Abstract
Let H and K be two infinite-dimensional complex Hilbert spaces, and fix two nonzero vectors h(0) is an element of H and k(0) is an element of K. Let L(H) (resp. L(K)) denote the algebra of all bounded linear operators on H (resp. on K), and let F-2(K) be the set of all operators in L(K) of rank at most two. We show that a map. from L(H) into L(K) such that its range contains F-2(K) satisfies
sigma(phi(T)phi(S)*+phi(S)*phi(T))(k(0)) = sigma(TS*+S*T)(h(0)), (T, S is an element of L(H)),
if and only if there exist a unitary operator U from H into K and a scalar alpha is an element of C such that Uh(0) = alpha k(0) and phi(T) = lambda UTU* for all T is an element of L(H), where lambda is a scalar of modulus 1.