SEVERAL PROBLEMS IN OPERATOR THEORY

WILLIAM ARVESON

1. Introduction

We discuss some problems and conjectures in higher dimensional operator theory. These all have something to do with the basic problem of developing an effective Fredholm theory of d-contractions, and completing the index theorem that was partially established in Theorem B of [Arv02], following up on [Arv00]. My opinion is that good progress on any one of these problems will be a significant advance, if not a breakthrough.

Most of what follows is an exposition of the theory of Dirac operators, Fredholmness, and index from scratch, in a form accessible to anyone with a good basic knowledge of operators on Hilbert spaces. The conjectures and problems will be found in Section 4.

2. Dirac Operators in dimension d

Let T̄ = (T_{1}, …, T_{d}) be a multioperator of complex dimension d, that is to say, a d-tuple of mutually commuting bounded operators acting on a common Hilbert space H. All geometric properties of T̄ are reflected in properties of its associated Dirac operator, and we begin by recalling the basic facts about Dirac operators from [Arv02].

The Dirac operator associated with a d-dimensional multioperator T̄ is constructed as follows. Let Z be a complex Hilbert space of dimension d, which of course we may take as ℂ^{d}. The exterior algebra over Z is defined as the direct sum of Hilbert spaces

ΛZ = Λ^{0}Z ⊕ Λ^{1}Z ⊕ ⋯ ⊕ Λ^{d}Z,

where Λ^{k}Z denotes the kth exterior power of Z. By definition, Λ^{0}Z = ℂ, and the last summand Λ^{d}Z is also isomorphic to ℂ. ΛZ is a 2^{d}-dimensional Hilbert space, and Λ^{k}Z is spanned by vectors of the form z_{1} ∧ z_{2} ∧ ⋯ ∧ z_{k}, where the inner product on Λ^{k}Z is determined by

⟨z_{1} ∧ ⋯ ∧ z_{k}, w_{1} ∧ ⋯ ∧ w_{k}⟩ = det(⟨z_{i}, w_{j}⟩)

the right side denoting the determinant of the k×k matrix of inner products ⟨z_{i}, w_{j}⟩. For each z ∈ Z, there is a creation operator C(z) that maps Λ^{k}Z to Λ^{k+1}Z, and acts on generators as follows

C(z)x_{1} ∧ ⋯ ∧ x_{k} = z ∧ x_{1} ∧ ⋯ ∧ x_{k}.

Date: 17 June, 2003.

C becomes a linear map of Z into B(ΛZ) that gives rise to an irreducible representation of the canonical anticommutation relations; more explicitly, C(Z) is an irreducible set of operators such that for all z, w ∈ Z one has

C(z)C(w) + C(w)C(z) = 0, C(z)C(w)^{∗} + C(w)^{∗}C(z) = ⟨z, w⟩1.

In a less coordinate-free form, one can choose an orthonormal basis e_{1}, …, e_{d} for Z, set c_{k} = C(e_{k}) for 1 ≤ k ≤ d, and one finds that

c_{i}c_{j} + c_{j}c_{i} = 0, c_{i}c_{j}^{∗} + c_{j}^{∗}c_{i} = δ_{ij}1.

An elementary discussion of these facts can be found in [Arv01].

Given a d-dimensional multioperator T̄ acting on a Hilbert space H, we can form the somewhat larger Hilbert space

H̃ = H ⊗ ΛZ = (H ⊗ Λ^{0}Z) ⊕ ⋯ ⊕ (H ⊗ Λ^{d}),

and the operator B defined on it by

B = T_{1} ⊗ c_{1} + ⋯ + T_{d} ⊗ c_{d},

where c_{k} = C(e_{k}) is as above. The fact is that B does not depend on the choice of basis in the sense that an operator B′ associated with a different basis e′_{1}, …, e′_{d} for Z must be naturally isomorphic to B (see [Arv02] for more detail). For purposes of these notes, we define the Dirac operator of T̄ to be the bounded self-adjoint operator D = B + B^{∗}. When it is necessary to refer to the underlying multioperator in the notation for Dirac operators we will do that by writing DT̄ rather than D.

