Product of Projections

Theorem
Let $H$ be a Hilbert space.

Let $P, Q$ be projections.

Then the following are equivalent:


 * $(1): \quad P Q$ is a projection
 * $(2): \quad P Q = Q P$
 * $(3): \quad P + Q - P Q$ 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 $(2)$
Let $P Q$ be a projection.

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


 * $P Q = \paren {P Q}^* = Q^* P^* = Q P$

where the penultimate equality follows from Adjoint of Composition of Linear Transformations is Composition of Adjoints.

$(2)$ implies $(1)$
Let $P Q = Q P$.

Then:
 * $\paren {P Q}^2 = P Q P Q = P^2 Q^2 = P Q$

as $P, Q$ are projections.

Hence $P Q$ is an idempotent.

Also from Adjoint of Composition of Linear Transformations is Composition of Adjoints:
 * $\paren {P Q}^* = Q^* P^* = Q P = P Q$.

Hence, by Characterization of Projections, statement $(4)$, $P Q$ 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 $P Q$ are all projections, and $P Q = Q P$.

Now compute:

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

From:
 * Adjoining is Linear
 * Adjoint of Composition of Linear Transformations is Composition of Adjoints:


 * $\paren {P + Q - P Q}^* = P^* + Q^* - Q^* P^* = P + Q - QP = P + Q - P Q$

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

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

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

Hence necessarily $P Q = Q P$.

Also see

 * Sum of Projections
 * Difference of Projections