Power bounded operators and the mean ergodic theorem for subsequences

Abstract

Let T be a power bounded Hilbert space operator without unimodular eigenvalues. We show that the subsequential ergodic averages N−1∑n=1NTan converge in the strong operator topology for a wide range of sequences (an), including the integer part of most of subpolynomial Hardy functions. Moreover, we show that the weighted averages N−1∑n=1Ne2πig(n)Tan also converge for many reasonable functions g. In particular, we generalize the polynomial mean ergodic theorem for power bounded operators due to ter Elst and the second author [16] to real polynomials and polynomial weights.