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 T-1 o T-2 o ... o T-k = Ti1Ti2 ...T-im + T-im ... 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 Phi : A(1) -> A(2) be a map with range containing all operators of rank at most three. It is shown that Phi satisfies that sigma(pi) (Phi(A(1)) o ... o Phi(A(k))) = sigma(pi) (A(1) o ... o A(k)) for all A(1),..., A(k), where sigma(pi) (A) stands for the peripheral spectrum of A, if and only if Phi is a Jordan isomorphism multiplied by an m-th root of unity.