Let $H$ be an Hilbert space. For $P\in B(H)$ (the set of bounded linear operators) projection (i.e. $P^2=P$) we say that $P$ is an orthogonal projection if $P(H)^{\perp}=(I-P)(H)$.
Now I have to prove three more characterizations of the orthogonal projection. That is, I have to prove the equivalence of:
- $P$ is an orthogonal projection.
- $P^{*}=P$.
- For all $x\in H$, $\langle Px,x \rangle\geq 0$.
- $P$ is normal.
I've managed $1)\Leftrightarrow 2)$, $2)\Rightarrow 3)$ but I can't seem to complete the equivalences. Can someone give me a hint? Thank you in advance.
Let $A:=P(H)$ be the image of the projection and $B:=\ker P=(I-P)(H)$ the direction of the projection. Then we have $H=A\oplus B$ with $P(a+b)=a$.
$\ $3. $\Rightarrow$ 1. : If $A\not\perp B$, then there are vectors $a\in A,\ b\in B$ such that $a\not\perp b$. Then
$$\langle P(a+\lambda b),\ a+\lambda b\rangle=\langle a,\,a+\lambda b\rangle=\|a\|^2+\lambda\langle a,b\rangle$$ $\quad$ can be easily made nonpositive for appropriate value of $\lambda$.
$\ $2. $\Rightarrow$ 4. is immediate.
$\ $4. $\Rightarrow$ 1. The adjoint operator $P^*$ is also a projection, with $\ker P^*=A^\perp$ and $P^*(H)=B^\perp$.
$\quad$ Moreover, $\ker M^*M=\ker M$ for any operator, so $P^*P=PP^*$ implies $\ker P=\ker P^*$.