Range and Kernel of Idempotent are Algebraically Complementary

Theorem

Let $H$ be a Hilbert space.

Let $A$ be an idempotent operator.

Then $\ker A$ and $\Rng A$ are algebraically complementary, that is:

$\ker A \cap \Rng A = \left({0}\right)$, the zero subspace

$\ker A + \Rng A = H$, where $+$ signifies setwise addition.

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Proof

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Sources

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