Banach algebra mappings preserving the invertibility of linear pencils

Abstract

Let $A$ and $B$ be complex unital Banach algebras, and let $\varphi, \psi: A \to B$ be surjective mappings. If $A$ is semisimple with an essential socle and $\varphi$ and $\psi$ preserves the invertibility of linear pencils in both directions, that is, for any $x, y \in A$ and $\lambda \in \mathbb{C}$, $\lambda x+y$ is invertible in $A$ if and only if $\lambda \varphi(x) + \psi(y)$ is invertible in $B$, then we show that there exists an invertible element $u$ in $B$ and a Jordan isomorphism $J: A \to B$ such that $\varphi(x) = \psi(x) = uJ(x)$ for all $x \in A$.