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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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