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 $\{n\in \ddot{I}‰:T^{n}x\in U\}$ has positive upper asymptotic density for all nonempty open sets $U\subseteq X$. It is known that $\mathrm{UFHC}(T)$ is a $G_{\hat{I}´\ddot{I}ƒ\hat{I}´}$-set which is either empty or contains a dense $G_{\hat{I}}´$-set. Using a purely topological proof, we improve it by showing that $\mathrm{UFHC}(T)$ is always a $G_{\hat{I}´\ddot{I}ƒ}$-set. Bonilla and Grosse-Erdmann asked in [Rev. Mat. Complut. \textbf{31} (2018), 673--711] whether $\mathrm{UFHC}(T)$ is always a $G_{\hat{I}}´$-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 $F_{\ddot{I}ƒ\hat{I}´}$-set (hence not $G_{\hat{I}}´$). 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.