A Lyapunov-type theorem from Kadison-Singer
Charles A. Akemann, Nik Weaver · Bulletin of the London Mathematical Society · 2014
Marcus, Spielman, and Srivastava [‘Interlacing families II: mixed characteristic polynomials and the Kadison–Singer problem’, Preprint, 2013, arXiv:1306.3969] recently solved the Kadison–Singer problem by showing that if u 1 , … , u m are column vectors in C d such that ∑ u i u i ∗ = I , then a set of indices S ⊆ { 1 , … , m } can be chosen so that ∑ i ∈ S u i u i ∗ is approximately 1 2 I , with the approximation good in operator norm to order ϵ 1 / 2 where ϵ = max ∥ u i ∥ 2 . We extend their result to show that every linear combination of the matrices u i u i ∗ with coefficients in [ 0 , 1 ] can be approximated in operator norm to order ϵ 1 / 8 by a matrix of the form ∑ i ∈ S u i u i ∗ .