Rudin's proof of invariant subspace existence

Question

I have questions about Rudin's proof of invariant subspace existence.

On page 327, point 12.27,

1. How does he get that $Tx=TE(\omega)x$, and
2. How does he know $E(\omega)$ is not the zero map?

Thanks.

Answer 1

1. This is a typo. The second $x$ should be a different letter. Since $x\in M=\mathrm{im}\,E(\omega)$, there is a $y\in H$ such that $E(\omega)y=x$. Therefore $$Tx=TE(\omega)y=E(\omega)Ty$$ so $Tx\in M$.

2. It is possible that $E(\omega)=0$, but the point is that there exists $\omega$ such that $E(\omega)\not=0$, because otherwise $E(\omega)=0$ for all Borel sets $\omega$ and therefore

$$T=\int_{\sigma(T)} \lambda\,dE(\lambda)=0$$