Self-adjoint extensions with compact resolvent
Yicao Wang
TL;DR
We characterize when a densely defined closed symmetric operator $T$ with equal deficiency indices admits a self-adjoint extension with compact resolvent: this occurs if and only if $D(T)$ endowed with the graph norm $\|\cdot\|_g$ embeds compactly into $H$. Using boundary triplets, Cayley transforms, and the Weyl function $M(\lambda)$, the paper derives a Krein-type resolvent formula and provides a complete unitary-parameterization of all compact-resolvent extensions: precisely those $T_U$ with $U\in \mathbb{U}(G)$ and $U-\mathrm{Id}$ compact. The results reveal the topological structure of the family of such extensions, linking to Grassmannians and the deformation retract of $\mathbb{GL}_c(G)$ and showing that most unitary perturbations near $\mathrm{Id}$ do not yield compact resolvent. The work thus unifies von Neumann theory with boundary-triplet methods to give an intrinsic criterion and a global description of good boundary conditions for compact resolvent extensions.
Abstract
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.