ABSTRACT

In this paper, we study the properties $\left(bz\right)$ and $\left(w_{{\pi }_{00}^a}\right),$ 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 ${\sigma }_{uf}\left(T\right)$ if and only if T possesses property $\left(bz\right).$ We also study the stability of these properties under several commuting perturbations, and we prove that if T possesses property $\left(bz\right)$ then $f\left(T\right)+R$ possesses property $\left(bz\right)$ for every Riesz operator R commuting with T and $f\in \text{Hol}\left(\sigma \right(T\left)\right).$ We also give an example which shows that the property $\left(w_{{\pi }_{00}^a}\right)$ is generally unstable under this type of perturbations, and we prove that if T possesses property $\left(w_{{\pi }_{00}^a}\right)$ then T + R possesses property $\left(w_{{\pi }_{00}^a}\right)$ ⇔ ${\pi }_{00}^a(T+R)\cap {\sigma }_a\left(T\right)\subset {\pi }_{00}^a\left(T\right).$ Analogous results are proved for the property $\left(g{w}_{{\pi }_{00}^a}\right)$ and some applications to the class of a-isoloid-type operators are given.

摘要

在本文中，我们研究了属性$\left(bz\right)$和$\left(w_{{\pi }_{00}^{\mathrm{一个}}}\right),$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${\sigma }_{你F}\left(吨\right)$当且仅当T拥有财产$\left(bz\right).$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$\left(bz\right)$然后$F\left(吨\right)+R$拥有财产$\left(bz\right)$对于每个 Riesz 算子R与T和$F\in \text{霍尔}\left(\sigma \right(吨\left)\right).$我们还举了一个例子来说明属性$\left(w_{{\pi }_{00}^{\mathrm{一个}}}\right)$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$\left(w_{{\pi }_{00}^{\mathrm{一个}}}\right)$那么T  +  R拥有属性$\left(w_{{\pi }_{00}^{\mathrm{一个}}}\right)$⇔${\pi }_{00}^{\mathrm{一个}}(吨+R)\cap {\sigma }_{\mathrm{一个}}\left(吨\right)\subset {\pi }_{00}^{\mathrm{一个}}\left(吨\right).$类似的结果也证明了性质$\left(G{w}_{{\pi }_{00}^{\mathrm{一个}}}\right)$并给出了a-isoroid-type算子类的一些应用。