Aleman, Hartz, McCarthy and Richter characterize interpolating sequences in complete Pick spaces

by Orr Shalit

The purpose of this post is to discuss the recent important contribution by Aleman, Hartz, McCarthy and Richter to the characterization of interpolating sequences (for multiplier algebras of certain Hilbert function spaces). Their recent paper “Interpolating sequences in spaces with the complete Pick property” was uploaded to the arxiv about two weeks ago, and, as usual, writing this post is meant mostly as a diversion for me (somewhere between doing “real” work and getting frustrated about the news), just giving some background and stating the main result. (Even more recently these four authors released yet another paper that looks very interesting – this one.)

1. Background – interpolating sequences

We will be working with the notion of Hilbert function spaces – also called reproducing Hilbert spaces (see this post for an introduction). Suppose that H is a Hilbert function space on a set X, and k its reproducing kernel. The Pick interpolation problem is the following:

Pick interpolation problem: given x_1, \ldots, x_n \in X and w_1, \ldots, w_n \in \mathbb{C}, does there exist a multiplier f \in Mult(H) of norm less than 1, such that f(x_i) = w_i for all i=1, \ldots, n?

Recall that H is said to have the Pick property, if the positive semi definiteness of the Pick matrix

\left[(1-w_i {w_j}^*) k(x_i,x_j) \right]_{i,j=1}^n

is a sufficient condition for the existence of a positive answer to the Pick interpolation problem (it is a necessary condition in any Hilbert function space).  Further, H is said to have the complete Pick property if, whenever the w_1, \ldots, w_n are matrices rather than numbers, then the same condition is sufficient for the existence of a matrix valued multiplier of norm less than or equal to 1 sending x_i to w_i.

The most familiar example of a Hilbert function space is the Hardy space H^2(\mathbb{D}) (the space of all holomorphic functions in the unit disc \mathbb{D} with square summable Taylor coefficients), and its multiplier algebra is H^\infty = H^\infty(\mathbb{D}); this multiplier algebra is simply the space of all bounded analytic functions on the disc (and the multiplier norm is the supremum norm), and therefore it exists as a respectable function algebra outside theory of Hilbert function spaces. (The Pick problem for the space H^2 becomes a question of interpolating with a bounded analytic function, and this problem was considered directly by Pick without any mention of Hilbert spaces. For a Hilbert function space theoretic proof, see this post.)

We come now to the interesting subject of interpolation sequences. Let {\bf x} = \{x_n\}_{n=1}^\infty be a sequence of points in X (note that unlike in Pick’s problem, this sequence is infinite). The sequence gives rise to a bounded map Ev_{\bf x} : Mult(H) \to \ell^\infty given by

EV_{\bf x}(f) = (f(x_1), f(x_2), \ldots).

The sequence {\bf x} is said to be an interpolating sequence for Mult(H) if this map is surjective. In other words, if for every bounded sequence of targets \{w_n\}_{n=1}^\infty, one can find a multiplier that interpolates all the nodes (x_1, w_1), (x_2, w_2), \ldots.

2. Characterization of interpolating sequences in complete Pick spaces

In 1958, Lennart Carleson characterized the interpolating sequences of H^\infty = Mult(H^2). His work shows that the following two conditions are necessary and sufficient for a sequence {\bf x} = \{x_n\}_{n=1}^\infty \subset \mathbb{D} to be an interpolating sequence for H = H^\infty:

Carleson measure condition (C): There exists a constant C such that for all f \in H,

\sum \|k_{x_n}\|^{-2}|f(x_n)|^2 \leq C\|f\|_{H}.

(Here, k_x denotes the kernel function at the point x.)

Separating sequence condition (S): There exists a constant c>0, such that for all m \neq n,

d_{H}(x_m,x_n) \geq c,

where d_{H} is the metric induced on X by H^2, given by

d_H(x,y) = \sqrt{1 - \frac{|\langle k_x, k_y \rangle|^2}{\|k_x\|^2 \|k_y\|^2}}.

