Lemma 43.8.4.label Let $H$ be a complex Hilbert space, $T: D(T) \to H$ be a densely defined, self-adjoint operator, then:
- (1)
For any $\xi \in D(T)$,
\[\norm{(T \pm i)\xi}_{H}^{2} = \norm{T\xi}_{H}^{2} + \norm{\xi}_{H}^{2} \ge \norm{\xi}_{H}^{2}\]In particular, $T \pm i$ is injective.
- (2)
$(T \pm i)(H) = H$.
In other words, $(T \pm i)$ is bijective onto $H$.
Proof. (1): For any $\xi \in D(T)$,
(2, closed range): For any $\seq{\xi_n}\subset D(T)$ and $\eta \in H$ with $(T + i)\xi_{n} \to \eta$, (1) implies that $\seq{\xi_n}\subset D(T)$ and $\seq{T\xi_n}$ are both Cauchy. Let $\xi = \limv{n}\xi_{n}$, then since $T$ is self-adjoint, it is closed, so $T\xi = \limv{n}T\xi_{n} \in T(H)$, and $\eta = (T + i)\xi \in (T+i)(H)$.
(2, dense range): Let $\eta \in H$, then the following are equivalent:
- (a)
$\eta \perp (T + i)(H)$.
- (b)
$\dpn{T\xi, \eta}{H}= \dpn{\xi, i\eta}{H}$ for all $\xi \in D(T)$.
- (c)
$\eta \in D(T^{*})$ with $T^{*}\eta = i\eta$.
- (d)
$\eta \in D(T)$ with $(T - i)\eta = 0$.
By (1), $\eta \perp (T+i)(H)$ if and only if $\eta = 0$.$\square$
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