A Note on Property (gb) and Perturbations

A Note on Property (gb) and Perturbations

An operator T∈ℬ(X) defined on a Banach space X satisfies property (gb) if the complement in the approximate point spectrum σa(T) of the upper semi-B-Weyl spectrum σSBF+-(T) coincides with the set Π(T) of all poles of the resolvent of T. In this paper, we continue to study property (gb) and the stabi...

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Main Authors: Qingping Zeng, Huaijie Zhong
Format: Article
Language:English
Published: Hindawi Limited 2012-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2012/523986
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spelling doaj-28dbe36866974207bcc5cdb5d39a287f2020-11-25T00:52:43ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/523986523986A Note on Property (gb) and PerturbationsQingping Zeng0Huaijie Zhong1School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, ChinaSchool of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, ChinaAn operator T∈ℬ(X) defined on a Banach space X satisfies property (gb) if the complement in the approximate point spectrum σa(T) of the upper semi-B-Weyl spectrum σSBF+-(T) coincides with the set Π(T) of all poles of the resolvent of T. In this paper, we continue to study property (gb) and the stability of it, for a bounded linear operator T acting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, and by quasinilpotent operators commuting with T. Two counterexamples show that property (gb) in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.http://dx.doi.org/10.1155/2012/523986
collection DOAJ
language English
format Article
sources DOAJ
author Qingping Zeng
Huaijie Zhong
spellingShingle Qingping Zeng
Huaijie Zhong
A Note on Property (gb) and Perturbations
Abstract and Applied Analysis
author_facet Qingping Zeng
Huaijie Zhong
author_sort Qingping Zeng
title A Note on Property (gb) and Perturbations
title_short A Note on Property (gb) and Perturbations
title_full A Note on Property (gb) and Perturbations
title_fullStr A Note on Property (gb) and Perturbations
title_full_unstemmed A Note on Property (gb) and Perturbations
title_sort note on property (gb) and perturbations
publisher Hindawi Limited
series Abstract and Applied Analysis
issn 1085-3375
1687-0409
publishDate 2012-01-01
description An operator T∈ℬ(X) defined on a Banach space X satisfies property (gb) if the complement in the approximate point spectrum σa(T) of the upper semi-B-Weyl spectrum σSBF+-(T) coincides with the set Π(T) of all poles of the resolvent of T. In this paper, we continue to study property (gb) and the stability of it, for a bounded linear operator T acting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, and by quasinilpotent operators commuting with T. Two counterexamples show that property (gb) in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.
url http://dx.doi.org/10.1155/2012/523986
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AT huaijiezhong anoteonpropertygbandperturbations
AT qingpingzeng noteonpropertygbandperturbations
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