Let H be an infinite-dimensional complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. The characteristics of a linear surjective map φ on B(H) preserving asymptotic unitary similarity in both directions are studied. By giving the definition of minimal asymptotic unitary similarity-invariant subspace and applying the closure of the unitary orbit of a linear operator, it is proved that the map φ has the following representation: φ(X)=cU*XU (X∈B(H)) or φ(X)=cU*XTU( X∈B(H)), where c is a non-zero constant, U∈B(H) is a unitary operator, and XT denotes the transpose of X relative to a given orthonormal basis for H.

Hilbert space; asymptotic unitary similarity; linear map; minimal asymptotic u-similarity; invariant subspace; unitary orbit