Let be a linear operator defined in a Banach space with norm . The main result:
If has an absolutely convergent Laplace transform , which is nonzero for every , and exists for every , then there exist a linear subspace and a norm such that is a Banach space, the restriction of on is a closed linear operator with densely defined domain and the Volterra equation of convolution type
admits a resolvent family of contractions on , i.e. a strongly continuous family of bounded linear operators defined in , which commutes with and satisfies the equation
and inequality . This Hille-Yosida space is maximal-unique in a certain sense. If is, in addition, a positive function, then generates a strongly continuous semigroup of contractions on .