Noncommutative Analysis

Tag: Drury-Arveson space

Great new set of lecture notes on the Drury-Arveson space

A few days ago Michael Hartz uploaded a very nice set of lectures notes on the Drury-Arveson space, called “An Invitation to the Drury-Arveson Space“. These notes are an expanded written version of the mini-course that he gave in the Focus Program on Analytic Function Spaces, which I blogged about a few months ago. I highly recommend these notes, they seem to me the best introduction to the subject (yes, even better than my own survey which is almost eight years old, and definitely better than my old series of blog posts, which I won’t even link to). If somebody wants to start working in the Drury-Arveson today, this seems like the right place to start.

Souvenirs from the children’s room, and the warmest recommendation for an online mini-course

Ilia Binder, Damir Kinzebulatov and Javad Mashreghi have organized a Focus Program on Analytic Function Spaces and their Applications at the Fields Institute, and this week, as part of this focus program, there was a Mini-course and Workshop on Drury-Arveson Space which I virtually attended (from the “children’s room” in our house, because that’s where we have the internet connection). The workshop is still not over, I have Ken Davidson’s talk to look forward to tonight.

I used to have a section in my blog Souvenirs from … where I would write about my favorite talks that I heard in recent conferences. This exercise helped concentrate during conferences (“hmmm, I wonder who’s going to be my souvenir?”) and also helped me get the most out of great talks after the conference (writing about stuff forces you to actually look up the paper or at least have another look at the notes you took during the talk). In fact, some of the souvenirs I brought home from conferences ended up becoming major parts of my own research program.

“Coming back” from the workshop on Drury-Arveson space, I can report that all the talks are recorded and can be found on the Fields Institute’s Youtube channel. To a certain extent that makes the task of reporting from conferences seem less needed.

Still, I will share my recommendations. And I want to give one very very warm recommendation for the Mini-course that Michael Hartz gave on the Drury-Arveson space. I have been to several minicourses in my life, and I also gave a couple, and I think that I have never seen a better prepared or more motivating mini-course. It was artful! Really, anybody going to work on Drury-Arveson space and the related operator theory should see it.

Here are the talks:

First lecture:

Mini-course on Drury-Arveson space, Lecture 1

Second Lecture:

Mini-course on Drury-Arveson space, Lecture 2

Third lecture:

Mini-course on Drury-Arveson space, Lecture 3

Since it was recorded, I can also put up here a link to my own talk at the workshop “Quotients of the Drury-Arveson space and their classification in terms of complex geometry”:

von Neumann’s inequality for row contractive matrix tuples

Michael Hartz, Stefan Richter and I recently uploaded our paper von Neumann’s inequality for row contractive matrix tuples to the arxiv.

The main result is the following.

We prove that for all d,n\in \mathbb{N}, there exists a constant C_{d,n} such that for every row contraction T consisting of d commuting n \times n matrices and every polynomial p, the following inequality holds:

 \|p(T)\| \le C_{d,n} \sup_{z \in \mathbb{B}_d} |p(z)| .

We then apply this result and the considerations involved in the proof to several open problems from the literature. I won’t go into that because I think that the abstract and introduction do a good job of explaining what we do in the paper. In this post I will write about how this collaboration with Michael and Stefan started, and give some heuristic explanation why our result is not trivial.

Read the rest of this entry »

New paper: Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball

Danny Ofek, Satish Pandey and I just uploaded our new paper “Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball” to the arxiv. This papers fits into my ongoing work on the isomorphism problem for complete Pick algebras, but it raises a very fundamental question that I think is worth highlighting.

As in other subjects of mathematics, when working on Hilbert function spaces, one sometimes asks very basic questions, such as: when are two Hilbert function spaces the same? what is the “true” set on which the functions in a RKHS are defined? (see Section 2 in this paper) or what information is encoded in a space or its multiplier algebra? (see the “road map” here). The underlying questions behind our new paper are when are two Hilbert function spaces “almost” the same and what happens if you change a Hilbert function space “just a little bit”? If these sound like interesting questions, then I suggest you take a look at the paper’s introduction.

