Toeplitz Operators with Essentially Radial Symbols

Abstract

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 𝑇_𝜙 when 𝜙∈𝐿²(Ω) is essentially radial and Ω is multiply-connected domains.

1. 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(𝐿²(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 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∈𝐿²(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 {√𝑘+1𝑧^𝑘}^∞_𝑘=0. 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.

2. 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 𝛾₁ is the boundary of the unbounded component of ℂ⧵Ω. Let Ω₁ be the bounded component of ℂ⧵𝛾₁, and Ω_𝑗(𝑗=2,…,𝑛) the unbounded component of ℂ⧵𝛾_𝑗, respectively, so that Ω=⋂^𝑛_𝑗=1Ω_𝑗.

For 𝑑𝜈=(1/𝜋)𝑑𝑥𝑑𝑦 we consider the usual 𝐿²-space 𝐿²(Ω)=𝐿²(Ω,𝑑𝜈). The Bergman space 𝐿²_𝑎(Ω,𝑑𝜈), consisting of all holomorphic functions which are 𝐿²-integrable, is a closed subspace of 𝐿²(Ω,𝑑𝜈) with the inner product given by𝑓,𝑔=_Ω𝑓(𝑧)𝑔(𝑧)𝑑𝜈(𝑧), (2.1) for 𝑓,𝑔∈𝐿²(Ω,𝑑𝜈). The Bergman projection is the orthogonal projection𝑃∶𝐿²(Ω,𝑑𝜈)⟶𝐿²_𝑎(Ω,𝑑𝜈); (2.2) it is well-known that for any 𝑓∈𝐿²(Ω,𝑑𝜈) we have𝑃𝑓(𝑤)=_Ω𝑓(𝑧)𝐾^Ω(𝑧,𝑤)𝑑𝜈(𝑧), (2.3) where 𝐾^Ω is the Bergman reproducing kernel of Ω. For 𝜑∈𝐿^∞(Ω,𝑑𝜈) the Toeplitz operator 𝑇_𝜑∶𝐿²_𝑎(Ω,𝑑𝜈)→𝐿²_𝑎(Ω,𝑑𝜈) is defined by 𝑇_𝜑=𝑃𝑀_𝜑 where 𝑀_𝜑 is the standard multiplication operator. A simple calculation shows that𝑇_𝜑𝑓(𝑧)=_Ω𝜑(𝑤)𝑓(𝑤)𝐾^Ω(𝑤,𝑧)𝑑𝜈(𝑤). (2.4) 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, 𝑘^Γ(z,𝑤)=𝐾^Γ(𝑧,𝑤)/𝐾^Γ(𝑤,𝑤)^1/2.

For any 𝐴∈ℬ(𝐿²_𝑎(Γ,𝑑𝜈)), the space of bounded operators on 𝐿²_𝑎(Γ,𝑑𝜈), we define 𝐴, the Berezin transform of 𝐴, by𝐴(𝑤)=𝐴𝑘^Γ_𝑤,𝑘^Γ_𝑤=_Γ𝐴𝑘^Γ_𝑤(𝑧)𝑘^Γ_𝑤(𝑧)𝑑𝜈(𝑧), (2.5) where 𝑘^Γ_𝑤(⋅)=𝐾^Γ(⋅,𝑤)𝐾^Γ(𝑤,𝑤)^−1/2.

If 𝜑∈𝐿^∞(Γ), then we indicate with the symbol 𝜑 the Berezin transform of the associated Toeplitz operator 𝑇_𝜑, and we have𝜑(𝑤)=_Γ𝜑(𝑧)𝑘^Γ_𝑤(𝑧)²𝑑𝜈(𝑧). (2.6) We remind the reader that it is well known that 𝐴∈𝒞^∞_𝑏(Γ) and we have ‖𝐴‖_∞≤‖𝐴‖_{ℬ(𝐿2(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=1𝐷_𝑖 where 𝐷₁={𝑧∈ℂ∶|𝑧|<1} and 𝐷_𝑗={𝑧∈ℂ∶|𝑧−𝑎_𝑗|>𝑟_𝑗} for 𝑗=2,…,𝑛. Here 𝑎_𝑗∈𝐷₁ 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.

