On the class $(W_{e})$-operators

TL;DR

This paper introduces and studies a new class of operators, $(W_{e})$, characterized by a specific relation between their essential spectrum and spectrum, extending the theory of spectral properties in operator analysis.

Contribution

It defines the $(W_{e})$ class of operators, explores conditions for hyponormal operators to belong to this class, and generalizes to the $(gW_{e})$ class within B-Fredholm theory.

Findings

Hyponormal operators generally do not belong to $(W_{e})$.

Additional conditions ensure hyponormal operators are in $(W_{e})$.

Characterization of $(B_{e})$ via localized SVEP.

Abstract

It is well known that an hyponormal operator satisfies Weyl's theorem. A result due to Conway shows that the essential spectrum of a normal operator $N$ consists precisely of all points in its spectrum except the isolated eigenvalues of finite multiplicity, that's $\sigma_{e}(N)=\sigma(N)\setminus E^0(N).$ In this paper, we define and study a new class named $(W_{e})$ of operators satisfying $\sigma_{e}(T)=\sigma(T)\setminus E^0(T),$ as a subclass of $(W).$ A countrexample shows generally that an hyponormal does not belong to the class $(W_{e}),$ and we give an additional hypothesis under which an hyponormal belongs to the class $(W_{e}).$ We also give the generalisation class $(gW_{e})$ in the contexte of B-Fredholm theory, and we characterize $(B_{e}),$ as a subclass of $(B),$ in terms of localized SVEP.