Positive Linear Maps on Second Symmetric Product Spaces

TL;DR

This paper characterizes positive linear maps on second symmetric product spaces and explores their preservation properties, providing new proofs and generalizations for automorphisms and matrix preservers in infinite dimensions.

Contribution

It offers a new characterization of positive maps on symmetric product spaces and extends representation theorems for automorphisms and linear preservers to infinite-dimensional settings.

Findings

Characterization of linear maps preserving positive decomposable vectors.

Alternative proof and generalization of automorphism representation theorem.

Preservation of decomposable vectors by Drazin inverse and Moore-Penrose inverse.

Abstract

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 $T\colon X^{(2)}\to X^{(2)}$ 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.