Tag: Hilbert function space

Three classification results in the theory of weighted hardy spaces in the ball – summary of summer project

Last month we had the Math Research Week here at the Technion, and I promised in a previous post to update if there would be any interesting results (see that post for background on the problems). Well, there are! I am writing this short post just to update as promised on the interesting results.

The two excellent students that worked with us – Danny Ofek and Gilad Sofer – got some nice results. They almost solved to a large extent the main problems mentioned in my earlier post. See this poster for a concise summary of the main results:

Danny and Gilad summarized their results in the following paper. Just take a look. They have some new results that I thought were true, they have some new results that I didn’t guess were true, and they also have some new and simplified proofs for a couple of known results. Their work fits in the long term research project to discover how the structure of Hilbert function spaces and their multiplier algebras encodes the underlying structures, and especially the geometry of sets in the unit disc or the unit ball. More on that soon!

Aleman, Hartz, McCarthy and Richter characterize interpolating sequences in complete Pick spaces

The purpose of this post is to discuss the recent important contribution by Aleman, Hartz, McCarthy and Richter to the characterization of interpolating sequences (for multiplier algebras of certain Hilbert function spaces). Their recent paper “Interpolating sequences in spaces with the complete Pick property” was uploaded to the arxiv about two weeks ago; here I will just give some background and state the main result. (Even more recently these four authors released yet another paper that looks very interesting – this one.)

1. Background – interpolating sequences

We will be working with the notion of Hilbert function spaces – also called reproducing Hilbert spaces (see this post for an introduction). Suppose that $H$ is a Hilbert function space on a set $X$, and $k$ its reproducing kernel. The Pick interpolation problem is the following:

Advanced Analysis, Notes 17: Hilbert function spaces (Pick’s interpolation theorem)

In this final lecture we will give a proof of Pick’s interpolation theorem that is based on operator theory.

Theorem 1 (Pick’s interpolation theorem): Let $z_1, \ldots, z_n \in D$, and $w_1, \ldots, w_n \in \mathbb{C}$ be given. There exists a function $f \in H^\infty(D)$ satisfying $\|f\|_\infty \leq 1$ and

$f(z_i) = w_i \,\, \,\, i=1, \ldots, n$

if and only if the following matrix inequality holds:

$\big(\frac{1-w_i \overline{w_j}}{1 - z_i \overline{z_j}} \big)_{i,j=1}^n \geq 0 .$

Note that the matrix element $\frac{1-w_i\overline{w_j}}{1-z_i\overline{z_j}}$ appearing in the theorem is equal to $(1-w_i \overline{w_j})k(z_i,z_j)$, where $k(z,w) = \frac{1}{1-z \overline{w}}$ is the reproducing kernel for the Hardy space $H^2$ (this kernel is called the Szego kernel). Given $z_1, \ldots, z_n, w_1, \ldots, w_n$, the matrix

$\big((1-w_i \overline{w_j})k(z_i,z_j)\big)_{i,j=1}^n$

is called the Pick matrix, and it plays a central role in various interpolation problems on various spaces.

I learned this material from Agler and McCarthy’s monograph [AM], so the following is my adaptation of that source.

(A very interesting article by John McCarthy on Pick’s theorem can be found here).

Advanced Analysis, Notes 16: Hilbert function spaces (basics)

In the final week of the semester we will study Hilbert function spaces (also known as reproducing kernel Hilbert spaces) with the goal of presenting an operator theoretic proof of the classical Pick interpolation theorem. Since time is limited I will present a somewhat unorthodox route, and ignore much of the beautiful function theory involved. BGU students who wish to learn more about this should consider taking Daniel Alpay’s course next semester. Let me also note the helpful lecture notes available from Vern Paulsen’s webpage and also this monograph by Jim Agler and John McCarthy (in this post and the next one I will refer to these as [P] and [AM] below).

(Not directly related to this post, but might be of some interest to students: there is an amusing discussion connected to earlier material in the course (convergence of Fourier series) here).

The remarkable Hilbert space H^2 (Part I – definition and interpolation theory)

This series of posts is based on the colloquium talk that I was supposed to give on November 20, at our department. As fate had it, that week studies were cancelled.

Several people in our department thought that it would be a nice idea if alongside the usual colloquium talks given by invited speakers which highlight their recent achievements, we would also have some talks by department members that will be more of an exposition to the fields they work in. So my talk was supposed to be an exposition to the setting in which much of the research I do goes on.

The topic of the “talk”  is the Hilbert space $H^2_d$. There will be three parts to this series:

1. Definition and interpolation theory.
2. Multivariate operator theory and model theory
3. Current research problems