Will $P$ be necessarily bounded?

Question

Let $\mathcal H$ be a Hilbert space. Let $P : \mathcal H \longrightarrow \mathcal H$ be a self-adjoint idempotent linear operator. Can $P$ be necessarily bounded?

I can't prove boundedness of $P$ from these assumptions. Can anybody please help me in this regard?

Thanks for your time.

Answer 1

Here is an extended hint. You're given a lot of assumptions to work with here; idempotent, meaning $P^2 = P$, and self adjoint, meaning $\langle Pa, b\rangle = \langle a, Pb\rangle$ for any vectors $a$, $b$. Imagine combining these assumptions - say, consider $\langle P^2a, b\rangle$. On the one hand, this should be equal to $\langle Pa, b\rangle$ by the idempotence. On the other, it should also be equal to $\langle PPa, b\rangle = \langle Pa, Pb \rangle$ by the self-adjointness. If you choose $a = b$, then the latter can be related to $\| Pa \|$. Finally, relate the latter to the former, and try using the cauchy schwartz inequality. Hope this helps!