If $A_i = A, 1 \leq i \leq m$ are square matrices of dimension $n$ we know $Tr(A_1...A_m) = \sum_{i=1}^n \lambda_i^m$ where $\lambda_i$ are the eigenvalues of $A$. If $A_1,A_2$ are symmetric or Hermitian with eigenvalues $\lambda_{j,1} \leq \cdots \leq \lambda_{j,n}, j = 1,2$, respectively, then $tr(A_1 A_2) \leq \sum_{i=1}^n \lambda_{1,i}\lambda_{2,i}$. More generally the John von Neumann inequality says if $A,B$ are complex with singular values $\sigma_{j,1} \leq \cdots \sigma_{j,n}, j=1,2$, respectively, then $tr(AB) \leq \sum_{i=1}^n \sigma_{1,i}\sigma_{2,i}$.
These are special cases, but I'd like to know if there's a general case when knowing the eigenvalues or singular values of $A_i$ tells about $Tr(A_1,...,A_m)$ in the form of equalities or inequalities