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 \(s\ge 2n\).

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Correspondence to Brian Simanek.

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Simanek, B. Hyponormal Toeplitz operators with non-harmonic algebraic symbol. Anal.Math.Phys. 9, 1613–1626 (2019). https://doi.org/10.1007/s13324-018-00279-2

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Keywords

  • Hyponormal operator
  • Toeplitz operator
  • Bergman space
  • Perturbation theory

Mathematics Subject Classification

  • 47B20
  • 47B35