Product of Projections

Theorem
Let $H$ be a Hilbert space.

Let $P, Q$ be projections.

Then the following are equivalent:


 * $(1): \qquad $PQ$ is a projection
 * $(2): \qquad $PQ = QP$
 * $(3): \qquad $P + Q - PQ$ is a projection

Necessary Condition
Suppose $PQ$ is a projection.

Then by Characterization of Projections, statement $(4)$, one has:


 * $PQ = \left({PQ}\right)^* = Q^* P^* = QP$

where the penultimate equality follows from Adjoint of Composition.

Sufficient Condition
Suppose that $PQ = QP$.

Then $\left({PQ}\right)^2 = PQPQ = P^2 Q^2 = PQ$ as $P, Q$ are projections.

Hence $PQ$ is an idempotent.

Also, note that $\left({PQ}\right)^* = Q^* P^* = QP = PQ$.

Hence, by Characterization of Projections, statement $(4)$, $PQ$ is a projection.

Also see

 * Sum of Projections
 * Difference of Projections