On the Inversion of the Laplace Transform of $C_0 $ Semigroups and Its Applications

Abstract

The main results are as follows; (a) The inversion of the Laplace transform of $C_0 $ semigroup representation in a Hilbert space H is generalized to all $x \in H$. (b) A full characterization is given in terms of the resolvent $R(\lambda ;A)$ of the infinitesimal generator A of a $C_0 $ semigroup $T(t)$ in a Hilbert space, which assures the continuity of $T(t)$ for $t > 0$ in the uniform operator topology.