On Koliha-Drazin invertible operators and Browder type theorems

TL;DR

This paper establishes new necessary and sufficient spectral conditions for Koliha-Drazin and Drazin invertibility of bounded operators, and explores their relation to Browder's theorems, enhancing understanding of spectral decompositions.

Contribution

It introduces novel spectral decomposition criteria for invertibility and analyzes operators satisfying Browder's theorems through spectral part relationships.

Findings

New spectral conditions for Koliha-Drazin invertibility

Characterization of operators satisfying Browder's theorems

Connections between spectral parts and invertibility properties

Abstract

Let $T$ be a bounded linear operator on a Banach space $X$. We give new necessary and sufficient conditions for $T$ to be Drazin or Koliha-Drazin invertible. All those conditions have the following form: $T$ possesses certain decomposition property and zero is not an interior point of some part of the spectrum of $T$. In addition, we study operators $T$ satisfying Browder\textquoteright s theorem, or a-Browder\textquoteright s theorem, by means of some relationships between diferent parts of the spectrum of $T$.