A Note on Property ( ð ‘” ð ‘ ) and Perturbations

Qingping Zeng and Huaijie Zhong

Abstract and Applied Analysis, 2012, vol. 2012, 1-10

Abstract:

An operator 𠑇 ∈ ℬ ( ð ‘‹ ) defined on a Banach space ð ‘‹ satisfies property ( ð ‘” ð ‘ ) if the complement in the approximate point spectrum 𠜎 ð ‘Ž ( 𠑇 ) of the upper semi-B-Weyl spectrum 𠜎 𠑆 ð µ ð ¹ âˆ’ + ( 𠑇 ) coincides with the set Î ( 𠑇 ) of all poles of the resolvent of 𠑇 . In this paper, we continue to study property ( ð ‘” ð ‘ ) and the stability of it, for a bounded linear operator 𠑇 acting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, and by quasinilpotent operators commuting with 𠑇 . Two counterexamples show that property ( ð ‘” ð ‘ ) in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:523986

DOI: 10.1155/2012/523986

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