On Toeplitz Operators on the Weighted Harmonic Bergman Space on the Upper Half-Plane

Abstract

We study Toeplitz operators on the weighted harmonic Bergman space on the upper half-plane. Two classes of symbols are considered here: symbols that depend only on the vertical variable and symbols that depend only on the angular variable. For the first case, we prove that Toeplitz operators with such kind of symbols generate a commutative $C^*$-algebra in every weighted harmonic Bergman space. This algebra is isomorphic to the algebra of all very slowly oscillating functions. On the other hand, Toeplitz operators whose symbols depend only on the angular variable generate a non commutative $C^*$-algebra which is isomorphic to the $C^*$-algebra of all $2\times 2$ matrix-valued continuous functions $(f_{ij}(t))$ defined on $[-\infty ,\infty ]$ and such that they satisfy $f_{12}(\pm \infty )=f_{21}(\pm \infty )=0$ and $f_{11}(\pm \infty )=f_{22}(\mp \infty )$.

1 Introduction

As usual, we denote by $\Pi$ the upper half-plane in the complex plane ${\mathbb {C}}$, i.e., $\Pi =\{z=x+iy\in {\mathbb {C}}: \mathrm{\, Im \,}z:= y \ge 0\}$, and we consider the normalized measure on it

$$\begin{aligned} d\mu (z)=\frac{1}{\pi } \frac{dx\;dy}{(2y)^2}=\frac{1}{2\pi i} \frac{d\bar{z} \; dz}{(2\mathrm{\, Im \,}z)^2}, \end{aligned}$$

where $z=x+iy$. For $\alpha \in (-1,\infty )$ let $L^2(\Pi ,d\mu _\alpha )$ be the space consisting of all measurable functions satisfying

$$\begin{aligned} \Vert f\Vert _{2,\alpha }=\left( \,\, \int _\Pi |f(z)|^2\; d\mu _\alpha (z)\right) ^{1/2}<\infty , \end{aligned}$$

where

$$\begin{aligned} d\mu _\alpha (z)=(\alpha +1)(2 \mathrm{\, Im \,}z)^\alpha \frac{1}{2\pi i} d\bar{z} dz=(\alpha +1)(2 \mathrm{\, Im \,}z)^{\alpha +2}d\mu (z). \end{aligned}$$

The weighted harmonic Bergman space $b^2_{\alpha }(\Pi )$ is the closed subspace of $L^2(\Pi ,d\mu _{\alpha })$ consisting of all complex-valued harmonic functions. It is represented as the direct sum of the Bergman and the anti-Bergman spaces and then, the orthogonal projection from $L^2(\Pi ,d\mu _{\alpha })$ onto $b^2_{\alpha }(\Pi )$, denoted by $Q_{\Pi }^{\alpha }$, is written in the following form:

$$\begin{aligned} Q_{\Pi }^{\alpha }=B_{\Pi }^{\alpha }+ {\widetilde{B}}_{\Pi }^{\alpha }, \end{aligned}$$

where $B_{\Pi }^{\alpha }$ is the Bergman projection and ${\widetilde{B}}_{\Pi }^{\alpha }$ is the anti-Bergman projection. With this fact in mind, it is proved in [1], for unweighted harmonic Bergman spaces, and in Sect. 3 of this work, for weighted harmonic Bergman spaces, that every Toeplitz operator acting on the harmonic Bergman space is represented in terms of Toeplitz operators acting on the Bergman space, Toeplitz operators acting on the anti-Bergman space and small Hankel operators. Despite this, Toeplitz operators acting on the harmonic Bergman space behave quite different from such kind of operators acting on the Bergman (or anti-Bergman) space. Indeed, for the unit disk, in [3] Guo and Zheng proved that every Fredholm Toeplitz operator with continuous symbol has index zero.

Concerning the Bergman space setting, commutative algebras of Toeplitz operators in every weighted Bergman space have been completely described [12], and the kind of symbols that produce these algebras obey very interesting geometric properties. There are three model classes of such symbols: elliptic, which is realized by radial symbols on the unit disk, parabolic, which is realized by symbols depending only on $y=\mathrm{Im }z$ on the upper half-plane, and hyperbolic, which is realized by symbols that depend only on the angular variable on the upper half-plane. Toeplitz operators with bounded radial symbols, acting on $b^{2}(\mathbb {D})$, are diagonal operators (see for example [7]). Thus, they generate a commutative $C^*$-algebra.

In [8] the authors studied Toeplitz operators, acting on the unweighted harmonic Bergman space on the unit disk. There, they found essential differences between the Bergman space setting and the harmonic Bergman space setting. Contrary to the case of Toeplitz operators acting on the Bergman space, functions invariant under hyperbolic transformations do not produce a commutative $C^*$-algebra of Toeplitz operators and even its Calkin algebra is not commutative.

This paper is devoted to the study of Toeplitz operators acting on the weighted harmonic Bergman space $b^2_{\alpha }(\Pi )$. The following two model classes of symbols are used

• 1. Parabolic, or symbols that depend only on the imaginary part of the variable.

• 2. Hyperbolic, or symbols that depend only on the angular variable.

The main results of this work are the following:

For case (1), we prove that the $C^*$-algebra generated by Toeplitz operators whose symbols depend only on the vertical variable is commutative. Furthermore, using a result from [4], we prove that it is isomorphic to the $C^*$-algebra $VSO({\mathbb {R}}_+)$ of very slowly oscillating functions, i.e., functions that are uniformly continuous with respect to the logarithmic metric $\rho (x,y)=|\ln x-\ln y|$ on ${\mathbb {R}}_+$.

For case (2), we prove that the $C^*$-algebra generated by all Toeplitz operators with homogeneous symbols is not commutative. In fact, independently of the value of $\alpha$, it is isomorphic and isometric to the $C^*$-algebra of all $2\times 2$ matrix-valued continuous functions $(f_{ij}(t))$ defined on $[-\infty ,\infty ]$ and such that they satisfy $f_{12}(\pm \infty )=f_{21}(\pm \infty )=0$ and $f_{11}(\pm \infty )=f_{22}(\mp \infty )$.

It is worth to mention that this work is not just an extension to weighted harmonic Bergman spaces of the results in [8]. Proof of Theorem 4.9, in [8], that describes the $C^*$-algebra ${\mathcal T}({\mathcal A}_{\infty })$ generated by Toeplitz operators with homogeneous symbols, was done by case by case quite lengthy calculations. For the weighted case these calculations would become too long. And, even when the description of ${\mathcal T}({\mathcal A}_{\infty })$ given here (see Theorem 3.11) is also based on the non commutative Stone–Weierstrass theorem, the arguments are easier, shorter and we would say, more natural. Furthermore, for case (2), the authors proved in [8] that Toeplitz operators with vertical symbols generate a commutative $C^*$-algebra, but they did not describe it. In this work, the complete description of this algebra is included.

2 Preliminaries

We begin with a review of some of the basic results on the weighted Bergman space on the upper half-plane (see, for example [12]). As usual, we denote by $\Pi$ the upper half-plane in the complex plane ${\mathbb {C}}$, i.e., $\Pi =\{z\in {\mathbb {C}}: \mathrm{\, Im \,}z\ge 0\}$. We consider the normalized measure

$$\begin{aligned} d\mu (z)=\frac{1}{\pi } \frac{dx\;dy}{(2y)^2}=\frac{1}{2\pi i} \frac{d\bar{z} \; dz}{(2\mathrm{\, Im \,}z)^2}, \end{aligned}$$

where $z=x+iy$. For $\alpha \in (-1,\infty )$ let $L^2(\Pi ,d\mu _\alpha )$ be the space consisting of all measurable functions satisfying

$$\begin{aligned} \Vert f\Vert _{2,\alpha }=\left( \int _\Pi |f(z)|^2\; d\mu _\alpha (z)\right) ^{1/2}<\infty , \end{aligned}$$

where

$$\begin{aligned} d\mu _\alpha (z)=(\alpha +1)(2 \mathrm{\, Im \,}z)^\alpha \frac{1}{2\pi i} d\bar{z} dz=(\alpha +1)(2 \mathrm{\, Im \,}z)^{\alpha +2}d\mu (z). \end{aligned}$$

Denote by $\langle \cdot ,\cdot \rangle _\alpha$ the inner product in $L^2(\Pi ,d\mu _\alpha )$ given by

$$\begin{aligned} \langle f,g\rangle _\alpha =\int _\Pi f(z)\overline{g(z)}d\mu _\alpha (z). \end{aligned}$$

The weighted Bergman space ${\mathcal A}_\alpha ^2(\Pi )$ on the upper half-plane is the closed subspace of $L^2(\Pi ,d\mu _\alpha )$ consisting of all analytic functions (see [12]). If $\alpha =0$ then ${\mathcal A}_\alpha ^2(\Pi )$ is the (unweighted) Bergman space on the upper half-plane ${\mathcal A}^2(\Pi )$.

The orthogonal projection from $L^2(\Pi ,d\mu _\alpha )$ onto ${\mathcal A}_\alpha ^2(\Pi )$ is denoted by $B_{\Pi }^\alpha$ and is given by the integral formula

$$\begin{aligned} B_{\Pi }^\alpha f(z)&= (\alpha +1)\int _\Pi f(w)\left( \frac{w-\overline{w}}{z-\overline{w}}\right) ^{\alpha +2} d\mu \nonumber \\&= i^{\alpha +2} \int _{\Pi }\frac{f(w)}{(z-\overline{w})^{\alpha +2}}\;d\mu _\alpha (w). \end{aligned}$$

(2.1)

The function

$$\begin{aligned} K_\alpha (z,w)=\frac{i^{\alpha +2}}{(z-\overline{w})^{\alpha +2}} \end{aligned}$$

(2.2)

is the weighted Bergman reproducing kernel (for details see for example [11] and [12]).

Some works (for example [6]) use the measure $dA_r$ given by

$$\begin{aligned} dA_r(z)=(2r+1)K(z,z)^{-r}dx\; dy, \end{aligned}$$

where $r>-\frac{1}{2}$ and $K(z,w)$ is the Bergman reproducing kernel given by $K(z,w)=\frac{-1}{\pi (z-\overline{w})^2}$. We note that $dA_r$ and $d\mu _\alpha$ generate equivalent norms when $\alpha =2r$. In fact,

$$\begin{aligned} dA_r(z)&= (2r+1)\pi ^{r}(2\mathrm{\, Im \,}z)^{2r}dxdy\\&= (2r+1)\pi ^{r-1}(2\mathrm{\, Im \,}z)^{2r}\frac{dxdy}{\pi }\\&= \pi ^{\frac{\alpha }{2}-1} (\alpha +1)(2 \mathrm{\, Im \,}z)^\alpha \frac{dxdy}{\pi }\\&= \pi ^{\frac{\alpha }{2}-1} d\mu _\alpha (z). \end{aligned}$$

For any function $a\in L_\infty (\Pi )$ the Toeplitz operator $T_a^{\alpha }$, with symbol $a$, acting on ${\mathcal A}_\alpha ^2(\Pi )$ is the operator defined by

$$\begin{aligned} T_a^\alpha (f)=B_\Pi ^\alpha af, \end{aligned}$$

(2.3)

for all $f\in {\mathcal A}_\alpha ^2(\Pi )$.

Let $J$ be the (nonlinear) operator of complex conjugation acting on $L^2(\Pi ,d\mu _\alpha )$, i.e.,

$$\begin{aligned} Jf={\bar{f}}. \end{aligned}$$

(2.4)

We note $JaJ=\bar{a}I$ for every $a\in L_\infty (\Pi )$.

Denote by $\widetilde{{\mathcal A}}^2_\alpha (\Pi )$ the subspace of $L^2(\Pi ,d\mu _\alpha )$ consisting of all anti-analytic functions in $\Pi$ (analytic with respect to $\bar{z}$). Since $\widetilde{{\mathcal A}}^2_\alpha (\Pi )$ is the image of the closed subspace ${\mathcal A}^2_\alpha (\Pi )$ under $J$, we have $\widetilde{{\mathcal A}}^2_\alpha (\Pi )$ is a closed subspace of $L^2(\Pi ,d\mu _\alpha )$. We shall prove that $\widetilde{{\mathcal A}}^2_\alpha (\Pi )$ has the reproducing kernel $\overline{K}_\alpha (z,w)$.

Proposition 2.1

The function

$$\begin{aligned} \overline{K}_\alpha (z,w)=\frac{(-i)^{\alpha +2}}{(\overline{z}-w)^{\alpha +2}} \end{aligned}$$

(2.5)

is the weighted anti-Bergman reproducing kernel.

