Hyponormal Toeplitz operators with non-harmonic algebraic symbol
Tóm tắt
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 $$s\ge 2n$$.
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