von Neumann’s mean ergodic theorem on complete random inner product modules

Research Article

Abstract

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on Lp(ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, Lp(ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and Lp(ℰ(E,H) the complete random normed module generated by Lp(ℰ, H).

Keywords

Random inner product module random normed module random unitary operator random contraction operator von Neumann’s mean ergodic theorem 

MSC

47A35 46H25 46C50 47B80 60G57 

References

  1. 1.
    Albanese A A, Bonet J, Ricker W J. C 0-semigroups and mean ergodic operators in a class of Fréchet spaces. J Math Anal Appl, 2010, 365: 142–157MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Beck A, Schwartz J T. A vector-valued random ergodic theorem. Proc Am Math Soc, 1957, 8: 1049–1059MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chen P D. The theory of random measures. Acta Math Sinica, 1976, 19: 210–216 (in Chinese)MathSciNetMATHGoogle Scholar
  4. 4.
    Dunford N, Schwartz J T. Linear Operators (I). New York: Interscience, 1957Google Scholar
  5. 5.
    Filipović D, Kupper M, Vogelpoth N. Separation and duality in locally L 0-convex modules. J Funct Anal, 2009, 256: 3996–4029MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Fonf V P, Lin M, Wojtaszczyk P. Ergodic characterizations of reflexivity of Banach spaces. J Funct Anal, 2001, 187: 146–162MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Guo T X. Extension theorems of continuous random linear operators on random domains. J Math Anal Appl, 1995, 193: 15–27MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Guo T X. The Radon-Nikodým property of conjugate spaces and the w*-equivalence theorem for w*-measurable functions. Sci China Ser A, 1996, 39: 1034–1041MathSciNetMATHGoogle Scholar
  9. 9.
    Guo T X. Module homomorphisms on random normed modules. Chinese Northeastern Math J, 1996, 12: 102–114MATHGoogle Scholar
  10. 10.
    Guo T X. Some basic theories of random normed linear spaces and random inner product spaces. Acta Anal Funct Appl, 1999, 1: 160–184MathSciNetMATHGoogle Scholar
  11. 11.
    Guo T X. The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure. Sci China Ser A, 2008, 51: 1651–1663MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Guo T X. Relations between some basic results derived from two kinds of topologies for a random locally convex module. J Funct Anal, 2010, 258: 3024–3047MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Guo T X. Recent progress in random metric theory and its applications to conditional risk measures. Sci China Ser A, 2011, 54(4): 633–660MATHCrossRefGoogle Scholar
  14. 14.
    Guo T X. The theory of module homomorphisms in complete random inner product modules and its applications to Skorohod’s random operator theory. PreprintGoogle Scholar
  15. 15.
    Guo T X, Li S B. The James theorem in complete random normed modules. J Math Anal Appl, 2005, 308: 257–265MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Guo T X, Shi G. The algebraic structure of finitely generated L 0(ℱ,K)-modules and the Helly theorem in random normed modules. J Math Anal Appl, 2011, 381: 833–842MATHCrossRefGoogle Scholar
  17. 17.
    Guo T X, Xiao H X, Chen X X. A basic strict separation theorem in random locally convex modules. Nolinear Anal: TMA, 2009, 71: 3794–3804MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Guo T X, You Z Y. The Riesz’s representation theorem in complete random inner product modules and its applications. Chin Ann of Math, Ser A, 1996, 17: 361–364 (in Chinese)MathSciNetMATHGoogle Scholar
  19. 19.
    Guo T X, Zeng X L. Random strict convexity and random uniform convexity in random normed modules. Nonlinear Anal, 2010, 73: 1239–1263MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Guo T X, Zhang X. Stone’s representation theorem on complete complex random inner product modules. PreprintGoogle Scholar
  21. 21.
    Petersen K. Ergodic Theory. Cambridge Studies in Advanced Mathematics 2. London-New York-New Rochelle-Melbourne-Sydney: Cambridge University Press, 1983Google Scholar
  22. 22.
    von Neumann J. Proof of the quasi-ergodic hypothesis. Proc Nat Acad Sci, 1932, 18: 70–82CrossRefGoogle Scholar
  23. 23.
    Yosida K, Kakutani S. Operator theoretical treatment of Markoff’s process and the mean ergodic theorem. Ann of Math, 1941, 42: 188–228MathSciNetCrossRefGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.LMIB and School of Mathematics and Systems ScienceBeihang UniversityBeijingChina

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