n-Weakly hypercyclic and n-weakly supercyclic operators

If X is a locally convex topological vector space over a scalar field F=R or C and if E is a subset of X, then we define E to be n-weakly dense in X if for every onto continuous linear operator F:X→Fn we have that F(E) is dense in Fn. If X is a Hilbert space, this is equivalent to requiring that E have a dense orthogonal projection onto every subspace of dimension n. We then consider continuous linear operators on X that have orbits or scaled orbits that are n-weakly dense in X. We show that on a separable Hilbert space there are non-trivial examples of such operators and establish many of their basic properties. A fundamental tool is Ballʼs solution of the complex plank problem which implies that certain sets are 1-weakly closed.