Suppose that is a reproducing kernel Hilbert space on a space (I will assume knowledge in reproducing kernel Hilbert spaces (aka Hilbert function spaces) – see this old post for a crash introduction to the subject. ) The theory of reproducing kernel Hilbert spaces makes a connection between function theory, on the one hand, and Hilbert spaces and operator theory, on the other. The function theory captures features of the space , while the Hilbert space allows us to attach various functional analytic structures. The goal of the project is to explore the relationship between the various ingredients appearing above: the geometry/topology of and its subsets, the function theory on , the Hilbert function space structure of subspaces of and of , and maybe also operator algebras acting on these subspaces.

**Circle of problems 1: When are Hilbert function spaces “the same”?**

Let be a sequence of positive numbers, and consider the space of power series

.

One can show that (under some mild assumption) every element in is an analytic function in the unit disc, and that when one defines the inner product

then is a RKHS. If for all then we get the Hardy space , if then we get the Bergman space .

The sequence determines the space , and one function theoretic aspect encoded by is the rate of convergence of the coefficients, and in turn the “regularity” of functions in the space at the boundary.

As a warmup one can try to understand for which sequences does this construction work, and what is the reproducing kernel. The main problems to treat here are then to find when these spaces are “the same” in some sense. We need to explain what we mean by “the same”.

All Hilbert spaces (of the same dimension) are the same: there is alway a unitary – a linear isomorphism that preserves inner products – between them. However, when speaking of Hilbert function spaces, we wish our linear isomorphisms to preserve also the function-space structure. So, we will consider “natural” maps between Hilbert spaces (and also between their multiplier algebras).

If () are Hilbert spaces on sets with kernels , then there are several kinds of natural linear maps that in some sense preserve the structure of these spaces as RKHSs.

In the case that , one kind of natural map is multiplication: , where is a function. Another natural map is a composition operator , where is a map. It is not a trivial matter to decide which maps give rise to well defined composition operators, and then it is also quite a serious problem to be able to determine properties of the composition operator from the properties of the map . Composition operators can also be considered between multiplier algebras, and it is an interesting problem, with no known general answer, whether composition operators between the RKHS correspond to composition operators between the multiplier algebras.

A somewhat more flexible kind of “natural” operator acting between Hilbert function spaces is a * weighted composition operator*, which is nothing but the composition of a a composition operator and a multiplication operator, that is, a map that has the form

.

Recalling that an RKHS on a set is determined by the kernel functions for , and is the closed linear span of these kernel functions, it is natural to look at maps that send kernel functions to kernel functions, or at least to scalar multiples of kernel functions. So, we consider “diagonal” operators of the form

(*)

where is a family of scalars, and . Let us say that a linear map is a an ** isomorphism of Hilbert function spaces** if it is a bounded bijective linear map of the form (*). If the map is isometric, then we say that it is an

I leave it to the reader to figure out what is the relation between weighted composition operators and operators of the form given by equation (*).

**Concrete problems:**

- For which sequences is a RKHS? What is the kernel? One can also ask, when does the Hilbert function space consist of continuous functions on the closed disc?
- For which of those sequences , are the composition operators (for ) automorphisms of (isometric or not)? DOes the situation change if we allow for weighted composition operators?
- Determine, given sequences and , whether and isometric as Hilbert function spaces? In particular, for what sequences is the generalized Hardy space isomorphic to the classical Hardy space?
- Questions 2 and 3 can also be asked for the multiplier algebras: when do the automorphisms of the disc give rise to automorphisms of the multiplier algebras? And under what circumstances are the multiplier algebras and (isometrically) isomorphic?

**Circle of problems 2: Geometry and Hilbert spaces structure.**

Let be a subset of (a set on which the RKHS with kernel lives). We can form . Our second circle of problems is around the question: *how does the geometry of reflect in the structure of *. We will focus on the case that is a finite set.

Sometimes, the set has some structure to begin with. For example, if is the Hardy space , then . has the natural Euclidean metric defined on it

as well as the *pseudohyperbolic metric*

which is in some sense more natural – for example, an analytic self map of the disc is a conformal automorphism if and only if it is an isometry with respect to the pseudohyperbolic metric.

However, whether or not carries some structure to begin with, the Hilbert function space induces various metrics on it. For example, we can define

.

Another, slightly different metric is given by

,

where is the orthogonal projection on the one dimensional subspace spanned by . It can be shown (really can!) that

.

It is interesting to note that in the case of the Hardy space, we get

(see this paper by Arcozzi, Rochberg, Sawyer and Wick for hints about the above two computation jumps). So this shows that was a useful choice of metric, or looking at it the other way around, that the pseudohyperbolic metric is a natural one for when studying the Hardy space.

**Concrete problems:**

- Determine when, given , are and isometrically isomorphic. Do the same for multiplier algebras.
- When and are finite, it is clear that and are isomorphic if and only if and have the same number of points in them. However, when and are infinite, then determining when the Hilbert function spaces are isomorphic is an interesting and difficult problem.
- The above two problems might have a more substantial answer if we consider a certain space, such as the Hardy space or the Bergman space . We actually know the answer for finite in the case of the Hardy space, but we think that it could be a great problem to start the project with. The case of the Bergman space, as far as I know, is open.
- There are higher dimensional versions of the above problems, where the hardy space on the disc is replaced by the
space on the unit ball in (see these two older posts: one, two), or where the Bergman space on the unit disc is replaced by the Bergman space on the unit ball.*Drury-Arveson* - Finally, there is a quantitative version of the above problems, that we may describe during the week, depending on the progress made.

