Toeplitz Operators with Essentially Radial Symbols

For Topelitz operators with radial symbols on the disk, there are important results that characterize boundedness, compactness, and its relation to the Berezin transform. The notion of essentially radial symbol is a natural extension, in the context of multiply-connected domains, of the notion of radial symbol on the disk. In this paper we analyze the relationship between the boundary behavior of the Berezin transform and the compactness of Tφ when φ ∈ L2 Ω is essentially radial and Ω is multiply-connected domains.


Introduction
Toeplitz operators are object of intense study. Many papers have been dedicated to the study of these concrete class of operators generating many interesting results. A very important tool to study the behavior of these operators is the Berezin transform. This tool is particularly relevant with its connections with quantum mechanics, especially in the case of the Toeplitz operators on the Segal-Bargmann space. In this case, they arises naturally as anti-Wick quantization operators, and there is a natural equivalence between Toeplitz operators and a generalization of pseudodifferential operators, the so-called Weyl's quantization.
In a fundamental paper, Axler and Zheng proved that, if S ∈ B L 2 D can be written as a finite sum of finite products of Toeplitz operators with L ∞ -symbols, then S is compact if and only if S has a Berezin transform which vanishes at the boundary of the disk D. As they expected, this result has been extended into several directions, and it has been proved even for operators which are not of the Toeplitz type. Therefore it has been an important open problem to characterize the class of operators for which the compactness is equivalent to the vanishing of the Berezin transform. Since there are operators which are not compact but have a Berezin transform which vanishes at the boundary, it is now clear that the two notions are 2 International Journal of Mathematics and Mathematical Sciences not equivalent. Moreover, it is possible to show that in the context of Toeplitz operators there are examples of unbounded symbols whose corresponding operators are bounded and even compact.
Recently, many papers have been written in the case when the operator has an unbounded radial symbol ϕ ∈ L 2 D . Of course, for a square-integrable symbol, the Toeplitz operator is densely defined but is not necessarily bounded. However, it is possible see 1 of Grudsky and Vasilevski, 2 of Zorboska, and 3 of Korenblum and Zhu to show that operators with unbounded radial symbols can have a very rich structure. Moreover, there is a very neat and elegant way to characterize boundedness and compactness. The reason being that the operators with radial symbols on the disk are diagonal operators. In this context the relation between compactness and the Berezin transform has been studied in depth, and interesting results have been established.
In a previous paper see 4 , the author showed that it is possible to extend the notion of radial symbol when Ω is a bounded multiply-connected domain in the complex plane C, whose boundary ∂Ω consists of finitely many simple closed smooth analytic curves γ j j 1, 2, . . . , n where γ j are positively oriented with respect to Ω and γ j ∩ γ i ∅ if i / j. The key ingredient for this extension is to observe two facts. The first fact is that the structure of the Bergman kernel suggests that there is in any planar domain an internal region that we can neglect when we are interested in boundedness and compactness of the Toeplitz operators with square integrable symbols. The second observation consists in exploiting the geometry of the domain and conformal equivalence in order to partially recover the notion of radial symbol. For this class of essentially radial symbols, the compactness and boundedness have been studied and necessary and sufficient conditions established. In this paper we carry forward our analysis by investigating the relationship between the compactness and the vanishing of the Berezin transform. It is important to observe that in the case of the disk the analysis uses the fact that the Berezin transform can be easily written in a simple way since we can write explicitly an orthonormal basis, namely the collection of functions In the case of a planar domain, this is not possible because it is very hard to construct explicitly an orthonormal basis for the Bergman space. However, it is possible to reach interesting results that fully extend what it is known in the case of the disk.
The paper is organized as follows. In Section 2 we describe the setting where we work, give the relevant definitions, and state our main result. In Section 3 we prove the main result and we study several important consequences.

