Nonlinear maps preserving similarity on B(H)

Let X be a real or complex Banach space with dimension at least 3, N1(X) be the family of all rank one nilpotent operators. We give the concrete form of every bijective map Φ:N1(X)→N1(X) such that T+S∈N1(X)⇔Φ(T)+Φ(S)∈N1(X). Based on this result, we characterize the surjective map Φ:B(H)→B(H)withT±S∼R⇔Φ(T)±Φ(S)∼Φ(R) for all T,S,R∈B(H), where H is a Hilbert space (real or complex) with dimension at least 3.