Uniqueness of a pre-generator for $C_0$-semigroup on a general locally convex vector space

TL;DR

This paper extends the uniqueness theorem for $C_0$-semigroups from Banach spaces to general locally convex spaces, establishing cores as the only domains of uniqueness and applying this to the mass transport equation.

Contribution

It generalizes Arendt's theorem to locally convex spaces and characterizes domains of uniqueness for $C_0$-semigroups in this broader setting.

Findings

Cores are the only domains of uniqueness for $C_0$-semigroups on locally convex spaces.

Provides a necessary and sufficient condition for the uniqueness of solutions to the mass transport equation.

Extends classical semigroup theory results to more general topological vector spaces.

Abstract

The main purpose is to generalize a theorem of Arendt about uniqueness of $C_0$-semigroups from Banach space setting to the general locally convex vector spaces, more precisely, we show that cores are the only domains of uniqueness for $C_0$-semigroups on locally convex spaces. As an application, we find a necessary and sufficient condition for that the mass transport equation has one unique $L^1(\R^d,dx)$ weak solution.