Hyponormality of block Toeplitz operators
type: Item
instance of: scholarly article
title: Hyponormality of block Toeplitz operators
zbMATH Open document ID: 1125.47019
author: Caxing Gu
author: Jacob Hendricks
author: Dan Rutherford
published in: Pacific Journal of Mathematics
publication date: 2007-02-26T00:00:00Z
review text: The authors study the hyponormality of block Toeplitz operators acting on the vector-valued Hardy space of the unit disk \(H^2(\mathbb C^n)\). The block Toeplitz operator with matrix symbol \(G\in L^{\infty}(\mathbb C^{n\times n})\) is defined by \(T_Gh=P(Gh)\), where \(P\) is the projection from \(L^2(\mathbb C^n)\) to \(H^2(\mathbb C^n)\). \textit{C.\,C.\thinspace Cowen} [Proc.\ Am.\ Math.\ Soc.\ 103, No.\,3, 809--812 (1988; Zbl 0668.47021)] characterized scalar hyponormal Toeplitz operators as follows: \(T_{\varphi}\) is hyponormal if and only if there exists a \(k\in H^{\infty}\) with \(\| k\|_{\infty}\leq 1\) satisfying \(\varphi-k\overline{\varphi}\in H^2\). The first author in [J.~Funct.\ Anal.\ 124, No.\,1, 135--148 (1994; Zbl 0815.47029)] extended Cowen's result to generalized Toeplitz operators whose symbols are operator-valued, under the extra assumption that the symbol is normal. In the present paper, it is shown that the block Toeplitz operator \(T_G\) is hyponormal if and only if \(G^*G=GG^*\) and there exists a matrix \(K(z)\in H^{\infty}(\mathbb C^{n\times n})\) such that \(\| K\|_{\infty}\leq 1\) and \(G-KG^*\in H^{\infty}(\mathbb C^{n\times n})\). It is, however, not easy to find a matrix \(K(z)\) satisfying these conditions. The authors show that if \(G(z)\) is a trigonometric matrix polynomial with an invertible leading coefficient, then finding such a \(K\) is equivalent to solving a matrix-valued Carathéodory interpolation problem.
reviewed by: Željko Čučković
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47B20
zbMATH DE Number: 5129572
MaRDI profile type: Publication
full work available at URL: pjm.2006.223.95
OpenAlex ID: W2005775069
DOI: 10.2140/PJM.2006.223.95
Recommended article: A HYPONORMAL TOEPLITZ COMPLETION PROBLEM
Recommended article: Jointly hyponormal block Toeplitz pairs with rational symbols
Recommended article: Hyponormality of bounded-type Toeplitz operators
Recommended article: Hyponormal block Toeplitz operators and the Carathéodory interpolation problems
Recommended article: Hyponormal Toeplitz operators on the polydisk
Recommended article: Hyponormality and subnormality of block Toeplitz operators
Recommended article: Block Toeplitz operators with rational symbols
Recommended article: A NOTE ON HYPONORMAL TOEPLITZ OPERATORS
Recommended article: Hyponormal trigonometric Toeplitz operators
Recommended article: Kernels of Hankel operators and hyponormality of Toeplitz operators
description: scientific article; zbMATH DE number 5129572