Haagerup Bound for Quaternionic Grothendieck Inequality

Abstract:

We present here several versions of the Grothendieck inequality over the skew eld 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 =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 $x {}_2 F_1(\frac{1}{2}, \frac{1}{2}; 3; x^2)$ has first positive Taylor coefficient and all other Taylor coefficients are nonpositive.