Cesàro bounded operators in Banach spaces

Abstract

We study several notions of boundedness for operators. It is known that any power bounded operator is absolutely Cesàro bounded and strongly Kreiss bounded (in particular, uniformly Kreiss bounded). The converses do not hold in general. In this note, we give examples of topologically mixing (hence, not power bounded) absolutely Cesàro bounded operators on ℓ^p(ℕ), 1 ≤ p < ∞, and provide examples of uniformly Kreiss bounded operators which are not absolutely Cesàro bounded. These results complement a few known examples (see [27] and [2]). We also obtain a characterization of power bounded operators which generalizes a result of Van Casteren [32]. In [2] Aleman and Suciu asked if every uniformly Kreiss bounded operator T on a Banach space satisfies that ${\lim _{n \to \infty }}\left\| {{{{T^n}} \over n}} \right\|\; = \;0.$. We solve this question for Hilbert space operators and, moreover, we prove that, if T is absolutely Cesàro bounded on a Banach (Hilbert) space, then ∥Tⁿ∥ = o(n) ($(\left\| {{T^n}} \right\| = o({n^{{1 \over 2}}}),$, respectively). As a consequence, every absolutely Cesàro bounded operator on a reflexive Banach space is mean ergodic.