Let $T: \ell_2 \to \ell_2$ be $$ T((x_n)_{n \in \mathbb{N}}) = \left(\frac{x_n}{n}\right)_{n \in \mathbb{N}}. $$
We have to determine if $T$ is a compact operator, self-adjoint and if it has eigenvalues.
By simple arithmetic, we can determine the eigenvalues $\lambda = 1/n$ for each $n$ and all the eigenvectors associated to each of them.
It is easy to see that the operator is self-adjoint because $\ell_2$ is a Hilbert space and $T$ is in $\mathcal{L}(\ell_2, \ell_2)$ and $\langle T(x), x\rangle < \infty$ for all $x \in \ell_2$.
On the other hand, I have no idea how to prove compactness of the operator. I guess I need to use some characterization of compact operators such as the one that links bounded sequences with convergent sequences.