HAL: hal-00697603, version 2

arXiv: 1205.3575

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v1 (2012-05-15)

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

Romuald Ernst 1

(2012-05-10)

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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Mathematics/Functional Analysis

Keyword(s): n-supercyclicity – n-supercyclic operators – strong n-supercyclicity – strongly n-supercyclic operators – Jordan decomposition – finite dimension – supercyclic operators

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From: Romuald Ernst <>

Submitted on: Tuesday, 15 May 2012 17:45:37

Updated on: Wednesday, 16 May 2012 08:54:30