Improved bounds in Weaver's KS_r conjecture for high rank positive semidefinite matrices

Recently Marcus, Spielman and Srivastava proved Weaver's KS_r conjecture, which gives a positive solution to the Kadison-Singer problem. In [13,10], Cohen and Brändén independently extended this result to obtain the arbitrary-rank version of Weaver's KS_r conjecture. In this paper, we present a new bound in Weaver's KS_r conjecture for the arbitrary-rank case. To do that, we introduce the definition of (k,m)-characteristic polynomials and employ it to improve the previous estimate on the largest root of the mixed characteristic polynomials. For the rank-one case, our bound agrees with the Bownik-Casazza-Marcus-Speegle bound when r=2 [8] and with the Ravichandran-Leake bound when r>2 [21]. For the higher-rank case, we sharpen the previous bounds from Cohen and Brändén.