Annals of Functional Analysis

Doubly stochastic operators with zero entropy

Bartosz Frej and Dawid Huczek

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study doubly stochastic operators with zero entropy. We generalize three famous theorems: Rokhlin’s theorem on genericity of zero entropy, Kushnirenko’s theorem on equivalence of discrete spectrum and nullity, and Halmos–von Neumann’s theorem on representation of maps with discrete spectrum as group rotations.

Article information

Ann. Funct. Anal., Volume 10, Number 1 (2019), 144-156.

Received: 20 March 2018
Accepted: 22 June 2018
First available in Project Euclid: 16 January 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35}
Secondary: 28D20: Entropy and other invariants 47A35: Ergodic theory [See also 28Dxx, 37Axx]

Markov operator entropy discrete spectrum Kushnirenko’s theorem Halmos–von Neumann’s theorem


Frej, Bartosz; Huczek, Dawid. Doubly stochastic operators with zero entropy. Ann. Funct. Anal. 10 (2019), no. 1, 144--156. doi:10.1215/20088752-2018-0015.

Export citation


  • [1] T. Downarowicz, Entropy in Dynamical Systems, New Math. Monogr. 18, Cambridge Univ. Press, Cambridge, 2011.
  • [2] T. Downarowicz and B. Frej, Measure-theoretic and topological entropy of operators on function spaces, Ergodic Theory Dynam. Systems 25 (2005), no. 2, 455–481.
  • [3] T. Eisner, B. Farkas, M. Haasse, and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Grad. Texts in Math. 272, Springer, Cham, 2015.
  • [4] R. Ellis, Distal transformation groups, Pacific J. Math. 8 (1958), 401–405.
  • [5] B. Frej and P. Frej, The Shannon-McMillan theorem for doubly stochastic operators, Nonlinearity 25 (2012), no. 12, 3453–3467.
  • [6] A. G. Kushnirenko, Metric invariants of entropy type (in Russian), Uspekhi Mat. Nauk 22, no. 5 (1967), 57–66; English translation in Russian Math. Surveys 22 (5) (1967), 53–61.
  • [7] B. Sz.-Nagy and C. Foiaş, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam 1970.
  • [8] V. A. Rokhlin, Entropy of metric automorphism, Dokl. Akad. Nauk 124 (1959), 980–983.
  • [9] A. M. Vershik, What does a typical Markov operator look like? (in Russian), Algebra i Analiz 17, no. 5 (2005), 91–104; English translation in St. Petersburg Math. J. 17 (2006), 763–772.