Ergodic theorems for Abelian semigroups

Abstract

This paper adapts the method of F. Riesz to the proof of certain general ergodic theorems for Abelian semi-groups of operators on a Banach space to itself. The main features of the method are that no measurability conditions are imposed on the semi-group under consideration and that consistent use of the second conjugate space and its compactness properties make it possible to replace the compactness conditions often imposed by a more natural restriction on the transforms of points. Theorem 1 and the various supplementary results include as special cases theorems of Lorch [10], Dunford [7], Yosida [15], F. Riesz [12], and Cohen [6]. It overlaps the work of Alaoglu and Birkhoff [3] at those points where they consider Abelian cases; for example, Corollary 8 is a great generalization of their Theorem 5. Section 1 contains some introductory material on conjugate spaces and adjoint operations. Section 2 introduces bounded Abelian semi-groups of operators and near invariance of a system of set functions on such a semi-group; this section also contains the principal theorem (Theorem 1) of the paper. The form of this theorem raises three questions ((A) to (C) at the beginning of ?3). The answer to (A) shows, among other things, that every Abelian semi-group has a property much like ergodicity in the sense of Alaoglu and Birkhoff; Theorem 3 is the main result here. The answer to (B) again indicates the importance of reflexivity in theorems of this type; Corollary 8 is one example. Two special cases of (C) give a generalization of Dunford's theorem (Theorem 5) and a theorem on bounded Abelian semi-groups of projections (Theorem 6) which has not, so far as I know, been considered before. 1. Some properties of Banach spaces. If B is a Banach space(2), let B* be the set of all linear-that is, additive and continuous-real-valued functions on B. If, for A in B*, |I|3 =SUpjjbJJ 0, k a positive integer, and I1, . , f3k in B*. With this topology B is a linear topological