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Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 210596, 15 pages
http://dx.doi.org/10.1155/2011/210596
Research Article

Toeplitz Operators on the Weighted Pluriharmonic Bergman Space with Radial Symbols

1School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028000, China

Received 3 June 2011; Revised 20 July 2011; Accepted 21 July 2011

Academic Editor: NatigΒ Atakishiyev

Copyright Β© 2011 Zhi Ling Sun and Yu Feng Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We construct an operator 𝑅 whose restriction onto weighted pluriharmonic Bergman Space 𝑏2πœ‡(𝔹𝑛) is an isometric isomorphism between 𝑏2πœ‡(𝔹𝑛) and 𝑙#2. Furthermore, using the operator 𝑅 we prove that each Toeplitz operator π‘‡π‘Ž with radial symbols is unitary to the multication operator 𝛾a,πœ‡I. Meanwhile, the Wick function of a Toeplitz operator with radial symbol gives complete information about the operator, providing its spectral decomposition.

1. Introduction

Let 𝔹𝑛 be the open unit ball in the complex vector space ℂ𝑛. For any 𝑧=(𝑧1,…,𝑧𝑛) and πœ‰=(πœ‰1,…,πœ‰π‘›) in ℂ𝑛, let βˆ‘π‘§β‹…πœ‰=𝑛𝑗=1π‘§π‘—πœ‰π‘—, where πœ‰π‘— is the complex conjugate of πœ‰π‘— and √|𝑧|=𝑧⋅𝑧. For a multi-index 𝛼=(𝛼1,…,𝛼𝑛) and 𝑧=(𝑧1,…,𝑧𝑛)βˆˆβ„‚π‘›, we write 𝑧𝛼=𝑧𝛼11⋯𝑧𝛼𝑛𝑛, where π›Όπ‘˜βˆˆπ‘+=β„•βˆͺ{0}, and |𝛼|=𝛼1+β‹―+𝛼𝑛 is its length, 𝛼!=𝛼1!⋯𝛼𝑛!.

The weighted pluriharmonic Bergman space 𝑏2πœ‡(𝔹𝑛) is the subspace of the weighted space 𝐿2πœ‡(𝔹𝑛) consisting of all pluriharmonic functions on 𝔹𝑛. A pluriharmonic function in the unit ball is the sum of a holomorphic function and the conjugate of a holomorphic functions. It is known that 𝑏2πœ‡(𝔹𝑛) is a closed subspace of 𝐿2πœ‡(𝔹𝑛) and hence is a Hilbert space. Let π‘„πœ‡π”Ήπ‘› be the Hilbert space orthogonal projection from 𝐿2πœ‡(𝔹𝑛) onto 𝑏2πœ‡(𝔹𝑛). For a function π‘’βˆˆπΏ2πœ‡(𝔹𝑛), the Toeplitz operator π‘‡π‘’βˆΆπ‘2πœ‡(𝔹𝑛)→𝑏2πœ‡(𝔹𝑛) with symbol 𝑒 is the linear operator defined by 𝑇𝑒𝑓=π‘„πœ‡π”Ήπ‘›(𝑒𝑓),π‘“βˆˆπ‘2πœ‡(𝔹𝑛).(1.1)

𝑇𝑒 is densely defined and not bounded in general.

The boundedness and compactness of Toeplitz operators on Bergman type spaces have been studied intensively in recent years. The fact that the product of two harmonic functions is no longer harmonic adds some mystery in the study of Toeplitz operators on harmonic Bergman space. Many methods which work for the operator on analytic Bergman spaces lost their effectiveness on harmonic Bergman space. Therefore new ideas and methods are needed. We refer to [1–3] for references about the results of Toeplitz operator on harmonic Bergman space. The paper [3] characterizes compact Toeplitz operators in the case of the unit disk 𝔻. In [2], the authors consider Toeplitz operators acting on the pluriharmonic Bergman space and study the problem of when the commutator or semicommutator of certain Toeplitz operators is zero. Lee [1] proved that two Toplitz operators acting on the pluriharmonic Bergman space with radial symbols and pluriharmonic symbol, respectively, commute only in an obvious case.

The authors in [4] analyze the influence of the radial component of a symbol to spectral, compactness and Fredholm properties of Toeplitz operators on Bergman space on unit disk 𝔻. In [5], they are devoted to study Toeplitz operators with radial symbols on the weighted Bergman spaces on the unit ball in ℂ𝑛.

In this paper, we will be concerned with the question of Toeplitz operators with radial symbols on the weighted pluriharmonic Bergman space. Based on the techniques in [4–6], we construct an operator 𝑅 whose restriction onto weighted pluriharmonic Bergman space 𝑏2πœ‡(𝔹𝑛) is an isometric isomorphism between 𝑏2πœ‡(𝔹𝑛) and 𝑙#2, and π‘…π‘…βˆ—=πΌβˆΆπ‘™#2βŸΆπ‘™#2,π‘…βˆ—π‘…=π‘„πœ‡π”Ήπ‘›βˆΆπΏ2πœ‡(𝔹𝑛)βŸΆπ‘2πœ‡(𝔹𝑛);(1.2) where 𝑙#2 is the subspace of 𝑙2. Using the operator 𝑅 we prove that each Toeplitz operator π‘‡π‘Ž with radial symbols is unitary to the multication operator π›Ύπ‘Ž,πœ‡πΌ acting on 𝑙#2. Next, we use the Berezin concept of Wick and anti-Wick symbols. It turns out that in our particular (radial symbols) case the Wick symbols of a Toeplitz operator give complete information about the operator, providing its spectral decomposition.

2. Pluriharmonic Bergman Space and Orthogonal Projection

We start this section with a decomposition of the space 𝐿2πœ‡(𝔹𝑛). Consider a nonnegative measurable function πœ‡(π‘Ÿ),π‘Ÿβˆˆ(0,1), such that mes{π‘Ÿβˆˆ(0,1)βˆΆπœ‡(π‘Ÿ)>0}=1, and ξ€œπ”Ήπ‘›||π‘†πœ‡(|𝑧|)𝑑𝑣(𝑧)=2π‘›βˆ’1||ξ€œ10πœ‡(π‘Ÿ)π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿ<∞,(2.1) where |𝑆2π‘›βˆ’1|=2πœ‹π‘›βˆ’(1/2)Ξ“βˆ’1(π‘›βˆ’(1/2)) is the surface area of unit sphere 𝑆2π‘›βˆ’1 and Ξ“(𝑧) is the Gamma function.

Introduce the weighted space 𝐿2πœ‡(𝔹𝑛)=π‘“βˆΆβ€–π‘“β€–2𝐿2πœ‡(𝔹𝑛)=ξ€œπ”Ήπ‘›||||𝑓(𝑧)2ξ‚Όπœ‡(|𝑧|)𝑑𝑣(𝑧)<∞,(2.2) where 𝑑𝑣(𝑧) is the usual Lebesgue volume measure and 𝐿2(𝑆2π‘›βˆ’1) is the space with the usual Lebesgue surface measure.

