Laplace and Laplace-Stieltjes Space

Suppose X is a Banach space and ƒ is a function from [0,∞) into the space of all (possibly unbounded) linear operators on X. We construct a maximal Banach subspace, Y, the Laplace space, continuously embedded in X, so that ƒ|Y is the Laplace transform of a strongly continuous family of contractions on Y, and a maximal Banach subspace, W, the Laplace-Stieltjes space, continuously embedded in X, so that the map s ↦ƒ(s)x is the Laplace-Stieltjes transform of a vector-valued measure ∀x∈W. Under appropriate conditions of ƒ, that are satisfied in the eases of most interest, the Laplace space contains all x so that the map s↦ƒ(s)x is a uniformly strongly continuous Laplace transform. Appropriate choices of ƒ yield maximal subspaces on which integrodifferential equations or abstract Cauchy problems, of arbitrarily high order, are well-posed. Other choices of ƒ produce the semisimplicity manifold and the maximal subspace on which an operator is well-bounded. In the former case, the space contains all initial data for which the abstract Cauchy problem has a solution that equals the Laplace-Stieltjes transform of a vector-valued measure.