Toeplitz Operators on Harmonic Fock Spaces with Radial Symbols

Abstract

The main aim of this paper is to study new features and specific properties of the Toeplitz operator with radial symbols in harmonic Fock spaces. A new spectral decomposition of a Toeplitz operator with Wick symbols is also established.

1. Introduction and Preliminaries

Let $\mathbb{C}$ denote the complex plane and $\mathrm{d}\mu =\frac{\mathrm{d}v\left(z\right)}{\pi e^{z\overline{z}}}$ be the Gaussian measure on $\mathbb{C}$, where $\mathrm{d}v\left(z\right)=\mathrm{d}x \mathrm{d}y$ is the standard Lebesgue plane measure on $\mathbb{C}\simeq {\mathbb{R}}^2$. Denote by $L^2(\mathbb{C},\mathrm{d}\mu )$ the Hilbert space of square Lebesgue integrable functions on $\mathbb{C}$ with $\mathrm{d}\mu$. The Fock space $F^2\left(\mathbb{C}\right)$ is the Hilbert space of analytic functions which belong to $L^2(\mathbb{C},\mathrm{d}\mu )$. In fact, the Fock space $F^2\left(\mathbb{C}\right)$ is a reproducing function space of the reproducing kernel

$$K_z\left(w\right)=e^{\overline{z}w}=\sum _{n=0}^{\infty }\frac{{\left(\overline{z}w\right)}^n}{n!}.$$

The harmonic Fock space $F_h^2\left(\mathbb{C}\right)$ is the subspace of $L^2(\mathbb{C},\mathrm{d}\mu )$ consisting of all harmonic functions on $\mathbb{C}$. As in the harmonic Bergman space, it is known (see, e.g., [1]) that

$$F_h^2\left(\mathbb{C}\right)=F^2\left(\mathbb{C}\right)+\overline{F^2\left(\mathbb{C}\right)},$$

where $\overline{F^2\left(\mathbb{C}\right)}=\left\{\overline{f}\right|f\in F^2\left(\mathbb{C}\right)\}$. The space $F_h^2\left(\mathbb{C}\right)$ is also a reproducing function space with the reproducing kernel

$$R_z\left(w\right)=K_z\left(w\right)+\overline{K_z\left(w\right)}-1,z,w\in \mathbb{C}.$$

The concept and properties of Fock space have been improved and generalized in many various different directions by several authors; for more details, see, e.g., [1,2,3,4] and the references therein.

Denote by P the orthogonal Bargmann projection of $L^2(\mathbb{C},\mathrm{d}\mu )$ into the Fock space $F^2\left(\mathbb{C}\right)$. Then we have

$$\left(P\phi \right)\left(z\right)=\langle \phi ,K_z\rangle$$

for $\phi \in L^2(\mathbb{C},\mathrm{d}\mu )$, where the notation $\langle ·,·\rangle$ denotes the standard inner product in $L^2(\mathbb{C},\mathrm{d}\mu )$.

Let Q be the harmonic Bargmann projection from $L^2(\mathbb{C},\mathrm{d}\mu )$ into harmonic Fock space $F_h^2\left(\mathbb{C}\right)$. For a function $a\in L^2(\mathbb{C},\mathrm{d}\mu )$, the Toeplitz operator $T_a:F_h^2\left(\mathbb{C}\right)\to F_h^2\left(\mathbb{C}\right)$ with symbol a is the linear operator defined by

$$T_{a}f=Q\left(af\right)=\langle af,R_z\rangle =P\left(af\right)+\overline{P\left(\overline{af}\right)}+P\left(af\right)\left(0\right),f\in F_h^2\left(\mathbb{C}\right),$$

where $T_a$ is densely defined and not bounded in general.

The theory of the Toeplitz operator stems from a series of ideas developed in the second half of the 20th century by mathematicians such as Otto Toeplitz, Hermann Hankel, Eberhard Hopf, Norbert Wiener and Gábor Szegö. In recent years, many scholars have obtained many important results for the Toeplitz operator $T_a$ and the Hankel operator in Fock space $F^2\left(\mathbb{C}\right)$. The boundedness, compactness, algebraic properties and the product properties on Fock-type spaces have been studied extensively. For more details, we refer the readers to, for example, [1,3,5,6,7,8,9,10,11,12,13,14,15]. Because the product of two harmonic functions is often no longer a harmonic function, it is more difficult to study Toeplitz operators in harmonic Fock space than in analytic Fock space. Many of the methods used for operators in the analytic Fock space lose their effectiveness in harmonic Fock space. Therefore, many scholars attempted to provide some new ideas and methods to overcome such situations and generalize Fock spaces. For example, in the paper [2], Chen et alia considered the Toeplitz operator $T_a$ in vector-valued generalized Fock spaces. In the paper [4], He and Wu characterized dual Toeplitz operators in the orthogonal complement of Fock–Sobolev spaces.

There have been some results of the Toeplitz operator $T_a$ in harmonic Bergman space; please refer to [16,17,18]. In the paper [16], Guo and Zheng characterized compact Toeplitz operators in the unit disk $\mathbb{D}$. In [18], by using the system of integral equations, Lee characterized the commuting Toeplitz operators of holomorphic symbols and pluriharmonic symbols in pluriharmonic Bergman space. In addition, Lee proved in [17] the commutativity of an operator with a radial symbol and an operator with a pluriharmonic symbol in pluriharmonic Bergman space. In the papers [7,19,20,21,22], several mathematicians analyzed the influence of the radial component of a symbol of the spectral, compactness and Fredholm properties of Toeplitz operators on Bergman space or Fock space. In the paper [21], Li and Lu characterized radial symbols in Bergman spaces over the polydisk. In [22], the first author of this paper and her coauthor gave some properties about Toeplitz operators in weighted pluriharmonic Bergman spaces with radial symbols.