The space 𝐿2(𝑆2π‘›βˆ’1) is the direct sum of mutually orthogonal spaces β„‹π‘˜, that is, 𝐿2𝑆2π‘›βˆ’1ξ€Έ=βˆžξΆπ‘˜=0β„‹π‘˜,(2.3) where β„‹π‘˜ denotes the space of spherical harmonics of order π‘˜. Meanwhile, each space β„‹π‘˜ is the direct sum (under the identification ℂ𝑛=ℝ2𝑛) of the mutually orthogonal spaces ℋ𝑝,π‘ž (see, e.g., [7]): β„‹π‘˜=𝑝+π‘ž=π‘˜π‘,π‘žβˆˆβ„€+𝐻𝑝,π‘ž,π‘˜βˆˆβ„€+,(2.4) where 𝐻𝑝,π‘ž, for each 𝑝,π‘ž=0,1,…, is the space of harmonic polynomials (their restrictions to the unit sphere) of complete order 𝑝 in the variable 𝑧 and complete order π‘ž in the conjugate variable 𝑧=(𝑧1,…,𝑧𝑛). Thus, we can get 𝐿2𝑆2π‘›βˆ’1ξ€Έ=𝑝,π‘žβˆˆβ„€+𝐻𝑝,π‘ž.(2.5)

The Hardy space 𝐻2(𝔹𝑛) in the unit ball 𝔹𝑛 is a closed subspace of 𝐿2(𝑆2π‘›βˆ’1). Denote by 𝑃𝑆2π‘›βˆ’1 the SzegΓΆ orthogonal projection of 𝐿2(𝑆2π‘›βˆ’1) onto the Hardy space 𝐻2(𝔹𝑛). It is well known that 𝐻2(𝔹𝑛)=βˆžβ¨π‘=0𝐻𝑝,0. The standard orthonormal base in 𝐻2(𝔹𝑛) has the form 𝑒𝛼(πœ”)=𝑑𝑛,π›Όπœ”π›Ό,𝑑𝑛,𝛼=ξƒŽ(π‘›βˆ’1+|𝛼|)!||𝑆2π‘›βˆ’1||(π‘›βˆ’1)!𝛼!for|𝛼|=0,1,….(2.6)

Fix an orthonormal basis {𝑒𝛼,𝛽(πœ”)}𝛼,𝛽, 𝛼,π›½βˆˆβ„€π‘›+, in the space 𝐿2(𝑆2π‘›βˆ’1) so that 𝑒𝛼,0(πœ”)≑𝑒𝛼(πœ”), 𝑒0,𝛼(πœ”)≑𝑒𝛼,0(πœ”)≑𝑒𝛼(𝑀), |𝛼|=0,1,….

Passing to the spherical coordinates in the unit ball we have 𝐿2πœ‡(𝔹𝑛)=𝐿2ξ€·(0,1),πœ‡(π‘Ÿ)π‘Ÿ2π‘›βˆ’1ξ€Έπ‘‘π‘ŸβŠ—πΏ2𝑆2π‘›βˆ’1ξ€Έ.(2.7) For any function 𝑓(𝑧)∈𝐿2πœ‡(𝔹𝑛) have the decomposition 𝑓(𝑧)=βˆžξ“||𝛽|||𝛼|+=0𝑐𝛼,𝛽(π‘Ÿ)𝑒𝛼,𝛽(π‘§πœ”),π‘Ÿ=|𝑧|,πœ”=π‘Ÿ,(2.8) with the coefficients 𝑐𝛼,𝛽(π‘Ÿ) satisfying the condition ‖𝑓‖2𝐿2πœ‡(𝔹𝑛)=βˆžξ“||𝛽|||𝛼|+=0ξ€œ10||𝑐𝛼,𝛽(||π‘Ÿ)2πœ‡(π‘Ÿ)π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿ<∞.(2.9)

According to the decomposition (2.7), (2.8) together with Parseval’s equality, we can define the unitary operator π‘ˆ1∢𝐿2ξ€·(0,1),πœ‡(π‘Ÿ)π‘Ÿ2π‘›βˆ’1ξ€Έπ‘‘π‘ŸβŠ—πΏ2𝑆2π‘›βˆ’1ξ€ΈβŸΆπΏ2ξ€·(0,1),πœ‡(π‘Ÿ)π‘Ÿ2π‘›βˆ’1ξ€Έπ‘‘π‘ŸβŠ—π‘™2≑𝑙2𝐿2ξ€·(0,1),πœ‡(π‘Ÿ)π‘Ÿ2π‘›βˆ’1,π‘‘π‘Ÿξ€Έξ€Έ(2.10)

by the rule π‘ˆ1βˆΆπ‘“(𝑧)β†’{𝑐𝛼,𝛽(π‘Ÿ)}, and ‖𝑓‖2𝐿2πœ‡(𝔹𝑛)=‖‖𝑐𝛼,𝛽‖‖(π‘Ÿ)2𝑙2𝐿2ξ€·(0,1),πœ‡(π‘Ÿ)π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿξ€Έξ€Έ=βˆžξ“|𝛼|+|𝛽|=0‖‖𝑐𝛼,𝛽‖‖(π‘Ÿ)2𝐿2ξ€·(0,1),πœ‡(π‘Ÿ)π‘Ÿ2π‘›βˆ’1ξ€Έπ‘‘π‘Ÿ.(2.11)

Let 𝑓(𝑧) be a pluriharmonic in the unit ball 𝔹𝑛 and write 𝑓=𝑔+β„Ž, where the functions 𝑔, β„Ž are holomorphic in 𝔹𝑛. Suppose 𝑔(𝑧)=βˆžξ“|𝛼|=0𝑐𝛼z𝛼,β„Ž(𝑧)=βˆžξ“|𝛽|=0𝑐𝛽𝑧𝛽(2.12)

are their power series representations of 𝑔 and β„Ž, respectively. We have 𝑓(𝑧)=βˆžξ“|𝛼|=0𝑐𝛼𝑧𝛼+βˆžξ“|𝛽|=0𝑐𝛽𝑧𝛽=βˆžξ“|𝛼|=0𝑐𝛼(π‘Ÿ)𝑒𝛼(πœ”)+βˆžξ“|𝛽|=0𝑐𝛽(π‘Ÿ)𝑒𝛽(πœ”),(2.13) where 𝑐𝛼(π‘Ÿ)=π‘π›Όπ‘‘βˆ’1𝑛,π›Όπ‘Ÿ|𝛼|, 𝑐𝛽(π‘Ÿ)=π‘π›½π‘‘βˆ’1𝑛,π›½π‘Ÿ|𝛽|, π‘Ÿ=|𝑧|, πœ”=(𝑧/π‘Ÿ).

Let 𝑏2πœ‡(𝔹𝑛) be the pluriharmonic Bergman space in 𝔹𝑛 from 𝐿2πœ‡(𝔹𝑛). Denote by π‘„πœ‡π”Ήπ‘› the pluriharmonic Bergman orthogonal projection of 𝐿2πœ‡(𝔹𝑛) onto the Bergman space 𝑏2πœ‡(𝔹𝑛). From the above it follows that to characterize a function 𝑓(𝑧)βˆˆπ‘2πœ‡(𝔹𝑛) and considering its decomposition according to (2.13), one can restrict to the function having the representation 𝑓(𝑧)=𝑔(𝑧)+β„Ž(𝑧)=βˆžξ“|𝛼|=0𝑐𝛼,0(π‘Ÿ)𝑒𝛼,0(πœ”)+βˆžξ“|𝛽|=0𝑐0,𝛽(π‘Ÿ)𝑒0,𝛽(πœ”).(2.14) Now let us take an arbitrary 𝑓(𝑧) from 𝑏2πœ‡(𝔹𝑛) in the form (2.14). It will satisfy the Cauchy-Riemann equations, that is, πœ•πœ•π‘§π‘˜1𝑔(𝑧)≑2ξ‚΅πœ•πœ•π‘₯π‘˜πœ•+π‘–πœ•π‘¦π‘˜ξ‚Άπ‘”(𝑧)=0,π‘˜=1,…,𝑛,π‘§βˆˆπ”Ήπ‘›,πœ•πœ•π‘§π‘˜1β„Ž(𝑧)≑2ξ‚΅πœ•πœ•π‘₯π‘˜πœ•βˆ’π‘–πœ•π‘¦π‘˜ξ‚Άβ„Ž(𝑧)=0,π‘˜=1,…,𝑛,π‘§βˆˆπ”Ήπ‘›.(2.15)

