Questions in linear recurrence I: The $$T\oplus T$$-recurrence problem

We study, for a continuous linear operator T acting on an F-space X, when the direct sum operator $$T\oplus T$$ T ⊕ T is recurrent on the direct sum space $$X\oplus X$$ X ⊕ 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 $$T\oplus T$$ T ⊕ T -recurrence problem in the negative way.