Indices, characteristic numbers and essential commutants of Toeplitz operators
type: Item
instance of: scholarly article
title: Indices, characteristic numbers and essential commutants of Toeplitz operators
zbMATH Open document ID: 1021.47018
DOI: 10.1007/BF02384493
author: Kunyu Guo
published in: Arkiv för Matematik
publication date: 2003-06-16T00:00:00Z
review text: Let \(\gamma(T)\) denote the characteristic number of an essentially normal operator \(T\) and let \(C^*(T)\) be the \(C^*\)-algebra generated by \(T\). It is shown that there exists a unilateral shift of multiplicity \(m\) in \(C^*(T)\) if and only if \(\gamma(T)|m\). Using this result the author answers (in the negative) a question of \textit{M. Engliš} [Ark. Mat. 30, 227-240 (1992; Zbl 0784.46036)] whether the essential commutants of unilateral and bilateral shifts are isomorphic as \(C^*\) algebras. Let \(B_n\) be the unit ball of \(C^n\). On the Bergman space \(L^2_a(B_n)\) a natural \(C^*\)-algebra is constructed. It is shown that its essential commutant is generated by Toeplitz operators with symmetric continuous symbols and all compact operators.
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 1928890
zbMATH Keywords: Toeplitz operators
zbMATH Keywords: unilateral shift
zbMATH Keywords: essential commutants
zbMATH Keywords: bilateral shifts
zbMATH Keywords: \(C^*\) algebras
zbMATH Keywords: Bergman space
reviewed by: Michal Zajac
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02384493
OpenAlex ID: W2050934792
description: scientific article; zbMATH DE number 1928890