Toeplitz Operators on Higher Cauchy-Riemann Spaces

Miroslav Engliš, Genkai Zhang

Received: January 13, 2017 Revised: June 7, 2017

Communicated by Patrick Delorme

Abstract. We develop a theory of Toeplitz, and to some extent Hankel, operators on the kernels of powers of the boundary d-bar operator, suggested by Boutet de Monvel and Guillemin, and on their analogues, somewhat better from the point of view of complex analysis, defined using instead the covariant Cauchy-Riemann operators of Peetre and the second author. For the former, Dixmier class membership of these Hankel operators is also discussed. Our main tool are the generalized Toeplitz operators (with pseudodifferential symbols), in particular there appears naturally the problem of finding parametrices of matrices of such operators in situations when the principal symbol fails to be elliptic.

2010 Mathematics Subject Classification: Primary 47B35; Secondary 32W25, 46E20, 32W10 Keywords and Phrases: Toeplitz operator, Hankel operator, Cauchy-Riemann operators

1. Introduction

Let Ω be a bounded domain in C^{d} and L_{hol}^{2}(Ω) the Bergman space of all holomorphic functions in L^{2}(Ω). For ϕ ∈ L^{∞}(Ω), the Toeplitz operator T_{ϕ} with symbol ϕ is the operator on L_{hol}^{2}(Ω) defined by

T_{ϕ}f = Π(ϕf), f ∈ L_{hol}^{2}(Ω),

where Π : L^{2}(Ω) → L_{hol}^{2}(Ω) is the orthogonal projection (the Bergman projection). Similarly, if Ω has smooth boundary ∂Ω, one has the Hardy space H^{2}(∂Ω) consisting of all functions in L^{2}(∂Ω) (with respect to the surface measure on ∂Ω) whose Poisson extension into Ω is holomorphic, and for ϕ ∈ L^{∞}(∂Ω) the Toeplitz operator T_{ϕ} on H^{2}(∂Ω) is defined by

Research supported by GA ČR grant no. 16-25995S, by RVO funding for IČ 67985840 and by Swedish Science Council (VR).

T_{ϕ}u = Π(ϕu), u ∈ H^{2}(∂Ω),

where Π : L^{2}(∂Ω) → H^{2}(∂Ω) is the orthogonal projection (the Szegö projection). There are also Hankel operators H_{ϕ} : L_{hol}^{2}(Ω) → L^{2}(Ω) ⊖ L_{hol}^{2}(Ω) and H_{ϕ} : H^{2}(∂Ω) → L^{2}(Ω) ⊖ H^{2}(∂Ω) defined as

H_{ϕ}f = (I − Π)(ϕf), H_{ϕ}u = (I − Π)(ϕu), respectively.

Toeplitz and Hankel operators, and their various generalizations, have been extensively studied for the last three decades, and have turned out to play important role in many subjects ranging from operator theory and complex function theory to geometry and mathematical physics, see e.g. [14], [16], [19],

[18] and the references therein for a sample. The spaces L_{hol}^{2}(Ω) on which T_{ϕ} and H_{ϕ} act can alternatively be characterized as the kernel of the operator \overline{∂} in L^{2}(Ω), where, as usual, \overline{∂} denotes the operator assigning to a function f on Ω the (0, 1)-form

\overline{∂}f := ∑_{j=1}^{d} \frac{∂f}{∂\overline{z}_{j}} d\overline{z}_{j}.

For d > 1, one has a similar characterization of the Hardy space H^{2}(∂Ω) as the kernel of the operator \overline{∂}_{b} in L^{2}(∂Ω), where for a function u on ∂Ω, \overline{∂}_{b}u is the restriction of du to the antiholomorphic complex tangent space T′′(∂Ω), consisting of all vectors X on ∂Ω of the form

X = ∑_{j=1}^{d} X_{j} \frac{∂}{∂\overline{z}_{j}} , X_{j} ∈ C,

which are tangent to ∂Ω. Fixing a positively-signed defining function ρ for Ω, so that ρ > 0 on Ω and ρ = 0 < |∇ρ| on ∂Ω, the last condition just means that Xρ = 0, and T′′(∂Ω) is spanned by the (linearly dependent) vector fields

(1)  L_{jk} := \frac{∂ρ}{∂\overline{z}_{j}} \frac{∂}{∂\overline{z}_{k}} − \frac{∂ρ}{∂\overline{z}_{k}} \frac{∂}{∂\overline{z}_{j}} ,  1 ≤ j < k ≤ d;

so \overline{∂}_{b}u = 0 means that L_{jk}u = 0 for all j, k. A generalization of the Hardy space due to Boutet de Monvel and Guillemin [3, §15.3] are the subspaces B_{m} in L^{2}(∂Ω), m = 1, 2, …, of all functions u annihilated by \overline{∂}_{b}^{m}, in the sense that

L_{j_{1}k_{1}}L_{j_{2}k_{2}} … L_{j_{m}k_{m}}u = 0 for all j_{1}, …, j_{m}, k_{1}, …, k_{m}.

Clearly B_{1} ⊂ B_{2} ⊂ ⋯ ⊂ L^{2}(∂Ω) and B_{1} = H^{2}(∂Ω). It was shown in [3] that microlocally, the associated “higher Szegö projectors” Π_{m} : L^{2}(∂Ω) → B_{m} are of the same type as Π, and, hence, in this sense, so are the corresponding Toeplitz operators T_{ϕ}^{B_{m}} : u ↦ Π_{m}(ϕu) on B_{m}. The drawback of the spaces B_{m}, however, is that in spite of the microlocal equivalence of the Szegö projectors just mentioned, they fail to be invariant under biholomorphic maps. For instance, in the simplest possible situation when Ω = B^{2}, the unit ball of C^{2}, one checks easily that the function \overline{z}_{2} belongs to B_{2} = Ker L_{12}^{2}, but if ϕ_{a} is the automorphism of B^{2} interchanging the origin with a point a ∈ B^{2}, then \overline{z}_{2} ∘ ϕ_{a} ∉ B_{2} if a ≠ 0. (See Section 3 below for the details.) The aim of this paper is, firstly, to show that in spite of not being biholomorphically invariant, the spaces B_{m}, m > 1, have Toeplitz and Hankel operators which behave very similarly as in the classical case m = 1; and secondly, to propose a different generalization, the so-called higher Cauchy-Riemann spaces C_{m}, m = 1, 2, …, which are well-behaved under biholomorphic maps, and study the associated Toeplitz operators. For the first part, we work out only the case of Ω = B^{2}, the unit ball in C^{2}. Our main result is the following.

Theorem 1. The Toeplitz operator T_{f}^{B_{2}} on B_{2}(∂B^{2}) is unitarily equivalent to the operator

[ \begin{matrix}T_{f} & 0 \\ 0 & T_{f} \end{matrix} ] + lower order term

on the direct sum H^{2}(∂B^{2}) ⊕ H^{2}(∂B^{2}).

Here the “lower order term” means a 2 × 2 matrix of generalized Toeplitz operators of order at most −\frac{1}{2}; see again Section 3 below for more details. As a corollary to the theorem, we also get a similar result for the product H∗ ; \overline{f}H_{g} of two Hankel operators on B_{2}(∂B^{2}), and a formula for the Dixmier trace of (H∗ ; \overline{f}H_{g})^{2} that can be compared to the one for ordinary Hankel operators H∗ ; \overline{f}H_{g} from [11]. Concerning the second part, consider, quite generally, a Kähler metric g_{j\overline{k}}

on Ω, and let g^{\overline{l}j} be the inverse matrix to g_{j\overline{k}} (so the Kähler form is given

by g^{\overline{l}k}d\overline{z}_{l} ∧ dz_{k}). The Cauchy-Riemann operator, introduced by Peetre (cf. [8] and [15]), is the map from functions into holomorphic vector fields on Ω given by

\overline{D}f := g^{\overline{l}k}\overline{∂}_{l}f,

where we have started to employ the Einstein summation convention of summing automatically over any index that occurs twice, and also to write for brevity \overline{∂}_{l} = ∂/∂\overline{z}_{l}. One can iterate this construction and set, for m = 1, 2, …,

\overline{D}^{m}f := g^{\overline{l}_{m}k_{m}} \overline{∂}_{l_{m}}g^{\overline{l}_{m−1}k_{m−1}} \overline{∂}_{l_{m−1}} … g^{\overline{l}_{1}k_{1}} \overline{∂}_{l_{1}}f.