Applying πœ•/πœ•π‘§π‘˜, πœ•/πœ•π‘§π‘˜ to 𝑔 and β„Ž, respectively, we haveπœ•πœ•π‘§π‘˜βˆžξ“|𝛼|=0𝑐𝛼,0(π‘Ÿ)𝑒𝛼,0(π‘§πœ”)=π‘˜2π‘Ÿβˆžξ“|𝛼|=0ξ‚΅π‘‘π‘π‘‘π‘Ÿπ›Ό,0(π‘Ÿ)βˆ’|𝛼|π‘Ÿπ‘π›Ό,0(ξ‚Άπ‘’π‘Ÿ)𝛼,0(πœ•πœ”),πœ•π‘§π‘˜βˆžξ“||𝛽||=0𝑐0,𝛽(π‘Ÿ)𝑒0,𝛽(πœ”)=π‘§π‘˜2π‘Ÿβˆžξ“||𝛽||=0ξ‚΅π‘‘π‘π‘‘π‘Ÿ0,𝛽||𝛽||(π‘Ÿ)βˆ’π‘Ÿπ‘0,𝛽𝑒(π‘Ÿ)0,𝛽(πœ”),(2.16)

where π‘˜=0,…,𝑛, and we come to the infinite system of ordinary linear differential equations π‘‘π‘π‘‘π‘Ÿπ›Ό,0(π‘Ÿ)βˆ’|𝛼|π‘Ÿπ‘π›Ό,0𝑑(π‘Ÿ)=0,|𝛼|=0,1,β€¦π‘π‘‘π‘Ÿ0,𝛽(||𝛽||π‘Ÿ)βˆ’π‘Ÿπ‘0,𝛽(||𝛽||π‘Ÿ)=0,=0,1,….(2.17)

Their general solution has the form 𝑐𝛼,0=π‘π›Όπ‘Ÿ|𝛼|=πœ†(𝑛,|𝛼|)𝑐𝛼,0π‘Ÿ|𝛼|, 𝑐0,𝛽=π‘π›½π‘Ÿ|𝛽|=πœ†(𝑛,|𝛽|)𝑐0,π›½π‘Ÿ|𝛽|, with βˆ«πœ†(𝑛,π‘š)=(10𝑑2π‘š+2π‘›βˆ’1πœ‡(𝑑)𝑑𝑑)βˆ’1/2. Hence, for any 𝑓(𝑧)βˆˆπ‘2πœ‡(𝔹𝑛) we have 𝑓(𝑧)=βˆžξ“|𝛼|=0𝑐𝛼,0πœ†(𝑛,|𝛼|)π‘Ÿ|𝛼|𝑒𝛼,0+βˆžξ“||𝛽||=0𝑐0,π›½πœ†ξ€·||𝛽||ξ€Έπ‘Ÿπ‘›,|𝛽|𝑒0,𝛽.(2.18)

And, it is easy to verify ‖𝑓‖2𝐿2πœ‡(𝔹𝑛)=βˆ‘βˆž|𝛼|=0|𝑐𝛼,0|2+βˆ‘βˆž|𝛽|=0|𝑐0,𝛽|2. Thus the image 𝑏21,πœ‡(𝔹𝑛)=π‘ˆ1(𝑏2πœ‡(𝔹𝑛)) is characterized as the closed subspace of 𝐿2ξ€·(0,1),πœ‡(π‘Ÿ)π‘Ÿ2π‘›βˆ’1ξ€Έπ‘‘π‘ŸβŠ—π‘™2=𝑙2𝐿2ξ€·(0,1),πœ‡(π‘Ÿ)π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿξ€Έξ€Έ(2.19) which consists of all sequences {𝑐𝛼,𝛽(π‘Ÿ)} of the form 𝑐𝛼,π›½βŽ§βŽͺ⎨βŽͺ⎩(π‘Ÿ)=πœ†(𝑛,|𝛼|)𝑐𝛼,0π‘Ÿ|𝛼|,||𝛽||πœ†ξ€·||𝛽||𝑐=0𝑛,0,π›½π‘Ÿ|𝛽|||𝛽||,|𝛼|=00,|𝛼|β‰ 0,β‰ 0.(2.20)

For each π‘šβˆˆβ„€+ introduce the function πœ‘π‘š(𝜌)=πœ†(𝑛,π‘š)1/π‘›ξ‚΅ξ€œπœŒ0π‘Ÿ2π‘š+2π‘›βˆ’1ξ‚Άπœ‡(π‘Ÿ)π‘‘π‘Ÿ1/2𝑛[],𝜌∈0,1.(2.21) Obviously, there exists the inverse function for the function πœ‘π‘š(𝜌) on [0,1], which we will denote by πœ™π‘š(π‘Ÿ). Introduce the operator ξ€·π‘’π‘šπ‘“ξ€Έβˆš(π‘Ÿ)=2π‘›πœ†πœ™(𝑛,π‘š)π‘šβˆ’π‘šξ€·πœ™(π‘Ÿ)π‘“π‘šξ€Έ(π‘Ÿ).(2.22)

By Proposition 2.1 in [5], the operator π‘’π‘š maps unitary 𝐿2((0,1),πœ‡(π‘Ÿ)π‘Ÿ2π‘›βˆ’1) onto 𝐿2((0,1),π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿ) in such a way that π‘’π‘š(πœ†(𝑛,π‘š)π‘Ÿπ‘šβˆš)=2𝑛,π‘šβˆˆβ„€+.(2.23)

Intoduce the unitary operator π‘ˆ2βˆΆπ‘™2𝐿2ξ€·(0,1),πœ‡(π‘Ÿ)π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿξ€Έξ€ΈβŸΆπ‘™2𝐿2ξ€·(0,1),π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿξ€Έξ€Έ,(2.24) where π‘ˆ2βˆΆξ€½π‘π›Ό,π›½ξ€ΎβŸΆπ‘’(π‘Ÿ)ξ€½ξ€·|𝛼|+|𝛽|𝑐𝛼,𝛽(π‘Ÿ).(2.25)

By (2.23), we can get the space 𝑏22,πœ‡=π‘ˆ2(𝑏21,πœ‡) coincides with the space of all sequences {π‘˜π›Ό,𝛽} for whichπ‘˜π›Ό,𝛽=⎧βŽͺ⎨βŽͺ⎩√2𝑛𝑐𝛼,0,||𝛽||√=02𝑛𝑐0,𝛽||𝛽||,|𝛼|=00,|𝛼|β‰ 0,β‰ 0.(2.26)

Let 𝑙0√(π‘Ÿ)=2𝑛. We have 𝑙0(π‘Ÿ)∈𝐿2((0,1),π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿ) and ‖𝑙0‖𝐿2((0,1),π‘Ÿ2π‘›βˆ’1)=1. Denote by 𝐿0 the one-dimensional subspace of 𝐿2((0,1),π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿ) generated by 𝑙0(π‘Ÿ). The orthogonal projection 𝑃0 of 𝐿2((0,1),π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿ) onto 𝐿0 has the form 𝑃0𝑓(π‘Ÿ)=βŸ¨π‘“,𝑙0βŸ©π‘™0=βˆšξ€œ2𝑛10βˆšπ‘“(𝜌)2π‘›πœŒ2π‘›βˆ’1π‘‘πœŒ.(2.27) Let 𝑑𝛼,𝛽=π‘˜π›Ό,𝛽(√2𝑛)βˆ’1. Denote by 𝑙#2 the subspace of 𝑙2 consisting of all sequences {𝑑𝛼,𝛽}. And let 𝑝# be the orthogonal projections of 𝑙2 onto 𝑙#2, then 𝑝#=πœ’+(𝛼,𝛽)𝐼, where πœ’+(𝛼,𝛽)=0, if |𝛼‖𝛽|>0 and πœ’+(𝛼,𝛽)=1, if |𝛼‖𝛽|=0.

