Integral Equations and Operator Theory - Volume 61

Integr. equ. oper. theory 61 (2008), 1–19 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/010001-19, published online March 13, 2008 DOI 10.1007/s00020-008-1569-6

Integral Equations and Operator Theory

The Local Form of Doubly Stochastic Maps and Joint Majorization in II1 Factors Mart´ın Argerami and Pedro Massey Abstract. We find a description of the restriction of doubly stochastic maps to separable abelian C ∗ -subalgebras of a II1 factor M. We use this local form of doubly stochastic maps to develop a notion of joint majorization between n-tuples of mutually commuting self-adjoint operators that extends those of Kamei (for single self-adjoint operators) and Hiai (for single normal operators) in the II1 factor case. Several characterizations of this joint majorization are obtained. As a byproduct we prove that any separable abelian C ∗ -subalgebra of M can be embedded into a separable abelian C ∗ -subalgebra of M with diffuse spectral measure. Mathematics Subject Classification (2000). Primary 46L51; Secondary 46L10. Keywords. Joint majorization, doubly stochastic map, convex hull, unitary orbit.

1. Introduction Majorization between self-adjoint operators in finite factors was introduced by Kamei [21] as an extension of Ando’s definition of majorization between selfadjoint matrices [4]. Later on, Hiai considered majorization in semifinite factors between self-adjoint and normal operators [15, 16]. The reason why majorization has attracted the attention of many researchers (see the discussion in [16] and the references therein) is that it provides a rather subtle way to compare operators that occurs naturally in many contexts (for example [5, 13, 14]). Recently, majorization has regained interest because of its relation with norm-closed unitary orbits of self-adjoint operators and conditional expectations onto abelian subalgebras [5, 6, 10, 14, 18, 19, 24, 26]. Supported in part by NSERC of Canada, CONICET (PIP 5272) and UNLP (11 X472) of Argentina .

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In this paper we define an extension of the notion of majorization between normal operators to that of joint majorization between n-tuples of commuting self-adjoint operators in a II1 factor (such extension is achieved in [22] for finitedimensional factors). Given a II1 factor M with faithful normalized trace τ , and a1 , . . . , an , b1 , . . . , bn ∈ Msa such that ai aj = aj ai , bi bj = bj bi for every i,j, we define the joint majorization (a1 , . . . , an ) ≺ (b1 , . . . , bn ) in terms of doubly stochastic maps between abelian von Neumann algebras (see Definition 4.4). The main result of the paper (Theorem 4.5) is a series of characterizations of joint majorization. In particular, we prove that the following statements are equivalent: (a1 , . . . , an ) ≺ (b1 , . . . , bn ); (a1 , . . . , an ) ∈ conv(UM (b1 , . . . , bn )); (a1 , . . . , an ) ∈ conv w (UM (b1 , . . . , bn )); There exists a unital trace-preserving completely positive map T such that ai = T (bi ), 1 ≤ i ≤ n; 5. There exists a unital trace-preserving positive map T such that ai = T (bi ) for every i ∈ {1, . . . , n}; 6. τ (f (a1 , . . . , an )) ≤ τ (f (b1 , . . . , bn )) for every continuous convex function f : Rn → R. 1. 2. 3. 4.

A key concept for obtaining these characterizations is what we call the “local form” of a doubly stochastic (DS) map (Theorem 3.7): for any DS map T on a II1 factor M and abelian C∗ subalgebras A and B such that T (B) ⊂ A, there exists a sequence ρr of maps approximating T pointwise in norm, i.e. lim T (b) − ρr (b) = 0, b ∈ B, r



αi ui · u∗i , αi ≥ 0, αi = 1, ui ∈ UM . where ρr = This local form of a DS map is obtained as the consequence of several results developed in Sections 3 and 6. The main two technical tools are results on convex combinations of partitions with constant trace (Proposition 3.3 and Lemmas 3.5 and 3.6), and the construction of separable abelian diffuse refinements of separable abelian C ∗ -subalgebras of a II1 factor M. This last construction could have interest on its own; it has been further developed in [23], and similar ideas have been used in [7, 8]. Some of the techniques we use appear to be new, even in the single element case. The paper is organized as follows. In Section 2 we recall some facts about abelian C ∗ -subalgebras of a II1 factor. In Section 3, after describing some technical results, we obtain a description of the local structure of doubly stochastic maps. In Section 4 we introduce and develop the notion of joint majorization between finite abelian families of self-adjoint operators in a II1 factor and we obtain several characterizations of this relation. Section 5 deals with joint unitary orbits of abelian families. Finally, in Section 6 we prove the results described in Section 3.

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2. Preliminaries Throughout the paper M will be a II1 factor with normalized faithful normal trace τ . The C ∗ -subalgebras of M are always assumed unital. The subspace of self-adjoint elements of M will be denoted by Msa , the unitary group by UM . As is customary we will denote conjugation by u ∈ UM as Adu , i.e. Adu (x) = uxu∗ . An abelian family (ai )ni=1 in Msa , is a finite family of mutually commuting selfadjoint operators in M. If (ai )ni=1 ⊆ Msa is an abelian family then C ∗ (a1 , . . . , an ) denotes the (unital) separable abelian C ∗ -subalgebra of M generated by a1 , . . . , an . If A is an arbitrary abelian C ∗ -subalgebra of M then Γ(A) denotes its space of characters, i.e. the set of nonzero *-homomorphisms γ : A → C endowed with the weak∗ -topology. It is well-known that the set Γ(A) is a compact Hausdorff space and that A C(Γ(A)), where C(Γ(A)) denotes the C ∗ -algebra of continuous functions on Γ(A). We will use 1 to denote the constant function and πi : Rn → R to denote the projection onto the ith coordinate. 2.1. Joint spectral measures and joint spectral distributions As we will consider a several-variable extension of continuous functional calculus, we state a few facts about it (a different description can be found in [27]). Let ∗ a ¯ = (ai )ni=1 be an then Γ(A) can nabelian familyn in Msa . If A = C (a1 , . . . , an ), be embedded in i=1 σ(ai ) ⊆ R . Indeed, the map Φ : Γ(A) → ni=1 σ(ai ) ⊆ Rn given by Φ(γ) = (γ(a1 ), . . . , γ(an )) is a continuous injection and therefore Γ(A) is homeomorphic to its image under this map; this image is called the joint spectrum of the family and we denote it by σ(¯ a). Note that A C(σ(¯ a)) as C ∗ -algebras, and so for each f ∈ C(σ(¯ a)) there exists a normal operator, denoted f (a1 , . . . , an ), that corresponds to f under the isomorphism. If A ⊆ M is a separable C ∗ -subalgebra then Γ(A) is metrizable and the representation C(Γ(A)) A ⊆ M induces a spectral measure EA [12, IX.1.14] that takes values on the lattice P(M) of projections of M. Let µA be the (scalar) regular Borel measure on Γ(A) defined by µA (∆) = τ (EA (∆)). The regularity of µA follows from the fact that every open  set is σ-compact [25, 2.18]. The map Λ : L∞ (Γ(A), µA ) → M given by Λ(h) = Γ(A) h dEA is a normal ∗ -monomorphism (note that in this case the weak∗ topology of L∞ (Γ(A), µA ), restricted to the unit ball, is metrizable) and we have  τ (Λ(h)) = h dµA , ∀h ∈ L∞ (Γ(A), µA ). (1) Γ(A)

We will consider the von Neumann algebra L∞ (A) := Λ(L∞ (Γ(A), µA )) ⊆ M. When a ¯ = (ai )ni=1 and A = C ∗ (a1 , . . . , an ), we call Ea¯ := EA and µa¯ := µA the joint spectral measure and joint spectral distribution of the abelian family a ¯ a), µa¯ ) → L∞ (A) the normal isomorphism defined and we denote by Λa¯ : L∞ (Γ(¯ above. It is straightforward to verify that Λa¯ (πi ) = ai , 1 ≤ i ≤ n. Recall that for each h ∈ L∞ (Γ(¯ a), µa¯ ) we write h(a1 , . . . , an ) for the operator Λa¯ (h) ∈ A. In the

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case of a single self-adjoint operator a ∈ Msa the measure µa is the usual spectral distribution of a [10]. In the particular case when x ∈ M is a normal operator, the real and imaginary parts of x are mutually commuting self-adjoint elements of M. Identifying the complex plane with R2 in the usual way, the spectrum of x coincides with the joint spectrum of the abelian pair (Re(x), Im(x)), and that the spectral measure of x coincides with the joint spectral measure of (Re(x), Im(x)). 2.2. Comparison of measures and diffuse measures ∼ (Rn ) the set of all regular finite positive Borel measures ν on We denote by M+  ∼ Rn with ζ dν(ζ) < ∞. We write ν(f ) = Rn f dν, for every ν ∈ M+ (Rn ) and every ν-integrable function f . ∼ (Rn ). Definition 2.1. Let µ, ν ∈ M+ 1. We write ν ∼ µ whenever ν(1) = µ(1) and ν(πj ) = µ(πj ) for every 1 ≤ j ≤ n; 2. we say that µ is
majorized by ν, and we write µ ≺ ν, if for every µ1 , . . . , µm ∈ m ∼ n ∼ n (R ) with M i=1 µi = µ there exist ν1 , . . . , νm ∈ M+ (R ) such that
+ m i=1 νi = ν, and νi ∼ µi , 1 ≤ j ≤ n. .

The relation ≺ in Definition 2.1 does not seem to be called “majorization” in the literature, but it will be a suitable name for us in the light of Theorem 4.5. ∼ Theorem 2.2. [3, I.3.2] Let µ, ν ∈ M+ (Rn ). Then µ ≺ ν if and only if µ(f ) ≤ ν(f ) for every continuous convex function f : Rn → R.

The next corollary is an immediate consequence of Theorem 2.2 and the identity (1). Corollary 2.3. Let a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 ⊂ Msa be two abelian families. Then µa¯ ≺ µ¯b if and only if τ (f (a1 , . . . , an )) ≤ τ (f (b1 , . . . , bn )) for every continuous convex function f : Rn → R. We end this section with the following elementary fact about diffuse (scalar) measures, i.e. measures where µ({x}) = 0 for every x. Lemma 2.4. Let K ⊂ Rn be compact and let µ be a regular diffuse Borel probability measure on K. Then for every α ∈ (0, 1) there exists a measurable set S ⊂ K such that µ(S) = α.

3. The local form of doubly stochastic maps This section deals with doubly stochastic maps (see Definition 3.1 below). These maps play an important role in the theory of majorization between self-adjoint operators (see for instance [1, 2, 15, 16]); they will also play a central role in majorization between abelian families in Section 4. For the sake of clarity, several proofs of technical results will be delayed until Section 6. Let A ⊆ M be an abelian C ∗ -subalgebra, and let EA and µA denote the spectral measure and the spectral distribution of A as defined in Section 2.1.

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Definition 3.1. 1. A linear map Φ : M → M is said to be doubly stochastic [15] if it is unital, positive, and trace preserving. The set of all doubly stochastic maps on M is denoted by DS(M); 2. if x ∈ Γ(A) is such that EA ({x}) = 0, we say that x is an atom for EA ; 3. the set of atoms of EA is denoted At(EA ); 4. we say that A is diffuse if At(EA ) = ∅; 5. the set D(M) is the convex semigroup D(M) = conv{Adu : u ∈ U(M)}. We will make use of the fact that DS(M) is compact in the BW topology [9], i.e. under pointwise strong convergence. Since µA = τ ◦EA , the faithfulness of the trace implies that At(µA ) =At(EA ). The following theorem states that spectral measures of a separable A can be refined in a coherent way. Theorem 3.2. Let A ⊆ M be a separable abelian C ∗ -subalgebra. Then there exists a ∈ Msa such that C ∗ (A, a) is abelian and diffuse.


Proof. (see Section 6).

Since the atoms of EA are in correspondence with the set of minimal projections of L∞ (A), Theorem 3.2 provides a way to embed A into a separable ˜ has no minimal projections (see C ∗ -subalgebra A˜ = C ∗ (A, a) such that L∞ (A) Remark 6.3 for further discussion). Any operator in a von Neumann algebra can be approximated in norm by linear combinations of projections. In the case of a II1 factor, an added requirement could be for the projections to have equal trace; with such requirement, the norm approximation is usually lost, and only norm-1 and norm-2 approximation can be achieved. What Proposition 3.3 shows is that norm approximation can still be obtained by taking convex combinations of linear combinations of orthogonal families of projections with equal trace. Proposition 3.3. Let B ⊂ M be a separable, diffuse, abelian C ∗ -subalgebra. Then there is an unbounded set M ⊆ N such that for every m ∈ M there exist k = k(m) t,m  ) = 1/m (1 ≤ i ≤ partitions of unity {qit,m }m i=1 ⊆ B ∩ M, 1 ≤ t ≤ k, with τ (qi m, 1 ≤ t ≤ k), and such that for each b ∈ B,  m  k   1   t,m t,m  (2) mτ (b qi ) qi lim  b −  = 0. m→∞   k t=1

i=1

Proof. (see Section 6).




Remark 3.4. For fixed m and partitions of unity {qit }m i=1 , 1 ≤ t ≤ k, the linear map m  k 1  t t m τ (b qi ) qi b → k t=1 i=1 is a contraction with respect to the operator norm.

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m Lemma 3.5. Let {pi }m i=1 , {qi }i=1 ⊆ M be partitions of unity such that τ (pi ) = 1 τ (qi ) = m , and let T ∈ DS(M). Let β1 , . . . , βm ∈ R. Then there exists ρ ∈ D(M) such that m  m   αi pi = ρ βi qi , (3)

where αi = m

i=1


m j=1

i=1

βj τ (T (qj ) pi ), 1 ≤ i ≤ m.

verify that Proof. Let γi,j = m τ (T (qj ) pi ) ≥ 0; it is then straightforward to
m γ = (γi,j ) ∈ Rm×m is a doubly stochastic matrix and that αi = j=1 γi,j βj for every i = 1, . . . , m. By Birkhoff’s Theorem [11] the doubly stochastic matrix (γi,j
) can be written as a convex
combination of permutation matrices, i.e. (γi,j ) = σ∈Sm ησ Pσ , where ησ ≥ 0, σ∈Sm ησ = 1 and Pσ is the m × m permutation matrix induced by σ ∈ Sm . Then we have   m    αi = γi,j βj = (γ · β)i = ησ Pσ · β = ησ βσ(i) , 1 ≤ i ≤ m. (4) j=1

σ∈Sm

i

σ∈Sm

The fact that M is a II1 factor and that the elements of the partitions {pi }i , {qi }i have the same trace guarantees the existence of unitaries uσ such that uσ qσ(i) u∗σ = pi , 1 ≤ i ≤ m, for every σ ∈ Sm . Indeed, if σ ∈ Sm , the equalities, τ (qσ(i) ) = τ (pi ) ∗ imply that there exist partial isometries vi,σ ∈ M such that vi,σ vi,σ = pi and
m ∗ vi,σ vi,σ = qσ(i) for i = 1, . . . , m. Thus uσ =
i=1 vi,σ ∈ M are the required unitaries. Using equation (4), and letting ρ(· ) = σ∈Sm ησ uσ (· ) u∗σ ∈ D(M),   m  m m      ∗ αi pi = ησ βσ(i) pi = ησ βσ(i) uσ qσ(i) uσ i=1

i=1

=

 σ∈Sm

σ∈Sm

ησ uσ

m 

σ∈Sm

 βi qi

i=1

u∗σ

=ρ

m  i=1

i=1

βi qi

 .


 Lemma 3.6. Let B ⊂ M be a separable C ∗ -subalgebra, and let {pi }m i=1 ⊆ B ∩ M be a partition of unity. Then there exists a sequence {ρi }i∈N ⊂ D(M) such that for every b ∈ B, if we let βi (b) = τ (b pi )/τ (pi ), then   m      lim ρj (b) − βi (b)pi  = 0. j→∞   i=1

Proof. (see Section 6).




Theorem 3.7 (The Local Form). Let A, B ⊆ M be separable abelian C ∗ -subalgebras and let T ∈ DS(M). Let S be the operator subsystem of B given by S = T −1 (A) ∩ B. Then there exists a sequence (ρr )r∈N ⊆ D(M) such that lim T (b) − ρr (b) = 0 r→∞ for every b ∈ S.

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Proof. First, note that we just have to prove the theorem for separable diffuse abelian C ∗ -subalgebras of M; indeed, assume it holds for such algebras and let A, B ⊆ M be arbitrary separable abelian C ∗ -subalgebras. Then, by Theorem 3.2 there exist separable diffuse abelian subalgebras A˜ and B˜ of M such that A ⊆ A˜ ˜ Thus we get a sequence {ρr }r∈N ⊆ D such that limr→∞ T (b) − and B ⊆ B. ˜ ∩ B. ˜ So we assume that A and B ρr (b) = 0, for every b ∈ T −1 (A) ∩ B ⊆ T −1 (A) are diffuse. By Proposition 3.3, there exists an unbounded set M ⊆ N and, for each j,m m   m ∈ M, k(m) partitions of unity {qij,m }m i=1 ⊆ B ∩ M and {pi }i=1 ⊆ A ∩ M (in order to simplify the notation we suppress the supra-index m and write qij , pji ), 1 ≤ j ≤ k, such that for every b ∈ T (A)−1 ∩ B and every r ∈ N, there exists m0 (r, b) ∈ M such that if m ≥ m0 we have  m    k   j j  1  1 b − (5) β q <  k j=1 i=1 i i    r  m   k   j j   1  1 T (b) − αi pi  < r  k   j=1 i=1

and

(6)

where βij = m τ (b qij ), αji = m τ (T (b) pji ), τ (pji ) = τ (qij ) = 1/m, (from the construction of such partitions it is evident that we can assume that both have the same unbounded set M and the same k(m) for every m ∈ M). Fix b ∈ B. Since T  = 1, it follows from (5) that   m   k   j  1  1 j T (b) − (7) βi T (qi )   ≤ r.  k j=1 i=1   Applying to (7) the fact that the linear map in Remark 3.4 is linear and contractive (with {pji }i as the partitions of unity), we get   k m   k m  k    j, t 1  j j  1 1 t   αi pi − 2 αi pi  ≤ , (8) k k  j=1 i=1  r t=1 i=1 j=1
m t where αj, = m l=1 βlj τ (T (qlj )pit ), and αji are as defined above. By Lemma 3.5 i there exists ρm j, t ∈ D(M) such that m  m   j j j, t t m αi pi = ρj, t βl ql , 1 ≤ j, t ≤ k. (9) i=1

l=1

Using (6), (8), and (9) we get  m   k  k   j j   2  1 T (b) − ρm βl ql  j, t  ≤ r,  2 k j=1 t=1   l=1

(10)

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By Lemma 3.6 there exist sequences (˜ ρjn )n∈N ⊆ D(M), 1 ≤ j ≤ k, independent of b, such that for every r ∈ N there exists n0 = n0 (r, b) such that if n ≥ n0 then   m  1    j j j (11) βl ql − ρ˜n (b) ≤ , 1 ≤ j ≤ k.   r  l=1

From (10) and (11), together with the fact that each ρ ∈ D(M) is contractive we get, for every n ≥ n0 (r, b)     k  k   3  m j ≤ , T (b) − 1 (12) ρ (˜ ρ (b)) j, t n  r  2 k j=1 t=1   Consider a dense countable subset {b1 , b2 , . . .} of B. Now define n(r), m(r) as n(r) = max{n0 (r, b1 ), . . . , n0 (r, br )}, m(r) = max{m0 (r, b1 ), . . . , m0 (r, br )}

m(r) and let ρr := k12 kj=1 kt=1 ρj, t ◦ ρ˜jn(r) ∈ D(M), where k = k(m(r)). Then, from the previous calculations, we see that T (bj ) − ρr (bj ) < 3/r whenever 1 ≤ j ≤ r. Let b ∈ B, and  > 0. Then there exists l ∈ N such that b − bl  < /3. If r > max{l, 9/}, then T (bl ) − ρr (bl ) < /3, and so T (b) − ρr (b) ≤ .
Corollary 3.8. Let T ∈ DS(M) and let (ai )ni=1 , (bi )ni=1 ⊆ Msa be abelian families such that T (bi ) = ai , 1 ≤ i ≤ n. Then there exists a sequence (ρr )r∈N ⊆ D(M) such that for 1 ≤ i ≤ n limr→∞ ai − ρr (bi ) = 0. Proof. Consider A = C ∗ (a1 , . . . , an ) and B = C ∗ (b1 , . . . , bn ), which are separable abelian C ∗ -subalgebras of M. Applying Theorem 3.7 to these algebras we get a sequence (ρr )r∈N ⊆ D(M) such that limr→∞ T (b) − ρr (b) = 0 for every b ∈ T −1 (A) ∩ B. By hypothesis bi ∈ T −1 (A) ∩ B and so T (bi ) − ρr (bi ) = ai − r → 0.
ρr (bi ) −

4. Doubly stochastic kernels and joint majorization We begin by introducing doubly stochastic kernels, which are a natural generalization of doubly stochastic matrices (see Example 4.2). We shall use them to define joint majorization in analogy with [22]. Definition 4.1. Let (X, µX ), (Y, µY ) be two probability spaces. A positive unital ∞ ∞ linear  map ν : L (Y, µY ) → L (X, µX ) is said to be a doubly stochastic kernel if X ν(1∆ ) dµX = µY (∆), for every µY -measurable set ∆ ⊆ Y . Doubly stochastic kernels between probability spaces are norm continuous and normal. Example 4.2. Let X and Y be compact spaces and let µX and µY be regular Borel probability measures in X and Y respectively. Consider D ∈ L1 (µX × µY ) and let

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 ν(f )(x) = Y D(x, y) f (y) dµY (y). Then ν : L∞ (X, µX ) → L∞ (Y, µY ) is a doubly stochastic kernel if and only if D(x, y) ≥ 0 (µX × µY )-a.e. and   D(x, y) dµX (x) = 1, µY -a.e., D(x, y) dµY (y) = 1, µX -a.e. X

Y

In particular, if µX = µY is a measure with finite support {xi }m i=1 and such that 1 µX ({xi }) = m for 1 ≤ i ≤ m then D is a doubly stochastic kernel if and only if the matrix (D(xi , xj ))i, j is an m × m doubly stochastic matrix. Proposition 4.3. Let a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 ⊆ Msa be abelian families. Then the following statements are equivalent: 1. There exists T ∈ DS(M) such that T (bi ) = ai , 1 ≤ i ≤ n. 2. There exists a doubly stochastic kernel ν : L∞ (σ(¯b), µ¯b ) → L∞ (σ(¯ a), µa¯ ) such that ν(πi ) = πi , 1 ≤ i ≤ n. Proof. Assume first that T (bi ) = ai , 1 ≤ i ≤ n, with T ∈ DS(M). Let A = C ∗ (a1 , . . . , an ) and B = C ∗ (b1 , . . . , bn ). As M is a finite von Neumann algebra, there exists a conditional expectation PA : M → L∞ (A) that commutes with τ . Then ν = Λ−1 a ¯ ◦ PA ◦ T ◦ Λ¯ b is the desired doubly stochastic kernel. Conversely, let us assume the existence of ν as in (2). Let PB : M → L∞ (B) be the conditional ◦ PB ∈ expectation onto L∞ (B) that commutes with τ . Define T = Λa¯ ◦ ν ◦ Λ¯−1 b
DS(M). Clearly T (bi ) = ai , 1 ≤ i ≤ n. Definition 4.4. Let a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 be two abelian families in Msa . We say that a ¯ is jointly majorized by ¯b (and we write a ¯ ≺ ¯b) if there exists a doubly stochastic kernel ν : L∞ (σ(¯b), µ¯b ) → L∞ (σ(¯ a), µa¯ ) such that ν(πi ) = πi , 1 ≤ i ≤ n. If (x1 , . . . , xn ) is a finite family in M, let UM (x1 , . . . , xn ) denote the joint unitary orbit of the family with respect to the unitary group UM of M, i.e. UM (x1 , . . . , xn ) = {(u∗ x1 u, . . . , u∗ xn u) : u ∈ UM }. We shall also consider the convex hull of the unitary orbit of a family (xi )ni=1 , conv(UM (xi )ni=1 ) = {(ρ(xi ))ni=1 , ρ ∈ D}. We denote by conv(UM (xi )ni=1 ), convw (UM (xi )ni=1 ) and conv1 (UM (xi )ni=1 ) the respective closures in the coordinate-wise norm topology, coordinate-wise weak operator topology, and coordinate-wise L1 topology. Theorem 4.5. Let a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 be abelian families in Msa . Then the following statements are equivalent: 1. a ¯ ≺ ¯b. 2. a ¯ ∈ conv(UM (¯b)). 3. a ¯ ∈ conv 1 (UM (¯b)). 4. a ¯ ∈ conv w (UM (¯b)). 5. µa¯ ≺ µ¯b . 6. There exists a completely positive T ∈ DS(M) such that ai = T (bi ), 1 ≤ i ≤ n.

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7. There exists T ∈ DS(M) such that ai = T (bi ), 1 ≤ i ≤ n. 8. τ (f (a1 , . . . , an )) ≤ τ (f (b1 , . . . , bn )) for every continuous convex function f : Rn → R. The proof of Theorem 4.5 will be split into several lemmas, and all the pieces will be put together at the end of the section. Remark 4.6. Let x ∈ M be a normal operator. Recall (see the last paragraph of Section 2.1) that there is a natural way to identify the usual spectral measure of x with that of the abelian pair (Re(x), Im(x)). If T ∈ DS(M), then since T is positive T (x) = y if and only if T (Re(x)) = Re(y) and T (Im(x)) = Im(y). Using Theorem 4.5, we see that if x, y ∈ M are normal operators then x ≺ y in the sense of [16] if and only if (Re(x), Im(x)) ≺ (Re(y), Im(y)) in the sense of Definition 4.4. Let PN denote the trace preserving conditional expectation onto the abelian von Neumann subalgebra N ⊆ M. Using Theorem 4.5 we can then obtain a generalization of Theorem 7.2 in [10]. Corollary 4.7. Let N ⊆ M be an abelian von Neumann subalgebra and let (bi )ni=1 ⊆ Msa be an abelian family. Then (PN (bi ))ni=1 ≺ (bi )ni=1 . In the remainder of this section we prove the implications needed to prove Theorem 4.5. The single variable case of the following lemma can be found in [16]. Lemma 4.8 ((4)⇒(6) in Theorem 4.5). Let a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 ⊆ Msa be abelian families and assume that a ¯ ∈ conv w (UM (¯b)). Then there exists a completely positive T ∈ DS(M) such that ai = T (bi ), 1 ≤ i ≤ n. weakly

Proof. Let {(bj1 , . . . , bjn )}j∈J ⊆ conv(UM (b1 , . . . , bn )) such that bji −−−−→ ai , 1 ≤ j

i ≤ n. Then there exists a sequence (ρj )j∈J ⊆ D(M) such that (bj1 , . . . , bjn ) = (ρj (b1 ), . . . , ρj (bn )), for every j ∈ J. Note that each ρj is a completely positive doubly stochastic map. Since the net {ρj }j∈J is norm bounded, it has an accumulation point in the BW topology [9], i.e. there exists a subnet (which we still call weakly

{ρj }j∈J ) and a completely positive map T : M → M such that ρj (x) −−−−→ T (x) j

if x ∈ M. By normality of the trace, T is trace preserving, positive and unital. weakly
Since ρj (bi ) = bji −−−−→ ai , we have T (bi ) = ai , 1 ≤ i ≤ n. j

Lemma 4.9 ((1)⇒(5) in Theorem 4.5). Let a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 ⊆ Msa be ¯ abelian families. If a ¯ ≺ b, then µa¯ ≺ µ¯b . Proof. The hypothesis a ¯ ≺ ¯b means that there exists a doubly stochastic kernel ∞ ∞ ¯ ) → L (σ(¯ a), µa¯ ) such that ν(πi ) = πi , 1 ≤ i ≤ n. Let ν1 , . . . , νm ∈ ν : L (σ(b), µ¯b
m ∼ M+ (Rn ) with j=1 νj = µa¯ . Define measures ν j by νj (∆) = νj (ν(1∆ )). By continuity of ν, νj (f ) = νj (ν(f )) for every f ∈ L∞ (σ(¯b), µ¯b ). So ν j (πi ) = νj (ν(πi )) = νj (πi ), 1 ≤ i ≤ n and 1 ≤ j ≤ m, and similarly νj (1) = ν j (1), so that νj ∼ ν j ,
m
 for 1 ≤ j ≤ m. Finally, m ¯ (ν(1∆ )) = µ¯ b (∆). So j=1 ν j (∆) = j=1 νj (ν(1∆ )) = µa
m  ν = µ . We conclude that µ ≺ µ .
¯ ¯ a ¯ b b j=1 j

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Lemma 4.10 ((5)⇒(1) in Theorem 4.5). Let a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 ⊂ Msa be abelian families. If µa¯ ≺ µ¯b , then there exists T ∈ DS(M) such that T (bi ) = ai , 1 ≤ i ≤ n. m(r)

a) with diam(∆rj ) Proof. By compactness we can consider partitions {∆rj }j=1 of σ(¯ r r less than 1/r for every 1 ≤ j ≤ m. Fix points x1 , . . . , xm(r) with xrj ∈ ∆rj and
define measures µrj by µrj (· ) = µa¯ (· ∩ ∆rj ). Then clearly j µrj = µa¯ . As µa¯ ≺ µ¯b
by hypothesis, there exist measures νjr with νjr ∼ µrj and j νjr = µ¯b . Let gjr be
the Radon-Nikodym derivatives gjr = dνjr /dµ¯b . Note that j gjr = 1 (µ¯b − a.e.). a) × σ(¯b) → R by Define a function Dr : σ(¯ 

m(r)

Dr (s, t) =

j=1

gjr (t) 1∆r (s). µa (∆rj ) j

We will use the kernels Dr to approximate T in the BW topology. Let us define a), µa¯ ) by νr : L∞ (σ(¯b), µ¯b ) → L∞ (σ(¯  b(t) Dr (s, t) dµ¯b (t). νr (b)(s) = σ(¯ b)

The map νr can be seen to be doubly stochastic using the equivalence µrj ∼ νjr . By Proposition 4.3 there is an associated sequence {Tr }r ⊂ DS(M) such that  Tr (bi ) = σ(¯a) νr (πi ) dEa¯ ∈ L∞ (A), 1 ≤ i ≤ n. The bounded net {Tr }r∈N has a subnet {Tk }k∈K that converges to a cluster point T ∈ DS(M) in the BW topology. Since this subnet is bounded, T (bi ) = w- limk∈K Tk (bi ) ∈ L∞ (A). We claim that T (bi ) = ai , 1 ≤ i ≤ n. To see this, since the net {Tk (bi )}k∈K is bounded, we just have to prove that lim τ (x Tk (bi )) = τ (x ai ), 1 ≤ i ≤ n, ∀x ∈ A. k

Equivalently, we have to show that for every continuous function f ∈ C(σ(¯ a)) and every i = 1, . . . , n,     f (s) Dk (s, t) πi (t) dµ¯b (t) dµa¯ (s) = f (s) πi (s) dµa¯ (s). lim k

σ(¯ a)

σ(¯ b)

σ(¯ a)

This can be seen by a standard approximation argument, using the uniform continuity of f , the fact that the diameters of ∆rj tend to 0 as r increases, and the
equivalence µrj ∼ νjr . Proof of Theorem 4.5. Proposition 4.3 shows the equivalence (7)⇔(1) and Corollary 3.8 is (7)⇒(2). The implication (2)⇒ (3)⇒(4) is trivial. Lemma 4.8 shows that (4)⇒(6), and it is clear that (6)⇒(7). Lemmas 4.9, 4.10 and Proposition 4.3 prove the equivalence (5)⇔(1). So we have that (1)-(7) are equivalent. Finally, Corollary 2.3 shows that (5)⇔(8).


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5. Joint unitary orbits of abelian families in Msa Given families a ¯ = (ai )ni=1 , ¯b = (bi )ni=1 ⊆ M, we say that a ¯ and ¯b are jointly approximately unitarily equivalent in M if a ¯ ∈ UM (¯b), that is if there exists a sequence of unitary operators (un )n∈N ⊆ M such that limn→∞ un bi u∗n − ai  = 0 for every i = 1 . . . , n. It is clear that this is an equivalence relation. Moreover, if a ¯ and ¯b are jointly approximately unitarily equivalent in M then a ¯ is an abelian family if and only if ¯b is. In [10] a characterization of approximately unitarily equivalence between selfadjoint operators is obtained in terms of the spectral distributions. The main result of this section, Theorem 5.1, characterizes this relation for abelian families in Msa . Theorem 5.1. Let a ¯ = (ai )ni=1 and ¯b = (bi )ni=1 ⊂ Msa be abelian families. Then the following statements are equivalent: 1. a ¯ and ¯b are jointly approximately unitary equivalent in M. 2. a ¯ ≺ ¯b and ¯b ≺ a ¯ 3. µa¯ = µ¯b 4. τ (f (a1 , . . . , an )) = τ (f (b1 , . . . , bn )) for every continuous convex function f : Rn → R. 5. τ (f (a1 , . . . , an )) = τ (f (b1 , . . . , bn )) for every continuous function f : Rn → R. Proof. By Theorem 4.5 we have (1)⇒(2) and (2)⇔(4). Moreover, (4) is equivalent to µa¯ (f ) = µ¯b (f ) for every convex function f . Then µa¯ (f ) = µ¯b (f ) for every continuous function f [3, Proposition I.1.1], and so µa¯ = µ¯b . Therefore (4)⇒(5)⇒(3). Again, by Theorem 4.5 (3)⇒(2) and so (2)-(5) are equivalent. Finally, we prove a) = supp µa¯ = supp µ¯b = σ(¯b) that (3)⇒(1). If we assume that µa¯ = µ¯b then σ(¯ and for every Borel set ∆ in σ(¯ a) we have τ (Ea¯ (∆)) = µa¯ (1∆ ) = µ¯b (1∆ ) = τ (E¯b (∆)).

(13)

Let  > 0. By compactness, choose Borel sets B1 , . . . , Bm to be a finite disjoint covering of σ(¯ a) = σ(¯b) such that there are points xj ∈ Bj with the property that |πi (λ) − πi (xj )| < /2 for every λ ∈ Bj , 1 ≤ i ≤ n, 1 ≤ j ≤ m. Then we get, using the Spectral Theorem,         m m            ai − πi (xj )Ea¯ (Bj )  < ,  bi − πi (xj )E¯b (Bj )  < 2  2     j=1 j=1 for i = 1, . . . , n. From equation (13) we get that τ (Ea¯ (Bj )) = τ (E¯b (Bj )) for every j = 1, . . . , m. As in the proof of Lemma 3.5, we get a unitary w ∈ U(M) such that w∗ E¯b (Bj )w = Ea¯ (Bj ) for every j. Then   m m   πi (xj )E¯b (Bj ) w = πi (xj )Ea¯ (Bj ). w∗  j=1

j=1

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Finally, for every i we have

      m     ∗ ∗    w bi w − ai  ≤  w bi − πi (xj )E¯b (Bj ) w   + 2 < .   j=1




Corollary 5.2. Let Θ be a ∗-automorphism of M. Then Θ|A is approximately inner for each separable abelian C ∗ subalgebra A ⊂ M. Proof. The uniqueness of the trace guarantees that Θ is trace-preserving. Being multiplicative, the range of an abelian set will be again abelian. So Θ is a DS map that takes an abelian family in M into another. Consider a countable dense subset {ai } of A, and use Theorem 5.1 to obtain unitaries un for each finite subset {a1 , . . . , an }. An /3 argument shows then that the sequence {Ad un } approximates Θ in all of A.
s

x) the closure in the coordinateGiven x ¯ = (xi )ni=1 ⊆ M we denote by UM (¯ wise strong operator topology. An immediate consequence of Theorem 5.1 is that the norm closure of the unitary orbit of a selfadjoint abelian family in a II1 factor is strongly closed. This generalizes [10, Theorem 5.4] and [26, Theorem 8.12(1)]: 

Corollary 5.3. Let a ¯ = (ai )ni=1 ⊆ Msa be an abelian family. Then UM (¯ a) s UM (¯ a) .

=

s Proof. Let ¯b = (bi )ni=1 ∈ UM (¯ a) . There exists a net (bj1 , . . . , bjn )j∈J ⊆ UM (¯ a) such that bji converges strongly to bi for each i = 1, . . . , n. Let f : Rn → R be a continuous function. Then τ (f (bj1 , . . . , bjn )) = τ (f (a1 , . . . , an )) for every j. Using [27, Lemma II.4.3] we conclude that τ (f (b1 , . . . , bn )) = τ (f (a1 , . . . , an )). So (5) of a). The other inclusion is trivial.
Theorem 5.1 implies that ¯b ∈ UM (¯

6. Some technical results In this section we prove several results presented in Section 3. First we show that any separable abelian C ∗ -subalgebra of M can be embedded into a separable diffuse abelian C ∗ -subalgebra. Then we prove some approximation results that hold for separable diffuse abelian C ∗ subalgebras of M. 6.1. Refinements of spectral measures We begin by recalling some elementary facts about inclusions of abelian C ∗ algebras. If A ⊆ B are unital abelian C ∗ -algebras, then the function Φ : Γ(B) → Γ(A) given by Φ(γ) = γ|A is a continuous surjection onto Γ(A). If we assume further that A ⊆ B ⊆ M are separable and that EA , EB denote their spectral measures, then EA = EB ◦ Φ−1 and µA = µB ◦ Φ−1 . Note that At(µA ) =At(EA ) where At(EA ) is
the set of atoms of the spectral measure EA (see the beginning of Section 3). Let x∈At(EA ) µA ({x}) be the total

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atomic mass of EA . Since µA is finite, the total atomic mass is bounded and thus the set of atoms is a countable set. The two results below lead to the proof of Theorem 3.2. Lemma 6.1. With the notations above, if x ∈ At (EB ) then Φ(x) ∈ At (EA ), and the total atomic mass of B is smaller that the total atomic mass of A. Proof. Let x ∈ At(EB ) and note that 0 < µB ({x}) ≤ µB (Φ−1 (Φ({x}))) = µA (Φ({x})), so Φ(x) ∈ At(EA ) = At(µA ). We consider the equivalence relation in At(EB ) induced by Φ, i.e. x ∼ y if Φ(x) = Φ(y). If Q ∈ Q := At(EB )/
∼ is such that Φ(x) = xQ for every x ∈ Q, then using that Q is countable we get x∈Q µB ({x}) = µB (Q) ≤ µB (Φ−1 ({xQ })) = µA ({xQ }). Therefore      µB ({x}) = µB ({x}) ≤ µA ({xQ }) ≤ µA ({x}). x∈At(EB )

Q∈Q x∈Q

Q∈Q

x∈At(EA )




Proposition 6.2. With the notations above, let x0 ∈ Γ(A) be an atom of EA and let α, β ∈ R with 0 < α < β. Then there exists a ∈ A ∩ Msa with [α, β] ⊆ σ(a) ⊆ [α, β] ∪ {0}, PR(a) = EA ({x0 }), and such that if B = C ∗ (A, a) ⊂ M, then EB has no atoms in the fibre Φ−1 (x0 ). ˜ Then pA˜ Proof. Let p = EA ({x0 }) and consider a masa A˜ ⊂ M such that A ⊂ A. 1 τ . It is well known that is a masa in the II1 factor pMp, where the trace is τp = τ (p) there exists a countably generated, non-atomic von Neumann subalgebra A˜0 of pA˜ such that there is a von Neumann algebra isomorphism Φ : L∞ ([0, 1], m) → A˜0 , 1 ˜ = Φ(id); with m the Lebesgue measure on [0, 1], and τp (Φ(f )) = 0 f dm. Put a it is clear that a ˜ has no atoms in its spectrum with the exception of 0, and that a +α p, so [α, β] ⊆ σ(a) ⊆ [α, β]∪{0}, Ea˜ ({0}) = 1−p, σ(a) = [0, 1]. Let a = (β −α)˜ PR(a) = p = EA ({x0 }). As p is a minimal projection in L∞ (A), for every b ∈ A we have pb = pbp = λb p and so ab = apb = λb pa = bpa = ba. Thus a ∈ A ∩ M. Let B = C ∗ (A, a) and let Φ : Γ(B) → Γ(A), Ψ : Γ(B) → Γ(C ∗ (a)) be the continuous surjections induced by the inclusions A ⊆ B and C ∗ (a) ⊆ B. Note that the restriction Ψ|Φ−1 (x0 ) is injective. Indeed, let x, y ∈ Φ−1 (x0 ) be such that Ψ(x) = Ψ(y), i.e. the restriction of the characters to C ∗ (a) coincide. Since Φ(x) = Φ(y) (= x0 ), the characters also coincide on A and therefore are equal as characters in B, since B is generated by A and C ∗ (a). On the other hand, if x ∈ Γ(B) is such that x(a) = 0, then Φ(x) = x0 . Indeed, assume that Φ(x) = x0 . Let f ∈ C(Γ(A)) with f (Φ(x)) = 0 and f (x0 ) = 1. So f ◦ Φ ≥ 1Φ−1 (x0 ) . But then   f ◦ Φ dEB ≥ 1Φ−1 (x0 ) dEB = EB (Φ−1 (x0 )) = EA ({x0 }) = p. Γ(B)

Γ(B)

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Note that if 0 ∈ σ(a) then it is an isolated point, and so in any case we have p ∈ C ∗ (a) ⊆ B. Then 0 = f ◦ Φ(x) ≥ x(p) ≥ 0, so x(p) = 0. Since 0 ≤ a ≤ β p, x(a) = 0 and the claim follows. Now let z ∈ Φ−1 (x0 ). If z(a) = 0, from the first part of the proof we deduce that Ψ−1 (Ψ(z)) = {z}. Therefore EB ({z}) = EA ({Ψ(z})) = 0, since Ψ(z)(a) = 0 and At(EA ) ⊆ {0}. If z(a) = 0, then {z} = =

Φ−1 (x0 ) \ {x ∈ Φ−1 (x0 ) : x(a) = 0} Φ−1 (x0 ) \ Ψ−1 ({x ∈ Γ(C ∗ (a)) : x(a) = 0})

and EB (Ψ−1 ({x ∈ Γ(C ∗ (a)) : x(a) = 0})) = = From this we conclude that EB ({z}) = 0.

EA ({x ∈ Γ(C ∗ (a)) : x(a) = 0}) EA ({x0 }) = EB (Φ−1 (x0 )).


Proof of Theorem 3.2. Recall that the set At(EA ) of atoms of EA is a (possibly infinite) countable set. If At(EA ) = ∅ then EA is already diffuse and we are done. Otherwise, let us enumerate At(EA ) = {xi : 1 ≤ i ≤ r}, where r ∈ N ∪ {∞}. For

1 1 , 1+ 2n−1 ]. Then Ii ∩ 1≤i =j≤r Ij = ∅ and ri=1 Ii ⊆ [1, 2]. 1 ≤ i ≤ r, let Ii = [1+ 2n For each i = 1, . . . , r there exists, by Proposition 6.2, ai ∈ A ∩ Msa such that PR(ai ) = EA ({xi }), Ii ⊆ σ(ai ) ⊆ Ii ∪ {0}, and such that EAi has no atoms

in the fibre Φ−1 the continuous surjection i (xi ), where Φi : Γ(Ai ) → A denotes
induced by the inclusion A ⊆ Ai := C ∗ (A, ai ). Let a = ri=1 ai ∈ A ∩ Msa (this sum converges strongly because the ranges of the operators ai are orthogonal and ai  ≤ 2 for every i). Then B = C ∗ (A, a) is an abelian subalgebra of M. We claim that the spectral measure EB of B has no atoms. Indeed, first note that 1Ii ∈ C(∪1≤j≤r Ij ) is a continuous function (because the distance between the sets Ii and ∪i =j Ij is positive); then, since 1Ii (a) = ai , it follows that Ai ⊂ B for every i = 1, . . . , r. Assume now that x ∈ At(Γ(B)) and let Φ : Γ(B) → Γ(A) be as before. By Lemma 6.1 there exists i ∈ {1, . . . , r} such that Φ(x) = xi ∈ At(EA ) . Since Φ = Φi ◦ Ψi , where Ψi : Γ(B) → Γ(Ai ) is the surjection induced by the inclusion Ai ⊆ B, we conclude that Ψi (x) ∈ Φ−1 i (xi ) is an atom of the measure EAi , again by Lemma 6.1. But this last assertion is a contradiction because by
construction there are no atoms in the fibre Φ−1 i (xi ).

Remark 6.3. Given an abelian C ∗ subalgebra A ⊂ M, a direct way to find an abelian C ∗ -subalgebra A ⊆ A˜ ⊂ M with diffuse spectral measure is to consider a masa in M that contains A. Theorem 3.2 shows that A˜ can be chosen separable (as a C ∗ -algebra) whenever A is separable. When this is the case, the character space of A˜ is metrizable, a fact that is crucial for our calculations. 6.2. Discrete approximations in separable diffuse abelian algebras Given a compact metric space it is always possible to find, using uniform continuity, discrete uniform approximations of continuous functions by linear combinations of characteristic functions of certain sets {Qi }m i=1 . But if we consider a measure on

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this space and we require equal measures for these sets, there might not be any good uniform approximation based on characteristic functions (even for measures of compact support in the real line). Proposition 3.3 is an intermediate solution to this problem. It was inspired by the proof of [16, Lemma 4.1]. The idea is to use convex combinations to “distribute” the part of the projections that does not fit in an equal measure partition. Proof of Proposition 3.3. As the space Γ(B) is a metrizable compact topological space, we consider a metric d in Γ(B) inducing its topology. Let r ∈ N; by ˜ i ) < 1 and ˜ i }k0 of Γ(B) with diamd (Q compactness, there exists a partition {Q i=1 r
k0 ˜ i ) = 1. Let m = m(r) be such that 1/m ≤ min{µB (Q ˜ j )2 : 1 ≤ j ≤ k0 }. µ ( Q B i=1 ˜ j ) = kj /m + δj with 0 ≤ Then for 1 ≤ j ≤ k0 there exists kj ∈ N such that µB (Q ˜ ˜ j )−1 , 1 ≤ j ≤ k0 }. δj < 1/m. If we let k˜ = k(r) = minj {kj } then k˜ ≥ max{µB (Q k j t ˜ ˜ } ˜ For t = 1, . . . , k0 , choose k˜ partitions {Q j,s s=0 of each Qj (1 ≤ t ≤ k), with t t t ˜ ˜ ˜ ˜1 , µB (Qj,s ) = 1/m if 1 ≤ s ≤ kj and µB (Qj,0 ) = δj , in such a way that Qj,0 ⊂ Q j, t ˜ Note that we can always make such a choice: using Lemma 2.4 choose 2 ≤ t ≤ k. ˜ 1 with µB (Q ˜ t ) = δj < 1/m, and then take a partition {Q ˜ t }kj of ˜t ⊆ Q Q j,0 j, t j,0 j,s s=1 ˜j \ Q ˜ t using again Lemma 2.4 (note that µB (Q ˜j \ Q ˜ t ) = kj /m). By this choice, Q j,0 j,0 ˜t ∩ Q ˜ t
= ∅ if t = t  . Q j,0 j,0 ˜ let Q ˜ t . Then ˜ t0,0 = ∪k0 Q For each t = 1, . . . , k, j,0

j=1

˜t ) = 1 − µB (Q 0,0

 kj j

m

=

k0  1 (m − kj ). m j=1

˜ t into n1 = m −
kj subsets {Q ˜ t }n1 of Finally, make partitions of each set Q 0,0 i i=1 j ˜ t }i , we end up with k˜ ˜ t }j, s ∪ {Q measure 1/m. By re-labeling the k˜ partitions {Q j,s i m m ˜ such that: }i=1 , for 1 ≤ t ≤ k, partitions {Qt, i t, m ˜ 1. µB (Q ) = 1/m, for every i ∈ {1, . . . , m}, t ∈ {1, . . . , k}; i

m 2. diamd (Qt, ) ≤ 1/r, if i > n1 ; i
m 3. if 1 ≤ i, i  ≤ n1 then Qt, ∩ Qti
, m = ∅ if i = i  or t = t  . i

m m }i=1 was done in such Note that the construction of the k partitions {Qt, i a way that the subsets that do not have small diameters are disjoint, even for different partitions. ˜ Let M = {m(r), r ≥ 1} and for every m = m(r) ∈ M let k(m) = k(r) as det,m t, m fined above and, for i, t, m, let qi = EB (Qi ). The set M is unbounded because ˜ the measure µB being diffuse makes limr→∞ m(r) = ∞, and so limr→∞ k(r) = ∞.  For each t = 1, . . . , k, {qit,m }m ⊂ B ∩ M is a partition of unity. i=1  Let b ∈ B,  > 0, and let f ∈ C(Γ(B)) be such that b = Γ(B) f dEB . Then, by compactness, there exists δ > 0 such that if Q ⊆ Γ(B) with diamd (Q) < δ then diam(f (Q)) < . Let r ∈ N be such that 1/r < δ and 2b/k(r) ≤ ; let

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m = m(r) ∈ M, and let βit,m = m τ (b qit,m ) = m translate into



Qt,m i

17

f dµB . Properties 1-3

1’. τ (qit,m ) = 1/m, for every i ∈ {1, . . . , m}, t ∈ {1, . . . , k}; 2’. if i > n1 , then |f (x) − βit,m | ≤ , ∀x ∈ Qt,m i ; t,m t
,m  3’. if 1 ≤ i, i ≤ n1 then qi ⊥ qi
if i = i  or t = t  . Therefore we have   k m  1   t,m t,m    βi qi  = b −   k t=1 i=1

=

   k m 1    t,m t,m  b− βi qi   k  t=1

i=1

  k  m  1    t,m (f − βi ) dEB   k  Qt,m i t=1 i=1

≤

  n1  k   1   (f − βit,m ) dEB  +   t,m  k Qi t=1 i=1

≤

  n1 k   2 b    t,m  qi  +    k  t=1 i=1

=

2b +  ≤ 2, k

where the first inequality is a consequence of 2’ and the last equality follows from 3’.
Proof of Lemma 3.6. Fix a norm dense subset B = (bj )j∈N ⊆ B. In the construction leading to Dixmier’s Theorem, a previous result [20, 8.3.4] asserts that for each j, there exists a sequence {ρnj }n∈N ⊆ D(M) such that for every 1 ≤ h ≤ j, n ρnj (bh ) − τ (bh ) I − → 0. For each j ∈ N, let n0 = n0 (j) ∈ N be such that if n ≥ n0 n (j)

then ρnj (bh ) − τ (bh ) I ≤ 1/j for 1 ≤ h ≤ j. If we let ρj = ρj 0 j

for j ∈ N, we

get ρj (bh ) − τ (bh ) I − → 0 for every h ∈ N. Since (bj )j∈N is norm dense in B we have limj ρj (b) − τ (b) I = 0 for every b ∈ B. For every i = 1, . . . , m, consider the factor pi Mpi with (normalized) trace τi (pi x) = τ (xpi )/τ (pi ) . By the Dixmier approximation property mentioned in the first paragraph, applied to the separable C ∗ -subalgebra pi B of the finite factor pi Mpi , there exists a sequence {ρij }j∈N ∈ D(pi Mpi ) such that limj→∞ ρij (pi b) − τi (pi b)pi  = 0, for every b ∈ B. For each ρ ∈ D(pi Mpi ), we can consider an extension ρ˜ ∈ D(M) as follows:
if ρ(pi b) = kh=1 λh uh b u∗h , with uh ∈ U(pi Mpi ), define ρ˜ ∈ D(M) by ρ˜(b) =
k m ˜h b u ˜∗h , where u ˜h = uh + (1 − pi ) ∈ U(M). If 1 ≤ i ≤ m set ρj = i=1 ρ˜ji h=1 λh u for j ≥ 1. It is easy to verify that if 1 ≤ i ≤ m then ρj (b pi ) = ρ˜ji (b pi ) for every

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b ∈ B. Thus, if b ∈ B, then     m m          βi (b)pi  =  ρ˜ji (b pi ) − τi (b pi )pi  −−−→ 0. ρj (b) −    j→∞  i=1

i=1


Ackowledgements. We wish to thank Professors D. Farenick and D. Stojanoff for their support and useful discussions regarding the material in this paper.

References [1] P.M. Alberti and A. Uhlmann, Stochasticity and partial order. Doubly stochastic maps and unitary mixing. Mathematische Monographien, 18. VEB Deutscher Verlag der Wissenschaften, Berlin, 1981. [2] P.M. Alberti, and A. Uhlmann, Dissipative motion in state spaces. Teubner-Texte zur Mathematik, 33. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1981. [3] E.M. Alfsen, Compact Convex Sets and Boundary Integrals, Springer-Verlag, New York, NY 1971. [4] T. Ando, Majorization, doubly stochastic matrices and comparison of eigenvalues, Lecture Notes, Hokkaido Univ., 1982. [5] J. Antezana, P. Massey, and D. Stojanoff, Jensen’s Inequality and Majorization, J. Math. Anal. App. 331 1 (2007), 297–307. [6] J. Antezana, P. Massey, M. Ruiz and D. Stojanoff, The Schur-Horn theorem for operators and frames with prescribed norms and frame operator, Illinois J. of Math. 51 2 (2007), 537–560. [7] M. Argerami and P. Massey, A Schur-Horn Theorem in II1 factors, Indiana Univ. Math. J., 56, 5 (2007), 2051–2060. [8] M. Argerami and P. Massey, A contractive version of a Schur-Horn theorem in II1 factors, J. Math. Anal. App., 337, 1 (2008), 231–238. [9] W. Arveson, Subalgebras of C ∗ -algebras Acta Math. 123 (1969), 141–224. [10] W. Arveson and R. Kadison, Diagonals of self-adjoint operators, In D. R. Larson D. Han, P. E. T. Jorgensen, editor, Operator theory, operator algebras and applications, Contemp. Math. Amer. Math. Soc., 2006. [11] G. Birkhoff, Three observations on linear algebra. (Spanish) Univ. Nac. Tucum´ an. Revista A. 5 (1946), 147–151. [12] J.B. Conway, A course in functional analysis, Springer-Verlag, New York, NY 1990. [13] T. Fack, Sur la notion de valeur caract´eristique, J. Operator Theory (1982), 307–333. [14] D.R. Farenick and S.M. Manjegani, Young’s Inequality in Operator Algebras, J. Ramanujan Math. Soc. 20, 2 (2005), 107–124. [15] F. Hiai, Majorization and Stochastic maps in von Neumann algebras, J. Math. Anal. Appl. 127, 1 (1987), 18–48. [16] F. Hiai, Spectral majorization between normal operators in von Neumann algebras, Operator algebras and operator theory (Craiova, 1989), 78–115, Pitman Res. Notes Math. Ser., 271, Longman Sci. Tech., Harlow, 1992.

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[17] F. Hiai and Y. Nakamura, Closed Convex Hulls of Unitary Orbits in von Neumann Algebras, Trans. Amer. Math. Soc. 323 (1991), 1–38. [18] R. Kadison, The Pythagorean theorem I: the finite case, Proc. N.A.S. (USA), 99, 7 (2002), 4178–4184. [19] R. Kadison, The Pythagorean theorem II: the infinite discrete case, Proc. N.A.S. (USA), 99, 8 (2002), 5217–5222. [20] R. Kadison and J. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. II, Academic Press, Orlando, Florida, 1986. [21] E. Kamei, Majorization in finite factors, Math. Japonica 28, 4 (1983), 495–499. [22] F.D. Mart´ınez Per´ıa, P. Massey, L. Silvestre, Weak matrix majorization. Linear Algebra Appl. 403 (2005), 343–368. [23] P. Massey, Refinements of spectral resolutions and modelling of o perators in II1 factors, J. Operator Theory, to appear. [24] A. Neumann, An infinite-dimensional version of the Schur-Horn convexity theorem, J. Funct. Anal. 161 (1999), 418–451. [25] W. Rudin, Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. [26] D. Sherman, Unitary orbits of normal operators in von Neumann algebras, Journal fur die Reine und Angewandte Mathematik 605 (2007), 95–132. [27] M. Takesaki, Theory of Operator Algebras I, Encyclopaedia of Mathematical Sciences V, Springer Verlag, 2nd printing of the First Edition 1979. Mart´ın Argerami Department of Mathematics and Statistics University of Regina Saskatchewan, S4S 0A2 Canada e-mail: [email protected] Pedro Massey Departamento de Matem´ atica Facultad de Ciencias Exactas Universidad Nacional de La Plata Argentina e-mail: [email protected] Submitted: October 29, 2006 Revised: December 22, 2007

Integr. equ. oper. theory 61 (2008), 21–43 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/010021-23, published online March 13, 2008 DOI 10.1007/s00020-008-1571-z

Integral Equations and Operator Theory

Resolvent Estimates for Operators Belonging to Exponential Classes Oscar F. Bandtlow Abstract. For a, α > 0 let E(a, α) be the set of all compact operators A on a separable Hilbert space such that sn (A) = O(exp(−anα )), where sn (A) denotes the n-th singular number of A. We provide upper bounds for the norm of the resolvent (zI − A)−1 of A in terms of a quantity describing the departure from normality of A and the distance of z to the spectrum of A. As a consequence we obtain upper bounds for the Hausdorff distance of the spectra of two operators in E(a, α). Mathematics Subject Classification (2000). Primary 47A10; Secondary 47B06, 47B07. Keywords. Resolvent growth, exponential class, departure from normality, bounds for spectral distance.

1. Introduction Let A and B be compact operators on a Hilbert space. It is known that if
A − B
is small then the spectra of A and B are close in a suitable sense (for example, with respect to the Hausdorff metric on the space of compact subsets of C). Just how close are they? Standard perturbation theory gives bounds in terms of quantities that require a rather detailed knowledge of the spectral properties of both operators, for example the norms of the resolvents of A and B along contours in the complex plane, which are difficult to obtain in practice. The main concern of this article is to derive an upper bound for the norm of the resolvent (zI − A)−1 of an operator A belonging to certain subclasses of compact operators in terms of simple, readily computable quantities, typically involving the distance of z to the spectrum of A and a number measuring the departure from normality of A. As a result, we obtain simple upper bounds for the Hausdorff distance of the spectra of two operators in these subclasses. Estimates of this type have previously been obtained for operators in the Schatten classes

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(see [Gil, Ban]) and more generally (but less sharp), for operators belonging to Φ-ideals (see [Pok]). These subclasses, termed exponential classes, are constructed as follows. For a, α > 0 let E(a, α) denote the collection of all compact operators on a separable Hilbert space for which sn (A) = O(exp(−anα )), where sn (A) denotes the n-th singular number of A. As we shall see, E(a, α) is not a linear space (see Remark 2.9), hence a fortiori not an operator ideal, and may thus be viewed as a slightly pathological object in this context. There is nevertheless compelling reason to consider these classes: on the one hand, the resolvent bounds given in [Ban, Gil, Pok], while applicable to operators in E(a, α), can be improved significantly (see Remark 3.2), the lack of linear structure posing almost no problem for the derivation of these improvements. On the other hand, operators belonging to exponential classes arise naturally in a number of different ways. For example, if A is an integral operator with real analytic kernel given as a function on [0, 1]d × [0, 1]d , then A ∈ E(a, 1/d) for some a > 0 (see [KR]). Other examples of operators in the exponential class E(a, 1/d) for some a > 0 include composition operators on Bergman spaces over domains in Cd whose symbols are strict contractions, or more generally transfer operators corresponding to holomorphic map-weight systems on Cd , the latter providing one of the motivations to look more closely into the properties of operators belonging to exponential classes (see Example 2.3 (iv) and [BanJ1, BanJ2, BanJ3]). This article is organised as follows. In Section 2 we define the exponential classes and study some of their properties. In particular, we shall give a sharp description of the behaviour of exponential classes under addition (see Proposition 2.8), and a sharp characterisation of the eigenvalue asymptotics of an operator in a given exponential class (see Proposition 2.10). Some of the arguments in this section rely on results concerning monotonic arrangements of sequences, which are presented in the Appendix. In Section 3 we will use techniques similar to those already employed in [Ban] to obtain resolvent estimates for operators in E(a, α) (see Theorem 3.13). In particular we shall give a sharp estimate for the growth of the resolvent of a quasi-nilpotent operator in E(a, α) (see Proposition 3.1). These estimates will then be used in the final section to deduce Theorem 4.2, which provides spectral variation and spectral distance formulae for operators in E(a, α). Notation 1.1. Throughout this article H and Hi will be assumed to be separable Hilbert spaces. We use L(H1 , H2 ) to denote the Banach space of bounded linear operators from H1 to H2 equipped with the usual norm and S∞ (H1 , H2 ) ⊂ L(H1 , H2 ) to denote the closed subspace of compact operators from H1 to H2 . We shall often write L or S∞ if the Hilbert spaces H1 and H2 are understood. For A ∈ S∞ (H1 , H2 ) we use sk (A) := inf {
A − F
: F ∈ L(H1 , H2 ), rank(F ) < k }

(k ∈ N)

to denote the k-th approximation number of A, and s(A) to denote the sequence {sk (A)}∞ k=1 . The spectrum and the resolvent set of A ∈ L(H, H) will be denoted by σ(A) ∞ and (A), respectively. For A ∈ S∞ (H, H) we let λ(A) = {λk (A)}k=1 denote the

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23

sequence of eigenvalues of A, each eigenvalue repeated according to its algebraic multiplicity, and ordered by magnitude, so that |λ1 (A)| ≥ |λ2 (A)| ≥ . . .. Similarly, ∞ we write |λ(A)| for the sequence {|λk (A)|}k=1 . We note that the approximation numbers coincide with the singular numbers, that is,
sk (A) = λk (A∗ A) (k ∈ N) , where A∗ ∈ L(H2 , H1 ) denotes the adjoint of A ∈ L(H1 , H2 ). For more information about these notions see, for example, [Pie, GK, DS, Rin].

2. Exponential classes Exponential classes arise by grouping together all operators A whose singular numbers sn (A) decay at a given (stretched) exponential rate, that is, sn (A) = O(exp(−anα )) for fixed a > 0 and α > 0. Our main concern in this section will be to investigate how these classes behave under addition and multiplication, and to determine the rate of decay of the eigenvalue sequence of an operator in a given class. Some of the arguments in this section depend on results concerning monotonic arrangements of sequences, which are discussed in the Appendix. Definition 2.1. Let a > 0 and α > 0. Then   E(a, α) := x ∈ CN : |x|a,α := sup |xn | exp(anα ) < ∞ , n∈N

and E(a, α; H1 , H2 ) :=



 A ∈ S∞ (H1 , H2 ) : |A|a,α := sup sn (A) exp(an ) < ∞ α

.

n∈N

are called exponential classes of type (a, α) of sequences and operators, respectively. The numbers |x|a,α and |A|a,α are called (a, α)-gauge or simply gauge of x and A, respectively. Whenever the Hilbert spaces are clear from the context, we suppress reference to them and simply write E(a, α) instead of E(a, α; H1 , H2 ). Remark 2.2. Note that E(a, α) is a Banach space when equipped with the gauge |·|a,α . On the other hand, the set E(a, α), the non-commutative analogue of E(a, α), is not even a linear space in general (see Proposition 2.8 and Remark 2.9 below). The reason for this is that if a sequence lies in E(a, α) then a rearrangement of this sequence need not; in particular E(a, α) is not a Calkin space in the sense of [Sim2, p. 26] (cf. also [Cal]). However, E(a, α; H, H) turns out to be a pre-ideal (see Remark 2.6). Operators belonging to exponential classes arise naturally in a number of different contexts.

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Example 2.3. (i) Let σ be a complex measure on the circle group T such that its Fourier transform satisfies | σ (n)| ≤ exp(−a|n|) (n ∈ Z) . It is not difficult to see that this is the case if and only if σ is absolutely continuous with respect to Haar measure on T and the corresponding Radon-Nikod´ ym derivative is holomorphic on T . Let L2 (T ) be the complex Hilbert space of square-integrable functions on T , with respect to Haar measure on T . Let A : L2 (T ) −→ L2 (T ) be the convolution operator Af = f ∗ σ . ∗σ  (Z) ∪ {0} = The spectrum of A is σ (Z) ∪ {0} and the spectrum of A∗ A equals σ
| σ (Z)| ∪ {0}, where d σ (t) = dσ(t−1 ) (cf. [BerF, p. 87]). Moreover, A is a compact operator and the non-zero eigenvalues of A are precisely the numbers σ  (n) for n ∈ Z. In order to locate A in the scale of exponential classes, we enumerate the eigenvalues of A as follows   (−1)n (2n + (−1)n − 1)  (n ∈ N). xn = σ 4 Then the sequence x belongs to the class E(a/2, 1) with |x|1/2,1 ≤ exp(a/2) since      (−1)n n σ (2n + (−1) − 1)   4     (−1)n  n  ≤ exp −a  (2n + (−1) − 1) 4 a

a

a

≤ exp − (n − 1) = exp exp − n . 2 2 2 By Corollary 5.4 the decreasing arrangement x(+) of x also belongs to E(a/2, 1) with |x(+) |1/2,1 ≤ exp(a/2). Thus s(A) = |λ(A)| ∈ E(a/2, 1) and it follows that A ∈ E(a/2, 1) with |A|a/2,1 ≤ exp(a/2). (ii) A variant of the above example is discussed by K¨onig and Richter [KR], who showed that if A is an integral operator on the space of Lebesgue squareintegrable functions on the d-dimensional unit-cube [0, 1]d whose kernel is real analytic on [0, 1]d × [0, 1]d , then A ∈ E(1/d). (iii) For a domain Ω ⊂ Rd with d > 1 let h2 (Ω) be the Bergman space of Lebesgue square-integrable harmonic functions on Ω. If Ω1 , Ω2 ⊂ Rd are two domains such that Ω2 is compactly contained in Ω1 , that is Ω2 is a compact subset of Ω1 , then the canonical embedding J : h2 (Ω1 ) → h2 (Ω2 ) given by Jf = f |Ω2 belongs to the exponential class E(1/(d − 1)). Moreover, for domains Ω1 and Ω2 with simple geometries it is possible to sharply locate J in an exponential class E(a, 1/(d − 1)) and calculate the corresponding (a, 1/(d − 1))-gauge of J exactly. See [BanC].

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(iv) For Ω ⊂ Cd a bounded domain, let L2Hol (Ω) denote the Bergman space of holomorphic functions which are square-integrable with respect to 2d-dimensional Lebesgue measure on Ω. Given a collection {φ1 , . . . , φK } of holomorphic maps φk : Ω → Ω and a collection {w1 , . . . , wK } of bounded holomorphic functions wk : Ω → C consider the corresponding linear operator A on L2Hol (Ω) given by Af :=

K 

wk · f ◦ φk .

k=1

If ∪k φk (Ω) is compactly contained in Ω (see the previous example for the definition) then A is a compact endomorphism of L2Hol (Ω) and A ∈ E(a, 1/d), where a depends on the geometry of Ω and ∪k φk (Ω) (see [BanJ3]). Operators of this type, known as transfer operators, play an important role in the ergodic theory of expanding dynamical systems due to the remarkable fact that their spectral data can be used to gain insight into geometric and dynamic invariants of a given expanding dynamical system (see [Rue]). As a consequence, it is of interest to determine spectral properties of these operators exactly, or at least to a given accuracy. The latter problem, namely that of calculating rigorous error bounds for spectral approximation procedures for these operators provided one of the main motivations to study operators in exponential classes (see [BanJ3]). We shall now study some of the properties of the classes E(a, α). First we note that if we order the indices (a, α) reverse lexicographically, that is, by defining (a, α) ≺ (a , α ) :⇔ (α < α ) or (α = α and a < a ), then we obtain the following inclusions. Proposition 2.4. Let a, a > 0 and α, α > 0. Then (i) (a, α) ≺ (a , α ) ⇔ E(a , α )  E(a, α); (ii) (a, α) ≺ (a , α ) ⇔ E(a , α )  E(a, α). Proof. The proof of (i) is straightforward and will be omitted. Assertion (ii) follows from (i) together with the observation that A ∈ E(a, α) iff s(A) ∈ E(a, α) and the fact that for every monotonically decreasing x ∈ E(a, α) there is a compact A with s(A) = x.
While E(a, α) is not, in general, a linear space, it does enjoy the following closure properties. Proposition 2.5. Let a, α > 0. If A ∈ L(H2 , H1 ), B ∈ E(a, α; H3 , H2 ), and C ∈ L(H4 , H3 ), then |ABC|a,α ≤
A
|B|a,α
C
. In particular, L(H2 , H1 ) E(a, α; H3 , H2 ) L(H4 , H2 ) ⊂ E(a, α; H4 , H1 ). Proof. Follows from sk (ABC) ≤
A
sk (B)
C
for k ∈ N (see [Pie, 2.2]).




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Remark 2.6. The proposition implies that L(H, H) E(a, α; H, H) L(H, H) ⊂ E(a, α; H, H) . Thus the classes E(a, α; H, H), while lacking linear structure, satisfy part of the definition of an operator ideal. In other words, E(a, α; H, H) is what is sometimes referred to as a pre-ideal (see, for example, [Nel]). We now consider in more detail the relation between different exponential classes under addition. We start with a general result concerning the singular numbers of a sum of operators. Proposition 2.7. Let Ak ∈ S∞ (H1 , H2 ) for 1 ≤ k ≤ K. Then K

 sn Ak ≤ Kσn (n ∈ N), k=1

where σ denotes the decreasing arrangement (see the Appendix) of the K singular number sequences s(A1 ), . . . , s(AK ). Proof. Set A := representations

K k=1

Ak . The compactness of the Ak means they have Schmidt

Ak =

∞ 

(k)

sl (Ak )al

(k)

⊗ bl ,

l=1 (k)

(k)

where {al }l∈N and {bl }l∈N are suitable orthonormal systems in H1 and H2 respectively. Here a ⊗ b, where a ∈ H1 and b ∈ H2 , denotes the rank-1 operator H1 → H2 given by (a ⊗ b)x = (x, a)H1 b . Let ν : N → N and κ : N → N be functions that effect the decreasing arrangement of the singular numbers of the Ak in the sense that σn = sν(n) (Aκ(n) ) (n ∈ N). Then A=

∞ 

(κ(n))

(κ(n))

σn aν(n) ⊗ bν(n) ,

n=1

which suggests defining, for each m ∈ N0 , the rank-m operator Fm : H1 → H2 by F0 := 0 , Fm :=

m  n=1

(κ(n))

(κ(n))

σn aν(n) ⊗ bν(n)

(m ∈ N).

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27

If x ∈ H1 and y ∈ H2 then |((A − Fm−1 )x, y)H2 | ≤

∞ 

    (κ(n)) (κ(n)) σn (x, aν(n) )H1 (bν(n) , y)H2 

n=m

≤ σm

∞      (κ(n)) (κ(n)) (x, aν(n) )H1 (bν(n) , y)H2 

n=1

= σm

∞  K      (k) (k) (x, al )H1 (bl , y)H2  k=1 l=1

  ∞  K  2  2   ∞     (k)  (k)  ≤ σm (x, al )H1   (bl , y)H2  k=1

≤ σm

K 

l=1

l=1


x
H1
y
H2

k=1

= σm K
x
H1
y
H2 . This estimate justifies the rearrangements (since the series are absolutely convergent) and also yields
A − Fm−1
≤ Kσm , from which the assertion follows.
Proposition 2.8. Suppose that An ∈ E(an , α; H1 , H2 ) for 1 ≤ n ≤ K. Let A := K K −1/α −α  ) . Then n=1 An and a := ( n=1 an  (i) A ∈ E(a , α) with |A|a
,α ≤ K max1≤n≤K |An |an ,α . In particular E(a1 , α) + · · · + E(aK , α) ⊂ E(a , α) . (ii) If both H1 and H2 are infinite-dimensional then the inclusion above is sharp in the sense that E(a1 , α) + · · · + E(aK , α) ⊂ E(b, α) , whenever b > a . Proof. Assertion (i) follows from Proposition 2.7 and Corollary 5.4 (i), which gives an upper bound on the rate of decay of the decreasing arrangement of the sequences s(A1 ), . . . , s(AK ). For the proof of (ii) define K sequences s(1) , . . . , s(K) by α s(k) n := exp(−ak n ) (n ∈ N) . (k)

It turns out that it suffices to exhibit K compact operators Ak with sn (Ak ) = sn such that s(A) is the decreasing arrangement of the sequences s(1) , . . . , s(K) . To see this, note that then Ak ∈ E(ak , α) for k ∈ {1, · · · , K}. At the same time sn (A) ≥ exp(−a (n + K)α ) by Corollary 5.4 (ii), so that A ∈ E(b, α) whenever b > a . In order to construct these operators we proceed as follows. Since each Hi was (i) assumed to be infinite-dimensional we can choose an orthonormal basis {hn }n∈N

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for each of them. For each k = 1, . . . , K, we now define a compact operator Ak : H1 → H2 by  (k) (2) s(n+(k−1))/K hn for n ∈ KN − (k − 1), (1) Ak hn := 0 for n ∈ KN − (k − 1). (k)

It is not difficult to see that sn (Ak ) = sn . Moreover, it is easily verified that (k) the singular numbers of A are precisely the numbers of the form sn with n ∈ N and k = 1, . . . , K. Thus, s(A) is the decreasing arrangement of the sequences s(1) , . . . , s(K) as required.
Remark 2.9. The proposition implies that E(a, α) + E(a, α) ⊂ E(2−α a, α), but E(a, α) + E(a, α) ⊂ E(a, α), because 2−α a < a. In particular, E(a, α) is not a linear space. The following result establishes a sharp bound on the eigenvalue decay rate in each exponential class. Proposition 2.10. Let a, α > 0 and A ∈ E(a, α; H, H). Then λ(A) ∈ E(a/(1 + α), α) with |λ(A)|a/(1+α),α ≤ |A|a,α . If H is infinite-dimensional, the result is sharp in the sense that there is an operator A ∈ E(a, α; H, H) such that λ(A) ∈ E(b, α) whenever b > a/(1 + α). Proof. If A ∈ E(a, α) then sk (A) ≤ |A|a,α exp(−ak α ). Using the multiplicative Weyl inequality [Pie, 3.5.1] we have |λk (A)|k ≤

k 

|λl (A)| ≤

l=1

k 

sl (A) ≤

l=1

k 

|A|a,α exp(−alα )

l=1 k

= |A|a,α exp(−a But

k l=1

lα ≥

k 0

k 

lα ).

(1)

l=1

xα dx =

1 α+1 , 1+α k

which combined with (1) yields

|λk (A)| ≤ |A|a,α exp(−ak α /(1 + α)). Sharpness is proved in several steps. We start with the following observation. Let τ1 ≥ . . . ≥ τN ≥ 0 be positive real numbers. Consider the matrix C(τ1 , . . . , τN ) ∈ L(CN , CN ) given by   0 0 τ1 . . . 0 0 0 ... 0 0     .. . . ..  . .. .. C(τ1 , . . . , τN ) :=  . .    0 0 . . . 0 τN −1  τN 0 . . . 0 0 It easy to see that sn (C(τ1 , . . . , τN )) = τn

Vol. 61 (2008)

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29

and |λ1 (C(τ1 , . . . , τN ))| = · · · = |λN (C(τ1 , . . . , τN ))| = (τ1 · · · τN )1/N . The desired operator is constructed as follows. Fix a > 0 and α > 0. Next choose a super-exponentially increasing sequence Nn , that is, Nn is increasing and limn→∞ Nn−1 /Nn = 0. For definiteness we could set Nn = exp(n2 ). Put N0 = 0 and define dn := Nn − Nn−1

(n ∈ N).

Define matrices An ∈ L(C , C ) by dn

dn

An = C(exp(−a(Nn−1 + 1)α ), . . . , exp(−a(Nn )α )). Then sk (An ) = exp(−a(Nn−1 + k)α ) (1 ≤ k ≤ dn ) and |λk (An )| = exp(−apα n ) (1 ≤ k ≤ dn ), where pn :=

1 dn

Put H :=

Nn 

lα.

l=Nn−1 +1 ∞ 

Cd n ,

n=1

and let A : H → H be the block-diagonal operator (Ax)n = An xn . Clearly, the singular numbers of A are given by sk (A) = exp(−ak α ) and the moduli of the eigenvalues are the numbers exp(−apα n ) occurring with multiplicity dn . Before checking that A has the desired properties we observe that pα n

1 = dn

Nn 

1 l ≤ dn α

l=Nn−1 +1



Nn +1

xα

Nn−1 +1

1 1 1 N α δn , (2) ((Nn + 1)α+1 − (Nn−1 + 1)α+1 ) = α + 1 dn α+1 n with limn→∞ δn = 1. The latter follows from the fact that the sequence Nn was chosen to be super-exponentially increasing. Suppose now that b > a/(α + 1). Since |λNn (A)| = exp(−apα n ) we have a a Nnα δn + bNnα ) = exp(Nnα (b − δn )). |λNn (A)| exp(bNnα ) ≥ exp(− α+1 α+1 Thus |λNn (A)| exp(bNnα ) → +∞ as n → ∞, =

which means that λ(A) ∈ E(b, α).




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Remark 2.11. Similar results have been obtained by K¨ onig and Richter [KR, Proposition 1], though without estimates on the gauge of λ(A).

3. Resolvent estimates In this section we shall derive an upper bound for the norm of the resolvent (zI − A)−1 of A ∈ E(a, α) in terms of the distance of z to the spectrum of A and the departure from normality of A, a number quantifying the non-normality of A. We shall employ a technique originally due to Henrici [Hen], who used it in a finite-dimensional context. The basic idea is to write A as a perturbation of a normal operator having the same spectrum as A by a quasi-nilpotent operator. A similar argument can be used to derive resolvent estimates for operators belonging to Schatten classes (see [Gil] (and references therein) and [Ban]). Following the idea outlined above we start with bounds for quasi-nilpotent operators. Proposition 3.1. Let a, α > 0. (i) If A ∈ E(a, α; H, H) is quasi-nilpotent, that is, σ(A) = {0}, then   (I − A)−1  ≤ fa,α (|A|a,α ),

(3)

+ where fa,α : R+ 0 → R0 is defined by

fa,α (r) =

∞ 

(1 + r exp(−anα )).

n=1

Moreover, fa,α has the following asymptoticss: log fa,α (r) ∼ a−1/α

α (log r)1+1/α as r → ∞. 1+α

(4)

(ii) If H is infinite-dimensional the estimate (3) is sharp in the sense that there is a quasi-nilpotent B ∈ E(a, α; H, H) such that   (5) log (I − zB)−1  ∼ log fa,α (|zB|a,α ) as |z| → ∞. Proof. Fix a, α > 0. (i) Since A is trace class and quasi-nilpotent a standard estimate (see, for example, [GGK, Chapter X, Theorem 1.1]) shows that ∞    (I − A)−1  ≤ (1 + sn (A)) . n=1

Thus

∞    (I − A)−1  ≤ (1 + |A|a,α exp(−anα )) = fa,α (|A|a,α ). n=1

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It remains to prove the growth estimate (4). We proceed by noting that fa,α extends to an entire function of genus zero with fa,α (0) = 1. Moreover, the maximum modulus of fa,α (z) for |z| = r equals fa,α (r). The growth of fa,α can thus be estimated by (6) N (r) ≤ log fa,α (r) ≤ N (r) + Q(r) ,  r −1  ∞ −2 where N (r) = 0 t n(t) dt, Q(r) = r r t n(t) dt and n(r) denotes the number of zeros of fa,α lying in the closed disk with radius r centred at 0 (see [Boa, p. 47]). Since n(r) = a−1/α (log+ r)1/α , where log+ (r) = max {0, log r} and · denotes the floor-function, we have, for r ≥ 1,  r −1/α N (r) = a t−1 (log t)1/α dt + O(log r) 1

=a while Q satisfies

−1/α

α (log r)1+1/α + O(log r) ; 1+α

Q(r) = O((log r)1/α ) as r → ∞ .

To see this, note that Q(r) ≤ a−1/α r



∞

t−2 (log t)1/α dt = a−1/α r



(7) (8)

∞

e−u u1/α du ;

(9)

log r

r

putting r = es it thus suffices to show that  ∞ e−u u1/α du = O(s1/α ) as s → ∞. es s

This, however, is the case since  ∞  ∞  s−1/α es e−u u1/α du = e−(u−s) (u/s)1/α du = s

∞

0

s

e−t (1 + t/s)1/α dt → 1

as s → ∞. Combining (8), (7) and (6) the growth estimate (4) follows. (ii) Since H is infinite-dimensional, we may choose an orthonormal basis {hn }n∈N . Define the operator B ∈ L(H, H) by Bhn := exp(−anα )hn+1

(n ∈ N).

It is not difficult to see that sn (B) = exp(−anα ) for n ∈ N, so that B ∈ E(a, α; H, H). Before we proceed let cn := and note that since

n 0

n 

kα

k=1

xα dx ≤

n

k=1

kα ≤

(n ∈ N0 ),  n+1 1

xα dx, we have

1 1 nα+1 ≤ cn ≤ (n + 1)α+1 (n ∈ N0 ). α+1 α+1 The operator B is quasi-nilpotent, since a
B n
= exp(−acn ) ≤ exp(− nα+1 ) , α+1

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1/n

which implies
B n
→ 0 as n → ∞.   In order to determine the asymptotics of log (I − zB)−1  we start by noting that ∞      (I − zB)−1 2 ≥ (I − zB)−1 h1 2 =
(zB)n h1
2 =

∞ 

|z|2n exp(−2acn ) ≥

n=0

∞ 

|z|2n exp(−2

n=0

where g(r) :=

∞  n=1

Thus

n=0

r2n exp(−2

a (n + 1)α+1 ) ≥ |z|−2 g(|z|), (10) α+1

a nα+1 ) (r ∈ R+ 0 ). α+1

  2 log fa,α (|zB|a,α ) ≥ 2 log (I − zB)−1  ≥ −2 log |z| + log g(|z|) ,

(11)

which shows that in order to obtain the desired asymptotics (5) it suffices to prove that α (12) (log r)1+1/α as r → ∞. log g(r) ∼ 2a−1/α α+1 In order to establish the asymptotics above we introduce the maximum term a nα+1 ) (r ∈ R+ µ(r) := max r2n exp(−2 (13) 0 ). 1≤n<∞ α+1 Since g extends to an entire function of finite order we have (see, for example, [PS, Problem 54]) log µ(r) ∼ log g(r) as r → ∞ , which implies that it now suffices to show that µ has the desired asymptotics α (log r)1+1/α as r → ∞. log µ(r) ∼ 2a−1/α (14) α+1 + We now estimate µ(r) for fixed r. Define the function mr : R+ 0 → R0 by a xα+1 + 2x log r). mr (x) = exp(−2 α+1

It turns out that mr has a maximum at xr = a−1/α (log r)1/α and that mr is monotonically increasing on (0, xr ) and monotonically decreasing on (xr , ∞). Thus log mr (xr − 1) ≤ log µ(r) ≤ log mr (xr ) .

(15)

Write xr − 1 = δr xr and note that δr → 1 as r → ∞. Observing that log mr (xr ) α = 2a−1/α α+1 (log r)1+1/α while   δrα+1 log mr (δr xr ) α log mr (xr − 1) −1/α + δr → 2a−1/α − = = 2a α+1 α+1 (log r)1+1/α (log r)1+1/α

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as r → ∞ we conclude, using (15), that (14) holds. This implies (12), which yields (5), as required.
Remark 3.2. (i) The bound for the growth of the resolvent of a quasi-nilpotent A ∈ E(a, α) given in the above proposition is an improvement compared to those obtainable from the usual estimates for operators in the Schatten classes. Indeed, if A belongs to the Schatten p-class (i.e. s(A) is p-summable) for some p > 0, then   (I − A)−1  ≤ fp (
A
) p

+ where
A
p denotes the Schatten p-(quasi) norm of A. Here, fp : R+ 0 → R0 is given by fp (r) = exp(ap rp + bp ) , where ap and bp are positive numbers depending on p, but not on A (see, for example, [Sim1] or [Ban, Theorem 2.1], where a discussion of the constants ap and bp can be found). (ii) Closer inspection of the proof yields the following explicit upper bound for fa,α   α log fa,α (r) ≤ a−1/α (log+ r)1+1/α + rΓ(1 + 1/α, log+ r) , 1+α

where Γ(β, s) denotes the incomplete gamma function  ∞ Γ(β, s) = exp(−t) tβ−1 dt . s

This follows from (6) together with the estimate n(r) ≤ a−1/α (log+ r)1/α . A simple consequence of the proposition is the following estimate for the growth of the resolvent of a quasi-nilpotent A ∈ E(a, α). Corollary 3.3. If A ∈ E(a, α) is quasi-nilpotent, then   (zI − A)−1  ≤ |z|−1 fa,α (|z|−1 |A|a,α ) for z = 0. The proposition above can be used to obtain growth estimates for the resolvents of any A ∈ E(a, α) by means of the following device. Theorem 3.4. Let A ∈ S∞ . Then A can be written as a sum A = D + N, such (i) (ii) (iii)

that D ∈ S∞ , N ∈ S∞ ; D is normal and λ(D) = λ(A); N and (zI − D)−1 N are quasi-nilpotent for every z ∈ (D) = (A).

Proof. See [Ban, Theorem 3.2]. This result motivates the following definition.




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Definition 3.5. Let A ∈ S∞ . A decomposition A=D+N with D and N enjoying properties (i–iii) of the previous theorem is called a Schur decomposition of A. The operators D and N will be referred to as the normal and the quasi-nilpotent part of the Schur decomposition of A, respectively. Remark 3.6. (i) The terminology stems from the fact that in the finite dimensional setting the decomposition in Theorem 3.4 can be obtained as follows: since any matrix is unitarily equivalent to an upper-triangular matrix by a classical result due to Schur, it suffices to establish the result for matrices of this form. In this case, simply choose D to be the diagonal part, and N the off-diagonal part of the matrix. (ii) The decomposition is not unique, not even modulo unitary equivalence: there is a matrix A with two Schur decompositions A = D1 + N1 and A = D2 + N2 such that N1 is not unitarily equivalent to N2 (see [Ban, Remark 3.5 (i)]). Note, however, that the normal parts of any two Schur decompositions of a given compact operator are always unitarily equivalent. Using the results in the previous section we are able to locate the position of the normal part and the quasi-nilpotent part of an operator A ∈ E(a, α) in the scale of exponential classes. Proposition 3.7. Let A ∈ E(a, α). If A = D + N is a Schur decomposition of A with normal part D and quasi-nilpotent part N , then (i) D ∈ E(a/(1 + α), α) with |D|a/(1+α),α ≤ |A|a,α ; (ii) N ∈ E(a , α) with |N |a
,α ≤ 2|A|a,α , where a = a(1 + (1 + α)1/α )−α . Proof. Since D is normal, its singular numbers coincide with its eigenvalues, which in turn coincide with the eigenvalues of A. Assertion (i) is thus a consequence of Proposition 2.10, while assertion (ii) follows from (i) and Proposition 2.8 by taking N = A − D.
Remark 3.8. Assertion (i) above is sharp in the following sense: there is A ∈ E(a, α) such that for any normal part D of A we have D ∈ E(b, α) whenever b > a/(1+α). This follows from the corresponding statement in Proposition 2.10 and the fact that all normal parts of A are unitarily equivalent. For later use we define the following quantities, originally introduced by Henrici [Hen]. Definition 3.9. Let a, α > 0 and define νa,α : S∞ → R+ 0 ∪ {∞} by νa,α (A) := inf { |N |a,α : N is a quasi-nilpotent part of A } . We call νa,α (A) the (a, α)-departure from normality of A.

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Remark 3.10. Henrici originally introduced this quantity for matrices and with the (a, α)-gauge of N replaced by the Hilbert-Schmidt norm. For a discussion of the case where the (a, α)-gauge is replaced by a Schatten norm and its uses to obtain resolvent estimates for Schatten class operators see [Ban]. The term ‘departure from normality’ is justified in view of the following characterisation. Proposition 3.11. Let A ∈ E(a, α). Then νa,α (A) = 0 ⇔ A is normal. Proof. Let νa,α (A) = 0. Then there exists a sequence of Schur decompositions with quasi-nilpotent parts Nn such that |Nn |a,α → 0. Thus
A − Dn
=
Nn
= s1 (Nn ) ≤ exp(−a)|N |a,α → 0, where Dn are the corresponding normal parts. Thus A is a limit of normal operators converging in the uniform operator topology and is therefore normal. The converse is trivial.
For a given A ∈ E(a, α) the departure from normality is difficult to calculate. The following simple but somewhat crude bound is useful in practice. Proposition 3.12. Let A ∈ E(a, α). Then νb,α (A) ≤ 2|A|a,α whenever b ≤ a(1 + (1 + α)1/α )−α . Proof. Follows from Proposition 3.7 together with the fact that |N |b,α ≤ |N |a
,α whenever b ≤ a .
We are now ready to deduce resolvent estimates for arbitrary A ∈ E(a, α). Using a Schur decomposition A = D + N with D normal and N quasi-nilpotent we consider A as a perturbation of D by N . Since D is normal we have   1 (zI − D)−1  = (z ∈ (D)), (16) d(z, σ(D)) where for z ∈ C and σ ⊂ C closed, d(z, σ) := inf |z − λ| λ∈σ

denotes the distance of z to σ. The influence of the perturbation N on the other hand, is controlled by Proposition 3.1. All in all, we have the following. Theorem 3.13. Let A ∈ E(a, α). If b ≤ a(1 + (1 + α)1/α )−α , then     1 νb,α (A) (zI − A)−1  ≤ fb,α . d(z, σ(A)) d(z, σ(A))

(17)

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Proof. Fix b ≤ a(1 + (1 + α)1/α )−α . Then there is a Schur decomposition of A with normal part D and quasi-nilpotent part N ∈ E(b, α) by Proposition 3.7. Since the bound above is trivial for z ∈ σ(A) we may assume z ∈ (A). As D and N stem from a Schur decomposition of A we see that (zI − D)−1 exists (because σ(A) = σ(D)) and that (zI − D)−1 N is quasi-nilpotent. Moreover   |N |b,α , |(zI − D)−1 N |b,α ≤ (zI − D)−1  |N |b,α = d(z, σ(D)) by (16) and Proposition 2.5. Thus (I − (zI − D)−1 N ) is invertible in L and     |N |b,α (I − (zI − D)−1 N )−1  ≤ fb,α , d(z, σ(D)) by Proposition 3.1. Now, since (zI − A) = (zI − D)(I − (zI − D)−1 N ), we conclude that (zI − A) is invertible in L and      (z − A)−1  ≤ (I − (zI − D)−1 N )−1  (zI − D)−1    1 |N |b,α ≤ fb,α . d(z, σ(D)) d(z, σ(D)) Taking the infimum over all Schur decompositions while using σ(A) = σ(D) once again the result follows.
Remark 3.14. (i) The estimate remains valid if νb,α (A) is replaced by something larger, for example by the upper bound given in Proposition 3.12. (ii) The estimate is sharp in the sense that if A is normal then (17) reduces to the sharp estimate (16).

4. Bounds for the spectral distance Using the resolvent estimates obtained in the previous section it is possible to give upper bounds for the Hausdorff distance of the spectra of operators in E(a, α). Recall that the Hausdorff distance Hdist (., .) is the following metric defined on the space of compact subsets of C ˆ 1 , σ2 ), d(σ ˆ 2 , σ1 )}, Hdist (σ1 , σ2 ) := max {d(σ where ˆ 1 , σ2 ) := sup d(λ, σ2 ) d(σ λ∈σ1

and σ1 and σ2 are two compact subsets of C. For A, B ∈ L we borrow terminology ˆ from matrix perturbation theory and call d(σ(A), σ(B)) the spectral variation of A with respect to B and Hdist (σ(A), σ(B)) the spectral distance of A and B. The main tool to bound the spectral variation is the following result, which is based on a simple but powerful argument usually credited to Bauer and Fike [BauF] who first employed it in a finite-dimensional context.

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Proposition 4.1. Let A ∈ S∞ . Suppose that there is a strictly monotonically in+ creasing surjection g : R+ 0 → R0 such that   (zI − A)−1  ≤ g(d(z, σ(A))−1 ) (∀z ∈ (A)). Then for any B ∈ L, the spectral variation of B with respect to A satisfies ˆ d(σ(B), σ(A)) ≤ h(
A − B
), + where h : R+ 0 → R0 is the function defined by

h(r) = (˜ g (r−1 ))−1 + and g˜ : R+ 0 → R0 is the inverse of the function g.

Proof. Let B ∈ L. In what follows we shall use the abbreviations d := d(z, σ(A)),

E := B − A.

Without loss of generality we may assume that E = 0. We shall first establish the following implication:   −1 (18) z ∈ σ(B) ∩ (A) ⇒
E
≤ (zI − A)−1  . To see this let z ∈ σ(B) ∩ (A) and suppose to the contrary that   (zI − A)−1 
E
< 1.     Then I −(zI −A)−1 E is invertible in L, so (zI −B) = (zI −A) I −(zI −A)−1 E is invertible in L. Hence z ∈ (B) which contradicts z ∈ σ(B). Thus the implication (18) holds. In order to prove the proposition it suffices to show that z ∈ σ(B) ⇒ d(z, σ(A)) ≤ h(
E
) ,

(19)

which is proved as follows. Let z ∈ σ(B). If z ∈ σ(A) there is nothing to prove. We may thus assume that z ∈ (A). Hence, by (18),   −1
E
≤ (zI − A)−1  ≤ g(d−1 ). Since g is strictly monotonically increasing, so is g˜. Thus g˜(
E
−1 ) ≤ d−1 , and hence −1

d(z, σ(A)) = d ≤ (˜ g (
E


))−1 = h(
E
) = h(
A − B
).




Using the proposition above together with the resolvent estimates in Theorem 3.13 we now obtain the following spectral variation and spectral distance formulae. Theorem 4.2. Let A ∈ E(a, α) and define a := a(1 + (1 + α)1/α )−α . (i) If B ∈ L and b > a , then  
A − B
ˆ d(σ(B), σ(A)) ≤ ν(A)b,α hb,α . νb,α (A)

(20)

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(ii) If B ∈ E(a, α) and b > a , then



Hdist (σ(A), σ(B)) ≤ mhb,α


A − B
m

 ,

(21)

where m := max {ν(A)b,α , ν(B)b,α }. + Here, the function hb,α : R+ 0 → R0 is given by

gb,α (r−1 ))−1 , hb,α (r) := (˜ + + + where g˜ : R+ 0 → R0 is the inverse of the function gb,α : R0 → R0 defined by

gb,α (r) := rfb,α (r) , and fa,α is the function defined in Proposition 3.1. Proof. To prove (i) fix b ≥ a . The assertion now follows from the previous proposition by noting that     1 νb,α (A) (zI − A)−1  ≤ gb,α νb,α (A) d(z, σ(A)) by Theorem 3.13. To prove (ii) fix b ≥ a . Then it is not difficult see that     m (zI − A)−1  ≤ 1 gb,α , m d(z, σ(A)) and

  (zI − B)−1  ≤ 1 gb,α m



m d(z, σ(B))

and the assertion follows as in the proof of (i).

 ,


Remark 4.3. (i) Note that limr↓0 hb,α (r) = 0, so the estimate for the spectral distance becomes small when
A − B
is small. In fact, it can be shown that  α/(1+α) 1+α log hb,α (r) ∼ −b1/(1+α) | log r|α/(1+α) as r ↓ 0. α This follows from the asymptotics in Proposition 3.1 together with the fact that if log f (r) ∼ a(log r)β , then log f˜(r) ∼ a−1/β (log r)1/β where f˜ is the inverse of f . (ii) It is not difficult to see, for example by arguing as in the proof of part (ii) of the theorem, that the inequalities (20) and (21) above remain valid if νb,α (A) or νb,α (B) is replaced by something larger — for example, by the upper bounds given in Proposition 3.12. (iii) Assertion (ii) of the theorem is sharp in the sense that if both operators are normal, then (ii) reduces to Hdist (σ(A), σ(B)) ≤
A − B
.

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5. Appendix: Monotone arrangements In this appendix we present a number of results concerning sequences and their arrangements used in Section 2. Let a : N → R be a sequence. Define
a
+ := sup an , n∈N


a
− := inf an . n∈N

For u : N → R call rank (u) := card { n ∈ N : un = 0 } . Definition 5.1. Let a : N → R be a sequence. Let extended real-valued sequences a(+) and a(−) be defined by ! := inf
a − u
+ : rank u < n , a(+) : N → R ∪ {−∞, ∞}, a(+) n a(−) : N → R ∪ {−∞, ∞},

! a(−) := sup
a − u
− : rank u < n . n

We call a(+) the decreasing arrangement of a, and a(−) the increasing arrangement of a. This terminology is justified in view of the fact that a(+) (respectively a(−) ) is a decreasing (respectively increasing) sequence. Moreover, if a is monotonically decreasing, then a(+) = a, and similarly for monotonically increasing sequences. More generally, we will consider monotone arrangements of collections of sequences by first amalgamating them into one sequence and then regarding the resulting monotone arrangement. A more precise definition is the following. (1)

(K)

Definition 5.2. Given K real-valued sequences {an }n∈N , . . . , {an }n∈N , define a new sequence a : N → R by (i)

a(k−1)K+i = ak

for k ∈ N and 1 ≤ i ≤ K .

We then call a(+) (a(−) ) the decreasing (increasing) arrangement of the K sequences a(1) , · · · , a(K) . Our application of decreasing arrangements will typically be to singular number sequences, all of which converge to zero at some stretched exponential rate. Technically and notationally it is preferable to work with the logarithms of reciprocals of such sequences, that is, increasing sequences converging to +∞ at some polynomial rate. The following is the main result of this appendix. Proposition 5.3. Let α > 0 and K ∈ N. Suppose that for each k ∈ {1, . . . , K} we are given a real sequence a(k) , a positive constant ak > 0, and a real number Ak .

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Let a(−) denote the increasing arrangement of the K sequences a(1) , . . . , a(K) , and define K

−α  −1/α c= ak . k=1

(i) If α a(k) n ≥ ak n + Ak

(∀n ∈ N, k ∈ {1, . . . , K}) ,

then a(−) ≥ cnα + min {A1 , . . . , AK } n

(∀n ∈ N) .

(ii) If α a(k) n ≤ ak n + Ak

(∀n ∈ N, k ∈ {1, . . . , K}) ,

then a(−) ≤ c(n + K)α + max {A1 , . . . , AK } n (k)

Proof. For k ∈ {1, . . . , K} set a0 functions

µk (r) := card

(∀n ∈ N) .

= −∞ and, for r ∈ R, define the counting " "

µ(r) := card

n ∈ N : a(k) n ≤ r ≤r n ∈ N : a(−) n

# # .

The following relations are easily verified. We have µ(r) =

K 

µk (r) ,

(22)

(∀r ∈ R) ,

(23)

(∀n ∈ N0 ) .

(24)

k=1

and (−)

aµ(r) ≤ r µ(a(−) n )

≥n

(i) Set C = min {A1 , . . . , AK }. Since for each k ∈ {1, . . . , K}, " # n ∈ N : a(k) ⊂ { n ∈ N : ak n α + C ≤ r } , n ≤r we have, for r ≥ C,

$

µk (r) ≤ card { n ∈ N : ak nα + C ≤ r } =

r−C ak

1/α % ,

where · denotes the floor function. Thus, using (22), we have µ(r) =

K  k=1

µk (r) ≤ (r − C)1/α

K  k=1

−1/α

ak

.

(25)

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Resolvent Estimates (−)

If n ∈ N then an

41

≥ C, so combining (25) with (24) gives 1/α (a(−) n − C)

K 

−1/α

≥ µ(a(−) n ) ≥ n,

ak

k=1

from which (i) follows. (ii) Set C = max {A1 , . . . , AK }. Since for each k ∈ {1, . . . , K}, " # , { n ∈ N : ak nα + C ≤ r } ⊂ n ∈ N : a(k) n ≤ r we have, for r ≥ C, $ 1/α % r−C = card { n ∈ N : ak nα + C ≤ r } ≤ µk (r) . ak Thus, using (22), we have µ(r) =

K 

µk (r) ≥

1/α

(r − C)

k=1

−1/α ak

−K.

(26)

k=1

Now fix n0 ∈ N. Choose r0 ≥ C such that n0 =

K 

1/α

(r0 − C)

K 

−1/α ak

−K.

(27)

k=1

From (26) and (27) we see that n0 ≤ µ(r0 ). Using (23), together with the fact that (−) n → an is monotonically increasing, now yields (−)

α a(−) n0 ≤ aµ(r0 ) ≤ r0 = c(n0 + K) + C .




Since n0 was arbitrary, (ii) follows.

Corollary 5.4. Let α > 0 and K ∈ N. Suppose that for each k ∈ {1, . . . , K} we are given a real sequence b(k) , a positive constant ak > 0, and a real number Bk . Let b(+) denote the decreasing arrangement of the K sequences b(1) , . . . , b(K) and define K

−α  −1/α c := ak . k=1

(i) If α b(k) n ≤ Bk exp(−ak n )

(∀n ∈ N, k ∈ {1, . . . , K}) ,

then ≤ B exp(−cnα ) b(+) n where B = max {B1 , . . . , BK }.

(∀n ∈ N) ,

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(ii) If α b(k) n ≥ Bk exp(−ak n )

(∀n ∈ N, k ∈ {1, . . . , K}) ,

then

b(+) ≥ B exp(−c(n + K)α ) n where B = min {B1 , . . . , BK }.

(∀n ∈ N) ,

Acknowledgements I would like to thank Oliver Jenkinson for the many discussions that helped to shape this article, in particular the results in Section 2 and in the Appendix. The research described in this article was partly supported by EPSRC Grant GR/R64650/01.

References [Ban]

O.F. Bandtlow (2004) Estimates for norms of resolvents and an application to the perturbation of spectra; Math. Nachr. 267, 3–11 [BanC] O.F. Bandtlow and C.-H. Chu (2006) Eigenvalue decay of operators on harmonic function spaces; preprint [BanJ1] O.F. Bandtlow and O. Jenkinson (2007) Explicit a priori bounds on transfer operator eigenvalues; Commun. Math. Phys. 276, 901–905 [BanJ2] O.F. Bandtlow and O. Jenkinson (2007) On the Ruelle eigenvalue sequence; to appear in Ergod. Th. & Dynam. Sys. [BanJ3] O.F. Bandtlow and O. Jenkinson (2007) Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions; to appear in Adv. Math. [BauF] F.L. Bauer and C.T. Fike (1960) Norms and exclusion theorems; Num. Math. 2, 42–53 [BerF] G. Bery and G. Forst (1975) Potential Theory on Locally Compact Groups; Heidelberg, Springer [Boa] R.P. Boas (1954) Entire Functions; New York, Academic Press [Cal] J.W. Calkin (1941) Two sided ideals and congruences in the ring of bounded operators in Hilbert space; Ann. Math. 42, 839–873 [DS] N. Dunford and J.T. Schwartz (1963) Linear Operators, Vol. 2; New York, Interscience [Gil] M.I. Gil’ (2003) Operator Functions and Localization of Spectra; Berlin, Springer [GGK] I. Gohberg, S. Goldberg and M.A. Kaashoek (1990) Classes of Linear Operators Vol. 1; Basel, Birkh¨ auser [GK] I. Gohberg and M.G. Krein (1969) Introduction to the Theory of Linear NonSelfadjoint Operators; Providence, AMS [Hen] P. Henrici (1962) Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices; Num. Math. 4, 24–40 [KR] H. K¨ onig and S. Richter (1984) Eigenvalues of integral operators defined by analytic kernels; Math. Nachr. 119, 141–155

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E. Nelimarkka (1982) On λ(P, N )-nuclearity and operator ideals; Math. Nachr. 99, 231–237 A. Pietsch (1988) Eigenvalues and s-Numbers; Cambridge, CUP A. Pokrzywa (1985) On continuity of spectra in norm ideals; Lin. Alg. Appl. 69, 121–130 G. P´ olya and G. Szeg¨ o (1976) Problems and Theorems in Analysis, Volume 2; Berlin, Springer J.R. Ringrose (1971) Compact Non-Self-Adjoint Operators; London, van Nostrand D. Ruelle (2004) Thermodynamic formalism: the mathematical structures of equilibrium statistical mechanics; Cambridge CUP B. Simon (1977) Notes on infinite determinants of Hilbert space operators; Adv. in Math., 24, 244–273 B. Simon (1979) Trace ideals and their applications; Cambridge, CUP

Oscar F. Bandtlow School of Mathematical Sciences Queen Mary, University of London London E3 4NS UK e-mail: [email protected] Submitted: December 20, 2006 Revised: December 17, 2007

Integr. equ. oper. theory 61 (2008), 45–62 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/010045-18, published online March 13, 2008 DOI 10.1007/s00020-008-1568-7

Integral Equations and Operator Theory

Semigroups of Composition Operators in BMOA and the Extension of a Theorem of Sarason O. Blasco, M.D. Contreras, S. D´ıaz-Madrigal, J. Mart´ınez and A.G. Siskakis Abstract. In this paper we deal with the maximal subspace in BM OA where a general semigroup of analytic functions on the unit disk generates a strongly continuous semigroup of composition operators. Particular cases of this question are related to a well-known theorem of Sarason about V M OA. Our results describe analytically that maximal subspace and provide a condition which is sufficient for the maximal subspace to be exactly V M OA. A related necessary condition is also proved in the case when the semigroup has an inner Denjoy-Wolff point. As a byproduct we provide a generalization of the theorem of Sarason. Mathematics Subject Classification (2000). Primary 30H05, 32A37, 47B33, 47D06; Secondary 46E15. Keywords. Semigroups, composition operators, BMOA, VMOA.

1. Introduction Let H(D) be the Fr´echet space of all analytic functions in the unit disk endowed with the topology of uniform convergence on compact subsets of D. A (one-parameter) semigroup of analytic functions is any continuous homomorphism Φ : t
→ Φ(t) = ϕt from the additive semigroup of nonnegative real numbers into the composition semigroup of all analytic functions which map D into D. In other words, Φ = (ϕt ) consists of analytic functions on D with ϕt (D) ⊂ D and for which the following three conditions hold: 1. ϕ0 is the identity in D, This research has been partially supported by the Ministerio de Educaci´ on y Ciencia projects n. MTM2006-14449-C02-01 and MTM2005-08350-C03-03 and by La Consejer´ıa de Educaci´ on y Ciencia de la Junta de Andaluc´ıa.

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2. ϕt+s = ϕt ◦ ϕs , for all t, s ≥ 0, 3. ϕt → ϕ0 , as t → 0, uniformly on compact subsets of D. It is well known that condition (3) above can be replaced by (3
) For each z ∈ D, ϕt (z) → z, as t → 0. Each such semigroup gives rise to a semigroup (Ct ) consisting of composition operators on H(D), f ∈ H(D). Ct (f ) := f ◦ ϕt , We are going to be interested in the restriction of (Ct ) to certain linear subspaces H(D). Given a Banach space X consisting of functions of H(D) and a semigroup (ϕt ), we say that (ϕt ) generates a semigroup of operators on X if (Ct ) is a well-defined strongly continuous semigroup of bounded operators in X. This exactly means that for every f ∈ X, we have Ct (f ) ∈ X for all t ≥ 0 and lim Ct (f ) − f X = 0.

t→0+

Thus the crucial step to showing that (ϕt ) generates a semigroup of operators in X is to pass from the pointwise convergence limt→0+ f ◦ ϕt (z) = f (z) on D to the convergence in the norm of X. This connection between composition operators and semigroups opens the possibility of studying spectral properties, operator ideal properties or dynamical properties of the semigroup of operators (Ct ) in terms of the theory of functions. Papers [1] and [12] can be considered as the starting points in this direction. Classical choices of X treated in the literature are the Hardy spaces H p , the disk algebra A(D), the Bergman spaces Ap , the Dirichlet space D and the chain of spaces Qp and Qp,0 which have been introduced recently and which include the spaces BM OA, Bloch as well as their “little oh” analogues. See [21] and [22] for definitions and basic facts of the spaces and [17], [18], and [20] for composition semigroups on these spaces. Very briefly, the state of the art is the following: (i) Every semigroup of analytic functions generates a semigroup of operators on the Hardy spaces H p (1 ≤ p < ∞), the Bergman spaces Ap (1 ≤ p < ∞), the Dirichlet space, and on the spaces VMOA and little Bloch. (ii) No non-trivial semigroup generates a semigroup of operators in the space H ∞ of bounded analytic functions. (iii) There are plenty of semigroups (but not all) which generate semigroups of operators in the disk algebra. Indeed, they can be well characterized in several analytical terms [4]. In this paper we concentrate on the space BM OA. As we will see, the strong continuity behavior differs notably from other known cases, since it depends heavily on the particular semigroup. This has led us to introduce the following notation: Given a semigroup (ϕt ) we denote by [ϕt , BM OA] the maximal closed linear subspace of BM OA such that (ϕt ) generates a semigroup of operators on it. The existence of such a maximal subspace, as well as analytical descriptions of it will be discussed in section two. In that section, we also present an alternative self-contained proof of the fact that

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47

every semigroup generates a semigroup of operators on V M OA. This in particular means that in our notation V M OA ⊆ [ϕt , BM OA] for every semigroup (ϕt ). It is important to underline that, in general, this inclusion can be proper. The chain of inclusions V M OA ⊆ [ϕt , BM OA] ⊆ BM OA leads us to wonder about those semigroups with an extreme character, that is, those giving equality V M OA = [ϕt , BM OA]

or

[ϕt , BM OA] = BM OA.

In section three we deal with the left hand equality V M OA = [ϕt , BM OA], and present a sufficient condition on the semigroup for this equality to hold. A different but closely related condition is shown to be necessary for semigroups with inner Denjoy-Wolff point. The conditions are in terms of the growth of the infinitesimal generator of (ϕt ) near the boundary of D. There is an important connection of the above results with a well-known theorem of D. Sarason which characterizes the space V M OA. Namely, Sarason [14], [15] proved Theorem A. (Sarason [14]) Suppose f ∈ BM OA; then the following are equivalent: 1. f ∈ V M OA. 2. limt→0+ f (eit z) − f BMOA = 0. 3. limt→0+ f (e−t z) − f BMOA = 0. In our notation this theorem can be written as V M OA = [eit z, BM OA] = [e−t z, BM OA]. In other words, the semigroups ϕt (z) = eit z of rotations and ϕt (z) = e−t z of dilatations with respect to the origin are left-extreme in the above sense. Our results provide many more nontrivial examples of semigroups of this type. We end this introduction by presenting a quick review of basic facts about a general semigroup of analytic functions (see [18]). The basic material about operator semigroups on Banach spaces can be found in [9, Chapter VIII]. If (ϕt ) is a semigroup, then each map ϕt is univalent. The infinitesimal generator of (ϕt ) is the function G(z) := lim

t→0+

ϕt (z) − z , z ∈ D. t

This convergence holds uniformly on compact subsets of D so G ∈ H(D). Moreover G satisfies ∂ϕt (z) ∂ϕt (z) G(ϕt (z)) = = G(z) , z ∈ D, t ≥ 0. (1.1) ∂t ∂z Further G has a unique representation G(z) = (bz − 1)(z − b)P (z), z ∈ D

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where b ∈ D and P ∈ H(D) with Re P (z) ≥ 0 for all z ∈ D. If G is not identically null, the couple (b, P ) is uniquely determined from (ϕt ) and the point b is called the Denjoy-Wolff point of the semigroup. We want to mention that this point plays a crucial role in the dynamical behavior of the semigroup (see [18], [5]). Note that for r sufficiently near to one, it is clear from the above representation of G that G has no zeros in the annulus r < |z| < 1, so 1/G is analytic on that annulus. This remark will be implicitly used throughout the paper.

2. Semigroups in BMOA For the sake of completeness and to fix notations, we begin with a quick review of basic properties of V M OA and BM OA. BM OA is the Banach space of all analytic functions in the Hardy space H 2 whose boundary values have bounded mean oscillation. There are many characterizations of this space but we will use the one in terms of Carleson measures (see [22, 11]). Namely, a function f ∈ H 2 belongs to BM OA if and only if there exists a constant C > 0 such that
|f
(z)|2 (1 − |z|2 )dA(z) ≤ C|I|, R(I)

for any arc I ⊂ ∂D, where R(I) is the Carleson rectangle determined by I, that is,   |I| iθ iθ R(I) := re ∈ D : 1 − < r < 1 and e ∈ I . 2π As usual, |I| denotes the length of I and dA(z) the normalized Lebesgue measure on D. The corresponding BM OA norm is 1/2 
1
2 2 f BMOA := |f (0)| + sup |f (z)| (1 − |z| )dA(z) . |I| R(I) I⊂∂D Trivially, each polynomial belongs to BM OA. The closure of all polynomials in BM OA is denoted by V M OA. Alternatively, V M OA is the subspace of BM OA formed by those f ∈ BM OA such that
1 |f
(z)|2 (1 − |z|2 )dA(z) = 0. lim |I|→0 |I| R(I) Particular and quite interesting examples of members of V M OA are provided by functions in the Dirichlet space D, which is the space of those functions f ∈ H(D) such that f
∈ L2 (D, dA). In fact, for every f ∈ D,

1
2 2 |f (z)| (1 − |z| )dA(z) ≤ 2 |f
(z)|2 dA(z) → 0 as |I| → 0. (2.1) |I| R(I) R(I) For more information on these Banach spaces, we refer the reader to the excellent monographs or [11] or [22].

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In our first result, we confirm the existence of a maximal closed linear subspace of BM OA on which a semigroup (ϕt ) generates a semigroup of operators. In this context, we recall that any analytic self map ϕ of the disk induces a bounded composition operator Cϕ (f ) = f ◦ ϕ on BM OA and there is a constant C > 0, not depending on ϕ, such that   1 Cϕ BMOA ≤ C 1 + log . (2.2) 1 − |ϕ(0)| Moreover Cϕ is bounded on V M OA if and only if ϕ ∈ V M OA, see [2] for details. Proposition 2.1. Let (ϕt ) be a semigroup of analytic functions. Then there exists a closed subspace Y of BM OA such that (ϕt ) generates a semigroup of operators on Y and such that any other subspace of BM OA with this property is contained in Y . Proof. Consider the linear subspace of BM OA defined by   Y := f ∈ BM OA : lim f ◦ ϕt − f BMOA = 0 . t→0+

Notice that supt∈[0,1] |ϕt (0)| = M < 1. Hence from (2.2), sup Ct BMOA ≤ C (1 − log(1 − M )) < +∞.

t∈[0,1]

This and the triangle inequality for norms shows that Y is a closed subspace of BM OA. Thus in order to prove that (ϕt ) generates a semigroup of operators in Y , it remains to check that if f ∈ Y , then Cs (f ) ∈ Y for all s ≥ 0. To see this let s, t ≥ 0, then   1 Cs (f ) ◦ ϕt − Cs (f )BMOA ≤ C 1 + log f ◦ ϕt − f BMOA → 0 1 − |ϕs (0)| as t → 0+ . Finally, if W is a subspace of BM OA such that (ϕt ) generates a semigroup of operators on W , then for any f ∈ W we have in particular lim f ◦ ϕt − f BMOA = 0,

t→0+

thus f ∈ Y and we conclude W ⊂ Y .




In what follows, this maximal subspace Y will be denoted as [ϕt , BM OA]. It is easy to see that if Z is any closed subspace of [ϕt , BM OA] which is invariant under (Ct ) (i.e. Ct (Z) ⊂ Z for every t ≥ 0), then (ϕt ) generates a semigroups of operators on Z. This maximal subspace can be also described directly in terms of the infinitesimal generator. Theorem 2.2. Let G be the infinitesimal generator of (ϕt ). Then, [ϕt , BM OA] = {f ∈ BM OA : Gf
∈ BM OA}.

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Proof. We may assume that (ϕt ) is not trivial. Denote by Γ the infinitesimal generator of the operator semigroup (Ct ) acting on the Banach space [ϕt , BM OA], and by D (Γ) its domain. We will show that if f ∈ D (Γ), then Gf
∈ BM OA. Indeed if f ∈ D (Γ), then Γ(f ) ∈ BM OA and   1   = 0. lim  (Ct (f ) − f ) − Γ(f )  t→0+ t BMOA Since convergence in the BM OA norm implies uniform convergence on compact subsets of D and therefore in particular pointwise convergence, for each z ∈ D we have f (ϕt (z)) − f (z) f (ϕt (z)) − f (ϕ0 (z)) = lim+ t t t→0 ∂f ◦ ϕt (z) ∂ϕ (z) t = = f
(ϕ0 (z)) = f
(z)G(z), ∂t ∂t

Γ(f )(z) = lim+ t→0

t=0

t=0

that is, Gf
= Γ(f ) ∈ BM OA, and thus D (Γ) ⊂ {f ∈ BM OA : Gf
∈ BM OA}. Taking closures and bearing in mind the fact from the general theory of operator semigroups that D (Γ) is dense in [ϕt , BM OA] we conclude [ϕt , BM OA] ⊆ {f ∈ BM OA : Gf
∈ BM OA}. Conversely, let f ∈ BM OA such that m := Gf
∈ BM OA. First of all, we assert that





(f ◦ ϕt ) (z) − f (z) =


0

t

(m ◦ ϕs )
(z)ds; for t ≥ 0, z ∈ D.

Indeed, G(z) ((f ◦ ϕt )
(z) − f
(z)) = f
(ϕt (z))G(z)ϕ
t (z) − m(z) ∂ϕt (z) − m(z) ∂t
t ∂(m ◦ ϕs )(z) ds = m(ϕt (z)) − m(z) = ∂s 0
t = G(z)m
(ϕs (z))ϕ
s (z)ds 0
t (m ◦ ϕs )
(z)ds. = G(z)

= f
(ϕt (z))

0

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Since G is not identically null this verifies our assertion. Next let I be an interval in ∂D and R(I) the corresponding Carleson rectangle. For 0 ≤ t ≤ 1 we have
2 |(f ◦ ϕt )
(z) − f
(z)| (1 − |z|2 )dA(z) R(I)


t 2
2 = (m ◦ ϕs ) (z)ds (1 − |z| )dA(z) 0 R(I) 
1 
2
t |(m ◦ ϕs ) (z)| ds (1 − |z|2 )dA(z) ≤


R(I)

0

where we have applied Cauchy-Schwarz in the inside integral. Dividing by |I|, taking sup and interchanging the integrals we have  12 
1 2

2 |(f ◦ ϕt ) (z) − f (z)| (1 − |z| )dA(z) sup |I| R(I) I⊆∂D  12 

1  1 2
2 ≤ sup t |(m ◦ ϕs ) (z)| ds (1 − |z| )dA(z) |I| R(I) I⊆∂D 0   12 

1 1 2 ≤ sup t |(m ◦ ϕs )
(z)| (1 − |z|2 )dA(z) ds |I| R(I) I⊆∂D 0 
1  12 2 ≤ t m ◦ ϕs BMOA ds 0 √ ≤ t sup m ◦ ϕs BMOA s∈[0,1]

√ ≤ tCmBMOA sup (1 − log(1 − |ϕs (0)|)) ≤C




√

s∈[0,1]

t,




where C > 0 is a certain constant not depending on t. Hence, √ Ct f − f BMOA ≤ |f (ϕt (0)) − f (0)| + C
t. Since limt→0+ ϕt (0) = 0 we find that limt→0+ Ct f − f BMOA = 0, hence f ∈ [ϕt , BM OA]. We have shown {f ∈ BM OA : Gf
∈ BM OA} ⊂ [ϕt , BM OA], and the desired inclusion follows by taking closures.
If (ϕt ) is a semigroup of analytic functions, then every composition operator Ct (f ) = f ◦ ϕt is bounded on V M OA. This is because each ϕt belongs to the Dirichlet space D (recall that ϕt is univalent) and therefore also in V M OA. Thus the composition semigroup (Ct ) consists of bounded operators on V M OA. Moreover (Ct ) is strongly continuous on V M OA for every semigroup (ϕt ). This was stated in [18] without proof, and is contained as a special case among the Qp,0 spaces in [20, Theorem 4.1]. A short proof goes as follows. Strong continuity

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requires that limt→0 f ◦ ϕt − f BMOA = 0 for each f ∈ V M OA. For a polynomial P we can write f ◦ ϕt − f BMOA ≤ f ◦ ϕt − P ◦ ϕt BMOA + P ◦ ϕt − P BMOA + P − f BMOA ≤ (Ct V MOA + 1)P − f BMOA + P ◦ ϕt − P BMOA . Since V M OA contains the polynomials as a dense set and since sup Ct V MOA < ∞,

0≤t<1

it suffices to show limt→0 P ◦ ϕt − P BMOA = 0 for each polynomial. This now follows from the inequality gBMOA ≤ gD between the V M OA-norm and the Dirichlet space norm which is valid for all g ∈ D, see (2.1), along with the fact that every semigroup generates a semigroup of operators on the Dirichlet space [17, Theorem 1]. We proceed however to provide an alternative direct proof of this result which does not use [17, Theorem 1] and which is based on the V M OA−H 1 duality. Recall that this duality is realized by the pairing,
f, g := lim

r→1

2π

0

f (reiθ )g(reiθ )

dθ , 2π

f ∈ V M OA, g ∈ H 1 .

If we restrict the choice of f and g, for example if both are chosen to lie in H 2 , then the pairing can be expressed by the Littlewood-Paley identity (see [22, 8.1.9]):
1 (2.3) f, g = f (0)g(0) + 2 f
(z)g
(z) log dA(z). |z| D Now, we present another formulation of this dual pair involving functions in spaces which will be more convenient for our purposes. Lemma 2.3. If f ∈ D and g ∈ H 1 , then
1 f, g = f (0)g(0) + 2 f
(z)g
(z) log dA(z). |z| D Proof. Select a sequence (gn ) in H 2 converging to g in H 1 . Using the LittewoodPaley identity, we have
1 f, gn = f (0)gn (0) + 2 f
(z)gn
(z) log dA(z) |z| D and f, g = limn→∞ f, gn .

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Now,


Semigroups in BMOA

53

1 dA(z) |z|
D 1 ≤ |f
(z)(gn
(z) − g
(z))| log dA(z) |z| |z|>1/2
1 + |f
(z)(gn
(z) − g
(z))| log dA(z) |z| |z|≤1/2  12 
 12 
2 2


2 ≤ C1 |f (z)| dA(z) |gn (z) − g (z)| (1 − |z|) dA(z) D
D 1 + |f
(z)(gn
(z) − g
(z))| log dA(z) |z| |z|≤1/2

n 1 For any function h = ∞ n=0 an z ∈ H , we obtain from Hardy’s inequality (see [10, Theorem 6.2]) that |f
(z)(gn
(z) − g
(z))| log


D

Hence,

|h
(z)|2 (1 − |z|)2 dA(z) ≤ C2


D

∞  |an |2 ≤ C2 h2H 1 . n n=0

(2.4)

2

|gn
(z) − g
(z)| (1 − |z|)2 dA(z) ≤ C2 gn − g2H 1 .

1 ∈ L1 (dA) and Lebesgue’s dominated Finally, applying the inclusion f
(z) log |z| convergence theorem, one also has
1
|f
(z)(gn
(z) − g
(z))| log dA(z) = 0. lim n→∞ |z|≤1/2 |z|

Theorem 2.4. Every semigroup (ϕt ) generates a semigroup of operators on V M OA. Proof. From the general theory of operator semigroups, a semigroup which is weakly continuous on a Banach space is in fact strongly continuous [19, p. 233]). Thus it suffices to prove that for each f ∈ V M OA we have w − lim+ Ct (f ) = f, t→0

where w− denotes the weak limit. In other words for each fixed f ∈ V M OA we want to prove lim+ Ct (f ), g = f, g t→0

for every g ∈ H 1 . Arguing as before about the density of polynomials in V M OA and the fact that sup0≤t<1 Ct V MOA < ∞ we see that it suffices to prove this for f = P a polynomial. Now, using again the Area Theorem (ϕt is univalent), we find

2 2

|P (ϕt (z))ϕt (z)| dA(z) ≤ P
∞ |ϕ
t (z)|2 dA(z) ≤ P
2∞ , D

D

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so we can apply Lemma 2.3. Therefore, if P a polynomial and g ∈ H 1 we deduce
1 P ◦ ϕt − P, g = (P (ϕt (0))−P (0))g(0) + 2 (P ◦ ϕt − P )
(z)g
(z) log |dA(z). |z| D For each δ > 0, we split the integral
1 |(P ◦ ϕt − P )
(z)| |g
(z) | log dA(z) |z| D
1

≤ |(P ◦ ϕt ) − P ) (z)||g (z) | log dA(z) |z| |z|>δ
1 + |(P ◦ ϕt − P )
(z)||g
(z) | log dA(z) = (1) + (2). |z| |z|≤δ 1 To estimate the first integral, we use the estimate log |z| ≈ 1 − |z| and apply Cauchy-Schwarz to obtain
(1) ≤ CP
∞ (|ϕ
t (z)| + 1)|g
(z)|(1 − |z|)dA(z) |z|>δ

1 1 2 ≤ CP
∞ (( |ϕ
t (z)| dA(z)) 2 + 1)( |g
(z)|2 (1 − |z|)2 dA(z)) 2 D |z|>δ
1 ≤ 2CP
∞ ( |g
(z)|2 (1 − |z|)2 dA(z)) 2 . |z|>δ

Now using (2.4) one has that g
(z)(1 − |z|) ∈ L2 (dA), which shows that given ε > 0 and, for all t > 0, there exists 0 < δ0 < 1 such that
1 (2.5) |(P ◦ ϕt ) − P )
(z)| |g
(z) | log |dA(z) < ε . |z| |z|>δ0 At the same time, for every z ∈ D, lim (P ◦ ϕt − P )
(z) = 0.

t→0

1 ∈ L1 (dA) (note that L2 (dA) ⊂ L1 (dA) Therefore, using the inclusion g
(z) log |z| 1 and again log |z| ≈ 1 − |z|) and the Lebesgue dominated convergence theorem, one concludes that
1 (2.6) lim |(P (ϕt ) − P )
(z)||g
(z) | log dA(z) = 0. t→0 |z|≤δ |z| 0




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3. VMOA and Maximal Subspaces This section is devoted to analyzing those semigroups of analytic functions (ϕt ) such that V M OA = [ϕt , BM OA]. Since V M OA is always contained in the subspace [ϕt , BM OA] we see that V M OA = [ϕt , BM OA] is equivalent to the following statement: if f ∈ BM OA, then f ∈ V M OA

if and only if

lim f ◦ ϕt − f BMOA = 0.

t→0+

It is easy to see that the inclusion V M OA ⊂ [ϕt , BM OA] can be proper. The easiest example of this type is the semigroup ϕt (z) = e−t z + 1 − e−t ,

t ≥ 0, z ∈ D.

For this semigroup the function f (z) = log(1 − z) ∈ BM OA \ V M OA satisfies   f ◦ ϕt − f BMOA = log(e−t (1 − z)) − log(1 − z)BMOA = log e−t −→ 0, thus f ∈ [ϕt , BM OA]. In fact it is easy to construct general examples of semigroups with this behavior. For instance take any starlike univalent function h : D → C with h(0) = 0 and h ∈ BM OA \ V M OA and define ϕt (z) = h−1 (e−t h(z)). Then h ◦ ϕt − hBMOA = |e−t − 1|hBMOA −→ 0, so that h ∈ [ϕt , BM OA] while h ∈ V M OA. The following theorem gives a sufficient condition on the infinitesimal generator which assures that V M OA = [ϕt , BM OA]. Theorem 3.1. Let (ϕt ) be a semigroup with infinitesimal generator G. Assume that for some 0 < α < 1, (1 − |z|)α = O (1) as |z| → 1. (3.1) G(z) Then V M OA = [ϕt , BM OA]. Proof. Since [ϕt , BM OA] = {f ∈ BM OA : Gf
∈ BM OA}, it suffices to show {f ∈ BM OA : Gf
∈ BM OA} ⊂ V M OA. Let g ∈ BM OA with Gg
∈ BM OA. First we choose indices p, p
such that 1/p + 1/p
= 1, 2 < p < ∞, and such that α < p1
< α + 12 . Hence α = p1
− ε with 0 < ε < 1/2. We use the usual notation 
2π 1/p iθ p dθ Mp (f, r) = |f (re )| 2π 0 and we have, taking into account that BM OA ⊂ H p , for 0 < δ ≤ r < 1, Mp (g
, r) = Mp (Gg


Cp 1 1 , r) ≤ Mp (Gg
, r)M∞ ( , r) ≤ , G G (1 − r)1/p
−ε

where Cp is a constant depending only on p.

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In order to show that g ∈ V M OA we will use the characterization of V M OA in terms of Carleson measures (see [22, 8.2.5 and 8.4.2]); it suffices to prove
2 2 2 (1 − |w| )(1 − |z| ) lim |g
(w)| dA(w) = 0. 2 |z|→1 D |1 − zw| Now let q = p/2 > 1 and apply H¨ older’s inequality for the pair of indices q, q
, that is, p2 + q1
= 1 to obtain
D

|g
(w)|2


1
2π 2 2 (1 − |w|2 )(1 − |z|2 )
iθ 2 (1 − r )(1 − |z| ) dA(w) = |g (re )| dθrdr |1 − zw|2 |1 − zre−iθ |2 0 0 
 q1


1 2π 2 q
2 q (1 − r ) (1 − |z| ) ≤ Mp2 (g
, r) dθ dr |1 − zre−iθ |2q
0 0
1 (1 − r) Mp2 (g
, r) dr = C(1 − |z|)Q(|z|), ≤ C(1 − |z|) 2− 1 0 (1 − |z|r) q


where the last inequality follows from the standard estimate (c > 0)
2π dθ 1 ≈ as |w| → 1 −iθ |1+c |1 − we (1 − |w|2 )c 0 (see, for example, [8, Exercise 2.1.4]) and Q(|z|) denotes the last integral. We now have
δ
1 (1 − r) (1 − r) 2
Q(|z|) = Mp (g , r) dr + Mp2 (g
, r) dr 2− q1
2− 1 0 δ (1 − |z|r) (1 − |z|r) q

δ
1
1 (1 − r)1−2/p +2ε 2
2 ≤ Mp (g , δ) dr 1 dr + Cp 2− 1 0 (1 − |z|r) q δ (1 − |z|r) q

1
1
1 (1 − r)1−2/p +2ε dr + C dr ≤ C1 2 1 2− 1 0 (1 − r) q 0 (1 − |z|r) q

1
(1 − r)1−2/p +2ε = C1
+ C2 dr 2− 1 0 (1 − |z|r) q
≤ C1
+ C2
(1 − |z|)−1+2ε where the last integral was calculated by integration by parts as in [16, Lemma 6]. For the sake of clearness, we recall that such an estimate is exactly
1 (1 − r)γ dr ≤ C(1 − ρ)1+γ−m , ρ ∈ (0, 1), m 0 (1 − ρr) where γ = 1 −

2 p


+ 2ε > −1, m = 2 −

1 q


> 1 + γ, and ρ = r|z|.

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Putting all these together we find

D

|g
(w)|

2

(1 − |w|2 )(1 − |z|2 ) dA(w) ≤ C(1 − |z|)Q(|z|) |1 − zw|2 ≤ C(1 − |z|)(C1
+ C2
(1 − |z|)−1+2ε ) ≤ C

max{(1 − |z|), (1 − |z|)2ε },


and the proof is complete. As an immediate corollary we have

Corollary 3.2. Suppose (ϕt ) is a semigroup whose infinitesimal generator G satisfies condition (3.1) of Theorem 3.1. Then for a function f ∈ BM OA the following are equivalent 1. f ∈ V M OA. 2. limt→0+ f ◦ ϕt − f BMOA = 0. Clearly the semigroups ϕt (z) = e−t z and ϕt (z) = eit z satisfy the condition (3.1) since, in both cases, the infinitesimal generator is G(z) = cz for a certain nonzero constant c. Thus Theorem 3.1 gives an alternative proof (with entirely different techniques, see also [22]) of Sarason’s result. But there is a plethora of different semigroups (ϕt ) for which V M OA = [ϕt , BM OA]. A specific class of examples of this type are given by the semigroups associated with the generators G(z) = −z(1 − z)α ,

0 < α < 1.

To appreciate the breadth of the theorem recall that infinitesimal generators of semigroups with Denjoy-Wolff point b = 0 have the form G(z) = −zP (z) where Re P (z) ≥ 0. By Schwarz’s lemma applied to 1/P which also has nonnegative real part, P (z) satisfies the growth condition 1 1 + |z| P (z) ≤ C2 1 − |z| as |z| → 1. Thus the most general infinitesimal generator of semigroups with inner DenjoyWolff point, fulfills the condition 1−|z| G(z) = O(1). Remark. Clearly, our O(1) condition is intimately related to the number and location of the zeros (in the angular sense) of the infinitesimal generator on the boundary of the unit disk. This topic has been partly analyzed in [6]. Another sufficient condition of the same nature which implies the conclusion of the theorem is
dA(z) < ∞, p |G(z)| |z|≥δ for some 0 < δ < 1 and p > 2. Indeed for δ < r < 1 we have
1 p (1 − r)Mp (1/G, r) ≤ Mpp (1/G, s)ds, r

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Blasco, Contreras, D´ıaz-Madrigal, Mart´ınez and Siskakis

so by the finiteness of the integral we have  Mp (1/G, r) = o

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1 1

(1 − r) p

.

Now, using the Hardy-Littlewood estimates (see [10, 5.9]) one has   CMp (1/G, r) 1 M∞ (1/G, r) ≤ =o , 1 2 (1 − r) p (1 − r) p and this little-oh condition implies our big-Oh condition. We now present a necessary condition for semigroups with inner Denjoy-Wolff point for V M OA = [ϕt , BM OA] to hold. Observe that this necessary condition is quite close to the sufficient condition of Theorem 3.1. Theorem 3.3. Let (ϕt ) be a semigroup with infinitesimal generator G and DenjoyWolff point b ∈ D. If V M OA = [ϕt , BM OA], then 1 − |z| = 0. |z|→1 G(z) lim

Proof. Without loss of generality, we may assume that b = 0. The infinitesimal generator then is G(z) = −zP (z), where P is analytic with Re P ≥ 0. If P is constant, the result is clear. Otherwise consider the function
z
z u 1 du = − du. m(z) = G(u) P (u) 0 0 Since Re(1/P ) ≥ 0 we have m is univalent (see [13, Proposition 1.10]) so, according to [11, Chap. IV, Exercise 25], we see that m ∈ BM OA if and only if m belongs to the Bloch space. Moreover, using [7, Chap. 17, Proposition 1.5], we deduce that m belongs to the Bloch space and thus m ∈ BM OA. Now observe that z ϕt (z) − z ϕt (z)ϕ
t (z) − = G(ϕt (z)) G(z) G(z)
t
t ∂ϕs (z) 1 ds = ϕ
s (z) ds, = G(z) 0 ∂s 0

(m ◦ ϕt )
(z) − m
(z) =

where we have used (1.1) twice. Hence
t 2
t

2
|(m ◦ ϕt ) (z) − m (z)| = ϕs (z) ds ≤ t |ϕ
s (z)|2 ds. 0

0

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Now let I ⊂ ∂D be an interval and R(I) the corresponding Carleson rectangle. We have
1 |(m ◦ ϕt )
(z) − m
(z)|2 (1 − |z|2 ) dA(z) |I| R(I)  
t
1
2 ≤ |ϕs (z)| ds (1 − |z|2 ) dA(z) t |I| R(I) 0
t
1 =t |ϕ
s (z)|2 (1 − |z|2 ) dA(z)ds |I| 0 R(I)
t
≤ Ct |ϕ
s (z)|2 dA(z) ds 0

≤ Ct

R(I)


t
0


= Ct

0

t

D

|ϕ
s (z)|2 dA(z) ds

[Area(ϕs (D))]2 ds ≤ C1 t2 .

Now the norm m ◦ ϕt − mBMOA equals 1/2 
1

2 2 |(m ◦ ϕt ) (z) − m (z)| (1 − |z| ) dA(z) |m(ϕt (0)) − m(0)| + sup |I| R(I) I⊂∂D ≤ |m(ϕt (0)) − m(0)| + C2 t, so limt→0 m ◦ ϕt − mBMOA = 0. Thus m ∈ [ϕt , BM OA] and by the hypothesis m ∈ V M OA. The following standard argument for functions in V M OA completes a−z the proof. For a ∈ D write φa (z) = 1−¯ az , then (1 − |a|2 )2 |m
(a)|2 = |(m ◦ φa )
(0)|2 ≤ m ◦ φa 2H 2
≤C |(m ◦ φa (z))
|2 (1 − |z|2 ) dA(z) D

(by the change of variables w = φa (z))
=C |m
(w)|2 (1 − |φa (w)|2 ) dA(w) D
(1 − |a|2 )(1 − |w|2 ) =C |m
(w)|2 dA(w) |1 − a ¯w|2 D and this last integral tends to 0 as |a| → 1 because m ∈ V M OA. It follows that 1 − |a| (1 − |a|)m
(a) = lim = 0. |a|→1 G(a) |a|→1 a lim




We end this paper by characterizing those semigroups (ϕt ) of linear fractional maps such that V M OA = [ϕt , BM OA]. Roughly speaking, we show that, when

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Blasco, Contreras, D´ıaz-Madrigal, Mart´ınez and Siskakis

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dealing with semigroups of linear fractional maps, those found by Sarason are the unique ones for which his theorem is true. For a detailed analysis of these types of semigroups in one and several variables we refer the reader to [3]. Proposition 3.4. Let (ϕt ) be a semigroup such that each iterate is a linear fractional map. Then, V M OA = [ϕt , BM OA] if and only if (ϕt ) has a fixed point in the unit disk but is without fixed points in the boundary of the unit disk. Proof. We freely use some results from [3]. In particular, we use that the infinitesimal generator of a semigroup of linear fractional maps is a polynomial of degree two. Assume that V M OA = [ϕt , BM OA]. First of all, we are going to prove that the Denjoy-Wolff point of the semigroup (ϕt ) must be in the unit disk. Let σ be the Koenigs or univalent map of the semigroup (see [17]). It is known that σ satisfies σ ◦ ϕt = σ + t, for all t ≥ 0. If the semigroup is hyperbolic, using [5, Theorem 2.1], [11, page 283] and [13, page 78], we conclude that σ ∈ BM OA \ V M OA. Moreover, we observe that t→0

σ ◦ ϕt − σBMOA = t −→ 0, and, therefore, σ ∈ [ϕt , BM OA]. Hence, the equality we are dealing with is impossible for this type of semigroup (we want to mention that this argument is indeed completely general, not only valid in the framework of semigroups of linear fractional maps). If the semigroup is parabolic, then σ is the Riemann map of a half-plane. In this case, taking c ∈ C\σ(D), we see that f (z) = Log(σ(z)−c) ∈ BM OA\V M OA and, arguing as in the hyperbolic case, we obtain f ∈ [ϕt , BM OA]. Therefore, for this type of semigroup the equality is also impossible and we conclude that the semigroup is necessarily elliptic (it has a fixed point in D). Therefore, we may assume in our proof that the semigroup has Denjoy-Wolff point in D. Now, since V M OA = [ϕt , BM OA], using Theorem 3.3, we have 1 − |z| = 0. |z|→1 G(z) lim

If the other fixed point is in the boundary, then the infinitesimal generator has the form G(z) = λ(z − a)(z − b), with a ∈ ∂D, b ∈ D and λ = 0. Taking limits, we obtain a contradiction. Likewise, assume now that (ϕt ) has no fixed point in the boundary of D. In this case, the infinitesimal generator fits the following scheme: G(z) = or G(z) =

λ(z − a)(z − b), with a ∈ C \ D, b ∈ D and λ = 0 λ(z − b), with b ∈ D and λ = 0.

Finally, apply Theorem 3.1




Vol. 61 (2008)

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References [1] E. Berkson and H. Porta, Semigroups of analytic functions and composition operators, Michigan Math. J. 25 (1978) 101–115. [2] P.S. Bourdon, J.A. Cima, and A.L. Matheson, Compact composition operators on BM OA, Trans. Amer. Math. Soc. 351 (1999) 2183–2196. [3] F. Bracci, M.D. Contreras, and S. D´ıaz-Madrigal, Infinitesimal generators associated with semigroups of linear fractional maps. J. Analyse Math. 102 (2007) 119–142. [4] M.D. Contreras and S. D´ıaz-Madrigal, Fractional iteration in the disk algebra: prime ends and composition operators, Revista Mat. Iberoamericana 21 (2005) 911–928. [5] M.D. Contreras and S. D´ıaz-Madrigal, Analytic flows in the unit disk: angular derivatives and boundary fixed points, Pacific J. Math. 222 (2005) 253–286. [6] M.D. Contreras, S. D´ıaz-Madrigal, and Ch. Pommerenke, On boundary critical points for semigroups of analytic functions, Math. Scand. 98 (2006) 125–142. [7] J.B. Conway, Functions of One Complex Variable II, Graduated Texts in Mathematics, vol. 159, Springer-Verlag, New York, 1995. [8] C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. [9] N. Dunford and J. T. Schwartz, Linear operators, Part I. Wiley Classics Library, John Wiley & Sons Inc., New York, 1988. [10] P.L. Duren, Theory of H p -spaces, Academic Press, New York, 1970. [11] J.B. Garnett, Bounded Analytic Functions, Pure and Applied Mathematics, Vol. 96, Academic Press, New York, 1981. ¨ [12] Ch. Pommerenke, Uber die subordination analytischer Funktionen, J. Reine Angew. Math. 218 (1965) 159–173. [13] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992. [14] D. Sarason, Function of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975) 391–405. [15] D. Sarason, Function Theory on the Unit Circle, Virginia Poly. Inst. and State Univ. Blacksburg, Virginia, 1979. [16] A. L. Shields and D. L. Williams, Bounded projections, duality and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162 (1971), 287–302. [17] A. G. Siskakis, Semigroups of composition operators on the Dirichlet space, Results Math. 30 (1996), 165–173. [18] A. G. Siskakis, Semigroups of composition operators on spaces of analytic functions, a review, Contemp. Math. 213 (1998) 229–252. [19] K. Yosida, Functional Analysis, Springer-Verlag, Sixth Edition, Fundamental Principles of Mathematical Sciences, vol. 123, Springer-Verlag, Berlin-New York, 1980. [20] K. J. Wirths and J. Xiao, Recognizing Qp,0 functions as per Dirichlet space structure, Bull. Belg. Math. Soc. 8 (2001), 47–59. [21] J. Xiao, Holomorphic Q Classes, Lecture Notes in Math. 1767, Springer-Verlag 2001. [22] K. Zhu, Operator Theory in Function spaces, Marcel Dekker, Inc., New York and Basel, 1990.

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O. Blasco Departamento de An´ alisis Matem´ atico Universidad de Valencia 46100 Burjassot Valencia Spain e-mail: [email protected] M.D. Contreras and S. D´ıaz-Madrigal Departamento de Matem´ atica Aplicada II Escuela T´ecnica Superior de Ingenieros Universidad de Sevilla Camino de los Descubrimientos, s/n 41092, Sevilla Spain e-mail: [email protected] [email protected] J. Mart´ınez Departamento de An´ alisis Matem´ atico Universidad de Valencia 46100 Burjassot Valencia Spain e-mail: [email protected] A.G. Siskakis Department of Mathematics Aristotle University of Thessaloniki 54006 Thessaloniki Greece e-mail: [email protected] Submitted: July 28, 2006 Revised: January 14, 2008

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Integr. equ. oper. theory 61 (2008), 63–75 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/010063-13, published online April 18, 2008 DOI 10.1007/s00020-008-1583-8

Integral Equations and Operator Theory

Spectra of Toeplitz Operators and Compositions of Muckenhoupt Weights with Blaschke Products Sergei Grudsky and Eugene Shargorodsky Abstract. We discuss the optimality of a sufficient condition from [12] for a point to belong to the essential spectrum of a Toeplitz operator with a bounded measurable coefficient. Our approach is based on a new sufficient condition for a composition of a Muckenhoupt weight with a Blaschke product to belong to the same Muckenhoupt class. Mathematics Subject Classification (2000). 47B35, 45E10, 30D50. Keywords. Toeplitz operators, cluster values, Blaschke products, Muckenhoupt weights.

1. Introduction and main results Let T = {ζ ∈ C : |ζ| = 1} be the unit circle. A number c ∈ C is called a (left, right) cluster value of a measurable function a : T → C at a point ζ ∈ T if 1/(a − c) ∈ L∞ (W ) for every neighbourhood (left semi-neighbourhood, right semi-neighbourhood) W ⊂ T of ζ. Cluster values are invariant under changes of the function on measure zero sets. We denote the set of all left (right) cluster values of a at ζ by a(ζ − 0) (by a(ζ + 0)), and use also the following notation a(ζ) = a(ζ − 0) ∪ a(ζ + 0), a(T) = ∪ζ∈T a(ζ). It is easy to see that a(ζ − 0), a(ζ + 0), a(ζ) and a(T) are closed sets. Hence they are all compact if a ∈ L∞ (T). Let H p (T), 1 ≤ p ≤ ∞ denote the Hardy space, that is H p (T) := {f ∈ Lp (T) : fn = 0 for n < 0}, where fn is the nth Fourier coefficient of f . Let T (a) : H p (T) → H p (T), 1 < p < ∞ denote the Toeplitz operator generated by a function a ∈ L∞ (T), i.e. T (a)f = P (af ), f ∈ H p (T), where P is the Riesz The first author was partially supported by CONACYT project U46936-F, Mexico.

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projection:


1 g(w) 1 dw, ζ ∈ T. P g(ζ) = g(ζ) + 2 2πi T w − ζ P : Lp (T) → H p (T), 1 < p < ∞ is a bounded projection and  +∞  +∞   n gn ζ gn ζ n . = P n=−∞

n=0

If a(ζ) consists of at most two points for each ζ ∈ T, in particular if a is continuous or piecewise continuous, then the spectrum of T (a) can be described in terms of a(ζ ± 0), ζ ∈ T (see [3, 4, 13]). This is no longer possible if a(ζ) is allowed to contain more than two points (see [2, 4.71–4.78] and [10]). It is no longer sufficient to know the values of a in this case, it is important to know “how these values are attained” by a. Since a complete description of the essential spectrum of T (a) in terms of the cluster values of a ∈ L∞ (T) is impossible, it is natural to try finding “optimal” sufficient conditions for a point λ to belong to the essential spectrum. Results of this sort were obtained in [11, 12]. In order to state them we need the following notation. Let K ⊂ C be an arbitrary compact set and λ ∈ C \ K. Then the set   w − λ  σ(K; λ) = w∈K ⊆T |w − λ| is compact as a continuous image of a compact set. Hence the set ∆λ (K) := T \ σ(K; λ) is open in T. So, ∆λ (K) is the union of an at most countable family of open arcs. We call an open arc of T p–large if its length is greater than or equal to p 2π max{p,q} , where q = p−1 , 1 < p < ∞. The following result has been proved in [12]. Theorem 1.1. Let 1 < p < ∞, a ∈ L∞ (T), λ ∈ C \ a(T) and suppose that, for some ζ ∈ T, (i) ∆λ (a(ζ − 0)) (or ∆λ (a(ζ + 0)) ) contains at least two p–large arcs, (ii) ∆λ (a(ζ + 0)) (or ∆λ (a(ζ − 0)) respectively ) contains at least one p–large arc. Then λ belongs to the essential spectrum of T (a) : H p (T) → H p (T). A weaker result (with ∆λ (a(ζ)) in place of ∆λ (a(ζ ± 0)) in condition (ii)) was proved in [11] where it was also shown that condition (i) is optimal in the following sense: for any compact K ⊂ C and λ ∈ C \ K such that ∆λ (K) contains at most one p–large arc there exists a ∈ L∞ (T) such that a(−1 ± 0) = a(T) = K and T (a) − λI : H r (T) → H r (T) is invertible for any r ∈ [min{p, q}, max{p, q}]. A question that has been open since [11] is whether or not condition (ii) can be dropped, i.e. whether condition (i) alone is sufficient for λ to belong to the essential spectrum of T (a) : H p (T) → H p (T). The following result gives a negative answer to this question.

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Theorem 1.2. There exists a ∈ L∞ (T) such that a(1 − 0) = {±1}, |a| ≡ 1, T (a) : H p (T) → H p (T) is invertible for any p ∈ (1, 2), and T (1/a) : H p (T) → H p (T) is invertible for any p ∈ (2, +∞). The proof of Theorem 1.2 relies on an argument which is related to the following question. Suppose v is an inner function, i.e. v is a nonconstant function in H ∞ (T) such that |v| = 1 almost everywhere on T. If b ∈ L∞ (T), then b◦v ∈ L∞ (T) and the question is whether or not the invertibility of T (b) : H p (T) → H p (T) implies that of T (b ◦ v) : H p (T) → H p (T). An equivalent form of this question is in terms Ap classes (see [1, Section 1]). We say that a measurable function ρ : T → [0, +∞] satisfies the Ap condition if p1 
q1 
1 1 sup ρp (ζ)|dζ| ρ−q (ζ)|dζ| = Cp < ∞, (1.1) |I| I |I| I I where I ⊂ T is an arbitrary arc and |I| denotes its length. The question is whether or not ρ ∈ Ap implies ρ ◦ v ∈ Ap . Although the answer is positive in the case p = 2 (see, e.g., [1, Section 2]), it turns out that for every p ∈ (1, +∞) \ {2} there exist a Blaschke product B and ρ ∈ Ap such that ρ ◦ B ∈ Ap (see [1, Theorem 9]). Equivalently, there exists b ∈ L∞ (T) such that T (b) : H p (T) → H p (T) is invertible, but T (b ◦ B) : H p (T) → H p (T) is not invertible (see [1, Theorem 12]). We prove a result in the opposite direction, namely we describe a class of Blaschke products for which the implications ρ ∈ Ap =⇒ ρ ◦ B ∈ Ap , T (b) : H p (T) → H p (T) is invertible =⇒ T (b ◦ B) : H p (T) → H p (T) is invertible do hold. Consider the Blaschke product ∞

 rk − eiθ B eiθ = , 1 − rk eiθ k=1

where rk ∈ (0, 1) and ∞ k=1 (1 − rk ) < 1.

θ ∈ [−π, π],

Theorem 1.3. Suppose r1 ≤ r2 ≤ · · · ≤ rn ≤ · · · , and 1 − rk+1 inf > 0. k≥1 1 − rk If ρ satisfies the Ap condition, then ρ ◦ B also satisfies the Ap condition.

(1.2)

(1.3)

Corollary 1.4. Let 1 < p < ∞, a ∈ L∞ (T), and let a Blaschke product B satisfy the conditions of Theorem 1.3. Then T (a) : H p (T) → H p (T) is invertible if and only if T (a ◦ B) : H p (T) → H p (T) is invertible. Proof. The invertibility of T (a ◦ B) implies that of T (a) according to [1, Theorem 12]. The opposite implication follows from Theorem 1.3 (see [1, Section 1]).


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2. Auxiliary results on inner and outer functions According to the canonical factorisation theorem (see, e.g., [5, Theorem 2.8]), any function from H p (T) \ {0} has a unique, modulo a constant factor, representation as the product of an outer function from H p (T) and an inner function. A function F ∈ H p (T) is called an outer function if 
π it 1 e +z F (z) = eic exp log φ(t) dt , |z| < 1, (2.1) 2π −π eit − z where c is a real number, φ ≥ 0, log φ ∈ L1 ([−π, π]), and φ ∈ Lp ([−π, π]). A function v ∈ H ∞ (T) is called an inner function if |v| = 1 almost everywhere on T. Any inner function v admits a unique factorisation of the form v(z) = eic B(z)S(z), where c is a real number, B is a Blaschke product zk zk − z B(z) = z m |zk | 1 − zk z k

with m ∈ N∪{0}, zk = rk exp(iθk ) = 0, θk ∈ (−π, π], rk = |zk | < 1, 1, and S is a singular inner function 
π it e +z S(z) = exp − dµ(t) it −π e − z

k (1−rk )

<

with a nonnegative measure µ which is singular with respect to the standard Lebesgue measure on [−π, π]. We are particularly interested in the case where v has a unique discontinuity at z = 1 and infinitely many zeros zk . In this case, limk→∞ zk = 1, the singular measure µ is supported by the point t = 0, and  z+1 S(z) = exp κ , κ = const > 0 z−1 (see [7, Ch. II, Theorems 6.1 and 6.2]). We will also assume that B(0) = 0. Then ∞

 zk zk − eiθ , θ ∈ [−π, π]. (2.2) B eiθ = |zk | 1 − zk eiθ k=1

Theorem 2.1. ([6, Theorem 2.8]) Suppose B has  the form (2.2) and limk→∞ zk = 1. Then one can choose a branch of arg B eiτ which is continuous and increasing on (0, 2π), and which satisfies the following condition



 lim arg B eiτ =: A+ < 0, lim arg B eiτ =: A− > 0. τ →0+0

τ →2π−0

Moreover, at lest one of these limits is infinite and   

 −2 (π + ϕ (θ)) + ϕ (θ) , θ ∈ (0, π],  k k θk ≥θ θk <θ

iθ   arg B e =   

  2 θk ≤θ (π − ϕk (θ)) − θk >θ ϕk (θ) , θ ∈ [−π, 0),

(2.3)

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where

Toeplitz Operators and Blaschke Products

 θ − θk ϕk (θ) = arctan εk cot , 2

εk =

1 − rk . 1 + rk

67

(2.4)

Theorem 2.2. (See [6, Theorem 2.10 and the end of the proof of Theorem 5.9].) Suppose a real valued function η is continuous on [−π, π] \ {0} and lim (η(t) ∓ π log |t|) = 0.

t→0±0

Then the function eiη admits the following representation

   eiη(t) = B eit g B eit d eit , t ∈ [−π, π], where g, d ∈ C(T), the index of g is 0, and B is the infinite Blaschke product with the zeros 2 − exp(−k + 1/2) . zk = 2 + exp(−k + 1/2) We finish this section with an example of an outer function which is used in the proof of Theorem 1.2. Example 2.3. Consider the function   1−z h(z) = exp −i c log i , 2 where c > 0 and log denotes the branch of logarithm which is analytic in the complex plane cut along (−∞, 0] and real valued on (0, +∞). It is clear that h is analytic inside the unit disk, and since  1−z > 0, |z| < 1, Im i 2 h satisfies the following estimate 1 < |h(z)| < ecπ ,

|z| < 1.

Hence h, 1/h ∈ H ∞ (T) and h is an outer function (see [7, Ch. II, Corollary 4.7]). It is also clear that h ∈ C ∞ (T \ {1}), and since i we have |h(eiθ )| =

θ θ 1 − eiθ = ei 2 sin , 2 2



  exp c θ2 ,

θ ∈ (0, π],





exp c θ2 + π , θ ∈ [−π, 0),    θ  iθ  arg h(e ) = −c log sin  . 2 

(2.5)

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3. Proof of Theorem 1.3 Suppose the conditions of Theorem 1.3 are satisfied and let

 A(θ) := arg B eiθ , A(±π) = 0. The proof of Theorem 1.3 relies upon analysis of the properties of A. The corresponding results are collected in the following two lemmas. Since A admits the representation (2.3), (2.4) (with θk = 0 for all k = 1, 2, . . . ), it is convenient to rewrite (1.3) in the following equivalent form εk+1 inf =: c0 > 0. (3.1) k≥1 εk Lemma 3.1. a) The derivative A is increasing on [−π, 0) and decreasing on (0, π]. b)1 c1 |A(θ)| , ∀θ ∈ [−π, π] \ {0}, ≤ A (θ) ≤ | sin θ| 4| sin θ2 | c)

A (θ/c) < c2 , ∀θ ∈ [−π, π] \ {0}, A (θ)

Proof. Let

c1 := min{c0 , ε1 }.

∀c > 1.

 θ Ak (θ) := arctan εk cot . 2

Then A(θ) = −2

∞ 

Ak (θ),

A (θ) = −2

k=1

∞ 

Ak (θ)

k=1

(see (2.3), (2.4)). a) Since −Ak (θ)

= =

1 2 =  2

1 + εk cot θ2 2 sin εk

, 2 (1 − ε2k ) sin2 θ2 + ε2k εk 2 sin2

θ 2

θ 2

εk

2  + εk cos θ2

A is increasing on [−π, 0) and decreasing on (0, π]. b) The equality −Ak (θ) = implies

1 We

εk 2 sin2

θ 2

εk cot θ2 1 1 =



  2 sin θ 1 + εk cot θ 2 1 + εk cot θ2 2

    Ak (θ)  1 uk    Ak (θ)  = | sin θ| (1 + u2 ) arctan uk , k

will not use the upper estimate for A
(θ).

uk = εk cot

|θ| . 2

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u u = lim = 1, 2 ) arctan u 2 ) arctan u u→0+0 (1 + u (1 + u u∈(0,+∞) sup

we get the second inequality in b). Let us prove the first one. It is clear that εk0 cot θ2 uk0 1 1 |θ| , uk0 = εk0 cot A (θ) ≥  =

sin θ 1 + εk cot θ 2 | sin θ| 1 + u2k0 2 0

2

for any k0 ∈ N. Let k0 be the smallest natural number such that uk0 ≤ 1. If k0 > 1, then (3.1) implies εk0 uk0 = ≤ uk0 ≤ 1. c0 ≤ εk0 −1 uk0 −1 Hence uk0 c0 ≥ 2 1 + uk0 2 and c0 c0 ≥ A (θ) ≥ . 2| sin θ| 4| sin θ2 | If k0 = 1, then ε1 ε1 ε1 1 . A (θ) ≥ 2 ≥

2 θ 2 θ ≥ θ 2 sin 2 1 + ε1 cot 4 sin 2 4| sin θ2 | 2

This proves the first inequality in b). c) Since sin ϑ ≤ c sin ϑc and cot ϑc > cot ϑ, ∀ϑ ∈ (0, π/2], we have 2

sin2 θ2 1 + εk cot θ2 Ak (θ/c) 2 = 2 < c .

θ Ak (θ) sin2 2c 1 + εk cot θ




2c

Lemma 3.2. Suppose ϑ0 , ϑ1 , ϑ2 ∈ [−π, π] \ {0}, signϑ0 = signϑ1 = signϑ2 , |ϑ0 | > |ϑ1 | > |ϑ2 |, and |A(ϑ1 ) − A(ϑ0 )| = 2π = |A(ϑ2 ) − A(ϑ1 )|. Then a) |ϑ0 − ϑ1 | ≤ c2 |ϑ0 |, where the constant c2 ∈ (0, 1) depends only on c1 from Lemma 3.1-b); b) |ϑ0 − ϑ1 | ≤ c3 , 1≤ |ϑ1 − ϑ2 | where c3 depends only on c1 . Proof. a) Let ϑ˜ ∈ (ϑ1 , ϑ0 ) be such that c1 . 4 ˜ ϑ0 ) such that Then, according to the mean value theorem, there exists ϑ∗ ∈ (ϑ,   c     ∗ ˜ 1 . A (ϑ ) ϑ − ϑ0  = 4 ˜ − A(ϑ0 )| = |A(ϑ)

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It follows from Lemma 3.1-b) that      c    ϑ  |ϑ | ϑ∗   c  1 ϑ∗  ϑ0 − ϑ˜ ≤ 1 =⇒ ϑ0 − ϑ˜ ≤ sin  ≤ sin 0  ≤ 0 . 4 2 2 2 4 sin 2      Since ϑ0 − ϑ˜ ≤ |ϑ0 |/2, the monotonicity of A implies |A(ϑ0 /2) − A(ϑ0 )| ≥

c1 . 4

Similarly |A(ϑ0 /2j ) − A(ϑ0 /2j−1 )| ≥

c1 , 4

j ∈ N.

Let M = [8π/c1 ] + 1. Then |A(ϑ0 /2M ) − A(ϑ0 )| =

M 

|A(ϑ0 /2j ) − A(ϑ0 /2j−1 )| ≥ M

j=1

8π c1 c1 > = 2π. 4 c1 4

M

Hence ϑ1 ∈ (ϑ0 /2 , ϑ0 ) and



|ϑ0 − ϑ1 | < |ϑ0 − ϑ0 /2M | = 1 − 2−M |ϑ0 |.

This proves a) with c2 = 1 − 2−M = 1 − 2−([8π/c1 ]+1) . b) According to the mean value theorem, there exist ϕ1 ∈ (ϑ1 , ϑ0 ) and ϕ2 ∈ (ϑ2 , ϑ1 ) such that |ϑ0 − ϑ1 | |A (ϕ2 )| =  . |ϑ1 − ϑ2 | |A (ϕ1 )| It follows from part a) that 1≥

ϕ2 ϑ2 ϑ2 ϑ1 > = ≥ (1 − c2 )2 = 2−2([8π/c1 ]+1) . ϕ1 ϑ0 ϑ1 ϑ0

It is now left to use Lemma 3.1-a), c). One can take c3 = 24([8π/c1 ]+1) .




Proof of Theorem 1.3. Let θj ∈ (−π, π] be the points such that A(θj ) = −2πj,

j = 0, ±1, ±2, . . .

(3.2)

and let Ij = γ (exp(iθj+1 ), exp(iθj )) ,

j = 0, ±1, ±2, . . . ,



where γ(ζ, ζ ) ⊂ T is the arc described by a point moving from ζ to ζ  in the counterclockwise direction. Any arc I ⊂ T admits the representation:        I˜j  , Ij   I = j∈J

j∈J˜

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where the set J is finite or infinite, the set J˜ contains at most two elements, and the arcs I˜j have one of the following forms: a) if J = ∅, then J˜ contains one element and   I˜j = γ exp(iθ˜j+1 ), exp(iθ˜j ) , where |A(θ˜j+1 ) − A(θ˜j )| < 4π; b) if J = ∅, then

    I˜j = γ exp(iθj ), exp(iθ˜j ) or γ exp(iθ˜j ), exp(iθj )

and |A(θj ) − A(θ˜j )| < 2π. Case a). Suppose J = ∅,   I = I˜j = γ exp(iθ˜j+1 ), exp(iθ˜j ) ,

|A(θ˜j+1 ) − A(θ˜j )| < 4π.

Since I may contain the point −1, but does not contain in our case the point 1, it is convenient to switch from the function A defined on [−π, π]\{0} to the following function defined on (0, 2π):  A(ψ), if ψ ∈ (0, π], A(ψ) = (3.3) A(ψ − 2π), if ψ ∈ (π, 2π). Let ψ0 < ψ1 be such that A(ψ0 ) = A(θ˜j+1 ) and A(ψ1 ) = A(θ˜j ). Using the change of variable u = A(ψ) we get
ψ1


 1 1 p ρ (B(ζ))|dζ| = ρp exp(iA(ψ)) dψ ∆p := |I| I ψ1 − ψ0 ψ0
A(ψ1 ) 1 du = ρp (exp(iu))  ψ1 − ψ0 A(ψ0 ) A (ψ(u))
maxψ∈[ψ0 ,ψ1 ] (A (ψ))−1 A(ψ1 ) p ≤ ρ (exp(iu))du. ψ1 − ψ0 A(ψ0 ) According to the mean value theorem there exists ψ ∗ ∈ (ψ0 , ψ1 ) such that A (ψ ∗ )(ψ1 − ψ0 ) = A(ψ1 ) − A(ψ0 ). It is now easy to derive from Lemmas 3.1 and 3.2 that  
A(ψ1 ) A (ψ ∗ ) 1 p ∆p ≤ ρ (exp(iu))du minψ∈[ψ0 ,ψ1 ] A (ψ) A(ψ1 ) − A(ψ0 ) A(ψ0 )
A(ψ1 ) c4 ρp (exp(iu))du, ≤ A(ψ1 ) − A(ψ0 ) A(ψ0 )

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where the constant c4 depends only on c1 from Lemma 3.1-b). Similarly,

A(ψ1 ) 1 c4 −q ρ (B(ζ))|dζ| ≤ ρ−q (exp(iu))du. |I| I A(ψ1 ) − A(ψ0 ) A(ψ0 ) Hence



p1 
1q 1 −q ρ (B(ζ))|dζ| ρ (B(ζ))|dζ| ≤ |I| I I  p1
A(ψ1 ) 1 p ρ (exp(iu))du × c4 A(ψ1 ) − A(ψ0 ) A(ψ0 )  q1 
A(ψ1 ) 1 ρ−q (exp(iu))du ≤ 2c4 Cp A(ψ1 ) − A(ψ0 ) A(ψ0 ) 1 |I| 




p

(see (1.1)). The factor 2 appears in the right-hand side because A(ψ1 ) − A(ψ0 ) may be larger than 2π but is less than 2 × 2π. Case b). Let J0 ⊂ Z be the smallest set such that  Ij . I⊆ j∈J0

It follows from Lemma 3.2-b) that   |Ij | ≤ c5 |Ij | ≤ c5 |I|, j∈J0

(3.4)

j∈J

where the constant c5 depends only on c1 from Lemma 3.1-b). Let us estimate
ρp (B(ζ))|dζ|. Λj,p = Ij

This is similar to but easier than the estimate for ∆p in the case a), because we do not need to deal with the function (3.3) now. Since A(θj ) − A(θj+1 ) = 2π, we have
c4 |Ij | −2πj c4 |Ij | ρpLp (T) . Λj,p ≤ ρp (exp(iu))du = 2π −2π(j+1) 2π Hence


I

ρp (B(ζ))|dζ| ≤




ρp (B(ζ))|dζ| =

 j∈J0

Ij


j∈J0

ρp (B(ζ))|dζ|

Ij

 c4 |Ij |  c4 c4 c5 ≤ |Ij | ≤ ρpLp(T) = ρpLp (T) ρpLp(T) |I| 2π 2π 2π j∈J0

(see (3.4)). Similarly

j∈J0


I

ρ−q (B(ζ))|dζ| ≤

c4 c5 −1 q ρ Lq (T) |I|. 2π

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Hence



1 |I|

Toeplitz Operators and Blaschke Products

73

p1 
1q 1 ρp (B(ζ))|dζ| ρ−q (B(ζ))|dζ| ≤ |I| I I c4 c5 ρLp (T) ρ−1 Lq (T) ≤ c4 c5 Cp . 2π







Remark 3.3. The proof of Theorem 1.3 can be easily extended to any inner function v such that arg v(eiτ ) has a continuous and increasing branch on (0, 2π), and A(θ) := arg v(eiθ ) has the following property maxθ∈[θj+1 ,θj−1 ] A (θ) ≤ m < +∞, minθ∈[θj+1 ,θj−1 ] A (θ)

∀j ∈ Z,

(3.5)

where θj ’s are defined by (3.2). Indeed, (3.5) is exactly what is needed for the case a) in the proof of Theorem 1.3. The case b) relies also on Lemma 3.2-b) which in turn follows from (3.5). The above applies for example to the singular inner function  ζ +1 S(ζ) = exp κ , κ = const > 0. ζ −1 Indeed, θ 2 and it is not difficult to see that (3.5) holds in this case. This corresponds to the case of the so-called periodic discontinuity which was considered in [9]. A(θ) = arg S(eiθ ) = −κ cot

4. Proof of Theorem 1.2 Proof. Let a0 ∈ L∞ (T) be defined by  τ a0 (eiτ ) = exp i , τ ∈ (0, 2π). 2 Then a0 is continuous on T \ {1}, a0 (1 ± 0) = ±1, T (a0 ) : H p (T) → H p (T) is invertible for any p ∈ (1, 2), and T (1/a0 ) : H p (T) → H p (T) is invertible for any p ∈ (2, +∞) (see [8, 9.3, 9.8] or [2, 5.39]). Let h0 = h exp −i π2 log 2 , where h is the function from Example 2.3 with π c = 2 . Then h0 (eit ) = |h(eit )| eiϕ(t) , t ∈ [−π, π], where    π t ϕ(t) = − log 2 sin  2 2 (see (2.5)). Let f be a 2π-periodic function such that f ∈ C ∞ ([−π, π] \ {0}), f (t) = ϕ(t) if −π/2 ≤ t < 0, and f (t) = −f (−t) if 0 < t ≤ π/2. Then

   (4.1) e2if (t) = B eit g B eit d eit , t ∈ [−π, π],

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where g, d ∈ C(T), the index of g is 0, and B is the infinite Blaschke product with the zeros 2 − exp(−k + 1/2) zk = 2 + exp(−k + 1/2) (see Theorem 2.2). Since the index of g is 0, there exists g0 ∈ C(T) such that g02 = g. Let d0 ∈ C(T) be such that d20 (eit ) = d(eit ) for t ∈ [−π/2, π/2], d0 (eit ) = 0 for t ∈ [−π, π] and the index of d0 is 0. Consider the function a ∈ L∞ (T) defined by     it it it



 g B e d e |h (e )| 0 0 0 . (4.2) a(eit ) = a0 B eit h0 (eit ) It follows from (4.1) and from the definition of the function f that a2 (eit ) = 1 if −π/2 ≤ t < 0. It is clear that the second factor in the right-hand side of (4.2) is continuous on {eit | − π/2 ≤ t < 0}, whereas the first one has infinitely many discontinuities in any left semi-neighbourhood of 1. Hence a takes values 1 and −1 in any left semi-neighbourhood of 1. So, a(1 − 0) = {±1}. The operator T (a±1 ) : H p (T) → H p (T) is invertible if and only if T (a±1 0 ◦B) : p H (T) → H p (T) is invertible (see, e.g., [6, Theorem 2.1, Propositions 2.3, 4.1 and p p 5.4]). The latter operator is indeed invertible because T (a±1 0 ) : H (T) → H (T) is invertible and B satisfies (1.3) (see Corollary 1.4).


References [1] A. B¨ ottcher and S. Grudsky, On the composition of Muckenhoupt weights and inner functions. J. Lond. Math. Soc., II. Ser. 58, No.1 (1998), 172-184. [2] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz Operators. Springer-Verlag, 1990. [3] K.F. Clancey, One dimensional singular integral operators on Lp . J. Math. Anal. Appl. 54 (1976), 522-529. [4] K.F. Clancey, Corrigendum for the article “One dimensional singular integral operators on Lp ”. J. Math. Anal. Appl. 99 (1984), 527-529. [5] P.L. Duren, Theory of H p Spaces. Academic Press, 1970, and Dover, 2000. [6] V. Dybin and S.M. Grudsky, Introduction to the theory of Toeplitz operators with infinite index. Birkh¨ auser Verlag, 2002. [7] J.B. Garnett, Bounded Analytic Functions. Academic Press, 1981. [8] I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations I & II. Birkh¨ auser Verlag, 1992. [9] S.M. Grudskij and A.B. Khevelev, On invertibility in L2 (R) of singular integral operators with periodic coefficients and a shift. Sov. Math. Dokl. 27 (1983), 486-489. [10] E. Shargorodsky, On singular integral operators with coefficients from Pn C. Tr. Tbilis. Mat. Inst. Razmadze 93 (1990), 52-66 (Russian).

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[11] E. Shargorodsky, On some geometric conditions of Fredholmity of one-dimensional singular integral operators. Integral Equations Oper. Theory 20, No.1 (1994), 119123. [12] E. Shargorodsky, A remark on the essential spectra of Toeplitz operators with bounded measurable coefficients. Integr. Equ. Oper. Theory 57 (2007), 127-132. [13] I. M. Spitkovskij, Factorization of matrix-functions belonging to the classes A˜n (p) and TL. Ukr. Math. J. 35 (1983), 383-388 (translation from Ukr. Mat. Zh. 35, No.4 (1983), 455-460). Sergei Grudsky Departamento de Matematicas CINVESTAV del I.P.N. Apartado Postal 14-740 07000, Mexico, D.F. Mexico e-mail: [email protected] Eugene Shargorodsky Department of Mathematics King’s College London Strand, London WC2R 2LS UK e-mail: [email protected] Submitted: July 9, 2007 Revised: March 3, 2008

Integr. equ. oper. theory 61 (2008), 77–102 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/010077-26, published online April 18, 2008 DOI 10.1007/s00020-008-1581-x

Integral Equations and Operator Theory

Non-Commutative Harmonic and Subharmonic Polynomials J. William Helton, Daniel P. McAllaster and Joshua A. Hernandez Dedicated to Israel Gohberg on the occasion of his 80th birthday.

Abstract. The paper introduces a notion of the Laplace operator of a polynomial p in noncommutative variables x = (x1 , . . . , xg ). The Laplacian Lap[p, h] of p is a polynomial in x and in a noncommuting variable h. When all variables commute we have Lap[p, h] = h2 ∆x p where ∆x p is the usual Laplacian. A symmetric polynomial in symmetric variables will be called harmonic if Lap[p, h] = 0 and subharmonic if the polynomial q(x, h) := Lap[p, h] takes positive semidefinite matrix values whenever matrices X1 , . . . , Xg , H are substituted for the variables x1 , . . . , xg , h. In this paper we classify all homogeneous symmetric harmonic and subharmonic polynomials in two symmetric variables. We find there are not many of them: for example, the span of all such subharmonics of any degree higher than 4 has dimension 2 (if odd degree) and 3 (if even degree). Hopefully, the approach here will suggest ways of defining and analyzing other partial differential equations and inequalities. Keywords. Non-commutative polynomials, Laplace operator.

1. Introduction In the introduction we shall make essential definitions, then state our main results. The rest of the paper proves them. 1.1. Definitions 1.1.1. Non-Commutative Polynomials. A non-commutative monomial m of degree d on the free variables (x1 , . . . , xg ) is a product xa1 xa2 · · · xad of these variables, corresponding to a unique sequence of ai of nonnegative integers, 1 ≤ ai ≤ g. We abbreviate this m = xw , where w is the d-tuple (a1 , . . . , ad ). The set of all All authors were partially supported by J.W. Helton’s grants from the NSF and the Ford Motor Co.. and J. A. Hernandez was supported by a McNair Fellowship.

