The constrained bilinear form and the C-numerical range

Abstract

Let V be an n-dimentional unitary space with inner product (·,·) and S the set {x∈V:(x, x)=1}. For any A∈Hom(V, V) and q∈C with {divides}q{divides}≤1, we define W(A:q)={(Ax, y):x, y∈S, (x, y)=q}. If q=1, then W(A:q) is just the classical numerical range {(Ax, x):x∈S}, the convexity of which is well known. Another generalization of the numerical range is the C-numerical range, which is defined to be the set WC(A)={tr(CU*AU):U unitary} where C∈Hom(V, V). In this note, we prove that W(A:q) is always convex and that WC(A) is convex for all A if rank C=1 or n=2. © 1984.