If $P$ is a projection operator, is $1-P$ also a projection operator?

Question

Show that if $P$ is a (hermitian) projection operator, so are (a) $1-P$ and (b) $$ U^{+}PU $$ for any operator $U$

Answer 1

Definition of projection operator is $P\circ P = P$, for (a) you can simply expand $(1-P)\circ (1-P)$ and find it holds true. You have to define the notation $U^+$ to get an answer for (b).

Answer 2

$$\begin{align} (1-P)^2(x) &=(1-P) \circ ((1-P)(x)) \\ \implies (1-P)^2(x) &=(1-P) \circ (x-P(x)) \\ \implies (1-P)^2(x) &=(x-P(x))-(P(x)-P^2(x)) \\ \end{align}$$ Now using the fact that P is a projection operator, i.e. $P^2=P$, we get: $$ \begin{align} \implies (1-P)^2(x)&=(1-P)(x)-(P(x)-P(x)) \\ \implies (1-P)^2(x)&=(1-P)(x)\\ \end{align} $$

Hence we get $(1-P)^2=(1-P)$, which implies $(1-P)$ is a projection operator.