Definition 43.8.2 (Adjoint).label Let $H$ be a complex Hilbert space, $T: D(T) \to H$ be a densely defined linear operator,
\[D(T^{*}) = \bracsn{\xi \in H| (\eta \mapsto \dpn{\xi, T\eta}{H}) \text{ is bounded on }D(T)}\]
and[1]
\[T^{*}: D(T^{*}) \to H \quad \dpn{T^*\xi, \eta}{H}= \dpn{\xi, T\eta}{H}\]
then $T^{*}$ is a closed operator, known as the adjoint of $T$.
Proof. Let $\seq{\xi_n}\subset D(T^{*})$ and $\xi, \eta \in H$ such that $\xi_{n} \to \xi$ and $T^{*}\xi_{n} \to \eta$, then for any $\theta \in D(T)$,
\[\dpn{\eta, \theta}{H}= \limv{n}\dpn{T^*\xi_n, \theta}{H}= \limv{n}\dpn{\xi_n, T\theta}{H}= \dpn{\xi, T\theta}{H}\]
As $D(T)$ is dense, $T \circ \xi \in D(T)^{*}$, and $\eta = T^{*}\xi$. Thus the graph of $T^{*}$ is closed.$\square$
- $\eta$ sampled from $D(T)$.keyboard_return
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