Here is the abstract:

In this paper we study the relationships between a reproducing kernel Hilbert space, its multiplier algebra, and the geometry of the point set on which they live. We introduce a variant of the Banach-Mazur distance suited for measuring the distance between reproducing kernel Hilbert spaces, that quantifies how far two spaces are from being isometrically isomorphic as reproducing kernel Hilbert spaces. We introduce an analogous distance for multiplier algebras, that quantifies how far two algebras are from being completely isometrically isomorphic. We show that, in the setting of finite dimensional quotients of the Drury-Arveson space, two spaces are “close” to one another if and only if their multiplier algebras are “close”, and that this happens if and only if the underlying point-sets are “almost congruent”, meaning that one of the sets is very close to an image of the other under a biholomorphic automorphism of the unit ball. These equivalences are obtained as corollaries of quantitative estimates that we prove.

Spaces of Dirichlet series with the complete Pick property (or: the Drury-Arveson space in a new disguise)

John McCarthy and I have recently uploaded a new version of our paper “Spaces of Dirichlet series with the complete Pick property” to the arxiv. I would like to advertise the central discovery of this paper here.

Recall that the Drury-Arveson space H^2_d is the reproducing kernel Hilbert space on the open unit ball of a d dimensional Hilbert space, with reproducing kernel

k(z,w) = \frac{1}{1 - \langle z, w \rangle}.

It has the remarkable universal property that every Hilbert function space with the complete Pick property is naturally isomorphic to the restriction of H^2_\infty to a subset of the unit ball (see Theorem 6 and its corollary in this post), and consequently, every complete Pick algebra is a quotient of the multiplier algebra \mathcal{M}_\infty = Mult(H^2_\infty). To the best of my knowledge, no other Hilbert function spaces with such a universal property have been studied.

John and I discovered another reproducing kernel Hilbert space that turns out to be “the same” as the Drury-Arveson space H^2_\infty. Since the space H^2_\infty as been so well studied, it interesting to discover a new incarnation. The really interesting part is that the space we discovered is a space of analytic functions on a half plane (that is, a space of functions in one complex variable), rather than a space of analytic functions in infinitely many variables on the unit ball of a Hilbert space.

To be precise, the spaces we consider are spaces of Dirichlet series \mathcal{H}, of the form

\mathcal{H} = \{f(s) = \sum_{n=1}^\infty \gamma_n n^{-s} : \sum |\gamma_n|^2 a_n^{-1} < \infty \}.

(Here a_n is a sequence of positive numbers). These are Hilbert function spaces on some half plane that have a kernel of the form k(s,u) = \sum a_n n^{-s-\bar u}.

We first answer the question which of these spaces \mathcal{H} have the complete Pick property. This problem has a simple solution (which has been anticipated by similar results on spaces on the disc): if we denote by g(s) = \sum a_n n^{-s} the “generating function” of the space, and if we write

\frac{1}{g(s)} = \sum c_n n^{-s},

then \mathcal{H} is a complete Pick space if and only if c_n \leq 0 for all n \geq 2.

After we know to tell when these spaces are complete Pick, it is natural to ask which complete Pick spaces arise like this? We do not give a complete answer, but our surprising discovery is that things can easily be cooked up so to obtain the Drury-Arveson space H^2_d, where d can be any cardinal number in \{1,2,\ldots, \infty\}. For example, \mathcal{H} turns out to be “the same” as H^2_\infty if the kernel k is given by

k(s,u) = \frac{P(2)}{P(2) - P(2+s+\bar u)},

where P(s) = \sum_{p} p^{-s} is the prime zeta function (the sum is taken over all primes p).

 Now, I have been a little vague about what it means that \mathcal{H} is “the same” as H^2_\infty. In fact, this is a subtle question, and we devote a part of our paper what it means for two Hilbert function spaces to be the same — something that has puzzled us for a while.

What does this appearance of Drury-Arveson space as a space of Dirichlet series mean? Can we use this connection to learn something new on multivariable operator theory, or on Dirichlet series? How did the prime zeta function smuggle itself into this discussion? This requires further thought.