What is the point, continuous and residual spectrum of the following operator?

$T: \ell^{2}\supset\text{dom}(T) \rightarrow \ell^{2}$, $(x\_{n})\_{n\in \mathbb{N}} \mapsto (nx\_{n})\_{n\in \mathbb {N}}$, where $\text{dom}(T) = \left\{(x\_{n})\_{n\in \mathbb{N}} \in \ell^{2} | 
 (nx\_{n})\_{n\in \mathbb {N}} \in \ell^{2} \right\}$

My thoughts so far: Since $T$ is self-adjoint it must be that $\sigma\_{r}(T) = \emptyset$. Now considering the point spectrum, I tried to find the eigenvalues of $T$: $Tx = \lambda x \iff (n-\lambda) x\_{n} = 0$. But then I would have $x\_{n}=1 $ for $n = \lambda$ and $x\_{n}=0$ otherwise, so the corresponding eigenvectors are not in $\text{dom}(T)$, right? So $\sigma\_{p}(T) = \emptyset$? Can anyone provide hints for $\sigma\_r(T)$ or $\sigma\_c(T)$, please?