On linear preservers of semipositive matrices

Abstract

Given proper cones $K_1$ and $K_2$ in ${\mathbb{R}}^n$ and ${\mathbb{R}}^m$, respectively, an $m\times n$ matrix $A$ with real entries is said to be semipositive if there exists a $x\in K_1^{\circ }$ such that $Ax\in K_2^{\circ }$, where $K^{\circ }$ denotes the interior of a proper cone $K$. This set is denoted by $S(K_1,K_2)$. We resolve a recent conjecture on the structure of into linear preservers of $S({\mathbb{R}}_{+}^n,{\mathbb{R}}_{+}^m)$. We also determine linear preservers of the set $S(K_1,K_2)$ for arbitrary proper cones $K_1$ and $K_2$. Preservers of the subclass of those elements of $S(K_1,K_2)$ with a $(K_2,K_1)$-nonnegative left inverse as well as connections between strong linear preservers of $S(K_1,K_2)$ with other linear preserver problems are considered.

51 pages. This version has two parts. The first one concerns resolving a conjecture on into linear preservers of semipositive matrices over the nonnegative orthants. The second part discusses linear preservers of semipositive matrices over proper proper cones. We have also included an Appendix that contains detailed calculations in the $3\times 3$ case