Generalized Drazin invertible elements relative to a regularity

Abstract

This paper is an attempt to give an axiomatic approach to the investigation of various kinds of generalizations of Drazin invertibility in Banach algebras. We shall say that an element a of a Banach algebra is generalized Drazin invertible relative to a regularity if there is such that and . The concept of Koliha-Drazin invertible elements, as well as some generalizations of this concept are described via the concept of generalized Drazin invertible elements relative to a regularity which satisfies two properties: (D1) if , p is an idempotent commuting with a and b, then ; (D2) if , then a is almost invertible. If a regularity satisfies the properties (D1) and (D2), we prove that is generalized Drazin invertible relative to if and only if 0 is not an accumulation point of . In particular we define and characterize generalized Drazin-T-Riesz invertible elements relative to an arbitrary (not necessarily bounded) Banach algebra homomorphism T and so extend the concept of generalized Drazin-Riesz invertible operators introduced in [Živković-Zlatanović SČ, Cvetković MD. Generalized Kato-Riesz decomposition and generalized Drazin-Riesz invertible operators. Linear Multilinear A. 2017;65(6):1171–1193]. Also we consider generalized Drazin invertibles relative to in the case when is the set of Drazin invertibles, as well as when is the set of Koliha-Drazin invertibles.