Numerical radius inequality for C0 contractions

We show that if A is a C0 contraction with minimal function ϕ such that w(A)=w(S(ϕ)), where w(·) denotes the numerical radius of an operator and S(ϕ) is the compression of the shift on H2⊖ϕH2, and B commutes with A, then w(AB)⩽w(A)‖B‖. This is in contrast to the known fact that if A=S(ϕ) (even on a finite-dimensional space) and B commutes with A, then w(AB)⩽‖A‖w(B) is not necessarily true. As a consequence, we have w(AB)⩽w(A)‖B‖ for any quadratic operator A and any B commuting with A.