C-Semigroups and the Cauchy Problem

We extend the definition of generator to C-semigroups that may not be exponentially bounded, where the range of C may not be dense. We then characterize linear operators, A, for which the associated abstract Cauchy problem has a unique solution, for every initial value in the domain of another operator, B, without assuming that the domain of A is dense, or the solutions are exponentially bounded. We also give Hille-Yosida type characterizations of generators that may not be densely defined.