Toeplitz Operators on the Weighted Pluriharmonic Bergman Space with Radial Symbols

Abstract

We construct an operator 𝑅 whose restriction onto weighted pluriharmonic Bergman Space 𝑏^2𝜇(𝔹^𝑛) is an isometric isomorphism between 𝑏^2𝜇(𝔹^𝑛) and 𝑙^#2. Furthermore, using the operator 𝑅 we prove that each Toeplitz operator 𝑇_𝑎 with radial symbols is unitary to the multication operator 𝛾_a,𝜇I. Meanwhile, the Wick function of a Toeplitz operator with radial symbol gives complete information about the operator, providing its spectral decomposition.

1. Introduction

Let 𝔹^𝑛 be the open unit ball in the complex vector space ℂ^𝑛. For any 𝑧=(𝑧₁,…,𝑧_𝑛) and 𝜉=(𝜉₁,…,𝜉_𝑛) in ℂ^𝑛, let 𝑧⋅𝜉=∑^𝑛𝑗=1𝑧_𝑗𝜉_𝑗, where 𝜉_𝑗 is the complex conjugate of 𝜉_𝑗 and |𝑧|=√𝑧⋅𝑧. For a multi-index 𝛼=(𝛼₁,…,𝛼_𝑛) and 𝑧=(𝑧₁,…,𝑧_𝑛)∈ℂ^𝑛, we write 𝑧^𝛼=𝑧^𝛼1₁⋯𝑧^𝛼𝑛_𝑛, where 𝛼_𝑘∈𝑍₊=ℕ∪{0}, and |𝛼|=𝛼₁+⋯+𝛼_𝑛 is its length, 𝛼!=𝛼₁!⋯𝛼_𝑛!.

The weighted pluriharmonic Bergman space 𝑏^2𝜇(𝔹^𝑛) is the subspace of the weighted space 𝐿^2𝜇(𝔹^𝑛) consisting of all pluriharmonic functions on 𝔹^𝑛. 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 𝑏^2𝜇(𝔹^𝑛) is a closed subspace of 𝐿^2𝜇(𝔹^𝑛) and hence is a Hilbert space. Let 𝑄^𝜇𝔹_𝑛 be the Hilbert space orthogonal projection from 𝐿^2𝜇(𝔹^𝑛) onto 𝑏^2𝜇(𝔹^𝑛). For a function 𝑢∈𝐿^2𝜇(𝔹^𝑛), the Toeplitz operator 𝑇_𝑢∶𝑏^2𝜇(𝔹^𝑛)→𝑏^2𝜇(𝔹^𝑛) with symbol 𝑢 is the linear operator defined by 𝑇_𝑢𝑓=𝑄^𝜇𝔹_𝑛(𝑢𝑓),𝑓∈𝑏^2𝜇(𝔹^𝑛).(1.1)

𝑇_𝑢 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 ℂ^𝑛.

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 𝑏^2𝜇(𝔹^𝑛) is an isometric isomorphism between 𝑏^2𝜇(𝔹^𝑛) and 𝑙^#2, and 𝑅_∗𝑅𝑅^∗=𝐼∶𝑙^#2⟶𝑙^#2,𝑅=𝑄^𝜇𝔹_𝑛∶𝐿^2𝜇(𝔹^𝑛)⟶𝑏^2𝜇(𝔹^𝑛);(1.2) where 𝑙^#2 is the subspace of 𝑙₂. Using the operator 𝑅 we prove that each Toeplitz operator 𝑇_𝑎 with radial symbols is unitary to the multication operator 𝛾_𝑎,𝜇𝐼 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 𝐿^2𝜇(𝔹^𝑛). Consider a nonnegative measurable function 𝜇(𝑟),𝑟∈(0,1), such that mes{𝑟∈(0,1)∶𝜇(𝑟)>0}=1, and _𝔹𝑛𝜇(|𝑧|)𝑑𝑣(𝑧)=||𝑆^2𝑛−1||¹⁰𝜇(𝑟)𝑟^2𝑛−1𝑑𝑟<∞,(2.1) where |𝑆^2𝑛−1|=2𝜋^𝑛−(1/2)Γ⁻¹(𝑛−(1/2)) is the surface area of unit sphere 𝑆^2𝑛−1 and Γ(𝑧) is the Gamma function.

Introduce the weighted space 𝐿^2𝜇(𝔹^𝑛)=𝑓∶‖𝑓‖^2𝐿_{2𝜇(𝔹𝑛)}=_𝔹𝑛||𝑓(𝑧)||²𝜇(|𝑧|)𝑑𝑣(𝑧)<∞,(2.2) where 𝑑𝑣(𝑧) is the usual Lebesgue volume measure and 𝐿₂(𝑆^2𝑛−1) is the space with the usual Lebesgue surface measure.

The space 𝐿₂(𝑆^2𝑛−1) is the direct sum of mutually orthogonal spaces ℋ_𝑘, that is, 𝐿₂𝑆^2𝑛−1=^∞_𝑘=0ℋ_𝑘,(2.3) where ℋ_𝑘 denotes the space of spherical harmonics of order 𝑘. Meanwhile, each space ℋ_𝑘 is the direct sum (under the identification ℂ^𝑛=ℝ^2𝑛) of the mutually orthogonal spaces ℋ_𝑝,𝑞 (see, e.g., [7]): ℋ_𝑘=_{𝑝+𝑞=𝑘𝑝,𝑞∈ℤ+}𝐻_𝑝,𝑞,𝑘∈ℤ₊,(2.4) where 𝐻_𝑝,𝑞, for each 𝑝,𝑞=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 𝑞 in the conjugate variable 𝑧=(𝑧₁,…,𝑧_𝑛). Thus, we can get 𝐿₂𝑆^2𝑛−1=_{𝑝,𝑞∈ℤ+}𝐻_𝑝,𝑞.(2.5)

The Hardy space 𝐻²(𝔹^𝑛) in the unit ball 𝔹^𝑛 is a closed subspace of 𝐿₂(𝑆^2𝑛−1). Denote by 𝑃_𝑆^2𝑛−1 the Szegö orthogonal projection of 𝐿₂(𝑆^2𝑛−1) onto the Hardy space 𝐻²(𝔹^𝑛). It is well known that 𝐻²(𝔹^𝑛)=⨁^∞_𝑝=0𝐻_𝑝,0. The standard orthonormal base in 𝐻²(𝔹^𝑛) has the form 𝑒_𝛼(𝜔)=𝑑_𝑛,𝛼𝜔^𝛼,𝑑_𝑛,𝛼=(𝑛−1+|𝛼|)!||𝑆^2𝑛−1||(𝑛−1)!𝛼!for|𝛼|=0,1,….(2.6)

Fix an orthonormal basis {𝑒_𝛼,𝛽(𝜔)}_𝛼,𝛽, 𝛼,𝛽∈ℤ^𝑛+, in the space 𝐿₂(𝑆^2𝑛−1) so that 𝑒_𝛼,0(𝜔)≡𝑒_𝛼(𝜔), 𝑒_0,𝛼(𝜔)≡𝑒_𝛼,0(𝜔)≡𝑒_𝛼(𝑤), |𝛼|=0,1,….

