Drazin and g-Drazin invertibility of combinations of three Banach algebra elements

Abstract

Consider a complex unital Banach algebra $\mathcal{A}.$ For $x_1,x_2,x_3\in \mathcal{A},$ in this paper, we establish that under certain assumptions on $x_1,x_2,x_3$, Drazin (resp. g-Drazin) invertibility of any three elements among $x_1,x_2,x_3$ and $x_1+x_2+x_3(\text{or }{x}_1{x}_2+x_1{x}_3+x_2{x}_3)$ ensure the Drazin (resp. g-Drazin) invertibility of the remaining one. As a consequence for two idempotents $p,q\in \mathcal{A},$ this result indicates the equivalence between Drazin (resp. g-Drazin) invertibility of $${\lambda }_{1}p+{\gamma }_{1}q-{\lambda }_{1}pq+{\lambda }_2\left(pqp-(pq{)}^2\right)+\cdots +{\lambda }_m\left((pq{)}^{m-1}p-(pq{)}^m\right)$$ and $${\lambda }_1-{\lambda }_{1}pq+{\lambda }_2\left(pqp-(pq{)}^2\right)+\cdots +{\lambda }_m\left((pq{)}^{m-1}p-(pq{)}^m\right),$$ where ${\gamma }_1,{\lambda }_i\in \mathbb{C}$ for $i=1,2,\cdots ,m,$ with ${\lambda }_1{\gamma }_1\ne 0.$ Furthermore, for $x_1,x_2$, we establish that the Drazin (resp. g-Drazin) invertibility of any two elements among $x_1,x_2$ and $x_1+x_2$ indicates the Drazin (resp. g-Drazin) invertibility of the remaining one, provided that $x_1{x}_2=\alpha (x_1+x_2)$ for some $\alpha \in \mathbb{C}$. Additionally, if it exists, we furnish a new formula to represent the Drazin (resp. g-Drazin) inverse of any element among $x_1,x_2$ and $x_1+x_2$, by using the other two elements and their Drazin (resp. g-Drazin) inverse.