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 Ton a Banach space we have parallel to T-n(I - T)parallel to -> 0 if and only if sigma(T) boolean AND T subset of{1}. The main result of the present paper gives a sharp estimate for the rateat which this decay occurs for operators on Hilbert space, assuming the growth of the resolvent norms parallel to R(e(i theta), T)parallel to as vertical bar theta vertical bar -> 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. (C) 2020 Elsevier Inc. All rights reserved.