Linear maps preserving equivalence or asymptotic equivalence on Banach space

Abstract

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

Keywords

• linear preservers
• asymptotic equivalence
• equivalence relations
• banach spaces
• 47b49