That’s all for now. The plan is the the students will think about these problems, choose one or two, or something related, and do their best in solving them. I hope to report on interesting discoveries after the week is over.

]]>Here is a link to the talk:

]]>

The conference will take place assuming that there will be an improvement in the global health situation, but I am already starting to think: what if not? We might move to an online format – no promises at this point.

]]>I will be talking mostly about these two papers of mine with co-authors: older one, newer one. Here is the title and abstract:

**Title: Matrix ranges, fields, dilations and representations**

**Abstract:** In my talk I will present several results whose unifying theme is a matrix-valued analogue of the numerical range, called the matrix range of an operator tuple. After explaining what is the matrix range and what it is good for, I will report on recent work in which we prove that there is a certain â€śuniversalâ€ť matrix range, to which the matrix ranges of a sequence of large random matrices tends to, almost surely. The key novel technical aspects of this work are the (levelwise) continuity of the matrix range of a continuous field of operators, and a certain quantitative matrix valued Hahn-Banach type separation theorem. In the last part of the talk I will explain how the (uniform) distance between matrix ranges can be interpreted equivalently as a â€śdilation distanceâ€ť, which can be interpreted as a kind of â€śrepresentation distanceâ€ť. These vague ideas will be illustrated with an application: the construction of a norm continuous family of representations of the noncommutative tori (recovering a result of Haagerup-Rordam in the d=2 case and of Li Gao in the d>2 case).

Based on joint works with Malte Gerhold, Satish Pandey and Baruch Solel.

]]>**Abstract:**

We study the space of all -tuples of unitaries using dilation theory and matrix ranges. Given two -tuples and generating C*-algebras and , we seek the minimal dilation constant such that , by which we mean that is a compression of some -isomorphic copy of . This gives rise to a metric

on the set of equivalence classes of -isomorphic tuples of unitaries. We also consider the metric

and

and we show the inequality

Let be the universal unitary tuple satisfying , where is a real antisymmetric matrix. We find that . From this we recover the result of Haagerup-Rordam and Gao that there exists a map such that and

Of special interest are: the universal -tuple of noncommuting unitaries , the -tuple of free Haar unitaries , and the universal -tuple of commuting unitaries . We obtain the bounds

From this, we recover Passer’s upper bound for the universal unitaries . In the case we obtain the new lower bound improving on the previously known lower bound .

]]>This talk is a short version of the talk I gave at the Besancon Functional Analysis Seminar last week; here are the slides for that talk.

]]>**Title**: CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: the Multi-Parameter Case and Beyond.

**Abstract**:Â We introduce a framework for studying dilations of semigroups of completely positive maps on von Neumann algebras. The heart of our method is the systematic use of families of Hilbert C*-correspondences that behave nicely with respect to tensor products: these are product systems, subproduct systems and superproduct systems. Although we developed our tools with the goal of understanding the multi-parameter case, they also lead to new results even in the well studied one parameter case. In my talk I will give a broad outline and a taste of the dividends our work.

The talk is based on a recent joint work with Michael Skeide.

**Assumed knowledge**: Completely positive maps and C*-algebras.

Feel free to write to me if you are interested in a link to the video talk.

]]>We would like to announce that the 48th Canadian Operator Symposium will be held online May 25 to May 29.Â Since many of the early summer Operator Algebra conferences have been cancelled and since we have the support and structural capabilities of the Fields Institute, our hope is to make the best of the current situation and provide a conference experience to the operator algebra community where researchers can present their research and can collaborate and socialize with others.

All talks will be given with Zoom (there are plenary speakers and there will be parallel session of contributed talks), and there will be “lunches” and “work rooms”. They say more details will be in the site soon. I plan to check it out.

]]>This is a quite short paper (200 full pages shorter than the paper I recently announced), which tells a simple and interesting story: we find that optimal constant , such that every pair of unitaries satisfying the q-commutation relation

dilates to a pair of **commuting** normal operators with norm less than or equal to (this problems is related to the “complex matrix cube problem” that we considered in the summer project half year ago and the one before). We provide a full solution. There are a few ramifications of this idea, as well as surprising connections and applications, so I invite you to check out the nice little introduction.

Michael Skeide and I have recently uploaded our new paper to the arxiv: CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond. In this gigantic (219 pages) paper, we propose a framework for studying the dilation theory of CP-semigroups parametrized by rather general monoids (i.e., semigroups…]]>

Reblogging the announcement of my recent giant paper with Michael Skeide. I want this to be at the head of my blog for a while.

Michael Skeide and I have recently uploaded our new paper to the arxiv: CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond. In this gigantic (219 pages) paper, we propose a framework for studying the dilation theory of CP-semigroups parametrized by rather general monoids (i.e., semigroups with unit), and we use this framework for obtaining new results regarding the possibility or impossibility of constructing or having a dilation, we use it also for obtaining new structural results on the â€śmechanicsâ€ť of dilations, and we analyze many examples using our tools. We present results that we have announced long ago, as well as some surprising discoveries.

This is an exciting moment for me, since we have been working on this project for more than a decade.

â€śExcuse me, did you really say *decade*?â€ť

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