Does the composition of orthogonal projections have norm $1$?

Question

Let $X$ a Hilbert space, and $P_1, P_2: X \to X$ be orthogonal projections.

Consider $P = P_2P_1P_2$. Is it always the case that $\|P\|=1$?

Certainly $\|P\| \leq \|P_2\|\|P_1\|\|P_2\| = 1$, but does the reverse inequality always hold?

Answer 1

It is not necessarily the case the $P$ has norm one, and in fact $P$ could even be the zero map. This happens for instance if $P_2=I-P_1$.

Answer 2

No; consider the simple case when $X = \mathbb{R}^2$ and $P_1$ is the projection onto the $x$-axis, and $P_2$ is the projection onto the $y$-axis. What is the composition of $P_1$ and $P_2$?