Toeplitz Operators with Radial Symbols on Weighted Pluriharmonic Bergman Spaces over Reinhardt Domains

Abstract

In this paper, we design an operator A restricted to a weighted pluriharmonic Bergman space $b_{\mu }^2\left(\mathrm{\Omega }\right)$ over the Reinhardt domains, with an isometric isomorphism between $b_{\mu }^2\left(\mathrm{\Omega }\right)$ and the subset of $l_2\left({\mathbb{Z}}^n\right)$. Furthermore, we show that Toeplitz operators $T_a$ with radial symbols are unitary to the multiplication operators ${\gamma }_{a}I$ on sequence space $l_2$ by using the operator A. The Wick function of a Toeplitz operator with a radial symbol provides some features to the operator, establishing its spectral decomposition. Finally, we specify the obtained results on the Reinhardt domains for the unit ball.

1. Introduction

For any $\rho =({\rho }_1,{\rho }_2,\dots ,{\rho }_n)$, ${\rho }_j>0$ for $j=1,2,\dots ,n$, the closed polydisc centered at the origin on ${\mathbb{C}}^n$ with polyradius $\rho$ is defined by

$$P(0,\rho )=\{c=(c_1,c_2,\dots ,c_n):|c_j|\le {\rho }_j\mathrm{for}j=1,2,\dots ,n\}.$$

Recall that an open domain $\mathrm{\Omega }\subset {\mathbb{C}}^n$ is called the complete Reinhardt domain centered at the origin if for any $c\in \mathrm{\Omega }$, the polydisc $P(0,\sigma (c\left)\right)$ is a nonempty subset of $\mathrm{\Omega }$, where

$$\sigma \left(c\right)=\left(\right|c_1|,|c_2|,\dots ,|c_n\left|\right).$$

The base of the Reinhardt domain $\mathrm{\Omega }$ is a subset $\sigma \left(\mathrm{\Omega }\right)$ of ${\mathbb{R}}_{+}^n$ defined by

$$\sigma \left(\mathrm{\Omega }\right)=\{\sigma \left(c\right):c=(c_1,c_2,\dots ,c_n)\in \mathrm{\Omega }\}.$$

If the set $log\sigma \left(\mathrm{\Omega }\right)$ is convex, then the Reinhardt domain $\mathrm{\Omega }$ is said to be logarithmically convex. The following useful characterizations were established in [1]:

Theorem 1

(see [1]). Let Ω be the Reinhardt domain. Then, the following are equivalent:

(1) 

Ω is logarithmically convex;

(2) 

Ω is a region of convergence of a power series;

(3) 

Ω is a domain of holomorphy.

Let set $\mathrm{\Omega }\subset {\mathbb{C}}^n$ denote the bounded holomorphy complete Reinhardt centered at the origin. As completed in [2], we introduce a nonnegative measurable weight function $\mu \left(r\right)=\mu (r_1,r_2,\dots ,r_n)$. For $r\in \sigma \left(\mathrm{\Omega }\right)$, we have

$${\int }_{\mathrm{\Omega }}\mu \left(\right|z\left|\right)\mathrm{d}V\left(z\right)={\left(2\pi \right)}^n{\int }_{\sigma \left(\mathrm{\Omega }\right)}\mu (r)r \mathrm{d}r<\infty ,$$

where $r \mathrm{d}r={\prod }_{k=1}^n(r_k \mathrm{d}{r}_k)$, $\left|z\right|=\left(\right|z_1|,|z_2|,\dots ,|z_n\left|\right)$, and $\mathrm{d}V\left(z\right)=\mathrm{d}{x}_1\mathrm{d}{y}_1\dots \mathrm{d}{x}_n\mathrm{d}{y}_n$ is the Lebesgue measure on the n-dimensional complex plane. We select the weight function $\mu \left(r\right)$ to be bounded in a neighborhood of the origin and to not vanish in this neighborhood.

Let $L_2(\mathrm{\Omega },\mu )$ be the weighted Hilbert space with the scalar product

$$\langle f,g\rangle ={\int }_{\mathrm{\Omega }}f\left(z\right)\overline{g\left(z\right)}\mu \left(\right|z\left|\right)\mathrm{d}V\left(z\right).$$

The weighted pluriharmonic Bergman space $b_{\mu }^2\left(\mathrm{\Omega }\right)$ is the subspace of $L_2(\mathrm{\Omega },\mu )$. As we all know that the space $b_{\mu }^2\left(\mathrm{\Omega }\right)$ is a closed subspace of $L_2(\mathrm{\Omega },\mu )$ and hence is a Hilbert space, we can check that $b_{\mu }^2\left(\mathrm{\Omega }\right)=A_{\mu }^2\left(\mathrm{\Omega }\right)+\overline{A_{\mu }^2\left(\mathrm{\Omega }\right)}$, where $A_{\mu }^2\left(\mathrm{\Omega }\right)$ is the weighted holomorphic Bergman space. See [3] for more information.

In the setting of the Bergman-type space, some researchers have undertaken a lot of work on the algebraic properties of Toeplitz operators and Hankel operators. Toeplitz operators on the harmonic Bergman space have many dissimilar properties compared to the Bergman space. The product of analytic functions is still analytic, but this is not true for harmonic functions, which creates some difficulties in the research. Therefore, we need to find more methods and tools. In their paper [4], Choe and Lee characterized the commutativity of Toeplitz operators on the harmonic Bergman space over a unit disk; in particular, they showed that two analytic Toeplitz operators commute only when their symbols and the constant 1 are linearly dependent. However, the commutativity of two analytic Toeplitz operators is always held on analytic Bergman space. Guo and Zheng [5] studied the Toeplitz algebra and the Hankel algebra by using the compactness of Toeplitz operators on the harmonic Bergman space. Lee and Zhu [6] investigated the commutativity and regularity of Toeplitz operators by applying certain integral and differential equations on the pluriharmonic Bergman space over the unit ball; the commutativity of the Toeplitz operators is similar to the case of the unit disk in [4]. Lee [7] proved the commutativity of Toeplitz operators $T_u$ with radial symbol u and Toeplitz operators $T_v$ with pluriharmonic symbol v; he drew particular attention to the fact that the two Toeplitz operators can commute only when at least one of the two symbol functions is a constant on a unit ball. Choe and Nam [8] characterized commuting and normal Toeplitz operators on the pluriharmonic Bergman space of the polydisk, which are similar to those in [7]. On the cutoff harmonic Bergman space over a unit disk, Yang et al. [9,10] separately considered certain algebraic properties of the Toeplitz and little Hankel operators.

In the papers [2,11,12], several mathematicians analyzed the effect of the radial component of a symbol function for the spectral, compactness, Fredholm properties, and $C^{*}$-algebra generated by certain Toeplitz operators on Bergman space. In the paper [13], Li and Lu characterized the related problems of radial operators and Toeplitz operators on the weighted Bergman spaces by using the $(m,\lambda )$-Berezin transform over the polydisk. In [14,15], Sun et al. studied some properties of Toeplitz operators with radial symbols on weighted pluriharmonic Bergman spaces and harmonic Fock spaces, respectively.

