Hypercyclic subsets

Abstract

We completely characterize the finite-dimensional subsets C of any separable Hilbert space for which the notion of C-hypercyclicity coincides with the notion of hypercyclicity, where an operator T on a topological vector space X is said to be C-hypercyclic if the set {Tnx, n ≥ 0, x ∈ C} is dense in X. We give a partial description for non-necessarily finite-dimensional subsets. We also characterize the finite-dimensional subsets C of any separable Hilbert space H for which the somewhere density in H of {Tnx, n ≥ 0, x ∈ C} 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.