are in H ∞ , φ(0) = 0; 2. ψ is an outer function, ψ(0) > 0; 3. |φ|2 + |ψ|2 = 1 on the boundary are called γ-generating. 1 A classical monograph on the subject is [22], where transformation operators are extensively used in spectral and scattering theory for Schr¨ odinger operator. Historical remarks are also given in Sect. 1.

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As we saw in Sect. 1, such pairs appear in the spectral analysis of CMV matrices. Proposition 2.2. To every γ-generating pair (φ, ψ) one can associate the family of functions (compare to (1.12)) ψ E −φ , E ∈ H ∞ , E ∞ ≤ 1. (2.1) ψ 1 − Eφ All the functions sE belong to the unit ball of L∞ . Moreover, the Fourier coefficients (sE )k with negative indices k are independent of E. sE =

Proof. The ﬁrst assertion follows from the relation 1 − |sE |2 =

(1 − |E|2 )(1 − |φ|2 ) . |1 − Eφ|2

Let s0 correspond to E = 0, then sE − s0 =

ψ ψ E −φ ψ2 E + φ= ∈ H ∞. 1 − Eφ ψ 1 − Eφ ψ

(2.2)

The following observation will be helpful later on. For each γgenerating pair (φ, ψ) and any Schur class function E (i.e., E ∈ H ∞ , E ∞ ≤ 1) the function DE (z) :=

ψ(z) 1 − E(z)φ(z)

(2.3)

2 is an outer function from H+ . Indeed, DE is the outer function (as a ratio of outer functions) from the Smirnov class, and

1 − |φ(t)|2 1 − |E(t)φ(t)|2 1 + E(t)φ(t) . ≤ = 2 2 |1 − E(t)φ(t)| |1 − E(t)φ(t)| 1 − E(t)φ(t) The right hand side is the boundary value of the Poisson integral of a ﬁnite positive measure, and so belongs to L1 (T). Adamyan–Arov–Krein (AAK) theory deals with the following Nehari problem [1–3,13]. |DE (t)|2 =

Problem 2.3. (Nehari) Given function h ∈ L∞ , h ∞ ≤ 1, describe the set of functions N (h) = {f ∈ L∞ : f ∞ ≤ 1, f − h ∈ H ∞ }, that is, the set of functions f ∈ L∞ such that the Fourier coefﬁcients (f )k = (h)k for k < 0. The Nehari problem is indeterminate (determinate) if it has inﬁnitely many solutions (a unique solution). It follows from Proposition 2.2 that the function s deﬁned in (1.12) is a unimodular solution of an indeterminate Nehari problem. By Proposition 2.2 for every γ-generating pair (φ, ψ) the functions {sE }, described in (2.1), are solutions of the Nehari problem, generated by s0 . However, formula (2.1) may not produce all the solutions of this problem.

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Definition 2.4. A γ-generating pair (φ, ψ), or simply a function φ, is called Arov-regular (see [4]) if {sE : E ∈ H ∞ and E ≤ 1} = N (s0 ). Definition 2.5. We say that a CM V matrix of the Szeg˝ o class is regular, if the associated function φ deﬁned by (1.4) and (1.2) is Arov-regular. An important result is proved in [2, Remark 4.1]. Theorem 2.6. (AAK) If φ is Arov-regular, then for every Schur class function E the measure σE

1 + E(z)φ(z) t+z = σE (dt) (2.4) 1 − E(z)φ(z) t−z T

is absolutely continuous. 2 2 → H− as For h ∈ L∞ , h ∞ ≤ 1, we deﬁne a Hankel operator H : H+ 2 , H = Hh = P− h|H+

h is called a symbol of H. Note that H ≤ h ∞ ≤ 1, 2 2 2 → H+ is H∗ = P+ h|H− , P+ (P− ) is the and the adjoint operator H∗ : H− 2 2 2 standard projection from L onto H+ (H− ). For f ∞ ≤ 1, Hf = H if and only if f ∈ N (h). A Hankel operator Hh is called indeterminate, if it has many symbols f with f ∞ ≤ 1.

Theorem 2.7. (Adamyan–Arov–Krein) The Nehari problem is indeterminate if and only if 2 , 1 ∈ (I − H∗ H)1/2 H+

H = Hh .

In this case the set N (h) is of the form ψ H E − φH ∞ N (h) = fE = : E ∈ H , E ∞ ≤ 1 , ψ H 1 − EφH

(2.5)

(2.6)

where (φH , ψH ) is a uniquely determined Arov-regular pair, ψH (0) > 0. The next theorem gives sufﬁcient conditions for regularity of φ. The second condition is known (see, e.g., [4,27]). For a weaker condition on |ψ|, which ensures regularity of φ (see [31]). Theorem 2.8. φ is Arov-regular if one of the following conditions holds 1. 2.

for a constant τ ∈ C, |τ | = 1, the measure στ , defined by (2.4), is 2 ; absolutely continuous and (1 − τ φ)ψ −1 ∈ H+ −1 2 ψ ∈ H+ .

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We consider a unimodular function

ψ 1 − τφ ψ τ −φ =τ . (2.7) 1 − τ φ ψ ψ 1 − τφ We associate an indeterminate Nehari problem to s with the Hankel operator H = Hs . By Theorem 2.7, s admits the representation s=

s=

ψ H E − φH ψ H 1 − EφH

(2.8)

with the Arov-regular pair (φH , ψH ) and the inner function E, so we can write s=E

ψ H 1 − E φH . ψ H 1 − E φH

Combining (2.7) and (2.8), we get G := E

1 − τφ ψH ψH 1 − τφ =τ . ψ 1 − EφH ψ 1 − EφH

(2.9)

2 It was mentioned above (see (2.3)) that ψH (1 − EφH )−1 ∈ H+ , so, due to assumption (1), G ∈ H 1 . At the same time G ∈ H 1 , so G is a constant function. Since E is the inner part of G, we have E = const. Using the normalization ψ(0) > 0, ψH (0) > 0, we get E = τ and τ G > 0. Next, by (2.9),

τG so, in particular,

ψH ψ = . 1 − τφ 1 − τ φH

ψ 2 ψH 2 = (τ G)2 1 − τ φH . 1 − τφ

In other words, 1 + τ φH 1 + τφ = (2.10) 1 − τφ 1 − τ φH almost everywhere on the unit circle. By assumption (1), στ is absolutely continuous, and, by Theorem 2.6, στ,H is absolutely continuous. Therefore, integration of (2.10) over the unit t+z circle T with the kernel t−z gives (τ G)2

(τ G)2

1 + τ φH (z) 1 + τ φ(z) = . 1 − τ φ(z) 1 − τ φH (z)

Since φ(0) = φH (0) = 0, we get that τ G = 1 and 1 + τφ 1 + τ φH = . 1 − τφ 1 − τ φH Therefore φ = φH , as claimed. (2). We choose τ = −α−1 . We show ﬁrst that σ corresponding to absolutely continuous. Indeed, by (1.11) 1 D(z) 1 − α−1 φ(z) = ∈ H 1 ⇒ R(z) = ∈ H 1. 1 + α−1 φ(z) ψ(z) 1 + α−1 φ(z)

1+τ φ 1−τ φ

is

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By the Cauchy integral formula,

R(z) = T

and

1 = R(0) =

R(t) m(dt), 1 − tz

R(t)m(dt) =

R(t)m(dt) = T

IEOT

T

w(t)m(dt), T

so σs = 0, as claimed. Next, by the assumption 1 + α−1 φ(z) 1 2 = ∈ H+ , D(z) ψ(z) and the second statement of the theorem follows from the ﬁrst one.

Definition 2.9. If φ is Arov-regular and E is a constant function, |E| = 1, then the function ψ E −φ sE = ψ 1 − Eφ is called canonical. Such sE is also called a canonical symbol of the associated Hankel operator H. Proposition 2.10. [1,2,19,27] Let s be a unimodular function on T. Then the following are equivalent 1. s is canonical, 2 2 is dense in H+ , 2. P+ s|H+ 2 2 (the space is of codimension one), P+ ts|H+ is not dense in H+ 3. sh+ = h− has only the trivial solution, sh+ = th− has a nontrivial solution (the space of solutions is of dimen2 . sion one), h± ∈ H± As a simple consequence of Proposition 2.10 we have Proposition 2.11. Let s be canonical, and N = 0 an integer. Then stN is noncanonical. Proof. Assume that both s and stN are canonical. Then without loss of generality we may assume that N > 0. By the second condition (3) the equa2 tion sh+ = th− has a nontrivial solution. Hence stN h+ = t1−N h− ∈ H− also has a nontrivial solution, which means that the ﬁrst condition in (3) fails for stN . So stN is noncanonical, which is a contradiction.

3. Uniqueness in the Inverse Scattering Problem We are interested in the following questions: given a unimodular solution s of an indeterminate Nehari problem, does there exist a CM V matrix A with this scattering function? Is such a matrix A unique? The main result of the section gives complete answers to these questions. Theorem 3.1. 1. Each regular CM V matrix A has absolutely continuous spectral measure σ(A), and its scattering function s is canonical.

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Let s be a canonical solution of an indeterminate Nehari problem, then there exists a unique CM V matrix A ∈ Szac , whose scattering function is s, moreover A is regular. Let s be a noncanonical unimodular solution of an indeterminate Nehari problem, then there exist infinitely many CM V matrices A ∈ Szac with scattering function s.

Proof. (1). Let A be regular, so φ in (1.12) is Arov-regular. By Theorem 2.6, σ(A) is absolutely continuous. By Definition 2.9, the function s, deﬁned by (1.12) with E = −α−1 , is canonical. (2). Since s is canonical, we have ψH E − φH , (3.1) ψ H 1 − EφH where E is a unimodular constant. Therefore, a solution of the inverse scattering problem can be chosen as ψH 2 ψH (z) m(dt). , σ(dt) := α−1 := −E, D(z) := 1 − EφH (z) 1 − EφH s=

Since

R(z) = T

1 − α−1 φH (z) t+z σ(dt) = , t−z 1 + α−1 φH (z)

the function φ associated with σ coincides with φH . Hence, φ is regular, as needed. Assume that there are two absolutely continuous CMV matrices A and A of Szeg˝o class with the scattering function s. The corresponding spectral measures are σ = |D|2 m and σ = |D |2 m,

2 |D| m(dt) = |D |2 m(dt) = 1, D(0) > 0, D (0) > 0. (3.2) T

T

Then we have s(t) = −α−1

D(t) D(t)

= −α−1

D (t) D (t)

(3.3)

and −α−1 D(t)s(t) = D(t)

− α−1 D (t)s(t) = D (t).

There exist two real nonzero constants α and α such that αD(0) + α D (0) = 0, Then 2 , h− = αD + α D ∈ H−

2 h+ = −α−1 αD − α−1 α D ∈ H+

is a solution of sh+ = h− . Since s is canonical, by Proposition 2.10, (3), this is a trivial solution. In other words, αD + α D = 0 identically. In view of (3.2), this yields D = D . The uniqueness follows.

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(3). If s is a noncanonical unimodular solution of an indeterminate Nehari problem, then in (3.1) E is a nonconstant inner function, and (3.1) can be rephrased as s=E

1 + τ E ψH 1 − EφH ψH 1 − EφH =τ , 1 + τ E ψ H 1 − EφH ψ H 1 − EφH

∀τ ∈ T.

Therefore, we get inﬁnitely many solutions of the inverse scattering problem |1 + τ E(0)| (1 + τ E(z))ψH (z) 2 ∈ H+ , 1 + τ E(0) 1 − E(z)φH (z) 1 + τ E(0) where kτ > 0 is chosen to make |D(t)|2 m(dt) = 1. It is veriﬁed by a

α−1 = −

τ + E(0)

,

D(z) = kτ

T

straightforward computation that indeed α−1 and |D| are different for different τ . Corollary 3.2. 1. Let A be a regular CM V matrix, let A1 ∈ Szac . If A and A1 have the same scattering function s then A1 = A. 2. Let A ∈ Szac be a nonregular CM V matrix with the scattering function s. Then there exist infinitely many CM V matrices in Szac with the same scattering function. Remark 3.3. As we saw earlier, the scattering function of every CMV matrix of Szeg˝o class is a unimodular solution of an indeterminate Nehari problem. As a byproduct of this section, we have shown that every unimodular solution of an indeterminate Nehari problem is the scattering function of a CM V matrix A ∈ Szac . We complete this section with a simple example, when the solution of the inverse scattering problem is not unique. Example 3.4. Let P (z) =

N

(z − tj ),

tj ∈ T

j=1

be a monic polynomial of degree N with all its zeros on T. For the measure σ(dt) = w(t)m(dt),

w(t) := c|P (t)|2 = c

N

|t − tj |2 ,

c > 0,

j=1

the Szeg˝o function D =

√ cP/P (0), and the scattering function is

s(t) = −α−1

D(t) D(t)

= −α−1 P (0)tN .

Thus for any two polynomials P1 , P2 with P1 (0) = P2 (0) we have s1 = s2 , and there is no uniqueness in the inverse scattering even for α−1 = −1. Note that s is not canonical.

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In the case N = 1 we have s = α−1 t1 t, and again there is no uniqueness.

4. The Schur Algorithm Given φ of the Schur class, φ(0) = 0, we deﬁne φ0 (z) = φ(z), zf0 (z) = φ0 and the sequence of (Schur class) functions {fn }n≥0 by fn+1 (z) =

fn (z) − fn (0) z(1 − fn (z)fn (0))

,

n = 0, 1, . . .

(4.1)

We also let φn (z) = zfn (z). This procedure is called the Schur algorithm. By the Geronimus theorem (see, e.g., [29, Theorem 3.1.4]) fn (0) = an := −α−1 αn ,

n = 0, 1 . . . .

(4.2)

If φ is of Szeg˝o class (see Sect. 1.2), then all the functions φn , deﬁned by the Schur algorithm, are also of Szeg˝ o class. So, we can deﬁne a sequence of γ-generating pairs (φn , ψn ). It is easy to see that {ψn } satisﬁes ψk+1 = ψk

ρk , 1 − ak fk

ψn = ψ

n−1

ρk , 1 − ak fk

k=0

ψ = ψ0 .

(4.3)

Indeed, for t ∈ T |ψk+1 (t)|2 = 1 − |fk+1 (t)|2 =

(1 − |fk (t)|2 )(1 − |αk |2 ) |ψk (t)|2 ρ2k = . |1 − ak fk (t)|2 |1 − ak fk (t)|2

It is also clear from (1.11) and (4.2) that ψn (0) = ψ(0)

n−1

ρ−1 k =

k=0

∞

ρ−1 k .

(4.4)

k=n

Lemma 4.1. Recurrences (4.1) and (4.3) can be put into the form ⎡ ⎤ ⎤ ⎡ 1 φn+1 φn 1 t −an 1 t 0 − − ψ ψ ψ ψ n+1 ⎦ n ⎦ = ⎣ n+1 ⎣ n . (4.5) φn+1 φn 1 1 ρn 0 1 −a t 1 −ψ n − ψn ψn+1 ψn+1 n Proof. By (4.1), (1 − an fn )φn+1 = fn − an , and (1 − an fn )(1 + an φn+1 ) = 1 − |an |2 = ρ2n . Therefore, ρ2n = 1 + an φn+1 . 1 − an fn Next, by (4.3), 1 ρn 1 1 + an φn+1 = = = ψn ψn+1 1 − an fn ψn+1 ρn which is (2, 2) entry of (4.5).

φn+1 1 an + ψn+1 ψn+1

1 , ρn

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Similarly, by (4.1), (1 − an fn )(φn+1 + an ) = tρ2n φn , and ρ2n φn = t(φn+1 + an ). 1 − an fn Therefore, by (4.3), ρn φn φn 1 t(φn+1 + an ) = = =t ψn ψn+1 1 − an fn ψn+1 ρn

φn+1 1 + an ψn+1 ψn+1

1 , ρn

which is (2, 1) entry of (4.5). Repeatedly applying (4.5) we get for n > j ⎤ ⎤ ⎛ ←− ⎡ 1 φj φn (n−j) 1 n−1 t − − t 0 ψj ⎦ ψn ⎦ ⎜ ⎣ ψj ⎣ ψn = ⎝ φ φn 1 1 0 1 −ψ − ψjj k=j −ak t ψn ψj n ⎡

−ak

1

⎞ 1⎟ ⎠. ρk (4.6)

We deﬁne

Pjj Qjj

and for n > j

Pnj (z) Qjn (z)

=

0 1

⎛ ←− n−1 z ⎜ =⎝ −ak z k=j

−ak 1

(4.7)

⎞

1⎟ 0 ⎠ 1 . ρk

(4.8)

Note that Pnj and Qjn are polynomials, deg Pnj ≤ n − j − 1,

deg Qjn ≤ n − j − 1,

Qjn (0) =

n−1

ρ−1 k > 0.

(4.9)

k=j

It is easily seen from (4.8) that ⎞ ⎛ ←− 0 n−1 j Pn Pn t −ak 1 ⎟ Pj0 ⎜ = ⎠ ⎝ −ak t 1 Q0j ρk Q0n Qjn k=j

0 . 1

Taking determinants we come to Pn0 (z)Qjn (z) − Q0n (z)Pnj (z) = z n−j Pj0 (z).

(4.10)

From (4.6) and (4.8) we have −

φj ψj

= tj−n

Pnj − φn Qjn , ψn

1 −Pnj φn + Qjn = . ψj ψn

(4.11)

Remark 4.2. Matrix products (4.8) arise in the Szeg˝ o recurrences for OPUC (see, e.g., [29, formula (1.5.35)]).

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We also deﬁne Enj =

Pnj Qjn

n ≥ j.

,

(4.12)

It is clear from (4.8) that Enj can be deﬁned recursively as Ejj = 0,

j En+1 =

tEnj − an 1 − an tEnj

,

n ≥ j,

(4.13)

so Enj ∞ < 1 for n ≥ j. Remark 4.3. Using the notations introduced just above we can rewrite (4.10) as En0 (z) − Enj (z) =

Pn0 (z)Qjn (z) − Q0n (z)Pnj (z) Q0n (z)Qjn (z)

=

z n−j Pj0 (z) Q0n (z)Qjn (z)

,

(4.14)

which implies, in view of (4.9), that the difference En0 (z) − Enj (z)

(4.15)

vanishes at the origin with order of at least n − j. The second equality in (4.11) also can be rewritten as ψn = −Pnj φn + Qjn = Qjn (1 − Enj φn ). ψj Hence, ψn 1 = Qjn ψj 1 − Enj φn

(4.16)

and, therefore, is a polynomial of degree at most n − j − 1. ψk Lemma 4.4. Let sk := − ψ φk . Then, for n ≥ j k

(n−j) ψn

sj = t

Enj − φn

ψ n 1 − Enj φn

(4.17)

and (cf. (2.2)) sj tn−j − sn =

ψn2 Enj 1 − Enj φn

∈ H ∞,

where Enj are defined as in (4.8)–(4.12) or (equivalently) by (4.13). 0 Proof. Apply (4.6) to . 1

(4.18)

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5. The Model Space and the Transformation Operator Let A ∈ Sz, (φ, ψ) be the corresponding γ-generating pair. Definition 5.1. We deﬁne the Faddeev–Marchenko space Mφ as the Hilbert space of analytic vector-functions F1 2 , F1 = F+ /ψ, F2 = F− /ψ, F± ∈ H± F2 with the inner product

F1 G1 = , G1 F2 G2 M φ

G2

T

1 s0

s0 1

F1 m(dt), F2

(5.1)

φ. where s0 = − ψ ψ We mention that Mφ is a functional model space for the CMV matrix A. More specifically, we can start with the de Branges–Rovnyak model space 2 , Kφ : F ± ∈ H ± −1 2

1 φ(t) F+ (t) F+ = m(dt) F+ (t) F− (t) F− K F− (t) φ(t) 1 φ T

and transform it as follows

= F+ (t) F− (t) T

1

−φ(t)

−φ(t)

1

F+ (t) F− (t)

m(dt) 1 − |φ(t)|2

F+ (t) m(dt) F+ (t) F− (t) F− (t) |ψ(t)|2 −φ(t) 1 T

! " F+ /ψ 1 s0 m(dt). = F+ /ψ F− /ψ s0 1 F− /ψ

=

−φ(t)

1

T

Proposition 5.2. The linear manifold

I h+ g+ = , h− g− H M φ

2 H+

2 H−

H∗ I

is contained in Mφ , and

g+ h+ , h− g−

,

L2

where H = Hs0 . 2 Proof. Let h± ∈ H± . Then

h+ = and

h+ h−

2

= Mφ

ψh+ , ψ

h− =

ψh− ψ

(|h+ |2 + |h− |2 + s0 h+ h− + s0 h+ h− ) m(dt)

T

= h+ 2 + h− 2 + s0 h+ , h− + s0 h+ , h− .

(5.2)

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But s0 h+ , h− = P− s0 h+ , h− = Hh+ , h− , so 2 h+ = h+ 2 + h− 2 + Hh+ , h− + Hh+ , h− , h− M φ

as claimed.

The next theorem was proved in [18,19,27] (see also [9]). 2 H+ Theorem 5.3. φ is Arov-regular if and only if the set is dense in Mφ . 2 H− Let A ∈ Sz, (φn , ψn ) be the sequence of γ-generating pairs related to the Schur algorithm (4.1). Lemma 5.4. The vectors

⎡ fn = ⎣

tn ψn φn ψn

⎤ ⎦

(5.3)

form an orthonormal system in the Faddeev–Marchenko space Mφ . Let Mφ,+ ⊥ be the subspace in Mφ spanned by those vectors, then Mφ,+ consists of func2 tions with F1 = 0, F2 ∈ H− . Proof. Due to recurrence (4.3), tn tn hn , = ψn ψ

φn φ hn = n , ψn ψ

hn ∈ H ∞ .

Using Lemma 4.4, we ﬁrst compute $⎤ ⎤ ⎡ ⎤ ⎡ # ⎡ φn φn tn φn ψn En tn 1 n s + t s + + 0 n 1 s0 ⎣ ψn ⎦ ψn ψn ψn ψn 1−En φn ⎦=⎣ ⎦ =⎣ φn φn φn ψn En s0 tn sn s0 1 + + + ψn

=

ψn

t

ψn

ψn

ψn 1−En φn En ψn 1−En φn

n

,

1−En φn

ψn

En = En0 .

Next, we assume that m ≥ n and compute ⎡ n ⎤ " 1 s ψt ! m 1 ψn En φm (m−n) ψ n φm t 0 ⎣ n ⎦ =t + . ψm φn ψm s0 1 ψ ψ m 1 − En φn m 1 − En φn ψ n

(5.4) Since En ∞ < 1 and (4.3) is in force, 1 ∈ H ∞, 1 − En φn

ψn ∈ H ∞. ψm

Hence (5.4) belongs to L∞ . In particular (n = m), this guarantees that fn ∈ Mφ . Furthermore, (5.4) implies

1 ψn En φm (m−n) ψ n m(dt) + m(dt) = δmn . fn , fm Mφ = t ψm 1 − En φn ψ m 1 − En φn T

T

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The ﬁrst assertion follows. To verify the second assertion, assume that vector

F+ /ψ F− /ψ

is

orthogonal to fn for all n = 0, 1, . . .. As above in (5.4) ⎡ n ⎤ ! " 1 s ψt 1 ψ ψn F − En F+ F− 0 ⎣ n ⎦ = F + tn n + , (5.5) φn ψ ψ s0 1 ψ 1 − En φn ψ 1 − En φn ψn

so

⎡

%

fn , ⎣

0=

+ T

F+ ψ F− ψ

⎤& ⎦

=

Mφ

F+ T

1 ψn tn m(dt) ψ 1 − En φn

ψn F − En m(dt). ψ 1 − En φn

(5.6)

2 , so The second term in the right hand side of (5.6) is zero, since F− ∈ H−

1 ψn F+ t−n m(dt) = 0, n = 0, 1, . . . . (5.7) ψ 1 − En φn T

If to the contrary F+ (t) =

' (F+ )j tj ,

(F+ )q = 0,

j≥q

(F+ )j is the jth Fourier coefﬁcient of F+ , then from (5.7) with n = q

1 ψq ψq (0) = 0. 0 = F+ t−q m(dt) = (F+ )q ψ 1 − Eq φq ψ(0) T

The contradiction shows that F+ = 0, and

1 s0 0 m(dt) = |F2 (t)|2 m(dt) < ∞. 0 F2 s0 1 F2 T

T

2 H− ,ψ

is outer, then, by Smirnov Since F2 is of the form F2 = F− /ψ, F− ∈ 2 . The proof is complete. maximum principle, F2 ∈ H− h+ 2 Corollary 5.5. , h+ ∈ H+ ⊂ Mφ,+ . −Hh+ 2 H+ Proof. By Proposition 5.2 the manifold is contained in Mφ . By (5.2), 2 H− 2 for all F2 ∈ H− & % (I − H∗ H)h+ 0 h+ 0 , = = 0, , F F2 −Hh+ 2 0 M 2 φ

L

and the result follows from the second assertion of Lemma 5.4.

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2 as Definition 5.6. We deﬁne a unitary operator L( from Mφ,+ onto H+

( n = tn . Lf The transformation L :

2 H+

→

(5.8)

2 H+

is deﬁned as h+ Lh+ = L( . −Hh+

(5.9)

L is called the transformation operator associated to the given sequence of Verblunsky coefﬁcients. Proposition 5.7. The following equality holds true I − H∗ H = L∗ L.

(5.10)

Proof. This follows from the unitarity of L( h+ h+ 2H 2 = 2 = (I − H∗ H)h+ , h+ . Lh+ 2H 2 = L( + + −Hh+ −Hh+ Mφ Equality (5.10) is called the Gelfand–Levitan–Marchenko (GLM) equation. Remark 5.8. Similar to Lemma 5.4 we can show that the system of vectors ⎤ ⎡ n 1 φ2n+1 t ψ2n tn ψ 2n+1 e2n = n φ2n , e2n+1 = ⎣ n+1 1 ⎦ , n ≥ 0 (5.11) t ψ t ψ 2n

2n+1

forms an orthonormal basis for Mφ . Similar to Definition 5.6 we can deﬁne ) the transformation M n 0 ) 2n = t , Me ) 2n+1 = n+1 Me . (5.12) 0 t ) transforms the basis (5.11), associated with the given CMV matrix A, M into the basis associated with the simplest CMV matrix (the one with φ = ) is called the transformation operator associ0, α−1 = −1). The operator M ated with the CMV matrix A. The transformation 2 2 H+ H+ → M: 2 2 H− H− ) is deﬁned as a restriction of M

2 H+ ) M = M 2 H−

Similar to (5.10) we can get

I H

H∗ I

(5.13)

= M∗ M.

However, it is more convenient for our purposes to use the operator L( rather ) than M.

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Proposition 5.9. L is a contraction. The matrix of L with respect to the basis {tk }k≥0 L = Lnm n,m≥0 ,

Lnm = Ltm , tn

is lower triangular. Proof. The ﬁrst assertion is straightforward from (5.10). For the second one tn we show that is in the span of {fk }k≥n . Indeed, by using the for−Htn mulae of Lemma 5.4 we get the following expression for the entries of L m m t t m n ( ( Lnm = Lt , t = L , Lfn = , fn −Htm −Htm Mφ ,+ % & m ψ tn 1−φn E t tm 1 s0 n n = = f , , ψn En −Htm s0 1 n L2 −Htm 1−φn En L2 ψ E ψ n n n = t m , tn − Htm , 1 − φn En L2 1 − φn En L2 The last term is zero, so ﬁnally ψn ψn Lnm = Ltm , tn = , tn−m = . (5.14) 1 − φn En 1 − En φn n−m L2 The latter is zero as long as m > n, as claimed. *∞ Since Lnn = ψn (0) = k=n ρk > 0, all diagonal entries of L are nonzero numbers. Therefore, the matrix of L has a formal inverse L−1 = L−1 nm . Theorem 5.10. The entries of the mth column of the matrix L−1 are the m Taylor coefficients of the function ψt m m t 1 L−1 = = . (5.15) n,m ψm n ψm n−m Proof. Since a product of the lower triangular matrices is a lower triangular one, need to show that for n ≥ j n ' 1 Lnm = δnj . (5.16) ψ j m−j m=j In view of (5.14) n ' m=j

Lnm

1 ψj

= m−j

ψn 1 ψj 1 − En φn

.

(5.17)

n−j

For n = j (5.16) is straightforward from (5.17). For n > j we turn to (4.14) and write 1 1 φn (En − Enj ) ψn ψn ψn − =− j ψj 1 − En φn ψj 1 − En φn ψj (1 − Enj φn )(1 − En φn ) =−

ψ n φn z n−j Pj ψj (1 − Enj φn )(1 − En φn )Qn Qjn

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497

z → 0.

We have n − j + 1 in the exponent since φn (0) = 0. Hence ψn ψn 1 1 = . ψj 1 − En φn n−j ψj 1 − Enj φn n−j On the other hand, (4.16) says that the right hand side of the above equation is zero, which proves (5.17). n

2 if and Proposition 5.11. The system of functions ψt n is a Riesz basis for H+ −1 2 only if the matrix L defines a bounded operator on , equivalently, L is 2 . an isomorphism of H+ n

2 and 2 , ψt n is a Riesz Proof. Due to the natural isomorphism between H+ 2 basis for H+ if and only if the columns of L−1 form a Riesz basis for 2 . In turn, the columns of L−1 form a Riesz basis for 2 if and only if both matrices L−1 and L deﬁne bounded operators on 2 . By Proposition 5.9 L is always a contraction.

As a straightforward corollary of Theorem 5.3 and the second part of Lemma 5.4, we get Theorem 5.12. φ is regular if and only if h+ 2 , h+ ∈ H+ −Hh+ is dense in Mφ,+ . In view of Definition 5.6 we get the following 2 if and only if φ is regular. Corollary 5.13. The range of L is dense in H+ 2 Theorem 5.14. L−1 is a bounded operator on H+ if and only if φ is regular and H < 1.

Proof. By the GLM Eq. (5.10), H < 1 if and only if L∗ L ≥ cI.

(5.18)

2 By Corollary 5.13, φ is regular if and only if the range of L is dense in H+ . −1 The latter along with (5.18) is equivalent to the boundedness of L .

6. The Helson–Szeg˝ o Class For a function u u=

∞ '

ck tk

k=−∞

the harmonic conjugate u ˜ is deﬁned as ∞ ' u ˜ = −i sign (k)ck tk , k=−∞

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so the function u + i˜ u is “analytic”. If u is real, then so is u ˜. Note that since sign (0) = 0, u ˜ does not depend on the constant Fourier coefﬁcient u0 . By ( the definition u ( = −u + u0 . Definition 6.1. We say that w is a positive Helson–Szeg˝ o function if it admits a representation of the form w = Ceu−˜v , u0 = v0 = 0,

u, v ∈ L∞ (real), C > 0,

ess sup v − ess inf v < π, (6.1)

where v˜ is the harmonic conjugate of v, u0 , v0 are the constant Fourier coefﬁcients. In this case we will say that the absolutely continuous measure σ(dt) = w(t)m(dt) ∈ HS. Unlike the standard convention v < π/2 we prefer to deal with ess sup v − ess inf v < π which is invariant under addition of any constant. Conversely, if the latter holds, then vc <

π , 2

vc := v +

ess sup v + ess inf v . 2

Definition 6.2. A positive function w is said to satisfy A2 (or Hunt—Muckenhoupt—Wheeden) condition if for all arcs I ⊂ T the following supremum is ﬁnite

1 w(t)m(dt). (6.2) supwI w−1 I < ∞, wI := |I| I I

Clearly w ∈ A2 if and only if 1/w ∈ A2 . The following classical theorem can be found, e.g., in [24, Lecture VIII]. Theorem 6.3. (Helson–Szeg˝o) The following conditions are equivalent 1. w is a positive Helson–Szego function (see Definition 6.1); 2. w satisfies the A2 condition (6.2); 2 2 and H−,w in L2w is positive: 3. the angle between H+,w 2

|g+ , g− w | ≤ β g+ 2w · g− 2w ,

β < 1.

2 2 is the closure of analytic polynomials in L2w , H−,w is the Here H+,w closure of conjugate-analytic polynomials that vanish at the origin. It is known that for w = |D|2 2 2 H+,w = D−1 H+ ,

2 H−,w =D

−1

2 H− .

Definition 6.4. We say that s ∈ HS if s is a canonical symbol of a Hankel operator H (see Definition 2.9) with H < 1. Definition 6.5. We say that a CMV matrix A is of Helson–Szeg˝o class (A ∈ HS) if L−1 is a bounded operator, where L is the transformation operator (5.9).

