Cross-positive linear maps, positive polynomials and sums of squares

TL;DR

This work investigates cross-positivity vs. complete cross-positivity of $\,\ast$-linear maps $\Phi:M_n(\mathbb{R})\to M_n(\mathbb{R})$, translating the problem into real algebraic geometry via the biquadratic form $p_{\Phi}(x,y)=y^T\Phi(xx^T)y$ and the variety $V(I)$ defined by $y^Tx=0$. It shows that completely cross-positive maps are rare among cross-positive maps by deriving a dimension-free bound $p_n\le (C n)^{-\frac12 \binom{n+1}{2}^2}$, implying $\lim_{n\to\infty} p_n=0$, and establishes accurate certificates for $n=3$ via Nichtnegativstellensätze. A dimension-robust Reverse Hölder inequality for bilinear biforms is developed and used to obtain Blekherman-type volume estimates, comparing sections of the nonnegative and SOS cones on the Stiefel-type variety, and yielding explicit bounds on the SOS-to-nonnegativity ratio. Finally, a randomized polynomial-time algorithm (based on semidefinite programming) is described to produce cross-positive maps that are not completely cross-positive, with a concrete $3\times3$ example illustrating the method. These results illuminate the gap between cross-positivity notions and provide practical tools to generate and certify proper cross-positive maps relevant to operator semigroups and affine processes on symmetric cones.

Abstract

A linear map $Φ$ between matrix spaces is called cross-positive if it is positive on orthogonal pairs $(U,V)$ of positive semidefinite matrices in the sense that $\langle U,V\rangle:=\text{Tr}(UV)=0$ implies $\langle Φ(U),V\rangle\geq0$, and is completely cross-positive if all its ampliations $I_n\otimes Φ$ are cross-positive. (Completely) cross-positive maps arise in the theory of operator semigroups, where they are sometimes called exponentially-positive maps, and are also important in the theory of affine processes on symmetric cones in mathematical finance. To each $Φ$ as above a bihomogeneous form is associated by $p_Φ(x,y)=y^TΦ(xx^T)y$. Then $Φ$ is cross-positive if and only if $p_Φ$ is nonnegative on the variety of pairs of orthogonal vectors $\{(x,y)\mid x^Ty=0\}$. Moreover, $Φ$ is shown to be completely cross-positive if and only if $p_Φ$ is a sum of squares modulo the principal ideal $(x^Ty)$. These observations bring the study of cross-positive maps into the powerful setting of real algebraic geometry. Here this interplay is exploited to prove quantitative bounds on the fraction of cross-positive maps that are completely cross-positive. Detailed results about cross-positive maps $Φ$ mapping between $3\times 3$ matrices are given. Finally, an algorithm to produce cross-positive maps that are not completely cross-positive is presented.

Theorems & Definitions (57)

• Theorem 2.1: SV70
• Corollary 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• proof
• Proposition 2.5
• proof
• Proposition 2.6
• proof : Proof of Proposition \ref{['prop:obviousPQccp']}