Page 17 - Textos de Matemática Vol. 44

NATA´LIA BEBIANO’S LEGACY 7
This theorem generalizes an earlier 1978 result of Marcus and Filipenko who had formulated it for the regions Wk(A) obtained by choosing above c = (1k , 0n−k ).
One corollary says that in the ‘generic case’ Wc(A) cannot have corners. Corollary. If A is not normal and all the components of c ∈ Rn are distinct,
then ∂Wc(A) is smooth.
[B86c, Analogies between ∆c(A) and Wc(A)]: Here as a further analogy with
the numerical range, the following results are found for corners of ∆c(A). Theorem. Let A,C ∈ Mn(C), C diagonal, and let z ∈ ∂∆c(A). If z ̸= 0 is a
corner, then z is a σ-point.
Corollary. Let C be an n×n normal matrix all whose eigenvalues are distinct.
If there exists a corner in ∂∆C(A), then A is normal.
3. The de Oliveira Marcus Conjecture and related Investigations
In the same year in which Nat´alia contributed so much to results on the numerical range, she also dedicated much of her time to a conjecture that was published a few years earlier by Graciano de Oliveira and is still open.
Namely, in 1982 Nat´alia’s Ph. D. supervisor, inspired by a result of Fiedler, conjectured that the determinant of the sum of two normal n × n matrices is contained in the convex hull of all determinants of sums of diagonal matrices of the respective same spectra; it was later found that Marcus had conjec- tured the same thing somewhat earlier. In other words, de Oliveira and Marcus conjectured that if A,B are normal matrices of spectra α1,...,αn;β1,...,βn, respectively, then
n det(A+B)∈∆:=conv{ (αi +βσ(i)):σ∈Sn}.
i=1
The conjecture became known as OMC; the mentioned result of Fiedler says
that the conjecture is true for the case that A and B are hermitian matrices. Nat´alia is co-author of a number in part very penetrating papers on the
sub ject.
[BQ85, OMC, corners]: In this paper with Jo˜ao Queir´o, Nat´alia shows that if z ∈ ∂∆ (the boundary of ∆) is a nonzero corner point, then there is a σ in Sn such that z = i(αj + βσj ). If one draws a picture, one easily sees that this is a clear hint at OMC. The proof in [BQ85] is a nice application of the formula
det(P + εQ) = detP(1 + εtrQP−1) + O(ε2)