Proof

Let $z\in \Pi$. The evaluation functional $g\mapsto g(z)$, defined in $\widetilde{{\mathcal A}}^2_\alpha (\Pi )$ is continuous. According to Riesz representation theorem there exists a unique element $\widetilde{K}_\alpha \in \widetilde{{\mathcal A}}^2_\alpha (\Pi )$ such that $g(z)=\langle g,\widetilde{K}_\alpha \rangle _\alpha$. On the other hand, for each $g\in {\mathcal A}^2(\Pi )$ it holds that ${\bar{g}}\in \widetilde{{\mathcal A}}^2_\alpha (\Pi )$. Since, $K_\alpha (z,w)$ is the Bergman reproducing kernel for ${\mathcal A}^2_\alpha (\Pi )$,

$$\begin{aligned} {\bar{g}}(z)&= \langle {\bar{g}},K_\alpha \rangle _\alpha ,\\ g(z)&= \overline{\langle {\bar{g}},K_\alpha \rangle _\alpha }\\&= \langle g,\overline{K}_\alpha \rangle _\alpha . \end{aligned}$$

And therefore, $\widetilde{K}_\alpha =\overline{K}_\alpha$. $\square$

The orthogonal projection ${\widetilde{B}}_{\Pi }^\alpha$ of $L^2(\Pi ,d\mu _\alpha )$ onto $\widetilde{{\mathcal A}}^2_\alpha (\Pi )$, is called the anti-Bergman projection and, from Proposition 2.1, it has the integral representation

$$\begin{aligned} {\widetilde{B}}_{\Pi }^\alpha f =(-i)^{\alpha +2}\int _{\Pi } \frac{f(w)}{(\bar{z}-w)^{\alpha +2}}\; d\mu _\alpha (w). \end{aligned}$$

(2.6)

By formulas (2.1), (2.4) and (2.6) we have

$$\begin{aligned} {\widetilde{B}}_{\Pi }^\alpha Jf&= {\widetilde{B}}^\alpha _\Pi {\bar{f}}= (-i)^{\alpha +2}\int _{\Pi } \frac{{\bar{f}}(w)}{(\bar{z}-w)^{\alpha +2}}\; d\mu _\alpha (w), \\ JB_\Pi ^\alpha f&= \overline{B_\Pi ^\alpha f} =(-i)^{\alpha +2}\int _{\Pi } \frac{{\bar{f}}(w)}{(\bar{z}-w)^{\alpha +2}}\; d\mu _\alpha (w). \end{aligned}$$

Hence

$$\begin{aligned} {\widetilde{B}}^\alpha _\Pi =JB_\Pi ^\alpha J. \end{aligned}$$

(2.7)

Let $a\in L_\infty (\Pi )$. The Toeplitz operator $\widetilde{T}_a^\alpha :\widetilde{{\mathcal A}}^2_\alpha (\Pi )\rightarrow \widetilde{{\mathcal A}}^2_\alpha (\Pi )$ is the operator defined by

$$\begin{aligned} \widetilde{T}_a^\alpha (g)={\widetilde{B}}_{\Pi }^\alpha (ag),\quad g\in \widetilde{{\mathcal A}}^2_\alpha (\Pi ). \end{aligned}$$

The small Hankel operator $H_a^\alpha :{\mathcal A}_\alpha ^2(\Pi )\rightarrow \widetilde{{\mathcal A}}^2_\alpha (\Pi )$ is defined by

$$\begin{aligned} H_a^\alpha f={\widetilde{B}}^\alpha _\Pi (af), \end{aligned}$$

(2.8)

for all $f\in {\mathcal A}_\alpha ^2(\Pi )$.

The weighted harmonic Bergman space $b^2_\alpha (\Pi )$ is the closed subspace of $L^2(\Pi ,d\mu _\alpha )$ consisting of all complex-valued harmonic functions. Recall that, if $h_1$ is a real-valued harmonic function, a harmonic conjugate for $h_1$ is a real-valued function $h_2$ such that the function $H:=h_1+ih_2$ is analytic. Moreover, if $h_1\in b^2_\alpha (\Pi )$ then the function $h_2$ belongs to $b^2_\alpha (\Pi )$, for more details see [9]. Hence, $H$ belongs to $b^2_\alpha (\Pi )$. Consequently, if $u=u_1+iu_2$ belongs to $b^2_\alpha (\Pi )$ then $u$ has the decomposition

$$\begin{aligned} u=f+{\bar{g}}, \end{aligned}$$

(2.9)

with $f,g\in {\mathcal A}_\alpha ^2(\Pi )$. In fact, $f=(U_1+U_2)/2$ and $g=(U_1-U_2)/2$, where $U_1=u_1+iv_1$ (respectively, $U_2=i(u_2+iv_2)$), and $v_1$ ($v_2$) is the harmonic conjugate for $u_1\,(u_2)$.

On the other hand, if $f$ is an analytic function in $L^1(\Pi , d\mu _\alpha )$, then

$$\begin{aligned} \int _\Pi f(w)d\mu _\alpha (w)=0. \end{aligned}$$

This is called the cancellation property and is proved in [5, Lemma 3.5].

Since the product of two analytic functions satisfies the cancellation property we have $\langle f,{\bar{g}}\rangle _\alpha =0$ for $f,g\in {\mathcal A}_\alpha ^2(\Pi )$. Then ${\mathcal A}_\alpha ^2(\Pi )$ and $\widetilde{{\mathcal A}}^2_\alpha (\Pi )$ are mutually orthogonal spaces. Using the decomposition (2.9) we have that

$$\begin{aligned} b^2_\alpha (\Pi )={\mathcal A}_\alpha ^2(\Pi )\oplus \widetilde{{\mathcal A}}^2_\alpha (\Pi ) \end{aligned}$$

(2.10)

and then, $b^2_\alpha (\Pi )$ is a Hilbert space with the inner product inherited from $L^2(\Pi ,d\mu _\alpha )$. From (2.10), the harmonic Bergman projection $Q^\alpha _\Pi$ from $L^2(\Pi ,d\mu _\alpha )$ onto $b^2_\alpha (\Pi )$ is given by the formula

$$\begin{aligned} Q^\alpha _\Pi =B_\Pi ^\alpha +{\widetilde{B}}^\alpha _\Pi . \end{aligned}$$

(2.11)

3 Toeplitz Operators on the Harmonic Bergman Space on the Upper Half-Plane

For any function $a\in L_\infty (\Pi )$ the Toeplitz operator ${\widehat{T}}_a^\alpha$ with symbol $a$, acting on $b_\alpha ^2(\Pi )$, is defined by

$$\begin{aligned} {\widehat{T}}_a^\alpha =Q^\alpha _\Pi aI=(B_\Pi ^\alpha +{\widetilde{B}}^\alpha _\Pi )aI. \end{aligned}$$

(3.1)

Toeplitz operators acting on the harmonic Bergman space can be represented in terms of Toeplitz operators and small Hankel operators as shown in Theorem 3.1. The unweighted version of this theorem is due to Choe and Nam, see [1].

Theorem 3.1

For $a\in L_\infty (\Pi ),$ the Toeplitz operator ${\widehat{T}}_a^\alpha$ has the following matrix representation

$$\begin{aligned} {\widehat{T}}_a^\alpha =\left( \begin{array}{c@{\quad }c} T_a^\alpha &{} JH^\alpha _{\bar{a}} J \\ H_a^\alpha &{} JT_{\bar{a}}^\alpha J \end{array}\right) \end{aligned}$$

in $b^2_\alpha (\Pi )={\mathcal A}_\alpha ^2(\Pi )\oplus \widetilde{{\mathcal A}}^2_\alpha (\Pi )$, where operators $T_a^\alpha$ and $H^\alpha _{\bar{a}}$ are given by (2.3) and (2.8), respectively.

Proof

Let $a\in L_\infty (\Pi )$ and $f\in {\mathcal A}_\alpha ^2(\Pi )$ then

$$\begin{aligned} {\widehat{T}}_a^\alpha f =B_\Pi ^\alpha af+{\widetilde{B}}_{\Pi }^\alpha af=T_a^\alpha f+H_a^\alpha f, \end{aligned}$$

i.e., ${\widehat{T}}_a^\alpha |_{{\mathcal A}_\alpha ^2(\Pi )}=T_a^\alpha +H_a^\alpha .$ We shall show that ${\widehat{T}}_a^\alpha |_{\widetilde{{\mathcal A}}^2_\alpha (\Pi )}=JH^\alpha _{\bar{a}} J+JT_{\bar{a}}^\alpha J$. Let $f,g,h\in {\mathcal A}_\alpha ^2(\Pi )$. Thus,

$$\begin{aligned} \langle {\widehat{T}}_a^\alpha {\bar{f}},g\rangle _\alpha&= \langle Q^\alpha _\Pi a{\bar{f}},g\rangle _\alpha =\langle a{\bar{f}},Q^\alpha _\Pi g\rangle _\alpha = \langle a{\bar{f}},g\rangle _\alpha =\langle {\bar{g}},\bar{a}f\rangle _\alpha \\&= \langle {\widetilde{B}}^\alpha _\Pi {\bar{g}},\bar{a}f\rangle _\alpha =\langle {\bar{g}},{\widetilde{B}}^\alpha _\Pi (\bar{a}f)\rangle _\alpha =\langle {\bar{g}},H^\alpha _{\bar{a}} f\rangle _\alpha . \end{aligned}$$

Furthermore, $\langle h,H^\alpha _{\bar{a}} f\rangle _\alpha =0$ since $h\in {\mathcal A}_\alpha ^2(\Pi )$ and $H^\alpha _{\bar{a}} f\in \widetilde{{\mathcal A}}^2_\alpha (\Pi )$. Then,

$$\begin{aligned} \langle {\widehat{T}}^{\alpha }_a{\bar{f}},g\rangle _\alpha =\langle {\bar{g}}+h,H^\alpha _{\bar{a}} f\rangle _\alpha . \end{aligned}$$

Analogously,

$$\begin{aligned} \langle {\widehat{T}}^{\alpha }_a{\bar{f}},\bar{h}\rangle _\alpha&= \langle Q^\alpha _\Pi a{\bar{f}},\bar{h}\rangle _\alpha =\langle a{\bar{f}},Q^\alpha _\Pi \bar{h}\rangle _\alpha = \langle a{\bar{f}},\bar{h}\rangle _\alpha =\langle h,\bar{a}f\rangle _\alpha \\&= \langle B_\Pi ^\alpha h,\bar{a}f\rangle _\alpha =\langle h,B_\Pi ^\alpha (\bar{a}f)\rangle _\alpha =\langle h,T_{\bar{a}}^\alpha f\rangle _\alpha \\&= \langle {\bar{g}}+h,T_{\bar{a}}^\alpha f\rangle _\alpha . \end{aligned}$$

So, we obtain

$$\begin{aligned} \langle {\widehat{T}}^{\alpha }_a{\bar{f}},g+\bar{h}\rangle _\alpha&= \langle {\bar{g}}+h,(H^\alpha _{\bar{a}} +T_{\overline{a}}^\alpha )f\rangle _\alpha \\&= \overline{ \langle (H^\alpha _{\bar{a}} +T^\alpha _{\overline{a}})f,{\bar{g}}+h\rangle _\alpha }= \langle \overline{(H^\alpha _{\bar{a}} +T^\alpha _{\overline{a}})f},g +\bar{h}\rangle _\alpha \\&= \langle J(H^\alpha _{\bar{a}} +T_{\overline{a}}^\alpha )J{\bar{f}},g+\bar{h}\rangle _\alpha , \end{aligned}$$

which finishes the proof. $\square$

3.1 Toeplitz Operators with Homogeneous Symbols

In this section we are concerned about Toeplitz operators, acting on the harmonic Bergman space whose symbols are homogeneous, i.e., they depend only on the angular variable.