Definition 2.1. Let Ω=⋂^𝑛_𝑖=1Ω_𝑖 be a canonical bounded multiply-connected domain. One says that the set of 𝑛+1 functions 𝔓={𝑝₀,𝑝₁,…,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 𝜖_𝑗>0 such that 𝑈_𝜖𝑗={𝜁∈Ω∶𝑟_𝑗<|𝜁−𝑎_𝑗|<𝑟_𝑗+𝜖_𝑗} and the support of 𝑝_𝑗 are contained in 𝑊_𝑗 and 𝑝_𝑗(𝜁)=1∀𝜁∈𝑈_𝜖𝑗; (2.7)(3)for 𝑗=1 there exists an open set 𝑊₁⊂Ω and an 𝜖₁>0 such that 𝑈_𝜖1={𝜁∈Ω∶1−𝜖₁<|𝜁|<1} and the support of 𝑝₁ are contained in 𝑊₁ and 𝑝₁(𝜁)=1∀𝜁∈𝑈_𝜖1; (2.8)(4)for every 𝑗,𝑘=1,…,𝑛, 𝑊_𝑗∩𝑊_𝑘=∅, the set Ω⧵(⋃^𝑛_𝑗=1𝑊_𝑗) is not empty and the function 𝑝₀(𝜁)=1∀𝜁∈^𝑛_𝑗=1𝑊_𝑗^𝑐𝑝∩Ω,₀(𝜁)=0∀𝜁∈𝑈_𝜖𝑘,𝑘=1,…,𝑛, (2.9)(5)for any 𝜁∈Ω the following equation ^𝑛_𝑘=0𝑝_𝑘(𝜁)=1 (2.10)holds.

We also need the following.

Definition 2.2. A function 𝜑∶Ω=⋂^𝑛_𝑖=1Ω_𝑖→ℂ is said to be essentially radial if there exists a conformally equivalent canonical bounded domain 𝐷=⋂^𝑛_𝑖=1𝐷_𝑖 such that, if the map Θ∶Ω→𝐷 is the conformal mapping from Ω onto 𝐷, then(1)for every 𝑘=2,…,𝑛 and for some 𝜖_𝑘>0, one has 𝜑∘Θ⁻¹(𝑧)=𝜑∘Θ⁻¹𝑧−𝑎_𝑘 (2.11) when 𝑧∈𝑈_𝜖𝑘={𝜁∈Ω∶𝑟_𝑘<|𝜁−𝑎_𝑘|<𝑟_𝑘+𝜖_𝑘},(2)for 𝑘=1 and for some 𝜖₁>0, one has 𝜑∘Θ⁻¹(𝑧)=𝜑∘Θ⁻¹(|𝑧|) (2.12) when 𝑧∈𝑈_𝜖1={𝜁∈Ω∶1−𝜖₁<|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 𝜑 is essentially radial via Θ∶⋂^𝑛_ℓ=1Ω_ℓ→⋂^𝑛_ℓ=1𝐷_ℓ. 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_ℓ=1Ω_ℓ→⋂^𝑛_ℓ=1𝐷_ℓ, we always assume that the 𝜕-partition is given on ⋂^𝑛_ℓ=1𝐷_ℓ and transferred to ⋂^𝑛_ℓ=1Ω_ℓ through Θ in the natural way.

Definition 2.3. If 𝜑∈𝐿²(Ω) is an essentially radial function via Θ∶⋂^𝑛_𝑖=1Ω_𝑖→⋂^𝑛_𝑖=1𝐷_𝑖, 𝜑_𝑗=𝜑⋅𝑝_𝑗 for any 𝑗=1,…,𝑛 where 𝔓={𝑝₀,𝑝₁,…,𝑝_𝑛} is a 𝜕-partition for Ω then one defines the 𝑛 sequences 𝑎_𝜑1=𝑎_𝜑1(𝑘)_𝑘∈ℤ+,𝑎_𝜑2=𝑎_𝜑2(𝑘)_𝑘∈ℤ+,…,𝑎_𝜑𝑛=𝑎_𝜑𝑛(𝑘)_𝑘∈ℤ+ (2.13) as follows: if 𝑗=2,…,𝑛, 𝑎_𝜑𝑗(𝑘)=𝑟_𝑗^∞_𝑟𝑗𝜑_𝑗∘Θ⁻¹𝑟_𝑗𝑠+𝑎_𝑗()𝑟𝑘+1^𝑗2𝑘+1𝑠^2𝑘+11𝑠²𝑑𝑠∀𝑘∈ℤ₊, (2.14) and if 𝑗=1, 𝑎_𝜑1(𝑘)=¹₀𝜑₁∘Θ⁻¹(𝑠)(𝑘+1)𝑠^2𝑘+1𝑑𝑠∀𝑘∈ℤ₊. (2.15)

At this point we can state the main result.

