Trichotomy for the orbits of a hypercyclic operator on a Banach space

We obtain a trichotomy for the orbits of a hypercyclic operator T T on a separable Banach space X X : (1) every vector is mean asymptotic to zero; (2) generic vectors are absolutely mean irregular; (3) every hypercyclic vector is mean divergent to infinity. Examples of weighted backward shifts on ℓ p ^p show that all three cases can happen.