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monomials in (x1 , . . . , xg ) is denoted as M and the set of indexes w is denoted W. Some notation is: |w| = d the length of w (w)i = ai the ith entry of w T w = (ad , . . . , a1 ) the transpose of w φ = () the empty word (word of length zero) The space of non-commutative polynomials p(x) = p(x1 , . . . , xg ) with real coefficients is denoted Rx and we express p as
p(x) = Am m. m∈M

An example of a non-commutative polynomial is p(x) = p(x1 , x2 ) = x21 x2 x1 + x1 x2 x21 + x1 x2 − x2 x1 + 7 (in commutative variables, this would be equivalent to 2x31 x2 + 7). T The transpose of a monomial m = xw is defined to be mT = xw . The  transpose of a polynomial p, denoted pT , is defined by p(x) = Am mT and m∈M

has the following properties: (1) (pT )T = p (2) (p1 + p2 )T = pT1 + pT2 (3) (αp)T = αpT (α ∈ R) (4) (p1 p2 )T = pT2 pT1 . In this paper, we shall consider primarily polynomials in symmetric variables. That is, we consider variables xi where xTi = xi . Then monomials satisfy (xa1 . . . xad−1 )T T = xad−1 . . . xa1 , which in other notation is (xw )T = xw . Symmetric (or selfadjoint) polynomials are those that are equal to their transposes. g 1.1.2. Evaluating Noncommutative Polynomials. Let (Rn×n sym ) denote the set of g-tuples (X1 , . . . , Xg ) of real symmetric n × n matrices. We shall be interested in evaluating a polynomial p(x) = p(x1 , . . . , xg ) that belongs to Rx at a tuple X = g (X1 , . . . , Xg ) ∈ (Rn×n sym ) . In this case p(X) is also an n×n matrix and the involution on Rx that was introduced earlier is compatible with matrix transposition, i.e.,

pT (X) = p(X)T , g where p(X)T denotes the transpose of the matrix p(X). When X ∈ (Rn×n sym ) is substituted into p the constant term p(0) of p(x) becomes p(0)In . Thus, for example, p(x) = 3 + x21 + 5x32 =⇒ p(X) = 3In + X12 + 5X23 .

A symmetric polynomial p ∈ Rx is matrix positive if p(X) is a positive g semidefinite matrix for each tuple X = (X1 , . . . , Xg ) ∈ (Rn×n sym ) . We emphasize that throughout this paper, unless otherwise noted, x1 , x2 , . . . , xn stand for variables and X1 , X2 , . . . , Xn stand for matrices (usually symmetric).

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1.1.3. Non-Commutative Differentiation. For our non-commutative purposes, we take directional derivatives in xi with regard to an indeterminate direction parameter h. d D[p(x1 , . . . , xg ), xi , h] := [p(x1 , . . . , (xi + th), . . . , xg )]|t=0 . (1) dt We say that this is the directional derivative of p(x) = p(x1 , . . . , xg ) in xi in the direction h. Note it is linear in h. For a detailed formal definition see [5], for more examples see [1]. Example 1.1. The directional derivative D[x21 x2 , x1 , h] d [(x1 + th)2 x2 ]|t=0 dt d = [x21 x2 + th x1 x2 + tx1 h x2 + t2 h2 x2 ]|t=0 dt = [h x1 x2 + x1 h x2 + 2th2 x2 ]|t=0 =

= h x1 x2 + x1 h x2 . As this example shows, the directional derivative of p on xi in the direction h is the sum of the terms produced by replacing one instance of xi with h. Lemma 1.1. The directional derivative of NC polynomials is linear, D[a p(x) + b q(x), xi , h] = a D[p(x), xi , h] + b D[q(x), xi , h] and respects transposes D[p(x)T , xi , h] = D[p(x), xi , h]T .


Proof. Straighforward.

1.1.4. Non-Commutative Laplacian and Subharmonicity. The Laplacian of a NC polynomial p(x) is defined as: Lap[p, h] := =

g


D[D[p(x), xi , h], xi , h]

i=1 g 2


d [p(x1 , . . . , (xi + th), . . . , xg )]|t=0 . 2 dt i=1

(2) (3)

Our notation is slightly inconsistent (but has advantages) in that the single letter x stands for g variables x1 , . . . , xg while h is a single variable. Note that Lap is linear in h. An NC polynomial is called harmonic if its Laplacian is zero, and subharmonic if its Laplacian is matrix-positive and purely subharmonic is used to describe a polynomial which is subharmonic but not harmonic - that is, having a nonzero, matrix-positive Laplacian.     Specialization of Lap[p, h], to commutative variables, is h2 ∆ p where ∆ p g    ∂xi xi p(x). Here p : Rn → R. is the standard Laplacian, namely, ∆ p := i=1

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1.2. Classification of Harmonics and Subharmonics in Two Variables For our special homogeneous polynomials on two variables, define γ := x1 + i x2

(4)

where i is the imaginary number. Theorem 1. The homogeneous noncommutative polynomials in two symmetric variables which are (1a) harmonic of degree d > 2 are exactly the linear combinations of Re(γ d )

and

Im(γ d ),

(1b) harmonic of degree d = 2 are exactly the linear combinations of Re(γ d )

and

Im(γ d )

and

x1 x2 ,

(note this includes x2 x1 ), (2a) subharmonic of degree 2d with d > 2 , are exactly the linear combinations: c0 [Re(γ d )]2 + c1 Re(γ 2d ) + c2 Im(γ 2d ) = c0 [Im(γ d )]2 + (c0 + c1 ) Re(γ 2d ) + c2 Im(γ 2d )

(5)

where c0 ≥ 0, (2b) symmetric subharmonics of degree 4, are exactly the linear combinations: f

= B1 (x41 − x21 x22 − x22 x21 + x42 ) + B2 (x31 x2 + x2 x31 − x2 x1 x22 − x22 x1 x2 ) (6) +B3 (x21 x2 x1 + x1 x2 x21 − x1 x32 − x32 x1 ) + B4 (x1 x2 x1 x2 + x2 x1 x2 x1 )

+B5 x1 x22 x1 + B6 x2 x21 x2

with coefficients satisfying the inequalities: (B1 + B6 )(B1 + B5 ) > (B3 − B2 )2 + (B1 + B4 )2 and

(7)

B1 + B6 > 0

(8)

(or, equivalently

B1 + B5 > 0).

(2c) All subharmonics of degree 2 are, A1 x21 + A2 x22 + A3 x1 x2 + A4 x2 x1 with A1 + A2 ≥ 0. (3) Pure subharmonics of odd degree do not exist. Note: all of these functions except for x1 x2 and x2 x1 in (1b) and in (2c) are symmetric. Proof. Most of the remainder of this paper is focused on proving this theorem. The proofs for the parts of the theorem are as follows:

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Part of theorem Section of the proof (1a.) 4.2.1 (1b.) 4.2.2 (2a.) 2.4 (2b.) 3.3 also Remark 2 (2c.) 4.2.2 (3.) 2.4
Remark 1. The following degree 3 polynomial p is unusual in that there is a region of X1 , X2 where Lap(p) is positive, but Lap(p) is not positive everywhere: A1 (x31 − x1 x22 − x2 x21 ) + A2 x2 x1 x2 + A3 x1 x2 x1 + A4 (x32 − x21 x2 − x2 x21 ) For this the region of subharmonicity is (A1 + A2 )x1 + (A3 + A4 )x2 > 0 and the region of harmonicity is A1 + A2 = A3 + A4 = 0. Of course, there is no homogeneous polynomial of degree three which is subharmonic over all values of x1 and x2 . 1.3. Subharmonics are All Built from Harmonics Our second main result is a general fact which holds in any number of variables: Theorem 2. Assume the harmonic polynomials homogeneous of degree d2 have a basis γ1 , . . . , γk with the independence property: there is a monomial wj in γj which does not occur in the other γ1 , . . . , γk . If p is a homogeneous symmetric subharmonic polynomial of degree even d, then p has the form p=

f
inite

ci RiT Ri

i

for some homogeneous harmonic functions Rj of degree

d 2

and real numbers cj .

Because of this, knowing all homogeneous subharmonics will likely occur once the harmonics are classified. Proof. The proof is found in §4.1.1.




1.4. Comparison with Commutative Subharmonic Polynomials The study of harmonic and subharmonic polynomials in commuting variables is classical. Harmonic commuting polynomials are classified in any number of variables and the have a close correspondence to spherical harmonics. A good reference on this is [7] §2.4, pp. 110–113. For two commuting variables, the homogeneous harmonic polynomials are those of the form, Re(x1 + Ix2 )n

and Im(x1 + Ix2 )n ,

so the commuting and noncommuting case are exactly parallel.

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1.5. Related Topics and Motivation Non-Commutative Convexity The non-commutative Hessian is defined as: d2 [p(x1 + tη1 , . . . , xg + tg ηg )]|t=0 . dt2 Note that this is composed of several independent direction parameters, ηi and that if p is a polynomial, then its Hessian is a polynomial in x and η which is homogeneous of degree 2 in η. A non-commutative polynomial is considered convex wherever its Hessian is matrix-positive. A polynomial p(x) = p(x1 , . . . , xd ) is geometrically convex if and only if, for g every X, Y ∈ (Rn×n sym ) ,    X +Y 1 p(X) + p(Y ) − p 2 2 NCHes[p(x1 , . . . , xg ), {x1 , η1 }, . . . , {xg , ηg }] :=

is positive-semidefinite. It is proved in [3] that convexity is equivalent to geometric convexity. A crucial fact regarding these polynomials (see [4]) is that they are all of degree two or less. Some excellent papers on noncommutative convexity are [8] [2]. The commutative analog of this “directional” Hessian is the quadratic function     η1 η1   .   .  . (9) H p  .  ·  ..    where H p is the Hessian matrix:  ∂x1 x1 p(x)  ..  .

ηg

··· .. . ∂xg x1 p(x) · · ·

ηg  ∂x1 xg p(x)  .. . .

(10)

∂xg xg p(x)

If this Hessian is positive semidefinite at all (x1 , . . . , xg ), then f is said to be convex. Non-Commutative Algebra in Engineering. Inequalities, involving polynomials in matrices and their inverses, and associated optimization problems have become very important in engineering. When such polynomials are matrix convex, local minima are global. This is extremely important in applications. Also, interior point numerical methods apply well to these. In the last few years, the approaches that have been proposed in the field of optimization and control theory based on linear matrix inequalities and semidefinite programming have become very important and promising, since the same framework can be used for a large set of problems. Matrix inequalities provide a nice setup for many engineering and related problems, and if they are convex the optimization problem is well behaved and interior point methods provide efficient algorithms which are effective on moderate sized

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problems. Unfortunately, the class of matrix convex noncommutative polynomials is very small; as already mentioned they are all of degree two or less [4]. Our original interest in subharmonic polynomials was to analyze conditions similar to convexity, though not as restrictive, in the hopes of finding much broader classes which still had nice properties. What we found (as reported here) was that subharmonic polynomials are (in two) variables a highly restricted class. Noncommutative Analysis. This article would come under the general heading of “free analysis”, since the setting is a noncommutative algebra whose generators are “free” of relations. This is a burdgeoning area, of which free probability is currently the largest component. The interested reader is referred to the web site [10] of American Institute of Mathematics, in particular it gives the findings of the AIM workshop in 2006 on free analysis. A fairly expository article describing noncommutative convexity, noncommutative semialgebraic geometry and relations to engineering is [6].

2. Existence Proofs Now we set about to prove Theorem 1. In this section, we show that the polynomials claimed to be harmonic and subharmonic are indeed. In section 4, we show that these are the only posibilities. 2.1. Product Rules for Derivatives To begin with, we will build up facts about derivatives. 2.1.1. Product Rule for First Derivatives. Lemma 2.1. The product rule for the directional derivative of NC polynomials is D[p1 p2 , xi , h] = D[p1 , xi , h] p2 + p1 D[p2 , xi , h].

(11)

Proof. The directional derivative D[m, xi , h] of a product m = m1 m2 of noncommutative monomials m1 and m2 is the sum of terms produced by replacing one instance of xi in m by h. This sum can be divided into two parts: µ1 , the sum of terms whose h lie in the first |m1 | letters, i.e. D[m1 , xi , h]m2 µ2 , the sum of terms whose h lie in the last |m2 | letters, i.e. m1 D[m2 , xi , h]. Therefore D[m1 m2 , xi , h] = D[m1 , xi , h]m2 + m1 D[m2 , xi , h]. We can extend this product rule to the product of any two non-commutative polynomials p1 and p2 as follows.

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D[p1 p2 , xi , h] = D


=




Am1 m1




m1 ∈Wp1




IEOT

  Am2 m2 , xi , h

m2 ∈Wp2

Am1 Am2 D[m1 m2 , xi , h]

m1 ∈Wp1 m2 ∈Wp2




=




Am1 Am2 D[m1 , xi , h] m2 + Am1 Am2 m1 D[m2 xi , h]

m1 ∈Wp1 m2 ∈Wp2

=D




Am1 m1 , xi , h

m1 ∈Wp1




Am2 m2 +

m2 ∈Wp2




Am1 m1 D

m1 ∈Wp1




 Am2 m2 , xi , h

m2 ∈Wp2

= D[p1 , xi , h] p2 + p1 D[p2 , xi , h].

(12)


2.1.2. The Laplacian of a Product. We now prove Theorem 1 part (2b). Lemma 2.2. The product rule for the Laplacian of NC polynomials is Lap[p1 p2 , h] = Lap[p1 , h] p2 + p1 Lap[p2 , h] + 2

g
  D[p1 , xi , h] D[p2 , xi , h] . i=1

As a consequence if p is harmonic, then Lap[pT p, h] = 2

g
  D[p, xi , h]T D[p, xi , h] .

(13)

i=1

Proof. Lap[p1 p2 , h] = =

g
i=1 g


D[D[p1 p2 , xi , h], xi , h]   D[p1 D[p2 , xi , h] + D[p1 , xi , h] p2 , xi , h]

i=1 g
 p1 D[D[p2 , xi , h], xi , h] + D[D[p1 , xi , h], xi , h] p2 = i=1

 + 2D[p1 , xi , h], D[p2 , xi , h]

= Lap[p1 , h] p2 + p1 Lap[p2 , h] + 2

g
  D[p1 , xi , h] D[p2 , xi , h] . i=1


2.2. Formulas Involving γ d and its Derivatives Recall from (4) that γ := x1 + ix2 . Note that γ = γ T and therefore that γ d = (γ d )T . So T  T  Im(γ d ) = Im((γ d )T ) = Im(γ d ). Re(γ d ) = Re((γ d )T ) = Re(γ d ) and

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This proves the (last) assertion in Theorem 1 that all but a few subharmonics on our list are symmetric. Lemma 2.3. The derivatives of of γ d exhibit the following symmetries. D[Re(γ d ), x1 , h] = D[Im(γ d ), x2 , h] and D[Re(γ d ), x2 , h] = −D[Im(γ d ), x1 , h]. Proof. The proof proceeds by induction. To begin, it is easily seen that D[Re(γ), x1 , h] = D[Im(γ), x2 , h] = h D[Im(γ), x1 , h] = −D[Re(γ), x2 , h] = 0. Assume that D[Re(γ d−1 ), x1 , h] = D[Im(γ d−1 ), x2 , h] D[Im(γ d−1 ), x1 , 1] = −D[Re(γ d−1 ), x2 , h]. Then D[ Re(γ d ), x1 , h] = D[x1 Re(γ d−1 ) − x2 Im(γ d−1 ), x1 , h] = x1 D[Re(γ d−1 ), x1 , h] + h Re(γ d−1 ) − x2 D[Im(γ d−1 ), x1 , h] D[Im(γ d ), x2 , h] = D[x1 Im(γ d−1 ) + x2 Re(γ d−1 ), x2 , h] = x1 D[Im(γ d−1 ), x2 , h] + x2 D[Re(γ d−1 ), x2 , h] + h Re(γ d−1 ), so D[Re(γ d ), x1 , h] = D[Im(γ d ), x2 , h] which satisfies the first half of our inductive hypothesis. For the next half compute D[ Re(γ d ), x2 , h] = D[x1 Re(γ d−1 ) − x2 Im(γ d−1 ), x2 , h] = x1 D[Re(γ d−1 ), x2 , h] − x2 D[Im(γ d−1 ), x2 , h] − h Im(γ d−1 ) D[Im(γ d ), x1 , h] = D[x1 Im(γ d−1 ) + x2 Re(γ d−1 ), x1 , h] = x1 D[Im(γ d−1 ), x1 , h] + h Im(γ d−1 ) + x2 D[Re(γ d−1 ), x2 , h], so D[Re(γ d ), x2 , h] = −D[Im(γ d ), x1 , h].




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2.3. Harmonics degree > 2: Proof of Theorem 1 part (1) Proof. Our proof will proceed by induction. Since the Laplacian of words of length 1 is zero, Lap[Re(γ), h] = Lap[x1 , h] = 0 and Lap[Im(γ), h] = Lap[x2 , h] = 0.

(14)

Now, assume that Lap[Re(γ d−1 ), h] = Lap[Im(γ d−1 ), h] = 0.

(15)

Re(γ d ) = Re((x1 + ix2 ) γ d−1 ) = x1 Re(γ d−1 ) − x2 Im(γ d−1 )

(16)

Im(γ d ) = Im((x1 + ix2 ) γ d−1 ) = x1 Im(γ d−1 ) + x2 Re(γ d−1 ).

(17)

Pushing ahead,

Applying our product rule to (16): Lap[Re(γ d ), h] = Lap[x1 Re(γ d−1 ), h] − Lap[x2 Im(γ d−1 ), h] = x1 Lap[Re(γ d−1 ), h] + Lap[x1 , h] Re(γ d−1 ) + 2D[x1 , x1 , h] D[Re(γ d−1 ), x1 , h] + 2D[x1 , x2 , h] D[Re(γ d−1 ), x2 , h] − x2 Lap[Im(γ d−1 ), h] − Lap[x2 , h] Im(γ d−1 ) − 2D[x2 , x1 , h] D[Im(γ d−1 ), x1 , h] − 2D[x2 , x2 , h] D[Im(γ d−1 ), x2 , h]. Use (14) and (15) to obtain that the Lap[] terms are 0, and that “cross partials are 0” to get Lap[Re(γ d ), h] = h D[Re(γ d−1 ), x1 , h] − h D[Im(γ d−1 ), x2 , h]. By symmetry Lemma 2.3, this means Re(Lap[γ d , h]) = Lap[Re(γ d ), h] = 0. By a similar argument, applying the product rule to (17), Im(Lap[γ d , h]) = Lap[Im(γ d ), h] = 0.




2.4. Subharmonics degree > 4: Proof of Theorem 1 (2a.) Proof. The product rule for the Laplacian of harmonics in Lemma 2.2 says  2  2 Lap[ Re(γ d ) , h] is a sum of squares. Thus we have shown Re(γ d ) is subharmonic. Now we prove the formula (5) relating subharmonics. We use γ 2d = (Re(γ d ) + i Im(γ d ))2 = (Re(γ d ))2 − (Im(γ d ))2 + i(Re(γ d )Im(γ d ) + Im(γ d ) Re(γ d )). Therefore

 2  2 Re(γ 2d ) = Re(γ d ) − Im(γ d ) .

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 2 c0 Re(γ d ) + c1 Re(γ 2d ) + c2 Im(γ 2d )  2 = c0 Im(γ d ) + (c0 + c1 ) (Re(γ 2d )) + c2 Im(γ 2d ),


which is (5).

Up to this point we have handled subharmonics of even degree. To see that there are no pure subharmonics of odd degree, note that the Laplacian L(x) of an odd degree polynomial is itself an odd degree polynomial which is matrix-positive. Consider L(tx) as t ∈ R approaches ±∞. Since the highest order terms dominate, the signs of these limits are opposite. Thus the highest order terms are 0.

3. Classification when Degree is Four or Less We handle now what appear to be special cases which are exceptions to the general degree > 4 theorem. 3.1. The Matrix Representation Important in our proofs for polynomials of low degree is a representation for polynomials q(x1 , . . . , xg )[h] which are homogeneous of degree 2 in h. Recall that often x stands for (x1 , . . . , xg ) and h is a single variable. In our notation q(x)[h] we use [ ] to distinguish the variable which is of degree 2. Any NC symmetric polynomial q in symmetric variables quadratic in h can be written n
n
n n

q(x)[h] = (h mi )T Zij (x) (h mj ) = (mi T h) Zij (h mj ) i=1 j=1

i=1 j=1

where mi are monomials in x and Zij (x) are polynomials in x. Define Z(x) as the N -by-N matrix of polynomials in x whose i, j th element is Zij , and define V (x)[h] as V (x)[h]T = h(m1 , m2 , . . . , mN ). We call Z the middle matrix for q and V its border vector. In this notation our representation is q(x, h) = V (x)[h]T Z(x)V (x)[h] We can and typically do take Z(x) to be symmetric. If the monomials mi in V (x)[h] do not repeat, then Z(x) is uniquely determined and is symmetric. Example 3.1. A “middle matrix” representation g = 2 3 x1 hx22 hx1 + hx1 x2 x1 h − hx1 hx22 − x22 hx1 h + 5 x1 x2 hx2 hx2 x1 T     0 0 −x1 x1 x2 x1 h h   hx1   0  3 x22 0 0      hx1  . = hx2 x1   0 0  hx2 x1  0 5 x2 −x1 hx22 0 0 0 hx22

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3.1.1. Positivity of q vs. Positivity of its Middle Matrix. A key fact is that positivity of q is equivalent to positivity of its middle matrix in the following sense. Lemma 3.1. Suppose q(x)[h] a symmetric NC polynomial in noncommuting varig ables pure quadratic in h and Z(x) is its middle matrix. If X ∈ (Rn×n sym ) and n×n g Z(X) 0, then q(X)[H] 0 for all H ∈ (Rsym ) . Conversely, if q(x)[h] is matrix positive; i.e., q(X)[H] 0, for every n and g X, H ∈ (Rn×n sym ) in the (non empty) positivity domain {X : f (X) 0} of some g polynomial f , then for each n and X ∈ (Rn×n sym ) , we have Z(X) 0 on {X : f (X) 0}. Proof. The first statement is evident. The converse is proved in [1] in Lemma 9.5 and Theorem 10.10 in [1]. for a cleaner proofs see [5] in particular Proposition 6.1.
3.2. The Zeroes Lemma The following is useful in our analysis of subharmonics. Lemma 3.2. Let S be any N × N symmetric matrix with entries in Rx. If there exists some diagonal entry Sii = 0 and corresponding off-diagonal entries T Sij = Sji = 0, then S is not matrix-positive semidefinite. Proof. Let ei and ej be standard basis vectors for RN (i.e. eTi A ej = aij ) and define v := β1 ei + β2 ej where β1 , β2 ∈ R. Then, v T S(x)v = ((Sij + Sji ) β1 + Sjj β2 ) β2 = (2 Sij β1 + Sjj β2 ) β2 Given β2 > 0, we can choose β1 such that 2 β1 Sij (X)β2 + β2 Sjj (X) β2 is neither a positive nor negative matrix.




This lemma is useful when applied to our matrix representation of the Laplacian of a symmetric NC polynomial . 3.3. The Laplacian of a Degree 4 Polynomial Proof. We begin with a parameterization the set of degree 4 homogeneous polynomials in symmetric free variables p = A1 x41 x21 + A2 (x31 x2 + x2 x31 ) + A3 (x21 x2 x1 + x1 x2 x21 ) + A4 (x21 x22 + x22 x21 ) + A5 (x1 x2 x1 x2 + x2 x1 x2 x1 ) + A6 x1 x22 x1 + A7 (x1 x32 + x32 x1 ) + A8 x2 x21 x2 + A9 (x2 x1 x22 + x22 x1 x2 ) + A10 x42 .

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We calculate the Laplacian of p: 2A1 (h2 x21 + h x1 h x1 + h x21 h + x1 h2 x1 + x1 h x1 h + x21 h2 )

+2A2 (h2 x1 x2 + h x1 h x2 + x1 h2 x2 + x2 h2 x1 + x2 h x1 h + x2 x1 h2 ) + 2 A3 (h2 x2 x1 + h x1 x2 h + h x2 h x1 + h x2 x1 h + x1 h x2 h + x1 x2 h2 )

+ 2 A4 (h2 x21 + h2 x22 + x21 h2 + x22 h2 ) + 2 A5 (h x1 h x1 + h x2 h x2 + x1 h x1 h + x2 h x2 h) + 2 A6 (h x22 h + x1 h2 x1 )

+ 2 A7 (h2 x2 x1 + h x2 h x1 + x1 h2 x2 + x1 h x2 h + x1 x2 h2 + x2 h2 x1 ) + 2 A8 (h x21 h + x2 h2 x2 )

+ 2 A9 (h2 x1 x2 + h x1 h x2 + h x1 x2 h + h x2 x1 h + x2 h x1 h + x2 x1 h2 ) + 2 A10 (h2 x22 + h x2 h x2 + h x22 h + x2 h2 x2 + x2 h x2 h + x22 h2 ). The directional Laplacian is quadratic in h, and so can be represented by border vector  T V (x)[h]T = h x1 h x2 h x21 h x1 x2 h x2 x1 h x22 h and middle matrix Z(x) which is  2 2 (A1+A8 )x1 +(A6+A10 )x2 +(A3+A9 )(x1 x2 +x2 x1 )

                     

(A1 +A5 )x1 +(A3 +A7 )x2

(A2 +A9 )x1 +(A5 +A10 )x2

(A1 + A5 ) x1 + (A3 + A7 ) x2

A1 + A6

A2 + A7

(A2 + A9 ) x1 + (A5 + A10 ) x2

A2 + A7

A1 + A4

A1+A4

A3+A7

A2+A9

A4+A10

0

0

0

0

A8 + A10

0

0

0

0

0

0

0

0

0

0

A3 + A7

0

0

0

0

0

0

A2 + A9

0

0

0

0

0

0

A4 + A10

0

0

0

0

0

0

            .           

g Assume that Z(X) is a positive semidefinite matrix for X ∈ (Rn×n sym ) . By the Zeroes Lemma, the zeroes on the last four diagonals force all entries in the last four rows or columns to be zero, that is:

A4 = −A1 ,

A10 = A1 ,

A9 = −A2 ,

A7 = −A3 .

(18)

Applying these conditions to the matrix above, and ignoring the rows and columns which are zero, we have:   2 2 (A1 + A8 ) x1 + (A1 + A6 ) x2 x2 )

+ (A2 − A3 )(x2 x1 − x1  (A1 +A5 )x1 (A1 +A5 )x2

(A1 +A5 )x1

(A1 +A5 )x2

A1 +A6

A2 −A3

A2 −A3

A1 +A8

 .

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This matrix can be simplified by substitution of reoccurring pairs by single letters: G = A1 + A6 ,

J = A2 − A3 ,

H = A1 + A8



to obtain

Hx21 − J(x1 x2 + x2 x1 ) + Gx22  Kx1 Kx2

K = A1 + A5 .  Kx2 J . H

Kx1 G J

We now find its noncommutative LDLT (Cholesky) decomposition to have D term equal to   G 0 0 2  0 H− J 0 G .  0

0

Hx21 +J(x1 x2 + x2 x1 )+Gx22 −

K 2 x21 G

−

(

JKx1 G

+Kx2 ) (

JKx1 G 2

+Kx2 )

H− JG

A reference is [1] which describes the NCAlgebra command, NCLDUDecomposition, we used to do this. We see there are three inequalities, which must be satisfied for Z(X) to be positive semidefinite. J2 G > 0, H − > 0, G 1 1 + KX2 ) ( JKX + KX2 ) K 2 X12 ( JKX G G − HX12 +J(X1 X2 +X2 X1 )+GX22 − > 0. 2 J G H− G The last condition is purely quadratic in X1 and X2 , and therefore has a middle matrix representation which we compute to be   2 2 2 2 G − H HK2 2 − KJ H + HK H2 (G− J )J (G− J )J   . 2 2 H + HK J − KH 2 H2 (G−

J

)J

G−

J

T

Again we perform the LDL decomposition:   2 0 G − K J2 H− JG 2    2 .  J− JK J 2 )G 2 2 2   (H− G 0 H − J JK2 2 − KG − 2 K (H−

G

)G

G−

2 H− J G

Although the inequality

 2 2 J − JKJ 2 J 2K 2 K2 (H− G )G − H− − >0 2 2 G (H − JG )G2 G − K J2 H−

(19)

G

is quite complicated, we can simplify it some by multiplying it by expressions which are known to be positive, such as: G,

H−

J2 , G

and G −

K2 2 H − JG

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which we encountered earlier. This gives a polynomial inequality equivalent to (19), which, after some simplification, gives us: (GH − J 2 − K 2 )2 > 0. Which, considering only the case of all real coefficients, is rather vacuous, informing us only that GH − J 2 − K 2 = 0. Bringing all our inequalities together (simplifying each as we did above), we obtain (I) G > 0,

(II) GH > J 2 ,

(III) GH > J 2 + K 2 ,

(IV ) GH = H 2 + K 2 .

Notice (III) implies (II) and (IV), thus reducing to (I) and (II). Therefore, we conclude that the set of polynomials making the Laplacian matrix “positive” is exactly those of the form: f

=

A1 (x41 − x21 x22 − x22 x21 + x42 ) + A2 (x31 x2 + x2 x31 − x2 x1 x22 − x22 x1 x2 )(20) +A3 (x21 x2 x1 + x1 x2 x21 − x1 x32 − x32 x1 ) + A5 (x1 x2 x1 x2 + x2 x1 x2 x1 )

+A6 x1 x22 x1 + A8 x2 x21 x2

with coefficients satisfying the inequalities: (III) =⇒ (A1 + A8 )(A1 + A6 ) > (A3 − A2 )2 + (A1 + A5 )2 and

(21)

(I) =⇒

(22)

A1 + A8 > 0

(or, equivalently A1 + A6 > 0).

For neatness, and to more clearly see the dimension of the space of subharmonics,we set B1 = A1 , B2 = A2 , B3 = A3 , B4 = A5 , B5 = A6 , B6 = A8 .


4. Uniqueness Proofs Now we set about to prove that the list of subharmonic and harmonic polynomials in Theorem 1 is complete. We do this, as is required, only for two variables but in the course of our proof we discover some promising recursions valid in any number of variables. 4.1. Even Degree Homogeneous p Given 0 < m < d a noncommutative polynomial p of degree d decompose it as
p= xt pt (x) + Λ (23) |t|=m

where deg Λ < m. Call the polynomial pt (x) the right neighbor of xt Lemma 4.1. If p is harmonic in any number of variables consider the right neighbor representation of p for any m; the right neighbor pt of each monomial xt of degree m is harmonic, that is, Lap(pt ) = 0. If p is subharmonic in any number of variables, if p is homogeneous of degree d then the right neighbor pt of each monomial xt of degree d2 is harmonic, that is, Lap(pt ) = 0.

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Proof. Apply the Laplacian to the right neighbor decomposition (23) of p and get from the product rule for the Laplacian (Lemma 2.2): Lap[p, h] =




xt Lap[pt (x), h]

(24)

|t|=m

+




Lap[xt , h]pt (x)

(25)

D(xt )[h]D(pt )[h]

(26)

|t|=m

+2




|t|=m

+Lap[Λ, h]. Suppose Lap[p, h] = 0. We shall now show that polynomial (24) is 0, (25) is 0, and (26) is 0. All terms of the polynomials (24), (25), and (26) have degree at least m, while deg Λ < m. Since the Laplacian of a polynomial respects degree, we have Lap(Λ) = 0. Next factor a given degree ≥ m monomial r into its m−front and back: namely, r = rf rb where rf has degree m. Consider the polynomial (24): the m−back of each monomial in it contains two h’s. Likewise the m−back of each monomial in (25) and (26) contains no h’s and one h respectively. Thus polynomials (24), (25), and (26) contain no monomials which cancel and since their sum is zero they must be zero. From (24) is 0 we immediately get Lap(pt ) = 0. This proves the first part of the lemma. Now to the subharmonic part. That Lap[p, h] is matrix positive implies that it is a sum of squares:
LTj (x)[h]Lj (x)[h] (27) Lap(p) = j

First observe that each Lj is linear in h. This is true since the highest degree in h monomial λ(x)[h] of Lj (x)[h] contributes a λ(x)[h]T λ(x)[h] to LTj (x)[h]Lj (x)[h] monomial which holds because its coefficient is positive and can not be cancelled out; likewise λ(x)[h]T λ(x)[h] appears in Lap(p). Thus the monomial λ(x)[h] has degree one in h. Because of equation (27) we can refer to each term of Lap(p) as having a first half and second half; each half has degree d2 . Also every term of Lap(p) has an h in its first half and also in its second half. However, if m = d2 all terms in (24) have two h’s in their second half and none in their first half. This contradicts the previous sentence; thus equation (24) is 0. Since we have factored out xt in the representation (24), their coefficients Lap(pt ) are 0 for all |t| = d2 .
4.1.1. Homogeneous Subharmonics are Sums of Products of Harmonics. In this section we prove under weak hypotheses that homogeneous subharmonics are sums of products of harmonics. A subharmonic polynomial of odd degree is harmonic, so is the product of itself and 1. Thus we restrict to even degree and prove the following.

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Proposition 4.1. Assume the harmonic polynomials homogeneous of degree d2 are the span of γ1 , . . . , γk . Assume there is a monomial wj in γj which does not occur in the other γ1 , . . . , γk . If p is subharmonic homogeneous of even degree d, then it has the form k


p=

φij γi γj

(28)

i,j=1

where each φij is a real number. Note further that for symmetric p we may take φij = φji . Let S denote the span of these symmetric subharmonics. . For example, This implies that S is a space of at most dimension k(k+1) 2 in two variables there are two independent homogeneous harmonic polynomials of degree other than 2, so dim S is at most 3 for all d = 4. For d = 4 we have dim S ≤ 6. Proof. Assume p is subharmonic homogeneous of degree d. Write down its right neighbor representation with m = d2 and use Lemma 4.1 to get Lap(pt ) = 0 for |t| = d2 . Thus pt =

k


µj (t)γj

j

for some numbers µj (t). Plug this into the decomposition


p= xt pt (x) = µj (t)xt γj |t|= d 2

to get p=


j

|t|= d 2

 




|t|= d 2

(29)

j

 µj (t)xt  γj =

k


pj (x)γj .

j

Now make a left neighbor decomposition of p which by the definition of the monomial w1 has the form p1 (x)w1 + G where all terms of G are without w1 on the right. The left neighbor version of Lemma 4.1 implies Lap(p1 (x)) = 0. Likewise each pj (x) is harmonic of degree d2 . This proves representation (28) for p.
Next we prove our representation of subharmonics stated in the introduction as Theorem 2. Proof. (Theorem 2) Now suppose p is symmetric. Proposition 4.1 says we can represent p as in equation (28). Note if u is harmonic then uT is harmonic and relabel and possibly expand (by taking transposes) the set γ1 , . . . , γk as s1 , . . . , sα , u1 , . . . , uβ , uT1 , . . . , uTβ

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α+2β where the si are symmetric polynomials. Set Ψ := {φ˜ij }i,j=1 where φ˜ij = φij for     u1 s1  ..   ..  i, j corresponding to an original γ and 0 otherwise. Now let s =  . , u =  . 





 T   s s  .  and v =  .. . Then p = u Ψ u and v v uTβ  T   s s p + pT = u Φ u p= 2 v v uT1

sα

uβ

Ψ + ΨT , a symmetric matrix as required. 2 Decompose the symmetric matrix Φ as Φ = N JN T where J is a diagonal matrix with ±1  or 0on the diagonal and N has real numbers as entries. Now, let s us put R = N T u. Then v  T    T   s s s s
p = u Φ u = u N JN T u = RT JR = ci RiT Ri . i v v v v where Φ =

where cj is ±1 or 0. The si , ui , uTi are harmonic, so their linear combinations Ri are harmonic.
An appealing, easily proved, formula is Lap(p) = 2

g


D[R, xj , h]T JD[R, xj , h].

j α+2β Clearly, if the matrix Φ := {φij }i,j=1 is positive semidefinite (or equivalently J has nonnegative entries), Lap(p) will be positive, so then p is subharmonic. Also we get even degree harmonics are sums and differences of squares of harmonics. It is not clear which differences make p harmonic or subharmonic. However, we conjecture A homogeneous symmetric subharmonic polynomial p of even degree d is a finite sum f
inite f
inite RiT Ri + H p= i



for some homogeneous harmonic functions Ri , H with Ri of degree degree d.

d 2

and H of

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At this point, we have finished discussing subharmonics, and will now turn our full attention to harmonic polynomials. 4.2. Uniqueness of Harmonics in Two Variables 4.2.1. Polynomials of degree Three and Larger. At this point, we have proved that there are harmonic polynomials of arbitrary degree. Working in two variables, we will now show that the polynomials Re γ d and Im γ d span all of the harmonics. This can be a helpful result, which, as yet, we have been unable to show for any higher number of variables. In fact, McAllaster has found, experimentally, that in three variables, the size of the basis of harmonic polynomials increases on the order of d2 (See [9]). Proposition 4.2. Let γ = x1 +ix2 , and Bd = {Re γ d , Im γ d }. Then Bd forms a basis for all harmonic polynomials which are homogeneous of degree d for any d > 2. To prove the proposition we need two lemmas. Lemma 4.2. In degree three, there are two linearly independent homogeneous harmonic polynomials whose span is all harmonic polynomials which are homogeneous of degree three. Lemma 4.3. Let β(x1 , x2 ) be harmonic and homogeneous of degree d. Then we may uniquely represent β as β(x1 , x2 ) = x1 f (x1 , x2 ) + x2 g(x1 , x2 ), where f and g are harmonic and homogeneous of degree d − 1 and D[f (x1 , x2 ), x1 , h] = −D[g(x1 , x2 ), x2 , h]. Proof. (Lemma 4.2) Every homogeneous polynomial of degree three has the form a1 x31 + a2 x21 x2 + a3 x1 x2 x1 + a4 x1 x22 + a5 x2 x21 + a6 x2 x1 x2 + a7 x22 x1 + a8 x32 and the Laplacian of this is a1 h2 x1 + a7 h2 x1 + a2 h2 x2 + a8 h2 x2 + a1 x1 h2 + a4 x1 h2 + a5 x2 h2 + a8 x2 h2 + a1 hx1 h + a6 hx1 h + a3 hx2 h + a8 hx2 h = (a1 + a7 )h2 x1 + (a2 + a8 )h2 x2 + (a1 + a4 )x1 h2 + (a5 + a8 )x2 h2 + (a1 + a6 )hx1 h + (a3 + a8 )hx2 h, so if we want the polynomial to be harmonic, we need each monomial of the Laplacian to be zero, so we need the equations a1 + a7 = 0, a2 + a8 = 0, a1 + a4 = 0, a5 + a8 = 0, a1 + a6 = 0, a3 + a8 = 0 to hold. This amounts to having the vector (a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 ) in the nullspace of the matrix   1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1   1 0 0 1 0 0 0 0   0 0 0 0 1 0 0 1 ,   1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

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which has the basis {(0, −1, −1, 0, −1, 0, 0, 1), (−1, 0, 0, 1, 0, 1, 1, 0)}, corresponding to polynomials x32 − x21 x2 − x2 x21 − x1 x2 x1 and −x31 + x1 x22 + x22 x1 + x2 x1 x2 . Hence there are exactly two linearly independent harmonic polynomials which are homogeneous of degree three.
Proof. (Lemma 4.3) Now, suppose that we are given a polynomial function β(x1 , x2 ) which is harmonic, homogeneous, and of degree d. Then, every monomial of β begins with either x1 or x2 , so we may uniquely represent β by the neighbor decomposition β(x1 , x2 ) = x1 f (x1 , x2 )+x2 g(x1 , x2 ). Now, by Lemma 4.1, we know that f and g are harmonic, and using the product rule for the Laplacian, we find: Lap[β, h] =Lap[x1 f (x1 , x2 ) + x2 g(x1 , x2 ), h] =(Lap[x1 , h]f (x1 , x2 ) + x1 Lap[f (x1 , x2 ), h] + 2(D[x1 , x1 , h]D[f (x1 , x2 ), x1 , h] + D[x1 , x2 , h]D[f (x1 , x2 ), x2 , h])) + (Lap[x2 , h]g(x1 , x2 ) + x2 Lap[g(x1 , x2 ), h] + 2(D[x2 , x1 , h]D[g(x1 , x2 ), x1 , h] + D[x2 , x2 , h]D[g(x1 , x2 ), x2 , h])) =2h(D[f (x1 , x2 ), x1 , h] + D[g(x1 , x2 ), x2 , h]) and since β is harmonic, this is zero, which gives: 0 = 2h(D[f (x1 , x2 ), x1 , h] + D[g(x1 , x2 ), x2 , h]). or more specifically, D[f (x1 , x2 ), x1 , h] = −D[g(x1 , x2 ), x2 , h].

(30)


Proof. (Prop. 4.2) First of all, we show that Re γ d and Im γ d are linearly independent: Suppose they are linearly dependent. Then a Re γ d = b Im γ d , where a = 0, b = 0, a, b ∈ R. But then, γ d = (x1 +ix2 )(R+iI), where R = Re γ d−1 , I = Im γ d−1 , so that a(x1 R − x2 I) = b(x1 I + x2 R). Now, we equate the terms starting with x1 and x2 , respectively, to get that aR = bI and −aI = bR. Then, we get that R = (b/a)I and I = (−b/a)R from the first and second equations, repectively. Puting this together, we get R = (b/a)I = (b/a)(−b/a)R, so that cancelling R, we get b2 /a2 = −1, or b2 = −a2 , which can happen only if a = b = 0, a contradiction. Now, we are going to prove the proposition by induction. First of all, Lemma 4.2 begins the induction. To prove the rest of the proposition, suppose that for degree d − 1 ≥ 3, we know that there are exactly two linearly independent polynomials which are harmonic and homogeneous. Then for degree d, we suppose that β is harmonic and homogeneous. Then β(x1 , x2 ) = x1 ϕ(x1 , x2 ) + x2 ψ(x1 , x2 ), where ϕ and ψ are both harmonic, and homogeneous of degree d − 1 (Lemma 4.3). Then by the induction hypothesis β(x1 , x2 ) = x1 (aϕ Re γ d−1 + bϕ Im γ d−1 ) + x2 (aψ Re γ d−1 + bψ Im γ d−1 ),

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but from Lemma 4.3, we know that we must have D[ϕ, x1 , h] + D[ψ, x2 , h] = 0 which is equivalent to saying that aϕ D[Re γ d−1 , x1 , h] + bϕ D[Im γ d−1 , x1 , h] + aψ D[Re γ d−1 , x2 , h] + bψ D[Im γ d−1 , x2 , h] = 0. Now, by applying the identities for the derivatives of Re γ d and Im γ d (see Lemma 2.3), we get the following: aϕ D[Re γ d−1 , x1 , h] + bϕ D[Im γ d−1 , x1 , h] − aψ D[Im γ d−1 , x1 , h] + bψ D[Re γ d−1 , x1 , h] = 0, so (aϕ + bψ )D[Re γ d−1 , x1 , h] + (bϕ − aψ )D[Im γ d−1 , x1 , h] = 0, which gives that 0=

D[(aϕ + bψ ) Re γ d−1 + (bϕ − aψ ) Im γ d−1 , x1 , h],

but if the derivative of a function with respect to x1 is zero, that function must be a polynomial in x2 , but we know that Re γ d−1 and Im γ d−1 are homogeneous of degree d − 1, so the function of x2 can only be cxd−1 for some constant c. That 2 is to say (aϕ + bψ ) Re γ d−1 + (bϕ − aψ ) Im γ d−1 = cxd−1 . 2 is not harmonic, it is not in the span of Re γ d−1 and Im γ d−1 , so Now since xd−1 2 c = 0. Therefore, 0 = (aϕ + bψ ) Re γ d−1 + (bϕ − aψ ) Im γ d−1 , which means that 0 = aϕ + bψ and 0 = bϕ − aψ , so aϕ = −bψ and aψ = bϕ . Using this, we get β(x1 , x2 )

= x1 (aϕ Re γ d−1 + bϕ Im γ d−1 ) + x2 (aψ Re γ d−1 + bψ Im γ d−1 ) = x1 (aϕ Re γ d−1 + aψ Im γ d−1 ) + x2 (aψ Re γ d−1 − aϕ Im γ d−1 ) = aϕ x1 Re γ d−1 + aψ x1 Im γ d−1 + aψ x2 Re γ d−1 − aϕ x2 Im γ d−1 = aϕ (x1 Re γ d−1 − x2 Im γ d−1 ) + aψ (x1 Im γ d−1 + x2 Re γ d−1 ) = aϕ Re γ d + aψ Im γ d ,

which implies that β is linearly dependent upon Re γ d and Im γ d . Hence Re γ d and Im γ d form a basis for all of the harmonic polynomials which are homogeneous of degree d.


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4.2.2. Degree Two Polynomials. The polynomials of degree two are a special case. This is because some terms of polynomials will vanish when the Laplacian is taken. Specifically, if we are given the general polynomial p = A1 x21 + A2 x22 + A3 x1 x2 + A4 x2 x1 , we find that the Laplacian is Lap(p) = A1 h2 + A2 h2 , meaning that the polynomial will be harmonic provided that A1 + A2 = 0 and subharmonic provided that A1 + A2 ≥ 0. This is the one case where the harmonic polynomial is not symmetric. Also, because the subharmonic polynomials are built up of harmonic polynomials of one half the degree, this means that it may be possible, in the degree four case alone, to create nonsymmetric subharmonic polynomials. Remark 2. We now show that this gives a 6 dimensional spanning set for the symmetric subharmonics of degree 4; denote these by S4 . We use Proposition 4.1 which says S4 is spanned by symmetrized products of the basis s =: x21 − x22 , u := x1 x2 , uT := x2 x1 . Thus we obtain s2 , su + uT s, suT + us, uu + uT uT , uT u, uuT . Note this is consistent with Theorem 1 part (2b) which implies the span of the degree 6 symmetric subharmonics has dimension. 4.3. Homogeneous Harmonics of Odd Degree The remainder of this section is not used in the rest of the paper, but Proposition 4.3 may be useful in further research on harmonics in many variables. What does the argument in Section 4.1.1 say about harmonics of odd degree? As we have already stated in §2.4, any subharmonic polynomial of odd degree is required to be harmonic. Given NC polynomial p decompose it as p=

g

|t|= d−1 2

xt xi pt,i (x)

(31)

i=1

and call the polynomial pt,i (x) the right neighbor of xt xi . Here we are assuming all terms of p have degree d−1 2 . Apply Lap to the right neighbor decomposition (31) of harmonic p and from the Laplacian Product Rule get 0 = Lap[p, h] = g

Lap[xt xi , h]pt,i (x) (32) i=1 |t|= d−1 2

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g



99

xt xi Lap[pt,i (x), h]

(33)

i=1 |t|= d−1 2

+2

g g


|t|= d−1 2

which is g



D[xt xi , xj , h]D[pt,i (x), xj , h]

(34)

i=1 j=1

 Lap[xt , h]xi + xt Lap[xi , h] + 2

i=1 |t|= d−1 2

g


 D[xt , xj , h]D[xi , xj , h] pt,i (x)

j=1

(35) +




g


xt xi Lap[pt,i (x), h]

(36)

i=1 |t|= d−1 2 g g


  D[xt , xj , h]xi + xt D[xi , xj , h] D[pt,i (x), xj , h]

+2

(37)

i=1 j=1 |t|= d−1 2

Finally it becomes g

  Lap[xt , h]xi + 2D[xt , xi , h]h pt,i (x)

(38)

i=1 |t|= d−1 2

+

+2

g

|t|= d−1 2

g



xt xi Lap[pt,i (x), h]

i=1 |t|= d−1 2



xt hD[pt,i (x), xi , h] +

i

g


(39) 

D[xt , xj , h]xi D[pt,i (x), xj , h] ,

(40)

j=1

which must be 0. The right half of each monomial in (38) contains no h’s, while in (39) and (40) each right half does; thus no term of (38) can be cancelled. We conclude (38) is 0. Similarly the right halves in (39) are the only right halves monomials which contain two h’s and so cannot be cancelled. Thus (39) is 0 and so we get Lap[pt,i , h] = 0 for each |t| = d−1 2 , i = 1, . . . , g. Use k
pt,i = µj (t, i)γj j

as before. Plug this into the decomposition

xt xi pt,i (x) = p=




|t|= d−1 2 ,i

j

|t|= d−1 2 ,i

µj (t, i)xt xi γj .

(41)

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to get p=




 

j






µj (t, i)xt xi  γj =


j

|t|= d−1 2 ,i

=

 k

j=1

   

  µj (t, i)xt  xi  γj i

 j,i

IEOT

p (x)xi

|t|= d−1 2

γj

i

Now make a left neighbor decomposition of p which by the definition γ1 has the form  
1,i p (x)xi γ1 + G i

where all terms of G are without γ1 on the right. The left handed version of Lemma 4.1 implies Lap(p1,i ) = 0 for each i = 1, . . . , g. We have proved the following: Proposition 4.3. Assume the harmonic polynomials homogeneous of degree d−1 2 are the span of γ1 , . . . , γk . Assume there is a monomial wj in γj which does not occur in the other γ1 , . . . , γk . If p is subharmonic homogeneous of odd degree d, then it is harmonic and has the form g k

p= φmij γm xi γj (42) i=1 m,j=1

where each φmij is a number. Question 1. What φmij make it 0? That is to say, what properties must φmij satisfy in order for p to be harmonic? We do a few calculations which might someday help with this question. Note the Laplacian of such a p is: Lap[p, h] =

g k



φmij Lap[γm xi γj , h]

i=1 m,j=1

=

 φmij Lap[γm , h]xi γj + γm Lap[xi , h]γj + γm xi Lap[γj , h]

g k

i=1 m,j=1

+2

g
l=1

=2

g k



(γm D[xi , xl , h]D[γj , xl , h] + D[γm , xl , h]xi D[γj , xl , h]  +D[γm , xl , h]D[xi , xl , h]γj )

g  
φmij γm hD[γj , xi , h]+D[γm , xi , h]hγj + D[γm , xl , h]xi D[γj , xl , h]

i=1 m,j=1

l=1

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As before cancellation cannot occur between terms with right halves containing two h’s, one h and no h’s. Thus, Lap[p, h] = 0 is equivalent to 
 g k k

γm h D φmij γj , xi , h = 0, and m=1 g k 



i=1

D


k

=1 i=1 j=1

 φmij γm , xi , h

hγj = 0

and





m=1

j=1 i=1 g
g
k


j=1

D


k



φmij γm , xl , h xi D γj , xl , h = 0.

m

Acknowledgments Thanks to Nick Slinglend and John Shopple for help with computations. Thanks to Professor Roger Howe for very helpful conversations about the classical commutative analog of the noncommutative results here.

References [1] Juan F. Camino, J.W. Helton, R.E. Skelton, and Jieping Ye. Matrix inequalities: a symbolic procedure to determine convexity automatically. Integral Equations Operator Theory, 46(4):399–454, 2003. [2] Frank Hansen. Operator convex functions of several variables. Publ. Res. Inst. Math. Sci., 33(3):443–463, 1997. [3] J.W. Helton and Orlando Merino. Sufficient Conditions for Optimization of Matrix Functions. CDC, 1998. pp. 1–5. [4] J.W. Helton and Scott A. McCullough. Convex noncommutative polynomials have degree two or less. Siam J. Matrix Anal. Appl, 25(4):1124–1139, 2004. [5] J.W. Helton, Scott A. McCullough, and Victor Vinnikov. Noncommutative convexity arises from linear matrix inequalities. J. Funct. Anal., 240(1):105–191, 2006. [6] J.W. Helton and M. Putinar. Positive polynomials in scalar and matrix variables, the spectral theorem and optimization. To appear (see arXiv: http://arxiv.org/abs/math.FA/0612103), 2006. p. 106. [7] Roger Howe and Eng-Chye Tan. Nonabelian harmonic analysis. Applications of SL(2, R). Universitext. Springer-Verlag, New York, 1992. [8] Frank Hansen and Jun Tomiyama. Differential analysis of matrix convex functions. Linear Algebra and its Applications, 2006. [9] Daniel P. McAllaster. Homogeneous harmonic noncommutative polynomials from degree 3 to 7 (includes a recurrence relation for higher degrees). http://math.ucsd.edu/~dmcallas/papers/, July 2004. [10] Dimitri Shlyakhtenko and Dan Voiculescu. Free analysis workshop summary: American institute of mathematics. http://www.aimath.org/pastworkshops/freeanalysis.html, 2006.

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J. William Helton, Daniel P. McAllaster and Joshua A. Hernandez University of California, San Diego Department of Mathematics La Jolla, CA, 92093-0112 USA e-mail: [email protected] [email protected] Submitted: May 24, 2007

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Integr. equ. oper. theory 61 (2008), 103–120 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/010103-18, published online April 18, 2008 DOI 10.1007/s00020-008-1575-8

Integral Equations and Operator Theory

Hyperinvariant Subspace Problem for Quasinilpotent Operators Hyoung Joon Kim Abstract. In this paper we consider the hyperinvariant subspace problem for quasinilpotent operators. Let (CRQ) denote the class of quasinilpotent quasiaffinities Q in L(H) such that Q∗ Q has an infinite dimensional reducing subspace M with Q∗ Q|M compact. It was known that if every quasinilpotent operator in (CRQ) has a nontrivial hyperinvariant subspace, then every quasinilpotent operator has a nontrivial hyperinvariant subspace. Thus it suffices to solve the hyperinvariant subspace problem for elements in (CRQ). The purpose of this paper is to provide sufficient conditions for elements in (CRQ) to have nontrivial hyperinvariant subspaces. We also introduce the notion of “stability” of extremal vectors to give partial solutions to the hyperinvariant subspace problem. Mathematics Subject Classification (2000). Primary 47A15. Keywords. Hyperinvariant subspace problem, quasinilpotent operators, extremal vectors.

1. Introduction Let H be a separable infinite dimensional complex Hilbert space and L(H) be the algebra of all bounded linear operators acting on H. The commutant of T , denoted by {T }
, is the algebra of all operators X in L(H) such that XT = T X. A closed subspace M ⊂ H is called a nontrivial hyperinvariant subspace for T if {0} = M = H and XM ⊆ M for each X ∈ {T }
. The hyperinvariant subspace problem is the question whether every operator in L(H)\C has a nontrivial hyperinvariant subspace. This problem remains still open, especially for quasinilpotent operators in L(H) i.e., operators Q such that σ(Q) = {0}, where σ(·) means spectrum. In this paper we consider the hyperinvariant subspace problem for quasinilpotent operators. We write (Q) for the class of quasinilpotent operators in L(H).

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We begin with: Definition 1.1. A quasinilpotent operator Q is called a universal quasinilpotent operator if there exists a sequence {Sm } of invertible operators such that −1 − Q
 → 0 for each Q
∈ (Q). Sm QSm −1





Let Sm = cSm /Sm  for c ≥ 0. Then Sm QSm − Q
 → 0 and Sm  = c. So in Definition 1.1, we can choose each Sm with arbitrary norm. Let N(H) and K(H) denote the set of all nilpotent operators and the ideal of compact operators, respectively. The following theorem asserts that if Q is a quasinilpotent operator / K(H) for all n ≥ 1, then Q is universal. By the well known such that Qn ∈ Lomonosov’s theorem, we know that every polynomially compact operator has a nontrivial hyperinvariant subspace. Thus it suffices to solve the hyperinvariant subspace problem for universal quasinilpotent operators.

Theorem 1.2. ([8, Theorem 1]) A quasinilpotent operator Q is a universal quasinilpotent operator if and only if it belongs to (Q)\[N(H) + K(H)]. Consequently if Q ∈ (Q) is not a universal quasinilpotent operator, then Q has a nontrivial hyperinvariant subspace. On the other hand, in [1], S. Ansari and P. Enflo introduced extremal vectors as a method of constructing hyperinvariant subspaces for a certain class of linear operators. As consequence of this technique, P. Enflo [5] gave the “two sequence” theorem which provided a contribution to the hyperinvariant subspace problem for quasinilpotent operators. In [9], I. Jung, E. Ko, and C. Pearcy obtained a better version of the above theorem modifying Enflo’s technique mentioned in [1]. By defining a subclass (CRQ) of quasinilpotent operators, the authors of [6] obtained a reduction theory of these operators aided by the theory of D. Herrero mentioned in [8]. An operator T ∈ L(H) is called quasiaffinity if it is a one-one mapping having dense range. Write (CRQ) for the class of quasinilpotent quasiaffinities Q in L(H) such that Q∗ Q has an infinite dimensional reducing subspace M with Q∗ Q|M compact. Theorem 1.3. ([6, Theorem 5.6]) If every element in (CRQ) has a nontrivial hyperinvariant subspace, then every quasinilpotent operator has a nontrivial hyperinvariant subspace. If we want to emphasize the subspace M in the definition of (CRQ), we write (CRQ)M for (CRQ). The compactness of Q∗ Q|M in the class (CRQ)M will be a core property of this paper. In Section 2, we use P. Enflo’s “extremal vectors” to obtain partial solutions for the hyperinvariant subspace problem. In Section 3, we define the notion of “stability” for extremal vectors and then give connections between this property and hyperinvariant subspaces of quasinilpotent operators and then prove that quasinilpotent operators with some stability conditions have nontrivial hyperinvariant subspaces.

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2. Extremal Vectors of (CRQ)M The following lemma plays an important role in the sequel. Lemma 2.1. Let M be a closed subspace of H and T ∈ L(H) be an operator such that T ∗ T = K ⊕ R on M ⊕ M ⊥ , where T  = 1 and K is a compact operator. If there exist sequences {sk } and {tk } satisfying (i) sk −→ s0 (= 0) weakly, tk −→ t0 (= 0) weakly and sk ∈ M for all k ≥ k0 (ii) Xsk , tk → 0 for all X ∈ {T }
, then T has a nontrivial hyperinvariant subspace. Proof. Observe that

 xk x if −→ weakly, then xk −→ x weakly and yk −→ y weakly. y yk Suppose that X is a contraction. Since XT  ≤ T , we have XT ∈ {T }
. Therefore XT sk , tk → 0. Let T = U |T | be the polar decomposition. Since

 x0 xk ∗ ∗ ∗ ∗ := U X tk −→ U X t0 =: weakly, yk y0 it follows that xk −→ x0 weakly. Since sk ∈ M and M reduces |T |, it follows that 1 XT sk , tk = sk , |T |U ∗ X ∗ tk = sk , |T |xk . Since |T |M = K 2 is compact, |T |xk converges to |T |x0 in norm. Therefore sk , |T |xk → s0 , |T |x0 = s0 , |T |U ∗ X ∗ t0 = XT s0 , t0 , so that XT sk , tk → XT s0 , t0 . Since XT sk , tk → 0, we have XT s0, t0 = 0. But since s0 and t0 are nonzero, N ≡ cl{T }
T s0 is a nontrivial hyperinvariant subspace for T .
The notion of extremal vectors was introduced by S. Ansari and P. Enflo [1]. Extremal vectors provide a tool to find hyperinvariant subspaces for compact and normal operators on Hilbert spaces. In this section, we introduce Enflo’s extremal vectors and consider the behavior of them for the class (CRQ)M . Assume that T has dense range and choose a unit vector x0 ∈ H and 0 < ε < 1. If F = {y ∈ H : T y − x0  ≤ ε}, then F is a nonempty, norm closed and convex set. So there exists a unique minimal vector y0 = y0 (x0 , ε) ∈ F. We say that y0 is the extremal(minimal) vector for T . In this case, T y0 − x0  = ε. Let yn = yn (x0 , ε) be the extremal vector for T n and for a notational convenience, write y0 := y1 . In [1], Enflo established an important equation on extremal vectors so called “Orthogonality Equation”: if T has dense range, then there exists a constant r < 0 satisfying T ∗ (T y0 − x0 ) = ry0 .

(2.1)

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Also it was shown in [1, Lemma 1] that if T is a quasinilpotent operator with dense range, then there exists a subsequence {ynk } such that lim

k→∞

ynk  = 0. ynk +1 

(2.2)

By using (2.1) and (2.2), Enflo constructed the following “two sequence” theorem. Lemma 2.2. ([5, Theorem 4.8], [9, Theorem 1.3]) Suppose T is a quasinilpotent quasiaffinity. Then there exist two sequences {sk } and {tk } that converge weakly to nonzero vectors s0 and t0 respectively, and Xsk , tk → 0 for all X ∈ {T }
. On the other hand, V. G. Troitsky [11] and I. Chalendar, J. Partington [3] introduced a notion of λ-extremal vectors which is a generalization of a notion of Enflo’s extremal vectors to arbitrary Banach spaces. Let X be a Banach space and T be a bounded linear operator on X. Assume that T has dense range and choose a unit vector x0 ∈ X and 0 < ε < 1. For λ ≥ 1, a vector y0 = y0 (x0 , ε, T ) is called a λ-extremal(minimal) vector if T y0 − x0  ≤ ε

and y0  ≤ λ inf{y : T y − x0  ≤ ε}.

For λ > 1, the existence of λ-extremal vectors is clear and obviously they are not unique. If λ = 1, then the λ-extremal vector in the Hilbert space setting is exactly the Enflo’s extremal vector. We now introduce a weak notion of a reducing subspace. Definition 2.3. We say that T ∈ L(H) has a weakly reducing subspace N if for some polynomial p satisfying p(0) = 0, p(T ) can be written as
 A B N , p(T ) = 0 C N⊥ where A, B and C satisfy: (1) ABX = BC for some strict contraction X ∈ {C}
. (2) B ∗ x ≤ cA∗ x for all x ∈ N and for some c > 0. We can easily find a quasinilpotent quasiaffinity which has a weakly reducing subspace.
  2A A Example 2.4. Let T = ∈ L(H H), where A is a quasinilpotent 0 A quasiaffinity. Then H has a weakly reducing subspace for T because Definition 2.3 is satisfied with X = 12 I, c = 1 and p(t) = t. We also see that T is a quasinilpotent quasiaffinity. We now have:

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Theorem 2.5. Suppose Q ∈ (CRQ)M . If either Q or Q∗ has a weakly reducing subspace N ⊆ M , then Q has a nontrivial hyperinvariant subspace. Proof. We first suppose that Q has a weakly reducing subspace N . Write
 A B N Q1 := p(Q) = . 0 C N⊥ Then Q1 n =




An 0

Bn Cn



N , N⊥

where Bn =

n 

An−i BC i−1 .

i=1

Since Q1 is also a quasinilpotent quasiaffinity, it follows that A is quasinilpotent and one-one. Furthermore, A has dense range because if x ∈ ker A∗ then, since B ∗ x ≤ cA∗ x, x ∈ ker B ∗ , and hence x ∈ ker Q∗1 , which forces x = 0 since Q1 is one-one. Choose a unit vector x0 ∈ N . Given 0 < ε < 1, let yn = yn (x0 , ε) be the extremal vector for An . By a similar argument to the proof in Lemma 2.2, we can choose {nk } such that {Ank ynk } and {Ank +1 ynk +1 } converge weakly and ynk  = 0. k→∞ ynk +1  lim

Let sk := Q1 nk +1 ynk ∈ N ⊆ M and tk := Q1 nk +1 ynk +1 − x0 ∈ N . Then {sk } and {tk } converge weakly to nonzero vectors s0 , t0 , respectively. If we can show Xsk , tk → 0 for all X ∈ {Q}
,

(2.3)

then the proof is complete because all assumptions of Lemma 2.1 are satisfied. For (2.3) we first argue that Bn∗ x ≤ dA∗ n x

(2.4)

for all x ∈ N , n ≥ 1 and for some d > 0. Indeed, since by assumption, ABX = BC for some strict contraction X ∈ {C}
, we have Bn =

n 

An−i BC i−1 = An−1 B

i=1

n 

X i−1 .

i=1

c Writing h := A∗ n−1 x ∈ N and d := , we can get, using the second 1 − X ∗ ∗ condition B h ≤ cA h, Bn∗ x ≤

1 − Xn ∗ B h ≤ dA∗ h = dA∗ n x. 1 − X

By the orthogonality equation (2.1), we have A∗ nk +1 (Ank +1 ynk +1 − x0 ) = rnk +1 ynk +1 . Define vnk +1 :=

1 B∗ (Ank +1 ynk +1 − x0 ) rnk +1 nk +1

(k = 1, 2, . . .).

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Then a straightforward calculation shows Q∗1 nk +1 tk

= rnk +1

For X ∈ {Q}
which is a contraction, let
 ynk +1 + wk , Xynk := αk vnk +1 Then




 ynk +1 . vnk +1

(2.5)


where wk ⊥

 ynk +1 . vnk +1

ynk 2 ≥ Xynk 2 = |αk |2 (ynk +1 2 + vnk +1 2 ) + wk 2 ,

which gives |αk | ≤

ynk  −→ 0. ynk +1 

(2.6)

Since XQ1 = Q1 X, we have


 ynk +1 , tk + Q1 nk +1 wk , tk . vnk +1
 ynk +1 nk +1 wk , tk = rnk +1 wk , By (2.5), we have Q1

= 0. Therefore vnk +1 Xsk , tk = αk Q1 nk +1

Xsk , tk = αk { Q1 nk +1 ynk +1 , tk + Q1 nk +1 vnk +1 , tk }. Now apply (2.4) with rn 1+1 (Ank +1 ynk +1 − x0 ) and nk + 1 in place of x and n, k respectively. Then we get vnk +1  ≤ dynk +1 . Thus by (2.5), | Q1 nk +1 vnk +1 , tk | = =

|rnk +1 |vnk +1 2 ≤ d2 |rnk +1 |ynk +1 2 d2 | Q1 nk +1 ynk +1 , tk |.

Therefore we have | Xsk , tk | ≤ |αk |(1 + d2 )Q1 nk +1 ynk +1 tk . But since Q1 nk +1 ynk +1  ≤ x0  + ε and tk  = ε, it follows from (2.6) that Xsk , tk → 0 for all X ∈ {Q}
, which proves (2.3). Therefore Q has a nontrivial hyperinvariant subspace. Similarly if Q∗ has a weakly reducing subspace M , then Q∗ has a nontrivial hyperinvariant subspace, and so does Q.
Corollary 2.6. Let Q ∈ (CRQ)M and N ⊆ M . If N reduces p(Q) for some polynomial p such that p(0) = 0, then Q has a nontrivial hyperinvariant subspace. Proof. Immediate from Theorem 2.5.




If p(Q) is of an upper triangular form, what can we say about a nontrivial
 A B N hyperinvariant subspace? Suppose Q ∈ (CRQ)M and p(Q) = 0 C N⊥ (N ⊆ M ) for some polynomial p satisfying p(0) = 0. Write Bn := PN p(Q)n (1 − PN ), where PN is the orthogonal projection onto N . Then we get:

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Corollary 2.7. With the above notation, if Bn∗ x ≤ dA∗ n x, where d > 0 for all x ∈ N and all n, then Q has a nontrivial hyperinvariant subspace. Proof. This follows at once from an analysis of the proof of Theorem 2.5.




In some sense, Corollary 2.7 says that if the upper-right entry Bn of the upper triangular form p(Q)n is sufficiently “small” then Q has a nontrivial hyperinvariant subspace. The following theorem extends Corollary 2.7 in the sense that if the skew diagonal entries of a four block form p(Q)n are sufficiently “small” then the conclusion remains still true. Theorem 2.8. Let Q ∈ (CRQ)M and N ⊆ M . Suppose p(Q) is of the form
 A B N p(Q) = C D N⊥ for some polynomial p satisfying p(0) = 0 and write
 N An Bn p(Q)n := . Cn Dn N⊥ If Cn x ≤ cAn x and Bn∗ x ≤ dA∗n x, where c > 0 and d > 0 for all x ∈ N and all n, then Q has a nontrivial hyperinvariant subspace. Proof. Write Q1 := p(Q). Since Bn∗ x ≤ dA∗n x, the same argument as the proof of Theorem 2.5 shows that An has dense range for all n. Choose  a unit

1


1 − δ2 ) be c2 +1 . Let yn = yn (x0 , zn (x0 , √δ2 ) be the extremal vector for An .

vector x0
∈ N and let x0 = δx0
with δ =

the extremal vector for Q1 n and zn = Then we claim that zn  ≥ yn
 for all n ≥ 1. 1 c2 +1 ,

Since Cn zn  ≤ cAn zn  and δ =
 2    n zn − x0
 ≤ Q1 0 = Since An zn − x0  =

(2.7)

by a straightforward calculation, we have An zn − x0
2 + c2 An zn 2 1 An zn − x0 2 + 1 − δ. δ

√δ , 2

it follows that
    δ  n zn  − x0  ≤ 1 − Q1 0 2

for all c > 0.

But since yn
is a minimal vector, we have yn
 ≤ zn . This proves (2.7). In a similar manner used in the proof of Theorem 2.5, we can choose {nk } such that {Q1 nk ynk
} and {Ank +1 znk +1 } converge weakly, nk+1 = nk + 1, and ynk
 = 0. k→∞ ynk +1
 lim

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Put

yn :=

IEOT

z n , n = nk + 1 yn
, otherwise

Then by (2.7), we have ynk +1
 ≤ znk +1  = ynk +1  and hence

ynk  = 0. k→∞ ynk +1  lim

Let sk := Q1 nk +1 ynk and tk := Ank +1 ynk +1 − x0 ∈ N ⊆ M . Then {sk } and {tk } converge weakly to nonzero vectors s0 and t0 , respectively. By the same argument as the proof of Theorem 2.5, since Ank +1 ynk +1  ≤ (1 + √12 )δ and tk  = √δ2 , we can see that Xsk , tk → 0 for all X ∈ {Q}
. Hence Y tk , sk → 0 for all Y ∈ {Q∗ }


and tk ∈ M.

This satisfies the assumptions of Lemma 2.1. Therefore Q∗ has a nontrivial hyperinvariant subspace, and so does Q.
A trivial example of Theorem 2.8 is the operator Q := A ⊕ A with a compact quasinilpotent quasiaffinity A. We would here give a nontrivial example of Theorem 2.8 which is not, in particular, applied by [4, Theorem 1.4]. Example 2.9. Let T ≡ Wα be the bilateral weighted shift on 2 (Z) with weight sequence α ≡ {2−|k| }. Then T is a compact quasinilpotent quasiaffinity. Write M := 2 (Z) and H := ⊕M . Define an operator Q on H by   1 ( 12 )2 T ( 12 )3 T · · · −T 2T  1T T 0 0 ···   12 2   ( ) T 0 T 0 ···  2   Q :=  ..   ( 1 )3 T  . 0 0 T  2  .. .. .. .. .. . . . . . Then Q is a bounded linear operator. We now claim that Q is a quasinilpotent quasiaffinity. Indeed,  2  T 0 0 ···  0 T2 0 ···    4 2 2 ..  Q = T ⊕ (IM ⊥ + S) ·  2  , . 0 0 T 3   .. .. .. .. . . . . where S ∈ L(⊕M ) is a block Hankel matrix which (i, j)-block entry is ( 12 )i+j IM . Since S ≤ 12 , we know that IM ⊥ + S is invertible. Thus Q2 is a quasinilpotent

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quasiaffinity, and hence so is Q. Moreover,
4 ∗  M T T 0 ∗ 3 Q Q= , 0 ∗ M⊥ so that Q∗ Q|M = 43 T ∗ T is compact. Therefore Q ∈ (CRQ)M . With the notations in Theorem 2.8, put p(t) = t and N = M . Since  n−1 √1 ( 4 ) 2 T ∗ n x if n is odd ∗ 3 3 Bn x = 0 if n is even and n

A∗n x = ( 43 )[ 2 ] T ∗ n x it follows that Bn∗ x ≤ √13 A∗n x for all x ∈ M we can see that Cn x ≤ √13 An x for all x ∈

(x ∈ M ), and all n. By the same argument M and all n. Therefore Q has a

nontrivial hyperinvariant subspace.

The remaining part of this section devotes to answer the question: when does Q have a weakly reducing subspace? We here find an interesting answer. To do so we need the following observation. Proposition 2.10. Suppose T and S are positive operators. If T + S is compact, then T and S are compact. Proof. Let K = T + S and assume K = 1. Then K is a compact positive operator. By the spectral theorem, there exists a finite dimensional subspace M such that for any ε > 0
 K1 0 M K := , where K1  = 1, K2  ≤ ε. 0 K2 M⊥
 
D −C M M A C and S := . Since B and E Write T := −C ∗ E M⊥ M⊥ C∗ B are positive, we have 1

B 2 2 = sup Bx, x ≤ sup ( Bx, x + Ex, x ) = K2  ≤ ε. x=1

x=1

1 2

Similarly, we have A 2 ≤ 1. Since T is positive, there exists a contraction 1 1 1 1 1 2 2 2 2 2 F
such that  C = A F B . Then C ≤ A SB  ≤ ε . Write T1 := 1 A 0 M . Then since B = B 2 2 ≤ ε, we have 0 0 M⊥
 0 C 2 T − T1  =  2 ≤ 2C2 + B2 ≤ ε(2 + ε). C∗ B Since T1 is a finite rank operator, it follows that T is compact, and so does S.


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Suppose N and N ⊥ are infinite dimensional subspaces of H. Let X ∈ L(N ⊥ ) be a universal quasinilpotent operator and Y ∈ L(N ) be any quasinilpotent operator. Then for any ε > 0, there exists an invertible operator S ∈ L(N ⊥ , N ) such that SXS −1 − Y  < ε. For each 0 < r < 1 and m ≥ 1, define 1 for 0 < |α| ≤ r}. Trm (X, Y ) := {Sα ∈ L(N ⊥ , N ) : Sα XSα−1 − αY  < m Evidently, Trm (X, Y ) is nonempty. Suppose that Q and p(Q) are of the form

 A D ∗ M and 0 E ∗ M⊥

B C



N , N⊥

respectively, for some polynomial p such that p(0) = 0 and N ⊆ M . Then since Q ∈ (CRQ)M , we have D∗ D + E ∗ E = K for some compact operator K. By Proposition 2.10, D∗ D and E ∗ E are compact, so are D and E. Let T ∈ L(M ) be the compression of p(Q) to M . Then by a direct calculation, we can see that T is a compact operator. Moreover, since N ⊆ M , A is a part of T and hence A is also a compact operator. Suppose C is not a universal quasinilpotent operator. Then by Theorem 1.2, C n is also a compact operator for some n ≥ 1. An operator p(Q)n+1 is of the form
n+1  n−1  A Bn+1 N n+1 p(Q) = , where B = An−i BC i + BC n . n+1 0 C n+1 N⊥ i=0

Since A and C n are compact, it follows that Bn+1 is compact. It means that p(Q)n+1 is compact and hence Q is polynomially compact. Therefore Q has a nontrivial hyperinvariant subspace. So we can assume that C is a universal quasinilpotent operator. In this case, Trm (C, A) is a nonempty set. Theorem 2.11. With the above notation, if there exists a uniformly bounded sequence {Sm } in Trm (C, A) such that {ASm } converges to B in the weak operator topology, then Q has a weakly reducing subspace N . Proof. Let En ∈ L(N ⊥ , N ) (n = 1, 2, . . .), P ∈ L(N ) and Q ∈ L(N ⊥ ). If {En } converges to zero in the weak operator topology, then {En∗ }, {P En }, and {En Q} ∗ converge to zero in the weak operator topology. Write Tm := Sm . Then Tm A∗ → B ∗ in the weak operator topology by the above observation. Since Tm  ≤ c for some c > 0, we have cA∗ xB ∗ x ≥ | Tm A∗ x, B ∗ x | −→ B ∗ x2 ,

x ∈ N.

It follows that for all x ∈ N , A x ≥ dB x, where d = 1c . On the other −1 − hand, since Sm ∈ Tm r (C, A), there exists a sequence {αm } such that Sm CSm 1 αm A < m . Since Sm is uniformly bounded, αm ASm − Sm C → 0 in norm. So by assumption we have Sm C − αm B → 0 in the weak operator topology and hence ∗

∗

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ASm C − αm AB → 0 in the weak operator topology. Choose a subsequence {mk } such that αmk converges to α with |α| ≤ r < 1. Then ASmk C → αAB in the weak operator topology. Moreover, by assumption, ASm C → BC in the weak operator topology. So we conclude that αAB = BC for |α| < 1. Therefore Q satisfies the conditions of Definition 2.3, so that Q has a weakly reducing subspace N .


3. The Stability of Extremal Vectors In this section, we study the behavior of extremal vectors. Suppose T has dense range and let yn = yn (x0 , ε) be the extremal vector for T n . Write zn := T n yn . Then zn lies in the boundary of the ball B(x0 , ε). If T is a quasinilpotent operator, a stability of zn can be used for solving the hyperinvariant subspace problem for T . we first define the strong stability for extremal vectors. Definition 3.1. Suppose T has dense range. Then T is said to be strongly stable (for x0 ) if there exist a unit vector x0 in H and 0 < ε < 1 such that T n yn ∈ ∨{x0 } for all n ≥ n0 , where the yn = yn (x0 , ε) are extremal vectors for T n . Also T is said to be strongly hyperstable if T is strongly stable for every unit vector in H. The following observation is elementary: Proposition 3.2. Suppose T has dense range. If y0 = y0 (x0 , ε) is an extremal vector for T and z := T y0 , then z ≥ x0 , z > z2 . Proof. By the orthogonality equation (2.1), T ∗ (T y0 − x0 ) = ry0 (r < 0). We thus have T T ∗ (x0 − z) = sz (s = −r > 0), (3.1) ∗ so that (x0 − z), T T (x0 − z) = s( x0 , z − z2 ) > 0 since x0 − z is nonzero and
T ∗ is one-one. Hence x0 z ≥ x0 , z > z2 . We would remark that if T n yn = cn x0 (n ≥ n0 ) with extremal vectors yn = yn (x0 , ε) for T n , then cn can be chosen as a unique value c regardless of n. Indeed, if zn := T n yn = cn x0 , then x0 , zn = c¯n > 0, and hence 0 < cn < 1 by Proposition 3.2. Since x0 − zn  = ε, we have cn = 1 − ε. Therefore if T is strongly stable, then T n yn = (1 − ε)x0 for all n ≥ n0 , i.e., T n yn are fixed. More generally, n0 yn0 . Then T n yn = cn z suppose ∨{T n yn }∞ n=n0 is one dimensional. Write z := T for some cn ∈ R. Consider a quadratic equation of t ∈ R: tz − x0 2 = ε2 . By Proposition 3.2, z, x0 > z2 > 0, so that the equation t2 z2 − 2t z, x0 + 1 − ε2 = 0 has distinct positive roots 1, α (1 < α). Since yn are minimal vectors, we conclude cn = 1, i.e., T n yn = z for all n ≥ n0 . Lemma 3.3. Let x0 ∈ N ⊆ H and z0 = z0 (x0 , ε) be an extremal vector for T . If N reduces T , then z0 ∈ N .

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 x N A 0 and let z := be an extremal vector for 0 0 B N⊥  y T . Suppose y = 0. Then z0  >  x0 . Observe

Proof. Write T :=

T z0 − x0 2 = Ax − x0 2 + By2 = ε2 , and hence

 
x    − x0  = Ax − x0  = ε2 − By2 ≤ ε. T 0   Since z0 is minimal, we have z0  ≤  x0 , a contradiction. Therefore we conclude y = 0.
The following theorem is a characterization of strongly stable operators. Theorem 3.4. Suppose T has dense range. Then T is strongly stable for x0 if and only if the operators in the family {T n T ∗ n }∞ n=n0 (some n0 ∈ N) have a common eigenvector x0 . Proof. If T is strongly stable, then T n yn = (1 − ε)x0 by the remark above Lemma 3.3. Using the equation (3.1), we have T n T ∗ n (x0 − (1 − ε)x0 ) = sn (1 − ε)x0

(sn = −rn > 0)

and hence T n T ∗ n x0 = λn x0 (λn = sn (1−ε) ). Conversely, suppose T n T ∗ n have a ε ∗n ∗n common eigenvector x0 . Let T = Un |T | be the polar decomposition of T ∗ n . Then T n = |T ∗ n |Un∗ . Choose 0 < ε < 1 and let zn = zn (x0 , ε) be the extremal vector for |T ∗ n |. Since ∨{x0 } reduces T n T ∗ n for all n ≥ n0 , zn = cn x0 for some cn by Lemma 3.3. So |T ∗ n |zn = cn λn x0 for some λn . Write yn := Un zn . Then we obtain that T n yn = |T ∗ n |zn ∈ ∨{x0 }. On the other hand, note that T n yn − x0  = |T ∗ n |zn − x0  = ε.

(3.2)

Since Un is an isometry for each n, we have yn  = zn . This implies that yn is the minimal vector satisfying (3.2) for each n, and hence yn is the extremal vector for T n . Therefore T is strongly stable for x0 .
Example 3.5. Let Q be the backward weighted shift with weights {αn } such that αn > 0 and αn is strictly decreasing to zero. Then Q is a quasinilpotent operator with dense range. Moreover, Q is strongly stable because Qn Q∗ n are diagonal operators, so every vector in the standard orthonormal basis of H is a common eigenvector of Qn Q∗ n . The above example is a typical example of a strongly stable quasinilpotent operator. In fact, diagonal operators and weighted shifts with dense range are strongly stable operators. The following corollary is a characterization of strongly hyperstable operators.

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Corollary 3.6. Suppose T  = 1. If T has dense range, then T is strongly hyperstable if and only if T ∗ is an isometry. In particular, if T is a quasiaffinity, then T is strongly hyperstable if and only if T is a unitary operator. Proof. Suppose T is strongly hyperstable and choose a unit vector x in H. Then by Theorem 3.4, x and T ∗ x are eigenvectors of T n T ∗ n for all n ≥ n0 (some n0 ∈ N). Then for some m ≥ n0 , we have T m+1 T ∗ m+1 x = νx (ν > 0) and T m T ∗ m (T ∗ x) = µT ∗ x (µ > 0). Therefore T m+1 T ∗ m+1 x = νx = µT T ∗ x, and hence T T ∗x = λx (λ = µν ). But since x ∈ H is arbitrary and T  = 1, it follows that T T ∗ = I and hence T ∗ is an isometry. Conversely, if T ∗ is an isometry, then T ∗ n is also an isometry for all n. Since T n T ∗ n = I, it follows from Theorem 3.4 that T is strongly stable for any unit vector in H, and hence T is strongly hyperstable. This proves the first assertion. The second assertion follows at once from the first.
Corollary 3.6 implies that quasinilpotent operators cannot be strongly hyperstable. On the other hand, we can find some equivalent condition of strong hyperstability. Suppose T has dense range and T  = 1. Let y0 = y0 (x0 , ε) be an extremal vector for T . Then by the orthogonality equation (2.1) we can easily show that T T ∗ = I ⇐⇒ T y0 ∈ ∨{x0 } for all x0 ∈ H So T is strongly hyperstable if and only if T y0 ∈ ∨{x0 } for each unit vector x0 in H, where y0 = y0 (x0 , ε) be the extremal vector for T . In [2], I. Chalendar and J. Partington found some stability of extremal vectors for certain operators. In particular, [2, Proposition 2.1] states that if T is a normal operator with dense range, then {T n yn } converges in norm for any x0 and ε. On the other hand, if we add a spectral condition to normal operators, we have some more for some fixed x0 and ε. Proposition 3.7. Suppose T is injective and σp (T ∗ T ) = ∅. If T is a quasinormal operator, then T ∗ is strongly stable. In particular, if T is normal then T and T ∗ are strongly stable. Proof. Let T = U |T | be the polar decomposition of T . We can see that T is quasinormal if and only if U and |T | commute. Since U is an isometry, it follows that T ∗ n T n = (T ∗ T )n for all n ≥ 1. Let λ ∈ σp (T ∗ T ) and x0 be an eigenvector for λ. Then λ = 0 by hypothesis and T ∗ n T n x0 = (T ∗ T )n x0 = λn x0 , that is, x0 is a common eigenvector of T ∗ n T n for all n ≥ 1. Therefore by Theorem 3.4, T ∗ is strongly stable. If T is normal then T ∗ n T n = T n T ∗ n for all n, so it follows that T
is strongly stable if and only if T ∗ is strongly stable. The spectral condition of Proposition 3.7 is essential. Suppose T is normal and σp (T ∗ T ) = ∅. Clearly, T is a quasiaffinity because T ∗ T = T T ∗ is one-one. Moreover, by the spectral mapping theorem for all n, we have σp (T ∗ n T n ) = σp ((T ∗ T )n ) = {λn : λ ∈ σp (T ∗ T )} = ∅ for all n.

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By Theorem 3.4, T ∗ is not strongly stable, and hence T is not. Moreover, there exists a hermitian operator which is not strongly stable as we see in the following example. Example 3.8. Let A be a multiplication operator on L2 [0, 1] by Aϕ(t) = tϕ(t). Clearly A is a hermitian operator. Also A∗ A = A2 has no eigenvalue because if A2 ϕ = λϕ, then (t2 − λ)ϕ = 0 a.e., so that ϕ = 0 a.e. Therefore A is a quasiaffinity but not strongly stable. From the proof of Proposition 3.7, we can see that the adjoint of a strongly stable normal operator is also strongly stable. On the other hand, we can easily construct a strongly stable operator whose adjoint is not. Let U be the backward shift. Since U n U ∗ n = I, U is clearly strongly stable. But U ∗ does not have dense range, thus we cannot consider the strong stability of U . However, with some condition, we meet a nontrivial question: If a quasiaffinity T is strongly stable, is T ∗ strongly stable? For example, if a quasiaffinity operator is strongly hyperstable, then its adjoint is also strongly hyperstable by Corollary 3.6. Definition 3.9. Suppose T has dense range. Then T is said to be stable (for N ) if there exists a unit vector x0 in H and 0 < ε < 1 such that N := {x0 }∨{T n yn }∞ n=n0 is a proper subspace in H for some n0 , where the yn = yn (x0 , ε) are extremal vectors for T n . In this case, we say that N is a stable space of T . Further, if N is finite dimensional then T is said to be finitely stable (for N ). If T is strongly stable then N = ∨{x0 }, so T is finitely stable. Now we try to obtain a characterization of stability as in Theorem 3.4. The following proposition is a sufficient condition for stability. Proposition 3.10. Suppose T has dense range. If the operators in the family {T n T ∗ n }∞ n=n0 (some n0 ∈ N) have a common reducing subspace M , then T is stable for N ⊆ M . Proof. Let T ∗ n = Un |T ∗ n | be the polar decomposition of T ∗ n . Choose x0 ∈ M and 0 < ε < 1. Let zn = zn (x0 , ε) be extremal vectors for |T ∗ n |. Since M reduces T n T ∗ n for each n, it follows from Lemma 3.3 zn ∈ M , and hence |T ∗ n |zn ∈ M . Write yn := Un zn . By the same argument as the proof of Theorem 3.4, we know that the yn are extremal vectors for T n . Moreover, since T n y n = |T ∗ n |zn ∈ M , we
have N := {x0 } ∨ {T n yn }∞ n=n0 ⊆ M . The following definition is a weakened version of the stability in Definition 3.9. Definition 3.11. Suppose T has dense range. Then T is said to be quasi-stable (for N ) if there exist a unit vector x0 in H, 0 < ε < 1 and a proper subspace N of H containing x0 such that d(T n yn , N ) := inf z∈N T n yn − z → 0, where the yn = yn (x0 , ε) are extremal vectors for T n . In this case, we say that N is a

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quasi-stable space of T . Furthermore, if N is finite dimensional, then T is said to be finitely quasi-stable (for N ). Evidently, if T is (finitely) stable, then T is (finitely) quasi-stable. It was known [6, Theorem 3.3] that if T is a quasinilpotent operator and there exists a finite dimensional subspace which reduces T n T ∗ n for all n ≥ 1, then T has a nontrivial hyperinvariant subspace. By Proposition 3.10, if the operators in the family {T nT ∗ n }∞ n=n0 (some n0 ∈ N) have a finite dimensional common reducing subspace, then T is finitely stable and hence T is finitely quasi-stable. However, the converse does not hold even for a hermitian operator. For example, let A be a multiplication operator in Example 3.8. By [2, Proposition 2.1], the operator A is finitely quasi-stable. Suppose for a fixed n, An A∗ n = A2n has a finite dimensional common reducing subspace N . Since A|N is a compact hermitian operator, A2n has an eigenvalue. However, in Example 3.8, Ak has no eigenvalue for all k, a contradiction. In this sense, the following theorem is a generalization of [6, Theorem 3.3]. Theorem 3.12. Suppose Q is a quasinilpotent quasiaffinity. If Q is finitely quasistable, then Q has a nontrivial hyperinvariant subspace. Proof. Since Q is a quasinilpotent quasiaffinity, we can find a subsequence {ynk } of {yn } satisfying ynk  =0 (3.3) lim k→∞ ynk +1  and both {T nk ynk } and {T nk +1 ynk +1 } converge weakly to some nonzero vectors. Write sk := Qnk ynk and tk := Q∗ (Qnk +1 ynk +1 − x0 ). Then {sk } and {tk } converge weakly to nonzero s0 and t0 , respectively. By (2.1) and (3.3), we have Xsk , tk → 0 for all X ∈ {Q}
. Since Q is finitely quasi-stable, there exists a finite dimensional space N such that sk − zk  −→ 0

for some zk ∈ N.

(3.4)

Since {sk } converges weakly to s0 , it follows that so does {zk }. Write Xsk , tk = Xzk , tk + X(sk − zk ), tk . Then by (3.4), X(sk − zk ), tk → 0 and hence Xzk , tk → 0. Moreover, since N is finite dimensional, {zk } converges to s0 in norm. So we can see that Xs0 , t0 = 0.
Therefore M ≡ cl{Q}
s0 is a nontrivial hyperinvariant subspace for Q. Corollary 3.13. If Q ∈ (CRQ)M is quasi-stable for N ⊆ M , then Q has a nontrivial hyperinvariant subspace. Proof. Immediate from a slight variation of the proof of Theorem 3.12.




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Next, we study a part of a strongly stable quasinilpotent operator. In general, we can find a strongly stable operator whose part is not strongly stable. However, we can ask an interesting question about strong stability: Suppose T ∈ L(H) has an invariant subspace M and let A := T |M . If T is strongly stable for x0 ∈ M , is A strongly stable for x0 ? For example, if T is hermitian, then the answer is affirmative. The above question is closely related to the model of quasinilpotent operators due to C. Foias and C. Pearcy ([7], [10]). Let k = {kn } be a positive sequence decreasing to zero. Define K ≡ Kk ∈ L(⊕H) by   0 k1 1H   0 k2 1H   . . Kk :=  .   . 0   .. . Then evidently Kk is a strongly stable quasinilpotent operator. The following is the model theory for quasinilpotent operators. Lemma 3.14. ([7, Theorem 1], [10, Theorem 9.6]) If T is a quasinilpotent operator, then there exists a decreasing sequence k = {kn } of nonnegative numbers converging to zero, an invariant subspace M of the operator Kk , and an invertible operator S : H → M such that ST S −1 = Kk |M . The above lemma says that every quasinilpotent operator is a part of some quasinilpotent backward weighted shift with
infinite multiplicity. In other words, if  M A B ˜= Q ∈ (Q) then there exists an extension Q such that A is sim0 C M⊥ ˜ is unitarily equivalent to Kk . Since the strong stability is invariant ilar to Q and Q ˜ is strongly stable. Now the following under unitary equivalence, it follows that Q question arises: Is A or A∗ strongly stable? If the answer is affirmative, then every quasinilpotent operator has a nontrivial hyperinvariant subspace. Unfortunately, ˜ is unitarily equivalent to we were unable to answer this question. However if Q Kk∗ , then we can get an affirmative answer with an additional condition on the norm. We conclude with: Theorem 3.15. Suppose T ∈ L(H) is of the form
 A B M T = , 0 C M⊥ where T is unitarily equivalent to Kk∗ . If A = T , then A∗ is strongly stable. Proof. Without loss of generality, suppose T  = k1 = 1 > k2 . Since T ∗ n T n is unitarily equivalent to Kkn Kk∗ n which is a diagonal operator, we can see that the operators in the family {T ∗n T n }∞ n=1 have common eigenspace L corresponding to

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the eigenvalue T n 2 for all n ≥ 1, that is, L = ∨{y ∈ H : T ∗ n T n y = T n 2 y, n = 1, 2, . . .}. We now claim that L ∩ M = {0}. (3.5) Indeed, since A = T  = 1, we can choose a unit vector x in M such that T x2 > 1 − ε for any ε > 0. Write x := sx0 + ty0 , where x0 and y0 are unit vectors in L and L⊥ , respectively and |s|2 + |t|2 = 1. Since x0 ∈ L and T  = 1, we have T ∗ T x0 = x0 , and hence T x0 , T y0 = x0 , y0 = 0. And since T x0 2 = x0 2 = 1 and T |L⊥  = k2 < 1, we have T x2 = |s|2 T x0 2 + |t|2 T y0 2 ≤ 1 − |t|2 (1 − k2 ). Therefore

1 − ε < T x2 ≤ 1 − |t|2 (1 − k2 ), which forces t = 0, and hence x = sx0 ∈ L ∩ M . This proves (3.5). Then for a unit vector x0 ∈ L ∩ M , we have
∗n n  A A x0 ∗n n T T x0 = = T n 2 x0 ∈ M. ∗

Thus x0 is a common eigenvector for A∗ n An for all n ≥ 1, and therefore A∗ is strongly stable.
Acknowledgment The author is thankful to the referee for many helpful comments.

References [1] S. Ansari and P. Enflo, Extremal vectors and invariant subspaces, Trans. Amer. Math. Soc. 350 (1998), 539-558. [2] I. Chalendar and J. Partington, Convergence properties of minimal vectors for normal operators and weighted shifts, Proc. Amer. Math. Soc. 133 (2005), 501-510. [3] I. Chalendar and J. Partington, Variations on Lomonosov’s theorem via the technique of minimal vectors, Acta. Sci. Math. (Szeged) 71 (2005), no. 3-4, 603-617. [4] R.G. Douglas and C. Pearcy, Hyperinvariant subspaces and transitive algebras, Michigan Math. J. 19 (1972), 1-12. [5] P. Enflo and V. Lomonosov, Some aspects of the invariant subspace problem, Handbook of the geometry of Banach spaces. v.1 (2001), 533-559. [6] C. Foias, I. Jung, E. Ko, and C. Pearcy, On quasinilpotent operators III, J. Operator Theory 54 (2005), 401-414. [7] C. Foias and C. Pearcy, A model for quasinilpotent operators, Michigan Math. J. 21 (1974), 399-404. [8] D. Herrero, Almost every quasinilpotent Hilbert space operator is a universal quasinilpotent, Proc. Amer. Math. Soc. 71 (1978), 212-216. [9] I. Jung, E. Ko, and C. Pearcy, On quasinilpotent operators, Proc. Amer. Math. Soc. 131 (2003), 2121-2127.

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[10] C. Pearcy, Some recent developments in operator theory, C.B.M.S. Regional Conference Series in Mathematics No. 36, Amer. Math. Soc., providence, 1978. [11] V.G. Troitsky, Minimal vectors in arbitrary Banach spaces, Proc. Amer. Math. Soc. 132 (2004), 1177-1180. Hyoung Joon Kim Department of Mathematics Seoul National University Seoul 151-742 Korea e-mail: [email protected] Submitted: March 27, 2007 Revised: January 8, 2008

Integr. equ. oper. theory 61 (2008), 121–145 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/010121-25, published online April 18, 2008 DOI 10.1007/s00020-008-1580-y

Integral Equations and Operator Theory

Approximate Factorization in Generalized Hardy Spaces Bebe Prunaru Abstract. In this paper we establish a connection between the approximate factorization property appearing in the theory of dual algebras and the spectral inclusion property for a class of Toeplitz operators on Hilbert spaces of vector valued square integrable functions. As an application, it follows that a wide range of dual algebras of subnormal Toeplitz operators on various Hardy spaces associated to function algebras have property (A1 (1)). It is also proved that the dual algebra generated by a spherical isometry (with a possibly infinite number of components) has the same property. One particular application is given to the existence of unimodular functions sitting in cyclic invariant subspaces of weak* Dirichlet algebras. Moreover, by this method we provide a unified approach to several Toeplitz spectral inclusion theorems. Mathematics Subject Classification (2000). Primary 47L45, 47B35; Secondary 47B20, 46E40, 46J10, 46J15, 46E30. Keywords. Dual algebra, function algebra, Hardy space, Toeplitz operator, spectral inclusion, spherical isometry.

1. Introduction Let {Ω, B, µ} be a σ-finite measure space and let D be a complex separable Hilbert space. Let L2 (µ.D) be the Hilbert space of all classes of D-valued, Bochner mea
surable functions x : Ω → D with Ω x(ζ)2 dµ(ζ) < ∞. If x, y ∈ L2 (µ.D) one denotes by x · y the integrable function defined by (x · y)(ζ) = x(ζ), y(ζ)D for ζ ∈ Ω, where ·, ·D stands for the scalar product in D. In the theory of dual algebras initiated by S. Brown in [14] it is often very useful to approximate functions in L1 (µ) by elements of the form x · y where x, y are vectors in a given subspace H ⊂ L2 (µ.D) (see Definition 2.1 below). The approximate factorization method initially devised in [15] has been developed in Research partially supported by grant CNCSIS GR202/2006 (cod 813).

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an abstract setting in [5] subsequently yielding spectacular results on the existence of invariant subspaces and the structure of contraction operators (see [4], [16], [17], [18]). The approximate factorization method has been also applied to other classes of operators associated to planar or n-dimensional domains (see [9], [11], [12], [26], [27]) and to algebras generated by commuting isometries [6]. In this paper, we establish and then exploit a connection between the approximate factorization property for a given subspace H ⊂ L2 (µ.D) and the spectral inclusion property for a class of Toeplitz-type operators Tφ on H with symbol φ ∈ L∞ (µ) defined in a natural way (see Definition 2.7) similar to that of the classical case of the Hardy space H 2 of the unit circle. More precisely, in the second section we show (cf. Lemma 2.8) that the spectral inclusion property R(φ) ⊂ σ(Tφ ) holds true for every φ ∈ L∞ (µ) (equivalently, the map φ → Tφ is isometric on L∞ (µ)) if and only if H has a certain localization property (see Definition 2.3) which in many cases is known to be, at its turn, equivalent to the approximate factorization property. On the other hand, the mapping φ → Tφ is shown to be isometric on L∞ (µ) precisely whenever the convex hull of the set {x · x : x ∈ H} is dense in the positive cone of L1 (µ) and this is most useful in concrete applications. This enables us to show in the third section that for large classes of function algebras A ⊂ C(X) on compact Hausdorff spaces their 2 (µ) with respect to suitably chosen measures on X do associated Hardy spaces HA have both the approximate factorization property and the spectral inclusion property for Toeplitz operators with symbol in L∞ (µ). As a consequence of these facts coupled with results from the theory of dual algebras, it will be shown that various types of nonselfadjoint dual operator algebras of Toeplitz operators have property (A1 ) that is every element in the predual is induced by a rank-one operator. As a particular application, we prove that if A ⊂ L∞ (µ) is a weak* Dirichlet algebra, then every cyclic A-invariant subspace H ⊂ L2 (µ.D) generated by a non-vanishing vector contains a vector u with u(ζ) = 1 a.e. This result is related to a well known theorem of H. Helson [32]. The fourth section contains applications to the case of a single subnormal operator S ∈ B(H) and to spherical isometries as well. Let N ∈ B(K) be its minimal normal extension and let µ be a scalar valued spectral measure for N . It is first observed that the Toeplitz operators associated to S (which have been previously studied by several authors) are, up to unitary equivalence, of the type introduced in the second section. It is then proved that the image of the restriction algebra R(N, H) through the Conway-Olin functional calculus has property (A1 ) provided the Toeplitz map φ → Tφ is isometric on L∞ (µ). For instance, this happens whenever the convex hull of the set {|f |2 : f ∈ R(N, H)} is weak* dense in the positive cone of L∞ (µ). We also show in this section that the dual algebra generated by a spherical isometry with a possibly infinite number of components has the property (A1 (1)). The proof is based on results obtained in the previous sections together with those obtained in [50]. The corresponding result for spherical isometries with a finite number of terms was proved in [26].

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In the fifth section we provide an equivalent characterization of planar measures µ for which the convex hull of the set {|f |2 : f ∈ P ∞ (µ)} is weak* dense in the positive cone of L∞ (µ) (P ∞ (µ) stands for the weak* closure in L∞ (µ) of all analytic polynomials). More precisely we show that µ has this property iff it vanishes on its Sarason hull Σ(µ). The proof makes use in an essential way of the D. Sarason’s analytic description of of P ∞ (µ). The last section contains a characterization of those compactly supported planar measures µ for which the space P 2 (µ), that is the closure in L2 (µ) of all analytic polynomials, has the approximate factorization property. It says that µ has this property if and only if it vanishes on the set of analytic bounded point evaluations of P 2 (µ). The proof relies heavily on J. Thomson’s analytic description of the spaces P 2 (µ). A common feature of the various lines of research in dual algebras theory has been the strong interplay between diverse areas of analysis and operator theory. For the first category, one should mention measure theory, analytic functions, function algebras and rational approximation. As for the second one, here we have, among others, dilation theory, functional models, functional calculus, representation theory and local spectral theory. The present paper has been intended to emphasize some of these connections, especially those related to function algebras and representation theory. We close this introductory section with a short list of abbreviations used in the text: • (AFP)=the approximate factorization property (Definition 2.1); • (ALP)=the approximate localization property (Definition 2.3); • (WDP)=the weak* denseness property (Definition 2.10).

2. Factorization and spectral inclusion In this section {Ω, B, µ} will be a fixed σ-finite measure space and D will be a complex separable Hilbert space. We shall denote by L∞ (µ, D) the space of all classes of D-valued Bochner measurable and essentially bounded functions defined on Ω. If σ ⊂ Ω then χσ will denote the characteristic function of σ. Definition 2.1. Let H ⊂ L2 (µ.D) be a closed subspace. Then H is said to have the approximate factorization property (AFP) if for each function f ∈ L1 (µ) and each ε > 0 there exist vectors x, y ∈ H such that f − x · y1 < ε and moreover 1/2 1/2 x ≤ f 1 and y ≤ f 1 . Remark 2.2. One can easily see that if H has this property and if f ∈ L1 (µ) is nonnegative, then one can find vectors x, y ∈ H as above such that x = y. Definition 2.3. We shall say that a closed subspace H ⊂ L2 (µ.D) has the approximate localization property (ALP) if for every measurable subset ω ⊂ Ω with 0 < µ(ω) < ∞ and for every ε > 0 there exists a vector x ∈ H such that χΩ\ω x < εχω x.

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It is not difficult to see that if H has the (AFP) then it has the (ALP) as well. Under some extra assumptions, the converse is also true but a lot more difficult to prove. This implication was initially proved in [5] under the assumption that the localizing vector x ∈ H appearing in Definition 2.3 can be chosen to be essentially bounded and to belong to any given finite codimensional subspace of H. In this case the resulting approximating vectors appearing in Definition 2.3 can be found so that they belong to any given finite codimensional subspace. This result was basic in the proofs of one of the main conjectures in the theory of dual algebras, namely that the dual algebra generated by an absolutely continuous Hilbert space contraction with isometric H ∞ -calculus has property (A1 ) (see [4] and [18]). On the same line, it was subsequently proved that every such operator is reflexive (see [16]). The essentially boundedness condition appearing in [5] was removed in [8], under the condition that H is separable. Now, even if one is not able to find the localizing vectors sitting in arbitrary finite codimensional subspaces one still can get approximating vectors, as the following result from [11] shows: Theorem 2.4 (cf. [11]). Suppose that the measure space {Ω, B, µ} is separable and that H ⊂ L2 (µ.D) is a closed subspace. If H has the (ALP) then it has the (AFP) as well. (Recall that {Ω, B, µ} is called separable if there exists a countable subfamily B  ⊂ B such that for every ω ∈ B there exists a sequence {ωn }n≥1 ⊂ B  with µ(ω ωn ) → 0. This is equivalent to saying that, the space L2 (µ) is separable.) A closer look at the proof of this theorem given in [11] reveals the fact that the separability assumption on the measure space {Ω, B, µ} is used only to reduce the problem, via the above mentioned theorem from [8], to the case when the bounded functions are dense in H. Therefore we can drop the separability condition, assuming instead the denseness of bounded functions and we get the following result which will be useful in cases when the underlying measure space may not be separable: Theorem 2.5. Let H ⊂ L2 (µ.D) be a closed subspace such that H ∩ L∞ (µ.D) is dense in H. If H has the (ALP) then it has the (AFP) as well. We also recall the following basic result from [6]. Theorem 2.6. Suppose H ⊂ L2 (µ.D) is a closed subspace with (AFP) and let ε > 0. Then for each nonnegative function f ∈ L1 (µ) there exists x ∈ H such that x(ζ)2 ≥ f (ζ) a.e. on Ω and moreover such that x2 ≤ f 1 + ε. Consequently, for each f ∈ L1 (µ) there are vectors x ∈ H and y ∈ L2 (µ.D) such that f = x · y and x2 ≤ f 1 + ε and similarly y2 ≤ f 1 + ε. Before going further, let us fix some notations. If K is any Hilbert space then B(K) will denote the space of all bounded linear operators on K. If ϕ ∈ L∞ (µ) we denote by R(ϕ) its essential range. We also denote by Mϕ the multiplication operator on L2 (µ.D) induced by ϕ. Let L1 (µ)+ denote the cone {f ∈ L1 (µ) : f ≥

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0 a.e. on Ω} and similarly for L∞ (µ)+ . We also denote by L1R (µ) the set of real 2 functions in L1 (µ) and similarly for L∞ R (µ). If x ∈ L (µ.D) we denote by Z(x) the zero set of x i.e. the largest set ω ∈ B (determined up to a set of measure zero) on which x vanishes almost everywhere. If E ⊂ L∞ (µ) is a linear subspace and x ∈ L2 (µ.D) we denote by [Ex] the closure in L2 (µ.D) of the linear manifold {f x : f ∈ E}. If x ∈ L2 (µ.D) we shall write |x|2 for the function ζ → x(ζ)2 . If C ⊂ V is a subset in some vector space V then convC will stand for the convex hull of C. If V is a Banach space then convC stands for the norm closure of convC. We now define the Toeplitz operators that will be relevant in our characterization of the spaces H ⊂ L2 (µ.D) with (ALP). Definition 2.7. If H ⊂ L2 (µ.D) is a closed subspace and ϕ ∈ L∞ (µ), we shall define the Toeplitz operator Tϕ on H by Tϕ x = PH (ϕx)

x∈H

where PH denotes the orthogonal projection of L2 (µ.D) onto H. It is obvious that the mapping ϕ → Tϕ is unital and positive hence contractive on L∞ (µ). The following lemma shows that this mapping is isometric precisely when H has the (ALP). Lemma 2.8. Let H ⊂ L2 (µ.D) be a closed subspace. Then the following are equivalent: (1) The space H has the (ALP). (2) For every ϕ ∈ L∞ (µ) we have R(ϕ) ⊂ σ(Tϕ ). (3) For every ϕ ∈ L∞ (µ) we have Tϕ  = ϕ∞ . (4) conv{|x|2 : x ∈ H} = L1 (µ)+ . Proof. (1) ⇒ (2). Assume first that H has the (ALP) and let ϕ ∈ L∞ (µ) such that Tϕ is invertible and let δ > 0 such that Tϕ x ≥ δx for all x ∈ H. It then follows that   |ϕ|2 |x|2 dµ ≥ δ 2 |x|2 dµ Ω

Ω

for every x ∈ H. We shall prove that |ϕ| ≥ δ µ-almost everywhere on Ω. Suppose on the contrary that there exist 0 < δ0 < δ and ω ⊂ Ω measurable such that 0 < µ(ω) < ∞, and |ϕ| ≤ δ0 almost everywhere on ω. Since H has the (ALP), there exists a sequence of unit vectors {xn } in H such that xn − χω xn  → 0. This, together with the fact that |ϕ| ≤ δ0 on ω implies that  2 δ ≤ lim sup |ϕ|2 |xn |2 dµ ≤ δ02 n→∞

ω

a contradiction. (2) ⇒ (3) is obvious. (3) ⇒ (1). Suppose that (3) holds true and let ω ⊂ Ω be a measurable set such that 0 < µ(ω) < ∞ and let ϕ = χω . It then follows from (3) that

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Tϕ  = ϕ∞ = 1. Therefore, there exists a sequence of unit vectors {xn } in H such that (χω xn , xn ) → 1 hence χΩ\ω xn  → 0. This proves that (1) holds true. (4) ⇒ (2). Let ϕ ∈ L∞ (µ) such that Tϕ is invertible and let δ > 0 such that Tϕ x ≥ δx for all x ∈ H. It then follows that   |ϕ|2 |x|2 dµ ≥ δ 2 |x|2 dµ Ω

Ω

2

1

for every x ∈ H. Since conv{|x| : x ∈ H} = L (µ)+ it follows that   2 2 |ϕ| gdµ ≥ δ gdµ Ω

Ω

1

for every g ∈ L (µ)+ therefore |ϕ| ≥ δ µ-almost everywhere on Ω. (2) ⇒ (4). Let ϕ ∈ L∞ R (µ) such that  ϕ|x|2 dµ ≥ 0 Ω

for every x ∈ H. It then follows that (Tϕ x, x) ≥ 0 for all x ∈ H. It then follows from (2) that R(φ) ⊂ [0, ∞) therefore ϕ ≥ 0 a.e. on Ω. The proof of this lemma is finished.
The following lemma shows that for cyclic subspaces of L2 (µ.D) the (ALP) is always equivalent to the (AFP). Lemma 2.9. Let E ⊂ L∞ (µ) be a linear subspace. If x ∈ L2 (µ.D) is a vector such that [Ex] has the (ALP) then [Ex] has the (AFP) as well. Proof. First of all, it is obvious that µ(Z(x)) = 0. For each ω ∈ B define  ν(ω) = |x|2 dµ. ω

Then {Ω, B, ν} is a finite measure space and we consider the unitary operator V : L2 (µ, D) → L2 (ν, D) defined by (V h)(ζ) = h(ζ)/x(ζ),

ζ ∈ Ω.

It is obvious that H has (AFP) if and only if G = V H has and similarly for (ALP). In addition, we see that the essentially bounded functions are norm dense in G because V (f x) ∈ L∞ (ν, D) for every f ∈ E. We can now apply Theorem 2.5 to infer that the space G has indeed the (AFP) hence H has (AFP) as well.
Definition 2.10. A linear subspace E ⊂ L∞ (µ) will be said to have the weak* denseness property (WDP) if conv{|f |2 : f ∈ E} is weak* dense in L∞ (µ)+ . We are now able to provide the following characterization of those subspaces E ⊂ L∞ (µ) for which every E-cyclic subspace of L2 (µ.D) with a non-vanishing generator has the (AFP).

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Proposition 2.11. Let E ⊂ L∞ (µ) be a linear subspace. The following assertions are equivalent; (1) E has the (WDP); (2) For every x ∈ L2 (µ.D) such that µ(Z(x)) = 0 the space [Ex] has the (ALP); (3) For every x ∈ L2 (µ.D) such that µ(Z(x)) = 0 the space [Ex] has the (AFP). Proof. (1) ⇒ (2). Let x ∈ L2 (µ.D) such that µ(Z(x)). It suffices, by Lemma 2.8 to show that conv{|f x|2 : f ∈ E} = L1 (µ)+ . Let ϕ ∈ L∞ R (µ) such that  ϕ|f |2 |x|2 dµ ≥ 0, f ∈ E. Ω

It then follows from (1) that  ϕh|x|2 dµ ≥ 0, Ω

h ∈ L∞ (µ)+ .

Since µ(Z(x)) = 0 this shows that ϕ(ζ) ≥ 0 a.e. (2) ⇒ (3) Follows from Lemma 2.9. (3) ⇒ (2) Obvious. (2) ⇒ (1). Let ϕ ∈ L1R (µ) such that  ϕ|f |2 dµ ≥ 0, f ∈ E. Ω

1 Then ϕ can be written as a product ϕ = ψu where ψ ∈ L∞ R (µ) and u ∈ L (µ)+ and we may also assume that the set {ζ ∈ Ω : u(ζ) = 0} has null measure. Let x ∈ L2 (µ.D) such that x(ζ)2 = u(ζ) a.e. on Ω. By assumption, the space [Ex] has the (ALP). It then follows from Lemma 2.8 that conv{|f x|2 : f ∈ E} = L1 (µ)+ . Since  ψ|f |2 |x|2 dµ ≥ 0, f ∈ E Ω

we get

 Ω

ψhdµ ≥ 0,

h ∈ L∞ (µ)+ ,

therefore ψ(ζ) ≥ 0 a.e. on Ω hence ϕ(ζ) ≥ 0 a.e. on Ω as well. This shows that (1) holds true. The proof is finished.
If E ⊂ L∞ (µ) is a linear subspace, then we shall say that a subspace H ⊂ L (µ.D) is E-invariant if ϕx ∈ H for every ϕ ∈ E and x ∈ H. The following result for the noncyclic case will be used in the next section. Its proof is similar to that of Proposition 2.11. 2

Proposition 2.12. Let E ⊂ L∞ (µ) be a linear subspace having the (WDP). Let H ⊂ L2 (µ.D) be a closed E-invariant subspace such that the map ϕ → Tϕ is faithful on L∞ (µ) (i.e. ϕ ≥ 0 and Tϕ = 0 implies ϕ = 0). Then this map is isometric on L∞ (µ) hence H has the (ALP). If moreover, the measure space is separable then H has the (AFP) as well.

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Proof. As in the proof of Proposition 2.11, it suffices to show that conv{|x|2 : x ∈ H} = L1 (µ)+ . Let ϕ ∈ L∞ R (µ) such that  ϕ|x|2 dµ ≥ 0, x ∈ H. Ω

Since H is E-invariant we get  ϕ|f |2 |x|2 dµ ≥ 0,

f ∈ E, x ∈ H.

Ω

Since E has the (WDP) we get  ϕg|x|2 dµ ≥ 0, Ω

g ∈ L∞ (µ)+ , x ∈ H.

Suppose now there exists ω ⊂ Ω measurable subset such that ϕ(ζ) < 0 a.e. on ω. It then follows from the above that x(ζ) = 0 a.e. on ω for all x ∈ H therefore Tψ = 0 for ψ = χω . This contradicts the hypothesis that the map ψ → Tψ is faithful on L∞ (µ).
For the rest of this section we shall assume that {Ω, B, µ} is a probability space and that D is a complex separable Hilbert space. We shall recall several definitions from the theory of dual algebras (see [7]). Suppose K is a complex Hilbert space. It is well known that B(K) can be identified with the dual space of the space C1 (K) of all trace-class operators on K via the pairing < T, L >= tr(T L),

T ∈ B(K), L ∈ C1 (K).

If x, y ∈ K let x ⊗ y ∈ C1 (K) denote the rank one operator defined by (x ⊗ y)(h) = (h, y)x,

h ∈ K.

If M ⊂ B(K) is a weak* closed subspace, then it follows from general principles that M can be identified with the dual space of the quotient space QM = C1 (K)/M⊥ where M⊥ is the preannihilator of M in C1 (K). If L ∈ C1 (K) we shall denote by [L]M its class in QM . Therefore the duality between M and QM is implemented via the pairing < T, [L]M >= tr(T L),

T ∈ M, [L]M ∈ QM .

It then follows that < T, [x ⊗ y]M >= (T x, y) for every T ∈ M. The space M is said to have property (A1 (r)) for some r ≥ 1, if for each [L]M ∈ QM and for each ε > 0 there exist vectors x, y ∈ K such that [L]M = [x ⊗ y]M and moreover, x · y ≤ (r + ε)[L]M . Recall also that M is called a dual algebra if it is closed under multiplication and contains the identity operator. The following easy lemma is well known and we omit its proof. Lemma 2.13. Suppose A ⊂ B(K) is a dual algebra and ρ : A → C is a multiplicative linear functional. Suppose moreover that there are vectors x, y ∈ K such that ρ(T ) = (T x, y),

T ∈ A.

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Then there exists a vector u ∈ K such that ρ(T ) = (T u, u),

T ∈ A.

If A ⊂ L∞ (µ) is a subalgebra and H ⊂ L2 (µ.D) is a closed subspace we denote T (A) = {Tφ : φ ∈ A} ⊂ B(H). Proposition 2.14. Let H ⊂ L2 (µ.D) be a closed subspace with the (AFP). Let A ⊂ L∞ (µ) be a weak* closed unital subalgebra such that H is A-invariant. Then the mapping ϕ → Tϕ implements a weak* continuous isometry from A onto a dual subalgebra of B(H) that has property (A1 (1)). Moreover if ρ : A → C is a weak* continuous multiplicative functional then there exists a vector x ∈ H such that  ϕ|x|2 dµ ρ(ϕ) = Ω

for every ϕ ∈ A. Proof. Since H has the (AFP) it has obviously the (ALP) therefore, according to Lemma 2.8 the mapping ϕ → Tϕ is an isometry on L∞ (µ). Moreover, since H is A-invariant, this mapping is multiplicative on A, therefore its image T (A) is a dual algebra on H. Let [L] ∈ QT (A) with [L] = 1 and let ε > 0. Then there exists f ∈ L1 (µ) such that  tr(Tφ L) = φf dµ for every φ ∈ A and moreover, f 1 ≤ 1 + ε/2. Since H has the (AFP) it follows from Theorem 2.6 that there exist vectors x ∈ H and y ∈ L2 (µ.D) such that f = x · y and moreover, such that x2 ≤ 1 + ε and y2 ≤ 1 + ε. Let z = PH y where PH is the orthogonal projection of L2 (µ.D) onto H. Since H is A-invariant and f = x · y we see that   φf dµ = φ(x · z)dµ for every φ ∈ A therefore [L] = [x ⊗ z] in QT (A) . This shows that the space T (A) has property (A1 (1)). The last assertion follows from the previous ones and Lemma 2.13.
Propositions 2.11, 2.12 and 2.14 yield the following: Theorem 2.15. Let A ⊂ L∞ (µ) be a unital weak* closed subalgebra. (1) If A has the (WDP) and x ∈ L2 (µ.D) such that µ(Z(x)) = 0 then the dual algebra T (A) acting on H = [Ax] has property (A1 (1)). (2) Suppose now that {Ω, B, µ} is separable and that A has the (WDP). Let H ⊂ L2 (µ.D) be a closed A-invariant subspace such that the map ϕ → Tϕ from L∞ (µ) to B(H) is faithful. Then this map is isometric on L∞ (µ) and the dual algebra T (A) ⊂ B(H) has property (A1 (1)).

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(3) If ρ : A → C is a weak* continuous multiplicative functional on A then in either case (1) or (2) there exists u ∈ H such that  ρ(ϕ) = ϕ|u|2 dµ for every ϕ ∈ A. Definition 2.16. Let A ⊂ L∞ (µ) be a subalgebra containing the constants. One 2 denotes by HA (µ) ⊂ L2 (µ) the norm closure of A in L2 (µ). This is usually called the Hardy space associated to A w.r.t. the measure µ. 2 (µ) is the cyclic A-invariant subspace of L2 (µ) generIt is obvious that HA ated by the constant function ϕ = 1. The following theorem gives a characterization of those Hardy spaces having the (AFP). Its proof follows immediately from Lemma 2.8 together with Theorem 2.6 and Proposition 2.14. We mention that the equivalence between (4) and (5) has been previously proved in [39].

Theorem 2.17. Let A ⊂ L∞ (µ) be a subalgebra containing the constants and let 2 (µ) be the Hardy space associated to A w.r.t. the measure µ. For each ϕ ∈ HA ∞ 2 L (µ) we denote by Tϕ the corresponding Toeplitz operator on HA (µ). Then the following are equivalent: 2 (1) The space HA (µ) has the (ALP). 2 (2) The space HA (µ) has the (AFP). (3) For every ϕ ∈ L∞ (µ) we have Tϕ  = ϕ∞ . (4) For every ϕ ∈ L∞ (µ) we have R(ϕ) ⊂ σ(Tϕ ). 2 (5) The set conv{|x|2 : x ∈ HA (µ)} is norm dense in L1 (µ)+ . If any of the above hold true then for each f ∈ L1 (µ)+ and each ε > 0 there 2 (µ) such that |x(ζ)|2 ≥ f (ζ) a.e. on Ω and x2 < f 1 +ε. Moreover, exists x ∈ HA in this case, the mapping ϕ → Tϕ implements a weak* continuous isometry from 2 (µ)) that has the weak* closure of A in L∞ (µ) onto a dual subalgebra of B(HA property (A1 (1)). Remark 2.18. It is easy to see that if A ⊂ L∞ (µ) is a unital subalgebra with the 2 (µ) satisfies condition (5) hence all the other in Theorem 2.17. (WDP) then HA Proposition 2.19. Let A ⊂ L∞ (µ) be a unital subalgebra. The following are equivalent: (1) A has the (WDP); (2) For every probability measure ν on B which is mutually absolutely continuous 2 w.r.t. µ the Hardy space HA (ν) has the (AFP). Proof. (1) ⇒ (2) Follows immediately from Theorem 2.17. (2) ⇒ (1) We shall use for this purpose Proposition 2.11. Let x ∈ L2 (µ) such that µ(Z(x)) = 0. Let us define the following measure  |x(ζ)|2 dµ(ζ), ω ∈ B. ν(ω) = ω

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Then ν is equivalent to µ and U : L2 (ν) → L2 (µ) defined as U (h) = hx is a 2 unitary operator that takes HA (ν) onto [Ax]. As in the the proof of Lemma 2.9 2 the space [Ax] has the (ALP) (or (AFP)) iff HA (ν) has the same property. Since by assumption, the latter has the (AFP) it follows that the former has it too. We can now apply Proposition 2.11 to deduce that A has the (WDP). This completes the proof.


3. Applications to function algebras There is a large array of examples of probability spaces {Ω, B, µ} and subalgebras A ⊂ L∞ (µ) for which one or the other properties appearing in the statements of Theorems 2.15 and 2.17 hold true. In order to describe some examples, we need to recall some definitions from the theory of uniform algebras. We refer to T. Gamelin s book [29] for (i)-(iii) below. Consider a compact Hausdorff space X and let C(X) be the Banach algebra of all continuous complex valued functions on X. Let also CR (X) be the real part of C(X) and C(X)+ be the cone of all nonnegative functions in C(X). Let A ⊂ C(X) be a closed subalgebra containing the constants. Denote by A−1 the group of all invertible elements of A and let ReA = {Ref : f ∈ A}. Let us point out that all the examples that follow are uniform algebras on X in the sense that they separate the points of X. However, there is no need to assume this fact a priori because it follows directly from their definitions. (i) A is a Dirichlet algebra (on X) if ReA is uniformly dense in CR (X). The standard example is the disc algebra, i.e. the algebra A(D) of all continuous functions on the closed unit disc that are analytic on its interior. It is wellknown that this algebra is Dirichlet when restricted to the unit circle. (ii) A is hypo-Dirichlet on X if the closure of ReA has finite codimension in CR (X) and moreover the real linear span of the set {log |φ| : φ ∈ A−1 } is dense in CR (X). For example, let K ⊂ C be a compact subset in the complex plane and let R(K) be the uniform closure in C(K) of the space of all rational functions with poles off K. If C\K has a finite number of components then R(K) is hypo- Dirichlet when restricted to ∂K. If C\K is connected then R(K) is even a Dirichlet algebra (the Walsh-Lebesgue Theorem). (iii) A is said to be logmodular if the set {log |φ| : φ ∈ A−1 } is uniformly dense in CR (X). Every Dirichlet algebra is logmodular. If m is the Lebesgue measure on the unit circle T and H ∞ (m) is the weak* closure in L∞ (m) of the analytic polynomials then it is known that this algebra is logmodular on the maximal ideal space of L∞ (m) and it is not a Dirichlet algebra on this space. More generally, it is known that the algebra H ∞ (G) of all bounded analytic functions on a bounded planar domain is logmodular on its Shilov boundary (see [30]). (iv) A is said to be approximating in modulus (cf. [31]) if the set {|f |2 : f ∈ A} is uniformly dense in C(X)+ . Every logmodular algebra, in particular

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every Dirichlet algebra (for instance the disc algebra) is approximating in modulus. According to [31] every unital subalgebra A ⊂ C(X) for which the set of unimodular functions in A (i.e. functions of constant modulus one) separates the points of X is approximating in modulus. For instance, the polydisk algebra A(Dk ) that is the uniform closure of all analytic polynomials is approximating in modulus when restricted to the k-torus Tk because its unimodular elements separate the points of Tk . Also it is known that if G ⊂ C is a bounded finitely connected domain with analytic boundary, then the algebra R(G) is approximating in modulus on ∂G ( see [28]). (v) A is said to be convexly approximating in modulus (see Def. 2.18 in [25]) if conv{|f |2 : f ∈ A} is uniformly dense in C(X)+ . It will be seen below that if K ⊂ C is a compact set such that R(K) is hypo-Dirichlet on ∂K then it is also convexly approximating in modulus. We do not know any example of a function algebra which is convexly approximating in modulus and is not approximating in modulus. Example 3.1. It is obvious that if A ⊂ C(X) is a convexly approximating in modulus subalgebra and if µ is any regular probability Borel measure on X then 2 (µ) A has the (WDP). Therefore Theorem 2.17 applies in this case, hence HA has the (AFP). Toeplitz spectral inclusion theorems for such algebras have been previously proved in [36] and [37]. Also it was shown in [39] that if a uniform algebra A ⊂ C(X) satisfies the Toeplitz spectral inclusion appearing in Theorem 2.17 for every Borel regular probability measure µ on X then it is convexly approximating in modulus. Remark 3.2. Consider a function algebra A ⊂ C(X) on some compact Hausdorff space and suppose that µ is the unique representing measure for some character of A. Let H ∞ (µ) be the weak* closure of A in L∞ (µ). It then follows from [35] that the image of H ∞ (µ) through the Gelfand transform of L∞ (µ) is logmodular on the maximal ideal space of L∞ (µ) and this implies that H ∞ (µ) has the (WDP). 2 In particular this shows that Toeplitz operators on HA (µ) with symbol in L∞ (µ) have the spectral inclusion property. This fact has been previously proved in [44]. Remark 3.3. In the case when m is the Lebesgue measure on the unit circle T and A is the disc algebra, assertion (4) in Theorem 2.17 is known as the HartmanWintner Theorem (see [24] for basics on Toeplitz operators on the unit circle). Example 3.4. In the case when A is the polydisk algebra, and mk is the Lebesgue 2 measure on Tk the fact that the corresponding Hardy space HA (mk ) has the (AFP) was proved in [12]. Theorem 2.17 together with the above remark that 2 (µ) has the (AFP) for every Borel A is approximating in modulus show that HA k probability measure µ on T . Example 3.5. It was shown in [22] that the ball algebra, i.e. the uniform closure of analytic polynomials restricted to the unit sphere S 2n−1 in Cn satisfies the Toeplitz spectral inclusion appearing in Theorem 2.17 for every Borel probability

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measure on the sphere. It then follows from the above mentioned result from [39] that this algebra is convexly approximating in modulus, therefore it also has the (WDP) with respect to any Borel probability measure µ on the sphere. This also 2 (µ) has the (AFP) for any such measure. In the particular shows that the space HA 2 case when µ is the area measure on S 2n−1 the fact that HA (µ) has the (AFP) was 2 proved in [12]. Since no nonzero vector in HA (µ) vanishes on a set of positive measure, it follows from Theorem 2.15 that in this particular case every closed 2 (µ) has the (AFP). This result has been previously A-invariant subspace of HA proved in [6]. Example 3.6. In order to describe a class of subalgebras A ⊂ L∞ (µ) with the (WDP), we need to recall the following definition (see [29]). If A ⊂ C(X) is a uniform algebra on a compact Hausdorff space then its Shilov boundary is the smallest compact subset K ⊂ X such that sup |f (λ)| = sup |f (λ)|

λ∈X

λ∈K

for every f ∈ A. For instance, it is well known that if A ⊂ C(X) is logmodular and more generally if it is convexly approximating in modulus then Sh(A) = X. Let {Ω, B, µ} be a probability space and let Y = M(L∞ (µ)) be the maximal ideal space of L∞ (µ). Let A ⊂ L∞ (µ) be a norm closed unital subalgebra. Suppose that A separates Y and that Sh(A) = Y where we identify A with its image through the Gelfand transform of L∞ (µ). Under these conditions, it turns out that A has the (WDP). For this purpose, let ν be a probability measure defined on B which is equivalent to µ. Since the assumption that Sh(A) = M(L∞ (µ)) is invariant under *-isomorphisms it follows that the same holds true with ν instead of µ. Now, an easy adaptation of the proof of Theorem 2.4 from [58] shows that the mapping 2 (ν)) is isometric on L∞ (ν). We can now apply Proposition 2.19 ϕ → Tϕ ∈ B(HA therefore A has the (WDP). Example 3.7. Let A ⊂ C(X) be a hypo-Dirichlet algebra and let ρ be a nonzero character of
A. Then there exists a unique representing measure µ for ρ such that log |ρ(f )| = log |f |dµ for every f ∈ A−1 (such measures are called Arens-Singer ∞ or logmodular representing measures). In this case, it is known that Sh(HA (µ)) = ∞ (µ) stands for the weak* closure of A in L∞ (µ) (see SecM(L∞ (µ)) where HA ∞ tion IV.8 in [29]). It then follows from the previous example that HA (µ) has the (WDP). Toeplitz spectral inclusion for hypo-Dirichlet algebras have been previously proved proved in [38]. Example 3.8. Suppose now that K ⊂ C is a compact set such that R(K) is hypoDirichlet on ∂K. In this case, it was proved in [11] that for every Borel probability 2 (µ) (usually denoted measure µ on ∂K and for A = R(K) the Hardy space HA 2 by R (K, µ)) has the (AFP). It then follows from Theorem 2.17 that the Toeplitz spectral inclusion holds true on this space. As a consequence, it turns out that the algebra R(K) is convexly approximating in modulus when restricted to ∂K. This result holds true in particular when the complement of K has a finite number

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of components. This has been previously proved by W. Mlak (see [2] and the references therein). In the case when K is the closure of a bounded domain D ⊂ C whose boundary consists of a finite number of non-intersecting analytic Jordan curves and µ is the harmonic measure on ∂D, the Toeplitz spectral inclusion was proved in [1]. Example 3.9. Here follows another class of function algebras for which Theorem 2.17 applies, as we shall see below. Recall (see [54]) that if {Ω, B, µ} is a probability space, then a subalgebra A ⊂ L∞ (µ) is said to be a weak* Dirichlet algebra if it satisfies the following conditions: (a) A functions;
contains
the constant
(b) ϕψdµ = ϕdµ · ψdµ for every ϕ, ψ ∈ A; (c) The space {ϕ + ψ : ϕ, ψ ∈ A} is weak* dense in L∞ (µ). It is known (cf. [34]) that every logmodular algebra is a weak* Diriclet subalgebra of L∞ (µ) whenever µ is the (unique) representing measure for an arbitrary character. Most of the basic results of the classical theory of Hardy spaces on the unit circle can be extended in the framework of the weak* Dirichlet algebras (see [29], [35], [34], [54]). Some remarkable examples of weak* Dirichlet algebras are provided by the algebras of bounded analytic functions associated to ergodic flows on probability spaces (see [43], [33], [59]). These include in particular the algebras of generalized analytic functions ([3]). In the latter case, it was proved by H. Helson [32] that every non-reducing invariant subspace contains a unimodular function. Some versions of this theorem have been subsequently obtained for arbitrary weak* Dirichlet algebras (see [45], [56]). Proposition 3.10. Let {Ω, B, µ} be a probability space and let A ⊂ L∞ (µ) be a weak* Dirichlet algebra. If D is a separable Hilbert space and H ⊂ L2 (µ.D) is a closed A- invariant subspace such that the mapping φ → Tφ is faithful on L∞ (µ) then there exists a vector u ∈ H such that u(ζ) = 1 µ-almost everywhere on Ω. In particular this holds true whenever H = [Ax] where x ∈ L2 (µ.D) is a vector such that µ(Z(x)) = 0. ∞ Proof. According to a result from [35] the weak* closure HA (µ) of A in L∞ (µ) is a logmodular algebra on the maximal ideal space of L∞ (µ) which in turn implies ∞ that the set {|ϕ|2 : ϕ ∈ HA (µ)} is uniformly dense in L∞ (µ)+ . This implies, via Theorem 2.15 that H has the (AFP). Moreover by the same theorem there exists a vector u ∈ H such that   2 ϕ|u| dµ = ϕdµ, ϕ ∈ A.

Since {ϕ+ ψ : ϕ, ψ ∈ A} is weak* dense in L∞ (µ) we see that the previous identity
holds true for every ϕ ∈ L∞ (µ). This implies that u(ζ) = 1 a.e. In the case when H is cyclic and dimD = 1 the above result was proved in [56].

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Corollary 3.11. If A ⊂ L∞ (µ) is a weak* Dirichlet algebra then its associated Hardy 2 space HA (µ) has the (AFP) therefore any of the assertions in Theorem 2.17 holds true.

4. Applications to subnormal operators The following result that will be useful in the sequel, is by no means new, however for completeness we include a proof. Lemma 4.1. Let {Ω, B, µ} be a separable probability space, let H be a separable Hilbert space and let Φ : L∞ (µ) → B(H) be a unital, weak* continuous and positive linear mapping. Then there exist a separable Hilbert space D and an isometry V : H → L2 (µ.D) such that Φ(ϕ) = V ∗ Mϕ V for every ϕ ∈ L∞ (µ), where Mϕ stands for the multiplication operator on L2 (µ.D) with symbol ϕ. Proof. Let π : L∞ (µ) → B(K) be the minimal Stinespring dilation of Φ (cf. [55]). This means that K is a Hilbert space such that H ⊂ K, π : L∞ (µ) → B(K) is a *-representation and Φ(ϕ) = PH π(ϕ)PH ∞ for every ϕ ∈ L (µ). Moreover, K is the smallest π(L∞ (µ))-invariant subspace containing H. Since the measure space is separable and H is separable and since π is minimal, it follows that K is a separable Hilbert space. Moreover, π is clearly weak* continuous, a fact that follows from minimality. It now follows from Proposition 2.7.4 in [51] that there exist a separable Hilbert space D and an isometry U : K → L2 (µ.D) such that Mϕ U = U π(ϕ) for every ϕ ∈ L∞ (µ). If V : H → L2 (µ.D) denotes the restriction of U to H then obviously Φ(ϕ) = V ∗ Mϕ V for every ϕ ∈ L∞ (µ).
We are now ready to prove the following result: Proposition 4.2. Let {Ω, B, µ} be a separable probability space, let H be a separable Hilbert space and let Φ : L∞ (µ) → B(H) be a unital, weak* continuous and positive linear mapping. Let A = {ϕ ∈ L∞ (µ) : Φ(ϕ)∗ Φ(ϕ) = Φ(|ϕ|2 )}. If Φ is isometric on L∞ (µ) then Φ(A) ⊂ B(H) is a dual algebra with property (A1 (1)). Proof. First, we recall (see [19]) that A can be equivalently described as A = {ϕ ∈ L∞ (µ) : Φ(ψϕ) = Φ(ψ)Φ(ϕ) for every ψ ∈ L∞ (µ)}. This shows that A is a weak* closed subalgebra of L∞ (µ) and that Φ is multiplicative on A. From Lemma 4.1 it follows that we can assume that there exists a separable Hilbert space D such that H ⊂ L2 (µ.D) and such that Φ(ϕ) = Tϕ for every ϕ ∈ L∞ (µ) where Tϕ is the Toeplitz operator on H (see Definition 2.7). Since Φ is isometric on L∞ (µ), it follows from Lemma 2.8 that the space H has the (ALP). Since the measure space is separable, H has also the (AFP) by Theorem 2.4. Now it can be easily seen that H is invariant for all operators T on L2 (µ.D) of the form T = Mϕ with ϕ ∈ A. Since the space H has the (AFP) it follows from Lemma 2.14

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that Φ(A) is a weak* closed algebra with property (A1 (1)). This finishes the proof of this proposition.
The following is an immediate consequence of Lemma 4.1, Theorem 2.15 an Proposition 4.2. Corollary 4.3. Let {Ω, B, µ} be a separable probability space, let H be a separable Hilbert space and let Φ : L∞ (µ) → B(H) be a unital, weak* continuous and positive linear mapping. Let A = {ϕ ∈ L∞ (µ) : Φ(ϕ)∗ Φ(ϕ) = Φ(|ϕ|2 )}. If Φ is faithful on L∞ (µ) and if A has the (WDP) then Φ is isometric on L∞ (µ) and the dual algebra Φ(A) has property (A1 (1)). Suppose now that S is a subnormal operator acting on a separable Hilbert space H (see [20] for a comprehensive treatment of these operators). Let N ∈ B(K) be its minimal normal extension. Let also µ be a scalar valued spectral measure for N and let π : L∞ (µ) → B(K) be the functional calculus for N . Then it can be easily shown that the map Φ : L∞ (µ) → B(H) defined by Φ(ϕ) = PH π(ϕ)PH for ϕ ∈ L∞ (µ) is faithful. This mapping has been studied before (see for instance [40] and [48]). In particular, [40] contains some equivalent characterizations of the cases in which this mapping is isometric. It can be easily seen that one of these conditions is precisely the approximate localization property (in a different formulation). Let R(N, H) = {ϕ ∈ L∞ (µ) : π(ϕ)H ⊂ H}. This is a weak* closed subalgebra of L∞ (µ) and it is called the restriction algebra. Moreover, Φ is isometric on R(N, H) (cf. [20]). It is also easy to see that R(N, H) = {ϕ ∈ L∞ (µ) : Φ(ϕ)∗ Φ(ϕ) = Φ(|ϕ|2 )}. The restriction algebra contains the space R∞ (σ(S), µ) that is the weak* closure in L∞ (µ) of R(σ(S)). In particular it contains the space P ∞ (µ) which is the weak* closure in L∞ (µ) of analytic polynomials. The following result is an immediate consequence of Corollary 4.3: Corollary 4.4. Suppose that S is a subnormal operator acting on a separable Hilbert space H and let N ∈ B(K) be its minimal normal extension and let µ be a scalar valued spectral measure for N . Let π : L∞ (µ) → B(K) be the functional calculus for N . Let Φ : L∞ (µ) → B(H) be the mapping defined by Φ(ϕ) = PH π(ϕ)PH for ϕ ∈ L∞ (µ). If Φ is isometric on L∞ (µ) (in particular if the restriction algebra R(N, H) ⊂ L∞ (µ) has the (WDP)) then Φ(R(N, H)) is a dual algebra with property (A1 (1)). The fact that the dual algebra generated by a subnormal operator has property (A1 (r)) for some r ≥ 1 was proved in [47].

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Definition 4.5. A sequence {Tn }n≥1 of commuting bounded operators on some complex Hilbert space H is said to be a spherical isometry if ∞ 

Tn∗ Tn = IH

n=1

in the weak operator topology, where IH stands for the identity operator on H. The following result, which is a particular case of a more general theorem from [50] will be used in the proof of Theorem 4.7 below. Lemma 4.6. Let {Tn }n≥1 be a spherical isometry on a complex Hilbert space H. There exist a Hilbert space K ⊃ H and a sequence {Nn }n≥1 of commuting normal operators on K such that {Nn }n≥1 is a spherical isometry and such that Nn H ⊂ H and Nn h = Tn h for all h ∈ H and all n ≥ 1. Moreover, if the normal extension is minimal, then the mapping Y → PH Y PH is an isometric isomorphism from the commutant of {Nn }n≥1 onto the set {X ∈ B(H) :

∞ 

Tn∗ XTn = X}.

n=1

Here PH denotes the orthogonal projection of K onto H. In particular the above mapping induces a dual algebras isomorphism between the dual algebra AN generated by {Nn }n≥1 and the dual algebra AT generated by {Tn }n≥1 . Theorem 4.7. Let {Tn }n≥1 be a spherical isometry on a separable Hilbert space H. Then the dual algebra generated by the set {Tn }n≥1 has property (A1 (1)). Proof. Let {Nn }n≥1 on K be the minimal normal extension of {Tn }n≥1 as in Lemma 4.6. Since K is a separable Hilbert space, there exist a separable probability measure space {Ω, B, µ} and a normal and isometric *-representation π : L∞ (µ) → B(K) whose image coincides with the von Neumann algebra W ∗ (N ) generated by {Nn }n≥1 in B(K). Let Φ : L∞ (µ) → B(H) be defined as Φ(f ) = PH π(f )PH for all f ∈ L∞ (µ). Then Φ is isometric, unital and weak* continuous. Moreover, if f ∈ L∞ (µ) is such that π(f ) ∈ AN then Φ(f )∗ Φ(f ) = Φ(|f |2 ). This is because H is invariant for π(f ). It then follows from Proposition 4.2 that AT is a subalgebra of a dual algebra with property (A1 (1)) hence AT itself has indeed the property (A1 (1)). This completes the proof.
For spherical isometries with a finite number of components, the result in Theorem 4.7 was proved in [26], see also [27]. It was also proved in [23] that for this class of spherical isometries the dual algebra AT is reflexive. As far as we know, it is an open question whether the latter result can be extended to spherical isometries with an infinite number of components.

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5. P ∞ (µ) spaces with the weak* denseness property In this section, by a measure we shall mean a compactly supported positive finite Borel measure in the complex plane. In what follows we shall provide an equivalent characterization of measures µ for which the algebra P ∞ (µ) has the (WDP). For this purpose we shall make use of the analytic structure of the space P ∞ (µ) that has been uncovered by D. Sarason in [53]. Suppose that µ is a positive, finite Borel measure in the complex plane with compact support supp(µ). It is then constructed in [53], by means of a transfinite ˜ µ ˜ (possibly induction process, a pair {K, ˜ } consisting of a compact planar set K ˜ having the following properties: empty) and a measure µ ˜ supported on K (1) µ − µ ˜⊥µ ˜; (2) P ∞ (µ) = L∞ (µ − µ ˜) ⊕ P ∞ (˜ µ); ˜ ˜ ˜ equals the closure of its (3) If K is nonempty then R(K) is a Dirichlet algebra, K ∞ ˜ interior, R(K) ⊂ P (˜ µ), and µ ˜ is the sum between µ|IntK˜ and the absolutely ˜ continuous part of µ|∂ K˜ w.r.t. the harmonic measure on ∂ K; ˜ = ∅ then IntK ˜ = ∅ and there exists a Banach (4) Assuming again that K ˜ → P ∞ (˜ µ) such that Φ(f ) = f for algebras isomorphism Φ : H ∞ (IntK) ˜ every f ∈ R(K). Moreover Φ is also a homeomorphism when both algebras are endowed with their weak* topologies. ˜ is called the Sarason hull of µ. Since R(K) ˜ is Dirichlet, The set Σ(µ) = IntK ˜ one knows that every component U of IntK is simply connected. Moreover, any conformal mapping φ of the unit disc onto U is a weak* generator of H ∞ (D) in the sense that the polynomials in φ are weak* dense in H ∞ (D). Simply connected regions that are images of weak* generators of H ∞ (D) have been described in [52]. For our purposes, we also need to recall two more results about the algebra P ∞ (µ). The first one provides a decomposition into antisymmetric summands: Theorem 5.1 (see [21] and [20]). Let µ be a measure, and let U1 , U2 , . . . be the connected components of its Sarason hull Σ(µ) (provided the latter is nonempty). Then µ can be decomposed as µ = µ0 + µ1 + . . . where µj are mutually singular measures with the following properties: (1) For every j ≥ 1 µ0 (Uj ) = 0, supp(µj ) ⊂ U j and µj restricted to ∂Uj is absolutely continuous w.r.t. the harmonic measure of Uj ; (2) P ∞ (µ) = L∞ (µ0 ) ⊕ (⊕j≥1 P ∞ (µj )); (3) For each j ≥ 1 the identity map extends to a dual algebras isomorphism between P ∞ (µj ) and H ∞ (Uj ); (4) If φj : D → Uj is any conformal mapping then φj is a weak* generator of H ∞ (D).

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Theorem 5.2 ([46]). Let µ be a measure such that P ∞ (µ) is antisymmetric (that is it does not contain non constant real functions) and let G = Σ(µ) be its Sarason hull. Then the following are equivalent: (1) µ(G) = 0; (2) If N is a normal operator on a separable Hilbert space K such that µ is a scalar valued spectral measure for N and if S is a subnormal operator for which N is the minimal normal extension, then every operator commuting with S can be extended to an operator commuting with N . In order to use these results we also need the following: Proposition 5.3. Let S ∈ B(H) be a subnormal operator on a separable Hilbert space and let N ∈ B(K) be its minimal normal extension. Let µ be a scalarvalued spectral measure for N . Suppose that the space P ∞ (µ) has the (WDP). Let X ∈ B(H) such that XS = SX. Then there exists a unique operator Y ∈ B(K) such that Y N = N Y , Y H ⊂ H and Y |H = X. Proof. For each h ∈ H let gh,h ∈ L1 (µ) such that  (f (N )h, h) = f gh,hdµ for every f ∈ L∞ (µ). Assume that X = 1. We will show first that f (N )Xh ≤ f (N )h for every f ∈ L∞ (µ) and every h ∈ H. Indeed this holds true for all f ∈ P ∞ (µ) which means that   2 |f | gXh,Xh dµ ≤ |f |2 gh,h dµ, f ∈ P ∞ (µ), h ∈ H. Since P ∞ (µ) has the (WDP) the above holds true for all f ∈ L∞ (µ). Now the rest of the proof is quite straightforward and goes as that of Lemma 4.1 in [42] (see also [41] for the case of subnormal representations of approximating in modulus function algebras). The uniqueness of the extension follows from the minimality together with the Fuglede-Putnam Theorem.
The main result of this section is the following: Theorem 5.4. Let µ be a compactly supported positive finite Borel measure in the plane and let G be its Sarason hull. Then the following are equivalent: (1) The space P ∞ (µ) has the (WDP); (2) µ(G) = 0. Proof. (1) ⇒ (2) In the case when P ∞ (µ) is antisymmetric this implication follows immediately from Theorem 5.2 together with Proposition 5.3. The general case follows now from the antisymmetric one and Theorem 5.1 and the fact that P ∞ (µ) has the (WDP) if and only if all its antisymmetric summands have the (WDP). ˜ be ˜. Let K (2) ⇒ (1) We may assume that P ∞ (µ) is pure, that is µ = µ ˜ the compact set appearing in Sarason’s Theorem for which G = Int(K). Since ˜ Because R(K) ˜ is Dirichlet on ∂ K ˜ it is in µ(G) = 0 we see that supp(µ) ⊂ ∂ K.

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particular approximating in modulus, therefore its weak* closure in L∞ (µ) has the ˜ ⊂ P ∞ (µ) we see that P ∞ (µ) has the (WDP) as well.
(WDP). Since R(K) The following result is proved in [13], see also [46] for a different proof: Theorem 5.5 (cf. [13]). Suppose µ is a measure such that P ∞ (µ) is antisymmetric and let G = Σ(µ). Then the following are equivalent: (1) The space {p + q : p, q analytic polynomials} is weak* dense in L∞ (µ); (2) µ(G) = 0. Using this theorem together with Theorem 5.4 and Theorem 5.1 one can now easily prove the following: Theorem 5.6. Let µ be a measure. Then P ∞ (µ) has the (WDP) if and only if the linear manifold {p + q : p, q polynomials} is weak* dense in L∞ (µ). Proof. We leave this proof to the reader.




6. P 2 (µ) spaces with the approximate factorization property Let µ be a positive finite Borel measure in the complex plane with compact support. Let P 2 (µ) denote the closure in L2 (µ) of all analytic polynomials. In this section we give an equivalent description of those measures µ for which the space P 2 (µ) has the (AFP). The fundamental result on the analytic structure of the spaces P 2 (µ) has been obtained by J.E. Thomson in [57]. We first recall a couple of definitions related to bounded point evaluations (see Sec. II.7 in [20]). A point λ ∈ C is said to be a bounded point evaluation for P 2 (µ) if the mapping p → p(λ) defined on polynomials extends to a continuous linear functional on P 2 (µ). In this case, for each f ∈ P 2 (µ) one denotes by fˆ(λ) the value of this extension applied to f . The set of all bounded point evaluations for P 2 (µ) is denoted by bpe(P 2 (µ)). A point λ ∈ C is called an analytic bounded point evaluation for P 2 (µ) if there exists an open disc U around λ such that U ⊂ bpe(P 2 (µ)) and moreover such that for each f ∈ P 2 (µ) the mapping ζ → fˆ(ζ) is analytic on U . The set of all analytic bounded point evaluations for P 2 (µ) is denoted by abpe(P 2 (µ)). Let Nµ denote the multiplication by z on L2 (µ) and let Sµ denote its restriction to P 2 (µ). The space P 2 (µ) is pure if it does not contain nontrivial L2 summand. The space P 2 (µ) is said to be irreducible if it does not contain nontrivial characteristic functions or, equivalently, if Sµ does not have nontrivial reducing subspaces. If G ⊂ C is a bounded open set then H ∞ (G) denotes the Banach algebra of all bounded analytic functions on G. The following theorem summarizes some of the basic results from [57] that will be needed in the sequel. These results hold true for any exponent t ∈ [1, ∞). Here we only consider the case t = 2.

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Theorem 6.1 (cf. [57]). Let µ be a compactly supported positive finite Borel measure in the complex plane and let supp(µ) denote its support. Then there exists a Borel partition {∆n }n≥0 of supp(µ) such that the space P 2 (µ) admits the direct sum decomposition P 2 (µ) = L2 (µ0 ) ⊕ (⊕n≥1 P 2 (µn )) (where µn is the restriction of µ to ∆n ) such that for each n ≥ 1 the space P 2 (µn ) has the following properties: (1) P 2 (µn ) is irreducible; (2) If Wn = abpe(P 2 (µn )) then Wn is a simply connected region and its closure contains ∆n ; (3) The mapping f → fˆ is one-to-one on P 2 (µn ). Moreover, f = fˆ µn -a.e. on Wn for every f ∈ P 2 (µn ); (4) The mapping f → fˆ implements an isometric isomorphism and a weak* homeomorphism between the Banach algebras P 2 (µn )∩L∞ (µn ) and H ∞ (Wn ). The next lemma reduces the study of (AFP) to irreducible P 2 (µ) spaces. Lemma 6.2. Let µ be a measure and let P 2 (µ) = L2 (µ0 ) ⊕ (⊕n≥1 P 2 (µn )) be its decomposition as in Theorem 6.1. Then the following are equivalent: (1) P 2 (µ) has the (AFP); (2) P 2 (µn ) has the (AFP) for every n ≥ 1.


f |p|2 dµn ≥ 0 for every Proof. (1) ⇒ (2) Let n ≥ 1 and let f ∈ L∞ R (µn ) such that polynomial p. Extend f with 0 on the complement of ∆n . Call g this extension.
Then g|p|2 dµ ≥ 0 for every polynomial p. Since P 2 (µ) has the (AFP) it follows from Theorem 2.17 that g ≥ 0 µ-a.e. therefore f ≥ 0 µn -a.e. It now follows again from the same theorem that the space P 2 (µn ) has the (AFP). (2) ⇒ (1) Suppose now that every space P 2 (µn ) has the (AFP). Let ∆ = supp(µ) and let ω ⊂ ∆ with µ(ω) > 0. Let ωn = ω ∩ ∆n for n ≥ 0. Let ε > 0. For each n ≥ 1 choose a nonzero vector xn ∈ P 2 (µn ) with xn  < 2−n such that   |xn |2 dµn ≤ ε |xn |2 dµn . ∆n \ωn

ωn

Since L2 (µ0 ) has obviously the (AFP) we may also find (provided L2 (µ0 ) = 0) a nonzero vector x0 ∈ L2 (µ0 ) having the same property above for n = 0. Let x = ⊕n≥0 xn . Then x ∈ P 2 (µ) and it is clear that   2 |x| dµ ≤ ε |x|2 dµ. ∆\ω

ω

This finishes the proof of this lemma. Our main result on P 2 (µ) spaces with the (AFP) is the following:




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Theorem 6.3. Let µ be a compactly supported Borel probability measure in the plane. Let G be the set of all analytic bounded point evaluations for the space P 2 (µ). Then the following are equivalent: (1) The space P 2 (µ) has the (AFP); (2) µ(G) = 0. Proof. We first assume that P 2 (µ) is irreducible. (1) ⇒ (2). Assume on the contrary that µ(G) > 0. Let K ⊂ G be a compact subset such that µ(K) > 0 and let χK ∈ L1 (µ) be the characteristic function of K. Then it follows from Theorem 2.6 that for each n ≥ 1 there exists xn ∈ P 2 (µ) such that |xn (ζ)|2 ≥ χK (ζ) µ-a.e. and moreover
xn 2 ≤ µ(K) + 1/n. In particular it follows that |xn (ζ)| ≥ 1 a.e. on K and G\K |xn (ζ)|2 dµ(ζ) ≤ 1/n for n ≥ 1. We may assume by passing to a subsequence that xn → y weakly in P 2 (µ(. It
then follows easily that G\K |y(ζ)|2 dµ(ζ) = 0 therefore y(ζ) = 0 µ-a.e. on G\K therefore yˆ = 0 µ-a.e. on G\K. On the other hand, since K ⊂ abpe(P 2 (µ)) we have that | xn (ζ)| ≥ 1 µ-a.e. on K. Since xn → y weakly in P 2 (µ) we have that limn→∞ xˆn (ζ) = yˆ(ζ) for every ζ ∈ G therefore | y (ζ)| ≥ 1 µ-a.e. on K. Since yˆ is analytic on G it follows that µ restricted to G\K is discrete. Since this holds true for every compact subset K ⊂ G with µ(K) > 0 we see that µ restricted to G is a discrete measure. In particular if K is a singleton, the previous argument shows that P 2 (µ) contains a nontrivial characteristic function, and this contradicts the fact that P 2 (µ) is irreducible. The conclusion is that µ(G) = 0. (2) ⇒ (1). Suppose now that µ(G) = 0. Let ψ : G → D be a conformal mapping onto the unit disc. Denote by ψ˜ ∈ P 2 (µ) ∩ L∞ (µ) the preimage of ψ under the isomorphism appearing in Theorem 6.1. It then follows from Lemma ˜ 2.2 in [49] that |ψ(ζ)| = 1 µ-a.e. on ∂G. Since the polynomials are weak* dense in H ∞ (D) it follows that the unital subalgebra generated by ψ is weak* dense in H ∞ (G). It then follows that the selfadjoint subalgebra A of L∞ (µ) generated by ψ˜ is weak* dense in L∞ (µ) hence also dense in L∞ (µ) with respect to the strong operator topology on B(L2 (µ)). Now every element h ∈ A can be written as h = uf where u, f ∈ P 2 (µ) ∩ L∞ (µ) and |u| = 1 µ-a.e. on ∂G. Since A is strongly dense in L∞ (µ) it then follows that the set {|f |2 : f ∈ P 2 (µ) ∩ L∞ (µ)} is weak* dense in L∞ (µ)+ . This shows that the algebra P 2 (µ) ∩ L∞ (µ) has the (WDP) hence the space P 2 (µ) has the (AFP) by Proposition 2.19. The proof of this theorem is thus completed in the case when P 2 (µ) is irreducible. The general case follows using Lemma 6.2 and the fact (see [57]) that abpe(P 2 (µ)) = ∪n≥1 abpe(P 2 (µn )).


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auser Verlag Basel/Switzerland 0378-620X/010147-2, published online April 18, 2008 DOI 10.1007/s00020-008-1591-8

Integral Equations and Operator Theory

Addendum to “The Extremal Truncated Moment Problem” Ra´ ul E. Curto, Lawrence A. Fialkow and H. Michael M¨oller Abstract. Due to a technical problem, we accidentally omitted a paragraph from the proof of one of the main results in [1]. In this Addendum we provide the portion of the proof that did not appear in [1]. Mathematics Subject Classification (2000). Primary 47A57, 44A60, 42A70, 30E05; Secondary 15A57, 15-04, 47N40, 47A20. Keywords. Extremal truncated moment problems, moment matrix extension, Riesz functional, real ideals, affine Hilbert function, Hilbert polynomial of a real ideal.

Due to a technical problem, we accidentally omitted a paragraph from the proof of one of the main results. In this Addendum we provide the portion of the proof that did not appear in [1]. For the sake of completeness, we restate Theorem 2.8. Theorem 2.8. For β ≡ β (2n) extremal, the following are equivalent: (i) β has a representing measure; (ii) β has a unique representing measure, which is rank M(n)-atomic; (iii) For some (respectively, for every) basis B of CM(n) , V ≡ VB [V] is invertible and µB is a representing measure for β; (iv) M(n) ≥ 0 and for some (respectively, for every) basis B of CM(n) , V ≡ VB [V] is invertible and µB is an interpolating measure for β; (v) β is consistent and M(n) ≥ 0; (vi) M(n) ≥ 0 has a flat extension M(n + 1); (vii) M(n) ≥ 0 has a unique flat extension M(n + 1). Proof of (iv) =⇒ (iii). We need to establish that the interpolating measure µB is a representing measure. To do this, it suffices to show that µB ≥ 0. For 1 ≤ k ≤ r, let Vk ≡ Vk (x) denote the matrix obtained from V by replacing wk (in column k) by the variable x, and let fk ∈ Pn be defined by fk (x) := det Vk (x). Clearly,

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fk (wj ) = δkj det V (1 ≤ k, j ≤ r). Now  
2 ˆ ˆ 0 ≤ M(n)fk , fk = Λβ (fk ) = fk2 dµB (since µB is interpolating) =

r 

ρj fk2 (wj ) = ρk (det V )2 ,

j=1

and since det V = 0, it follows that ρk ≥ 0. (Since card supp µB = r, it then follows that ρk > 0 (1 ≤ k ≤ r).) This establishes (iii).


References [1] R. Curto, L. Fialkow and M. M¨ oller, The extremal truncated moment problem, Integral Equations Operator Theory 60 (2008), 177–200. Ra´ ul E. Curto Department of Mathematics The University of Iowa Iowa City, IA 52242-1419 USA e-mail: [email protected] Lawrence A. Fialkow Department of Computer Science State University of New York New Paltz, NY 12561 USA e-mail: [email protected] H. Michael M¨ oller FB Mathematik der Universit¨ at Dortmund 44221 Dortmund Germany e-mail: [email protected]

Integr. equ. oper. theory 61 (2008), 149–158 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/020149-10, published online March 11, 2008 DOI 10.1007/s00020-008-1578-5

Integral Equations and Operator Theory

Doubly-Invariant Subspaces for the Shift on the Vector-Valued Sobolev Spaces of the Disc and Annulus I. Chalendar and J. R. Partington Abstract. This paper characterizes the closed doubly shift-invariant subspaces of the Sobolev space of vector-valued functions defined on the unit circle. Partial results are obtained on the analogous problem for the Sobolev space on the boundary of the annulus. Mathematics Subject Classification (2000). Primary 46E20, 47B38; Secondary 30D55, 47A15. Keywords. Shift operator, Sobolev space.

1. Introduction The purpose of this paper is to study the shift operator S (multiplication by the independent variable) on certain Sobolev spaces, of which formal definitions will be given later in the paper. These spaces consist of vector-valued functions on the boundary of the annulus A = {z ∈ C : r0 < |z| < 1}, where r0 is a positive real number less than unity. In particular, we are interested in the invariant subspaces of S. We consider also the Hardy–Sobolev spaces of vector-valued functions on the annulus. In the case of scalar-valued Hardy spaces on the annulus there has been much significant work [14, 6, 12, 15, 1], and some extensions to the vector-valued case were recently obtained in [2]. However, as a preliminary step, we need also to study certain spaces of functions defined on the unit disc in C. The function space W 1,2 (T) denotes the set of functions f defined on T = {z ∈ C
: |z| = 1} for which there exists a sequence (an )n∈Z in C satisfying
f (eit ) = n∈Z an eint and f 2W 1,2 (T) := |a0 |2 + n∈Z n2 |an |2 < ∞. The functions f in W 1,2 (T) have their derivatives with respect to t, denoted by ft , in L2 (T) and

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W 1,2 (T) is a Hilbert space endowed with the scalar product defined by  2π 1 f, gW 1,2 (T) = ft (eit )gt (eit ) dt + f(0) g(0). 2π 0 The Hardy–Sobolev space H 1,2 (D) is the subspace of W 1,2 (T) consisting of functions whose Poisson extension to the disc D = {z ∈ C : |z| < 1}
is analytic; ∞ n equivalently, it may be regarded as the space of Taylor series f : z →  n=0 an z
∞ 2 2 2 2 for which the norm given by f  = |a0 | + n=1 n |an | is finite. For 0 < r0 < 1 we  \ r0 D), consisting of functions with an shall also be interested in the space H 1,2 (C
0  analytic extension to C\r0 D, represented by a Laurent series f : z → n=−∞ an z n with finite norm given by f 2 = |a0 |2 +

−1  n=−∞

n2 r02n |an |2 .

Denote by S the operator of multiplication by z on W 1,2 (T). Clearly S is invertible and S −1 f (eit ) = e−it f (eit ) for f ∈ W 1,2 (T). A closed subspace M in W 1,2 (T) is said to be invariant for S if SM ⊂ M, doubly invariant for S if M is both invariant for S and S −1 and reducing for S if M is both invariant for S and S ∗ . For f ∈ W 1,2 (T), 1. IS [f ] will denote the smallest closed subspace M in W 1,2 (T) containing f and invariant for S. 2. DS [f ] will denote the smallest closed subspace M in W 1,2 (T) containing f and doubly invariant for S. 3. RS [f ] will denote the smallest closed subspace M in W 1,2 (T) containing f and reducing for S. Let E be a closed subset of T. Denote by MW 1,2 (T) (E) the closed doubly-invariant subspace of W 1,2 (T) defined by MW 1,2 (T) (E) = {f ∈ W 1,2 (T) : f|E = 0}. Such subspaces are called spectral. It can happen that spectral subspaces completely characterize the doubly-invariant subspaces, as in the Wiener theorem asserting that a every doubly-invariant subspace of L2 (T) has the form χE L2 (T) for some measurable set E ⊂ T (see [5, 10, 11]). We employ an analogous notation for subspaces of W 1,2 (T, Cm ); in general we use lower case letters for scalar functions and capital letters for vector-valued functions. In particular,     (f1 )t f1     for F =  ...  in W 1,2 (∂A, Cm ), (F )t denotes  ... . fm

(fm )t

The structure of this paper is as follows. In Section 2 we characterize the doubly-invariant subspaces of W 1,2 (T, Cm ), first in the known case m = 1, and

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then for general m. The case when m = 2 and the subspace is the graph of an operator is of particular importance, as this enables a characterization of unbounded shift-invariant operators. The Sobolev space (and Hardy–Sobolev space) of the annulus are much harder to analyse: in Section 3 we show that under some circumstances the doubly-invariant subspaces are spectral as in the case of the disc (i.e., determined by the set on which the functions in the subspace vanish), although in general this is not the case.

2. Doubly-invariant subspaces in W 1,2 (T, Cm ) 2.1. Scalar case The next result was announced in [8], having been obtained by a different method based on the approximation of the functions in the Sobolev space defined on the real axis. We give a direct proof, which we later apply to the vectorial case. Some related results, in the context of spectral synthesis, have been given by Netrusov (see [9, 4]). Proposition 2.1. Let K be a closed doubly-invariant subspace in W 1,2 (T). Then there exists a function φ ∈ W 1,2 (T) such that K = DS (φ) and moreover K = {f ∈ W 1,2 (T) : f|Z(φ) = 0}, where Z(φ) = {eit ∈ T : φ(eit ) = 0}. In other words the closed doubly-invariant subspace in W 1,2 (T) are spectral. Proof. Let PK denote the orthogonal projection from W 1,2 (T) onto K, and take φ = PK 1, where 1 is the function identically equal to 1 on T. Obviously, if φ = 1, then K = W 1,2 (T). Suppose now that φ = 1. Then it follows that φW 1,2 (T) = 1 and thus  = 0. 1 − φ(0) (2.1) Since φ = PK 1, φ ∈ K and 1 − φ is orthogonal to K. Since K is doubly-invariant it follows that (2.2) S n φ, 1 − φW 1,2 (T) = 0, n ∈ Z. Now, (2.2) means that  2π  int it int it it (ine φ(e ) + e φt (e ))(−φt (e )) dt + 0

0

2π

 φ(eit )eint (1 − φ(0)) dt = 0,

for all n ∈ Z, and then we get:  0

2π

−ine

int

it

φ(e )φt

 (eit ) dt

=

2π

e 0

int

it

φt (e )φt

 (eit ) dt−

2π

 φ(eit )eint (1− φ(0)) dt

0

for all n ∈ Z. It follows that φφt is absolutely continuous and  (φφt )t = φt φt − φ(1 − φ(0).

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Since (φt )t = −φtt , we then obtain,  −φφtt = −φ(1 − φ(0)).

(2.3)

 = 0 on T \ Z(φ). Combining (2.3) and (2.1) we have φtt = 1 − φ(0) Thus the circle can be written as the union of a closed set on which φ = 0 and a countable union of intervals, on each of which φ has constant second derivative. In order to check that K = DS (φ), take f ∈ K orthogonal to DS (φ). Then S n φ, f W 1,2 (T) = 0, n ∈ Z.

(2.4)

Moreover, since f ∈ K, it follows that S n f, 1 − φW 1,2 (T) = 0, n ∈ Z.

(2.5)

1,2

In the sequel we will show that if f ∈ W (T) satisfies (2.4) and (2.5), then f is identically equal to 0. Using (2.4) we get:  2π  2π (ineint φ(eit ) + eint φt (eit ))ft (eit ) dt + φ(eit )eint f(0) dt = 0, n ∈ Z, 0

0

and then, for all n ∈ Z, we get:   2π −ineint φ(eit )ft (eit ) dt = 0

2π 0

eint (φt (eit )ft (eit ) + φ(eit )f(0)) dt.

It follows that φft is absolutely continuous and (φft )t = φt ft + φf(0). We then obtain, φftt = φf(0), which means that ftt = f(0) on T \ Z(φ).

(2.6)

Using a similar calculation, (2.5) implies that f φt is absolutely continuous and  = 0. f (φtt − (1 − φ(0)))

(2.7)

We claim now that f = 0 on Z(φ). Recall that on each “component interval”,  = (a, b), forming a component of the open set T\Z(φ), we may write φtt = 1− φ(0) c = 0, say (a.e.), and so, integrating and recalling that φ vanishes at the end points, we find that c for a ≤ t ≤ b. φ(eit ) = (t − a)(t − b) 2 Note that φt (a+) = −c(b − a)/2 and φt (b−) = c(b − a)/2. Take z ∈ Z(φ). If z ∈ int Zφ , then φtt = 0 on an interval including z and so f (z) = 0 by continuity and (2.7). Otherwise, suppose now that z ∈ ∂Z(φ) and f (z) = 0, so f (eit ) = 0 on some nontrivial interval (z − δ, z + δ). Now f φt is known to be absolutely continuous, and the condition on f implies that φt must be continuous on (z − δ, z + δ). But this is absurd, since there is either a lefthand endpoint a of a component interval (a, b) lying in [z, z + δ) or a right-hand endpoint b lying in (z − δ, z]. Suppose the first case (the second is similar): then

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φt (a+) = −c(b − a)/2; on the other hand, either there is an interval (a − η, a] ⊆ Z(φ) (so that φt (a−) = 0 and φt is discontinuous at a) or there is a sequence of component intervals (an , bn ) with bn → a− (possibly bn = a for each n). In this case φt (bn −) = c(bn − an )/2, which cannot converge to −c(b − a)/2. Moreover, by (2.6), ftt equals a unique constant on T\Z(φ), and since φtt has the same property, there exists λ ∈ C such that f = λφ. Finally φ, λφW 1,2 (T) = 0 so λ = 0. Therefore f is identically equal to 0 and thus K = DS (φ). It remains to check that K = M(Z(φ))(= {f ∈ W 1,2 (T) : f|Z(φ) = 0}). Obviously, K is a subset of M(Z(φ)). In order to prove the converse inclusion, take f ∈ M(Z(φ)) satisfying (2.4). Then f|Z(φ) = 0 and ftt = f(0) on T \ Z(φ) (see (2.6). It follows that there exists λ ∈ C such that f = λφ and as previously φ, λφW 1,2 (T) = 0 implies that λ = 0. Therefore f is identically equal to 0 and thus K = M(DS (φ)).
2.2. Vectorial case Let m be a positive integer. Theorem 2.2. Let K be a closed doubly-invariant subspace in W 1,2 (T, Cm ). Then, for almost all eiw ∈ T, there exists a subspace I(eiw ) of Cm such that K = {F ∈ W 1,2 (T, Cm ) : F (eiw ) ∈ I(eiw )}. Proof. Let PK : W 1,2 (T, Cm ) → K denote the orthogonal projection onto K. Let {e1 , . . . , em } denote the canonical basis of Cm . For r ∈ Z and 1 ≤ k ≤ m, let Ψr,k denote the vector-valued function defined on T by z −→ PK z r ek . For each w ∈ T, let I(eiw ) denote the linear span of the set of vectors {Ψr,k (eiw ) : r ∈ Z, 1 ≤ k ≤ m}. Note that I(eiw ) is closed as a subspace of finite dimension. Define P (eiw ) to be the orthogonal projection from Cm onto I(eiw ). It is clear that K ⊂ {F ∈ W 1,2 (T, Cm ) : F (eiw ) ∈ I(eiw )} and that {F ∈ W 1,2 (T, Cm ) : F (eiw ) ∈ I(eiw )} is both closed and doublyinvariant. If the two spaces were unequal, then there would exist F ∈ W 1,2 (T, Cm ) such that F (eiw ) ∈ I(eiw ) and with F orthogonal to S n Ψr,k for all n, r ∈ Z and k ∈ {1, . . . , m}. Thus we get, for all n, r ∈ Z and k ∈ {1, . . . , m}:  2π  2π (eint Ψr,k (eit ))t , Ft (eit )Cm dt + (eint Ψr,k (eit )), F(0)Cm dt = 0. 0

0

It follows that for all n, r ∈ Z and k ∈ {1, . . . , m}:  2π eint (inΨr,k (eit ), Ft (eit )Cm + (Ψr,k (eit ))t , Ft (eit )Cm 0

+ Ψr,k (eit ), F(0)Cm ) dt = 0.

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So Ψr,k , Ft  is absolutely continuous and (Ψr,k , Ft )t = (Ψr,k )t , Ft  + Ψr,k , F (0). Or, writing G = Ft − tF (0), we have Ψr,k , Gt = (Ψr,k )t , G, implying that the the directional derivative Ψr,k , Gt  exists and is zero almost everywhere; hence F, Gt  also exists and is zero a.e. So  2π F (eit ), Gt (eit ) dt = 0. 0

That is,

 0

2π

F, Ftt − F (0) dt

=

[F, Ft ]2π 0 

=

−

0

2π

 −

0

2π

|Ft |2 dt − 2π|F (0)|2

|Ft |2 dt − 2π|F(0)|2 = 0.

Thus F is constant and F (0) = 0. Hence, F is identically equal to 0 and K = {F ∈ W 1,2 (T, Cm ) : F (eiw ) ∈ I(eiw )}.




Remark 2.3. If we define P (eiw ) to be the orthogonal projection from Cm onto I(eiw ), P is a measurable function of eiw but it does not even need to be continuous. Indeed, let m ≥ 2 and consider I(eiw ) to be the linear span of the set of vectors e1 and e1 + (eiw − 1)e2 , where {e1 , . . . , em } denote the canonical basis of Cm . Then clearly P (eiw ) is the identity map whenever eiw = 1 and P (eiw ) is the projection onto e1 otherwise. Note that eiw −→ e1 and eiw −→ e1 + (eiw − 1)e2 belong to W 1,2 (T, Cm ). One application of these results is to the graphs of operators. We recall that, for Banach spaces X and Y the graph of a (possibly unbounded) operator T : D(T ) → Y , with domain D(T ) ⊂ X , is the subspace G(T ) = {(x, y) : x ∈ D(T ), y = T x} ⊂ X × Y. Theorem 2.2 can be used to obtain information on the structure of doublyinvariant graphs; however, this does not seem to give the full analogue of the known result for L2 (T, C2 ), which may be found in [7] and [11, Sec. 3.2].

3. Doubly-invariant subspaces in the Sobolev space in the annulus Let r0 be a positive real number less than unity and let A be the annulus defined by A = {z ∈ C : r0 < |z| < 1}. The boundary ∂A of A consists of two circles T and r0 T.

