Additive maps preserving similarity or asymptotic similarity on

Let H be an infinite-dimensional complex separable Hilbert space and denote the algebra of all bounded linear operators acting on H. We show that an additive continuous surjective map Φ on is asymptotic similarity preserving if and only if it is similarity preserving, and in turn, if and only if there exist a scalar c and an invertible bounded linear or conjugate linear operator A on H such that either Φ(T)=cATA −1 for all T or Φ (T)=cAT*A −1 for all T.