It turns out that (\overline{D}^{m}f)^{k_{m}...k_{1}} is symmetric in the indices k_{m}, …, k_{1} [15], and in fact coincides with the contravariant derivative f^{/k_{m}...k_{1}} with respect to the Hermitian connection [8]. The m-th Cauchy-Riemann space C_{m} is, by definition, the kernel of \overline{D}^{m}:

C_{m} := \{f : \overline{D}^{m}f = 0 on Ω\}.

Clearly C_{1} comprises precisely of holomorphic functions, and is also independent of the metric g_{j\overline{k}}. For m > 1, C_{m} depends on g_{j\overline{k}} (although this fact is not reflected by the notation). Now there are various holomorphically invariant Kähler metrics associated to a given bounded strictly pseudoconvex domain with smooth boundary, such as the Bergman metric, the Poincare (Kähler-Einstein) metric, the metric coming from the invariant Szegö kernel, and so forth. Taking any of these for g_{j\overline{k}}, by the very nature of their construction the spaces C_{m}, unlike B_{m}, will be invariant under biholomorphisms. Choosing a (positive smooth) weight w on Ω, let C_{m,w} := C_{m} ∩ L^{2}(Ω, w) (since differential operators are closed, this is a closed subspace of L^{2}(Ω, w)), and let T_{ϕ}^{(m,w)} : f ↦ Π^{(m,w)}(ϕf), where Π^{(m,w)} : L^{2}(Ω, w) → C_{m,w} is the orthogonal projection, be the associated Toeplitz operator on C_{m,w} with symbol ϕ ∈ L^{∞}(Ω). (Thus T_{ϕ}^{(m,w)} again depends also on the choice of the metric g_{j\overline{k}}, although this is not reflected by the notation.) We will actually assume that Ω is bounded, strictly pseudoconvex and with smooth boundary, that ϕ ∈ C^{∞}(\overline{Ω}) is smooth on the closure \overline{Ω} of Ω, that the weight w is of the form

w = ρ^{ν}, ν ∈ R,

where ρ is a (fixed) positively-signed defining function for Ω and ν is large enough; and that g_{j\overline{k}} is given by a Kähler potential Ψ,

g_{j\overline{k}} = ∂_{j}\overline{∂}_{k}Ψ,

where Ψ is of the form

Ψ ≈ ∑_{j=0}^{∞}(ρ^{M} log ρ)^{j}η_{j}, η_{j} ∈ C^{∞}(\overline{Ω}),

with an integer M ≥ 2; see Section 4 for the details. Note that all the metrics g_{j\overline{k}} mentioned in the penultimate paragraph are of this kind.

Theorem 2. Let Ω, w and g_{j\overline{k}} be as stated above. Assume that ν > 1 and that −ρ is strictly plurisubharmonic near ∂Ω and |∂ρ| = 1 on ∂Ω. Then the Toeplitz operator T_{ϕ}^{(2,w)} is unitarily equivalent to the operator

(2)  T_{ϕ}^{(2,w)} ≅ ⨁_{j=0}^{d} T_{ϕ|∂Ω} + lower order term

on the direct sum ⊕_{j=0}^{d}H^{2}(∂Ω) of (d + 1) copies of H^{2}(∂Ω).

Here the “lower order term” means a (d + 1) × (d + 1) matrix of sums of generalized Toeplitz operators of orders and −\frac{1}{2} and −1, see Section 4 for the details. The hypothesis ν > 1 is needed to have C_{2,w} nontrivial: for ν ≤ 1, C_{2,w} contains just the function constant zero. For general m ≥ 2, (2) holds too, except that the hypothesis on ν becomes ν > 2m − 3 and instead of d + 1 copies of T_{ϕ|∂Ω}, one gets \binom{d+m−1}{m−1} copies (and matrices of the corresponding size). Our main tool are Toeplitz operators with pseudodifferential symbols, or generalized Toeplitz operators, on H^{2}(∂Ω) whose theory was worked out by Boutet de Monvel and Guillemin in [3]. The proof of Theorem 1 follows what is now already a more or less standard Ansatz (cf. e.g. [2] and [11]) once we identify B_{2} explicitly by parameterizing it by two copies of H^{2}(∂Ω). To some extent this is also true for Theorem 2, however the main difficulty there is that we are confronted with inverting a matrix of generalized Toeplitz operators whose principal symbol is not invertible (i.e. the matrix is not elliptic). The proof of Theorem 1 is presented in Section 3, after reviewing various prerequisites in Section 2. The proof of Theorem 2 occupies Section 4, while the simplest case of the ball is worked out in detail in Section 5. As already introduced above, we write simply ∂_{j}, \overline{∂}_{j} for ∂/∂z_{j} and ∂/∂\overline{z}_{j}, respectively. Throughout the paper, abusing the notation slightly, we will also denote the restriction ϕ|_{∂Ω} of a function ϕ ∈ C^{∞}(\overline{Ω}) to ∂Ω just again by ϕ.

2. Background

2.1 Pseudodifferential operators. Throughout the rest of this paper, Ω will be a bounded strictly pseudoconvex domain in C^{d}, d > 1, with smooth (= C^{∞}) boundary, and ρ a positively signed defining function for Ω, i.e. ρ ∈ C^{∞}(\overline{Ω}), ρ > 0 on Ω, and ρ = 0 < |∇ρ| on ∂Ω. Denote by η the restriction to ∂Ω of the 1-form Im(−∂ρ) = (\overline{∂}ρ − ∂ρ)/2i. The strict pseudoconvexity of Ω guarantees that the half-line bundle

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

is a symplectic submanifold of the cotangent bundle T^{∗}(∂Ω). By a classical (or polyhomogeneous) pseudodifferential operator (ψdo for short) P on ∂Ω of order m we will mean a ψdo whose total symbol in any local coordinate chart has an asymptotic expansion p(x, ξ) ∼ ∑_{j=0}^{∞} p_{m−j}(x, ξ), where p_{m−j} is C^{∞} in x, ξ and positively homogeneous of degree m − j in ξ for |ξ| > 1. Here m can be any real number, and “∼” means that the difference p − ∑_{j=0}^{k−1} p_{m−j} should belong to the Hörmander class S^{m−k}, for each k = 0, 1, 2, . . .. The function p_{m}(x, ξ) is called the (leading or principal) symbol of P, denoted σ_{m}(P) (or just σ(P) if the order m is clear from the context), and the set of all ψdo’s of order m will be denoted by Ψ^{m}. Operators in ⋂_{m∈R} Ψ^{m} are the smoothing operators, i.e. those with C^{∞} Schwartz kernel; and we will write P ∼ Q if P − Q is smoothing.

Unless explicitly stated otherwise, all ψdo’s henceforth will be classical. In Section 4, we will also need ψdo’s with symbols whose degrees of homogeneity go down by \frac{1}{2} instead of 1; for the purposes of this paper, we will call these demiclassical ψdo’s. In other words, a demi-classical ψdo of order m is the sum of a classical ψdo of order m and a classical ψdo of order m − \frac{1}{2}.

2.2 Generalized Toeplitz operators. For Q ∈ Ψ^{m} the generalized Toeplitz operator (or gTo for short) T_{Q} is defined as

T_{Q} = ΠQ|_{H^{2}(∂Ω)}.

Alternatively, one can view T_{Q} as the operator

T_{Q} = ΠQΠ

on all of L^{2}(∂Ω). In both cases, T_{Q} is a densely defined operator (its domain contains the Sobolev space W^{m}(∂Ω)), and extends to a continuous map from the Sobolev space W^{s}(∂Ω) into W_{hol}^{s−m}(∂Ω), for any s ∈ R. Generalized Toeplitz operators are known to enjoy the following properties.

(P1) They form an algebra, i.e. T_{P} T_{Q} = T_{R} for some ψdo R.

(P2) In fact, for any T_{Q} there exists a ψdo P of the same order as Q such

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

on the half-line bundle Σ. One can thus define unambiguously the order of T_{Q} as ord(T_{Q}) := inf{ord(P) : T_{P} = T_{Q}}, and the (principal) symbol σ(T_{Q}) := σ(Q)|_{Σ} if ord(Q) = ord(T_{Q}). (The symbol is undefined if ord(T_{Q}) = −∞.)

