On linear maps preserving generalized invertibility and related properties
type: Item
instance of: scholarly article
title: On linear maps preserving generalized invertibility and related properties
zbMATH Open document ID: 1178.47023
author: Nadia Boudi
author: Youness Hadder
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-06-24T00:00:00Z
review text: The general idea of linear preserver problems is to describe all maps \(\varphi: {\mathcal A}\to{\mathcal B}\) between Banach algebras which preserve some class of elements or some relation. The most famous is probably Kaplansky's problem which asks whether any surjective unital linear map \(\varphi\) between semi-simple Banach algebras which preserves invertibility has to be a Jordan homomorphism, i.e., satisfy \(\varphi(x^2)=\varphi(x)^2\) for all \(x\). Denote, respectively, by \({\mathcal B}(H)\), \({\mathcal F}(H)\) and \({\mathcal K}(H)\) the algebras of bounded, finite-rank and compact operators on the Hilbert space \(H\), and by \(\mathcal{C}(H):=\mathcal{B}(H)/\mathcal{K}(H)\) the Calkin algebra. Recall that an operator is called Fredholm if \(A+{\mathcal K}(H)\) is invertible in \({\mathcal C}(H)\) and semi-Fredholm if \(A+{\mathcal K}(H)\) is either left or right-invertible in \({\mathcal C}(H)\). For \(A\in {\mathcal B}(H)\), an element \(B\in{\mathcal B}(H)\) is called a generalized inverse of \(A\) if \(ABA=A\) and \(BAB=B\). In the paper under review, the authors consider the case of linear surjective maps \(\varphi:{\mathcal B}(H)\to {\mathcal B}(H)\) which preserve generalized invertibility. Their main theorem asserts that, if there is a semi-Fredholm operator \(U\) such that \(\varphi(U)\) is semi-Fredholm (this is the case, in particular, if \(\varphi\) is unital), then \(\varphi({\mathcal F}(H))\subset {\mathcal F}(H)\), \(\phi({\mathcal K}(H))\subset {\mathcal K}(H)\), and the induced map \(\widetilde{\varphi}: {\mathcal C}(H)\to {\mathcal C}(H)\) is of the form \(\widetilde{\varphi}(x)=a\psi(x)\), where \( a \in {\mathcal C}(H)\) is invertible and \(\psi\) is a Jordan homomorphism. This situation is slightly more general than the one treated in [\textit{M. Mbekhta} and \textit{P. Šemrl}, ``Linear maps preserving semi-Fredholm operators and generalized invertibility'', Linear Multilinear Algebra 57, No. 1, 55--64 (2009; Zbl 1201.47015) and \text…
reviewed by: Line Baribeau
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 5292284
zbMATH Keywords: generalized inverse
zbMATH Keywords: semi-Fredholm operators
zbMATH Keywords: Jordan homomorphism
zbMATH Keywords: linear preserver problems
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.03.066
OpenAlex ID: W1989127849
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DOI: 10.1016/J.JMAA.2008.03.066
description: scientific article; zbMATH DE number 5292284