We study the problem of the boundedness and compactness of ๐‘‡๐œ™ when ๐œ™โˆˆ๐ฟ2(ฮฉ) and ฮฉ 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 ๐‘‡๐œ™.

1. Introduction

Let ฮฉ be a bounded multiply-connected domain in the complex plane โ„‚, whose boundary ๐œ•ฮฉ consists of finitely many simple closed smooth analytic curves ๐›พ๐‘—(๐‘—=1,2,โ€ฆ,๐‘›) where ๐›พ๐‘— are positively oriented with respect to ฮฉ and ๐›พ๐‘—โˆฉ๐›พ๐‘–=โˆ… if ๐‘–โ‰ ๐‘—. We also assume that ๐›พ1 is the boundary of the unbounded component of โ„‚โงตฮฉ. Let ฮฉ1 be the bounded component of โ„‚โงต๐›พ1, and ฮฉ๐‘—(๐‘—=2,โ€ฆ,๐‘›) the unbounded component of โ„‚โงต๐›พ๐‘—, respectively, so that ฮฉ=โˆฉ๐‘›๐‘—=1ฮฉ๐‘—.

For ๐‘‘๐œˆ=(1/๐œ‹)๐‘‘๐‘ฅ๐‘‘๐‘ฆ, we consider the usual ๐ฟ2-space ๐ฟ2(ฮฉ)=๐ฟ2(ฮฉ,๐‘‘๐œˆ). The Bergman space ๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ), consisting of all holomorphic functions which are ๐ฟ2-integrable, is a closed subspace of ๐ฟ2(ฮฉ,๐‘‘๐œˆ) with the inner product given by ๎€œโŸจ๐‘“,๐‘”โŸฉ=ฮฉ๐‘“(๐‘ง)๐‘”(๐‘ง)๐‘‘๐œˆ(๐‘ง)(1.1) for ๐‘“,๐‘”โˆˆ๐ฟ2(ฮฉ,๐‘‘๐œˆ). The Bergman projection is the orthogonal projection ๐‘ƒโˆถ๐ฟ2(ฮฉ,๐‘‘๐œˆ)โŸถ๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ).(1.2) It is well-known that for any ๐‘“โˆˆ๐ฟ2(ฮฉ,๐‘‘๐œˆ), we have ๎€œ๐‘ƒ๐‘“(๐‘ค)=ฮฉ๐‘“(๐‘ง)๐พฮฉ(๐‘ง,๐‘ค)๐‘‘๐œˆ(๐‘ง),(1.3) where ๐พฮฉ is the Bergman reproducing kernel of ฮฉ. For ๐œ‘โˆˆ๐ฟโˆž(ฮฉ,๐‘‘๐œˆ), the Toeplitz operator ๐‘‡๐œ‘โˆถ๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ)โ†’๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ) is defined by ๐‘‡๐œ‘=๐‘ƒ๐‘€๐œ‘, where ๐‘€๐œ‘ is the standard multiplication operator. A simple calculation shows that ๐‘‡๐œ‘๎€œ๐‘“(๐‘ง)=ฮฉ๐œ‘(๐‘ค)๐‘“(๐‘ค)๐พฮฉ(๐‘ค,๐‘ง)๐‘‘๐œˆ(๐‘ค).(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 ๐œ‘โˆˆ๐ฟ2(ฮฉ,๐‘‘๐œˆ) for the Toeplitz operators ๐‘‡๐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 ๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ). In Section 4, we prove the main result and study several important consequences.

2. Preliminaries

Let ฮฉ be the bounded multiply-connected domain given at the beginning of Section 1, that is, ฮฉ=โˆฉ๐‘›๐‘—=1ฮฉ๐‘—, where ฮฉ1 is the bounded component of โ„‚โงต๐›พ1, and ฮฉ๐‘—(๐‘—=2,โ€ฆ,๐‘›) is the unbounded component of โ„‚โงต๐›พ๐‘—. We use the symbol ฮ” to indicate the punctured disk {๐‘งโˆˆโ„‚โˆฃ0<|๐‘ง|<1}. Let ฮ“ be any one of the domains ฮฉ,ฮ”,ฮฉ๐‘—(๐‘—=2,โ€ฆ,๐‘›).

We call ๐พฮ“(๐‘ง,๐‘ค) the reproducing kernel of ฮ“ and we use the symbol ๐‘˜ฮ“(๐‘ง,๐‘ค) to indicate the normalized reproducing kernel, that is, ๐‘˜ฮ“(๐‘ง,๐‘ค)=๐พฮ“(๐‘ง,๐‘ค)/๐พฮ“(๐‘ค,๐‘ค)1/2.

For any ๐ดโˆˆโ„ฌ(๐ฟ2๐‘Ž(ฮ“,๐‘‘๐œˆ)), we define ๎‚๐ด, the Berezin transform of ๐ด, by ๎‚๐ด(๐‘ค)=โŸจ๐ด๐‘˜ฮ“๐‘ค,๐‘˜ฮ“๐‘ค๎€œโŸฉ=ฮ“๐ด๐‘˜ฮ“๐‘ค(๐‘ง)๐‘˜ฮ“๐‘ค(๐‘ง)๐‘‘๐œˆ(๐‘ง),(2.1) where ๐‘˜ฮ“๐‘ค(โ‹…)=๐พฮ“(โ‹…,๐‘ค)๐พฮ“(๐‘ค,๐‘ค)โˆ’1/2.

If ๐œ‘โˆˆ๐ฟโˆž(ฮ“), then we indicate with the symbol ๎‚๐œ‘ the Berezin transform of the associated Toeplitz operator ๐‘‡๐œ‘, and we have ๎€œ๎‚๐œ‘(๐‘ค)=ฮ“||๐‘˜๐œ‘(๐‘ง)ฮ“๐‘ค(||๐‘ง)2๐‘‘๐œˆ(๐‘ง).(2.2) We remind the reader that it is well known that ๎‚๐ดโˆˆ๐’žโˆž๐‘(ฮ“), and we have โ€–๎‚๐ดโ€–โˆžโ‰คโ€–๐ดโ€–โ„ฌ(๐ฟ2(ฮฉ)). 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 ฮฉ bounded multiply-connected domain in the complex plane โ„‚, whose boundary ๐œ•ฮฉ consists of finitely many simple closed smooth analytic curves ๐›พ๐‘—(๐‘—=1,2,โ€ฆ,๐‘›), 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 ๐ท=โˆฉ๐‘›๐‘–=1๐ท๐‘– where ๐ท1={๐‘งโˆˆโ„‚โˆถ|๐‘ง|<1} and ๐ท๐‘—={๐‘งโˆˆโ„‚โˆถ|๐‘งโˆ’๐‘Ž๐‘—|>๐‘Ÿ๐‘—} for ๐‘—=2,โ€ฆ,๐‘›. Here ๐‘Ž๐‘—โˆˆ๐ท1 and 0<๐‘Ÿ๐‘—<1 with |๐‘Ž๐‘—โˆ’๐‘Ž๐‘˜|>๐‘Ÿ๐‘—+๐‘Ÿ๐‘˜ if ๐‘—โ‰ ๐‘˜ and 1โˆ’|๐‘Ž๐‘—|>๐‘Ÿ๐‘—. Before we state the main results of this paper we need to give a few definitions.

Definition 2.1. Let ฮฉ=โˆฉ๐‘›๐‘–=1ฮฉ๐‘– be a canonical bounded multiply-connected domain. We say that the set of ๐‘›+1 functions ๐”“={๐‘0,๐‘1,โ€ฆ,๐‘๐‘›} is a ๐œ•-partition for ฮฉ if(1)for every ๐‘—=0,1,โ€ฆ,๐‘›,๐‘๐‘—โˆถฮฉโ†’[0,1] is a Lipschitz, ๐ถโˆž-function,(2)for every ๐‘—=2,โ€ฆ,๐‘›, there exists an open set ๐‘Š๐‘—โŠ‚ฮฉ and an ๐œ–๐‘—>0 such that ๐‘ˆ๐œ–๐‘—={๐œโˆˆฮฉโˆถ๐‘Ÿ๐‘—<|๐œโˆ’๐‘Ž๐‘—|<๐‘Ÿ๐‘—+๐œ–๐‘—}, and the support of ๐‘๐‘— is contained in ๐‘Š๐‘— and ๐‘๐‘—(๐œ)=1,โˆ€๐œโˆˆ๐‘ˆ๐œ–๐‘—,(2.3)(3)for ๐‘—=1, there exists an open set ๐‘Š1โŠ‚ฮฉ and an ๐œ–1>0 such that ๐‘ˆ๐œ–1={๐œโˆˆฮฉโˆถ1โˆ’๐œ–1<|๐œ|<1} and the support of ๐‘1 is contained in ๐‘Š1 and ๐‘1(๐œ)=1,โˆ€๐œโˆˆ๐‘ˆ๐œ–1,(2.4)(4)for every ๐‘—,๐‘˜=1,โ€ฆ,๐‘›,๐‘Š๐‘—โˆฉ๐‘Š๐‘˜=โˆ…, the set โ‹ƒฮฉโงต(๐‘›๐‘—=1๐‘Š๐‘—) is not empty and the function ๐‘0๎ƒฉ(๐œ)=1,โˆ€๐œโˆˆ๐‘›๎š๐‘—=1๐‘Š๐‘—๎ƒช๐‘๐‘โˆฉฮฉ,0(๐œ)=0,โˆ€๐œโˆˆ๐‘ˆ๐œ–๐‘˜,๐‘˜=1,โ€ฆ,๐‘›,(2.5)(5) for any ๐œโˆˆฮฉ, the following equation: ๐‘›๎“๐‘˜=0๐‘๐‘˜(๐œ)=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 ๐‘0 is different from zero.

Definition 2.2. A function ๐œ‘โˆถฮฉ=โˆฉ๐‘›๐‘–=1ฮฉ๐‘–โ†’โ„‚ is said to be essentially radial if there exists a conformally equivalent canonical bounded domain ๐ท=โˆฉ๐‘›๐‘–=1๐ท๐‘–, such that if the map ฮ˜โˆถฮฉโ†’๐ท is the conformal mapping from ฮฉ onto ๐ท, then(1)for every ๐‘˜=2,โ€ฆ,๐‘› and for some ๐œ–๐‘˜>0, we have ๐œ‘โˆ˜ฮ˜โˆ’1(๐‘ง)=๐œ‘โˆ˜ฮ˜โˆ’1๎€ท||๐‘งโˆ’๐‘Ž๐‘˜||๎€ธ,(2.7) when ๐‘งโˆˆ๐‘ˆ๐œ–๐‘˜={๐œโˆˆฮฉโˆถ๐‘Ÿ๐‘˜<|๐œโˆ’๐‘Ž๐‘˜|<๐‘Ÿ๐‘˜+๐œ–๐‘˜},(2)for ๐‘˜=1 and for some ๐œ–1>0, we have ๐œ‘โˆ˜ฮ˜โˆ’1(๐‘ง)=๐œ‘โˆ˜ฮ˜โˆ’1(|๐‘ง|),(2.8) when ๐‘งโˆˆ๐‘ˆ๐œ–1={๐œโˆˆฮฉโˆถ1โˆ’๐œ–1<|๐œ|<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 ๐œ‘ is essentially radial via ฮ˜โˆถโˆฉ๐‘›โ„“=1ฮฉโ„“โ†’โˆฉ๐‘›โ„“=1๐ทโ„“.

