  The Katznelson-Tzafriri theorem is a central result in the asymptotic theory
of discrete operator semigroups. It states that for a power-bounded operator
$T$ on a Banach space we have $||T^n(I-T)\|\to0$ if and only if
$\sigma(T)\cap\mathbb{T}\subseteq\{1\}$. The main result of the present paper
gives a sharp estimate for the rate at which this decay occurs for operators on
Hilbert space, assuming the growth of the resolvent norms
$\|R(e^{i\theta},T)\|$ as $|\theta|\to0$ satisfies a mild regularity condition.
This significantly extends an earlier result by the second author, which
covered the important case of polynomial resolvent growth. We further show
that, under a natural additional assumption, our condition on the resolvent
growth is not only sufficient but also necessary for the conclusion of our main
result to hold. By considering a suitable class of Toeplitz operators we show
that our theory has natural applications even beyond the setting of normal
operators, for which we in addition obtain a more general result.
