REMARKS ON RATES OF CONVERGENCE OF POWERS OF CONTRACTIONS

We prove that if the numerical range of a Hilbert space contraction $T$ is in a certain closed convex set of the unit disk which touches the unit circle only at 1, then $|T^n(I-T)| =\mathcal O(1/n^{\beta})$ with $\beta \in [\frac{1}{2}, 1)$. For normal contractions the condition is also necessary. Another sufficient condition for $\beta=\frac{1}{2}$, necessary for $T$ normal, is that the numerical range of $T$ be in a disk ${z: |z-\delta| \le 1-\delta}$ for some $\delta \in (0,1)$. As a consequence of results of Seifert, we obtain that a power-bounded $T$ on a Hilbert space satisfies $|T^n(I-T)| = \mathcal O(1/n^{\beta})$ with $\beta \in (0,1]$ if and only if $\sup_{1<|\lambda| <2} |\lambda -1|^{1/\beta} |R(\lambda,T)|< \infty$. When $T$ is a contraction on $L_2$ satisfying the numerical range condition, it is shown that $T^nf /n^{1-\beta}$ converges to 0 a.e. with a maximal inequality, for every $f \in L_2$. An example shows that in general a positive contraction $T$ on $L_2$ may have an $f \ge 0$ with $\limsup T^nf/\log n \sqrt{n} =\infty$ a.e.