Passing to the spherical coordinates in the unit ball we have 𝐿^2𝜇(𝔹^𝑛)=𝐿₂(0,1),𝜇(𝑟)𝑟^2𝑛−1𝑑𝑟⊗𝐿₂𝑆^2𝑛−1.(2.7) For any function 𝑓(𝑧)∈𝐿^2𝜇(𝔹^𝑛) have the decomposition 𝑓(𝑧)=^∞_||𝛽|_||𝛼|+=0𝑐_𝛼,𝛽(𝑟)𝑒_𝛼,𝛽(𝜔),𝑟=|𝑧|,𝜔=𝑧𝑟,(2.8) with the coefficients 𝑐_𝛼,𝛽(𝑟) satisfying the condition ‖𝑓‖^2𝐿_{2𝜇(𝔹𝑛)}=^∞_||𝛽|_||𝛼|+=0¹⁰||𝑐_𝛼,𝛽(𝑟)||²𝜇(𝑟)𝑟^2𝑛−1𝑑𝑟<∞.(2.9)

According to the decomposition (2.7), (2.8) together with Parseval’s equality, we can define the unitary operator 𝑈₁∶𝐿₂(0,1),𝜇(𝑟)𝑟^2𝑛−1𝑑𝑟⊗𝐿₂𝑆^2𝑛−1⟶𝐿₂(0,1),𝜇(𝑟)𝑟^2𝑛−1𝑑𝑟⊗𝑙₂≡𝑙₂𝐿₂(0,1),𝜇(𝑟)𝑟^2𝑛−1𝑑𝑟,(2.10)

by the rule 𝑈₁∶𝑓(𝑧)→{𝑐_𝛼,𝛽(𝑟)}, and ‖𝑓‖^2𝐿_{2𝜇(𝔹𝑛)}=‖‖𝑐_𝛼,𝛽(𝑟)‖‖^2𝑙_{2𝐿2(0,1),𝜇(𝑟)𝑟2𝑛−1𝑑𝑟}=^∞_|𝛼|+_|𝛽|=0‖‖𝑐_𝛼,𝛽(𝑟)‖‖^2𝐿_{2(0,1),𝜇(𝑟)𝑟2𝑛−1𝑑𝑟}.(2.11)

Let 𝑓(𝑧) be a pluriharmonic in the unit ball 𝔹^𝑛 and write 𝑓=𝑔+ℎ, where the functions 𝑔, ℎ are holomorphic in 𝔹^𝑛. Suppose 𝑔(𝑧)=^∞_|_𝛼|=0𝑐_𝛼z^𝛼,ℎ(𝑧)=^∞_|_𝛽|=0𝑐_𝛽𝑧^𝛽(2.12)

are their power series representations of 𝑔 and ℎ, respectively. We have 𝑓(𝑧)=^∞_|_𝛼|=0𝑐_𝛼𝑧^𝛼+^∞_|_𝛽|=0𝑐_𝛽𝑧^𝛽=^∞_|_𝛼|=0𝑐_𝛼(𝑟)𝑒_𝛼(𝜔)+^∞_|_𝛽|=0𝑐_𝛽(𝑟)𝑒_𝛽(𝜔),(2.13) where 𝑐_𝛼(𝑟)=𝑐_𝛼𝑑^{−1𝑛,𝛼}𝑟^|𝛼|, 𝑐_𝛽(𝑟)=𝑐_𝛽𝑑^{−1𝑛,𝛽}𝑟^|𝛽|, 𝑟=|𝑧|, 𝜔=(𝑧/𝑟).

Let 𝑏^2𝜇(𝔹^𝑛) be the pluriharmonic Bergman space in 𝔹^𝑛 from 𝐿^2𝜇(𝔹^𝑛). Denote by 𝑄^𝜇𝔹_𝑛 the pluriharmonic Bergman orthogonal projection of 𝐿^2𝜇(𝔹^𝑛) onto the Bergman space 𝑏^2𝜇(𝔹^𝑛). From the above it follows that to characterize a function 𝑓(𝑧)∈𝑏^2𝜇(𝔹^𝑛) and considering its decomposition according to (2.13), one can restrict to the function having the representation 𝑓(𝑧)=𝑔(𝑧)+ℎ(𝑧)=^∞_|_𝛼|=0𝑐_𝛼,0(𝑟)𝑒_𝛼,0(𝜔)+^∞_|_𝛽|=0𝑐_0,𝛽(𝑟)𝑒_0,𝛽(𝜔).(2.14) Now let us take an arbitrary 𝑓(𝑧) from 𝑏^2𝜇(𝔹^𝑛) in the form (2.14). It will satisfy the Cauchy-Riemann equations, that is, 𝜕𝜕𝑧_𝑘𝑔(𝑧)≡12𝜕𝜕𝑥_𝑘+𝑖𝜕𝜕𝑦_𝑘𝑔(𝑧)=0,𝑘=1,…,𝑛,𝑧∈𝔹^𝑛𝜕,𝜕𝑧_𝑘ℎ(𝑧)≡12𝜕𝜕𝑥_𝑘−𝑖𝜕𝜕𝑦_𝑘ℎ(𝑧)=0,𝑘=1,…,𝑛,𝑧∈𝔹^𝑛.(2.15)

