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

Volume 220 / 2014

Romuald Ernst Studia Mathematica 220 (2014), 15-53 MSC: Primary 47A16; Secondary 15A21. DOI: 10.4064/sm220-1-2


We prove that on $\mathbb{R}^N$, there is no $n$-supercyclic operator with $1\leq n< \lfloor {(N+1)}/{2}\rfloor$, i.e. if $\mathbb{R}^N$ has an $n$-dimensional subspace whose orbit under $T\in\mathcal{L}(\mathbb{R}^N)$ is dense in $\mathbb{R}^N$, then $n$ is greater than $\lfloor{(N+1)}/{2}\rfloor$. Moreover, this value is optimal. We then consider the case of strongly $n$-supercyclic operators. An operator $T\in\mathcal{L}(\mathbb{R}^N)$ 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 dimensions.


  • Romuald ErnstClermont Université, UniversitéBlaise Pascal
    Laboratoire de Mathématiques
    bp 10448, F-63000 Clermont-Ferrand, France
    CNRS, UMR 6620
    Laboratoire de Mathématiques
    F-63177 Aubière, France

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