Passing to polar coordinates we have the following tensor product

$$\begin{aligned} L^2(\Pi ,d\mu _\alpha )=L^2({\mathbb {R}}_+,r^{\alpha +1}dr)\otimes L^2\left( [0,\pi ], {\tiny \frac{1}{\pi }} 2^{\alpha }(\alpha +1)\sin ^{\alpha }\theta d\theta \right) . \end{aligned}$$

(3.2)

The equations

$$\begin{aligned} \frac{\partial }{\partial \bar{z}}f(z)=0,\quad \frac{\partial }{\partial z}g(z)=0, \end{aligned}$$

(3.3)

characterize the functions in ${\mathcal A}_\alpha ^2(\Pi )$ and $\widetilde{{\mathcal A}}^2_\alpha (\Pi )$, respectively within all functions in $L^2(\Pi ,d\mu _\alpha )$. In polar coordinates we rewrite the Eq. (3.3) by

$$\begin{aligned} \left( r\frac{\partial }{\partial r}+i\frac{\partial }{\partial \theta }\right) \phi (r,\theta )=0,\quad \left( r\frac{\partial }{\partial r}-i\frac{\partial }{\partial \theta }\right) \psi (r,\theta )=0. \end{aligned}$$

Consider the Mellin transform $M: L^2({\mathbb {R}}_+,r^{\alpha +1}dr)\rightarrow L^2({\mathbb {R}})$ given by the formula

$$\begin{aligned} (Mf)(x)=\frac{1}{\sqrt{2\pi }}\int _{{\mathbb {R}}_+} r^{-ix+\frac{\alpha }{2}}\;f(r)\;dr. \end{aligned}$$

The inverse Mellin transform $M^{-1}:L^2({\mathbb {R}})\rightarrow L^2({\mathbb {R}}_+,r^{\alpha +1}dr)$ has the form

$$\begin{aligned} (M^{-1}f)(r)=\frac{1}{\sqrt{2\pi }}\int _{{\mathbb {R}}} r^{ix-\frac{\alpha }{2}-1}\;f(x)\;dx. \end{aligned}$$

Following [12], introduce the unitary operator

$$\begin{aligned} U_1&= \frac{1}{\sqrt{\pi }}(M\otimes I) : L^2({\mathbb {R}}_+,r^{\alpha +1}dr)\otimes L^2\left( [0,\pi ],\frac{1}{\pi }2^\alpha (\alpha +1) \sin ^\alpha \theta d\theta \right) \\&\rightarrow L^2({\mathbb {R}})\otimes L^2\left( [0,\pi ],2^\alpha (\alpha +1) \sin ^\alpha \theta d\theta \right) . \end{aligned}$$

Theorem 3.2

[12] The unitary operator $U_1$ is an isometric isomorphism from the space $L^2(\Pi ,d\mu _\alpha )$ onto $L^2({\mathbb {R}})\otimes L^2\left( [0,\pi ],\frac{1}{\pi }2^\alpha (\alpha +1) \sin ^\alpha \theta d\theta \right)$ under which the weighted Bergman space ${\mathcal A}_{\alpha }^2(\Pi )$ is mapped onto

$$\begin{aligned} {\mathcal A}_{1,\alpha }^2=\left\{ f(\lambda )\vartheta _\alpha (\lambda )e^{-(\lambda +(1+\frac{\alpha }{2})i)\theta }:\; f(\lambda )\in L^2({\mathbb {R}})\right\} , \end{aligned}$$

(3.4)

where

$$\begin{aligned} \vartheta _\alpha (\lambda )=\left( 2^\alpha (\alpha +1)\int _0^\pi e^{-2\lambda \theta } \sin ^\alpha \theta d\theta \right) ^{-\frac{1}{2}}= \frac{\left| \Gamma \left( \frac{\alpha +2}{2}+i\lambda \right) \right| }{\sqrt{\pi }\;\Gamma (\alpha +2)^{\frac{1}{2}}} e^{\frac{\lambda \pi }{2}}. \end{aligned}$$

(3.5)

An analogous result holds for the weighted anti-Bergman space.

Theorem 3.3

The unitary operator $U_1$ is an isometric isomorphism of the space $L^2(\Pi ,d\mu _\alpha )$ onto $L^2({\mathbb {R}})\otimes L^2\left( [0,\pi ],\frac{1}{\pi }2^\alpha (\alpha +1) \sin ^\alpha \theta d\theta \right)$ under which the weighted anti-Bergman space $\widetilde{{\mathcal A}}_{\alpha }^2(\Pi )$ is mapped onto

$$\begin{aligned} \widetilde{{\mathcal A}}_{1,\alpha }^2=\left\{ g(\lambda )\vartheta _\alpha (-\lambda )e^{(\lambda +(1+\frac{\alpha }{2})i)\theta }:\; g(\lambda )\in L^2({\mathbb {R}})\right\} . \end{aligned}$$

(3.6)

Proof

Recall that the equation

$$\begin{aligned} \left( r\frac{\partial }{\partial r}-i\frac{\partial }{\partial \theta }\right) \psi (r,\theta )=0 \end{aligned}$$

characterizes each function $\psi$ in $\widetilde{{\mathcal A}}_{\alpha }^2(\Pi )$ within all functions in $L^2(\Pi ,d\mu _\alpha )$. The operator $r\frac{\partial }{\partial r}-i\frac{\partial }{\partial \theta }$ is unitarily equivalent to the operator

$$\begin{aligned} U_1\left( r\frac{\partial }{\partial r}-i\frac{\partial }{\partial \theta } \right) U_1^*=i\left[ \left( \lambda +\frac{\alpha }{2}+1\right) I- \frac{\partial }{\partial \theta }\right] . \end{aligned}$$

Thus, the image of $\widetilde{{\mathcal A}}_{\alpha }^2(\Pi )$ under the operator $U_1$ is

$$\begin{aligned} \widetilde{{\mathcal A}}_{1,\alpha }^2=\left\{ \psi (\lambda ,\theta ):\; \left[ \left( \lambda +\frac{\alpha }{2}+1\right) I-\frac{\partial }{\partial \theta }\right] \psi (\lambda ,\theta )=0\right\} . \end{aligned}$$

The general solution to the differential equation

$$\begin{aligned} \left[ \left( \lambda +\frac{\alpha }{2}+1\right) I- \frac{\partial }{\partial \theta }\right] \psi (\lambda ,\theta )=0 \end{aligned}$$

is

$$\begin{aligned} \psi (\lambda ,\theta )=h(\lambda )e^{\left( \lambda +(1+ \frac{\alpha }{2})i\right) \theta }, \end{aligned}$$

(3.7)

where $h(\lambda )\in L^2({\mathbb {R}})$. Formula (3.7) can be rewritten in the following form

$$\begin{aligned} \psi (\lambda , \theta )=g(\lambda )\vartheta _\alpha (-\lambda )e^{(\lambda +(1+\frac{\alpha }{2})i)\theta }, \end{aligned}$$

where $\vartheta _\alpha$ is given by (3.5) and $g(\lambda )\in L^2({\mathbb {R}})$. Furthermore,

$$\begin{aligned} \Vert \psi \Vert _{{\mathcal A}_{1,\alpha }^2}=\Vert g\Vert _{L^2({\mathbb {R}})}. \end{aligned}$$

$\square$

Denote by $B_1^\alpha$ the orthogonal projection from $L^2({\mathbb {R}})\otimes L^2\left( [0,\pi ],2^\alpha (\alpha +1)\right.$ $\left. \sin ^\alpha \theta d\theta \;\right)$ onto ${\mathcal A}_{1,\alpha }^2$ and by ${\widetilde{B}}^\alpha _1$ the corresponding orthogonal projection from $L^2({\mathbb {R}})\otimes L^2\left( [0,\pi ],2^\alpha (\alpha +1) d\theta \right)$ onto $\widetilde{{\mathcal A}}_{1,\alpha }^2$.

Corollary 3.4

The following equations hold

$$\begin{aligned} (B_1^\alpha \phi )(\lambda ,\theta )&= \left( U_1\;B_\Pi ^\alpha \; U_1^*\phi \right) (\lambda ,\theta )\\&= 2^\alpha (\alpha +1)\vartheta _\alpha (\lambda ) e^{-(\lambda +(1+\frac{\alpha }{2})i)\eta }\int _0^\pi \phi (\lambda ,\eta )\vartheta _\alpha (\lambda ) e^{-(\lambda -(1+\frac{\alpha }{2})i)\eta } \sin ^\alpha \eta \; d\eta ,\\ ({\widetilde{B}}_1^\alpha \psi )(\lambda ,\theta )&= \left( U_1\;{\widetilde{B}}_\Pi ^\alpha \; U_1^*\psi \right) (\lambda ,\theta )\\&= 2^\alpha (\alpha +1)\vartheta _\alpha (-\lambda ) e^{(\lambda +(1+\frac{\alpha }{2})i)\theta }\int _0^\pi \psi (\lambda ,\eta )\vartheta _\alpha (-\lambda ) e^{(\lambda -(1+\frac{\alpha }{2})i)\eta } \sin ^\alpha \eta \; d\eta , \end{aligned}$$

where $\vartheta _\alpha$ is given by (3.5).

Introduce the isometric embeddings $R_0\!:\! L^2({\mathbb {R}})\rightarrow {\mathcal A}_{1,\alpha }^2,\,{\widetilde{R}}_0\!:\! L^2({\mathbb {R}})\rightarrow \widetilde{{\mathcal A}}_{1,\alpha }^2$ with ${\mathcal A}_{1,\alpha }^2,\widetilde{{\mathcal A}}_{1,\alpha }^2\subset L^2({\mathbb {R}})\otimes L^2\left( [0,\pi ],2^\alpha (\alpha +1) \sin ^\lambda \theta d\theta \right) ,$ defined by

$$\begin{aligned} \nonumber (R_0f)(\lambda ,\theta )&= f(\lambda )\vartheta _\alpha (\lambda ) \; e^{-\left( \lambda +(1+\frac{\alpha }{2})i\right) \theta }, \\ ({\widetilde{R}}_0g)(\lambda ,\theta )&= g(\lambda )\vartheta _\alpha (-\lambda )e^{\left( \lambda +(1+\frac{\alpha }{2})i\right) \theta }. \end{aligned}$$

(3.8)

The adjoint operators

$$\begin{aligned} R_0^*, {\widetilde{R}}_0^*: L^2({\mathbb {R}})\otimes L^2\left( [0,\pi ],2^\alpha (\alpha +1) \sin ^\alpha \theta d\theta \right) \rightarrow L^2({\mathbb {R}}), \end{aligned}$$

are given by

$$\begin{aligned} \nonumber (R_0^*\phi )(\lambda )&= 2^\alpha (\alpha +1)\vartheta _\alpha (\lambda )\; \int _{0}^{\pi }\; \phi (\lambda ,\theta )e^{-[\lambda -(1+\frac{\alpha }{2})i)] \theta }\sin ^\alpha \theta \; d\theta , \\ ({\widetilde{R}}^*_0\psi )(\lambda )&= \; 2^\alpha (\alpha +1)\vartheta _\alpha (-\lambda )\; \int _{0}^{\pi }\; \psi (\lambda ,\theta )e^{[\lambda -(1+\frac{\alpha }{2})i)] \theta }\sin ^\alpha \theta \; d\theta . \end{aligned}$$

These operators satisfy the following statements

$$\begin{aligned}&\nonumber R_0^*R_0= I :L^2({\mathbb {R}})\rightarrow L^2({\mathbb {R}}),\\&{\widetilde{R}}_0^*{\widetilde{R}}_0= I :L^2({\mathbb {R}})\rightarrow L^2({\mathbb {R}}),\\&\nonumber R_0R_0 ^*= B_1^\alpha :L^2({\mathbb {R}})\otimes L^2\left( [0,\pi ],2^\alpha (\alpha +1) \sin ^\alpha \theta d\theta \right) \rightarrow {\mathcal A}_{1,\alpha }^2,\\&\nonumber {\widetilde{R}}_0{\widetilde{R}}^*_0 = {\widetilde{B}}^\alpha _1 : L^2({\mathbb {R}})\otimes L^2\left( [0,\pi ],2^\alpha (\alpha +1) \sin ^\alpha \theta d\theta \right) \rightarrow \widetilde{{\mathcal A}}_{1,\alpha }^2. \end{aligned}$$

(3.9)

The operators

$$\begin{aligned} R_\alpha =R_0 ^*U_1\quad \text {and}\quad {\widetilde{R}}_\alpha ={\widetilde{R}}_0 ^*U_1 \end{aligned}$$

(3.10)

transform the space $L^2(\Pi ,d\mu _\alpha )$ onto $L^2({\mathbb {R}})$. Furthermore $R_\alpha |_{{\mathcal A}_\alpha ^2(\Pi )}:{\mathcal A}_\alpha ^2(\Pi )\rightarrow L^2({\mathbb {R}})$ is an isometric isomorphism as well as ${\widetilde{R}}_\alpha |_{\widetilde{{\mathcal A}}_\alpha ^2(\Pi )}:\widetilde{{\mathcal A}}_\alpha ^2(\Pi )\rightarrow L^2({\mathbb {R}})$. Moreover, the following equations hold

$$\begin{aligned}&\nonumber R_\alpha R_\alpha ^*=I:L^2({\mathbb {R}})\rightarrow L^2({\mathbb {R}}),\\&\nonumber {\widetilde{R}}_\alpha {\widetilde{R}}_\alpha ^*=I :L^2({\mathbb {R}})\rightarrow L^2({\mathbb {R}}),\\&R_\alpha ^*R_\alpha = B_\Pi ^\alpha :L^2(\Pi , d\mu _\alpha )\rightarrow {\mathcal A}_\alpha ^2(\Pi ),\\&\nonumber {\widetilde{R}}_\alpha ^*{\widetilde{R}}_\alpha ={\widetilde{B}}^\alpha _\Pi :L^2(\Pi , d\mu _\alpha )\rightarrow \widetilde{{\mathcal A}}_\alpha ^2(\Pi ). \end{aligned}$$

(3.11)

Let ${\mathcal A}_\infty$ be the algebra of all bounded measurable homogeneous functions on $\Pi$ of order zero or functions depending only on the angular variable $\theta$ and such that the limits

$$\begin{aligned} a_0=\lim _{\theta \rightarrow 0} a(\theta ),\quad a_\pi =\lim _{\theta \rightarrow \pi }a(\theta ) \end{aligned}$$

(3.12)

exist.

Theorem 3.5

[12] Let $a(\theta )\in {\mathcal A}_\infty$, the Toeplitz operator $T_a^\alpha$, acting on ${\mathcal A}_\alpha ^2(\Pi )$, is unitarily equivalent to the operator of multiplication $\gamma _{a,\alpha }I= R_\alpha T_a^\alpha R_\alpha ^*$ acting on the space $L^2({\mathbb {R}})$. The function $\gamma _{a,\alpha }(\lambda )$ is given by

$$\begin{aligned} \nonumber \gamma _{a,\alpha }(\lambda )&= 2^\alpha (\alpha +1)\; \vartheta _\alpha ^2(\lambda )\;\int _0^\pi a(\theta )e^{-2\lambda \theta } \sin ^\alpha \theta d\theta ,\\&= \left( \,\int _0^\pi e^{-2\lambda \theta }\sin ^\alpha \theta d\theta \right) ^{-1}\int _0^\pi a(\theta )e^{-2\lambda \theta } \sin ^\alpha \theta d\theta , \lambda \in {\mathbb {R}}. \end{aligned}$$

(3.13)

An analogous result holds for Toeplitz operators acting on the weighted anti-Bergman space on the upper half-plane.