Theorem 2.4. Let 𝜑∈𝐿²(Ω) be an essentially radial function via Θ∶⋂^𝑛_ℓ=1Ω_ℓ→⋂^𝑛_ℓ=1𝐷_ℓ and 𝜑_𝑗=𝜑⋅𝑝_𝑗 for any 𝑗=1,…,𝑛 where 𝔓={𝑝₀,𝑝₁,…,𝑝_𝑛} is a 𝜕-partition for Ω. If the operator 𝑇_𝜑∶𝐿²_𝑎(Ω,𝑑𝜈)→𝐿²_𝑎(Ω,𝑑𝜈) is bounded and if for any 𝑗=1,…,𝑛 the sequence 𝑎_𝜑𝑗={𝑎_𝜑𝑗(𝑘)}_𝑘∈ℤ+ satisfies the following sup_𝑘∈ℤ+(𝑘+1)𝑎_𝜑𝑗(𝑘)−𝑘𝑎_𝜑𝑗(𝑘−1)<∞, (2.16) then the operator 𝑇_𝜑∶𝐿²_𝑎(Ω,𝑑𝜈)→𝐿²_𝑎(Ω,𝑑𝜈) is compact if and only if lim_{𝑤→𝜕Ω}𝑇_𝜑(𝑤)=0. (2.17)

3. 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∏^𝑚𝑗_𝑘𝑇_{𝜑𝑖,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 multiply-connected planar domain [6]. In this section we try to characterize the compactness in terms 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, then𝑇_𝜑(𝑧)=1−|𝑧|²²(𝑛+1)𝑇_𝜑𝑒_𝑛,𝑒_𝑛|𝑧|^2𝑛, (3.1) where, by definition,𝑒_𝑛(𝑧)=√𝑛+1𝑧^𝑛∀𝑛∈ℤ₊. (3.2) 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_𝜑𝑒_𝑛,𝑒_𝑛⟩|𝑧|^2𝑛=0 implies lim_𝑛→∞⟨𝑇_𝜑𝑒_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 multiply-connected 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∶𝐿²(Δ)→𝐿²(Ω₁) such that 𝐼𝐿²_𝑎(Δ)=𝐿²_𝑎Ω₁. (3.3) Moreover, for any 𝑝≥2 one has that 𝐿^𝑝_𝑎(Δ)=𝐿^𝑝_𝑎(Ω₁), and, for any (𝑧,𝑤)∈Δ², the Bergman kernels 𝐾^Δ and 𝐾^Ω1 satisfy the following equation: 𝐾^Δ(𝑧,𝑤)=𝐾^Ω1(𝑧,𝑤). (3.4)

Proof. Suppose that 𝑓∈𝐿²_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 𝑓(𝑧)=^∞_𝑛=−∞𝑎_𝑛𝑧^𝑛. (3.5) This implies that |𝑓(𝑧)|²=∑^∞_{𝑛,𝑚=−∞}𝑎_𝑛𝑎_𝑚𝑧^𝑛𝑧^𝑚; therefore we have _Δ𝑓(𝑧)²𝑑𝜈(𝑧)=_Δ^∞_{𝑛,𝑚=−∞}𝑎_𝑛𝑎_𝑚𝑧^𝑛𝑧^𝑚(𝑧)=𝑑𝜈_{02𝜋10∞𝑛,𝑚=−∞}𝑎_𝑛𝑎_𝑚𝑟^𝑛+𝑚+1𝑒^{𝑖(𝑛−𝑚)𝜃}=𝑑𝑟𝑑𝜃_{∞𝑛,𝑚=−∞}𝑎_𝑛𝑎_𝑚0^2𝜋𝑒^{𝑖(𝑛−𝑚)𝜃}𝑑𝜃¹₀𝑟^𝑛+𝑚+1𝑑𝑟=2𝜋^∞_𝑛=−∞𝑎_𝑛²¹₀𝑟^2𝑛+1𝑑𝑟=2𝜋^∞_𝑛≠−1𝑎_𝑛²𝑟^2𝑛+22𝑛+2¹₀+𝑎₋₁²¹₀1𝑟𝑑𝑟. (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 Ω₁. We define ℐ∶𝐿²(Δ)⟶𝐿²Ω₁ (3.7) in this way: if 𝑔∈𝐿²(Δ), then ℐ𝑔(𝑧)=𝑔(𝑧) if 𝑧≠0 and ℐ𝑔(0)=_Δ𝑔(𝑧)𝑑𝜈(𝑧). (3.8) Then ℐ𝑔∈𝐿²(Ω₁) and ‖ℐ𝑔‖_Ω1=‖𝑔‖_Δ. If 𝑓∈𝐿²_𝑎(Δ), we have just shown that ℐ𝑓∈𝐿²_𝑎(Ω₁). Clearly ℐ is injective and surjective, in fact if 𝐺∈𝐿²(Ω₁), then 𝑔=𝐺_|Δ is an element of 𝐿²(Δ) and ℐ(𝑔)=𝐺. Then ℐ is an isomorphism of 𝐿²(Δ) onto 𝐿²(Ω₁) and ℐ(𝐿²_𝑎(Δ))=𝐿²_𝑎(Ω₁). 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(Ω₁).
Finally, it is easy to verify that for any 𝑓,𝑔∈𝐿²_𝑎(Δ) we have 𝑓,𝑔_Δ=ℐ𝑓,ℐ𝑔_Ω1, (3.9) and this fact implies, by the definition of the Bergman reproducing kernel, that 𝐾^Δ(𝑧,w)=𝐾^Ω1(𝑧,𝑤), (3.10) for any (𝑧,𝑤)∈Δ².

