The authors derive Markov-Bernstein type inequalities of a very general form. They show that such inequalities hold in weighted \(L^2\)-spaces not only for derivatives but also for any linear operator on \(P\), the space of polynomials with complex coefficients, when the measure is a positive Borel measure. Specifically, for any linear operator \(T\) on the space \(P\) and any integer \(n\geq 0\), there is a positive constant \(\gamma_n(T)\), independent of \(p(x)\), such that \(\| Tp\| \leq\gamma_n(T) \| p\|\) where \(\deg(P) \leq n\). Here, \(\| p\|: =\{\int^{+\infty}_{-\infty} | p(x)|^2 d\mu (x)\}^{1/2}\) and \(\mu(x)\) is an increasing function on the line with finite moments of all orders. They find the optimal value \(\Gamma_n(T)\) for \(\gamma_n(T)\) and estimate its value. Examples are given when \(T\) is a differentiation or difference operator and the measure is orthogonalized for classical or discrete orthogonal polynomials.