Convergence of the lacunary ergodic Cesàro averages

Let T be a positive linear operator with positive inverse. We consider in this paper the ergodic Cesàro-α averages An,αf, 0<α⩽1, and the ergodic Cesàro-α maximal operator associated to T. For Lebesgue spaces Lp(ν), it is known that the good range for the convergence of the Cesàro-α averages and the boundedness of the maximal operator is 1/α