\(\Delta _{{{\varvec{i}}}_{g}}\)-invertible operators I

Abstract

Inspired by generalized invertibility, Drazin invertibility and some recent works (Lahmar and Skhiri in Pseudo-generalized inverse I, 2022; Pseudo-generalized inverse II, 2022; Results Math 78:24, 2023; Acta Sci. Math. (Szeged) 89:389–411, 2023), this paper explores a novel class of operators called \(\Delta _{{{\varvec{i}}}_{g}}\)-invertible operators including all the mentioned concepts. Due to this extension, we successfully introduce a new concept of inverse that extends various inverse concepts, such as Drazin inverse, group inverse, Moore-Penrose inverse and DPG inverses, establishing a unified framework for all these concepts. We discuss several properties of this new inverse such as its uniqueness and outer inverse nature. In the context of Hilbert space, we present a specific case in which several properties of the Moore-Penrose inverse remain valid. As an application of our new concept, we prove various properties and perturbation results related to the pseudo-generalized invertibility introduced in (Lahmar and Skhiri in Pseudo-generalized inverse I, 2022).