$n$-supercyclic and strongly $n$-supercyclic operators in finite dimension

Romuald Ernst (1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

We prove that on $\mathbb{R}^n$, there is no $N$-supercyclic operator with $1\leq N< \lfloor \frac{n+1}{2}\rfloor$ i.e. if $\mathbb{R}^n$ has an $N$ dimensional subspace whose orbit under $T$ is dense in $\mathbb{R}^n$, then $N$ is greater than $\lfloor\frac{n+1}{2}\rfloor$. Moreover, this value is optimal. We then consider the case of strongly $N$-supercyclic operators. An operator $T$ is strongly $N$-supercyclic if $\mathbb{R}^n$ has an $N$-dimensional subspace whose orbit under $T$ is dense in $\mathbb{P}_N(\mathbb{R}^n)$, the $N$-th Grassmannian. We prove that strong $N$-supercyclicity does not occur non-trivially in finite dimension.

Mots clés: n-supercyclicity; n-supercyclic operators; strong n-supercyclicity; strongly n-supercyclic operators; Jordan decomposition; finite dimension; supercyclic operators

Domaines: Analyse fonctionnelle [math.FA]

Romuald Ernst. $n$-supercyclic and strongly $n$-supercyclic operators in finite dimension. 2012. ⟨hal-00697603v1⟩