Γ-supercyclicity

Stéphane Charpentier (1) , Romuald Ernst (1) , Quentin Menet (2)

1 I2M - Institut de Mathématiques de Marseille
2 LML - Laboratoire de Mathématiques de Lens

We characterize the subsets $\Gamma$ of $\C$ for which the notion of $\Gamma$-supercyclicity coincides with the notion of hypercyclicity, where an operator $T$ on a Banach space $X$ is said to be $\Gamma$-supercyclic if there exists $x\in X$ such that $\overline{\text{Orb}}(\Gamma x, T)=X$. In addition we characterize the sets $\Gamma \subset \C$ for which, for every operator $T$ on $X$, $T$ is hypercyclic if and only if there exists a vector $x\in X$ such that the set $\text{Orb}(\Gamma x, T)$ is somewhere dense in $X$. This extends results by Le\'on-M\"uller and Bourdon-Feldman respectively. We are also interested in the description of those sets $\Gamma \subset \C$ for which $\Gamma$-supercyclicity is equivalent to supercyclicity.

Mots clés: Hypercyclicity; Supercyclicity

Domaines: Analyse fonctionnelle [math.FA]

Stéphane Charpentier, Romuald Ernst, Quentin Menet. Γ-supercyclicity. Journal of Functional Analysis, 2016, 270 (12), pp.4443-4465. ⟨10.1016/j.jfa.2016.03.005⟩. ⟨hal-01199885⟩