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arXiv · 2212.00208

Haagerup bound for quaternionic Grothendieck inequality

Shmuel Friedland·Zehua Lai·Lek-Heng Lim
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Abstract

We present here several versions of the Grothendieck inequality over the skew field of quaternions: The first one is the standard Grothendieck inequality for rectangular matrices, and two additional inequalities for self-adjoint matrices, as introduced by the first and the last authors in a recent paper. We give several results on ``conic Grothendieck inequality'': as Nesterov $\pi/2$-Theorem, which corresponds to the cones of positive semidefinite matrices; the Goemans--Williamson inequality, which corresponds to the cones of weighted Laplacians; the diagonally dominant matrices. The most challenging technical part of this paper is the proof of the analog of Haagerup result that the inverse of the hypergeometric function $x {}_2F_1(\frac{1}{2}, \frac{1}{2}; 3; x^2)$ has first positive Taylor coefficient and all other Taylor coefficients are nonpositive.

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Shmuel Friedland, Zehua Lai, Lek-Heng Lim. 2022-12-01. Haagerup bound for quaternionic Grothendieck inequality. https://arxiv.org/abs/2212.00208

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