Constants in the estimates of the rate of convergence in von Neumann's ergodic theorem with continuous time

Estimates for the rate of convergence in ergodic theorems are necessarily spectral. We find the equivalence constants relating the polynomial rate of convergence in von Neumann's mean ergodic theorem with continuous time and the polynomial singularity at the origin of the spectral measure of the function averaged over the corresponding dynamical system. We also estimate the same rate of convergence with respect to the decrease rate of the correlation function. All results of this article have obvious exact analogs for the stochastic processes stationary in the wide sense.