Geometric & Functional Analysis GAFA

, Volume 5, Issue 1, pp 1–13 | Cite as

Unimodular eigenvalues and linear chaos in Hilbert spaces

  • E. Flytzanis


For linear operatorsT in a complex separable Hilbert spaceH we consider the problem of existence of invariant Gaussian measuresm:mT−1=m. We relate the size of the unimodular point spectrum ofT to mixing properties of the measure preserving transformations defined byT with respect to such invariant measures, and we draw some conclusions concerning orbit structure properties ofT.


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  1. [A]
    J. Aaronson, Category theorems for some ergodic multiplier properties, Israel J. Math 51 (1985), 1–12.Google Scholar
  2. [B]
    B. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, North Holland (1988).Google Scholar
  3. [Be]
    A. Beck, Eigenoperators of ergodic transformations, T.A.M.S. 9 (1960), 118–129.Google Scholar
  4. [CFomS]
    I. P. Cornfeld, S.V. Fomin, Ya. G. Sinai, Ergodic Theory, Springer-Verlag (1980).Google Scholar
  5. [D]
    R. Devaney, Chaotic Dynamical Systems, Addison-Wesley (1989).Google Scholar
  6. [Di]
    J. Dixmier, Les Algebres d'Operateurs dans l'Espace Hilbertien, Gauthier-Villars (1975), 139–156.Google Scholar
  7. [F]
    E. Flytzanis, Invariant measure for linear operators, Springer Lecture Notes in Mathematics 709 (1979), 103–110.Google Scholar
  8. [FK]
    E. Flytzanis, L. Kanakis, Measure preserving composition operators, J. Funct. Anal. 73:1 (1987), 113–121.Google Scholar
  9. [Fo]
    S. Foguel, Invariant subspaces of m.p.t., Israel J. Math. 2 (1964), 198–200.Google Scholar
  10. [Fu]
    H. Furstenberg, IP-systems in ergodic theory. Contem. Math. 28 (1984), 131–148.Google Scholar
  11. [GSh]
    G. Godefroy, L.H. Shapiro, Operators with dense invariant cyclic vector manifolds 98:2 (1991), 229–269.Google Scholar
  12. [Ku]
    H.H. Kuo, Gaussian Measures in Banach Spaces, Springer Lecture Notes in Mathematics 463 (1975), 1–35.Google Scholar
  13. [L]
    L.D. Landau, On the problem of turbulence, Akad. Nauk, Diklady 44 (1934), 339–344.Google Scholar
  14. [N]
    I.N. Nikolskaya, Geometric properties of system of characteristic vectors and point spectra of linear operators, Fun. Ann. Appl. 4:3 (1970), 105–106.Google Scholar
  15. [P]
    K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press (1967).Google Scholar
  16. [RT]
    D. Ruelle, F. Takens, On the nature of turbulence, Comm. Math. Physics 20 (1971), 167–192.Google Scholar

Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • E. Flytzanis
    • 1
  1. 1.Athens University of Economics and BusinessAthensGreece

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