Some Properties of N-supercyclic Operators

Let T be a continuous linear operator on a Hausdorff topological vector space X over the field C. We show that if T is N -supercyclic, i.e., if X has an N dimensional subspace whose orbit under T is dense in X , then T ∗ has at most N eigenvalues (counting geometric multiplicity). We then show that N -supercyclicity cannot occur nontrivially in the finite dimensional setting: the orbit of an N dimensional subspace cannot be dense in an N + 1 dimensional space. Finally, we show that a subnormal operator on an infinitedimensional Hilbert space can never be N -supercyclic.