Hankel Operators and the Dixmier Trace

on Strictly Pseudoconvex Domains

Miroslav Engliš, Genkai Zhang

Received: November 20, 2009 Revised: June 11, 2010

Communicated by Patrick Delorme

Abstract. Generalizing earlier results for the disc and the ball, we give a formula for the Dixmier trace of the product of 2n Hankel operators on Bergman spaces of strictly pseudoconvex domains in C^{n}. The answer turns out to involve the dual Levi form evaluated on boundary derivatives of the symbols. Our main tool is the theory of generalized Toeplitz operators due to Boutet de Monvel and Guillemin.

2000 Mathematics Subject Classification: Primary 32A36; Secondary 47B35, 47B06, 32W25 Keywords and Phrases: Dixmier trace, Toeplitz operator, Hankel operator, Bergman space, Hardy space, strictly pseudoconvex domain, pseudodifferential operator, Levi form

1. Introduction

Let Ω be a bounded strictly pseudoconvex domain in C^{n} with smooth boundary, and L_{hol}^{2}(Ω) the Bergman space of all holomorphic functions in L^{2}(Ω). For a bounded measurable function f on Ω, the Toeplitz and the Hankel operator with symbol f are the operators T_{f} : L_{hol}^{2}(Ω) → L_{hol}^{2}(Ω) and H_{f} : L_{hol}^{2}(Ω) → L^{2}(Ω) ⊖ L_{hol}^{2}(Ω), respectively, defined by

(1)  T_{f}g := Π(fg), H_{f}g := (I − Π)(fg),

where Π : L^{2}(Ω) → L_{hol}^{2}(Ω) is the orthogonal projection. It has been known for some time that for f holomorphic and n > 1, the Hankel operator H\overline{f} belongs to the Schatten ideal S^{p} if and only if f is in the diagonal Besov space B^{p}(Ω) and p > 2n, or f is constant (so H_{f} = 0) and p ≤ 2n; see Arazy, Fisher and Peetre [1] for Ω = B^{n}, the unit ball of C^{n}, and Li and Luecking [21] for general smoothly bounded strictly pseudoconvex domains Ω. This phenomenon is called a cutoff at p = 2n. In dimension n = 1, the situation is slightly different, in that the cutoff occurs not at p = 2 but at p = 1. One can rephrase the above results also in terms of membership in the Schatten classes of the commutators [T\overline{f}, T_{g}] := T\overline{f}T_{g} − T_{g}T\overline{f} of Toeplitz operators. In fact, it is immediate from (1) that

Research supported by GA AV ČR grant no. IAA100190802, Czech Ministry of Education research plan no. MSM4781305904, and the Swedish Research Council (VR).

T_{\overline{f}g} − T_{g}T\overline{f} = H∗ ; \overline{g}H\overline{f},

and also that T\overline{f}T_{g} = T_{\overline{f}g} if f or g is holomorphic; thus for holomorphic f and g

[T\overline{f}, T_{g}] = H∗ ; \overline{g}H\overline{f}.

In any case, it follows that there are no nonzero trace-class Hankel operators H\overline{f}, with f holomorphic, if n = 1, and similarly the product H_{\overline{f}_{1}}^{∗}H_{\overline{f}_{2}} … H_{\overline{f}_{2n−1}}^{∗}H_{\overline{f}_{2n}} = [T_{\overline{f}_{2}}, T_{f_{1}}] … [T_{\overline{f}_{2n}}, T_{f_{2n−1}}], with f_{1}, …, f_{2n} holomorphic, is never trace-class if n > 1. In particular, there is no hope for n > 1 of having an analogue of the well-known formula for the unit disc,

(2)  tr[T\overline{f}, T_{f}] = ∫_{D} |f′(z)|^{2} dm(z)

expressing the trace of the commutator [T\overline{f}, T_{f}] as the square of the Dirichlet norm of the holomorphic function f, which is one of the best known Moebius invariant integrals. (This formula actually holds for Toeplitz operators on any Bergman space of a bounded planar domain, if the Lebesgue area measure dm(z) is replaced by an appropriate measure associated to the domain, see [2].) A remarkable substitute for (2) on the unit ball B^{n} is the result of Helton and Howe [19], who showed that for smooth functions f_{1}, …, f_{2n} on the closed ball, the complete anti-symmetrization [T_{f_{1}}, T_{f_{2}}, …, T_{f_{2n}}] of the 2n operators T_{f_{1}}, …, T_{f_{2n}} is trace-class and

tr[T_{f_{1}}, T_{f_{2}}, …, T_{f_{2n}}] = ∫_{B^{n}} df_{1} ∧ df_{2} ∧ ⋯ ∧ df_{2n}.

There is, however, a generalization of (2) to the unit ball B^{n}, n > 1, in a different direction — using the Dixmier trace. This may be notable especially in view of the prominent applications of the Dixmier trace in noncommutative differential geometry [9]. Namely, it was shown by the present authors and Guo [12] that for f_{1}, …, f_{n} and g_{1}, …, g_{n} smooth on the closed ball, the product [T_{f_{1}}, T_{g_{1}}] … [T_{f_{n}}, T_{g_{n}}] belongs to the Dixmier class S^{Dixm} and has Dixmier trace equal to

(3)  Tr_{ω}([T_{f_{1}}, T_{g_{1}}] … [T_{f_{n}}, T_{g_{n}}]) = \frac{1}{n!} ∫_{∂B^{n}} ∏_{j=1}^{n}\{f_{j}, g_{j}\}_{b} dσ,

where dσ is the normalized surface measure on ∂B^{n} and {f, g}_{b} is the “boundary Poisson bracket” given by

\{f, g\}_{b} := ∑_{j=1}^{n} (\frac{∂f}{∂\overline{z}_{j}} \frac{∂g}{∂z_{j}} − \frac{∂f}{∂z_{j}} \frac{∂g}{∂\overline{z}_{j}} ) − (\overline{R}fRg − Rf\overline{R}g),

with \overline{R} := ∑_{j=1}^{n} \overline{z}_{j} \frac{∂}{∂\overline{z}_{j}} and R := ∑_{j=1}^{n} z_{j} \frac{∂}{∂z_{j}} the anti-holomorphic and the holomorphic part of the radial derivative, respectively. In particular, for f holomorphic on B^{n} and smooth on the closed ball, (H∗ ; \overline{f}H\overline{f})^{n} = [T\overline{f}, T_{f}]^{n} ∈ S^{Dixm} and

Tr_{ω}((H∗ ; \overline{f}H\overline{f})^{n}) = \frac{1}{n!} ∫_{∂B^{n}} ( ∑_{j=1}^{n} |\frac{∂f}{∂z_{j}} |^{2} − |Rf|^{2})^{n} dσ.

Note that for n = 1 the right-hand side vanishes, in accordance with the fact that in dimension 1 the cutoff occurs at p = 1 instead of p = 2n = 2; in fact, it was shown by Rochberg and the first author [13] that for n = 1 actually |H_{f}| = (H_{f}^{∗}H_{f})^{1/2}, rather than H_{f}^{∗}H_{f}, is in the Dixmier class for any f ∈ C^{∞}(\overline{D}), and

Tr_{ω}(|H_{f}|) = ∫_{∂D} |\overline{∂}f| dσ,

so, in particular,

Tr_{ω}(|H\overline{f}|) = ∫_{∂D} |f′| dσ = ∥f′∥_{H_{1}}

for f ∈ C^{∞}(\overline{D}) holomorphic on D, where H^{1} denotes the Hardy 1-space on the unit circle. In this paper, we generalize the result of [12] to arbitrary bounded strictly pseudoconvex domains Ω with smooth boundary. Our result is that for any 2n functions f_{1}, g_{1}, …, f_{n}, g_{n} ∈ C^{∞}(\overline{Ω}),

(4) Tr_{ω}(H_{f_{1}}^{∗}H_{g_{1}} … H_{f_{n}}^{∗}H_{g_{n}}) = \frac{1}{n!(2π)^{n}} ∫_{∂Ω} ∏_{j=1}^{n} L(\overline{∂}_{b}g_{j}, \overline{∂}_{b}f_{j}) η ∧ (dη)^{n−1},

where \overline{∂}_{b} stands for the boundary \overline{∂}-operator [14], η ∧(dη)^{n−1} is a certain measure on ∂Ω, and L stands for the dual of the Levi form on the anti-holomorphic tangent bundle; see §§ 2 and 4 below for the details. In contrast to [12], where we were using the so-called pseudo-Toeplitz operators of Howe [18], our proof here relies on Boutet de Monvel’s and Guillemin’s theory of Toeplitz operators on the Hardy space H^{2}(∂Ω) with pseudodifferential symbols. (This is also the approach used in [13], however the situation Ω = D treated there is much more manageable.)

