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

- Series
- Analysis Seminar
- Time
- Wednesday, September 28, 2016 - 14:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Wing Li – 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.