### 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).

#### 1. Rochberg’s talk – the embedding dimension of the Dirichlet space, via metric considerations

The title of Rochberg’s talk was “The Dirichlet Space as a Quotient”, and here is a link to the video (and you can find the slides of the talk on that page too). It is a really lovely talk, and I really reccommend it; on the other hand it is longish (41 minutes), so I will try here to briefly describe the problem and Rochberg’s approach.

Recall that every complete Pick [7.2] reproducing kernel Hilbert space can be identified with the restriction of Drury-Arveson space [3] $H^2_d$ (where $d$ is either an integer or $\infty$) to a sub-variety of the unit ball $\mathbb{B}_d$ [7.3]. (numbers appearing in square brackets denote section numbers in this survey paper)

For the sake of the talk Rochberg concentrated on a particular reproducing kernel Hilbert space – the Dirichlet space. As a set of functions, the Dirichlet space $\mathcal{D}$ is the space of analytic functions $f$ on the unit disc $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$ such that

$\int_{\mathbb{D}} |f'(z)|^2 d \lambda(z) < \infty$.

Thus it is the set of analytic functions $f$ on the disc such that the area of $f(\mathbb{D})$ is finite. An inner product can be defined on this space such that it is the Hilbert function space with kernel

$k^\mathcal{D}(z,w) = -\frac{\log(1-z\bar w )}{z \bar w}$ .

This is a complete Pick reproducing kernel Hilbert space. By the facts recalled above, one may find a cardinal number $d$ and a variety $V \subseteq \mathbb{B}_d$ such that the $\mathcal{D}$ is isometrically isomorphic (via a natural map) to $H^2_d \big|_V$. The question is: how small can $d$ be? Can we take $d$ to be finite?

In the talk Rochberg shows that we can’t take $d$ to be finite. This is not a surprise, but the interesting thing is the method used, which is deeper than the problem it solves here. The main point is that on every reproducing kernel Hilbert space $H$ on a set $X$, one may define a metric using the kernel functions in the following way:

$\delta^H(x,y) = \sqrt{1 - |\langle k_x, k_y \rangle|}$.

(Here $k_x$ denotes the normalised kernel function at the point $x$). For example, if one takes $H = H^2_d$ (the Drury-Arveson space in dimension $d$) then

$\delta^{H^2_d}(x,y) = d_{ph}(x,y)$

where the right hand side denotes the pseudo-hyperbolic metric. In the important case $d=1$ we have

$d_{ph}(z,w) = \big| \frac{z-w}{1-z \bar w} \big|$ .

Now if $\mathcal{D}$ “embeds into” $H^2_d$ for $d$ finite, then this must give rise to an isometry between the disc $\mathbb{D}$ with the metric $\delta^{\mathcal{D}}$ and the variety $V \subset \mathbb{B}_d$ with the metric $\delta^{H^2_d}$.

Rochberg shows that this impossible in the following way. First, as a motivation, he considers the infinitesimal Riemannian metric induced by $\delta^{\mathcal{D}}$, and calculates that the curvatures tends to $- \infty$ as you approach the boundary. In contrast, the infinitesimal metric generated by $\delta^{H^2_d}$ is the usual (Bergman-Poincare) hyperbolic metric in the ball and has constant curvature $-1$. This leads to the heuristic that the length of the circumference of a circle will grow too quickly (relative to the Dirichlet metric) to be embed in a circle of same length in pseudo-hyperbolic metric.

The heuristic is then made precise: Rochberg calculates the number of points separated by a given length on a circle of a certain radius, and shows that these have to be mapped to a sphere of a certain radius, but the points then – forced to be separated by that distance – do not fit in the sphere!

Although my colleagues and I have been playing around with the pseudo-hyperbolic metric and the implications for the existence of isomorphisms between multiplier algebras (see Theorem 6.2, Remark 6.3 and Example 6.4 in this paper), we have not been able to make clever and delicate calculations of this type.

#### 2. Klep’s talk – on commuting dilations of non commuting matrices

The title of Klep’s talk was “Commuting dilations and linear positivstellensätze” based on this paper by Helton-Klep-McCullough-Schweighofer. As you can see if you clicked on the link, the paper is 77 pages long. Here is a link to the video, which is 29 minutes long. I have been studying the work of Helton, McCullough, Klep and others on free “real” algebraic geometry for some time (this is the topic of our learning seminar this semester), but it seems that these guys do not stop to let us catch up with them, no, they come up with really great new ideas. The beauty is that lots of their machinery is like bread and butter for operator theorists, but they handle it with such ingenuity so that their work is both digestible and delicious.

