## Tag: Reproducing kernel Hilbert space

### 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:

### Souvenirs from the Rocky Mountains

I recently returned from the Workshop on Multivariate Operator Theory at Banff International Research Station (BIRS). BIRS is like the MFO (Oberwolfach): a mathematical resort located in the middle of a beautiful landscape, to where mathematicians are invited to attend/give talks, collaborate, interact, catch up with old friends, make new friends, have fun hike, etc.

As usual I am going over the conference material the week after looking for the most interesting things to write about. This time there were two talks that stood out from my perspective, the one by Richard Rochberg (which was interesting to me because it is on a problem that I have been thinking a lot about), and the one by Igor Klep (which was fascinating because it is about a subject I know little about but wish to learn). There were some other very nice talks, but part of the fun is choosing the best; and one can’t go home and start working on all the new ideas one sees.

A very cool feature of BIRS is that now they automatically shoot the talks and put the videos online (in fact the talks are streamed in real time! If you follow this link at the time of any talk you will see the talk; if you follow the link at any other time it is even better, because there is a webcam outside showing you the beautiful surroundings.

I did not give a talk in the workshop, but I prepared one – here are the slides on the workshop website (best to download and view with some viewer so that the talk unfolds as it should). I also wrote a nice “take home” that would be probably (hopefully) what most people would have taken home from my talk if they heard it, if I had given it. The talk would have been about my recent work with Evgenios Kakariadis on operator algebras associated with monomial ideals (some aspects of which I discussed in a previous post), and here is the succinct Summary (which concentrates on other aspects).  Read the rest of this entry »

### 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