Commuting Toeplitz operators on weighted harmonic Bergman spaces and hyponormality on the Bergman space of the punctured unit disk

1.   Introduction

Finding commuting Toeplitz operators is a question of interest for researchers working on Toeplitz operators. The most important results on this subject are from the work of Axler and Cuckovic ^[1] in the case of the Bergman space, and Choe and Lee in the case of the harmonic Bergman space ^[2]. The results of Axler and Cuckovic are generalized by the authors to the case of the weighted Bergman space ^[3]. In the first part, we generalize most of the results of Choe and Lee to the case of weighted harmonic Bergman spaces. In the second part, we give a sufficient condition for hyponormality on newly considered Bergman spaces on the punctured unit disk. Hyponormality has seen growing interest the last few decades. The first important result is the following.

Theorem 1.1. Let $f$, $g$ be bounded and analytic in the unit disk $D$ with $f^{\prime}\in H^{2}$. If $T_{f+\overline{g}}$ is hyponormal on the Bergman space, then $g^{\prime}\in H^{2}$ and $|g^{\prime}|\leq |f^{\prime}|$ a.e in the unit circle.

This can be found in ^[4]. Ahern and Cuckovic generalized this result in ^[5] by weakening the assumption on the derivative. Their result itself is generalized to weighted Bergman spaces in ^[6].

Theorem 1.2. Let $f$, $g$ be bounded and analytic in the unit disk $D$. Assume $f^{\prime}\in H^{2}(I)$, where $I$ is an open arc of the unit circle. If $T_{f+\overline{g}}$ is hyponormal on the weighted Bergman space $L_{a, \omega}^{2}$, where $\omega(r) = (\alpha+1)(1-r^{2})^{\alpha}$, $\alpha > -1$, then, $g^{\prime}\in H^{2}(I)$ and $|g^{\prime}|\leq |f^{\prime}|$ a.e on $I$.

2.   Commuting Toeplitz operators on the weighted harmonic Bergman space

Denote by $D$ the unit disk and consider the Hilbert space $L^{2}\left(D, d\mu_{\alpha }\right)$ of measurable functions on $D$ such that

The closed subspace of $L^{2}\left(D, d\mu _{\alpha }\right)$ consisting of harmonic functions is denoted by $L_{h}^{2}$. If the weighted Bergman space is denoted by $L_{a, \omega }^{2}$ where $\omega (r) = (\alpha +1)(1-r^{2})^{\alpha }$, then $L_{h}^{2} = L_{a, \omega }^{2}\oplus \overline{zL_{a, \omega }^{2}}$. As in the case of unweighted spaces ($\omega = 1$) ^[2], the reproducing kernel is given by

In what follows, we consider commuting Toeplitz operators on $L_{h}^{2}$ with harmonic symbols $\varphi$ and $\psi$. We denote by $P$ the projection of $L^{2}\left(D, d\mu _{\alpha }\right)$ onto $L_{a, \omega}^{2}$, and by $Q$ the projection onto $L_{h}^{2}$. We have $Q(\varphi) = P(\varphi)+\overline{P(\overline{\varphi })}-P(\varphi)(0)$, where $P(\varphi)(0) = \int_{D}\varphi dA$. We give a generalization of some of the results in ^[2], shown in the case of the unweighted harmonic Bergman space, to the case of weighted harmonic Bergman spaces with radial weights $\omega (r) = (\alpha+1)(1-r^{2})^{\alpha }$, where $\alpha > -1$. Our first main result is the following theorem.

Theorem 2.1. Let $\varphi$ and $\psi$ be in $L_{h}^{2}$ and not both conjugate holomorphic. If $T_{\varphi }T_{\psi } = T_{\psi }T_{\varphi }$, then $\frac{\partial \psi }{\partial z} = c\frac{\partial \varphi }{\partial z}$ for some constant $c$.

The proof is based on the properties of the projection. Here is a summary of the proof of Theorem 2.1. Set $\varphi = \varphi_{1}+\overline{\varphi_{2}}$, $\psi = \psi_{1}+\overline{\psi_{2}}$. Then,

By evaluation at $\overline{w}$, and after making extensive use of some properties of the Bergman space projection we get $\varphi_{1}P\left(\overline{w}\psi_{1}\right) = \psi_{1}P\left(\overline{w}\varphi_{1}\right)$. By Proposition 2.2, this is reduced to $\psi_{1} = c\varphi_{1}$.

We start by verifying these properties.

