Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem

TL;DR

This paper proves key theorems related to the Kadison--Singer problem using interlacing families of polynomials, analyzing mixed characteristic polynomials to establish paving bounds and solve longstanding conjectures.

Contribution

It introduces the use of mixed characteristic polynomials and interlacing families to prove theorems that imply a positive solution to the Kadison--Singer problem, including explicit paving bounds.

Findings

Proved Weaver's conjecture $KS_{2}$

Computed explicit paving bounds for matrices

Established a positive solution to the Kadison--Singer problem

Abstract

We use the method of interlacing families of polynomials introduced to prove two theorems known to imply a positive solution to the Kadison--Singer problem. The first is Weaver's conjecture $KS_{2}$ \cite{weaver}, 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.