Recently I was reading a book "Operator Function and system" written by Nikolski. There I found this statement.
Let $\mu$ be a finite positive measure on the unit circle in complex plane which is also singular w.r.t Lebesgue measure on the circle. Now consider $L^2(\mu)$ and consider the operator $M : L^2(\mu) \to L^2(\mu)$ defined by $(Mf)(z) = z f(z) $ for $z$ in unit cicle. Then every invariant subspace for $M$ is also reducing subspace for $M$.
Is there any simple, elementary way to prove the result?
If the measure $\mu$ is supported on finitely many point, then I have some idea how to prove the result. But in general, I have no idea.
Any answer or reference will be very helpful.