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Abstract
Let A be a unital complex Banach algebra and p, q be two idempotents in A. In this paper, we prove that the Drazin (resp. generalized Drazin) invertibility of αp+βq−γpq is independent of the choice of scalars α,β∈C∖{0} and γ∈C, we also give a simple representation formula for the Drazin inverse of αp+βq. As an application, we study the left (resp. right) Drazin invertibility and the property of being Browder of linear combinations of two idempotent operators on a Banach space.
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