In this paper, inspired by the above, we are committed to investigating the problem of Toeplitz operators with radial symbols in a harmonic Fock space $F_h^2\left(\mathbb{C}\right)$. Basing our work on the techniques used in [7,19,20,22], we will construct an operator R whose restriction in harmonic Fock space $F_h^2\left(\mathbb{C}\right)$ is an isometric isomorphism between $F_h^2\left(\mathbb{C}\right)$ and ${\ell }_2$, that is,

$$R{R}^{*}=I:{\ell }_2\to {\ell }_2$$

and

$$R^{*}R=Q:L^2(\mathbb{C},\mathrm{d}\mu )\to F_h^2\left(\mathbb{C}\right),$$

where I is the identity operator. Employing the operator R, we will prove that each Toeplitz operator $T_a$ with radial symbols is unitary to the multiplication operator ${\gamma }_{a}I$. Making use of the Berezin concept of Wick and anti-Wick symbols (see [1,5,23]), we will show that, in our particular case (radial symbols), the Wick symbols of a Toeplitz operator give complete information about the operator and provide its spectral decomposition.

2. New Results for Harmonic Fock Spaces and Related Operators

We now recall some notations, definitions and well-known facts about harmonic Fock space. Let $f\left(z\right)\in F_h^2\left(\mathbb{C}\right)$ and write $f=g+\overline{h}$. The harmonic Fock space $F_h^2\left(\mathbb{C}\right)$ can be described as the closure in $L^2(\mathbb{C},\mathrm{d}\mu )$ of the set of all smooth functions satisfying the equations

$$\frac{\partial }{\partial \overline{z}}g=\frac{1}{2}\left(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y}\right)g=0$$

and

$$\frac{\partial }{\partial z}\overline{h}=\frac{1}{2}\left(\frac{\partial }{\partial x}-i\frac{\partial }{\partial y}\right)\overline{h}=0,$$

where $z=x+yi$.

As performed in [7], introduce the unitary operator

$$U_1:L_2(\mathbb{C},\mathrm{d}\mu )\to L_2\left({\mathbb{R}}^2,\mathrm{d}x \mathrm{d}y\right)$$

with the rule

$$\left(U_1\phi \right)(x,y)=\frac{\phi (x+yi)}{{\pi }^{1/2}{e}^{(x^2+y^2)/2}}=\frac{\phi \left(z\right)}{{\pi }^{1/2}{e}^{z\overline{z}/2}}.$$

Then the image $F^{\left(1\right)}=U_1\left(F_h^2\left(\mathbb{C}\right)\right)$ of the harmonic Fock space $F_h^2\left(\mathbb{C}\right)$ is the closure of the set of all smooth functions $f=g+\overline{h}$ in $L^2({\mathbb{R}}^2,\mathrm{d}x \mathrm{d}y)$, which satisfies the equations

$$D^{\left(1\right)}g=U_1\frac{\partial }{\partial \overline{z}}{U}_1^{-1}g=\left(\frac{\partial }{\partial \overline{z}}+\frac{z}{2}\right)g=\frac{1}{2}\left(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y}+x+yi\right)g=0$$

and

$$D^{\left(1\right)}\overline{h}=U_1\frac{\partial }{\partial z}{U}_1^{-1}\overline{h}=\left(\frac{\partial }{\partial z}+\frac{\overline{z}}{2}\right)\overline{h}=\frac{1}{2}\left(\frac{\partial }{\partial x}-i\frac{\partial }{\partial y}+x-yi\right)\overline{h}=0.$$

Passing to polar coordinates in ${\mathbb{R}}^2$, we have

$$L_2({\mathbb{R}}^2,\mathrm{d}x \mathrm{d}y)=L_2({\mathbb{R}}_{+},r\mathrm{d}r)\otimes L_2([0,2\pi ),\mathrm{d}\alpha )=L_2({\mathbb{R}}_{+},r\mathrm{d}r)\otimes L_2\left(S^1,\frac{\mathrm{d}t}{ti}\right),$$

where $S^1$ is the unit circle and

$$\frac{\mathrm{d}t}{ti}=\frac{\mathrm{d}{e}^{i\alpha }}{i{e}^{i\alpha }}=\mathrm{d}\alpha =\left|\mathrm{d}t\right|$$

is the element of length. In addition,

$$\frac{\partial }{\partial \overline{z}}+\frac{z}{2}=\frac{e^{i\alpha }}{2}\left(\frac{\partial }{\partial r}+i\frac{1}{r}\frac{\partial }{\partial \alpha }+r\right)=\frac{t}{2}\left(\frac{\partial }{\partial r}-\frac{t}{r}\frac{\partial }{\partial t}+r\right)$$

and

$$\frac{\partial }{\partial z}+\frac{\overline{z}}{2}=\frac{1}{2e^{i\alpha }}\left(\frac{\partial }{\partial r}-i\frac{1}{r}\frac{\partial }{\partial \alpha }+r\right)=\frac{1}{2t}\left(\frac{\partial }{\partial r}+\frac{t}{r}\frac{\partial }{\partial t}+r\right).$$

As performed in [7], introduce the unitary operator

$$U_2=I\otimes \mathcal{F}:L_2({\mathbb{R}}_{+},r\mathrm{d}r)\otimes L_2\left(S^1\right)\to L_2({\mathbb{R}}_{+},r\mathrm{d}r)\otimes {\ell }_2={\ell }_2\left(L_2({\mathbb{R}}_{+},r\mathrm{d}r)\right)$$

and the discrete Fourier transform $\mathcal{F}:L_2\left(S^1\right)\to {\ell }_2$ with

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

The inverse of $\mathcal{F}$ is given by

$${\mathcal{F}}^{*}={\mathcal{F}}^{-1}:\left\{c_n\right\}\to f=\frac{1}{\sqrt{2\pi }}\sum _{n=-\infty }^{\infty }{c}_n{t}^n.$$

