Additive property of pseudo Drazin inverse of elements in a Banach algebra

TL;DR

This paper investigates the properties of the pseudo Drazin inverse in Banach algebras, establishing conditions under which sums and differences of pseudo Drazin invertible elements are also invertible, and providing explicit formulas.

Contribution

It introduces new criteria for the pseudo Drazin invertibility of sums and differences of elements in Banach algebras, extending previous results to non-commutative cases.

Findings

Sum of pseudo Drazin invertible elements is invertible iff 1 + a^d b is invertible.

Explicit formula for (a + b)^d is derived.

Pseudo Drazin invertibility of a - b under weakened commutativity condition.

Abstract

We study properties of pseudo Drazin inverse in a Banach algebra with unity 1. If $ab=ba$ and $a,b$ are pseudo Drazin invertible, we prove that $a+b$ is pseudo Drazin invertible if and only if $1+a^\ddag b$ is pseudo Drazin invertible. Moreover, the formula of $(a+b)^\ddag$ is presented . When the commutative condition is weaken to $ab=\lambda ba ~(\lambda \neq 0)$, we also show that $a-b$ is pseudo Drazin invertible if and only if $aa^\ddag(a-b)bb^\ddag$ is pseudo Drazin invertible.