In fact, it turns out that for any classical pseudodifferential operator Q on ∂Ω of order −n, the corresponding Hardy-Toeplitz operator T_{Q} belongs to the Dixmier class and

(5)  Tr_{ω}(T_{Q}) = \frac{1}{n!(2π)^{n}} ∫_{∂Ω} σ_{−n}(Q)(x, η(x)) η(x) ∧ (dη(x))^{n−1},

where σ_{−n}(Q) is the principal symbol of Q, and η is a certain 1-form on ∂Ω; see again §2 below for the details. In particular, in view of the results of Guillemin [16] [17], this means that on Toeplitz operators T_{Q} of order ≤ −n, the Dixmier trace Tr_{ω} T_{Q} coincides with the residual trace Tr_{Res} T_{Q}, a quantity constructed using the meromorphic continuation of the ζ function of T_{Q} (Wodzicki [24], Boutet de Monvel [7], Ponge [23], Lesch [20], Connes [9]). We recall the necessary prerequisites on the Dixmier trace, Hankel operators and the Boutet de Monvel-Guillemin theory in Section 2. The proofs of (5) and (4) appear in Sections 3 and 4, respectively. Some concluding comments are assembled in the final Section 5. Throughout the paper, we will denote Bergman-space Toeplitz operators by T_{f}, in order to distinguish them from the Hardy-space Toeplitz operators T_{f} and T_{Q}. Since Hankel operators on the Hardy space never appear in this paper, Hankel operators on the Bergman space are denoted simply by H_{f}.

2. Background

2.1 Generalized Toeplitz operators. Let r be a defining function for Ω, that is, r ∈ C^{∞}(\overline{Ω}), r < 0 on Ω, and r = 0, ∥∂r∥ > 0 on ∂Ω. Denote by η the restriction to ∂Ω of the 1-form Im(∂r) = (∂r − \overline{∂}r)/2i. The strict pseudoconvexity of Ω guarantees that η is a contact form, i.e. the half-line bundle

Σ := {(x, ξ) ∈ T^{∗}(∂Ω) : ξ = tη_{x}, t > 0}

is a symplectic submanifold of T^{∗}(∂Ω). Equip ∂Ω with a measure with smooth positive density, and let L^{2}(∂Ω) be the Lebesgue space with respect to this measure. The Hardy space H^{2}(∂Ω) is the subspace in L^{2}(∂Ω) of functions whose Poisson extension is holomorphic in Ω; or, equivalently, the closure in L^{2}(∂Ω) of C_{hol}^{∞}(∂Ω), the space of boundary values of all the functions in C^{∞}(\overline{Ω}) that are holomorphic on Ω. (In dimensions greater than 1, H^{2}(∂Ω) can also be characterized as the null-space of the \overline{∂}_{b}-operator, which will appear in Section 4 further on.) We will also denote by W^{s}(∂Ω), s ∈ R, the Sobolev spaces on ∂Ω, and by W_{hol}^{s}(∂Ω) the corresponding subspaces of nontangential boundary values of functions holomorphic in Ω. (Thus W^{0}(∂Ω) = L^{2}(∂Ω) and W_{hol}^{0}(∂Ω) = H^{2}(∂Ω).) Unless otherwise specified, by a pseudodifferential operator or Fourier integral operator (ΨDO or FIO for short) on ∂Ω we will always mean an operator which is “classical”, i.e. whose total symbol (or amplitude) in any local coordinate system has an asymptotic expansion

p(x, ξ) ∼ ∑_{j=0}^{∞} p_{m−j}(x, ξ),

where p_{m−j} is C^{∞} in x, ξ, and is positive homogeneous of degree m − j in ξ for |ξ| > 1. Here j runs through nonnegative integers, while m can be any integer; and the symbol “∼” means that the difference between p and ∑_{j=0}^{k−1} p_{m−j} should belong to the Hörmander class S^{m−k}, for each k = 0, 1, 2, . . .. The set of all classical ΨDOs on ∂Ω as above (i.e. of order m) will be denoted by Ψ_{cl}^{m}; and we set, as usual, Ψ_{cl} := ⋃_{m∈Z} Ψ_{cl}^{m} and Ψ^{−∞} := ⋂_{m∈Z} Ψ_{cl}^{m}. The operators in Ψ^{−∞} are precisely the smoothing operators, i.e. those given by a C^{∞} Schwartz kernel; and for any P, Q ∈ Ψ_{cl}, we will write P ∼ Q if P − Q is smoothing. Note that if P ∈ Ψ_{cl}^{m}, then P is continuous from W^{s}(∂Ω) into W^{s−m}(∂Ω), for any s ∈ R. For Q ∈ Ψ_{cl}^{m}, the generalized Toeplitz operator T_{Q} : W_{hol}^{m}(∂Ω) → H^{2}(∂Ω) is defined as

T_{Q} = ΠQ,

where Π : L^{2}(∂Ω) → H^{2}(∂Ω) is the orthogonal projection (the Szegö projection). Alternatively, one may view T_{Q} as the operator

T_{Q} = ΠQΠ

on all of W^{m}(∂Ω). Actually, T_{Q} maps continuously W^{s}(∂Ω) into W_{hol}^{s−m}(∂Ω), for each s ∈ R, because Π is bounded on W^{s}(∂Ω) for any s ∈ R (see [6]). It is known that the generalized Toeplitz operators T_{P}, P ∈ Ψ_{cl}, have the following properties.

(P1) They form an algebra which is, modulo smoothing operators, locally

isomorphic to the algebra of classical ΨDOs on R^{n}. (P2) In fact, for any T_{Q} there exists a ΨDO P of the same order such that

T_{Q} = T_{P} and P Π = ΠP. (P3) If P, Q are of the same order and T_{P} = T_{Q}, then the principal symbols

σ(P) and σ(Q) coincide on Σ. One can thus define unambiguously the order of a generalized Toeplitz operator as ord(T_{Q}) := min{ord(P) : T_{P} = T_{Q}}, and its principal symbol (or just “symbol”) as σ(T_{Q}) := σ(Q)|_{Σ} if ord(Q) = ord(T_{Q}). (The symbol is undefined if ord(T_{Q}) = −∞.)

(P4) The order and the symbol are multiplicative: ord(T_{P} T_{Q}) = ord(T_{P}) +

ord(T_{Q}) and σ(T_{P} T_{Q}) = σ(T_{P})σ(T_{Q}). (P5) If ord(T_{Q}) ≤ 0, then T_{Q} is a bounded operator on L^{2}(∂Ω); if ord(T_{Q}) <

0, then it is even compact. (P6) If Q ∈ Ψ_{cl}^{m} and σ(T_{Q}) = 0, then there exists P ∈ Ψ_{cl}^{m−1} with T_{P} = T_{Q}.

In particular, if T_{Q} ∼ 0, then there exists a ΨDO P ∼ 0 such that T_{Q} = T_{P}.

