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

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Ernst R.

Studia Mathematica

2014

tom 220

nr 1

15-53

We prove that on $ℝ^{N}$, there is no n-supercyclic operator with 1 ≤ n < ⌊(N + 1)/2⌋, i.e. if $ℝ^{N}$ has an n-dimensional subspace whose orbit under $T ∈ 𝓛(ℝ^{N})$ is dense in $ℝ^{N}$, then n is greater than ⌊(N + 1)/2⌋. Moreover, this value is optimal. We then consider the case of strongly n-supercyclic operators. An operator $T ∈ 𝓛(ℝ^{N})$ is strongly n-supercyclic if $ℝ^{N}$ has an n-dimensional subspace whose orbit under T is dense in $ℙₙ(ℝ^{N})$, the nth Grassmannian. We prove that strong n-supercyclicity does not occur non-trivially in finite dimensions.

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1 Studia Mathematica

1 Ernst R.

1 2014

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