Page 25 - Textos de Matemática Vol. 44

NATA´LIA BEBIANO’S LEGACY 15
relative entropy of Umegaki. In the last section a Peierls-Bogolyubov inequality which says
tr(eA+B) ≥ tr(eA)etr BeA/ tr(eA)
for A, B hermitian is shown equivalent to the thermodynamic inequality.
[BNPLS05, Versions of Schur, Ky Fan, and Rayleigh Ritz type for J-hermitian matrices]: Recall the theorems of Schur, Ky Fan, and Rayleigh Ritz that say for hermitian A = (aij ) of spectrum α1 ≥ α2 ≥ · · · ≥ αn respectively, that (a11,...,ann) ≼ (α1,...,αn); that α1 = maxx∗x=1 x∗Ax and αn = minx∗x=1 x∗Ax; and that as [x1|...|xk] range over the first k columns of a unitary matrix, one has
kk max x∗jAxj = αj.
j=1 j=1
Let 0 < r < n and let J = Ir ⊕ −In−r. Such matrices define ‘indefinite inner products’ [x,y] = y∗Jx in face of which it is natural to call a matrix S for which for all x, y ∈ Cn [Sx, y] = [x, Sy], J-hermitian, and a matrix U for which for all such x, y there holds [U x, U y] = [x, y], J -unitary. Equivalently S is J-hermitian if JS∗J = S, and U is J-unitary if U∗JU = J.
For any two n-by-n matrices A and C, define the J,C-numerical range of A as the set
WCJ(A) = {tr(CU−1AU) : U ∈ Mn(C) is J-unitary }.
One now sees that, in analogy to the conventional tracial range, we have with J = diag(ε1,...,εn) for J-hermitian J-unitarily diagonalizable C with eigen- values c1, ..., cn that
rn
WCJ(A)={ ci[Axi,xi]− ci[Axi,xi]:[xi,xl]=δilεi}.
i=1 i=r+1
Let σJ±(A) = {eigenvalues of A with eigenvectors x such that x∗Jx = ±1}. A J-hermitian A is J-unitarily diagonalizable iff every eigenvalue of A belongs to σJ−(A)∪σJ+(A); or equivalently if it is diagonalizable by a J-unitary matrix. The spectrum of a J-hermitian matrix is symmetric to the real axis. Assume now that σJ+(A) = {α1 ≥ ... ≥ αr} and σJ−(A) = {αr+1 ≥ ... ≥ αn}. The eigenvalues of A are said to not interlace if αr > αr+1 or αn > α1.
Assume now that C is another J-hermitian J-unitarily diagonalizable matrix and assume that similarly as for A, σJ+(C) = {c1 ≥ ... ≥ cr}, and σJ−(C) = {cr+1 ≥ ... ≥ cn}.