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 $x_1+x_2+x_3\text{ }(\text{or }x_1x_2+x_1x_3+x_2x_3)$ 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 $$\lambda_1p+\gamma_1q-\lambda_1pq+\lambda_2\left(pqp-(pq)^2\right)+\cdots+\lambda_m\left((pq)^{m-1}p-(pq)^m\right)$$ and $$\lambda_1-\lambda_1pq+\lambda_2\left(pqp-(pq)^2\right)+\cdots+\lambda_m\left((pq)^{m-1}p-(pq)^m\right),$$ where $\gamma_1,\lambda_i\in\mathbb{C}$ for $i=1,2,\cdots,m,$ with $\lambda_1\gamma_1\neq0.$ 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 $x_1x_2=\alpha(x_1+x_2)$ 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.