Let $T$ be a bounded linear operator on a Hilbert space $H$. Let there exists a positive square root of $T^{*}T$. Let us call it $P$. Then is it true that range of $P$ is a closed subspace of $H$? How can I show that?
Otherwise how an I show that $H$ is direct sum of range of $P$ and its orthogonal complement?
Should I show that $P$ is a projection? For that, I will have to show that $P$ is self-adjoint and idempotent. But I can't see $P$ being idempotent.
Please help