Applying 𝜕/𝜕𝑧_𝑘, 𝜕/𝜕𝑧_𝑘 to 𝑔 and ℎ, respectively, we have𝜕𝜕𝑧_{𝑘∞|𝛼|=0}𝑐_𝛼,0(𝑟)𝑒_𝛼,0(𝜔)=𝑧^𝑘2𝑟^∞_|𝛼|=0𝑑𝑑𝑟𝑐^𝛼,0(𝑟)−|𝛼|𝑟𝑐^𝛼,0𝜕(𝑟)𝑒_𝛼,0(𝜔),𝜕𝑧_{𝑘∞||𝛽||=0}𝑐_0,𝛽(𝑟)𝑒_0,𝛽(𝜔)=𝑧_𝑘2𝑟^∞_||𝛽||=0𝑑𝑑𝑟𝑐^0,𝛽(𝑟)−||𝛽||𝑟𝑐^0,𝛽(𝑟)𝑒_0,𝛽(𝜔),(2.16)

where 𝑘=0,…,𝑛, and we come to the infinite system of ordinary linear differential equations 𝑑𝑑𝑟𝑐^𝛼,0(𝑟)−|𝛼|𝑟𝑐^𝛼,0𝑑(𝑟)=0,|𝛼|=0,1,…𝑑𝑟𝑐^0,𝛽(𝑟)−||𝛽||𝑟𝑐^0,𝛽(𝑟)=0,||𝛽||=0,1,….(2.17)

Their general solution has the form 𝑐_𝛼,0=𝑏_𝛼𝑟^|𝛼|=𝜆(𝑛,|𝛼|)𝑐_𝛼,0𝑟^|𝛼|, 𝑐_0,𝛽=𝑏_𝛽𝑟^|𝛽|=𝜆(𝑛,|𝛽|)𝑐_0,𝛽𝑟^|𝛽|, with 𝜆(𝑛,𝑚)=(∫¹⁰𝑡^{2𝑚+2𝑛−1}𝜇(𝑡)𝑑𝑡)^−1/2. Hence, for any 𝑓(𝑧)∈𝑏^2𝜇(𝔹^𝑛) we have 𝑓(𝑧)=^∞_|_𝛼|=0𝑐_𝛼,0𝜆(𝑛,|𝛼|)𝑟^|𝛼|𝑒_𝛼,0+^∞_||_𝛽||=0𝑐_0,𝛽𝜆𝑛,||𝛽||𝑟^|𝛽|𝑒_0,𝛽.(2.18)

And, it is easy to verify ‖𝑓‖^2𝐿_{2𝜇(𝔹𝑛)}=∑^∞|𝛼|=0|𝑐_𝛼,0|²+∑^∞|𝛽|=0|𝑐_0,𝛽|². Thus the image 𝑏^21,𝜇(𝔹^𝑛)=𝑈₁(𝑏^2𝜇(𝔹^𝑛)) is characterized as the closed subspace of 𝐿₂(0,1),𝜇(𝑟)𝑟^2𝑛−1𝑑𝑟⊗𝑙₂=𝑙₂𝐿₂(0,1),𝜇(𝑟)𝑟^2𝑛−1𝑑𝑟(2.19) which consists of all sequences {𝑐_𝛼,𝛽(𝑟)} of the form 𝑐_𝛼,𝛽(𝑟)=⎧⎪⎨⎪⎩𝜆(𝑛,|𝛼|)𝑐^𝛼,0𝜆𝑛,||𝛽||𝑐_0,𝛽𝑟^|𝛼|,||𝛽||=00,𝑟^|𝛽|,||𝛼𝛼||=≠00,||𝛽||≠0.(2.20)

For each 𝑚∈ℤ₊ introduce the function 𝜑_𝑚(𝜌)=𝜆(𝑛,𝑚)^1/𝑛^𝜌0𝑟^{2𝑚+2𝑛−1}𝜇(𝑟)𝑑𝑟^1/2𝑛,𝜌∈[0,1].(2.21) Obviously, there exists the inverse function for the function 𝜑_𝑚(𝜌) on [0,1], which we will denote by 𝜙_𝑚(𝑟). Introduce the operator 𝑢_𝑚𝑓(𝑟)=√2𝑛𝜆(𝑛,𝑚)𝜙^𝑚−𝑚(𝑟)𝑓𝜙_𝑚(𝑟).(2.22)

By Proposition 2.1 in [5], the operator 𝑢_𝑚 maps unitary 𝐿₂((0,1),𝜇(𝑟)𝑟^2𝑛−1) onto 𝐿₂((0,1),𝑟^2𝑛−1𝑑𝑟) in such a way that 𝑢_𝑚(𝜆(𝑛,𝑚)𝑟^𝑚)=√2𝑛,𝑚∈ℤ₊.(2.23)

Intoduce the unitary operator 𝑈₂∶𝑙₂𝐿₂(0,1),𝜇(𝑟)𝑟^2𝑛−1𝑑𝑟⟶𝑙₂𝐿₂(0,1),𝑟^2𝑛−1𝑑𝑟,(2.24) where 𝑈₂∶𝑐_𝛼,𝛽(𝑟)⟶𝑢_{|𝛼|+|𝛽|}𝑐_𝛼,𝛽(𝑟).(2.25)

By (2.23), we can get the space 𝑏^22,𝜇=𝑈₂(𝑏^21,𝜇) coincides with the space of all sequences {𝑘_𝛼,𝛽} for which𝑘_𝛼,𝛽=⎧⎪⎨⎪⎩√√2𝑛𝑐_𝛼,0,||𝛽||=00,2𝑛𝑐_0,𝛽,||𝛼𝛼||=≠00,||𝛽||≠0.(2.26)

Let 𝑙₀(𝑟)=√2𝑛. We have 𝑙₀(𝑟)∈𝐿₂((0,1),𝑟^2𝑛−1𝑑𝑟) and ‖𝑙₀‖_{𝐿2((0,1),𝑟2𝑛−1)}=1. Denote by 𝐿₀ the one-dimensional subspace of 𝐿₂((0,1),𝑟^2𝑛−1𝑑𝑟) generated by 𝑙₀(𝑟). The orthogonal projection 𝑃₀ of 𝐿₂((0,1),𝑟^2𝑛−1𝑑𝑟) onto 𝐿₀ has the form 𝑃₀𝑓(𝑟)=⟨𝑓,𝑙₀⟩𝑙₀=√2𝑛¹⁰𝑓(𝜌)√2𝑛𝜌^2𝑛−1𝑑𝜌.(2.27) Let 𝑑_𝛼,𝛽=𝑘_𝛼,𝛽(√2𝑛)⁻¹. Denote by 𝑙^#2 the subspace of 𝑙₂ consisting of all sequences {𝑑_𝛼,𝛽}. And let 𝑝^# be the orthogonal projections of 𝑙₂ onto 𝑙^#2, then 𝑝^#=𝜒₊(𝛼,𝛽)𝐼, where 𝜒₊(𝛼,𝛽)=0, if |𝛼‖𝛽|>0 and 𝜒₊(𝛼,𝛽)=1, if |𝛼‖𝛽|=0.

