Article,
Interlacing Families II: Mixed Characteristic Polynomials and the Kadison--Singer Problem
Annals of Mathematics, (July 2015)
DOI: 10.4007/annals.2015.182.1.8
Abstract
We use the method of interlacing polynomials introduced in our previous
article to prove two theorems known to imply a positive solution to the
Kadison–Singer problem. The first is Weaver’s conjecture KS2, which is
known to imply Kadison–Singer via a projection paving conjecture of Akemann
and Anderson. The second is a formulation due to Casazza et al. of
Anderson’s original paving conjecture(s), for which we are able to compute
explicit paving bounds. The proof involves an analysis of the largest roots
of a family of polynomials that we call the “mixed characteristic polynomials”
of a collection of matrices.