Theorem 3.6

Let $a(\theta )\in {\mathcal A}_\infty .$ The Toeplitz operator $\widetilde{T}_a^\alpha$ acting on $\widetilde{{\mathcal A}}_\alpha ^2(\Pi )$ is unitarily equivalent to the multiplication operator $\widetilde{\gamma }_{a,\alpha }I= {\widetilde{R}}_\alpha \widetilde{T}_a^\alpha {\widetilde{R}}_\alpha ^*$ on the space $L^2({\mathbb {R}})$. The function $\widetilde{\gamma }_{a,\alpha }(\lambda )$ is given by

$$\begin{aligned} \widetilde{\gamma }_{a,\alpha }(\lambda )=\gamma _{a,\alpha }(-\lambda ). \end{aligned}$$

(3.14)

From (3.12) it follows that, for $a(\theta )\in {\mathcal A}_\infty$, the functions $\gamma _{a,\alpha }$ and $\widetilde{\gamma }_{a,\alpha }$ satisfy

$$\begin{aligned} \nonumber \lim \limits _{\lambda \rightarrow \infty }\gamma _{a,\alpha }(\lambda )&= \lim \limits _{\lambda \rightarrow -\infty }\widetilde{\gamma }_{a,\alpha }(\lambda )=\lim _{\theta \rightarrow 0} a(\theta )=a_0,\\ \lim \limits _{\lambda \rightarrow -\infty }\gamma _{a,\alpha }(\lambda )&= \lim \limits _{\lambda \rightarrow \ \infty }\widetilde{\gamma }_{a,\alpha }(\lambda )\ =\lim _{\theta \rightarrow \pi } a(\theta )=a_\pi . \end{aligned}$$

(3.15)

Consider the unitary operator $V_\alpha : L^2(\Pi , d\mu _\alpha )\oplus L^2(\Pi , d\mu _\alpha )\rightarrow L^2({\mathbb {R}})\oplus L^2({\mathbb {R}})$ given by

$$\begin{aligned} V_\alpha =\left( \begin{array}{c@{\quad }c} R_\alpha &{} 0 \\ 0 &{} {\widetilde{R}}_\alpha \\ \end{array} \right) . \end{aligned}$$

Its restriction $V_\alpha |_{b_\alpha ^2(\Pi )}:b_\alpha ^2(\Pi )={\mathcal A}_\alpha ^2(\Pi )\oplus \widetilde{{\mathcal A}}_\alpha ^2(\Pi ) \rightarrow L^2({\mathbb {R}})\oplus L^2({\mathbb {R}})$ is an isometric isomorphism. The adjoint operator $V_\alpha ^*$ is an isometric isomorphism from $L^2({\mathbb {R}})\oplus L^2({\mathbb {R}})$ onto $b_\alpha ^2(\Pi )$. Using (3.11) we get

$$\begin{aligned} V_\alpha V_\alpha ^*&= \left( \begin{array}{c@{\quad }c} \; I\; &{} 0 \\ 0 &{} \;I\; \\ \end{array} \right) : L^2({\mathbb {R}})\oplus L^2({\mathbb {R}})\rightarrow L^2({\mathbb {R}}) \oplus L^2({\mathbb {R}}), \\ V_\alpha ^*V_\alpha&= \left( \begin{array}{c@{\quad }c} B^\alpha _\Pi &{} 0 \\ 0 &{} {\widetilde{B}}^\alpha _\Pi \\ \end{array} \right) : L^2(\Pi ,d\mu _\alpha )\oplus L^2(\Pi , d\mu _\alpha )\rightarrow b^2_\alpha (\Pi )={\mathcal A}_\alpha ^2(\Pi )\oplus \widetilde{{\mathcal A}}_\alpha ^2(\Pi ). \end{aligned}$$

Theorem 3.7

Let $a(\theta )\in {\mathcal A}_\infty$. Then$,$ the Toeplitz operator ${\widehat{T}}_a^\alpha$ acting on $b^2_\alpha (\Pi )$ is unitarily equivalent to the multiplication operator by the matrix $A_a^{\alpha }$ acting on $L^2({\mathbb {R}})\oplus L^2({\mathbb {R}})$. The matrix $A_a^{\alpha }$ is given by

$$\begin{aligned} A_a^{\alpha }(\lambda )=2^\alpha (\alpha +1)\;\vartheta ^2_\alpha (\lambda )\;\left( \,\int _0^\pi a(\theta ) \; a_{ij}(\theta ,\lambda )\sin ^\alpha \theta d\theta \right) , \end{aligned}$$

(3.16)

where

$$\begin{aligned}&a_{11}(\theta ,\lambda )=e^{-2\lambda \theta },\quad a_{12}(\theta ,\lambda )=e^{-\lambda \pi }e^{(\alpha +2)i\theta },\\&a_{21}(\theta ,\lambda )=\overline{a_{12}(\theta ,\lambda )},\quad a_{22}(\theta ,\lambda )=e^{-2\lambda \pi }a_{11}(\theta ,-\lambda ), \end{aligned}$$

and $\vartheta _\alpha$ is given by (3.5).

Proof

We shall prove that

$$\begin{aligned} A_a^{\alpha }(\lambda )I=V_\alpha {\widehat{T}}_aV^*_\alpha =\left( \begin{array}{c@{\quad }c} \gamma _{a,\alpha } (\lambda ) &{} \nu _{a,\alpha } (\lambda ) \\ \bar{\nu }_{\bar{a},\alpha }(\lambda ) &{} \widetilde{\gamma }_{a,\alpha }(\lambda ) \\ \end{array} \right) I, \end{aligned}$$

(3.17)

where $\gamma _{a,\alpha }(\lambda ),\,\widetilde{\gamma }_{a,\alpha }(\lambda )$ are given by (3.13) and (3.14) respectively, and $\nu _{a,\alpha }(\lambda )$ is given by

$$\begin{aligned} \nu _{a,\alpha }(\lambda )&= 2^\alpha (\alpha +1) \vartheta _\alpha (\lambda )\vartheta _\alpha (-\lambda )\int _0^{\pi } a(\theta )\; e^{(\alpha +2)i\theta }\sin ^\alpha \theta d\theta \\ \nonumber&= 2^\alpha (\alpha +1)\vartheta _\alpha ^2(\lambda ) e^{-\lambda \pi }\int _0^{\pi } a(\theta )\; e^{(\alpha +2)i\theta }\sin ^\alpha \theta d\theta ,\quad \lambda \in {\mathbb {R}}. \end{aligned}$$

(3.18)

Let $a\in {\mathcal A}_\infty$, using the matrix representation for the Toeplitz operator ${\widehat{T}}_a^\alpha$ given in Theorem 3.1, we get

$$\begin{aligned} V_\alpha {\widehat{T}}_a^\alpha V^*_\alpha =\left( \begin{array}{c@{\quad }c} R_\alpha T_a^\alpha R^*_\alpha &{} R_\alpha JH_{\bar{a}}^\alpha J{\widetilde{R}}^*_\alpha \\ {\widetilde{R}}_\alpha H_a^\alpha R^*_\alpha &{} {\widetilde{R}}_\alpha JT_{\bar{a}}^\alpha J{\widetilde{R}}^*_\alpha \end{array}\right) . \end{aligned}$$

Let us analyze the entries of the above matrix. From Theorem 3.5, $R_\alpha T_a^\alpha R^*_\alpha =\gamma _{a,\alpha }(\lambda ) I$. Using (2.7) and (2.8)

$$\begin{aligned} R_\alpha JH_{\bar{a}}^\alpha J{\widetilde{R}}^*_\alpha&= R_\alpha J{\widetilde{B}}_\Pi ^\alpha \bar{a} J{\widetilde{R}}^*_\alpha = R_\alpha B_\Pi ^\alpha J\bar{a} J{\widetilde{R}}^*_\alpha \\&= R_\alpha B_\Pi ^\alpha a{\widetilde{R}}^*_\alpha . \end{aligned}$$

From (3.11) it follows that

$$\begin{aligned} R_\alpha B_\Pi ^\alpha a(\theta ) {\widetilde{R}}^*_\alpha&= R_\alpha (R^*_\alpha R_\alpha ) a(\theta ) {\widetilde{R}}^*_\alpha = R_\alpha a(\theta ) {\widetilde{R}}_\alpha ^*\\&= R_0^*(M\otimes I)a(\theta )(M^{-1}\otimes I){\widetilde{R}}_0= R_0^*a(\theta ){\widetilde{R}}_0. \end{aligned}$$

On the other hand,

$$\begin{aligned} \left( R_0^*a(\theta ){\widetilde{R}}_0 f\right) (\lambda ,\theta )&= 2^\alpha (\alpha +1) \vartheta _\alpha (\lambda )\\&\times \int _0^\pi f(\lambda )\vartheta _\alpha (-\lambda ) e^{[\left( \lambda +(1+\frac{\alpha }{2}) i\right) -\left( \lambda -(1+ \frac{\alpha }{2})i\right) ]\theta }a(\theta ){\sin }^\alpha \theta \;d\theta \\&= 2^\alpha (\alpha +1)\vartheta _\alpha (\lambda ) \vartheta _\alpha (-\lambda )f(\lambda )\int _0^\pi a(\theta )e^{\left( 2+\alpha \right) i\theta } {\sin }^\alpha \theta \; d\theta \\&= f(\lambda )\;2^\alpha (\alpha +1) e^{-\lambda \pi }\;\vartheta _\alpha ^2(\lambda )\int _0^\pi a(\theta )e^{\left( 2+\alpha \right) i\theta }{\sin }^\alpha \theta \; d\theta \\&= f(\lambda )\;\nu _{a,\alpha }(\lambda ). \end{aligned}$$

Analogously,

$$\begin{aligned} {\widetilde{R}}_\alpha {\widetilde{B}}_\Pi a(\theta )R^*_\alpha&= {\widetilde{R}}_\alpha ({\widetilde{R}}^*_\alpha {\widetilde{R}}_\alpha ) a(\theta )R^*_\alpha = {\widetilde{R}}_\alpha a(\theta )R_\alpha ^* \\&= {\widetilde{R}}_0^*(M\otimes I)a(\theta )(M^{-1}\otimes I)R_0\\&= {\widetilde{R}}_0^*a(\theta )R_0=\bar{\nu }_{a,\alpha }(\lambda )I. \end{aligned}$$

Now, using (2.7) and (3.11)

$$\begin{aligned} {\widetilde{R}}_\alpha JT_{\bar{a}}^\alpha J{\widetilde{R}}^*_\alpha&= {\widetilde{R}}_\alpha JB_\Pi ^{\alpha } \bar{a}J{\widetilde{R}}^*_\alpha ={\widetilde{R}}_\alpha JB_\Pi ^\alpha Ja{\widetilde{R}}^*_\alpha \\&= {\widetilde{R}}_\alpha {\widetilde{B}}_\Pi ^{\alpha } a{\widetilde{R}}^*_\alpha ={\widetilde{R}}_\alpha ({\widetilde{R}}^*_\alpha {\widetilde{R}}_\alpha ) a{\widetilde{R}}^*_\alpha ={\widetilde{R}}_\alpha a{\widetilde{R}}^*_\alpha \\&= {\widetilde{R}}_0^*(M\otimes I)a(\theta )(M^{-1}\otimes I){\widetilde{R}}_0\\&= {\widetilde{R}}_0^*a(\theta ){\widetilde{R}}_0. \end{aligned}$$

This implies that,

$$\begin{aligned} \left( {\widetilde{R}}_0^*a(\theta ){\widetilde{R}}_0g\right) (\lambda ,\theta )&= 2^\alpha (\alpha +1) \vartheta _\alpha (-\lambda )\\&\times \int _{0}^{\pi }g(\lambda ) \vartheta _\alpha (-\lambda )e^{\left( \lambda + (1+\frac{\alpha }{2})i+\lambda -(1+ \frac{\alpha }{2})i)\right) \theta }a(\theta )\sin ^\alpha \theta \; d\theta \\&= g(\lambda )\;2^\alpha \;(\alpha +1)\vartheta _\alpha ^2 (-\lambda )\int _{0}^{\pi }a(\theta )\;e^{2\lambda \theta }\sin ^\alpha \theta \; d\theta \\&= g(\lambda )\;\widetilde{\gamma }_{a,\alpha }(\lambda ). \end{aligned}$$

$\square$

We denote by ${\mathcal T}({\mathcal A}_\infty )$ the $C^*$-algebra generated by all operators ${\widehat{T}}_a^\alpha$, where $a\in {\mathcal A}_\infty$.

