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

本文试图给出一种公理化的方法来研究 Banach 代数中 Drazin 可逆性的各种推广。我们将说Banach 代数的元素a$\mathcal{A}$是相对于正则性的广义 Drazin 可逆$\mathcal{R}$如果有$b\in \mathcal{A}$这样$Ab=bA,bAb=b$和${\sigma }_{\mathcal{R}}(A-AbA)\subset \left\{0\right\}$. Koliha-Drazin 可逆元素的概念，以及这个概念的一些推广是通过广义 Drazin 可逆元素相对于正则性的概念来描述的$\mathcal{R}$满足两个属性： (D1) 如果$A,b\in \mathcal{R}$, p是与a和b 的幂等交换，则$Ap+b(\mathrm{1个}-p)\in \mathcal{R}$; (D2) 如果$A\in \mathcal{R}$，那么a几乎是可逆的。如果有规律$\mathcal{R}$满足性质（D1）和（D2），我们证明$A\in \mathcal{A}$是广义的 Drazin 可逆的$\mathcal{R}$当且仅当 0 不是${\sigma }_{\mathcal{R}}\left(A\right)$. 特别地，我们定义和描述了相对于任意（不一定有界）Banach 代数同态T的广义 Drazin- T -Riesz 可逆元素，因此扩展了 [Živković-Zlatanović SČ, Cvetković MD 中引入的广义 Drazin-Riesz 可逆算子的概念。广义 Kato-Riesz 分解和广义 Drazin-Riesz 可逆算子。线性多线性 A. 2017;65(6):1171–1193]。我们还考虑广义 Drazin 可逆相对于$\mathcal{R}$在这种情况下$\mathcal{R}$是 Drazin 可逆集合，以及当$\mathcal{R}$是 Koliha-Drazin 可逆集合。