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 ฮ˜โˆถโˆฉ๐‘›โ„“=1ฮฉโ„“โ†’โˆฉ๐‘›โ„“=1๐ทโ„“, we always assume that the ๐œ•-partition is given on โˆฉ๐‘›โ„“=1๐ทโ„“ and transferred to โˆฉ๐‘›โ„“=1ฮฉโ„“ through ฮ˜ in the natural way.

At this point, we can state the main result.

Theorem 2.3. Let ๐œ‘โˆˆ๐ฟ2(ฮฉ) be an essentially radial function via ฮ˜โˆถโˆฉ๐‘›โ„“=1ฮฉโ„“โ†’โˆฉ๐‘›โ„“=1๐ทโ„“, if one defines ๐œ‘๐‘—=๐œ‘โ‹…๐‘๐‘—, where ๐‘—=1,โ€ฆ,๐‘› and ๐”“ is a ๐œ•-partition for ฮฉ, then the following are equivalent: (1)the operator ๐‘‡๐œ‘โˆถ๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ)โŸถ๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ)(2.9) is bounded (compact).(2)for any ๐‘—=1,โ€ฆ,๐‘› the sequences ๐›พ๐œ‘๐‘—={๐›พ๐œ‘๐‘—(๐‘š)}๐‘šโˆˆโ„•are in โ„“โˆž(โ„ค+)(๐‘0(โ„ค+)) where, by definition, if ๐‘—=2,โ€ฆ,๐‘›, ๐›พ๐œ‘๐‘—(๐‘š)=๐‘Ÿ๐‘—๎€œโˆž๐‘Ÿ๐‘—๐œ‘๐‘—โˆ˜ฮ˜โˆ’1๎‚€๐‘Ÿ๐‘—(2๐‘š+1)/2(๐‘š+1)๐‘ 1/2(๐‘š+1)+๐‘Ž๐‘—๎‚1๐‘ 2๐‘‘๐‘ ,โˆ€๐‘šโˆˆโ„ค+,(2.10) and if ๐‘—=1๐›พ๐œ‘1๎€œ(๐‘š)=10๐œ‘1โˆ˜ฮ˜โˆ’1๎€ท๐‘ 1/2(๐‘š+1)๎€ธ๐‘‘๐‘ ,โˆ€๐‘šโˆˆโ„ค+.(2.11)

3. The Structure of ๐ฟ2๐‘Ž(ฮฉ) and Some Estimates about the Bergman Kernel

From now on, we will assume that ฮฉ=โˆฉ๐‘›๐‘—=1ฮฉ๐‘— where ฮฉ1={๐‘งโˆˆโ„‚โˆถ|๐‘ง|<1} and ฮฉ๐‘—={๐‘งโˆˆโ„‚โˆถ|๐‘งโˆ’๐‘Ž๐‘—|>๐‘Ÿ๐‘—} for ๐‘—=2,โ€ฆ,๐‘›. Here, ๐‘Ž๐‘—โˆˆฮฉ1 and 0<๐‘Ÿ๐‘—<1 with |๐‘Ž๐‘—โˆ’๐‘Ž๐‘˜|>๐‘Ÿ๐‘—+๐‘Ÿ๐‘˜ if ๐‘—โ‰ ๐‘˜ and 1โˆ’|๐‘Ž๐‘—|>๐‘Ÿ๐‘—. We will indicate with the symbol ฮ”0,1 the punctured disk ฮฉ1โงต{0}.

With the symbols ๐พฮฉ๐‘—(๐‘ง,๐‘ค),๐พฮฉ(๐‘ง,๐‘ค),๐พฮ”(๐‘ง,๐‘ค), we denote the Bergman kernel on ฮฉ๐‘—,ฮฉ, and ฮ”, 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 ฮ”0,1 and the disk ฮฉ1, we have ๐ฟ๐‘๐‘Ž(ฮ”0,1)=๐ฟ๐‘๐‘Ž(ฮฉ1), if ๐‘โ‰ฅ2, and, for any (๐‘ง,๐‘ค)โˆˆฮ”2,๐พฮ”(๐‘ง,๐‘ค)=๐พฮฉ1(๐‘ง,๐‘ค) (see [7, 8]). This fact has an important and simple consequence. In fact, if we consider ฮ”๐‘Ž,๐‘Ÿ={๐‘งโˆˆโ„‚โˆถ0<|๐‘งโˆ’๐‘Ž|<๐‘Ÿ} and ๐‘‚๐‘Ž,๐‘Ÿ={๐‘งโˆˆโ„‚โˆถ|๐‘งโˆ’๐‘Ž|>๐‘Ÿ}, we can conclude that ๐พ๐‘‚๐‘Ž,๐‘Ÿ๐‘Ÿ(๐‘ง,๐‘ค)=2๎‚€๐‘Ÿ2โˆ’(๐‘งโˆ’๐‘Ž)โ‹…๎‚(๐‘คโˆ’๐‘Ž)2,โˆ€(๐‘ง,๐‘ค)โˆˆ๐‘‚๐‘Ž,๐‘Ÿร—๐‘‚๐‘Ž,๐‘Ÿ.(3.1)

To see this, we use the well-known fact that the reproducing kernel of the unit disk is given by (1โˆ’๐‘ง๐‘ค)โˆ’2, therefore we have ๐พฮ”0,11(๐‘ง,๐‘ค)=๎€ท1โˆ’๐‘งโ‹…๐‘ค๎€ธ2,โˆ€(๐‘ง,๐‘ค)โˆˆฮ”0,1ร—ฮ”0,1.(3.2) This implies, by conformal mapping, that the reproducing kernel of ฮ”๐‘Ž,๐‘Ÿ is ๐พฮ”๐‘Ž,๐‘Ÿ๐‘Ÿ(๐‘ง,๐‘ค)=2๎‚€๐‘Ÿ2โˆ’(๐‘งโˆ’๐‘Ž)โ‹…๎‚(๐‘คโˆ’๐‘Ž)2,โˆ€(๐‘ง,๐‘ค)โˆˆฮ”๐‘Ž,๐‘Ÿร—ฮ”๐‘Ž,๐‘Ÿ.(3.3) Now, we define ๐œ‘โˆถฮ”๐‘Ž,๐‘Ÿโ†’๐‘‚๐‘Ž,๐‘Ÿ by ๐œ‘(๐‘ง)=(๐‘งโˆ’๐‘Ž)โˆ’1๐‘Ÿ2+๐‘Ž,(3.4) and we use the well-known fact that the Bergman kernels of ฮ”๐‘Ž,๐‘Ÿ and ๐œ“(ฮ”๐‘Ž,๐‘Ÿ)=๐‘‚๐‘Ž,๐‘Ÿ are related via ๐พ๐‘‚๐‘Ž,๐‘Ÿ(๐œ‘(๐‘ง),๐œ‘(๐‘ค))๐œ‘๎…ž(๐‘ง)๐œ‘๎…ž(๐‘ค)=๐พฮ”๐‘Ž,๐‘Ÿ(๐‘ง,๐‘ค)(3.5) to obtain that ๐พ๐‘‚๐‘Ž,๐‘Ÿ๐‘Ÿ(๐‘ง,๐‘ค)=2๎‚€๐‘Ÿ2โˆ’(๐‘งโˆ’๐‘Ž)โ‹…๎‚(๐‘คโˆ’๐‘Ž)2,โˆ€(๐‘ง,๐‘ค)โˆˆ๐‘‚๐‘Ž,๐‘Ÿร—๐‘‚๐‘Ž,๐‘Ÿ.(3.6) Since ฮฉ1=๐‘‚0,1 and, for ๐‘—=2,โ€ฆ,๐‘›,๐‘‚๐‘Ž๐‘—,๐‘Ÿ๐‘—=ฮฉ๐‘—, then the last equations implies that ๐พฮฉ11(๐‘ง,๐‘ค)=๎€ท1โˆ’๐‘งโ‹…๐‘ค๎€ธ2,๐พฮฉ๐‘—๐‘Ÿ(๐‘ง,๐‘ค)=2๐‘—๎‚€๐‘Ÿ2๐‘—โˆ’๎€ท๐‘งโˆ’๐‘Ž๐‘—๎€ธโ‹…๎€ท๐‘คโˆ’๐‘Ž๐‘—๎€ธ๎‚2(3.7) if ๐‘—=2,โ€ฆ,๐‘›.

We also note that if we define ๐ธฮฉ(๐‘ง,๐‘ค)=๐พฮฉ(๐‘ง,๐‘ค)โˆ’๐‘›๎“๐‘—=1๐พฮฉ๐‘—(๐‘ง,๐‘ค),(3.8) we can prove the following.

Lemma 3.1. (1) ๐ธฮฉ is conjugate symmetric about z and w. For each ๐‘คโˆˆฮฉ,๐ธฮฉ(โ‹…,๐‘ค) is conjugate analytic on ฮฉ and ๐ธฮฉโˆˆ๐ถโˆž(ฮฉร—ฮฉ).
(2) There are neighborhoods ๐‘ˆ๐‘— of ๐œ•ฮฉ๐‘—(๐‘—=1,โ€ฆ,๐‘›) and a constant ๐ถ>0 such that ๐‘ˆ๐‘—โˆฉ๐‘ˆ๐‘˜ is empty if ๐‘—โ‰ ๐‘˜ and ||๐พฮฉ(๐‘ง,๐‘ค)โˆ’๐พฮฉ๐‘—||(๐‘ง,๐‘ค)<๐ถ,(3.9) for ๐‘งโˆˆฮฉ and ๐‘คโˆˆ๐‘ˆ๐‘—.
(3) ๐ธฮฉโˆˆ๐ฟโˆž(ฮฉร—ฮฉ).

Proof. (1) Since the Bergman kernels ๐พฮฉ and ๐พฮฉ๐‘— have these properties (see [9]), by the definition of ๐ธฮฉ, we get (1).
(2) The proof is given in [7, 8].
(3) Using the fact that ๐พฮฉ11(๐‘ง,๐‘ค)=๎€ท1โˆ’๐‘งโ‹…๐‘ค๎€ธ2,๐พฮฉ๐‘—๐‘Ÿ(๐‘ง,๐‘ค)=2๐‘—๎‚€๐‘Ÿ2๐‘—โˆ’๎€ท๐‘งโˆ’๐‘Ž๐‘—๎€ธโ‹…๎€ท๐‘คโˆ’๐‘Ž๐‘—๎€ธ๎‚2,(3.10) for ๐‘—=2,...,๐‘› and (1) and (2), we get (3).

