## Category: Complex variables

### 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.

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### The disc trick (and some other cute moves)

This post is about a chain of little tricks that I discovered with collaborators and used in several papers. It is just a collection of simple moves that lets you deduce the existence of a zero preserving map of a certain class between two gauge invariant spaces, given the existence of a map from that class (things will be very clear soon, I hope). These tricks were later used by some other people, who applied it in different settings.

I am writing this post as notes for my upcoming Pizza & Beer seminar talk. The section at the end of the notes contains references and links to papers where this was used.

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### One of the most outrageous open problems in operator/matrix theory is solved!

I want to report on a very exciting development in operator/matrix theory: the von Neumann inequality for $3 \times 3$ matrices has been shown to hold true. I learned this from a recent paper (with the irresistible title) “The von Neumann inequality for $3 \times 3$ matrices“, posted on the arxiv by Greg Knese. In this paper, Knese explains how the solution of this outstanding open problem follows from results in a paper by Lukasz Kosinski, “The three point Nevanlinna-Pick problem in the polydisc” that appeared on the arxiv about a half a year ago. Beautifully, and not surprisingly, the solution of this operator/matrix theoretic problem follows from deep new facts in complex function theory in several variables.

To recall the problem, let us denote $\|A\|$ the operator norm of a matrix $A$, and for every polynomial $p$ in $d$ variables we denote by $\|p\|_\infty$ the supremum norm

$\|p\|_\infty = \sup_{|z_i|\leq 1} |p(z_1, \ldots, z_d)|$.

A matrix $A$ is said to be a contraction (or contractive) if $\|A\| \leq 1$.

We say that $d$ commuting contractions $A_1, \ldots, A_d$ satisfy von Neumann’s inequality if

(*)  $\|p(A_1,\ldots, A_d)\| \leq \|p\|_\infty$

for every polynomial $p$ in $d$ variables.

It was known since the 1960s that (*) holds when $d \leq 2$. Moreover, it was known that for $d \geq 3$, there are counter examples, consisting of $d$ contractive $4 \times 4$ matrices that do not satisfy von Neumann’s inequality. On the other hand, it was known that (*) holds for any $d$ if the matrices $A_1, \ldots, A_d$ are of size $2 \times 2$. Thus, the only missing piece of information was whether or not von Neumann’s inequality holds or not for three or more contractive $3 \times 3$ matrices. To stress the point: it was not known whether or not von Neumann’s inequality holds for three three-by-three matrices. The problem in this form has been open for 15 years  – but the problem is much older: in 1974 Kaiser and Varopoulos came up with a $5 \times 5$ counter-example, and since then both the $3 \times 3$  and the $4 \times 4$ cases were open until Holbrook in 2001 found a $4 \times 4$ counter example. You have to agree that this is outrageous, perhaps even ridiculous, I mean, three $3 \times 3$ matrices, come on!

In Knese’s paper this story and the positive solution to the problem is explained very clearly and succinctly, and is recommended reading for any operator theorist. One has to take on faith the paper of Kosinski which, as Knese stresses, is where the major new technical advance has been made (though one should not over-stress this fact, because tying things together, the way Knese has done, requires a deep understanding of this problem and of the various ingredients). To understand Kosinki’s paper would require a greater investment of time, but it appears that the paper has already been accepted for publication, so I am quite confident and happy to see this problem go down.

### 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.