The analogue of the Riesz-Dunford functional calculus has been introduced and studied recently as well as the theory of semigroups and groups of linear quaternionic operators. In this paper we suppose that $T$ is the infinitesimal generator of a strongly continuous group of operators $(\mathcal{ Z}_T(t))_{ t \in \mathbb{ R}}$ and we show how we can define bounded operators $f(T)$, where $f$ belongs to a class of functions which is larger than the class of slice regular functions, using the quaternionic Laplace-Stieltjes transform. This class will include functions that are slice regular on the $S$-spectrum of $T$ but not necessarily at infinity. Moreover, we establish the relation of $f(T)$ with the quaternionic functional calculus and we study the problem of finding the inverse of $f(T)$.
Categories: math.SP