Essentially self-adjoint operator

Let $T$ be a densely defined symmetric operator on a complex Hilbert space. It is essentially self-adjoint if its closure $\overline T$ is a self-adjoint operator. Equivalently, $T$ has exactly one self-adjoint extension, namely $\overline T$ . The definition refers to the operator together with its specified dense domain: the same differential expression on different initial domains can be essentially self-adjoint, admit many self-adjoint extensions, or admit none. Symmetry alone does not imply essential self-adjointness because the domains of $\overline T$ and $T^*$ may differ.

Deficiency-space criteria

Essential self-adjointness is equivalent to vanishing of both deficiency spaces:

\ker(T^*-i)=\{0\},\qquad \ker(T^*+i)=\{0\}.

Equivalently, each of the ranges $\operatorname{Ran}(T+i)$ and $\operatorname{Ran}(T-i)$ is dense in $H$ . Surjectivity of these ranges is a criterion for an already closed self-adjoint operator; density is the correct criterion for the unclosed operator $T$ . These are the deficiency-space criteria for essential self-adjointness.

Cores and closure

If $A$ is self-adjoint and $D\subseteq\mathcal D(A)$ is dense in $\mathcal D(A)$ for the graph norm, then $D$ is a core for $A$ , and the restriction $A|_D$ is essentially self-adjoint with closure $A$ . This is the standard way differential and geometric operators initially defined on test sections recover a canonical self-adjoint operator. Ordinary Hilbert-space density of $D$ is not enough; graph-norm density is required.

Examples and domain sensitivity

On $L^2(\mathbb R)$ , the momentum operator $-i\,d/dx$ with initial domain $C_c^\infty(\mathbb R)$ is essentially self-adjoint. On $L^2(0,1)$ , the same expression on $C_c^\infty(0,1)$ is symmetric but not essentially self-adjoint; different boundary conditions produce distinct self-adjoint extensions. This contrast shows why a formal adjoint calculation does not settle essential self-adjointness.

References

Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1972. Publisher record. Relevant: Chapter VIII on symmetric operators, deficiency indices, and essential self-adjointness.
Konrad Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer, 2012. DOI record. Relevant: Chapters 1, 3, and 13 on closed operators, self-adjointness criteria, and extension theory.