On the g$\pi$-Hirano invertibility in Banach algebras

Abstract

In a Banach algebra, we introduce a new type of generalized inverse called g$\pi$-Hirano inverse. Firstly, several existence criteria and the equivalent definition of this inverse are investigated. Then, we discuss the relationship between the g$\pi$-Hirano invertibility of $a$, $b$ and that of the sum $a+b$ under some weaker conditions.
Finally, as applications to the previous additive results, some equivalent characterizations for the g$\pi$-Hirano invertibility of the anti-triangular matrix over Banach algebras are obtained.