Let R be a bounded operator on a Hilbert Space, H.
I am trying to show that if X is a closed subspace of H such that
\begin{align*} x + Rx \in X \ \ \ \text{and} \ \ \ x - Rx \in X^{\perp} \end{align*}
for every $x \in H$, then $R = R^*$ and $R^2 = I$.
I am not really sure how to show either of the conclusions. Any help is much appreciated.
Since $x=\frac {x+Rx} 2 +\frac {x-Rx} 2$, the first term is in $X$ and the second term in $X^{\perp}$ it follows that $\frac {x+Rx} 2=Px$ where $P$ is the projection on $X$. Hence $R=2P-I$. It follows immediately that $R^{*}=R$ and $R^{2}=(2P-I)^{2}=4P^{2}-4P+I=I$.