Let $A$ be a non-empty subset of a Hilbert space $H$. Suppose that $T$ is a linear operator on $H$ such that $T(H) \subseteq A$ and, for every $x \in H, (x-Tx) \perp A$. Then
- $T$ is bounded.
- $A$ is a closed linear subspace.
- $T$ is the orthogonal projection onto $A$.
Using the facts that $T(H) \subseteq A$ and, for every $x \in H, (x-Tx) \perp A$ I've been able to prove that $T$ is self-adjoint. With that, and the Closed Graph Theorem, I was also able to show that $T$ is bounded.
I'm stuck with (2). For some reason I don't see what characterizes $A$; I don't even see why is it a linear subspace. Having (2), my plan is to prove that
$ Tx = \begin{cases} x & \text{for } x \in A \\ 0 & \text{for } x \notin A \end{cases} \tag{*} $
since that would imply that, necessarily, $T$ is the orthogonal projection onto $A$.
Any hint, idea, comment, etc. on (2) and (*) will be appreciated.