(P4) ord(T_{P} T_{Q}) = ord(T_{P}) + ord(T_{Q}), σ(T_{P} T_{Q}) = σ(T_{P})σ(T_{Q}), and σ([T_{P} , T_{Q}]) = \frac{1}{i}\{σ(T_{P}), σ(T_{Q})\}_{Σ} where \{·, ·\}_{Σ} denotes the Poisson bracket on Σ.

(P5) If ord(T_{Q}) ≤ 0, then T_{Q} is bounded on L^{2}(∂Ω); if ord(T_{Q}) < 0, it is

even compact. (P6) If Q ∈ Ψ^{m} and σ_{m}(Q)|_{Σ} = 0, then there exists P ∈ Ψ^{m−1} with T_{P} =

T_{Q}. If T_{Q} ∼ 0, then there exists P ∼ 0 such that T_{P} = T_{Q}. (P7) One says that T_{Q} is elliptic if σ(T_{Q}) does not vanish. Then T_{Q} has a

parametrix, i.e. there exists a gTo T_{P}, with ord(T_{P}) = − ord(T_{Q}) and σ(T_{P}) = σ(T_{Q})^{−1}, such that T_{P} T_{Q} ∼ T_{Q}T_{P} ∼ I_{H^{2}(∂Ω)}.

(P8) If an elliptic gTo T_{P} is in addition positive self-adjoint as an operator on H^{2}(∂Ω), then its complex power T_{P}^{z}, z ∈ C, defined using the spectral theorem, is again a gTo, of order z ord(T_{P}) and with symbol equal to σ(T_{P})^{z}; in particular, the inverse T_{P}^{−1} and the positive square roots T_{P}^{1/2}, T_{P}^{−1/2} are gTo’s.

We refer to the book [3], especially its Appendix, and to the paper [2] for the proofs and for additional information on generalized Toeplitz operators.

In addition to classical ψdo’s and gTo’s, we will also need the more general class Ψ_{log} of log-classical (or log-polyhomogeneous) ψdo’s and gTo’s, whose total symbol in any local coordinate chart satisfies

p(x, ξ) − ∑_{j=0}^{k−1} p_{m−j}(x, ξ) ∈ S^{m−k+ϵ} ∀ϵ > 0, k = 0, 1, 2, …,

where p_{m−j} are of the form

(3)  p_{m−j}(x, ξ) = ∑_{k=0}^{k_{j}} p_{m−j,k}(x, \frac{ξ}{|ξ|})|ξ|^{m−j}(log |ξ|)^{k}

for |ξ| > 2, for some (finite) integers k_{j}. Such ψdo’s arise naturally as logarithms of complex powers of elliptic classical ψdo’s, and similarly for the corresponding gTo’s. The properties (P1)–(P8) above remain in force for logclassical gTo’s, except in (P7), (P8) and the first part of (P5) one must assume that k_{0} = 0 (i.e. that the principal symbol is log-free). The reader is referred e.g. to [10], and the references therein, for the details. Again, in Section 4 we will also need the demi-classical analogues of log-pluri-homogeneous ψdo’s and gTo’s, i.e. with symbols whose degrees of homogeneity go down by \frac{1}{2} instead of 1; everything above extends also to this case.

2.3 Boutet de Monvel calculus. Let K denote the Poisson extension operator, i.e. K solves the Dirichlet problem

(4)  ΔKu = 0 on Ω, Ku|_{∂Ω} = u.

(Thus K acts from functions on ∂Ω into functions on Ω. Here Δ is the ordinary Laplacian.) By the standard elliptic regularity theory [13], K is continuous from W^{s}(∂Ω) into W^{s+ \frac{1}{2}} (Ω), for any real s; 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 a positive selfadjoint elliptic ψdo on ∂Ω of order −1. We have by definition Λ^{−1}K^{∗}K = I_{L^{2}(∂Ω)}; comparing this with (4) we see that the restriction

γ := Λ^{−1}K^{∗}|_{Ran K}

is the operator of “taking the boundary values” of a harmonic function. The operators

Λ_{w} := K^{∗}wK,

with w a smooth function on \overline{Ω}, are governed by a calculus developed by Boutet de Monvel in [1]. For typographical reasons, we will often write Λ[w] instead of Λ_{w}. It was shown that for w of the form

w = ρ^{s}g, g ∈ C^{∞}(\overline{Ω}), s > −1,

Λ[w] is a ψdo on ∂Ω of order −s − 1, with principal symbol

(5)  σ(Λ_{w})(x, ξ) = \frac{Γ(s + 1)|η_{x}|^{s}}{2|ξ|^{s+1}} g(x).

More generally, Λ[gρ^{s}(log ρ)^{k}] is a log-classical ψdo on ∂Ω whose leading symbol p_{−s−1}(x, ξ) has the form (3) with k_{0} = k; see e.g. [12].

2.4 The Levi form. We denote by T′′ ≡ T′′(∂Ω) ⊂ T (∂Ω) ⊗ C the antiholomorphic complex tangent space to ∂Ω, i.e. elements of T′′_{x}, x ∈ ∂Ω, are vectors X = ∑_{j=1}^{d} X_{j} \frac{∂}{∂\overline{z}_{j}}, X_{j} ∈ C, such that Xρ = 0. (This notation follows [4, p. 141].) The holomorphic complex tangent space T′ is defined similarly, and the whole complex tangent space T (∂Ω) ⊗ C is spanned by T′, T′′ and the vector

E := ∑_{j=1}^{d} \frac{∂ρ}{∂\overline{z}_{j}} \frac{∂}{∂z_{j}} − \frac{∂ρ}{∂z_{j}} \frac{∂}{∂\overline{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^{d} (the right-hand side is independent of the choice of such extension). Recall that the Levi form is the Hermitian form on T′ defined by

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

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

L′′(X, Y) := − ∑_{j,k=1}^{d} \frac{∂^{2}ρ}{∂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

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

By the usual formalism, L′′ induces a positive definite Hermitian form (or, perhaps more appropriately, a positive definite Hermitian bivector) on the dual space T^{′′∗} of T′′; we denote it by L. Namely, for 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})}.

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

dη = −i∂\overline{∂}ρ = −i ∑_{k,l=1}^{d} \frac{∂^{2}ρ}{∂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′′. Let us define E_{T} ∈ T′ + T′′ by

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. For f, g ∈ C^{∞}(∂Ω), if we denote by f, g also the corresponding functions on the half-line bundle Σ constant on each fiber, then one has the following formula for their Poisson bracket:

(6)  \frac{1}{i}\{f, g\}_{Σ} = \frac{L(\overline{∂}_{b}f, \overline{∂}_{b}\overline{g}) − L(\overline{∂}_{b}g, \overline{∂}_{b}\overline{f})}{t},  ξ = tη_{x}, t > 0.

In particular, the right-hand side gives the symbol σ_{−1}([T_{f}, T_{g}]) of the commutator of two Toeplitz operators, by the property (P4). One can also show that

(7)  σ_{−1}(T_{fg} − T_{f}T_{g}) = \frac{1}{t}L(\overline{∂}_{b}g, \overline{∂}_{b}\overline{f}).

More generally, identifying — once for all — the half-line bundle Σ with ∂Ω × R_{+} via the map (x, tη_{x}) ↦ (x, t), let F, G be the functions on Σ 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

(8) \{F, G\}_{Σ} = t^{−k−m−1}(iL(\overline{∂}_{b}f, \overline{∂}_{b}\overline{g}) − iL(\overline{∂}_{b}g, \overline{∂}_{b}\overline{f}) + mgE_{⊥}f − kfE_{⊥}g).

See [11, Corollary 8] for (6) and (8), from which (7) follows in the same way as in the proof of Theorem 9 there.

2.5 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

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

With the norm defined as the l^{∞}-norm of the left-hand side, S^{Dixm} becomes a Banach space. 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 [7] for details and for discussion of the role of these functionals. It is a result of Connes [6] that if Q is a ψdo on a compact manifold M of real dimension n and ord(Q) = −n, then Q ∈ S^{Dixm} and

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.) It was shown in [11] that for Toeplitz operators T_{Q} on ∂Ω, the “right” order for T_{Q} to belong to S^{Dixm} is not − dim_{R} ∂Ω = −(2d − 1), but rather − dim_{C} Ω = −d. Namely, if T is a generalized Toeplitz operator on H^{2}(∂Ω) of order −d, then T ∈ S^{Dixm}, T is measurable, and

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

See Theorem 3 in [11].