Inspired by the above-mentioned results, in this paper, we will research the problems of the Toeplitz operators with radial symbols on the weighted pluriharmonic Bergman space over the Reinhardt domains. The main objective of this paper is to extend a part of the results from the weighted holomorphic Bergman space $A_{\mu }^2\left(\mathrm{\Omega }\right)$ to the weighted pluriharmonic Bergman space $b_{\mu }^2\left(\mathrm{\Omega }\right)$. This method is based on a set of newly added differential equation operators. Meanwhile, the Toeplitz operators with radial and separately radial symbols on the Bergman type can generate $C^{*}$-algebras; for example, see [2,12]. In [16], Quiroga-Barranco further studied the commutation of $C^{*}$-algebra generated by Toeplitz operators with radial and separately radial symbols. The results of this paper can lay a foundation for further research with respect to the algebraic properties on harmonic Bergman spaces. Based on the techniques in [2,14,15,17,18], in Section 2, we will construct an operator A whose restriction onto the weighted pluriharmonic Bergman space $b_{\mu }^2\left(\mathrm{\Omega }\right)$ over the Reinhardt domains is an isometric isomorphism between $b_{\mu }^2\left(\mathrm{\Omega }\right)$ and the subset $l_2\left({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n\right)$ of $l_2\left({\mathbb{Z}}^n\right)$ with

$$A{A}^{*}=I\mathrm{and}{A}^{*}A=P_{\mathrm{\Omega },\mu },$$

where $I:l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)\to l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)$ is an identity operator and $P_{\mathrm{\Omega },\mu }$ is the orthogonal projection of $L_2(\mathrm{\Omega },\mu )$ onto $b_{\mu }^2\left(\mathrm{\Omega }\right)$. We will show that each Toeplitz operator $T_a$ with radial symbols on $b_{\mu }^2\left(\mathrm{\Omega }\right)$ is a unitary equivalent to the multiplication operator ${\gamma }_{a}I$ acting on $l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)$. In Section 3, we will use the Wick and anti-Wick functions of a Toeplitz operator with a radial symbol to give complete theory and provide its spectral decomposition. Finally, in Section 4, we specify the above-obtained results on the Reinhardt domains for the unit ball.

2. Weighted Pluriharmonic Bergman Spaces on Reinhardt Domains and Related Operators

We decompose the space $L_2(\mathrm{\Omega },\mu )$ at the beginning of this section. As carried out in [2], passing to $z_j=t_j{r}_j$, where $t_j\in \mathbb{T}=S^1$ for $j=1,2,\dots ,n$, and under the identification

$$z=(z_1,z_2,\dots ,z_n)=(t_1{r}_1,t_2{r}_2,\dots ,t_n{r}_n)=(t,r),$$

where $r=(r_1,r_2,\dots ,r_n)\in \sigma \left(\mathrm{\Omega }\right)$ and $t=(t_1,t_2,\dots ,t_n)\in {\mathbb{T}}^n$, we obtain

$$\mathrm{d}V\left(z\right)=\prod _{j=1}^n{\displaystyle \frac{\mathrm{d}{t}_j}{i{t}_j}}\prod _{j=1}^n\left(r_j\mathrm{d}{r}_j\right)$$

and

$$\mathrm{\Omega }={\mathbb{T}}^n\times \sigma \left(\mathrm{\Omega }\right),$$

where $i=\sqrt{-1}$ is the imaginary unit. Equivalently,

$$L_2(\mathrm{\Omega },\mu )=L_2\left({\mathbb{T}}^n\right)\otimes L_2(\sigma \left(\mathrm{\Omega }\right),\mu ),$$

where

$$L_2\left({\mathbb{T}}^n\right)=L_2\left(\mathbb{T},{\displaystyle \frac{\mathrm{d}{t}_1}{i{t}_1}}\right)\otimes L_2\left(\mathbb{T},{\displaystyle \frac{\mathrm{d}{t}_2}{i{t}_2}}\right)\otimes \cdots \otimes L_2\left(\mathbb{T},{\displaystyle \frac{\mathrm{d}{t}_n}{i{t}_n}}\right),$$

and

$$\mathrm{d}\mu =\mu (r_1,r_2,\dots ,r_n)\prod _{j=1}^n\left(r_j\mathrm{d}{r}_j\right)(\mathrm{the}\mathrm{measure}\mathrm{d}\mu \mathrm{in}{L}_2(\sigma \left(\mathrm{\Omega }\right),\mu )).$$

Let $f\left(z\right)\in b_{\mu }^2\left(\mathrm{\Omega }\right)$ and write $f\left(z\right)=g\left(z\right)+\overline{h\left(z\right)}$ with $g,h\in A_{\mu }^2\left(\mathrm{\Omega }\right)$. The weighted pluriharmonic Bergman space $b_{\mu }^2\left(\mathrm{\Omega }\right)$ can be described as the closure in $L_2(\mathrm{\Omega },\mu )$ of the set of all smooth functions, satisfying the equations

$${\displaystyle \frac{\partial }{\partial {\overline{z}}_k}}g\left(z_k\right)={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial x_k}}+i{\displaystyle \frac{\partial }{\partial y_k}}\right)g\left(z_k\right)=0$$

and

$${\displaystyle \frac{\partial }{\partial z_k}}\overline{h\left(z_k\right)}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial x_k}}-i{\displaystyle \frac{\partial }{\partial y_k}}\right)\overline{h\left(z_k\right)}=0,$$

where $z_k=x_k+y_{k}i$ for $k=1,2,\dots ,n$. Alternatively, in the polar coordinates, we have

$${\displaystyle \frac{\partial }{\partial {\overline{z}}_k}}g\left(z_k\right)={\displaystyle \frac{t_k}{2}}\left({\displaystyle \frac{\partial }{\partial r_k}}-{\displaystyle \frac{t_k}{r_k}}{\displaystyle \frac{\partial }{\partial t_k}}\right)g\left(z_k\right)=0$$

and

$${\displaystyle \frac{\partial }{\partial z_k}}\overline{h\left(z_k\right)}={\displaystyle \frac{1}{2t_k}}\left({\displaystyle \frac{\partial }{\partial r_k}}+{\displaystyle \frac{t_k}{r_k}}{\displaystyle \frac{\partial }{\partial t_k}}\right)\overline{h\left(z_k\right)}=0$$

for $k=1,2,\dots ,n$.

