# 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:

- An idempotent operator is an orthogonal projector if and only if it is self-adjoint.
- An idempotent operator is an orthogonal projector if and only if its norm is 1 (proof).