Noncommutative Analysis

Tag: Reproducing kernel Hilbert space

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 H^2, whose multiplier algebra is H^\infty, the algebra of bounded holomorphic functions. In particular, the norm of a multiplier on H^2 is the pointwise supremum norm. 

For general reproducing kernel Hilbert spaces, the multiplier norm can be computed by testing positivity of n \times n matrices analogous to the classical Pick matrix. For H^2, n=1 suffices. I will talk about when it suffices to consider matrices of bounded size n. 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.

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:

Read the rest of this entry »

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 »