Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that $\langle Be_j,e_k \rangle = 0$ if $\left| j - k \right| > n$ for some fixed $n < m$. Let $\{\phi_j,\rho_j\}$ be the eigenbasis of $B$ such that $\rho_j > \rho_{j + 1}$ and $\langle \phi_j,e_k\rangle = 0$ if $\vert j - k \vert > n$.
I want to estimate the eigenvalues $\{y_j\}$ (with $y_j > y_{j + 1})$ of $ABA$ (which we know exists). It is obviously true that $y_j \leq \rho_j \lambda_1^2$. Can we show that $y_j \leq \rho_j \lambda_{min\{j - n, 1\}}^2$? The same question can also be asked for real self adjoint compact operators.