Closed symmetric operators and deficiency data

Assume the Axiom of Choice and Countable Choice. Let $H$ be a complex Hilbert space and let $S:D(S)\to H$ be a densely defined, closed, symmetric linear operator. The domain $D(S)$ is part of the operator data. Its adjoint is the operator $S^{\ast }:D(S^{\ast })\to H$ characterized by $⟨Sf,g⟩=⟨f,S^{\ast }g⟩$ for every $f\in D(S)$ and $g\in D(S^{\ast })$, using an inner product linear in its first argument. A graph core for $S$ is a linear subspace $C\subseteq D(S)$ dense for the graph norm $\mathrm{\parallel }f{\mathrm{\parallel }}_S=(\mathrm{\parallel }f{\mathrm{\parallel }}^2+\mathrm{\parallel }Sf{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}$; equivalently, the graph of the restriction $S{\mathrm{\mid }}_C$ is dense in the graph of $S$.

Fix a positive scalar $E_{\ast }$ and write $I_H$ for the identity on $H$. The scaled deficiency subspaces of $S$ at $E_{\ast }$ are $N_{+}(S;E_{\ast }):=\ker (S^{\ast }-i{E}_{\ast }{I}_H)$ and $N_{-}(S;E_{\ast }):=\ker (S^{\ast }+i{E}_{\ast }{I}_H)$. Their deficiency indices are $n_{\pm }(S;E_{\ast }):=\dim N_{\pm }(S;E_{\ast })$, where Hilbert dimension means the cardinality of an orthonormal basis as in the standard deficiency-index definition. The spaces are closed: if $g_k\in N_{\pm }(S;E_{\ast })$ and $g_k\to g$, then $S^{\ast }{g}_k=\pm i{E}_{\ast }{g}_k\to \pm i{E}_{\ast }g$, so closedness of $S^{\ast }$ gives $g\in D(S^{\ast })$ and $S^{\ast }g=\pm i{E}_{\ast }g$. The dimension convention is well defined under Choice by the cited deficiency-index definition.

For any specified linear trace map $\gamma :D(S^{\ast })\to B$ into a vector space $B$ and specified linear subspace $\mathrm{\Lambda }\subseteq B$, the restriction of $S^{\ast }$ to $\{f\in D(S^{\ast }):\gamma (f)\in \mathrm{\Lambda }\}$ is the boundary realization specified by $(\gamma ,\mathrm{\Lambda })$. Its domain and trace data are part of the operator realization; the differential expression alone is not.

Symmetric, self-adjoint and essentially self-adjoint operators

Adjoint of a densely defined operator

Unbounded linear operators: domain, graph and extension

The adjoint is well defined, closed, and reverses inclusions

Deficiency subspaces and deficiency indices