The strong closure of the simultaneous similarity orbit of some pairs of finite rank operators
type: Item
instance of: scholarly article
title: The strong closure of the simultaneous similarity orbit of some pairs of finite rank operators
zbMATH Open document ID: 1318.47024
author: José Barría
published in: Linear Algebra and its Applications
publication date: 2013-02-21T00:00:00Z
review text: Let \(A\) and \(B\) be operators on a separable Hilbert space \(\mathcal H\). Denote by \(\mathcal S(A)\) the similarity orbit of \(A\), that is, the set of all operators \(W^{-1}AW\), where \(W\) is an invertible operator on \(\mathcal H\), and by \(\mathcal S(A,B)\) the simultaneous orbit of the pair \((A,B)\), that is, the set of all pairs \((W^{-1}AW, W^{-1}BW)\) where \(W\) is an invertible operator on \(\mathcal H\). Denote by \(\mathcal L(\mathcal H)\) the algebra of all operators on \(\mathcal H\). On \(\mathcal L(\mathcal H)\), consider the strong operator topology. Let \(\bar{\mathcal S}(A)\) and \(\bar{\mathcal S}(A,B)\) denote the strong closure of \(\mathcal S(A)\) and \(\mathcal S(A,B)\), respectively. Let \(d(A,B)\) denote the dimension of the span of \(({\text{ker A}})^\perp\) and \(({\text{ker B}})^\perp\). ``Let \(\mathcal C(A,B)\) be the set of pairs \((X,Y)\) in \(\bar{\mathcal S}(A)\times\bar{\mathcal S}(B)\) such that \(d(X,Y)\leq d(A,B)\). In [the author, Linear Algebra Appl. 432, No. 8, 1873--1877 (2010; Zbl 1182.47022)] it was shown that \(\bar{\mathcal S}(A,B)\subset \mathcal C(A,B)\) if \(A\) and \(B\) are finite rank operators. In the paper we study the equality \(\bar{\mathcal S}(A,B) = \mathcal C(A,B)\).'' The following theorem is the main result of the paper. Theorem. Let \(A_1\) and \(A_2\) be finite rank operators on \(\mathcal H\). Then \(\bar{\mathcal S}(A,B) = \mathcal C(A,B)\) if and only if \((\mathrm{ran} A)\cap(\mathrm{ran} B)=\{0\}\).
reviewed by: Ignat Domanov
Mathematics Subject Classification ID: 47A58
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 6138290
zbMATH Keywords: finite rank operator
zbMATH Keywords: similarity orbit
zbMATH Keywords: strong closure
MaRDI profile type: Publication
full work available at URL: j.laa.2012.11.018
OpenAlex ID: W4206743339
cites work: On the strong closure of the simultaneous similarity orbit of a pair of finite rank operators
cites work: Most Similarity Orbits are Strongly Dense
DOI: 10.1016/J.LAA.2012.11.018
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description: scientific article; zbMATH DE number 6138290