B-Fredholm and Drazin invertible operators through localized SVEP

TL;DR

This paper explores the relationships between various spectra of bounded linear operators on Banach spaces, establishing equalities up to the set where the operator lacks the single-valued extension property, and applies these results to generalized Weyl's theorem.

Contribution

It proves the equality of the left Drazin spectrum with the left B-Fredholm spectrum and the semi-essential approximate point spectrum with the left Drazin spectrum, up to the set where SVEP fails.

Findings

Equality of left Drazin and B-Fredholm spectra up to S(T)

Equality of semi-essential approximate point spectrum and left Drazin spectrum up to S(T)

Applications to generalized Weyl's theorem for operator matrices

Abstract

Let $X$ a Banach space and $T$ a bounded linear operator on $X.$ We denote by $S(T)$ the set of all $\lambda \in \cit$ such that $T$ does not have the single-valued extension property at $\lambda$. In this note we prove equality up to $S(T)$ between the left Drazin spectrum and the left B-Fredholm spectrum and between the semi-essential approximate point spectrum and the left Drazin spectrum. As applications we investigate generalized Weyl's theorem for operator matrices and multipliers operators.