TITLE: Action of a projection on a pairwise orthogonal projections.
QUESTION [5 upvotes]: Let us consider two projections $e_1$ and $e_2$ on the Hilbert space $H$ with $e_1e_2=0$. Does there exists any projection $p$ with 
$$\overline{pe_1(H)}=\overline{pe_2(H)}=\overline{p(e_1+e_2)(H)}\neq0$$

REPLY [2 votes]: Wlog we can assume that
$$e_1=\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}
\mbox{ and }
e_2=\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix}$$
with respect to a certain orthogonal decomposition of $H=H_1\oplus H_2\oplus H_3$.
Then it suffices to set 
$$p=\begin{pmatrix}1/2&1/2&0\\1/2&1/2&0\\0&0&0\end{pmatrix}.$$