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 $N^{-1}{\sum }_{n=1}^N{T}^{a_n}$ converge in the strong operator topology for a wide range of sequences $(a_n)$, including the integer part of most of subpolynomial Hardy functions. Moreover, we show that the weighted averages $N^{-1}{\sum }_{n=1}^N{e}^{2\pi ig(n)}{T}^{a_n}$ 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 \cite{tEM} to real polynomials and polynomial weights.

24 pages; small changes, referee's comments incorporated (in particular, Example 3.4 (b) added)