It is not difficult to see that

$$(I\otimes \mathcal{F})\frac{t}{2}\left(\frac{\partial }{\partial r}-\frac{t}{r}\frac{\partial }{\partial t}+r\right)\left(I\otimes {\mathcal{F}}^{-1}\right){\left\{c_n\left(r\right)\right\}}_{n\in {\mathbb{Z}}_{+}}={\left\{\frac{1}{2}\left(\frac{\partial }{\partial r}-\frac{n-1}{r}+r\right)c_{n-1}\left(r\right)\right\}}_{n\in {\mathbb{Z}}_{+}}$$

and

$$(I\otimes \mathcal{F})\frac{1}{2t}\left(\frac{\partial }{\partial r}+\frac{t}{r}\frac{\partial }{\partial t}+r\right)(I\otimes {\mathcal{F}}^{-1}){\left\{c_n\left(r\right)\right\}}_{n\in {\mathbb{Z}}_{-}}={\left\{\frac{1}{2}\left(\frac{\partial }{\partial r}+\frac{n+1}{r}+r\right)c_{n+1}\left(r\right)\right\}}_{n\in {\mathbb{Z}}_{-}},$$

where ${\mathbb{Z}}_{+}=\left\{0\right\}\cup \mathbb{N}$ and ${\mathbb{Z}}_{-}=\mathbb{Z}\setminus {\mathbb{Z}}_{+}$. Thus, the image $F^{\left(2\right)}=U_2\left(F^{\left(1\right)}\right)$ of the space $F^{\left(1\right)}$ can be described as the subspace of $L_2({\mathbb{R}}_{+},r\mathrm{d}r)\otimes l_2$, which is the closure of all sequences ${\left\{c_n\left(r\right)\right\}}_{n\in \mathbb{Z}}$ with smooth components satisfying the equations

$$\frac{1}{2}\left(\frac{\partial }{\partial r}-\frac{n}{r}+r\right)c_n\left(r\right)=0,n\in {\mathbb{Z}}_{+}$$

(1)

and

$$\frac{1}{2}\left(\frac{\partial }{\partial r}-\frac{n}{r}+r\right)c_{-n}\left(r\right)=0,n\in {\mathbb{Z}}_{+}.$$

(2)

Equations (1) and (2) are easy to solve and their general solutions have the form

$$c_n\left(r\right)=c_n^{\prime }\frac{r^n}{e^{r^2/2}}=c_n\sqrt{\frac{2}{n!}}\frac{r^n}{e^{r^2/2}}$$

(3)

and

$$c_{-n}\left(r\right)=c_{-n}^{\prime }\frac{r^n}{e^{r^2/2}}=c_{-n}\sqrt{\frac{2}{n!}}\frac{r^n}{e^{r^2/2}}.$$

(4)

Each function $c_n\left(r\right)$ must be in $L_2({\mathbb{R}}_{+},r\mathrm{d}r)$. Therefore, the space $F^{\left(2\right)}$ coincides with the space of all two-sided sequences ${\left\{c_n\left(r\right)\right\}}_{n\in \mathbb{Z}}$ with

$$c_n\left(r\right)=\begin{array}{cc}\hfill c_n\sqrt{\frac{2}{n!}}\frac{r^n}{e^{r^2/2}},& n\in {\mathbb{Z}}_{+}\\ \hfill c_n\sqrt{\frac{2}{\left|n\right|!}}\frac{r^{\left|n\right|}}{e^{r^2/2}},& n\in {\mathbb{Z}}_{-}\end{array}$$

and

$$\parallel {\left\{c_n\left(r\right)\right\}}_{n\in \mathbb{Z}}\parallel =∥{\left(\sum _{n\in \mathbb{Z}}{\left|c_n\right|}^2\right)}^{1/2}∥={\parallel {\left\{c_n\right\}}_{n\in \mathbb{Z}}\parallel }_{l_2}.$$

For each $n\in \mathbb{Z}$, as performed in [7], introduce the unitary operator

$$u_n:L_2({\mathbb{R}}_{+},\mathrm{d}r)\to L_2({\mathbb{R}}_{+},r\mathrm{d}r)$$

with the rule

$$\left(u_{n}f\right)\left(r\right)={\omega }_n\left(r\right)f\left({\alpha }_n\left(r\right)\right),$$

where

$${\omega }_n\left(r\right)=r^n\sqrt{\frac{1}{\left|n\right|!}{\sum }_{k=0}^{\left|n\right|}\frac{r^{2k}}{k!}}\mathrm{and}{\alpha }_n\left(r\right)=r^2-ln\sum _{k=0}^{\left|n\right|}\frac{r^{2k}}{k!}.$$

(5)

As performed in [7], define the unitary operator

$$U_3:{\ell }_2\left(L_2({\mathbb{R}}_{+},r\mathrm{d}r)\right)\to L_2\left({\mathbb{R}}_{+}\right)\otimes {\ell }_2$$

as

$$U_3:{\left\{c_n\left(r\right)\right\}}_{n\in \mathbb{Z}}\mapsto {\left\{\left(u_{\left|n\right|}^{-1}{c}_n\right)\left(r\right)\right\}}_{n\in \mathbb{Z}}.$$

The space $F^{\left(3\right)}=U_3\left(F^{\left(2\right)}\right)$ coincides with the space of all sequences ${\left\{d_n\left(r\right)\right\}}_{n\in \mathbb{Z}}$, where

$$d_n\left(r\right)=u_{\left|n\right|}^{-1}\left(c_n\sqrt{\frac{2}{\left|n\right|!}}\frac{r^{\left|n\right|}}{e^{r^2/2}}\right)=\frac{c_n}{e^{r/2}},n\in \mathbb{Z}.$$

(6)

Let $l_0\left(r\right)=\frac{1}{e^{r/2}}$. Then ${\ell }_0\left(r\right)\in L_2\left({\mathbb{R}}_{+}\right)$ and $\parallel {\ell }_0\left(r\right)\parallel =1$. Denote by $L_0$ the one-dimensional subspace of $L_2\left({\mathbb{R}}_{+}\right)$ generated by $l_0\left(r\right)$. Then the one-dimensional projection $P_0$ of $L_2\left({\mathbb{R}}_{+}\right)$ onto $L_0$ is of the form

$$\left(P_{0}f\right)\left(r\right)=\langle f,l_0\rangle l_0={\int }_{{\mathbb{R}}_{+}}\frac{f\left(\rho \right)}{e^{(r+\rho )/2}}\mathrm{d}\rho .$$

(7)

Now, $F^{\left(3\right)}=L_0\otimes {\ell }_2$ and the orthogonal projection $P^{\left(3\right)}$ of

$${\ell }_2\left(L_2\left({\mathbb{R}}_{+}\right)\right)=L_2\left({\mathbb{R}}_{+}\right)\otimes {\ell }_2$$

onto $F^{\left(3\right)}$ is obviously of the form $P^{\left(3\right)}=P_0\otimes I$.

The above work leads to the following theorem:

Theorem 1. 

