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 backwar