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Define W 1,2 (rT) as the set of functions f defined on rT satisfying f (reit ) = n int and n∈Z bn r e  f 2W 1,2 (rT) := |b0 |2 + n2 r2n |bn |2 < ∞. n∈Z 1,2

Define W (∂A) as the set of functions f on ∂A of the form f = f1 ⊕ f0 with f1 ∈ W 1,2 (T) and f0 ∈ W 1,2 (r0 T). The subspace H 1,2 (∂A) set of functions f0 in W 1,2 (T) ⊕
is theint
f = f1 n⊕int 1,2 it it W (rT) where f1 (e ) = n∈Z an e , f0 (r0 e ) = n∈Z an r0 e and  f 2H 1,2 (∂A) := 2|a0 |2 + n2 (1 + r02n )|an |2 < ∞. n∈Z

Note that H 1,2 (∂A) can be isometrically with the set H 1,2 (A) of func
identified n tions analytic in A of the form f (z) = n∈Z an z with  2|a0 |2 + n2 (1 + r02n )|an |2 < ∞. n∈Z

We shall start with a factorization theorem for functions in H 1,2 (A). Similar formulae have been found useful in H 2 (A) (see for example [6], [3], [1, Thm. 2.1]), and our method of proof is similar to that of [3]. Theorem 3.1. Let f ∈ H 1,2 (A). Then there exist m ∈ Z,f1 ∈ H 1,2 (D) and f0 ∈  \ r0 D) such that f (z) = z m f0 (z)f1 (z). Moreover, the zeros of f0 and f1 H 1,2 (C can be taken to lie in A.  \ r0 D) be the Blaschke product associated Proof. Let r1 ∈ (r0 , 1) and B0 ∈ H ∞ (C with the zeros of f in {z ∈ C : r0 < |z| ≤ r1 } and B1 ∈ H ∞ (D) be the Blaschke product associated with the zeros of f in {z ∈ C : r1 < |z| < 1}. Then B1fB0 ∈ H 2 (A) and B1fB0 has no zero.  Now, there exists m ∈ Z such that h := log B1 Bf0 zm is single-valued. Note

h(z)
+∞ 1 n that h ∈ H 1 (A), and h(z) = n=−∞ an z , where an = 2iπ Γ z n+1 dz for any circle Γ lying in A. Moreover, there exists C > 0 such that |an | ≤ C for
n ≥ 0 and |an | ≤ rCn for n ≤ 0. It follows that h(z) = h1 (z)+h0 (z) where h1 (z) := n≥0 an z n 0
 \ r0 D. is holomorphic in D, and h0 (z) := an z n is holomorphic in C n<0

Set f1 (z) = B1 (z)eh1 (z) and f0 (z) = B0 (z)eh0 (z) . Note that f1 is holomorphic in D and z −m f (z)e−h0 (z) . f1 (z) = B0 (z) Since f ∈ W 1,2 (T), and since z → z −m , z → e−h0 (z) and z → B11(z) are holomorphic in a neighbourhood of T, it is clear that f1 ∈ H 1,2 (D).  \ r0 D and Moreover, f0 is holomorphic in C f0 (z) =

z −m f (z)e−h1 (z) . B1 (z)

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Let us return to the analysis of the structure of the doubly-invariant subspaces. In general the doubly-invariant subspaces are not spectral. Indeed, for example DS (1) = H 1,2 (A) and thus DS (1) is not spectral. In order to present our next result, we introduce the following definition. Definition 3.2. A function F ∈ W 1,2 (∂A, Cm ) is bounded below if there exists δ > 0 such that F (ξ)Cm ≥ δ almost everywhere on ∂A. Proposition 3.3. Suppose that F ∈ W 1,2 (∂A, Cm ) satisfies Ft ∞ < ∞ and F is essentially bounded below. Then DS (F ) = H 1,2 (∂A)F . Proof. We need to show that H 1,2 (∂A)F is closed, in which case it will be DS (F ),
N being limits of expressions of the form −N an z n F (z). Suppose that {gn F } is a Cauchy sequence in W 1,2 (∂A, Cm ), where gn ∈ H (∂A). Now (gn F )t = (gn )t F + gn Ft , in the sense that gn F is absolutely continuous and is the integral of the L2 (∂A, Cm ) function (gn )t F + gn Ft . 1,2

Since {gn F } converges in L2 (∂A, Cm ) and F is bounded below, we get convergence of {gn } in L2 (∂A, Cm ). Hence we have L2 (∂A, Cm ) convergence of {gn Ft } as Ft is in L∞ (∂A, Cm ). By subtraction, we have L2 (∂A, Cm ) convergence of {(gn )t F }, and hence of {(gn )t }. Hence {gn } is Cauchy in H 1,2 (∂A), which is enough to show that
H 1,2 (∂A)F is closed. One circumstance under which a subspace DS (f ) will be spectral is given by the following result. Theorem 3.4. Take f = f1 ⊕f0 in W 1,2 (∂A), and suppose that f1 = 0 or f0 = 0 on an interval of positive measure. Then DS (f ) = DS (f1 ) ⊕ DS (f0 ) and thus DS (f ) is spectral. Proof. Clearly DS (f ) ⊂ DS (f1 ) ⊕ DS (f0 ), so we need to show that if g = g1 ⊕ g0 ∈ DS (f1 ) ⊕ DS (f0 ) then it is in DS (f ). Suppose to fix ideas that f = 0 on an arc Z = {r0 eit : π − α ≤ t ≤ π + α} with 0 < α < π. It will be clear how the proof can be adapted in general. We have a sequence of pairs of trigonometric polynomials {pn ⊕ qn } with pn f1 − g1 W 1,2 (T) → 0, and qn f2 − g2 W 1,2 (r0 T) → 0. If we can find a sequence of polynomials in z and 1/z, say {rn (z)} with rn − pn W 1,∞ (T) → 0 and rn − qn W 1,∞ (r0 T\Z) → 0, then we shall also have rn f − gW 1,2 (∂A) → 0, and the result follows. We are grateful to the referee for the following conclusion to the proof, which is somewhat simpler than our original argument using Mergelyan’s theorem. We apply Runge’s theorem [13, Thm. 13.6] to a small neighbourhood U of ∂A \ Z with two components, not containing 0. This gives us a sequence {rn } such

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that rn − pn → 0 uniformly on an open subset containing T and rn − qn → 0 uniformly on an open subset containing r0 T \ Z. By an easy estimate involving Cauchy’s integral formula, this implies also that rn − pn → 0 uniformly on T and
rn − qn → 0 uniformly on r0 T \ Z. Same arguments imply the following result. Corollary 3.5. Take F = F1 ⊕ F0 ∈ W 1,2 (∂A, Cm ) and suppose that F1 = 0 or F0 = 0 on an interval of positive measure. Then DS (F ) = DS (F1 ) ⊕ DS (F0 ) and thus DS (F ) is spectral. We conclude with the following question: in [14], it was shown that if f = f1 ⊕ f0 ∈ L2 (∂A) satisfies f1 is not log-integrable on T or f0 is not log-integrable on r0 T, then DS (f ) = DS (f1 ) ⊕ DS (f0 ). Is a similar property still true in analytic Sobolev spaces? Acknowledgements The authors are grateful to the EPSRC for financial support, and they thank Janyne and Claude Charlier for providing such a pleasant environment in which to do research. They also wish to express their gratitude to the referee for reading the manuscript very carefully and helping them to remove a number of errors.

References [1] A. Aleman and S. Richter. Simply invariant subspaces of H 2 of some multiply connected regions. Integral Equations Operator Theory, 24(2):127–155, 1996. Erratum. Ibid, 29(4):501–504, 1997. [2] I. Chalendar, N. Chevrot, and J. R. Partington. Invariant subspaces for the shift on the vector-valued L2 space of an annulus. Journal of Operator Theory, to appear, 2007. [3] K. C. Chan and A. L. Shields. Zero sets, interpolating sequences, and cyclic vectors for Dirichlet spaces. Michigan Math. J., 39(2):289–307, 1992. [4] L. I. Hedberg and Yu. Netrusov. An axiomatic approach to function spaces, spectral synthesis, and Luzin approximation. Mem. Amer. Math. Soc., 188(882), 2007. [5] H. Helson. Lectures on invariant subspaces. Academic Press, New York, 1964. [6] D. Hitt. Invariant subspaces of H2 of an annulus. Pacific J. Math., 134(1):101–120, 1988. [7] B. Jacob and J. R. Partington. Graphs, closability, and causality of linear timeinvariant discrete-time systems. Internat. J. Control, 73(11):1051–1060, 2000. [8] L. Khanin. Spectral synthesis of ideals in algebras of functions having generalized derivatives. Russian Math. Surveys, 39:167–168, 1984. [9] Yu. V. Netrusov. Spectral synthesis in a Sobolev space generated by an integral metric. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 217(Issled. po Linein. Oper. i Teor. Funktsii. 22):92–111, 221, 1994. Translation in J. Math. Sci. (New York) 85 (1997), no. 2, 1814–1826.

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[10] N. K. Nikolski. Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, Vol. 92. American Mathematical Society, Providence, RI, 2002. Translated from the French by A. Hartmann. [11] J. R. Partington. Linear Operators and Linear Systems. Cambridge University Press, 2004. London Math. Soc. Student Texts 60. [12] H. L. Royden. Invariant subspaces of H p for multiply connected regions. Pacific J. Math., 134(1):151–172, 1988. [13] W. Rudin. Real and complex analysis. McGraw–Hill, third edition, 1986. [14] D. Sarason. The H p spaces of an annulus. Mem. Amer. Math. Soc., 56, 1965. [15] D. V. Yakubovich. Invariant subspaces of the operator of multiplication by z in the space E p in a multiply connected domain. J. Soviet Math., 61(2):2046–2056, 1992. I. Chalendar Institut Girard Desargues UFR de Math´ematiques Universit´e Lyon 1 69622 Villeurbanne Cedex France e-mail: [email protected] J. R. Partington School of Mathematics University of Leeds Leeds LS2 9JT U.K. e-mail: [email protected] Submitted: February 12, 2007 Revised: February 6, 2008

Integr. equ. oper. theory 61 (2008), 159–165 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/020159-7, published online April 17, 2008 DOI 10.1007/s00020-008-1584-7

Integral Equations and Operator Theory

Xia Spectrum for Some Class of Operators Muneo Ch¯o, Tadasi Huruya and Kˆotarˆo Tanahashi Dedicated to Professor Robin Harte on his seventieth birthday

Abstract. In this paper, we introduce Xia spectra of n-tuples of operators satisfying |T 2 | ≥ U |T 2 |U ∗ for the polar decomposition of T = U |T | and we extend Putnam’s inequality to these tuples [7]. Mathematics Subject Classification (2000). Primary 47B20; Secondary 47A10. Keywords. Hilbert space, operator, Putnam’s inequality, Taylor spectrum, Xia spectrum, joint approximate point spectrum.

1. Introduction Let H be a complex separable Hilbert space and B(H) be the set of all bounded linear operators on H. For an operator T ∈ B(H) is called p-hyponormal if (T ∗ T )p ≥ (T T ∗ )p . If p = 1, T is called hyponormal, and p = 12 , T is called semi-hyponormal. For an operator T ∈ B(H), we denote the spectrum of T by σ(T ). D. Xia, in [6], introduced semi-hyponormal tuples and extended the Putnam’s inequality to semi-hyponormal tuples. In this paper, we introduce an operator T such that |T 2 | ≥ U |T 2 |U ∗ = (|T ∗ ||T |2 |T ∗ |)1/2 , where T = U |T | is the polar decomposition of T . By Lemma 1 (iii) of [4](p.159), it holds that U |T 2 |U ∗ = (|T ∗ ||T |2 |T ∗ |)1/2 for any operator T = U |T |. If an operator T = U |T | satisfies T 2 = 0 and |T 2 | ≥ U |T 2 |U ∗ , then we say it power pusedo-semihyponormal and extend the Putnam’s inequality to these operators. In section 3, we give a power pusedo-semihyponormal operator T which is not paranormal. Let U = (U1 , . . . , Un ) be an commuting n-tuple of unitary operators and, for an operator T ∈ B(H), we denote the (n + 1)-tuple (U1 , . . . , Un , T ) by (U, T ). An operator Qj (j = 1, . . . , n) on B(H) is defined by Qj T = T − Uj T Uj∗

(T ∈ B(H)).

This research is partially supported by Grant-in-Aid Research No.17540176.

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Let A ∈ B(H) and A ≥ 0. (U, A) is called a semi-hyponormal tuple if Qj1 · · · Qjm A ≥ 0 for all 1 ≤ j1 < · · · < jm ≤ n. If (U, A) is a semi-hyponormal tuple, then Uj A is semi-hyponormal for every j (j = 1, . . . , n). For an operator T ∈ B(H), if Sj± (T ) = s - lim (Uj−n T Ujn ) n→±∞

Sj± (T )

exist, then the operators are called the polar symbols of T . If Uj A is semi± hyponormal, then Sj (A) exist (cf.[8]). For 0 ≤ k ≤ 1, we denote (kSj+ + (1 − k)Sj− )T = kSj+ (T ) + (1 − k)Sj− (T ). Let k = (k1 , . . . , kn ) ∈ [0, 1]n and (U, A) be a semi-hyponormal tuple. Then the generalized polar symbols Ak of A are defined by Ak =

n
j=1

(kj Sj+ + (1 − kj )Sj− )A.

Then we have the following theorem. Theorem 1.1. (Proposition, [1]) With the above notations, (U, Ak ) is a commuting (n + 1)-tuple for every k ∈ [0, 1]n . We denote the Taylor spectrum of a commuting m-tuple S = (S1 , . . . , Sm ) of operators by σT (S). For an m-tuple S = (S1 , . . . , Sm ), let σja (S) be the joint approximate point spectrum of S, i.e., the set of all points (z1 , . . . , zm ) for which there exits a sequence {xk } of unit vectors such that lim (Sj − zj )xk  = 0 (j = 1, . . . , m).

k→∞

It is well known that σT (S) = σja (S) if S is a commuting m-tuple of normal operators. Hence, from Theorem A, it holds that σT (U, Ak ) = σja (U, Ak ) for every k ∈ [0, 1]n (cf. [2],[6]). Next, D. Xia in [7] defined the joint spectrum for a non-commuting (n+1)-tuple (U, A) = (U1 , . . . , Un , A) as follows: Let T = {z ∈ C : |z| = 1} and let E(·) be the spectral measure of U. For z = (z1 , . . . , zn ) ∈ σT (U), the set of all products ∆ = γ1 × · · · × γn of open arcs γj ⊂ T, containing zj (j = 1, . . . , n), is denoted by Γ(z). The set  σ(E(∆)AE(∆))} {(z, r) : z ∈ σT (U), r ∈ ∆∈Γ(z)

is called the Xia spectrum of (U, A) and we denote it by σX (U, A). And D. Xia proved the following Theorem 1.2. (Theorem 2, [7]) Let (U, A) be a semi-hyponormal tuple. Then  σX (U, A) = σja (U, Ak ). k∈[0,1]n

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Let mj (j = 1, . . . , n) be the normalized Haar measure in the unit circle T, i.e., 1 dθj (eiθj ∈ T) 2π and m = m1 × · · · × mn × dr, where dr is the Lebesgue measure on R. Then also D. Xia proved the following dmj =

Theorem 1.3. (Theorem 5, [7]) Let (U, A) be a semi-hyponormal tuple. Then   1  Q1 · · · Qn A  ≤ m(σX (U, A)) = dθ1 · · · dθn dr. · · · (2π)n σX (U,A)

2. Xia spectrum for new class of operator Let T = (T1 , T2 , . . . , Tn ) be a doubly commuting n-tuple of operators Tj = Uj |Tj | with unitary Uj which satisfies |Tj2 | ≥ Uj |Tj2 |Uj∗ for every j (j = 1, 2, . . . , n). Let A = |T12 | · · · |Tn2 |. Then A ≥ 0 since (|T12 |, . . . , |Tn2 |) is a commuting n-tuple of positive operators. Let U = (U1 , . . . , Un ). Since U = (U1 , . . . , Un ) is a commuting n-tuple of unitary operators and
Qj A = ( |Ti2 | )( |Tj2 | − Uj |Tj2 |Uj∗ ), i=j

we have Qj1 · · · Qjm A =








|Ti2 |




(|Ti2 |



−

Ui |Ti2 |Ui∗ )

≥0

i=j1 ,...,jm

i=j1 ,...,jm

for all 1 ≤ j1 < · · · < jm ≤ n. Therefore, we have that (U, A) is a semi-hyponormal tuple and n
Q1 · · · Qn A = (|Tj2 | − Uj |Tj2 |Uj∗ ). j=1

Therefore, we have the following theorems. Theorem 2.1. Let T = (T1 , . . . , Tn ) be a doubly commuting n-tuple of power pusedo-semihyponormal operators Tj = Uj |Tj | with unitary Uj (j = 1, . . . , n). Let U = (U1 , . . . , Un ) and A = |T12 | · · · |Tn2 |. Then  σja (U, Ak ). σX (U, A) = k∈[0,1]n

Theorem 2.2. Let T = (T1 , . . . , Tn ) be a doubly commuting n-tuple of power pusedo-semihyponormal operators Tj = Uj |Tj | with unitary Uj (j = 1, . . . , n). Let U = (U1 , . . . , Un ) and A = |T12 | · · · |Tn2 |. Then   n
1 || (|Tj2 | − Uj |Tj2 |Uj∗ )|| ≤ dθ1 · · · dθn dr. · · · (2π)n σX (U,A) j=1

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3. Examples Let A, B, C ≥ 0 be 2 × 2 defined by  0 B  0  0 T =  ..  .  0  .. . Then

√       |T 2 | =       

2 ∞ 2 matrices. Let T be an operator on (⊕∞ 0 C ) ⊕ (⊕0 C )

BA2 B 0 0 0 .. . 0 0 .. . 

B 0  0  0  . |T | =   ..  0  0  .. . and



C 0  0  0  ∗ . |T | =   ..  0  0  .. .

0 0 A 0 .. .

0 0 0 A .. .

0 0 0 0 .. .

··· ··· ··· ···

C 0 0 0 .. .

0 0 0 0 .. .

0 .. .

0 .. .

0 .. .

···

0 .. .

0 0 .. .

0 0 0 0

0 A2 0 0 .. .

0 0 A2 0 .. .

0 0 0 A2 .. .

··· ··· ··· ··· .. .

0 0 .. .

0 0

0 0

··· ··· .. .

0 A 0 0 .. .

0 0 A 0

0 0 0 A

··· ··· ··· ··· .. .

0 0 0 0 .. .

0 0 0 0 .. .

0 0 0 0

0 0 .. .

0 0 .. .

0 0

··· ··· .. .

C 0 .. .

0 0 .. .

0 0 .. . 0 0 0 0

0 0 0 0 .. .

0 0 0 0 .. .

√ CB 2 C 0 .. .

0 B 0 0 .. .

0 0 0 0 A 0 0 A

··· ··· ··· ··· .. .

0 0 0 0 .. .

0 0 0 0 .. .

0 0 .. .

0 0

··· ··· .. .

0 0 .. .

0 0 0 0 .. .

0 0

 ··· ···  ···  ··· . ..  .  0 ···  .. . 0 0 0 0

0 0 0 0

0 0 0 0 .. .

 ··· ···  ···  ···  ..  , .  0 ···  0 ···  .. .

0 0 0 0

 ··· ···  ···  ···  ..  .  0 ···  0 ···  .. . 0 0 0 0

 ··· ···  ···  ···  ..  . .  0 ···  0 ···  .. . 0 0 0 0

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Now, let  A=

3 2

Xia Spectrum for Some Class of Operators

 0

0

3 4

, Q = 1 2



1 1

1 1

163

 , B = b · Q and C = c · Q (0 < b, c).

Then (1) |T 2 | ≥ |T ∗ |2 implies √ √ 3 2 · b · Q = BA2 B ≥ C 2 = c2 · Q and A2 ≥ B 2 = b2 · Q. 4 √

Hence 0 < b ≤ 1 and 0 < c2 ≤ 3 4 2 · b. Also, (2) |T | ≥ |T ∗ | implies A ≥ B = b · Q ≥ C = c · Q. √ √ Hence 0 < c ≤ b ≤ 6( 2 − 1) = 1.0146 . . . . Since √ CB 2 C √ 0 0 0 ··· 2B  0 BA 0 0 ···  2  0 ··· 0 0 A  2  0 0 0 A ···  ∗ 2 ∗ 12 . . .. (|T ||T | |T |) =  .. ..  .   0 0 0 0 · · ·   0 0 0 0 ···  .. .. .. . . .

0 0 0 0 .. .

0 0 0 0 .. .

0 0 0 0

0 0 .. .

0 0 0 0 .. .

 ··· ···  ···  ···  ..  , .  0 ···  0 ···  .. .

0 0 0 0

1

(3) |T 2 | ≥ (|T ∗ ||T |2 |T ∗ |) 2 implies √ √ √ 2 3 2 A ≥ BA2 B = · b · Q ≥ CB 2 C = bc · Q. 4 √

√

Hence 0 < b ≤ 2 3 2 = 0.943 . . . and 0 < c ≤ 3 4 2 . This example shows that there is no inclusional relation for (1), (2), (3). √ 2 2 For example, let b = 3 . Then T satisfies (1), (3) and does not satisfy (2) if √ √ 2 2 3 2 3 < c ≤ 1, and T satisfies (3) and does not satisfy (1), (2) if 1 < c ≤ 4 . Next, let vectors 0 and x be  where x1 =

0 = (0, 0, . . .) and x = (x1 , 0, 0, . . .),  √ 1 . Then 0 ⊕ x = 2 and 1 T (0 ⊕ x) = (Cx1 , 0, 0, . . .) ⊕ (0, 0, 0, . . .).

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Since



0  0   AB   0  2 . T =  ..   0   0  .. . we have

0 0 0 A2 .. .

0 0 0 0 .. .

0 0 0 0 .. .

··· ··· ··· ··· .. .

0 BC 0 0 .. .

0 0 .. .

0 0

0 0

··· ··· .. .

0 0 .. .

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0 0 0 0 .. .

0 0 0 0

0 0 0 0 .. .

 ··· ···  ···  ···  ..  , .  0 ···  0 ···  .. . 0 0 0 0

T 2 (0 ⊕ x) = (0, BCx1 , 0, 0, . . .) ⊕ (0, 0, 0, . . .). 

Since Cx1 = c

1 1



 and BCx1 = bc

1 1

 ,

hence C2c2 = T (0 ⊕ x)2 ≤ T 2 (0 ⊕ x) · 0 ⊕ x = 2bc implies c ≤ b. √ √ Therefore, T is not paranormal if b < c. Let b = 2 3 2 and c = 3 4 2 . Then since b < c, the operator T is a power pusedo-semihyponormal operator which is not paranormal.

References [1] M. Ch¯ o and T. Huruya, Putnam’s inequality for p-hyponormal n-tuples, Glasgow Math. J. 41(1999), 13-17. [2] R. Curto, On the connectedness of invertible n-tuples, Indiana Univ. Math. J. 29(1980), 393-406. [3] R. Curto, P. Muhly and D. Xia, A trace estimate for p-hyponormal operators, Integral Equations and Operator Theory 6(1983), 507-514. [4] T. Furuta, Invitation to linear operators, Taylor and Francis, 2001. [5] C. R. Putnam, Commutation properties of Hilbert space operators, Springer-Verlag, 1967. [6] J. L. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal. 6(1970), 172-191. [7] D. Xia, On the semi-hyponormal n-tuple of operators, Integral Equations and Operator Theory 6(1983), 879-898. [8] D. Xia, Spectral Theory of Hyponormal Operators, Birkh¨ auser 1983. Muneo Ch¯ o Department of Mathematics Kanagawa University Yokohama, 221-8686 Japan e-mail: [email protected]

Vol. 61 (2008)

Xia Spectrum for Some Class of Operators

Tadasi Huruya Faculty of Education and Human Sciences Niigata University Niigata 950-2181 Japan e-mail: [email protected] Kˆ otarˆ o Tanahashi Department of Mathematics Tohoku College of Pharmacy Sendai 981-8558 Japan e-mail: [email protected] Submitted: August 28, 2007 Revised: March 3, 2008

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Integr. equ. oper. theory 61 (2008), 167–186 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/020167-20, published online April 17, 2008 DOI 10.1007/s00020-008-1579-4

Integral Equations and Operator Theory

Composition Operators on the Space of Bounded Harmonic Functions Jun Soo Choa, Kei Ji Izuchi and Shˆ uichi Ohno Abstract. We study composition operators on the space of bounded harmonic functions on the open unit disk. The principal goal of this paper is to provide criteria for determining the essential norm of difference of two composition operators. Mathematics Subject Classification (2000). Primary 47B33; Secondary 46J15. Keywords. Composition operators, the space of harmonic functions, essential norm, difference of composition operators.

1. Introduction Throughout this paper h∞ denotes the Banach space of all bounded harmonic functions f on the open unit disk D := {z ∈ C : |z| < 1}, with the supremum norm f ∞ = sup{|f (z)| : z ∈ D}. We denote by S(D) the set of analytic self-maps of D. For every ϕ ∈ S(D), the composition operator Cϕ , defined by Cϕ f = f ◦ ϕ, is easily seen to be bounded on h∞ as well as on the space H ∞ of bounded analytic functions on D. Composition operators have been investigated mainly in spaces of analytic functions to characterize the operator-theoretic behavior of Cϕ in terms of the function-theoretic properties of the symbol ϕ ; see the books [2], [11] and [13] for a thorough treatment on such classical settings. In a recent paper, MacCluer, Zhao and the third author [10] have obtained results concerning the topological structure of composition operators on H ∞ . After this work, H ∞ has attracted much attention in the The first author is partially supported by KRF-2002-070-C00004. The second author is partially supported by Grant-in-Aid for Scientific Research (No.16340037), Japan Society for the Promotion of Science. The third author is partially supported by Grant-in-Aid for Scientific Research (No.17540169), Japan Society for the Promotion of Science.

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study of this area; see, for example, [5], [8], [9] and [12]. In particular, Gorkin, Mortini and Su´ arez [5, Theorem 6] gave upper and lower bounds for the essential norm of difference of two composition operators on H ∞ , where the setting is the unit ball of Cn (n ≥ 1). A refined partial result on the essential norm of the composition difference on H ∞ has recently obtained by Hosokawa and the second author [7]. It seems difficult to compute the exact value of the essential norm of the composition difference on H ∞ . In this paper we study properties of composition operators on h∞ , and provide the exact value of the essential norm of difference of any two composition operators on h∞ . The next section outlines the prerequisites, and in Section 3 we give elementary results of composition operators on h∞ . In Sections 4 and 5 we give lower and upper bounds for the essential norm Cϕ − Cψ e of the difference Cϕ − Cψ on h∞ whenever ϕ, ψ ∈ S(D). Summing these estimates, in Section 6, we completely describe Cϕ − Cψ e and present some examples of analytic self-maps ϕ, ψ attaining the essential norm.

2. Prerequisites In this section, we collect some materials and facts that will be needed in the sequel. 2.1. Harmonic functions For f ∈ h∞ , there exists a boundary function f ∗ on the unit circle ∂D := {|z| = 1} defined by f ∗ (eiθ ) = lim f (reiθ ) a.e.. ∗

∞

r→1 ∞

It is well known that {f ; f ∈ h } = L (∂D) and
2π dθ f ∗ (eiθ )Pz (eiθ ) f (z) = 2π 0 where Pz is the Poisson kernel for the point z in D. We denote by B(H ∞ ) and B(h∞ ) the closed unit balls of H ∞ and h∞ , respectively. It is known (e.g., see [4, p. 42]) that if we define ρ(z, w) = |z − w|/|1 − zw|, the pseudo-hyperbolic distance between z and w in D, then sup f ∈B(H ∞ )

|f (z) − f (w)| =

2ρ(z, w)  . 1 + 1 − ρ(z, w)2

Write 2t √ , 0 ≤ t ≤ 1. 1 + 1 − t2 Then λ is an increasing function on [0, 1], λ(0) = 0, λ(1) = 2, and λ(t) =

sup f ∈B(H ∞ )

|f (z) − f (w)| = λ(ρ(z, w)).

(2.1)

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In what follows, we use the usual notation
2π dθ |f (eiθ )| f 1 = 2π 0 for f ∈ L1 (∂D). The next identity, which will be useful in our approach, is also taken from [4, p. 42]. Lemma 2.1. Pz − Pw 1 = 2 − (4/π) cos−1 ρ(z, w). Lemma 2.2.

sup |f (z) − f (w)| = Pz − Pw 1 ≥ λ(ρ(z, w)) for z, w ∈ D.

f ∈B(h∞ )

Proof. If f ∈ B(h∞ ), it is clear that  
2π   f ∗ (Pz − Pw )dθ/2π  ≤ Pz − Pw 1 . |f (z) − f (w)| =  0

On the other hand, there exists a function g ∈ B(h∞ ) with g ∗ (Pz −Pw ) = |Pz −Pw | a.e. on ∂D. Thus g(z) − g(w) = Pz − Pw 1 . As a consequence, we get sup f ∈B(h∞ )

|f (z) − f (w)| = Pz − Pw 1 .

The second assertion follows from (2.1) and the containment relation B(H ∞ ) ⊂
B(h∞ ). 2.2. The maximal ideal space of H ∞ We denote by M (H ∞ ) and M (L∞ ) the set of nonzero multiplicative linear functionals on H ∞ and L∞ , respectively. We consider the weak∗ -topology for M (H ∞ ) and M (L∞ ). It is well known that with the weak∗ -topology, M (H ∞ ) and M (L∞ ) are compact Hausdorff spaces and D ⊂ M (H ∞ ). Usually we call M (H ∞ ) and M (L∞ ) with the weak∗ -topology the maximal ideal spaces of H ∞ and L∞ (∂D). The famous “corona theorem” due to Carleson [1] states that D is a dense subset of M (H ∞ ), which will be used for our developments. We identify a function f in H ∞ and L∞ (∂D) with its Gelfand transform, that is, f (m) = m(f ) for m ∈ M (H ∞ ) or m ∈ M (L∞ ). Therefore we regard f as a continuous function on M (H ∞ ) and M (L∞ ), respectively. We may think of M (L∞ ) as a closed subset of M (H ∞ ), and M (L∞ ) is the Shilov boundary for H ∞ . For each x ∈ M (H ∞ ), there exists a unique probability measure µx on M (L∞ ) satisfying
f (x) = f dµx M(L∞ )

for every f ∈ H ∞ . See [3, 6] for these subjects. For g ∈ L∞ (∂D), define gˆ on M (H ∞ ) by
gˆ(x) = gdµx , x ∈ M (H ∞ ). M(L∞ )

(2.2)

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Then gˆ is a continuous function on M (H ∞ ), see Garnett [4, p. 375]. In case f ∈ H ∞ , we easily see that f∗ coincides with the Gelfand transform of f . Let m be the probability measure on M (L∞ ) satisfying





2π

gdm = M(L∞ )

0

g(eiθ )

dθ , 2π

g ∈ L∞ (∂D).

The following three lemmas will be used in the proof of Theorem 5.1. Lemma 2.3. Let f ∈ h∞ . Then f∗ = f on D, and f∗ on M (H ∞ ) is a continuous extension of f on D. Proof. We need only verify the first assertion. When z ∈ D, we know that Pz dm = dµz . Hence

2π ∗ iθ iθ dθ = f (e )Pz (e ) f ∗ Pz dm f (z) = 2π 0 M(L∞ )
= f ∗ dµz = f∗ (z). M(L∞ )




Let x ∈ M (H ∞ ) and ϕ ∈ S(D). It is easy to see that H ∞  f → (f ◦ ϕ)(x) is a nonzero multiplicative linear functional of H ∞ . Hence there exists a point ϕ(x) ˜ ∈ M (H ∞ ) such that (f ◦ ϕ)(x) = f (ϕ(x)) ˜

(2.3)

for every f ∈ H ∞ . Lemma 2.4. Let ϕ ∈ S(D) and x ∈ M (H ∞ ). Then ϕ, ˜ defined by (2.3), is a continuous map from M (H ∞ ) to M (H ∞ ). Moreover if |ϕ(x)| < 1, we have ϕ(x) ˜ = ϕ(x). Proof. Let f ∈ H ∞ and {ζα }α be a net in M (H ∞ ) such that ζα → ζ ∈ M (H ∞ ). Then it is immediate that f (ϕ(ζ ˜ α )) = (f ◦ ϕ)(ζα ) → (f ◦ ϕ)(ζ) = f (ϕ(ζ)). ˜ ˜ in M (H ∞ ). This shows ϕ(ζ ˜ α ) → ϕ(ζ) Suppose that |ϕ(x)| < 1. By the corona theorem, there exists a net {zα }α in D such that zα → x in M (H ∞ ). It follows that ϕ(zα ) → ϕ(x) ∈ D. But it is obvious that ϕ(z) ˜ = ϕ(z) for z ∈ D. Therefore ϕ(zα ) = ϕ(z ˜ α ) → ϕ(x) ˜ and ϕ(x) ˜ = ϕ(x).
Lemma 2.5. Let f ∈ h∞ and ϕ ∈ S(D). Then ((f ◦ ϕ)∗ )ˆ= f∗ ◦ ϕ˜ on M (H ∞ ).

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Proof. Let z ∈ D. Then we see that ((f ◦ ϕ)∗ )ˆ(z) = = =

(f ◦ ϕ)(z)

=

f (ϕ(z)) f (ϕ(z)) ˜ ∗  f (ϕ(z)) ˜

=

˜ (f∗ ◦ ϕ)(z).

by Lemma 2.3 by Lemma 2.4 by Lemma 2.3

Hence ((f ◦ ϕ)∗ )ˆ= f∗ ◦ ϕ˜ on D. By the continuity of functions we are considering,
this shows that ((f ◦ ϕ)∗ )ˆ= f∗ ◦ ϕ˜ on M (H ∞ ).

3. Elementary properties In this section we gather elementary properties concerning the space h∞ and composition operators on h∞ . The following is easy to check. Lemma 3.1. Every uniformly bounded subset of h∞ is a normal family. Using this, we can have a characterization of compactness of composition operators on h∞ . Proposition 3.2. Let ϕ ∈ S(D). Then the following are equivalent: (i) Cϕ is compact on h∞ . (ii) Cϕ is compact on H ∞ . (iii) ϕ∞ < 1. Proof. The implication (i)⇒(ii) is clear, and the equivalence of (ii) and (iii) is well known. Also we can get the implication (iii)⇒(i) by an easy application of Lemma 3.1.
The next result is often called the weak convergence theorem, which can be easily proved by a standard argument. Lemma 3.3. Let ϕ, ψ ∈ S(D). Then Cϕ − Cψ is compact on h∞ if and only if (Cϕ − Cψ )fn ∞ → 0 for every bounded sequence {fn }n in h∞ which converges to zero uniformly on every compact subset of D. The following is a more concrete characterization of compact difference of two composition operators on h∞ . Proposition 3.4. Let ϕ, ψ ∈ following are equivalent: (i) Cϕ − Cψ is compact on (ii) Cϕ − Cψ is compact on (iii) lim ρ(ϕ(z), ψ(z)) = |ϕ(z)|→1

S(D), ϕ = ψ and ϕ∞ = ψ∞ = 1. Then the h∞ . H ∞. lim ρ(ϕ(z), ψ(z)) = 0.

|ψ(z)|→1

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Proof. The implication (i)⇒(ii) is clear, and the equivalence of (ii) and (iii) is shown in [9, p.1767] (see also Theorem 6 of [5]). We now use Lemma 3.3 to prove the implication (iii)⇒(i). To verify this, we let {fn }n be a sequence in B(h∞ ) such that fn converges to zero uniformly on every compact subset of D. Then
2π dθ |(Cϕ − Cψ )fn (z)| ≤ |fn∗ (eiθ )||Pϕ(z) (eiθ ) − Pψ(z) (eiθ )| 2π 0 ≤ fn∗ ∞ Pϕ(z) − Pψ(z) 1 ≤ 2 − (4/π) cos−1 ρ(ϕ(z), ψ(z))

by Lemma 2.1.

By the condition (iii), for any ε > 0, there is a number δ, 0 < δ < 1, such that |ϕ(z)| > δ ⇒ ρ(ϕ(z), ψ(z)) < ε and |ψ(z)| > δ ⇒ ρ(ϕ(z), ψ(z)) < ε. Since {fn }n converges to zero uniformly on compact subsets of D, for all n sufficiently large we have |(Cϕ − Cψ )fn (z)| < ε for all z with |ϕ(z)| ≤ δ and |ψ(z)| ≤ δ. Thus (Cϕ − Cψ )fn ∞ < ε for sufficiently large n. Therefore (i) holds by Lemma 3.3.
We make a comment on a connection between compact difference and the maximal ideal space M (H ∞ ). For this, we let ϕ, ψ ∈ S(D) with ϕ∞ = 1. If Cϕ − Cψ is compact on h∞ , then ψ∞ = 1 and by Proposition 3.4 lim ρ(ϕ(z), ψ(z)) =

|ϕ(z)|→1

lim ρ(ϕ(z), ψ(z)) = 0.

|ψ(z)|→1

By an application of the corona theorem, together with the fact |ϕ(z) − ψ(z)| ≤ 2ρ(ϕ(z), ψ(z)), one can then easily show that {x ∈ M (H ∞ ) : |ϕ(x)| = 1} = {x ∈ M (H ∞ ) : |ψ(x)| = 1} and ϕ = ψ on {x ∈ M (H ∞ ) : |ϕ(x)| = 1}. We give, with few preliminaries, some observations on the component problem of composition operators on h∞ . For ϕ, ψ ∈ S(D), the operator Cϕ − Cψ has two norms Cϕ − Cψ h∞ and Cϕ − Cψ H ∞ on h∞ and H ∞ , respectively. By Lemmas 2.1 and 2.2, we obtain the following identities of the norm of composition difference: Cϕ − Cψ h∞ = sup Pϕ(z) − Pψ(z) 1 z∈D



and

4 cos−1 ρ(ϕ(z), ψ(z)) π z∈D The above identities motivate the following definitions: Cϕ − Cψ h∞ = sup

2−

 .

(3.1)

λ∞ (ϕ, ψ) = lim sup Pϕ(z) − Pψ(z) 1

(3.2)

σ∞ (ϕ, ψ) = lim sup ρ(ϕ(z), ψ(z)).

(3.3)

|(ϕψ)(z)|→1

and |(ϕψ)(z)|→1

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The quantity σ∞ (ϕ, ψ) is recently used by Hosokawa and the second author [7] to give upper and lower bounds for the essential norm of composition difference on H ∞ . In this paper, we use the quantity λ∞ (ϕ, ψ) to estimate the essential norm of composition difference on h∞ . We remark that Lemma 2.2 immediately provides a relationship between the quantities λ∞ (ϕ, ψ) and σ∞ (ϕ, ψ). Lemma 3.5. Let ϕ, ψ ∈ S(D) with ϕψ∞ = 1. Then 4 cos−1 σ∞ (ϕ, ψ) , π λ(σ∞ (ϕ, ψ)) ≤ λ∞ (ϕ, ψ), and σ∞ (ϕ, ψ) = 1 if and only if λ∞ (ϕ, ψ) = 2. λ∞ (ϕ, ψ) = 2 −

We now show that path components are the same in each other in the sets C(H ∞ ) and C(h∞ ) of composition operators on H ∞ and h∞ , respectively. Theorem 3.6. Let ϕ, ψ ∈ S(D). Then Cϕ and Cψ are in the same path component of C(H ∞ ) if and only if both are in the same path component of C(h∞ ). Proof. The “only if” part needs proof. Assume that Cϕ and Cψ are in the same path component of C(H ∞ ). Let [0, 1]  t → Cϕt denote a continuous arc in C(H ∞ ) joining Cϕ and Cψ , such that Cϕ0 = Cϕ and Cϕ1 = Cψ . Then, given ε > 0, there exists δ > 0 such that Cϕs − Cϕt H ∞ < ε

whenever |s − t| < δ.

By (2.1), Cϕs − Cϕt H ∞

= sup λ(ρ(ϕs (z), ϕt (z))) z∈D

2ρ(ϕs (z), ϕt (z))  1 + 1 − ρ(ϕs (z), ϕt (z))2 z∈D ≥ sup ρ(ϕs (z), ϕt (z)),

= sup z∈D

so that sup ρ(ϕs (z), ϕt (z)) < ε z∈D

if |s − t| < δ.

On the other hand, the formula (3.1) yields   4 cos−1 ρ(ϕs (z), ϕt (z)) Cϕs − Cϕt h∞ = sup 2 − π z∈D 4 = 2 − cos−1 (sup ρ(ϕs (z), ϕt (z))). π z∈D These observations insure that the map t → Cϕt is also a continuous arc joining
Cϕ to Cψ in C(h∞ ). This finishes the proof. The above theorem, combined to Theorem 4.1 in [9], yields the following.

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Corollary 3.7. Let ϕ ∈ S(D). Then the following are equivalent: (i) Cϕ is not isolated in C(h∞ ). (ii) Cϕ is not isolated in C(H ∞ ).  2π (iii) 0 log(1 − |ϕ∗ (eiθ )|)dθ/2π > −∞.

4. Lower bounds for essential norms In this section, we will explore lower bounds for the essential norm of difference of two composition operators on h∞ . Recall that the essential norm of a bounded linear operator T on h∞ is defined as T e = inf{T + K : K is compact on h∞ }. We start with a technical lemma, which follows easily from basic properties of Poisson kernels. Lemma 4.1. Let {zj }j and {wj }j be sequences in D such that zj → α and wj → β and |α| = |β| = 1. Let U be an open subset of ∂D such that {α, β} ⊂ U . Then
2π dθ = 0. χ(∂D\U) |Pzj − Pwj | lim j→∞ 0 2π Consequently, if lim Pzj − Pwj 1 exists, then j→∞




lim

j→∞

0

2π

χU |Pzj − Pwj |

dθ = lim Pzj − Pwj 1 . 2π j→∞

Theorem 4.2. Let ϕ, ψ ∈ S(D). If ϕψ∞ = 1, then Cϕ − Cψ e ≥ λ∞ (ϕ, ψ). Proof. The assertion is trivial if λ∞ (ϕ, ψ) = 0, so we assume that λ∞ (ϕ, ψ) > 0. By the definition of λ∞ (ϕ, ψ), there exists a sequence {zj }j in D such that |zj | → 1, |(ϕψ)(zj )| → 1, and Pϕ(zj ) − Pψ(zj ) 1 → λ∞ (ϕ, ψ) as j → ∞. We may assume that ϕ(zj ) = ψ(zj ) for every j, ϕ(zj ) → α, |α| = 1, and ψ(zj ) → β, |β| = 1. We now use induction to show that we can obtain a subsequence {zjn }n of {zj }j and a sequence of open subsets {Un }n of the unit circle ∂D satisfying the following conditions: {α, β} ⊂ Un+1 ⊂ Un ,




2π

0

and




2π

χUn |Pϕ(zjn ) − Pψ(zjn ) |

dθ 1 > λ∞ (ϕ, ψ)(1 − ), 2π n

(4.1) (4.2)

dθ 1 > λ∞ (ϕ, ψ)(1 − ) (4.3) 2π n 0 for every n. It is clear that (4.2) is implied by (4.3), but we added the condition (4.2) here to understand the argument below more clearly. χ(Un \Un+1 ) |Pϕ(zjn ) − Pψ(zjn ) |

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Let U1 = ∂D and j1 = 1. Since ϕ(zj1 ) = ψ(zj1 ),


2π

0

χU1 |Pϕ(zj1 ) − Pψ(zj1 ) |

dθ 1 > 0 = λ∞ (ϕ, ψ)(1 − ). 2π 1

We can take an open subset U2 of ∂D such that {α, β} ⊂ U2 ⊂ U1 and


2π 0

χU1 \U2 |Pϕ(zj1 ) − Pψ(zj1 ) |

dθ 1 > λ∞ (ϕ, ψ)(1 − ). 2π 1

Let N be an integer greater than 1. We assume that {zjn }1≤n≤N and {Un }1≤n≤N satisfy the conditions (4.1)–(4.3). Then by (4.2) for n = N , we get


2π

0

χUN |Pϕ(zjN ) − Pψ(zjN ) |

dθ 1 > λ∞ (ϕ, ψ)(1 − ). 2π N

Also, by (4.1) and (4.3) for n = N , there is an open subset UN +1 of ∂D such that {α, β} ⊂ UN +1 ⊂ UN and


2π

0

χ(UN \UN +1 ) |Pϕ(zjN ) − Pψ(zjN ) |

dθ 1 > λ∞ (ϕ, ψ)(1 − ). 2π N

By Lemma 4.1,
0

2π

χUN +1 |Pϕ(zj ) − Pψ(zj ) |

dθ → λ∞ (ϕ, ψ) 2π

as j → ∞. Hence there exists a positive integer jN +1 such that
0

2π

χUN +1 |Pϕ(zjN +1 ) − Pψ(zjN +1 ) |

dθ 1 > λ∞ (ϕ, ψ)(1 − ). 2π N +1

This completes the induction. For each positive integer n, there exists a function gn in L∞ (∂D) with gn (Pϕ(zjn ) − Pψ(zjn ) ) = |Pϕ(zjn ) − Pψ(zjn ) |

(4.4)

a.e. on ∂D. Note that |gn | = 1 a.e. on ∂D. Now we define the new functions hn by hn = gn χ(Un \Un+1 ) .

(4.5)

It is then clear that hn ∞ = 1 and hn → 0 weakly in L∞ (∂D), and hence ˆ n → 0 weakly in h∞ . Now let K be an arbitrary compact operator on h∞ . Then h

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ˆ n ∞ → 0. Therefore K h ˆ n ∞ Cϕ − Cψ + K ≥ lim sup (Cϕ − Cψ + K)h n→∞

ˆ n ∞ = lim sup (Cϕ − Cψ )h n→∞

ˆ n (ϕ(zj )) − h ˆ n (ψ(zj ))| ≥ lim sup |h n n n→∞
2π  dθ  hn (Pϕ(zjn ) − Pψ(zjn ) ) = lim sup  2π n→∞ 0
2π dθ = lim sup χ(Un \Un+1 ) |Pϕ(zjn ) − Pψ(zjn ) | 2π n→∞ 0 ≥ λ∞ (ϕ, ψ) by (4.3), where the last equality follows from (4.4) and (4.5). This shows that Cϕ − Cψ e ≥
λ∞ (ϕ, ψ). Theorem 4.3. Let ϕ, ψ ∈ S(D). If there is a sequence {zj } in D such that |ϕ(zj )| → 1 and lim sup|ψ(zj )| < 1, then Cϕ − Cψ e ≥ 1. j→∞

Proof. The idea of the proof is the same as the one of Theorem 4.2, so we give only its outline. By our assumption, there exists a sequence {zj }j in D such that ϕ(zj ) → α with |α| = 1 and ψ(zj ) → β with |β| < 1. Then by Lemma 4.1, there exists a sequence of open subsets {Ui }i of ∂D such that α ∈ Ui+1 ⊂ Ui ,
2π dθ → 1 as j → ∞ for each i, χUi Pϕ(zj ) 2π 0


2π

χ∂D\Ui Pϕ(zj )

0

and


0

2π

χUi Pψ(zj )

dθ → 0 as j → ∞ for each i, 2π

dθ → 0 as i → ∞ for each j. 2π

By considering subsequences {zjn }n of {zj }j and {Uin }n of {Ui }i , we may assume that
2π dθ → 1, (4.6) χ(Uin \Uin+1 ) Pϕ(zjn ) 2π 0 and


0

2π

χ(Uin \Uin+1 ) Pψ(zjn )

dθ → 0, 2π

as n → ∞. Also, there is a sequence {gn } in L∞ (∂D) satisfying gn (Pϕ(zjn ) − Pψ(zjn ) ) = |Pϕ(zjn ) − Pψ(zjn ) |

(4.7)

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ˆ n → 0 weakly in h∞ . Thus, for any a.e. on ∂D. Let hn = gn χ(Uin \Uin+1 ) . Then h compact operator K on h∞ , we have ˆ n ∞ Cϕ − Cψ + K ≥ lim sup (Cϕ − Cψ + K)h n→∞
2π dθ ≥ lim sup χ(Uin \Uin+1 ) |Pϕ(zjn ) − Pψ(zjn ) | 2π n→∞ 0 =1 by (4.6) and (4.7). This proves Cϕ − Cψ e ≥ 1.




5. Upper bounds for essential norms For g ∈ L∞ (∂D), we define a bounded linear operator Mg on h∞ as
2π dθ (Mg f )(z) = g(eiθ )f ∗ (eiθ )Pz (eiθ ) 2π 0 for f ∈ h∞ and z ∈ D, that is, Mg f is the harmonic extension of gf ∗ into D. If we identify a function f in h∞ with its boundary function f ∗ in L∞ (∂D), the operator Mg is just a multiplication operator on L∞ (∂D). For ϕ ∈ S(D), we define EL∞ (|ϕ|) = {x ∈ M (L∞ ) : |ϕ(x)| = 1}. For each x ∈ M (H ∞ ), we recall that there exists a unique probability measure µx on M (L∞ ) satisfying
f (x) = f dµx M(L∞ )

for every f ∈ H ∞ . We denote by supp µx the closed support set of µx . For a subset E of M (H ∞ ), we denote by E the closure of E in M (H ∞ ). Theorem 5.1. Let ϕ ∈ S(D) and g ∈ L∞ (∂D). Then the following conditions are equivalent: (i) Mg Cϕ is a compact operator on h∞ . (ii) lim gˆ(z) = 0. |ϕ(z)|→1

(iii) g = 0 on EL∞ (|ϕ|). Proof. We write K = Mg Cϕ for notational simplicity. At first we show the implication (i)⇒(ii). We shall prove this by a contradiction argument. Assume that there is a sequence {zk }k in D satisfying |ϕ(zk )| → 1 and gˆ(zk ) → c = 0. Further we may assume that ϕ(zk ) → a ∈ ∂D. Let fn (z) =

(z + a)n . (2a)n

Then fn ∈ H ∞ ⊂ h∞ , fn ∞ = 1, fn (a) = 1, and fn → 0 uniformly on every compact subset of D. Let ζ be a cluster point of {zk }k in M (H ∞ ), and choose a

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net {zkα }α in {zk }k with zkα → ζ. Then kα → ∞, gˆ(ζ) = c, and by (2.3) and Lemma 2.4, (fn ◦ ϕ)(ζ) = fn (ϕ(ζ)) ˜ = lim fn (ϕ(zkα )) = fn (a) = 1. kα →∞




This shows

M(L∞ )

(fn ◦ ϕ)(ξ)dµζ (ξ) = 1.

Using fn ◦ ϕ∞ ≤ 1, we obtain fn ◦ ϕ = 1

on supp µζ .

∞

∗

(5.1) ∞

Since fn ◦ ϕ ∈ H , we note that fn ◦ ϕ = (fn ◦ ϕ) on M (L ). Hence Kfn ∞

≥ |((Kfn )∗ )ˆ(ζ)| 
   =  g(ξ)(fn ◦ ϕ)(ξ)dµζ (ξ) M(L∞ ) 
   =  g(ξ)dµζ (ξ) by (5.1)

by (2.2)

M(L∞ )

= |ˆ g (ζ)| = |c|. Thus Kfn ∞ ≥ |c| > 0 for every n. On the other hand, since K is compact, there exists a subsequence {fm }m of {fn }n satisfying Kfm −f ∞ → 0 for some f ∈ h∞ . Note that (ϕ∗ +a)m /(2a)m → 0 a.e. on ∂D. Hence, for each z ∈ D, we have by the Lebesgue dominated convergence theorem
2π g(eiθ )(fm ◦ ϕ)∗ (eiθ )Pz (eiθ )dθ/2π (Kfm )(z) = 0


= → 0

0

2π

g(eiθ )(ϕ∗ (eiθ ) + a)m /(2a)m Pz (eiθ )dθ/2π as m → ∞.

As a consequence, we get f = 0 and Kfm ∞ → 0. This is a contradiction. Secondly we shall also prove the implication (ii)⇒(iii) by contradiction. Assume that g ≡ 0 on EL∞ (|ϕ|), and choose a y ∈ M (L∞ ) satisfying |ϕ(y)| = 1 and g(y) = 0. By the corona theorem, there is a net {zα }α in D with zα → y. We then see that ϕ(zα ) → ϕ(y) and gˆ(zα ) → gˆ(y) = g(y). This contradicts (ii). Lastly we show the implication (iii)⇒(i). Suppose that g = 0 on EL∞ (|ϕ|). To prove that K is compact on h∞ , suppose {fj }j is a sequence of functions in B(h∞ ) such that fj → 0 uniformly on each compact subset of D. It suffices to show that Kfj ∞ → 0 as j → ∞. If EL∞ (|ϕ|) = M (L∞ ), then g = 0, so that Mg Cϕ = 0. Hence we may assume that EL∞ (|ϕ|) = M (L∞ ). Let x ∈ M (L∞ ) with |ϕ(x)| < 1. Then by Lemmas 2.3–2.5, ((fj ◦ ϕ)∗ )ˆ(x) = fj (ϕ(x)) for every j. Hence ((fj ◦ ϕ)∗ )ˆ(x) → 0 as j → ∞. This, combined with the fact that ((fj ◦ ϕ)∗ )ˆ = (fj ◦ ϕ)∗ on M (L∞ ), shows that (fj ◦ ϕ)∗ converges uniformly to zero on every

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compact subset of M (L∞ ) \ EL∞ (|ϕ|). Now using g = 0 on EL∞ (|ϕ|), it is not difficult to show that Kfj ∞ = g(fj ◦ ϕ)∗ ∞ → 0 as j → ∞, which is what we wanted.




In the rest of this section, we will write f U = sup |f (ζ)| ζ∈U

∞

whenever f ∈ L (∂D) and U is a subset of M (L∞ ). We now turn to the upper estimate of Cϕ −Cψ e . We first handle a relatively easy case. Theorem 5.2. Let ϕ, ψ ∈ S(D). If ϕψ∞ < 1, then Cϕ − Cψ e ≤ 1. Proof. If ϕ∞ < 1 or ψ∞ < 1, the assertion is immediate from Proposition 3.2. Hence we may assume that ϕ∞ = ψ∞ = 1. Also, noting that EL∞ (|ϕ|) ∩ EL∞ (|ψ|) = ∅ since ϕψ∞ < 1, we can take open and closed subsets Uϕ and Uψ of M (L∞ ) such that EL∞ (|ϕ|) ⊂ Uϕ ,

EL∞ (|ψ|) ⊂ Uψ ,

and (5.2) Uϕ ∩ Uψ = ∅. Here we used the well-known fact (e.g., see [6, p. 170]) that M (L∞ ) is extremely disconnected, that is, the closure of every open subset of M (L∞ ) is also open. By Theorem 5.1, Mχ(M (L∞ )\Uϕ ) Cϕ − Mχ(M (L∞ )\Uψ ) Cψ

is compact. So, for any f ∈ B(h∞ ), we have

(Cϕ − Cψ − Mχ(M (L∞ )\Uϕ ) Cϕ + Mχ(M (L∞ )\Uψ ) Cψ )f ∞ =

(I − Mχ(M (L∞ )\Uϕ ) )Cϕ f − (I − Mχ(M (L∞ )\Uψ ) )Cψ f ∞

=

MχUϕ Cϕ f − MχUψ Cψ f ∞

≤

max{(f ◦ ϕ)∗ Uϕ , (f ◦ ψ)∗ Uψ }

≤

1,

by (5.2)

where I is the identity operator. This proves Cϕ − Cψ e ≤ 1.




Next, we prove the following. Theorem 5.3. Let ϕ, ψ ∈ S(D) with ϕψ∞ = 1. Then we have the following: (i) If EL∞ (|ϕψ|) = EL∞ (|ϕ|), then Cϕ − Cψ e ≤ max {1, λ∞ (ϕ, ψ)}. (ii) If EL∞ (|ϕ|) = EL∞ (|ψ|), then Cϕ − Cψ e ≤ λ∞ (ϕ, ψ). To prove this, we need some lemmas. Lemma 5.4. Let ϕ, ψ ∈ S(D) with ϕψ∞ = 1. If EL∞ (|ϕψ|) is not an open and closed subset of EL∞ (|ϕ|), then λ∞ (ϕ, ψ) = 2.

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Proof. Note that EL∞ (|ϕψ|) is a closed Gδ -subset of M (L∞ ). By our assumption, we can choose a sequence {xn }n in EL∞ (|ϕ|) \ EL∞ (|ϕψ|) satisfying {xn }n \ {xn }n ⊂ EL∞ (|ϕψ|). Then |ψ(xn )| < 1 and |ψ(xn )| → 1 as n → ∞. For each xn , there exists a net {zn,α }α in D satisfying zn,α → xn as α → ∞. Hence |ϕ(zn,α )| → 1 and |ψ(zn,α )| → |ψ(xn )| < 1, and so Pϕ(zn,α) − Pψ(zn,α ) 1 → 2 as α → ∞. It follows that for each n, there exists αn such that |ϕ(zn,αn )| > 1 − 1/n,

|ψ(zn,αn )| > |ψ(xn )| − 1/n,

and Pϕ(zn,αn ) − Pψ(zn,αn ) 1 > 2 − 1/n. These show that |ϕ(zn,αn )| → 1, |ψ(zn,αn )| → 1, and Pϕ(zn,αn ) − Pψ(zn,αn ) 1 → 2.


Therefore, by (3.2) we get λ∞ (ϕ, ψ) = 2.

Lemma 5.5. Let f ∈ h∞ and U be an open and closed subset of M (L∞ ). Then f ∗ U = lim sup |f (z)|. χ ˆU (z)→1

Proof. Let f and U be as in the hypothesis, and let x ∈ U . By the corona theorem there is a net {zα }α in D such that zα → x in M (H ∞ ). Noting that f∗ = f ∗ on M (L∞ ), we then have f (zα ) = f∗ (zα ) → f∗ (x) = f ∗ (x). Meanwhile, using the fact that a set is open and closed if and only if its characteristic function is continuous, we get ˆU (x) = χU (x) = 1. χ ˆU (zα ) → χ Combining these yields

f ∗ U ≤ lim sup |f (z)|. χ ˆU (z)→1

ˆU (zn ) → To obtain the reverse inequality, we take a sequence {zn }n in D satisfying χ 1 and f (zn ) → a. Let ζ be a cluster point of {zn }n in M (H ∞ ). Then χ ˆU (ζ) = 1 and f∗ (ζ) = a. Since
f ∗ dµζ , a = f∗ (ζ) = M(L∞ )

∗

we get |a| ≤ f supp µζ . Since 1=χ ˆU (ζ) =


M(L∞ )

χU dµζ ,

we must have supp µζ ⊂ U . Hence |a| ≤ f ∗ U , so that lim sup |f (z)| ≤ f ∗ U .

χ ˆU (z)→1

Therefore the lemma follows.




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Lemma 5.6. Let ϕ, ψ ∈ S(D) with ϕψ∞ = 1, and let U be the family of open and closed subsets U of M (L∞ ) with EL∞ (|ϕψ|) ⊂ U . Then λ∞ (ϕ, ψ) = inf lim sup Pϕ(z) − Pψ(z) 1 . U∈U χ ˆU (z)→1

Proof. Let U ∈ U. By (3.2), λ∞ (ϕ, ψ) = lim sup Pϕ(z) − Pψ(z) 1 ,

(5.3)

|(ϕψ)(z)|→1

so we can choose a sequence {zn }n in D such that |(ϕψ)(zn )| → 1 and Pϕ(zn ) − Pψ(zn ) 1 → λ∞ (ϕ, ψ).

(5.4)

Let ζ be a cluster point of {zn }n in M (H ∞ ). Then |(ϕψ)(ζ)| = 1 and 

  1= ϕψ dµζ  ≤ |ϕψ| dµζ ≤ 1, M(L∞ )

M(L∞ )

so that supp µζ ⊂ EL∞ (|ϕψ|) ⊂ U . Hence χ ˆU (ζ) = 1. This implies that χ ˆU (zn ) → 1. Therefore, it follows from (5.4) that λ∞ (ϕ, ψ) ≤ inf lim sup Pϕ(z) − Pψ(z) 1 . U∈U χ ˆU (z)→1

(5.5)

If λ∞ (ϕ, ψ) = 2, this inequality yields the desired assertion. So we assume that λ∞ (ϕ, ψ) < 2, and we prove the reverse inequality of (5.5). Fix a point ζ in M (H ∞ ) \ D and choose a net {zα }α in D with zα → ζ. Since EL∞ (|ϕψ|) is a closed Gδ -subset of M (L∞ ), there exists a function F in L∞ (∂D) such that F = 1 on EL∞ (|ϕψ|)

and 0 < F < 1 on M (L∞ ) \ EL∞ (|ϕψ|).

If Fˆ (zα ) → 1, then Fˆ (ζ) = 1 and supp µζ ⊂ EL∞ (|ϕψ|). Since λ∞ (ϕ, ψ) < 2, we have σ∞ (ϕ, ψ) < 1 by Lemma 3.5. This shows that ϕ = ψ on EL∞ (|ϕψ|). For, if ϕ(x) = ψ(x) for some x ∈ EL∞ (|ϕψ|), there exists a net {wα }α in D satisfying wα → x, and then ρ(ϕ(wα ), ψ(wα )) → 1. This contradicts σ∞ (ϕ, ψ) < 1. Since supp µζ ⊂ EL∞ (|ϕψ|), ϕ(ζ) = ψ(ζ). If |ϕ(ζ)| < 1, then we have Pϕ(zα ) − Pψ(zα ) 1 → 0. Accordingly, by recalling (5.3), we get lim sup Pϕ(z) − Pψ(z) 1 ≤ lim sup Pϕ(z) − Pψ(z) 1 = λ∞ (ϕ, ψ). Fˆ (z)→1

(5.6)

|(ϕψ)(z)|→1

∞ Since EL∞ (|ϕψ|) is a Gδ -subset ∞ of M (L ), there is a sequence of open and closed subsets {Un }n in U with n=1 Un = EL∞ (|ϕψ|). Here we shall prove that Fˆ (zα ) → 1 if and only if χ ˆUn (zα ) → 1 as α → ∞ for each n. To show this, suppose that Fˆ (zα ) → 1. Then Fˆ (ζ) = 1, so that supp µζ ⊂ EL∞ (|ϕψ|) ⊂ Un . Hence χ ˆUn (ζ) = 1. This implies that χ ˆUn (zα ) → 1 as α → ∞. Suppose that Fˆ (zα ) → 1. Then 0 < Fˆ (ζ) < 1, so that supp µζ ⊂ EL∞ (|ϕψ|). It follows that

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there exists a positive integer n with supp µζ ⊂ Un . Hence χ ˆUn (ζ) < 1, so that χ ˆUn (zα ) → χ ˆUn (ζ) = 1 as α → ∞. Thus we get lim sup Pϕ(z) − Pψ(z) 1 = inf lim sup Pϕ(z) − Pψ(z) 1 . n χ ˆUn (z)→1

Fˆ (z)→1

Therefore, we have by (5.6) inf lim sup Pϕ(z) − Pψ(z) 1 ≤ λ∞ (ϕ, ψ).

U∈U χ ˆU (z)→1




This, combined with (5.5), proves our assertion.

Proof of Theorem 5.3. If λ∞ (ϕ, ψ) = 2, each assertion is trivial, so we assume that λ∞ (ϕ, ψ) < 2. Then by Lemma 5.4, EL∞ (|ϕψ|) is an open and closed subset of EL∞ (|ϕ|) and EL∞ (|ψ|), respectively. (i) Suppose that EL∞ (|ϕψ|) = EL∞ (|ϕ|). Then there exist open and closed subsets Uϕ and Uψ of M (L∞ ) such that

(5.7) EL∞ (|ϕ|) ⊂ Uϕ , Uϕ ∩ EL∞ (|ψ|) \ EL∞ (|ϕψ|) = ∅, and



EL∞ (|ψ|) ⊂ Uψ , Uψ ∩ EL∞ (|ϕ|) \ EL∞ (|ϕψ|) = ∅. (5.8) By Theorem 5.1, both Mχ(M (L∞ )\Uϕ ) Cϕ and Mχ(M (L∞ )\Uψ ) Cψ are compact. Write U = Uϕ ∩Uψ . Then, as in the proof of Theorem 5.2, we find that for any f ∈ B(h∞ ), 

  Cϕ − Cψ − Mχ Cϕ + Mχ Cψ f  ∞ ∞ (M (L

)\Uϕ )

(M (L

)\Uψ )

∞

≤ MχUϕ Cϕ f − MχUψ Cψ f ∞ = χUϕ (f ◦ ϕ)∗ − χUψ (f ◦ ψ)∗ L∞ (∂D)

≤ max 1, (f ◦ ϕ)∗ − (f ◦ ψ)∗ U   = max 1, lim sup |f (ϕ(z)) − f (ψ(z))|

by Lemma 5.5

χ ˆU (z)→1

≤ max

  1, lim sup Pϕ(z) − Pψ(z) 1 . χ ˆU (z)→1

Hence Cϕ − Cψ e ≤ max

  1, lim sup Pϕ(z) − Pψ(z) 1 .

(5.9)

χ ˆU (z)→1

This holds for U = Uϕ ∩ Uψ for every Uϕ and Uψ satisfying (5.7) and (5.8). Therefore (5.9) holds as well for every open and closed subset U ∈ U, where U is the family of open and closed subsets of M (L∞ ) containing EL∞ (|ϕψ|). The estimate (i) follows now from Lemma 5.6. (ii) Suppose that EL∞ (|ϕ|) = EL∞ (|ψ|) (and hence EL∞ (|ϕψ|) = EL∞ (|ϕ|)). In this case, using the same reasoning as done in the proof of (i), both operators Mχ(M (L∞ )\U ) Cϕ are shown to be compact and   Cϕ − Cψ − Mχ

and Mχ(M (L∞ )\U ) Cψ

(M (L∞ )\U )

 Cϕ + Mχ(M (L∞ )\U ) Cψ f ∞

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≤ lim sup Pϕ(z) − Pψ(z) 1 χ ˆU (z)→1

for every U ∈ U, provided f ∈ B(h∞ ). Therefore the estimate (ii) also follows from Lemma 5.6.


6. The main theorem Combining the estimates given in Sections 4 and 5, we obtain the following complete characterization of Cϕ − Cψ e on h∞ . Theorem 6.1. Let ϕ, ψ ∈ S(D). (i) If max{ϕ∞ , ψ∞ } < 1, then Cϕ − Cψ e = 0. (ii) If max {ϕ∞ , ψ∞ } = 1 and ϕψ∞ < 1, then Cϕ − Cψ e = 1. (iii) If ϕψ∞ = 1 and EL∞ (|ϕ|) = EL∞ (|ψ|), then Cϕ − Cψ e = λ∞ (ϕ, ψ). (iv) If ϕψ∞ = 1 and EL∞ (|ϕψ|) = EL∞ (|ϕ|), then Cϕ − Cψ e = max{1, λ∞ (ϕ, ψ)}. Proof. (i) follows from Proposition 3.2. (ii) follows from Theorems 4.3 and 5.2. (iii) follows from Theorems 4.2 and 5.3(ii). (iv) follows from Theorems 4.2, 4.3 and 5.3(i).




Remark. The case (iv) above actually covers the case EL∞ (|ψ|) = EL∞ (|ϕ|) (with ϕψ∞ = 1); for the latter case can only happen if EL∞ (|ϕψ|) = EL∞ (|ϕ|)

or EL∞ (|ϕψ|) = EL∞ (|ψ|).

We here present some examples concerning (iii) and (iv) of Theorem 6.1. Example 6.2. This example comes from [9, pp. 1770–1771]. Let ϕ ∈ B(H ∞ ) be such that ϕ∞ = 1, and that
2π dθ > −∞. log(1 − |ϕ(eiθ )|) 2π 0 There exists an outer function ω ∈ H ∞ such that |ω| = 1 − |ϕ| a.e. on ∂D, see [6]. Let t be a number with 0 < t < 1. There exists a sequence {zn }n in D satisfying 1≤

1 − |ϕ(zn )| 1 < |ω(zn )| t

and |ϕ(zn )| → 1.

We may assume that 1 − |ϕ(zn )| → Reiθ1 , ω(zn )

1≤R≤

1 t

and ϕ(zn ) → eiθ2 .

Letting θ3 = θ1 + θ2

and ψt (z) = ϕ(z) + teiθ3 ω(z),

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we have ϕψt ∞ = 1 and EL∞ (|ϕ|) = EL∞ (|ψt |). By [9, p. 1771], we also have

t ≤t<1 ρ ϕ(z), ψt (z) ≤ 1 + |ϕ(z)| − |tϕ(z)| and



ρ ϕ(zn ), ψt (zn ) ≥

t2 . 2 − t2

By (3.3), it follows that t2 ≤ σ∞ (ϕ, ψt ) ≤ t < 1 2 − t2 which, by Lemma 3.5, yields 2

t 4 cos−1 2−t 4 cos−1 t 2 2− ≤ λ∞ (ϕ, ψt ) ≤ 2 − . π π Therefore, we obtain by Theorem 6.1(iii)

2−

4 cos−1 π

t2 2−t2

≤ Cϕ − Cψt e = λ∞ (ϕ, ψt ) ≤ 2 −

4 cos−1 t . π




Example 6.3. Let Ω be the simply connected domain defined by

Ω = D ∩ z = x + iy : y < (x − 1)2 and y > −(x − 1)2 which has a cusp at the point 1. Consider a Riemann map ψ from D onto Ω, with ψ(1) = 1. Clearly ψ ∈ B(H ∞ ). For each t > 0, define ϕt ∈ B(H ∞ ) by ϕt (z) =

ψ(z) + t . 1+t

Then ϕt ψ∞ = 1, EL∞ (|ϕt |)
EL∞ (|ψ|), and ψ = ϕt on EL∞ (|ϕt |). Also, we have (ϕt ψ)(z) → 1 if and only if z → 1. A manipulation shows

 ϕt (z) − ψ(z)  t . (6.1) ρ ϕt (z), ψ(z) =   =  1−ψ(z)  1 − ψ(z)ϕt (z) 1 + t + ψ(z)  1−ψ(z)

From the definitions of Ω and ψ, we can see that lim ψ(z)

(6.2)

z→1

1 − ψ(z) = 1. 1 − ψ(z)

Combining (6.1), (6.2) and (3.3), we get

σ∞ (ϕt , ψ) = lim ρ ϕt (z), ψ(z) =

t 2+t

z→1

which, by Lemma 3.5, gives λ∞ (ϕt , ψ) = 2 −

4 cos−1 π

t 2+t

.

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√ Note that λ∞ (ϕt , ψ) ≥ 1 if and only if t ≥ 2/( 2 − 1). Therefore, we finally use Theorem 6.1(iv) to conclude that √ Cϕt − Cψ e = 1 if and only if 0 < t ≤ 2/( 2 − 1) and Cϕt − Cψ e = 2 − √ for t ≥ 2/( 2 − 1).

4 cos−1

t 2+t

π


Acknowledgments The first and the third authors thank Professor K. J. Izuchi for his hospitality while they visited Department of Mathematics, Niigata University, where the work for part of this paper was done.

References [1] L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962), 547–559. [2] C.C. Cowen and B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. [3] T.W. Gamelin, Uniform Algebras, Prentice Hall, New Jersey, 1969. [4] J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. [5] P. Gorkin, R. Mortini and D. Su´ arez, Homotopic composition operators on H ∞ (B n ), Contemp. Math. 328 (2003), 177–188. [6] K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, New Jersey, 1962. [7] T. Hosokawa and K. Izuchi, Essential norms of differences of composition operators on H ∞ , J. Math. Soc. Japan 57 (2005), 669–690. [8] T. Hosokawa, K. Izuchi and S. Ohno, Topological structure of the space of weighted composition operators on H ∞ , Integral Equations Operator Theory 53 (2005), 509– 526. [9] T. Hosokawa, K. Izuchi and D. Zheng, Isolated points and essential components of composition operators on H ∞ , Proc. Amer. Math. Soc. 130 (2002), 1765–1773. [10] B.D. MacCluer, S. Ohno and R. Zhao, Topological structure of the space of composition operators on H ∞ , Integral Equations Operator Theory 40 (2001), 481–494. [11] J.H. Shapiro, Composition Operators and Classical Function Theory, SpringerVerlag, New York, 1993. [12] L. Zheng, The essential norms and spectra of composition operators on H ∞ , Pacific J. Math. 203 (2002), 503–510. [13] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990.

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Jun Soo Choa Department of Mathematics Education Sungkyunkwan University Jongro-gu, Seoul 110-745 Korea e-mail: [email protected] Kei Ji Izuchi Department of Mathematics Niigata University Niigata 950-2181 Japan e-mail: [email protected] Shˆ uichi Ohno Nippon Institute of Technology Miyashiro, Minami-Saitama 345-8501 Japan e-mail: [email protected] Submitted: January 22, 2007 Revised: February 25, 2008

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Integr. equ. oper. theory 61 (2008), 187–210 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/020187-24, published online April 17, 2008 DOI 10.1007/s00020-008-1586-5

Integral Equations and Operator Theory

Transitive Spaces of Operators Kenneth R. Davidson, Laurent W. Marcoux and Heydar Radjavi Abstract. We investigate algebraic and topological transitivity and, more generally, k-transitivity for linear spaces of operators. In finite dimensions, we determine minimal dimensions of k-transitive spaces for every k, and find relations between the degree of transitivity of a product or tensor product on the one hand and those of the factors on the other. We present counterexamples to some natural conjectures. Some infinite dimensional analogues are discussed. A simple proof is given of Arveson’s result on the weak-operator density of transitive spaces that are masa bimodules. Mathematics Subject Classification (2000). 15A04, 47A15, 47A16, 47L05. Keywords. Transitive subspace, k-transitive, k-separating.

A collection E of operators from a vector space X to another vector space Y is said to be transitive if, given nonzero x ∈ X and y ∈ Y, there is an element A ∈ E such that Ax = y. If E has some additional structure imposed on it, for example, if it is a ring or an algebra with X = Y, then a great deal is implied by the transitivity assumption. This paper represents an effort to extract as much information as possible from transitivity (and its strengthening given below) from the linear structure on E alone. This paper could well have been written twenty years ago. Indeed, it is in the spirit of Azoff’s paper [2] which explores transitive, reflexive, and elementary spaces and the relations between these properties. When we examine infinite dimensional analogues, the Toeplitz operators are a natural source of examples. The paper of Azoff and Ptak [3] contains some related results about the space of Toeplitz operators. Let H and K be Hilbert spaces. A subspace L of B(H, K) is k-transitive provided that for every choice of k linearly independent vectors x1 , . . . , xk in H and arbitrary vectors y1 , . . . , yk in K, there is an A ∈ M so that Axi = yi for 1 ≤ i ≤ k. Thus transitivity coincides with 1- transitivity. L is topologically ktransitive if this can be done approximately; that is, for each ε > 0 there exists A ∈ L such that Axi − yi  < ε, 1 ≤ i ≤ k. In the finite dimensional setting, Authors partially supported by NSERC grants.

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the two notions coincide. As with transitivity, if k = 1, we shall abbreviate the notation and refer to a space as being topologically transitive. Starting with finite dimensions, we present a number of positive results and a lot of counterexamples to natural conjectures. Among other things, we determine the minimal dimension of k-transitive subspaces of B(H, K) in terms of the dimensions of H and K. We consider spans of products as well as tensor products of spaces, and study the relations between their degree of transitivity and those of their constituent spaces. We investigate the relations between the minimal and maximal ranks present in a transitive subspace of Mn . In particular, we show that such a subspace must contain invertible elements. In the infinite dimensional setting, there are fewer positive results and more counterexamples. We are able to extend some of the finite dimensional results. We provide a simple proof of a result of Arveson [1] that a topologically transitive subspace of B(H, K) which is a masa bimodule is wot-dense in B(H, K). Any transitive subalgebra of Mn = B(Hn ), the space of n × n complex matrices, is equal to all of Mn by Burnside’s Theorem. The situation is completely different for subspaces. For every 0 ≤ k < min{m, n}, there are k- transitive subspaces of Mmn which are not (k + 1)-transitive (see Example 1.4). In infinite dimensions, a topologically transitive operator algebra has no proper invariant subspaces. Whether it is wot-dense in B(H) is the famous Transitive Algebra Problem, a generalization of the Invariant Subspace Problem. Again, if we consider subspaces, there are many proper k-transitive wot-closed subspaces. It is a well-known result of Azoff [2] that a subspace L of Mmn = B(Hn , Hm ) is k-transitive if and only if the pre-annihilator L⊥ contains no non-zero elements of rank at most k. Here (Mmn )∗ is identified with Mnm equipped with the trace norm via the bilinear pairing A, T  = Tr(AT ).
k ∗ Indeed suppose that 0 = T = i=1 xi yi belongs to L⊥ , where the vectors xi are linearly independent. Then the k-tuple (Ax1 , . . . , Axk ) is orthogonal to (y1 , . . . , yk ) for all A ∈ L. Thus L is not k-transitive. Conversely, if L is not k-transitive, then for some linearly independent set x1 , . . . , xn , the k-tuples k (Ax1 , . . . , Axk ) span a proper subspace of Hm , and thus is orthogonal to a non-zero
k vector (y1 , . . . , yk ). Reversing the argument shows that T = i=1 xi yi∗ belongs to L⊥ . In the infinite-dimensional setting, we identify (B(H, K))∗ with the space C1 (K, H) of trace class operators from K to H via the same bilinear pairing. The above Theorem of Azoff then applies to topological k-transitivity, and shows that a subspace L of B(H, K) is topologically k-transitive if and only if the pre-annihilator L⊥ of L contains no non-zero elements of rank less than or equal to k. We end this introduction with a simple but handy observation which will be used implicitly throughout this paper, namely: if L ⊆ B(H, K) is topologically k-transitive, and if P ∈ B(H) and Q ∈ B(K) are projections which satisfy min(rank P, rank Q) ≥ k, then QLP ⊆ B(P H, QK) is also topologically

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k-transitive. (This follows immediately from Azoff’s Theorem.) If, furthermore, P and Q are finite rank, then QLP is in fact k-transitive. We are indebted to the referee for providing us with a number of very useful references and helpful comments – in particular, the argument of Theorem 2.4.

1. Dimension In this section, we find the minimal dimension of a k-transitive subspace of Mmn . By Azoff’s Theorem as outlined above, if k ≥ min(m, n), then L⊥ = 0 and hence L = Mmn . As such, we always assume that k < min(m, n). Lemma 1.1. For every p ≥ 1 and every 0 ≤ k ≤ p, the diagonal subalgebra Dp of Mp contains a subspace of dimension p − k containing no non-zero element of rank at most k. Proof. It suffices to choose p − k diagonal operators with the property that when restricted to any p − k diagonal entries, they are linearly independent. For then a linear combination which has p − k zeros must be zero. An example of such a
sequence is Dj = diag(1j , 2j , . . . , pj ) for 0 ≤ j < p − k. The special case of the following result for k = 1 was established by Azoff [2]. Theorem 1.2. The minimal dimension of a k-transitive subspace of Mmn is k(m + n − k). Proof. Using Azoff’s Theorem, we search instead for the maximal dimension of the pre-annihilator of a k-transitive subspace L. This subspace L⊥ cannot intersect the closed variety Rk of matrices of rank at most k except in {0}. This variety has dimension k(m+n−k) (see [7, Prop. 12.2]). It follows that dim L⊥ +dim Rk ≤ mn. Hence dim L ≥ dim Rk = k(m + n − k). On the other hand, consider the subspace N of Mnm which has zeros on diagonals of length p ≤ k and has dimension p − k on the diagonals with p > k entries such that the rank of any non-zero element on one such diagonal is always at least k + 1. This is possible by Lemma 1.1. There are m + n − 1 diagonals, and 2k of them have length at most k. So the dimension of N is mn − (m+n−1−2k)k − 2

k 

p = mn − (m+n−1−2k)k − (k+1)k

p=1

= mn − k(m + n − k) Thus L = N ⊥ , the annihilator of N , has dimension k(m + n − k). Consider a non-zero element N of N . It must be non-zero on some diagonal. Let p0 be the shortest non-zero diagonal. Consider the square submatrix containing the p0 th diagonal as its main diagonal. This submatrix is triangular, and hence its rank is at least as great as the rank of the diagonal, which is at least k+1. Hence N contains no non-zero elements of rank at most k. Therefore L is k-transitive.


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Example 1.3. By the above theorem, any proper (n − 1)-transitive subspace L of Mn necessarily has dimension n2 − 1. It is then not hard to see that we can find invertible elements S, T ∈ Mn so that SLT = sln , the space of trace zero matrices in Mn . Since multiplying a k-transitive subspace M of B(H, K) by invertibles S ∈ B(K, K ) and T ∈ B(H , H) yields a k-transitive subspace of B(H , K ), we shall think of M and SMT as being equivalent insofar as transitivity is concerned. A similar statement holds for topologically k-transitive spaces. Example 1.4. As mentioned in the introduction, there are k-transitive subspaces of B(H, K) which are not (k + 1)-transitive provided that k < min{dim H, dim K}. For example, if H = Hn and K = Hm , then it suffices to consider R ∈ Mnm of rank k + 1. With M = span{R}, L = M⊥ ⊆ Mmn is such an example. This example can easily be adapted to the infinite-dimensional setting to produce subspaces L ⊆ B(H, K) which are topologically k-transitive but not topologically (k + 1)-transitive. Example  As shown in Example 3.4 of [2], the space Tn of all Toeplitz matrices  1.5. T = ti−j in Mn is a transitive subspace of dimension 2n − 1. It is routine to verify that the pre-annihilator of Tn consists of those matrices in Mn whose entries along any diagonal sum to zero. The rank of such a matrix is at least as big as the length of the smallest non-zero diagonal, which is at least two if the matrix is non-zero. By Theorem 1.2, this is the minimal possible dimension of a transitive subspace of Mn . As pointed out by the referee, there are a number of other papers which investigate the relationship between the dimension of a linear subspace S of matrices and the ranks of its members. For example, H. Flanders [6] has shon that if S ⊆ Mn is a linear space and the rank of every element of S is at most k, then dim S ≤ kn, and he classified those spaces S of maximal dimension. More recently, R. Meshuˇ lam and P. Semrl [8] have shown that if min {rankS : 0 = S ∈ S} > dim S, then S is reflexive as a space of operators. In other words, S⊥ is spanned by its rank one elements.

2. Dually Transitive In finite dimensions, we can consider both L and L⊥ as spaces of matrices. So we can ask whether both can be k-transitive. Dimension arguments show that this requires the space to be sufficiently large. Proposition 2.1. There is a subspace L of M4 such that both L and L⊥ are transitive.

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Proof. It suffices to exhibit a transitive subspace L so that neither L nor L⊥ contains a rank one element. Let Φ be a linear bijection of M2 onto itself with four distinct eigenvalues whose corresponding eigenvectors all have rank 2. Define    A Φ(B) L= : A, B ∈ M2 . B A  ∗  xu xv ∗ Suppose that L contains a rank 1 element , where x, y, u, v are yu∗ yv ∗ vectors in C2 . Then comparing the diagonal entries shows that xu∗ = yv ∗ , so that xv ∗ is a multiple of yu∗ , say xv ∗ = λyu∗ . So now a comparison of the off-diagonal entries shows that Φ(yu∗ ) = λyu∗ , contrary to fact. Therefore L contains no rank 1 elements. The analysis of L⊥ is similar since Φt has the same eigenvector property.
a b  d 2c Example 2.2. For example, let Φ = . Then b a c d     a b −h −g a b h 2g  c c d f e  d −f −e  .  L=  e f a b  and L⊥ =  e f −a −b  g/2 h −c −d g h c d Here is a general technique modelled on the previous example. √ Theorem 2.3. If 1 ≤ k < n/(2 + 2), then M2n contains a subspace L such that both L and L⊥ are k-transitive. Proof. The proof of Theorem 1.2 shows that there is a subspace N of Mn with dimension√(n − k)2 which contains no non-zero elements of rank k or less. Since n > (2 + 2)k, it is easy to check that (n − k)2 ≥ n2 /2. Thus there is an injective linear map T : Mn /N → N . Let Φ = JT Q, where Q is the quotient map of Mn onto Mn /N and J is the injection of N into Mn . Observe that for A ∈ Mn , rank A ≥ k + 1

or

rank Φ(A) ≥ k + 1.

Indeed, if Φ(A) = 0, then it has rank at least k + 1. If Φ(A) = 0, then A ∈ N ; so it has rank at least k + 1. Define a subspace L of M2n consisting of all elements of the form   A B Φ(B) Φ(A) where A, B ∈ Mn are arbitrary. A non-zero element of L has either A or B nonzero. So at least one of the four matrix entries has rank at least k +1. In particular, L⊥ is k transitive.  t  Φ (X) −Y Note that L⊥ consists of all matrices of the form where X, Y ∈ Φt (Y ) −X Mn are arbitrary. So the same argument shows that L⊥ contains no non-zero elements of rank at most k. Therefore L is k-transitive.


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We thank the referee for the following argument which significantly sharpens the above result. Theorem 2.4. There is a subspace L of Mmn such that if L is k-transitive and L⊥ is l- transitive if and only if k, l ≤ min{m, n} and k(m+n−k)+l(m+n−l) ≤ mn. Proof. Recall that for 1 ≤ t ≤ min {m, n}, Rt denotes the variety of m×n matrices whose rank does not exceed t, and that dim Rt = t(m+ n− t). The subspace L has the property that L ∩ Rl = {0} and L⊥ ∩ Rk = {0}. Thus dim L + dim Rl ≤ mn, and dim L⊥ + dim Rk ≤ mn. Therefore mn = dim L + dim L⊥ ≤ 2mn − dim Rl − dim Rk . Hence we require dim Rk + dim Rl ≤ mn. Conversely, suppose that dim Rk + dim Rl ≤ mn. Let d = dim Rk = k(m + n−k); and look for a subspace L of dimension d as above. Among the Grassmanian space Gd of d- dimensional subspaces of Mmn , the subset of subspaces L with L ∩ Rl = {0} is Zariski open and non-empty. In particular, it is a dense open subset in the usual topology. (We postpone this argument to the end of the proof.) Similarly, L⊥ belongs to Gmn−d , the space of (mn − d)-dimensional subspaces of Mnm . Since (mn − d) + dim Rk = mn, the properaty that L⊥ ∩ Rk = {0} is also generic. So this also corresponds to a dense open set of subspaces L of Gd . Thus generically, both properties hold and many such subspaces exist. Finally, we give the geometric argument to explain the genericity. Consider the projective variety Rl in the projective space Pmn−1 = Mmn . Form the space Gd × Pmn−1 and consider the subset Σ = {(G, P ) : P ∈ G}. Then Σ is Zariski closed because it is the set of elements of rank (at most) d. (Here we may consider Gd and Pmn−1 as the spaces of projections in L(Mmn ) of ranks d and 1, respectively. Then the rank of (G, P ) is just its bona fide rank in M12 (L(Mmn )). ) Now Gd × Rl is a subset of Gd × Pmn−1 , and thus (Gd × Rl ) ∩ Σ is Zariski closed in Gd × Rl . Project onto the first coordinate, and note that this is precisely {G ∈ Gd : G ∩ Rl = ∅}; i.e., the set of subspaces L containing a non-zero element of rank at most l. This is therefore Zariski closed and clearly it is a proper subspace (see Theorem 1.2). So the complement is Zariski open. Since Gd is an irreducible variety, it is open and dense.
Corollary 2.5. There exists√a subspace L ⊆ Mn such that L and L⊥ are k-transitive if and only if k < n/(2 + 2). Proof. When m = n and k = l, the inequality of Theorem 2.4 reduces to n2 − 4kn + 2k 2 ≥ 0,

√ √ or 2(k − n)2 ≥ n2 . Since k ≤ n/2, we obtain k ≤ n(1 − 1/ 2) = n/(2 + 2). This is never an equality.


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Example 2.6. Theorem 2.4 above implies in particular that there should exist a subspace L ⊆ M3,4 of dimension 6 so that L and L⊥ are transitive. Let us exhibit such an example. Consider     a b c 2d  L =  e f a b  : a, b, c, d, e, f ∈ C .   d c f e Observe that if L contains a rank one matrix L, then the compression P LQ of that matrix to any 2 × 2 block must have determinant equal to zero. We consider two cases:      f a   b 2d      = 0 yields Case 0: a = 0. From  = 0, we get f = 0. Then  c f   f b       a 2d   e b   b c  =0      = 0, gives e = 0; and  = 0 gives c = 0;  b = 0. So  d e  d e  c f  shows that d = 0. Thus L =     0.    a c   = 0, we get ec = 1; and  e f  = 0 yields Case 1: a = 1. From    d c   e a   c 2d   = 0 leads to the erroneous conclusion 1 = ec = 2df = df = 1. Therefore  f e  2, a contradiction. Thus L does not contain a rank one operator. A similar analysis may be applied to    u y −x        v z −w   L⊥ =  , : u, v, w, x, y, z ∈ C w −u −z       1  −y 2 x −v to prove that L⊥ does not contain any rank one matrices. From this we conclude that L and L⊥ are both transitive.

3. Tensor Products In this section, we consider tensor products of k-transitive subspaces. The first lemma is well known, but is included for the convenience of the reader. Lemma 3.1. Let L and M be subspaces of Mlp and Mmn respectively. Then (L ⊗ M)⊥ = L⊥ ⊗ Mnm + Mpl ⊗ M⊥ . Proof. Clearly L⊥ ⊗ Mnm + Mpl ⊗ M⊥ is contained in (L ⊗ M)⊥ . The other containment follows from a simple dimension argument. Indeed, let dim L = d1 and dim M = d2 . Then dim L ⊗ M = d1 d2 , from which we deduce that dim(L ⊗ M)⊥ = lpmn − d1 d2 . Now dim(L⊥ ⊗ Mnm ) = (lp − d1 )mn and dim(Mpl ⊗ M⊥ ) = lp(mn − d2 ). Since L⊥ ⊗ Mnm ∩ Mpl ⊗ M⊥ = L⊥ ⊗ M⊥

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has dimension (lp − d1 )(mn − d2 ), it follows that dim(L⊥ ⊗ Mnm + Mpl ⊗ M⊥ ) = (lp − d1 )mn + lp(mn − d2 ) − (lp − d1 )(mn − d2 ) = lpmn − d1 d2 .


From this the result follows easily.

The main theorem of this section shows that under additional hypotheses, tensoring preserves some level of transitivity. Theorem 3.2. Suppose that a subspace L ⊂ Mlp is k-transitive and is spanned by its elements of rank at most r. If a subspace M ⊂ Mmn is rk-transitive, then L ⊗ M is k-transitive.   Proof. Let T = Tji ∈ Mpl ⊗ Mnm , considered as a p × l matrix with coefficients Tji ∈ Mnm. Itfollows from Lemma 3.1 that (L ⊗ M)⊥ = L⊥ ⊗ Mnm + Mpl ⊗ M⊥ . Now T = Tji ∈ L⊥ ⊗ Mnm = (L ⊗ Mnm )⊥ if and only if p l  

  lij Tji = 0 for all L = lij ∈ L,

i=1 j=1

while T ∈ Mpl ⊗ M⊥ = (Mpl ⊗ M)⊥ if and only if each Tji ∈ M⊥ .     We may decompose T = R + S where R = Rji ∈ L⊥ ⊗ Mnm and S = Sji ∈ Mpl ⊗ M⊥ .   For L = lij ∈ L, consider p l  

lij Tji =

i=1 j=1

p l  

lij (Rji + Sji ) =

i=1 j=1

p l  

lij Sji ∈ M⊥ .

(1)

i=1 j=1

Suppose that rank T = 1. Then there are vectors uj and vi for 1 ≤ j ≤ p and 1 ≤ i ≤ l so that Tji = uj vi∗ . Now if L is rank one, then there are scalars xi and yj so that lij = xi yj . In this case, p l   i=1 j=1

lij Tji =

p l   i=1 j=1

xi yj uj vi∗ =

p  j=1

yj u j

l  

xi vi

∗

i=1

is a rank one matrix. (As pointed out by the referee, this step is similar to the proof that the set of operators of rank 1 is stable under Schur multiplication.) It is then easy to see that if rank T ≤ k and rank L ≤ r, then this sum has rank at most rk. Clearly it suffices to satisfy equation (1) for a spanning subset of L. So we may suppose that each L has rank at most r. Consider an element T ∈ (L ⊗ M)⊥ with rank at most k. The analysis of the previous paragraph yields an element of

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M⊥ with rank at most rk. As M is rk-transitive, this means that these sums are all zero: p l     lij Sji = 0 for all L = lij ∈ L. i=1 j=1

Consequently, S ∈ L⊥ ⊗ Mnm . Since R is also in this set, we conclude that T ∈ L⊥ ⊗ Mnm . But L⊥ admits no non-zero operators of rank at most k, and thus neither does L⊥ ⊗ Mnm . To see this, think of such matrices as n × m matrices with coefficients in L⊥ . Any non-zero coefficient results in rank at least k + 1. Hence (L ⊗ M)⊥ contains no non-zero elements of rank at most k, and therefore L ⊗ M is k-transitive.
We shall revisit this result in the infinite-dimensional setting in Section 6 (see Theorem 6.12). Definition 3.3. A subspace L ⊂ Mmn is fully k-transitive if L ⊗ M is k-transitive whenever M is k-transitive. It is clear that the tensor product of fully k-transitive spaces is again fully k-transitive. The following corollary yields a large class of fully k-transitive subspaces. This is an immediate application of Theorem 3.2 taking r = 1. Corollary 3.4. If L ⊂ Mlp is a k-transitive subspace which is spanned by its rank one elements, then L is fully k-transitive. Another easy consequence uses the fact that L ⊂ Mlp is always spanned by elements of rank at most min{l, p}. Corollary 3.5. If L ⊂ Mlp is a k-transitive subspace and M ⊂ Mmn is min{kl, kp}transitive, then L ⊗ M is k-transitive. Fully k-transitive spaces have a certain permanence. Proposition 3.6. If L ⊂ Mlp is fully k-transitive, and P, Q are idempotents in Ml and Mp respectively, then P LQ ⊂ B(QHp , P Hl ) is fully k-transitive. Proof. Let M ⊂ Mmn be a k-transitive space. If A ∈ (P LQ)⊥ , then 0 = Tr(P LQA) = Tr(LQAP ) for all L ∈ L. So QAP ∈ L⊥ , where QAP is just A considered as an element of Mlp rather than B(P Hl , QHp ). Therefore (P LQ ⊗ M)⊥ = (P LQ)⊥ ⊗ Mnm + QMpl P ⊗ M⊥ ⊂ L⊥ ⊗ Mnm + Mpl ⊗ M⊥ = (L ⊗ M)⊥ The right hand side contains no rank k matrices, and so neither does the left side. Therefore P LQ is fully k-transitive.


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Example 3.7. The space of Toeplitz matrices Tm is fully transitive because the rank  one matrices ai−j for a ∈ C span Tm . To see this, just observe that the entries on the first row and column determine T , and that this may be any vector in C2m−1 . The rank one matrices mentioned above correspond to the vectors (ak )|k|<m . Any choice of 2m − 1 distinct non-zero values of a yields a basis. Example 3.8. The space sld ⊂ Md from Example 1.3 is fully (d−1)- transitive since it is spanned by the rank one matrices {Eij : i = j} and {E11 + E1j − Ej1 − Ejj : 2 ≤ j ≤ m}. this space as those Example 3.9. Consider  L= sld ⊗ Mm ⊂ Mdm . We may think of
d d × d matrices A = Aij with coefficients in Mm such that i=1 Aii = 0. Then L⊥ = (sld )⊥ ⊗ Mm = CId ⊗ Mm . Thus the minimum rank of a non-zero element of L⊥ is d. Therefore L is (d − 1)-transitive. It is spanned by its rank one elements since both sld and Mm are. Hence L is fully (d − 1)-transitive. Example 3.10. There are transitive spaces which are not fully transitive. Consider the space L from Example 2.2. Evidently, the smallest rank of a non-zero element of L is 2. We will show that (L⊥ ⊗ L⊥ )⊥ contains a rank one. By symmetry, it follows that neither L ⊗ L nor L⊥ ⊗ L⊥ is transitive. By Lemma 3.1, (L⊥ ⊗ L⊥ )⊥ = L ⊗ M4 + M4 ⊗ L. To find a rank 1 in this space, we look for matrices A, . . . , H in M4 , Xij ∈ L and vectors ui , vj ∈ C4 for 1 ≤ i, j ≤ 4 so that     A B H 2G  C D F E   Xij + ui vj∗      E F A B  =  G H C D This is equivalent to solving the system u1 v1∗ − u3 v3∗ = X33 − X11 ∈ L

u2 v4∗ − u3 v1∗ = X31 − X24 ∈ L

u2 v1∗ − u4 v3∗ = X43 − X21 ∈ L

u1 v3∗ − u4 v2∗ = X42 − X13 ∈ L

u1 v2∗ − u3 v4∗ = X34 − X12 ∈ L u2 v2∗ − u4 v4∗ = X44 − X22 ∈ L

u2 v3∗ − u3 v2∗ = X32 − X23 ∈ L u1 v4∗ − 2u4 v1∗ = 2X41 − X14 ∈ L

Let e1 , . . . , e4 be the standard basis for C4 . One can check that         e4 v1 e4 u1     u2  e3    =   and v2  =  e3  v3  −e2  u3  e2  u4 e1 v4 −e1 is a non-trivial solution.

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4. Spans of products While transitive subspaces are not algebras, an algebra can be obtained by taking spans of products. Thus in the matrix case, one eventually obtains Mn . In many cases, this happens very quickly. In particular, the order of transitivity increases quickly. Proposition 4.1. If L ⊂ Ml is a transitive subspace which is spanned by its rank one elements, then span L2 = Ml . Proof. L has no kernel; so we may select l non-zero rank one elements Ri = xi yi∗ in L such that {y1 , . . . , yl } forms a basis for Hl . Every matrix in Ml may be written as
l
l T = i=1 ui yi∗ . Choose Ai ∈ L so that Ai xi = ui . Then T = i=1 Ai Ri belongs to span L2 .
Example 4.2. Consider the space L from Example 2.2.    a1 b1 h1 2g1 a2  c1 d1 f1 e1   c2   X =  e1 f1 a1 b1  and Y =  e2 g1 h1 c1 d1 g2

Let b2 d2 f2 h2

h2 f2 a2 c2

 2g2 e2   b2  d2

be arbitrary elements of L. Let Z = XY , and consider the diagonal elements zii of Z, 1 ≤ i ≤ 4. Then z11 = a1 a2 + b1 c2 + h1 e2 + 2g1 g2 z22 = c1 b2 + d1 d2 + f1 f2 + e1 h2 z33 = e1 h2 + f1 f2 + a1 a2 + b1 c2 z44 = 2g1 g2 + h1 e2 + c1 b2 + d1 d2 . Thus z11 + z22 = z33 + z44 for all Z = [zij ] ∈ L2 . Since I ∈ L, span{L, L2 } = span L2 = M4 . and D3 , while Note that span{L, L2 , L3 } = span L3 = M4 . The following concept is a substantial weakening of the notion of k-transitivity. Definition 4.3. A subspace L ⊂ B(H, K) is k-separating if for every set x1 , . . . , xk of independent vectors in H, there is an L ∈ L so that Lxi = 0 for 1 ≤ i ≤ k − 1 and Lxk = 0. The simple result below shows why the property of being k-separating is nice to have when considering products of spaces. Lemma 4.4. Suppose that L1 , L2 are subspaces of Mn with L1 transitive and L2 k-separating. Then span L1 L2 is k-transitive. Proof. Let x1 , . . . , xk be independent vectors, and let vectors y1 , . . . , yk be given. Use the k-separating property to select elements B1 , . . . , Bk in L2 such that Bi xj = δij zi , where zi are non-zero vectors. By the transitivity of L1 , select elements

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A1 , . . . , Ak in L1 so that Ai zi = yi . Then ki=1 Ai Bi belongs to span L1 L2 and takes xi to yi for 1 ≤ i ≤ k. Therefore this space is k-transitive.
Since Mn is the unique n-transitive subspace of itself, we obtain: Corollary 4.5. Suppose that L1 , L2 are subspaces of Mn with L1 transitive and L2 n-separating. Then span L1 L2 = Mn . Lemma 4.6. Let 1 ≤ k < min{m, n}. Then a k-transitive subspace L of Mmn is (k + 1)-separating. Proof. Let x1 , . . . , xk+1 be linearly independent in Cn ; and set X = span{x1 , . . . , xk+1 }. Then the restriction M = L|X ⊂ Mm,k+1 is k-transitive. By Theorem 1.2, dim M ≥ k(m + 1). The subspace of M which vanishes on span{x1 , . . . , xk } has km linear conditions imposed, and hence it has dimension at least k. Thus there
are elements of M which annihilate x1 , . . . , xk and are non-zero on xk+1 . Example 4.7. The set Tn of n × n Toeplitz matrices (n ≥ 3) is an example of a space which is 2-separating but not 3-separating. Indeed, the fact that Tn is 1-transitive implies that it is 2-separating by the above result. On the other hand, if the first and last columns of a matrix T in Tn are both zero, then all entries of T are zero, so that Tn is not 3-separating. Example 4.8. The converse of Lemma 4.6 is false. If M is a k-transitive but not (k + 1)-transitive subspace of Mmn , consider   x where x is an arbitrary row vector the subspace L of Mm+1,n of the form L = M in Cn and M ∈ M. Then this is an n-separating space which is k-transitive but not (k + 1)-transitive. In particular, if we take M = {0}, then L is n-separating but is not even 1-transitive. Theorem 4.9. Suppose that subspaces K ⊂ Mmn and L ⊂ Mnp are k-transitive and l-transitive respectively. Then the product span KL is min{k + l, m, p}-transitive. Proof. We know that K⊥ contains no non-zero element of rank at most k; and L⊥ contains no non-zero element of rank at most l. Assume first that l < min{n, p}. Suppose that A ∈ (KL)⊥ satisfies 1 ≤ rank A ≤ k + l. Then 0 = Tr(KLA) for all K ∈ K and L ∈ L. Hence LA ∈ K⊥ for all L ∈ L. As L is (l + 1)-separating by Lemma 4.6, select L ∈ L which is non-zero on some vector in the range of A and annihilates min{l, rank A − 1} independent vectors in the range of A. Then 1 ≤ rank LA ≤ (k + l) − l = k. This contradicts the k-transitivity of K. Thus span KL is (k+l)-transitive if k+l < min{m, p}. But if k + l ≥ min{m, p}, then this shows that (KL)⊥ = {0}. Hence span KL = Mmp is min{m, p}-transitive.

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We obtain a similar conclusion if k < min{m, n}. If k = min{m, n} and l = min{n, p}, then K = Mmn and L = Mnp . Thus span KL = Mmp is min{m, p} transitive.
For M a subspace of Mn , let M∗ = {M ∗ : M ∈ M}, again considered as a subspace of Mn . Proposition 4.10. If L ⊂ Mn is transitive and spanned by its rank r elements and M∗ ⊂ Mn is r-separating, then span LM = Mn . Proof. If L ∈ L has rank at most r, then for any vector
u ∈ LH and 0 = x ∈ H, we will show that ux∗ ∈ LM. Indeed, we may write L = si=1 ui vi∗ where u1 = u, s ≤ r and {v1 , . . . , vs } are linearly independent. Select M ∈ M so that M ∗ v1 = x and M ∗ vi = 0 for 2 ≤ i ≤ s. Then s s   LM = ui vi∗ M = ui (M ∗ vi )∗ = ux∗ . i=1

i=1

As the ranges of elements (of rank at most r) of L span H, the result follows.
If M ⊆ Mn is r-transitive for some r ≥ 1, then so is M∗ . This is easily seen by considering Azoff’s characterisation of r-transitivity in terms of the preannihilator of M. It then follows from Lemma 4.6 that M∗ is (r + 1)-separating. When r = 0, no such statement holds. So we obtain: Corollary 4.11. If L ⊂ Mn is transitive and spanned by its rank r elements and M ⊂ Mn is max{r − 1, 1}-transitive, then span LM = Mn . This allows an extension of Proposition 4.1 and Example 4.2. Corollary 4.12. If L ⊂ Mn is transitive and is spanned by its rank r elements, then span Lr+1 = Mn . Proof. By Theorem 4.9, span Lr is r-transitive. So by the previous lemma,
span Lr+1 = Mn .

5. Invertibles Proposition 5.1. If L ⊂ Mn is a subspace consisting of singular matrices, then L is not transitive. Proof. Let k be the largest rank of an element of L, and fix A ∈ L with rank A = k. Transitivity is unchanged if L is multiplied on either side by invertible operators. So after such a change, we may suppose that A = A2 = A∗ is a projection. Decompose H = AH  ⊕ (I −  A)H. With thisdecomposition,  each element L0 E L0 + tIk E L ∈ L has the form L  . Since L + tA = , the 1, 1 entry F L1 F L1 is invertible for large t, and so it factors as     L0 +tIk 0 0 Ik (L0 +tIk )−1 E Ik . F In−k 0 L1 −F (L0 +tIk )−1 E 0 In−k

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From this, it follows that k ≥ rank(L + tA) = k + rank(L1 − F (L0 + tIk )−1 E). Therefore L1 = F (L0 + tIk )−1 E. As the right side tends to 0 as t → ∞, L1 = 0 for all L ∈ L. This shows that (I − A)L(I − A) = 0 and so L is not transitive.
Proposition 5.2. If L ⊂ Mn is transitive, let r be the minimal rank of non-zero elements of L and let s be the largest rank of singular elements of L. Then r+s ≥ n. Proof. Let F ∈ L with rank F = r. Suppose first that there is an invertible element A ∈ L such that 0 = λ ∈ σ(A−1 F ). Then 0 = det(λI − A−1 F ) = det(A−1 ) det(λA − F ). As det(A−1 ) = 0, λA − F is singular and thus has rank at most s. But clearly it has rank at least n − r. So r + s ≥ n. Otherwise, for every invertible A in L, A−1 F is nilpotent. By our Proposition 5.1, L contains invertible elements. Select an invertible B ∈ L so that among the elements of the form A−1 F for invertible A in L, the operator F0 = B −1 F is nilpotent of the greatest index, say m+ 1. That is, F0m = 0 = F0m+1 = (A−1 F )m+1 for all A ∈ L which are invertible. For any L ∈ L, and sufficiently small µ, the operator B − µL is invertible. So  m+1  m 0 = (B − µL)−1 F0 = (I − µB −1 L)−1 F0 (I − µB −1 L)−1 F0
Therefore expanding (I − µB −1 L)−1 = k≥0 µk (B −1 L)k ,  m  k = µ Xk . 0 = F0 (I − µB −1 L)−1 F0 k≥0

All coefficients of this power series must vanish, and in particular 0 = X1 = F0 B −1 LF0m + F02 B −1 LF0m−1 + · · · + F0m B −1 LF0 . Multiply on the left by F0m−1 to obtain F0m B −1 LF0m = 0 for all L ∈ L. This means that L Ran F0m ⊂ ker F0m B −1 . Both Ran F0m and ker F0m B −1 = B ker F0m are proper subspaces, so this contradicts the transitivity of L. This contradiction establishes the result.


6. Infinite dimensional results In this section we examine to what degree the results of the previous sections extend to the infinite dimensional setting. As we shall see, there are more negative results than positive results. We begin with an infinite dimensional version of Proposition 5.1. Recall that if T ∈ B(H), then an element λ of the spectrum σ(T ) of T is  called a Riesz point if λ is an isolated point of σ(T ), and if k≥1 ker(λI − T )k is finite dimensional. In particular, λ is not an element of the essential spectrum σe (T ) of T .

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Theorem 6.1. Let L be a subspace of B(H) consisting of singular operators. If L contains an operator A with 0 as a Riesz point of its spectrum, then L is not topologically transitive.   A0 0 Proof. After a similarity, A  where H = M ⊕ M⊥ and M is the 0 A1 p ⊥ spectral subspace  ker A for all sufficiently large p. So A1 is invertible in B(M ), 0 0 and Ap  . 0 Ap1   L0 E Any L ∈ L has the form L  . As in the proof of Proposition 5.1, F L1 p the 2, 2 entry of Lt := L + tA is invertible for all t sufficiently large. Thus Lt is Fredholm of index 0. Since 0 ∈ σ(Lt ), it must have non-trivial kernel. However, Lt factors as above as     IM E(L1 +tAp1 )−1 L0 −E(L1 +tAp1 )−1 F 0 0 IM . 0 IM⊥ F L1 +tAp1 0 IM⊥ The middle factor must have kernel. Letting t → ∞ shows that L0 is singular. This shows that PM LPM consists of singular matrices. Hence it is not transitive by Proposition 5.1. Since the rank of PM is finite, L is not topologically transitive, as observed in the last paragraph of the introduction.
Example 6.2. The set K of compact operators is transitive and singular. To get even closer to the hypotheses of Theorem 6.1, consider the set L = CS ∗ + K, where S ∗ is the backward shift. Every element is singular, and there are elements A ∈ L such that 0 ∈ σe (A) and ker A = 0. Nevertheless, this is a transitive space. The spaces T and H of Toeplitz and of Hankel operators appear frequently as counterexamples to possible extensions of our finite dimensional results. Let dm denote normalized Lebesgue measure on the unit circle T = {z ∈ C : |z| = 1}. The set {en = einθ }n∈Z forms an orthonormal basis for the Hilbert space L2 (T) = L2 (T, dm). Let us denote by M∞ (T) the space {Mf : f ∈ L∞ (T, dm)} of multiplication operators on L2 (T). It is well-known and routine to verify that T = [tij ] ∈ M∞ (T) if and only if T ∈ B(L2 (T)) and tij = ti+k j+k for all i, j, k ∈ Z. We denote by H 2 (T) the Hardy space span{en }∞ n=0 of analytic functions in L2 (T). If P denotes the orthogonal projection of L2 (T) onto H 2 (T), then the Toeplitz operators are the elements of T = P M∞ |H 2 (T) , and with Q = (I − P ), the set of Hankel operators is H = QM∞ |H 2 (T) . That the Toeplitz operators are topologically transitive may be found in [3]. We include a slightly different proof to that found there. Example 6.3. The spaces H and T of Hankel and Toeplitz operators are topologically transitive.

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We show that the space M∞ P is topologically transitive. Since T and H are compressions of M∞ P , it immediately follows that they too are topologically transitive. Consider A ∈ (M∞ P )⊥ with rank A = 1. Then A = f g ∗ for some f ∈ H 2 (T) and g ∈ L2 (T). Since A = 0, neither f nor g is zero. The condition A ∈ (M∞ P )⊥ implies that tr(Mh P A) = tr(Mh f g ∗ ) = hf, g = 0 for all h ∈ L∞ (T). That is,  ∞ 1 ∞ T hf g = 0 for all h ∈ L (T). Since f g ∈ L (T) = (L (T))⊥ , it follows that f g = 0 a.e.. The classical F. and M. Riesz Theorem (see [5, Theorem 6.13]) asserts that as 0 = f ∈ H 2 (T), the set {z ∈ T : f (z) = 0} has measure 0. From this it follows that g = 0 a.e., a contradiciton. Thus M∞ P is topologically transitive. We observe that an analogous argument may be used to establish the fact that QM∞ is also topologically transitive. Note also that Te0 = {Th e0 = P h : h ∈ L∞ (T)} is dense in H 2 (T), but is not everything. Thus T is an example of a topologically transitive space which is not transitive. Definition 6.4. A subspace L ⊆ B(H1 , H2 ) is said to be totally separating if L is k-separating for all k ≥ 2. There is no point in defining totally topologically transitive in the analogous way, because this would just say that L is sot-dense in B(H). A natural modification of Example 4.8 shows that for each k ≥ 1 there are totally separating spaces which are topologically k-transitive but not topologically (k + 1)-transitive. Proposition 6.5. Let L1 and L2 be subspaces of B(H), and suppose that L1 is topologically transitive. (a) If L2 is k-separating for some k ≥ 1, then span L1 L2 is topologically ktransitive. (b) If L2 is totally separating, then span L1 L2 is dense in the strong operator topology on B(H). Proof. (a) Choose x1 , x2 , ..., xk ∈ H linearly independent. Let ε > 0, and choose y1 , y2 , ..., yk ∈ H arbitrary. Since L2 is k-separating, we can find operators L1 , L2 , ..., Lk ∈ L2 so that Li xi = 0, but Li xj = 0 for all 1 ≤ i = j ≤ k. Since L1 is topologically transitive, we can find K1 , K2 , ..., Kk ∈ L1 with Kj (Lj xj ) − yj  < ε/k, 1 ≤ j ≤ k.
Let T = kj=1 Kj Lj ∈ L1 L2 . Then n  Ki Li )xj − yj  = Kj Lj xj − yj  < ε/k, T xj − yj  = ( i=1

for each 1 ≤ j ≤ k. Since {xj }kj=1 linearly independent and {yj }kj=1 are arbitrary, span L1 L2 is topologically k-transitive. (b) By part (a), span L1 L2 is topologically k-transitive for all k ≥ 1. By the comments preceding the proposition, this says that span L1 L2 is dense in the strong operator topology in B(H).


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Example 6.6. The Hankel operators H and the Toeplitz operators T are products of two topologically 1-transitive spaces, but they are not topologically 2-transitive nor even 2-separating. Indeed, H = (QM∞ )(M∞ P ) and T = (P M∞ )(M∞ P ). In finite dimensions, we have seen that the product of two transitive spaces is 2-transitive. We have also seen that in the finite dimensional setting, 1-transitive spaces are automatically 2-separating. A typical operator in H admits an infinite matrix representation of the form   a1 a2 a3 ... a2 a3 ...    H = a3 ... ... ...   .. . 2 ∞ 2 ⊥ relative to the bases {en }∞ n=0 for H (T) and {en }n=−1 for (H (T)) . Therefore He0 = 0 implies H = 0, and hence H is neither 2-separating nor topologically 2-transitive. Similarly, if T ∈ T and T en = 0, then T ei = 0 for 0 ≤ i < n. So T is also neither 2-separating nor topologically 2-transitive. This can be contrasted with Example 4.7 where it is shown that Tn is 2-separating but not 3-separating.

The following technical result will be used in the proof of Proposition 6.8. Lemma 6.7. Suppose L2 ⊆ B(H) contains a sequence (Fm )∞ m=1 of operators with 1) rank Fm = m, m ≥ 1; 2) ker  Fm+1 ⊆ ker Fm for all m ≥ 1; and 3) m≥1 ker Fm = {0}. If L1 is topologically transitive, then span L1 L2 is dense in the weak operator topology on B(H). Proof. Choose a sequence (Fm )∞ m=1 ⊆ L2 as in the statement of the Lemma. For each m ≥ 1, let Hm = (ker Fm )⊥ , so that Hm ⊆ Hm+1 , and dim Hm = m for all m. Fix e1 ∈ H1 with e1  = 1, and for m ≥ 2, choose em ∈ Hm  Hm−1 with em  = 1. The third hypothesis above guarantees that {em }∞ m=1 spans H, and thus forms an orthonormal basis for H. Our goal is to show that if Pm is the wot orthogonal projection of H onto Hm , then span L1 L2 contains B(H)Pm for all m ≥ 1. Since the latter set is clearly dense in the weak operator topology, so is the former. Let T ∈ B(H) be arbitrary, and set ε > 0. Now em ∈ Hm , so zm := Fm em = 0. Fix Rm ∈ L1 so that Rm Fm em − T em < ε/m. Next, em−1 ∈ Hm−1 , so Fm−1 em−1 = 0. Fix Rm−1 ∈ L so that Rm−1 zm−1 − (T em−1 − Rm Fm em−1 ) < ε/m. Observe that Rm−1 Fm−1 em = 0, so that (Rm−1 Fm−1 + Rm Fm )em − T em  < ε/m.

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More generally, having chosen Rm , Rm−1 , ..., Rm−k , we can choose Rm−(k+1) in L1 so that m      Rm−(k+1) Fm−(k+1) em−(k+1) − T − Rj Fj em−(k+1)  < ε/m. j=m−k


m

It follows that Qm = j=1 Rj Fj satisfies Qm er −T er  < ε/m for 1 ≤ r ≤ m. Since Qm = Qm Pm , Qm −T Pm < ε. Finally, since Qm belongs to span L1 L2 and wot ε > 0 is arbitrary, B(H)Pm ⊂ span L1 L2 . Therefore span L1 L2 = B(H).
Proposition 6.8. Suppose that L ⊆ B(H) is topologically transitive, and let H ⊆ B(H) denote the space of Hankel operators. Then span LH

wot

wot

= span HL

= B(H).

2

In particular, span H is weak operator dense in B(H). Proof. For each n ≥ 1, the rank n operators  1 1 1 ... 1 1 1 ... 1 0   ..  Fn =  . 1 1 ...  1 0 0 ...

0 0 0 ...

 ... ...   ...    

lie in H and satisfy the conditions of Lemma 6.7. Thus we may conclude that wot span LH = B(H). Now L topologically transitive implies that Lt is topologically transitive. Since H = Ht , it follows that (HL)t = Lt H is weak operator dense in B(H), wot whence span HL = B(H).
Let us next consider span T2 . Let Eij = ei e∗j , i, j ≥ 0 denote the matrix units of B(H 2 (T)). Letting S = P Mz |H 2 (T) ∈ T, S is unitarily equivalent to the unilateral forward shift, and a routine calculation reveals that for i, j ≥ 0, Ei,j = S i (I − SS ∗ )(S ∗ )j = S i (S ∗ )j − S i+1 (S ∗ )j+1 . Since S k , (S ∗ )l ∈ T for all k, l ≥ 0, Eij ∈ span T2 for all i, j ≥ 0. Thus the norm closure of span T2 contains all compact operators, and is therefore transitive. So far we have not been able to determine whether L ⊆ B(H 2 (T)) topologiwot cally transitive implies that span LT = B(H). As we have seen in Section 4, if L ⊆ Mn is transitive, then span Lr = Mn for some 1 ≤ r ≤ n. It would be interesting to find estimates for κn := min{1 ≤ r ≤ n : span Lr = Mn for all L ⊆ Mn transitive}. In particular, is (κn )∞ n=1 bounded? In the infinite dimensional setting, does there always exist some r ≥ 1 so that

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L ⊆ B(H) topologically transitive implies that span Lr = B(H)? More generally, does there exist s ≥ 1 so that if L1 , L2 , ..., Ls are topologically transitive wot subspaces of B(H), then span L1 L2 · · · Ls = B(H)? Observe that iif L is a unital subalgebra of B(H), then Ls = L for all s ≥ 1, and so an affirmative answer to this question would resolve the Transitive Algebra Problem. However, since we are only asking for each Lj to be a topologically transitive subspace, it seems plausible that one might be able to find a counterxample for each s ≥ 1. Finally, by Example 6.6, if such an s exists, then s ≥ 3. Example 6.9. A subspace of B(H) can be topologically (n − 1)-transitive, but not n-separating. Let {en = z n : n ∈ Z} be the standard orthonormal basis for L2 (T), and the space Ln ⊆ B(Wn , L2 (T)) of operators set Wn = span{e1 , e2 , ..., en }. Consider
n of the form A = [aij ] where k=1 ai+k,k = 0, i ∈ Z. To see that Ln is not nseparating, it suffices to observe that if the first (n − 1) columns of Ln are zero, then the last column of Ln is necessarily zero as well. The proof that Ln is topologically (n − 1)-transitive relies upon the structure of (Ln )⊥ . We may, in a manner analogous to that used in our analysis of the finite dimensional setting, identify (L with the set of trace class operators B =
nn)⊥
[bij ] ⊆ B(Wn , L2 (T)) which satisfy i=1 j∈Z aij bji = 0 for all A = [aij ] ∈ Ln . A routine calculation then shows that B ∈ (Ln )⊥ implies that bk,i+k = b1,i+1 for all 2 ≤ k ≤ n. If we think of the rows L2 (T),  then this says that B ∈  of B as 2vectors inn−1 t f for some f ∈ L2 (T). If Ln (Ln )⊥ if and only if B = f zf z f ... z were not topologically (n − 1)-transitive, then we could find such a B = 0 with rank B ≤ n −
1. In particular, rank B t =
rank B ≤ n − 1 and so ker B t
= 0. Choose n t a vector 0 = i=1 αi ei ∈ ker B . Then ( ni=1 αi z i−1 )f = 0 a.e. Since ni=1 αi z i−1 is a non-trivial polynomial, it has at most finitely many zeroes. Therefore f = 0 a.e., contradicting B = 0. Hence Ln is topologically (n − 1)-transitive. Example 6.10. The intersection of a descending sequence of wot-closed transitive spaces need not be topologically transitive. As before, we let {en : n ∈ Z} be the standard basis for L2 (T). Let Rn denote the orthogonal projection of L2 (T) onto span{ek : |k| ≤ n}. Then Rn M∞ |Rn L2 (T) ⊆ B(Rn L2 (T)) is clearly unitarily equivalent to the Toeplitz matrices on H2n+1 , and so it is transitive (see Example 1.5). Let Rn = {X ∈ B(L2 (T)) : Rn XRn ∈ Rn M∞ Rn }. Then Rn is transitive and wot-closed. Indeed, with respect to the L2 (T) = Rn L2 (T) ⊕  decomposition  X1 X2 (Rn L2 (T))⊥ , an element of Rn looks like where X1 ∈ Rn M∞ |Rn L2 (T) X3 X4 and the other entries are arbitrary. As the matrix entries are independent and each corner is transitive, it followseasily that Rn is transitive. ∞ ∞ Observe, however, that ∞ n=1 Rn = M (T). Since M (T) has many proper closed invariant subspaces, it is not topologically transitive. So the intersection of a descending sequence of transitive spaces need not be topologically transitive.

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Note that there are limits to the decreasing intersection of transitive spaces  L = n≥1 Ln . For if P and Q are rank n projections, then P Ln Q is a transitive subspace of B(QH, P H) for all n, and so has dimension at least 2n − 1. Thus the same is true for the intersection. Moreover since a decreasing sequence of subspaces of a finite dimensional space is eventually constant, we see that P LQ is transitive whenever P and Q are finite rank. Our example shows that this estimate is sharp because the compression using P H = QH = span{ei : 0 ≤ i < n} yields Tn as the intersection; and it has dimension exactly 2n − 1. Theorem 6.11. Suppose that L, M ⊂ B(H) are topologically transitive. If L is contained in the wot-closed span of its rank one elements, then the norm closure of span LM is transitive. Proof. For each rank one element uv ∗ ∈ L, LM contains uv ∗ M = u(M ∗ v)∗ for all M ∈ M. By the topological transitivity of M, the norm closure of LM contains ue∗j where {ej } is an orthonormal basis for H. As L is topologically transitive and wot-spanned by rank ones, the collection of such vectors u densely spans H, from which it follows that the norm closure of span LM contains the compact operators.
We now give the infinite dimensional analogue of Theorem 3.2. The proof is different, and provides an alternate proof in the finite dimensional case. Theorem 6.12. Let L ⊂ B(H1 , H2 ) and M ⊂ B(K1 , K2 ). Suppose that L is topologically k-transitive, that it is contained in the wot-closed span of its elements of rank at most r, and that M is topologically rk-transitive. Then the spatial tensor product L ⊗ M is topologically k-transitive. Proof. Let {Lp : p ∈ P} be elements of L of rank at most r with wot-closed span containing all of L. Suppose that R ∈ B(H2 ⊗ K2 , H1 ⊗ K1 ) has rank at most k and is orthogonal to L ⊗ M in the trace pairing. This is equivalent to Lp ⊗ M, R = 0 for all p ∈ P and all M ∈ M.
r Fix p ∈ P and write Lp = s=1 us vs∗ . Also let e be any fixed unit vector in H1 . We calculate 0 = Lp ⊗ M, R =

r 

(us vs∗ ⊗ I)(I ⊗ M ), R

s=1

=

r 

(us e∗ ⊗ I)(I ⊗ M )(evs∗ ⊗ I), R

s=1

=

r 

I ⊗ M, (evs∗ ⊗ I)R(us e∗ ⊗ I)

s=1

= I ⊗ M, ee∗ ⊗

r  s=1

Rs 

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= ee∗ ⊗ M, ee∗ ⊗

r 

Rs  = M,

s=1

r 

207

Rs 

s=1

where Rs ∈ B(K2 , K1 ) is given by ee∗ ⊗ Rs
:= (evs∗ ⊗ I)R(us e∗ ⊗ I); and therefore r is an operator of rank at k. Thus s=1 Rs has rank at most rk and is
most r orthogonal to M. Hence s=1 Rs = 0. Now let P ∈ B(K1 ) and Q ∈ B(K2 ) and observe that r    0 = (I ⊗ P ) ee∗ ⊗ Rs (I ⊗ Q) s=1

=

r 

(evs∗ ⊗ I)(I ⊗ P )Rs (I ⊗ Q)(us e∗ ⊗ I)

s=1

Therefore reversing the computation above using (I ⊗ P )R(I ⊗ Q) in place of R yields for all p ∈ P and M ∈ M that 0 = Lp ⊗ M, (I ⊗ P )R(I ⊗ Q) = Lp ⊗ I, (I ⊗ M P )R(I ⊗ Q). For convenience, we assume that K1 and K2 have the same dimension, so that there is a unitary operator U in B(K1 , K2 ). Now take P and Q to be rank one projections P = f f ∗ and Q = gg ∗ . Choose Mn ∈ M so that Mn f → U f . Then 0 = Lp ⊗ I, (I ⊗ Mn P )R(I ⊗ Q) = Lp ⊗ I, (I ⊗ Mn f f ∗ )R(I ⊗ gg ∗ ) → Lp ⊗ I, (I ⊗ U f f ∗ )R(I ⊗ gg ∗ ) = Lp ⊗ g(g ∗ U f )(U f )∗ , (I ⊗ U f f ∗ )R(I ⊗ gg ∗ ) = (g ∗ U f )Lp ⊗ gf ∗ , (I ⊗ f f ∗ )R(I ⊗ gg ∗ ) = (g ∗ U f )Lp ⊗ gf ∗ , Rf,g ⊗ f g ∗  = (g ∗ U f )Lp , Rf,g  where Rf,g ⊗ f g ∗ := (I ⊗ f f ∗ )R(I ⊗ gg ∗ ) defines Rf,g ∈ B(K2 , K1 ). The pairs f, g with g ∗ U f = 0 are dense in K1 × K2 ; so we conclude that Lp , Rf,g  = 0 for all pairs. Evidently rank Rf,g ≤ rank R ≤ k, and it is orthogonal to L. Hence Rf,g = 0 for all pairs. That is, (I ⊗ f f ∗ )R(I ⊗ gg ∗ ) = 0 for all unit vectors f and g. This clearly implies that R = 0. Hence L ⊗ M is topologically k-transitive.
As we did in the finite-dimensional setting, we may define a subspace L ⊆ B(H1 , H2 ) to be fully k-transitive if the spatial tensor product L⊗M is k-transitive for all k-transitive subspaces M of B(K1 , K2 ). The analogue of Corollary 3.4 follows as before. Corollary 6.13. If L ⊆ B(H1 , H2 ) is a k-transitive subspace which is contained in the wot−closed span of its rank one elements, then L is fully k-transitive.

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If the subspace has additional structure, such as being a module over a masa, then stronger results may hold. For example, a non-trivial result of Arveson [1] (see also [4, Theorem 15.9]) shows: Theorem 6.14. Let Di be masas in B(Hi ). A topologically transitive subspace L of B(H1 , H2 ) which is a D2 –D1 bimodule is wot-dense in B(H1 , H2 ). We provide a new, more elementary proof. Actually Arveson’s proof works for the weak-∗ topology, whereas this proof is only valid for the wot-topology.
m Lemma 6.15. Let (Xi , µi ) be regular Borel measures. Let k(x, y) = i=1 αi (x)βi (y) ∈ L2 (µ1 ×µ2 ) where αi ∈ L2 (µ1 ) and βi ∈ L2 (µ2 ). For any a in the essential range of k and any ε > 0, there is a measurable rectangle A1 × A2 with 0 < µi (Ai ) < ∞ such that |k(x, y) − a| < ε for all (x, y) ∈ A1 × A2 . Proof. Choose a measurable rectangle Y1 × Y2 of finite positive measure on which k is bounded and still has a in its essential range. It is a standard argument to approximate each αi and βi uniformly (and in L2 ) by simple functions on Y1 and Y2 respectively. Combining these simple functions allows us to approximate k χY1 ×Y2 uniformly by a finite linear combination of characteristic functions of measurable rectangles. We may then pick a rectangle on which k takes values close to a.
Remark 6.16. The lemma fails for arbitrary functions in L2 (µ1 × µ2 ). For example. take µ1 = µ2 to be Lebesgue measure on [0, 1]. Let A be a compact nowhere dense subset of [0, 1] with positive measure. Then k(x, y) = χA (x−y) has 1 in its essential range. However if k = 1 on a measurable rectangle A1 × A2 , then A1 − A2 ⊂ A is nowhere dense. It is a well known fact that the difference of two measurable sets of positive measure has interior. So A1 × A2 has measure 0. Proof of Theorem 6.14. By the spectral theorem for masas, we may suppose that there are regular Borel spaces (Xi , µi ) so that Di are unitarily equivalent to L∞ (µi ) acting by multiplication on Hi = L2 (µi ). If L is not wot-dense, then there is a finite rank operator F ∈ L⊥ . Moreover, it is evident that L⊥ is a D1 –D2 bimodule. Our goal is to show that using F and the bimodule property, we may find a rank one element of L⊥ . This will contradict topological transitivity. as an integral operator with kernel k(x, y) =
m Observe that F may be written 2 2 i=1 αi (x)βi (y) where αi ∈ L (µ1 ) and βi ∈ L (µ2 ). Since F = 0, there is a non-zero value a in the essential range of k. By Lemma 6.15, there is a measurable rectangle A1 × A2 of finite non-zero measure so that |k(x, y) − a| < |a|/2 for all (x, y) ∈ A1 × A2 . Let h(x, y) = χA1 ×A2 k(x, y)−1 . Then h ∈ L∞ (µ1 × µ2 ). Hence
k h is a limit in ∞ L (µ1 ×µ2 ) of a sequence of simple functions of the form hk = m j=1 fkj (x)gkj (y). It follows by routine calculations that mk  j=1

Mfkj χA1 F Mgkj χA2

has kernel

mk  j=1

fkj (x)k χA1 ×A2 gkj (y).

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This converges in L2 (µ1 × µ2 ) to χA1 ×A2 , and thus the corresponding operators converge in norm to the rank one integral operator with kernel χA1 ×A2 . This produces a rank one element of L⊥ .
The following result is very easy. Recall that a masa is atomic if it consist of all diagonal operators with respect to some orthonormal basis. Proposition 6.17. Suppose that a topologically transitive subspace L ⊆ B(H) is a wot = B(H). left or right module over an atomic masa D. Then L Proof. First we suppose that L is a right D-module. Let D be diagonal with respect to {en : n ≥ 1}; and let Pn = en e∗n . Since L is topologically transitive, LPn = B(H)Pn . Summing over n yields a wot-dense subspace. By considering Lt which is also topologically transitive, we obtain the left D-module case.
Example 6.18. A right or left module over a non-atomic masa need not be wotdense. Example 6.3 showed that M∞ P ⊂ B(H 2 , L2 (T)) is topologically transitive. It is evidently a left M∞ module, and a proper wot-closed subspace. The adjoint P M∞ ⊂ B(L2 (T), H 2 ) is similarly a right M∞ module, and a proper wot-closed subspace. Proposition 6.19. Suppose that L is a left module over a masa D, and that L is wot wot = span L2 = B(H). topologically transitive. Then span LD Proof. Evidently span LD is a D-bimodule. So it is wot-dense by Arveson’s Theorem. Therefore span L2

wot

= span LDL

wot

wot

= span B(H)L

= B(H).




References [1] W.B. Arveson, Operator algebras and invariant subspaces, Ann. Math. 100 (1974), 433–532. [2] E.A. Azoff, On finite rank operators and preannihilators, Mem. Amer. Math. Soc. 64 (1986), no. 357. [3] E.A. Azoff and M. Ptak, A dichotomy for linear spaces of Toeplitz operators, J. Funct. Anal. 156 (1998), 411–428. [4] K.R. Davidson, Nest Algebras, Pitman Research Notes in Mathematics Series 191, Longman Scientific and Technical Pub. Co., London, New York, 1988. [5] R.G. Douglas, Banach algebra techniques in operator theory, Academic Press, New York, 1972. [6] H. Flanders, On spaces of linear transformations with bounded rank, J. London Math. Soc. 37 (1962), 10–16. [7] J. Harris, Algebraic Geometry, A First Course, Springer–Verlag, New York, 1992.

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ˇ [8] R. Meshulam and Peter Semrl, Minimal rank and reflexivity of operator spaces, Proc. Amer. Math. Soc. 135 (2007), 1839–1842. [9] H. Radjavi and P. Rosenthal, Invariant Subspaces, Second Edition, Dover Publications, Inc., Mineola, New York, 2003. Kenneth R. Davidson, Laurent W. Marcoux and Heydar Radjavi Department of Pure Mathematics University of Waterloo Waterloo, ON N2L–3G1 Canada e-mail: [email protected] [email protected] [email protected] Submitted: February 21, 2007 Revised: January 25, 2008

Integr. equ. oper. theory 61 (2008), 211–239 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/020211-29, published online April 17, 2008 DOI 10.1007/s00020-008-1582-9

Integral Equations and Operator Theory

Two-Isometries on Pontryagin Spaces Christian Hellings Abstract. A model for a class of cyclic analytic 2-isometries acting on Pontryagin spaces is given, generalizing a Hilbert space version given by S. Richter. Furthermore, an example is constructed showing that, unlike in the Hilbert space case, a cyclic analytic 2-isometry need not have a cyclic vector in the orthogonal complement of its range. Mathematics Subject Classification (2000). Primary 47B50; Secondary 47A45, 47B32, 46C20, 30H05, 31C25. Keywords. 2-isometry, Pontryagin space, Kre˘ın space, Dirichlet space.

1. Introduction A 2-isometry is a continuous linear operator T on a Kre˘ın space satisfying the identity (1) T ∗2 T 2 − 2T ∗T + I = 0. First studied by J. Agler on Hilbert spaces (see Agler and Stankus [1, 2, 3]), this class of operators generalizes the concept of an isometry in a certain sense. The shift operator on the Dirichlet space is an important example of a 2-isometry. In an analysis of the invariant subspaces of this operator, S. Richter [20] developed a model for all cyclic analytic 2-isometries acting on Hilbert spaces as multiplication by the independent variable on the D(µ) spaces, a family of Dirichlet-like spaces of analytic functions on the unit disk. S. McCullough and L. Rodman [17] have considered 2-isometries in indefinite inner product spaces. (See related work in [16], [18].) Additionally, certain Pontryagin space generalizations of the Dirichlet space appear in the de Branges theory of coefficient estimates of univalent functions, and the shift operator is a cyclic analytic 2-isometry on those spaces as well. (See L. de Branges [8, 9], and also G. Christner, Kin Y. Li, and J. Rovnyak [11] and M. Rosenblum and J. Rovnyak [22].) In this paper, we investigate the extent to which Richter’s model can be generalized to the Pontryagin space setting.

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A key property of cyclic analytic 2-isometries that Richter used in his Hilbert space model is that they always have a cyclic vector in the orthogonal complement of the range. In Section 6, we give an example showing that this is not necessarily true on Pontryagin spaces. Thus, the structure of these operators may be richer in the indefinite setting. If we take this property as hypothesis, however, then a model theorem like that of Richter’s can be established. In Sections 3 and 4, we consider a family of Pontryagin spaces that generalizes Richter’s D(µ) spaces and includes the spaces in de Branges’ work. In these spaces, the operation of multiplication by the independent variable is a cyclic analytic 2-isometry (with a cyclic vector in the orthogonal complement of the range.) We show in Section 5 that these operators account for all cyclic analytic 2-isometries (with the additional hypothesis) acting on Pontryagin spaces. Definitions For terminology and definitions concerning Kre˘ın spaces, we follow Dritschel and Rovnyak [13]. Recall that a Kre˘ın space is an indefinite inner product space (K,
·, ·) that admits a decomposition of the form K = K+ ⊕K− , where (K± , ±
·, ·) are Hilbert spaces. Such a decomposition is called a fundamental decomposition of K, and it is in general not unique. Given any particular decomposition, we can form an associated Hilbert space, which we denote by |K|, which is defined to be (K+ ,
·, ·) ⊕ (K− , −
·, ·). Although different decompositions yield different associated Hilbert spaces, they all induce the same topology on K, and this is the topology we use. A Kre˘ın space K is a Pontryagin space if dim K− is finite in some fundamental decomposition. When it is, dim K− is the same for all decompositions, and this dimension is called the negative index of K. n An operator T ∈ L(K) is said to be analytic if ∩∞ n=0 T K = {0}. T ∈ L(K) is cyclic if there exists a v ∈ K such that {v, T v, T 2v, . . .} spans a dense subspace of K.

2. Some Basic Properties of 2-Isometries Let K be a Kre˘ın space, and suppose that T ∈ L(K) is a 2-isometry. Using (1), we see that 
2 (2) T x, T 2 x − 2
T x, T x +
x, x = 0 for each x ∈ K. In fact, by polarization, this inner product characterization is a sufficient condition for an operator to be a 2-isometry. We define T = T ∗ T − I. Equation (1) can be factored to arrive at another characterization of 2-isometries T ∗ T T = T .

(3)

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Additionally, one can verify by induction that T ∗n T n − I = n T .

(4)

Agler and Stankus [1] have shown that 2-isometries on Hilbert spaces are always expansive (
T x, T x ≥
x, x for all x). This is not generally true in an indefinite setting. (For example, any 2-isometry on the antispace of a Hilbert space is contractive.) However, the Agler/Stankus proof carries through under an additional condition. Theorem 1. Suppose K is a Kre˘ın space, and T ∈ L(K) is a 2-isometry. If there is a non-negative integer N such that T N K is a non-negative subspace (i.e.,
x, x ≥ 0 for all x ∈ T N K), then T is expansive. Proof. Suppose x ∈ K. By (4),
T n x, T n x −
x, x = n (
T x, T x −
x, x) . Rearranging terms, we have n−1 1 n
T x, T n x +
x, x =
T x, T x . n n Suppose that for all n ≥ N ,
T n x, T n x ≥ 0. Then
T x, T x ≥ n−1 n
x, x for big enough n. Letting n approach infinity then gives the desired result.
The next result shows that, on a Pontryagin space, the condition of the previous theorem is always satisfied when the 2-isometry is analytic. The proof uses Pontryagin’s Theorem: Given a dense subspace M of a Pontryagin space K, one can always find a fundamental decomposition K = K+ ⊕K− such that K− ⊂ M. (See p.164 of [13].) Theorem 2. Suppose K is a Pontryagin space, and T ∈ L(K) is an analytic 2-isometry. Then there is a non-negative integer N and a fundamental decomposition K = K+ ⊕ K− such that T N K ⊂ K+ . Therefore, T N K is a Hilbert space, and T is expansive. Proof. Equation (1) can be factored as (2T ∗ − T ∗2 T )T = I. So T has a left inverse. Therefore, {T n K} is a nested sequence of closed subspaces. Since T is analytic, n ⊥ ∞ n ⊥ K = (∩∞ n=0 T K) = ∪n=0 (T K) . n ⊥ is a dense subspace. By Pontryagin’s Theorem, (See [13].) Thus, ∪∞ n=0 (T K) there exists a fundamental decomposition K = K+ ⊕ K− such that K− is a subset n ⊥ of ∪∞ n=0 (T K) . n ⊥ ∞ {(T K) }n=0 is an increasing sequence of subspaces. Since K− has finite dimension, there must be an N such that K− ⊂ (T N K)⊥ . Consequently, T N K ⊂ K+ . Thus T N K is a closed subspace of a Hilbert space, so it is Hilbert as well. By Theorem 1, T is expansive.


The extension of Richter’s model to the Pontryagin space setting will require that we establish some properties of the subspace ker T .

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Theorem 3. Suppose that K is a Pontryagin space and T ∈ L(K) is an analytic 2-isometry. Then ker T is a Hilbert space and is invariant under T . Furthermore, T acts isometrically on every vector of ker T . Proof. The operator T = T ∗ T − I is selfadjoint, so it has a Bogn´ar-Kr´ amli factorization T = DD∗ for some Kre˘ın space D and some injective operator D ∈ L(D, K). Since T is expansive by Theorem 2, D is a Hilbert space. (See Theorem 2.1 of [13], or [12].) Now suppose x ∈ ker T . Then, by (3),
D∗ T x, D∗ T xD =
T T x, T xK =
T ∗ T T x, xK =
T x, xK = 0. Thus, D∗ T x = 0. But T = DD∗ and D is injective, so ker T = ker D∗ . Therefore, T x ∈ ker T . So ker T is invariant under T . For any x ∈ ker T ,
T x, T x−
x, x =
T x, x = 0. So T acts isometrically on this subspace. By Theorem 2, there is an N and a fundamental K = K+ ⊕ K−
decomposition  such that T N K ⊂ K+ . So if x ∈ ker T ,
x, x = T N x, T N x ≥ 0. Thus, ker T is a non-negative closed subspace. Let |K| be the Hilbert space associated with the above decomposition. Now T is expansive, so T N is too. So there is a δ > 0 such that T N x |K| ≥ δ x |K| for all non-negative vectors x. (See Theorem VII.5.2 of [6].) Thus we have, for every x ∈ ker T ,

x, x = T N x, T N x = T N x 2|K| ≥ δ 2 x 2|K| . This shows that ker T is actually uniformly positive. This suffices to conclude that it is a Hilbert space. (See Definition 1.4 of [13].)


3. Indefinite Local Dirichlet Spaces Let D denote the open unit disk in the complex plane. For any f ∈ H 2 and ζ ∈ ∂D, we let f (ζ) denote the non-tangential limit of f (z) as z approaches ζ, if it exists. For a fixed ζ ∈ ∂D, the local Dirichlet space at ζ is defined to be the set of (ζ) functions f in H 2 such that f (ζ) exists and f (·)−f also belongs to H 2 . Note ·−ζ that this includes all polynomials, indeed all functions analytic on D. For any α ∈ R, we will refer to this space as Dζ,α when it is equipped with the (possibly indefinite) inner product   f (·) − f (ζ) g(·) − g(ζ) , .
f, gDζ,α = α
f, gH 2 + ·−ζ ·−ζ H2 2   (ζ)  Let us write Dζ (f ) =  f (·)−f  2 . Then the self-inner product of a function ·−ζ H f ∈ Dζ,α is
f, f Dζ,α = α f 2H 2 + Dζ (f ). (5)

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When α is negative, the inner product of Dζ,α is indeed indefinite. The constant functions, for example, have negative self-inner product. When α > 0, the inner product is positive definite. In the α = 1 case, these spaces can be viewed as the building blocks from which Richter’s D(µ) spaces are constructed. Sarason has also studied them in [23]. Similarly, we will construct an indefinite generalization of the D(µ) spaces from these Dζ,α spaces in Section 4. Also, the example in Section 6 of new 2-isometry behavior on a Pontryagin space will be constructed on a variant of Dζ,α . Lemma 4. Suppose ζ ∈ ∂D. If fn → f and fn (ζ) → f (ζ).

fn (·)−fn (ζ) ·−ζ

→

f (·)−f (ζ) ·−ζ

in H 2 , then

Proof. Let S denote the shift operator on H 2 . Then fn (ζ) = fn − (S − ζI)

f − f (ζ) fn − fn (ζ) → f − (S − ζI) = f (ζ). ·−ζ ·−ζ




Lemma 5. Suppose ζ ∈ ∂D. If f (z) is a function analytic on D, and if zf (z) ∈ Dζ,α , then f (z) ∈ Dζ,α . Proof. If zf (z) belongs to Dζ,α , then it belongs to H 2 , and therefore so does f (z). (ζ) is an H 2 function. Then But also, zf (z)−ζf z−ζ   zf (z) − ζf (ζ) f (z) − f (ζ) ¯ =ζ − f (z) ∈ H 2 . z−ζ z−ζ Thus, f ∈ Dζ,α .




In Theorem 8, we will determine the values of α for which the space Dζ,α is actually a Pontryagin space. We will make use of the following lemma, which gives a formula for Dζ (f ) for functions analytic on D in terms of the Fourier coefficients of f . It is essentially the same calculation as that found in Proposition 2.2 of [21]. Lemma 6. If f (z) =

∞

n=0

an z n is analytic on D, then

Dζ (f ) =

∞ ∞

min(m, n)am an ζ m−n .

m=1 n=1 (ζ) Proof. The function g(z) = f (z)−f is analytic on D if f is, so its Maclaurin z−ζ series converges absolutely and uniformly there. This justifies the interchange of sum and integral in the following calculation.

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  ∞  n=0 an (eint − ζ n ) 2 dt    2π  eit − ζ 0     −int

∞ 2π eimt − ζ m e − ζ −n dt = am an eit − ζ e−it − ζ −1 2π 0 m,n=1   n −int 

2π  −m imt ∞ ζ e ζ e −1 − 1 dt am an ζ m−n = ζ −1 eit − 1 ζe−it − 1 2π 0 m,n=1

2π

Dζ (f ) =

=

∞

am an ζ m−n

0

m,n=1

=

∞

am an ζ m−n

m,n=1

=

∞

=

=

m,n=1 ∞

2π m−1

eikt ζ −k

k=0

m−1 n−1

ζ l−k

am an ζ m−n

m−1 n−1



n−1

e−ilt ζ l

l=0 2π

ei(k−l)t

0

k=0 l=0

m,n=1 ∞



dt 2π

dt 2π

ζ l−k δk,l

k=0 l=0



min(m,n)−1

am an ζ m−n

1

k=0

am an ζ m−n min(m, n).

m,n=1




In light of this formula, we define, for any positive integer n and ζ ∈ ∂D, the hermitian matrix   1 ζ ζ2 ζ3 ··· ζ n−2 ζ n−1  ζ 2 2ζ 2ζ 2 ··· 2ζ n−3 2ζ n−2     2  2ζ 3 3ζ ··· 3ζ n−4 3ζ n−3   ζ  3  2  2ζ 3ζ 4 ··· 4ζ n−5 4ζ n−4  (6) Aζn =  ζ .  ..  .. .. .. .. .. ..  .  . . . . . .   n−3 n−4 n−5  n−2  2ζ 3ζ 4ζ ··· n−1 (n − 1)ζ  ζ n−1 n−2 n−3 n−4 ζ 2ζ 3ζ 4ζ · · · (n − 1)ζ n In general, the entry of Aζn in the i-th row and j-th column is (Aζn )i,j = min(i, j)ζ i−j .

(7)

Lemma 7. For any ζ ∈ ∂D, the set of eigenvalues of Aζn is     πj 1 2 1 + tan : j = 1, 2, . . . , n . 4 2n + 1 Proof. Fix ζ ∈ ∂D, and let pn (z) be the characteristic polynomial of Aζn . We will derive a recurrence relation for pn (z). Let W be the n × n elementary matrix

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217

consisting of ones on the main diagonal, −ζ in the (n − 1)st row, nth column entry, and zeros elsewhere. Then det W = 1. So the characteristic polynomial of Aζn is pn (z) = det(Aζn − zI) = det(W ∗ (Aζn − zI)W )   1−z ζ ζ2 ζ3 ··· ζ n−2 0 2 n−3  ζ 2−z 2ζ 2ζ ··· 2ζ 0   2   ζ n−4 2ζ 3−z 3ζ ··· 3ζ 0    2  3  n−5  2ζ 3ζ 4 − z ··· 4ζ 0  = det  ζ .. ..  .. .. ..  ..  ..  . . . .  . . .  n−2  n−3 n−4 n−5 ζ 2ζ 3ζ 4ζ ··· n − 1− z ζz  ζz 1 − 2z 0 0 0 0 ···  2 ··· ζ n−3 1−z ζ ζ  ζ 2−z 2ζ ··· 2ζ n−4  2  ζ 2ζ 3 − z ··· 3ζ n−5  = (1 − 2z)pn−1 (z) − ζz det  .. . . . ..  .. .. .. . .   n−3 n−4 n−5 ζ 2ζ 3ζ ··· n − 2 − z n−2 n−3 n−4 ζ 2ζ 3ζ · · · (n − 2)ζ

 0 0  0  ..  .   0 ζz

= (1 − 2z)pn−1 (z) − z pn−2 (z). 2

The determinant in the third line is calculated by expanding along the bottom row, and the determinant of the resulting submatrix in the fourth line is found by expanding along the last column. We claim that  n  1 2n + 1 Pn (z) = n (8) (1 − 4z)k 2k + 1 4 k=0

satisfies this recurrence relation. We compute (1 − 2z)Pn−1 (z) − z 2 Pn−2 (z)   n−1  n−2  1 2n − 3 1 2n − 1 = (1 − 2z) n−1 (1 − 4z)k − z 2 n−2 (1 − 4z)k 2k + 1 2k + 1 4 4 k=0 k=0     n−1 1 2n − 1 1 1 + (1 − 4z) = (1 − 4z)k n−1 2k + 1 2 2 4 k=0    n−2  1 1 1 1 2n − 3 2 − (1 − 4z) + (1 − 4z) − (1 − 4z)k 2k + 1 16 8 16 4n−2 k=0

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 n−1 n  2n − 1 2n − 1 1 1 k + (1 − 4z) (1 − 4z)k 2k + 1 2k − 1 2 · 4n−1 2 · 4n−1 k=0 k=1   n−2 n−1 2n − 3 1 2n − 3 1 − n (1 − 4z)k + (1 − 4z)k 2k + 1 2k − 1 4 2 · 4n−1 k=0 k=1  n  2n − 3 1 − n (1 − 4z)k 2k − 3 4 k=2   1 1 2n + 1 = n (2n + 1) + n (1 − 4z) 3 4 4      n−2   2n − 1 2n − 1 2n − 3 1 + n 2 +2 − 2k + 1 2k − 1 2k + 1 4 k=2     2n − 3 2n − 3 +2 − (1 − 4z)k 2k − 1 2k − 3   1 2n + 1 + n (1 − 4z)n−1 + (1 − 4z)n 4 2n − 1  n  1 2n + 1 = n (1 − 4z)k 2k + 1 4 =

k=0

= Pn (z). It is straightforward to verify that P1 (z) and P2 (z) are the characteristic polynomials for Aζ1 and Aζ2 , respectively: P1 (z) = 1 − z = det(Aζ1 − zI) = p1 (z), P2 (z) = 1 − 3z + z 2 = det(Aζ2 − zI) = p2 (z). Therefore, Pn (z) is the characteristic polynomial for Aζn for every n.  (j) πj 1 is a zero of Pn (z). The Finally, we will show that ξn = 4 1 + tan2 2n+1 (1)

(2)

(n)

n numbers ξn , ξn , . . . , ξn are distinct, and so they will account for all n eigenvalues of Aζn . πij ω −1 πj Fix n. Let tj = i tan 2n+1 . By noting that tj = ωjj +1 , where ωj = e 2n+1 , one finds that  2n+1 tj + 1 = −1. tj − 1 Then we have   1 (1 − t2j ) Pn (ξn(j) ) = Pn 4  n  1 2n + 1 2k = n t 2k + 1 j 4 k=0

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=

=

2n+1

1 22n tj 1

219

  2n + 1 k tj k

k=0 k odd 2n+1 

22n+1 tj

k=0

  2n+1 2n + 1 2n + 1 k 2n−k+1 k tj tj + (−1) k k k=0

  (tj + 1)2n+1 + (tj − 1)2n+1 = 2n+1 2 tj   2n+1 tj + 1 1 2n+1 +1 = 2n+1 (tj − 1) 2 tj tj − 1 1

= 0.




In private communications, Stefan Maurer has observed that the characteristic polynomials (8) can be written in terms of Jacobi polynomials:  −1 n − 12 ( 1 ,− 1 ) pn (z) = (4z)n Pn 2 2 ((2z)−1 − 1). n See Chihara [10], for example, for definitions and notation. Notice that all the eigenvalues of Aζn are greater than 14 . Furthermore, one can find arbitrarily many of them close to the value 14 by choosing n big enough. More precisely, for any fixed positive integer M , the M th smallest eigenvalue  (M) πM converges to 14 as n goes to infinity. ξn = 14 1 + tan2 2n+1 N Consider a polynomial p(z) = n=0 an z n . Let vp be the column vector whose nth entry is an . Using Lemma 6 and equations (5) and (7), it is evident that   αI1×1 0
p, pDζ,α = vp∗ (9) vp . 0 αIN ×N + AζN Theorem 8. Suppose ζ ∈ ∂D. When α > 0, Dζ,α is a Hilbert space. When − 41 < α < 0, Dζ,α is a Pontryagin space of negative index 1, and a fundamental decomposition is Dζ,α = zDζ,α ⊕span{1}. In both cases, the polynomials are dense, and the operation of evaluation at any w ∈ D ∪ {ζ} is continuous. When α < − 41 , Dζ,α is not a Pontryagin space. (Whether or not Dζ,α is a Kre˘ın space when α < − 14 is an open question.) Proof. First assume α > − 41 . We will show that zDζ,α is a Hilbert space. Suppose g ∈ zDζ,α , g = 0. We write g(z) = zf (z) for some f ∈ Dζ,α . There is a sequence {pn } of polynomials converging to f in H 2 such that f (z) − f (ζ) pn (z) − pn (ζ) → z−ζ z−ζ in H 2 . (This follows from the fact that polynomials are dense in the Hilbert space Dζ,1 . See [20].) Let qn (z) = zpn (z). Then {qn } is a sequence of polynomials in

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zDζ,α converging to g in H 2 . Also, zpn (z) − ζpn (ζ) pn (z) − pn (ζ) qn (z) − qn (ζ) = =z + pn (ζ). z−ζ z−ζ z−ζ So, by Lemma 4, qn (z) − qn (ζ) f (z) − f (ζ) f (z) − f (ζ) →z + f (ζ) = z−ζ z−ζ z−ζ in H 2 . Thus,
qn , qn Dζ,α →
g, gDζ,α .

Since all the eigenvalues of αIN ×N +AζN are greater than α+1/4 by Lemma 7, equation (9) yields   1
qn , qn Dζ,α ≥ α +

qn 2H 2 . 4

Letting n go to infinity, we find that
g, gDζ,α ≥ (α + 14 ) g 2H 2 > 0. This holds for all g ∈ zDζ,α . Therefore zDζ,α is a strictly positive subspace. Now suppose that {gn } is a Cauchy sequence in zDζ,α . Then

gm − gn 2H 2 ≤ (α + 1/4)−1
gm − gn , gm − gn Dζ,α → 0 as m and n increase. So the sequence converges in H 2 to some g ∈ zH 2 . Now Dζ (gm − gn ) =
gm − gn , gm − gn Dζ,α − α gm − gn 2H 2 must be Cauchy as well. Thus, {gn } is actually Cauchy in Dζ,1 , a known Hilbert space. So the sequence in fact converges in Dζ,1 , and the limit must be g, which evidently is in Dζ . Lemma 5 implies that g ∈ zDζ,α . Thus, gn → g in zDζ,α as well. It has therefore been shown that zDζ,α is complete. If g ∈ zDζ,α , then   g(z) − g(ζ) ,0 = 0.
g, 1Dζ,α = α
g, 1H 2 + z−ζ H2 So if α = 0, Dζ,α = zDζ,α ⊕ span{1}. Also,
1, 1Dζ,α = α 1 2H 2 + Dζ (1) = α. So Dζ,α is a Hilbert space if α > 0, and it is a Pontryagin space of negative index one if − 14 < α < 0. The above argument shows that the polynomials in zDζ,α are in fact dense in zDζ,α . The same is trivially true for the subspace of constants. So if we consider the Hilbert space structure on Dζ,α associated with this particular fundamental decomposition, it is evident that polynomials are dense in Dζ,α . Finally, the above arguments also show that any convergent sequence in zDζ,α is also convergent in H 2 . Thus, evaluation at any given w ∈ D is continuous on zDζ,α It follows that it is continuous on all of Dζ,α . Similarly, it has been shown that any convergent sequence in zDζ,α satisfies the hypotheses of Lemma 4, and so evaluation at ζ is continuous on zDζ,α , and hence on Dζ,α . Now suppose α < − 14 . Let  > 0 be small enough so that α + 1/4 +  < 0. Let M be an arbitrary positive integer. Using Lemma 7, we can choose an N such that

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Two-Isometries on Pontryagin Spaces

at least M distinct eigenvalues of the matrix AζN are in the interval Thus there are at least M distinct eigenvalues of the matrix   αI1×1 0 BN = 0 αIN ×N + AζN

221

1

1 4, 4

 + .

less than α+1/4+. Label these eigenvalues a1 , a2 , . . . , aM , and choose corresponding eigenvectors u1 , u2 , . . . , uM . The eigenvalues are distinct, so these vectors are pairwise orthogonal. For each j = 1, 2, . . . , M , let qj be the polynomial whose coefficients are the components of uj . Then for j = k,
qj , qk Dζ,α = u∗k · (BN uj ) = aj u∗k · uj = 0,
qj , qj Dζ,α = u∗j · (BN uj ) = aj |uj |2 < 0. Thus, the span of {q1 , q2 , . . . , qM } is a negative subspace of dimension M . Since M can be chosen arbitrarily large here, the inner product does not have a finite
number of negative squares, and Dζ,α can not be Pontryagin. We denote the operation of multiplication by the independent variable on Dζ,α by Mz : (Mz f )(z) = zf (z). The calculations that show that this is a 2-isometry are essentially the same as in the α = 1 case in [21]. The identity f (z) − f (ζ) zf (z) − ζf (ζ) =z + f (ζ). z−ζ z−ζ shows that Mz f belongs to Dζ,α if f does. So the Closed Graph Theorem implies that Mz ∈ L(Dζ,α ). Furthermore, since Mz is an isometry on H 2 , we see that Dζ (Mz f ) = Dζ (f ) + |f (ζ)|2 ,

(10)


Mz f, Mz f Dζ,α =
f, f Dζ,α + |f (ζ)|2 .

(11)

which immediately yields In fact, a similar calculation gives
Mz f, Mz gDζ,α =
f, gDζ,α + f (ζ)g(ζ).

(12)

Theorem 9. Suppose that α ∈ (−1/4, 0) ∪ (0, ∞). Then Mz ∈ L(Dζ,α ) is a cyclic, analytic 2-isometry. Proof. For any f ∈ Dζ,α ,
2  Mz f, Mz2 f D − 2
Mz f, Mz f Dζ,α +
f, f Dζ,α ζ,α   =
Mz (Mz f ), Mz (Mz f )Dζ,α −
Mz f, Mz f Dζ,α   −
Mz f, Mz f Dζ,α −
f, f Dζ,α = |ζf (ζ)|2 − |f (ζ)|2 = 0.

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In this computation, we have used (11) twice. Thus, Mz is a 2-isometry. It is clearly analytic. By Theorem 8, polynomials are dense in Dζ,α , so Mz is cyclic, with the constant functions as cyclic vectors.
Sarason [23, Proposition 2] has identified Dζ,1 as an instance of a de BrangesRovnyak space H(b); that is, a reproducing kernel Hilbert space of analytic functions on the unit disk for which the kernel functions take the form K(w, z) =

1 − b(w)b(z) 1 − wz ¯

for w, z ∈ D and for some function b in the closed unit ball of H ∞ . In this case, the function b is bζ (z) = where γ =

¯ (1 − γ)ζz , ¯ 1 − γ ζz

√ 3− 5 2 .

(α)

We can generalize this result to Dζ,α for any α ∈ (−1/4, 0) ∪ (0, ∞). Let γ0 (α) be the root of the equation z = α(z − 1)2 that lies in the unit disk, and let γ1 (α) (α) be the other root. (Both roots are real, and γ0 γ1 = 1. They are negative when α is negative.) Define (α) ¯ (1 − γ0 )ζz (α) bζ (z) = . (13) (α) ¯ 1 − γ0 ζz This function is always bounded for α > − 14 . When α is positive, it is actually in the closed unit ball of H ∞ , but when α is negative, it is only bounded by 1 (4α + 1)− 2 . Theorem 10. Suppose ζ ∈ ∂D, and α ∈ (− 14 , 0) ∪ (0, ∞). Then the reproducing kernel for Dζ,α is (α)

(α)

Qζ (w, z) =

  (α) Also, for any f ∈ Dζ,α , f (ζ) = f, qζ (α)

(α)

(α)

1 − bζ (w) bζ (z) α(1 − wz) ¯ Dζ,α

qζ (z) = Qζ (ζ, z) =

.

(14)

, where 1 (α) ¯ α(1 − γ0 ζz)

.

(α)

(15)

Proof. Let Qζ (w, z) be the reproducing kernel of Dζ,α at w ∈ D. We will show that it is in fact equal to the expression in (14). Note that if we let k(w, z) = 1−1wz ¯ , 2 the reproducing kernel of H , then w ¯ k(w, z) − k(w, ζ) = k(w, z) ∈ H 2 . z−ζ 1 − wζ ¯

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So k(w, ·) ∈ Dζ,α . Thus we compute, for any w, z ∈ D,   (α) k(z, w) = Qζ (w, ·), k(z, ·) Dζ,α   (α) (α)   Qζ (w, ·) − Qζ (w, ζ) 1 z (α) = α Qζ (w, ·), k(z, ·) , k(z, ·) + ¯ 1 − wz ¯ ·−ζ H2 1 − ζz H2   (α) (α) Qζ (w, z) − Qζ (w, ζ) z 1 (α) = αQζ (w, z) + . ¯ 1 − wz ¯ z−ζ 1 − ζz (α)

Solving for Qζ (w, z), we get (α) Qζ (w, z)

(α) ¯ ¯ − 1)2 − ζz(1 − wz)Q ¯ (ζz ζ (w, ζ) = . ¯ − 1)2 − ζz) ¯ (1 − wz)(α( ¯ ζz

(α)

(16) (α)

The number Qζ (w, ζ) is determined by the requirement that Qζ (w, ·) have no (α)

poles in the unit disk. The denominator of (16) has a zero at z = ζγ0 ∈ D, and (α) (α) so we choose Qζ (w, ζ) so that the numerator has a zero at z = ζγ0 too. This gives  2 (α) γ0 − 1 1 (α)   =  . Qζ (w, ζ) = (α) (α) (α) 1 − γ0 ζ w γ0 ¯ α 1 − γ0 ζ w ¯ We replace this into (16) and use the factorization      (α) ¯ ¯ − 1 2 − ζz ¯ = α 1 − γ (α) ζz ¯ α ζz 1 − γ ζz 0

1

to arrive at ¯ − 1)2 − (ζz (α)

Qζ (w, z) = =

=

=

=

¯ ¯  ζz(1−wz) (α) α 1−γ0 ζ w ¯

¯ − 1)2 − ζz) ¯ (1 − wz)(α( ¯ ζz   2  (α) ¯ −1 ¯ 1 − γ0 ζ w α ζz ¯ − ζz(1 − wz) ¯     (α) (α) ¯ (α) ¯ α2 (1 − wz) ¯ 1 − γ0 ζ w¯ 1 − γ0 ζz 1 − γ1 ζz     (α) ¯ + γ (α) wz ¯ − 1 2 1 − γ (α) ζ w¯ − ζz ¯ + ζ¯wz ¯ 2 α ζz 0 0 ¯ − γ0 wz     (α) (α) ¯ (α) ¯ α2 (1 − wz) ¯ 1 − γ0 ζ w ¯ 1 − γ0 ζz 1 − γ1 ζz         (α) ¯ ¯ − 1 2 1 − γ (α) ζ w¯ − ζz ¯ 1 − γ (α) ζ w¯ − γ (α) wz α ζz 1 − γ1 ζz 0 0 0 ¯     (α) (α) ¯ (α) ¯ 1 − γ1 ζz α2 (1 − wz) ¯ 1 − γ0 ζ w¯ 1 − γ0 ζz        (α) (α) ¯ ¯ − 1 2 − ζz ¯ − γ (α) wz 1 − γ0 ζ w¯ α ζz ¯ 1 − γ ζz 0 1     (α) (α) (α) ¯ ¯ α2 (1 − wz) ¯ 1 − γ ζ w¯ 1 − γ ζz 1 − γ ζz 0

0

1

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      (α) (α) ¯ (α) ¯ (α) (α) ¯ α 1 − γ0 ζ w¯ 1 − γ0 ζz ¯ 1 − γ1 ζz 1 − γ1 ζz − γ0 wz     = (α) (α) ¯ (α) ¯ α2 (1 − wz) ¯ 1 − γ0 ζ w ¯ 1 − γ0 ζz 1 − γ1 ζz    (α) (α) ¯ (α) ¯ − γ0 wz α 1 − γ0 ζ w¯ 1 − γ0 ζz    = (α) (α) ¯ α2 (1 − wz) ¯ 1 − γ0 ζ w ¯ 1 − γ0 ζz 1− =

(α)

γ0 wz ¯    (α) (α) ¯ α 1−γ0 ζ w ¯ 1−γ0 ζz

α(1 − wz) ¯ 

1−



=

 (α) 2 wz ¯ 1−γ0   (α) (α) ¯ ¯ 1−γ0 ζz 1−γ0 ζ w

α(1 − wz) ¯



1−



ζw ¯

(α) 1−γ0 ζ w ¯

=

=

(α)

1−γ0



(α)

1−γ0

·



¯ ζz

(α) ¯ 1−γ0 ζz

α(1 − wz) ¯

1−

(α) (α) bζ (w)bζ (z)

α(1 − wz) ¯

.

Now by Theorem at ζ is continuous, so there must be some   8, evaluation (α) (α) vector qζ such that f, qζ = f (ζ) for every f ∈ Dζ,α . Then Dζ,α



(α)

f, qζ

 Dζ,α

  (α) = f (ζ) = lim f (w) = lim f, Qζ (w, ·) w→ζ

w→ζ

(α)

for all f ∈ Dζ,α , if w approaches ζ non-tangentially. So qζ and therefore the pointwise limit, every z ∈ D, it is easily seen that

Dζ,α

is the weak limit,

(α) of Qζ (w, ·) as w → ζ non-tangentially. (α) (α) 1 Qζ (w, z) → Qζ (ζ, z) = . (α) ¯ ζz) α(1−γ 0

But for


4. The D(µ, α) and P(µ, α) Spaces Now suppose µ is a positive finite Borel measure on ∂D. We define  

D(µ) = f ∈ H 2 : Dζ (f ) dµ(ζ) < ∞ .

(17)

∂D

Non-tangential limits of functions in D(µ) exist at [µ]-almost every point of ∂D. In fact, Richter and Sundberg [21, Corollary 2.3] have shown that the boundary function f (ζ) belongs to L2 (µ).

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D(µ) is a vector space, and it can be given one of a family of (possibly indefinite) inner products. For any α ∈ R, define 

 f (·) − f (ζ) g(·) − g(ζ) ,
f, gµ,α = α
f, gH 2 + dµ(ζ) (18) ·−ζ ·−ζ ∂D H2 for f, g ∈ D(µ). When D(µ) is equipped with this inner product, we will refer to it as D(µ, α). If µ is taken to be a unit point mass at some ζ ∈ ∂D, then D(µ, α) reduces to the local Dirichlet space Dζ,α . When µ is taken to be Lebesgue measure, D(µ, 1) can be identified with the classical Dirichlet space. (See [21].) When α is positive, D(µ, α) is a Hilbert space. The α = 1 case gives exactly the D(µ) spaces of Richter. (Of course, when α > 0, D(µ, α) is just a scaled version of D(µ/α, 1).) When α is negative, the inner product is indefinite. A complete description of which measures µ and real numbers α result in D(µ, α) being a Pontryagin space (or a Kre˘ın space) is not yet known; however, some partial results can be established. (See [14] for details.) • Every measure µ has a range of negative values of α for which D(µ, α) is a Pontryagin space. In particular, Theorem 8 can be used to show that if −µ(∂D)/4 < α < 0, then D(µ, α) is a Pontryagin space with negative index one. • When µ is Lebesgue measure, D(µ, α) is a Pontryagin space for all negative non-integer values of α. (When α is a negative integer, the inner product is degenerate.) In this case, D(µ, α) can be naturally identified with the generalized Dirichlet spaces Dν of de Branges ([8, 9]). • If µ is greater than a constant multiple of Lebesgue measure, D(µ, α) is a Pontryagin space for all values of α for which the inner product is nondegenerate. Equation (10) can be integrated with respect to µ to give



Dζ (Mz f ) dµ(ζ) = Dζ (f ) dµ(ζ) + f 2L2(µ) . ∂D

(19)

∂D

For any f ∈ D(µ), the terms on the right side of this equation are finite, so Mz f ∈ D(µ). This equation then gives
Mz f, Mz f µ,α =
f, f µ,α + f 2L2(µ) .

(20)

Theorem 11. Suppose µ and α are such that D(µ, α) is a Kre˘ın space. Then Mz ∈ L(D(µ, α)) is an expansive analytic 2-isometry. Proof. It has been pointed out that D(µ, α) is closed under Mz . By the Closed Graph Theorem, then, Mz is a continuous operator on D(µ, α). It is clearly an

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analytic operator. Equation (20) shows that Mz is expansive, and we have
2  Mz f, Mz2 f µ,α − 2
Mz f, Mz f µ,α +
f, f µ,α = (
Mz (Mz f ), Mz (Mz f )µ,α −
Mz f, Mz f µ,α ) − (
Mz f, Mz f µ,α −
f, f µ,α ) =

ζf (ζ) 2L2 (µ)

−

f (ζ) 2L2 (µ)

= 0.


Thus, Mz is a 2-isometry.

In the main theorem of Section 5, an isomorphism is constructed on the set of polynomials. In order that this isomorphism might be lifted to the entire D(µ, α) space, it is desired that the polynomials form a dense subspace. When µ and α are such that D(µ, α) is a Pontryagin space, this is indeed the case. Theorem 12. Suppose µ and α are such that D(µ, α) is a Pontryagin space. Then the polynomials are dense in D(µ, α). Proof. Let f ∈ D(µ, α). It must be shown that f can be approximated by polynomials in the unique strong topology induced by any of the associated fundamental decompositions of D(µ, α). The operator Mz is an expansive analytic 2-isometry on this Pontryagin space, so by Theorem 2, there exists a non-negative integer N and a fundamental decomposition D(µ, α) = K+ ⊕ K− (where K+ is a Hilbert space and K− is the anti-space of a Hilbert space) such that MzN D(µ, α) ⊂ K+ . Thus, we write f = q + k, where q is a polynomial of degree less than N , and k ∈ MzN D(µ, α). ∈ H 2 }. Since k ∈ D(µ), µ(∂D − Z) = 0. Let Z = {ζ ∈ ∂D : k(z)−k(ζ) z−ζ For 0 < r < 1, let kr be defined by kr (z) = k(rz). By a theorem of Sarason [23, Proposition 3], Dζ (kr ) ≤ Dζ (k) and Dζ (kr − k) → 0 for each ζ ∈ Z. Thus, by Dominated Convergence,

Dζ (kr − k) dµ(ζ) → 0. ∂D

Both k and kr belong to the Hilbert space MzN D(µ, α) ⊂ K+ . We have

Dζ (kr − k) dµ(ζ) → 0.

kr − k 2K+ =
kr − k, kr − kµ,α = α kr − k 2H 2 + ∂D

Let  > 0 be given. Then there is an r ∈ (0, 1) such that kr − k K+ < /2. The function kr is analytic on the closed unit disk. Let pM be the M th partial sum of its power series. By Lemma 6, Dζ (kr − pM ) → 0 as M → ∞, for every ζ ∈ ∂D. Again by Dominated Convergence,

Dζ (kr − pM ) dµ(ζ) → 0. ∂D

But all the partial sums pM are in MzN D(µ, α) ⊂ K+ as well. Thus

kr − pM 2K+ =
kr − pM , kr − pM µ,α → 0

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227

as M → ∞. So there exists a particular M such that kr − pM K+ < /2. So set p = q + pM . Then p is a polynomial, and

f − p |D(µ,α)| = k − pM K+ ≤ k − kr K+ + kr − pM K+ < . Thus, the polynomials are dense in D(µ, α).




Corollary 13. Suppose µ and α are such that D(µ, α) is a Pontryagin space. Then Mz ∈ L(D(µ, α)) is a cyclic analytic 2-isometry. Any nonzero constant function is a cyclic vector. It is currently unknown whether polynomials are dense in D(µ, α) when it is only a Kre˘ın space, even if the inner product has only finitely many negative squares on the polynomials. For this reason, we make the following definition. Definition 14. Let P(µ, α) denote the completion of the polynomials in the inner product of D(µ, α), whenever that inner product is non-degenerate and has a finite number of negative squares on the polynomials. We note that such a completion always exists; see Theorem 2.5 of [15]. A complete characterization of the P(µ, α) spaces is unknown. However, by Theorem 12, whenever D(µ, α) is a Pontryagin space, it can be identified with P(µ, α). The Mz operator will still be an expansive cyclic 2-isometry on P(µ, α), provided that it can be extended continuously from the polynomials to all of P(µ, α). The necessary condition for such an extension is that, in some associated Hilbert norm, there be a constant C > 0, depending on µ and α, such that zp ≤ C p

for every polynomial p.

5. A Model for a Class of 2-Isometries We now proceed with the Pontryagin space generalization of Richter’s model for cyclic analytic 2-isometries. Theorem 15. Suppose T is a cyclic analytic 2-isometry on a Pontryagin space K, with the range of T a Kre˘ın subspace, and with a cyclic vector in K  T K. Then there is a positive finite Borel measure µ on ∂D, and a real non-zero α, such that Mz is a continuous analytic operator on P(µ, α), and T is unitarily equivalent to Mz . The proof follows the same outline as in the original Hilbert space model. The 2-isometry induces a certain cyclic subnormal operator (actually an isometry), and then an appeal to Bram’s Theorem is made. Bram’s Theorem states that every cyclic subnormal operator is unitarily equivalent to multiplication by the variable on the closure of the polynomials in some L2 (µ) space (with the cyclic vector corresponding to the constant function 1.) (See [7].) It has been established in Section 2 that an analytic 2-isometry on a Pontryagin space is expansive and that ker T is a Hilbert space. These facts are needed in the proof.

