Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self‐Adjoint Operators

Let A1 and A2 be standard real Jordan algebras of self‐adjoint operators on complex Hilbert spaces H1 and H2, respectively. For k ≥ 2, let (i1, …, im) be a fixed sequence with i1, …, im∈{1, …, k} and assume that at least one of the terms in (i1, …, im) appears exactly once. Define the generalized Jordan product T1∘T2∘⋯∘Tk=Ti1Ti2⋯Tim+Tim⋯Ti2Ti1 on elements in Ai. Let Φ:A1→A2 be a map with the range containing all rank‐one projections and trace zero‐rank two self‐adjoint operators. We show that Φ satisfies that σπ(Φ(A1)∘⋯∘Φ(Ak)) = σπ(A1∘⋯∘Ak) for all A1, …, Ak, where σπ(A) stands for the peripheral spectrum of A, if and only if there exist a scalar c ∈ {−1,1} and a unitary operator U : H1 → H2 such that Φ(A) = cUAU* for all A∈A1, or Φ(A) = cUAtU* for all A∈A1, where At is the transpose of A for an arbitrarily fixed orthonormal basis of H1. Moreover, c = 1 whenever m is odd.