Additive Drazin inverse preservers
type: Item
instance of: scholarly article
title: Additive Drazin inverse preservers
zbMATH Open document ID: 1131.47035
published in: Linear Algebra and its Applications
publication date: 2007-09-14T00:00:00Z
review text: Let \(H\) be a real or complex Hilbert space and denote by \(B(H)\) the algebra of all bounded linear operators acting on \(H\). An element \(T\in B(H)\) is called Drazin invertible if there exists an element \(T^D\in B(H)\) and a positive integer \(k\) such that \[ TT^D=T^DT, \quad T^DTT^D=T^D, \quad T^{k+1}T^D=T^k. \] The operator \(T^D\) is unique and called the Drazin inverse of \(T\). The author characterizes the additive maps \(\phi:B(H)\to B(K)\) (\(H,K\) being infinite-dimensional real or complex Hilbert spaces) which preserve the Drazin inverse in the sense that \(\phi(T^D)=\phi(T)^D\) holds for every Drazin invertible operator \(T\in B(H)\). It is proved that if the range of \(\phi\) contains every rank-one idempotent in \(B(K)\) and \(\phi\) does not annihilate all rank-one idempotents in \(B(H)\), then \(\phi\) is of one of the forms \[ \phi(T)=\xi ATA^{-1}, \quad A\in B(H), \] \[ \phi(T)=\xi AT^{tr}A^{-1}, \quad A\in B(H), \] where \(\xi=\pm 1\) and \(A:H\to K\) is a bounded linear or conjugate-linear bijection. The finite-dimensional case is also considered.
reviewed by: Lajos Molnár
Mathematics Subject Classification ID: 47B49
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 5190951
zbMATH Keywords: additive preservers
zbMATH Keywords: Drazin inverse of operators
author: Jianlian Cui
MaRDI profile type: Publication
full work available at URL: j.laa.2007.05.020
OpenAlex ID: W2045337745
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DOI: 10.1016/J.LAA.2007.05.020
description: scientific article; zbMATH DE number 5190951