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If $P$ in $L(X,Y)$ and $Q$ in $L(Y,Z)$ are projections, Can we conclude that $QP$ is also a projection ? Thank you !

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What is your definition of projection? $P^2 = P$? Or are you also assuming $P$ is a self-adjoint operator? Except that these notions don't seem to make sense when $P \in L(X,Y)$. – Christopher A. Wong Mar 13 '13 at 20:45
A projection $X \to Y$ is an operator isomorphic to the projection $Y \oplus Z \to Y$ for some $Z$. – Martin Brandenburg Mar 13 '13 at 21:45
@Martin, what does isomorphism of operators mean in this context? – Gerry Myerson Mar 13 '13 at 23:08

Even for orthogonal projections on a Hilbert space the answer is NO: Consider in $\mathbb R^2$ the orthogonal projection $P$ on the $x$-axis and $Q$ the orthogonal projection onto the diagonal. Draw a picture to see what happens with the orbits $\lbrace (PQ)^n x: n\in\mathbb N_0\rbrace$.

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