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A Generalization of Strongly Preserver Problems of Drazin Invertibility

Mourad Oudghiri , $icon-email$ Khalid Souilah

Abstract

Let $ϕ$ be an additive map between unital complex Banach algebras such that $ϕ (1)$ is invertible. We show that $ϕ$ satisfies $ϕ (a^{D}) ϕ (b)^{D} = ϕ (a)^{D} ϕ (b^{D})$ for every Drazin invertible elements $a, b$ if and only if $ϕ (1)^{- 1} ϕ$ 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.

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A Generalization of Strongly Preserver Problems of Drazin Invertibility

Abstract