Generalized bicircular projections via the operator equation \(\alpha X^2AY^2+\beta XAY+A=0\)
type: Item
instance of: scholarly article
title: Generalized bicircular projections via the operator equation \(\alpha X^2AY^2+\beta XAY+A=0\)
zbMATH Open document ID: 1153.47014
author: Dijana Ilišević
published in: Linear Algebra and its Applications
publication date: 2008-10-06T00:00:00Z
review text: A linear projection \(P\) on a complex Banach space \(E\) is called a generalized bicircular projection if, for some complex number \(\lambda \neq 1\) of modulus 1, the operator \(P+\lambda (I-P)\) is an isometry. Transformations of this kind have been rather extensively studied recently. Motivated by that study and the structure of surjective isometries on operator algebras, the author considers the following problem. Given a linear projection \(P\) on the space \(B(E)\) of all bounded linear operators and a complex number \(\lambda \neq 1\) of modulus 1, when does the mapping \(P+\lambda (I-P)\) equal one of the following transformations: \(A\mapsto UAV\), \(A\mapsto UA^*V\), \(A\mapsto UA^{t}V\)? Here, in the first case \(E\) is an arbitrary Banach space and \(U,V:E\to E\) are given bounded linear operators, in the second case \(E\) is a Hilbert space and \(U,V\) are conjugate-linear, while in the third case \(E\) is a Hilbert space and \(U,V\) are linear. Based on the solution of a certain operator equation, the structure of the transformation \(P\) is completely described in all those cases and some applications related to preserver problems are also given.
reviewed by: Lajos Molnár
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 47A62
zbMATH DE Number: 5349167
zbMATH Keywords: generalized bicircular projections
zbMATH Keywords: isometries
zbMATH Keywords: preservers
MaRDI profile type: Publication
full work available at URL: j.laa.2008.05.035
OpenAlex ID: W2046185278
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DOI: 10.1016/J.LAA.2008.05.035
description: scientific article; zbMATH DE number 5349167