ON THE ALMOST EVERYWHERE CONVERGENCE OF ERGODIC AVERAGES FOR POWER-BOUNDED OPERATORS ON LP-SUBSPACES

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