We observe that we can choose ๐‘…๐‘—>๐‘Ÿ๐‘— for ๐‘—=2,โ€ฆ,๐‘› and ๐‘…1<1 such that ๐บ๐‘—={๐‘งโˆถ๐‘Ÿ๐‘—<|๐‘งโˆ’๐‘Ž๐‘—|<๐‘…๐‘—}(๐‘—=2,โ€ฆ,๐‘›) and ๐บ1={๐‘งโˆถ๐‘…1<|๐‘ง|<1}, then we have ๐บ๐‘—โŠ‚๐‘ˆ๐‘—, where ๐‘ˆ๐‘— is the same as in Lemma 3.1. We also have the following.

Lemma 3.2. There are constants ๐’Ÿ>0 and โ„ณ>0 such that (1)for any (๐‘ง,๐‘ค)โˆˆ๐บ๐‘–ร—ฮฉโˆชฮฉร—๐บ๐‘–, one has ||๐พฮฉ||||๐พ(๐‘ง,๐‘ค)<๐ทฮฉ๐‘—||,||๐พ(๐‘ง,๐‘ค)ฮฉ๐‘—||<||๐พ(๐‘ง,๐‘ค)ฮฉ||(๐‘ง,๐‘ค)+โ„ณ,(3.11)(2)for any ๐‘งโˆˆฮฉ, one has ๐พฮฉ๐‘—(๐‘ง,๐‘ง)<๐พฮฉ(๐‘ง,๐‘ง).

Proof. By the explicit formula of the Bergman kernels ๐พฮฉ๐‘–, there are constants ๐ถ๐‘– and ๐‘€๐‘– such that ||๐พฮฉ๐‘–||(๐‘ง,๐‘ค)โ‰ฅ๐ถ๐‘–,(3.12) for (๐‘ง,๐‘ค)โˆˆ(๐บ๐‘–ร—ฮฉ)โˆช(ฮฉร—๐บ๐‘–) and ||๐พฮฉ๐‘–||(๐‘ง,๐‘ค)โ‰ค๐‘€๐‘–(3.13) if (๐‘ง,๐‘ค)โˆ‰๐บ๐‘–ร—๐บ๐‘– for ๐‘–=1,2,โ€ฆ,๐‘›. From the last Lemma, it follows that ||๐พฮฉ||โ‰ค||๐พ(๐‘ง,๐‘ค)ฮฉ๐‘–||๎‚ต๐ถ(๐‘ง,w)+๐ถโ‰ค1+๐ถ๐‘–๎‚ถ||๐พฮฉ๐‘–||,||๐พ(๐‘ง,๐‘ค)ฮฉ๐‘–||โ‰ค||๐พ(๐‘ง,๐‘ค)ฮฉ||+||๐ธ(๐‘ง,๐‘ค)ฮฉ||+๎“(๐‘ง,๐‘ค)๐‘—โ‰ ๐‘–||๐พฮฉ๐‘—||<||๐พ(๐‘ง,๐‘ค)ฮฉ||(๐‘ง,๐‘ค)+โ€–๐ธฮฉโ€–โˆž+๎“๐‘–โ‰ ๐‘—๐‘€๐‘—,(3.14) whenever (๐‘ง,๐‘ค)โˆˆ(๐บ๐‘–ร—ฮฉ)โˆช(ฮฉร—๐บ๐‘–). If we call ๐’Ÿ the biggest number among {1+๐ถ/๐ถ๐‘—} and we let โ„ณ=โ€–๐ธฮฉโ€–โˆž+โˆ‘๐‘›๐‘—=1๐‘€๐‘—, 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 ฮฉ is actually the intersection of other domains, that is, ฮฉ=โˆฉ๐‘›๐‘—=1ฮฉ๐‘—. This also suggests that we should look for a representation of the elements of ๐ฟ2๐‘Ž(ฮฉ) that reflects this fact. For this reason, we give the following.

Definition 3.3. Given ฮฉ=โˆฉ๐‘›๐‘—=1ฮฉ๐‘— with ฮฉ1={๐‘งโˆˆโ„‚โˆถ|๐‘ง|<1} and ฮฉ๐‘—={๐‘งโˆˆโ„‚โˆถ|๐‘งโˆ’๐‘Ž๐‘—|>๐‘Ÿ๐‘—}, for any ๐‘“โˆˆ๐ฟ2๐‘Ž(ฮฉ), we define ๐‘›+1 functions ๐‘ƒ0๐‘“,๐‘ƒ1๐‘“,๐‘ƒ2๐‘“,โ€ฆ,๐‘ƒ๐‘›๐‘“ as follows: if ๐‘งโˆˆฮฉ, then we set, for ๐‘—=1, ๐‘ƒ11๐‘“(๐‘ง)=โ‹…๎€œ2๐œ‹๐‘–ฬ‚๐›พ1๐‘“(๐œ)๐œโˆ’๐‘ง๐‘‘๐œ,(3.15) for ๐‘—=2,3,โ€ฆ,๐‘›, ๐‘ƒ๐‘—1๐‘“=โ‹…๎€œ2๐œ‹๐‘–ฬ‚๐›พ๐‘—๐‘“(๐œ)1๐œโˆ’๐‘ง๐‘‘๐œโˆ’โ‹…๎€œ2๐œ‹๐‘–ฬ‚๐›พ๐‘—๐‘“(๐œ)๐‘‘๐œ,(3.16) and for ๐‘—=0, ๐‘ƒ0๐‘“=๐‘›๎“๐‘—=2๎ƒฉ1โ‹…๎€œ2๐œ‹๐‘–ฬ‚๐›พ๐‘—๎ƒช1๐‘“(๐œ)๐‘‘๐œ๐‘งโˆ’๐‘Ž๐‘—,(3.17) where ฬ‚๐›พ๐‘—(๐‘—=1,โ€ฆ,๐‘›) are the circles which center at ๐‘Ž๐‘—(๐‘Ž1=0) and lie in ๐บ๐‘— (see Lemma 3.2), respectively, so that ๐‘ง is exterior to ฬ‚๐›พ๐‘—(๐‘—=2,โ€ฆ,๐‘›) and interior to ฬ‚๐›พ1.

It is important that the reader notices that the Cauchy theorem implies that our definition is independent from how we choose ฬ‚๐›พ1,โ€ฆ,ฬ‚๐›พ๐‘›. Moreover, it is important to notice that the domains of the functions ๐‘ƒ2๐‘“,โ€ฆ,๐‘ƒ๐‘›๐‘“ are actually the sets ฮฉ2,โ€ฆ,ฮฉ๐‘›. In the next Lemma, we give more information about this representation.

Lemma 3.4. For ๐‘“โˆˆ๐ฟ2๐‘Ž(ฮฉ), one can write it uniquely as ๐‘“(๐‘ง)=๐‘›๎“๐‘—=1๎€ท๐‘ƒ๐‘—๐‘“๎€ธ๎€ท๐‘ƒ(๐‘ง)+0๐‘“๎€ธ(๐‘ง),(3.18) with ๐‘ƒ๐‘—๐‘“โˆˆ๐ฟ2๐‘Ž(ฮฉ๐‘—),๐‘ƒ0๐‘“โˆˆ๐ฟ2๐‘Ž(ฮฉ)โˆฉ๐ถโˆž(ฮฉ),๐‘ƒ๐‘˜(๐‘ƒ๐‘—๐‘“)=0 if ๐‘—โ‰ ๐‘˜, and moreover, there exists a constant ๐‘€1 such that, for ๐‘—=0,1,โ€ฆ,๐‘›, one has โ€–โ€–๐‘ƒ๐‘—๐‘“โ€–โ€–ฮฉโ‰คโ€–โ€–๐‘ƒ๐‘—๐‘“โ€–โ€–ฮฉ๐‘—โ‰ค๐‘€1โ€–๐‘“โ€–ฮฉ.(3.19) In particular, if ๐‘“โˆˆ๐ฟ2๐‘Ž(ฮฉ๐‘–), then ๐‘ƒ๐‘–๐‘“=๐‘“ and โ€–๐‘“โ€–ฮฉ๐‘–โ‰ค๐‘€1โ€–๐‘“โ€–ฮฉ,(3.20) for ๐‘–=1,โ€ฆ,๐‘›.

