Generalized Drazin-Riesz invertible elements in a semi-simple Banach algebra

Abstract

We extend the notion of generalized Drazin-Riesz inverse introduced for bounded linear operators in Živković-Zlatanović and Cvetković (Linear Multilinear Algebra 65(6):1171–1193, 2017) to elements in a complex unital semi-simple Banach algebra. Several characterizations and properties of generalized Drazin-Riesz invertible elements are given. In particular, we extend those of Abad and Zguitti (Ann Funct Anal 12:Article number: 55, 2021), Djordjević and Stanimirovic (Czech Math J 51(3):617–634, 2001) and Živković-Zlatanović and Cvetković (2017).