Thus, for the space H = H^2, Carleson’s interpolation theorem can be stated as:

(C) + (S) \Leftrightarrow interpolating .

Intuitively speaking, the conditions make sure that the sequence {\bf x} is spread out well enough in X. It is clear that if a certain point is repeated, say x_m = x_n, then one cannot assign arbitrary values at these points, i.e., one must have w_m = w_n for interpolation to be possible. Thus interpolation is impossible if x_m = x_n for some m \neq n. Moreover, if the points x_n have a subsequence that converges to a point in x \in X, then the values of the corresponding w_ns will also have to converge (in most multiplier algebras of interest, the functions are continuous). The condition (S) makes sure that this does not happen; in fact this condition is somewhat stronger than just requiring that the sequence has no convergent subsequence in X.

The condition (C) is a little harder to visualize. One should think of it as guaranteeing that the sequence x_n escapes to “infinity” fast enough. In the case of the Hardy space on the disc H^2 = H^2(\mathbb{D}), k_w(z) = \frac{1}{1 - z \overline{w}}, so that \|k_{x_n}\|^{-2} = 1 - |x_n|^2, so the “Carleson measure” condition shows that the points x_n in the disc have to move towards the unit circle (which is infinity in a certain sense), and the points have to move to circle fast enough so that the series converges.

The convergence of the series itself (that is, condition (C)) is not a strong enough condition, since it does not rule out that every point is repeated twice. On the other hand, the separation condition (condition (S)) is also not strong enough, as examples show. The two conditions put together turn out to be equivalent to that the sequence is interpolating.

In the case of the Hardy space on the disc, Carleson gave more geometric characterization of  interpolating sequences; the characterization mentioned above is in a form that allows for a generalization to Hilbert function spaces on other sets.

It can be shown, that for every Hilbert function space

interpolating \Rightarrow (C) + (S)

and for several Hilbert function spaces with the complete Pick property it was shown that the converse holds:

(C) + (S) \Rightarrow interpolating .

In their monograph, Agler and McCarthy asked whether (C) + (S) \Rightarrow interpolating for all Hilbert functions spaces with the complete Pick property. About 15 years later, the paper “Interpolating sequences in spaces with the complete Pick property” finally answers this question in the affirmative.

What happened in these fifteen years that allowed the solution of this open problem? Well, some mathematicians working in the field became older and wiser. Also, some new mathematicians joined the field. But I guess that the authors would agree that the really dramatic move forward is the solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava (see this wiki entry for background and links). That is, most of the machinery needed for the proof was there at the hands of the masters of the subject, and the paving theorem/conjecture (which is equivalent to the Kadison-Singer problem), together with some tricks of the trade, allows the authors to split every sequence that satisfies (C) and (S) into a union of finitely many interpolating sequences. From this they are able to construct a right inverse to EV_{\bf x}. After all the preliminaries, this is explained in the paper in slightly more than one page.

A reader familiar with Hilbert function spaces, who takes the paving theorem on faith, can understand the proof immediately after reading Chapter 9 in Agler and McCarthy’s monograph. For a very readable exposition of Kadison-Singer including a proof of the paving theorem (following the work of Marcus-Spielman-Srivastava) see Terry Tao’s blog.

3. Closing remarks

Besides the interest in seeing an open problem in the field solved, I thought that it was cool that it was by an application of the solution to the Kadison-Singer problem. It is nice that a breakthrough result has continuing effect on mathematics, and is not just a Big Event and a dead end.

The paper of Aleman, Hartz, McCarthy and Richter contains some other interesting topics that I did not mention here, such as interpolating sequences between different Hilbert function spaces, and existence of non-essentially normal multipliers (in many reasonable complete Pick spaces). There is one result that they apparently obtained but which will appear in a forthcoming paper: this is a proof of the above characterization of interpolating sequences for the Drury-Arveson space H^2_d (for d< \infty), which is independent of the Marcus-Spielman-Srivastava theorem and the various equivalent formulations of the Kadison-Singer problem. That is worth looking forward to.