The unitary operator $U=U_3{U}_2{U}_1$ gives an isometric isomorphism of the space $L_2(\mathbb{C},\mathrm{d}\mu )$ onto $L_2\left({\mathbb{R}}_{+}\right)\otimes {\ell }_2$ such that the following statements hold:

(a) 

The harmonic Fock space $F_h^2\left(\mathbb{C}\right)$ is mapped onto $L_0\otimes l_2$ by $U:F_h^2\left(\mathbb{C}\right)\to L_0\otimes {\ell }_2$, where $L_0$ is the one-dimensional subspace of $L_2\left({\mathbb{R}}_{+}\right)$ generated by the function ${\ell }_0\left(r\right)=\frac{1}{e^{r/2}}$.

(b) 

The harmonic Bargmann projection is the unitary equivalent of $UQ{U}^{-1}=P_0\otimes I$, where $P_0$ is the one-dimensional projection of $L_2\left({\mathbb{R}}_{+}\right)$ onto $L_0$.

Proof. 

First, we prove (a). For any $f\in F_h^2\left(\mathbb{C}\right)$ and $f=g+\overline{h}$, according to the definitions of $U_1$, $U_2$ and $U_3$, as well as Equations (3)–(7), we have

$$\begin{array}{cc}\hfill U_3{U}_2{U}_1:\left(g+\overline{h}\right)& \mapsto U_3{U}_2\frac{1}{{\pi }^{1/2}{e}^{z\overline{z}/2}}\left(g+\overline{h}\right)\hfill \\ \hfill & \mapsto U_3\left({\left\{c_n\left(r\right)\right\}}_{n\in {\mathbb{Z}}_{+}}+{\left\{c_{-n}\left(r\right)\right\}}_{n\in {\mathbb{Z}}_{+}}\right)\hfill \\ \hfill & \mapsto {\left\{\frac{c_n}{e^{r/2}}\right\}}_{n\in \mathbb{Z}}\hfill \\ \hfill & \mapsto L_0\otimes {\ell }_2,\hfill \end{array}$$

which shows (a).

To see (b), according to the definitions of Q and U, we obtain

$$\begin{array}{cc}\hfill UQ{U}^{-1}:L_2\left({\mathbb{R}}_{+}\right)\otimes {\ell }_2& \mapsto UQ:L_2(\mathbb{C},\mathrm{d}\mu )\hfill \\ \hfill & \mapsto U:F_h^2\left(\mathbb{C}\right)\hfill \\ \hfill & \mapsto L_0\otimes {\ell }_2.\hfill \end{array}$$

So, it follows from Equation (7) that

$$P_0\otimes I:L_2\left({\mathbb{R}}_{+}\right)\otimes {\ell }_2\mapsto L_0\otimes {\ell }_2,$$

which shows (b). The proof of Theorem 1 is completed. □

Introduce the isometric imbedding

$$R_0:l_2\to L_2\left({\mathbb{R}}_{+}\right)\otimes {\ell }_2$$

with the rule

$$R_0:\left\{c_n\right\}\mapsto {\left\{c_n{\ell }_0\left(r\right)\right\}}_{n\in \mathbb{Z}}.$$

The adjoint operator $R_0^{*}:L_2\left({\mathbb{R}}_{+}\right)\otimes {\ell }_2\to {\ell }_2$ is given by

$$R_0^{*}:{\left\{c_n\left(r\right)\right\}}_{n\in \mathbb{Z}}\mapsto {\left\{{\int }_{{\mathbb{R}}_{+}}\frac{c_n\left(\rho \right)}{e^{\rho /2}}\mathrm{d}\rho \right\}}_{n\in \mathbb{Z}}={\left\{c_n\right\}}_{n\in \mathbb{Z}},$$

$$R_0^{*}{R}_0=I:{\ell }_2\to {\ell }_2,$$

and

$$R_0{R}_0^{*}=P^{\left(3\right)}:L_2\left({\mathbb{R}}_{+}\right)\otimes {\ell }_2\to F^{\left(3\right)}=L_0\otimes {\ell }_2.$$

Now, the operator $R=R_0^{*}U$ maps the space $L_2(\mathbb{C},\mathrm{d}\mu )$ onto ${\ell }_2$ and the restriction

$${R|}_{F_h^2\left(\mathbb{C}\right)}:F_h^2\left(\mathbb{C}\right)\to {\ell }_2$$

is an isometric isomorphism. The adjoint operator

$$R^{*}=U^{*}{R}_0:l_2\to F_h^2\left(\mathbb{C}\right)\subset L_2(\mathbb{C},\mathrm{d}\mu )$$

is an isometric isomorphism of $l_2$ in the subspace $F_h^2\left(\mathbb{C}\right)$ of the space $L_2(\mathbb{C},\mathrm{d}\mu )$.

Remark 1. 

It is not difficult to see that

$$R{R}^{*}=I:{\ell }_2\to {\ell }_2$$

and

$$R^{*}R=Q:L_2(\mathbb{C},\mathrm{d}\mu )\to F_h^2\left(\mathbb{C}\right).$$

Theorem 2. 

The isometric isomorphism $R^{*}=U^{*}{R}_0:l_2\to F_h^2\left(\mathbb{C}\right)$ is given by

$$R^{*}:\left\{c_n\right\}\mapsto \sum _{n\in {\mathbb{Z}}_{+}}\frac{c_n}{\sqrt{\left|n\right|!}}{z}^n+\sum _{n\in {\mathbb{Z}}_{-}}\frac{c_n}{\sqrt{\left|n\right|!}}{\overline{z}}^{\left|n\right|}.$$

Proof. 

