Schur product of matrices and numerical radius (range) preserving maps

Let F(A) be the numerical range or the numerical radius of a square matrix A. Denote by A ∘ B the Schur product of two matrices A and B. Characterizations are given for mappings on square matrices satisfying F(A ∘ B) = F(ϕ(A) ∘ ϕ(B)) for all matrices A and B. Analogous results are obtained for mappings on Hermitian matrices.