Linear Preservers of Semipositivity over Proper Cones

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 necessa