Averaging theorems for linear operators in compact groups and semigroups

The Weyl criterion for uniform distribution of a sequence has an especially simple form in compact abelian groups. The authors use this and the structure of compact monothetic groups and semigroups to generalise the convergence, under certain compactness conditions, of the operator averages: $n^{-1} \sum_{k=1}^n T^k → P (n→ ∞)$ where P is a projection associated with the eigenvalue one of T