Almost everywhere convergence of powers of some positive L_p contractions

We extend the solution of Burkholder's conjecture for products of conditional expectations, obtained by Delyon and Delyon for L₂ and by Cohen for L_p, 1<p<∞, to the context of Badea and Lyubich: Let T be a finite convex combination of operators T_j which are products of finitely many conditional expectations. Then Tⁿf converges a.e. for every f∈L_p, 1<p<∞, with sup_n|Tⁿf|∈L_p. The proof uses the work of Le Merdy and Xu on positive L_p 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∈Z has bounded angular ratio, then for every positive invertible isometry S of an L_p space (1<p<∞), the operator T=∑k∈ZakSk is a positive L_p contraction such that for every f∈L_p, Tⁿf converges a.e. and sup_n|Tⁿf|∈L_p. If {a_k} is supported on N, the same result is true when S is only a positive contraction of L_p. Similar results are obtained for μ-averages of bounded continuous representations of a σ-compact LCA group by positive operators in one L_p space, 1<p<∞. For a positive contraction T on L_p which satisfies Ritt's condition and f∈(I-T)^αL_p (0<α<1) we prove that n^αTⁿf→0 a.e., and sup_nn^α|Tⁿf|∈L_p.