Definition:Complementary Idempotent

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Let $\HH$ be a Hilbert space.

Let $A$ be an idempotent operator on $\HH$.

Then the complementary idempotent (operator) of $A$ is the bounded linear operator $I - A$, where $I$ is the identity operator on $H$.

Complementary Projection

If $A$ is a projection, $I - A$ is called the complementary projection.

This name is justified by Complementary Projection is Projection.

Also see

The name is appropriate, by Complementary Idempotent is Idempotent and the fact that $I - \paren {I - A} = A$.