Let $\phi\in\ell^\infty(\mathbb{N})$. For $p\in[1,\infty]$, define
$$M_\phi:\ell^p\to\ell^p,\quad f\mapsto\phi f.$$
Use spectral theory to show that, if $M_\phi$ is compact, then $\phi\in c_0$. Here all sequence spaces are over the field $\mathbb{C}$.
I have no idea how to apply spectral theory, I know how to prove the claim without using them.
If $M_\phi$ is compact, its spectrum is countable and has only zero as a limit point. Since $e_k$ are eigenvectors for $M_\phi$, with eigenvalues $\phi(k)$, it follows that $\phi(k) \to 0$.