Domains of uniqueness for C0-semigroups on the dual of a Banach space

Abstract

Let ( X,\|\:.\:\|) be a Banach space. In general, for a C0-semigroup on ( X,\|\:.\:\|), its adjoint semigroup is no longer strongly continuous on the dual space ( X*,\|\:.\:\|*). Consider on X* the topology of uniform convergence on compact subsets of ( X,\|\:.\:\|) denoted by C( X*, X), for which the usual semigroups in literature becomes C0-semigroups. The main purpose of this paper is to prove that only a core can be the domain of uniqueness for a C0-semigroup on ( X*, C( X*, X)). As application, we show that the generalized Schr\"odinger operator AVf=1/2 f+b·∇ f-Vf, f∈ C0∞(d), is L∞(d,dx)-unique. Moreover, we prove the L1(d,dx)-uniqueness of weak solution for the Fokker-Planck equation associated with AV.