Maps preserving the local reduced minimum modulus

Abstract

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H. For any operator T∈B(H) and unit vector x∈H, let γ(T,x) denote the local reduced minimum modulus of T at x. In this paper, we characterize surjective maps on B(H) such that, for all S,T∈B(H) and all unit vectors x∈H, one hasγ(T−S,x)=0 if and only if γ(Φ(T)−Φ(S),x)=0. We also describe the form of all maps on B(H) preserving the product of operators of zero local reduced minimum moduli. Furthermore, some consequences and open problems are also discussed.