Let $({ \cal X},\|\:.\:\|)$ be a Banach space. In general, for a $C_0$-semigroup \semi on $({ \cal X},\|\:.\:\|)$, its adjoint semigroup \semia is no longer strongly continuous on the dual space $({ \cal X}^{ *},\|\:.\:\|^{ *})$. Consider on ${ \cal X}^{ *}$ the topology of uniform convergence on compact subsets of $({ \cal X},\|\:.\:\|)$ denoted by ${ \cal C}({ \cal X}^{ *},{ \cal X})$, 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 $({ \cal X}^{ *},{ \cal C}({ \cal X}^{ *},{ \cal X}))$. As application, we show that the generalized Schr\"odinger operator ${ \cal A}^Vf={ 1/2}\Delta f+b\cdot\nabla f-Vf$, $f\in C_0^\infty(\R^d)$, is $L^\infty(\R^d,dx)$-unique. Moreover, we prove the $L^1(\R^d,dx)$-uniqueness of weak solution for the Fokker-Planck equation associated with ${ \cal A}^V$.
Categories: math-ph math.MP