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arXiv · 2405.08615

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

Rounak Biswas·Falguni Roy
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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 $$λ_1p+γ_1q-λ_1pq+λ_2\left(pqp-(pq)^2\right)+\cdots+λ_m\left((pq)^{m-1}p-(pq)^m\right)$$ and $$λ_1-λ_1pq+λ_2\left(pqp-(pq)^2\right)+\cdots+λ_m\left((pq)^{m-1}p-(pq)^m\right),$$ where $γ_1,λ_i\in\mathbb{C}$ for $i=1,2,\cdots,m,$ with $λ_1γ_1\neq0.$ 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 $x_1x_2=α(x_1+x_2)$ for some $α\in\mathbb{C}$. 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.

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Rounak Biswas, Falguni Roy. 2024-05-14. Drazin and g-Drazin invertibility of combinations of three Banach algebra elements. https://arxiv.org/abs/2405.08615

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