International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 164843 | https://doi.org/10.1155/2011/164843

Roberto C. Raimondo, "Toeplitz Operators on the Bergman Space of Planar Domains with Essentially Radial Symbols", International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 164843, 26 pages, 2011. https://doi.org/10.1155/2011/164843

Toeplitz Operators on the Bergman Space of Planar Domains with Essentially Radial Symbols

Academic Editor: B. N. Mandal
Received25 Feb 2011
Revised05 Jun 2011
Accepted06 Jun 2011
Published22 Aug 2011

Abstract

We study the problem of the boundedness and compactness of π‘‡πœ™ when πœ™βˆˆπΏ2(Ξ©) and Ξ© is a planar domain. We find a necessary and sufficient condition while imposing a condition that generalizes the notion of radial symbol on the disk. We also analyze the relationship between the boundary behavior of the Berezin transform and the compactness of π‘‡πœ™.

1. Introduction

Let Ξ© be a bounded multiply-connected domain in the complex plane β„‚, whose boundary πœ•Ξ© consists of finitely many simple closed smooth analytic curves 𝛾𝑗(𝑗=1,2,…,𝑛) where 𝛾𝑗 are positively oriented with respect to Ξ© and π›Ύπ‘—βˆ©π›Ύπ‘–=βˆ… if 𝑖≠𝑗. We also assume that 𝛾1 is the boundary of the unbounded component of ℂ⧡Ω. Let Ξ©1 be the bounded component of ℂ⧡𝛾1, and Ω𝑗(𝑗=2,…,𝑛) the unbounded component of ℂ⧡𝛾𝑗, respectively, so that Ξ©=βˆ©π‘›π‘—=1Ω𝑗.

For π‘‘πœˆ=(1/πœ‹)𝑑π‘₯𝑑𝑦, we consider the usual 𝐿2-space 𝐿2(Ξ©)=𝐿2(Ξ©,π‘‘πœˆ). The Bergman space 𝐿2π‘Ž(Ξ©,π‘‘πœˆ), consisting of all holomorphic functions which are 𝐿2-integrable, is a closed subspace of 𝐿2(Ξ©,π‘‘πœˆ) with the inner product given by ξ€œβŸ¨π‘“,π‘”βŸ©=Ω𝑓(𝑧)𝑔(𝑧)π‘‘πœˆ(𝑧)(1.1) for 𝑓,π‘”βˆˆπΏ2(Ξ©,π‘‘πœˆ). The Bergman projection is the orthogonal projection π‘ƒβˆΆπΏ2(Ξ©,π‘‘πœˆ)⟢𝐿2π‘Ž(Ξ©,π‘‘πœˆ).(1.2) It is well-known that for any π‘“βˆˆπΏ2(Ξ©,π‘‘πœˆ), we have ξ€œπ‘ƒπ‘“(𝑀)=Ω𝑓(𝑧)𝐾Ω(𝑧,𝑀)π‘‘πœˆ(𝑧),(1.3) where 𝐾Ω is the Bergman reproducing kernel of Ξ©. For πœ‘βˆˆπΏβˆž(Ξ©,π‘‘πœˆ), the Toeplitz operator π‘‡πœ‘βˆΆπΏ2π‘Ž(Ξ©,π‘‘πœˆ)→𝐿2π‘Ž(Ξ©,π‘‘πœˆ) is defined by π‘‡πœ‘=π‘ƒπ‘€πœ‘, where π‘€πœ‘ is the standard multiplication operator. A simple calculation shows that π‘‡πœ‘ξ€œπ‘“(𝑧)=Ξ©πœ‘(𝑀)𝑓(𝑀)𝐾Ω(𝑀,𝑧)π‘‘πœˆ(𝑀).(1.4) For square-integrable symbols, the Toeplitz operator is densely defined but is not necessarily bounded; therefore, the problem of finding necessary and sufficient conditions on the function πœ‘βˆˆπΏ2(Ξ©,π‘‘πœˆ) for the Toeplitz operators π‘‡πœ‘ to be bounded or compact is a natural one, and it has been studied by many authors. Several important results have been established when the symbol has special geometric properties. In fact, in the context of radial symbols on the disk, many papers have been written with quite surprising results (see [1] of Grudsky and Vasilevski, [2] of Zorboska, and [3] of Korenblum and Zhu) showing that operators with unbounded radial symbols can have a very rich structure. In fact, in the case of a continuous symbol, the compactness of the Toeplitz operators depends only on the behavior of the symbol on the boundary of the disk and this is similar to what happens in the Hardy space case, even though in the case of Bergman space, the Toeplitz operator with continuous radial symbol is a compact perturbation of a scalar operator and in the Hardy space case a Toeplitz operator with radial symbol is just a scalar operator. In the case of unbounded radial symbols, a pivotal role is played by the fact that in the Bergman space setting, contrary to the Hardy space setting, there is an additional direction that Grudsky and Vasileski term as inside the domain direction: symbols that are nice with respect to the circular direction may have very complicated behavior in the radial direction. Of course, in the context of arbitrary planar domains, it is not possible to use the notion of radial symbol. We go around this difficulty by making two simple observations. To start, it is necessary to notice 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 Toeplitz operators with square integrable symbols, therefore the inside the domain direction counts up to a certain point. 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 these reasons, we study the problem for planar domains when the Toeplitz operator symbols have an almost-radial behavior and, for this class, we give a necessary and sufficient condition for boundedness and compactness. We also address the problem of the characterization of compactness by using the Berezin transform. In fact, under a growth condition for the almost-radial symbol, we show that the Berezin transform vanishes to the boundary if and only if the operator is compact.

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 collect results about the Bergman kernel for a planar domain and the structure of 𝐿2π‘Ž(Ξ©,π‘‘πœˆ). In Section 4, we prove the main result and study several important consequences.

2. Preliminaries

Let Ξ© be the bounded multiply-connected domain given at the beginning of Section 1, that is, Ξ©=βˆ©π‘›π‘—=1Ω𝑗, where Ξ©1 is the bounded component of ℂ⧡𝛾1, and Ω𝑗(𝑗=2,…,𝑛) is the unbounded component of ℂ⧡𝛾𝑗. We use the symbol Ξ” to indicate the punctured disk {π‘§βˆˆβ„‚βˆ£0<|𝑧|<1}. Let Ξ“ be any one of the domains Ξ©,Ξ”,Ω𝑗(𝑗=2,…,𝑛).

We call 𝐾Γ(𝑧,𝑀) the reproducing kernel of Ξ“ and we use the symbol π‘˜Ξ“(𝑧,𝑀) to indicate the normalized reproducing kernel, that is, π‘˜Ξ“(𝑧,𝑀)=𝐾Γ(𝑧,𝑀)/𝐾Γ(𝑀,𝑀)1/2.

For any π΄βˆˆβ„¬(𝐿2π‘Ž(Ξ“,π‘‘πœˆ)), we define 𝐴, the Berezin transform of 𝐴, by 𝐴(𝑀)=βŸ¨π΄π‘˜Ξ“π‘€,π‘˜Ξ“π‘€ξ€œβŸ©=Ξ“π΄π‘˜Ξ“π‘€(𝑧)π‘˜Ξ“π‘€(𝑧)π‘‘πœˆ(𝑧),(2.1) where π‘˜Ξ“π‘€(β‹…)=𝐾Γ(β‹…,𝑀)𝐾Γ(𝑀,𝑀)βˆ’1/2.

If πœ‘βˆˆπΏβˆž(Ξ“), then we indicate with the symbol ξ‚πœ‘ the Berezin transform of the associated Toeplitz operator π‘‡πœ‘, and we have ξ€œξ‚πœ‘(𝑀)=Ξ“||π‘˜πœ‘(𝑧)Γ𝑀(||𝑧)2π‘‘πœˆ(𝑧).(2.2) We remind the reader that it is well known that ξ‚π΄βˆˆπ’žβˆžπ‘(Ξ“), and we have β€–ξ‚π΄β€–βˆžβ‰€β€–π΄β€–β„¬(𝐿2(Ξ©)). It is possible, in the case of bounded symbols, to give a characterization of compactness using the Berezin transform (see [4, 5]).

We remind the reader that any Ξ© bounded multiply-connected domain in the complex plane β„‚, whose boundary πœ•Ξ© consists of finitely many simple closed smooth analytic curves 𝛾𝑗(𝑗=1,2,…,𝑛), is conformally equivalent to a canonical bounded multiply-connected domain whose boundary consists of finitely many circles (see [6]). This means that it is possible to find a conformally equivalent domain 𝐷=βˆ©π‘›π‘–=1𝐷𝑖 where 𝐷1={π‘§βˆˆβ„‚βˆΆ|𝑧|<1} and 𝐷𝑗={π‘§βˆˆβ„‚βˆΆ|π‘§βˆ’π‘Žπ‘—|>π‘Ÿπ‘—} for 𝑗=2,…,𝑛. Here π‘Žπ‘—βˆˆπ·1 and 0<π‘Ÿπ‘—<1 with |π‘Žπ‘—βˆ’π‘Žπ‘˜|>π‘Ÿπ‘—+π‘Ÿπ‘˜ if π‘—β‰ π‘˜ and 1βˆ’|π‘Žπ‘—|>π‘Ÿπ‘—. Before we state the main results of this paper we need to give a few definitions.

