Additivity of Jordan multiplicative maps on Jordan operator algebras

Let H be a Hilbert space and N a nest in H. Denote by S-a(H) the Jordan ring of all self-adjoint operators on H and AlgN the nest algebra associated to M. We show that a bijective map Phi: S-a(H) -> S-a(H) satisfying (1) Phi(ABA) = Phi(A)Phi(B)Phi(A) for every pair of A, B, or (2) Phi(AB + BA) = Phi(A)Phi(B) + Phi(B)Phi(A) for every pair of A, B, or (3) Phi(1/2(AB + BA)) = 1/2(Phi(A)Phi(B)) + Phi(B)Phi(A)) for every pair of A, B must be additive, that is, a Jordan ring isomorphism. We also show that if a bijective map Phi : AlgN AlgN satisfies the Jordan multiplicativity of the form (2) or (3), then (P must be a Jordan isomorphism. Moreover, such Jordan multiplicative maps are characterized completely.