Almost everywhere convergence of powers of some positive contractions

We extend the solution of Burkholderʹs conjecture for products of conditional expectations, obtained by Delyon and Delyon for L 2 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 T n f converges a.e. for every f ∈ L p , 1 < p < ∞ , with sup n ⁡ | T n f | ∈ L p . The proof uses the work of Le Merdy and Xu on positive L p contractions satisfying Rittʹs 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 { a k } k ∈ Z has bounded angular ratio, then for every positive invertible isometry S of an L p space ( 1 < p < ∞ ), the operator T = ∑ k ∈ Z a k S k is a positive L p contraction such that for every f ∈ L p , T n f converges a.e. and sup n ⁡ | T n f | ∈ 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 Rittʹs condition and f ∈ ( I − T ) α L p ( 0 < α < 1 ) we prove that n α T n f → 0 a.e., and sup n ⁡ n α | T n f | ∈ L p .