Linear maps preserving the minimum and surjectivity moduli of Hilbert space operators

Let B(H) be the algebra of all bounded linear operators on a complex infinite-dimensional Hilbert space H. For every T∈B(H), let m(T) and q(T) denote the minimum modulus and surjectivity modulus of T respectively. Let ϕ:B(H)→B(H) be a surjective linear map. In this paper, we prove that the following assertions are equivalent:(i)m(T)=m(ϕ(T)) for all T∈B(H),(ii)q(T)=q(ϕ(T)) for all T∈B(H),(iii)there exist two unitary operators U,V∈B(H) such that ϕ(T)=UTV for all T∈B(H). This generalizes the result of Mbekhta [7, Theorem 3.1] to the non-unital case.