Laplace transforms of vector-valued functions with growth ω and semigroups of operators

For an arbitrary continuous, increasing function ω : [0, ∞) ↦ C of finite exponential type, we characterize Laplace-Stieltjes transforms of Banach-space-valued functions that are O(ω) , and use this to establish a Hille-Yosida type theorem for strongly continuous semigroups that are O(ω) . Corollaries include characterizing generators of strongly continuous semigroups that are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$O\left( {\exp \left( {dt^{\tfrac{1}{d}} } \right)} \right)$$ \end{document}, for d > 1 , in addition to the already known examples of exponential or polynomial growth.