On the property $(Z_{E_a})$

TL;DR

This paper introduces and characterizes new spectral properties $(Z_{ ext{E}_a})$ and $(Z_{ ext{Pi}_a})$ for bounded linear operators on Banach spaces, exploring their stability under Riesz operator perturbations.

Contribution

It defines the properties $(Z_{ ext{E}_a})$ and $(Z_{ ext{Pi}_a})$, provides their characterization via localized single valued extension property, and studies their behavior under commuting Riesz operator perturbations.

Findings

Characterization of $(Z_{E_a})$ and $(Z_{ ext{Pi}_a})$ properties.

Analysis of stability under Riesz operator perturbations.

Illustrative examples of operator classes.

Abstract

The paper introduces the notion of properties $(Z_{\Pi_a})$ and $(Z_{E_a})$ as variants of Weyl's theorem and Browder's theorem for bounded linear operators acting on infinite dimensional Banach spaces. A characterization of these properties in terms of localized single valued extension property is given, and the perturbation by commuting Riesz operators is also studied. Classes of operators are considered as illustrating examples.