Equivalency of Drazin and g-Drazin invertibility of elements in a Banach algebra

Abstract

Consider a complex unital Banach algebra A. For x1,x2,x3?A, 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(orx1x2+x1x3+x2x3) ensure the Drazin (resp. g-Drazin) invertibility of the remaining one. As a consequence for two idempotents p,q?A, this result indicates the equivalence between Drazin (resp. g-Drazin) invertibility of (Formula presented.) and (Formula presented.) where ?1,?i?C for i=1,2,?,m, with ?1?1?0; which extend the work of Barraa and Benabdi [1]. 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=?(x1+x2) for some ??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. © The Indian National Science Academy 2025.