On the complexity of upper frequently hypercyclic vectors

Abstract

Given a continuous linear operator $T:X\to X$, where $X$ is a topological vector space, let $\mathrm{UFHC}(T)$ be the set of upper frequently hypercyclic vectors, that is, the set of vectors $x \in X$ such that $\{n \in \omega: T^nx \in U\}$ has positive upper asymptotic density for all nonempty open sets $U\subseteq X$. It is known that $\mathrm{UFHC}(T)$ is a $G_{\delta\sigma\delta}$-set which is either empty or contains a dense $G_{\delta}$-set. Using a purely topological proof, we improve it by showing that $\mathrm{UFHC}(T)$ is always a $G_{\delta\sigma}$-set. Bonilla and Grosse-Erdmann asked in [Rev. Mat. Complut. \textbf{31} (2018), 673--711] whether $\mathrm{UFHC}(T)$ is always a $G_{\delta}$-set. We answer such question in the negative, by showing that there exists a continuous linear operator $T$ for which $\mathrm{UFHC}(T)$ is not a $F_{\sigma\delta}$-set (hence not $G_\delta$). In addition, we study the [non-]equivalence between (the ideal versions of) upper frequently hypercyclicity in the product topology and upper frequently hypercyclicity in the norm topology.