## Category: Complex variables

### 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 contractive if $\|A\| \leq 1$.

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

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

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.

### 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.

### Advanced Analysis, Notes 19: The holomorphic functional calculus II (definition and basic properties)

In this post we continue our discussion of the holomorphic functional calculus for elements of a Banach algebra (or operators). The beginning of this discussion can be found in Notes 18. Read the rest of this entry »

### Spectral sets and distinguished varieties in the symmetrized bidisc

In this post I will write about a new paper, “Spectral sets and distinguished varieties in the symmetrized bidisc“, that Sourav Pal and I posted on the arxiv, and give the background to understand what we do in that paper.

### A sneaky proof of the maximum modulus principle

The April 2013 issue of the American Mathematical Monthly has just appeared, and with it my small note “A Sneaky Proof of the Maximum Modulus Principle”. Here is a link to the current issue on the journal’s website, and here is a link to a version of the paper on my homepage. As the title suggest, the note contains a new proof — which I find extremely cool — for the maximum modulus principle from the theory of complex variables. The cool part is that the proof is based on some basic linear algebra. The note is short and very easy, and I am not going to say anything more about the proof, except that it relates to some of my “real” research (the way in which it relates can be understood by reading the Note and its references).

I am writing this post not only to publicize this note, but also to record somewhere my explanation why I have been behaving in a sneaky fashion. Indeed, this is the first paper that I wrote which I did not post on the Arxiv. Why?

Unlike research journals, the American Mathematical Monthly is a journal which has, if I am not mistaken, actual subscribers. I mean real people, some of them perhaps old school (like myself), and I could see them waiting to receive their copy in the mailbox, and then when the new issue finally arrives they gently open the envelope — or perhaps they tear it open, depending on their custom — after which they sit down and browse through the fresh issue. I could believe that there are such persons (for I myself am such a person) that do not look at the online version of the journal even though they have access, because that would spoil their fun with the paper copy which is to arrive a few days later.

Now I wouldn’t like to spoil a small pleasure of a subscriber, somewhere out there. So I did not post the Note on the Arxiv, lest it pop up on somebody’s mailing list. “Oh, this I have already seen…”. I shall not be resposible for such spoilers! So I decided to keep my note relatively secret, putting it on my homepage, but putting off the Arxiv until the journal really gets published and all the physical copies are safely in the mailboxes of all subscribers. I made this decision about a year ago from now, and to tell the truth I felt that a year is a terribly long time to wait. In the end, this year appears much much shorter from this end than from the other one.

(I guess that it does not matter much if I put it on the Arxiv now: in the meanwhile I discovered that google scholar has managed to figure out that such a note exists on somebody’s webpage. Probably I will post it on the Arxiv, for the sake of all things being in good order).

### Advanced Analysis, Notes 17: Hilbert function spaces (Pick’s interpolation theorem)

In this final lecture we will give a proof of Pick’s interpolation theorem that is based on operator theory.

Theorem 1 (Pick’s interpolation theorem): Let $z_1, \ldots, z_n \in D$, and $w_1, \ldots, w_n \in \mathbb{C}$ be given. There exists a function $f \in H^\infty(D)$ satisfying $\|f\|_\infty \leq 1$ and

$f(z_i) = w_i \,\, \,\, i=1, \ldots, n$

if and only if the following matrix inequality holds:

$\big(\frac{1-w_i \overline{w_j}}{1 - z_i \overline{z_j}} \big)_{i,j=1}^n \geq 0 .$

Note that the matrix element $\frac{1-w_i\overline{w_j}}{1-z_i\overline{z_j}}$ appearing in the theorem is equal to $(1-w_i \overline{w_j})k(z_i,z_j)$, where $k(z,w) = \frac{1}{1-z \overline{w}}$ is the reproducing kernel for the Hardy space $H^2$ (this kernel is called the Szego kernel). Given $z_1, \ldots, z_n, w_1, \ldots, w_n$, the matrix

$\big((1-w_i \overline{w_j})k(z_i,z_j)\big)_{i,j=1}^n$

is called the Pick matrix, and it plays a central role in various interpolation problems on various spaces.

I learned this material from Agler and McCarthy’s monograph [AM], so the following is my adaptation of that source.

(A very interesting article by John McCarthy on Pick’s theorem can be found here).