Additive property of pseudo Drazin inverse of elements

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 $ab = \lambda ba$,
we also show that $a -b$ is pseudo Drazin invertible if and only if $aa^\ddag (a-b)bb^\ddag$ is pseudo
Drazin invertible.