### abstract

- 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 ) ‖ = O ( 1 / n β ) with β ∈ [ 1 2 , 1 ) . For normal contractions the condition is also necessary. Another sufficient condition for β = 1 2 , necessary for T normal, is that the numerical range of T be in a disk { z : | z − δ | ≤ 1 − δ } for some δ ∈ ( 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 ) ‖ = O ( 1 / n β ) with β ∈ ( 0 , 1 ] if and only if sup 1 | λ | 2 | λ − 1 | 1 / β ‖ R ( λ , T ) ‖ ∞ . When T is a contraction on L 2 satisfying the numerical range condition, it is shown that T n f / n 1 − β converges to 0 a.e. with a maximal inequality, for every f ∈ L 2 . An example shows that in general a positive contraction T on L 2 may have an f ≥ 0 with lim sup T n f / log n n = ∞ a.e.