Journal of Operator Theory

Volume 79, Issue 2, Spring 2018 pp. 287-300.

A new necessary condition for the hyponormality of Toeplitz operators on the Bergman space

Authors: Zeljko Cuckovic (1) and Raul E. Curto (2)
Author institution:(1) Department of Mathematics, University of Toledo, Toledo, Ohio 43606, U.S.A.
(2) Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242, U.S.A.

Summary: A well known result of C. Cowen states that, for a symbol $\varphi \in L^{\infty }, \varphi \equiv \overline{f}+g \;(f,g\in H^{2})$ , the Toeplitz operator $T_{\varphi }$ acting on the Hardy space of the unit circle is hyponormal if and only if $f=c+T_{\overline{h}}g,$ for some $c\in {\mathbb C}$ , $h\in H^{\infty }$ , $\| h \| _{\infty}\leqslant 1.$ In this note we consider possible versions of this result in the {\it Bergman} space case. Concretely, we consider Toeplitz operators on the Bergman space of the unit disk, with symbols of the form $\varphi \equiv \alpha z^n+\beta z^m +\gamma \overline z ^p + \delta \overline z ^q,$ where $\alpha, \beta, \gamma, \delta \in \mathbb{C}$ and $m,n,p,q \in \mathbb{Z}_+$ , $m < n$ and $p < q$ . By studying the asymptotic behavior of the action of $T_{\varphi}$ on a particular sequence of vectors, we obtain a sharp inequality involving the above mentioned data. This inequality improves a number of existing results, and it is intended to be a precursor of basic necessary conditions for joint hyponormality of tuples of Toeplitz operators acting on Bergman spaces in one or several complex variables.

DOI: http://dx.doi.org/10.7900/jot.2017feb15.2152
Keywords: hyponormality, Toeplitz operators, Bergman space, commutators

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