Shift Invariant Subspaces, An Introduction

 Note: this was originally a thread on my twitter

Ok, I need to compile some knowledge in my brain, so let's talk about shift invariant subspaces!

First, we need to recall the Hardy space on the disk. This is a Hilbert space of analytic functions with square summable Taylor coefficients.

The Hardy space can be thought of either as functions analytic on the disk, or as a subspace of \(L^2(\mathbb{T})\) consisting of functions with all negative coefficients zero. Either way, power series are life, and we care about the coefficients.

 

There are lots of interesting bounded linear operators (or "operators" if you're me) on \(H^2\), but here I'll talk about two of the most important: the shift and backward shift.

The shift is multiplication by z. It gets its name from what that would do to the Taylor coefficients: it shifts them over by one. The shift is an isometry! Putting more zeros at the front isn't going to change the square sum of a sequence.

 

The backward shift does the opposite! You lose information in this case, because the constant term vanishes. The backward shift is a contraction! (Technically the shift is too, but its isometry-ness is more exciting)

 

One of the big things we're interested in in operator theory is Invariant Subspaces
In particular, we'd like to know what subspaces are invariant for the shift and the backward shift.

In order to talk about this, we need to define two delightfully named classes of functions: inner functions and outer functions.

Inner functions are those with boundary values of modulus 1 almost everywhere. Inner functions can also be subcategorized as either Blaschke* products or singular inner functions.
*Blaschke was a Nazi. We should rename these. 

Outer functions can be defined as an integral, but we'll have a nice classification of them shortly.

It turns out that every Hardy space function can be factored as (Blaschke)(singular inner)(outer) where the Blaschke part contains all the information about the zeros.

(I haven't seen it written anywhere, but I'm pretty sure this is where the terms "inner" and "outer" come from: inner functions have zeros in the disk, outer functions can only have zeros outside the disk. I'll come back to that second part.)

Now, bringing this back to invariant subspaces: it's an important theorem due to Beurling that all proper shift invariant subspaces of the Hardy space are of the form \(uH^2\) where \(u\) is an inner function. 

As a corollary to this, we can see that functions that are cyclic (for the shift) are precisely the outer functions. (This is the thing about outer functions having zeros outside the disk: any zeros in the disk will still be there after multiplication by a polynomial, so you can't get all of \(H^2\)) 

Another thing that's cool about our shift invariant subspaces is that they're Reproducing Kernel Hilbert Spaces! (Because the Hardy space is)

The reproducing kernel for our invariant subspace is exactly what you would expect: the reproducing kernel for the Hardy space multiplied appropriately by our inner function.

So we've looked at subspaces of \(H^2\) that are invariant under the shift, but what about spaces invariant under the BACKWARD shift?????

That's the next post ;)