Observe that 𝑏22,πœ‡=𝐿0βŠ—π‘™#2 and the orthogonal projection 𝐡2 of 𝑙2𝐿2ξ€·(0,1),π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿξ€Έξ€Έβ‰‘πΏ2ξ€·(0,1)π‘Ÿ2π‘›βˆ’1ξ€Έπ‘‘π‘ŸβŠ—π‘™2(2.28)

onto 𝑏22,πœ‡ has the form 𝐡2=𝑃0βŠ—π‘#. This leads to the following theorem.

Theorem 2.1. The unitary operator π‘ˆ=π‘ˆ1π‘ˆ2 gives an isometric isomorphism of the space 𝐿2πœ‡(𝔹𝑛) onto 𝑙2(𝐿2((0,1),π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿ))≑𝐿2((0,1),π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿ)βŠ—π‘™2 such that
(1) the pluriharmonic Bergman space 𝑏2πœ‡(𝔹𝑛) is mapped onto 𝐿0βŠ—π‘™#2, π‘ˆβˆΆπ‘2πœ‡(𝔹𝑛)⟢𝐿0βŠ—π‘™#2,(2.29) where 𝐿0 is the one-dimensional subspace of 𝐿2((0,1),π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿ), generated by the function 𝑙0√(π‘Ÿ)=2𝑛;
(2) the pluriharmonic Bergman projection π‘„πœ‡π”Ήπ‘› is unitary equivalent to π‘ˆπ‘„πœ‡π”Ήπ‘›π‘ˆβˆ’1=𝑃0βŠ—π‘#,(2.30) where 𝑃0 is the one-dimensional projection (2.27) of 𝐿2((0,1),π‘Ÿ2π‘›βˆ’1π‘‘π‘Ÿ) onto 𝐿0.

Introduce the operator 𝑅0βˆΆπ‘™#2⟢𝐿2ξ€·(0,1),π‘Ÿ2π‘›βˆ’1ξ€Έπ‘‘π‘ŸβŠ—π‘™2(2.31)

by the rule 𝑅0βˆΆξ€½π‘‘π›Ό,π›½ξ€ΎβŸΆπ‘™0𝑑(π‘Ÿ)𝛼,𝛽.(2.32)

The mapping 𝑅0 is an isometric embedding, and the image of 𝑅0 coincides with the space 𝑏22,πœ‡. The adjoint operator π‘…βˆ—0∢𝐿2ξ€·(0,1),π‘Ÿ2π‘›βˆ’1ξ€Έπ‘‘π‘ŸβŠ—π‘™2βŸΆπ‘™#2(2.33)

is given by π‘…βˆ—0βˆΆξ€½π‘π›Ό,π›½ξ€ΎβŸΆξ‚»πœ’(π‘Ÿ)+ξ€œ(𝛼,𝛽)10𝑐𝛼,π›½βˆš(𝜌)2π‘›πœŒ2π‘›βˆ’1ξ‚Ό,π‘…π‘‘πœŒβˆ—0𝑅0=πΌβˆΆπ‘™#2βŸΆπ‘™#2,𝑅0π‘…βˆ—0=𝐡2∢𝐿2ξ€·(0,1),π‘Ÿ2π‘›βˆ’1ξ€Έπ‘‘π‘ŸβŠ—π‘™2βŸΆπ‘22,πœ‡.(2.34)

Meanwhile the operator 𝑅=π‘…βˆ—0π‘ˆ maps the space 𝐿2πœ‡(𝔹𝑛) onto 𝑙#2, and its restriction π‘…βˆ£π‘2πœ‡(𝔹𝑛)βˆΆπ‘2πœ‡(𝔹𝑛)βŸΆπ‘™#2(2.35)

is an isometric isomorphism. The adjoint operator π‘…βˆ—=π‘ˆβˆ—π‘…0βˆΆπ‘™#2βŸΆπ‘2πœ‡(𝔹𝑛)βŠ‚πΏ2πœ‡(𝔹𝑛)(2.36)

is isometric isomorphism of 𝑙#2 onto the subspace 𝑏2πœ‡(𝔹𝑛) of 𝐿2πœ‡(𝔹𝑛).

Remark 2.2. We have π‘…π‘…βˆ—=πΌβˆΆπ‘™#2βŸΆπ‘™#2,π‘…βˆ—π‘…=π‘„πœ‡π”Ήπ‘›βˆΆπΏ2πœ‡(𝔹𝑛)βŸΆπ‘2πœ‡(𝔹𝑛).(2.37)

Theorem 2.3. The isometric isomorphism π‘…βˆ—=π‘ˆβˆ—π‘…0βˆΆπ‘™#2→𝑏2πœ‡(𝔹𝑛) is given by π‘…βˆ—βˆΆξ€½π‘‘π›Ό,π›½ξ€ΎβŸΌβˆžξ“|𝛼|=0πœ†(𝑛,|𝛼|)𝑐𝛼,0π‘Ÿ|𝛼|𝑒𝛼,0(πœ”)+βˆžξ“||𝛽||=1πœ†ξ€·||𝛽||𝑐𝑛,0,π›½π‘Ÿ|𝛽|𝑒0,𝛽(πœ”).(2.38)

Proof. Let {𝑑𝛼,𝛽}βˆˆπ‘™#2, we can get π‘…βˆ—=π‘ˆβˆ—1π‘ˆβˆ—2𝑅0βˆΆξ€½π‘‘π›Ό,π›½ξ€ΎβŸΌπ‘ˆβˆ—1π‘ˆβˆ—2βˆšξ‚€ξ‚†2𝑛𝑑𝛼,𝛽=π‘ˆβˆ—1ξ€·ξ€½πœ†(𝑛,|𝛼|)𝑐𝛼,0π‘Ÿ|𝛼|ξ€Ύ+ξ€½πœ†ξ€·||𝛽||𝑐𝑛,0,π›½π‘Ÿ|𝛽|=ξ€Ύξ€Έβˆžξ“|𝛼|=0πœ†(𝑛,|𝛼|)𝑐𝛼,0π‘Ÿ|𝛼|𝑒𝛼,0(πœ”)+βˆžξ“||𝛽||=1πœ†ξ€·||𝛽||𝑐𝑛,0,π›½π‘Ÿ|𝛽|𝑒0,𝛽(πœ”).(2.39)

