Questions in linear recurrence I:The T ⊕ T - recurrence problem

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