Definition 2.1. Let Ξ©=βˆ©π‘›π‘–=1Ω𝑖 be a canonical bounded multiply-connected domain. We say that the set of 𝑛+1 functions 𝔓={𝑝0,𝑝1,…,𝑝𝑛} is a πœ•-partition for Ξ© if(1)for every 𝑗=0,1,…,𝑛,π‘π‘—βˆΆΞ©β†’[0,1] is a Lipschitz, 𝐢∞-function,(2)for every 𝑗=2,…,𝑛, there exists an open set π‘Šπ‘—βŠ‚Ξ© and an πœ–π‘—>0 such that π‘ˆπœ–π‘—={πœβˆˆΞ©βˆΆπ‘Ÿπ‘—<|πœβˆ’π‘Žπ‘—|<π‘Ÿπ‘—+πœ–π‘—}, and the support of 𝑝𝑗 is contained in π‘Šπ‘— and 𝑝𝑗(𝜁)=1,βˆ€πœβˆˆπ‘ˆπœ–π‘—,(2.3)(3)for 𝑗=1, there exists an open set π‘Š1βŠ‚Ξ© and an πœ–1>0 such that π‘ˆπœ–1={𝜁∈Ω∢1βˆ’πœ–1<|𝜁|<1} and the support of 𝑝1 is contained in π‘Š1 and 𝑝1(𝜁)=1,βˆ€πœβˆˆπ‘ˆπœ–1,(2.4)(4)for every 𝑗,π‘˜=1,…,𝑛,π‘Šπ‘—βˆ©π‘Šπ‘˜=βˆ…, the set ⋃Ω⧡(𝑛𝑗=1π‘Šπ‘—) is not empty and the function 𝑝0(𝜁)=1,βˆ€πœβˆˆπ‘›ξšπ‘—=1π‘Šπ‘—ξƒͺπ‘π‘βˆ©Ξ©,0(𝜁)=0,βˆ€πœβˆˆπ‘ˆπœ–π‘˜,π‘˜=1,…,𝑛,(2.5)(5) for any 𝜁∈Ω, the following equation: π‘›ξ“π‘˜=0π‘π‘˜(𝜁)=1.(2.6) holds.

We need to point out two facts about the definition above: (i) that near each connected component of the boundary there is only one function which is different from zero (note that this implies that the function must be equal to 1), and (ii) far away from the boundary only the function 𝑝0 is different from zero.

Definition 2.2. A function πœ‘βˆΆΞ©=βˆ©π‘›π‘–=1Ω𝑖→ℂ is said to be essentially radial if there exists a conformally equivalent canonical bounded domain 𝐷=βˆ©π‘›π‘–=1𝐷𝑖, such that if the map Ξ˜βˆΆΞ©β†’π· is the conformal mapping from Ξ© onto 𝐷, then(1)for every π‘˜=2,…,𝑛 and for some πœ–π‘˜>0, we have πœ‘βˆ˜Ξ˜βˆ’1(𝑧)=πœ‘βˆ˜Ξ˜βˆ’1ξ€·||π‘§βˆ’π‘Žπ‘˜||ξ€Έ,(2.7) when π‘§βˆˆπ‘ˆπœ–π‘˜={πœβˆˆΞ©βˆΆπ‘Ÿπ‘˜<|πœβˆ’π‘Žπ‘˜|<π‘Ÿπ‘˜+πœ–π‘˜},(2)for π‘˜=1 and for some πœ–1>0, we have πœ‘βˆ˜Ξ˜βˆ’1(𝑧)=πœ‘βˆ˜Ξ˜βˆ’1(|𝑧|),(2.8) when π‘§βˆˆπ‘ˆπœ–1={𝜁∈Ω∢1βˆ’πœ–1<|𝜁|<1}.

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 πœ‘ is essentially radial via Ξ˜βˆΆβˆ©π‘›β„“=1Ξ©β„“β†’βˆ©π‘›β„“=1𝐷ℓ.

Before we proceed, the reader should notice that the definition, in the case of the disk, just says that, when we are near to the boundary, the values depend only on the distance from the center of the disk, so the function is essentially radial. In the general case, to formalize the fact that the values depend essentially on the distance from the boundary, we can simplify our analysis if we use the fact that this type of domain is conformally equivalent to a canonical bounded multiply-connected domain whose boundary consists of finitely many circles. For this type of domain the idea of essentially radial symbol is quite natural. For this reason, we use this simple geometric intuition to give the general definition.

Before we state the main result, we stress that in what follows, when we are working with a general multiply-connected domain and we have a conformal equivalence Ξ˜βˆΆβˆ©π‘›β„“=1Ξ©β„“β†’βˆ©π‘›β„“=1𝐷ℓ, we always assume that the πœ•-partition is given on βˆ©π‘›β„“=1𝐷ℓ and transferred to βˆ©π‘›β„“=1Ξ©β„“ through Θ in the natural way.

At this point, we can state the main result.

Theorem 2.3. Let πœ‘βˆˆπΏ2(Ξ©) be an essentially radial function via Ξ˜βˆΆβˆ©π‘›β„“=1Ξ©β„“β†’βˆ©π‘›β„“=1𝐷ℓ, if one defines πœ‘π‘—=πœ‘β‹…π‘π‘—, where 𝑗=1,…,𝑛 and 𝔓 is a πœ•-partition for Ξ©, then the following are equivalent: (1)the operator π‘‡πœ‘βˆΆπΏ2π‘Ž(Ξ©,π‘‘πœˆ)⟢𝐿2π‘Ž(Ξ©,π‘‘πœˆ)(2.9) is bounded (compact).(2)for any 𝑗=1,…,𝑛 the sequences π›Ύπœ‘π‘—={π›Ύπœ‘π‘—(π‘š)}π‘šβˆˆβ„•are in β„“βˆž(β„€+)(𝑐0(β„€+)) where, by definition, if 𝑗=2,…,𝑛, π›Ύπœ‘π‘—(π‘š)=π‘Ÿπ‘—ξ€œβˆžπ‘Ÿπ‘—πœ‘π‘—βˆ˜Ξ˜βˆ’1ξ‚€π‘Ÿπ‘—(2π‘š+1)/2(π‘š+1)𝑠1/2(π‘š+1)+π‘Žπ‘—ξ‚1𝑠2𝑑𝑠,βˆ€π‘šβˆˆβ„€+,(2.10) and if 𝑗=1π›Ύπœ‘1ξ€œ(π‘š)=10πœ‘1βˆ˜Ξ˜βˆ’1𝑠1/2(π‘š+1)𝑑𝑠,βˆ€π‘šβˆˆβ„€+.(2.11)

3. The Structure of 𝐿2π‘Ž(Ξ©) and Some Estimates about the Bergman Kernel

From now on, we will assume that Ξ©=βˆ©π‘›π‘—=1Ω𝑗 where Ξ©1={π‘§βˆˆβ„‚βˆΆ|𝑧|<1} and Ω𝑗={π‘§βˆˆβ„‚βˆΆ|π‘§βˆ’π‘Žπ‘—|>π‘Ÿπ‘—} for 𝑗=2,…,𝑛. Here, π‘Žπ‘—βˆˆΞ©1 and 0<π‘Ÿπ‘—<1 with |π‘Žπ‘—βˆ’π‘Žπ‘˜|>π‘Ÿπ‘—+π‘Ÿπ‘˜ if π‘—β‰ π‘˜ and 1βˆ’|π‘Žπ‘—|>π‘Ÿπ‘—. We will indicate with the symbol Ξ”0,1 the punctured disk Ξ©1⧡{0}.

With the symbols 𝐾Ω𝑗(𝑧,𝑀),𝐾Ω(𝑧,𝑀),𝐾Δ(𝑧,𝑀), we denote the Bergman kernel on Ω𝑗,Ξ©, and Ξ”, respectively.

