ps-Drazin Inverses in Banach Algebras

Abstract

An element $a$ in a ring $R$ has ps-Drazin inverse if there exists
$p^2=p\in comm^2(a)$ such that $(a-p)^k\in J(R)$ for some $k\in
{\Bbb N}$. Let $R$ be a ring, and let $a,b\in R$ have ps-Drazin
inverses. If $a^2b=aba$ and $b^2a=bab$, we prove that $ab\in R$
has ps-Drazin inverse and $a+b\in R$ has ps-Drazin inverse if and
only if $1+a^db\in R$ has ps-Drazin inverse. As applications, we
present various conditions under which a $2\times 2$ matrix over a
Banach algebra has ps-Drazin inverse.