An Ergodic Theorem for a Noncommutative Semigroup of Linear Operators

Introduction. The question of ergodicity of a semigroup of bounded linear operators on a Banach space has been reduced, by Alaoglu and Birkhoff [l],1 Day [2, 3], and Eberlein [4], to the study firstly of the ergodicity of the semigroup itself and secondly, of the ergodicity of each element of the Banach space with respect to this ergodic semigroup. In the case of a bounded and commutative semigroup operating on a reflexive Banach space, the ergodicity has been established [2, Corollary 6], [4, Corollary 5.1, Theorem 5.5]. For noncommutative semigroups, Alaoglu and Birkhoff have proved ergodicity by restricting the Banach space [l, Theorem 6]. A general discussion of the noncommutative case is given by Day [3]. The following is an attempt to prove ergodicity, in the strong topology, for a certain noncommutative case, where we use a weakened form of the commutative law. Our method uses a fixed point theorem of Kakutani [S], and a theorem about the group of cluster points of a directed subsemigroup, which we have developed elsewhere [ó]. It should be observed, however, that the theorem which is proved here, though being similar to, is not a generalization of the results mentioned above. In fact it is apparently independent of them, because we postulate here a compactness in the strong topology of operators; whereas in the analogous theorems of Day and Eberlein, a kind of weak compactness is used, by considering bounded semigroups acting on reflexive spaces, a natural legacy from the study of Hubert space. However our theorem is a generalization of a theorem of Yosida [7, Theorem 2],