We prove a van der Corput-type lemma for power bounded Hilbert space operators. As a corollary we show that N−1∑Nn=1Tp(n)N−1∑n=1NTp(n) converges in the strong operator topology for all power bounded Hilbert space operators T and all polynomials p satisfying p(N0)⊂N0p(N0)⊂N0. This generalizes known results for Hilbert space contractions. Similar results are true also for bounded strongly continuous semigroups of operators.
