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arXiv · 1711.10932

Hypercyclic subsets

S. Charpentier·R. Ernst
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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.

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S. Charpentier, R. Ernst. 2017-11-29. Hypercyclic subsets. https://arxiv.org/abs/1711.10932

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