Resolvent conditions and growth of powers of operators

Abstract: Following Berm\'udez et al. (ArXiv: 1706.03638v1), we study the rate of growth of the norms of the powers of a linear operator, under various resolvent conditions or Ces`aro boundedness assumptions. We show that $T$ is power-bounded if (and only if) both $T$ and $T^*$ are absolutely Ces`aro bounded. In Hilbert spaces, we prove that if $T$ satisfies the Kreiss condition, $|T^n|=O(n/\sqrt {\log n})$; if $T$ is absolutely Ces`aro bounded, $|T^n|=O(n^{1/2 -\varepsilon})$ for some $\varepsilon >0$ (which depends on $T$); if $T$ is strongly Kreiss bounded, then $|T^n|=O((\log n)^\kappa)$ for some $\kappa >0$. We show that a Kreiss bounded operator on a reflexive space is Abel ergodic, and its Ces`aro means of order $\alpha$ converge strongly when $\alpha >1$.