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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, $Se_n=e_{n+1}$.


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up vote 1 down vote accepted

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.

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Well, I like your answer and I know Beurling's is like the ultimate answer to all kinds of problems concerning invariant spaces of shifts. But I am hoping to see something more 'elementary', like some examples involving only sequences. – Hui Yu Jun 26 '12 at 0:32
@HuiYu Chances are that if the speaker had an example in terms of sequences that could be easily described, he would have put it on the slide. – user31373 Jun 26 '12 at 13:25
Haha~~That's true! Thanks! – Hui Yu Jun 26 '12 at 13:36

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