(Non-)weakly mixing operators and hypercyclicity sets
Annales de l'Institut Fourier, Volume 59 (2009) no. 1, p. 1-35

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

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. Certains de nos résultats font intervenir des propriétés de lacunarité de type “Sidon” pour les suites d’entiers.

DOI : https://doi.org/10.5802/aif.2425
Classification:  47A16,  37B99,  11B99
Keywords: Hypercyclic operators, weak mixing, Sidon sequences
@article{AIF_2009__59_1_1_0,
author = {Bayart, Fr\'ed\'eric and Matheron, \'Etienne},
title = {(Non-)weakly mixing operators and hypercyclicity sets},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {59},
number = {1},
year = {2009},
pages = {1-35},
doi = {10.5802/aif.2425},
mrnumber = {2514860},
zbl = {1178.47003},
language = {en},
url = {http://www.numdam.org/item/AIF_2009__59_1_1_0}
}

(Non-)weakly mixing operators and hypercyclicity sets. Annales de l'Institut Fourier, Volume 59 (2009) no. 1, pp. 1-35. doi : 10.5802/aif.2425. http://www.numdam.org/item/AIF_2009__59_1_1_0/

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