Journal of Operator Theory

Volume 54, Issue 2, Fall 2005 pp. 261-267.

Cowen sets for Toeplitz operators with finite rank selfcommutators

Authors: Woo Young Lee
Author institution: Department of Mathematics, Seoul National University, Seoul 151--742, Korea

Summary: Cowen's theorem states that if $\varphi\in L^\infty(\Bbb{T})$ then the Toeplitz operator $T_{\varphi}$ is hyponormal if and only if the following ``Cowen'' set \begin{equation*} {\mathcal E}(\varphi)=\{k\in H^{\infty}({\Bbb T}):\|k\|_{\infty}\leqslant1\quad \mathrm{and}\quad \varphi-k{\overline\varphi}\in H^{\infty}({\Bbb T})\} \end{equation*} is nonempty. In this paper we give a complete description on the Cowen set $\mathcal{E}(\varphi)$ if the selfcommutator $[T_\varphi^*, T_\varphi]$ is of finite rank. In particular, it is shown that the solution for the cases where $\varphi$ is of bounded type has a connection with a $H^\infty$ optimization problem.

Keywords: Cowen sets, $H^\infty$ optimization problem, Toeplitz operators, Hankel operators, hyponormal, bounded type, Carathéodory-Schur interpolation problem.

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