Uniform convergence on subspaces in von Neumann's ergodic theorem with continuous time

Abstract

Power uniform (in the operator norm) convergence on vector subspaces
with their own norms in von Neumann's ergodic theorem with
continuous time is considered. All possible exponents of the
considered power convergence are found; for each of these exponents,
spectral criteria for such convergence are given and a complete
description of all such subspaces is obtained. Uniform convergence
over the entire space takes place only in trivial cases, which
explains the interest in the uniform convergence just on subspaces.

In addition, along the way, the old convergence rate estimates in the von Neumann ergodic theorem for (semi)flows are generalized and refined.