Page 30 - Textos de Matemática Vol. 44

20 A. KOVACˇEC
Theorem. Let J = diag(1, −1). The J-numerical range VJ (A) of A ∈ M2(C) is bounded by a possibly degenerated hyperbola. Its foci are the eigenvalues of λ1,λ2 of JA, and its nontransversal axis has length
|λ1|2 + |λ2|2 − Tr(A∗JAJ).
The proof is done step by step first for a special matrix and then reducing the general case to the special one.
Recall that the sets Wc(A) and ∆c(A) are elliptical discs. In Section 4, it is shown, analogously, that VJ,c(A) and ∆J,c(A) are bounded by branches of possibly degenerate hyperbolas with foci γ1λi − γ2λj and (γ1 + λi)(γ2 − λj), where {i, j} = {1, 2}, respectively. Both hyperbolas have a nontransverse axis of length
|γ1 + γ2| |λ1|2 + |λ2|2 − tr(A∗JAJ).
Sections 5 and 6 study the precise conditions under which the sets VJ,c(A) and DJ,c(A) are singletons or subsets of the real line. In Section 7 it is shown that if z is a corner point of VJ,c(A) or DJ,c(A), then the matrix associated to z will have a block diagonal structure. In other words, results of the Marcus-Filipenko type related in Section 2 of this survey are obtained. Section 8 concludes the paper with the description of an anything-but-trivial algorithm that was implemented in MATLAB in order to plot an approximation for VJ (A). This yields beautiful illustrations.
[BdPT09, indefinite numerical range, algebraic curves, Kippenhahn]: One of the highlights of the paper mentioned before was the Murnaghan-Kippenhahn-type description of the support lines of an indefinite numerical range. In the current paper the authors give a theorem allowing to pass to the point equation of the corresponding curve. This step is taken in analogy to a step of Fiedler in 1981 who gave a new deduction of the point equation for the boundary generating curve of the classical numerical range using the original Kippenhahn result.
The authors use a notation partially different from the previous one; we will accompany them in this choice, but it will be clear that largely the same concepts are involved.
Let r,J be given. If r ̸= 0,n then we have an underlying indefinite inner product [x,y]. Given a matrix A ∈ Mn we define HJ = 1(A + JA∗J) and
2
KJ = 1 (A − JA∗J). The matrices HJ and KJ are J-hermitian, and A =
2i
HJ + iKJ is the J-cartesian decomposition. The J-numerical range of A is
WJ(A) = {x∗JAx : x ∈ Cn,x∗Jx ̸= 0}; x∗Jx
it would be VJ(JA) in the notation of the preceding paper. The J-numerical range is invariant under the J-unitary group: WJ(A) = WJ(U−1AU).