N-weakly Supercyclic Matrices

We define an operator to n-weakly hypercyclic if it has an orbit that has a dense projection onto every n-dimensional subspace. Similarly, an operator is n-weakly supercyclic if it has a scaled orbit that has a dense projection onto every n-dimensional subspace. In this paper, we show the following results: (i) There are no n-weakly hypercyclic matrices on \({\mathbb{R}^n}\) or \({\mathbb{C}^n}\). (ii) There are no 2-weakly supercyclic matrices on \({\mathbb{C}^n}\) for n ≥ 2. (iii) There are no 3-weakly supercyclic matrices on \({\mathbb{R}^n}\) for n ≥ 3; and (iv) there are 2-weakly supercyclic matrices on \({\mathbb{R}^n}\) if and only if n is even. Finally, we show that there is an onto isometry on \({\ell^2_\mathbb{R}(\mathbb{N})}\) that is 2-weakly supercyclic, but not 3-weakly supercyclic and also give some examples involving tuples of matrices. We conclude with some questions.