We establish an inversion formula of Post-Widder type for ${\lambda }^{\alpha }$-multiplied vector-valued Laplace transforms (α > 0). This result implies an inversion theorem for resolvents of generators of α-times integrated families (semigroups and cosine functions) which, in particular, provides a unified proof of previously known inversion formulae for α-times integrated semigroups.

Suppose A is an injective linear operator on a Banach space that generates a uniformly bounded strongly continuous semigroup ${e^{tA}}_{t\ge 0}$. It is shown that $A^{-1}$ generates an $O(1+\tau )A{(1-A)}^{-1}$-regularized semigroup. Several equivalences for $A^{-1}$ generating a strongly continuous semigroup are given. These are used to generate sufficient conditions on the growth of ${e^{tA}}_{t\ge 0}$, on subspaces, for $A^{-1}$ generating a strongly continuous semigroup, and to show that the inverse of -d/dx on the closure of its image in L¹([0,∞)) does not generate a strongly...