, Volume 7, Issue 3, pp 161–175 | Cite as

A Tauberian Theorem for Ergodic Averages, Spectral Decomposability, and the Dominated Ergodic Estimate for Positive Invertible Operators

  • Earl Berkson
  • T.A. Gillespie


Suppose that (Ω,μ) is a σ-finite measure space, and 1 < p < ∞. Let T:Lp(μ → L p(μ) be a bounded invertible linear operator such that T and T−1 are positive. Denote by \({\mathfrak{E}}\)n(T) the nth two-sided ergodic average of T, taken in the form of the nth (C,1) mean of the sequence {Tj+T−j}j =1. Martín-Reyes and de la Torre have shown that the existence of a maximal ergodic estimate for T is characterized by either of the following two conditions: (a) the strong convergence of En(T)n=1; (b) a uniform App estimate in terms of discrete weights generated by the pointwise action on Ω of certain measurable functions canonically associated with T. We show that strong convergence of the (C,2) means of {Tj+T−j}j=1 already implies (b). For this purpose the (C,2) means are used to set up an `averaged' variant of the requisite uniform Ap weight estimates in (b). This result, which can be viewed as a Tauberian-Type replacement of (C,1) means by (C,2) means in (a), leads to a spectral-theoretic characterization of the maximal ergodic estimate by application of a recent result of the authors establishing the strong convergence of the (C,2)-weighted ergodic means for all trigonometrically well-bounded operators. This application also serves to equate uniform boundedness of the rotated Hilbert averages of T with the uniform boundedness of the ergodic averages En(T)n = 1.

positive operator ergodic averages (C 2) summability maximal operator trigonometrically well-bounded operator Ap weight condition 


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  1. 1.
    Abramovich, Yu.A., Veksler, A.I., Koldunov, A.V. On operators preserving disjointness, Soviet Math. Dokl., 20 (1979), 1089–1093 (1980).Google Scholar
  2. 2.
    Aliprantis, C.D. and Burkinshaw, O., Positive Operators, Pure and Applied Math. 119, Academic Press, Orlando, Florida, 1985.Google Scholar
  3. 3.
    Berkson, E. and Gillespie, T.A. AC functions on the circle and spectral families, Journal of Operator Theory, 13 (1985), 33–47.Google Scholar
  4. 4.
    Berkson, E. and Gillespie, T.A. Fourier series criteria for operator decomposability, Integral Equations and Operator Theory, 9 (1986), 767–789.Google Scholar
  5. 5.
    Berkson, E. and Gillespie, T.A. Steckin's theorem, transference, and spectral decompositions, J. Functional Analysis, 70 (1987), 140–170.Google Scholar
  6. 6.
    Berkson, E. and Gillespie, T.A. Mean–boundedness and Littlewood–Paley for separationpreserving operators, Trans. Amer. Math. Soc., 349 (1997), 1169–1189.Google Scholar
  7. 7.
    Berkson, E. and Gillespie, T.A. Spectral integration from dominated ergodic estimates, Ill. J. Math., 43 (1999), 500–519.Google Scholar
  8. 8.
    Berkson, E. and Gillespie, T.A. Spectral decompositions, ergodic averages, and the Hilbert transform, Studia Math., 144 (2001), 39–61.Google Scholar
  9. 9.
    Berkson, E., Gillespie, T.A. and Muhly, P.S. Abstract spectral decompositions guaranteed by the Hilbert transform, Proc. London Math. Soc. (3), 53 (1986), 489–517.Google Scholar
  10. 10.
    Dowson, H.R. Spectral Theory of Linear Operators, London Math. Soc. Monographs, No. 12, Academic Press, New York, 1978.Google Scholar
  11. 11.
    Goldberg, R. Methods of Real Analysis, Blaisdell, New York, 1965.Google Scholar
  12. 12.
    Kan, C.-H. Ergodic properties of Lamperti operators, Canadian J. Math., 30 (1978), 1206–1214.Google Scholar
  13. 13.
    Martín-Reyes, F.J. and de la Torre, A. The dominated ergodic theorem for invertible, positive operators, Semesterbericht Funktionalanalysis Tübingen, Sommersemester 1985, pp. 143–150.Google Scholar
  14. 14.
    Martín-Reyes, F.J. and de la Torre, A. The dominated ergodic estimate for mean bounded, invertible, positive operators, Proc. Amer. Math. Soc., 104 (1988), 69–75.Google Scholar
  15. 15.
    Sato, R. A remark on the ergodic Hilbert transform,Math. J. Okayama Univ., 28 (1986), 159–163.Google Scholar
  16. 16.
    Zygmund, A. Trigonometric Series, Second Edition, vol. 1, Cambridge Univ. Press, Cambridge, 1959.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Earl Berkson
    • 1
  • T.A. Gillespie
    • 2
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaU.S.A
  2. 2.Department of Mathematics and StatisticsUniversity of EdinburghEdinburghScotland

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