(P7) We will say that a generalized Toeplitz operator T_{Q} of order m is elliptic if σ(T_{Q}) does not vanish. Then T_{Q} has a parametrix, i.e. there exists a Toeplitz operator T_{P} of order −m, with σ(T_{P}) = σ(T_{Q})^{−1}, such that T_{Q}T_{P} ∼ I_{H^{2}(∂Ω)} ∼ T_{P} T_{Q}.

We refer to the book [5], especially its Appendix, and to the paper [4] (which we have loosely followed in this section) for the proofs and additional information on generalized Toeplitz operators.

2.2 The Poisson operator. Let K denote the Poisson extension operator on Ω, i.e. K solves the Dirichlet problem

(6)  ΔKu = 0 on Ω,  Ku|_{∂Ω} = u.

(Thus K acts from functions on ∂Ω into functions on Ω. Here Δ is the ordinary Laplace operator.) By the standard elliptic regularity theory (see e.g. [22]), K acts continuously from W^{s}(∂Ω) onto the subspace W_{harm}^{s+1/2}(Ω) of all harmonic functions in W^{s+1/2}(Ω). In particular, it is continuous from L^{2}(∂Ω) into L^{2}(Ω), and thus has a continuous Hilbert space adjoint K^{∗} : L^{2}(Ω) → L^{2}(∂Ω). The composition

K^{∗}K =: Λ

is known to be an elliptic positive ΨDO on ∂Ω of order −1. We have

(7)  Λ^{−1}K^{∗}K = I_{L^{2}(∂Ω)},

while

KΛ^{−1}K^{∗} = Π_{harm},

the orthogonal projection in L^{2}(Ω) onto the subspace L_{harm}^{2}(Ω) of all harmonic functions. (Indeed, from (7) it is immediate that the left-hand side acts as the identity on the range of K, while it trivially vanishes on Ker K^{∗} = (Ran K)^{⊥}.) Comparing (7) with (6), we also see that the restriction

γ := Λ^{−1}K^{∗}|_{L_{harm}^{2}(Ω)}

is the operator of “taking the boundary values” of a harmonic function. Again, by elliptic regularity, γ extends to a continuous operator from W_{harm}^{s}(Ω) onto W^{s−1/2}(∂Ω), for any s ∈ R, which is the inverse of K. The operators

Λ_{w} := K^{∗}wK,

with w a smooth function on \overline{Ω}, are governed by a calculus developed by Boutet de Monvel [3]. It was shown there that for w of the form

(8)  w = r^{m}g,  m = 0, 1, 2, …, g ∈ C^{∞}(\overline{Ω}),

Λ_{w} is a ΨDO on ∂Ω of order −m − 1, with symbol

(9)  σ(Λ_{w})(x, ξ) = \frac{(−1)^{m}m!}{2|ξ|^{m+1}} g(x) ∥η_{x}∥^{m}.

(In particular, σ(Λ)(x, ξ) = 1/2|ξ|.) By abstract Hilbert space theory, K has, as an operator from L^{2}(∂Ω) into L^{2}(Ω), the polar decomposition

(10)  K = U(K^{∗}K)^{1/2} = UΛ^{1/2},

where U is a partial isometry with initial space \overline{Ran K^{∗}} = (Ker K)^{⊥} and final space \overline{Ran K}; that is, U is a unitary operator from L^{2}(∂Ω) onto L_{harm}^{2}(Ω). The operators γ, K and U = KΛ^{−1/2} can be used to “transfer” operators on L_{harm}^{2}(Ω) ⊂ L^{2}(Ω) into operators on L^{2}(∂Ω). The following proposition appears as Proposition 8 in [11]; we reproduce its (short) proof here for completeness.

Proposition 1. γΠK = T_{Λ}^{−1}ΠΛ.

Proof. Set Π_{Λ} := KT_{Λ}^{−1}ΠΛγ, an operator on L_{harm}^{2}(Ω); we need to show that Π_{Λ} = Π|_{L_{harm}^{2}}. Since T_{Λ}^{−1}ΠΛ acts as the identity on the range of Π, it is immediate that Π_{Λ}^{2} = Π_{Λ}; furthermore, Π_{Λ} = KT_{Λ}^{−1}ΠK^{∗} = KΠT_{Λ}^{−1}ΠK^{∗} is evidently self-adjoint. Thus Π_{Λ} is the orthogonal projection in L_{harm}^{2}(Ω) onto Ran Π_{Λ}. But

Ran Π_{Λ} = (Ker Π_{Λ})^{⊥} = (Ker KΠT_{Λ}^{−1}ΠK^{∗})^{⊥} = (Ker T_{Λ}^{−1/2}ΠK^{∗})^{⊥}
= (Ker ΠK^{∗})^{⊥} = \overline{Ran KΠ} = \overline{KH^{2}(∂Ω)}
= \overline{W_{hol}^{1/2}(Ω)} = L_{hol}^{2}(Ω).

So, indeed, Π_{Λ} = Π. □

Similarly to (10), the bounded (in fact — since Λ is of order < 0 — even compact) operator Λ^{1/2}Π on L^{2}(∂Ω) has polar decomposition

Λ^{1/2}Π = W (ΠΛΠ)^{1/2} = W T_{Λ}^{1/2},

where W is a partial isometry with initial space \overline{Ran ΠΛ^{1/2}} = H^{2}(∂Ω) and final space \overline{Ran Λ^{1/2}Π} = \overline{Λ^{1/2}H^{2}(∂Ω)}; in particular,

(11)  W^{∗}W = I on H^{2}(∂Ω).

The following proposition is analogous to Corollary 9 of [11].

Proposition 2. Let w ∈ C^{∞}(\overline{Ω}) be of the form (8). Then

U^{∗}T_{w}U = W T_{Λ}^{−1/2}T_{Λ_{w}}T_{Λ}^{−1/2}W^{∗} = W T_{Q_{w}}W^{∗},

where Q_{w} is a ΨDO on ∂Ω of order −m with

σ(Q_{w})(x, ξ)|_{Σ} = \frac{(−1)^{m}m!}{|ξ|^{m}} g(x) ∥η_{x}∥^{m}.

Proof. By Proposition 1, ΠK = KT_{Λ}^{−1}ΠΛ = KΠT_{Λ}^{−1}ΠΛ; hence

U^{∗}T_{w}U = Λ^{−1/2}K^{∗}ΠwΠKΛ^{−1/2}
= Λ^{1/2}ΠT_{Λ}^{−1}ΠK^{∗}wKΠT_{Λ}^{−1}ΠΛ^{1/2}
= Λ^{1/2}ΠT_{Λ}^{−1}ΠΛ_{w}ΠT_{Λ}^{−1}ΠΛ^{1/2}
= Λ^{1/2}ΠT_{Λ}^{−1}T_{Λ_{w}}T_{Λ}^{−1}ΠΛ^{1/2}
= W T_{Λ}^{−1/2}T_{Λ_{w}}T_{Λ}^{−1/2}W^{∗},

proving the first equality. The second equality follows from (9) and the properties (P1) and (P4). □

2.3 The Dixmier trace. Recall that if A is a compact operator acting on a Hilbert space then its sequence of singular values {s_{j}(A)}_{j=1}^{∞} is the sequence of eigenvalues of |A| = (A^{∗}A)^{1/2} arranged in nonincreasing order. In particular if A ≫ 0 this will also be the sequence of eigenvalues of A in nonincreasing order. For 0 < p < ∞ we say that A is in the Schatten ideal S_{p} if {s_{j}(A)} ∈ l^{p}(Z_{>0}). If A ≫ 0 is in S_{1}, the trace class, then A has a finite trace and, in fact, tr(A) = ∑_{j} s_{j}(A). If however we only know that

s_{j}(A) = O(j^{−1}) or that

S_{k}(A) := ∑_{j=1}^{k} s_{j}(A) = O(log(1 + k))

then A may have infinite trace. However in this case we may still try to compute its Dixmier trace, Tr_{ω}(A). Informally Tr_{ω}(A) = lim_{k} \frac{1}{log k} S_{k}(A) and this will actually be true in the cases of interest to us. We begin with the definition. Select a continuous positive linear functional ω on l^{∞}(Z_{>0}) and denote its value on a = (a_{1}, a_{2}, ...) by Lim_{ω}(a_{k}). We require of this choice that Lim_{ω}(a_{k}) = lim a_{k} if the latter exists. We require further that ω be scale invariant; a technical requirement that is fundamental for the theory but will not be of further concern to us. Let S^{Dixm} be the class of all compact operators A which satisfy

(12)  (\frac{S_{k}(A)}{log(1 + k)}) ∈ l^{∞}.

