On the Drazin inverse of the linear combinations of two idempotents in a complex Banach algebra

Let AA be a complex Banach algebra with unit and let p,qp,q be two idempotents in AA. For α,β∈C\{0}α,β∈C\{0}, the authors obtain an explicit representation formula for the Drazin inverse of αp+βqαp+βq and give the upper bound of the corresponding Drazin index by using the presented method. As a consequence, the main previously published results on Drazin inverse of αp+βqαp+βq are corollaries of our theorem. Finally, the authors apply the theorem to an example which cannot be solved by previously known results.