Let $\Omega$ a bounded domain in $\mathbb{R}^n$ and let $A$ a compact self-adjoint and positive operator defined from $L^2(\partial\Omega)$ to itself.
Let $\lambda_n$ the decreasing sequence of eigenvalues of $A$ and let $u_n$ be an orthonormal sequence, such that $\lVert u_n \rVert=1$ for all $n$.
It is possible to prove that $$ \langle Au_n,u_n \rangle \geq K\lambda_n \quad \text{for $n\gg 1$} $$ ?