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The return times and the Wiener—Wintner property for mean-bounded positive operators in Lp

Published online by Cambridge University Press:  19 September 2008

I. Assani
I. Assani
Affiliation:
Department of Mathematics, The University of North Carolina at Chapel Hill, CB #3250 Phillips Hall, Chapel Hill, NC 27514, USA
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Abstract

We prove the following two results for mean-bounded positive operators on Lp(µ) (1<p>∞).

(1) If (X, , µ, ϕ) is a dynamical system and f ∈ L∞ (X) then the sequence f(ϕn x) is a.e. a universal good sequence for mean-bounded positive operators in Lp. (Return times property.)

(2) If T is a mean-bounded positive operator on LP(X, , µ) and f ∈ Lp (µ) then the sequence Tnf)(x) is a.e. a universal good sequence for all dynamical systems (Y, , v,S) in L∞(v). A corollary of (2) is a Wiener-Wintner property for mean-bounded positive operators on Lp.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 1 , March 1992 , pp. 1 - 12
Copyright
Copyright © Cambridge University Press 1992

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References

REFERENCES

[1a]Assani, I.. Dominated estimates for positive operators. Proc. Amer. Math. Soc. 104 (1988), 1131–1133.CrossRefGoogle Scholar
[1b]Assani, I.. Universal weights from dynamical systems to mean bounded positive operators in L p. Proc. Conf. Almost Everywhere II.(Academic:New York,1991) pp. 9–16.CrossRefGoogle Scholar
[2]Assani, I. & Wós, J.. An equivalent measure for nonsingular transformations and application. Studia Mathematica 97 (1990), 1–12.CrossRefGoogle Scholar
[3]Baxter, J. R. & Olsen, J. H.. Weighted and subsequential ergodic theorems. Canad. J. Math. 35 (1983), 145–166.CrossRefGoogle Scholar
[4]Bellow, A. & Losert, V.. The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences. Trans. Amer. Math. Soc. 288 (1985), 307–346.CrossRefGoogle Scholar
[5]Bourgain, J., Furstenberg, H., Katznelson, Y. & Ornstein, D.. Appendix to J. Bourgain's paper: Pointwise ergodic theorems for arithmetic sets. Publ. I.H.E.S. 69 (1989), 42–45.Google Scholar
[6]Brunei, A.. Le théoréme ergodique ponctuel pour les opérateurs positifs à moyennes bornées dan L p. Preprint.Google Scholar
[7]Emilion, R.. Mean bounded operators and mean ergodic theorems. J. Funct. Anal. 61 (1985), 1–14.CrossRefGoogle Scholar
[8]Martin-Reyes, F. & de la Torre, A.. The dominated ergodic estimate for mean bounded, invertible, positive operators. Proc. Amer. Math. Soc. 104 (1988), 69–75.CrossRefGoogle Scholar
[9]Krengel, U.. Ergodic Theorems, de Gruyter, 1985.CrossRefGoogle Scholar
[10]Nikishin, E. M.. Resonance theorems and superlinear operators. (Russian) Usp. Mat. Nauk. 25 (1970), 129–191.Google Scholar
Translated in Russian Math. Surveys 25 no. 6 (1970), 125–187.Google Scholar
[11]Wiener, N. & Wintner, A.. Harmonic analysis and ergodic theory. Amer. J. Math. 63 (1941), 415–426.CrossRefGoogle Scholar