TY - JOUR
AU - Bermúdez, T.
AU - Bonilla, A.
AU - Müller, V.
AU - Peris, A.
T1 - Cesàro bounded operators in Banach spaces
LA - eng
PY - 2020
SP - 187
EP - 206
T2 - Journal d'Analyse Mathematique
SN - 1565-8538
VL - 140
IS - 1
PB - Hebrew University Magnes Press
AB - 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 ℓp(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) ((∥Tn∥=o(n12), respectively). As a consequence, every absolutely Cesàro bounded operator on a reflexive Banach space is mean ergodic.
DO - 10.1007/s11854-020-0085-8
UR - https://portalciencia.ull.es/documentos/5fa288e02999524084dd7391
DP - Dialnet - Portal de la Investigación
ER -