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

TL;DR

This paper introduces the gπ-Hirano inverse in Banach algebras, explores its existence, properties, and relationships, and applies these findings to anti-triangular matrices, highlighting differences from classical inverses.

Contribution

It defines a new type of generalized inverse, the gπ-Hirano inverse, and investigates its properties and applications in Banach algebras, extending classical inverse concepts.

Findings

Established existence criteria for the gπ-Hirano inverse.

Analyzed the relationship between invertibility of elements and their sums.

Provided characterizations for the inverse of anti-triangular matrices.

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.In particular, some results in this paper are different from the corresponding ones of classical generalized inverses, such as Drazin inverse and generalized Drazin inverse.