Additive results of the Drazin inverse for bounded linear operators
type: Item
instance of: scholarly article
title: Additive results of the Drazin inverse for bounded linear operators
zbMATH Open document ID: 1306.47004
author: Yunfeng Shi
author: Alatancang
author: Junjie Huang
published in: Complex Analysis and Operator Theory
publication date: 2014-08-29T00:00:00Z
review text: Let \(\mathcal X\) be a Banach space and \(A\) be a bounded linear operator on \(\mathcal X\). If there exists a bounded linear operator \(T\) on \(\mathcal X\) such that \(AT=TA\), \(AT^2=T\) and \(A^{k+1}T=A^k\), where \(k\) is the index of \(A\), then such a \(T\) is called the Drazin inverse of \(A\). This generalized inverse is unique if it exists and is denoted by \(A^D\). However, it may not exist. There is a well known result of \textit{M. P. Drazin} [Am. Math. Mon. 65, 506--514 (1958; Zbl 0083.02901)] for the Drazin inverse of a sum of two operators, viz., \((P+Q)^D=P^D+Q^D\) whenever \(P\) and \(Q\) are Drazin invertible operators satisfying the conditions \(PQ=QP=0\). The authors of the paper under review show that \(P+Q\) is Drazin invertible if \(P^2Q+PQ^2=0\), \(P^3Q=0=PQ^3\) and \(PQ\) is Drazin invertible. In such a case, a rather complicated representation is proved for the Drazin inverse of \(P+Q\).
reviewed by: K. C. Sivakumar
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 6336134
zbMATH Keywords: Drazin inverse
zbMATH Keywords: additive result
zbMATH Keywords: bounded linear operators
MaRDI profile type: Publication
full work available at URL: s11785-013-0307-5
OpenAlex ID: W2162519601
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DOI: 10.1007/S11785-013-0307-5
description: scientific article; zbMATH DE number 6336134