Lemma 39.3.2.label Let $H$ be a complex Hilbert space, $M \subset H$ be a closed subspace, $P \in B(H)$ be the orthogonal projection onto $M$, and $T \in B(H)$, then the following are equivalent:
- (1)
$T(M) \subset M$.
- (2)
$PTP = TP$.
- (3)
$T^{*}PTP = T^{*}TP$.
Proof. (3) $\Leftarrow$ (1): Let $\xi \in H$, then
\begin{align*}\dpn{T^*TP\xi, P\xi}{H}&= \dpn{TP\xi, TP\xi}{H}= \norm{TP\xi}_{H}^{2} \\ \dpn{T^*PTP\xi, P\xi}{H}&= \dpn{PTP\xi, TP\xi}{H}= \dpn{PTP\xi, PTP\xi}{H}= \norm{PTP\xi}_{H}^{2}\end{align*}
As $\norm{TP\xi}_{H} = \norm{PTP\xi}_{H}$, $TP\xi \in H$, so $T(M) \subset M$.$\square$
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