Let $T$ be the unit circle and $H^1=\{f\in L^1(T): \int_0^{2\pi} f(e^{it})\chi_n(e^{it})dt=0 \text{ for } n>0\}$ where $\chi_n(e^{it})=e^{int}$. Let $M$ be a closed subspace of $H^1$. Then $\chi_1 M\subset M$ if and only if $M=\phi H^1$ for some inner function $\phi$.
We say $\psi$ is an inner function if $\psi\in H^\infty$ and $|\psi|=1$ a.e.
This is a problem from Banach algebra Techniques in Operator theory by Ronald Douglas.
I was able to show that if $M=\phi H^1$ for some inner function $\phi$ then $\chi_1M\subset M$. For the other direction, I tried going through Beurling's theorem but I get stuck.
I also tried writing $M$ as $M_1M_2$ where $M_1$ and $M_2$ are both subsets of $H^2$ but that got me nowhere.