Operators on wighted spaces of holomorphic functions

Müller

et al. 2020

JAMA

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 [24] and [4]). In [4] Aleman and Suciu ask if every uniformly Kreiss bounded operator T on a Banach spaces satisfies that lim n T n n 1 2 . 4. There exist Kreiss bounded operators which are not Cesàro bounded, and conversely [28].

Section: Corollary 25 Any Uniformly Kreiss Bounded Operator On a Hisupporting

confidence: 55%

“…It is worth to mention that results of this type already appear in the PhD Thesis of María José Beltrán Meneu [5], provided by the fourth author (see Section 3.7 in [5]), and in [4].…”

Section: Corollary 27 Any Absolutely Cesàro Bounded Operator On a Ref...mentioning

confidence: 66%

Section: On Finite-dimensional Hilbert Spaces the Classes Of Uniformmentioning

confidence: 99%

“…Examples of absolutely Cesàro bounded mixing operators on ℓ p (N) are given in [20] (see Section 3.7 in [5]), [17], and [10] (see [11]).…”

Section: Inmentioning

confidence: 99%

See 2 more Smart Citations

Cesàro bounded operators in Banach spaces

Bermúdez

Müller

et al. 2020

JAMA

“…As consequence of Corollary 2.2, Corollary 2.3 and [10, Proposition 3.2], we obtain the following result. Results according to previous ones are shown in [3,4,5] for α = 1.…”

Section: Examples and Growth Orders Of Absolutely Cesàro Bounded Oper...mentioning

confidence: 64%

Growth orders and ergodicity for absolutely Cesàro bounded operators

Abadías

Linear Algebra and its Applications

2019

In this paper, we extend the concept of absolutely Cesàro boundedness to the fractional case. We construct a weighted shift operator belonging to this class of operators, and we prove that if T is an absolutely Cesàro bounded operator of order α with 0 < α ≤ 1, then T n = o(n α ), generalizing the result obtained for α = 1. Moreover, if α > 1, then T n = O(n). We apply such results to get stability properties for the Cesàro means of bounded operators.2010 Mathematics Subject Classification. 47A35, 47A10, 47B99.

Mean Li-Yorke chaos in Banach spaces

Bernardes

Journal of Functional Analysis

Peris

2020

We investigate the notion of mean Li-Yorke chaos for operators on Banach spaces. We show that it differs from the notion of distributional chaos of type 2, contrary to what happens in the context of topological dynamics on compact metric spaces. We prove that an operator is mean Li-Yorke chaotic if and only if it has an absolutely mean irregular vector. As a consequence, absolutely Cesàro bounded operators are never mean Li-Yorke chaotic. Dense mean Li-Yorke chaos is shown to be equivalent to the existence of a dense (or residual) set of absolutely mean irregular vectors. As a consequence, every mean Li-Yorke chaotic operator is densely mean Li-Yorke chaotic on some infinite-dimensional closed invariant subspace. A (Dense) Mean Li-Yorke Chaos Criterion and a sufficient condition for the existence of a dense absolutely mean irregular manifold are also obtained. Moreover, we construct an example of an invertible hypercyclic operator T such that every nonzero vector is absolutely mean irregular for both T and T −1 . Several other examples are also presented. Finally, mean Li-Yorke chaos is also investigated for C 0 -semigroups of operators on Banach spaces. 1 1 2010 Mathematics Subject Classification: 47A16, 37D45.