Two variable shift operators

This is my first proper blog post that wasn't just adapted from a twitter thread! Yay!

In my last post, I talked about model spaces: the subspaces of the Hardy space $H^2$ that are shift invariant under the backward shift. One of the things I've been getting into lately is looking at model spaces in two variables, where the situation is more complicated.

As a quick refresher, we're living on the Hardy space of the disk, consisting of analytic functions with square summable coefficients. Helpfully, this also corresponds to (radial limits) of integrals around a circle, so we can think of the Hardy space as a subspace of $L^2(\mathbb{T})$.

 

Because these functions are analytic, we're thinking about them as a power series, so we can think about multiplication by $z$ as shifting the coefficients, hence the name "the shift."

If we can move coefficients one way, we might want to move them the other way, and this is the backwards shift:

 

Now, I want to talk about things in two variables!

First, I need to choose what two variable analog of the disk I want to use. From my point of view there are a couple of plausible options: the bidisk ${\mathbb{D}}^2=\{(z_1,z_2)\in \mathbb{C}:|z_j|<1,j=1,2\}$, and the (complex) 2-ball ${\mathbb{B}}_2=\{(z_1,z_2)\in \mathbb{C}:|z_1{|}^2+|z_2{|}^2<1\}$.

These actually are different, and I'm going to stick with the bidisk here.

We want an analog of the Hardy space of the disk, and delightfully, things generalize nicely: require analytic functions with square summable coefficients, and that corresponds to doing double integrals on the (distinguished) boundary ${\mathbb{T}}^2$. 

You may have noticed that this is a double sum, and if you're thinking shift thoughts, you may be slightly distressed about that. You're correct: things are more complicated in two variables! We have two different shifts (multiplication by $z_1$ and $z_2$) and they don't *really* correspond to shifting coefficients anymore. I find it helpful to write out the two variable monomials in this triangle:



So with this triangle, we think about multiplication by $z_1$ as shifting coefficients up and to the right (so that what was the 1-coefficient is now the $z_1$-coefficient, and what was the $z_1^2{z}_2$-coefficient is now the $z_1^3{z}_2$-coefficient).

We can think of the "backward shifts" similarly where as soon as you "run out" of $z_1$s you just remove that coefficient.

Now we want to think about invariant subspaces, so we should talk about what that means in this context, since we have two different shifts, and for this, let's think back to what invariant subspaces are in the one variable case. In one variable, when I ask for a subspace invariant under the shift, I mean that if I multiply any function in the subspace by any polynomial, I stay in the subspace:



In two variables, I have two shifts, so I can think of invariant subspaces as spaces that are invariant under multiplication by any *two variable* polynomial.

If we tiptoe into algebra, we can see that examples of invariant spaces aren't hard to find: close up (under the norm of $H^2({\mathbb{D}}^2)$) an ideal the polynomial ring of $\mathbb{C}[z_1,z_2]$. Because of the definition of ideals, the ideal is closed under multiplication by polynomials, so the norm closure will be as well. Ideals can be generated by one or several functions and the minimal number of functions necessary to generate an invariant subspace is called the rank of the subspace.

In the one variable case, Beurling's theorem tells us that shift invariant subspaces have the form $u{H}^2(\mathbb{D})$ where $u$ is an inner function, that is, that $u$ has radial limit value 1 almost everywhere on the unit circle. We can define inner functions similarly on the bidisk as functions that have value 1 almost everywhere on the distinguished boundary of the bidisk (the 2-torus ${\mathbb{T}}^2$). However, in two variables, Beurling's theorem no longer holds (and in fact, fails quite badly)!

There are a couple of examples we can use to demonstrate this. The first comes from the relatively simple ideal $[z_1-z_2]$. This is not an inner function and also can't be factored into an inner function times a Hardy space function (you can find a proof of this as Theorem 2 in "The Failure of Interior Exterior Factorization in the Polydisc and the Ball" by Rubel and Shields, 1971).

There are some more dramatic examples due to Rudin in "Function Theory in Polydiscs" (1969). This one is Theorem 4.4.2 where he constructs an invariant subspace that isn't formed from a finitely generated ideal (this image and the next are from Yang's survey paper linked below, since the typesetting is nicer):

Rudin also shows (Theorem 4.4.1) that there exists an invariant subspace that contains NO bounded functions other than zero. In the one variable case, all shift invariant subspaces are of the form $u{H}^2$ where $u$ is an inner function (and hence bounded), but the subspace here can certainly not be generated by an inner function!

Despite this, it's still worth talking about so called "Beurling-type" submodules $u{H}^2({\mathbb{D}}^2)$ ($u$ inner) and I'll go into that more in the future!

Thanks very much to Kelly Bickel and Alan Sola for answering questions and for references!
Here is the arxiv version of Yang's extremely helpful and well written survey paper!