Some important operators were introduced in [2] and extended as follows. Recall that

$$\mathcal{F}:f\mapsto c_n={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{S^1}f\left(t\right)t^{-n}{\displaystyle \frac{dt}{it}},n\in \mathbb{Z},$$

is the Fourier transform $\mathcal{F}:L_2\left(\mathbb{T}\right)\to l_2\left(\mathbb{Z}\right)$ and

$${\mathcal{F}}^{-1}(\mathrm{inverse}\mathrm{operator}\mathrm{of}\mathcal{F})={\mathcal{F}}^{*}:{\left\{c_n\right\}}_{n\in \mathbb{Z}}\mapsto f\left(t\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}\sum _{n\in \mathbb{Z}}{c}_n{t}^n.$$

(1)

It is known that the operator $\mathcal{F}$ is unitary. In [2], the operator $U_1$ was introduced as

$$U_1=(\mathcal{F}\otimes I){\displaystyle \frac{t}{2}}\left({\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{t}{r}}{\displaystyle \frac{\partial }{\partial t}}\right)\left({\mathcal{F}}^{-1}\otimes I\right),$$

where

$$U_1:l_2\otimes L_2((0,1),r\mathrm{d}r)\to l_2\otimes L_2((0,1),r \mathrm{d}r)$$

such that

$$U_1:{\left\{c_n\left(r\right)\right\}}_{n\in \mathbb{Z}}\mapsto {\left\{{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r}}-{\displaystyle \frac{n-1}{r}}\right)c_{n-1}\left(r\right)\right\}}_{n\in \mathbb{Z}}.$$

In this paper, we define the extended operator $U_2$ by

$$U_2=(\mathcal{F}\otimes I){\displaystyle \frac{1}{2t}}\left({\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{t}{r}}{\displaystyle \frac{\partial }{\partial t}}\right)\left({\mathcal{F}}^{-1}\otimes I\right)$$

such that

$$U_2:{\left\{c_n\left(r\right)\right\}}_{n\in \mathbb{Z}}\mapsto {\left\{{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{n+1}{r}}\right)c_{n+1}\left(r\right)\right\}}_{n\in \mathbb{Z}}.$$

Denote ${\mathcal{F}}_{\left(n\right)}=\underset{\underset{n}{︸}}{\mathcal{F}\otimes \mathcal{F}\otimes \cdots \otimes \mathcal{F}}$. As carried out in [2], the unitary operator U is given by the formula

$$U={\mathcal{F}}_{\left(n\right)}\otimes I:L_2\left({\mathbb{T}}^n\right)\otimes L_2(\sigma \left(\mathrm{\Omega }\right),\mu )\to l_2\left({\mathbb{Z}}^n\right)\otimes L_2(\sigma \left(\mathrm{\Omega }\right),\mu ).$$

(2)

Let $b_1^2$ be the image of the operator U acting on the pluriharmonic Bergman space. Thus, the set $b_1^2$ is the closed subsequences of $l_2\left({\mathbb{Z}}^n\right)\otimes L_2(\sigma \left(\mathrm{\Omega }\right),\mu )$, which consists of all sequences ${\left\{c_q\left(r\right)\right\}}_{q\in {\mathbb{Z}}^n}$ with $r=(r_1,r_2,\dots ,r_n)\in \sigma \left(\mathrm{\Omega }\right)$, satisfying the following differential equations

$${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r_k}}-{\displaystyle \frac{q_k}{r_k}}\right)c_{(q_1,q_2,\dots ,q_n)}(r_1,r_2,\dots ,r_n)=0q_k\in {\mathbb{Z}}_{+}$$

and

$${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r_k}}+{\displaystyle \frac{q_k}{r_k}}\right)c_{(q_1,q_2,\dots ,q_n)}(r_1,r_2,\dots ,r_n)=0q_k\in {\mathbb{Z}}_{-}$$

for $k=1,2,\dots ,n$, where ${\mathbb{Z}}_{-}=\mathbb{Z}\setminus {\mathbb{Z}}_{+}$ and ${\mathbb{Z}}_{+}=\mathbb{N}\cup \left\{0\right\}$. The general solutions of these equations have the following forms

$$c_q\left(r\right)={\alpha }_q{c}_q{r}^q,q=(q_1,q_2,\dots ,q_n)\in {\mathbb{Z}}_{+}^n$$

and

$$c_q\left(r\right)={\alpha }_q{c}_q{r}^{\left|q\right|},q=(q_1,q_2,\dots ,q_n)\in {\mathbb{Z}}_{-}^n,$$

where $c_q\in \mathbb{C}$, $r^q=r_1^{q_1}{r}_2^{q_2}\cdots r_n^{q_n}$, $r^{\left|q\right|}=r_1^{|q_1|}{r}_2^{|q_2|}\cdots r_n^{|q_n|}$, and ${\alpha }_q={\alpha }_{\left|q\right|}$ is given by

$$\begin{array}{cc}\hfill {\alpha }_q& ={\left({\int }_{\sigma \left(\mathrm{\Omega }\right)}{r}^{2\left|q\right|}\mu \left(r\right)r \mathrm{d}r\right)}^{-1/2}\hfill \\ \hfill & ={\left({\int }_{\sigma \left(\mathrm{\Omega }\right)}{r}_1^{2|q_1|}{r}_2^{2|q_2|}\dots r_n^{2|q_n|}\mu (r_1,r_2,\dots ,r_n)\prod _{k=1}^n\left(r_k\mathrm{d}{r}_k\right)\right)}^{-1/2}.\hfill \end{array}$$

(3)

Since $c_q\left(r\right)\in L_2(\sigma \left(\mathrm{\Omega }\right),\mu )$, the subspace $b_1^2\subset l_2\left({\mathbb{Z}}^n\right)\otimes L_2(\tau \left(D\right),\mu )$ corresponds to the space of all sequences

$$c_q\left(r\right)=\{\begin{array}{cc}{\alpha }_q{c}_q{r}^{\left|q\right|},\hfill & q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n\hfill \\ 0,\hfill & \hfill q\in {\mathbb{Z}}^n\setminus ({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)\end{array}$$

and furthermore

$$\parallel {\left\{c_q\left(r\right)\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}{\parallel }_{l_2\left({\mathbb{Z}}^n\right)\otimes L_2(\sigma \left(\mathrm{\Omega }\right),\mu )}={\parallel {\left\{c_q\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}\parallel }_{l_2\left({\mathbb{Z}}^n\right)}.$$

The isometric operator $A_0$ is defined by

$$A_0:l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)\mapsto l_2\left({\mathbb{Z}}^n\right)\otimes L_2(\sigma \left(\mathrm{\Omega }\right),\mu )$$

and

$$A_0:{\left\{c_q\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}\mapsto c_q\left(r\right)=\{\begin{array}{cc}{\alpha }_q{c}_q{r}^{\left|q\right|},\hfill & q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n\hfill \\ 0,\hfill & \hfill q\in {\mathbb{Z}}^n\setminus ({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n).\end{array}$$

(4)

The adjoint operator $A_0^{*}:l_2\left({\mathbb{Z}}^n\right)\otimes L_2(\sigma \left(\mathrm{\Omega }\right),\mu )\mapsto l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)$ has the form

$$A_0^{*}:{\left\{c_q\left(r\right)\right\}}_{q\in {\mathbb{Z}}^n}\mapsto {\left\{{\alpha }_q{\int }_{\sigma \left(\mathrm{\Omega }\right)}{r}^{\left|q\right|}\mu (r_1,r_2,\dots ,r_n)\prod _{k=1}^n\left(r_k\mathrm{d}{r}_k\right)\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}.$$

(5)

Remark 1.

It is not difficult to show the following facts:

(i) 

Let $I:l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)\to l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)$ be an identity operator on $l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)$. Then, we have $R_0^{*}{R}_0=I$.

