Extreme rays of certain cones of Hermitian forms

Abstract:

Let $\mathcal {H}$ be the real vector space of hermitian forms on a finite-dimensional complex vector space $V$. For $f \in \mathcal {H}$ we denote by $\mathcal {H}(f)$ the closed convex cone in $\mathcal {H}$ consisting of forms $g$ such that $g(x,x) \geqslant 0$ for all $x$ satisfying $f(x,x) \geqslant 0$. Unless $f \leqslant 0$ and $f \ne 0$, the cone $\mathcal {H}(f)$ contains no nonzero subspaces of $\mathcal {H}$. Assuming that this is the case, we determine the extreme rays of $\mathcal {H}(f)$. The same problem is solved for real and quaternionic spaces.