Acta Mathematica Vietnamica
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A Generalization of Strongly Preserver Problems of Drazin Invertibility
Mourad Oudghiri
,
Khalid Souilah
Abstract
Let $\phi$ be an additive map between unital complex Banach algebras such that $\phi(1)$ is invertible. We show that $\phi$ satisfies $\phi(a^D)\phi(b)^D = \phi(a)^D\phi(b^D)$ for every Drazin invertible elements $a, b$ if and only if $\phi(1)^{− 1}\phi$ is a Jordan homomorphism and $\phi(1)$ commutes with the range of $\phi$. 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.