Beurling's theorem characterize the closed subspaces $M\subset H^2$ of the Hardy space, which are invariant under the shift operator $Sf(z):=zf(z)$, as spaces of the form $\varphi H^2 $ where $\varphi$ is an inner function.
For $H^1$ the analogous theorem is due to Rudin and De Leeuw [Extreme points and extremum problems in $H_1$].
A quick search in the literature suggests that no similar theorem is known for closed, shift invariant subspaces of the Hardy spaces $H^p, p\neq 1,2$.
I was wondering which is the state of the art in this problem. In particular:
Is it known if there exists a closed invariant subspace of $H^p$ which is not of the form $\varphi H^p$ for some inner function $\varphi$ ?
