Hypercyclic subsets

TL;DR

This paper characterizes finite-dimensional subsets of separable Hilbert spaces where A-hypercyclicity matches hypercyclicity, and explores conditions under which the density of orbits implies hypercyclicity, advancing understanding of hypercyclic operators.

Contribution

It provides a complete characterization of finite-dimensional subsets for A-hypercyclicity and hypercyclicity implications, improving previous results and addressing open questions.

Findings

Characterized finite-dimensional subsets where A-hypercyclicity equals hypercyclicity.

Identified conditions where orbit density implies hypercyclicity for finite-dimensional subsets.

Extended understanding of hypercyclic operators beyond previous partial results.

Abstract

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 $\ge$ 0, x $\in$ 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 $\ge$ 0, x $\in$ 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.