The Core Inverse in a Banach Algebra with Involution

TL;DR

This paper investigates the properties of the core inverse in Banach algebras with involution, providing new additive results, necessary and sufficient conditions for invertibility of sums, and applications to block matrices.

Contribution

It introduces new additive results and conditions for the core inverse in Banach algebras with involution, extending existing theorems to block matrices.

Findings

Necessary and sufficient conditions for the sum of two core invertible elements to be core invertible.

Application of results to block complex matrices.

Generalization of known results in the literature.

Abstract

We present new additive results for the core inverse in a Banach algebra with involution. We obtain necessary and sufficient conditions under which the sum of two core invertible elements in a Banach algebra with involution is core invertible. Then we apply our results to block complex matrices and obtain certain conditions under which a block complex is core invertible. These generalize many known results, e.g.,~\cite[Theorem 4.3]{XCZ}, ~\cite[Theorem 2.5]{XS}.