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Some new examples of universal hypercyclic operators in the sense of Glasner and Weiss

Author: Sophie Grivaux
Journal: Trans. Amer. Math. Soc. 369 (2017), 7589-7629
MSC (2010): Primary 47A16, 37A35, 47A35, 47B35, 47B37
Published electronically: March 6, 2017
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Abstract: A bounded operator $ A$ on a real or complex separable infinite-dimensional Banach space $ Z$ is universal in the sense of Glasner and Weiss if for every invertible ergodic measure-preserving transformation $ T$ of a standard Lebesgue probability space $ (X,\mathcal {B},\mu )$, there exists an $ A$-invariant probability measure $ \nu $ on $ Z$ with full support such that the two dynamical systems $ (X,\mathcal {B},\mu ;T)$ and $ (Z,\mathcal {B}_{Z},\nu ;A)$ are isomorphic. We present a general and simple criterion for an operator to be universal, which allows us to characterize universal operators among unilateral or bilateral weighted shifts on $ \ell _{p}$ or $ c_{0}$, to show the existence of universal operators on a large class of Banach spaces and to give a criterion for universality in terms of unimodular eigenvectors. We also obtain similar results for operators which are universal for all ergodic systems (not only for invertible ones) and study necessary conditions for an operator on a Hilbert space to be universal.

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Additional Information

Sophie Grivaux
Affiliation: CNRS, Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France

Keywords: Universal hypercyclic operators, ergodic theory of linear dynamical systems, frequently hypercyclic operators, isomorphisms of dynamical systems
Received by editor(s): October 12, 2014
Received by editor(s) in revised form: October 3, 2015, and October 13, 2015
Published electronically: March 6, 2017
Additional Notes: This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01)
Article copyright: © Copyright 2017 American Mathematical Society

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