Hypercyclic subsets

S. Charpentier (1) , Romuald Ernst (2)

1 I2M - Institut de Mathématiques de Marseille
2 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

We completely characterize the finite dimensional subsets A of any separable Hilbert space for which the notion of A-hypercyclicity coincides with the notion of hypercyclicity, where an operator T on a topological vector space X is said to be A-hypercyclic if the set {T n x, n ≥ 0, x ∈ A} is dense in X. We give a partial description for non necessarily finite dimensional subsets. We also characterize the finite dimensional subsets A of any separable Hilbert space H for which the somewhere density in H of {T n x, n ≥ 0, x ∈ A} implies the hypercyclicity of T. We provide a partial description for infinite dimensional subsets. These improve results of Costakis and Peris, Bourdon and Feldman, and Charpentier, Ernst and Menet, and answer a number of related open questions.

Domaines: Analyse fonctionnelle [math.FA]

S. Charpentier, Romuald Ernst. Hypercyclic subsets. 2018. ⟨hal-01651264v4⟩