  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 $\alpha >1$.