With the norm defined as the l^{∞}-norm of the left-hand side of (12), S^{Dixm} becomes a Banach space [15]. For a positive operator A ∈ S^{Dixm}, we define the Dixmier trace of A, Tr_{ω}(A), as Tr_{ω}(A) = Lim_{ω}(\frac{S_{k}(A)}{log(1+k)}). Tr_{ω}(·) is then extended by linearity to all of S^{Dixm}. Although this definition does depend on ω the operators A we consider are measurable, that is, the value of Tr_{ω}(A) is independent of the particular choice of ω. We refer to [9] for details and for discussion of the role of these functionals. It is a result of Connes [8] that if Q is a ΨDO on a compact manifold M of real dimension n and ord(Q) = −n, then Q ∈ S^{Dixm} and

(13)  Tr_{ω}(Q) = \frac{1}{n!(2π)^{n}} ∫_{(T ^{∗}M)_{1}} σ(Q).

(Here (T^{∗}M)_{1} denotes the unit sphere bundle in the cotangent bundle T^{∗}M, and the integral is taken with respect to a measure induced by any Riemannian metric on M; since σ(Q) is homogeneous of degree −n, the value of the integral is independent of the choice of such metric.) In the next section, we will see that for Toeplitz operators T_{Q} on ∂Ω, Ω ⊂ C^{n}, the “right” order for T_{Q} to belong to S^{Dixm} is not − dim_{R} ∂Ω = −(2n − 1), but − dim_{C} Ω = −n.

3. Dixmier trace of generalized Toeplitz operators

Let T be a positive self-adjoint generalized Toeplitz operator on ∂Ω of order 1 with σ(T) > 0. Let 0 < λ_{1} ≤ λ_{2} ≤ λ_{3} ≤ … be the points of its spectrum (counting multiplicities) and let N(λ) denote the number of λ_{j}’s less than λ. It was shown in Theorem 13.1 in [5] that as λ → +∞,

(14)  N(λ) = \frac{vol(Σ_{T})}{(2π)^{n}}λ^{n} + O(λ^{n−1}),

where Σ_{T} is the subset of Σ where σ(T) ≤ 1, and vol(Σ_{T}) is its symplectic volume. Using properties of generalized Toeplitz operators, it is easy to derive from here the formula for the Dixmier trace.

Theorem 3. Let T be a generalized Toeplitz operator on H^{2}(∂Ω) of order −n. Then T ∈ S^{Dixm}, and

Tr_{ω}(T) = \frac{1}{n!(2π)^{n}} ∫_{∂Ω} σ(T)(x, η_{x}) η ∧ (dη)^{n−1}.

In particular, T is measurable.

Proof. As the Dixmier trace is defined first on positive operators and then extended to all of S^{Dixm} by linearity, while T may be split into its real and imaginary parts each of which can be expressed as a difference of two positive generalized Toeplitz operators of the same order, it is enough to prove the assertion when T is positive self-adjoint with σ(T) > 0. Then T is elliptic, and it follows from Seeley’s theorem on complex powers of ΨDO’s and from the property (P2) that T^{−1/n} is also a generalized Toeplitz operator, with symbol σ(T)^{−1/n} and of order 1 (see [10], Proposition 16, for the detailed argument). Thus the eigenvalues λ_{1} ≤ λ_{2} ≤ … of T^{−1/n} satisfy (14). Consequently,

S_{k}(T) = ∑_{j=1}^{k} s_{j}(T) = ∑_{j=1}^{k} λ_{j}^{−n} = ∫_{[λ_{1},λ_{k}]} λ^{−n} dN(λ)

= ∫_{[λ_{1},λ_{k}]} (\frac{c}{N(λ)} + O(N(λ)^{−1− \frac{1}{n}} )) dN(λ)

= ∫_{1}^{k} (\frac{c}{N} + O(N^{−1− \frac{1}{n}})) dN

= c log k + O(1).

Here we have temporarily denoted c := (2π)^{−n} vol(Σ_{T^{−1/n}}). Dividing by log(k + 1) and letting k tend to infinity, it follows that T ∈ S^{Dixm} and

(15)  Tr_{ω}(T) = lim_{k→∞} \frac{S_{k}(T)}{log(k + 1)} = c.

Let us parameterize Σ as (x, tη_{x}) with x ∈ ∂Ω, t > 0. The subset Σ_{T^{−1/n}} is then characterized by

σ(T)(x, tη_{x})^{−1/n} ≤ 1, or t ≤ σ(T)(x, η_{x})^{1/n}.

A routine computation, which we postpone to the next lemma, shows that the symplectic volume on Σ with respect to the above parameterization is given by \frac{t^{n−1}}{(n−1)!} dt ∧ η(x) ∧ (dη(x))^{n−1}. Consequently,

vol(Σ_{T^{−1/n}}) = ∫_{∂Ω} ∫_{0}^{σ(T)(x,η_{x})^{1/n}} \frac{t^{n−1}}{(n − 1)!} dt ∧ η ∧ (dη)^{n−1}

= \frac{1}{n!} ∫_{∂Ω} σ(T)(x, η_{x}) η ∧ (dη)^{n−1}.

Combining this with (15) and the definition of c, the assertion follows. □

Remark 4. Observe that, in analogy with (13), the last integral is independent of the choice of the defining function. Indeed, if r is replaced by gr, with g > 0 on ∂Ω, then η = Im(∂r) is replaced by gη (since ∂(gr) = g∂r on the set where r = 0), and η ∧ (dη)^{n−1} by gη ∧ (g dη + dg ∧ η)^{n−1} = g^{n} η ∧ (dη)^{n−1} (because η ∧ η = 0); as σ(T)(x, ξ) is homogeneous of degree −n in ξ, the integrand remains unchanged. □

Lemma 5. With respect to the parameterization Σ = {(x, tη_{x}) : x ∈ ∂Ω, t > 0}, the symplectic form on Σ is given by

ω = t dη + dt ∧ η = d(tη).

Consequently, the symplectic volume in the (x, t) coordinates is given by

\frac{ω^{n}}{n!} = \frac{t^{n−1}}{(n − 1)!} dt ∧ η ∧ (dη)^{n−1}.

We are supplying a proof of this simple fact below, since we were unable to locate it in the literature (though we expect that it must be at least implicitly contained e.g. somewhere in [5]).

Proof. Recall that if (x_{1}, x_{2}, …, x_{2n−1}) is a real coordinate chart on ∂Ω and (x, ξ) the corresponding local coordinates for a point (x; ξ_{1}dx_{1} + ⋯ + ξ_{2n−1}dx_{2n−1}) in T^{∗}∂Ω, then the form α = ξ_{1}dx_{1} +⋯+ξ_{2n−1}dx_{2n−1} is globally defined and the symplectic form is given by ω = dα = dξ_{1} ∧dx_{1} +⋯+dξ_{2n−1} ∧ dx_{2n−1}. Since exterior differentiation commutes with restriction (or, more precisely, with the pullback j^{∗} under the inclusion map j : Σ → T^{∗}∂Ω), it follows that the symplectic form ω_{Σ} = j^{∗}ω on Σ is given by ω_{Σ} = d(j^{∗}α). As in our case j^{∗}α = tη, the first formula follows. (We will drop the subscript Σ from now on.) The second formula is immediate from the first since η ∧ η = 0 and (dη)^{n} = 0. □

The following corollary is immediate upon combining Theorem 3 and Proposition 2.

