About the uniqueness of generalized Drazin $T$-Riesz inverses

Abstract

For two complex unital Banach algebras $\mathcal{A}$ and $\mathcal{B}$ and a homomorphism algebra $T: \mathcal{A} \rightarrow \mathcal{B}$ having the strong Riesz property, we propose a new formula for generalized Drazin $T$-Riesz inverses, and we give a new Laurent expansion for the resolvent of a given generalized Drazin $T$-Riesz invertible element. Most importantly, necessary and sufficent conditions to have the uniqueness of generalized Drazin $T$-Riesz inverse are given.