Some applications of two completely copositive maps

Abstract

A linear map $\Phi :{\mathbb{M}}_n\to {\mathbb{M}}_k$ is called completely copositive if the resulting matrix $[\Phi (A_{j,i}){]}_{i,j=1}^m$ is positive semidefinite for any integer $m$ and positive semidefinite matrix $[A_{i,j}{]}_{i,j=1}^m$. In this paper, we present some applications of the completely copositive maps $\Phi (X)=(\mathrm{tr}X)I+X$ and $\Psi (X)=(\mathrm{tr}X)I-X$. Some new extensions about traces inequalities of positive semidefinite $3\times 3$ block matrices are included.

8 pages