An explicit example of an invariant halfspace of the unilateral shift?

Question

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!

Answer 1

Naturally, we should use the Beurling theorem on invariant subspaces. Let $\theta$ be an infinite Blaschke product. The space of functions of the form $\theta f$, $f\in H^2(\mathbb D)$, is infinite dimensional and also has infinite codimension. The former is obvious and the latter is because you can knock out the zeros of $\theta$ one by one.

I'm not sure if this qualifies as an explicit example. The criteria that make an example explicit were not made explicit.