Page 21 - Textos de Matemática Vol. 44

NATA´LIA BEBIANO’S LEGACY 11
The proof is quite involved; the authors consider expressions of the form ZS (ε) = tr(AeiεS CU e−iεS ) with CU = U C U ∗ , A without loss of generality diagonal, S n × n hermitian and use of it up to the 4th power of ε in the development
ZS(ε)=tr(ACU)+···+ 1ε4tr(A[S[S[S[S,CU]]]])+··· 4!
and six lemmas. A testimony of the authors’ technical skills.
Among the open questions the authors pose, is whether there exist in
∂WC(A) nondifferentiable points other than corners. 5. Matrix inequalities
Nat´alia has also shown a keen interest in matrix inequalities. Proof of this are, beyond her own papers, some of her students’ theses like those of Rute Lemos and Ricardo Teixeira. In this section we report results on matrix in- equalities.
In 1937 von Neumann and in 1993 Miranda and Thompson determined ex- tremal results for the trace trAUBV as U,V range both over the unitary or both over the special orthogonal group, respectively.
Let A, B, be square complex matrices of singular values α1 ≥ ... ≥ αn ≥ 0 andβ1 ≥...≥βn ≥0.Thenwehave
i. max{|trAUBV|:U,V ∈U(n)}= ni=1αiβi; (v.Neumann)
ii. max{trAUBV : U,V ∈ SO(n)} = n−1 αiβi + signdet(AB)αnβn. (Mi-
i=1
randa -Thompson)
In two papers in 1997 and 1998, Nat´alia and coauthors sought to get analo- gous results for the determinant function replacing the trace.
Define the two sets
D(A, B) = {det(A′ + B′) : A′, B′ ∈ Mn(C) of singular values αi, βi of A, B}
={eiφdet(A0 +UB0V):U,V ∈U(n),φ∈R},
where A0 = diag(α1,··· ,αn) and B0 = diag(β1,··· ,βn), and for A,B ∈
Mn(R), define
R(A,B)={det(A+UBV):U,V ∈SO(n)}.
It is clear that D(A, B) is an annulus (possibly degenerate to disk or point)
in the complex plane, while R(A, B) is a subset of the real line.