Observe that 𝑏^22,𝜇=𝐿₀⊗𝑙^#2 and the orthogonal projection 𝐵₂ of 𝑙₂𝐿₂(0,1),𝑟^2𝑛−1𝑑𝑟≡𝐿₂(0,1)𝑟^2𝑛−1𝑑𝑟⊗𝑙₂(2.28)

onto 𝑏^22,𝜇 has the form 𝐵₂=𝑃₀⊗𝑝^#. This leads to the following theorem.

Theorem 2.1. The unitary operator 𝑈=𝑈₁𝑈₂ gives an isometric isomorphism of the space 𝐿^2𝜇(𝔹^𝑛) onto 𝑙₂(𝐿₂((0,1),𝑟^2𝑛−1𝑑𝑟))≡𝐿₂((0,1),𝑟^2𝑛−1𝑑𝑟)⊗𝑙₂ such that
(1) the pluriharmonic Bergman space 𝑏^2𝜇(𝔹^𝑛) is mapped onto 𝐿₀⊗𝑙^#2, 𝑈∶𝑏^2𝜇(𝔹^𝑛)⟶𝐿₀⊗𝑙^#2,(2.29) where 𝐿₀ is the one-dimensional subspace of 𝐿₂((0,1),𝑟^2𝑛−1𝑑𝑟), generated by the function 𝑙₀(𝑟)=√2𝑛;
(2) the pluriharmonic Bergman projection 𝑄^𝜇𝔹_𝑛 is unitary equivalent to 𝑈𝑄^𝜇𝔹_𝑛𝑈⁻¹=𝑃₀⊗𝑝^#,(2.30) where 𝑃₀ is the one-dimensional projection (2.27) of 𝐿₂((0,1),𝑟^2𝑛−1𝑑𝑟) onto 𝐿₀.

Introduce the operator 𝑅₀∶𝑙^#2⟶𝐿₂(0,1),𝑟^2𝑛−1𝑑𝑟⊗𝑙₂(2.31)

by the rule 𝑅₀∶𝑑_𝛼,𝛽⟶𝑙₀(𝑟)𝑑_𝛼,𝛽.(2.32)

The mapping 𝑅₀ is an isometric embedding, and the image of 𝑅₀ coincides with the space 𝑏^22,𝜇. The adjoint operator 𝑅^∗0∶𝐿₂(0,1),𝑟^2𝑛−1𝑑𝑟⊗𝑙₂⟶𝑙^#2(2.33)

is given by 𝑅^∗0∶𝑐_𝛼,𝛽(𝑟)⟶𝑅𝜒₊(𝛼,𝛽)¹⁰𝑐_𝛼,𝛽(𝜌)√2𝑛𝜌^2𝑛−1𝑑𝜌_∗0,𝑅₀𝑅₀=𝐼∶𝑙^#2⟶𝑙^#2,𝑅^∗0=𝐵₂∶𝐿₂(0,1),𝑟^2𝑛−1𝑑𝑟⊗𝑙₂⟶𝑏^22,𝜇.(2.34)

Meanwhile the operator 𝑅=𝑅^∗0𝑈 maps the space 𝐿^2𝜇(𝔹^𝑛) onto 𝑙^#2, and its restriction 𝑅∣𝑏^2𝜇(𝔹^𝑛)∶𝑏^2𝜇(𝔹^𝑛)⟶𝑙^#2(2.35)

is an isometric isomorphism. The adjoint operator 𝑅^∗=𝑈^∗𝑅₀∶𝑙^#2⟶𝑏^2𝜇(𝔹^𝑛)⊂𝐿^2𝜇(𝔹^𝑛)(2.36)

is isometric isomorphism of 𝑙^#2 onto the subspace 𝑏^2𝜇(𝔹^𝑛) of 𝐿^2𝜇(𝔹^𝑛).

Remark 2.2. We have 𝑅𝑅^∗=𝐼∶𝑙^#2⟶𝑙^#2,𝑅^∗𝑅=𝑄^𝜇𝔹_𝑛∶𝐿^2𝜇(𝔹^𝑛)⟶𝑏^2𝜇(𝔹^𝑛).(2.37)

Theorem 2.3. The isometric isomorphism 𝑅^∗=𝑈^∗𝑅₀∶𝑙^#2→𝑏^2𝜇(𝔹^𝑛) is given by 𝑅^∗∶𝑑_𝛼,𝛽⟼^∞_|_𝛼|=0𝜆(𝑛,|𝛼|)𝑐_𝛼,0𝑟^|𝛼|𝑒_𝛼,0(𝜔)+^∞_||_𝛽||=1𝜆𝑛,||𝛽||𝑐_0,𝛽𝑟^|𝛽|𝑒_0,𝛽(𝜔).(2.38)

Proof. Let {𝑑_𝛼,𝛽}∈𝑙^#2, we can get 𝑅^∗==𝑈𝑈^∗1_∗1𝑈^∗2𝑅₀∶𝑑_𝛼,𝛽⟼𝑈^∗1𝑈^∗2√2𝑛𝑑_𝛼,𝛽=𝜆(𝑛,|𝛼|)𝑐_𝛼,0𝑟^|𝛼|+𝜆𝑛,||𝛽||𝑐_0,𝛽𝑟^|𝛽|_∞|_𝛼|=0𝜆(𝑛,|𝛼|)𝑐_𝛼,0𝑟^|𝛼|𝑒_𝛼,0(𝜔)+^∞_||_𝛽||=1𝜆𝑛,||𝛽||𝑐_0,𝛽𝑟^|𝛽|𝑒_0,𝛽(𝜔).(2.39)