In order to better explain our intuition, we remind the reader that we proved that, if 𝜑∈𝐿²(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={𝑝₀,𝑝₁,…,𝑝_𝑛} is a 𝜕-partition for Ω, then the fact that the operator 𝑇_𝜑∶𝐿²_𝑎(Ω,𝑑𝜈)→𝐿²_𝑎(Ω,𝑑𝜈) is bounded (compact) is equivalent to fact, that for any 𝑗=1,…,𝑛, the operators 𝑇_𝜑𝑗∶𝐿²_𝑎(Ω_𝑗,𝑑𝜈)→𝐿²_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∶𝐿²_𝑎(𝐷,𝑑𝜈)→𝐿²_𝑎(𝐷,𝑑𝜈) is compact;(2)for any 𝑗=1,…,𝑛 one has lim_𝑘→∞𝑎_𝜑𝑗(𝑘)=0. (3.11)

Proof. Since 𝜑∈𝐿^∞(𝐷), we know that the operator 𝑇_𝜑∶𝐿²_𝑎(Ω,𝑑𝜈)→𝐿²_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_𝜑𝑗∶𝐿²(𝐷_𝑗,𝑑𝜈)→𝐿²_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_{𝑎𝑗,𝑟𝑗}={𝑧∈ℂ∶0<|𝑧−𝑎_𝑗|<𝑟_𝑗} and the maps Δ^𝛼_0,1−→Δ−→_{𝑎𝑗,𝑟𝑗}^𝛽𝐷_𝑗 where 𝛼(𝑧)=𝑎_𝑗+𝑟_𝑗𝑧 and 𝛽(𝑤)=(𝑤−𝑎_𝑗)⁻¹𝑟²_𝑗+𝑎_𝑗 and we use Proposition  1.1 in [8], we can claim that 𝑇_𝜑𝑗=𝑉⁻¹_𝛽∘𝛼𝑇_{𝜑𝑗∘𝛽∘𝛼}𝑉_𝛽∘𝛼 where 𝑉_𝛽∘𝛼∶𝐿²(Δ_0,1)→𝐿²(𝐷_𝑗) is an isomorphism of the Hilbert spaces. Therefore 𝑇_𝜑𝑗 is compact if and only if 𝑇_{𝜑𝑗∘𝛽∘𝛼} is compact. We also notice that the previous theorem implies that function {√𝑘+1𝑧^𝑘} is an orthonormal basis for 𝐿²(Δ_0,1), and this, in turn, implies that the compactness of the operator 𝑇_{𝜑𝑗∘𝛽∘𝛼} is equivalent to the fact that for the sequence 𝑎_𝜑𝑗={𝑎_𝜑𝑗(𝑘)}_𝑘∈ℕ we have lim_𝑘→∞𝑎_𝜑𝑗(𝑘)=0 where, by definition, 𝑎_𝜑𝑗(𝑘)=_Δ0,1𝜑_𝑗∘𝛽∘𝛼(𝑧)(𝑘+1)𝑧^𝑘𝑧^𝑘𝑑𝑧∀𝑚∈ℤ₊. (3.12) To complete the proof we observe that, since 𝜑_𝑗 is radial and 𝛽∘𝛼(𝑟)=𝑟⁻¹𝑟_𝑗+𝑎_𝑗, then, after a change of variable, we can rewrite the last integral, and hence the formula 𝑎_𝜑𝑗(𝑘)=𝑟_𝑗^∞_𝑟𝑗𝜑_𝑗𝑟_𝑗𝑠+𝑎_𝑗()𝑟𝑘+1^𝑗2𝑘+1𝑠^2𝑘+11𝑠²𝑑𝑠∀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_ℓ=1Ω_ℓ→⋂^𝑛_ℓ=1𝐷_ℓ, 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∶𝐿²_𝑎(Ω,𝑑𝜈)→𝐿²_𝑎(Ω,𝑑𝜈) is compact;(2)for any 𝑗=1,…,𝑛 one has lim_𝑘→∞𝑎_𝜑𝑗(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_Θ𝑓=Θ^′⋅𝑓∘Θ is an isometry from 𝐿²(𝐷) onto 𝐿²(Ω) (see [8, Proposition  1.1]). In particular we have 𝑉_Θ𝑃^𝐷=𝑃^Ω𝑉_Θ and this implies that 𝑉_Θ𝑇_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_{𝜑𝑗∘Θ−1}∶𝐿²_𝑎(𝐷_𝑗,𝑑𝜈)→𝐿²_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 lim_𝑘→∞𝑎_𝜑𝑗(𝑘)=0, (3.15) where, by definition, if 𝑗=2,…,𝑛, 𝑎_𝜑𝑗(𝑘)=𝑟_𝑗^∞_𝑟𝑗𝜑_𝑗∘Θ⁻¹𝑟_𝑗𝑠+𝑎_𝑗()𝑟𝑘+1^𝑗2𝑘+1𝑠^2𝑘+11𝑠²𝑑𝑠∀𝑘∈ℤ₊, (3.16) and if 𝑗=1, 𝑎_𝜑1(𝑘)=¹₀𝜑₁∘Θ⁻¹(𝑠)(𝑘+1)𝑠^2𝑘+1𝑑𝑠∀𝑘∈ℤ₊. (3.17)

