Converse of commutative projections.

Question

Let $X$ be a norm space on $F(F=R\ or\ C)$, if $P$ and $Q$ are two projections of $X$, and $P,Q$ are commutative, i.e., $PQ=QP$, then we know $PQ$ is the projection onto $im P\bigcap im Q$.
I wonder whether the converse is true. That is, if $im PQ=im QP$, then do we have $PQ=QP$? It seems that this could be right, but i cannot deduce it simply from the properties of projections.
I can get that $im P\bigcap im Q=im PQ=imQP$ and $QP=PQP,PQ=QPQ$, but what then?
If it is not true, could anyone please give a counter example?

Answer 1

In general, the statement is false. Let $p,q$ be projections in $\mathbb{R}^2$ with kernels $\mathbb{R}(1,1)$ and $\mathbb{R}(0,1)$ onto $\mathbb{R}(1,0)$. Then $p,q,pq,qp$ have the same image $\mathbb{R}(1,0)$. However $pq(1,1)=p(1,0)=(1,0)$ while $qp(1,1)=q(0,0)=(0,0)$ so $p,q$ do not commute.

Answer 2

If two orthgonal projections have the same image, then they are equal: Let $P$ and $Q$ be same projections such that $\text{im } P=\text{im }Q$. Then $\text{im } P\subseteq\text{im }Q$ implies $QP=P$ and $\text{im } P\supseteq \text{im }Q$ implies $PQ=Q$. Taking adjoints in the latter $QP=Q$. Hence $P=QP=Q$.