Numerical range and compact convex sets

Let K be a compact convex subset of the plane, μ be a regular Borel measure with support K and N _μ be the multiplication operator on L ²(μ). In this article we show that (overline{W}(N_{mu})), the closure of numerical range of N _μ , is K. Also we prove that if K has uncountable many extreme points then the Berberian Hilbert space extension of L ²(μ) is non separable.