We point out that this definition of Dirac operator is less comprehensive than the one in [Arv02], since we have not been careful to state the precise relations that exist between B (and therefore D) and the the anticommuting operators 1_{H} ⊗c_{1}, …, 1_{H} ⊗c_{d}. This is discussed in Section 2 of [Arv02]. But for our purposes here, the shorter definition given above will suffice. At the same time, one should keep in mind that with a more comprehensive definition of Dirac operator, the theory of d-dimensional multioperators becomes identical with the theory of d-dimensional Dirac operators in the sense that every abstract Dirac operator is associated with a multioperator, and two multioperators S̄ and T̄ are unitarily equivalent iff their Dirac operators DS̄ and DT̄ are isomorphic (see Theorem A of [Arv02]).

3. Fredholm Multioperators and index

A multioperator T̄ = (T_{1}, …, T_{d}) is said to be Fredholm if its Dirac operator D is a Fredholm operator. Since D is self-adjoint, this simply means that the range of D is closed and its kernel is finite-dimensional. The index of a self-adjoint Fredholm operator is of course zero; but in this case there is additional structure available that allows one to make a more subtle definition of index that will be essential in what follows. We now discuss these issues.

Suppose that the operators T_{1}, …, T_{d} act on a Hilbert space H, so that D acts on H̃ = H ⊗ ΛZ. There is a natural ℤ_{2}-grading of H̃ that makes D into an operator of odd degree, and which is defined as follows. Consider the two subspaces of H̃ defined by

H̃_{+} = ∑_{k even} H ⊗ Λ^{k}Z,  H̃_{−} = ∑_{k odd} H ⊗ Λ^{k}Z.

These spaces are mutually orthogonal and sum to H̃, hence they provide H̃ with a ℤ_{2}-grading. Elements of H̃_{+} (resp. H̃_{−}) are called even vectors (resp. odd vectors). Since both B and its adjoint map even vectors to odd ones and odd ones to even ones, it follows that D does the same, and in particular the restriction D_{+} of D to H̃_{+} defines a bounded operator from H̃_{+} to H̃_{−}. The orthogonal decomposition H̃ = H̃_{+} ⊕ H̃_{−} gives rise to a 2 × 2 matrix representation

D = (\begin{matrix}0 & D_{+}^{∗} \\ D_{+} & 0 \end{matrix}) .

It is an instructive exercise to show when D is a Fredholm operator, the range of D_{+} is a closed subspace of H̃_{−} of finite codimension, so that D_{+} can be regarded as a Fredholm operator from H̃_{+} to H̃_{−}. We can now define the index of D as follows

ind D = dim ker D_{+} − dim coker D_{+} = dim(H̃_{+} ∩ ker D) − dim(H_{−}/DH_{+}).

The index is an important integer invariant for Fredholm multioperators. By analogy with the work of Atiyah and Singer, one might hope that the computation of the index of Fredholm multioperators will lead to important relations between the geometric and analytic properties of multioperators, and perhaps even feed back into basic issues of algebraic geometry. Thus, one would like to have sharp tools for a) determining when a given Dirac operator is Fredholm and b) computing the index in terms of concrete geometric properties of its underlying multioperator.

Some progress has been made in the direction of b), and we will describe that below. However, the problem a) of proving that natural examples of multioperators are Fredholm remains largely open. It is the latter problem that we want to discuss here in more detail.

Here is a simple and useful sufficient condition.

Proposition 3.1. Let T̄ = (T_{1}, …, T_{d}) be a multioperator with the following two properties:

(i) T̄ is essentially normal in that all self-commutators T_{k}T_{j}^{∗} −T_{j}^{∗}T_{k} are compact, 1 ≤ j, k ≤ d.

(ii) T_{1}T_{1}^{∗} + ⋯ + T_{d}T_{d}^{∗} is a Fredholm operator. Then the Dirac operator of T̄ is Fredholm.