Corollary 3.8

The $C^*$-algebra ${\mathcal T}({\mathcal A}_\infty )$ is isomorphic and isometric to the algebra generated by all the multiplication operators by matrices $A_a^{\alpha }(\lambda )$ acting on $L^2({\mathbb {R}})\oplus L^2({\mathbb {R}})$. The isomorphism is given by the following transform of the generators:

$$\begin{aligned} {\widehat{T}}_a^\alpha \mapsto A_a^{\alpha }(\lambda ) I, \end{aligned}$$

where the matrix $A_a^{\alpha }(\lambda )$ is given by (3.17).

In [10] the asymptotic behavior of the Gamma function $|\Gamma (x+iy)|$, when $|y|$ tends to infinity, is studied. It is proved there that

$$\begin{aligned} |\Gamma (x+iy)|=\sqrt{2\pi }\; |y|^{x-1/2}\; e^{-x-|y|\pi /2}\left( 1+O\left( \frac{1}{|y|}\right) \right) \end{aligned}$$

as $|y|\rightarrow \infty$. Then, for $|\lambda |$ big enough

$$\begin{aligned} \vartheta _\alpha (\lambda )\;\vartheta _\alpha (-\lambda )&= \frac{\left| \Gamma (\frac{\alpha +2}{2}+i\lambda )\right| ^2}{\pi \Gamma (\alpha +2)}\approx \frac{\left( \sqrt{2\pi }\; |\lambda |^{\frac{\alpha +2}{2}-\frac{1}{2}}\; e^{-\frac{\alpha +2}{2}-\frac{|\lambda |\pi }{2}} \right) ^2}{\pi \Gamma (\alpha +2)}\\&= \frac{2\; |\lambda |^{\alpha +1}\; e^{-(\alpha +2+|\lambda |\pi )}}{\Gamma (\alpha +2)}= \frac{2\;e^{-(\alpha +2)} |\lambda |^{\alpha +1} e^{-|\lambda |\pi }}{\Gamma (\alpha +2)}. \end{aligned}$$

Thus,

$$\begin{aligned} \lim _{\lambda \rightarrow \pm \infty }\vartheta _\alpha (\lambda ) \;\vartheta _\alpha (-\lambda )=0. \end{aligned}$$

Using last result in (3.18) and the fact that every function $a\in {\mathcal A}_{\infty }$ is bounded, we get

$$\begin{aligned} \lim _{\lambda \rightarrow \pm \infty }\nu _{a,\alpha }(\lambda )=0. \end{aligned}$$

Analogously

$$\begin{aligned} \lim _{\lambda \rightarrow \pm \infty }\bar{\nu }_{\bar{a},\alpha }(\lambda )=0. \end{aligned}$$

From this it follows that the matrix function $A_a^{\alpha }(\cdot )$, defined by (3.16), is diagonal at the points $-\infty$ and $\infty$.

Lemma 3.9

• (1) There are two functions $a, b \in {\mathcal A}_\infty ,$ depending on the parameter $\alpha ,$ such that $A_a^{\alpha }(\lambda ), A_b^\alpha (\lambda )$ generate $M_2({\mathbb {C}})$ for each $\lambda \in {\mathbb {R}}.$

• (2) For $\lambda \in \{\pm \infty \},$ any pair of functions $a,b \in {\mathcal A}_\infty$ such that the vector $(a_0, a_{\pi })$ is not a scalar multiple of $(b_0, b_{\pi })$ are such that $A_a^{\alpha }(\lambda ), A_b^\alpha (\lambda )$ generate the algebra of all $2\times 2$ diagonal matrices.

Proof

To prove (1) recall that any two matrices $A,B \in M_2({\mathbb {C}})$ generate $M_2({\mathbb {C}})$ if and only if the commutator $[A,B]=AB-BA$ is invertible. Straightforward calculations show that the commutator $[A_a^{\alpha }(\lambda ), A_b^\alpha (\lambda )]$ is given by the matrix

$$\begin{aligned}{}[A_a^{\alpha }(\lambda ), A_b^\alpha (\lambda )]=f(\lambda )\;\left( \,\,\int _0^\pi \int _0^\pi a(\theta ) b(\varphi )\; M_{ij}^\alpha (\theta ,\varphi )\sin ^\alpha \theta \sin ^\alpha \varphi \;d\varphi \;d\theta \right) , \end{aligned}$$

where $f(\lambda )=2^\alpha (\alpha +1)\vartheta _\alpha ^2(\lambda )$ and $M_{ij}^\alpha , i,j \in \{1,2\}$ are given by

$$\begin{aligned} M_{11}^\alpha (\theta ,\varphi )&= 2ie^{-2\lambda \pi } \sin [(\alpha +2)(\theta -\varphi )],\\ M_{12}^\alpha (\theta ,\varphi )&= e^{-\lambda \pi } \left[ e^{(\alpha +2)i\varphi }\left( e^{-2\lambda \theta } -e^{2\lambda (\theta -\pi )}\right) -e^{(\alpha +2)i\theta } \left( e^{-2\lambda \varphi }-e^{2\lambda (\varphi -\pi )}\right) \right] ,\\ M_{21}^\alpha (\theta ,\varphi )&= -\overline{M_{12}^\alpha (\theta ,\varphi )},\\ M_{22}^\alpha (\theta ,\varphi )&= -M_{11}^\alpha (\theta ,\varphi ). \end{aligned}$$

Thus, the determinant of the matrix $[A_a^{\alpha }(\lambda ), A_b^\alpha (\lambda )]$ is given by

$$\begin{aligned} \det [A_a^{\alpha }(\lambda ),A_b^\alpha (\lambda )] =f(\lambda )^2\left[ 4(I_1^\alpha (\lambda ))^2+ \left| I_2^\alpha (\lambda )\right| ^2\right] \ge 0, \end{aligned}$$

where

$$\begin{aligned} I_1^\alpha (\lambda )&= e^{-2\lambda \pi }\int _0^\pi \int _0^\pi a(\theta )b(\varphi )\sin [(\alpha +2)(\theta -\varphi )] \sin ^\alpha \theta \sin ^\alpha \varphi d\varphi \;d\theta ,\\ I_2^\alpha (\lambda )&= e^{-\lambda \pi }\int _0^\pi \int _0^\pi a(\theta )b(\varphi )\left[ e^{(\alpha +2)i\varphi } \left( e^{-2\lambda \theta }-e^{2\lambda (\theta -\pi )}\right) \right. \\&-\left. e^{(\alpha +2)i\theta }\left( e^{-2\lambda \varphi } -e^{2\lambda (\varphi -\pi )}\right) \right] \sin ^\alpha \theta \sin ^\alpha \varphi d\varphi \;d\theta . \end{aligned}$$

Consider the functions $a_0(\theta )=\chi _{\big (0, \frac{\pi }{2(\alpha +2)}\big )}(\theta )(\alpha +2) \cos (\alpha +2)\theta \sin ^{-\alpha }\theta$ and $b_0(\theta )=\chi _{\big (0,\frac{\pi }{2(\alpha +2)} \big )}(\theta )(\alpha +2)\sin (\alpha +2)\theta \sin ^{-\alpha }\theta$, where $\theta \in (0,\pi )$. Then we have,

$$\begin{aligned} I_1^{\alpha }(\lambda )=e^{-2\lambda \pi }\left( \frac{1}{4}- \frac{\pi ^2}{16}\right) . \end{aligned}$$

(3.19)

Since $f(\lambda )\not =0$ for every $\lambda \in {\mathbb {R}}$, (3.19) implies that $\det [A_a^{\alpha }(\lambda ), A_b^\alpha (\lambda )]>0$ for every $\lambda \in {\mathbb {R}}$. Thus, the commutator $[A_a^{\alpha }(\lambda ), A_b^\alpha (\lambda )]$ is invertible and then $A_{a_0}^{\alpha }(\lambda ), A_{b_0}^\alpha (\lambda )$ generate $M_2({\mathbb {C}})$.

To prove (2) note that,

$$\begin{aligned} A_{a,\alpha }(\pm \infty )=\left( \begin{array}{c@{\quad }c} \lim \nolimits _{\lambda \rightarrow \pm \infty } \gamma _{a,\alpha }(\lambda ) &{} 0 \\ 0&{} \lim \nolimits _{\lambda \rightarrow \pm \infty } \widetilde{\gamma }_{a,\alpha }(\lambda ) \\ \end{array} \right) . \end{aligned}$$

From which any pair of functions $a,b \in {\mathcal A}_\infty$ such that the vector $(a_0, a_{\pi })$ is not a scalar multiple of $(b_0, b_{\pi })$ generate all $2\times 2$ diagonal matrices. As an example of such kind of functions we fix $\theta _0\in (0,\pi )$ and let $a_0(\theta )=\chi _{(0,\theta _0)}(\theta )$ and $b_0(\theta )=\chi _{(\theta _0,\pi )}(\theta )$.

$\square$

In order to prove the main result of this section, we recall some results from [2, Chapter 11].

A $C^*$-algebra ${\mathcal A}$ is called type I or postliminal or GCR if for each irreducible representation $(\pi ,H)$ of ${\mathcal A},{\mathcal K}(H)\subset \pi ({\mathcal A})$, where ${\mathcal K}(H)$ denote the ideal of all compact operators defined in $H$.

Let ${\mathcal A}$ be a $C^*$-algebra, and ${\mathcal B}$ a $C^*$-subalgebra of ${\mathcal A}$. Then ${\mathcal B}$ is said to be a rich $C^*$-subalgebra of ${\mathcal B}$ if the following conditions are satisfied.

• (i) For every irreducible representation $\pi$ of ${\mathcal A},\,\pi |_{{\mathcal B}}$ is irreducible.

• (ii) If $\pi$ and $\pi '$ are inequivalent irreducible representation of ${\mathcal A}$, then $\pi |_{{\mathcal B}}$ and $\pi '|_{{\mathcal B}}$ are inequivalent.

So, we have the next result which is an extension of the Stone–Weierstrass theorem to $C^*$-algebras of type I.

Proposition 3.10

In a type I $C^*$-algebra ${\mathcal A},$ every rich $C^*$-subalgebra is equal to ${\mathcal A}.$

Now, let ${\mathfrak {S}}_{\mathbb {R}}$ be the $C^*$-algebra of all $2\times 2$ matrix-valued continuous functions $(f_{ij}(t))$ defined on $[-\infty ,\infty ]$ and such that they satisfy $f_{12}(\pm \infty )=f_{21}(\pm \infty )=0$ and $f_{11}(\pm \infty )=f_{22}(\mp \infty )$. Note that ${\mathfrak {S}}_{\mathbb {{\mathbb {R}}}}$ is a type I $C^*$-algebra. Now we are ready to formulate the main result of this section.

Theorem 3.11

The $C^*$-algebra ${\mathcal T}({\mathcal A}_\infty )$ is isomorphic and isometric to the algebra ${\mathfrak {S}}_{\mathbb {{\mathbb {R}}}}.$

Proof

By Theorem 3.8, it is sufficient to prove that the algebra generated by all multiplication operators by matrices $A_a^{\alpha }(\lambda )$, defined in $L^2({\mathbb {R}})\oplus L^2({\mathbb {R}})$ is isomorphic and isometric to the algebra ${\mathfrak {S}}_{\mathbb {{\mathbb {R}}}}$. To do so, we shall prove that it is a rich $C^*$-subalgebra of ${\mathfrak {S}}_{\mathbb {{\mathbb {R}}}}$.

We begin by characterizing all irreducible representations of ${\mathfrak {S}}_{\mathbb {{\mathbb {R}}}}$. Any element $F\in {\mathfrak {S}}_{\mathbb {{\mathbb {R}}}}$ can be regarded as a $2\times 2$ matrix $(f_{ij})$ of complex-valued continuous functions defined in $[-\infty ,\infty ]$, satisfying $f_{12}(\pm \infty )=f_{21}(\pm \infty )=0$. Then, for each $\lambda \in {{\mathbb {R}}}$ we obtain a two-dimensional irreducible representation $\pi _\lambda$ of ${\mathfrak {S}}_{\mathbb {{\mathbb {R}}}}$ by the evaluation at $\lambda$, i.e., $\pi _\lambda (F)=F(\lambda )$. Associated with the points $\lambda =\pm \infty$ we have two one-dimensional irreducible representations: $\pi _{1}^{\pm }(F)=f_{11}(\pm \infty )$. These are the only inequivalent irreducible representations of ${\mathfrak {S}}_{\mathbb {{\mathbb {R}}}}$.

From Lemma 3.9 it follows that every irreducible representation of ${\mathfrak {S}}_{\mathbb {{\mathbb {R}}}}$, restricted to the algebra generated by all multiplication operators by matrices $A_a^{\alpha }(\lambda )$, is also irreducible.

