Strong transitivity properties for operators

Given a Furstenberg family $\mathcal{F}$ of subsets of $\mathbb{N}$, an operator $T$ on a topological vector space $X$ is called $\mathcal{F}$-transitive provided for each non-empty open subsets $U$, $V$ of $X$ the set $\{n\in {\mathbb{Z}}_{+}:T^n(U)\cap V\mathrm{\ne }\mathrm{\varnothing }\}$ belongs to $\mathcal{F}$. We classify the topologically transitive operators with a hierarchy of $\mathcal{F}$-transitive subclasses by considering families $\mathcal{F}$ that are determined by various notions of largeness and density in ${\mathbb{Z}}_{+}$.