Nondifferentiable boundary points of the higher numerical range

AbstractLet A be an n-square complex matrix. Every nondifferentiable point on ∂Wm(A), the boundary of the mth numerical range of A, is a sum of m eigenvalues of A. This generalizes a theorem of W. F. Donoghue. Moreover, if sufficiently many sums of m eigenvalues of A occur on ∂Wm(A), then A is normal. From these results it follows that if ∂Wm(A) is a convex polygon with sufficiently many vertices, then A is normal