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”:

Seminar talk by Pandey – Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball

In our next Operator Algebras/Operator Theory Seminar, Satish Pandey will present our recently published online paper (together with Danny Ofek and myself) “Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball” (arxiv version).

Time: 15:30-16:30

Date: May 6th, 2021

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

Abstract:

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 one of the underlying point sets is 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.

This is joint work with Danny Ofek and Orr Shalit.

If you are interested in the zoom link, let me know.

Seminar talk by Hartz: How can you compute the multiplier norm?

Happy new year!

Next Thursday, January 7th, 2021, Michael Hartz will speak in our Operator Algebras and Operator Theory seminar.

Title: How can you compute the multiplier norm?

Time: 15:30-16:30

Zoom link: Email me.

Abstract:

Multipliers of reproducing kernel Hilbert spaces arise in various contexts in operator theory and complex analysis. A basic example is the Hardy space , whose multiplier algebra is , the algebra of bounded holomorphic functions. In particular, the norm of a multiplier on is the pointwise supremum norm. 

For general reproducing kernel Hilbert spaces, the multiplier norm can be computed by testing positivity of matrices analogous to the classical Pick matrix. For , suffices. I will talk about when it suffices to consider matrices of bounded size . Moreover, I will explain how this problem is related to subhomogeneity of operator algebras.

This is joint work with Alexandru Aleman, John McCarthy and Stefan Richter

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.