Maps preserving maximal numerical range of operator products

Abstract

Let H be a complex Hilbert space and denote by B(H) the algebra of all linear bounded operators on H. For any T ∈ B(H), denote by V0(T) its maximal numerical range. We prove that if ϕ : B(H) −→ B(H) is a surjective map such that V0(ϕ(T)ϕ(S) = V0(T S)) (resp. V0(ϕ(T)ϕ(S) ∗) = V0(T S∗)) for any T, S ∈ B(H) then there exists a unitary operator U ∈ B(H) such that ϕ(T) = ±UT U∗ (resp. ϕ(T) = λUT U∗ with |λ| = 1) for  any T ∈ B(H).