On hyponormal Toeplitz operators with polynomial and circulant-type symbols
type: Item
instance of: scholarly article
title: On hyponormal Toeplitz operators with polynomial and circulant-type symbols
zbMATH Open document ID: 0899.47013
DOI: 10.1007/BF01191430
author: Douglas Farenick
author: Woo Young Lee
published in: Integral Equations and Operator Theory
publication date: 1997-01-01T00:00:00Z
review text: A Toeplitz operator \(T_\varphi\) with symbol \(\varphi\in L^\infty(\mathbb{T})\) -- \(\mathbb{T}\) denotes the unit circle -- is known to be hyponormal if and only if there exists a function \(k\in H^\infty(\mathbb{T})\) such that \(\| k\|_\infty\leq 1\) and \(\varphi- k\overline{\varphi}\in H^\infty(\mathbb{T})\). This is used to investigate the case, when \(\varphi\) is a trigonometric polynomial. If the positive and negative coefficients of \(\varphi\) are composed by a certain ``circular'' condition from given complex numbers \(a_0,\dots, a_N\), then for the hyponormality of \(T_\varphi\) the zeroes of the analytic polynomial generated by the ``essential'' ones among the numbers \(a_0,\dots, a_N\) are to examine. They determine the above function \(k\), which is unique in this case, and the rank of the selfcommutator of \(T_\varphi\) can be computed from the number of zeroes inside and outside the unit circle.
Mathematics Subject Classification ID: 47B20
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 1113519
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: hyponormality
reviewed by: Gerhard Garske
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01191430
OpenAlex ID: W1971224268
description: scientific article; zbMATH DE number 1113519