We consider the class of resolvent estimators of the corrélation operator ruling the functional autoregressive processes introduced by Mas, A. ([10] [11]). Under mild conditions on smoothing parameter, we establish exponential bounds and almost sure convergence of the resolvent estimators as well as convergence rates improving the existing results. As a conséquence we dérivé asymptotic results on the resolvent predictors. Numerical studies illustrate the performance of the resolvent predictors giving a comparison with other existing prédiction methods both on simulated and real functional data sets showing compétitive results.