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In view of Theorem 5.14, A ∈ HS if and only of φ is regular and H < 1. Such functions φ are called strongly regular. They ﬁrst appeared in [3] and they were extensively studied in [4–8]. Strongly regular functions form a proper subclass of the regular ones. As a consequence of the regularity of φ, all the CM V matrices of the Helson–Szeg˝o class have absolutely continuous spectral measures. The main result of this section is Theorem 6.6. The following equivalences hold true A ∈ HS ⇐⇒ σ ∈ HS ⇐⇒ s ∈ HS. Moreover, there is a one-to-one correspondence between HS classes of CM V matrices (Verblunsky coefficients), spectral (probability) measures, and scattering functions. Proof. A ∈ HS =⇒ s ∈ HS. By Definition 6.5, A ∈ HS means that L−1 is a bounded operator. By Theorem 5.14, the boundedness of L−1 is equivalent to the regularity of φ and H < 1. Since φ is regular, then, by Definition 2.9, s is canonical. Therefore, s ∈ HS. σ ∈ HS =⇒ s ∈ HS. Recall also that spectral measure in our context is always a probability measure. Hence, w=

1 − |φ|2 1 − α−1 φ = 1 + α−1 φ |1 + α−1 φ|2

−1 φ with absolutely continuous 1−α 1+α−1 φ . Assumption that w ia a positive Helson– Szeg˝o function implies that w is a Szeg˝o function. Therefore,

w=

|ψ|2 = |D|2 , |1 + α−1 φ|2

where

D=

ψ 1 + α−1 φ

2 In view of Theorem 6.3(2), and since D is outer, we get that 1/D ∈ H+ . Since 1−α−1 φ we also have that 1+α−1 φ is absolutely continuous, then, by Theorem 2.8(1), 2 2 and h− ∈ H− we have φ is regular. Therefore, s is canonical. For h+ ∈ H+ that D h+ , h− |Hh+ , h− | = |sh+ , h− | = D 1 = Dh , Dh (6.3) + − = |Dh+ , Dh− |w−1 . 2 |D|

Since w−1 = |D|−2 ∈ A2 , then, by Theorem 6.3, |Dh+ , Dh− |w−1 ≤ β Dh+ w−1 Dh− w−1 = β h+ h− ,

β < 1. (6.4)

Therefore, H < 1. Hence, s ∈ HS. s ∈ HS =⇒ A ∈ HS and σ is a probability measure, σ ∈ HS. Let s be a canonical symbol of a Hankel operator H with H < 1: ψH E − φ H , ψ H 1 − φH E with E unimodular constant and φH Arov-regular. By Theorem 3.1, there exists a unique CM V matrix A ∈ Szac whose scattering function is s, s=

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moreover this A is regular. α−1 and the (probability) spectral density w are given by α−1 = −E,

D=

ψH ; 1 − EφH

w = |D|2 .

The Verblunsky coefﬁcients of A are the Schur parameters of φH (ζ)/ζ. Since φH is regular and H < 1, then Theorem 5.14 implies that L−1 is bounded, i.e., A ∈ HS. 2 2 For h+ ∈ H+ and h− ∈ H− we have that |Hh+ , h− | ≤ β h+ h− ,

β < 1.

(6.5)

In view of (6.3) and by Theorem 6.3, (6.5) implies (6.4). Therefore, |D|2 ∈ A2 , meaning that σ ∈ HS. Remark 6.7. The connection between strong regularity and A2 condition was observed and studied by Arov and Dym [7,8]. They also extensively used that in their study of inverse spectral problems for canonical systems of differential equations. Remark 6.8. Theorem 6.6 is contained in a preliminary version of the paper (see [16, Theorem 4.5, Proposition 4.7]). It was recently observed in [11, Theorem 6.3], that operator L has a multiplicative structure. This observation gives hope that the boundedness condition on L−1 may be restated as a constructive condition on the Verblunsky coefﬁcients via convergence of inﬁnite products (series). Definition 6.9. We say that s is a unimodular Helson–Szeg˝ o function if it admits a representation of the form s = cei(˜u+v) , u, v ∈ L∞ (real), u0 = v0 = 0, |c| = 1,

ess sup v − ess inf v < π, (6.6)

where u ˜ is the harmonic conjugate of u, u0 , v0 are the constant Fourier coefﬁcients. Theorem 6.10. Canonical symbols of Hankel operators with H < 1 are exactly unimodular Helson–Szeg˝ o functions. Proof. Let s be a canonical symbol of the Hankel operator H with H < 1, then, by Theorem 6.6, the unique w ∈ A2 , equivalently, w is of the form (6.1). Then w = |D|2 , where (taking into account our normalization u0 = v0 = 0) u + i˜ u + i(v + i˜ v) D = D(0) exp , D(0) > 0. (6.7) 2 Therefore, s = −α−1

i(u+v) ˜ D = −α−1 e , D

D(0) > 0,

and s is a canonical symbol of the Hankel operator H with H < 1.

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Conversely, let s be a unimodular Helson–Szeg˝ o function, i.e., it is of the form (6.6). Then i(u+v) ˜

s = ce

=c

D , D

where |c| = 1, D can be chosen as in (6.7). The corresponding w = |D|2 is of the form (6.1). Therefore, w ∈ A2 and, by Theorem 6.6, s is a canonical solution of the Nehari problem with H < 1. Remark 6.11. In terms of representation (6.6), the unique solution of the inverse scattering problem is given as

α−1 = −c, w = Ceu−˜v , w(t)m(dt) = 1. T

7. The Golinskii–Ibragimov Class 1/2

Definition 7.1. A function g is in the Besov class B2 g=

∞ ' n=−∞

1/2

Obviously, g ∈ B2

gn tn ,

∞ '

if

|n||gn |2 < ∞.

(7.1)

n=−∞ 1/2

if and only if the harmonic conjugate g˜ ∈ B2 .

Our arguments depend upon some classical results, mostly due to Peller [25] and Khrushchev and Peller [20] (see also [26]). 1/2

1/2

Theorem 7.2. [29, Proposition 6.1.11]. If g ∈ B2 , g is real, then eig ∈ B2 as well. The preceding theorem has a converse: 1/2

Theorem 7.3. [25]. Every unimodular function s in the Besov class B2 of the form s = tN eig ,

is

(7.2)

1/2

where g is real, g ∈ B2 , N is an integer called the index of s. N is determined uniquely, and g is determined up to an additive constant from 2πZ. 1/2

Theorem 7.4. [25]. Every function g in the Besov class B2 representation of the form

has a

g = g1 + g˜2 , 1/2

where g1 and g2 are continuous functions in the Besov class B2 . If g is real, then g1 and g2 are also real. Since every function continuous on the unit circle T can be approximated by polynomials (in t and 1/t), the sup-norm of g1 or g2 can be made arbitrarily small.

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Theorem 7.5. [20, Corollary 1.7, p. 72]. Let s be a unimodular function. Let 2 2 2 : H+ → H+ . If Ts = P+ s|H+ ker Ts = ker Ts∗ = {0},

(7.3)

then the operators Hs∗ Hs and Hs∗ Hs are unitarily equivalent. The equivalence is achieved by the unitary factor U in the polar decomposition of Ts

Ts = U Ts∗ Ts . We start with the following 1/2

Lemma 7.6. Let s be a unimodular function in the Besov class B2 of the index N . Then s is a unimodular Helson–Szeg˝ o function if and only if N = 0. 1/2

Proof. By Theorem 7.3, s = tN eig , g is real, g ∈ B2 . By Theorem 7.4 the o function (see (6.6)), so by function sˆ = eig is a unimodular Helson–Szeg˝ Theorem 6.10 sˆ is canonical. If N = 0, then, by Proposition 2.11, s = tN sˆ is not canonical, so, by Theorem 6.10, s is not a unimodular Helson–Szeg˝ o function. If N = 0, then s = sˆ is a unimodular Helson–Szeg˝ o function. The proof is complete. Definition 7.7. We deﬁne Golinskii–Ibragimov (GI) classes of CM V matrices (Verblunsky coefﬁcients), spectral measures and scattering functions as follows (1) The GI class of CM V matrices ∞ '

n|an |2 < ∞,

equivalently

n=0

∞

ρnn < ∞.

(7.4)

n=0

We will also write A ∈ GI. (2) The GI class of spectral measures consists of absolutely continuous measures with density w of the form w = eg , where g is a real func1/2 tion in B2 . We will write σ ∈ GI. We will also say that the spectral data {σ, α−1 } ∈ GI if σ ∈ GI. (3) The GI class of scattering functions is the class of functions s of the form 1/2 s = eig , where g is a real function in B2 . We will also write s ∈ GI. Lemma 7.8. For GI classes of CM V matrices (Verblunsky coefficients), spectral data and scattering functions the following inclusions hold true GI ⊂ HS. Proof. Inclusion GI ⊂ HS for spectral measures and for scattering functions follows from Theorem 7.4. To prove the inclusion for CMV matrices we show that (7.4) implies boundedness of L−1 . Let Lm be the m × m principal block of the inﬁnite matrix L. Then the inverse matrix (Lm )−1 will be the m × m principal block of the inﬁnite matrix L−1 (Lm )−1 = (L−1 )m .

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Due to this equality, we will use the notation L−1 m . Note that Lm is a contraction. Indeed, for m a ﬁnite vector of length m, Lm m ≤ L m ≤ m . Therefore, L−1 m is an expansion −1 L−1∗ m Lm ≥ Im .

Now we get an upper bound on

−1 L−1∗ m Lm .

(7.5)

Due to (7.5) +

,2 1 −1 −1∗ −1 −1 2 Im L−1∗ m Lm ≤ det(Lm Lm )Im = | det Lm | Im = ψk (0) k=0 ⎞2 ⎞2 ⎞2 ⎛ ⎛ ⎛ m ∞ ∞ ∞ ∞ 1 1 1 ⎠ Im ≤ ⎝ ⎠ Im = ⎝ ⎠ Im . =⎝ j+1 ρj ρj ρ j j=0 k=0 j=k k=0 j=k m

Since the bound does not depend on m, we get that the matrix L−1 deﬁnes a bounded operator on 2 . The inclusion follows. Theorem 7.9. The following equivalences hold true A ∈ GI ⇐⇒ σ ∈ GI ⇐⇒ s ∈ GI. Moreover, there is a one-to-one correspondence between GI classes of CM V matrices (Verblunsky coefficients), spectral (probability) measures, and scattering functions. Proof. σ ∈ GI ⇐⇒ s ∈ GI is straightforward. s deﬁnes σ and α−1 uniquely since s is canonical (see Theorem 3.1). A ∈ GI =⇒ s ∈ GI. We consider the m × m principal block of the GLM Eq. (5.10) Im − (H∗ H)m = (L∗ L)m ≥ L∗m Lm .

(7.6)

We take the determinant of the both sides to get ∗

| det Lm |2 ≤ det(Im − (H∗ H)m ) ≤ e−tr(H As we saw above

+

m

| det Lm |2 = =⎝

.

(7.7)

,2 ψk (0)

k=0

⎛

H)m

m ∞

⎞2

⎛

ρj ⎠ ≥ ⎝

k=0 j=k

∞ ∞

k=0 j=k

⎞2

⎛

ρj ⎠ = ⎝

∞

⎞2 ⎠ > 0. ρj+1 j

j=0

The latter bound is independent of m. This and (7.7) imply that ⎞ ⎛ ∞ j+1 tr(H∗ H)m ≤ −2 log ⎝ ρj ⎠ =⇒ tr(H∗ H) < ∞. j=0

(7.8)

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The trace is computed in terms of s as follows ∞ ∞ ' ' H∗ Htk , tk = Htk 2 tr(H∗ H) = k=0

k=0

∞ −k−1 ' '

=

−1 '

2

|cn | =

k=0 n=−∞

|n||cn |2 ,

n=−∞

where cn are the Fourier coefﬁcients of s. Therefore, 1/2

P− s ∈ B2 .

(7.9)

We show that actually 1/2

s ∈ B2 . We are going to apply Theorem 7.5. To this end we need to check (7.3). The 2 such that kernel of Ts consists of the functions h+ ∈ H+ 2 sh+ = h− ∈ H− .

, we get that Since s = −α−1 D D −α−1 Dh+ = Dh− . 1 1 , the right-hand one is in H− . Therefore, both The left-hand side is in H+ sides equal 0. Hence, the kernel of Ts is trivial

Ker Ts = {0}. The kernel of

Ts∗

consists of the functions h+ ∈ sh+ = h− ∈

(7.10) 2 H+

such that

2 H− .

Since s is canonical, Proposition 2.10 implies that this equation has only a trivial solution. Therefore, the kernel of Ts∗ is trivial Ker Ts∗ = {0}.

(7.11)

Due to (7.10) and (7.11) Theorem 7.5 applies and we get that Hs∗ Hs and Hs∗ Hs are unitarily equivalent. Therefore, the eigenvalues of the operators Hs∗ Hs and Hs∗ Hs coincide. The latter and (7.8) imply that tr(Hs∗ Hs ) = tr(Hs∗ Hs ) < ∞. 1/2 B2 .

(7.12) 1/2 B2 .

Hence, P− s ∈ We combine this with (7.9) to get that s ∈ Since s is a unimodular Helson–Szeg˝ o function, then it has zero index by Lemma 7.6. s ∈ GI =⇒ A ∈ GI. By Lemma 7.8 s ∈ GI =⇒ s ∈ HS. Then, by Theorem 6.6, there exists a unique CMV matrix A with this scattering function and the corresponding operator L−1 is bounded. The latter allows us to rewrite the GLM equation (5.10) as (I − H∗ H)−1 = (L∗ L)−1 = L−1 L−1∗ .

(7.13)

Note that the ﬁrst equality in (7.13) makes sense once H < 1, while the second does for the HS class only! We set (I − H∗ H)−1 =: I + Δ,

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where Δ ≥ 0. tr(H∗ H) < ∞ if and only if trΔ < ∞. Let Δm be m × m principal block of Δ (in the basis tn ). Then −1∗ Im + Δm = (L−1 L−1∗ )m = L−1 m Lm .

(7.14)

The second equality here (compare with the inequality in (7.6)) holds true since now the left factor L−1 is lower triangular and the right factor L−1∗ is upper triangular. From (7.14) we get 2 | det L−1 m | = det(I + Δm ).

Since 1 ≤ det(I + Δm ) ≤ etrΔm ≤ etrΔ ,

(7.4) follows.

Remark 7.10. As we showed in the proof of Theorem 7.9, if L−1 is bounded, then the following version of Widom’s formula holds true det(I − H∗ H) =

∞

2(j+1)

ρj

.

j=0

For the original Widom’s formula see [32], also [29, Theorem 6.2.13]. Remark 7.11. The equivalence A ∈ GI ⇐⇒ σ ∈ GI is the celebrated strong Szeg˝o Theorem (in Ibragimov’s version, [15]). For a detailed exposition see [29, Chapter 6], where several independent proofs are presented. Theorem 7.9 suggests an alternative proof of this fundamental result via the scattering theory for CMV matrices. Remark 7.12. In the late 60s Ibragimov and Solev in their study of classes of Gaussian stationary processes (see [17, Chapter 4.4]) came up with the class of spectral measures of the form σ(dt) = w(t)m(dt),

w(t) = |P (t)|2 eh(t) ,

(7.15)

where P is a polynomial of degree N with all its zeros on the unit circle, and 1/2 h is a real function from B2 . They proved that the scattering functions of 1/2 the measures deﬁned in (7.15) are exactly unimodular functions s from B2 with ind s = N . Note that in this class a solution of the inverse scattering problem is not unique. A description of the corresponding CMV matrices (similar to (7.4)) is not known. Example 7.13. This example shows that the inclusion GI ⊂ HS is proper. We consider the Jacobi weight for the unit circle 1 2 that enters the theory several times. First, for the choices of the parameters γ1 = 0, γ2 = 2 and γ1 = 2, γ2 = 0 we get two different weights w± = o functions D± (z) = C 1/2 (1 ± z)2 , that have the same C|t ± 1|4 with the Szeg˝ w(t) = C|t − 1|2γ1 |t + 1|2γ2 ,

D(z) = C 1/2 (1 − z)γ1 (1 + z)γ2 ,

γ1,2 > −

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scattering function s = t2 . Next, w ∈ A2 if and only if |γk | < 1/2. Finally, the Verblunsky coefﬁcients were computed in [14] an = −

γ1 − (−1)n γ2 , n + 1 + γ1 + γ2

n = 0, 1, . . .

so w is never in GI unless γ1 = γ2 = 0.

References [1] Adamjan, V., Arov, D., Kre˘ın, M.: Infinite Hankel matrices and generalized problems of Caratheodory-Fej´er and F. Riesz. Funkcional Anal. i Priloˇzen 2(1), 1–19 (1968) (Russian) [2] Adamjan, V., Arov, D., Kre˘ın, M.: Infinite Hankel matrices and generalized Carath´eodory-Fej´er and I. Schur problems. Funkcional Anal. i Priloˇzen. 2(4), 1–17 (1968) (Russian) [3] Adamjan, V., Arov, D., Kre˘ın, M.: Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem. Mat. Sb. (N.S.) 86(128), 34–75 (1971) (Russian) [4] Arov, D.Z.: Regular and singular J-inner matrix functions and corresponding extrapolation problems. Funktsional Anal. i Prilozhen. 22(1), 57–59 (1988) (Russian). Translation in Funct. Anal. Appl. 22(1), 46–48 (1988) [5] Arov, D.Z.: Regular J-inner matrix-functions and related continuation problems. In: Helson, H., Sz.-Nagy, B., Vasilescu, F.-H. (eds.) Linear Operators in Functions Spaces, vol. OT 43, pp. 63–87. Birkh¨ auser-Verlag, Basel (1990) [6] Arov, D.Z., Dym, H.: J-inner matrix functions, interpolation and inverse problems for canonical systems. I. Foundations. Integral Equ. Oper. Theory 29(4), 373–454 (1997) [7] Arov, D., Dym, H.: On matricial Nehari problems, J-inner matrix functions and the Muckenhoupt condition. J. Funct. Anal. 181, 227–299 (2001) [8] Arov, D., Dym, H.: Criteria for the strong regularity of J-inner functions and γ-generating matrices. J. Math. Anal. Appl. 280, 387–399 (2003) [9] Ball, J., Kheifets, A.: The inverse commutant lifting problem: characterization of associated Redheffer linear-fractional maps (2010). arXiv:1004.0447v1 [math.FA] [10] Cantero, M.J., Moral, L., Vel´ azquez, L.: Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl. 362, 29– 56 (2003) [11] Dubovoy, V., Fritzsche, B., Kirstein, B.: Description of Helson-Szeg˝ o measures in terms of the Schur parameter sequences of associated Schur functions (2010). arXiv:1003.1670v4 [math.FA] [12] Faddeev, L.D.: Properties of the S-matrix of the one-dimensional Schr¨ odinger equation. Trudy Mat. Inst. Steklov. 73, 314–336 (1964) (Russian) [13] Garnett, J.B.: Bounded Analytic Functions, revised 1st edn. Graduate Texts in Mathematics, vol. 236, xiv+459 pp. Springer, New York (2007) [14] Golinskii, B.L.: On asymptotic behavior of the prediction error. Probab. Theory Appl. 19(4), 224–239 (1974) (Russian)

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[15] Golinskii, B.L., Ibragimov, I.A.: On Szeg˝ o’s limit theorem. Math. USSR Izv. 5, 421–444 (1971) [16] Golinskii, L., Kheifets, A., Peherstorfer, F., Yuditskii, P.: FaddeevMarchenko scattering for CMV matrices and the Strong Szeg˝ o theorem (2008). arXiv:0807.4017v1 [math.SP] [17] Ibragimov, I.A., Rozanov, Yu.A.: Gaussian Stochastic Processes. Nauka, Moscow (1970) [18] Kheifets, A.: On regularization of γ-generating pairs. J. Funct. Anal. 130, 310– 333 (1995) [19] Kheifets, A.: Nehari’s interpolation problem and exposed points of the unit ball in the Hardy space H 1 . In: Proceedings of the Ashkelon Workshop on Complex Function Theory (1996), pp. 145–151, Israel Mathematical Conference Proceedings, vol. 11. Bar-Ilan University, Ramat Gan (1997) [20] Khrushchev, S., Peller, V.: Hankel operators, best approximations, and stationary Gaussian processes. Uspekhi Mat. Nauk 37(1), 53–124 (1982). English Transl. in Russ. Math. Surv. 37(1), 61–144 (1982) [21] Killip, R., Nenciu, I.: CMV: the unitary analogue of Jacobi matrices. Commun. Pure Appl. Math. 60, 1148–1188 (2007) [22] Marchenko, V.: Sturm-Liouville Operators and Applications. Birkh¨ auser Verlag, Basel (1986) [23] Marchenko, V.: Nonlinear Equations and Operator Algebras. Mathematics and its Applications (Soviet Series), vol. 17. D. Reidel Publishing Co., Dordrecht (1988) [24] Nikolskii, N.: Treatise on the Shift Operator. Springer, Berlin (1986) [25] Peller, V.: Hankel operators of class Sp and their applications (rational approximation, Gaussian processes, the problem of majorizing operators). Mat. Sbornik 113, 538–581 (1980). English Transl. in Math. USSR Sbornik 41, 443– 479 (1982) [26] Peller, V.: Hankel Operators and Their Applications. Springer, Berlin (2003) [27] Sarason, D.: Exposed points in H 1 . Operaor Theory: Advances and Applications, vol. 41, pp. 485–496. Birkh¨ auser, Basel (1989) [vol. 48, pp. 333–347 (1990)] ¨ [28] Schur, I.: Uber Potenzreihn, die im Innern des Einheitskreises beschr¨ ankt sind, I, II. J. Reine Angew. Math. 147, 205–232 (1917) [148, 122–145 (1918)] [29] Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory; Part 2: Spectral Theory. AMS Colloquium Series. AMS, Providence (2005) [30] Simon, B.: CMV matrices: five years after. J. Comput. Appl. Math. 208, 120– 154 (2007) [31] Volberg, A., Yuditskii, P.: Remarks on Nehari’s problem, matrix A2 condition, and weighted bounded mean oscillation. Am. Math. Soc. Transl. 226(2), 239– 254 (2009) [32] Widom, H.: Asymptotic behavior of block Toeplitz matrices and determinants, II. Adv. Math. 21, 1–29 (1976)

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L. Golinskii Mathematics Division Institute for Low Temperature Physics and Engineering 47 Lenin Ave., Kharkov 61103, Ukraine e-mail: [email protected] A. Kheifets (B) Department of Mathematics University of Massachusetts Lowell Lowell, MA 01854, USA e-mail: Alexander [email protected] P. Yuditskii and F. Peherstorfer Institute for Analysis Johannes Kepler University of Linz 4040 Linz, Austria e-mail: [email protected] Received: October 29, 2009. Revised: December 22, 2010.

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Integr. Equ. Oper. Theory 69 (2011), 509–533 DOI 10.1007/s00020-011-1868-1 Published online February 15, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

Linear Maps Preserving Numerical Radius on Nest Algebras Fangyan Lu Abstract. A linear map φ of operator algebras is said to preserve numerical radius ( or to be a numerical radius isometry) if w(φ(A)) = w(A) for all A in its domain algebra, where w(A) stands for the numerical radius of A. In this paper, we prove that a surjective linear map φ of the nest algebra AlgN onto itself preserves numerical radius if and only if there exist a unitary U and a complex number ξ of modulus one such that φ(A) = ξU AU ∗ for all A ∈ AlgN , or there exist a unitary U , a conjugation J and a complex number ξ of modulus one such that φ(A) = ξU JA∗ JU ∗ for all A ∈ AlgN . Mathematics Subject Classification (2010). Primary 47L35, 47B48; Secondary 47A12. Keywords. Numerical radius, numerical range, isometry, nest algebra.

1. Introduction The isometries of Banach algebras are always of interest. In [12], Kadison characterized all operator norm isometries of von-Neumann algebras. The structure of operator norm isometries of nest algebras was obtained in [15] and [1] independently. Also, Moore and Trent [16] and Solel [21] characterized operator norm isometries of reflexive algebras with completely distributive commutative lattice and of CSL algebras, respectively. All results above said that such an isometry is a Jordan isomorphism multiplied by a unitary. Besides the operator norm, there is another important norm on operators, which is equivalent to the operator norm. This is the numerical radius. The study of the numerical radius, the related numerical range, and their generalizations has a long and distinguished history (e.g., see [9] and [11, Chapter 1]). The subject is related and has applications to many different branches of pure and applied science. In [2,3], Chan proved that if A is a unital C∗ -algebra then a surjective numerical radius isometry of A is a Jordan isomorphism multiplied by a fixed unitary element in the center of A. This This work was supported by NSFC (No. 10771154).

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is also true for weakly continuous, surjective numerical radius isometries of atomic nest algebras [4,5,14]. Roughly speaking, nest algebras are reflexive algebras with ordered lattices. In some sense, nest algebras lie at the “opposite pole” (among reflexive algebras) from von Neumann algebras. The purpose of this paper is to characterize surjective numerical radius isometries of nest algebras. Unlike [4,5], we do not require isometries to be weakly continuous and do not require nest algebras to be atomic. Also, our approach is very different from that in [4,5]. Though the numerical radius w(·) and the operator norm · are equivalent, they have many differences. Due to the Sz.-Nagy’s dilation theorem, the operator norm satisfies the von-Neumann inequality: p(A) ≤ p∞ := max{|p(z)| : z ∈ C, |z| = 1} for every contraction operator A and every polynomial p. Using this fact, Arazy and Solel [1] proved that each operator norm isometry of operator algebras is a Jordan isomorphism followed by a unitary multiplication. On the other hand, if we let p(z) = 1 + 5z − 12 z 2 and suppose that an operator A satisfies A2 = 0 and w(A) = 1, then w(p(A)) = w(I + 5A) = 6 > 11 2 = p∞ . This simple example shows that the numerical radius does not satisfy the von-Neumann inequality. Even so, numerical radius isometries are often proved to be operator norm isometries. Where shall one find their connections? We observe that w(A) = 12 A if A2 = 0 (cf. Proposition 2.8). So, on the nilpotent set whose image is also nilpotent, a numerical radius isometry is an operator norm isometry. One of the keys in this paper is to find subsets which are nilpotent and whose images are nilpotent. Then we can use some ideas in the study of operator norm isometries. Especially, we shall use some ideas of Moore and Trent [15] they gave an elementary proof of characterizing operator norm isometries of nest algebras. To achieve our goal, another of the keys is to reduce the question to that of characterizing the linear maps that preserve the closure of numerical range. Those maps are special numerical radius isometries: carry the diagonal of the algebra in question onto itself. Then by a result of Pellegrini [19] we can identify the behavior of maps on the diagonal. The paper is organized as follows. In Sect. 2, we collect and give some basic properties of numerical radius and nest algebras. Section 3 is the crucial part of the paper, in which we give a complete description of linear maps of nest algebras which preserve the closure of numerical range (see Theorem 3.1). In Sect. 4, we get the main theorem of characterizing numerical radius isometries of nest algebras ( see Theorem 4.1).

2. Preliminaries Throughout the paper, H is a Hilbert space over the complex field C with the inner product (·, ·). By B(H) we denote the algebra of all linear bounded operators on H. The term projection will mean“ self-adjoint projection”. For a projection P, P ⊥ denotes the projection I − P . For convenience, we shall

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disregard the distinction between a projection and its range. For x, y ∈ H, the operator x ⊗ y is given by z → (z, y)x, which has rank one if and only if both x and y are non-zero. Before doing others, we present an interesting property concerning the operator norm. Proposition 2.1. Let x and y be unit vectors in H. Let A be an operator in B(H) with A = 1. Let d be in R with 0 ≤ d ≤ 1. Suppose that λA + x ⊗ y = max{|1 + λ|, d} for all λ ∈ C. Then d = 0. Proof. See the proof of [15, Proposition 9].

2.1. Numerical Range and Numerical Radius For an operator A ∈ B(H), the numerical range and the numerical radius are defined by W (A) = {(Ax, x) : x ∈ H, x = 1} and w(A) = sup{|λ| : λ ∈ W (A)}, respectively. We first recall some basic results on numerical range and numerical radius that are useful in our study. One may see [9,10] and Chapter 1 of [11] for more information. Proposition 2.2. ([10]). Let A ∈ B(H). (1) W (A) = W (U AU ∗ ) for any unitary U . (2) W (λA) = λW (A) for any λ ∈ C. (3) W (λI + A) = λ + W (A) for any λ ∈ C. (4) The spectrum of A is contained in W (A), the closure of W (A). Proposition 2.3. ([9,10]). The numerical range of A ∈ B(H) is always conλ1 b vex. In particular, if A ∈ M2 (C) is unitarily similar to , then W(A) 0 λ2 is an elliptical disk with λ1 and λ2 as foci, and length of minor axis equal to |b|, where M2 (C) denotes the 2 × 2 complex matrix algebra. Proposition 2.4. ([10]). Let A ∈ B(H). Then W (A) = {λ} if and only if A = λI. Proposition 2.5. ([10,11]). Suppose that N ⊆ H is a closed subspace and A ∈ B(H). Then W (PN A|N ) ⊆ W (A) and w(PN A|N ) ≤ w(A), where PN is the projection onto N . Proposition 2.6. ([11]). Suppose that A ∈ Mn (C) such that A + A∗ has λ2 and λ1 as the largest and smallest eigenvalues, respectively. Then [λ1 , λ2 ] = {z + z¯ : z ∈ W (A)}. Proposition 2.7. ([10]). If A ∈ B(H) is unitarily similar to A1 ⊕ A2 , then W (A) is equal to the convex hull of W (A1 ) ∪ W (A2 ). Proposition 2.8. ([9]). Let A ∈ B(H). (1) w(A) ≤ A ≤ 2w(A). (2) If A is normal then w(A) = A. (3) If A2 = 0 then w(A) = 12 A.

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Proposition 2.9. Let A be in B(H) and a be a positive number. If there exists a positive b such that w(λI +A)2 ≤ |λ|2 +a holds for all scalars λ with |λ| ≥ b, then A = 0. Proof. With unit vectors x, we have |λ + (Ax, x)|2 ≤ w(λI + A)2 ≤ |λ|2 + a. From this, we get ¯ 2Reλ(Ax, x) ≤ a. Putting λ = t(Ax, x) for positive number t with t|(Ax, x)| > b, we get 2tRe|(Ax, x)|2 ≤ a. This is impossible unless (Ax, x) = 0. So w(A) = 0 and then A = 0.

We close this subsection by computing the numerical radius of a rank-2 operator. We shall use [x1 , x2 , . . . , xn ] to denote the linear space spanned by vectors x1 , x2 , . . . , xn . Example 2.10. Let {x, y, u, v} be a set of unit vectors. Suppose that x⊥y and [x, y]⊥[u, v]. Then w(x ⊗ u + v ⊗ y) = 12 1 + |(u, v)|. Hence, if w(x ⊗ u + √ v ⊗ y) = 22 then u and v are linearly dependent. Proof. Let A = x ⊗ u + v ⊗ y. Then (A + A∗ )2 = x ⊗ x + y ⊗ y + u ⊗ u + v ⊗ v + (v, u)x ⊗ y + (u, v)y ⊗ x. So A + A∗ 2 = (A + A∗ )2 ≥ x ⊗ x + y ⊗ y + (v, u)x ⊗ y + (u, v)y ⊗ x = 1 + |(u, v)|. Hence 2w(A) ≥ w(A + A∗ ) = A + A∗ ≥

1 + |(u, v)|.

To prove the converse inequality, we write v = au + bv , where a = (v, u), |a|2 + |b|2 = 1 and v is a unit vector which is orthogonal to [x, y, u]. For any unit vector z, we can write z = cx + dy + eu + f v + z , where z is a vector orthogonal to [x, y, u, v]. Then 1 ≥ |c|2 + |d|2 + |e|2 + |f |2 2

2

1 |a| + 1 + |a| 1 + |a|

= |c| + (|a| + (1 − |a|))|d| + |e|2 + |f |2 |a| 1 |c||e| + |d||e| + 1 − |a||d||f | ≥2 1 + |a| 1 + |a| 2 (|c||e| + |a||d||e| + |b||d||f |). = 1 + |a|

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Therefore

1 |(Az, z)| = |¯ ce + ad¯ e + bdf¯| ≤ 1 + |a| 2 for any unit vector z. Thus w(A) ≤ 12 1 + |(u, v)|. We are done.

2.2. Nest Algebras A nest N of H is a strongly closed lattice of projections on H which contains the zero operator 0 and the identity I and is totally ordered. The associated nest algebra denoted by AlgN is the algebra of operators that leave each element in N invariant, that is, AlgN = {T ∈ B(H) : T E = ET E for each E ∈ N }. ⊥

⊥

⊥

(2.1)

If we let N = {E : E ∈ N }, then N is also a nest and AlgN = (AlgN )∗ . The von-Neumann algebra (AlgN ) ∩ (AlgN )∗ is called the diagonal of AlgN , which is equal to the linear span of self-adjoint operators in AlgN and also to the commutant N of N . The double commutant N of N is called the core of N , which is a commutative von Neumann algebra generated by N . Let N be a nest. For E ∈ N , we define E− = ∨{F ∈ N : F < E} and

⊥

E+ = ∧{F ∈ N : F > E}.

Here, for a family P of projections, ∨{P : P ∈ P} is the smallest projection whose range contains the range of each projection in P, and ∧{P : P ∈ P} is the largest projection whose range is contained in the range of each projection in P. Obviously, E− ≤ E ≤ E+ . It is not difficult to verify E = ∨{F ∈ N : F− < E} and E = ∧{F− : F ∈ N , F > E}. This is so-called the completely distributivity of nests [6]. If E, F ∈ N with E < F , then the projection E − F is called an interval of N . A minimal ⊥ for some interval is called an atom of N . Any atom takes the form EE− E ∈ N and vice visa. Moreover, atoms of N are atoms of the von-Neumann algebra N and vice visa. For more information on nest algebras, we refer to [6]. One of the most important tools in the study of nest algebras is the set of rank-one operators. The following characterization was first established by Ringrose. Proposition 2.11. ([20]). Let N be a nest. Then x ⊗ y belongs to AlgN if and ⊥ . only if there is E ∈ N such that x ∈ E and y ∈ E− The following result shows that nest algebras are rich in rank-one operators, which was first proved by Erdos. Proposition 2.12. ([7]). Let N be a nest. Then the linear span of rank-one operators in AlgN is weakly dense. Proposition 2.13. Let AlgN be a nest algebra. (1). Suppose that A, B ∈ AlgN satisfy A(AlgN )B = {0}. Let E = ∨{F ∈ N : AF = 0}. Then AE = 0 and E ⊥ B = 0.

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Suppose that P, Q ∈ AlgN are projections which satisfy P (AlgN )Q = {0} and Q(AlgN )P = {0}. Then P = 0 or Q = 0. Suppose that P ∈ AlgN is a projection for which P (AlgN )P ⊥ (AlgN ) P = {0}. Then P is an interval of N . Suppose that P ∈ AlgN is a projection for which P (AlgN )P ⊥ (AlgN ) P = {0} and P ⊥ (AlgN )P (AlgN )P ⊥ = {0}. Then P ∈ N or P ⊥ ∈ N .

Proof. (1) That AE = 0 is obvious. For F ∈ N with F > E, by the definition of E, we can pick a vector x in F such that Ax = 0. For y ∈ F−⊥ , since x ⊗ y ∈ AlgN by Proposition 2.11, it follows that A(x ⊗ y)B = 0. So B ∗ y = 0 for all y ∈ F−⊥ . Now from the completely distributivity which implies E ⊥ = ∨{F−⊥ : F ∈ N , F > E}, we conclude that B ∗ E ⊥ = 0. (2) By (1), there exists E, F ∈ N such that P E = QE ⊥ = QF = P F ⊥ = 0. If E ≤ F , then Q = QE+QE ⊥ = 0; if E ≥ F , then P = P F +P F ⊥ = 0. (3) By (1), there exists an E in N such that P E = 0 and E ⊥ P ⊥ (AlgN )P = {0}. Hence by (1) again, E ⊥ P ⊥ F = 0 and F ⊥ P = 0 for some F ∈ N with F ≥ E. Now P = (E + (F − E) + F ⊥ )P = (F − E)P = F − E. (4)

By (3), P = F1 − E1 and P ⊥ = F2 − E2 , where Ei , Fi ∈ N with Ei ≤ Fi , i = 1, 2. Then E1 E2 = (P + P ⊥ )E1 E2 = (F1 − E1 + F2 − E2 )E1 E2 = 0. So E1 = 0 or E2 = 0. Thus P = F1 or P ⊥ = F2 .

The study of maps on nest algebras is often reduced to the study of maps that preserve rank-one operators. The proof of the following proposition is routine (see, for example, [15,18,22]), we leave it to reader. Proposition 2.14. Let Pi and Qi be projections, i = 1, 2. Let φ be a surjective linear operator norm isometry from span{x ⊗ y : x ∈ P1 , y ∈ P2 } onto span{x ⊗ y : x ∈ Q1 , y ∈ Q2 } which preserves rank one operators in both directions. Then one of the following holds. (1). There exist linear unitaries S : P1 → Q1 and T : P2 → Q2 such that φ(x ⊗ y) = Sx ⊗ T y (2).

for all x ∈ P1 and y ∈ P2 . There exist conjugate linear unitaries S : P1 → Q2 and T : P2 → Q1 such that φ(x ⊗ y) = T y ⊗ Sx for all x ∈ P1 and y ∈ P2 .

3. Linear Maps Preserving the Closure of Numerical Range In this section, we characterize linear maps of nest algebras which preserve the closure of numerical range. Recall that a conjugation J on H is a conjugative linear isometric map on H such that J 2 = I. The main result is as follows.

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Theorem 3.1. Suppose that N is a nest of a Hilbert spaces H. Let φ : AlgN → AlgN be a surjective linear map such that W (φ(A)) = W (A) for all A ∈ AlgN . Then φ has one of the following forms. 1. 2.

There exists a unitary U ∈ B(H) such that φ(A) = U AU ∗ for all A ∈ AlgN . There exist a unitary U ∈ B(H) and a conjugation J on H such that φ(A) = U JA∗ JU ∗ for all A ∈ AlgN .

If N is trivial, i.e. N = {0, I}, then AlgN = B(H). By the main theorem in [17], the theorem is true. In the sequel, we assume that N is non-trivial. The proof is not short. For clarity of exposition, we shall organize the proof in several subsections. 3.1 We first remark that φ(I) = I since an operator is I if and only if the closure of its numerical range is the set {1}. It is well-known that an operator A is self-adjoint if and only if W (A) ⊆ R. It follows that φ maps the set of self-adjoint operators in AlgN onto itself. Thus, since the diagonal N of AlgN is the linear span of self-adjoint operators in AlgN , we know that φ(N ) = N . Then the restriction of φ to N is a linear bijection of a vonNeumann which preserves the closure of numerical range. By [19, Theorem 3.1], it is a C ∗ -Jordan isomorphism, that is, it preserves powers of elements, as well as preserving adjoints. Thus φ|N preserves the commutativity and hence φ(N ) = N . Therefore, φ|N is a ∗−automorphism of N . Note that the set of atoms of N coincides with the set of atoms of N . Thus we have Lemma 3.2. φ maps the set of atoms of N onto itself. Since φ|N is a C∗ -Jordan isomorphism, we know that if P ∈ N is a projection, so are φ(P ) and φ−1 (P ). In the sequel, we shall use Pˆ and P˜ to denote φ(P ) and φ−1 (P ), respectively. Lemma 3.3. Let P be an atom of N and suppose that x is in P . Then φ(x⊗x) is of rank one. Proof. Assume that x = 1. Since P is an atom, it follows from Proposition 2.11 and Lemma 3.2 that x ⊗ x is in N and Pˆ is an atom. Let Q = φ(x ⊗ x). Since x ⊗ x is a subprojection of P and φ|N preserves the commutativity, it follows that Q is a subprojection of the atom Pˆ . Let y be a unit vector in Q. Then R = φ−1 (y ⊗ y) is a subprojection of x ⊗ x. Consequently, R = x ⊗ x and φ(x ⊗ x) = y ⊗ y. 3.2 2 Let P1 be a projection in AlgN and set P2 = I − P1 . Then AlgN = j,k=1 Pj (AlgN )Pk . This is so-called the Peirce decomposition of AlgN . We now investigate the behavior of φ on the each summand. Lemma 3.4. Let P be a projection in AlgN and suppose that A in AlgN satisfies A = P AP . Then φ(A) = Pˆ φ(A)Pˆ .

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Proof. With the decomposition I = Pˆ + Pˆ ⊥ , we write φ(A) as a two-by-two matrix: S1 S2 φ(A) = . S3 S4 Then, for λ ∈ C, φ(λP

⊥

S1 + A) = S3

S2 . λ + S4

So by Proposition 2.5 w(λ + S4 )2 ≤ w(φ(λP ⊥ + A))2 = w(λP ⊥ + A)2 ≤ λP ⊥ + A2 = max{|λ|2 , A2 }. By Proposition 2.9, S4 = 0. For unit vectors u ∈ Pˆ and c = (S3 u, v). We also let a ⊥ A(t) := Qφ(tP + A)|Q = c

and v ∈ Pˆ ⊥ , we let a = (S1 u, u), b = (S2 v, u) Q = u ⊗ u + v ⊗ v. Then for t ∈ R, b ia ib ⊥ and B(t) := Qφ(tP +iA)|Q = . t ic t It is easily seen that A(t)+A(t)∗ has an eigenvaluet+a1 + (t − a1 )2 +|b+¯ c|2 ∗ and that B(t) + B(t) has an eigenvalue t − a2 + (t + a2 )2 + |b − c¯|2 , where a1 and a2 are the real part and the imaginary part of a, respectively. So, for t > A, since w(φ(tP ⊥ + A)) = w(tP ⊥ + A) = t and w(φ(tP ⊥ + iA)) = w(tP ⊥ + iA) = t, by Proposition 2.5 we have t + a1 + (t − a1 )2 + |b + c¯|2 ≤ w(A(t) + A(t)∗ ) ≤ 2w(A(t)) ≤ 2t, and t − a2 +

(t + a2 )2 + |b − c¯|2 ≤ w(B(t) + B(t)∗ ) ≤ 2w(B(t)) ≤ 2t.

Thus we have b + c¯ = 0 and b − c¯ = 0. Therefore, for all u ∈ Pˆ and v ∈ Pˆ ⊥ , ((S2 ± S3∗ )v, u) = (S2 u, v) ± (S3 u, v) = 0 and hence S2 ± S3∗ = 0. Consequently, S2 = S3∗ = 0. We are done.

Lemma 3.5. Let P be a projection in AlgN and suppose that A in AlgN satisfies A = P AP ⊥ . Then φ(A) = Pˆ φ(A)Pˆ ⊥ + Pˆ ⊥ φ(A)Pˆ . Proof. Represent

φ(A) =

S1 S3

S2 S4

under the decomposition I = Pˆ + Pˆ ⊥ . Then for λ ∈ C, λ + S 1 S2 φ(λP + A) = . S3 S4 Thus w(λ + S1 )2 ≤ w(φ(λP + A))2 = w(λP + A)2 ≤ λP + A2 = |λ|2 + A2 . So S1 = 0 by Proposition 2.9. Similarly, S4 = 0, completing the proof.

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3.3 If A and φ(A) are both nilpotent (of order 2), then φ(A) = 2w(φ(A)) = 2w(A) = A by Proposition 2.8. So we shall find nilpotent subsets whose images under φ are also nilpotent so that φ acts as an operator norm isometry. For E ∈ N , E(AlgN )E ⊥ is obviously nilpotent. However we do not know whether its image is nilpotent at present. Luckily, many subsets of it are as required. Recall that P˜ = φ−1 (P ) for a projection P . Lemma 3.6. Let E and F be in N . Suppose that x ⊗ y is in F˜ E(AlgN )E ⊥ F˜ ⊥ ˆ φ(x ⊗ y)F ⊥ E ˆ ⊥ or φ(x ⊗ y) = or in F˜ ⊥ E(AlgN )E ⊥ F˜ . Then φ(x ⊗ y) = EF ⊥ ⊥ ˆ F φ(x ⊗ y)F E. ˆ E Proof. Assume x = y = 1. Using Lemma 3.5 twice, we get ˆ ˆ⊥ + E ˆ ⊥ φ(x ⊗ y)E)F ˆ ⊥. φ(x ⊗ y) = F (Eφ(x ⊗ y)E From this we see φ(x ⊗ y)2 = 0. By Proposition 2.8, φ(x ⊗ y) = 2w(φ(x ⊗ y)) = 2w(x ⊗ y) = x ⊗ y = 1. ˆ ⊥ and B = E ˆ Since E ˆ is a ˆ ⊥ F φ(x ⊗ y)F ⊥ E. ˆ φ(x ⊗ y)F ⊥ E Let A = EF ˆ selfadjoint operator, it is in the diagonal N . Hence E commutes with F . So φ(x ⊗ y) = A + B. Thus max{A, B} = φ(x ⊗ y) = 1. Assume that A = 1. Choose D ∈ AlgN such that φ(D) = A. Applying Lemma 3.5 to φ−1 , we get D = EDE ⊥ . Hence D = A = 1 by Proposition 2.8. For λ ∈ C, it is obvious that φ(x ⊗ y + λD) = (1 + λ)A + B. Since x ⊗ y + λD and (1 + λ)A + B are both nilpotent, we have x ⊗ y + λD = (1 + λ)A + B = max{|1 + λ|, B}. ˆ φ(x ⊗ By Proposition 2.1, we know that B = 0 and then φ(x ⊗ y) = EF ⊥ ˆ⊥ y)F E . Lemma 3.7. Let Pi and Qi be projections in AlgN , i = 1, 2, which satisfy (1). (2). (3).

P1 P2 = 0 and Q1 Q2 = 0; φ−1 (R) ∈ P1 (AlgN )P2 for all rank-one operators R in Q1 (AlgN )Q2 ; u ⊗ v ∈ Q1 (AlgN )Q2 for all u ∈ Q1 , v ∈ Q2 .

Suppose that x⊗y is a rank-one operator in P1 (AlgN )P2 for which φ(x⊗y) ∈ Q1 (AlgN )Q2 . Then φ(x ⊗ y) is of rank one. Proof. The argument is similar to the proof of [15, Lemma 13]. We shall omit some details and only reduce the argument to the operator norm isometry situation. Assume x = y = 1. Let B = φ(x ⊗ y). Then B = Q1 BQ2 . Since B 2 = (x ⊗ y)2 = 0, it follows that B = 2w(B) = 2w(x ⊗ y) = 1. 1 , choose a unit vector h ∈ Q2 such that Bh 2 ≥ 1 − . For ≤ 36 By the condition (3), Bh ⊗ h is in Q1 (AlgN )Q2 . Then by the condition (2) we can choose an operator D in P1 (AlgN )P2 such that φ(D ) = Bh ⊗ h . Since D and Bh ⊗ h are both nilpotent, we have D = Bh ⊗ h =

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√ Bh ≥ 1 − . Now let t be a positive number. Obviously φ(x ⊗ y − tD ) = B − tBh ⊗ h . Since the operators involved are both nilpotent, we have x ⊗ y − tD 2 = B − tBh ⊗ h 2 . √ From this one can assert that |(D x, y)| ≥ 1 − 6 . Using the assertion above, one can conclude that B ∗ B (in fact, it is equal to 1) is an eigenvalue of B ∗ B. Let a unit vector h be the corresponding eigenvector. Then h ∈ Q2 and Bh2 = (B ∗ Bh, h) = h2 = 1. Choose R in P1 (AlgN )P2 such that φ(R) = Bh ⊗ h. Then R = 1. Moreover, for all λ ∈ C, we have λR + x ⊗ y = 2w(λR + x ⊗ y) = 2w(λBh ⊗ h + B) = λBh ⊗ h + B. Let T be a unitary such that T Bh = h. Note that T Bh = Bh = 1 = T B. Then with the space decomposition [h] ⊕ [h]⊥ , we have 1 0 1 0 h⊗h= and T B = . 0 0 0 S Thus λR + x ⊗ y = T (λBh ⊗ h + B) = λh ⊗ h + T B = max{|1 + λ|, S}. It follows from Proposition 2.1 that S = 0. Consequently, B is of rank one. Lemma 3.8. Let E and F be in N . Suppose that 0 = x ⊗ y is in E F˜ (AlgN )F˜ ⊥ E ⊥ or in F˜ ⊥ E(AlgN )E ⊥ F˜ . Then φ(x ⊗ y) is of rank one. ˆ φ(x ⊗ y)F ⊥ E ˆ⊥. Proof. By Lemma 3.6, we can suppose that φ(x ⊗ y) = EF −1 −1 ˆ ⊥ ˆ⊥ Applying Lemma 3.5 to φ , we get φ (EF (AlgN )F E ) ⊆ E(AlgN )E ⊥ . ˆ and Now Lemma 3.7 can apply by letting P1 = E, P2 = E ⊥ , Q1 = EF ˆ⊥F ⊥. Q2 = E Lemma 3.9. Let E and F be in N . Then one of the following holds. ˆ φ(x ⊗ y)F ⊥ E ˆ ⊥ for all x ∈ F˜ E and y ∈ E ⊥ F˜ ⊥ . (1). φ(x ⊗ y) = EF ˆ ⊥ F φ(x ⊗ y)F ⊥ E ˆ for all x ∈ F˜ E and y ∈ E ⊥ F˜ ⊥ . (2). φ(x ⊗ y) = E Proof. Suppose on the contrary that there are non-zero vectors x, u ∈ E F˜ and y, v ∈ E ⊥ F˜ ⊥ such that ˆ⊥ ˆ φ(x ⊗ y)F ⊥ E φ(x ⊗ y) = EF and ˆ ˆ ⊥ F φ(u ⊗ v)F ⊥ E. φ(u ⊗ v) = E As for u ⊗ y, we can suppose, without loss of generality, that ˆ⊥. ˆ φ(u ⊗ y)F ⊥ E φ(u ⊗ y) = EF Then ˆ + Eφ(u ˆ ˆ⊥. ˆ ⊥ φ(u ⊗ v)E ⊗ y)E φ(u ⊗ (y + v)) = E This conflicts with the rank-oneness of φ(u ⊗ (y + v)).

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Lemma 3.10. Let E be in N . Then one of the following holds. ˆ ⊥ for all F ∈ N and all rank one operators ˆ φ(R)F ⊥ E (1). φ(R) = EF ⊥ ˜⊥ ˜ R ∈ F E(AlgN )E F . ˆ for all F ∈ N and all rank one operators ˆ ⊥ F φ(R)F ⊥ E (2). φ(R) = E ⊥ ⊥ R ∈ F˜ E(AlgN )E F˜ . Proof. Suppose on the contrary that the lemma is not true. Note, by Lemma 3.5, that for any F ∈ N and A ∈ F˜ E(AlgN )E ⊥ F˜ ⊥ there holds ˆ⊥ + E ˆ ˆ φ(A)F ⊥ E ˆ ⊥ F φ(A)F ⊥ E. φ(A) = EF ˆ = 0 for some ˆ ⊥ F1 φ(R1 )F ⊥ E So there exist F1 and F2 in N such that E 1 ⊥ ˜⊥ ˆ ⊥ = 0 for ˜ ˆ rank one operator R1 ∈ F1 E(AlgN )E F1 and EF2 φ(R2 )F2⊥ E ⊥ ⊥ some rank one operator R2 ∈ F˜2 E(AlgN )E F˜2 . This implies that neither of E F˜1 , E F˜2 , E ⊥ F˜1⊥ and E ⊥ F˜2⊥ is zero. Moreover, by Lemma 3.9, ˆ F˜1 ERE ⊥ F˜ ⊥ )E ˆ ⊥ = 0 and E ˆ ⊥ φ(F˜2 ERE ⊥ F˜ ⊥ )E ˆ = 0 for all rank one Eφ( 1 2 operators R ∈ AlgN . Without loss of generality, suppose that F1 < F2 . Then F˜1 < F˜2 . Choose non-zero vectors x ∈ F˜1 E and y ∈ F˜2⊥ E ⊥ . Then x ⊗ y is in (F˜1 E(AlgN )E ⊥ F˜1⊥ ) ∩ (F˜2 E(AlgN )E ⊥ F˜2⊥ ). Thus we would have ˆ ˆ⊥ + E ˆ ⊥ φ(x ⊗ y)E ˆ = 0. φ(x ⊗ y) = Eφ(x ⊗ y)E This conflicts with the fact x ⊗ y = 0.

Similarly, we have Lemma 3.11. Let E be in N . Then one of the following holds: ˆ φ(R)F ⊥ E ˆ ⊥ for all F ∈ N and all rank one operators (1). φ(R) = EF ⊥ ⊥˜ ˜ R ∈ F E(AlgN )E F . ˆ ⊥ F φ(R)F ⊥ E ˆ for all F ∈ N and all rank one operators (2). φ(R) = E ⊥ ⊥˜ ˜ R ∈ F E(AlgN )E F . The next goal is to show φ(N ) = N or φ(N ) = N ⊥ . For this, we distinguish several cases according to Lemma 3.10 and Lemma 3.11. 3.4 Case that Lemma 3.10(1) and Lemma 3.11(1) hold Lemma 3.12. Suppose that Lemma 3.10(1) and Lemma 3.11(1) hold. Let E ˆ ⊥ = 0. ˆ ⊥ F = 0 or EF be in N . Then for any F ∈ N , either E ˆ ⊥ F = 0 and EF ˆ ⊥ = 0 for some Proof. Suppose on the contrary that E ⊥ ˆ F and v ∈ EF ˆ ⊥ . Then u ⊗ v ∈ F ∈ N . Take non-zero vectors u ∈ E ⊥ ⊥ ˆ F (AlgN )F E. ˆ Applying Lemma 3.6 and Lemma 3.8 to φ−1 , we know that E −1 φ (u ⊗ v) is a rank one operator in E F˜ (AlgN )F˜ ⊥ E ⊥ or E F˜ ⊥ (AlgN )F˜ E ⊥ . ˆ ⊥ (u ⊗ v)E ˆ=E ˆ ⊥ φ(φ−1 (u ⊗ v))E ˆ = 0. In the either cases, we have u ⊗ v = E This contradiction completes the proof. Proposition 3.13. Suppose that Lemma 3.10(1) and Lemma 3.11(1) hold. Then φ(N ) ⊆ N .

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Proof. For E ∈ N , define ˆ ⊥ N = 0}. F = ∨{N ∈ N : E ˆ ⊥ F = 0. For N ∈ N with N > F , since E ˆ ⊥ N = 0, by Lemma 3.12 Then E ⊥ ⊥ ˆ ˆ + = 0. So we have EN = 0. It follows that EF ˆ = EF ˆ + = E(F ˆ + (F+ − F )) = EF ˆ + E(F ˆ + − F ) = F + E(F ˆ + − F ). E ˆ + − F ) = F+ − F or 0, we have E ˆ = F+ or F , completing Hence, since E(F the proof. 3.5 Case that Lemma 3.10(2) and Lemma 3.11(2) hold By an argument similar to that in the above case, we have Proposition 3.14. Suppose that Lemma 3.10(2) and Lemma 3.11(2) hold. Then φ(N ) ⊆ N ⊥ . 3.6 Case that Lemma 3.10(1) and Lemma 3.11(2) hold In this subsection, we assume that Lemma 3.10(1) and Lemma 3.11(2) hold. Throughout this subsection, we shall fix a projection E in N . Let F be in N . Then our assumptions are as follows: ˆ ⊥ for all rank one operators ˆ φ(R)F ⊥ E (A1) φ(R) = EF ⊥ ˜⊥ ˜ R ∈ F E(AlgN )E F ; ˆ ⊥ F φ(R)F ⊥ E ˆ for all rank one operators (A2) φ(R) = E ⊥ R ∈ F˜ E(AlgN )E ⊥ F˜ . Lemma 3.15. Let F be in N . ˆ (AlgN )F ⊥ E ˆ ⊥ is of rank one. Then φ−1 (G) is of (1). Suppose that G ∈ EF −1 −1 rank one and φ (G) = F˜ Eφ (G)E ⊥ F˜ ⊥ ˆ is of rank one. Then φ−1 (G) is of ˆ ⊥ F (AlgN )F ⊥ E (2). Suppose that G ∈ E −1 ⊥ −1 ˜ rank one and φ (G) = F Eφ (G)E ⊥ F˜ Proof. (1) Applying Lemma 3.8 and Lemma 3.6 to φ−1 , we know that φ−1 (G) is of rank one and that either φ−1 (G) = F˜ Eφ(G)E ⊥ F˜ ⊥ or φ−1 (G) = F˜ ⊥ Eφ−1 (G)E ⊥ F˜ . If the latter case occurs, we would have ˆ ⊥ φ(φ−1 (G))E ˆ = 0 by assumption (A2), a contraG = φ(φ−1 (G)) = E diction. (2) Similarly. ˆ N⊥ Lemma 3.16. Let N and M be in N with N < M and suppose that EM ⊥ ⊥ ⊥ ˆ N = 0 or E ˆ M = 0. has dimension greater than one. Then either E ˜N ˜ ⊥ has dimension greater than one by Lemmas Proof. First we note that E M 3.2 and 3.3 ˆ ⊥ M ⊥ = 0. For ˆ ⊥ N = 0 and E Suppose now on the contrary that E ⊥ ⊥ ˜⊥ ˜ ˜ rank one operators A ∈ E M N (AlgN )E M , by the assumption (A1) and Lemma 3.4 we have ˜ ⊥N ˜ ⊥) ˜ AE ⊥ M ˜ ⊥EM φ(A) = φ(N ˜ ⊥ )N ⊥ ˜ EAE ⊥ M = N ⊥ φ(M ˆ N ⊥ φ(A)E ˆ⊥M ⊥ = EM

(3.1)

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˜ , by the assumption ˜N ˜ ⊥ (AlgN )E ⊥ N and for rank one operators B ∈ E M (A2) and Lemma 3.4 we have ˜) ˜ EN ˜ ⊥BN ˜ E⊥M φ(B) = φ(M ⊥ ⊥ ˜ )M ˜ EBE N = M φ(N ˆ ⊥ N φ(B)EM ˆ N ⊥. =E

(3.2)

Also, by Lemma 3.15 and applying Lemma 3.4 to φ−1 , for rank one operators ˆ N ⊥ (AlgN )E ˆ ⊥ M ⊥ we have C ∈ EM ˜⊥ ˜N ˜ ⊥ φ−1 (C)E ⊥ M φ−1 (C) = E M

(3.3)

ˆ N ⊥ we have ˆ ⊥ N (AlgN )EM and for rank one operators D ∈ E ˜ φ−1 (D)E ⊥ N ˜. ˜ ⊥M φ−1 (D) = E N

(3.4)

Note that both φ(A) and φ−1 (C) are of rank one. Thus, from (3.1) and ˜ ⊥M ˜ (AlgN ) (3.3) we know that φ maps the set of rank one operators in E N ⊥ ˜⊥ ⊥ ⊥ ˆ ˆ E M onto the set of rank one operators in EM N (AlgN )E M ⊥ . Moreover, the restriction of φ to the linear span of rank one operators in ˜ (AlgN )E ⊥ M ˜ ⊥ is an operator norm isometric map. By Proposi˜ ⊥M EN tion 2.14, one of the following holds: ˜N ˜ ⊥ → EM ˆ N ⊥ and T1 : (a) there exist bijective linear isometries S1 : E M ˜⊥ → E ˆ ⊥ M ⊥ such that E⊥M φ(x ⊗ y) = S1 x ⊗ T1 y ˜ and y ∈ E ⊥ M ˜ ⊥. ˜ ⊥M for all x ∈ E N ˜N ˜⊥ → (b) there exist bijective conjugative linear isometries T1 : E M ⊥ ⊥ ⊥ ⊥ ⊥ ˆ M and S1 : E M ˜ → EM ˆ N such that E φ(x ⊗ y) = S1 y ⊗ T1 x ˜ ⊥. ˜N ˜ ⊥ and y ∈ E ⊥ M for all x ∈ E M Also, by (3.2) and (3.4) one of the following holds: (c)

˜N ˜⊥ → E ˆ ⊥ N and S2 : there exist bijective linear isometries T2 : E M ⊥ ˜ ⊥ ˆ N such that E N → EM φ(x ⊗ z) = T2 x ⊗ S2 z

˜. ˜N ˜ ⊥ and z ∈ E ⊥ N for all x ∈ E M ˜N ˜⊥ → (d) there exist conjugative bijective linear isometries S2 : E M ⊥ ⊥ ˜ ⊥ ˆ ˆ EM N and T2 : E N → E N such that φ(x ⊗ z) = T2 z ⊗ S2 x ˜. ˜N ˜ ⊥ and z ∈ E ⊥ N for all x ∈ E M We now distinguish four cases in order to arrive a contradiction. Note that the range of T1 and the range of T2 are always orthogonal, and that the range of Si and the range of Tj are always orthogonal for 1 ≤ i, j ≤ 2.

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Case 1. (a) and (c) holds. ˜ ⊥ and z0 ∈ E ⊥ N ˜ . Then for each unit vector Fix unit vectors y0 ∈ E ⊥ M ⊥ ˜ ˜ x ∈ E M N , we have √ 2 1 w(T2 x ⊗ S2 z0 + S1 x ⊗ T1 y0 ) = w(x ⊗ z0 + x ⊗ y0 ) = x ⊗ (y0 + z0 ) = . 2 2 ˜N ˜ ⊥. By Example 2.10, S2 z0 and S1 x are linearly dependent for each x ∈ E M ⊥ ˜ ˜ This is impossible because of the dimension of E M N greater than one. Case 2. (a) and (d) holds. ˜ ⊥ and z0 ∈ E ⊥ N ˜ . Then for each As before, fix unit vectors y0 ∈ E ⊥ M ˜N ˜ ⊥ , we have unit vector x ∈ E M √ 2 w(S1 x ⊗ T1 y0 + T2 z0 ⊗ S2 x) = w(x ⊗ y0 + x ⊗ z0 ) = . 2 ˜N ˜ ⊥ there exists a scalar ξ(x) such that By Example 2.10, for each x ∈ E M ˜N ˜ ⊥ . Then for x2 ∈ E M ˜N ˜⊥ S1 x = ξ(x)S2 x. Fix a non-zero vector x1 ∈ E M being independent of x1 , we have that ξ(x1 + x2 )S2 x1 + ξ(x1 + x2 )S2 x2 = S1 (x1 + x2 ) = ξ(x1 )S2 x1 + ξ(x2 )S2 x2 . From the independence of vectors, we have ξ(x1 ) = ξ(x1 + x2 ) = ξ(x2 ). Thus iS1 x2 = S1 (ix2 ) = ξ(ix2 )S2 (ix2 ) = ξ(x1 )S2 (ix2 ) = −iξ(x1 )S2 x2 = −iS1 x2 . This is an obvious contradiction. Case 3. (b) and (d) holds. By an argument similar to that in Case 1, we can also get a contradiction. Case 4. (b) and (c) holds. By an argument we know that S1 y and S2 z are always linearly depen˜ ⊥ and z ∈ E ⊥ N ˜ . This is impossible unless both E ⊥ M ˜⊥ dent for all y ∈ E ⊥ M ⊥ ˜ and E N are one-dimensional ˜ ⊥ and E ⊥ N ˜ are of dimension one. Then Now suppose that both E ⊥ M ⊥ ˜⊥ they are atoms. Let y in E M be a unit vector. By Lemma 3.3 we can ˆ ⊥ M ⊥ . For each suppose that φ(y ⊗ y) = u ⊗ u for some unit vector u ∈ E ⊥ ˜ ˜ x ∈ E M N , since ˜N ˜ ⊥ + y ⊗ y), ˜N ˜ ⊥ + y ⊗ y)(x ⊗ y)(E M x ⊗ y = (E M it follows from Lemma 3.4 and Eq. (3.1) that ˆ N ⊥ φ(x ⊗ y)E ˆ ⊥ M ⊥ (EM ˆ N ⊥ + u ⊗ u) ˆ N ⊥ + u ⊗ u)EM φ(x ⊗ y) = (EM = φ(x ⊗ y)u ⊗ u. This conflicts with the assumption that (b) holds. The proof is complete.

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Similarly, we have ˆ ⊥M N ⊥ Lemma 3.17. Let N and M be in N with N < M and suppose that E ⊥ ˆ ˆ has dimension greater than one. Then either EN = 0 or EM = 0. ˆ N⊥ Lemma 3.18. Let N and M be in N with N < M and suppose that EM ⊥ ⊥ ⊥ ˆ ˆ is an atom. If one of E N and E M contains an atom, then the other is zero. ˜N ˜ ⊥ is also an atom by Lemma 3.2. Let x be a unit Proof. Note first that E M ˜N ˜ ⊥ . Then by Lemma 3.3 we can suppose that vector in E M φ(x ⊗ x) = u ⊗ u ˆ N ⊥. for some unit vector u in EM ˜ contains an atom P while Suppose on the contrary that E ⊥ N ⊥ ˜⊥ ˜ ⊥ = {0} or E M = 0. By Proposition 2.13 (2), either P (AlgN )E ⊥ M ⊥ ˜⊥ ⊥ ˜⊥ E M (AlgN )P = {0}. We shall consider only the case E M (AlgN )P = {0}. The proof for the other case is similar. ˜ ⊥ (AlgN )P = {0}. Then by Proposition 2.12 we can Now suppose E ⊥ M ˜ ⊥ (AlgN )P . Assume y = z = 1. take a rank one operator z ⊗y from E ⊥ M By Lemma 3.3 we may suppose that φ(y ⊗ y) = v ⊗ v ˆ⊥N . for some unit vector v in Pˆ ⊆ E Now since x ⊗ y = (x ⊗ x + y ⊗ y)(x ⊗ y)(x ⊗ x + y ⊗ y), by Lemma 3.4 and Eq. (3.2) we have ˆ ⊥ N φ(x ⊗ y)EM ˆ N ⊥ (u ⊗ u + v ⊗ v) φ(x ⊗ y) = (u ⊗ u + v ⊗ v)E = (v ⊗ v)φ(x ⊗ y)(u ⊗ u) ˆ N⊥ ˆ ⊥ N (AlgN )EM = αv ⊗ u ∈ E for some α ∈ C. Moreover, |α| = 1 because of φ(x ⊗ y) = 1 by Proposition 2.8. Since ˜ ⊥ )(x ⊗ z)(x ⊗ x + E ⊥ M ˜ ⊥ ), x ⊗ z = (x ⊗ x + E ⊥ M by Lemma 3.4 and Eq. (3.1) we have that ˆ ⊥ M ⊥ )EM ˆ N ⊥ φ(x ⊗ z)E ˆ ⊥ M ⊥ (u ⊗ u + E ˆ⊥M ⊥) φ(x ⊗ z) = (u ⊗ u + E = (u ⊗ u)φ(x ⊗ z) ˆ⊥M ⊥ ˆ N ⊥ (AlgN )E = u ⊗ w ∈ EM ˆ ⊥ M ⊥ . Moreover w = 1 because of φ(x ⊗ z) = 1 for some vector w in E by Proposition 2.8. ˜ and ˜ ⊥ (AlgN )M Finally, since z ⊗ y ∈ M ˜ ⊥ + y ⊗ y), ˜ ⊥ + y ⊗ y)(z ⊗ y)(M z ⊗ y = (M

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by Lemmas 3.4 and 3.5 we get φ(z ⊗ y) = (M ⊥ + v ⊗ v)M φ(z ⊗ y)M ⊥ (M ⊥ + v ⊗ v) = (v ⊗ v)φ(z ⊗ y) ˆ ⊥ N (AlgN )M ⊥ = v ⊗ w ∈ E for some unit vector w in M ⊥ . Further, noting that u ⊗ w + v ⊗ w ∈ M (AlgN )M ⊥ , it follows from Proposition 2.8 that √ u ⊗ w + v ⊗ w = 2w(u ⊗ w + v ⊗ w ) = 2w(x ⊗ z + z ⊗ y) = 2. ¯ for some β ∈ C with |β| = 1. Thus Hence w = βw φ(z ⊗ y) = βv ⊗ w. For b, c, d ∈ C, let T (b, c, d) = bx ⊗ y + cx ⊗ z + dz ⊗ y and S(b, c, d) = bαv ⊗ u + cu ⊗ w + dβv ⊗ w. Then φ(T (b, c, d)) = S(b, c, d). One can easily verify that T (b, c, d) = A(b, c, d) ⊕ 0 under the decomposition H = [x, y, z] ⊕ [x, y, z]⊥ and that S(b, c, d) = B(b, c, d) ⊕ 0 under the decomposition H = [u, v, w] ⊕ [u, v, w]⊥ . Here ⎡ ⎤ 0 b c A(b, c, d) = ⎣ 0 0 0 ⎦ 0 d 0 ⎡ ⎤ 0 0 c quadand B(b, c, d) = ⎣ bα 0 dβ ⎦ . 0 0 0 By Proposition 2.7, W (T (b, c, d)) = W (A(b, c, d)) and W (S(b, c, d)) = W (B(b, c, d)). Hence W (A(b, c, d)) = W (B(b, c, d)). It follows from Proposition 2.6 that A(b, c, d) + A(b, c, d)∗ and B(b, c, d) + B(b, c, d)∗ have the same largest eigenvalues. A computation shows that the characteristic equations of ⎡

0 A(b, c, d) +A(b, c, d)∗ = ⎣ ¯b c¯

b 0 d

⎤ c d¯⎦ 0

⎡

and

¯bα 0 ¯ ∗ ⎣ B(b, c, d) +B(b, c, d) = bα 0 c¯ d¯β¯

are ¯ =0 λ3 − (|b|2 + |c|2 + |d|2 )λ − (¯bcd + b¯ cd) and ¯ d¯ + α λ3 − (|b|2 + |c|2 + |d|2 )λ − (αβbc ¯ β¯b¯ cd) = 0, respectively. So for all b, c, d ∈ C, ¯bcd + b¯ ¯ d¯ + α cd¯ = αβbc ¯ β¯b¯ cd. Now putting b = c = d = 1 in the above equation, we get αβ¯ + α ¯ β = 2, and putting b = i and c = d = 1, we get αβ¯ − α ¯ β = 0.