Proof. Let ๐‘“ be any function analytic on ฮฉ. For any ๐‘งโˆˆฮฉ, let ๐›พ๐‘–(๐‘–=1,โ€ฆ,๐‘›) be the circles which center at ๐‘Ž๐‘–(๐‘Ž1=0) and lie in ๐บ๐‘–, respectively, so that ๐‘ง is exterior to ๐›พ๐‘–(๐‘–=2,โ€ฆ,๐‘›) and interior to ๐›พ1. Using Cauchy's Formula, we can write ๐‘“(๐‘ง)=๐‘›๎“๐‘—=11โ‹…๎€œ2๐œ‹๐‘–๐›พ๐‘—๐‘“(๐œ)๐œโˆ’๐‘ง๐‘‘๐œ.(3.21) Let ๐‘“๐‘—(1๐‘ง)=โ‹…๎€œ2๐œ‹๐‘–๐›พ๐‘—๐‘“(๐œ)๐œโˆ’๐‘ง๐‘‘๐œ.(3.22) By Cauchy's Formula, the value ๐‘“๐‘—(๐‘ง) does not depend on the choice of ๐›พ๐‘— if 1โ‰ค๐‘—โ‰ค๐‘› and โˆ‘๐‘“(๐‘ง)=๐‘›๐‘—๐‘“๐‘—(๐‘ง). Of course, each ๐‘“๐‘— is well defined for all ๐‘งโˆˆฮฉ๐‘— and analytic in ฮฉ๐‘—. In addition, if ๐‘—โ‰ 1, we have that ๐‘“๐‘—(๐‘ง)โ†’0 as |๐‘ง|โ†’โˆž. Writing the Laurent expansion at ๐‘Ž๐‘— of ๐‘“๐‘—, we have ๐‘“1(๐‘ง)=โˆž๎“๐‘˜=0๐›ผ1,๐‘˜๐‘ง๐‘˜,(3.23) and, for ๐‘—โ‰ 1, ๐‘“๐‘—(๐‘ง)=โˆ’โˆž๎“๐‘˜=โˆ’1๐›ผ๐‘—,๐‘˜๎€ท๐‘งโˆ’๐‘Ž๐‘—๎€ธ๐‘˜,(3.24) and these series converge to ๐‘“๐‘— uniformly and absolutely on any compact subset of ฮฉ๐‘—, respectively. We remark that the coefficients are given by the following formula: ๐›ผ๐‘—,๐‘˜=1๎€œ2๐œ‹๐‘–๐›พ๐‘—๐‘“(๐œ)๎€ท๐œโˆ’๐‘Ž๐‘—๎€ธ๐‘˜+1๐‘‘๐œ,(3.25) where ๐‘˜โ‰ฅ0 if ๐‘—=1 and ๐‘˜โ‰คโˆ’1 if ๐‘—โ‰ 1 and ๐›พ๐‘—โŠ‚๐บ๐‘—,1โ‰ค๐‘—โ‰ค๐‘›. Moreover, if ๐‘“ is holomorphic in some ฮฉ๐‘— and ๐‘“(๐‘ง)โ†’0 as |๐‘ง|โ†’โˆž when ๐‘–โ‰ 1, then ๐›ผ๐‘—๐‘˜=0 for all ๐‘—โ‰ ๐‘– by Cauchy's theorem and, therefore, ๐‘“๐‘—=0.
Now, we define ๐‘ƒ1๐‘“=๐‘“1 and ๐‘ƒ๐‘—๐‘“(๐‘ง)=โˆ’โˆž๎“๐‘˜=โˆ’2๐›ผ๐‘—๐‘˜๎€ท๐‘งโˆ’๐‘Ž๐‘—๎€ธ๐‘˜,(3.26) for ๐‘—=2,3,โ€ฆ,๐‘› and ๐‘ƒ0๐‘“(๐‘ง)=๐‘›๎“๐‘—=2๐›ผ๐‘—,โˆ’1๎€ท๐‘งโˆ’๐‘Ž๐‘—๎€ธโˆ’1,(3.27) then โˆ‘๐‘“(๐‘ง)=๐‘›๐‘–=0๐‘ƒ๐‘–๐‘“(๐‘ง) for all ๐‘งโˆˆฮฉ and ๐‘ƒ๐‘˜(๐‘ƒ๐‘—๐‘“)=0 if 0โ‰ ๐‘˜โ‰ ๐‘—โ‰ 0 as we have proved above.
We claim that ๐‘“โˆˆ๐ฟ2๐‘Ž(ฮฉ) implies that ๐‘ƒ๐‘–๐‘“โˆˆ๐ฟ2๐‘Ž(ฮฉ๐‘—) for ๐‘—=1,2,โ€ฆ,๐‘›, respectively. Indeed, since each annulus ๐บ๐‘— is contained in ฮฉ,๐‘“โˆˆ๐ฟ2๐‘Ž(ฮฉ) implies that ๐‘“ is an element of ๐ฟ2๐‘Ž(๐บ๐‘–) for all ๐‘–=1,2,โ€ฆ,๐‘›.
For any fixed ๐‘–, note that ๐‘ƒ๐‘—๐‘“(0โ‰ ๐‘—โ‰ ๐‘–) and ๐‘ƒ0๐‘“โˆ’๐›ผ๐‘—,โˆ’1โ‹…(๐‘งโˆ’๐‘Ž๐‘—)โˆ’1 are analytic on ๐บ๐‘–โˆช(โ„‚/ฮฉ๐‘–) and lim|๐‘ง|โ†’โˆž๐‘ƒ๐‘—๐‘“(๐‘ง)=0 for ๐‘—โ‰ 1. Expanding them as Laurent series, it follows that:(1)If ๐‘–=1, then ๐‘ƒ๐‘—โˆ‘๐‘“=+โˆž๐‘˜=1๐›ฝ๐‘—๐‘˜/๐‘ง๐‘˜ for ๐‘—โ‰ 1,(2)If ๐‘–โ‰ 1, then ๐‘ƒ๐‘—๐‘“(๐‘ง)=+โˆž๎“๐‘˜=0๐›ฝ๐‘—๐‘˜๎€ท๐‘งโˆ’๐‘Ž๐‘–๎€ธ๐‘˜,(3.28) for 0โ‰ ๐‘—โ‰ ๐‘– and ๐‘ƒ0๐‘“(๐‘ง)=+โˆž๎“๐‘˜=0๐›ฝ0๐‘˜๎€ท๐‘งโˆ’๐‘Ž๐‘–๎€ธ๐‘˜+๐›ผ๐‘–,โˆ’1๐‘งโˆ’๐‘Ž๐‘–.(3.29) It is obvious that, in any case, these series converge uniformly and absolutely on ๐บ๐‘–. Observing that each ๐บ๐‘– is an annulus at ๐‘Ž๐‘–, we have, by direct computation, that โŸจ๐‘“,๐‘“โŸฉ๐บ๐‘–โ‰ฅโŸจ๐‘ƒ๐‘–๐‘“,๐‘ƒ๐‘–๐‘“โŸฉ๐บ๐‘–+||๐›ผ๐‘–,โˆ’1||2๎€ทln๐‘…๐‘–โˆ’ln๐‘Ÿ๐‘–๎€ธ(3.30) if ๐‘–โ‰ 1 and โŸจ๐‘“,๐‘“โŸฉ๐บ1โ‰ฅโŸจ๐‘ƒ1๐‘“,๐‘ƒ1๐‘“โŸฉ๐บ1.(3.31) Therefore, for any ๐‘–=1,โ€ฆ,๐‘›, there exists a constant ๐‘€๎…ž such that โ€–โ€–๐‘ƒ๐‘–๐‘“โ€–โ€–๐บ๐‘–โ‰คโ€–๐‘“โ€–๐บ๐‘–โ‰คโ€–๐‘“โ€–ฮฉ,||๐›ผ(โˆ—)๐‘–,โˆ’1||โ‰ค๐‘€๎…žโ‹…โ€–๐‘“โ€–ฮฉ.(โˆ—โˆ—) From the definition of ๐‘ƒ๐‘—๐‘“, we derive โ€–โ€–๐‘ƒ1๐‘“โ€–โ€–2๐บ1=+โˆž๎“0||๐›ผ1๐‘˜||2๎€ท1โˆ’๐‘…12๐‘˜+2๎€ธ,โ€–โ€–๐‘ƒ๐‘˜+1๐‘–๐‘“โ€–โ€–2๐บ๐‘–=โˆ’โˆž๎“๐‘˜=โˆ’2|๐›ผ|2๐‘–๐‘˜๎€ท๐‘Ÿ๐‘–2๐‘˜+2โˆ’๐‘…๐‘–2๐‘˜+2๎€ธ,๐‘˜+1(3.32) for ๐‘–=2,โ€ฆ,๐‘›. The convergence of these series is guaranteed by the conditions โˆ— and โˆ—โˆ—. Since ๐‘…1<1 and ๐‘Ÿ๐‘–<๐‘…๐‘–, it follows that ๐‘ƒ๐‘–๐‘“โˆˆ๐ฟ2๐‘Ž(ฮฉ๐‘–) and โ€–โ€–๐‘ƒ1๐‘“โ€–โ€–2ฮฉ1=+โˆž๎“0||๐›ผ1๐‘˜||2,โ€–โ€–๐‘ƒ๐‘˜+1๐‘–๐‘“โ€–โ€–2ฮฉ๐‘–=โˆ’โˆž๎“๐‘˜=โˆ’2||๐›ผ1๐‘˜||2๐‘Ÿ๐‘–2๐‘˜+2,๐‘˜+1(3.33) for ๐‘–=2,โ€ฆ,๐‘›. Comparing the expression of โ€–๐‘ƒ๐‘–๐‘“โ€–ฮฉ๐‘– with the expression of โ€–๐‘ƒ๐‘–๐‘“โ€–๐บ๐‘–, it follows that โ€–๐‘ƒ๐‘–๐‘“โ€–ฮฉ๐‘–<๐‘€โ‹…โ€–๐‘ƒ๐‘–๐‘“โ€–๐บ๐‘– for some constant ๐‘€ for ๐‘–=1,โ€ฆ,๐‘›. Hence, โ€–๐‘ƒ๐‘–๐‘“โ€–ฮฉ๐‘–<๐‘€โ‹…โ€–๐‘ƒ๐‘–๐‘“โ€–ฮฉ. Moreover, if we define ๐‘€๎…ž๎…ž=Max{โ€–(๐‘งโˆ’๐‘Ž๐‘–)โˆ’1โ€–ฮฉ}, from the inequalities โ€–๐‘ƒ๐‘–๐‘“โ€–๐บ๐‘–โ‰คโ€–๐‘“โ€–๐บ๐‘–โ‰คโ€–๐‘“โ€–ฮฉ and |๐›ผ๐‘–,โˆ’1|โ‰ค๐‘€๎…žโ‹…โ€–๐‘“โ€–ฮฉ and the definition of ๐‘ƒ0, it follows that โ€–๐‘ƒ0๐‘“โ€–ฮฉโ‰ค๐‘›โ‹…๐‘€๎…žโ‹…๐‘€๎…ž๎…žโ‹…โ€–๐‘“โ€–ฮฉ.
If ๐‘“โˆˆ๐ฟ2๐‘Ž(ฮฉ๐‘–) for some ๐‘–โˆˆ{1,2,โ€ฆ,๐‘›}, note that lim๐‘“(๐‘ง)=0 as |๐‘ง|โ†’โˆž for ๐‘–โ‰ 1, then ๐‘“(๐‘ง)=๐‘ƒ๐‘–๐‘“(๐‘ง)+๐›ผ๐‘–,โˆ’1(๐‘งโˆ’๐‘Ž๐‘–)โˆ’1 if ๐‘–โ‰ 1 and ๐‘ƒ1๐‘“=๐‘“ if ๐‘–=1. For ๐‘–โ‰ 1, since ๐‘“โˆˆ๐ฟ2๐‘Ž(ฮฉ๐‘–)โŠ‚๐ฟ2๐‘Ž(ฮฉ) implies that ๐‘ƒ๐‘–๐‘“โˆˆ๐ฟ2๐‘Ž(ฮฉ๐‘–), then ๐›ผ๐‘–,โˆ’1โ‹…(๐‘งโˆ’๐‘Ž๐‘–)โˆ’1โˆˆ๐ฟ2๐‘Ž(ฮฉ๐‘–). We must have ๐›ผ๐‘–,โˆ’1=0 and, consequently, ๐‘ƒ0๐‘“=0. Hence, in any case, ๐‘“โˆˆ๐ฟ2๐‘Ž(ฮฉ๐‘–) implies ๐‘“=๐‘ƒ๐‘–๐‘“ and ๐‘ƒ๐‘—๐‘“=0 if ๐‘–โ‰ ๐‘—, and this remark completes our proof.

Lemma 3.5. If {๐‘“๐‘›} is a bounded sequence in ๐ฟ2๐‘Ž(ฮฉ) and ๐‘“๐‘›โ†’0 weakly in ๐ฟ2๐‘Ž(ฮฉ), then ๐‘ƒ๐‘—๐‘“๐‘›โ†’0 weakly on ๐ฟ2๐‘Ž(ฮฉ๐‘—) for ๐‘—=1,โ€ฆ,๐‘› and ๐‘ƒ0๐‘“๐‘›โ†’0 uniformly on ฮฉ.

