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

Romuald Ernst

Studia Mathematica (2014)

  • Volume: 220, Issue: 1, page 15-53
  • ISSN: 0039-3223

Abstract

top
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 ∈ ( 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 ∈ ( ℝ N ) is strongly n-supercyclic if ℝ N has an n-dimensional subspace whose orbit under T is dense in ℙ ₙ ( ℝ N ) , the nth Grassmannian. We prove that strong n-supercyclicity does not occur non-trivially in finite dimensions.

How to cite

top

Romuald Ernst. "n-supercyclic and strongly n-supercyclic operators in finite dimensions." Studia Mathematica 220.1 (2014): 15-53. <http://eudml.org/doc/285856>.

@article{RomualdErnst2014,
abstract = {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 $T ∈ (ℝ^\{N\})$ 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 $T ∈ (ℝ^\{N\})$ 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.},
author = {Romuald Ernst},
journal = {Studia Mathematica},
keywords = {hypercyclic; supercyclic; -supercyclic; strongly -supercyclic; finite dimension},
language = {eng},
number = {1},
pages = {15-53},
title = {n-supercyclic and strongly n-supercyclic operators in finite dimensions},
url = {http://eudml.org/doc/285856},
volume = {220},
year = {2014},
}

TY - JOUR
AU - Romuald Ernst
TI - n-supercyclic and strongly n-supercyclic operators in finite dimensions
JO - Studia Mathematica
PY - 2014
VL - 220
IS - 1
SP - 15
EP - 53
AB - 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 $T ∈ (ℝ^{N})$ 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 $T ∈ (ℝ^{N})$ 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.
LA - eng
KW - hypercyclic; supercyclic; -supercyclic; strongly -supercyclic; finite dimension
UR - http://eudml.org/doc/285856
ER -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.