(ii) 

Let $P_1:l_2\left({\mathbb{Z}}^n\right)\otimes L_2(\sigma \left(\mathrm{\Omega }\right),\mu )\to b_1^2$ denote the orthogonal projection. Then, we have $A_0{A}_0^{*}=P_1$.

Theorem 2.

If the operator $A=A_0^{*}U$ is a mapping from $L_2(\mathrm{\Omega },\mu )$ to $l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)$, then the restriction acting on $b_{\mu }^2$ space

$${R|}_{b_{\mu }^2}:b_{\mu }^2\to l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)$$

is an isometric isomorphism.

Proof. 

This follows from applying (2), (4), and (5) straightforwardly. □

Corollary 1.

The adjoint operator

$$A^{*}=U^{*}{A}_0:l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)\to b_{\mu }^2\left(\mathrm{\Omega }\right)\subset L_2(\mathrm{\Omega },\mu )$$

is the isometric isomorphism of $l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)$ onto the space $b_{\mu }^2\left(\mathrm{\Omega }\right)$.

Remark 2.

It is not difficult to verify the following:

(i) 

Let $I:l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)\to l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)$ be an identity operator on $l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)$. Then, we have $R{R}^{*}=I$.

(ii) 

Let $P_{\mathrm{\Omega },\mu }:L_2(\mathrm{\Omega },\mu )\to b_{\mu }^2\left(\mathrm{\Omega }\right)$ denote the orthogonal projection. Then, we have $A^{*}A=P_{\mathrm{\Omega },\mu }$.

Theorem 3.

The adjoint operator

$$A^{*}=U^{*}{A}_0:l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)\to b_{\mu }^2$$

has the following form

$$A^{*}:{\left\{c_q\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}\mapsto {\displaystyle \frac{1}{{\left(2\pi \right)}^{n/2}}}\left(\sum _{q\in {\mathbb{Z}}_{+}^n}{\alpha }_q{c}_q{z}^q+\sum _{q\in {\mathbb{Z}}_{-}^n}{\alpha }_q{c}_q{\overline{z}}^{\left|q\right|}\right).$$

(6)

Proof. 

Let $\left\{c_p\right\}\in l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)$. According to Corollary 1, as well as the Equations (1), (2), and (4), we obtain

$$A^{*}=U^{*}{A}_0:{\left\{c_q\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}\mapsto U^{*}\left({\left\{{\alpha }_q{c}_q{r}^q\right\}}_{q\in {\mathbb{Z}}_{+}^n}+{\left\{{\alpha }_q{c}_q{r}^{\left|q\right|}\right\}}_{q\in {\mathbb{Z}}_{-}^n}\right)$$

where

$$\begin{array}{cc}\hfill U^{*}\left({\left\{{\alpha }_q{c}_q{r}^q\right\}}_{q\in {\mathbb{Z}}_{+}^n}+{\left\{{\alpha }_q{c}_q{r}^{\left|q\right|}\right\}}_{q\in {\mathbb{Z}}_{-}^n}\right)& ={\displaystyle \frac{1}{{\left(2\pi \right)}^{n/2}}}\left(\sum _{q\in {\mathbb{Z}}_{+}^n}{\alpha }_q{c}_q{\left(rt\right)}^q+\sum _{q\in {\mathbb{Z}}_{-}^n}{\alpha }_q{c}_q{\left(\overline{t}r\right)}^{\left|q\right|}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{{\left(2\pi \right)}^{n/2}}}\left(\sum _{q\in {\mathbb{Z}}_{+}^n}{\alpha }_q{c}_q{z}^q+\sum _{p\in {\mathbb{Z}}_{-}^n}{\alpha }_q{c}_q{\overline{z}}^{\left|q\right|}\right).\hfill \end{array}$$

The proof is completed. □

Corollary 2.

The inverse isometric isomorphism operator $A:b_{\mu }^2\to l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)$ of $A^{*}$ is denoted by

$$\begin{array}{cc}\hfill A:g+\overline{h}\to & {\displaystyle \frac{{\alpha }_q}{{\left(2\pi \right)}^{n/2}}}({\left\{{\int }_{\mathrm{\Omega }}g\left(z\right){\overline{z}}^q\mu \left(\right|z\left|\right)\mathrm{d}v\left(z\right)\right\}}_{q\in {\mathbb{Z}}_{+}^n}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\left\{{\int }_{\mathrm{\Omega }}\overline{h\left(z\right)}{z}^{\left|q\right|}\mu \left(\right|z\left|\right)\mathrm{d}v\left(z\right)\right\}}_{q\in {\mathbb{Z}}_{-}^n}),\hfill \end{array}$$

(7)

where $g,h\in A_{\mu }^2\left(\mathrm{\Omega }\right)$.

3. Study of Toeplitz Operators with Separately Radial Symbols on ${\mathit{b}}_{\mathit{\mu }}^{\mathit{2}}\left(\mathrm{\Omega }\right)$

A function $a\left(z\right)$ with $z\in \mathrm{\Omega }$ is said to be separately radial if it satisfies

$$a\left(z\right)=a\left(r\right)=a(r_1,r_2,\dots ,r_n).$$

In other words, the function $a\left(z\right)$ depends only on the radial components of $z=(t_1{r}_1,t_2{r}_2,\dots ,t_n{r}_n)$.

Theorem 4.

Let $a=a\left(r\right)$ be a bounded measurable separate from the radial function. Then, the Toeplitz operator $T_a$ acting on $b_{\mu }^2\left(\mathrm{\Omega }\right)$ is the unitary equivalent to the multiplication operator ${\gamma }_{a}I=A{T}_a{A}^{*}$ acting on $l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)$, where A and $A^{*}$ are given by (6) and (7), respectively. The sequence ${\gamma }_a={\left\{{\gamma }_a\left(q\right)\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}$ is

$${\gamma }_a\left(q\right)={\alpha }_q^2{\int }_{\sigma \left(\mathrm{\Omega }\right)}a\left(r\right)r^{2\left|q\right|}\mu (r_1,r_2,\dots ,r_n)\prod _{k=1}^n\left(r_k\mathrm{d}{r}_k\right),q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n,$$

(8)

in which ${\alpha }_q$ is showed by (3).

Proof. 