Let $\left\{c_n\right\}\in {\ell }_2$. Then we have

$$\begin{array}{cc}\hfill R^{*}& =U_1^{*}{U}_2^{*}{U}_3^{*}{R}_0:{\left\{c_n\right\}}_{n\in \mathbb{Z}}\mapsto U_1^{*}{U}_2^{*}{U}_3^{*}\left({\left\{\frac{c_n}{e^{r/2}}\right\}}_{n\in \mathbb{Z}}\right)\hfill \\ \hfill & =U_1^{*}{U}_2^{*}\left({\left\{c_n\sqrt{\frac{2}{\left|n\right|!}}\frac{r^{\left|n\right|}}{e^{r^2/2}}\right\}}_{n\in \mathbb{Z}}\right)\hfill \\ \hfill & =U_1^{*}\left(\frac{1}{\sqrt{2\pi }}\frac{1}{e^{r^2/2}}\sum _{n\in \mathbb{Z}}{c}_n\sqrt{\frac{2}{\left|n\right|!}}{r}^{\left|n\right|}{t}^n\right)\hfill \\ \hfill & =\sum _{n\in {\mathbb{Z}}_{+}}\frac{c_n}{\sqrt{\left|n\right|!}}{z}^n+\sum _{n\in {\mathbb{Z}}_{-}}\frac{c_n}{\sqrt{\left|n\right|!}}{\overline{z}}^{\left|n\right|}.\hfill \end{array}$$

The proof of Theorem 2 is thus complete. □

Corollary 1. 

A function

$$f\left(z\right)=\sum _{n\in \mathbb{Z}}{a}_n{z}^n$$

belongs to the harmonic Fock space $F_h^2\left(\mathbb{C}\right)$ if and only if

$$\sum _{n\in \mathbb{Z}}|a_n{|}^2\left|n\right|!<\infty$$

and

$$\parallel f\left(z\right)\parallel ={\left(\sum _{n\in \mathbb{Z}}|a_n{|}^2\left|n\right|!\right)}^{1/2}.$$

Corollary 2. 

The inverse isomorphism $R:F_h^2\left(\mathbb{C}\right)\to {\ell }_2$ is given by

$$R:f\left(z\right)\mapsto {\left\{\frac{1}{\sqrt{\left|n\right|!}}{\int }_{\mathbb{C}}f\left(z\right){\overline{z}}^n\mathrm{d}\mu \left(z\right)\right\}}_{n\in \mathbb{Z}}.$$

3. Toeplitz Operators with Radial Symbols

Denote by $L_1^{\infty }\left({\mathbb{R}}_{+},\frac{1}{e^{r^2}}\right)$ the linear space of all measurable functions $a\left(r\right)$ on ${\mathbb{R}}_{+}$ for which the integrals

$${\int }_{{\mathbb{R}}_{+}}\frac{\left|a\left(r\right)\right|r^{\left|n\right|}}{e^{r^2}}\mathrm{d}r<\infty ,n\in \mathbb{Z}$$

are finite. In this section, we investigate Toeplitz operators with symbols from $L_1^{\infty }\left({\mathbb{R}}_{+},\frac{1}{e^{r^2}}\right)$ acting in the harmonic Fock space $F_h^2\left(\mathbb{C}\right)$.

Theorem 3. 

Let $a=a\left(r\right)$ belong to $L_1^{\infty }\left({\mathbb{R}}_{+},\frac{1}{e^{r^2}}\right)$. Then the Toeplitz operator $T_a$ acting in harmonic Fock space $F_h^2\left(\mathbb{C}\right)$ is the unitary equivalent of the multiplication operator ${\gamma }_{a}I$ acting on ${\ell }_2$. The sequence ${\gamma }_a={\left\{{\gamma }_a\left(n\right)\right\}}_{n\in \mathbb{Z}}$ is given by

$${\gamma }_a\left(n\right)=\frac{1}{\left|n\right|!}{\int }_{{\mathbb{R}}_{+}}\frac{a\left(\sqrt{r}\right)r^{\left|n\right|}}{e^r}\mathrm{d}r,n\in \mathbb{Z}.$$

Proof. 

The operator $T_a$ is obviously the unitary equivalent of the operator

$$\begin{array}{cc}\hfill R{T}_a{R}^{*}& =RQaQ{R}^{*}\hfill \\ \hfill & =R(R^{*}R)a(R^{*}R)R^{*}\hfill \\ \hfill & =(R{R}^{*})Ra{R}^{*}(R{R}^{*})\hfill \\ \hfill & =Ra{R}^{*}\hfill \\ \hfill & =R_0^{*}{U}_3{U}_2{U}_{1}a\left(r\right)U_1^{-1}{U}_2^{-1}{U}_3^{-1}{R}_0\hfill \\ \hfill & =R_0^{*}{U}_3(I\otimes \mathcal{F})a\left(r\right)(I\otimes {\mathcal{F}}^{-1})U_3^{-1}{R}_0\hfill \\ \hfill & =R_0^{*}{U}_3\left\{a\left(r\right)\right\}{U}_3^{-1}{R}_0\hfill \\ \hfill & =R_0^{*}\left\{a\left({\alpha }_n^{-1}\left(r\right)\right)\right\}{R}_0,\hfill \end{array}$$

where the function ${\alpha }_n\left(r\right)$ is given by (5) and ${\alpha }_n^{-1}\left(r\right)$ is the inverse of ${\alpha }_n\left(r\right)$. Therefore, it follows that

$$\begin{array}{cc}\hfill R_0^{*}\left\{a\left({\alpha }_n^{-1}\left(r\right)\right)\right\}{R}_0{\left\{c_n\right\}}_{n\in \mathbb{Z}}& ={\left\{{\int }_{{\mathbb{R}}_{+}}a\left({\alpha }_n^{-1}\left(r\right)\right)\frac{c_n}{e^r}\mathrm{d}r\right\}}_{n\in \mathbb{Z}}\hfill \\ \hfill & ={\left\{c_n{\int }_{{\mathbb{R}}_{+}}a\left({\alpha }_n^{-1}\left(r\right)\right)\frac{1}{e^r}\mathrm{d}r\right\}}_{n\in \mathbb{Z}}\hfill \\ \hfill & ={\left\{c_n{\int }_{{\mathbb{R}}_{+}}\frac{a\left(r\right){\alpha }_n^{\prime }\left(r\right)}{e^{{\alpha }_n\left(r\right)}}\mathrm{d}r\right\}}_{n\in \mathbb{Z}}\hfill \\ \hfill & ={\left\{\frac{2c_n}{\left|n\right|!}{\int }_{{\mathbb{R}}_{+}}\frac{a\left(r\right)r^{2\left|n\right|+1}}{e^{r^2}}\mathrm{d}r\right\}}_{n\in \mathbb{Z}}\hfill \\ \hfill & ={\left\{\frac{c_n}{\left|n\right|!}{\int }_{{\mathbb{R}}_{+}}\frac{a\left(\sqrt{r}\right)r^{\left|n\right|}}{e^r}\mathrm{d}r\right\}}_{n\in \mathbb{Z}}\hfill \\ \hfill & ={\left\{r_a\left(n\right)c_n\right\}}_{n\in \mathbb{Z}}.\hfill \end{array}$$

Theorem 3 is proved. □

Theorem 4. 

