Linear maps preserving equivalence or asymptotic equivalence on Banach space

Abstract

Abstract Let X X be a complex Banach space with dimension at least two and B ( X ) B\left(X) the algebra of all bounded linear operators on X X . We show that a bijective linear map Φ \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 T and S S such that either Φ ( A ) = T A S \Phi \left(A)=TAS or Φ ( A ) = T A * S \Phi \left(A)=T{A}^{* }S for all A ∈ B ( X ) A\in B\left(X) .