(Non-)weakly mixing operators and hypercyclicity sets

Bayart, Frédéric; Matheron, Étienne

doi:10.5802/aif.2425

(Non-)weakly mixing operators and hypercyclicity sets
[Opérateurs (non) faiblement mélangeants et ensembles d’hypercyclicité]

Bayart, Frédéric ¹ ; Matheron, Étienne ²

¹ Université Blaise Pascal Laboratoire de Mathématiques Campus universitaire des Cézeaux 63177 Aubières Cedex (France)
² Université d’Artois Laboratoire de Mathématiques de Lens Rue Jean Souvraz S.P. 18 62307 Lens (France)

Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 1-35.

Résumé
Abstract

On étudie la fréquence d’hypercyclicité des opérateurs hypercycliques non faiblement mélangeants. On montre en particulier qu’il est possible de construire sur l’espace $ℓ^{1}$ des opérateurs non faiblement mélangeants de fréquence d’hypercyclicité arbitrairement grande. On obtient un résultat analogue (mais plus faible) sur $c_{0}$ ou $ℓ^{p}$ , $1 < p < \infty$ . Certains de nos résultats font intervenir des propriétés de lacunarité de type “Sidon” pour les suites d’entiers.

We study the frequency of hypercyclicity of hypercyclic, non–weakly mixing linear operators. In particular, we show that on the space $ℓ^{1} (ℕ)$ , any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for $c_{0} (ℕ)$ or $ℓ^{p} (ℕ)$ , $1 < p < \infty$ . Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.

MR Zbl

DOI : 10.5802/aif.2425

Classification : 47A16, 37B99, 11B99
Keywords: Hypercyclic operators, weak mixing, Sidon sequences
Mot clés : opérateurs hypercycliques, opérateurs faiblement mélangeants, ensembles d’hypercyclicité, suites de Sidon

Affiliations des auteurs :

Bayart, Frédéric ¹ ; Matheron, Étienne ²

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Bayart, Frédéric; Matheron, Étienne. (Non-)weakly mixing operators and hypercyclicity sets. Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 1-35. doi : 10.5802/aif.2425. https://www.numdam.org/articles/10.5802/aif.2425/

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