Noncommutative Analysis

Tag: multivariable operator theory

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.

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 »

The remarkable Hilbert space H^2 (part III – three open problems)

This is the last in the series of three posts on the d–shift space, which accompany/replace the colloquium talk I was supposed to give. The first two parts are available here and here. In this post I will discuss three open problems that I have been thinking about, which are formulated within the setting of H^2_d.

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The remarkable Hilbert space H^2 (Part II – multivariable operator theory and model theory)

This post is the second post in the series of posts on the d–shift space, a.k.a. the Drury–Arveson space, a.k.a. H^2_d (see this previous post about the space H^2).

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