Additive property of pseudo Drazin inverse of elements in a Banach algebra
Huihui Zhu, Jianlong Chen · arXiv (Cornell University) · 2013
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 $1+a^‡b$ is pseudo Drazin invertible. Moreover, the formula of $(a+b)^‡$ is presented . When the commutative condition is weaken to $ab=λba ~(λ eq 0)$, we also show that $a-b$ is pseudo Drazin invertible if and only if $aa^‡(a-b)bb^‡$ is pseudo Drazin invertible.