2024/05/14 by Rounak Biswas, Biswas, Rounak, Falguni Roy +1
Computer Science · Mathematics · #15A09 #17C27 #32A65 #47A10 #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2405.08615
openalex publication_date 2024/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a complex unital Banach algebra A. For x1,x2,x3\inA, in this paper, we establish that under certain assumptions on x1,x2,x3, Drazin (resp. g-Drazin) invertibility of any three elements among x1,x2,x3 and x1+x2+x3 (or x1x2+x1x3+x2x3) ensure the Drazin (resp. g-Drazin) invertibility of the remaining one. As a consequence for two idempotents p,q\inA, this result indicates the equivalence between Drazin (resp. g-Drazin) invertibility of λ1p+γ1q-λ1pq+λ2(pqp-(pq)2)+⋯+λm((pq)m-1p-(pq)m) and λ1-λ1pq+λ2(pqp-(pq)2)+⋯+λm((pq)m-1p-(pq)m), where γ1,λi∈ℂ for i=1,2,⋯,m, with λ1γ1≠0. Furthermore, for x1,x2, we establish that the Drazin (resp. g-Drazin) invertibility of any two elements among x1,x2 and x1+x2 indicates the Drazin (resp. g-Drazin) invertibility of the remaining one, provided that x1x2=α(x1+x2) for some α∈ℂ. Additionally, if it exists, we furnish a new formula to represent the Drazin (resp. g-Drazin) inverse of any element among x1,x2 and x1+x2, by using the other two elements and their Drazin (resp. g-Drazin) inverse.