3. Operators on B_{m}

In this section we consider Toeplitz and Hankel operators on the Boutet de Monvel-Guillemin spaces B_{m} associated to “higher Szegö projectors”. We deal in detail with the case m = 2 on the unit ball B^{2} of C^{2}, and at the end discuss what happens for m > 2 and general domains. Let thus Ω = B^{2}, with the usual defining function ρ(z) = 1 − |z|^{2}. The antiholomorphic complex tangent space T′′ is then one-dimensional, spanned by the single vector field

\overline{Z} := L_{12} = z_{1}\overline{∂}_{2} − z_{2}\overline{∂}_{1} on ∂B^{2}.

Its adjoint with respect to the inner product in L^{2}(∂B^{2}) equals (by Stokes’ theorem)

Z := \overline{z}_{2}∂_{1} − \overline{z}_{1}∂_{2}.

The spaces B_{m}, m = 1, 2, …, are given simply by

B_{m} = L^{2}(∂B^{2}) ∩ Ker \overline{Z}^{m}.

As already noted, B_{1} = H^{2}(∂B^{2}) ≡ H^{2}. Let

H_{0}^{2} := {f ∈ H^{2} : f(0) = 0},

where, abusing notation slightly, we denote by f also the holomorphic extension of f ∈ H^{2} into B^{2}.

Proposition 3. Every function in B_{2} can uniquely be written in the form f + Zg, with f ∈ H^{2} and g ∈ H_{0}^{2}.

Proof. By a simple computation

(9)  \overline{Z}Z(z_{1}^{m}z_{2}^{n}) = (m + n)z_{1}^{m}z_{2}^{n} = R(z_{1}^{m}z_{2}^{n}),

where R := z_{1}∂_{1} + z_{2}∂_{2} is the holomorphic radial derivative. Letting S : z_{1}^{m}z_{2}^{n} ↦ z_{1}^{m}z_{2}^{n}/(m+n) stand for the inverse of R on H_{0}^{2}, we thus have \overline{Z}ZSh = h for all h ∈ H_{0}^{2}. Also, by direct check, \overline{z}_{2}∂_{1}S and \overline{z}_{1}∂_{2}S are both bounded from H_{0}^{2} into L^{2}(∂B^{2}), hence so is ZS. Now if u ∈ B_{2}, then \overline{Z}u =: h must be a function in H^{2}. By Stokes’ theorem,

h(0) = − ∫_{∂B^{2}} h = − ∫_{∂B^{2}} \overline{Z}u = − ∫_{∂B^{2}} u(\overline{Z}1) = 0, so in fact h ∈ H_{0}^{2}. Hence \overline{Z}(u − ZSh) = 0, so u − ZSh is holomorphic and belongs to L^{2}(∂B^{2}), i.e. u − ZSh ∈ H^{2}. Taking f = u − ZSh, g = Sh thus yields the desired decomposition. Uniqueness is immediate from the fact that \overline{Z}(f + Zg) = \overline{Z}Zg = Rg together with the injectivity of R on H_{0}^{2}. □

The last proof actually shows that

A : f ⊕ g ↦ f + Zg, f ∈ H^{2}, g ∈ H_{0}^{2},

is a densely defined closed operator mapping its domain H^{2} ⊕ SH_{0}^{2} bijectively onto B_{2}. By abstract operator theory, we have the polar decomposition

A = U(A^{∗}A)^{1/2},

where U = A(A^{∗}A)^{−1/2} is a partial isometry with initial space \overline{Ran A^{∗}} = (Ker A)^{⊥} = H^{2} ⊕ H_{0}^{2} and final space \overline{Ran A} = B_{2}; that is, U is a unitary map of H^{2} ⊕ H_{0}^{2} onto B_{2}, and UU^{∗} = Π_{2} is the orthogonal projection of L^{2}(∂B^{2}) onto B_{2}. For f ∈ L^{∞}(∂B^{2}), the Toeplitz operator T_{f}^{B_{2}} — which, throughout this section, we will abbreviate just to T_{f} — can thus be written as

T_{f} = UU^{∗}fUU^{∗},

and is therefore unitarily equivalent to the operator

U^{∗}fU = (A^{∗}A)^{−1/2}A^{∗}fA(A^{∗}A)^{−1/2}

on H^{2} ⊕ H_{0}^{2}. Denote by Π_{0} = Π − ⟨·, 1⟩1 the orthogonal projection of H^{2} onto H_{0}^{2}; note that Π − Π_{0} is a smoothing operator (its Schwartz kernel equals constant 1). The complex normal vector field

E = ∑_{j=1}^{2}(\overline{z}_{j}\overline{∂}_{j} − z_{j}∂_{j})

is tangential to ∂B^{2} and E|_{H^{2}} = −R, i.e. T_{E} = −R. It follows that R is an elliptic gTo of order 1 with principal symbol |ξ|/|η_{x}|; and, hence, also its parametrix Π_{0}SΠ_{0} is a gTo, of order −1 and with principal symbol |η_{x}|/|ξ|. Consequently (see Proposition 16 in [9] for detailed argument), the square root

Ř := Π_{0}S^{−1/2}Π_{0} : z_{1}^{m}z_{2}^{n} ↦ \{ \begin{matrix}(m + n)^{−1/2}z_{1}^{m}z_{2}^{n}, & m + n > 0, \\ 0, & m = n = 0, \end{matrix}

is a gTo of order −\frac{1}{2} with symbol |η_{x}|^{1/2}/|ξ|^{1/2}. After these preliminaries, we are ready for the main result of this section.

Theorem 4. For f ∈ C^{∞}(∂B^{2}),

(10)  U^{∗}fU = [ \begin{matrix}T_{f} & T_{fZ} Ř \\ ŘT_{\overline{Z}f} & ŘT_{\overline{Z}fZ} Ř \end{matrix} ] .

Here the off-diagonal entries are of order −\frac{1}{2}, while ŘT_{\overline{Z}fZ} Ř − T_{f} is of or-

der −1; consequently, T_{f} ≅ U^{∗}fU = [ \begin{matrix}T_{f} & 0 \\ 0 & T_{f} \end{matrix} ] + A where A is a 2 × 2 matrix

of gTo’s of orders at most −\frac{1}{2}.

Here and below, T_{\overline{Z}f} stands for the operator u ↦ Π\overline{Z}(fu), i.e. f is to be understood as a multiplier; we will write T_{(\overline{Z}f)} for the operator T_{g} with the function g = \overline{Z}f (i.e. u ↦ Π(u\overline{Z}f)).

Proof. Writing f ⊕ g ∈ H^{2} ⊕ H_{0}^{2} as the column vector [ \begin{matrix}f \\ g \end{matrix} ], we have A =

[I, Z], so

A^{∗}A = [ \begin{matrix}I & ΠZΠ_{0} \\ Π_{0}\overline{Z}Π & Π_{0}\overline{Z}ZΠ \end{matrix} ] = [ \begin{matrix}I & 0 \\ 0 & Π_{0}RΠ_{0} \end{matrix} ] ,

since \overline{Z}Z = R while \overline{Z}Π = ΠZ = 0. It follows that (A^{∗}A)^{−1/2} = [ \begin{matrix}I & 0 \\ 0 & Ř \end{matrix} ].

Next,

A^{∗}fA = [ \begin{matrix}I \\ Π_{0}\overline{Z} \end{matrix} ] f[I, Z] = [ \begin{matrix}T_{f} & ΠfZΠ_{0} \\ Π_{0}\overline{Z}fΠ & Π_{0}\overline{Z}fZΠ_{0} \end{matrix} ] ,

and so

U^{∗}fU = (A^{∗}A)^{−1/2}A^{∗}fA(A^{∗}A)^{−1/2} = [ \begin{matrix}T_{f} & T_{fZ} Ř \\ ŘT_{\overline{Z}f} & ŘT_{\overline{Z}fZ} Ř \end{matrix} ] ,

proving (10). We have T_{\overline{Z}f} = T_{(\overline{Z}f)} + T_{f\overline{Z}} = T_{(\overline{Z}f)}, since \overline{Z}Π = 0; similarly

(11)  T_{fZ} = T_{Zf} − T_{(Zf)} = −T_{(Zf)},

since ΠZ = 0. Thus T_{\overline{Z}f} and T_{fZ} are gTo’s of order 0, hence T_{fZ} Ř and ŘT_{\overline{Z}f} are indeed of order −\frac{1}{2}. Finally, by the Leibniz rule and (11),

(12)  T_{\overline{Z}fZ} = T_{f\overline{Z}Z} + T_{(\overline{Z}f)Z} = T_{f}R − T_{(Z\overline{Z}f)} = −T_{f}T_{E} − T_{(Z\overline{Z}f)}

is a gTo of order 1, so that ŘT_{\overline{Z}fZ} Ř is of order 0, with principal symbol σ(Ř)^{2}σ(−T_{E})σ(T_{f}) = σ(T_{f}). Thus ŘT_{\overline{Z}fZ} Ř − T_{f} is of order −1, and the second part of the theorem follows. □

Clearly, the second part of the last theorem is precisely the statement of Theorem 1 from the Introduction. Using the standard relation

(13)  H∗ ; \overline{f}H_{g} = T_{fg} − T_{f}T_{g},

one can also get information about the “higher Hankel” operators H_{f} ≡ H_{f}^{(2)} : u ↦ (I − Π_{2})(fu), u ∈ B_{2}.

