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 \(T\oplus T\) is recurrent on the direct sum space \(X\oplus X\). 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.