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2015 Property $(aBw)$ and perturbations
Qingping Zeng
Ann. Funct. Anal. 6(2): 212-223 (2015). DOI: 10.15352/afa/06-2-18


A bounded linear operator $T$ acting on a Banach space $X$ satisfies property $(aBw)$, a strong version of a-Weyl's theorem, if the complement in the approximate point spectrum $\sigma_{a}(T)$ of the upper semi-B-Weyl spectrum $\sigma_{USBW}(T)$ is the set of all isolated points of approximate point spectrum which are eigenvalues of finite multiplicity. In this paper we investigate the property $(aBw)$ in connection with Weyl type theorems. In particular, we show that $T$ satisfies property $(aBw)$ if and only if $T$ satisfies a-Weyl's theorem and $\sigma_{USBW}(T)=\sigma_{USW}(T)$, where $\sigma_{USW}(T)$ is the upper semi-Weyl spectrum of $T$. The preservation of property $(aBw)$ is also studied under commuting nilpotent, quasi-nilpotent, power finite rank or Riesz perturbations. The theoretical results are illustrated by some concrete examples.


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Qingping Zeng. "Property $(aBw)$ and perturbations." Ann. Funct. Anal. 6 (2) 212 - 223, 2015.


Published: 2015
First available in Project Euclid: 19 December 2014

zbMATH: 1252.47012
MathSciNet: MR3292527
Digital Object Identifier: 10.15352/afa/06-2-18

Primary: 47A10
Secondary: 47A53 , 47A55

Keywords: a-Weyl's theorem , generalized a-Browder's theorem , perturbation , Property $(aBw)$

Rights: Copyright © 2015 Tusi Mathematical Research Group


Vol.6 • No. 2 • 2015
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