Resolvent conditions and growth of powers of operators

Following Bermúdez et al. [5], we study the rate of growth of the norms of the powers of a linear operator, under various resolvent conditions or Cesàro boundedness assumptions. In Hilbert spaces, we prove that if T satisfies the Kreiss condition, T n = O(n/ √ log n); if T is absolutely Cesàro bounded, T n = O(n 1/2−ε) for some ε > 0 (which depends on T); if T is strongly Kreiss bounded, then T n = O((log n) κ) for some κ > 0. We show that a Kreiss bounded operator on a reflexive space is Abel ergodic, and its Cesàro means of order α converge strongly when α > 1