# Hyponormal Toeplitz operators with non-harmonic algebraic symbol

Hyponormal Toeplitz operators with non-harmonic algebraic symbol Given a bounded function $$\varphi$$ φ on the unit disk in the complex plane, we consider the operator $$T_{\varphi }$$ T φ , defined on the Bergman space of the disk and given by $$T_{\varphi }(f)=P(\varphi f)$$ T φ ( f ) = P ( φ f ) , where P denotes the orthogonal projection to the Bergman space in $$L^2({\mathbb {D}},dA)$$ L 2 ( D , d A ) . For algebraic symbols $$\varphi$$ φ , we provide new necessary conditions on $$\varphi$$ φ for $$T_{\varphi }$$ T φ 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 $$T_{z^n+C|z|^s}$$ T z n + C | z | s to be hyponormal. This condition is also sufficient if $$s\ge 2n$$ s ≥ 2 n . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# Hyponormal Toeplitz operators with non-harmonic algebraic symbol

, Volume 9 (4) – Jan 8, 2019
14 pages

/lp/springer-journals/hyponormal-toeplitz-operators-with-non-harmonic-algebraic-symbol-4XTiTM9emJ
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-018-00279-2
Publisher site
See Article on Publisher Site

### Abstract

Given a bounded function $$\varphi$$ φ on the unit disk in the complex plane, we consider the operator $$T_{\varphi }$$ T φ , defined on the Bergman space of the disk and given by $$T_{\varphi }(f)=P(\varphi f)$$ T φ ( f ) = P ( φ f ) , where P denotes the orthogonal projection to the Bergman space in $$L^2({\mathbb {D}},dA)$$ L 2 ( D , d A ) . For algebraic symbols $$\varphi$$ φ , we provide new necessary conditions on $$\varphi$$ φ for $$T_{\varphi }$$ T φ 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 $$T_{z^n+C|z|^s}$$ T z n + C | z | s to be hyponormal. This condition is also sufficient if $$s\ge 2n$$ s ≥ 2 n .

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Jan 8, 2019