As suggested in a comment below my previous question here I'm going to ask a similar question but under a more specific hypothesis.
Let $\Omega\subset\mathbb{R}^n$, $n\geq 2$ be a bounded domain with smooth boundary $\partial\Omega$. Let $A: H\to H$ and $B: H\to H$ be two compact (self-adjoint) positive definite operators defined on $H=L^2(\partial\Omega)$. Let $(x_n)_n$ the sequence of eigenfunctions of $A$.
Since $B$ is compact and $(x_n)_n$ is a basis for $H$ then $$ \lim_{n\to\infty} \langle B(x_n),x_n\rangle=0 $$ Assuming $\mathcal{R}(A^{1/2}) \bigcap \mathcal{R}(B^{1/2})= \{0\}$ is it possible to prove that $$ \lim_{n\to\infty} \frac{\langle A(x_n),x_n\rangle}{\langle B(x_n),x_n\rangle}=0\,? $$