Positive maps of second-order cones
Résumé
Let $n \in \mathbb N$ and let $L_n \subset \mathbb R^n$ be the $n$-dimensional second order cone, or Lorentz cone. A linear map $M$ from $\mathbb R^m$ to $\mathbb R^n$ is called positive if $M[L_m] \subset L_n$. The set of positive maps forms a convex cone, the positive cone. Its dual cone, the separable cone, is the convex hull of tensor products $x \otimes y$, where $x \in L_n$, $y \in L_m$. The structure of the positive and the separable cones is the subject of the present contribution. We study the action of the automorphism group of the positive cone and provide a complete classification of its orbits by constructing explicitly canonical forms. This enables us to identify and to classify the extreme rays of the positive cone, as well as their dual counterparts, the largest faces of the separable cone. We also characterize positive maps which are scalable to doubly stochastic ones.