Proof. Since D = D^{∗}, it suffices to show that D^{2} is a Fredholm operator. To that end, consider B = T_{1} ⊗ c_{1} + ⋯ + T_{d} ⊗ c_{d}. Since T_{j} commutes with T_{k} and c_{j} anticommutes with c_{k}, a straightforward computation shows that B^{2} = 0. Hence

D^{2} = (B + B^{∗})^{2} = B^{∗}B + BB^{∗} = ∑_{k,j=1}^{d} T_{k}^{∗}T_{j} ⊗ c_{k}^{∗}c_{j} + ∑_{k,j=1}^{d} T_{j}T_{k}^{∗} ⊗ c_{j}c_{k}^{∗}.

Using the anticommutation relations c_{j}c_{k}^{∗} = δ_{jk}1 − c_{k}^{∗}c_{i}, we can write the second term on the right as

A ⊗ 1 − ∑_{k,j=1}^{d} T_{j}T_{k}^{∗} ⊗ c_{k}^{∗}c_{j},

where A = T_{1}T_{1}^{∗} + ⋯ + T_{d}T_{d}^{∗}, so that

D^{2} = A ⊗ 1 + ∑_{k,j=1}^{d}(T_{k}^{∗}T_{j} − T_{j}T_{k}^{∗}) ⊗ c_{k}^{∗}c_{j}.

Since A ⊗ 1 is a Fredholm operator by (ii) and each summand in the second term is compact by (i), it follows that D^{2} is invertible modulo compacts. □

Remark 3.2 (Commutativity vs. Noncommuativity). It has become fashionable these days to look in the most noncommutative directions for results that generalize previously known commutative ones. The reader will note, however, that we have used commutativity in an essential way in the preceding proof to conclude that B^{2} = 0. In turn, the fact that B^{2} = 0 is critical for developing the homology and cohomology of multioperators; indeed, B is nothing other than the coboundary operator for the Koszul complex of T̄. If one gives up commutativity of the operators T_{1}, …, T_{d} then one must also give up the fact that H ⊗ ΛZ is a complex.

4. Finite Rank d-contractions

Let d be a positive integer. A d-contraction is a multioperator T̄ = (T_{1}, …, T_{d}) acting on a Hilbert space H that defines a row contraction in the sense that

T_{1}T_{1}^{∗} + ⋯ + T_{d}T_{d}^{∗} ≤ 1.

The rank of T̄ is the rank of the defect operator 1 − T_{1}T_{1}^{∗} − ⋯ − T_{d}T_{d}^{∗}, and T̄ is said to be pure if the completely positive operator map defined by

ϕ(X) = T_{1}XT_{1}^{∗} + ⋯ + T_{d}XT_{d}^{∗}, X ∈ B(H)

satisfies ϕ^{n}(1) ↓ 0 as n → ∞.

A principal theme of [Arv98], [Arv00], and [Arv02] has been to understand the nature of pure finite rank d-contractions. In particular, we have focused on the following three related questions. When is such a T̄ Fredholm? How does one compute the index of DT̄? How is the index related to concrete geometric properties of T̄? The papers [Arv00] and [Arv02] give various partial answers to the second two of these three questions. However, they do not address the first, and that is the question we want to discuss here: Which pure finite rank d-contractions are Fredholm? While it is conceivable that they all are, approaching the question in that generality appears today to be well out of reach. Instead, we fix attention on certain sub-questions which seem accessible, which have a genuine connection with algebraic geometry, and for which there is evidence that the answers are positive.

In order to discuss these issues, we first point out that finite rank pure d-contractions can all be realized as compressions of finite-multiplicity dshifts to quotients (see [Arv98]). This is a significant reduction, since the properties of such compressions can be approached in very concrete terms. We now describe these examples of d-contractions in more detail.

Let Z be a d-dimensional Hilbert space. The symmetric Fock space over Z will be denoted H_{d}^{2}, since it can be identified with the completion of the algebra of polynomials ℂ[z_{1}, …, z_{d}] with respect to a natural inner product. Choosing an orthonormal basis e_{1}, …, e_{d} for Z, one can form a row contraction S̄ = (S_{1}, …, S_{d}) of operators on H_{d}^{2}, where S_{k} denotes symmetric tensoring by e_{k}; equivalently, S_{k} can be interpreted as the operator that multiplies polynomials by the kth coordinate variable z_{k}. Let r be a positive integer and let r ·H_{d}^{2} be the direct sum of r copies of H_{d}^{2}. There is a natural way to define an action of S_{1}, …, S_{d} on r · H_{d}^{2} simply by increasing the multiplicity by a factor of r. We refer to that universal d-tuple as the d-shift of multiplicity r.

