Almost everywhere convergence of powers of some positive [formula omitted] contractions

Abstract

We extend the solution of Burkholder's conjecture for products of conditional expectations, obtained by Delyon and Delyon for L2 and by Cohen for Lp, 1<p<∞, to the context of Badea and Lyubich: Let T be a finite convex combination of operatorsTjwhich are products of finitely many conditional expectations. ThenTnfconverges a.e. for everyf∈Lp, 1<p<∞, withsupn⁡|Tnf|∈Lp. The proof uses the work of Le Merdy and Xu on positive Lp contractions satisfying Ritt's resolvent condition. As another application of the work of Le Merdy and Xu, we extend a result of Bellow, Jones and Rosenblatt, proving that if a probability{ak}k∈Zhas bounded angular ratio, then for every positive invertible isometry S of anLpspace (1<p<∞), the operatorT=∑k∈ZakSkis a positiveLpcontraction such that for everyf∈Lp, Tnfconverges a.e. andsupn⁡|Tnf|∈Lp. If{ak}is supported onN, the same result is true when S is only a positive contraction ofLp. Similar results are obtained for μ-averages of bounded continuous representations of a σ-compact LCA group by positive operators in one Lp space, 1<p<∞. For a positive contraction T on Lp which satisfies Ritt's condition and f∈(I−T)αLp (0<α<1) we prove that nαTnf→0 a.e., and supn⁡nα|Tnf|∈Lp.