Extreme Positive Operators from 2 × 2 to 3 × 3 Hermitian Matrices

Let $E_n$ be the real ordered space of all $n{\times}n$ Hermitian Matrices and let T be a positive linear operator from $E_2$ to $E_3$. We prove in this paper that T is extreme if and only if T is unitarily equivalent to a map of the form $S_z$ for some $z{\in}C^2$ where $S_z$ is defined by $S_z(xx^*)=ww^*$, $w_i=x_iz_i$ for i = 1, 2 and $w_3=0$.