I'm learning about model spaces

Note: this was originally a thread on my twitter.

Last time I talked about subspaces of the Hardy space that are invariant under the shift operator. Now let's talk about the backward shift!

(Most of the discussion here comes from the excellent "Introduction to Model Spaces and their Operators" by Garcia, Mashreghi, and Ross, which I cannot recommend highly enough.)

Our shift invariant subspaces are of the form \(uH^2\) where u is an inner function

The backward shift is the adjoint of the shift, so in some sense, invariant subspaces for the backward shift should be "opposite" of those of the shift. In fact, we define "model spaces" as the orthogonal complement in \(H^2\) of \(uH^2\)!

As a reminder of your linear algebra, the orthogonal complement of a subspace is everything orthogonal to every element of that subspace. That is, if you take the inner product of anything in a subspace V with anything in the orthogonal complement V-perp, you get zero

 

It's actually not TOO hard to show that these model spaces are precisely the backward shift invariant subspaces in H^2: orthogonality is defined as inner product zero, and adjoints are defined using the inner product. Then we can use Beurling's theorem to get the inner function part!

It can be hard to think about which functions are in a given model space. Our Beurling-type subspaces \(uH^2\) are easier to talk about because we have inner/outer factorization, but orthogonal complements can be harder to see explicitly.

We do have another way to characterize our model spaces, remembering that the Hardy space can be thought of as a subspace of L^2(\T) and taking conjugates:

 

As with our shift invariant subspaces, our model spaces are also Reproducing Kernel Hilbert Spaces, and again, the reproducing kernel is exactly what you would want!
(I'm being terrible and changing notation slightly here: \(c_w\) is the Cauchy kernel, which is the reproducing kernel for \(H^2\).)

 

This actually points to one way to generalize model spaces! Here, we're requiring u to be an inner function (modulus 1 on the circle), but if we replace it with a function b that is modulus bounded by 1, we can build an RKHS with that kernel: this is called a de Branges-Rovnyak space! That's a topic for another time, but it's neat :)

Let's now look at a couple of examples of nice (finite dimensional) model spaces.

First: all monomials \(z^N\) are inner functions and give nice invariant subspaces and model spaces:

Finite Blaschke* Products also give finite dimensional model spaces, and these are a classic example that I'll come back to when we start talking about generalizing model spaces to two variables.

Model spaces are "the opposite of shift invariant" so it can be interesting to see how the shift acts inside the model space, if we can figure out how to keep things IN this not shift invariant space. To talk about this, let's talk about orthogonal projections.

You know (and hopefully love) orthogonal projections from your Calc 3 or Linear Algebra class:

Generalizing this to a (separable, complex) Hilbert space H with a subspace M, we can use the orthogonal decomposition of H into M and M-perp, and think of a matrix representation of an operator T on H:

If our operator R behaves nicely (stays being in the corner of the block matrix) when you plug things into (analytic) polynomials, we then say that R is a compression of T (or T is a dilation of R). (Sometimes, this definition wiggles a little)

WE want to talk about how the shift operator acts on model spaces (which are very much not shift invariant: that is, applying the shift to an element of a model space doesn't leave it in the model space in general.)

So we want to *compress* the shift to our model space! Conveniently, the adjoint of the compression of the shift still works like the backward shift

 

It can also be shown that the compressed shift operator is cyclic in our model space. That is, there is a vector such that repeated applications of the compressed shift span the space. (It's a convenient vector too!)

(This is good because the shift operator is cyclic in H^2, so we would hope that that property would carry down).

Ok, so why do we care about these (beyond the fact that invariant subspaces are cool)? It comes back to why we call them "model spaces."

First, remember that unitary operators are *nice*: the spectral theorem gives a good way to talk about them concretely (you might know the spectral theorem as a generalization of diagonalization from linear algebra).

It's also true that any contractive linear operator on a Hilbert space can be decomposed into the direct sum of a unitary operator and a completely non unitary (CNU) contraction. (Remember our compressions/dilations from earlier).

 

Since unitary operators are *nice,* if we want to understand the operator, we just need to understand the CNU part.

It turns out that with a few extra assumptions, the operator will be unitarily equivalent to a compression of the shift for some model space.

The space contains the information about the contraction: it "models" it.

This is the Sz.Nagy-Foias model and it gets into a lot of other things that I'm just learning, but that I think are cool!

People also study Clark measures and Clark operators (which are rank one unitary perturbations of the compressed shift. This is the next thing I need to learn, but it's not the next thing we'll be talking about!

Next I'm going to talk about how we generalize these model spaces and compressions of the shift to the two variable case. Things are… difficult, so get hyped!