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