A probabilistic version of the Frequent Hypercyclicity Criterion

Sophie Grivaux¹

¹Laboratoire Paul Painlevé, UMR 8524 Université des Sciences et Technologies de Lille Bâtiment M2, Cité Scientifique 59655 Villeneuve d'Ascq Cedex, France

Studia Mathematica, Tome 176 (2006) no. 3, pp. 279-290

Résumé

For a bounded operator $T$ on a separable infinite-dimensional Banach space $X$, we give a “random” criterion not involving ergodic theory which implies that $T$ is frequently hypercyclic: there exists a vector $x$ such that for every non-empty open subset $U$ of $X$, the set of integers $n$ such that $T^{n}x$ belongs to $U$, has positive lower density. This gives a connection between two different methods for obtaining the frequent hypercyclicity of operators.

Détail

DOI : 10.4064/sm176-3-5

Keywords: bounded operator separable infinite dimensional banach space random criterion involving ergodic theory which implies frequently hypercyclic there exists vector every non empty subset set integers belongs has positive lower density gives connection between different methods obtaining frequent hypercyclicity operators