Theorem 3.4. Let 𝜑∈𝐿²(𝐷) be an essentially radial function, if one defines 𝜑_𝑗=𝜑⋅𝑝_𝑗 where 𝑗=1,…,𝑛 and 𝔓 is a 𝜕-partition for Ω and the operator 𝑇_𝜑∶𝐿²_𝑎(𝐷,𝑑𝜈)→𝐿²_𝑎(𝐷,𝑑𝜈) is bounded (compact) and if for any 𝑗=1,…,𝑛 the sequences 𝑎_𝜑𝑗={𝑎_𝜑𝑗(𝑘)}_𝑘∈ℕ are such that sup_𝑘∈ℕ(𝑘+1)𝑎_𝜑𝑗(𝑘)−𝑘𝑎_𝜑𝑗(𝑘−1) (3.18) is finite, then the operator 𝑇_𝜑∶𝐿²_𝑎(𝐷,𝑑𝜈)→𝐿²_𝑎(𝐷,𝑑𝜈) is compact if and only if lim_{𝑤→𝜕𝐷}𝑇_𝜑(𝑤)=0. (3.19)

Proof. We know that the operator under examination is bounded (compact) if and only if for any 𝑗=1,…,𝑛 the operators 𝑇_𝜑𝑗∶𝐿²𝐷_𝑗,𝑑𝜈⟶𝐿²_𝑎𝐷_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_{𝑎𝑗,𝑟𝑗}={𝑧∈ℂ∶0<|𝑧−𝑎_𝑗|<𝑟_𝑗} and the following maps Δ^𝛼_0,1⟶Δ⟶_{𝑎𝑗,𝑟𝑗}^𝛽𝐷_𝑗, (3.21) where 𝛼(𝑧)=𝑎_𝑗+𝑟_𝑗𝑧 and 𝛽(𝑤)=(𝑤−𝑎_𝑗)⁻¹𝑟²_𝑗+𝑎_𝑗 and we use Proposition  1.1 in [8], we can claim that 𝑇_𝜑𝑗=𝑉⁻¹_𝛽∘𝛼𝑇_{𝜑𝑗∘𝛽∘𝛼}𝑉_𝛽∘𝛼, (3.22) where 𝑉_𝛽∘𝛼∶𝐿²(Δ_0,1)→𝐿²(𝐷_𝑗) is an isomorphism of Hilbert’s spaces. Therefore 𝑇_𝜑𝑗 is compact if and only if 𝑇_{𝜑𝑗∘𝛽∘𝛼} is compact. Since 𝑇_{𝜑𝑗∘𝛽∘𝛼}∶𝐿²_𝑎(Δ_0,1)→𝐿²_𝑎(Δ_0,1) and {√𝑘+1𝑧^𝑘}^∞_𝑘=0 is an orthonormal basis, a simple calculation shows that 𝑎_𝜑𝑗(𝑘)=𝑇_{𝜑𝑗∘𝛽∘𝛼}√𝑘+1𝑧^𝑘,√𝑘+1𝑧^𝑘; (3.23) therefore our assumption on 𝑎_𝜑𝑗={𝑎_𝜑𝑗(𝑘)}_𝑘∈ℕ that sup_𝑘∈ℤ+(𝑘+1)𝑎_𝜑𝑗(𝑘)−𝑘𝑎_𝜑𝑗(𝑘−1)<∞ (3.24) implies (see [9, Theorem  6]) that the compactness of 𝑇_{𝜑𝑗∘𝛽∘𝛼} is equivalent to the fact that the Berezin transform vanishes at the boundary. Since the case 𝑗=1 is immediate, we can conclude, from what we proved so far, that for any 𝑗=1,2,…,𝑛 we have that compactness is equivalent to the fact that lim_{𝑧→𝜕𝐷ℓ}𝜑_ℓ^𝐷ℓ(𝑧)=0. To complete the proof we set, for any 𝑗=1,…,𝑛, 𝒮_𝑗={𝑤∈Ω∣𝑝_𝑗(𝑤)=1} where {𝑝₁,…,𝑝_𝑛} is in the 𝜕-partition for Ω. By definition of 𝜕-partition, it follows that 𝒮_𝑗∩𝒮_𝑖=∅ is 𝑗≠𝑖, and we can write𝜑(𝑧)=𝑇_𝜑𝑘^𝐷_𝑧,𝑘^𝐷_𝑧=_𝐷𝜑(𝑤)𝑘^𝐷_𝑧(𝑤)²𝑑𝑤=_𝒮𝑗𝜑_ℓ(𝑤)𝑘^𝐷_𝑧(𝑤)²𝑑𝑤+_{Ω∩𝒮𝑐𝑗}𝜑_ℓ(𝑤)𝑘^𝐷_𝑧(𝑤)²𝑑𝑤. (3.25) Since for any ℓ=1,…,𝑛 the quantity ∫_𝒮ℓ𝜑_ℓ(𝑤)|𝑘^𝐷_𝑧(𝑤)|²𝑑𝑤 can be written as _𝒮ℓ𝜑_ℓ(𝑤)𝐾^𝐷ℓ_𝑧²𝐾^𝐷ℓ_𝑧²₂𝐾^𝐷ℓ_𝑧²₂𝐾^𝐷_𝑧²₂𝑑𝑤+_𝒮ℓ𝜑_ℓ(𝑤)𝑘^𝐷_𝑧(𝑤)²−𝐾^𝐷ℓ_𝑧²𝐾^𝐷_𝑧²₂𝑑𝑤, (3.26) we observe that the function 𝜃_ℓ(𝑧,𝑤)=(|𝑘^𝐷_𝑧(𝑤)|²−(𝐾^𝐷ℓ_𝑧)²/‖𝐾^𝐷_𝑧‖²₂) is bounded on the set 𝒮_ℓ and vanishes at the boundary.
