Continuous semigroups on ordered Banach spaces

AbstractLet (B, B+, ∥ · ∥) denote a Banach space B, ordered by a proper norm-closed convex cone B+, with a Riesz norm ∥ · ∥, and define the canonical half-norm N associated with B+ by N(a)=inf{∥a+b∥;bϵB+}. The analogs of the Hille-Yosida and Feller-Miyadera-Phillips theorems characterizing the generators H of C0- or C0∗-semigroups S = {St}t ⩾ 0 of positive operators, i.e., operators such that StB+⊆B+, are proved. In these theorems conditions of norm-dissipativity, e.g., ∥(I + αH) a ∥ ⩾ ∥ a ∥, α > 0, a ϵ D(H) are replaced by N-dissipativity, i.e., N((I + αH)a) ⩾ N(a), α > 0, a ϵ D(H)