Theorem 5. For f, g ∈ C^{∞}(∂B^{2}), (14)

\begin{matrix}U^{∗}H∗ ; \overline{f}H_{g}U = \\ [ \begin{matrix}T_{fg} − T_{f}T_{g} − T_{fZ} Ř^{2}T_{\overline{Z}g} & (T_{fgZ} − T_{f}T_{gZ} − T_{fZ} Ř^{2}T_{\overline{Z}gZ})Ř \\ Ř(T_{\overline{Z}fg} − T_{\overline{Z}f}T_{g} − T_{\overline{Z}fZ} Ř^{2}T_{\overline{Z}g}) & Ř(T_{\overline{Z}fgZ} − T_{\overline{Z}f}T_{gZ} − T_{\overline{Z}fZ} Ř^{2}T_{\overline{Z}gZ})Ř \end{matrix} ] . \end{matrix}

Here the orders of the entries are at most [ \begin{matrix}−2 & −\frac{3}{2} \\ −\frac{3}{2} & −1 \end{matrix} ], with

σ_{−1}(Ř(T_{\overline{Z}fgZ} − T_{\overline{Z}f}T_{gZ} − T_{\overline{Z}fZ} Ř^{2}T_{\overline{Z}gZ})Ř) = −2(Zf)(\overline{Z}g).

Proof. The formula (14) follows directly from (10) and (13). In the upper left corner, σ(T_{fg}) = fg = σ(T_{f}T_{g}), so T_{fg} − T_{f}T_{g} is of order −1, and so is T_{fZ} Ř^{2}T_{\overline{Z}g} since T_{fZ} and T_{\overline{Z}g} are of order 0 by (11) while Ř is of order −\frac{1}{2}. In the bottom right corner T_{\overline{Z}fgZ} has order 1 and symbol fg/σ(Ř)^{2} by (12), and so does T_{\overline{Z}fZ} Ř^{2}T_{\overline{Z}gZ} (for the same reason); so their difference is of order 0, while T_{\overline{Z}f} and T_{gZ} are also of order 0 by (11); so the whole entry has the same order as Ř^{2}, i.e. −1. Finally, in the upper right corner, σ(T_{fgZ}) = −Z(fg) and σ(T_{f}T_{gZ}) = −f(Zg) by (11), so σ(T_{fgZ} − T_{f}T_{gZ}) = −(Zf)g; while σ(T_{fZ} Ř^{2}T_{\overline{Z}gZ}) = −(Zf)σ(Ř)^{2} \frac{g}{σ(Ř)^{2}} = −(Zf)g as well, all these gTo’s

being of order 0. Hence T_{fgZ} − T_{f}T_{gZ} − T_{fZ} Ř^{2}T_{\overline{Z}gZ} is of order at most −1, and the whole entry is of order at most −1 + ord(Ř) = −\frac{3}{2}. We claim that σ_{−1}(T_{fg}−T_{f}T_{g}−T_{fZ} Ř^{2}T_{\overline{Z}g}) = 0, so the upper left corner in (14) is in fact of order −2 (at most). Note first of all that L′′(\overline{Z}, \overline{Z}) = 1, whence by a short computation Z′′_{\overline{∂}_{b}g} = −(Z\overline{g})\overline{Z} and L′′(\overline{∂}_{b}f, \overline{∂}_{b}g) = −(\overline{Z}f)(Z\overline{g}). Therefore by (7),

(15)  σ_{−1}(T_{fg} − T_{f}T_{g}) = −\frac{1}{t}(Zf)(\overline{Z}g),  ξ = tη_{x}.

Consequently,

σ_{−1}(T_{fg} − T_{f}T_{g} − T_{fZ} Ř^{2}T_{\overline{Z}g})

= σ_{−1}(T_{fg} − T_{f}T_{g} + T_{(Zf)}Ř^{2}T_{(\overline{Z}g)}) by (11)

= −\frac{1}{t}(Zf)(\overline{Z}g) + (Zf)σ_{−1}(Ř^{2})(\overline{Z}g) by (15)

= 0 since σ(Ř^{2}) = \frac{1}{t},

proving the claim. It remains to compute the principal symbol of the bottom right corner in (14). Now by (11) and (12) again

T_{\overline{Z}fgZ} − T_{\overline{Z}f}T_{gZ} − T_{\overline{Z}fZ} Ř^{2}T_{\overline{Z}gZ}

= T_{−fgE−(Z\overline{Z}fg)} + T_{(\overline{Z}f)}T_{(Zg)} − T_{−fE−(Z\overline{Z}f)}Ř^{2}T_{−gE−(Z\overline{Z}g)}
= −T_{fg}T_{E} − T_{(Z\overline{Z}f)g+(Zf)(\overline{Z}g)+(\overline{Z}f)(Zg)+f(Z\overline{Z}g)} + T_{(\overline{Z}f)}T_{(Zg)}

− T_{fE}Ř^{2}T_{gE} − T_{fE}Ř^{2}T_{(Z\overline{Z}g)} − T (Z\overline{Z}f)Ř^{2}T_{gE} − T_{(Z\overline{Z}f)}Ř^{2}T_{(Z\overline{Z}g)}.

The last summand is of order −1, while −T_{fE}Ř^{2} = −T_{f}T_{E}Ř^{2} = T_{f}RŘ^{2} = T_{f} −T_{f}(Π−Π_{0}) ∼ T_{f}, and similarly σ_{0}(−Ř^{2}T_{gE}) = σ_{0}(−T_{gE}Ř^{2}) = σ_{0}(T_{g}) = g. Thus

σ_{0}(T_{\overline{Z}fgZ} − T_{\overline{Z}f}T_{gZ} − T_{\overline{Z}fZ} Ř^{2}T_{\overline{Z}gZ})

= σ_{0}(−T_{fgE} − T_{fE}Ř^{2}T_{gE}) − (Zf)(\overline{Z}g)

= σ_{−1}(T_{fg} + T_{fE}Ř^{2}T_{g})σ_{1}(−T_{E}) − (Zf)(\overline{Z}g)

= σ_{−1}(T_{fg} − T_{f}T_{g})σ_{1}(−T_{E}) − (Zf)(\overline{Z}g)

= −\frac{(Zf)(\overline{Z}g)}{t}t − (Zf)(\overline{Z}g) by (15)

= −2(Zf)(\overline{Z}g),

completing the proof. □

Corollary 6. For f, g ∈ C^{∞}(∂B^{2}), (H∗ ; \overline{f}H_{g})^{2} ∈ S^{Dixm}, is measurable, and

tr_{ω}(H∗ ; \overline{f}H_{g})^{2} = 4 tr_{ω}(H∗ ; \overline{f}H_{g})^{2} = 2 ∫_{∂B^{2}}(Zf)^{2}(\overline{Z}g)^{2} dσ̃,

where dσ̃ stands for the normalized surface measure on ∂B^{2}.

Proof. Immediate from the last theorem and Theorem 11 in [11]. □

Remark. By a similar computation as above, one can show that the principal symbol of the upper right corner in (14) is

σ_{−3/2}((T_{fgZ} − T_{f}T_{gZ} − T_{fZ} Ř^{2}T_{\overline{Z}gZ})Ř) = (Z^{2}f)(\overline{Z}g).

