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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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 $\lbrace (PQ)^n x: n\in\mathbb N_0\rbrace$.