Through the Equations (1) and (2), with the aid of Remark 2, and in view of Theorem 2, we derive that the operator $T_a$ is equivalent to the operator

$$\begin{array}{cc}\hfill A{T}_a{A}^{*}& =A{P}_{\mathrm{\Omega },\mu }a{P}_{\mathrm{\Omega },\mu }{A}^{*}\hfill \\ \hfill & =A\left(A^{*}A\right)a\left(A^{*}A\right)A^{*}\hfill \\ \hfill & =Aa{A}^{*}\hfill \\ \hfill & =A_0^{*}Ua\left(r\right)U^{-1}{A}_0\hfill \\ \hfill & =A_0^{*}\left({\mathcal{F}}_{\left(n\right)}\otimes I\right)a\left(r\right)\left({\mathcal{F}}_{\left(n\right)}^{-1}\otimes I\right)A_0\hfill \\ \hfill & =A_0^{*}a\left(r\right)A_0.\hfill \end{array}$$

The direct computation gives

$$\begin{array}{cc}\hfill A_0^{*}a\left(r\right)A_0{\left\{c_q\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}& =A_0^{*}{\left\{a\left(r\right){\alpha }_q{c}_q{r}^{\left|q\right|}\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}\hfill \\ \hfill & =\left\{{\alpha }_q{\int }_{\sigma \left(\mathrm{\Omega }\right)}{r}^{\left|q\right|}a\left(r\right){\alpha }_q{c}_q{r}^{\left|q\right|}\mu (r_1,r_2,\dots ,r_n)\prod _{k=1}^n{r}_k\mathrm{d}{r}_k\right\}\hfill \\ \hfill & ={\left\{{\gamma }_a\left(q\right)c_q\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n},\hfill \end{array}$$

in which

$${\gamma }_a\left(q\right)={\alpha }_q^2{\int }_{\sigma \left(\mathrm{\Omega }\right)}{r}^{2\left|q\right|}a\left(r\right)\mu (r_1,r_2,\dots ,r_n)\prod _{k=1}^n\left(r_k\mathrm{d}{r}_k\right),q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n.$$

The proof is completed. □

Obviously, the system of functions

$${\left\{e_q\right\}}_{q\in {\mathbb{Z}}_{+}^n}\cup {\left\{\overline{e_q}\right\}}_{q\in {\mathbb{Z}}_{-}^n}$$

(9)

forms an orthonormal base in $b_{\mu }^2\left(\mathrm{\Omega }\right)$, where $e_q\left(z\right)={\displaystyle \frac{{\alpha }_q}{{\left(2\pi \right)}^{n/2}}}{z}^q$ and $\overline{e_q}\left(z\right)={\displaystyle \frac{{\alpha }_q}{{\left(2\pi \right)}^{n/2}}}{\overline{z}}^{\left|q\right|}$.

Corollary 3.

If $a\left(r\right)$ is a bounded measurable separately radial function, then the Toeplitz operator $T_a$ with symbol $a\left(r\right)$ is diagonal with respect to the orthogonal base in (9), satisfying

$$T_a{e}_q={\gamma }_a\left(q\right)e_q,q\in {\mathbb{Z}}_{+}^n$$

(10)

and

$$T_a\overline{e_q}={\gamma }_a\left(q\right)\overline{e_q},q\in {\mathbb{Z}}_{-}^n.$$

(11)

Remark 3.

It is worth noting that the Equations (10) and (11) still hold for the measurable unbounded separately radial symbol function $a=a\left(r\right)\in L_1(\sigma \left(\mathrm{\Omega }\right),\mu )$.

In the following results, we show that the densely defined Toeplitz operators (i.e., defined on the finite linear combinations of elements of the base (9)) can be extended to bounded operators on whole $b_{\mu }^2\left(\mathrm{\Omega }\right)$ if and only if ${\gamma }_a={\left\{{\gamma }_a\left(q\right)\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}$ are bounded.

Corollary 4.

Let $a=a\left(r\right)\in L_1(\sigma \left(\mathrm{\Omega }\right),\mu )$ be a separately radial function; then, the Toeplitz operator $T_a$ with symbol $a\left(r\right)$ is bounded on $b_{\mu }^2\left(\mathrm{\Omega }\right)$ if and only if

$${\gamma }_a={\left\{{\gamma }_a\left(q\right)\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}\in l_{\infty },$$

and in this case

$$\parallel T_a\parallel =\underset{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}{sup}\left|{\gamma }_a\left(q\right)\right|.$$

Recall that the Toeplitz operator $T_a$ is compact if and only if ${\left\{{\gamma }_a\left(q\right)\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}\in c_0$, that is,

$$\underset{q\to \infty }{lim}{\gamma }_a\left(q\right)=0.$$

The spectrum of the bounded Toeplitz operator $T_a$ is given by

$$sp{T}_a={\overline{\left\{{\gamma }_a\left(q\right)\right\}}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}$$

and its essential spectrum corresponds to the set of all limit points of the sequence ${\left\{{\gamma }_a\left(q\right)\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}$.

Let H be a Hilbert space and ${\left\{{\phi }_g\right\}}_{g\in E}$ be a subset of elements of H parameterized by elements g of some set E with a measure $\mathrm{d}\mu$ (see [12,17] for more details). Then, ${\left\{{\phi }_g\right\}}_{g\in E}$ is a system of coherent states if

$$({\phi }_1,{\phi }_2)={\int }_E({\phi }_1,{\phi }_g)\overline{({\phi }_2,{\phi }_g)}\mathrm{d}\mu$$

for all ${\phi }_1,{\phi }_2\in H$.

Define an isomorphic inclusion $V:H\to L_2\left(E\right)$ by the rule

$$V:{\phi }_h\in H\to f=f_h\left(g\right)=({\phi }_h,{\phi }_g)\in L_2\left(E\right).$$

Then, we have $({\phi }_1,{\phi }_2)=\langle f_1,f_2\rangle$, in which $(·,·)$ and $\langle ·,·\rangle$ are the scalar products on H and $L_2\left(E\right)$, respectively, with $f_h\left(g\right)=\overline{f_g\left(h\right)}$.

Let $H_2\left(E\right)=V\left(H\right)\subset L_2\left(E\right)$, and the operator P denote the orthogonal projection of $L_2\left(E\right)$ onto $H_2\left(E\right)$. A function $a\left(g\right),g\in E$ is said to be the anti-wick symbol of an operator $T:H\to H$ if operator $VT{V}^{-1}|H_2\left(E\right)$ is the Toeplitz operator

$$T_{a\left(g\right)}=Pa\left(g\right)I|H_2\left(E\right):H_2\left(E\right)\to H_2\left(E\right)$$

with the symbol $a\left(g\right)$.

As carried out in [12,17], we introduce an operator $T:H\to H$ to show the Wick function

$$\tilde{a}(g,h)={\displaystyle \frac{(T{\phi }_h,{\phi }_g)}{({\phi }_h,{\phi }_g)}},g,h\in E.$$

(12)

If the operator T has an anti-Wick symbol, that is, $VT{V}^{-1}=T_{a\left(g\right)}$ for some functions $a=a\left(g\right)$, then we have

$$\tilde{a}(g,h)={\displaystyle \frac{\langle T_a{f}_h,f_g\rangle }{\langle f_h,f_g\rangle }},g,h\in E$$

and

$$\begin{array}{cc}\hfill \left(T_{a}f\right)\left(g\right)& ={\int }_{E}a\left(t\right)f\left(t\right)f_t\left(g\right)\mathrm{d}\mu \left(t\right)\hfill \\ \hfill & ={\int }_{E}a\left(t\right)f_t\left(g\right)\mathrm{d}\mu \left(t\right){\int }_{E}f\left(h\right)f_h\left(t\right)\mathrm{d}\mu \left(h\right)\hfill \\ \hfill & ={\int }_{E}f\left(h\right)\mathrm{d}\mu \left(h\right){\int }_{E}a\left(t\right)f_t\left(g\right)f_h\left(t\right)\mathrm{d}\mu \left(t\right)\hfill \\ \hfill & ={\int }_{E}f\left(h\right){\displaystyle \frac{f_h\left(g\right)}{\langle f_h,f_g\rangle }}\mathrm{d}\mu \left(h\right){\int }_{E}a\left(t\right)f_h\left(t\right)\overline{f_g\left(t\right)}\mathrm{d}\mu \left(t\right)\hfill \\ \hfill & ={\int }_E\tilde{a}(g,h)f\left(h\right)f_h\left(g\right)\mathrm{d}\mu \left(h\right).\hfill \end{array}$$