Let $\lambda _{0}\ne \lambda _1$ be two points in $[-\infty ,\infty ]$. Consider the representations $\pi _{\lambda _{0}},\pi _{\lambda _{1}}$ and the restrictions $\pi _{\lambda _0}^{\prime }:=\pi _{\lambda _{0}}|_{{\mathcal T} ({\mathcal A}_\infty )},\pi _{\lambda _1}^{\prime }:=\pi _{\lambda _{1}}|_{{\mathcal T}({\mathcal A}_\infty )}$. We shall prove that $\pi _{\lambda _0}^{\prime }$ and $\pi _{\lambda _1}^{\prime }$ are not equivalent. This fact is very easy to prove in the following cases: (1) when $\lambda _0\in {\mathbb {R}}$ and $\lambda _1\in \{\pm \infty \}$ and (2) when $\lambda _0,\lambda _1\in \{\pm \infty \}$. Consider that both $\lambda _0$ and $\lambda _1$ are real numbers. In such a case $\pi _{\lambda _0}^{\prime }$ and $\pi _{\lambda _1}^{\prime }$ are two-dimensional irreducible representations.

Now, if $a_0\in {\mathcal A}_\infty$ with $a_0(\theta )=\chi _{(\frac{\pi }{4},\frac{3\pi }{4})}(\theta )\; \sin ^{-\alpha }\theta$ then the Toeplitz operator $A_{a_0}^{\alpha }(\lambda )$ [see (3.16)] with symbol $a_0$ is given by

$$\begin{aligned} 2^\alpha \vartheta ^2_\alpha (\lambda )(\alpha +1)\; \left( \begin{array}{c@{\quad }c} \frac{1}{\lambda }e^{-\lambda \pi }\sinh \left( \frac{\lambda \pi }{2}\right) &{}\frac{-1}{2+\alpha }e^{-\lambda \pi +\frac{i\pi \alpha }{4}} \left( 1+e^{\frac{i\pi \alpha }{2}}\right) \\ \frac{-1}{2+\alpha }e^{-\lambda \pi -\frac{3i\pi \alpha }{4}} \left( 1+e^{\frac{i\pi \alpha }{2}}\right) &{} \frac{1}{\lambda }e^{-\lambda \pi }\sinh \left( \frac{\lambda \pi }{2}\right) \\ \end{array} \right) . \end{aligned}$$

The last matrix has the eigenvalues

$$\begin{aligned} x_1(\lambda )&= \frac{e^{-\lambda \pi }\left[ -2\lambda \cos \left( \frac{\pi \alpha }{4}\right) +(\alpha +2) \sinh \left( \frac{\lambda \pi }{2}\right) \right] }{\lambda (\alpha +2)},\\ x_2(\lambda )&= \frac{e^{-\lambda \pi }\left[ \;2\lambda \; \cos \left( \frac{\pi \alpha }{4}\right) +(\alpha +2)\sinh \left( \frac{\lambda \pi }{2}\right) \right] }{\lambda (\alpha +2)}. \end{aligned}$$

Fig. 1

Function $x_1(\lambda )$

For $\alpha =4k+2$ with $k\in \mathbb {Z}$, we have

$$\begin{aligned} x_1(\lambda )=x_2(\lambda )=\frac{e^{-\lambda \pi }\sinh \left( \frac{\pi \lambda }{2}\right) }{\lambda }. \end{aligned}$$

Note that the function $x_1(\lambda )$ is a decreasing function (see Fig. 1). Then, for $\lambda _0\not =\lambda _1$ we have that

$$\begin{aligned} \mathrm{Sp } \left( A_{a_0}^{\alpha }(\lambda _0)\right) \not =\mathrm{Sp } \left( A_{a_0}^{\alpha }(\lambda _1)\right) , \end{aligned}$$

where $\mathrm{Sp }(.)$ denotes the spectrum of the corresponding operator. Hence, the representations $\pi _{\lambda _0}^{\prime }$ and $\pi _{\lambda _1}^{\prime }$ are inequivalent irreducible representations.

Now, we consider the case $\alpha \not =4k+2$ with $k\in \mathbb {Z}$. It follows that $x_1(\lambda )\not =x_2(\lambda )$. We suppose that for $\lambda _0\not =\lambda _1$ and that

$$\begin{aligned} \mathrm{Sp } \left( A_{a_0}^{\alpha }(\lambda _0)\right) =\mathrm{Sp } \left( A_{a_0}^{\alpha }(\lambda _1)\right) . \end{aligned}$$

(3.20)

and we shall obtain a contradiction. Since $x_1$ and $x_2$ are decreasing functions, (3.20) holds if only if $x_1(\lambda _0)=x_2(\lambda _1)$ and $x_2(\lambda _0)=x_1(\lambda _1)$. Then,

$$\begin{aligned} x_1(\lambda _0)-x_2(\lambda _0)=x_2(\lambda _1)-x_1(\lambda _1). \end{aligned}$$

This fact implies that

$$\begin{aligned} \frac{-4e^{-\lambda _0\pi }\cos \left( \frac{\pi \alpha }{4} \right) }{\alpha +2}=\frac{4e^{-\lambda _1\pi }\cos \left( \frac{\pi \alpha }{4} \right) }{\alpha +2}. \end{aligned}$$

Thus $e^{-\lambda _1\pi }+e^{-\lambda _0\pi }=0$, which is a contradiction. Consequently, the representations $\pi _{\lambda _0}^{\prime }$ and $\pi _{\lambda _1}^{\prime }$ are inequivalent irreducible representations. This completes the proof.

$\square$

Theorem 3.11 shows that the description of the algebra ${\mathcal T}({\mathcal A}_{\infty })$ does not depend on the parameter of the weight ($\alpha$).

3.2 Toeplitz Operators Whose Symbols Depend only on the Imaginary Part of $z$

In this part of the paper we study Toeplitz operators acting on the weighted harmonic Bergman space on the upper half-plane with bounded symbols that depend only on the imaginary part of the variable. Using some results from [4] we describe the $C^*$-algebra generated by all Toeplitz operators with such kind of symbols. To do so, we recall some results from [12].

The equations

$$\begin{aligned} 2\frac{\partial }{\partial \bar{z}}f=\left( \frac{\partial }{\partial x}+i\frac{\partial }{\partial y}\right) f=0,\quad 2\frac{\partial }{\partial z}g=\left( \frac{\partial }{\partial x}-i\frac{\partial }{\partial y}\right) g=0, \end{aligned}$$

characterize all functions in ${\mathcal A}^2_\alpha (\Pi )$ and $\widetilde{{\mathcal A}}^2_\alpha (\Pi )$, respectively within $L^2(\Pi ,d\mu _\alpha )$. The Fourier transform $F:L^2({\mathbb {R}})\rightarrow L^2({\mathbb {R}})$ is given by

$$\begin{aligned} (Ff)(x)=\frac{1}{\sqrt{2\pi }}\int _{{\mathbb {R}}} e^{-ix\xi }\;f(\xi )~d\xi . \end{aligned}$$

Introduce the unitary operator

$$\begin{aligned} U_1=\frac{1}{\sqrt{\pi }} (F\otimes I): L^2(\Pi , d\mu _\alpha )\rightarrow L^2({\mathbb {R}},dx)\otimes L^2({\mathbb {R}}_+,(\alpha +1)(2y)^\alpha dy). \end{aligned}$$

Theorem 3.12

[12] The operator $U_1$ is an isometric isomorphism under which the weighted Bergman space ${\mathcal A}_{\alpha }^2(\Pi )$ is mapped onto

$$\begin{aligned} {\mathcal A}_{1,\alpha }^2(\Pi )=\left\{ \chi _+(x)\theta _\alpha (x)f(x)e^{-xy}:\; f\in L^2({\mathbb {R}})\right\} , \end{aligned}$$

(3.21)

where $\chi _+(x)$ is the characteristic function of ${\mathbb {R}}_+$ and

$$\begin{aligned} \theta _\alpha (x)=\left( (\alpha +1)\int _{\mathbb {R_+}} e^{-2x\nu }(2\nu )^\alpha d\nu \right) ^{-\frac{1}{2}}= \left( \frac{2x^{\alpha +1}}{\Gamma (\alpha +2)}\right) ^{\frac{1}{2}},\quad x\ge 0.\qquad \end{aligned}$$

(3.22)

The orthogonal projection $B_{1}^{\alpha }$ from $L^{2}({\mathbb {R}}, dx)\otimes L^{2}({\mathbb {R}}_{+}, (\alpha +1){(2y)}^{\alpha }dy)$ onto ${\mathcal {A}}_{1,\alpha }^{2}(\Pi )$ has the form $B_{1}^{\alpha }=U_{1}B_{\Pi }U_{1}^{-1}$.

An analogous result holds for the weighted anti-Bergman space.

Theorem 3.13

The unitary operator $U_1$ is an isometric isomorphism under which the weighted anti-Bergman space $\widetilde{{\mathcal A}}_{\alpha }^2(\Pi )$ is mapped onto

$$\begin{aligned} \widetilde{{\mathcal A}}_{1,\alpha }^2(\Pi )=\left\{ \chi _-(x)\theta _\alpha (-x)g(x)e^{xy}:\; g\in L^2({\mathbb {R}})\right\} , \end{aligned}$$

where $\chi _-(x)$ is the characteristic function of ${\mathbb {R}}_-.$

Proof

The image, under the operator $U_1$, of the weighted anti-Bergman space $\widetilde{{\mathcal A}}_{1,\alpha }^2(\Pi ):=U_1(\widetilde{{\mathcal A}}_\alpha ^2(\Pi ))$ consists of all functions $\varphi (x,y)$ satisfying the equation

$$\begin{aligned} \left( U_1\frac{\partial }{\partial z}U_1^{-1}\varphi \right) =\frac{i}{2}\left( x-\frac{\partial }{\partial y}\right) \varphi =0. \end{aligned}$$

(3.23)

The general solution of (3.23) has the form

$$\begin{aligned} \varphi (x,y)=\psi (x) e^{xy}. \end{aligned}$$

The condition $\varphi \in L^2(\Pi ,d\mu _\alpha )$ implies that the space $\widetilde{{\mathcal A}}_{1,\alpha }^2(\Pi )$ consists of all functions of the form

$$\begin{aligned} \varphi _0(x,y)=\chi _-(x)\theta _\alpha (-x)g(x)e^{xy}, \end{aligned}$$

(3.24)

where $\theta _\alpha$ is given by (3.22) and $g(x)\in L^2({\mathbb {R}})$. Furthermore,

$$\begin{aligned} \Vert \varphi _0(x,y)\Vert _{\widetilde{{\mathcal A}}_{1,\alpha }^2}=\Vert g(x)\Vert _{L^2({\mathbb {R}}_-)}. \end{aligned}$$

The orthogonal projection ${\widetilde{B}}_1^\alpha : L^{2}({\mathbb {R}}, dx)\otimes L^{2}({\mathbb {R}}_{+}, (\alpha +1){(2y)}^{\alpha }dy) \rightarrow \widetilde{{\mathcal A}}_{1,\alpha }^2(\Pi )$ has the form

$$\begin{aligned} {\widetilde{B}}_1^\alpha =U_1 {\widetilde{B}}_\Pi ^\alpha U_1^{-1}. \end{aligned}$$

It is easy to see that $U_1^{-1} {\widetilde{B}}_1^\alpha \varphi =\frac{1}{\sqrt{\pi }}\langle \varphi ,(F\otimes I)\overline{ K_\alpha (z,w)} \rangle$. Now we observe that

$$\begin{aligned} \overline{ K_\alpha (z,w)}&=\frac{1}{\left( i\bar{z}-iw \right) ^{\alpha +2}}=\frac{1}{\left( \eta -iw+i\zeta \right) ^{\alpha +2}}, \quad z=\zeta +i\eta . \end{aligned}$$

The Fourier transform (with respect to $\zeta$) of the function $\overline{ K_\alpha (z,w)}$ is equal to

$$\begin{aligned} \frac{\sqrt{2\pi }\;\chi _-(\zeta )\;(-\zeta )^{\alpha +1}\; e^{(\eta -iw)\zeta }}{\;\Gamma (\alpha +2)}. \end{aligned}$$

It follows from (3.22) that

$$\begin{aligned} (F\otimes I)\overline{ K_\alpha (z,w)}=\sqrt{\frac{\pi }{2}}\;\chi _-(\zeta )\;\theta _\alpha ^2(-\zeta )\; e^{(\eta -iw)\zeta }. \end{aligned}$$

Then,

$$\begin{aligned} U_1^{-1} {\widetilde{B}}_1^\alpha \varphi&= \frac{1}{\sqrt{\pi }}\left\langle \varphi ,\sqrt{\frac{\pi }{2}}\;\chi _-(\zeta )\;\theta _\alpha ^2(-\zeta )\; e^{(\eta -iw)\zeta }\right\rangle \\&= \frac{1}{\sqrt{2}}\int _{{\mathbb {R}}} \int _{{\mathbb {R}}_+}\varphi (\zeta ,\eta ) \;\chi _-(\zeta )\;\theta _\alpha ^2(-\zeta )\; e^{(\eta +i\bar{w})\zeta }(\alpha +1)(2\eta )^{\alpha }d\eta d\zeta \\&= \frac{1}{\sqrt{2}}\int _{{\mathbb {R}}}\int _{{\mathbb {R}}_+} \varphi (\zeta ,\eta )\;\chi _-(\zeta )\;\theta _\alpha ^2(-\zeta )\; e^{[\eta +i(u-iv)]\zeta }(\alpha +1)(2\eta )^{\alpha }d\eta d\zeta \\&= \frac{1}{\sqrt{2\pi }} \int _{{\mathbb {R}}} e^{iu\zeta }\left( \sqrt{\pi }\chi _-(\zeta )\; \theta _\alpha ^2(-\zeta )\int _{{\mathbb {R}}_+} \varphi (\zeta ,\eta )e^{(\eta +v)\zeta }(\alpha \!+\!1)(2\eta )^{\alpha }d\eta \right) d\zeta \\&= (F^{-1}\otimes I)\left( \sqrt{\pi }\;\chi _-(\zeta ) \;\theta _\alpha ^2(-\zeta )e^{v\zeta }\int _{{\mathbb {R}}_+} \varphi (\zeta ,\eta )e^{\eta \zeta }(\alpha +1)(2\eta )^{\alpha }d\eta \right) . \end{aligned}$$

