Applications of the Choquet boundary to resolvents

Abstract

Let X be a compact Hausdorff space, and suppose that for each λ > 0 there is a linear transformation R _λ : C(X) → C(X) such that R _λ ≧ 0 (i.e., R _λ f ≧ 0 whenever f ≧ 0) and R _λ1 = 1/λ. We call the family of operators R _λ(λ > 0) a resolvent if the following identity is valid for all λ, λ′ > 0: R _λ′ - R _λ = (λ - λ′)R _λ′ R _λ.