Given any closed invariant subspace M ⊆ r · H_{d}^{2} for the latter, one can form a quotient Hilbert space H = (r · H_{d}^{2})/M and a d-contraction T̄ on H by compressing the action of the d-shift or multiplicity r to the quotient ℂ[z_{1}, …, z_{d}]-module (r · H_{d}^{2})/M. It is not hard to see that such a multioperator T̄ is a pure d-contraction of rank at most r; indeed, the rank will be r except in the presence of simple degeneracies. We are interested in determining whether such multioperators T̄ are Fredholm, for various classes of invariant subspaces M.

The simplest versions of this general problem connect with algebraic geometry, and are described as follows. Let r be a positive integer. We may consider r-tuples of polynomials as elements of r·H_{d}^{2}, and in an obvious sense such elements are vector-valued polynomials. Choose a finite set p_{1}, …, p_{s} of vector-valued polynomials in r · ℂ[z_{1}, …, z_{d}], such that each p_{i} is homogeneous of some degree n_{i} - i.e., each of the r components of p_{i} is a homogeneous polynomial of the same degree n_{i}. Let

M = [f_{1}p_{1} + ⋯ + f_{s}p_{s} : f_{1}, …, f_{s} ∈ ℂ[z_{1}, …, z_{d}]]

be the invariant subspace of r · H_{d}^{2} generated by p_{1}, …, p_{s}.

Conjecture 1. Let H = (r · H_{d}^{2})/M be the quotient Hilbert space, and let T̄ = (T_{1}, …, T_{d}) be the d-contraction defined on H by compressing the multiplicity r d-shift. Then the self-commutators T_{j}T_{k}^{∗} − T_{k}^{∗}T_{j}, 1 ≤ j, k ≤ d, belong to the Schatten-von Neumann class L^{p} for every p > d.

Remark 4.1 (Consequences of Conjecture 1). Assuming that Conjecture 1 is true, it follows that the Dirac operators of such multioperators T̄ are all Fredholm. Indeed, since T_{1}T_{1}^{∗}+⋯+T_{d}T_{d}^{∗} = 1−F where the defect operator F is positive of rank at most r, and since all self-commutators are compact, Proposition 3.1 applies in a straightforward manner.

In turn, the fact that DT̄ is Fredholm can be applied to strengthen the index theorem proved in [Arv02] as follows. In [Arv00] we introduced a curvature invariant K(T̄) that can be associated with any pure finite-rank d-contraction T̄, and showed that it occupies a position analogous to the average Gaussian curvature of a compact oriented Riemannian manifold. Thanks in part to work of Greene, Richter and Sundberg [GRS02], it is now known that K(T̄) is always an integer. But what does that integer represent? It had been shown earlier in [Arv00] that K(T̄) frequently coincides with the Euler characteristic of a certain finitely generated algebraic module associated with T̄, but it was also known that there are examples for which K(T̄) is not the Euler characteristic of that module.

In [Arv02] a new approach was initiated, in which one sought to relate the value of K(T̄) to the index of a Dirac operator. The results apply to graded d-contractions - essentially, those associated with Conjecture 1. The key formula is Theorem B of [Arv02], which makes the following assertion about the d-contractions T̄ of Conjecture 1 and their associated Dirac operators D: Both ker D_{+} and ker D_{+}^{∗} are finite dimensional, and

(4.1)  (−1)^{d}K(T̄) = dim ker D_{+} − dim ker D_{+}^{∗}.

Unlike the earlier result that related K(T̄) to an Euler characteristic, there are no known exceptions to this formula, even for nongraded d-contractions.

