Category: Hilbert function spaces

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

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.

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!

Summer project 2020 – Hilbert function spaces of analytic functions in a complex variable

In the week of September 6-11 the Math Department at the Technion will again host the “Math Research Week“, or what we refer to as the the “summer projects week”. As in previous years, I will be offering a project, and this year, with the help of Ran Kiri and Satish Pandey, it will be a project on Hilbert function spaces. See here for the abstract. The purpose of this post is to collect my thoughts and my plans for this project.

Read the rest of this entry »

Topics in Operator Theory, Lecture 4: Pick interpolation via commutant lifting

Finally we reached the point where we can apply the general theory that we developed in the last two weeks to obtain an interesting application to function theory, namely, the Pick interpolation theorem. Read the rest of this entry »