Interrelation between the convergence rates in von Neumann's and Birkhoff's ergodic theorems

In the $L_p$ spaces, $1<p<\infty$, we prove some inequalities for discrete and continuous times that make it possible to obtain the convergence rate in Birkhoff's theorem in the presence of bounds on the convergence rate in von Neumann's ergodic theorem belonging to a sufficiently large rate range. The exact operator analogs of these inequalities for contraction semigroups in $L_p$ are given. These results also have the obvious exact analogs in the class of wide-sense stationary stochastic processes.