Preliminaries
Let Ω be a bounded multiply-connected domain in the complex plane C, whose boundary ∂Ω consists of finitely many simple closed smooth analytic curves γ j j 1, 2, . . . , n where γ j are positively oriented with respect to Ω and γ j ∩ γ i ∅ if i / j. We also assume that γ 1 is the boundary of the unbounded component of C \ Ω. Let Ω 1 be the bounded component of C \ γ 1 , and Ω j j 2, . . . , n the unbounded component of C \ γ j , respectively, so that Ω n j 1 Ω j . For dν 1/π dx dy we consider the usual a Ω, dν , consisting of all holomorphic functions which are L 2 -integrable, is a closed subspace of L 2 Ω, dν with the inner product given by International Journal of Mathematics and Mathematical Sciences 3 for f, g ∈ L 2 Ω, dν . The Bergman projection is the orthogonal projection We use the symbol Δ to indicate the punctured disk {z ∈ C | 0 < |z| < 1}. Let Γ be any one of the domains Ω, Δ, Ω j j 2, . . . , n . We call K Γ z, w the reproducing kernel of Γ, and we use the symbol k Γ z, w to indicate the normalized reproducing kernel; that is, then we indicate with the symbol ϕ the Berezin transform of the associated Toeplitz operator T ϕ , and we have We remind the reader that it is well known that A ∈ C ∞ b Γ and we have A ∞ ≤ A B L 2 Ω . It is possible, in the case of bounded symbols, to give a characterization of compactness using the Berezin transform see 5, 6 .
We remind the reader that any Ω bounded multiply-connected domain in the complex plane C, whose boundary ∂Ω consists of finitely many simple closed smooth analytic curves γ j j 1, 2, . . . , n , is conformally equivalent to a canonical bounded multiply-connected domain whose boundary consists of finitely many circles see 7 . This means that it is possible to find a conformally equivalent domain D n i 1 D i where D 1 {z ∈ C : |z| < 1} and D j {z ∈ C : |z−a j | > r j } for j 2, . . . , n. Here a j ∈ D 1 and 0 < r j < 1 with |a j −a k | > r j r k if j / k and 1 − |a j | > r j . Before we state the main result of this paper, we need to give a few definitions.

International Journal of Mathematics and Mathematical Sciences
Definition 2.1. Let Ω n i 1 Ω i be a canonical bounded multiply-connected domain. One says that the set of n 1 functions P {p 0 , p 1 , . . . , p n } is a ∂-partition for Ω if 1 for every j 0, 1, . . . , n, p j : Ω → 0, 1 is a Lipschitz, C ∞ -function; 2 for every j 2, . . . , n there exists an open set W j ⊂ Ω and an j > 0 such that U j {ζ ∈ Ω : r j < |ζ − a j | < r j j } and the support of p j are contained in W j and p j ζ 1 ∀ζ ∈ U j ; 2.7 3 for j 1 there exists an open set W 1 ⊂ Ω and an 1 > 0 such that U 1 {ζ ∈ Ω : 1 − 1 < |ζ| < 1} and the support of p 1 are contained in W 1 and 5 for any ζ ∈ Ω the following equation We also need the following.
when z ∈ U k {ζ ∈ Ω : r k < |ζ − a k | < r k k }, 2 for k 1 and for some 1 > 0, one has International Journal of Mathematics and Mathematical Sciences 5 The reader should note that, in the case where it is necessary to stress the use of a specific conformal equivalence, we will say that the map ϕ is essentially radial via Θ : n 1 Ω → n 1 D . Moreover, we stress that in what follows, when we are working with a general multiply-connected domain and we have a conformal equivalence Θ : n 1 Ω → n 1 D , we always assume that the ∂-partition is given on n 1 D and transferred to n 1 Ω through Θ in the natural way. any j 1, . . . , n where P {p 0 , p 1 , . . . , p n } is a ∂-partition for Ω then one defines the n sequences a ϕ 1 a ϕ 1 k k∈Z , a ϕ 2 a ϕ 2 k k∈Z , . . . , a ϕ n a ϕ n k k∈Z 2.13 as follows: if j 2, . . . , n, 2.14 and if j 1, At this point we can state the main result.

