Orthogonality of almost eiegnvectors of normal operator Suppose $\mathcal{H}$ is an innfinite dimensional, separable Hilbert space and $A : \mathcal{H}\rightarrow \mathcal{H}$
a bounded normal operator, i.e. $AA^{\ast} = A^{\ast}A$. Prove that if $\lambda$ and $\mu$  are complex
numbers such that $\lambda\neq\mu$    and $f_{n},g_{n}$ are sequences in $\mathcal{H}$ such that $\|f_n\| = \|g_n\| = 1$
and 
$$\|Af_n-\lambda f_n\|\rightarrow 0,\|Ag_n-\mu g_n\|\rightarrow 0 $$
 as $n\rightarrow \infty$ , then $\langle f_{n},g_{n}\rangle\rightarrow 0$ as  $n\rightarrow \infty$ 
 A: For a given $\nu\in\mathbb{C}$ denote $A_\nu=A-\nu I$. 
1) One can check that if $A$ is normal so does $A_\nu$. Now, for any $x\in \mathcal{H}$, then
$$
\Vert A_\nu x\Vert^2
=\langle A_\nu x,A_\nu x\rangle
=\langle  x,A_\nu^*A_\nu x\rangle
=\langle  x,A_\nu A_\nu^* x\rangle
=\langle  A_\nu^*x,A_\nu^* x\rangle
=\Vert A_\nu^*x\Vert^2
$$
This means that 
$$
\lim\limits_{n\to\infty}A_\nu x_n=0\quad\mbox{ if and only if }\quad\lim\limits_{n\to\infty}A_\nu^* x_n=0
$$
2) Moreover if $\lim\limits_{n\to\infty}A_\nu x_n=0$ and $\Vert y_n\Vert\leq C$ for all $n\in\mathbb{C}$, then
$$
\lim\limits_{n\to\infty}|\langle A_\nu x_n,y_n\rangle|
\leq\lim\limits_{n\to\infty}\Vert A_\nu x_n\Vert\Vert y_n\Vert
\leq C\lim\limits_{n\to\infty}\Vert A_\nu x_n\Vert=0
$$
so
$$
\lim\limits_{n\to\infty}\langle A_\nu x_n,y_n\rangle=0
$$
3) Now we return to the original problem. Using result of previous paragraphs we can say that
$$
\begin{align}
\lambda\lim\limits_{n\to\infty}\langle f_n,g_n\rangle
&=\lim\limits_{n\to\infty}\langle \lambda f_n,g_n\rangle\\
&=\lim\limits_{n\to\infty}(\langle A f_n,g_n\rangle - \langle A_\lambda f_n,g_n\rangle)\\
&=\lim\limits_{n\to\infty}\langle f_n, A^*g_n\rangle - \lim\limits_{n\to\infty}\langle A_\lambda f_n,g_n\rangle\\
&=\lim\limits_{n\to\infty}(\langle f_n, A_\mu^*g_n\rangle+\langle f_n, \overline{\mu}g_n\rangle)\\
&=\lim\limits_{n\to\infty}\mu\langle f_n, g_n\rangle\\
&=\mu\lim\limits_{n\to\infty}\langle f_n, g_n\rangle\\
\end{align}
$$
Since $\lambda\neq\mu$ we conclude
$$
\lim\limits_{n\to\infty}\langle f_n,g_n\rangle=0
$$
