Is a orthogonal projection in a Hilbert space automatically selfadjoint?

Question

Let $P$ be a self adjoint projector on a Hilbert space $H$ i.e. $P: H \rightarrow H$ is linear and continuous, $P^*=P$ and $P^2=P$

Then $P$ is also an orthogonal projection i.e. $\mathrm{ran}P=\mathrm{ran}(I-P)^\perp$.

Assume $Q: H \rightarrow H$ is linear and continuous, $Q^2=Q$ and $\mathrm{ran}Q=\mathrm{ran}(I-Q)^\perp$.

Is $Q$ selfadjoint?

Answer 1

Your assumption says that for all $\xi, \eta \in H$, we have $$0=\langle Q \xi, \eta-Q\eta\rangle\iff \langle Q\xi, \eta\rangle = \langle Q\xi, Q\eta\rangle.$$ Therefore, we find $$\langle Q\xi, \eta\rangle = \langle Q\xi, Q\eta\rangle = \overline{\langle Q\eta, Q\xi\rangle}= \overline{\langle Q\eta, \xi\rangle}= \langle \xi, Q\eta\rangle$$ whence $Q= Q^*$.