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?