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In addition, two other conditions are taken as hypothesis in Theorem 15. First, although the range of T is always closed, we require that it be a Kre˘ın subspace. In general, a linear subspace M of a Kre˘ın space K is a Kre˘ın subspace if and only if K = M ⊕ M⊥ . This is also equivalent to the existence of an orthogonal projection whose range is M. (See Theorem 1.3 of [13].) It is known that the range of a (continuous) isometry on any Kre˘ın space is a Kre˘ın subspace (see Corollary 1.9 of [13]). Whether or not this is true for 2-isometries is unknown, but it turns out to be equivalent to the invertibility of T ∗ T , as shown in the more general Lemma 16 below. This is useful, because we then have a specific left inverse for T , L = (T ∗ T )−1 T ∗ , which plays a special role. Note that the invertibility of T ∗ T is automatic if K is Hilbert, because T ∗ T ≥ I. (In a Pontryagin space, though, a non-negative operator may have spectra on the negative real axis.) The second condition hypothesized in Theorem 15 is that the cyclic analytic 2-isometry have a cyclic vector in the orthogonal complement of the range. Richter ([19, Theorem 1]) was able to prove that this is always true when T acts on a Hilbert space. It turns out that this is not always the case on a Pontryagin space, and we give an example of this in Section 6. However, on the D(µ, α) and P(µ, α) Pontryagin spaces, the orthogonal complement of the range of Mz is the set of constant functions, which are cyclic vectors. Thus, this hypothesis is a necessary one. Lemma 16. Suppose K is a Kre˘ın space and T ∈ L(K) is injective. Then the range of T is a Kre˘ın subspace of K if and only if T ∗ T is invertible. When this holds, the orthogonal projection onto the range of T is given by T L, where L = (T ∗ T )−1 T ∗ . Proof. Let M be the range of T , and suppose that M is a Kre˘ın subspace. Then K = M ⊕ M⊥ , where M⊥ = ker T ∗ . Now if T ∗ T y = 0 for some y ∈ K, then T y ∈ M⊥ . But T y belongs to M also. Thus, T y = 0. Since T is injective, y = 0. Therefore, T ∗ T is injective. Now T ∗ T is selfadjoint, so (T ∗ T K)⊥ = ker T ∗ T = {0}. Thus T ∗ T is also surjective. Therefore, T ∗ T is invertible. Now suppose that T ∗ T is invertible. Let L = (T ∗ T )−1 T ∗ . T L is a selfadjoint idempotent, and its range is contained in the range of T . Also, if x = T y for some y ∈ K, then T Lx = T LT y = T y = x. So the ranges of T and T L are the same. Hence, T L is an orthogonal projection onto the range of T . Therefore, the range of T is a Kre˘ın subspace.
The following two lemmas provide the connection between 2-isometries and the D(µ, α) inner product. The calculations are essentially the same as those used by Richter in the Hilbert space case. Lemma 17 is proved in Theorem 4.1 of [20] using a different but equivalent formula for the integral of Dζ (f ). We provide a direct proof here. Lemma 18 is a slight generalization of Lemma 2.1 of [20].

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For any polynomial p(z) =

N

n=0

229

an z n , we define

p(k) (z) =

N

an z n .

n=k

Lemma 17. For any polynomial p,

∞
 Dζ (p) dµ(ζ) = p(k) , p(k) L2 (µ) .

∂D

Proof. Let p(z) =

M

n=0

k=1

an z n . Then

  it  p(e ) − p(ζ) 2 dt   Dζ (p) dµ(ζ) =  eit − ζ  2π dµ(ζ) ∂D ∂D 0 2

2π  M ijt j   j=0 aj (e − ζ )  dt dµ(ζ) =    2π  eit − ζ ∂D 0  2 

2π  j−1 M  ilt j−l−1  dt  = aj e ζ   2π dµ(ζ) ∂D 0 j=1 





2π

l=0

=

j−1 M i−1

ai aj

∂D

i,j=1 k=0 l=0

=

j−1 M i−1



δk,l ai aj

=

δk,l

k,l=0

=

M

ai aj

ζ i−k−j+l dµ(ζ)

∂D

ζ i−j dµ(ζ)

∂D

 M 2    m am ζ  dµ(ζ)   ∂D 

M

k=1

=

ζ i−k−j+l dµ(ζ)



ai aj

dt dµ(ζ) 2π

∂D

i=k+1 j=l+1 M

M−1

k=0 i,j=k+1

=

M

ei(k−l)t ζ i−k−j+l

0



i,j=1 k=0 l=0 M−1

2π

M k=1

m=k


 p(k) , p(k) L2 (µ) .


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Lemma 18. Suppose that T is a 2-isometry on a Pontryagin space K, and the range of T is a Kre˘ın subspace. If x ∈ K  T K and p is any polynomial, then


p(T )x, p(T )x = QLk p(T )x, QLk p(T )x + T p(k) (T )x, p(k) (T )x k≥0

=

k≥1


x, x p 2H 2


 + T p(k) (T )x, p(k) (T )x ,

(21) (22)

k≥1

where L = (T ∗ T )−1 T ∗ and Q = I − T L. Proof. By Lemma 16, L exists, and Q and T L are complementary projections. Then
p(T )x, p(T )x =
Qp(T )x, Qp(T )x +
T Lp(T )x, T Lp(T )x =
Qp(T )x, Qp(T )x 

 T Lk p(T )x, T Lk p(T )x − T Lk+1 p(T )x, T Lk+1 p(T )x + k≥1

=
Qp(T )x, Qp(T )x 
 + (I − T L)Lk p(T )x, (I − T L)Lk p(T )x k≥1



 + T Lk p(T )x, T Lk p(T )x − Lk p(T )x, Lk p(T )x

=
Qp(T )x, Qp(T )x 

 QLk p(T )x, QLk p(T )x + T Lk p(T )x, Lk p(T )x + k≥1



 = QLk p(T )x, QLk p(T )x + T Lk p(T )x, Lk p(T )x k≥0

k≥1

k≥0

k≥1



 = QLk p(T )x, QLk p(T )x + T T k Lk p(T )x, T k Lk p(T )x . The last equality follows from (3). Note that Lk T n x = 0 if k > n, since LT = I and x ∈ ker T ∗ = ker L. If k ≤ n, then T k Lk T n x = T n x. Therefore, T k Lk p(T )x = p(k) (T )x. Equation (21) then follows. Furthermore, if k < n, then QLk T n x = QT n−k x = 0, since Q is the projection onto the orthogonal complement of the range of T . Also, QLk T k x = Qx = x.  n k Therefore, if p(z) = N n=0 an z , then QL p(T )x = ak x. Thus,
 |ak |2
x, x = p 2H 2
x, x . QLk p(T )x, QLk p(T )x = k≥0

Equation (22) follows from this.

k≥0




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231

Corollary 19. Suppose T is an analytic 2-isometry on a Pontryagin space K, with the range of T a Kre˘ın subspace, and with a cyclic vector in K  T K. Then K  T K is a one-dimensional subspace, and it is negative if the negative index of K is at least one. Proof. Let x ∈ K  T K be a cyclic vector for T . Then K is the closed linear span of {x, T x, T 2x, . . .}. Let M be the closed linear span of {T x, T 2x, . . .}. It has codimension no more than one. But M is contained in the range of T , so K  T K ⊂ M⊥ . Thus, K  T K has dimension at most one. But the range of T is not all of K, since T is analytic. Thus, K  T K is one-dimensional. Suppose x were not a negative vector. In (21), Q is the projection onto KT K, and so

p(T )x, p(T )x ≥ T p(k) (T )x, p(k) (T )x ≥ 0 k≥1

for all polynomials p, since T ≥ 0 (Theorem 2). But vectors of this form are dense in K, since x is cyclic. This contradicts the assumption that K has negative index at least one.
Proof of Theorem 15. By Theorem 3, ker T is a Hilbert space, and it is closed in K, so it is a Kre˘ın subspace of K. Let M0 be the orthogonal complement of ker T , and equip it with the inner product
f, gM0 =
T f, gK for all f, g ∈ M0 . By Theorem 2, this is a strictly positive inner product space, so we form the Hilbert space completion M. Let P denote the orthogonal projection of K onto M0 , and let T be the operator on M which agrees with P T on the dense subspace M0 . Let x ∈ M0 . Using the fact that P T = T P = T , we have   T x, T x =
T P T x, P T xK M0

=
T ∗ T T x, xK =
T x, xK =
x, xM0 .

Therefore, T ∈ L(M) is an isometry. Let x be a cyclic vector of T in K  T K. It will be shown that P x is cyclic for T . Note that since ker T is invariant under T (by Theorem 3), P T P = P T . So for any polynomial q, q(T )P x = P q(T )P x = P q(T )x. Therefore, if y ∈ M0 ,

y − q(T )P x 2M =
T (y − P q(T )x), y − P q(T )xK =
T (y − q(T )x), y − q(T )xK (since T (I − P ) = 0). As in the proof of Theorem 3, we factor T = DD∗ , where D ∈ L(D, K) for some Hilbert space D. Thus, y−q(T )P x M = D∗ (y−q(T )x) D ≤ D∗

y−q(T )x |K|. Since x is cyclic for T , this can be made arbitrarily small by choosing the right

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! polynomial q. Thus, y ∈ T n P x. Since M0 is dense in M, this proves that P x is cyclic for T . Being an isometry, T is subnormal. Therefore, Bram’s Theorem [7] implies the existence of a positive finite Borel measure µ on the spectrum of T ’s minimal normal extension (which, since T is an isometry, is contained in ∂D), such that T is unitarily equivalent to Mz on P 2 (µ). In this equivalence, the cyclic vector P x corresponds to the constant function 1. Therefore, for any polynomial q,

q(T )P x M = q L2 (µ) . Thus,
T q(T )x, q(T )xK =
T P q(T )x, P q(T )xK =
T P q(T )P x, P q(T )P xK = q(T )P x 2M = q L2 (µ) . Using this in conjunction with (22) and Lemma 17, we get
 T p(k) (T )x, p(k) (T )x K
p(T )x, p(T )xK =
x, x p 2H 2 + k≥1

=


x, x p 2H 2

+



p(k) L2 (µ) (23)

k≥1

=
x, x p 2H 2 + =
p, pµ,α ,

∂D

Dζ (p(k) ) dµ(ζ)

where α =
x, x. Note that α is positive if K is actually a Hilbert space, and it is strictly negative otherwise by Corollary 19. Let κ be the negative index of K. The subspace {p(T )x} is dense in K, so it has κ negative squares. By (23), then, the polynomials have κ negative squares in the D(µ, α) inner product. Therefore, P(µ, α) is defined, and it has the same number of negative squares as K. The linear relation {(p, p(T )x) : p polynomial} ⊂ P(µ, α)×K is isometric by (23). So the closure of this linear relation is the graph of a unitary operator U ∈ L(P(µ, α), K) (see Theorem 1.4.2 (3) of [4]). So U p = p(T )x for all polynomials p. Using Theorem 2, K can be fundamentally decomposed as K = K+ ⊕ K− , such that T N K ⊂ K+ for some N . This induces a fundamental decomposition P(µ, α) = (U −1 K+ ) ⊕ (U −1 K− ). If p(z) = z N g(z) for some polynomial g, then p(T )x ∈ K+ , so p ∈ U −1 K+ . Using Hilbert norms associated with these decompositions, we have

p 2|P(µ,α)| =
p, pµ,α =
p(T )x, p(T )xK = T p(T )x 2|K| for all such polynomials p. Therefore,

zp 2|P(µ,α)| = T p(T )x 2|K| ≤ T 2 p(T )x 2|K| = T 2 p 2|P(µ,α)| .

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Now T N K has finite codimension, and {p(T )x : p = z N g, gpolynomial} is dense in T N K. Thus the orthogonal complement of {p : p = z N g, g polynomial} in P(µ, α) is finite dimensional. It is clear, then, that there is a constant C > 0 such that

zp 2|P(µ,α)| ≤ C p 2|P(µ,α)| for all polynomials. Therefore, since polynomials are dense in P(µ, α), Mz is a continuous operator on all of P(µ, α). Furthermore, for any polyomial p, U Mz p = U (zp) = T p(T )x = T U (p). Thus, U Mz = T U . Finally,
Mz must be analytic, because T is.

6. A Variation on the Local Dirichlet Spaces In Theorem 1 of [19], Richter proved the nontrivial result that every cyclic analytic 2-isometry on a Hilbert space has a cyclic vector in the orthogonal complement of its range. This property had to be taken as an explicit hypothesis in Theorem 15. In this section, it is shown that there is no way around this. We give an example of a cyclic analytic 2-isometry on a Pontryagin space with no cyclic vector in the orthogonal complement of the range. We consider a Pontryagin space with a reproducing kernel that is a variation of the kernel in the local Dirichlet space Dζ,α . Define (α)

(α)

bζ (z) =

¯ (1 − γ1 )ζz , (α) ¯ 1 − γ ζz

(24)

1

(α)

(α)

Qζ (w, z) = (α)

(α)

1 − bζ (w) bζ (z) α(1 − wz) ¯

.

(25)

(α)

The difference between bζ here and bζ in (13) is that we have replaced the root of the equation z = α(z − 1)2 that lies inside the unit disk with the one that lies (α) (α) outside of it. This means that bζ has a simple pole in the unit disk at z = ζγ0 . (α)

Actually, bζ

(α)

(α)

= bζ /Bζ , where (α)

Bζ

(α)

= ζ¯

z − ζγ0

(α)

is a single Blaschke factor. Thus, Qζ

(26)

(α) ¯ 1 − γ0 ζz

is the reproducing kernel for a unique

Pontryagin space Dζ,α , which consists of functions analytic on the unit disk except (α) (α) for possibly a pole at z = ζγ0 . Furthermore, due to this factorization of bζ (z), Dζ,α decomposes as where Cζ,α

Dζ,α = Dζ,α ⊕ Cζ,α , is the unique Pontryagin space having reproducing kernel (α)

(α) Cζ (w, z)

=

(α) bζ (w)

1 − Bζ (w)

−1

Bζ (z)−1 (α)

α(1 − wz) ¯

(α)

bζ (z).

(27)

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See Chapter 4 of Alpay, Dijksma, Rovnyak, and de Snoo [4], and in particular Theorem 4.2.3, for details on such factorizations of reproducing kernel Pontryagin spaces. Moreover, using Theorem 10 of [5], one sees that Cζ,α is just the one-dimensional space of multiples of the function (α)

bζ (z) , (α) ¯ 1 − γ ζz 1

and we have



(α)

(α)

bζ (z) bζ (z) s , t (α) ¯ (α) ¯ 1 − γ1 ζz 1 − γ1 ζz

 Cζ,α

α = st¯ 2 .  (α) 1 − γ1

(28) (α)

Thus, if α is positive, then Cζ,α is a negative space, since |γ1 | > 1. But if α is negative (but still greater than −1/4), then it is a positive space. Recall that the negative index of Dζ,α is zero if α > 0, and one if −1/4 < α < 0. But Cζ,α ∩ Dζ,α = {0}, so by Theorem 8 of [5], the negative index of Dζ,α is the sum of the negative indices of Cζ,α and Dζ,α . Thus, the negative index of Dζ,α is one in all cases. Proposition 20.Evaluation at ζ ∈ ∂D is continuous on Dζ,α , and for all f ∈ Dζ,α ,  (α) f (ζ) = f, qζ , where Dζ,α

1

(α)

qζ (z) =

(α) ¯ α(1 − γ1 ζz)

.

(29)

Proof. By Theorem 8, evaluation at ζ is continuous on Dζ,α . The space Cζ,α is (α) (α) ¯ are one-dimensional, and functions which are multiples of bζ (z)/(1 − γ1 ζz) certainly defined at ζ. Thus, continuity holds on all of Dζ,α . Just as in the proof of Theorem 10, the Riesz vector for evaluation at ζ must be the pointwise limit of (α) Qζ (w, z) as w approaches ζ nontangentially. This gives (29).
In order to show that Mz is a well-defined operator on Dζ,α , we must show that it maps Cζ,α back into Dζ,α . This follows from the calculation (α)

zbζ (z) (α) ¯ 1 − γ1 ζz

(α)

=

(α)

(z − ζγ0 )bζ (z) (α) ¯ 1 − γ1 ζz

(α) (α)

+

ζγ0 bζ (z) (α) ¯ 1 − γ1 ζz (α)

=

(α) (α) −ζγ0 bζ (z)

+

(α) ζγ0

(30)

bζ (z) . (α) ¯ 1 − γ ζz 1

which belongs to Dζ,α . (The first term is analytic on D and so belongs to Dζ,α , and the second term clearly belongs to Cζ,α .) We can conclude that Mz ∈ L(Dζ,α ) by the Closed Graph Theorem.

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It turns out that the formula (11), which allowed us to conclude that Mz was a 2-isometry on Dζ,α , still holds true in Dζ,α . Lemma 21. For all f ∈ Dζ,α ,
Mz f, Mz f Dζ,α =
f, f Dζ,α + |f (ζ)|2 . Proof. In the subsequent calculations, we use (12) several times. We also repeat(α) (α) edly use the fact that γ0 and γ1 are roots of the equation α(1 − z)2 = z, as well as these two easily verifiable identities:    (α) (α) (α) (α) γ0 γ1 = 1 1 − γ1 = −1 α 1 − γ0   (α) (α) (α) First, note that bζ (z) = αζ¯ 1 − γ0 zqζ (z). So   (α) (α) bζ , bζ

Dζ,α

 2   (α) (α) (α) Mz qζ , Mz qζ = α2 1 − γ0 Dζ,α   2   (α) (α) (α) (α) qζ , qζ = α2 1 − γ0 − |qζ (ζ)|2 Dζ,α   (α) (α) (α) 2 qζ (ζ) + |qζ (ζ)| = αγ0    2  (α) (α) (α) γ1 − 1 + γ1 − 1 = αγ0    (α) (α) (α) = αγ0 γ1 γ1 − 1   (α) = α γ1 − 1 .

For any f ∈ Dζ,α , write (α)

f (z) = t

bζ (z) (α) ¯ 1 − γ1 ζz

+ g(z),

where t is a scalar and g ∈ Dζ,α . We then have, using (28),   (α) (α) bζ (z) bζ (z) ,t
f, f Dζ,α = t +
g, gDζ,α (α) ¯ (α) ¯ 1 − γ ζz 1 − γ ζz 1

2

=

α|t|

(α)2

1 − γ1

Cζ,α

1

+
g, gDζ,α .

Also, using (30), (α)

(α)

zf (z) = tζγ0

bζ (z) (α) ¯ 1 − γ1 ζz

(α) (α)

− tζγ0 bζ (z) + zg(z),

(31)

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and so we have  (α) (α) bζ (z) (α) bζ (z) , tζγ0 (α) ¯ (α) ¯ 1 − γ1 ζz 1 − γ1 ζz Cζ,α   (α) (α) (α) (α) + zg − tζγ0 bζ (z), zg − tζγ0 bζ (z)



(α) tζγ0


zf, zf Dζ,α =

=

(α)2 α|t|2 γ0 (α)2 1 − γ1

− But notice that   (α) bζ , zg

Dζ,α

+
zg, zgDζ,α

¯ (α) t¯ζγ 0



(α) zg, bζ

  (α) (α) bζ , zg − tζγ0

Dζ,α

Dζ,α

 Dζ,α

+

(α)2 α|t|2 γ0

  (α) γ1 − 1 .

   (α) (α) zqζ , zg = αζ¯ 1 − γ0 Dζ,α      (α) (α) (α) ¯ qζ , g = αζ 1 − γ0 + qζ (ζ) g(ζ) Dζ,α      (α) (α) = αζ¯ 1 − γ0 g(ζ) + γ1 − 1 g(ζ)   (α) (α) γ1 g(ζ) = αζ¯ 1 − γ0   (α) = αζ¯ γ1 − 1 g(ζ).

So we have
zf, zf Dζ,α =

(α)2

α|t|2 γ0 1−

(α)2 γ1

(α)

+
zg, zgDζ,α − αtγ0 (α)

− αt¯γ0



(α)

γ1

=

(α)2

1 − γ1

(α)

γ1

 − 1 g(ζ)

   (α)2 (α) γ1 − 1 − 1 g(ζ) + α|t|2 γ0

(α)2

α|t|2 γ0





+
g, gDζ,α + |g(ζ)| − αt 1 − 2

(α) γ0



(32)

g(ζ)

    (α) (α)2 (α) g(ζ) + α|t|2 γ0 γ1 − 1 . − αt¯ 1 − γ0

Subtracting (31) from (32), we get
zf, zf Dζ,α −
f, f Dζ,α (α)2

=

α|t|2 γ0

(α)2

1 − γ1

  (α) + |g(ζ)|2 − αt 1 − γ0 g(ζ)

    (α) (α)2 (α) g(ζ) + α|t|2 γ0 γ1 − 1 − − αt¯ 1 − γ0

α|t|2 (α)2

1 − γ1

.

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Comparing this to  2   t   2 + g(ζ) |f (ζ)| =    1 − γ (α)  1 2      (α) + g(ζ) = −αt 1 − γ0  2     (α) (α) (α) g(ζ) + |g(ζ)|2 = α2 |t|2 1 − γ0 − αt 1 − γ0 g(ζ) − αt¯ 1 − γ0     (α) (α) (α) g(ζ) + |g(ζ)|2 , = αγ0 |t|2 − αt 1 − γ0 g(ζ) − αt¯ 1 − γ0 we see that the claim will be proved if (α)

(α)2

=

αγ0

(α)2

(α)

γ1

 −1 −

α

(33)

This is verified algebraically using the above mentioned identities.




1−

(α)2 γ1

+ αγ0



.

αγ0

(α)2

1 − γ1

Theorem 22. Mz ∈ L(Dζ,α ) is a cyclic, analytic 2-isometry, and its range is a Kre˘ın subspace. However, Dζ,α  Mz Dζ,α contains no cyclic vector. Proof. With the result of Lemma 21 in hand, the proof that Mz is a 2-isometry is identical to that in Theorem 9. Moreover, Dζ,α consists of functions which are analytic in a neighborhood of zero. No such function could be in the range of Mzn for every n. Thus, Mz is analytic. (α) We claim that qζ is a cyclic vector for Mz . First, observe that (α) (α) ¯ (α) αqζ (z) − αγ1 ζz qζ (z) = 1.

!∞

Mzn qζ . Hence all polynomials are in this closed span as well. Since !∞ (α) polynomials are dense in Dζ,α , we have Dζ,α ⊂ n=0 Mzn qζ . However, equality So 1 ∈

n=0

(α)

(α)

∈ / Dζ,α since it has a pole in the disk. But the dimension of ! (α) n (α) Dζ,α  Dζ,α = Cζ,α is one. Thus, Dζ,α = ∞ is cyclic. n=0 Mz qζ , and so qζ The calculation does not hold, for qζ

(α)   bζ (z) (α) (α) (α) ¯ ¯ zqζ (z) + γ1 ζz = αζ 1 − γ0 (α) ¯ (α) ¯ 1 − γ1 ζz 1 − γ1 ζz (α)

bζ (z)

shows that Cζ,α is contained in the range of Mz . Thus the range of Mz is Mz Dζ,α ⊕ Cζ,α , which is a Kre˘ın subspace. The orthogonal complement of the range, then, is the one-dimensional space of constant functions. These can not be cyclic vectors,
for they only generate Dζ,α , not all of Dζ,α .

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References [1] J. Agler and M. Stankus, m-isometric transformations of Hilbert space. I, Integral Equations Operator Theory 21 (1995), no. 4, 383–429. MR 96j:47008a [2]

, m-isometric transformations of Hilbert space. II, Integral Equations Operator Theory 23 (1995), no. 1, 1–48. MR 96j:47008b

[3]

, m-isometric transformations of Hilbert space. III, Integral Equations Operator Theory 24 (1996), no. 4, 379–421. MR 96j:47008c

[4] D. Alpay, A. Dijksma, J. Rovnyak, and H. S. V. de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Birkh¨ auser Verlag, Basel, 1997. MR 2000a:47024 [5]

, Reproducing kernel Pontryagin spaces, Holomorphic spaces (Berkeley, CA, 1995), Oper. Theory Adv. Appl., vol. 96, Cambridge Univ. Press, Cambridge, 1998, pp. 425–444. MR 99g:46028

[6] J. Bogn´ ar, Indefinite inner product spaces, Springer-Verlag, New York, 1974, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78. MR 57 #7125 [7] J. Bram, Subnormal operators, Duke Math. J. 22 (1955), 75–94. MR 16,835a [8] L. de Branges, Unitary linear systems whose transfer functions are Riemann mapping functions, Operator theory and systems (Amsterdam, 1985), Oper. Theory Adv. Appl., vol. 19, Birkh¨ auser, Basel, 1986, pp. 105–124. [9]

, Underlying concepts in the proof of the Bieberbach conjecture, Proceedings of the International Congress of Mathematicians (Berkeley, California, 1986), Amer. Math. Soc., Providence, RI, 1988, pp. 25–42.

[10] T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York, 1978, Mathematics and its Applications, Vol. 13. MR 58 #1979 [11] G. Christner, Kin Y. Li, and J. Rovnyak, Julia operators and coefficient problems, Nonselfadjoint operators and related topics (Beer Sheva, 1992), Birkh¨ auser, Basel, 1994, Oper. Theory Adv. Appl., vol. 73, pp. 138–181. MR 95m:47058 [12] T. Constantinescu and A. Gheondea, Representations of Hermitian kernels by means of Kre˘ın spaces, Publ. Res. Inst. Math. Sci. 33 (1997), no. 6, 917–951. [13] M. A. Dritschel and J. Rovnyak, Operators on indefinite inner product spaces, Lectures on operator theory and its applications (Waterloo, ON, 1994), Fields Institute Monographs, vol. 3, Amer. Math. Soc., Providence, RI, 1996, pp. 141–232. MR 96k:47001 [14] C. Hellings, Two-isometries on Pontryagin spaces, Ph.D. thesis, University of Virginia, Charlottesville, VA, August 2000. [15] I. S. Iokhvidov, M. G. Kre˘ın, and H. Langer, Introduction to the spectral theory of operators in spaces with an indefinite metric, Akademie-Verlag, Berlin, 1982. MR 85g:47050 [16] S. A. McCullough and L. Rodman, Two-selfadjoint operators in Kre˘ın spaces, Integral Equations Operator Theory 26 (1996), no. 2, 202–209. MR 98b:47047 [17]

, Hereditary classes of operators and matrices, Amer. Math. Monthly 104 (1997), no. 5, 415–430. MR 98g:47021

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, Normal generalized selfadjoint operators in Krein spaces, Linear Algebra Appl. 283 (1998), no. 1-3, 239–245. MR 2000a:47077 S. Richter, Invariant subspaces of the Dirichlet shift, J. Reine Angew. Math. 386 (1988), 205–220. MR 89e:47048 , A representation theorem for cyclic analytic two-isometries, Trans. Amer. Math. Soc. 328 (1991), no. 1, 325–349. MR 92e:47052 S. Richter and C. Sundberg, A formula for the local Dirichlet integral, Michigan Math. J. 38 (1991), no. 3, 355–379. MR 92i:47035 M. Rosenblum and J. Rovnyak, Topics in Hardy classes and univalent functions, Birkh¨ auser Verlag, Basel, 1994. D. Sarason, Local Dirichlet spaces as de Branges-Rovnyak spaces, Proc. Amer. Math. Soc. 125 (1997), no. 7, 2133–2139. MR 98h:46023

Christian Hellings Gwynedd-Mercy College 1325 Sumneytown Pike P.O. Box 901 Gwynedd Valley, PA 19437 U.S.A. e-mail: [email protected] Submitted: June 26, 2007 Revised: February 25, 2008

Integr. equ. oper. theory 61 (2008), 241–279 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/020241-39, published online April 17, 2008 DOI 10.1007/s00020-008-1589-2

Integral Equations and Operator Theory

Operators with Singular Trace Conditions on a Manifold with Edges D. Kapanadze, B.-W. Schulze and J. Seiler Abstract. We establish a new calculus of pseudodifferential operators on a manifold with smooth edges and study ellipticity with extra trace and potential conditions (as well as Green operators) at the edge. In contrast to the known scenario with conditions of that kind in integral form we admit in this paper ‘singular’ trace and Green operators. In contrast to standard conditions in the theory of elliptic boundary value problems (like Dirichlet or Neumann conditions) our singular trace conditions, in general, do not act on functions that are smooth up to the boundary, but admit a more general asymptotic structure. Their action is now associated with the Laurent coefficients of the meromorphic Mellin transforms of functions with respect to the half-axis variable, the distance to the edge. Keywords. Pseudodifferential operator, ellipticity, Green operator, singular trace condition, manifold with edge.

1. Introduction The construction of a pseudodifferential algebra containing both the classical boundary value problems (such as the Dirichlet or Neumann problem for the Laplacian) as well as their parametrices leads in case of a (compact) manifold M with smooth boundary to Boutet de Monvel’s algebra [1] (see also the work of Vishik and Eskin [4], [9]). The operators in this calculus have block-matrix form,
 A K : T Q

C ∞ (M, F ) C ∞ (M, E) ⊕ ⊕ −→ , C ∞ (∂M, J− ) C ∞ (∂M, J+ )

(1.1)

(which continuously extend to Sobolev spaces), where E, F and J− , J+ are vector bundles over M and the boundary
∂M  , respectively. Classical boundary value problems correspond to operators

A T

and dim J− = 0, where A is a differential

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operator and T the boundary condition, now called a trace operator. ShapiroLopatinskij elliptic problems have a parametrix of the form (A K), where K is a potential operator and A = r+ P e+ + G with a (singular) Green operator G and a classical pseudodifferential operator P on the double 2M of M that has the transmission property with respect to the boundary of M (moreover, e+ denotes the operator of extension by zero from M to 2M , while r+ denotes restriction from 2M to int M ). Elements of Boutet de Monvel’s algebra are filtered by their order µ ∈ Z (which is simply the usual order of the (pseudo)differential part in the upper left corner) and posses a so-called type, which is a non-negative integer d ∈ N0 . The type can be different from zero only for trace operators and (singular) Green operators. For example, taking k-times the derivative in normal direction followed by restriction to the boundary and application of a pseudodifferential operator on the boundary yields a trace operator of type d = k + 1. Another more complicated task is that of developing an elliptic theory for differential operators on a manifold M with edges. Roughly speaking, outside a lower dimensional smooth stratum Y (the edge), M is a smooth manifold, while each point of Y has a neighbourhood homeomorphic to U × X ∆ , where U is an open subset of Y and X ∆ is a cone with smooth closed cross section X. Note that a manifold with boundary is a particular case of a manifold with edge, namely one with X being a point and thus X ∆ = R+ , the normal to the boundary. Typical differential operators on M are edge degenerate, i.e. they have a specific singular structure close to the edge (see Section 3 for more details). As it turns out, elliptic edge degenerate differential operators in general do not have the Fredholm property (in suitable edge Sobolev spaces), but one has to impose additional trace and potential conditions over the edge, possibly even simultaneously. In other words, developing a Fredholm theory for edge degenerate operators leads to a pseudodifferential calculus of operators in block-matrix form similarly as in (1.1), the so-called edge algebra. This calculus has been developed by the second author in [10], [11], [12]. A detailed exposition may be found in [3]; a short overview we provide in Section 8 of this paper. As is known from the authors’ joint works [13] or [14] the analogy between the respective calculi for boundary value problems and ‘edge problems’ goes very far. But there are also essential differences: While Boutet de Monvel’s calculus requires the transmission property, the edge calculus admits general pseudodifferential symbols smooth up to the boundary. In contrast to standard elliptic trace conditions (e.g. of Dirichlet or Neumann type) at the boundary which make sense on solutions of sufficiently high Sobolev regularity, the solutions to edge problems generally do not admit traces of that kind. On the other hand, solutions to edge problems may have asymptotics at the edge, that can be regarded as a substitute of Taylor asymptotics at the boundary (i.e., smoothness up to the boundary). It is now natural and desirable to organise a calculus of edge degenerate operators with trace conditions based on those general asymptotics (called singular trace conditions) rather than Taylor asymptotics.

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In the present paper we just establish such an extension of the edge algebra by our new singular trace and Green operators. We study compositions of the corresponding edge operators, formulate ellipticity with singular trace, etc., conditions at the edge, and obtain parametrices within the calculus.

2. Symbols of Boutet de Monvel’s algebra As noted in the introduction, boundary value problems with the transmission property may be regarded as a model for calculi with singular trace (and also Green) operators. Let us describe this on the level of boundary symbols in a halfspace Ω × R+ with an open set Ω ⊂ Rq . The variable in Ω we will denote by y, the one of R+ by t. Let p(y, t, η, τ ) be a (classical) symbol of integer order µ with the transmission µ property at t = 0 (the symbols of that kind form a closed subspace Str (Ω × µ µ q+1 ∞ q+1 q+1 R+ × R ) of C (Ω × R+ , Scl (R )) with Scl (R ) being the space of classical symbols of order µ with constant coefficients). Assume that p is independent of t for t > T for some T > 0. Let us set (2.1) op+ (p)(y, η) = r+ op(p)(y, η)e+    i(t−t
)τ p(y, t, η, τ )u(t ) dt d¯τ, d¯τ = (2π)−1 dτ, where e+ for op(p)(y, η)u (t) = e denotes the operator of extension by zero from R+ to R and r+ the restriction from R to R+ of distributions (in (2.1) on the right hand side we tacitly insert a symbol p on Ω × R × Rq+1 with p = p|Ω×R+ ×Rq+1 ; however, since the operator is independent of the choice of p we simply write p). As is known, (2.1) defines a family of continuous operators op+ (p)(y, η) : H s (R+ ) −→ H s−µ (R+ ),

s > − 12 ,

where H s (R+ ) := H s (R)|R+ with H s (R) being the standard Sobolev space of smoothness s on R. Defining for λ > 0 κλ : H s (R+ ) −→ H s (R+ ),

1

(κλ u)(t) = λ 2 u(λt),

(2.2)

we obtain a strongly continuous group of isomorphisms on the space H s (R+ ), i.e., i) κ1 = 1 and κλ κδ = κλδ for all λ, δ > 0, ii) λ → κλ u : [0, ∞) → H s (R+ ) is continuous for any u ∈ H s (R+ ). In particular, these operators are unitary on L2 (R+ ) (with the standard scalar product). An essential observation is that op+ (p)(y, η) is an operator-valued symbol in the following sense. If E is a Hilbert space and {κλ }λ∈R+ a strongly continuous group of isomorphisms on E, we say that E is endowed with a group action. Let us set η = (1 + |η|2 )1/2 .  with group actions {κλ }λ∈R+ and Definition 2.1. Given Hilbert spaces E and E {˜ κλ }λ∈R+ , respectively,  (2.3) S µ (Ω × Rq ; E, E)

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 such that will denote the space of all functions a(y, η) ∈ C ∞ (Ω × Rq , L (E, E)) sup

y∈K,η∈Rq

α β η−µ+|α| ˜ κ−1  < ∞ η Dη Dy a(y, η)κη L (E,E)

for every compact K ⊂ Ω and all multi-indices α, β ∈ Nq . Note that we obtain an equivalent definition of (2.3) when we replace η by, for instance, a smooth function η → [η] that is strictly positive and satisfies [η] = |η| for |η| > C for a C > 0. In the sequel, for convenience, we assume C = 1. Remark 2.2. By standard estimates of strongly continuous semi-groups, a group action {κλ }λ∈R+ on E satisfies κλ L (E) ≤ C max(λ, λ−1 )K

for all λ > 0

with suitable constants C, K ≥ 0. Consequently, 



 (1) → S µ (Ω × Rq ; E, E)  → S µ+K+K (Ω × Rq ; E, E)  (1) , S µ−K−K (Ω × Rq ; E, E)  are equipped with the where the subscript (1) here indicates that both E and E trivial group action κ ≡ 1.  is called (twisted) homoA function a(µ) (y, η) ∈ C ∞ (Ω × (Rq \{0}), L (E, E)) geneous in η = 0 of order µ if ˜λ a(µ) (y, η)κ−1 a(µ) (y, λη) = λµ κ λ

for all (y, η) ∈ Ω × (Rq \{0}),

λ > 0. (2.4)

Note that when χ(η) is an arbitrary zero-excision function on R (i.e., χ is smooth, vanishes in a neighbourhood of the origin, and equals 1 for |η| > R for some R > 0)  if a(µ) is homogeneous in the former we have χ(η)a(µ) (y, η) ∈ S µ (Ω × Rq ; E, E) sense. This gives rise to µ  Scl (Ω × Rq ; E, E), (2.5) q

the subspace of (2.3) of all elements a(y, η) which admit an asymptotic expansion into terms of the kind χ(η)a(µ−j) (y, η), with homogeneous functions a(µ−j) (y, η) of order µ − j, j ∈ N. In this case we let σ∧ (a)(y, η) = a(µ) (y, η) denote the homogeneous principal symbol.  is a Remark 2.3. In view of Remark 2.2, a function a ∈ C ∞ (Ω × Rq , L (E, E)) µ  if and only if there exists a sequence a(µ−j) symbol in the class Scl (Ω × Rq ; E, E) of homogeneous components such that to any given M ≥ 0 there exists an L ∈ N such that a(y, η) −

L 

 (1) , χ(η)a(µ−j) (y, η) ∈ S −M (Ω × Rq ; E, E)

j=0

where the subscript (1) has the same meaning as in Remark 2.2.

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The concept of operator-valued symbols in the above sense is very close to the  = C and the group actions are trivial (i.e., identity opscalar case where E = E erators for all λ > 0). Nevertheless there are beautiful (and sometimes surprising) examples of such operator-valued symbols, as we shall see below. µ (Ω × R+ × Rq+1 ) (independent of t for large Example. For every p(y, t, η, τ ) ∈ Str t) we have

1 s>− . 2 denotes the homogeneous principal part of p of order µ in

op+ (p)(y, η) ∈ S µ (Ω × Rq ; H s (R+ ), H s−µ (R+ )), Moreover, if p(µ) (η, τ ) = 0, then

p(µ) (y, η) := op+ (p(µ) |t=0 )(y, η),

(y, η) ∈ Ω × (Rq \{0})

is homogeneous in the sense of the relation (2.4).  to be a Fr´echet space with It is also necessary for our purposes to admit E group action: Let  = lim j E E ←−j∈N  j with continuous embeddings E  j+1 → be a projective limit of Hilbert spaces E j 0 0    E → . . . → E such that E is endowed with a group action {κλ }λ∈R+ that  j for every j. Then E  is said to be restricts to a group action {κλ |E j }λ∈R+ on E µ j ) endowed with the group action {κλ }λ∈R+ . We have the spaces S (Ω × Rq ; E, E (cl)

for all j, and µ  = lim S µ (Ω × Rq ; E, E j ) S(cl) (Ω × Rq ; E, E) ←− (cl) j∈N

is, by definition, the projective limit of these spaces (subscript ‘(cl)’ means that we are talking about the classical or the general case). A As is known, the parametrix of an elliptic boundary value problem T in the half-space Ω × R+ , with an elliptic differential operator A and a trace operator T which satisfies the Shapiro-Lopatinskij condition with respect to A (for instance, the Dirichlet or the Neumann problem for the Laplacian), can be expressed within Boutet de Monvel’s calculus of pseudodifferential boundary value problems. Besides the symbols of Example 2, this calculus also contains so-called Green symbols. Such symbols have an order µ ∈ R and a so-called type d ∈ N. They have the form of 2 × 2 block matrices
l  d  ∂t 0 gl (y, η) (2.6) g(y, η) = g0 (y, η) + 0 0 l=1

where ∂t indicates differentiation in the direction normal to the boundary, and, with suitable j+ , j− ∈ N0 , ν gl (y, η) ∈ Scl (Ω × Rq ; L2 (R+ ) ⊕ Cj− , S (R+ ) ⊕ Cj+ ),

ν = µ − l,

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is a 2 × 2 block matrix symbol of order ν, whose pointwise adjoint satisfies ν gl (y, η)∗ ∈ Scl (Ω × Rq ; L2 (R+ ) ⊕ Cj+ , S (R+ ) ⊕ Cj− ).

Here, S (R+ ) = limk∈N t−k H k (R+ ) and L2 (R+ ) are endowed with the group ←− action {κλ }λ∈R+ from (2.2), while the spaces L2 (R+ ) ⊕ CN and S (R+ ) ⊕ CN are endowed with {κλ ⊕ idCN }λ∈R+ . Writing g(y, η) = (gij (y, η))i,j=1,2 , the component g21 (the left lower corner) is a so-called trace symbol of order µ and type d, while g12 is a potential symbol of order µ. The associated operators Op(g) (the pseudodifferential operator with respect to the Fourier transform in the y-variables), occasionally also denoted by Opy (g), are generated in the parametrix construction of elliptic boundary value problems. For example, Green’s function of the Dirichlet problem for the Laplacian ∆ is, locally near the boundary, of the form E + Op(g11 ) for a fundamental solution E of ∆ and a Green symbol g11 (y, η) of type 0. Moreover, the potential operator in the solution of the Dirichlet problem is, locally near the boundary, of the form Op(g12 ) (clearly, these relations are true modulo smoothing operators in Boutet de Monvel’s calculus, cf. [1]). Remark 2.4. The amplitude functions of boundary value problems in Ω × R+ have the form
+  op (p)(y, η) 0 a(y, η) = + g(y, η) (2.7) 0 0 where op+ (p) is as in Remark 2, and the second summand, given by (2.6) belongs to µ Scl (Ω × Rq ; H s (R+ ) ⊕ Cj− , S (R+ ) ⊕ Cj+ ), for all s > d − 12 . From (2.6) we see that trace and Green operators of type 0 are of integral form (in contrast to operators of positive type that are combined with differentiations in normal direction and the restriction to the boundary). As noted in the introduction, the calculus of edge problems of [12] has a similar structure, but the corresponding analogues of trace and Green operators of positive type are still missing. In the present paper we introduce them as so-called singular operators while the former ones from the edge calculus are of integral form and regarded as regular operators.

3. Symbolic structure of edge-degenerate differential operators We now start discussing a category of operator-valued symbols that contribute to the symbolic calculus of operators on a manifold with edge. By definition (cf. Section 6.1 below) such a manifold is locally, near the edge, represented by a wedge X  × Ω,

X  = (R+ × X)/({0} × X),

with an edge Ω (an open set in Rq ) and a model cone X  for a closed smooth Riemannian manifold X. The half-space case just corresponds to dim X = 0.

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An edge-degenerate differential operator A on an open stretched wedge X ∧ × Ω,

X ∧ = R+ × X,

in the splitting of variables (r, x, y) has the form  ajα (r, y)(−r∂r )j (rDy )α A = r−µ

(3.1)

j+|α|≤µ

with coefficients ajα ∈ C ∞ (R+ × Ω, Diff µ−(j+|α|) (X)), where Diff ν (X) is the space of all differential operators of order ν with smooth coefficients on X. Since the main aspects concern a neighbourhood of r = 0 we assume that the coefficients ajα are independent of r for r > R for some R > 0. We have A = Opy (a) for the operator-valued amplitude function  a(y, η) = r−µ ajα (r, y)(−r∂r )j (rη)α . (3.2) j+|α|≤µ

We now fix an adequate scale of spaces such that (3.2) is an operator-valued symbol in the sense of (2.3). To this end we let ω denote a cut-off function on the half-axis, i.e. ω ∈ C0∞ (R+ ) and ω ≡ 1 near r = 0. Definition 3.1. Let s ∈ N, γ ∈ R, and n := dim X. We denote by Ks,γ (X ∧ ) the space of all u(r, x) ∈ L2loc (R+ × X, drdx) such that n

r 2 −γ (r∂r )k Dxα (ωu)(r, x) ∈ L2 (R+ × X, drdx)

for all k + |α| ≤ s,

where Dxα for α = (α1 , . . . , αn ) denotes any composition v1α1 . . . vnαn of vector fields vj on X, and s (1 − ω)u ∈ Hcone (X ∧ ), the standard Sobolev space of smoothness s ∈ R on the infinite cone X ∧ (cf. [11], p. 151, for details). Note that for X = S n (the unit sphere in Rn+1 ) we have s (1 − ω)Hcone (X ∧ ) = (1 − ω)H s (Rn+1 ).

The definition of Ks,γ (X ∧ ) for general s ∈ R follows by duality (with respect to the K0,0 -scalar product) and interpolation. The spaces Ks,γ (X ∧ ) can be equipped with norms in terms of Hilbert space scalar products. In particular, we have n K0,0 (X ∧ ) = r− 2 L2 (R+ × X, drdx). We shall endow each Ks,γ (X ∧ ) with the group action defined by (κλ u)(r, x) = λ

n+1 2

u(λr, x),

λ > 0.

(3.3)

For references below we also form the Fr´echet spaces S γ (X ∧ ) = {u ∈ K∞,γ (X ∧ ) : (1 − ω)u ∈ S (R+ , C ∞ (X))}, endowed with the same group action.

(3.4)

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Remark 3.2. For purposes below we also consider the spaces Ks,γ;g (X ∧ ) := [r]−g Ks,γ (X ∧ ),

g ∈ R,

endowed with the group action (κgλ u)(r, x) = λg+

n+1 2

u(λr, x),

λ > 0.

Setting K s,γ (X ∧ ) = Ks,γ;s−γ (X ∧ ), we have that K s,γ (X ∧ ) = rγ K s,0 (X ∧ ) for every s, γ ∈ R.

(3.5)

Proposition 3.3. Given an edge-degenerate differential operator A, the associated operator family (3.2) represents an operator-valued symbol a(y, η) ∈ S µ (Ω × Rq ; Ks,γ (X ∧ ), Ks−µ,γ−µ (X ∧ )) for every s, γ ∈ R. The family of operators  σ∧ (A)(y, η) = r−µ ajα (0, y)(−r∂r )j (rη)α

(3.6)

j+|α|≤µ

is (twisted) homogeneous of order µ, i.e., σ∧ (A)(y, λη) = λµ κλ σ∧ (A)(y, η)κ−1 λ for all λ > 0. Remark 3.4. The operator family (3.2) also represents an operator-valued symbol a(y, η) ∈ S µ (Ω × Rq ; Ks,γ;g (X ∧ ), Ks−µ,γ−µ;g (X ∧ )) for every s, γ, g ∈ R, where we now refer to the group action {κgλ }λ∈R+ in both spaces, cf. Remark 3.2. In particular, we have a(y, η) ∈ S µ (Ω × Rq ; K s,γ (X ∧ ), K s−µ,γ−µ (X ∧ )) for every s, γ ∈ R. For convenience, we shall formulate most of our results for the case g = 0 but already mention here that they extend to the case of arbitrary weight g ∈ R. Let σψ (A)(r, x, y, , ξ, η) denote the usual homogeneous principal symbol of the operator (3.1) of order µ. Together with (3.6) we then have the principal symbol of A, (3.7) σ(A) := (σψ (A), σ∧ (A)), which controls the ellipticity in the edge calculus. The operator A is called σψ -elliptic if it is elliptic as usual and if, in addition, its rescaled symbol   σ ψ (A)(x, y, , ξ, η) := rµ σψ (A)(r, x, y, r−1 , ξ, r−1 η)  r=0

does not vanish for (, ξ, η) = 0. The ellipticity of A with respect to both components of (3.7) also requires the bijectivity of σ∧ (A)(y, η) : Ks,γ (X ∧ ) −→ Ks−µ,γ−µ (X ∧ )

(3.8)

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for all (y, η) ∈ Ω×(Rq \{0}). However, this cannot be expected to hold true without additional information. It is known that the σψ -ellipticity of A entails the Fredholm property of (3.8) for all weights γ ∈ R \ D, for a discrete set D = D(y) ⊂ R. The (necessary and sufficient) condition for the Fredholm property is that the subordinate principal conormal symbol σM (A)(y, z) =

µ 

aj0 (0, y)z j : H s (X) −→ H s−µ (X)

j=0

is invertible for all z with Re z = n+1 2 − γ. Similarly as for boundary value problems, the idea of ellipticity of edge problems (in the ‘usual’ sense, cf. [12], [3]) is to fill up the Fredholm family (3.8) by finite rank operators to a 2 × 2 block matrix family of isomorphisms Ks,γ (X ∧ ) Ks−µ,γ−µ (X ∧ ) ⊕ ⊕ −→ (3.9) σ∧ (A)(y, η) : Cj− Cj+
 A K with σ∧ (A)(y, η) as the upper left corner, where A = is a corresponding T Q edge problem with the homogeneous principal edge symbol (3.9). Edge symbols are then 2 × 2 block matrix symbols
 a(y, η) + m(y, η) 0  a(y, η) = + g(y, η) ∈ S µ (Ω × Rq ; E, E) 0 0  = Ks−µ,γ−µ (X ∧ ) ⊕ Cj+ , where g(y, η) is a sowith E = Ks,γ (X ∧ ) ⊕ Cj− and E called Green symbol and m(y, η) a smoothing Mellin symbol, such that A = Op(a). For details see the appendix, Section 8.

4. Spaces with asymptotics 4.1. Cone Sobolev spaces Beside the spaces Ks,γ (X ∧ ) that we defined in Definition 3.1, we are also interested in subspaces consisting of functions that have asymptotics for r → 0. In this connection let P = {(pj , mj , Lj ) : j = 0, . . . , N } for (pj , mj , Lj ) ∈ C × N × C ∞ (X),

(4.1)

∞

with N ∈ N ∪ {∞} and each Lj is a finite dimensional subspace of C (X). We

l and, in case N = ∞, that Re pj → −∞ for also assume that pk = pl for k = j → ∞. Definition 4.1. Let γ ∈ R and 0 < θ ∈ R ∪ {∞}. A discrete asymptotic type associated with the weight data (γ, θ) is a set P as in (4.1) that satisfies

n+1 πC P := {pj : j = 0, . . . , N } ⊂ z : n+1 2 − γ − θ < Re z < 2 − γ . The set of all such P we shall denote by As(γ, θ).

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For P ∈ As(γ, θ) with θ < ∞ we form the space mj N    EP (X ∧ ) = u(r, x) = ω(r) cjk (x)r−pj logk r : cjk ∈ Lj for all j, k , (4.2) j=0 k=0

which is of finite dimension and contained in S γ (X ∧ ). We equip EP (X ∧ ) with a norm by fixing an isomorphism EP (X ∧ ) ∼ = L0 ⊕ . . . ⊕ L0 ⊕ . . . ⊕ LN ⊕ . . . ⊕ LN ∼ = Cι(P ) ,

 

  (m0 +1)-times

ι(P ) =

(mN +1)-times

(4.3)

N 

(mj + 1) dim Lj .

j=0

For θ < ∞ and s ∈ R let us set 1

Kθs,γ (X ∧ ) = limj∈N Ks,γ+θ− 1+j (X ∧ ). (4.4) ←− This Fr´echet space is regarded as the subspace of functions which are flat of order θ with respect to the reference weight γ. Definition 4.2. Let s, γ ∈ R and P ∈ As(γ, θ) with θ < ∞. Then we define KPs,γ (X ∧ ) = Kθs,γ (X ∧ ) + EP (X ∧ ). Note that the sum in the previous definition is a direct sum. In the case P ∈ As(γ, ∞), we set Pk = {(p, m, L) ∈ P :

n+1 2

− γ − k < Re p} ∈ As(γ, k),

k ∈ N,

and then define (X ∧ ), SPγ (X ∧ ) = S γ (X ∧ ) ∩ KP∞,γ (X ∧ ). (4.5) KPs,γ (X ∧ ) = limk∈N KPs,γ k ←− All these spaces with asymptotics we endow with the group action {κλ }λ∈R+ from (3.3). Note that in the previous constructions, we also allow P ∈ As(γ, θ) to be the empty set. In this case, obviously, KPs,γ (X ∧ ) = Kθs,γ (X ∧ ). To unify notation, in this case we write Sθγ (X ∧ ) := SPγ (X ∧ ). 4.2. Edge Sobolev spaces We shall now introduce Sobolev spaces on Rq × X ∧ that will serve as the local models for corresponding Sobolev spaces on a manifold with edge. Definition 4.3. Let E be a Hilbert space with group action {κλ }λ∈R+ . Then the socalled abstract edge Sobolev space W s (Rq , E), s ∈ R, is defined as the completion of S (Rq , E) with respect to the norm
  12 2   dη [η]2s κ−1 u  (η) , u W s(Rq ,E) = [η] E where u (η) = (F u)(η) is the Fourier transform of u.

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The space W s (Rq , E) is then a Hilbert space with scalar product    u, vW s (Rq ,E) = [η]2s κ−1 (η), κ−1 (η) E dη. [η] u [η] v In the special case of κλ = idE for all λ > 0, the edge Sobolev space W s (Rq , E) coincides with H s (Rq , E), the usual E-valued Sobolev space on Rq . Note that replacing in the previous definition [η] by η yields an equivalent norm on the space W s (Rq , E) and that s q s q L := F −1 κ−1 [η] F : W (R , E) −→ H (R , E)

(4.6)

is an isomorphism for each real s. Remark 4.4. If {κλ }λ∈R+ is a group action on E and κλ L (E) ≤ C max(λ−1 , λ)K for all λ > 0, cf. Remark 2.2, then W s+K (Rq , E) → H s (Rq , E) → W s−K (Rq , E). Analogous constructions make sense for a Fr´echet space E = limj∈N E j with ←− group action, cf. the notation after Remark 2. In this case we have the spaces W s (Rq , E) = lim W s (Rq , E j ) ←− j∈N

and a corresponding isomorphism (4.6). Example. Choosing E = Ks,γ (X ∧ ) or E = KPs,γ (X ∧ ) with the group action from (3.3), we obtain so-called weighted edge Sobolev spaces (with asymptotics) W s,γ (Rq × X ∧ ) := W s (Rq , Ks,γ (X ∧ )), WPs,γ (Rq × X ∧ ) := W s (Rq , KPs,γ (X ∧ )). Remark 4.5. We can also form W s (Rq , Ks,γ;g (X ∧ )) with the spaces Ks,γ;g (X ∧ ) and the corresponding group action from Remark 3.2. In particular, for the case g = s − γ we obtain the spaces W s,γ (Rq × X ∧ ) := W s (Rq , K s,γ (X ∧ )). It then can be proved (cf. [16], [6]) that, for any cut-off function ω(r) ∈ C ∞ (R+ ), ω W s,γ (Rq × X ∧ ) = rγ ω W s,0 (Rq × X ∧ ). In our calculus we assume from now on, for convenience, that g = 0. However, note that the (analogous) results remain true for arbitrary g, especially, for the case g = s − γ. For purposes below we recall a continuity result of operators between abstract edge Sobolev spaces.  be spaces endowed with group actions {κλ }λ∈R+ and Proposition 4.6. Let E and E  for Ω = Rq be {˜ κλ }λ∈R+ , respectively. Moreover, let a(y, η) ∈ S µ (Ω × Rq ; E, E) independent of y for large |y|. Then  Op(a) : S (Rq , E) −→ S (Rq , E),

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and Op(a) extends for each s ∈ R to a continuous operator  Op(a) : W s (Rq , E) −→ W s−µ (Rq , E). If E is the direct sum E = E0 + E1 of two closed subspaces, where {κλ }λ∈R+ restricts to a group action on E0 but E1 is not necessarily invariant under the group action, we obtain by (4.6) W s (Rq , E) = W s (Rq , E0 ) + V s (Rq , E1 ), V s (Rq , E1 ) := L−1 H s (Rq , E1 ).

(4.7)

This is also a direct decomposition into closed subspaces. If E0 is orthogonal to E1 , (4.7) is also an orthogonal decomposition. In fact, if b1 is the (orthogonal) projection in E onto E1 , then −1 (κ[η] b1 κ−1 P1 := Fη→y [η] )F = Op(p1 ),

p1 (η) = κ[η] b1 κ−1 [η] ,

(4.8)

is the (orthogonal) projection onto V s (Rq , E1 ) along W s (Rq , E0 ). Example. Let 0 < σ < ∞ and S ∈ As(γ − σ, σ) a finite asymptotic type. Then E1 := ES (X ∧ ) is not invariant under κλ from (3.3). Taking either E0 = Ks,γ (X ∧ ) or E0 = KPs,γ (X ∧ ) and E = E0 + E1 , the above construction yields spaces W s,γ (Rq × X ∧ )S = W s (Rq , Ks,γ (X ∧ )) + V s (Rq , ES (X ∧ )), WPs,γ (Rq × X ∧ )S = W s (Rq , KPs,γ (X ∧ )) + V s (Rq , ES (X ∧ )).

(4.9)

Since ES (X ∧ ) ⊂ K∞,γ−σ (X ∧ ), these are both subspaces of W s,γ−σ (Rq × X ∧ ). In the situation of the previous example, we shall derive in Proposition 5.9 more precise information about the structure of the corresponding projection P1 from (4.8) and its symbol p1 (η). Remark 4.7. In our formalism we also want to admit the choice σ = 0. Then S is s,γ s,γ q ∧ q ∧ the empty set and we simply have W(P ) (R × X )S = W(P ) (R × X ). Let us stress that the difference between the regular and singular part P and S, respectively, of the asymptotic data is only determined by the position of the weight line, i.e., the choice of the reference weight γ such that Γ n+1 −γ has empty 2 intersection with both πC P and πC S. Changing γ amounts to changing the regular q ∧ and singular part, for example, WPs,γ (Rq × X ∧ )S = WPs,γ−σ ∪S (R × X ) when we replace γ by γ − σ. However, once the choice of γ is fixed, the regular and singular part are treated in our calculus in a completely different way.

5. The class of edge symbols of non-trivial type The aim of the present section is to introduce our new calculus with singular trace and Green operators, locally on a stretched wedge Ω × X ∧ . The global situation of a manifold with edges is studied in Section 6.

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Throughout the section we fix a weight γ ∈ R, real numbers µ and σ, τ ≥ 0, and endow C with the trivial group action, κλ ≡ 1 for all λ > 0, while all spaces on X ∧ carry the group action from (3.3). Also, we fix asymptotic types S = {(pj , mj , Lj )}j=0,...,N , πC S ⊂ z : n+1 2 − γ < Re z <

n+1 2

− (γ − σ) ,

T = {(pj , mj , Lj )}j=0,...,N
, πC T ⊂ z : n+1 2 − (γ − µ) < Re z <

n+1 2

(5.1)

− (γ − µ − τ ) .

We write Ks,γ (X ∧ )S := Ks,γ (X ∧ ) + ES (X ∧ ), and set, for each η ∈ Rq , p1,S (η) = κ[η] bS κ−1 [η] ,

p0,S (η) = 1 − p1,S (η),

(5.2)

with bS being the projection in Ks,γ (X ∧ )S on ES (X ∧ ) along Ks,γ (X ∧ ), cf. (4.8). Analogous notation we use for spaces and projections associated with T and γ − µ. Maybe it would be more precise to include in the notation of p0,S (η) and p1,S (η) the smoothness-parameter s ∈ R. However, we shall not do so, since we may consider bS as the restriction to Ks,γ (X ∧ )S of the map u + v → v : K−∞,γ (X ∧ ) ⊕ ES (X ∧ ) → ES (X ∧ ). Hence pj,S (η) for small s restricts to the corresponding pj,S (η) for larger s. 5.1. Trace and potential symbols We begin with the intoduction and analysis of so-called singular trace symbols. Definition 5.1. A singular trace symbol of order ν (with respect to S and the weight datum γ) is an element t1 (y, η) ∈

∩ Sclν (Ω × Rq ; Ks,γ (X ∧)S , C)

s∈R

with the property that t1 (y, η) vanishes on Ks,γ (X ∧ ) for all (y, η). Note that when t1 (y, η) vanishes on Ks,γ (X ∧ ) for all (y, η), then also the homogeneous components t1,(ν−l) (y, η) vanish on Ks,γ (X ∧ ) for all l ∈ N. In fact, we have t1,(ν) (y, η) = lim λ−ν t1 (y, λη)κλ λ→∞

which yields the result for l = 0. In a similar manner we can argue for arbitrary l ∈ N. Singular trace symbols can be characterised in a more explicit way. But before we derive this description (cf. Proposition 5.4, below), let us first illustrate how the standard trace symbols of positive type in Boutet de Monvel’s algebra can be interpreted in this context.

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Example. Consider the case X ∧ = R+ (i.e., dim X = 0) such that Ω × R+  (y, r) may be interpreted as the local model of a smooth manifold with boundary. It is known that for any γ > 12 with γ ∈ / 12 + N we have the identity S = {(j, 0) : j = 0, . . . , [γ − 12 ]};

H γ (R+ ) = Kγ,γ (R+ ) + ES (R+ ),

here we write [t] = max{m ∈ Z : m ≤ t} if t is a real number. ES (R+ ) consists of all finite Taylor polynomials [γ− 12 ]

u(r) = ω(r)



cj r j ,

cj ∈ C,

j=0

and has dimension [s − 12 ] + 1. Let us now fix 0 ≤ k ≤ [γ − 12 ] and define dk  t1 : H γ (R+ ) −→ C, t1 u = k  u. dr r=0 Obviously, t1 vanishes on Kγ,γ (R+ ). We consider t1 as a symbol independent of (y, η). It then satisfies the homogeneity relation 1

t1 u = λk+ 2 t1 κ−1 λ u k+ 12

which shows that t1 ∈ Scl

for all λ > 0,

(Ω × Rq ; Kγ,γ (R+ )S , C). Even more,

t1 ∈

∩ S k+ s∈R cl

1 2

(Rq ; Ks,γ (R+ )S , C),

(5.3)

by extending (for s < γ) t1 by vanishing on Ks,γ (R+ ). Thus we are in the above situation, here for σ = γ. Let us assume for the moment that the asymptotic type S only consists of a single triple (p, m, L) with p ∈ C satisfying Re p < n+1 2 − (γ − σ). Then  m  j ϕj (x) log r : ϕj ∈ L . ES (X ∧ ) = ω(r) r−p j=0

We shall now make use of the Mellin transform. It is defined by  ∞ dr rz v(r) (M v)(z) = r 0 for v ∈ C0∞ (R+ ) and can be extended to various other spaces (for the basic properties of the Mellin transform we refer to [3]). For example, if ω1 is an arbitrary cut-off function then  

s M ω1 : Ks,γ (X ∧ )S −→ A Re z > n+1 (5.4) 2 − γ \{p}, H (X) , where A(U, F ) denotes the holomorphic functions on U with values in F . In fact, if u ∈ Ks,γ (X ∧ )S , then v := M (ω1 u) has a pole in p of multiplicity m + 1. More precisely, if u(r, x) = ω(r) r−p ϕ(x) logk r, then the principal part of v is (−1)k k!ϕ(x)(z − p)−(k+1) . This allows us to compose (5.4) with the map  1 (z − p)k v(z) dz, 0≤k≤m (5.5) Bp,k : v → 2πi |z−p|<ε

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where ε > 0 is chosen sufficiently small. The map Bp,k computes the Laurent coefficient of v at (z − p)−(k+1) which belongs to the space L. Now the following observation is obvious: Lemma 5.2. Let S be as in (5.1). Then any function t(y, η) ∈ C ∞ (Ω × Rq , L (Ks,γ (X ∧ )S , C)) with t(y, η)|Ks,γ (X ∧ ) ≡ 0 has a unique representation as t(y, η) =

mj N  

djk (y, η) ◦ Bpj ,k ◦ M ω1

j=0 k=0

with djk (y, η) ∈ C ∞ (Ω × Rq , L∗j ), where L∗j denotes the dual space of Lj . In fact, given t(y, η), then   (−1)k djk (y, η) ϕ = ϕ ∈ Lj . t(y, η) ω(r) r−pj ϕ logk r , k! Of course, the previous lemma has a corresponding formulation if we replace the parameter-space Ω × Rq by another one. A direct consequence of Lemma 5.2 is that singular trace symbols of order −∞ associated with the asymptotic type S are precisely of the form mj N   1    djk (y, η) (z − pj )k [M (ω1 u)](z) dz (5.6) t(y, η)u = 2πi |z−pj |=ε j=0 k=0

 π S (Rq , L∗j ). Moreover, if t1 (y, η) is as in Definition 5.1 with djk (y, η) ∈ C (Ω) ⊗ and t1,(α) (y, η) is a homogeneous component, then ∞

α− n+1 2

t1,(α) (y, η)u =|η|

mj N  

(α)

η djk (y, |η| )×

j=0 k=0

  1    r (z − pj )k Mr→z (ω1 u)( |η| ) (z) dz × 2πi |z−pj |=ε α− n+1 2

=|η|

mj N  

(5.7)

(α)

η djk (y, |η| )×

j=0 k=0

  1  |η|z (z − pj )k [M (ω1 u)](z) dz × 2πi |z−p|=ε (α)

for functions djk (y, η) ∈ C ∞ (Ω × S q−1 , L∗j ), where S q−1 denotes the unit-sphere in Rq . This follows by using the homogeneity relation η t1,(α) (y, η) = |η|α t1,(α) (y, |η| ) κ−1 |η|

together with the formula from Lemma 5.2 in the version for the unit-sphere. In fact, it is also true that if we define t1,(α) (y, η) by (5.7) with arbitrary functions (α) djk (y, η) ∈ C ∞ (Ω×S q−1 , L∗j ), then we obtain that t1,(α) (y, η) is a smooth function

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on Ω×(Rq \{0}) with values in L (Ks,γ (X ∧ )S , C) that vanishes on Ks,γ (X ∧ ) and is twisted homogeneous of order α. The homogeneity is clear, the smoothness follows from the next result. Lemma 5.3. Let S be as in (5.1) and t : Ω × (Rq \ {0}) → L (Ks,γ (X ∧ )S , C) such that t(y, η)|Ks,γ (X ∧ ) ≡ 0. Then t(y, η) is a smooth function if and only if so is s(y, η) = t(y, η)κ−1 |η| . Proof. Let bS : Ks,γ (X ∧ )S → ES (X ∧ ) be the canonical projection along Ks,γ (X ∧ ). By the vanishing condition and the κλ -invariance of Ks,γ (X ∧ ) it is straightforward to verify that t(y, η) = t(y, η)κ−1 |η| bS κ|η| = s(y, η)(bS κ|η| ),

s(y, η) = t(y, η) κ−1 |η| .

So it remains to show that bS κλ ∈ L (ES (X ∧ )) depends smoothly on λ > 0, since from this it follows its smoothness as a function with values in L (Ks,γ (X ∧ )S ). However, if u(r, x) = ω(r)r−p ϕ(x) logj r, then (bS κλ u)(r, x) = ω(r)(λr)−p ϕ(x) logj (λr) = ω(r) r−p

j 

cl (λ)ϕ(x) log l r

(5.8)

l=0

 with cl (λ) = jl λ−p logj−l λ. This at once implies the requested smoothness.




There is an obvious analogous version of Lemma 5.3 by replacing Ω×(Rq \{0}) by Ω × Rq and κ|η| by κ[η] or κη . The above observations together with Remark 2.3 now lead to the announced explicit description of singular trace symbols. For this, let χ(η) be a zero excision function on Rq . Proposition 5.4. A function t1 (y, η) ∈ C ∞ (Ω×Rq , L (Ks,γ (X ∧ )S , C)) is a singular trace symbol of order ν in the sense of Definition 5.1, if and only if there exists a sequence of homogeneous components t1,(ν−l) , l ∈ N, of the form (5.7) (with α replaced by ν − l) such that for each given M ≥ 0 there exists an L ∈ N such that L  the difference t1 (y, η) − χ(η)t1,(ν−l) (y, η) is of the form (5.6) with all djk (y, η) l=0

belonging to S −M (Ω × Rq ). Before defining our new class of trace symbols let us recall the structure of a trace symbol from the standard edge algebra: It is a symbol t0 (y, η) ∈

∩

s∈R

ν Scl (Ω × Rq ; Ks,γ (X ∧ ), C),

such that the (pointwise) formal adjoint symbol satisfies, for some ε > 0, ν t0 (y, η)∗ ∈ Scl (Ω × Rq ; C, Sε−γ (X ∧ )).

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Definition 5.5. A trace symbol of order ν in the local edge calculus on X ∧ × Ω (with respect to S and the weight-datum γ) is an operator family t(y, η) such that t(y, η) ∈

∩

s∈R

ν Scl (Ω × Rq ; Ks,γ (X ∧ )S , C),

and t(y, η) has a representation t(y, η) = t0 (y, η) p0,S (η) + t1 (y, η) with a ν-th order singular trace symbol t1 (y, η) and a t0 (y, η) being a trace symbol in the standard edge symbol algebra as described above. If t1 (y, η) ≡ 0, we call t(y, η) also a regular trace symbol. Theorem 5.6. Let t(y, η) be a trace symbol as in Definition 5.5 for Ω = Rq , and let t(y, η) be independent of y for large |y|. Then Op(t) induces continuous operators Op(t) : W s (Rq , Ks,γ (X ∧ )S ) → H s−ν (Rq ),

s ∈ R,

(5.9)

and we have

Op(t1 )u = 0 for all u ∈ W s (Rq , Ks,γ (X ∧ )). (5.10) Moreover, the singular trace operators are localised at the edges (in contrast to the regular trace operators), i.e., if v ∈ W s (Rq , Ks,γ (X ∧ )S ) vanishes in a neighbourhood of r = 0 then Op(t1 )v = 0.

Proof. Property (5.9) follows from Proposition 5.9 (see below) and Proposition 4.6, noting that W s (Rq , C) = H s (Rq ). Relation (5.10) holds true, since t1 (y, η) u(η) ≡ 0 for any u ∈ S (Rq , Ks,γ (X ∧ )). For the latter statement note that the vanishing of
v near r = 0 implies that, in fact, v ∈ W s (Ω, Ks,γ (X ∧ )). Then apply (5.10). We finish this section by defining potential symbols. For their definition it will be convenient to use the notation Sεγ−µ (X ∧ )T = Sεγ−µ (X ∧ ) ⊕ ET (X ∧ ),

ε > 0.

(5.11)

Definition 5.7. A potential symbol of order ν (with respect to T and the weightdatum γ − µ) is a symbol ν (Ω × Rq ; C, Sεγ−µ (X ∧ )T ) k(y, η) ∈ Scl

(5.12)

for some ε = ε(k) > 0. If we write k(y, η) = p0,T (η)k(y, η) + p1,T (η)k(y, η) then the first summand is a potential symbol from the usual edge-algebra associated with the weight-datum γ − µ. Also potential symbols can be characterized in a more explicit way. In fact, symbols (5.12) are precisely those of the form C  c → [η]

n+1 2

 k(y, η; r[η], x) c ∈ Sεγ−µ (X ∧ )T

with a symbol kernel ν ∧   π Sεγ−µ (X(r,x) k(y, η; r, x) ∈ Scl (Ω × Rq ) ⊗ )T .

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5.2. Parameter-dependent families of projections We now investigate more deeply the pseudodifferential structure of the family of projections p1,S (η), cf. (5.2). Let us define, for 0 ≤ j ≤ N and 0 ≤ k ≤ mj , zero order symbols by  k n+1 (−1) 1 tjk (η)u = [η]− 2 (z − pj )k M (ω1 u(r/[η]))(z) dz. k! 2πi |z−pj |=ε Using the unitary isomorphism ES (X ∧ ) ∼ = Cι(S) , see (4.3), this gives rise to a map t(η) = tS (η) := (tjk (η)) 0≤j≤N, : Ks,γ (X ∧ )S −→ Cι(S) . 0≤k≤mj

0 q s,γ We have t(λη) = t(η)κ−1 (X ∧ )S , Cι(S) ) λ for λ ≥ 1, |η| ≥ 1, hence t(η) ∈ Scl (R ; K (in fact, each component of t(η) is a singular trace symbol). Moreover, let us define a (system of) potential symbols

k(η) = kS (η) : Cι(S) −→ Ks,γ (X ∧ )S by setting, for each c = (cjk ) 0≤j≤N, ∈ Cι(S) , 0≤k≤mj

k(η)c = [η]

n+1 2

ω(r[η])

mj N  

cjk (r[η])−pj logk (r[η])

j=0 k=0

(on the right-hand side we have identified cjk with an element in Lj , using the identification (4.3)). We then have k(λη) = κλ k(η) for λ ≥ 1, |η| ≥ 1 which entails 0 k(η) ∈ Scl (Rq ; Cι(S) , Ks,γ (X ∧ )S ). By construction, t(η)k(η) = idCι(S) for all η ∈ Rq , hence k(η)t(η) : Ks,γ (X ∧ )S −→ Ks,γ (X ∧ )S

(5.13)

is a family of continuous projections. Lemma 5.8. If k(η) and t(η) are constructed as above, then p0,S (η) = 1 − k(η) t(η)

for all η ∈ Rq .

(5.14)

Proof. By conjugation with κ[η] the claim is equivalent to showing that bS = (κ−1 [η] k(η))(t(η)κ[η] ) for each η. However, this is true, since the first factor on the right-hand side is the map mj N   c = (cjk ) → ω(r) cjk r−pj logk r, j=0 k=0

while the second factor equals (tjk ) with  (−1)k 1 (z − pj )k M (ω1 u)(z) dz. tjk u = k! 2πi |z−pj |=ε




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By straightforward calculation, for any u ∈ Ks,γ (X ∧ ), mj N     p0,S (η) u + ω(r) cjk r−pj logk r j=0 k=0 mj N    cjk r−pj logk r. = u + ω(r) − ω(r[η])

(5.15)

j=0 k=0

From this formula and by Lemma 5.8 we deduce the following: 0 Proposition 5.9. Let s ∈ R. Then p0,S (η) ∈ Scl (Rq ; Ks,γ (X ∧ )S , Ks,γ (X ∧ )) and

Op(p0,S ) : W s (Rq , Ks,γ (X ∧ )S ) −→ W s (Rq , Ks,γ (X ∧ )S ) is the projection onto W s (Rq , Ks,γ (X ∧ )) along V s (Rq , ES (X ∧ )). Moreover, −|α|

Dηα p0,S (η) ∈ Scl (cf. (3.4)), and

(Rq ; Ks,γ (X ∧ )S , S ∞ (X ∧ ))

Dηα p0,S (η)

vanishes on K

s,γ

for all |α| ≥ 1

(5.16)

∧

(X ) for each η.

In the terminology introduced in the next section, (5.16) means that Dηα p0,S (η) is a singular Green symbol of order −|α| whenever |α| ≥ 1. 5.3. Regular and singular Green symbols Besides trace and potential symbols there is another category of so-called Green symbols needed for our new calculus. They are build up from regular and singular Green symbols. Definition 5.10. A singular Green symbol of order ν (with respect to S, T , and the weight-datum (γ, γ − µ, θ)) is a symbol g1 (y, η) ∈

∩

s∈R

ν Scl (Ω × Rq ; Ks,γ (X ∧ )S , Sεγ−µ (X ∧ )T )

(see (5.11) for the notation) that vanishes on Ks,γ (X ∧ ) for all (y, η), and where ε = ε(g1 ) > 0. Along the lines of the material from Section 5.1, singular Green symbols have a more explicit representation: If g1 (y, η) is as in Definition 5.10, all homogeneous components g1,(ν−j) , j ∈ N, pointwise vanish on Ks,γ (X ∧ ). If S is as in (5.1) then g1,(ν−j) (y, η) = |η|

ν−j

mj N  

η κ|η| ◦ ejk (y, |η| ) ◦ Bpj ,k ◦ Mγ−σ− n2 ◦ κ−1 |η| ◦ ω1

j=0 k=0

: Ks,γ (X ∧ )S → Sεγ−µ (X ∧ )T , where Bp,k is as in the formula (5.5) and with uniquely determined functions ejk (y, η) ∈ C ∞ (Ω × S q−1 , L (Lj , Sεγ−µ (X ∧ )T )). Identifying ejk (y, η) with an

L∗j -valued

function

∗  π Sεγ−µ (X ∧ )T ⊗  π L∗j , (y, η; r, x) ∈ C ∞ (Ω × S q−1 ) ⊗ ljk

(5.17)

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we obtain [g1,(ν−j) (y, η)u](r, x) =|η|

n+1 2 +ν−j

mj N  

η ∗ ljk (y, |η| ; r|η|, x)×

j=0 k=0

    1  (z − pj )k Mr
→z (κ−1 ω1 u) (z)dz . × |η| 2πi |z−pj |=ε (5.18) In other words, each homogeneous component of a singular Green symbol has a unique representation as a finite linear combination of summands that are the (pointwise) composition of homogeneous components of a singular trace symbol and a potential symbol. Definition 5.11. A Green symbol of order ν (with respect to S, T , and the weightdatum (γ, γ − µ, θ)) in the local edge calculus on X ∧ × Ω is any operator function

∩ Sclν (Ω × Rq ; Ks,γ (X ∧ )S , K∞,γ−µ (X ∧ )T )

g(y, η) ∈

s∈R

which has a representation g(y, η) = g0 (y, η) p0,S (η) + g1 (y, η), where g1 (y, η) is a singular Green symbol and, for some ε = ε(g0 ) > 0, g0 (y, η) ∈ g0 (y, η)∗ ∈

∩ Sclν (Ω × Rq ; Ks,γ (X ∧ ), Sεγ−µ (X ∧ )T ),

s∈R

∩ Sclν (Ω × Rq ; Ks,−γ+µ+τ (X ∧ ), Sε−γ (X ∧ )),

s∈R

where ∗ refers to the pointwise adjoint with respect to the K0,0 (X ∧ ) scalar product. If g1 (y, η) ≡ 0, we call g(y, η) a regular Green symbol. Note that if g0 (y, η) is as in the previous definition, then p0,T (η) g0 (y, η) ∈ RνG (Ω × Rq ; (γ, γ − µ)) is a Green symbol from the standard edge algebra, cf. Section 8. Example. The pointwise composition g(y, η) := k(y, η) t(y, η) of a ν0 -th order trace symbol and a ν1 -th order potential symbol in the sense of Definitions 5.5 and 5.7, respectively, is a Green symbol of order ν = ν0 + ν1 in the sense of the Definition 5.11. We shall also consider vector-valued variants of trace, potential, and Green symbols, i.e. t(y, η) : Ks,γ (X ∧ , Ck )S −→ Cj+ , k(y, η) : Cj− −→ Ks−µ,γ−µ (X ∧ , Cl )T , g(y, η) : Ks,γ (X ∧ , Ck )S −→ Ks−µ,γ−µ (X ∧ , Cl )T

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k

where Ks,γ (X ∧ , Ck ) = ⊕ Ks,γ (X ∧ ), and analogously for the other spaces. This is j=1

achieved by considering matrices of corresponding size, whose entries are all trace, potential, or Green symbols of same order and with respect to the same data. Definition 5.12. Let 1 ≤ k, l ∈ N and j+ , j− ∈ N (possibly also 0). Then RνG (Ω × Rq ; (γ, γ − µ), (k, l; j− , j+ ))S,T denotes the space of all (2 × 2)-block matrix symbols
g (y, η) g(y, η) = 11 g21 (y, η)



Ks−ν,γ−µ (X ∧ , Cl )T Ks,γ (X ∧ , Ck )S ⊕ −→ ⊕ , : Cj− Cj+ (5.19) where g11 (y, η) is an (l × k)-matrix of ν-th order Green symbols, g21 (y, η) is an (j+ × k)-matrix of ν-th order trace symbols, g12 (y, η) is an (l × j− )-matrix of ν-th order potential symbols (all with respect to S, T and the weight-datum (γ, γ − µ), ν (Ω × Rq ). and g22 (y, η) is an (j+ × j− )-matrix of scalar symbols from Scl g12 (y, η) g22 (y, η)

To have a convenient terminology at hand, we shall call symbols (5.19) again Green symbols. Note that they are particular operator-valued symbols ν g(y, η) ∈ Scl (Ω × Rq ; Ks,γ (X ∧ , Ck )S ⊕ Cj− , S γ−µ (X ∧ , Cl )T ⊕ Cj+ ).

(5.20)

5.4. The full symbol class and its calculus For the following recall that Ks−µ,γ−µ (X ∧ ) is a subspace of Ks−µ,γ−µ (X ∧ )T . Definition 5.13. Let ν ∈ R with µ − ν ∈ N. The space Rν (Ω × Rq ; (γ, γ − µ), (k, l; j− , j+ ))S,T consists of all symbols a(y, η) of the form
a0 (y, η) p0,S (η) a(y, η) = 0

 Ks−ν,γ−µ (X ∧ , Cl )T Ks,γ (X ∧ , Ck )S 0 ⊕ ⊕ −→ + g(y, η) : 0 Cj− Cj+ (5.21) with an arbitrary standard edge symbol a0 (y, η) ∈ Rν (Ω × Rq ; (γ, γ − µ), (k, l)) (cf. Section 8), and some Green symbol g(y, η) ∈ RνG (Ω × Rq ; (γ, γ − µ), (k, l, j− , j+ ))S,T . Observe that then, for every s ∈ R, Rν (Ω × Rq ;(γ, γ − µ), (k, l; j− , j+ ))S,T ⊂ ⊂ S ν (Ω × Rq ; Ks,γ (X ∧ , Ck )S ⊕ Cj− , Ks−ν,γ−µ (X ∧ , Cl )T ⊕ Cj+ ).

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Thus if Ω = Rq and a(y, η) as in (5.21) is independent of y for large |y|, then it induces, for each s ∈ R, continuous operators W s−ν (Rq , Ks−ν,γ−µ (X ∧ , Cl )T ) W s (Rq , Ks,γ (X ∧ , Ck )S ) −→ . ⊕ ⊕ Op(a) : H s (Rq , Cj− ) H s−ν (Rq , Cj+ ) Definition 5.14. For a(y, η) ∈ Rµ (Ω × Rq ; (γ, γ − µ), (k, l; j− , j+ ))S,T we set
 σ∧ (a0 )(y, η) σ∧ (p0,S )(η) 0 σ∧ (a)(y, η) = + σ∧ (g)(y, η), 0 0 where σ∧ (a0 ) is the standard principal edge symbol (cf. Section 8) and σ∧ (p0,S )(η) := 1 − κ|η| bS κ−1 |η| . Moreover, σ∧ (g)(y, η) denotes the homogeneous principal symbol of g(y, η) as a classical symbol of order µ, cf. (5.20). We call σ∧ (a)(y, η) the principal edge symbol of a(y, η). From this definition if follows that σ∧ (a)(y, λη) = λµ diag(κλ , idCj+ ) σ∧ (a)(y, η) diag(κ−1 λ , idCj− ) as a family of operators Ks,γ (X ∧ , Ck )S ⊕ Cj+ → Ks−µ,γ−µ (X ∧ , Cl )T ⊕ Cj− with n+1 the usual group action defined by (κλ u)(r, x) = λ 2 u(λr, x) for λ ∈ R+ . Definition 5.15. Let a(y, η) ∈ Rµ (Ω×Rq ; (γ, γ −µ), (k, l; j− , j+ ))S,T be as in (5.21) (with ν = µ). Then we define the conormal symbol of a as σM (a)(y, z) = σM (a0 )(y, z), where σM (a0 ) denotes the standard conormal symbol, cf. Section 8. Proposition 5.16. If a(y, η) ∈ Rν (Ω × Rq ; (γ, γ − µ), (k, l; j− , j+ ))S,T then ∂ηα ∂yβ a(y, η) ∈ Rν−|α| (Ω × Rq ; (γ, γ − µ), (k, l; j− , j+ ))S,T for any multi-indices α, β ∈ Nq . Theorem 5.17. For j ∈ N let aj (y, η) ∈ Rν−j (Ω × Rq ; (γ, γ − µ), (k, l; j− , j+ ))S,T such that the ε-weights involved in the Green symbol parts of each aj are independent of j. Then there exists a symbol a(y, η) ∈ Rν (Ω×Rq ; (γ, γ−µ), (k, l; j−, j+ ))S,T such that, for all N ∈ N, a(y, η) −

N −1 

aj (y, η) ∈ Rν−N (Ω × Rq ; (γ, γ − µ), (k, l; j− , j+ ))S,T .

j=0

Proposition 5.18. The pointwise composition of l; j− ,  j+ ))S,R a(y, η) ∈ Rν (Ω × Rq ; (γ, γ − µ), (k,  and

 (y, η) ∈ Rν (Ω × Rq ; (γ − µ, γ − µ − µ a ), ( l, l;  j+ , j+ ))R,T

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yields a symbol  (y, η)a(y, η) ∈ Rν+ν (Ω × Rq ; (γ, γ − µ − µ a ), (k, l; j− , j+ ))S,T . Theorem 5.19. Let the notation be as in the previous Proposition 5.18 with Ω = Rq  (y, η) and a(y, η) be independent of y for large |y|. Then there exists a and both a symbol ), (k, l; j− , j+ ))S,T , ( a#a)(y, η) ∈ Rν+ν (Rq × Rq ; (γ, γ − µ − µ  and a, such that Op( the so-called Leibniz-product of a a)Op(a) = Op( a#a). If ν = µ and ν = µ  then a#a)(y, η) = σ∧ ( a)(y, η)σ∧ (a)(y, η), σ∧ ( σM ( a#a)(y, z) = (T µ σM ( a))(y, z) σM (a)(y, z), where T σ acts on functions by shifting the argument, (T σ f )(z) = f (z + σ). The proofs of Propositions 5.16, 5.18 and Theorems 5.17, 5.19 are somewhat lengthy and laborious, nevertheless elementary in the sense that they only rely on the correponding properties of standard edge symbols and general operator-valued symbols (and (5.16) for Proposition 5.16). Therefore we omit these proofs.

6. Operators of non-trivial type on manifolds with edges Let us begin this section by introducing a useful notation that we shall use frequently throughout the sequel: If u, v are real-valued, continuous functions on a topological space, we shall write u ≺ v if v ≡ 1 on an open neighborhood containing the support of u. 6.1. Manifolds with edges and Sobolev spaces The analysis of our classes of edge-pseudodifferential operators takes place on (the interior of) a smooth manifold with boundary M, that near the boundary has the structure of a fibre bundle over a smooth closed base space and where the fibre is a cone over a smooth closed manifold (both base space and cross section of the cone without boundary). For simplicity, we shall assume in this paper that this bundle is trivial. In case of trivial cross section (i.e., a point) we obtain a usual manifold with smooth boundary. More precisely, there exists a homeomorphism of a neighborhood V of ∂M to Y × [0, 1) × X, where X and Y are smooth closed manifolds, that restricts to a diffeomorphism between V \ ∂M and Y × (0, 1) × X. This gives rise to a splitting of coordinates that we shall keep fixed from now on. By identifying points (y, 0, x) and (y, 0, x ) for any x, x ∈ X, we may obtain from M a topological space M which is regarded as a manifold with edge Y and model-cone X ∧ . In order to emphasise this geometric structure, we shall often write M instead of M. We shall now define weighted edge Sobolev spaces with and without asymptotics. To this end we assume that the asymptotic type S is as in (5.1) but,

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additionally satisfies the so-called shadow condition: If (p, m, L) ∈ S and Re p−1 ≥ n+1  ∈ S with p = p − 1, m ≤ m  This p, m,  L)  and L ⊂ L. 2 − γ then there exists ( property ensures, in particular, that the operator of multiplication by any function ϕ ∈ S (R+ ) maps Ks,γ (X ∧ )S into itself. Let us fix a covering of Y by coordinate systems χj : U j ⊂ Y → Ω j ⊂ R q ,

j = 1, . . . , N,

C0∞ (Uj )

such that ϕ1 , . . . , ϕN form a partition and a system of functions ϕj , ψj ∈ of unity, and ϕj ≺ ψj for each 1 ≤ j ≤ N . Moreover, ω ∈ C0∞ ([0, 1[) is a cutoff function, and we consider (1 − ω) as a function on M that vanishes near the boundary as well as a function on the double 2M (by extension by 0). Definition 6.1. The space W s,γ;g (M ) for g, s, γ ∈ R consists of all distributions u on M \ ∂M such that (1 − ω)u ∈ H s (2M) and χj∗ (ϕj ωu) ∈ W s (Rq , Ks,γ;g (X ∧ ))

for all 1 ≤ j ≤ N

(6.1)

(cf. Remark 4.5), where χj∗ denotes the push-forward of distributions from Uj × (0, 1) × X to Ωj × (0, 1) × X under the map (y, r, x) → (χj (y), r, x). In an analogous manner we define W s,γ;g (M )S and subspaces Wεs,γ;g (M )(S) for some ε > 0. Up to equivalence of norms, the previous definitions are independent of the involved data (cut-off function, partition of unity, etc.), see Theorems 4 and 19 in Section 3.2.5 of [11]. Clearly Definition 6.1 has a straightforward extension to sections of vector-bundles E over M , yielding a scale W s,γ;g (M, E). For simplicity of the presentation, however, we shall restrict ourselves to trivial bundles, i.e., k

W s,γ;g (M, Ck )(S) = ⊕ W s,γ;g (M )(S) j=1

and analogously subspaces with asymptotics Q ∈ As(γ, θ). In the following, we again restrict ourselves to the case g = 0, having in mind that the (analogous) results remain valid also for arbitrary g, in particular, for g = s − γ. 6.2. Global projections in edge Sobolev spaces with asymptotics In the construction of our new symbol algebra we made often use of the canonical splitting W s (Rq , Ks,γ (X ∧ , Ck )S ) = W s (Rq , Ks,γ (X ∧ , Ck )) ⊕ V s (Rq , ES (X ∧ , Ck )) and the corresponding projection Op(p0,S ). Passing to the manifold with edges W , we canonically can speak of W s,γ (M, Ck )S and its subspace W s,γ (M, Ck ), but there is not such a canonical choice of a complementing space. To overcome this problem, we shall prove in this subsection the following Theorem 6.2. Theorem 6.2. There exists an operator P0,S = P0,S (k) having the following properties, simultaneously for all s ∈ R:

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a) P0,S ∈ L (W s,γ (M, Ck )S ) is a projection onto W s,γ (M, Ck ). Moreover, P0,S is a pseudodifferential operator in the following sense: b) If ϕ, ψ ∈ C ∞ (M) are located in a chart U × [0, 1) × X with coordinate map χ : U → Ω ⊂ Rq , then ϕ P0,S ψ = ϕ χ∗ Op(p0,S + r) ψ, where p0,S (η) is as in (5.2) (with [η] now referring to the local expression for a smoothed norm function on the cotangent bundle of the edge Y ), −1 r(y, η) ∈ Scl (Ω × Rq ; Ks,γ (X ∧ , Ck )S , S ∞ (X ∧ , Ck )),

and r(y, η) ≡ 0 on Ks,γ (X ∧ , Ck ). c) If ω1 , ω2 ∈ C ∞ ([0, 1[) are cut-off functions with ω2 ≺ ω1 then (1 − ω1 ) P0,S ω2 , ω2 P0,S (1 − ω1 ) : W s,γ (M, Ck )S −→ W ∞,∞ (M, Ck ), both vanishing on W s,γ (M, Ck ). d) If ω1 , ω2 ∈ C ∞ ([0, 1[) are cut-off functions and ϕ1 , ϕ2 ∈ C ∞ (Y ) have disjoint support, then (ω1 ϕ1 ) P0,S (ω2 ϕ2 ) : W s,γ (M, Ck )S −→ W ∞,∞ (M, Ck ), vanishing on W s,γ (M, Ck ). In b), χ∗ denotes the operator pull-back under the map (y, r, x) → (χ(y), r, x). By virtue of property a), we have that P0,S ϕ = ϕ whenever ϕ ∈ C ∞ (M) vanishes to infinite order at the boundary (respectively to sufficiently high order, depending on the asymptotic type S). Property b) of the previous theorem says that locally, near the edge, P0,S coincides with Op(p0,S ), cf. Proposition 5.9, modulo a singular Green operator of order −1, that additionally generates infinite flatness. In particular, the local edge symbols σ∧ (p0,S )(y, η) globally lead to a vector-bundle homomorphism σ∧ (P0,S ) : (T ∗ Y \ 0) × Ks,γ (X ∧ , Ck )S −→ (T ∗ Y \ 0) × Ks,γ (X ∧ , Ck )S .

(6.2)

Note that different choices of projections, say P0,S and P0,S , having the properties stated in the theorem give rise to the same principal edge symbol (6.2). Moreover, P0,S − P0,S : W s,γ (M, Ck )S −→ W s−1,∞ (M, Ck ), vanishing on W s,γ (M, Ck ). Proof of Theorem 6.2. Without loss of generality we assume k = 1. Recall that we have identified a collar neighbourhood of ∂M in M with Y × N

[0, 1[×X. Let us fix a covering Y = ∪ Ui with coordinate charts χi : Ui → Ωi ⊂ Rq i=1

and a subordinate partition of unity {ϕi ∈ C0∞ (Ui ) : i = 1, . . . , N }. Moreover choose ψi ∈ C0∞ (Ui ) with ϕi ≺ ψi , and let ω, ω  ∈ C0∞ ([0, 1[) be cut-off functions

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(viewed as functions on M, supported near the boundary) with ω ≺ ω  . Then we set N  P0,S = (ωϕi ) χi∗ Op(p0,i ) (ψi ω  ) + (1−ω), p0,i (y, η) = 1−κ[η] bS κ−1 [η] , (6.3) i=1

with [η] denoting the expression in local coordinates y ∈ Ωi of a smoothed norm function on the cotangent-bundle over Y , and bS being the projection in Ks,γ (X ∧ )S onto ES (X ∧ ) along Ks,γ (X ∧ ), cf. (5.2). By construction, it is clear that P0,S = 1 on W s,γ (M ) and that P0,S from s,γ W (M )S to W s,γ (M ) is onto. Since W s,γ (M ) is a closed subspace of W s,γ (M )S , we obtain a). For b) we first have a look how an operator of the form Op(p0,i ) behaves under coordinate changes (in the edge variable). By the standard formula of expressing a pseudodifferential operator in new coordinates,  1   (Dηα p)(y, t χ (y)ξ) Dzα eiκ(y,z)ξ z=y (χ∗ p)(x, ξ)x=χ(y) ∼ α! α with κ(y, z) = χ(z)−χ(y)−χ (y)(z −y), applied to p0,i and using (5.16), we obtain that Op(p0,i ) on (Ωi ∩ Ω) × X ∧ under the coordinate change χi ◦ χ transforms to Op(p0,S + r) with p0,S and r as described in the statement. Thus, choosing a cut-off function ω0 with ω0 ≺ ω and using P0,S (1 − ω0 ) = (1 − ω0 ), we find ϕ P0,S ψ = ϕ P0,S ω0 ψ + ϕ (1 − ω0 ) ψ = ϕ χ∗ Op(p0,S + r) ω0 ψ + ϕ (1 − ω0 ) ψ. By the properties of p0,S and r we find ϕ χ∗ Op(p0,S +r) (ω0 −1) ψ = −ϕ (1−ω0 ) ψ and this obviously yields ϕ P0,S ψ = ϕ χ∗ Op(p0,S + r) ψ. c), d) First of all, observe that all operators in question vanish on W s,γ (M ), since on that space P0,S coincides with the identity operator. For the same reason, we even have ω2 P0,S (1 − ω1 ) = 0 on W s,γ (M )S . By definition of P0,S in (6.3), the statement will follow if we verify corresponding properties for the local operators R1 := (1 − ω1 ) Op(p0,i ) ω2 and R2 := ϕ 1 Op(p0,i ) ϕ 2 , where ϕ 1 , ϕ 2 ∈ C ∞ (Ωi ) have disjoint support. To show this let, without loss of generality, Ωi = Rq and [·] be independent of y for large |y|. By using the explicit representation (5.15), and writing (1 − ω1 ) = tN (t−N (1 − ω1 )) for arbitrary N ∈ N, it is straightforward to see that R1 = Op(r1 ) with a symbol r1 ∈ S −N (Rq × Rq ; Ks,γ (X ∧ )S , S ∞ (X ∧ )). This yields R1 : W s,γ (Rq × X ∧ )S → W ∞,∞ (Rq × X ∧ ). The same is true for R2 , using integration by parts and (5.16).




Remark 6.3. Adapting methods and results of [2] that show that, for a smooth com1 pact manifold Ω with boundary, H s (Ω)/H0s (Ω) is isomorphic to H s (∂Ω, C[s− 2 ] ), where H0s (Ω) denotes the closure of C0∞ (int Ω) in H s (Ω), one can show that  ∼ H s (Y, Ck ⊗ Cι(S) ), ∼ W s,γ (M, Ck )S W s,γ (M, Ck ) = ker P0,S = where ι(S) is the dimension of ES (X ∧ ), cf. (4.3).

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6.3. Global regular and singular Green operators The full algebra of operators on a manifold with edges will consist, roughly speaking, of pseudodifferential operators that locally near the edge can be represented as an operator with symbol from the edge symbol algebra introduced in Definition 5.13. There are also non-local operators in this algebra which we shall introduce first. Definition 6.4. A smoothing operator (associated with asymptotic types S, T , vector bundles Ck , Cl , J− , J+ , and with weight-data (γ, γ − µ)) is an operator C such that, for all s ∈ R, C = C0


P0,S 0


0 C11 + 1 C21

 W s,γ (M, Ck )S W ∞,γ−µ (M, Cl )T 0 ⊕ ⊕ −→ : 0 H s (Y, J− ) H ∞ (Y, J+ )

(6.4)

where a) P0,S is a projection as in Theorem 6.2, b) both C11 and C21 vanish on W s,γ (M, Ck ), and there exists an ε = ε(C11 ) > 0 such that C11 : W s,γ (M, Ck ) → Wε∞,γ−µ (M, Cl )T , c) for any r ∈ R W r,γ (M, Ck ) Wε∞,γ−µ (M, Cl )T ⊕ C0 : ⊕ −→ H r (Y, J− ) H ∞ (Y, J+ ) for some ε = ε(C0 ) > 0, while its adjoint C0∗ induces maps C0∗

Wε∞,−γ (M, Ck ) W r,−γ+µ+τ (M, Cl ) ⊕ ⊕ −→ . : H r (Y, J+ ) H ∞ (Y, J− )

The space of all such operators we shall denote by Y −∞ (M ; (γ, γ − µ), (k, l; J− , J+ ))S,T . If in (6.4) we can choose C11 = 0 and C21 = 0, we call
C0 a regular  smoothing 0 C11 C12 operator. If C0 is represented as a block matrix, C0 = , and we can 0 0 C21 C22 0 0 = 0 and C21 = 0, we call C a singular smoothing operator. choose in (6.4) both C11 Let us remark that – after fixing the choice of P0,S – the decomposition (6.4) uniquely determines C0 , C11 , and C21 . N

For the definition of global Green operators let Y = ∪ Ui with coordinate i=1

charts χi : Ui → Ωi ⊂ Rq over which J± is trivial with corresponding trivializations χi,± : J± Ui −→ Ωi × Cj± . Let ϕ1 , . . . , ϕN be a corresponding partition of unity on Y and ψi ∈ C0∞ (Ui ) such that ϕi ≺ ψi . Moreover, let ω, ω  ∈ C0∞ ([0, 1)) be cut-off functions with ω ≺ ω .

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Definition 6.5. A Green operator of order ν ∈ R (associated with asymptotic types S, T , vector bundles Ck , Cl , J− , J+ , and with weight-data (γ, γ − µ)) is any operator G of the form
  N
 ωϕi ◦ χ∗i 0 (χi )∗ ◦ ωψi 0 G= Op(gi ) + C (6.5) 0 ϕi ◦ χ∗i,+ 0 (χi,− )∗ ◦ ψi i=1

(χ∗ and χ∗ denote the pull-back and push-forward, respectively, of sections under χ), where C is a smoothing operator in the sense of Definition 6.4 and each gi is a ν-th order Green symbol as in Definition 5.12, gi ∈ RνG (Ωi × Rq ; (γ, γ − µ), (k, l; j− , j+ ))S,T . The space of all such operators is denoted by ν (M ; (γ, γ − µ), (k, l; J− , J+ ))S,T YG

6.4. The algebra of edge operators on W Definition 6.6. Let µ, ν ∈ R with µ − ν ∈ N. The space Y ν (M ; (γ, γ − µ), (k, l; J− , J+ ))S,T consists of all operators
 W s−µ,γ−µ (M, Cl )T W s,γ (M, Ck )S A P0,S 0 −→ A= ⊕ ⊕ + G: 0 0 H s (Y, J− ) H s−µ (Y, J+ )

(6.6)

ν where P0,S is as in Theorem 6.2, G ∈ YG (M ; (γ, γ − µ), (k, l; J− , J+ ))S,T , and A is an operator from the standard edge algebra Y ν (M ; (γ, γ − µ), (k, l)), cf. Section 8.

Remark 6.7. Throughout the paper we admit also the case σ = τ = 0, i.e., that the asymptotic types S and T disappear, cf. Remark 4.7. In that case we recover the standard edge algebra Y ν (M ; (γ, γ − µ), (k, l; J− , J+ )). Similarly as in (6.5), each operator A ∈ Y ν (M ; (γ, γ − µ), (k, l; J− , J+ ))S,T can be expressed locally near the edge in terms of edge symbols belonging to Rν (Ωi × Rq ; (γ, γ − µ), (k, l; j− , j+ ))S,T . Definition 6.8. Let A ∈ Y µ (M ; (γ, γ − µ), (k, l; J− , J+ ))S,T be as in (6.6) with ν = µ. Let us denote by πY : T ∗ Y \ 0 → Y the canonical projection. Then the principal edge symbol of A is the vector-bundle homomorphism
 σ∧ (A) σ∧ (P0,S ) 0 σ∧ (A) = + σ∧ (G) : 0 0  s,γ ∧ k   s−µ,γ−µ ∧ l  (6.7) (X , C )T K (X , C )S K ∗  ∗    ⊕ ⊕ πY −→ πY , s ∈ R, J− J+ where σ∧ (A) is the principal edge symbol of standard edge operators, cf. Section 8, and σ∧ (P0,S ) is as in (6.2). Moreover, the conormal symbol of A is, by definition, σM (A)(y, z) = σM (A)(y, z) : H s (X) −→ H s−µ (X), where y ∈ Y and z ∈ C with Re z =

n+1 2

− γ, cf. Section 8.

s ∈ R,

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Occasionally, we prefer to write the action of the principal edge symbol in each fibre, i.e., Ks,γ (X ∧ , Ck )S Ks−µ,γ−µ (X ∧ , Cl )T ⊕ −→ ⊕ , σ∧ (A)(y, η) : J−,y J+,y

(y, η) ∈ T ∗ Y \0. (6.8)

Besides principal edge and conormal symbol we have the standard interior pseudodifferential symbol   σψ (A) := σψ (A) ∈ C ∞ T ∗ (M \ Y ) \ 0 , cf. (8.8), and the rescaled symbol

  σ ψ (A) := σ ψ (A) ∈ C ∞ (T ∗ ∂M × R) \ 0 ,

cf. (8.9). Based on the algebra property of the standard edge algebra and the composition result Theorem 5.19, one can show the following: l; J− , J+ ))S,R and A ∈ Y ν (M ; (γ − Theorem 6.9. Let A ∈ Y ν (M ; (γ, γ − µ), (k,  µ, γ − µ − µ ), ( l, l; J+ , J+ ))R,T . Then  ∈ Y ν+ν (M ; (γ, γ − µ − µ AA ), (k, l; J− , J+ ))S,T . Moreover, in case ν = µ and ν = µ ,   σ∧ (AA)(y, η) = σ∧ (A)(y, η)σ∧ (A)(y, η),   z) = (T µ σM (A))(y, z)σM (A)(y, z), σM (AA)(y, where T σ is the shift operator, (T σ f )(z) = f (z + σ). Similarly, using Theorem 5.17, we have: ), (k, l; J− , J+ ))S,T such Theorem 6.10. For j ∈ N let Aj ∈ Y µ−j (M ; (γ, γ − µ − µ that the ε-weights involved in the Green parts of each Aj do not dependent on j. Then there exists an operator A ∈ Y µ (M ; (γ, γ − µ − µ ), (k, l; J− , J+ ))S,T such that, for all N ∈ N, A−

N −1 

Aj ∈ Y µ−N (M ; (γ, γ − µ − µ ), (k, l; J− , J+ ))S,T .

j=0

7. Ellipticity and parametrix In this section we introduce the notion of ellipticity in the edge calculus with singular trace and Green operators and show that elliptic operators possess a parametrix within the calculus. As a preliminary step, we consider in Section 7.1 the subalgebra of operators which coincide with the identity modulo Green operators. The general case is then treated in the subsequent Section 7.2.

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Definition 7.1. An operator A ∈ Y µ (M, (γ, γ−µ), (k, k; J− , J+ ))S,T is called elliptic if ψ (A) are pointwise invertible on T ∗ (M \Y )\0 and (T ∗ ∂M×R)\0, (i) σψ (A) and σ respectively, (ii) σ∧ (A)(y, η) defines isomorphisms (6.8) for all (y, η) ∈ T ∗ Y \0 and some s ∈ R.
 A11 A12 Note that if A = is elliptic, then A21 A22 σ∧ (A11 )(y, η) : Ks,γ (X ∧ , Ck )S −→ Ks−µ,γ−µ (X ∧ , Ck )T

(7.1)

is a family of Fredholm operators of index j+ − j− , where j± denotes the fibre dimensions of J± . Under the ellipticity assumption (i), the Fredholm property of (7.1) in turn is equivalent to the bijectivity of the subordinate conormal symbol σM (A)(y, z) : H s (X) → H s−µ (X),

s ∈ R,

for all y ∈ Y and z ∈ Γ n+1 −γ . In particular, it does not depend on the choice of S 2 and T , and the conormal symbol might have non-bijectivity points on the weight line Γ n+1 −(γ−σ) (with σ corresponding to the type S). 2

7.1. The inverse of identity plus Green symbol For the following statement we use the notation I = diag(1, 1). 0 Proposition 7.2. Let G ∈ YG (M, (γ, γ), (k, k; J, J))S,S such that σ∧ (I + G)(y, η) = I + σ∧ (G)(y, η) is an isomorphism for each y and η = 0. Then there exists a 0 (M, (γ, γ), (k, k; J, J))S,S such that C ∈ YG

(I + σ∧ (G)(y, η))−1 = I + σ∧ (C)(y, η)

for all (y, η) ∈ T ∗ Y \0.

Proof. For notational simplicity let k = 1. Without loss of generality, we may assume γ = 0. Let us write
1 + G11 I + σ∧ (G)(y, η) =: G21

 K0,0 (X ∧ )S K0,0 (X ∧ )S G12 ⊕ ⊕ (y, η) : −→ G22 J J

(7.2)

and let us suppress (y, η) from the notation for a while. Since the following considerations are local in (y, η), we assume that J = Cj for some j ∈ N. Let bS be the projection in K0,0 (X ∧ )S onto ES (X ∧ ) along K0,0 (X ∧ ), cf. (5.2) 
K0,0 (X ∧ ) 1 − bS 0,0 ∧ ⊕ is an isomorphism with : K (X )S → and (5.14). Then bS ∧ E (X ) S   1 − bS 0 0 and form the isomorphism the inverse (l0 l1 ). We then set B :=  bS 0 1

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B(I + σ∧ (G))B−1 which equals  

b0 (1 + G11 )l0 b1 G21 l0 G21 l0

b0 (1 + G11 )l1 b1 G21 l1 G21 l1

K0,0 (X ∧ ) K0,0 (X ∧ ) ⊕ ⊕ b0 G12 b1 G12  : ES (X ∧ ) −→ ES (X ∧ ) . G22 ⊕ ⊕ Cj Cj 

(7.3)

Using an isomorphism ES (X ∧ ) ∼ = Cι(S) , cf. (4.3), and writing G := b0 G11 l0 , we transform (7.3) to the form  K0,0 (X ∧ )
K0,0 (X ∧ ) 1 + G H12 ⊕ ⊕ −→ . (7.4) : H21 H22 Cι(S)+j Cι(S)+j Now let (y0 , η0 ) ∈ S ∗ Y (the unit cosphere bundle) be fixed. Since isomorphisms form an open set and each matrix can be approximated arbitrarily well by an  22 such invertible one, there exists an invertible (ι(S) + j) × (ι(S) + j)-matrix H that
 K0,0 (X ∧ ) K0,0 (X ∧ ) 1 + G H12 ⊕ ⊕ −→ (7.5)  22 : H21 H Cι(S)+j Cι(S)+j is pointwise invertible in an open neighborhood of (y0 , η0 ) in S ∗ Y . From the decomposition



  −1 H21  −1 1 + G H12 1 0 0 (1 + G) − H12 H 1 H12 H 22 22  −1 H21 1  22 = 0 ˜ 22 H H21 H 1 0 H 22 we deduce that the second term on the right-hand side is invertible, since all the other block-matrices are. In particular, ˜ −1 H21 , D := G − H12 H 1 + D : K0,0 (X ∧ ) −→ K0,0 (X ∧ ), 22

is invertible. But now D is (the symbol of) a regular Green operator, i.e. a Green operator from the standard edge calculus. Then it is known that (1+D)−1 = 1+M with a (symbol of a) regular Green operator. This yields # −1 "
 −1 1 + G H12 1+M −(1 + M )H12 H 22 =  22  −1 H21 (1 + M )  −1 H21 H −H H 22 22
 1 + M N12 =: . N21 N22


 1 + G H12 1 + M N12 1 0 = H21 H22 N21 N22 B R with an invertible matrix R, and it follows that −1



1 + M N12 1 0 1+K 1 + G H12 = =: H21 H22 N21 N22 L21 −R−1 B R−1 Then

 L12 , L22 (7.6)

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where the functions K and Lij are smooth near (y0 , η0 ), and have the properties that K(y, η), K(y, η)∗ : K0,0 (X ∧ ) −→ Sε0 (X ∧ ), L12 (y, η), L21 (y, η)∗ : Cι(S)+j −→ Sε0 (X ∧ ) for some ε > 0. Since the local inverses are unique, we thus obtain an inverse of the form (7.6) globally on S ∗ Y , having the mapping properties just described. It then remains to set 

1 + K L12 1 + C11 C12 = B−1 B C21 C22 L21 L22 and to define σ∧ (C)(y, η) by
 1 + C11 C12 I + σ∧ (C)(y, η) = diag(κ|η| , 1) (y, η/|η|) diag(κ−1 |η| , 1). C21 C22 By using a zero excision function χ(η), we then find a corresponding operator C having the constructed principal edge symbol.
7.2. The parametrix construction Theorem 7.3. Let A ∈ Y µ (M, (γ, γ − µ), (k, k, J− , J+ ))S,T be elliptic. Then A has a parametrix P ∈ Y −µ (M, (γ − µ, γ), (k, k, J+ , J− ))T,S , i.e., AP − I ∈ Y −∞ (M, (γ − µ, γ − µ), (k, k, J+ , J+ ))T,T , PA − I ∈ Y −∞ (M, (γ, γ), (k, k, J− , J− ))S,S , where I = diag(1, 1) is the identity operator. In particular, A induces Fredholm operators W s−µ,γ−µ (M, Ck )T W s,γ (M, Ck )S ⊕ ⊕ −→ , A: H s (Y, J− ) H s−µ (Y, J+ )

s ∈ R.

Proof. Let us construct a left parametrix P; the construction of a right-parametrix is similar. Then P is a two-sided parametrix, and we obtain the Fredholm property of A, since the remainders are compact operators in the respective Sobolev spaces. For the construction of P we assume for simplicity that k = 1 (the general case is completely analogous). Writing A as a block-matrix A = (Aij )i,j=1,2 with A11 = AP0,S , cf. the formulas (6.6) and (7.1), condition (ii) of Definition 7.1 yields a family of Fredholm operators σ∧ (A)(y, η) : Ks,γ (X ∧ ) −→ Ks−µ,γ−µ (X ∧ ),

(y, η) ∈ T ∗ Y \ 0,

of index l+ − l− for l+ = j+ + ι(T ) and l− = j− + ι(S). Using the technique of constructing a parametrix in the standard edge calculus, we find an elliptic operator B ∈ Y −µ (M, (γ − µ, γ), (1, 1; L+ , L− )) for L+ = J+ ⊕ Cι(T ) and L− = J− ⊕ Cι(S) , which has the following properties: The upper left corner B of B, viewed as an

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element of L−µ (M \ Y ), is a parametrix of the elliptic operator A ∈ Lµ (M \ Y ), and (T µ σM (B))σM (A) = 1. Moreover, the operator B is Fredholm as a map W s−µ,γ−µ (M ) W s,γ (M ) ⊕ ⊕ B: −→ . H s−µ (Y, J+ ) ⊕ H s−µ (Y, Cι(T ) ) H s (Y, J− ) ⊕ H s (Y, Cι(S) ) A simple algebraic argument, using the isomorphisms ∼ W s,γ (M )S , W s,γ (M ) ⊕ H s (Y, Cι(S) ) = W s−µ,γ−µ (M ) ⊕ H s−µ (Y, Cι(T ) ) ∼ = W s−µ,γ−µ (M )T , cf. Remark 6.3, allows us to reorganise the operator B to an operator W s,γ (M )S W s−µ,γ−µ (M )T ⊕ ⊕ −→ P0 : H s−µ (Y, J+ ) H s (Y, J− ) with P0 ∈ Y −µ (M, (γ − µ, γ), (1, 1; J− , J+ ))T,S and such that 0 G := P0 A − I ∈ YG (M, (γ, γ), (1, 1; J− , J− ))S,S .

Both P0 and A are elliptic; thus also I + G is elliptic. By Proposition 7.2 we find 0 an operator C ∈ YG (M, (γ, γ), (1, 1; J− , J− ))S,S such that (I + C)(I + G) = I + K

for a

−1 K ∈ YG (M, (γ, γ), (1, 1; J− , J− ))S,S .

Setting P1 = (I + C)P0 we thus obtain P1 A = I + K. Applying Theorem 6.10, we can form the asymptotic sum L :=

∞ 

−1 (−1)j Kj ∈ YG (M, (γ, γ), (1, 1; J−, J− ))S,S .

j=1

Then P := (I+L )P1 is the desired left parametrix. In the latter steps we employed Theorem 6.9 several times.
7.3. Concluding remarks Comparing the spaces Y µ (M, (γ, γ − µ), (k, l, J− , J+ ))S,T and Y µ (M, (γ, γ − µ), (k, l, J− , J+ )), the additional ingredients of the new calculus are Green operators (in block-matrix sense) associated with discrete asymptotic data in the weight strips  n+1 n+1 − γ < Re z < −γ+σ z: 2 2 and  n+1 n+1 − (γ − µ) < Re z < − (γ − µ) + τ , z: 2 2 respectively. Such operators are able to ‘reproduce’ non-smoothing Mellin operators in the upper left corner when they are given on the line Re z = n+1 2 −γ+σ instead of Re z = n+1 − γ (the latter is the case in both variants of the edge 2 calculus). This is due to the fact that differences of Mellin operators for different

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weight lines (and with specific meromorphic symbols) generate Green operators of the singular category, cf. [3]. It would be interesting to study subalgebras of our calculus with meromorphic Mellin symbols in which the Mellin operators refer to Re z = n+1 2 − γ + σ instead of Re z = n+1 − γ. This would require a careful choice of admitted Mellin symbols 2 and a detailed analysis of the asymptotic data. The paper [8] of Liu and Witt can be interpreted in this spirit. One may expect many other interesting examples of such subcalculi of edge problems, for instance, algebras generated by boundary value problems without the transmission property at the boundary, with (principal) interior symbols of / 2Z (the case µ ∈ 2Z corresponds to the case with the form |ξ|µ for some µ ∈ transmission property). An edge algebra interpretation of boundary value problems (without transmission property) is given in [13]. Such investigations could deepen the insight into the structure of parametrices of mixed elliptic problems, but the details might be quite voluminous (see, for instance, [7] or [6]).

8. Appendix: The standard calculus for manifolds with edges In this appendix we shall give a brief summary of the so-called edge algebra. For a reference of the here presented material we refer the reader to [3],[5], and [15]. In the following Ω ⊂ Rq denotes an open set, and X is a closed smooth manifold of dimension n ∈ N0 . We shall denote by Lµcl (X) the Fr´echet space of classical pseudodifferential operators on X, and by Lµcl (X; Λ) the space of parameterdependent operators, where Λ = Rl for some l ∈ N. More precisely, the local symbols of parameter-dependent operators satisfy estimates of the form |∂xβ ∂ξα ∂λγ a(x, ξ, λ)| ≤ C (1 + |ξ| + |λ|)µ−|α|−|γ| , and they have asymptotic expansions in components that are positively homogeneous in (ξ, λ). In short, (ξ, λ) is treated as a covariable. 8.1. Mellin symbols and Mellin pseudodifferential operators For γ ∈ R and ε > 0 let us set

  S(γ,) = {z ∈ C :  12 − γ − Re z  < ε},

i.e. S(γ,ε) is the open, vertical strip of width ε around the line Re z =

1 2

− γ.

µ Definition 8.1. Let γ ∈ R, µ ∈ R ∪ {−∞} and ε > 0. The space M(γ,ε) (X; Λ) µ consists of all holomorphic functions h : S(γ− n2 ,ε) → Lcl (X; Λ) such that

hδ (λ, ) := h(λ,

n+1 2

− γ + δ + i) ∈ Lµcl (X; Λ × R )

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uniformly in δ ∈ (−ε, ε). Since this is a Fr´echet space in a canonical way, it is µ meaningful to speak of C ∞ (Ω × R+ , M(γ,ε) (X; Λ)). We set Mγµ (X; Λ) = C ∞ (Ω × R+ , Mγµ (X; Λ)) =

µ ∪ M(γ,ε) (X; Λ),

ε>0

µ ∪ C ∞ (Ω × R+ , M(γ,ε) (X; Λ)),

ε>0

and µ ∩ M(γ,ε) (X; Λ),

MOµ (X; Λ) =

ε>0

C ∞ (Ω × R+ , MOµ (X; Λ)) =

ε>0

µ ∩ C ∞ (Ω × R+ , M(γ,ε) (X; Λ)).

The definitions are analogous for the spaces depending only on Ω × R+ and those independent of the parameter λ ∈ Λ. A symbol h(y, r, λ, z) belonging to C ∞ (Ω × R+ , Mγµ (X; Λ)) induces a family of Mellin pseudodifferential operators on the infinite cone X ∧ = R+ × X by  1 γ− n 2 [opM (h)(y, λ)u](r) := r−z h(y, r, λ, z)(M u)(z) dz 2πi Re z= n+1 2 −γ for u ∈ C0∞ (X ∧ ). Here, we have identified C0∞ (X ∧ ) with C0∞ (R+ , C ∞ (X)), and M denotes the Mellin transform. 8.2. Green and smoothing Mellin symbols For the following definition recall the definition of cone Sobolev spaces Ks,γ (X ∧ ) from Section 2 and its group action κλ . µ (Ω × Rq ; (γ, γ  )) Definition 8.2. Let γ, γ  ∈ R and µ ∈ R ∪ {−∞}. The space RG consists of all operator-valued symbols (cf. Section 2)


µ (Ω × Rq ; K0,γ (X ∧ ), K0,γ (X ∧ )) g(y, η) ∈ Scl

such that there exists an ε > 0 with g(y, η) ∈ ∗

g(y, η) ∈




∩ Sclµ (Ω × Rq ; Ks,γ (X ∧ ), S γ +ε (X ∧ )),

s∈R




∩ Sclµ (Ω × Rq ; Ks,−γ (X ∧ ), S −γ+ε (X ∧ )).

s∈R

In the previous definition, ∗ refers to the pointwise adjoint with respect to the scalar product of K0,0 (X ∧ ) = L2 (X ∧ , r−n drdx), which allows an identification of K−s,−γ (X ∧ ) and the dual space of Ks,γ (X ∧ ). µ Definition 8.3. Let γ ∈ R and µ ∈ R ∪ {−∞}. The space RM+G (Ω × Rq ; (γ, γ − µ)) consists of all operator-valued symbols of the form γ− n 2

m(y, η) + g(y, η) = ω0 (r[η]) r−µ opM µ (Ω ∈ RG ∞

(h)(y) ω1 (r[η]) + g(y, η), ∞

(8.1)

(Ω, Mγ−∞ (X)),

× R ; (γ, γ − µ)) is a Green symbol, h ∈ C where g(y, η) and ω0 , ω1 ∈ C ([0, 1[) are arbitrary cut-off functions. For a real number ν ∈ R with 1 ≤ µ − ν ∈ N we set q

ν ν RM+G (Ω × Rq ; (γ, γ − µ)) = RG (Ω × Rq ; (γ, γ − µ)).

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It can be shown that then, for each s ∈ R, ν ν RM+G (Ω × Rq ; (γ, γ − µ)) ⊂ Scl (Ω × Rq ; Ks,γ (X ∧ ), K∞,γ−µ (X ∧ )).

If (m + g)(y, η) is as in (8.1), its homogeneous principal symbol is defined as γ− n 2

σ∧ (m + g)(y, η) = ω0 (r|η|) r−µ opM

η = 0, (8.2) where σ∧ (g)(y, η) denotes the homogeneous principal symbol of g(y, η). This symbol we shall refer to as the principal edge symbol of (m + g)(y, η). The (principal) conormal symbol of (m + g)(y, η) is, by definition, (h)(y) ω1 (r|η|) + σ∧ (g)(y, η),

σM (m + g)(y, z) = h(y, z) ∈ C ∞ (Ω, Mγ−∞ (X)).

(8.3)

8.3. Edge amplitude functions Definition 8.4. Let γ ∈ R and µ, ν ∈ R with µ− ν ∈ N. Then Rν (Ω× Rq ; (γ, γ − µ)) denotes the space of so-called edge symbols  γ−n/2 a(y, η) = σ r−ν opM (h)(y, η) + m(y, η) + g(y, η) σ0 + + (1 − σ) op(p)(y, η) (1 − σ1 ), where the notation has the following meaning: a) b) c) d)

σ, σ0 , σ1 ∈ C0∞ ([0, 1[) are cut-off functions with σ1 ≺ σ ≺ σ0 , h(y, r, η, z) =  h(y, r, rη, z) with  h ∈ C ∞ (Ω × R+ , MOν (X; Rqη )), ν (Ω × Rq ; (γ, γ − µ)), (m + g)(y, η) ∈ RM+G p(y, r, η, ) ∈ C ∞ (Ω × R+ , Lνcl (X; Rqη × R )), independent of r for large r. Edge symbols are particular operator-valued symbols, namely Rν (Ω × Rq ; (γ, γ − µ)) ⊂ S ν (Ω × Rq ; Ks,γ (X ∧ ), Ks−ν,γ−µ (X ∧ )).

In case ν = µ we define the principal edge symbol of a(y, η) by σ∧ (a)(y, η) = r−µ opM

γ−n/2

(h∧ )(y, η) + σ∧ (m + g)(y, η),

h∧ (y, r, z, η) =  h(y, 0, rη, z),

(8.4)

cf. (8.2), and the conormal symbol of a(y, η) by σM (a)(y, z) =  h(y, 0, z, 0) + σM (m + g)(y, z),

(8.5)

cf. (8.3). There is an obvious generalization to (l × k)-matrices of such edge symbols, yielding the space Rν (Ω × Rq ; (γ, γ − µ), (k, l)).

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8.4. The algebra of edge pseudodifferential operators on M In the following we let M be a manifold with edge Y and model cone X ∧ as it is described in Section 6. For γ ∈ R and µ, ν ∈ R with µ − ν ∈ N, Y ν (M, (γ, γ − µ), (k, l))

(8.6)

denotes the space of ν-th order edge operators (with respect to (k, l) and the weight-data (γ, γ − µ)). Modulo global smoothing operators they are obtained by pasting together (using a partition of unity on M ) usual pseudodifferential operators of order ν on the interior of M with operators that are localized near the edge and that are defined locally by means of pseudodifferential operators with operator-valued symbols from Rν (Ω × Rq ; (γ, γ − µ), (k, l)). We shall omit the details here. The space of global smoothing operators, Y −∞ (M, (γ, γ − µ), (k, l)), consists of all operators G : W 0,γ (M, Ck ) → W 0,γ−µ (M, Cl ) such that there exists an ε > 0 with G : W s,γ (M, Ck ) −→ W ∞,γ−µ+ε (M, Cl ), G∗ : W s,−γ+µ (M, Cl ) −→ W ∞,−γ+ε (M, Ck ),

(8.7)

for each s ∈ R, where ∗ refers to the adjoint with respect to the scalar product of W 0,0 (M ) that admits an identification of the dual space of W s,σ (M ) with W −s,−σ (M ). Each A ∈ Y ν (M, (γ, γ − µ), (k, l)) induces continuous operators W s,γ (M, Ck ) −→ W s−ν,γ−µ (M, Cl ),

s ∈ R.

8.5. Principal symbols As a matter of fact, Y ν (M, (γ, γ − µ), (k, l)) ⊂ Lνcl (int M ; Ck , Cl ) for int M := M \ Y , i.e., any edge operator is a usual pseudodifferential operator on the interior of M . In particular, with each A ∈ Y µ (M, (γ, γ − µ), (k, l)) we can associate the standard homogeneous principal symbol σψ (A) ∈ C ∞ (T ∗ int M \ 0, L (Ck , Cl )).

(8.8)

In local coordinates, corresponding to the splitting of coordinates (y, r, x) near the edge, the homogeneous principal symbol has a special ‘degenerate’ form namely σψ (A)(y, r, x, η, , ξ) = r−µ p(µ) (y, r, x, rη, r, ξ), with a symbol p(µ) (y, r, x, η, , ξ) homogeneous in ( η , , ξ). Removing this degeneracy leads to σ ψ (A)(y, x, η, , ξ) = p(µ) (y, 0, x, η, , ξ) = lim rµ σψ (A)(y, r, x, r−1 η, r−1 , ξ). r→0+

Globally, this yields the so-called rescaled symbol σ ψ (A) ∈ C ∞ ((T ∗ (Y × X) × R) \ 0, L (Ck , Cl )).

(8.9)

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Moreover, induced by the local amplitude functions of Definition 8.4, we have the principal edge symbol σ∧ (A) ∈ C ∞ (T ∗ Y \ 0, L (Ks,γ (X ∧ , Ck ), Ks−µ,γ−µ (X ∧ , Cl )).

(8.10)

8.6. Ellipticity An operator A ∈ Y µ (M, (γ, γ − µ), (k, k; J− , J+ )) (cf. the notation of Remark 6.7) is said to be elliptic, if σψ (A), σ ψ (A), and σ∧ (A) satisfy the conditions of Definition 7.1. Theorem 8.5. An elliptic operator A ∈ Y µ (M, (γ, γ − µ), (k, k; J− , J+ )) possesses a parametrix P ∈ Y −µ (M, (γ − µ, γ), (k, k; J+ , J− )), i.e. AP − I and PA − I are of order −∞ in the spaces belonging to the data ((γ − µ, γ − µ), (k, k; J+ , J+ )) and ((γ, γ), (k, k; J− , J− )), respectively. For a proof of this theorem see [3], for example.

References [1] L. Boutet de Monvel. Boundary problems for pseudo-differential operators. Acta Math. 126 (1971), 11-51. [2] N. Dines, B.-W. Schulze. Mellin-edge-representations of elliptic operators. Math. Methods Appl. Sci. 28 (2005), 2133-2172. [3] Ju.V. Egorov, B.-W. Schulze. Pseudo-differential operators, singularities, applications. Operator Theory: Advances and Applications 93, Birkh¨ auser Verlag, 1997. [4] G.I. Eskin. Boundary value problems for elliptic pseudodifferential equations. Math. Monographs 52, Amer. Math. Soc., 1980. [5] J.B. Gil, B.-W. Schulze, and J. Seiler. Cone pseudodifferential operators in the edge symbolic calculus. Osaka J. Math. 37 (2000), 221-260. [6] G. Harutjunjan, B.-W. Schulze. Elliptic Mixed, Transmission, and Singular Crack Problems. Tracts in Math. 4, World Scientific, 2008. [7] D. Kapanadze, B.-W. Schulze. Crack theory and edge singularities. Mathematics and its Applications 561, Kluwer Academic Publishing, 2003. [8] X. Liu, I. Witt. Pseudodifferential calculi on the half-line respecting prescribed asymptotic types. Integral Equations Opeartor Theory 49 (2004), 473-497. [9] M.I. Vishik, G.I. Eskin, Convolution equations in a bounded region, Uspekhi Mat. Nauk 20 (1965), 89-152. [10] S. Rempel, B.-W. Schulze. Parametrices and boundary symbolic calculus for elliptic boundary problems without the transmission property. Math. Nachr. 105 (1982), 45-149. [11] B.-W. Schulze. Pseudo-differential operators on manifolds with singularities. NorthHolland, 1991. [12] B.-W. Schulze. Pseudo-differential operators on manifolds with edges. In Symp. “Partial Differential Equations”, Holzhau 1988. Teubner-Texte Math. 112, pp. 259-287. Teubner Verlag, 1989.

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[13] B.-W. Schulze, J. Seiler. The edge algebra structure of boundary value problems. Ann. Global Anal. Geom. 22 (2002), 197-265. [14] B.-W. Schulze, J. Seiler. Pseudodifferential boundary value problems with global projection conditions. J. Funct. Anal. 206 (2004), 449-498. [15] J. Seiler. Pseudodifferential calculus on manifolds with non-compact edges. Dissertation, University of Potsdam, 1997. [16] N. Tarkhanov. Harmonic integrals on domains with edges. Preprint 2002/02, University of Potsdam. D. Kapanadze A. Razmadze Mathematical Institute Academy of Sciences of Georgia M. Alexidze Str. 1 Tbilisi 0193 Georgia e-mail: [email protected] B.-W. Schulze Universit¨ at Potsdam Institut f¨ ur Mathematik Postfach 601553 14415 Potsdam Germany e-mail: [email protected] J. Seiler Leibniz Universit¨ at Hannover Institut f¨ ur Angewandte Mathematik Welfengarten 1 30167 Hannover Germany e-mail: [email protected] Submitted: August 28, 2007 Revised: February 23, 2008

Integr. equ. oper. theory 61 (2008), 281–298 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/020281-18, published online April 17, 2008 DOI 10.1007/s00020-008-1588-3

Integral Equations and Operator Theory

Unbounded Toeplitz Operators Donald Sarason In memory of Paul R. Halmos

Abstract. This partly expository article develops the basic theory of unbounded Toeplitz operators on the Hardy space H 2 , with emphasis on operators whose symbols are not square integrable. Unbounded truncated Toeplitz operators on coinvariant subspaces of H 2 are also studied. Mathematics Subject Classification (2000). Primary 47B35; Secondary 30D50. Keywords. Toeplitz operator, truncated Toeplitz operator, Smirnov class, de Branges–Rovnyak space.

1. Introduction Unbounded Toeplitz operators on the Hardy space H 2 of the unit disk D arise often, usually with symbols belonging to L2 of ∂D. There are natural questions, though, which lead to Toeplitz operators having more general symbols — I mention in particular interesting work of Henry Helson [4], Daniel Su´ arez [10], and Steven Seubert [9]. This being so, it seems worthwhile to gather the basic properties of these more general unbounded Toeplitz operators in a single source. That is the aim of the present paper. Section 2 is a short review of unbounded Toeplitz operators with L2 symbols, which are conveniently defined by means of the Cauchy integral. Sections 3 and 4 contain preliminary material on, respectively, the Smirnov class and de Branges– Rovnyak spaces. Section 5 is devoted to analytic Toeplitz operators, the case of interest to Helson, and Section 6 to coanalytic Toeplitz operators, the case of interest to Su´ arez and Seubert. I have not attempted here to go substantially beyond those two cases, as far as symbols outside of L2 are concerned, because I am not aware of any study in which that has been necessary. Nevertheless, one can ask just how much generality is possible. The question is posed in the concluding Section 8.

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Section 7 concerns compressions of unbounded Toeplitz operators to invariant subspaces of the backward shift on H 2 , which were of interest to Su´ arez and Seubert. There is a large overlap between this exposition and the cited papers of Helson, Su´arez and Seubert. Some of the results of those authors will be mentioned in the course of this study. I am grateful to Israel Gohberg for suggesting that I submit a paper in memory of Paul Halmos to IEOT. Halmos was responsible for my first encounter with Toeplitz operators, when I was his student in Ann Arbor, a mathematical lifetime ago. The Halmos influence permeates my entire mathematical life. Gohberg’s pioneering work and incisive questions have been a continual source of inspiration in the Toeplitz community. Notations • • • • • • • • •

The inner product and norm in L2 will be denoted by
·, · and  · 2 . The orthogonal projection of L2 onto H 2 will be denoted by P+ . The unilateral shift operator on H 2 will be denoted by S. A function defined in D will be identified notationally with its boundary function. The mth Fourier coefficient of a function f in L1 will be denoted by f
(m). The identity operator will be denoted by I; the space on which it acts will be clear from the context. The domain and graph of an operator T will be denoted by D(T ) and G(T ), respectfully. The standard tensor notation will be used for operators of rank one: for v and w vectors in a Hilbert space H, the operator v ⊗ w is defined by (v ⊗ w)x =
x, wH v. H(D) denotes the space of functions holomorphic in D, H(D) the space of functions holomorphic in neighborhoods of D.

2. L2 Symbols The Cauchy integral of a function f in L1 of ∂D is the function Kf defined in D by  1 f (eiθ ) dθ. (Kf )(z) = 2π ∂D 1 − e−iθ z If f is in L2 , then Kf equals P+ f , the projection of f onto H 2 . The map f → Kf is continuous relative to the weak topology of L1 and the topology of locally uniform convergence of H(D). For ϕ a function in L2 , the Toeplitz operator Tϕ on H 2 is defined by Tϕ f = K(ϕf ), with domain D(Tϕ ) = {f ∈ H 2 : K(ϕf ) ∈ H 2 }. It is a closed, densely defined operator, bounded if and only if ϕ is bounded. The domain D(Tϕ ) always contains H ∞ .

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One place Toeplitz operators with L2 symbols arise is in the invertibility criterion of A. Devinatz and H. Widom (see, e.g., [6], p. 250). The Devinatz– Widom analysis shows that if a bounded Toeplitz operator is invertible then its inverse, while not in general a Toeplitz operator, can be written as the product of an analytic Toeplitz operator times a coanalytic Toeplitz operator, both factors having L2 symbols, but possibly being unbounded.

3. The Smirnov Class We recall that the Nevanlinna class N consists of the holomorphic functions in D that are quotients of functions in H ∞ , and the Smirnov class N + consists of such quotients in which the denominators are outer functions. The representation of Smirnov-class functions given by the following proposition underlies much of what follows. Proposition 3.1. A nonzero function ϕ of Smirnov class can be written uniquely as ϕ = b/a, where a and b are in the unit ball of H ∞ , a is an outer function, a(0) > 0, and |a|2 + |b|2 = 1 on ∂D. The representation ϕ = b/a of the proposition will be termed the canonical representation of ϕ. It would not be surprising if the proposition could be found somewhere in the literature, because pairs a, b satisfying the given conditions arise in several connections. However, I have no recollection of seeing it stated explicitly. At any rate, the proposition is easy to prove. Proof. Suppose ϕ = ψ/χ, where ψ and χ are in H ∞ , ψ = 0, and χ is outer. If the required pair a, b exists then, because of the condition that |a|2 + |b|2 = 1 on ∂D, the function a must satisfy |χ|2 (3.1) |a|2 = |ψ|2 + |χ|2 on ∂D. The function |χ|2 is log-integrable because χ is in H ∞ , and hence |ψ|2 +|χ|2 is also log-integrable. The function on the right side of equality (3.1) is thus logintegrable, so there is a unique outer function a that satisfies (3.1) on ∂D and is positive at the origin. For the function b = aϕ we then have, on ∂D,    |ψ|2 |χ|2 1 + = 1, |a|2 + |b|2 = |ψ|2 + |χ|2 |χ|2 and the existence of the representation ϕ = b/a is established. Uniqueness holds because a is uniquely determined by (3.1).
Remark 3.2. If the Smirnov function ϕ is rational, then the functions a and b in the representation of ϕ are rational. In fact, suppose ϕ = p/q, where p and q are relatively prime polynomials, q has no roots in D, and q(0) > 0. As in the preceding proof, we have |q|2 |a|2 = 2 |p| + |q|2

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on ∂D. The function |p|2 + |q|2 is, on ∂D, a nonnegative trigonometric polynomial. By the well-known Fej´er–Riesz theorem, there is a polynomial r without roots in D, and with r(0) > 0, such that |r|2 = |p|2 + |q|2 on ∂D. We get a = q/r, a rational function. The functions a and b can be determined explicitly in simple cases. In the following examples, the functions a corresponding to several choices of ϕ are listed, but the computational details are omitted.



ϕ(z)

a(z)

1+z 1−z

1−z 2

1+z 1−z

2

1 1−z z (1 − z)2 1 γ =1+ √ 2 2

  (1 − z)2 1 √ 1+ √ 2 (z 2 + 3 + 2 2)  √  z−1 1+ 5 √ 2 z − 3+ 5 2

2

|γ|(1 − z) (γ defined below) (γ − 2)(γ − z)     √ √ 1 1 + √ 17 − 1 − i 17 + 1 2 2 2

4. De Branges–Rovnyak Spaces The needed information on these spaces can be found in [7]; the spaces were first studied by L. de Branges and J. Rovnyak [2]. They are associated with functions in the unit ball of H ∞ . Of interest here is the case where those functions are not extreme points of the unit ball. Let b be a nonconstant function in the unit ball of H ∞ , but not an extreme point. The de Branges–Rovnyak space H(b) is by definition the range of the operator (I − Tb T¯b )1/2 with the range norm, i.e., the norm that makes (I − Tb T¯b )1/2 an isometry of H 2 onto H(b). (Note that (I − Tb T¯b )1/2 is injective.) The norm and inner product in H(b) will be denoted by  · b and
·, ·b . Since b is not an extreme point of the unit ball of H ∞ , there is a unique outer function a in H ∞ such that a(0) > 0 and |a|2 + |b|2 = 1 on ∂D. The function b/a is in the Smirnov class N + ; according to Proposition 3.1, the general nonconstant function in N + arises in this way. The following proposition gives an alternative description of H(b).

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Proposition 4.1 ([7], p. 24). The H 2 function f belongs to H(b) if and only if there is a function g in H 2 such that T¯b f = Ta¯ g. In that case f 2b = f 22 + g22 . Note that the function g in the proposition is unique, because Ta¯ has a trivial kernel (a being an outer function). The inclusion Ta¯ H 2 ⊂ H(b) is a consequence of Proposition 4.1 since, for h in H 2 , T¯b Ta¯ h = Ta¯ T¯b h. (In fact, Ta¯ H 2 is dense in H(b) ([7], p. 25).) One also has the inclusion aH 2 ⊂ Ta¯ H 2 : for h in H 2 , Ta h = Ta¯ Ta/¯a h.

5. Analytic Toeplitz Operators We start with a function ϕ that is holomorphic in D and define Tϕ to be the operator of multiplication by ϕ on the domain D(Tϕ ) = {f ∈ H 2 : ϕf ∈ H 2 }. Clearly, Tϕ is a closed operator. Lemma 5.1. If D(Tϕ ) is nontrivial then ϕ is in the Nevanlinna class N . Proof. In fact, suppose D(Tϕ ) contains the nonzero function f . Then ϕ = ϕf /f is the ratio of two H 2 functions, hence the ratio of two functions in N , hence in N.
Lemma 5.2. If D(Tϕ ) is dense in H 2 then ϕ is in the Smirnov class N + . Proof. Let ϕ = ψ/χ be in N , where ψ and χ are functions in H ∞ whose inner factors are relatively prime. Suppose f is in D(Tϕ ) and let g = ϕf . Then ψf = χg. Let ψ0 be the outer factor of ψ and χ1 the inner factor of χ. Because χ1 and the inner factor of ψ are relatively prime, the equality ψf = χg implies that ψ0 f is in χ1 H 2 , and hence that f is in χ1 H 2 . Thus D(Tϕ ) ⊂ χ1 H 2 , and since D(Tϕ ) is dense in H 2 we can conclude that χ1 is constant. So χ is an outer function, i.e., ϕ is in N + .
The converse of Lemma 5.2 is immediate: If ϕ = ψ/χ is as above, with χ outer, then D(Tϕ ) contains χH 2 and so is dense in H 2 . Proposition 5.3. Let ϕ be a nonzero function in N + , with canonical representation ϕ = b/a. Then D(Tϕ ) = aH 2 . Proof. In fact, the inclusion aH 2 ⊂ D(Tϕ ) is clear (as just noted). Suppose f is in D(Tϕ ). We have, on ∂D,  2 f  |b|2 |f |2 2   − |f |2 , |ϕf | = = a |a|2 implying that f /a is in L2 . Since f /a is in N + it follows that f /a is in H 2 , giving the inclusion D(Tϕ ) ⊂ aH 2 .
Tϕ∗

Since for ϕ in N + the operator Tϕ is densely defined and closed, its adjoint is densely defined and closed.

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Proposition 5.4. Let ϕ be a nonzero function in N + with canonical representation ϕ = b/a. Then D(Tϕ∗ ) = H(b), and the projection of G(Tϕ∗ ) onto H(b) is an isometry relative to the norms  · 2 and  · b . Proof. In fact, by Proposition 5.3 we have G(Tϕ ) = {ah ⊕ bh : h ∈ H 2 }, which we can write in operator form as   Ta H 2. G(Tϕ ) = Tb Therefore G(Tϕ )⊥ = ker(Ta¯ T¯b ) = {g ⊕ f ∈ H 2 ⊕ H 2 : Ta¯ g + T¯b f = 0}. It follows that G(Tϕ∗ ) = {f ⊕ g ∈ H 2 ⊕ H 2 : (−g) ⊕ f ∈ G(T )⊥ } = {f ⊕ g ∈ H 2 ⊕ H 2 : T¯b f = Ta¯ g}, and the desired conclusions follow immediately from Proposition 4.1.




Let ϕ be as in the preceding proposition. From the inclusion aH 2 ⊂ H(b) noted in Section 4 one sees that D(Tϕ ) ⊂ D(Tϕ∗ ). Also D(Tϕ∗ ) contains H(D), the space of functions holomorphic in neighborhoods of D ([7], p. 26). The space H(b) is S ∗ -invariant, as one can see from Proposition 4.1. (The ∗ S -invariance is in fact a property of general de Branges–Rovnyak spaces ([7], p. 11). From the description of G(Tϕ∗ ) obtained in the preceding proof, one sees arez proved in [10] that Tϕ∗ with ϕ that if f is in D(Tϕ∗ ) then Tϕ∗ S ∗ f = S ∗ Tϕ∗ f . Su´ + in N is the general closed densely defined operator that commutes with S ∗ . If T is such an operator then the subspace G(T )⊥ of H 2 ⊕ H 2 is invariant under S ⊕ S. Su´ arez’s proof relies on the generalization of Beurling’s theorem to shift-invariant subspaces of H 2 ⊕ H 2 . Proposition 5.4 suggests that, for ϕ in N + , the conjugate-analytic Toeplitz operator Tϕ should be defined by Tϕ = Tϕ∗ . Rather than adopting this definition at this point, we take up coanalytic Toeplitz operators ab initio in the next section.

6. Coanalytic Toeplitz Operators

∞ As in Section 5, we start with a function ϕ that is holomorphic in D. Let n=0 γn z n be the power series of ϕ centered at the origin. The Toeplitz matrix associated with Tϕ is the lower-triangular Toeplitz matrix built from the coefficients of the power series; its (m, n)th entry for m ≥ n is γm−n . Given a function f in D(Tϕ ), one obtains the column representing Tϕ f with respect to the standard basis for H 2 through multiplication of the column representing f by the Toeplitz matrix. Because of lower triangularity, the preceding multiplication makes perfect sense for any f in H 2 (and of course more generally).

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We consider now the adjoint of the Toeplitz matrix associated with ϕ, the upper-triangular Toeplitz matrix whose (m, n)th entry for m ≤ n is γ n−m . Unless ϕ is in H 2 , the product of this matrix with the column representing an arbitrary ∞ function f in H 2 , whose mth entry is given formally by the series n=0 γ n f
(m+n) (m = 0, 1, . . . ), need not be well defined. It is well defined, though, if the function f is a polynomial, hence for a dense set of functions in H 2 . More is true.
Lemma 6.1. If f is in H(D), then each series ∞ n=0 γ n f (m + n) (m = 0, 1, . . . ) converges absolutely, and the function represented by the power series  ∞ ∞ m
is in H(D). m=0 n=0 γ n f (m + n) z Proof. To prove the lemma, suppose the function f is in H(RD), where R > 1. Fix  > 0 such that (1 + )2 /R < 1. Since lim sup |f
(m)|1/m ≤ 1/R, we have

m |f
(m)| ≤ const. 1+ . Since lim sup |γn |1/n ≤ 1, we also have |γn | ≤ const.(1 + R n ) . Therefore  m+n ∞ ∞   1+ |γ n f
(m + n)| ≤ const. (1 + )n R n=0 n=0  m  n ∞  1+ (1 + )2 = const. R R n=0 ≤ const. The series

∞




n=0 |γ n f (m m m

(1 + )m . Rm

+ n)| thus converge absolutely, with sums bounded  by ∞ ∞
(m + n) z m f γ m=0 n=0 n

const.(1 + ) /R , implying that the power series

has radius of convergence at least R/(1 + ). Since  can be taken arbitrarily small, the radius of convergence is at least R.
We let Tϕ0 be the operator with domain H(D) that maps the function f in

 ∞
(m + n) z m , which is also in H(D) by γ f H(D) to the function ∞ m=0 n=0 n

the last lemma. Thus D(Tϕ0 ) is dense in H 2 . The kernel function in H 2 for the evaluation functional at the point λ of D will be denoted, as usual, by kλ : kλ (z) = 1/(1 − λz). Lemma 6.2. Tϕ0 kλ = ϕ(λ)kλ .

n Proof. In fact, since
kλ (n) = λ , the mth power series coefficient of Tϕ0 kλ equals ∞  n=0

m+n

γnλ

=λ

m

∞ 

n γ n λ = ϕ(λ)
kλ (m).

n=0

Since D(Tϕ0 ) is dense in H 2 , the adjoint (Tϕ0 )∗ is defined. Lemma 6.3. If D((Tϕ0 )∗ ) is nontrivial, then ϕ is in the Nevanlinna class N .




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Proof. Suppose the nonzero function f belongs to D((Tϕ0 )∗ ), and let g = (Tϕ0 )∗ f . Then, by Lemma 6.2, for λ in D, g(λ) =
g, kλ  =
f, Tϕ0 kλ  =
f, ϕ(λ)kλ  = ϕ(λ)f (λ). Hence ϕ = g/f , showing that ϕ is in N .




Lemma 6.4. If Tϕ0 is closable, then ϕ is in the Smirnov class N + . In that case Tϕ∗ is the closure of Tϕ0 . Proof. In fact, if Tϕ0 is closable then its adjoint (Tϕ0 )∗ is densely defined, and of course closed. By the proof of Lemma 6.3, (Tϕ0 )∗ = Tϕ , so ϕ is in N + by Lemma 5.2. Let ϕ be in N + . To prove Tϕ∗ is the closure of Tϕ0 , it suffices to prove the graphs G(Tϕ∗ ) and G(Tϕ0 ) have the same orthogonal complement. The argument here is again the same as in the proof of Lemma 6.3. The orthogonal complement of G(Tϕ∗ ) is {(−ϕf ) ⊕ f : f ∈ D(Tϕ )}. Suppose g ⊕ f is orthogonal to G(Tϕ0 ). Then for λ in D, by Lemma 6.2, 0 =
g ⊕ f, kλ ⊕ Tϕ0 kλ  = g(λ) + ϕ(λ)f (λ). Hence g = −ϕf .




With the preceding lemmas in mind, for ϕ in N + we now define Tϕ to be Tϕ∗ . The inclusion D(Tϕ ) ⊂ D(Tϕ ) was noted in Section 5. We note also that, for g in D(Tϕ ), we have Tϕ g = P+ (ϕg). In fact, for any f in D(Tϕ ),
Tϕ g, f  =
g, ϕf  =
ϕg, f  =
P+ (ϕg), f , from which the asserted equality follows. Having defined Tϕ and Tϕ for ϕ in N + , we can go slightly beyond the cases of analytic and conjugate-analytic symbols by defining the operator Tϕ+ϕ as: D(Tϕ+ϕ ) = D(Tϕ ), Tϕ+ϕ f = Tϕ f + Tϕ f for f in D(Tϕ ). We have Tϕ+ϕ f = P+ ((ϕ + ϕ)f ) for f in D(Tϕ ), according to remarks in the last paragraph. Hence Tϕ+ϕ is bounded if and only if Re ϕ is bounded. In any case the operator Tϕ+ϕ is symmetric. Helson in his paper [4] studied Tϕ for ϕ a nonconstant function in N + that is real valued on ∂D. In this case Tϕ is by itself symmetric. Examples of such functions ϕ are i(1 + z)/(1 − z), z/(1 − z)2 (the Koebe function), positive powers of the preceding functions, and the composite of such a function with a nonconstant inner function. Helson proved that Tϕ has finite deficiency indices if and only if ϕ is a rational function, and that all pairs of deficiency indices arise. If ϕ is a nonconstant function in N + that is real valued on ∂D, then ϕ = ϕ on ∂D, but Tϕ = Tϕ . For a Toeplitz operator whose symbol is not in L2 , the operator need not be determined by the behavior of its symbol on ∂D. We note for future use the following property. Proposition 6.5. If ϕ is in N + , ψ is in H ∞ , and f is in D(Tϕ ), then Tϕ Tψ f = Tϕψ f = Tψ Tϕ f .

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Proof. In fact, the domain D(Tϕ ), being a de Branges–Rovnyak space, is invariant under (bounded) conjugate-analytic Toeplitz operators (see, for example, [7], p. 11). Hence Tψ f is in D(Tϕ ). For g in D(Tϕ ),
Tϕ Tψ f, g =
Tψ f, ϕg =
f, ψϕg =
Tψϕ f, g, which shows that Tϕ Tψ f = Tϕψ f . Moreover, if g is in D(Tϕ ) then so is ψg, so that
Tψ Tϕ f, g =
Tϕ f, ψg =
f, ϕψg =
Tϕψ f, g implying that Tψ Tϕ f = Tψϕ f .




7. Truncations Let u be a nonconstant inner function. The S ∗ -invariant subspace H 2  uH 2 will be denoted, as is commonly done, by Ku2 . In the paper [8] the author studied truncations (i.e., compressions) of Toeplitz operators to Ku2 , concentrating on the case of bounded truncations. Here it will be shown how one can truncate Tϕ and Tϕ to Ku2 , for ϕ in N + . Some preliminaries are needed. They are presented in greater detail in [8]. The kernel function in Ku2 for the evaluation functional at the point λ of D will be denoted by kλu ; it is given by kλu (z) = (1 − u(λ)u(z))/(1 − λz). The space Ku2 carries a natural symmetry (i.e., an antiunitary involution) C, defined by (Cf )(z) = u(z)¯ zf (z) (z ∈ ∂D). The function Cf will be denoted, when convenient, by f. In particular,  kλu (z) = (u(z) − u(λ))/(z − λ). The compression of the shift operator S to Ku2 will be denoted by Su . Its adjoint Su∗ is the restriction of S ∗ to Ku2 . One has the relation I − Su Su∗ = k0u ⊗ k0u . The orthogonal projection on L2 with range Ku2 will be denoted by Pu . Like P+ , it can be extended to a map from L1 into H(D), continuous with respect to the weak topology of L1 and the topology of locally uniform convergence of H(D). In fact, as was the case with P+ , the projection Pu is an integral operator, its kernel being the reproducing kernel for Ku2 :    1 1 − u(eiθ )u(z) iθ dθ, f ∈ L2 , z ∈ D. (Pu f )(z) = f (e ) 2π ∂D 1 − e−iθ z For f in L1 , we define Ku f to be the function in H(D) given by the expression on the right side of the preceding equality. For ϕ a function in L2 , the truncated Toeplitz operator Aϕ with symbol ϕ is defined by Aϕ = Ku (ϕf ) with domain D(Aϕ ) = {f ∈ Ku2 : Ku f ∈ Ku2 }. The operator Aϕ is closed (by a standard argument) and densely defined (since Ku2 ∩ H ∞ ⊂ D(Aϕ )).

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In contrast to the Toeplitz case, the symbol of a truncated Toeplitz operator is highly nonunique. It is proved in [8] (Theorem 3.1) that Aϕ = 0 if and only if 2 ¯H (a result originating with K. Izuchi [5]). It is an open ϕ belongs to uH 2 + u question whether every bounded truncated Toeplitz operator possesses a bounded symbol. If ϕ is in L2 and Aϕ is bounded, then A∗ϕ = Aϕ , and Aϕ is C-symmetric: CAϕ C = A∗ϕ (Lemma 2.1 of [8], originally due to S. R. Garcia and M. Putinar [3]). For the remainder of this section we let ϕ denote a function in N + , with canonical representation ϕ = b/a. We define the truncation Aϕ of Tϕ to be the restriction Tϕ | D(Tϕ ) ∩ Ku2 . Proposition 7.1. Aϕ is closed and densely defined. It maps its domain into Ku2 . Proof. We know from Proposition 5.4 that D(Tϕ ) is the de Branges–Rovnyak space H(b). The proof of Proposition 7.1 relies on the description of H(b) given by Proposition 4.1. We first assume f is a function in H(b) ∩ Ku2 and show that then Tϕ f is in 2 Ku . This will prove Aϕ maps its domain into Ku2 . By Proposition 4.1 there is a function g in H 2 such that T¯b f = Ta¯ g. The proof of Proposition 5.4 shows that Tϕ f = g, so it only remains to show that g is in Ku2 , which amounts to showing that Tu¯ g = 0. We have Ta¯ Tu¯ g = Tu¯ Ta¯ g = Tu¯ T¯b f = T¯b Tu¯ f, and Tu¯ f = 0 because f is in Ku2 . Since a is an outer function, the operator Ta¯ has a trivial kernel, so Tu¯ g = 0, as desired. Next we show that D(Aϕ ) is dense in Ku2 . The inclusion Ta¯ H 2 ⊂ H(b) was noted in Section 4, so, since Ku2 is Ta¯ -invariant, it will suffice to prove that Ta¯ Ku2 is dense in Ku2 . But Ta¯ Ku2 = Aa¯ Ku2 , and A∗a¯ = Aa , the compression of Ta to Ku2 . The orthogonal complement of Aa¯ Ku2 in Ku2 is thus the kernel of Aa , which is trivial because a is outer (any function in the kernel must be in Ku2 and in uH 2 ). Thus D(Aϕ ) is dense in Ku2 . Finally, we show that Aϕ is closed. Let the sequence (fn )∞ 1 in D(Aϕ ) converge converge in norm to the in norm to the function f , and let the sequence (Aϕ fn )∞ 1 function g. Letting gn = Aϕ fn , we have, by Proposition 5.4 and its proof, T¯b fn = Ta¯ gn , which in the limit as n → ∞ gives T¯b f = Ta¯ g. Thus f is in H(b) ∩ Ku2 =
D(Aϕ ) and Tϕ f = g (again by Proposition 5.4 and its proof). Su´ arez in his paper [10] characterized the closed densely defined operators on Ku2 that commute with Su∗ . The characterization is somewhat indirect. The graph of such an operator is an invariant subspace of the backward shift on H 2 ⊕ H 2 . By the extension to multiple shifts of Beurling’s theorem, the orthogonal complement of the graph is determined by a two-by-two matrix inner function. Su´ arez’s basic theorem describes the inner functions that arise.

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Su´ arez showed that not all of his operators have the form Aϕ with ϕ in H 2 . Similarly, one can show that not all of them have the form Aϕ with ϕ in N + . I hope to return to these matters in a subsequent publication. Seubert in his paper [9] examined closed, densely defined, dissipative operators on Ku2 that commute with Su∗ . (By a dissipative operator is meant here an operator A on a Hilbert space H satisfying Im
Af, f H ≥ 0 for all f in D(A).) Such an operator is the infinitesimal generator of a contractive C0 -semigroup of operators on Ku2 that commutes with Su∗ . Using Su´arez’s characterization and the semigroup connection, Seubert proved that his operators are the operators Aϕ for functions ϕ in N + satisfying Im ϕ ≤ 0. As preparation he showed how to define Aϕ and Aϕ for the general function ϕ in N + , ground we are in the process of retrodding, along a modified course. We define Aϕ to be the transform of Aϕ under the symmetry C. Thus, D(Aϕ ) = CD(Aϕ ) = {f : Cf ∈ H(b) ∩ Ku2 }, and Aϕ f = CAϕ Cf for f in D(Aϕ ). Because Aϕ is closed and densely defined, so is Aϕ . We shall show that Aϕ and Aϕ are adjoints of each other. For this we recall the inclusion Aa¯ Ku2 ⊂ D(Aϕ ) (see the proof of Proposition 7.1) and define the restriction operators A0ϕ = Aϕ | Aa¯ Ku2 and A0ϕ = Aϕ | Aa Ku2 . Lemma 7.2. Aϕ = (A0ϕ )∗ , Aϕ = (A0ϕ )∗ . Proof. It will suffice to establish the first equality, because an application of the symmetry C reduces the second equality to the first one. We show first that (A0ϕ )∗ is an extension of Aϕ . For f in D(Aϕ ) and h in Ku2 ,
Aϕ f, Aa¯ h =
CAϕ Cf, Aa¯ h =
CAa¯ h, Aϕ Cf  =
Aa Ch, Aϕ Cf  =
Ch, Aa¯ Aϕ Cf  =
Ch, A¯b Cf  =
CA¯b Cf, h =
Ab f, h =
f, A¯b h =
f, A0ϕ Aa¯ h. (Note that by Proposition 6.5, Aa¯ Aϕ = A¯b |D(Aϕ ) and Aϕ Aa¯ = A¯b .) This establishes the inclusion Aϕ ⊂ (A0ϕ )∗ . To establish the reverse inclusion, suppose f is in D(A0ϕ )∗ , and let g = (A0ϕ )∗ f . Then for h in Ku2 ,
g, Aa¯ h =
(A0ϕ )∗ f, Aa¯ h =
f, A0ϕ Aa¯ h =
f, A¯b h. It follows that Ab f = Aa g, and hence that A¯b Cf = Aa¯ Cg. The last equality tells us that Cf is in H(b), and so is in D(Aϕ ), with Aϕ Cf = Cg. Therefore f is in
D(Aϕ ) and g = Aϕ f , which gives the inclusion (A0ϕ )∗ ⊂ Aϕ . Proposition 7.3. The operators Aϕ and Aϕ are adjoints of each other. They are the respective closures of A0ϕ and A0ϕ .

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Proof. Define the operator W on Ku2 ⊕ Ku2 by W (f ⊕ g) = g ⊕ −f . Thus, the graph of the adjoint of a densely defined operator on Ku2 is the image under W of the orthogonal complement of the graph of the operator. By Lemma 7.2, we have the decompositions Ku2 ⊕ Ku2 = G(A0ϕ ) ⊕ W G(Aϕ ) = G(Aϕ ) ⊕ W G(A0ϕ ), as well as the orthogonality of the subspaces G(A0ϕ ) and W G(A0ϕ ). Suppose the vector f ⊕ g in Ku2 ⊕ Ku2 is orthogonal to the span of the last two subspaces. Then f ⊕ g belongs to W G(Aϕ ) ∩ G(Aϕ ), so f is in D(Aϕ ) with g = Aϕ f , and g is in D(Aϕ ) with f = −Aϕ g. Therefore (see Proposition 6.5) Aa¯ g = Aa¯ Aϕ f = A¯b f, Aa f = −Aa Aϕ g = −Ab g. From the equality Aa f = −Ab g we conclude that the function af + bg is in uH 2 . g = Ab f, so the function a g − bf Applying C to the equality Aa¯ g = A¯b f , we get Aa  2 is also in uH . Therefore the function f(af + bg) + g(a g − bf) = a(f f + g g) g must also be in uH 1 . is in uH 1 . Since a is an outer function, the function f f + g On ∂D we have z |g(z)|2 . f (z)f(z) + g(z) g(z) = u(z)¯ z |f (z)|2 + u(z)¯ So for f f + g g to belong to uH 1 the function |f |2 + |g|2 must belong to H01 , and so must be the zero function. Hence f = g = 0, and we can conclude that the mutually orthogonal subspaces G(A0ϕ ) and W G(A0ϕ ) span Ku2 ⊕ Ku2 .

The equalities G(Aϕ ) = G(A0ϕ ) and G(Aϕ ) = G(A0ϕ ) are now immediate, as is the equality Aϕ = A∗ϕ .


Under what conditions is the operator Aϕ (or equivalently Aϕ ) bounded? An obvious sufficient condition for boundedness is that ϕ be bounded, but of course that is not a necessary condition. Because Aϕ is closed and densely defined, a necessary and sufficient condition is that D(Tϕ ) = Ku2 , in other words, that Ku2 ⊂ H(b). Proposition 7.4. The operator Aϕ is bounded if and only if dist(b, uH ∞ ) < 1. Proof. By a well-known range inclusion criterion of R. G. Douglas (see, for example, [7], p. 2), the inclusion Ku2 ⊂ H(b) is equivalent to the existence of a constant c ≥ 1 such that I − Tu Tu¯ ≤ c(I − Tb T¯b ). The last inequality holds if and only if, for all f in Ku2 , f 22 ≤ c
(I − Tb T¯b )f, f  = cf 22 − cA¯b f 22 , 

which can be rewritten A¯b f 22

≤

c−1 c



f 22 .

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Hence Aϕ is bounded if and only if A¯b  < 1, which by the (scalar) commutant lifting theorem happens if and only if dist(b, uH ∞ ) < 1.
The next result is a sufficient condition for boundedness. Proposition 7.5. If the functions a and u form a corona pair then Aϕ is bounded. Proof. Two functions in H ∞ are said to form a corona pair if they satisfy the hypothesis in L. Carleson’s corona theorem, which in the case of interest here says that |a| + |u| is bounded away from 0 in D. Assuming this is so, Carleson’s theorem says there exist functions ψ and χ in H ∞ such that ψa + χu = 1. Then Aψ Aa = I, so Aa is invertible, and hence so is Aa¯ . Since Aa¯ Ku2 ⊂ H(b), the equality D(Aϕ ) = Ku2 follows, and with it the boundedness of Aϕ .
To conclude this section I’ll present a pair of examples which expose an anomaly one encounters in trying to go beyond the analytic and coanalytic cases of unbounded truncated Toeplitz operators on Ku2 . The anomaly is related to the notion of angular derivative, a brief discussion of which is in order. Further details, including proofs, can be found, for example, in [7] and [8]. The function u is said to have an angular derivative in the sense of Carath´eodory (an ADC) at the point η of ∂D if u has a unimodular nontangential limit u(η) at η, and u has a nontangential limit u (η) at η (equivalently, (u(z) − u(η))/(z − η) → u (η) as z → η nontangentially). In that case u (η) = 0, and ηu (η)/u(η) > 0. A necessary and sufficient condition for u to have an ADC at η is that every function in Ku2 have a nontangential limit at η. When that happens, the evaluation functional on Ku2 induced by the nontangential limit at η is bounded; the corresponding kernel function is the function kηu (z) = (1 − u(η)u(z))/(1 − ηz). The kηu (z) = (u(z) − u(η))/(z − η) transform of kηu under the symmetry C is given by  (from which one sees that  kηu = ηu(η)kηu ). One has the relation Su kηu = ηkηu − ηk0u ([8], p. 496). Another necessary and sufficient condition for u to have an ADC at η is that u(η) exist as a nontangential limit and be of unit modulus, and that the function (u(z) − u(η))/(z − η) belong to H 2 . A closely related criterion due to P. R. Ahern and D. N. Clark [1] is needed for the first example below. Lemma 7.6. The function u has an ADC at the point η of ∂D if and only if the function k0u lies in the range of the operator Su − ηI. Proof. In fact, if u has an ADC at η then, by the relation mentioned at the end of the paragraph prior to the preceding one, (Su − ηI)kηu = −ηk0u . The condition is thus necessary. To prove the condition is sufficient, suppose there is a function f in Ku2 such that Su f − ηf = k0u . Since SKu2 is contained in the span of u and Ku2 , we then have (z − η)f (z) = 1 + cu(z) with c a constant. In this equality, the function on the left side has the nontangential limit 0 at η, so c = 0, and u has the nontangential limit −1/c at η. Thus |c| ≥ 1. But the strict inequality |c| > 1 is impossible, because

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the function (1 + cu(z))/(z − η) = f (z) is in L2 on ∂D. Hence |c| = 1, so that also |u(η)| = 1. We can now write f (z) = −u(η)(u(z) − u(η))/(z − η), and it follows by the criterion mentioned in the paragraph preceding the statement of the lemma that u has an ADC at η.
Example 1. We fix a point η in ∂D and let ϕ(z) = 1/(1 − ηz). On ∂D we have 1 1 + 1 − ηz 1 − η¯ z z 2 − ηz − η¯ = 1. = |1 − ηz|2

ϕ(z) + ϕ(z) =

Hence Aϕ+ϕ is the identity operator on Ku2 . It will be shown that D(Aϕ ) ∩ D(Aϕ ) is dense in Ku2 , in other words, the operator Aϕ + Aϕ is densely defined. We shall see, however, that the equality Aϕ + Aϕ = Aϕ+ϕ | D(Aϕ ) ∩ D(Aϕ ) fails when u has an ADC at η. For the case η = 1, the function a in the canonical representation ϕ = b/a is written down in Section 3. For general η one has   √  √  1+ 5 1+ 5 z−η −η √ √ a(z) = , b(z) = . 2 2 z − 3+ 5 z − 3+ 5 2

2

As the function b is bounded away from 0 in D, the functions a and b, in a trivial way, form a corona pair. This is known to imply that H(b) = Ta¯ H 2 (see [7], p. 62). We thus have D(Aϕ ) = H(b) ∩ Ku2 = Ta¯ H 2 ∩ Ku2 . But if h is a function in H 2 such that Ta¯ h is in Ku2 , then 0 = Tu¯ Ta¯ h = Ta¯ Tu¯ h, which because Ta¯ has a trivial kernel implies Tu¯ h = 0, i.e., h is in Ku2 . We thus have D(Aϕ ) = Aa¯ Ku2 , and so also D(Aϕ ) = Aa Ku2 . (In the present case, Aϕ = A0ϕ and Aϕ = A0ϕ .) Moreover, the function a(z)/(z − η) is an invertible function in H ∞ , so Aa equals Su − ηI times an invertible operator. Thus D(Aϕ ) = (Su − ηI)Ku2 , and so also D(Aϕ ) = (Su∗ − ηI)Ku2 . The argument that D(Aϕ ) ∩ D(Aϕ ) is dense in Ku2 will use the fact that the operator Su − ηI is injective, which can be seen as follows. If f is a function in Ku2 such that Su f = ηf , then f 2 = Su f 2 = Pu Sf 2 ≤ Sf 2 = f 2 . The inequality above is strict unless Sf is in Ku2 , which forces Su f to equal Sf , and hence forces f to be 0 (since S − ηI is obviously injective). Consider now two nonzero functions g and h in Ku2 satisfying Su g − ηg = Su∗ h − ηh.

(7.1)

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Applying Su to the preceding equality, and using the relation Su Su∗ = I −(k0u ⊗k0u ), we get (Su − ηI)Su g = Su Su∗ h − ηSu h

= h − h(0)k0u − ηSu h

= −η(Su h − ηh) − h(0)k0u .

It follows that

(Su − ηI)(Su g + ηh) = −h(0)k0u . (7.2) From this point the reasoning depends on whether or not u has an ADC at

η. Case 1. u does not have an ADC at η. In this case we conclude from (7.2) and Lemma 7.2 that h(0) = 0. Since (7.2) in the present case then reduces to (Su − ηI)(Su g + ηh) = 0, it follows by the injectivity of Su − ηI that Su g + ηh = 0, which with (7.1) tells us that also Su∗ h + ηg = 0. Consider now any function h in Ku2 satisfying h(0) = 0, and define the function g to be −ηSu∗ h. Then 0 = Su (Su∗ h + ηg) = h − h(0)k0u + ηSu g = η(Su g + ηh). The pair of equalities Su∗ h + ηg = 0, Su g + ηh = 0 combine to give Su g − ηg = Su∗ h − ηh. We can conclude that D(Aϕ ) ∩ D(Aϕ ) = {S ∗ h − ηh : h ∈ Ku2 , h(0) = 0}. Suppose the function h0 in Ku2 is orthogonal to D(Aϕ ) ∩ D(Aϕ ). Then for all h in Ku2 satisfying h(0) = 0, 0 =
h0 , Su∗ h − ηh =
(Su − ηI)h0 , h, implying that (Su − ηI)h0 is a scalar multiple of k0u . By Lemma 7.2 and the injectivity of Su − ηI, then, h0 = 0. We can conclude that D(Aϕ ) ∩ D(Aϕ ) is dense in Ku2 . Now let the function f belong to D(Aϕ ) ∩ D(Aϕ ), say f = (Su − ηI)g = (Su∗ − ηI)h, where h and g are as in the preceding discussion. Rewriting the last equalities as f = −η(I − ηSu )g = −η(I − ηSu∗ )h, we see that Aϕ f = −ηg, Aϕ f = −ηh. Hence Aϕ f + Aϕ f = −ηg − ηh. As noted above, Su g + ηh = 0, so we have in fact Aϕ f + Aϕ f = Su g − ηg = f . In the case at hand, where u does not have an ADC at η, the operator Aϕ+ϕ agrees with Aϕ + Aϕ on D(Aϕ ) ∩ D(Aϕ ), as expected. Case 2. u has an ADC at η. In this case (7.2), together with the injectivity of Su − ηI and the equality Su kηu − ηkηu = −ηk0u , gives Su g + ηh = ηh(0)kηu , and hence also Su∗ h+ ηg = ηh(0)kηu (by (7.1)).

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Consider now any function h in Ku2 , and define the function g by ηg = −Su∗ h + ηh(0)kηu .

(7.3)

It is asserted that then Su g − ηg = Su∗ h − ηh. In fact, applying Su to (7.3), we obtain ηSu g = −Su Su∗ h − nh(0)Su kηu

= −h + h(0)k0u − ηh(0)(ηkηu − ηk0u ) = −h + h(0)kηu ,

so that Su g +ηh = ηh(0)kηu = Su∗ h+ηg. The asserted equality Su g −ηg = Su∗ h−ηh, follows. We can conclude that D(Aϕ ) ∩ D(Aϕ ) = (Su∗ − ηI)Ku2 = D(Aϕ ). Conjugation of the last equality gives D(Aϕ ) ∩ D(Aϕ ) = D(Aϕ ). So in fact we have D(Aϕ ) = D(Aϕ ). In particular, D(Aϕ ) ∩ D(Aϕ ) is dense in Ku2 . Let f belong to D(Aϕ ) = D(Aϕ ), say f = Su g − ηg = Su∗ h − ηh. As in Case 1, we have Aϕ f + Aϕ f = −ηg − ηh. Combining this with the equality Su g + ηh = ηh(0)kηu , we get Aϕ f + Aϕ f = Su g − ηg − ηh(0)kηu = f − ηh(0)kηu .

(7.4)

We also have f (η) =
f, kηu  =
Su∗ h − ηh, kηu  =
h, Su kηu  − ηh(η)

=
h, ηkηu − ηk0u  − ηh(η) = −ηh(0).

Hence (7.4) can be written Aϕ f + Aϕ f = f − f (η)kηu . We see that Aϕ + Aϕ coincides in its domain with the operator I − (kηu ⊗ kηu ). In particular, it does not coincide there with Aϕ+ϕ . Example 2. This example is closely related to Example 1. We assume the function u has an ADC at the point η of ∂D, and we let ϕ(z) = −u(η)u(z)/(1 − ηz). The function a in the canonical representation ϕ = b/a is the same as in Example 1, while b here equals −u(η)u times the b from Example 1. Since b is divisible by u we have D(Tϕ ) = H(b) ⊃ Ku2 ([7], p. 10), so D(Aϕ ) = Ku2 , and also D(Aϕ ) = Ku2 . In fact, because u divides ϕ, Aϕ = Aϕ = 0.

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On the other hand, on ∂D, ϕ(z) + ϕ(z) = − =

u(η)u(z) u(η)u(z) − 1 − ηz 1 − η¯ z

1 − u(η)u(z) 1 − u(η)u(z) 1 1 + − − 1 − ηz 1 − η¯ z 1 − ηz 1 − η¯ z

= kηu (z) + kηu (z) − 1. It is proved in [8] (p. 502) that Aku +ku −1 = kηu ⊗ kηu . In the present case, therefore, η η Aϕ+ϕ = Aϕ + Aϕ .

8. A Problem Characterize the closed densely defined operators T on H 2 with the properties (i) D(T ) is S-invariant, and S ∗ T S = T | SD(T ), and (ii) if f is in D(T ) and f (0) = 0, then S ∗ f is in D(T ). Properties (i) and (ii) are possessed by Tϕ for ϕ in L2 , as well as by Tϕ and Tϕ for ϕ in N + . These Toeplitz operators are determined by their symbols. Is every closed densely defined operator on H 2 satisfying (i) and (ii) determined in some sense by a symbol? Can one characterize the infinite Toeplitz matrices that represent closed densely defined operators on H 2 satisfying (i) and (ii)?

References [1] P. R. Ahern and D. N. Clark, Radial limits and invariant subspaces. Amer. J. Math. 92 (1970), 332–342; MR 0262511 (41#7117). [2] L. de Branges and J. Rovnyak, Square Summable Power Series. Holt, Rinehart and Winston, New York–Toronto. Ont.–London, 1966; MR 0215065 (35#5909) [3] S. R. Garcia and M. Putinar, Complex symmetric operators and applications. Trans. Amer. Math. Soc. 358 (2006), 1285–1315; MR 2187654 (2006j;47036) [4] H. Helson, Large analytic functions. Linear Operators in Function Spaces (Timi¸soara, 1988), 209–216, Oper. Theory Adv. Appl. 43 Birkh¨ auser, Basel, 1990; MR 1090128 (92c:30038). [5] K. Izuchi, Personal communication. [6] N. K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Hardy, Hankel, and Toeplitz. Mathematical Sureys and Monographs, 92. American Mathematical Society, Providence, RI, 2002; MR 1864396 (2003i:47001a). [7] D. Sarason, Sub-Hardy Hilbert spaces in the unit disk. John Wiley & Sons, Inc., New York, 1994; MR 1289670 (96k:46039). [8] D. Sarason, Algebraic properties of truncated Toeplitz operators. Operators and Matrices 1 (2007), 491–526. [9] S. M. Seubert, Unbounded dissipative compressed Toeplitz operators. J. Math. Anal. Appl. 290 (2004), 132–146; MR 2032231 (2004i:47053).

