Weak-Star Limits of Polynomials and Their Derivatives

Abstract

Let μ and v be regular finite Borel measures with compact support in the real line M. and define the differential operator D :L ^∞(μ}) → L ^∞(v) with domain equal to the polynomials P by Dp = p′. In this paper we will characterize the weak-star closure of the graph of D in ^∞(μ) ⊕ ^∞(y). As a consequence we will characterize when D is closable (i.e. the weak-star closure of G contains no non-zero elements of the form 0 ⊕ g) and when g is weak-star dense in L^∞(μ) ⊕ L ^∞(v). We will also consider the same problem where μ and v are measures supported on the unit circle T.