Product of Projections

Theorem
Let $H$ be a Hilbert space.

Let $P, Q$ be projections.

Then the following are equivalent:


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

Proof
The proof proceeds by first showing that $(1)$ is equivalent to $(2)$.

Then, these are combined and shown equivalent to $(3)$.

$(1)$ implies $(2)$
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.

$(2)$ implies $(1)$
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.

$(1), (2)$ imply $(3)$
The above establishes that assuming either of $(1)$ and $(2)$ yields both to hold.

So assuming $(1)$, $P, Q$ and $PQ$ are all projections, and $PQ = QP$.

Now compute:

It follows that $P + Q - PQ$ is an idempotent.

Observe from Adjoining is Linear and Adjoint of Composition:


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

Now applying Characterization of Projections, statement $(4)$, conclude that $P + Q - PQ$ is a projection.

$(3)$ implies $(2)$
Let $P + Q - PQ$ be a projection.

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

Hence necessarily $PQ = QP$.

Also see

 * Sum of Projections
 * Difference of Projections