Sweeping Out Properties of Operator Sequences

Akcoglu, Mustafa A. ; Ha, Dzung M. ; Jones, Roger L.

Canadian journal of mathematics, Tome 49 (1997) no. 1, pp. 3-23

Résumé

Let Lp = Lp (X, μ), 1 ≤ p ≤ ∞, be the usual Banach Spaces of real valued functions on a complete non-atomic probability space. Let (T 1, . . . ,T K ) be L 2-contractions. Let 0 < ε < δ ≤ 1. Call a function f a δ-spanning function if ‖f‖2 = 1 and if ‖Tkf - Q k-1 Tkf‖2 ≥ δ for each k = 1, . . . ,K, where Q 0 = 0 and Qk is the orthogonal projection on the subspace spanned by (T 1 f , . . . ,Tkf). Call a function h a (δ, ε) -sweeping function if ‖h‖∞ ≤ 1, ‖h‖1 < ε, and if max1≤k≤K |Tk h| > 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 , T 1 f , . . . ,TKf) is normal, then h = ((fΛM)Ꮩ(-M)/M is a (δ, ε)-sweeping function, for some M > 0. Furthermore, if Tk s 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 (Ti 1 , . . . ,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.

Détail

DOI : 10.4153/CJM-1997-001-3

Mots-clés : 28D99, 60F99, Strong and δ -sweeping out, Gaussian distributions, Bourgain’s entropy theorem

Bibliographie

[1] 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] 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] 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] 4. Bellow, A., On bad universal sequences in ergodic theory (II), Springer Verlag Lecture Notes in Math.. 1033(1983). Google Scholar

[5] 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] 6. Bellow, A. , Two problems, Proc. Oberwolfach Conference on Measure Theory (June 1981), Springer Lecture Notes in Math. 945(1982). Google Scholar

[7] 7. Bellow, A. and Jones, R., A Banach Principle for L1, Adv. Math., to appear. Google Scholar

[8] 8. Bellow, A., Jones, R. and Rosenblatt, J., Convergence for moving averages, Ergodic Theory Dynamical Systems. 10(1990), 43–62. Google Scholar

[9] 9. Bourgain, J., On the maximal theorem for certain subsets of the integers, Israel J. Math. 61(1988), 39–72. Google Scholar

[10] 10. Bourgain, J., Almost sure convergence and bounded entropy, Israel J. Math. 63(1988), 79–97. Google Scholar

[11] 11. del Junco, A. and Rosenblatt, J., Counterexamples in ergodic theory and number theory, Math. Ann. 245(1979), 185–197. Google Scholar

[12] 12. Feller, W., An introduction to Probability theory and its applications, vol. 1, 2nd edition, John Wiley &amp; Sons, Inc., 1957. Google Scholar

[13] 13. Ha, D., Operators with Gaussian distribution property, L2-entropy, and almost everywhere convergence, Ph.D. Thesis, University of Toronto, 1994. Google Scholar

[14] 14. Jones, R., A remark on singular integrals with complex homogeneity, Proc. Amer. Math. Soc. 114(1992), 763–768. Google Scholar

[15] 15. Jones, R. and Wierdl, M., Convergence and divergence of ergodic averages, Ergodic Theory Dynamical System. 14(1994), 515–535. Google Scholar

[16] 16. Krengel, U., On the individual ergodic theorem for subsequences, Ann. Math. Stat. 42(1971), 1091.ndash;1095. Google Scholar

[17] 17. Rosenblatt, J., Ergodic group actions, Arch.Math. 47(1986), 263–269. Google Scholar

[18] 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] 19. Rudin, W., An arithmetic property of Riemann sums, Proc. Amer.Math. Soc. 15(1964), 321–324. Google Scholar