  We study, for a continuous linear operator $T$ on an F-space $X$, when the
direct sum operator $T\oplus T$ is recurrent on $X\oplus X$. In particular: we
establish the analogous notion, for recurrence, to that of (topological)
weak-mixing for transitivity/hypercyclicity, namely quasi-rigidity; and we
construct a recurrent but not quasi-rigid operator on each separable
infinite-dimensional Banach space, solving the $T\oplus T$-recurrence problem
in the negative way. The quasi-rigidity notion is closely related to the dense
lineability of the set of recurrent vectors, and using similar conditions we
study the lineability and dense lineability properties for the set of
$\mathcal{F}$-recurrent vectors, under very weak assumptions on the Furstenberg
family $\mathcal{F}$.
