Let $H$ be a linear subspace of the space of Hermitian $n\times n$ matrices. Is there a good characterization of those $H$ such that every $A\in H$ has at least $k$ positive and $k$ negative eigenvalues?

For $k=1$ a nice characterization is the following: there is a positive definite matrix $B$ orthogonal to $H$ (w.r.t. the scalar product $(A,B)=\mathrm{tr}(AB)$), or equivalently there exists a basis of $\mathbb C^n$ such that all matrices in $H$ have zero trace.

Even for $k=2$ I was not able to find any good characterization.