We have not tried to compute σ_{−2}(T_{fg} − T_{f}T_{g} − T_{fZ} Ř^{2}T_{\overline{Z}g}), which is probably going to be more tricky (but is of no relevance from the point of view of e.g. Corollary 6). □ The last theorem and corollary extend also to the spaces B_{m}(∂B^{2}) = Ker \overline{Z}^{m} ∩ L^{2}(∂B^{2}) for m > 2. Indeed, generalizing (9), one has

\overline{Z}^{m}Z^{m} = m!R(R − 1) … (R − m + 1),

and as before one concludes that

B_{m} = H^{2} + ZH_{0}^{2} + Z^{2}H_{00} + ⋯ + Z^{m−1}H_{0^{m−1}}^{2},

where we denoted H_{0^{j}}^{2} := {f ∈ H^{2} : ∂^{α}f(0) = 0 ∀|α| < j}; and following the same argument as in the proof of Theorem 5, one gets a unitary equivalence of the Toeplitz operator T_{f}^{B_{m}} on B_{m} to a certain m × m matrix of gTo’s on H^{2}. We are leaving the details to the interested reader.

As already mentioned, what we perceive as the main drawback of the spaces B_{m} — despite their similarity to the ordinary H^{2} from the “microlocal” point of view — is that they fail to be invariant under biholomorphic equivalence. In fact, assume that ϕ = (ϕ_{1}, ϕ_{2}) is a biholomorphic automorphism of B^{2}. Then by the chain rule

\overline{Z}(f ∘ ϕ) = ∑_{k=1}^{2}(\overline{∂}_{k}f) ∘ ϕ · \overline{Z}\overline{ϕ_{k}}.

Since 0 = \overline{Z}1 = \overline{Z}(ϕ_{1}\overline{ϕ}_{1} + ϕ_{2}\overline{ϕ}_{2}) = ϕ_{1}\overline{Z}\overline{ϕ}_{1} + ϕ_{2}\overline{Z}\overline{ϕ}_{2}, we see that

\overline{Z}(f ∘ ϕ) = a_{ϕ} · (\overline{Z}f) ∘ ϕ,

where a_{ϕ} = −\frac{\overline{Z}\overline{ϕ}_{1}}{ϕ_{2}} = \frac{\overline{Z}\overline{ϕ}_{2}}{ϕ_{1}} . Hence

\overline{Z}^{2}(f ∘ ϕ) = \overline{Z}[a_{ϕ} · (\overline{Z}f) ∘ ϕ] = a_{ϕ}^{2} · (\overline{Z}^{2}f) ∘ ϕ + \overline{Z}a_{ϕ} · (\overline{Z}f) ∘ ϕ.

Consequently, f ∈ B_{2} ⟹ f ∘ ϕ ∈ B_{2} would mean that \overline{Z}a_{ϕ} · (\overline{Z}f) ∘ ϕ = 0 ∀f ∈ B_{2}, so — taking f so that \overline{Z}f are e.g. the coordinate functions — \overline{Z}a_{ϕ} = 0, whence \overline{Z}^{2}\overline{ϕ}_{1} = \overline{Z}^{2}\overline{ϕ}_{2} = 0. The last condition is easily seen to be fulfilled only if ϕ is an affine map, showing that B_{2} ∘ ϕ ⊄ B_{2} for other ϕ. Yet another drawback of the spaces B_{m} is their dependence on the — in some sense arbitrary — choice of the special vector fields L_{jk}, 1 ≤ j < k ≤ d, in (1) (reducing to just L_{12} = \overline{Z} for d = 2). Namely, the “natural” definition would rather be

B̃_{m} := \{u ∈ L^{2}(∂Ω) : \overline{X}_{1} … \overline{X}_{m}u = 0
for any C^{∞} sections X_{1}, …, X_{m} of T′′(∂Ω)},

i.e. \overline{X}_{1} … \overline{X}_{m}u = 0 for any m-tuple of anti-holomorphic complex tangential vector fields \overline{X}_{1}, …, \overline{X}_{m} on ∂Ω. However, unfortunately, one has B̃_{m} = H^{2}(∂Ω) for all m ≥ 1. In fact, e.g. for Ω = B^{2} and m = 2 again, the condition f ∈ B̃_{2} means that

\overline{Z}a\overline{Z}f = 0 ∀a ∈ C^{∞}(∂Ω),

or (\overline{Z}a)(\overline{Z}f)+a\overline{Z}^{2}f = 0 for all a. Taking a = 1 gives \overline{Z}^{2}f = 0, so (\overline{Z}a)(\overline{Z}f) = 0 for all a; and taking a = Zz_{j}, so that \overline{Z}a = Rz_{j} = z_{j}, then yields z_{j}\overline{Z}f = 0 ∀j. Thus \overline{Z}f = 0 and f ∈ H^{2}(∂B^{2}), as claimed. We therefore proceed to describe a different variant of the spaces B_{m}, which does not suffer from the above deficiencies.

4. Toeplitz operators on C_{m}

Throughout this section, we assume that Ω is a bounded strictly pseudoconvex domain in C^{d}, d > 1, with smooth boundary ∂Ω. We fix once for all a positively-signed defining function ρ for Ω, i.e. ρ ∈ C^{∞}(\overline{Ω}) satisfies ρ > 0 on Ω and ρ = 0 < |∇ρ| on ∂Ω. Assume that Ω is equipped with a Kähler metric

g_{j\overline{k}} = ∂_{j}\overline{∂}_{k}Ψ,

where Ψ is a real-valued strictly-plurisubharmonic function (Kähler potential) on Ω, which we assume to be of the form

(16)  Ψ ≈ log \frac{1}{ρ} + ∑_{j=0}^{∞}(ρ^{M} log ρ)^{j}η_{j}

with an integer M ≥ 2 and η_{j} ∈ C^{∞}(\overline{Ω}). Here “≈” is to be understood in the sense of “resolution of singularities”, i.e. it means that the difference Ψ−log \frac{1}{ρ} −

∑_{j=1}^{N−1} should belong to C^{MN−1}(\overline{Ω}) and vanish on ∂Ω to order MN − 1. It is known that e.g. the Bergman metric on Ω is of this form (with M = d+1), as is the “Szegö metric” corresponding to Ψ(z) = \frac{1}{d} log K_{Sz}(z, z) where K_{Sz} is the invariantly defined Szegö kernel on Ω (then M = d), and likewise the Poincare metric (i.e. the Kähler-Einstein metric) on Ω corresponding to Ψ = log \frac{1}{u} with u the solution of the Monge-Ampere equation (then M = d + 1 again); see for instance the survey [5] and the references therein. Finally, we equip Ω with the weight

w = ρ^{ν}, ν ∈ R,

where ν will be sufficiently large as precised further below. As described in the introduction, we then have the kernels of powers of the Cauchy-Riemann operator

C_{m} := \{f on Ω : \overline{D}^{m}f = 0\}

and the associated “higher Cauchy-Riemann spaces”

C_{m,w} := C_{m} ∩ L^{2}(Ω, w),

with their Toeplitz and Hankel operators

T_{ϕ}^{(m,w)} : u ↦ Π^{(m,w)}(ϕu),  H_{ϕ}^{(m,w)} : u ↦ (I − Π^{(m,w)})(ϕu),  u ∈ C_{m,w},

where ϕ ∈ L^{∞}(Ω) and Π^{(m,w)} : L^{2}(Ω, w) → C_{m,w} is the orthogonal projection. We will usually write just T_{ϕ}, H_{ϕ} instead of T_{ϕ}^{(m,w)}, H_{ϕ}^{(m,w)} if there is no danger of confusion. Clearly T_{ϕ} and H_{ϕ} have the usual properties of Toeplitz and Hankel operators, namely they depend linearly on ϕ, T_{1} = I, T_{ϕ}^{∗} = T\overline{ϕ}, and ∥T_{ϕ}∥ ≤ ∥ϕ∥_{∞} and similarly for H_{ϕ} (by Cauchy-Schwarz). In the context of bounded symmetric domains, the next assertion is proved as Proposition 2.4 in Shimura [17].

Proposition 7. One has f ∈ C_{2} if and only if f can be written in the form

(17)  f = h^{k}∂_{k}Ψ + h

where h^{k}, k = 1, …, d, and h are holomorphic functions; the representation (17) is unique. More generally, f ∈ C_{m} if and only if f can be written in the form

f = ∑_{j=0}^{m−1} h^{k_{1}...k_{j}}Ψ_{k_{1}} … Ψ_{k_{j}},

with h^{k_{1}...k_{j}}, 1 ≤ k_{1} ≤ ⋯ ≤ k_{j} ≤ d, holomorphic on Ω; and this representation is unique.

