Journal of Operator Theory

Volume 56, Issue 1, Summer 2006 pp. 47-58.

Hyponormal Toeplitz operators with rational symbols

Authors: In Sung Hwang (1) and Woo Young Lee (2)
Author institution: (1) Department of Mathematics, Institute of Basic Sciences, Sungkyunkwan University, Suwon 440-746, Korea
(2) Department of Mathematics, Seoul National University, Seoul 151-742, Korea

Summary: In this paper we consider the self-commutators of Toeplitz operators

$T_{\varphi}$ with rational symbols

$\varphi$ using the classical Hermite-Fej\' er interpolation problem. Our main theorem is as follows. Let

$\varphi = \overline{g}+ f \in L^{\infty}$ and let

$f= \theta \overline{a}$ and

$g =\theta \overline{b}$ , where

$\theta$ is a finite Blaschke product of degree

$d$ and

$a,b\in \mathcal H(\theta):=H^2\ominus \theta H^2$ . Then

$\mathcal H (\theta)$ is a reducing subspace of

$[T_{\varphi}^* , T_{\varphi}]$ , and

$[T_{\varphi}^* , T_{\varphi}]$ has the following representation relative to the direct sum

$\mathcal{H}(\theta)\oplus \mathcal{H}(\theta)^\perp$ :

$[T_{\varphi}^* , T_{\varphi}]= A(a)^* W M(\varphi) W^* A(a)\,\oplus\, 0_\infty,$ where

$A(a):=P_{\mathcal{H}(\theta)}M_a \mid_{\mathcal H (\theta)}$ (

$M_a$ is the multiplication operator with symbol

$a$ ),

$W$ is the unitary operator from

$\mathbb{C}^d$ onto

$\mathcal{H}(\theta)$ defined by

$W:=(\phi_1,\ldots,\phi_d)$ (

$\{\phi_j\}$ is an orthonormal basis for

$\mathcal{H}(\theta)$ ), and

$M(\varphi)$ is a matrix associated with the classical Hermite-Fej\' er interpolation problem. Hence, in particular,

$T_\varphi$ is hyponormal if and only if

$M(\varphi)$ is positive. Moreover the rank of the self-commutator

$[T_\varphi^*, T_\varphi]$ is given by

$\text{\rm{rank}}\,[T_\varphi^*, T_\varphi]=\text{\rm{rank}}\,M(\varphi)$ .

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