In order to gain more information about the kernel of a planar domain, it is important to remind the reader that for the the punctured disk Ξ”0,1 and the disk Ξ©1, we have πΏπ‘π‘Ž(Ξ”0,1)=πΏπ‘π‘Ž(Ξ©1), if 𝑝β‰₯2, and, for any (𝑧,𝑀)βˆˆΞ”2,𝐾Δ(𝑧,𝑀)=𝐾Ω1(𝑧,𝑀) (see [7, 8]). This fact has an important and simple consequence. In fact, if we consider Ξ”π‘Ž,π‘Ÿ={π‘§βˆˆβ„‚βˆΆ0<|π‘§βˆ’π‘Ž|<π‘Ÿ} and π‘‚π‘Ž,π‘Ÿ={π‘§βˆˆβ„‚βˆΆ|π‘§βˆ’π‘Ž|>π‘Ÿ}, we can conclude that πΎπ‘‚π‘Ž,π‘Ÿπ‘Ÿ(𝑧,𝑀)=2ξ‚€π‘Ÿ2βˆ’(π‘§βˆ’π‘Ž)⋅(π‘€βˆ’π‘Ž)2,βˆ€(𝑧,𝑀)βˆˆπ‘‚π‘Ž,π‘ŸΓ—π‘‚π‘Ž,π‘Ÿ.(3.1)

To see this, we use the well-known fact that the reproducing kernel of the unit disk is given by (1βˆ’π‘§π‘€)βˆ’2, therefore we have 𝐾Δ0,11(𝑧,𝑀)=ξ€·1βˆ’π‘§β‹…π‘€ξ€Έ2,βˆ€(𝑧,𝑀)βˆˆΞ”0,1Γ—Ξ”0,1.(3.2) This implies, by conformal mapping, that the reproducing kernel of Ξ”π‘Ž,π‘Ÿ is πΎΞ”π‘Ž,π‘Ÿπ‘Ÿ(𝑧,𝑀)=2ξ‚€π‘Ÿ2βˆ’(π‘§βˆ’π‘Ž)⋅(π‘€βˆ’π‘Ž)2,βˆ€(𝑧,𝑀)βˆˆΞ”π‘Ž,π‘ŸΓ—Ξ”π‘Ž,π‘Ÿ.(3.3) Now, we define πœ‘βˆΆΞ”π‘Ž,π‘Ÿβ†’π‘‚π‘Ž,π‘Ÿ by πœ‘(𝑧)=(π‘§βˆ’π‘Ž)βˆ’1π‘Ÿ2+π‘Ž,(3.4) and we use the well-known fact that the Bergman kernels of Ξ”π‘Ž,π‘Ÿ and πœ“(Ξ”π‘Ž,π‘Ÿ)=π‘‚π‘Ž,π‘Ÿ are related via πΎπ‘‚π‘Ž,π‘Ÿ(πœ‘(𝑧),πœ‘(𝑀))πœ‘ξ…ž(𝑧)πœ‘ξ…ž(𝑀)=πΎΞ”π‘Ž,π‘Ÿ(𝑧,𝑀)(3.5) to obtain that πΎπ‘‚π‘Ž,π‘Ÿπ‘Ÿ(𝑧,𝑀)=2ξ‚€π‘Ÿ2βˆ’(π‘§βˆ’π‘Ž)⋅(π‘€βˆ’π‘Ž)2,βˆ€(𝑧,𝑀)βˆˆπ‘‚π‘Ž,π‘ŸΓ—π‘‚π‘Ž,π‘Ÿ.(3.6) Since Ξ©1=𝑂0,1 and, for 𝑗=2,…,𝑛,π‘‚π‘Žπ‘—,π‘Ÿπ‘—=Ω𝑗, then the last equations implies that 𝐾Ω11(𝑧,𝑀)=ξ€·1βˆ’π‘§β‹…π‘€ξ€Έ2,πΎΞ©π‘—π‘Ÿ(𝑧,𝑀)=2π‘—ξ‚€π‘Ÿ2π‘—βˆ’ξ€·π‘§βˆ’π‘Žπ‘—ξ€Έβ‹…ξ€·π‘€βˆ’π‘Žπ‘—ξ€Έξ‚2(3.7) if 𝑗=2,…,𝑛.

We also note that if we define 𝐸Ω(𝑧,𝑀)=𝐾Ω(𝑧,𝑀)βˆ’π‘›ξ“π‘—=1𝐾Ω𝑗(𝑧,𝑀),(3.8) we can prove the following.

Lemma 3.1. (1) 𝐸Ω is conjugate symmetric about z and w. For each π‘€βˆˆΞ©,𝐸Ω(β‹…,𝑀) is conjugate analytic on Ξ© and 𝐸Ω∈𝐢∞(Ω×Ω).
(2) There are neighborhoods π‘ˆπ‘— of πœ•Ξ©π‘—(𝑗=1,…,𝑛) and a constant 𝐢>0 such that π‘ˆπ‘—βˆ©π‘ˆπ‘˜ is empty if π‘—β‰ π‘˜ and ||𝐾Ω(𝑧,𝑀)βˆ’πΎΞ©π‘—||(𝑧,𝑀)<𝐢,(3.9) for π‘§βˆˆΞ© and π‘€βˆˆπ‘ˆπ‘—.
(3) 𝐸Ω∈𝐿∞(Ω×Ω).

Proof. (1) Since the Bergman kernels 𝐾Ω and 𝐾Ω𝑗 have these properties (see [9]), by the definition of 𝐸Ω, we get (1).
(2) The proof is given in [7, 8].
(3) Using the fact that 𝐾Ω11(𝑧,𝑀)=ξ€·1βˆ’π‘§β‹…π‘€ξ€Έ2,πΎΞ©π‘—π‘Ÿ(𝑧,𝑀)=2π‘—ξ‚€π‘Ÿ2π‘—βˆ’ξ€·π‘§βˆ’π‘Žπ‘—ξ€Έβ‹…ξ€·π‘€βˆ’π‘Žπ‘—ξ€Έξ‚2,(3.10) for 𝑗=2,...,𝑛 and (1) and (2), we get (3).

We observe that we can choose 𝑅𝑗>π‘Ÿπ‘— for 𝑗=2,…,𝑛 and 𝑅1<1 such that 𝐺𝑗={π‘§βˆΆπ‘Ÿπ‘—<|π‘§βˆ’π‘Žπ‘—|<𝑅𝑗}(𝑗=2,…,𝑛) and 𝐺1={π‘§βˆΆπ‘…1<|𝑧|<1}, then we have πΊπ‘—βŠ‚π‘ˆπ‘—, where π‘ˆπ‘— is the same as in Lemma 3.1. We also have the following.

Lemma 3.2. There are constants π’Ÿ>0 and β„³>0 such that (1)for any (𝑧,𝑀)βˆˆπΊπ‘–Γ—Ξ©βˆͺΩ×𝐺𝑖, one has ||𝐾Ω||||𝐾(𝑧,𝑀)<𝐷Ω𝑗||,||𝐾(𝑧,𝑀)Ω𝑗||<||𝐾(𝑧,𝑀)Ξ©||(𝑧,𝑀)+β„³,(3.11)(2)for any π‘§βˆˆΞ©, one has 𝐾Ω𝑗(𝑧,𝑧)<𝐾Ω(𝑧,𝑧).

Proof. By the explicit formula of the Bergman kernels 𝐾Ω𝑖, there are constants 𝐢𝑖 and 𝑀𝑖 such that ||𝐾Ω𝑖||(𝑧,𝑀)β‰₯𝐢𝑖,(3.12) for (𝑧,𝑀)∈(𝐺𝑖×Ω)βˆͺ(Ω×𝐺𝑖) and ||𝐾Ω𝑖||(𝑧,𝑀)≀𝑀𝑖(3.13) if (𝑧,𝑀)βˆ‰πΊπ‘–Γ—πΊπ‘– for 𝑖=1,2,…,𝑛. From the last Lemma, it follows that ||𝐾Ω||≀||𝐾(𝑧,𝑀)Ω𝑖||𝐢(𝑧,w)+𝐢≀1+𝐢𝑖||𝐾Ω𝑖||,||𝐾(𝑧,𝑀)Ω𝑖||≀||𝐾(𝑧,𝑀)Ξ©||+||𝐸(𝑧,𝑀)Ξ©||+(𝑧,𝑀)𝑗≠𝑖||𝐾Ω𝑗||<||𝐾(𝑧,𝑀)Ξ©||(𝑧,𝑀)+β€–πΈΞ©β€–βˆž+𝑖≠𝑗𝑀𝑗,(3.14) whenever (𝑧,𝑀)∈(𝐺𝑖×Ω)βˆͺ(Ω×𝐺𝑖). If we call π’Ÿ the biggest number among {1+𝐢/𝐢𝑗} and we let β„³=β€–πΈΞ©β€–βˆž+βˆ‘π‘›π‘—=1𝑀𝑗, then we get the first claimed estimate. The proof of (2) can be found in [8, 10].

It is clear from what we wrote so far that we put a strong emphasis on the fact that the domain under analysis Ξ© is actually the intersection of other domains, that is, Ξ©=βˆ©π‘›π‘—=1Ω𝑗. This also suggests that we should look for a representation of the elements of 𝐿2π‘Ž(Ξ©) that reflects this fact. For this reason, we give the following.

