TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Closability is equivalent to density of the adjoint domain

Statement

Assume Countable Choice. Let $T$ be a densely defined linear operator on $H$ . Then $T$ is closable if and only if $D (T^{*})$ is dense in $H$ . In that case $\overline{T} = T^{* *}$ and $T^{*} = {\overline{T}}^{*}$ ; moreover if $T \subseteq T^{*}$ , then $T$ is closable.

Facts & Assumptions

[A1]

For densely defined $T$ the adjoint is defined by $⟨ T x, y ⟩ = ⟨ x, T^{*} y ⟩$ for $x \in D (T)$ , $y \in D (T^{*})$ , and $Γ (T^{*}) = {(y, w) : ⟨ T x, y ⟩ = ⟨ x, w ⟩ for all x \in D (T)}$ . Putting $W (x, y) : = (- y, x)$ defines a bijective isometry of $H \oplus H$ with $W^{2} = - I$ , and $W$ maps orthocomplements to orthocomplements: $W (M^{⊥}) = (W M)^{⊥}$ (Adjoint of a densely defined operator, Hilbert space, Orthogonality and the orthogonal complement).

[A2]

If $M$ is a linear subspace of the Hilbert space $H \oplus H$ , then $M^{⊥ ⊥} = \overline{M}$ (The double orthogonal complement of a subspace is its closure).

[A3]

For a densely defined $T$ the operator $T^{*}$ is closed. If in addition $D (T^{*})$ is dense, then $T^{* *}$ is defined, is closed, and contains $T$ (The adjoint is well defined, closed, and reverses inclusions, Adjoint of a densely defined operator).

[A4]

$T$ is closable when it has a closed extension; if $T$ is closable then $\overline{T}$ is the least closed extension of $T$ and $T \subseteq \overline{T}$ (Closure of a closable operator, Densely defined, closed and closable operators, and cores).

Proof

technique · direct

Given: A densely defined linear operator $T$ on $H$ .

1.1

By [A1] we have $Γ (T^{*}) = W Γ (T)^{⊥}$ : indeed $(y, w) \in Γ (T^{*})$ means $⟨ T x, y ⟩ = ⟨ x, w ⟩$ for all $x \in D (T)$ , which is exactly $⟨ (x, T x), W (y, w) ⟩ = 0$ for all $x \in D (T)$ , that is $W (y, w) ⊥ Γ (T)$ .

2.1

Assume in addition that $D (T^{*})$ is dense, so that $T^{* *}$ is defined. Replacing $T$ by $T^{*}$ in step 1.1 gives $Γ (T^{* *}) = W Γ (T^{*})^{⊥}$ , so by step 1.1 and unitarity of $W$ , with $W (M^{⊥}) = (W M)^{⊥}$ and $W^{2} = - I$ , one has $Γ (T^{* *}) = W (W Γ (T)^{⊥})^{⊥} = W W Γ (T)^{⊥ ⊥} = - \overline{Γ (T)} = \overline{Γ (T)}$ .

step 1.1A1A2

2.2

Conversely assume $T$ is closable, and let $\overline{T}$ be its closure, a closed densely defined operator with $Γ (T) \subseteq Γ (\overline{T})$ and $\overline{Γ (T)} = Γ (\overline{T})$ by [A4]. Then $D (\overline{T}^{*})$ is dense: if $y ⊥ D (\overline{T}^{*})$ then $(y, 0) ⊥ Γ (\overline{T}^{*})$ , so by step 1.1 applied to $\overline{T}$ and closedness of $\overline{T}$ we get $(y, 0) \in Γ (\overline{T}^{*})^{⊥} = W Γ (\overline{T})^{⊥ ⊥} = W Γ (\overline{T})$ , that is $(y, 0) = W (x, \overline{T} x) = (- \overline{T} x, x)$ for some $x \in D (\overline{T})$ , forcing $x = 0$ and $y = 0$ .

A1A2A4step 1.1

3.1

Under the hypothesis of step 2.1 the set $\overline{Γ (T)}$ is the graph of the operator $T^{* *}$ , which is closed by [A3]; hence $T$ has a closed extension and is closable, and its closure is $T^{* *}$ by minimality in [A4].

step 2.1A3A4

3.2

With $T$ closable as in step 2.2, apply step 1.1 to $T$ and to $\overline{T}$ : using $\overline{Γ (T)} = Γ (\overline{T})$ and [A2] one gets $Γ (T^{*}) = W Γ (T)^{⊥} = W {\overline{Γ (T)}}^{⊥} = W Γ (\overline{T})^{⊥} = Γ (\overline{T}^{*})$ , so $T^{*} = \overline{T}^{*}$ and $D (T^{*}) = D (\overline{T}^{*})$ is dense by step 2.2.

A1A2step 1.1step 2.2

4.1

The two implications are steps 3.1 (density of $D (T^{*})$ gives closability, with closure $T^{* *}$ ) and 3.2 (closability gives density of $D (T^{*})$ , and $T^{*} = \overline{T}^{*}$ ). If $T \subseteq T^{*}$ , then $T^{*}$ is a closed extension of $T$ by [A3], so $T$ is closable.

step 3.1step 3.2∎

Depends on

Used by

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)