Definition:Idempotent Operator

Definition
Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be a bounded linear operator.

Then $A$ is said to be idempotent, or an idempotent operator, iff $A^2 = A$.

Also known as
Some sources refer to this concept as a projection.

However, another common convention (especially when dealing with Hilbert spaces) is to demand self-adjointness as well.

In doing this, one arrives at what is called a projection on ProofWiki in this context.

Also see

 * Complementary Idempotent