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Abstract We study, for a continuous linear operator T acting on an F-space X, when the direct sum operator T T T ⊕ T is recurrent on the direct sum space X 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 T T ⊕ T -recurrence problem in the negative way.
Grivaux et al. (Mon,) studied this question.