Nonlinear maps preserving asymptotic equivalence on B(X)

Abstract

Let X be an infinite-dimensional complex Banach space. By B(X) we denote the algebra of all bounded linear operators on X. It is shown that if Φ : B(X) → B(X) is a surjective map satisfying A + B is asymptotically equivalent to C if and only if Φ(A) + Φ(B) is asymptotically equivalent to Φ(C), then either there exist bijective continuous linear or conjugate-linear maps T, S : X → X such that Φ(A) = TAS for every A ∈ B(X), or there exist bijective continuous linear or conjugate-linear maps T : X* → X and S : X → X * such that Φ(A) = TA* S for every A ∈ B(X).