The Toeplitz operator $T_a$ with radial symbol $a=a\left(r\right)$ belonging to $L_1^{\infty }\left({\mathbb{R}}_{+},\frac{1}{e^{r^2}}\right)$ is bounded on $F_h^2\left(\mathbb{C}\right)$ if and only if ${\gamma }_a={\left\{{\gamma }_a\left(n\right)\right\}}_{n\in \mathbb{Z}}\in l_{\infty }$ and $\parallel T_a\parallel ={sup}_{n\in \mathbb{Z}}\left|{\gamma }_a\left(n\right)\right|$.

Moreover, the Toeplitz operator $T_a$ is compact if and only if ${lim}_{n\to \infty }{\gamma }_a\left(n\right)=0$.

Proof. 

This follows directly from Theorem 3. □

Proposition 1 below implies that a Toeplitz operator $T_a$ with symbol

$$a\left(r\right)\in L_1^{\infty }\left({\mathbb{R}}_{+},\frac{1}{e^{r^2}}\right)$$

is a well-defined linear operator with a dense domain.

Proposition 1. 

Let $F_0^2\left(\mathbb{C}\right)$ and $\overline{F_0^2\left(\mathbb{C}\right)}$ denote the set of all polynomials on z and $\overline{z}$, respectively. If $p\left(z\right)+\overline{p\left(z\right)}\in F_0^2\left(\mathbb{C}\right)+\overline{F_0^2\left(\mathbb{C}\right)}$ and $a\left(r\right)\in L_1^{\infty }\left({\mathbb{R}}_{+},\frac{1}{e^{r^2}}\right)$, then

$$T_a(p+\overline{p})\in F_0^2\left(\mathbb{C}\right)+\overline{F_0^2\left(\mathbb{C}\right)}\subset F_h^2\left(\mathbb{C}\right).$$

Proof. 

When

$$p_1\left(z\right)=p\left(z\right)+\overline{p\left(z\right)}=\sum _{n=0}^m{c}_n{z}^n+\sum _{n=0}^m{c}_n{\overline{z}}^n\in F_0^2\left(\mathbb{C}\right)+\overline{F_0^2\left(\mathbb{C}\right)}$$

and $a\left(r\right)\in L_1^{\infty }\left({\mathbb{R}}_{+},\frac{1}{e^{r^2}}\right)$, we have

$$\begin{array}{cc}\hfill \left(T_a{p}_1\right)\left(z\right)& =\frac{1}{\pi }{\int }_{{\mathbb{R}}^2}\frac{\left(a\left(\right|\xi \left|\right)p\left(\xi \right)+\overline{p\left(\xi \right)}\right)\left(e^{\overline{\xi }z}+e^{\xi \overline{z}}-1\right)}{e^{{\left|\xi \right|}^2}}\mathrm{d}v\left(\xi \right)\hfill \\ \hfill & =\frac{1}{\pi i}{\int }_{{\mathbb{R}}_{+}}\left[{\int }_{S^1}\left(\sum _{n=0}^m{c}_n{r}^n{t}^n\right)e^{\left(rz\right)/t}\frac{\mathrm{d}t}{t}\right]\frac{a\left(r\right)r}{e^{r^2}}\mathrm{d}r\hfill \\ \hfill & +\frac{1}{\pi i}{\int }_{{\mathbb{R}}_{+}}\left[{\int }_{S^1}\left(\sum _{n=0}^m\frac{c_n{r}^n}{t^n}\right)e^{\left(r\overline{z}\right)t}\frac{\mathrm{d}t}{t}\right]\frac{a\left(r\right)r}{e^{r^2}}\mathrm{d}r-c_0{\gamma }_a\left(0\right)\hfill \\ \hfill & =\sum _{n=0}^m\frac{1}{\pi i}{\int }_{{\mathbb{R}}_{+}}{\int }_{S^1}{t}^n\left(\sum _{k=0}^{\infty }\frac{r^k{z}^k}{k!t^k}\right)\frac{\mathrm{d}t}{t}\frac{a\left(r\right)r^{n+1}}{e^{r^2}}\mathrm{d}r\hfill \\ \hfill & +\sum _{n=0}^m\frac{1}{\pi i}{\int }_{{\mathbb{R}}_{+}}{\int }_{S^1}\frac{1}{t^n}\left(\sum _{k=0}^{\infty }\frac{r^k{\overline{z}}^k{t}^k}{k!}\right)\frac{\mathrm{d}t}{t}\frac{a\left(r\right)r^{n+1}}{e^{r^2}}\mathrm{d}r-c_0{\gamma }_a\left(0\right)\hfill \\ \hfill & =\sum _{n=0}^m{c}_n(z^n+{\overline{z}}^n)\left(\frac{2}{n!}{\int }_{{\mathbb{R}}_{+}}\frac{a\left(r\right)r^{2n+1}}{e^{r^2}}\right)\mathrm{d}r-c_0{\gamma }_a\left(0\right)\hfill \\ \hfill & =\sum _{n=0}^m{c}_n(z^n+{\overline{z}}^n){\gamma }_a\left(n\right)-c_0{\gamma }_a\left(0\right).\hfill \end{array}$$

Accordingly, we acquire

$$T_a{p}_1\in F_0^2\left(\mathbb{C}\right)+\overline{F_0^2\left(\mathbb{C}\right)}\subset F_h^2\left(\mathbb{C}\right)$$

and the set $F_0^2\left(\mathbb{C}\right)+\overline{F_0^2\left(\mathbb{C}\right)}$ is the domain for each Toeplitz operator $T_a$ with symbol $a\left(r\right)$. The proof of Proposition 1 is thus complete. □

By Proposition 1, the Toeplitz operator $T_a$ with symbol $a\left(r\right)\in L_1^{\infty }\left({\mathbb{R}}_{+},\frac{1}{e^{r^2}}\right)$ has a bounded extension to the whole space $F_h^2\left(\mathbb{C}\right)$ if and only if the sequence $\left\{{\gamma }_a\left(n\right)\right\}$ is bounded.

