On the almost everywhere convergence of ergodic averages for power-bounded operators on LP-subspaces

Abstract

Let X be a closed subspace of LP(μ), where μ is an arbitrary measure and 1<p<∞. For an invertible power-bounded linear operator U: X→X and n=1,2,..., letA(n) and ℋ(n) denote the discrete ergodic averages and Hilbert transform truncates defined by U. We extend to this setting the μ-a. e. convergence criteria forA(n) and ℋ(n) which V. F. Gaposhkin and R. Jajte introduced for unitary operators on L2(μ). Our methods lift the setting from X to lp, where classical harmonic analysis and interpolation can be applied to suitable square functions.