A DICHOTOMY THEOREM FOR THE ADJOINT OF A SEMIGROUP OF OPERATORS

Let T(t) be a C0-semigroup of linear operators on a Banach space X, and let X(X) , resp. X., denote the closed subspaces of X* consisting of all functionals x* such that the map t --> T*(t)x* is strongly continuous for t > 0 , resp. t greater-than-or-equal-to 0. Theorem. Every nonzero orbit of the quotient semigroup on X*/X(X) is nonseparably valued. In particular, orbits in X*/X. are either zero for t > 0 or nonseparable. It also follows that the quotient space X*/X(X) is either zero or nonseparable. If T(t) extends to a C0-group, then X*/X. is either zero or nonseparable. For the proofs we make a detailed study of the second adjoint of a C0-semigroup.