Range of projections

Question

Let $H$ be a Hilbert space and $P,Q$ be two projections in $\mathcal{B}(H)$. If $P\leq Q$, i.e., $\langle Px,x\rangle\leq \langle Qx,x\rangle$ for all $x\in H$, then show that $\text{range } P\subseteq \text{range }Q$.

My attempt: We need to show that $QP=Q$. I am able to show that $P=PQP$ but cannot conclude $QP=Q$ from this.

Edit: Let $x\in\text{range } P$. Then $Px=x$. Since $P\leq Q$, so $\langle x,x\rangle=\langle Px,x\rangle\leq \langle Qx,x\rangle$. Hence $x\leq Qx$. How to show that $Qx\leq x$?

Answer 1

You actually need to show that $QP=P$.

Hint: $$P-PQP=P(1-Q)P=[(1-Q)P]^*[(1-Q)P].$$