One of the most important (and very well-known) result in the study of the spectrum of Hermitian matrices is Horn's conjecture (or theorem?), which provides a complete answer to the following problem:

Let $A$ and $B$ be $n\times n$ Hermitian matrices. Determine all the possible values of the eigenvalues of $A+B$, given the eigenvalues of $A$ and $B$.

Given this result,
it seems a natural to wonder if there exists (or could possibly exist) such characterizations for other *combinations* Hermitian matrices.

For instance,
if by *combination* we mean the tensor product,
then the answer is trivial:
the eigenvalues of $A\otimes B$ is simply the set of all products of the eigenvalues of $A$ and $B$.

However, if by combination we mean the Hadamard (or Schur) product $$(A\circ B)_{ij}=A_{ij}B_{ij},$$ the answer is seemingly nontrivial, as with $A+B$.

Letting $\lambda_1(A\circ B),\ldots,\lambda_n(A\circ B)$ denote the eigenvalues of $A\circ B$, we observe that $$\lambda_1(A\circ B)+\cdots+\lambda_n(A\circ B)=\sum_{i=1}^nA_{ii}B_{ii},$$ and thus the set of possible eigenvalues of $A\circ B$ lies in a hyperplane $H$ determined by the diagonal entries of $A$ and $B$, which are themselves (to some extent) determined by the eigenvalues of $A$ and $B$.

My question is thus the following:

Does there exist conjectures or results on what the set of possible eigenvalues of $A\circ B$ might look like in the hyperplane $$x_1+\cdots+x_n=\sum_{i=1}^nA_{ii}B_{ii},$$ or is there a reason to believe that there is little hope to find a satisfying description of such a subset?