The constrained bilinear form and the C-numerical range

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 $\text{q∈}\text{C}$ with ∣q∣?1, we define

$\text{W(A:q)={(Ax, y):x, y∈S, (x, y)=q}}$

. 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

$\text{W}{}_{\mathrm{C}}\text{(A)={}\text{tr}\text{(CU}{}^1\text{AU):U}\text{unitary}\text{}}$

where C∈Hom(V, V). In this note, we prove that W(A:q) is always convex and that W_C(A) is convex for all A if rank C=1 or n=2.