Sum of Projections/Binary Case

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Theorem

Let $H$ be a Hilbert space.

Let $P, Q$ be projections.


Then $P + Q$ is a projection if and only if $\operatorname{ran} P \perp \operatorname{ran} Q$.

Here, $\operatorname{ran}$ denotes range, and $\perp$ denotes orthogonality.


Proof

Necessary Condition

Suppose $P + Q$ is a projection. Then:

\(\displaystyle P + Q\) \(=\) \(\displaystyle \left({P + Q}\right)^2\) $P + Q$ is an idempotent
\(\displaystyle \) \(=\) \(\displaystyle P^2 + PQ + QP + Q^2\)
\(\displaystyle \) \(=\) \(\displaystyle P + PQ + QP + Q\) $P$ and $Q$ are idempotents
\(\displaystyle \iff \ \ \) \(\displaystyle PQ + QP\) \(=\) \(\displaystyle 0\)


Now suppose that $h \in \operatorname{ran} Q$; say $h = Qq$ for $q \in H$.

Then it follows that $Qh = QQq = Qq = h$ as $Q$ is idempotent.

It follows that $0 = PQh + QPh = Ph + QPh$.


From Characterization of Projections, statement $(6)$, $\left\langle{QPh, Ph}\right\rangle_H \ge 0$.

Next, observe $0 = \left\langle{Ph + Qph, Ph}\right\rangle_H = \left\langle{Ph, Ph}\right\rangle_H + \left\langle{QPh, Ph}\right\rangle_H$.

As the second term is non-negative, the first is non-positive; it follows that $Ph = \mathbf{0}_H$ from the definition of the inner product.


Hence $h \in \operatorname{ker} P = \left({\operatorname{ran} P}\right)^\perp$.

It follows that $\operatorname{ran} Q \perp \operatorname{ran} P$, as asserted.

$\Box$


Sufficient Condition

Suppose that $\operatorname{ran} P \perp \operatorname{ran} Q$.

Then as $P, Q$ are projections, have:

$\operatorname{ran} P \subseteq \operatorname{ker} Q$
$\operatorname{ran} Q \subseteq \operatorname{ker} P$

That is, for all $h \in H$, one has $QPh = PQh = \mathbf{0}_H$. Hence:

\(\displaystyle \left({P + Q}\right)^2\) \(=\) \(\displaystyle P^2 + PQ + QP = Q^2\)
\(\displaystyle \) \(=\) \(\displaystyle P + Q\) $P$ and $Q$ are idempotents


That is, $P + Q$ is an idempotent.

Furthermore, by Adjoining is Linear, have:

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

where the latter follows from Characterization of Projections, statement $(4)$.

This same statement implies that $P+Q$ is also a projection.

$\blacksquare$


Also see


Sources