(Non-)weakly mixing operators and hypercyclicity sets

Frédéric Bayart[1]; Étienne Matheron[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 (2009)

  • Volume: 59, Issue: 1, page 1-35
  • ISSN: 0373-0956

Abstract

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We study the frequency of hypercyclicity of hypercyclic, non–weakly mixing linear operators. In particular, we show that on the space , any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for or , . Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.

How to cite

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Bayart, Frédéric, and Matheron, Étienne. "(Non-)weakly mixing operators and hypercyclicity sets." Annales de l’institut Fourier 59.1 (2009): 1-35. <http://eudml.org/doc/10391>.

@article{Bayart2009,
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&lt;p&lt;\infty $. Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.},
affiliation = {Université Blaise Pascal Laboratoire de Mathématiques Campus universitaire des Cézeaux 63177 Aubières Cedex (France); Université d’Artois Laboratoire de Mathématiques de Lens Rue Jean Souvraz S.P. 18 62307 Lens (France)},
author = {Bayart, Frédéric, Matheron, Étienne},
journal = {Annales de l’institut Fourier},
keywords = {Hypercyclic operators; weak mixing; Sidon sequences; hypercyclic operators},
language = {eng},
number = {1},
pages = {1-35},
publisher = {Association des Annales de l’institut Fourier},
title = {(Non-)weakly mixing operators and hypercyclicity sets},
url = {http://eudml.org/doc/10391},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Bayart, Frédéric
AU - Matheron, Étienne
TI - (Non-)weakly mixing operators and hypercyclicity sets
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 1
SP - 1
EP - 35
AB - 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&lt;p&lt;\infty $. Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.
LA - eng
KW - Hypercyclic operators; weak mixing; Sidon sequences; hypercyclic operators
UR - http://eudml.org/doc/10391
ER -

References

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  1. S. Ansari, Hypercyclic and cyclic vectors, J. Funct. Anal. 128 (1995), 374-383 Zbl0853.47013MR1319961
  2. F. Bayart, S. Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358 (2006), 5083-5117 Zbl1115.47005MR2231886
  3. F. Bayart, É. Matheron, Hypercyclic operators failing the Hypercyclicity Criterion on classical Banach spaces, J. Funct. Anal. 250 (2007), 426-441 Zbl1131.47006MR2352487
  4. J. Bès, A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), 94-112 Zbl0941.47002MR1710637
  5. A. Bonilla, K.-G. Grosse-Erdmann, Frequently hypercyclic operators and vectors, Ergodic Theory Dynam. Systems 27 (2007), 383-404 Zbl1119.47011MR2308137
  6. P. S. Bourdon, N. S. Feldman, Somewhere dense orbits are everywhere dense, Indiana Univ. Math. J. 52 (2003), 811-819 Zbl1049.47002MR1986898
  7. G. Costakis, M. Sambarino, Topologically mixing hypercyclic operators, Proc. Amer. Math. Soc. 132 (2004), 385-389 Zbl1054.47006MR2022360
  8. M. De La Rosa, C. J. Read, A hypercyclic operator whose direct sum is not hypercyclic Zbl1193.47014
  9. H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, (1981), Princeton University Press Zbl0459.28023MR603625
  10. E. Glasner, Ergodic theory via joinings, 101 (2003), American Mathematical Society Zbl1038.37002MR1958753
  11. E. Glasner, B. Weiss, On the interplay between mesurable and topological dynamics, 1B (2006), Elsevier B. V. Zbl1130.37303MR2186250
  12. S. Grivaux, Hypercyclic operators, mixing operators, and the bounded steps problem, J. Operator Theory 54 (2005), 147-168 Zbl1104.47010MR2168865
  13. K.-G. Grosse-Erdmann, A. Peris, Frequently dense orbits, C. R. Acad. Sci. Paris 341 (2005), 123-128 Zbl1068.47012MR2153969
  14. H. Halberstam, K. F. Roth, Sequences, (1983), Springer-Verlag Zbl0498.10001MR687978
  15. A. Peris, L. Saldivia, Syndetically hypercyclic operators, Integral Equations Operator Theory 51 (2005), 275-281 Zbl1082.47004MR2120081
  16. I. Z. Rusza, An infinite Sidon sequence, J. Number Theory 68 (1998), 63-71 Zbl0927.11005MR1492889

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