⎤ c dβ ⎦ 0

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Those two equations give αβ¯ = α ¯ β = 1. Thus, for all b, c, d ∈ C, ¯bcd + b¯ cd¯ = bcd¯ + ¯b¯ cd. But this is impossible. For example, if we let b = i, c = 1 + i and d = 1 − i then ¯bcd + b¯ cd¯ = 0 while bcd¯ + ¯b¯ cd = −4. The proof is complete. Similarly, we have ˆ ⊥M N ⊥ Lemma 3.19. Let N and M be in N with N < M and suppose that E ⊥ ˆ and EM ˆ is an atom. If one of EN contains an atom, then the other is zero. Proposition 3.20. Suppose that Lemma 3.10(1) and Lemma 3.11(2) hold. Let ˆ ∈ N or E ˆ⊥ ∈ N . E be in N . Then E Proof. Suppose on the contrary that the proposition is not true. Then ˆ ˆ = {0} or E ˆ ⊥ (AlgN )E ˆ by Proposition 2.13(4), E(AlgN )E ⊥ (AlgN )E ⊥ ⊥ ˆ ˆ (AlgN )E = {0}. Without loss of generality, assume that E (AlgN ) ˆ ˆ ⊥ = {0}. Then there are rank one operator x1 ⊗ y1 and x2 ⊗ y2 E(AlgN )E ˆ ⊥ (x1 ⊗ y1 )E(x ˆ 2 ⊗ y2 )E ˆ ⊥ = 0. By Proposition 2.11, in AlgN such that E there exists Li ∈ N such that xi ∈ Li and yi ∈ (Li )⊥ − , i = 1, 2. Thus, ⊥ ⊥ ⊥ ⊥ ˆ ˆ ˆ E L1 = 0, EL2 (L1 )− = 0 and E (L2 )− = 0. Define ˆ ⊥ L1 (L1 )⊥ = 0, L1 , if E − K1 = ˆ ⊥ L1 (L1 )⊥ = 0 (L1 )− , if E − ⊥ ˆ L2 , if E L2 (L2 )⊥ − = 0, quadand K2 = ˆ ⊥ L2 (L2 )⊥ (L2 )− , if E − = 0. ˆ ⊥ K1 = 0, EK ˆ 2 K ⊥ = 0 and E ˆ ⊥ K ⊥ = 0. Then it is not difficult to verify that E 1 2 Now let ˆ ⊥ K = 0} and M = ∧{K ∈ N : E ˆ ⊥ K ⊥ = 0}. N = ∨{K ∈ N : E ˆ ⊥ M ⊥ = 0, and N < K1 < K2 < M . Moreover, ˆ ⊥ N = 0 and E Then E ⊥ ˆ ⊥ L = 0, EKL ˆ for any N < L ≤ K1 and K2 ≤ K < M , we have E = 0 ⊥ ⊥ ⊥ ˆ K = 0. Thus EKL ˆ and E is of dimension one by Lemma 3.16. Hence ˆ − N ⊥ is of dimension one and then it is an atom. By the definition of N EM + ˆ + N ⊥ = 0 and EM ˆ M ⊥ = 0. So, if we denote the and M , we know that EN − ⊥ ⊥ ˆ atom EM− N+ by F F− with N ≤ F− < F ≤ M , we have that ˆ = N + F F ⊥ + M ⊥. E −

(3.5) ˆ⊥

ˆ is an interval, it follows that E ˆ ∈ N or E ∈ N . This conflicts with If E ˆ ⊥ (AlgN )E(AlgN ˆ ˆ ⊥ = {0}. So by Proposition 2.13(3), the assumption E )E ˆ ˆ ⊥ (AlgN )E ˆ = {0}. E(AlgN )E Then a similar argument yields ⊥ ˆ ⊥ = L + GG⊥ E −+K

(3.6) ˆ⊥

ˆ nor E is an for some L, K, G ∈ N with L ≤ G− < G ≤ K. Since neither E ⊥ ˆ ˆ interval and E E = 0, it follows from Eq. (3.5) and Eq. (3.6) that one of the following holds:

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N = 0, K = I, M = I, L = 0; M = I, L = 0, N = 0, K = I.

ˆ⊥ = L + ˆ = F F−⊥ + M ⊥ and E First suppose that (i) holds. Then E ⊥ ˆ ˆ These together with E + E = I yield that either L = G− < G = F− < F = M or L = F− < F = G− < G = M . If the former case ˆ ⊥ = G ∈ N , a contradiction. If the latter case occurs, then occurs, then E ⊥ ˆ = FF ,E ˆ ⊥ M F ⊥ = GG⊥ and EM ˆ ⊥ = M ⊥ = 0, which conflicts with EF − − Lemma 3.19. Similarly, if (ii) holds, we can also get a contradiction. The proof is complete. GG⊥ −.

3.7 Case that Lemma 3.10(2) and Lemma 3.11(1) hold By an argument similar to that in the above case, we have Proposition 3.21. Suppose that Lemma 3.10(2) and Lemma 3.11(1) hold. Let ˆ ∈ N or E ˆ⊥ ∈ N . E be in N . Then E 3.8 So far, we have exhausted all of the possible combinations of Lemma 3.10 and Lemma 3.11. We therefore conclude the following result. ˆ ∈ N or E ˆ⊥ ∈ N . Proposition 3.22. Let E be in N . Then E Further, we have Proposition 3.23. φ(N ) = N or φ(N ) = N ⊥ . ˆ and Fˆ are comProof. Let E and F be non-trivial projections in N . Then E parable. This observation together with Proposition 3.22 shows that either ˆ Fˆ ∈ N or E ˆ ⊥ , Fˆ ⊥ ∈ N . Thus we have shown that φ(N ) ⊆ N or φ(N ) ⊆ E, ⊥ N . The same is also true for φ−1 , and a consideration of the possible combinations completes the proof. 3.9 Let M be a maximal abelian self-adjoint algebra (masa) containing N . Then M ⊆ N . Note that φ|N is a C∗ -Jordan isomorphism. Arguing as between [15, Proposition 10] and [15, Proposition 11], we have Proposition 3.24. Exactly one of the following holds. 1. φ(N ) = N . In this case, there exists a unitary operator V such that V ∗ N V = N , the map ψ(A) = V φ(A)V ∗ is a surjective linear map of AlgN onto itself, and such that ψ(M ) = M for each M ∈ M. 2. φ(N ) = N ⊥ . In this case, there exist a unitary operator V and a conjugation J such that V ∗ N V = N ⊥ , JEJ = E for each E ∈ N , and the map ψ(A) = V (Jφ(A)∗ J)V ∗ is a surjective linear map of AlgN onto itself, and such that ψ(M ) = M for each M ∈ M. We now consider the linear bijection ψ, just appeared in Proposition 3.24, of AlgN onto itself which preserves the closure of the numerical range and satisfies ψ(M ) = M for each M ∈ M. For E ∈ N and A ∈ AlgN , by Lemmas 3.4 and 3.5 we have ψ(EAE ⊥ ) = Eψ(A)E ⊥ .

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Applying Lemma 3.7, one can see that ψ maps the set of operators of rank one in E(AlgN )E ⊥ onto itself for all E ∈ N . Arguing as in the proof [15, Proposition 16], there exist surjective linear ⊥ isometric maps U1 : I− → I− and V1 : (0)⊥ + → (0)+ such that ψ(x ⊗ y) = U1 x ⊗ V1 y for all E ∈ N and all x ∈ E, y ∈ E ⊥ . Moreover, U1 E = E and V1 E ⊥ = E ⊥ for all E ∈ N \ {0, I}. Let E and F be in N such that (0) < E < F < I. For unit vectors x ∈ E, y ∈ F − E, z ∈ F ⊥ , we have ψ(x ⊗ y) = U1 x ⊗ V1 y ψ(y ⊗ z) = U1 y ⊗ V1 z So

√

2 2 and hence, by Example 2.10, U1 y = ηV1 y for some η ∈ C with |η| = 1. Set w(U1 x ⊗ V1 y + U1 y ⊗ V1 z) = w(x ⊗ y + y ⊗ z) =

T = x ⊗ y + x ⊗ z + y ⊗ z. Then ψ(T ) = U1 x ⊗ V1 y + U1 x ⊗ V1 z + ηV1 y ⊗ V1 z. A computation shows that W (T ) = W (A) and W (ψ(T )) = W (B), where ⎡ ⎤ ⎡ ⎤ 0 1 1 0 1 1 A = ⎣ 0 0 1 ⎦ and B = ⎣ 0 0 η ⎦ 0 0 0 0 0 0 The characteristic equation of the matrix ⎡ ⎤ 0 1 1 A + A∗ = ⎣ 1 0 1 ⎦ 1 1 0 is λ3 − 3λ − 2 = 0 and the characteristic equation of the matrix ⎡ ⎤ 0 1 1 B + B∗ = ⎣ 1 0 η ⎦ 1 η¯ 0 is λ3 − 3λ − η − η¯ = 0. Since A + A∗ and B + B ∗ have the same largest eigenvalues, we get η + η¯ = 2. So η = 1. So U1 y = V1 y for all y ∈ F − E whenever (0) < E < F < I. Taking the limit, we get U1 y = V1 y for all y ∈ I− − (0)+ . Now let ⊥ . U = U1 ⊕ V1 I−

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Then U is a unitary operator in N and for E ∈ N , x ∈ E, y ∈ E ⊥ ψ(x ⊗ y) = U1 x ⊗ V1 y = U x ⊗ U y = U (x ⊗ y)U ∗ . Consider the linear map ψ˜ defined by ˜ ψ(A) = U ∗ ψ(A)U, A ∈ AlgN . Then ψ˜ is a linear bijection of AlgN which preserves the closure of the numerical range. Moreover, ˜ ⊗ y) = x ⊗ y for all E ∈ N , x ∈ E, y ∈ E ⊥ . (P1). ψ(x ˜ (P2). For E ∈ N , ψ(E) = U ∗ ψ(E)U = U ∗ EU = U ∗ U E = E (as U ∈ N ). ˜ (P3). For E ∈ N , ψ(E(AlgN )E ⊥ ) = E(AlgN )E ⊥ . Lemma 3.25. Let A be in AlgN such that A = EAE ⊥ for some E ∈ N . Then ˜ ψ(A) = A. ˜ = A by (P1). Now Proof. If one of E and E ⊥ is finite-dimensional, then ψ(A) suppose that both E and E ⊥ are infinite-dimensional. Let T ∈ B(H) be a unitary operator that maps E onto E ⊥ . Let B = ψ(A). Then B ∈ E(AlgN )E ⊥ by (P3). For a finite rank operator K ∈ E(AlgN )E ⊥ , since A + K and ψ(A + K) = B + K are both in E(AlgN )E ⊥ , we have A + K = B + K and hence T A + T K = T B + T K. For unit vectors x ∈ E and scalars λ ∈ C, let K = λx ⊗ T x − x ⊗ xB(E ⊥ − T x ⊗ T x) − T ∗ (E ⊥ − T x ⊗ T x)T BT x ⊗ T x. Then T B + T K = ((BT x, x) + λ)T x ⊗ T x + (E ⊥ − T x ⊗ T x)T B(E ⊥ − T x ⊗ T x). So for λ with |(T Bx, x) + λ| > T B, we have T A + λT x ⊗ T x − T x ⊗ xB(E ⊥ − T x ⊗ T x) −(E ⊥ − T x ⊗ T x)T BT x ⊗ T x = |(BT x, x) + λ| and then |(BT x, x) + λ| ≥ (T A + λT x ⊗ T x − T x ⊗ xB(E ⊥ − T x ⊗ T x) −(E ⊥ − T x ⊗ T x)T BT x ⊗ T x)T x = T AT x + λT x − T BT x + (BT x, x)T x = (A − B)T x + ((BT x, x) + λ)x. From this, we get (A − B)T x = 0 for all x ∈ E. Thus (A − B)T E = 0 and hence A − B = 0, completing the proof. Lemma 3.26. Let P ∈ AlgN be a non-zero projection. Suppose that, for each E ∈ N , either P E = 0 or P E ⊥ = 0. Then P is an atom.

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Proof. Let F = ∨{E ∈ N : P E = 0}. Then P F = 0. Moreover, for E ∈ N with E > F, P E = 0 and therefore P E ⊥ = 0. Consequently, P (F+ )⊥ = 0. So P = P (F+ − F ) = F+ − F , completing the proof. ˜ Lemma 3.27. ψ(D) = D for all D ∈ N . ˜ ) = P for each projection P in N . This is Proof. We first show that ψ(P ⊥ true if P or P is an atom by (P2). Suppose now that neither P nor P ⊥ is an atom. Then by Lemma 3.26, there exist E and F in N for which P E = 0, P E ⊥ = 0, P ⊥ F = 0 and P ⊥ F ⊥ = 0. Let x ∈ P E and y ∈ P E ⊥ be unit vectors. Then for t > 2, we have ˜ ⊥ ) + tx ⊗ y) = w(P ⊥ + tx ⊗ y) = t . w(ψ(P 2 So ˜ ⊥ )x, x) + (ψ(P ˜ ⊥ )y, y) + t ˜ ⊥ ) + tx ⊗ y)(x + y), x + y)| = (ψ(P t ≥ |((ψ(P and hence (ψ(P ⊥ )x, x) = (ψ(P ⊥ )y, y) = 0 ˜ ⊥ )P E = ψ(P ˜ ⊥ )P E ⊥ = 0 and ˜ ⊥ ) is positive. Consequently, ψ(P since ψ(P ⊥ ⊥ ˜ then ψ(P )P = 0. Similarly, using P F = 0 and P ⊥ F ⊥ = 0, we get ˜ ) = P. ˜ )P ⊥ = 0. So ψ(P ψ(P ˜ N is a C∗ -Jordan isomorphism of the von Neumann algebra N Now ψ| onto itself and is therefore a weakly continuous map on N . By the above paragraph, it fixes all projection in N . So it must be the identity map on N , completing the proof. Proposition 3.28. ψ(A) = U AU ∗ for each A ∈ AlgN . ˜ Proof. Let A be in AlgN . Let D = ψ(A) − A. For E ∈ N , by Lemma 3.25 we have ⊥ ⊥ ˜ ˜ − EAE ⊥ = ψ(EAE ) − EAE ⊥ = EAE ⊥ − EAE ⊥ = 0. EDE ⊥ = E ψ(A)E So D ∈ N . Then by Lemma 3.27, ˜ ψ(A)) ˜ ˜ ψ( = ψ(A) + D = A + 2D, ˜ and so w(A + 2D) = w(ψ(A)) = w(A + D) = w(A). By induction we have w(A + nD) = w(A) for all n ∈ N. This obviously gives D = 0, completing the proof. 3.10 We are now ready to prove the main result in this section. Proof of Theorem 3.1. If φ(N ) = N , then, by Proposition 3.24, φ(A) = V ∗ ψ(A)V for all A ∈ AlgN . Here V is a unitary operator, ψ is a linear bijection of AlgN onto itself which preserves the closure of the numerical range and satisfies ψ(M ) = M for each M ∈ M. Hence, by Proposition 3.28, φ(A) = V ∗ U AU ∗ V for all A ∈ AlgN , where U is a unitary operator. If φ(N ) = N ⊥ , then by Proposition 3.24, φ(A) = JV ∗ ψ(A)∗ V J for all A ∈ AlgN . Here V is a unitary operator, J is a conjugation, ψ is a linear bijection of AlgN onto itself which preserves the closure of the numerical

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range and satisfies ψ(M ) = M for each M ∈ M. Hence by Proposition 3.28, φ(A) = (JV ∗ U J)JA∗ J(JU ∗ V J) for all A ∈ AlgN , where U is a unitary operator. Noting that JV ∗ U J is a linear unitary operator, we are done.

4. Linear Maps Preserving Numerical Radius The main result in this paper reads as follows. Theorem 4.1. Suppose that N is a nest of a Hilbert space H. Let φ : AlgN → AlgN be a surjective linear map such that w(φ(A)) = w(A) for all A ∈ AlgN . Then φ has one of the following forms. 1. There exist a unitary U ∈ B(H) and a complex number ξ of modulus one such that φ(A) = ξU AU ∗ for all A ∈ AlgN . 2. There exist a unitary U ∈ B(H), a conjugation J on H, and a complex number ξ of modulus one such that φ(A) = ξU JA∗ JU ∗ for all A ∈ AlgN . This theorem follows immediately from Theorem 3.1 and the following two propositions. Proposition 4.2. Let N be a nest on a Hilbert space H. Then, an operator A in AlgN is a scalar multiple of the identity if and only if for each B ∈ AlgN , there exists a scalar η with |η| = 1 such that w(A + ηB) = w(A) + w(B). Proof. We first prove two claims. Claim 1. Let E be in N and suppose that the rank of E ⊥ is greater than 1. Then |(Ax, x)| = w(A) for all unit vectors x ∈ E. Furthermore, EAE = μE for some μ ∈ C with |μ| = w(A) Let x be a unit vector in E. Then we can choose a unit vector y in E ⊥ such that (A∗ x, y) = 0. Since B = x ⊗ y ∈ AlgN , there exists a scalar η with |η| = 1 such that w(A + ηB) = w(A) + w(B). Suppose that {zn } is a sequence of unit vectors in H such that lim |((A + ηB)zn , zn )| = w(A + ηB).

n→∞

By taking a subsequence, we may suppose that lim (Azn , zn ) = c and

n→∞

lim (Bzn , zn ) = d.

n→∞

Then we have |c| ≤ w(A), |d| ≤ w(B), |c + ηd| = w(A) + w(B). So we have |c| = w(A) and |d| = w(B), that is, 1 . n→∞ n→∞ 2 Write zn = an x + bn y + vn , where vn is a vector which is orthogonal to x and y. Then sequences {an } and {bn } are both bounded by 1 and we can suppose that limn→∞ an = a and limn→∞ bn = b. Then 1 = lim |(Bzn , zn )| = lim |an ||bn | = |a||b|. n→∞ 2 n→∞ lim |(Azn , zn )| = w(A)

and

lim |(Bzn , zn )| = w(B) =

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√

Since |a|2 + |b|2 ≤ 1, it follows that |a| = |b| = 22 . Hence limn→∞ vn = 0 and therefore |(A(ax + by), ax + by)| = w(A). Using (Ay, x) = (y, A∗ x) = 0 and (Ax, y) = 0 (as Ax ∈ E), we get 1 |(Ax, x) + (Ay, y)|. 2 This forces that |(Ax, x)| = |(Ay, y)| = w(A). So we have W (EA|E ) ⊆ {λ ∈ C : |λ| = w(A)}. It follows from the convexity of the numerical ranges that W (EA|E ) is singleton. Hence EAE = μE for some μ ∈ C with |μ| = w(A), establishing the claim. w(A) = |(A(ax + by), ax + by)| =

Claim 2. Suppose that P is an atom of N . Then |(Ax, x)| = w(A) for all x ∈ P . Furthermore, P AP = μP for some μ ∈ C with |μ| = 1. Let x be a unit vector in P . Since B = x ⊗ x ∈ AlgN , there exists a scalar η with |η| = 1 such that w(A + ηB) = w(A) + w(B). Suppose that {zn } is a sequence of unit vectors in H such that lim |((A + ηB)zn , zn )| = w(A + ηB).

n→∞

By taking a subsequence, we may get lim |(Azn , zn )| = w(A)

n→∞

and

lim |(Bzn , zn )| = w(B) = 1

n→∞

The latter is equivalent to lim |(x, zn )| = 1.

n→∞

Since all the vectors involved are unit vectors, there must exist a complex number α with |α| = 1 such that zn → αx. Hence w(A) = lim |(Azn , zn )| = |(Ax, x)| n→∞

and it follows from the convexity that P AP = μP for some μ ∈ C with |μ| = w(A). Now we complete the proof by distinguishing cases. Case 1: I− = I, Then, for any E ∈ N with E < I, the rank of E ⊥ is greater than one. Thus, it follows from Claim 1 that Ax and x are linearly dependent for all x ∈ E. Note that in the present situation the linear manifold span{z ∈ E : E ∈ N , E < I} is dense in H. So A is a scalar multiple of I. Case 2: F = I− < I. By Claim 2, F ⊥ AF ⊥ = μF ⊥ some μ ∈ C with |μ| = w(A). Case 2.1: F− = F . Then the same argument as in Case 1 shows that F AF = λF for some λ ∈ C with |λ| = w(A). Since F− = F , we can take a projection E in N such that E < F and the rank of E is greater than 1. Then there exists a scalar γ such that E ⊥ AE ⊥ = γE ⊥ . Consequently, μ = γ = λ. So A = μI + B, where B ∈ F (AlgN )F ⊥ . Furthermore, since W (B) is a disk

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centered at the original, we have |μ| = w(A) = w(μI + B) = |μ| + w(B). So B = 0 and then A = μI. Case 2.2: G = F− < F . Then the rank of G⊥ is greater than 1. It follows from Claim 1 that GAG = λG for some λ ∈ C with |λ| = w(A). Also, (F − G)A(F − G) = γ(F − G) some γ ∈ C with |γ| = w(A) by Claim 2. If C = GAF ⊥ = 0, then there exist unit vectors u ∈ F ⊥ and v ∈ G such that e = (Au, v) = 0. Let λ C λ e D= and M = . 0 μ 0 μ Then W (M ) ⊆ W (D) ⊆ W (A). Since W (M ) is an elliptical disk with λ and μ as foci, and the length of minor axis equal to |e|, it follows that w(A) = |λ| < w(M ) ≤ w(A), a contradiction. So GAF ⊥ = 0. Similarly, GA(F −G) = (F − G)AF ⊥ = 0. Consequently A = λG + γ(F − G) + μF ⊥ . Now take unit vectors x in G and y ∈ F − G. By the proof of Claim 1, we see that |(A(ax + by), ax + by)| = w(A) = |λ| 2

2

where |a| = |b| = λ = μ. So A = λI.

1 2.

So

1 2 |λ

+ γ| = |λ|. Consequently, λ = γ. Similarly,

Proposition 4.3. Let A and B be linear unital subspaces of B(H). Let ψ : A → B be a linear bijection which satisfies ψ(I) = I and w(ψ(A)) = w(A) for all A ∈ A. Then ψ preserves the closure of the numerical range. Proof. The proof is standard; the readers can refer [4,5,14].

References [1] Arazy, J., Solel, B.: Isometries of non-selfadjoint operator algebras.. J. Funct. Anal. 90, 284–305 (1990) [2] Chan, J.T.: Numerical radius preserving operators on B(H). Proc. Amer. Math. Soc. 123, 1437–1439 (1995) [3] Chan, J.T.: Numerical radius preserving operators on C∗ -algebras. Arch. Math. (Basel) 70, 486–488 (1998) [4] Cui, J., Hou, J.: Linear maps preserving the closure of numerical range on nest algebras with maximal atomic nest. Integr. Equ. Oper. Theory 46, 253– 266 (2003) [5] Cui, J., Hou, J.: Non-linear numerical radius isometries on atomic nest algebras and diagonal matrices. J. Funct. Anal. 206, 414–448 (2004) [6] Davidson, K.R.: Nest Algebra. Ritman Research Notes in Mathematics, Vol. 191. Longman, London (1988) [7] Erdos, J.A.: Operators of finite rank in nest algebras. J. London Math. Soc. 43, 391–397 (1968) [8] Gao, M.: Numerical range preserving linear maps and spectrum preserving elementary operators on B(H). Chin. Ann. Math. 14, 295–301 (1993) [9] Gustafson, K.E., Rao, D.K.M.: Numerical range: the field of values of linear operators and matrices. Springer, New york (1997)

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[10] Halmos, P.R.: A Hilbert space problem book 2nd ed. Springer, New York (1982) [11] Horn, R.A., Johnson, C.R.: Topics in matrix Analysis. Cambridge University Press, New York (1991) [12] Kadison, R.V.: Isometries of operator algebras. Ann. Math. 54, 325–338 (1951) [13] Li, C.K.: Linear operators preserving the numerical radius of matrices. Proc. Am. Math. Soc. 99, 601–608 (1987) [14] Li, C.K., Semrl, P., Soares, G.: Linear operators preserving the numerical range (radius) on triangular matrices. Linear Multilinear Algebra 48, 281–292 (2001) [15] Moore, R.L., Trent, T.T.: Isometries of nest algebras. J. Funct. Anal. 86, 180– 209 (1989) [16] Moore, R.L., Trent, T.T.: Isometries of certain reflective operator algebras. J. Funct. Anal. 98, 437–471 (1991) [17] Omladic, M.: On operators preserving the numerical range. Linear Algebra Appl. 134, 31–51 (1990) [18] Omladic, M., Semrl, P.: Additive mappings preserving operators of rank one. Linear Algebra Appl. 182, 239–256 (1993) [19] Pellegrini, V.: Numerical range preserving operators on a Banach algebra. Studia Math. 54, 143–147 (1975) [20] Ringrose, J.R.: On some algebras of operators II. Proc. London Math. Soc. 16, 385–402 (1966) [21] Solel, B.: Isometries of CSL algebras. Trans. Am. Math. Soc. 332, 595– 606 (1992) [22] Wei, S., Hou, S.: Rank preserving linear maps on nest algebras. J. Oper. Theory 39, 207–217 (1998) Fangyan Lu (B) Department of Mathematics Soochow University Suzhou 215006 People’s Republic of China e-mail: [email protected] Received: April 28, 2010. Revised: January 16, 2011.

Integr. Equ. Oper. Theory 69 (2011), 535–555 DOI 10.1007/s00020-010-1853-0 Published online February 5, 2011 c Springer Basel AG 2011

Integral Equations and Operator Theory

The Generalised Dyson Circular Unitary Ensemble: Asymptotic Distribution of the Eigenvalues at the Origin of the Spectrum Philippe Rambour and Abdellatif Seghier Abstract. The ﬁrst part of this paper is devoted to the study of ΦN the orthogonal polynomials on the circle, with respect to a weight of type f = (1 − cos θ)α c where c is a sufﬁciently smooth function and ∗(p) α > − 12 . We obtain an asymptotic expansion of the coefﬁcients ΦN (1) ∗ ∗ N¯ 1 for all integer p where ΦN is deﬁned by ΦN (z) = z ΦN ( z ) (z = 0). These results allow us to obtain an asymptotic expansion of the associated Christofel–Darboux kernel, and to compute the distribution of the eigenvalues of a family of random unitary matrices. The proof of the results related to the orthogonal polynomials are essentially based on the inversion of the Toeplitz matrix associated to the symbol f . Mathematics Subject Classification (2010) . Primary 47B39; Secondary 47BXX. Keywords. Random unitary matrices, orthogonal polynomials, Fisher-Hartwig conjecture.

1. Introduction It is a well known fact that random matrices are characterized by the distribution of their eigenvalues. For the case of random unitary matrices the most studied case is the Dyson circular unitary ensemble, where the density of the vector (θ1 , θ2 , . . . , θN ) of eigenvalue angles is given, for a N × N matrix (see [8–10,13,30,31]) as: PN (θ1 , . . . , θN ) = cN

1≤j
|eiθj − eiθk |2 .

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On the other hand for the Dyson generalized circular unitary ensemble this density is (see [22,29]): PN (θ1 , . . . , θN ) = f (θj ) |eiθj − eiθk |2 , (1) 1≤j≤N

1≤j
where f is a positive integrable function on T. In the case of the Dyson circular unitary ensemble the correlation function is related to the orthogonal polynomials with respect to the Haar measure on T, but in the case of Dyson generalized circular unitary ensemble this function is written by means of the Christoffel-Darboux kernel KN associated to the orthogonal polynomials (Φn )1≤n≤N with respect to the weight f . We deﬁne generally KN by (see [27]):

KN (eiθ , eiθ ) =

f (eiθ )

f (eiθ )

N −1

1 Φm (eiθ )Φm (eiθ ) h m=0 m

with π

f (eiθ )Φm (eiθ )Φm (eiθ )dθ = hm .

−π

If z = 0, and θ = θ , we can write ∗ iθ ∗ iθ ) − ΦN (eiθ )ΦN (eiθ ) 1 ΦN (e )ΦN (e iθ iθ iθ iθ KN (e , e ) = f (e ) f (e ) hN (1 − ei(θ −θ) ) ¯ N 1 . For the unitary case the link between this kernel where Φ∗N (z) = z N Φ z and the correlation function is given in the proof of Theorem 2. On the other hand if we denote by DN (f ) the determinant of a (N +1)× (N + 1) Toeplitz matrix (see the Eq. (8) for a review) with a singular symbol, (N ) it is well known that the correlations functions Rk are closely related with the asymptotic behaviour of DN (f ) when N goes to the inﬁnity (the study of this asymptotic is the aim of the Fisher-Hartwig conjecture see, for instance, [1,4,5,11]). Then we have the fundamental relation ([3,12,13,27,31]) N 1 iθμ f (e ) |eiθμ − eiθν |2 dθ1 · · · dθN (2) DN (f ) = (2π)N N ! ν=1 ]−π,π]N

1≤μ≤ν≤N

that gives us, with the definition of the correlation functions [see the Eqs. (23) and (24)], and using the equality (N )

Rk (θ1 , . . . , θN ) = det[KN (θi , θj )1≤i≤k,1≤j≤k ] (N ) R (θ1 , . . . , θk )dθ1 · · · dθk ]−π,π]k k DN (f ) = (2π)N N (N − 1) · · · (N − k + 1) det[KN (θi , θj )1≤i≤k,1≤j≤k ]dθ1 · · · dθk ]−π,π]k . = (2π)N N (N − 1) · · · (N − k + 1)

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These relations show that the Fisher-Hartwing determinants and the Christoffel Darboux kernels are closely related. In this work we consider the integrals I k det[KN (θi , θj )1≤i≤k,1≤j≤k ]dθ1 · · · dθk where IN is a little N neighbourhood around zero. But Theorem 1 in the paper of Martinez-Finkel stein et al. [18] provides the integrals ]−π,π]\I k det[KN (θi , θj )1≤i≤k,1≤j≤k ]dθ1 N · · · dθk . For all random unitary matrices an important question is the study of the eigenvalue distribution when N goes to inﬁnity. In fact this problem can be subdivided into two parts for the bulk and the edge of the spectrum. For the Gaussian Hermitian matrices we must consider the eigenvalue distribution in ]0, +∞[, and this same distribution in a neighborhood of zero or inﬁnity. For the unitary matrices we want to know the distribution of the eigenvalues angles in ], 2π − [ ( > 0) or in ]−, [ (see [28–30]). It is a well known result that for the case of random matrices which are in the Gaussian Unitary Ensemble the problem of the eigenvalues distribution is closely related with the theory of orthogonal polynomials and with a knowledge of Christoffel Darboux kernel ([2,6,7,16]. . . can be consulted). For the edge of the spectrum Kuijlaars and Vanlessen [16] study a family of Hermitian random matrices which induces a probability density function on the eigenvalues x1 , x2 , . . . , xN given by P (N ) (x1 , x2 , . . . , xN ) =

N 1 wN (xj ) |xi − xj |2 Zn j=1 i<j

where wN is the following weight wN (x) = |x|2α e−N V (x) , forx ∈ R, and α > − 21 and V a real valued function with enough increase at inﬁnity, for example a polynomial of even degree with positive leading coefﬁcient. The authors show that the Christoffel Darboux kernel KN associated to wN veriﬁes α α

u v u v 1 0 KN , = Jα (u, v) + O as n → ∞ (3) N ψ(0) N ψ(0) N ψ(0) N where J0α is the universal local eigenvalue correlation of the origin spectrum, which is described with usual Bessel kernels and ψ the density of the equilibrium measure for V . In this paper we obtain a similar result but with a class of random circular unitary matrices (Dyson generalized ensemble). We study the orthogonal polynomials with respect to the weight f (eiθ ) = |1 − eiθ |2α c(eiθ )

(4)

where c is a regular function on T and with α ∈ ] − 12 , +∞[. In previous papers we have obtained an asymptotic expansion, when N goes to inﬁnity, of the coefﬁcients of these orthogonal polynomials and in the present work we deduce from this asymptotic the eigenvalue distribution around 1. As far as we know the limiting kernel that we obtain in Theorem 2 has not previously appeared in the mathematical literature related to the random circular unitary ensemble. But in a physical frame the random matrices the

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density of which is

|1 − eiθj |2α c(eiθj )

1≤j≤N

IEOT

|eiθj − eiθk |2

1≤j
corresponds to a model describing a disordered conductor (see [20] or [21]). ∗(p) Our ﬁrst result gives us an asymptotic expansion of ΦN (1) (refer to Theorem 1), for a ﬁxed integer p. Lastly we obtain an asymptotic expression of the Christoffel-Darboux kernel with respect to the symbol |1−eiθ |2α c(eiθ ). If −π < u < v ≤ π and u = v, we have KN (eiu/N , eiv/N ) =

N (uv)α ψ(α, −u)ψ(α, v) − ψ(α, u)ψ(α, −v)ei(v−u) + o(N ) Γ2 (α)c1 (1) i(u − v)

(5)

where ψ is a function which is given in what follows (see Theorem 2). For a matrix which belongs to the circular unitary ensemble it is a well known result that the asymptotic of the probability that a given interval contains a given number of eigenvalue angles is the same as in the case of Gaussian unitary ensemble. By comparing the Eqs. (5) and (3) we see that the asymptotic of the two Christoffel-Darboux kernels bears some formal analogy in spite of some differences, but we have not obtained a universality which would be controlled by the parameter α as it is expected in [21]. N the increasing sequence of the eigenvalue Denote now by θ1N , . . . , θN angles of a random unitary N × N matrix U . We suppose that the eigenvalues angles are in ]− π, π] and that the distribution of the eigenvalues of U is as in (1) with a weight f such f (eiθ ) = |1 − eiθ |2α c(eiθ ) (see (4)). Putting FN =

N i=1

δθiN

we obtain (Theorem 2) i)

u v , = m = F∞ (a, b, m) lim P FN N →+∞ N N where F∞ is a function deﬁned by the determinant of a Fredholm operator with kernel K = limN →+∞ KNN . ii) if q is an integer greatest or equal to 2 we have u v , q = m = 0. lim P FN q N →+∞ N N

Heuristically this means that we have few eigenvalues of the form eiθN with θN = O( N1 ) when N goes to inﬁnity, but for q > 1 there are no eigenvalues = O( N1q ). eiθN with θN The ﬁrst part of our results is an extension (for a certain part) of the Martinez-Finkelstein, Mac-Laughin and Saﬀ work. In [18] they study a family ˜ N with respect to the weight of orthogonal polynomials Φ m W (z) = w(z) |z − ak |2βk , z ∈ T, (6) k=1

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where w is a sufﬁciently smooth function, βk > − 12 , k = 1, . . . , m and where W satisﬁes the Szeg¨o conditions W (z)dz > 0 and ln(W (z))dz > −∞. (7) T

T

Martinez-Finkelstein, Mac-Laughin and Saﬀ give the asymptotic behavior of ˜ N (z) and study the position of the zeros of this polynomial in the complex Φ plane. Their main tool is the Riemann-Hilbert method which we did not use here.