Corollary 6. Assume that f ∈ C^{∞}(\overline{Ω}) vanishes at ∂Ω to order n. Then T_{f} belongs to the Dixmier class, is measurable, and

Tr_{ω}(T_{f}) = \frac{1}{n!(4π)^{n}} ∫_{∂Ω} 𝔑^{n}f \frac{η ∧ (dη)^{n−1}}{∥η∥^{n}} ,

where 𝔑 denotes the interior unit normal derivative.

4. Dixmier trace for products of Hankel operators

It is known [5] that the symbol of the commutator of two generalized Toeplitz operators is given by the Poisson bracket (with respect to the symplectic structure of Σ) of their symbols:

σ([T_{P} , T_{Q}]) = \frac{1}{i}\{σ(T_{P}), σ(T_{Q})\}_{Σ}.

We need an analogous formula for the semi-commutator T_{P Q}−T_{P} T_{Q} of two generalized Toeplitz operators. Not surprisingly, it turns out to be given (at least in the cases of interest to us) by an appropriate “half” of the Poisson bracket.

Let us denote by T′′ ⊂ T ∂Ω ⊗ C the anti-holomorphic complex tangent space to ∂Ω, i.e. the elements of T′′_{x}, x ∈ ∂Ω, are the vectors ∑_{j=1}^{n} a_{j} \frac{∂}{∂\overline{z}_{j}}, a_{j} ∈ C, such that ∑_{j} a_{j} \frac{∂r}{∂\overline{z}_{j}} (x) = 0. (This notation follows [6], p. 141.) On the open subset U_{m} of ∂Ω where \frac{∂r}{∂\overline{z}_{m}} ≠ 0 (as m ranges from 1 to n, these subsets cover all of ∂Ω), T′′ is spanned by the n − 1 vector fields

\overline{R}_{j} := \frac{∂}{∂\overline{z}_{j}} − \frac{∂r/∂\overline{z}_{j}}{∂r/∂\overline{z}_{m}} \frac{∂}{∂\overline{z}_{m}} , j ≠ m.

(Thus \overline{R}_{j} depends also on m, although this is not reflected by the notation.) The (similarly defined) holomorphic complex tangent space T′ is, analogously, spanned on U_{m} by the n − 1 vector fields

R_{j} := \frac{∂}{∂z_{j}} − \frac{∂r/∂z_{j}}{∂r/∂z_{m}} \frac{∂}{∂z_{m}} , j ≠ m,

while the whole complex tangent space T ∂Ω ⊗ C is spanned there by the R_{j}, \overline{R}_{j} and

E := ∑_{j=1}^{n} \frac{∂r}{∂z_{j}} \frac{∂}{∂\overline{z}_{j}} − \frac{∂r}{∂\overline{z}_{j}} \frac{∂}{∂z_{j}}

(the “complex normal” direction). The boundary d-bar operator \overline{∂}_{b} : C^{∞}(∂Ω) → C^{∞}(∂Ω, T^{′′∗}) is defined as the restriction

\overline{∂}_{b}f := df|_{T ′′},

or, more precisely, \overline{∂}_{b}f = df̃|_{T ′′} for any smooth extension f̃ of f to a neighbourhood of ∂Ω in C^{n} (the right-hand side is independent of the choice of such extension). On U_{m}, T^{′′∗} admits d\overline{z}_{j}|_{T ′′}, j ≠ m, as a basis and

\overline{∂}_{b}f = ∑_{j} \overline{R}_{j}f d\overline{z}_{j}|_{T ′′}.

Under our parameterization of Σ by (x, t) ∈ ∂Ω × R_{+}, the tangent bundle T Σ is identified with T ∂Ω × R, being spanned at each (x, tη_{x}) ∈ Σ by \overline{R}_{j}, R_{j}, E and the extra vector T := \frac{∂}{∂t}. Recall that the Levi form L′ is the Hermitian form on T′ defined by

L′(X, Y) := ∑_{j,k=1}^{n} \frac{∂^{2}r}{∂z_{j}∂\overline{z}_{k}} X_{j}\overline{Y}_{k} if X = ∑_{j} X_{j} \frac{∂}{∂z_{j}} , Y = ∑_{k} Y_{k} \frac{∂}{∂z_{k}} .

The strong pseudoconvexity of Ω means that L′ is positive definite. Similarly, one has the positive-definite Levi form L′′ on T′′ defined by

L′′(X, Y) := ∑_{j,k=1}^{n} \frac{∂^{2}r}{∂z_{k}∂\overline{z}_{j}} X_{j}\overline{Y}_{k} if X = ∑_{j} X_{j} \frac{∂}{∂\overline{z}_{j}} , Y = ∑_{k} Y_{k} \frac{∂}{∂\overline{z}_{k}} .

In terms of the complex conjugation X ↦ \overline{X} given by \overline{X_{j} \frac{∂}{∂z_{j}}} = \overline{X}_{j} \frac{∂}{∂\overline{z}_{j}}, mapping T′ onto T′′ and vice versa, the two forms are related by

(16)  L′′(X, Y) = L′(\overline{Y} , \overline{X})  ∀X, Y ∈ T′′.

By the usual formalism, L′′ induces a positive definite Hermitian form^{1} on the dual space T^{′′∗} of T′′; we denote it by L. Namely, if L′′ is given by a matrix 𝕃 with respect to some basis {e_{j}}, then L is given by the inverse matrix 𝕃^{−1} with respect to the dual basis {ê_{k}} satisfying ê_{k}(e_{j}) = δ_{jk}. An alternative description is the following. For any α ∈ T^{′′∗}, let Z′′_{α} ∈ T′′ be defined by

L′′(X, Z′′_{α}) = α(X) ∀X ∈ T′′.

(This is possible, and Z′′_{α} is unique, owing to the non-degeneracy of L′′. Note that α ↦ Z′′_{α} is conjugate-linear.) Then

L(α, β) = L′′(Z′′_{β}, Z′′_{α}) = α(Z′′_{β}) = \overline{β(Z′′_{α})}.

Let, in particular, Z′′_{f} := Z′′_{\overline{∂}_{b}f}, so that

L′′(X, Z′′_{f}) = \overline{∂}_{b}f(X) ∀X ∈ T′′,

and denote by Z′_{f} ∈ T′ the similarly defined holomorphic vector field satisfying

L′(Y, Z′_{f}) = ∂_{b}f(Y) ∀Y ∈ T′,

where ∂_{b}f := df|_{T ′}. Set

Z_{f} := i(\overline{Z′′_{f}} − \overline{Z′_{f}}) ∈ T′ + T′′.

These objects are related to the symplectic structure of Σ as follows. Note that

dη = i∂\overline{∂}r = i ∑_{k,l=1}^{n} \frac{∂^{2}r}{∂z_{k}∂\overline{z}_{l}} dz_{k} ∧ d\overline{z}_{l},

hence

dη(X′ + X′′, Y′ + Y′′) = iL′(X′, \overline{Y′′}) − iL′(Y′, \overline{X′′})

for all X′, Y′ ∈ T′ and X′′, Y′′ ∈ T′′. It follows that dη is a non-degenerate skew-symmetric bilinear form on T′ + T′′, and

(17)  dη(X, Z_{f}) = Xf  ∀X ∈ T′ + T′′.

1or, perhaps more appropriately, a positive definite Hermitian bivector

Indeed,

dη(X′ + X′′, Z_{f}) = iL′(X′, \overline{−i\overline{Z′_{f}}}) − iL′(i\overline{Z′′_{f}}, \overline{X′′})
= L′(X′, Z′_{f}) + L′′(X′′, Z′′_{f})
= ∂_{b}f(X′) + \overline{∂}_{b}f(X′′) = df(X′ + X′′).

