Banach algebra mappings preserving the invertibility of linear pencils

Let A and B be complex unital Banach algebras, and let phi, 0: A -+ B be surjective mappings. If A is semisimple with an essential socle and phi and 0 together preserve the invertibility of linear pencils in both directions, that is, for any x, y is an element of A and lambda is an element of C, lambda x + y is invertible in A if and only if lambda phi(x) + 0(y) is invertible in B, then we show that there exists an invertible element u in B and a Jordan isomorphism J : A -+ B such that phi(x) = 0(x) = uJ(x) for all x is an element of A. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).