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[10] D. Su´ arez, Closed commutants of the backward shift operator. Pacific J. Math. 179 (1997), 371–396; MR 1452540 (99a:47050). Donald Sarason Department of Mathematics University of California Berkeley, CA 94720-3840 U.S.A. e-mail: [email protected] Submitted: February 13, 2008

Integr. equ. oper. theory 61 (2008), 299–323 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/030299-25, published online April 17, 2008 DOI 10.1007/s00020-008-1590-9

Integral Equations and Operator Theory

Beurling-type Representation of Invariant Subspaces in Reproducing Kernel Hilbert Spaces Christoph Barbian Abstract. By Beurling’s theorem, the orthogonal projection onto an invariant subspace M of the Hardy space H 2 (D) on the unit disk can be represented as PM = Mφ Mφ∗ where φ is an inner multiplier of H 2 (D). This concept can be carried over to arbitrary Nevanlinna-Pick spaces but fails in more general settings. This paper introduces the notion of Beurling decomposable subspaces. An invariant subspace M of a reproducing kernel Hilbert space will be called Beurling decomposable if there exist (operator-valued) multipliers φ1 , φ2 such that PM = Mφ1 Mφ∗1 −Mφ2 Mφ∗2 and M = ran Mφ1 . We characterize the finitecodimensional and the finite-rank Beurling decomposable subspaces by means of their core function and core operator. As an application, we show that in many analytic Hilbert modules H, every finite-codimensional submodule M
can be written as M = ri=1 pi · H with suitable polynomials pi . Mathematics Subject Classification (2000). Primary 47B32; Secondary 47A13, 47A15. Keywords. Reproducing kernel Hilbert space, invariant subspace, core function, core operator, analytic Hilbert module.

1. Introduction In many areas of analysis, reproducing kernel Hilbert spaces and their multipliers play an important role. Probably the best understood reproducing kernel Hilbert spaces are the Hardy space H 2 (D) and the Bergman space L2a (D) over the open unit disk in C. The unilateral shift on H 2 (D), that is, the multiplication by the independent variable z, is one of the few operators whose lattice of invariant subspaces is completely known. By Beurling’s theorem, a subspace M of H 2 (D) is invariant

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for Mz precisely if it is of the form φ · H 2 (D) for some inner function φ, or equivalently, if the orthogonal projection on M can be represented as PM = Mφ Mφ∗ with some function φ ∈ H ∞ (D). Similar results in the more general setting of Nevanlinna-Pick (NP) spaces were proved by McCullough and Trent [28], and Greene, Richter and Sundberg [21]. Recall that a reproducing kernel Hilbert space over an arbitrary set D is by definition a Nevanlinna-Pick space if its reproducing kernel K has no zeroes and 1 if the function 1 − K is positive definite. Clearly, the Hardy space H 2 (D) and also the Arveson-Drury space H(Bd ) over the complex unit ball Bd are examples of NP spaces (see [8] for details). Theorem 1 (McCullough, Trent; Greene, Richter, Sundberg). Suppose that H is an NP space and that M ⊂ H is an invariant subspace (that is, M is closed and invariant for all multipliers). Then there exist a Hilbert space E and a multiplier φ : D → L(E, C) of H such that PM = Mφ Mφ∗

(and, in particular, M = ran Mφ )

(1.1)

holds. In the classical case H = H 2 (D), one can choose E = C, and the multiplier φ automatically is an inner function in H ∞ (D). Therefore, the above theorem contains Beurling’s invariant subspace theorem as a particular case. While the strategy of looking for factorizations of the form (1.1) is perfect for the representation of invariant subspaces in NP spaces, it fails in more general spaces such as the Bergman space L2a (D) over the complex disk or the multivariable Hardy and Bergman spaces on the open unit ball or polydisk in Cd . This becomes clear by the following result, which can be obtained by inspecting the proof of Theorem 1. Theorem 2. Suppose that H is a reproducing kernel Hilbert space over an arbitrary set D and that the reproducing kernel K of H has no zeroes. Suppose that M is an invariant subspace of H and let KM denote the reproducing kernel of M . Then PM can be factorized as in (1.1) if and only if the core function GM = KKM of M is positive definite. In this case, every factorization GM (z, w) = φ(z)φ(w)∗ (1) yields a multiplier φ : D → L(E, C) with PM = Mφ Mφ∗ . Note that the core function used in the preceding theorem is nothing but the Berezin transform of the orthogonal projection PM (cf. [12], [13]). Core functions were studied by Guo in [22], [23], and also appear as the root function in [32]. The above Beurling-type result for NP spaces then can be rephrased by saying that in NP spaces, every invariant subspace automatically has a positive definite core function. This behaviour essentially characterizes the class of NP spaces: if H is a reproducing kernel Hilbert space, which is normalized at some point z0 ∈ D in the sense that K(·, z0 ) ≡ 1, then the core function of the invariant subspace 1 Mz0 = {f ∈ H ; f (z0 ) = 0} is given by GMz0 = 1 − K . Hence, if all invariant

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subspaces of H have a positive core function, then H necessarily is a NevanlinnaPick space. So in non-NP spaces, we cannot expect all invariant subspaces to have a positive core function. As observed by Guo [22], the extreme case occurs in the Bergman space setting, where no non-trivial invariant subspace has a positive definite core function. Motivated by these observations, we consider in the present paper invariant subspaces M of reproducing kernel Hilbert spaces with the property that the orthogonal projection onto M allows a more general representation of the form PM = Mφ1 Mφ∗1 − Mφ2 Mφ∗2 with suitable multipliers φi : D → L(Ei , C) such that in addition M = ran Mφ1 holds. Such spaces will be called Beurling decomposable. From the definition, it is immediately clear that the class of Beurling decomposable subspaces contains all subspaces with a positive definite core function, and it is the aim of this paper to show that many non-NP spaces arising in applications have a rich supply of Beurling decomposable subspaces. To use the concept of the core function, we shall require in the sequel that the kernel of the underlying reproducing kernel Hilbert space H ⊂ CD has no zeroes. Furthermore, we shall always assume that the functions K(·, w), w ∈ D, are multipliers of H. Finally, we suppose that the inverse kernel admits a representation of the form 1 = β(z)β(w)∗ (1) − γ(z)γ(w)∗ (1) K(z, w) with suitable multipliers β : D → L(B, C) and γ : D → L(C, C). We shall see that Nevanlinna-Pick spaces as well as the standard reproducing kernel Hilbert spaces on bounded symmetric domains in Cd (which form our basic examples) fulfill these conditions. The first main result of this paper (Theorem 3.1) gives a characterization of Beurling decomposable subspaces by means of their core function. Theorem 3. A closed subspace M of H is Beurling decomposable if and only if its core function can be written as GM (z, w) = φ1 (z)φ1 (w)∗ (1) − φ2 (z)φ2 (w)∗ (1) with multipliers φi ∈ M(H ⊗ Ei , H). The non-trivial ’if’ part of the preceding theorem will be used to show (Theorem 4.4) that, for a large class of examples, all finite-codimensional invariant subspaces are Beurling decomposable. Theorem 4. Suppose that D ⊂ Cd is a bounded symmetric domain with rank r and characteristic multiplicities a, b. Let H = Hν be one of the standard reproducing kernel Hilbert spaces, with ν ≥ r−1 2 a + 1. Then all finite-codimensional invariant subspaces of H are Beurling decomposable.

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We shall see in Section 3 that in general not all invariant subspaces are Beurling decomposable. In fact, every Beurling decomposable subspace contains non-trivial multipliers and hence, well-known examples in [23] and [30] imply that even in very familiar spaces not all invariant subspaces are Beurling decomposable. In the particular case of the Bergman space on the unit disk, we give a complete characterization of the Beurling decomposable subspaces: We show (Theorem 5.1) that an invariant subspace M of L2a (D) is Beurling decomposable precisely if its extremal function is bounded. This implies in particular that L2a (D) contains infinite-codimensional invariant subspaces which are Beurling decomposable. Using the Beurling decomposability of finite-codimensional invariant subspaces, together with a technique going back to Douglas, Paulsen, Sah and Yan [16], which was extended by Guo [15], we prove that for a natural class of analytic reproducing kernel Hilbert spaces on domains D ⊂ Cd , the right essential spectrum σre (Mz ) of the multiplication tuple Mz = (Mz1 , . . . , Mzd ) coincides with the topological boundary of D (Proposition 4.5). As an application, we show that for the same class of reproducing kernel Hilbert spaces, every finite-codimensional invariant subspace is algebraically generated by finitely many polynomials (Corollary 4.6). These results apply in particular to the standard reproducing kernel Hilbert spaces Hν (with ν ≥ r−1 2 a + 1 and ν integer) on bounded symmetric domains. Theorem 5. Suppose that D ⊂ Cd is a bounded symmetric domain of rank r, with characteristic multiplicites a, b. Let H = Hν be one of the standard reproducing kernel Hilbert spaces Hν , with ν ≥ r−1 2 a + 1 and ν integer. Then the finite-codimensional invariant subspaces of H are exactly the closed subspaces M
r of the form i=1 pi · H, where r ∈ N and p = (p1 , . . . , pr ) is a tuple of polynomials with Z(p) ⊂ D. Furthermore, if M is a finite-codimensional invariant subspace of H, then the polynomials p1 , . . . , pr can always be chosen as a generating set of the ideal M ∩ C[z] ⊂ C[z]. Similar results are due to Axler and Bourdon [9] in the setting of Bergman spaces over certain domains in the complex plane, and to Putinar for Bergman spaces over strictly pseudoconvex domains with smooth boundary (cf. Theorem 8.3.1 in [18]). Our result also extends the well-known Ahern-Clark-type result of Douglas, Paulsen, Sah and Yan (Corollary 2.8 in [16]), which states that (for a more general class of function spaces H) every finite-codimensional subspace of H
r is of the form M = i=1 pi · H, where p = (p1 , . . . , pr ) is a tuple of polynomials with Z(p) ⊂ D. Different representation results for finite-codimensional invariant subspaces of Bergman spaces over certain domains in Cd can also be found in [9] and [11].

2. Preliminaries A Hilbert space H of complex-valued functions on an arbitrary set D is called a reproducing kernel Hilbert space if all evaluation functionals

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δz : H → C , f → f (z) (z ∈ D) are continuous. In this case there exists a unique function (the reproducing kernel of H) K : D × D → C such that the function K(·, z) belongs to H for all z ∈ D and satisfies f, K(·, z) = f (z) (f ∈ H). It is easy to see that K is a positive definite function in the sense that, for all finite sequences z1 , . . . , zn in D, the matrices [K(zi , zj )]i,j are positive semidefinite. It is a well-known fact (see [7] for more information) that, for every positive definite function F , one can construct a unique reproducing kernel Hilbert space F ⊂ CD whose reproducing kernel is given by F . We call F the reproducing kernel Hilbert space associated to F . We shall write F ≤ G to indicate that G − F is positive definite. In this way we obtain a partial ordering on the set of all positive definite functions on D. Suppose that F1 , F2 : D × D → C are positive definite functions. Then F1 and F2 are said to be disjoint if the only positive definite function F which satisfies F ≤ F1 and F ≤ F2 is F = 0. It can be shown (see [31] for details) that F1 and F2 are disjoint if and only if the associated reproducing kernel Hilbert spaces F1 and F2 have trivial intersection, that is, F1 ∩ F2 = {0}. The following lemma provides a useful tool to decide whether or not a given function f : D → C belongs to a reproducing kernel Hilbert space. Lemma 2.1. Let H ⊂ CD denote a reproducing kernel Hilbert space with reproducing kernel K. For a function f : D → C, the following assertions are equivalent: (i) f belongs to H. (ii) There exists a real number c ≥ 0 such that the function D × D → C , (z, w) → c2 K(z, w) − f (z)f (w) is positive definite. In this case, f is the minimum of all constants c satisfying (ii). A proof of this well-known result can be found in [14]. A Kolmogorov factorization of a positive definite function F is a pair (E, d) consisting of a Hilbert space E and a function d : D → L(E, C) such that  E = {d(w)∗ (1) ; w ∈ D} and F (z, w) = d(z)d(w)∗ (1) holds for all z, w ∈ D. Obviously, the reproducing kernel Hilbert space F associated to F and the mapping d : D → L(F , C) , z → δz , define a possible Kolmogorov factorization of F . If E is a Hilbert space and H is a reproducing kernel Hilbert space with kernel K, then HE will denote the Hilbert space of all functions f : D → E such that for every x ∈ E the function fx : D → C , fx (z) = f (z), x

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belongs to H and such that f 2 =



IEOT

fei 2 < ∞

i

holds for some (equivalently every) orthonormal basis (ei )i of E. One easily verifies that the above norm · on HE does not depend on the choice of the orthonormal basis. The space HE can also be thought of as the reproducing kernel Hilbert space with operator-valued kernel K · 1E . We refer to [14] for further treatment of vectorvalued reproducing kernel Hilbert spaces. It is quite standard to show that there exists a unique isometric isomorphism U : H ⊗ E → HE

with

U (f ⊗ x) = f · x (f ∈ H, x ∈ E)

between the Hilbertian tensor product H ⊗ E and HE . In the sequel, we will use this identification without further mentioning. Assume now that H is a reproducing kernel Hilbert space with kernel K and that E, E∗ are arbitrary Hilbert spaces. In this setting, a function φ : D → L(E, E∗ ) is called an L(E, E∗ )-valued multiplier of H if for every function f ∈ H ⊗ E, the pointwise product φ · f belongs to H ⊗ E∗ . The collection of all such multipliers will be denoted by M(H ⊗ E, H ⊗ E∗ ) and of course, we use the abbreviation M(H) = M(H, H). A standard application of the closed graph theorem shows that each φ ∈ M(H ⊗ E, H ⊗ E∗ ) defines a bounded linear operator Mφ : H ⊗ E → H ⊗ E∗ , f → φ · f. Obviously, the operator norm of L(H ⊗ E, H ⊗ E∗ ) induces a (semi-) norm · M on the space M(H ⊗ E, H ⊗ E∗ ), the so-called multiplier norm. It is not hard to see that, if · M happens to be a norm (which is the case if K(z, z) > 0 for all z ∈ D), then M(H ⊗ E, H ⊗ E∗ ) is a Banach space. It is a well-known fact that the functions K(·, w) (w ∈ D) are eigenfunctions for the adjoints of multiplication operators. More generally, if φ belongs to M(H ⊗ E, H ⊗ E∗ ), then the equality Mφ∗ (K(·, w)x) = K(·, w)(φ(w)∗ x) holds for all x ∈ E∗ and w ∈ D. For a multiplier φ ∈ M(H ⊗ E, H), we obtain the formula   Mφ Mφ∗ K(·, w) (z) = φ(z)φ(w)∗ (1) K(z, w) (z, w ∈ D) which will be frequently used in this paper. Lemma 2.2. Let H be a reproducing kernel Hilbert space with kernel K and let E, E∗ be arbitrary Hilbert spaces. For a function φ : D → L(E, E∗ ), the following are equivalent: (i) φ belongs to M(H ⊗ E, H ⊗ E∗ ). (ii) There exists a real number c ≥ 0 such that D × D → L(E∗ ) , (z, w) → K(z, w)(c2 − φ(z)φ(w)∗ ) is an operator-valued positive definite function. In this case Mφ is the minimum of all constants c satisfying (ii).

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Analogously to the scalar definition, a function F : D × D → L(E) is called positive definite if, for all finite sequences z1 , . . . , zn , the matrix [F (zi , zj )]i,j is a positive operator on E n . A more general form of this result treating the case of arbitrary vector-valued reproducing kernel Hilbert spaces and their multipliers can be found in [14]. Next we recall the concept of subordinate kernels which was introduced in [7] and refined in [10]. In this context, a kernel simply is a complex-valued function on D × D. A kernel is called positive, if it is a positive definite function. A kernel L is said to be hermitian if L(z, w) = L(w, z) holds for all z, w ∈ D. Now let K : D × D → C denote a positive kernel and let H be the associated reproducing kernel Hilbert space. A kernel L : D × D → C is said to be subordinate to K (L ≺ K) if there exists a (necessarily unique) operator T ∈ L(H) such that L(z, w) = T K(·, w), K(·, z) (z, w ∈ D). In this case, T is called the representing operator for L. We write S(K) for the set of all kernels that are subordinate to K. Note that a subordinate kernel is hermitian (positive) if and only if its representing operator is self-adjoint (positive). Furthermore, every hermitian kernel in S(K) can be written as a difference of two positive kernels in S(K), and S(K) is the linear span of its positive kernels. To prove this, observe that the analogous statements are true in L(H). If L ≺ K is a positive kernel, one may ask for the relation between the associated reproducing kernel Hilbert spaces. The following lemma answers this question. Lemma 2.3. Let K, L : D × D → C denote positive kernels and let H, L be the associated reproducing kernel Hilbert spaces. Then the following are equivalent: (i) L is subordinate to K. (ii) There exists a real number c ≥ 0 such that cK − L is a positive kernel. (iii) L is continuously embedded in H. (iv) L is a linear subspace of H. If in this case, T ∈ L(H) is the the (positive) representing operator of L, then 1 L = ran T 2 . Proof. For the sake of completeness, we include a proof of this well-known fact. Suppose that L is subordinate to K with representing operator T . Then we can choose c ≥ 0 such that c1H − T is a positive operator. Consequently, cK − L is a positive kernel. Now fix a function f ∈ L with f L = 1. By Lemma 2.1, the kernel cK(z, w) − f (z)f (w) = (cK(z, w) − L(z, w)) + (L(z, w) − f (z)f (w)) is positive, √ and another application of Lemma 2.1 yields that f belongs to H with f H √ ≤ c. Therefore, L is contained in H and the inclusion mapping has norm at most c. If L is contained in H and the inclusion mapping i : L → H is bounded, then it is easy to verify that i∗ K(·, w) = L(·, w)

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holds for all w ∈ D and therefore L is subordinate to K and is represented by the operator ii∗ ∈ L(H). This settles the equivalence of (i)–(iii). A simple application of the closed graph theorem furnishes the equivalence of (iii) and (iv). Now let T ∈ L(H) denote the (positive) representing operator for L. The identity 1 1 L(·, w), L(·, z) L = L(z, w) = T 2 K(·, w), T 2 K(·, z) H , valid for all z, w ∈ D, implies that there exists a unitary operator 1

1

α : L → ran T 2 with αL(·, w) = T 2 K(·, w). The calculation 1

T 2 αL(·, w), K(·, z) = = =

T K(·, w), K(·, z)

L(z, w) iL(·, w), K(·, z) (z, w ∈ D)

1 2

proves that i = T α. Finally, the observation that 1

1

1

1

i(L) = T 2 α(L) = T 2 (ran T 2 ) = ran T 2 completes the proof.




Of central importance for this paper are those positive kernels which can be factorized by multipliers. Lemma 2.4. Let K : D × D → C be a positive kernel and let H be the associated reproducing kernel Hilbert space. For a positive kernel G : D×D → C, the following assertions are equivalent: (i) G · K ∈ S(K). (ii) G · L ∈ S(K) for all L ∈ S(K). (iii) There exists a Hilbert space E and a multiplier φ ∈ M(H ⊗ E, H) such that G(z, w) = φ(z)φ(w)∗ (1) holds for all z, w ∈ D. In this case, if G denotes the reproducing kernel Hilbert space associated to G, then G is contained in M(H). Furthermore, the set of all positive kernels G satisfying the equivalent conditions above, is closed under pointwise addition and multiplication. Proof. By choosing a Kolmogorov factorization (E, φ) of G and using Lemma 2.3, the equivalence of (i) and (iii) becomes a reformulation of Lemma 2.2. Now suppose that (i) holds. Since every kernel S(K) can be written as a linear combination of positive kernels in S(K), it suffices to show that G · L ∈ S(K) holds for all positive L ∈ S(K). To this end, let c, c be positive constants such that cK − G · K and c K − L are positive. Then cc K − G · L = c (cK − G · K) + G · (c K − L) is positive definite as a sum and product of positive definite functions. Hence G · L belongs to S(K). The implication (ii) to (i) is obvious.

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We are now going to prove the inclusion G ⊂ M(H). Choose a positive number c such that cK − G · K is positive and let φ be a function in G with φ G = 1. Since by Lemma 2.1, the kernel K(z, w)(c − φ(z)φ(w)) = (cK(z, w) − K(z, w)G(z, w)) + K(z, w)(G(z, w) − φ(z)φ(w)) is positive, Lemma 2.2 ensures that φ is a multiplier of H. To prove the final assertion, fix two positive kernels G1 , G2 satisfying (i). Obviously (G1 + G2 ) · K = G1 · K + G2 · K belongs to S(K), since S(K) is a linear space. Now choose positive constants ci such that ci K − Gi · K are positive. Then c1 c2 K − G1 · G2 · K = c1 (c2 K − G2 · K) + G2 · (c1 K − G1 · K) is positive as well. Hence (G1 · G2 ) · K ∈ S(K).




3. Beurling decomposition of subspaces Throughout this section, let H ⊂ CD be a reproducing kernel Hilbert space with reproducing kernel K such that K has no zeroes. Furthermore, we suppose that the inverse kernel admits a representation of the form 1 = β(z)β(w)∗ (1) − γ(z)γ(w)∗ (1) (z, w ∈ D) (3.1) K(z, w) with multipliers β ∈ M(H ⊗ B, H) and γ ∈ M(H ⊗ C, H), where B, C are appropriate Hilbert spaces. Since the functions β(·)β(w)∗ (1) and γ(·)γ(w)∗ (1) are 1 complex-valued multipliers, the functions K(·,w) belong to M(H) for all w ∈ D. 1 In particular, since 1 = K(·,w) · K(·, w) for some w ∈ D, the space H contains the constant functions. Finally, we require that also the functions K(·, w) are multipliers. We will now discuss three classes of spaces which fulfill these requirements. Example 1. (a) Suppose that K is a Nevanlinna-Pick kernel. This means by definition that 1 is positive definite. Choose a K has no zeroes and that the kernel 1 − K 1 Kolmogorov factorization (C, γ) of 1 − K . Since the kernel D × D → L(C) , (z, w) → K(z, w)(1 − γ(z)γ(w)∗ ) = 1 is positive, Lemma 2.2 implies that γ is a multiplier with multiplier norm less or equal to 1. Since 1 = 1 − γ(z)γ(w)∗ (1) K(z, w) holds for all z, w ∈ D, the inverse kernel has a representation of the form 1 < 1 holds for all w ∈ D, we conclude that (3.1). Since γ(w) 2 = 1 − K(w,w) for w ∈ D, the function φw : D → C , φw (z) = γ(z)γ(w)∗ (1)

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belongs
to M(H), with multiplier norm strictly less than 1. Therefore the ∞ n series n=0 φw converges (absolutely) in M(H). On the other hand, the series converges pointwise to K(·, w). Consequently, the functions K(·, w) are multipliers for all w ∈ D. A simple argument shows that the class of kernels we consider is closed under pointwise multiplication. Hence products of Nevanlinna-Pick kernels belong to this class as well. (b) Assume that D is a bounded domain in Cd and that K is sesquianalytic on D × D, or equivalently, that H consists of holomorphic functions on D. Let us suppose further that the coordinate functions zi (1 ≤ i ≤ d)) are multipliers on H such that the Taylor spectrum of the commuting tuple Mz = (Mz1 , . . . , Mzd ) ∈ L(H)d is contained in D. Finally, we suppose that 1 K is defined and sesquianalytic on an open neighbourhood of D × D. In [10] (proof of Theorem 3.3) it is shown that every sesquianalytic kernel on a domain is subordinate to the reproducing kernel of some weighted Bergman 1 is sesquianalytic space. Since we can find a domain U ⊃ D such that K 1 on U × U , the hermitian kernel K can be written as a difference of two positive definite sesquianalytic kernels defined on U × U . Taking Kolmogorov factorizations of these positive kernels, we obtain functions β and γ which satisfy the identity (3.1) and, in addition, are analytic on U . The assumption on the spectrum of Mz guarantees that every operator-valued function which is analytic on a neighbourhood of D, belongs to M(H) (see for example [5] for a proof). Thus, the functions β, γ are in fact multipliers of H. Therefore a decomposition of the form (3.1) automatically exists in this situation. The condition that the functions K(·, w) are multipliers of H probably is not automatically fulfilled in this situation. The following example suggests that this condition depends in an intimate way on symmetry properties of D. (c) We now focus on reproducing kernel Hilbert spaces over bounded symmetric domains in Cd . To this end, we fix a Cartain domain in Cd of rank r and characteristic multiplicities a, b. Let us denote by h the Jordan triple determinant of D and let H = Hν be the reproducing kernel Hilbert space associated to the kernel K(z, w) = Kν (z, w) = h(z, w)−ν , where ν is in the Wallach set of D. It is well known that K has no zeroes and H contains the constant functions. It is shown in [17] that, under the additional hypothesis that ν ≥ r−1 2 a + 1, the inverse kernel admits a representation of the form (3.1). For ν in the continuous Wallach set (this means ν > r−1 2 a), the functions K(·, w) are multipliers for all w ∈ D. In fact, it is proved in [6] that the Taylor spectrum of the tuple Mz is D. Therefore, by the same argument as in the previous example, it suffices to show that K(·, w) is analytic on an open neighbourhood of D. To see this, fix w ∈ D and choose a real number 0 < ρ < 1 such that wρ ∈ D. By homogeneous expansion, it can easily be checked that K satisfies the equation K(z, w) = K(ρz, wρ ) for all

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z ∈ D. Obviously the right-hand side defines an analytic extension of K(·, w) on the set ρ1 D which is an open neighbourhood of D. Hence (at least for ν ≥ r−1 2 a + 1), the spaces Hν fulfill the conditions at the beginning of this section. Following [22] we define the core function and the core operator of a closed subspace of H. To this end, note first that every closed subspace M of H is a reproducing kernel Hilbert space itself. In the following, the reproducing kernel of M will be denoted by KM , and it is very easy to see that KM (z, w) = PM K(·, w), K(·, z)

holds for all z, w ∈ D. The quotient GM = KKM is called the core function of M . The representing operator of GM is called the core operator of GM , and denoted 1 by ∆M ∈ L(H) (note that by Lemma 2.4, GM = K · L ≺ K holds for all kernels L ≺ K and, in particular, for KM ). The rank of M is defined to be the rank of ∆M , that is, rank M = rank ∆M = dim ran ∆M . In many cases, the core operator can be expressed in a very concrete form. Example 2. 1 is a polynomial in z and w, (a) Suppose that D is an open set in Cd and that K  1 (z, w) = cα,β z α w β . K α,β

Assume further that the coordinate functions zi (1 ≤ i ≤ d) are multipliers of H. Let Mz denote the commuting tuple (Mz1 , . . . , Mzd ). Then  cα,β Mzα PM Mz∗ β ∆M = α,β

is the core operator of a given subspace M of H. Note that the identity GM + GM ⊥ = 1 holds for every closed subspace M of H, and let PC denote the orthogonal projection onto the one-dimensional subspace of all constant functions in H. Since the constant kernel 1 is represented by the operator 1 2 PC , we infer that ∆M + ∆M ⊥ = 1 2 PC . This observation and the above formula for ∆M show that the finite dimension of M or M ⊥ implies that both ∆M and ∆M ⊥ have finite rank. (b) Suppose that D is a bounded symmetric domain in Cd and adopt the notations of Example 1. In view of the Faraut-Koranyi formula  1 = (−ν)m Km (z, w) (z, w ∈ D) K(z, w) m (see [19] for details), we show that  ∆M = (−ν)m Km (LMz , RMz∗ )(PM ) m

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∗ (at least if ν ≥ r−1 2 a + 1). In the above expression, LMz and RMz denote the tuples of left and right multiplications with the operators Mzi and Mz∗i , respectively. Since the kernels Km are polynomials in z and w, the terms of the series are well defined. Moreover, Km is positive definite and hence

0 ≤ Km (LMz , RMz∗ )(PM ) ≤ Km (LMz , RMz∗ )(1H ). The convergence of the series above now follows directly by a result in [17], where it is shown that the series  |(−ν)m | Km (LMz , RMz∗ )(1H ) m

converges (for ν ≥

r−1 2 a

+ 1).

We now turn to the study of invariant subspaces of H. A closed subspace M of H will be called K-invariant if it is invariant under multiplication by all 1 . As usual, M is said to be invariant if φ · M ⊂ M for functions K(·, w) and K(·,w) all φ ∈ M(H). As indicated in the introduction, an invariant subspace M of H is called Beurling decomposable if there exist Hilbert spaces E1 , E2 and multipliers φ1 ∈ M(H ⊗ E1 , H), φ2 ∈ M(H ⊗ E2 , H) such that PM = Mφ1 Mφ∗1 − Mφ2 Mφ∗2

and

ran Mφ1 = M.

In this case, the pair (φ1 , φ2 ) is called a Beurling decomposition of M . Let M be a Beurling decomposable subspace of H. The obvious identity KM (z, w) = (φ1 (z)φ1 (w)∗ (1) − φ2 (z)φ2 (w)∗ (1))K(z, w)

(3.2)

valid for all z, w ∈ D, shows that the functions KM (·, w) are multipliers of H, since we supposed K(·, w) ∈ M(H) for all w ∈ D. Hence the set M ∩ M(H) is dense in M . An example given by Rudin (cf. [30], Theorem 4.1.1) shows that there exists an invariant subspace of the Hardy space H 2 (D2 ) over the bidisk which does not contain any nonzero multiplier φ ∈ M(H 2 (D2 )) = H ∞ (D2 ). So in general, we cannot expect all invariant subspaces to be Beurling decomposable. From (3.2), we see immediately that GM (z, w) = φ1 (z)φ1 (w)∗ (1) − φ2 (z)φ2 (w)∗ (1) holds for all z, w ∈ D. Equivalently, GM can be written as the difference of two positive kernels G1 , G2 satisfying K · Gi ≺ K for i = 1, 2. As we shall see in the following theorem, the existence of such a decomposition is basically sufficient for the Beurling decomposability of M . Theorem 3.1. Let M be a closed subspace of H which is K-invariant. Then M is Beurling decomposable if and only if there exist positive kernels G1 , G2 on D such that (i) GM = G1 − G2 (ii) K · Gi ≺ K for i = 1, 2.

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Furthermore, G1 and G2 can always be chosen disjoint. If G1 , G2 are disjoint, then any pair of Kolmogorov factorizations φ1 : D → L(E1 , C)

,

φ2 : D → L(E2 , C)

of G1 and G2 defines a Beurling decomposition of M . Proof. Suppose that M is Beurling decomposable. Then the above discussion proves the existence of positive kernels G1 , G2 satisfying conditions (i) and (ii). In order to prove the opposite direction, let us first point out that we may assume G1 , G2 to be disjoint. In fact, one can show that the set {G : D × D → C ; 0 ≤ G ≤ G1 , G2 } is inductively ordered (see [4] or [31] for details). Let Gmax be a maximal element in this set and write G1 = G1 − Gmax

G2 = G2 − Gmax .

and

By construction, G1 , G2 are disjoint positive kernels which satisfy condition (i). Since K · Gi ≺ K · Gi ≺ K holds for i = 1, 2, condition (ii) is fulfilled as well. Thus let us suppose that G1 and G2 are disjoint. Choose functions φ1 : D → L(E1 , C) ,

φ2 : D → L(E2 , C)

such that G1 (z, w) = φ1 (z)φ1 (w)∗ (1)

G2 (z, w) = φ2 (z)φ2 (w)∗ (1)

and

holds for all z, w ∈ D. Condition (ii) guarantees that φ1 , φ2 are in fact multipliers. It follows that   Mφ1 Mφ∗1 − Mφ2 Mφ∗2 K(·, w), K(·, z) = (G1 (z, w) − G2 (z, w))K(z, w) =

KM (z, w)

=

PM K(·, w), K(·, z) (z, w ∈ D),

and therefore that

Mφ1 Mφ∗1 − Mφ2 Mφ∗2 = PM . It remains to show that ran Mφ1 = M . To this end, we note that G1 , G2 belong 1 · K belongs to S(K). to S(K) by Lemma 2.4, since the constant kernel 1 = K Let ∆1 , ∆2 ∈ L(H) denote the (positive) representing operators for G1 , G2 . Since G1 , G2 are disjoint, the associated reproducing kernel Hilbert spaces G1 and G2 have trivial intersection. By Lemma 2.3 we obtain that 1

1

ran ∆12 ∩ ran ∆22 = {0} and hence that ran ∆1 ∩ ran ∆2 = {0}. Now it is an elementary exercise to verify that the ranges of ∆1 , ∆2 must necessarily be contained in the closure of the range of ∆M = ∆1 − ∆2 . Since all the functions 1 · KM (·, w) (w ∈ D) ∆M K(·, w) = GM (·, w) = K(·, w)

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are contained in M , it follows that ran ∆M ⊂ M and hence that ran ∆1 ⊂ ran ∆M ⊂ M. Therefore also the functions G1 (·, w) = ∆1 K(·, w) are contained in M for all w ∈ D. Using the K-invariance of M , we see that Mφ1 Mφ∗1 K(·, w) = G1 (·, w)K(·, w) ∈ M for every w ∈ D. Thus ran Mφ1 ⊂ M . The opposite inclusion is easier to prove. First, it is elementary to show and well known that for Hilbert spaces H1 , H2 , H and operators A1 ∈ L(H1 , H), A2 ∈ L(H2 , H) with A1 A∗1 ≥ A2 A∗2 , there exists a contraction C ∈ L(H1 , H2 ) with CA∗1 = A∗2 . In view of A1 A∗1 − A2 A∗2 = A1 (1H1 − C ∗ C)A∗1 , it is obvious that ran A1 A∗1 − A2 A∗2 ⊂ ran A1 . To prove that M ⊂ ran Mφ1 , it suffices to apply this remark with A1 = Mφ1 and A2 = Mφ2 .
Corollary 3.2. For every λ ∈ D, the invariant subspace Mλ = {f ∈ H ; f (λ) = 0} = {K(·, λ)}⊥ is Beurling decomposable. Proof. An easy calculation shows that K(z, λ)K(w, λ) GMλ (z, w) = 1 − K(λ, λ)K(z, w)     K(z, λ)K(w, λ) K(z, λ)K(w, λ) ∗ ∗ γ(z)γ(w) (1) − β(z)β(w) (1) = 1+ K(λ, λ) K(λ, λ) holds for all z, w ∈ D. Since the function K(·, λ) is a multiplier of H, this furnishes the desired decomposition of GMλ .
The spaces Mλ considered above have codimension one and form, in some sense, the simplest type of invariant subspaces of H. Now is natural to examine arbitrary subspaces of finite codimension. Proposition 3.3. If M ⊂ H is a finite-codimensional subspace of H which is Kinvariant, then the following assertions are equivalent: (i) M ⊥ ⊂ M(H). (ii) M is Beurling decomposable. Proof. Let M be Beurling decomposable. As we have observed earlier, KM (·, w) is a multiplier for every w ∈ D. As the functions K(·, w) are supposed to belong to M(H), the functions KM ⊥ (·, w) = K(·, w) − KM (·, w)

(w ∈ D)

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define multipliers as well. Thus M ⊥ , being the linear span of the KM ⊥ (·, w), is a subset of M(H). Suppose conversely that M ⊥ ⊂ M(H). Choose an orthonormal ⊥ basis (ui )m i=1 of M , and note that KM ⊥ (z, w) = PM ⊥ K(·, w), K(·, z) =

m 

ui (z)ui (w)

(z, w ∈ D).

i=1

As the functions ui are all multipliers, Lemma 2.4 yields K · KM ⊥ ∈ S(K). We define B(z, w) = β(z)β(w)∗ (1) and C(z, w) = γ(z)γ(w)∗ (1). As B and C are positive kernels with K · B, K · C ∈ S(K), another application of Lemma 2.4 proves that the decomposition KM ⊥ = (1 + KM ⊥ · C) − (KM ⊥ · B). K fulfills the hypotheses of Theorem 3.1. GM = 1 −




Later we shall see that in many cases of practical interest, condition (i) of the above proposition is automatically fulfilled for all finite-codimensional invariant subspaces. We conclude this section by giving a characterization of Beurling decomposability for finite-rank subspaces. Let M be a Beurling decomposable subspace. From the definitions, it is clear that all functions GM (·, w) = ∆M K(·, w) (w ∈ D) belong to M(H). Moreover, the range of the core operator ∆M consists of multipliers. In order to prove this, we choose G1 , G2 as in Theorem 3.1 and operators ∆1 , ∆2 ∈ L(H) representing G1 , G2 . Let G1 , G2 denote the associated kernel spaces and note that by Lemma 2.4 and Lemma 2.3, 1

ran ∆i ⊂ ran ∆i2 = Gi ⊂ M(H)

(i = 1, 2).

Hence ran ∆M ⊂ ran ∆1 + ran ∆2 ⊂ M(H). For finite-rank invariant subspaces M , the condition ran ∆M ⊂ M(H) is also sufficient for the Beurling decomposability of M . Proposition 3.4. Let M be a closed subspace of H which is K-invariant. Suppose that M has finite rank. Then M is Beurling decomposable if and only if ran ∆M is contained in M(H). In this case, for every decomposition GM = G1 − G2 with disjoint positive kernels G1 , G2 ∈ S(K), it follows that K · Gi ≺ K for i = 1, 2. In particular, there exist multipliers φ1 , . . . , φs , ψ1 , . . . , ψt ∈ ran ∆M (s + t = rank M ) such that s t   Mφi Mφ∗i − Mψj Mψ∗j PM = i=1

and M=

j=1 s  i=1

φi · H.

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Proof. Suppose that the inclusion ran ∆M ⊂ M(H) holds. Fix an arbitrary decomposition GM = G1 − G2 with disjoint positive kernels G1 , G2 ∈ S(K). Let ∆M = ∆1 − ∆2 denote the corresponding decomposition of ∆M . As seen in the proof of Theorem 3.1, the disjointness of G1 , G2 and the finite rank of ∆M imply that ran ∆1 ∩ ran ∆2 = {0} and ran ∆M = ran ∆1 + ran ∆2 . Since in particular ran ∆i ⊂ M(H), there exist multipliers φ1 , . . . , φs and ψ1 , . . . , ψt (s + t = rank M ) with s t   φi ⊗ φi and ∆2 = ψj ⊗ ψj . ∆1 = i=1

j=1

Since G1 (z, w) = ∆1 K(·, w), K(·, z) =

s 

φi (z)φi (w),

i=1


and analogously G2 (z, w) = tj=1 ψj (z)ψj (w), an application of Lemma 2.4 shows that K · Gi ∈ S(K) for i = 1, 2. Hence G1 and G2 are disjoint kernels satisfying the hypotheses of Theorem 3.1. But then the Beurling decomposability of M and all remaining assertions follow directly from Theorem 3.1.


4. Application to analytic Hilbert modules Throughout this section, we fix a bounded open set D ⊂ Cd and suppose that H ⊂ O(D) is an analytic Hilbert module in the sense of [15] having some additional properties which allow us to apply the results of the preceding section. To be more precise, we shall suppose that (A) H is a reproducing kernel Hilbert space containing the constant functions; (B) H is a C[z]-module, or equivalently, the coordinate functions zi (1 ≤ i ≤ d) are multipliers of H; (C) the polynomials are dense in H; (D) there are no points z ∈ Cd \D for which the mapping C[z] → C , p → p(z) extends to a continuous linear form on all of H. In the language of [15] this means that the set of virtual points of H coincides with D. In [15] a Hilbert space H ⊂ O(D) satisfying the above conditions is called an analytic Hilbert module. To be able to apply the results of Section 3 we require in addition that: 1 admits (E) the reproducing kernel K of H has no zeroes and the inverse kernel K a decomposition of the form (3.1) with suitable multipliers β, γ. (F) the Taylor spectrum σ(Mz ) of the tuple Mz = (Mz1 , . . . , Mzd ) ∈ L(H)d is contained in D; (G) for all z ∈ D, there exist open neighbourhoods U ⊂ D of z and V of D such that K|U×D admits a zero-free sesquianalytic extension to U × V .

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Although these conditions seem to be rather technical, they are general enough to cover in particular the standard reproducing kernel Hilbert spaces on bounded symmetric domains. Example 3. Suppose that D is a bounded symmetric domain with rank r and characteristic multiplicities a, b and that ν is in the continuous Wallach set of D, that is, ν > r−1 2 a. It is well known that the reproducing kernel Hilbert spaces Hν contain the polynomials as a dense subset. By a recent result of Arazy and Zhang [6], the coordinate functions are multipliers of Hν . For the special case that Hν is the Bergman space on D, it is shown in [24] that there are no virtual points outside D. But it is easy to see that the given proof remains valid for all ν > r−1 2 a. That condition (E) is satisfied (at least if ν ≥ r−1 a + 1) was observed earlier in Example 2 1. According to [6], the Taylor spectrum of Mz is D. To show that condition (G) is fulfilled, we fix z ∈ D and a positive number 0 < ρ < 1 such that zρ ∈ D. If Kν : D × D → C denotes the reproducing kernel of Hν , then the function 1 ζ ρD × D → C , (ζ, ω) → Kν ( , ρω) ρ ρ is a zero-free sesquianalytic extension of Kν |ρD×D . This can be seen by use of the Faraut-Koranyi expansion  Kν (z, w) = (ν)m Km (z, w) (z, w ∈ D), m

where the sum ranges over all signatures m of length r, the numbers (ν)m are the generalized Pochhammer symbols and the functions Km are the reproducing kernels of the homogeneous spaces Pm of the Peter-Weyl decomposition  Hν = Pm . m

Hence, the spaces Hν (for ν ≥

r−1 2 a

+ 1) satisfy the conditions (A)-(G).

We collect some consequences of our hypotheses. As mentioned earlier, every function φ ∈ O(D) automatically is a multiplier of H and the equality Mφ = φ(Mz ) holds, where the right-hand side is formed with the help of Taylor’s functional calculus. A proof of this fact can be found in [5]. When dealing with analytic Hilbert modules, there is a natural notion of submodules. A linear subspace M of H is called a submodule of H if it is closed in H and a submodule of H as a C[z]-module (in other words, a common invariant subspace of the tuple Mz ). Of course, this concept differs from the definition of invariant subspaces as given before. Obviously, every invariant subspace is a submodule, but the converse is not necessarily true. However, because of condition (F ) every finite-codimensional submodule M of H automatically is an O(D)-submodule of H and hence K-invariant by conditions (G) and (E). To see this, first note that by Theorem 2.2.5 in [15], the canonical mapping C[z]/(M ∩ C[z]) → H/M , [p] → [p]

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is an isomorphism of (finite dimensional) linear spaces and the inclusion σp (Mz , C[z]/(M ∩ C[z])) ⊂ D holds. Therefore we have σ(Mz , H/M ) = σ(Mz , C[z]/(M ∩ C[z])) = σp (Mz , C[z]/(M ∩ C[z])) ⊂ D and, by Lemma 2.2.3 in [18], we obtain σ(Mz |M ) ⊂ σ(Mz ) ∪ σ(Mz , H/M ) = D = σ(Mz ). It is a well-known property of the analytic functional calculus (see Lemma 2.5.8 in [18]) that in this case, M is invariant for φ(Mz ), whenever φ is analytic on an open neighbourhood of σ(Mz ). Before we proceed, we formulate the concept of ’higher order kernels’. Lemma 4.1. For every multiindex α ∈ Nd0 and every w ∈ D, there exists a unique (α) function Kw ∈ O(D) satisfying (α)

Dα f (w) = f, Kw

for all f ∈ H. If ((w1 , α1 ), . . . , (wm , αm )) are pairwise different, then the functions (α ) (α ) Kw11 , . . . , Kwmm are linearly independent in H. Proof. Since the inclusion mapping H → O(D) is continuous, the higher order point evaluation (α) : H → C , f → Dα f (w) δw defines a continuous linear functional for every α ∈ Nd0 and w ∈ D. Hence (α) (α) ∗ Kw = δw (1) is the unique function in H with (α)

Dα f (w) = f, Kw

for all functions f ∈ H. Let us observe that ∗

(α) (α) (α) (z) = δw (1), K(·, z) = 1, δw K(·, z) = (Dα K(·, z))(w) Kw (α)

for all z, w ∈ D and α ∈ Nd0 . It remains to show that the functions Kw belong to O(D). By assumption (G), there exist open neighbourhoods V of D and U ⊂ D of w such that K|U×D extends to a sesquianalytic function H : U × V → C. But then h : V˜ → O(U ) , z → H(·, z), defined on the set V˜ = {z ; z ∈ V }, is analytic as a function with values in the Fr´echet space O(U ). Since continuous linear maps preserve analyticity, it follows that the function V → C , z → (Dα H(·, z))(w) (α)

is analytic again and, as seen above, extends the function Kw .

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(α )

To see that the functions Kwi i (1 ≤ i ≤ m) are linearly independent, choose polynomials p1 , . . . , pm such that 1 if i=j . Dαi pj (wi ) = 0 else The observation that m m   αi (αi ) cj = ci D pj (wi ) = pj , ci K w

(1 ≤ j ≤ m) i i=1

i=1

holds for any choice of complex numbers c1 , . . . , cm , completes the proof.




The following definitions are, up to a slight
reformulation, taken from [15]. Let w ∈ D be arbitrary. For a polynomial p = α cα z α ∈ C[z] set  (p) (α) Kw = cα K w . α

Then (p)

= f, Kw



cα Dα f (w)

α

for f ∈ H, and the mapping (p) γw : C[z] → H , p → Kw

is antilinear and one-to-one by the preceding lemma. Let M be a submodule of H. Then −1 (M ⊥ ) ⊂ C[z] Mw = γw is a linear subspace and the enveloping space of M defined by Mwe = (γw (Mw ))⊥ ⊂ H is a submodule containing M . We refer to [15] for more details. For an arbitrary subspace N of H, we denote by Z(N ) the zero variety of N , that is, Z(N ) = {z ∈ D ; f (z) = 0 for all f ∈ N }. Now consider a finite-codimensional submodule M of H. Then the zero sets of the enveloping spaces Mwe have a very simple structure. More precisely, we observe that {w} if w ∈ Z(M ) e Z(Mw ) = ∅ else holds for all w ∈ D. To prove this, we suppose first that z ∈ Z(Mwe ). Then the function K(·, z) is contained in γw (Mw ) = γw (Mw ) since Mw has finite dimension (α) by hypothesis. Therefore K(·, z) is a linear combination of the elements Kw and e hence z = w. This proves the inclusion Z(Mw ) ⊂ {w}. For obvious reasons, we have Z(Mwe ) ⊂ Z(M ). So it remains to show that w ∈ Z(Mwe ) whenever w ∈ Z(M ). But w ∈ Z(M ) is equivalent to 1 ∈ Mw which implies K(·, w) ∈ γw (Mw ). Hence w ∈ Z(Mwe ).

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The following result completely describes the finite-codimensional submodules of H by means of the enveloping spaces Mwe and appears as Corollary 2.2.6 in [15] . Lemma 4.2. Suppose M is a finite-codimensional submodule of H. Then we have 1. Z(M ) is a finite subset of D. 2. M = w∈Z(M) Mwe .
3. dim M ⊥ = w∈Z(M) dim Mw . This enables us to prove that, for every finite-codimensional submodule of M , the orthogonal complement of M consists entirely of multipliers. Other variants of this result have been proved by Guo and Zheng (cf. [24], or Corollaries 2.5.3 and 2.5.4 in [15]). Proposition 4.3. Assume that M is a finite-codimensional submodule of H. Then the inclusions M ⊥ ⊂ O(D) ⊂ M(H) hold. Proof. Assume first that Z(M ) = {w} for some w ∈ D. By Lemma 4.2, we obtain (p) M = Mwe = (γw (Mw ))⊥ , and therefore M ⊥ = γw (Mw ). Since every Kw belongs to O(D) by Lemma 4.1, it follows that ran γw ⊂ O(D), . If Z(M ) is arbitrary, then for every w ∈ Z(M ), the subspace Mwe is a finite-codimensional submodule with Z(Mwe ) = {w}, and thus (Mwe )⊥ ⊂ O(D). Another application of Lemma 4.2 yields  (Mwe )⊥ ⊂ O(D). M⊥ = w∈Z(M)


The main result of this section can now be stated. Theorem 4.4. Suppose that M is a finite-codimensional submodule of H. Then M is Beurling decomposable. If in addition M has finite rank, then there exist multipliers φ1 , . . . , φs and ψ1 , . . . , ψt (s + t = rank M ) such that PM =

s 

Mφi Mφ∗i −

i=1

and M=

t 

Mψj Mψ∗j

j=1 s 

φi · H.

i=1

The functions φ1 , . . . , φs and ψ1 , . . . , ψt can be chosen in O(D). Proof. By Propositions 4.3 and 3.3, the space M is Beurling decomposable. Choose an orthonormal basis e1 , . . . , er of M ⊥ . Since KM ⊥ (z, w) =

r  i=1

ei (z)ei (w)

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holds for all z, w ∈ D, it is clear that the functions KM ⊥ (·, w) belong to O(D) 1 for all w ∈ D. Since by condition (G) the functions K(·,w) belong to O(D) for all w ∈ D, we infer that GM (·, w) = 1 −

1 · KM ⊥ (·, w) K(·, w)

belongs to O(D) for all w ∈ D. Now suppose in addition that M has finite rank. Then ran ∆M , being the linear span of the functions GM (·, w), is contained in O(D). By Proposition 3.4, there are multipliers φ1 , . . . , φs and ψ1 , . . . , ψt in ran ∆M allowing the claimed representations of PM and M .
As an application, we compute the right essential spectrum σre (Mz ) of the commuting tuple Mz . Recall that the right essential spectrum of a commuting tuple T ∈ L(H)d is the set of all λ ∈ Cd for which the last cohomology group in the Koszul complex of λ − T has infinite dimension. Equivalently, λ ∈ Cd is not in the right essential spectrum of T exactly if the row operator (T1 , . . . , Td ) ∈ L(H d , H) has finite-codimensional range. Proposition 4.5. Suppose that the inverse kernel σre (Mz ) = ∂D.

1 K

is a polynomial in z, w. Then

Proof. First of all, observe that σre (Mz ) ⊂ σ(Mz ) ⊂ D. We are now going to prove that σre (Mz ) ∩ D = ∅. To this end, fix λ ∈ D and let Mλ be the finite-codimensional submodule Mλ = {f ∈ H ; f (λ) = 0} = {K(·, λ)}⊥ . By Example 2, the submodule Mλ has finite rank, and Theorem 4.4 shows that there exist multipliers φ1 , . . . , φs ∈ O(D), such that Mλ =

s 

φi · H.

i=1

The row operator (Mφ1 , . . . , Mφs ) ∈ L(Hs , H) consequently has finite-codimensional range. This means that 0 is not in the right essential spectrum of the commuting tuple Mφ = (Mφ1 , . . . , Mφs ) ∈ L(H)s . By the spectral mapping theorem for the right essential spectrum (Corollary 2.6.9 in [18]), we have σre (Mφ ) = φ(σre (Mz )). Since φ(λ) = 0, it follows that λ ∈ / σre (Mz ). This proves that σre (Mz ) ⊂ ∂D. Suppose conversely that λ is in the boundary of D. Then λ is not a virtual point of H. As observed in [15], this is equivalent to the fact that the maximal

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ideal of C[z] at λ is dense in H, in other words d 

(λi − Mzi )H =

i=1

d  (λi − Mzi )C[z] = H. i=1

Assume now that λ ∈ / σre (Mz ). Then the space d 

(λi − Mzi )H ⊂ H

i=1

is closed and therefore equals H. Since the surjectivity spectrum is closed, there exists some r > 0 such that d 

(µi − Mzi )H = H

i=1

holds for all µ ∈ Cd with |µ − λ| < r. Hence there would have to be a point µ ∈ D
with 1 ∈ di=1 (µi − Mzi )H. This contradiction completes the proof.
1 We point out that the condition that K is a polynomial in z and w is fulfilled by many spaces arising in applications. In the case that H = Hν is one of the standard reproducing kernel Hilbert spaces over a bounded symmetric domain in Cd , it can easily be verified that the inverse kernel is a polynomial in z and w if and only if ν is an integer. As an application of Proposition 4.5, we prove the following supplement to the Ahern-Clark type result stated in [16] and [15] (Theorem 2.2.3). 1 Corollary 4.6. Suppose that K is a polynomial in z and w. Then the finitecodimensional submodules of H are exactly the closed subspaces M of the form
r M = i=1 pi · H, where p = (p1 , . . . , pr ) is a tuple of polynomials with Z(p) ⊂ D.

Proof. Suppose that M is a finite-codimensional submodule of H. By Theorem 2.2.3 in [15], the intersection M ∩ C[z] is a finite-codimensional ideal in C[z] with Z(I) ⊂ D and M
= I. Now we choose a generating set p = (p1 , . . . , pr ) of I and r claim that M = i=1 pi · H. Since M =I=

r  i=1

pi · C[z] =

r 

pi · H,

i=1

it suffices to show that the row operator (Mp1 , . . . , Mpr ) ∈ L(Hr , H) has closed range. But this is obvious, because Z(p) = Z(I) ⊂ D and σre (Mz ) = ∂D, and hence 0∈ / σre (Mp1 , . . . , Mpr ) = p(σre (Mz )).
The proof shows that the polynomials p1 , . . . , pr can be chosen as a generating set of the Ideal M ∩ C[z]. If in particular d = 1, then we can achieve that r = 1.

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Note also that, under the same hypotheses, Gleason’s problem can be solved in H. Recall that Gleason’s problem is, for a given function f ∈ H and λ ∈ D, to find functions g1 , . . . , gd ∈ H satisfying f (z) − f (λ) =

d 

(zi − λi )gi (z) (z ∈ D).

i=1

To solve Gleason’s problem, it is therefore sufficient to apply Corollary 4.6 to the submodule Mλ = {h ∈ H ; h(λ) = 0}.

5. Beurling decomposable subspaces of the Bergman space In this final section, we characterize the Beurling decomposable subspaces of the Bergman space L2a (D). Recall that, given an invariant subspace M of L2a (D), there exists a unique solution gM of the extremal problem sup{Re f (n) (0) ; f ∈ M, f ≤ 1}, known as the extremal function of M . It is easy to see that gM is always a unit vector in M  zM . For further details on the extremal function, the reader is referred to [25]. Theorem 5.1. Suppose that M is an invariant subspace of L2a (D). Then M is Beurling decomposable if and only if gM ∈ H ∞ (D). In this case, M = ran gM . Proof. We start with the following two observations: First, for every invariant subspace M of L2a (D), we have PMzM = (∆M )+ , where ∆M = (∆M )+ − (∆M )− is the canonical spectral decomposition of ∆M . In fact, this follows easily from Theorem 0.6 in [27]. Secondly, if an invariant subspace M of L2a (D) contains a nonzero bounded holomorphic function, then the index of M , defined as dim M  zM , equals one. This follows immediately from Theorem 1.1 in [20] and the fact that, for invariant subspaces M of L2a (D), the function D → N0 , λ → dim M  (z − λ)M is constant. Now suppose that M is Beurling decomposable. As we observed earlier, this implies that M ∩ H ∞ (D) is dense in M . By the preceding remarks, M has index one. Therefore, (∆M )+ = PMzM = gM ⊗ gM . Since M is Beurling decomposable, we have ran ∆M ⊂ H ∞ (D). Hence, gM ∈ ran(∆M )+ ⊂ ran ∆M ⊂ H ∞ (D). Conversely, if gM is bounded, then M has index one due to our initial observations. Hence (∆M )+ = gM ⊗ gM . Write G1 (z, w) = gM (z)gM (w) and let G2 be the positive kernel represented by (∆M )− . Then clearly GM = G1 −G2 is a disjoint decomposition of GM . Since K ·G1 ≺ K, we infer that also K ·G2 = K ·G1 −KM ≺ K. By Theorem 3.1, we conclude that M is Beurling decomposable. Since the chosen kernels G1 , G2 are disjoint, they induce a Beurling decomposition of M . This means in particular that M = ran gM .


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This result allows us to give examples of Beurling decomposable subspaces of infinite codimension. To this end, it suffices to exhibit an invariant subspace M with infinite zero set Z(M ) and bounded extremal function gM . In view of Corollary 2.4 in [2], every subspace of the form MA = {f ∈ L2a (D) ; f (an ) = 0 for all n ∈ N}, where A = (an )n is a Frostman sequence (see [26] for the definition and properties of Frostman sequences), certainly satisfies these conditions.

References [1] P. Ahern and D.N. Clark, Invariant subspaces and analytic continuations in several variables, J. Math. Mech. 19 (1969), 963-969 [2] A. Aleman and S. Richter, Single point extremal functions in Bergman-type spaces, Indiana Univ. Math. J. 51 (2002), 581-605 [3] A. Aleman, S. Richter and C. Sundberg, Beurling’s theorem for the Bergman space, Acta Math. 177 (1996), 275-310 [4] D. Alpay, Some remarks on reproducing kernel Krein spaces, Rocky Mt. J. Math. 21 (1991), 1189-1205 [5] C. Ambrozie and J. Eschmeier, A commutant lifting theorem on analytic polyhedra, Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 67 (2005), 83-108 [6] J. Arazy and G. Zhang, Homogeneous multiplication operators on bounded symmetric domains, J. Funct. Anal. 202 (2003), 44-66 [7] N. Aronszajn, Theory of reproducing kernels, Trans. Am. Math. Soc. 68 (1950), 337404 [8] W. Arveson, Subalgebras of C ∗ -algebras. III. Multivariable operator theory, Acta Math. 181 (1998), 159-228 [9] S. Axler and P. Bourdon, Finite-codimensional invariant subspaces of Bergman spaces, Trans. Amer. Math. Soc. 306 (1988), 805-817 [10] F. Beatrous Jr. and J. Burbea, Positive Definiteness and its Applications to Interpolation Problems for Holomorphic Functions, Trans. Am. Math. Soc. 284 (1984), 247-270 [11] H. Bercovici, A question on invariant subspaces of Bergman spaces, Proc. Amer. Math. Soc. 103 (1988), 759-760 [12] F.A. Berezin, Covariant and contravariant symbols of operators, Math. USSR Izv. 6 (1972), 1117-1151 [13] F.A. Berezin, Quantization, Math. USSR Izv. 8 (1974), 1109-1163 [14] J. Burbea and P. Masani, Banach and Hilbert spaces of vector-valued functions. Their general theory and applications to holomorphy, Research Notes in Mathematics 90, Boston-London-Melbourne, Pitman Advanced Publishing Program (1984) [15] X. Chen and K. Guo, Analytic Hilbert Modules, Chapman & Hall/CRC (2003) [16] R.G. Douglas, V.I. Paulsen, C.H. Sah and K. Yan, Algebraic reduction and rigidity for Hilbert modules, Amer. J. Math. 117 (1995), 75-92

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[17] M. Englis, Some Problems in Operator Theory on Bounded Symmetric Domains, Act. Appl. Math. 81 (2004), 51-71 [18] J. Eschmeier and M. Putinar, Spectral decompositions and analytic sheaves, London Mathematical Society Monographs, Clarendon Press, Oxford (1996) [19] J. Faraut and A. Koranyi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), 64-89 [20] J. Gleason, S. Richter and C. Sundberg, On the index of invariant subspaces in spaces of analytic functions of several complex variables, J. Reine Angew. Math. 587 (2005), 49-76 [21] D. Greene, S. Richter and C. Sundberg, The structure of inner multipiers on spaces with complete Nevanlinna Pick kernels, J. Funct. Anal. 194 (2002), 311-331 [22] K. Guo, Defect operators, defect functions and defect indices for analytic submodules, J. Funct. Anal. 213 (2000), 380-411 [23] K. Guo and R. Yang, The core function of submodules over the bidisk, Indiana Univ. Math. J. 53 (2004), 205-222 [24] K. Guo and D. Zheng, Invariant subspaces, quasi-invariant subspaces, and Hankel operators, J. Funct. Anal. 187 (2001), 308-342 [25] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces, Graduate Texts in Math. 199, Springer Verlag, New York (2000) [26] S.V Khrushch¨ev and S.A. Vinogradov, Inner functions and multipliers of Cauchy type integrals, Ark. Mat. 19 (1981), 23-42 [27] S. McCullough and S. Richter, Bergman-type reproducing kernels, contractive divisors, and dilations, J. Funct. Anal. 190 (2002), 447-480 [28] S. McCullough and T.T. Trent, Invariant Subspaces and Nevanlinna-Pick Kernels, J. Funct. Anal. 178 (2000), 226-249 [29] P. Quiggin, For which reproducing kernel Hilbert spaces is Pick’s theorem true?, Integral Equations Oper. Theory 16 (1993), 244-266 [30] W. Rudin, Function theory in polydiscs, Mathematics Lecture Notes Series. New York – Amsterdam, W.A. Benjamin, Inc.(1969) [31] L. Schwartz, Sous-espaces d’espaces vectoriels topologiques et noyaux associ´es. (Noyaux reproduisants.), J. Anal. Math. 13 (1964), 115-256; J. Funct. Anal. 178 (2000), 226-249 [32] R. Yang and K. Zhu, The root operator on invariant subspaces of the Bergman space, Illinois J. Math. 47 (2003), 1227-1242 Christoph Barbian Fachrichtung Mathematik Universit¨ at des Saarlandes Postfach 15 11 50 D-66041 Saarbr¨ ucken Germany e-mail: [email protected] Submitted: September 19, 2007

Integr. equ. oper. theory 61 (2008), 325–340 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/030325-16, published online June 20, 2008 DOI 10.1007/s00020-008-1592-7

Integral Equations and Operator Theory

On Uniform Exponential Stability of Exponentially Bounded Evolution Families C. Bu¸se, A.D.R. Choudary, S.S. Dragomir and M.S. Prajea Abstract. A result of Barbashin ([1], [15]) states that an exponentially bounded evolution family {U (t, s)}t≥s≥0 defined on a Banach space and satisfying some measurability conditions is uniformly exponentially stable if and only if for some 1 ≤ p < ∞, we have that:
t ||U (t, s)||p ds < ∞. sup t≥0

0

Actually the Barbashin result was formulated for non-autonomous differential equations in the framework of finite dimensional spaces. Here we replace the above ”uniform” condition be a ”strong” one. Among others we shall prove that the evolution family {U (t, s)}t≥s≥0 is uniformly exponentially stable if there exists a non-decreasing function φ : R+ → R+ with φ(r) > 0 for all r > 0 such that for each x∗ ∈ X ∗ , one has:
t sup φ(||U (t, s)∗ x∗ ||)ds < ∞. t≥0

0

In particular, the family U is uniformly exponentially stable if and only if for some 0 < p < ∞ and each x∗ ∈ X ∗ , the inequality
t sup ||U (t, s)∗ x∗ ||p ds < ∞ t≥0

0

is fulfilled. The latter result extends a similar one from the recent paper [4]. Related results for periodic evolution families are also obtained. Mathematics Subject Classification (2000). Primary 47D06, Secondary 35B35. Keywords. Operator semigroups, rearrangement function space, evolution families of bounded linear operators, uniform exponential stability.

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1. Introduction The uniform exponential stability of an evolution family U = {U (t, s) : t ≥ s} of bounded linear operators acting on a Banach space X has been intensively investigated and several important characterizations in terms of integral conditions with respect to the first variable are known. See [9], [10], [16], [17], [8], [19] and the references therein. See also [7], [6], for recent developments. In particular, a result like this pointed out that the family U is uniformly exponentially stable if the norm of each its trajectory (t → U (t, s)x, t ≥ s, x ∈ X) belongs to a Banach function space E having some natural properties. Next we remind some characterizations of uniform exponential stability in terms of integral conditions with respect to the second variable. Recall here Barbashin’s result: Let X(·) be the unique solution of the Cauchy Problem  U˙ (t) = A(t)U (t), t ≥ 0 (A(t), 0, I) U (0) = I. where A(t) and U (t) are quadratic matrixes, I is the identity matrix of the same order as A(t) and the map t → A(t) is continuous. If
t ||X(t)X −1 (ξ)||dξ < ∞, sup t≥0

0

then there exist the positive constants N and ν such that ||X(t)X −1 (s)|| ≤ N e−ν(t−s) for all t ≥ s ≥ 0. See [[1], Theorem 5.1]. As was noticed in [15], this can be reformulated in terms of evolution families as follows: Let U = {U (t, s) : t ≥ s ≥ 0} be an exponentially bounded evolution family of bounded linear operators acting on a Banach space X such that for each t > 0 the map s → ||U (t, s)|| : [0, t] → R is measurable. The following two statements are equivalent: (i) The family U is uniformly exponentially stable. (ii) U satisfies the uniform Barbashin condition, that is, 1/p 
t p sup ||U (t, s)|| ds < ∞. t≥0

0

for some 1 ≤ p < ∞. We remark that in this context U (t, s) needs not be an invertible operator. The following result of Rolewicz’s type for evolution families defined on the entire real line can be found in [5], Theorem 4.1: Let φ : R + → R + be a nondecreasing function such that φ(t) > 0 for all t > 0 and U = {U (t, s) : t ≥ s} be an exponentially bounded evolution family of bounded linear operators on X. We assume that the function s → ||U (t, s)|| : (−∞, t] → R+ ,

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On Uniform Exponential Stability

is measurable for all t ∈ R . If
sup t∈R

t

−∞

327

φ(||U (t, s)||)ds < ∞,

then the family U is uniformly exponentially stable. Under similar assumptions it is shown that a q-periodic evolution family U is uniformly exponentially stable if
∞ φ(||U (0, −t)||)dt < ∞. 0

See [5], Theorem 4.2, for details. All the discussion above refer to the case where the topology of B(X) is generated by the operatorial norm. In the reversible case the solution of the Cauchy Problem (−A(t)∗ , 0, I ∗ ) is [(X(·))∗ ]−1 and the elements of the evolution family associated to (−A(t)∗ )t∈R are defined by V (t, s) = [(X(t)X −1 (s))∗ ]−1 , t ≥ s. On the other hand it is clear that the finite dimensional autonomous system x(t) ˙ = Ax(t)

t∈R

(A)

is asymptotically stable or equivalently uniformly exponentially stable if and only if the system (−A∗ ) is uniformly exponentially unstable. At least formally, such result can be also stated in the non-autonomous case, so it is naturally to ask if the integral conditions
t sup t≥0

||U (t, s)∗ x∗ ||ds < ∞,

for all x∗ ∈ X ∗

0

imply the uniform exponential stability of the family U. Results of this type has been stated in [4]. This paper is an attempt to prove generalized Rolewicz conditions for stability to non-autonomous systems.

2. Notations and preliminary results Let X be a real or complex Banach space and X ∗ its dual space. By B(X) will denote the Banach algebra of all linear bounded operators acting on X. The norms on X, X ∗ and B(X) will be denoted by the symbol || · ||. Let R+ := [0, ∞) and J either of R or R+ . By ∆J will denote the set of all pairs (t, s) ∈ J × J with t ≥ s, and let ∆∗J := ∆J \ {(t, t) : t ∈ J}. By an evolution family of bounded linear operators acting on X we will mean a family U = {U (t, s) : (t, s) ∈ ∆J } ⊂ B(X) which verifies the following two conditions: 1. U (t, t) = I the identity of B(X)- for all t ∈ J; and 2. U (t, s)U (s, r) = U (t, r) for all t, s, r ∈ J with t ≥ s ≥ r.

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We make the convention that U (t, τ ) = 0 whenever t < τ. Let q > 0. We say that an evolution family is q-periodic if U (t, s) = U (t + q, s + q) for all (t, s) ∈ ∆J . An evolution family is called strongly continuous if for each x ∈ X the maps τ → U (τ, s)x : [s, t] → X and s → U (t, s)x : [s, t] → X are continuous for any pair (t, s) ∈ ∆J . An evolution family is exponentially bounded if there exist ω ∈ R and Mω ≥ 1 such that ||U (t, s)|| ≤ Mω eω(t−s) for all (t, s) ∈ ∆J .

(2.1)

If the evolution family U is exponentially bounded then we may choose a positive ω such that (2.1) holds. An evolution family is uniformly exponentially stable if there exists a negative ω such that the relation (2.1) is fulfilled. If an evolution family U satisfies the convolution condition U (t, s) = U (t − s, 0) for all (t, s) ∈ ∆J then the one parameter evolution family T := {U (t, 0), t ≥ 0} is a semigroup of operators on X. If T is strongly continuous then it is exponentially bounded, that is there exist a real numbers ω and Mω ≥ 1 such that ||T (t)|| ≤ Mω eωt for all t ≥ 0. The converse statement is not true but the exponentially bounded semigroups possess a certain type of measurability. Precisely we shall use the following well-known proposition whose proof may be found for example in [19]. Proposition 1. If a one parameter semigroup {T (t)}t≥0 is exponentially bounded then for each x ∈ X, the map t → ||T (t)x|| is measurable. Let {T (t)}t≥0 be a strongly continuous one parameter semigroup on a Banach space X and T∗ = {T (t)∗ }t≥0 the associated one parameter dual semigroup on X ∗ . It is known that the dual semigroup T∗ may be not strongly continuous but it is exponentially bounded because ||T (t)|| = ||T (t)∗ || for all t ≥ 0. Then for each x∗ ∈ X ∗ the map t → ||T (t)∗ x∗ || is measurable. In the next we give some details about Banach function spaces over J. Let (J, L, m) be the Lebesgue measure space and let M(J) be the linear space of all measurable functions f : J → R identifying the functions which are equal almost everywhere on J. Let ρ : M(J) → [0, ∞] with the following properties: (N1) ρ(f ) = 0 if and only if f = 0. (N2) ρ(af ) = |a|ρ(f ) for any real scalar a and any function f ∈ M(J) with ρ(f ) < ∞. (N3) ρ(f + g) ≤ ρ(f ) + ρ(g) for all f, g ∈ M(J). Let E = Eρ = E(J) be the set of all f ∈ M(J) for which ||f ||E := ρ(f ) < ∞. It is clear that (E, || · ||) is a normed linear space which has nondecreasing norm, that is (N4) If f ∈ E, g ∈ E and |f | ≤ |g| almost everywhere then ||f ||E ≤ ||g||E .

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329

Such a norm is called function norm. The associate function norm ρ : M(J) → [0, ∞] of the function norm ρ is given by
 ρ (g) := sup{ f (t)g(t)dt : f ∈ M(J), ρ(f ) ≤ 1} J

and the associate function space is E  (J) = {g ∈ M : ρ (g) < ∞}. In particular, we have the H¨ older inequality, that is:   
    f (t)g(t)dt ≤ ||f ||E(J) ||g||E
(J) .     J

When (E, || · ||) is complete we say that it is a Banach function space over J. As is usually by χA (·) shall denote the characteristic function of the Lebesgue measurable set A ⊂ J and assume that each Banach function space over J contains the characteristic functions of the finite subintervals of J. Two positive functions defined on J are called equal-measurable if for each t > 0 the sets {f > t} and {g > t} have the same measure. A Banach function space E(J) is called rearrangement invariant if its function norm is rearrangement invariant, i.e. if ρ(f ) = ρ(g) for every two equal-measurable functions in E(J). For more details about the function spaces and the rearrangement invariant Banach spaces we refer to [13], [2]. We recall some well-known facts about Orlicz spaces as Banach function space over J. For more details, see [11], [12], [14], [2] and the references therein. Let Φ : [0, ∞) → [0, ∞] be a convex, non-decreasing function such that Φ(0) = Φ(0+) = 0, and Φ is not identically with 0 or with ∞ on (0, ∞). We denote by LΦ (R+ ) the set of all functions f for which there exists a positive real scalar k such ∞ that 0 Φ(k|f (t)|)dt < ∞. It is easily to see that LΦ becomes linear space with the usual operations. The Luxemburg norm of a function f ∈ LΦ (R+ ) is defined by
∞

ρΦ (f ) := inf{k > 0 :

0

Φ(k −1 |f (t)|)dt ≤ 1}.

Similar definitions may be given for Orlicz spaces over R. Every Orlicz space is a rearrangement invariant function space. Its dual space is a rearrangement invariant function space as well. The dual space E  of an Orlicz space LΦ need not be an Orlicz space, but if the function Φ satisfies the ∆2 - condition, that is there exists a positive constant C such that Φ(2t) ≤ CΦ(t) for all t ≥ 0,





then the dual space (LΦ ) (J) is also an Orlicz space over J. Moreover, (LΦ (J))
can be identified by LΦ (J), where


Φ (t) := sup(ts − Φ(s)) s∈J

is the Legendre transform of Φ.

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3. Uniform Boundedness Theorem 1. Let E(J) be a rearrangement invariant Banach function space over J and let U = {U (t, s) : (t, s) ∈ ∆J } be an evolution family such that the functions χ[s,t] (·)||U (·, s)x|| and χ[s,t] (·)||U (t, ·)∗ x∗ || are measurable for each pair (t, s) ∈ ∆J each x ∈ X and each x∗ ∈ X ∗ . The following two statements are equivalent: (i) The evolution family U is uniformly bounded, that is sup (t,s)∈∆J

||U (t, s)|| = MJ < ∞.

(3.1)

(ii) There exist positive constants ME and ME
such that   χ[s,t](·) ||U (·, s)x|| E(J)   sup ≤ ME ||x||, ∀x ∈ X χ[s,t] (·) (t,s)∈∆∗ E(J)

J

and sup

(t,s)∈∆∗ J

  χ[s,t] (·)||U (t, ·)∗ x∗ ||
E (J)   ≤ ME
||x||, χ[s,t] (·)


∀x∗ ∈ X ∗ .

E (J)

Proof. The implication (i)⇒(ii) is an obvious consequence of the fact that ||U (t, s)|| = ||U (t, s)∗ ||. We can choose ME = ME
= MJ in order to finish the proof. The our next goal is to prove the implication (ii)⇒(i). Let x ∈ X, x∗ ∈ X ∗ and (t, s) ∈ ∆∗J . Using the H¨older inequality we get:
t ∗ (t − s)| x , U (t, s)x| = | x∗ , U (t, τ )U (τ, s)x|dτ s
≤ χ[s,t] (τ )||U (t, τ )∗ x∗ ||χ[s,t] (τ )||U (τ, s)x||dτ J     ≤ χ[s,t] (·)||U (t, ·)∗ x∗ ||E
(J) χ[s,t] (·)||U (·, τ )x||E(J)     χ[s,t] (·)||U (·, τ )x|| χ[s,t] (·)||U (t, ·)∗ x∗ ||
E(J) E (J)     = (t − s) · χ[s,t] (·) χ[s,t] (·)
E(J) E (J) ≤ (t − s)M M  ||x||||x∗ ||, where the identity

    χ[s,t] (·)
· χ[s,t] (·) = t − s, E (J) E(J)

which is fulfilled in rearrangement invariant function space, was used. We can conclude that sup ||U (t, s)|| ≤ ME ME
< ∞.
(t,s)∈∆J

Remark 1. The assumption that E(J) is a rearrangement invariant Banach function space over J, in the Theorem 1 above, may be replaced with the more general

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assumption that the dual pair (E(J), E  (J)) of Banach function spaces has the property that for each (t, s) ∈ ∆J the following inequality is fulfilled:     χ[s,t] (·)
· χ[s,t]  ≤ (t − s). (3.2) E (J) E(J) A concrete example for such a situation is described as follows. Let 1 < p < ∞ and let E1,p be the set of all (classes) of complex-valued measurable functions f defined almost everywhere on R+ for which 1/p 
t+1 |f (τ )|p dτ < ∞. ||f ||1,p := sup t≥0

t

It is well-known that E1,p endowed with || · ||1,p is a Banach function space whose dual space is E1,q with 1/p + 1/q = 1. Moreover after a simple calculus we obtain:  1 (t − s) p if t − s < 1 ||χ[s,t] (·)||(1,p) = 1, if t − s ≥ 1. Now it is clear that the space E1,p is not rearrangement invariant while that the condition (3.2) is fulfilled. Thus the following result is stated. If an exponentially bounded evolution family U = {U (t, s) : t ≥ s ≥ 0} satisfies the measurability conditions from Theorem 1 and if for each x ∈ X and each x∗ ∈ X ∗ the two inequalities
τ +1 χ[s,t] (ρ)||U (ρ, s)x||p dρ < ∞ sup sup t≥s≥0 τ ≥0

and

τ




τ +1

sup sup

t≥s≥0 τ ≥0

τ

χ[s,t] (ρ)||U (t, ρ)∗ x∗ ||q dρ < ∞

hold true, then the family U is uniformly bounded. In the case of semigroups Theorem 1 yields the following result: Corollary 1. Let T = {T (t)}t≥0 be an exponentially bounded one parameter semigroup on a Banach space X. The following two statements are equivalent: (i) The semigroup T is uniformly bounded, that is sup ||T (t)|| = M < ∞; t≥0

(ii) There exist a rearrangement invariant space E(J) and positive constants ME and ME
such that   χ[0,t] (·)||T (·)x|| E(J)   sup ≤ M ||x||, ∀x ∈ X (3.3) χ[0,t] (·) t>0 E(J)

and

  χ[0,t] ||T (·)∗ x∗ ||
E (J)   sup ≤ ME
||x∗ ||,   χ (·) t>0 [0,t]
E (J)

∀x∗ ∈ X ∗ .

(3.4)

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Proof. The measurability of the maps τ → ||T (τ )x|| and τ → ||T (τ )∗ x∗ || follows from Proposition 1. This makes possible to apply Theorem 1.
It is worth to notice that the implication (ii) ⇒ (i) in Corollary 1 fails if one of the relations (3.3) and (3.4) is dropped. Here is an example adapted from [18]. Let X = L2 (R) and g : R → R+ be the function given by g(s) = (1 + |s|)1/4 . For each t ≥ 0 and each f ∈ X let us consider the bounded linear operator T (t) defined by: g(t + s) f (t + s), s ∈ R, t ≥ 0. (T (t)f )(s) = g(s) The family T = {T (t)}t≥0 is a strongly continuous semigroup of bounded linear operators on X and ||T (t)|| = (1 + t)1/4 , so that T is unbounded. However, the semigroup T verifies the relation (3.3) for E(J) := L2 (J).

4. Uniform exponential stability In this section we use only Banach function space E(J) whose dual space E  (J) verifies the condition   lim χ[0,t] (·)E
(J) = ∞. t→∞

In order to prove the first result of this section we need a preparation: Lemma 1. Let U = {U (t, s), (t, s) ∈ ∆J } be an exponentially bounded evolution family of bounded linear operators on a Banach space X. If there exists a function g : R+ → R+ such that inf g(t) < 1 and ||U (t, s)|| ≤ g(t − s), for all t ≥ s ∈ J,

t>0

then the family U is uniformly exponentially stable. For a proof of the above lemma we refer to [[3], Lemma 4]. Theorem 2. Let U = {U (t, s) : (t, s) ∈ ∆J } be an exponentially bounded evolution family on a Banach space X and E(J) be a rearrangement invariant Banach function space over J, such that   lim χ[0,r] (·)E
(J) = ∞. r→∞

The following statements are equivalent: (i) The family U is uniformly exponentially stable. (ii) There exists a positive constant M  (J) such that for each (s, t) ∈ ∆J and each x∗ ∈ X ∗ the map χ[s,t] (·)||U (t, ·)∗ x∗ || defines an element of the space E  (J) and   (4.1) sup χ[s,t] (·)||U (t, ·)∗ x∗ ||E
(J) ≤ M  (J)||x∗ ||. (t,s)∈∆J

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Proof. (i)⇒(ii). Let N > 0 and ν > 0 such that ||U (t, s)∗ x∗ || ≤ N e−ν(t−s) ||x∗ || ∀(t, s) ∈ ∆J ,

∀x∗ ∈ X ∗ .

The dual space E  (J) is rearrangement invariant function space as well. Then it contains the set L1 (J) ∩ L∞ (J). In particular, the function χ[0,∞) (·)e−ν· belongs to E  (J). A simple calculus points out that the functions χ[s,t] (·)e−ν(t−·) and χ[s,t] (t − ·)χ[0,∞) (·)e−ν· are equal-measurable in E  (J). Then for each (s, t) ∈ ∆J we have successively:     ∗  −ν(t−·)  χ[s,t] (·)||U (t, ·)∗ x∗ ||
≤ N ||x || (·)e χ 
[s,t] E (J) E (J)   ∗  = N ||x || χ[s,t] (t − ·)χ[0,∞) (·)e−ν· E
(J)   ≤ N ||x∗ || χ[0,∞) (·)e−ν· E
(J) . The proof of the implication (ii) ⇒ (i) will be divided in two steps. Step 1. Let s ≥ 0 and x∗ ∈ X ∗ be fixed, t ≥ s + 1 and let ω > 0 and Mω ≥ 1 such that (2.1) is fulfilled. For each τ ∈ [s, t] one has: e−ω χ[s,s+1] (τ )| x∗ , U (t, s)x| ≤

e−ω(τ −s) | U (t, τ )∗ x∗ , U (τ, s)x|

≤

e−ω(τ −s) ||U (τ, s)||||x||||U (t, τ )∗ x∗ ||

≤

Mω ||x||||U (t, τ )∗ x∗ ||.

As consequence, for each u ∈ J, it follows: e−ω χ[s,s+1] (u)| x∗ , U (t, s)x| ≤ Mω χ[s,t] (u)||U (t, u)∗ x∗ ||. Passing to the norms into E  (J) and using the assumptions, we get:   e−ω |χ[0,1] (·)|E
(J) | x∗ , U (t, s)x| = e−ω χ[s,s+1] (·)E
(J) | x∗ , U (t, s)x|   ≤ Mω χ[s,t] (·)||U (t, ·)∗ x∗ ||E
(J) ≤ Mω M  (J)||x∗ ||. We can conclude that the evolution family is bounded. Step 2. Let MJ := sup ||U (t, s)|| < ∞, u ∈ J and (s, t) ∈ ∆J . Successively (t,s)∈∆J

one has: χ[s,t] (u)| x∗ , U (t, s)x|

χ[s,t] (u)||U (t, u)∗ x∗ ||||U (t, s)x|| MJ ||x||χ[s,t] (u)||U (t, u)∗ x∗ ||.

≤ ≤

Passing again to the norms obtain:   χ[s,t] (·)
| x∗ , U (t, s)x| E (J)

≤

  MJ ||x|| χ[s,t] (·)U (t, ·)∗ x∗ ||E
(J)

≤

MJ M  (J)||x||||x∗ ||.

Finally we obtain the estimation: | x∗ U (t, s)x| ≤

MJ (M  (J)||x||||x∗ || + 1)   . 1 + χ[s,t] (·)E
(J)

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Now we can use the Lemma 1 above, to finish the proof.

Sufficiency of the Theorem 2 above, can be stated also for Banach spaces E(J) with the property that for each s ∈ J the map   h → χ[s,s+h] (·)E
(J) : R+ → R+ is nondecreasing. The following example describes this situation. Let a > 0, 1 < p < ∞ and 1 < q < ∞ such that 1/p + 1/q = 1. With E1,p,a shall denote the set of all (classes) of complex-valued measurable functions defined almost everywhere on R+ such that 1/p 
t+1 −pas p e |f (s)| ds < ∞. ||f ||1,p,a := sup t≥0

t

Then (E1,p,a ) = E1,q,−a . With these notations the next result reads as follows: Proposition 2. Any exponentially bounded evolution family U = {U (t, s)}t≥s≥0 which verifies the inequality
τ +1 sup sup eqaρ χ[s,t] (ρ)||U (t, ρ)∗ x∗ ||q dρ < ∞ for all x∗ ∈ X ∗ t≥s≥0 τ ≥0

τ

is uniformly exponentially stable. Corollary 2. Let T = {T (t)}t≥0 be an exponentially bounded semigroup of linear operators on a Banach space X and let E(R+ ) be a rearrangement invariant Banach function space over R+ whose dual space has the property that   lim χ[0,t] (·)E
(R ) = ∞. t→∞

+

The following two statements are equivalent: 1. The semigroup T is uniformly exponentially stable. 2. There exists a positive constant M  such that

||T (·)∗ x∗ || E
(R+ ) ≤ M  ||x∗ ||,

for all x∗ ∈ X ∗ .

Proof. The measurability of the map ||T (·)∗ x∗ || follows by the Proposition 1 above. For each positive s and every t ≥ s, one has:     χ[s,t] (·)||T (t − ·)∗ x∗ ||
≤ χ[0,t] (·)||T (t − ·)∗ x∗ ||E
(R+ ) E (R+ )   = χ[0,t] (t − ·)||T (·)∗ x∗ ||
≤

||T (·)∗ x∗ || E
(R+ ) .

E (R+ )

Therefore the map χ[s,t] (·)||T (t − ·)∗ x∗ || defines an element of the space E  (R+ ). Now we can apply Theorem 2 above, to prove that the semigroup T is uniformly exponentially stable.


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The next result is an extension to dual evolution families of a well-known Barbashin-Rolewicz’s theorem. The last part of its proof is based on a result due to van Neerven, see Lemma 3.2.1 from [20]. With our notations this result can be reformulated as follows. Lemma 2. Let φ : [0, ∞) → [0, ∞) be a nondecreasing function. Then there exists an Orlicz space LΦ , over J, with limt→∞ ρΦ (χ[0,t] (·)) = ∞, which has the property that f ∈ LΦ whenever f is a bounded function defined on J satisfying the inequality
φ(|f (s)|)ds < ∞. J

Actually van Neerven was proved such lemma only for J = R+ but our case can be easily recovered by his proof. Theorem 3. Let φ : R+ → R+ be a nondecreasing function such that φ(t) > 0 for all t > 0 and let U = {U (t, s)}t≥s be an exponentially bounded evolution family of bounded linear operators acting on a Banach space X. If there exists a positive constant Mφ such that
∞ sup φ(χ[s,t] (τ )||U (t, τ )∗ x∗ ||dτ ) ≤ Mφ < ∞ x∗ ∈ X ∗ , ||x∗ || ≤ 1, (4.2) (t,s)∈∆J

s

then the family U is uniformly exponentially stable. Proof. Clearly, φ(0) = 0. We may assume that φ(1) = 1 (by replacing the function φ with an appropriate multiple of it). Let N > Mφ , s ∈ J, t ≥ s + N, x ∈ X, x = 0, x∗ ∈ X ∗ , ||x∗ || ≤ 1 and s ≤ τ ≤ t. Successively one has: e−ωN χ[s,s+N ] (τ )| x∗ , U (t, s)x| ≤ e−ω(τ −s) χ[s,s+N ] (τ )||U (t, τ )∗ x∗ ||||U (τ, s)x|| ≤ Mω ||x||χ[s,s+N ] (τ )||U (t, τ )∗ x∗ || ≤ Mω ||x||χ[s,t] (τ )||U (t, τ )∗ x∗ ||. Using the properties of φ, we get: N φ(e−ωN Mω−1 ||x||−1 | x∗ , U (t, s)x|)
∞ φ(e−ωN Mω−1 ||x||−1 χ[s,s+N ] (τ )| x∗ , U (t, s)x|)dτ = s
∞ ≤ φ(χ[s,t] (τ )||U (t, τ )∗ x∗ ||)dτ ≤ Mφ . s

From the inequality φ(e−ωN Mω−1 ||x||−1 | x∗ , U (t, s)x|) ≤ we infer that ||U (t, s)|| ≤ Mω eωN .

Mφ < 1 = φ(1) N

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In the case s ≤ t ≤ s + N, we obtain the same estimate for the norm of U (t, s), by using the exponential boundedness of the family U. The above Lemma 2 assures the existence of an Orlicz space LΦ (J) = E  (J) verifying the condition: lim ρΦ (χ[0,r] (·)) = ∞,

r→∞

which contains the map χ[s,t] (·)||U (t, ·)∗ x∗ || for each pair (t, s) ∈ ∆J . Moreover the map Φ can be chosen less than or equal to φ, so the inequality (4.2) still fulfills with Φ instead of φ. Using the uniform boundedness principle we shall recapture an inequality of type (4.1) and then can apply Theorem 2 above to finish the proof.
Corollary 3. An exponentially bounded evolution family {U (t, s)}t≥s≥0 is uniformly exponentially stable if and only if there exist p ∈ (0, ∞) such that for every x∗ ∈ X ∗ of norm less than or equals one, one has:
t sup ||U (t, s)∗ x∗ ||p ds < ∞. t≥0

0

Proof. Follows by the previous Theorem 3 for φ(t) = tp .




5. The periodic case In this section we refer to the case J = R only. This restriction does not affect the generality because every q-periodic evolution family U defined on ∆R+ can ˜ defined on ∆R . Moreover be extended to a unique q-periodic evolution family U ˜ has the same U is uniformly exponentially stable if and only if its extension U property. Theorem 4. Let q > 0 and U = {U (t, s) : (t, s) ∈ ∆J } be an exponentially bounded and q-periodic evolution family of bounded linear operators acting on a Banach space X which has the property that for each x∗ ∈ X ∗ the function ||U (0, −·)∗ x∗ || is locally measurable on J. If φ : R+ → R+ is a nondecreasing function such that φ(t) > 0 for each t > 0 and
∞ φ(||U (0, −t)∗ x∗ ||)dt = Kφ < ∞, x∗ ∈ X ∗ , ||x∗ || ≤ 1, 0

then the family U is uniformly exponentially stable. Proof. We divide the proof into three steps. Step 1. We shall prove that lim ||U (0, −t)∗ x∗ || = 0. Indeed, assuming the t→∞ contrary, it would exists a x∗ ∈ X ∗ such that lim sup ||U (0, −t)∗ x∗ || > 0. t→∞

(5.1)

Let ω > 0 and Mω ≥ 1 such that ||U (t, s)|| ≤ Mω eω(t−s) for every pair (t, s) ∈ ∆J . According to (5.1) we can choose a positive δ and a sequence (tν ) of positive real

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numbers with tν+1 − tν > 1/ω such that ||U (0, −tν+1 )∗ x∗ || > δ. Let n ≥ 2 be a fixed natural number and let k be a natural number such that k + 1 ≤ n. For each t ∈ [tk+1 − 1/ω, tk+1 ] we have that U (0, −t)U (−t, −tk+1 ) = U (0, −tk+1 ), so that U (0, −tk+1 )∗ x∗ = U (−t, −tk+1 )∗ U (0, −t)x∗ . Therefore δ ≤ ||U (0, −tk+1 )∗ x∗ || ≤ ≤

||U (−t, −tk+1 )∗ ||||U (0, −t)∗ x∗ || M e||U (0, −t)∗x∗ ||,

and thus

n−1 δ ) ≤ φ(||U (0, −t)∗ x∗ ||), for all t ∈ An := [tk+1 − 1/ω, tk+1 ]. k=1 Me Consequently for any natural number n ≥ 2, we obtain
∞
∞ δ δ m(An )φ( φ( φ(||U (0, −t)∗ x∗ ||) ≤ Kφ , )= )χAn (t)dt ≤ Me M e 0 0 φ(

which is a contradiction. Step 2. Here we shall prove that the family U is bounded. For this to be end let M := sup ||U (0, −t)∗ || < ∞. We shall analyze two cases. t≥0

Case 1. When there exists k = 0, 1, 2, . . . such that s ≤ kq ≤ t < (k + 1)q. Then we have that U (t, s) = U (t, kq)U (kq, s) = U (t − kq, 0)U (0, s − kq) and so ||U (t, s)|| = ≤

||U (t, s)∗ || ||U (0, s − kq)∗ ||||U (t − kq, 0)∗ || ≤ M eωq .

Case 2. When s ≤ −kq ≤ t ≤ −(k + 1)q. Using the evolution property and periodicity we get U (t, s) = U (t, −kq)U (−kq, s) and so ||U (t, s)||

= ||(U (t + kq, 0)U (0, s + kq))∗ || ≤ ||U (0, s + kq)∗ ||||U (t + kq, 0)∗ || ≤ M Mω eωq .

In the rest of the cases shall obtain that ||U (t, s)|| ≤ Mω eωq . Step 3. Let sup ||U (t, s)|| ≤ K < ∞ and let E(J) be an Orlicz space containt≥s

ing the map χ[0,t] (·)||U (·, −t)∗ x∗ || and having the property that   lim χ[0,t] (·)E
(J) = ∞. t→∞

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For each t ≥ 0 and each x∗ ∈ X ∗ , successively one has:     χ[0,t] (·) ||U (0, −t)∗ x∗ || = χ[0,t] (·)||(U (0, −·)U (−·, −t))∗ ||
E (J)   = χ[0,t] (·)||U (−·, −t)∗ U (0, −·)∗ ||
  ≤ K χ[0,t] (·)||U (0, −·)∗ x∗ ||E
(J) ≤ KME ||x∗ ||.

E (J)

Using the uniform bounded-ness principle we get: K(ME + 1)  ||U (0, −t)|| = ||U (0, −t)∗ || ≤  χ[0,t] (·)
+ 1 . E (J)

(5.2)

Let now t > s + q and let k be the largest integer number such that t ≥ kq > s. Then there exists ρ ∈ [0, q) such that t = kq + ρ. In the view of (5.2), we get: ||U (t, s)|| ≤ ||U (t, kq)||||U (kq, s)|| ≤ ≤

K||U (0, s − kq)|| K 2 (ME +1)

χ[0,t−s−ρ] (·) E
(J) +1

Now we can apply Lemma 1 above, to finish the proof.




The R+ -version of the Theorem 3 says that if U is an exponentially bounded evolution family and there exist a positive constant N such that for each x∗ ∈ X ∗ one has
t

sup t≥0

0

φ(||U (t, s)∗ x∗ ||)ds ≤ N ||x∗ ||

then U is uniformly exponentially stable. In the periodic case this result can be improved as follows. Corollary 4. Let U = {U (t, s) : (t, s) ∈ ∆J } be an exponentially bounded and q-periodic evolution family on a Banach space X, such that for each positive t and each x∗ ∈ X ∗ , the function ||U (t, ·)∗ x∗ || is measurable on the interval [0, t]. Let φ : R+ → R+ be a nondecreasing function with the property that φ(t) > 0 for every t > 0. Under these assumptions and if there exists a positive constant N such that
nq φ(||U (nq, s)∗ x∗ ||)ds ≤ N ||x∗ || sup n∈N

0

then the family U is uniformly exponentially stable. Proof. Successively one has: N ||x∗ || ≥ sup

n∈N

 nq 0

φ(||U (nq, s)∗ x∗ ||)ds

= = =

 nq

φ(||U (0, s − nq)∗ x∗ ||)ds 0 n∈N  nq sup 0 φ(||U (0, −t)∗ x∗ ||dt n∈N ∞ ∗ ∗ 0 φ(||U (0, −t) x ||)dt. sup

Using the previous Theorem 4 follows that the family U is uniformly exponentially stable.


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References [1] E.A. Barbashin, Introduction in the theory of stability, Izd. Nauka, Moscow, 1967 (Russian). [2] C. Bennett and S. Sharpley, Interpolation of Operators, Academic Press, Boston 1988. [3] C. Bu¸se, On the Perron-Bellman theorem for evolutionary processes with exponential growth in Banach spaces, New Zealand Journal of Mathematics 27 (1998), 183–190. [4] C. Bu¸se, M. Megan, M.S. Prajea and P. Preda, The strong variant of a Barbashin’s theorem on stability of solutions for non-autonomous differential equations in Banach spaces, Integral Equations and Operator Theory 59 no. 4 (2007), 491–500. [5] C. Bu¸se, S.S. Dragomir, A theorem of Rolewicz’s type in solid function spaces, Glasgow Math. Journal, 44 (2002), 125–135. [6] C. Bu¸se, C.P. Niculescu, An ergodic characterization of uniformly exponentially stable evolution families, Bull. Math. Soc. Sci. Math. Roum. 51 (99) no. 4 (2008), to appear. [7] C. Bu¸se, M.S. Prajea, Asymptotic behavior of discrete and continuous semigroups on Hilbert space , Bull. Math. Soc. Sci. Math. Roum. 51 (99) no. 2 (2008), 123–135. [8] C. Chicone, Yuri Latushkin, Evolution Semigrous in Dynamical Systems and Differential Equations, Amer. Math. Soc., Math. Surv. and Monographs 70, 1999. [9] Y.L. Daletkii, M.G. Krein, Stability of Solutions of Differential Equations in Banach Spaces, Izd. Nauka, Moskwa, 1970. [10] R. Datko, Uniform asymptotic stability of evolutionary processes in Banach space, SIAM J. Math. Analysis 3 (1973), 428–445. [11] A.Yu. Karlovich, L. Malingranda, On the interpolation constant for Orlicz spaces, Proc. Amer. Math. Soc. 129 no. 9 (2001), 2727-2739 (electronic). [12] M.A. Krasnoselskii, Ya.B. Rutiskii, Convex functions and Orlicz spaces, Fizmatgiz, Moscow 1958 (Russian); English translation: Noordhoff Ltd. Groningen, 1961. [13] S.G. Krein, Yu.I. Petunin and E.M. Semeonov, Interpolation of linear operators, Transl. Math. Monogr. 54 (Amer. Math. Soc., Providence, 1982). [14] L. Maligranda, Orlicz spaces and interpolation, Sem. Math. 5, Dep. Mat. Univ. Estadul de Campinas, Campinas SP, Brazil, 1989. [15] M. Megan, Proprietes qualitatives des systems lineaires controles dans les espaces de dimension infinie, Monographies Mathematiques (Timisoara, 1988). [16] A. Pazy, Semigroups of linear operators and applications to partial differential equations (Springer-Verlag, 1983). [17] S. Rolewicz, On uniform N-equistability, Journal of Math. Anal. Appl. 115 (1986), 434–441. [18] J.A. van Casteren, Operators similar to unitary or selfadjoint ones, Pacific Journal of Mathematics 104 (1983), 241–255. [19] J.M.A.M. van Neerven, Exponential stability of operators and operator semigroups, Journal of Functional Analysis 130 no. 2 (1995), 293–309. [20] J.M.A.M. van Neerven, The Asymptotic Behaviour of Semigroups of Linear Operators, Birkh¨ auser, Basel, 1996.

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C. Bu¸se Department of Mathematics Faculty of Mathematics and Computer Science West University of Timisoara Bd. V. Parvan No. 4 300223 Timisoara Romania Current address: Government College University Abdus Salam School of Mathematical Sciences Lahore Pakistan e-mail: [email protected] A.D.R. Choudary Government College University Abdus Salam School of Mathematical Sciences Lahore Pakistan e-mail: [email protected] S.S. Dragomir School of Computer Science and Mathematics Victoria University of Technology PO Box 14428, MCMC 8001 Victoria Australia e-mail: [email protected] URL: http://rgmia.vu.edu.au/SSDragomirWeb.html M.S. Prajea National College “Traian” Bd. Carol I, No. 6 Drobeta Turnu Severin Romania e-mail: [email protected] Submitted: July 22, 2007 Revised: May 29, 2008

IEOT

Integr. equ. oper. theory 61 (2008), 341–364 0378-620X/030341-24, DOI 10.1007/s00020-008-1593-6 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Integral Equations and Operator Theory

Regularities in Noncommutative Banach Algebras Anar Dosiev Abstract. In this paper we introduce regularities and subspectra in a unital noncommutative Banach algebra and prove that there is a correspondence between them similar to the commutative case. This correspondence involves a radical on a class of Banach algebras equipped with a subspectrum. Taylor and Slodkowski spectra for noncommutative tuples of bounded linear operators are the main examples of subspectra in the noncommutative case. Mathematics Subject Classification (2000). Primary 47A13; Secondary 46H30, 17B30. Keywords. Regularities, subspectrum, characters.

1. Introduction The regularities play an important role in the (joint) spectral theory of the Banach algebra framework. They generalize the (joint) invertibility in a Banach algebra. It is well known [13] that there is a close relationship between the spectral systems and regularities. In [15] the characterization of those regularities in a commutative Banach algebra related to subspectra has been proposed. A subspectrum in the sense of Zelazko [22] on a commutative Banach algebra A is a set-valued mapping over all tuples in A with the properties to be compact and polynomial spectral mapping. In this paper we introduce a subspectrum on a unital (noncommutative) Banach algebra based upon the properties to be compact and spectral mapping with respect to the noncommutative polynomials (see below Section 3), and establish a correspondence between them and regularities. In the noncommutative case, a subspectrum on A can be determined in terms of Lie algebras generated by the tuples a = (a1 , . . . , ak ) ∈ Ak in A using a fixed Banach space representation α : A → B (X) [8]. To conduct that approach, one might demand a restrictive condition concerning the Banach algebra A. We meet with the known phenomena [8] when a tuple of noncommutative polynomials p(a) = (p1 (a), . . . , pm (a)) ∈ Am in

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elements of a k-tuple a ∈ Ak generating a finite dimensional nilpotent Lie subalgebra L (a) ⊆ A, may generate an infinite dimensional Lie subalgebra L (p (a)) ⊆ A. To avoid these type of problems, we shall assume that A is a nilpotent Lie algebra, that is, its Lie algebra structure determined by the Lie multiplication [a, b] = ab − ba, a, b ∈ A, is nilpotent (see Section 6). Such algebra A admits sufficiently many subspectra. So are Slodkowski, Taylor spectra σπ,n (a) = σπ,n (π(a1 ), . . . , π (ak )) , σδ,n (a) = σδ,n (π(a1 ), . . . , π (ak )) ,

n ≥ 0,

and Harte type spectrum σR (a) for tuples a in A. Thus if τ is one of these spectra, then τ (a) is a nonempty compact subset in Ck for a k-tuple a in A, τ (x) is a subset of the usual spectrum σ (x) for a singleton x ∈ A, and if p (a) is a m-tuple of noncommutative polynomials in elements of a k-tuple a, then τ (p (a)) = p (τ (a)) (see [8] and Proposition 6.5). The assumption on A to be a nilpotent Lie algebra is also sustained by the noncommutative functional calculus problem [9]. In Proposition 6.6 we show that the closed associative envelopes of a supernilpotent Lie subalgebra g (that is, its commutator [g, g] consists of nilpotent elements) possess that property. But an operator tuple a in B (X) generating a supernilpotent Lie subalgebra g ⊆ B (X) admits [9], [6], [7] a noncommutative holomorphic functional calculus in a neighborhood of the Taylor spectrum σT (a), which extends Taylor functional calculus [19]. Thus a noncommutative Banach algebra A which is nilpotent as a Lie algebra has all the favorable spectral properties just as commutative Banach algebras. A regularity R in a unital Banach algebra A is defined as a nonempty subset R ⊆ A such that ab ∈ R iff a, b ∈ R (see Section 4). Each regularity automatically involves a set KR of characters ϕ of A such that R ∩ ker (ϕ) = ∅, and the closed two-sided ideal
R (A) = {ker (ϕ) : ϕ ∈ KR } called the R-radical of A. The set R# = A\



{ker (ϕ) : ϕ ∈ KR }

is called an envelope of R. A regularity R having an open proper envelope R# is of importance in our consideration. Such regularities appear when we deal with subspectra. Namely, a subspectrum τ on A associates a regularity Rτ in A given by the rule / τ (a)} . Rτ = {a ∈ A : 0 ∈ In this case, Rτ is a nonempty open proper subset in A and Rτ# = Rτ (see [15] for the commutative case). A key role in the noncommutative case plays the τ -radical Radτ A of A associated to a subspectrum τ on a Banach algebra A. According to the definition Radτ A = {a ∈ A : τ (a) = {0}} .

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It is proved (see Corollary 4.9) that Radτ A is a closed two-sided ideal in A which contains the Jacobson radical Rad A, and A is commutative modulo Radτ A. Moreover, τ determines a subspectrum τ ∼ on the quotient algebra A/ Radτ A. Thus τ ∼ is a subspectrum in the sense of Zelazko on the commutative (semisimple) Banach algebra A/ Radτ A. Furthermore, τ generates a compact subset Kτ of the character space Char (A) of A such that τ (a1 , . . . , ak ) = {(ϕ (a1 ) , . . . , ϕ (ak )) : ϕ ∈ Kτ } for a k-tuple (a1 , . . . , ak ) in A. The process of generating regularities from subspectra can be reversed (see Section 5). Namely, fix a regularity R in A with its open proper envelope R# ; one may define a Harte type spectrum σR on A by the rule   σR (a) = λ ∈ Ck : A (a − λ) ∩ R# = ∅ , where a is a k-tuple in A and A (a − λ) is the left ideal in A generated by the tuple a − λ. One can prove that the left ideal in the definition of σR (a) can be replaced with the right ideal (a − λ) A generated by a − λ, and σR is a subspectrum on A. Moreover, RσR = R# and the σR -radical is reduced to the R-radical, that is, RadσR A = R (A) . Thus the correspondence τ → Rτ between subspectra on A and regularities in A has a right inverse R → σR . Furthermore τ ⊆ σRτ . In the commutative case that relation has been observed in [15]. Note that, in general τ = σRτ . We investigate that difference in Section 6 by proposing necessary and sufficient condition when the latter inclusion turns out to be an equality.

2. Preliminaries All considered linear spaces are assumed to be complex and C denotes the field of all complex numbers. For a unital associative algebra A, Rad (A) denotes its Jacobson radical and A∗ the space of all linear functionals. A unital algebra homomorphism λ : A → C is said to be a character of A, and the set of all characters of A is denoted by Char (A). If S is a subset of an associative algebra A, then A (S) (respectively, (S) A) denotes the left (respectively, right) ideal in A generated by S. The group of all invertible elements in A is denoted by G (A). If A is a Banach algebra, then as it is well known [4, 1.2], G (A) is an open subset in A and Char (A) is a compact space with respect to the weak∗ -topology in the space of all bounded linear functionals on A. We use the denotation σ (a) for the spectrum of an element a ∈ A. The Banach algebra of all bounded linear operators on a Banach space X is denoted by B (X). If π : A → B is an algebra homomorphism, then π (n) : An → B n denotes the mapping π (n) (a1 , . . . , an ) = (π (a1 ) , . . . , π (an )) between the n-tuples in A and B.

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The following assertion is a well known [14] fact. Theorem (Gleason, Kahane, Zelazko). Let A be a unital Banach algebra and let ϕ : A → C be a linear functional such that ϕ (1) = 1 and ϕ (a) = 0 for all a ∈ G (A). Then ϕ ∈ Char (A). Now let Fn (e) be the free associative algebra generated by n elements e = (e1 , . . . , en ) .

 Each its element p (e) is a noncommutative polynomial p (e) = ν αν eν , where αν ∈ C and eν = ej1 · · · ejk for a finite sequence ν = (j1 , . . . , jk ) of elements from the set {1, . . . , n}. For a n-tuple a = (a1 , . . . , an ) ∈ An in a unital associative algebra A, we have a well defined algebra homomorphism Γa : Fn (e) → A such that Γa (ei ) = ai for all i. If p (e) ∈ Fn (e) is a free polynomial, then Γa (p (e)) = p (a) is the same polynomial in A taken by a. Indeed,     Γa (p (e)) = Γa αν eν = αν aν = p (a) . ν

ν

We say that Γa is a polynomial calculus for the tuple a. Similarly, each element λ = (λ1 , . . . , λn ) ∈ Cn determines a character Γλ : Fn (e) → C such that Γλ (ei ) = λi for all i. We say that Γλ is a point calculus. Actually, each character of Fn (e) is a point calculus. We put p (λ) = Γλ (p (e)) . If p (e) = (p1 (e) , . . . , pm (e)) is a m-tuple in Fn (e) and λ ∈ Cn , then we write p (λ) to indicate the m-tuple (p1 (λ) , . . . , pm (λ)) ∈ Cm in C. Now let A be a unital Banach algebra and let B be a unital subalgebra of A. Consider the family IA (B) of all left ideals I in B such that I ∩ G (A) = ∅. The following assertion was proved in [10] (see also [23]). Theorem 2.1. If aBa−1 ⊆ B for all a ∈ B ∩ G (A), then IA (B) possesses the projection property, that is, for each mutually commuting k-tuple b = (b1 , . . . , bk ) in I ∈ IA (B) and c ∈ B commuting with b there correspond λ ∈ C and J ∈ IA (B) such that (b1 , . . . , bk , c − λ) ∈ J k+1 . We shall apply (as in [15]) Theorem 2.1 to the following particular case. Let A = C (K) be the Banach algebra of all complex continuous functions on a compact topological space K furnished with the uniform norm f ∞ = sup {|f (x)| : x ∈ K}, and let B be a unital subalgebra of A. For a commutative tuple a ∈ B k we put   τ (a) = λ ∈ Ck : B (a − λ) ∩ G (C (K)) = ∅ , (2.1) which is a compact subset in Ck . On account of Theorem 2.1, we infer that τ possesses the projection property, that is, if a = (a1 , . . . , ak+1 ) ∈ B k+1 is a k + 1tuple and a = (a1 , . . . , ak ), then τ (a ) = π (τ (a)), where π : Ck+1 → Ck is the

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canonical projection onto the first k coordinates. Actually, the projection property involves (see [10]) the polynomial spectral mapping property τ (p (a)) = p (τ (a)) , where p is a family of polynomials in several complex variables. In this case it is said that τ is a subspectrum on B (see below Section 3). Thus (2.1) determines a subspectrum on B. This type of subspectra were characterized by A. Wawrzynczyk in [23]. Finally, if A is an associative algebra, then it is also a Lie algebra with respect to the canonical Lie multiplication [a, b] = ab − ba, a, b ∈ A. To indicate this Lie algebra structure we use the denotation Alie , thus Alie is the same algebra A considered with respect to the Lie multiplication called the attendant Lie algebra. Let us recall that a Lie algebra is said central series  (n)

to be nilpotent if its lower L(n−1) (n) (k) L ) is vanishing, that is, L (where L = L, L = {0} for a n∈N certain k. Thus each operator ad x : L → L, (ad x) (y) = [x, y] (x ∈ L) of the adjoint representation is nilpotent. If k = 1, the Lie algebra L is commutative. A finite-dimensional nilpotent Lie algebra L with L(2) = {0} is called a Heisenberg algebra. A typical example is a Lie algebra g with a basis e1 , e2 , e3 such that (1) [e1 , e2 ] = i. Further, note that Alie = [A, A] = A(1) and  e3 and [e i , e3 ] = 0 for all

(n) (n−1) = A, A(n−1) = A(n) , n > 1. Alie = A, Alie Let A be a unital associative algebra. A subalgebra B ⊆ A is said to be an inverse closed subalgebra if any invertible in A element of B is invertible in B. Since the inverse closed subalgebras are stable with respect to arbitrary intersections, it can be defined an inverse closed envelope of a subset in A. The following assertion is well known [20], [1]. Lemma (Turovskii). Let A be a unital Banach algebra which is the closure of the inverse closed envelope of a (not necessarily finite-dimensional) nilpotent Lie algebra. Then A is commutative modulo its Jacobson radical Rad A. In particular, so is a Banach algebra A with its nilpotent attendant Lie algebra Alie . Note that the closed associative envelope in A generated by Alie is obviously reduced to the whole algebra A. Therefore if Alie is a nilpotent Lie algebra, then A as a closed associative envelope of Alie is commutative modulo the Jacobson radical thanks to Turovskii’s lemma.

3. Subspectra In this section we consider purely algebraic case. We introduce a subspectrum in a noncommutative algebra and show that it generates a subspectrum on the quotient algebra modulo suitable radical.

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Let A be a unital associative algebra. As in the commutative case [21], a subspectrum τ on A is a mapping which associates to every k-tuple a = (a1 , . . . , ak ) ∈ k

σ (ai ) and it possesses Ak a nonempty compact set τ (a) ⊆ Ck such that τ (a) ⊆ i=1

the spectral mapping property τ (p (a)) = p (τ (a))

(3.1) m

for an m-tuple p (e) = (p1 (e) , . . . , pm (e)) ∈ Fk (e) . Of course, we have assumed that the usual spectrum σ (a) of each element a ∈ A is nonvoid. That is true whenever A is a Banach algebra. Note that the equality (3.1) establishes a relation between the polynomial calculus Γa and point calculi Γλ , λ ∈ τ (a). Namely,     (m) p (τ (a)) = {p (λ) : λ ∈ τ (a)} = Γλ (p (e)) : λ ∈ τ (a) = τ Γ(m) (p (e)) . a For subspectra τ and σ on A we put τ ⊆ σ if τ (a) ⊆ σ (a) for all tuples a in A. Now let τ be a subspectrum on A. We put Radτ (A) = {a ∈ A : τ (a) = {0}} . We say that Radτ (A) is the τ -radical in A. Lemma 3.1. Let τ be a subspectrum on A. Then Radτ (A) is a two-sided ideal in A and the quotient algebra A/ Radτ (A) is commutative. Moreover, Rad (A) ⊆ Radτ (A) , and the inclusion turns out to be an equality whenever τ (a) = σ (a) for all a ∈ A. Proof. By assumption, τ (a) ⊆ σ (a) is a nonempty subset for each a ∈ A. Therefore τ (a) = {0} if σ (a) = {0}. Take a ∈ Rad (A). Then λ − a is invertible in A for all λ, λ = 0 (see [5, 8.6.3]). It follows that σ (a) = {0}, that is, a ∈ Radτ (A). Thus Rad (A) ⊆ Radτ (A). Take a, b ∈ Radτ (A). Then τ (a + b) = {λ + µ : (λ, µ) ∈ τ (a, b)} and τ (a, b) ⊆ τ (a) × τ (b) = {0}. Whence τ (a + b) = {0} and a + b ∈ Radτ (A). Using a similar argument, we conclude that ca, ac ∈ Radτ (A) for any c ∈ A. Thus Radτ (A) is a two-sided ideal in A. Now take a, b ∈ A and let p (e1 , e2 ) = e1 e2 − e2 e1 ∈ F2 (e). Then p (a, b) = [a, b] ∈ A and τ (p (a, b)) = p (τ (a, b)) = {λµ − µλ : (λ, µ) ∈ τ (a, b)} = {0} . Hence [a, b] ∈ Radτ (A) and therefore A is commutative modulo Radτ (A).




Consider the quotient linear mapping πτ : A → A/ Radτ (A) πτ (a) = a∼ . (k)

∼ If a = (a1 , . . . , ak ) is a k-tuple in A, then a∼ = (a∼ 1 , . . . , ak ) = πτ (a) is a k-tuple in A/ Radτ (A).

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∼ Lemma 3.2. Let τ be a subspectrum on A. Then to each k-tuple a∼ = (a∼ 1 , . . . , ak ) in A/ Radτ (A) there corresponds a subset τ ∼ (a∼ ) ⊆ Ck such that

τ ∼ (a∼ ) = τ (b) ∼ for a k-tuple b = (b1 , . . . , bk ) ∈ Ak with a∼ i = bi for all i.

Proof. One has to prove that τ (a + x) = τ (a) for any k-tuple x = (x1 , . . . , xk ) in Radτ (A). Take λ ∈ τ (a). Then (λ, µ) ∈ τ (a, x) for some µ ∈ Ck . But µ ∈ τ (x) ⊆ k

τ (xi ) = {0} and λ+µ ∈ τ (a + x), that is, λ ∈ τ (a + x). Thus τ (a) ⊆ τ (a + x).

i=1

Since −x ∈ Radτ (A)k , it follows that τ (a + x) ⊆ τ (a + x − x) = τ (a). It remains to put τ ∼ (a∼ ) = τ (a).
Lemma 3.3. Let τ be a subspectrum on A. Then τ ∼ (a∼ ) ⊆ σ (a∼ ) for a singleton a ∈ A, where σ (a∼ ) is the usual spectrum of a∼ in the algebra A/ Radτ (A). Proof. If λ ∈ / σ (a∼ ), then (a − λ) b = 1 + x for some x ∈ Radτ (A) and b ∈ A. But τ ((a − λ) b) = τ (1 + x) = {1} and τ ((a − λ) b) = {zw : (z, w) ∈ τ (a − λ, b)} . Then 0 ∈ / τ (a − λ), for in the contrary case (0, w) ∈ τ (a − λ, b) for some w ∈ C, which in turn implies that 0 = 0w ∈ τ ((a − λ) b). So, λ ∈ / τ (a). On account of Lemma 3.2, τ ∼ (a∼ ) = τ (a) ⊆ σ (a∼ ). Whence λ ∈ / τ ∼ (a∼ ).
Using Lemmas 3.2 and 3.3, we obtain that ∼ τ ∼ (a∼ 1 , . . . , ak ) = τ (a1 , . . . , ak ) ⊆

k 

τ (ai ) =

i=1

k 

τ (a∼ i ) ⊆

i=1

k 

σ (a∼ i )

i=1

is a nonempty compact subset. Theorem 3.4. Let τ be a subspectrum on a unital associative algebra A. The correspondence τ ∼ over all tuples in A/ Radτ (A) is a subspectrum on the commutative algebra A/ Radτ (A). ∼ ∼ Proof. Take a k-tuple (a∼ 1 , . . . , ak ) in A/ Radτ (A) and a noncommutative  a = ν polynomial p (e) = ν αν e ∈ Fk (e). Then    ν ν αν (a∼ ) = αν πτ(k) (a) = αν πτ (aν ) p (a∼ ) = Γa∼ (p (e)) =

 = πτ

ν



 αν aν

ν

ν

∼

= πτ (Γa (p (e))) = p (a) .

ν

If p (e) = (p1 (e) , . . . , pm (e)) is a m-tuple in Fk (e), then τ ∼ (p (a∼ )) = τ ∼ (p (a)∼ ) = τ (p (a)) = p (τ (a)) = p (τ ∼ (a∼ )) . It remains to appeal to Lemmas 3.1, 3.2 and 3.3.




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In particular, if τ is a subspectrum on a Banach algebra A and Radτ (A) is closed, then τ ∼ is a subspectrum on the commutative Banach algebra A/ Radτ (A). In the next section we prove that Radτ (A) is closed for each subspectrum on a Banach algebra A. Definition 3.5. Let τ be a subspectrum on a unital associative algebra A. We say that a subspace I ⊆ A is τ -singular if 0 ∈ τ (c) for any tuple c ∈ I k , k ∈ N. A linear functional ϕ ∈ A∗ is said to be a τ -singular if its kernel ker (ϕ) is a τ -singular subspace in A. The set of all τ -singular functionals on A is denoted by Kτ . The concept of a τ -singular subspace is motivated by the key reasoning in [15, Lemma 2.3] for the commutative Banach algebra case. The following simple lemma will be useful later. Lemma 3.6. If I ⊆ A is a τ -singular subspace, then 1 ∈ / I. Proof. Being τ (1) ⊆ σ (1) = {1} a nonempty subset, we conclude that τ (1) = {1}. Then 0 ∈ / τ (1) and therefore 1 ∈ / I.
Proposition 3.7. Each τ -singular subspace in A is annihilated by a τ -singular functional on A. Proof. Let I is a τ -singular subspace in A. One has to prove that ϕ (I) = {0} for a certain τ -singular functional ϕ ∈ A∗ . Consider a family E of all τ -singular subspaces F ⊆ A such that I ⊆ F , that is, 0 ∈ τ (c) for any tuple c = (c1 , . . . , ck ) ∈ F k , k ∈ N (Definition 3.5). If {Fα } is a linearly ordered family in E, then ∪α Fα ∈ E. By Zorn’s lemma, there is a maximal element in E, say J. By Lemma 3.6, 1 ∈ / J. Let us prove that A = J ⊕C1. If the latter does not hold, then Cu∩(J ⊕ C1) = {0} for a certain u ∈ A, that is, J ∩ (Cu ⊕ C1) = {0}. Let c = (c1 , . . . , ck ) ∈ J k be a k-tuple in J, and let (c, u) = (c1 , . . . , ck , u) ∈ Ak+1 . Since 0 ∈ τ (c), it follows that (0, λ) ∈ τ (c, u) or 0 ∈ τ (c, u − λ) for some λ ∈ C. Thus K (c) = {µ ∈ C : 0 ∈ τ (c, u − µ)} is a nonempty compact subset in C. Using (3.1) (namely, the Projection Property), we obtain that K (c, b) ⊆ K (c) ∩ K (b) for all tuples c, b in J. Hence there is a common point λ0 ∈ K (c) for all tuples c in J. Put x = u − λ0 1 ∈ Cu ⊕ C1. Thus x∈ / J and 0 ∈ τ (c, x) for all tuples c in J. Consider the subspace J = J ⊕ Cx ⊆ A. k If c + ξx = (c1 + ξ1 x, . . . , ck + ξk x) ∈ J (herein ξ = (ξ1 , . . . , ξk ) ∈ Ck ) is a k-tuple in J, then τ (c + ξx) = {λ + µξ : (λ, µ) ∈ τ (c, x)} , which in turn implies that 0 ∈ τ (c + ξx). Thus J ∈ E and J = J, a contradiction. Consequently, A = J ⊕ C1, that is, J = ker (ϕ) for some ϕ ∈ A∗ . But ϕ is a τ -singular functional, for J ∈ E. It remains to note that I ⊆ J.
Now let τ be a subspectrum on A and let Rτ = {a ∈ A : 0 ∈ / τ (a)} . Note that Rτ ∩ ker (ϕ) = ∅ for each ϕ ∈ Kτ .

(3.2)

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Corollary 3.8. A\Rτ =



349

{ker (ϕ) : ϕ ∈ Kτ }.

Proof. Take a ∈ A\Rτ . Then 0 ∈ τ (a). Consider the subspace Ca in A generated by a and let ξa = (ξ1 a, . . . , ξk a) be a k-tuple in Ca, where ξ = (ξ1 , . . . , ξk ) ∈ Ck . Let pi (e) = ξi e, 1 ≤ i ≤ k, be polynomials in F1 (e), and let p (e) = k (p1 (e) , . . . , pk (e)) ∈ F1 (e) . Then τ (ξa) = τ (p (a)) = p (τ (a)) = {p (λ) : λ ∈ τ (a)} = {λξ : λ ∈ τ (a)} ⊆ Ck . In particular, 0 ∈ τ (ξa), that is, Ca is a τ -singular subspace in A. By Proposition 3.7, a ∈ ker (ϕ) for some τ -singular functional ϕ ∈ A∗ . Thus a ∈ ker (ϕ) for some ϕ ∈ Kτ .
Corollary 3.9. If τ is a subspectrum on A, then
Radτ (A) = {ker (ϕ) : ϕ ∈ Kτ } . Proof. Take a ∈ / Radτ (A). Then λ ∈ τ (a) for some nonzero λ ∈ C, that is, 0 ∈ τ (a − λ). The latter means that a − λ ∈ A\Rτ . Using Corollary 3.8, infer that a − λ ∈ ker (ϕ) for some ϕ ∈ Kτ . It follows that ϕ (a) = λϕ (1) = 0 by virtue of Lemma 3.6. Thereby a ∈ / ker (ϕ). So,
{ker (ϕ) : ϕ ∈ Kτ } ⊆ Radτ (A) . Conversely, take a ∈ / ker (ϕ) for some ϕ ∈ Kτ . Then ϕ (a) = 0 and   −1 −1 = ϕ (a) − ϕ (a) ϕ (1) ϕ (1) = 0, ϕ a − ϕ (a) ϕ (1) −1

that is,  a − ϕ (a) ϕ (1) ∈ ker (ϕ) (see Lemma 3.6). Since ϕ ∈ Kτ , it follows that −1 −1 0 ∈ τ a − ϕ (a) ϕ (1) . This in turn implies that ϕ (a) ϕ (1) ∈ τ (a), that is, 
τ (a) = {0} or a ∈ / Radτ (A). Thus Radτ (A) ⊆ {ker (ϕ) : ϕ ∈ Kτ }. Corollary 3.10. Let τ be a subspectrum on A. Then τ ∼ is a subspectrum on the algebra A/ Radτ (A) with the properties πτ (Rτ ) = Rτ ∼ and πτ∗ (Kτ ∼ ) = Kτ , where ∗

πτ∗ : (A/ Radτ (A)) → A∗ is the dual of the quotient mapping πτ : A → A/ Radτ (A). Proof. Using Lemma 3.1, infer that A/ Radτ (A) is commutative. Moreover, τ ∼ is a subspectrum on A/ Radτ (A) as we have shown in Theorem 3.4. The equality πτ (Rτ ) = Rτ ∼ follows directly from (3.2) and Lemma 3.2. Now take ψ ∈ Kτ ∼ and put ϕ = ψ · πτ . If a is a k-tuple in ker (ϕ), then a∼ is a k-tuple in ker (ψ) and 0 ∈ τ ∼ (a∼ ) = τ (a) by virtue of Lemma 3.2. Thus ϕ is a τ -singular functional. Therefore πτ∗ (Kτ ∼ ) ⊆ Kτ . Conversely, take ϕ ∈ Kτ . By ∗ Corollary 3.9, ϕ = ψ · π for some ψ ∈ (A/ Radτ (A)) . Again by Lemma 3.2, ψ is ∼ τ -singular.


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A result of Zelazko [22] asserts that for each subspectrum τ on a commutative Banach algebra a unique compact subset K ⊆ Char (B) such  B there corresponds  that τ (a) = ϕ(k) (a) : ϕ ∈ K for any k-tuple a in B. In the pure algebraic context this result has the following generalization. Theorem 3.11. Let τ be a subspectrum on a unital associative algebra A. Then τ (a1 , . . . , ak ) = {(ϕ (a1 ) , . . . , ϕ (ak )) : ϕ ∈ Kτ } −1

for all tuples (a1 , . . . , ak ) in A, where ϕ = ϕ (1)

ϕ (see Lemma 3.6).

Proof. Let a = (a1 , . . . , ak ) and take µ ∈ τ (a). Then 0 ∈ τ (a − µ). Consider a subspace F in A generated by a − µ. Each element of F has the form p (a − µ) for some p (e) ∈ Fk (e) such that p (0) = 0. Using the Spectral Mapping Property (3.1), we conclude that F is a τ -singular subspace in A. On account of Proposition 3.7, F ⊆ ker (ϕ) for some τ -singular functional ϕ ∈ A∗ . Thus ϕ (ai − µi ) = 0 or ϕ (ai ) = µi ϕ (1), 1 ≤ i ≤ k. It follows that ϕ (ai ) = µi or ϕ(k) (a) = µ. Conversely, take ϕ ∈ Kτ . Then ai − ϕ (ai ) ∈ ker  (ϕ) for all i. Being ϕ a τ -singular functional, we deduce that 0 ∈ τ a − ϕ(k) (a) or ϕ(k) (a) ∈ τ (a). Thus τ (a) =  (k)  ϕ (a) : ϕ ∈ Kτ .


4. Regularities In this section we introduce regularities in a unital associative algebra and investigate their properties. Let A be a unital associative algebra and let R be a nontrivial subset in A. As in [15], we introduce the envelope R# of R in A as  A\R# = {ker (ϕ) : ϕ ∈ A∗ , R ∩ ker (ϕ) = ∅} . By its very definition, R ⊆ R# . Lemma 4.1. R## = R# . Proof. Since R# ⊆ R## , it suffices to prove that R## ⊆ R# . Take a ∈ A\R# . Then a ∈ ker (ϕ), R ∩ ker (ϕ) = ∅, for some ϕ ∈ A∗ . But ker (ϕ) ⊆ A\R# , that is,
ker (ϕ) ∩ R# = ∅. The latter in turn implies that a ∈ A\R## . The following definition plays a key role in our consideration. Definition 4.2. Let A be a unital algebra. A nonempty subset R ⊆ A is said to be a regularity in A if it possesses the following property ab ∈ R

iff a, b ∈ R.

The following two assertions provide us with examples of regularities. Lemma 4.3. The set G (A) of all invertible elements in a unital algebra A is a regularity in A whenever A is commutative modulo its Jacobson radical. Moreover, # G (A) = G (A) if A is a Banach algebra.

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Proof. If a, b ∈ A, then σ (ab) = σ ((ab)∼ ), where (ab)∼ is the image of ab in ∼ the commutative algebra A/ Rad (A). But (ab) = a∼ b∼ . Therefore, 0 ∈ / σ (ab) ∼ ∼ iff both a and b are invertible in A/ Rad (A). Thus 0 ∈ / σ (a∼ ) = σ (a) and 0∈ / σ (b∼ ) = σ (b), that is, a, b ∈ G (A). Thus G (A) is a regularity in A. Now assume that A is a Banach algebra. Take a ∈ / G (A). Then a∼ is not invertible in A/ Rad (A) and therefore belongs to a maximal ideal I of A/ Rad (A). But I = ker (φ) for some character φ of the commutative algebra A/ Rad (A) [4, 1.3.2]. It follows that a ∈ ker (ϕ), where ϕ = φπ ∈ Char (A). But G (A) ∩ ker (ϕ) =
∅. Therefore a ∈ / G (A)# . Thus G (A) = G (A)# . Proposition 4.4. Let τ be a subspectrum on A and let Rτ = {a ∈ A : 0 ∈ / τ (a)} (see (3.2)). Then Rτ is a regularity in A and Rτ = Rτ# . Proof. With τ (0) = {0} in mind, infer 0 ∈ / Rτ . Moreover, 1 ∈ Rτ , for τ (1) ⊆ σ (1) = {1}. Thus ∅ = Rτ = A. Using Corollary 3.8, we obtain that  {ker (ϕ) : ϕ ∈ Kτ } . A\Rτ =  But {ker (ϕ) : ϕ ∈ Kτ } ⊆ A\Rτ# , that is, A\Rτ ⊆ A\Rτ# . Thus Rτ = Rτ# . Since τ (ab) = {λµ : (λ, µ) ∈ τ (a, b)}, it follows that 0 ∈ / τ (ab) iff 0 ∈ / τ (a) and 0 ∈ / τ (b). Whence Rτ is a regularity in A.
Now let us prove the main result of this section. Theorem 4.5. Let R be a regularity in A. Then G (A) ⊆ R. Moreover, if A is a Banach algebra and ϕ ∈ A∗ is such that R ∩ ker (ϕ) = ∅, then ker (ϕ) = ker (φ) for some φ ∈ Char (A). In particular,  {ker (ϕ) : ϕ ∈ Char (A) , R ∩ ker (ϕ) = ∅} . A\R# = Proof. Take a ∈ R. Then a = a · 1 ∈ R and therefore 1 ∈ R. Further, if a ∈ G (A), then 1 = aa−1 ∈ R, which in turn implies that a ∈ R. Thus G (A) ⊆ R. Now suppose A is a Banach algebra and let ϕ ∈ A∗ such that R ∩ ker (ϕ) = ∅. Taking into account that G (A) ⊆ R, we obtain G (A) ∩ ker (ϕ) = ∅. Using the Gleason-Kahane-Zelazko theorem, we deduce that φ = ϕ (1)−1 ϕ ∈ Char (A). But ker (ϕ) = ker (φ). Finally, the union over all the characters indicated above belongs to A\R# . Conversely, take a ∈ A\R# . Then a ∈ ker (ϕ) and R ∩ ker (ϕ) = ∅ for some
functional ϕ ∈ A∗ . But ker (ϕ) = ker (φ) for some φ ∈ Char (A). Corollary 4.6. If R is a regularity in a Banach algebra A, then so is R# . / R# , then ab ∈ ker (ϕ) for some ϕ ∈ Char (A), Proof. Take a, b ∈ R# . If ab ∈ R ∩ ker (ϕ) = ∅, by virtue of Theorem 4.5. Then 0 = ϕ (ab) = ϕ (a) ϕ (b) . Hence a or b belongs to ker (ϕ), that is, a or b does not belong R# , a contradiction.

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Conversely, if a ∈ / R# or b ∈ / R# , then ϕ (ab) = ϕ (a) ϕ (b) = 0 for some ϕ ∈ Char (A), R ∩ ker (ϕ) = ∅ (Theorem 4.5), that is, ab ∈ / R# . Whence R# is a regularity in A.
Now let τ be a subspectrum on a unital Banach algebra A and let Kτ ⊆ Char (A) be the set of all τ -singular characters of A. Note that Kτ ⊆ Kτ and ker (ϕ) ∩ Rτ = ∅ for all ϕ ∈ Kτ (see Section 3). Corollary 4.7. Let τ be a subspectrum on a unital Banach algebra A. If ϕ ∈ Kτ , then ker (ϕ) = ker (φ) for some φ ∈ Kτ . In particular,  A\Rτ = {ker (ϕ) : ϕ ∈ Kτ } . Moreover, Rτ is open in A. Thus Rτ has an open proper envelope Rτ# which coincides with itself. Proof. Take ϕ ∈ Kτ . Using Proposition 4.4 and Theorem 4.5, we infer that ker (ϕ) = ker (φ) for some φ ∈  Char (A). But ker (ϕ) is a τ -singular subspace, therefore φ ∈ Kτ . Thus A\Rτ = {ker (ϕ) : ϕ ∈ Kτ }. Finally, take a ∈ Rτ . Then sa = min {|λ| : λ ∈ τ (a)} > 0 and if b < sa , then a + b ∈ Rτ .
Corollary 4.8. Let τ be a subspectrum on a unital Banach algebra A. Then   τ (a) = ϕ(k) (a) : ϕ ∈ Kτ for all tuples a ∈ Ak , k ∈ N. Moreover, Kτ is a compact subspace in Char (A). −1

Proof. First note that φ (x) = φ (1) φ (x) = φ (x) for all φ ∈ Char (A) and x ∈ A. Thus     ϕ(k) (a) : ϕ ∈ Kτ ⊆ ϕ(k) (a) : ϕ ∈ Kτ . On account of Theorem 3.11, it suffices to prove that for any ϕ ∈ Kτ there corresponds φ ∈ Kτ such that ϕ = φ. By Corollary 4.7, ker (ϕ) = ker (φ) for some −1 φ ∈ Kτ . Then φ = αϕ for some α ∈ C. But 1 = φ (1) = αϕ (1), that is, α = ϕ (1) (Lemma 3.6). Thus φ = ϕ. Now let us prove that Kτ is compact with respect to the topology inherited from Char (A). It suffices to prove Kτ is closed in Char (A). Take ϕ ∈ Char (A) \Kτ . Then ker (ϕ) is not τ -singular, that is, 0 ∈ / τ (b) for some (say k-)tuple b in ker (ϕ). Let ε > 0 be the distance between the origin and τ (b) in Ck , and let   Ub,η (ϕ) = φ ∈ Char (A) : max |φ (bi ) − ϕ (bi )| < η 1≤i≤k

−1/2

be a neighborhood of ϕ in Char (A), where η = (2k) Then max |φ (bi )| < (2k)−1/2 ε. Moreover,

ε. Take φ ∈ Ub,η (ϕ).

/ τ (b) , (φ (b1 ) , . . . , φ (bk )) ∈ k 2 2 2 for in the contrary case we would have ε ≤ i=1 |φ (bi )| ≤ k max |φ (bi )| < ε2 /2. It follows that φ ∈ / Kτ . Thus Ub,η (ϕ) ∩ Kτ = ∅.


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Corollary 4.9. Let τ be a subspectrum on a unital Banach algebra A. Then
{ker (ϕ) : ϕ ∈ Kτ } . Radτ (A) = In particular, Radτ (A) is a closed two-sided ideal in A. Proof. It suffices to apply Corollary 3.9 and Corollary 4.7.




Corollary 4.10. Let τ be a subspectrum on a unital Banach algebra A. Then τ ∼ is a subspectrum on the commutative Banach algebra A/ Radτ (A) and πτ∗ (Kτ ∼ ) = Kτ . Proof. Using Lemma 3.1 and Corollary 4.9, we infer that A/ Radτ (A) is a commutative Banach algebra. Moreover, τ ∼ is a subspectrum on A/ Radτ (A) as we have shown in Lemma 3.2. Now take ψ ∈ Kτ ∼ . Then ψ ∈ Char (A/ Radτ (A)) such that ψ is τ ∼ -singular. Put ϕ = πτ∗ (ψ) = ψ·πτ . Evidently, ϕ ∈ Char (A) and ϕ ∈ Kτ by virtue of Corollary 3.10. So ϕ ∈ Kτ . Conversely, take ϕ ∈ Kτ . By Corollary 4.9, ϕ = ψ · πτ , ψ ∈ Char (A/ Radτ (A)) . Again by Corollary 3.10, ψ ∈ Kτ ∼ . Thus πτ∗ (Kτ ∼ ) = Kτ .




5. Subspectrum associated to a regularity in a Banach algebra In this section we construct a subspectrum called Harte type spectrum by means of a regularity in a Banach algebra. That will reverse (in a certain sense) the process of creating regularities from subspectra. Let R be a regularity in a unital Banach algebra A and let KR = {ϕ ∈ Char (A) : R ∩ ker (ϕ) = ∅} . According to Theorem 4.5, A\R# =



{ker (ϕ) : ϕ ∈ KR } .

(5.1)

Let us introduce a closed two-sided ideal
R (A) = {ker (ϕ) : ϕ ∈ KR } in A. Evidently, Rad A ⊆ R (A). Moreover, since ϕ ([x, y]) = 0, x, y ∈ A, ϕ ∈ Char (A), it follows that A/R (A) is a semisimple commutative Banach algebra. Lemma 5.1. If R has an open proper envelope R# in A, then KR is a nonempty compact space. Proof. Since R# is proper, it follows that A/R# = ∅ and therefore KR = ∅. It remains to prove that KR is a closed subset in Char (A). Take a net {ϕι } in KR which tends to ϕ ∈ Char (A). If ϕ ∈ / KR , then ϕ (x) = 0 for a certain x ∈ R. Note that x−ϕι (x) ∈ ker (ϕι ) and ker (ϕι )∩R# = ∅ (see Lemma 4.1). Thus {x − ϕι (x)} is a net in the closed set A\R# and it tends to x. Consequently, x ∈ A\R# and thereupon x ∈ / R, a contradiction. Thus ϕ ∈ KR and KR is closed in Char (A).


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Now let a = (a1 , . . . , ak ) ∈ Ak be a k-tuple in A. It determines a continuous mapping a k ) : K R → Ck ,  a = ( a1 , . . . , 

 a (ϕ) = ϕ(k) (a) = (ϕ (a1 ) , . . . , ϕ (ak )) .

a : Char (A) → Ck , Actually, it is the restriction to KR of the continuous mapping  (k)  a (ϕ) = ϕ (a) determined by the Gelfand transform. Put a) , τR (a) = im ( which is a nonempty compact subset in Ck thanks to Lemma 5.1. Lemma 5.2. Let a ∈ Ak and let x ∈ R (A)k . Then τR (a + x) = τR (a).

  Proof. Note that x  (ϕ) = 0 for all ϕ ∈ KR . Then τR (a + x) = im a
+x =


im ( a+x ) = im ( a) = τR (a).

Proposition 5.3. Let R be a regularity in a Banach algebra A whose envelope is open and proper. Then the correspondence τR over all tuples in A is a subspectrum on A. Moreover, RτR = R# and RadτR A = R (A). Proof. Take a k-tuple a in A, and m-tuple p (e) in Fk (e). Then p (a) is a m-tuple in A and a (ϕ)) , . . . , pm ( a (ϕ))) p (a) (ϕ) = ϕ(m) (p (a)) = (ϕ (p1 (a)) , . . . , ϕ (pm (a))) = (p1 ( = p ( a (ϕ)) . Therefore

  τR (p (a)) = im p (a) = p (im ( a)) = p (τR (a)) ,

that is, (3.1) holds. Further, RτR = {a ∈ A : 0 ∈ / τR (a)} = {a ∈ A :  a (ϕ) = 0, ϕ ∈ KR } = {a ∈ A : ϕ (a) = 0, ϕ ∈ KR } = {a ∈ A : a ∈ / ker (ϕ) , ϕ ∈ KR }  # = A\ {ker (ϕ) : ϕ ∈ KR } = R , that is, RτR = R# . Finally, RadτR A = {a ∈ A : τR (a) = {0}} = {a ∈ A : ϕ (a) = 0, ϕ ∈ KR }
= {ker (ϕ) : ϕ ∈ KR } = R (A) , that is, R (A) = RadτR A.




By Lemma 3.1, Corollary 4.9 and Proposition 5.3, infer that A/R (A) is a commutative semisimple Banach algebra. The mapping T : A → C (KR ) ,

T (a) =  a,

 a (ϕ) = ϕ (a) ,

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is a contractive homomorphism. Evidently, ker (T ) = R (A) and it can be factored as a composition of the quotient mapping A → A/R (A) and a contractive homomorphism A/R (A) → C (KR ) (see Lemma 5.2). Denote the range of T by B, which is a unital subalgebra in C (KR ).   Lemma 5.4. T R# = B ∩ G (C (KR )). Proof. If a ∈ R# , then  a (ϕ) = ϕ (a) = 0 for all ϕ ∈ KR , that is, the function  a is a is invertible in C (KR ), then ϕ (a) = 0 for all invertible in C (KR ). Conversely, if  ϕ ∈ KR . It follows that / τR (a), that is, a ∈ RτR . Using Proposition 5.3, infer  0 ∈ that a ∈ R# . Thus T R# = B ∩ G (C (KR )).
For a k-tuple  a in B we set   γR ( a) = λ ∈ Ck : B ( a − λ) ∩ G (C (KR )) = ∅ . Note that γR is a subspectrum on B (see (2.1)). Lemma 5.5. If R is a regularity in a unital Banach algebra A, then R# = R# + R (A). In particular, if A (M ) (respectively, (M ) A) is the left ideal (respectively, right) in A generated by a subset M ⊆ A, then A (M )∩R# = ∅ iff (M ) A∩R# = ∅. Proof. If a + x ∈ / R# for some a ∈ R# and x ∈ R (A), then ϕ (a + x) = 0 for some ϕ ∈ KR thanks to (5.1). Thereby ϕ (a) = 0, that is, a ∈ A/R# . So, R# = R# + R (A). Now let M be a subset in A such that A (M ) ∩ R# = ∅. Then (M ) A ⊆ A (M ) + [A, M ] ⊆ A (M ) + R (A) (see Corollary 4.9). If x ∈ (M ) A ∩ R# , then x = y + z for some y ∈ A (M ) and z ∈ R (A). It follows that y = x − z ∈ R# + R (A) = R# , a contradiction.
Lemma 5.6. If ϕ is a γR -singular functional on B, then A (a) ∩ R# = ∅ for any tuple a in ker (ϕT ). Proof. If a is a k-tuple in ker (ϕT ), then so is  a in ker (ϕ). Being ϕ a γR -singular, a), that is, B ( a) ∩ G (C we obtain that 0 ∈ γR (  (KR )) = ∅. The latter in turn a) ∩ G (C (KR )) thanks to implies that A (a) ∩ R# = ∅, for T A (a) ∩ R# = B ( Lemmas 5.4 and 5.5.
Now let R be a regularity in a unital Banach algebra A whose envelope R# is open and proper. By a Harte type spectrum σR associated with R we mean a set-valued function over all tuples in A determined by the rule   σR (a) = λ ∈ Ck : A (a − λ) ∩ R# = ∅ for a k-tuple a in A. Using Lemma 4.1, we deduce that   σR# (a) = λ ∈ Ck : A (a − λ) ∩ R## = ∅   = λ ∈ Ck : A (a − λ) ∩ R# = ∅ = σR (a) .

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Furthermore, the left ideal A (a − λ) generated by a − λ in the definition of σR (a) can be replaced with the right one as follows from Lemma 5.5. If R = G (A) the set σR (a) is the known [13, 1.8.1] Harte spectrum of the tuple a. Theorem 5.7. Let R be a regularity in a Banach algebra A whose envelope R# is open and proper. Then σR (a) = τR (a) for any k-tuple a in A. In particular, σR is a subspectrum on A. Moreover, RσR = R# and RadσR A = R (A) . Proof. Take  a (ϕ) ∈ τR (a), where ϕ ∈ KR . Since ϕ is a character of A, it follows that A (a −  a (ϕ)) ⊆ ker (ϕ). Therefore A (a −  a (ϕ)) ∩ R# = ∅, that is,  a (ϕ) ∈ σR (a). Hence τR (a) ⊆ σR (a). Conversely, assume that 0 ∈ σR (a). Then A (a) ∩ R# = ∅. Using Lemmas 5.4 and 5.5, we infer that B ( a) ∩ G (C (KR )) =∅.Whence 0 ∈ γR ( a). Since γR is a subspectrum on B, we deduce that 0 ∈ γR b for any tuple in the subspace in B generated by  a. Thus the subspace generated by  a a is a tuple in ker (φ) for is γR -singular. Using Proposition 3.7, we obtain that  some γR -singular functional φ ∈ B ∗ . According to Lemma 5.6, A (b) ∩ R# = ∅ for any tuple b in ker (ϕ), where ϕ = φT . In particular, ker (ϕ) ∩ R = ∅. By Theorem 4.5, one can assume that ϕ ∈ Char (A), that is, ϕ ∈ KR . Moreover, a is a tuple in ker (ϕ). Therefore 0 =  a (ϕ) ∈ τR (a), that is, σR (a) ⊆ τR (a). Thus σR (a) = τR (a). It remains to apply Proposition 5.3.


6. Slodkowski and Harte type spectra In this section we compare various subspectra and investigate when a subspectrum on a Banach algebra is reduced to the Harte type spectrum. Let A be a unital Banach algebra. In Section 4, a correspondence τ → Rτ between subspectra on A and regularities in A has been proposed. Further, in Section 5, we have considered a correspondence R → σR between regularities and Harte type spectra, which can regarded as a right inverse of the first one, since RσR = R# by virtue of Theorem 5.7. Thus RσR = R whenever R# = R. Theorem 6.1. Let τ be a subspectrum on a unital Banach algebra A. Then τ ⊆ σRτ . Moreover, τ = σR for a regularity R that has an open and proper envelope R# iff each ϕ ∈ Char (A) with ker (ϕ) ∩ R = ∅ is a τ -singular functional on A.

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Proof. Put KRτ = {ϕ ∈ Char (A) : ker (ϕ) ∩ Rτ = ∅} and let a be a k-tuple in A. By Theorem 5.7, σRτ (a) = τRτ (a) =  a (KRτ ). Moreover,  A/Rτ# = {ker (ϕ) : ϕ ∈ KRτ } thanks to Theorem 4.5. But Rτ# =Rτ due to Proposition 4.4. Furthermore, on account of Corollary 4.7, A\Rτ = {ker (ϕ) : ϕ ∈ Kτ }, where Kτ ⊆ KRτ is the subset of all τ -singular functionals. Using Corollary 4.8, infer that   a (Kτ ) ⊆  a (KRτ ) = σRτ (a) , τ (a) = ϕ(k) (a) : ϕ ∈ Kτ =  that is, τ ⊆ σRτ . Now assume that τ = σR for some regularity R in A that has an open and proper envelope R# . Take ϕ ∈ KR . We want to prove that ϕ ∈ Kτ . If a is a k-tuple in ker (ϕ), then 0 = ϕ(k) (a) =  a (ϕ) ∈ τR (a) = σR (a) = τ (a) by virtue of Theorem 5.7. Thus 0 ∈ τ (a) for each tuple a in ker (ϕ), that is, ϕ is τ -singular. Further, if ϕ ∈ Kτ , then 0 ∈ τ (a) for each a ∈ ker (ϕ). It follows that 0 ∈ σR (a), that is, A (a) ∩ R# = ∅. In particular, a ∈ / R. Thus ker (ϕ) ∩ R = ∅, which means that ϕ ∈ KR . Thus KR = Kτ . Conversely, assume that Kτ = KR . Then a (KR ) =  a (Kτ ) = τ (a) σR (a) =  by virtue of Theorem 5.7 and Corollary 4.8. Thus σR = τ .




Note that the inclusion τ ⊆ σRτ stated in Theorem 6.1 can be proper, that is, there are Harte type spectrum σR and subspectrum τ with the same regularity R (that is, Rτ = R) such that τ = σR . That can be characterized in terms of characters. Consider a regularity R in A that has an open and proper envelope R# . Then we have a nonempty compact subset KR ⊆ Char (A) (see Lemma 5.1) of all the σR -singular functionals (see Theorem 5.7). For a closed subset K ⊆ KR we define its A-rationally convex hull in Char (A) as     = ϕ ∈ Char (A) : ker (ϕ) ⊆ K {ker (φ) : φ ∈ K} .   Note that {ker (φ) : φ ∈ K} ⊆ {ker (φ) : φ ∈ KR } = A\R# . It follows that  that is, K  ⊆ KR . In terms of the Gelfand transform ker (ϕ) ∩ R = ∅ for all ϕ ∈ K, we have     = ϕ ∈ Char (A) : ∀x ∈ A, x ∈ ker (ϕ) =⇒ x ∈ K {ker (φ) : φ ∈ K} = {ϕ ∈ Char (A) : ∀x ∈ A,

x  (ϕ) = 0 =⇒

0∈x  (K)}  (see [15], [23]). Since Rτ = R, it follows that A\R = {ker (ϕ) : ϕ ∈ Kτ } (see Corollary 4.7). But  A\R ⊇ A\R# = {ker (ϕ) : ϕ ∈ KR } .

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{ker (φ) : φ ∈ Kτ } for all ϕ ∈ KR . Thus τ = KR K

(6.1)

for a subspectrum with the regularity R. Theorem 6.2. Let R be a regularity in A that has an open and proper envelope  = KR . Then R# . Assume that K is a nonempty closed subset in KR such that K # there is a subspectrum τ on A such that Rτ = R , Kτ = K and τ ⊆ σR . Namely, τ (a) =  a (K) for a tuple a in A. Moreover, τ = σR iff K =KR . Proof. Evidently, the relation τ (a) =  a (K) determines a subspectrum on A (see the proof of Proposition 5.3). Moreover, / τ (a)} = {a ∈ A : 0 ∈ / a (K)} Rτ = {a ∈ A : 0 ∈  = {a ∈ A : ϕ (a) = 0, ϕ ∈ K} = A\ {ker (ϕ) : ϕ ∈ K}     = A\ {ker (ϕ) : ϕ ∈ KR } ker (ϕ) : ϕ ∈ K = A\ = R# . By Theorem 6.1, τ ⊆ σR# = σR . Now let us prove that Kτ = K. Clearly K ⊆ Kτ . Take ϕ ∈ Kτ . So, ϕ is a τ -singular character. We shall show that ϕ belongs to the closure of K. Take a neighborhood   Ua,ε (ϕ) = φ ∈ Char (A) : max |φ (ai ) − ϕ (ai )| < ε 1≤i≤k

of ϕ in Char (A), where a = (a1 , . . . , ak) is a k-tuple in A. Obviously, a − ϕ(k) (a) is a k-tuple in ker (ϕ). Therefore 0 ∈ τ a − ϕ(k) (a) , which in turn implies that   φ(k) a − ϕ(k) (a) = 0 for some φ ∈ K. Thus φ(k) (a) = ϕ(k) (a). It follows that φ ∈ Ua,ε (ϕ) ∩ K for any ε. Taking into account that K is closed, infer that ϕ ∈ K. Finally, if K is a proper subset in KR , then Kτ = KR , for K = Kτ as we have just proven. By Theorem 6.1, τ = σR . Conversely, if τ = σR , then τ (a) = σR (a) for a k-tuple a in A. It follows that 0 ∈ σR (b) \τ (b) for a k-tuple b in A. According k to Theorem 5.7, b ∈ ker (ϕ) for some ϕ ∈ KR . But 0 ∈ / τ (b). It follows that ϕ is / K.
not τ -singular, that is, ϕ ∈ / Kτ . Thereby ϕ ∈ Thus each regularity may generate a family of subspectra different from Harte type spectrum. Let us illustrate this by an example. Example. Let A be the algebra of all continuous functions on the closed ball   2 2 B (0, 1) = (z, w) ∈ C2 : |z| + |w| ≤ 1

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centered at the origin that are holomorphic on its interior, and let B (0, 1) be the unit open ball in C2 centered at the origin. The algebra A furnished with the uniform norm on B (0, 1) is a commutative semisimple Banach algebra. Moreover, Char (A) = B (0, 1) . Take a closed subset K ⊆B (0, 1) containing the topological (or Shilov) boundary  = B (0, 1). Indeed, first note that of B (0, 1). Then K B (0, 1) \K ⊆B (0, 1)  Assume that (z, w) ∈ by assumption. Take (z, w) ∈ B (0, 1). Prove that (z, w) ∈ K. / K. Then (z, w) ∈ B (0, 1). If f (z, w) = 0 for a function f ∈ A, then f (z0 , w0 ) = 2 2 0 for some (z0 , w0 ), |z0 | + |w0 | = 1, by the known property of holomorphic  Thus functions. But (z0 , w0 ) ∈ K, therefore 0 ∈ f (K). It follows that (z, w) ∈ K.  = B (0, 1). Further, demonstrate that if τ is a subspectrum on A associated K with K (see Theorem 6.2), then K is exactly the set of all τ -singular characters. Indeed, take (a, b) ∈ B (0, 1) which is a τ -singular character. We have to prove that (a, b) ∈ K. It suffices to assume that (a, b) ∈ B (0, 1). Consider the polynomials p (z, w) = z − a and q (z, w) = w − b. Obviously, p (a, b) = q (a, b) = 0. Since (a, b) is a τ -singular character, it follows that 0 ∈ τ (p, q) = {(p (z, w) , q (z, w)) : (z, w) ∈ K} . Then p (z0 , w0 ) = q (z0 , w0 ) = 0 for some (z0 , w0 ) ∈ K. Whence z0 = a and w0 = b, or (a, b) = (z0 , w0 ) ∈ K. The example can be modified by extending the boundary as in [18]. Now we apply Theorem 6.2 to demonstrate a difference between Slodkowski and Harte type spectra. Let us start with simple assertions. Lemma 6.3. Let α : A → B be a unital algebra homomorphism between unital Banach algebras A and B, and let R be a regularity in B. Then so is α−1 (R) and   # α−1 (R) ⊆ α−1 R# .    #   In particular, α−1 R# is a regularity in A such that α−1 R# = α−1 R# . Moreover, if R has a proper envelope R# in B, then α−1 (R) has a proper envelope too. Proof. Take a, b ∈ A. Then ab ∈ α−1 (R) iff α (ab) = α (a) α (b) ∈ R which in turn is possible (see Definition 4.2) iff both α (a) , α (b) ∈ R, that is, a, b ∈ α−1 (R). / R# . So, ψ (α (a)) = 0 for some Further, take a ∈ A\α−1 R# . Then α (a) ∈ ∗ ψ ∈ B , ker (ψ) ∩ R = ∅. It follows that a ∈ ker (ϕ) and ker (ϕ) ∩ α−1 (R) = ∅,   # # where ϕ = ψα. Thus a ∈ / α−1 (R) and therefore α−1 (R) ⊆ α−1 R# .

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  Using Corollary 4.6, infer that R# is a regularity in B. Therefore α−1 R# is a regularity in A. Moreover, on account of the inclusion that we have just proven and Lemma 4.1, we deduce that    #     α−1 R# ⊆ α−1 R# ⊆ α−1 R## = α−1 R# ,  #   that is, α−1 R# = α−1 R# .   Finally, if R# = B, then 0 ∈ / R# and therefore 0 ∈ / α−1 R# . Since   α−1 (R)# ⊆ α−1 R# , it follows that α−1 (R)# is proper.
Corollary 6.4. (see [15, Proposition 3.3 ]) If α : A → B is a bounded algebra homomorphism and R is a regularity in B such that R = R# is open, then # α−1 (R) = α−1 (R) is an open regularity too. Now assume that A is a unital Banach algebra such that its attendant Lie algebra Alie is nilpotent, and let α : A → B (X) be a unital bounded algebra homomorphism, that is, a bounded representation of A on the Banach space X. Put Rα = α−1 (G (B (X))) . (6.2) If B is the closure of the inverse closed envelope of the nilpotent Lie algebra α(A) in B(X), then B is an inverse closed Banach algebra, which is commutative modulo its Jacobson radical thanks to Turovskii’s lemma. By Lemma 4.3, the set G (B) is a regularity in B and G (B) = G (B)# . Furthermore, Rα = α−1 (B ∩ G(B(X)) = α−1 (G(B)). Using Lemma 6.3 and Corollary 6.4, infer that Rα is a regularity in # is an open proper subset in A. A such that Rα = Rα If a is a s-tuple in A, then the Lie subalgebra L (a) in Alie generated by a is nilpotent. Being a finitely generated nilpotent Lie algebra, L (a) has a finite m dimension. In particular, if p (e) ∈ Fk (e) is a m-tuple of polynomials, then L (p (a)) is a finite dimensional nilpotent Lie subalgebra in Alie . Consider a unital bounded representation α : A → B (X) and a s-tuple a in A. Then π|L(a) : L (a) → B (X) isa Lie representation. Using the Koszul complex generated by the L (a) module X, π|L(a) , it is defined a family of Slodkowski spectra {σπ,k (a) , σδ,k (a) : k ≥ 0} (see [8]), which are compact subsets in Ck (see [3]). So, we have a family S = {σπ,k , σδ,k : k ≥ 0} of set-valued functions over all tuples in A. Proposition 6.5. Each τ ∈ S is a subspectrum on A. Proof. Fix a s-tuple a in A and let A (L (a)) be the associative subalgebra in A generated by the nilpotent Lie algebra L (a). The algebra A (L (a))  furnished  with the finest locally convex topology is dominating over the module X, π|L(a) in the sense of [8, Definition 4], we write   A (L (a))  X, π|L(a) .

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Taking into account that A (L (a)) is comprising polynomials in elements of L (a), we may apply the noncommutative spectral mapping theorems from [8]. As we have confirmed above each tuple p (a) in A (L (a)) generates a finite-dimensional nilpotent Lie subalgebra L (p (a)), therefore τ (p (a)) = p (τ (a)) for all τ ∈ S, due to [8, Propostion 6 and Corollary 8]. Thus τ is a subspectrum on A.
Take τ ∈ S and x ∈ A. Then τ (x) = σ (α (x)) for all k > 1. Therefore / τ (x)} = {x ∈ A : 0 ∈ / σ (α (x))} Rτ = {x ∈ A : 0 ∈ = {x ∈ A : α (x) ∈ G (B (X))} = α−1 (G (B (X))) = Rα (see (6.2)). Thus Rα = Rτ for all τ ∈ S. By Theorem 6.1, τ ⊆ σRα . Moreover,   τ = KRα using Corollary 4.8, infer that τ (a) = ϕ(s) (a) : ϕ ∈ Kτ . Therefore K by (6.1). In particular, we have a chain K0 ⊆ K1 ⊆ · · · ⊆ Kn ⊆ Kn+1 ⊆ · · · n = KRα for all n ≥ 0, where of increasing compact subsets in KRα such that K Kn = Kσπ,n . The known [16] example by Z. Slodkowski shows that spectra σπ,n are different for a Hilbert space representation α of a commutative Banach algebra. So, if σπ,n = σπ,n+1 , then Kn = Kn+1 and Kn turns out to be a nonempty proper n = KRα by virtue of Theorem 6.2. closed subset in KRα such that K Finally, let us consider the Taylor spectrum σT which is defined as σT (a) = σπ,n (a) if a is a n-tuple in A. By Proposition 6.5, σT is a subspectrum on A, therefore   σT (a) = ϕ(n) (a) : ϕ ∈ KσT ,  where KσT is a closed subset in KRα such that K σT = KRα . Moreover, Kn ⊆ KσT for all n. But again KσT may be a proper subset of KRα as shows the example in [2] by R. Berntzen and A. Soltysiak. Namely, there are commuting Banach space operators a, b ∈ B (X) such that σG(B(X)) (a, b) is not contained in σT (a, b). If A is the closed unital associative subalgebra in B (X) generated by a and b, and α is the identical representation A → B (X), then σG(B(X)) (a, b) ⊆ σA∩G(B(X)) (a, b) . Therefore σA∩G(B(X)) (a, b) is not contained in σT (a, b). Thus KσT is a proper  closed subset in KRα such that K σT = KRα . We end the paper by proposing an example of a noncommutative Banach algebra A with its nilpotent attendant Lie algebra Alie . That will demonstrate a gap between commutative and noncommutative cases. For the sake of generality, we consider the case of an Arens-Michael (locally multiplicatively associative) algebra

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reducing it to a Banach algebra. Fix a Heisenberg algebra g with its generators e1 , e2 , e3 (see Section 2). Thus [e1 , e2 ] = e3 and [ei , e3 ] = 0 for all i. Let A be an Arens-Michael algebra contained the Heisenberg algebra g as a Lie subalgebra (in Alie ) and the associative subalgebra in A generated by g is dense in it. If U (g) is the universal enveloping algebra of g, then the canonical embedding ι : g → A is extended up to a canonical algebra homomorphism  ι : U (g) → A with the dense range. Proposition 6.6. If ker ( ι) = {0}, then e3 is a nilpotent element in A. In particular, Alie is a nilpotent Lie algebra. Proof. Let C = im ( ι). By assumption, C is a dense subalgebra in A. One can easily (k) verify that the k-th term Clie of the lower central series of the Lie subalgebra Clie (k)

(k)

is contained in Cek3 , k ∈ N. Moreover, Alie is included into the closure Clie . To establish that Alie is nilpotent, it suffices to prove that e3 is nilpotent in A. Let {·ν : ν ∈ Λ} be a family of multiplicative seminorms on A defining its locally convex topology and let Iν = {a ∈ A : aν = 0}. Then Iν is a twosided ideal in A and A/Iν is a normed algebra with respect to the quotient norm induced by ·ν . Let Aν be its norm-completion. The family of Banach algebras {Aν } generates an inverse system and its inverse limit is topologically isomorphic to A [12, 5.2.17]. It is obvious that the associative subalgebra in Aν generated by the nilpotent Lie subalgebra πν (g) is dense in Aν , where πν : A → Aν is the canonical map, ν ∈ Λ. By Turovskii’s lemma, Aν is commutative modulo its Jacobson radical, thereupon πν (e3 ) is a quasinilpotent element in Aν . On that account we conclude that σ (e3 ) = {0}, for  σ (e3 ) = σ (πν (e3 )) ν∈Λ

(see [12, 5.2.12]). To prove that e3 is nilpotent, one suffices to demonstrate that q (e3 ) = 0 for a certain nonzero polynomial q (z) of one complex variable z [11, Problem 97]. By assumption  ι (p) = 0 for some nonzero p ∈ U (g). By Poincare-BirkhoffWitt theorem, n  p= pm (e1 , e2 ) em 3 m=0

for some polynomials pm = pm (z, w) in two complex variables. Assume that i = max {deg (pm )}, where deg (pm ) is the degree (maximum of the homogeneous degrees) of pm . Put i = deg (pm1 ) = · · · = deg (pms ) for some mj , 0 ≤ m1 < · · · < ms ≤ n. Consider the polynomial pm1 . Then pm1 (e1 , e2 ) = λkm1 ek1 ei−k + qm1 (e1 , e2 ) 2 such that λkm1 = 0 and qm1 (z, w) is a polynomial without the monomial z k wi−k , for some k. Let λkmj be the coefficient of z k wi−k in pmj (z, w), 2 ≤ j ≤ s. So

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pm1 (e1 , e2 ) = λkmj ek1 ei−k + qmj (e1 , e2 ), where qmj (z, w) is a polynomial without 2 k i−k z w . Consider a linear operator k

k

Ti,k = (−1) (ad e2 ) (ad e1 )

i−k

: U (g) → U (g) .

Note that     Ti,k es1 et2 er3 = Ti,k es1 et2 er3 =

s!t! es−k et−i+k ei+r 2 3 (s − k)! (t − i + k)! 1

s t r (here we have assumed that e−q p = 0, p = 1, 2, q ∈ N), thereby Ti,k (e1 e2 e3 ) = 0 only when s ≥ k and t ≥ i − k. It follows that         + Ti,k qmj (e1 , e2 ) = λkmj Ti,k ek1 ei−k Ti,k pmj (e1 , e2 ) = Ti,k λkmj ek1 ei−k 2 2

= k! (i − k)!λkmj ei3 , for all j, 1 ≤ j ≤ s. Thus Ti,k (p) =

s  j=1

s   m  i+m Ti,k pmj (e1 , e2 ) e3 j = k! (i − k)! λkmj e3 j .

s

j=1 i+m

We set q = k! (i − k)! j=1 λkmj e3 j , which is a nonzero polynomial in U (g). ι) ⊆ Prove that q (e3 ) = 0 in A. Being a two-sided ideal in U (g), the subspace ker ( U (g) is invariant with respect to the operator Ti,k . With p ∈ ker ( ι) in mind, infer that q = Ti,k (p) ∈ ker ( ι). Therefore q (e3 ) =  ι (q) = 0. Thus e3 is a nilpotent
element in A. It follows that Alie is a nilpotent Lie algebra. The assertion stated in Proposition 6.6 can be proved for arbitrary nilpotent Lie algebra. If A is a closed associative envelope of a finite-dimensional nilpotent Lie algebra g and all elements from [g, g] are nilpotent in A, then Alie is nilpotent. We omit the details. Acknowledgment I wish to thank A. Soltysiak and A. Wawrzynczyk for sending me papers [2], [15], [10] and [23]. The author also thanks the referee for the proposed comments.

References [1] D. Beltita, M. Sabac, Lie algebra of bounded operators, Birkh¨auser, Basel, 2001. [2] R. Berntzen, A. Soltysiak, The Harte spectrum is not contained in the Taylor spectrum, Comment. Math. Prace Mat. 38 (1998) 29-35. [3] E. Boasso, Dual properties and joint spectra for solvable Lie algebras of operators, J. Operator Theory 33 (1995) 105-116. [4] N. Bourbaki, Spectral Theory, MIR (Moscow) 1972. [5] N. Bourbaki, Algebra, modules, rings, forms, Nauka (Moscow) 1966. [6] A. A. Dosiev, Formally-radical functions in elements of a nilpotent Lie algebra and noncommutative localizations, Algebra Colloquium (to appear).

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[7] A. A. Dosiev, Local left invertibility of operator tuples and noncommutative localizations, J. K-theory (to appear). [8] A. A. Dosiev, Cartan-Slodkowski spectra, splitting elements and noncommutative spectral mapping theorems, J. Funct. Anal. 230 (2) (2006) 446-493. [9] A. A. Dosiev, Cohomology of Sheaves of Fr´echet algebras and spectral theory, Funct. Anal. and its Appl. 39 (3) (2005) 225-228. [10] A. V. Gonzalez , A. Wawrsynczyk, The projection property of a family of ideals in subalgebras of Banach algebras, Bol. Soc. Mat. Mexicana (3) 8 (2002) (2), 155-160. [11] R. P. Halmos, A Hilbert space problem book, Springer-Verlag, New York (1982). [12] A. Ya. Helemskii, Banach and polinormed algebras: general theory, representations, homology, Moscow, Nauka, 1989. [13] V. M¨ uller, Spectral Theory of Linear Operators and Spectral systems in Banach algebras, Operator Theory: Adv. Appl. 139, Birkh¨ auser, Basel, 2003. [14] W. Rudin, Functional Analysis, Inter. Ser. Pure Appl. Math. 1991. [15] J. Saul, C. Orozco, A. Wawrzynczyk, Regularities and subspectra for commutative Banach algebras, Inter. J. Math. and Math. Sciences. 15 (2005) 2399-2407. [16] Z. Slodkowski, An infinite family of joint spectra, Stud. Math. LXI (1977), 239-255. [17] Z. Slodkowski, W. Zelazko, On joint spectra of commuting families of operators, Stud. Math. 50 (1974) 127-148. [18] A. Soltysiak, The strong spectral extension property does not imply the multiplicative Hahn-Banach property, Stud. Math. 153 (3) (2002) 297-301. [19] J. L. Taylor, A general framework for a multi-operator functional calculus, Adv. Math. 9 (1972), 183-252. [20] Yu. V. Turovskii, On spectral properties of elements of normed algebras and invariant subspaces, Funktsional. Anal. i Prilozhen. 18 (2) (1984) 77-78. [21] W. Zelazko, Banach algebras, Elsevier, Amsterdam; PWN, Warsaw, 1973. [22] W. Zelazko, An axiomatic approach to joint spectra, Studia Math. 64 (3) (1979) 249-261. [23] A. Wawrzynczyk, On subspectra generated in subalgebras, Bull. London Math. Soc. 35 (2003) 367-372. Anar Dosiev Middle East Technical University NCC G¨ uzelyurt, KKTC Mersin 10 Turkey e-mail: [email protected] Submitted: September 28, 2006 Revised: February 12, 2008

Integr. equ. oper. theory 61 (2008), 365–399 0378-620X/030365-35, DOI 10.1007/s00020-008-1597-2 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Integral Equations and Operator Theory

Scattering in a Forked-Shaped Waveguide Y. Latushkin and V. Pivovarchik Abstract. We consider wave scattering in a forked-shaped waveguide which consists of two finite and one half-infinite intervals having one common vertex. We describe the spectrum of the direct scattering problem and introduce an analogue of the Jost function. In case of the potential which is identically equal to zero on the half-infinite interval, the problem is reduced to a problem of the Regge type. For this case, using Hermite-Biehler classes, we give sharp results on the asymptotic behavior of resonances, that is, the corresponding eigenvalues of the Regge-type problem. For the inverse problem, we obtain sufficient conditions for a function to be the S-function of the scattering problem on the forked-shaped graph with zero potential on the half-infinite edge, and present an algorithm that allows to recover potentials on the finite edges from the corresponding Jost function. It is shown that the solution of the inverse problem is not unique. Some related general results in the spectral theory of operator pencils are also given. Mathematics Subject Classification (2000). Primary 34A55, 34B24; Secondary 43L20,34C25. Keywords. Inverse problems on graphs, operator pencils, Jost and HermiteBiehler functions, Sturm-Liouville problem, Dirichlet boundary conditions, S-function, Marchenko equation, Regge problem.

1. Introduction Scattering problems on graphs have been considered in many publications, see, for example, [3, 6, 7, 8, 11, 13, 14, 19, 20, 25, 26, 45], because of their general importance and, in particular, because of their significance in the theory of electronic micro-schemes [1, 12]. The corresponding inverse problems have been solved in This work was supported by the grant UM1-2567-OD-03 from the Civil Research and Development Foundation (CRDF). YL was partially supported by the NSF grants 0338743, 0354339 and 0754705, by the Research Board and Research Council of the University of Missouri, and by the EU Marie Curie “Transfer of Knowledge” program.

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[13, 20, 39, 41]. However, the problem of characterizing scattering data, i.e., the Sfunction, normal eigenvalues (often referred to as the energies of bound states), and normalizing constants, usually appears to be rather complicated (see [13, 20, 41]). In this paper we treat both direct and inverse scattering problems for the case of a simple forked-shaped graph having one half-infinite and two finite edges. The most complete results are obtained in case of the potential which is identically equal to zero on the half-infinite edge of the graph. In this case we give sufficient conditions for a set of data to be scattering data and show that these conditions are close to be necessary. Although the scattering theory for the forked-shaped graph shares many common features with the classical theory for the half-axis, it turns out that the situation considered in the current paper is essentially more complex than classical and exhibits numerous new effects. One of them is the possible presence of real eigenvalues (the bound states embedded in continuous spectrum, in terms of quantum mechanics). Another complication is the nonuniqueness of solutions of the inverse problem as described in Section 4 below. Yet another new effect is related to location of zeros of the Jost function, and occurs even in case of zero potential on the half-infinite edge. The latter setting corresponds to the case of a finitely supported potential in classical scattering theory where it suffices to know only a (meromorphic) scattering function to be able to recover the potential uniquely. In contrast to the classical case, for scattering on the forkedshaped graph with zero potential on the half-infinite edge, one can not claim that a meromorphic scattering function determines potentials on the finite edges uniquely even assuming that the corresponding Jost function has no real zeros (except, maybe, a simple zero at the origin); in addition, one needs to suppose that the Jost function has no pure imaginary zeros symmetric about the origin. The following spectral problem describes one-dimensional scattering of a quantum particle when the way of propagation is a graph which consists of two finite and one half-infinite intervals (edges) having one common vertex: yj

+ (λ2 − qj (x))yj = 0,

x ∈ [0, a],

y3



x ∈ [0, ∞),

2

+ (λ − q3 (x))y3 = 0,

j = 1, 2,

(1.1) (1.2)

y1 (λ, a) = y2 (λ, a) = y3 (λ, 0),

(1.3)

y1
(λ, a)

(1.4)

+

y2
(λ, a)

−

y3
(λ, 0)

= 0,

y1 (λ, 0) = 0,

(1.5)

y2 (λ, 0) = 0.