In fact, to prove this we remind the reader that there exists an isomorphism 𝐼∶𝐿²(Δ_0,1)→𝐿²(Ω₁) such that 𝐼(𝐿²_𝑎(Δ_0,1))=𝐿²_𝑎(Ω₁) and the Bergman kernels 𝐾^Δ and 𝐾^Ω1 satisfy the following equation 𝐾^Δ(𝑧,𝑤)=𝐾^Ω1(𝑧,𝑤). If we define Δ_𝑎,𝑟={𝑧∈ℂ∶0<|𝑧−a|<𝑟} and 𝑂_𝑎,𝑟={𝑧∈ℂ∶|𝑧−𝑎|>𝑟}, then 𝐾^{𝑂𝑎,𝑟}(𝑧,𝑤)=𝑟²/(𝑟²−(𝑧−𝑎)⋅(𝑤−𝑎))² for all (𝑧,𝑤)∈𝑂_𝑎,𝑟×𝑂_𝑎,𝑟. The well-known fact that the reproducing kernel of the unit disk is given by (1−𝑧𝑤)⁻² implies that 𝐾^Δ0,1(𝑧,𝑤)=1/(1−𝑧⋅𝑤)²forall(𝑧,𝑤)∈Δ_0,1×Δ_0,1 therefore, by conformal mapping, that the reproducing kernel of Δ_𝑎,𝑟 is 𝐾^Δ𝑎,𝑟(𝑧,𝑤)=𝑟²/(𝑟²−(𝑧−𝑎)⋅(𝑤−𝑎))² for all (𝑧,𝑤)∈Δ_𝑎,𝑟×Δ_𝑎,𝑟. If we define 𝜙∶Δ_𝑎,𝑟→𝑂_𝑎,𝑟 by 𝜙(𝑧)=(𝑧−𝑎)⁻¹𝑟²+𝑎 and using the fact that 𝐾^{𝑂𝑎,𝑟}(𝜙(𝑧),𝜙(𝑤))𝜙^′(𝑧)𝜙^′(𝑤)=𝐾^Δ𝑎,𝑟(𝑧,𝑤), we obtain that 𝐾^{𝑂𝑎,𝑟}(𝑧,𝑤)=𝑟²/(𝑟²−(𝑧−𝑎)⋅(𝑤−𝑎))² for all (𝑧,𝑤)∈𝑂_𝑎,𝑟×𝑂_𝑎,𝑟. Since Ω₁=𝑂_0,1 and, for 𝑗=2,…,𝑛, 𝑂_{𝑎𝑗,𝑟𝑗}=Ω_𝑗, then we prove that 𝐾^𝐷𝑗(𝑧,𝑤)=𝑟²_𝑗/(𝑟²_𝑗−(𝑧−𝑎_𝑗)⋅(𝑤−𝑎_𝑗))² if 𝑗=2,…,𝑛.
Hence, for the function 𝜃_ℓ(𝑧,𝑤), we have 𝜃_ℓ(=∑𝜁,𝑧)^𝑛𝑚=0𝐾^𝐷_𝑚(𝜁,𝑧)𝐾^𝐷_𝑧²₂²−𝐾^𝐷ℓ_𝑧²𝐾^𝐷_𝑧²₂=𝐾^𝐷ℓ_𝑧²𝐾^𝐷_𝑧²₂1+^𝑛_𝑚≠ℓ𝐾^𝐷_𝑚(𝜁,𝑧)𝐾^𝐷ℓ_𝑧(𝜁,𝑧)²−𝐾^𝐷ℓ_𝑧²𝐾^𝐷_𝑧²₂=𝐾^𝐷ℓ_𝑧²𝐾^𝐷_𝑧²₂1+^𝑛_{𝑚≠ℓ𝑚>0}𝑟²_𝑚/𝑟²_𝑚−𝑧−𝑎_𝑚⋅𝜁−𝑎_𝑚²𝑟²_ℓ/𝑟²_ℓ−𝑧−𝑎_ℓ⋅𝜁−𝑎_ℓ²+𝐾^𝐷₀(𝜁,𝑧)𝐾^𝐷ℓ_𝑧(𝜁,𝑧)²−𝐾^𝐷ℓ_𝑧²𝐾^𝐷_𝑧²₂. (3.27) Hence it follows that 𝜃_ℓ(𝑧,𝑤) is bounded on the set 𝒮_ℓ and goes to zero as it approaches the boundary. Since 𝜑∈𝐿²(𝐷) and 𝐷 has a finite measure, we can conclude that, by the dominated convergence theorem, we have lim_{𝑧→𝜕𝐷ℓ𝒮ℓ}𝜑_ℓ(𝑤)𝑘^𝐷_𝑧(𝑤)²−𝐾^𝐷ℓ_𝑧²𝐾^𝐷_𝑧²₂𝑑𝑤=0. (3.28) Now we observe that Lemma  2 in [4] implies that lim_{𝑧→𝜕𝐷ℓ𝐷∩𝒮ℓ}𝜑_ℓ(𝑤)𝐾^𝐷ℓ_𝑧𝐾^𝐷ℓ_𝑧2²𝐾^𝐷ℓ_𝑧2𝐾^𝐷_𝑧2²𝑑𝑤 (3.29) goes to zero if and only lim_{𝑧→𝜕𝐷ℓ}𝜑_ℓ^𝐷ℓ(𝑧)=0, and a simple calculation shows that _{Ω∩𝒮𝑐𝑗}𝜑_ℓ(𝑤)𝑘^𝐷_𝑧(𝑤)²𝑑𝑤=0. (3.30) Hence, as a consequence, we have shown that, if the conditions in the hypothesis hold, then lim_{𝑧→𝜕𝐷ℓ}𝜑^𝐷(𝑧)=0, (3.31) and this completes the proof since 𝜕𝐷=⋃^𝑛₁𝜕𝐷_ℓ.

