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Hyponormal Toeplitz operators with non-harmonic algebraic symbol

  • Brian SimanekEmail author
Article
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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 \(s\ge 2n\).

Keywords

Hyponormal operator Toeplitz operator Bergman space Perturbation theory 

Mathematics Subject Classification

47B20 47B35 

Notes

Compliance with Ethical Standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Baylor Math DepartmentBaylor UniversityWacoUSA

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