Meredith's Math Blog

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   I...

 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...

 Note: this was originally a thread on my twitter Let's talk about a weird connection between functional analysis and number theory: Dirichlet Series!  (Full Disclosure: I'm not a number theorist, not even an analytic number theorist, but spaces of  functions are my jam.) The most famous Dirichlet series is the Riemann zeta function, but we can generalize by changing the  coefficients:   To help get our hands on these, let's consider a very simple case where we don't have to be distressed about convergence: \(2^{-s}\). Looking at how this maps the plane helps me think about things like periodicity (which gets complicated when looking at a full DS).   Complex TAYLOR series converge in disks in \(\mathbb{C}\). Dirichlet series converge in right half planes of \(\mathbb{C}\). Show this by showing that if a DS converges at a point \(s_0\), it converges in a sector, then union those to get a half plane.     So instead of a radius of convergence li...

 Note: this was originally a thread on my twitter I'm assuming the reader is already familiar with Hilbert spaces. If you need a refresher, here's my post about them. We'll start out in the world of a generic Hilbert space and talk about dual spaces and bounded linear functionals. Conveniently, this is sort of what my linear algebra students are learning about right now! A linear functional is a linear transformation from a vector space to the underlying field: for example, remember \(\ell^2(\mathbb{C})\) :     There are lots of examples of linear functionals, but we'll care about the ones that are *bounded*, that is, the sup of the image of the functional is finite: There's a nice theorem that says that for linear functionals, being bounded is equivalent to being continuous and to being continuous at 0.   The space of continuous linear functionals on a normed vector space is called the dual of the space and has some other nifty properties that I'm not going t...