Definition 37.2.3 (Reducing Subspace).label Let $H$ be a complex Hilbert space, $M \subset H$ be a closed subspace, $P \in B(H)$ be the orthogonal projection onto $P$, and $T \in B(H)$, then the following are equivalent:
- (1)
$T(M) \subset M$ and $T^{*}(M) \subset M$.
- (2)
$TP = PT$.
If the above holds, then $M$ is a reducing subspace of $T$.
Proof, [Corollary 18.3, Zhu93]. (1) $\Rightarrow$ (2): By Lemma 37.2.2, $PTP = TP$ and $PT^{*}P = T^{*}P$. Thus $TP = PTP = PT$.
(2) $\Rightarrow$ (1): Since $TP = PT$, $PTP = PT$, and $T(M) \subset M$ by Lemma 37.2.2. Similarly, $T^{*}P = PT^{*}$ implies that $PT^{*}P = PT^{*}$, and $T^{*}(M) \subset M$ by Lemma 37.2.2.$\square$
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