Maps preserving peripheral spectrum of generalized Jordan products of operators

Abstract

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 $${T_1} \circ {T_2} \circ \cdots \circ {T_k} = {T_{{i_1}}}{T_{{i_2}}} \cdots {T_{{i_m}}} + {T_{{i_m}}} \cdots {T_{{i_2}}}{T_{{i_1}}}$$ 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 Φ: A 1 → A 2 be a map with range containing all operators of rank at most three. It is shown that Φ satisfies that σ π (Φ(A 1) ○ · · · ○ Φ(A k )) = σ π (A1 ○ ··· ○ A k ) for all A 1,..., A k , where σ π (A) stands for the peripheral spectrum of A, if and only if Φ is a Jordan isomorphism multiplied by an m-th root of unity.