Remarks on rates of convergence of powers of contractions

摘要

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 parallel to T-n(I - T)parallel to = O(1(beta)) with beta is an element of [1/2, 1). For normal contractions the condition is also necessary. Another sufficient condition for beta = 1/2, necessary for T normal, is that the numerical range of T be in a disk {z : vertical bar z - delta vertical bar <= 1 - delta} for some delta is an element of (0,1). As a consequence of results of Seifert, we obtain that a power-bounded T on a Hilbert space satisfies parallel to T-n (I - T)parallel to = O(1(beta)) with beta is an element of (0,1] if and only if suP(1= 0 with lim sup T-n f / log n root n = infinity a.e. (C) 2015 Elsevier Inc. All rights reserved.