A Generalization of Strongly Preserver Problems of Drazin Invertibility

Abstract

Let ϕ be an additive map between unital complex Banach algebras such that ϕ(1) is invertible. We show that ϕ satisfies ϕ(aD)ϕ(b)D = ϕ(a)Dϕ(bD) for every Drazin invertible elements a, b if and only if ϕ(1)− 1ϕ is a Jordan homomorphism and ϕ(1) commutes with the range of ϕ. A similar result is established for group invertible elements, and more explicit forms of such maps are given in the context of the algebra of all bounded linear operators on a complex Banach space.