  We completely characterize the finite dimensional subsets A of any separable
Hilbert space for which the notion of A-hypercyclicity coincides with the
notion of hypercyclicity, where an operator T on a topological vector space X
is said to be A-hypercyclic if the set {T n x, n $\ge$ 0, x $\in$ A} is dense
in X. We give a partial description for non necessarily finite dimensional
subsets. We also characterize the finite dimensional subsets A of any separable
Hilbert space H for which the somewhere density in H of {T n x, n $\ge$ 0, x
$\in$ A} implies the hypercyclicity of T. We provide a partial description for
infinite dimensional subsets. These improve results of Costakis and Peris,
Bourdon and Feldman, and Charpentier, Ernst and Menet, and answer a number of
related open questions.