Definition 3.3. Given Ξ©=βˆ©π‘›π‘—=1Ω𝑗 with Ξ©1={π‘§βˆˆβ„‚βˆΆ|𝑧|<1} and Ω𝑗={π‘§βˆˆβ„‚βˆΆ|π‘§βˆ’π‘Žπ‘—|>π‘Ÿπ‘—}, for any π‘“βˆˆπΏ2π‘Ž(Ξ©), we define 𝑛+1 functions 𝑃0𝑓,𝑃1𝑓,𝑃2𝑓,…,𝑃𝑛𝑓 as follows: if π‘§βˆˆΞ©, then we set, for 𝑗=1, 𝑃11𝑓(𝑧)=β‹…ξ€œ2πœ‹π‘–Μ‚π›Ύ1𝑓(𝜁)πœβˆ’π‘§π‘‘πœ,(3.15) for 𝑗=2,3,…,𝑛, 𝑃𝑗1𝑓=β‹…ξ€œ2πœ‹π‘–Μ‚π›Ύπ‘—π‘“(𝜁)1πœβˆ’π‘§π‘‘πœβˆ’β‹…ξ€œ2πœ‹π‘–Μ‚π›Ύπ‘—π‘“(𝜁)π‘‘πœ,(3.16) and for 𝑗=0, 𝑃0𝑓=𝑛𝑗=21β‹…ξ€œ2πœ‹π‘–Μ‚π›Ύπ‘—ξƒͺ1𝑓(𝜁)π‘‘πœπ‘§βˆ’π‘Žπ‘—,(3.17) where ̂𝛾𝑗(𝑗=1,…,𝑛) are the circles which center at π‘Žπ‘—(π‘Ž1=0) and lie in 𝐺𝑗 (see Lemma 3.2), respectively, so that 𝑧 is exterior to ̂𝛾𝑗(𝑗=2,…,𝑛) and interior to ̂𝛾1.

It is important that the reader notices that the Cauchy theorem implies that our definition is independent from how we choose ̂𝛾1,…,̂𝛾𝑛. Moreover, it is important to notice that the domains of the functions 𝑃2𝑓,…,𝑃𝑛𝑓 are actually the sets Ξ©2,…,Ω𝑛. In the next Lemma, we give more information about this representation.

Lemma 3.4. For π‘“βˆˆπΏ2π‘Ž(Ξ©), one can write it uniquely as 𝑓(𝑧)=𝑛𝑗=1𝑃𝑗𝑓𝑃(𝑧)+0𝑓(𝑧),(3.18) with π‘ƒπ‘—π‘“βˆˆπΏ2π‘Ž(Ω𝑗),𝑃0π‘“βˆˆπΏ2π‘Ž(Ξ©)∩𝐢∞(Ξ©),π‘ƒπ‘˜(𝑃𝑗𝑓)=0 if π‘—β‰ π‘˜, and moreover, there exists a constant 𝑀1 such that, for 𝑗=0,1,…,𝑛, one has ‖‖𝑃𝑗𝑓‖‖Ω≀‖‖𝑃𝑗𝑓‖‖Ω𝑗≀𝑀1‖𝑓‖Ω.(3.19) In particular, if π‘“βˆˆπΏ2π‘Ž(Ω𝑖), then 𝑃𝑖𝑓=𝑓 and ‖𝑓‖Ω𝑖≀𝑀1‖𝑓‖Ω,(3.20) for 𝑖=1,…,𝑛.

Proof. Let 𝑓 be any function analytic on Ξ©. For any π‘§βˆˆΞ©, let 𝛾𝑖(𝑖=1,…,𝑛) be the circles which center at π‘Žπ‘–(π‘Ž1=0) and lie in 𝐺𝑖, respectively, so that 𝑧 is exterior to 𝛾𝑖(𝑖=2,…,𝑛) and interior to 𝛾1. Using Cauchy's Formula, we can write 𝑓(𝑧)=𝑛𝑗=11β‹…ξ€œ2πœ‹π‘–π›Ύπ‘—π‘“(𝜁)πœβˆ’π‘§π‘‘πœ.(3.21) Let 𝑓𝑗(1𝑧)=β‹…ξ€œ2πœ‹π‘–π›Ύπ‘—π‘“(𝜁)πœβˆ’π‘§π‘‘πœ.(3.22) By Cauchy's Formula, the value 𝑓𝑗(𝑧) does not depend on the choice of 𝛾𝑗 if 1≀𝑗≀𝑛 and βˆ‘π‘“(𝑧)=𝑛𝑗𝑓𝑗(𝑧). Of course, each 𝑓𝑗 is well defined for all π‘§βˆˆΞ©π‘— and analytic in Ω𝑗. In addition, if 𝑗≠1, we have that 𝑓𝑗(𝑧)β†’0 as |𝑧|β†’βˆž. Writing the Laurent expansion at π‘Žπ‘— of 𝑓𝑗, we have 𝑓1(𝑧)=βˆžξ“π‘˜=0𝛼1,π‘˜π‘§π‘˜,(3.23) and, for 𝑗≠1, 𝑓𝑗(𝑧)=βˆ’βˆžξ“π‘˜=βˆ’1𝛼𝑗,π‘˜ξ€·π‘§βˆ’π‘Žπ‘—ξ€Έπ‘˜,(3.24) and these series converge to 𝑓𝑗 uniformly and absolutely on any compact subset of Ω𝑗, respectively. We remark that the coefficients are given by the following formula: 𝛼𝑗,π‘˜=1ξ€œ2πœ‹π‘–π›Ύπ‘—π‘“(𝜁)ξ€·πœβˆ’π‘Žπ‘—ξ€Έπ‘˜+1π‘‘πœ,(3.25) where π‘˜β‰₯0 if 𝑗=1 and π‘˜β‰€βˆ’1 if 𝑗≠1 and π›Ύπ‘—βŠ‚πΊπ‘—,1≀𝑗≀𝑛. Moreover, if 𝑓 is holomorphic in some Ω𝑗 and 𝑓(𝑧)β†’0 as |𝑧|β†’βˆž when 𝑖≠1, then π›Όπ‘—π‘˜=0 for all 𝑗≠𝑖 by Cauchy's theorem and, therefore, 𝑓𝑗=0.
Now, we define 𝑃1𝑓=𝑓1 and 𝑃𝑗𝑓(𝑧)=βˆ’βˆžξ“π‘˜=βˆ’2π›Όπ‘—π‘˜ξ€·π‘§βˆ’π‘Žπ‘—ξ€Έπ‘˜,(3.26) for 𝑗=2,3,…,𝑛 and 𝑃0𝑓(𝑧)=𝑛𝑗=2𝛼𝑗,βˆ’1ξ€·π‘§βˆ’π‘Žπ‘—ξ€Έβˆ’1,(3.27) then βˆ‘π‘“(𝑧)=𝑛𝑖=0𝑃𝑖𝑓(𝑧) for all π‘§βˆˆΞ© and π‘ƒπ‘˜(𝑃𝑗𝑓)=0 if 0β‰ π‘˜β‰ π‘—β‰ 0 as we have proved above.
We claim that π‘“βˆˆπΏ2π‘Ž(Ξ©) implies that π‘ƒπ‘–π‘“βˆˆπΏ2π‘Ž(Ω𝑗) for 𝑗=1,2,…,𝑛, respectively. Indeed, since each annulus 𝐺𝑗 is contained in Ξ©,π‘“βˆˆπΏ2π‘Ž(Ξ©) implies that 𝑓 is an element of 𝐿2π‘Ž(𝐺𝑖) for all 𝑖=1,2,…,𝑛.
For any fixed 𝑖, note that 𝑃𝑗𝑓(0≠𝑗≠𝑖) and 𝑃0π‘“βˆ’π›Όπ‘—,βˆ’1β‹…(π‘§βˆ’π‘Žπ‘—)βˆ’1 are analytic on 𝐺𝑖βˆͺ(β„‚/Ω𝑖) and lim|𝑧|β†’βˆžπ‘ƒπ‘—π‘“(𝑧)=0 for 𝑗≠1. Expanding them as Laurent series, it follows that:(1)If 𝑖=1, then π‘ƒπ‘—βˆ‘π‘“=+βˆžπ‘˜=1π›½π‘—π‘˜/π‘§π‘˜ for 𝑗≠1,(2)If 𝑖≠1, then 𝑃𝑗𝑓(𝑧)=+βˆžξ“π‘˜=0π›½π‘—π‘˜ξ€·π‘§βˆ’π‘Žπ‘–ξ€Έπ‘˜,(3.28) for 0≠𝑗≠𝑖 and 𝑃0𝑓(𝑧)=+βˆžξ“π‘˜=0𝛽0π‘˜ξ€·π‘§βˆ’π‘Žπ‘–ξ€Έπ‘˜+𝛼𝑖,βˆ’1π‘§βˆ’π‘Žπ‘–.(3.29) It is obvious that, in any case, these series converge uniformly and absolutely on 𝐺𝑖. Observing that each 𝐺𝑖 is an annulus at π‘Žπ‘–, we have, by direct computation, that βŸ¨π‘“,π‘“βŸ©πΊπ‘–β‰₯βŸ¨π‘ƒπ‘–π‘“,π‘ƒπ‘–π‘“βŸ©πΊπ‘–+||𝛼𝑖,βˆ’1||2ξ€·lnπ‘…π‘–βˆ’lnπ‘Ÿπ‘–ξ€Έ(3.30) if 𝑖≠1 and βŸ¨π‘“,π‘“βŸ©πΊ1β‰₯βŸ¨π‘ƒ1𝑓,𝑃1π‘“βŸ©πΊ1.(3.31) Therefore, for any 𝑖=1,…,𝑛, there exists a constant π‘€ξ…ž such that ‖‖𝑃𝑖𝑓‖‖𝐺𝑖≀‖𝑓‖𝐺𝑖≀‖𝑓‖Ω,||𝛼(βˆ—)𝑖,βˆ’1||β‰€π‘€ξ…žβ‹…β€–π‘“β€–Ξ©.(βˆ—βˆ—) From the definition of 𝑃𝑗𝑓, we derive ‖‖𝑃1𝑓‖‖2𝐺1=+βˆžξ“0||𝛼1π‘˜||2ξ€·1βˆ’π‘…12π‘˜+2ξ€Έ,β€–β€–π‘ƒπ‘˜+1𝑖𝑓‖‖2𝐺𝑖=βˆ’βˆžξ“π‘˜=βˆ’2|𝛼|2π‘–π‘˜ξ€·π‘Ÿπ‘–2π‘˜+2βˆ’π‘…π‘–2π‘˜+2ξ€Έ,π‘˜+1(3.32) for 𝑖=2,…,𝑛. The convergence of these series is guaranteed by the conditions βˆ— and βˆ—βˆ—. Since 𝑅1<1 and π‘Ÿπ‘–<𝑅𝑖, it follows that π‘ƒπ‘–π‘“βˆˆπΏ2π‘Ž(Ω𝑖) and ‖‖𝑃1𝑓‖‖2Ξ©1=+βˆžξ“0||𝛼1π‘˜||2,β€–β€–π‘ƒπ‘˜+1𝑖𝑓‖‖2Ω𝑖=βˆ’βˆžξ“π‘˜=βˆ’2||𝛼1π‘˜||2π‘Ÿπ‘–2π‘˜+2,π‘˜+1(3.33) for 𝑖=2,…,𝑛. Comparing the expression of ‖𝑃𝑖𝑓‖Ω𝑖 with the expression of ‖𝑃𝑖𝑓‖𝐺𝑖, it follows that ‖𝑃𝑖𝑓‖Ω𝑖<𝑀⋅‖𝑃𝑖𝑓‖𝐺𝑖 for some constant 𝑀 for 𝑖=1,…,𝑛. Hence, ‖𝑃𝑖𝑓‖Ω𝑖<𝑀⋅‖𝑃𝑖𝑓‖Ω. Moreover, if we define π‘€ξ…žξ…ž=Max{β€–(π‘§βˆ’π‘Žπ‘–)βˆ’1β€–Ξ©}, from the inequalities ‖𝑃𝑖𝑓‖𝐺𝑖≀‖𝑓‖𝐺𝑖≀‖𝑓‖Ω and |𝛼𝑖,βˆ’1|β‰€π‘€ξ…žβ‹…β€–π‘“β€–Ξ© and the definition of 𝑃0, it follows that ‖𝑃0π‘“β€–Ξ©β‰€π‘›β‹…π‘€ξ…žβ‹…π‘€ξ…žξ…žβ‹…β€–π‘“β€–Ξ©.
If π‘“βˆˆπΏ2π‘Ž(Ω𝑖) for some π‘–βˆˆ{1,2,…,𝑛}, note that lim𝑓(𝑧)=0 as |𝑧|β†’βˆž for 𝑖≠1, then 𝑓(𝑧)=𝑃𝑖𝑓(𝑧)+𝛼𝑖,βˆ’1(π‘§βˆ’π‘Žπ‘–)βˆ’1 if 𝑖≠1 and 𝑃1𝑓=𝑓 if 𝑖=1. For 𝑖≠1, since π‘“βˆˆπΏ2π‘Ž(Ω𝑖)βŠ‚πΏ2π‘Ž(Ξ©) implies that π‘ƒπ‘–π‘“βˆˆπΏ2π‘Ž(Ω𝑖), then 𝛼𝑖,βˆ’1β‹…(π‘§βˆ’π‘Žπ‘–)βˆ’1∈𝐿2π‘Ž(Ω𝑖). We must have 𝛼𝑖,βˆ’1=0 and, consequently, 𝑃0𝑓=0. Hence, in any case, π‘“βˆˆπΏ2π‘Ž(Ω𝑖) implies 𝑓=𝑃𝑖𝑓 and 𝑃𝑗𝑓=0 if 𝑖≠𝑗, and this remark completes our proof.

