Hyponormal Toeplitz Operators on the Bergman Space of the Disk

TL;DR

This paper investigates the hyponormality of specific Toeplitz operators on the Bergman space of the disk, focusing on the impact of the symbol's parameters and correcting previous norm estimates.

Contribution

It provides new bounds on the symbol's coefficient for hyponormality and corrects a prior typo related to the commutator norm of certain Toeplitz operators.

Findings

Determined bounds for |a(t)| ensuring hyponormality for large t

Corrected previous errors in the norm calculation of the commutator

Enhanced understanding of symbol parameter effects on operator hyponormality

Abstract

We consider Toeplitz operators with bounded symbol acting on the Bergman space of the unit disk and assess their hyponormality. We will mainly be concerned with the symbol $\varphi(z)=z^{n}|z|^{2s}+a(t)\bar{z}^{m}|z|^{2t}$, where $s$ and $t$ are positive real numbers and $m$ and $n$ are natural numbers. The main goal is to understand how large $|a(t)|$ can be for this operator to be hyponormal and we will answer this question for large values of $t$. We also correct a typo from a 2019 paper of Fleeman and Liaw concerning the norm of the commutator of the Toeplitz operator with symbol $z^m\bar{z}^n$ when $m>n$.