Positive Linear Maps on Second Symmetric Product Spaces

Let $X^{(2)}$ denote the second symmetric product space of a partially ordered vector space $X$, endowed with the projective cone. A characterization of linear maps $T\text{\hspace{0.17em}⁣}:X^{(2)}\to X^{(2)}$ which preserve the set of all positive decomposable vectors, is proved. As applications of this result, an alternative proof, as well as an infinite dimensional generalization, of a representation theorem for (i) automorphisms on the completely positive cone and (ii) linear preservers of CP-rank-1 matrices, are presented. It is also shown that if $T$ preserves the set of all decomposable vectors, then so does the Drazin inverse, $T^D$ (if it exists). The case of the Moore-Penrose inverse is also investigated.