Numerical radius inequality for C₀ contractions

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