Lemma 37.7.4.label Let $H$ be a complex Hilbert space and $P, Q \in B(H)$ be projections, then:
- (1)
$\ker(PQ) = \ker(Q) + \ker(P) \cap Q(H)$.
- (2)
If $PQ = QP$, then $PQ$ is a projection with $PQ(H) = P(H) \cap Q(H)$.
Proof. (1): Let $x \in \ker(PQ)$, then $Q(x) \in \ker(P)$, so $x = Q(x) + (1 - Q)(x) \in \ker(Q) + \ker(P) \cap Q(H)$.
(2): Since $PQ = QP$, $(PQ)^{2} = P^{2}Q^{2} = PQ$ and $(PQ)^{*} = Q^{*}P^{*} = QP = PQ$, $PQ$ is a projection. As $PQ(H) = P(Q(H)) \subset P(H)$ and $PQ(H) = Q(P(H)) \subset Q(H)$, $PQ(H) \subset P(H) \cap Q(H)$. On the other hand, $PQ$ is the identity on $P(H) \cap Q(H)$, so $PQ(H) = P(H) \cap Q(H)$.$\square$
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