Generalized Drazin invertible elements relative to a regularity
Snežana Č. Živković-Zlatanović
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 A is generalized Drazin invertible relative to a regularity R if there is b∈A such that ab=ba, bab=b and σR(a−aba)⊂{0}. 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 R which satisfies two properties: (D1) if a,b∈R, p is an idempotent commuting with a and b, then ap+b(1−p)∈R; (D2) if a∈R, then a is almost invertible. If a regularity R satisfies the properties (D1) and (D2), we prove that a∈A is generalized Drazin invertible relative to R if and only if 0 is not an accumulation point of σR(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 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 R in the case when R is the set of Drazin invertibles, as well as when R is the set of Koliha-Drazin invertibles.