A GENERALIZATION OF THE 0-NUMERICAL RANGE
Let H be a complex Hilbert space. Given a bounded linear operator A on H, we describe the set R n (A) = fV AW : V;W : C n ! H;V V = W W = In;V W = 0g. It is shown that the closed matricial convex hull of R n (A) is a closed ball of radius minfkA I k : 2 Cg centered at the origin.
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 ∣q∣⩽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 W C (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.
3×3 Orthostochastic matrices and the convexity of generalized numerical ranges
Let U3 be the set of all 3 × 3 unitary matrices, and let A and B be two 3 × 3 complex norḿal matrices. In this note, the authors first give a necessary and sufficient condition for a 3 × 3 doubly stochastic matrix to be orthostochastic and then use this result to consider the structure of the sets W (A) = {Diag UAU∗ : U ∈ U3} and W(A,B) = {Tr UAU∗ B: U ∈ U3}, where ∗ denotes the transpose conjugate.