Let $mu$(x) be an increasing function on the real line with finite moments of all oeders. We show that for any linear operator T on the space of polynomials and any interger n $geq$ 0, there is a constant $gamma n(T)geq0$, independent of p(x), such that $parallel T_pparallelleqgamma n(T)parallel Pparallel$, for any polynomial p(x) of degree $leq$ n, where We find a formular for the best possible value $Gamma_n(T);of;gamma n(T)$ and estimations for $Gamma_n(T)$. We also give several illustrating examples when T is a differentiation or a difference operator and $dmu$(x) is an orthogonalizing measure for classical or discrete orthogonal polynomials.