Some Ergodic Theorems for One-Parameter Semigroups of Operators

If ^ = {7^: t ^0} is a one-parameter semigroup of operators on a Banach space X, an element x of X is called ergodic if TtX has a generalized limit as t -> oo. It is shown, for a wide class of semigroups, that the use of Abel or Cesaro limits, and of weak or strong convergence, leads to four equivalent definitions of ergodicity. When the resolvent operator of G has suitable compactness properties, every element of X is ergodic. The ergodic properties of G can be completely determined when its infinitesimal generator is known. Some of these results can be extended to more generaltypes of weak convergence in X, and this leads to a discussion of ergodic properties of the semigroup adjoint to G