A characterization of the class of partial isometries
type: Item
instance of: scholarly article
title: A characterization of the class of partial isometries
zbMATH Open document ID: 1270.47001
author: Maryam Khosravi
published in: Linear Algebra and its Applications
publication date: 2012-08-03T00:00:00Z
review text: Let \(S\) be a self-adjoint invertible operator acting on a Hilbert space. The Corach-Porta-Recht inequality states that \(\|SXS^{-1} + S^{-1}XS\| \geq 2\|X\|\) holds true for every bounded linear operator~\(X\). Several authors have considered the case of equality and, more generally, characterized subclasses of normal operators by inequalities or equalities in the spirit of the Corach-Porta-Recht inequality. The following is part of the main result of this paper. Suppose that \(S\) is a closed range operator such that the range of the adjoint \(S^{\ast}\) is the range of \(S\). Denote by \(P\) the orthogonal projection onto the range of \(S\). Then \(S\) is a nonzero real multiple of a normal partial isometry if and only if \(\|S^{\ast}XS^{+} + S^{+}XS^{\ast}\| = 2\|PXP\|\) for all \(X\), if and only if \(\|S^{\ast}\oplus S^{+} + S^{+}\oplus S^{\ast}\|_{\lambda} = 2\). Here, \(S^{+}\) is the Moore-Penrose inverse of \(S\) and \(\|\cdot\|_{\lambda}\) denotes the injective norm on the tensor product.
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 47B15
zbMATH DE Number: 6063153
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: operator inequalities
zbMATH Keywords: closed range operators
zbMATH Keywords: partial isometries
reviewed by: Cătălin Badea
MaRDI profile type: Publication
full work available at URL: j.laa.2012.04.025
OpenAlex ID: W2010127450
cites work: Operator inequalities related to the Corach-Porta-Recht inequality
cites work: An operator inequality
cites work: On generalized inverses in C*-algebras
cites work: Corach-Porta-Recht inequality for closed range operators
cites work: On the norm of symmetrised two-sided multiplications
cites work: On the injective norm and characterization of some subclasses of normal operators by inequalities or equalities
cites work: Rank one operators and norm of elementary operators
cites work: Some results related to the Corach-Porta-Recht inequality
DOI: 10.1016/J.LAA.2012.04.025
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description: scientific article; zbMATH DE number 6063153