Therefore,

$$\begin{aligned} ({\widetilde{B}}_1^\alpha \varphi )(u,v)=\chi _-(u)\; \theta _\alpha ^2(-u)e^{uv}\int _{{\mathbb {R}}_+} \varphi (u,\eta )e^{u\eta }(\alpha +1)(2\eta )^{\alpha }d\eta . \end{aligned}$$

Thus for each function $\varphi \in L^2(\Pi ,d\mu _\alpha ),\,({\widetilde{B}}_1^\alpha \varphi )(u,v)$ has the form (3.24) with

$$\begin{aligned} g(u)=\theta _\alpha (-u)\int _{{\mathbb {R}}_+} \varphi (u,\eta )e^{u\eta }(\alpha +1)(2\eta )^{\alpha }d\eta . \end{aligned}$$

In addition, for each function $\varphi _0$ of the form (3.24) we have

$$\begin{aligned} ({\widetilde{B}}_1^\alpha \varphi _0)(u,v)=\varphi _0(u,v). \end{aligned}$$

In fact,

$$\begin{aligned} ({\widetilde{B}}_1^\alpha \varphi _0)(u,v)&= \chi _-(u)\theta ^2_\alpha (-u)e^{uv}\int _{{\mathbb {R}}_+}\left( \chi _-(u)\theta _\alpha (-u)g(u)e^{u\eta }\right) e^{u\eta } (\alpha +1)(2\eta )^{\alpha }d\eta \\&= \chi _-(u)\theta ^3_\alpha (-u)e^{uv}g(u)\theta ^{-2}_\alpha (-u)\\&= \varphi _0(u,v). \end{aligned}$$

So $\widetilde{{\mathcal A}}_{1,\alpha }^2(\Pi )$ coincides with the space of all functions of the form (3.24). $\square$

Following [12], introduce the unitary operator

$$\begin{aligned} U_{2,\alpha }:L^2({\mathbb {R}},dx)\otimes L^2({\mathbb {R}}_+,(\alpha +1)(2y)^\alpha dy)\rightarrow L^2({\mathbb {R}},dx)\otimes L^2({\mathbb {R}}_+,dy) \end{aligned}$$

by the following formula

$$\begin{aligned} (U_{2,\alpha }\varphi )(x,y)=\frac{1}{\theta _\alpha (|x|)}e^{-\frac{y}{2}+|x|\beta (|x|,y)}\varphi (x,\beta (|x|,y)), \end{aligned}$$

where for each fixed $x>0$ the function $\beta (x,y)$ is the inverse function to

$$\begin{aligned} \gamma (x,y)=-\ln \left\{ \theta _\alpha ^2(x)(\alpha +1)\int _{y}^{\infty } (2\eta )^\alpha e^{-2x\eta } d\eta \right\} , \end{aligned}$$

i.e., $\beta (x,\gamma (x,y))=y$ for $x>0.$ The inverse operator

$$\begin{aligned} U_{2,\alpha }^{-1}: L^2({\mathbb {R}},dx)\otimes L^2({\mathbb {R}}_+,dy)\rightarrow L^2({\mathbb {R}},dx)\otimes L^2({\mathbb {R}}_+,(\alpha +1)(2y)^{\alpha }dy) \end{aligned}$$

is given by

$$\begin{aligned} (U_{2,\alpha }^{-1}\varphi )(x,y)= \theta _\alpha (|x|) e^{\gamma (|x|,y)/2-|x|y}\varphi (x,\gamma (|x|,y)). \end{aligned}$$

Let $L_0$ be the one-dimensional subspace of $L^2({\mathbb {R}}_+,dy)$ generated by the unitary element $\ell _0(y)=e^{-y/2}$. We denote by $P_0$ the one-dimensional projection from $L^2({\mathbb {R}}_+,dy)$ onto $L_0$. It is given by the following formula

$$\begin{aligned} (P_0\psi )(y)=e^{-y/2}\int _{{\mathbb {R}}_+} \psi (v) e^{-v/2}dv. \end{aligned}$$

Proposition 3.14

[12] The image ${\mathcal A}_{2,\alpha }^2(\Pi )=U_{2,\alpha }({\mathcal A}_{1,\alpha }^2(\Pi ))$ is the set of all functions of the form

$$\begin{aligned} \varphi (x,y)=\chi _+(x) f(x) e^{-y/2},\quad f\in L^2({\mathbb {R}}). \end{aligned}$$

The operator $B_2^\alpha =U_{2,\alpha } B_1^\alpha U_{2,\alpha }^{-1}$ is the orthogonal projection of $L^2({\mathbb {R}}, dx) \otimes L^2 ({\mathbb {R}}_+, dy)$ onto ${\mathcal A}_{2,\alpha }^2(\Pi )$ and coincides with the projection $\chi _+I\otimes P_0$.

Now let us state the analogous result for the anti-Bergman space.

Proposition 3.15

The image $\widetilde{{\mathcal A}}_{2,\alpha }^2(\Pi )=U_{2,\alpha }\big (\widetilde{{\mathcal A}}_{1,\alpha }^2(\Pi )\big )$ is the set of all functions of the form

$$\begin{aligned} \psi (x,y)=\chi _-(x) g(x) e^{-y/2},\quad g\in L^2({\mathbb {R}}). \end{aligned}$$

The operator ${\widetilde{B}}_2^\alpha =U_{2,\alpha } {\widetilde{B}}_1^\alpha U_{2,\alpha }^{-1}$ is the orthogonal projection of $L^2({\mathbb {R}},dx) \otimes L^2 ({\mathbb {R}}_+, dy)$ onto $\widetilde{{\mathcal A}}_{2,\alpha }^2(\Pi )$ and coincides with the projection $\chi _-I\otimes P_0$.

Proof

For each $g\in L^2({\mathbb {R}})$, we have

$$\begin{aligned} U_{2,\alpha }: \chi _-(x)\theta _\alpha (-x) g(x) e^{xy}\mapsto \chi _-(x)g(x)e^{-y/2}. \end{aligned}$$

Note that ${\widetilde{B}}_2^\alpha =U_{2,\alpha } {\widetilde{B}}_1^\alpha U_{2,\alpha }^{-1}$ is the orthogonal projection of $L^2({\mathbb {R}},dx) \otimes L^2 ({\mathbb {R}}_+, dy)$ onto $\widetilde{{\mathcal A}}_{2,\alpha }^2(\Pi )$. We calculate ${\widetilde{B}}_2^\alpha$:

$$\begin{aligned} ({\widetilde{B}}_2^\alpha \varphi )(x,y)&= U_{2,\alpha } \left( \chi _-(x)\theta ^3_\alpha (x)e^{xy} \int _{{\mathbb {R}}_+}e^{\frac{1}{2}\gamma (-x,\eta )} \varphi (x,\gamma (-x,\eta ))e^{2x\eta }(\alpha +1)(2\eta )^{\alpha }d\eta \right) \\&= \chi _-(x)\theta ^2_\alpha (-x)e^{-y/2}\int _{{\mathbb {R}}_+} e^{\frac{1}{2}\gamma (-x,\eta )}\varphi \left( x,\gamma (-x,\eta ) \right) e^{2x\eta }(\alpha +1)(2\eta )^{\alpha }d\eta \\&= \chi _-(x)e^{-y/2}\int _{{\mathbb {R}}_+} e^{-\nu /2}\varphi (x,\nu ) d\nu . \end{aligned}$$

$\square$

Using Theorems 3.12, 3.13 and Propositions 3.14, 3.15 we get the following result.

Theorem 3.16

The unitary operator $\widehat{U}_\alpha =U_{2,\alpha }U_1$ is an isometric isomorphism of $L^2(\Pi ,d\mu _\alpha )$ under which

• 1. [12] the weighted Bergman space ${\mathcal A}_{\alpha }^2(\Pi )$ is transformed in $L^2({\mathbb {R}}_+)\otimes L_0$,

• 2. the weighted anti-Bergman space $\widetilde{{\mathcal A}}_\alpha ^2(\Pi )$ is transformed in $L^2({\mathbb {R}}_-)\otimes L_0$.

Introduce the isometric embeddings $R_0\!:\! L^2({\mathbb {R}}_+)\rightarrow L^2({\mathbb {R}})\otimes L^2({\mathbb {R}}_+)$, ${\widetilde{R}}_0 : L^2({\mathbb {R}}_-)\rightarrow L^2({\mathbb {R}})\otimes L^2({\mathbb {R}}_+)$ defined by

$$\begin{aligned} (R_0f)(x,y)&= \chi _+(x)f(x)\ell _0(y),\nonumber \\ ({\widetilde{R}}_0g)(x,y)&= \chi _-(x)g(x)\ell _0(y), \end{aligned}$$

(3.25)

the functions $f$ and $g$ on the right hand side of (3.25) are extended to elements of $L^2({\mathbb {R}})$ by setting $f(x)=0$ for $x<0$ and $g(x)=0$ for $x>0$. The adjoint operators $R_0^*\!:\!L^2(\Pi , d\mu _\alpha )\rightarrow L^2({\mathbb {R}}_+),$ ${\widetilde{R}}^*_0\!:\!L^2(\Pi , d\mu _\alpha )\rightarrow L^2({\mathbb {R}}_-)$, are given by

$$\begin{aligned} (R_0^*\phi )(x)&= \chi _+(x)\; \int _{{\mathbb {R}}_+}\; \phi (x,\eta )\ell _0(\eta )\; d\eta ,\\ ({\widetilde{R}}^*_0\psi )(x)&= \chi _-(x)\; \int _{{\mathbb {R}}_+}\; \phi (x,\eta )\ell _0(\eta )\; d\eta . \end{aligned}$$

It is easy to prove the following equalities

$$\begin{aligned}&\nonumber R_0^*R_0= I :L^2({\mathbb {R}}_+) \rightarrow L^2({\mathbb {R}}_+),\\&\nonumber {\widetilde{R}}_0^*{\widetilde{R}}_0 = I :L^2({\mathbb {R}}_-)\rightarrow L^2({\mathbb {R}}_-),\\&R_0R_0 ^*= B_2^\alpha : L^2(\Pi , d\mu _\alpha )\rightarrow L^2({\mathbb {R}}_+)\otimes L_0,\\&\nonumber {\widetilde{R}}_0\tilde{R}^*_0 = {\widetilde{B}}_2^\alpha : L^2(\Pi , d\mu _\alpha )\rightarrow L^2({\mathbb {R}}_-)\otimes L_0. \end{aligned}$$

(3.26)

The operators $R_\alpha =R_0 ^*\widehat{U}_\alpha$ and ${\widetilde{R}}_\alpha ={\widetilde{R}}_0^*\widehat{U}_\alpha$ transform the space $L^2(\Pi ,d\mu _\alpha )$ onto $L^2({\mathbb {R}}_+)$ and $L^2({\mathbb {R}}_-)$, respectively. Furthermore, $R_\alpha |_{{\mathcal A}_\alpha ^2(\Pi )}:{\mathcal A}_\alpha ^2(\Pi )\rightarrow L^2({\mathbb {R}}_+)$ and ${\widetilde{R}}_\alpha |_{\widetilde{{\mathcal A}}_\alpha ^2 (\Pi )}:\widetilde{{\mathcal A}}_\alpha ^2 (\Pi )\rightarrow L^2({\mathbb {R}}_-)$ are isometric isomorphisms. Moreover,

$$\begin{aligned}&\nonumber R_\alpha R_\alpha ^*= I:L^2({\mathbb {R}}_+)\rightarrow L^2({\mathbb {R}}_+),\\&\nonumber {\widetilde{R}}_\alpha \tilde{R}_\alpha ^*= I :L^2({\mathbb {R}}_-)\rightarrow L^2({\mathbb {R}}_-),\\&R_\alpha ^*R_\alpha = B_\Pi ^\alpha :L^2(\Pi ,d\mu _\alpha )\rightarrow {\mathcal A}_\alpha ^2(\Pi ),\\&\nonumber {\widetilde{R}}_\alpha ^*{\widetilde{R}}_\alpha = {\widetilde{B}}_\Pi ^{\alpha }:L^2(\Pi , d\mu _\alpha )\rightarrow \widetilde{{\mathcal A}}_\alpha ^2(\Pi ). \end{aligned}$$

(3.27)

The following two theorems are the main tools to study Toeplitz operators on the weighted harmonic Bergman space whose symbols depend only on the imaginary part of $z$. Let $a=a(\mathrm{\, Im \,}(z))=a(y)\in L_\infty ({\mathbb {R}}_+)$.

