A note on numerical range and product of matrices

Let A be any n×n complex matrix. Let σ(A) and W(A) denote respectively the spectrum and the numerical range of A. It is well-known that if A is a multiple of a positive semidefinite matrix, then σ(AB)⊆W(A)W(B) for all n×n complex matrices B. We show that the converse is also true.