Pseudo core invertibility in Banach *-algebras and its applications

TL;DR

This paper investigates the conditions under which the sum of two pseudo core invertible elements in a Banach *-algebra remains pseudo core invertible, extending previous results and applying them to block matrices.

Contribution

It provides new additive results and necessary and sufficient conditions for pseudo core invertibility in Banach *-algebras, including applications to block matrices.

Findings

Derived new additive results for pseudo core invertibility.

Established necessary and sufficient conditions for sums of pseudo core invertible elements.

Extended previous work on pseudo core invertibility to broader contexts.

Abstract

We present new additive results for the pseudo core inverse in a Banach algebra with involution. The necessary and sufficient conditions under which the sum of two pseudo core invertible elements in Banach *-algebra is pseudo core invertible are obtained. As an application, the pseudo core invertibility for block complex matrices is investigated. These extend the main results of pseudo core invertibility of Gao and Chen [Comm. Algebra, 46(2018), 38--50].