Components of topological uniform descent resolvent set and local spectral theory
type: Item
instance of: scholarly article
title: Components of topological uniform descent resolvent set and local spectral theory
zbMATH Open document ID: 1280.47019
author: Qiaofen Jiang
author: Shifang Zhang
author: Huaijie Zhong
published in: Linear Algebra and its Applications
publication date: 2013-01-16T00:00:00Z
review text: Let \(T\) be a bounded linear operator acting on an infinite-dimensional complex Banach space \(X\). \(T\) is said to have topological uniform descent (TUD) for order \(d\in\mathbb{N}\) if, for \(n\geq d\), \(R(T)+N(T^n)=R(T)+N(T^d)\) and \(R(T^n)\) is closed in the operator range topology of \(R(T^d)\), where \(R(T)\) and \(N(T)\) denote the range and null space of \(T\), respectively. The topological uniform descent spectrum is defined by \(\sigma_{ud}(T)=\{\lambda\in\mathbb{C}\,:\,\lambda I-T\text{ does not have TUD}\}\). In this paper, the authors prove that the topological uniform descent spectrum \(\sigma_{ud}(T)\) is a compact subset of the usual spectrum and satisfies the spectral mapping theorem. They investigate the component of \(\rho_{ud}(T)=\mathbb{C}\setminus\sigma_{ud}(T)\) using tools from local spectral theory. Some applications to Drazin invertibility are given.
reviewed by: H. Zguitti
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A11
zbMATH DE Number: 6125854
zbMATH Keywords: Banach space
zbMATH Keywords: topological uniform descent
zbMATH Keywords: single-valued extension property
zbMATH Keywords: spectrum
MaRDI profile type: Publication
full work available at URL: j.laa.2012.10.004
OpenAlex ID: W1989335430
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DOI: 10.1016/J.LAA.2012.10.004
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description: scientific article; zbMATH DE number 6125854