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

- 41 Downloads

## Abstract

Suppose that (Ω,μ) is a σ-finite measure space, and 1 < p < ∞. Let T:L^{p}(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 *n*th two-sided ergodic average of T, taken in the form of the *n*th (C,1) mean of the sequence {T^{j}+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 E_{n}(T)_{n=1}^{∞}; (b) a uniform *A*_{p}*p* 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 {T^{j}+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 *A*_{p} 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 E_{n}(T)_{n = 1}^{∞}.

*A*

_{p}weight condition

## Preview

Unable to display preview. Download preview PDF.

## References

- 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.Aliprantis, C.D. and Burkinshaw, O.,
*Positive Operators,*Pure and Applied Math. 119, Academic Press, Orlando, Florida, 1985.Google Scholar - 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.Berkson, E. and Gillespie, T.A. Fourier series criteria for operator decomposability,
*Integral Equations and Operator Theory*,**9**(1986), 767–789.Google Scholar - 5.Berkson, E. and Gillespie, T.A. Steckin's theorem, transference, and spectral decompositions,
*J. Functional Analysis*,**70**(1987), 140–170.Google Scholar - 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.Berkson, E. and Gillespie, T.A. Spectral integration from dominated ergodic estimates,
*Ill. J. Math.*,**43**(1999), 500–519.Google Scholar - 8.Berkson, E. and Gillespie, T.A. Spectral decompositions, ergodic averages, and the Hilbert transform,
*Studia Math.*, 144 (2001), 39–61.Google Scholar - 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.Dowson, H.R.
*Spectral Theory of Linear Operators*, London Math. Soc. Monographs, No. 12, Academic Press, New York, 1978.Google Scholar - 11.Goldberg, R.
*Methods of Real Analysis,*Blaisdell, New York, 1965.Google Scholar - 12.Kan, C.-H. Ergodic properties of Lamperti operators,
*Canadian J. Math.*,**30**(1978), 1206–1214.Google Scholar - 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.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.Sato, R. A remark on the ergodic Hilbert transform,
*Math. J. Okayama Univ.*,**28**(1986), 159–163.Google Scholar - 16.Zygmund, A. Trigonometric Series, Second Edition, vol. 1, Cambridge Univ. Press, Cambridge, 1959.Google Scholar