Continuous surjectivemaps preserving projections of Jordan products on the space of self-adjoint operators

Let B-s(H)be the real linear space of all self-adjoint operators on a complex Hilbert space H with dim H >= 3. It is proved that a continuous surjective map phi on B-s(H) preserves nonzero projections of Jordan products of two operators in both directions if and only if there exist a unitary or an anti-unitary operator U on H and a constant lambda with lambda(2) = 1 such that phi(A) = lambda U *AU for all A is an element of B-s(H).