Generalized Drazin invertible elements relative to a regularity.

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 $ \mathcal {A} $ A is generalized Drazin invertible relative to a regularity $ \mathcal {R} $ R if there is $ b\in \mathcal {A} $ b ∈ A such that $ ab=ba,\ bab=b $ ab = ba , bab = b and $ \sigma _{\mathcal {R}}(a-aba)\subset \{0\} $ σ R (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 $ \mathcal {R} $ R which satisfies two properties: (D1) if $ a, b\in \mathcal {R} $ a , b ∈ R , p is an idempotent commuting with a and b, then $ ap+b(1-p)\in \mathcal {R} $ ap + b (1 − p) ∈ R ; (D2) if $ a\in \mathcal {R} $ a ∈ R , then a is almost invertible. If a regularity $ \mathcal {R} $ R satisfies the properties (D1) and (D2), we prove that $ a\in \mathcal {A} $ a ∈ A is generalized Drazin invertible relative to $ \mathcal {R} $ R if and only if 0 is not an accumulation point of $ \sigma _{\mathcal {R}}(a) $ σ 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 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 $ \mathcal {R} $ R in the case when $ \mathcal {R} $ R is the set of Drazin invertibles, as well as when $ \mathcal {R} $ R is the set of Koliha-Drazin invertibles. [ABSTRACT FROM AUTHOR]