2. Main Results 2.1. Notations and Definitions Let f be a function that belongs to L1 (T). We denote by TN (f ) the (N + 1) × (N + 1) Toeplitz matrix such that TN (f ) = f(i − j) (8) 0≤i,j≤N

where f(k) =

π

f (eiθ )e−ikθ

−π

dθ , 2π

k∈Z

(9)

is the Fourier coefﬁcient of f . The link between the Toeplitz matrices and orthogonal polynomials is well known (see, for instance, [17]). A very important relation is Φ∗N (z) =

N (TN (f ))−1 k+1,1 k=0

(TN (f ))−1 1,1

zk ,

|z| = 1.

(10)

The polynomials Φ∗n (TN (f ))−1 1,1 are often called predictor polynomials. As we can see in the previous formula their coefﬁcients are, except for a normal−1 ization, the entries of the ﬁrst column of TN (f ) . In a precedent work we have given an asymptotic of these coefﬁcients in statement (see [24,25] and also the Theorems 1, 4, 5). On the other hand the relation

1 ∗ N¯ ΦN (z) = z ΦN , (11) z implies that the coefﬁcients of the orthogonal polynomial are, always except −1 for a normalization, the entries of the last column of TN (f ) . At last for all k in Z we use the notation χk : eiθ → χk (θ) = eikθ . We denote by H+ (resp.H− ) the space ˆ H+ = {h ∈ L2 (T)/h(s) = 0, s < 0} H− = L2 (T) H+ . Let π+ and π− be the orthogonal projections from L2 (T) onto H+ and H− .

540

P. Rambour and A. Seghier

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It is said that a real numbers sequence (μj )j∈Z is a Beurling class if we have

⎧ j∈Z ⎨ (i) μj ≥ 1, j∈Z (ii) μj = μ−j , ⎩ (iii) μj+k ≤ μj μk , j, k ∈ Z.

(12)

Then a Beurling class Wμ with respect to the weight μ = (μj ) is deﬁned as Wμ = {w ∈ L1 (T) : μj |w(j)| < ∞}. (13) j∈Z

Let f be the symbol f (eiθ ) = |1 − eiθ |2α c(eiθ ) where c is a positive function on T and belonging to a Beurling class Wμ with μj ≥ 32 for all j ∈ Z. Then there exist two functions gα , c1 ∈ H+ analytic in {z/|z| < 1} such that f = |gα |2 ,

c = c1 c¯1 .

(α) (βk )

the Fourier coefﬁcients of 1/gα , and to reduce the Lastly we denote by −1 notations we assume in all this work that β0α = 1, that implies (TN (f ))1,1 = 1 + O( N1 ) (see [23,24]). 2.2. Statement of the Main Results 2.2.1. Orthogonal Polynomials. As it has been said in the introduction, we (p) (p) have to know precisely the quantities ΦN (1) and (Φ∗N ) (1) to obtain an asymptotic expansion of the Christoffel-Darboux kernel. This is the aim of Theorem 1. Despite the fact that we treat only the case with one singularity we can observe that these result are an extension of a part of [18], since in this last (p) (p) paper the value of ΦN (1) and (Φ∗N ) (1) cannot be easily evaluated for a sufﬁciently large p. Moreover we have here a less demanding hypothesis on the regular function c. The reason while we assume μj ≥ 32 can be found in [24,25] ([14] can also be consulted). Theorem 1. Let TN (f ) be a Toeplitz matrix with symbol f |1 − χ|2α c where α ∈ R∗ , α > − 12 and where the function c > 0 belongs to a Beurling class Wμ with μj ≥ 32 for all j ∈ Z. Then for an integer j ≥ 0 and for α > − 12 we have 1 Γ(α + j)Γ(α + 1) (j) + o(N α+j ). (Φ∗N ) (1) = N α+j Γ(2α + j + 1) Γ(α)c1 (1) 2.2.2. Random Unitary Matrices. Let UN be a N × N random unitary matrix. Denote by θiN 1 ≤ i ≤ N the eigenvalue angles of UN with −π < N ≤ π. For this paper we will assume that the vector θ1N ≤ θ2N ≤ · · · ≤ θN (θ1 , . . . , θN ) has the probability density PN (θ1 , θ2 , . . . , θN ) =

N j=1

f (eiθj )

N j
|eiθj − eiθi |2

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541

with f (eiθ ) = |1 − eiθ |2α c(eiθ ) and where c belongs to a Beurling class Wμ N with μj ≥ 32 for all j ∈ Z. Then we put FN = i=1 δθiN where δθiN denotes the Dirac measure with support θiN . In what follows we assume that u and v are two reals such that −π − 12 . Define the two following functions ∀u ∈ R

ψ(α, u) =1 F1 (α, 2α + 1, iu)B(α, α + 1), √ τ (α, u) = πΓ(α + 1)(2u)−1/2−α eiu Jα+1/2 (u).

With the previous hypotheses we can write, for all non-negative integer m u v (−1)m d m , =m = det (Id − γK)|L2 (I) γ=1 lim P FN N →+∞ N N m! dγ (14) with I = [u, v] and where K is the Fredholm operator with kernel K defined as • if θ = θ θα θα B(α, α + 1) (1 F1 (α, 2α + 1, −iθ)1 F1 (α, 2α + 1, iθ ) (θ, θ ) → 2 Γ (α)c(1) i(θ − θ) − 1 F1 (α, 2α + 1, iθ)1 F1 (α, 2α + 1, −iθ )ei(θ −θ) , • if θ = θ , θ → −2 (ψ(α, θ)τ (α, −θ)) + |ψ(α, θ)|2

θ2α Γ2 (α)c(1)

.

Remark 1. We have the result u v lim P FN , = 0 = det[(Id − K)|L2 (I) ] N →+∞ N N

(15)

Theorem 3. For all the non-negative integer p and m such that p > 1 and N > m ≥ 1 and for all real α with α > −1 2 we have, always with the same hypotheses that in Theorem 2 u v (1−p) = m = o N . (16) , lim P FN N →+∞ Np Np

3. Proofs of the Results 3.1. Proof of Theorem 1 First we have to recall the two following results (see [14,24,25]). Lemma 1. Assume always that the function c > 0 belongs to a Beurling class Wμ with μj ≥ 32 for all j ∈ Z. If Nk → 0 for N → +∞ we have for all α > − 21 1. −1

(α)

(TN (f ))k+1,1 = βk (1 + o(1)) uniformly in k for k ∈ [0, [N ]] for a sufficiently small .

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2.

3.

IEOT

1 −1 (TN (f ))N −k+1,1 = O max N (α−1) , N uniformly in k for k ∈ [0, [N ]] for a sufficiently small . For all j ∈ {0, 1} we can write c1 (1) (Φ∗N )

(j)

(1) = N α+j

Γ(α + j)Γ(α + 1) + o(N α+j ). Γ(2α + 1)(Γ(α)

(17)

Remark 2. We can observe that the third statement can be obtained by evaluating the limit when θ → 1 in the Theorem 9 of [19]. Theorem 4. −1

c1 (1) (TN (f ))[N x]+1,1 = Kα (x)N α−1 + o(N α−1 ), N → ∞

(18)

uniformly in x for x ∈ [δ1 , δ2 ] for 0 < δ1 < δ2 < 1 and where Kα (x) =

1 α−1 x (1 − x)α . Γ(α)

It is clear that these terms are also in the coefﬁcients of the polynomial Φ∗N . For ΦN we deduce this terms from (11). Expression of (Φ∗N )(p) (1), for 1 < p.

N In the following of the proof we put Φ∗N (eiθ ) = l=0 βN,l (eiθ )l . Let p > 1 an integer. Using always the same notations as previously, we can write (Φ∗N )(p) (eiθ ) =

N

l(l − 1) · · · (l − p + 1)βN,l (eiθ )(l−p) .

l=p

Put (Φ∗N )(p) (1) ∼ S1 + S2 + S3 with [N ]

S1 =

l=0

N −[N ]

lp βN,l ,

S2 =

lp βN,l ,

l=[N ]+1

N

S3 =

lp βN,l ,

l=N −[N ]+1

+

and → 0 . Using the Lemma 1 we obtain ⎞ ⎛ [N ] (α) p l (βl ⎠ S1 ∼ ⎝ l=0

= O(N α+p α+p ) = o(N α+p ) and S3 = o(N α+p ). Using Theorem 4 we easily obtain ⎛ 1 ⎞ 1 N α+p ⎝ tα+p−1 (1 − t)α dt⎠ + o(N α+p ), S2 = Γ(α)c1 (1) 0

as we have obviously S3 = o(N α+p ) this ends the proof of this case.

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3.2. Proof of Theorem 2 3.2.1. Determinant of Fredholm Operators in L2 (T). For (θ, θ ) in ]− π, π] × ]− π, π] we put

k(θ, θ ) =

N

φm (θ)fm (θ )

m=1

where, for all integer m, the functions φm and fm are in L2 (T). If X is an open interval in ]−π, π], we can also deﬁne an operator K on L2 (X) by Kx (θ) = k(θ, θ )x(θ )dθ X 2

where x ∈ L (X). Then it is known (see [15] or [26]) that det(Id + K) is given by: N (−1)m det(k(θi , θj )i,j=1,...,m dθ1 · · · dθm det(Id + K) = 1 + m! m=1 Xm

that can also be written as: N (−1)m det(Id + K) = det(k(θi , θj )i,j=1,...,m dθ1 · · · dθm m! m=0 (with the convention that

Xm

h(t)dt = 1). Then the trace of K is given by: TrK = k(θ, θ)dθ.

X0

X

Now we consider (Kn )n∈N a sequence of such operators with limit equal to A. We can deﬁne det(Id + A) = lim det(I + Kn ) n→∞

and

TrA = lim Tr(Kn ). n→∞

It is known that if A is such an operator (see [15]) the function λ → det(I + λA) is a complex analytic function which is equal to the series det(Id + λA) = 1 + where cn

⎛

∞ cn (A) n λ n! n=1

Tr(A) n−1 0 ⎜ Tr(A2 ) Tr(A) n − 2 ⎜ ⎜ . . . ⎜ . . . cn (A) = det ⎜ ⎜ ⎜ . . . ⎜ ⎝ Tr(An−1 ) Tr(An−2 ) Tr(An−3 ) Tr(An−1 ) Tr(An−2 ) Tr(An )

··· ··· . . . ··· ···

⎞ 0 0 0 0 ⎟ ⎟ . . ⎟ ⎟ . . ⎟ ⎟ . . ⎟ ⎟ Tr(A) 1 ⎠ Tr(A2 ) Tr(A)

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P. Rambour and A. Seghier

That we can also write

IEOT

∞ (−1)m+1 m m Tr(A )λ . det(Id + λA) = exp m! m=1

Then we can claim the result Property 1. The operator Id + A is invertible if and only if det(Id + A) = 0. 3.2.2. Orthogonal Polynomials and Correlation Functions: the Case α > 0. Consider the equality Φ∗N (eiθ ) =

N (eiθ − 1)j

j!

j=0

(Φ∗N )(j) (1).

For a ﬁxed integer k0 we have Φ∗N (eiθ )

=

k0 (eiθ − 1)j j=0

j!

(Φ∗N )(j) (1)

α

+N O

|θk0 +1 | (k0 + 1)!

.

If we remark that l(l − 1)(l − 2) · · · (l − p + 1) < lp the proof of Theorem 1 allows us to obtain, for j > k0 , ⎛ 1 ⎞ iθ/N α+j−1 j α j α (e N − 1) |iθ) | x (1 − x) ⎝ (Φ∗N )(j) (1) ≤ dx + o(N α )⎠ j! c1 (1)Γ(α) j! 0

uniformly in j. Always using Theorem 1 we obtain Γ(α)c(1)Φ∗N (eiθ/N ) ⎛ ⎞ k0 +1

k0 1 α+j−1 j |θ x (iθ) | ⎠. (1 − x)α dx + O = Nα ⎝ j! (k + 1)! 0 j=0 0

This last equality is also

⎛ 1 ⎞ Γ(α)c1 (1)Φ∗N (eiθ/N ) = N α ⎝ eiθx xα−1 (1 − x)α dx + R(k0 )⎠.

(19)

0

with k0 +1

+∞ 1 α+j−1 |θ x (iθ)j | (1 − x)α dx + O R(k0 ) = − j! (k0 + 1)! k0 +1 0 k0 +1

|θ | =O . (k0 + 1)! Let be a positive real number. We can clearly choose k0 such that |R(k0 )| < . We can ﬁnally write: ⎛ 1 ⎞ (20) Γ(α)c1 (1)Φ∗N (eiθ/N ) = N α ⎝ eiθx xα−1 (1 − x)α dx + o(1)⎠. 0

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Asymptotic Distribution of the Eigenvalues

The same proof gives us

⎛

Γ(α)c1 (1)ΦN (eiθ/N ) = N α ⎝

1

545

⎞ eiθ(1−x) xα−1 (1 − x)α dx + o(1)⎠.

(21)

0

Consider now the Christoffel-Darboux kernel KN deﬁned by

KN (eiθ , eiθ ) =

f (eiθ )

f (eiθ )

N −1

1 Φm (eiθ )Φm (eiθ ) h m m=0

where hm is such that π f (eiθ )Φm (eiθ )Φn (eiθ )dθ = hm δm,n . −π

Then we can write, for θ = θ (see [27]) Φ∗ (eiθ )Φ∗N (eiθ ) − ΦN (eiθ )ΦN (eiθ ) 1 KN (eiθ , eiθ ) = f (eiθ ) f (eiθ ) N . hN (1 − ei(θ −θ) ) For θ = θ we can differentiate, with respect to θ , the function Fθ given by

Fθ (θ ) = eiN θ Φ∗N (eiθ )Φ∗N (eiθ ) − eiN θ Φ∗N (eiθ )Φ∗N (eiθ ). −iN θ Fθ (θ) and Then we have KN (eiθ , eiθ ) = h1N f (eiθ ) f (eiθ ) −e i

1 dΦ∗ (eiθ ) KN (eiθ , eiθ ) = f (eiθ )N |Φ∗ (eiθ )|2 − 2 Φ∗N (eiθ ) N . hN dθ Using formulas (19) and (21) we can write, for u = v and with θ =

u N,

θ =

Γ2 (α)c1 (1)KN (eiu/N , eiv/N ) K(u, v) + o(1) 1 2α N f (eiu/N ) f (eiv/N ) = hN 1 − ei(v/N −u/N ) where 1 K(u, v) =

−iux α−1

e

x

α

1

(1 − x) dx

0

eivx xα−1 (1 − x)α dx

0

1 −

−iu(1−x) α−1

e 0

x

α

1

(1 − x) dx

eiv(1−x) xα−1 (1 − x)α dx.

0

That we can also write as K(u, v) = ψ(α, −u)ψ(α, v) − ψ(α, u)ψ(α, −v)ei(v−u) with 1 ψ(α, u) = 0

xα−1 (1 − x)α eiux dx =1 F1 (α, 2α + 1, iu)B(α, α + 1).

v N

(22)

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P. Rambour and A. Seghier √

If we replace for u = v,

IEOT

f and (1 − ei(u/N −v/N ) ) by an equivalent, we obtain, always

Γ2 (α)c(1)KN (eiu/N , eiv/N ) =

K(u, v) −1 + o(N ). N (uv)α hN i(v − u)

To ﬁnd the value of KN (eiu/N , eiv/N ) for u = v we can differentiate with respect of v the function Gu deﬁned by Gu (v) = eiu ψ(α, −u)ψ(α, v) − eiv ψ(α, u)ψ(α, −v), and iu/N

lim KN (e

v→u

iv/N

,e

1 )= N (u)2α hN

e−iu i

Gu (u).

That gives us Γ2 (α)c1 (1)KN (eiu/N , eiu/N ) = where

1 N u2α j(α, u) + o(N ) hN

j(α, u) = −2 (ψ(α, u)τ (α, −u)) + |ψ(α, u)|2

and 1 τ (α, u) =

xα (1 − x)α eiux dx =

√ πΓ(α + 1)(2u)−1/2−α eiu Jα+1/2 (u).

0

On the other hand we also have (see [17] for the definition of hm ) π hm = f (eiθ )|ΦN (eiθ )|2 dθ, −π

that means −1

hm = (Tm (f ))1,1 . But we have stated in a previous work [23,24] that

α2 −1 α 2 |β0 | 1 − ∼ (Tm (f ))1,1 . N Using the hypothesis β0α = 1 we can lastly conclude, for u = v, Γ2 (α)c(1)KN (eiu/N , eiv/N ) = N (uv)α

K(u, v) + o(N ), i(u − v)

and for u = v Γ2 (α)c(1)KN (eiu/N , eiu/N ) = N u2α −2 (ψ(α, u)τ (α, −u)) + |ψ(α, u)|2 + o(N ). (N )

denote the correlation function deﬁned by 1 (N ) PN (θ1 , . . . , θN )dθk+1 · · · dθN Rk (θ1 , θ2 , . . . , θk ) = (N − k)!

Let Rk

]−π,π[N −k

(23)

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Asymptotic Distribution of the Eigenvalues

547

with N

PN (θ1 , . . . , θN ) =

|1 − eiθj |2α c(θj )

j=1 (N )

It is well known that Rk

|eiθj − eiθk |2 .

(24)

j
satisfy the fundamental relation

Property 2. For any interval I included in ]−π, π[ and for any integer m ∈ {0, . . . , N } we have P (|{i ≤ N, =

(−1) m

θi ∈ I}| = m) (−1)k (N ) Rk (θ1 · · · θk )dθ1 · · · dθk . (k − m)!

N

m

k=m

(25)

Ik

Classical properties of orthogonal polynomials and determinants give us (see [7]) (N )

Rk (θ1 · · · θk ) = (det[KN (θi , θj )i,j=1···k ]) . Then we can also formulate the property (2) as θi ∈ I}| = m) (−1)k det[KN (θi , θj )i,j=1···k ]dθ1 · · · dθk . (k − m)!

P (|{i ≤ N, =

N m

(−1) m

k=m

(26)

Ik

u v For −π < u < v < π we put I = [u, v] and IN = [ N , N ]. The previous equality becomes

θi ∈ IN }| = m) (−1)k det[KN (θi , θj )i,j=1···k ]dθ1 · · · dθk . (k − m)!

P (|{i ≤ N, =

N m

(−1) m

k=m

(27)

k IN

Putting ui = N θi we obtain θi ∈ IN }| = m) (−1)k N −k det[KN (ui /N, uj /N )i,j=1···k ]du1 · · · duk , (k − m)!

P (|{i ≤ N, m

=

(−1) m

N

k=m

Ik

(28) or else P (|{i ≤ N, =

(−1) m

m

θi ∈ IN }| = m) N (−1)k det[HN (ui , uj )i,j=1···k ]dk (u), (k − m)!

k=m

Ik

where HN (u, v) =

K(u, v) 1 (uv)α + o(1), Γ2 (α)c(1) i(u − v)

(29)

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P. Rambour and A. Seghier

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for u = v and HN (u, u) =

1 Γ2 (α)c(1)

u2α −2 (ψ(α, u)τ (α, u)) + |ψ(α, u)|2 + o(1)

for u = v. We can now express the sum N (−1)k det[HN (ui , uj )i,j=1···k ]du1 · · · duk k! k=0

Ik

in terms of an operator. We have N (−1)k det[HN (ui , uj )i,j=1···k ]du1 · · · duk , = det[(I − |HN |)|L2 (I) ] (30) k! k=0

Ik

On the other hand, for any γ ∈ R from a classical formula for trace class operators (see [26]) N (−γ)k det[HN (ui , uj )i,j=1···k ]du1 · · · duk , = det[(Id − γHN )|L2 (I) ]. k! k=0

Ik

(31) Now differentiating m times with respect to γ N (−1)k γ k−m det[HN (ui , uj )i,j=1···k ]du1 · · · duk (k − m)! k=m

Ik

dm = m det[(Id − γHN )|L2 (I) ]. dγ Now put γ = 1 to obtain the following formula N (−1)k det[HN (ui , uj )i,j=1···k ]du1 · · · duk ) (k − m)! k=m k I

m d det(Id − γHN ) γ=1 . = dγ m

(32)

(33)

When N goes to inﬁnity we obtain lim P (|{i ≤ N θi ∈ IN }|) = m} m (−1)m d = det[(Id − γK)|L2 (I) ]γ=1 . m! dγ

N →+∞

(34)

This ends the proof of Theorem 2. 3.2.3. Orthogonal Polynomials and Correlations Functions: the Case α < 0. Of course this proof uses exactly the same methods as that of the case α > 0. We still write N (eiθ − 1)j ∗ (j) Φ∗N (eiθ ) = (ΦN ) (1). j! j=0

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549

As for the case α > 0, Theorem 1 gives us, for j ≥ 1 (eiθ/N − 1)j ∗ (j) N α (iθ)j (ΦN ) (1) = j! c1 (1)Γ(α)

1 0

xα+j−1 (1 − x)α dx + o(N α ). j!

(35)

In the case j = 0, we have to compute again Φ∗N (1), since a mere evaluation of limits is not sufﬁcient to obtain the need expression with an integral, like (35). We have to use the following result, more precise than Lemma 1. Theorem 5 consider the asymptotic behavior of the coefﬁcients of the −1 “edge” the ﬁrst column of the inverse matrix (TN (f )) when − 12 < α ≤ 12 (see [14]). Theorem 5. Let TN (f ) be a Toeplitz matrix with symbol f |1 − χ|2α c where α ∈ R∗ , |α| < 12 and where the function c > 0 belongs to a Beurling class Wμ k with μj ≥ 32 for all j ∈ Z. For all k ∈ N, such that limN →∞ N = 0 we have i)

−1

(TN (f ))k+1,1 =

(α)

βk

ii)

−1

(TN (f ))N +1−k,1 =

−

α2 (α+1) βk (1 + o(1)) , N

c(1) (α+1) β c¯1 (1) k

α (1 + o(1)) , N

N →∞

(36)

N → ∞,

(37)

uniformly in k for k ∈ [0, [N ]] for a sufficiently small . N Put Φ∗N (eiθ ) = l=0 βN,l (eiθ )l . Write Φ∗N (1) = S1 + S2 + S3 with S1 = [N ] N −[N ] N + l=N −[N ]+1 βN,l , with → 0 . Using l=0 βN,l , S2 = l=[N ]+1 βN,l , S3 = Theorem 5 we obtain ⎛ ⎞

[N ] (α) α2 (α+1) l ⎠ β S1 = ⎝ βl − +O N l N2 l=0

l α2 (α+2) [N ] (α+1) = β[N ] − β O − . N [N ] N N As (α+1)

βl

(α+2)

βl

1 lα + O(lα ) and Γ(α + 1)c1 (1) 1 lα+1 + O(lα + 1) = Γ(α + 2)c1 (1) =

we can conclude

1 α2 α 2 S1 = [N ] 1 − + O(1) Γ(α + 1)c1 (1) α+1

or S1 =

1 [N ]α + o(N α ). Γ(α + 1)c1 (1)

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P. Rambour and A. Seghier

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Now we consider the next sum S2 . Using Theorem 4 we have

α N −[N ] l α−1 1 l S2 ∼ . 1− Γ(α)c1 (1) N N l=[N ]+1

We can now write S2 =

⎛

1

1 ⎜ Nα ⎝ Γ(α)c1 (1)

tα−1 ((1−t)α −1) dt +

([N ]+1)/N

⎞ ⎛ 1 1 1 N α ⎝ tα−1 ((1 − t)α − 1) dt + ⎠ = Γ(α)c1 (1) α

⎞

1

⎟ tα−1 dt⎠ +o(N α )

([N ]+1)/N

0

[N ]α + o(N α ) − αΓ(α)c1 (1)

Since we have βk = O

(α+1)

βN −k N

for k ∈ [N − [N ]] we obtain as previously

S3 = O(N α α+1 ) = o(N α ). Hence Φ∗N (1) = S1 + S2 + S3

⎞ ⎛ 1 1 1 N α ⎝ tα−1 ((1 − t)α − 1) dt + ⎠ + o(N α ). = Γ(α)c1 (1) α 0

In the case j = 0, we can observe that the function: ⎛ 1 ⎞ 1 α → ⎝ tα−1 ((1 − t)α − 1) dt + ⎠ α 0

∗

on C . Since we have the well known formula for α > 0: is1 analytic α−1 t ((1 − t)α − 1) + α1 dt = Γ(α)Γ(α+1) we can write, using the analicity Γ(2α+1) 0 ∗ on C ⎞ ⎛ 1 1 ⎝ tα−1 ((1 − t)α − 1) dt + ⎠ = Γ(α)Γ(α + 1) α Γ(2α + 1) 0

and also c1 (1)Φ∗N (1) = N α

Γ(α + 1) . Γ(2α + 1)

Now using a same proof as previously, we obtain N

Γ(α)c1 (1) ∗ iθ/N ΦN (e )= α N j=1

1 0

1 + 0

xα+j−1 (iθ)j (1 − x)α dx j!

xα−1 ((1 − x)α − 1) dx +

1 + o(1). α

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Asymptotic Distribution of the Eigenvalues

551

This leads us to conclude

⎛ 1 Γ(α)c1 (1)Φ∗N (eiθ/N ) = N α ⎝ xα−1 (eiθx − 1)(1 − x)α dx 0

1 +

xα−1 ((1 − x)α − 1)dx +

0

or else

⎛

Γ(α)c1 (1)Φ∗N (eiθ/N ) = N α ⎝

1 0

⎞ 1 + o(1)⎠ α

(38)

⎞ 1 xα−1 eiθx (1 − x)α − 1 dx + + o(1)⎠. α (39)

¯ N ( 1 ) we obtain: Using (39) and Φ∗N (z) = z N Φ z ⎞ ⎛ 1 1 Γ(α)c1 (1)ΦN (eiθ/N ) = N α eiθ ⎝ (e−iθx (1 − x)α − 1)xα−1 dx + + o(1)⎠. α 0

(40) It is now clear that the function ⎛ 1 ⎞ 1 α → ⎝ xα−1 eiθx (1 − x)α − 1 dx + ⎠ α 0

∗

is analytical on C . as for α > 0 we have 1

xα−1 eiθx (1 − x)α dx =1 F1 (α, 2α + 1, iθ)B(α; α + 1)

0

we have for α ∈] − 12 , 0[ ⎞ ⎛ 1 ⎝ xα−1 eiθx (1 − x)α − 1 dx + 1 ⎠ =1 F1 (α, 2α + 1, iθ)B(α; α + 1). α 0

˜ N the Christoffel-Darboux kernel. We put again θ = u , We now denote by K N v and θ = N , and we use formulas (38) and (40). Then we obtain, by the same methods as previously, and for u = v ˜ N (eiu/N , eiv/N ) Γ2 (α)c(1)K ˜ K(u, v) + o(N 2α ) 1 2α = N f (eiu/N ) f (eiv/N ) hN 1 − ei(u/N −v/N ) with ˜ −u)ψ(α, ˜ v) − ψ(α, ˜ u)ψ(α, ˜ −v)ei(v−u) , ˜ K(u, v) = ψ(α,

(41)

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and 1 ˜ u) = ψ(α,

xα−1 ((1 − x)α e−iux − 1)dx +

0

1 . α

√ If we replace f and (1 − ei(u/N −v/N ) ) by an equivalent we have, for u = v ˜ ˜ N (eiu/N , eiv/N ) = 1 N (−uv)α K(u, v) + o(N ). Γ2 (α)c(1)K hN i(u − v) the case u = v can be obtained by differentiation. The end of the proof is the same as Theorem 2. 3.3. Proof of Theorem 3 In the ﬁrst part of this demonstration we assume α > 0. We can write for p>1 p

Φ∗N (eiθ/N ) =

p N (eiθ/N − 1)j ∗ (j) (ΦN ) (1). j! j=0

Theorem 1 gives us p

N α (iθ)j 1 (eiθ/N − 1)j ∗ (j) (ΦN ) (1) = j(p−1) j! c1 (1)Γ(α) N This last equation implies Γ(α)c1 (1)Φ∗N (eiθ/N

p

1 0

xα+j−1 (1 − x)α dx + o(N α ). j!

⎛

⎞ N 1 α+j−1 p−1 j x (iθ/N ) (1 − x)α dx + o(1)⎠ . ) = Nα ⎝ j! j=0 0

That is also Γ(α)c1 (1)Φ∗N (eiθ/N

p

⎛ 1 ⎞ p−1 ) = N α ⎝ eiθx/N xα−1 (1 − x)α dx + o(1)⎠. (42) 0

And we obtain:

⎛ p

Γ(α)c1 (1)ΦN (eiθ/N ) = N α ⎝

1

⎞ eiθ(1−x)/N

p−1

xα−1 (1 − x)α dx + o(1)⎠.

0

(43) These equalities give us the following asymptotic expansion p p 1 2α Γ2 (α)c(1)KN (eiu/N , eiv/N ) = N f (eiu/N p ) f (eiv/N p ) hN K(u/N p−1 , v/N p−1 ) + o(N 2α ) × . (44) 1 − ei(u/N p −v/N p )

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As previously we have to compute P (|{i ≤ N, θi ∈ IN p }|) = m} for all integers m, p ≥ 1 and with IN p = [ Nup , Nvp ]. For −π < u < v < π we put I = [u, v], and IN p = [ Nup , Nvp ], we have P (|{i ≤ N,

θi ∈ IN p }| = m) (−1)k det[KN (θi , θj ]i,j=1···k ]dk (θ). (k − m)!

N m

(−1) m

=

k=m

(45)

k IN

p

Putting ui = N θi we have P (|{i ≤ N, =

θi ∈ IN }| = m) N (−1)k I k N −pk det[KN (ui /N p , uj /N p ]i,j=1···k ]dk (u) , (k − m)!

(−1)m m

k=m

or P (|{i ≤ N, m

−1 = m

θi ∈ IN }| = m)

N k=m

N

k k(1−p) (−1)

Ik

det[HN (ui /N p−1 , uj /N p−1 )i,j=1···k ]dk (u) , (k − m)!

For all non-negative integers k ∈ [1, N ] the quantity det[HN (ui /N p−1 , uj /N p−1 )i,j=1···k ] goes to zero when N goes to inﬁnity. That gives us the expected result. The proof for α < 0 is obviously the same.

