This is an announcement for the paper “Cesàro bounded operators in Banach spaces” by Teresa Bermúdezhttps://arxiv.org/find/math/1/au:+Bermudez_T/0/1/0/all/0/1, Antonio Bonillahttps://arxiv.org/find/math/1/au:+Bonilla_A/0/1/0/all/0/1, Vladimir Müllerhttps://arxiv.org/find/math/1/au:+Muller_V/0/1/0/all/0/1, Alfredo Perishttps://arxiv.org/find/math/1/au:+Peris_A/0/1/0/all/0/1.

Abstract: 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 $\ell_p(\mathbb{N}), 1\leq p<\infty$, 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 $\lim_n|T_n/n|=0$. 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 $|T_n|=o(n)$ ($|T_n|=o(n^{1/2})$, 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.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1706.03638