SCHWEITZER 2009 Problem 11

Denote by $H_n$ the linear space of $n\times n$ self-adjoint complex matrices, and by $P_n$ the cone of positive-semidefinite matrices in this space. Let us consider the usual inner product on $H_n$ $$\langle A,B\rangle = {\rm tr} AB (A,B\in H_n)$$ and its derived metric. Show that every $\phi: P_n\to P_n$ isometry (that is a not necessarily surjective, distance preserving map with respect to the above metric) can be expressed as $$\phi(A) = UAU^* + X (A\in H_n)$$ or $$\phi(A) = UA^TU^* + X (A\in H_n)$$ where $U$ is an $n\times n$ unitary matrix, $X$ is a positive-semidefinite matrix, and $^T$ and $^*$ denote taking the transpose and the adjoint, respectively.