Maps preserving the spectrum of generalized Jordan product of operators

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