Weak-star limits on polynomials and their derivatives

William T. Ross, University of Richmond
Joseph A. Ball

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.