Mixed determinants and the Kadison–Singer problem

Ravichandran, Mohan; Leake, Jonathan

doi:10.1007/s00208-020-01986-7

Cited by 9 publications

(9 citation statements)

References 17 publications

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“…Our bound in (1.9) coincides with that of [RL20] for each r ≥ 2 when k = 1. In particular, our bound is also the same with that of [BCMS19] when r = 2 and k = 1.…”

supporting

confidence: 67%

“…Ravichandran and Leake used the k-characteristic polynomial to prove Anderson's paving formulation of Kadison-Singer problem [RL20]. It is easy to see the relationship between χ k and ψ k,m :…”

Section: A New Formula For Mixed Characteristic Polynomialsmentioning

confidence: 99%

“…Ravichandran and Srivastava used the mixed determinantal polynomial to provide a simultaneous paving of a k-tuple of zero-diagonal Hermitian matrices [RL20]. According to (3.2), we immediately have the following identity:…”

Section: A New Formula For Mixed Characteristic Polynomialsmentioning

confidence: 99%

“…Ravichandran and Leake in [RL20] adapted the method of interlacing families to directly prove Anderson's paving conjecture, which is well-known to be equivalent to Weaver's KS r conjecture. They showed that, for any integer r ≥ 2 and any real number 0 < ε ≤ (r−1) 2 r 2 , if…”

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confidence: 99%

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Improved bounds in Weaver's ${\rm KS}_r$ conjecture for high rank positive semidefinite matrices

Xu,

Zhu

2021

Preprint

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Recently Marcus, Spielman and Srivastava proved Weaver's KSr conjecture, which gives a positive solution to the Kadison-Singer problem. In [Coh16, Brä18], Cohen and Brändén independently extended this result to obtain the arbitrary-rank version of Weaver's KSr conjecture. In this paper, we present a new bound in Weaver's KSr 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's bound when r = 2 [BCMS19] and with the Ravichandran-Leake's bound when r > 2 [RL20]. For the higher-rank case, we sharpen the previous bounds from Cohen and from Brändén.

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“…Our bound in (1.9) coincides with that of [RL20] for each r ≥ 2 when k = 1. In particular, our bound is also the same with that of [BCMS19] when r = 2 and k = 1.…”

supporting

confidence: 67%

Section: A New Formula For Mixed Characteristic Polynomialsmentioning

confidence: 99%

Section: A New Formula For Mixed Characteristic Polynomialsmentioning

confidence: 99%

mentioning

confidence: 99%

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Improved bounds in Weaver's ${\rm KS}_r$ conjecture for high rank positive semidefinite matrices

Xu,

Zhu

2021

Preprint

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“…Closely related to the mixed determinant is the following generalization of the characteristic polynomial of a single matrix to a tuple of k matrices, which we call the mixed determinantal polynomial (introduced in [Rav16]) or MDP in short.…”

Section: Definition 4 Given a K Tuple Of Matricesmentioning

confidence: 99%

Asymptotically Optimal Multi-Paving

Ravichandran

Srivastava

2019

International Mathematics Research Notices

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Anderson's paving conjecture, now known to hold due to the resolution of the Kadison-Singer problem asserts that every zero diagonal Hermitian matrix admits non-trivial pavings with dimension independent bounds. In this paper, we develop a technique extending the arguments of Marcus, Spielman and Srivastava in their solution of the Kadison-Singer problem to show the existence of non-trivial pavings for collections of matrices. We show that given zero diagonal Hermitian contractions A (1) , · · · , A (k) ∈ M n (C) and ǫ > 0, one may find a pavingAs a consequence, we get the correct asymptotic estimates for paving general zero diagonal matrices; zero diagonal contractions can be (O(ǫ −2 ), ǫ) paved. As an application, we give a simplified proof wth slightly better estimates of a theorem of Johnson, Ozawa and Schechtman concerning commutator representations of zero trace matrices. IntroductionAnderson's Paving conjecture [And79] asserts that for every ǫ > 0 there is an integer r such that every zero diagonal complex n × n matrix M admits a paving X 1 ∪ . . . ∪ X r = [n], which satisfies:where the P X i are diagonal projections. This conjecture, which implies a positive solution to the Kadison-Singer Problem, is proven in [MSS15] with the dependence r = (6/ǫ) 8 which arises as follows. First, the "one-sided" boundis obtained with r = (6/ǫ) 2 for Hermitian M , via the method of interlacing families of polynomials. Then this is extended to a two-sided bound by taking a product of pavings for M and −M , and finally to non-Hermitian M = A+iB by taking yet another product of two-sided pavings of A and B. Each product increases r quadratically, and in general this approach allows one to simultaneously pave any k Hermitian matrices in the one-sided sense with the dependence (6/ǫ) 2k . Our main theorem says, Theorem 1 Given zero diagonal Hermitian contractions A (1) , · · · , A (k) ∈ M n (C) and ǫ > 0, there exists a paving X 1 ∐ · · · ∐ X r = [n] where r ≤ 18kǫ −2 such that,

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Upper and Lower Bounds for Matrix Discrepancy

Xie

Zhu

2022

J Fourier Anal Appl

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Mixed determinants and the Kadison–Singer problem

Cited by 9 publications

References 17 publications

Improved bounds in Weaver's ${\rm KS}_r$ conjecture for high rank positive semidefinite matrices

Improved bounds in Weaver's ${\rm KS}_r$ conjecture for high rank positive semidefinite matrices

Asymptotically Optimal Multi-Paving

Upper and Lower Bounds for Matrix Discrepancy

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