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

A ⁎-linear map Φ between matrix spaces is cross-positive if it is positive on orthogonal pairs $(U,V)$ of positive semidefinite matrices in the sense that $〈U,V〉:=\mathrm{tr}\left(UV\right)=0$ implies $〈\Phi \left(U\right),V〉\ge 0$, and is completely cross-positive if all its ampliations $I_n\otimes \Phi$ 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 $p_{\Phi }(x,y)=y^T\Phi \left(x{x}^T\right)y$. Then Φ is cross-positive if and only if $p_{\Phi }$ is nonnegative on the variety of pairs of orthogonal vectors $\left\{\right(x,y\left)\right|x^{T}y=0\}$. Moreover, Φ is shown to be completely cross-positive if and only if $p_{\Phi }$ is a sum of squares modulo the principal ideal $\left(x^{T}y\right)$. 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.