(13)

Lemma 1

([2] (Lemma 7.1)). The Bergman kernel $K_{\mathrm{\Omega }}$ of the domain Ω admits the representation

$$K_{\mathrm{\Omega }}(z,\zeta )={\displaystyle \frac{1}{{\left(2\pi \right)}^n}}\sum _{q\in {\mathbb{Z}}_{+}^n}{\alpha }_q^2{z}^q{\overline{\zeta }}^q,$$

in which the coefficient ${\alpha }_q,q\in {\mathbb{Z}}_{+}^n$ is shown by (3).

Since $b_{\mu }^2\left(\mathrm{\Omega }\right)=A_{\mu }^2\left(\mathrm{\Omega }\right)+\overline{A_{\mu }^2\left(\mathrm{\Omega }\right)}$, there is the relation

$$\begin{array}{cc}\hfill R_{\mathrm{\Omega },\zeta }\left(z\right)& =K_{\mathrm{\Omega }}(z,\zeta )+\overline{K_{\mathrm{\Omega }}(z,\zeta )}-{\displaystyle \frac{1}{{\left(2\pi \right)}^n}}{\alpha }_{\theta }^2\hfill \\ \hfill & ={\displaystyle \frac{1}{{\left(2\pi \right)}^n}}\left[\sum _{q\in {\mathbb{Z}}_{+}^n}{\alpha }_q^2{z}^q{\overline{\zeta }}^q+\sum _{q\in {\mathbb{Z}}_{-}^n}{\alpha }_q^2{\overline{z}}^{\left|q\right|}{\zeta }^{\left|q\right|}\right],\hfill \end{array}$$

(14)

between pluriharmonic Bergman kernel $R_{\mathrm{\Omega },z}$ and the Bergman kernel $K_{\mathrm{\Omega }}(z,\zeta )$, where $\theta =(\underset{\underset{n}{︸}}{0,0,\dots ,0})$. For $f\in b_{\mu }^2\left(\mathrm{\Omega }\right)$, the reproducing property

$$f\left(z\right)=\left(P_{\mathrm{\Omega },\mu }f\right)\left(z\right)={\int }_{\mathrm{\Omega }}f\left(\zeta \right)R_{\mathrm{\Omega },\zeta }\left(z\right)\mu \left|\zeta \right|\mathrm{d}V\left(\zeta \right)$$

shows that the system of the functions $R_{\mathrm{\Omega },\zeta }\left(z\right)$ for $\zeta \in \mathrm{\Omega }$ forms a system of coherent states in the space $b_{\mu }^2\left(\mathrm{\Omega }\right)$. In our context, we have $E=\mathrm{\Omega }$, $\mathrm{d}\mu =\mu \left(\right|z\left|\right)\mathrm{d}V\left(z\right)$, $H=H_2\left(E\right)=b_{\mu }^2\left(\mathrm{\Omega }\right)$, $L_2\left(E\right)=L_{\mu }^2\left(\mathrm{\Omega }\right)$, and ${\phi }_g=f_g=R_{\mathrm{\Omega },g}$, where $g=z\in \mathrm{\Omega }$.

Lemma 2.

Let $T_a$ be the Toeplitz operator with a radial symbol $a=a\left(r\right)$. Then, the Wick function (12) is given by

$$R_{\mathrm{\Omega },w}^{-1}\left(z\right){\displaystyle \frac{1}{{\left(2\pi \right)}^n}}\left(\sum _{q\in {\mathbb{Z}}_{+}^n}{\alpha }_q^2{\overline{w}}^q{z}^q{\gamma }_a\left(q\right)+\sum _{q\in {\mathbb{Z}}_{-}^n}{\alpha }_q^2{w}^{\left|q\right|}{\overline{z}}^{\left|q\right|}{\gamma }_a\left(q\right)\right).$$

Proof. 

By the Equations (13) and (14), we have

$$\begin{array}{cc}\hfill \tilde{a}(z,w)& ={\displaystyle \frac{\langle T_a{R}_{\mathrm{\Omega },w},R_{\mathrm{\Omega },z}\rangle }{\langle R_{\mathrm{\Omega },w},R_{\mathrm{\Omega },z}\rangle }}\hfill \\ \hfill & =R_{\mathrm{\Omega },w}^{-1}\left(z\right)\langle a{R}_{\mathrm{\Omega },w},R_{\mathrm{\Omega },z}\rangle \hfill \\ \hfill & =R_{\mathrm{\Omega },w}^{-1}\left(z\right)〈a\left(r\right)\left(\sum _{q\in {\mathbb{Z}}_{+}^n}{\alpha }_q^2{\overline{w}}^q{\overline{\zeta }}^q+\sum _{q\in {\mathbb{Z}}_{-}^n}{\alpha }_q^2{w}^{\left|q\right|}{\overline{\zeta }}^{\left|q\right|}\right),\sum _{q\in {\mathbb{Z}}_{+}^n}{\alpha }_q^2{\overline{z}}^q{\overline{\zeta }}^q+\sum _{q\in {\mathbb{Z}}_{-}^n}{\alpha }_q^2{z}^{\left|q\right|}{\overline{\zeta }}^{\left|q\right|}〉\hfill \\ \hfill & =R_{\mathrm{\Omega },w}^{-1}\left(z\right)\left(\sum _{q\in {\mathbb{Z}}_{+}^n}{\alpha }_q^4{\overline{w}}^q{z}^q\langle {\zeta }^q,{\zeta }^q\rangle +\sum _{q\in {\mathbb{Z}}_{-}^n}{\alpha }_q^4{w}^{\left|q\right|}{\overline{z}}^{\left|q\right|}〈{\overline{\zeta }}^{\left|q\right|},{\overline{\zeta }}^{\left|q\right|}〉\right)\hfill \\ \hfill & =R_{\mathrm{\Omega },w}^{-1}\left(z\right){\displaystyle \frac{1}{{\left(2\pi \right)}^n}}\left(\sum _{q\in {\mathbb{Z}}_{+}^n}{\alpha }_q^2{\overline{w}}^q{z}^q{\gamma }_a\left(q\right)+\sum _{q\in {\mathbb{Z}}_{-}^n}{\alpha }_q^2{w}^{\left|q\right|}{\overline{z}}^{\left|q\right|}{\gamma }_a\left(q\right)\right).\hfill \end{array}$$

