A dichotomy theorem for the adjoint of a semigroup of operators

Abstract:

Let $T(t)$ be a ${C_0}$-semigroup of linear operators on a Banach space $X$, and let ${X^ \otimes }$, resp. ${X^ \odot }$, denote the closed subspaces of ${X^{\ast }}$ consisting of all functionals ${x^{\ast }}$ such that the map $t \mapsto {T^{\ast }}(t){x^{\ast }}$ is strongly continuous for $t > 0$, resp. $t \geqslant 0$. Theorem. Every nonzero orbit of the quotient semigroup on ${X^{\ast }}/{X^ \otimes }$ is nonseparably valued. In particular, orbits in ${X^{\ast }}/{X^ \odot }$ are either zero for $t > 0$ or nonseparable. It also follows that the quotient space ${X^{\ast }}/{X^ \otimes }$ is either zero or nonseparable. If $T(t)$ extends to a ${C_0}$-group, then ${X^{\ast }}/{X^ \odot }$ is either zero or nonseparable. For the proofs we make a detailed study of the second adjoint of a ${C_0}$-semigroup.