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New approach to a-Weyl's theorem through localized SVEP and Riesz-type perturbations
Linear and Multilinear Algebra ( IF 0.9 ) Pub Date : 2020-10-22 , DOI: 10.1080/03081087.2020.1833823
Kaoutar Ben Ouidren 1 , Hassan Zariouh 2
Affiliation  

ABSTRACT

In this paper, we study the properties (bz) and (wπ00a), which we had introduced in [Ben Ouidren K, Zariouh H. New approach to a-Weyl's theorem and some preservation results. Rend Circ Mat Palermo. doi: 10.1007/s12215-020-00525-2], for an operator having the SVEP on the complementary of distinguished parts of its spectrum. We prove in particular, that a bounded linear operator T acting on a Banach space has the SVEP on the complementary of σuf(T) if and only if T possesses property (bz). We also study the stability of these properties under several commuting perturbations, and we prove that if T possesses property (bz) then f(T)+R possesses property (bz) for every Riesz operator R commuting with T and fHol(σ(T)). We also give an example which shows that the property (wπ00a) is generally unstable under this type of perturbations, and we prove that if T possesses property (wπ00a) then T + R possesses property (wπ00a)π00a(T+R)σa(T)π00a(T). Analogous results are proved for the property (gwπ00a) and some applications to the class of a-isoloid-type operators are given.



中文翻译:

通过局部 SVEP 和 Riesz 型扰动实现 a-Weyl 定理的新方法

摘要

在本文中,我们研究了属性(bz)(wπ00一个),我们在 [Ben Ouidren K, Zariouh H. 中介绍了 a-Weyl 定理的新方法和一些保存结果。撕裂 Circ Mat 巴勒莫。doi: 10.1007/s12215-020-00525-2],适用于在其频谱的显着部分互补上具有 SVEP 的运营商。我们特别证明了作用于 Banach 空间的有界线性算子T的 SVEP 在σF()当且仅当T拥有财产(bz).我们还研究了这些性质在几种通勤扰动下的稳定性,并证明了如果T具有性质(bz)然后F()+R拥有财产(bz)对于每个 Riesz 算子RTF霍尔(σ()).我们还举了一个例子来说明属性(wπ00一个)在这种类型的扰动下通常是不稳定的,我们证明如果T具有性质(wπ00一个)那么T  +  R拥有属性(wπ00一个)π00一个(+R)σ一个()π00一个().类似的结果也证明了性质(Gwπ00一个)并给出了a-isoroid-type算子类的一些应用。

更新日期:2020-10-22
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