Functions of the infinitesimal generator of a strongly continuous quaternionic group

The quaternionic analogue of the Riesz-Dunford functional calculus and the theory of semigroups and groups of linear quaternionic operators have recently been introduced and studied. In this paper, we suppose that T is the quaternionic infinitesimal generator of a strongly continuous group of operators (Z_T(t)t∈R and we show how we can define bounded operators f(T), where f belongs to a class of functions that is larger than the one to which the quaternionic functional calculus applies, using the quaternionic Laplace-Stieltjes transform. This class includes functions that are slice regular on the S-spectrum of T but not necessarily at infinity. Moreover, we establish the relation between f(T) and the quaternionic functional calculus and we study the problem of finding the inverse of f(T).