Uniform Convergence on Subspaces in von Neumann Ergodic Theorem with Discrete Time

We consider the power-law uniform (in the operator norm) convergence on vector subspaces with their own norms in the von Neumann ergodic theorem with discrete time. All possible exponents of the considered power-law convergence are found; for each of these exponents, spectral criteria for such convergence are given and the complete description of all such subspaces is obtained. Uniform convergence on the whole space takes place only in the trivial cases, which explains the interest in uniform convergence precisely on subspaces. In addition, by the way, old estimates of the rates of convergence in the von Neumann ergodic theorem for measure-preserving mappings are generalized and refined.