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

Abstract

A linear map $\Phi$ between matrix spaces is called cross-positive if it is positive on orthogonal pairs $(U,V)$ of positive semidefinite matrices in the sense that $⟨U,V⟩:=\text{Tr}(UV)=0$ implies $⟨\Phi (U),V⟩\ge 0$, 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 $\Phi$ as above a bihomogeneous form is associated by $p_{\Phi }(x,y)=y^T\Phi (x{x}^T)y$. Then $\Phi$ is cross-positive if and only if $p_{\Phi }$ is nonnegative on the variety of pairs of orthogonal vectors $\{(x,y)\mid x^{T}y=0\}$. Moreover, $\Phi$ is shown to be completely cross-positive if and only if $p_{\Phi }$ is a sum of squares modulo the principal ideal $(x^{T}y)$. 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 $\Phi$ mapping between $3\times 3$ matrices are given. Finally, an algorithm to produce cross-positive maps that are not completely cross-positive is presented.

44 pages