Maps preserving the local spectrum of skew-product of operators

Let H and H be infinite dimensional complex Hilbert spaces and let B(H) be the algebra of all bounded linear operators on H. Let sigma(T)(h) denote the local spectrum of an operator T is an element of B(H) at any vector h is an element of H, and fix two nonzero vectors h(0) is an element of H and k(0) is an element of H. We show that if a map phi : B(H) -> B(K) has a range containing all operators of rank at most two and satisfies sigma(TS*) (h(0)) = sigma(phi)(T)phi(S)* (k(0)) for all T, S is an element of B(H), then there exist two unitary operators U and V in B(H,K) such that Uh(0) = alpha k(0) for some nonzero alpha is an element of C and phi (T) = UTV* for all T is an element of B(H). We also described maps phi : B(H) -> B(K) satisfying sigma(TS*T)(h(0)) = sigma phi(T) phi(S)*phi(T) (k(0)) for all T, S is an element of B(H), and with the range containing all operators of rank at most four. (C) 2015 Elsevier Inc. All rights reserved.