(1.6)

Here, λ is a complex spectral parameter, and the potentials are assumed to be realvalued and satisfy qj (x) ∈ L2 (0, a) for j = 1, 2 and xq3 (x) ∈ L1 (0, ∞) ∩ C[0, ∞). The essential spectrum of the operator corresponding to (1.1)–(1.6) covers the positive semi-axis (and thus, since we use λ2 as the spectral parameter in (1.1)–(1.2), the essential spectrum of problem (1.1)–(1.6) covers the real axis). In this paper we show that there may be only a finite number of normal eigenvalues of (1.1)–(1.6) lying on the imaginary axis, and a finite or infinite number of

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eigenvalues that belong to the essential spectrum. We construct an analogue of the S-function of classical quantum scattering theory (see [35] or [31, Chap.3]), also known as the coefficient of reflection in the theory of mechanical or electromagnetic wave propagation. Also, assuming that the potential q3 (x) is identically equal to zero for x ∈ [0, ∞), we solve the corresponding inverse problem; i.e., the problem of recovering the potentials q1 (x) and q2 (x) for a given S(λ). The paper is organized as follows. In Section 2 we first describe properties of a differential operator, A, corresponding to the boundary value problem (1.1)–(1.6) when all three potentials qj (x) are, generally, nonzero, see Theorem 2.2. Next, we introduce an analogue of the Jost function for this boundary value problem, that is, a function E(λ) whose zeros in the lower half-plane are the normal eigenvalues of the problem, see Theorem 2.3. Also, we count in Theorem 2.4 the number of the normal eigenvalues for (1.1)–(1.6) via the number of negative eigenvalues of the Sturm-Liouville problems on each edge of the waveguide. These results use machinery from [5, 21, 22] related to the Nevanlinna, or R-functions. Finally, we introduce an analogue of the scattering, or S-function, for (1.1)–(1.6). In Section 3 we specialize to the case when the potential q3 (x) is identically equal to zero on the half-infinite edge of the waveguide. Under this condition, the zeros of the Jost function E0 (−λ) for (1.1)–(1.6) with q3 (x) = 0, x ∈ [0, ∞), constitute the spectrum of a boundary value problem of the Regge type, cf. [46]. First, we give a description of this spectrum in Theorem 3.3 using some abstract results from the theory of linear operator pencils proved in Appendix A. Next, we derive in Lemma 3.4 a representation for the Jost function E0 (−λ) that allows us to obtain some preliminary information on asymptotics of its zeros in Lemma 3.6. Using this information and some more abstract results from Appendix A, we give in Theorem 3.7 a complete description of the geometric structure of the spectrum. In addition, we are able to prove in Theorem 3.14 that the Jost function belongs to the class of shifted symmetric generalized Hermite-Biehler functions. This fact has a number of consequences; the most notable is that the zeros of the “even” and “odd” parts of the Jost function interlace, which eventually helps to describe the asymptotic behavior of the zeros in Theorem 3.15. The information about the asymptotic behavior is, in fact, used in the sequel to setup the inverse problem for (1.1)–(1.6) with q3 (x) = 0, x ∈ [0, ∞). The inverse problem for (1.1)–(1.6) with q3 (x) = 0, x ∈ [0, ∞), is solved in Section 4. First of all, we describe a way of recovering E0 (−λ) from a given S(λ). Next, given a function E0 (−λ), we show how to recover potentials qj (x), j = 1, 2, in a way that E0 (−λ) becomes the corresponding Jost function for (1.1)–(1.6) with q3 (x) = 0, x ∈ [0, ∞). Two results of this type are proved. In a “simpler” Theorem 4.3 we are given an entire function E0 (λ) that has a rather special representation which allows us to use the classical inverse problem results from [31, Chap.3]. In a much more involved Theorem 4.4 the given entire function E0 (−λ) from the shifted symmetric Hermite-Biehler class is assumed to have a more general representation that resembles the representation in Lemma 3.4 used to treat the direct problem. First, we describe the asymptotic behavior of zeros of the “even” and “odd” parts

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of E0 (−λ) which, again, matches the behavior seen in the direct problem. Next, we use a general fact about the Hermite-Biehler functions (see Lemma 3.11 proved in Appendix B) to show that the behavior of the zeros of the “even” and “odd” parts matches the conditions needed to apply an inverse problem result from [38], thus enabling us to recover the potentials. Finally, in Appendix A we prove several abstract spectral results for operator pencils having some independent interest besides applications to the boundary value problem (1.1)–(1.6), and in Appendix B we collect necessary information on the Hermite-Biehler functions, give the proof of Lemma 3.11, and formulate the result from [38] used in the inverse problem part of the current paper.

2. Direct problem: general case For an operator A on a Hilbert space, we let D(A), ρ(A) and σ(A) denote its domain, resolvent set and spectrum. We refer to [16, Sec.I.2] for the definition of normal (that is, isolated Fredholm) eigenvalues, and denote by σ0 (A) the set of normal eigenvalues of A and by σess (A) = σ(A)\σ0 (A) the essential spectrum. Some standard notions from the spectral theory of operator pencils are collected in Appendix A. At this point we recall that the spectrum of any selfadjoint operator A coincides with its approximative spectrum, see, e.g., [9, p.118], where the latter ∞ is defined as the set of λ ∈ C such that there exists a sequence {fn }n=1 in D(A), called the approximate sequence for λ, with the properties ||fn || = 1 and (λI − ∞ A)fn → 0 as n → ∞. If the sequence {fn }n=1 is compact, then λ is either a normal eigenvalue, or an eigenvalue that belongs to the essential spectrum (in the latter case, in quantum mechanics, λ is called a bound state embedded into the continuous spectrum). We denote
by L11 (0, ∞) the class of functions f (x) ∈ ∞ L1 (0, ∞) with the finite first moment 0 x|f (x)|dx, and by C([0, ∞)) the class of continuous functions.   On the Hilbert space L2 (0, a) L2 (0, a) L2 (0, ∞) of vector-valued functions (yj (x))3j=1 we introduce an operator, A, related to the boundary value problem (1.1)–(1.6), acting as A(yi (x))3j=1 = (−yj

(x)+qj (x)yj (x))3j=1 with the domain  D(A) = (yj )3j=1 : yj (x) ∈ W22 (0, a), yj (0) = 0, j = 1, 2, (2.1) y3 (x) ∈ L2 (0, ∞), −y3

+ q3 (x)y3 (x) ∈ L2 (0, ∞),

 y1 (a) = y2 (a) = y3 (0), y1
(a) + y2
(a) − y3
(0) = 0 , where W22 is the usual Sobolev space. We identify the spectrum of the operator pencil λ2 I − A with the spectrum of the boundary value problem (1.1)–(1.6), i.e., λ ∈ C is called an eigenvalue of (1.1)–(1.6) if and only if λ2 is an eigenvalue of A. Hypothesis 2.1. Assume that the real-valued potentials qj (x), j = 1, 2, 3, satisfy conditions qj (x) ∈ L2 (0, a), j = 1, 2, and q3 (x) ∈ L11 (0, ∞) ∩ C[0, ∞).

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Theorem 2.2. Assume Hypothesis 2.1. Then the following assertions hold: (i) The operator A is self-adjoint, and is bounded from below, that is, A ≥ −βI, where I is the identity operator and β > 0. (ii) σess (A) = [0, ∞). (iii) The eigenvalues of A on the essential spectrum are simple. Proof. First, we claim that A is symmetric. Indeed, for Y = (yj (x))3j=1 ∈ D(A) and Z = (zj (x))3j=1 ∈ D(A), integrating by parts, we obtain: 

(AY, Z) =

 ∞ y2

z2 dx − y3

z3 dx 0 0 0  a  ∞  a q1 y1 z1 dx + q2 y2 z2 dx + q3 y3 z3 dx + a

y1

z1 dx −



0

a

0

0

= −y1
(a)z1 (a) − y2
(a)z2 (a) + y3
(0)z1 (0)  a  a  ∞ + y1
z1
dx + y2
z2
dx + y3
z3
dx 0 0 0  a  ∞  a q1 y1 z1 dx + q2 y2 z2 dx + q3 y3 z3 dx. + 0

0

0

Since Y ∈ D(A) and Z ∈ D(A), we have z1 (a) = z2 (a) = z3 (0) and y1
(a) + y2
(a) − y3
(0) = 0, and therefore  a  a  ∞



(AY, Z) = y1 z1 dx + y2 z2 dx + y3
z3
dx 0 0 0  a  ∞  a q1 y1 z1 dx + q2 y2 z2 dx + q3 y3 z3 dx. (2.2) + 0

0

0

Another integration by parts yields  a  a  ∞



(AY, Z) = − y1 z1 dx − y2 z2 dx − y3 z3

dx 0 0 0  a  a  ∞ + q1 y1 z1 dx + q2 y2 z2 dx + q3 y3 z3 dx = (Y, AZ), 0

0

0

proving the claim. Letting Z = Y in (2.2), we obtain  a  a  ∞ (AY, Y ) = |y1
|2 dx + |y2
|2 dx + |y3
|2 dx 0 0 0  a  ∞  a 2 2 q1 |y1 | dx + q2 |y2 | dx + q3 |y3 |2 dx. + 0

0

(2.3)

0

Using the description of the domain of A∗ , as in [47, Sec.7.5], it follows that A is self-adjoint.

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The operator A is a self-adjoint extension of the operator A0 defined by the formula A0 (yi (x))3j=1 = (−yj

(x) + qj (x)yj (x))3j=1 with the domain  (2.4) D(A0 ) = (yj )3j=1 : yj (x) ∈ W22 (0, a), yj (0) = 0, j = 1, 2, y3 (x) ∈ L2 (0, ∞), −y3

+ q3 (x)y3 (x) ∈ L2 (0, ∞),

 y1 (a) = y2 (a) = y3 (0) = y1
(a) = y2
(a) = y3
(0) = 0 . The operator A0 is the direct sum of symmetric, closed, and bounded from below operators, cf., e.g., [34, Thm.V.19.5]. Therefore, A0 is also symmetric, closed, and bounded from below (that is, A0 ≥ −β1 I for some β1 > 0). Furthermore, using Theorem 16 in [34, Sec.IV], we conclude that the part of the spectrum of A located below −β1 consists of no more than finite number of normal eigenvalues. To prove assertion (ii), for any given λ2 ≥ 0 we construct an approxi(n) mate sequence Yn (x) in D(A) for λ2 by letting Yn (x) = (1/3)(yj (x))3j=1 , where (n)

(n)

we choose y3 (x) = n−1/4 exp(−n−1 x2 + iλx) and yj (x) ∈ W22 (0, a) so that
a (n) (n) (n) (n) yj (0) = 0, 0 |yj (x)|2 dx → 0, (yj )

+ (λ2 − qj (x))yj L2 (0,a) → 0 as (n)

(n)

n → ∞, j = 1, 2, and, in addition, such that y1 (a) = y2 (a) = n−1/4 and (n) (n) (y1 )
(a) + (y2 )
(a) − iλn−1/4 = 0. This yields the inclusion σess (A) ⊇ [0, ∞). The inverse inclusion holds by Weyl’s theorem since A is a relatively compact perturbation of the operator corresponding to the boundary value problem (1.1)–(1.6) with all three identically zero potentials. To prove assertion (iii), we remark that if λ2 ≥ 0 is an eigenvalue of A, then the trivial solution y3 (x) = 0 is the only solution of (1.2) that belongs to L2 (0, ∞). Therefore, for the corresponding eigenvector Y (x) = (yj (x))3j=1 of A one has y3 (x) = 0, and consequently y1 (x) and y2 (x) satisfy the conditions y1 (a) = y1 (0) = y2 (a) = y2 (0) = 0 and y1
(a) + y2
(a) = 0.
Below, we will use some special solutions of the differential equations (1.1)– (1.2). If j = 1, 2 and λ ∈ C, then we let sj (λ, x) denote the solution of (1.1) which satisfies the conditions sj (λ, 0) = s
j (λ, 0) − 1 = 0, and let cj (λ, x) denote the solution of (1.1) which satisfies the conditions cj (λ, 0) − 1 = c
j (λ, 0) = 0. The functions sj (λ, x) and cj (λ, x) form a fundamental system of solutions of (1.1), and thus for any solution yj (x) of (1.1) there exist some constants aj , bj such that yj (x) = aj sj (λ, x) + bj cj (λ, x),

x ∈ [0, a],

j = 1, 2,

λ ∈ C.

(2.5)

The Jost solutions of equation (1.2) will be denoted by e(λ, x), Im λ ≥ 0, and e(−λ, x), Im λ ≤ 0; we recall from [31, Sec.3.1] that the Jost solutions can be represented as  ∞ K(x, t)eiλt dt, Im λ ≥ 0, (2.6) e(λ, x) = eiλx + x  ∞ K(x, t)e−iλt dt, Im λ ≤ 0, (2.7) e(−λ, x) = e−iλx + x

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where K(x, t) is the integral kernel of a transformation operator that satisfies some well-known properties listed, e.g, in [31, Lem.3.1.1]. Moreover, the function e(λ, x) is analytic in the open upper half-plane {λ ∈ C : Im λ > 0} and continuous in the closed upper half-plane {λ ∈ C : Im λ ≥ 0}, see [31, Lem.3.1.3]. If Im λ = 0, then both Jost solutions e(±λ, x) are defined; moreover, if Im λ = 0 and λ = 0, then the functions e(−λ, x) and e(λ, x) form a fundamental system of solutions of equations (1.2), cf. [31, Lem.3.1.3], and thus if y3 (x) is a solution of (1.2), then for some constants a3 , b3 one has: y3 (x) = a3 e(−λ, x) + b3 e(λ, x),

x ∈ [0, ∞),

Im λ = 0,

λ = 0.

(2.8)

Using notation just introduced, we define the following function E(λ) which is analytic in the open upper half-plane {λ ∈ C : Im λ > 0} and continuous in the closed upper half-plane {λ ∈ C : Im λ ≥ 0}: E(λ) = s
1 (λ, a)s2 (λ, a)e(λ, 0) + s1 (λ, a)s
2 (λ, a)e(λ, 0) − s1 (λ, a)s2 (λ, a)e
(λ, 0),

Im λ ≥ 0.

(2.9)

Also, since sj (−λ, x) = sj (λ, x) for j = 1, 2, we remark that E(−λ) = s
1 (λ, a)s2 (λ, a)e(−λ, 0) + s1 (λ, a)s
2 (λ, a)e(−λ, 0) − s1 (λ, a)s2 (λ, a)e
(−λ, 0),

Im λ ≤ 0,

(2.10)

and the function E(−λ) is analytic in the open lower half-plane and continuous in the closed lower half-plane. Theorem 2.3. Assume Hypothesis 2.1. Then: (i) The set of normal eigenvalues of problem (1.1)–(1.6) which are located in the open lower half-plane coincides with the set of zeros of the function E(−λ) located in the open lower half-plane. In addition, these zeros belong to the imaginary axis. (ii) The geometric multiplicity of any normal eigenvalue does not exceed 2. Proof. Let us determine the normal eigenvalues of the boundary value problem (1.1)–(1.6) that belong to the open lower half plane, and correspond to solutions (yj (x))3j=1 of (1.1)–(1.6) from D(A). For this, see (2.5), we note that the functions yj (x) = aj sj (λ, x), j = 1, 2, satisfy conditions (1.5),(1.6). Next, for Im λ < 0, we need to consider two linearly independent solutions of (1.2). One of these two solutions, e(−λ, x), is given by (2.7). The second linearly independent solution will be denoted by e˜(−λ, x); this is the solution with the asymptotics e˜(−λ, x) = eiλx (1 + o(1))

as x → ∞.

(2.11)

The solution e˜(−λ, x) exists by Theorem 7 in [34, Sec.VII.2] and grows exponentially as x → ∞. Then every solution y3 (λ, x) of (1.2) is of the form y3 (λ, x) = a3 e(−λ, x) + b3 e˜(−λ, x). Since we are looking for a square summable solution y3 (λ, x), we must have b3 = 0. Substituting yj (x) = aj sj (λ, x), j = 1, 2 and y3 (λ, x) = a3 e(−λ, x) in the boundary conditions (1.3)–(1.4), we obtain a 3 × 3 system of algebraic equations for aj , j = 1, 2, 3. This system has a nonzero solution

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if and only if λ is a normal eigenvalue of A with Im λ < 0. In turn, this happens if and only if λ is a root of the equation  s (λ,a) −s (λ,a)  0 1 2 s (λ,a) 0 −e(−λ,0) E(−λ) := det 1
= 0. (2.12)

s1 (λ,a) s2 (λ,a) −e (−λ,0)

This proves the first part of assertion (i). Also, the geometric multiplicity of the eigenvalue λ is equal to the dimension of the null-space of the matrix in (2.12). Since the rank of this matrix is at least one, assertion (ii) follows. Since the eigenvalues of problem (1.1)–(1.6) are square roots of the eigenvalues of the self-adjoint operator A, the eigenvalues of (1.1)–(1.6) located in the open lower half-plane must be pure imaginary, finishing the proof of assertion (i).
Theorem 2.4. Assume Hypothesis 2.1. Then the number n of normal eigenvalues of the boundary value problem (1.1)–(1.6) located in the open lower half-plane (counting their multiplicities) satisfies the inequalities: n1 + n2 + n3 ≤ n ≤ n1 + n2 + n3 + 1,

(2.13)

where nj is the number of negative eigenvalues of the problem yj

+ (λ − qj (x))yj = 0,

yj (0) = yj (a) = 0,

j = 1, 2,

and n3 is the number of normal negative eigenvalues of the problem y3

+ (λ − q3 (x))y3 = 0,

y3 (0) = 0.

Proof. As we have seen in Theorem 2.3, the normal eigenvalues of A located in the open lower half-plane coincide with squares of the zeros of the function E(−λ) defined in (2.10); moreover, the multiplicities of the eigenvalues coincide with the multiplicities of the zeros. We introduce the function Ξ(λ) = −

E(−λ) , s1 (λ, a)s2 (λ, a)e(−λ, 0)

Im λ ≤ 0.

(2.14)

For β given in Theorem 2.2, we fix√ any β1 > β and introduce a new spectral parameter τ by the formula √ λ = τ − β1 , where we select the branch of the square root such that Im τ − β1 ≥ 0 for Im τ ≥ 0. Using (2.10), equation (2.14) reads as follows: s
[τ, a] s
2 [τ, a] e
[−τ, 0] − + , (2.15) Ξ[τ ] = − 1 s1 [τ, a] s2 [τ, a] e[−τ, 0] √ where we use notation f [±τ, x] = f (± τ − β1 , x). We claim that Ξ[τ ] is a Nevanlinna function (an R-function, in the terminology of [22]; in particular, Ξ[τ ] maps the open upper half-plane into itself). To prove the claim, we remark that a sum of Nevanlinna functions is again a Nevanlinna function. That −s
j [τ, a]/sj [τ, a], j = 1, 2, are Nevanlinna functions was proved in Lemma 2.3 of [22] for β1 = 0. The same proof works for β1 = 0. It remains to show

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that e
[τ, 0]/e[τ, 0] is a Nevanlinna function. We evaluate first its imaginary part:  e
[τ, 0] 1  e
[τ, 0] e
[τ, 0] e
[−τ, 0]e[−τ, 0] − e
[−τ, 0]e[−τ, 0] Im = − = . + e[τ, 0] 2i e[τ, 0] 2i |e[−τ, 0]|2 e[τ, 0] (2.16) Next, we substitute e[−τ, x] in equation (1.2), and multiply it by e[−τ, x] to infer e

[−τ, x]e[−τ, x] + (τ − β1 − q(x))e[−τ, x]2 = 0. Taking the imaginary part of this 2 equation, we have e

[τ, x]e[τ, x] − e

[τ, x]e[τ, x] + (τ − τ )|e[τ,  x]| = 0. Integrating, we finally have −e
[τ, 0]e[τ, 0] + e
[τ, 0]e[τ, 0] = 2iIm τ

∞

0

|e[τ, x]|2 dx. Using

arguments similar to [22, Lem.2.3], we conclude that the last term in (2.15) is indeed a Nevanlinna function, thus proving the claim. It follows from the claim, see, e.g., [5, Thm.II.3.1], that the real poles of Ξ[τ ] are simple, and there is at least one zero between any two neighboring poles. Also, it is clear that −1/Ξ[τ ] is a Nevanlinna function as well. Thus, all zeros of Ξ[τ ] are simple and zeros and poles of Ξ[τ√ ] interlace. Consequently, the poles and zeros of Ξ(λ), which lie on the interval (−i β1 , 0) of the imaginary axis, also interlace. Now Theorem 2.4 is proved as soon as the following assertion is verified: The smallest pole of Ξ[τ ] is smaller than the smallest zero of Ξ[τ ]. To prove this, let τ0 denote the smallest pole of the function Ξ[τ ], that is, by (2.14), the smallest zero of the function s1 [τ, a]s2 [τ, a]e[−τ, 0]. We claim that lim Ξ[τ ] = −∞ while

τ →−∞

lim

τ →τ0 , τ <τ0

Ξ[τ ] = −∞,

(2.17)

which implies the required assertion. The first formula in claim (2.17) follows from (2.15) and the asymptotic properties as τ → −∞ of the functions sj [τ, a], j = 1, 2, and e[−τ, 0] and the derivatives of these functions using formulas (2.6) and (3.13), (3.15). To prove the second formula in (2.17), we consider the case when s1 [τ0 , a] = 0 (the cases when τ0 is a zero of the function s2 [τ, a] or e[−τ, 0] are similar). Using (2.15) and writing s1 [τ, a] = s˙ 1 [τ0 , a](τ − τ0 ) + o(τ − τ0 ) as τ → τ0 , τ < τ0 (here “dot” denotes d/dτ ), we see that the second formula in (2.17) follows from the inequality s˙ 1 [τ0 , a]s
1 [τ0 , a] > 0.

(2.18)

It remains to prove (2.18). Differentiating s

1 [τ, x]+(τ −β1 )s1 [τ, x]−q1 (x)s1 [τ, x] = 0, we infer s˙

1 [τ, x]+(τ −β1 )s˙ 1 [τ, x]+s1 [τ, x]−q1 (x)s˙ 1 [τ, x] = 0. These two equations yield s

1 [τ, x]s˙ 1 [τ, x] − s˙

1 [τ, x]s1 [τ, x] = (s1 [τ, x])2 . Since the left-hand side of the last formula is equal to (s
1 [τ, x]s˙ 1 [τ, x] − s˙
1 [τ, x]s1 [τ, x])
, integrating with respect to x from 0 to a and noting that s1 [τ0 , 0] = s˙ 1 [τ0 , 0] = 0 and s1 [τ0 , a] = 0, we have (2.18), finishing the proof of Theorem 2.4.
Our next goal is to introduce an analogue of the scattering, or S-function for the boundary value problem (1.1)–(1.6), cf., e.g., [31, Lem.3.1.5]. The importance of this function in the classical case of scattering on the half-axis is well-known: indeed, the phase-shift, that is, the argument of the unitary S-function, is known

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to be a measurable quantity, see, e.g., [31, 35]. To define the S-function S(λ) for Im λ = 0, consider a triple (yj (x))3j=1 of solutions of (1.1)–(1.2) that satisfy all four conditions (1.3)–(1.6). Formula (2.8) for y3 (x) shows that, up to an independent of x multiple, y3 (x) could be written as y3 (x) = e(λ, x) − S(λ)e(−λ, x) for real λ = 0. Here, the function S(λ), called the S-function, should be chosen in a way that yj (x), j = 1, 2, 3, satisfy conditions (1.3)–(1.6). Since yj (x), j = 1, 2, must satisfy (1.5),(1.6), using (2.5) we have yj (x) = aj sj (λ, x), j = 1, 2. Substituting this and y3 (x) = e(λ, x) − S(λ)e(−λ, x) in (1.3) and (1.4), we obtain a 3 × 3 system of equations with unknowns a1 , a2 and S(λ). Solving this system, we arrive at the following formula for the S-function: S(λ) =

E(λ) , E(−λ)

Im λ = 0.

(2.19)

The denominator E(−λ) of this ratio is the analogue of the Jost function of classical scattering theory on the half-axis, cf. [31, Lem.3.1.5]. As we have seen in Theorem 2.2, similarly to the classical case, the zeros of E(−λ) in the lower half-plane coincide with the normal eigenvalues of (1.1)–(1.6).

3. Direct problem: zero half-line potential In this section we consider the case when the potential is identically equal to zero on the semi-infinite part of the waveguide, that is, throughout, we impose the following conditions on the potentials. Hypothesis 3.1. Assume that q3 (x) = 0 for all x ∈ [0, ∞) and that q1 and q2 are real-valued and satisfy qj (x) ∈ L2 (0, a), j = 1, 2. Under these assumptions, assertion (ii) in Theorem 2.3 can be refined as follows. Proposition 3.2. If q3 (x) = 0, x ∈ [0, ∞), then the operator A may have only simple normal eigenvalues. Proof. Under our assumption, e(−λ, x) = e−iλx and therefore 0) = 1 and  s (λ,a)e(−λ, −s2 (λ,a) 0 1 −1 e
(−λ, 0) = −iλ. Using this in (2.12), we see that rank s1
(λ,a)
0 ≥ 2 s1 (λ,a) s2 (λ,a) iλ 



s (λ,a) 0 s (λ,a) −1
since if det s1
(λ,a) iλ = 0, then det s1
(λ,a) iλ = 0. 1

1

In the case when q3 (x) = 0, x ∈ [0, ∞), the Jost function E(−λ) defined in (2.10) will be denoted by E0 (−λ), and could be simplified. Indeed, substituting e(−λ, x) = e−iλx in (2.10) we obtain: E0 (−λ) = s1 (λ, a)s
2 (λ, a) + s
1 (λ, a)s2 (λ, a) + iλs1 (λ, a)s2 (λ, a), λ ∈ C.

(3.1)

We remark that E0 (λ) is symmetric, that is, one has: E0 (−λ) = E0 (λ),

λ ∈ C.

(3.2)

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The scattering function defined in (2.19) can be expressed as follows: S(λ) =

E0 (λ) , E0 (−λ)

λ ∈ C.

(3.3)

We note that if q3 (x) = 0 for x ∈ [0, ∞), then S(λ), λ ∈ C, is a meromorphic function. The Jost function E0 (−λ) given in (3.1) is related to the following boundary value problem of the Regge type (cf. [46]): yj

+ (λ2 − qj (x))yj = 0,

x ∈ [0, a],

j = 1, 2,

(3.4)

yj (λ, 0) = 0, j = 1, 2,

(3.5)

y1 (λ, a) = y2 (λ, a), y1
(λ, a) + y2
(λ, a) = −iλy1 (λ, a).

(3.6) (3.7)

This problem was considered in [40] for the case when all eigenvalues are located in the upper half-plane. The set of zeros of E0 (−λ), located in the open lower halfplane, coincides with the part of the spectrum in the open lower half-plane of the Regge-type problem (3.4)–(3.7). Indeed, to see this, let us notice that, because of q3 (x) = 0, x ∈ [0, ∞), an eigenvalue λ of (3.4)–(3.7) with Im λ < 0 should have an eigenvector (yj (λ, x))3j=1 with y3 (λ, x) = Ce−iλx . Substituting this into (1.3) and (1.4) we obtain equalities y1 (λ, a) = y2 (λ, a) = C and y1
(λ, a) + y2
(λ, a) = −iλC, which is equivalent to (3.6), (3.7). On the Hilbert space H = L2 (0, a) ⊕ C ⊕ L2 (0, a) we introduce operators A0 and A00 , acting by the formulae   

   y1 (x) y1 (x) −y1 (x) + q1 (x) y1 (x) , y1
(a) + y2
(a) A0 y1 (a) = A00 y1 (a) =  (3.8)

y2 (x) y2 (x) −y2 (x) + q2 (x) y2 (x) with the domains given as follows (we use to denote transposed vectors):  (3.9) D(A0 ) = (y1 (x), y1 (a), y2 (x)) ∈ H : yj (x) ∈ W22 (0, a), j = 1, 2,  y1 (a) = y2 (a), y1 (0) = y2 (0) = 0 ,  D(A00 ) = (y1 (x), y1 (a), y2 (x)) ∈ H : yj (x) ∈ W22 (0, a),

j = 1, 2,  y1 (a) = y2 (a) = y1
(a) = y2
(a) = 0, y1 (0) = y2 (0) = 0 .

(3.10)

By [34, Chap.5], A00 is a closed symmetric minimal and bounded from below (cf. [34, Thm.V.19.5]) operator with the defect indices (4, 4), while A0 is a self-adjoint extension of A00 . Hence, the spectrum of A0 consists only of normal eigenvalues, and has no more than finitely many negative eigenvalues. Moreover, there exists a positive constant β such that A0 + βI > 0 and  inverse operator

I 0 0

0 0the 0 (A0 + βI)−1 is compact. We introduce the operators K = 0 I 0 and P = 0 0 0 0 00 0 0I so that P ≥ 0, K ≥ 0 and P + K = I, and consider the quadratic operator pencil L (λ) = λ2 P − iλK − A0

(3.11)

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with the domain D(L(λ)) = D(A0 ) which is independent of λ and dense in H. We collected in Appendix A several definitions and proved some abstract results from the spectral theory of operator pencils needed in the sequel. We remark that the operator pencil (3.11) satisfies Hypothesis A.1 imposed in Appendix A. Also, we identify the spectrum of the boundary value problem in (3.4)–(3.7) with the spectrum of the operator pencil L(λ) introduced in (3.11). Theorem 3.3. Assume Hypothesis 3.1. Then: (i) The spectrum of (3.4)–(3.7) consists only of normal eigenvalues. (ii) The geometric multiplicity of each of the eigenvalues is 1. (iii) The spectrum of (3.4)–(3.7) is symmetric with respect to the imaginary axis, and symmetrically located eigenvalues have equal algebraic multiplicities. (iv) The part of the spectrum of (3.4)–(3.7) in the open lower half-plane lies on the imaginary axis. (v) The spectrum of (3.4)–(3.7) in the open lower half-plane is semi-simple. (vi) The total algebraic multiplicity of the spectrum of (3.4)–(3.7) in the open lower half-plane coincides with that of the following Dirichlet problem: y

+ (λ2 − q(x))y = 0,

x ∈ [0, 2a],

y(λ, 0) = y(λ, 2a) = 0,

(3.12)

where q(x) = q1 (x) if x ∈ [0, a) and q(x) = q2 (2a − x) if x ∈ [a, 2a]. Proof. To prove (i), we apply Theorem 4.2 of [15, Chap.XI] to the operator pencil 1

1

− (A0 + βI)− 2 L(λ)(A0 + βI)− 2 = I − β(A0 + βI)−1 1

1

1

1

+ i(A0 + βI)− 2 K(A0 + βI)− 2 − λ2 (A0 + βI)− 2 P (A0 + βI)− 2 which has the same spectrum as L(λ). Assertion (ii) follows since there exists only one linearly independent solution of (3.12). Assertion (iii) holds due to the symmetry of the problem (recall that the functions qj (x) are real-valued). Assertion (iv) follows from Lemma A.3. Assertion (v) is a particular case of Lemma A.4. Since the square of the spectrum of problem (3.12) is, in fact, equal to the spectrum of
the operator pencil λP − A0 , Assertion (vi) follows from Corollary A.9. For j = 1, 2, we will use the following integral representations (see [31, Sec.1.2], in particular, formula (1.2.11) therein):  x Kj (x, t)λ−1 sin λtdt (3.13) sj (λ, x) = λ−1 sin λx + 0  x (Kj )t (x, t)λ−2 cos λtdt, (3.14) = λ−1 sin λx − Kj (x, x)λ−2 cos λx + 0  x
−1 sj (λ, x) = cos λx + Kj (x, x)λ sin λx + (Kj )x (x, t)λ−1 sin λtdt, (3.15) 0

where we let Kj (x, t) = 0 for |t| > |x|, and, otherwise, Kj (x, t) = Rj (x, t) − Rj (x, −t),

(3.16)

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and Rj (x, t) is the unique solution of the following integral equation:  x+t  x+t  x−t 2 2 2 1 qj (α)dα + dα qj (α + β)Rj (α + β, α − β)dβ. (3.17) Rj (x, t) = 2 0 0 0 If qj (x) ∈ W21 (0, a), j = 1, 2, then, integrating by parts in (3.14) and (3.15), sj (λ, x) = λ−1 sin λx − Kj (x, x)λ−2 cos λx (3.18)  x + (Kj )t (x, x)λ−3 sin λx − (Kj )tt (x, t)λ−3 sin λtdt, s
j (λ, x)

σ

−1

0

= cos λx + Kj (x, x)λ sin λx − (Kj )x (x, x)λ−2 cos λx  x + (Kj )xt (x, t)λ−2 cos λtdt.

(3.19)

0

Let L denote the class of entire functions of exponential type no greater than σ which belong to L2 (−∞, ∞) for real values of the argument. Lemma 3.4. If
Hypothesis 3.1 holds, E0 (λ) is defined in (3.1), and Fj are defined a by Fj = (1/2) 0 qj (x) dx, j = 1, 2, then: (i) The following representation holds: E0 (λ) = λ−1 sin 2λa − (F1 + F2 )λ−2 cos 2λa + ψ0 (λ)λ−2 − i(λ−1 sin2 λa − (F1 + F2 )/2 · λ−2 sin 2λa + ψ1 (λ)λ−2 ),

(3.20)

where ψk (λ) ∈ L2a , k = 0, 1. (ii) If qj (x) ∈ W21 (0, a), j = 1, 2, then E0 (λ) = λ−1 sin 2λa − (F1 + F2 )λ−2 cos 2λa  + (K1 )t (a, a) + (K2 )t (a, a) − (K1 )x (a, a) (3.21) − (K2 )x (a, a) − 2F1 F2 λ−3 sin 2λa + ψ0 (λ)λ−3  − i λ−1 sin2 λa − (F1 + F2 )/2 · λ−2 sin 2λa   + (K1 )t (a, a) + (K2 )t (a, a) λ−3 sin2 λa + F1 F2 λ−3 cos2 λa + ψ1 (λ)λ−3 sin λa + ψ2 (λ)λ−4 ,

where

ψk (λ) ∈ L2a , k = 0, 1, 2.

Proof. We obtain assertion (i) by substituting (3.13)–(3.16) with x = a in (3.1), and assertion (ii) by substituting (3.18) and (3.19) in (3.1) and taking into account
a that 0 f (t) sin λtdt ∈ La whenever f ∈ L2 (0, a) by the Paley-Wiener theorem.
In what follows we will use notation E0 (λ) 2 cos λa + i sin λa  S(λ) = · . E0 (−λ) 2 cos λa − i sin λa Corollary 3.5. Assume Hypothesis 3.1 and λ ∈ R. Then: (i) |E0 (−λ) − λ−1 (sin 2λa + i sin2 λa)| = O(|λ|−2 ) as λ → ±∞.

(3.22)

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 |S(λ) − 1| = O(|λ|−1 ) as λ → ±∞.

(ii)




Proof. Assertions (i) and (ii) follow directly from (3.20).

Next, we will describe the spectrum of (3.4)–(3.7), that is, zeros of E0 (−λ). Lemma 3.6. Assume Hypothesis 3.1. Then: (i) The set Λ = {λk }∞ k=−∞,k=0 of zeros of the function E0 (−λ) is contained in the horizontal strip |Im λ| ≤ M for some M > 0. (ii) The zeros of the function E0 (−λ) satisfy λ−k = −λk for all not pure imaginary λk , and the sequence Λ can be split into two subsequences, ∞ Λ = {λ2k−1 }∞ k=−∞ ∪ {λ2k }k=−∞,k=0 , with the following asymptotic behavior: (0)

λ2k−1 = λ2k−1 + o(1), (0)

(0)

λ2k = λ2k + o(1)

as

(0)

|k| → ∞, (0)

where

(3.23)

(0)

λ2k−1 = (π(2k − 1) + i ln 3)(2a)−1 , λ2k = πk/a, λ−k = −λk , k = 1, 2, . . . . (3.24) Proof. Suppose there exists a subsequence {λkm } of the sequence {λk } such that Im λkm → ∞ as m → ∞. Then (3.20) implies  E0 (−λkm ) + (4iλkm )−1 exp(−2iλkm a) = o |λkm |−1 exp(2|Im λkm a|) , m → ∞, contradicting the identity E0 (−λkm ) = 0 and proving that the set {Im λk } is bounded from above. Similarly, it is bounded from below, and thus assertion (i) holds. Turning to the proof of assertion (ii), we temporarily introduce the function (0) E00 (−λ) = λ−1 sin 2λa + i sin2 λa whose zeros form the sequence {λk }∞ k=−∞,k=0 given in (3.24). Comparing E00 (−λ) and (3.20), we conclude that there exist constants C > 0 and ε > 0 such that the inequality |E0 (−λ) − E00 (−λ)| < C|λ|−2 holds for all λ ∈ Π, where Π = {λ : |Im λ| ≤ M + ε,|λ| ≥ ε}. For every r ∈ (0, ε) one can find a d > 0 such that sin 2λa + i sin2 λa > d for all λ ∈ Π\ ∪k Ck , (0) where Ck are the disks of radii r centered at λk . Consequently, we have the inequalities |E00 (−λ)| > d/|λ| > C/|λ|2 > |E0 (−λ) − E00 (−λ)| for all λ ∈ {λ : λ ∈ Π\ ∪k Ck , |λ| > C/d}. Since r can be chosen arbitrary small, we can apply Rouch´e (0) Theorem to conclude that λk − λk = o(1) as |k| → ∞.
In fact, the spectrum of the boundary value problem (3.4)–(3.7) admits even more detailed description given next. Theorem 3.7. Assume Hypothesis 3.1. The spectrum of problem (3.4)–(3.7) is equal to Λ(1) ∪ Λ(2) , where the sequences (1)

(2)

Λ(1) = {λk : k = ±1, ±2, . . . }, Λ(2) = {λl

: l = ±1, ±2, · · · ± p}, p ≤ ∞,

satisfy the following properties: (1) All but finitely many elements of the sequence Λ(1) belong to the open upper half-plane; the number of the elements of Λ(1) that belong to the closed lower half-plane will be denoted by κ1 .

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(2) All κ1 elements of the sequence Λ(1) that belong to the closed lower half-plane are purely imaginary and occur only once; if κ1 ≥ 1, then we denote these (1) (1) elements by λ−j = −i|λ−j |, j = 1, . . . , κ1 , and enumerate them such that (1)

(1)

|λ−j | < |λ−(j+1) | for j = 1, . . . , κ1 − 1.

(1)

(3) If κ1 ≥ 1, then the complex conjugates, i|λ−j |, j = 1, . . . , κ1 , of the elements listed in item (2) do not belong to the sequence Λ(1) (with a possible exception (1) of λ−1 = 0). (1) (1) (4) If κ1 ≥ 2, then the interval (i|λ−j |, i|λ−(j+1) |), j = 1, . . . , κ1 − 1, of the (5)

(6) (7) (8) (9)

imaginary axis contains an odd number of elements of the sequence Λ(1) . (1) (1) If |λ−1 | > 0, then the interval (0, i|λ−1 |) of the imaginary axis either contains no elements of the sequence Λ(1) , or contains an even number of elements of this sequence. (1) If κ1 ≥ 1, then the interval (i|λ−κ1 |, i∞) of the imaginary axis contains an odd number of elements of the sequence Λ(1) . If κ1 = 0, then the sequence Λ(1) has an even number of elements with positive imaginary parts. (2) The numbers (λl )2 are real for all l = ±1, ±2, · · · ± p, p ≤ ∞. (2) The numbers (λl )2 can be enumerated such that (2)

(2)

(2)

(2)

2 2 2 (2) 2 (λ1 )2 < (λ2 )2 < · · · < (λ(2) κ2 ) < 0 ≤ (λκ2 +1 ) < (λκ2 +2 ) < · · · < (λp ) , (2)

(2)

where λ−l = −λl , l = ±1, ±2, · · · ± p, p ≤ ∞. Proof. Let us define the function E0 (−λ, η), η ∈ [0, 1], by E0 (−λ, η) = s1 (λ, a)s
2 (λ, a) + s
1 (λ, a)s2 (λ, a) + iηλs1 (λ, a)s2 (λ, a), and analyze the behavior of its zeros when the parameter η changes from 0 to 1. When η = 0, then all zeros of E0 (−λ, 0) are real or pure imaginary because they are the eigenvalues of problem (3.12). Among them we select those for which (2) either E0 (−λ, 0) = s1 (λ, a) = 0 or E0 (−λ, 0) = s2 (λ, a) = 0, denote them by λl , (1) and put them in the sequence Λ(2) . All other zeros, denoted by λk , will form the sequence Λ(1) . The zeros of E0 (−λ, η) from the sequence Λ(2) do not move when η (1) (1) changes from 0 to 1. If λk = λk (0) belongs to the sequence Λ(1) , then the zeros (1) λk (η) have the following property: For all η > 0, if −iτ ∈ Λ(1) for some τ > 0 is one of these zeros, then iτ is not a zero from the sequence Λ(1) . To finish the proof, we use the symmetry of the problem, and Lemmas A.3, A.4, A.8.
Theorem 3.7 can be also deduced from more general results in [44]. Corollary 3.8. Assume Hypothesis 3.1. Then: (1) The S-function (3.3) is a meromorphic function in C which is continuous on R and has no real zeros.

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(2) The set of poles of S(λ) is the set of zeros of E0 (−λ) excluding real zeros and imaginary zeros symmetric about the origin; this set satisfies properties (1)–(7) in Theorem 3.7 for Λ(1) . Proof. Represent the numerator and the denominator of the fraction S(λ) defined in (3.3) as products of linear terms corresponding to their zeros. The terms that correspond to the real and pure imaginary symmetric about the origin zeros of E0 (−λ) in the denominator of S(λ) will cancel the terms in the numerator of S(λ) that correspond to the zeros of E0 (λ).
We will involve in the discussion the class of Hermite-Biehler functions and its modifications, cf. [5, 21, 22]. As we will see below, the Jost function for problem (1.1)–(1.6) with q3 (x) = 0, x ∈ [0, ∞), belongs to an appropriately modified class of the Hermite-Biehler functions. The main advantage of this is that we can establish important interlacing properties of zeros of the “even” and “odd” parts of the Jost function. The importance of these properties becomes especially transparent in the next section where they are used to obtain results on the inverse problem. We recall definitions from [27, p.307] and [27, p.313]. Definition 3.9. An entire function ω(λ) with no zeros in the closed lower half-plane and satisfying the condition |ω(λ)/¯ ω(λ)| < 1 for all Im λ > 0 is called a HermiteBiehler function (for short, an HB-function, or a function of the HB-class). An entire function ω(λ) with no zeros in the open lower half-plane and satisfying the condition |ω (λ)/¯ ω (λ)| ≤ 1 for all Im λ > 0 is called a generalized Hermite-Biehler function (for short, HB-function). Here, ω ¯ (λ) denotes the entire function obtained from ω (λ) by replacing the coefficients in its Tailor series by their complex-conjugates, i.e. ω(λ) = ω(λ). Definition 3.10. A Hermite-Biehler function ω(λ) (resp., a generalized HermiteBiehler function) is called symmetric or a function of the SHB-class (resp., SHBclass) if ω(−λ) = ω(λ). For a symmetric function ω(λ) one has the following representations: ˆ ˜ 2 ), ω(λ) = P (λ) + iQ(λ) = P (λ) + iλQ(λ) = P˜ (λ2 ) + iλQ(λ

(3.25)

ˆ where P (λ) and Q(λ) are real (that is, having real values for real λ’s) and even ˜ as follows: functions. Here, we introduce the functions P˜ and Q P˜ (λ2 ) = P (λ),

˜ 2 ) = Q(λ). ˆ Q(λ

(3.26)

The proof of the following lemma can be found in Appendix B. Lemma 3.11. If an entire function ω(λ) = P (λ) + iQ(λ) of form (3.25) belongs to ˜ the class SHB (respectively, SHB), then the entire function P˜ (λ) + iQ(λ) with P˜ ˜ given in (3.26), belongs to the class HB (respectively, HB). and Q

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˜ 2 ) belongs to the Definition 3.12. Assume that the function ω(λ) = P˜ (λ2 ) + iλQ(λ ˜ 2 + c) with some c > 0 is SHB-class. Then the function ωc (λ) = P˜ (λ2 + c) + iλQ(λ called a shifted symmetric Hermite-Biehler function (for short, SHBc -function). Definition 3.13. (see [28]) An entire function ω(λ) of exponential type σ > 0 is said to be of sine-type if there exist positive constants h, m, and M such that for |Im λ| ≥ h the inequalities m ≤ |ω(λ)|e−σ|Im λ| ≤ M are satisfied. Theorem 3.14. Assume Hypothesis 3.1. Then the function E0 (λ) given in (3.1) belongs to SHB c . Proof. Consider (1.1)–(1.6) with the potentials q3 (x) = 0, x ∈ [0, ∞), and q1 (x) = (0) (0) q1 (x) − c and q2 (x) = q2 (x) − c, where c is a real parameter independent on x and selected such that the operator A0 corresponding to the “shifted” potentials (0) qj (x), j = 1, 2, is strictly positive. In this case the spectrum of problem (3.12) is real. Therefore, according to assertion (vi) of Theorem 3.3, the spectrum of problem (3.4)–(3.7) lies in the closed upper half-plane. Temporarily denote by (0) (0) E0 (−λ) the function computed by (3.1) but with qj (x) replaced by qj (x). We (0)

(0)

will prove first that E0 (−λ) ∈ SHB. It follows from Lemma 3.4 that λE0 (−λ) (0) is a sine-type n function. Then this function can be represented as λE0 (−λ) = λClimn→∞ k=−n (1 − λ/λk ), cf. [29, p.88]. Now by Theorem 6 of [27, Chap.VII] (0)

we obtain that E0 (−λ) ∈ HB (we can not claim that this function belongs to HB because it may have zeros on the real axis). Moreover, due to the symmetry (0)

(0)

(0)

E0 (−λ) = E0 (λ), cf. (3.2), we conclude that E0 (−λ) ∈ SHB. Next, passing to the case c = 0, we rewrite (3.4) as (0)

yj

+ (λ2 + c − qj (x))yj = 0,

j = 1, 2,

(3.27)

and notice that the function E0 (−λ), corresponding to problem (3.27) with the
boundary conditions (3.5)–(3.7), belongs to SHB c . Introduce the following “even” and “odd” parts of the function E0 (−λ):   ϕe (λ) = E0 (λ) + E0 (−λ) /2, ϕo (λ) = E0 (−λ) − E0 (λ) /(2i), (3.28) ϕˆo (λ) = λ−1 ϕo (λ),

λ ∈ C.

(3.29)

Due to (3.2), it follows that the functions ϕe (λ) and ϕo (λ) are real-valued for λ ∈ R. Let us denote by {µk }∞ −∞,k=0 the set of zeros of the function ϕe (λ) and by {θk }∞ the set of zeros of the function λ−1 ϕo (λ). The enumeration is −∞,k=0 symmetric with respect to the origin, i.e. µ−k = −µk , µ2k < µ2k+1 and θ−k = −θk , 2 θk2 ≤ θk+1 . We recall notation Fj , j = 1, 2, from Lemma 3.4. Theorem 3.15. Assume Hypothesis 3.1. Then: (1) All zeros µn and θn are simple, and either real or pure imaginary. 2 2 (2) For every k > 1 either θk−1 < µ2k < θk2 or µ2k−1 < θk−1 = µ2k = θk2 < µ2k .

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(3) The sequence {µk }∞ k=−∞,k=0 has the following asymptotic behavior: F1 + F2 γk πk + + , as 2a πk k is a sequence from 2 .

µk = where {γk }∞ −∞,k=0

|k| → ∞,

(3.30)

(4) The sequence {θk }∞ −∞,k=0 can be split in two subsequences so that (1)

(2)

∞ ∞ {θk }∞ −∞,k=0 = {νk }−∞,k=0 ∪ {νk }−∞,k=0

and the following asymptotic relations hold: (1)

(1)

β F1 πk + + k , a πk k (2) β F2 πk + + k , = a πk k

νk = (2)

νk

as

|k| → ∞,

(3.31)

as

|k| → ∞,

(3.32)

(j)

where {βk }∞ −∞,k=0 is a sequence from 2 , j = 1, 2. Proof. Substituting (3.1) in (3.28)–(3.29), we infer: ϕe (λ) = s
1 (λ, a)s2 (λ, a) + s
2 (λ, a)s1 (λ, a),

(3.33)

ϕˆo (λ) = s1 (λ, a)s2 (λ, a).

(3.34)

The functions ϕe (λ) and ϕˆo (λ) are real, and therefore all µk and θk are real or pure imaginary. The set of zeros of ϕe (λ) coincides with the spectrum of problem (3.12), or, which is the same, with the spectrum of the problem yj

+ λ2 yj − qj (x)y = 0,

= 1, 2,

y1 (λ, a) = y2 (λ, a),

0,

yj (λ, 0) = 0, j
y1 (λa ) + y2
(λ, a) =

and, using [31, Thm.3.4.1], we obtain (3.30). Similarly, (3.34) implies (3.31)–(3.32). Due to Theorem 3.14, the function E0 (−λ) = ϕe (λ) + iλϕˆo (λ) belongs to SHB c . Then E0 (−λ) = ϕ˜e (λ2 ) + iλϕ˜o (λ2 ) where we define ϕ˜e (λ2 ) = ϕe (λ) and ϕ˜o (λ2 ) = ϕˆo (λ). Clearly, there exists a constant c > 0 such that the function ϕ˜e (λ2 − c) + iλϕ˜o (λ2 − c) belongs to SHB, and according to Lemma 3.11, we have ϕ˜e (λ − c) + iϕ˜o (λ − c) ∈ HB. Thus, we can apply Theorem 3’ in [27, Sec.VII.2], cf. also Appendix B, and obtain the inequality 2 ≤ µ2k ≤ θk2 ≤ µ2k+1 ≤ · · · . · · · ≤ θk−1

(3.35)

If µk = θn for some k = 0 and n, that is, if ϕe (θn ) = ϕo (θn ) = 0, then either s1 (θn , a) = 0 or s2 (θn , a) = 0. Suppose that s1 (θn , a) = 0; then from (3.33) we obtain s
1 (θn , a)s2 (θn , a) = 0. Consequently, s2 (θn , a) = 0, and θn is a double zero. Assertions (1) and (2) follow. Now assertion (3) follows from the fact that µk are the eigenvalues of problem (3.12). Statement (4) follows from the fact that the set of zeros of sj (λ, a) coincides with the spectrum of the Dirichlet problem yj

+ λ2 yj − qj (x)yj = 0, yj (λ, 0) = yj (λ, a) = 0.


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Lemma 3.16. Assume Hypothesis 3.1. The function E0 (−λ) can be represented as  (3.36) E0 (λ) = g1 (λ)(3g2 (λ) − g2 (−λ)) − g1 (−λ)(g2 (λ) − g2 (−λ)) /(4iλ), where the functions gj (λ) belong to SHB c , and are given by   gj (λ) = eiλa 1 − iFj λ−1 + ξj (λ)λ−1 with some ξj (λ) ∈ La , j = 1, 2.

(3.37)

Proof. For j = 1, 2 we denote gj (λ) = s
j (λ, a) + iλsj (λ, a).

(3.38)

s
j (λ, a)

= (gj (λ)+gj (−λ))/2, sj (λ, a) = (gj (λ)−gj (−λ))/(2iλ) Then, substituting in (3.1), we obtain (3.36). Representation (3.37) follows by substituting (3.14) and (j) (3.15) in (3.38). It is well known that the squares of zeros (νk )2 of the functions (j) sj (λ, a) interlace with the squares of zeros (γk )2 of the function s
j (λ, a) in the fol(j)

(j)

(j)

lowing strict sense (see [31, Sec.3.4]): (γ1 )2 < (ν1 )2 < (γ2 )2 < · · · .√Therefore, there exists a constant c such that for the zeros of the functions s
j ( λ2 − c, a) √ and sj ( λ2 − c, a) the following inequalities hold:     (j) (j) (j) (j) · · · − (ν−1 )2 + c < − (γ−1 )2 + c < 0 < (γ1 )2 + c < (ν1 )2 + c < · · · . ∞ 2 Also, we know from [31] that sj (λ, a) = a k=1 πa2 k2 (νk2 − λ2 ) and s
j (λ, a) =  2 ∞ 2a (γk2 − λ2 ). Therefore, k=1 (2k−1)π  ∞ a2 sj ( λ2 − c, a) = a (ν 2 + c − λ2 ), k=1 π 2 k 2 k 2  ∞  2a s
j ( λ2 − c, a) = (γk2 + c − λ2 ). k=1 (2k − 1)π √ √ Thus, the function s
j ( λ2 − c, a) + iλsj ( λ2 − c, a) satisfies the conditions of Corollary B.3 and therefore belongs to the class HB. Since it is symmetric, it also belongs to SHB. As a result, we have the inclusion gj (λ) ∈ SHBc .


4. Inverse problem In this section we consider the problem of recovering the potentials q1 (x) and q2 (x) from scattering data assuming that the potential q3 (x) is identically equal to zero on the semi-infinite part of the wave-guide. In fact, in Theorems 4.3 and 4.4 we show how to recover the potentials as soon as we are given a function E0 (λ) with the properties similar to the properties of the Jost function discussed in the previous sections. Before proceeding with the solution of the inverse problem when E0 (−λ) is given, we make the following remark: Even in the case when q3 (x) = 0, x ∈ [0, ∞), the Jost function E0 (−λ) is not uniquely determined by the scattering function S(λ) as long as E0 (λ) is allowed to have zeros on the real axis or pairs of pure

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imaginary zeros symmetric about the real axis. To illustrate this, let us suppose the λk is a real zero of E0 (λ). Then, due to the symmetry (3.2), −λk is also a zero of E0 (λ), and ±λk are zeros of E0 (−λ) as well. Cancellation of the corresponding factors in the fraction S(λ) = E0 (λ)/E0 (−λ) shows that the scattering function S(λ) does not change as long as we move zeros of E0 (λ) along the real axis in a symmetric fashion. Similarly, we can achieve the same cancellation effect if we suppose that E0 (λ) has two symmetrically located pure imaginary zeros λk = i|λk | and λ−k = −i|λk |; indeed, in this case we can move λk and λ−k along the imaginary axis preserving the symmetry |λk | = |λ−k | and having S(λ) unchanged. However, if we exclude these possibilities, that is, if we assume a’priori that E0 (−λ) may have only a single simple zero at the origin and does not have any other real zeros nor any pairs of symmetric about the origin pure imaginary zeros, then the Jost function is uniquely determined by the scattering function. Indeed, the S-function is meromorphic due to q3 (x) = 0, x ∈ [0, ∞), and, under the a’priori assumptions above, it is clear that the zeros of S(λ) are the zeros of E0 (λ), see (3.3). The function λE0 (−λ) is a sine-type function and therefore, see [29], the set of its zeros together with their asymptotics uniquely determine E0 (λ), cf. Corollary 3.8. Passing to the solution of the inverse problem, we will now describe the properties of a meromorphic function S(λ) that enable us to construct a function E0 (λ) having the same properties as the Jost function. Given a function S(λ), we  define, cf. (3.22), the function S(λ) by the formula 2 cos λa + i sin λa  . S(λ) = S(λ) · 2 cos λa − i sin λa Hypothesis 4.1. Assume that S(λ) is a meromorphic in C function that satisfies the following conditions: (a) (b)

S(−λ) = 1/S(λ) and S(−λ) = S(λ) for λ ∈ C.   S(λ) − 1 = O(|λ|−1 ) as λ → ±∞, λ ∈ R.

Proposition 4.2. Assume that S(λ) satisfies Hypothesis 4.1, let Λ denote the set of poles of S(λ), and, in addition, assume one of the following conditions: (i) If S(0) = 1, then the set Λ = {λk }∞ k=−∞,k=0 has properties (1)–(7) of Theo(0)

rem 3.7 for Λ(1) , and satisfies the asymptotic relations (3.23) with λ2k−1 and (0)

λ2k given in formula (3.24). (ii) If S(0) = −1, then the set Λ ∪ {0} = {λk }∞ k=−∞,k=0 has properties (1)–(7) of Theorem 3.7 for Λ(1) , and satisfies relations (3.23)–(3.24). Then there exists a unique entire function E0 (λ) of exponential type 2a which has no real zeros (except, maybe, a simple zero at the origin), has no pairs of symmetric about the origin pure imaginary zeros, and satisfies the relations S(λ) =  E0 (λ)/E0 (−λ) and E0 (−λ) − λ−1 (sin 2λa + i sin2 λa) = O(|λ|−2 ) as λ → ±∞. In addition, E0 (−λ) ∈ SHBc .

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Proof. We know that S(λ) as a meromorphic function, and thus we know its zeros and poles. Let us denote the poles  of S(λ) by λk . Then in case (i) we define ∞ E0 (−λ) as the product E0 (−λ) = k=−∞,k=0 (1 − λ/λk ). Using assertion (b) in Hypothesis 4.1, we conclude that λE0 (−λ) is a sine-type function, because  (0) −1 (sin 2λa + i sin2 λa), where C the function λ ∞ k=−∞,k=0 (1 − λ/λk ) = C(2a) is  a constant, is a sine-type function (see [29]). In case (ii) we define E0 (−λ) = λ λk ∈Λ (1 − λ/λk ) , and conclude again that λE0 (−λ) is a sine-type function. The
proof of Φ(λ) ∈ SHBc is similar to that of Propositions 4.8 and 4.9 in [43]. Next, we will discuss the solution of the inverse problem of recovering the potentials for the boundary value problem (1.1)–(1.2) with q3 (x) = 0, x ∈ [0, ∞), given a function E0 (−λ). Let us consider the following set of triples of real-valued potentials: Q = {(qj (x))3j=1 : qj (x) ∈ L2 (0, a), j = 1, 2, q3 (x) = 0, x ∈ [0, ∞)}. Theorem 4.3. Assume that E0 (−λ) is an entire function of exponential type 2a which satisfies the following conditions: (1) E0 (−λ) can be represented in form (3.36), where gj (λ) ∈ SHB c , j=1,2; (2) gj (λ) can be represented in form (3.37), where Fj are some real constants and ξj (λ) ∈ La , j = 1, 2. Then there exists a unique triple (qj (x))3j=1 ∈ Q such that the S-function for the boundary value problem (1.1)–(1.6) with the potentials qj (x) is given by (3.3). Proof. We will prove that there exist real-valued potentials qj (x) ∈ L2 (0, a), j = 1, 2, such that the function gj (λ) is the Jost function of the problem   yj

+ λ2 − qˆj (x) yj = 0, x ∈ [0, ∞), yj (λ, 0) = 0, where qˆj (x) = qj (a − x) if x ∈ [0, a) and qˆj (x) = 0 if x ∈ [a, ∞). For this, let us introduce the functions gje (λ) = (gj (λ) + gj (−λ))/2,

gjo (λ) = (gj (λ) − gj (−λ))/(2i).

(4.1)

Substituting (3.37) in (4.1), we obtain gje (λ) = cos λa + Fj λ−1 sin λa + (ξj (λ) + ξj (−λ))/2,

(4.2)

gjo (λ)

(4.3)

−1

= sin λa − Fj λ

cos λa + (ξj (λ) − ξj (−λ))/(2i).

(j)

(j)

e ∞ We denote by {µk }∞ −∞,k=0 the set of zeros of gj (λ) and by {νk }−∞ the set of o zeros of gj (λ). It follows from (4.3) and [31, Lem.3.4.2] applied for the interval (j)

[0, a] that {νk }∞ k=−∞,k=0 satisfy (3.31)–(3.32), and from (4.2) and [31, Lem.3.4.2] that

(j) {µk }∞ −∞,k=0

have the following asymptotics:

(j)

(j)

µk = π(k − 1/2)/a − Fj (πk)−1 + k −1 γk , (j) {γk }∞ −∞,k=0

as |k| → ∞,

(4.4)

where ∈ 2 . It follows from (4.2) and (4.3) that gje (λ) and λ−1 gjo (λ) are even functions. The condition gj (λ) ∈ SHBc means that there exists c ∈ R

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such that g˜je (λ2 − c) + iλ˜ gjo (λ2 − c) ∈ SHB, where g˜je (λ2 ) = gje (λ), g˜jo (λ2 ) = √ √ λ−1 gjo (λ). By Lemma 3.11 we obtain gje ( λ2 − c) + igjo ( λ2 − c) ∈ HB. By The (j) orem B.1, the zeros {± (µk )2 + c}∞ ˜je (λ2 − c) and the zeros k=1 of the function g  (j) {± (νk )2 + c}∞ ˜jo (λ2 − c) interlace: k=1 of the function g    (j) (j) (j) 0 < (µ1 )2 + c < (ν1 )2 + c < (µ2 )2 + c < · · · . Consequently,

(j)

(j)

(j)

(µ1 )2 < (ν1 )2 < (µ2 )2 < · · · .

(j) {µk }∞ −∞,k=0

(4.5)

(j) {νk }∞ −∞,k=0

and satisfy all conditions of TheoNow the sequences rem 3.4.1 in [31]. By this theorem, there exists a unique pair of real-valued poten(j) tials qj (x) ∈ L2 (0, a), j = 1, 2, such that the set {νk }∞ −∞,k=0 coincides with the spectrum of the Dirichlet problem yj

+ (λ2 − qj (x))yj = 0, x ∈ [0, a], yj (λ, a) = yj (λ, 0) = 0,

j = 1, 2,

(4.6)

(j)

while {µk }∞ −∞,k=0 is the spectrum of the Dirichlet - Neumann problem yj

+ (λ2 − qj (x))yj = 0,

x ∈ [0, a],

yj (λ, a) = yj
(λ, 0) = 0,

j = 1, 2.

Due to (3.1), the triple (q1 (x), q2 (x), q3 (x) ≡ 0), just constructed, generates the S-function by formula (3.3). Uniqueness follows from [31, Thm.3.4.1].
The next theorem gives even more explicit sufficient conditions for the ratio E0 (λ)/E0 (−λ) to be an S-function for (1.1)–(1.6) with q3 (x) = 0, x ∈ [0, ∞). Theorem 4.4. Let E0 (−λ) be a given entire function of exponential type 2a which satisfies the following conditions: (1) E0 (−λ) ∈ SHB c ; (2) E0 (−λ) is of the form E0 (−λ) = λ−1 sin 2λa − P1 λ−2 cos 2λa + P2 λ−3 sin 2λa + φ1 (λ)λ−3   + i λ−1 sin2 λa − P1 /2 · λ−2 sin 2λa + P3 λ−3 cos2 λa + P4 λ−3 sin2 λa + iφ2 (λ)λ−3 sin λa + iφ3 (λ)λ−4 ,

(4.7)

where Pk ∈ R, k = 1, . . . , 4, are given constants such that P12 > 4P3 , and φl (λ) are given function such that φl (−λ) = φl (λ), l = 1, 2, 3, and φ1 (λ), φ3 (λ) ∈ L2a , φ2 (λ) ∈ La . Then there exists a triple (q1 (x), q2 (x), q3 (x) ≡ 0) ∈ Q such that the S-function for (1.1)–(1.6) is furnished by (3.3) with the given function E0 (−λ). Proof. Substituting (4.7) in (3.28), we compute: ϕe (λ) = λ−1 sin 2λa − P1 λ−2 cos 2λa + P2 λ−3 sin 2λa + ψ1 (λ)λ−3 , −1

ϕo (λ) = λ

2

−2

sin λa − P1 /2 · λ −3

+ P3 λ

2

sin 2λa

−3

cos λa + P4 λ

(4.8)

sin2 λa + ψ2 (λ)λ−3 sin λa + ψ3 (λ)λ−4 . (4.9)

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Let us denote by {µk }∞ k=−∞,k=0 the set of zeros of the function ϕe (λ) and by ∞ {νk }k=−∞ the set of zeros of the function ϕo (λ). Condition (1) of Theorem 4.4 implies that all zeros of the functions ϕe (λ) and ϕo (λ) are real or pure imaginary. We enumerate them in the following way: µ−k = −µk , (µk )2 ≤ (µk+1 )2 , and ν−k = −νk , (νk )2 < (νk+1 )2 for k = 0, and ν0 = 0. An application of [38, Lem.2.1] ∞ shows that the sequence {µk }k=−∞,k=0 has the following asymptotic behavior: µk = πk/(2a) + P1 (πk)−1 + γk k −2 ,

as |k| → ∞,

(4.10)

where {γk }∞ k=−∞,k=0 is a sequence from 2 . The asymptotic behavior of the se∞ quence {νk }k=−∞,k=0 is described next. Proposition 4.5. The sequence {νk }∞ k=−∞,k=0 satisfies the relation ν−k = −νk for all k = 0, and can be represented as a union of two subsequences, (1) (2) ∞ {νk }∞ k=−∞, k=0 and {νk }k=−∞, k=0 , such that (1)

(1)

β F1 πk − + k2 , a πk k (2) β πk F2 = − + k2 , a πk k

νk = (2)

νk

as

|k| → ∞,

(4.11)

as

|k| → ∞,

(4.12)

  where F1 = (−P1 + (P1 )2 − 4P3 )/2, F2 = (−P1 − (P1 )2 − 4P3 )/2, and the (j) sequence {βk }∞ k=−∞,k=0 belongs to 2 , j = 1, 2. Proof. We consider λ−1 ϕo (λ) as a perturbation of the following function:   −1 2 −1 λ−1 ϕ(0) λ sin λa − P1 λ−2 sin λa cos λa + P3 λ−3 cos2 λa o (λ) = λ (4.13)    = λ−1 sin λa + F1 λ−2 cos λa λ−1 sin λa + F2 λ−2 cos λa . Clearly, the set of zeros of this function can be split in two subsequences with the following asymptotic behavior: (01)

νk

(01) −2

= πk/a − F1 (πk)−1 + βk (0j)

k

(02)

, νk

(02) −2

= πk/a − F2 (πk)−1 + βk

k

as |k| → ∞, where {βk }∞ k=−∞,k=0 ∈ 2 , j = 1, 2. It follows that for any ρ ∈ (0, (F1 − F2 )/2) there exists a k1 (ρ) ∈ N such that for each k > k1 (ρ) the disc of radius ρ(πk)−1 centered at πk/a − F1 (πk)−1 or πk/a − F2 (πk)−1 contains (0) exactly one simple zero of the function λ−1 ϕo (λ). Let us introduce a variable τ = τ (k, ρ, θ) by the formula τ = πk/a − F1 (πk)−1 + ρeiθ (πk)−1 , where ρ > 0 and (0) θ ∈ [0, 2π). First, let us estimate |τ ϕ0 (τ )| from below. The inequalities   2   sin τ a + (−1)k (aF1 + aρeiθ )(πk)−1  ≤ Ck1 (ρ) k −3 , k ∈ N,   cos τ a − (−1)k  ≤ Ck1 k −2

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hold uniformly with respect to θ ∈ [0, 2π) and k > k1 ∈ N with a positive constant Ck1 . Therefore,  −1  τ sin τ a + F1 τ −2 cos τ a − (−1)k a2 ρeiθ (πk)−2  < C˜k2 k −4 , (4.14)   −1 τ sin τ a + F2 τ −2 cos τ a − (−1)k a2 (F2 − F1 + ρeiθ )(πk)−2  < C˜k2 k −4 (4.15) uniformly with respect to θ ∈ [0, 2π) for k > k2 ∈ N, where C˜k2 > 0. Using (4.13)–(4.15), we obtain     −1 (0) (4.16) τ ϕ0 (τ ) − a4 ρeiθ (F2 − F1 + ρeiθ )(πk)−2 /k −2  < Ck03 k −6 for k > k3 . Since |F2 − F1 | > ρ, we conclude that for some C˜k04 > 0 and k > k4 the following inequality holds: (0)

|τ −1 ϕ0 (τ )| > C˜k04 k −4 .

(4.17)

(0)

Next, let us estimate |τ −1 (ϕ0 (τ ) − ϕ0 (τ ))| from above:   (0) |τ −1 (ϕ0 (τ ) − ϕ0 (τ ))| = P4 τ −4 sin2 τ a + ψ2 (τ )τ −3 sin λa + ψ1 (τ )τ −4  < βk (ρ)k −4 ,

(4.18)

where βk → 0 as k → +∞. By (4.18), for any fixed ρ < |F2 − F1 |/2 uniformly with respect to θ ∈ [0, 2π) there exists Ck5 = Ck5 (ρ) ∈ (0, C˜k04 ) such that (0)

|τ −1 (ϕ0 (τ ) − ϕ0 (τ ))| < Ck5 k −4

(4.19)

for k > k5 . Comparing (4.17) with (4.19) we obtain (0)

(0)

|τ −1 (ϕ0 (τ ) − ϕ0 (τ ))| < |τ −1 ϕ0 (τ )|.

(4.20)

Now Rouch´e Theorem implies that for k > k2 (ρ) every disc of radius ρ centered at πk/a − F1 /(πk) contains exactly one simple zero of the function λ−1 ϕo (λ). We can choose ρ arbitrary small to achieve the following: (1)

(1)

(1)

(2)

(2)

νk = πk/a − F1 (πk)−1 + κk k −1 , where κk = o (1) as |k| → ∞.

(4.21)

Similarly, we obtain (2)

νk = πk/a − F2 (πk)−1 + κk k −1 , where κk = o (1) as |k| → ∞.

(4.22)

(1) (1) (νk )−1 ϕo (νk )

= 0 and make use of Let us substitute now (4.21) in the equation (4.10) and (4.13). Then we obtain:  (1) −1 (1) (1) (1) sin νk a + F1 (νk )−2 cos νk a λ−1 ϕ(0) o (λ) = (νk )  (1) (1) (1) (1) (4.23) × (νk )−1 sin νk a + F2 (νk )−2 cos νk a (1)

(1)

(1)

(1)

(1)

+ ψ2 (νk )(νk )−4 sin νk a + ψ1 (νk )(νk )−5 = 0.

(4.24)

Substituting (4.21), we also have   (1) (1) κk a/(πk 2 ) + O(k −3 ) (F2 − F1 )a2 (−1)k (πk)−2 + O(k −3 ) = βk k −5 ,

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(j)

where {βk }∞ k=−∞ ∈ 2 , yielding (4.11). The proof of (4.12) is similar.




Returning to the proof of Theorem 4.4, we remark that the function E0 (−λ) belongs to SHBc , and thus, using Lemma 3.11, the sequences {µk }∞ k=−∞,k=0 and ∞ {νk }k=−∞,k=0 interlace in the following sense: −∞ < (µ1 )2 < (ν1 )2 < (µ2 )2 < (ν2 )2 < · · · . 2

(4.25) 2

Adding a sufficiently large positive constant c to each (µk ) and (νk ) we obtain for (˜ µk )2 = (µk )2 + c and (˜ νk )2 = (νk )2 + c the inequalities 0 < (˜ µ1 )2 < (˜ ν1 )2 < (j) (˜ µ2 )2 < (˜ ν2 )2 < . . . . Let us defineµ ˜k and ν˜k for k = ±1, ±2, . . . as follows:  (j) (j) µ|k| )2 and ν˜±|k| = ± (˜ ν|k| )2 , j = 1, 2. Due to (4.10), (4.11) and µ ˜±|k| = ± (˜ (4.12), these sequences have the following asymptotics: µ ˜k = πk/(2a) + P˜1 (πk)−1 + γk k −2 , (1) ν˜k (2) ν˜k

−1

= πk/a − F˜1 (πk)

+

= πk/a − F˜2 (πk)−1 +

(1) βk k −2 , (2) βk k −2 ,

as |k| → ∞, as |k| → ∞, as |k| → ∞,

where P˜1 = P1 − ca, F˜j = Fj − ca/2, j = 1, 2, and thus P˜1 = F˜1 + F˜2 , F˜1 = F˜2 . (1) We observe that the three sequences, {˜ µk }∞ νk }∞ k−∞,k=0 , {˜ k=−∞,k=0 , and (2)

{˜ νk }∞ k=−∞,k=0 , satisfy conditions of Theorem 2.1 in [38]. For reader’s convenience, this theorem is also recorded as Theorem B.4 in Appendix B below. Using Theorem B.4, we conclude that there exists a unique pair of real potentials q˜1 (x) and (1) (2) µk }∞ νk }∞ νk }∞ q˜2 (x) such that the sequences {˜ k−∞,k=0 , {˜ k=−∞,k=0 , and {˜ k=−∞,k=0 constitute, respectively, the spectra of the following three problems: y

+ (λ2 − q˜(x))y = 0, x ∈ [0, 2a], y(λ, 0) = y(λ, 2a) = 0; y

+ (λ2 − q˜1 (x))y = 0, x ∈ [0, a], y(λ, 0) = y(λ, a) = 0; y

+ (λ2 − q˜2 (x))y = 0, x ∈ [0, a], y(λ, 0) = y(λ, a) = 0.

(4.26)

(4.27) (4.28)

Here, q˜(x) = q˜1 (x) for x ∈ [0, a] and q˜(x) = q˜2 (2a−x) for x ∈ [a, 2a]. Consequently, ∞  4a2  2 2 (˜ µ (4.29) ϕ˜e (λ) = 2a ) − λ k k=1 π 2 k 2 is, in fact, equal to s(λ, 2a), where s(λ, x) is the solution of equation (4.26) that satisfies the conditions s(λ, 0) = s
(λ, 0) − 1 = 0 (indeed, (4.29) is a modification for the interval [0, 2a] of formula (3.4.15) in [31]). Similarly, the expressions ∞  ∞    a2  (1) 2 a2  (2) 2 2 2 ϕ˜01 (λ) = a ) − λ (λ) = a ) − λ (˜ ν and ϕ ˜ (˜ ν 02 k k π2 k2 π2 k2 k=1

k=1

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are equal, respectively, to s1 (λ, a) and s2 (λ, a), where s1 (λ, x), respectively, s2 (λ, x) is the solution of equation (4.27), respectively, equation (4.28) that satisfies the conditions sj (λ, 0) = s
j (λ, 0) − 1 = 0, j = 1, 2. According to (3.1), (3.33) and (3.34) this implies that s(λ, 2a) = s
1 (λ, a)s2 (λ, a) + s1 (λ, a)s
2 (λ, a), that is, that ˜0 (λ) = ϕ˜e (λ) + iλϕ˜01 (λ)ϕ˜02 (λ) is the Jost function for the problem the function E yj

+ (λ2 − q˜j (x))yj = 0, y3



2

+ λ y3 = 0,

x ∈ [0, a],

j = 1, 2,

x ∈ [0, ∞),

y1 (λ, a) = y2 (λ, a) = y3 (λ, 0), y1
(λ, a) + y2
(λ, a) − y3
(λ, 0) = 0, y1 (λ, 0) = y2 (λ, 0) = 0. (1)

(1)

νk )2 → (νk )2 , Now let us make the inverse transformation (˜ µk )2 → (µk )2 , (˜ (2) 2 (2) 2 (˜ νk ) → (νk ) . The corresponding functions are defined as follows: ∞  4a2  ∞  4a2  2 2 2 2 (µ = 2a (˜ µ , ϕe (λ) := 2a ) − λ ) − c − λ k k k=1 π 2 k 2 k=1 π 2 k 2 ∞  a2  (1) ∞  a2  (1) 2 2 2 2 ϕ01 (λ) := a (ν = a (˜ ν , ) − λ ) − c − λ k k k=1 π 2 k 2 k=1 π 2 k 2     2 2 ∞ ∞ a a (2) (2) ϕ02 (λ) := a (νk )2 − λ2 =a (˜ νk )2 − λ2 . 2 2 k=1 π k k=1 π 2 k 2 Therefore, the function E0 (λ) = ϕe (λ) + iλϕ01 (λ)ϕ02 (λ) is the Jost function for the boundary value problem (1.1)–(1.6) with q3 (x) = 0, x ∈ [0, ∞). Thus, qj (x) = q˜j (x) − c, j = 1, 2, are the potentials that we had to construct in Theorem 4.4.
Remark 4.6. Even if we assume a’priori that the Jost function E0 (λ) has no real zeros (with a possible exception of a simple zero at the origin) and no pairs of pure imaginary zeros, then, given a function S(λ), the choice of the pair of potentials (1) νk }∞ (qj (x))2j=1 is not unique because the choice of the sequences {˜ −∞,k=0 and (2)

{˜ νk }∞ µk }∞ −∞,k=0 is not unique. However, as soon as the three spectra, {˜ −∞,k=0 , (1)

(2)

νk }∞ {˜ νk }∞ −∞,k=0 , and {˜ −∞,k=0 , are fixed (and do not intersect), the procedure of recovering q˜1 (x) and q˜2 (x) from the three spectra, as described in [38], gives a unique pair (˜ qj (x))2j=1 , and thus a unique pair (qj (x))2j=1 .

Appendix A. In this section we prove several abstract results from spectral theory of operator pencils mainly used in the proof of Theorem 3.3 (but also of some independent interest). First, we recall some terminology (for more details see, e.g., [32, Sec.11]). Let L(λ) be a pencil of linear operators acting on a separable complex Hilbert space H with the domain D(L) independent of λ, and let B(H) denote the set of bounded operators on H. The set (L) of λ ∈ C such that L(λ)−1 ∈ B(H) is called the resolvent set of the operator pencil L(λ), and the set σ(L) = C\(L) is called

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the spectrum of L(λ). A number λ0 ∈ C is called an eigenvalue of L(λ) if there exists a nonzero vector y0 ∈ D(L) (called an eigenvector) such that L(λ0 )y0 = 0. Nonzero vectors y1 , y2 , · · · , yp−1 are called associated vectors if n  1 dk L(λ) |λ=λ0 yn−k = 0, k! dλk

n = 1, · · · , p − 1.

(A.1)

k=0

The number p is called the length of the chain composed of the eigenvector and associated vectors. An eigenvalue is called semisimple if it does not have associated vectors. The geometric multiplicity of an eigenvalue is defined as the maximal number of the corresponding linearly independent eigenvectors. The algebraic multiplicity is defined as the maximal value of the sum of the lengths of chains corresponding to the linearly independent eigenvectors. An eigenvalue is called isolated if it has a punctured neighborhood contained in the resolvent set. An isolated eigenvalue λ0 of finite algebraic multiplicity is called normal if the subspace Im L(λ0 ) is closed. The set of normal eigenvalues of L(λ) is denoted by σ0 (L). In what follows we consider the quadratic operator pencil L(λ) = λ2 M − iλK − A and, throughout, we assume that K and M are bounded operators on H, that is, K, M ∈ B(H), and A is a closed operator with the domain D(A) dense in H. As usual, the domain of the pencil is chosen to be D(L(λ)) = D(M ) ∩ D(K) ∩ D(A) = D(A), and is independent of λ. Hypothesis A.1. Assume that: (i) M ≥ 0, K ≥ 0, and A = A∗ ≥ −βI for some positive β. (ii) If β1 > β, then the inverse operator (A + β1 I)−1 is a compact operator. (iii) Ker A ∩ Ker K ∩ Ker M = {0}. Under Hypothesis A.1, the spectrum of L(λ) consists of normal eigenvalues only. This follows, in fact, from well-known results on analytic functions with values in the set of Fredholm operators on H, see, e.g., [15, Cor.XI.8.4]. Our first result is a generalization of Conclusion 2.40 in [23]. Theorem A.2. Assume Hypothesis A.1. Then: (1) If A ≥ 0, then the spectrum of L(λ) is located in the closed upper half-plane. (2) If A ≥ βI for some β > 0, and K > 0, then the spectrum of L(λ) is located in the open upper half-plane. (3) If A ≥ βI for some β > 0, and λ2 M y − Ay = 0 for all real λ and all nonzero y ∈ Ker K, then the spectrum of L(λ) is located in the open upper half-plane. Proof. Let y0 = 0 be an eigenvector of L(λ) corresponding to an eigenvalue λ0 . Then the equality (L(λ0 )y0 , y0 ) = 0 implies: ((Re λ0 )2 − (Im λ0 )2 )(M y0 , y0 ) + Im λ0 (Ky0 , y0 ) − (Ay0 , y0 ) = 0,

(A.2)

Re λ0 (2Im λ0 (M y0 , y0 ) − (Ky0 , y0 )) = 0.

(A.3)

If Re λ0 = 0, then (M y0 , y0 ) = 0 by (iii) in Hypothesis A.1, and the inequality Im λ0 ≥ 0 follows from (A.3) and (i) in Hypothesis A.1. If Re λ0 = 0, then (A.2)

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implies Im λ0 (Ky0 , y0 ) = (Im λ0 )2 (M y0 , y0 ) + (Ay0 , y0 ) ≥ 0 by (i) in Hypothesis A.1 and the assumption A ≥ 0. Then (Ky0 , y0 ) = 0 by (iii) in Hypothesis A.1, and thus Im λ0 ≥ 0, proving assertion (1). Next, assume A > βI for some β > 0 and K > 0, and suppose that Im λ0 = 0 in (A.2) and (A.3). If Re λ0 = 0, then (A.2) implies (Ay0 , y0 ) = 0, contradicting positivity of A. If Re λ0 = 0, then (A.3) implies (Ky0 , y0 ) = 0, contrary to K > 0, proving assertion (2). If Im λ0 = 0, then (A.3) implies y0 ∈ Ker K because equality Re λ0 = 0 by (A.2) contradicts positivity of A. But then L(λ0 )y0 = λ20 M y0 − Ay0 = 0, contradicting the assumptions and proving (3).
If A is not assumed to be nonnegative, then L(λ) might have eigenvalues in the open lower half-plane; they are located as follows. Lemma A.3. Assume Hypothesis A.1. Then: (1) The part of the spectrum of L(λ), located in the open lower half-plane, belongs to the imaginary axis. (2) If K > 0, then the part of the spectrum of L(λ), located in the closed lower half-plane, belongs to the imaginary axis. Proof. Let y0 = 0 be an eigenvector of L(λ) corresponding to an eigenvalue λ0 with Im λ0 < 0. Then for Re λ0 = 0 equation (A.3) implies (M y0 , y0 ) = (Ky0 , y0 ) = 0, and, consequently, M y0 = Ky0 = 0. Then L(λ0 )y0 = Ay0 = 0, contradicting (iii) in Hypothesis A.1 and thus proving assertion (1). If K > 0, then for Im λ0 ≤ 0 the equality Re λ0 = 0 follows from (A.3) as above, proving assertion (2).
Lemma A.4. Assume Hypothesis A.1. Then: (1) All nonzero eigenvalues of L(λ) in the closed lower half-plane are semisimple. (2) If K > 0 on Ker A, then all eigenvalues of L(λ), located in the closed lower half-plane, are semisimple. Proof. Let λ0 be an eigenvalue of L(λ) located in the open lower half-plane, let y0 = 0 be a corresponding eigenvector, and suppose that there exists a nonzero associated vector y1 . Then, using (A.1), we compute: λ20 M y1 − iλ0 Ky1 − Ay1 + 2λ0 M y0 − iKy0 = 0.

(A.4)

Multiplying (A.4) by y0 we infer: ((λ20 M − iλ0 K − A)y1 , y0 ) + ((2λ0 M − iK)y0 , y0 ) = 0.

(A.5)

Since λ0 is pure imaginary by Lemma A.3, we have from (A.5): (y1 , (λ20 M − iλ0 K − A)y0 ) + ((2λ0 M − iK)y0 , y0 ) = 0,

(A.6)

which implies, taking the imaginary part, that ((2Im λ0 M − K)y0 , y0 ) = 0.

(A.7)

Now Im λ0 < 0 implies (M y0 , y0 ) = (Ky0 , y0 ) = 0, yielding M y0 = Ky0 = 0. In this case L(λ0 )y0 = −Ay0 = 0 and, consequently, y0 ∈ Ker M ∩ Ker K ∩ Ker A. Then, due to (iii) in Hypothesis A.1, we have y0 = 0, a contradiction.

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Next, let λ0 = 0 be a real eigenvalue of L(λ). Then (A.3) implies (Ky0 , y0 ) = 0, and, consequently, Ky0 = 0 and (λ20 M − A)y0 = 0. Then (A.6) implies that 2λ0 (M y0 , y0 ) = 0, yielding M y0 = 0. Hence, using Ky0 = 0, we obtain Ay0 = 0, which contradicts (iii) in Hypothesis A.1 again, proving assertion (1). To prove assertion (2), we need to show that if λ0 = 0 is an eigenvalue, then it is semisimple. But if λ0 = 0, then y0 ∈ Ker A and (A.4) can be written as Ay1 + iKy0 = 0. Multiplying this by y0 , we have (Ay1 , y0 ) + i(Ky0 , y0 ) = (y1 , Ay0 ) + i(Ky0 , y0 ) = i(Ky0 , y0 ) = 0, in contradiction with K > 0 on Ker A.




Sometimes, it is more convenient to deal with bounded operator pencils. Assuming A ≥ −βI > −β1 I for some β > 0, we introduce the auxiliary bounded op˜ erator pencil L(λ) = L(λ)(A+β1 I)−1 . The next lemma follows form [32, Lem.20.1]. ˜ Lemma A.5. If A ≥ −βI > −β1 I for some β > 0, then σ(L(λ)) = σ(L(λ)). Next, we introduce the family L(λ, η) = λ2 M − iληK − A of operator pencils depending on a parameter η ∈ C so that L(λ, 1) = L(λ). Lemma A.5 enables us to use for the unbounded operator pencil L(λ, η) the results of [10] (see also [17, 24]) established for bounded operator pencils. Adapted to the current discussion, these results can be summarized as follows. Theorem A.6. Assume Hypothesis A.1. Given η0 ∈ C, let Ω be a connected domain in C containing only one eigenvalue λ0 of the pencil L(λ, η0 ). Let yl0 , l = 1, . . . , , denote linearly independent eigenvectors corresponding to the eigenvalue λ0 of the pencil L(λ, η0 ), and let pl , l = 1, . . . , , denote the length of the chain composed of the eigenvector yl0 and its associated vectors. Finally, let m denote the algebraic multiplicity of the eigenvalue λ0 . Then there exist numbers  > 0 and m0 ∈ N such that m0 ≤ m and for each η from the neighborhood {η ∈ C : |η − η0 | < } of η0 the following assertions hold: (1) L(λ, η) has exactly m0 different eigenvalues in the domain Ω. These eigenvalues can be arranged in groups λlj (η), where j = 1, . . . , pl and l = 1, . . . ,   such that l=1 pl = m0 . The groups can be chosen in the way that the functions λl1 (η), λl2 (η),. . . ,λlpl (η), that belong to the same group, correspond to the complete set of pl branches of the multi-valued function η 1/pl . Moreover, these eigenvalues can be represented as the following series: ∞ λlj (η) = λ0 + alk (((η − η0 )1/pl )j )k , j = 1, . . . , pl , (A.8) k=1

where alk ∈ C and ((η − η0 )1/pl )j , j = 1, . . . , pl , denotes the j-th branch of the multi-valued function (η − η0 )1/pl . (2) A basis in the eigenspace corresponding to λlj (η) can be chosen as follows: ∞ (q) (q) (q) ylk (((η − η0 )1/pl )j )k , j = 1, . . . , pl , q = 1, . . . , αl , (A.9) ylj (η) = yl0 + k=1

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where αl is the geometric multiplicity of the eigenvalue λlj (η), and the vectors (q) yl0 , q = 1, . . . , αl , belong to the eigenspace of L(λ, η0 ) corresponding to the eigenvalue λ0 . This theorem is a generalization of a well-known theorem on expansions for analytic functions in multi-valued case, cf. [33, Thm.13.3.6]. Corollary A.7. Suppose that assumptions of Theorem A.6 hold. If λ0 is a semisimple eigenvalue of L(λ, η0 ), then formulae (A.8) and (A.9) assume the form ∞ alk (η − η0 )k , l = 1, . . . , , (A.10) λl (η) = λ0 + k=1  ∞ (q) (q) (q) ylk (η − η0 )k , q = 1, . . . , αl . (A.11) yl (η) = yl0 + k=1

Lemma A.8. Suppose that the assumptions of Theorem A.6 hold, and let λk (η) with λk (0) = iτ , where τ ∈ R, be an eigenvalue of L(λ, η). Then: (1) Re λ˙ k (0) = 0 and Im λ˙ k (0) ≥ 0, where “dot” denotes d/dη. (2) If τ < 0, then Re λ˙ k (η) = 0 and Im λ˙ k (η) ≥ 0 for all η ≥ 0. (3) If 0 is an eigenvalue of L(λ, η) for some η ≥ 0, then it is an eigenvalue for all η ≥ 0. The algebraic multiplicity of the zero eigenvalue for η = 0 is even, and if it is denoted by 2κ, then for all η > 0 the algebraic multiplicity of the zero eigenvalue is equal to κ. Proof. Let η0 ∈ [0, 1] and let λ0 with Re λ0 = 0 and Im λ0 < 0 be an eigenvalue of L(λ, η0 ). Due to Lemma A.4 this eigenvalue is semi-simple. Then (A.8) and (A.9) (q) can be written as (A.10) and (A.11). Taking the η-derivative in L(λl (η), η)yl (η) = (q) 0 and multiplying the resulting equation by yl , we infer for η = η0 : (q)

al1 =

(q)

(q)

(q)

(q)

iλ0 (Kyi0 , yi0 )

(q)

(q)

2λ0 (M yl0 , yl0 ) − iη0 (Kyl0 , yl0 )

=

(q)

iτ (Kyl0 , yl0 )

(q)

(q)

(q)

(q)

2τ (M yl0 , yl0 ) − η0 (Kyl0 , yl0 )

.

It is clear that Re al1 = 0 and Im al1 ≥ 0 for η0 = 0 and for η0 ≥ 0 and τ < 0.
The total algebraic multiplicity of the part of the spectrum of L(λ) in a domain n Ω is defined as k=1 mk , where mk , k = 1, . . . , n, are the algebraic multiplicities of all n eigenvalues located in Ω. The following fact is a consequence of Corollary A.7, Lemma A.8, Theorem A.2, Lemma A.3, and Lemma A.4. Corollary A.9. (1) Assume that M ≥ 0, K ≥ 0 and M + K > βI for some β > 0. Then the total algebraic multiplicity of the part of the spectrum of L(λ) located in the open lower half-plane coincides with the total algebraic multiplicity (which is equal to the geometric multiplicity) of the negative spectrum of A. (2) If, in addition, K > 0, then the total algebraic multiplicity of the part of the spectrum of L(λ) located in the closed lower half-plane coincides with the total algebraic multiplicity of the nonnegative spectrum of A.

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This fact (under different assumptions) was proved in [36] and [37]; for other versions of this result see [2, 4, 18, 48] and review [42].

Appendix B. The main objective of this section is to prove Lemma 3.11. We will use the following theorem (see Theorem 3 in [27, Sec.VII.2]). Theorem B.1. Assume that ω(λ) = P (λ) + iQ(λ), where P (λ) and Q(λ) are real entire functions, and suppose that ∞ P (λ) = Aeu(λ) (λ − a0 ) (1 − λ/ak )epk (λ/ak ) , u(0) = 0, k=−∞,k=0 ∞ (1 − λ/bk )epk (λ/bk ) , v(0) = 0 Q(λ) = Bev(λ) (λ − b0 ) k=−∞,k=0

are their expansions in infinite products. Then the function ω(λ) belongs to the class HB if and only if the following conditions hold: (a) The zeros ak and bk of the functions P (λ) and Q(λ) interlace, that is: bk < ak < bk+1 ,

k = ±1, ±2, . . . ,

and

a−1 < 0 < b1 .

(B.1)

(b) The real entire functions u(λ) and v(λ) and the exponents pk (λ/ak ) and pk (λ/bk ) satisfy the condition ∞ (pk (λ/ak ) − pk (λ/bk )) = 0. u(λ) − v(λ) + k=−∞

(c) The constants A and B have opposite signs. Remark B.2. We note a misprint in assertion (c) of Theorem 3 in [27, Sec.7.2], where the clause “same signs” should be replaced by the clause “opposite signs”. The proof of Theorem 3 in [27, Sec.7.2] also gives the following corollary. Corollary B.3. Assume that ω(λ) = P (λ) + iQ(λ), where P (λ) and Q(λ) are real entire functions having the following expansions into infinite products: ∞ P (λ) = Aeu(λ) (1 − λ/ak )epk (λ/ak ) , u(0) = 0, k=1 ∞ (1 − λ/bk )epk (λ/bk ) , v(0) = 0. Q(λ) = Bev(λ) k=1

Then ω(λ) belongs to the class HB if and only if the following conditions hold: (a) The zeros ak and bk of the functions P (λ) and Q(λ) interlace: ak−1 < bk−1 < ak < bk ,

k = 2, 3, . . . .

(b) The entire real-valued functions u(λ) and v(λ) and the exponents pk (λ/ak ) and pk (λ/bk ) satisfy the condition ∞ (pk (λ/ak ) − pk (λ/bk )) = 0. u(λ) − v(λ) + k=1

(c) The constants A and B have the same sign.

(B.2)

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We are ready to prove Lemma 3.11. Proof. Using the symmetry of the given function ω(λ) ∈ SHB, we enumerate the zeros of Q(λ) so that b0 = 0, b−k = −bk ; then a0 > 0 due to (B.1). Changing the ˜k = ak for k < 0 and a ˜k = ak−1 for k > 0, and using numeration of ak by letting a the symmetry a ˜−k = −˜ ak , we obtain: 2 ∞ 2 2 P (λ) = −a0 Aeu(λ ) (1 − λ2 /˜ a2k )epk (λ /˜ak ) , u(0) = 0, k=1 2 ∞ 2 2 Q(λ) = Bλev(λ ) (1 − λ2 /b2k )epk (λ /bk ) , v(0) = 0, k=1

˜1 < so that the following statements hold: (1) The zeros a ˜k and bk interlace: 0 < a b1 < a ˜2 < b2 < ...; (2) The entire real-valued functions u(λ) and v(λ) and the exponents pk (λ2 /˜ ak ) and pk (λ/bk ) satisfy the condition ∞   2 pk (λ2 /˜ ak ) − pk (λ2 /bk ) = 0; u(λ ) − v(λ2 ) + 2 k=1

(3) The constants −a0 A and B have the same sign. Therefore, we infer ∞ 2 P˜ (λ) = −a0 Aeu(λ) (1 − λ/˜ a2k )epk (λ/ak ) , u(0) = 0, k=−∞,k=0 ∞ 2 v(λ) ˜ Q(λ) = Be (1 − λ/b2k )epk (λ/bk ) , v(0) = 0, k=1

and an application of Corollary B.3 concludes the proof of Lemma 3.11.




The proof of the following theorem can be found in [38, Thm.2.1]. (1)

(2)

∞ ∞ Theorem B.4. Assume that {µk }∞ k=−∞,k=0 , {νk }k=−∞,k=0 , {νk }k=−∞,k=0 are three sequences of real numbers satisfying the relations µk < µk+1 , νk < νk+1 , (1) (1) (1) (2) (2) (2) νk < νk+1 , µ−k = µk , ν−k = −νk , ν−k = νk for k = 1, 2, . . . , and having the asymptotic properties given in (3.30)–(3.32), where P1 , F1 and F2 are real constants satisfying the inequality F1 = F2 and equality F1 + F2 = P1 . Also, assume (1) (2) ∞ that {νk }∞ k=−∞,k=0 ∩ {νk }k=−∞,k=0 = ∅, and the squares of the elements of the (1)

(2)

∞ ∞ ∞ sequences {µk }∞ k=−∞,k=0 and {νk }k=−∞,k=0 = {νk }k=−∞,k=0 ∪ {νk }k=−∞,k=0 2 2 2 2 interlace as follows: 0 < (µ1 ) < (ν1 ) < (µ2 ) < (ν2 ) < . . . . Then there exists a unique real-valued potential q(x) ∈ L2 (0, 2a) such that the three sequences (1) ∞ (2) ∞ {µk }∞ k=−∞,k=0 , {νk }k=−∞,k=0 and {νk }k=−∞,k=0 , respectively, constitute the spectra of the three boundary value problems (3.12), (4.6) for j = 1, and (4.6) for j = 2, respectively, where the potentials qj in (4.6) defined via this q(x) by q1 (x) = q(x) and q2 (x) = q(2a − x), x ∈ [0, a].

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[3] V. M. Adamyan and B. P. Pavlov, Zero-radius potentials and M. G. Kre˘ın’s formula for generalized resolvents. J. Soviet Math. 42 (1988), 1537–1550. [4] V. M. Adamyan and V. N. Pivovarchik, On the spectra of some classes of quadratic operator pencils. Oper. Theory Adv. Appl. 106 (1998), 23–36. [5] F. V. Atkinson, Discrete and continuous boundary problems. Academic Press, 1964. [6] J. von Below, Can one hear the shape of a network? Lect. Notes Pure Appl. Math. 219, 19–36, Dekker, New York, 2001. [7] R. Carlson, Inverse eigenvalue problems on directed graph. Trans. AMS, 351 (1999), 4069–4088. [8] R. Carlson, Hill’s equation for a homogeneous tree. Electr. JDE. 23 (1997), 1–30. [9] D. E. Edmuns and W. D. Evans, Spectral theory and differential operators. Claredon Press, Oxford, 1989. [10] V. N. Eni, On stability of the root number of analytic operator-function and on perturbations of its characteristic numbers and eigenvectors. Doklady AN SSSR 173 (1967), 1251–1254. [11] P. Exner and R. Gawlista, Band spectra of rectangular graph superlattices. Phys. Rev. B 53 (1996), 4275–4286. [12] P. Exner and P. Seba, A new type of quantum interference transistor. Phys. Lett. A 129:8,9 (1988), 477–480. [13] N. I. Gerasimenko, Inverse scattering problem on a noncompact graph. Teoretich. Matem. Fisika 75 (1988), 187–200. [14] N. I. Gerasimenko and B. S. Pavlov, Scattering problem on noncompact graphs. Teoretich. Matem. Fisika 74 (1988), 345–359. [15] I. C. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of linear operators. I. Oper. Theory Adv. Appl. 49, Birkh¨ auser Verlag, Basel, 1990. [16] I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators. AMS, Providence, 1969. [17] I. C. Gohberg and E. I. Sigal, The root multiplicity of the product of meromorphic operator functions. Mat. Issled. 6 (1971), 33–50, 158. [18] R. O. Griniv and A. A. Shkalikov, On operator pencils arising in the problem of beam oscillations with internal damping. Math. Notes 56 (1994), 114–131. [19] B. Gutkin and U. Smilansky, Can one hear the shape of a graph? J. Phys. A Math. Gen. 34 (2001), 6061–6068. [20] M. S. Harmer, Inverse scattering for the matrix Schr¨ odinger operator and Schr¨ odinger operator on graphs with general self-adjoint boundary conditions. ANZIAM J. 44 (2002), 161–168. [21] I. S. Kac and M. G. Krein, R-functions – analytic functions mapping the upper half-plane into itself. AMS Translations, Ser. 2, 103 (1974), 1–18. [22] I. S. Kac and M. G. Krein, On the spectral function of the string. AMS Translations, Ser. 2, 103 (1974), 19–102. [23] M. G. Krein and H. Langer, On some mathematical principles in the linear theory of damped oscillations of continua. I. Integral Eqns. Oper. Theory 1 (1978), 364–399.

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[46] T. Regge, Construction of potential from resonances. Nuovo Cimento, 9, No. 3 and No. 5 (1958), 491–503, 671–679. [47] R. D. Richtmyer, Principles of advanced mathematical physics-I. Springer, 1978. [48] A. A. Shkalikov, Operator pencils arising in elasticity and hydrodynamics: the instability index formula. Oper. Theory Adv. Appl. 87 (1996), 358–385. Y. Latushkin Department of Mathematics University of Missouri Columbia, MO 65203 USA e-mail: [email protected] V. Pivovarchik Department of Applied Mathematics and Informatics South-Ukrainian State Pedagogical University Staroportofrankovskaya str. 26 65020 Odessa Ukraine e-mail: [email protected] Submitted: June 20, 2007

Integr. equ. oper. theory 61 (2008), 401–412 0378-620X/030401-12, DOI 10.1007/s00020-008-1595-4 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Integral Equations and Operator Theory

Discontinuity of the Lempert Function and the Kobayashi–Royden Metric of the Spectral Ball Nikolai Nikolov, Pascal J. Thomas and W
lodzimierz Zwonek Abstract. Some results on the discontinuity properties of the Lempert function and the Kobayashi pseudometric in the spectral ball are given. Mathematics Subject Classification (2000). Primary 32F45; Secondary 32A07. Keywords. Spectral Nevanlinna–Pick problem, spectral ball, symmetrized polydisc, Lempert function, Kobayashi–Royden pseudometric.

1. Introduction and results Let Mn be the set of all n × n complex matrices. For A ∈ Mn denote by sp(A) and r(A) = max |λ| the spectrum and the spectral radius of A, respectively. λ∈sp(A)

The spectral ball Ωn is the set Ωn = {A ∈ Mn : r(A) < 1}. The Nevanlinna–Pick problem in Ωn (or the spectral Nevanlinna–Pick problem) is the following one: given N points a1 , . . . , aN in the unit disk D ⊂ C and N matrices A1 , . . . , AN ∈ Ωn decide whether there is a holomorphic map ϕ ∈ O(D, Ωn ) such that ϕ(aj ) = Aj , 1 ≤ j ≤ N . This problem has been studied by many authors; we refer the reader to [1, 2, 3, 4, 6, 7] and the references there. The study of the spectral Nevanlinna–Pick problem in the case N = 2 reduces to the computation of the Lempert function of the spectral ball. Recall that for a domain D ⊂ Cm the Lempert function of the domain D is defined as follows: lD (z, w) := inf{|α| : ∃ϕ ∈ O(D, D) : ϕ(0) = z, ϕ(α) = w}, z, w ∈ D. The infinitesimal version of the above problem, the so-called spectral Carath´eodory–Fej´er problem is the following one: given N + 1 matrices A0 , . . . , AN This work was initialized during the stay of the first and second named authors at the Jagiellonian University, Krak´ ow in October, 2006, supported by the EGIDE program. They wish to thank both institutions. The third author was supported by the KBN research grant No. 1 PO3A 005 28.

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in Mn decide whether there is a map ϕ ∈ O(D, Ωn ) such that Aj = ϕ(j) (0), 0 ≤ j ≤ N . This problem has been studied in [13]. The study of the spectral Carath´eodory-Fej´er problem in the case N = 1 reduces to the computation of the Kobayashi–Royden pseudometric of the spectral ball. Recall that for a domain D ⊂ Cm the Kobayashi–Royden pseudometric is defined as follows: κD (z; X) := inf{|α| : ∃ϕ ∈O(D, D) : ϕ(0) = z, αϕ (0) = X}, z ∈ D, X ∈ Cm . The functions lD and κD , where D is a domain in some Cn , as examples of the so-called holomorphically invariant functions (or pseudometrics) are studied in detail in many textbooks. All the properties of these functions that we use in the paper may be found in [14]. In particular, recall that both functions lD and κD are always upper semicontinuous and in the case when D is taut (for definition see also [14]) they are both continuous. Recall also that if D is a bounded domain then lD (w, z) > 0, κD (w; X) > 0, w, z ∈ D, w = z, X ∈ Cn , X = 0. In this note we point out some of the instability phenomena of both spectral problems which complicate their study. First note that if we replace each of the matrices in the spectral Nevanlinna– Pick problem by similar ones then we do not change its solution.1 A natural reduction of the problems is then to associate to each matrix its spectrum, or, in order to deal with n-tuples of complex numbers, the coefficients of its characteristic polynomial n
(−1)j ej (A)tn−j , PA (t) := det(tI − A) = tn + j=1

where I ∈ Mn is the unit matrix, and ej (A) := ej (λ1 , . . . , λn ) :=




λk1 . . . λkj

1≤k1 <···
are the elementary symmetric functions of the eigenvalues λ1 , . . . , λn of A. Put E = (e1 , . . . , en ). We shall consider E as a map either from O(Mn , Cn ), or from O(Cn , Cn ). The set Gn := {E(A) : A ∈ Ωn } is called the symmetrized n-disk, and has been widely studied; we refer the reader to [3, 7, 15, 16, 17] and references there. We recall a few definitions from linear algebra. Definition 1. Given a matrix A ∈ Mn , the commutant of A is C(A) := {M ∈ Mn : M A = AM }, 1 Indeed,

˜j , 1 ≤ j ≤ N . Then A ˜j = assume that ϕ ∈ O(D, Ωn ), ϕ(aj ) = Aj and Aj ∼ A ˆ ˆ ˆj ∈ Mn . In the standard way we may find ϕ eAj Aj e−Aj for some A ˆ ∈ O(D, Mn ) with ϕ(a ˆ j) = ˆj , 1 ≤ j ≤ N. Then ϕ ˜ := eϕˆ ϕe−ϕˆ ∈ O(D, Ωn ) and ϕ(a ˜ j ) = Aj . A

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and the set of polynomials in A, P(A) ⊂ C(A) is given by P(A) := {M ∈ Mn : M = p(A), for some p ∈ C[X]}. Definition 2. Given (a0 , . . . , an−1 ) ∈ Cn , the associated companion matrix is   0 −a0  ..   1 0  .     . .. ..  . . 1     . . . 0 −a   n−2 1 −an−1 The companion matrix associated to a matrix A is the one associated to the coefficients of its characteristic polynomial, namely we set aj = (−1)n−j en−j (A), 0 ≤ j ≤ n − 1. We now gather some useful facts from linear algebra, all well known except perhaps for those involving differentiation. In the proposition below (and in the sequel), E∗,A denotes the Fr´echet derivative of E at A and (ΦA )∗,B denotes the Fr´echet derivative of the mapping ΦA at B. Proposition 3. A matrix A ∈ Mn with the following equivalent properties is called non-derogatory. 1. A is similar to its companion matrix. 2. There exists a cyclic vector for A. 3. The characteristic and minimal polynomials of A coincide. 4. Different blocks in the Jordan normal form of A correspond to different eigenvalues (that is, each eigenspace is of dimension exactly 1). 5. C(A) = P(A). 6. rank(E∗,A ) = n. 7. dim C(A) = n. −1 8. If ΦA : M−1 n −→ Mn , where Mn stands for the set of invertible matrices, is defined by ΦA (P ) := P −1 AP , then rank((ΦA )∗,In ) = n2 − n (its maximal possible value). Most of those properties can be found in [11, pp. 135–147]; more precise references and (easy) complements are given in Section 4. Recall that the j-th coordinate of E∗,A (B) is the sum of all j ×j determinants obtained by taking a principal j × j submatrix of A and replacing one column by the corresponding entries of B. In particular, the first coordinate of E∗,A (B) equals tr(B). Denote by Cn the set of all non-derogatory matrices in Ωn . Obviously Cn is an open and dense subset of Ωn . Note that if A1 , . . . , AN belong to Cn , then any mapping ϕ ∈ O(D, Gn ) with ϕ(αj ) = E(Aj ) can be lifted to a mapping ϕ˜ ∈ O(D, Ωn ) with ϕ(α ˜ j ) = Aj , 1 ≤ j ≤ N (see [1]). This means that in a generic case the spectral Nevanlinna–Pick

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problem for Ωn with dimension n2 can be reduced to the standard Nevanlinna–Pick problem for Gn with dimension n. As a consequence of the existence of the lifting above, we have the equality lΩn (A, B) = lGn (E(A), E(B)),

A, B ∈ Cn .

(1)

Note that Gn is a taut domain (cf. [9], [15]). In particular, there always exist extremal discs for lGn ; moreover, lGn is a continuous function (see e.g. [14]). Thus the spectral Nevanlinna–Pick problem with data (α1 , A1 ), (α1 , A2 ) ∈ D × Cn is solvable if and only if    α1 − α2  . lGn (E(A), E(B)) ≤ m(α1 , α2 ) :=  1 − α1 α2  An explicit formula for lG2 is found in [3]. The proof there is based on studying the complex geodesics of G2 . It turns out that tanh−1 lG2 coincides with the Carath´eodory distance of G2 . On the other hand, G2 cannot be exhausted by domains biholomorphic to convex domains (see [5], [8]). So G2 serves as the first counterexample to converse of the Lempert theorem (cf. [15]). In spite of this phenomenon, tanh−1 lGn , does not even satisfy the triangle inequality for n > 2, that is, it does not coincide with the Kobayashi distance of Gn for n > 2 (see [17]). The behavior of lΩn is much more complicated when one of the arguments is derogatory. However, if A is a scalar matrix, say A = tI, t ∈ D, then (cf. [1]) lΩn (tI, B) = max m(t, λ). λ∈sp(B)

(2)

To prove (2), observe first that B → (B − tI)(I − tB)−1 is an automorphism of Ωn . So we may assume that t = 0. Then it remains to make use of the fact that lΩn (0, B) equals the Minkowski function of the balanced domain Ωn at B, that is, r(B). Since the 2 × 2 derogatory matrices are scalar, we also get that the function lΩ2 is not continuous at the point (A, B) if and only if one of the matrices is scalar, say A, and the other one has two distinct eigenvalues (see [6]). In this case even lΩ2 (·, B) is not continuous at A (but lΩ2 (A, ·) is continuous at B). We shall show that this phenomenon extends to Ωn . Proposition 4. For B ∈ Cn and t ∈ D the following conditions are equivalent: (i) the eigenvalues of B are equal; (ii) the function lΩn is continuous at the point (tI, B); (iii) the function lΩn (·, B) is continuous at the point tI. At the infinitesimal level of the Kobayashi–Royden pseudometric, for A ∈ Cn and B ∈ Mn one has that (see Theorem 2.1 in [13]) κΩn (A; B) = κGn (E(A); E∗,A (B)).

(3)

Since κΩn (A; B) ≥ κGn (E(A); E∗,A (B)) for (A; B) ∈ Ωn × Mn , κΩn is an upper semicontinuous function and κGn is continuous (because Gn is a taut domain), we get that κΩn is a continuous function at any point (A; B) ∈ Cn × Mn .

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The things are more complicated if A ∈ Cn . Proposition 5. For B ∈ Mn and t ∈ D the following conditions are equivalent: (i) the eigenvalues of B are equal; (ii) the function κΩn is continuous at the point (tI; B); (iii) the function κΩn (·; B) is continuous at the point tI. Note that, similarly to the equality (2), one has that κΩn (tI; B) =

r(B) . 1 − |t|2

(4)

As a consequence of our considerations, we may also identify in a simple way ˆ n of Ωn . the convex hull Ω ˆ n = {A ∈ Mn : |tr(A)| < n}. Proposition 6. Ω We now turn to analyzing the failure of hyperbolicity of Ωn . Observe first that if sp(A) = sp(B) then E(A) =  E(B) and hence lΩn (A, B) ≥ lGn (E(A), E(B)) > 0. Then as a consequence of the proof of Lemma 13 in [9] we have the following Proposition 7. For any A, B ∈ Ωn the equality lΩn (A, B) = 0 holds if and only if sp(A) = sp(B). Moreover, in this case there is a ϕ ∈ O(C, Ωn ) with ϕ(0) = A, ϕ(1) = B and sp(ϕ(λ)) = sp(A) for any λ ∈ C. It is natural to consider the infinitesimal version of this proposition. First, note that the equality (4) implies that if A ∈ Ωn is a scalar matrix and B ∈ Mn , then κΩn (A; B) = 0 if and only if sp(B) = {0}. Furthermore, if A is scalar and sp(B) = {0}, then the linear mapping p : λ → A + λB has the following properties: p(0) = A, p (0) = B and sp(p(λ)) = sp(A) for any λ ∈ C. On the other hand, the equality (3) implies that if A ∈ Cn and B ∈ Mn , then κΩn (A; B) = 0 if and only if E∗,A (B) = 0. Consider the analytic set LA := {C ∈ Ωn : sp(C) = sp(A)}. Since A ∈ Cn , we get that rank(E∗,A ) = n. An application of the Implicit Function Theorem shows that LA is smooth near A, and that the kernel of E∗,A coincides with the tangent cone TA of LA at A. Hence there is r > 0 and ϕ ∈ O(rD, Ωn ) such that ϕ(0) = A, ϕ (0) = B and sp(ϕ(λ)) = sp(A) for any λ ∈ rD. These two observations let us state the following Conjecture 8. If A ∈ Ωn , B ∈ Mn and κΩn (A; B) = 0, then there is a polynomial mapping p : C → Ωn of degree at most n with p(0) = A, p (0) = B and sp(p(λ)) = sp(A) for any λ ∈ C. To support this conjecture, we shall prove it for n = 2, and also give the following weaker result.

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Proposition 9. If A ∈ Cn , B ∈ Mn and κΩn (A; B) = 0, then there is a mapping ϕ ∈ O(C, Ωn ) with ϕ(0) = A, ϕ (0) = B and sp(ϕ(λ)) = sp(A) for any λ ∈ C. The rest of the paper is organized as follows. The proof of Proposition 4 is given in Section 4. Section 5 contains the proofs of Propositions 5, 6 and 9, as well as Conjecture 8 for n = 2. The proof of Proposition 3 is discussed in Section 4.

2. Proof of Proposition 4 We shall need the following Proposition 10.

(i) If A, B ∈ Ωn , then

lGn (E(A), E(B)) ≤ lΩn (A, B) ≤ min max m(λj , µπ(j) ), π

1≤j≤n

where sp(A) = {λ1 , . . . , λn }, sp(B) = {µ1 , . . . , µn }, and the minimum is taken over all permutations π of {1, . . . , n}. 1−α z−α (ii) Let B ∈ O(D, Ωn ). If the eigenvalues of B(z) have the form eiθ 1−αjj 1−zαjj , where αj ∈ C, 1 ≤ j ≤ n, then lGn (E(B(z)), E(B(w))) = lΩn (B(z), B(w)) = m(z, w). (iii) If B ∈ Ωn and t ∈ D, then the eigenvalues of B are equal if and only if lGn (E(tI), E(B)) = max m(t, λ). λ∈sp(B)

Remark. One may conjecture that Proposition 10 (ii) describes all the possibilities for the equality lGn (E(A), E(B)) = min

max m(λ, µ).

λ∈sp(A) µ∈sp(B)

Assuming Proposition 10 (iii), we are ready to prove Proposition 4. The implication (ii)⇒(iii) is trivial. For the rest of the proof we may assume that t = 0. We shall show that (i)⇒(ii) for any B ∈ Ωn . Let (Aj ) → 0 and (Bj ) → B. Then, by Proposition 10 (iii) and (2), lΩn (Aj , Bj ) ≥ lGn (E(Aj ), E(Bj )) → lGn (0, E(B)) = r(B) = lΩn (0, B). Thus the function lΩn (·, B) is lower semicontinuous at the point (0, B). Since it is (always) upper semicontinuous, we conclude that it is continuous at this point. It remains to prove that (iii)⇒(i). Since Cn is a dense subset in Ωn , we may find Cn ⊃ (Aj ) → 0. Then, by (2) and (1), r(B) = lΩn (0, B) ← lΩn (Aj , B) = lGn (E(Aj ), E(B)) → lGn (0, E(B)) and hence lGn (0, E(B)) = r(B). It follows by Proposition 10 (iii) that the eigenvalues of B are equal. This completes the proof of Proposition 4.


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Proof of Proposition 10 (i). The first inequality is trivial. To prove the second one recall that (see the footnote on page 2) lΩn (A, B) = lΩn (A , B  ),

A ∼ A, B  ∼ B.

So we may assume A = (ajk ) and B = (bjk ) are Jordan matrices with max m(ajj , bjj ) = s := min max m(λj , µπ(j) ). π

1≤j≤n

1≤j≤n

Let s1 > s. Then we may choose ϕjj ∈ O(D, D) such that ϕjj (0) = ajj and ϕjj (s1 ) = bjj . For ζ ∈ C set 0, j>k ϕjk (ζ) = b −a ajk + jk s1 jk ζ, j < k. Now ϕ = (ϕjk ) ∈ O(D, Ωn ) which shows that lΩn (A, B) < s1 . Since s1 > s was arbitrary, we are done.
Remark. Obvious modifications in the above proof imply Proposition 7. Proof of Proposition 10 (ii). It follows by Theorem 5.2 in [7].




Proof of Proposition 10 (iii). If the eigenvalues of B are equal, say to λ, then lGn (E(tI), E(B)) = m(t, λ) by Proposition 10 (ii) with αj = 0 (or, directly, considering the mapping ζ → E(ζ, . . . , ζ) shows that second inequality in Proposition 10 (i) becomes equality). To prove the converse, we shall need the following Lemma 11. Let ε1 , . . . , εn ∈ T = ∂D be pairwise different points. Then for any λ1 , . . . , λn ∈ D, there are β ∈ D and a Blaschke product B of order ≤ n with B(0) = 0, B(ε1β) = λ1 , . . . , B(εn β) = λn . Assuming Lemma 11, we shall complete the proof of Proposition 10√(iii). We may assume that t = 0. Let λ1 , . . . , λn be the eigenvalues of B. Set n 1 = {ε1 , . . . , εn }. Let β ∈ D and B be as in Lemma 11. Consider the mapping



ζ → fB (ζ) := E(B(ε1 n ζ), . . . , B(εn n ζ)) √ (where n ζ is chosen arbitrarily). It is easy to see that fB ∈ O(D, Gn ), fB (0) = 0, fB (β n ) = E(B). Hence lGn (0, E(B)) ≤ |β|n . It remains to prove that if |β|n ≥ max |λj |, then λ1 = · · · = λn . We may assume that

1≤j≤n

B(ζ) = ζ

a0 ζ k + a1 ζ k−1 + · · · + ak , a ¯k ζ k + a ¯k−1 ζ k−1 + · · · + a ¯0

where a0 = 1 and k ≤ n−1. Then for 1 ≤ j ≤ n one has that |β|n ≥ |λj | = |B(εj β)| and hence |β|n−1 |ak (εj β)k + ak−1 (εj β)k−1 + · · · + a0 | ≥ |a0 (εj β)k + a1 (εj β)k−1 + · · · + ak |.

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Squaring both sides of this inequality, we get that k
|β|2n−2 ( |as |2 |β|2s + 2 s=0

≥

k





p

ap as β s β εs−p ) j

0≤p<s≤k

|as |2 |β|2(k−s) + 2

s=0




ap as β k−p β

k−s s−p εj .

0≤p<s≤k

Summing these inequalities for j = 1, . . . , n we get that |β|2n−2

k
s=0

|as |2 |β|2s ≥

k


|as |2 |β|2(k−s) ,

s=0

that is, k


|as |2 (|β|2(n+s−1) − |β|2(k−s) |) ≥ 0.

s=0

Since k ≤ n−1, then k−s < n+s−1 if s > 0 and hence as = 0. On the other hand, a0 = 1 and thus k = n − 1. It follows that B(z) = z n and then λ1 = · · · = λn .
Proof of Lemma 11. Let S be the set of all β ∈ D for which the Nevanlinna– Pick problem with data (0, 0), (ε1 β, λ1 ), . . . , (εn β, λn ) is solvable. We may assume that 0 ∈ S; otherwise, the identity mapping does the job. Then we must have λj , 1 ≤ j ≤ n. The existence of such a f ∈ O(D, D) with f (εj β) = λj := εj β function is equivalent to the semi-positivity of the matrix A(β) = [aj,k (β)]nj,k=1 , ¯ 1 − λj λ k . Observe that aj,k (·), j = k, is bounded on D. On where aj,k (β) = 1 − εj εk |β|2 the other hand, lim aj,j (β) = +∞. Thus the matrix A(β) is (strictly) positive for β→T

β near T. Since 0 ∈ S, it follows that S is a proper non-empty (circular) closed subset of D. So there is a boundary point β0 ∈ D of S. Then A(β0 ) is not strictly positive which means that m = rank(A(β0 )) is not maximal, that is, m < n. This implies that the respective Nevanlinna–Pick problem has a unique solution and it ˜ is a Blaschke product B˜ of order m (cf. [10]). It remains to set B(ζ) = ζ B(ζ).


3. Proofs of Propositions 5, 6, 9 and Conjecture 8 for n = 2 Proof of Proposition 5. The implication (ii)⇒(iii) is trivial. For the rest of the proof we may suppose as above that t = 0.

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We shall show that (i)⇒(ii) for any B ∈ Ωn . Let (Aj ) → 0 and (Bj ) → B. 1 Then, by (4) and the equality κGn (0; e1 ) = (cf. [17]), n κΩn (Aj ; Bj ) ≥ κGn (E(Aj ); E∗,Aj (Bj )) → κGn (0; E∗,0 (B)) |tr(B)| = r(B) = κΩn (0; B). n Thus, the function κΩn is lower semicontinuous at the point (0; B). Since it is (always) upper semicontinuous, we conclude that it is continuous at this point. It remains to prove that (iii) ⇒ (i). Since Cn is a dense subset of Ωn , we may find Cn ⊃ (Aj ) → 0. Then, by (4) and (3), = κGn (0; tr(B)e1 ) =

r(B) = κΩn (0; B) ← κΩn (Aj ; B) = κGn (E(Aj ); E∗,Aj (B)) → κGn (0; E∗,0 (B)) = Hence r(B) =

|tr(B)| . n

|tr(B)| , that is, the eigenvalues of B are equal. n




Proof of Proposition 6. Since Ωn is a balanced domain, we have that the limit kΩ (0, tA) exists and equals hΩˆ n (A) (cf. Proposition 4.3.11 (a) in [15]), where lim n t
0 |t| ˆ n and the Kobayashi distance of Ωn , hΩˆ n and kΩn are the Minkowski function of Ω respectively. On the other hand, since kΩn is a continuous function, the density of Cn in Ωn and the equality (1) imply that kΩn (A, B) = kGn (E(A)), E(B)),

A, B ∈ Ωn .

It follows that hΩˆ n (A) = lim

t
0

lim

t
0

kΩn (0, tA) = |t|

kGn (0, E(tA)) kG (0, t · tr(A)e1 + o(t)) = lim n . t
0 |t| |t|

Denote by κ ˆGn (0; ·) the Kobayashi–Buseman metric of Gn at 0, that is, the largest norm bounded above by κGn (0; ·). Since Gn is a taut domain, we have that (see [18]) kG (0, t · tr(A)e1 + o(t)) = |tr(A)|ˆ κGn (0; e1 ). lim n t
0 |t| 1 Making use of the equality κ ˆ Gn (0; e1 ) = (cf. [17]), we get that n ˆ n = {A ∈ Mn : h ˆ (A) = |tr(A)| < 1}. Ω
Ωn n Remark. An algebraic approach in the proof of Proposition 6 also works.

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Proof of Conjecture 8 for n = 2. If A is derogatory, it is scalar. Then sp(B) = 0 and so the linear mapping λ → A + λB does the job. Let A be non-derogatory. Choose r > 0 and ϕ ∈ O(rD, Ωn ) such that ϕ(0) = A, ϕ (0) = B and sp(ϕ(λ)) = sp(A) for any λ ∈ rD. Then ϕ(λ) = A + λB + λ2 ψ(λ),

ψ ∈ O(rD, Mn ).

The Taylor expansion shows that the condition sp(ϕ) = sp(A), that is tr(ϕ) = tr(A) and det ϕ = det A, is equivalent to tr(B) = tr(ψ) = 0 and f (A, B) = det B + f (A, ψ) = f (B, ψ) = det ψ = 0, where f (C, D) = c11 d22 + c22 d11 − c12 d21 − c21 d12 ,

C, D ∈ M2

Observe that the quadratic mapping λ → A + λB + λ2 ψ(0) satisfies the same conditions. Therefore it has the desired properties.
Proof of Proposition 9. By (3), the equality κΩn (A; B) is equivalent to E∗,A (B) = 0. By property (8) in Proposition 3 and its proof, we have a matrix Y ∈ Mn such that −Y A + AY = B. Then the mapping λ → e−λY AeλY satisfies all the required properties.


4. Appendix: Proof of Proposition 3 In [11, definition 3.2.4.1, p. 135], property (4) is taken as defining nonderogatory matrices. The fact that (4) implies (5) is [11, Theorem 3.2.4.2, p. 135]. The converse implication is stated in [11, p. 137], and proved in [12, Corollary 4.4.18, p. 275]. The fact that (7) is equivalent to (4) is part of [12, Theorem 4.4.17, p. 275]. The equivalence between (1) and (3) is [11, Theorem 3.3.15, p. 147]. The equivalence of those properties with (4) is left as an exercise immediately after this. The form of a companion matrix shows that the first basis vector is cyclic. Conversely, if one has a cyclic vector the space is generated by its first n iterates, which form a basis in which the matrix will take the companion form. So (2) is equivalent to (1). We now move on to the statements about ranks. First note that ΦA (I + H) := (I + H)−1 A(I + H) = A + (−HA + AH) + O(H 2 ), so dim Ker((ΦA )∗,In ) = dim C(A), and, by the rank theorem, (8) is equivalent to (7). The comment about maximality follows from [12, Theorem 4.4.17(d), p. 275]. To study E∗,A , first note that for P ∈ M−1 n , E∗,A (H) = E∗,P −1 AP (P −1 HP ), so rank(E∗,A ) is preserved when we pass to a similar matrix. Thus, if A verifies (1), we may suppose then that it is a companion matrix. Choose H = (hi,j ) such

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that hi,j = 0, 1 ≤ j ≤ n − 1. Then A + H is also a companion matrix, and E when restricted to that set is a linear map in the last column. Then the mapping E∗,A (H) = (−hn,n , hn−1,n , . . . , (−1)n−1 h1,n ), is onto Cn . So (6) follows by (1). To complete the proof of Proposition 3, it is enough to show (6) implies (4). Given any λ ∈ C, let Aλ := A − λIn . Then PA (X) = PAλ (X + λ), so that E(A) is a polynomial expression (involving the parameter λ) of the components of E(Aλ ). Therefore rank(E∗,A ) ≤ rank(E∗,Aλ ). Suppose now that property (6) holds and (4) does not. Let λ be an eigenvalue such that dim Ker(A − λIn ) ≥ 2. Choose a basis of Cn containing a basis of Ker(A − λIn ). In this basis, the matrix A − λIn transforms into a matrix with at least two columns which are identically zero, and therefore En (A − λIn + H) is a polynomial containing only monomials of degree at least 2 in the hi,j . This implies that (En )∗,Aλ = 0 and therefore rank(E∗,A ) ≤ rank(E∗,Aλ ) ≤ n − 1, which is a contradiction.




Acknowledgment. We wish to thank the referee for his useful comments.

References [1] J. Agler, N. J. Young, The two-point spectral Nevanlinna–Pick problem, Integral Equations and Operator Theory 37 (2000), 375–385. [2] J. Agler, N. J. Young, The two-by-two spectral Nevanlinna–Pick problem , Trans. Amer. Math. Soc. 356 (2004), 573–585. [3] J. Agler, N. J. Young, The hyperbolic geometry of the symmetrized bidisc, J. Geom. Anal. 14 (2004), 375–403. [4] H. Bercovici, C. Foia¸s, A. Tannenbaum, A spectral commutant lifting theorem, Trans. Amer. Math. Soc. 325 (1991), 741–763. [5] C. Costara, The symmetrized bidisc and Lempert’s theorem, Bull. London Math. Soc. 36 (2004), 656–662. [6] C. Costara, The 2 × 2 spectral Nevanlinna-Pick problem J. London Math. Soc. 71 (2005), 684–702. [7] C. Costara, On the spectral Nevanlinna–Pick problem, Studia Math. 170 (2005), 23– 55. [8] A. Edigarian, A note on Costara’s paper, Ann. Polon. Math. 83 (2004), 189–191. [9] A. Edigarian, W. Zwonek, Geometry of the symmetrized polydisc, Arch. Math. (Basel) 84 (2005), 364–374. [10] J. B. Garnett, Bounded analytic functions, Academic Press, New York, 1981. [11] R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, New York, Melbourne, 1985. [12] R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, New York, Melbourne, 1991.

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[13] H.-N. Huang, S. A. M. Marcantognini, N. J. Young, The spectral Carath´ eodory-Fej´er problem, Integral Equation and Operator Theory 56 (2006), 229–256. [14] M. Jarnicki, P. Pflug, Invariant distances and metrics in complex analysis, de Gruyter Exp. Math. 9, de Gruyter, Berlin, New York, 1993. [15] M. Jarnicki, P. Pflug, Invariant distances and metrics in complex analysis–revisited, Diss. Math. 430 (2005), 1–192. [16] N. Nikolov, P. Pflug, P. J. Thomas, W. Zwonek, Estimates of the Carath´ eodory metric on the symmetrized polydisc, J. Math. Anal. Appl. 341 (2008), 140–148. [17] N. Nikolov, P. Pflug, W. Zwonek, The Lempert function of the symmetrized polydisc in higher dimensions is not a distance, Proc. Amer. Math. Soc. 135 (2007), 29212928. [18] M.-Y. Pang, On infinitesimal behavior of the Kobayashi distance, Pacific J. Math. 162 (1994), 121–141. Nikolai Nikolov Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev 8 1113 Sofia Bulgaria e-mail: [email protected] Pascal J. Thomas Laboratoire Emile Picard, UMR CNRS 5580 Universit´e Paul Sabatier 118 Route de Narbonne F-31062 Toulouse Cedex France e-mail: [email protected] W
lodzimierz Zwonek Instytut Matematyki Uniwersytet Jagiello´ nski Reymonta 4 30-059 Krak´ ow Poland e-mail: [email protected] Submitted: February 21, 2007 Revised: May 27, 2008

Integr. equ. oper. theory 61 (2008), 413–422 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/030413-10, published online April 17, 2008 DOI 10.1007/s00020-008-1585-6

Integral Equations and Operator Theory

Spectralizable Operators Vladimir A. Strauss and Carsten Trunk Abstract. We introduce the notion of spectralizable operators. A closed operator A in a Hilbert space is called spectralizable if there exists a non-constant polynomial p such that the operator p(A) is a scalar spectral operator in the sense of Dunford. We show that such operators belongs to the class of generalized spectral operators and give some examples where spectralizable operators occur naturally. Mathematics Subject Classification (2000). Primary 47B40; Secondary 47A60, 47B50. Keywords. Generalized spectral operators, spectral functions, definitizable operators.

1. Introduction Spectral operators (see [11]) are operators which possess a spectral resolution with properties comparable to the spectral function of self-adjoint operators in Hilbert spaces. In particular, this resolution is bounded, that is, a spectral operator has no spectral singularities. One of the important directions of the development of modern operator theory is related to find spectral resolutions for more general classes of operators. Operators with spectral singularities belong to the class of the so-called generalized spectral operators [9]. This class is actively investigated [2, 3, 4, 5, 13, 24, 25, 26]. It is well known that for many concrete operator classes these resolutions exist in a generalized sense only thanks to some spectral singularities. In the well-known monograph of Colojoar˘ a and Foia¸s [9] it is shown (Chapter 5, Corollary 5.7) that J-unitary and J-self-adjoint operators in Pontryagin spaces are examples of generalized spectral operators. This is based on two following facts: 1. a π-self-adjoint J-non-negative operator represents a generalized scalar spectral operator with the unique singularity in zero; Vladimir Strauss gratefully acknowledges support by DFG, Grant No. TR 903/3-1.

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2. for every π-self-adjoint operator A there exists a non-constant polynomial p that maps A to a π-non-negative operator. Let us note that this is also correct for so-called definitizable operators in Krein spaces. For the basic theory of definitizable operators we refer to [20]. Another class of operators with spectral singularities was considered in a paper by Naimark [27]. There non-self-adjoint differential operators of second order on the semi-axis are studied. These investigations were continued by Lyance in [21] and [22]. There is a large number of papers connected with these problems including different kinds of differential and difference operators from mathematical physics and other areas. We mention here only [8, 12, 19, 23, 28] The aim of our work is the investigation of the class of spectralizable operators. Definition 1.1. A bounded operator A in a Hilbert space is called a spectralizable operator if there exists a non-constant polynomial p such that the operator p(A) is a scalar spectral operator. The polynomial p is then called a spectralizing polynomial for A. Spectralizable operators arise in many different problems, see, e.g., [7, 14, 15, 16, 31, 32, 33]. We mention that the term spectralizable was used first in [30] in a very special setting, where, in particular, one can find an example of a non-self-adjoint difference operator with a self-adjoint square. For operators with an identity iteration, see, e.g., [18]. In [6] operators are studied which have the property that the closure of its square is similar to some self-adjoint operator. In this work we study spectral properties of spectralizable operators. It is easy to see (by some simple examples) that a spectralizable operator has in general a spectral function with singularities. There is some kind of similarity between these two classes of spectralizable and definitizable operators. This gives the expectation that the theory of spectralizable operators can be developed in a similar direction as the theory of definitizable operators. We proceed as follows. In Section 2 we provide the main definitions and in 3 we prove the main result of this paper, i.e. that spectralizable operators possesses an eigen spectral function. In Section 4 we give an example of a spectralizable operator with an unbounded spectral function and in Section 5 we give some more examples of spectralizable operator. Finally, in Section 6 we construct with the help of the results from Section 3 an eigen spectral function for bounded operators A in Krein spaces which have the property that p(A) is J-non-negative. For this we do not assume that A is J-self-adjoint or J-unitary.

2. Main Definitions Let A be a bounded operator in a Hilbert space H. Denote, as usual, by L(H) the set of all bounded operators in H.

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Spectral operators (see [11]) possess a spectral function with properties comparable to the spectral function of self-adjoint operators in Hilbert spaces. We introduce now the notion of a spectral function with a set of peculiar points. Definition 2.1. Let Λ = {λk }n1 be a finite set of complex numbers and let RΛ := {X ⊂ C : X is a Borel set with ∂X ∩ Λ = ∅} ,

(2.1)

where ∂X is the boundary of X in C. Let E : RΛ → L(H) be a countably additive (with respect to weak topology) function, that maps RΛ to a commutative algebra of projections with E(C) = I. The function E is called a spectral function on C with the set Λ of peculiar points. Definition 2.2. A spectral function E with the set Λ of peculiar points is called the eigen spectral function of an operator A if the following holds for all X ∈ RΛ . a) E(X)A = AE(X), σ(A|E(X)H ) ⊂ X; b) if X ∩ Λ = ∅ then the operator AE(X) is a scalar spectral operator and
ξE(dξ); AE(X) = X

c) if X ∩ Λ = ∅ then AE(X) is not a scalar spectral operator. The following is a consequence of [13]. Theorem 2.3. The eigen spectral function with the set Λ of peculiar points of an operator is unique. Later, in Section 4, we will give an example where the spectral function is unbounded in a neighbourhood of a peculiar point. For this situation we introduce the following notion. Definition 2.4. Let E be a eigen spectral function of A with the set of peculiar points Λ. If λ ∈ Λ then λ will be called a peculiarity or a peculiar point (of A). The peculiarity λ is called regular if for a fixed neighbourhood X, X ∈ RΛ , the operator family {E(X ∩ Y )}Y ∈RΛ is bounded. The peculiarity λ is called singular in the opposite case. Let us note that the notion of regular and singular peculiarity does not depend on the choice of X.

3. Main Results In this section we show that spectralizable operators possesses an eigen spectral function. It is the main result of our paper.

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Theorem 3.1. Let A be a spectralizable operator and let p be its spectralizing polynomial. Then A has the eigen spectral function E with a finite number of peculiarities, moreover the peculiar set Λ is a subset of the set of roots of p , where p is the derivative of p. Proof. If p equals a constant then A is a scalar spectral operator and Theorem 3.1 is proved. Therefore, we assume that p is not equal to a constant. We divide the proof in several steps. 1. Let ξ0 ∈ σ(A) with p (ξ0 ) = 0. We show that there exists a neighborhood U (ξ0 ) of ξ0 with the properties p (ξ) = 0 for every ξ ∈ U (ξ0 ), where U (ξ0 ) is the closure of U (ξ0 ) and there is a projection E A (U (ξ0 )) commuting with A such that A|E A (U(ξ0 )H is a scalar spectral operator with

(3.1)

σ(A|E A (U(ξ0 ))H ) ⊂ U (ξ0 ) and σ(A|(I−E A (U(ξ0 )))H ) ⊂ C\U (ξ0 ). Let V (ξ0 ) be an open ball with centre ξ0 and p |V (ξ0 ) = 0. Due to the inverse function theorem there are a neighborhood W (p(ξ0 )) of the point p(ξ0 ) and a continuous function p inverse to p, p : W (p(ξ0 )) → V (ξ0 )

with p (p(ξ0 )) = ξ0 .

As an abbreviation we will write in the following W := W (p(ξ0 )). The pre-image of the polynomial p, p−1 (W ) = {x ∈ C : p(x) ∈ W }, has at most finitely many connected components. Hence, by choosing V (ξ0 ) sufficiently small, it is no restriction to assume that the closure of p (W ) is isolated in the closure of p−1 (W ). Let E p(A) be the spectral resolution of p(A). Then (see [11], Corollary XV.3.7) A commutes with E p(A) , therefore the subspace E p(A) (W )H is invariant with respect to A. So, due to the theorem of spectral mapping we obtain   σ(A|E p(A) (W )H ) ⊂ p−1 σ(p(A)|E p(A) (W )H ) ⊂ p−1 (W ). and

  σ(A|(I−E p(A) (W ))H ) = p−1 σ(p(A)|(I−E p(A) (W ))H ) ⊂ p−1 (C\W ). Denote by Qw , Qw : E p(A) (W )H → E p(A) (W )H, the Riesz-Dunford projection of the operator A|E p(A) (W )H in E p(A) (W )H which corresponds to the spectral set σ(A|E p(A) (W )H ) ∩ p (W ). Now we define U (ξ0 ) := p (W )

and E A (U (ξ0 )) := Qw E p(A) (W ).

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By the construction of E A (U (ξ0 )) the polynomial p is a one-to-one map of the set σ(A|E A (U(ξ0 ))H ) onto the set σ(p(A)|E p(A) (W )H ). As p is analytic on W we have A|E A (U(ξ0 ))H = p (p(A)|E A (U(ξ0 ))H ). Since p(A)|E A (U(ξ0 ))H is a spectral operator of type 0 (i.e., a scalar spectral operator, see [11], Section XV.5 for details), A|E A (U(ξ0 ))H is also a scalar spectral operator (see [11, Corollary XV.5.7]) and (3.1) is proved. 2. The spectrum of A is a compact set, therefore it is sufficient to define the eigen spectral function E of A for bounded sets. We set Λ0 := {ξ ∈ C : p (ξ) = 0} and define RΛ0 in the same way as in (2.1) where we replace Λ by Λ0 . Let X ∈ RΛ0 be a bounded set which does not contain any zeros of p . Then its closure X has the same property and X is a compact set. Therefore, there exists finitely many points ξ1 , . . . , ξN for some N ∈ N and neighborhoods U (ξj ) of ξj , j = 1, . . . , N , which satisfy (3.1) with X ⊂ ∪N j=1 U (ξj ). As A|E A (U(ξj ))H , 1 ≤ j ≤ N , is a spectral operator, the projection E A (U (ξj ) ∩ U (ξk )) is defined for 1 ≤ k ≤ N and E A (U (ξj ) ∪ U (ξk )) := E A (U (ξj )) + E A (U (ξk )) − E A (U (ξj ) ∩ U (ξk )) is defined as usual. In a similar way, we define E A (∪N j=1 U (ξj )). Then the operator A A|E A (∪N is a spectral operator. Therefore E (X) is defined for all X ∈ j=1 U(ξj ))H RΛ0 with X ∩ Λ0 = ∅.  3. Let X ∈ RΛ0 and let {λl }L l=1 , L > 0, be the set of roots of p in X, where all roots are different, i.e. we don’t take into account the multiplicity of the roots. Let Z(λj ), 1 ≤ j ≤ L, be a bounded neighborhood of the point λj such that every pair of the closure of these neighborhoods contains disjoint elements. Then Z := X \ ∪L j=1 Z(λj ) ∈ RΛ0

with Z ∩ Λ0 = ∅.

By the second step of this proof, A|E A (Z)H is a spectral operator. Denote by E A (∪L j=1 Z(λj )) the Riesz-Dunford projector of the operator A|(I−E A (Z))H corresponding to the spectral set ∪L j=1 Z(λj ). Then we define E A (X) := E A (Z) + E A (∪L j=1 Z(λj )). 4. By the first three steps, E A is a spectral function with the set Λ0 of peculiar points. Moreover, a), b) and c) of Definition 2.2 are satisfied for all X ∈ RΛ0 . If for some λ ∈ Λ0 there exists a neighborhood U (λ) of λ such that there is a projection E A (U (λ)) such that A|E A (U(λ)H is a scalar spectral operator with σ(A|E A (U(λ))H ) ⊂ U (λ), then, using the construction above, E can be extended to all Borel sets containing λ in their boundary in such a way that a), b) and c) of Definition 2.2 remains valid. Hence, we choose Λ to be the set of all ξ ∈ Λ0 with the property that there exists no neighborhood U (ξ) of ξ such that there

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exists a projection E A (U (λ)) with A|E A (U(λ)H is a scalar spectral operator with
σ(A|E A (U(λ))H ) ⊂ U (λ). Theorem 3.1 is proved.

4. Singularities of the spectral function It is shown in Theorem 3.1 that a spectralizable operator has a spectral function with peculiar points. In this section we give a simple example for a singular peculiar point. Example. Let H be the Hilbert space of all square summable sequences, H = l2 (N). Define C : H → H by C(xn )n∈N = (n−2 xn )n∈N and set

 B :=

We have 2



B =

0 C

I 0

C 0

0 C

 .  ,

hence B2 is a self-adjoint operator and B is spectralizable with spectralizing polynomial p(t) = t2 . Therefore it follows from Theorem 3.1 that zero is the only possible peculiar point. We will show that the spectral function is unbonded near zero. Denote by en , n ∈ N, the sequence (δkn )k∈N . A simple computation shows   1 1 en  ± ∈ σp (B) with eigenvector . ± n1 en n 1 + n12 The spectral projection E([m−1 , 1]), m ∈ N, of B corresponding to the interval [m−1 , 1] is given by    m  1 (xn )n∈N 2 (xn en + nyn en )  E([m−1 , 1]) = . 1 −1 (yn )n∈N xn en + yn en 2 n n=1

From this it is easily seen that the norm of E([m−1 , 1]) tends to infinity as m → ∞.

5. Examples It turns out that a large class of operators is spectralizable. We illustrate this with some examples. Example. Let A be a bounded self-adjoint operator in some Hilbert space H. Let B and C be bounded operators in H which commutes with A such that the operators A2 + BC

and A2 + CB

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are scalar spectral operators. An easy calculation shows that  2  2  A B 0 A + BC = C −A 0 A2 + CB in the Hilbert space H × H, that is,   A B is a spectralizable operator. C −A Example. Let A0 ∈ L(H) be a uniformly positive self-adjoint operator in some Hilbert space H. Define

 0 I 1 A0 := . −A0 −A02 An easy calculation shows that A30

=



1

A0 A02

0

0

A0 A02

.

1

It is a self-adjoint operator in the Hilbert space H × H and, hence, the operator A0 is spectralizable. We mention that this operator is a special case of (in general unbounded) operators considered in [14], [15], [32] and [33]. Example. Let A1 ∈ L(H) be a self-adjoint operator in some Hilbert space H. Define   0 I . A1 := −A1 −I An easy calculation shows that A21 + A1 =



−A1 0

0 −A1

 ,

which is a self-adjoint operator in the Hilbert space H×H. Hence, A1 is a spectralizable operator. If, in addition, A1 is uniformly positive, then, as in the example above, the operator A1 again fulfills the asumptions of [14], [15], [32] and [33]. Example. Let B ∈ L(H) be a self-adjoint operator in some Hilbert space H. Define   0 i(B + 1) A := . i(B + 1) B An easy calculation shows that  −(B + 2)(B + 1)2 A3 + 2A2 + A = 0

0 −(B + 2)(B + 1)2

 ,

which is a self-adjoint operator in the Hilbert space H × H. Hence, A is a spectralizable operator.

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6. Operators in Krein spaces There is a well developed theory of J-self-adjoint and J-unitary operators in Krein spaces (see [1] and [20] for details). Recall that a bounded self-adjoint operator A in a Krein space (H, [., .]) is called definitizable if there exists a non-zero polynomial p such that [p(A)x, x] ≥ 0 for all x ∈ H. (6.1) Then the spectrum of A is real or its non-real part consists of a finite number of points, cf. [20]. A definitizable operator possesses a spectral function defined on the ring generated by all connected subsets of R whose endpoints do not belong to some finite set of so-called critical points (see [20]). In what follows, we show that bounded operators in Krein spaces, which are not necessarily J-self-adjoint or J-unitary, satisfying (6.1) possess a eigen spectral function with a set of peculiar points in the sense of Definition 2.2. Theorem 6.1. Let A be a bounded operator in a Krein space (H, [., .]). Assume that there is a polynomial p such that the operator p(A) is J-non-negative. Then A has the eigen spectral function E with finite number of peculiarities. Moreover the peculiar set Λ is a subset of the union of the set of roots of p and the set of roots of p , where p is the derivative of p. Proof. Let us consider the J-non-negative operator p(A) and its spectral function E p(A) (see [20]). As it is well known, for every positive  > 0 the operator p(A)|(I−E p(A) ([−,]))H is a scalar spectral operator, hence, we can apply to it Theorem 3.1. Thus, we need only to show how to construct projections corresponding to the pre-image of p of small neighborhoods of zero. Let us consider the set U := {ξ ∈ C : |p(ξ)| < } with  > 0. For every  this set is bounded, so its closure is compact. Moreover the number of roots (without multiplicity) does not exceed the degree of the polynomial p. There exists δ > 0 such that for every 0 <  < δ the set U represents an union of disjoint neighborhoods of roots of p. Each neighborhood contains only one root. Then for each neighborhood we can define a Riesz-Dunford projector similar as in step 2 of the proof of Theorem 3.1. By a reasoning similar to the steps 3 and 4 of the proof of Theorem 3.1, we construct the eigen spectral function E.
Corollary 6.2. Let A be a bounded operator and p as in Theorem 6.1. Let E be the eigen spectral function of A with the set of peculiar points Λ. If the closed set X ∈ RΛ is such that p(ξ) > 0 for every ξ ∈ X and X ∩ Λ = ∅, then the subspace E(X)H is uniformly positive, that is (E(X)H, [., .]) is a Hilbert space.. Proof. The properties of the spectral function E p(A) of the J-non-negative operator p(A) implies that (E p(A) (p(X))H, [., .]) is a Hilbert space, cf. [20]. By construction, E(X)H is a subspace of E p(A) (p(X))H.


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References [1] Azizov, T.Ya.; Iokhvidov, I.S., Linear Operators in Spaces with an Indefinite Metric, John Wiley & Sons, Ltd., Chichester, 1989. [2] Albrecht, E., On some classes of generalized spectral operators, Arch. Math. 30 (1978), 297-303. [3] Allakhverdiev, D.E.; Akhmedov, A.M., Some classes of generalized spectral operators and their applications, Math. USSR, Sb. 67 (1990), 43-63. [4] Azzouni, A., Spectral operators with critical points, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sr. 35(83) (1991), 3-17 (French). [5] Bacalu, I., S-decomposable operators in Banach spaces, Rev. Roum. Math. Pures Appl. 20 (1975), 1101-1107. [6] Binding, P.; Strauss, V., On operators with spectral square but without resolvent points, Can. J. Math. 57 (2005), 61-81. [7] Chen, G.; Russell, D., A mathematical model for linear elastic systems with structural damping, Q. Appl. Math. 39 (1982), 433-454. [8] Cheremnyh, E.V., Spectral analysis of certain non-self-adjoint difference operators, Ukr. Mat. Zhournal 35 (1983), 393-398 [9] Colojoar˘ a, I.; Foia¸s, C., Theory of generalized spectral operators, Gordon and Breach, 1968. [10] Cleaver, Ch.E., A characterization of spectral operators of finite type, Compos. Math. 26 (1973), 95-99. [11] Dunford N.; Schwartz J. T., Linear Operators. Part III. Spectral Theory, John Wiley & Sons, 1971. [12] Gesztesy, F.; Tkachenko, V., When is a non-self-adjoint Hill operator a spectral operator of scalar type?, C. R., Math., Acad. Sci. Paris 343 (2006), 239-242. [13] Harvey, B.N., Spectral operators with critical points, Am. J. Math. 96 (1974), 41-61. [14] Jacob, B.; Trunk, C., Location of the spectrum of operator matrices which are associated to second order equations, Operators and Matrices 1 (2007), 45-60. [15] Jacob, B.; Morris, K.; Trunk, C., Minimum-phase infinite-dimensional second-order systems, IEEE Trans. Automat. Control 52 (2007), 1654-1665. [16] Jacob, B.; Trunk, C.; Winklmeier, M., Analyticity and Riesz basis property of semigroups associated to damped vibrations, to appear in Journal of Evolution Equations, DOI 10.1007/s00028-007-0351-6. [17] Jonas, P., On the functional calculus and the spectral function for definitizable operators in Krein space, Beitr. Anal. 16 (1981), 121-135. [18] Karapetiants, N.K.; Samko, S.G., Equations with an Involutive Operators and Their Applications, Birkh¨ auser, Boston, 2001. [19] Kuperin, Yu.; Naboko, S.; Romanov R., Spectral analysis of the transport operator: a functional model approach, Indiana Univ. Math. J. 51 (2002), 1389-1425. [20] Langer, H., Spectral functions of definitizable operators in Krein space, Lect. Not. in Math. 948 (1982), 1-46. [21] Lyance, V.E., A differential operator with spectral singularities, I, Am. Math. Soc., Transl., II. 60 (1967), 185-225.

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[22] Lyance, V.E., A differential operator with spectral singularities, II, Am. Math. Soc., Transl., II. 60 (1967), 227-283. [23] Naboko, S.; Romanov, R., Spectral singularities, Sz¨ okefalvi-Nagy-Foias model and the spectral analysis of the Boltzmann operator, Oper. Theory: Adv. Appl. 127 (2001), 473-490. [24] Nagy, B., Residually spectral operators, Acta Math. Acad. Sci. Hungar. 35 (1980), 37-48. [25] Nagy, B., Operators with spectral singularities, J. Oper. Theory 15 (1986), 307-325. [26] Nagy, B., Spectral measures with singularities, Acta Math. Hungar. 49 (1987), 51-64. [27] Naimark, M.A., Investigation of the spectrum and the expansion in eigenvalues of a non-selfajoint operator of second order on a semi-axis, Trudy Mosk. Mat. Obsch. 3 (1954), 181-270 (Russian). [28] Pavlov, B.S., Spectral analysis of a dissipative singular Schr¨odinger operator in terms of a functional model, Encycl. Math. Sci. 65 (M. A. Shubin ed.); Springer-Verlag, Berlin, 1996, 87–153. [29] Strauss, V.A., Integro-polynomial representation of regular functions of an operator whose spectral function has critical points, Dokl. Akad. Nauk Ukrain. SSR. Ser. A 8 (1986), 26-29 (Russian). [30] Strauss, V., On simultaneously definitizable and spectralizable operators in Krein space, Math. Nachr. 245 (2002), 167-184. [31] Trunk, C., Spectral theory for operator matrices related to models in mechanics, to appear in Math. Notes. [32] Tucsnak, M.; Weiss, G., How to get a conservative well-posed system out of thin air, Part I, ESAIM Control Optim. Calc. Var. 9 (2003), 247-274. [33] Tucsnak, M.; Weiss, G., How to get a conservative well-posed system out of thin air, Part II, SIAM J. Control Optim. 42 (2003), 907-935. Vladimir A. Strauss Department of Pure and Applied Mathematics Sim´ on Bol´ıvar University Sartenejas-Baruta, Apartado 89.000 Caracas Venezuela e-mail: [email protected] Carsten Trunk Department of Mathematics Technische Universit¨ at Ilmenau Postfach 100565 D-98684 Ilmenau Germany e-mail: [email protected] Submitted: September 29, 2007

Integr. equ. oper. theory 61 (2008), 423–432 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/030423-10, published online April 17, 2008 DOI 10.1007/s00020-008-1587-4

Integral Equations and Operator Theory

Domination in the Bergman Space and Korenblum’s Constant Chunjie Wang Abstract. Let a ∈ C and m be a positive integer. We obtain a condition under which z m + a ≺ z(1 + az m ). As an application we get an upper bound on Korenblum’s constant γ : γ ≤ 0.6778994 . . .. We also give another proof of a well-known result concerning the domination problem in the Bergman space. Mathematics Subject Classification (2000). Primary 30C80; Secondary 30H05. Keywords. Domination, Bergman space, Korenblum’s constant.

1. Introduction and main results Let C denote the complex plane, D the open unit disk, and L2 (D) the Hilbert space of all complex-valued measurable functions f (z) in D with the norm
  12
f
= |f (z)|2 dA(z) , D

where dA(z) denotes the normalized Lebesgue area measure. The Bergman space A2 (D) is defined to be the subspace of L2 (D) consisting of functions analytic on D. Let H ∞ be the space of all bounded analytic functions on D, with
h
∞ = sup{|h(z)| : z ∈ D}. For G, F ∈ L2 (D), we say that G is dominated by F if
Gh
≤
F h
for all h ∈ H ∞ , and we write G ≺ F . Let a ∈ C and m be a positive integer. In this paper we consider conditions under which z m + a ≺ z(1 + az m ). Our main result is the following. This work was supported by NNSF of China No. 10601025 and the Natural Science Foundation of Hebei Province of China (No. 07M001).

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Theorem 1.1. Let a ∈ C and m be a positive integer. Suppose that one of the following conditions is satisfied:  m−2 , (a) m ≥ 3 and |a| ≤ 2m−2 (b) m = 2 and |a| ≤ 13 .

Then

 D

|(z m + a)h(z)|2 dA(z) ≤

 D

|z(1 + az m )h(z)|2 dA(z)

(1.1)

holds for any function h analytic on D. The paper is organized as follows. Section 2 is devoted to the proof of Theorem 1.1. In section 3, as an application of Theorem 1.1, we get an upper bound on Korenblum’s constant γ : γ ≤ 0.6778994 . . .. In section 4 we give another proof of a well-known result concerning the domination problem in the Bergman space.

2. The proof of Theorem 1.1 To prove Theorem 1.1, we need the following lemma (see [2]). Lemma 2.1. If f (z) =

∞ 

ak z k ∈ A2 (D), then

k=0



∞  |ak |2 k+1


f
=

 12 .

k=0

Proof of Theorem 1.1. Let h(z) = Define

∞

k=0 ck z

k

be the power series expansion of h.

f (z) = (z m + a)h(z), g(z) = z(1 + az m )h(z), where a ∈ C and m ∈ N. Then f (z) = a

g(z) =

m−1 

k=0 m−1 

ck z k +

∞ 

(ck + ack+m )z k+m ,

k=0 ∞ 

ck z k+1 +

k=0

(ack + ck+m )z k+m+1 .

k=0

If
g
= ∞, then there is nothing to prove. Thus we can assume that
g
< ∞.

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It follows from Lemma 2.1 that

= =


f
2 −
g
2



m−1 ∞  |a|2 |ck + ack+m |2 1 |ack + ck+m |2 − − |ck |2 + k+1 k+2 k+m+1 k+m+2 k=0 k=0
 ∞  (m + 1)|a|2 m−1 − |ck |2 (k + 1)(k + m + 2) (k + 2)(k + m + 1) k=0 ∞ 

+

k=0

2Re(ack ck+m ) . (k + m + 1)(k + m + 2)

(2.1)

Notice that for any nonnegative integer k and any positive number b, 2Re(ack ck+m ) ≤ |a|b

k + 2m |a| k + m |ck |2 + |ck+m |2 . k+m b k + 2m

Then we have ∞  k=0 ∞ 

2Re(ack ck+m ) (k + m + 1)(k + m + 2)

∞  |ck |2 |ck |2 |a| k k + 2m + k + m (k + m + 1)(k + m + 2) b k + m (k + 1)(k + 2) k=0 k=m
 ∞  |a|k k + 2m ≤ + |ck |2 |a|b . (k + m)(k + m + 1)(k + m + 2) b(k + m)(k + 1)(k + 2)

≤

|a|b

k=0

Thus we obtain
|ck |2 (k + 1)(k + m + 2)
f
−
g
≤ (m + 1)|a|2 − (m − 1) (k + 1)(k + m + 2) (k + 2)(k + m + 1) k=0  (k + 1)(k + 2m) |a|k(k + m + 2) + |a|b + . (k + m)(k + m + 1) b(k + m)(k + 2) 2

2

∞ 

Denote the expression in the bracket above by dk . We proceed to show that dk ≤ 0, which will complete the proof of Theorem 1.1.   m−2 2 (a) Note that m ≥ 3 and |a| ≤ 2m−2 . Choose b = (m−1)(m−2) . Then |a|b ≤

1 , m−1

|a| m−2 ≤ . b 2

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Hence
 m m−2 − (m − 1) 1 − 2m − 2 (k + 2)(k + m + 1)

 1 m(m − 1) m−2 2m + 1− + 1− m−1 (k + m)(k + m + 1) 2 (k + m)(k + 2) m m(m − 2) m(m − 1) − − = (k + 2)(k + m + 1) (k + m)(k + m + 1) (k + m)(k + 2) = 0.

dk ≤ (m + 1)

(b) Note that m = 2 and |a| ≤ 13 . Choose b = 1. Then

 2 1 1 2 − 1− + 1− 3 (k + 2)(k + 3) 3 (k + 2)(k + 3) 
1 4 + 1− 3 (k + 2)2 4 4 − = 3(k + 2)(k + 3) 3(k + 2)2 < 0.

dk ≤

This completes the proof of Theorem 1.1.




3. An upper bound on Korenblum’s constant An important result in the theory of Bergman spaces is the so-called Korenblum’s maximum principle, which states that, there is a constant c ∈ (0, 1) such that whenever |f (z)| ≤ |g(z)| (f, g ∈ A2 (D)) in the annulus c < |z| < 1, then
f
≤
g
. The sharp(i.e. the largest) value of c above is called Korenblum’s constant, and denoted by γ. First conjectured by Korenblum [4], the maximum principle was proved by Hayman [1] in 1999, and a lower bound on γ was given: γ ≥ 0.04. Later Hinkkanen [3] improved Hayman’s result and proved that γ ≥ 0.15724 . . . . Recently Schuster [7] using M¨obius pseudodistance has shown that γ ≥ 0.21. And Wang [10] has proved that γ ≥ 0.25018. On the other hand, an upper bound on γ can be found from Martin’s example (see [4] or [8]): γ < 0.70450 . . .. Wang [8] gave an upper bound on γ: γ < 0.69472. The best upper bound on Korenblum’s constant γ until now is γ ≤ 0.67795 (see [9]). As an application of Theorem 1.1 we get a slightly improved upper bound on Korenblum’s constant γ : γ ≤ 0.6778994 . . .. Now we prove the following results.

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Theorem 3.1. Suppose that m ≥ 4 is an integer, a ≥ 0 and b = f (z) = g(z) = Then
f
>
g
if and only if a >





2 (m−1)(m−2) .

Let

a + zm , (1 − bz m )2 z(1 + az m ) . (1 − bz m )2

m−2 2m−2 .

Proof. Note that ∞

 1 = (k + 1)z k , (1 − z)2

z ∈ D.

(3.1)

k=0

We have f (z) = a +

∞ 

(kbk−1 + a(k + 1)bk )z mk ,

k=1

g(z) = z +

∞ 

((k + 1)bk + akbk−1 )z mk+1 .

k=1

It follows from Lemma 2.1 and (3.1) that, when b =  m−2 2m−2 ,



2 (m−1)(m−2)

and a =


f
2 −
g
2

 ∞
((k + 1)bk + akbk−1 )2 1  (kbk−1 + a(k + 1)bk )2 − =a − + 2 mk + 1 mk + 2 k=1
 ∞ 1  2k−2 mk + 1 mk + 2 = a2 − + b − 2 (m − 1)2 2(m − 1)(m − 2) k=1  ∞ ∞   1 1 2 2k−2 2k−2 m(m − 3) =a − + kb −2 b 2 2(m − 1)2 (m − 2) k=1 k=1 
1 m(m − 3) 1 2 2 − =a − + 2 2(m − 1)2 (m − 2) (1 − b2 )2 1 − b2 = 0. 2

of a on From (2.1) one can see that
f
2 −
g
2 is an increasing function  m−2 m−2 [0, ∞). Hence when a > 2m−2 we have
f
>
g
, and when 0 ≤ a < 2m−2 we have
f
<
g
. This completes the proof of Theorem 3.1.


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Theorem 3.2. Suppose that m ≥ 4 is an integer. Let f (z) = g(z) =  where a =

m−2 2m−2

 and b =

a + zm , (1 − bz m )2 z(1 + az m ) , (1 − bz m )2

2 (m−1)(m−2)

. Then
f
=
g
and |f (z)| ≤ |g(z)| in

c < |z| < 1, where c is the real root in (0, 1) of the equation a + z m = z(1 + az m ).

(3.2)

In particular, when m = 10 we have c = 0.6778994 . . .. Proof. It follows from Theorem 3.1 that
f
=
g
. On the other hand, it is easy to see that

m

f (z)

= a+r

ϕ(r) = max . r(1 + arm ) |z|=r g(z) (z) In particular, ϕ(c) = ϕ(1) = 1. Since fg(z) is analytic in c ≤ |z| ≤ 1, the classical maximum principle implies that |f (z)| ≤ |g(z)| in c < |z| < 1. Note that equation (3.2) has a unique real root c in (0, 1). Using Mathematica we obtain that, when m = 10, c = 0.6778994 . . .. This completes the proof of Theorem 3.2.


Note that Theorem 3.2 gives an upper bound on γ : γ ≤ 0.6778994 . . .. Moreover, using Mathematica to solve equation (3.2) for 3 ≤ m ≤ 14, we obtain the following results. m

a

c

m

a  7 1 1 3 9 2 16  1 2 0.7848811 . . . 10 4 3 3 3 9 0.7215112 . . . 11 5 8  20 2 5 0.6961732 . . . 12 6 5  11 5 11 0.6849591 . . . 13 7 12   24 3 6 0.6800255 . . . 14 8 7 13 As for m ≥ 15, it follows from equation (3.2) that   1 − cm−1 m−2 13 c≥c ≥ =a= m+1 1−c 2m − 2 28

c 0.6781693 . . . 0.6778994 . . . 0.6784598 . . . 0.6794453 . . . 0.6806307 . . . 0.6818882 . . .

= 0.6813851 . . . .

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Thus for all functions f (z) and g(z) (which depend on m) defined in Theorem 3.2, c = 0.6778994 . . . is the smallest root. Is it the sharp value of Korenblum’s constant γ?

4. Another proof of a well-known result The following proposition is a direct consequence of the definition of domination (see [6]). Proposition 4.1. If G, F ∈ L2 (D) and G ≺ F , then G ◦ φ ≺ F ◦ φ for all M¨ obius transformations φ on D. Denote Ba (z) = (z + a)/(1 + az) for a ∈ D. In 1990 Korenblum [5] proved the following theorem(see also [6]). Theorem 4.2. Let a ∈ D and α > 0. Then |Ba |α((1+|a|)/(1−|a|)) ≺ |z|α . Using only a power series expansion technique as in the proof of Theorem 1.1 we can prove the following well-known result(see [6] Corollary 1, p.154), which is a consequence of Theorem 4.2 and Proposition 4.1. Theorem 4.3. Let a ∈ D and define α = (1 − |a|)/(1 + |a|), β = (1 + |a|)/(1 − |a|). Then |z|β ≺ |Ba | ≺ |z|α . ∞ Proof. For any function h analytic on D, let h(z) = k=0 ck z k be the power series expansion of h. Define f (z) = (z + a)h(z), g(z) = |z|α (1 + az)h(z), where α ≥ 0 and a ∈ D. Then f (z) = ac0 +

∞ 

(ck−1 + ack )z k ,

k=1

(1 + az)h(z) = c0 +

∞ 

(ck + ack−1 )z k .

k=1

It follows from Lemma 2.1 that
f
2 = |a|2 |c0 |2 +

∞  |ck−1 + ack |2 k=1

Note that 1 2π

 0

2π

|(1 + az)h(z)|2 = |c0 |2 +

k+1 ∞  k=1

.

|ck + ack−1 |2 r2k ,

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where z = reiθ . We have
g
2 =

∞

 |ck + ack−1 |2 |c0 |2 + . α+1 k+α+1 k=1

Thus we obtain 2

2


f
−
g
=

∞ 

2

|ck |

k=0

+α

∞  k=0




(1 + α)|a|2 1−α − (k + 1)(k + α + 2) (k + 2)(k + α + 1)



2Re(ack ck+1 ) . (k + 2)(k + α + 2)

Noticing that |2Re(ack ck+1 )| ≤ |a||ck |2 we have

k+2 k+1 + |a||ck+1 |2 , k+1 k+2

∞

 2Re(ac c

k k+1 )

(k + 2)(k + α + 2) k=0  ∞
 1 2k +2 2k + 1 ≤ |a| + |ck+1 | |ck | k+1 k + 2 (k + 2)(k + α + 2) k=0
 ∞  1 k = |a| + |ck |2 . (k + 1)(k + α + 2) (k + 1)2 (k + α + 1) k=0

If α ≤

1−|a| 1+|a| ,

then |a| ≤

1−α 1+α .

Thus
∞  (1 + α)|a|2 1−α − |ck |2
f
2 −
g
2 ≤ (k + 1)(k + α + 2) (k + 2)(k + α + 1) k=0  α|a| α|a|k + + (k + 1)(k + α + 2) (k + 1)2 (k + α + 1)
∞  1−α 1+α 21−α − |ck | ≤ 1 + α (k + 1)(k + α + 2) (k + 2)(k + α + 1) k=0  α αk + + . (k + 1)(k + α + 2) (k + 1)2 (k + α + 1)

Denote the expression in the bracket of the last inequality above by ek . A simple calculation shows that
 k+α+1 k+1 αk 1 − (1 + α) + ek = (k + 1)(k + α + 1) k + α + 2 k+2 k+1
 1 1 α − = (k + 1)(k + α + 1)(k + 2) k + α + 2 k + 1 ≤ 0.

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Domination in the Bergman Space

Similarly, if α = ∞ 

1+|a| 1−|a| ,




then |a| =

α−1 α+1 .

431

Thus

(1 + α)|a|2 1−α − (k + 1)(k + α + 2) (k + 2)(k + α + 1) k=0  α|a| α|a|k − − (k + 1)(k + α + 2) (k + 1)2 (k + α + 1)
∞  |ck |2 (k + 1)(k + α + 2) = (1 + α)|a|2 + (α − 1) (k + 1)(k + α + 2) (k + 2)(k + α + 1) k=0  α|a|k(k + α + 2) −α|a| − . (k + 1)(k + α + 1)


f
2 −
g
2 ≥

|ck |2

Denote the expression in the bracket of the last identity above by fk . A simple calculation shows that


α−1 α fk = α − 1 + (α + 1) 1 − α+1 (k + 2)(k + α + 1)

 α+1 −α − α 1 − (k + 1)(k + α + 1)
 1 1 α(α − 1) − = k+α+1 k+1 k+2 ≥ 0. This completes the proof of Theorem 4.3.




References [1] W. K. Hayman, On a conjecture of Korenblum, Analysis (Munich) 19(1999), 195-205. [2] H. Hedenmalm, Recent progress in the function theory of the Bergman space, pp. 35-50 in Holomorphic spaces, edited by S. Axler, J. E. McCarthy and D. Sarason, Mathematical Sciences Research Institute Publications 33, Cambridge University Press, 1998. [3] A. Hinkkanen, On a maximum principle in Bergman space, J. Anal. Math. 79(1999), 335-344. [4] B. Korenblum, A maximum principle for the Bergman space, Publ. Mat. 35(1991), 479-486. [5] B. Korenblum, Transformation of zero sets by contractive operators in the Bergman space, Bull. Sci. Math. 114(1990), 385-394. [6] B. Korenblum and K. Richards, Majorization and domination in the Bergman space, Proc. Amer. Math. Soc. 117 (1993), 153-158. [7] A. Schuster, The maximum principle for the Bergman space and the M¨ obius pseudodistance for the annulus, Proc. Amer. Math. Soc. 134(2006), 3525-3530. [8] C. Wang, Refining the constant in a maximum principle for the Bergman space, Proc. Amer. Math. Soc. 132(2004), 853-855.

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[9] C. Wang, On Korenblum’s constant, J. Math. Anal. Appl. 296(2004), 262-264. [10] C. Wang, On Korenblum’s maximum principle, Proc. Amer. Math. Soc. 134(2006), 2061-2066. Chunjie Wang Department of Mathematics Hebei University of Technology Tianjin 300130 P. R. China Mathematics Research Center of Hebei Province Shijiazhuang 050016 P. R. China e-mail: [email protected] Submitted: October 17, 2007 Revised: February 20, 2008

Integr. equ. oper. theory 61 (2008), 433–448 0378-620X/030433-16, DOI 10.1007/s00020-008-1594-5 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Integral Equations and Operator Theory

The Riemann Hypothesis, the Generalized Riemann Hypothesis, and the Ces´aro Operator Robert Whitley Abstract. A result due to Nyman establishes the equivalence of the Riemann hypothesis with the density of a set of functions in L2 [0, 1]. Here a large class of analytic functions is considered, which includes the Riemann zeta function and the Dirichlet L-functions as well as functions not given by a Dirichlet series. For each such function φ(s) there is an associated integral operator T on L2 [0, 1] such that φ(s) has no zeros in Re(s) > 1/2 iff the operator T has dense range iff a specified set of functions is dense in L2 [0, 1]. Mathematics Subject Classification (2000). Primary 11M26, 45P05, 47G10. Keywords. Riemann hypothesis, generalized Riemann hypothesis, BeurlingNyman density theorem.

Nyman [22] (Beurling [8], [12], [6]) established a remarkable equivalence of the Riemann hypothesis with an approximation problem in L2 [0, 1]. Let ρ(x) = x − [x],

(1)

where [x] denotes the greatest integer less than or equal to x, and define B to be the set of functions f(x) which are finite sums of the form

ak θk = 0. (2) f (x) = ak ρ(θk /x) where 0 < θk ≤ 1 and Neyman’s theorem states that the Riemann zeta function has no zeros in the half plane Re(s) > 1/2 if and only if B is dense in L2 [0, 1]. Beurling’s result in [8] is more general, that B is dense in Lp [0, 1] iff the zeta function has no zeros with real part greater than 1/p; the result for p = 2 is given in [22], and these results are often referred to under the joint name Beurling-Nyman. Here only the case p = 2 will be considered. Research related to this result is discussed in the useful comprehensive summary [3]. That the density of B implies the Riemann hypothesis is easy; the converse is in Beurling’s own words [8] “less trivia”, as quoted in [5] where the point is

434

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made that the “stark contrast” in difficulty of the proofs of the sufficiency and necessity of the density condition calls out for a more informative proof. The proof given here answers that call, makes the equivalence transparent, and introduces a method of some generality. In what follows, L2 [0, 1] will denote the Hilbert space of complex-valued square integrable functions on [0,1] with the standard inner product < f, g >= 1 1 f (x)g(x) dx and derived norm f  =< f, f > 2 [28]. It will be convenient to 0 extend the domain of each f in L2 [0, 1] by defining f(t)=0 for t > 1. Each density theorem will be formulated as a statement concerning the density of the range of a specific integral operator on L2 [0, 1], which for kernels which are a.e. continuous is equivalent to the density of a set like B in Beurling’s theorem. For example, for the Dirichlet L-function L(χ, s), with non-principal character χ, the generalized Riemann hypothesis, [19], or extended Riemann hypothesis [21], that L(χ, s) has no zeros in Re(s) > 1/2, will be shown to be equivalent to the question of whether a specific integral operator on L2 [0, 1] has a dense range. A detailed analysis of the operator Tρ below is given in [1]. An interesting extension of the Beurling-Nyman density theorem has also been given in [25], [26] for the Dirichlet L-functions and Dirichlet series which include the Selberg class [19, Chap. 8]. The setting for the Beuring invariant subspace theorems used here is H 2 (D), while in [4] and [26] it is H 2 of a half-plan, which amounts to a change of variable. The results in this paper apply to a class of analytic functions which is much larger than those given by a Dirichlet series.

1. Operators and Density Theorems Consider the Dirichlet series φ(s) =

∞
an 1

ns

.

(3)

A standard method for obtaining an analytic continuation for the series is the application of partial summation to represent the sum as an integral [2, Theorem 4.2], [19, 2.1.5]. If the sum function
an (4) S(t) = n≤t

is bounded, the result is:



∞

S(t) dt, (5) ts+1 1 and the integral furnishes an analytic continuation to Re(s) > 0. An important example is the Dirichlet L-function, defined [2, Chapter 6] for χ a non-principal character modulo m by φ(s) = s

L(χ, s) =

∞
χ(n) 1

ns

,

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The Riemann Hypothesis and the Ces´ aro Operator

435

for which holds with φ(s) = L(χ, s), the sum function S(t) being bounded (5) m because 1 χ(n) = 0 [2, Theorem 6.10]. The series for the Riemann-zeta function ∞
1 . ζ(s) = s n 1

has an unbounded sum function S(t) = [t]. One approach [19, ex. 2.1.6] is to write S(t) = t − (t − [t]) = t − ρ(t) and obtain  ∞ s ρ(t) ζ(s) = −s dt, (6) s+1 s−1 t 1 which gives a meromorphic continuation of ζ(s) from Re(s) > 1 to Re(s) > 0. The integrals above are each of the same general form (5), which suggests considering the integral  ∞ κ(t) φ(s) = s dt, for κ in L∞ [1, ∞) , (7) s+1 t 1 and there is no harm in setting κ(t) = 0 for 0 ≤ t < 1 . The class of analytic functions which correspond to functions κ in L∞ [1, ∞) is much larger than the subclass associated with a Dirichlet series, for in that case on [1, ∞) κ is periodic and constant between consecutive integers. (Note that under the appropriate change of variable, φ(s)/s is the Laplace transform of κ(ey ), and [31, page 315] gives necessary and sufficient conditions for φ(s)/s to be the Laplace transform of a bounded function.) Let x in (0,1] be fixed and make the change of variable t=x/y in the integral obtaining:  1  x xs s−1 φ(s). (8) κ(x/y)y dy = κ(x/y)y s−1 dy = s 0 0 For the L-function,  1 xs L(χ, s) (9) S(x/t)ts−1 dt = s 0  with S(t) = n≤t χ(n), while for the zeta function,  x  1  1 s−1 s−1 s ρ(x/t)t dt = s ρ(x/t)t dt − s ρ(x/t)ts−1 dt. 0

0

x

Using ρ(u) = u for 0 ≤ u < 1 in the second integral  1 xs x − ζ(s), ρ(x/t)ts−1 dt = s−1 s 0 which holds in {s : Re(s) > 0} − {1}. Define the following integral operators on L2 [0, 1]:  1 Tρ f (x) = ρ(t/x)f (t) dt, 0

(10)

(11)

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Whitley

 Tκ f (x) =

1

κ(t/x)f (t) dt,

(12)

0

for κ(t) in L∞ (0, ∞) and vanishing on (0,1). A special case is  1 Tχ f (x) = S(t/x)f (t) dt, with S(t) =

IEOT

(13)

0

 n≤t

χ(n) for χ a non-principal character modulo m.