The proof is completed. □

Denote the one-dimensional subspaces of $b_{\mu }^2\left(\mathrm{\Omega }\right)$, generated by element ${\left\{e_q\right\}}_{q\in {\mathbb{Z}}_{+}^n}$ and ${\left\{\overline{e_q}\right\}}_{q\in {\mathbb{Z}}_{-}^n}$, and by $L_{\mu }$ and $\overline{L_{\mu }}$, respectively. The orthogonal projections $P_{\mu }:b_{\mu }^2\left(\mathrm{\Omega }\right)\to L_{\mu }$ and $\overline{P_{\mu }}:b_{\mu }^2\left(\mathrm{\Omega }\right)\to \overline{L_{\mu }}$ are of the forms

$$\left(P_{\mu }f\right)\left(z\right)=\langle f,e_q\rangle e_q=e_q{\int }_{\mathrm{\Omega }}f\left(w\right)\overline{e_q\left(w\right)}\mu \left(\right|w\left|\right)\mathrm{d}V\left(w\right),q\in {\mathbb{Z}}_{+}^n$$

(15)

and

$$\left(\overline{P_{\mu }}f\right)\left(z\right)=\langle f,\overline{e_q}\rangle \overline{e_q}=\overline{e_q}{\int }_{\mathrm{\Omega }}f\left(w\right)\overline{e_q\left(w\right)}\mu \left(\right|w\left|\right)\mathrm{d}V\left(w\right),q\in {\mathbb{Z}}_{-}^n.$$

(16)

Theorem 5.

Let $T_a$ be a bounded Toeplitz operator having the radial symbol $a\left(r\right)$. Writing the Toeplitz operator $T_a$ in the form of an operator with a Wick symbol gives the following spectral decomposition of the operator $T_a$:

$$T_a=\sum _{q\in {\mathbb{Z}}_{+}^n}{\gamma }_a\left(q\right)P_{\mu }+\sum _{q\in {\mathbb{Z}}_{-}^n}{\gamma }_a\left(q\right)\overline{P_{\mu }}.$$

Proof. 

Using (13), (15), and (16) and applying Lemma 2, we gain

$$\begin{array}{cc}\hfill \left(T_{a}f\right)\left(z\right)& ={\int }_{\mathrm{\Omega }}\tilde{a}(z,w)f\left(w\right)R_{\mathrm{\Omega },w}\left(z\right)\mu \left(\right|w\left|\right)\mathrm{d}V\left(w\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{{\left(2\pi \right)}^n}}{\int }_{\mathrm{\Omega }}\left(\sum _{q\in {\mathbb{Z}}_{+}^n}{\alpha }_q^2{\overline{w}}^q{z}^q{\gamma }_a\left(q\right)+\sum _{q\in {\mathbb{Z}}_{-}^n}{\alpha }_q^2{w}^{\left|q\right|}{\overline{z}}^{\left|q\right|}{\gamma }_a\left(q\right)\right)f\left(w\right)\mu \left(\right|w\left|\right)\mathrm{d}V\left(w\right)\hfill \\ \hfill & =\sum _{q\in {\mathbb{Z}}_{+}^n}{\gamma }_a\left(q\right)P_{\mu }f\left(z\right)+\sum _{q\in {\mathbb{Z}}_{-}^n}{\gamma }_a\left(q\right)\overline{P_{\mu }}f\left(z\right).\hfill \end{array}$$

Thus, our conclusion is proved. □

4. Toeplitz Operator with Separately Radial Functions on ${\mathit{b}}_{\mathit{\lambda }}^{\mathit{2}}\left({\mathbb{B}}^{\mathit{n}}\right)$

Let the unit ball ${\mathbb{B}}^n$ be a Reinhardt domain; the base $\sigma \left({\mathbb{B}}^n\right)$ is defined by

$$\sigma \left({\mathbb{B}}^n\right)=\{\rho =({\rho }_1,{\rho }_2,\dots ,{\rho }_n)=\left(\right|z_1|,|z_2|,\dots ,|z_n\left|\right):{\rho }^2={\rho }_1^2+{\rho }_2^2+\cdots +{\rho }_n^2\in [0,1]\}\subset {\mathbb{R}}_{+}^n.$$

Let the function of weight (see, e.g., [18]) be

$${\mu }_{\lambda }\left(\right|z\left|\right)=c_{\lambda }{(1-|z|}^2{)}^{\lambda },$$

and the normalizing constant be

$$c_{\lambda }={\displaystyle \frac{\mathrm{\Gamma }(\lambda +n+1)}{{\pi }^n\mathrm{\Gamma }(\lambda +1)}}.$$

Meanwhile, the classical Euler gamma function $\mathrm{\Gamma }\left(z\right)$ can be defined (see [19] (Chapter 3) and [20] (Section 1)) by

$$\mathrm{\Gamma }\left(z\right)=\underset{n\to \infty }{lim}{\displaystyle \frac{n!n^z}{{\prod }_{k=0}^n(z+k)}},z\in \mathbb{C}\setminus \{0,-1,-2,\dots \}.$$

Thus, the element ${\mu }_{\lambda }\left(\right|z\left|\right)\mathrm{d}V\left(z\right)$ acting on ${\mathbb{B}}^n$ is a probability measure.

The pluriharmonic Bergman space $b_{\lambda }^2\left({\mathbb{B}}^n\right)$ is the subspace of the Lebesgue space $L_2({\mathbb{B}}^n,{\mu }_{\lambda })$ consisting of all complex-valued pluriharmonic functions on ${\mathbb{B}}^n$. One can check that $b_{\lambda }^2\left({\mathbb{B}}^n\right)=A_{\lambda }^2\left({\mathbb{B}}^n\right)+\overline{A_{\lambda }^2\left({\mathbb{B}}^n\right)}$, in which $A_{\lambda }^2\left({\mathbb{B}}^n\right)$ represents the holomorphic functions of $L_2({\mathbb{B}}^n,{\mu }_{\lambda })$. The orthogonal projection $P_{\lambda }:L_2({\mathbb{B}}^n,{\mu }_{\lambda })\to b_{\lambda }^2\left({\mathbb{B}}^n\right)$ satisfies

$$P_{\lambda }\phi (z,\overline{z})={\int }_{B^n}\phi \left(\xi \right)\left[k_{\lambda }(z,\xi )+\overline{k_{\lambda }(z,\xi )}-1\right]{\mu }_{\lambda }\left(\right|\xi \left|\right)\mathrm{d}V\left(\xi \right),$$

where

$$k_{\lambda }(z,\xi )={\displaystyle \frac{1}{{\left(1-{\sum }_{k=1}^n{z}_k\overline{{\xi }_k}\right)}^{n+1+\lambda }}}.$$

Introduce in ${\mathbb{C}}^n$ the polar coordinates

$$z=(z_1,z_2,\dots ,z_n)=(t_1{\rho }_1,t_2{\rho }_2,\dots ,t_n{\rho }_n)=(t,\rho ),$$

where $t=(t_1,t_2,\dots ,t_n)\in {\mathbb{T}}^n$ and $\rho =({\rho }_1,{\rho }_2,\dots ,{\rho }_n)\in \sigma \left({\mathbb{B}}^n\right)$. We show

$${\mathbb{B}}^n={\mathbb{T}}^n\times \sigma \left({\mathbb{B}}^n\right)$$

and

$$L_2({\mathbb{B}}^n,{\mu }_{\lambda })=L_2\left({\mathbb{T}}^n\right)\otimes L_2(\sigma \left({\mathbb{B}}^n\right),\mu ),$$

in which

$$L_2\left({\mathbb{T}}^n\right)=\underset{j=1}{\overset{n}{⨂}}{L}_2\left(\mathbb{T},{\displaystyle \frac{\mathrm{d}{t}_j}{i{t}_j}}\right).$$

