Journal of Operator Theory

Volume 59, Issue 1, Winter 2008 pp. 69-80.

Quasihyponormal Toeplitz operators

Authors: S.C. Arora (1) and Geeta Kalucha (2)
Author institution: (1) Department of Mathematics, University of Delhi, Delhi-110007, India
(2) Department of Mathematics, PGDAV College, University of Delhi, Delhi-110065, India

Summary: Motivated by a question on subnormal Toeplitz subnormal operators raised in 1970 by P.R.~Halmos, we show that there exist quasihyponormal Toeplitz operators which are neither hyponormal nor analytic. In addition, for

$\varphi \in L^{\infty}(\mathbb{T})$ and letting

$\varphi = f+\overline{g}$ , where

$f$ and

$g$ are in

$H^2$ , we show that the Toeplitz operator

$T_{\varphi}$ is quasihyponormal if and only if

$P(g\overline{f}) = c + T_{\overline{h}}f\overline{f}$ for some constant

$c$ and some function

$h \in H^{\infty}(\mathbb{D})$ with

$\|h\|_{\infty} \leqslant 1$ . Finally, we also show that the problem of quasihyponormality for Toeplitz operators with (trigonometric) polynomial symbols can be reduced to the classical Schur's algorithm in function theory.

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