Algebra matrix and similarity classification of operators
type: Item
instance of: scholarly article
title: Algebra matrix and similarity classification of operators
zbMATH Open document ID: 1117.47027
author: Zilong Zhang
author: Yu Cheng Li
published in: Science in China. Series A
publication date: 2007-01-24T00:00:00Z
review text: A bounded linear operator \(A\) on a complex separable Hilbert space is said to be a Cowen--Douglas operator of index \(n\) (\(n\) is a given positive integer) if the spectrum of \(A\) contains a complex domain \(\Omega\) such that the operators \(A-z,\;z\in\Omega\), are onto and their kernels have dimension \(n\) and generate a dense subspace. \(A\) is called strongly irreducible if its commutant algebra \({\mathcal A}'(A)\) has no nontrivial idempotents. By Gelfand representation theory and the Silov idempotents theorem, the authors obtain a central idempotents theorem for unital semisimple \(n\)-homogeneous Banach algebras. In view of [\textit{C.\,L.\thinspace Jiang}, Can.\ J.\ Math.\ 56, No.\,4, 742--775 (2004; Zbl 1068.47044)] and as an application of this central idempotent theorem, it is finally shown that two strongly irreducible Cowen--Douglas operators \(A\) and \(B\) are similar if and only if \({\mathcal A}'(A\oplus B)\) has no nontrivial central idempotent elements.
Mathematics Subject Classification ID: 47B99
Mathematics Subject Classification ID: 46H99
Mathematics Subject Classification ID: 47A65
zbMATH DE Number: 5118278
zbMATH Keywords: Cowen-Douglas operator
zbMATH Keywords: Silov idempotents theorem
zbMATH Keywords: strongly irreducible operator
zbMATH Keywords: algebra matrix
zbMATH Keywords: similarity of operators
MaRDI profile type: Publication
full work available at URL: s11425-006-0398-9
OpenAlex ID: W1945616007
cites work: Similarity Classification of Cowen-Douglas Operators
cites work: Complex geometry and operator theory
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DOI: 10.1007/S11425-006-0398-9
description: scientific article; zbMATH DE number 5118278