Lemma 3.5. If {𝑓𝑛} is a bounded sequence in 𝐿2π‘Ž(Ξ©) and 𝑓𝑛→0 weakly in 𝐿2π‘Ž(Ξ©), then 𝑃𝑗𝑓𝑛→0 weakly on 𝐿2π‘Ž(Ω𝑗) for 𝑗=1,…,𝑛 and 𝑃0𝑓𝑛→0 uniformly on Ξ©.

Proof. By the previous Lemma, we know that the linear transformations {𝑃𝑗} are bounded operators, then 𝑓𝑛→0 weakly in 𝐿2π‘Ž(Ξ©) implies that 𝑃𝑗𝑓𝑛→0 weakly on 𝐿2π‘Ž(Ω𝑗) for 𝑗=1,…,𝑛. For the same reason, 𝑃0𝑓𝑛→0 weakly in 𝐿2π‘Ž(Ξ©) and then 𝑃0𝑓𝑛(𝜁)β†’0 for any 𝜁∈Ω. Since 𝑃0π‘“π‘š=𝑛𝑖=2𝛼𝑖,βˆ’1(π‘š)ξ€·πœβˆ’π‘Žπ‘–ξ€Έ,(3.34) by the estimates given in the last lemma, we have that |𝛼𝑖,βˆ’1(π‘š)|<π‘€β€–π‘“π‘šβ€–Ξ©. The boundedness of {β€–π‘“π‘šβ€–Ξ©} implies that the family of continuous functions {𝑃0π‘“π‘š} is uniformly bounded and equicontinuous on Ξ©, then, by Arzela-Ascoli's Theorem, we have that 𝑃0π‘“π‘šβ†’0 uniformly on Ξ©.

4. Canonical Multiply-Connected Domains and Essentially Radial Symbols

In this section, we investigate, with the help of the results established in the previous section, necessary and sufficient conditions on the essentially radial function πœ‘βˆˆπΏ2(Ξ©,π‘‘πœˆ) for the Toeplitz operator π‘‡πœ‘ to be bounded or compact.

Before we state the next Theorem, we remind the reader that 𝐾Ω(𝜁,𝑧)=𝐸Ω(𝜁,𝑧)+𝑛ℓ=1𝐾Ωℓ(𝜁,𝑧),(4.1) where 𝐸Ω∈𝐿∞(Ω×Ω) and, for all β„“=1,…,𝑛, we have 𝐾Ωℓ(𝜁,𝑧)=𝐾Ωℓ(𝜁,𝑧),βˆ€πœ,π‘§βˆˆΞ©Γ—Ξ©,(4.2) where 𝐾Ωℓ is the reproducing kernel of Ξ©β„“. If we use the symbol 𝐾Ω0 to indicate 𝐸Ω, we can write 𝐾Ω(𝜁,𝑧)=𝑛ℓ=0𝐾Ωℓ(𝜁,𝑧).(4.3) We also remind the reader that if 𝐼∢𝐿2π‘Ž(Ξ©)→𝐿2π‘Ž(Ξ©) is the identity operator, then 𝐼=𝑛ℓ=0𝑃ℓ,(4.4) where π‘ƒβ„“βˆΆπΏ2π‘Ž(Ξ©)→𝐿2π‘Ž(Ξ©) is a bounded operator for all β„“=0,1,…,𝑛 with π‘ƒβ„“π‘“βˆˆπΏ2π‘Ž(Ξ©β„“) if β„“=1,…,𝑛 and 𝑃0π‘“βˆˆπ’žβˆž(Ξ©) and π‘ƒπ‘˜π‘ƒβ„“=0 if π‘˜β‰ β„“ (see Lemma 3.4).

In order to make our notation a little simpler, when we use a kernel operator we will denote it by the name of its kernel function. For example, the Bergman projection will be denoted by the symbol 𝐾Ω.

We are now in a position to prove the following result.

Lemma 4.1. Let πœ‘βˆˆπΏ2(𝐷) be an essentially radial function where 𝐷=βˆ©π‘›π‘—=1𝐷𝑗 with 𝐷1={π‘§βˆˆβ„‚βˆΆ|𝑧|<1} and 𝐷𝑗={π‘§βˆˆβ„‚βˆΆ|π‘§βˆ’π‘Žπ‘—|>π‘Ÿπ‘—} for 𝑗=2,…,𝑛. If one defines πœ‘π‘—=πœ‘β‹…π‘π‘— where 𝑗=1,…,𝑛 and 𝔓={𝑝0,𝑝1,…,𝑝𝑛} is a πœ•-partition for 𝐷,then the following are equivalent: (1)the operator π‘‡πœ‘βˆΆπΏ2π‘Ž(𝐷,π‘‘πœˆ)⟢𝐿2π‘Ž(𝐷,π‘‘πœˆ)(4.5) is bounded (compact);(2)for any 𝑗=1,…,𝑛, the operators π‘‡πœ‘π‘—βˆΆπΏ2π‘Žξ€·π·π‘—ξ€Έ,π‘‘πœˆβŸΆπΏ2π‘Žξ€·π·π‘—ξ€Έ,π‘‘πœˆ(4.6) are bounded (compact).

