Optimal rates of decay in the Katznelson-Tzafriri theorem for operators on Hilbert spaces

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 ||Tn(I−T)‖→0 if and only if σ(T)∩T⊆{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(eiθ,T)‖ as |θ|→0 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.