In a recent talk, A. Popov stated the following fact
The unilateral shift on $\ell^2$ has invariant halfspaces.
Halfspaces are closed subspaces whose dimension and codimension are both infinite.
He did not prove it. I know that unilateral shift has many invariant halfspaces, but all the examples I know are finite dimensional. Thus I wonder whether somebody can give an explicit invariant halfspace of the unilateral shift.
Just to be precise, I am asking about the forward shift, that is, $Se_n=e_{n+1}$.
Thanks!