Corollary 2.4. The inverse isomorphism π‘…βˆΆπ‘2πœ‡(𝔹𝑛)→𝑙#2 is given by ξ€½π‘‘π‘…βˆΆπœ‘(𝑧)βŸΌπ›Ό,𝛽=ξ‚»(√2𝑛)βˆ’1π‘˜π›Ό,𝛽,(2.40) where 𝑐𝛼,0=βŸ¨πœ‘,Μƒπ‘’πœ‡π›Ό,0⟩=πœ†(𝑛,|𝛼|)𝑑𝑛,π›Όβˆ«π”Ήπ‘›πœ‘(𝑧)𝑧𝛼𝑑𝑣(𝑧), 𝑐0,𝛽=βŸ¨πœ‘,Μƒπ‘’πœ‡0,π›½βŸ©=πœ†(𝑛,|𝛽|)𝑑𝑛,π›½βˆ«π”Ήπ‘›πœ‘(𝑧)𝑧𝛽𝑑𝑣(𝑧), |𝛼|,|𝛽|βˆˆβ„€+, and {Μƒπ‘’πœ‡π›Ό,0}∞|𝛼|=0βˆͺ{Μƒπ‘’πœ‡0,𝛽}∞|𝛽|=1 is the standard basis for the pluriharmonic Bergman space 𝑏2πœ‡(𝔹𝑛); that is, Μƒπ‘’πœ‡π›Ό,0=𝑑𝑛,π›Όπœ†(𝑛,|𝛼|)𝑧𝛼,Μƒπ‘’πœ‡0,𝛽=𝑑𝑛,π›½πœ†ξ€·||𝛽||𝑛,𝑧𝛽.(2.41)

3. Toeplitz Operator with Radial Symbols on 𝑏2πœ‡(𝔹𝑛)

In this section we will study the Toeplitz operators π‘‡π‘Ž=π‘„πœ‡π”Ήπ‘›π‘ŽβˆΆπœ‘βˆˆπ‘2πœ‡(𝔹𝑛)β†¦π‘„πœ‡π”Ήπ‘›π‘Žπœ‘βˆˆπ‘2πœ‡(𝔹𝑛) with radial symbols π‘Ž=π‘Ž(π‘Ÿ).

Theorem 3.1. Let π‘Ž(π‘Ÿ) be a measurable function on the segment [0,1]. Then the Toeplitz operator π‘‡π‘Ž acting on 𝑏2πœ‡(𝔹𝑛) is unitary equivalent to the multication operator π›Ύπ‘Ž,πœ‡πΌ acting on 𝑙#2. The sequence π›Ύπ‘Ž,πœ‡={πœ’+(𝛼,𝛽)π›Ύπ‘Ž,πœ‡(|𝛼|+|𝛽|)} is given by π›Ύπ‘Ž,πœ‡(π‘š)=πœ†2ξ€œ(𝑛,π‘š)10π‘Ž(π‘Ÿ)π‘Ÿ2π‘š+2π‘›βˆ’1πœ‡(π‘Ÿ)π‘‘π‘Ÿ,π‘šβˆˆπ‘+.(3.1)

Proof. By means of Remark 2.2, the operator π‘‡π‘Ž is unitary equivalent to the operator π‘…π‘‡π‘Žπ‘…βˆ—=π‘…π‘„πœ‡π”Ήπ‘›π‘Žπ‘„πœ‡π”Ήπ‘›π‘…βˆ—ξ€·π‘…=π‘…βˆ—π‘…ξ€Έπ‘Žξ€·π‘…βˆ—π‘…ξ€Έπ‘…βˆ—=ξ€·π‘…π‘…βˆ—ξ€Έπ‘…π‘Žπ‘…βˆ—π‘…π‘…βˆ—=π‘…π‘Žπ‘…βˆ—=π‘…βˆ—0π‘ˆ2π‘ˆ1π‘Ž(π‘Ÿ)π‘ˆ1βˆ’1π‘ˆ2βˆ’1𝑅0=π‘…βˆ—0π‘ˆ2π‘Ž(π‘Ÿ)π‘ˆ2βˆ’1𝑅0=π‘…βˆ—0ξ€½πœ’+ξ€·πœ™(𝛼,𝛽)π‘Ž|𝛼|+|𝛽|𝑅(π‘Ÿ)ξ€Έξ€Ύ0.(3.2) Further, let {𝑑𝛼,𝛽} be a sequence from 𝑙#2. By (2.21), we have π‘…βˆ—0ξ€½πœ’+ξ€·πœ™(𝛼,𝛽)π‘Ž|𝛼|+|𝛽|𝑅(π‘Ÿ)ξ€Έξ€Ύ0𝑑𝛼,𝛽=π‘…βˆ—0ξ‚†βˆš2𝑛𝑑𝛼,π›½πœ’+ξ€·πœ™(𝛼,𝛽)π‘Ž|𝛼|+|𝛽|=ξ‚»ξ€œ(π‘Ÿ)10πœ’+ξ€·πœ™(𝛼,𝛽)π‘Ž|𝛼|+|𝛽|ξ€Έ(π‘Ÿ)2𝑛𝑑𝛼,π›½π‘Ÿ2π‘›βˆ’1ξ‚Ό=ξ‚»πœ’π‘‘π‘Ÿ+(𝛼,𝛽)𝑑𝛼,π›½ξ€œ10π‘Ž(𝑦)π‘‘πœ‘||𝛽||2𝑛|𝛼|+ξ‚Ό=ξ‚»πœ’(𝑦)+(𝛼,𝛽)𝑑𝛼,π›½πœ†2ξ€·||𝛽||ξ€Έξ€œπ‘›,|𝛼|+10π‘Ž(𝑦)𝑦2(|𝛼|+|𝛽|)+2π‘›βˆ’1ξ‚Ό=ξ€½πœ’πœ‡(𝑦)𝑑𝑦+(𝛼,𝛽)𝑑𝛼,π›½π›Ύπ‘Ž,πœ‡ξ€·||𝛽||.|𝛼|+ξ€Έξ€Ύ(3.3)

Corollary 3.2. (i) The Toeplitz operator π‘‡π‘Ž with measurable radial symbol π‘Ž(π‘Ÿ) is bounded on 𝑏2πœ‡(𝔹𝑛) if and only if supπ‘šβˆˆβ„€+|π›Ύπ‘Ž,πœ‡(π‘š)|<∞. Moreover, β€–β€–π‘‡π‘Žβ€–β€–=supπ‘šβˆˆβ„€+||π›Ύπ‘Ž,πœ‡(||π‘š).(3.4)
(ii) The Toeplitz operator π‘‡π‘Ž is compact if and only if limπ‘šβ†’βˆžπ›Ύπ‘Ž,πœ‡(π‘š)=0. The spectrum of the bounded operator π‘‡π‘Ž is given by spπ‘‡π‘Ž=ξ€½π›Ύπ‘Ž,πœ‡(π‘š)βˆΆπ‘šβˆˆβ„€+ξ€Ύ,(3.5) and its essential spectrum ess-spπ‘‡π‘Ž coincides with the set of all limits points of the sequence {π›Ύπ‘Ž,πœ‡(π‘š)}π‘šβˆˆβ„€+.

Let 𝐻 be a Hilbert space and {πœ‘π‘”}π‘”βˆˆπΊ a subset of elements of 𝐻 parameterized by elements 𝑔 of some set 𝐺 with measure π‘‘πœ‡.

