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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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/523986 |
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| _version_ | 1832568201576185856 |
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| author | Qingping Zeng Huaijie Zhong |
| author_facet | Qingping Zeng Huaijie Zhong |
| author_sort | Qingping Zeng |
| collection | DOAJ |
| 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. |
| format | Article |
| id | doaj-art-28dbe36866974207bcc5cdb5d39a287f |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-28dbe36866974207bcc5cdb5d39a287f2025-02-03T00:59:34ZengWileyAbstract 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 |
| spellingShingle | Qingping Zeng Huaijie Zhong A Note on Property (gb) and Perturbations Abstract and Applied Analysis |
| title | 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_short | A Note on Property (gb) and Perturbations |
| title_sort | note on property gb and perturbations |
| url | http://dx.doi.org/10.1155/2012/523986 |
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