## Tag: Drury-Arveson space

### 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.

### The isomorphism problem: update

Ken Davidson, Chris Ramsey and I recently uploaded a new version of our paper “Operator algebras for analytic varieties” to the arxiv. This is the second paper that was affected by a discovery of a mistake in the literature, which I told about in the previous post. Luckily, we were able to save all the results in that paper, but had to work a a little harder than what we thought was needed in our earlier version. The isomorphism problem for complete Pick algebras (which I like to call simply “the isomorphism problem”) has been one of my favorite problems during the last five years. I wrote four papers on this problem, with five co-authors. I want to give a short road-map to my work on this problem. Before I do so, here is  link to the talk that I will give in IWOTA 2014 about this stuff. I think (hope) this talk is a good introduction to the subject. The problem is about the classification of a large class of non-selfadjoint operator algebras – multiplier algebras of complete Pick spaces – which can also be realized as certain algebras of functions on analytic varieties. These algebras all have the form

$M_V = Mult(H^2_d)\big|_V$

where $V$ is a subvariety of the unit ball and $Mult(H^2_d)$  denotes the multiplier algebra of Drury-Arveson space (see this survey), and therefore $M_V$ is the space of all restrictions of multipliers to $V$. The hope is to show that the geometry of the variety $V$ is a complete invariant for the algebras $M_V$, in various senses that will be made precise below.

### Souvenirs from the Black Forest

Last week I attended a workshop titled “Hilbert modules and complex geometry” in MFO (Oberwolfach). In this post I wish to tell about some interesting things that I have learned. There were many great talks to choose from. Below is a sample, in short form, with links.