Reducing essential eigenvalues in the boundary of the numerical range

We examine a purely geometric property of a point in the boundary of the numerical range of a (Hilbert space) operator that implies that such a point is a reducing essential eigenvalue of the given operator. Roughly speaking, such a property means that the boundary curve of the numerical range has infinite curvature at that point (we must exclude however linear verteces because they may be reducing eigenvalues without being reducing essential eigenvalues). This result allows us to give an elegant proof of a conjecture of Joel Anderson: {\it A compact perturbation of a scalar multiple of the identity operator can not have the closure of its numerical range equal to half a disk (neither equal to any acute circular sector).}