##
**(Non-)weakly mixing operators and hypercyclicity sets.**
*(English)*
Zbl 1178.47003

Let \(X\) be an infinite-dimensional separable Banach space. For any \(x\in X\) and any nonempty open set \(V\subset X\), define the set \(N(x,V):=\{n\in\mathbb{N}: T^n(x)\in V\}\). An operator \(T\in L(X)\) is called hypercyclic if there exists \(x\in X\) such that \(N(x,V)\neq\emptyset\) for any nonempty open subset \(V\) of \(X\). Computing the frequency in which every nonempty open subset is visited by a hypercyclic vector \(x\), F. Bayart and S. Grivaux introduced the notion of frequent hypercyclicity [Trans. Am. Math. Soc. 358, No. 11, 5083–5117 (2006; Zbl 1115.47005)]. A hypercyclic operator \(T\in L(X)\) is said to be frequently hypercyclic if there is a hypercyclic vector \(x\in X\) such that for every nonempty open set \(V\) of \(X\), the set \(N(x,V)\) can be enumerated as an increasing sequence \((n_k)\) with \(n_k=\mathcal{O}(k)\) as \(k\rightarrow\infty\).

An operator \(T\) is said to be weakly mixing if \(T\oplus T\) is hypercyclic on \(X\oplus X\). K.-G. Grosse-Erdmann and A. Peris proved that every frequently hypercyclic operator is weakly mixing [C. R., Math., Acad. Sci. Paris 341, No. 2, 123–128 (2005; Zbl 1068.47012)]. On the other hand, De la Rosa and Read have solved a long-standing question posed by D. A. Herrero [J. Funct. Anal. 99, No. 1, 179–190 (1991; Zbl 0758.47016)], proving that there are hypercyclic operators which are not weakly mixing [M. De La Rosa and C. Read, J. Oper. Theory 61, No. 2, 369–380 (2009; Zbl 1193.47014); see also J. Funct. Anal. 250, No. 2, 426–441 (2007; Zbl 1131.47006)]. In fact, this question results to be equivalent to see wheter every hypercyclic operator verifies the Hypercyclicity Criterion [J. Bès and A. Peris, J. Funct. Anal. 167, No. 1, 94–112 (1999; Zbl 0941.47002)].

In the paper under review, the authors make a deep analysis of the constructions shown in [Zbl 1193.47014] and in [Zbl 1131.47006] in order to quantify how big the sets \(N(x,V)\) can be for a weakly mixing operator, and also for a hypercyclic operator which is not weakly mixing. In these constructions, the arithmetic properties of the sets \(N(x,V)\) play an important role, such as Sidon-type properties. The authors also present examples in the Fréchet space setting.

An operator \(T\) is said to be weakly mixing if \(T\oplus T\) is hypercyclic on \(X\oplus X\). K.-G. Grosse-Erdmann and A. Peris proved that every frequently hypercyclic operator is weakly mixing [C. R., Math., Acad. Sci. Paris 341, No. 2, 123–128 (2005; Zbl 1068.47012)]. On the other hand, De la Rosa and Read have solved a long-standing question posed by D. A. Herrero [J. Funct. Anal. 99, No. 1, 179–190 (1991; Zbl 0758.47016)], proving that there are hypercyclic operators which are not weakly mixing [M. De La Rosa and C. Read, J. Oper. Theory 61, No. 2, 369–380 (2009; Zbl 1193.47014); see also J. Funct. Anal. 250, No. 2, 426–441 (2007; Zbl 1131.47006)]. In fact, this question results to be equivalent to see wheter every hypercyclic operator verifies the Hypercyclicity Criterion [J. Bès and A. Peris, J. Funct. Anal. 167, No. 1, 94–112 (1999; Zbl 0941.47002)].

In the paper under review, the authors make a deep analysis of the constructions shown in [Zbl 1193.47014] and in [Zbl 1131.47006] in order to quantify how big the sets \(N(x,V)\) can be for a weakly mixing operator, and also for a hypercyclic operator which is not weakly mixing. In these constructions, the arithmetic properties of the sets \(N(x,V)\) play an important role, such as Sidon-type properties. The authors also present examples in the Fréchet space setting.

Reviewer: Jose A. Conejero (Valencia)

### MSC:

47A16 | Cyclic vectors, hypercyclic and chaotic operators |

37B99 | Topological dynamics |

11B99 | Sequences and sets |

### Citations:

Zbl 1115.47005; Zbl 1068.47012; Zbl 0758.47016; Zbl 1131.47006; Zbl 0941.47002; Zbl 1193.47014
PDF
BibTeX
XML
Cite

\textit{F. Bayart} and \textit{É. Matheron}, Ann. Inst. Fourier 59, No. 1, 1--35 (2009; Zbl 1178.47003)

### References:

[1] | Ansari, S., Hypercyclic and cyclic vectors, J. Funct. Anal., 128, 2, 374-383, (1995) · Zbl 0853.47013 |

[2] | Bayart, F.; Grivaux, S., Frequently hypercyclic operators, Trans. Amer. Math. Soc., 358, 11, 5083-5117, (2006) · Zbl 1115.47005 |

[3] | Bayart, F.; Matheron, É., Hypercyclic operators failing the hypercyclicity criterion on classical Banach spaces, J. Funct. Anal., 250, 426-441, (2007) · Zbl 1131.47006 |

[4] | Bès, J.; Peris, A., Hereditarily hypercyclic operators, J. Funct. Anal., 167, 1, 94-112, (1999) · Zbl 0941.47002 |

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

[6] | Bourdon, P. S.; Feldman, N. S., Somewhere dense orbits are everywhere dense, Indiana Univ. Math. J., 52, 3, 811-819, (2003) · Zbl 1049.47002 |

[7] | Costakis, G.; Sambarino, M., Topologically mixing hypercyclic operators, Proc. Amer. Math. Soc., 132, 2, 385-389, (2004) · Zbl 1054.47006 |

[8] | De La Rosa, M.; Read, C. J., A hypercyclic operator whose direct sum is not hypercyclic · Zbl 1193.47014 |

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

[10] | Glasner, E., Ergodic theory via joinings, 101, (2003), American Mathematical Society · Zbl 1038.37002 |

[11] | Glasner, E.; Weiss, B., On the interplay between mesurable and topological dynamics, 1B, (2006), Elsevier B. V. · Zbl 1130.37303 |

[12] | Grivaux, S., Hypercyclic operators, mixing operators, and the bounded steps problem, J. Operator Theory, 54, 1, 147-168, (2005) · Zbl 1104.47010 |

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

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

[15] | Peris, A.; Saldivia, L., Syndetically hypercyclic operators, Integral Equations Operator Theory, 51, 2, 275-281, (2005) · Zbl 1082.47004 |

[16] | Rusza, I. Z., An infinite sidon sequence, J. Number Theory, 68, 63-71, (1998) · Zbl 0927.11005 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.