Definition:Idempotent Operator

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

• Definition:Complementary Idempotent

Sources

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