# Definition:Idempotent Operator

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## Definition

Let $\HH$ be a Hilbert space.

Let $A \in \map B \HH$ be a bounded linear operator.

Then $A$ is said to be **(an) idempotent (operator)** if and only if $A^2 = A$.

## 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 also that it be Hermitian.

In doing this, one arrives at what is called a projection on $\mathsf{Pr} \infty \mathsf{fWiki}$ in this context.

## Also see

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) ... (previous) ... (next) $\text {II}.3.1$