Extremal Growth of Powers of Operators Satisfying Resolvent Conditions of Kreiss-Ritt Type

Let E be a compact subset of the unit circle. We determine the extremal rate of growth of (∥Tn∥)n⩾1 for Banach-space operators T satisfying the resolvent condition∥(T−λI)−1∥⩽const.dist(λ,E)(∣λ∣1) . This includes, as extreme cases, the Kreiss condition E=T and the Ritt condition E={1}. For intermediate sets E, the cardinality, the measure and the Hausdorff dimension of E all play a rôle in determining the growth of ∥Tn∥. As a by-product, we also obtain lower bounds for the Taylor coefficients of functions f holomorphic on the unit disk and satisfying∣f(z)∣⩾1dist(z,E)(∣z∣<1) .