DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedaudited 2026-09-22

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Symmetric, self-adjoint and essentially self-adjoint operators

Definition

Assume Countable Choice. Let $T$ be a densely defined linear operator on $H$ .

$T$ is symmetric when $T \subseteq T^{*}$ , that is, $D (T) \subseteq D (T^{*})$ and $T x = T^{*} x$ for all $x \in D (T)$ ; equivalently $⟨ T x, y ⟩ = ⟨ x, T y ⟩$ for all $x, y \in D (T)$ .
$T$ is self-adjoint when $T = T^{*}$ , that is, the domains and the values agree.
$T$ is essentially self-adjoint when its closure $\overline{T}$ is self-adjoint; by Closability is equivalent to density of the adjoint domain this presupposes that $T$ is closable.

Consequences, with proofs. These are part of the content of the definition.

A symmetric $T$ is closable. $T \subseteq T^{*}$ and $T^{*}$ is closed (The adjoint is well defined, closed, and reverses inclusions), so $T^{*}$ is a closed extension of $T$ . In particular the closure $\overline{T}$ exists and $\overline{T} \subseteq T^{*}$ .
The closure of a symmetric operator is symmetric. If $T \subseteq T^{*}$ , the inclusion reversal of The adjoint is well defined, closed, and reverses inclusions applied to $T \subseteq T^{*}$ gives $(T^{*})^{*} \subseteq T^{*}$ , that is $T^{* *} \subseteq T^{*}$ . Applying it once more to the inclusion $T^{* *} \subseteq T^{*}$ gives $(T^{*})^{*} \subseteq (T^{* *})^{*}$ , that is $T^{* *} \subseteq T^{* * *}$ . Since $\overline{T} = T^{* *}$ and $(\overline{T})^{*} = T^{* * *}$ (Closability is equivalent to density of the adjoint domain), this reads $\overline{T} \subseteq (\overline{T})^{*}$ : the closure is symmetric. (Each application is legitimate because the adjoint is defined once its operator is densely defined, and $D (T^{* *})$ is dense because $T \subseteq T^{* *} = \overline{T}$ , so that $D (T^{* *})$ contains the dense domain $D (T)$ .)
A self-adjoint operator has no proper symmetric extension. If $T = T^{*}$ and $T \subseteq S \subseteq S^{*}$ , then $S^{*} \subseteq T^{*}$ by inclusion reversal, so $T \subseteq S \subseteq S^{*} \subseteq T^{*} = T$ and all inclusions are equalities.
A self-adjoint operator is closed, being equal to the adjoint $T^{*}$ of the densely defined $T$ , and an essentially self-adjoint operator has exactly one self-adjoint extension, namely $\overline{T}$ : a self-adjoint extension $S$ of $T$ is closed, hence contains the least closed extension $\overline{T}$ , and then $S \subseteq S^{*} \subseteq (\overline{T})^{*} = \overline{T}$ by item 2 and symmetry, so $S = \overline{T}$ .

Depends on

Used by

Existence of self-adjoint extensions is equality of deficiency indices Corollary
Unitary groups converge under strong resolvent convergence Corollary
A symmetric closed operator that is not self-adjoint Counterexample
An everywhere-defined closed operator on a Banach space is bounded Counterexample
Cayley transform of a self-adjoint operator Definition
Deficiency subspaces and deficiency indices Definition
Discrete and essential spectrum of a self-adjoint operator Definition
Norm and strong resolvent convergence Definition
Pure point, absolutely continuous and singular continuous spectral subspaces Definition
Relative compactness with respect to an operator Definition
Periodic derivative and its unitary translation group Example
A self-adjoint operator generates a strongly continuous unitary group Lemma
Spectral form domain and core of a semibounded operator Lemma
The generator of a unitary group is closed and skew-adjoint Lemma
The resolvent star algebra is dense in C₀(R) Lemma
Cayley correspondence between self-adjoint operators and unitaries Theorem
Kato-Rellich theorem Theorem
Min-max principle below the essential spectrum Theorem
Range criterion for self-adjointness Theorem
Resolvent of a self-adjoint operator: nonreal resolvents and the estimate Theorem
Spectral theorem for unbounded self-adjoint operators (PVM form) Theorem
Stone's theorem: unitary groups and self-adjoint generators Theorem
Unbounded Borel functional calculus: domains, products, spectral mapping Theorem
Weyl's theorem: invariance of the essential spectrum Theorem

Dependency tree · two levels

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Sources

Dana P. Williams, Lecture Notes on the Spectral Theorem (standard reference, not scraped)
Gerald Teschl, Mathematical Methods in Quantum Mechanics, second edition (standard reference, not scraped)
Theo Buehler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)