Cesàro bounded operators in Banach spaces

2017/06/12 by Bermúdez, Teresa, Bonilla, Antonio, Müller, Vladimir +1 · 2 citations
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1706.03638

Abstract

We study several notions of boundedness for operators. It is known that any power bounded operator is absolutely Cesàro 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 Cesàro bounded operators on ℓ^p(ℕ), 1≤ p < ∞, which are not power bounded, and provide examples of uniformly Kreiss bounded operators which are not absolutely Cesàro bounded. These results complement very limited number of known examples (see \citeShi and \citeAS). In \citeAS Aleman and Suciu ask if every uniformly Kreiss bounded operator T on a Banach spaces satisfies that lim_n‖ (Tⁿ)/(n)‖=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(n^(1)/(2)), respectively). As a consequence, every absolutely Cesàro bounded operator on a reflexive Banach space is mean ergodic, and there exist mixing mean ergodic operators on ℓ^p(ℕ), 1< p

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• Kreiss bounded and uniformly Kreiss bounded operators
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