no. 1
(Non-)weakly mixing operators and hypercyclicity sets
Bayart, Frédéric1; Matheron, Étienne2
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, Volume 59 (2009) no. 1, pp. 1-35.

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

DOI: 10.5802/aif.2425
Classification: 47A16, 37B99, 11B99
Keywords: Hypercyclic operators, weak mixing, Sidon sequences
Mots-clés : opérateurs hypercycliques, opérateurs faiblement mélangeants, ensembles d’hypercyclicité, suites de Sidon

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) 
@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},
     pages = {1--35},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {1},
     year = {2009},
     doi = {10.5802/aif.2425},
     mrnumber = {2514860},
     zbl = {1178.47003},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2425/}
}
TY  - JOUR
AU  - Bayart, Frédéric
AU  - Matheron, Étienne
TI  - (Non-)weakly mixing operators and hypercyclicity sets
JO  - Annales de l'Institut Fourier
PY  - 2009
SP  - 1
EP  - 35
VL  - 59
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2425/
DO  - 10.5802/aif.2425
LA  - en
ID  - AIF_2009__59_1_1_0
ER  - 
%0 Journal Article
%A Bayart, Frédéric
%A Matheron, Étienne
%T (Non-)weakly mixing operators and hypercyclicity sets
%J Annales de l'Institut Fourier
%D 2009
%P 1-35
%V 59
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2425/
%R 10.5802/aif.2425
%G en
%F AIF_2009__59_1_1_0
Bayart, Frédéric; Matheron, Étienne. (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. https://aif.centre-mersenne.org/articles/10.5802/aif.2425/

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

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

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

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

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

Cited by Sources: