The remarkable Hilbert space H^2 (part III – three open problems)

by Orr Shalit

This is the last in the series of three posts on the d–shift space, which accompany/replace the colloquium talk I was supposed to give. The first two parts are available here and here. In this post I will discuss three open problems that I have been thinking about, which are formulated within the setting of H^2_d.

1. Essential normality of graded quotients

Let \mathbb{C}[z] = \mathbb{C}[z_1, \ldots, z_d] denote the ring of polynomials in d-variables, and let I be an ideal in \mathbb{C}[z]. Since the polynomials form a dense subspace in H^2_d, the ideal I can also be considered as a subspace of H^2_d. In fact, I is a joint invariant subspace for S = (S_1, \ldots, S_d). The closure of I, M := \overline{I}, is also an invariant subspace for S. Denote by Z = (Z_1, \ldots, Z_d) the compression of S to M^\perp, meaning that Z_i = P_{M^\perp} S_i \big|_{M^\perp}, or equivalently Z_i^* = S_i^*\big|_{M^\perp}, for all i.

The tuple Z is a row contraction consisting of commuting operators. Moreover, these operators satisfy

p(Z) = 0 \Leftrightarrow p \in I .

In a sense we have here a very concrete row contraction which is, in a sense, a universal model for row contractions “satisfying the relations in I“. The construction is indeed pretty concrete, but it turns out that the analysis of Z as a tuple of operators on a Hilbert space is complicated.

Analysis of operators can sometimes become simpler under the assumption that there is some underlying symmetry, and one of the simplest and most naturally occurring symmetry in our context is the natural action of the circle \mathbb{T} on the polynomials given by

\Gamma(\lambda) f(z) = f(\lambda z) .

The ring of polynomials naturally breaks up into a direct sum of eigenspaces

\mathbb{C}[z] = \mathbb{H}_0 \oplus \mathbb{H}_1 \oplus \mathbb{H}_2 \oplus \ldots ,

where \mathbb{H}_k is equal to the joint eigenspace

\{f \in \mathbb{C}[z] : \forall \lambda \in \mathbb{T} . \Gamma(\lambda) f = \lambda^k f\} .

A polynomial p is said to be homogeneous if it lies in one of the spaces \mathbb{H}_k (it is then said to be of degree k). An ideal I is said to be homogeneous if it is generated by homogeneous polynomials. Homogeneous ideals are in many ways more tractable then arbitrary ones, and from now on we will concentrate on the homogeneous case.

Let I be a homogeneous ideal in \mathbb{C}[z], and as above form M = \overline{I}, and let Z be the compression of S to M^\perp. The open problem I present is a simplified version of what is known as Arveson’s Conjecture.

Arveson’s Conjecture: Under the above assumption, the row contraction Z = (Z_1, \ldots, Z_d) is essentially normal, meaning that for all i,j, the commutator [Z_i, Z_j^*] := Z_i Z_j^* - Z_j^* Z_i is compact. 

As I said, this is a simplified version. For an up–to–date account of this conjecture see this paper of Kennedy and myself, and the references therein.

This problem looks like a decent problem in operator theory, but why is it important? Arveson raised this problem in connection to an invariant which he introduced for row contractions, called the curvature invariant. In this paper (arxiv) Arveson introduced the curvature invariant, and in this one (arxiv) he made his conjecture. The point of the conjecture is that the curvature invariant is indeed invariant under unitary equivalence and finite rank perturbations, but it remained an open question whether or not it is invariant under compact perturbations. If the conjecture is true, then for graded commuting row contractions the curvature invariant would serve as some kind of index, and would be invariant under compact perturbations, similarity and homotopy.

It was quickly observed that Arveson’s conjecture is related to K-homology, and it being true would give a concrete way of constructing K_1 elements for certain varieties. In this paper (arxiv) Douglas takes Arveson’s conjecture several steps forward (as a conjecture) and says that his conjecture, if it is true, would be “a new kind of index theorem”.

2. The stable division property

Let I be a homogeneous ideal in \mathbb{C}[z]. By Hilbert’s basis theorem we know that there exists a finite set of generators for I, meaning finitely many polynomials f_1, \ldots, f_k \in I such that for all h \in I there are g_1, \ldots, g_k \in \mathbb{C}[z] such that

(*)  h = g_1 f_1 + \ldots + g_k f_k .

But let us not forget that \mathbb{C}[z] sits inside H^2_d, and thus inherits a norm structure. Hilbert’s basis theorem, naturally, carries no information about the norms of the elements g_i f_i involved in the equality (*).

Definition: Let I be an ideal in \mathbb{C}[z]. I is said to have the stable division property if it has a set of generators f_1, \ldots, f_k \in I and a constant C such that for all h \in I and for all h \in I there are g_1, \ldots, g_k \in \mathbb{C}[z] such that (*) holds together with \sum \|g_i f_i \| \leq C \|h\|

I introduced the stable division property in this paper (arxiv), where it was proved that homogeneous ideals in two variables, and also ideals generated by monomials, have the stable division property. The following theorem, which explains what led me to introduce this notion, is proved in that same paper.

Theorem: If an ideal I has the stable division property then the tuple Z formed by compressing S to \overline{I}^\perp, is essentially normal. 

Although this Arveson’s conjecture was the motivation for introducing the notion of stable division, im my mind the latter is not less interesting than the former. Indeed, it should be considered as an effective version of Hilbert’s basis theorem, and thus is fundamental. For me, one of the reasons that Arveson’s conjecture is interesting is that if it were true then that would provide evidence that every homogeneous ideal has the stable division property.

Open problem: Does every homogeneous ideal have the stable division property?

3. The isomorphism problem for complete Pick algebras

The last open problem is the isomorphism problem for complete Pick algebras. In my cancelled colloquium talk (which didn’t happen yet, and on which these three posts are based) I will surely like to discuss it. But on this blog I already discussed it previously, in this post. Perhaps now it will be easier to understand (and more interesting).