Idempotent Operators

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Let $H$ be a Hilbert space $H$.

A linear operator $P: H \to H$ is called idempotent if

$P^2 = P$

or equivalently

$P x = x$ for $x \in \Rng P$.

An idempotent operator is called a projector or orthogonal projector if

$\forall x \in H: P x - x \perp \Rng P$.

Especially in the context of linear algebra, many texts refer to all idempotent operators as "projectors" and use the same definition as above only for "orthogonal projectors." In such texts, idempotent operators that are not orthogonal projectors may be called oblique projectors.

Orthogonal projectors are extremely important in applied linear algebra and spectral theory. They can be characterized in several additional ways: