(Non-)weakly mixing operators and hypercyclicity sets

Bayart, Frédéric; Matheron, Étienne

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

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 ${\ell }^1$ des opérateurs non faiblement mélangeants de fréquence d’hypercyclicité arbitrairement grande. On obtient un résultat analogue (mais plus faible) sur $c_0$ ou ${\ell }^p$, $1<p<\infty$. 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 ${\ell }^1\left(\mathbb{N}\right)$, any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for $c_0\left(\mathbb{N}\right)$ or ${\ell }^p\left(\mathbb{N}\right)$, $1<p<\infty$. Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.

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

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