Range of Idempotent is Kernel of Complementary Idempotent

Theorem

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.

Let $A$ be an idempotent operator.

Then:

$\Rng A = \map \ker {I - A}$

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Corollary 1

$\ker A = \Rng {I - A}$

Corollary 2

Then:

$\Rng A$ is a closed linear subspace of $\HH$.

Proof

If $h \in \map \ker {I - A}$, we have:

$\map {\paren {I - A} } h = {\mathbf 0}_\HH$

That is:

$h - A h = {\mathbf 0}_\HH$

so that:

$A h = h$

and so:

$h \in \Rng A$

So we have:

$\map \ker {I - A} \subseteq \Rng A$

Now let $h \in \Rng A$.

Then there exists $k \in \HH$ such that $h = A k$.

Then we have $h - A h = h - A^2 k$.

Since $A$ is an idempotent operator, we have:

$A^2 k = A k = h$

Hence:

$h - A h = {\mathbf 0}_\HH$

So:

$h \in \map \ker {I - A}$

We therefore obtain:

$\Rng A \subseteq \map \ker {I - A}$

So:

$\Rng A = \map \ker {I - A}$

$\blacksquare$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.3.2 \ \text {(b)}$