Proof. By the previous Lemma, we know that the linear transformations {๐‘ƒ๐‘—} are bounded operators, then ๐‘“๐‘›โ†’0 weakly in ๐ฟ2๐‘Ž(ฮฉ) implies that ๐‘ƒ๐‘—๐‘“๐‘›โ†’0 weakly on ๐ฟ2๐‘Ž(ฮฉ๐‘—) for ๐‘—=1,โ€ฆ,๐‘›. For the same reason, ๐‘ƒ0๐‘“๐‘›โ†’0 weakly in ๐ฟ2๐‘Ž(ฮฉ) and then ๐‘ƒ0๐‘“๐‘›(๐œ)โ†’0 for any ๐œโˆˆฮฉ. Since ๐‘ƒ0๐‘“๐‘š=๐‘›๎“๐‘–=2๐›ผ๐‘–,โˆ’1(๐‘š)๎€ท๐œโˆ’๐‘Ž๐‘–๎€ธ,(3.34) by the estimates given in the last lemma, we have that |๐›ผ๐‘–,โˆ’1(๐‘š)|<๐‘€โ€–๐‘“๐‘šโ€–ฮฉ. The boundedness of {โ€–๐‘“๐‘šโ€–ฮฉ} implies that the family of continuous functions {๐‘ƒ0๐‘“๐‘š} is uniformly bounded and equicontinuous on ฮฉ, then, by Arzela-Ascoli's Theorem, we have that ๐‘ƒ0๐‘“๐‘šโ†’0 uniformly on ฮฉ.

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 ๐œ‘โˆˆ๐ฟ2(ฮฉ,๐‘‘๐œˆ) for the Toeplitz operator ๐‘‡๐œ‘ to be bounded or compact.

Before we state the next Theorem, we remind the reader that ๐พฮฉ(๐œ,๐‘ง)=๐ธฮฉ(๐œ,๐‘ง)+๐‘›๎“โ„“=1๐พฮฉโ„“(๐œ,๐‘ง),(4.1) where ๐ธฮฉโˆˆ๐ฟโˆž(ฮฉร—ฮฉ) and, for all โ„“=1,โ€ฆ,๐‘›, we have ๐พฮฉโ„“(๐œ,๐‘ง)=๐พฮฉโ„“(๐œ,๐‘ง),โˆ€๐œ,๐‘งโˆˆฮฉร—ฮฉ,(4.2) where ๐พฮฉโ„“ is the reproducing kernel of ฮฉโ„“. If we use the symbol ๐พฮฉ0 to indicate ๐ธฮฉ, we can write ๐พฮฉ(๐œ,๐‘ง)=๐‘›๎“โ„“=0๐พฮฉโ„“(๐œ,๐‘ง).(4.3) We also remind the reader that if ๐ผโˆถ๐ฟ2๐‘Ž(ฮฉ)โ†’๐ฟ2๐‘Ž(ฮฉ) is the identity operator, then ๐ผ=๐‘›๎“โ„“=0๐‘ƒโ„“,(4.4) where ๐‘ƒโ„“โˆถ๐ฟ2๐‘Ž(ฮฉ)โ†’๐ฟ2๐‘Ž(ฮฉ) is a bounded operator for all โ„“=0,1,โ€ฆ,๐‘› with ๐‘ƒโ„“๐‘“โˆˆ๐ฟ2๐‘Ž(ฮฉโ„“) if โ„“=1,โ€ฆ,๐‘› and ๐‘ƒ0๐‘“โˆˆ๐’žโˆž(ฮฉ) and ๐‘ƒ๐‘˜๐‘ƒโ„“=0 if ๐‘˜โ‰ โ„“ (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 ๐พฮฉ.

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

Lemma 4.1. Let ๐œ‘โˆˆ๐ฟ2(๐ท) be an essentially radial function where ๐ท=โˆฉ๐‘›๐‘—=1๐ท๐‘— with ๐ท1={๐‘งโˆˆโ„‚โˆถ|๐‘ง|<1} and ๐ท๐‘—={๐‘งโˆˆโ„‚โˆถ|๐‘งโˆ’๐‘Ž๐‘—|>๐‘Ÿ๐‘—} for ๐‘—=2,โ€ฆ,๐‘›. If one defines ๐œ‘๐‘—=๐œ‘โ‹…๐‘๐‘— where ๐‘—=1,โ€ฆ,๐‘› and ๐”“={๐‘0,๐‘1,โ€ฆ,๐‘๐‘›} is a ๐œ•-partition for ๐ท,then the following are equivalent: (1)the operator ๐‘‡๐œ‘โˆถ๐ฟ2๐‘Ž(๐ท,๐‘‘๐œˆ)โŸถ๐ฟ2๐‘Ž(๐ท,๐‘‘๐œˆ)(4.5) is bounded (compact);(2)for any ๐‘—=1,โ€ฆ,๐‘›, the operators ๐‘‡๐œ‘๐‘—โˆถ๐ฟ2๐‘Ž๎€ท๐ท๐‘—๎€ธ,๐‘‘๐œˆโŸถ๐ฟ2๐‘Ž๎€ท๐ท๐‘—๎€ธ,๐‘‘๐œˆ(4.6) are bounded (compact).

Proof. Let {๐‘0,๐‘1,โ€ฆ,๐‘๐‘›} be a partition of the unit on ๐ท=โˆฉ๐‘›๐‘—=1๐ท๐‘—, which is a canonical domain. Now, we notice that for all ๐‘“โˆˆ๐ฟ2(๐ท) and for all ๐‘คโˆˆ๐ท, we have the following: ๐‘‡๐œ‘๎€œ๐‘“(๐‘ค)=๐ท๐œ‘(๐‘ง)๐‘“(๐‘ง)๐พ๐ท(=๐‘ง,๐‘ค)๐‘‘๐‘ฃ(๐‘ง)๐‘›๎“๐‘—=0๎€œ๐ท๐œ‘(๐‘ง)๐‘“(๐‘ง)๐พ๐ท๐‘—(=๐‘ง,๐‘ค)๐‘‘๐‘ฃ(๐‘ง)๐‘›๎“๐‘›๐‘—=0๎“๐‘˜=0๎€œ๐ท๐œ‘(๐‘ง)๐‘๐‘˜(๐‘ง)๐‘“(๐‘ง)๐พ๐ท๐‘—=(๐‘ง,๐‘ค)๐‘‘๐‘ฃ(๐‘ง)๐‘›๎“๐‘›๐‘—=0๎“๐‘˜=0๐‘‡๐‘—๐‘˜๐‘“(๐‘ค),(4.7) where, by definition, we have ๐‘‡๐‘—๐‘˜๎€œ๐‘“(๐‘ค)=๐ท๐œ‘(๐‘ง)๐‘๐‘˜(๐‘ง)๐พ๐ท๐‘—(๐‘ง,๐‘ค)๐‘“(๐‘ง)๐‘‘๐‘ฃ(๐‘ค)๐‘‘๐‘ฃ(๐‘ง).(4.8)

Claim 1. The operator ๐‘‡๐‘—0 is Hilbert-Schmidt for any ๐‘—=0,1,โ€ฆ,๐‘›.

Proof. We observe that, by definition, we have ๐‘‡๐‘—0๎€œ๐‘“(๐‘ค)=๐ท๐œ‘(๐‘ง)๐‘0(๐‘ง)๐พ๐ท๐‘—(๐‘ง,๐‘ค)๐‘“(๐‘ง)๐‘‘๐‘ฃ(๐‘ง),(4.9) therefore, if we define โ„1=๎€๐ท||๐œ‘(๐‘ง)๐‘0(๐‘ง)๐พ๐ท๐‘—(||๐‘ง,๐‘ค)2๐‘‘๐‘ฃ(๐‘ง)๐‘‘๐‘ฃ(๐‘ค),(4.10) we have โ„1=๎€œ๐ท||๐œ‘(๐‘ง)๐‘0||(๐‘ง)2๎‚ต๎€œ๐ท||๐พ๐ท๐‘—||(๐‘ง,๐‘ค)2๎‚ถโ‰ค๎€œ๐‘‘๐‘ฃ(๐‘ค)๐‘‘๐‘ฃ(๐‘ง)๐ท||๐œ‘(๐‘ง)๐‘0||(๐‘ง)2||๐พ๐ท๐‘—||โ‰ค๎‚ต(๐‘ง,๐‘ง)๐‘‘๐‘ฃ(๐‘ง)Max๐‘งโˆˆsupp(๐‘0)||๐‘0||(๐‘ง)2๐พ๐ท๐‘—๎‚ถ๎€œ(๐‘ง,๐‘ง)๐ท||๐œ‘๐‘—||(๐‘ง)2โ‰ค๎‚ต๐‘‘๐‘ฃ(๐‘ง)Max๐‘งโˆˆsupp(๐‘0)||๐‘0||(๐‘ง)2๐พ๐ท๐‘—(๎‚ถ๐‘ง,๐‘ง)โ‹…โ€–๐œ‘โ€–2๐ท,2<โˆž.(4.11) This implies that for any ๐‘ก=0,1,โ€ฆ,๐‘›, ๐‘‡๐‘ก0 is Hilbert-Schmidt. Therefore, the operator ๐‘›๎“๐‘ก=0๐‘‡๐‘ก0(4.12) is Hilbert-Schmidt, and this completes the proof of the claim.

Claim 2. The operator ๐‘‡0๐‘˜ is Hilbert-Schmidt for any ๐‘˜=0,1,โ€ฆ,๐‘›.

Proof. We observe that, by definition, we have ๐‘‡0๐‘˜๎€œ๐‘“(๐‘ค)=๐ท๐œ‘(๐‘ง)๐‘๐‘˜(๐‘ง)๐พ๐ท0(๐‘ง,๐‘ค)๐‘“(๐‘ง)๐‘‘๐‘ฃ(๐‘ง),(4.13) therefore, if we define โ„2=๎€๐ท||๐œ‘(๐‘ง)๐‘๐‘˜(๐‘ง)๐พ๐ท0(||๐‘ง,๐‘ค)2๐‘‘๐‘ฃ(๐‘ง)๐‘‘๐‘ฃ(๐‘ค),(4.14) we have โ„2=๎€๐ท||๐œ‘(๐‘ง)๐‘0||(๐‘ง)2||๐พ๐ท0(||๐‘ง,๐‘ค)2โ‰ค๎‚ต๐‘‘๐‘ฃ(๐‘ค)๐‘‘๐‘ฃ(๐‘ง)Max(๐‘ง,๐‘ค)โˆˆ๐ทร—๐ท||๐พ๐ท0||(๐‘ง,๐‘ค)2๎‚ถ๎€œโ‹…๐‘ฃ(๐ท)โ‹…๐ท||๐œ‘(๐‘ง)๐‘0||(๐‘ง)2โ‰ค๎‚ต๐‘‘๐‘ฃ(๐‘ง)Max(๐‘ง,๐‘ค)โˆˆ๐ทร—๐ท||๐พ๐ท0||(๐‘ง,๐‘ค)2๎‚ถโ‹…๐‘ฃ(๐ท)โ‹…โ€–๐œ‘โ€–2๐ท,2<โˆž.(4.15) This implies that for any ๐‘ก=0,1,โ€ฆ,๐‘›, ๐‘‡0๐‘ก is Hilbert-Schmidt. Therefore, the following ๐‘›๎“๐‘ก=0๐‘‡0๐‘ก(4.16) is Hilbert-Schmidt, and this completes the proof of the claim.