References [1] Basor, E.L.: Asymptotic formulas for Toeplitz determinants. Trans. Am. Math. Soc. 239, 33–65 (1978) [2] Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. Math. 150, 185–266 (1999) [3] B¨ ottcher, A.: The Onsager formula, the Fisher-Hartwig conjecture, and their inﬂuence on research into Toeplitz operators. J. Stat. Phys. 78, 575–588 (1995) [4] B¨ ottcher, A., Silbermann, B.: Toeplitz matrices and determinants with FisherHartwig symbols. J. Funct. Anal. 63, 178–214 (1985) [5] B¨ ottcher, A., Silbermann, B.: Toeplitz operators and determinants generated by symbols with one Fisher-Hartwig singularity. Math. Nachr. 127, 95–124 (1986) [6] Deift, P.A., McLaughlin, K.T.R., Kriecherbauer, T., Venakides, S., Zhou, X.: A Riemann-Hilbert approach to asymptotic questions for orthogonal polynomials. J. Approx. Theory 95, 388–475 (1998) [7] Deift, P.A.: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. AMS, New York (1998) [8] Dyson, F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys., 3, 1191–1198 (1962)

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[9] Dyson, F.J.: Statistical theory of the energy levels of complex systems, i–iii. J. Math. Phys. 3, 140–156, 157–165, 166–175 (1962) [10] Dyson, F.J.: The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechnics. J. Math. Phys. 3, 1199–1215 (1962) [11] Fisher, M.E., Hartwig, R.E.: Determinants: some applications, theorems, and conjectures. Adv. Chem. Phys. 15, 333–353 (1968) [12] Forrester, P.J., Frankel, N.E.: Applications and generalizations of Fisher-Hartwig asymptotics. J. Math. Phys. 45, 2003–2028 (2004) [13] Johansson, K.: On random matrices from the compact classical groups. Ann. Math. 145, 519–545 (1997) [14] Kateb, D., Rambour, P., Seghier, A.: Asymptotic behavior of the predictor polynomial associated to regular symbols. Pr´epublications de l’Universit´e Paris-Sud (2003) [15] Krupnik, N., Gohberg, I., Goldberg, S.: Operator theory advances and applications. In: Traces and Determinants of Linear Operators, vol. 116. Birk¨ auser Verlag, Basel (2000) [16] Kuijlaars, A.B.J, Vanlessen, M.: Universality for eigenvalue correlation at the origin of the spectrum. Comm. Math. Phys. 243, 163–191 (2003) [17] Landau, H.J.: Maximum entropy and the moment problem. Bull. Am. Math. Soc. 16(1), 47–77 (1987) [18] Martinez-Finkelshtein, A., McLaughlin, K.T.R, Saﬀ, E.B.: Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle. Internet Mathematical Research Notices (2006) [19] Martinez-Finkelshtein, A., McLaughlin, K.T.R, Saﬀ, E.B.: Szeg¨ o orthogonal polynomal with respect to an analytic weight: canonical representation and strong asymptotic. Constr. Approx. 24, 319–363 (2006) [20] Muttalib, K.A., Ismail, M.E.H.: Impact of localization on Dyson’s circular ensemble. J. Phys. A Math. Gen. 28, 541–548 (1995) [21] Nagao, T.: Universal correlations near a singularity of random matrix spectrum. J. Phys. Soc. Jpn. 64, 3675–3681 (1995) [22] Nagao, T., Wadati, M.: An integration method on generalized circular ensembles. J. Phys. Soc. Jpn. 61, 1903–1909 (1992) [23] Rambour, P., Seghier, A.: Inversion asymptotique des matrices de Toeplitz ´ a symboles singuliers, Extension d’un r´esultat de H, Kesten. Pr´epublications de l’Universit´e Paris-sud (2003) [24] Rambour, P., Seghier, A.: Inverse asymptotique des matrices de Toeplitz de < α ≤ 12 , et noyaux int´egraux. Bull. Des. Sci. symbole (1 − cos θ)α f1 , −1 2 Math. 134, 155–188 (2008) [25] Rambour, P., Seghier, A.: Inversion des matrices de Toeplitz dont le symbole admet un z´ero d’ordre fractionnaire positif, valeur propre. arXiv:1005.4073 (2010) [26] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, I–IV. Academic Press, New York (1997) [27] Szeg¨ o, G.: Orthogonal Polynomials. 3rd edn. American Mathematical Society, Colloquium ` publication, Providence (1967) [28] Tracy, C.A., Widom, H.: Level-spacing distibutions and the airy kernel. Commun. Math. Phys. 159, 151–174 (1994)

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[29] Tracy, C.A., Widom, H.: Correlation functions, cluster functions and spacing distribution for random matrices. J. Stat. Phys. 92, 809–835 (1999) [30] Tracy, C.A., Widom, H.: Universality of the distribution functions of random matrix theory. CRM Proc. 26, 1251–1264 (2000) [31] Weyl, H.: The Classical Groups: Their Invariants and Representations. Princeton University Press, Princeton (1946) Philippe Rambour (B) and Abdellatif Seghier Universit´e de Paris Sud Bˆ atiment 425 91405 Orsay Cedex France e-mail: [email protected]; [email protected] Received: June 1, 2010. Revised: November 10, 2010.

Integr. Equ. Oper. Theory 69 (2011), 557–566 DOI 10.1007/s00020-010-1837-0 Published online October 30, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Norm of an Integral Operator Related to the Harmonic Bergman Projection Congwen Liu and Lifang Zhou Abstract. The norm of the operator f (y)dVα (y) , Tα f (x) := (1 − 2x · y + |x|2 |y|2 )(n+α)/2 Bn

p

n

acting on L (B , dVα ), is determined to be Tα Lpα →Lpα =

Γ(n/2 + α + 1)Γ((α + 1)/p)Γ((α + 1)/q) . Γ((n + α)/2)Γ(α/2 + 1)Γ(α + 1)

for a range of p and α. Mathematics Subject Classification (2010). Primary 47B38, 47G10; Secondary 32A25. Keywords. Integral operators, harmonic Bergman projection, Lp space, operator norm, hypergeometric functions.

1. Introduction Let Bn denote the open unit ball in Rn for n ≥ 2. For α > −1 let Vα be the measure on Bn deﬁned by dVα (x) =

Γ(n/2 + α + 1) (1 − |x|2 )α dx, π n/2 Γ(α + 1)

where dx denotes the Lebesgue measure on Rn . In case α = 0, Vα is just the normalized Lebesgue measure on Bn , which we shall simply denote by V . It will be convenient to write dVα (x) = cα (1 − |x|2 )α dV (x) with cα =

Γ(n/2 + α + 1) . Γ(n/2 + 1)Γ(α + 1)

This work was supported by the National Natural Science Foundation of China grant 10601025 and 10771201.

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As usual, Lpα := Lp (Bn , Vα ) denotes the space of p-integrable functions on Bn with respect to the measure Vα . In this short note we are concerned with the action of the operator f (y)dVα (y) (1.1) Tα f (x) := (1 − 2x · y + |x|2 |y|2 )(n+α)/2 Bn

p

n

on L (B , dVα ). This operator arises naturally when looking at harmonic analysis questions on the unit ball of Rn as harmonic Bergman-type operators are dominated by it. Recall that the harmonic Bergman projections Πα , from L2α to its closed subspace consisting of harmonic functions, is given by Πα f (x) = f (y)Rα (x, y)dVα (y), Bn

with Rα (x, y) =

∞ Γ(k + n/2 + α + 1)Γ(n/2) k=0

Γ(n/2 + α + 1)Γ(k + n/2)

Zk (x, y),

where Zk (x, y) denotes extended zonal harmonics of degree k. It is well known that C |Rα (x, y)| ≤ (1 − 2x · y + |x|2 |y|2 )(n+α)/2 for some constant C = C(n, α) > 0. Hence |Πα f | ≤ CTα |f |. It is also well known that Tα is bounded on Lpα for any ﬁxed α > −1, 1 < p < ∞, and this in turn implies the boundedness of Πα on Lpα . See [1–3,7] for the details and [6] for the original idea. Moreover, Choe et al. [1], obtained the following sharp norm estimate: Theorem A ([1, Theorem 2.2]). Given ν > 0, there is a constant Cν > 0 such that p2 p2 ≤ Tα Lpα →Lpα ≤ Cν (1.2) Cν−1 p−1 (α + 1)(p − 1) for 1 −1, Tα comparable to p2 /(p − 1). This is in the spirit of a result of Zhu [9] dealing with holomorphic Bergman projections. Actually, the true harmonic analog of Zhu’s result was also given in [1]: Πα Lpα →Lpα ≈ p2 /(p − 1). Note that (1.2) also provides some information on the behavior of Tα p as α → −1, that is, T p grows at most like (α + 1)−1 as α Lα →Lp Lp α →Lα α α → −1. However, the authors did not know whether such behavior with parameter α is sharp. The purpose of this note is to answer this question in the aﬃrmative. Moreover, we determine explicitly the norm for a range of values of p and α, while for another range we give explicit (not necessarily accurate) bounds of the norm. Before we state our theorems, it is convenient to establish some notation. p Lp α →Lα

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Figure 1. The shaded regions stand for ΩI , on which Tα Lpα →Lpα is exactly determined Notation 1. We split the region {(p, α) : 1 −1} into three parts (see Fig. 1): ΩI := {(p, α) : 1 2, −1 < α ≤ 2/(p − 2)} {(p, α) : p > 2, α ≥ ((n − 2)p + 2)/(p − 2)}, ΩII := {(p, α) : 1 2, 2/(p − 2) < α < ((n − 2)p + 2)/(p − 2)}. Our main result is the following: Theorem 1.1. Let ΩI , ΩII , ΩIII be as above and 1/p + 1/q = 1. Then (i) For (p, α) ∈ ΩI , we have Tα Lpα →Lpα = (ii)

Γ(n/2 + α + 1)Γ((α + 1)/p)Γ((α + 1)/q) . Γ((n + α)/2)Γ(α/2 + 1)Γ(α + 1)

(1.3)

For (p, α) ∈ ΩII ∪ ΩIII , we have Γ(n/2 + α + 1)Γ((α + 1)/p)Γ((α + 1)/q) Γ((n + α)/2)Γ(α/2 + 1)Γ(α + 1) Γ(n/2 + α + 1)Γ((α + 1)/ max{p, q}) . ≤ Tα Lpα →Lpα ≤ Γ(α + 1)Γ(n/2 + (α + 1)/ max{p, q})

(1.4)

As a corollary, we obtain an aﬃrmative answer to the aforementioned question. Having in mind that (p, α) ∈ ΩI when α is near −1, and Γ(z) ≈ z −1

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as z → 0, it is easy to see from (1.3) that Tα Lpα →Lpα is comparable to (α + 1)−1 for any ﬁxed 1 −1. However, we have not been able to show this and leave it as an open question for future research. With some minor modiﬁcations, our proof also applies to the following, somewhat more general, class of integral operators: (1 − |y|2 )β f (y)dVα (y) Tα,β f (x) := . (1.5) (1 − 2x · y + |x|2 |y|2 )(n+α+β)/2 Bn

We now state without proof our second main result (for simplicity, we consider the case α = 0 only). Theorem 1.2. Suppose that 1 ≤ p < ∞. (i)

If 1/p − 1 < β ≤ 2/p or β ≥ 2/p + n − 2, then we have T0,β Lp →Lp =

(ii)

Γ(n/2 + 1)Γ(1/p)Γ(β + 1 − 1/p) . Γ((n + β)/2)Γ(β/2 + 1)

(1.6)

If 2/p < β < 2/p + n − 2 then we have Γ(n/2 + 1)Γ(1/p)Γ(β + 1 − 1/p) Γ(n/2 + 1)Γ(1/p) ≤ T0,β Lp →Lp ≤ Γ((n + β)/2)Γ(β/2 + 1) Γ(n/2 + 1/p)

Note that when n = 2, Theorem 1.2 reads: for all 1 ≤ p < ∞ and β > 1/p − 1, T0,β Lp →Lp =

Γ(1/p)Γ(β + 1 − 1/p) . Γ2 (β/2 + 1)

(1.7)

In particular, when β = 2, this gives BLp →Lp =

π p+1 , 2 p sin(π/p)

1 < p ≤ ∞,

where B is the Berezin transform on the unit disk given by (1 − |z|2 )2 Bf (z) := dA(w). |1 − z w| ¯4

(1.8)

(1.9)

D

So, Theorem 1.2 can be regarded as a higher dimensional generalization of the main result in [4].

2. Preliminaries Here we collect some preliminary facts used in the sequel. In the paper, the classical notation 2F1 (α, β; γ; z) denotes the Gauss hypergeometric function 2F1 (α, β; γ; z) =

∞ (α)k (β)k z k (γ)k k!

k=0

(2.1)

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with γ = 0, −1, −2, . . ., where (α)0 = 1,

(α)k = α(α + 1) . . . (α + k − 1)

for k ≥ 1.

We list a few formulas for easy reference (see [5, Chapter II]): 2F1

(α, β; γ; 1) =

Γ(γ)Γ(γ − α − β) , Γ(γ − α)Γ(γ − β)

Re (γ − α − β) > 0.

(2.2)

(α, β; γ; z) = (1 − z)γ−α−β 2F1 (γ − α, γ − β; γ; z). (2.3) d αβ (2.4) 2F1 (α, β; γ; z) = 2F1 (α + 1, β + 1; γ + 1; z). dz γ 1 Γ(γ) tλ−1 (1 − t)γ−λ−1 2F1 (α, β; λ; tz) dt, 2F1 (α, β; γ; z) = Γ(λ)Γ(γ − λ) 2F1

0

Re γ > Re λ > 0; | arg(1 − z)| < π; z = 1.

(2.5)

Lemma 2.1. Suppose Re δ > 0 and Re (λ + δ − α − β) > 0. Then 1

tλ−1 (1 − t)δ−1 2F1 (α, β; λ; t) dt =

0

Γ(λ)Γ(δ)Γ(λ + δ − α − β) . (2.6) Γ(λ + δ − α)Γ(λ + δ − β)

Proof. Note that, under the assumption of the lemma, both sides of (2.5) are continuous at z = 1. The lemma then follows by letting z → 1 and applying (2.2). Lemma 2.2. Let β > −1 and λ ∈ R. We have Bn

(1 − |y|2 )β dV (y) (1 − 2x · y + |x|2 |y|2 )2λ =

Γ(n/2 + 1)Γ(1 + β) n n (2.7) + 1; + 1 + β; |x|2 2F1 λ, λ − Γ(n/2 + 1 + β) 2 2

for all x ∈ Bn . Proof. We shall use the following formula from [8, Lemma 2.1] Sn−1

n n dσ(ζ) 2 λ, λ − , + 1; ; |x| = F 2 1 |x − ζ|2λ 2 2

x ∈ Bn ,

(2.8)

where σ denotes the normalized surface-area measure on the unit sphere Sn−1 .

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Integrating in polar coordinates and using (2.8), we get (1 − |y|2 )β dV (y) (1 − 2x · y + |x|2 |y|2 )2λ n B ⎛ ⎞ 1 dσ(ζ) ⎠ dr = n rn−1 (1 − r2 )β ⎝ |rx − ζ|2λ 0

=

n 2

1 0

Sn−1

n n rn/2−1 (1 − r)β 2F1 λ, λ − + 1; ; r|x|2 dr. 2 2

Now (2.7) follows from the formula (2.5).

The following result, usually called Schur’s test, is a very eﬀective tool in proving the Lp -boundedness of integral operators. See, for example, [10]. Lemma 2.3. Suppose that (X, μ) is a σ-finite measure space and K(x, y) is a nonnegative measurable function on X × X and T the associated integral operator T f (x) = K(x, y)f (y)dμ(y). X

Let 1 < p < ∞ and 1/p + 1/q = 1. If there exist a positive constant C and a positive measurable function u on X such that K(x, y)u(y)q dμ(y) ≤ Cu(x)q X

for almost every x in X and K(x, y)u(x)p dμ(x) ≤ Cu(y)p X

for almost every y in X, then T is bounded on Lp (X, dμ) with T ≤ C.

3. Proof of Theorem 1.1 We divide the argument into two propositions. In the ﬁrst one we show the upper bound, while the second contains a proof of the lower bound. Let ΩI , ΩII and ΩIII be as in Notation 1. For convenience, we set ⎧ Γ(n/2 + α + 1)Γ((α + 1)/p)Γ((α + 1)/q) ⎪ ⎪ , (p, α) ∈ ΩI , ⎪ ⎪ ⎪ Γ((n + α)/2)Γ(α/2 + 1)Γ(α + 1) ⎪ ⎪ ⎪ ⎨ Γ(n/2 + α + 1)Γ((α + 1)/q) , (p, α) ∈ ΩII , C(p, α) := ⎪ Γ(α + 1)Γ(n/2 + (α + 1)/q) ⎪ ⎪ ⎪ ⎪ ⎪ Γ(n/2 + α + 1)Γ((α + 1)/p) ⎪ ⎪ , (p, α) ∈ ΩIII . ⎩ Γ(α + 1)Γ(n/2 + (α + 1)/p)

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Proposition 3.1. With the notation above, we have Tα Lpα →Lpα ≤ C(p, α).

(3.1)

We begin by considering the function α+1 α n−α α+1 n α+1 − , + − 1; + ; t . (3.2) Φp,α (t) := 2F1 p 2 2 p 2 p Introducing the temporary notation Γ(n/2 + (α + 1)/p)Γ((α + 1)/q) A(p, α) = , Γ((n + α)/2)Γ(α/2 + 1) we have the following Lemma 3.2. (i) If (p, α) ∈ ΩI , then for 0 ≤ t < 1 1 ≤ Φp,α (t) ≤ A(p, α) and 1 ≤ Φq,α (t) ≤ A(q, α). (ii)

If (p, α) ∈ ΩII , then for 0 ≤ t < 1 A(q, α) ≤ Φq,α (t) ≤ 1 ≤ Φp,α (t) ≤ A(p, α).

(iii)

(3.3) (3.4)

If (p, α) ∈ ΩIII , then for 0 ≤ t < 1 A(p, α) ≤ Φp,α (t) ≤ 1 ≤ Φq,α (t) ≤ A(q, α).

(3.5)

Proof. We only prove the ﬁrst assertion, the other two follow the same lines. We ﬁrst note, by the definition and (2.2), that Φp,α (0) = 1,

Φp,α (1) = A(p, α).

(3.6)

Using (2.4) and (2.3), we have −1 α+1 α n α+1 dΦp,α n−α α+1 (t) = − + −1 + dt p 2 2 p 2 p α+1 α n−α α+1 n α+1 − + 1, + ; + + 1; t × 2F1 p 2 2 p 2 p −1 α+1 α n α+1 n−α α+1 = − + −1 + p 2 2 p 2 p n + α α n α + 1 α−(α+1)/p , + 1; + + 1; t . ×(1 − t) 2F1 2 2 2 p Note that the last hypergeometric function is positive on the interval [0, 1), since its Taylor coefﬁcients are all positive. Thus, α+1 α n−α α+1 − + −1 ≥ 0. Φp,α (t) is nondecreasing on [0, 1) ⇐⇒ p 2 2 p But it is easy to check that ⎧ α+1 α n−α α+1 ⎪ ⎪ − + − 1 ≥ 0, ⎪ ⎨ p 2 2 p (p, α) ∈ ΩI ⇐⇒ ⎪ α+1 α n−α α+1 ⎪ ⎪ − + − 1 ≥ 0. ⎩ q 2 2 q We conclude that when (p, α) ∈ ΩI , the functions Φp,α (t) and Φq,α (t) are both nondecreasing on [0, 1). Now (3.3) follows in view of (3.6).

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Proof of Proposition 3.1. By Schur’s test, with K(x, y) = (1 − 2x · y + |x|2 |y|2 )−(n+α)/2 and u(x) = (1 − |x|2 )−(α+1)/(pq) , we only need to show (1 − |y|2 )−(α+1)/p dVα (y) ≤ C(p, α)(1 − |x|2 )−(α+1)/p (3.7) (1 − 2x · y + |x|2 |y|2 )(n+α)/2 Bn

holds for all x ∈ B and (1 − |x|2 )−(α+1)/q dVα (x) ≤ C(p, α)(1 − |y|2 )−(α+1)/q (1 − 2x · y + |x|2 |y|2 )(n+α)/2

(3.8)

Bn

holds for all y ∈ B. We only prove (3.7), the proof of (3.8) being similar. Using (2.7), we obtain (1 − |y|2 )−(α+1)/p dVα (y) (1 − 2x · y + |x|2 |y|2 )(n+α)/2 Bn

(1 − |y|2 )α−(α+1)/p dV (y) (1 − 2x · y + |x|2 |y|2 )(n+α)/2 n B n+α α n α+1 Γ(n/2 + 1)Γ((α + 1)/q) 2 = cα , + 1; + ; |x| . 2F1 Γ(n/2 + (α + 1)/q) 2 2 2 q

= cα

By (2.3), and in view of (3.2), this equals Γ(n/2 + α + 1)Γ((α + 1)/q) Φq,α (|x|2 ) (1 − |x|2 )−(α+1)/p . Γ(α + 1)Γ(n/2 + (α + 1)/q) (3.7) now follows by applying Lemma 3.2, and the proposition is proved. Proposition 3.3. For α > −1 and 1 0, 2 (−α−1)/p f (x) = λ(1) , (1 − |x| ) 2 (−α−1)/q g (y) = λ(2) |y|2/p+(2α+2)/q , (1 − |y| )

where = λ(1) λ(2)

cα

=

Γ(n/2 + 1)Γ() Γ(n/2 + )

−1/p ,

(n/2)Γ(n/2 + (q/p) + α + 1)Γ() cα Γ(n/2 + (q/p) + α + 1 + )

It is easy to check that f p = g q = 1.

(3.9) −1/q .

(3.10)

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Applying Lemma 2.2 and integrating in polar coordinates, we obtain ⎛ ⎞ f (x)dV (x) α ⎝ ⎠ g (y)dVα (y) (1 − 2x · y + |x|2 |y|2 )(n+α)/2 Bn Bn (1) (2) 2 Γ(n/2 + 1)Γ(/p + (α + 1)/q) = λ λ cα (1 − |y|2 )(−α−1)/q+α Γ(n/2 + /p + (α + 1)/q) Bn n + α α n α+1 , + 1; + + ; |y|2 dV (y) ×|y|2/p+2(α+1)/q 2F1 2 2 2 p q 1 Γ(n/2 + 1)Γ(/p + (α + 1)/q) n = λ(1) λ(2) c2 rn/2+/p+(α+1)/q−1 2 α Γ(n/2 + /p + (α + 1)/q) 0 n+α α n α+1 , + 1; + + ; r dr ×(1 − r)(−α−1)/q+α 2F1 2 2 2 p q Γ(n/2 + 1)Γ(/p + (α + 1)/q)Γ(/q + (α + 1)/p)Γ() n . = λ(1) λ(2) c2 2 α Γ( + α/2 + 1)Γ( + (n + α)/2) In the last equality we have used (2.6). Having in mind that ⎫ ⎧ ⎛ ⎞ ⎬ ⎨ f (x)dVα (x) ⎝ ⎠ g(y)dVα (y) , Tα Lpα →Lpα = sup ⎭ 2 2 (n+α)/2 (1−2x · y+|x| |y| ) f Lp =1 ⎩ α n n gLq =1 α

B

B

we ﬁnally obtain Tα Lpα →Lpα ≥

Γ(n/2 + α + 1)Γ(/p + (α + 1)/q)Γ(/q + (α + 1)/p) Γ( + α/2 + 1)Γ( + (n + α)/2)Γ(α + 1) 1/p 1 Γ( + n/2) Γ((q/p + 1) + n/2 + α + 1) q × Γ(n/2) Γ((q/p) + n/2 + α + 1)

The proposition now follows by letting → 0+ .

Acknowledgements We thank Jiansong Deng for his help in drawing the ﬁgure.

References [1] Choe, B.R., Koo, H., Nam, K.: Optimal norm estimate of operators related to the harmonic bergman projection on the ball. Tohoku Math. J. (2) 62, 357–374 (2010) [2] Coifman, R.R., Rochberg, R.: Representation theorems for holomorphic and harmonic functions in Lp . Asterisque 77, 12–66 (1980) [3] Djrbashian, A.E., Shamoian, F.A.: Topics in the Theory of Apα Spaces. BSB B. G. Teubner Verlagsgesellschaft, Leipzig (1988)

566

C. Liu and L. Zhou

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[4] Dostani´c, M.: Norm of the Berezin transform on Lp spaces. J. d’Analyse Math. 104, 13–23 (2008) [5] Erd´elyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. I. McGraw-Hill, New York (1973) [6] Forelli, F., Rudin, W.: Projections on spaces of holomorphic functions in balls. Indiana Univ. Math. J. 24, 593–602 (1974) [7] Jevti´c, M., Pavlovi´c, M.: Harmonic Bergman functions on the unit ball in Rn . Acta Math. Hungr. 85, 81–96 (1999) [8] Liu, C., Peng, L.: Boundary regularity in the Dirichlet problem for the invariant Laplacian Δγ on the unit real ball. Proc. Am. Math. Soc. 132(11), 3259–3268 (2004) [9] Zhu, K.: A sharp norm estimate of the Bergman projection on Lp spaces. Contemp. Math. 404, 199–205 (2006) [10] Zhu, K.: Operator Theory in Function Spaces, 2nd edn. Mathematical Sureys and Monographs 138, American Mathematical Society, Providence (2007) Congwen Liu (B) and Lifang Zhou Department of Mathematics University of Science and Technology of China Hefei, 230026 Anhui People’s Republic of China e-mail: [email protected]; [email protected] Received: June 20, 2010. Revised: October 8, 2010.

Integr. Equ. Oper. Theory 69 (2011), 567–600 DOI 10.1007/s00020-010-1843-2 Published online November 25, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Construction of the Solution of the Inverse Spectral Problem for a System Depending Rationally on the Spectral Parameter, Borg–Marchenko-Type Theorem and Sine-Gordon Equation Alexander Sakhnovich Abstract. Weyl theory for a non-classical system depending rationally on the spectral parameter is treated. Borg–Marchenko-type uniqueness theorem is proved. The solution of the inverse problem is constructed. An application to sine-Gordon equation in laboratory coordinates is given. Mathematics Subject Classification (2010). Primary 34B07, 34A55; Secondary 34B20, 35Q51. Keywords. Weyl theory, inverse problem, operator identity, spectral parameter, rational dependence, sine-Gordon equation.

1. Introduction Canonical systems d w(x, λ) = iλJH(x)w(x, λ), dx

H ≥ 0,

J=

0 In , In 0

(1.1)

where H are 2n × 2n matrix functions and In is the n × n identity matrix, are classical objects of analysis, which include Dirac systems, matrix string equations and Schr¨ odinger equations as particular cases. For the literature on canonical systems see, for instance, the books [3,12,21,43] and various references in the papers [25,26,31–33,37]. We shall consider systems of the form A. Sakhnovich’s work was supported by the Austrian Science Fund (FWF) under Grant no. Y330.

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A. Sakhnovich

y (x, λ) = i

m

bk (λ − dk )−1 (βk (x)∗ βk (x)) y(x, λ),

IEOT bk = ±1,

x ∈ [0, ∞),

k=1

(1.2)

d , βk (x) = βk1 (x) βk2 (x) are C2 -valued differentiable vector where y = dx functions such that

sup βk (x) < ∞,

0<x<∞

βk (x)βk (x)∗ ≡ 1,

1 ≤ k ≤ m,

(1.3)

and C is the complex plane. We shall treat also a somewhat wider class of systems (1.2), such that the vector functions βk satisfy relations sup βk (x) < ∞

0<x
for all 0 < l < ∞,

βk (x)βk (x)∗ ≡ 1,

1 ≤ k ≤ m. (1.4)

Systems (1.2) generalize a subclass of canonical systems for the important case of several poles dp with respect to the spectral parameter λ. See, for instance, interesting papers [13,44,50] on systems with rational dependence on λ. A system of the form (1.2), where m = 2, can be treated as an auxiliary system for the sine-Gordon equation in laboratory coordinates (see Introduction in [29] and Sect. 6 here). System (1.2), where m = 2, was studied in [29]. Here we essentially develop the results from [29]. In particular, we consider the case m ≥ 2, construct the general solution of the inverse problem, and get applications to the sine-Gordon equation. We always assume that dk = dk = dp

for k = p,

1 ≤ k, p ≤ m,

(1.5)

where dk is the complex conjugate to dk . The 2 × 2 matrix function w(x, λ) satisfying (1.2) and the normalization condition, that is, w (x, λ) = i

m

bk (λ − dk )−1 βk (x)∗ βk (x)w(x, λ),

w(0, λ) = I2

(1.6)

k=1

is called the fundamental solution of (1.2). Different generalizations of the notion of a Weyl function are based on the asymptotics of the fundamental solution (see e.g. [4,5,11,18,22,27,28,34,36,48,50]). A kth Weyl–Titchmarsh function φk (λ) of the system (1.2) was introduced in [29] on the complex disk 1 bk DM := λ ∈ C : λ − dk − i < . M M Here, it is more convinient to change variables and use the functions ϕk (μ) = φk (λ(μ)), where λ = dk +

bk , 2μ

μ=

bk . 2(λ − dk )

(1.7)

In view of (1.7) the inequality μ < −M/4 is equivalent to the relation λ ∈ DM .

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Definition 1.1. A function ϕk (μ) is called a W Tk function of the system (1.2) with the properties (1.4) if and only if there exists an M = Mk > 0 such that ϕk is holomorphic on the half-plane μ < −M/4 and for all x ∈ [0, ∞)

ϕk (μ)

< ∞. w (x, λ(μ)) (1.8) sup

1 μ<−M/4 We shall use the notation ζ = μ, η = μ (μ = ζ + iη). Here is the real part, is the imaginary part, and the real axis will be denoted by R. When the conditions (1.3) hold, the analyticity of ϕk follows automatically from (1.8). The direct Weyl–Titchmarsh theory for m = 2 (two poles) was treated and the uniqueness of the solution of the inverse problem was proved in [29]. Here we shall construct this unique solution of the inverse spectral problem (m ≥ 2). Starting from the seminal work [24] by Krein structured operators were successfully used to solve inverse spectral problems. In the cases of Krein or self-adjoint Dirac-type systems these were operators with difference kernels [2,14,24,38,41,43]. A somewhat more complicated structured operators will appear in this article. An important series of papers by Gesztesy, Simon and coauthors on the high energy asymptotics of the Weyl functions and local Borg–Marchenkotype uniqueness results has initiated a growing interest in this important domain (see [6,8–10,16,17,19,26,38,46,47] and references therein). The Weyl–Titchmarsh theory for a non-self-adjoint case (the skew-self-adjoint Dirac type system) has been studied in [7,20,34,35] and the Borg–Marchenko-type results for this system have been published in [39]. Using the procedure to construct the solution of the inverse problem, we obtain here a Borg–Marchenko-type theorem for another interesting non-self-adjoint case, namely, for system (1.2). Finally, an application to the boundary value problem for sine-Gordon equation in laboratory coordinates (the case of bounded solution) will be given. Some preliminary results on the existence and uniqueness of the W Tk functions, representation of the fundamental solutions, and uniqueness of solution of the inverse problem are given in the next Sect. 2. The structured operators, which are necessary to construct the solution of the inverse problem, are studied in Sect. 3. The construction of the solution of the inverse problem and Borg–Marchenko-type theorem are contained in Sect. 4. The notion of the Weyl set is introduced in Sect. 5, and the sine-Gordon equation is treated in Sect. 6.

2. Preliminaries To make our paper self-contained we shall formulate in this section some results from [29]. We formulate them for m ≥ 2 as the proofs for the case m ≥ 2 are similar to the proofs for the case m = 2 treated in [29]. The existence and uniqueness of the Weyl functions are stated in Theorem 2.2 [29].

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Theorem 2.1. Let (1.2) be a system with coefﬁcients βk which are absolutely continuous vector functions satisfying (1.3) and the additional condition βk1 (0) = 0

(1 ≤ k ≤ m).

(2.1)

Then there exist unique W Tk -functions ϕk (1 ≤ k ≤ m) of the system (1.2). To connect W Tk -functions with the solutions from L2 , similar to the classical Weyl functions, let w be the fundamental solution of system (1.2). For ﬁxed k (which we sometimes omit in the notations), we deﬁne W (x, μ) = Wk (x, μ) := e−ixμ Q(x)w (x, λ(μ)) ,

(2.2)

where μ and λ are connected by the formula (1.7) and Q is the 2 × 2 matrix function given by βk1 (x) βk2 (x) Q(x) = Qk (x) := , x ∈ [0, ∞). (2.3) −βk2 (x) βk1 (x) Here βkj denote the entries of βk , that is, βk = [βk1 βk2 ]. By (1.3) and (2.3) the function Q(x) is unitary-valued, that is, Q(x)∗ Q(x) = Q(x)Q(x)∗ = I2 ,

x ∈ [0, ∞).

(2.4)

From the second relation in (1.3) and formulas (1.7) and (2.3) it follows that bk 1 0 i Qk (x)βk (x)∗ βk (x)Qk (x)∗ = 2iμ , λ = λ(μ). (2.5) 0 0 λ − dk By (1.6), (2.2), and (2.5) the matrix function W satisﬁes the system 1 0 W (x, μ) = (iμj + ξ(x, μ)) W (x, μ), j = , x ∈ [0, ∞), (2.6) 0 −1 where

⎛

ξ(x, μ) = Qk (x)Qk (x)∗ + iQk (x) ⎝

bp βp (x)∗ βp (x) p=k

λ − dp

⎞ ⎠ Qk (x)∗ .

(2.7)

From (2.2), (2.4), (2.6), and (2.7) it follows that W (x, μ)∗ W (x, μ) = Q(0)∗ Q(0) = I2 ,

W (0, μ) = Q(0).

