ERGODICITY AND DIFFERENCES OF FUNCTIONS ON SEMIGROUPS

Iseki [11] defined a general notion of ergodicity suitable for functio ns to phi J --> X where J is an arbitrary abelian semigroup and X is a Banach space. In this paper we develop the theory of such functions, showing in particular that it fits the general framework established b y Eberlein [9] for ergodicity of semigroups of operators acting on X. Moreover, let A be a translation invariant closed subspace of the spac e of all bounded functions from J to X. We prove that if A contains th e constant functions and cp is an ergodic function whose differences l ie in A then phi epsilon A. This result has applications to spaces of sequences facilitating new proofs of theorems of Gelfand and Katznelso n-Tzafriri [12]. We also obtain a decomposition for the space of ergod ic vectors of a representation T : J --> L(X) generalizing results kno wn for the case J = Z(+). Finally, when J is a subsemigroup of a local ly compact abelian group G, we compare the Iseki integrals with the be tter known Cesaro integrals.