Additive and multiplicative properties of Drazin inverse under new weakly commutativity condition

Given a complex Banach space $X$, let $\mathcal{B}(X)$ be the collection of all bounded linear operators on $X\mathrm{.}$ For $A,B\in \mathcal{B}(X)$ we define $A,B$ are $A$-weakly commutative if there exists $C\in \mathcal{B}(X)$ satisfying $$AB=CA\text{ and }BA=AC\mathrm{.}$$ The objective of this paper is to study the Drazin invertibility of $A+B$ and $AB$, when $A,B\in \mathcal{B}(X{)}^D$ are $A\text{ and }B$-weakly commutative. Consequently, this extends some of the additive and multiplicative results established by Huanyin and Marjan Sheibani (Linear and Multilinear Algebra $\mathbf{\text{70.1}}$ (2022): 53-65) for a distinct family of elements. Moreover, we also establish these additive and multiplicative properties for g-Drazin inverses in a complex Banach algebra.