The unitary orbit of strongly irreducible operators in the nest algebra with well-ordered set
type: Item
instance of: scholarly article
title: The unitary orbit of strongly irreducible operators in the nest algebra with well-ordered set
zbMATH Open document ID: 0888.47024
author: You Qing Ji
author: Chunlan Jiang
author: Zong Yao Wang
published in: Michigan Mathematical Journal
publication date: 1998-01-14T00:00:00Z
review text: For a nest algebra \(N\) of operators on a Hilbert space let \(\tau(\mathcal N)\) be the set of all operator \(T\) such that \(TN=N\) for all \(N\in\mathcal N\). An operator \(T\) is strongly irreducible if it does not commute with any nontrivial idempotent. Let \(\Lambda(T)\) denote the spectral picture of \(T\) and \(S(T)^-\) the closure of the similarity orbit of \(T\). The authors prove that if \(\mathcal N\) is maximal and well-ordered and \(T\in\tau(\mathcal N)\) has connected spectrum then there exists a strongly irreducible operator \(A\) in \(\tau(\mathcal N)\) with the same spectral picture as \(T\) and \(T\in S(A)^-\). Moreover, they prove that if \(\mathcal N\) is well ordered with finite dimensional atoms then the closure of the unitary orbit of all strongly irreducible elements of \(\tau(\mathcal N)\) consists of all \(T\) such that its usual and Weyl spectrum are connected and ind\((T-\lambda)\geq 0\) for all \(\lambda\) such that \(T-\lambda\) is semi-Fredholm. As a corollary they prove that, for good domains, if \(T\) is a Cowen-Douglas operator on \(\Omega\) and its spectrum is the closure of \(\Omega\) then for every \(\varepsilon>0\) there is a compact operator \(K\) with \(|K|<\varepsilon\) such that \(T+K\) is strongly irreducible.
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 47A53
zbMATH DE Number: 1046969
zbMATH Keywords: nest algebra
zbMATH Keywords: strongly irreducible
zbMATH Keywords: idempotent
zbMATH Keywords: spectral picture
zbMATH Keywords: similarity orbit
zbMATH Keywords: Weyl spectrum
zbMATH Keywords: semi-Fredholm
zbMATH Keywords: Cowen-Douglas operator
reviewed by: Gustavo Corach
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005622
OpenAlex ID: W2014364305
DOI: 10.1307/MMJ/1029005622
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description: scientific article; zbMATH DE number 1046969