Drazin spectrum of operator matrices on the Banach space
type: Item
instance of: scholarly article
title: Drazin spectrum of operator matrices on the Banach space
zbMATH Open document ID: 1157.47004
author: Shifang Zhang
author: Qiaofen Jiang
author: Huaijie Zhong
published in: Linear Algebra and its Applications
publication date: 2008-10-06T00:00:00Z
review text: An operator \(T:X\rightarrow X\) on a complex Banach space \(X\) is called Drazin invertible if there exists another operator \(S\) on \(X\) such that \(ST=TS\), \(STS=S\) and \(T^{k+1}S=T^k\) for some nonnegative integer \(k\). The Drazin spectrum of \(T\) is the set \(\sigma_D(T) :=\{\lambda\in \mathbb{C} : \lambda - T\text{ is not Drazin invertible}\}\). Given two complex Banach spaces \(X\) and \(Y\), the authors consider operators on the product space \(X \times Y\) defined by a \(2\times 2\) upper triangular matrix \(M_C\) with \(A\) and \(B\) in the diagonal and \(C\) in the upper right entry. They show that \(\sigma_D(A)\cup \sigma_D(B)=\sigma_D(M_C) \cup W\), where \(W\) is the union of certain holes in \(\sigma_D(M_C)\) contained in \(\sigma_D(A)\cap \sigma_D(B)\). Moreover, they study the set \(\bigcap_{C\in B(Y,X)}\sigma_D(M_C)\).
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 5349171
zbMATH Keywords: operator matrices
zbMATH Keywords: Drazin spectrum
zbMATH Keywords: Drazin inverse
reviewed by: M. González
MaRDI profile type: Publication
full work available at URL: j.laa.2008.06.002
OpenAlex ID: W2026316659
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DOI: 10.1016/J.LAA.2008.06.002
description: scientific article; zbMATH DE number 5349171