Claim 3. The operator ๐‘‡๐‘–๐‘— is Hilbert-Schmidt if ๐‘–โ‰ ๐‘—โ‰ 0 and ๐‘—,๐‘–=1,โ€ฆ,๐‘›.

Proof. We observe that ๐‘‡๐‘—๐‘˜๎€œ๐‘“(๐‘ค)=๐ท๐œ‘(๐‘ง)๐‘๐‘˜(๐‘ง)๐พ๐ท๐‘—(๐‘ง,๐‘ค)๐‘“(๐‘ง)๐‘‘๐‘ฃ(๐‘ค)๐‘‘๐‘ฃ(๐‘ง).(4.17) To start, we give the following: ๐’ฉ๐‘—๐‘–(๐‘ง,๐‘ค)def=๐œ‘๐‘—(๐‘ง)โ‹…๐พ๐ท๐‘–(๐‘ง,๐‘ค).(4.18) We will show that Fubini theorem and the properties of the ๐œ•-partition imply that ๎€๐ท||๐’ฉ๐‘—๐‘–||(๐‘ง,๐‘ค)2๐‘‘๐‘ฃ(๐‘ค)๐‘‘๐‘ฃ(๐‘ง)<โˆž.(4.19) In fact, we have ๎€๐ท||๐’ฉ๐‘—๐‘–||(๐‘ง,๐‘ค)2=๎€œ๐ท๎‚ต๎€œ๐ท||๐’ฉ๐‘—๐‘–||(๐‘ง,๐‘ค)2๎‚ถ=๎€๐‘‘๐‘ฃ(๐‘ค)๐‘‘๐‘ฃ(๐‘ง)๐ท||๐œ‘๐‘—||(๐‘ง)2||๐พ๐ท๐‘–||(๐‘ง,๐‘ค)2=๎€œ๐‘‘๐‘ฃ(๐‘ค)๐‘‘๐‘ฃ(๐‘ง)๐ท||๐œ‘๐‘—||(๐‘ง)2๎‚ต๎€œ๐ท||๐พ๐ท๐‘–||(๐‘ง,๐‘ค)2๎‚ถ=๎€œ๐‘‘๐‘ฃ(๐‘ค)๐‘‘๐‘ฃ(๐‘ง)๐ท||๐œ‘๐‘—||(๐‘ง)2๐พ๐ท๐‘–=๎€œ(๐‘ง,๐‘ง)๐‘‘๐‘ฃ(๐‘ง)๐ท||||๐œ‘(๐‘ง)2||๐‘๐‘—||(๐‘ง)2๐พ๐ท๐‘–โ‰ค๎‚ต(๐‘ง,๐‘ง)๐‘‘๐‘ฃ(๐‘ง)Max๐‘งโˆˆsupp(๐‘๐‘—)||๐‘๐‘—||(๐‘ง)2๐พ๐ท๐‘–๎‚ถ(๐‘ง,๐‘ง)โ‹…โ€–๐œ‘โ€–2๐ท,2<โˆž.(4.20)
Therefore, we can write that ๐‘‡๐œ‘=๐’ฆ+๐‘›๎“โ„“=1๐‘‡๐œ‘โ„“โ„“,(4.21) where ๐’ฆ is a compact operator.
We also observe that Lemma 3.4 implies that ๐‘‡๐œ‘โ„“โ„“=โˆ‘๐‘›๐‘—=0๐‘‡๐œ‘โ„“โ„“๐‘ƒ๐‘—, and we prove that the operator ๐‘‡๐œ‘โ„“โ„“๐‘ƒ๐‘— is compact if ๐‘—โ‰ โ„“ and ๐‘—,โ„“=1,โ€ฆ,๐‘›.

Proof. In order to simplify the notation, we define the operator ๐‘…๐‘—,โ„“=๐‘‡๐œ‘โ„“โ„“๐‘ƒ๐‘—=๐พ๐ทโ„“๐‘€๐œ‘๐‘โ„“๐‘ƒ๐‘—. To prove our statement, it is enough to prove that if we take a bounded sequence {๐‘“๐‘›} in ๐ฟ2(๐ท) such that ๐‘“๐‘›โ†’0 weakly, then we can prove that โ€–๐‘…๐‘—,โ„“๐‘“๐‘›โ€–2โ†’0. We know that the continuity of ๐‘ƒโ„“ implies that ๐‘ƒ๐‘—๐‘“๐‘˜โ†’0 weakly on ๐ป2(๐ท๐‘™), and {โ€–๐‘ƒ๐‘—๐‘“๐‘˜โ€–๐ทโ„“} is bounded by Lemma 3.5. Since it is a sequence of holomorphic functions, we know that {๐‘ƒ๐‘—๐‘“๐‘˜} is uniformly bounded on any compact subset of ๐ทโ„“. Therefore, the sequence {๐‘ƒ๐‘—๐‘“๐‘˜} is a normal family of functions. Since ๐‘ƒ๐‘—๐‘“๐‘˜(๐œ)โ†’0 for any ๐œโˆˆ๐ท๐‘—, then ๐‘ƒ๐‘—๐‘“๐‘˜ converges uniformly on any compact subset of ๐ท๐‘— and consequently on ๐น=supp(๐‘โ„“). To complete the proof, we remind the reader that if we define the operators ๐‘„โ„“โˆถ๐ฟ2(๐ท)โ†’๐ฟ2(๐ท), for โ„“=1,2,โ€ฆ,๐‘›, in this way ๐‘„โ„“๎€œ๐‘“(๐‘ง)=๐ท||๐พ๐‘“(๐œ)๐ทโ„“(||๐œ,๐‘ง)๐‘‘๐‘ฃ(๐œ).(4.22) It is possible to prove, with the help of Schur's test (see [11] ), that ๐‘„โ„“ is a bounded operator (see [5]). Now, we observe that ||๐‘…๐‘—,โ„“๐‘“๐‘˜||๎€ฝ||๐‘ƒ(๐œ)โ‰คSup๐‘—๐‘“๐‘˜||๎€พโ‹…||๐‘„(๐œ)โˆถ๐œโˆˆ๐น๐‘—๎€ท||๐’ณ๐น๐œ‘๐‘๐‘ ||๎€ธ||,(๐œ)(4.23) then, by using the fact that ๐‘„โ„“ is bounded, we have โ€–โ€–๐‘…๐‘—,โ„“๐‘“๐‘˜โ€–โ€–๐ท๎€ฝ||๐‘ƒโ‰คSup๐‘—๐‘“๐‘˜||๎€พโ€–โ€–๐œ‘(๐œ)โˆถ๐œโˆˆ๐นโ‹…๐‘€โ‹…1๐‘๐‘ โ€–โ€–๐ท,2โŸถ0,(4.24) and this completes the proof of our claim. Notice also that using the same strategy, we can prove that each ๐‘‡๐œ‘โ„“โ„“๐‘ƒ0 is compact.

Therefore, we have ๐‘‡๐œ‘=๐’ฆ+๐‘›๎“โ„“=1๐‘‡๐œ‘โ„“โ„“=๐’ฆ+๐พ1+๐‘›๎“โ„“=1๐‘‡๐œ‘โ„“โ„“๐‘ƒโ„“,(4.25) where ๐’ฆ,๐พ1 are compact operators. Since ๐‘ƒ2๐‘ก=๐‘ƒ๐‘ก,๐‘ƒ๐‘ก๐‘ƒ๐‘ =0 and if ๐‘—โ‰ โ„“, then ๐‘‡๐œ‘ is bounded (compact) if and only if the operators ๐‘‡๐œ‘โ„“โ„“๐‘ƒโ„“ are bounded (compact) operators.

Since ๐‘ƒโ„“๐ฟ2๐‘Ž(๐ท)=๐ฟ2๐‘Ž(๐ทโ„“), then it follows that the operator ๐‘‡๐œ‘โ„“โ„“๐‘ƒโ„“ is bounded (compact) if and only if ๐‘‡๐œ‘โ„“โ„“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 ๐œ‘โˆˆ๐ฟ2(๐ท) be an essentially radial function where ๐ท=โˆฉ๐‘›๐‘—=1๐ท๐‘— with ๐ท1={๐‘งโˆˆโ„‚โˆถ|๐‘ง|<1} and ๐ท๐‘—={๐‘งโˆˆโ„‚โˆถ|๐‘งโˆ’a๐‘—|>๐‘Ÿ๐‘—} for ๐‘—=2,โ€ฆ,๐‘›. If one defines ๐œ‘๐‘—=๐œ‘โ‹…๐‘๐‘— where ๐‘—=1,โ€ฆ,๐‘› and ๐”“={๐‘0,๐‘1,โ€ฆ,๐‘๐‘›} is a ๐œ•-partition for ๐ทthen the following are equivalent: (1) the operator ๐‘‡๐œ‘โˆถ๐ฟ2๐‘Ž(๐ท,๐‘‘๐œˆ)โŸถ๐ฟ2๐‘Ž(๐ท,๐‘‘๐œˆ)(4.26) is bounded (compact).(2) for any ๐‘—=1,โ€ฆ,๐‘›, the sequences ๐›พ๐œ‘๐‘—={๐›พ๐œ‘๐‘—(๐‘š)}๐‘šโˆˆโ„•are in โ„“โˆž(โ„ค+)(๐‘0(โ„ค+)) where, by definition, if ๐‘—=2,โ€ฆ,๐‘›๐›พ๐œ‘๐‘—(๐‘š)=๐‘Ÿ๐‘—๎€œโˆž๐‘Ÿ๐‘—๐œ‘๐‘—๎‚€๐‘Ÿ๐‘—(2๐‘š+1)/2(๐‘š+1)๐‘ 1/2(๐‘š+1)+๐‘Ž๐‘—๎‚1๐‘ 2๐‘‘๐‘ โˆ€๐‘šโˆˆโ„ค+,(4.27) and for ๐‘—=1, ๐›พ๐œ‘1๎€œ(๐‘š)=10๐œ‘1๎€ท๐‘ 1/2(๐‘š+1)๎€ธ๐‘‘๐‘ ,โˆ€๐‘šโˆˆโ„ค+.(4.28)

