n-SUPERCYCLIC AND STRONGLY n-SUPERCYCLIC OPERATORS IN FINITE DIMENSIONS

Abstract. We prove that on RN, there is no n-supercyclic operator with 1 ≤ n < bN+1 2 c i.e. if RN has an n-dimensional subspace whose orbit under T ∈ L(RN) is dense in RN, then n is greater than bN+1 2 c. Moreover, this value is optimal. We then consider the case of strongly n-supercyclic operators. An operator T ∈ L(RN) is strongly n-supercyclic if RN has an n-dimensional subspace whose orbit under T is dense in Pn(RN), the n-th Grassmannian. We prove that strong n-supercyclicity does not occur non-trivially in finite dimension. Let T be a continuous linear operator on a Banach space X. The orbit of a set E ∈ X under T is defined by O(E, T): = ∪n∈Z+Tn(E). Many authors have already studied some density properties of such orbits for different original sets E. If E is a singleton and O(E, T) is dense in X, then T is said to be hypercyclic