Domains of uniqueness for $C_0$-semigroups on the dual of a Banach space

Abstract

Let $(\mathcal{X},\Vert .\Vert )$ be a Banach space. In general, for a $C_0$-semigroup \semi on $(\mathcal{X},\Vert .\Vert )$, its adjoint semigroup \semia is no longer strongly continuous on the dual space $({\mathcal{X}}^{\ast },\Vert .{\Vert }^{\ast })$. Consider on ${\mathcal{X}}^{\ast }$ the topology of uniform convergence on compact subsets of $(\mathcal{X},\Vert .\Vert )$ denoted by $\mathcal{C}({\mathcal{X}}^{\ast },\mathcal{X})$, for which the usual semigroups in literature becomes $C_0$-semigroups. The main purpose of this paper is to prove that only a core can be the domain of uniqueness for a $C_0$-semigroup on $({\mathcal{X}}^{\ast },\mathcal{C}({\mathcal{X}}^{\ast },\mathcal{X}))$. As application, we show that the generalized Schrödinger operator ${\mathcal{A}}^{V}f=1/2\Delta f+b\cdot \nabla f-Vf$, $f\in C_0^{\infty }({\mathbb{R}}^d)$, is $L^{\infty }({\mathbb{R}}^d,dx)$-unique. Moreover, we prove the $L^1({\mathbb{R}}^d,dx)$-uniqueness of weak solution for the Fokker-Planck equation associated with ${\mathcal{A}}^V$.