Abstract and Applied Analysis

A Note on Property ( g b ) and Perturbations

Qingping Zeng and Huaijie Zhong

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An operator T ( X ) defined on a Banach space X satisfies property ( g b ) if the complement in the approximate point spectrum σ a ( T ) of the upper semi-B-Weyl spectrum σ S B F + - ( T ) coincides with the set Π ( T ) of all poles of the resolvent of T . In this paper, we continue to study property ( g b ) 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 ( g b ) in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.

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Abstr. Appl. Anal., Volume 2012 (2012), Article ID 523986, 10 pages.

First available in Project Euclid: 14 December 2012

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Zeng, Qingping; Zhong, Huaijie. A Note on Property $(gb)$ and Perturbations. Abstr. Appl. Anal. 2012 (2012), Article ID 523986, 10 pages. doi:10.1155/2012/523986. https://projecteuclid.org/euclid.aaa/1355495824

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