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 Chypercyclic if the set {T n x, 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 {T n x, 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.2010 Mathematics Subject Classification. 47A16.

Characterization of Diskcyclic Operators

“…Later, in 2020 Charpentier and Ernst in [4] gave a complete characterization of 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 T n x : n ≥ 0, x ∈ C is dense in X. They gave a partial description for non necessarily finite dimensional subsets.…”