Complementary Idempotent of Complementary Idempotent is Idempotent

Theorem

Let $\HH$ be a Hilbert space.

Let $I$ be an identity operator on $\HH$.

Let $A$ be an idempotent operator.

Let $B$ be the complementary idempotent of A.

Then the complementary idempotent of $B$ is $A$.

Proof

From Complementary Idempotent is Idempotent the complementary idempotent of $B$ is well-defined.

Let $C$ be the complementary idempotent of $B$.

We have:

\(\ds C\)

\(=\)

\(\ds I - B\)

Definition of Complementary Idempotent

\(\ds \)

\(=\)

\(\ds I - \paren{I - A}\)

Definition of Complementary Idempotent

\(\ds \)

\(=\)

\(\ds A\)

$\blacksquare$