Haagerup bound for quaternionic Grothendieck inequality

TL;DR

This paper extends the Grothendieck inequality to quaternionic matrices, providing new bounds and inequalities for various matrix cones, including positive semidefinite and diagonally dominant matrices, with technical advances on hypergeometric functions.

Contribution

It introduces several versions of the Grothendieck inequality over quaternions, including for self-adjoint matrices and conic cases, and proves a key hypergeometric function property.

Findings

Extended Grothendieck inequalities to quaternionic matrices.

Established bounds for positive semidefinite and Laplacian cones.

Proved a novel property of hypergeometric function coefficients.

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.