I know that if one has a densely-defined operator $T$ with compact resolvent $(T-\lambda)^{-1}$ (for some $\lambda\in\mathbb{C}$) then its spectrum $\sigma(T)$ comprises a countable set of isolated eigen-values of finite multiplicity, accumulating at no finite point.
What I don't know (and can't seem to find anywhere) is whether there is some sort of converse implication in the case of $T$ being a self-adjoint operator on a Hilbert space. Can anybody offer an answer to the title question? If positively, please may I have a reference?
For $\lambda$ in the resolvent set, $(T-\lambda)^{-1}$ is the limit in the operator norm topology of the following finite rank operators: $$K_n = \int_{|\sigma(T)| \le n} \dfrac{1}{\lambda-t} \, dP(t)$$ where $P$ is the projection valued measure such that $T = \int_{\sigma(T)} t \, dP(t)$. These are finite rank since the range of $K_n$ is just the union of finitely many finite-dimensional eigenspaces. $$ (T-\lambda)^{-1}-K_n = \int_{|\sigma(T)|>n} \dfrac{1}{\lambda-t} \, dP(t)$$ which goes to $0$ as $n \to \infty$.