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Abstract and Applied Analysis
Volume 2012, Article ID 523986, 10 pages
http://dx.doi.org/10.1155/2012/523986
Research Article

A Note on Property ( ๐‘” ๐‘ ) and Perturbations

School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China

Received 14 April 2012; Accepted 26 July 2012

Academic Editor: Sergey V. Zelik

Copyright © 2012 Qingping Zeng and Huaijie Zhong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An operator ๐‘‡ โˆˆ โ„ฌ ( ๐‘‹ ) defined on a Banach space ๐‘‹ satisfies property ( ๐‘” ๐‘ ) if the complement in the approximate point spectrum ๐œŽ ๐‘Ž ( ๐‘‡ ) of the upper semi-B-Weyl spectrum ๐œŽ ๐‘† ๐ต ๐น โˆ’ + ( ๐‘‡ ) coincides with the set ฮ  ( ๐‘‡ ) of all poles of the resolvent of ๐‘‡ . In this paper, we continue to study property ( ๐‘” ๐‘ ) and the stability of it, for a bounded linear operator ๐‘‡ acting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, and by quasinilpotent operators commuting with ๐‘‡ . Two counterexamples show that property ( ๐‘” ๐‘ ) in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.