Self-adjoint extensions with compact resolvent

Let $T$ be a densely defined closed symmetric operator with equal deficiency indices in a separable complex Hilbert space $H$. In this paper, we prove that $T$ has a self-adjoint extension with compact resolvent if and only if the domain $D(T)$ of $T$ is compactly embedded in $H$ w.r.t. the graph norm on $D(T)$. If it is the case, we also prove that all self-adjoint extensions with compact resolvent can be parameterized by unitary operators $U$ on a certain Hilbert space such that $U-Id$ is compact.