Canonical Multiply-Connected Domains and Essentially Radial Symbols
We concentrate on the relationship between compact Toeplitz operators and the Berezin transform. As we said in the introduction, Axler and Zheng have proved see 5 that if D is the disk, S m i m j k T ϕ i,k , where ϕ i,k ∈ L ∞ D , then S is compact if and only if its Berezin transform vanishes at the boundary of the disk. Their fundamental result has been extended in several directions, in particular when Ω is a general smoothly bounded multiplyconnected planar domain 6 . In this section we try to characterize the compactness in terms 6 International Journal of Mathematics and Mathematical Sciences of the Berezin transform. In the next theorem, under a certain condition, we will show that the Berezin transform characterization of compactness still holds in this context.
In the case of the disk, it is possible to show that when the operator is radial then its Berezin transform has a very special form. In fact, if ϕ : D → C is radial, then T ϕ z 1 − |z| 2 2 n 1 T ϕ e n , e n |z| 2n , 3.1 where, by definition, Therefore to show that the vanishing of the Berezin transform implies compactness is equivalent, given that T ϕ is diagonal and to show that lim |z| → 1 1 − |z| 2 2 n 1 T ϕ e n , e n |z| 2n 0 implies lim n → ∞ T ϕ e n , e n 0, Korenblum and Zhu realized this fact in their seminal paper 3 , and, along this line, more was discovered by Zorboska see 2 and Grudsky and Vasilevski see 1 .
In the case of a multiply-connected domain, it is not possible to write things so neatly; however, we can exploit our estimates near the boundary to use similar arguments. In fact, for an essentially radial function, the values depend essentially on the distance from the boundary. Moreover, we can simplify our analysis if we use the fact that every multiplyconnected domain is conformally equivalent to a canonical bounded multiply-connected domain whose boundary consists of finitely many circles. It is important to stress that in the case of essentially radial symbol it is possible to exploit what has been done in the case of the disk, but the operator is not a diagonal operator, and the Berezin transform is not particularly simple to write in an explicit way.
In what follows the punctured disk Δ {z ∈ C | 0 < |z| < 1} plays a very important role; for this reason we need the following. Theorem 3.1. There exists an isomorphism I : L 2 Δ → L 2 Ω 1 such that Moreover, for any p ≥ 2 one has that L p a Δ L p a Ω 1 , and, for any z, w ∈ Δ 2 , the Bergman kernels K Δ and K Ω 1 satisfy the following equation: Proof. Suppose that f ∈ L 2 a Δ ; this means that f is holomorphic on Δ, then we can write down the Laurent expansion of f about 0, and we have

3.6
The last equation, together with the fact that f is square-integrable, implies that a n 0 if n ≤ −1. Then we can conclude that f has an holomorphic extension on Ω 1 . We define in this way: if g ∈ L 2 Δ , then Ig z g z if z / 0 and Ig 0 Δ g z dν z .

