## Abstract and Applied Analysis

### A Note on Property $(gb)$ and Perturbations

#### Abstract

An operator $T\in \scr B(X)$ defined on a Banach space $X$ satisfies property $(gb)$ if the complement in the approximate point spectrum ${\sigma }_{a}(T)$ of the upper semi-B-Weyl spectrum ${\sigma }_{SB{F}_{+}^{-}}(T)$ coincides with the set $\Pi (T)$ 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.

#### Article information

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

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.aaa/1355495824

Digital Object Identifier
doi:10.1155/2012/523986

Mathematical Reviews number (MathSciNet)
MR2965440

Zentralblatt MATH identifier
1252.47012

#### Citation

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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