Laplace Transforms and Generators of Semigroups of Operators

In this paper, a characterization for continuous functions on $(0,\infty)$ to be the Laplace transforms of $f\in L^{\infty}(0,\infty)$ is obtained. It is also shown that the vector-valued version of this characterization holds if and only if the underlying Banach space has the Radon-Nikodým property. Using these characterizations, some results, different from that of the Hille-Yosida theorem, on generators of semigroups of operators are obtained.