Linear dynamics and recurrence properties defined via essential idempotents of $\beta \mathbb{N}$

Consider $\mathcal{F}$ a non-empty set of subsets of $\mathbb{N}$. An operator $T$ on $X$ satisfies property ${\mathcal{P}}_{\mathcal{F}}$ if for any $U$ non-empty open set in $X$, there exists $x\in X$ such that $\{n\in \mathbb{N}:T^{n}x\in U\}\in \mathcal{F}$. Let $\overline{\mathcal{B}\mathcal{D}}$ the collection of sets in $\mathbb{N}$ with positive upper Banach density. Our main result is a characterization of sequence of operators satisfying property ${\mathcal{P}}_{\overline{\mathcal{B}\mathcal{D}}}$, for which we have used a strong result of Bergelson and Mccutcheon in the vein of Szemer'{e}di’s theorem. It turns out that operators having property ${\mathcal{P}}_{\overline{\mathcal{B}\mathcal{D}}}$ satisfy a kind of recurrence described in terms of essential idempotents of $\beta \mathbb{N}$. We will also discuss the case of weighted backward shifts. Finally, we obtain a characterization of reiteratively hypercyclic operators.