Theorem 3.17

[12] The Toeplitz operator $T_a^\alpha$ acting on ${\mathcal A}_\alpha ^2(\Pi )$ is unitarily equivalent to the multiplication operator $\gamma _{a,\alpha }I= R_\alpha T_a^{\alpha }R^*_\alpha$, acting on $L^2({\mathbb {R}}_+)$. The function $\gamma _{a,\alpha }(x)$ is given by the formula

$$\begin{aligned} \gamma _{a,\alpha }(x)=\frac{x^{\alpha +1}}{\Gamma (\alpha +1)} \int _{{\mathbb {R}}_+} a\left( \frac{t}{2}\right) \; t^{\alpha } e^{-tx} dt,\quad x\in {\mathbb {R}}_+. \end{aligned}$$

(3.28)

An analogous result holds for Toeplitz operators acting on the anti-Bergman space.

Theorem 3.18

The Toeplitz operator $\widetilde{T}_a^\alpha$ acting on $\widetilde{{\mathcal A}}_\alpha ^2(\Pi )$ is unitarily equivalent to the multiplication operator $\widetilde{\gamma }_{a,\alpha }I= {\widetilde{R}}_\alpha \widetilde{T}_a^\alpha {\widetilde{R}}^*_\alpha ,$ acting on $L^2({\mathbb {R}}_-),$ where $\widetilde{\gamma }_{a,\alpha }$ is given by the formula

$$\begin{aligned} \widetilde{\gamma }_{a,\alpha }(x)=\gamma _{a,\alpha }(-x). \end{aligned}$$

(3.29)

Proof

It follows from (3.27) that

$$\begin{aligned} {\widetilde{R}}_\alpha \widetilde{T}_a^\alpha {\widetilde{R}}^*_\alpha&= {\widetilde{R}}_\alpha ({\widetilde{B}}_\Pi ^{\alpha }a{\widetilde{B}}_\Pi ^{\alpha }) {\widetilde{R}}^*_\alpha \\&= {\widetilde{R}}_\alpha ({\widetilde{R}}_\alpha ^* {\widetilde{R}}_\alpha )a({\widetilde{R}}_\alpha ^*{\widetilde{R}}_\alpha ) {\widetilde{R}}^*_\alpha \\&= {\widetilde{R}}_\alpha a {\widetilde{R}}^*_\alpha \\&= {\widetilde{R}}^*_0U_{2,\alpha } U_1a(y)U_1^{-1} U_{2,\alpha }^{-1}{\widetilde{R}}_0\\&= {\widetilde{R}}^*_0U_{2,\alpha }a(y) U_{2,\alpha }^{-1}{\widetilde{R}}_0\\&= {\widetilde{R}}^*_0a\left( \beta (|x|,y)\right) {\widetilde{R}}_0. \end{aligned}$$

Let $g\in L^2({\mathbb {R}}_-)$. Then,

$$\begin{aligned} \left( {\widetilde{R}}_0a\left( \beta (|x|,y)\right) {\widetilde{R}}^*_0g\right) x&= \chi _-(x)\int _{{\mathbb {R}}_+}\; \left( a\left( \beta (|x|,\eta )\right) \chi _-(x)g(x) e^{-\frac{\eta }{2}}\right) e^{-\frac{\eta }{2}}\; d\eta \\&= \chi _-(x)g(x)\int _{{\mathbb {R}}_+}\;a \left( \beta (|x|,\eta )\right) e^{-\eta }\; d\eta \\&= \chi _-(x)g(x)\theta _\alpha ^2(-x)(\alpha +1) \int _{{\mathbb {R}}_+}\;a\left( t\right) (2t)^{\alpha }e^{2tx} \; dt\\&= \chi _-(x)g(x)\frac{(-x)^{\alpha +1}}{\Gamma (\alpha +1)}\int _{{\mathbb {R}}_+}\;a \left( \frac{t}{2}\right) t^{\alpha }e^{tx}\; dt\\&= g(x)\;\gamma _{a,\alpha }(-x). \end{aligned}$$

$\square$

The operator

$$\begin{aligned} W_\alpha =\left( \begin{array}{c@{\quad }c} R_\alpha &{} 0 \\ 0 &{} {\widetilde{R}}_\alpha \\ \end{array} \right) \end{aligned}$$

transforms the space $L^2(\Pi ,d\mu _\alpha )\oplus L^2(\Pi ,d\mu _\alpha )$ onto $L^2({\mathbb {R}}_+)\oplus L^2({\mathbb {R}}_-)$ and its restriction $W|_{ b^2_\alpha (\Pi )={\mathcal A}^2_\alpha (\Pi )\oplus \widetilde{{\mathcal A}}^2_\alpha (\Pi )}:b^2_\alpha (\Pi )\rightarrow L^2({\mathbb {R}}_+)\oplus L^2({\mathbb {R}}_-)$ is an isometric isomorphism. Its adjoint operator $W^*_\alpha$ is an isomorphism from $L^2({\mathbb {R}}_+)\oplus L^2({\mathbb {R}}_-)$ onto $b^2_\alpha (\Pi )$. As a consequence of (3.27) we get

$$\begin{aligned} W_\alpha W_\alpha ^*&= \left( \begin{array}{c@{\quad }c} \; I\; &{} 0 \\ 0 &{} \;I\; \\ \end{array} \right) : L^2({\mathbb {R}}_+)\oplus L^2({\mathbb {R}}_-)\rightarrow L^2({\mathbb {R}}_+)\oplus L^2({\mathbb {R}}_-), \\ W_\alpha ^*W_\alpha&= \left( \begin{array}{c@{\quad }c} B_\Pi ^\alpha &{} 0 \\ 0 &{} {\widetilde{B}}_\Pi ^\alpha \\ \end{array} \right) : L^2(\Pi ,d\mu _\alpha )\oplus L^2(\Pi ,d\mu _\alpha )\rightarrow b^2_\alpha (\Pi ). \end{aligned}$$

Theorem 3.19

Let $a=a(y)\in L_\infty ({\mathbb {R}}_+)$. Then the Toeplitz operator ${\widehat{T}}_a^\alpha$ acting on $b^2_\alpha (\Pi )$ is unitarily equivalent to the multiplication operator

$$\begin{aligned} \gamma _{a,\alpha }(|x|)I, \end{aligned}$$

(3.30)

acting on $L^2({\mathbb {R}})$. The function $\gamma _{a,\alpha }(x)$ is given by (3.28).

Proof

Let $a\in L_\infty ({\mathbb {R}}_+)$ then

$$\begin{aligned} \nonumber W_\alpha {\widehat{T}}_a^\alpha W^*_\alpha&= \left( \begin{array}{c@{\quad }c} R_\alpha &{} 0 \\ 0 &{} {\widetilde{R}}_\alpha \\ \end{array} \right) \left( \begin{array}{c@{\quad }c} T_a^\alpha &{} JH_{\bar{a}}^\alpha J \\ H_a^\alpha &{} J(T_a^\alpha )^*J \end{array}\right) \left( \begin{array}{c@{\quad }c} R^*_\alpha &{} 0 \\ 0 &{} {\widetilde{R}}^*_\alpha \\ \end{array} \right) \\&= \left( \begin{array}{c@{\quad }c} R_\alpha T_a^\alpha R^*_\alpha &{} R_\alpha JH_{\bar{a}}^\alpha J{\widetilde{R}}^*_\alpha \\ {\widetilde{R}}_\alpha H_a^\alpha R^*_\alpha &{} {\widetilde{R}}_\alpha J(T_a^\alpha )^*J{\widetilde{R}}^*_\alpha \end{array}\right) . \end{aligned}$$

(3.31)

Using Theorem 3.17 we get

$$\begin{aligned} R_\alpha T_a^\alpha R^*_\alpha =\gamma _{a,\alpha }(x) I. \end{aligned}$$

From Eqs. (2.7), (2.8) and (3.27) we have that

$$\begin{aligned} R_\alpha J H_{\bar{a}}^\alpha J{\widetilde{R}}^*_\alpha&= R_\alpha J{\widetilde{B}}_\Pi ^\alpha \bar{a} J{\widetilde{R}}^*_\alpha = R_\alpha B_\Pi ^\alpha J\bar{a} J{\widetilde{R}}^*_\alpha \\&= R_\alpha B_\Pi ^\alpha a{\widetilde{R}}^*_\alpha = R_\alpha (R^*_\alpha R_\alpha )a{\widetilde{R}}^*_\alpha \\&= R_0^*\widehat{U}_\alpha a(y)\widehat{U}^*_\alpha {\widetilde{R}}_0\\&= R_0^*U_{2,\alpha } U_1a(y)U_1^{-1}U_{2,\alpha }^{-1} {\widetilde{R}}_0\\&= R_0^* a\left( \beta (|x|,y)\right) {\widetilde{R}}_0. \end{aligned}$$

Let $g\in L^2({\mathbb {R}}_-)$. Then,

$$\begin{aligned} \left( R_\alpha JH_{\bar{a}}^\alpha J{\widetilde{R}}^*_\alpha g\right) (x)&= \chi _+(x)\int _{{\mathbb {R}}_+} \left( a\left( \beta (|x|,\eta )\right) \chi _-(x)g(x) e^{-\frac{\eta }{2}}\right) e^{-\frac{\eta }{2}}d\eta \\&= 0. \end{aligned}$$

In a similar way to the previous case, we can show that ${\widetilde{R}}_\alpha H_a^\alpha R^*_\alpha =0$ and that ${\widetilde{R}}_\alpha J(T_a^\alpha )^*J{\widetilde{R}}^*_\alpha =\widetilde{\gamma }_{a,\alpha }I$, where the function $\widetilde{\gamma }_{a,\alpha }$ is given by (3.14). Therefore, the Toeplitz operator ${\widehat{T}}_{a}^\alpha$ is unitarily equivalent to the multiplication operator

$$\begin{aligned} \left( \begin{array}{c@{\quad }c} \gamma _{a,\alpha }(x) &{} 0 \\ 0 &{} \gamma _{a,\alpha }(-x) \\ \end{array} \right) I= \gamma _{a,\alpha }(|x|) I, \end{aligned}$$

acting on $L^2({\mathbb {R}}_+)\oplus L^2({\mathbb {R}}_-)=L^2({\mathbb {R}})$. $\square$

Denote by ${\mathcal B}_{\infty }$ the algebra of all bounded functions, defined in $\Pi$, that depend only on the vertical variable and by ${\mathcal T}({\mathcal B}_{\infty })$ the $C^*$-algebra generated by all Toeplitz operators whose symbols are in ${\mathcal B}_{\infty }$. From last theorem ${\mathcal T}({\mathcal B}_{\infty })$ is commutative. And, due to (3.30) it is isomorphic to the $C^*$-algebra generated by the set

$$\begin{aligned} \Gamma :=\{\gamma _{a,\alpha }: a\in L_{\infty }({\mathbb {R}}_+)\}. \end{aligned}$$

This algebra was studied in [4], for the unweighted case ($\alpha =0$). The description of the algebra generated by $\Gamma$ does not depend on the weight and the proof of it has already been done by the authors of [4].

Recall that the logarithmic metric $\rho : {\mathbb {R}}_+\times {\mathbb {R}}_+\rightarrow [0,\infty )$ on the positive half-line, defined by

$$\begin{aligned} \rho (x,y):=|\ln (x)-\ln (y)|, \end{aligned}$$

is invariant under dilations.

A bounded function $f:{\mathbb {R}}_+\rightarrow {\mathbb {C}}$ is called very slowly oscillating if it is uniformly continuous with respect to the metric $\rho$ or, equivalently, if the composition $f\circ \exp$ is uniformly continuous with respect to the usual metric on ${\mathbb {R}}$. Denote by $VSO({\mathbb {R}}_+)$ the set of such functions. It is proved in [4] that $VSO({\mathbb {R}}_+)$ is a closed $C^*$-subalgebra of the algebra $C_b({\mathbb {R}}_+)$ of all bounded and continuous functions defined in ${\mathbb {R}}_+$, with pointwise operations.

Theorem 3.20

[4] We have that $\overline{\Gamma }=VSO({\mathbb {R}}_+)$.

Using last theorem and the fact that $VSO({\mathbb {R}}_+)$ is a $C^*$-algebra we can formulate the main theorem of this section.

Theorem 3.21

The $C^*$-algebra ${\mathcal T}({\mathcal B}_{\infty })$, generated by all Toeplitz operators ${\widehat{T}}_a^\alpha$ with $a\in {\mathcal B}_{\infty }$, is isomorphic and isometric to $VSO({\mathbb {R}}_+)$.

Observe that the description given in last theorem does not depend on the parameter $\alpha .$ Furthermore, it is very interesting and unusual that the $C^*$-algebra generated by $\Gamma$ coincides with the closure $\overline{\Gamma }$. It implies that the set of initial generators of ${\mathcal T}({\mathcal B}_{\infty })$, i.e., all Toeplitz operators ${\widehat{T}}^{\alpha }_a$ with $a\in {\mathcal B}_{\infty }$, is dense in ${\mathcal T}({\mathcal B}_{\infty })$.