Continuous extension of a densely parameterized semigroup

Abstract

Let $\mathcal{S}$ be a dense sub-semigroup of ℝ+, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of operators on X over $\mathcal{S}$ can be extended to a weakly continuous semigroup over ℝ+. We obtain similar results for nonlinear, nonexpansive semigroups as well. As a corollary we characterize all densely parametrized semigroups which are extendable to semigroups over ℝ+.