Can we give bounds or infer any type of information about the spectrum of the point-wise product of two matrices $A$ and $B$ given knowledge about the spectrum of $A$ and $B$ ?
Are there non trivial conditions on $A$ and $B$ that would make reasoning about these easier ?
The inequality on the product of eigenvalues at this section on Wikipedia reproduced here:
$$\det({A} \circ {B}) \ge \det({A}) \det({B}).$$
is already a nice result.
$$ \operatorname{rank}(A \circ B) \leq \operatorname{rank}(A) \operatorname{rank}(B) $$
as well. Are there other known inequalities involving eigenvalues ? Can one infer anything about individual eigenvalues ? Maybe the dominant ones ?
Edit : There are nice inequalities on the spectral radius in this answer https://math.stackexchange.com/a/2706288/1049002 and in the comment below. This question might be a bit of a duplicate of that one but maybe some new proofs were found specifically for eigenvalues and eigenvectors since.
Edit 2 : This answer https://scicomp.stackexchange.com/a/29071 claims that not much can be known in general. [Edit 3 : but it does provide a link to a paper that seems to show some inequalities.]