Maps preserving peripheral spectrum of generalized Jordan products of operators

Let X 1 and X 2 be complex Banach spaces with dimension at least three, A 1 and A 2 be standard operator algebras on X 1 and X 2, respectively. For k . 2, let (i 1, i 2,..., i m ) be a finite sequence such that {i 1, i 2,..., i m} = {1, 2,..., k} and assume that at least one of the terms in (i 1,..., i m) appears exactly once. Define the generalized Jordan product T1.T2...Tk=Ti1Ti2.Tim+Tim.Ti2Ti1 on elements in A i . This includes the usual Jordan product A 1 A 2 + A 2 A 1, and the Jordan triple A 1 A 2 A 3 + A 3 A 2 A 1. Let .: A 1 . A 2 be a map with range containing all operators of rank at most three. It is shown that . satisfies that . . (.(A 1) . · · · . .(A k )) = . . (A1 . ··· . A k ) for all A 1,..., A k , where . . (A) stands for the peripheral spectrum of A, if and only if . is a Jordan isomorphism multiplied by an m-th root of unity.