Proof. In the previous theorem, we proved that the operator under examination is bounded (compact) if and only if for any ๐‘—=1,โ€ฆ,๐‘› the operators ๐‘‡๐œ‘๐‘—โˆถ๐ฟ2๎€ท๐ท๐‘—๎€ธ,๐‘‘๐œˆโŸถ๐ฟ2๐‘Ž๎€ท๐ท๐‘—๎€ธ,๐‘‘๐œˆ(4.29) are bounded (compact). If ๐‘—=2,โ€ฆ,๐‘›, we observe that if we consider the following sets ฮ”0,1={๐‘งโˆˆโ„‚โˆถ0<|๐‘งโˆ’๐‘Ž|<1} and ฮ”๐‘Ž๐‘—,๐‘Ÿ๐‘—={๐‘งโˆˆโ„‚โˆถ0<|๐‘งโˆ’๐‘Ž๐‘—|<๐‘Ÿ๐‘—} and the following maps ฮ”๐›ผ0,1โˆ’โˆ’โˆ’โ†’ฮ”๐‘Ž๐‘—,๐‘Ÿ๐‘—๐›ฝโˆ’โˆ’โˆ’โ†’๐ท๐‘—,(4.30) where ๐›ผ(๐‘ง)=๐‘Ž๐‘—+๐‘Ÿ๐‘—๐‘ง and ๐›ฝ(๐‘ค)=(๐‘คโˆ’๐‘Ž๐‘—)โˆ’1๐‘Ÿ2๐‘—+๐‘Ž๐‘— and we use Proposition 1.1 in [8], we can claim that ๐‘‡๐œ‘๐‘—=๐‘‰โˆ’1๐›ฝโˆ˜๐›ผ๐‘‡๐œ‘๐‘—โˆ˜๐›ฝโˆ˜๐›ผ๐‘‰๐›ฝโˆ˜๐›ผ,(4.31) where ๐‘‰๐›ฝโˆ˜๐›ผโˆถ๐ฟ2(ฮ”0,1)โ†’๐ฟ2(๐ท๐‘—) is an isomorphism of Hilbert spaces. Therefore, ๐‘‡๐œ‘๐‘— is bounded (compact) if and only if ๐‘‡๐œ‘๐‘—โˆ˜๐›ฝโˆ˜๐›ผ is bounded (compact). We also know that this, in turn, is equivalent to the fact that the sequence ๐›พ๐œ‘๐‘—=๎‚†๐›พ๐œ‘๐‘—๎‚‡(๐‘š)๐‘šโˆˆโ„•(4.32) is in โ„“โˆž(โ„ค+)(๐‘0(โ„ค+)), where ๐›พ๐œ‘๐‘—๎€œ(๐‘š)=10๐œ‘๐‘—๎€ท๐‘Ÿโˆ˜๐›ฝโˆ˜๐›ผ1/2(๐‘š+1)๎€ธ๐‘‘๐‘Ÿ,โˆ€๐‘šโˆˆโ„ค+.(4.33) To complete the proof, we observe that since ๐œ‘๐‘— is radial and ๐›ฝโˆ˜๐›ผ(๐‘Ÿ)=๐‘Ÿโˆ’1๐‘Ÿ๐‘—+๐‘Ž๐‘— then, after a change of variable, we can rewrite the last integral, and therefore the formula ๐›พ๐œ‘๐‘—(๐‘š)=๐‘Ÿ๐‘—๎€œโˆž๐‘Ÿ๐‘—๐œ‘๐‘—๎‚€๐‘Ÿ๐‘—(2๐‘š+1)/2(๐‘š+1)๐‘ 1/2(๐‘š+1)+๐‘Ž๐‘—๎‚1๐‘ 2๐‘‘๐‘ ,โˆ€๐‘šโˆˆโ„ค+(4.34) must hold for any ๐‘—=2,โ€ฆ,๐‘›. The case ๐‘—=1 is immediate.

Now, we can prove the following.

Theorem 4.3. Let ๐œ‘โˆˆ๐ฟ2(ฮฉ) be an essentially radial function via the conformal equivalence ฮ˜โˆถฮฉโ†’๐ท, define ๐œ‘๐‘—=๐œ‘โ‹…๐‘๐‘— where ๐‘—=1,โ€ฆ,๐‘› and ๐”“ is a ๐œ•-partition for ฮฉ, then the following conditions are equivalent: (1)the operator ๐‘‡๐œ‘โˆถ๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ)โŸถ๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ)(4.35) is bounded (compact);(2)for any ๐‘—=1,โ€ฆ,๐‘›, the sequences ๐›พ๐œ‘๐‘—={๐›พ๐œ‘๐‘—(๐‘š)}๐‘šโˆˆโ„•are in โ„“โˆž(โ„ค+)(๐‘0(โ„ค+)) where, by definition, if ๐‘—=2,โ€ฆ,๐‘›๐›พ๐œ‘๐‘—(๐‘š)=๐‘Ÿ๐‘—๎€œโˆž๐‘Ÿ๐‘—๐œ‘๐‘—โˆ˜ฮ˜โˆ’1๎‚€๐‘Ÿ๐‘—(2๐‘š+1)/2(๐‘š+1)๐‘ 1/2(๐‘š+1)+๐‘Ž๐‘—๎‚1๐‘ 2๐‘‘๐‘ ,โˆ€๐‘šโˆˆโ„ค+,(4.36) and for ๐‘—=1๐›พ๐œ‘1๎€œ(๐‘š)=10๐œ‘1โˆ˜ฮ˜โˆ’1๎€ท๐‘ 1/2(๐‘š+1)๎€ธ๐‘‘๐‘ ,โˆ€๐‘šโˆˆโ„ค+.(4.37)

Proof. We know that ฮฉ is a regular domain, and therefore if ฮ˜ is a conformal mapping from ฮฉ onto ๐ท then the Bergman kernels of ฮฉ and ฮ˜(ฮฉ)=๐ท, are related via ๐พ๐ท(ฮ˜(๐‘ง),ฮ˜(๐‘ค))ฮ˜๎…ž(๐‘ง)ฮ˜๎…ž(๐‘ค)=๐พฮฉ(๐‘ง,๐‘ค), and the operator ๐‘‰ฮ˜๐‘“=ฮ˜๎…žโ‹…๐‘“โˆ˜ฮ˜ is an isometry from ๐ฟ2(๐ท) onto๐ฟ2(ฮฉ) (see Proposition 1.1 in [8]). In particular, we have ๐‘‰ฮ˜๐‘ƒ๐ท=๐‘ƒฮฉ๐‘‰ฮ˜ and this implies that ๐‘‰ฮ˜๐‘‡๐œ‘=๐‘‡๐œ‘โˆ˜ฮ˜โˆ’1๐‘‰ฮ˜. Therefore, the operator ๐‘‡๐œ‘ is bounded (compact) if and only if the operator ๐‘‡๐œ‘โˆ˜ฮ˜โˆ’1โˆถ๐ฟ2(๐ท,๐‘‘๐œˆ)โ†’๐ฟ2๐‘Ž(๐ท,๐‘‘๐œˆ) is bounded (compact). In the previous theorem we proved that the operator in exam is bounded (compact) if and only if for any ๐‘—=1,โ€ฆ,๐‘› the operators ๐‘‡๐œ‘๐‘—โˆ˜ฮ˜โˆ’1โˆถ๐ฟ2๐‘Ž๎€ท๐ท๐‘—๎€ธ,๐‘‘๐œˆโŸถ๐ฟ2๐‘Ž๎€ท๐ท๐‘—๎€ธ,๐‘‘๐œˆ(4.38) are bounded (compact). Hence, we can conclude that the operator is bounded (compact) if and only if for any ๐‘—=1,โ€ฆ,๐‘› the sequences ๐›พ๐œ‘๐‘—={๐›พ๐œ‘๐‘—(๐‘š)}๐‘šโˆˆโ„• are in โ„“โˆž(โ„ค+)(๐‘0(โ„ค+)) where, by definition, if ๐‘—=2,โ€ฆ,๐‘›, we have ๐›พ๐œ‘๐‘—(๐‘š)=๐‘Ÿ๐‘—๎€œโˆž๐‘Ÿ๐‘—๐œ‘๐‘—โˆ˜ฮ˜โˆ’1๎‚€๐‘Ÿ๐‘—(2๐‘š+1)/2(๐‘š+1)๐‘ 1/2(๐‘š+1)+๐‘Ž๐‘—๎‚1๐‘ 2๐‘‘๐‘ ,โˆ€๐‘šโˆˆโ„ค+,(4.39) and for ๐‘—=1, ๐›พ๐œ‘1๎€œ(๐‘š)=10๐œ‘1โˆ˜ฮ˜โˆ’1๎€ท๐‘ 1/2(๐‘š+1)๎€ธ๐‘‘๐‘ ,โˆ€๐‘šโˆˆโ„ค+,(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 ๐‘—=2,โ€ฆ,๐‘›, we define ๐ต๐œ‘๐‘—(๐‘ )=๐‘Ÿ๐‘—๎€œ๐‘ ๐‘Ÿ๐‘—๐œ‘๐‘—โˆ˜ฮ˜โˆ’1๎‚€๐‘Ÿ๐‘—1/2๐‘ฅ1/2+๐‘Ž๐‘—๎‚1๐‘ฅ2๐‘‘๐‘ฅ,(4.41) and for ๐‘—=1, we set ๐ต๐œ‘1๎€œ(๐‘ )=1๐‘ ๐œ‘1โˆ˜ฮ˜โˆ’1๎€ท๐‘ฅ1/2๎€ธ๐‘‘๐‘ฅ.(4.42)

We obtain the following useful theorem.