Theorem 1. The operators Tρ and Tκ are bounded, indeed compact, linear operators on L2 [0, 1]. The adjoint operators are given by  1 Tρ∗ g(x) = ρ(x/t)g(t) dt, (14) 0

Tκ∗ g(x) =



1

κ(x/t)g(t) dt.

(15)

0

For Re(s) > 1/2, the function ts−1 belongs to L2 [0, 1] and Tρ∗ ts−1 (x) =

xs x − ζ(s), s−1 s

(16)

xs φ(s), s

(17)

xs L(χ, s). s

(18)

Tκ∗ ts−1 (x) = Tχ∗ ts−1 (x) =

Proof. Since the kernels are bounded, they are square integrable, and therefore the operators are compact linear operators on L2 [0, 1] [28, Lemma 11.10]. The equations for the adjoint operators are found by interchanging the order of integration in the inner product < T f, g >. The equations for T ∗ ts−1 hold because of how the kernels have been chosen.
Corollary 1. If Tκ has a dense range, then φ(s) is not zero for Re(s) > 1/2. Define Mx to be the subspace of L2 [0, 1] of those functions perpendicular to x:  1 Mx = {f : xf (x) dx = 0}. (19) 0

If Tρ Mx is dense in L2 [0, 1], then the zeta function ζ(s) is not zero for Re(s) > 1/2. x Proof. If ζ(s) = 0, Re(s) > 1/2, then for g(t) = ts−1 , Tρ∗ g(x) = s−1 . Since Tρ∗ g is a scalar multiple of the function x, for any f belonging to the subspace Mx , < g, Tρ f >=< Tρ∗ g, f >= 0, and Tρ Mx is not dense. The argument for Tκ is similar.


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As Beurling notes in [8], Corollary 1 shows that to establish the Riemann hypotheses it sufficient to show that the function e0 (t) = 1 (extended to be zero for t > 1) belongs to the closure of Tρ Mx and, given the Beurling-Nyman theorem it is also necessary; similarly L(χ, s) has no zeros in Re(s) > 1/2 iff e0 belongs to the closure of the range of Tχ , using the density theorem established below. Corollary 1 corresponds to the trivial direction of Beurling’s theorem, the converse is the ”less trivial” result. An interesting Dirichlet series with a bounded  (−1)n−1 sum function is the alternating zeta-function ζa (s) = ∞ [30, page 16]. n=1 ns 1 Since ζa (s) = (1 − 2s−1 )ζ(s), the functions ζa (s) and ζ(s) have the same number of zeros, presumably none, in 1/2 < Re(s) < 1. But ζa (s) has infinitely many zeros on the line Re(s) = 1, and thus the closure of the range of the corresponding operator has infinite co-dimension. Although the question of whether or not the ranges of the operators Tρ and Tχ are dense is deep, implying the Riemann hypothesis and the generalized Riemann hypothesis, the dual question is easy to answer. The operators Tρ∗ and Tχ∗ have dense range; consequently the operators Tρ and Tχ are one-to-one. The density of the range of Tρ∗ follows from equation (16), since that equation shows that (Tρ∗ ntn )(x) converges to the function x in L2 [0, 1] as n tends to infinity, and then a further application of (16) shows that the dense span of the powers x, x2 , x3 , . . . is in the closure of the range of Tρ∗ , using the fact that ζ(n) = 0 for n = 2, 3, . . . . The same argument applies to Tχ , using L(χ, n) = 0 for n = 2, 3, . . . [20, Theorem 10.3]. That a non-zero operator of type Tκ is one-to-one follows similarly with an application of the M¨ untz-Szasz theorem [27] showing that the powers ts with φ(s) not zero form a set with dense span. These results will also follow easily from later considerations. Theorem 2. Let f be any function in L2 [0, 1]. (a) Suppose that κ in L∞ [1, ∞) is continuous a.e. Then Tκ f is continuous on (0, 1]. If in addition, κ is periodic with period p on [1, ∞), then Tκ f (x) is 1 continuous on [0, 1] if defined at 0 by Tκ f (0) = c0 0 f (t) dt where c0 =  1 p+1 κ(t) dt. p 1 (b) Tρ f (x) is continuous on (0, 1] and is continuous at 0 if defined there by 1 Tρ f (0) = 12 0 f (t) dt. (c) The adjoint operators map L2 [0, 1] into functions continuous on [0, 1]. Proof. Let x be any point in (0,1] and {xn } a sequence converging to x. Then κ(t/xn ) converges to κ(t/x) for almost all t and the continuity of Tκ f at x follows from the Lebesgue dominated convergence theorem. For the point x=0, if κ is also periodic on [1, ∞) as is the sum function for L(χ, t), continuity at zero follows from a lemma due to Fej´er ([29, ex. 24, page 312]), which states that for a function π(t), bounded and measurable on R and periodic with period p, and any function

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f belonging to L1 (R),    p   1 lim f (t)π(τ t) dt = f (t) dt π(t) dt . τ →∞ R p 0 R To apply this lemma to Tκ , κ having period p on [1, ∞), extend κ to κ ˆ on R with period p on R and write  1  1 lim κ(t/x)f (t) dt = lim κ ˆ (τ t)f (t) dt x→0





1/τ

= lim

τ →∞

τ →∞

x

κ ˆ (τ t)f (t) dt + lim

τ →∞

0

0



1

κ(τ t)f (t) dt = lim

τ →∞

1/τ

1

κ(τ t)f (t) dt. 0

The proof for Tρ is similar. The continuity for the adjoint operators follow from dominated convergence for 0 < x ≤ 1 and the assumption that κ is continuous a.e. and continuity at 0 follows from the continuity of ρ and κ at 0.
The next Theorem establishes the relation between the density of the subspace B of Beurling’s result and density of the operator range, and the generalization to Tκ operators. Theorem 3. (a) If κ is continuous a.e. the range of Tκ has closure R(Tκ ) = sp(κ(θ/x) : 0 < θ ≤ 1). (b) Beurling’s subspace B of functions of the form (2) and Tρ Mx have the same closure in L2 [0, 1]. (c) The range of Tρ , R(Tρ ), has closure R(Tρ ) = sp(ρ(θ/x) : 0 < θ ≤ 1). Proof. Suppose that g is perpendicular to Tρ Mx . Since < Tρ∗ g, Mx >= 0, Tρ∗ g(t) ⊥⊥ = sp(x), and Tρ∗ g(t) = αt, for some scalar α. For a belongs to Mx ⊥ = {x} function h belonging to B,
h(x) = an ρ(θn /x), (20)  1


an g(x)ρ(θn /x) dx = an Tρ∗ g(θn ) = α an θn = 0. (21) < g, h >= 0

Hence B ⊂ Tρ Mx . To be precise, this argument involving the point values Tρ∗ g(θn ) implicitly uses the fact that Tρ∗ g is continuous on (0,1]. Then since the integral 1 g(x)ρ(θ/x) dx and the function αθ are equal in L2 [0, 1], that is equal a.e., and 0 are continuous, they are equal for all θ in (0,1], as the proof requires. Conversely, suppose that g in L2 [0, 1] is perpendicular to B, and consider the function in B given by h(x) = ρ(a/x) − aρ(1/x). (22) Then   1

0 =< g, h >= 0

g(x)ρ(a/x) dx − a

1

g(x)ρ(1/x) dx, 0

(23)

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that is Tρ∗ g(a) = a(Tρ∗ g)(1) = αa, where α = Tρ∗ g(1), and g is also perpendicular to Tρ Mx . Thus Tρ Mx ⊂ B. The arguments for (a) and (c) are similar. For example, a function g in L2 [0, 1] 1 is perpendicular to the range of Tρ iff Tρ∗ g = 0 iff 0 ρ(x/t)g(t) dt = 0 iff g(t) is perpendicular to each function ρ(θ/t) for all 0 < θ ≤ 1 (here continuity is used) iff g is perpendicular to sp(ρ(θ/x) : 0 < θ ≤ 1).
One striking consequence of the Beuling-Nyman theorem is that the Riemann hypothesis is equivalent to a real variable density problem; note the titles of the papers [5] and [7]. This is so because Tρ Mx is dense iff dense for the operator acting on the inner product space L2R [0, 1] of all real-valued functions in L2 [0, 1]. This is also true for Tκ if κ is real-valued, and so is true for the operator Tχ in the case that χ is a real-valued character modulo m.

2. The Ces´aro Operator on L2 [0, 1] The Ces´aro operator Hf (x) =

1 x



x

f (t) dt

(24)

0

was shown by Hardy to be a bounded linear operator on L2 [0, 1] [15, 9.9.1, pg. 244], also see [12, pg. 279] and [27, pg. 72, ex. 14], and the interesting historical article [17]. That the operator U = I − H ∗ is unitarily equivalent to the unilateral shift is known [9], [10], as referenced in [24, pg. 21]. For completeness and to explicitly obtain useful details, a proof follows. Theorem 4. Set U = I − H ∗ and therefore U ∗ = I − H. Then: (a) U ∗ U = I. (b) U U ∗ = I − P1 , where P1 is the orthogonal projection of L2 [0, 1] onto the subspace spanned by the constant function e0 = 1. (c) For n = 0, 1, 2, . . . , define en = U n e0 . This sequence is an orthonormal basis for L2 [0, 1] with respect to which U is (unitarily equivalent to) the unilateral shift. Proof. Compute HH ∗ f by interchanging the order of integration:    1 x ∗ 1 x 1 f (t) dt dw HH ∗ f (x) = H f (w)dw = x 0 x 0 w t  x  x  t  1 1 1 dw dt + dw dt f (t) f (t) = xt xt x 0 0 0  1  x 1 f (t) dt + f (t) dt = H ∗ f (x) + Hf (x). = t x x 0

(25) (26) (27)

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In the other order  1  1  w Hf (w) 1 ∗ H Hf (x) = dw = f (t) dt (28) 2 w x x w 0  1  1  1  x 1 1 f (t) dw dt + f (t) dw dt (29) = 2 2 0 x w x t w  1  x  1 1 1 = f (t)( − 1) dt + f (t)( − 1) dt = Hf (x) + H ∗ f (x) − f (t) dt. x t 0 x 0 (30) Items (a) and (b) of the theorem follow from setting U = I − H ∗ and using (27) and (30). Consider the sequence of functions {en } defined in the theorem. From < en+1 , en+1 >=< U en , U en >=< U ∗ U en , en >=< en , en > it follows that en  = 1 for all n = 0, 1, . . . . Orthogonality follows from (a) and (b). It remains to show that {en } form a basis for L2 [0, 1]. A formula for the {en } can be established: n  
n logk (x) en (x) = . (31) k k! k=0

This is true for n = 0. Assume (31) holds for n. Then n   n  
n logk (x)
n logk+1 (x) en+1 = U en = en (x) − H ∗ (en )(x) = + (32) k k (k + 1)! k! k=0 k=0   n   n logk (x) logn+1 (x)
n + =1+ + (33) k k−1 (n + 1)! k! k=1  n  logn+1 (x)
n + 1 logk (x) + (34) =1+ k (n + 1)! k! k=1

which agrees with (31) with n+1 in place of n. Make the change of variable u = −log(x) in the integral which defines the inner product for L2 [0, 1]:  1  ∞ < f, g >= f (x)g(x) dx = f (e−u )g(e−u )e−u du. (35) 0

0

Equation (35) establishes a Hilbert space equivalence between L2 [0, 1] and the Hilbert space L2 ((0, ∞), e−u du) of those functions on (0, ∞) whose modulus is square integrable with respect to the measure e−u du. With this change of variable, en (x) becomes n  
n (−u)k −u . (36) en (e ) = k k! k=0

Equation (36) defines the n-th Laguerre polynomial Ln (u) [31, IV.17]. The set {Ln (u)} is a basis of orthogonal polynomials for the space L2 ((0, ∞), e−u du);

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therefore the set {en (x)} is a basis for L2 [0, 1]. Note that in [10] the completeness of the Laguerre polynomials is obtained as a consequence of properties of the shift. The action of U on the basis shows that it is equivalent to the unilateral shift.
The comparison of 1 U ∗ ts−1 (x) = (1 − )xs−1 s with the equations for the adjoint operators in Theorem 1 indicates a useful modification. Theorem 5. The following operators are bounded linear operators on L2 [0, 1]: ∗ Tρ/t g(x) =

1 ∗ T g(x) x ρ

(37)

∗ Tκ/t g(x) =

1 ∗ T g(x), x κ

(38)

and

and the corresponding  Tρ/t f (x) = and

ρ(t/x)

f (t) dt t

(39)

κ(t/x)

f (t) dt. t

(40)

0

 Tκ/t f (x) =

1

1

0

Proof. Since κ(t) is zero on (0,1)  1 x ∗ |Tκ/t g(x)| = | κ(x/t)g(t) dt| ≤ κ∞ H|g|(x). x 0 ∗ Since Hardy’s operator H is bounded, Tκ/t is bounded and therefore so is Tκ/t . In ∗ particular, Tχ/t and Tχ/t are bounded. Since ρ(t) = t for 0 ≤ t < 1, ∗ g(x)| |Tρ/t

and

∗ Tρ/t

1 ≤| x





x 0

ρ(x/t)g(t) dt| + |

and Tρ/t are bounded.

1

x

g(t) dt| ≤ H|g|(x) + H ∗ |g|(x) t


Lemma 1. (a) The ranges of the operators Tκ and Tκ/t , have the same closure in L2 [0, 1]. (b) Let M1 be the set of all functions in L2 [0, 1] which are perpendicular to the constant function 1 (= e0 ). The sets Tρ/t M1 and Tρ Mx have the same closure.

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∗ Proof. Since Tκ∗ and Tκ/t obviously have the same null space, ∗ ) = N (T ∗ )⊥ = N (T ∗ )⊥ = R(T ∗ ). R(Tκ/t κ κ κ/t

If f (t) in L2 [0, 1] vanishes on [0, δ] for some 0 < δ < 1 and belongs to M1 , then f (t) belongs to Mx and Tρ/t f (t) = Tρ f (t) t t . Since functions such as f are dense in M1 , the closure of Tρ/t M1 is contained in the closure of Tρ Mx . The reverse inclusion follows since if f (t) belongs to Mx , tf (t) belongs to M1 .
To motivate the consideration of these new operators consider Tκ/t . ∗ s−1 s−1 From Tκ/t t (x) = φ(s) and U ∗ ts−1 (x) = (1 − 1s )xs−1 it follows that s x ∗ ∗ ∗ ∗ s−1 [Tκ/t U − U Tκ/t ](t ) = 0 Since the operator in brackets vanishes on the dense subspace containing the powers of t it is identically zero. Taking adjoints, Tκ/t commutes with the unilateral shift U. One consequence is that the range of Tκ/t is an invariant subspace of the shift. A famous result of Beurling’s is the characterization of the closed invariant subspaces of the shift in the Hardy-Hilbert space H 2 (D), which strongly suggests moving from the Hilbert space L2 [0, 1] to H 2 (D); a useful reference is the recent [18].

3. Analytic Toeplitz Operators Any f in L2 [0, 1] can be expanded in term of the basis {en } of Theorem 4: f (x) =

∞


an en (x) with

n=0

∞


|an |2 = f 2 < ∞.

(41)

n=0

The map taking f (x) to fˆ(z) defined by fˆ(z) =

∞


an z n

(42)

n=0

corresponds to a standard method of defining the Hardy space H 2 (D) of functions fˆ(z), analytic on the unit disk D = {z : |z| < 1}, namely by means of the analytic extension of an orthonormal expansion to D [16], [14]. The map f → fˆ is a Hilbert space isometry of L2 [0, 1] onto H 2 (D). The novelty here is the choice of basis {en }, and the relation of that basis to the operator U . Define the following map, which associates a function of s in Re(s) > 1/2, with a function f in L2 [0, 1] by: f˜(s) =< sts−1 , f > . Apply this map to the real-valued orthonormal basis {en }: e˜0 (s) = 1 and, inductively, e˜n+1 (s) =< sts−1 , en+1 >=< sts−1 , U en >=< sU ∗ ts−1 , en > 1 1 = (1 − ) < sts−1 , en >= (1 − )e˜n (s). s s

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The Riemann Hypothesis and the Ces´ aro Operator

Thus e˜n (s) = (1 − 1s )n e˜0 , and  1 ∞ ∞

1 ˜ sts−1 an en (t) dt = an e˜n (s) = fˆ(1 − ). f (s) = s 0 n=0 n=0

443

(43)

This suggests the change of variable 1 (44) z =1− , s which maps the half-plane Re(s) > 12 onto the unit disk D = {z : |z| < 1}. Then ∞
1 )= f˜(s) = f˜( an z n = fˆ(z) 1−z n=0

(45)

with s and z related by (44). Note that the tilde map, absent the factor s, is the Mellin transform of f and the integral Tκ/t a Mellin convolution. Theorem 6. Let T be a bounded linear operator on L2 [0, 1] which commutes with U. Then the operator Tˆ defined on H 2 (D) by Tˆfˆ(z) = T f(z) is an analytic Toeplitz operator, defined on H 2 (D) as multiplication by the H ∞ (D) function ψ(z) = T

e0 (z). m Proof. If f (x) = 0 an en (x) in L2 [0, 1], then T f (x) =

m


an T e n =

m


0

0

Set h = T e0 , and compute  1  s−1 n n  st U h(t) dt = U h(s) = 0

an T U n e 0 =

0

1

m


an U n T e 0 .

0

1 ˜ s(U ∗ )n (ts−1 )h(t) dt = (1 − )n h(s). s

n h(z) = z n h(z) and Thus U

T f(z) = fˆ(z)T

e0 (z).

(46)

Since Tˆ commutes with multiplication by z, by virtue of the isomorphisms, it is an analytic Toeplitz operator and T

e0 belongs to H ∞ (D)
The operator Tκ/t commutes with U and T
κ/t e0 is easy to compute: s−1 , Tκ/t e0 > . T κ/t e0 (s) =< st

The complex conjugate of this inner product is: φ(s) . s Taking the complex conjugate and passing to the variable z, 1 T
). κ/t e0 (z) = (1 − z)φ( 1−z ∗ < sts−1 , Tκ/t e0 >=< Tκ/t sts−1 , e0 >=

(47)

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Since Tρ/t does not commute with U, more must be said. Note that s−1  , Tρ/t f (t) > T ρ/t f (s) =< st

which, because the operator has a real-valued kernel, equals  1 s 1 s−1 ∗ s−1 f (t) dt − ζ(s)f˜(s). < st , Tρ/t f (t) >=< sTρ/t t , f (t) >= s−1 0 s Changing to the variable z 1 fˆ(0)
T − (1 − z)ζ( )fˆ(z). ρ/t f (z) = z 1−z

(48)

Note that the pole of ζ(s) at s = 1 cancels out the apparent singularity in the first 1 term of (48) at z = 0, and z1 − (1 − z)ζ( 1−z ) is in H ∞ (D) from (48). Theorem 7. (a) Suppose that κ(t) does not vanish a.e. in a neighborhood of 1. Then the operator Tκ has a dense range iff the function φ(s) has no zeros in Re(s) > 1/2. In particular, for a Dirichlet series (3) with κ the bounded sum S(t), if the coefficient a1 of the Dirichlet series is not zero, then the operator Tκ has a dense range iff the function φ(s) has no zeros in Re(s) > 1/2. Thus the operator Tχ has a dense range iff the Dirichlet L-function L(χ, s), for χ a non-principal character modulo m, has no zeros in Re(s) > 1/2. (b) If κ(t) does vanish a.e. in a neighborhood of 1, then the range of Tκ is not dense. In particular if the coefficient a1 = 0 in a Dirichlet series with bounded sum function S(t), then for κ(t) = S(t), the range of Tκ is not dense. (c) The subspace Tρ Mx is dense iff ζ(s) has no zeros in Re(s) > 1/2. 2 Proof. By Theorem 6 the range of T

κ/t is ψ(z)H (D). Beurling’s characterization of the invariant subspaces of the shift [16], [13] shows that the closure of this invariant subspace has the form:

B0 (z)S0 (z)H 2 (D).

(49)

The factor B0 (z) is the Blaschke product consisting of the zeros of ψ(z), or is 1 if there are no zeros, so the Blaschke product is 1 iff φ(s) has no zeros in Re(s) > 1/2, see equation (47). The second factor S0 (z) is a singular function and as such has the form:    iθ e +z dµ(θ) S0 (z) = exp − iθ Γ e −z where Γ = {z : |z| = 1} is the boundary of D and µ is a positive measure on Γ which is singular with respect to Lebesgue measure on Γ. The function T e0 (s) is analytic on Re(s) > 0 and thus ψ(z) is analytic on the open arc Γ − {1}, as is S0 , from which it follows that the support of µ is contained in the one point set {1} [11, page 19], [16, page 68], [13, Theorem 6.3]. Hence, for some positive number α, 1+z

S0 (z) = e−α 1−z .

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Consider the operator Bτ , occurring in [8], defined on L2 [0, 1] for 0 < τ < 1 by Bτ f (x) = f (x/τ ); because of the convention that all f in L2 [0, 1] vanish off [0,1], Bτ f (x) = 0 for x > τ . Transfer this operator to H 2 (D):  1  τ s−1  st Bτ f (t) dt = sts−1 f (t/τ ) dt. Bτ f (s) = 0

0

Make the change of variable w = t/τ ,  1 s sws−1 f (w) dw = τ s f˜(s). =τ 0

Write this as eslog(τ ) f˜(s) = e(2s−1)log(τ )/2 elog(τ )/2 f˜(s). In terms of z, 1+z 1/2 −a 1−z

e B fˆ(z), τ f (z) = τ

with α = −log(τ )/2. To within the multiplicative constant τ 1/2 , this is the form of the singular factor S0 (z). Thus if there is a singular factor with support {1} in the representation of an = M, invariant subspace M in H 2 (D), then the pre-image N of that subspace, N in L2 [0, 1] is contained in the range of Bτ for some τ in (0,1), and thus all the functions in N vanish on the interval [τ, 1]. Compute  1 1 κ(t/x) dt. Tκ/t e0 (x) = t x For 1/2 < x < 1, make the change of variable w = t/x  1/x 1 Tκ/t e0 (x) = κ(w) dw, w 1 which does not vanish for all x in (1/2, 1) ∩ (τ, 1), unless κ(w) = 0 a.e. on a neighborhood of one (noting that κ(t) = 0 for 0 ≤ t < 1). Thus the singular function is not present in the representation of the subspace. Hence the range of Tκ/t is dense, equivalently the range of Tκ is dense, iff φ(s) has no zeros in Re(s) > 1/2. For a Dirichlet series, κ(w) = a1 for 1 ≤ w < 2 and κ does not vanish in a neighborhood of 1 if a1 is not zero; for the L-functions, a1 = χ(1) = 1. On the other hand, if κ(w) = 0 a.e. on (1, 1 + δ], δ > 0, then for any f in L2 [0, 1] and 1/2 < x < 1,  1/x κ(w) f (xw) dw = 0 Tκ/t f (x) = w 1 for x in (1/2, 1) ∩ (1/(1 + δ), 1) and the range of Tκ/t , equivalently the range of Tκ , cannot be dense. 1 The subspace M1 , associated with Tρ/t , consists of all f with 0 f (t) dt = 0 and the image of this set under the hat map is the set of all f in H 2 (D) which

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vanish at zero, i.e. zH 2 (D). From equation (48), 1 (50) )H 2 (D); T
ρ/t M1 = z(z − 1) ζ( 1−z the term z in the factor does not produce a zero because it is canceled by the pole of the zeta function at 1. When the closure of this invariant subspace of U is written as a Blaschke product times a singular function times H 2 (D), the zeros of the Blaschke product correspond exactly to the zeros of ζ(s) in Re(s) > 1/2 and is equal to 1 if there are no such zeros. As proved above, if the representation of the closure of the invariant subspace T
ρ/t M1 has a singular factor, then all the functions in Tρ/t M1 must vanish on [τ, 1] for some 0 < τ < 1. But the function f (t) = 2t − 1 belongs to M1 and for this f and 1/2 < x < 1  1  1 1 x t Tρ/t f (x) = (2t − 1) dt + ( − 1)(2 − ) dt = 2x − 2 − log(x) x 0 t x x which does not vanish for 1/2 < x < 1. Hence Tρ/t M1 is dense, equivalently Tρ Mx , is dense iff the range of Tρ is dense by Corollary 2 iff ζ(s) has no zero in Re(s) > 1/2. Equation (50) and the argument above show that Tρ/t U is an analytic Toeplitz operator on H 2 (D) which has dense range iff ζ(s) has no zeros in Re(s) > 1/2. 1 Note that U does not commute with Tρ/t ; from (48), U Tρ/t f (x) = 0 f (t) dt + Tρ/t U f (z).
Corollary 2. The subspace Tρ Mx is dense in L2 [0, 1] iff Tρ has dense range. Proof. Suppose that Tρ Mx is not dense. By Theorem 3 and Theorem 7, the Riemann Hypothesis is false and there is an complex number s0 with Re(s0 ) > 1/2 and ζ(s0 ) = 0. Then Tρ∗ (s0 − 1)ts0 −1 (x) = x and taking the complex conjugate Tρ∗ (s0 −1)ts0 −1 = x. Let h = (s0 −1)ts0 −1 −(s0 −1)ts0 −1 . Then Tρ∗ h = 0, and h is not zero, for if it were 0 =< h, 1 >= −2iIm( s10 ) which is false since zeta has no real zeros in Re(s) > 1/2. Hence {0} = N (Tρ ) = R(Tρ )⊥ and the range of
Tρ is not dense.  Theorem 3 and Corollary 2 show that the condition ak θk = 0 for B in equation (2) can be omitted and still have a density problem which is equivalent to the Riemann hypothesis. I thank the referee for his advice. This paper would not have been written without Marin Schechter’s encouragement or without the support and encouragement of Ted Hromadka.

References [1] Alc´ antara-Bode, J., An integral formulation of the Riemann hypothesis, Integral Equations and Operator Theory 17 (1993) 151-168. [2] Apostol, T. M., Introduction to Analytic Number Theory, Springer-Verlag, 1976.

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[3] Balazard, M., Completeness problems and the Riemann hypothesis: An annotated Bibliography, in Number Theory for the Millennium I, A. K. Peters, Natwick, MA, (2002) 21-48. [4] Balazard, M. and Saias, E. The Nyman-Beurling equivalent form for the Riemann Hypothesis, Expo. Math. 18 (2000) 131-138. [5] B´ aez-Duarte, L., Arithmetical aspects of Beurling’s real variable formulation of the Riemann hypothesis, arXiv:Math /0011254 (Nov 2000), 21 pages. [6] B´ aez-Duarte, L., and C´ amera, G. A., Horizontal growth of harmonic functions, Stud. Appl. Math. 98 (1997) 195-202. [7] Bercovici, H. and Foias, C. A real variable restatement of Riemann’s hypothesis, Israel J. of Math. 48 (1984) 57-68. [8] Beurling, A. A closure problem related to the Riemann zeta-function, Proc. of the Nat. Acad. Sci. 41 (1955), 312-314. [9] de Branges, L., Some Hilbert spaces of entire functions III, Trans. Amer. Math. Soc. 100 (1961) 73-115. [10] Brown, A, Halmos, P, and Shields, A., Ces´ aro Operators, Acta Sci. Math. (Szeged) 26 (1965) 125-137. [11] Cima, J. and Ross, W., The Backward Shift on the Hardy Space, American Math. Soc., 2000. [12] Donoghue, W. F., Distributions and Fourier Transforms, Academic Press, 1969. [13] Garnett, J., Bounded Analytic Functions, Springer-Verlag, Revised 1st Ed, 2007. [14] Halmos, P., A Hilbert Space Problem Book, 2nd Edition, Springer-Verlag, 1982. [15] Hardy, G. H., Littlewood, J.E., and Polya, G., Inequalities, Cambridge University Press, 1967. [16] Hoffman, K., Banach Spaces of Analytic Functions, Dover Publications (reprint), 1988. [17] Kufner, A., Maligranda, L., and Persson, L-E., The prehistory of the Hardy inequality, American Math. Monthly 113 (2006) 715-732. [18] Mart´ınez-Alvenda˜ no, R. and Rosenthal, P., An Introduction to Operators on the Hardy-Hilbert Space, Springer-Verlag, 2007. [19] Murty, M. R., Problems in Analytic Number Theory, Springer-Verlag, 2001. [20] Nathanson, M. B., Elementary Methods in Number Theory Springer-Verlag, 2000. [21] Narkiewicz, W., The Development of Prime Number Theory, Springer-Verlag, 2000. [22] Nyman, B., On some groups and semigroups of translations, Thesis, Uppsala, 1950. [23] Radjavi, H. and Rosenthal, P., Invariant Subspaces, Dover Publications (Reprint), 2003. [24] Rosenblum, M. and Rovnyak, J., Hardy Classes and Operator Theory, Dover Publications (Reprint), 1997. [25] de Roton, A., G´en´eralisation du crit`ere de Beurling et Nyman pour l’hypoth`ese de Riemann, C. R. Acad. Sci. Paris, Ser. I 340 (2005) 191-194. [26] de Roton, A., G´en´eralisation du crit`ere de Beurling et Nyman pour l’hypoth`ese de Riemann, Trans. Amer. Math. Soc. 359 (2007) 6111-6126. [27] Rudin, W., Real and Complex Analysis, 3rd Edition, McGraw-Hill, 1987.

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[28] Schechter, M., Principles of Functional Analysis, 2nd edition, American Math. Society, 2002. [29] Stromberg, K. R., An Introduction to Classical Real Analysis, Wadsworth, 1981. [30] Titchmarsh, E. and Heath-Brown, D. The Theory of the Riemann Zeta-Function, 2nd edition, Oxford University Press, 1986. [31] Widder, D., The Laplace Transform, Princeton University Press, 1972. Robert Whitley P.O. Box 11133 Bainbridge Island, WA 98110 USA e-mail: [email protected]. Submitted: January 24, 2007 Revised: February 26, 2008

Integr. equ. oper. theory 61 (2008), 449–475 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/040449-27, published online July 25, 2008 DOI 10.1007/s00020-008-1600-y

Integral Equations and Operator Theory

Sharp Two Weight Inequalities for Commutators of Riemann-Liouville and Weyl Fractional Integral Operators Ana L. Bernardis and Mar´ıa Lorente +,k Abstract. Let b be a BMO function, 0 < α < 1 and Iα,b the commutator of order k for the Weyl fractional integral. In this paper we prove weighted strong type (p, p) inequalities (p > 1) and weighted endpoint estimates (p = 1) for +,k and for the pairs of weights of the type (w, Mw), where w the operator Iα,b is any weight and M is a suitable one-sided maximal operator. We also prove +,k p that, for A+ ∞ weights, the operator Iα,b is controlled in the L (w) norm by a composition of the one-sided fractional maximal operator and the one-sided Hardy-Littlewood maximal operator iterated k times. These results improve those obtained by an immediate application of the corresponding two-sided results and provide a different way to obtain known results about the operators +,k . The same results can be obtained for the commutator of order k for the Iα,b −,k . Riemann-Liouville fractional integral Iα,b

Mathematics Subject Classification (2000). Primary 26A33; Secondary 42B25. Keywords. Weighted inequalities, Riemann-Liouville and Weyl fractional integrals, commutators.

1. Introduction We are interested in two weight inequalities of the type

|T f |p w ≤ C |f |p MT w, 1 < p < ∞,

(1.1)

with no a priori assumption on the weight w and where MT is some maximal operator associated with the operator T . This research has been partially supported by Spanish goverment Grant MTM2005-8350-C0302. The first author was also supported by CONICET, ANPCyT and CAI+D-UNL. The second author was also supported by Junta de Andaluc´ıa Grant FQM 354.

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This type of inequalities have been studied by several authors. For example, if T is a Calder´ on-Zygmund singular integral operator, C. P´erez ([16]) proved (1.1) with MT = M [p]+1 , where M k is the Hardy-Littlewood maximal operator M iterated k times and [p] is the integer part of p. Later on, P´erez ([20]) generalized the result in [16] obtaining (1.1) for T equal to the commutator of the Calder´ onZygmund singular integral of order k and in this case MT = M [(k+1)p]+1 . For the one-sided singular integrals, i.e., singular integrals operators with kernels supported in (−∞, 0) or (0, ∞), inequalities like (1.1) were proved in [25]. In this case, if the kernel has support in (−∞, 0), (1.1) can be obtained with MT = (M − )[p]+1 , where M − is the one-sided Hardy-Littlewood maximal operator. This result was generalized in [8] for the commutator of the one-sided singular integral of order k with MT = (M − )[(k+1)p]+1 . In the case where T is the fractional integral operator, C. P´erez ([19]) proved (1.1) with MT = Mαp (M [p] ), where Mαp is the fractional maximal operator. This result was generalized in [2] to the commutator of the fractional integral of order k, obtaining in this case that MT = Mαp (M [(k+1)p] ). The result in [2] was obtained in the general context of the spaces of homogeneous type. The first purpose of this paper is to prove an inequality like (1.1) for the commutators of the one-sided fractional integrals (Riemann-Liouville and Weyl fractional integrals). For 0 < α < 1, b ∈ BM O and k = 0, 1, . . . , the k-th order commutators of the one-sided fractional integrals are defined by
x f (y) −,k Iα,b f (x) = (b(x) − b(y))k dy (x − y)1−α −∞ and




∞

f (y) dy. (y − x)1−α x Let us notice that when k = 0 the above operators are the Riemann-Liouville and Weyl fractional integral operators, respectively. Applying Theorem 1.2 in [2] with X = R, d(x, y) = |x−y| and µ the Lebesgue −,k +,k measure, we obtain (1.1) for T = Iα,b and T = Iα,b with MT = Mαp (M [(k+1)p] ). On the other hand, inequality (1.1) was proved in [24] for T equal to the one-sided − − fractional integral Iα+ and MT = Mαp (M [p] ), where Mαp is the one-sided maximal fractional operator (see Theorem 1.4 in [24]). Our result improves these because we obtain a smaller operator MT in the right hand side of (1.1). Precisely, we shall prove the following theorem. +,k f (x) Iα,b

=

(b(x) − b(y))k

Theorem 1.1. Let w be any weight, 0 < α < 1, 1 < p < ∞ and k ∈ N ∪ {0}. If b ∈ BM O then there exists a positive constant C such that

+,k − |Iα,b f (x)|p w(x)dx ≤ Cbkp |f (x)|p Mαp ((M − )[(k+1)p] w)(x)dx. (1.2) BMO R

R

In this paper, every one-sided result has a corresponding one reversing the orientation of the real line.

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Let us observe that from Theorem 1.1 with k = 0, we also improve Theorem 1.3 in [24]. In fact, from (1.2) for k = 0, by using duality and the trivial boundedness Mα− f (x) ≤ Iα− f (x) we get the following dual inequality for the non linear operator Mα−


− p
− − [p] 1−p
[Mα f (x)] [Mαp ((M ) w)(x)] dx ≤ C |f (x)|p w(x)1−p dx. R

R

In a similar way as [20], we can show that (1.2) is sharp, in the sense that we cannot replace [(k + 1)p] by [(k + 1)p] − 1. In fact, as in [20], we can construct a counterexample by taking f = w = χ(0,1) and b(x) = log |x|. One of the main results for proving the above theorem is a pointwise equivalence between two type of maximal operators: the composition of one-sided maximal operators Mα− ((M − )k ) and a one-sided fractional maximal operator associ− ated with the mean Luxemburg norm in the Orlicz spaces, Mα,φ , with φk (t) = k t[log(e + t)]k (see Section 2 for the corresponding definitions). This equivalence was previously proved in [25] for the case α = 0, and in [2] for 0 < α < 1 but in the two-sided case. The proof for 0 < α < 1 in the one-sided case is not an obvious generalization of the case α = 0 not even in the two-sided case. Section 3 is devoted to show how to adapt the two-sided argument to the one-sided context. From now on, we shall denote φr (t) = t[log(e + t)]r , for r > 0. As a consequence of the above mentioned equivalence we can obtain the − − w instead of Mαp ((M − )[(k+1)p] w). On inequality in Theorem 1.1 with Mαp,φ [(k+1)p] the other hand, from the proof of Theorem 1.1 it is easy to see that we can obtain a sharper estimate, namely

+,k − |Iα,b f (x)|p w(x)dx ≤ Cbkp |f (x)|p Mαp,φ w(x)dx, (1.3) BMO (k+1)p−1+δ R

R

for any δ > 0 and with C depending on δ. The arguments used to prove Theorem 1.1 are similar to the ones used in [20] and later on in [2]. As in those articles we need to prove previously a one-sided version of the weighted norm inequality between the commutator and a suitable maximal operator. Concretely, we shall prove the following result. Theorem 1.2. Let 0 < p < ∞, 0 < α < 1 and k ∈ N ∪ {0}. If w ∈ A+ ∞ and b ∈ BM O then there exists a constant C depending on k and on the A+ ∞ constant of w, such that

+,k kp + p |Iα,b f (x)| w(x)dx ≤ CbBMO [Mα,φ f (x)]p w(x)dx, k R

R

for all f such that the left hand side of the previous inequality is finite. Just reversing the orientation of R in the proof of the equivalence between − + and Mα− ((M − )k ) we get that Mα,φ ≈ Mα+ ((M + )k ). Then, we can write Mα,φ k k the inequality in the above theorem as

+,k kp p |Iα,b f (x)| w(x)dx ≤ CbBMO [Mα+ ((M + )k f )(x)]p w(x)dx. (1.4) R

R

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The case k = 0 of (1.4) was proved by F.J. Mart´ın-Reyes, L. Pick and A. de la Torre in [11]. Let us observe that Theorem 1.2 also improves the result obtained as a consequence of the results in [2] for the one-sided commutators (see also [4] for the case k = 1) in two ways: putting in the right hand side a smaller operator and allowing a wider class of weights. On the other hand, Theorem 1.2 implies another proof of the weighted strong +,k obtained in [9] (see Theorem 2.9 in [9]). (p, q) inequality for the commutators Iα,b In fact, we can obtain the following corollary. Corollary 1.3. Let 0 < α < 1, 1 < p < q < ∞ such that p1 − 1q = α, b ∈ BM O, and k ∈ N ∪ {0}. Let w be a weight such that w ∈ A+ (p, q), that is, there exists C > 0 such that 1/p

x 1/q 
x+h
1 1 wq w−p ≤ C, h x−h h x +,k for all h > 0 and x ∈ R. Then the operator Iα,b satisfies the strong (p, q) inequality


R

+,k q q |Iα,b f| w




1/q ≤C

R

p

|f | w

p

1/p .

The proof of the corollary follows easily from (1.4), the strong (p, q) inequality for Mα+ (see [1]) and the weighted Lp boundedness of the maximal operator M + for + + q  A+ p weights. In fact, notice that w ∈ A (p, q) is equivalent to w ∈ Aβ , β = 1+q/p q + + p + (therefore w ∈ A∞ ) and that w ∈ A (p, q) implies w ∈ Ap . Now we turn our attention to the case p = 1 of the inequality (1.1). In particular we shall study the following endpoint inequality
{|T f |>λ}




w≤C

ψ

|f | λ

 MT w,

λ > 0,

(1.5)

where ψ is a Young function. When T is the commutator of the Calder´on-Zygmund singular integral of order k, (1.5) was proved in [21] with ψ(t) = φk (t) and MT = Mφk+ε for any ε > 0 and with a constant depending on ε. The case k = 0 was previously proved in [16]. When T is the fractional integral, two different versions of (1.5) were obtained in [3]. Both with ψ(t) = t but in one of them MT = Mα (Mφε ) for any ε > 0 and in the other one MT w(x) = Mα w(x) + |x|α M w(x). The second purpose of this paper is to obtain an inequality like (1.5) for the commutators of the one-sided fractional integral operator. In the following theorem we state our result.

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Theorem 1.4. Let 0 < α < 1, b ∈ BM O, and k = 0, 1, . . . . Then  

||b||kBMO |f (x)| − w≤C φk w(x)) dx, (Mφ−k w(x) + Mαp,φ k+ε +,k λ {x∈R:|Iα,b f (x)|>λ} R ε for any weight w, λ > 0, ε > 0, 1 < p < 1 + k+1 and where the constant C depends on ε and p.

We shall prove the above theorem following the arguments in [21], that is, by using a Calder´on-Zygmund decomposition and an induction argument. It is easy to see that from the above theorem and the corresponding one for the commutator −,k k Iα,b we obtain a similar result for the two-sided commutators Iα,b in one dimension. In this case the weight in the right hand side will be Mφk w + Mαp,φk+ε w. From Theorem 1.2 in [2], applying the same type of Calder´ on-Zygmund decomposition as in [21] and following the same steps that in the proof of Theorem 1.4 we can prove the following result in dimension greater than one (the details are left to the reader). Theorem 1.5. Let 0 < α < 1, b ∈ BM O, and k = 0, 1, . . . . Then  

||b||kBMO |f (x)| w≤C φk (Mφk w(x) + Mαp,φk+ε w(x)) dx, k f (x)|>λ} λ {x:|Iα,b Rn ε and where the constant C depends for any weight w, λ > 0, ε > 0, 1 < p < 1 + k+1 on ε and p.

Let us notice that when we formally consider α = 0 in the above theorem we recover the corresponding result in [21] for the commutators of the Calder´ onZygmund singular integral. On the other hand, the above theorem in the case k = 0 gives (1.5) with ψ(t) = t and MT = M + Mαp,φε . This result is an endpoint inequality different than the ones in [3]. As far as we know, our results are not comparable with those in [3]. The article is organized in the following way: in Section 2 we give some definitions and preliminaries. Section 3 is devoted to prove the pointwise equivalence between the one-sided maximal operators previously mentioned in this introduction. In Section 4 we shall prove Theorems 1.1 and 1.2, while Theorem 1.4 will be proved in Section 5.

2. Definitions and preliminaries The one-sided fractional maximal operators Mα+ and Mα− , 0 ≤ α < 1 are defined for locally integrable functions f by
x+h
x 1 1 Mα+ f (x) = sup 1−α |f (y)| dy and Mα− f (x) = sup 1−α |f (y)| dy. h>0 h h>0 h x x−h When α = 0 in the above operators we get the one-sided Hardy-Littlewood maximal operators and we denote them simply with M + and M − .

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− The good weights for M + and M − are the one-sided weights A+ p and Ap introduced by E. Sawyer [26]: w ∈ A+ p if there exists a constant Cp < ∞ such that for all a < b < c  

p−1 b c 1 1−p
w(x) dx w(x) dx ≤ Cp . (c − a)p a b − + In the case p = 1, w ∈ A+ 1 if M w(x) ≤ C1 w(x). The class A∞ is defined as the + + + − union of all the Ap classes, A∞ = ∪p≥1 Ap . The classes Ap are defined in a similar way. (See [26], [10], [12] for more definitions and results.)

We shall use two results of the one-sided weights. It is not difficult to see that (Mα− f )δ ∈ A+ 1 for any 0 ≤ α, δ < 1 and all locally integrable functions f such that Mα− f < ∞ a.e. (see [15] for the two-sided case). The other result is the following: 1−r if w ∈ A+ ∈ A− r , for all r > 1 (see [26] or [10]). 1 then w Let us recall some of the needed background on Orlicz spaces. The reader is referred to [23] and [13] for a complete account of this topic. A function φ : [0, ∞) → [0, ∞) is a Young function if it is continuous, convex and increasing satisfying φ(0) = 0 and φ(t) → ∞ as t → ∞. Given a Young function φ, we define the φ-mean Luxemburg norm of a function f on I by  
  |f | 1 φ ||f ||φ,I = inf λ > 0 : ≤1 . (2.1) |I| I λ It is well known that if φ(t) ≤ Cψ(t) for all t ≥ t0 then ||f ||φ,I ≤ C||f ||ψ,I , for all intervals I and functions f . Thus, the behavior of φ(t) for t ≤ t0 is not important. If φ ≈ ψ, that is there are constants t0 , c1 , c2 > 0 such that c1 φ(t) ≤ ψ(t) ≤ c2 φ(t) for t ≥ t0 , the latter estimate implies that ||f ||φ,I ≈ ||f ||ψ,I . Each Young function φ has an associated complementary Young function φ˜ satisfying t ≤ φ−1 (t)φ˜−1 (t) ≤ 2t, for all t > 0. There is a generalization of H¨ older’s inequality
1 |f g| ≤ ||f ||φ,I ||g||φ,I (2.2) ˜ . |I| I A further generalization of H¨older’s inequality (see [13]) that will be useful later is the following: If φ, ψ and ϕ are Young functions and φ−1 (t)ψ −1 (t) ≤ ϕ−1 (t) then f gϕ,I ≤ 2f φ,I gψ,I . (2.3) A generalization of Young inequality states that if φ−1 (t)ψ −1 (t) ≤ ϕ−1 (t) for t > 0, then ϕ(st) ≤ φ(s) + ψ(t), (2.4) for all s, t > 0.

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For each locally integrable function f and 0 ≤ α < 1, the one-sided fractional maximal operators associated to the Young function φ are defined by + Mα,φ f (x) = sup(b − x)α f φ,(x,b) x
When α = 0 we use the notation

Mφ+

− and Mα,φ f (x) = sup(x − a)α f φ,(a,x). a<x

instead of

+ M0,φ

− (Mφ− instead of M0,φ ).

We shall need the following results about these maximal operators. First,  1 notice that if φ(t) = t then ||f ||φ,I = |I| |f | and let us recall that φr (t) = I t[log(e + t)]r . For every 0 ≤ α < 1 and every l ≤ s, we get that − − − f (x) ≤ Mα,φ f (x) ≤ Mα,φ f (x). Mα− f (x) = Mα,φ 0 s l

(2.5)

− Splitting the family of intervals in the definition of Mα,φ in two families, those intervals with measure smaller than one and the rest, we can easily prove that, for any Young function φ and for every p > 1, − − f (x) ≤ Mαp,φ f (x) + Mφ− f (x). Mα,φ

(2.6)

On the other hand, if I = (a, b) and I − = (c, a) with |I| = |I − |, for any Young − function φ and nonnegative function f with Mα,φ f (x) < ∞ a.e., we get that − |I|α Mφ− (f χR\(I − ∪I) )(y) ≤ CMα,φ (f χR\(I − ∪I) )(b),

a.e y ∈ I,

(2.7)

− − Mα,φ (f χR\(I − ∪I) )(y) ≈ inf Mα,φ (f χR\(I − ∪I) )(z),

a.e y ∈ I.

(2.8)

and z∈I

− and These results follow easily from the definition of the maximal functions Mα,φ keepping in mind which is the support of f χR\(I − ∪I) .

Let us recall that a locally integrable function b belongs to BMO = BM O(R)
1 |b(x) − bI | dx < ∞, bBMO = sup I |I| I where the supremum runs over all intervals I ⊂ R and where bI stands for the average of b over I. It is easy to prove that a function b is in BM O if for each interval I there exists a constant c(I) such that
1 |b(x) − c(I)| dx < ∞. sup I |I| I

if

Further, this supremum is comparable to ||b||BMO . We state some results about BM O functions that we shall use in the article (for the proofs see for example [27]). If b ∈ BM O then there exists an absolute constant C such that, for any interval I = (x, x + h) and j ≥ 0, |bIj − bI | ≤ C(j + 1)||b||BMO , j

where Ij = (x + 2 h, x + 2 Ij = (x − 2j+1 h, x − 2j h).

j+1

(2.9)

h). Obviously, the same holds if I = (x − h, x) and

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By applying the John-Nirenberg inequality we get the following facts: (i) For each p, 1 < p < ∞, there exists a constant Cp such that 1/p 
1 p sup |b(x) − bI | dx ≤ Cp ||b||BMO . |I| I I

(2.10)

(ii) If b ∈ BM O then there exists a constant C such that for every interval I,  
|b(x) − bI | 1 exp dx < ∞. (2.11) |I| I C||b||BMO ˜ = exp(t) As a consequence of (2.11) we get that for φ(t) ||b − bI ||φ,I ˜ ≤ C||b||BMO .

(2.12)

For a locally integrable funtion f we define the one-sided sharp maximal function as + 

1 x+2h 1 x+h +,# f (x) = sup f dy, f (y) − M h x+h h>0 h x where z + = max(z, 0). It is proved in [12] that
1 x+h +,# f (x) ≤ sup inf (f (y) − a)+ dy M a∈R h h>0 x


x+2h 1 + + (a − f (y)) dy . h x+h

(2.13)

Given an operator T , we use the notation T(δ) (f ), 0 < δ < 1, for the operator [T (|f |δ )]1/δ .

3. Equivalence between two maximal operators This section is devoted to prove the following equivalence between two one-sided maximal operators. Theorem 3.1. Let 0 ≤ α < 1, k ∈ N and φk (t) = t[log(e + t)]k . Then, there exist constants C1 , C2 > 0 such that − Mα− ((M − )k f )(x) ≤ C1 Mα,φ f (x) k

(3.1)

− Mα,φ f (x) ≤ C2 Mα− ((M − )k f )(x), k

(3.2)

and for every x ∈ R. To adapt the two-sided arguments in the proof of (3.2) we shall need the following two lemmas.

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Lemma 3.2. Let 0 ≤ α < 1 and k ∈ N. For each interval I = (a, x) let us consider the decomposition I = Ik− ∪ Ik+ such that Ik− = (a, a + 2−k |I|) and Ik+ = (a + 2−k |I|, x), and let us define the maximal operator − Nα,φ f (x) = sup |Ik− |α ||f ||φk ,I − . k k

I=(a,x)

Then, there exists a constant C depending on k such that − Mα,φ f (x) ≤ C k

k  j=0

− Nα,φ f (x), j

− − simply denotes Mα,φ . where Nα,φ 0 0

Proof. First, notice that for every γ > 1 there exists a constant Cγ,k depending on γ and k such that

φk (γt) ≤ Cγ,k

k−1 

φj (t) + γφk (t).

(3.3)

j=0

In fact, φk (γt) = ≤

γt[log(e + γt)]k ≤ γt[log γ + log(e + t)]k γt

k 

cj,k [log γ]k−j [log(e + t)]j

j=0

≤

max {γcj,k [log γ]k−j }

0≤j≤k−1

k−1 

φj (t) + γφk (t),

j=0

where cj,k are constants proceeding from the Newton’s formula. Now, by using (3.3) with γ = γk = 2k /(2k − 1) we shall prove that 

k−1 

γk ||f ||φk ,I ≤ C 

j=0

 ||f ||φj ,I + ||f ||φk ,I −  + ||f ||φk ,I + , k

k

(3.4)

with C = max{Cγk ,k , 1}. In fact, if we denote by µ the right hand side of (3.4), using (3.3) with γ = γk , the convexity of the functions φj , j = 1, · · · , k and (2.1) we have

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1 |I|




I

φk ≤

γk |f | µ

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k−1
|f | |f | |f | Cγk ,k  γk γk φj φk φk + + |I| j=0 I µ |I| Ik− µ |I| Ik+ µ


  k−1 Cγk ,k  1 |f | ≤ ||f ||φj ,I φj µ j=0 |I| I ||f ||φj ,I   
γk ||f ||φk ,I − 1 |f | k + φk 2k µ ||f ||φk ,I − |Ik− | Ik− k   
||f ||φk ,I + 1 |f | k + φk + + µ ||f ||φk ,I + |Ik | Ik k k−1 Cγk ,k j=0 ||f ||φj ,I + γk 2−k ||f ||φk ,I − + ||f ||φk ,I + k k ≤ 1. ≤ µ Then, using again (2.1) we get (3.4). Now, let us observe that by (3.4) |I|α ||f ||φk ,I

≤

k−1 C  α C |I|α |I| ||f ||φj ,I + |I|α ||f ||φk ,I − + ||f ||φk ,I + k k γk γk γk j=0

≤

C γk

k−1  j=0

− Mα,φ f (x) + j

C2kα − − Nα,φk f (x) + γkα−1 Mα,φ f (x). k γk

Taking supremum on I = (a, x) and using that γk > 1 we get that   k−1  − − − Mα,φ f (x) ≤ C  Mα,φ f (x) + Nα,φ f (x) . j k k j=0

− is equal to To finish the proof of the lemma notice that, by definition, Nα,φ 0 − − Mα,φ and that M is pointwise equivalent to the one-sided fractional maximal α,φ0 0 operator Mα− . Then, clearly, the lemma holds for k = 1. For general k ∈ N the lemma follows by applying an induction argument over k.


Lemma 3.3. Let I = (a, b) be a fix interval and let I − = (a, (a + b)/2). Then, there exists a constant C such that
1 f ≤ C|{x ∈ I : M − (f χI − )(x) > λ}|, (3.5) λ {x∈I − :f (x)>λ}  for any λ ≥ |I1− | I − f and all nonnegative integrable functions f . Proof. It is well known (see for example [6], pp. 423) that

1 1 f≤ f = |{x : M − f (x) > λ}|. λ {x:f (x)>λ} λ {x:M − f (x)>λ}

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Applying the above result to f χI − we get that
1 f ≤ C|{x : M − (f χI − )(x) > λ}|. λ {x∈I − :f (x)>λ} − Now, the lemma follows  since if x < a then M (f χI − )(x) = 0 and if x > b then 1 − M (f χI − )(x) ≤ |I − | I − f ≤ λ.


Proof of Theorem 3.1. Without loss of generality we may assume that f ≥ 0. To prove (3.1) let us consider an interval I = (a, x). Notice that by inequality (4.4) in [2]

1 1 (M − )k f (y) dy ≤ M k f (y) dy  C||f ||φk ,I , (3.6) |I| I |I| I for any f such that supp(f ) ⊂ I. Now, writting f = f1 + f2 where f1 = f χ2I with 2I = (2a − x, x) we get that


|I|α−1 (M − )k f ≤ |I|α−1 (M − )k f1 + |I|α−1 (M − )k f2 = A + B. I

I

By (3.6) A ≤ C|2I|α−1


2I

I

− (M − )k f1 (y) dy ≤ C|2I|α ||f ||φk ,2I ≤ CMα,φ f (x). k

Using the equivalence for α = 0 (see Proposition 1 in [25]) and (2.7) we get that
− − f2 (x) ≤ CMα,φ f (x). B ≤ |I|α−1 Mφ−k−1 f2 (y) dy ≤ CMα,φ k−1 k I

Then (3.1) follows taking supremum on a < x. To prove (3.2) we proceed as in [22] (see also [2]). By Lemma 3.2 it suffices to show that − f (x) ≤ CMα− ((M − )k f )(x), Nα,φ k for any k ∈ Z. Let I = (a, x) be any interval. Notice that it is enough to show that there is a constant Ck such that
Ck (M − )k f (y) dy. (3.7) f φk ,I − ≤ k |I| I  1 Let λk = λk (f ) = |I| (M − )k f. To prove (3.7) we shall show that there is a I constant Ck > 1 such that
1 f φk ( ) ≤ 1. (3.8) − Ck λk |Ik | Ik− To prove (3.8) we shall use induction on k and the formula
∞
Φ(|f |) dν = Φ (λ)ν({x ∈ I : |f (x)| > λ}) dλ, I

a

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which holds for any increasing continuously differentiable function Φ and where a is such that Φ(a) = 0. In fact, for k = 1 and g = C1fλ1 we have  

f 1 1 φ1 g[log(e + g)] = C1 λ1 |I1− | I1− |I1− | I1−
∞ 1 1 g({x ∈ I1− : g(x) > λ}) dλ = − |I1 | 1−e e + λ 
1

∞ 1 1 + g(x) dx dλ = e + λ {x∈I1− :g(x)>λ} |I1− | 1−e 1 =

I + II.

−

Since f (y) ≤ M f (y) a.e., I

≤ =


log(1 + e)g(I1− ) log(1 + e) f (y) dy = |I1− | |I1− |C1 λ1 I1−  2 log(1 + e) I − f (y) dy 2 log(1 + e) 1  < 1, ≤ C1 C1 I M − f (y) dy

if we choose C1 > 2 log(1 + e). On the other hand, by Lemma 3.3, since λ > 1 > 1 − g there exists a constant C such that |I1− | I1

∞ 1 1 II = g(x) dx dλ |I1− | 1 e + λ {x∈I1− :g(x)>λ}
∞ C λ ≤ |{x ∈ I : M − g(x) > λ}| dλ |I1− | 1 e + λ
∞ C |{x ∈ I : M − g(x) > λ}| dλ ≤ |I1− | 0

2C 2C 2C M − g(x) dx = M − f (x) dx = < 1, ≤ |I| I λ1 C1 |I| I C1 provided that C1 > 2C. Thus we have proved the case k = 1. Suppose that k > 1 and the result holds for k − 1. If g = Ckfλk then  

f 1 1 φ g[log(e + g)]k = k Ck λk |Ik− | Ik− |Ik− | Ik−
∞ [log(e + λ)]k−1 k g({x ∈ Ik− : g(x) > λ}) dλ = e+λ |Ik− | 1−e 
1
∞ k + = I + II. = |Ik− | 1−e 1 Notice that, I

≤

2k [log(1 + e)]k [log(1 + e)]k −  g(I ) = k Ck I (M − )k f |Ik− |


Ik−

f≤

2k [log(1 + e)]k < 1, Ck

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if we choose Ck > 2k [log(1 + e)]k . Notice that by this election of Ck ,

461 1 |Ik− |

 Ik−

g ≤ 1.

Then, applying Lemma 3.3 we get

∞ [log(e + λ)]k−1 k II = g(x) dx dλ e+λ |Ik− | 1 {x∈Ik− :g(x)>λ}
∞ λ[log(e + λ)]k−1 kC − |{x ∈ Ik−1 : M − g(x) > λ}|dλ ≤ e+λ |Ik− | 1
∞ kC − [log(e + λ)]k−1 |{x ∈ Ik−1 : M − g(x) > λ}|dλ ≤ |Ik− | 1
∞ kC − ≤ φk−1 (λ)|{x ∈ Ik−1 : M − g(x) > λ}|dλ |Ik− | 0
kC ≤ φk−1 (M − g) − |Ik− | Ik−1  
M −f 2kC ≤ φk−1 . − − Ck λk (f ) |Ik−1 | Ik−1 Let us observe that λk (f ) = λk−1 (M − f ). Then, choosing Ck > 2kCCk−1 , using that the function φk−1 is convex and the induction hypothesis we obtain that  
Ck−1 1 M −f II ≤ 2kC φ k−1 − − Ck |Ik−1 Ck−1 λk−1 (M − f ) | Ik−1 ≤

2kC

Ck−1 < 1. Ck

In this way, inequality (3.8) is proved and so is inequality (3.2).




4. Proof of Theorems 1.1 and 1.2 We begin by proving the following pointwise estimate. Lemma 4.1. Let 0 < α < 1, b ∈ BM O, k ∈ N and 0 < δ < < 1. Then there exists a constant C such that  k−1   +,# +,k +,m k−m + + k M(δ) (Iα,b f )(x) ≤ C bBMO M( ) (Iα,b f )(x) + bBMO Mα,φk f (x) , m=0

for x ∈ R and f ≥ 0. Proof. As in [5] (see also [19]), writing b(x) − b(y) = (b(x) − λ) − (b(y) − λ), where λ is an arbitrary constant, we can obtain the following decomposition +,k f (x) = Iα+ ((b − λ)k f )(x) + Iα,b

k−1  m=0

+,m Cm,k (b(x) − λ)k−m Iα,b f (x).

(4.1)

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Observe that 0 < δ < 1 implies ||a|δ − |c|δ | ≤ |a − c|δ for a, c ∈ R. Then, given x ∈ R and h > 0 and taking into account (2.13), it is enough to show that for some constant a depending on x and h, there exists C > 0 such that 1/δ 
1 x+2h +,k δ |Iα,b f (y) − a| dy h x  k−1   +,m k−m + + k ≤ C bBMO M( ) (Iα,b f )(x) + bBMO Mα,φk f (x) . (4.2) m=0

Now, let us fix x and h > 0 and let J = [x, x + 4h]. Then we write f = f1 + f2 ,  x+4h 1 where f1 = f χJ . Taking λ = bJ = 4h b, a = Iα+ ((b − bJ )k f2 )(x + 2h) and x using (4.1) we have that 
1/δ 1 x+2h +,k δ |Iα,b f (y) − a| dy ≤ I1 + I2 + I3 , (4.3) h x where

1/δ 
1 x+2h (k−m)δ +,m δ I1 = C |b(y) − bJ | |Iα,b f (y)| dy , h x m=0 
1/δ 1 x+2h + I2 = C |Iα ((b − bJ )k f1 )(y)|δ dy h x k−1 

and


1/δ 1 x+2h + k + k δ I3 = |Iα ((b − bJ ) f2 )(y) − Iα ((b − bJ ) f2 )(x + 2h)| dy . h x

Choosing 1 < r < /δ, using H¨ older’s inequality and (2.10), it follows that  1/δr

1/δr
k−1  1 x+2h
1 x+2h +,m δr (k−m)δr |b − bJ | |Iα,b f | I1 ≤ C h x h x m=0 ≤

C

k−1 

+,m + bk−m BMO M(δr) (Iα,b f )(x)

m=0

≤

C

k−1 

+,m + bk−m BMO M( ) (Iα,b f )(x).

m=0

Now, we estimate I2 . Since Iα+ is of weak type (1, (1 − α)−1 ), Kolmogorov’s inequality and (2.2) with φk (t) = t[log(e + t)]k and φ˜k (t) = exp(t1/k ) yield
x+4h C |b − bJ |k f (y)dy I2 ≤ h1−α x ≤ Chα |b − bJ |k φ˜k ,J f φk ,J .

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Then, by (2.12) we get that I2

+ + k Cb − bJ kφ,J ˜ Mα,φk f (x) ≤ CbBMO Mα,φk f (x).

≤

Notice that, by Jensen’s inequality 

   1 x+2h  ∞ k α−1 α−1 I3 ≤ (b(t) − bJ ) f (t) (t − y) − (t − (x + 2h)) dt dy  h x x+4h j+1
∞
  1 x+2h  x+2 h ≤ |b(t) − bJ |k |f (t)| (t − y)α−1 − (t − (x + 2h))α−1  dtdy. h x j j=2 x+2 h Now, by using the mean value theorem we have that
x+2j+1 h ∞  α−1 j(α−2) I3 ≤ Ch 2 |b(t) − bJ |k |f (t)| dt. x+2j h

j=2

Let Jj = [x + 2j h, x + 2j+1 h]. Then I3

≤

Chα−1

∞ 

2j(α−2)

+Ch

x+2j+1 h

x+2j h

j=2 α−1




∞ 

2

j(α−2)

|b(t) − bJj |k |f (t)| dt k

|bJj − bJ |

j=2




x+2j+1 h

x+2j h

|f (t)| dt = I + II.

Observe that from the generalized H¨older’s inequality and (2.12) we obtain I

≤ Chα

∞  j=2

2j(α−1) ||b − bJj ||kφ,J ˜ j ||f ||φk ,Jj

≤ C||b||kBMO

∞  j=2

+ 2−j (2j h)α ||f ||φk ,Jj ≤ C||b||kBMO Mα,φ f (x). k

On the other hand, using (2.9) it is easy to see that  
∞ x+2j+1 h  α−1 j(α−2) II ≤ Ch 2 |f (t)| dt (j + 1)k ||b||kBMO j=2

≤

x+2j h

C||b||kBMO Mα+ f (x).

Putting together the above estimates we are done.




Proof of Theorem 1.2. We shall prove the theorem proceeding by induction on k. As we mentioned in the introduction, the case k = 0 was proved in [11]. So let us assume that the theorem is true for all j ≤ k − 1 and let us prove the case j = k. Applying Theorem 4 in [12] and Lemma 4.1 we have that, for every δ small enough

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and any with δ < < 1, +,k ||Iα,b f ||p,w

+,k +,# +,k + ≤ ||M(δ) (Iα,b f )||p,w ≤ C||M(δ) (Iα,b f )||p,w

≤ C

k−1  j=0

+,j + + k ||b||k−j BMO ||M( ) (Iα,b f )||p,w + C||b||BMO ||Mα,φk f ||p,w .

+,k + Notice that the condition M(δ) (Iα,b f ) ∈ Lp (w) in Theorem 4 ([12]) is satisfied. In

+,k fact, observe that we work with functions f such that Iα,b f ∈ Lp (w). On the other + + hand, since w ∈ A∞ , there exists r > 1 such that w ∈ Ar . Then, for all δ > 0 small +,k + enough we have that r < pδ and thus, w ∈ A+p . Therefore M(δ) (Iα,b f ) ∈ Lp (w). δ

Now, we choose > δ such that r < p . So that w ∈ A+p and we get 

p +,j +,j  +,j p +,j + (Iα,b f )||pp,w = [M + (|Iα,b f | ] w ≤ C [|Iα,b f | ] w = C||Iα,b f ||pp,w . ||M( ) R

R

Then, by recurrence and taking into account (2.5), +,k f ||p,w ||Iα,b

≤

C

k−1  j=0

≤

C

k−1  j=0

≤

+,j + k ||b||k−j BMO ||Iα,b f ||p,w + C||b||BMO ||Mα,φk f ||p,w

j + + k ||b||k−j BMO ||b||BMO ||Mα,φj f ||p,w + C||b||BMO ||Mα,φk f ||p,w

+ C||b||kBMO ||Mα,φ f ||p,w . k




Proof of Theorem 1.1. The proof of this theorem is similar to the corresponding one in [2] and follows the lines of P´erez’s articles (see for example [20]), but we include it for the sake of completeness. First, let us observe that we only need to consider the case αp < 1 (see, for example, the beginning of the proof of Theorem 1.2 in [2]). By a duality argument, it is enough to show that



−,k p
− − [(k+1)p] 1−p
|Iα,b f | (Mαp (M ) w) ≤C |f |p w1−p . (4.4) R

R

As mentioned in Section 2, for 0 < α < 1 and 0 ≤ δ < 1 the function (Mα− g)δ   belongs to A+ 1 . Thus, choosing r > p and δ = (p − 1)/(r − 1), 1−r  p
−1
− − − [Mαp ((M − )[(k+1)p] w)(x)]1−p = [Mαp ((M − )[(k+1)p] w)(x)] r−1 ∈ A− r ⊂ A∞ . Applying Theorem 1.2 with the orientation reversed we get


−,k − |Iα,b f (x)|p [Mαp ((M − )[(k+1)p] w)(x)]1−p dx


− − f (x)]p [Mαp ((M − )[(k+1)p] w)(x)]1−p dx, ≤ C [Mα,φ k

(4.5)

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and then, by Theorem 3.1, it is enough to show that



− − p
1−p
f (x)] [M w(x)] dx ≤ C |f (x)|p w(x)1−p dx. [Mα,φ αp,φ[(k+1)p] k Defining g = f w−1/p , the above inequality may be stated as


− − 1/p p
1−p
[Mα,φ (gw )(x)] [M w(x)] dx ≤ C |g(x)|p dx. αp,φ[(k+1)p] k Now, we shall use that, for large t, φ−1 k (t) ≈

t [log(e + t)]k

=


t1/p × t1/p [log(e + t)](p−1+ )/p k+(p−1+ )/p [log(e + t)]

= ψk−1 (t) × ϕ−1 (t),





where ψk (t) ≈ tp [log(e + t)](k+1)p−1+ and ϕ(t) ≈ tp [log(e + t)]−(1+(p −1) ) (see [14]). Thus, by (2.3), (x − a)α gw1/p φk ,(a,x) ≤ C(x − a)α gϕ,(a,x)w1/p ψk ,(a,x) . Choosing > 0 so that (k + 1)p − 1 + = [(k + 1)p] we have that  1/p . (x − a)α gw1/p φk ,(a,x) ≤ Cgϕ,(a,x) (x − a)αp wφ[(k+1)p] ,(a,x) Therefore − − (gw1/p )(x) ≤ CMϕ− g(x)[Mαp,φ w(x)]1/p . Mα,φ k [(k+1)p]

Moreover, since ϕ satisfies condition Bp
(that is, there is a positive constant c ∞ such that c tϕ(t) p
+1 dt < ∞), applying Theorem 1.7 in [17] we get that


− − 1/p p
1−p
[Mα,φk (gw )(x)] [Mαp,φ[(k+1)p] w(x)] dx ≤ C |Mϕ− g(x)|p dx

≤ C |Mϕ g(x)|p dx

≤ C |g(x)|p dx, where Mϕ g(x) = supx∈I ||g||ϕ,I . This concludes the proof of the theorem.




Remark 4.2. Observe that from the proof of Theorem 1.1 we can obtain the following sharper inequality

+,k − |Iα,b f (x)|p w(x)dx ≤ Cbkp |f (x)|p Mαp,φ w(x)dx, BMO η R

R

with η = (k + 1)p− 1 + ε, ε > 0, and where the constant C depends on ε. In fact, to
− see this we only need to show that (Mαp,φ w)1−p ∈ A− ∞ . Let us sketch the proof. η

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− − First, notice that Mαp,φ w ≈ [Mα,ψ (w1/p )]p , where ψη (t) = tp [log(e + t)]η ≈ η η − (w1/p )]δ ∈ A+ φη (tp ). Then, if we prove that [Mα,ψ 1 for any δ ∈ (0, 1), we will get η





− − w]1−p ≈ [Mα,ψ (w1/p )]−p ∈ A− that [Mαp,φ ∞ since η η





− − − [Mα,ψ (w1/p )]−p = {[Mα,ψ (w1/p )]δ }1−(1+p /δ) ∈ A− 1+p
/δ ⊂ A∞ . η η − (w1/p )]δ ∈ A+ Observe that, the fact [Mα,ψ 1 follows trivially from the inequalities η − − C1 Mα,ψ w(x) ≤ Mα− (Mψ−η w)(x) ≤ C2 Mα,ψ w(x). η η

(4.6)

To prove the first inequality of (4.6) we define the maximal operator − Nα,ψ w(x) = sup |I − |α ||w||ψη ,I − , η I=(a,x)

−

p

where I = (a, x − |I|/2 ). As in Lemma 3.2 we can prove that there exists a constant C such that − − Mα,ψ w(x) ≤ CNα,ψ w(x). (4.7) η η In fact, notice that ψ0p (2t) = 2p ψ0p (t), ψη (2t) ≤ 2p (log 2)η ψ0 (t) + 2p ψη (t) for 0 < η ≤ 1, and, in general, if k < η ≤ k + 1, with k ∈ N,   k k−1    ψj (t) + ψη+j−k (t) + 2p ψη (t). ψη (2t) ≤ C   j=0

j=0

Then, following the arguments in the proof of Lemma 3.2 with γk = 2 and using that αp ∈ (0, 1) we get (4.7). Now, let I + = I \ I − . Then
||w||ψη ,(a,y) dy |I − |α ||w||ψη ,I − ≤ |I − |α |I + |−1 +
I ≤ 2p |I|α−1 Mψ−η w(y) dy ≤ 2p Mα− (Mψ−η w)(x). I

Putting together the above inequalities and (4.7) we get the desired inequality. The second inequality in (4.6) follows as in the proof of (3.1). In fact, taking into account (2.7), we only need to show that

1 1 − M w≤ Mψη w ≤ C||w||ψη ,I , |I| I ψη |I| I for any function w with support in I. The last inequality follows, with standard arguments, by using a weak type inequality of Mψη (see for example [2]).

5. Proof of Theorem 1.4 Proof. Without loss of generality we may assume that f ≥ 0, f ∈ L1 (R) and ||b||BMO = 1. Let λ > 0 and let Ij = (aj , bj ) be the connected components of

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Ω = {x ∈ R : M + f (x) > λ}. Then Ω = ∪j Ij

and

467




1 |Ij |

f =λ

(5.1)

Ij

 x+h (see for example [6] p. 423). If x ∈ / Ω, then h1 x f ≤ λ for all h > 0, and therefore f (x) ≤ λ a.e x ∈ R \ Ω. Let us write f = g + h, with g defined by  f (x), x∈R\Ω g(x) = x ∈ Ij fIj ,   1 where fI = |I| j hj (x), with hj (x) = (f (x) − fIj )χIj (x). I f and h(x) = Observe that g(x) ≤ λ, a.e.. Let us define ˜ = ∪j (I − ∪ Ij ) = ∪j I˜j , Ω j where Ij− = (cj , aj ) and |Ij− | = |Ij |. We will use the notation w∗ (x) = w(x)χR\Ω˜

and wj (x) = w(x)χR\I˜j .

Now, we prove the theorem proceeding by induction on k. We start by proving the case k = 0. Notice that ˜ : |Iα+ g(x)| > λ/2}) + w(Ω) ˜ w({x ∈ R : |Iα+ f (x)| > λ}) ≤ w({x ∈ R \ Ω ˜ : |I + h(x)| > λ/2}) + w({x ∈ R \ Ω α = I + II + III. Given ε > 0 we choose p such that 1 < p < 1 + ε. We apply (1.3) with δ = ε + 1 − p > 0. Then we have that
C I≤ p |I + g(x)|p w∗ (x) dx λ R α

C C − − ∗ ≤ p [g(x)]p Mαp,φ w (x) dx ≤ g(x)Mαp,φ w∗ (x) dx ε ε λ R λ R

C C − ∗ ≤ f (x)Mαp,φ w (x) dx + fI M − wj (x) dx. ε λ R\Ω λ j Ij j αp,φε It is clear that we only have to estimate the second term. By (2.8) we get that
Ij

− fIj Mαp,φ wj (x) dx ε


≤

 f (x)dx

Ij


≤

f (x) dx


≤



Ij

Ij

1 |Ij |


Ij

− Mαp,φ wj (x) dx ε

− inf Mαp,φ wj (z) ε

z∈Ij

− f (x)Mαp,φ wj (x) dx. ε

(5.2)

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Hence, I ≤

C λ

IEOT



− f (x)Mαp,φ w(x) dx. Now, we shall estimate II. Notice that ε  ˜ = w(∪j I˜j ) ≤ II = w(Ω) [w(Ij− ) + w(Ij )]. R

j

For each j we have w(Ij− )

w(Ij− ) 1 = |Ij− | λ


Ij

C f (x) dx ≤ λ




f (x)M − w(x) dx.

Ij

If we now use that M + is of weak-type (1,1) with respect to the pair of weights (w, M − w) (see [26] or [10]) we get that
 C w(Ij ) = w(Ω) ≤ f (x)M − w(x) dx. λ R j   C Then II ≤ λ R f (x)M − w(x) dx. To estimate III we use the fact that Ij hj = 0 and we obtain

C C + III ≤ |I hj (x)|w(x) dx ≤ |I + hj (x)|wj (x) dx λ j R\Ω˜ α λ j R\I˜j α 
  
hj (y) C  cj  hj (y)  = − dy  wj (x) dx   λ j −∞  Ij (y − x)1−α (aj − x)1−α  
∞
   1 C 1   wj (x) dx dy, |hj (y)| − ≤   1−α 1−α λ j Ij (y − x) (aj − x) m=0 Im,j where Im,j = (aj − 2m+1 rj , aj − 2m rj ) with rj = |Ij |. Now, using the mean value theorem for each y ∈ Ij we get that  

  1 1 rj   wj (x) dx ≤ C − wj (x) dx  1−α (aj − x)1−α  (2m rj )2−α Im,j Im,j (y − x)
y rj ≤ C m 2−α wj (x) dx (2 rj ) aj −2m+1 rj ≤ C2−m Mα− wj (y). Then


C III ≤ |hj (y)|Mα− wj (y) dy λ j Ij

C C − f (y)Mα wj (y) dy + fI M − wj (y) dy. ≤ λ j Ij λ j Ij j α

Using (2.8) as in (5.2) in the second term of the last inequality,

C C III ≤ f (y)Mα− wj (y) dy ≤ f (y)Mα− w(y) dy. λ j Ij λ R

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Therefore, collecting the estimates for I, II and III and using (2.6) we get the theorem in the case k = 0. Let now k ∈ N and suppose that the theorem is true for j < k. Then with the same notation as in the proof of the case k = 0, +,k ˜ : |I +,k g(x)| > λ/2}) + w(Ω) ˜ f (x)| > λ}) ≤ w({x ∈ R \ Ω w({x ∈ R : |Iα,b α,b

˜ : |I +,k h(x)| > λ/2}) + w({x ∈ R \ Ω α,b = I + II + III. ε Given ε > 0 we choose p such that 1 < p < 1 + k+1 and we apply (1.3) with δ = ε − (k + 1)(p − 1) > 0. Then, as in the case k = 0, we obtain that

I≤

C λ


R

− f (x)Mαp,φ w(x) dx k+

and II ≤

C λ




f (x)M − w(x) dx.

R

To estimate III we write 

+,k Iα,b hj (x) =

j



(b(x) − bIj )k Iα+ hj (x) +

j

+



  Iα+ (b − bIj )k hj (x)

j k−1 

+,l Ck,l Iα,b



 (b − bIj )k−l hj (x).

j

l=1

The above decomposition follows from (4.1) as in [21]. Then ˜ :| III ≤ w({x ∈ R \ Ω



(b(x) − bIj )k Iα+ hj (x)| >

j

˜ :| + w({x ∈ R \ Ω

 j

˜ :| + w({x ∈ R \ Ω

k−1 

  λ Iα+ (b − bIj )k hj (x)| > }) 6 +,l Ck,l Iα,b

b

 j

l=1 a

λ }) 6

c

= (III) + (III) + (III) .

 λ (b − bIj )k−l hj (x)| > }) 6

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In a similar way as in the estimate of III for k = 0, we get that
C (III)a ≤ |b(x) − bIj |k |Iα+ hj (x)|w(x) dx λ j R\Ω˜  


  C hj (y) hj (y)  k ≤ |b(x) − bIj |  dy − dy  w(x) dx 1−α   Ij (y − x)1−α λ j R\Ω˜ Ij (aj − x)

  C (y − x)α−1 − (aj − x)α−1  |b(x) − bIj |k wj (x) dx dy ≤ |hj (y)| λ j Ij R\I˜j
∞
   C (y − x)α−1 − (aj − x)α−1  |b(x) − bIj |k wj (x) dx dy, |hj (y)| ≤ λ j Ij m=0 Im,j where Im,j = (aj − 2m+1 rj , aj − 2m rj ). Using again the mean value theorem, we get  
  1 1   |b(x) − bIj |k wj (x) dx −  (y − x)1−α  1−α (a − x) j Im,j
rj ≤ C m 2−α |b(x) − bIm,j |k wj (x) dx (2 rj ) Im,j
rj k + C m 2−α |bIj − bIm,j | wj (x) dx (2 rj ) Im,j = (III)a1 + (III)a2 .

1/k

By the generalized H¨older’s inequality with φk (t) = t[log(e+t)]k and φ˜k (t) ≈ et and using (2.12) we get

(III)a1 ≤ C

rj ||b||kBMO ||wj ||φk ,Im,j m (2 rj )1−α

≤C

1 M − wj (y), 2m α,φk

for all y ∈ Ij . Now, applying (2.9) we get that (III)a2 ≤ C

(m + 1)k − Mα wj (y), 2m

for all y ∈ Ij . Then  ∞ 
∞  1  (m + 1)k − C − (III) ≤ |hj (y)| M wj (y) + Mα wj (y) dy λ j Ij 2m α,φk 2m m=0 m=1

C C − − |hj (y)|Mα,φ w (y) dy ≤ f (y)Mα,φ w(y) dy, ≤ j k k λ j Ij λ R a

,

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where the last inequality follows as in the estimation of III in the case k = 0. To estimate (III)b we shall use the case k = 0 and we obtain
C − (III)b ≤ |b(x) − bIj |k |f (x) − fIj |[Mαp,φ wj (x) + M − wj (x)] dx ε λ j Ij
C − − [inf Mαp,φ w + inf M w ] |b(x) − bIj |k f (x)dx ≤ j j ε Ij λ j Ij Ij
C − − + [inf Mαp,φ w + inf M w ] |b(x) − bIj |k fIj dx j j ε Ij λ j Ij Ij =(III)b1 + (III)b2 . To estimate (III)b2 we will use (2.8) and (2.10). Then we have

C 1 − − k (III)b2 ≤ [inf Mαp,φ w + inf M w ] |b(x) − b | f (y)dydx j j Ij ε Ij λ j |Ij | Ij Ij Ij

C 1 − |b(x) − bIj |k dx f (y)[Mαp,φ wj (y) + M − wj (y)]dy ≤ ε λ j |Ij | Ij Ij
C − ||b||kBMO f (y)[Mαp,φ wj (y) + M − wj (y)]dy ≤ ε λ j Ij
C − ≤ f (y)[Mαp,φ w(y) + M − w(y)]dy. ε λ R To estimate (III)b1 we use again the generalized H¨older’s inequality for φk and φ˜k
C − − (III)b1 ≤ [inf Mαp,φ w + inf M w ] |b(x) − bIj |k f (x)dx j j ε Ij λ j Ij Ij ≤ ≤

C [inf M − wj + inf M − wj ]|Ij |||(b − bIj )k ||φ˜k ,Ij ||f ||φk ,Ij Ij λ j Ij αp,φε

 C − ||b||kBMO [inf Mαp,φ wj + inf M − wj ]|Ij |||f ||φk ,Ij . ε Ij Ij λ j

Now, the inequality ||f ||φk ,I

  
|f | µ ≤ inf µ + φk , µ>0 |I| I µ

(see [7] and also [23]) and (5.1) gives us that  


|f | 1 µ 1 |Ij |||f ||φk ,Ij ≤ |Ij | inf µ + φk µ>0 λ λ |Ij | Ij µ      



f f f 1 φk f+ φk φk ≤ |Ij | + = ≤2 . λ λ Ij λ λ Ij Ij Ij

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Then

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 f (x) − wj (x) + M − wj (x)]dx [Mαp,φ ε λ I j j  
f (x) − ≤ C φk w(x) + M − w(x)]dx. [Mαp,φ ε λ

(III)b1 ≤ C




φk

To conclude the proof we only have to estimate (III)c where we will use the induction argument: ˜ :| (III)c ≤ w({x ∈ R \ Ω

k−1 

+,l Ck,l Iα,b

l=1

˜ :| + w({x ∈ R \ Ω

k−1 

  λ }) (b − bIj )k−l f χIj (x)| > 12 j

+,l Ck,l Iα,b

 j

l=1

=

(III)c1

+

 λ }) (b − bIj )k−l fIj χIj (x)| > 12

(III)c2 .

By induction, (III)c1 ≤C

l=1

≤C

R

φl 

k−1 
l=1

≤C



k−1 


j

k−1  l=1

j

Ij

[inf Ij

f (x)  λ 

φl

j

 − (b(x) − bIj )k−l χIj (x) [Mφ−l w∗ (x) + Mαp,φ w∗ (x)]dx l+ε

 f (x) − k−l (b(x) − bIj ) wj (x)]dx [Mφ−l wj (x) + Mαp,φ l+ε λ

Mφ−l wj

+ inf Ij

Now observe that φ−1 k (t) ≈

− Mαp,φ l+ε

t [log(e+t)]k




wj ]

Ij

φl

 f (x) k−l (b(x) − bIj ) dx. λ

and φ˜k−1 (t) ≈ [log(e + t)]k . Then

−1 φk−1 (t)φ˜−1 k−l (t) ≤ C φl (t).

Using (2.4), (2.11) and (5.1) we get that  
f (x) (b(x) − bIj )k−l dx φl λ Ij  

  f (x) φk φ˜k−l (b(x) − bIj )k−l dx dx + C ≤C λ Ij Ij  
f (x) ≤C φk dx + C|Ij | λ Ij    


f (x) f (x) f (x) dx ≤ C φk φk ≤C dx + C dx. λ λ λ Ij Ij Ij

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Commutators of One-sided Fractional Integrals

473

Then (III)c1

≤C

k−1  l=1

≤C

j

[inf Ij



k−1 
l=1

j

Mφ−l wj

Ij

φk

+ inf

f (x) λ

Ij



− Mαp,φ l+ε




wj ]

Ij

φk

f (x) λ

 dx

− w(x)]dx. [Mφ−l w(x) + Mαp,φ l+ε

(III)c2

The term is controlled in the same way, just observe that by (2.11) and Jensen’s inequality    
fIj fIj (b(x) − bIj )k−l dx ≤ C|Ij |φk φl + C|Ij | λ λ Ij  
f (x) φk ≤C dx. λ Ij Then, using (2.5), (III)c ≤ C




φk

f (x) λ



− w(x)] dx. [Mφ−k−1 w(x) + Mαp,φ k−1+ε

By (2.5) again and (2.6) we have that  
f (x) − III ≤ C φk w(x) + Mφ−k w(x)] dx. [Mαp,φ k+ε λ Now, putting together the estimates of I, II and III, we are done.




References [1] K. F. Andersen and E. T. Sawyer, Weighted norm inequalities for the RiemannLiouville and Weyl fractional integral operators, Trans. Amer. Math. Soc. 308 (1988) 547-558. [2] A. Bernardis, S. Hartzstein and G. Pradolini, Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type, J. Math. Anal. Appl. 322 (2006) 825-846. [3] M. J. Carro, C. P´erez, F. Soria and J. Soria, Maximal functions and control of weighted inequalities for the fractional integral operator, Indiana Univ. Math. J. 54 (2005) 627-644. [4] D. Cruz-Uribe, SFO, J.M. Martell and C. P´erez, Extrapolation from A∞ weights and applications, J. Funct. Anal. 213 (2004) 412-439. [5] J. Garc´ıa-Cuerva, E. Harboure, C. Segovia and J.L. Torrea, Weighted norm inequalities for commutators of strongly singular integrals, Indiana Univ. Math. J. 40 (1991) 1397-1420. [6] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York, Heidelberg and Berlin,1965.

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[7] M. A. Krasnosel’ski˘ı and Ya. B. Ruticki˘ı, Convex Functions and Orlicz Spaces, P. Noordhoff, Groningen, 1961. [8] M. Lorente and M.S. Riveros, Weighted inequalities for commutators of one-sided singular integrals, Comment. Math. Univ. Carolinae 43 (1) (2002) 83-101. [9] M. Lorente and M.S. Riveros, Weights for commutators of the one-sided discrete square function, the Weyl fractional integral and other one-sided operators, Proc. Royal Soc. Edinb. 135A, (2005), 845-862. [10] F.J. Mart´ın-Reyes, P. Ortega and A. de la Torre, Weighted inequalities for one-sided maximal functions, Trans. Amer. Math. Soc. 319 (2) (1990) 517-534. [11] F.J. Mart´ın-Reyes, L. Pick and A. de la Torre, A+ ∞ condition, Canad. J. Math. 45 (1993) 1231-1244. [12] F.J. Mart´ın-Reyes and A. de la Torre, One Sided BM O Spaces, J. London Math. Soc. 2 (49) (1994) 529-542. [13] R. O’Neil, Fractional integration in Orlicz spaces, Trans. Amer. Math. Soc. 115, (1963) 300-328. [14] R. O’Neil, Integral transforms and tensor products on Orlicz spaces and Lp,q spaces, J. Anal. Math. 21 (1968) 1-276. [15] C. P´erez, Two weighted norm inequalities for Riesz potentials and uniform Lp weighted Sobolev inequalities, Indiana Univ. Math. J. 39 (1) (1990) 31-44. [16] C. P´erez, Weighted norm inequalities for singular integral operators, J. London Math. Soc. 49 (1994) 296-308. [17] C. P´erez, On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted Lp -spaces with different weights, Proc. London Math. Soc. 71 (3) (1995) 135-157 [18] C. P´erez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal. 128 (1995) 163–185. [19] C. P´erez, Sharp Lp -weighted Sobolev inequalities, An. Inst. Fourier, Grenoble, 45 (1995) 809-824. [20] C. P´erez, Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function, J. Fourier Anal. Appl. 3 (6) (1997) 743-756. [21] C. P´erez, G. Pradolini, Sharp weighted endpoint estimates for commutators of singular integrals, Michigan Math. J 49 (1) (2001) 23–37. [22] C. P´erez and R. Wheeden, Uncertainty principle estimates for vector fields, J. Funct. Anal. 181 (2001) 146-188. [23] M. Rao and Z.D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics 146, Marcel Dekker, Inc., New York, 1991. [24] L. de Rosa, Two weight norm inequalities for fractional one-sided maximal and integral operators, Comment. Math. Univ. Carolin. 47 (1) (2006) 35-46. [25] M. S. Riveros, L. de Rosa and A. de la Torre, Sufficient Conditions for one-sided Operators, J. Fourier Anal. Appl. 6 (2000) 607-621. [26] E. Sawyer, Weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986) 53-61.

Vol. 61 (2008)

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475

[27] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, New York, 1986. Ana L. Bernardis IMAL-CONICET G¨ uemes 3450 3000 Santa Fe Argentina e-mail: [email protected] Mar´ıa Lorente Departamento de An´ alisis Matem´ atico Facultad de Ciencias Universidad de M´ alaga Spain e-mail: [email protected] Submitted: January 29, 2007 Revised: March 25, 2008

Integr. equ. oper. theory 61 (2008), 477–492 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/040477-16, published online July 25, 2008 DOI 10.1007/s00020-008-1601-x

Integral Equations and Operator Theory

Boundedness of Commutators of Marcinkiewicz Integral with Rough Variable Kernel Yanping Chen and Yong Ding Abstract. In this paper the authors give the L2 (Rn ) (n ≥ 2) boundedness of the commutator of Marcinkiewicz integral with rough variable kernel, which is an extension of the result in [5] . Mathematics Subject Classification (2000). 42B20, 42B25. Keywords. Commutator of singular integral, rough variable kernel, BMO.

1. Introduction Let S n−1 be the unit sphere in Rn (n ≥ 2) with normalized Lebesgue measure dσ. A function Ω(x, z) defined on Rn × Rn is said to be in L∞ (Rn ) × Lq (S n−1 ), q ≥ 1, if Ω(x, z) satisfies the following conditions: (1) for any x, z ∈ Rn and λ > 0, Ω(x, λz) = Ω(x, z);
 1/q (2) ΩL∞ (Rn )×Lq (S n−1 ) := supx∈Rn S n−1 |Ω(x, z  )|q dσ(z  ) < ∞, where z z  = |z| , for any z ∈ Rn \ {0}. In 1955, Calder´ on and Zygmund [1] investigate the L2 boundedness of the singular integral T with variable kernel. They found that these operators connect closely with the problem about the second order linear elliptic equations with variable coefficients, let Ω(x, z  ) ∈ L∞ (Rn ) × Lq (S n−1 ) satisfy  Ω(x, z  ) dσ(z  ) = 0 for any x ∈ Rn . (1.1) S n−1

The singular integral operator with variable kernel is defined by  Ω(x, x − y) f (y) dy. T f (x) = p.v. |x − y|n n R Research supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025).

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In [1], Calder´ on and Zygmund obtained the following result (see also [2]): Theorem A. If Ω(x, z  ) ∈ L∞ (Rn ) × Lq (S n−1 ), q > 2(n − 1)/n, satisfies (1.1), then there is a constant C > 0 such that T f L2 ≤ Cf L2 . On the other hand, it is well known that the commutator of the Calder` onZygmund singular integral operator T and a BM O(Rn ) function plays an important role in characterizing Hardy space H 1 (Rn ) and studying the regularity of the solution of the second order elliptic equations (see [3], [4], for example). To study the interior W 2,2 estimates for nondivergence elliptic second order equation with discontinuous coefficients, in 1991, Chiarenza, Frasca and Longo [3] gave the L2 (Rn ) boundedness of the commutator Tb,k with variable kernel for k = 1, which is defined by  Ω(x, x − y) Tb,k f (x) = p.v. (b(x) − b(y))k f (y) dy, |x − y|n Rn where k ∈ N and b ∈ BM O(Rn ). That is,  1 b∗ := sup |b(y) − bQ |dy < ∞, Q |Q| Q where the supremum is taken over all cubes Q in Rn and bQ = In [3], the authors gave the following result:

1 |Q|

 Q

b(x)dx.

Theorem B. Suppose that Ω(x, z  ) ∈ L∞ (Rn ) × C ∞ (S n−1 ) satisfying (1.1). Then there is a constant C > 0 such that Tb,1 f L2 ≤ Cb∗ f L2 . Notice that a stronger smoothness is assumed on the kernel function Ω in the conditions of Theorem B. On the other hand, in 2004, Ding, Lin and Shao [5] defined and proved the L2 boundedness of the Marcinkiewicz integral operator with variable kernel:  ∞ 1/2 dt |Ft (x)|2 3 µ(f )(x) = t 0 and

 Ft (x) =

|x−y|≤t

Ω(x, x − y) dy. |x − y|n−1

Theorem C. ([5]) If Ω(x, z  ) ∈ L∞ (Rn ) × Lq (S n−1 ), q > 2(n − 1)/n, satisfies (1.1), then there is a constant C > 0 such that µf L2 ≤ Cf L2 . In this paper, we will discuss the L2 boundedness of commutator of the rough Marcinkiewicz integral with variable kernel. For k ∈ N and b ∈ BM O(Rn ), the k-th order commutator of the Marcinkiewicz integral µ with variable kernel is defined by  ∞ 1/2 2 dt |Ft;b,k (x)| 3 µb,k f (x) = t 0

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Boundedness of Commutators

and

 Ft;b,k (x) =

|x−y|≤t

479

Ω(x, x − y) (b(x) − b(y))k dy. |x − y|n−1

When k = 0, the Marcinkiewicz integral µ with convolution kernel, that is, Ω(x, z  ) = Ω(z  ), was defined first by Stein in [8]. Historically, the k-th order commutator µb,k with convolution kernel was studied by many authors (see [10], [6], for example). We will show that the condition Ω(x, z  ) ∈ L∞ (Rn ) × Lq (S n−1 ) is also sufficient for the L2 (Rn ) boundedness for the commutator µb,k . Theorem 1.1. Suppose that Ω(x, z  ) ∈ L∞ (Rn ) × Lq (S n−1 ) satisfies (1.1) for q > 2(n − 1)/n, and b ∈ BM O. Then there is a constant C > 0 such that µb,k f L2 ≤ Cbk∗ f L2 . Remark 1.2. For k = 0, Theorem 1.1 is just Theorem C. Hence our result is also an extension of the main result in [5].

2. Some lemmas In this section, we give some lemmas which will be used in the proof of Theorem 1.1. Lemma 2.1. ([9]) Let n ≥ 2, and f ∈ L1 (Rn ) ∩ L2 (Rn ) has the form f (x) = f0 (|x|)P (x), where P (x) is a solid spherical harmonic of degree m. Then the Fourier transform of f has the form f = F0 (|x|)P (x), where  ∞ F0 (r) = 2πi−m r−[(n+2m−2)/2] f0 (s)J(n+2m−2)/2 (2πrs)s(n+2m)/2 ds, 0

r = |x|, and Jv is the Bessel function. Lemma 2.2. Suppose that 0 < β < 1, α ∈ Z, m ∈ N. Denote by Hm the space of surface spherical harmonics of degree m on S n−1 with its dimension Dm . m {Ym,j }D j=1 denotes the normalized complete system in Hm . Let σα,t,m,j (x) = (2α t)−1

Ym,j (x ) χ{0<|x|≤2α t} (x). |x|n

Then −λ−1+β/2 |σ
min{2α t|ξ|, |2α tξ|−β/2 }|Ym,j (ξ  )|, α,t,m,j (ξ)| ≤ Cm

(2.1)

−λ−1 |Ym,j (ξ  )|, |σ
α,t,m,j (ξ)| ≤ Cm

(2.2)

α |∇σ
α,t,m,j (ξ)| ≤ C2 t,

(2.3)

where λ =

n−2 2

and ξ  =

ξ |ξ| .

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Proof. First we give the estimate of |∇σ
α,t,m,j (ξ)|. Since  Ym,j (x ) −2πix·ξ α −1 e dx σ
α,t,m,j (ξ) = (2 t) n−1 0<|x|≤2α t |x| α  2 t  = (2α t)−1 Ym,j (x )e−2πirx ·ξ dσ(x )dr. 0

S n−1

Since Ym,j L2 (S n−1 ) = 1, then we get |∇σ
α,t,m,j (ξ)|

≤ (2α t)−1 ≤



2α t



|Ym,j (x )| dσ(x )rdr

0 S n−1 C2α tYm,j L2 (S n−1 )

≤ C2α t.

Set Pm,j (x) = Ym,j (x )|x|m . Then Pm,j is a solid spherical harmonic of degree m and we have σα,t,m,j (x) = (2α t)−1 |x|−n+1−m Pm,j (x)χ{|x|≤2α t} (x). Obviously, Ψ0 (|x|) := (2α t)−1 |x|−n+1−m χ{|x|≤2α t} (x) is a radial function in x. Using Lemma 2.1, we have  m σ
α,t,m,j (ξ) = ψ0 (|ξ|)Pm,j (ξ) = Ym,j (ξ )|ξ| ψ0 (|ξ|),

where ψ0 (r)

= 2πi−m r−[(n+2m−2)/2]

 0

∞

(2.4)

Ψ0 (s)J(n+2m−2)/2 (2πrs)s(n+2m)/2 ds,

= 2πi−m (2α t)−1 r−[(n+2m−2)/2]  2α t × s−n+1−m J(n+2m−2)/2 (2πrs)s(n+2m)/2 ds 0  2π2α tr J(n+2m−2)/2 (s) = (2π)n/2 i−m r−m (2π2α tr)−1 ds. s(n−2)/2 0 From this and (2.4) we have n/2 −m σ
i (2π2α t|ξ|)−1 Ym,j (ξ  ) α,t,m,j (ξ) = (2π)



2π2α t|ξ| 0

J(n+2m−2)/2 (s) ds. (2.5) s(n−2)/2

Let λ = (n − 2)/2. First we estimate |σ
α,t,m,j (ξ)|. Now we consider three special cases, namely m+λ m+λ , 3◦ 2α t|ξ| ≥ . 4π 4π Case 1◦ . When 2α t|ξ| ≤ 1, the classical formula (see [11, p.48])  1   (s/2)m+λ   |Jm+λ (s)| =  (1 − r2 )m+λ−1/2 eitr dr Γ(m + λ + 1/2)Γ(1/2) −1 (s/2)m+λ . ≤C Γ(m + λ + 1/2) 1◦ 2α t|ξ| ≤ 1,

2◦ 1 < 2α t|ξ| <

Applying Stirling formula, we get for x > 1 √ √ 2πxx−1/2 e−x ≤ Γ(x) ≤ 2 2πxx−1/2 e−x .

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Thus

Boundedness of Commutators

481

 Jm+λ (s)  ds (2π2 t|ξ|) sλ 0  2π2α t|ξ| C ≤ (2π2α t|ξ|)−1 m+λ sm ds 2 Γ(m + λ + 1/2) 0 1 C · (2π2α t|ξ|)m ≤ m+λ 2 Γ(m + λ + 1/2) m + 1 2α t|ξ| (2π2α t|ξ|)m−1 √ ≤C m + 1 2m+λ 2π(m + λ + 1/2)m+λ e−m−λ (2π)m em+λ ≤ C2α t|ξ| m+λ · 2 (m + 1)(m + λ + 1/2)m+λ em+λ (2π)m ≤ C2α t|ξ|m−λ−1 m+λ · 2 (m + λ + 1/2)m (2eπ)m ≤ C2α t|ξ|m−λ−1 (e/2)λ (2m + 2λ)m −λ−1 α ≤ Cm 2 t|ξ|.   

−1 

α

Case 2◦ . When 1 < 2α t|ξ| <

2π2α t|ξ|

m+λ 4π ,

  α −1  (2π2 t|ξ|) 

we have

 Jm+λ (s)  ds sλ 0  2π2α t|ξ| C ≤ (2π2α t|ξ|)−1 m+λ sm ds 2 Γ(m + λ + 1/2) 0 1 C · (2π2α t|ξ|)m ≤ m+λ 2 Γ(m + λ + 1/2) m + 1 (2π2α t|ξ|)m 1 √ ≤C m + 1 2m+λ 2π(m + λ + 1/2)m+λ e−m−λ 1 (m + λ)m em+λ ≤ C m+λ m · 2 2 (m + λ + 1/2)m+λ (m + 1) em eλ ≤ C(m + λ)−λ−1 m λ 4 2 ≤ C(m + λ)−λ−1 ≤ Cm−1−λ+β/2 (2α t|ξ|)−β/2 . 2π2α t|ξ|

α Case 3◦ . When m+λ 4π < 2 t|ξ|, we have (see [1, Lemma 2])  b   Jm+λ (s)   ds ≤ Cm−λ , for any 0 ≤ b ≤ ∞.  sλ 0 Then we have   2π2α t|ξ|   Jm+λ (s)  α −1  (2π2 t|ξ|)  ds ≤ Cm−1−λ+β/2 (2α t|ξ|)−β/2 . sλ

(2.6)

0

Thus from (2.5) and the above three cases , we get −λ−1+β/2 |σ
min{2α t|ξ|, |2α tξ|−β/2 }|Ym,j (ξ  )|. α,t,m,j (ξ)| ≤ Cm −λ−1 |Ym,j (ξ  )|. |σ
α,t,m,j (ξ)| ≤ Cm




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Lemma 2.3. For 0 < δ < ∞, m ∈ N and j = 1, . . . , Dm , take Γδ,m,j ∈ C0∞ (Rn ) such that supp(Γδ,m,j ) ⊂ {δ/2 ≤ |ξ| ≤ 2δ}. Let Tδ,m,j be the multiplier operators defined by  T
j = 1, . . . , Dm . δ,m,j f (ξ) = Γδ,m,j (ξ)f (ξ), Moreover, for b ∈ BM O and k ∈ N, denote by Tδ,m,j; b,k f (x) = Tδ,m,j ((b(x) − b(·))k f )(x) the k-th order commutator of Tδ,m,j and  1/2 Dm 2 (Tδ,m,j; b,k f (x)) . Tδ,m; b,k f (x) = j=1

If for some constant 0 < β < 1, Υδ,m,j satisfies the following conditions: |Γδ,m,j (ξ)| ≤ Cm−λ−1+β/2 min{δ, δ −β/2}|Ym,j (ξ  )|,

(2.7)

|Γδ,m,j (ξ)| ≤ Cm−λ−1 |Ym,j (ξ  )|,

(2.8)

|∇Γδ,m,j (ξ)| ≤ C,

(2.9)

then for any fixed 0 < v < 1, there exists a positive constant C = C(n, k, v) such that Tδ,m; b,k f L2 ≤ Cm(−1+β/2)v min{δ v , δ −βv/2 }bk∗ f L2 . Proof. We may assume that b∗

= 1. Taking a C0∞ (Rn ) radial function φ, such that suppφ ⊂ {1/2 ≤ |x| ≤ 2} and l∈Z φ(2−l |x|) = 1 for any |x| > 0.

Denote φ0 (x) = 0l=−∞ φ(2−l |x|) and φl (x) = φ(2−l |x|) for positive integer l. ∨ Then φ0 ∈ S(Rn ) and suppφ0 ⊂ {x : 0 < |x| ≤ 2}. Let Kδ,m,j (x) = (Γδ,m,j ) (x), l (x) = Kδ,m,j (x)φl (x) for the inverse Fourier transform of Γδ,m,j . Denote Kδ,m,j l = 0, 1, . . . , we have ∞ l Kδ,m,j (x) = Kδ,m,j (x). l=0 l l l and Tδ,m,j;b,k the convolution operator with kernel Kδ,m,j and Denote by Tδ,m,j l and b, respectively. Then by the Minkowski the k-th order commutator of Tδ,m,j inequality  Dm  ∞ 2 1/2   l Tδ,m;B,k f L2 = Tδ,m,j;b,k f (x) dx  R j=1 l=0 Dm  ∞  n

≤ =

l=0 ∞

Rn j=1

2 1/2  l  Tδ,m,j;b,k f (x) dx

l Tδ,m;b,k f L2 ,

l=0

where l Tδ,m;b,k f (x) =

 Dm j=1

2  δ,m,j;b,k f (x)

 l T

1/2 .

(2.10)

Vol. 61 (2008)

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483

By (2.10) it is easy to see that for any fixed 0 < v < 1, if there exists γ > 0 such that l f L2 ≤ Cm(−1+β/2)v 2−lγ min{δ v , δ −βv/2 }f L2 , (2.11) Tδ,m;b,k ∞ n then Lemma 2.3 follows. To do this, for l ≥ 0, we decompose R = d=−∞ Qd , where Qd s are non-overlapping cubes with side length 2l . Set fd = f χQd . Then ∞

f (x) =

a.e. x ∈ Rn .

fd (x),

d=−∞ l l ) ⊂ {x : |x| ≤ 2l+2 }. It is obvious that supp(Tδ,m,j;b,k fd ) ⊂ Since supp(Kδ,m,j l ∞ 10nQd, and that the supports of {Tδ,m,j;b,k fd }d=−∞ have bounded overlaps. So we have the following almost orthogonality property: ∞

l Tδ,m,j;b,k f 2L2 ≤ C

l Tδ,m,j;b,k fd 2L2 .

d=−∞

Thus

l Tδ,m;b,k f 2L2

Dm = j=1 T l f 2 2

∞ δ,m,j;b,k

Dm Ll ≤ C d=−∞ j=1 Tδ,m,j;b,k fd 2L2

∞ l = C d=−∞ Tδ,m;b,k fd 2L2 .

Hence, it is suffices to verify (2.11) for the function f with suppf ⊂ Q, where Q has its side length 2l . Choose ϕ ∈ C0∞ (Rn ), 0 ≤ ϕ ≤ 1, ϕ is identically one  = 200nQ, and b = |Q| −1 b(y) dy. Let on 50nQ, and suppϕ ⊂ 100nQ. Set Q Q Q b = (b(x) − b )ϕ(x). It is easy to see Q

l Tδ,m,j;b,k f (x) =

k

l Ckw bw (x)Tδ,m,j ( bk−w f )(x).

w=0

Denote l Tδ,m f (x)

=

 Dm

  l Tδ,m,j f (x)2

1/2 ,

j=1

then we have l Tδ,m;b,k f 2L2

2 Dm   k  w w l bk−w f )(x) dx  C (x)T ( b k δ,m,j   Rn j=1 w=0 2  Dm k   w  l k−w  ≤C f )(x) dx b (x)Tδ,m,j (b 

=

≤C =C

Rn j=1 w=0 k  w

w=0 k w=0

Rn

2

|b (x)|

Dm

(2.12) l |Tδ,m,j ( bk−w f )(x)|2

j=1

l  bw Tδ,m ( bk−w f )2L2 .

dx

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Thus by (2.12) we need only show that for the function f supported in Q with side length 2l l  bw Tδ,m ( bk−w f )L2 ≤ Cm(−1+β/2)v 2−lγ min{δ v , δ −βv/2 }f L2 . (2.13) 

Let us first show that for g ∈ Lq (Rn ), 1 < q  ≤ 2 (hence 2 ≤ q < ∞), 0 < t < 1 (−2+β)(1−t)

2tl

2

2tλ

2

−(1− q )+ q n(1− q ) q ≤ C2− q m δ 2(1−t)
 × min{δ, δ −β } q gLq .

l gLq Tδ,m

(2.14)

l , we have In fact, by the definition of Tδ,m,j  2 1/2   Dm l l |Kδ,m,j (x − y)||g(y)| dy |Tδ,m g(x)| ≤

 ≤

 Dm

Rn

 ≤

Rn

j=1

j=1  Dm

Rn

l |Kδ,m,j (x − y)|2 2
l |K δ,m,j (x)|

|g(y)| dy

1/2 dx gL1 .

j=1

Since




l
 K δ,m,j (x) = Kδ,m,j ∗ φl (x)

= Rn

Dm

Γδ,m,j (x − y)φl (y) dy,

(2.15)

|Ym,j (x )|2 ∼ m2λ (see [2, p.225, (2.6)]), we get 2 1/2   Dm        ≤ dx gL1  n Γδ,m,j (x − y)φl (y) dy  n R R j=1 1/2    Dm 2 ≤ |Γδ,m,j (x − y)| φl (y) dy dx gL1

by (2.8) and the fact l g(x)| |Tδ,m

1/2

j=1

Rn



Rn

j=1

 Dm

≤ δ/2<|x|<2δ

≤m

−λ−1



|Γδ,m,j (x)|

j=1

δ/2<|x|<2δ

≤ Cm−1 δ n gL1 ,

2

 Dm

1/2



dx φl L1 gL1 2

|Ym,j (x )|

1/2 dx gL1

j=1

i.e.

l gL∞ ≤ Cm−1 δ n gL1 . (2.16) Tδ,m   On the other hand, note that Rn φ(η) dη = φ(0) = 0, then by (2.15) and (2.9) we get 
l  (x)| ≤ |(Γδ,m,j (x − 2−l y) − Γδ,m,j (x))||φ(y)| dy |K δ,m,j Rn   ≤ C2−l ∇Γδ,m,j L∞ |y||φ(y)| dy ≤ C2−l . Rn

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Thus, by the Plancherel theorem and the fact Dm ∼ m2λ (see [2, p.226, (2.8)]), we have  l Tδ,m gL2

≤

Dm

Rn

2
l |K g(ξ)|2 dξ δ,m,j (ξ)| |

1/2

≤ C2−l mλ gL2 .

Applying the Plancherel theorem again, and by (2.15), (2.7) and

D m

|Ym,j (x )|2 ∼

j=1

m2λ , we obtain l Tδ,m g2L2

(2.17)

j=1

 ≤

Dm

Rn



≤C

2
l |K g(ξ)|2 dξ δ,m,j (ξ)| |

j=1 Dm

Rn

|Γδ,m,j ∗ φl (ξ)|2 | g(ξ)|2 dξ

j=1

2 1/2 2

 Dm       ≤C Γδ,m,j (ξ − y)φl (y) dy  | g (ξ)|2 dξ  Rn Rn j=1 2 1/2    Dm 2  ≤C |Γδ,m,j (ξ − y)| |φl (y)| dy | g (ξ)|2 dξ 

Rn

Rn

Rn

Rn


j=1 2 ≤ m−2λ−2+β min{δ, δ −β/2 } 2 1/2    Dm × |Ym,j ((ξ − y) )|2 |φl (y)| dy | g (ξ)|2 dξ ≤ Cm


−2+β

j=1 2 min{δ, δ −β/2} φl 2L1 g2L2 .

That is, l gL2 ≤ Cm(−1+β/2) min{δ, δ −β/2}gL2 . Tδ,m

(2.18)

Hence, by (2.17) and (2.18), for any 0 < t < 1,
1−t β l gL2 ≤ C2−tl mtλ m(−1+ 2 )(1−t) min{δ, δ −β/2} gL2 . Tδ,m

(2.19)

Thus we obtain (2.14) by interpolating between (2.16) and (2.19). Let us return to the proof of (2.13). For 2 < q1 , q2 < ∞ with 1/q1 +1/q2 = 1/2, by (2.14) and the fact 1/σ ≤ C2nl/σ  bw Lσ ≤ Cbw ∗ |Q|

we have

for 1 < σ < ∞,

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Chen and Ding

l  bw Tδ,m ( bk−w f )L2

l ≤  bw Lq1 Tδ,m ( bk−w f )Lq2 − 2tl n(1−

2

)

(−2+β)(1−t)

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−(1−

2

)+ 2tλ

2

2tλ

q2 q2 q2 q2 ≤ C2 q2 δ m 2(1−t)
 × min{δ, δ −β/2 } q2  bw Lq1  bk−w f Lq2 2tl

2

(−2+β)(1−t)

−(1− q )+ q q2 2 2 ≤ C2− q2 δ n(1− q2 ) m 2(1−t)
 × min{δ, δ −β/2 } q2  bw Lq1  bk−w L2q2 /(q2 −2) f L2 2tl

2

(−2+β)(1−t)

2

2tλ

2

−(1− q )+ q n(1− q ) q2 2 2 δ 2 ≤ C2− q2 +nl(1− q2 ) m 2(1−t)
 −β/2 q2 × min{δ, δ } f L2 .

(2.20) For any fixed 0 < v < 1, we choose q2 > 2 and sufficiently close to 2, t > 0 but sufficiently close to 0, such that q2 and t satisfy: 2t/q2 > n(1 − 2/q2 ),

β(1 − t)/q2 > n(1 − 2/q2 ) + vβ/2 + 2tλβ/q2 .

Then there exists γ > 0, independent of l, such that 2−2tl/q2 +nl(1−2/q2 ) = 2−γl . We also can get m(−2+β)(1−t)/q2 −(1−2/q2 )+2tλ/q2 ≤ m(−1+β/2)v . If δ ≥ 1, then by (2.20) l  bw Tδ,m ( bk−w f )L2

≤ Cm(−1+β/2)v 2−lγ δ n(1−2/q2 ) δ −β(1−t)/q2 f L2 ≤ Cm(−1+β/2)v 2−lγ δ −βv/2 f L2 .

If 0 < δ < 1, then by (2.20) l  bw Tδ,m ( bk−w f )L2

≤ Cm(−1+β/2)v 2−lγ δ n(1−2/q2 ) δ 2(1−t)/q2 f L2 ≤ Cm(−1+β/2)v 2−lγ δ v f L2 .

Therefore we get (2.13) and the proof of Lemma 2.3 is complete.




Remark 2.4. When k = 0, Lemma 2.3 holds also. Lemma 2.5. ([7]) Let ψ ∈ C0∞ (n ≥ 1) be a radial function such that suppψ ⊂ {1/2 ≤ |ξ| ≤ 2} and ψ 3 (2−l ξ) = 1, |ξ| = 0. l∈Z

Define the multiplier operator Sl by −l   S l f (ξ) = ψ(2 ξ)f (ξ),

and Sl2 by Sl2 f (x) = Sl (Sl f )(x). For b ∈ BM O and nonnegative integer k, denote 2 ) the k-th order commutator of Sl (resp. Sl2 ). Then for 1 < p < by Sl;b,k (resp. Sl;b,k ∞, 


   2 1/2  (i)  |S f |  p ≤ C(n, k, p)bkBMO f Lp ; l;b,k l∈Z 


 L  2 2 1/2  (ii)  |S f |  ≤ C(n, k, p)bkBMO f Lp . l;b,k l∈Z Lp

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487

3. Proof of Theorem 1.1 As in [2], by a limit argument , we may reduce the proof of Theorem 1.1 to the case of f ∈ C0∞ (Rn ) and Dm

Ω(x, z  ) =

am,j (x)Ym,j (z  )

m≥0 j=1 

is a finite sum. Notice that Ω(x, z ) satisfies (1.1), so a0,j ≡ 0. Denote  1/2 Dm am,j (x) . am (x) = |am,j (x)|2 and bm,j (x) = am (x) j=1 Then

Dm

b2m,j (x) = 1,

(3.1)

j=1

and Ω(x, z  ) =



am (x)



µm,j;b,k f (x) =

0

∞

   

|x−y|≤t

bm,j (x)Ym,j (z  ).

j=1

m≥1

If we write

Dm

2 1/2  dt Ym,j (x − y) k (b(x) − b(y)) f (y)dy  3 , n−1 |x − y| t

then by using H¨ older’s inequality twice and (3.1), we have (µb,k f (x))2 2  ∞    dt Ω(x, x − y) k   = (b(x) − b(y)) f (y)dy   t3 n−1 0 |x−y|≤t |x − y| 2  ∞  Dm   dt Ym,j (x − y) k  = am (x) bm,j (x) (b(x) − b(y)) f (y)dy  3  n−1 |x − y| t 0 |x−y|≤t m≥1  j=1  ≤ a2m (x)m−ε mε m≥1

∞ 

m≥1

2  dt Ym,j (x − y) k × bm,j (x) (b(x) − b(y)) f (y)dy  3 n−1 |x − y| t 0 |x−y|≤t j=1     ∞  Dm ≤ a2m (x)m−ε mε b2m,j (x) 

 

m≥1 Dm 

Dm

m≥1

0

j=1

2  dt Ym,j (x − y) k  (b(x) − b(y)) f (y)dy  t3 n−1 |x − y| j=1 |x−y|≤t     Dm ≤ a2m (x)m−ε · mε (µm,j;b,k f (x))2 , ×

m≥1

  

m≥1

j=1

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where 0 < ε < 1. By [2, p.230, (4.4)] , if we take ε sufficiently close to 1, then 

a2m (x)m−ε



1/2 ≤C

S n−1

m≥1

1/q |Ω(x, z )| dσ(z ) ≤ CΩL∞ (Rn )×Lq (S n−1 ) 

q



(3.2) for q > 2(n − 1)/n. Let µm;b,k f (x) =

 Dm

2

1/2

|µm,j;b,k f (x)|

.

j=1

Applying the Minkowski inequality and (3.2), for q > 2(n − 1)/n and 0 < ε < 1 which close to 1 sufficiently, we get µb,k f 2L2 ≤ CΩ2L∞ (Rn )×Lq (S n−1 ) mε µm;b,k f 2L2 . (3.3) m≥1

If we can show that for some 0 < β < (1 − ε)/2, such that µm;b,k f 2L2 ≤ Cm−2+2β f 2L2 ,

(3.4)

then from (3.3) and (3.4) we get immediately the conclusion of Theorem 1.1. Hence, it remains to show (3.4) to prove Theorem 1.1. Now we prove (3.4). Without loss of generality, we may assume that b∗ = 1. Let ψ ∈ C0∞ (Rn ) be a radial function such that 0 ≤ ψ ≤ 1, suppψ ⊂ {1/2 ≤ |ξ| ≤ 2} and ψ 2 (2−l ξ) = 1, |ξ| = 0. l∈Z

Define the multiplier operator Sl by −l   S l f (ξ) = ψ(2 ξ)f (ξ).

Write σα,t,m,j (x) = (2α t)−1

Ym,j (x ) χ{0<|x|≤2α t} (x) |x|n−1

for α ∈ Z, m = 1, 2, . . . , and j = 1, . . . , Dm . Set Γα,t,m,j (ξ) = σ
α,t,m,j (ξ),

Γlα,t,m,j (ξ) = Γα,t,m,j (ξ)ψ(2α−l ξ).

Denote by Fα,t,m,j the convolution operator whose kernel is σα,m,j,t and Fα,t,m,j;b,k l the kth order commutator of Fα,t,m,j . Define the operator Fα,t,m,j by l
f (ξ) = Γl  Fα,t,m,j α,t,m,j (ξ)f (ξ).

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Boundedness of Commutators

489

l l Let Fα,t,m,j;b,k denote the k-th order commutator of Fα,t,m,j . Then   2α+1 1/2 dt µm,j;b,k f (x) = |Ft,m,j;b,k f (x)|2 3 t 2α 1/2  α∈Z  2 2 dt = |Fα,t,m,j;b,k f (x)| t 1 α∈Z  2  2 dt 1/2   2 = F (S f )(x) .   α,t,m,j;b,k l−α t 1 α∈Z

l∈Z

With the aid of the formula k Cks (b(x) − b(z))s (b(z) − b(y))k−s , (b(x) − b(y))k =

x, y, z ∈ Rn ,

s=0

the Fubini theorem and a trivial computation gives that 2 2 Fα,t,m,j;b,k (Sl−α f )(x) = (Fα,t,m,j Sl−α )b,k f (x) −

k

2 Cks Fα,t,m,j;b,k−s (Sl−α;b,s f )(x).

s=1

On the other hand, we see that if s > 0 and f h ∈ C0∞ (Rn ), then  2 h(x)Sl−α;b,s f (x) dx l∈Z

= =

Rn

s

w=0 s

Csw Csw

w=0

 l∈Z

Rn

l∈Z

Rn



2 h(x)(b(x) − bQ )w Sl−α ((bQ − b(·))s−w f )(x) dx

h(x)(b(x) − bQ )w (bQ − b(x))s−w f (x) dx = 0,

where Q is a cube with large diameters such that supp f and supp h are both contained in Q, and bQ is the mean value of b on Q. Thus in this case, 2 Sl−α;b,s f (x) = 0, a, e, x ∈ Rn .

Therefore by the Minkowski inequality, 2 dt 1/2   2   2 µm,j;b,k f (x) ≤ . (Fα,t,m,j Sl−α )b,k f (x) t 1 α∈Z

l∈Z

Then µm;b,k f 22

=

Dm  j=1

≤

Dm  j=1

≤

Rn



Rn l∈Z 2

 l∈Z

|µm,j;b,k f (x)|2 dx

1

2 dt   2 dx )b,k f (x) (Fα,t,m,j Sl−α t 1 α∈Z Dm  2 dt   2 )b,k f (x) dx . (Fα,t,m,j Sl−α t j=1 2

Rn α∈Z

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Observe that 2 (Fα,t,m,j Sl−α )b,k f (x) =

k

l Csk Fα,t,m,j;b,s (Sl−α;b,k−s f )(x).

s=0

Then µm;b,k f 22  2 ≤

Dm  k 2 dt   l Csk Fα,t,m,j;b,s (Sl−α;b,k−s f )(x) dx  t n R α∈Z j=1 s=0 l∈Z 1   Dm  k 2 dt 2  l  ≤C Fα,t,m,j;b,s (Sl−α;b,k−s f )(x) dx . t 1 Rn s=0

(3.5)

α∈Z j=1

l∈Z

Let l f (x) = Fα,t,m;b,s

 Dm

l |Fα,t,m,j;b,s f (x)|2

1/2 .

j=1

Thus

µm;b,k f 22  2 ≤C l∈Z

Define the operator

1

k  2 dt  l  Fα,t,m;b,s (Sl−α;b,k−s f )(x) dx . t n R s=0

(3.6)

α∈Z

l F α,t,m,j

by

−α l
f (ξ) = Γl F α,t,m,j ξ)f(ξ). α,t,m,j (2 l l Let F α,t,m,j;b,k denote the k-th order commutator of F α,t,m,j . From Lemma 2.2, we can get for t ∈ [1, 2]

|Γα,t,m,j (ξ)| ≤ Cm−λ−1+β/2 min{2α |ξ|, (2α |ξ|)−β/2 }|Ym,j (ξ  )|, |Γα,t,m,j (ξ)| ≤ Cm−λ−1 |Ym,j (ξ  )|, |∇Γα,t,m,j (ξ)| ≤ C2α . Thus supp(Γlα,t,m,j (2−α ·)) ⊂ {2l−1 ≤ |ξ| ≤ 2l+1 }, |Γlα,t,m,j (2−α ξ)| ≤ Cm−λ−1+β/2 min{2l , 2−βl/2 }|Ym,j (ξ  )|, |Γlα,t,m,j (2−α ξ)| ≤ Cm−λ−1 |Ym,j (ξ  )|, |∇Γlα,t,m,j (2−α ξ)| ≤ C. Let l F α,t,m;b,s f (x) =

 Dm

l |F α,t,m,j;b,s f (x)|2

1/2

j=1

Using Lemma 2.3 and Remark 2.1 with δ = 2l , we know for any fixed 0 < v < 1 and nonnegative integer s l f 2 ≤ Cm(−1+β/2)v 2−β|l|v/2 f 2 . F α,t,m;b,s

Vol. 61 (2008)

Boundedness of Commutators

491

So we can find 0 < v0 < 1, such that v0 (−1 + β/2) ≤ −1 + β. Then l F α,t,m;b,s f 2 ≤ Cm−1+β 2−β|l|v0 /2 f 2 .

Which by dilation-invariance implies l Fα,t,m;b,s f 2 ≤ Cm−1+β 2−β|l|v0 /2 f 2 .

Then by Lemma 2.4 (i), we get  k  2  l  Fα,t,m;b,s (Sl−α;b,k−s f )(x) dx Rn s=0 α∈Z

≤ m−2+2β 2−β|l|v0

k

Sl−α;b,k−s f 22

s=0 α∈Z

≤ m−2+2β 2−β|l|v0 f 22 . Then by (3.6), we get

µm;b,k f 22 ≤ m−2+2β f 22 . Thus (3.4) is proved.

References [1] A. Calder´ on and A. Zygmund, On a problem of Mihlin, Trans. Amer. Math. Soc., 78 (1955), 209-224. [2] A. Calder´ on and A. Zygmund, On singular integrals with variable kernels, Applicable Anal. 7 (1977/78), 221-238. [3] F. Chiarenza, M. Frasca and P. Longo, Interior W 2,p estimates for nondivergence elliptic equations with discontinuous coefficiens, Ric. Math. XL (1991), 149-168. [4] G. Di Fazio and M. A. Ragusa, Interior estimates in morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. of Funct. Analysis. 112 (1993), 241-256. [5] Y. Ding, C. Lin and S. Shao, On the Marcinkiewicz integral with variable kernel , Indiana Univ. Math. J. 53 (2004), 805-821. [6] Y. Ding, S. Lu and K. Yabuta, On commutators of Marcinkiewicz integrals with rough kernel , J. Math. Anal. Appl. 275 (2002), 60-68. [7] G. Hu, Lp (Rn ) boundedness for the commutator of a homogeneous singular integral operator, Studia Math. 154 (2003), 13-27. [8] E. Stein, On the function of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430-466. [9] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N. J., 1971. [10] A. Torchinsky and S. Wang, A note on the Marcinkiewicz integral, Colloquium Math, 60/61(1990), 235-243. [11] G. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, 1922.

492

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Yanping Chen Department of Mathematics and Mechanics Applied Science School University of Science and Technology Beijing Beijing, 100083 The People’s Republic of China e-mail: [email protected] Yong Ding School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education Beijing, 100875 The People’s Republic of China e-mail: [email protected] Submitted: March 27, 2007.

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Integr. equ. oper. theory 61 (2008), 493–509 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/040493-17, published online July 25, 2008 DOI 10.1007/s00020-008-1604-7

Integral Equations and Operator Theory

Convolution-Dominated Operators on Discrete Groups Gero Fendler, Karlheinz Gr¨ochenig and Michael Leinert Abstract. We study infinite matrices A indexed by a discrete group G that are dominated by a convolution operator in the sense that |(Ac)(x)| ≤ (a ∗ |c|)(x) for x ∈ G and some a ∈
1 (G). This class of “convolution-dominated” matrices forms a Banach-∗-algebra contained in the algebra of bounded operators on
2 (G). Our main result shows that the inverse of a convolution-dominated matrix is again convolution-dominated, provided that G is amenable and rigidly symmetric. For abelian groups this result goes back to Gohberg, Baskakov, and others, for non-abelian groups completely different techniques are required, such as generalized L1 -algebras and the symmetry of group algebras. Mathematics Subject Classification (2000). Primary 47B35; Secondary 43A20. Keywords. Groups of polynomial growth, convolution, symmetric Banach algebras, inverse-closed, generalized L1 -algebra.

1. Introduction Is the off-diagonal decay of an infinite matrix inherited by its inverse matrix? This question arises in many problems in numerical analysis and approximation theory and its solution has many applications in frame theory and pseudodifferential operators and wireless communications. See [5, 11, 13, 18, 29, 30] for a sample of papers. The study of the off-diagonal decay has two distinct facets, namely the rate of the off-diagonal decay and the nature of the underlying index set. Usually the index set is (a subset of) Zd and the focus is on obtaining various forms of offdiagonal decay conditions. For instance, it is known that polynomial decay and subexponential decay are preserved under matrix inversion [18, 15]. In general, the preservation of off-diagonal decay under inversion depends also on the index set. For instance, in the theory of Calder` on-Zygmund operators, K. G. was supported by the Marie-Curie Excellence Grant MEXT-CT 2004-517154.

494

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the index set consists of all dyadic cubes. On this index set the quality of the off-diagonal decay is not necessarily preserved, and as a consequence the inverse of a Calder` on-Zygmund operator need not be a Calder` on-Zygmund operator [25, 32]. Thus the interaction between the precise form of off-diagonal decay and the index set plays a decisive role. This observation is implicit in [18, 15, 31]. In [18, 15] it was mentioned (without explicit proof) that polynomial or subexponential decay are preserved under inversion whenever the index set of the matrix class possesses a metric with a polynomial growth condition. We study the interaction between the decay conditions and the index set in the context of non-commutative harmonic analysis. Precisely, the index set will be a discrete (non-Abelian) group, e.g., a finitely generated discrete group of polynomial growth. We then investigate the class of convolution-dominated matrices, which are described by a specific type of off-diagonal decay. Convolutiondominated matrices over the index set Zd were introduced by Gohberg, Kashoeck, and Woerdeman [10] as a generalization of Toeplitz matrices, and they showed that this class of matrices was closed under inversion. Similar results and generalizations were obtained independently by Kurbatov [19], Baskakov [2]. Sometime later Sj¨ ostrand [29] rediscovered their results, gave a completely different proof, and used it in the context of a deep theorem about pseudodifferential operators. We consider matrices indexed by a discrete group G: every operator on
2 (G) is described
by a matrix A with entries A(x, y), x, y ∈ G by the usual action (Ac)(x) = y∈G A(x, y)c(y) on a sequence c ∈
2 (G). We will consider mostly groups of polynomial growth. A finitely generated group is of polynomial growth, ∞ if there exists a finite set U ⊆ G, such that n=1 U n = G and card U n ≤ CnD for some constants C, D > 0. Our main theorem reads as follows. Theorem 1. Let G be a discrete finitely generated group of polynomial growth. If a matrix A indexed by G satisfies the off-diagonal decay condition |A(x, y)| ≤ a(xy −1 ), x, y ∈ G for some a ∈
1 (G) and A is invertible on
2 (G), then there exists b ∈
1 (G) such that |A−1 (x, y)| ≤ b(xy −1 ), x, y ∈ G. We will extend this result and also consider the situation where
1 (G) is replaced by the weighted algebra
1 (G, ω) for certain weight functions on G. This weighted case is easier and follows from Theorem 1 by standard methods. To put Theorem 1 into a bigger context, let us consider the case A(x, y) = a(xy −1 ) for a sequence a ∈
1 (G). This matrix A corresponds to the convolution operator Ac = a ∗ c on
2 (G). Even this case is highly non-trivial. Theorem 1 implies the symmetry of the group algebra
1 (G), i.e., the spectrum of positive elements a∗ ∗ a is contained in [0, ∞) for all a ∈
1 (G). This fact is of course well known, but its proof requires the combination of two landmark results of harmonic analysis, namely Gromov’s characterization of finitely generated groups of polynomial growth as finite extensions of nilpotent groups and Hulanicki’s result that discrete nilpotent groups are symmetric [16, 17]. Convolution-dominated matrices on groups of polynomial growth occur implicitly in Sun’s remarkable work [31]. His conditions on the off-diagonal decay

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495

are somewhat complicated and exclude the basic case of
1 -decay. In view of the relation with the symmetry of groups, this omission is not surprising. If G is a discrete Abelian group G, the proof of the main theorem is based on an idea of de Leeuw [4]: to every operator on G one can assign an operatorvalued Fourier series and then classical Fourier series arguments, such as Wiener’s Lemma, can be applied. This approach is championed in [2, 10]. For a non-Abelian group as the index set, these ideas break down completely, and a new approach is required. Our key idea is to replace the Fourier series arguments by methods taken from Leptin’s investigation of generalized L1 algebras [20, 21, 22]. The main insight is that the algebra of convolution-dominated matrices can be identified with a generalized L1 -algebra in the sense of Leptin. This observation allows us to translate the original problem about matrix inversion into a problem of abstract harmonic analysis. The analysis of generalized L1 -algebras was advanced by Leptin and Poguntke [20, 21, 22, 24] and has produced deep results. In fact, we will resort to their representation theoretic results and to the concept of the “rigid symmetry” of Banach algebras and apply these at a crucial point. The relation between a “simple” matrix problem and the theory of generalized L1 -algebras may seem surprising at first glance, but it is exactly this connection that allows us to use the power of non-commutative harmonic analysis to solve the problem. Let us mention that a similar theory can be established for convolution-dominated integral operators. This generalization is more technical and will be dealt with in a subsequent paper. The paper is organized as follows: in Section 2 we give a formal definition of the algebra of convolution-dominated operators and identify it as a generalized L1 -algebra. In Section 3 we prove the symmetry of this algebra, and in Section 4 we treat the related concept of inverse-closedness. In particular, we prove Theorem 1. In Section 5 we treat the weighted case and characterize all weights for which the generalized weighted L1 -algebra is symmetric. Acknowledgement: We would like to thank Marc Rieffel for his useful comments and questions on an early draft of the paper.

2. The Algebra of Convolution-Dominated Operators as a Twisted L1 -Algebra Let G be a discrete group. For x ∈ G we denote the operator of left translation on
1 (G) and on
2 (G) by λ(x), i.e. if f ∈
1 (G) or f ∈
2 (G), then λ(x)f (y) = f (x−1 y), x, y ∈ G. By B(
2 (G)) we denote the algebra of bounded operators on
2 (G). For an operator A :
2 (G) →
2 (G) let A(x, y) =< Aδy , δx >, x, y ∈ G, be its matrix, where < , > is the inner product of the Hilbert space
2 (G) and δx (x) = 1 and δx (z) = 0 for z = x.

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Definition 1. The operator A is called convolution-dominated, in short notation A ∈ CD(G), if there exists a sequence a ∈
1 (G) such that |A(x, y)| ≤ a(xy −1 ),

∀x, y ∈ G.

We define the norm of A as an element in CD(G) by A 1 := inf{ a
1 : a ∈
1 (G), |A(x, y)| ≤ a(xy −1 ) ∀x, y ∈ G}. By choosing a(z) to be the supremum of the entries of A on the z-th diagonal, namely a(z) = sup{x,y: xy−1 =z} |A(x, y)|, we see that  sup |A(x, y)| < ∞. A 1 = z∈G

{x,y: xy −1 =z}

To shed light on this definition, consider the action of the operator or matrix A on a finitely supported vector c and take absolute values:   A(x, y)c(y)| ≤ a(xy −1 ) |c(y)| = (a ∗ |c|)(x) . (1) |(Ac)(x)| = | y∈G

y∈G

Thus A is dominated pointwise by a convolution operator, whence our terminology. Clearly, if a ∈
1 (G), i.e., if A ∈ CD(G), then A is bounded on
2 (G), and the operator norm on
2 (G), in fact on all
p (G), 1 ≤ p ≤ ∞, is majorized by the A 1 norm. If we consider the composition of two convolution-dominated operators A and B, then we obtain similarly |(ABc)(x)| ≤ (a ∗ b ∗ |c|)(x) , and therefore the operator AB is again convolution-dominated and we obtain that AB 1 ≤ A 1 B 1 , because
1 (G) is a convolution algebra. We may summarize these observations as follows. Lemma 1. The space CD(G) is a Banach ∗-algebra with respect to composition of operators and taking the adjoint operator as involution. Moreover, CD(G) is continuously embedded into B(
2 (G)). Our first goal is to represent CD(G) as a generalized L1 -algebra in the sense of Leptin [20]. Consider the C ∗ -algebra
∞ (G) with pointwise multiplication and complex conjugation as involution. This algebra is isometrically represented as an algebra of multiplication operators on
2 (G) by Dm f (x) = m(x)f (x), where x ∈ G, f ∈
2 (G), m ∈
∞ (G). Analogously, we define an operator Dzm by Dzm = λ(z) ◦ Dm . As is easily seen, the matrix of Dzm has the entries Dzm (x, y) = m(y)δz (xy −1 ) .

(2)

Whereas the matrix of the multiplication operator D is a diagonal matrix, the matrix of Dzm is non-zero only on the z-th side-diagonal. Since every matrix can be written as the sum of its side-diagonals, every operator is a sum of the elementary m

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operators Dzm . This simple observation is crucial for the analysis of convolutiondominated operators. Next we study how the operators Dzm behave under composition: if v, w ∈ G and m, n ∈
∞ (G), then  m m (Dvn ◦ Dw )(x, y) = Dvn (x, z)Dw (z, y) z∈G

=



n(z)δv (xz −1 )m(y)δw (zy −1 )

z∈G

=



n(zy)δv (xy −1 z −1 )m(y)δw (z)

(3)

z∈G

=

n(wy)m(y)δv (xy −1 w−1 )

=

n(wy)m(y)δvw (xy −1 )

=

Dvww−1

(T

n) m

(x, y) .

In the last equality we have set Ty n(z) = n(y −1 z) for n ∈
∞ (G). We use a notation different from λ, because Ty :
∞ (G) →
∞ (G) is a C ∗ automorphism of the algebra
∞ (G) and the mapping y → Ty defines a homomorphism of the group G into the group of C ∗ -automorphisms of
∞ (G). Using this homomorphism, we may now form the twisted L1 -algebra L =
1 (G,
∞ (G), T ) in the sense of Leptin [20, 21, 22]. The underlying Banach space of L is the space of
∞ (G)valued absolutely summable sequences on G, but we will often interpret it as the projective tensor product ˆ
∞ (G).
1 (G,
∞ (G)) =
1 (G) ⊗ Thus for an element f ∈
1 (G,
∞ (G)) we denote its value in
∞ (G) by f (x), x ∈ G, and we write f (x)(z) or f (x, z) for the value of this
∞ -function at z ∈ G. The twisted convolution of h, f ∈ L is defined by  Ty h(xy)f (y −1 ), for x ∈ G , h  f (x) = y∈G

and the involution of h ∈ L by h∗ (x) = Tx−1 h(x−1 ), for x ∈ G . An element f ∈ L may be represented uniquely as  f= mz δz , z∈G

where mz = f (z) ∈
∞ (G). By using mz as the z-th side-diagonal of a matrix, we define a map R :
1 (G,
∞ (G), T ) → B(
2 (G)) (4) by  Dzmz . (5) Rf = z∈G

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Proposition 1. The map R :
1 (G,
∞ (G), T ) → CD(G) is an isometric ∗-isomorphism.

Proof. Let f = z∈G mz δz and h = z∈G nz δz ∈ L. By (2) we have  Rf CD = Dzmz 1 z∈G



=

sup

−1 z∈G {x,y:xy =z}



=

|Dzmz (x, y)|

sup |mz (y)| = f
1 (G,
∞ (G)) . y

z∈G

Thus R is an isometry. The twisted convolution of f and h may be computed as follows:   (h  f )(x, z) = Ty h(xy)f (y −1 ) (z) y∈G



=

h(xy, y −1 z)f (y −1 , z)

y∈G



=

nxy (y −1 z)my−1 (z)

(6)

y∈G



=

lv (z)δv (x)

v

where



lv =

Ty nvy my−1 ∈
∞ (G) .

y∈G

By comparison, the composition of the corresponding operators Rf and Rh (matrix multiplication) yields that   mw Drnr ◦ Dw Rf ◦ Rh = r

=



w (Tw−1 nr ) mw Drw

=



r,w

where lv =



l

Dvv

v

Tw−1 nr mw =

{r,w: rw=v}



Ty nvy my−1 = lv .

y∈G

Thus Rf ◦ Rh = R(f  h) and R is an algebra homomorphism. The involution of f as above is given by f ∗ (x, z) = = =

Tx−1 mx−1 (z) mx−1 (xz)  lv (z)δv (x), v

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where lv (z) = Tv−1 mv−1 (z). By comparison, the adjoint of a single side-diagonal operator is (Dvmv )∗ (x, y)

= Dvmv (y, x) = mv (x)δv (yx−1 ) = mv (x)δv−1 (xy −1 ) = mv (v −1 y)δv−1 (xy −1 ) v mv = DvT−1 (x, y) .

These equalities imply that  Dvmv )∗ (

=

v



v mv DvT−1

v

=



T

Dv v−1

mv−1

= R(f ∗ ) ,

v

and so R preserves the involution. Finally, from the definition of A 1 and the equalities (2) and (5) one sees that R is onto.


3. Symmetry of the Twisted L1 -Algebra Recall that a Banach algebra A with isometric involution is called symmetric if the spectrum of every positive element is contained in the non-negative reals, i. e. sp(a∗ a) ⊂ [0, ∞) for all a ∈ A. For various abstract characterizations of symmetry see [3, Section 41] or [24]. Furthermore, a locally compact group G is called symmetric, if its convolution algebra L1 (G) is symmetric. Various classes of groups are known to be symmetric: (a) locally compact Abelian groups, (b) compact groups, (c) finite extensions of discrete nilpotent groups, (d) compactly generated groups of polynomial growth, (e) compact extensions of locally compact nilpotent groups, and others. See [23]. For the groups of the classes (a) — (c) Leptin and Poguntke [24] have shown that they satisfy an even stronger property, namely that of rigid symmetry. This means ˆ is symmetric. that for every C ∗ -algebra C the projective tensor product L1 (G)⊗C Later Poguntke [27] showed that all nilpotent locally compact groups are rigidly symmetric. Our goal is to show that the twisted L1 -algebra L =
1 (G,
∞ (G), T ) of a rigidly symmetric discrete group G is symmetric and hence that the algebra of convolution-dominated operators CD(G) is also symmetric. To this end we define a map 2 ˆ (G)) Q :
1 (G,
∞ (G), T ) →
1 (G)⊗B(


by f=

 v

δv ⊗ mv →

 v

δv ⊗ Dvmv .

(7) (8)

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Proposition 2. The map Q is an isometric ∗-isomorphism of
1 (G,
∞ (G), T ) onto 2 ˆ a closed subalgebra of
1 (G)⊗B(
(G)). ˆ E, Proof. The proof rests on the isometrical identification
1 (G, E) =
1 (G) ⊗ which holds for any Banach space E [6, Ch. VIII.1.]. It follows that for f =
v δv ⊗ mv ∈ L   mv ∞ = Dvmv B(
2 (G)) f 1 = v



=



v

δv ⊗

Dvmv


1 (G)⊗B(
2 (G)) . ˆ

v

Thus Q is an isometry. Let h =




⊗ nv , then by (6)  hf = δv ⊗ l v ,

where lv =




v δv

v y∈G (Ty nvy )my −1 .

Q(h  f ) =



Hence

δv ⊗ Dvlv

v

=



=

mw δz δw ⊗ Dznz Dw

z,w

=

(

mw Dznz Dw

{z,w:zw=v}

v





δv ⊗



 mw δz ⊗ Dznz )( δw ⊗ D w ) = Q(h)Q(f ) . w

z∈G

Similarly one computes that Q intertwines the involutions. In fact  δv∗ ⊗ (Dvmv )∗ Q(f )∗ = v

=



v mv δv−1 ⊗ DvT−1

v

=



T

δv ⊗ Dv v−1

mv−1

= Q(f ∗ ) .

v −1

Thus Q is a ∗-homomorphism. Since Q is an isometry, the image of Q is a closed ˆ B(
2 ).
subalgebra of
1 ⊗ Since symmetry is inherited by closed subalgebras, we obtain the following consequence. Corollary 1. Let G be a discrete rigidly symmetric group. Then
1 (G,
∞ (G), T ) and CD(G) are symmetric Banach ∗-algebras.

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4. Inverse Closedness Given two Banach algebras A ⊆ B with common identity, A is inverse-closed in B, if a ∈ A and a−1 ∈ B ⇒ a−1 ∈ A . This notion occurs under many names: one also says that A is a spectral subalgebra or a local subalgebra of B. The pair (A, B) is called a Wiener pair by Naimark [26]. An important property of an inverse-closed subalgebra A is that it possesses the same holomorphic functional calculus as B. Inverse-closedness is usually proved by means of Hulanicki’s Lemma [17]. Let rA (a) denote the spectral radius of a in the algebra A. If rA (a) = rB (a) for all a = a∗ ∈ A, then we have equality of the spectra spA (a) = spB (a) for all a ∈ A. Consequently, if B is symmetric, then A is also symmetric. For this version of Hulanicki’s lemma, see [9, Lemma 3.1 and 6.1] and [12, Lemma 5.1] for an elementary proof. Our goal is to show that the algebra of convolution-dominated matrices CD(G) is inverse-closed in B(
2 (G)). For this purpose we consider two natural unitary representations of the twisted L1 -algebra L. The first representation is the so-called D-regular representation of L. Recall that D : m → Dm is a faithful representation of the C ∗ -algebra
∞ by multiplication operators in B(
2 (G)). Then as in Leptin [22, §3] the D-regular representation λD of L =
1 (G,
∞ , T ) on the Hilbert space
2 (G,
2 (G)) is defined by  λD (f )ξ(x) = DTy f (xy) ξ(y −1 ), ξ ∈
2 (G,
2 (G)), f ∈ L. y∈G

One easily verifies that this defines indeed a ∗-representation of L. The second representation is the mapping R : L → CD(G) ⊂ B(
2 (G)) introduced in (5). By Proposition 1, R is also a ∗-representation of L on
2 (G). We call this representation the canonical representation of L. Proposition 3. The D-regular representation λD of L is a multiple of the canonical representation R. Hence R(f ) = λD (f ) for all f ∈ L. Proof. We identify
2 (G,
2 (G)) with
2 (G×G). Let Rω
be the extension of R from
2 (G) to
2 (G × G) by letting the operators R(f ) = y∈G λ(y) ◦ Df (y) , f ∈ L, act in the first coordinate only, i.e., for ξ ∈
2 (G × G)  Rω (f )ξ(x, z) = f (y)(y −1 x)ξ(y −1 x, z). (9) y∈G

Next we define a candidate for an intertwining operator between the D-regular representation and the card(G)-multiple Rω of the canonical representation by Sξ(x, z) = ξ(xz, z), where ξ ∈
2 (G × G). Then on the one hand we have S[Rω (f )ξ](x, z) =

 y∈G

f (y)(y −1 xz)ξ(y −1 xz, z).

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On the other hand λD (f )(Sξ)(x, z)

=



IEOT

(Ty f (xy))(z)(Sξ)(y −1 , z)

y∈G

=



(Tx−1 y f (y))(z)(Sξ)(y −1 x, z)

y∈G

=



f (y)(y −1 xz)(Sξ)(y −1 x, z)

y∈G

=



f (y)(y −1 xz)ξ(y −1 xz, z).

y∈G

Consequently, λD (f )(Sξ) = SRω (f )ξ

(10)

2

2

for all f ∈ L and ξ ∈
(G × G). Since S is unitary on
(G × G), λ equivalent.

D

and R are
ω

To deal with inverse-closedness, we need to compare several norms on L and CD(G). Let . ∗ be the largest C ∗ norm on L. By a theorem of Ptak [28] a Banach ∗-algebra A is symmetric, if and only if the largest C ∗ -seminorm · ∗ on A satisfies a∗ a ∗ = rA (a∗ a) for all a ∈ A. See also [3, §41 Corollary 8]. As a first consequence of Proposition 3 we identify the largest C ∗ -norm on CD(G). Corollary 2. Let G be an amenable discrete group, then the largest C ∗ norm on L equals the operator norm on CD(G). Proof. Since G is amenable, it follows from [22, Satz 6] of Leptin that for the representation D of
∞ (G) the D-regular representation λD defines the largest C ∗ norm on L. Therefore we obtain f ∗ = λD (f ) = R(f ) B(
2 (G))

for every f ∈ L ,


where the last equality follows from Proposition 3.

Proposition 4. Let G be a discrete, amenable, and rigidly symmetric group. Then rL (f ∗ f ) = rCD(G) (R(f )∗ R(f )) = R(f ) 2B(L2 (G))

for all f ∈ L.

(11)

Proof. Since L and CD(G) are symmetric by Corollary 1, Ptaks theorem [28] implies that f 2∗ = rL (f ∗ f ) = rCD(G) (R(f )∗ R(f )). Since Corollary 2 says that f ∗ f ∗ = R(f )∗ R(f ) B(
2 ) , we obtain the identity (11).
Theorem 2. Let G be a discrete, amenable, and rigidly symmetric group. If f ∈ L is such that R(f ) ∈ CD(G) has an inverse in B(
2 (G)) then f −1 exists in L and R(f −1 ) = R(f )−1 is in CD(G).

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Proof. If f ∈ L is hermitian, i.e. f = f ∗ , then by Proposition 4 rL (f )2 = rL (f ∗ f ) = R(f ) 2B(L2 (G)) . [9, Lemma 6.1 and 3.1] imply that spL (f ) = spB(
2 (G)) (R(f )),

∀f ∈ L.

2

Thus the invertibility of R(f ) in B(
(G)) implies the invertibility of f in L.




By writing Theorem 2 explicitly as a statement about the off-diagonal decay of an invertible matrix, we recover Theorem 1 of the introduction. Corollary 3. Let G be a discrete, amenable, and rigidly symmetric group (for instance, a finitely generated group of polynomial growth). If a matrix A indexed by G satisfies the off-diagonal decay condition |A(x, y)| ≤ a(xy −1 ) for some a ∈
1 (G) and A is invertible on
2 (G), then there exists b ∈
1 (G) such that |A−1 (x, y)| ≤ b(xy −1 ). A slight variation yields the following result of which previous versions have been quite useful in time-frequency analysis [7]. Corollary 4. Assume that A ∈ CD(G) and that A = A∗ . Then the following are equivalent: (i) A is invertible on
2 (G). (ii) A is invertible on
p (G) for all p, 1 ≤ p ≤ ∞. (iii) A is invertible on
p (G) for some p, 1 ≤ p ≤ ∞. Proof. (i) ⇒ (ii) Recall that every matrix A ∈ CD(G) is bounded on all
p (G), 1 ≤ p ≤ ∞ by (1). Thus if A ∈ CD(G) is invertible on
2 (G), then by Theorem 1 A−1 ∈ CD(G) and thus A−1 is invertible on
p (G) for arbitrary p, 1 ≤ p ≤ ∞. The implication (ii) ⇒ (iii) is obvious. (iii) ⇒ (i) Assume that A is invertible on some
p (G). Then the adjoint  operator A∗ = A is invertible on the dual space
p (G), where p = p/(p − 1) is the conjugate index. By interpolation we obtain that A is invertible on the
interpolation space
2 (G). Remark. The hypotheses on the group G are almost sharp. To see this, let λ(f ) denote the convolution operator c → λ(f )c = f ∗ c acting on
p (G), and let sp
p (f ) the spectrum of λ(f ) as an operator acting on
p (G). Then sp
p (f ) = sp
2 (f ) for all p ∈ [1, ∞], if and only if the group G is amenable and symmetric [1, 14]. Thus amenability and symmetric are necessary in Theorem 2. We do not know whether we can replace the rigid symmetry by symmetry, because it is an open problem whether every symmetric group is rigidly symmetric [27]. We emphasize once more that all discrete finitely generated groups of polynomial growth satisfy the hypotheses of amenability and rigid symmetry. These groups are finite extensions of some discrete nilpotent group by Gromov’s result [16], and thus they are rigidly symmetric by [24] and [27].

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5. Symmetry of weighted algebras In this section we extend the results about the symmetry of convolution-dominated operators to the weighted case. A function ω : G → [1, ∞) is called a weight on G, if it fulfills ω(xy) −1

ω(x

)

≤ ω(x)ω(y), = ω(x),

∀x, y ∈ G

∀x ∈ G

ω(e) = 1 . Given a weight ω we let
1 (G,ω) be the weighted
1 -algebra on G. Using weights, one can model stronger decay conditions on convolution-dominated operators as follows. Definition 2. An operator A on
2 (G) is called ω-convolution-dominated, A ∈ CD(G, ω) in short, if there exists an a ∈
1 (G,ω) such that |A(x, y)| ≤ a(xy −1 ),

∀x, y ∈ G.

We define its norm as A ω := inf{ a
1(G,ω) : a ∈
1 (G,ω), |A(x, y)| ≤ a(xy −1 ) ∀x, y ∈ G}. As in the unweighted case, we may write the norm as  A ω = sup |A(x, y)| ω(z) < ∞ . z∈G

{x,y: xy −1 =z}

Thus an operator A is in CD(G, ω), if it is dominated by a convolution operator in
1 (G, ω) in the sense that |Ac(x)| ≤ (a ∗ |c|)(x) for some a ∈
1 (G, ω). Since
1 (G,ω) is a convolution algebra, the space of ω-convolution-dominated operators CD(G, ω) is a Banach ∗-algebra with respect to composition of operators and the usual involution of operators in B(
2 (G)). Furthermore, CD(G, ω) ⊆ CD(G) ⊆ B(
2 (G)). For the study of CD(G, ω), we consider the weighted, twisted L1 -algebra Lω =
1 (G, ω,
∞ (G), T ), which is defined as a subalgebra of L endowed with the norm  f Lω = f (x) ∞ ω(x) . x∈G

Since Lω is a subalgebra of L, all algebraical relations are preserved and the results of Sections 2 and 3 carry over to Lω after a slight modification of the norm computations. Proposition 5. Let Rω and Qω denote the restrictions of the maps R and Q defined in (4) and (7) from L to Lω =
1 (G, ω,
∞ (G), T ). Then Rω :
1 (G, ω,
∞ (G), T ) → CD(G, ω) is an isometric ∗-isomorphism and 2 ˆ Qω :
1 (G, ω,
∞ (G), T ) →
1 (G,ω)⊗B(
(G))

is an isometric ∗-isomorphism onto a closed ∗-subalgebra.

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We are interested in the symmetry of the weighted
1 -algebra. This forces us to impose some conditions of subexponential growth on the weight. Definition 3. (a) A weight ω is said to satisfy the GRS-condition (GelfandRaikov-Shilov condition) if lim ω(xn )1/n = 1

for all x ∈ G.

n→∞

(b) A weight ω is said to satisfy the UGRS-condition (the uniform GRS-condition), if for some generating subset U of G containing the identity element lim sup ω(y)1/n = lim

n→∞ y∈U n

sup

n→∞ x1 ,...,xn ∈U

ω(x1 x2 . . . xn )1/n = 1 .

The GRS-condition is a necessary condition for the spectral identity r
1 (f ) = r
1ω (f ) in weighted group algebras, and hence for the symmetry of
1 (G, ω) [8]. If G is a compactly generated locally compact group of polynomial growth, then the GRS-condition is also sufficient for the symmetry of
1 (G, ω). In this case, the UGRS-condition with a relatively compact set U is also equivalent to the GRScondition by the results in [8]. However, if G is not compactly generated, the UGRS-condition may be a stronger assumption on the weight. We emphasize that in Definition 3, U need not be finite. As a example consider the group Z2 and the weight ω(k1 , k2 ) = (1 + |k1 |)s , k1 , k2 ∈ Z, s > 0. This weight satisfies the GRS-condition and the UGRS-condition with the generating set {−1, 0, 1} × Z. Theorem 3. Let G be a rigidly symmetric, amenable, discrete group. If the weight ω satisfies the UGRS-condition and the condition sup x∈U n \U n−1

w(x) ≤ C

inf

x∈U n \U n−1

w(x) ,

(12)

2 2 ˆ ˆ then
1 (G,ω)⊗B(
(G)) is inverse-closed in
1 (G)⊗B(
(G)) and hence symmetric.

ˆ B(
2 (G)) is Proof. By the assumption on G we know that the algebra B =
1 (G) ⊗ ˆ B(
2 (G)) is a subalgebra of B, by [9, Lemmas 3.1 symmetric. Since A =
1 (G,ω) ⊗ and 6.1], we need only show the equality of the both spectral radii on the latter algebra. Since for f ∈ A   f (x) B(
2 (G)) ≤ f (x) B(
2 (G)) ω(x) = f A , f B = x∈G

x∈G

the spectral radius formula implies that rB (f ) ≤ rA (f )

for all f ∈ A.

Thus it suffices to show the converse inequality. To this end we define a weight v on Z by v(n) = sup ω(y), y∈U |n|

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where U is a generating set, containing the identity element, such that limn→∞ supy∈U n ω(y)1/n = 1. By induction one finds an estimation for the norm of the n-th convolution power f (n) of f ∈ A:   ··· f (x1 ) f (x2 ) . . . f (xn ) ω(x1 . . . xn ) . (13) f (n) A ≤ G

G

 n   n−1 Since G = ∞ as a disjoint union (where U 0 = ∅), we may split n=1 U \ U each sum accordingly. This yields f (n) A ≤ ∞  ≤





···

k1 ,k2 ,...,kn =1 U k1 \U k1 −1

f (x1 ) . . . f (xn ) ω(x1 . . . xn ) .

U kn \U kn −1

If xj ∈ U kj \ U kj −1 , then x1 . . . xn ∈ U k1 +···+kn and so the weight is majorized by ω(x1 . . . xn ) ≤

sup y∈U k1 +···+kn

ω(y) = v(k1 + · · · + kn ) .


Set bk := U k \U k−1 f (x) and b = (bk )k∈N . Then clearly we have f B = b
1 and condition (12) implies that C −1 b
1v ≤ f Lω ≤ b
1v . For the convolution powers of f we obtain that ∞  f (n) A ≤ bk1 bk2 . . . bkn v(k1 + k2 + . . . kn ) = b(n)
1 (Z,v) < ∞ . k1 ,k2 ,...,kn =1

By its definition the weight v on Z satisfies the GRS-condition, and
1 (Z,v) is symmetric by [8, Lemma 3.2]. Hence rA (f ) =

1/n

1/n

lim f (n) A ≤ lim b(n)
1 (Z,v)

n→∞

n→∞

=

r
1 (Z,v) (b) = r
1 (Z) (b) = b
1

=

f B .

So for all k ∈ N we have 1/k

rA (f ) = rA (f (k) )1/k ≤ f (k) B , and by letting k → ∞ we obtain the required inequality rA (f ) ≤ rB (f ).




Combining Proposition 5 and Theorem 3, we obtain the symmetry of the weighted convolution-dominated operator algebras CD(G, ω). Corollary 5. Under the same assumptions on G and ω as in Theorem 3, the algebra CD(G, ω) is symmetric. Moreover, the Theorem 3 combined with Theorem 2 shows that for f ∈ Lω : rLω (f )

=

rA (Qω (f )) = rB (Q(f ))

=

rL (f ) = rB(
2 (G)) (R(f )) = rB(
2 (G)) (Rω (f )).

Using Hulanicki’s Lemma in the form of [9, Lemma 6.1 and 3.1] we conclude as in the proof of Theorem 2 that CD(G, ω) is inverse-closed in B(
2 (G)).

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Corollary 6. Impose the same assumptions on G and ω as in Theorem 3. If f ∈ Lω is such that Rω (f ) ∈ CD(G, ω) has an inverse in B(
2 (G)) then f −1 exists in Lω , and Rω (f −1 ) = Rω (f )−1 is in CD(G, ω). For a single matrix Corollary 6 can be recast once again as a statement about the preservation of the off-diagonal decay by the inverse. Corollary 7. Impose the same assumptions on G and ω as in Theorem 3. If a matrix A on G satisfies the off-diagonal decay condition |A(x, y)| ≤ a(xy −1 ), ∀x, y ∈ G, for some a ∈
1 (G, ω) and A is invertible on
2 (G), then there exists some b ∈
1 (G, ω), such that |A−1 (x, y)| ≤ b(xy −1 ), ∀x, y ∈ G. Remark. The proof of Thm. 3 is similar to the one of [8, Thm. 3.3]. However, the proof given there works only under an additional assumption on the weight, such as (12), the result remains correct as a consequence of the main result in [8].

References [1] B. A. Barnes. When is the spectrum of a convolution operator on Lp independent of p? Proc. Edinburgh Math. Soc. (2), 33(2):327–332, 1990. [2] A. G. Baskakov. Wiener’s theorem and asymptotic estimates for elements of inverse matrices. Funktsional. Anal. i Prilozhen., 24(3):64–65, 1990. [3] F. Bonsall and J. Duncan. Complete normed algebras. Springer Verlag, 1973. [4] K. deLeeuw. An harmonic analysis for operators. I. Formal properties. Illinois J. Math., 19(4):593–606, 1975. [5] S. Demko, W. F. Moss, and P. W. Smith. Decay rates for inverses of band matrices. Math. Comp., 43(168):491–499, 1984. [6] J. Diestel, J. J. Uhl. Vector Measures. Math. Surveys 15, Amer. Math. Soc., Providence, RI, 1977. [7] H. G. Feichtinger and K. Gr¨ ochenig. Gabor frames and time-frequency analysis of distributions. J. Functional Anal., 146(2):464–495, 1997. [8] G. Fendler, K. Gr¨ ochenig, and Leinert. Symmetry of weighted L1 -algebras and the GRS-condition. Bull. London Math. Soc., 38:625–635, 2006. [9] G. Fendler, K. Gr¨ ochenig, M. Leinert, J. Ludwig, and C. Molitor-Braun. Weighted group algebras on groups of polynomial growth. Math. Z., 245:791–821, 2003. [10] I. Gohberg, M. A. Kaashoek, and H. J. Woerdeman. The band method for positive and strictly contractive extension problems: an alternative version and new applications. Integral Equations Operator Theory, 12(3):343–382, 1989. [11] K. Gr¨ ochenig. Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl., 10(2), 2004. [12] K. Gr¨ ochenig. Composition and spectral invariance of pseudodifferential operators on modulation spaces. J. Anal. Math., 98:65–82, 2006. [13] K. Gr¨ ochenig. Time-frequency analysis of Sj¨ ostrand’s class. Revista Mat. Iberoam., 22(2):703–724, 2006.

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[14] K. Gr¨ ochenig and M. Leinert. Wiener’s lemma for twisted convolution and Gabor frames. J. Amer. Math. Soc., 17:1–18, 2004. [15] K. Gr¨ ochenig and M. Leinert. Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices. Trans. Amer. Math. Soc., 358(6):2695– 2711 (electronic), 2006. ´ [16] M. Gromov. Groups of polynomial growth and expanding maps. Inst. Hautes Etudes Sci. Publ. Math., 53(1):53–78, 1981. [17] A. Hulanicki. On the spectrum of convolution operators on groups with polynomial growth. Invent. Math., 17:135–142, 1972. [18] S. Jaffard. Propri´et´es des matrices “bien localis´ees” pr`es de leur diagonale et quelques applications. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 7(5):461–476, 1990. [19] V. G. Kurbatov. Algebras of difference and integral operators. Funktsional. Anal. i Prilozhen., 24(2):87–88, 1990. [20] H. Leptin. Verallgemeinerte L1 -Algebren und projektive Darstellungen lokal kompakter Gruppen I. Invent. Math., 3:257–281, 1967. [21] H. Leptin. Verallgemeinerte L1 -Algebren und projektive Darstellungen lokal kompakter Gruppen II. Inventiones Math., 4:68–86, 1967. [22] H. Leptin. Darstellungen verallgemeinerter L1 -Algebren. Inventiones Math., pages 192–215, 1968. [23] H. Leptin. The structure of L1 (G) for locally compact groups. In Operator algebras and group representations, Vol. II (Neptun, 1980), volume 18 of Monogr. Stud. Math., pages 48–61. Pitman, Boston, MA, 1984. [24] H. Leptin and D. Poguntke. Symmetry and non-symmetry for locally compact groups. J. Funct. Anal., 33:119–134, 1979. [25] Y. Meyer. Ondelettes et op´erateurs. II. Hermann, Paris, 1990. Op´erateurs de Calder´ on-Zygmund. [Calder´ on-Zygmund operators]. [26] M. A. Na˘ımark. Normed algebras. Wolters-Noordhoff Publishing, Groningen, third edition, 1972. Translated from the second Russian edition by Leo F. Boron, WoltersNoordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics. [27] D. Poguntke. Rigidly symmetric L1 -group algebras. Seminar Sophus Lie, 2:189–197, 1992. [28] V. Pt´ ak. On the spectral radius in Banach algebras with involution. Bull. London Math. Soc., 2:327–334, 1970. [29] J. Sj¨ ostrand. Wiener type algebras of pseudodifferential operators. In S´eminaire sur ´ ´ les Equations aux D´eriv´ees Partielles, 1994–1995, Exp. No. IV, 19 pages. Ecole Polytech., Palaiseau, 1995. [30] T. Strohmer. Pseudodifferential operators and Banach algebras in mobile communications. Appl. Comput. Harmon. Anal., 20(2):237–249, 2006. [31] Q. Sun. Wiener’s lemma for infinite matrices. Trans. Amer. Math. Soc., 359(7):3099– 3123 (electronic), 2007. [32] P. Tchamitchian. Calcul symbolique sur les op´erateurs de Calder´ on-Zygmund et er. I Math., 303(6):215– bases inconditionnelles de L2 (R). C. R. Acad. Sci. Paris S´ 218, 1986.

Vol. 61 (2008)

Convolution-Dominated Operators

Gero Fendler Finstertal 16 D-69514 Laudenbach Germany e-mail: [email protected] Karlheinz Gr¨ ochenig Fakult¨ at f¨ ur Mathematik Universit¨ at Wien Nordbergstrasse 15 A-1090 Wien Austria e-mail: [email protected] Michael Leinert Institut f¨ ur Angewandte Mathematik Universit¨ at Heidelberg Im Neuenheimer Feld 294 D-69120 Heidelberg Germany e-mail: [email protected] Submitted: January 2, 2008.

509

Integr. equ. oper. theory 61 (2008), 511–547 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/040511-37, published online July 25, 2008 DOI 10.1007/s00020-008-1602-9

Integral Equations and Operator Theory

Carleson Measures for the Bloch Space ´ Daniel Girela, Jos´e Angel Pel´aez, Fernando P´erez-Gonz´alez and Jouni R¨atty¨a Abstract. In this paper we study the positive Borel measures µ on the unit disc D in C for which the Bloch space B is continuously included in Lp (dµ), 0 < p < ∞. We call such measures p-Bloch-Carleson measures. We give two conditions on a measure µ in terms of certain logarithmic integrals one of which is a necessary condition and the other a sufficient condition for µ being a p-Bloch-Carleson measure. We also give a complete characterization of the p-Bloch-Carleson measures within certain special classes of measures. It is also shown that, for p > 1, the p-Bloch-Carleson measures
are exactly those for f (w) which the Toeplitz operator Tµ , defined by Tµ (f )(z) = D (1−wz) 2 dµ(w) (f ∈


L1 (dµ), z ∈ D), maps continuously Lp (dµ) into the Bergman space A1 , 1p + 1 = 1. Furthermore, we prove that if p > 1, α > −1 and ω is a weight which p
satisfies the Bekoll´e-Bonami Bp,α -condition, then the measure µα,p defined by dµα,p (z) = (1 − |z|2 )α ω(z)dA(z) is a p-Bloch-Carleson-measure. ∞ We also consider the Banach space Hlog of those functions f which are 

1 , as |z| → 1. The Bloch space analytic in D and satisfy |f (z)| = O log 1−|z| ∞ ∞ and study is contained in Hlog . We describe the p-Carleson measures for Hlog weighted composition operators and a class of integration operators acting ∞ continuously in this space. We determine which of these operators map Hlog p to the weighted Bergman space Aα (p > 0, α > −1) and show that they are automatically compact.

Mathematics Subject Classification (2000). 30H05, 32A36, 30D45. Keywords. Bloch function, Bergman spaces, Bekoll´e-Bonami weights, Carleson measures, interpolating sequences, weighted composition operators, integration operators.

This research is partially supported by several grants from “the Ministerio de Educaci´ on y Ciencia, Spain” (MTM2005-07347, MTM2007-60854, MTM2006-26627-E, MTM2007-30904-E and Ingenio Mathematica (i-MATH) No. CSD2006-00032); from “La Junta de Andaluc´ıa” (FQM210 and P06-FQM01504); from “the Academy of Finland” (210245) and from the European Networking Programme “HCAA” of the European Science Foundation.

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1. Introduction and main results Let D denote the open unit disc of the complex plane C, D = {z ∈ C : |z| < 1} and let Hol(D) be the space of all analytic functions in D. If 0 < r < 1 and f ∈ Hol(D), we set  1/p  2π 1 |f (reit )|p dt , 0 < p < ∞, Mp (r, f ) = 2π 0 M∞ (r, f ) = sup |f (z)|. |z|=r

For 0 < p < ∞, α > −1 the Bergman space Apα is the set of all f ∈ Hol(D) such that  |f (z)|p dAα (z) < ∞, D

where dAα (z) = π −1 (1 + α)(1 − |z|2 )α dx dy = π −1 (1 + α)r(1 − r2 )α dr dθ. We shall write simply dA(z) for dA0 (z) and Ap for the classical Bergman space Ap0 . We mention [17] and [25] as general references for the theory of Bergman spaces. The Bloch space B consists of those f ∈ Hol(D) for which ρB (f ) = sup(1 − |z|2 )|f  (z)| < ∞. def

z∈D

The Bloch space is a Banach space with the norm  · B defined by f B = |f (0)| + ρB (f ),

f ∈ B.

A general reference for the Bloch space is [3]. If I ⊂ ∂D is an interval, |I| will denote the length of I. The Carleson square S(I) is defined as |I| ≤ r < 1} . 2π If s > 0 and µ is a positive Borel measure in D, we shall say that µ is an s-Carleson measure if there exists a positive constant C such that S(I) = {reit : eit ∈ I,

µ (S(I)) ≤ C|I|s ,

1−

for any interval I ⊂ T.

An 1-Carleson measure will be simply called a (classical) Carleson measure. If X is a subspace of Hol(D), 0 < p < ∞, and µ is a positive Borel measure in D, µ is said to be a p-Carleson measure for the space X if X ⊂ Lp (dµ). For a large class of spaces X ⊂ Hol(D) a characterization of the p-Carleson measures for the space X is known and such a characterization is useful in the study of the boundedness and the compactness of operators acting on X. Denote by H p (0 < p ≤ ∞) the classical Hardy spaces of analytic functions in D (see [15]). Carleson [11] characterized the p-Carleson measures for the Hardy space H p . Namely, he proved that H p ⊂ Lp (dµ), 0 < p < ∞, if and only if µ is a Carleson measure. The q-Carleson measures for the space H p were characterized by Duren [14] in the case 0 < p < q < ∞, and by Luecking [28] in the case 0 < q < p < ∞ (see also the recent paper [9]). Luecking [26, 29], characterized the q-Carleson measures for the

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space Apα , 0 < p, q < ∞. A good number of results about p-Carleson measures for Besov spaces and spaces of Dirichlet type of analytic functions have been obtained by different authors (see, e. g., [4], [5], [20], [21], [22], [33], [39], [42], [43], and [44]). Our main purpose in this paper is studying the Carleson measures for the Bloch space. A p-Carleson measure for the Bloch space will be simply called a p-Bloch-Carleson measure. Using the closed graph theorem, we see that µ is a p-Bloch-Carleson measure if and only if there exists a positive constant such that  |f (z)|p dµ(z) ≤ Cf pB (1.1) D

for all f ∈ B. Let us remark that since all constant functions are in B, a p-BlochCarleson measure is necessarily a finite measure. ∞ Let Hlog denote the space of those functions f ∈ Hol(D) such that M∞ (r, f ) =   1 ∞ ∞ O log 1−r , as r → 1. The space Hlog is a Banach space with the norm || · ||Hlog ∞ = sup defined by ||f ||Hlog z∈D

|f (z)| e log 1−|z|

< ∞.

It is clear that if 0 < p < ∞ and µ is a positive Borel measure in D such that p   1 dµ(z) < ∞, (1.2) log 1 − |z| D ∞ then µ is a p-Carleson measure for the space Hlog . ∞ Since it is also clear that B ⊂ Hlog , it follows that (1.2) implies that µ is a p-Bloch-Carleson measure. On the other hand, using an argument of Arazy, Fisher and Peetre [4] we shall prove that if µ is a p-Bloch-Carleson measure then p/2   1 dµ(z) < ∞. (1.3) log 1 − |z| D

That is, we have: Theorem 1.1. Suppose that 0 < p < ∞ and µ is a positive Borel measure in D. Then: (i) If µ satisfies (1.2), then µ is a p-Bloch-Carleson measure. (ii) If µ is a p-Bloch-Carleson measure, then µ satisfies (1.3). It is natural to ask whether either (1.2) or (1.3) is equivalent to saying that µ is a p-Bloch-Carleson measure. We recall that Ramey and Ullrich [35] proved that there exist two Bloch functions f1 and f2 such that (1 − |z|2 ) (|f1 (z)| + |f2 (z)|) ≥ 1,

for all z ∈ D.

If we were able to find two functions f1 , f2 ∈ B such that |f1 (z)| + |f2 (z)| ≥ log

1 , 1 − |z|

z ∈ D,

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then condition (1.2) would characterize p-Bloch-Carleson measures. However, such a pair of functions does not exist: Its existence would be in contradiction with the estimate  1/2  1 Mp (r, f ) = O log , as r → 1, (1.4) 1−r valid for all f ∈ B and 0 < p < ∞, which was proved by Clunie and MacGregor [12] and Makarov [30]. In any case, one may ask the following questions. Question 1.1. Do there exist two functions f1 , f2 ∈ B such that  1/2 1 , z ∈ D? |f1 (z)| + |f2 (z)| ≥ log 1 − |z| ∞ Question 1.2. Do there exist two functions f1 , f2 ∈ Hlog such that

|f1 (z)| + |f2 (z)| ≥ log

1 , 1 − |z|

z ∈ D?

We do not know the answer to Question 1.1. However, we will show that the answer to Question 1.2 is affirmative. ∞ Theorem 1.2. There exist two functions f1 , f2 ∈ Hlog such that

|f1 (z)| + |f2 (z)| ≥ log

1 , 1 − |z|

z ∈ D.

As a consequence we shall deduce that the p-Carleson measures for the space ∞ are characterized by condition (1.2): Hlog Theorem 1.3. Suppose that 0 < p < ∞ and µ is a positive Borel measure in D. ∞ if and only if it satisfies (1.2). Then µ is a p-Carleson measure for the space Hlog Theorem 1.2 and Theorem 1.3 will be proved in Section 8 where we shall ∞ which may be of independent interest. prove other results about the space Hlog ∞ In particular, we shall characterize the membership in Hlog of a class of functions given by power series with large gaps. These results will be applied in Section 9 to ∞ study the boundedness and compactness of a number of operators acting on Hlog . In particular, we shall give a complete characterization of the weighted composition ∞ operators which map Hlog continuously into Apα and will show that they are all compact. Back to the Bloch space, the estimate (1.4) might suggest that the condition (1.3) characterizes the p-Bloch-Carleson measures. However, we shall prove that this is not the case at all. In fact, we shall see that neither the converse of (i) nor the converse of (ii) is true in general. On the other hand, we shall find large classes of Borel measures µ in D for which (1.3) or (1.2) is equivalent to saying that µ is a p-Bloch-Carleson measure. All of this will be found in Section 5. More precisely, the following results related to condition (1.3) will be proved there.

