n-supercyclic and strongly n-supercyclic operators in finite dimensions

摘要

We prove that on R~N there is no n-supercyclic operator with 1 ≤ n < [(N + 1)/2], i.e. if R~N has an n-dimensional subspace whose orbit under T∈L(R~N) is dense in R~N, then n is greater than [(N + 1)/2J. Moreover, this value is optimal. We then consider the case of strongly n-supercyclic operators. An operator T ∈ L(R~N) is strongly n-supercyclic if R~N has an n-dimensional subspace whose orbit under T is dense in P_n(R~N), the nth Grassmannian. We prove that strong n-supercyclicity does not occur non-trivially in finite dimensions.