Rates of decay in the classical Katznelson-Tzafriri theorem

Given a power-bounded operator $T$, the theorem of Katznelson and Tzafriri states that $\|T^n(I-T)\|\to0$ as $n\to\infty$ if and only if the spectrum $\sigma(T)$ of $T$ intersects the unit circle $\mathbb{T}$ in at most the point 1. This paper investigates the rate at which decay takes place when $\sigma(T)\cap\mathbb{T}=\{1\}$. The results obtained lead in particular to both upper and lower bounds on this rate of decay in terms of the growth of the resolvent operator $R(\mathrm{e}^{\mathrm{i}\theta},T)$ as $\theta\to0$. In the special case of polynomial resolvent growth, these bounds are then shown to be optimal for general Banach spaces but not in the Hilbert space case.