Mixed Determinants and the KadisonSinger problem
Abstract
We adapt the arguments of Marcus, Spielman and Srivastava in their proof of the KadisonSinger problem to prove improved paving estimates. Working with Anderson's paving formulation of KadisonSinger instead of Weaver's vector balancing version, we show that the machinery of interlacing polyomials due to Marcus, Spielman and Srivastava works in this setting as well. The relevant expected characteristic polynomials turn out to be related to the so called "mixed determinants" that have been carefully studied by Borcea and Branden. This technique allows us to show that any projection with diagonal entries strictly less than $\frac{1}{4}$ can be two paved, matching recent results of Bownik, Casazza, Marcus and Speegle, though our estimates are asymptotically weaker. We also show that any projection with diagonal entries at most $\frac{1}{2}$ can be four paved, yielding improvements over currently known estimates. We also relate the problem of finding optimal paving estimates to bounding the root intervals of a natural one parameter deformation of the characteristic polynomial of a matrix that turns out to have some remarkable combinatorial properties.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.04195
 Bibcode:
 2016arXiv160904195L
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Combinatorics;
 Mathematics  Operator Algebras
 EPrint:
 26 pages, no figures