Stable division and essential normality: the non-homogeneous and quasi homogeneous cases

by Orr Shalit

Update (January 29, 2016): paper revised, see this post


Several months ago Shibananda Biswas (henceforth: Shibu) and I posted to the arxiv our paper “Stable division and essential normality: the non-homogeneous and quasi homogeneous cases“. I was a little too busy to write about it at the time, but now that it is summer it seems like a good time to do it, since I am too busy, and I need a break from work. Nothing like going back and thinking about papers you have already written when you are overwhelmed by your current project.

Anyway, the main problem the paper I wrote with Shibu deals with, is the essential normality of submodules of various Hilbert modules (closely related to the Drury-Arveson module that I wrote about in the past: one, two, three, or if you are really asking for trouble, look at this survey). This paper is highly technical, and I want to try to explain it in a non-technical fashion.

1. Background

Recall some definitions. Let H be a reproducing kernel Hilbert space on the unit ball \mathbb{B}_d \subset \mathbb{C}^d. In the paper we deal with the spaces \mathcal{H}^{(t)}, (t \geq -d) which are determined by the kernel

k(z,w) = \frac{1}{(1 - \langle z,w \rangle)^{d+t+1}} \,\, , \,\, z,w \in \mathbb{B}_d.

The precise form of the kernel is used in calculations, but is not crucial for the explanation of what the paper is about. These reproducing kernel Hilbert spaces have a natural action of the polynomials on them, which make them a module over the ring \mathbb{C}[z]= \mathbb{C}[z_1, \ldots, z_d]. In particular, there is a d-tuple of operators S = (S_1, \ldots, S_d) given by

(*) [S_i f](z) = z_i f(z).

Given an ideal I \triangleleft \mathbb{C}[z], the closure [I] of I is clearly an invariant subspace for S. Equivalently, M = [I] is a submodule of the \mathbb{C}[z]-module H. We denote the restriction of S to M by T, that is, T_i = S_i \big|_M for i=1, \ldots, d. Then the action if T_i on M is also given by equation (*) above.

Multiplication operators as these are a typical example of a naturally occurring commuting tuple of operators that are not normal. In trying to analyse the operator theoretic and operator algebraic properties of tuples of operators, a natural question is whether they are a normal tuple (that is: a tuple of commuting normals), and if not, “how close” is this tuple to being normal? This is a natural question because normal tuples are “completely understood” (I hate saying that!), by Gelfand theory (or spectral theory, if you prefer to call it that).

This “how close” can be quantified by looking at the cross commutators

[T_i , T_j^*] = T_i T_j^* - T_j^* T_i

and checking “how small” they are. For example, if [T_i, T_j^*] is compact for all i,j ,  then we say that T is essentially normal. If [T_i, T_j^*] is in the Schatten p-class (meaning that |[T_i, T_j^*]|^p is a trace class operator, or equivalently, that |[T_i, T_j^*]|^{p/2} is Hilbert-Schmidt) then we say that T is p-essentially normal. Sometimes one says that M is essentially normal (or p-essentially normal) when it is clear that this really means that T is essentially normal.

Now, it is true that the tuple S defined above is p-essentially normal for every p>d, for many natural choices of space H, (in particular this is true for the spaces \mathcal{H}^{(t)}). But note that once you restrict to an invariant subspace, this may stop being true. Arveson conjectured that in the natural examples of spaces H (such as the spaces \mathcal{H}^{(t)}), if I is a homogeneous ideal, then T is also p-essentially normal for all p>d. On the one hand, this conjecture has not yet been fully verified for all homogeneous ideals, although a lot of partial results are available (a rather detailed account of existing results are available in the introduction of our paper). On the other hand, it is now widely believed that homogeneity is not essential to the problem, and if the zero variety of the polynomial ideal I satisfies that it crosses the boundary of \mathbb{B}_d “transversally” then T should still be p-essentially normal for p>d (for example, in the paper which I wrote about in the previous post, Matt Kennedy and I verified some consequences of the essential normality conjecture for ideals whose zero set (roughly) touches the sphere only when it crosses it). On the yet other hand, there is an example of a submodule M (not arising as the closure of a polynomial ideal) such that the restriction T of S to M is not essentially normal.

(Below, we will just say “essentially normal” instead of “p-essentially normal for every p>d“).

2. Stable division

previous paper that I wrote a few years ago connected the essential normality conjecture to a problem in polynomial division.

Definition: An ideal I \triangleleft \mathbb{C}[z] is said to have the stable division property if there exists a constant C and polynomials f_1, \ldots, f_k \in I such that for every polynomial h \in I, one can find polynomials g_1, \ldots, g_k \in \mathbb{C}[z] such that

 (*) h = \sum g_i f_i

with the norm control

\sum \|g_i f_i \| \leq C \|h\| .

My favourite open problem is: do all ideals have the stable division property? I will be happy enough to know the answer for homogeneous ideals. In that paper from a few years ago I proved that homogeneous ideals generated by monomials and that ideals in two variables have the stable division property. Matt Kennedy gave some improvements in a subsequent paper. I tore my hair out trying to go beyond these results, but nothing much has come from this effort.

The reason I came up with this notion of stable division is the observation that homogeneous ideals that have the stable division property, also satisfy Arveson’s essential normality conjecture. The proof of this observation (that’s in  Section 4 of this paper) uses the homogeneity of the ideal twice: first it uses  uses an important result of Guo and Wang (Section 2 in this paper) – that all principal ideals generated by a single homogeneous polynomial satisfy the conjecture, and second, the graded structure of the ideal is used to apply Guo and Wang’s result together with the stable division property to obtain essential normality to an arbitrary homogeneous ideal with the stable division property.

3. The new results

Since the problems of essential normality as well as stable division are both interesting also in the non-homogeneous case, it is natural to ask whether stable division implies essential normality also for arbitrary ideals. For this we need first a version of Guo and Wang’s result, that says that principal ideals are essentially normal also in the non-homogeneous case. This is furnished by this paper of Fang and Xia, which achieves this goal when the original Hilbert spaces are taken to be H = \mathcal{H}^{(t)} for t > -2 (in a subsequent paper they improve this slightly, but I don’t want to go into details). In fact, when Shibu and I saw Fang and Xia’s preprint we realised that we should be able to use it to prove that stable division implies essential normality in the non-homogeneous case, at least for the spaces which they treat.

But one then has to think how to apply this together with the stable division property, and this turns out to be rather tricky; eventually we found a (clever!) way to do it. One new feature of our proof is that it forces us to introduce a more flexible notion : approximate stable division property. It is like the stable division property (see the Definition above) but instead of equality in (*) we are willing to settle for almost equality, that is, one should be able to make \|h - \sum g_i f_i\| as small as possible (perhaps one will not be able to get equality with polynomials). The main result in our paper is that the approximate stable division property implies essential normality.

To be precise, we prove that if t>-3, and if I has the stable division property with respect to the \mathcal{H}^{(t+1)} norm, then the closure of I in \mathcal{H}^{(t)} is p-essentially normal for all p>d.

Now that we know that (approximate) stable division implies essential normality, we must find new examples of classes of ideals that have the (approximate) stable division property. Unfortunately, it is really hard to find examples – and even harder to find counter examples. As a secondary result, Shibu and I prove that all quasi homogeneous ideals in two variables have the stable division property.

Together with our main result, this is used to provide a new proof to the fact that quasi homogeneous ideals in two variables satisfy Arveson’s conjecture (this has been obtained in the past by Douglas and Sarkar by different methods).