Corollary 3. 

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

$$sp{T}_a=\overline{\{{\gamma }_a\left(n\right):n\in \mathbb{Z}\}}$$

and its essential spectrum $esssp{T}_a$ coincides with the set of all limit points of the sequence ${\left\{{\gamma }_a\left(n\right)\right\}}_{n\in \mathbb{Z}}$.

4. Properties of Toeplitz Operators with Radial Symbols

We start with conditions which guarantee the boundedness or compactness of Toeplitz operators with radial symbols from $L_1^{\infty }\left({\mathbb{R}}_{+},\frac{1}{e^{r^2}}\right)$.

Theorem 5. 

Let $a\left(r\right)\in L_1^{\infty }\left({\mathbb{R}}_{+},\frac{1}{e^{r^2}}\right)$. Then the Toeplitz operator $T_a$ is bounded on $F_h^2\left(\mathbb{C}\right)$ if one of the following statements holds:

(1) 

The relation $a\left(r\right)\in L_1^{\infty }\left({\mathbb{R}}_{+}\right)$ is valid.

(2) 

The sequence ${\gamma }_a^{\left(1\right)}={\left\{{\gamma }_a^{\left(1\right)}\left(n\right)\right\}}_{n\in \mathbb{Z}}$ is bounded, where

$${\gamma }_a^{\left(1\right)}\left(n\right)=\frac{1}{\left|n\right|!}{\int }_{{\mathbb{R}}_{+}}\frac{\left|a\left(\sqrt{r}\right)\right|r^{\left|n\right|}}{e^r}\mathrm{d}r.$$

(3) 

The function

$$B\left(r\right)={\int }_r^{\infty }\left|a\left(\sqrt{r}\right)\right|r^{\left|n\right|}{e}^{r-u}\mathrm{d}u$$

is bounded.

Proof. 

The first statement is well known. Let the second statement hold. Then we have

$$\begin{array}{cc}\hfill |{\gamma }_a\left(n\right)|& \le \frac{1}{\left|n\right|!}\left|{\int }_{{\mathbb{R}}_{+}}\frac{a\left(\sqrt{r}\right)r^{\left|n\right|}}{e^r}\mathrm{d}r\right|\hfill \\ \hfill & \le \frac{1}{\left|n\right|!}{\int }_{{\mathbb{R}}_{+}}\frac{\left|a\left(\sqrt{r}\right)\right|r^{\left|n\right|}}{e^r}\mathrm{d}r\hfill \\ \hfill & ={\gamma }_a^{\left(1\right)}\left(n\right).\hfill \end{array}$$

Accordingly, the second statement is proved.

Finally, integrating by parts yields

$$\begin{array}{cc}\hfill |{\gamma }_a\left(n\right)|& \le \frac{1}{\left|n\right|!}\left|{\int }_{{\mathbb{R}}_{+}}\frac{a\left(\sqrt{r}\right)r^{\left|n\right|}}{e^r}\mathrm{d}r\right|\hfill \\ \hfill & =\frac{1}{\left|n\right|!}\left|{\int }_{{\mathbb{R}}_{+}}{r}^{\left|n\right|}\mathrm{d}\left[{\int }_r^{\infty }\frac{a\left(\sqrt{u}\right)}{e^u}\mathrm{d}u\right]\right|\hfill \\ \hfill & =\frac{1}{\left(\right|n|-1|)!}\left|{\int }_{{\mathbb{R}}_{+}}\frac{r^{\left|n\right|-1}}{e^r}\left[{\int }_r^{\infty }a\left(\sqrt{u}\right)e^{r-u}\mathrm{d}u\right]\mathrm{d}r\right|\hfill \\ \hfill & =\frac{1}{\left(\right|n|-1)!}\left|{\int }_{{\mathbb{R}}_{+}}\frac{r^{\left|n\right|-1}}{e^r}B\left(r\right)\mathrm{d}r\right|\hfill \\ \hfill & =\left|{\gamma }_B\left(\right|n|-1)\right|.\hfill \end{array}$$

Further applying the second statement to the function $B\left(r\right)$ leads to the third statement. Theorem 5 is thus proved. □

Theorem 6. 

Let $a\left(r\right)\in L_1^{\infty }\left({\mathbb{R}}_{+},\frac{1}{e^{r^2}}\right)$. Then the Toeplitz operator $T_a$ is compact on $F_h^2\left(\mathbb{C}\right)$ if one of the following statements holds:

(1) 

The limit ${lim}_{r\to \infty }a\left(r\right)=0$ is valid.

(2) 

The limits

$$\underset{n\to \infty }{lim}{\gamma }_a^{\left(1\right)}\left(n\right)=\underset{n\to \infty }{lim}\frac{1}{\left|n\right|!}{\int }_{{\mathbb{R}}_{+}}\frac{|a\left(\sqrt{r}\right)|r^{\left|n\right|}}{e^r}\mathrm{d}r=0$$

are valid.

(3) 

The limits

$$\underset{r\to \infty }{lim}B\left(r\right)=\underset{r\to \infty }{lim}{\int }_r^{\infty }\left|a\left(\sqrt{r}\right)\right|r^n{e}^{r-u}\mathrm{d}u=0$$

are valid.

Proof. 

This follows directly from Theorem 5. □

A Toeplitz operator $T_a$ with a symbol $a=a\left(z\right)$ acting in the harmonic Fock space $F_h^2\left(\mathbb{C}\right)$ is an operator with the anti-Wick symbol $a=a\left(z\right)$. The function $\tilde{a}(z,\overline{z})$ is called a Wick symbol of an operator T if this operator acts on $F_h^2\left(\mathbb{C}\right)$ as follows:

$$\begin{array}{cc}\hfill (Tf)\left(z\right)& =\frac{1}{\pi }{\int }_{\mathbb{C}}f\left(\zeta \right)\left[\frac{\tilde{a}\left(z,\overline{\zeta }\right)}{e^{\overline{\zeta }(\zeta -z)}}+\frac{\tilde{a}\left(z,\overline{\zeta }\right)}{e^{\zeta (\overline{\zeta }-\overline{z})}}\right]\mathrm{d}v\left(\zeta \right)\hfill \\ \hfill & =\frac{1}{\pi }{\int }_{\mathbb{C}}f\left(\zeta \right)\left[\frac{\tilde{a}\left(z,\overline{\zeta }\right)}{e^{\overline{\zeta }z}}+\tilde{a}\left(z,\overline{\zeta }\right)e^{\zeta \overline{z}}\right]\mathrm{d}\mu \left(\zeta \right).\hfill \end{array}$$

