Riesz* Homomorphisms on the copositive Cone

For a cone $K\subseteq \mathbb{R}^n$, a real symmetric matrix $A$ is called $K$-copositive if $x^\top A x\geq 0$ for every $x\in K.$ This class of matrices plays a central role in copositive optimization and linear complementarity problems. However, a complete characterization of linear maps that preserve the $K$-copositive cone is unknown, even for $K:=\mathbb{R}^n_+$. In this paper, we develop a new approach to copositivity preservers that uses only order-theoretic arguments. We consider a smaller class of copositivity preservers, called Riesz* homomorphisms, and develop a general technique to deduce the structure of these preservers directly from a representation theorem of Riesz* homomorphisms on $S_n$. Following are the main outcomes of this paper: 1) We obtain a representation theorem for Riesz* homomorphisms on the partially ordered vector space of all real symmetric matrices endowed with the cone of all $K$-copositive matrices. 2) As a corollary of our representation theorem, we recover the main results of [Shitov, Proc. Amer. Math. Soc., 2021] and [Gowda et al., Linear Alg. Appl., 2013], providing a unified framework for studying cone automorphisms. 3) We introduce the notion of a $(K_1,K_2)$-unisigned matrix $P\in M_{m\times n}$, defined by the algebraic condition $P[K_1]\subseteq K_2\cup (-K_2)$, for cones $K_1\subseteq \mathbb{R}^n$ and $K_2\subseteq \mathbb{R}^m$. We also provide a characterization of such matrices. 4) We prove that a linear map of the standard form ($A\mapsto P^\top AP$; for $P\in M_{m\times n}$) preserves copositivity if and only if $P$ is $(K_2,K_1)$-unisigned, correcting a recent characterization of such maps preserving the $\mathbb{R}^n_+$-copositivity.