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

TL;DR

This paper investigates conditions under which the Drazin and g-Drazin invertibility of certain combinations of three elements in a Banach algebra imply the invertibility of the remaining element, providing new formulas and equivalences.

Contribution

It establishes new invertibility criteria for combinations of three elements in Banach algebras, including explicit formulas for their Drazin and g-Drazin inverses.

Findings

Invertibility of three elements implies the fourth under certain conditions.

Equivalence of invertibility for specific combinations of two idempotents.

New formulas for Drazin and g-Drazin inverses of sums of two 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.