Proposition 2.2. Let $\varphi \in L_{a, \omega }^{2}$. The projection onto $L_{a, \omega }^{2}$ satisfies the following properties:

(ⅰ) $z^{2}\frac{d}{dz}P\left((1-|w|^{2})\varphi \right)+(\alpha +2)zP\left((1-|w|^{2})\varphi\right) = (\alpha +1)z\varphi (z)$.

(ⅱ) $z^{2}\frac{d}{dz}P\left((1-|w|^{2})\overline{\varphi }\right)+(\alpha+2)zP\left((1-|w|^{2})\overline{\varphi }\right) = (\alpha +1)z\overline{\varphi }(0)$.

(ⅲ) $P\left(|w|^{2}\varphi \right)(z) = \varphi (z)-\frac{(\alpha +1)}{z^{\alpha +2}}\int_{0}^{z} u^{\alpha +1}\varphi(u) du$.

(ⅳ) $P\left(|w|^{2}\overline{\varphi }\right) = \overline{P\left(|w|^{2}\overline{\varphi }\right)(0)} = \frac{\alpha +1}{\alpha +2}\overline{\varphi }(0)$.

Proof. For (ⅰ) we have

and, by differentiating under the integral sign, we get

We deduce

where the last equality holds because $K_{z}(w) = \frac{1}{(1-z\overline{w})^{\alpha +3}}$ is the reproducing kernel of the weighted Bergman space with a weight $\omega _{2}(r) = (\alpha +2)(1-r^{2})^{\alpha +1} dA(z)$ ^[7]. We obtain Eq (ⅱ) in a similar manner. For Eq (ⅲ), we formally solve the differential equation

which gives

and

Similarly, from (ⅱ), by integration we get

thus

□

This leads to the following properties of the projection.

Lemma 2.3. Let $\varphi \in L_{a, \omega }^{2}$ satisfy $\varphi (0) = 0$. The following properties hold:

$1)$ $P\left(\overline{w}\varphi \right)(z) = \frac{\varphi (z)}{z}-\frac{\alpha +1}{z^{\alpha +2}}\int_{0}^{z}u^{\alpha }\varphi(u)\ du$.

$2)$ $P\left(w\overline{\varphi }\right)(z) = \frac{1}{\alpha +2}\overline{\varphi^{\prime }(0)}$.

Proof. Since $\varphi (0) = 0$, $\varphi (z) = z\psi(z)$. From (ⅲ) and (ⅳ) of Proposition 2.2, we get

□

The following properties of the projection of $L^{2}\left(D, d\mu _{\alpha }\right)$ are easy to check. The first one is a consequence of the known property of Toeplitz operators $T_{f}T_{g} = T_{fg}$ on $L_{a, \omega }^{2}$ if $g$ is analytic or $f$ is conjugate analytic.

Lemma 2.4. For $\varphi$ and $\psi \in L_{a, \omega}^{2}$, we have the following equalities:

(ⅰ) $P\left(\overline{\varphi }P(\overline{w}\psi)\right)(z) = P\left(\overline{\varphi }\overline{w}\psi \right)(z)$.

(ⅱ) $P\left(\varphi \overline{P(\overline{w}\psi }\right)(0) = P\left(\varphi w\overline{\psi }\right)(0)$.

Proof. We verify (ⅱ):

□

This leads to the following theorem.

Theorem 2.5. Let $\varphi$ and $\psi$ be in $L_{h}^{2}$. If $T_{\varphi}T_{\psi} = T_{\psi}T_{\varphi}$, then, there exists a constant $\lambda$ such that $\frac{\partial \psi}{\partial z} = \lambda \frac{\partial \varphi}{\partial z}$.

Proof. We have $\varphi = \varphi _{1}+\overline{\varphi _{2}}$, and $\psi = \psi _{1}+\overline{\psi _{2}}$, where $\varphi _{1}$, $\varphi _{2}$, $\psi _{1}$ and $\psi _{2}\in L_{a, \omega }^{2}$. We may assume

We evaluate both sides of the equality $T_{\varphi}T_{\psi} = T_{\psi}T_{\varphi}$ at the function $\overline{w}$. We make use of the following equalities, where Lemmas 2.3 and 2.4 are used.

Similarly, we compute

Thus, the equality

leads to

We deduce that

which, by Proposition 2.2, gives

which can be written as

where

This shows that

and, consequently,

for some constant $c$. □

Corollary 2.6. Let $\varphi$, $\psi$ be in $L_{a, \omega }^{2}$. The equality $T_{\varphi}T_{\psi } = T_{\psi }T_{\varphi }$ holds if and only if $\psi = c\varphi +d$ for some constants $c$ and $d$.

