Some convexity theorems for the generalized numerical ranges
Chi-Kwong Li · Linear and Multilinear Algebra · 1996
Let Mn be the algebra n × n complex matrices, where n ≥ 2. Given a nonscalar matrix C ∊ Mn , the C-numerical range of A ∊ Mn is defined by If rank (C − γI) for some γ ∊ C(which is always true when n = 2) then W(C:A) be written as a + bW (q:A) for some a,b ∊ C and q ∊ C and q ∊[0,1], where is the q-numerical range of A. We give short proofs for the faets that W(q :A) is convex for all A ∊ Mn , and that W(C : A) is an elliptical disk if A,C ∊ M2 . These results have been proved by Tsing and Nakazato, respectively, by some very involved computational methods.