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Frédéric Bayart; Étienne Matheron
(Non-)weakly mixing operators and hypercyclicity sets
(Opérateurs (non) faiblement mélangeants et ensembles d’hypercyclicité)
Annales de l'institut Fourier, 59 no. 1 (2009), p. 1-35, doi: 10.5802/aif.2425
Article: subscription required (your ip address: 50.19.155.235) | Reviews MR 2514860 | Zbl 1178.47003
Class. Math.: 47A16, 37B99, 11B99
Keywords: Hypercyclic operators, weak mixing, Sidon sequences
[1] S. Ansari, “Hypercyclic and cyclic vectors”, J. Funct. Anal. 128 (1995) no. 2, p. 374-383
Article | MR 1319961 | Zbl 0853.47013
[2] F. Bayart & S. Grivaux, “Frequently hypercyclic operators”, Trans. Amer. Math. Soc. 358 (2006) no. 11, p. 5083-5117
Article | MR 2231886 | Zbl 1115.47005
[3] F. Bayart & É. Matheron, “Hypercyclic operators failing the Hypercyclicity Criterion on classical Banach spaces”, J. Funct. Anal. 250 (2007), p. 426-441
Article | MR 2352487 | Zbl 1131.47006
[4] J. Bès & A. Peris, “Hereditarily hypercyclic operators”, J. Funct. Anal. 167 (1999) no. 1, p. 94-112
Article | MR 1710637 | Zbl 0941.47002
[5] A. Bonilla & K.-G. Grosse-Erdmann, “Frequently hypercyclic operators and vectors”, Ergodic Theory Dynam. Systems 27 (2007), p. 383-404
Article | MR 2308137 | Zbl 1119.47011
[6] P. S. Bourdon & N. S. Feldman, “Somewhere dense orbits are everywhere dense”, Indiana Univ. Math. J. 52 (2003) no. 3, p. 811-819
Article | MR 1986898 | Zbl 1049.47002
[7] G. Costakis & M. Sambarino, “Topologically mixing hypercyclic operators”, Proc. Amer. Math. Soc. 132 (2004) no. 2, p. 385-389
Article | MR 2022360 | Zbl 1054.47006
[8] M. De La Rosa & C. J. Read, “A hypercyclic operator whose direct sum is not hypercyclic”, Journal of Operator Theory, to appear
[9] H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, 1981 MR 603625 | Zbl 0459.28023
[10] E. Glasner, Ergodic theory via joinings, Mathematical Surveys and Monographs 101, American Mathematical Society, 2003 MR 1958753 | Zbl 1038.37002
[11] E. Glasner & B. Weiss, On the interplay between mesurable and topological dynamics, Handbook of dynamical systems 1B, Elsevier B. V., 2006, 597–648 MR 2186250 | Zbl 1130.37303
[12] S. Grivaux, “Hypercyclic operators, mixing operators, and the bounded steps problem”, J. Operator Theory 54 (2005) no. 1, p. 147-168 MR 2168865 | Zbl 1104.47010
[13] K.-G. Grosse-Erdmann & A. Peris, “Frequently dense orbits”, C. R. Acad. Sci. Paris 341 (2005), p. 123-128 MR 2153969 | Zbl 1068.47012
[14] H. Halberstam & K. F. Roth, Sequences, Springer-Verlag, 1983 MR 687978 | Zbl 0498.10001
[15] A. Peris & L. Saldivia, “Syndetically hypercyclic operators”, Integral Equations Operator Theory 51 (2005) no. 2, p. 275-281
Article | MR 2120081 | Zbl 1082.47004
[16] I. Z. Rusza, “An infinite Sidon sequence”, J. Number Theory 68 (1998), p. 63-71
Article | MR 1492889 | Zbl 0927.11005
Frédéric Bayart; Étienne Matheron
(Non-)weakly mixing operators and hypercyclicity sets
(Opérateurs (non) faiblement mélangeants et ensembles d’hypercyclicité)
Annales de l'institut Fourier, 59 no. 1 (2009), p. 1-35, doi: 10.5802/aif.2425
Article: subscription required (your ip address: 50.19.155.235) | Reviews MR 2514860 | Zbl 1178.47003
Class. Math.: 47A16, 37B99, 11B99
Keywords: Hypercyclic operators, weak mixing, Sidon sequences
Résumé - 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.
Bibliography
Article | MR 1319961 | Zbl 0853.47013
[2] F. Bayart & S. Grivaux, “Frequently hypercyclic operators”, Trans. Amer. Math. Soc. 358 (2006) no. 11, p. 5083-5117
Article | MR 2231886 | Zbl 1115.47005
[3] F. Bayart & É. Matheron, “Hypercyclic operators failing the Hypercyclicity Criterion on classical Banach spaces”, J. Funct. Anal. 250 (2007), p. 426-441
Article | MR 2352487 | Zbl 1131.47006
[4] J. Bès & A. Peris, “Hereditarily hypercyclic operators”, J. Funct. Anal. 167 (1999) no. 1, p. 94-112
Article | MR 1710637 | Zbl 0941.47002
[5] A. Bonilla & K.-G. Grosse-Erdmann, “Frequently hypercyclic operators and vectors”, Ergodic Theory Dynam. Systems 27 (2007), p. 383-404
Article | MR 2308137 | Zbl 1119.47011
[6] P. S. Bourdon & N. S. Feldman, “Somewhere dense orbits are everywhere dense”, Indiana Univ. Math. J. 52 (2003) no. 3, p. 811-819
Article | MR 1986898 | Zbl 1049.47002
[7] G. Costakis & M. Sambarino, “Topologically mixing hypercyclic operators”, Proc. Amer. Math. Soc. 132 (2004) no. 2, p. 385-389
Article | MR 2022360 | Zbl 1054.47006
[8] M. De La Rosa & C. J. Read, “A hypercyclic operator whose direct sum is not hypercyclic”, Journal of Operator Theory, to appear
[9] H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, 1981 MR 603625 | Zbl 0459.28023
[10] E. Glasner, Ergodic theory via joinings, Mathematical Surveys and Monographs 101, American Mathematical Society, 2003 MR 1958753 | Zbl 1038.37002
[11] E. Glasner & B. Weiss, On the interplay between mesurable and topological dynamics, Handbook of dynamical systems 1B, Elsevier B. V., 2006, 597–648 MR 2186250 | Zbl 1130.37303
[12] S. Grivaux, “Hypercyclic operators, mixing operators, and the bounded steps problem”, J. Operator Theory 54 (2005) no. 1, p. 147-168 MR 2168865 | Zbl 1104.47010
[13] K.-G. Grosse-Erdmann & A. Peris, “Frequently dense orbits”, C. R. Acad. Sci. Paris 341 (2005), p. 123-128 MR 2153969 | Zbl 1068.47012
[14] H. Halberstam & K. F. Roth, Sequences, Springer-Verlag, 1983 MR 687978 | Zbl 0498.10001
[15] A. Peris & L. Saldivia, “Syndetically hypercyclic operators”, Integral Equations Operator Theory 51 (2005) no. 2, p. 275-281
Article | MR 2120081 | Zbl 1082.47004
[16] I. Z. Rusza, “An infinite Sidon sequence”, J. Number Theory 68 (1998), p. 63-71
Article | MR 1492889 | Zbl 0927.11005
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310