We prove a van der Corput-type lemma for power bounded Hilbert space operators. As a corollary we show that \(N^{-1}\sum _{n=1}^N T^{p(n)}\) converges in the strong operator topology for all power bounded Hilbert space operators T and all polynomials p satisfying \(p(\mathbb {N}_0)\subset \mathbb {N}_0\). This generalizes known results for Hilbert space contractions. Similar results are true also for bounded strongly continuous semigroups of operators.