An Inversion Formula for Laplace Transforms and Semi-Groups of Linear Operators

Abstract

This paper is primarily concerned with the problem of determining necessary and sufficient conditions that a closed linear operator be the infinitesimal generator of a semi-group of linear bounded transformations. The first results in this direction were published independently by E. Hille [3] and K. Yosida [8] in 1948. They found necessary and sufficient conditions on the resolvent R(X; U) of an operator U in order that U generate a semi-group T(s) such that 1f T(s) 11 f exp (ws) for some real w. Hille has recently given another proof of this theorem [4] based entirely on the Post-Widder inversion formula for Laplace transforms. Yosida, in his proof, introduced the approximating semi-groups