  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
$N^{-1}\sum_{n=1}^N e^{2\pi i g(n)}T^{a_n}$ 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.
