Generalized Drazin invertible elements relative to a regularity

This paper is an attempt to give an axiomatic approach to the inves-tigation of various kinds of generalizations of Drazin invertibility in Banach algebras. We shall say that an element a of a Banach alge-bra A is generalized Drazin invertible relative to a regularity R if there is b is an element of A such that ab = ba, bab = b and sigma R(a - aba) subset of {0}. The concept of Koliha-Drazin invertible elements, as well as some generalizations of this concept are described via the concept of gen-eralized Drazin invertible elements relative to a regularity R which satisfies two properties: (D1) if a,b is an element of R, p is an idempotent com-muting with a and b, then ap + b(1 - p) is an element of R; (D2) if a is an element of R, then a is almost invertible. If a regularity R satisfies the properties (D1) and (D2), we prove that a is an element of A is generalized Drazin invertible rel-ative to R if and only if 0 is not an accumulation point of sigma R(a). 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 gen-eralized Drazin-Riesz invertible operators introduced in [Zivkovic-Zlatanovic SC, Cvetkovic MD. Generalized Kato-Riesz decomposition and generalized Drazin-Riesz invertible operators. Linear Multilin-ear A. 2017;65(6):1171-1193]. Also we consider generalized Drazin invertibles relative to R in the case when R is the set of Drazin invertibles, as well as when R is the set of Koliha-Drazin invertibles.