Convergence rates in mean ergodic theorems for Cesàro bounded operators

Abstract We establish the (polynomially) logarithmic decay of ergodic means of Cesàro bounded operators of any fractional order, under convergence of the one-sided ergodic Hilbert transform. This extends the theorem of Gomilko, Haase and Tomilov for power bounded operators. We also improve the polynomial decay of means involved in the fractional Poisson equation. The theorems are obtained as an application of a general result, also proved here, about rates of decay of means for Cesàro bounded operators.