Discrete Spectrum Theorem

Part of the Texts and Readings in Mathematics book series (volume 6)


We know that if two measure preserving automorphisms σ and τ are metrically isomorphic then the associated unitary operators U σ and U τ are unitarily equivalent. Let us say that σ and τ are spectrally isomorphic if U σ and U τ are unitarily equivalent. If σ and τ are spectrally isomorphic and σ is ergodic then τ is ergodic, because σ is ergodic if and only if 1 is a simple eigenvalue of U σ hence also of U τ , which in turn implies the ergodicity of τ. Similarly the mixing and weak mixing properties are invariant under spectral isomorphism. The question whether spectrally isomorphic measure preserving automorphisms are also metrically isomorphic has a negative answer. However in some special cases the answer is affirmative. One such situation is when U σ and U τ admit a complete set of eigenfunctions, σ and τ being ergodic and defined on a standard probability space.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P. Billingsley. Ergodic Theory and Information, John Wiley and Sons, 1965.zbMATHGoogle Scholar
  2. [2]
    P. Halmos and J. von Neumann. Operator Methods in Classical Mechanics, II, Ann. Math., 43(1942), 332–350, 1942, John von Neumann: Collected Works Vol. IV, Pergamon, 251–269.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    D. Ornstein. Bernoulli Shifts with Same Entropy are Isomorphic, Advances in Mathematics, 5 (1970), 337–352.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    D. Ornstein. Ergodic Theory, Randomness and Dynamical Systems, Yale University Press, 1974.zbMATHGoogle Scholar
  5. [5]
    H. L. Royden. Real Analysis, 3rd Edition, MacMillan Publishing Co., 1989.zbMATHGoogle Scholar

Copyright information

© Hindustan Book Agency 2013

Authors and Affiliations

  1. 1.University of MumbaiIndia

Personalised recommendations