Proof. Let {𝑝0,𝑝1,…,𝑝𝑛} be a partition of the unit on 𝐷=βˆ©π‘›π‘—=1𝐷𝑗, which is a canonical domain. Now, we notice that for all π‘“βˆˆπΏ2(𝐷) and for all π‘€βˆˆπ·, we have the following: π‘‡πœ‘ξ€œπ‘“(𝑀)=π·πœ‘(𝑧)𝑓(𝑧)𝐾𝐷(=𝑧,𝑀)𝑑𝑣(𝑧)𝑛𝑗=0ξ€œπ·πœ‘(𝑧)𝑓(𝑧)𝐾𝐷𝑗(=𝑧,𝑀)𝑑𝑣(𝑧)𝑛𝑛𝑗=0ξ“π‘˜=0ξ€œπ·πœ‘(𝑧)π‘π‘˜(𝑧)𝑓(𝑧)𝐾𝐷𝑗=(𝑧,𝑀)𝑑𝑣(𝑧)𝑛𝑛𝑗=0ξ“π‘˜=0π‘‡π‘—π‘˜π‘“(𝑀),(4.7) where, by definition, we have π‘‡π‘—π‘˜ξ€œπ‘“(𝑀)=π·πœ‘(𝑧)π‘π‘˜(𝑧)𝐾𝐷𝑗(𝑧,𝑀)𝑓(𝑧)𝑑𝑣(𝑀)𝑑𝑣(𝑧).(4.8)

Claim 1. The operator 𝑇𝑗0 is Hilbert-Schmidt for any 𝑗=0,1,…,𝑛.

Proof. We observe that, by definition, we have 𝑇𝑗0ξ€œπ‘“(𝑀)=π·πœ‘(𝑧)𝑝0(𝑧)𝐾𝐷𝑗(𝑧,𝑀)𝑓(𝑧)𝑑𝑣(𝑧),(4.9) therefore, if we define ℐ1=𝐷||πœ‘(𝑧)𝑝0(𝑧)𝐾𝐷𝑗(||𝑧,𝑀)2𝑑𝑣(𝑧)𝑑𝑣(𝑀),(4.10) we have ℐ1=ξ€œπ·||πœ‘(𝑧)𝑝0||(𝑧)2ξ‚΅ξ€œπ·||𝐾𝐷𝑗||(𝑧,𝑀)2ξ‚Άβ‰€ξ€œπ‘‘π‘£(𝑀)𝑑𝑣(𝑧)𝐷||πœ‘(𝑧)𝑝0||(𝑧)2||𝐾𝐷𝑗||≀(𝑧,𝑧)𝑑𝑣(𝑧)Maxπ‘§βˆˆsupp(𝑝0)||𝑝0||(𝑧)2πΎπ·π‘—ξ‚Άξ€œ(𝑧,𝑧)𝐷||πœ‘π‘—||(𝑧)2≀𝑑𝑣(𝑧)Maxπ‘§βˆˆsupp(𝑝0)||𝑝0||(𝑧)2𝐾𝐷𝑗(𝑧,𝑧)β‹…β€–πœ‘β€–2𝐷,2<∞.(4.11) This implies that for any 𝑑=0,1,…,𝑛, 𝑇𝑑0 is Hilbert-Schmidt. Therefore, the operator 𝑛𝑑=0𝑇𝑑0(4.12) is Hilbert-Schmidt, and this completes the proof of the claim.

Claim 2. The operator 𝑇0π‘˜ is Hilbert-Schmidt for any π‘˜=0,1,…,𝑛.

Proof. We observe that, by definition, we have 𝑇0π‘˜ξ€œπ‘“(𝑀)=π·πœ‘(𝑧)π‘π‘˜(𝑧)𝐾𝐷0(𝑧,𝑀)𝑓(𝑧)𝑑𝑣(𝑧),(4.13) therefore, if we define ℐ2=𝐷||πœ‘(𝑧)π‘π‘˜(𝑧)𝐾𝐷0(||𝑧,𝑀)2𝑑𝑣(𝑧)𝑑𝑣(𝑀),(4.14) we have ℐ2=𝐷||πœ‘(𝑧)𝑝0||(𝑧)2||𝐾𝐷0(||𝑧,𝑀)2≀𝑑𝑣(𝑀)𝑑𝑣(𝑧)Max(𝑧,𝑀)βˆˆπ·Γ—π·||𝐾𝐷0||(𝑧,𝑀)2ξ‚Άξ€œβ‹…π‘£(𝐷)⋅𝐷||πœ‘(𝑧)𝑝0||(𝑧)2≀𝑑𝑣(𝑧)Max(𝑧,𝑀)βˆˆπ·Γ—π·||𝐾𝐷0||(𝑧,𝑀)2⋅𝑣(𝐷)β‹…β€–πœ‘β€–2𝐷,2<∞.(4.15) This implies that for any 𝑑=0,1,…,𝑛, 𝑇0𝑑 is Hilbert-Schmidt. Therefore, the following 𝑛𝑑=0𝑇0𝑑(4.16) is Hilbert-Schmidt, and this completes the proof of the claim.

Claim 3. The operator 𝑇𝑖𝑗 is Hilbert-Schmidt if 𝑖≠𝑗≠0 and 𝑗,𝑖=1,…,𝑛.

Proof. We observe that π‘‡π‘—π‘˜ξ€œπ‘“(𝑀)=π·πœ‘(𝑧)π‘π‘˜(𝑧)𝐾𝐷𝑗(𝑧,𝑀)𝑓(𝑧)𝑑𝑣(𝑀)𝑑𝑣(𝑧).(4.17) To start, we give the following: 𝒩𝑗𝑖(𝑧,𝑀)def=πœ‘π‘—(𝑧)⋅𝐾𝐷𝑖(𝑧,𝑀).(4.18) We will show that Fubini theorem and the properties of the πœ•-partition imply that 𝐷||𝒩𝑗𝑖||(𝑧,𝑀)2𝑑𝑣(𝑀)𝑑𝑣(𝑧)<∞.(4.19) In fact, we have 𝐷||𝒩𝑗𝑖||(𝑧,𝑀)2=ξ€œπ·ξ‚΅ξ€œπ·||𝒩𝑗𝑖||(𝑧,𝑀)2ξ‚Ά=𝑑𝑣(𝑀)𝑑𝑣(𝑧)𝐷||πœ‘π‘—||(𝑧)2||𝐾𝐷𝑖||(𝑧,𝑀)2=ξ€œπ‘‘π‘£(𝑀)𝑑𝑣(𝑧)𝐷||πœ‘π‘—||(𝑧)2ξ‚΅ξ€œπ·||𝐾𝐷𝑖||(𝑧,𝑀)2ξ‚Ά=ξ€œπ‘‘π‘£(𝑀)𝑑𝑣(𝑧)𝐷||πœ‘π‘—||(𝑧)2𝐾𝐷𝑖=ξ€œ(𝑧,𝑧)𝑑𝑣(𝑧)𝐷||||πœ‘(𝑧)2||𝑝𝑗||(𝑧)2𝐾𝐷𝑖≀(𝑧,𝑧)𝑑𝑣(𝑧)Maxπ‘§βˆˆsupp(𝑝𝑗)||𝑝𝑗||(𝑧)2𝐾𝐷𝑖(𝑧,𝑧)β‹…β€–πœ‘β€–2𝐷,2<∞.(4.20)
Therefore, we can write that π‘‡πœ‘=𝒦+𝑛ℓ=1π‘‡πœ‘β„“β„“,(4.21) where 𝒦 is a compact operator.
We also observe that Lemma 3.4 implies that π‘‡πœ‘β„“β„“=βˆ‘π‘›π‘—=0π‘‡πœ‘β„“β„“π‘ƒπ‘—, and we prove that the operator π‘‡πœ‘β„“β„“π‘ƒπ‘— is compact if 𝑗≠ℓ and 𝑗,β„“=1,…,𝑛.

