Linear Preservers of Semipositivity over Proper Cones

Abstract

We classify all real linear operators on $M_{m,n}(\mathbb{R})$, with $m,n\ge2$, that preserve semipositivity between two prescribed pairs of proper cones, without assuming invertibility. Every such operator has product form $L(A)=XAY$ or degenerate form $L(A)=Ψ(u^TA)$, possibly both. We give necessary and sufficient cone conditions for each form; the degenerate condition is expressed through a linear matrix pencil on the dual of the target output cone. The degenerate preservers are exactly those of operator rank $n$, and every preserver of larger rank has product form. We also classify all linear preservers from $M_{2,2}(\mathbb{R})$ to $M_{1,3}(\mathbb{R})$ over the nonnegative orthants, obtaining three disjoint classes, including two rank-three classes outside product form.