Dzung M. Ha

Sweeping out properties of operator sequences

Let Lp ≥ Lp(X,ñ), 1 p 1, be the usual Banach Spaces of real valued functions on a complete non-atomic probability space. Let (T1, . . . , TK) be L2contractions. Let 0 Ú ¢ Ú é 1. Call a function f a é-spanning function if kfk2 ≥ 1 and if kTkf Qk 1Tkfk2 1⁄2 é 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 (é, ¢) -sweeping function if khk1 1, khk1 Ú ¢, and if max1 k K jTkhj Ù é ¢ on a set of measure greater than 1 ¢. The following is the main technical result, which is obtained by elementary estimates. There is an integer K ≥ K(¢, é) 1⁄2 1 such that if f is a é-spanning function, and if the joint distribution of (f , T1f , . . . , TK f ) is normal, then h ≥ (f^M)_( M) ÛM is a (é, ¢)-sweeping function, for some M Ù 0. Furthermore, if Tks are the averages of operators induced by the iterates of a measure 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 1⁄2 1 there is a subsequence (Ti1 , . . . , TiK ) of length K, and a é-spanning function fK for this subsequence. Then for each ¢ Ù 0 there is a function h, 0 h 1, khk1 Ú ¢, such that lim supi Tih 1⁄2 é 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. The first author was partially supported by an NSERC Grant. The second author was partially supported by an NSERC Grant. The third author was partially supported by an NSF Grant. Received by the editors May, 1995. AMS subject classification: Primary: 28D99; Secondary: 60F99.