Journal of Operator Theory

Volume 87, Issue 1, Winter 2022 pp. 113-136.

Common hypercyclic vectors for unilateral weighted shifts on $\ell^2$

Authors: Konstantinos Beros (1), Paul B. Larson (2)
Author institution:(1) Department of Mathematics, Miami University, Oxford, OH 45056, U.S.A.
(2) Department of Mathematics, Miami University, Oxford, OH 45056, U.S.A.

Summary: Each $w \in \ell^\infty$ defines a bacwards weighted shift $B_w : \ell^2 \rightarrow \ell^2$ . A vector $x \in \ell^2$ is \textit{hypercyclic} for $B_w$ if the set of forward iterates of $x$ is dense in $\ell^2$ . For each such $w$ , the set $\mathsf{HC} (w)$ consisting of all vectors hypercyclic for $B_w$ is $G_{\delta}$ . The set of \textit{common hypercyclic vectors} for a set $W \subseteq \ell^\infty$ is the set $\mathsf{HC}^* (W) = \bigcap\limits_{w \in W} \mathsf{HC} (w)$ . We show that $\mathsf{HC}^* (W)$ can be made arbitrarily complicated by making $W$ sufficiently complex, and that even for a $G_\delta$ set $W$ the set $\mathsf{HC}^* (W)$ can be non-Borel. Finally, by assuming the continuum hypothesis or Martin's axiom, we are able to construct a set $W$ such that $\mathsf{HC}^* (W)$ does not have the property of Baire.

DOI: http://dx.doi.org/10.7900/jot.202jul23.2345
Keywords: weighted shift, hypercyclic vectors, continuum hypothesis, Martin's axiom

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