Our next main result is based on the following lemmas.

Lemma 2.7. Let $\varphi$, $\psi$ be in $L_{a, \omega }^{2}$. Then we have the following identity:

Proof. Use power series and the fact that, for $\varphi$, $\psi$ in $L_{a, \omega}^{2}$, with $\varphi = {\sum\limits_{n\geq 0}}\varphi_{n} z^{n}$, $\psi = {\sum\limits_{n\geq 0}}\psi _{n} z^{n}$, we have the inner product property ^[7]

□

Lemma 2.8. Let $\varphi$, $\psi$ be in $L_{a, \omega }^{2}$ and satisfy $\varphi (0) = \psi(0) = 0$. If $T_{\varphi }T_{\overline{\psi }} = T_{\overline{\psi }}T_{\varphi }$ on $L_{h}^{2}$, then the following equality holds

for any $n\in\mathbb{N}$, where $\Psi (w) = \frac{\alpha +1}{w^{\alpha +1}}\int_{0}^{w}u^{\alpha }\psi(u) du$.

Proof. From $T_{\varphi }T_{\overline{\psi }} = T_{\overline{\psi }}T_{\varphi }$, we deduce $T_{\psi }T_{\overline{\varphi }} = T_{\overline{\varphi }}T_{\psi }$, which, by evaluation at $\overline{w}$ and from Eq (2.1) by taking $\varphi = \varphi _{2}$, $\varphi _{1} = 0$, $\psi = \psi _{1}$, and $\psi_{2} = 0$ gives

This gives

Using the identity $w P\left(\psi \overline{w}\right) = \psi -\Psi$, we get

which can be written as

Using Lemma 2.7, we get

After a simplification using Eq (2.2), we deduce

□

This leads to the second main result. The proof, being similar to the case of the unweighted harmonic Bergman space ^[2], is omitted.

Theorem 2.9. Assume $f\in L_{h}^{2}$. Then, $T_{f}$ is normal if and only if $f(D)$ is contained in a line in the complex plane. If $f\in L_{a, \omega }^{2}$, then $T_{f}$ is normal if and only if $f$ is constant.

3.   Hyponormality on the weighted Bergman space of the punctured unit disk

Let $m$ be a nonnegative integer, $\beta$ be in $(m-1, m]$, and $\alpha > -1$ be a real number. Let $d\mu _{\alpha, \beta }(z) = \frac{1}{B(\alpha +1, \beta+1)}|z|^{2\beta }(1-|z|^{2})^{\alpha }dA(z)$ be the normalized Lebesgue measure on the unit disk $D$. The space $L^{2}(D, d\mu _{\alpha, \beta })$ is the Hilbert space of measurable functions $f$ on $D$ such that

We consider holomorphic functions $g$ on $D^{\ast } = D-\{0\}$ that satisfy

In ^[8], it is shown that this space, denoted by $A_{\alpha, \beta }^{2}$, is a closed Hilbert space of $L^{2}\left(D, d\mu _{\alpha, \beta }\right)$. If $f = {\sum\limits_{n\geq -m}}a_{n}z^{n}$ and $g = {\sum\limits_{n\geq -m}}b_{n}z^{n}$, then the inner product is given by

and its natural orthonormal basis is $\left\{e_{n}(z) = \sqrt{\frac{B(\alpha+1, \beta +1)}{B(\alpha +1, \beta +n+1)}}z^{n}, \ n\geq -m\right\}$. We set $\rho _{n} = \frac{B(\alpha +1, \beta +n+1)}{B(\alpha +1, \beta +1)}$, $n\geq -m$, and recall that a Toeplitz operator $T_{\varphi }$, for $\varphi$ bounded measurable on $D$, is defined on $A_{\alpha, \beta }^{2}$ by $T_{\varphi }(f) = P(\varphi f)$, where $P$ is the orthogonal projection onto $A_{\alpha, \beta }^{2}$. Hankel opertaors on $A_{\alpha, \beta }^{2}$ are defined by $H_{\varphi}(f) = (I-P)(\varphi f)$. The following properties of Toeplitz operators, known in the case of the weighted Bergman space $A_{\alpha }^{2}$ $(\beta = 0)$, hold also on $A_{\alpha, \beta }^{2}$.

Proposition 3.1. ^[8] Let $\varphi$ and $\psi$ be bounded measurable on $D$. The following properties hold:

(ⅰ) $T_{\varphi +\psi } = T_{\varphi }+T_{\psi }$.