Let us define E_{T} ∈ T′ + T′′ by

(18)  dη(X, E_{T}) = dη(X, E)  ∀X ∈ T′ + T′′

(again, this is possible and unambiguous by virtue of the non-degeneracy of dη on T′ + T′′), and set

E_{⊥} := \frac{E − E_{T}}{η(E)} = \frac{E − E_{T}}{i∥η∥^{2}} .

The vector field E_{⊥} is usually called the Reeb vector field, and is defined by the conditions η(E_{⊥}) = 1, i_{E_{⊥}}dη = 0.

Proposition 7. Let f, g ∈ C^{∞}(∂Ω), and let F, G be the functions on Σ ≅ ∂Ω × R_{+} given by

F (x, t) = t^{−k}f(x), G(x, t) = t^{−m}g(x).

Then the Poisson bracket of F and G is given by

{F, G}_{Σ} = t^{−k−m−1}(Z_{f}g + mgE_{⊥}f − kfE_{⊥}g).

Proof. Recall that the Hamiltonian vector field H_{F} of F is the pre-dual of dF with respect to the symplectic form ω_{Σ} ≡ ω on Σ, namely

ω(X, H_{F}) = dF (X) = XF, ∀X ∈ T Σ.

Since F = t^{−k}f(x), we have dF = t^{−k}df − kt^{−k−1}fdt, so

(19)  H_{F} = t^{−k}H_{f} − kt^{−k−1}fH_{t}.

We claim that

(20)  H_{t} = E_{⊥},  H_{f} = \frac{1}{t}Z_{f} − E_{⊥}f T.

We check the formula for H_{t}, i.e.

ω(X, H_{t}) = dt(X) ∀X ∈ T Σ.

For X = T,

ω(T, E_{⊥}) = \frac{1}{η(E)}(tdη + dt ∧ η)(T, E − E_{T})

= \frac{1}{η(E)} dt ∧ η(T, E − E_{T}) = \frac{η(E) − η(E_{T})}{η(E)}

= 1 = dt(T),

since η vanishes on T′ + T′′ ∋ E_{T}. Similarly, for X = X′ ∈ T′,

ω(X′, E_{⊥}) = \frac{1}{η(E)} t dη(X′, E_{⊥})

vanishes by the definition (18) of E_{T}, and so does dt(E_{⊥}) since E_{⊥} contains no t-differentiations. Analogously for X = X′′ ∈ T′′. Finally, for X = E we have

ω(E, E_{⊥}) = −\frac{1}{η(E)}ω(E, E_{T}) = −\frac{1}{η(E)} t dη(E, E_{T})

= −\frac{1}{η(E)} t dη(E_{T} , E_{T}) = 0 = dt(E),

where in the third equality we have used (18) for X = E_{T}. Next we check the formula for H_{f}. For X = T, both ω(X, H_{f}) and df(X) are zero. For X ∈ T′ + T′′, we have ω(X, T) = dt ∧ η(X, T) = −η(X) = 0 and the equality follows by (17). Finally for X = E

ω(E, H_{f}) = t dη(E, \frac{1}{t}Z_{f}) + dt ∧ η(E, −E_{⊥}f T)
= dη(E_{T} , Z_{f}) + η(E)E_{⊥}f
= E_{T} f + η(E)E_{⊥}f by (17)
= E_{T} f + (E − E_{T})f,

which indeed coincides with df(E) = Ef. By (20) and (19), we thus get

H_{F} = t^{−k−1}Z_{f} − t^{−k}E_{⊥}f T − kt^{−k−1}fE_{⊥}.

Consequently,

{F, G}_{Σ} = ω(H_{F} , H_{G}) = H_{F} G
= t^{−k−m−1}Z_{f}g + mt^{−k−m−1}gE_{⊥}f − kt^{−k−m−1}fE_{⊥}g,

and the assertion follows. □

Corollary 8. Let f, g ∈ C^{∞}(∂Ω), and denote by f, g also the corresponding functions on Σ ≅ ∂Ω × R_{+} constant on each fiber. Then

\{f, g\}_{Σ} = \frac{1}{t}Z_{f}g = i \frac{L(\overline{∂}_{b}f, \overline{∂}_{b}\overline{g}) − L(\overline{∂}_{b}g, \overline{∂}_{b}\overline{f})}{t}.

Proof. Immediate upon taking m = k = 0 in the last proposition, and observing that

\frac{1}{i}Z_{f}g = \overline{Z′′_{f}}g − \overline{Z′_{f}}g = dg(\overline{Z′′_{f}}) − dg(\overline{Z′_{f}})
= ∂_{b}g(\overline{Z′′_{f}}) − \overline{∂}_{b}g(\overline{Z′_{f}}) = \overline{\overline{∂}_{b}\overline{g}(Z′′_{f})} − \overline{∂}_{b}g(\overline{Z′_{f}})
= \overline{L(\overline{∂}_{b}\overline{g}, \overline{∂}_{b}f)} − L(\overline{∂}_{b}g, \overline{∂}_{b}\overline{f}),

since \overline{Z′_{f}} = Z′′ ; \overline{f} by virtue of (16). □

We are now ready to state the main result of this section and, in some sense, of this paper.

Theorem 9. Let U, W have the same meaning as in Proposition 2. Then for f, g ∈ C^{∞}(\overline{Ω}),

U^{∗}(T_{f\overline{g}} − T\overline{g}T_{f})U = W T_{Q}W^{∗},

where T_{Q} is a generalized Toeplitz operator on ∂Ω of order −1 with principal symbol

(21)  σ(T_{Q})(x, tη_{x}) = \frac{1}{t} L(\overline{∂}_{b}f, \overline{∂}_{b}g)(x).

Proof. By Proposition 2,

U^{∗}(T_{f\overline{g}} − T\overline{g}T_{f})U = W (T_{Q_{f\overline{g}}} − T_{Q\overline{g}}T_{Q_{f}})W^{∗},

where T_{Q_{f}} = T_{Λ}^{−1/2}T_{Λ_{f}} T_{Λ}^{−1/2} is a generalized Toeplitz operator of order 0 with symbol σ(T_{Q_{f}})(x, ξ) = f(x). By (P1) and (P4), the expression T_{Q_{f\overline{g}}} − T_{Q\overline{g}}T_{Q_{f}} is thus a generalized Toeplitz operator T_{Q} of order 0 with symbol σ(T_{Q}) = σ(T_{Q_{f\overline{g}}}) − σ(T_{Q\overline{g}})σ(T_{Q_{f}}) = f\overline{g} − \overline{g}f = 0; thus by (P6), it is indeed, in fact, a generalized Toeplitz operator of order −1. It remains to show that its symbol, which we denote by ρ(f, g), is given by (21). By the general theory, ρ(f, g) is given by a local expression, i.e. one involving only finitely many derivatives of f and g at the given point, and linear in f and \overline{g}. (Indeed, the proof of Proposition 2.5 in [5] shows that the construction, for a given ΨDO Q, of the ΨDO P from property (P2), i.e. such that T_{Q} = T_{P} and [P, Π] = 0, is completely local in nature, so the total symbol of the P corresponding to Q = Λ_{f} is given by local expressions in terms of the total symbol of Λ_{f}, hence, by local expressions in terms of f; the claim thus follows from the product formula for the symbol of ΨDOs.) It is therefore enough to show that

(22)  ρ(f, g) = \frac{1}{t}L(\overline{∂}_{b}f, \overline{∂}_{b}g)

for functions f, g of the form u\overline{v}, with u, v holomorphic on \overline{Ω}. ^{2} Next, if u and v are holomorphic on \overline{Ω}, then T\overline{v}T_{f} = T_{\overline{v}f} and T_{f}T_{u} = T_{fu} for any f; consequently, using Proposition 2 and (11),

U^{∗}(T_{uf\overline{vg}} − T\overline{vg}T_{uf})U = U^{∗}T\overline{v}(T_{f\overline{g}} − T\overline{g}T_{f})T_{u}U
= U^{∗}T\overline{v}UU^{∗}(T_{f\overline{g}} − T\overline{g}T_{f})UU^{∗}T_{u}U
= W T_{Q\overline{v}}W^{∗}W (T_{Q_{f\overline{g}}} − T_{Q\overline{g}}T_{Q_{f}})W^{∗}W T_{Q_{u}}W^{∗}
= W T_{Q\overline{v}}(T_{Q_{f\overline{g}}} − T_{Q\overline{g}}T_{Q_{f}})T_{Q_{u}}W^{∗}.

