Maps preserving the spectrum of generalized Jordan product of operators

Let A(1), A(2) be standard operator algebras on complex Banach spaces X(1),X(2), respectively, For k \u3e= 2, let (i(1),...,i(m)) be a sequence with terms chosen from {1,..., k}, and define the generalized Jordan product T(1) o ... o T(k) = T(i1) ... T(im) + T(im) ... T(i1) on elements in A(i). This includes the usual Jordan product A(1) o A(2) = A(1)A(2) + A(2)A(1), and the triple {A(1), A(2), A(3)} = A(1)A(2)A(3) + A(3)A(2)A(1). AS-some that at least one of the terms in (i(1),....,i(m)) appears exactly once. Let a map Phi : A(1) -\u3e A(2) satisfy that s (Phi (A(1)) o ... o Phi (A(k))) = sigma (A(1) o ... o A(k)), whenever any one of A(1),...,A(k) has rank at most one. It is shown in this paper that if the range of 0 contains all operators of rank at most three, then (P 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. (C) 2009 Elsevier Inc. All rights reserved