As an exercise for myself, I will now write a summary of what I understand HKMS do in their paper (I did not read their paper from start to end, this summary is based on the talk, on the introduction to their paper, and on some hopping back and forth I did through their paper). The work is related to free analysis, or NC function theory, so close to the work of Agler-McCarthy which I described in a previous post.

linear matrix inequality (LMI) is an expression of the form

(*) $L(x) := I_d - \sum_{j=1}^g A_j x_j \geq 0$,

where $A_1, \ldots , A_g$ are symmetric $d \times d$ matrices. The integer $d$ is called the size of L. A $g$-tuple $X = (X_1, \ldots, X_g)$ of symmetric $n \times n$ matrices is said to satisfy the LMI (*) if when plugging $X$ into $L$ one obtains a positive semi-definite matrix, namely

$L(X) := I_d \otimes I_n - \sum_{j=1}^g A_j \otimes X_j \geq 0$ .

The set of symmetric $n \times n$ matrices satisfying the LMI (*) is denoted by $D_L(n)$. The set $D_L(1)$ is what is known as a spectrahedron, and arises in many practical problems because sectrahedra are the basic sets in which semi-definite programming (a major branch of optimisation) is carried out. The collection of sets $D_L = \cup_{n=1}^\infty D_L(n)$ is called a free spectrahedron. Let’s make a standing assumption that all spectrahedra occurring here are bounded.

The spectrahedral inclusion problem is the problem of determining whether $D_L(1) \subseteq D_{\tilde{L}}(1)$, where $\tilde{L}$ is given by $\tilde{L} = I - \sum \tilde{A}_j x_j$ (we denote the size of $\tilde{L}$ by $\tilde{d}$). It turns out that the spectrahedral inclusion problem is important in optimisation theory – let’s just believe that (HKMS paper has references to the relevant papers); however it is provenly intractable (a special case – the matrix cube problem – is known to be NP hard). On the other hand, Helton-Klep-McCullough showed in previous papers (see this and this) that the problem of determining whether the free spectrahedra are contained one in the other, that is, whether $D_L \subseteq D_{\tilde{L}}$, is effectively solvable (the inclusion means that $D_L(n) \subseteq D_{\tilde{L}}(n)$ holds for all $n$).

Summary up to here. People want to answer the question: “when does the containment

$D_L(1) \subseteq D_{\tilde{L}}(1)$

hold?” People know how to answer the question: “when does the containment

$D_L \subseteq D_{\tilde{L}}$

hold?” Now, in general, $D_L \subseteq D_{\tilde{L}}$ obviously implies that $D_L(1) \subseteq D_{\tilde{L}}(1)$, but the converse implication is false. The purpose of the paper is to study a variant of the converse implication, thereby giving some kind of relaxation of the spectrahedral inclusion problem which is solvable.

The result that I found interesting, is that given an LMI as in (*) above, there is, for every $\tilde{d}$, a constant $\tau(L)(\tilde{d})$, such that whenever $\tilde{L}$ defines an LMI of size $\tilde{d}$, we have that

$D_L(1) \subseteq D_{\tilde{L}}(1)$

implies

$\tau(L)(\tilde{d}) D_L \subseteq D_{\tilde{L}}$ .

The authors also estimate values for $\tau(L)(\tilde{d})$ in important cases, and in fact obtain an exact a sharp optimal exact value that works when $L$ defines the so called matrix cube. What really caught my attention is that the authors show that these constants $\tau(L)$ are actually related to dilations in the following sense.

Theorem 1: Let $d \in \mathbb{N}$. There exists a constant $\sigma$, a Hilbert space $H$ and an isometry $V : \mathbb{R}^d \rightarrow H$, and a family of commuting selfadjoint operators $C_d \subseteq B(H)$, such that for every symmetric $A \in M_n(\mathbb{R})$, there exists some $T \in C_d$ such that $T$ dilates $\sigma A$, that is:

$\sigma A = V^* T V$.

Moreover, the authors calculate the best value for $\sigma$ (it is less than $1$, of course). The construction of the dilating space and the isometry is explicit ($H = L^2(O_d, \mathbb{R}^d)$, where $O_d$ is the group of orthogonal $d \times d$ matrices, and the isometry $V$ just sends a vector $v \in \mathbb{R}^d$ to the constant function $v$), and is of a different nature than the usual things people do in classical dilation theory. The dilation is not a “power dilation” – these authors live in a convex world.

It follows from Theorem 1 (as well as some other considerations) that given $L$ and $n$, there exists a constant $\tau$ such that for every $X \in D_L(n)$ there is a tuple $T$ of commuting selfadjoint operators on some Hilbert space $H$ with joint spectrum contained in $D_L(1)$ such that

(**) $\tau X = V^* T V$,

where $V$ is an embedding of $\mathbb{R}^n$ into $H$.

Theorem 2: Let $L$ define an LMI as in (*). Let $\tau(L)(n)$ be the biggest constant $\tau$ such that (**) holds. Then for any other LMI defined by $\tilde{L}$ of size $\tilde{d}$

$D_{L}(1) \subseteq D_{\tilde{L}}(1)$

implies

$\tau(L)(\tilde{d})D_L \subseteq D_{\tilde{L}}$.

Being a devoted dilation theorist, I am always happy to see applications of dilations to other problems, so this all looks very interesting.