By (P4) we see that

ρ(uf, vg) = u ρ(f, g) \overline{v}.

Since also

L(\overline{∂}_{b}uf, \overline{∂}_{b}vg) = u L(\overline{∂}_{b}f, \overline{∂}_{b}g) \overline{v}

(because \overline{∂}_{b}(uf) = u \overline{∂}_{b}f for holomorphic u), it in fact suffices to prove (22) when f, g are both conjugate-holomorphic, i.e. \overline{∂}_{b}\overline{f} = \overline{∂}_{b}\overline{g} = 0. However, in that case T_{f\overline{g}} = T_{f}T\overline{g}, so, using again Proposition 2 and (11),

U^{∗}(T_{f\overline{g}} − T\overline{g}T_{f})U = U^{∗}[T_{f}, T\overline{g}]U = [U^{∗}T_{f}U, U^{∗}T\overline{g}U]
= [W T_{Q_{f}} W^{∗}, W T_{Q\overline{g}}W^{∗}] = W [T_{Q_{f}} , T_{Q\overline{g}}]W^{∗},

implying that

ρ(f, g) = σ([T_{Q_{f}} , T_{Q\overline{g}}])

= \frac{1}{i}\{σ(T_{Q_{f}}), σ(T_{Q\overline{g}})\}_{Σ}

= \frac{1}{i}\{f, \overline{g}\}_{Σ}

= \frac{L(\overline{∂}_{b}f, \overline{∂}_{b}g) − L(\overline{∂}_{b}\overline{g}, \overline{∂}_{b}\overline{f})}{t} by Corollary 8

= \frac{1}{t} L(\overline{∂}_{b}f, \overline{∂}_{b}g),

completing the proof. □

Remark 10. It seems much more difficult to obtain a formula for the symbol of T_{P Q} − T_{P} T_{Q} for general ΨDOs P and Q. □ We are now ready to prove the main result on Dixmier traces.

2In fact, even holomorphic polynomials u, v would do.

Theorem 11. Let f_{1}, g_{1}, …, f_{n}, g_{n} ∈ C^{∞}(\overline{Ω}). Then the operator

H = H_{g_{1}}^{∗}H_{f_{1}}H_{g_{2}}^{∗}H_{f_{2}} … H_{g_{n}}^{∗}H_{f_{n}}

on L_{hol}^{2}(Ω) belongs to the Dixmier class, and

(23) Tr_{ω}(H) = \frac{1}{n!(2π)^{n}} ∫_{∂Ω} L(\overline{∂}_{b}f_{1}, \overline{∂}_{b}g_{1}) … L(\overline{∂}_{b}f_{n}, \overline{∂}_{b}g_{n}) η ∧ (dη)^{n−1}.

In particular, H is measurable. Proof. Denote, for brevity, V_{j} := T_{Λ}^{−1/2}(T_{Λ_{f_{j} \overline{g}_{j}}} −T_{Λ_{\overline{g}_{j}}} T_{Λ}^{−1}T_{Λ_{f_{j}}})T_{Λ}^{−1/2}. We have seen in the last theorem that H_{g_{j}}^{∗}H_{f_{j}} = T_{\overline{g}_{j}f_{j}} − T_{\overline{g}_{j}}T_{f_{j}} satisfies

U^{∗}H_{g_{j}}^{∗}H_{f_{j}}U = W V_{j}W^{∗}

and that V_{j} is a generalized Toeplitz operator of order −1 with symbol given by σ(V_{j})(x, tη_{x}) = \frac{1}{t}L(\overline{∂}_{b}f_{j}, \overline{∂}_{b}g_{j}). By iteration and using (11), it follows that

U^{∗}H_{g_{1}}^{∗}H_{f_{1}}H_{g_{2}}^{∗}H_{f_{2}} … H_{g_{n}}^{∗}H_{f_{n}}U = W V_{1}V_{2} … V_{n}W^{∗} = W V W^{∗},

where V := V_{1}V_{2} … V_{n} is a generalized Toeplitz operator of order −n with symbol σ(V)(x, tη_{x}) = t^{−n} ∏_{j=1}^{n} L(\overline{∂}_{b}f_{j}, \overline{∂}_{b}g_{j}). An application of Theorem 3 completes the proof. □ Corollary 12. Let f be holomorphic on Ω and C^{∞} on \overline{Ω}. Then |H\overline{f}|^{2n} is in the Dixmier class, measurable, and

Tr_{ω}(|H\overline{f}|^{2n}) = \frac{1}{n!(2π)^{n}} ∫_{∂Ω} L(\overline{∂}_{b}\overline{f}, \overline{∂}_{b}\overline{f})^{n} η ∧ (dη)^{n−1}.

By standard matrix algebra, one has^{3}

L(\overline{∂}_{b}f, \overline{∂}_{b}g) = [ \begin{matrix}\overline{∂} & g̃ \\ 0 \end{matrix} ]^{∗} [ \begin{matrix}∂\overline{∂}r & \overline{∂}r \\ ∂r & 0 \end{matrix} ]^{−1} [ \begin{matrix}\overline{∂} & f̃ \\ 0 \end{matrix}] ,

⟨A^{−1}P v, P w⟩ = [ \begin{matrix}w \\ 0 \end{matrix} ]^{∗} [ \begin{matrix}X & u \\ u^{∗} & 0 \end{matrix} ]^{−1} [ \begin{matrix}v \\ 0 \end{matrix} ] .

[ \begin{matrix}A & b \\ c^{∗} & d \end{matrix} ], with b, c ∈ C^{n}, d ∈ C. Then

[ \begin{matrix}X & u \\ u^{∗} & 0 \end{matrix} ] = [ \begin{matrix}A & b & 0 \\ c^{∗} & d & 1 \\ 0 & 1 & 0 \end{matrix} ] = [ \begin{matrix}1 & 0 & b \\ 0 & 1 & d \\ 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}A & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ] [ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ c^{∗} & 0 & 1 \end{matrix} ] ,

3Let, quite generally, X be an operator on C^{n}, u ∈ C^{n}, and denote by A the compression of X to the orthogonal complement u^{⊥} of u, i.e. A = P X|_{Ran P} where P : C^{n} → u^{⊥} is the orthogonal projection. Assume that A is invertible. Then the block matrix [ \begin{matrix}X & u \\ u^{∗} & 0 \end{matrix} ] ∈ C^{(n+1)×(n+1)} is invertible, and for any v, w ∈ C^{n}, Indeed, switching to a convenient basis we may assume that u = [0, …, 0, 1]^{t}. Write X =

where f̃, g̃ are any smooth extensions of f, g ∈ C^{∞}(∂Ω) to a neighbourhood of ∂Ω. In particular, for Ω = B^{d}, the unit ball, with the defining function r(z) = |z|^{2} − 1, we obtain

