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

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.