Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T15:13:16.394Z Has data issue: false hasContentIssue false
  • English
  • Français

Mixing operators with prescribed unimodular eigenvalues

Part of: Miscellaneous topics of analysis in the complex domain General theory of linear operators Ergodic theory

Published online by Cambridge University Press:  28 December 2020

H.-P. BEISE ,
L. FRERICK  and
J. MÜLLER Open the ORCID record for J. MÜLLER [Opens in a new window]
H.-P. BEISE
Affiliation:
Fachbereich Informatik, Hochschule Trier, D-54293Trier, Germany (e-mail:H.Beise@inf.hochschule-trier.de)
L. FRERICK
Affiliation:
Fachbereich IV Mathematik, Universität Trier, D-54286Trier, Germany (e-mail:frerick@uni-trier.de)
J. MÜLLER*
Affiliation:
Fachbereich IV Mathematik, Universität Trier, D-54286Trier, Germany (e-mail:frerick@uni-trier.de)
*
e-mail:jmueller@uni-trier.de
Get access Rights & Permissions [Opens in a new window]

Abstract

For arbitrary closed countable subsets Z of the unit circle examples of topologically mixing operators on Hilbert spaces are given which have a densely spanning set of eigenvectors with unimodular eigenvalues restricted to Z. In particular, these operators cannot be ergodic in the Gaussian sense.

Keywords

Taylor shiftunimodular eigenvaluesmixing operatorrational approximation

MSC classification

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators
Secondary: 30E10: Approximation in the complex domain 37A25: Ergodicity, mixing, rates of mixing
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 1 , January 2022 , pp. 1 - 8
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bayart, F. and Grivaux, S.. Frequently hypercyclic operators. Trans. Amer. Math. Soc. 358 (2006), 5083–5117.CrossRefGoogle Scholar
Bayart, F. and Grivaux, S.. Invariant Gaussian measures for operators on Banach spaces and linear dynamics. Proc. Lond. Math. Soc. 94 (2007), 181–210.CrossRefGoogle Scholar
Bayart, F. and Matheron, É.. Dynamics of Linear Operators. Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
Bayart, F. and Matheron, É.. Mixing operators and small subsets of the circle. J. Reine Angew. Math. 715 (2016), 75–123.Google Scholar
Beise, H.-P., Meyrath, T. and Müller, J.. Mixing Taylor shifts and universal Taylor series. Bull. Lond. Math. Soc. 47 (2015), 136–142.CrossRefGoogle Scholar
Beise, H.-P. and Müller, J.. Generic boundary behaviour of Taylor series in Hardy and Bergman spaces. Math. Z. 284 (2016), 1185–1197.CrossRefGoogle Scholar
Gaier, D.. Vorlesungen über Approximation im Komplexen. Birkhäuser, Basel, 1980.CrossRefGoogle Scholar
Grosse-Erdmann, K.-G. and Peris Manguillot, A.. Linear Chaos. Springer, London, 2011.CrossRefGoogle Scholar
Grivaux, S.. A new class of frequently hypercyclic operators. Indiana Univ. Math. J. 60 (2011), 1177–1201.CrossRefGoogle Scholar
Grivaux, S.. Ten questions in linear dynamics. Études Opératorielles (Banach Center Publications, 112). Institute of Mathematics of the Polish Academy of Sciences, Warsaw, 2017, pp. 143–151.Google Scholar
Grivaux, S., Matheron, E. and Menet, Q.. Linear dynamical systems on Hilbert spaces. Mem. Amer. Math. Soc., to appear.Google Scholar
Hedberg, L. I.. Approximation in the mean by analytic functions. Trans. Amer. Math. Soc. 163 (1972), 157–171.CrossRefGoogle Scholar
Menet, Q.. Linear Chaos and frequent hypercyclicity. Trans. Amer. Math. Soc. 369 (2017), 4977–4994.CrossRefGoogle Scholar
Mergelyan, S. N.. On the completeness of systems of analytic functions. Uspekhi Mat. Nauk (N. S.) 8 (1953), 3–63 (in Russian). Engl. transl. Trans. Amer. Math. Soc. 19 (1962), 109–166.Google Scholar
Luecking, D. H. and Rubel, L. A.. Complex Analysis: A Functional Analysis Approach. Springer, New York, 1984.CrossRefGoogle Scholar
Müller, J. and Thelen, M.. Dynamics of the Taylor shift on Bergman spaces. Preprint, 2020, arXiv:2001.02952.Google Scholar