C-Cosine Functions and the Abstract Cauchy Problem, I*

If A is the generator of an exponentially bounded C-cosine function on a Banach space X, then the abstract Cauchy problem ACP. for A has a unique solution for every pair x, y. of initial values from l yA.y1C X.. The main result is a characterization of the generator of a C-cosine function, which may not be exponentially bounded and may have a nondensely defined generator, in terms of the associated ACP.