Page 28 - Textos de Matemática Vol. 44

18 A. KOVACˇEC
If S is the identity matrix In, then VS(A) is exactly the classical numerical range W(A). It is known that in general VS(A) is not bounded, closed or convex. The C-numerical range WC(A) for n-by-n matrices A and C is also generalized to the S, C-tracial range
VS,C(A) = {tr(CU∗AU) : U∗SU = S}.
The latter reverts to the former if S = I. Some studies of the S, C-determinantal
range of A, that is
DS,C(A) = {det(C + U∗AU) : U∗SU = S}
are carried through.
Since S is hermitian, it is easy to see that defining the two sets VS±(A) =
{x∗Ax:x∗Sx=±1},onehasV+ (A)=V−(A)andVS(A)=V+(A)∪V+ (A). −SS S−S
As might be suspected from writing VS(A) as the evidently disjoint union of two sets, one can prove that VS(A) is p-convex; that is given two points x,y ∈ VS(A), VS(A) either contains the segment [x,y] or the relatively open set lx,y \ [x, y] in the line lx,y through x and y. Well known properties of the usual numerical range easily generalize to VS(.). In particular we have the following elementary ones: VS(A) = VS(U∗AU) for U ∈ U(n) (unitary invariance), VS(A∗) = VS(A), in particular VS(A) ⊆ R if A is hermitian, VS(A + B) ⊆ VS(A) + VS(B), VS(A) = {λ} iff S ̸= 0 and A = λS.
The S,C-tracial range has similar properties with subtle differences: VS,C(A) = VS,C(U∗AU) for all U with U∗SU = S;
VS,C (A)(αS + βA) = α tr(SC) + βVS,C (A) for α, β ∈ C;
VS,C∗(A∗) = VS,C(A); VS,C(A) = VS−1,A(C) whenever S is nonsingular.
If C = diag(γ1, · · · , γn), one writes in analogy with notation used for the clas- sical numerical range, VS,c(A) for VS,C(A).
A complex n × n matrix U is pseudounitary of signature (r, n − r) with 0 ≤ r ≤ n if it preserves the quadratic form
q(x)=|x1|2 +···+|xr|2 −|xr+1|2 −···−|xn|2,
i.e. if q(Ux) = q(x). Evidently these matrices form the group Ur,n−r or J-unitary matrices. The Sylvester inertia guarantees for any nonsingular her- mitian n × n matrix S, that R∗SR = J for some r and nonsingular R. This implies that to study the class of sets VS(A), A ∈ Mn is to study the class of sets VJ(A), A ∈ Mn.
In Section 2, the boundary generating curve of VS(A) is obtained, extending the cited theorem of Murnaghan and Kippenhahn as follows.