Theorem 4.4. Let ๐œ‘โˆˆ๐ฟ2(ฮฉ) be an essentially radial function via the conformal equivalence ฮ˜โˆถฮฉโ†’๐ท. If one defines ๐œ‘๐‘—=๐œ‘โ‹…๐‘๐‘— where ๐‘—=1,โ€ฆ,๐‘› and ๐”“ is a ๐œ•-partition for ฮฉ, then for the operator ๐‘‡๐œ‘โˆถ๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ)โ†’๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ) the following hold true: (1)if for any ๐‘—=1,โ€ฆ,๐‘›|||๐ต๐œ‘๐‘—|||๎€ท๐‘Ÿ(s)=๐‘‚๐‘—๎€ธโˆ’๐‘ as๐‘ โŸถ๐‘Ÿ๐‘—,(4.43) then ๐‘‡๐œ‘ is bounded;(2)if for any ๐‘—=1,โ€ฆ,๐‘›|||๐ต๐œ‘๐‘—|||๎€ท๐‘Ÿ(๐‘ )=๐‘œ๐‘—๎€ธโˆ’๐‘ as๐‘ โŸถ๐‘Ÿ๐‘—,(4.44) then ๐‘‡๐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 ๐‘—=1,โ€ฆ,๐‘› the sequences ๐›พ๐œ‘๐‘—={๐›พ๐œ‘๐‘—(๐‘š)}๐‘šโˆˆโ„• are in โ„“โˆž(โ„ค+)(๐‘0(โ„ค+)) where, by definition, if ๐‘—=2,โ€ฆ,๐‘›, ๐›พ๐œ‘๐‘—(๐‘š)=๐‘Ÿ๐‘—๎€œโˆž๐‘Ÿ๐‘—๐œ‘๐‘—โˆ˜ฮ˜โˆ’1๎‚€๐‘Ÿ๐‘—(2๐‘š+1)/2(๐‘š+1)๐‘ 1/2(๐‘š+1)+๐‘Ž๐‘—๎‚1๐‘ 2๐‘‘๐‘ โˆ€๐‘šโˆˆโ„ค+,(4.45) and for ๐‘—=1๐›พ๐œ‘๐‘—๎€œ(๐‘š)=10๐œ‘1โˆ˜ฮ˜โˆ’1๎€ท๐‘ 1/2(๐‘š+1)๎€ธ๐‘‘๐‘ โˆ€๐‘šโˆˆโ„ค+,(4.46) and, in virtue of [1]'s main result, it is true that ๐›พ๐œ‘๐‘—={๐›พ๐œ‘๐‘—(๐‘š)}๐‘šโˆˆโ„• are in โ„“โˆž(โ„ค+) if for any ๐‘—=1,โ€ฆ,๐‘›, |||๐ต๐œ‘๐‘—|||๎€ท๐‘Ÿ(๐‘ )=๐‘‚๐‘—๎€ธโˆ’๐‘ as๐‘ โŸถ๐‘Ÿ๐‘—,(4.47) and ๐›พ๐œ‘๐‘—={๐›พ๐œ‘๐‘—(๐‘š)}๐‘šโˆˆโ„• are in ๐‘0(โ„ค+)) if for any ๐‘—=1,โ€ฆ,๐‘›|||๐ต๐œ‘๐‘—|||๎€ท๐‘Ÿ(๐‘ )=๐‘œ๐‘—๎€ธโˆ’๐‘ as๐‘ โŸถ๐‘Ÿ๐‘—.(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 ๐œ‘โˆˆ๐ฟ2(ฮฉ) be an essentially radial function via the conformal equivalence ฮ˜โˆถฮฉโ†’๐ท. If we define ๐œ‘๐‘—=๐œ‘โ‹…๐‘๐‘— where ๐‘—=1,โ€ฆ,๐‘› and ๐”“ is a ๐œ•-partition for ฮฉ and if ๐œ‘โ‰ฅ0 a.e. in ฮฉ, then for the operator ๐‘‡๐œ‘โˆถ๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ)โ†’๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ), the following hold true: (1)๐‘‡๐œ‘ is bounded if and only if |||๐ต๐œ‘๐‘—|||๎€ท๐‘Ÿ(๐‘ )=๐‘‚๐‘—๎€ธโˆ’๐‘ as๐‘ โŸถ๐‘Ÿ๐‘—,(4.49) for any ๐‘—=1,โ€ฆ,๐‘›,(2)๐‘‡๐œ‘ is compact if and only if |||๐ต๐œ‘๐‘—|||๎€ท๐‘Ÿ(๐‘ )=๐‘œ๐‘—๎€ธโˆ’๐‘ as๐‘ โŸถ๐‘Ÿ๐‘—,(4.50) for any ๐‘—=1,โ€ฆ,๐‘›.

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 ๐‘‡๐œ‘โˆถ๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ)โ†’๐ฟ2๐‘Ž(ฮฉ,๐‘‘๐œˆ) has an essentially radial positive symbol ๐œ‘โ‰ฅ0 such that for some โ„“=1,โ€ฆ,๐‘›, the following lim๐›ฟโ†’0๎‚ตinfdist(๐‘ง,๐œ•ฮฉโ„“)<๐›ฟ๎‚ถ๐œ‘(๐‘ง)=โˆž(4.51) holds, then the operator ๐‘‡๐œ‘ is unbounded. Moreover, if ๐‘‡๐œ‘ is bounded and the symbol is an unbounded essentially radial function, then it must be true that around any ๐œ•ฮฉโ„“, the symbol has an oscillating behavior.

In order to present an application, we consider a family of examples. Let us consider the case where ฮฉ=โˆฉ๐‘›๐‘—=1ฮฉ๐‘— with ฮฉ1={๐‘งโˆˆโ„‚โˆถ|๐‘ง|<1} and ฮฉ๐‘—={๐‘งโˆˆโ„‚โˆถ|๐‘งโˆ’๐‘Ž๐‘—|>๐‘Ÿ๐‘—} for ๐‘—=2,โ€ฆ,๐‘›. Let ๐œ‘โˆˆ๐ฟ2(ฮฉ) be a function that can be written in the following way: ๐œ‘=๐‘›๎‘โ„“=1๐œ‘(โ„“),(4.52) where, for any โ„“=1,2,โ€ฆ,๐‘›,๐œ‘(โ„“) is radial, that is, ๐œ‘(โ„“)=๐œ‘(โ„“)(|๐‘งโˆ’๐‘Žโ„“|) and satisfies inf||๐‘งโˆ’๐‘Žโ„“||>๐‘Ÿโ„“+๐œ–โ„“๐œ‘๎€ท||(โ„“)๐‘งโˆ’๐‘Žโ„“||๎€ธ=๐‘šโ„“>0,sup||๐‘งโˆ’๐‘Žโ„“||>๐‘Ÿโ„“+๐œ–โ„“๐œ‘๎€ท||(โ„“)๐‘งโˆ’๐‘Žโ„“||๎€ธ=๐‘€โ„“<โˆž(4.53) if โ„“=2,โ€ฆ,๐‘› and inf|๐‘ง|<1โˆ’๐œ–1๐œ‘(1)(|๐‘ง|)=๐‘š1>0,sup||๐‘งโˆ’๐‘Žโ„“||<1โˆ’๐œ–1๐œ‘(1)(|๐‘ง|)=๐‘€1<โˆž.(4.54) if โ„“=1. As a consequence of our results, we can conclude that (1)๐‘‡๐œ‘ is bounded if there exists a constant ๐’ž1 such that for any ๐‘—=2,โ€ฆ,๐‘›, limsup๐‘ โ†’๐‘Ÿโ„“||||๐‘Ÿโ„“๐‘ โˆ’๐‘Ÿโ„“๎€œ๐‘ ๐‘Ÿโ„“๐œ‘๎€ท๐‘Ÿ(โ„“)โ„“1/2๐‘ฅ1/2+๐‘Žโ„“๎€ธ1๐‘ฅ2||||๐‘‘๐‘ฅ<๐’ž1,limsup๐‘ โ†’1||||1๎€œ1โˆ’๐‘ 1๐‘ ๎€ท๐‘ฅ๐œ‘(โ„“)1/2๎€ธ||||๐‘‘๐‘ฅ<๐’ž1,(4.55) for any ๐‘—=1,(2)๐‘‡๐œ‘ is compact if for any ๐‘—=2,โ€ฆ,๐‘›lim๐‘ โ†’๐‘Ÿโ„“๐‘Ÿโ„“๐‘ โˆ’๐‘Ÿโ„“๎€œ๐‘ ๐‘Ÿโ„“๎€ท๐‘Ÿ๐œ‘(โ„“)โ„“1/2๐‘ฅ1/2+๐‘Žโ„“๎€ธ1๐‘ฅ2๐‘‘๐‘ฅ=0,lim๐‘ โ†’11๎€œ1โˆ’๐‘ 1๐‘ ๎€ท๐‘ฅ๐œ‘(โ„“)1/2๎€ธ๐‘‘๐‘ฅ=0,(4.56)for ๐‘—=1.

It is also possible to show that the sufficient conditions may fail, but the operator is still bounded or even compact. In fact, we can show that given any planar bounded multiply-connected domain ฮฉ, whose boundary ๐œ•ฮฉ consists of finitely many simple closed smooth analytic curves, there exist unbounded functions ๐œ‘โˆˆ๐ฟ2(ฮฉ) such that ๐‘‡๐œ‘ is compact even when the sufficient conditions are not satisfied. To prove this claim, we observe that for the domain ฮฉ there exists a conformally equivalent domain ๐ท=โˆฉ๐‘›๐‘–=1๐ท๐‘– where ๐ท1={๐‘งโˆˆโ„‚โˆถ|๐‘ง|<1} and ๐ท๐‘—={zโˆˆโ„‚โˆถ|๐‘งโˆ’๐‘Ž๐‘—|>๐‘Ÿ๐‘—} for ๐‘—=2,โ€ฆ,๐‘› where ๐‘Ž๐‘—โˆˆ๐ท1 and 0<๐‘Ÿ๐‘—<1 with |๐‘Ž๐‘—โˆ’๐‘Ž๐‘˜|>๐‘Ÿ๐‘—+๐‘Ÿ๐‘˜ if ๐‘—โ‰ ๐‘˜ and 1โˆ’|๐‘Ž๐‘—|>๐‘Ÿ๐‘—. If we denote with the symbol ฮจโˆถฮฉโŸถ๐ท(4.57) the conformal equivalence between ฮฉ and ๐ท, then we can define, on ฮฉ, the map ๐œ‘๐ฎ,๐ฏ=๐‘›๎‘โ„“=1๐œ‘๐‘ขโ„“,๐‘ฃโ„“(โ„“),(4.58) where, for any โ„“, we have ๐œ‘๐‘ขโ„“,๐‘ฃโ„“๎€ท(โ„“)(๐‘ง)=โˆ’1โˆ’๐‘ขโ„“๎€ธ๎‚€๎€ทฮจ1โˆ’dist(๐‘ง),๐œ•๐ทโ„“๎€ธ2๎‚โˆ’๐‘ขโ„“๎‚€๎€ทฮจsin1โˆ’dist(๐‘ง),๐œ•๐ทโ„“๎€ธ2๎‚โˆ’๐‘ฃโ„“+๐‘ฃโ„“๎‚€๎€ท1โˆ’distฮจ(๐‘ง),๐œ•๐ทโ„“๎€ธ2๎‚โˆ’๐‘ฃโ„“โˆ’๐‘ขโ„“๎‚€๎€ทcos1โˆ’distฮจ(๐‘ง),๐œ•๐ทโ„“๎€ธ2๎‚โˆ’๐‘ฃโ„“,(4.59) where ๐‘โ„“,๐‘Žโ„“โˆˆ(0,โˆž). It is very easy to see that if we denote with ๐’ฌโ„“=๐‘ข๎€ฝ๎€ทโ„“,๐‘ฃโ„“๎€ธ