Stieltjes transforms defined by C0-semigroups

In this paper we use the resolvent semigroup associated to a C0-semigroup to introduce the vector-valued Stieltjes transform defined by a C0-semigroup. We give new results which extend known ones in the case of a scalar generalized Stieltjes transform. We work with the vector-valued Weyl fractional calculus to present a deep connection between both concepts.