Problem: $P_1$ and $P_2$ are two self-adjoint projections. Is it true that $\mbox{R} (P_1) \subseteq \mbox{R} (P_2) \Longrightarrow P_1 \leq P_2$? (Note that R() indicates the range).

I believe it is not true, but I cannot find any counterexamples or way to disprove.
One fact that might be useful to consider is that for a self-adjoint projection $\mbox{R}(P) = \mbox{N}(P)^\perp$

2. Originally Posted by DJDorianGray
Problem: $P_1$ and $P_2$ are two self-adjoint projections. Is it true that $\mbox{R} (P_1) \subseteq \mbox{R} (P_2) \Longrightarrow P_1 \leq P_2$? (Note that R() indicates the range).

I believe it is not true, but I cannot find any counterexamples or way to disprove.
One fact that might be useful to consider is that for a self-adjoint projection $\mbox{R}(P) = \mbox{N}(P)^\perp$
If $\mbox{R} (P_1) \subseteq \mbox{R} (P_2)$ then $P_2P_1 = P_1$. Taking adjoints, you see that $P_1P_2= P_1$. Thus $P_1$ and $P_2$ commute, and it follows (as you can easily check) that $(P_2 - P_1)^2 = P_2 - P_1$. Thus $P_2 - P_1$ is also a selfadjoint projection and is therefore positive. Hence $P_1 \leqslant P_2$.