(ⅱ) $\left(T_{\varphi }\right)^{\ast } = T_{\overline{\varphi }}$.

(ⅲ) $T_{\psi }T_{\varphi } = T_{\varphi \psi }$ if $\varphi$ analytic or $\psi$ is conjugate analytic on $D$.

A bounded operator $S$ on a Hilbert space is hyponormal if $S^{\ast}S-SS^{\ast }\geq 0$. In what follows, we consider hyponormality of Toeplitz operator $T_{z^{q}+\overline{\psi}}$ on $A_{\alpha, \beta}^{2}$ where $\psi$ is a polynomial. We show a sufficient condition in this case. As in the case of the classical unweighted Bergman space $A_{0, 0}^{2}$ ^[4], hyponormality is expressed in various equivalent forms which are listed in the following proposition.

Proposition 3.2. ^[4] The following statements are equivalent:

(a) $T_{\varphi +\overline{\psi }}$ is hyponormal.

(b) $T_{\overline{\psi }}T_{\psi }-T_{\psi }T_{\overline{\psi }}\leq T_{\overline{\varphi }}T_{\varphi }-T_{\varphi} T_{\overline{\varphi }}$.

(c) $\left(H_{\overline{\psi }}\right)^{\ast }H_{\overline{\psi }}\leq \left(H_{\overline{\varphi }}\right)^{\ast} H_{\overline{\varphi }}$.

(d) $H_{\overline{\psi }} = C H_{\overline{\varphi }}$, where $C$ is bounded of norm less than or equal to one.

3.1.   The main result

We consider the case $\alpha = 3$. Let $l$ and $q$ be two integers such that $q\geq m+1$. Denote by $N$ the smallest integer such that $N\geq \max \left(6q+1, 8, \frac{1}{2}(q^{2}-1)\right)$. Our main result is the following theorem.

Theorem 3.3. Let $\left(\lambda _{l}\right)_{N\leq l\leq N_{1}}$ be any finite sequence of complex numbers such that ${\sum\limits_{N\leq l\leq N_{1}}}|\lambda _{l}|\leq 1$, and set $g = {\sum\limits_{N\leq l\leq N_{1}}}\lambda _{l}\frac{q}{l}z^{l}$. Then, the operator $T_{z^{q}+\overline{g}}$ is hyponormal on $A_{\alpha, \beta }^{2}$ i.e $T_{\overline{g}}T_{g}-T_{g}T_{\overline{g}}\leq T_{\overline{z}^{q}}T_{z^{q}}-T_{z^{q}}T_{\overline{z}^{q}}$.

In the case $\alpha = 3$, we have

We start by computing the matrix of $T_{\overline{z}^{q}}T_{z^{q}}-T_{z^{q}}T_{\overline{z}^{q}}$ in the orthonomal basis $\left\{e_{n}, \ n\geq -m\right\}$.

Lemma 3.4. The matrix of $T_{\overline{z}^{q}}T_{z^{q}}-T_{z^{q}}T_{\overline{z}^{q}}$ is diagonal and is given by

Proof. We clearly have

Since

necessarily,

and

We deduce $c = \frac{\rho _{i}}{\rho _{i-q}},$ and

□

This leads to the following proposition on which our main result is based.

Proposition 3.5. Let $l$ and $q$ be two integers with $q\geq m+1$, $l\geq \max \left(6q+1, \frac{1}{2}(q^{2}-1), 8\right)$. Then, $T_{z^{q}+\delta \overline{z}^{l}}$ is hyponormal if and only if $|\delta |\leq \frac{q}{l}$.

Proof. Hyponormality is equivalent to the following three inequalities:

The first inequality (3.1) takes the form

Since the right hand side of this inequality decreases with $i$, inequality (3.1) is equivalent to

Inequality (3.2) can be written as

A computation reduces the above inequality to

where

and

As mentioned before, $Q_{l, q, i}$ is a decreasing function of $i$ and it is not difficult to see that $R_{q, i}$ is also decreasing. Thus, we get that inequality (3.2) is equivalent to

or, equivalently,

Since $Q_{l, q, i}$ is decreasing, $Q_{l, q, l-m-1}\leq Q_{l, q, q-m-1}$. Thus, we have

That is, if inequality (3.2) is satisfied, so is inequality (3.1). Inequality (3.3) takes the form

To simplify notation, set $A_{i}: = (\beta +i+1)$. A computation leads to the following inequality for $i\geq l-m$:

After simplification of

we get

where the functions $F_{1}$, $F_{2}$, and $F_{3}$ are as follows:

As seen before, $Q_{l, q, i}$ is a decreasing function of $i$. We will show that $F(i)$ is also a decreasing function. A computation gives

where we have

It suffices to verify the inequalities

Note $l-m\geq q-m+1\geq 2$. We show inequality (3.5) first. It takes the form

From the expressions of $F_{1}(i)$ and $F_{2}(i)$, we see that, since $\beta+i+1\geq q$ for $i\geq l-m$,

and thus inequality (3.5) holds. Inequality (3.6) can be reduced to

An elementary computation gives

where

and

The function $\Phi _{2}(i)$ is defined by

where

$P_{2}^{1, 4}$ and $P_{2}^{3, 4}$ are defined in a similar manner.

