Hyponormal Toeplitz operators with non-harmonic algebraic symbol

Abstract

Given a bounded function $\varphi$ on the unit disk in the complex plane, we consider the operator $T_{\varphi }$, defined on the Bergman space of the disk and given by $T_{\varphi }(f)=P(\varphi f)$, where P denotes the orthogonal projection to the Bergman space in $L^2({\mathbb {D}},dA)$. For algebraic symbols $\varphi$, we provide new necessary conditions on $\varphi$ for $T_{\varphi }$ to be hyponormal, extending recent results of Fleeman and Liaw. Our approach is perturbative and aims to understand how small changes to a symbol preserve or destroy hyponormality of the corresponding operator. We consider both additive and multiplicative perturbations of a variety of algebraic symbols. One of our main results provides a necessary condition on the complex constant C for the operator $T_{z^n+C|z|^s}$ to be hyponormal. This condition is also sufficient if $s\ge 2n$.