Inequalities for eigenvalues of sums of self-adjoint operators and related intersection problems (Part II)

Analysis Seminar
Wednesday, September 28, 2016 - 2:05pm
1 hour (actually 50 minutes)
Skiles 005
Georgia Tech
Consider Hermitian matrices A, B, C  on an n-dimensional Hilbert space such that C=A+B. Let a={a_1,a_2,...,a_n}, b={b_1, b_2,...,b_n}, and c={c_1, c_2,...,c_n} be sequences of eigenvalues of A, B, and C counting multiplicity, arranged in decreasing order.  Such a triple of real numbers (a,b,c) that satisfies the so-called Horn inequalities, describes the eigenvalues of the sum of n by n Hermitian matrices. The Horn inequalities is a set of inequalities conjectured by A. Horn in 1960 and later proved by the work of Klyachko and Knutson-Tao. In these two talks, I will start by discussing some of the history of Horn's conjecture and then move on to its more recent developments. We will show that these inequalities are also valid for selfadjoint elements in a finite factor,  for types of torsion modules over division rings,  and for  singular values for products of matrices, and how additional information can be obtained whenever a Horn inequality saturates. The major difficulty in our argument is the proof that certain generalized Schubert cells have nonempty intersection. In the finite dimensional case, it follows from the classical intersection theory. However, there is no readily available intersection theory for von Neumann algebras. Our argument requires a good understanding of the combinatorial structure of honeycombs, and produces an actual element in the intersection algorithmically, and it seems to be new even in finite dimensions. If time permits, we will also discuss some of the intricate combinatorics involved here. In addition, some recent work and open questions will also be presented.