# Weak-star limits on 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.

*This paper has been withdrawn.*