Small compact perturbation of strongly irreducible operators.
type: Item
instance of: scholarly article
title: Small compact perturbation of strongly irreducible operators.
zbMATH Open document ID: 1040.47001
DOI: 10.1007/BF01212703
author: Chunlan Jiang
author: You Qing Ji
published in: Integral Equations and Operator Theory
publication date: 2002-01-01T00:00:00Z
review text: Let \(\mathcal H\) be a complex, separable Hilbert space of infinite dimension. An element \(T\) in \(\mathcal L(\mathcal H)\), the algebra of all linear and bounded operators in \(\mathcal H\), is called strongly irreducible, if it is not similar to any reducible operator. It is easy to see that for any compact \(K\in \mathcal L(\mathcal H)\) the operator \(T+K\) is not strongly irreducible provided the spectrum of \(T\) is not connected. In this article, it is shown that for any \(T\in\mathcal L(\mathcal H)\) with connected spectrum there is a compact \(K\in \mathcal L(\mathcal H)\) of arbitrarily small norm such that \(T+K\) is strongly irreducible. In the second main result, it is assumed that \(T\) belongs to the \(n\)-th Cowen-Douglas complex \(\mathcal B_n(\Omega)\) for some open, bounded and connected subset \(\Omega\) of the complex plane. The authors prove that in this case \(K\) may be choosen such that \(T+K\) also belongs to \(\mathcal B_n(\Omega)\).
reviewed by: André Noll
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 1890589
zbMATH Keywords: perturbation
zbMATH Keywords: linear operators
zbMATH Keywords: irreducibility
zbMATH Keywords: complex Hilbert space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1890589