### 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.

Suppose that $H$ is a reproducing kernel Hilbert space on a space $X$ (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 $X$, 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 $X$ and its subsets, the function theory on $X$, the Hilbert function space structure of subspaces of $H$ and of $H$, and maybe also operator algebras acting on these subspaces.

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

Let $w = (w_n)_{n=0}^\infty$ be a sequence of positive numbers, and consider the space of power series

$H_w(D) = \{f (z) = \sum_{n=0}^\infty a_n z^n : \sum |a_n|^2 w_n < \infty \}$.

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

$\langle \sum a_n z^n , \sum b_n z^n \rangle = \sum w_n a_n \overline{b_n}$

then $H_w(D)$ is a RKHS. If $w_n = 1$ for all $n$ then we get the Hardy space $H^2(D)$, if $w_n = (n+1)^{-1}$ then we get the Bergman space $L^2_a(D)$.

The sequence $w$ determines the space $H_w(D)$, and one function theoretic aspect encoded by $w$ 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 $H_i$ ($i = 1,2$) are Hilbert spaces on sets $X_i$ with kernels $k^i$, 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 $X_1 = X_2 = X$, one kind of natural map is multiplication: $M_f h : h \mapsto fh$, where $f : X \to \mathbb{C}$ is a function. Another natural map is a composition operator $C_\varphi(h) = h \circ \varphi$, where $\varphi : X_2 \to X_1$ 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 $C_\varphi$ from the properties of the map $\varphi$. 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 $A : H_1 \to H_2$ that has the form

$Th = M_f h \circ \varphi$.

Recalling that an RKHS $H_i$ on a set $X_i$ is determined by the kernel functions $K^i_x$ for $x \in X_i$, and is the closed linear span of these kernel functions, it is natural to look at maps $T : H_1 \to H_2$ that send kernel functions to kernel functions, or at least to scalar multiples of kernel functions. So, we consider “diagonal” operators of the form

(*) $TK^1_x = \lambda_x K^2_{\varphi(x)}$

where $\{\lambda_x\}_{x\in X_1}$ is a family of scalars, and $\varphi : X_1 \to X_2$. Let us say that a linear map $T : H_1 \to H_2$ is a an isomorphism of Hilbert function spaces if it is a bounded bijective linear map of the form (*). If the map $T$ is isometric, then we say that it is an isometric isomorphism of Hilbert functions spaces (this is somewhat different from the definition in Exercise 6.4.5 in my book, where only maps of the form (*) with $\lambda_x = 1$ were considered).

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:

1. For which sequences $w = (w_n)$ is $H_w(D)$ a RKHS? What is the kernel? One can also ask, when does the Hilbert function space consist of continuous functions on the closed disc?
2. For which of those sequences $w = (w_n)$, are the composition operators $C_\varphi$ (for $\varphi \in Aut(D)$) automorphisms of $H_w(D)$ (isometric or not)? DOes the situation change if we allow for weighted composition operators?
3. Determine, given sequences $w = (w_n)$ and $w' = (w_n')$, whether $H_w(D)$ and $H_{w'}(D)$ isometric as Hilbert function spaces? In particular, for what sequences is the generalized Hardy space isomorphic to the classical Hardy space?
4. 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 $H_w(D)$ and $H_{w'}(D)$ (isometrically) isomorphic?

Circle of problems 2: Geometry and Hilbert spaces structure.

Let $E$ be a subset of $X$ (a set on which the RKHS $H$ with kernel $K$ lives). We can form $H_E = \overline{span}\{k_x : x \in E\}$. Our second circle of problems is around the question: how does the geometry of $E$ reflect in the structure of $H_E$. We will focus on the case that $E$ is a finite set.

Sometimes, the set $X$ has some structure to begin with. For example, if $H$ is the Hardy space $H^2(D)$, then $X = D$. $D$ has the natural Euclidean metric defined on it

$d(z,w) = |z - w|$

as well as the pseudohyperbolic metric

$\rho(z,w) = \frac{|z-w|}{|1-z\overline{w}|}$

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

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

$dist(x,y) = \|\hat{K}_x - \hat{K}_y\|$.

Another, slightly different metric is given by

$\delta_H(x,y) = \|P_x - P_y\|$,

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

$\delta_H(x,y) = \sqrt{1 - |\langle \hat{K}_x,\hat{K}_y\rangle|^2}$.

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

$\delta_H(z,w) = \sqrt{1 - \frac{(1-|z|^2)(1-|w|^2}{|1-z\overline{w}|^2}} =...= \rho(z,w)$

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

Concrete problems:

1. Determine when, given $E, F \subseteq X$, are $H_E$ and $H_F$ isometrically isomorphic. Do the same for multiplier algebras.
2. When $E$ and $F$ are finite, it is clear that $H_E$ and $H_F$ are isomorphic if and only if $E$ and $F$ have the same number of points in them. However, when $E$ and $F$ are infinite, then determining when the Hilbert function spaces are isomorphic is an interesting and difficult problem.
3. The above two problems might have a more substantial answer if we consider a certain space, such as the Hardy space $H^2(D)$ or the Bergman space $L^2_a(D)$. We actually know the answer for finite $E,F$ 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.
4. There are higher dimensional versions of the above problems, where the hardy space $H^2(D)$ on the disc is replaced by the Drury-Arveson space $H^2_d$ on the unit ball in $\mathbb{C}^d$ (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.
5. 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.