(24)  L(\overline{∂}_{b}f, \overline{∂}_{b}g) = ∑_{j=1}^{n} \frac{∂f̃}{∂\overline{z}_{j}} \overline{\frac{∂g̃}{∂\overline{z}_{j}}} − \overline{R}f̃\overline{\overline{R}g̃},

where \overline{R} := ∑_{j=1}^{n} \overline{z}_{j} \frac{∂}{∂\overline{z}_{j}} is the anti-holomorphic radial derivative. One also easily checks that η ∧ (dη)^{n−1} = (2π)^{n} dσ where dσ is the normalized surface measure on ∂B^{n}. The last two theorems thus recover, as they should, the results from [12] (Theorem 4.4 — which is the formula (3) above — and Corollary 4.5 there). Note also that for n = 1, the expression (24) vanishes; in this case U^{∗}H_{g}^{∗}H_{f}U is thus in fact of order not −1 but −2 (so that |H_{g}^{∗}H_{f}|^{1/2} is in the Dixmier class rather than H_{g}^{∗}H_{f}), and some additional work is needed to compute the symbol (and, from it, the Dixmier trace); see [13]. Finally, we pause to remark that the value of the integral (23) remains unchanged under biholomorphic mappings, as well as changes of the defining function. Indeed, if r is replaced by gr, with g > 0 on ∂Ω, then T′′ and \overline{∂}_{b} are unaffected, while the Levi form L on T′′ gets multiplied by g. Hence its dual L gets multiplied by g^{−1}, and as η ∧ (dη)^{n−1} transforms into g^{n}η ∧ (dη)^{n−1} (cf. Remark 4), the integrand in (23) does not change. Similarly, if ϕ : Ω_{1} → Ω_{2} is a biholomorphic map and r is a defining function for Ω_{2}, one can choose ϕ∘r as the defining function for Ω_{1}; then it is immediate, in turn, that ϕ sends T′ into T′ and T′′ into T′′, and that it transforms each of η, η ∧ (dη)^{n−1}, \overline{∂}_{b}, ∂_{b}, L and L into the corresponding object on the other domain. Hence L(\overline{∂}_{b}f, \overline{∂}_{b}g) = (ϕ_{∗}L)(ϕ^{∗}\overline{∂}_{b}f, ϕ^{∗}\overline{∂}_{b}g) = L(\overline{∂}_{b}(f ∘ ϕ), \overline{∂}_{b}(g ∘ ϕ)) and, finally, ϕ^{∗}(∏_{j} L(\overline{∂}_{b}f_{j}, \overline{∂}_{b}g_{j}) η ∧ (dη)^{n−1}) = ∏_{j} L(\overline{∂}_{b}(f_{j} ∘ ϕ), \overline{∂}_{b}(g_{j} ∘ ϕ)) η ∧ (dη)^{n−1}, proving the claim. Note that e.g. even in the formula (3) for Ω = B^{n}, the invariance of the value of the integral under biholomorphic self-maps of the ball is definitely not apparent.

5. Concluding remarks

5.1 Manifolds. The results in this paper should all be generalizable to arbitrary strictly pseudoconvex manifolds.

[ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ c^{∗} & 0 & 1 \end{matrix} ] [ \begin{matrix}X & u \\ u^{∗} & 0 \end{matrix} ]^{−1} [ \begin{matrix}1 & 0 & b \\ 0 & 1 & d \\ 0 & 0 & 1 \end{matrix} ] = [ \begin{matrix}A & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ]^{−1} = [ \begin{matrix}A^{−1} & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ] ,

whence and the claim follows.

The formula for L(\overline{∂}_{b}f, \overline{∂}_{b}g) is obtained upon taking X = 𝕃, u = \overline{∂}r, v = \overline{∂}f and w = \overline{∂}g.

5.2 Residual trace. Comparing Theorem 3 with the results of Guillemin [16] [17], we see that the Dixmier trace for generalized Toeplitz operators coincides (possibly up to different normalization) with the residual trace of Wodzicki, Guillemin, Manin and Adler. This is completely analogous to the situation for ΨDOs, cf. Connes [8], Theorem 1.

5.3 Nonsmooth symbols. For the unit disc D in C, the analogue of Corollary 12 is

Tr_{ω}(|H\overline{f}|) = ∫_{∂D} |f′(e^{iθ})| \frac{dθ}{2π}

for f holomorphic on D and smooth on \overline{D}; see [13]. It was shown in [13] that the smoothness assumption can be dispensed with: namely, for f holomorphic on D, |H\overline{f}| ∈ S^{Dixm} if and only if f′ belongs to the Hardy 1-space H^{1}(∂D), and then

Tr_{ω}(|H\overline{f}|) = ∥f′∥_{H^{1}}.

We expect that the same situation prevails also for general domains Ω of the kind studied in this paper, in the following sense. For f holomorphic on Ω, denote

L_{f}(z) := [ \begin{matrix}∂f \\ 0 \end{matrix} ]^{∗} [ \begin{matrix}∂\overline{∂}r & ∂r \\ \overline{∂}r & 0 \end{matrix} ]^{−1} [ \begin{matrix}∂f \\ 0 \end{matrix} ] (z).

This is a smooth function defined in some neighbourhood of ∂Ω in \overline{Ω}, whose boundary values coincide with L(\overline{∂}_{b}\overline{f}, \overline{∂}_{b}\overline{f}) if f is smooth up to the boundary.

Conjecture. Let f be holomorphic on Ω. Then |H\overline{f}|^{2n} ∈ S^{Dixm} if and only if

∥f∥_{L} := lim sup_{ϵ↘0} (\frac{1}{n!(2π)^{n}} ∫_{r=−ϵ} |L_{f}|^{n} |η ∧ (dη)^{n−1}|)^{1/2n}

is finite, and then

Tr_{ω}(|H\overline{f}|^{2n}) = ∥f∥_{L}^{2n}.

The proof for the disc went by showing first that ∥f′∥_{H_{1}} actually dominates the S^{Dixm} norm of |H\overline{f}|; the result then followed from the one for f ∈ C^{∞}(\overline{D}) by a straightforward approximation argument. This approach might also work for general domains Ω (with ∥f∥_{L} and |H\overline{f}|^{2n} replacing ∥f′∥_{H^{1}} and |H\overline{f}|), but the techniques for doing so (estimates for the oscillation of f′ on Carleson-type rectangles, etc.) are outside the scope of this paper.

5.4 Higher type. The generalized Toeplitz operators on H^{2}(∂Ω) of higher type m, m = 1, 2, …, are defined as T_{Q}^{(m)} = Π_{m}QΠ_{m}, where Q is a ΨDO on ∂Ω as before and Π_{m} is the orthogonal projection in L^{2}(∂Ω) onto the subspace H_{(m)}^{2}(∂Ω) of functions annihilated by the m-th symmetric power of \overline{∂}_{b}; in other words,

H_{(m)}^{2}(∂Ω) = closure of \{f ∈ C^{∞}(∂Ω) : \overline{R}_{j_{1}} \overline{R}_{j_{2}} … \overline{R}_{j_{m}}f = 0 ∀j_{1}, j_{2}, …, j_{m}\}.

For m = 1, this recovers the ordinary Szegö projector Π and the generalized Toeplitz operators discussed so far. As shown in §15.3 of [5], the projectors Π_{m} have almost the same microlocal description as Π, so it is conceivable that our results could also be extended to these higher type Toeplitz operators.

5.5 Weighted spaces. Our methods also work, with only minimal modifications, for L_{hol}^{2}(Ω) replaced by the weighted Bergman spaces L_{hol}^{2}(Ω, |r|^{ν}) ⊂ L^{2}(Ω, |r|^{ν}), with any ν > −1. The formulas in Theorems 9 and 11, and in Corollary 12, remain unchanged (i.e. they do not depend on ν). Finally, it is immediate from Theorem 3, the property (P4) and the proof of Theorem 9 that the formulas in Theorem 11 and Corollary 12 also remain valid for Hankel operators on the Hardy space H^{2}(∂Ω).

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Miroslav Engliš

Mathematics Institute

Silesian University at Opava Na Rybníčku 1 74601 Opava Czech Republic and Mathematics Institute Žitná 25 11567 Prague 1 Czech Republic englis@math.cas.cz Genkai Zhang Department of Mathematical Sciences Chalmers TH and Göteborg University SE-412 96 Göteborg Sweden genkai@chalmers.se