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Sweeping Out Properties of Operator Sequences

Published online by Cambridge University Press:  20 November 2018

Mustafa A. Akcoglu ,
Dzung M. Ha  and
Roger L. Jones
Mustafa A. Akcoglu
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 1A1
Dzung M. Ha
Affiliation:
Department of Mathematics and Statistics, Sultan Qaboos University, Sultanate of Oman
Roger L. Jones
Affiliation:
Department of Mathematics, DePaul University, 2219 N. Kenmore, Chicago IL, USA 60614
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Abstract

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Let Lp = Lp(X, μ), 1 ≤ p ≤ ∞, be the usual Banach Spaces of real valued functions on a complete non-atomic probability space. Let (T1, . . . ,TK) be L2-contractions. Let 0 < ε < δ ≤ 1. Call a function f a δ-spanning function if ‖f‖2 = 1 and if ‖Tkf - Qk-1Tkf‖2 ≥ δ for each k = 1, . . . ,K, where Q0 = 0 and Qk is the orthogonal projection on the subspace spanned by (T1f , . . . ,Tkf). Call a function h a (δ, ε) -sweeping function if ‖h‖∞ ≤ 1, ‖h‖1 < ε, and if max1≤k≤K|Tkh| > δ-ε on a set of measure greater than 1 - ε. The following is the main technical result, which is obtained by elementary estimates. There is an integer K = K(δ, ε) ≥ 1 such that if f is a δ-spanning function, and if the joint distribution of (f , T1f , . . . ,TKf) is normal, then h = ((fΛM)Ꮩ(-M)/M is a (δ, ε)-sweeping function, for some M > 0. Furthermore, if Tks are the averages of operators induced by the iterates of ameasure preserving ergodic transformation, then a similar result is true without requiring that the joint distribution is normal. This gives the following theorem on a sequence (Ti) of these averages.Assume that for each K ≤ 1 there is a subsequence (Ti1 , . . . ,Tik) of length K, and a δ-spanning function fK for this subsequence. Then for each ε > 0 there is a function h, 0 ≥ h ≥ 1, ‖h‖1 < ε, such that lim supi Tih ≤ δ a.e.. Another application of the main result gives a refinement of a part of Bourgain’s “Entropy Theorem”, resulting in a different, self contained proof of that theorem.

Keywords

28D9960F99Strong and δ-sweeping outGaussian distributionsBourgain’s entropy theorem
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 1 , 01 February 1997 , pp. 3 - 23
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Akcoglu, M., Bellow, A., Jones, R., Losert, V., Reinhold, K. andM.Wierdl, The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters, Ergodic Theory Dynamical Systems, to appear.Google Scholar
2. Akcoglu, M., del Junco, A., Lee, W., A solution to a problem of A. Bellow. In: Almost Everywhere Convergence II, (eds. Bellow, A. and Jones, R.), Academic Press, 1991. 1–7.Google Scholar
3. Akcoglu, M., Ha, M.D. and Jones, R.L., Divergence of Ergodic Averages. In: Topological vector spaces, algebras and related areas, (eds. Anthony To-Ming Lau and Ian Tweddle), Pitman Research Notes inMathematics Series 316, Longman, 1994. 175–192.Google Scholar
4. Bellow, A., On bad universal sequences in ergodic theory (II), Springer Verlag Lecture Notes in Math.. 1033(1983).Google Scholar
5. Bellow, A. , Sur la structure des suites mauvaises universelles en theorie ergodique, Comptes Rendus Acad. Sci. Paris. 294(1982), 55–58.Google Scholar
6. Bellow, A. , Two problems, Proc. Oberwolfach Conference on Measure Theory (June 1981), Springer Lecture Notes in Math. 945(1982).Google Scholar
7. Bellow, A. and Jones, R., A Banach Principle for L1, Adv. Math., to appear.Google Scholar
8. Bellow, A., Jones, R. and Rosenblatt, J., Convergence for moving averages, Ergodic Theory Dynamical Systems. 10(1990), 43–62.Google Scholar
9. Bourgain, J., On the maximal theorem for certain subsets of the integers, Israel J. Math. 61(1988), 39–72.Google Scholar
10. Bourgain, J., Almost sure convergence and bounded entropy, Israel J. Math. 63(1988), 79–97.Google Scholar
11. del Junco, A. and Rosenblatt, J., Counterexamples in ergodic theory and number theory, Math. Ann. 245(1979), 185–197.Google Scholar
12. Feller, W., An introduction to Probability theory and its applications, vol. 1, 2nd edition, John Wiley & Sons, Inc., 1957.Google Scholar
13. Ha, D., Operators with Gaussian distribution property, L2-entropy, and almost everywhere convergence, Ph.D. Thesis, University of Toronto, 1994.Google Scholar
14. Jones, R., A remark on singular integrals with complex homogeneity, Proc. Amer. Math. Soc. 114(1992), 763–768.Google Scholar
15. Jones, R. and Wierdl, M., Convergence and divergence of ergodic averages, Ergodic Theory Dynamical System. 14(1994), 515–535.Google Scholar
16. Krengel, U., On the individual ergodic theorem for subsequences, Ann. Math. Stat. 42(1971), 1091.ndash;1095.Google Scholar
17. Rosenblatt, J., Ergodic group actions, Arch.Math. 47(1986), 263–269.Google Scholar
18. Krengel, U., Universally bad sequences in ergodic theory. In: Almost EverywhereConvergence II, (eds.Bellow, A. and Jones, R.), Academic Press, 1991. 227–245.Google Scholar
19. Rudin, W., An arithmetic property of Riemann sums, Proc. Amer.Math. Soc. 15(1964), 321–324.Google Scholar