Then {πœ‘π‘”}π‘”βˆˆπΊ is called a system of coherent states, if for all πœ‘βˆˆπ», β€–πœ‘β€–2ξ€œ=(πœ‘,πœ‘)=𝐺||(πœ‘,πœ‘π‘”)||2π‘‘πœ‡,(3.6)

or equivalently, if for all πœ‘1,πœ‘2∈𝐻, ξ€·πœ‘1,πœ‘2ξ€Έ=ξ€œπΊξ€·πœ‘1,πœ‘π‘”ξ€Έξ€·πœ‘2,πœ‘π‘”ξ€Έπ‘‘πœ‡.(3.7)

We define the isomorphic inclusion π‘‰βˆΆπ»β†’πΏ2(𝐺) by the rule ξ€·π‘‰βˆΆπœ‘βˆˆπ»βŸΌπ‘“=𝑓(𝑔)=πœ‘,πœ‘π‘”ξ€ΈβˆˆπΏ2(𝐺).(3.8)

By (3.7) we have (πœ‘1,πœ‘2)=βŸ¨π‘“1,𝑓2⟩, where (β‹…,β‹…) and βŸ¨β‹…,β‹…βŸ© are the scalar products on 𝐻 and 𝐿2(𝐺), respectively, and π‘“β„Ž(𝑔)=𝑓𝑔(β„Ž). Let 𝐻2(𝐺)=𝑉(𝐻)βŠ‚πΏ2(𝐺). A function π‘“βˆˆπΏ2(𝐺) is an element of 𝐻2(𝐺) if and only if for all β„ŽβˆˆπΊ, βŸ¨π‘“,π‘“β„ŽβŸ©=𝑓(β„Ž). The operator ∫(𝑃𝑓)(𝑔)=𝐺(πœ‘π‘‘,πœ‘π‘”)𝑓(𝑑)π‘‘πœ‡(𝑑) is the orthogonal projection of 𝐿2(𝐺) onto 𝐻2(𝐺).

The function π‘Ž(𝑔), π‘”βˆˆπΊ, is called the anti-Wick (or contravariant) symbol of an operator π‘‡βˆΆπ»β†’π» if π‘‰π‘‡π‘‰βˆ’1∣𝐻2(𝐺)=π‘ƒπ‘Ž(𝑔)𝑃=π‘ƒπ‘Ž(𝑔)𝐼∣𝐻2(𝐺)∢𝐻2(𝐺)⟢𝐻2(G),(3.9)

or, in other terminology, the operator π‘‰π΄π‘‰βˆ’1∣𝐻2(𝐺) is the Toeplitz operator π‘‡π‘Ž(𝑔)=π‘ƒπ‘Ž(𝑔)𝐼∣𝐻2(𝐺)∢𝐻2(𝐺)⟢𝐻2(𝐺)(3.10) with the symbols π‘Ž(𝑔).

Given an operator π‘‡βˆΆπ»β†’π», introduce the (Wick) function ξ€·Μƒπ‘Ž(𝑔,β„Ž)=π‘‡πœ‘β„Ž,πœ‘π‘”ξ€Έξ€·πœ‘β„Ž,πœ‘π‘”ξ€Έ,𝑔,β„ŽβˆˆπΊ.(3.11)

If the operator 𝑇 has an anti-Wick symbols, that is, π‘‰π‘‡π‘‰βˆ’1=π‘‡π‘Ž(𝑔) for some function π‘Ž=π‘Ž(𝑔), then Μƒπ‘Ž(𝑔,β„Ž)=βŸ¨π‘‡π‘Žπ‘“β„Ž,π‘“π‘”βŸ©βŸ¨π‘“β„Ž,π‘“π‘”βŸ©,𝑔,β„ŽβˆˆπΊ.(3.12)

And the operator π‘‡π‘Ž admits the following representation in terms of its Wick function: ξ€·π‘‡π‘Žπ‘“ξ€Έ(ξ€œπ‘”)=πΊπ‘Ž(𝑑)𝑓(𝑑)𝑓𝑑(ξ€œπ‘”)π‘‘πœ‡(𝑑)=πΊπ‘Ž(𝑑)𝑓𝑑(ξ€œπ‘”)π‘‘πœ‡(𝑑)𝐺𝑓(β„Ž)π‘“β„Ž(=ξ€œπ‘‘)π‘‘πœ‡(β„Ž)πΊξ€œπ‘“(β„Ž)π‘‘πœ‡(β„Ž)πΊπ‘Ž(𝑑)𝑓𝑑(𝑔)π‘“β„Ž=ξ€œ(𝑑)π‘‘πœ‡(𝑑)𝐺𝑓𝑓(β„Ž)π‘‘πœ‡(β„Ž)β„Ž(𝑔)ξ«π‘“β„Ž,π‘“π‘”ξ¬ξ€œπΊπ‘Ž(𝑑)π‘“β„Ž(𝑑)𝑓𝑔=ξ€œ(𝑑)π‘‘πœ‡(𝑑)πΊΜƒπ‘Ž(𝑔,β„Ž)𝑓(β„Ž)π‘“β„Ž(𝑔)π‘‘πœ‡(β„Ž).(3.13) Interchanging the integrals above, we understand them in a weak sense.

The restriction of the function Μƒπ‘Ž(𝑔,β„Ž) onto the diagonal ξ€·Μƒπ‘Ž(𝑔)=Μƒπ‘Ž(𝑔,𝑔)=π‘‡πœ‘π‘”,πœ‘π‘”ξ€Έξ€·πœ‘π‘”,πœ‘π‘”ξ€Έ,π‘”βˆˆπΊ,(3.14)

is called the Wick (or covariant or Berezin) symbols of the operator π‘‡βˆΆπ»β†’π».

The Wick and anti-Wick symbols of an operator π‘‡βˆΆπ»β†’π» are connected by the Berezin transform ξ€·Μƒπ‘Ž(𝑔)=Μƒπ‘Ž(𝑔,𝑔)=π‘‡πœ‘π‘”,πœ‘π‘”ξ€Έξ€·πœ‘π‘”,πœ‘π‘”ξ€Έ=βŸ¨π‘‡π‘Žπ‘“π‘”,π‘“π‘”βŸ©βŸ¨π‘“π‘”,π‘“π‘”βŸ©=∫𝐺||π‘“π‘Ž(𝑑)𝑔||(𝑑)2π‘‘πœ‡(𝑑)∫𝐺||𝑓𝑔||(𝑑)2π‘‘πœ‡(𝑑).(3.15)

The pluriharmonic Bergman reproducing kernel in the space 𝑏2πœ‡(𝔹𝑛) has the form 𝑅𝑧(𝑀)=𝐾𝑧(𝑀)+𝐾𝑧(𝑀)βˆ’π‘‘2𝑛,πŸ˜πœ†2(𝑛,0)=βˆžξ“|𝛼|=0Μƒπ‘’πœ‡π›Ό(𝑀)Μƒπ‘’πœ‡π›Ό(𝑧)+βˆžξ“|𝛼|=0Μƒπ‘’πœ‡π›Ό(𝑧)Μƒπ‘’πœ‡π›Ό(𝑀)βˆ’π‘‘2𝑛,πŸ˜πœ†2(𝑛,0),(3.16) where 𝛼=𝟘=(0,…,0). For π‘“βˆˆπ‘2πœ‡(𝔹𝑛), the reproducing property 𝑄𝑓(𝑧)=πœ‡π”Ήπ‘›π‘“ξ€Έ(ξ€œπ‘§)=𝔹𝑛𝑓(𝑀)𝑅𝑧(𝑀)πœ‡(|𝑀|)𝑑𝑣(𝑀)(3.17)

shows that the system of functions 𝑅𝑧(𝑀), π‘€βˆˆπ”Ήπ‘›, forms a system of coherent states in the space 𝑏2πœ‡(𝔹𝑛). In our context, we have 𝐺=𝔹𝑛, π‘‘πœ‡=πœ‡(|𝑧|)𝑑π‘₯𝑑𝑦, 𝐻=𝐻2(𝐺)=𝑏2πœ‡(𝔹𝑛), 𝐿2(𝐺)=𝐿2πœ‡(𝔹𝑛), πœ‘π‘”=𝑓𝑔=𝑅𝑔, where 𝑔=π‘§βˆˆπ”Ήπ‘›.