Proof. In order to simplify the notation, we define the operator 𝑅𝑗,β„“=π‘‡πœ‘β„“β„“π‘ƒπ‘—=πΎπ·β„“π‘€πœ‘π‘β„“π‘ƒπ‘—. To prove our statement, it is enough to prove that if we take a bounded sequence {𝑓𝑛} in 𝐿2(𝐷) such that 𝑓𝑛→0 weakly, then we can prove that ‖𝑅𝑗,ℓ𝑓𝑛‖2β†’0. We know that the continuity of 𝑃ℓ implies that π‘ƒπ‘—π‘“π‘˜β†’0 weakly on 𝐻2(𝐷𝑙), and {β€–π‘ƒπ‘—π‘“π‘˜β€–π·β„“} is bounded by Lemma 3.5. Since it is a sequence of holomorphic functions, we know that {π‘ƒπ‘—π‘“π‘˜} is uniformly bounded on any compact subset of 𝐷ℓ. Therefore, the sequence {π‘ƒπ‘—π‘“π‘˜} is a normal family of functions. Since π‘ƒπ‘—π‘“π‘˜(𝜁)β†’0 for any πœβˆˆπ·π‘—, then π‘ƒπ‘—π‘“π‘˜ converges uniformly on any compact subset of 𝐷𝑗 and consequently on 𝐹=supp(𝑝ℓ). To complete the proof, we remind the reader that if we define the operators π‘„β„“βˆΆπΏ2(𝐷)→𝐿2(𝐷), for β„“=1,2,…,𝑛, in this way π‘„β„“ξ€œπ‘“(𝑧)=𝐷||𝐾𝑓(𝜁)𝐷ℓ(||𝜁,𝑧)𝑑𝑣(𝜁).(4.22) It is possible to prove, with the help of Schur's test (see [11] ), that 𝑄ℓ is a bounded operator (see [5]). Now, we observe that ||𝑅𝑗,β„“π‘“π‘˜||ξ€½||𝑃(𝜁)≀Supπ‘—π‘“π‘˜||ξ€Ύβ‹…||𝑄(𝜁)βˆΆπœβˆˆπΉπ‘—ξ€·||π’³πΉπœ‘π‘π‘ ||ξ€Έ||,(𝜁)(4.23) then, by using the fact that 𝑄ℓ is bounded, we have ‖‖𝑅𝑗,β„“π‘“π‘˜β€–β€–π·ξ€½||𝑃≀Supπ‘—π‘“π‘˜||ξ€Ύβ€–β€–πœ‘(𝜁)βˆΆπœβˆˆπΉβ‹…π‘€β‹…1𝑝𝑠‖‖𝐷,2⟢0,(4.24) and this completes the proof of our claim. Notice also that using the same strategy, we can prove that each π‘‡πœ‘β„“β„“π‘ƒ0 is compact.

Therefore, we have π‘‡πœ‘=𝒦+𝑛ℓ=1π‘‡πœ‘β„“β„“=𝒦+𝐾1+𝑛ℓ=1π‘‡πœ‘β„“β„“π‘ƒβ„“,(4.25) where 𝒦,𝐾1 are compact operators. Since 𝑃2𝑑=𝑃𝑑,𝑃𝑑𝑃𝑠=0 and if 𝑗≠ℓ, then π‘‡πœ‘ is bounded (compact) if and only if the operators π‘‡πœ‘β„“β„“π‘ƒβ„“ are bounded (compact) operators.

Since 𝑃ℓ𝐿2π‘Ž(𝐷)=𝐿2π‘Ž(𝐷ℓ), then it follows that the operator π‘‡πœ‘β„“β„“π‘ƒβ„“ is bounded (compact) if and only if π‘‡πœ‘β„“β„“is bounded (compact).

We are finally, with the help of [1]'s main result, in a position to prove the main result of this paper.

Theorem 4.2. Let πœ‘βˆˆπΏ2(𝐷) be an essentially radial function where 𝐷=βˆ©π‘›π‘—=1𝐷𝑗 with 𝐷1={π‘§βˆˆβ„‚βˆΆ|𝑧|<1} and 𝐷𝑗={π‘§βˆˆβ„‚βˆΆ|π‘§βˆ’a𝑗|>π‘Ÿπ‘—} for 𝑗=2,…,𝑛. If one defines πœ‘π‘—=πœ‘β‹…π‘π‘— where 𝑗=1,…,𝑛 and 𝔓={𝑝0,𝑝1,…,𝑝𝑛} is a πœ•-partition for 𝐷then the following are equivalent: (1) the operator π‘‡πœ‘βˆΆπΏ2π‘Ž(𝐷,π‘‘πœˆ)⟢𝐿2π‘Ž(𝐷,π‘‘πœˆ)(4.26) is bounded (compact).(2) for any 𝑗=1,…,𝑛, the sequences π›Ύπœ‘π‘—={π›Ύπœ‘π‘—(π‘š)}π‘šβˆˆβ„•are in β„“βˆž(β„€+)(𝑐0(β„€+)) where, by definition, if 𝑗=2,…,π‘›π›Ύπœ‘π‘—(π‘š)=π‘Ÿπ‘—ξ€œβˆžπ‘Ÿπ‘—πœ‘π‘—ξ‚€π‘Ÿπ‘—(2π‘š+1)/2(π‘š+1)𝑠1/2(π‘š+1)+π‘Žπ‘—ξ‚1𝑠2π‘‘π‘ βˆ€π‘šβˆˆβ„€+,(4.27) and for 𝑗=1, π›Ύπœ‘1ξ€œ(π‘š)=10πœ‘1𝑠1/2(π‘š+1)𝑑𝑠,βˆ€π‘šβˆˆβ„€+.(4.28)

Proof. In the previous theorem, we proved that the operator under examination is bounded (compact) if and only if for any 𝑗=1,…,𝑛 the operators π‘‡πœ‘π‘—βˆΆπΏ2𝐷𝑗,π‘‘πœˆβŸΆπΏ2π‘Žξ€·π·π‘—ξ€Έ,π‘‘πœˆ(4.29) are bounded (compact). If 𝑗=2,…,𝑛, we observe that if we consider the following sets Ξ”0,1={π‘§βˆˆβ„‚βˆΆ0<|π‘§βˆ’π‘Ž|<1} and Ξ”π‘Žπ‘—,π‘Ÿπ‘—={π‘§βˆˆβ„‚βˆΆ0<|π‘§βˆ’π‘Žπ‘—|<π‘Ÿπ‘—} and the following maps Δ𝛼0,1βˆ’βˆ’βˆ’β†’Ξ”π‘Žπ‘—,π‘Ÿπ‘—π›½βˆ’βˆ’βˆ’β†’π·π‘—,(4.30) where 𝛼(𝑧)=π‘Žπ‘—+π‘Ÿπ‘—π‘§ and 𝛽(𝑀)=(π‘€βˆ’π‘Žπ‘—)βˆ’1π‘Ÿ2𝑗+π‘Žπ‘— and we use Proposition 1.1 in [8], we can claim that π‘‡πœ‘π‘—=π‘‰βˆ’1π›½βˆ˜π›Όπ‘‡πœ‘π‘—βˆ˜π›½βˆ˜π›Όπ‘‰π›½βˆ˜π›Ό,(4.31) where π‘‰π›½βˆ˜π›ΌβˆΆπΏ2(Ξ”0,1)→𝐿2(𝐷𝑗) is an isomorphism of Hilbert spaces. Therefore, π‘‡πœ‘π‘— is bounded (compact) if and only if π‘‡πœ‘π‘—βˆ˜π›½βˆ˜π›Ό is bounded (compact). We also know that this, in turn, is equivalent to the fact that the sequence π›Ύπœ‘π‘—=ξ‚†π›Ύπœ‘π‘—ξ‚‡(π‘š)π‘šβˆˆβ„•(4.32) is in β„“βˆž(β„€+)(𝑐0(β„€+)), where π›Ύπœ‘π‘—ξ€œ(π‘š)=10πœ‘π‘—ξ€·π‘Ÿβˆ˜π›½βˆ˜π›Ό1/2(π‘š+1)ξ€Έπ‘‘π‘Ÿ,βˆ€π‘šβˆˆβ„€+.(4.33) To complete the proof, we observe that since πœ‘π‘— is radial and π›½βˆ˜π›Ό(π‘Ÿ)=π‘Ÿβˆ’1π‘Ÿπ‘—+π‘Žπ‘— then, after a change of variable, we can rewrite the last integral, and therefore the formula π›Ύπœ‘π‘—(π‘š)=π‘Ÿπ‘—ξ€œβˆžπ‘Ÿπ‘—πœ‘π‘—ξ‚€π‘Ÿπ‘—(2π‘š+1)/2(π‘š+1)𝑠1/2(π‘š+1)+π‘Žπ‘—ξ‚1𝑠2𝑑𝑠,βˆ€π‘šβˆˆβ„€+(4.34) must hold for any 𝑗=2,…,𝑛. The case 𝑗=1 is immediate.

Now, we can prove the following.