Now we can prove the following.

Theorem 3.5. Let 𝜑∈𝐿²(Ω) be an essentially radial function via Θ∶⋂^𝑛_ℓ=1Ω_ℓ→⋂^𝑛_ℓ=1𝐷_ℓ and 𝜑_𝑗=𝜑⋅𝑝_𝑗 for any 𝑗=1,…,𝑛 where 𝔓={𝑝₀,𝑝₁,…,𝑝_𝑛} is a 𝜕-partition for Ω. If the operator 𝑇_𝜑∶𝐿²_𝑎(Ω,𝑑𝜈)→𝐿²_𝑎(Ω,𝑑𝜈) is bounded and if for any 𝑗=1,…,𝑛 the sequence 𝑎_𝜑𝑗={𝑎_𝜑𝑗(𝑘)}_𝑘∈ℤ+ satisfies the following sup_𝑘∈ℤ+(𝑘+1)𝑎_𝜑𝑗(𝑘)−𝑘𝑎_𝜑𝑗(𝑘−1)<∞, (3.32) then the operator 𝑇_𝜑∶𝐿²_𝑎(Ω,𝑑𝜈)→𝐿²_𝑎(Ω,𝑑𝜈) is compact if and only if lim_{𝑤→𝜕Ω}𝑇_𝜑(𝑤)=0. (3.33)

