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 ‖Tn(I−T)‖=O(1/nβ)‖Tn(I−T)‖=O(1/nβ) with β∈[12,1). For normal contractions the condition is also necessary. Another sufficient condition for β=12, necessary for T normal, is that the numerical range of T   be in a disk {z:|z−δ|≤1−δ}{z:|z−δ|≤1−δ} for some δ∈(0,1)δ∈(0,1). As a consequence of results of Seifert, we obtain that a power-bounded T   on a Hilbert space satisfies ‖Tn(I−T)‖=O(1/nβ)‖Tn(I−T)‖=O(1/nβ) with β∈(0,1]β∈(0,1] if and only if sup1<|λ|<2⁡|λ−1|1/β‖R(λ,T)‖<∞sup1<|λ|<2⁡|λ−1|1/β‖R(λ,T)‖<∞. When T   is a contraction on L2L2 satisfying the numerical range condition, it is shown that Tnf/n1−βTnf/n1−β converges to 0 a.e. with a maximal inequality, for every f∈L2f∈L2. An example shows that in general a positive contraction T   on L2L2 may have an f≥0f≥0 with limsupTnf/log⁡nn=∞ a.e.