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