(2.8)

The analog of Theorem 2.4 [29], which is formulated below, states that ϕ (μ) Wk (x, μ)[ k ] ∈ L22 . 1 Theorem 2.2. Let (1.2) be a system with coefﬁcients βk which are absolutely continuous vector functions satisfying (1.3) and (2.1). Then the W Tk -functions ϕk are unique functions such that for some Mk > 0 and all μ satisfying inequality μ < −Mk we have ∞ ϕk (μ) ∗ [ϕk (μ) 1]Wk (x, μ) Wk (x, μ) dx < ∞. (2.9) 1 0

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We shall need some details from the proof of Theorem 2.2 [29] (Theorem 2.1 here) in our further considerations. Notice that the Dirac-type system can be written down in the form (2.6), where ξ does not depend on μ. Similar to the Dirac-type system case [34], choose a value M > 0 such that 1 (2.10) ξ(x, μ) < M. sup 4 x∈[0,∞),μ<−M/4 By (2.6) and (2.10) one can see that for μ < −M/4 we have [29]: d R(x, μ) > 0, R(x, μ) := Q(0)W (x, μ)∗ jW (x, μ)Q(0)∗ . dx Now, put A(x, μ) = Ak (x, μ) = {Ajp (x, μ)}2j,p=1 := Q(0)Wk (x, μ)−1 .

(2.11)

(2.12)

According to the second relation in (2.8) and to (2.11) we have R(x, μ) > j or, equivalently, A(x, μ)∗ jA(x, μ) < j. Thus, the linear fractional transformation ψk (l, μ) =

A11 (l, μ)θ(μ) + A12 (l, μ) , A21 (l, μ)θ(μ) + A22 (l, μ)

|θ(μ)| ≤ 1,

l > 0,

μ < −

M , 4 (2.13)

where θ is a holomorphic parameter function, is well-deﬁned, and |ψk (l, μ)| < 1.

(2.14)

The class of functions ψk (l, ·) given by (2.13) is denoted by Nk (l). Using (2.11), it is shown in [29] that Nk (l1 ) ⊂ Nk (l2 ) for l1 > l2 . For each l > 0 and for each μ ( μ < − M 4 ) the values of ψk (l, μ) (ψk ∈ Nk (l)) can be parametrized ψk (l, μ) = ρ1 (l, μ)−1/2 Θ(l, μ)ρ2 (l, μ)−1/2 + ρ0 (l, μ), l ∈ (0, ∞), −1 −1 −1 R12 , ρ1 = R11 , ρ2 = R21 R11 R12 − R22 , ρ0 = −R11

(2.15) (2.16)

where |Θ(l, μ)| ≤ 1 and Rjp are the entries of R = {Rjp }2j,p=1 . The set of values of ψk (l, μ) (ψk ∈ Nk (l)) coincides with the disk on the right-hand side of (2.15), that is, the values of ψk form the so called Weyl disks. The functions ρ1 (l)−1/2 and ρ2 (l)−1/2 are decreasing, and for ρ1 we have M + μ → ∞, when l → ∞. (2.17) ρ1 (l) ≥ 1 − 2l 4 Therefore, the intersection of the Weyl disks in (2.15) is a Weyl point, that is, there is only one function ψk (μ), which belongs to all Nk (l): Nk (l) =: ψk (·). (2.18) l<∞

To study the asymtotics of ψk we need the representation of the fundamental solution W from Theorem 2.1 [29]: Theorem 2.3. Let βk (x) be absolutely continuous C2 -valued vector functions on the interval [0, l] (0 < l < ∞) satisfying relations sup βk (x) < ∞,

0<x
βk (x)βk (x)∗ ≡ 1,

1 ≤ k ≤ m.

(2.19)

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Then W (x, μ) (x ∈ [0, l]) of the form (2.2) admits a representation ⎛ ⎞ −1 b k Wk (x, μ)Q(0)∗ = eiμxj ⎝Dk (x) + Dp (x)⎠ μ− 2(dp − dk ) p=k

x eiμu Nk (x, u) du

+ −x

+

μ−

p=k −2

+ O(μ

bk 2(dp − dk )

−1 x eiμu Np (x, u) du −x

),

(2.20)

for μ = ζ + iη, η = 0, |ζ| → ∞, where Ds (1 ≤ s ≤ m) are continuous diagonal matrix functions, Dk∗ = Dk−1 , and m sup Ns (x, u) < ∞. (2.21) |u|≤x≤l

s=1

From the representation (2.20), after some calculations one gets lim ψk (μ) = 0 (2.22) μ→−∞

uniformly with respect to μ (see formula (2.29) in [29]). As (2.22) holds and such that βk1 (0) = 0, there is a value M ≤ −M . (2.23) |βk2 (0)ψk (μ) + βk1 (0)| > ε > 0, μ < −M 4 by the formula The W Tk -function is deﬁned in the domain μ < −M ϕk (μ) =

βk1 (0)ψk (μ) − βk2 (0) . βk2 (0)ψk (μ) + βk1 (0)

(2.24)

Finally, using (2.15) the estimate M (2.25) 4 was proved for the arbitrary ψk , ψ˘k ∈ Nk (l) (see Lemma 2.3 [29]). This estimate, in its turn, helped to prove that the solution of the inverse problem is unique (Theorem 3.1 [29]): |ψk (l, μ) − ψ˘k (l, μ)| ≤ 2 exp ((i(μ − μ) + M/2) l) ,

μ < −

Theorem 2.4. For given W Tk -functions ϕk (1 ≤ k ≤ m) there is at most one system (1.2) satisfying conditions (1.3) and (2.1). In this paper we shall construct also the solution of the following inverse problem. Definition 2.5. The inverse spectral problem for system (1.2), (1.4) is the problem to recover the system, that is, to recover the matrix function ⎤ ⎡ β1 (x) β(x) = ⎣ · · · ⎦, βm (x)

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such that the relations (1.4) and (2.1) hold and that the given functions ϕk are system’s W Tk -functions. We call β the potential of system (1.2). The corresponding uniqueness theorem was proved in [29] using Theorem 2.3. Theorem 2.6. For given functions ϕk (1 ≤ k ≤ m), which admit asymptotic representations M (2.26) ϕk (μ) = ck + O(μ−1 ), ck ∈ C, μ < − , μ → ∞, 4 there is at most one system (1.2) satisfying conditions (1.4) and (2.1) and such that the functions ϕk are the system’s W Tk -functions. Moreover, it follows from (2.26) that ck = −βk2 (0)/βk1 (0).

(2.27)

3. S-nodes Interesting developments of the classical results on the inverse problems were obtained using the notion of the S-node (see [41,43] and references therein). The non-self-adjoint systems were also recovered from their Weyl functions using S-nodes [34,36,39]. Here, we shall consider S-nodes corresponding to system (1.2). By {H1 , H2 } we denote the class of the linear bounded operators acting from the Hilbert space H1 into the Hilbert space H2 , diag means 2 2 diagonal matrix, x and A0 (l) ∈ {Lm (0, l), Lm (0, l)} is an integration operator: (A0 f )(x) = 0 f (u) du, x ∈ [0, l]. As in the expression A0 f above, sometimes we omit l in our notations. Later we shall also denote by A0 (l) the operator of integration in L2 (0, l). We denote the identity operator by I. It is always clear from the context, where I is acting. Now, we introduce three bounded operators A(l), S(l), and Π(l). Operators A(l) have simple structure and do not depend on the choice of βk (x): Af = A(l)f = Df + iBA0 (l)f, D = diag{d1 , d2 , . . . , dm },

A(l) ∈ {L2m (0, l), L2m (0, l)}, B = diag{b1 , b2 , . . . , bm }.

(3.1) (3.2)

Operator Π is an operator of multiplication by the m × 2 matrix function [Φ1 (x) Φ2 (x)]: g Π(l) 1 = g1 Φ1 (x) + g2 Φ2 (x), Π(l) ∈ {C2 , L2m (0, l)}, (3.3) g2 where the entries of the absolutely continuous column vector functions Φp (p = 1, 2) are denoted by Φkp , and we require Φk1 (x) ≡ 1,

Φk2 (x) ∈ L2 (0, l),

1 ≤ k ≤ m.

(3.4)

Later we shall recover Φ2 from the Weyl functions. Operator S ∈ {L2m (0, l), L2m (0, l)} is chosen so that it satisﬁes the operator identity AS − SA∗ = iΠΠ∗ . Therefore we say that A, S, and Π form an S-node.

(3.5)

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Proposition 3.1. Let operators A and Π be given by equalities (3.1)–(3.3), and let Π satisfy (3.4). Then the unique bounded operator S, which satisﬁes (3.5), has the form + Sf = B Df

l s(x, u)f (u) du,

s(x, u) = {skp (x, u)}m k,p=1 ,

(3.6)

0

where = Im + diag{|Φ12 (0)|2 , |Φ22 (0)|2 , . . . , |Φm2 (0)|2 }. D

(3.7)

The entries of s(x, u) on the main diagonal are deﬁned by the equalities x+u v+x−u v+u−x bk skk (x, u) = Φk2 Φk2 dv. (3.8) 2 2 2 |x−u|

The oﬀdiagonal (k = p) entries of s(x, u) are deﬁned by the equalities 1 (λI − Ak )−1 [1 Φk2 ] (x) (λI − Ap )−1 [1 Φp2 ] (u)∗ dλ, skp (x, u) = 2π Γ

(3.9) where Γ = {λ : |λ − dk | = ε > 0} is anticlockwise oriented, ε < |dk − dp |, I is the identity operator, Ak = dk I + ibk A0 , and we use A0 here to denote the integration operator in L2 (0, l). Proof. Write down S in the matrix form S = {Skp }m k,p=1 ,

Skp ∈ {L2 (0, l), L2 (0, l)}.

Then identity (3.5) takes the form g A0 Skk +Skk A∗0 = bk Πk Π∗k , Πk 1 = g1 +g2 Φk2 (x), g2

Πk ∈ {C2 , L2 (0, l)}; (3.10)

Ak Skp −

Skp A∗p

=

iΠk Π∗p ,

Ak = dk I + ibk A0 ,

k = p.

(3.11)

A∗0 T

= Q for Q of the form The bounded solution T of the equation T A0 + l Q = 0 q(x, t) · dt is constructed in Theorem 1.3 (p.11) [42]. After easy transformations we derive from this result the solution of (3.10) too. Namely, we have ⎛ ⎞ x+u l ∂ ⎜ bk d ⎟ (1 + Υ(v, x, u)) dv ⎠ f (u) du, Skk f = ⎝ 2 dx ∂u 0

|x−u|

Υ(v, x, u) := Φk2 ((v + x − u)/2) Φk2 ((v − x + u)/2).

(3.12)

From (3.12), taking into account the second relation in (3.4), we derive (3.6)– (3.8). To get (3.9) rewrite (3.11) in the form (λI − Ak )−1 Skp − Skp (λI − A∗p )−1 = i(λI − Ak )−1 Πk Π∗p (λI − A∗p )−1 , (3.13)

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and notice that σ(Ak ) = dk and σ(A∗p ) = dp , where dk = dp and σ means spectrum. So, we recover Skp by integration of the both parts of (3.13) in the small neighborhood of dk : 1 Skp = (λI − Ak )−1 Πk Π∗p (λI − A∗p )−1 dλ. (3.14) 2π Γ

In other words we have l skp (x, u) · du,

Skp =

(3.15)

0

where skp satisﬁes (3.9).

Remark 3.2. It is easy to check the explicit formula for the resolvent of Ak : (λI − Ak )−1 f (x) = (λ − dk )−1 f (x) x ibk (x − u) −2 + ibk (λ − dk ) exp f (u) du. (3.16) λ − dk 0

Denote by Pr (r ≤ l) the orthogonal projector from L2m (0, l) onto that is, let Pr ∈ {L2m (0, l), L2m (0, r)} and let (Pr f )(x) = f (x) for x ∈ (0, r). Notice that Pr A(l) = A(r)Pr . Therefore, we get L2m (0, r),

A(r)Pr SPr∗ − Pr SPr∗ A(r)∗ = iΠ(r)Π(r)∗ ,

(3.17)

by applying Pr from the left and Pr∗ from the right to the both parts of (3.5). Remark 3.3. According to (3.17), the unique operator S(r) satisfying the identity A(r)S(r) − S(r)A(r)∗ = iΠ(r)Π(r)∗ ,

r < l,

(3.18)

is given by the formula S(r) =

Pr SPr∗

+ = BD

r s(x, u) · du,

(3.19)

0

means the operator of multiplication where s does not depend on r and B D by the matrix B D. We shall need some properties of S(l). Proposition 3.4. The operator S constructed in Proposition 3.1 is self-adjoint, boundedly invertible and S −1 admits a triangular factorization x −1 ∗ − 12 S = V BV, (V f )(x) = D f (x) + V (x, u)f (u) du, (3.20) 0

where 21 Tr (r, u) V (r, u) = B D

(r ≥ u),

(3.21)

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and Tr is the matrix kernel of the integral operator −1

T (r) = S(r)

−1

= BD

r Tr (x, u) · du.

+

(3.22)

0

Proof. The operator S is self-adjoint as the unique solution of (3.5). (One could also prove it by (3.6)–(3.9).) The invertibility of S is proved by contradiction. Suppose that S is not invertible. In view of the special structure (3.6) of S, it means that S has an eigenvector f = 0, such that Sf = 0. Taking into account identity (3.5) and equality Sf = 0, we derive (f, ΠΠ∗ f )L2 = 0, where (·, ·)L2 denotes the scalar product in L2m (0, l). It is immediate that Π∗ f = 0. Apply the both parts of (3.5) to f and use the equalities Sf = 0, Π∗ f = 0 to obtain SA∗ f = 0. So, from Sf=0 it follows that SA∗ f = 0. In other words, we have SL=0 for the linear span L of the vectors (A∗ )k f (k ≥ 0). Therefore, we have dim L < ∞. As A∗ L ⊆ L and dim L < ∞, there is an eigenvector g of A∗ : A∗ g = cg, g = 0, and g ∈ L. Hence, by the definition of Ak in (3.11), cgk , gk = 0. This is there is an eigenvector of integration in L2 (0, l): A∗0 gk = impossible, and so we come to a contradiction, that is, S is invertible. In view of (3.18) and (3.19) the invertibility of the operators Pr SPr∗ (r < l) is proved quite similar to the invertibility of S. By (3.8) and (3.9) the function s(x, u) is continuous. Thus, the factorization conditions from “result 2” − 12 S D − 12 . Hence, the Section IV.7 [21] are fulﬁlled for S−1 , where S = B D factorization formula for S −1 in (3.20), the second relation in (3.20), and equality (3.21) follow. Remark 3.5. Let the conditions of Proposition 3.1 be fulﬁlled and put ⎤ ⎡ β1 (x) β(x) = ⎣ · · · ⎦ = (V Φ)(x), Φ(x) := [Φ1 (x) Φ2 (x)] (0 ≤ x ≤ l), (3.23) βm (x) where V is applied to Φ(x) columnwise. In other words, we have V Πg = β(x)g

(g ∈ C2 ).

(3.24)

Then the matrix functions βk satisfy the second relation in (1.4), that is, βk βk∗ ≡ 1. Indeed, from (3.5) and the ﬁrst equality in (3.20), it follows that V ∗ BV A − A∗ V ∗ BV = iV ∗ BV ΠΠ∗ V ∗ BV, V AV

−1

∗ −1

B − B(V )

∗

∗

∗

i.e., ∗

(3.25)

A V = iV ΠΠ V .

By the definitions (3.1) and (3.20) of A and V , the operator V AV −1 B is lower x triangular and has the form DB + 0 γ(x, u)·du. The operator B(V ∗ )−1 A∗ V ∗ is upper triangular. Hence, one can derive the kernel γ of the integral term of V AV −1 B from (3.24) and (3.26). We get (V AV

−1

x f )(x) = Df (x) + iβ(x) 0

β(u)∗ Bf (u) du.

(3.26)

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By (3.26) it is immediate that x V A = DV + iβ(x)

β(u)∗ BV · du.

(3.27)

0

Rewrite (3.27) as the equality of the kernels of the corresponding integral operators: x − 12 iD B + V (x, u)D + i V (x, v)B dv u

⎛

= DV (x, u) + iβ(x) ⎝β(u)∗ B D

− 12

x +

⎞ β(v)∗ BV (v, u) dv ⎠.

(3.28)

u

When x = u, the equality of the main diagonals of the both sides of (3.28) implies βk βk∗ ≡ 1. Now, introduce the transfer matrix functions in the Lev Sakhnovich form [40,41,43]: −1

wA (r, λ) = I2 − iΠ(r)∗ S(r)−1 (A(r) − λI)

Π(r).

(3.29)

The following lemma is essential for the solution of the inverse problem. Lemma 3.6. Let the conditions of Proposition 3.1 be fulﬁlled, and let the S-node be given by the formulas (3.1)–(3.4) and (3.6)–(3.9). Then we have d wA (r, λ) = iβ(r)∗ B(λIm − D)−1 β(r)wA (r, λ). dr

(3.30)

Proof. First, introduce several notations. Let P1 (r, δ) and P2 (r, δ) denote ortoprojectors from L2m (0, r + δ) on L2m (0, r) and L2m (r, r + δ), respectively. That is, let P1 (r, δ)f ∈ L2m (0, r), P2 (r, δ)f ∈ L2m (r, r + δ), and (P1 (r, δ)f ) (x) = f (x), 0 < x < r;

(P2 (r, δ)f ) (x) = f (x), r < x < r + δ. (3.31)

For operators K acting in L2m (0, r + δ) we put Kjp := Pj (r, δ)KPp (r, δ)∗ (j, p = 1, 2). In particular, we use notations T (u) := S(u)−1 ,

T22 := P2 (r, δ)T (r + δ)P2 (r, δ)∗ .

(3.32)

According to [40] (see also Theorem 2.1 from Chapter 1 in [43]) we have wA (r + δ, λ) − wA (r, λ) = −iΠ(r + δ)∗ S(r + δ)−1 (A22 − λI)−1 −1 × T22 P2 (r, δ)S(r + δ)−1 Π(r + δ)wA (r, λ). (3.33)

Using formula (3.23) and the ﬁrst equality in (3.20), we rewrite (3.33) in the form r+δ wA (r + δ, λ) − wA (r, λ) = i β(x)∗ (Zδ β)(x) dx wA (r, λ), (3.34) r

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where the operator Zδ ∈ {L2m (r, r + δ), L2m (r, r + δ)} is given by the formula −1 P2 (r, δ)V ∗ B. Zδ = BV (λI − A22 )−1 T22

(3.35)

It is easy to see that Zδ − B(λI − D)−1 is an integral operator, and we shall show below that the kernel of this operator is bounded: −1

Zδ = B(λI − D)

r+δ + zδ (x, u) · du,

sup zδ (x, u) < ∞,

(3.36)

r

where r ≤ x, u ≤ r + δ ≤ l. For that purpose notice that the kernel s(x, u) is continuous, and according to Section IV.7 [21] the kernel Tr (x, u) is continuous with respect to r, x, and u (x, u ≤ r ≤ l) too. Hence, the functions V (x, u) and β(x) are continuous, and we also have sup s(x, u) < ∞,

sup Tr (x, u) < ∞.

x,u≤l

(3.37)

x,u≤r≤l

From the definition (3.32) we get −1 −1 T22 = S22 − S21 S11 S12 ,

(3.38)

−1 and so, by (3.37) the kernel of the integral term of T22 is bounded. In view −1 of (3.16), for the k-th entry of (λI − A22 ) f we have (λI − A22 )−1 f k (x) = (λ − dk )−1 f (x) x ibk (x − u) −2 + ibk (λ − dk ) exp f (u) du, (3.39) λ − dk r

and the kernel of (λI − A22 )−1 − (λI − D)−1 is bounded for any ﬁxed λ (λ = dk ) too. Therefore, for Zδ given by (3.35) the formula (3.36) is true. Recall that β is continuous. Thus, from (3.34) and (3.36) it follows that lim δ −1 (wA (r + δ, λ) − wA (r, λ)) = iβ(r)∗ B(λIm − D)−1 β(r)wA (r, λ).

δ→+0

(3.40) Quite similar one can prove that lim δ −1 (wA (r, λ) − wA (r − δ, λ)) = iβ(r)∗ B(λIm − D)−1 β(r)wA (r, λ).

δ→+0

(3.41) By (3.40) and (3.41) equality (3.30) holds.

Remark 3.7. It is easy to see that definition (3.29) implies limr→0 wA (r, λ) = I2 . Hence, the matrix function wA , which is treated in Lemma 3.6, is the fundamental solution of the system (1.2) corresponding to β(x) = (V Φ)(x).

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4. Inverse Problems: Construction of the Solution According to formulas (2.14), (2.23), and (2.24), one can choose a sufﬁciently large value M > 0, so that all the W Tk -functions (1 ≤ k ≤ m) are welldeﬁned and bounded in the half-plane μ < −M/4. Definition 4.1. The bounded matrix function ϕ(μ) = col[ϕ1 (μ), sup μ<−M/4

ϕ2 (μ), . . . , ϕ(μ) < ∞,

ϕm (μ)],

(4.1) (4.2)

where col means column, is called the Weyl function of system (1.2). One of our main results is the next theorem. Theorem 4.2. System (1.2) satisfying conditions (1.3) and (2.1) is uniquely recovered from its Weyl function. To recover (1.2) on an arbitrary ﬁxed interval [0, l] we use the following procedure. First, introduce the column vector function Φ2 (x) (0 < x < ∞) by the Fourier transform: ∞ M i μ! dζ μ = ζ + iη, η < − Φ2 (x) = μ−1 eiμx ϕ . (4.3) 2π 2 2 −∞

That is, we deﬁne Φ2 on (0, l) and (0, ∞) via norm limit l.i.m. in Lm 2 : a i −ηx μ! Φ2 (x) = e dζ. (4.4) l.i.m.a→∞ μ−1 eiζx ϕ 2π 2 −a

Here the right-hand side of (4.4) equals 0 for x < 0. Next, substitute Φ2 (x) into formulas (3.7)–(3.9) to introduce operator S ∈ {L2m (0, l), L2m (0, l)} of the form (3.6). Apply formulas (3.20)–(3.22) to recover operator V from the operators S(r) = Pr SPr∗ (r ≤ l). Then, the matrix functions βk (x) and therefore, system (1.2) is recovered by the formula (3.23), that is, β(x) = V [Φ1 (x) Φ2 (x)], where Φ1 is given by the ﬁrst relation in (3.4). Proof. Step 1. Let system (1.2) satisfy conditions (1.3) and (2.1) and let ϕ be the Weyl function of system (1.2). According to the inequality (4.2), the vector function Φ2 is well-deﬁned by (4.3) and does not depend on the choice M of η < − M 4 . So, we can ﬁx some η < − 4 . To show that Φ2 is absolutely continuous, introduce functions βk1 (0)ψ"k (μ) − βk2 (0) A12 (l, μ) ∈ Nk (l), ϕ . (4.5) "k (μ) = ψ"k (μ) := A22 (l, μ) βk2 (0)ψ"k (μ) + βk1 (0) By (2.8) and (2.12) we have A(l, μ) = Qk (0)Wk (l, μ)∗ . Hence, by Theorem 2.3 the matrix function ψ"k admits representation ψ"k (μ) = dk22 (l)

!−1 l −l

e−iμ(l+u) Nk21 (l, u) du + g1 (μ),

(4.6)

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where dk22 is the corresponding entry of the diagonal unitary matrix Dk , Nk21 is an entry of the bounded matrix function Nk , and the function g1 (ζ + iη) ∈ L1 (−∞, ∞) ∩ L∞ (−∞, ∞) with respect to the variable ζ. (Recall that L∞ is the space of bounded functions.) Taking into account (2.23) and (2.25), without loss of generality we can assume |βk2 (0)ψ"k (μ) + βk1 (0)| > ε > 0,

μ < −

M . 4

(4.7)

In view of relations (4.5)–(4.7) we get ϕ "k (μ) = −

βk2 (0) |βk1 (0)|2 − |βk2 (0)|2 " + ψk (μ) + g2 (μ), βk1 (0) βk1 (0)2

(4.8)

where g2 (ζ + iη) ∈ L1 (−∞, ∞) ∩ L∞ (−∞, ∞). Finally, notice that uniformly for x on the intervals [δ, l] we have i lim a→∞ 2π

a

μ−1 eiμx dζ = −1

μ = ζ + iη, η < −

M 2

.

(4.9)

M μ = ζ + iη, η < − , 2

(4.10)

−a

From (4.6), (4.8), and (4.9) it follows that the function " k2 (x) := i Φ 2π

∞

μ−1 eiμx ϕ "k

−∞

μ! dζ 2

is absolutely continuous and " k2 (x)| < ∞. sup |Φ

(4.11)

" k2 (x) = Φk2 (x) for 0 ≤ x ≤ l, Φ

(4.12)

0<x
Next, we shall show that

where Φk2 denotes the k-th entry of the Cm -valued vector function Φ2 . The proof of (4.12) requires some considerations. According to the definitions "k , respectively, we have (2.24) and (4.5) of ϕk and ϕ ! |βk1 (0)|2 + |βk2 (0)|2 ψk (μ) − ψ"k (μ) ! ! . (4.13) ϕk (μ) − ϕ "k (μ) = βk2 (0)ψk (μ) + βk1 (0) βk2 (0)ψ"k (μ) + βk1 (0) >0 In view of (2.23), (2.25), and (4.7), formula (4.13) implies for some C, M that "k (μ)| < C(l)e2ηl , |ϕk (μ) − ϕ

(l) < − η < −M

M 4

(μ = ζ + iη).

(4.14)

Taking into account (4.2) and (4.14), one can see that functions μ−1 ϕk μ2 (l) and "k μ2 are holomorphic in the half-plane η = μ < −2M and μ−1 ϕ (l). belong to L2 (−∞, ∞) with respect to ζ = μ for any ﬁxed η < −2M

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Therefore, according to Theorem V [30] these functions admit Fourier representations a μ! −1 = l.i.m.a→∞ e−iμx e2xM f (x) dx, f ∈ L2 (0, ∞), (4.15) μ ϕk 2 0

μ−1 ϕ "k

μ! 2

a = l.i.m.a→∞

e−iμx e2xM f"(x) dx,

f" ∈ L2 (0, ∞).

(4.16)

0

Using Plansherel’s theorem and formulas (4.3) and(4.10), we express f (x) and " k2 (x), respectively, and obtain: f"(x) in (4.15) and (4.16) via Φk2 (x) and Φ ∞ μ! i ϕk = e−iμx Φk2 (x) dx, e−2xM Φk2 (x) ∈ L2 (0, ∞), (4.17) μ 2 0

μ! i ϕ "k = μ 2

∞

" k2 (x) dx, e−iμx Φ

" k2 (x) ∈ L2 (0, ∞). e−2xM Φ

(4.18)

0

Consider now the entire matrix function l ! iμl " k2 (x) dx Y0 (μ) = e e−iμx Φk2 (x) − Φ 0

l =

! " k2 (x) dx. eiμ(l−x) Φk2 (x) − Φ

(4.19)

0

we have From (4.17)–(4.19) it follows that for μ = ζ + iη, η < −2M ∞ ! " k2 (x)| dx |Y0 (μ)| ≤ eη(x−l)) |Φk2 (x)| + |Φ l −ηl

+e

i μ! μ !! . "k μ ϕk 2 − ϕ 2

(4.20)

" k2 ∈ L2 (0, ∞). Hence, from (4.14) and (4.20) we derive Recall that Φk2 , Φ sup −ε η<−2M

|Y0 (μ)| < ∞

(ε > 0),

lim |Y0 (μ)| = 0 (μ = ζ + iη). (4.21)

η→−∞

By the definition (4.19) and by the ﬁrst relation in (4.21), the entire functon Y0 is bounded in C. So, in view of the second relation in (4.21) we have Y0 = 0, and equality (4.12) is immediate. From (4.11) and (4.12) we get the second relation in (3.4). Therefore, Π satisﬁes the conditions of Proposition 3.1, and the S-node and the operator V in our theorem are well-deﬁned. As the conditions of Propositions 3.1 and 3.4 ˘ are satisﬁed, the matrix function β(x) = (V Φ)(x) is well-deﬁned and unique. To prove the theorem means to prove the equalities β˘k (x)∗ β˘k (x) ≡ βk (x)∗ βk (x)

(1 ≤ k ≤ m).

(4.22)

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Step 2. By (4.10), taking into account (4.6), (4.8), and (4.9), we have " k2 (x) eηx Φ

1 = − l.i.m.a→∞ 2π

a iζx

e

−a

μ ! βk2 (0) + ϕ "k dζ, 2 βk1 (0)

" (x) ∈ L2 (0, ∞) ∩ L∞ (0, ∞), eηx Φ k2 Introduce a 2 × 2 matrix function: Ω2 (μ)] = Q(0)∗

Ω(μ) = [Ω1 (μ)

η<−

M . 2

exp{−2irμ} ψ"k (μ) . 0 1

(4.23) (4.24)

(4.25)

First, let us show that for r ≤ l and for sufﬁciently large M the inequalities sup μ<−M/4

wA (r, λ(μ)) Ω1 (μ) < ∞,

sup μ<−M/4

wA (r, λ(μ)) Ω2 (μ) < ∞ (4.26)

are true. Indeed, by Lemma 3.6 wA satisﬁes system (1.2), where we substitute β˘k instead of βk . Hence, for sufﬁciently large M we get d ((exp{−2ir(μ − μ)})wA (r, λ)∗ wA (r, λ)) = exp{−2ir(μ − μ)} dr ! ! (4.27) × wA (r, λ)∗ 4η I2 − β˘k (r)∗ β˘k (r) + g3 (r, λ) wA (r, λ), λ = λ(μ),

(η = μ < −M/4),

g3 (r, λ) < CI2

for some C > 0, which does not depend on r. As η(I2 − β˘k (r)∗ β˘k (r)) ≤ 0, formula (4.27) implies sup μ<−M/4

exp{−2irμ}wA (r, λ) < ∞,

(4.28)

that is, the ﬁrst inequality in (4.26) holds. Next, let us prove that ϕ "k (μ) sup wA (r, λ) < ∞ (r ≤ l). 1 μ<−M/4

(4.29)

For that purpose consider wA again and notice that from the operator identity (3.18) and definition (3.29) it follows [41] that wA (r, λ)∗ wA (r, λ) = I2 − i(λ − λ)Π(r)∗ A(r)∗ −1 −1 −λI S(r)−1 (A(r) − λI) Π(r). (4.30) By (3.4) and (3.16) we derive the equality: −1

(Ak (r) − λI)

[Φk1 ⎡

Φk2 ]

= −2bk μe2iμx ⎣1 Φk2 (0) +

x 0

⎤ e−2iμu Φk2 (u) du⎦,

(4.31)

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which helps to estimate the right-hand side of (4.30). According to (4.18) and (4.24) the equality ∞ " (u) du " ϕ "k (μ) = −Φk2 (0) − e−2iμu Φ (4.32) k2 0

is true. In view of (4.12), (4.31), and (4.32) we have ∞ ϕ "k (μ) −1 2iμx " (u) du. (Ak (r) − λI) Πk e−2iμu Φ = 2bk μe k2 1

(4.33)

x

By (4.24) and (4.33) for sufﬁciently large M we get

"k (μ)

(Ak (r) − λI)−1 Πk ϕ

≤ C(r)|μ/η|.

1

(4.34)

Moreover, for sufﬁciently large M the functions ϕ "k (μ) and resolvents −1 (Ap (r) − λI) (p = k) are uniformly bounded in the domain μ < −M/4. Therefore, the inequalities (4.34) imply

"k (μ)

≤ C1 (r)|μ/η|.

(A(r) − λI)−1 Π ϕ (4.35)

1 bk Notice that for λ = dk + 2μ we have |λ − λ| = −η|μ|−2 . Hence, from (4.30) and (4.35) follows (4.29). By (4.5) we have !−1 ϕ "k (μ) ψ"k (μ) βk2 (0)ψ"k (μ) + βk1 (0) . (4.36) = Q(0)∗ 1 1

Substitute (4.36) into (4.29) and use (2.14) to derive the second relation in (4.26). Thus, (4.26) is valid. Now, let us show that sup μ<−M/4

w (r, λ(μ)) Ω(μ) < ∞.

(4.37)

Here, the inequality sup μ<−M/4

exp{−2irμ}w(r, λ) < ∞,

λ = λ(μ)

(4.38)

is proved similar to (4.28). As ϕ of the form (4.1) is the Weyl function, so ϕk satisﬁes (1.8). The inequality

ϕ "k (μ)

< ∞ (r ≤ l)

(4.39) sup w(r, λ)

1 μ<−M/4

follows for sufﬁciently large M from (1.8), (4.14), and (4.38). In view of (2.14) and (4.36), formula (4.39) yields sup μ<−M/4

w(r, λ)Ω2 (μ) < ∞.

By (4.38) and (4.40) inequality (4.37) is valid.

(4.40)

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Step 3. Here, using (4.26) and (4.37) we shall show that sup |μ|>M/4

wA (r, λ)w(r, λ)−1 < ∞.

(4.41)

First, taking into account (2.2) and (2.20), we obtain w(r, λ)Q(0)∗ = eirμ Q(r)∗ W (r, μ)Q(0)∗ = Q(r)∗ Dk (r)diag{e2irμ , 1} + g4 (μ), (4.42) where the 2 × 2 matrix function g4 (ζ + iη) belongs to L22×2 (−∞, ∞) for each ﬁxed η < −M/4, that is, the entries of g4 belong L2 . According to (2.14) and " + iη), where η < −M/4, is bounded and belongs to (4.6) the function ψ(ζ L2 (−∞, ∞) with respect to ζ. Therefore, formulas (4.25) and (4.42) imply that for η < −M/4 we have w(r, λ)Ω(μ) = Q(r)∗ Dk (r) + g5 (μ),

g5 (ζ + iη) ∈ L22×2 (−∞, ∞).

