Nondifferentiable boundary points of the higher numerical range

Abstract

Let A be an n-square complex matrix. Every nondifferentiable point on ∂W m ( 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 ∂W m ( A), then A is normal. From these results it follows that if ∂W m ( A) is a convex polygon with sufficiently many vertices, then A is normal.