Corollary 2.4. The inverse isomorphism 𝑅∶𝑏^2𝜇(𝔹^𝑛)→𝑙^#2 is given by 𝑅∶𝜑(𝑧)⟼𝑑_𝛼,𝛽=(√2𝑛)⁻¹𝑘_𝛼,𝛽,(2.40) where 𝑐_𝛼,0=⟨𝜑,𝑒̃^𝜇𝛼,0⟩=𝜆(𝑛,|𝛼|)𝑑_𝑛,𝛼∫_𝔹𝑛𝜑(𝑧)𝑧^𝛼𝑑𝑣(𝑧), 𝑐_0,𝛽=⟨𝜑,𝑒̃^𝜇0,𝛽⟩=𝜆(𝑛,|𝛽|)𝑑_𝑛,𝛽∫_𝔹𝑛𝜑(𝑧)𝑧^𝛽𝑑𝑣(𝑧), |𝛼|,|𝛽|∈ℤ₊, and {𝑒̃^𝜇𝛼,0}^∞|𝛼|=0∪{𝑒̃^𝜇0,𝛽}^∞|𝛽|=1 is the standard basis for the pluriharmonic Bergman space 𝑏^2𝜇(𝔹^𝑛); that is, 𝑒̃^𝜇𝛼,0=𝑑_𝑛,𝛼𝜆(𝑛,|𝛼|)𝑧^𝛼,𝑒̃^𝜇0,𝛽=𝑑_𝑛,𝛽𝜆𝑛,||𝛽||𝑧^𝛽.(2.41)

3. Toeplitz Operator with Radial Symbols on 𝑏^2𝜇(𝔹^𝑛)

In this section we will study the Toeplitz operators 𝑇_𝑎=𝑄^𝜇𝔹_𝑛𝑎∶𝜑∈𝑏^2𝜇(𝔹^𝑛)↦𝑄^𝜇𝔹_𝑛𝑎𝜑∈𝑏^2𝜇(𝔹^𝑛) with radial symbols 𝑎=𝑎(𝑟).

Theorem 3.1. Let 𝑎(𝑟) be a measurable function on the segment [0,1]. Then the Toeplitz operator 𝑇_𝑎 acting on 𝑏^2𝜇(𝔹^𝑛) is unitary equivalent to the multication operator 𝛾_𝑎,𝜇𝐼 acting on 𝑙^#2. The sequence 𝛾_𝑎,𝜇={𝜒₊(𝛼,𝛽)𝛾_𝑎,𝜇(|𝛼|+|𝛽|)} is given by 𝛾_𝑎,𝜇(𝑚)=𝜆²(𝑛,𝑚)¹⁰𝑎(𝑟)𝑟^{2𝑚+2𝑛−1}𝜇(𝑟)𝑑𝑟,𝑚∈𝑍₊.(3.1)

Proof. By means of Remark 2.2, the operator 𝑇_𝑎 is unitary equivalent to the operator 𝑅𝑇_𝑎𝑅^∗=𝑅𝑄^𝜇𝔹_𝑛𝑎𝑄^𝜇𝔹_𝑛𝑅^∗=𝑅𝑅^∗𝑅𝑎𝑅^∗𝑅𝑅^∗=𝑅𝑅^∗𝑅𝑎𝑅^∗𝑅𝑅^∗=𝑅𝑎𝑅^∗=𝑅^∗0𝑈₂𝑈₁𝑎(𝑟)𝑈¹⁻¹𝑈²⁻¹𝑅₀=𝑅^∗0𝑈₂𝑎(𝑟)𝑈²⁻¹𝑅₀=𝑅^∗0𝜒₊(𝛼,𝛽)𝑎𝜙_{|𝛼|+|𝛽|}(𝑟)𝑅₀.(3.2) Further, let {𝑑_𝛼,𝛽} be a sequence from 𝑙^#2. By (2.21), we have 𝑅^∗0=𝜒𝑅_+∗0(𝛼,𝛽)𝑎𝜙_{|𝛼|+|𝛽|}(𝑟)𝑅₀𝑑_𝛼,𝛽=√2𝑛𝑑_𝛼,𝛽𝜒₊(𝛼,𝛽)𝑎𝜙_{|𝛼|+|𝛽|}(𝑟)₁₀=𝜒𝜒₊(𝛼,𝛽)𝑎𝜙_{|𝛼|+|𝛽|}(𝑟)2𝑛𝑑_𝛼,𝛽𝑟^2𝑛−1𝑑𝑟₊(𝛼,𝛽)𝑑_𝛼,𝛽¹⁰=𝜒𝑎(𝑦)𝑑𝜑^{||𝛽||2𝑛|𝛼|+}(𝑦)₊=𝜒(𝛼,𝛽)𝑑_𝛼,𝛽𝜆²𝑛,|𝛼|+||𝛽||¹⁰𝑎(𝑦)𝑦^{2(|𝛼|+|𝛽|)+2𝑛−1}𝜇(𝑦)𝑑𝑦₊(𝛼,𝛽)𝑑_𝛼,𝛽𝛾_𝑎,𝜇|𝛼|+||𝛽||.(3.3)

Corollary 3.2. (i) The Toeplitz operator 𝑇_𝑎 with measurable radial symbol 𝑎(𝑟) is bounded on 𝑏^2𝜇(𝔹^𝑛) if and only if sup_𝑚∈ℤ+|𝛾_𝑎,𝜇(𝑚)|<∞. Moreover, ‖‖𝑇_𝑎‖‖=sup_𝑚∈ℤ+||𝛾_𝑎,𝜇(𝑚)||.(3.4)
(ii) The Toeplitz operator 𝑇_𝑎 is compact if and only if lim_𝑚→∞𝛾_𝑎,𝜇(𝑚)=0. The spectrum of the bounded operator 𝑇_𝑎 is given by sp𝑇_𝑎=𝛾_𝑎,𝜇(𝑚)∶𝑚∈ℤ₊,(3.5) and its essential spectrum ess-sp𝑇_𝑎 coincides with the set of all limits points of the sequence {𝛾_𝑎,𝜇(𝑚)}_𝑚∈ℤ+.

Let 𝐻 be a Hilbert space and {𝜑_𝑔}_𝑔∈𝐺 a subset of elements of 𝐻 parameterized by elements 𝑔 of some set 𝐺 with measure 𝑑𝜇.

Then {𝜑_𝑔}_𝑔∈𝐺 is called a system of coherent states, if for all 𝜑∈𝐻, ‖𝜑‖²=(𝜑,𝜑)=_𝐺||(𝜑,𝜑_𝑔)||²𝑑𝜇,(3.6)

or equivalently, if for all 𝜑₁,𝜑₂∈𝐻, 𝜑₁,𝜑₂=_𝐺𝜑₁,𝜑_𝑔𝜑₂,𝜑_𝑔𝑑𝜇.(3.7)

We define the isomorphic inclusion 𝑉∶𝐻→𝐿₂(𝐺) by the rule 𝑉∶𝜑∈𝐻⟼𝑓=𝑓(𝑔)=𝜑,𝜑_𝑔∈𝐿₂(𝐺).(3.8)

By (3.7) we have (𝜑₁,𝜑₂)=⟨𝑓₁,𝑓₂⟩, where (⋅,⋅) and ⟨⋅,⋅⟩ are the scalar products on 𝐻 and 𝐿₂(𝐺), respectively, and 𝑓_ℎ(𝑔)=𝑓_𝑔(ℎ). Let 𝐻₂(𝐺)=𝑉(𝐻)⊂𝐿₂(𝐺). A function 𝑓∈𝐿₂(𝐺) is an element of 𝐻₂(𝐺) if and only if for all ℎ∈𝐺, ⟨𝑓,𝑓_ℎ⟩=𝑓(ℎ). The operator (𝑃𝑓)(𝑔)=∫_𝐺(𝜑_𝑡,𝜑_𝑔)𝑓(𝑡)𝑑𝜇(𝑡) is the orthogonal projection of 𝐿₂(𝐺) onto 𝐻₂(𝐺).

