Need help in proving $P$ is a projection.

Question

A bounded linear operator $P:\mathbb H\rightarrow\mathbb H$ on a Hilbert space $\mathbb H$ is a projection iff $P$ is Self-adjoint and idempotent

Proof:Initially,i started by assuming $P$ to be a projection and successfully shown that it is self-adojoint and idempotent.

I'm not getting the converse of it.

Please help in proving the converse part

Answer 1

Assume that $P^2 = P^* = P$.

Notice that $Px=x \iff x \in \operatorname{Im} P$. If $Px = x$ then clearly $x \in \operatorname{Im} P$. Conversely, if $y \in \operatorname{Im} P$ then $\exists x \in H$ such that $Px = y$. We have

$$Py = P^2x = Px = y$$

Then notice that $\operatorname{Im} P$ is closed. Namely, if $(Px_n)_n$ is a sequence in $\operatorname{Im} P$ converging to $y \in H$, then by applying $P$ we get $Px_n = P^2x_n \to Py$ so $Py = y$ which implies $y \in \operatorname{Im} P$.

Finally, in general we have $H = \ker P \oplus \overline{\operatorname{Im} P^*}$, but since $P^* = P$ and $\operatorname{Im} P$ is closed we get $H = \ker P \oplus \operatorname{Im} P$.

Then for $x_1 \in \ker P$, $x_2 \in \operatorname{Im} P$ we have $$P(x_1+x_2) = Px_1 + Px_2 = x_2$$

so $P$ is precisely the orthogonal projection onto the closed subspace $\operatorname{Im} P$ of $H$.