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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.

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  • $\begingroup$ The fact that $B$ has band structure does not imply anything on the support of the eigenfunctions $\phi_j$, so typically there won't be $\phi_j$'s with the properties you require. $\endgroup$ Feb 10, 2015 at 21:45
  • $\begingroup$ What do you mean by the support of $\phi_j$? Do you mean the condition $\langle \phi_j,e_k\rangle = 0$ if $\vert j - k \vert > n$? It is already given. We need to estimate the eigenvalues. Maybe we wont get the estimate as sharp as one I am proposing but still, how much can the first estimate be improved(say by using min max principle)? $\endgroup$
    – Madhuresh
    Feb 11, 2015 at 5:14

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