(4.43)

After some evident change of variables in Theorem VIII from [30], by (4.37) and (4.43) we can apply it to w(r, λ)Ω(μ) − Q(r)∗ Dk (r). That is, we derive for η < −M/4 the representation ∞

∗

w(r, λ)Ω(μ) = Q(r) Dk (r) +

e−ixμ f (x) dx,

eηx f (x) ∈ L22×2 (−∞, ∞).

0

(4.44) In view of (4.26) and (4.44) for sufﬁciently large M we have sup μ<−M/4

wA (r, λ)w(r, λ)−1 < ∞.

(4.45)

According to (1.6) and Lemma 3.6 the equalities w(r, λ)∗ = w(r, λ)−1 ,

wA (r, λ)∗ = wA (r, λ)−1

(4.46)

1 = det wA (r, λ). det wA (r, λ)∗

(4.47)

hold. In particular, we have 1 = det w(r, λ), det w(r, λ)∗

Recall also that w and wA are 2 × 2 matrices, and so (4.46) and (4.47) yield −1 = (det w(r, λ)) jJw(r, λ)Jj w(r, λ) = w(r, λ)∗

(4.48)

wA (r, λ) = (det wA (r, λ)) jJwA (r, λ)Jj.

(4.49)

Here we put n = 1 in the definition of J, that is, 0 1 1 0 J= , j= . 1 0 0 −1

(4.50)

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From (1.3) and (1.6) it follows that ⎛ r ⎞ m det w(r, λ) = exp ⎝i Tr bk (λ − dk )−1 βk (x)∗ βk (x) dx⎠

k=1

0

= exp ir

m

−1

bk (λ − dk )

,

(4.51)

k=1

where Tr means trend. In a similar way from Lemma 3.6 and Remark 3.5 we get m −1 det wA (r, λ) = exp ir bk (λ − dk ) . (4.52) k=1

Finally, substitute (4.48)–(4.52) into (4.45) to obtain sup μ<−M/4

wA (r, λ)w(r, λ)−1 < ∞.

(4.53)

Using (1.7) one can see that inequalities (4.45) and (4.53) imply (4.41). Step 4. Taking into account (1.6), (1.7), (4.46), and Lemma 3.6, it is easy to see that for K(r, μ) := wA (r, λ(μ)) w (r, λ(μ))

−1

(4.54)

we have lim

R→∞

1 ln ln sup K(r, μ) ≤ 1. ln R |μ|=R

(4.55)

In view of (4.41) and (4.55), we can apply to K(r, μ) the Phragmen-Lindel¨ of theorem. Thus, we see that K(r, μ) is uniformly bounded for sufﬁciently large |ζ| on the strip |η| ≤ M/4. Using this and inequality (4.41), we have sup K(r, μ) < ∞

(4.56)

|μ|>M1

for some M1 > 0. Let us switch to the variable λ. From (1.7), (4.54), and (4.56) we derive that λ) := wA (r, λ)w(r, λ)−1 K(r, is bounded in the deleted neighborhood of λ = dk . That is, by Theorem λ) is holomorphic at λ = dk . Since 11.4 from [45] the matrix function K(r, λ) is an entire function with respect to the k (1 ≤ k ≤ m) is arbitrary, K(r, variable λ. Moreover, it easy to see that lim wA (r, λ) = lim w(r, λ) = lim wA (r, λ)w(r, λ)−1 = I2 .

λ→∞

λ→∞

λ→∞

(4.57)

By Liouville’s theorem, it follows from (4.57) that the entire function wA w−1 is constant, that is, wA (r, λ) ≡ w(r, λ) Identity (4.58) implies (4.22).

(0 ≤ r ≤ l).

(4.58)

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The construction of the solution of the inverse problem, which is described in Definition 2.5, is given below. Theorem 4.3. Let a vector function ϕ(μ) be holomorphic in the half-plane μ < − M 4 and admit there an asymptotic representation 1 α1 M +O (4.59) , μ < − , μ → ∞. ϕ(μ) = α0 + 2 μ μ 4 Then ϕ is a Weyl function of the unique system (1.2) satisfying conditions (1.4) and (2.1). To recover (1.2) on an arbitrary ﬁxed interval [0, l] one can use the same procedure as in Theorem 4.2. First, introduce the column vector function Φ2 (x) (0 < x < ∞) by the Fourier transform (4.3) or, equivalently, (4.4). Here the right-hand side of (4.4) equals 0 for x < 0. Next, substitute Φ2 (x) into formulas (3.7)–(3.9) to introduce operator S ∈ {L2m (0, l), L2m (0, l)} of the form (3.6). Apply formulas (3.20)–(3.22) to recover operator V from the operators S(r) = Pr SPr∗ (r ≤ l). Then, the matrix functions βk (x) and therefore, system (1.2) is recovered using the formula (3.23), that is, β(x) = V [Φ1 (x) Φ2 (x)], where Φ1 is given by the ﬁrst relation in (3.4). Proof. Step 1. Let us show that the potential β = V [Φ1 Φ2 ] is differentiable and relations (1.4) and (2.1) hold. According to (4.9) and to the asymptotic relation (4.59), the vector function Φ2 is well-deﬁned by (4.3) and does not depend on the choice of M η < −M 2 . So, we can ﬁx some η < − 2 . Moreover, one can easily see that Φ2 (x) is twice differentiable and eηx Φ2 (x) ∈ L2m (0, ∞). Therefore, the matrix function s(x, u) given by (3.8) and (3.9) is continuous. So, Tr (x, u) given by (3.22) is continuous with respect to r, x, and u [21]. The product of the right-hand sides of the ﬁrst equality in (3.6) (for l = r) and of the formula (3.22) equals Im , which can be written down as the equality −1 + B DT r (x, u) + s(x, u)B D

r s(x, t)Tr (t, u) dt = 0

(x, u ≤ r)

(4.60)

0

for the kernels of the integral operators. In view of the continuity of s and Tr , equality (4.60) is true pointwise. Changing the order of multiplication of the right-hand sides of (3.6) (for l = r) and (3.22) we get also −1 s(x, u) + Tr (x, u)B D + BD

r Tr (x, t)s(t, u) dt = 0 (x, u ≤ r). (4.61) 0

From (3.20), (3.21), and (3.23) it follows that − 12 [Φ1 (r) β(r) = D

12 Φ2 (r)] + B D

r Tr (r, u)[Φ1 (u) 0

Φ2 (u)] du.

(4.62)

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Using (4.60)–(4.62), one can show that − 2 [0 β (r) = D 1

2 (Y1 + Y2 + Y3 ), Φ2 (r)] + B D 1

(4.63)

where Y1 = Tr (r, r)Φ(r), Φ(r) = [Φ1 (r) Φ2 (r)], r ∗ S(r)−1 s(x, r) Φ(x) dx, Y2 = −Tr (r, r) −1 Y3 = −B D

r

0

S(r)−1

0

∗ ∂ s(x, u) Φ(x) dx, ∂u u=r

(4.64) (4.65)

(4.66)

−1

and S(r) is applied to the matrix functions columnwise. Indeed, let us prove that r d Tr (r, u)Φ(u) du = Y1 + Y2 + Y3 . dr

(4.67)

0

It is immediate that for δ > 0 we have r+δ r Tr+δ (r + δ, u)Φ(u) du − Tr (r, u)Φ(u) du 0

0 r+δ = Tr+δ (r + δ, u)Φ(u) du r

r (Tr+δ (r + δ, u) − Tr+δ (r, u)) Φ(u) du

+ 0

r (Tr+δ (r, u) − Tr (r, u)) Φ(u) du.

+

(4.68)

0

As Tr (x, u) is continuous, we get lim δ

−1

δ→0

r+δ Tr+δ (r + δ, u)Φ(u) du = Y1 .

(4.69)

r

Substitute r + δ instead of r into (4.60) to obtain the equality r+δ (x, u) + −1 + B DT s(x, u)B D

r+δ s(x, t)Tr+δ (t, u) dt = 0, 0

which can be rewritten as −1 − S(r)Tr+δ (x, u) = −s(x, u)B D

r+δ s(x, t)Tr+δ (t, u) dt r

(4.70)

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for all ﬁxed values of u (u ≤ r). According to (4.70) we have ⎛ −1 Tr+δ (x, r + δ) − Tr+δ (x, r) = −S(r)−1 ⎝ (s(x, r + δ) − s(x, r)) B D ⎞ r+δ + s(x, t) (Tr+δ (t, r + δ) − Tr+δ (t, r)) dt⎠.

(4.71)

r

Recall that Φ is twice differentiable. Hence, for u ≥ r and x ≤ r the matrix ∂ function s(x, u) is differentiable with respect to u and ∂u s(x, u) is continuous with respect to x and u. Therefore, formula (4.71) implies −1 −1 ∂ −1 L2 = 0. s(x, u) BD lim δ (Tr+δ (x, r + δ)−Tr+δ (x, r)) + S(r) δ→0 ∂u u=r (4.72) Here X(x)L2 denotes the maximum of the norms in L2m (0, r) of the columns of the matrix function X. Take into account that T = T ∗ , and thus Tr (x, u) = Tr (u, x)∗ . So, it follows from (4.66) and (4.72) that lim δ

−1

δ→0

r (Tr+δ (r + δ, x) − Tr+δ (r, x)) Φ(x) dx = Y3 .

(4.73)

0

In a similar to (4.71) way, by (4.60) and (4.70) we have −1

Tr+δ (x, r) − Tr (x, r) = −S(r)

r+δ s(x, t)Tr+δ (t, r) dt.

(4.74)

r

From (4.74) it follows that lim δ −1

δ→0

r (Tr+δ (r, x) − Tr (r, x)) Φ(x) dx = Y2 .

(4.75)

0

According to (4.68), (4.69), (4.73), and (4.75) we get ⎛ r+δ ⎞ r −1 ⎝ Tr+δ (r + δ, u)Φ(u) du − Tr (r, u)Φ(u) du⎠ = Y1 + Y2 + Y3 . lim δ δ→0

0

0

(4.76) Using (4.61), in a similar way one obtains ⎛ r ⎞ r−δ Tr−δ (r − δ, u)Φ(u) du⎠ = Y1 + Y2 + Y3 . lim δ −1 ⎝ Tr (r, u)Φ(u) du − δ→0

0

0

(4.77) From (4.76) and (4.77), the equality (4.67) is immediate. Finally, by (4.62) and (4.67) we have (4.63).

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In view of (4.63)–(4.66), β is differentiable and the ﬁrst relations in (1.4) are valid. According to Remark 3.5 the second relations in (1.4) are fulﬁlled − 12 Φ(0) and so, taking into account (3.4), we too. By (4.62) we have β(0) = D − 1 get βk1 (0) = 1 + |Φk2 (0)|2 2 = 0, that is, the inequalities (2.1) are true. Step 2. Now, let us show that the matrix function β, which is constructed using (3.23), provides the solution of the inverse problem. In view of Definition 1.1, Lemma 3.6, and Remark 3.7 it will sufﬁce to show that for r ≤ l and 1 ≤ k ≤ m we have

ϕk (μ)

< ∞, λ = dk + bk . w (4.78) (r, λ) sup

A

1 2μ μ<−M/4 Taking into account (4.59) one can see that the vector function μ−1 ϕ μ2 is 2 holomorphic in the half-plane η = μ < − M 2 and belongs to Lm (−∞, ∞) M with respect to ζ = μ for any ﬁxed η < − 2 . Therefore, similar to the proof of Theorem 4.2 we apply Theorem V from [30] and derive: a 1 μ! −1 = l.i.m.a→∞ e−iμx e 2 xM f (x) dx, f ∈ L2m (0, ∞). (4.79) μ ϕ 2 0

Using Plansherel’s theorem and formula (4.3), we express f (x) in (4.79) via Φ2 (x) and obtain: ∞ 1 i μ! ϕ = e−iμx Φ2 (x) dx, e− 2 xM Φ2 (x) ∈ L2m (0, ∞). (4.80) μ 2 0

It follows also that the right-hand side of (4.4) equals 0 for x < 0, as stated in the theorem. The equality in (4.80) is true pointwise. Recall that according to (4.59) and (4.3), Φ2 is differentiable and eηu Φ2 (u) ∈ L2m (0, ∞) for η < − M 2 . Hence, we rewrite (4.80) as: ∞ ϕ (μ) = −Φ2 (0) − e−2iμu Φ2 (u) du. (4.81) 0

Without loss of generality we shall choose M for the half-plane η < −M/4, where ϕ(μ) is treated, so that M u Φ2 (u) ∈ L2m (0, ∞), M > 4 max |dp − dj |−1 , (4.82) exp ε − j=p 2 sup ϕ(μ) < ∞. (4.83) η≤−M/4

In view of (3.10), (4.31), and (4.81) we have ∞ ϕk (μ) −1 2iμx (Ak (r) − λI) Πk e−2iμu Φk2 (u) du. = 2bk μe 1

(4.84)

x

By (4.84) and the ﬁrst relation in (4.82) we get

#

(Ak (r) − λI)−1 Πk ϕk (μ) ≤ C(r)|μ|/ |η|.

1

(4.85)

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By the second relation in (4.82) the inequality |λ − dp | > |dk − dp |/2 is true −1 bk for p = k, λ = dk + 2μ , and η < −M/4, that is, the resolvents (Ap (r) − λI) (p = k) are bounded. Therefore, the inequalities (4.83) and (4.85) imply

#

(A(r) − λI)−1 Π ϕk (μ) ≤ C1 (r)|μ|/ |η|. (4.86)

1 As |λ − λ| = −η|μ|−2 , from (4.30), (4.83), and (4.86) the inequality ϕk (μ) ∗ sup[ϕk (μ) 1]wA (r, λ) wA (r, λ) <∞ 1 is immediate. Thus, (4.78) is valid, and the solution of the inverse problem can be obtained via (3.23). The uniqueness of the solution of the inverse problem is stated in Theorem 2.6. In a way similar to [38,39], the procedure to solve the inverse problem grants also a Borg–Marchenko-type theorem. Theorem 4.4. Let the Cm -valued vector functions ϕ(μ, 1) and ϕ(μ, 2) be holomorphic in the half-plane μ < −M/4 (M > 0) and satisfy (4.59). Suppose that on some ray c μ = μ (c = c, μ < −M/4) we have ϕ(μ, 1) − ϕ(μ, 2) = e−2iμl O(1)

for |μ| → ∞.

(4.87)

Then ϕ(μ, 1) and ϕ(μ, 2) are Weyl functions of systems (1.2) with potentials β(x, 1) and β(x, 2), respectively, which satisfy (1.4), (2.1) and the additional equality β(x, 1) ≡ β(x, 2)

(0 < x < l).

(4.88)

Proof. According to Theorem 4.3 the functions ϕ(μ, 1) and ϕ(μ, 2) are Weyl functions of systems (1.2) with potentials β(x, 1) and β(x, 2), which satisfy (1.4) and (2.1). Denote by Φ2 (x, j) the matrix function generated via (4.3) by ϕ(μ, j) (j = 1, 2), and introduce the matrix function l (exp 2iμ(l − u))) (Φ2 (u, 1) − Φ2 (u, 2)) du.

ν(μ) :=

(4.89)

0

Next, we shall show that ν(μ) = 0. Indeed, in view of (4.81) and (4.87) it is immediate that Φ2 (0, 1) = Φ2 (0, 2). Hence, from (4.81) and (4.89) we derive 2iμl

ν(μ) = e

∞ (ϕ(μ, 2) − ϕ(μ, 1)) +

(exp 2iμ(l − u)) (Φ2 (u, 1) − Φ2 (u, 2)) du. l

(4.90) It is immediate from (4.89) that ν(μ) is bounded on the line μ = −M/4. Using (4.87) and (4.90) we see that ν(μ) is bounded on the ray c μ =

μ ( μ < −M/4). Thus, by the Phragmen–Lindel¨ of theorem ν(μ) is bounded in the half-plane μ ≤ −M/4. Moreover, by (4.89) ν(μ) is bounded

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for μ > −M/4, that is, ν(μ) is an entire function bounded on C. Therefore, ν(μ) is a constant. As limη→∞ ν(μ) = 0, so we obtain ν(μ) ≡ 0, that is Φ2 (x, 1) ≡ Φ2 (x, 2)

(0 < x < l).

(4.91)

By the procedure to solve the inverse problem (see Theorem 4.3) formula (4.91) implies (4.88).

5. Weyl Set Condition (2.1) is not necessary in the considerations of Sects. 2, 3 and 4. Taking into account the identity in (1.3), one can choose two sets of natural numbers N1 and N2 , so that βk1 (0) = 0 for k ∈ N1 , βk2 (0) = 0 for k ∈ N2 , N1 ∩ N2 = ∅, N1 ∪ N2 = {1, 2, . . . , m}.

(5.1) (5.2)

The procedure to solve the inverse problem is easily modiﬁed for that case. Such a modiﬁcation is discussed below. We also introduce a notion of the Weyl set, and the system (1.2), which satisﬁes (1.3), is recovered from its Weyl set. Definition 5.1. The set κ of the pairs (5.3) κ = {βk (0), ψk (μ) | 1 ≤ k ≤ m}, where ψk is given by (2.18), is called a Weyl set of the system (1.2), which satisﬁes (1.3). ˘ coincide and the vecIf the functions ψk in the Weyl sets κ and κ ˘ respectively, satisfy the equalities β˘k (0) = tors βk (0) and β˘k (0) in κ and κ, ˘ as κ and κ ˘ correspond to ck βk (0), |ck | = 1 (1 ≤ k ≤ m), we say that κ = κ, the same system. After we exclude this arbitrariness, the Weyl set is uniquely deﬁned by system (1.2), (1.3). (See the construction of ψk in Sect. 2 and the proof of Theorem 2.2 in [29].) The next lemma is proved in a quite similar way to Lemma 3.6. Lemma 5.2. Let the S-node be given by the formulas (3.1)–(3.3) and (3.5), where Φk1 (x) ≡ 1, Φk2 (x) ≡ 1,

Φk2 (x) ∈ L2 (0, l)

for k ∈ N1 ,

(5.4)

∈ L (0, l)

for k ∈ N2 .

(5.5)

Φk1 (x)

2

Deﬁne the reductions A(r), S(r) and Π(r) (r < l) of the operators from this S-node in the same way as it was done after Remark 3.2 in Sect. 3. Then wA of the form (3.29) satisﬁes (3.30), where β is given by (3.23) and the operator V in (3.23) is obtained via (3.19)–(3.22). Remark 5.3. The operator S is uniquely recovered from the operator identity (3.5), the corresponding formulas from Proposition 3.1 are easily modiﬁed for the more general case (5.4), (5.5). In view of Lemma 5.2, the proof of the next theorem is similar to the proof of Theorem 4.2.

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Theorem 5.4. Let system (1.2) satisfy conditions (1.3) and let N1 and N2 be chosen so that (5.1) and (5.2) hold. Then, system (1.2) is uniquely recovered from its Weyl set. To recover (1.2) on an arbitrary ﬁxed interval [0, l] we use the following procedure. First, the entries of the column vector functions Φ1 (x) and Φ2 (x) (0 < x < ∞) are introduced by the equalities: ∞ μ! i dζ for k ∈ N1 ; (5.6) μ−1 eiμx ϕk Φk1 (x) ≡ 1, Φk2 (x) = 2π 2 Φk1 (x) =

ϕk (μ) =

i 2π

∞

−∞

μ−1 eiμx φk

−∞

βk1 (0)ψk (μ) − βk2 (0) , βk2 (0)ψk (μ) + βk1 (0)

μ! dζ, 2

Φk2 (x) ≡ 1

φk (μ) =

βk2 (0)ψk (μ) + βk1 (0) , βk1 (0)ψk (μ) − βk2 (0)

for k ∈ N2 ; (5.7)

(5.8)

where μ = ζ + iη and −η > 0 is sufﬁciently large. Next, we introduce the S-node by formulas (3.1)–(3.3) and (3.5). The is modiﬁed: operator S has the form (3.6), where the definition (3.7) of D =D 1 + D 2, D

p = diag{|Φ1p (0)|2 , |Φ2p (0)|2 , . . . , |Φmp (0)|2 } D

Finally, we put

(p = 1, 2). (5.9)

⎡

⎤ β˘1 (x) ˘ β(x) = ⎣ · · · ⎦ = (V Φ)(x) β˘m (x)

(0 ≤ x ≤ l),

(5.10)

where the operator V is obtained via (3.19)–(3.22). The equalities β˘k (x)∗ β˘k (x) ≡ βk (x)∗ βk (x) (1 ≤ k ≤ m).

(5.11)

are valid, that is, system (1.2) is recovered by the procedure, which is described above.

6. Sine-Gordon Equation The initial value problem for the sine-Gordon equation in the light cone coor∂ ω) was treated in [1]. The initial value probdinates ωxt = sin ω (ωx := ∂x lem (with initial conditions tending to zero) for the sine-Gordon equation in laboratory coordinates ωxx − ωtt = sin ω

(6.1)

was investigated by Faddeev, Takhtajan and Zakharov (see [49] and further references in [15]). Notice also that the Goursat problem for the equation ωxt = sin ω, which is treated on the characteristics t = 0 and x = −∞ in [23], is equivalent to the Cauchy problem for Eq. (6.1). In this section we consider (6.1) under boundary conditions ω(0, t) = ω0 (t) and ωx (0, t) = ω1 (t).

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(The boundary value problem is clearly equivalent to the initial value problem after the change of variables and the change ω → ω + π.) We do not require that ω tends to zero and only the boundedness of ωx and ωt is needed. Equation (6.1) admits a zero curvature representation Gt (x, t, λ) − Fx (x, t, λ) + G(x, t, λ)F (x, t, λ) − F (x, t, λ)G(x, t, λ) = 0. (6.2) We can modify the auxiliary systems wx (x, t, λ) = G(x, t, λ)w(x, t, λ),

wt (x, t, λ) = F (x, t, λ)w(x, t, λ), (6.3)

so that they have the form (1.2). Namely, put G(x, t, λ) = i

2

bk (λ − dk )−1 (βk (x, t)∗ βk (x, t)) ,

k=1

d1 = −d2 = 1, F (x, t, λ) = i

2

b1 = b2 = 1,

(6.4)

bk (λ − dk )−1 (βk (x, t)∗ βk (x, t)) ,

k=1

d1 = −d2 = 1,

b1 = −b2 = 1,

(6.5)

where 1 β1 (x, t) = √ [1 2

i eiω(x,t)/2 ]q(x, t),

1 β2 (x, t) = √ [1 ie−i ω(x,t)/2 ]q(x, t), 2 (6.6)

the 2 × 2 matrix function q satisﬁes the equations ˘ t)q(x, t), qx (x, t) = G(x, ˘ := −i G

qt (x, t) = F˘ (x, t)q(x, t),

ωt 1 ω! j + sin J , 4 2 2

q(0, 0) = I2 ,

ωx 1 ω! F˘ := −i j + cos Jj, 4 2 2

(6.7) (6.8)

and J, j are given in (4.50). It is easily checked that the sine-Gordon equation ˘ t − F˘x + G ˘ F˘ − F˘ G ˘ =0 (6.1) is equivalent to the compatibility condition G of the equations (6.7). Moreover, direct calculation shows that relations (6.6) and (6.7) imply (6.2), which is the compatibility condition for (6.3). Thus, if (6.1) holds, equations (6.3) are compatible. Introduce the 2 × 2 matrix functions Z(x, t, λ), Z(t, λ) := Z(0, t, λ) = {Zij (t, λ)}2i,j=1 ,

and Y (x, t, λ) = {Yij (x, t, λ)}2i,j=1

by the equations Yx (x, t, λ) = G(x, t, λ)Y (x, t, λ),

Y (0, t, λ) ≡ I2 ;

Zt (x, t, λ) = F (x, t, λ)Z(x, t, λ),

Z(x, 0, λ) ≡ I2 .

(6.9)

The matrix functions Qk (x, t) (k = 1, 2) are connected with βk (x, t) by the equalities (2.3). According to (2.3) and (6.6)–(6.8) the boundary conditions ω(0, t) = ω0 (t),

ωx (0, t) = ω1 (t)

(−∞ < t < ∞)

(6.10)

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uniquely deﬁne F˘ (0, t), q(0, t), βk (0, t) and Qk (0, t). If we recover also ψk (t, μ) for each −∞ < t < ∞, we have a Weyl set for each t. First, we recover F (0, t, λ) and Z(t, λ), using formulas (6.5) and (6.9), and put Uk (x, t, μ) := exp{(−1)k itμ}Qk (x, t)Z(x, t, λ)Q(x, 0)∗ ,

μ = (2(λ − dk ))−1 , (6.11)

Uk (t, μ) =

{ujp (t, μ, k)}2j,p=1

:= Uk (0, t, μ).

(6.12)

The matrix functions Uk (t, μ) are uniquely recovered from (6.10) too. Theorem 6.1. Let the function ω(x, t) have continuous second derivatives in the semi-plane x ≥ 0 and satisfy the sine-Gordon equation (6.1) and boundary conditions (6.10). Assume also that sup(|ωx (x, t)| + |ωt (x, t))|) < ∞.

(6.13)

x≥0

Then, cos ω(x, t) (x ≥ 0) is uniquely recovered from (6.10). For this purpose construct Uk (t, μ) (k = 1, 2) using (6.5)–(6.12). There is M1 > 0, such that for − μ > M1 we have u12 (t, μ, 1) , ψ1 (0, μ) = − lim t→∞ u11 (t, μ, 1)

u12 (t, μ, 2) ψ2 (0, μ) = − lim . (6.14) t→−∞ u11 (t, μ, 2)

The functions ψk (t, μ) are given by the formulas u11 (t, μ, k)ψk (0, μ) + u12 (t, μ, k) . ψk (t, μ) = u21 (t, μ, k)ψk (0, μ) + u22 (t, μ, k)

(6.15)

By formulas (2.3), (6.6)–(6.8), (6.14) and (6.15) we recover the Weyl set for each t. Finally, we recover the functions βk (x, t) (up to factors ck (x, t) such that |ck | = 1, ck (0, t) = 1 ) using Theorem 5.4. It follows that cos ω(x, t) = 2β1 (x, t)β2 (x, t)∗ β2 (x, t)β1 (x, t)∗ − 1.

(6.16)

Proof. Step 1. In this step we shall prove (6.15). Note that as ω has continuous second derivatives, so according to (6.4)–(6.8) the matrix functions G and F are continuously differentiable. Therefore, the formula (1.6) on p. 168 in [43] implies: Y (x, t, λ) = Z(x, t, λ)Y (x, 0, λ)Z(t, λ)−1 .

(6.17)

By (1.6), (2.4), and (6.9) we have w(x, t, λ) = Y (x, t, λ), Q(x, t)∗ = Q(x, t)−1 ,

Y (x, t, λ)∗ = Y (x, t, λ)−1 ,

Z(x, t, λ)∗ = Z(x, t, λ)−1 . (6.18)

Hence, taking into account (2.2) and (2.12) we have A(r, t, μ) = eirμ Q(0, t)Y (r, t, λ)∗ Q(r, t)∗ .

(6.19)

From (6.11), (6.12), (6.17), and (6.19) it follows that Ak (r, t, μ) = Uk (t, μ)Ak (r, 0, μ)Uk (r, t, μ)−1 .

(6.20)

Vol. 69 (2011) System Depending Rationally on Spectral Parameter In view of (6.9) and (6.11) it is easy to see that ∂ ∂ k+1 Uk (r, t, μ) = (−1) Qk (r, t) Qk (r, t)∗ iμj + ∂t ∂t βp (r, t)∗ βp (r, t) k ∗ + (−1) iQk (r, t) Qk (r, t) Uk (r, t, μ), λ − dp

595

(6.21)

where k and p take values 1 and 2, p = k. According to (6.7) and (6.8) the ∂ ∂ (q ∗ q) = 0 and ∂x (q ∗ q) = 0 are true and q(0, 0) = I2 . Therefore, equalities ∂t it is immediate that q is unitary: q(x, t)∗ q(x, t) ≡ I2 . By (6.6)–(6.8), (6.13) and (6.22) we have

∂

sup βk (x, t)

< ∞ (x ≥ 0, −∞ < t < ∞), ∂t

(6.22)

k = 1, 2.

(6.23)

Taking into account (6.21) and (6.23), in a way similar to the proof of (2.11) we derive ∂ (−1)k+1 (Uk (r, t, μ)∗ jUk (r, t, μ)) ∂t ! $ Uk (r, t, μ)∗ Uk (r, t, μ) > 0 (6.24) ≥ i(μ − μ) − M $ > 0 and μ < −M $/2. From (6.24) it follows that for some M U1 (r, t, μ)∗ jU1 (r, t, μ) > j

for t > 0;

U2 (r, t, μ)∗ jU2 (r, t, μ) > j

for t < 0. (6.25)

−1

From the ﬁrst inequality in (6.25) we have (U1 (r, t, μ)∗ ) jU1 (r, t, μ)−1 < j, θ(μ) −1 [θ(μ)∗ 1] (U1 (r, t, μ)∗ ) jU1 (r, t, μ)−1 < 0 for t > 0, |θ| ≤ 1. 1 (6.26) In a similar way, from the second inequality in (6.25) we derive ∗ ∗ −1 −1 θ(μ) [θ(μ) 1] (U2 (r, t, μ) ) jU2 (r, t, μ) < 0 for t < 0, |θ| ≤ 1. 1 (6.27) By (6.26) and (6.27) we obtain the inequalities |χ1 (r, t, μ)| < 1 for t > 0;

|χ2 (r, t, μ)| < 1 for t < 0

$ M μ < − 2

(6.28) for the functions χk (r, t, μ) :=

(Uk−1 )11 (r, t, μ)θ(μ) + (Uk−1 )12 (r, t, μ) , (Uk−1 )21 (r, t, μ)θ(μ) + (Uk−1 )22 (r, t, μ)

k = 1, 2; |θ| ≤ 1.

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In view of (2.13) and (2.18) we have A11 (r, t, μ)θ(μ) + A12 (r, t, μ) , |θ(μ)| ≤ 1, ψk (t, μ) = lim r→∞ A21 (r, t, μ)θ(μ) + A22 (r, t, μ)

μ < −

M . 4 (6.29)

From (6.20) it follows that the linear fractional transformation on the righthand side of (6.29) can be written down as the superposition of the three linear fractional transformations, the ﬁrst of which transforms θ into χk . Using (6.28) we see that the second transformation transforms χk into ψk (r, 0, μ) for k = 1 and t > 0 as well as for k = 2 and t < 0. In the limit, it follows from (6.20), (6.28), and (6.29) that (6.15) is true for k = 1, t > 0 and for $ M k = 2, t < 0 (−η > M0 = max( M 2 , 4 )). To prove (6.15) for k = 1, t < 0 and for k = 2, t > 0 rewrite (6.20) in the form Ak (r, t, μ)Uk (r, t, μ) = Uk (t, μ)Ak (r, 0, μ),

(6.30)

and use the inequalities U1 (r, t, μ)∗ jU1 (r, t, μ) < j

for t < 0;

U2 (r, t, μ)∗ jU2 (r, t, μ) < j

for t > 0, (6.31)

which are immediate from (6.24). By (6.31) one can see that $ M , μ < − 2

|χ ˘1 (r, t, μ)| < 1

for t < 0;

|χ ˘2 (r, t, μ)| < 1 for t > 0

(6.32) where χ ˘k (r, t, μ) :=

(Uk )11 (r, t, μ)θ(μ) + (Uk )12 (r, t, μ) , (Uk )21 (r, t, μ)θ(μ) + (Uk )22 (r, t, μ)

k = 1, 2; |θ| ≤ 1.

Now, consider the linear fractional transformations of θ, where the coefﬁcients are the entries of the left-hand side and right-hand side of (6.30), respectively. These linear fractional transformations coinside, and in the limit (as r tends to inﬁnity) we obtain (6.15) for k = 1, t < 0 and for k = 2, t > 0. Thus, (6.15) is proved. Step 2. By (2.14) we have |ψk (t, μ)| < 1 ( μ < − M 4 ). Hence, in view of (6.15) we obtain [ψk (0, μ)∗

ψk (0, μ) 1]Uk (t, μ) jUk (t, μ) 1 ∗

≤ 0.

(6.33)

Vol. 69 (2011) System Depending Rationally on Spectral Parameter

597

$

Recall that (6.15) holds for − μ > M0 ≥ M 2 . From (6.24) it follows that for − μ > M1 = 2ε + M0 (ε > 0) the inequalities t

∗

U1 (t, μ) jU1 (t, μ) − j ≥ ε

U1 (s, μ)∗ U1 (s, μ) ds

(t > 0),

(6.34)

U2 (s, μ)∗ U2 (s, μ) ds

(t < 0)

(6.35)

0

0

∗

U2 (t, μ) jU2 (t, μ) − j ≥ ε t

are true. According to (6.33)–(6.35), for any μ such that − μ > M1 we have ∞ [ψ1 (0, μ)∗ 0

0 −∞

[ψ2 (0, μ)∗

ψ1 (0, μ) 1]U1 (s, μ) U1 (s, μ) 1 ∗

ψ2 (0, μ) 1]U2 (s, μ) U2 (s, μ) 1 ∗

ds < ∞,

(6.36)

ds < ∞.

(6.37)

By (6.13), (6.21), (6.36), and (6.37) the inequalities

ψ1 (0, μ)

ψ2 (0, μ)

< ∞ (6.38) sup U1 (t, μ)

U2 (t, μ)

< ∞, sup

1 1 t>0 t<0 are valid. Inequalities (6.34) and (6.35) imply that |u11 (t, μ, 1)|2 > 1 + εt (t > 0),

|u11 (t, μ, 2)|2 > 1 − εt

(t < 0). (6.39)

From (6.38) and (6.39) follows (6.14). In view of (6.6) and (6.22) we obtain 2β1 β2∗ = 1 + cos ω + i sin ω,

i.e.,

Hence, the equality (6.16) is immediate.

2|β1 β2∗ |2 = 1 + cos ω.

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