The functions $\Phi_{3}$ and $\Phi_{4}$ are defined in a similar way. The function $\Psi$ is given by

with

and

and similar definitions for $\Psi_{3}$ and $\Psi_{4}$. Next, we simplify the right hand side of inequality (3.7). A tedious, but elementary, computation gives

where

with

and similar definitions for $\Delta_{3}(i)$ and $\Delta_{4}(i)$. The function $\Omega (i)$ is given by

with $\Omega _{1}(i)$ and $\Omega _{2}(i)$ defined by

where

and $\Omega _{1, 4}(i)$ defined in a similar way. We show

and, since $l\geq \frac{1}{2}(q^{2}-1)$,

and similarly

Thus,

Similar inequalities lead to

and we have

for $i\geq l-m$. Next, we show $(l^{2}+q^{2})\Psi _{1}(i)\leq \Omega_{1}(i)$ for $i\geq l-m$. We clearly have

Similarly, we have

We deduce that inequality (3.6) holds and that $F(i)$ is decreasing. Thus, inequality (3.3) holds if and only if $|\delta |\leq \sqrt{ {\lim_{i\rightarrow \infty }}F(i)} = \frac{q}{l}$. We finally show that

From inequality (3.7), we have

To simplify notation, we set $B_{i}: = (\beta +i+3)(\beta +i+2)(\beta+i+1)(\beta +i)$. Inequality (3.8) takes the form

and we have

where

and, with $A_{i}: = \beta +i+1$ as before,

and

Since

we get

We see that inequality (3.8) is equivalent to

We show that if $l\geq N$, then

We then rewrite this last inequality as

Since $\beta \in (m-1, m]$, by assumptions on $l$ we have

Thus, inequality (3.9) is satisfied if

Since by assumptions on $l$ and $q$,

it is enough to have

and this is satisfied by assumptions on $l$. Hence, inequality (3.9) holds. We conclude that if $q\geq m+1$ and $l\geq \max \left(6q+1, \frac{1}{2}(q^{2}-1), 8\right)$, $T_{z^{q}+\delta \overline{z}^{l}}$ is hyponormal if and only if $|\delta |\leq \frac{q}{l}$. The proof is complete. □

It is easy to see that if $a$ and $b$ are complex numbers satisfying $|a|+|b|\leq 1$, and if $T_{\varphi +\overline{\psi _{1}}}$ and $T_{\varphi +\overline{\psi _{2}}}$ are hyponormal then, $T_{\varphi +a\overline{\psi _{1}}+b\overline{\psi _{2}}}$ is hyponormal. This leads to our main result, where $N$ is defined as earlier.

Theorem 3.6. Let $\left(\lambda _{l}\right)_{N\leq l\leq N_{1}}$ be any finite sequence of complex numbers such that ${\sum\limits_{N\leq l\leq N_{1}}}|\lambda _{l}|\leq 1$, and set $g = {\sum\limits_{N\leq l\leq N_{1}}}\lambda _{l}\frac{q}{l}z^{l}$. Then, we have $T_{\overline{g}}T_{g}-T_{g}T_{\overline{g}}\leq T_{\overline{z}^{q}}T_{z^{q}}-T_{z^{q}}T_{\overline{z}^{q}}$ on $A_{\alpha, \beta}^{2}$.

4.   Conclusions

The other results of Choe and Lee ^[2] can be generalized and this could be addressed in a future work. Hyponormality on the punctured disk is a very recent subject of interest and much remains to be done on this. We hope this paves the way for further works.

Author contributions

All authors contributed equally and significantly to the study conception and design, material preparation, data collection and analysis. The first draft of the manuscript was written by H. Sadraoui, and all authors read and approved the final manuscript.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

The authors would like to extend their sincere appreciation to Researchers Supporting Project number (RSPD2024R1112), King Saud University, Riyadh, Saudi Arabia.

Conflicts of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.