EVERY WEAKLY SEQUENTIALLY HYPERCYCLIC SHIFT IS NORM HYPERCYCLIC

Such a vector x is called a weakly sequentially hypercyclic vector for T. If there is an orbit orb(T, x) that is dense in X in the weak (respectively, norm) topology of X. we say that T is weakly hypercyclic (respectively, norm hypercyclic), and that x is a weakly (respectively, norm) hypercyclic vector for T. Clearly if T is norm hypercyclic, then T is weakly sequentially hypercyclic, which in turn implies T is weakly hypercyclic. Similarly, an operator T on a Banach space X is (a) norm supercyclic (b) weakly supercyclic (c) weakly sequentially supercyclic, provided there exists a vector x so that Orb(span(x),T) = {XTkx : X e C, k > 0} is (a) norm dense, (b) weakly dense, (c) weakly sequentially dense in X, respectively. It is not known whether an operator can be weakly sequentially hypercyclic