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 φ∈L∞,φ≡¯f+g(f,g∈H2), the Toeplitz operator Tφ acting on the Hardy space of the unit circle is hyponormal if and only if f=c+T¯hg, for some c∈C, h∈H∞, ‖ 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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