ON THE TOPOLOGY INDUCED BY THE ADJOINT OF A SEMIGROUP OF OPERATORS

The adjoint of a C0-semigroup on a Banach space X induces a locally convex sigma(X, X.)-topology in X, which is weaker than the weak topology of X. In this paper we study the relation between these two topologies. Among other things a class of subsets of X is given on which they coincide. As an application, an Eberlein-Shmulyan type theorem is proved for the sigma (X, X.)-topology and it is sho that the uniform limit of sigma (X, X.)-compact operators is sigma (X, X.)-compact. Finally our results are applied to the problem when the Favard class of a semigroup equals the domain of the infinitesimal generator.