Lemma 3.3. Let π‘‡π‘Ž be the Toeplitz operator with a radial symbol π‘Ž=π‘Ž(π‘Ÿ). Then the corresponding Wick function (3.11) has the form Μƒπ‘Ž(𝑧,𝑀)=π‘…π‘€βˆ’1(𝑧)βˆžξ“|𝛼|=0Μƒπ‘’πœ‡π›Ό(𝑀)Μƒπ‘’πœ‡π›Ό(𝑧)π›Ύπ‘Ž,πœ‡(|𝛼|)+βˆžξ“|𝛼|=0Μƒπ‘’πœ‡π›Ό(𝑧)Μƒπ‘’πœ‡π›Ό(𝑀)π›Ύπ‘Ž,πœ‡(|𝛼|)βˆ’π‘‘2𝑛,πŸ˜πœ†2(𝑛,0)π›Ύπ‘Ž,πœ‡ξƒͺ(0).(3.18)

Proof. By (3.11) and (3.16), we have Μƒπ‘Ž(𝑧,𝑀)=βŸ¨π‘‡π‘Žπ‘…π‘€,π‘…π‘§βŸ©βŸ¨π‘…π‘€,π‘…π‘§βŸ©=π‘…π‘€βˆ’1(𝑧)βŸ¨π‘Žπ‘…π‘€,π‘…π‘§βŸ©=π‘…π‘€βˆ’1ξ‚€(𝑧)βŸ¨π‘ŽπΎπ‘€,πΎπ‘§ξ‚¬π‘ŽβŸ©+𝐾𝑀,πΎπ‘§ξ‚­βˆ’π‘‘4𝑛,πŸ˜πœ†4(𝑛,0)βŸ¨π‘Ž,1⟩=π‘…π‘€βˆ’1(𝑧)βˆžξ“|𝛼|=0Μƒπ‘’πœ‡π›Ό(𝑀)Μƒπ‘’πœ‡π›Όξ«(𝑧)π‘ŽΜƒπ‘’πœ‡π›Ό,Μƒπ‘’πœ‡π›Όξ¬+βˆžξ“|𝛼|=0Μƒπ‘’πœ‡π›Ό(𝑧)Μƒπ‘’πœ‡π›Όξ«(𝑀)π‘ŽΜƒπ‘’πœ‡π›Ό,Μƒπ‘’πœ‡π›Όξ¬βˆ’π‘‘4𝑛,πŸ˜πœ†4ξƒͺ(𝑛,0)βŸ¨π‘Ž,1⟩=π‘…π‘€βˆ’1(𝑧)βˆžξ“|𝛼|=0Μƒπ‘’πœ‡π›Ό(𝑀)Μƒπ‘’πœ‡π›Ό(𝑧)π›Ύπ‘Ž,πœ‡(|𝛼|)+βˆžξ“|𝛼|=0Μƒπ‘’πœ‡π›Ό(𝑧)Μƒπ‘’πœ‡π›Ό(𝑀)π›Ύπ‘Ž,πœ‡(|𝛼|)βˆ’π‘‘2𝑛,πŸ˜πœ†2(𝑛,0)π›Ύπ‘Ž,πœ‡ξƒͺ.(0)(3.19)

Denote by πΏπœ‡π›Ό the one-dimensional subspace of 𝑏2πœ‡(𝔹𝑛) generated by the base element Μƒπ‘’πœ‡π›Ό(𝑧), |𝛼|βˆˆβ„€+. Then the one-dimensional projection π‘ƒπœ‡π›Ό of 𝑏2πœ‡(𝔹𝑛) onto πΏπœ‡π›Ό has obviously the form π‘ƒπœ‡π›Όξ«π‘“=𝑓,Μƒπ‘’πœ‡π›Όξ¬Μƒπ‘’πœ‡π›Ό=Μƒπ‘’πœ‡π›Ό(ξ€œπ‘§)𝔹𝑛𝑓(𝑀)Μƒπ‘’πœ‡π›Ό(𝑀)πœ‡(|𝑀|)𝑑𝑣(𝑀).(3.20) In the similar method, πΏπœ‡π›Ό denote the one-dimensional subspace of 𝑏2πœ‡(𝔹𝑛) generated by the base element Μƒπ‘’πœ‡π›Ό(𝑧). Let π‘ƒπœ‡π›Ό be the projection from 𝑏2πœ‡(𝔹𝑛) onto πΏπœ‡π›Ό, and the projection can be rewritten as π‘ƒπœ‡π›Όξ‚¬π‘“(𝑧)=𝑓,Μƒπ‘’πœ‡π›Όξ‚­Μƒπ‘’πœ‡π›Ό(𝑧)=Μƒπ‘’πœ‡π›Όξ€œ(𝑧)𝔹𝑛𝑓(𝑀)Μƒπ‘’πœ‡π›Ό(𝑀)πœ‡(|𝑀|)𝑑𝑣(𝑀).(3.21)

Theorem 3.4. Let π‘‡π‘Ž be a bounded Toeplitz operator having radial symbol π‘Ž(π‘Ÿ). Then one can get the spectral decomposition of the operator π‘‡π‘Ž: π‘‡π‘Ž=βˆžξ“|𝛼|=0π›Ύπ‘Ž,πœ‡(|𝛼|)π‘ƒπœ‡π›Ό+βˆžξ“|𝛼|=0π›Ύπ‘Ž,πœ‡(|𝛼|)π‘ƒπœ‡π›Όβˆ’π›Ύπ‘Ž,πœ‡(0)π‘ƒπœ‡πŸ˜.(3.22)

Proof. According to (3.13), (3.20), (3.21), and Lemma 3.3, we get ξ€·π‘‡π‘Žπ‘“ξ€Έ(ξ€œπ‘§)=π”Ήπ‘›Μƒπ‘Ž(𝑧,𝑀)𝑓(𝑀)𝑅𝑀(=ξ€œπ‘§)πœ‡(|𝑀|)𝑑𝑣(𝑀).π”Ήπ‘›ξƒ©βˆžξ“|𝛼|=0Μƒπ‘’πœ‡π›Ό(𝑀)Μƒπ‘’πœ‡π›Ό(𝑧)π›Ύπ‘Ž,πœ‡(|𝛼|)+βˆžξ“|𝛼|=0Μƒπ‘’πœ‡π›Ό(𝑧)Μƒπ‘’πœ‡π›Ό(𝑀)π›Ύπ‘Ž,πœ‡(|𝛼|)βˆ’π‘‘2𝑛,πŸ˜πœ†2(𝑛,0)π›Ύπ‘Ž,πœ‡ξƒͺ=(0)𝑓(𝑀)πœ‡(|𝑀|)𝑑𝑣(𝑀).βˆžξ“|𝛼|=0π›Ύπ‘Ž,πœ‡(|𝛼|)π‘ƒπœ‡π›Όπ‘“(𝑧)+βˆžξ“|𝛼|=0π›Ύπ‘Ž,πœ‡(|𝛼|)π‘ƒπœ‡π›Όπ‘“(𝑧)βˆ’π›Ύπ‘Ž,πœ‡(0)π‘ƒπœ‡πŸ˜π‘“(𝑧).(3.23)

