Linear maps preserving equivalence or asymptotic equivalence on Banach space

Let X be a complex Banach space with dimension at least two and B(X) the algebra of all bounded linear operators on X . We show that a bijective linear map Phi preserves asymptotic equivalence if and only if it preserves equivalence, and in turn, if and only if there exist invertible bounded linear operators T and S such that either Phi(A) = TAS Phi or Phi(A) = TA(& lowast;)S for all A is an element of B(X)