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_Θ𝑓=Θ^′⋅𝑓∘Θ is an isometry from 𝐿²(𝐷) onto 𝐿²(Ω) (see [8, Proposition  1.1]). In particular we have 𝑉_Θ𝑃^𝐷=𝑃^Ω𝑉_Θ and this implies that 𝑉_Θ𝑇_𝜑=𝑇_{𝜑∘Θ−1}𝑉_Θ. Therefore the operator 𝑇_𝜑 is bounded (compact) if and only if the operator 𝑇_{𝜑∘Θ−1}∶𝐿²(𝐷,𝑑𝜈)→𝐿²_𝑎(𝐷,𝑑𝜈) 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 𝑇_{𝜑𝑗∘Θ−1}∶𝐿²_𝑎(𝐷_𝑗,𝑑𝜈)→𝐿²_𝑎(𝐷_𝑗,𝑑𝜈) are bounded (compact). Hence, since the sequences 𝑎_𝜑𝑗={𝑎_𝜑𝑗(𝑚)}_𝑚∈ℕ satisfy the stated properties, we can conclude that the operators 𝑇_{𝜑𝑗∘Θ−1}∶𝐿²_𝑎(𝐷_𝑗,𝑑𝜈)→𝐿²_𝑎(𝐷_𝑗,𝑑𝜈) are compact if and only if for any 𝑗=1,…,𝑛 we have lim_{𝑧→𝜕𝐷𝑗}𝜑_𝑗∘Θ^𝐷−1𝑗(𝑧)=0. (3.34) Therefore it follows that lim_{𝑧→𝜕𝐷}𝜑∘Θ^𝐷−1(𝑧)=0, (3.35) and, since Θ is a conformal mapping, this implies that lim_{𝑧→𝜕Ω}𝜑^Ω(𝑧)=0. (3.36)

Finally, we also observe that as a simple consequence we obtain the following.

Theorem 3.6. Let 𝜑∈𝐿²(Ω) be an essentially radial function via Θ∶⋂^𝑛_ℓ=1Ω_ℓ→⋂^𝑛_ℓ=1𝐷_ℓ and 𝜑_𝑗=𝜑⋅𝑝_𝑗 for any 𝑗=1,…,𝑛 where 𝔓={𝑝₀,𝑝₁,…,𝑝_𝑛} is a 𝜕-partition for Ω. If the operator 𝑇_𝜑∶𝐿²_𝑎(Ω,𝑑𝜈)→𝐿²_𝑎(Ω,𝑑𝜈) is bounded and if for any 𝑗=1,…,𝑛 the sequence 𝑎_𝜑𝑗={𝑎_𝜑𝑗(𝑘)}_𝑘∈ℤ+ satisfies the following sup_𝑘∈ℤ+𝑘𝑎_𝜑𝑗(𝑘)−𝑎_𝜑𝑗(𝑘−1)<∞, (3.37) then the operator 𝑇_𝜑∶𝐿²_𝑎(Ω,𝑑𝜈)→𝐿²_𝑎(Ω,𝑑𝜈) is compact if and only if lim_{𝑤→𝜕Ω}𝑇_𝜑(𝑤)=0. (3.38)

Finally, we observe that it is also to recover as corollary the following.

Corollary 3.7. Let 𝜑∈𝐿²(Ω) be an essentially radial symbol via the conformal equivalence Θ∶Ω→𝐷. If one defines 𝜑_𝑗=𝜑⋅𝑝_𝑗 where 𝑗=1,…,𝑛 and 𝔓 is a 𝜕-partition for Ω. Let us assume that 𝛾_𝜙𝑗={𝛾_𝜙𝑗(𝑚)}_𝑚∈ℕ is in ℓ_∞(ℤ₊) and that there is a constant 𝐶₃ such that for 𝑗=2,…,𝑛sup_{𝜏∈[𝑎𝑗+𝑟𝑗,∞)}𝜑_𝑗∘Θ(𝜏)−𝜏−𝑎_𝑗𝜏−𝑟_𝑗−𝑎_𝑗^𝜏_{𝑎𝑗+𝑟𝑗}𝜑_𝑗𝑦𝑟∘Θ^𝑗𝑦−𝑎_𝑗²𝑑𝑦<𝒞₃ (3.39) and for 𝑗=1sup_𝜏∈[0,1]𝜑₁(𝜏)−1∘Θ1−𝜏¹_𝜏𝜑₁∘Θ(𝑠)𝑑𝑠<𝒞₃. (3.40) Then the operator 𝑇_𝜑∶𝐿²_𝑎(Ω,𝑑𝜈)→𝐿²_𝑎(Ω,𝑑𝜈) is compact if and only if lim_{𝑤→𝜕Ω}𝑇_𝜑(𝑤)=0. (3.41)

The last corollary was also proved, in different way, in [4].