3.8
Then Ig ∈ L 2 Ω 1 and Ig Ω 1 g Δ . If f ∈ L 2 a Δ , we have just shown that If ∈ L 2 a Ω 1 . Clearly I is injective and surjective, in fact if G ∈ L 2 Ω 1 , then g G |Δ is an element of L 2 Δ and I g G. Then I is an isomorphism of L 2 Δ onto L 2 Ω 1 and I L 2 a Δ L 2 a Ω 1 . Moreover, observing that p > 2 implies f Δ,2 ≤ f Δ,p for any f ∈ L p Δ , we conclude that L p a Δ L p a Ω 1 . Finally, it is easy to verify that for any f, g ∈ L 2 a Δ we have and this fact implies, by the definition of the Bergman reproducing kernel, that for any z, w ∈ Δ 2 .
In order to better explain our intuition, we remind the reader that we proved that, if ϕ ∈ L 2 D is an essentially radial function where Ω is a bounded multiply-connected domain and if we define ϕ j ϕ · p j where j 1, . . . , n where P {p 0 , p 1 , . . . , p n } is a ∂-partition for Ω, then the fact that the operator T ϕ : L 2 a Ω, dν → L 2 a Ω, dν is bounded compact International Journal of Mathematics and Mathematical Sciences is equivalent to fact, that for any j 1, . . . , n, the operators T ϕ j : L 2 a Ω j , dν → L 2 a Ω j , dν are bounded compact see 4 .
We start our investigation by focusing our attention on the case of bounded symbols. In fact, we prove the following. Theorem 3.2. Let ϕ ∈ L ∞ D be an essentially radial function, if one defines ϕ j ϕ · p j where j 1, . . . , n and P is a ∂-partition for D. Then for the bounded operator T ϕ the following are equivalent: 1 the operator T ϕ : L 2 a D, dν → L 2 a D, dν is compact; 2 for any j 1, . . . , n one has Proof. Since ϕ ∈ L ∞ D , we know that the operator T ϕ : L 2 a Ω, dν → L 2 a Ω, dν is bounded, and we know that the boundedness compactness is equivalent to the fact that for any j 1, . . . , n the operators T ϕ j : L 2 D j , dν → L 2 a D j , dν are bounded compact . If j 2, . . . , n, we observe that if we consider the following sets Δ 0,1 {z ∈ C : 0 < |z − a| < 1} and Δ a j ,r j {z ∈ C : 0 < |z − a j | < r j } and the maps Δ 0,1 α → Δ a j ,r j β → D j where α z a j r j z and β w w − a j −1 r 2 j a j and we use Proposition 1.1 in 8 , we can claim that T ϕ j is an isomorphism of the Hilbert spaces. Therefore T ϕ j is compact if and only if T ϕ j •β•α is compact. We also notice that the previous theorem implies that function { √ k 1z k } is an orthonormal basis for L 2 Δ 0,1 , and this, in turn, implies that the compactness of the operator T ϕ j •β•α is equivalent to the fact that for the sequence a ϕ j {a ϕ j k } k∈N we have lim k → ∞ a ϕ j k 0 where, by definition, To complete the proof we observe that, since ϕ j is radial and β • α r r −1 r j a j , then, after a change of variable, we can rewrite the last integral, and hence the formula a ϕ j k r j ∞ r j ϕ j r j s a j k 1 r 2k 1 j s 2k 1 1 s 2 ds ∀m ∈ Z 3.13 must hold for any j 2, . . . , n. For the case j 1 the proof is similar. Now we can prove the following.

Theorem 3.3.
Let ϕ ∈ L ∞ Ω be an essentially radial function via Θ : n 1 Ω → n 1 D , if one defines ϕ j ϕ · p j where j 1, . . . , n and P is a ∂-partition for Ω. Then for the bounded operator T ϕ the following are equivalent: 1 the operator T ϕ : L 2 a Ω, dν → L 2 a Ω, dν is compact; 2 for any j 1, . . . , n one has lim k → ∞ a ϕ j k 0.
3.14 Proof. We know that Ω is a regular domain, and therefore if Θ is a conformal mapping from Ω onto D then the Bergman kernels of Ω and Θ Ω D are related via K D Θ z , Θ w Θ z Θ w K Ω z, w and the operator V Θ f Θ · f • Θ is an isometry from L 2 D onto L 2 Ω see 8, Proposition 1.1 . In particular we have V Θ P D P Ω V Θ and this implies that V Θ T ϕ T ϕ•Θ −1 V Θ . Therefore the operator T ϕ is bounded if and only if for any j 1, . . . , n the operators T ϕ j •Θ −1 : L 2 a D j , dν → L 2 a D j , dν are bounded compact . Hence we can conclude that the operator is bounded compact if for any j 1, . . . , n we have Proof. We know that the operator under examination is bounded compact if and only if for any j 1, . . . , n the operators T ϕ j : L 2 D j , dν −→ L 2 a D j , dν 3.20 are bounded compact . If j 2, . . . , n, we observe that if we consider the following sets Δ 0,1 {z ∈ C : 0 < |z − a| < 1} and Δ a j ,r j {z ∈ C : 0 < |z − a j | < r j } and the following maps