A Dichotomy Theorem for the Adjoint of a Semigroup of Operators

Let T{t) be a Co-semigroup of linear operators on a Banach space X, and let 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 > 0. Theorem. Every nonzero orbit of the quotient semigroup on 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® is either zero or nonseparable. If T{t) extends to a Co-group, then X*/X® is either zero or nonseparable. For the proofs we make a detailed study of the second adjoint of a Cosemigroup.