Here we have introduced the shorthand

Ψ_{k} := ∂_{k}Ψ,

and also started using the (Einstein) summation convention of automatically summing over any index that appears twice in the formula.

Proof. For m = 2, we have by the definition of \overline{D}

\overline{D}(h^{k}Ψ_{k}) = g^{\overline{l}m}\overline{∂}_{l}(h^{k}Ψ_{k}) = g^{\overline{l}m}h^{k}(\overline{∂}_{l}Ψ_{k}) = g^{\overline{l}m}h^{k}g_{k\overline{l}} = δ_{k}^{m}h^{k} = h^{m}

since h^{k} are holomorphic. Thus indeed \overline{D}^{2}(h^{k}Ψ_{k}) = 0, and since \overline{D}h = 0, any f of the form (17) belongs to C_{2}. Conversely, given f ∈ C_{2}, we must have \overline{D}f = h^{k} for some h^{k} ∈ Ker \overline{D} i.e. for some holomorphic functions h^{k}, k = 1, …, d, on Ω. Then by the above computation, \overline{D}(f − h^{k}Ψ_{k}) = 0, i.e. h := f − h^{k}Ψ_{k} is a holomorphic function, so f is of the form (17). Finally, if f is of the form (17) and f = 0, then h^{k} = \overline{D}f = 0, hence h = f = 0, proving uniqueness. The proof for general m is the same. □

Note as a corollary that if f ∈ C_{m} and g is holomorphic, then gf ∈ C_{m}. It follows, in particular, that for g holomorphic, the Toeplitz operator T_{g} is just the operator of “multiplication by g”, and in fact (18)

T_{ϕ}T_{g} = T_{ϕg}, T\overline{g}T_{ϕ} = T_{\overline{g}ϕ}, H_{g} = 0, for g holomorphic and any ϕ.

As in Section 3, our strategy now will be to transfer the Toeplitz operators T_{ϕ} to (the direct sum of copies of) the Hardy space. To avoid too many indices, we will again deal only with the case m = 2. Let κ : (⨁_{j=1}^{d} H^{2}(∂Ω)) ⊕ H^{2}(∂Ω) → L^{2}(Ω, w) be the operator defined by

(19)  κ [ \begin{matrix}u_{j} \\ u \end{matrix} ] = ∑_{j=1}^{d} Ψ_{j}Ku_{j} + Ku

where K is the Poisson extension operator from §2.3. Take again the polar decomposition of κ,

κ = U(κ^{∗}κ)^{1/2},

where U is a partial isometry with initial space \overline{Ran κ^{∗}} = (Ker κ)^{⊥} = 0^{⊥} = ⊕^{d+1}H^{2}(∂Ω) and final space \overline{Ran κ}, i.e. U is a unitary operator from ⊕^{d+1}H^{2}(∂Ω) onto C_{2,w} by the last proposition. Also, U^{∗}U = I on ⊕^{d+1}H^{2}(∂Ω) while UU^{∗} = Π^{(2,w)}, the projection onto C_{2,w}. The Toeplitz operator T_{ϕ} = UU^{∗}ϕ|_{Ran UU^{∗}} is therefore unitarily equivalent to the operator

(20)  U^{∗}ϕU = (κ^{∗}κ)^{−1/2}κ^{∗}ϕκ(κ^{∗}κ)^{−1/2}

on the direct sum ⊕^{d+1}H^{2}(∂Ω) of d + 1 copies of H^{2}(∂Ω).

Lemma 8. Let ν > 1. Writing as in (19) the elements of ⊕^{d+1}H^{2}(∂Ω) as

column vectors [ \begin{matrix}u_{j} \\ u \end{matrix} ] = [u_{1}, u_{2}, …, u_{d}, u]^{t}, we have

(21)  κ^{∗}ϕκ = [ \begin{matrix}T_{Λ[\overline{Ψ}_{k}ϕwΨ_{j}]} & T_{Λ[\overline{Ψ}_{k}ϕw]} \\ T_{Λ[ϕwΨ_{j}]} & T_{Λ[ϕw]} \end{matrix} ] .

(So here the right-hand side is a (d + 1) × (d + 1) matrix of gTo’s on H^{2}(∂Ω), with j = 1, …, d the column index and k = 1, …, d the row index.)

Proof. For any u, v ∈ C_{hol}^{∞}(\overline{Ω}), we have

⟨ϕKu, Kv⟩_{L^{2}(Ω,w)} = ⟨wϕKu, Kv⟩_{L^{2}(Ω)}
= ⟨K^{∗}wϕKu, v⟩_{L^{2}(∂Ω)}
= ⟨Λ[ϕw]u, v⟩_{L^{2}(∂Ω)}
= ⟨T_{Λ[ϕw]}u, v⟩_{H^{2}(∂Ω)},

and similarly for \overline{Ψ}_{k}ϕwΨ_{j}, \overline{Ψ}_{k}ϕw and ϕwΨ_{j} in the place of ϕw. By (19), the claim follows. □

For brevity, let us denote the collection of all functions in C^{∞}(Ω) of the form ∑_{j=0}^{∞} η_{j}(ρ^{m} log ρ)^{j} as in (16) by A_{M} (thus Ψ − log \frac{1}{ρ} ∈ A_{M}), and also denote

ρ_{j} := ∂_{j}ρ.

By the Leibniz rule, we have ρΨ_{j} ∈ ρ_{j}+ρA_{M−1} ⊂ A_{M}. From the facts reviewed in §2.3, we thus conclude that for ϕ ∈ C^{∞}(\overline{Ω}), Λ[ϕwΨ_{j}] = Λ[ϕρ^{ν−1}(ρΨ_{j})] is an operator in Ψ_{log}^{−ν}, with log terms appearing only at orders −ν − M and lower (in particular, there is no log term in the leading symbol), and with principal symbol σ_{−ν}(Λ[ϕwΨ_{j}]) = \frac{Γ(ν)|η|^{ν−1}ϕρ_{j}}{2|ξ|^{ν}}. Similarly for the other entries in (21),

σ_{−ν}(Λ[\overline{Ψ}_{k}ϕw]) = \frac{Γ(ν)|η|^{ν−1}\overline{ρ}_{k}ϕ}{2|ξ|^{ν}}, σ_{1−ν}(Λ[\overline{Ψ}_{k}ϕwΨ_{j}]) = \frac{Γ(ν − 1)|η|^{ν−2}\overline{ρ}_{k}ϕρ_{j}}{2|ξ|^{ν−1}} ,

σ_{−ν−1}(Λ[ϕw]) = \frac{Γ(ν + 1)|η|^{ν}ϕ}{2|ξ|^{ν+1}}.

Documenta Mathematica 22 (2017) 1081–1116

In particular, for ϕ = 1, the operator κ^{∗}κ has for its entries gTo’s of orders

[ \begin{matrix}1 − ν & −ν \\ −ν & −ν − 1 \end{matrix} ], with leading symbol σ_{1−ν}(κ^{∗}κ) = \frac{Γ(ν−1)|η|^{ν−2}}{2|ξ|^{ν−1}} [ \begin{matrix}\overline{ρ}_{k}ρ_{j} & 0 \\ 0 & 0 \end{matrix} ].

This is obviously not elliptic, so it is not clear at first whether (κ^{∗}κ)^{−1}, not to say (κ^{∗}κ)^{−1/2}, are given by generalized Toeplitz operators. Our next task is to show that in fact they are; the main role in this result is played by the “sub-principal” order terms of κ^{∗}κ. Denote by Q = [Q_{kj}]_{j,k=1}^{d} the d × d matrix of gTo’s

(22)  Q_{kj} := T_{Λ[\overline{Ψ}_{k}wΨ_{j}]} − T_{Λ[\overline{Ψ}_{k}w]}T_{Λ[w]}^{−1}T_{Λ[wΨ_{j}]}

(where as before j is the column index and k the row index). Since T_{Λ[w]} is elliptic of order −ν − 1, it follows from the formulas for symbols above that Q_{kj} are gTo’s of order 1 − ν (for ν > 1), with principal symbols

σ_{1−ν}(Q_{kj}) = σ_{1−ν}(Λ[\overline{Ψ}_{k}wΨ_{j}]) − \frac{σ_{−ν}(Λ[\overline{Ψ}_{k}w])σ_{−ν}(Λ[wΨ_{j}])}{σ_{−ν−1}(Λ[w])} |_{Σ}

= \frac{Γ(ν − 1)|η|^{ν−2}}{2ν|ξ|^{ν−1}} \overline{ρ}_{k}ρ_{j}. (23)

Denote

(24)  Z_{kj} := Q_{kj} − \frac{1}{ν} T_{\overline{ρ}_{k}}T_{Λ[ρ^{ν−2}]}T_{ρ_{j}}.