The Wick and anti-Wick symbols of the same operator are connected by the formula

$$\tilde{a}(z,\overline{z})=\frac{1}{\pi }{\int }_{\mathbb{C}}\frac{a\left(\zeta \right)}{e^{(z-\zeta )(\overline{z}-\overline{\zeta })}}\mathrm{d}v\left(\zeta \right).$$

Denote by $\mathcal{M}$ the linear subspace of $L_1^{\infty }\left({\mathbb{R}}_{+},\frac{1}{e^{r^2}}\right)$ such that for each $a\left(r\right)\in \mathcal{M}$ the Toeplitz operator $T_{a\left(r\right)}$ is bounded on $F_h^2\left(\mathbb{C}\right)$. Denote by $\mathcal{T}\left(\mathcal{M}\right)$ the $C^{*}$-algebra generated by all Toeplitz operators $T_a$ with symbols $a\in \mathcal{M}$.

The system of functions

$$L_n\left(z\right)=\{\begin{array}{cc}\hfill \frac{z^n}{\sqrt{n!}},& n\in {\mathbb{Z}}_{+}\\ \hfill \frac{{\overline{z}}^{\left|n\right|}}{\sqrt{\left|n\right|!}},& n\in {\mathbb{Z}}_{-}\end{array}$$

is an orthonormal basis for the harmonic Fock space $F_h^2\left(\mathbb{C}\right)$. Denote by $L_{\left(n\right)}$ the one-dimensional space generated by the function $l_{\left(n\right)}\left(z\right)$. The orthogonal projection $P_{\left(n\right)}:F_h^2\left(\mathbb{C}\right)\to L_{\left(n\right)}$ is obviously of the forms

$$\left(P_{\left(n\right)}f\right)\left(z\right)=\langle f\left(\zeta \right),l_{\left(n\right)}\left(z\right)\rangle l_{\left(n\right)}\left(z\right)=\frac{z^n}{n!}\frac{1}{\pi }{\int }_{\mathbb{C}}\frac{f\left(\zeta \right){\overline{\zeta }}^n}{e^{{\left|\zeta \right|}^2}}\mathrm{d}v\left(\zeta \right),n\in {\mathbb{Z}}_{+}$$

and

$$\left(P_{\left(n\right)}f\right)\left(z\right)=\langle f\left(\zeta \right),l_{\left(n\right)}\left(z\right)\rangle l_{\left(n\right)}\left(z\right)=\frac{{\overline{z}}^{\left|n\right|}}{\left|n\right|!}\frac{1}{\pi }{\int }_{\mathbb{C}}\frac{f\left(\zeta \right){\zeta }^{\left|n\right|}}{e^{{\left|\zeta \right|}^2}}\mathrm{d}v\left(\zeta \right),n\in {\mathbb{Z}}_{-}.$$

They are Toeplitz operators with the symbol from $\mathcal{M}$.

For any $n\in \mathbb{Z}$, the one-dimensional space $L_{\left(n\right)}$ is an eigenspace for the Toeplitz operator $T_a$ with $a\left(r\right)\in \mathcal{M}$, and the corresponding eigenvalue is equal to ${\gamma }_a\left(n\right)$.

Theorem 7. 

Let $a\left(r\right)\in \mathcal{M}$. Writing the Toeplitz operator $T_a$ in the form of an operator with a Wick symbol gives the spectral decomposition of the operator $T_a$:

$$T_a=\sum _{n=-\infty }^{\infty }{\gamma }_a\left(n\right)P_{\left(n\right)}.$$

Proof. 

For $n\in \mathbb{Z}$ and $f\in F_h^2\left(\mathbb{C}\right)$, consider the operator with the Wick symbol of the form $p_{\left(n\right)}(z,\overline{z})=e^{z\overline{z}}\frac{z^n{\overline{z}}^n}{n!}$. Then

$$\begin{array}{cc}\hfill \left(T_{a}f\right)\left(z\right)& =\frac{1}{\pi }{\int }_{\mathbb{C}}f\left(\zeta \right)\left[\frac{\tilde{a}\left(z,\overline{\zeta }\right)}{e^{\overline{\zeta }(\zeta -z)}}+\frac{\tilde{a}\left(z,\overline{\zeta }\right)}{e^{\zeta (\overline{\zeta }-\overline{z})}}\right]\mathrm{d}v\left(\zeta \right)\hfill \\ \hfill & =\frac{1}{\pi }\sum _{n=0}^{\infty }{\gamma }_a\left(n\right)\frac{z^n}{n!}{\int }_{\mathbb{C}}\frac{{\overline{\zeta }}^{n}f\left(\zeta \right)}{e^{|\overline{\zeta }{|}^2}}\mathrm{d}v\left(\zeta \right)+\frac{1}{\pi }\sum _{n=-\infty }^0{\gamma }_a\left(n\right)\frac{{\overline{z}}^{\left|n\right|}}{\left|n\right|!}{\int }_{\mathbb{C}}\frac{{\zeta }^{\left|n\right|}f\left(\zeta \right)}{e^{|\overline{\zeta }{|}^2}}\mathrm{d}v\left(\zeta \right)\hfill \\ \hfill & =\sum _{n=-\infty }^{\infty }{\gamma }_a\left(n\right)\left(P_{\left(n\right)}f\right)\left(z\right).\hfill \end{array}$$

The required proof is complete. □

5. Conclusions

This paper is devoted to studying specific properties (such as the boundedness, compactness, algebraic properties, spectral decomposition and others) of the Toeplitz operator with radial symbols in harmonic Fock spaces. On the basis of analytic functions theory, we present several problems of harmonic functions and expand the scope of the past study. In summary, new important results and features for Toeplitz operators with radial symbols in harmonic Fock spaces are established (see Theorems 5–7). We believe that these newly discovered results will help us study the problems in pluriharmonic Fock spaces or polydisk Fock spaces in future studies.