Theorem 4.3. Let πœ‘βˆˆπΏ2(Ξ©) be an essentially radial function via the conformal equivalence Ξ˜βˆΆΞ©β†’π·, define πœ‘π‘—=πœ‘β‹…π‘π‘— where 𝑗=1,…,𝑛 and 𝔓 is a πœ•-partition for Ξ©, then the following conditions are equivalent: (1)the operator π‘‡πœ‘βˆΆπΏ2π‘Ž(Ξ©,π‘‘πœˆ)⟢𝐿2π‘Ž(Ξ©,π‘‘πœˆ)(4.35) is bounded (compact);(2)for any 𝑗=1,…,𝑛, the sequences π›Ύπœ‘π‘—={π›Ύπœ‘π‘—(π‘š)}π‘šβˆˆβ„•are in β„“βˆž(β„€+)(𝑐0(β„€+)) where, by definition, if 𝑗=2,…,π‘›π›Ύπœ‘π‘—(π‘š)=π‘Ÿπ‘—ξ€œβˆžπ‘Ÿπ‘—πœ‘π‘—βˆ˜Ξ˜βˆ’1ξ‚€π‘Ÿπ‘—(2π‘š+1)/2(π‘š+1)𝑠1/2(π‘š+1)+π‘Žπ‘—ξ‚1𝑠2𝑑𝑠,βˆ€π‘šβˆˆβ„€+,(4.36) and for 𝑗=1π›Ύπœ‘1ξ€œ(π‘š)=10πœ‘1βˆ˜Ξ˜βˆ’1𝑠1/2(π‘š+1)𝑑𝑠,βˆ€π‘šβˆˆβ„€+.(4.37)

Proof. We know that Ξ© is a regular domain, and therefore if Θ is a conformal mapping from Ξ© onto 𝐷 then the Bergman kernels of Ξ© and Θ(Ξ©)=𝐷, are related via 𝐾𝐷(Θ(𝑧),Θ(𝑀))Ξ˜ξ…ž(𝑧)Ξ˜ξ…ž(𝑀)=𝐾Ω(𝑧,𝑀), and the operator π‘‰Ξ˜π‘“=Ξ˜ξ…žβ‹…π‘“βˆ˜Ξ˜ is an isometry from 𝐿2(𝐷) onto𝐿2(Ξ©) (see Proposition 1.1 in [8]). In particular, we have π‘‰Ξ˜π‘ƒπ·=π‘ƒΞ©π‘‰Ξ˜ and this implies that π‘‰Ξ˜π‘‡πœ‘=π‘‡πœ‘βˆ˜Ξ˜βˆ’1π‘‰Ξ˜. Therefore, the operator π‘‡πœ‘ is bounded (compact) if and only if the operator π‘‡πœ‘βˆ˜Ξ˜βˆ’1∢𝐿2(𝐷,π‘‘πœˆ)→𝐿2π‘Ž(𝐷,π‘‘πœˆ) is bounded (compact). In the previous theorem we proved that the operator in exam is bounded (compact) if and only if for any 𝑗=1,…,𝑛 the operators π‘‡πœ‘π‘—βˆ˜Ξ˜βˆ’1∢𝐿2π‘Žξ€·π·π‘—ξ€Έ,π‘‘πœˆβŸΆπΏ2π‘Žξ€·π·π‘—ξ€Έ,π‘‘πœˆ(4.38) are bounded (compact). Hence, we can conclude that the operator is bounded (compact) if and only if for any 𝑗=1,…,𝑛 the sequences π›Ύπœ‘π‘—={π›Ύπœ‘π‘—(π‘š)}π‘šβˆˆβ„• are in β„“βˆž(β„€+)(𝑐0(β„€+)) where, by definition, if 𝑗=2,…,𝑛, we have π›Ύπœ‘π‘—(π‘š)=π‘Ÿπ‘—ξ€œβˆžπ‘Ÿπ‘—πœ‘π‘—βˆ˜Ξ˜βˆ’1ξ‚€π‘Ÿπ‘—(2π‘š+1)/2(π‘š+1)𝑠1/2(π‘š+1)+π‘Žπ‘—ξ‚1𝑠2𝑑𝑠,βˆ€π‘šβˆˆβ„€+,(4.39) and for 𝑗=1, π›Ύπœ‘1ξ€œ(π‘š)=10πœ‘1βˆ˜Ξ˜βˆ’1𝑠1/2(π‘š+1)𝑑𝑠,βˆ€π‘šβˆˆβ„€+,(4.40) and this completes the proof.

We now introduce a set of functions that will allow us to further explore the structure of Toeplitz operators with radial-like symbols. For 𝑗=2,…,𝑛, we define π΅πœ‘π‘—(𝑠)=π‘Ÿπ‘—ξ€œπ‘ π‘Ÿπ‘—πœ‘π‘—βˆ˜Ξ˜βˆ’1ξ‚€π‘Ÿπ‘—1/2π‘₯1/2+π‘Žπ‘—ξ‚1π‘₯2𝑑π‘₯,(4.41) and for 𝑗=1, we set π΅πœ‘1ξ€œ(𝑠)=1π‘ πœ‘1βˆ˜Ξ˜βˆ’1ξ€·π‘₯1/2𝑑π‘₯.(4.42)

We obtain the following useful theorem.

Theorem 4.4. Let πœ‘βˆˆπΏ2(Ξ©) be an essentially radial function via the conformal equivalence Ξ˜βˆΆΞ©β†’π·. If one defines πœ‘π‘—=πœ‘β‹…π‘π‘— where 𝑗=1,…,𝑛 and 𝔓 is a πœ•-partition for Ξ©, then for the operator π‘‡πœ‘βˆΆπΏ2π‘Ž(Ξ©,π‘‘πœˆ)→𝐿2π‘Ž(Ξ©,π‘‘πœˆ) the following hold true: (1)if for any 𝑗=1,…,𝑛|||π΅πœ‘π‘—|||ξ€·π‘Ÿ(s)=π‘‚π‘—ξ€Έβˆ’π‘ asπ‘ βŸΆπ‘Ÿπ‘—,(4.43) then π‘‡πœ‘ is bounded;(2)if for any 𝑗=1,…,𝑛|||π΅πœ‘π‘—|||ξ€·π‘Ÿ(𝑠)=π‘œπ‘—ξ€Έβˆ’π‘ asπ‘ βŸΆπ‘Ÿπ‘—,(4.44) then π‘‡πœ‘ is compact.

Proof. To prove the first, we observe that our main theorem implies that the boundedness (compactness) of the operator is equivalent to the fact that for any 𝑗=1,…,𝑛 the sequences π›Ύπœ‘π‘—={π›Ύπœ‘π‘—(π‘š)}π‘šβˆˆβ„• are in β„“βˆž(β„€+)(𝑐0(β„€+)) where, by definition, if 𝑗=2,…,𝑛, π›Ύπœ‘π‘—(π‘š)=π‘Ÿπ‘—ξ€œβˆžπ‘Ÿπ‘—πœ‘π‘—βˆ˜Ξ˜βˆ’1ξ‚€π‘Ÿπ‘—(2π‘š+1)/2(π‘š+1)𝑠1/2(π‘š+1)+π‘Žπ‘—ξ‚1𝑠2π‘‘π‘ βˆ€π‘šβˆˆβ„€+,(4.45) and for 𝑗=1π›Ύπœ‘π‘—ξ€œ(π‘š)=10πœ‘1βˆ˜Ξ˜βˆ’1𝑠1/2(π‘š+1)ξ€Έπ‘‘π‘ βˆ€π‘šβˆˆβ„€+,(4.46) and, in virtue of [1]'s main result, it is true that π›Ύπœ‘π‘—={π›Ύπœ‘π‘—(π‘š)}π‘šβˆˆβ„• are in β„“βˆž(β„€+) if for any 𝑗=1,…,𝑛, |||π΅πœ‘π‘—|||ξ€·π‘Ÿ(𝑠)=π‘‚π‘—ξ€Έβˆ’π‘ asπ‘ βŸΆπ‘Ÿπ‘—,(4.47) and π›Ύπœ‘π‘—={π›Ύπœ‘π‘—(π‘š)}π‘šβˆˆβ„• are in 𝑐0(β„€+)) if for any 𝑗=1,…,𝑛|||π΅πœ‘π‘—|||ξ€·π‘Ÿ(𝑠)=π‘œπ‘—ξ€Έβˆ’π‘ asπ‘ βŸΆπ‘Ÿπ‘—.(4.48)

It is also useful to observe that in the case of a positive symbol, we can prove that the condition above is necessary and sufficient. In fact (see [1]), we have the following.

Theorem 4.5. Let πœ‘βˆˆπΏ2(Ξ©) be an essentially radial function via the conformal equivalence Ξ˜βˆΆΞ©β†’π·. If we define πœ‘π‘—=πœ‘β‹…π‘π‘— where 𝑗=1,…,𝑛 and 𝔓 is a πœ•-partition for Ξ© and if πœ‘β‰₯0 a.e. in Ξ©, then for the operator π‘‡πœ‘βˆΆπΏ2π‘Ž(Ξ©,π‘‘πœˆ)→𝐿2π‘Ž(Ξ©,π‘‘πœˆ), the following hold true: (1)π‘‡πœ‘ is bounded if and only if |||π΅πœ‘π‘—|||ξ€·π‘Ÿ(𝑠)=π‘‚π‘—ξ€Έβˆ’π‘ asπ‘ βŸΆπ‘Ÿπ‘—,(4.49) for any 𝑗=1,…,𝑛,(2)π‘‡πœ‘ is compact if and only if |||π΅πœ‘π‘—|||ξ€·π‘Ÿ(𝑠)=π‘œπ‘—ξ€Έβˆ’π‘ asπ‘ βŸΆπ‘Ÿπ‘—,(4.50) for any 𝑗=1,…,𝑛.

Proof. The proof is an immediate consequence of Theorem 3.5 in [1] and the theorem above.

There are a few useful observations that we can make at this point. If the Toeplitz operator π‘‡πœ‘βˆΆπΏ2π‘Ž(Ξ©,π‘‘πœˆ)→𝐿2π‘Ž(Ξ©,π‘‘πœˆ) has an essentially radial positive symbol πœ‘β‰₯0 such that for some β„“