Suppose $P$, $Q$ are two non-trivial projections in $B(H)$, can we deduce that $P\leq Q$ or $Q\leq P$?
2026-03-27 13:27:20.1774618040
Comparison of projections in $B(H)$
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$P \leq Q$ iff $\langle Px , x \rangle \leq \langle Qx , x \rangle $ and this implies $Px=0$ whenever $Qx=0$. This is equiavlent to the statement that the range of $P$ is contained in the range of $Q$. Just take any two closed subspaces neither of which is conatined in the other. If $P$ and $Q$ are the projections on these then $P\leq Q$ and $Q \leq P$ are both false.