A van der Corput-type lemma for power bounded operators

We prove a van der Corput-type lemma for power bounded Hilbert space operators. As a corollary we show that N-1∑n=1NTp(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^{-1}\sum _{n=1}^N T^{p(n)}$$\end{document} converges in the strong operator topology for all power bounded Hilbert space operators T and all polynomials p satisfying p(N0)⊂N0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbb {N}_0)\subset \mathbb {N}_0$$\end{document}. This generalizes known results for Hilbert space contractions. Similar results are true also for bounded strongly continuous semigroups of operators.