Extrernal growth of powers of operators satisfying resolvent conditions of Kreiss-Ritt type

摘要

Let E be a compact subset of the unit circle. We determine the extremal rate of growth of (parallel toT(n)parallel to)(ngreater than or equal to1) for Banach-space operators T satisfying the resolvent condition parallel to(T-lambdaI)(-1)parallel toless than or equal toconst/dist(lambda,E) (lambda>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 role in determining the growth of parallel toT(n)parallel to. As a by-product, we also obtain lower bounds for the Taylor coefficients of functions f holomorphic on the unit disk and satisfying (z)greater than or equal to1/dist(z,E) (z<1). (C) 2002 Elsevier Science (USA). [References: 11]