The generalized and pseudo $n$-strong Drazin inverse of the sum of elements in Banach algebras

TL;DR

This paper introduces generalized and pseudo n-strong Drazin inverses in Banach algebras, establishing conditions, additive properties, and new variants like weighted inverses, extending recent mathematical results.

Contribution

It provides necessary and sufficient conditions for generalized and pseudo n-strong Drazin invertibility, and generalizes recent results with new weighted inverse concepts.

Findings

Established conditions for g$n$s- and p$n$s-invertibility.

Proved additive properties of these inverses.

Defined and characterized weighted versions of these inverses.

Abstract

In this paper, we begin by introducing some necessary and sufficient conditions for generalized $n$-strong Drazin invertibility (g$n$s-invertibility) and pseudo $n$-strong Drazin invertibility (p$n$s-invertibility) of an element in a Banach algebra for $n\in\mathbb{N}$. Subsequently, these results are utilized to prove some additive properties of g$n$s (p$n$s)-Drazin inverse in a Banach algebra. This process produces a generalization of some recent results of H Chen, M Sheibani (Linear and Multilinear Algebra \textbf{70.1} (2022): 53-65) for g$n$s and p$n$s-Drazin inverse. Furthermore, we define and characterize weighted g$n$s and weighted p$n$s-Drazin inverse in a Banach algebra.