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 (ZT (t))t2R 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 ).
D. Alpay, F. Colombo, J. Gantner and D. Kimsey. Functions of the infinitesimal generator of a strongly continuous quaternionic group. Analysis and Applications vol. 15 (2017), 279-311.