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Note on Spectra of Non-Selfadjoint Operators Over Dynamical Systems

Part of: General theory of linear operators Special classes of linear operators

Published online by Cambridge University Press:  15 February 2018

Siegfried Beckus ,
Daniel Lenz ,
Marko Lindner  and
Christian Seifert
Siegfried Beckus
Affiliation:
Mathematisches Institut, Friedrich-Schiller-Universität, Jena, 07743, Jena, Germany (siegfried.beckus@uni-jena.de; daniel.lenz@uni-jena.de)
Daniel Lenz
Affiliation:
Mathematisches Institut, Friedrich-Schiller-Universität, Jena, 07743, Jena, Germany (siegfried.beckus@uni-jena.de; daniel.lenz@uni-jena.de)
Marko Lindner
Affiliation:
Technische Universität Hamburg-Harburg, Institut für Mathematik, 21073 Hamburg, Germany (marko.lindner@tuhh.de; christian.seifert@tuhh.de)
Christian Seifert*
Affiliation:
Technische Universität Hamburg-Harburg, Institut für Mathematik, 21073 Hamburg, Germany (marko.lindner@tuhh.de; christian.seifert@tuhh.de)
*
*Corresponding author.
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Abstract

We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.

Keywords

minimal dynamical systempseudo-ergodicityspectrum

MSC classification

Primary: 47A10: Spectrum, resolvent 47A35: Ergodic theory 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 2 , May 2018 , pp. 371 - 386
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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