The value π›Ύπ‘Ž,πœ‡(|𝛼|) depends only on |𝛼|. Collecting the terms with the same |𝛼| and using the formula ξ€·π‘§β‹…π‘€ξ€Έπ‘š=|𝛼|=π‘šπ‘š!𝑧𝛼!𝛼𝑀𝛼(3.24)

we obtain Μƒπ‘Ž(𝑧,𝑀)=π‘…π‘€βˆ’1(𝑧)βˆžξ“π‘š=0𝑙(π‘š,𝑛)π›Ύπ‘Ž,πœ‡(π‘š)ξ€·ξ€·π‘§β‹…π‘€ξ€Έπ‘š+ξ€·π‘€β‹…π‘§ξ€Έπ‘šξ€Έβˆ’π‘‘2𝑛,πŸ˜πœ†2(𝑛,0)π›Ύπ‘Ž,πœ‡ξƒ­(0),(3.25) where (𝑙(π‘š,𝑛)=(π‘š+π‘›βˆ’1)!/|𝑆2π‘›βˆ’1|π‘š!(π‘›βˆ’1)!)πœ†2(𝑛,π‘š). The orthogonal projection of 𝑏2πœ‡(𝔹𝑛) onto the subspace generated by all element Μƒπ‘’πœ‡π›Ό with |𝛼|=π‘š, π‘šβˆˆβ„€+ can be written as ξ‚€π‘ƒπœ‡(π‘š)𝑓(ξ€œπ‘§)=𝑙(π‘š,𝑛)𝔹𝑛𝑓(𝑀)π‘§β‹…π‘€ξ€Έπ‘šπœ‡(|𝑀|)𝑑𝑣(𝑀);(3.26)

similarly, ξ‚€π‘ƒπœ‡(π‘š)𝑓(ξ€œπ‘§)=𝑙(π‘š,𝑛)𝔹𝑛𝑓(𝑀)π‘€β‹…π‘§ξ€Έπ‘šπœ‡(|𝑀|)𝑑𝑣(𝑀)(3.27)

denotes the orthogonal projection from 𝑏2πœ‡(𝔹𝑛) onto the subspace generated by all elements Μƒπ‘’πœ‡π›Ό with |𝛼|=π‘š. Therefore, (3.22) has the form π‘‡π‘Ž=βˆžξ“π‘š=0π›Ύπ‘Ž,πœ‡(π‘š)π‘ƒπœ‡(π‘š)+βˆžξ“π‘š=0π›Ύπ‘Ž,πœ‡(π‘š)π‘ƒπœ‡(π‘š)βˆ’π›Ύπ‘Ž,πœ‡(0)π‘ƒπœ‡πŸ˜.(3.28)

In view of (3.25), we can get the following useful corollary.

Corollary 3.5. Let π‘‡π‘Ž be a bounded Toeplitz operator having radial symbol π‘Ž(π‘Ÿ). Then the Wick symbol of the operator π‘‡π‘Ž is radial as well and is given by the formula Μƒπ‘Ž(𝑧)=Μƒπ‘Ž(π‘Ÿ)=π‘…π‘§βˆ’12(𝑧)βˆžξ“π‘š=0𝑙(π‘š,𝑛)π›Ύπ‘Ž,πœ‡(π‘š)π‘Ÿ2π‘šβˆ’π‘‘2𝑛,πŸ˜πœ†2(𝑛,0)π›Ύπ‘Ž,πœ‡ξƒͺ(0),(3.29) where 𝑅𝑧(𝑧)=2βˆžβˆ‘π‘š=0𝑙(π‘š,𝑛)π‘Ÿ2π‘šβˆ’π‘‘2𝑛,πŸ˜πœ†2(𝑛,0).

In terms of Wick function the composition formula for Toeplitz operators is quite transparent.

Corollary 3.6. Let π‘‡π‘Ž, 𝑇𝑏 be the Toeplitz operators with the Wick function Μƒπ‘Ž(𝑧,𝑀)=π‘…π‘€βˆ’1(𝑧)βˆžξ“π‘š=0𝑙(π‘š,𝑛)π›Ύπ‘Ž,πœ‡(π‘š)ξ€·ξ€·π‘§β‹…π‘€ξ€Έπ‘š+ξ€·π‘€β‹…π‘§ξ€Έπ‘šξ€Έβˆ’π‘‘2𝑛,πŸ˜πœ†2(𝑛,0)π›Ύπ‘Ž,πœ‡ξƒ­,Μƒ(0)𝑏(𝑧,𝑀)=π‘…π‘€βˆ’1(𝑧)βˆžξ“π‘š=0𝑙(π‘š,𝑛)𝛾𝑏,πœ‡(π‘š)ξ€·ξ€·π‘§β‹…π‘€ξ€Έπ‘š+ξ€·π‘€β‹…π‘§ξ€Έπ‘šξ€Έβˆ’π‘‘2𝑛,πŸ˜πœ†2(𝑛,0)𝛾𝑏,πœ‡ξƒ­,(0)(3.30) respectively. Then the Wick function ̃𝑐(𝑧,𝑀) of the composition 𝑇=π‘‡π‘Žπ‘‡π‘ is given by ̃𝑐(𝑧,𝑀)=π‘…π‘€βˆ’1(𝑧)βˆžξ“π‘š=0𝑙(π‘š,𝑛)𝛾𝑏,πœ‡(π‘š)π›Ύπ‘Ž,πœ‡(π‘š)ξ€·ξ€·π‘§β‹…π‘€ξ€Έπ‘š+ξ€·π‘€β‹…π‘§ξ€Έπ‘šξ€Έβˆ’π‘‘2𝑛,πŸ˜πœ†2(𝑛,0)𝛾𝑏,πœ‡(0)π›Ύπ‘Ž,πœ‡ξƒ­(0).(3.31)

Proof. According to Lemma 3.3 and (3.25), we have ̃𝑐(𝑧,𝑀)=βŸ¨π‘‡π‘Žπ‘‡π‘π‘…π‘€,π‘…π‘§βŸ©βŸ¨π‘…π‘€,π‘…π‘§βŸ©=π‘…π‘€βˆ’1(𝑧)βŸ¨π‘‡π‘π‘…π‘€,π‘Žπ‘…π‘§βŸ©=π‘…π‘€βˆ’1ξ€œ(𝑧)𝔹𝑛𝑇𝑏𝑅𝑀(𝑒)𝑅𝑧(𝑒)π‘Ž(|𝑒|)πœ‡(|𝑒|)𝑑𝑣(𝑒)=π‘…π‘€βˆ’1ξ€œ(𝑧)π”Ήπ‘›βŸ¨π‘‡π‘π‘…π‘€,π‘…π‘’βŸ©π‘…π‘§(𝑒)π‘Ž(|𝑒|)πœ‡(|𝑒|)𝑑𝑣(𝑒)=π‘…π‘€βˆ’1(𝑧)βˆžξ“|𝛼|=0Μƒπ‘’πœ‡π›Ό(𝑀)π‘’πœ‡π›Ό(𝑧)𝛾𝑏,πœ‡(|𝛼|)π›Ύπ‘Ž,πœ‡(|𝛼|)+βˆžξ“|𝛼|=0Μƒπ‘’πœ‡π›Ό(𝑧)π‘’πœ‡π›Ό(𝑀)𝛾𝑏,πœ‡(|𝛼|)π›Ύπ‘Ž,πœ‡(|𝛼|)βˆ’π‘‘2𝑛,πŸ˜πœ†2(𝑛,0)𝛾𝑏,πœ‡(0)π›Ύπ‘Ž,πœ‡ξƒͺ(0)=π‘…π‘€βˆ’1(𝑧)βˆžξ“π‘š=0𝑙(π‘š,𝑛)𝛾𝑏,πœ‡(π‘š)π›Ύπ‘Ž,πœ‡(π‘š)ξ€·ξ€·π‘§β‹…π‘€ξ€Έπ‘š+ξ€·π‘€β‹…π‘§ξ€Έπ‘šξ€Έβˆ’π‘‘2𝑛,πŸ˜πœ†2(𝑛,0)𝛾𝑏,πœ‡(0)π›Ύπ‘Ž,πœ‡ξƒ­.(0)(3.32)

Acknowledgment

This work was supported by the National Science Foundation of China (Grant no. 10971020).

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