The measure $\mathrm{d}\mu$ of $L_2(\sigma \left({\mathbb{B}}^n\right),\mu )$ has the form

$$\mathrm{d}\mu ={\mu }_{\lambda }\left(\rho \right)\prod _{j=1}^n\left({\rho }_j\mathrm{d}{\rho }_j\right)=c_{\lambda }{\left(1-{\rho }^2\right)}^{\lambda }\prod _{j=1}^n\left({\rho }_j\mathrm{d}{\rho }_j\right).$$

The unitary operator $U_{\lambda }$ is given by

$$U_{\lambda }={\mathcal{F}}_{\left(n\right)}\otimes I:L_2\left({\mathbb{T}}^n\right)\otimes L_2(\sigma \left({\mathbb{B}}^n\right),\mu )\to l_2\left({\mathbb{Z}}^n\right)\otimes L_2(\sigma \left({\mathbb{B}}^n\right),\mu ).$$

To calculate the constant ${\alpha }_q$ for $q=(q_1,q_2,\dots ,q_n)\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n$, see (3), consider the integral

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{B}}^n}|z^{|q|}{|}^2{\mu }_{\lambda }(|z|)\mathrm{d}V\left(z\right)& ={\int }_{{\mathbb{B}}^n}|z_1^{2|q_1|}\parallel z_2^{2|q_2|}|\dots |z_n{|}^{2|q_n|}|{\mu }_{\lambda }(|z|)\mathrm{d}V\left(z\right)\hfill \\ \hfill & ={\int }_{{\mathbb{T}}^n}\prod _{i=1}^n{\displaystyle \frac{\mathrm{d}{t}_k}{i{t}_k}}{\int }_{\sigma \left({\mathbb{B}}^n\right)}{\rho }_1^{2|q_1|}{\rho }_2^{2|q_2|}\dots {\rho }_n^{2|q_n|}{\mu }_{\lambda }\left(\rho \right)\prod _{k=1}^n\left({\rho }_k\mathrm{d}{\rho }_k\right)\hfill \\ \hfill & ={\left(2\pi \right)}^n{\alpha }_q^{-2}.\hfill \end{array}$$

On the other hand, by virtue of [18] (Lemma 1.11), we have

$${\int }_{{\mathbb{B}}^n}|z^{|q|}{|}^2{\mu }_{\lambda }(|z|)\mathrm{d}V\left(z\right)={\displaystyle \frac{\left|q\right|!\mathrm{\Gamma }(n+\lambda +1)}{\mathrm{\Gamma }(n+\lambda +1+{\sum }_{i=1}^n|q_i\left|\right)}},$$

where $\left|q\right|!=|q_1|!|q_2|!\dots |q_n|!$, that is,

$${\alpha }_q={\left[{\displaystyle \frac{{\left(2\pi \right)}^n\mathrm{\Gamma }(n+\lambda +1+{\sum }_{i=1}^n|q_i\left|\right)}{\left|q\right|!\mathrm{\Gamma }(n+\lambda +1)}}\right]}^{1/2}.$$

(17)

Applying (17) and Theorem 4 for the Reinhardt domain of unit ball ${\mathbb{B}}^n$ leads to the following theorem.

Theorem 6.

Let $T_a$ be a bounded Toeplitz operator with measurable separately radial function $a=a\left(\rho \right)$. Then, the operator $T_a$ acting on $b_{\lambda }^2\left({\mathbb{B}}^n\right)$ is the unitary equivalent to the multiplication operator ${\gamma }_{a}I=A{T}_a{A}^{*}$ acting on $l_2({\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n)$, and the sequence ${\gamma }_{a,\lambda }={\left\{{\gamma }_{a,\lambda }\left(q\right)\right\}}_{q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n}$ is given by

$$\begin{array}{cc}\hfill {\gamma }_{a,\lambda }\left(q\right)& ={\displaystyle \frac{{\left(2\pi \right)}^n\mathrm{\Gamma }(n+\lambda +1+{\sum }_{i=1}^n|q_i\left|\right)}{\left|q\right|!\mathrm{\Gamma }(n+\lambda +1)}}{\int }_{\sigma \left({\mathbb{B}}^n\right)}a\left(\rho \right){\rho }^{2\left|q\right|}{\left(1-{\rho }^2\right)}^{\lambda }\prod _{k=1}^n\left({\rho }_k\mathrm{d}{\rho }_k\right)\hfill \\ \hfill & ={\displaystyle \frac{{\pi }^n\mathrm{\Gamma }(n+\lambda +1+{\sum }_{i=1}^n|q_i\left|\right)}{\left|q\right|!\mathrm{\Gamma }(n+\lambda +1)}}{\int }_{∆\left({\mathbb{B}}^n\right)}a\left(\sqrt{\rho }\right){\rho }^{\left|q\right|}{\left[1-\left(\sum _{i=1}^n{\rho }_i\right)\right]}^{\lambda }\prod _{k=1}^n\mathrm{d}{\rho }_k\hfill \end{array}$$

for $q\in {\mathbb{Z}}_{+}^n\cup {\mathbb{Z}}_{-}^n$, where

$$∆\left({\mathbb{B}}^n\right)=\{\rho =({\rho }_1,{\rho }_2,\dots ,{\rho }_n):{\rho }_1+{\rho }_2+\dots +{\rho }_n\in [0,1],{\rho }_j\ge 0,j=1,2,\dots ,n\}$$

and $\sqrt{\rho }=(\sqrt{{\rho }_1},\sqrt{{\rho }_2},\cdots ,\sqrt{{\rho }_n})$.

5. Conclusions

In this paper, we show that a new operator A restricted to a weighted pluriharmonic Bergman space $b_{\mu }^2\left(\mathrm{\Omega }\right)$ over the Reinhardt domains is an isometric isomorphism between $b_{\mu }^2\left(\mathrm{\Omega }\right)$ and the subset of $l_2\left({\mathbb{Z}}^n\right)$; see Theorems 2 and 3. We showed that each Toeplitz operator $T_a$ with radial symbols is unitary to the multiplication operator ${\gamma }_{a}I$ acting on $l_2$ and established the fact that the Wick function gives some new features for the operator by providing its spectral decomposition; see Theorems 4 and 5. Finally, we proved the case of obtained results on the Reinhardt domains for the unit ball, see Theorem 6. On the basis of the theory of analytic functions, we characterized algebraic propertiesof of Toeplitz operator $T_a$ with radial symbols in the setting of pluriharmonic Bergman space. We believe that our results in this paper will assist us in further obtaining novel expression results related to the $C^{*}$-algebra generated by Toeplitz operators with radial symbols on harmonic Bergman spaces. The conclusion of this paper can provide a basis for the progressive expansion and development of the content of operator algebra.