The function 𝑎(𝑔), 𝑔∈𝐺, is called the anti-Wick (or contravariant) symbol of an operator 𝑇∶𝐻→𝐻 if 𝑉𝑇𝑉⁻¹∣𝐻₂(𝐺)=𝑃𝑎(𝑔)𝑃=𝑃𝑎(𝑔)𝐼∣𝐻₂(𝐺)∶𝐻₂(𝐺)⟶𝐻₂(G),(3.9)

or, in other terminology, the operator 𝑉𝐴𝑉⁻¹∣𝐻₂(𝐺) is the Toeplitz operator 𝑇_𝑎(𝑔)=𝑃𝑎(𝑔)𝐼∣𝐻₂(𝐺)∶𝐻₂(𝐺)⟶𝐻₂(𝐺)(3.10) with the symbols 𝑎(𝑔).

Given an operator 𝑇∶𝐻→𝐻, introduce the (Wick) function 𝑎̃(𝑔,ℎ)=𝑇𝜑^ℎ,𝜑_𝑔𝜑_ℎ,𝜑_𝑔,𝑔,ℎ∈𝐺.(3.11)

If the operator 𝑇 has an anti-Wick symbols, that is, 𝑉𝑇𝑉⁻¹=𝑇_𝑎(𝑔) for some function 𝑎=𝑎(𝑔), then 𝑎̃(𝑔,ℎ)=⟨𝑇^𝑎𝑓_ℎ,𝑓_𝑔⟩⟨𝑓_ℎ,𝑓_𝑔⟩,𝑔,ℎ∈𝐺.(3.12)

And the operator 𝑇_𝑎 admits the following representation in terms of its Wick function: 𝑇_𝑎𝑓(𝑔)==_𝐺𝑎(𝑡)𝑓(𝑡)𝑓_𝑡(𝑔)𝑑𝜇(𝑡)=_𝐺𝑎(𝑡)𝑓_𝑡(𝑔)𝑑𝜇(𝑡)_𝐺𝑓(ℎ)𝑓_ℎ(𝑡)𝑑𝜇(ℎ)_𝐺=𝑓(ℎ)𝑑𝜇(ℎ)_𝐺𝑎(𝑡)𝑓_𝑡(𝑔)𝑓_ℎ(𝑡)𝑑𝜇(𝑡)_𝐺𝑓(ℎ)𝑑𝜇(ℎ)𝑓^ℎ(𝑔)𝑓_ℎ,𝑓_𝑔^𝐺=𝑎(𝑡)𝑓_ℎ(𝑡)𝑓_𝑔(𝑡)𝑑𝜇(𝑡)_𝐺𝑎̃(𝑔,ℎ)𝑓(ℎ)𝑓_ℎ(𝑔)𝑑𝜇(ℎ).(3.13) Interchanging the integrals above, we understand them in a weak sense.

The restriction of the function 𝑎(̃𝑔,ℎ) onto the diagonal 𝑎̃(𝑔)=𝑎̃(𝑔,𝑔)=𝑇𝜑^𝑔,𝜑_𝑔𝜑_𝑔,𝜑_𝑔,𝑔∈𝐺,(3.14)

is called the Wick (or covariant or Berezin) symbols of the operator 𝑇∶𝐻→𝐻.

The Wick and anti-Wick symbols of an operator 𝑇∶𝐻→𝐻 are connected by the Berezin transform 𝑎̃(𝑔)=𝑎̃(𝑔,𝑔)=𝑇𝜑^𝑔,𝜑_𝑔𝜑_𝑔,𝜑_𝑔=⟨𝑇^𝑎𝑓_𝑔,𝑓_𝑔⟩⟨𝑓_𝑔,𝑓_𝑔⟩=∫^𝐺𝑎(𝑡)||𝑓_𝑔(𝑡)||²𝑑𝜇(𝑡)∫_𝐺||𝑓_𝑔(𝑡)||²𝑑𝜇(𝑡).(3.15)

The pluriharmonic Bergman reproducing kernel in the space 𝑏^2𝜇(𝔹^𝑛) has the form 𝑅_𝑧(𝑤)=𝐾_𝑧(𝑤)+𝐾_𝑧(𝑤)−𝑑^2𝑛,𝟘𝜆²(𝑛,0)=^∞_|_𝛼|=0𝑒^𝜇𝛼̃(𝑤)𝑒^𝜇𝛼̃(𝑧)+^∞_|_𝛼|=0𝑒^𝜇𝛼̃(𝑧)𝑒^𝜇𝛼̃(𝑤)−𝑑^2𝑛,𝟘𝜆²(𝑛,0),(3.16) where 𝛼=𝟘=(0,…,0). For 𝑓∈𝑏^2𝜇(𝔹^𝑛), the reproducing property 𝑓(𝑧)=𝑄^𝜇𝔹_𝑛𝑓(𝑧)=_𝔹𝑛𝑓(𝑤)𝑅_𝑧(𝑤)𝜇(|𝑤|)𝑑𝑣(𝑤)(3.17)

shows that the system of functions 𝑅_𝑧(𝑤), 𝑤∈𝔹^𝑛, forms a system of coherent states in the space 𝑏^2𝜇(𝔹^𝑛). In our context, we have 𝐺=𝔹^𝑛, 𝑑𝜇=𝜇(|𝑧|)𝑑𝑥𝑑𝑦, 𝐻=𝐻₂(𝐺)=𝑏^2𝜇(𝔹^𝑛), 𝐿₂(𝐺)=𝐿^2𝜇(𝔹^𝑛), 𝜑_𝑔=𝑓_𝑔=𝑅_𝑔, where 𝑔=𝑧∈𝔹^𝑛.

