Range of Idempotent is Kernel of Complementary Idempotent

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Theorem

Let $H$ be a Hilbert space.

Let $A$ be an idempotent operator.

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

Corollary 1

Furthermore:

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

Corollary 2

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

Proof

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Sources

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