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Positivity

, 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
Article
  • 41 Downloads

Abstract

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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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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