Self-adjoint extensions with compact resolvent

TL;DR

This paper characterizes when a symmetric operator has a self-adjoint extension with compact resolvent, linking it to the compact embedding of its domain and parametrizing such extensions via unitary operators close to the identity.

Contribution

It establishes a necessary and sufficient condition for the existence of self-adjoint extensions with compact resolvent based on domain embedding, and provides a parametrization of all such extensions.

Findings

Self-adjoint extension with compact resolvent exists iff domain is compactly embedded.

All such extensions are parametrized by unitary operators differing from identity by a compact operator.

The characterization links operator domain properties to spectral properties.

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.