Maps preserving the spectrum of generalized Jordan product of operators

Let A1,A2A1,A2 be standard operator algebras on complex Banach spaces X1,X2X1,X2, respectively. For k⩾2k⩾2, let (i1,…,im)(i1,…,im) be a sequence with terms chosen from {1,…,k}{1,…,k}, and define the generalized Jordan productT1∘⋯∘Tk=Ti1⋯Tim+Tim⋯Ti1on elements in AiAi. This includes the usual Jordan product A1∘A2=A1A2+A2A1A1∘A2=A1A2+A2A1, and the triple {A1,A2,A3}=A1A2A3+A3A2A1{A1,A2,A3}=A1A2A3+A3A2A1. Assume that at least one of the terms in (i1,…,im)(i1,…,im) appears exactly once. Let a map Φ:A1→A2Φ:A1→A2 satisfy thatσ(Φ(A1)∘⋯∘Φ(Ak))=σ(A1∘⋯∘Ak)σ(Φ(A1)∘⋯∘Φ(Ak))=σ(A1∘⋯∘Ak)whenever any one of A1,…,AkA1,…,Ak has rank at most one. It is shown in this paper that if the range of ΦΦ contains all operators of rank at most three, then ΦΦ must be a Jordan isomorphism multiplied by an mmth root of unity. Similar results for maps between self-adjoint operators acting on Hilbert spaces are also obtained.