Page 19 - Textos de Matemática Vol. 44

NATA´LIA BEBIANO’S LEGACY 9
by functional calculus. Now a certain well known result on weak majorization says
(β−1α ,...,β−1α ) ≼ (η ,...,η ) ≼ (β−1α ,...,β−1α ). 11 nnw1 nw1n n1
This implies
β−1αj ≤tr(HK−1)≤ β−1αn−j+1. jj
jj
Since the function R≥0 ∋ x → x − arctan x is convex increasing, a theorem of Tomi´c see [MO], yields the claim.
[dPB86, OMC, a central article]: Again, in her ‘annus mirabilis’ 1986 Nat´alia co-authors with Jo˜ao de Providˆencia a paper that goes far beyond all of the above and lends much additional credibility to OMC.
Theorem. The determinant of the sum of a positive definite hermitian matrix and a skew hermitian is contained in the convex hull of the associated σ-points.
Proof. For outlining the road to a proof, let w.l.o.g. A = diag(α1, · · · , αn) with αi > 0 and B = iC = idiag(γ1,··· ,γn), CU = UCU∗ and zU = det(A+ iCU ), with U ∈ U (n).
First, for hermitian S and zU ̸= 0, and for all small real ε, a complicated formula of the type
det(A + eiεS CU e−iεS ) = det(A + iCU )(1 + tr(...))
is derived, in which in ‘...’, (A + iCU ) and the commutator [S, CU ] figure promi- nentlyseveraltimes. NowletzU lieonasupportinglineof∆iC(A)perpendic- ular to the direction eiφ. If we rotate ∆iC(A) by an angle −φ, the supporting line perpendicular to the direction eiφ becomes parallel to the y-axis and it follows that the function
ε → Re(e−iφ det(A + ieiεS CU e−iεS ))
is stationary for ε = 0. This together with the above formula can be used to
derive for θ = arg(zU ) − φ the commutation relation [A,(A+iCU)−1eiθ +(A−iCU)−1e−iθ]=0.
This and rather technical calculations running over six pages (one of them contains only five words and one line of text) leads to the paper’s Theorem 2.2: IfzU ̸=0liesonasupportinglineof∆iC(A)eitherzU isaσ-pointorzU belongs to a regular curve contained locally in ∆iC(A) and having zero curvature at zU.
Now fix a complex number w and consider a point zU so that r = |zU − w| is maximum. Then evidently the disk B(w,r) contains ∆iC(A) and zU lies on