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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

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