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Theorem 1.4. Suppose that 0 < p < ∞, 0 < r < 1 and µ is a positive Borel measure in D for which there exists a positive constant C such that µ (∆(λ1 , r)) ≤ Cµ (∆(λ2 , r))

(1.5)

for every pair of points λ1 , λ2 ∈ D with |λ1 | = |λ2 |. Then µ is a p-Bloch-Carleson measure if and only if µ satisfies (1.3). Here, ∆(a, r) denotes the pseudohyperbolic disc of center a ∈ D and radius r. We remark that if µ is radial, that is, dµ(reiθ ) = dν(r), θ ∈ [0, 2π), for a certain positive Borel measure ν in (0, 1), then µ satisfies condition (1.5) of Theorem 1.4. Hence, we have: Corollary 1.1. Suppose that 0 < p < ∞ and µ is a positive Borel measure in D which is radial, that is, dµ(reiθ ) = dν(r), θ ∈ [0, 2π), for a certain positive Borel measure ν in (0, 1). Then µ is a p-Bloch-Carleson measure if and only if µ satisfies (1.3). We shall also find a large class of discrete measures µ in D for which µ is p-Bloch-Carleson measure if and only if it satisfies condition (1.2). One of our results of this kind is the following. Theorem 1.5. Suppose that 0 < p < ∞ and {ak }∞ k=1 is a sequence of positive numbers. Let ∞ µ= ak δ1−e−k , k=1

where, as usual, δzk denotes the point mass at zk . Then the following assertions are equivalent:

∞ p (i) k=1 ak k < ∞. (ii) µ satisfies (1.2). (iii) µ is a p-Bloch-Carleson measure. Corollary 1.2. Suppose that 0 < p < ∞ and {ak }∞ k=1 is a sequence of positive numbers such that ∞ ∞ ak k p/2 < ∞, and ak k p = ∞. Set µ = measure.

∞ k=1

k=1

k=1

ak δ1−e−k . Then µ satisfies (1.3) and it is not a p-Bloch-Carleson

The discrete measure µ constructed in Theorem 1.5 has its support contained in a radius. In Theorem 5.1 we shall construct sequences {zk } in D having every

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point ξ ∈ ∂D as an accumulation point

and with the property that if {ak } is a sequence of positive numbers and µ = ak δzk , then µ is a p-Bloch-Carleson measure ⇔ µ satisfies (1.2). Section 6 will be devoted to study the relationship between α-Carleson measures and p-Bloch-Carleson measures. We shall prove the following result. Theorem 1.6. (a) If µ is an α-Carleson measure in D for some α > 1, then µ is a p-Bloch-Carleson measure for all p > 0. (b) For every p ∈ (0, ∞) there exists a classical Carleson measure µ in D which is not a p-Bloch-Carleson measure. More precisely, given p ∈ (0, ∞) there exists a function g ∈ Hol(D) such that the measure µg,p in D given by dµg,p (z) = (1−|z|2)p−1 |g  (z)|p dA(z) is not a p-Bloch-Carleson measure but is a classical Carleson measure. (c) For every p ∈ (0, ∞) there exists a p-Bloch-Carleson measure which is not an α-Carleson measure for any α > 0. Next we shall restrict ourselves to the case p > 1. For this range of values of p we shall use duality arguments to obtain results which are stronger than those in Theorem 1.1. Hence, assume that 1 < p < ∞ and let p be conjugate exponent of p, that 1 is, p + p1
= 1. Let µ be a positive Borel measure in D. The duality relation 
∗ Lp (dµ) = Lp (dµ) immediately yields that  µ satisfies (1.2) ⇔ L (dµ) ⊂ L log p


1

 e dµ . 1 − |z|

(1.6)

If µ is a finite positive Borel measure in D, the Toeplitz operator Tµ is defined by

 Tµ (f )(z) =

D

f (w) dµ(w), (1 − wz)2

z ∈ D,

f ∈ L1 (dµ).

(1.7)

Zhu [49, Chapter 6] proved that whenever 1 < p < ∞, Tµ maps Ap continuously into itself if and only if µ is a 2-Carleson measure. Zhao [48] gave a sufficient condition on µ for Tµ to be continuous from A1 into itself. We shall prove the following result. Theorem 1.7. Suppose that 1 < p < ∞ and let µ be a positive Borel measure in D. Let p be conjugate exponent of p, that is, 1p + p1
= 1. Then the following conditions are equivalent: (a) µ is a p-Bloch-Carleson measure.
(b) The operator Tµ maps Lp (dµ) continuously into the Bergman space A1 . A (non-necessarily radial) positive Borel measurable function ω defined in D will be called a weight. For a weight ω, the weighted Bergman space Apα (ω)

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consists of those f ∈ Hol(D) such that  def p ||f ||Apα (ω) = |f (z)|p ω(z) dAα (z) < ∞. D

Definition 1.1. Suppose that p > 1 and α > −1. We shall say that the weight ω satisfies the Bekoll´e-Bonami Bp,α -condition if there exists a constant Cω > 0 such that   p−1  1 ω(z) dAα (z) ω(z) 1−p dAα (z) ≤ Cω Aα (S(I))p S(I)

S(I)

for every interval I ⊂ ∂D. This condition was introduced in [8] and [7] and allows us to identify the dual space of Apα (ω)(see [8] or [27]). The radial weight ω(z) = (1 − |z|)γ satisfies the condition Bp,α if and only if −1 − α < γ < (1 + α)(p − 1) (see [5, p. 445] for a similar statement). One can also check that the non-radial weight ω(z) = |1 − z|γ satisfies the condition Bp,α for γ ∈ (−2 − α, (2 + α)(p − 1)) \ {−1, p − 1}. Arcozzi, Rochberg and Sawyer [5] obtained a characterization of the pCarleson measures for the weighted Besov space   def 2 p−2  p Bp (ω) = f ∈ Hol(D) : (1 − |z| ) |f (z)| ω(z) dA(z) < ∞ D

if ω satisfies a certain regularity condition and the weight (1−|z|2 )p−2 ω(z) satisfies the condition Bp,0 . We shall prove the following result. Theorem 1.8. Suppose that p > 1, α > −1 and ω is a weight which satisfies the Bekoll´e-Bonami Bp,α -condition. Then, the measure µα,p defined by dµα,p (z) = (1 − |z|2 )α ω(z)dA(z) is a p-Bloch-Carleson measure, or, equivalently, B ⊂ Apα (ω) .

(1.8)

We remark that we do not assume any regularity condition on ω. It is also worth noticing that (1.8) is a generalization of the well-known inclusion B ⊂ Apα . We close this section noting that, as usual, we shall be using the convention that C will denote a positive constant which may be different at each occurrence.

2. Proof of Theorem 1.1 We only have to prove (ii). As mentioned above, this will be done by following a reasoning due to Arazy, Fisher and Peetre. Let L denote the class of those f ∈ Hol(D) which are given by a power series with Hadamard gaps, that is, there exists λ > 1 such that f is of the form ∞ ak z nk (z ∈ D), with nk+1 ≥ λnk for all k. (2.1) f (z) = k=0

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It is well known (see [3]) that if f ∈ L is as in (2.1), then f ∈ B ⇔ sup |ak | < ∞. k≥0

∞ k+1 Thus, if λ ≥ 2 is an integer and f (z) = k=0 z λ , z ∈ D, then f ∈ B. This implies that there exists a constant C > 0, depending only on λ, such that

∞ λk+1 1 ≤ C log 1−r , for all r ∈ [0, 1). The opposite inequality is also true. k=0 r Lemma 2.1. Let 1 < λ < ∞ and 0 < r0 < 1. Then there exists a positive constant C, depending only on λ and r0 , such that ∞

k+1

rλ

1 1−r

≥ C log

k=0

for all r ∈ [r0 , 1). Proof. Take λ and r0 as in the statement. Fix r ∈ (r0 , 1) and define x+1

φ(x) = rλ

, x > 0.

Since φ is a decreasing function, ∞ k=0

and the change of variable

k+1

rλ



∞

≥

x+1

rλ

dx,

0

  y x = logλ − −1 log r 2

and the elementary inequality log 1r ≤ 1−r r0 , r0 < r < 1, yield  ∞  ∞  ∞ x+1 1 dy dy 1 ≥ rλ dx = e−y e−y −1 log λ y log λ y 2 0 −λ log r λ(1−r )r0  2 rλ 0 dy 1 ≥ 2λ −1 y 2 r0 λ(1−r )r0 log λe 1 1 . ≥ 2λ log 1−r log λe r0




Proof of Theorem 1.1 (ii). Assume that µ is a p-Bloch-Carleson measure. Then there exists a positive constant C such that (1.1) is satisfied for all f ∈ B. Choose

∞ λk now f (z) = k=0 z , where λ ≥ 2 is a fixed natural number. Then f ∈ B. Replace f by fθ (z) = f (eiθ z), integrate with respect to θ, use Fubini’s theorem and Zygmund’s result on gap series (see Theorem 8.20 on p. 215 of Volume I of

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[50]) to obtain

p  2π

∞

k k 1

≥ z λ eiθλ dθ dµ(z)

D 2π 0 k=0 p 2   ∞ 2λk |z| dµ(z) 

Cf pB

D

≥

k=0

  ∞ D

λk+1

 p2

|z|

dµ(z),

k=0




and the assertion follows by Lemma 2.1.

3. The conformally invariant Bloch space In this section we shall focus in one of the most relevant properties of the Bloch space, that of being a conformally invariant space. For a ∈ D, we let the M¨obius map ϕa : D → D be defined by ϕa (z) =

a−z , 1 − az

z ∈ D.

Then ϕa is an involutive conformal mapping from D onto itself. Let Aut(D) denote the group of all conformal mappings from D onto itself. It is well known that Aut(D) coincides with the set of all M¨obius transformations from D onto itself: Aut(D) = M¨ob(D) = {λϕa : a ∈ D, |λ| = 1}. A space X of analytic functions in D, equipped with a semi-norm ρ, is said to be conformally invariant or M¨ obius invariant if whenever f ∈ X, then also f ◦ ϕ ∈ X for any ϕ ∈ Aut(D) and, moreover, ρ(f ◦ ϕ) ≤ Cρ(f ) for some positive constant C and all f ∈ X. The Bloch space is conformally invariant: ρB (f ◦ ϕ) = ρB (f ),

for all f ∈ Hol(D) and all ϕ ∈ Aut(D).

We remark that Rubel and Timoney [36] proved that if X is any “reasonable” M¨ obius invariant space of analytic functions in D, then it is continuously contained in the Bloch space. Let us be more precise. Definition 3.1. Let X be a space of analytic functions in D. A nonzero linear functional L on X is said to be decent if there exists a positive constant M and a compact subset K of D such that |L(f )| ≤ M sup |f (z)|, z∈K

for all f ∈ X.

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We remark that for any given a ∈ D, the point evaluation functional La defined by La (f ) = f (a) is a decent linear functional in Hol(D). Likewise, for n = 1, 2, . . . , the functional La,n defined by La,n (f ) = f (n) (a) is also a decent linear functional in Hol(D). The above mentioned result of Rubel and Timoney is the following one. Theorem A. Let X be a M¨ obius-invariant linear space of analytic functions in D equipped with the M¨ obius-invariant seminorm . If there exists a decent linear functional L on X which is continuous with respect to , then X ⊂ B and there exists A > 0 such that B (f ) ≤ A(f ),

for all f ∈ X.

There are a lot of characterizations of the Bloch space. We shall see that some of them can be obtained using Theorem A. In order to do so we need some further results and concepts. Lemma 3.1. Let µ be a positive Borel measure in D. Then the following assertions are equivalent: (i) µ is a finite measure. (ii) There is a positive constant C such that  (1 − |z|2 )p |f  (z)|p dµ(z) ≤ CB (f )p , for all f ∈ B D

Proof. The implication (i) ⇒ (ii) follows directly from the definition of B. Suppose (ii). As we mentioned above, there exist two functions f1 , f2 ∈ B such that (1 − |z|2 ) (|f1 (z)| + |f2 (z)|) ≥ 1, for all z ∈ D. Then, we have   ∞ > (1 − |z|2 )p |f1 (z)|p dµ(z) + (1 − |z|2 )p |f2 (z)|p dµ(z) D D 2 p  p  = (1 − |z| ) (|f1 (z)| + |f2 (z)|p ) dµ(z) D  p ≥ Cp (1 − |z|2 )p (|f1 (z)| + |f2 (z)|) dµ(z) D

≥ Cp µ (D) .


Hence, (i) holds.

Definition 3.2. Suppose that 0 < p < ∞ and let µ be a positive Borel measure in D. We let M¨ ob (Dp (µ)) denote the space of those functions f ∈ Hol(D) such that 1/p 

p def  2 p

M¨ob(Dp (µ)) = sup (1 − |z| ) (f ◦ ϕ) (z) dµ(z) < ∞. ϕ∈M¨ ob(D)

D

obius invariant space with the M¨ obius-invariant It is clear that M¨ ob (Dp (µ)) is a M¨ seminorm M¨ob(Dp (µ)) .

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Using Theorem A and Lemma 3.1 we obtain the following characterization of the Bloch space. Theorem 3.1. Suppose that 0 < p < ∞ and let µ be a positive and finite Borel measure in D such that there exists a decent linear functional L on M¨ ob (Dp (µ)) continuous with respect the seminorm M¨ob(Dp (µ)) . Then, B = M¨ ob (Dp (µ)) , and M¨ob(Dp (µ)) B (f ),

f ∈ B.

Theorem 3.1 includes as particular cases some well known results. Let us mention some of them. Fix α > −1 and 0 < p < ∞. Note that the weighted-area-measure Aα satisfies the conditions for the measure which appears in Theorem 3.1. Indeed, it is a finite measure and the functional L defined by L(f ) = f  (0) is a decent linear functional in Hol(D) and there exists a positive constant Cp,α such that  1 r(1 − r2 )α Mpp (r, f  ) dr |L(f )|p = |f  (0)|p ≤Cp,α 0

≤Cp,α pM¨ob(Dp (Aα )) (f ), for all f ∈ M¨ ob (Dp (Aα )). Thus, Theorem 3.1 implies that B = M¨ ob (Dp (Aα )) 

and pB (f ) sup a∈D

D

p

 (1 − |z|2 )p+α (f ◦ ϕa ) (z) dA(z).

(3.1)

Taking p = 2 in (3.1) we obtain that  1/2   B (f ) sup (1 − |z|2 )s | (f ◦ ϕa ) (z)|2 dA(z) , for all s > 1. a∈D

D

This is equivalent to saying that B = Qs ,

s > 1,

a result which was proved by Xiao [45] for s = 2 and by Aulaskari and Lappan [6] for all s > 1. We refer to the monographs [46] and [47] for the theory of the Qs -spaces. Taking α = 0 in (3.1), we easily obtain that  1/p    2 2 2 p−2  p f ∈ B ⇔ sup 1 − |ϕa (z)| (1 − |z| ) |f (z)| dA(z) , a∈D

D

a result which was proved by Stroethoff [40].

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4. The Bloch space and hyperbolic geometry The pseudohyperbolic metric in the unit disc will be denoted by δ:

z−w

, z, w ∈ D. δ(z, w) = |ϕw (z)| =

1 − wz The Schwarz-Pick lemma tells us that δ(f (z), f (w)) ≤ δ(z, w) whenever f is an analytic self-map of D and z, w ∈ D. Also, equality holds whenever f is a disc automorphism, i.e., δ is invariant under the conformal maps of the disc onto itself. The pseudohyperbolic disc of center a and radius r (a ∈ D, 0 < r < 1) is the set ∆(a, r) = {z ∈ D : δ(a, z) < r}. It coincides with the Euclidean disc whose (Euclidean) radius R and center c are (see, e. g., p. 40 of [17]): R=

1 − |a|2 r, 1 − r2 |a|2

c=

1 − r2 a. 1 − r2 |a|2

A sequence {ak }∞ k=1 ⊂ D is said to be uniformly discrete if it is separated in the pseudohyperbolic metric, that is, if there exists a positive constant γ such that δ(aj , ak ) ≥ γ,

k = 1, 2, . . . .

Uniformly discrete sequences play an important role in different and interesting questions of the theory of Bergman spaces, (see, e. g., [16], section 2.11 of [17], [18] and [23]). An important subclass of uniformly discrete sequences is that of uniformly separated sequences or interpolating sequences. A sequence {ak } of points in D is uniformly separated if there exists a positive constant γ such that 

aj − ak



δ(aj , ak ) ≥ γ, k = 1, 2, . . . .

1 − aj ak = j=k

j=k

We recall that Carleson [11] (see also [15, Chapter 9]) proved that these are the universal interpolating sequences for the space H ∞ of bounded analytic functions in D. Definition 4.1. Suppose that 0 < r < 1 and let {λk } be a sequence of points in D. We shall say that {λk } is an r-dense sequence if D = ∪k ∆(λk , r). If in addition there exists a positive integer N such that no point z ∈ D belongs to more that N of the dilated discs ∆(λk , 1+r 2 ), we say that {λk } is an r-dense sequence of finite order. The smallest N with this property will be called the order of {λk }. The following result is stated as Lemma 12 on p. 62 of [17]. Proposition A. For each r ∈ (0, 1) there exists a sequence of points in D which is an r-dense sequence of finite order. Using this and some other results one can prove the following:

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Proposition 4.1. Suppose that 0 < r < 1 and let µ be a positive Borel measure in D. (i) Let α > 1. Then, µ is an α-Carleson measure if and only if there exists a constant Cr > 0 such that µ (∆(a, r)) ≤ Cr (1 − |a|)α ,

a ∈ D.

(4.1)

(ii) Suppose that s > 0 and µ is an s-logarithmic Carleson measure, that is, there exists a positive constant C such that µ (S(I)) ≤

C|I| 2 , logs |I|

for any interval I ⊂ ∂D.

(4.2)

Then there exists a positive constant Cr such that µ (∆(a, r)) ≤

Cr (1 − |a|) , 2 logs 1−|a|

a ∈ D.

(4.3)

Proposition 4.1 (i) for α = 2 is included in [17, Theorem 14, p. 62]. The argument used there in the proof (see [17, pp. 65-66]) can be used to prove (i) for any α > 1 and (ii) (see Lemma 9.5 of [20]). We omit the details. Let β denote the hyperbolic distance in D: β(z, w) =

1 1 + δ(z, w) log . 2 1 − δ(z, w)

Clearly, the hyperbolic metric is also M¨ obius invariant. We recall the following characterizations of Bloch functions (see, e. g. [10], [40] or [49, Chapter 5]). Theorem B. Let 0 < r < 1. For a function f ∈ Hol(D) the following quantities are equivalent: (i) ρB (f ). def

(ii)  (f ) = supz∈D supw∈∆(z,r) |f (z) − f (w)|. def

(iii) , (f ) = supz,w∈D

|f (z)−f (w)| . β(z,w)

Consequently, f ∈ B ⇔  (f ) < ∞ ⇔ , (f ) < ∞. Using these results and r-dense sequences of finite order we can give a “discrete formulation” of the concept p-Bloch-Carleson measures. Proposition 4.2. Suppose that p ∈ (0, ∞), r ∈ (0, 1) and µ is a finite positive Borel measure. Let {λk } ⊂ D be an r-dense sequence of finite order. Then µ is a p-BlochCarleson measure if and only if there exists a positive constant C = C(r, µ) such that ∞ |f (λk )|p µ (∆(λk , r)) ≤ C||f ||pB (4.4) k=1

for all f ∈ B.

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Proof. Suppose f ∈ B and {λk } is an r-dense sequence of finite order. If µ is a p-Bloch-Carleson measure in D, then Theorem B gives ∞ |f (λk )|p µ (∆(λk , r)) =

k=1 ∞  k=1



≤C

|f (λk )|p dµ(z) ∆(λk ,r)

∞ 

∆(λk ,r)

k=1

 ≤C

|f (z) − f (λk )| dµ(z) + p



p

 (f )

∞

µ (∆(λk , r)) +

k=1

∞  k=1



∞ 

|f (z)| dµ(z) p

∆(λk ,r)

k=1



|f (z)| dµ(z) p

∆(λk ,r)

    p p ≤ C  (f ) µ (D) + |f (z)| dµ(z) D

≤ C||f ||pB . Conversely, if µ satisfies (4.4), bearing in mind again Theorem B, we deduce  |f (z)|p dµ(z) D

≤

∞  k=1



≤C

|f (z)|p dµ(z)

∆(λk ,r)

∞ 

k=1

 ≤C



|f (z) − f (λk )| dµ(z) + p

∆(λk ,r) p

 (f )

∞

k=1

µ (∆(λk , r)) +

k=1

∞ 

∞

 |f (λk )| dµ(z)

∆(λk ,r)

p



|f (λk )| µ (∆(λk , r)) p

k=1

≤ C ( (f )p µ (D) + ||f ||pB ) ≤ C||f ||pB .




5. Special classes of p-Bloch-Carleson measures In this section we shall prove Theorems 1.4 and 1.5 and some other related results. In connection with Proposition A, we shall prove next that a certain particular sequence is an r-dense sequence of finite order, for all r sufficiently close to 1. This sequence will be used in the proof of Theorem 1.4. Lemma 5.1. Let  def  zk,j = 1 − 2−k exp



 2πj i , 2k

k = 0, 1, 2 . . .

(5.1)

j = 0, 1, . . . , 2 − 1. k

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Then the following assertions hold: (i) {zk,j } is a uniformly discrete sequence with constant γ ≥ 13 . def

(ii) {zk,j } is an r-dense sequence of finite order for all r ≥ r0 = √982 . (iii) Let Ik,j be the smallest arc on the circumference {z ∈ D : |z| = 1 − 2−k } from zk,j to zk,j+1 , then δ(z, zk,j ) ≤ r0 for all z ∈ Ik,j . Proof. The assertion in Part(i) follows from Theorem 3 in [18]. To prove (ii), for given z = reiθ ∈ D, take k ∈ N and j ∈ {0, 1, . . . 2k − 1} such that   2πj 2π(j + 1) , 1 − 2−k ≤ r < 1 − 2−(k+1) and θ ∈ . 2k 2k Then, setting rk = 1 − 2−k , we have 1 = 1 − δ(z, zk,j )2

(1 − rrk )2 + 4rrk sin2

=1+



θ−θk,j 2

(1 − r2 )(1 − rk2 ) (r − rk )2 + 4rrk sin2





θ−θk,j 2



(1 − r2 )(1 − rk2 )

2−2k−2 + π 2 2−2k+2 2−k−1 2−k −1 ≤ 1 + 2 + 23 π 2 ≤ 82 ≤1+

that is, δ(z, zk,j ) ≤ √982 , and therefore {zk,j } is an r-dense sequence for all r ≥ √982 . Now, bearing in mind (i) and Lemma 1 of [18], we see that if r ≥ r0 and N (r) ≥

196 , (1 − r)(3 + r)

  then no point z ∈ D belongs to more that N (r) of the dilated discs ∆ zk,j , 1+r . 2 So we have (ii). (iii) follows easily from the proof of (ii). This finishes the proof.
The following elementary lemma is well known (see, e. g. [17, Chapter 4]. Lemma A. Let 0 < r < 1. Then there exist positive constants C1 , C2 C3 and C4 which depend only on r such that, if a ∈ D and z ∈ ∆(a, r), then 1 − |a|2 ≤ C1 (1 − |z|2 ) ≤ C2 |1 − az| ≤ C3 A (∆(a, r))1/2 ≤ C4 (1 − |a|2 ).

(5.2)

Proof of Theorem 1.4. Let p, r, µ and A be as in Theorem 1.4. Theorem 1.1 (ii) shows that if µ is a p-Bloch-Carleson measure then it satisfies (1.3). Suppose now that µ satisfies (1.3). Let {zk,j } be the sequence constructed in Lemma 5.1. Using Lemma 5.1 and Proposition 4.2, we see that it suffices to prove

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that k

∞ 2 −1

|f (zk,j )|p µ (∆(zk,j , r)) ≤ C||f ||pB ,

f ∈ B.

(5.3)

k=0 j=0

Hence, take f ∈ B and set Rk = 1 − 2−k

k = 0, 1, . . .

Theorem B implies 2 −1  1 = |f (Rk eit )|p dt 2π j=0 Ik ,j k

Mpp (Rk , f )

≥C

k 2 −1

|f (zj,k )|p |Ik,j | −  (f )p

(5.4)

j=0 k

2 −1 1 ≥C k |f (zj,k )|p −  (f )p . 2 j=0

Consequently, using (1.5), (5.4), Lemma A, Theorem B and (1.4), we obtain k

∞ 2 −1

|f (zk,j )|p µ (∆(zk,j , r))

k=0 j=0

≤C ≤C

∞ k=0 ∞

µ (∆(Rk , r))

k 2 −1

|f (zk,j )|p

j=0

  µ (∆(Rk , r)) 2k Mpp (Rk , f ) +  (f )p

k=0



≤C

∞

µ (∆(Rk , r))

≤C

Mpp (Rk , f ) + ||f ||pB

j=0

k=0



k 2 −1

≤ C||f ||pB 

Mpp (Rk , f )µ (∆(zk,j , r)) + ||f ||pB

≤

 log

∞ 2 −1  k

  log D

−1 ∞ 2 k=0 j=0

k=0 j=0

C||f ||pB

µ (∆(Rk , r)) 2k  

k

k=0 j=0





k=0

k

∞ 2 −1

∞

∆(zk,j ,r)

e 1 − |z|

e 1 − |z|

p/2

(5.5)

µ (∆(zk,j , r)) 

dµ(z) + µ (D)



p/2 dµ(z) + µ (D)

≤ C||f ||pB . Hence, (5.3) holds.




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Now we turn to consider condition (1.2). Proof of Theorem 1.5. (i) ⇔ (ii) is obvious. (ii) ⇒ (iii) follows from Theorem 1.1. 1 , z ∈ D. Then f ∈ B and (iii) implies that (iii) ⇒ (i). Set f (z) = log 1−z p f ∈ L (µ). Hence,

p  ∞

log 1 dµ(z) = ak k p . ∞>

1 − z D k=1




This finishes the proof.

As we mentioned in Section 1, the family of discrete measures µ constructed in Theorem 1.5 for which (ii) and (iii) are equivalent, are supported in a radius. It is natural to ask whether a condition “close” to this is needed to make such a construction. We shall show that the answer to this question is negative in a very strong sense. Theorem 5.1. Suppose that 0 < p < ∞. For every η > 1 there is a sequence (η) {zk }∞ and with the k=1 ⊂ D having every point ξ ∈ ∂D as an accumulation point

∞ property that if {ak }∞ is a sequence of positive numbers and µ = (η) , k=1 k=1 ak δzk then the following statements are equivalent: (i) µ satisfies (1.2). (ii) µ is a p-Bloch-Carleson measure. (η)

The sequences {zk } which appears in the statement of Theorem 5.1 are interpolating sequences for the Bloch space, which, according to the equivalence (i) ⇔ (iii) in Theorem B are defined as follows: Definition 5.1. A sequence of points {zn } ⊂ D is said to be an interpolating sequence for the Bloch space if for any sequence of complex numbers {wn } for which there exists a constant C > 0 such that |wn − wm | ≤ Cβ(zn , zm ),

n, m = 1, 2 . . . ,

one can find a function f ∈ B such that f (zn ) = wn ,

n = 1, 2 . . . .

Boe and Nicolau [10] obtained the following characterization of interpolating sequences for the Bloch space (see also [31]). Theorem C. A sequence of points {zn } ⊂ D is an interpolating sequence for the Bloch space if and only if the following two conditions hold: (a) {zn } is the union of two uniformly discrete sequences. (b) There exist constants M > 0 and α ∈ (0, 1) such that   # zk ∈ S(I) : 2−N −1 |I| < 1 − |zk | < 2−N |I| ≤ M 2αn , N = 1, 2, . . . , for any interval I ⊂ ∂D.

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The sequence {zk,j } constructed in Lemma 5.1 is uniformly discrete but it is not an interpolating sequence for the Bloch space. Next, we modify this sequence to obtain an interpolating sequence for the Bloch space. Lemma 5.2. For η > 1 set    2πj (η) def  −ηk exp i , zk,j = 1 − 2 2k

k = 0, 1, 2 . . .

(5.6)

j = 0, 1, . . . , 2k − 1. (η)

The sequence {zk,j } is an interpolating sequence for the Bloch space. (η)

Proof. Arguing as in the proof of Theorem 3 of [18] we can show that {zk,j } is a uniformly discrete sequence. Hence, it remains to prove that it satisfies condition (b) of Theorem C. Let I ⊂ ∂D be an interval, and denote Sk = {z ∈ C : |z| = 1 − 2−ηk },

k = 2, 3, . . . . (η)

Let γk be the number of points of the sequence {zk,j } in the arc Sk ∩ S(I). It is easy to see that γk ≤ C 2k |I|. On the other hand, 2−N −1 |I| < 1 − |zk,j | < 2−N |I|  N  N +1  2 2 ⇔ η −1 log2 < k < η −1 log2 . |I| |I|  N  N +1  For simplicity, set AN = η −1 log2 2|I| , BN = η −1 log2 2 |I| . Then it follows that   (η) (η) # zk,j ∈ S(I) : 2−N −1 |I| < 1 − |zk,j | < 2−N |I| γk ≤ C|I| 2k ≤ (η)

AN ≤k≤BN

≤C |I|2

BN

= C |I|



AN ≤k≤BN

2

N +1

1/η

|I|

1

≤ C 2N/η |I|1− η ≤ C 2N/η .

(η)

Since η > 1, this shows that {zk,j } satisfies condition (b) of Theorem C and finishes the proof.
(η)

Lemma 5.3. Suppose that η > 1 and let {zk,j } be the sequence defined in (5.6). Then there exists a positive constant C = C(η) such that



  1 1

(η) (η) ≤ Cβ z (5.7) − log , z

log

k,j n,l (η) (η)

1 − |zk,j | 1 − |zn,l |

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for all k, n, j, l with k, n ∈ {0, 1, 2 . . . }, j ∈ {0, 1, . . . , 2k − 1} and l ∈ {0, 1, . . . , 2n − 1}. Proof. Without loss of generality we may assume that k > n. Then,

(η)

1 − |zn,l | 1 1

− log .

log

= log (η) (η) (η)

1 − |zk,j | 1 − |zn,l | 1 − |zk,j | 2

1+x So, using the elementary inequality 1+x 1−x2 ≤ 1−x (0 < x < 1), it is clear that it suffices to prove that there exists a positive constant C = C(η) such that  C (η) (η) (η) 1 + 2 (zn,l , zk,j ) 1 − |zn,l | ≤ (η) (η) (η) 1 − |zk,j | 1 − 2 (zn,l , zk,j ) (5.8) C  (η) (η) 2 (η) (η) 2 |1 − zk,j zn,l | + |zk,j − zn,l |  , = (η) (η) (1 − |zk,j |2 )(1 − |zn,l |2

whenever k > n, j = 0, 1, . . . , 2k − 1, and l = 0, 1, . . . , 2n − 1. Now, take C ∈ N such that   C 1 1 − η > 1. 2 2

(5.9)

Using (5.9) and the inequality (1 + b)C ≥ 1 + Cb, (b ≥ 0) we deduce that  C (η) (η) (η) (η) |1 − zk,j zn,l |2 + |zk,j − zn,l |2   (η) (η) (1 − |zk,j |2 )(1 − |zn,l |2 )  2  2 C (η) (η) (η) (η) 1 − |z ||z | + |z | − |z | k,j n,l k,j n,l   ≥  (η) 2 (η) 2 (1 − |zk,j | )(1 − |zn,l | )   = 1 + 2



(η)

(η)

(1 − |zk,j |2 )(1 − |zn,l |2 ) 

≥ 1 + 2C

2 (η) (η) |zk,j | − |zn,l |

2 (η) (η) |zk,j | − |zn,l | (η)

(η)

(1 − |zk,j |2 )(1 − |zn,l |2 )

C

(5.10)

 

,

k > n, j = 0, 1, . . . , 2k − 1, l = 0, 1, . . . , 2n − 1. Putting together (5.8) and (5.10) we see that it suffices to prove that  2 (η) (η) (η) |zk,j | − |zn,l | 1 − |zn,l | ≤ 1 + 2C , (5.11) (η) (η) (η) 1 − |zk,j | (1 − |zk,j |2 )(1 − |zn,l |2 )

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whenever k > n, j = 0, 1, . . . , 2k − 1 and l = 0, 1, . . . , 2n − 1. Now,  2 (η) (η) (η) (η) |zk,j | − |zn,l | |zk,j | − |zn,l | (5.11) ⇔ ≤ 2C (η) (η) (η) 1 − |zk,j | (1 − |zk,j |2 )(1 − |zn,l |2 )   (η) (η) (η) (η) (η) ⇔ (1 + |zk,j |)(1 + |zn,l |)(1 − |zn,l |) ≤ 2C |zk,j | − |zn,l | . Notice that (5.9) implies that the last inequality is true if k > n. Thus, (5.11) holds for the values of k, n, l, j under consideration.
Theorem 5.1 is a consequence of the following result. (η)

Proposition 5.1. Suppose that η > 1 and let {zk,j } be the sequence defined in (5.6). Let {ak,j : k = 0, 1, . . . , j = 0, 1, . . . 2k − 1} be a sequence of positive numbers and set k

µ=

∞ 2 −1

ak,j δz(η) . k,j

k=0 j=0

Then the following conditions are equivalent:

∞ p 2k −1 (i) k=0 k j=0 ak,j < ∞. (ii) µ satisfies (1.2). (iii) µ is a p-Bloch-Carleson measure. Proof. We observe that   D

1 log 1 − |z|

p



k

dµ(z) =

∞ 2 −1

ak,j

log

k=0 j=0 p

= (η log 2)

∞ k=0

k

p

p

1 (η)

1 − |zk,j |

k −1 2

ak,j ,

j=0

which proves (i) ⇔ (ii). (ii) ⇒ (iii) follows from Theorem 1.1. (iii) ⇒ (ii). Denote (η)

wk,j = log

1 (η)

1 − |zk,j |

k, n = 0, 1, 2 . . . j = 0, 1, . . . , 2k − 1,

l = 0, 1, . . . , 2n − 1.

Using Lemma 5.1 and Lemma 5.3 we see that there exists a function F ∈ B (η) (η) such that F (zk,j ) = wk,j for k, n = 0, 1, 2 . . . , j = 0, 1, . . . , 2k − 1 and, l = 0, 1, . . . , 2n − 1. Using (iii), we obtain

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 ∞>

=

k

|F (z)| dµ(z) = p

D

k ∞ 2 −1

k=0 j=0

k=0 j=0

 ak,j

∞ 2 −1

log

1

  p

(η) ak,j F zk,j p p

(η)

1 − |zk,j |

531

= (η log 2)

∞

k

k=0

p

k −1 2

ak,j ,

j=0

which proves (ii) and finishes the proof.




6. Relationship between α-Carleson measures and p-Bloch-Carleson measures In this section we shall prove Theorem 1.6. Its part (a) is a simple consequence of the following result. Theorem 6.1. Suppose that 0 < p < ∞ and s > p + 1. Let µ be a positive Borel measure in D which is an s-logarithmic Carleson measure, then µ is a p-BlochCarleson measure. Proof. Take r ∈ (0, 1) and let {λk }∞ k=1 ⊂ D be an r-dense sequence of finite order. Using Lemma A and Proposition 4.1, we obtain p   e dµ(z) log 1 − |z| D  p ∞  e ≤ dµ(z) log 1 − |z| k=1 ∆(λk ,r) p ∞  e ≤C µ (∆(λk , r)) log 1 − |λk | k=1 p   ∞ e µ (∆(λk , r)) log dA(z) ≤C m (∆(λk , r)) ∆(λk ,r) 1 − |z| k=1  p−s ∞  1 e ≤C dA(z) log 1 − |z| 1 − |z| k=1 ∆(λk ,r)  p−s  1 e dA(z) ≤C log 1 − |z| D 1 − |z| <∞, which, using Theorem 1.1, implies that µ is a p-Bloch-Carleson measure.




Theorem 1.6 (b) for 2 < p < ∞ follows from the constructions made in the proof of Theorem 2.1 of [21]. Indeed, let f and g be the functions constructed there. Then f ∈ B and the measure µg,p in D given by dµg,p (z) = (1 − |z|2 )p−1 |g  (z)|p dA(z) is a classical Carleson measure and satisfies

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 D

|f (z)|p dµg,p (z) = ∞.

Hence µg,p is not a p-Bloch-Carleson measure. Next we shall give a proof of Theorem 1.6 (b) valid for all p > 0. Proof of Theorem 1.6 (b). Take p ∈ (0, ∞) and set p 1 1+ . λ= p 2

2k Set f (z) = ∞ k=0 z , (z ∈ D). Then f ∈ B and it is well known that  M2 (r, f ) ≥ C log

1 1−r

1/2

1 < r < 1. 2

,

(6.1)

Since f is given by a power series with Hadamard gaps, using Theorem 8.25 in chapter V of Vol. I of [50], we see that there exist two absolute constants A > 0 and B > 0 such that for every r ∈ (0, 1) the set Er = {t ∈ [0, 2π] : |f (reit )| > BM2 (r, f )} has Lebesgue measure greater than or equal to A, |Er | ≥ A,

0 < r < 1.

(6.2)

Define 

φ(r) =

1

(1 − r) log



eλ 1−r

λ ,

0 ≤ r < 1.

(6.3)

Then φ is an increasing function defined in (0, 1) and, since λp > 1,  1 (1 − r)p−1 φp (r) dr < ∞. 0

Using Theorem 3.3 and Theorem 3.2 of [21], we see that there exists a function g ∈ Hol(D) which is given by a power series with Hadamard gaps and such that the measure µg,p is a classical Carleson measure and satisfying M2 (r, g  ) ≥ φ(r),

r ∈ (0, 1).

Now, we assert that there exists a positive constant C such  |g  (reit )|p dt ≥ CM2p (r, g  ), 0 < r < 1

(6.4)

(6.5)

Er

(see the last paragraph of p. 421 of [21] for the case 2 < p < ∞, and Lemma 1 of [24] for the case 0 < p ≤ 2). Bearing in mind the definition of the sets Er

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(0 < r < 1) and using (6.5), (6.1), (6.4) and the definition of λ, we obtain  |f (z)|p dµg,p (z) D  = (1 − |z|2 )p−1 |g  (z)|p |f (z)|p dA(z) D





1

≥C

|g  (reit )|p |f (reit )|p dtdr

(1 − r)

p−1

1/2  1

Er



|g  (reit )|p dtdr

(1 − r)p−1 M2p (r, f )

≥C 1/2 1

 ≥C

Er

(1 − r)p−1 M2p (r, f )M2p (r, g  ) dr

1/2



1

(1 − r)

≥C

p−1

1/2  1

≥C 1/2



1

=C 1/2



 log

1 1−r

dr

1 (1 − r) log 1−r

 p2

φp (r) dr

pλ− p2

dr   1 (1 − r) log 1−r

= ∞. Since f ∈ B, this implies that µg,p is not a p-Bloch-Carleson measure.




Proof of Theorem 1.6 (c). Take p > 0 and s > p + 1. Set µ=

∞ 1 δ −k . k s 1−e k=1

Using Theorem 1.5, we see that µ is a p-Bloch-Carleson measure. For 0 < h < 1, set Ih = {eit : − h2 < t < h2 }. We have  1−s 1 1 µ (S(Ih )) = log . ks h 2π k≥log

(6.6)

h

1 1−s log h hα

( ) Since → ∞, as h → 0, for every α > 0, (6.6) implies that µ is not an α-Carleson measure for any α > 0.


7. The case p > 1 Proof of Theorem 1.7. We shall use an argument which is closely related to that used by Arcozzi, Rochberg and Sawyer in the proof of [5, Theorem 7]. Let p, p

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  and µ be as in Theorem 1.7. Les us recall that B = A1 under the pairing ·, ·A2 defined by  f (z)g(z) dA(z) f, gA2 = D


 (see [49, Theorem 5. 1. 4]). We also have Lp (dµ) = Lp (dµ) under the pairing ·, ·L2 (dµ) ,

 f, gL2 (dµ) =

f (z)g(z) dµ(z). D

Now, µ is a p-Bloch-Carleson measure if and only if B ⊂ Lp (dµ) and, by the closed graph theorem, this is the same as saying that the inclusion map Id is continuous from B to Lp (dµ). This is equivalent to saying that the map T defined by T f, gA2 = f, gL2 (dµ)

(7.1)




is continuous from Lp (dµ) to A1 . Summarizing, we have The measure µ is a p-Bloch-Carleson measure ⇔


the operator T defined by (7.1) is continuous from Lp (dµ) to A1 .

(7.2)

Now, since the Bergman kernel K(z, w) =

1 , (1 − zw)2

z, w ∈ D,

reproduces each f ∈ A2 (see, e. g., [49, p. 49]), (7.1) gives     g(w) f (z)g(z) dµ(z) = f (z) dA(w) dµ(z) (1 − wz)2 D D D   f (z) = g(w) dµ(z) dA(z) 2 D (1 − zw) D = g(w)T f (w) dA(w). D

Thus

 T f (w) = D

f (z) dµ(z), (1 − zw)2

f ∈ A2 , w ∈ D.

That is, T = Tµ and the theorem follows from (7.2).




Now we turn to work with weights ω which satisfy the Bekoll´e-Bonami Bp,α condition. As mentioned in Section 1, this condition allows us to determine the dual space of the space Ap (ω). Indeed, we have (see [8] or [27]): Theorem D. Suppose that p > 1, α > −1 and ω is a weight which satisfies the Bekoll´e-Bonami Bp,α condition. Then, the dual space of Apα (ω) can be identified

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535

 

with Apα ω 1−p under the pairing 

 f, gA2α =

D

f (z)g(z) dAα (z) = lim− r→1

D

f (z)g(rz) dAα (z).

(7.3)

This result implies the following one. Lemma 7.1. Suppose that p > 1, α > −1 and ω is a weight  which  satisfies the

Bekoll´e-Bonami Bp,α condition. Then, the dual space of Apα ω 1−p can be identified with Apα (ω) under the pairing defined in (7.3). Proof. Using the fact that (1 − p)(1 − p) = 1, we easily deduce that ω satisfies the
Bekoll´e-Bonami Bp,α condition if and only if ω 1−p satisfies the condition Bp
,α . Then, using Theorem D it follows that  
  1−p  p 1−p
p 1−p
ω Aα ω = Aα = Apα (ω) under the A2α -pairing. This finishes the proof.




We shall also need the following result (see [25, Theorem 1.21]). Theorem E. Suppose that α > −1 and ω is a weight which satisfies the Bekoll´eBonami Bp,α condition. Then, the dual space of A1α can be identified with B under the pairing defined in (7.3). Proof of Theorem 1.8. Using Lemma 7.1, Theorem E and writting Id = T  for an appropriate operator T , we have that B ⊂ Apα (ω) ⇔ Id : B → Apα (ω) is bounded  
  
⇔ T  : A1α → Apα ω 1−p is bounded  

⇔ T : Apα ω 1−p → A1α is bounded, where T is determined by the identity T (f ), gA2α = f, T  (g)A2α = f, gA2α . It follows that T = Id , that is,  

B ⊂ Apα (ω) ⇔ Apα ω 1−p ⊂ A1α .

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Let us prove that this last inclusion holds. Let f ∈ Apα ω 1−p , then by H¨older’s inequality   1−p
p
−1 |f (z)| dAα (z) = |f (z)|ω p
(z)ω p
(z) dAα (z) D D 1−p
1 = |f (z)|ω p
(z)ω p (z) dAα (z) D






≤ ||f ||p p


Aα (ω 1−p
)

D

1/p ω(z) dAα (z) < ∞,

since ω satisfies the Bekoll´e-Bonami Bp,α condition. This finishes the proof.




∞ 8. Some results on the space Hlog

Our main objective in this section is to give a proof of Theorem 1.3. In order to ∞ do so we shall start proving a number of results on the space Hlog which may be of independent interest. The following result is a direct consequence of Cauchy’s integral formula.

∞ ∞ Proposition 8.1. If f (z) = k=0 ak z k ∈ Hlog , then sup k≥2

|ak | < ∞. log k

(8.1)

∞ We observe that (8.1) is not sufficient to assert that f ∈ Hlog , even for f ∈ L,

∞ 1/2 2k as the example f (z) = k=1 k z shows. However, we shall prove that condition ∞ (8.1) does imply that f ∈ Hlog if f belongs to a certain subclass of L.

∞ Theorem 8.1. Let f ∈ Hol(D) with the power series expansion f (z) = k=0 ak z nk and suppose that there exist λ > 1 and A > 1 such that

nλk ≤ nk+1 ≤ nAλ k ,

for all k.

(8.2)

Then, the following conditions are equivalent: ∞ (i) f ∈ Hlog .

(ii) supk≥0

|ak | log nk

< ∞.

Proof. (i) ⇒ (ii) follows by Proposition 8.1. (ii) ⇒ (i). Take λ > 1 and A > 1 such that (8.2) holds. Let us assume, without loss of generality, that n0 ≥ 1. It is clear that it suffices to prove that there exists a positive constant C = C(λ), such that ∞ k=1

(log nk )rnk ≤ C log

e , 1−r

0 ≤ r < 1.

(8.3)

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We recall that there exists C = C(λ) > 0 such that ∞

k

rλ ≤ C log

k=0

e , 1−r

0 ≤ r < 1.

(8.4)

It follows from (8.2) that log nk+1 − log nk ≥ (λ − 1) log nk ,

k ∈ N,

(8.5)

and log nk+1 ≤ Aλ, k ∈ N, (8.6) log nk For k = 0, 1, 2, ..., set I(k) = [nk , nk+1 ) and let βk denote the number of terms of the sequence {λj n0 }∞ j=0 which are contained in I(k). Then, we claim that log nk+1 − log nk ≤ (βk + 1) log λ.

(8.7)

jk ,

Indeed, let jk and be the smallest and the biggest non-negative integer j such that λj n0 ∈ I(k). That is, 



λjk −1 n0 < nk ≤ λjk n0 ≤ λjk +1 n0 ≤ . . . λjk n0 < nk+1 ≤ λjk +1 n0 . Notice that βk = jk + 1 − jk and

     log nk+1 − log nk ≤ log λjk +1 n0 − log λjk −1 n0 = (jk + 2 − jk ) log λ = (βk + 1) log λ,

as claimed. Putting together (8.6), (8.5), (8.7) and (8.4), we deduce that ∞

log nk rnk =

k=1

∞

log nk+1 rnk+1

k=0

≤ Aλ

∞

log nk rnk+1

k=0 ∞

Aλ (log nk+1 − log nk ) rnk+1 λ−1 k=0 ∞  ∞ Aλ log λ nk+1 nk+1 ≤ βk r + r λ−1 k=0 k=0 ∞  ∞ k ≤C rλ n0 + rnk+1 ≤

k=0

e , ≤ C log 1−r which gives (8.3) and finishes the proof.

k=0




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We can now prove Theorem 1.2 with an argument similar to that used by Ramey and Ullrich in the proof of Proposition 5.4 of [35].

∞ qj Proof of Theorem 1.2. Let f (z) = j=0 q j z q , where q is a large positive integer ∞ . We first show that to be determined. Theorem 8.1 implies that f ∈ Hlog |f (z)| ≥ C log

1 , 1 − |z|

k

if 1 − q −q ≤ |z| ≤ 1 − q −q

k+ 1 2

.

(8.8)

We have that |f (z)| ≥ q k |z|q

qk

k−1

−

q j |z|q

qj

∞

−

j=0

q j |z|q

qj

(8.9)

j=k+1

= I(z) − II(z) − III(z),

z ∈ D.

qk

Take a point z as in (8.8) and let x = |z|q . Then, if q is sufficiently large  qk −q

 q 1 1 ≤x≤ 3 2 It follows that I(z) ≥ and

k−1

II(z) ≤

k+ 1 2



.

(8.10)

qk 3

(8.11)

qj ≤

j=0

qk . q−1

(8.12)

In order to estimate III(z) we observe that if j ≥ k + 1, qq

j+1

j

− qq ≥ qq

k+2

− qq

k+1

.

Then, bearing in mind (8.10) we have that  k+2  j ∞  k+1 qk+1 −qq qq III(z) ≤ q k+1 |z|q q|z| j=0 qk+1

≤

q k+1 |z|q qk+2 −q qk+1 ] 1 − q|z|[q

≤q

k



1 − qx

qxq q (q

k+2 −qk )

−q(q



≤ qk



q

 q

1−q

(qk+1 −qk )

1 2

 1 q



k+1 −qk )

qk+1 −q

2 qk+2 −q

k+ 1 2

k+ 1 2







−q

(8.13)



qk+1 −q

k+ 1 2



 

.

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  1 Since, limq→∞ q k+1 − q k+ 2 = ∞, we see that by taking q sufficiently large, (8.11), (8.12) and (8.13) give |f (z)| ≥

qk 1 1 ≥ 1 , log 4 1 − |z| 2 4q log q

k

if 1 − q −q ≤ |z| ≤ 1 − q −q

k+ 1 2

.

∞ 1 In a similar way it can be proved that if g(z) = j=0 q j+ 2 z nj where nj is 1 the integer closest to q j+ 2 , then k+ 1 k+1 1 , if 1 − q −q 2 ≤ |z| ≤ 1 − q −q . |g(z)| ≥ C1 log 1 − |z| Now the theorem follows by taking (z ∈ D),

f1 (z) = A + B f (z) and f2 (z) = C g(z),




for some appropriate positive constants A, B, C.

Proof of Theorem 1.3. We have already noticed that if µ satisfies (1.2) then it is ∞ (see Section 1). a p-Carleson measure for the space Hlog ∞ . Using Theorem 1.2, we Suppose now that µ is p-Carleson measure for Hlog ∞ can take f1 , f2 ∈ Hlog satisfying

|f1 (z)| + |f2 (z)| ≥ log

1 , 1 − |z|

z ∈ D.

Using this and the simple fact that (a + b)p ≤ 2p (ap + bp ), whenever a, b ≥ 0, we deduce that p    1 p dµ(z) ≤ (|f1 (z)| + |f2 (z)|) dµ(z) log 1 − |z| D D    p p ≤C |f1 (z)| dµ(z) + |f2 (z)| dµ(z) < ∞.

D

D




∞ 9. Operators acting on Hlog

For g analytic in D, the multiplication operator Mg is defined by def

Mg (f )(z) = g(z)f (z),

f ∈ Hol(D), z ∈ D.

(9.1)

If X and Y are two spaces of analytic functions in D, we let M (X, Y ) denote the space of all multipliers from X to Y , i e., M (X, Y ) = {g ∈ Hol(D) : Mg (X) ⊂ Y } . If ϕ is an analytic self-map of D, the composition operator Cϕ is defined by Cϕ f = f ◦ ϕ,

f ∈ Hol(D).

(9.2)

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More generally, given ϕ an analytic self-map of D and u an analytic function in D, the weighted composition operator Cϕu with symbols u and ϕ is defined as follows Cϕu (f )(z) = u(z)(f ◦ ϕ)(z),

f ∈ Hol(D), z ∈ D.

(9.3)

We remark that the weighted composition operators contain as particular cases the multiplication operators (taking ϕ equal to the identity mapping) and the composition operators (taking u ≡ 1). There is a very extense literature on weighted composition operators. Les us just mention that two excelent monographs on composition operators are those of Shapiro [37] and Cowen and MacCluer [13]. The question of boundedness and compactness of these operators has been studied in many function spaces. We shall use the results obtained in Section 8 to ∞ to the weighted Bergman study the boundedness and compactness of Cϕu from Hlog spaces Apα . We shall prove the following result. Theorem 9.1. Let ϕ be an analytic self-map of D, u an analytic function in D, p > 0 and α > −1. Then the following assertions are equivalent: ∞ to Apα . (1) Cϕu is a continuous operator from Hlog  p  e |u(z)|p log dAα (z) < ∞. (2) 1 − |ϕ(z)| D   p e (3) lim |u(z)|p log dAα (z) = 0. r→1 |ϕ(z)|>r 1 − |ϕ(z)| ∞ (4) Cϕu is a compact operator from Hlog to Apα .

We shall prove first that (1) ⇔ (2) ⇔ (3). Notice that the implication (2) ⇒ (3) follows from the dominated convergence theorem, while the implication (3) ⇒ (2) is obvious. The implication (2) ∞ ⇒ (1) follows directly from the definition of the space Hlog . Proof of (1) ⇒ (2). Suppose (1). We use Theorem 1.2 to pick two functions f1 , f2 ∈ ∞ Hlog such that |f1 (z)| + |f2 (z)| ≥ log

1 , 1 − |z|

z ∈ D.

(9.4)

Now, (1) implies that  p

D

|u(z)|p |(fj ◦ ϕ)(z)| dAα (z) < ∞,

j = 1, 2.

(9.5)

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Using (9.4), the elementary inequality (a + b)p ≤ 2p (ap + bp ) (a, b ≥ 0) and (9.5), we obtain  p  1 p |u(z)| log dAα (z) 1 − |ϕ(z)| D ≤ |u(z)|p (|(f1 ◦ ϕ)(z)| + |(f2 ◦ ϕ)(z)|)p dAα (z) D  ≤ Cp |u(z)|p (|(f1 ◦ ϕ)(z)|p + |(f2 ◦ ϕ)(z)|p ) dAα (z) < ∞.

D




The implication (4) ⇒ (1) is obvious. Hence, it only remains to prove that (3) ⇒ (4). We shall also use the following result of Tjani [41, Lemma 3.7]. Lemma B. Let X and Y be two Banach spaces (or complete metric spaces) of analytic functions on D, and let T : X → Y be a linear operator. Suppose that the following conditions are satisfied: (a) The point evaluation functionals on Y are bounded. (b) For every bounded sequence in X, there is a subsequence which converges uniformly to an element of X on compact subsets of D. (c) If {fn } ⊂ X converges uniformly to zero on compact subsets of D, then {T (fn )} converges uniformly to zero on compact subsets of D. Then T is a compact operator from X to Y if and only if for any bounded sequence {fn } in X such that fn → 0 uniformly on compact subsets of D, the sequence {T (fn )} converges to zero in the norm (or in the metric) of Y . Proof of the implication (3) ⇒ (4) in Theorem 9.1. Suppose (3). It is clear that we are in the conditions of Lemma B. Hence, it suffices to show that if {fn } is a ∞ bounded sequence in Hlog such that fn → 0, as n → ∞, uniformly on compact subsets of D, then Cϕu (fn )Apα → 0, as n → ∞. Hence, take such a sequence {fn }. We have  |Cϕu (fn )(z)|p dAα (z) {|ϕ(z)|>r}

 |u(z)| log

p e dAα (z), ≤ 1 − |ϕ(z)| {|ϕ(z)|>r}  p  e ≤C |u(z)|p log dAα (z), for all n. 1 − |ϕ(z)| {|ϕ(z)|>r} ||fn ||pH ∞ log



p

Take ε > 0. The last inequality and (3) imply that there exists r0 ∈ (0, 1) such that  ε |Cϕu (fn )(z)|p dAα (z) < , for all n. 2 {|ϕ(z)|>r0 }

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On the other hand, since fn converges to zero uniformly on compact subsets of D, there exists N ∈ N such that  ε , for all n ≥ N . |Cϕu (fn )(z)|p dAα (z) ≤ 2 {|ϕ(z)|≤r0 } Then, it follows that uCϕ (fn )Apα → 0 as n → ∞. This finishes the proof.




As we mentioned above composition operators are particular cases of weighted composition operators. Consequently, we can state the following result. Corollary 9.1. Let ϕ be an analytic self-map of D, p > 0 and α > −1. Then the following assertions are equivalent: ∞ to Apα . (1) Cϕ is a continuous operator from Hlog p   e log dAα (z) < ∞. (2) 1 − |ϕ(z)| D   p e (3) lim dAα (z) = 0. log r→1 |ϕ(z)|>r 1 − |ϕ(z)| ∞ (4) Cϕ is a compact operator from Hlog to Apα . A similar statement is valid for multiplication operators. However, we shall obtain some further results for these operators as well as for a class of integration operators. Given g ∈ Hol(D) integration operator Tg is defined as follows:  z Tg (f )(z) = f (ξ)g  (ξ) dξ, f ∈ Hol(D), z ∈ D. (9.6) 0

The operators Tg have been studied in a number of papers and contain as special cases a number of important operators such as the integration operator (Tg with g(z) = z) and the Ces` aro operator (Tg with g(z) = log(1/(1 − z))). Let us mention that Pommerenke [34] proved that Tg is bounded in the Hardy space H 2 if and only if g ∈ BM OA, Aleman and Cima characterized in [1] those g ∈ Hol(D) for which Tg maps H p into H q , Aleman and Siskakis [2] studies the operators Tg acting on Bergman spaces and the first two authors of this paper studied them acting on spaces of Dirichlet type in [22]. We have the following result. Theorem 9.2. Suppose that 0 < p < ∞ and −1 < α < ∞ and let g ∈ Hol(D). Then the following conditions are equivalent:   ∞ , Apα . (1) g ∈ M Hlog  p
e (2) D |g(z)|p log 1−|z| (1 − |z|)α dA(z) < ∞.   p
e (3) D |g  (z)|p log 1−|z| (1 − |z|)α+p dA(z) < ∞. ∞ to Apα . (4) Tg is a bounded operator from Hlog ∞ to Apα . (5) The multiplication operator Mg is a compact operator from Hlog ∞ p (6) The operator Tg is a compact operator from Hlog to Aα .

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Proof. Theorem 9.1 shows that (1) ⇔ (2) ⇔ (5). The equivalence (2) ⇔ (3) follows from results of Siskakis (see Theorem 1.1 and Example 3.1 of [38]) and of Pavlovi´c and Pel´ aez [32, Theorem 1.1]. A result of Hardy and Littlewood asserts that for 0 < p < ∞ and α > −1   p α p |g(z)| (1 − |z|) dA(z) |f (0)| + |g  (z)|p (1 − |z|)α+p dA(z) D

D

for all g ∈ Hol(D) (see Chapter 5 of [15] for information and references and Theorem 6 of [19] for a proof). Using this we have ∞ ) ⊂ Apα Tg (Hlog  ∞ ⇔ |(1 − |z|)α+p |f (z)g  (z)|p dA(z) < ∞, for all f ∈ Hlog D

∞ ⇔ (1 − |z|)α+p |g  (z)|p dA(z) is a p − Hlog -Carleson measure  p  e ⇔ |g  (z)|p (1 − |z|)α+p log dA(z) < ∞. 1 − |z| D

This shows that (3) ⇔ (4). The implication (6) ⇒ (4) is obvious. Thus, it only remains to prove that (3) ⇒ (6). Proof of (3) ⇒ (6). The proof is similar to that of the implication (3) ⇒ (4) in Theorem 9.1. Suppose (3). By Lemma B, it suffices to show that if {fn } is a bounded ∞ such that fn → 0, as n → ∞, uniformly on compact subsets of sequence in Hlog D, then Tg (fn )Apα → 0, as n → ∞. (9.7) Now, using the above mentioned result of Hardy and Littlewood, this is equivalent to saying that  |fn (z)g  (z)|p (1 − |z|)α+p dA(z) → 0, as n → ∞. (9.8) D

Hence, take such a sequence {fn }. Notice that, by the dominated convergence theorem, (iii) implies  p  e  p |g (z)| log (1 − |z|)α+p dA(z) = 0. (9.9) lim r→1 {|z|>r} 1 − |z| We have  {|z|>r}

≤

|fn (z)g  (z)|p (1 − |z|)α+p dA(z) 

||fn ||pH ∞ log 

≤C

{|z|>r}

 |g (z)| log 

{|z|>r}

 |g  (z)|p log

p

e 1 − |z|

p e (1 − |z|)α+p dA(z) 1 − |z| p (1 − |z|)α+p dA(z), for all n.

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Take ε > 0. The last inequality and (9.9) imply that there exists r0 ∈ (0, 1) such that  ε |fn (z)g  (z)|p (1 − |z|)α+p dA(z) < , for all n. 2 {|z|>r0 } On the other hand, since fn converges to zero uniformly on compact subsets of D, there exists N ∈ N such that  ε , for all n ≥ N . |fn (z)g  (z)|p (1 − |z|)α+p dA(z) ≤ 2 {|z|≤r0 } Then, (9.8) follows. This finishes the proof.




References [1] A. Aleman and J. A. Cima, An integral operator on H p and Hardy’s inequality, J. Anal. Math. 85 (2001), 157–176. [2] A. Aleman and A. G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997), no. 2, 337–356. [3] J. M. Anderson, J. Clunie and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12–37. [4] J. Arazy, S. D. Fisher and J. Peetre, M¨ obius invariant function spaces, Reine Angew. Math. 363 (1985), 110–145. [5] N. Arcozzi, R. Rochberg and E. Sawyer, Carleson measures for analytic Besov spaces, Rev. Mat. Iberoamericana 18 (2002), 443–510. [6] R. Aulaskari and P. Lappan, Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal, sl In: Complex Analysis and its Applications, Pitman Research Notes in Mathematics 305 (1994), 136–146. [7] D. Bekoll´e, In´egalit´es ´ a poids pour le projecteur de Bergman dans la boule unit´e de Cn (French), Studia Math. 71 (1981/82), 305–323. [8] D. Bekoll´e and A. Bonami, In´egalit´es ` a poids pour le noyau de Bergman (French), C. R. Acad. Sci. Paris S´er. A-B 286 (1978), no. 18, 775–778. [9] O. Blasco and H. Jarchow, A note on Carleson measures for Hardy spaces, Acta Sci. Math. (Szeged) 71 (2005), no. 1–2, 371–389. [10] B. Boe and A. Nicolau, Interpolation by functions in the Bloch space, J. Anal. Math. 94 (2004), 171–194. [11] L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921–930. [12] J. G. Clunie and T. H. MacGregor, Radial growth of the derivative of univalent functions, Comment. Math. Helv. 59 (1984), 362–365. [13] C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. [14] P. L. Duren, Extension of a Theorem of Carleson, Bull. Amer. Math. Soc. 75 (1969), 143–146.

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[15] P. L. Duren, Theory of H p Spaces, Academic Press, New York-London 1970. Reprint: Dover, Mineola, New York 2000. [16] P. L. Duren and A.P. Schuster, Finite union of interpolation sequences, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2609–2615. [17] P. L. Duren and A. P. Schuster, Bergman Spaces, Math. Surveys and Monographs, Vol. 100, American Mathematical Society: Providence, Rhode Island, 2004. [18] P. L. Duren, A. P. Schuster and D. Vukoti´c, On uniformly discrete sequences in the disk. In: Quadrature domains and their applications, The Harold S. Shapiro Anniversary Volume (P. Ebenfelt, B. Gustafsson, D. Khavinson, and M. Putinar, editors) 131–150, Oper. Theory Adv. Appl., 156, Birkh¨ auser, Basel, 2005. [19] T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38 (1972), 756–765. [20] D. Girela, M. Pavlovi´c and J. A. Pel´ aez, Spaces of analytic functions of Hardy-Bloch type, J. Anal. Math. 100 (2006), 53–81. [21] D. Girela and J. A. Pel´ aez, Carleson measures for spaces of Dirichlet type, Integral Equations and Operator Theory 55 (2006), no. 3, 415–427. [22] D. Girela and J. A. Pel´ aez, Carleson measures, multipliers and integration operators for spaces of Dirichlet type, J. Funct. Analysis 241 (2006), no. 1, 334–358. [23] D. Girela, J. A. Pel´ aez and D. Vukoti´c, Uniformly discrete sequences in regions with tangential approach to the unit circle, Complex Variables and Elliptic Functions 52 (2007), no. 2-3, 161–173. [24] D. Girela and M. A. Rodr´ıguez, Sharp estimates on the radial growth of the derivative of bounded analytic fucntions, Complex Variables Theory Appl. 28 (1996), 271–283. [25] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman Spaces, Graduate Texts in Mathematics, Vol. 199, Springer: New York, Berlin, etc. 2000. [26] D. H. Luecking, Forward and reverse inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math., 107 (1985), 85–111. [27] D. H. Luecking, Representation and duality in weighted spaces of analytic functions, Indiana Univ. Math. 42 (1985), no. 2, 319–336. [28] D. H. Luecking, Embedding derivatives of Hardy spaces into Lebesgue spaces, Proc. London Math. Soc. 63 (1991), 565–619. [29] D. H. Luecking, Embedding theorems for spaces of analytic functions via Khinchine’s inequality, Michigan Math. J. 40 (1993), no. 2, 333–358. [30] N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. (3) 51 (1985), 369–384. [31] D. Pascuas, A note on interpolation by Bloch functions, Proc. Amer. Math. Soc. 135 (2007), no. 7, 2127–2130. [32] M. Pavlovi´c and J. A. Pel´ aez, An equivalence for weighted integrals of an analytic function and its derivative, to appear in Math. Nach.

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[33] F. P´erez-Gonz´ alez and J. R¨ atty¨ a, Forelli-Rudin estimates, Carleson measures and F (p, q, s)-functions, J. Math. Anal. Appl. 315, (2006), 394–414. [34] Ch. Pommerenke, Schlichte Funktionen und analytische Funktionen von beschr¨ ankter mittlerer Oszillation, Comment. Math. Helv. 52 (1977), 122–129. [35] W. Ramey and D. Ullrich, Bounded mean oscillation of Bloch pull-backs, Math. Ann. 291 (1991), no. 4, 591–606. [36] L. E. Rubel and R. M. Timoney, An extremal property of the Bloch space, Proc. Amer. Math. Soc. 75 (1979), no. 1, 45–49. [37] J. H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. [38] A. Siskakis, Weighted integral and conjugate functions in the unit disk, Acta. Sci. Math. (Szeged) 66 (2000), 651–664. [39] D. A. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980), no. 1, 113–139. [40] K. Stroethoff, Nevalinna-type characterizations for the Bloch space and related spaces, Proc. Edin. Math. Soc. 33 (1990), no. 1, 123–141. [41] M. Tjani, Compact composition operators on Besov spaces, Trans. Amer. Math. Soc. 355, (2003), no. 11, 4683–4698. [42] I. E. Verbitsky, Multipliers in spaces with “fractional” norms, and inner functions (in russian), Sibirsk. Mat. Zh., 26 (1985), 51–72, 221. Translated in Siber. J. of Mathematics, 26 (1985), 198–216. [43] S. A. Vinogradov, Multiplication and division in the space of analytic functions with area integrable derivative, and in some related spaces (in russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222 (1995), Issled. po Linein. Oper. i Teor. Funktsii 23, 45–77, 308; translation in J. Math. Sci. (New York) 87 (1997), no. 5, 3806–3827. [44] Z. Wu, Carleson measures and multipliers for Dirichlet spaces, J. Funct. Anal. 169 (1999), 148-163. [45] J. Xiao, Carleson measure, atomic decomposition and free interpolation from Bloch space, Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), no. 1, 35–46. [46] J. Xiao, Holomorphic Q classes, Lecture Notes in Mathematics 1767, Springer-Verlag, Berlin 2001. auser Verlag, Basel [47] J. Xiao, Geometric Qp functions, Frontiers in Mathematics. Birkh¨ 2006. [48] R. Zhao, On logarithmic Carleson measures, Acta Sci. Math. (Szeged) 69 (2003), 605–618. [49] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990. [50] A. Zygmund, Trigonometric Series, Vol. I and Vol. II, Second edition, Camb. Univ. Press, Cambridge, 1959.

Vol. 61 (2008)

Carleson Measures for the Bloch Space

Daniel Girela Departamento de An´ alisis Matem´ atico Facultad de Ciencias Universidad de M´ alaga 29071 M´ alaga Spain e-mail: [email protected] ´ Jos´e Angel Pel´ aez Departamento de An´ alisis Matem´ atico Universidad de C´ ordoba Edificio Einstein Campus de Rabanales 14014 C´ ordoba Spain e-mail: [email protected] Fernando P´erez-Gonz´ alez Departamento de An´ alisis Matem´ atico Universidad de La Laguna 38271 La Laguna Tenerife Spain e-mail: [email protected] Jouni R¨ atty¨ a University of Joensuu Mathematics P. O. Box 111 80101 Joensuu Finland e-mail: [email protected] Submitted: December 3, 2007.

547

Integr. equ. oper. theory 61 (2008), 549–557 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/040549-9, published online July 4, 2008 DOI 10.1007/s00020-008-1598-1

Integral Equations and Operator Theory

Conditions Implying Self-adjointness of Operators In Ho Jeon, In Hyoun Kim, Kotaro Tanahashi and Atsushi Uchiyama Abstract. In this paper, we show that if T ∈ B(H) is a quasi-class A operator and S is an arbitrary operator for which 0 ∈ / W (S) and ST = T ∗ S, then T is self-adjoint, and we also show that quasisimilar quasi-class A operators have equal spectra and essential spectra. Mathematics Subject Classification (2000). Primary 47B20; Secondary 47A53. Keywords. Self-adjoint operator, class A operator, quasi-class A operator, quasisimilar, essential spectrum.

1. Introduction Throughout this paper let H and K be separable complex Hilbert spaces with inner product
·, ·, let B(H, K) denote the set of bounded linear operators from H to K, and abbreviate B(H, H) to B(H). We denote the spectrum, the essential spectrum and the numerical range of an operator T ∈ B(H) by σ(T ), σe (T ) and W (T ), respectively. The norm closure of a subspace M of H is denoted by M. We denote the kernel and the range of an operator T ∈ B(H) by ker(T ) and ran(T ), respectively. Recall [1] that an operator T ∈ B(H) is called p-hyponormal if (T ∗ T )p −(T T ∗)p ≥ 0 for some 0 < p ≤ 1. If p = 1, then T is said to be hyponormal and if p = 12 , then T is said to be semi-hyponormal. It is well known that any p-hyponormal operator is q-hyponormal for q ≤ p by L¨ owner’s theorem. Recall [10, 11] that an operator T ∈ B(H) is called quasi-class A operator if T ∗ |T 2 |T ≥ T ∗ |T |2 T . Quasi-class A operator is an extension of p-hyponormal operator, class A operator (i.e., |T 2 | − |T |2 ≥ 0)(see [3, 7]) and p-quasihyponormal operator (i.e., T ∗ (|T |2p − |T ∗ |2p )T ≥ 0)(see, [2, 6, 12]). A. Aluthge [1], T. Furuta, M. Ito and T. Yamazaki [7], S. C. Arora and P. Arora [2], and the authors [10] introduced This work of the second author was supported by the University of Incheon Research Grant in 2008.

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p-hyponormal operator, class A operator, p-quasihyponormal operator and quasiclass A operator, respectively. It was well known that p-hyponormal operator =⇒ class A operator =⇒ quasi-class A operator, p-hyponormal operator =⇒ p-quasihyponormal operator =⇒ quasi-class A operator, and these operators share many interesting properties with hyponormal operators. In general, the conditions S −1 T S = T ∗ and 0 ∈ / W (S) do not imply that T is normal. For example (see [21]), if T = SB, where S is positive and invertible, B / W (S), is self-adjoint, and S and B do not commute, then S −1 T S = T ∗ and 0 ∈ but T is not normal. Therefore the following question arises naturally: Question 1.1. Suppose that T is an operator for which there is an operator S with 0 ∈ / W (S) such that S −1 T S = T ∗ . When does it follow that necessarily T is normal ? In section 2, we show that if T is a quasi-class A operator and S is an arbitrary operator for which 0 ∈ / W (S) and ST = T ∗ S, then T is a self-adjoint. An operator X ∈ B(H, K) is called a quasiaffinity if X is injective and has dense range. Two operators S ∈ B(H) and T ∈ B(K) are called quasisimilar if there exist quasiaffinities X ∈ B(H, K) and Y ∈ B(K, H) for which XS = T X and SY = Y T . Quasisimilarity was first introduced by B. Sz-Nagy and C. Foias [15]. Quasisimilarity coincides with similarity in finite dimensional spaces, but in infinite dimensional spaces it is a much weaker relation. In fact, similarity preserves the spectrum and essential spectrum of an operator, but this fails to be true for quasisimilarity. What condition should be imposed on two quasisimilar operators S ∈ B(H) and T ∈ B(K) to insure the equality relation σ(S) = σ(T ) and σe (S) = σe (T ) ? In section 3, we show that quasisimilar quasi-class A operators have equal spectra and essential spectra.

2. Operators similar to their adjoints In this section we consider the extension of results of I. H. Sheth and I. H. Kim. In 1966, I. H. Sheth [18] showed that if T is a hyponormal operator and S −1 T S = T ∗ for any operator S, where 0 ∈ / W (S), then T is self-adjoint. Very recently, I. H. Kim [14] extended the result of Sheth to the class of p-hyponormal operators and also showed that if T is a (p, k)-quasihyponormal operator (i.e., T ∗ k (|T |2p −|T ∗ |2p )T k ≥ 0) and S ∗ is a p-hyponormal operator, and if T X = XS, where X : K → H is an injective bounded linear operator with dense range, then T is a normal operator unitarily equivalent to S. In this section, we show that if T is a quasi-class A

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operator and S is an arbitrary operator for which 0 ∈ / W (S) and ST = T ∗ S, then T is self-adjoint. Lemma 2.1. ([21, Theorem 1]) If T ∈ B(H) is any operator such that S −1 T S = T ∗ , where 0 ∈ / W (S), then σ(T ) ⊆ R. In Lemma 2.1, the condition, 0 ∈ / W (S), is essential. For example ([12, Example 1]), let W is the bilateral shift on l2 which is defined by W en = en+1 , where 2 {en }∞ n=−∞ is the canonical orthonormal basis for l , and let S be the unitary op−1 ∗ erator defined by Sen = e−n . Then S W S = W , but the spectrum of W is not real. Actually, the spectrum of W is the unit circle. Lemma 2.2. ([3, Theorem 2.1]) Let T = U |T | be the polar decomposition of a class A operator T . Then |T |U |T | is semi-hyponormal and σ(|T |U |T |) = {r2 eiθ : reiθ ∈ σ(T )}. Lemma 2.3. For any operator T ∈ B(H)
1
∗ 2  12 ≥ |T |2 ⇔ |T ∗ ||T |2 |T ∗ | 2 ≥ |T ∗ |2 . T |T | T Proof. Let T ∗ = U ∗ |T ∗ | be the polar decomposition of T ∗ . Suppose that
∗ 2  12 T |T | T ≥ |T |2 . Then

∗ 1 1 |T ||T |2 |T ∗ | 2 = U U ∗ |T ∗ ||T |2 |T ∗ | 2 U U ∗
1 = U U ∗ |T ∗ ||T |2 |T ∗ |U 2 U ∗
1 = U T ∗ |T |2 T 2 U ∗ ≥ U |T |2 U ∗ = |T ∗ |2 .
1 Conversely, suppose that |T ∗ ||T |2 |T ∗ | 2 ≥ |T ∗ |2 . Then
∗ 2  12
1 T |T | T = U ∗ |T ∗ ||T |2 |T ∗ |U 2
1 = U ∗ |T ∗ ||T |2 |T ∗ | 2 U ≥ U ∗ |T ∗ |2 U = |T |2 .




We can extend the result of I. H. Kim ([14, Theorem 1]) to the class A operators. In the proof, we use the M. Cho and T. Yamazaki’s argument([3]). Theorem 2.4. Let T be a class A operator. If S is an arbitrary operator for which 0∈ / W (S) and ST = T ∗ S, then T is self-adjoint.

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Proof. Since σ(S) ⊆ W (S) for any operator S and 0 ∈ / W (S), S is invertible and hence ST = T ∗ S becomes T = S −1 T ∗ S. Therefore σ(T ) ⊆ R by Lemma 2.1. Let T = U |T | be the polar decomposition of T . Then |T |U |T | is semi-hyponormal and σ(|T |U |T |) = {r2 eiθ : reiθ ∈ σ(T )} by Lemma 2.2 because T is class A operator. On the other hand, since T is class A operator, the following inequalities hold by Lemma 2.3:
1
1 |T 2 | = T ∗ |T |2 T 2 ≥ |T |2 ⇔ |T ∗ ||T |2 |T ∗ | 2 ≥ |T ∗ |2 . Using the M. Ito and T. Yamazaki’s result [9, Theorem 1(i)]: r  p+r  p r
r p r  p+r p B2A B2 ≥ B r ⇒ Ap ≥ A 2 B r A 2 , where A and B are positive operators, p ≥ 0 and r ≥ 0, we have
1 |T |2 ≥ |T ||T ∗|2 |T | 2 . Therefore, we have    ∗ 2 1  2 1 |T | − |T |2  ≤  (T |T | T ) 2 − (|T ||T ∗ |2 |T |) 2    1 1  = ((|T |T )∗ (|T |T )) 2 − ((|T |T )(|T |T )∗) 2  . Since σ(T ) is real, σ(|T |T ) is also real. Thus m2 (σ(|T |T )) = 0, where m2 is the planer Lebesgue measure. Applying the Putnam’s inequality for semi-hyponormal operators:   1 1 1 drdθ,

(T ∗ T ) 2 − (T T ∗) 2 ≤ 2π σ(T ) we have |T 2 | = |T |2 because |T |T = |T |U |T | is semi-hyponormal by Lemma 2.2. Now let   A B T = on ran(T ) ⊕ ker(T ∗ ) 0 0 be a 2 × 2 matrix representation of T , and let P be the orthogonal projection T ∗)P = 0. onto ran(T ). Then since T ∗ (T ∗ T − T T ∗)T = 0, we have P (T ∗ T − T S1 S2 Therefore, A∗ A − AA∗ = BB ∗ and hence A is hyponormal. Let S = . S3 S4 / W (S1 ) and S1 A = A∗ S1 . Since Then since 0 ∈ / W (S) and ST = T ∗ S, we have 0 ∈ ∗ ∗ ∗ A is hyponormal, A   is selfadjoint and hence B = 0 because A A − AA = BB . A 0 Hence T = is selfadjoint.
0 0 Lemma 2.5. ([11, Theorem 2.2]) Let T be a quasi-class A operator and T not have a dense range. Then T has the following matrix representation:   A B T = on ran(T ) ⊕ ker(T ∗ ), 0 0 where A is a class A operator. Furthermore, σ(T ) = σ(A) ∪ {0}.

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The following is an extension of Theorem 2.4 and I. H. Kim ([14, Corollary 5]) to the quasi-class A operators. Theorem 2.6. If T is a quasi-class A operator and S is an arbitrary operator for which 0 ∈ / W (S) and ST = T ∗ S, then T is self-adjoint. Proof. Since T is quasi-class A operator, we have the following matrix representation by Lemma 2.5:   A B T = on ran(T ) ⊕ ker(T ∗ ), 0 0   S1 S2 where A is class A operator and σ(T ) = σ(A) ∪ {0}. Let S = . Then S3 S4 / W (S1 ) and S1 A = A∗ S1 . Therefore A from 0 ∈ / W (S) and ST = T ∗ S, we have 0 ∈ is self-adjoint by Theorem 2.4. Now let P be the orthogonal projection of H onto ran(T ). Then we have   A 0 = T P = P T P, 0 0 1

P |T 2 |P = P (T ∗ T ∗ T T ) 2 P 1

≤ (P T ∗ T ∗ T T P ) 2 by Hansen’s inequality 1

= (P T ∗ P T ∗ T P T P ) 2  2  A 0 = 0 0 and

 2 A P |T | P = P T T P = 0 2

∗

 0 . 0

Since T is quasi-class A operator,  2   2 A A 0 2 2 ≥ P |T |P ≥ P |T | P = 0 0 0 and hence we may write 2



On the other hand, let |T | =

E F∗

 C . D

A2 C∗

|T | = 

 0 , 0

 F . Then G

   1
E |A| 0 = P |T |2 P 2 ≥ P |T |P = 0 0 0 and

 |A| P (T T ) P ≥ P (T P T ) P = 0 ∗

1 2

∗

1 2

0 0

 0 . 0



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 |A| F . By straight forward calculation we have F∗ G  2  2   A A + FF∗ AB |A|F + F G 2 = |T | = . B ∗ A |B|2 F ∗ |A| + GF ∗ F ∗ F + G2   |A| 0 Therefore, F = 0 and AB = 0, and hence we have |T | = . Now let 0 |B|   V X . Then T S −1 = S −1 T ∗ and 0 ∈ / W (S −1 ) because 0 ∈ / W (S) and S −1 = Y Z / W (Z). Hence Z is invertible. By ST = T ∗ S. Thus T S −1 T = S −1 |T |2 and 0 ∈ straight forward calculation we have     V A2 X|B|2 AV A + BY A AV B + BY B −1 −1 2 . = T S T = S |T | = Y A2 Z|B|2 0 0 Hence E = |A| and |T | =

Therefore Z|B|2 = 0, which completes the proof.




3. Equality of essential spectra of quasisimilar quasi-class A operators Using the Fuglede-Putnam commutativity theorem R. G. Douglas ([5, Lemma 4.1]) showed that quasisimilar normal operators are unitarilly equivalent, and therefore have equal spectra and essential spectra. S. Clary ([4, Theorem 2]) proved that quasisimilar hyponormal operators have equal spectra and he asked whether quasisimilar hyponormal operators also have essential spectra. Later L. R. Williams ([20, Theorem 3], [21, Theorem 1]) showed that two quasisimilar quasinormal operators and under certain conditions two quasisimilar hyponormal operators have equal essential spectra, B. C. Gupta ([8, Theorem 4]) showed that quasisimilar biquasitriangular and quasisimilar k-quasihyponormal operators have equal essential spectra and M. Raphael ([17]) showed that two quasisimilar operators having Bishop’s property (β) have equal essential spectra. On the other hand, L. Yang ([22, Theorem 2.10])proved that quasisimilar M -hyponormal operators have equal essential spectra, and R. Yingbin and Y. Zikun ([23, Corollary 12]) showed that quasisimilar p-hyponormal operators have also equal spectra and essential spectra. Very recently, I. H. Jeon, J. I. Lee and A. Uchiyama ([12, Theorem 5]) showed that quasisimilar injective p-quasihyponormal operators have equal spectra and essential spectra and A. H. Kim and I. H. Kim ([13]) showed that quasisimilar (p, k)-quasihyponormal operators have equal spectra and essential spectra. In this section we show that quasisimilar quasi-class A operators have equal spectra and essential spectra, respectively. An operator T has the (Bishop’s) property (β) at λ ∈ C if for every open neighborhood D ⊂ C of λ and every vector-valued analytic functions fn : D → H (n = 1, 2, . . .) for which (T − µ)fn (µ) → 0 uniformly on every compact subset

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of D, fn (µ) → 0 uniformly in norm on every compact subset of D. The property (β) plays an important role in the study of spectral properties of operators. Lemma 3.1. ([3, Theorem 3.1]) If T belongs to class A, then T has property (β). Lemma 3.2. ([13, Corollary 7]) If σe (A) ∩ σe (B) has no interior points, then for every C ∈ L(K, H), σe (MC ) = σe (A) ∪ σe (B),   A C where A ∈ L(H), B ∈ L(K) and MC = . 0 B Theorem 3.3. If S ∈ L(H) and T ∈ L(K) are quasisimilar quasi-class A operators, then σ(S) = σ(T ) and σe (S) = σe (T ). Proof. Suppose that X ∈ L(H, K) and Y ∈ L(K, H) are injective operators with dense range such that XS = T X and SY = Y T . If the range of S is dense, then T X = XS implies that the range of T is also dense. Therefore S and T are quasisimilar class A operators, and hence σ(S) = σ(T ) and σe (S) = σe (T ) by Lemma 3.1 because two quasisimilar operators having property (β) have equal spectra and essential spectra. If instead the range of S is not dense, then SY = Y T implies that the range of T is not dense. Therefore by Lemma 2.5, S and T have the following matrix representations:   S1 S2 on ran(S) ⊕ ker(S ∗ ), S= 0 0 and

 T1 T = 0

T2 0



on ran(T ) ⊕ ker(T ∗ )

where S1 and T1 are class A operators. Since quasisimilar class A operators have equal spectra and essential spectra, in view of Lemma 2.5 and Lemma 3.2, it suffices to show that S1 and T1 are quasisimilar. If we let X1 and Y1 denote the restrictions of X and Y given by X1 = X |ran(S) : ran(S) → ran(T ) and Y1 = Y |ran(T ) : ran(T ) → ran(S), then X1 and Y1 are injective and have dense range. Now for any x ∈ ran(S), X1 S1 x = XSx = T Xx = T1 X1 x and for any y ∈ ran(T ), Y1 T1 y = Y T y = SY y = S1 Y1 y. Hence S1 and T1 are quasisimilar, which completes the proof.
From above theorem we have the generalization of I.H. Jeon, J.I. Lee and A. Uchiyama’s result. Actually, we can drop the injective condition. Corollary 3.4. ([12, Theorem 5]) If S ∈ L(H) and T ∈ L(K) are quasisimilar p-quasihyponormal operators, then they have equal spectra and essential spectra.

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References [1] A. Aluthge, On p-hyponormal operators for 0 < p < 1 Integral Equations and Operator Theory 13 (1990), 307–315. [2] S. C. Arora and P. Arora, On p-quasihyponormal operators for 0 < p < 1 Yokohama Math. J. 41 (1993), 25–29. [3] M. Cho and T. Yamazaki, An operator transform from class A to the class of hyponormal operators and its application Integral Equations and Operator Theory 53 (2005), 497–508. [4] S. Clary, Equality of spectra of quasisimilar hyponormal operators Proc. Amer. Math. Soc. 53 (1975), 88–90. [5] R. G. Douglas, On the operator equation S ∗ XT = X and related topics Acta Sci. Math.(Szeged) 30 (1969), 19–32. [6] B. P. Duggal, Hereditarily normaloid operators, Extracta Math. 20 (2005), 203–217. [7] T. Furuta, M. Ito and T. Yamazaki, A subclass of paranormal operators including class of log-hyponormal and several related classes Scientiae Mathematicae 1 (1998), 389-403. [8] B. C. Gupta, Quasisimilarity and k-quasihyponormal operators Math. Today 3 (1985), 49–54.  r  r r p+r ≥ Br [9] M. Ito and T. Yamazaki, Relations between two inequalities B 2 Ap B 2 r   p p p+r and Ap ≥ A 2 B r A 2 and their applications Integral Equations and Operator Theory 44 (2002), 442–450. [10] B. P. Duggal, I. H. Jeon and I. H. Kim, On Weyl’s theorem for quasi-class A operators J. Kor. Math. Soc. 43 (2006), 899–909. [11] I. H. Jeon and I. H. Kim, On operators satisfying T ∗ |T 2 |T ≥ T ∗ |T |2 T Linear Alg. Appl. 418 (2006), 854–862. [12] I. H. Jeon, J. I. Lee and A. Uchiyama, On p-quasihyponormal operators and quasisimilarity Mathematical Inequalities and Applications 6 (2003), 309–315. [13] A. H. Kim and I. H. Kim, Essential spectra of quasisimilar (p, k)-quasihyponormal operators J. Inequalities and Applications 9 (2006), 1–7. [14] I. H. Kim, The Fuglede-Putnam theorem for (p, k)-quasihyponormal operators J. Inequalities and Applications 9 (2006), 397–403. [15] B. Sz-Nagy and C. Foias, Harmonic analysis of operators on Hilbert spaces North Holland-American Elsevier, New York, 1970. [16] C. R. Putnam, An inequality for the area of hyponormal spectra Math. J. 116 (1970), 323–330. [17] M. Raphael, Quasisimilarity and essential spectra for subnormal operators Indiana Univ. Math. J. 31(2) (1982), 243–246. [18] I. H. Sheth, On hyponormal operators Proc. Amer. Math. Soc. 17 (1966), 998–1000. [19] L. R. Williams, Equality of essential spectra of quasisimilar quasinormal operators J. Operator Theory 3 (1980), 57–69. [20] L. R. Williams, Equality of essential spectra of certain quasisimilar seminormal operators Proc. Amer. Math. Soc. 78(2) (1980), 203–209.

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[21] J. P. Williams, Operators similar to their adjoints Proc. Amer. Math. Soc. 20 (1969), 121–123. [22] L. Yang, Quasisimilarity of hyponormal and subdecomposable operators J. Functional Analysis 112 (1993), 204–217. [23] R. Yingbin and Y. Zikun, Spectral structure and subdecomposability of p-hyponormal operators Proc. Amer. Math. Soc. 128 (1999), 2069–2074. In Ho Jeon Department of Mathematics Education Seoul National University of Education Seoul 137–742 Korea e-mail: [email protected] In Hyoun Kim Department of Mathematics University of Incheon Incheon 402-749 Korea e-mail: [email protected] Kotaro Tanahashi Department of Mathematics Tohoku Pharmaceutical University Sendai 981-8558 Japan e-mail: [email protected] Atsushi Uchiyama Department of Mathematics Sendai National College of Technology Sendai 989-3124 Japan e-mail: [email protected] Submitted: October 12, 2006. Revised: March 25, 2008.

Integr. equ. oper. theory 61 (2008), 559–572 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/040559-14, published online July 25, 2008 DOI 10.1007/s00020-008-1603-8

Integral Equations and Operator Theory

Seminorm Related to Banach-Saks Property and Real Interpolation of Operators Andrzej Kryczka Abstract. We introduce the arithmetic separation of a sequence—a geometric characteristic for bounded sequences in a Banach space which describes the Banach-Saks property. We define an operator seminorm vanishing for operators with the Banach-Saks property. We prove quantitative stability of the seminorm for a class of operators acting between lp -sums of Banach spaces. We show logarithmically convex-type estimates of the seminorm for operators interpolated by the real method of Lions and Peetre. Mathematics Subject Classification (2000). Primary 46B70, 47A30; Secondary 47B10. Keywords. Banach-Saks property, real interpolation method, spreading model.

1. Introduction The Banach-Saks property is related to Mazur’s theorem saying that for every sequence (xn ) weakly convergent to x in a Banach space, there exists a sequence of convex combinations of (xn ) tending to x in norm. Banach and Saks proved that in the space Lp with 1 < p < ∞ the convex combinations can be chosen to be the arithmetic means. The similar properties are investigated both for spaces and operators. We say that a bounded linear operator T : X → Y acting between Banach spaces has the Banach-Saks (BS) property, if every bounded sequence (xn ) in X contains a subsequence (x
n ) such that the Ces`aro means of (T x
n ) converge in Y . If every weakly null sequence (xn ) in X contains a subsequence (x
n ) such that the Ces` aro means of (T x
n ) converge in Y , we say that T has the weak BanachSaks (WBS) property. We say that T has the alternate signs Banach-Saks (ABS) property, if every bounded sequence (xn ) in X contains a subsequence (x
n ) such that the Ces` aro means of ((−1)n T x
n ) converge in Y . A Banach space X is said to have the BS, WBS or ABS property if the corresponding property is possessed by the identity operator IX .

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In this article we deal with the quantitative versions of several problems related to the stability of the BS property. Let us recall the main qualitative results. Partington [17] proved that the BS property is preserved by passing from a Banach space X to the lp -sum of X with 1 < p < ∞. The stability of the BS property under interpolation was investigated by Beauzamy [5] who proved that if (A0 , A1 ) is an interpolation pair with continuous injection I : A0 → A1 , then I has the BS property if and only if the real interpolation space Aθ,p with respect to (A0 , A1 ) has the BS property for all 0 < θ < 1 and 1 < p < ∞. In particular, this served to show that every operator with the BS property factors through a space with the same property. Later Heinrich [14] proved that if the canonical embedding I : A0 ∩ A1 → A0 + A1 has the BS property, then so has the real interpolation space Aθ,p with respect to (A0 , A1 ) for all 0 < θ < 1 and 1 < p < ∞. This result shows that the BS property is inherited by real interpolation spaces: if A0 or A1 has the BS property, then so has Aθ,p . In a quantitative approach we usually consider operator seminorms which measure the deviation from a given operator property. In some cases, the results obtained in this way may also provide qualitative information, for example, on the heredity of a given property under interpolation. The quantitative stability under interpolation of operator properties was studied mainly for strong and weak compactness (see [9], [15] and the references given there). Some other properties were investigated by Cobos, Manzano and Mart´ınez in [10] and [11]. The authors examined the case of the so-called outer [1] and inner [19] measures defined by Astala and Tylli for bounded linear operators. Each of these measures is given by the uniform formula with respect to certain classes of operators, in particular, to those with the BS property. In this paper we treat the BS property individually and we introduce a new seminorm which measure the deviation of operators from the BS property. The approach presented here is inspired by the results obtained for a measure of weak noncompactness in [15]. Our starting point is a modification of Beauzamy’s geometric description of spaces which do not have the BS property. The description we propose is based on the separation of successive arithmetic means on equipollent blocks of a sequence. This seems to be a close counterpart of James’ [13] characterization of weak noncompactness where the separation of successive convex combinations of a sequence is used (see also [15]). The new condition enables to define an operator seminorm related to the BS property with regular properties. We show a quantitative version of Partington’s result. This in turn enables to show logarithmically convex (up to a constant) estimates of the seminorm for operators interpolated by the real method of Lions and Peetre. In particular, the inequalities we show give another proof of the heredity of the BS property under real interpolation. The space of all bounded linear operators acting between Banach spaces X and Y will be denoted by L(X, Y ). The subspace of L(X, Y ) consisting of all operators having the BS property will be denoted by BS(X, Y ). We will write B(X) for the open unit ball of a Banach space X. The cardinality of a subset A ⊂ N = {1, 2, 3, . . .} will be denoted by |A|.

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2. BS property and arithmetic separation The major results on the Banach-Saks properties were obtained with the use of the Ramsey methods (see for example [3], [6], [7], [8], [12] and [18]). One of the concepts used in this context is Brunel and Sucheston’s spreading model. This will be a key tool of our investigations as well. The next result is due to Brunel and Sucheston [7]. We follow the formulation given in [3]. Proposition 2.1. Let (xn ) be a bounded sequence in a Banach space X. There exist a subsequence (x
n ) of (xn ) and a seminorm L in the set S of all finite sequences of scalars (real or complex), with the following property: for every ε > 0 and every a = (a1 , . . . , am ) ∈ S
there exists v ∈ N such that, if v ≤ n1 < . . . < nm , then

 m ai x
 − L(a)
< ε. ni i=1 If a bounded sequence (xn ) has no Cauchy subsequence and (x
n ) is the subsequence of (xn ) given by Proposition 2.1, then the formula a1 x
1 + · · · + am x
m E = L(a),

a = (a1 , . . . , am ) ∈ S,

defines a norm in the linear span of {x
n : n ∈ N}. Let E be the completion of the space span{x
n } in the norm ·E . The space E is called the spreading model of X built on the sequence (xn ). The sequence (x
n ) is called the fundamental sequence of the spreading model E. The norm under spreading;  in the space E is invariant  that is, a1 x
1 + · · · + am x
m E = a1 x
n1 + · · · + am x
nm E for all n1 < . . . < nm . The operator seminorm we introduce in this paper is based on a geometric description of Banach spaces which do not have the BS property. Beauzamy [3] proved that a Banach space X does not have the BS property if and only if there exist δ > 0 and a bounded sequence (xn ) in X such that for all subsequences (x
n ) of (xn ) and all k, m ∈ N with 1 ≤ k ≤ m,    k m 1     

xn − xn  ≥ δ.  m  n=1

n=k+1

We introduce a characteristic of a sequence (xn ) related to the BS property, which is a counterpart of the separation sep(xn ) = inf m=n xm − xn  (see [2]) and ∞ the convex separation csep(xn ) = inf m dist(conv{xn }m n=1 , conv{xn }n=m+1 ) (see [15]) of (xn ) used in the measures of strong and weak noncompactness, respectively. A slight modification of Beauzamy’s condition based on this characteristic will turn out very useful in our applications. Definition 2.2. Let (xn ) be a bounded sequence in a Banach space X. Define the arithmetic separation of (xn ) by    1      xn − xn  , asep(xn ) = inf  m  n∈A

n∈B

the infimum being taken over all m ∈ N and all finite subsets A, B ⊂ N with |A| = |B| = m and max A < min B.

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Proposition 2.3. Let X, Y be Banach spaces. An operator T ∈ L(X, Y ) does not have the BS property if and only if there exists a bounded sequence (xn ) in X such that asep(T xn ) > 0. Proof. If T has the BS property, then every bounded sequence (xn ) in X contains a subsequence (x
n ) such that the Ces`aro means of (T x
n ) converge in Y . Since for every m,      m 2m m 2m 1    1    1   



2 T xn − T xn  =  T xn − T xn  ≥ asep(T xn ), m   m 2m n=1 n=1 n=1 n=m+1 we have asep(T xn ) = 0. Assume that T does not have the BS property. If T is not weakly compact, by James’ result of [13], there exist δ > 0 and a bounded sequence (xn ) in X such that for every m, ∞ dist(conv{T xn }m n=1 , conv{T xn }n=m+1 ) ≥ δ.

In particular, asep(T xn ) > 0. If T is weakly compact, we may assume that T does not have the ABS property (we repeat for operators the reasoning of [6, Proposition II.4.1] concerning the equivalence of the ABS and BS properties in reflexive spaces). Then, by Theorem II.2 of [3] adapted for operators, there is a bounded sequence (zn ) in X such that the fundamental sequence of the spreading model built on (T zn ) is equivalent to the unit vector basis of l1 . By Theorems III.1 and IV.2 of [3], there exist δ > 0 and a bounded sequence (xn ) in X such that for all finite subsets A ⊂ N and all sequences of signs (n ), n = ±1 for all n,    1     n T xn  ≥ δ.   |A|  n∈A

In particular,

   1      T xn − T xn  ≥ 2δ  m  n∈A

n∈B

for all m ∈ N and all subsets A, B ⊂ N with |A| = |B| = m and max A < min B. Thus asep(T xn ) > 0.
Corollary 2.4. A Banach space X does not have the BS property if and only if there exists a bounded sequence (xn ) in X such that asep(xn ) > 0.

3. Stabilization of arithmetic means A key role in our study will play passing from a bounded sequence (xn ) to certain sequence (yn ) of successive arithmetic means for (xn ) with control of the arithmetic separation asep(yn ).

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We will use the following relation: (yn ) is a sequence of successive arithmetic means (sam) for (xn ), if there exist m ∈ N and a sequence (An ) of finite subsets of N with max An < min An+1 and |An | = m for all n, such that yn =

1  xk , m

n ∈ N.

k∈An

Since all sets An are equipollent, the relation sam is transitive. If (yn ) is a sequence of sam for (xn ), in particular, if (yn ) is a subsequence of (xn ), then asep(xn ) ≤ asep(yn ). Beauzamy [3] proved that a Banach space X does not have the ABS property if and only if for every ε > 0 there exists a bounded sequence (xn ) in X such that for all finite subsets A ⊂ N and all sequences of signs (n ) with n = ±1 for all n,     1    1−ε≤ n xn  ≤ 1 + ε.   |A| n∈A

The proof of this fact joints a few geometric properties, in particular, that of Rosenthal’s [18] result saying that if a Banach space X does not have the WBS property, then there exist δ > 0 and a bounded sequence (xn ) in X such that for all k ∈ N, all subsets A ⊂ N with |A| = 2k and k ≤ min A, and all sequences of scalars (cn ),        δ |cn | ≤  cn xn  .   n∈A

n∈A

We use similar arguments to show the next proposition (see the proof of Theorem II.2 of [3]). Proposition 3.1. Let (xn ) be a bounded sequence in a Banach space X. Then for every ε > 0 there exists a sequence (yn ) of sam for (xn ) such that for all m ∈ N and all subsets A, B ⊂ N with |A| = |B| = m and max A < min B,     1     yn − yn  ≤ asep(yn ) + ε.   m n∈A

n∈B

Proof. Fix ε > 0. If (xn ) contains a Cauchy subsequence (yn ), then asep(yn ) = 0. Ignoring a finite number of terms of (yn ), we see that (yn ) satisfies the assertion. Assume that (xn ) has no Cauchy subsequence. Let a subsequence (x
n ) of (xn ) be the fundamental sequence of the spreading model E built on (xn ). Considering (x
n ) as a sequence in E, we put K = asep(x
n ). Then there exists m  2m 0 0 1 

x − x , n1 < . . . < n2m0 , z= m0 i=1 ni i=m +1 ni 0

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such that K ≤ zE ≤ K + ε/4. Set a(m) = m−1 (1, . . . , 1, −1, . . . , −1) ∈ R2m for all m ∈ N, where 1 and −1 appear m times each. Put zn =

m0 1  x
, m0 i=1 (n−1)m0 +i

n ∈ N.

Since ·E is invariant under spreading, zn − zk E = zE = L(a(m0 )) for all n = k, where L is given by Proposition 2.1. Let A = {n1 , . . . , nm } and B = {nm+1 , . . . , n2m } be subsets of N such that max A < min B. Then    m  1    1    zni − znm+i  ≤ K + ε , zn − zn  ≤ K≤ E  m m i=1 4 n∈A

n∈B

E

and consequently, for all m ∈ N, ε K ≤ L(a(mm0 )) ≤ K + . 4 Let k ∈ N. Applying Proposition 2.1 for ε/4 and all vectors a(mm0 ) with m ≤ 2k , we get nk such that if A, B ⊂ N with |A| = |B| = m ≤ 2k , max A < min B and nk ≤ min A, then
 

 1 
ε  

 zn − zn  − L(a(mm0 ))
< .

 m
4  n∈A

n∈B

Clearly, we may assume that nk < nk+1 for all k. It follows that for the sequence (zk
) with zk
= znk , all A, B ⊂ N with |A| = |B| = m ≤ 2k , max A < min B and k ≤ min A,      ε ε  1 

 zn − zn  ≤ K + . K− ≤  4 m 2 n∈A

n∈B

Let A = {n1 , . . . , nm } and B = {nm+1 , . . . , n2m } be (naturally ordered) subsets of N such that max A < min B. Put A1 = {n ∈ A : n ≥ log2 m}. Clearly, A1 is nonempty and A1 = {ns+1 , . . . , nm }, where 0 ≤ s ≤ log2 m. Put B1 = {nm+s+1 , . . . , n2m }. Then       ε 

 zn − zn  ≥ (m − s) K −    4 n∈A1

and, if s ≥ 1,

n∈B1

         ε

  ≤ s K + . z − z n n  2 n∈A\A1  n∈B\B1

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It follows that    1     

zn − zn   m  n∈A n∈B        1         1  



  ≥ zn − zn  −  zn − zn  m  m  n∈A1 n∈B1 n∈A\A1 n∈B\B1 ε   s ε ≥K− + 2K + . 4 m 4 Let us choose m1 ∈ N such that m−1 s(2K + ε/4) ≤ ε/4 for all m ≥ m1 . Then for every m ≥ m1 ,      ε  ε 1 

 K− ≤ zn − zn  ≤ K + .  2 m 2 n∈A

Put yn =

n∈B

m1 1  z
, m1 i=1 (n−1)m1 +i

n ∈ N.

Thus (yn ) is a sequence of sam for (xn ). Since for all m ∈ N and all subsets A, B ⊂ N with |A| = |B| = m and max A < min B,      ε  1   K+ ≥ yn − yn   2 m n∈A n∈B    m1 m1  1    ε  

= z(n−1)m1 +i − z(n−1)m1 +i  ≥ K − ,  mm1  2 i=1 i=1 n∈A

n∈B




the assertion is proved.

In particular, Proposition 3.1 and Corollary 2.4 imply the following counterpart for the BS property of Beauzamy’s characterization of spaces which do not have the ABS property. Corollary 3.2. A Banach space X does not have the BS property if and only if for every ε > 0 there exists a bounded sequence (xn ) in X such that for all m ∈ N and all subsets A, B ⊂ N with |A| = |B| = m and max A < min B,    1      1−ε≤ xn − xn  ≤ 1 + ε. m  n∈A

n∈B

We will also need a result analogous to Proposition 3.1 for the following auxiliary quantity related to the arithmetic means for (xn ). Definition 3.3. Let (xn ) be a bounded sequence in a Banach space X. Define    1     φam (xn ) = inf  xn  ,   |A| n∈A

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the infimum being taken over all finite subsets A ⊂ N. Proposition 3.4. Let (xn ) be a bounded sequence in a Banach space X. Then for every ε > 0 there exists a sequence (yn ) of sam for (xn ) such that for all finite subsets A ⊂ N,    1     xn  ≤ φam (yn ) + ε.   |A|  n∈A

The proof is almost the same as for Proposition 3.1. We leave it to the reader.

4. Seminorm related to BS property We introduce a seminorm for bounded linear operators which can be regarded as a measure of deviation from the BS property. Definition 4.1. Let X, Y be Banach spaces and T ∈ L(X, Y ). Define ΦBS (T ) = sup {asep(T xn ) : (xn ) ⊂ B(X)} . Proposition 4.2. ΦBS is a seminorm in L(X, Y ) such that ΦBS (T ) = 0 if and only if T ∈ BS(X, Y ). Proof. It is clear that ΦBS (λT ) = |λ| ΦBS (T ) for all scalars λ. We show subadditivity of ΦBS . Let ε > 0, (xn ) ⊂ B(X) and S, T ∈ L(X, Y ). We apply Proposition 3.1 for (Sxn ). There exists a sequence (x
n ) of sam for (xn ) such that for the sequence (Sx
n ) of sam for (Sxn ),    1     

Sxn − Sxn  ≤ asep(Sx
n ) + ε  m  n∈A

n∈B

for all subsets A, B ⊂ N with |A| = |B| = m and max A < min B. Now we apply Proposition 3.1 for (T x
n ). We obtain a sequence (x

n ) of sam for (x
n ) such that for all subsets A, B ⊂ N with |A| = |B| = m and max A < min B,     1     T x

n − T x

n  ≤ asep(T x

n ) + ε.   m n∈A

n∈B

By transitivity of the relation sam, asep((S + T )xn ) ≤ asep((S + T )x

n )    1    



 ≤ (S + T )xn − (S + T )xn  m  n∈A n∈B        1   1        Sx

n − Sx

n  +  T x

n − T x

n  ≤  m  m n∈A

n∈B

n∈A

n∈B

≤ asep(Sx
n ) + asep(T x

n ) + 2ε ≤ ΦBS (S) + ΦBS (T ) + 2ε.

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Since ε > 0 and (xn ) ⊂ B(X) are arbitrary, we obtain the desired inequality ΦBS (S + T ) ≤ ΦBS (S) + ΦBS (T ). By Proposition 2.3 and the positive homogeneity of ΦBS , it follows that
ΦBS (T ) = 0 if and only if T ∈ BS(X, Y ). The discrete real interpolation method of Lions and Peetre is based on the lp -sums of Banach spaces. The BS property of this sort of spaces was studied by Partington [17] who proved that lp (X) with 1 < p < ∞ has the BS property if and only if so has X. Recall that given a Banach space X, by lp (X) we mean the Banach space of all sequences x = (x(i)) such that x(i) ∈ X for every i ∈ N and  ∞     x(i)X ei  < ∞, xlp (X) =    i=1

lp

where (ei ) is the unit vector basis of lp . In Lions and Peetre’s interpolation, we deal with lp (X), where the families x = (x(i))i∈Z are indexed by integers. Since both spaces are isometrically isomorphic, the next two results remain valid also for this case. The proof of the next lemma is in part inspired by the proof of Theorem 3 of [17]. Lemma 4.3. Let X be a Banach space and (xn ) a bounded sequence in lp (X), 1 < p < ∞. Then for every ε > 0 there exist r ∈ N and a sequence (yn ) of sam for (xn ) such that for all m ∈ N and all subsets A, B ⊂ N with |A| = |B| = m and max A < min B,  ∞       1         yn (i) − yn (i)  ei  < ε.    m   i=r+1 n∈A

n∈B

X

lp

∞ Proof. Let xn = (xn (i)) ∈ lp (X) and tn = i=1 xn (i)X ei ∈ lp . Since lp has the BS property, by Erd¨ os-Magidor’s theorem in [12], there exists a subsequence (t
n ) of (tn ) such that the Ces`aro means of all subsequences of (t
n ) converge to the same limit t in lp . It is easily seen that φam (sn − t) = 0 for every sequence (sn ) of sam for (t
n ). By Proposition 3.4, there exists a sequence (sn ) of sam for (t
n ) such that for every finite subset A ⊂ N,     ε  −1   s n − t < . |A|   4 n∈A

lp

There exist k0 ∈ N and a sequence (An ) of finite subsets of N with |An | = k0 and max An < min An+1 for all n, such that 1 
sn = tk . k0 k∈An

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Let (yn ) be the corresponding sequence of sam for (xn ). We take the subsequence ∞ (x
n ) of (xn ) such that t
n = i=1 x
n (i)X ei and we put 1 
yn = xk . k0 k∈An ∞  ∞ Let t = i=1 αi ei and let r ∈ N satisfy  i=r+1 αi ei l < ε/4. Then for all m ∈ N p and all subsets A, B ⊂ N with |A| = |B| = m and max A < min B,  ∞       ε 1  1    x
k (i)X − αi ei  < .    2m k 4 0 i=r+1 n∈A∪B

It follows that

k∈An

lp

    ∞    ε 1  1   
xk (i)X ei  < .    2m k 2 0 i=r+1 n∈A∪B

k∈An

lp

From this and by the hyperorthogonality of the basis (ei ),  ∞       1  1   
ε > 2 xk (i)X ei    2m k0 i=r+1 n∈A∪B k∈An lp     ∞    1    ≥ yn (i)X ei    m i=r+1 n∈A∪B lp      ∞          1  ≥ yn (i) − yn (i)  ei  ,    m  i=r+1

n∈A

n∈B

X

lp




which is our claim.

The next result is a quantitative generalization for operators of Partington’s theorem of [17]. Theorem 4.4. Let X, Y be Banach spaces and 1 < p < ∞. If T ∈ L(X, Y ) and if T ∈ L(lp (X), lp (Y )) is given by T x = (T x(i)) for every x = (x(i)), then ΦBS (T ) = ΦBS (T ). Proof. The inequality ΦBS (T ) ≤ ΦBS (T ) is obvious, since lp (X) and lp (Y ) contain isometric copies of X and Y , respectively. Fix ε > 0. There exists a sequence (xn ) ⊂ B(lp (X)) such that ΦBS (T ) − ε ≤ asep(T xn ). We apply Lemma 4.3 for the sequence (T xn ). There exist r ∈ N and a sequence (x
n ) of sam for (xn ) such that for the sequence (T x
n ) of sam for (T xn ), and for all m ∈ N and all subsets A, B ⊂ N with |A| = |B| = m and max A < min B, we have   ∞        1       T x
n (i) − T x
n (i)  ei  < ε.   m    i=r+1 n∈A

n∈B

X

lp

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In particular, if we replace (x
n ) by any sequence of sam for (x
n ), the estimate remains valid. We choose a subsequence (x

n ) of (x
n ) such that for each 1 ≤ i ≤ r the limit λi = limn x

n (i)X exists and x

n (i)X < λi + ε/r for every n. We put −1 vn (i) = (λi + ε/r) T x

n (i). Then (vn (i)) ⊂ T (B(X)) for every 1 ≤ i ≤ r. We now apply Proposition 3.1 for the sequence (vn (1)). In this way we obtain a sequence (x1n ) of sam for (x

n ) such that for the sequence (vn1 (1)) of sam for −1 (vn (1)), where vn1 (1) = (λ1 + ε/r) T x1n (1), we have    1        1 1 vn (1) − vn (1)  ≤ asep vn1 (1) + ε  m  n∈A

n∈B

Y

for all subsets A, B ⊂ N with |A| = |B| = m and max A < min B. We put −1 vn1 (i) = (λi + ε/r) T x1n (i) for i = 1. We proceed in this way for coordinates i = 2, . . . , r, consecutively. In the kth step, we apply Proposition 3.1 for the sequence (vnk−1 (k)). We obtain a sequence k k−1 (xkn ) of sam for (xk−1 (k)), n ) such that for the sequence (vn (k)) of sam for (vn −1 k k where vn (k) = (λk + ε/r) T xn (k), we have    1        k k vn (k) − vn (k)  ≤ asep vnk (k) + ε  m  n∈A

n∈B

Y

for all subsets A, B ⊂ N with |A| = |B| = m and max A < min B. Again we put −1 vnk (i) = (λi + ε/r) T xkn (i) for i = k. In this way, all sequences (vnr (i)) are built on the common sequence (xrn ) of sam for (xn ). Moreover,     1     r r vn (i) − vn (i)  ≤ asep (vnr (i)) + ε, 1 ≤ i ≤ r,   m n∈A

n∈B

Y

for all subsets A, B ⊂ N with |A| = |B| = m and max A < min B. It follows that asep(T xn ) ≤ asep(T xrn )  r       1         r r ≤ T xn (i) − T xn (i)  ei  + ε   m   i=1 n∈A n∈B Y lp   r        λ + ε/r    i    vnr (i) − vnr (i)  ei  + ε =   m    i=1 n∈A n∈B Y lp      r
   
 ε
 1  r  
r vn (i) − vn (i)  + ε ≤
λi +
ei  max   1≤i≤r  m   r i=1 n∈A n∈B lp Y   1/p−1 r max {asep (vn (i)) + ε} + ε. ≤ 1 + εr 1≤i≤r

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We choose 1 ≤ j ≤ r such that asep (vnr (j)) = max1≤i≤r asep (vnr (i)). Since is a sequence of sam for (vn (j)), we have (vnr (j)) ⊂ T (B(X)) and consequently,   ΦBS (T ) − 2ε ≤ 1 + εr1/p−1 (ΦBS (T ) + ε) . (vnr (j))

Since ε > 0 was chosen arbitrary, we obtain ΦBS (T ) ≤ ΦBS (T ).




5. Interpolation results  = (A0 , A1 ) of Banach spaces is called an interpolation pair, if A0 and A1 A pair A are linearly and continuously embedded in a common Hausdorff topological vector  = A0 ∩ A1 , Σ(A)  = A0 + A1 are Banach spaces with norms space V . Then ∆(A) a∆(A)  = max{aA0 , aA1 },

aΣ(A)  = inf{a0 A0 + a1 A1 : a0 + a1 = a}.

We deal with Lions and Peetre’s [16] discrete method of construction of real interpolation spaces. For 0 < θ < 1 and 1 < p < ∞, let    : a Aθ,p = a ∈ Σ(A) Aθ,p < ∞ , where

  aAθ,p = inf max (2iθ a0 (i))lp (A0 ) , (2i(θ−1) a1 (i))lp (A1 ) ,

the infimum being taken over all families (a0 (i)) ⊂ A0 and (a1 (i)) ⊂ A1 with a0 (i) + a1 (i) = a for all i ∈ Z. The Banach space Aθ,p with norm ·Aθ,p is called  = (A0 , A1 ). a real interpolation space with respect to A If a ∈ Aθ,p , then i(θ−1) aAθ,p ≤ 2θ(1−θ)(2iθ a0 (i))1−θ a1 (i))θlp (A1 ) lp (A0 ) (2

for all families (a0 (i)) ⊂ A0 and (a1 (i)) ⊂ A1 with a0 (i) + a1 (i) = a for all i ∈ Z.  ⊂ Aθ,p ⊂ Σ(A)  with continuous inclusions. Moreover, ∆(A) Let Aθ,p , Bθ,p be interpolation spaces with respect to the interpolation pairs  = (A0 , A1 ), B  = (B0 , B1 ). For a linear operator T : Σ(A)  → Σ(B),  we write A  → B,  if for j = 0, 1, the restriction T |Aj is a bounded operator into Bj . In T: A the main result of this paper, we show a counterpart for the seminorm ΦBS of the  → B:  following inequality valid for every T : A T : Aθ,p → Bθ,p  ≤ 2θ(1−θ) T : A0 → B0 1−θ T : A1 → B1 θ . Theorem 5.1. Let Aθ,p , Bθ,p with 0 < θ < 1 and 1 < p < ∞ be real interpolation  = (A0 , A1 ), B  = (B0 , B1 ). Then for spaces with respect to interpolation pairs A   every T : A → B, θ ΦBS (T : Aθ,p → Bθ,p ) ≤ 2θ(1−θ)Φ1−θ BS (T : A0 → B0 )ΦBS (T : A1 → B1 ).

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Proof. Fix ε > 0. Let (an ) be a sequence in B(Aθ,p ). For each an there exist xjn = (2i(θ−j) ajn (i))i∈Z ∈ B(lp (Aj )), j = 0, 1, such that a0n (i) + a1n (i) = an for all i ∈ Z. Set yjn = (2i(θ−j) T ajn (i))i∈Z for j = 0, 1 and every n ∈ N. Proceeding just as in the proof of subadditivity of ΦBS in Proposition 4.2, we may assume that for all subsets A, B ⊂ N with |A| = |B| = m and max A < min B,    1      yjn − yjn  ≤ asep(yjn ) + ε, j = 0, 1.  m  n∈A

n∈B

lp (Bj )

Let T j : lp (Aj ) → lp (Bj ), j = 0, 1, be defined as T in Theorem 4.4. Then yjn = T xjn . It follows that    1      T an − T an  asep(T an ) ≤  m  n∈A

n∈B

  1−θ      θ(1−θ)  1 ≤2 y0n − y0n    m n∈A

≤2

θ(1−θ)

n∈B

1−θ

Bθ,p

lp (B0 )

  θ  1     y1n − y1n    m n∈A

n∈B

lp (B1 )

θ

(asep(y0n ) + ε) (asep(y1n ) + ε)  1−θ  θ ΦBS (T 1 ) + ε . ≤ 2θ(1−θ) ΦBS (T 0 ) + ε By Theorem 4.4, ΦBS (T j ) = ΦBS (T : Aj → Bj ), j = 0, 1. Since ε > 0 and (an ) ⊂ B(Aθ,p ) are arbitrary, we obtain the conclusion.
A qualitative result on the heredity of the BS property by operators was given by Beauzamy in the case of interpolation pairs with continuous injections (see [4, Theorem III.3.3]). By Theorem 5.1, we obtain qualitative results without restrictions on interpolation pairs. In the following corollary, the claim concerning spaces can also be derived from Heinrich’s result mentioned in the Introduction. Corollary 5.2. If T : A0 → B0 or T : A1 → B1 has the BS property, then so has T : Aθ,p → Bθ,p for all 0 < θ < 1 and 1 < p < ∞. In particular, if A0 or A1 has the BS property, then so has Aθ,p .

References [1] K. Astala and H.-O. Tylli, Seminorms related to weak compactness and to Tauberian operators. Math. Proc. Cambridge Philos. Soc., 107 (1990), 367–375. [2] J.M. Ayerbe Toledano, T. Dom´ınguez Benavides and G. L´ opez Acedo, Measures of noncompactness in metric fixed point theory. Birkh¨ auser, Basel 1997. [3] B. Beauzamy, Banach-Saks properties and spreading models. Math. Scand., 44 (1979), 357–384. [4] B. Beauzamy, Espaces d’interpolation r´ eels: Topologie et g´eom´etrie. Lecture Notes in Mathematics, 666, Springer, Berlin 1978.

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[5] B. Beauzamy, Propri´ et´e de Banach-Saks. Studia Math., 66 (1980), 227–235. [6] B. Beauzamy, J.-T. Laprest´e, Mod`eles ´etal`es des espaces de Banach. Hermann, Paris 1984. [7] A. Brunel and L. Sucheston, On B-convex Banach spaces. Math. Systems Theory, 7 (1974), 294–299. [8] A. Brunel and L. Sucheston, On J-convexity and some ergodic super-properties of Banach spaces. Trans. Am. Math. Soc., 204 (1975), 79-90. [9] F. Cobos, P. Fern´ andez-Mart´ınez and A. Mart´ınez, Interpolation of the measure of non-compactness by the real method. Studia Math., 135 (1999), 25–38. [10] F. Cobos, A. Manzano and A. Mart´ınez, Interpolation theory and measures related to operator ideals. Quart. J. Math. Oxford Ser. (2), 50 (1999), 401–416. [11] F. Cobos and A. Mart´ınez, Extreme estimates for interpolated operators by the real method. J. London Math. Soc. (2), 60 (1999), 860–870. [12] P. Erd¨ os and M. Magidor, A note on regular methods of summability and the BanachSaks property. Proc. Amer. Math. Soc., 59 (1976), 232–234. [13] R.C. James, Weak compactness and reflexivity. Israel J. Math., 2 (1964), 101-119. [14] S. Heinrich, Closed operator ideals and interpolation. J. Funct. Anal., 35 (1980), 397–411. [15] A. Kryczka, S. Prus and M. Szczepanik, Measure of weak noncompactness and real interpolation of operators. Bull. Austral. Math. Soc., 62 (2000), 389–401. [16] J.-L. Lions and J. Peetre, Sur une classe d’espaces d’interpolation. Inst. Hautes ´ Etudes Sci. Publ. Math., 19 (1964), 5–68. [17] J.R. Partington, On the Banach-Saks property. Math. Proc. Cambridge Philos. Soc., 82 (1977), 369–374. [18] H.P. Rosenthal, Weakly independent sequences and the Banach-Saks property. In Durham symposium on the relations between infinite-dimensional and finitedimensional convexity. Bull. London Math. Soc., 8 (1976), 1–33. [19] H.-O. Tylli, The essential norm of an operator is not self-dual. Israel J. Math., 91 (1995), 93–110. Andrzej Kryczka Institute of Mathematics M. Curie-Sk
lodowska University 20-031 Lublin Poland e-mail: [email protected] Submitted: November 26, 2007. Revised: June 16, 2008.

Integr. equ. oper. theory 61 (2008), 573–591 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/040573-19, published online July 25, 2008 DOI 10.1007/s00020-008-1609-2

Integral Equations and Operator Theory

Hankel and Toeplitz Transforms on H 1: Continuity, Compactness and Fredholm Properties M. Papadimitrakis and J. A. Virtanen Abstract. We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H 1 and give a new proof of the old result stating that the Hankel operator Ha is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for Ha to be compact on H 1 . The Fredholm properties of Toeplitz operators on H 1 are studied for symbols in a Banach algebra similar to C + H ∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H 1 and H 2 . Mathematics Subject Classification (2000). Primary 47B35; Secondary 30D50, 30D55, 47A53. Keywords. Hankel operators, Toeplitz operators, boundedness, compactness, Fredholmness, Hardy space H 1 , bounded mean oscillation.

1. Introduction and main results Let D = {z ∈ C : |z| < 1} be the unit disk of the complex plane C and T = ∂D = {ζ ∈ C : |ζ| = 1} be the unit circle. The usual Lebesgue spaces for T are denoted by Lp = Lp (T) and we write f (ζ) ∼

+∞


f(n)ζ n

n=−∞

The first author was partially supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) and by the Greek Research Program “Pythagoras 2” (75% European funds and 25% National funds). The second author was fully supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) while he visited the first author at the University of Crete and later by the Academy of Finland Project 207048.

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for the Fourier series of a function f in L1 . The Hardy spaces for T are defined by   H p = f ∈ Lp : f(n) = 0 for n < 0   and their variants by H0p = f ∈ Lp : f(n) = 0 for n ≤ 0 . We also define the   spaces H p = f ∈ Lp : f(n) = 0 for n > 0 and the corresponding variants H0p =   f ∈ Lp : f(n) = 0 for n ≥ 0 . The M. Riesz Theorem says that the Riesz projection P , defined by P f (ζ) ∼

+∞


f(n)ζ n

n=0

+∞

for every f (ζ) ∼ n=−∞ f(n)ζ n , is a bounded operator Lp → H p when 1 < p < ∞; note, however, that the operator P is not bounded either on L1 or L∞ . We +∞ also define a related operator P1 : Lp → H0p by P1 f (ζ) ∼ n=1 f(n)ζ n and denote −1 the complementary projection of P by Q : Lp → H0p , Qf (ζ) ∼ n=−∞ f(n)ζ n . We say that P f is the analytic part and Qf is the antianalytic part of f . The Toeplitz operator Ta with symbol a ∈ L2 is defined by Ta f = P (af ) and the Hankel operator Ha by Ha f = P (aJf ), where J is the “flip operator” defined by Jf (ζ) = ζf (ζ) ∼

+∞


f(−n − 1)ζ n .

n=−∞

Both operators Ta and Ha are obviously well defined for analytic polynomials, i.e. N for finite sums f (ζ) = n=0 f(n)ζ n . The set of analytic polynomials is dense in each H p (1 ≤ p < +∞) and there are classical results which specify, for every particular value of p, the necessary and sufficient conditions on the symbol a so that these operators are extended as bounded or even compact operators on H p . It is easy to see that Ta is not compact whenever a is not the zero function. The situation is described by the following Theorems 1.1–1.5. Theorem 1.1. Let 1 < p < +∞. Then Ta is bounded on H p if and only if a ∈ L∞ . Theorem 1.2. (Nehari, for p = 2) Let 1 < p < +∞. Then Ha is bounded on H p if and only if P1 a ∈ BMO. Theorem 1.3. (Hartman, for p = 2) Let 1 < p < +∞. Then Ha is compact on H p if and only if P1 a ∈ VMO. In the case of the space H 1 the results are slightly more complicated.

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Theorem 1.4. (Stegenga, 1976, for real or antianalytic a; Janson-Peetre-Semmes, 1984; Tolokonnikov, 1987; Cima-Stegenga, 1987) The Toeplitz operator Ta with symbol a is bounded on H 1 if and only if a ∈ L∞ and Qa ∈ BMOlog . Theorem 1.5. (Janson-Peetre-Semmes, 1984; Tolokonnikov, 1987; Cima-Stegenga, 1987) The Hankel operator Ha is bounded on H 1 if and only if P1 a ∈ BMOlog . The purpose of the first part of the article is to give a new proof of Theorem 1.5 together with a precise estimate of the operator norm of Ha and to prove the analogous result about the compactness of Ha , that is, we prove the following two theorems. Theorem 1.6. The Hankel operator Ha is bounded on H 1 if and only if P1 a ∈ BMOlog , in which case Ha H 1 →H 1

 P1 aBMOlog ,

where A  B means that there are two positive numerical constants c1 and c2 so A that c1 ≤ B ≤ c2 for all values of the independent variables in A and B. Theorem 1.7. The Hankel operator Ha is compact on H 1 if and only if P1 a ∈ VMOlog . The second part of this article deals with spectral properties of Toeplitz operators. The case of continuous symbols was recently studied in [14]. Here we consider symbols that are not necessarily continuous. The motivation comes from the well-known result on the Fredholm properties of Toeplitz operators on H p (1 < p < ∞) with a ∈ C + H ∞ , due to Douglas [6] when p = 2. This suggests the following theorem, which is indeed the best we can hope for because of the differences in boundedness and compactness of the operators determined by the underlying spaces H 1 and H p . Theorem 1.8. Let a ∈ V + H ∞ ∩ BMOlog := V + (H ∞ ∩ BMOlog ), where V = C ∩ VMOlog . Then the following conditions are equivalent: (1) Ta is Fredholm on H 1 , that is, ker Ta and coker Ta are both of finite dimension; (2) a is invertible in the algebra V + H ∞ ∩ BMOlog ; (3) a is bounded away from zero, that is, there are  > 0 and δ > 0 such that |a(z)| ≥ 

for 1 − δ < |z| < 1,

where a(z) for z ∈ D is defined via the harmonic extension—see (1) below; in this case for any 1 − δ < r < 1 Ind Ta := dim ker Ta − dim coker Ta = − ind ar where ar (ζ) = a(rζ) for all ζ ∈ T and ind ar is the winding number of the function ar .

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2. Preliminaries In this section we consider some (known) results from harmonic analysis. The Poisson extension of f ∈ L1 at z ∈ D is given by  1 − |z|2 1 f (ζ) |dζ| (1) f (z) = 2π T |ζ − z|2 and the Szeg¨ o projection of f at z by 1 P f (z) = 2πi

 T

f (ζ) dζ. ζ −z

For 1 ≤ p < +∞ and every f ∈ Lp the limit limr→1− f (rζ) = f (ζ) holds for almost every ζ ∈ T and also in the Lp sense. Since f (z) is a harmonic function of z ∈ D, it is also called the harmonic extension of f in D. On the other hand, for 1 ≤ p < +∞ and every f ∈ Lp , the limit lim P f (rζ)

r→1−

exists for almost every ζ ∈ T and, when 1 < p < +∞, this limit is equal to P f (ζ) (where P is the Riesz projection) in both the almost everywhere sense and in the Lp sense. In the case p = 1, the limit P f (ζ) = limr→1− P f (rζ) serves as the definition of the function P f which, as is well known, belongs to the space L1,w of weak-L1 functions. In all cases P f (z) is an analytic function of z ∈ D. If 1 < p < +∞ and f ∈ Lp , the Poisson extension of P f ∈ Lp at every z ∈ D is equal to P f (z):  1 − |z|2 1 P f (ζ) |dζ|, P f (z) = 2π T |ζ − z|2 while if 1 ≤ p < +∞ and f ∈ H p , then (obviously) P f (ζ) = f (ζ) and   1 f (ζ) 1 − |z|2 1 P f (z) = f (z) = dζ f (ζ) |dζ| = 2π T |ζ − z|2 2πi T ζ − z for every z ∈ D. We next consider the space of functions of bounded mean oscillation and its important (logarithmic) subspaces. A function f is in BMO if f ∈ L1 and  1 |f (ζ) − fI ||dζ| < +∞, f ∗ = sup I |I| I  1 where the supremum is taken over all arcs I of T, fI = |I| I f (ζ)|dζ| and |I| is the length of I. The space BMO is a Banach space under the norm f BMO = |f(0)|+f ∗. We also have the space BMOA of analytic functions in BMO, defined as BMOA = BMO ∩H 1 = {f ∈ BMO : f(n) = 0 for n < 0}. It is well known that L∞ ⊆ BMO ⊆ Lp for every p < +∞ and that for every f ∈ BMO   12 1 − |z|2 f ∗  sup |f (ζ) − f (z)|2 |dζ| . (2) 2 |ζ − z| z∈D T

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The subspace VMO of BMO contains by definition all functions f ∈ L1 for which  1 lim sup |f (ζ) − fI ||dζ| = 0. δ→0+ I, |I|<δ |I| I We also define VMOA = VMO ∩H 1 . The space VMO is the closure in the space BMO of the set of all polynomials (or, equivalently, of all continuous functions). Also, f ∈ BMO belongs to VMO if and only if limr→1− fr − f BMO = 0, where the function fr is defined by fr (ζ) = f (rζ). Somewhat less known are the spaces BMOlog and VMOlog and their variants BMOAlog and VMOAlog . These are defined as follows. A function f is in BMOlog if f ∈ L1 and  log 4π |I| f ∗∗ = sup |f (ζ) − fI ||dζ| < +∞, |I| I I where, again, the supremum is taken over all arcs I of T. The space BMOlog is a Banach space under the norm f BMOlog = |f(0)| + f ∗∗ . We define BMOAlog = BMOlog ∩H 1 . It is obvious that BMOlog ⊆ BMO. The following estimate   12 2 1 − |z|2 sup log2 |f (ζ) − f (z)|2 |dζ| (3) f ∗∗  2 2 1 − |z| T |ζ − z| z∈D where f ∈ BMOlog , requires a similar consideration as in the case of the space BMO, starting with the analogue of the John-Nirenberg theorem. The proofs do not seem to have been recorded anywhere but they are almost straightforward and, in any case, these facts have been used many times in the literature. The logarithmic Lipschitz space Liplog is defined by 

4 |f (ζ) − f (η)| < ∞ . Liplog = f : T → C : sup log |ζ − η| ζ,η∈T This is a space of continuous functions under the norm 4 f Liplog = |f(0)| + sup log |f (ζ) − f (η)|. |ζ − η| ζ,η∈T The space Liplog is continuously imbedded in BMOlog and the main result of [9] is: (4) BMOlog = {f + P g : f, g ∈ Liplog } . In particular, if h ∈ BMOlog , there are f, g ∈ Liplog such that h = f + P g and f Liplog + gLiplog ≤ c hBMOlog . where c is a positive numerical constant. The subspace VMOlog of BMOlog contains by definition all functions f ∈ L1 for which  log 4π |I| lim sup |f (ζ) − fI ||dζ| = 0. δ→0+ I, |I|<δ |I| I

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We also define VMOAlog = VMOlog ∩H 1 . The following two results will be needed several times. Theorem 2.1. For the logarithmic VMO space, we have the following characterization: VMOlog = {f + P g : f, g ∈ liplog } , (5) where liplog stands for the so-called vanishing logarithmic Lipschitz space defined by 

4 |f (ζ) − f (η)| = 0 . liplog = f ∈ Liplog : lim sup log δ→0+ |ζ−η|<δ |ζ − η| Theorem 2.2. For f ∈ BMOlog , the following conditions are equivalent: (1) f ∈ VMOlog ; (2) limη→1 τη f − f BMOlog = 0, where τη f (ζ) = f (ζη) for η, ζ ∈ T; (3) limr→1− fr − f BMOlog = 0, where fr (ζ) = f (rζ) for ζ ∈ T. The following descriptions are also useful: {a ∈ L∞ : Qa ∈ BMOlog } = Liplog +H ∞

(6)

and

(7) {a ∈ L∞ : Qa ∈ VMOlog } = liplog +H ∞ . These can be verified by means of the characterizations in (4) and (5); for example, if a = l + h ∈ liplog +H ∞ , then a ∈ L∞ and Qa = Ql ∈ VMOlog , and conversely if a ∈ L∞ and Qa ∈ VMOlog , then Qa = f + P g for some f, g ∈ liplog , so Qa = Qf , which implies that a − f ∈ H ∞ and we can write a = f + (a − f ) ∈ liplog +H ∞ . z For each arc I we define S(I) = {z ∈ D : 0 < 1 − |z| < |I| 2π , |z| ∈ I}, called the Carleson “square” with base I. A positive Borel measure µ in D is called a Carleson measure if µ(S(I)) < +∞, sup |I| I where the supremum is taken over all arcs I of T. It is known that µ is a Carleson measure if and only if there is a constant c so that    |f (z)|2 dµ(z) ≤ c |f (ζ)|2 |dζ|, f ∈ L2 , (8) D

T

and that, if c is the smallest constant for which this inequality holds, c



sup I

µ(S(I)) . |I|

(9)

In this connection, we have a function f ∈ L1 in BMO if and only if the Borel measure |∇f (z)|2 (1 − |z|2 )dm(z), where dm is the area measure, is a Carleson measure and    12 1 sup |∇f (z)|2 (1 − |z|2 )dm(z) . (10) f ∗  I |I| S(I)

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Similarly, f ∈ L1 is in VMO if and only if   1 lim sup |∇f (z)|2 (1 − |z|2 )dm(z) δ→0+ I, |I|<δ |I| S(I)

579

= 0.

(11)

Of course, in the case of f ∈ H 1 we may replace ∇f (z) by f  (z) in the above characterizations of BMO and VMO. Analogously, for functions f in BMOlog , we have   12 log2 4π |I| f ∗∗  sup |∇f (z)|2 (1 − |z|2 )dm(z) . (12) |I| I S(I) Note also that there exists a positive numerical constant c so that for every f ∈ BMOA and every z ∈ D: |f (z)| ≤

cf BMO log

2 . 1 − |z|2

(13)

Conversely, there exists a positive numerical constant c so that for every z ∈ D there exists an f ∈ BMOA with f BMO

=

|f (z)| ≥ c log

1,

2 . 1 − |z|2

(14)

Also, if f ∈ VMOA then lim

|z|→1−

|f (z)| 2 log 1−|z| 2

=

0.

(15)

Finally, we shall use the inequality | f, g | ≤

cf H 1 gBMOA ,

where the binary form ·, · is defined by  1 f, g = lim fr (ζ)g(ζ)|dζ| = r→1− 2π T

1 r→1− 2πi



lim

f (rζ)g(ζ)ζdζ. T

The Fefferman-Stein duality which is induced by this binary form says that BMOA is isomorphic to (H 1 )∗ . It is also true that, under the same binary form, H 1 is isomorphic to (VMOA)∗ .

3. Proof of Theorem 1.6 Proof. Before proceeding to the proof, note that the part a − P1 a of a plays no role in the Hankel operator Ha . Indeed, for all analytic polynomials f the function (a − P1 a)Jf is antianalytic and, hence, Ha f = HP1 a f . We may thus suppose in all that follows that a = P1 a or in other words that  a(n) = 0 ,

n ≤ 0.

(16)

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We recall that BMOA is isomorphic to the dual space of H 1 and it is easy to see that, formally at least, the dual operator to Ha on H 1 is Ha on BMOA. This means that Ha f, g =

f, Ha g

for all analytic polynomials f and all g ∈ BMOA. Hence, we need to prove that Ha is bounded on BMOA if and only if a ∈ BMOlog and that Ha BMOA→BMOA

 aBMOlog

under the assumption (16). Sufficiency. Let a ∈ BMOlog satisfy (16) and take an arbitrary f ∈ BMOA. Then   1 a(ζ)ζf (ζ) b(ζ)g(ζ) 1 dζ = dζ, Ha f (z) = 2πi T ζ − z 2πi T ζ − z where we set b(ζ) = ζa(ζ) and g(ζ) = f (ζ). It is obvious that b ∈ BMOAlog with bBMOlog  aBMOlog and that g ∈ BMOA with gBMO = f BMO . Throughout, the symbol c denotes a numerical constant, not necessarily the same at each occurrence. We have  b(ζ)g(ζ) 1 (Ha f ) (z) = dζ 2πi T (ζ − z)2   b(ζ)(g(ζ) − g(z)) b(ζ) 1 1 dζ + g(z) dζ = 2πi T (ζ − z)2 2πi T (ζ − z)2  (b(ζ) − b(z))(g(ζ) − g(z)) 1 dζ + g(z)b (z). = 2πi T (ζ − z)2 Applying the Cauchy-Schwarz inequality together with (2) and (3), we get   |b(ζ) − b(z)|2 |g(ζ) − g(z)|2 |(Ha f ) (z)|2 ≤ c |dζ| |dζ| + c|g(z)|2 |b (z)|2 2 |ζ − z| |ζ − z|2 T T 1 ≤ cb2∗∗ g2∗ + c|g(z)|2 |b (z)|2 . 2 2 2 (1 − |z| ) log2 1−|z| 2 This, for every arc I of T, implies   1 |(Ha f ) (z)|2 (1 − |z|2 )dm(z) |I| S(I)   1 2 2 1 dm(z) ≤ cb∗∗g∗ 2 2 2 |I| S(I) (1 − |z| ) log 1−|z|2   1 |g(z) − g(zI )|2 |b (z)|2 (1 − |z|2 )dm(z) +c |I| S(I)   2 1 +c|g(zI )| |b (z)|2 (1 − |z|2 )dm(z) |I| S(I) = A + B + C,

(17)

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where zI is the point in the middle of the internal side of S(I), defined by 1−|zI | = |I| zI 2π and |zI | =midpoint of I. Let us first estimate the term A. A direct calculation of the integral, using polar coordinates, gives A ≤

c

1 b2∗∗g2∗ log 4π |I|

≤

c

1 a2BMOlog f 2BMO . log 4π |I|

(18)

Observing that |1 − zI z|  |I| for all z ∈ S(I) and considering the Borel measure dµ(z) which is equal to |b (z)|2 (1−|z|2 )dm(z) on S(I) and equal to zero on D\S(I), we find using (8) and (9) that   |g(z) − g(zI )|2  B ≤ c|I| |b (z)|2 (1 − |z|2 )dm(z) 2 |1 − z z| I S(I)  µ(S(J)) |g(ζ) − g(zI )|2 ≤ c|I| sup |dζ| |J| |1 − zI ζ|2 J T  µ(S(J)) 1 − |zI |2 |g(ζ) − g(zI )|2 |dζ| ≤ c sup |J| |ζ − zI |2 J T µ(S(J)) . (19) ≤ cg2∗ sup |J| J Estimating

µ(S(J)) |J|

µ(S(J)∩S(I)) , |J|

=

we observe that we need only consider arcs J

≤ µ(S(I)) having nonempty intersection with I. In the case |J| > |I|, µ(S(J)) |J| |I| . If |J| ≤ |I|, then J ⊆ 3I, where 3I is the arc with the same midpoint as I and with length three times the length of I. Hence, in both cases we get using (12)   µ(S(J)) 1 ≤ sup sup |b (z)|2 (1 − |z|2 )dm(z) |J| |J| J J⊆3I S(J) 1 2 ≤ c sup 2 4π b∗∗ log J⊆3I |J| ≤

c

1 b2∗∗ . log2 4π |I|

≤

c

1 b2∗∗g2∗ log2 4π |I|

≤

c

1 a2BMOlog f 2BMO . log2 4π |I|

Therefore (19) implies B

(20)

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Finally, (12) and (13) imply c b2∗∗ g2BMO

≤

C

c a2BMOlog f 2BMO .

≤

(21)

Now, estimates (18), (20) and (21) together with (17) imply   1 |(Ha f ) (z)|2 (1 − |z|2 )dm(z) ≤ c a2BMOlog f 2BMO |I| S(I) and, taking the supremum over all arcs I and using (10), Ha f ∗ On the other hand,

≤ c aBMOlog f BMO .

 |Ha f (ζ)|2 |dζ| = c |a(ζ)|2 |f (ζ)|2 |dζ| T T 12   12  |a(ζ)|4 |dζ| |f (ζ)|4 |dζ| ≤ c

2 |H a f (0)|



≤ c

≤

T c a2BMO f 2BMO

T

≤ c a2BMOlog f 2BMO .

The last two estimates show that Ha f BMO

≤ c aBMOlog f BMO

and hence Ha BMOA→BMOA

≤

c aBMOlog .

Necessity, step 1. Here we make the a priori assumption that a ∈ BM Olog (and, that a satisfies (16)) and we set b(ζ) = ζa(ζ) as before. If |b(0)| ≥ 12 bBMOlog , then Ha BMOA→BMOA

≥ Ha 1BMO = bBMO 1 ≥ |b(0)| ≥ bBMOlog 2 ≥ caBMOlog .

If |b(0)| < 12 bBMOlog , then b∗∗ ≥ arc I such that ca2BMOlog ≤ cb2BMOlog ≤

log2

1 2 bBMOlog 4π |I|

|I|

  S(I)

and based on (12) we find an

|b (z)|2 (1 − |z|2 )dm(z).

(22)

Through (14) we find an f ∈ BMOA and the corresponding g(ζ) = f (ζ) so that f BMO

= 1,

c log

4π ≤ |f (zI )| = |g(zI )|. |I|

(23)

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583

The trivial variant of inequality (17) together with the estimates (18), (20), (22) and (23) imply Ha 2BMOA→BMOA

≥ Ha f 2BMO   1 ≥ |(Ha f ) (z)|2 (1 − |z|2 )dm(z) |I| S(I)  1 ≥ C −A−B ≥ c 1−c a2BMOlog . log 4π |I|

Hence, if |I| is smaller than a certain positive numerical constant we find that Ha BMOA→BMOA

≥

caBMOlog .

On the other hand, if |I| is larger than the same positive numerical constant, then Ha 2BMOA→BMOA

≥ Ha 12BMO = b2BMO   1 ≥ c |b (z)|2 (1 − |z|2 )dm(z) |I| S(I)   log2 4π |I| |b (z)|2 (1 − |z|2 )dm(z) ≥ c |I| S(I) ≥ ca2BMOlog .

We conclude that if a is assumed to be in BMOlog and satisfy (16) then Ha BMOA→BMOA

≥ caBMOlog

and, by the usual duality, Ha H 1 →H 1

≥

caBMOlog .

Lemma 3.1. If Ha is bounded on H 1 , then for every f ∈ H 1 and all r < 1 we have Har f = r(Ha fr )r . Proof. The operator Har is bounded on H 1 since ar is smooth. Verifying the equality involves a straightforward calculation using Fourier series.
Necessity, step 2. Applying the a priori estimate of step 1 to the functions ar we have Har H 1 →H 1

≥

car BMOlog .

(24)

The lemma of Fatou with (12) implies that aBMOlog

≤

lim inf ar BMOlog . r→1−

Now, Lemma 3.1 implies that for all f ∈ H 1 Har f H 1

≤

rHa fr H 1

≤

Ha H 1 →H 1 fr H 1

≤

Ha H 1 →H 1 f H 1

(25)

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IEOT

and hence Har H 1 →H 1

≤

Ha H 1 →H 1 .

(26)

Relations (24), (25) and (26) complete the necessity part of Theorem 1.6.


4. Proof of Theorem 1.7 Proof. Let a ∈ VMOlog satisfy (16). Let r < 1 and take fn ∈ H 1 with fn H 1 ≤ 1. Choosing a subsequence, we may assume that there is a function f ∈ H 1 so that (fn )r → f in H 1 . Since Ha is bounded, we get Ha (fn )r → Ha f in H 1 and, hence, Har fn = r Ha (fn )r r → r(Ha f )r in H 1 . Therefore, Har is compact on H 1 . Finally, Har − Ha H 1 →H 1 ≤ car − aBMOlog → 0 as r → 1− and, hence, Ha is compact on H 1 . Let a ∈ BMOlog satisfy (16). It is a consequence of the proof of Theorem 1.6 that Ha is bounded on VMOA. Indeed, taking any f ∈ VMOA, (17) together with (12), (15), (18) and (20) imply that   1 sup |(Ha f ) (z)|2 (1 − |z|2 )dm(z) I, |I|<δ |I| S(I) ≤c

|f (zI )|2 1 2 2 2 4π aBMOlog f BMO + caBMOlog sup 2 4π log δ I, |I|<δ log |I|

→0 as δ → 0+. Therefore, (11) implies that Ha f ∈ VMOA. Now, if we assume that Ha is compact on H 1 then it is also compact on VMOA, since H 1 is isomorphic to (VMOA)∗ . To get a contradiction we suppose that a does not belong to VMOlog . Then there exist some δ > 0 and rn → 1− such that arn − aBMOlog

≥ δ.

This implies Harn − Ha BMOlog

≥ cδ

and we can choose fn ∈ H 1 with fn H 1 ≤ 1, so that cδ

≤

≤

Harn fn − Ha fn H 1

 rn Ha (fn )rn rn − Ha fn H 1 

 Ha (fn )rn rn − Ha fn H 1 + o(1)

  Ha (fn )rn rn − (Ha fn )rn H 1 + (Ha fn )rn − Ha fn H 1 + o(1)

≤

Ha (fn )rn − Ha fn H 1 + (Ha fn )rn − Ha fn H 1 + o(1).

= =

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585

Taking a subsequence, if necessary, we may assume that there is a v ∈ H 1 so that Ha fn → v in H 1 . Therefore

 cδ ≤ Ha (fn )rn − fn H 1 + (Ha fn )rn − vrn H 1 + vrn − vH 1 +v − Ha fn H 1 + o(1)

 = Ha (fn )rn − fn H 1 + o(1). If we choose hn ∈ VMOA with hn BMOA = 1 and 

   cHa (fn )rn − fn H 1 ≤ Ha (fn )rn − fn , hn , we have cδ

≤

   Ha (fn )rn − fn , hn + o(1)

= =

(fn )rn − fn , Ha hn + o(1) fn , (Ha hn )rn − Ha hn + o(1).

Since Ha is compact on VMOA, taking a subsequence once more we see that there is a w ∈ VMOA so that Ha hn → w in VMOA. Hence cδ

≤

fn , (Ha hn )rn − wrn + fn , wrn − w + fn , w − Ha hn + o(1)

≤

c(Ha hn )rn − wrn BMO + cwrn − wBMO + cw − Ha hn BMO + o(1)

≤ =

cwrn − wBMO + o(1) o(1).

This is false and hence a ∈ VMOlog .




5. Fredholmness of Toeplitz operators We start by proving the equivalence of the criteria (2) and (3) for Fredholmness in Theorem 1.8. We use the symbol χn for the functions χn (ζ) = ζ n ,

ζ ∈ T.

Lemma 5.1. The functions in V + H ∞ ∩ BMOlog can be approximated in the space L∞ ∩ BMOlog by functions of the form χn h with n ≥ 0 and h ∈ H ∞ ∩ BMOlog . Proof. Let v + b ∈ V + H ∞ ∩ BMOlog . According to (3) of Theorem 2.2, there are trigonometric polynomials pk such that v − pk L∞ ∩BMOlog → 0. Since   pk + b ∈ χn h : n ≥ 0, h ∈ H ∞ ∩ BMOlog , the proof is complete.




Proposition 5.2. Let a ∈ V + H ∞ ∩ BMOlog . Then a is invertible in V + H ∞ ∩ BMOlog if and only if a is bounded away from zero, that is, there are  > 0 and δ > 0 such that |a(z)| ≥  for 1 − δ < |z| < 1.

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Proof. If a is invertible in V + H ∞ ∩ BMOlog , then it is obviously invertible in C + H ∞ and thus bounded away from zero according to [7, Theorem 6.45]. By the preceding lemma, there are N → +∞ and corresponding hN ∈ H ∞ ∩ BMOlog such that a − χN hN L∞ ∩BMOlog → 0. By [7, Theorem 6.45], hN is invertible in H ∞ with N sufficiently large. As hN ∈ BMOlog , so is its inverse. Thus, χN hN is invertible in V + H ∞ ∩ BMOlog . Now χ−N hN −1 → a−1 in
L∞ ∩ BMOlog and so a−1 is in the closed space V + H ∞ ∩ BMOlog . It remains to show that the two conditions above are indeed sufficent and necessary for Fredholmness. This follows from Theorem 5.6 and Proposition 5.9 below. Let us first consider two basic results for quite general symbols that are needed in what follows. Proposition 5.3. Let a, b ∈ L∞ ∩ BMOlog . Then Tab = Ta Tb + Ha H˜b ,

(27)

Hab = Ta Hb + Ha T˜b ,

(28)

where ˜b(ζ) = b(1/ζ), ζ ∈ T. Proof. See, e.g., [3, Proposition 2.14].




The next theorem gives a necessary condition for Fredholmness—cf. the wellknown theorem of Simonenko in the case of 1 < p < ∞. Theorem 5.4. Let a ∈ L∞ ∩ BMOlog . If Ta is Fredholm, then ess inf ζ∈T |a(ζ)| > 0; in particular, the symbol is invertible in L∞ ∩ BMOlog . Proof. Suppose that Ta is Fredholm but ess inf ζ∈T |a(ζ)| = 0. We consider a small  > 0 and decompose a = u + iv into real and imaginary parts. Define u = max(u, ) + min(u, −) and v by the analogous formula. Now the function a = u + iv is equal to 0 on a set of positive measure and a − a L∞ ∩BMOlog → 0 as  → 0+. This implies that Ta − Ta
 → 0 as  → 0+ and, hence, that Ta
is Fredholm if  is small enough. If Ta
f = P (a f ) = 0, then Q(a f ) = a f − P (a f ) = 0 on a set of positive measure and, hence, Q(a f ) = 0. Therefore, a f = 0 and, if  is small enough (so that a = 0) we find that f = 0 on a set of positive measure. This implies that f = 0 and we conclude that Ta
is one-to-one. The same is true for the dual operator (Ta
)∗ = Ta˜
. Therefore, Ta
is invertible. Since Ta
is invertible, there is some f so that Ta
f = P (a f ) = 1. Then Q(a f ) = a f −P (af ) = −1 on a set of positive measure and, hence, Q(a f ) = −1 which is clearly impossible.


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Remark 5.5. We do not know whether Fredholmness of Ta , when a ∈ L∞ and Qa ∈ BMOlog , implies invertibility of the symbol in this symbol class, which is optimal in the sense of boundedness. We next turn our attention to the relation between the symbol class V + H ∞ ∩ BMOlog and the space   A1 = a ∈ L∞ ∩ BMOlog : Ha ∈ K(H 1 ) according to the following result. Theorem 5.6. A1 = V + H ∞ ∩ BMOlog . Proof. If a = v + h for some v ∈ V and h ∈ H ∞ ∩ BMOlog , then Ha = H v + H h = H v , which is compact according to Theorem 1.7. On the other hand, if Ha is compact, then P1 a ∈ VMOlog according to Theorem 1.7. Therefore, (7) implies that a ∈ liplog +H ∞ ∩ BMOlog ⊆ V + H ∞ ∩ BMOlog . 1

Proposition 5.7. The space A is a closed subalgebra of L

∞




∩ BMOlog .

Proof. The fact that the space is an algebra follows from Proposition 5.3. Suppose that an → a in L∞ ∩ BMOlog with an ∈ A1 . Then Ha − Han  = Ha−an  ≤ c P1 (a − an )BMOlog → 0 (see Theorem 1.6). Thus, Ha is compact.




Corollary 5.8. The space V + H ∞ ∩ BMOlog is a Banach algebra. Proof. This is immediate from the preceding two results. It can also be proved directly.
The proof of the following theorem is based on an argument of B¨ ottcher and Silberman [2, ch. IV] when 1 < p < ∞. Proposition 5.9. Let a ∈ A1 . Then Ta is Fredholm on H 1 if and only if a is invertible in A1 . Proof. If a is invertible, then formula (27) shows that Ta−1 is a regularizer of Ta , and so Ta is Fredholm. If Ta is Fredholm, then Theorem 5.4 implies that a is invertible in L∞ ∩ BMOlog . Since Ta has a regularizer, say R, we can write RTa = I + K, where K is compact. Therefore, by (28), 0 = Haa−1 = Ta Ha−1 + Ha Ta˜−1 . This implies and, hence, a−1 ∈ A1 .

Ha−1 = −KHa−1 − RHa Ta˜−1


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6. Index formula For analytic symbols, the Fredholm properties of Toeplitz operators on H 1 are well understood: Theorem 6.1. For a ∈ H ∞ , the Toeplitz operator Ta on H 1 is Fredholm if and only if a is bounded away from zero, in which case Ind Ta = − ind ar .


Proof. See [15, Theorem 10].

Our aim in this section is to show that the preceding formula also holds for invertible symbols in the algebra V + H ∞ ∩ BMOlog . We start with a preliminary lemma. Lemma 6.2. If v ∈ V and f ∈ L∞ ∩ BMOlog , then (vf )r − vr fr L∞ ∩BMOlog −→ 0

(29)

as r → 1. Proof. From Lemma 2.61 of [3] it follows that (vf )r − vr fr ∞ → 0. Therefore, it is enough to show that (vf )r − vr fr BMOlog → 0. Also, since (vf )r − vr fr ∗∗ ≤ (vf )r − vfr ∗∗ + vfr − vr fr ∗∗ , and v − vr BMOlog → 0 according to Theorem 2.2, it is sufficient to prove that (vf )r − vfr ∗∗ → 0. For a function g : T → C, we write gη (ζ) = g(ζη) when ζ, η ∈ T. Then

 1 − r2 1 (vf )r (ζ) − (vfr )(ζ) = (vη (ζ) − v(ζ))fη (ζ)|dη| 2π T |1 − rη|2 and we need to estimate the expression  log 4π |I| J := |(vf )r (ζ) − (vfr )(ζ) − ((vf )r − vfr )I | |dζ| |I| I    log 4π  1 1 − r2  |I|  (vη (ζ) − v(ζ))fη (ζ) =  2 |I| I 2π T |1 − rη|   1 − (vη (θ) − v(θ))fη (θ)|dθ| dη  |dζ| |I| I    2π  log 4π 1 1 1 − r2  |I| (vη (ζ) − v(ζ))fη (ζ) ≤ |I| |I| I |1 − rη|2  I 2π 0   − (vη (θ) − v(θ))fη (θ)|dθ||dη||dζ|.

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Write (vη (ζ) − v(ζ))fη (ζ) − (vη (θ) − v(θ))fη (θ)     = (vη (ζ) − v(ζ))fη (ζ) − (vη − v)I fη (ζ) + (vη − v)I fη (ζ) − (vη − v)I fη (θ)   + (vη − v)I fη (θ) − (vη (θ) − v(θ))fη (θ) =: J1 + J2 + J3 . (30) Then J≤

 log 4π |I| |I|

I

1 2π

 0

2π

1 |I|

 I

1 − r2 (|J1 | + |J2 | + |J3 |)|dθ||dη||dζ|. |1 − rη|2

Let us first consider J1 . We have   2π  log 4π 1 1 1 − r2 |I| |J1 | |dθ||dη||dζ| |I| |I| I |1 − rη|2 I 2π 0  4π  f ∞ 2π 1 − r2 log |I| |vη (ζ) − v(ζ) − (vη − v)I | |dζ||dη| ≤ 2π |1 − rη|2 |I| 0 I  f ∞ 2π 1 − r2 ≤ vη − v∗∗ |dη| 2π |1 − rη|2 0  f ∞ 1 − r2 = vη − v∗∗ |dη| 2 2π 0<|1−η|<δ |1 − rη|  f ∞ 1 − r2 + vη − v∗∗ |dη|. 2 2π δ<|1−η|<2 |1 − rη| Now given  > 0, there is δ > 0 (according to Theorem 2.2) and r < 1 such that the above sum of two integrals can be estimated above by f ∞ f ∞ 2π + 2 v∗∗ . 2π 2π Similarly the part made of J3 can be shown to be as small as we wish provided that r is sufficiently close to 1. It remains to consider J2 . Note that (vη − v)I fη (ζ) − (vη − v)I fη (θ) = (vη − v)I (fη (ζ) − (fη )I ) + (vη − v)I ((fη )I − fη (θ)) and that it is sufficient to consider only one of the terms in the equality above. Since, by the choice of δ > 0 and r < 1,  2π  log 4π 1 1 − r2 |I| |vη − v|I |fη (ζ) − (fη )I | |dζ||dη| 2π 0 |1 − rη|2 |I| I  f ∗∗ 1 − r2 ≤ ( |vη − v|I |dη| 2 2π 0<|1−η|<δ |1 − rη|  1 − r2 + |vη − v|I |dη| 2 δ<|1−η|<2 |1 − rη|

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can be made arbitrarily small, the proof is complete.

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Theorem 6.3. If a, b ∈ V + H ∞ ∩ BMOlog , then (ab)r − ar br L∞ ∩BMOlog −→ 0 as r → 1 Proof. The statement follows easily from the preceding lemma and the fact (hg)r = hr gr for h, g ∈ H ∞ .
Proof of the index formula. According to Lemma 5.1 and the general theory of Fredholm operators, there is a function χn h (n ≥ 0 and h ∈ H ∞ ) that has the same index as a and generates a Toeplitz operator that is Fredholm of the same index as Ta . Using Theorems 6.1 and 6.3, and well-known properties of the index (of Fredholm operators and of continuous functions), and [14, Lemma 5] saying that Ind Tχn = −n, we can conclude that for r sufficiently close to 1, we have Ind Tχn h = Ind Tχn + Ind Th

(Atkinson)

= −n − ind hr = − ind((χn )r hr ) = − ind(χn h)r = − ind ar .




Proof of Theorem 1.8. Indeed this is an immediate consequence of Propositions 5.2 and 5.9, Theorem 5.6, and the preceding proof of the index formula.


References [1] A. B¨ ottcher and Y. Karlovich, Carleson curves, Muckenhoupt weights, and Toeplitz operators. Progress in Mathematics 154, Birkh¨ auser Verlag, Basel, 1997. [2] A. B¨ ottcher and B. Silbermann, Invertibility and asymptotics of Toeplitz matrices. Akademie-Verlag, Berlin, 1983. [3] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz operators. 2nd edition, SpringerVerlag, Berlin and Heidelberg, 2006. [4] J. Cima and D. Stegenga, Hankel operators on H p . Analysis at Urbana. Vol. 1: Analysis in function spaces, London Mathematical Society Lecture Note Series, 137, Cambridge University Press, Cambridge (1989), 133–150. [5] L. A. Coburn, Weyl’s Theorem for non-normal operators. Michigan Math. J. 13 (1966), 285–286. [6] R. G. Douglas, Toeplitz and Wiener-Hopf operators in H ∞ + C. Bull. Amer. Math. Soc. 74 (1968), 895–899. [7] R. G. Douglas, Banach algebra techniques in operator theory. 2nd edition, SpringerVerlag, New York, 1998. [8] J. B. Garnett, Bounded Analytic Functions. Revised 1st edition, Springer-Verlag, 2007. [9] S. Janson, On functions with conditions on the mean oscillation. Ark. Mat. 14, N2 (1976), 189–196.

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[10] S. Janson, J. Peetre and S. Semmes, On the action of Hankel and Toeplitz operators on some function spaces. Duke Math. J. 51, no. 4 (1984), 937–958. [11] I. Katznelson, An introduction to harmonic analysis. 3rd edition, Cambridge University Press, New York, 2004. [12] D. A. Stegenga, Bounded Toeplitz operators on H 1 and applications of the duality between H 1 and the functions of bounded mean oscillation. Amer. J. of Math. 98, no. 3 (1976), 573–589. [13] V. A. Tolokonnikov, Hankel and Toeplitz operators in Hardy spaces (Russian. English summary). Investigations on linear operators and the theory of functions, XIV, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 141 (1985), 165–175. (English translation in J. Sov. Math. 37 (1987), 1359–1364.) [14] J. A. Virtanen, Fredholm theory of Toeplitz operators on the Hardy space H 1 . Bull. London Math. Soc. 38 (2006), 143–155. [15] D. Vukoti´c, Analytic Toeplitz operators on the Hardy space H p : a survey. Bull. Belg. Math. Soc. 10 (2003), 101–113. M. Papadimitrakis Department of Mathematics University of Crete Knossos Avenue 71409 Iraklion Greece e-mail: [email protected] J. A. Virtanen Department of Mathematics University of Crete Knossos Avenue 71409 Iraklion Greece Current address: Department of Mathematics University of Helsinki Gustaf H¨ allstr¨ omin katu 2 b 00014 Helsinki Finland e-mail: [email protected] Submitted: May 8, 2008. Revised: May 15, 2008.

Integr. equ. oper. theory 61 (2008), 593–598 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0378-620X/040593-6, published online July 4, 2008 DOI 10.1007/s00020-008-1596-3

Integral Equations and Operator Theory

On the Cowen-Douglas Class for Banach Space Operators Marcus Carlsson Abstract. In [3], M. J. Cowen and R. G. Douglas prove that the adjoint of a Hilbert space operator T is in the class Bn (Ω) if and only if T is unitarily equivalent with the operator Mz on a Hilbert space H of Cn -valued analytic functions, where Mz denotes the operator of multiplication by the independent variable. The proof involves holomorphic vector bundles and Grauert’s theorem. In this paper we use a theorem by I. Gohberg and L. Rodman [4] to give a more elementary proof of this fact, which also works for Banach space operators. Mathematics Subject Classification (2000). 47B32. Keywords. Vector-valued analytic functions, operator models, shift operator, Cowen-Douglas class.

1. Introduction Let B be a Banach space and let L(B) denote the set of linear bounded operators on B. Also, let Ω be a domain in C and let z : Ω → Ω be the identity function, i.e. z(ζ) = ζ for all ζ ∈ Ω. If the elements of B consists of analytic functions on Ω and if B is such that f ∈ B implies zf ∈ B, then we will let Mz denote the operator of multiplication by z. i.e. Mz f = zf for all f ∈ B. Given a Hilbert space H, a number n ∈ N and a bounded connected domain Ω in C, it follows by the work of Cowen and Douglas in [3] that if T ∈ L(H) satisfies T − λ is bounded below for each λ ∈ Ω.

(1.1)

codim Ran (T − λ) = n for each λ ∈ Ω.

(1.2)

∩λ∈Ω Ran (T − λ) = {0}.

(1.3)

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then there exists a Hilbert space A, whose elements are Cn -valued analytic functions, and a unitary map U : H → A such that U T = Mz U.

(1.4)

Given λ ∈ Ω let eλ denote the point evaluation at λ, i.e. eλ (f ) = f (λ), ∀f ∈ A. The space A will be such that ∀λ ∈ Ω the point evaluations are continuous and surjective. If f ∈ A and f (λ) = 0 for some λ ∈ Ω, then

f ∈A z−λ

(1.5) (1.6)

Note that (1.6) can be reformulated as Ran (Mz − λ) = Ker eλ and that (1.5) and (1.6) imply that Mz − λ is a Fredholm operator for all λ ∈ Ω with ind (Mz − λ) = −n. By a Banach space of Cn -valued analytic functions on Ω we shall mean a Banach space A such that Mz is in L(A) and such that (1.5) and (1.6) hold. The main theorem is the following. Theorem 1.1. Let B be a Banach space, let Ω be a connected domain and let T ∈ L(B) be such that (1.1)–(1.3) hold. Then there exist a Banach space A of Cn -valued analytic functions on Ω, and an isometric isomorphism U : B → A, such that Mz U = U T . For applications of this result in the case when σ(T ) = D, (the unit disc in C), see [1] concerning boundary behavior of functions in A, and see [2] concerning the Fredholm index of T restricted to invariant subspaces.

2. Main To prove Theorem 1.1, we shall need the following result, which easily follows from Theorem 1.1 in [4]. We let Mn,n denote the set of complex n × n-matrices. Theorem 2.1. Let {λj }∞ j=1 be a set of distinct points in a domain Ω ⊂ C, with the accumulation points on the boundary of Ω. For each j let qj ∈ N and Vj,0 , . . . , Vj,qj ∈ Mn,n be given. Then there exists an Mn,n −valued analytic function M on Ω such that det (M (ζ)) = 0 for all ζ ∈ Ω \ {λj }∞ j=1 and M (ζ) −

qj
k=0

has a zero of order at least qj + 1 at λj .

Vj,k (ζ − λj )k

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Proof. By a well known theorem of Weierstrass, (see [5]), there exists a C-valued analytic function φ on Ω that has no zeros in Ω\{λj }∞ j=1 , and a zero of multiplicity qj + 1 at each λj . For each j ∈ N let ∞


φ(ζ) =

aj,k (ζ − λj )k

k=qj +1

∞ be the series expansion of φ at λj and let k=−qj −1 Uj,k (ζ − λj )k be the Laurent   qj k series expansion of k=0 Vj,k (ζ − λj ) /φ(ζ). Clearly
aj,l Uj,m = Vj,k l+m=k

for all k = 0 . . . qj , (where the sum is taken only over those indices such that aj,l Uj,m has been defined). By Theorem 1.1 in [4] there exists a meromorphic −1 Mn,n -valued function W on Ω such that {λj }∞ j=1 are the only poles of W or W and such that for every j the function −1


W (ζ) −

Uj,k (ζ − λj )k

k=−qj −1

is analytic in a neighborhood of λj for all j ∈ N. It is now easy to see that M = φW


is the desired function.

We will write f ≡ 0 if f is a function that is equal to zero at all points in its domain of definition. Corollary 2.2. Let {λj }∞ j=1 be a set of distinct points in a domain Ω ⊂ C, with the accumulation points on the boundary of Ω. For each j let Aj be an Mn,n -valued  analytic function defined in a neighborhood ∆j of λj such that det Aj (·) ≡ 0. Then there exists an Mn,n −valued analytic function M on Ω such that (i) det (M (ζ)) = 0 for all ζ ∈ Ω \ {λj }∞ j=1 . (ii) For each j the function M (·)A−1 (·) extends by continuity to an invertible j matrix at λj . is Proof. By Cramer’s rule and the assumption det Aj (·) ≡ 0 it follows that A−1 j meromorphic. Let ∞
Vj,k (ζ − λk )k Aj (ζ) = k=0

be the series expansion of Aj at λj and let A−1 j (ζ) =

∞
k=−qj

Uj,k (ζ − λk )k

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be the Laurent series expansion of A−1 at λj . Then j
Vi,k Uj,k = 0

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(2.1)

i+j=p

 for −qj ≤ p < 0 and i+j=0 Vi,k Uj,k = I. By Theorem 2.1 there exists an analytic Mn,n -valued function M on Ω such that det (M (ζ)) = 0 for all ζ ∈ Ω \ {λj }∞ j=1 and such that for every j the function M (ζ) −

qj


Vj,k (ζ − λj )k

k=0

has a zero of order at least q + 1 at λk . Using (2.1) it is easily seen that M satisfies (i) and (ii).
Proof of Theorem 1.1. Pick x1 , . . . , xn ∈ B such that Span {xi }ni=1  Ran (T − ζ0 ) = B.

(2.2)

holds for some ζ0 ∈ Ω, (where A  B = B means that A ∩ B = 0 and A + B = B). Let λ ∈ Ω be arbitrary but fixed, let Lλ be a left-inverse of T − λ and set −1 ∆λ = {ζ ∈ Ω : |ζ − λ| < Lλ }. It is easily seen that ∞  
(T − ζ) = I, (ζ − λ)i Li+1 λ i=0

for all ζ ∈ ∆λ , where I denotes the identity operator. It follows that the operator Pλ,ζ defined by ∞  
Pλ,ζ = (T − ζ) (ζ − λ)i Li+1 λ i=0

2 satisfies Pλ,ζ = Pλ,ζ and it is easily seen that

Ran Pλ,ζ = Ran (T − ζ).

(2.3)

Thus Pλ,ζ is a projection onto Ran (T − ζ). Set Kλ = Ker Lλ and note that dim Kλ = n by condition (1.2). Clearly Kλ ⊂ Ker Pλ,ζ which by equality (2.3) and condition (1.2) immediately implies that in fact Kλ = Ker Pλ,ζ .

(2.4)

Kλ  Ran (T − ζ) = B

(2.5)

As Pλ,ζ is a projection we have for all ζ ∈ ∆λ . Let kλ1 , . . . , kλn ∈ B ∗ be such that kλ1 |Kλ , . . . , kλn |Kλ , (i.e. their restrictions to Kλ ), form a basis for Kλ∗ . Define the matrix-valued function Aλ in ∆λ by  1  kλ ((I − Pλ,ζ )x1 ) . . . kλ1 ((I − Pλ,ζ )xn )   .. .. .. (2.6) Aλ (ζ) =  . . . . n n kλ ((I − Pλ,ζ )x1 ) . . . (kλ (I − Pλ,ζ )xn )

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Given ζ ∈ ∆λ , it follows by standard linear algebra that det Aλ (ζ) = 0

(2.7)

− Pλ,ζ )xi }ni=1

holds if and only if {(I is a linearly independent set, (cf. (2.4)). But by (2.3), (2.4) and (2.5) the latter is true if and only if Span {xi }ni=1  Ran (T − ζ) = B.

(2.8)

Now, assume that the above construction has been carried out around each point λ ∈ Ω. By the equivalence of (2.7) and (2.8) and the analyticity of det Aλ (·) it follows that if det Aλ1 (·) ≡ 0 and ∆λ1 ∩ ∆λ2 = ∅ for some λ1 , λ2 ∈ Ω, then det Aλ2 (·) ≡ 0 as well. By (2.2) and the assumption that Ω is connected we conclude that det Aλ (·) ≡ 0 for all λ ∈ Ω. By this observation it follows that the set Λ ⊂ Ω where (2.8) fails to hold is a discrete set with all accumulation points on ∂Ω. For any x ∈ B and all ζ ∈ Ω \ Λ we now define the coefficients c1 (x, ζ), . . . , cn (x, ζ) ∈ C by demanding that x−

n


ci (x, ζ)xi ∈ Ran (T − ζ).

i=1

By equality (2.8) the ci ’s are uniquely determined by the above equation. Let C(x, ·) be the Cn -valued function on Ω \ Λ given by   c1 (x, ·)   C(x, ·) =  ...  cn (x, ·) and for fixed λ ∈ Ω define the function Bλ (x, ·) on ∆λ by   1 kλ ((I − Pλ,ζ )x)   .. Bλ (x, ζ) =  . . kλn ((I − Pλ,ζ )x) Then Bλ (x, ζ) = Aλ (ζ)C(x, ζ) for all ζ ∈ ∆λ \ Λ, which by Cramer’s rule implies that C(x, ·) is a meromorphic function in Ω with all poles in Λ. Let {λj } be an enumeration of the points in Λ and apply Corollary 2.2 with Aj = Aλj to get an Mn,n -valued analytic function M on Ω such that det (M (ζ)) = 0 for all ζ ∈ Ω \ Λ and such that each M A−1 λj extends  to an analytic function at λj with det (M A−1 )(λ )) =  0. Given any x ∈ B we j λj now define the Cn -valued function xˆ via x ˆ(·) = M (·)C(x, ·). It is clear that x ˆ is analytic in Ω \ Λ and that for all ζ ∈ Ω \ Λ we have x ˆ(ζ) = 0 if and only if x ∈ Ran (T − ζ).

(2.9)

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Moreover, as x ˆ(·) = (M A−1 ˆ is analytic at each λj )(·)Bλj (x, ·) in ∆λj it follows that x λj as well, and by (2.3)-(2.5) and the definition of Bλj (x, ·) we conclude that (2.9) holds for all ζ ∈ Ω. Now, let A be the set A = {ˆ x : x ∈ B} and let U : B → A be the map U (x) = xˆ. As U (T x)(ζ) = U ((T − ζ)x)(ζ) + U (ζx)(ζ) = 0 + U (ζx)(ζ) = ζU (x)(ζ) we see that U T = Mz U. By (1.3) and (2.9) it follows that U is injective, and hence A becomes a Banach space with the norm ˆ x = x . It is clear that A satisfies (1.5). Finally, if U (x)(ζ) = 0 then there exists a y ∈ B such that x = (T − ζ)y, (by (2.9)), and thus U (x) = U ((T − ζ)y) = (Mz − ζ)U (y). Hence (1.6) holds, and the proof is complete.




References [1] M. Carlsson, Boundary behavior in Hilbert spaces of vector-valued analytic functions. Journal of Func. Anal., no 247/1, 2007. [2] M. Carlsson, On the index of invariant subspaces in Hilbert spaces of vector valued analytic functions. Preprint. [3] M.J. Cowen and R.G. Douglas, Complex geometry and operator theory. Acta Math. Vol. 141 (1978), no. 3-4, p. 187–261. [4] I. Gohberg and L. Rodman, Interpolation and local data for meromorphic matrix and operator functions. Integr. equ. oper. theory, no 9, 1986. [5] S. Lang, Complex analysis. Graduate texts in mathematics, Springer Verlag, 1999. Marcus Carlsson Department of Mathematics Purdue University 150 N. University Street West Lafayette, IN 47907-2067 USA e-mail: [email protected] Submitted: May 28, 2007. Revised: April 30, 2008.