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arXiv : 1205.3575

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

Romuald Ernst 1

(10/05/2012)

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.

1 :	Laboratoire de Mathématiques
	CNRS : UMR6620 – Université Blaise Pascal - Clermont-Ferrand II

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Mathématiques/Analyse fonctionnelle

Mots Clés : n-supercyclicity – n-supercyclic operators – strong n-supercyclicity – strongly n-supercyclic operators – Jordan decomposition – finite dimension – supercyclic operators

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Contributeur : Romuald Ernst <>

Soumis le : Mardi 15 Mai 2012, 17:45:37

Dernière modification le : Mercredi 16 Mai 2012, 08:54:30