Page 27 - Textos de Matemática Vol. 44

NATA´LIA BEBIANO’S LEGACY 17
6. The indefinite numerical range
The trend that transpired in the last paper continued in recent years. Nat´alia grew increasingly interested in extending results for concepts whose definition depends on the classical inner product to analogue concepts defined for the indefinite inner product [x,y] = y∗Jx we defined above. The last two papers we will consider here are proof of this. We will use below again (as does Nat´alia) the not(at)ions J = Ir ⊕ −In−r, J-hermitian and J-unitary. The letter r has the indicated reserved meaning.
Although the most elegant proofs nowadays conceal it, already in the early days of the research into numerical ranges, the boundary curve of the classical numerical range was object of study. Toeplitz’ result that this curve is convex precedes Hausdorff’s stronger 1919 result that W(A) itself is convex. Also, in the same 1932 paper in which Murnaghan proved his elliptical range theorem for 2 × 2 matrices, he also showed that the supporting lines of W (A) satisfy a homogenous equation of degree n. If we write A in its unique cartesian decom- position, i.e. in the form A = H + iK with hermitian matrices H and K, then the triples (u, v, w) that describe supporting lines ux + vy + wz = 0 of W (A) in homogeneous coordinates (x : y : z), satisfy det(uH + vK + wI) = 0. In 1951 Kippenhahn completed this observation by showing that the boundary curve of W(A) is in fact described by the preceding equation. In other words, ∂W(A) is described by this equation as a line curve: as a curve that is the envelope of its tangents the coefficients of whose equations are described algebraically. Projective geometers know ways of how to pass from the description of a curve as a line curve to a description as a point curve; that is how to pass from an algebraic description of the lines in the form l(u, v, w) = 0 to a description of the form p(x, y, z) = 0 with l, p homogeneous polynomials whose degrees define respectively class and order of the curve. In 1981 Fiedler gave an alternative way in the case of the Murnaghan-Kippenhahn result.
These developments are reflected in what Nat´alia did with some of her Ph.D. students for the indefinite numerical range.
[BLdPS04, numerical range associated with indefinite inner product space, Kip- penhahn Murnaghan boundary]: In this long paper of eight sections and thirty pages, Nat´alia initiates with coauthors her study of generalizations of the nu- merical range on indefinite inner product spaces and gives a description of the supportlines of the indefinite numerical range similar to Murnaghan’s for the classical case.
For an n-by-n complex matrix A and Hermitian matrix S, the S-numerical range of A is by definition the set VS(A) = {x∗Ax/x∗Sx : x ∈ Cn, x∗Sx ̸= 0}.