Generalized Drazin invertibility and local spectral theory
type: Item
instance of: scholarly article
title: Generalized Drazin invertibility and local spectral theory
zbMATH Open document ID: 1422.47037
published in: Monatshefte für Mathematik
publication date: 2017-04-11T00:00:00Z
review text: In the present paper, the author generalizes the results of \textit{P. Aiena} and \textit{S. Triolo} [J. Math. Anal. Appl. 435, No. 1, 414--424 (2016; Zbl 1339.47006)]. The author proves that, if \(A\in B(\mathcal{X})\) is an upper triangular Banach space operator with diagonal \((A_{1},A_{2})\), \(A_{1}\) is invertible and \(A_{2}\) is quasinilpotent, then \(A^{-1}_{1}\oplus A_{2}\) satisfies either of the single valued extension property, Dunford's condition (C), Bishop's property \((\beta)\), decomposition property \(\delta\) or is decomposable if and only if \(A_{1}\) has the respective property. The operator \(A^{-1}_{1}\oplus0\) is subscalar, left polaroid, right polaroid if and only if \(A_{1}\) is. As a consequence, the author deduces that, for the Drazin invertible operators \(A,\) with Drazin inverse \(B\), the operator \(B\) satisfies any one of the precedent properties if and only if \(A\) does.
reviewed by: F. Lombarkia
Mathematics Subject Classification ID: 47B40
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A11
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 6704120
zbMATH Keywords: Banach space
zbMATH Keywords: generalized Drazin invertible
zbMATH Keywords: local spectral properties (C), \((\beta)\) and \(\delta\)
zbMATH Keywords: Eschmeier-Putinar-Bishop property \((\beta)\)
zbMATH Keywords: left polar
zbMATH Keywords: right polar
zbMATH Keywords: polar
author: B. P. Duggal
MaRDI profile type: Publication
full work available at URL: s00605-016-0898-9
OpenAlex ID: W2309524573
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DOI: 10.1007/S00605-016-0898-9
description: scientific article; zbMATH DE number 6704120