Hyponormal Toeplitz operators with non-harmonic symbol acting on the Bergman space

TL;DR

This paper investigates hyponormal Toeplitz operators with non-harmonic polynomial symbols on the Bergman space, exploring their properties, historical context, and extending understanding of hyponormality beyond analytic symbols.

Contribution

It provides new results on hyponormal Toeplitz operators with non-harmonic symbols and examines hyponormality behavior extending beyond traditional analytic cases.

Findings

Results for hyponormality with non-harmonic polynomial symbols

Extension of Putnam's inequality to non-analytic symbols

Identification of unusual hyponormality behaviors

Abstract

The Toeplitz operator acting on the Bergman space $A^{2}(\mathbb{D})$, with symbol $\varphi$ is given by $T_{\varphi}f=P(\varphi f)$, where $P$ is the projection from $L^{2}(\mathbb{D})$ onto the Bergman space. We present some history on the study of hyponormal Toeplitz operators acting on $A^{2}(\mathbb{D})$, as well as give results for when $\varphi$ is a non-harmonic polynomial. We include a first investigation of Putnam's inequality for hyponormal operators with non-analytic symbols. Particular attention is given to unusual hyponormality behavior that arises due to the extension of the class of allowed symbols.