Lemma 3.3. Let 𝑇_𝑎 be the Toeplitz operator with a radial symbol 𝑎=𝑎(𝑟). Then the corresponding Wick function (3.11) has the form 𝑎̃(𝑧,𝑤)=𝑅^𝑤−1(𝑧)^∞_|_𝛼|=0𝑒^𝜇𝛼̃(𝑤)𝑒^𝜇𝛼̃(𝑧)𝛾_𝑎,𝜇(|𝛼|)+^∞_|_𝛼|=0𝑒^𝜇𝛼̃(𝑧)𝑒^𝜇𝛼̃(𝑤)𝛾_𝑎,𝜇(|𝛼|)−𝑑^2𝑛,𝟘𝜆²(𝑛,0)𝛾_𝑎,𝜇(0).(3.18)

Proof. By (3.11) and (3.16), we have 𝑎̃(𝑧,𝑤)=⟨𝑇^𝑎𝑅_𝑤,𝑅_𝑧⟩⟨𝑅_𝑤,𝑅_𝑧⟩=𝑅^𝑤−1=𝑅_𝑤−1(𝑧)⟨𝑎𝑅_𝑤,𝑅_𝑧⟩=𝑅_𝑤−1(𝑧)⟨𝑎𝐾_𝑤,𝐾_𝑧⟩+𝑎𝐾_𝑤,𝐾_𝑧−𝑑^4𝑛,𝟘𝜆⁴(𝑛,0)⟨𝑎,1⟩(𝑧)^∞_|_𝛼|=0𝑒^𝜇𝛼̃(𝑤)𝑒^𝜇𝛼̃(𝑧)𝑎𝑒^𝜇𝛼̃,𝑒^𝜇𝛼̃+^∞_|_𝛼|=0=𝑅_𝑤−1𝑒^𝜇𝛼̃(𝑧)𝑒^𝜇𝛼̃(𝑤)𝑎𝑒^𝜇𝛼̃,𝑒^𝜇𝛼̃−𝑑^4𝑛,𝟘𝜆⁴(𝑛,0)⟨𝑎,1⟩(𝑧)^∞_|_𝛼|=0𝑒^𝜇𝛼̃(𝑤)𝑒^𝜇𝛼̃(𝑧)𝛾_𝑎,𝜇(|𝛼|)+^∞_|_𝛼|=0𝑒^𝜇𝛼̃(𝑧)𝑒^𝜇𝛼̃(𝑤)𝛾_𝑎,𝜇(|𝛼|)−𝑑^2𝑛,𝟘𝜆²(𝑛,0)𝛾_𝑎,𝜇(0).(3.19)

Denote by 𝐿^𝜇𝛼 the one-dimensional subspace of 𝑏^2𝜇(𝔹^𝑛) generated by the base element 𝑒^𝜇𝛼̃(𝑧), |𝛼|∈ℤ₊. Then the one-dimensional projection 𝑃^𝜇𝛼 of 𝑏^2𝜇(𝔹^𝑛) onto 𝐿^𝜇𝛼 has obviously the form 𝑃^𝜇𝛼𝑓=𝑓,𝑒^𝜇𝛼̃𝑒^𝜇𝛼̃=𝑒^𝜇𝛼̃(𝑧)_𝔹𝑛𝑓(𝑤)𝑒^𝜇𝛼̃(𝑤)𝜇(|𝑤|)𝑑𝑣(𝑤).(3.20) In the similar method, 𝐿^𝜇𝛼 denote the one-dimensional subspace of 𝑏^2𝜇(𝔹^𝑛) generated by the base element 𝑒^𝜇𝛼̃(𝑧). Let 𝑃^𝜇𝛼 be the projection from 𝑏^2𝜇(𝔹^𝑛) onto 𝐿^𝜇𝛼, and the projection can be rewritten as 𝑃^𝜇𝛼𝑓(𝑧)=𝑓,𝑒^𝜇𝛼̃𝑒^𝜇𝛼̃(𝑧)=𝑒^𝜇𝛼̃(𝑧)_𝔹𝑛𝑓(𝑤)𝑒^𝜇𝛼̃(𝑤)𝜇(|𝑤|)𝑑𝑣(𝑤).(3.21)

Theorem 3.4. Let 𝑇_𝑎 be a bounded Toeplitz operator having radial symbol 𝑎(𝑟). Then one can get the spectral decomposition of the operator 𝑇_𝑎: 𝑇_𝑎=^∞_|_𝛼|=0𝛾_𝑎,𝜇(|𝛼|)𝑃^𝜇𝛼+^∞_|_𝛼|=0𝛾_𝑎,𝜇(|𝛼|)𝑃^𝜇𝛼−𝛾_𝑎,𝜇(0)𝑃^𝜇𝟘.(3.22)

Proof. According to (3.13), (3.20), (3.21), and Lemma 3.3, we get 𝑇_𝑎𝑓(𝑧)==_𝔹𝑛𝑎̃(𝑧,𝑤)𝑓(𝑤)𝑅_𝑤(𝑧)𝜇(|𝑤|)𝑑𝑣(𝑤)._𝔹𝑛^∞_|𝛼|=0𝑒^𝜇𝛼̃(𝑤)𝑒^𝜇𝛼̃(𝑧)𝛾_𝑎,𝜇(|𝛼|)+^∞_|_𝛼|=0−𝑑_2𝑛,𝟘𝑒^𝜇𝛼̃(𝑧)𝑒^𝜇𝛼̃(𝑤)𝛾_𝑎,𝜇(|𝛼|)=𝜆²(𝑛,0)𝛾_𝑎,𝜇(0)𝑓(𝑤)𝜇(|𝑤|)𝑑𝑣(𝑤)._∞|_𝛼|=0𝛾_𝑎,𝜇(|𝛼|)𝑃^𝜇𝛼𝑓(𝑧)+^∞_|_𝛼|=0𝛾_𝑎,𝜇(|𝛼|)𝑃^𝜇𝛼𝑓(𝑧)−𝛾_𝑎,𝜇(0)𝑃^𝜇𝟘𝑓(𝑧).(3.23)

