$P_1, P_2$ orthogonal projections. Is the range of $P_1\circ P_2$ always closed?

Question

Let $H$ be a Hilbert space and $P_1,P_2$ be orthogonal projections. Does $P_1\circ P_2$ always have closed range? First I thought that this could be shown by showing first that orthogonal projections are closed mappings, but I don't think that this is true anymore. Is the above statement correct?

Answer 1

In $\ell^2$, let $P_1$ be the orthogonal projection on those $x = (x_1, x_2, \ldots)$ with $x_{n} = 0$ for $n$ even, and $P_2$ the orthogonal projection on those $x$ with $x_{2n}=n x_{2n-1}$ for all $n$. We have $$ \eqalign{P_1(x)_{2n-1} &= x_{2n-1}\cr P_1(x)_{2n} &= 0\cr P_2(x)_{2n-1} &= \frac{x_{2n-1} + n x_{2n}}{n^2+1}\cr P_2(x)_{2n} &= \frac{n x_{2n-1} + n^2 x_{2n}}{n^2+1}\cr}$$ Thus $$\eqalign{(P_1 \circ P_2(x))_{2n-1} &= \dfrac{x_{2n-1} + n x_{2n}}{n^2+1}\cr (P_1 \circ P_2(x))_{2n} &= 0\cr} $$ It's easy to show that $P_1 \circ P_2$ does not have closed range. In fact, its range includes all $x$ with only finitely many nonzero entries, all with odd index, but all its terms have $x_n = O(1/n)$.