Nonconvexity of the Generalized Numerical Range Associated with the Principal Character

Abstract

It is known that W H (A) is convex if m = 1 or if m = n = 2. We show that there exist normal matrices A for which W H (A) is not convex if 3 ≤ m ≤ n. Moreover, for m = 2 < n, we prove that a normal matrix A with eigenvalues lying on a straight line has convex W H (A) if and only if νA is Hermitian for some nonzero ν ∈ C. These results extend those of Hu, Hurley and Tam, who studied the special case when 2 ≤ m ≤ 3 ≤ n and H = S m .