Ces\`aro bounded operators in Banach spaces

We study several notions of boundedness for operators. It is known that any power bounded operator is absolutely Ces\`aro bounded and strong Kreiss bounded (in particular, uniformly Kreiss bounded). The converses do not hold in general. In this note, we give examples of topologically mixing absolutely Ces\`aro bounded operators on $\ell^p(\mathbb{N})$, $1\le p < \infty$, which are not power bounded, and provide examples of uniformly Kreiss bounded operators which are not absolutely Ces\`aro bounded. These results complement very limited number of known examples (see \cite{Shi} and \cite{AS}). In \cite{AS} Aleman and Suciu ask if every uniformly Kreiss bounded operator $T$ on a Banach spaces satisfies that $\lim_n\| \frac{T^n}{n}\|=0$. We solve this question for Hilbert space operators and, moreover, we prove that, if $T$ is absolutely Ces\`aro bounded on a Banach (Hilbert) space, then $\| T^n\|=o(n)$ ($\| T^n\|=o(n^{\frac{1}{2}})$, respectively). As a consequence, every absolutely Ces\`aro bounded operator on a reflexive Banach space is mean ergodic, and there exist mixing mean ergodic operators on $\ell^p(\mathbb{N})$, $1< p <\infty$. Finally, we give new examples of weakly ergodic 3-isometries and study numerically hypercyclic $m$-isometries on finite or infinite dimensional Hilbert spaces. In particular, all weakly ergodic strict 3-isometries on a Hilbert space are weakly numerically hypercyclic. Adjoints of unilateral forward weighted shifts which are strict $m$-isometries on $\ell ^2(\mathbb{N})$ are shown to be hypercyclic.