(Non-)weakly mixing operators and hypercyclicity sets

[Opérateurs (non) faiblement mélangeants et ensembles d’hypercyclicité]

Bayart, Frédéric ; Matheron, Étienne

Annales de l'Institut Fourier, Tome 59 (2009), p. 1-35 / Harvested from Numdam

Résumé

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.

Abstract

We study the frequency of hypercyclicity of hypercyclic, non–weakly mixing linear operators. In particular, we show that on the space $\ell ^1(\mathbb{N})$, any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for $c_0(\mathbb{N})$ or $\ell ^p(\mathbb{N})$, $1<p<\infty $. Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.

Détail

Publié le : 2009-01-01

MR 2514860 | Zbl 1178.47003

DOI : https://doi.org/10.5802/aif.2425
Classification:  47A16,  37B99,  11B99
Mots clés: opérateurs hypercycliques, opérateurs faiblement mélangeants, ensembles d’hypercyclicité, suites de Sidon

Bibliographie

[1] Ansari, S. Hypercyclic and cyclic vectors, J. Funct. Anal., Tome 128 (1995) no. 2, pp. 374-383 | Article | MR 1319961 | Zbl 0853.47013

[2] Bayart, F.; Grivaux, S. Frequently hypercyclic operators, Trans. Amer. Math. Soc., Tome 358 (2006) no. 11, pp. 5083-5117 | Article | MR 2231886 | Zbl 1115.47005

[3] Bayart, F.; Matheron, É. Hypercyclic operators failing the Hypercyclicity Criterion on classical Banach spaces, J. Funct. Anal., Tome 250 (2007), pp. 426-441 | Article | MR 2352487 | Zbl 1131.47006

[4] Bès, J.; Peris, A. Hereditarily hypercyclic operators, J. Funct. Anal., Tome 167 (1999) no. 1, pp. 94-112 | Article | MR 1710637 | Zbl 0941.47002

[5] Bonilla, A.; Grosse-Erdmann, K.-G. Frequently hypercyclic operators and vectors, Ergodic Theory Dynam. Systems, Tome 27 (2007), pp. 383-404 | Article | MR 2308137 | Zbl 1119.47011

[6] Bourdon, P. S.; Feldman, N. S. Somewhere dense orbits are everywhere dense, Indiana Univ. Math. J., Tome 52 (2003) no. 3, pp. 811-819 | Article | MR 1986898 | Zbl 1049.47002

[7] Costakis, G.; Sambarino, M. Topologically mixing hypercyclic operators, Proc. Amer. Math. Soc., Tome 132 (2004) no. 2, pp. 385-389 | Article | MR 2022360 | Zbl 1054.47006

[8] De La Rosa, M.; Read, C. J. A hypercyclic operator whose direct sum is not hypercyclic (Journal of Operator Theory, to appear)

[9] Furstenberg, H. Recurrence in ergodic theory and combinatorial number theory, Princeton University Press (1981) | MR 603625 | Zbl 0459.28023

[10] Glasner, E. Ergodic theory via joinings, American Mathematical Society, Mathematical Surveys and Monographs, Tome 101 (2003) | MR 1958753 | Zbl 1038.37002

[11] Glasner, E.; Weiss, B. On the interplay between mesurable and topological dynamics, Elsevier B. V., Handbook of dynamical systems, Tome 1B (2006) (597–648) | MR 2186250 | Zbl 1130.37303

[12] Grivaux, S. Hypercyclic operators, mixing operators, and the bounded steps problem, J. Operator Theory, Tome 54 (2005) no. 1, pp. 147-168 | MR 2168865 | Zbl 1104.47010

[13] Grosse-Erdmann, K.-G.; Peris, A. Frequently dense orbits, C. R. Acad. Sci. Paris, Tome 341 (2005), pp. 123-128 | MR 2153969 | Zbl 1068.47012

[14] Halberstam, H.; Roth, K. F. Sequences, Springer-Verlag (1983) | MR 687978 | Zbl 0498.10001

[15] Peris, A.; Saldivia, L. Syndetically hypercyclic operators, Integral Equations Operator Theory, Tome 51 (2005) no. 2, pp. 275-281 | Article | MR 2120081 | Zbl 1082.47004

[16] Rusza, I. Z. An infinite Sidon sequence, J. Number Theory, Tome 68 (1998), pp. 63-71 | Article | MR 1492889 | Zbl 0927.11005