Let's say we have two Hermitian matrices $A$ and $B$. We know the eigenvalues of both $A$ and $B$. We also know that $A - B$ is traceless, meaning:
$$ Tr(A - B) = 0. $$
A piece of additional information that we know is, the eigenvalues of $A - B$ are all the same, up to change of sign. More specifically, if $\lambda$ is an eigenvalue of $A - B$, then $-\lambda$ is also an eigenvalue with the same multiplicity.
With these pieces of information, what can we say about $\lambda$? I am aware of this mathoverflow question along with the Weyl inequalities. However, I believe there is something stronger that could be said in this specific situation. Could you please point to some directions? Thanks!