On the closed generalized Drazin-Riesz invertible operators and C₀-semigroups

This paper is a continuation of our paper [Med. J. Math 19 (2022) 31] in which we extended the notion of generalized Drazin–Riesz invertible operators to closed operators. We establish here, results relating the notion of closed generalized Drazin–Riesz invertibility with the theory of [Formula: see text]-semigroups. First, we generalize results obtained in the bounded case [Abad and Zguitti, Ann. Funct. Anal. 12 (2021) 55] to the context of closed operators. Second, we investigate when an infinitesimal generator (generator) [Formula: see text] of a given [Formula: see text]-semigroup is closed generalized Drazin–Riesz invertible. An application to [Formula: see text]-groups and abstract second-order differential equations is proposed, and an example of a [Formula: see text]-group with closed generalized Drazin–Riesz invertible generator is given.