The value 𝛾_𝑎,𝜇(|𝛼|) depends only on |𝛼|. Collecting the terms with the same |𝛼| and using the formula 𝑧⋅𝑤^𝑚=_|𝛼|=𝑚𝑚!𝛼!𝑧^𝛼𝑤^𝛼(3.24)

we obtain 𝑎̃(𝑧,𝑤)=𝑅^𝑤−1(𝑧)^∞_𝑚₌₀𝑙(𝑚,𝑛)𝛾_𝑎,𝜇(𝑚)𝑧⋅𝑤^𝑚+𝑤⋅𝑧^𝑚−𝑑^2𝑛,𝟘𝜆²(𝑛,0)𝛾_𝑎,𝜇(0),(3.25) where (𝑙(𝑚,𝑛)=(𝑚+𝑛−1)!/|𝑆^2𝑛−1|𝑚!(𝑛−1)!)𝜆²(𝑛,𝑚). The orthogonal projection of 𝑏^2𝜇(𝔹^𝑛) onto the subspace generated by all element 𝑒^𝜇𝛼̃ with |𝛼|=𝑚, 𝑚∈ℤ₊ can be written as 𝑃^𝜇(𝑚)𝑓(𝑧)=𝑙(𝑚,𝑛)_𝔹𝑛𝑓(𝑤)𝑧⋅𝑤^𝑚𝜇(|𝑤|)𝑑𝑣(𝑤);(3.26)

similarly, 𝑃^𝜇(𝑚)𝑓(𝑧)=𝑙(𝑚,𝑛)_𝔹𝑛𝑓(𝑤)𝑤⋅𝑧^𝑚𝜇(|𝑤|)𝑑𝑣(𝑤)(3.27)

denotes the orthogonal projection from 𝑏^2𝜇(𝔹^𝑛) onto the subspace generated by all elements 𝑒^𝜇𝛼̃ with |𝛼|=𝑚. Therefore, (3.22) has the form 𝑇_𝑎=^∞_𝑚₌₀𝛾_𝑎,𝜇(𝑚)𝑃^𝜇(𝑚)+^∞_𝑚₌₀𝛾_𝑎,𝜇(𝑚)𝑃^𝜇(𝑚)−𝛾_𝑎,𝜇(0)𝑃^𝜇𝟘.(3.28)

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

Corollary 3.5. Let 𝑇_𝑎 be a bounded Toeplitz operator having radial symbol 𝑎(𝑟). Then the Wick symbol of the operator 𝑇_𝑎 is radial as well and is given by the formula 𝑎̃(𝑧)=𝑎̃(𝑟)=𝑅^𝑧−1(𝑧)2^∞_𝑚₌₀𝑙(𝑚,𝑛)𝛾_𝑎,𝜇(𝑚)𝑟^2𝑚−𝑑^2𝑛,𝟘𝜆²(𝑛,0)𝛾_𝑎,𝜇(0),(3.29) where 𝑅_𝑧(𝑧)=2∑^∞_𝑚=0𝑙(𝑚,𝑛)𝑟^2𝑚−𝑑^2𝑛,𝟘𝜆²(𝑛,0).

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

Corollary 3.6. Let 𝑇_𝑎, 𝑇_𝑏 be the Toeplitz operators with the Wick function 𝑎̃(𝑧,𝑤)=𝑅^𝑤−1(𝑧)^∞_𝑚₌₀𝑏̃(𝑧,𝑤)=𝑅_𝑤−1𝑙(𝑚,𝑛)𝛾_𝑎,𝜇(𝑚)𝑧⋅𝑤^𝑚+𝑤⋅𝑧^𝑚−𝑑^2𝑛,𝟘𝜆²(𝑛,0)𝛾_𝑎,𝜇(0),(𝑧)^∞_𝑚₌₀𝑙(𝑚,𝑛)𝛾_𝑏,𝜇(𝑚)𝑧⋅𝑤^𝑚+𝑤⋅𝑧^𝑚−𝑑^2𝑛,𝟘𝜆²(𝑛,0)𝛾_𝑏,𝜇(0),(3.30) respectively. Then the Wick function 𝑐̃(𝑧,𝑤) of the composition 𝑇=𝑇_𝑎𝑇_𝑏 is given by 𝑐̃(𝑧,𝑤)=𝑅^𝑤−1(𝑧)^∞_𝑚₌₀𝑙(𝑚,𝑛)𝛾_𝑏,𝜇(𝑚)𝛾_𝑎,𝜇(𝑚)𝑧⋅𝑤^𝑚+𝑤⋅𝑧^𝑚−𝑑^2𝑛,𝟘𝜆²(𝑛,0)𝛾_𝑏,𝜇(0)𝛾_𝑎,𝜇(0).(3.31)

Proof. According to Lemma 3.3 and (3.25), we have 𝑐̃(𝑧,𝑤)=⟨𝑇^𝑎𝑇_𝑏𝑅_𝑤,𝑅_𝑧⟩⟨𝑅_𝑤,𝑅_𝑧⟩=𝑅^𝑤−1=𝑅_𝑤−1(𝑧)⟨𝑇_𝑏𝑅_𝑤,𝑎𝑅_𝑧⟩=𝑅_𝑤−1(𝑧)_𝔹𝑛𝑇_𝑏𝑅_𝑤(𝑢)𝑅_𝑧(𝑢)𝑎(|𝑢|)𝜇(|𝑢|)𝑑𝑣(𝑢)=𝑅_𝑤−1(𝑧)_𝔹𝑛⟨𝑇_𝑏𝑅_𝑤,𝑅_𝑢⟩𝑅_𝑧(𝑢)𝑎(|𝑢|)𝜇(|𝑢|)𝑑𝑣(𝑢)(𝑧)^∞_|_𝛼|=0𝑒^𝜇𝛼̃(𝑤)𝑒^𝜇𝛼(𝑧)𝛾_𝑏,𝜇(|𝛼|)𝛾_𝑎,𝜇(|𝛼|)+^∞_|_𝛼|=0−𝑑_2𝑛,𝟘𝑒^𝜇𝛼̃(𝑧)𝑒^𝜇𝛼(𝑤)𝛾_𝑏,𝜇(|𝛼|)𝛾_𝑎,𝜇(|𝛼|)=𝑅_𝑤−1𝜆²(𝑛,0)𝛾_𝑏,𝜇(0)𝛾_𝑎,𝜇(0)(𝑧)^∞_𝑚₌₀𝑙(𝑚,𝑛)𝛾_𝑏,𝜇(𝑚)𝛾_𝑎,𝜇(𝑚)𝑧⋅𝑤^𝑚+𝑤⋅𝑧^𝑚−𝑑^2𝑛,𝟘𝜆²(𝑛,0)𝛾_𝑏,𝜇(0)𝛾_𝑎,𝜇(0).(3.32)

Acknowledgment

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