However, note that Theorem B makes no assertion about whether or not the range of D_{+} is closed, and if the range of D_{+} is not closed then the right side of (4.1) is unstable. But if Conjecture 1 is true, then the range of D_{+} must be closed and the index formula (4.1) appears as a conventional assertion about the index of a Fredholm operator. Such a strengthening of Theorem B can now be seen as a full counterpart of the Gauss-Bonnet-Chern formula in its modern dress as an index theorem [GM91].

There are other consequences as well, relating to the stability of the curvature invariant under compact perturbations and homotopy, that we will not describe here. See Section 4 of [Arv02].

Remark 4.2 (Evidence for the truth of Conjecture 1). All concrete examples, for which the truth of Conjecture 1 can be decided, support it. A variety of such examples is described in [Arv02]. In this remark we describe another family of examples that also support Conjecture 1. These results are unpublished, and we give no details beyond pointing out the existence of such examples. Though this class of examples has little geometric interest (their associated algebraic varieties are trivial), the conclusion is significant from the point of view of operator theory. A monomial in ℂ[z_{1}, …, z_{d}] is a homogeneous polynomial of the particular form

p(z_{1}, …, z_{d}) = z_{1}^{k_{1}}z_{2}^{k_{2}} ⋯ z_{d}^{k_{d}}

where k_{1}, …, k_{d} are nonnegative integers. A vector-valued monomial is an element p of r·ℂ[z_{1}, …, z_{k}] of the form (p_{1}, p_{2}, …, p_{r}) where each component p_{i} is a scalar multiple of the same monomial z_{1}^{k_{1}} ⋯ z_{d}^{k_{d}}. The positive result that we want to report is that Conjecture 1 is true provided that each of the generators p_{1}, …, p_{s} of M is a vector-valued monomial.

Moving on to more general conjectures, we propose:

Conjecture 2. Let p_{1}, …, p_{s} be a finite set of vector polynomials in r · ℂ[z_{1}, …, z_{d}], not necessarily homogeneous, let M be the closed invariant subspace of r · H_{d}^{2} that they generate, and let T̄ be the compression of the multiplicity r shift to the quotient Hilbert space (r · H_{d}^{2})/M. Then the self-commutators T_{j}T_{k}^{∗} − T_{k}^{∗}T_{j} are all compact.

Assuming that this conjecture is true, the argument given above shows that all such compressed d-tuples T̄ are Fredholm. Such examples are not “graded” as were the ones discussed in the context of Conjecture 1, and we do not know if the index formula (4.1) holds for ungraded Hilbert modules. But there is enough evidence to lead us to:

Conjecture 3. Assuming that Conjecture 2 is true, then the index formula (4.1) holds in that setting as well.

Finally, let us skip directly to the most general question, about which we have the least evidence. It is conceivable (perhaps even likely) that for some invariant subspaces M ⊆ H_{d}^{2}, the compression of the d-shift to H_{d}^{2}/M does not have compact self-commutators. While this would be unfortunate, the index theory can be salvaged if the answer to the following question is yes.

Problem 4. Is the Dirac operator of every pure finite-rank d-contraction a Fredholm operator?

References

[Arv98] W. Arveson. Subalgebras of C^{∗}-algebras III: Multivariable operator theory. Acta Math., 181:159–228, (1998). arXiv:funct-an/9705007.

[Arv00] W. Arveson. The curvature invariant of a Hilbert module over ℂ[z_{1}, …, z_{d}]. J. Reine Angew. Mat., 522:173–236, (2000).

[Arv01] W. Arveson. The canonical anticommutation relations. unpublished: available from /~arveson, May (2001).

[Arv02] W. Arveson. The Dirac operator of a commuting d-tuple. Jour. Funct. Anal., 189:53–79, (2002).

[GM91] J. E. Gilbert and A. M. Murray. Clifford algebras and Dirac operators in harmonic analysis, volume 26 of Studies in Advanced Mathematics. Cambridge University Press, Cambridge, UK, (1991).

[GRS02] D. Greene, S. Richter, and C. Sundberg. The structure of inner multipliers on spaces with complete Nevanlinna-Pick kernels. J. Funct. Anal., 194(2):311–331, (2002).

   E-mail address: arveson@math.berkeley.edu