In view of (23) and (5), we have σ_{1−ν}(Z_{kj}) = 0, so in fact Z_{kj} is a gTo of order at most −ν. In addition to our positively signed defining function ρ, we will also use the negatively signed defining function r := −ρ, and denote again by

r_{j} := ∂_{j}r = −ρ_{j}, r\overline{k} := \overline{∂}_{k}r = −\overline{ρ}_{k}, r_{j\overline{k}} := ∂_{j}\overline{∂}_{k}r = −∂_{j}\overline{∂}_{k}ρ

its partial derivatives as indicated.

Proposition 9. Assume that ν > 1. Then there exists a function c ∈ C^{∞}(∂Ω) such that, using again the identification (x, tη_{x}) ∈ Σ ≅ ∂Ω × R_{+} ∋ (x, t),

(25)  σ_{−ν}(Z_{kj}) =

\frac{σ(T_{Λ[ρ^{ν−1}]})}{ν − 1}(r_{j\overline{k}} + \frac{|η|}{ν}r\overline{k}L(\overline{∂}_{b}r_{j}, \overline{∂}_{b} \frac{1}{|η|}) + \frac{|η|}{ν}r_{j}L(\overline{∂}_{b} \frac{1}{|η|}, \overline{∂}_{b}r\overline{k}) + cr\overline{k}r_{j}).

Proof. Denote, quite generally, for ϕ, ψ ∈ C^{∞}(∂Ω),

(26) σ_{−ν}(T_{Λ[\overline{ϕ}ρ^{ν−2}ψ]} − T_{Λ[\overline{ϕ}ρ^{ν−1}]}T_{Λ[ρ^{ν}]}^{−1}T_{Λ[ρ^{ν−1}ψ]} − \frac{1}{ν} T\overline{ϕ}T_{Λ[ρ^{ν−2}]}T_{ψ}) =: Q(\overline{ϕ}, ψ).

Again, first of all, by the same computation as in (23), we see that the operator on the left-hand side of (26) is a gTo of order 1 − ν with σ_{1−ν} vanishing, hence, indeed, is in fact a gTo of order −ν, so the σ_{−ν} in (26) makes sense. Next, from the relation ⟨T_{Λ[w]}u, v⟩ = ∫_{Ω} wu\overline{v} dz, u, v ∈ H^{2}(∂Ω), and the fact that T_{f} is just the operator of “multiplication by f” when f is holomorphic, we have

(27)  T_{Λ[wf]} = T_{Λ[w]}T_{f}, T_{Λ[\overline{g}w]} = T\overline{g}T_{Λ[w]}

for holomorphic f, g and arbitrary w. For our bilinear form Q from (26), this means that

Q(\overline{g}\overline{ϕ}, ψf) = \overline{g}Q(\overline{ϕ}, ψ)f for f, g holomorphic.

Since, by general theory, Q(\overline{ϕ}, ψ) depends only on finitely many derivatives of ϕ and ψ (see the bottom of p. 616 in [11] for the detailed argument), it is therefore enough to evaluate Q(\overline{ϕ}, ψ) for \overline{ϕ} = f and ψ = \overline{g} with f, g holomorphic. In that case we have, using (27) again,

Q(f, \overline{g}) = σ_{−ν}(T_{Λ[fρ^{ν−2}\overline{g}]} − T_{Λ[fρ^{ν−1}]}T_{Λ[ρ^{ν}]}^{−1}T_{Λ[ρ^{ν−1}\overline{g}]} − \frac{1}{ν} T_{f}T_{Λ[ρ^{ν−2}]}T\overline{g})

= σ_{−ν}(T\overline{g}T_{[ν−2]}T_{f} − T_{[ν−1]}T_{f}T_{[ν]}^{−1}T\overline{g}T_{[ν−1]} − \frac{1}{ν} T_{f}T_{[ν−2]}T\overline{g}),

where, for typographical reasons, we have started writing just [ν] for Λ[ρ^{ν}]. Abusing notation slightly, we will also write \{·, ·\} for \frac{1}{i}\{·, ·\}_{Σ}, so that (P4) reads simply σ([T_{P} , T_{Q}]) = {σ(T_{P}), σ(T_{Q})}. Then

σ_{−ν}(T\overline{g}T_{[ν−2]}T_{f} − T_{f}T_{[ν−2]}T\overline{g}) = \{g, [ν − 2]\}f + [ν − 2]\{\overline{g}, f\} + \{[ν − 2], f\}\overline{g},

where we started denoting by [ν] also the symbol of T_{[ν]}. Similarly,

σ_{−ν−1}(T_{[ν−1]}T_{f} − T_{f}T_{[ν−1]}) = {[ν − 1], f},
σ_{−ν−1}(T\overline{g}T_{[ν−1]} − T_{[ν−1]}T\overline{g}) = \{\overline{g}, [ν − 1]\},

and, consequently,

Q(f, \overline{g}) = \{g, [ν − 2]\}f + [ν − 2]\{\overline{g}, f\} + \{[ν − 2], f\}\overline{g}

− \{[ν − 1], f\}\frac{[ν − 1]}{[ν]}\overline{g} − \{\overline{g}, [ν − 1]\}\frac{[ν − 1]}{[ν]}f

+ σ_{ν}(T_{f}T_{[ν−2]}T\overline{g} − T_{f}T_{[ν−1]}T_{[ν]}^{−1}T_{[ν−1]}T\overline{g} − \frac{1}{ν} T_{f}T_{[ν−2]}T\overline{g}).

The last summand equals just fQ(1, 1)\overline{g}. Since [ν] = \frac{Γ(ν+1)}{2|η|t^{ν+1}} by (5), we have

\frac{[ν−1]}{[ν]} = \frac{t}{ν}. By the Leibniz rule for the Poisson bracket

\{\overline{g}, ab\} = a\{\overline{g}, b\} + \{\overline{g}, a\}b,

we have \{\overline{g}, t^{ν}\} = νt^{ν−1}\{\overline{g}, t\}, and so, using again (5),

\{\overline{g}, [ν − 2]\} = \frac{Γ(ν − 1)}{2}\{\overline{g}, \frac{t^{1−ν}}{|η|} \}

= \frac{Γ(ν − 1)}{2|η|}(1 − ν)t^{−ν}\{\overline{g}, t\} + \frac{Γ(ν − 1)}{2}t^{1−ν}\{\overline{g}, \frac{1}{|η|}\},

\frac{t}{ν} \{\overline{g}, [ν − 1]\} = \frac{Γ(ν)}{2ν|η|}(−ν)t^{−ν}\{\overline{g}, t\} + \frac{Γ(ν)}{2ν}t^{1−ν}\{\overline{g}, \frac{1}{|η|}\},

whence

\{\overline{g}, [ν − 2]\} − \frac{[ν − 1]}{[ν]}\{\overline{g}, [ν − 1]\} = \frac{[ν − 2]|η|}{ν}\{\overline{g}, \frac{1}{|η|}\}.

Similarly for the corresponding brackets with f. Finally by (6), when either f or g is holomorphic, we have

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

Putting everything together, we thus arrive at

(28)  Q(f, \overline{g}) =

[ν − 2]\frac{f \frac{|η|}{ν}L(\overline{∂}_{b}\overline{g}, \overline{∂}_{b} \frac{1}{|η|}) + \overline{g} \frac{|η|}{ν}L(\overline{∂}_{b} \frac{1}{|η|}, \overline{∂}_{b}\overline{f}) + L(\overline{∂}_{b}\overline{g}, \overline{∂}_{b}\overline{f})}{t} + f\overline{g}Q(1, 1),

and so

(29)  Q(\overline{ϕ}, ψ) =

[ν − 2]\frac{\overline{ϕ}\frac{|η|}{ν}L(\overline{∂}_{b}ψ, \overline{∂}_{b} \frac{1}{|η|}) + ψ \frac{|η|}{ν}L(\overline{∂}_{b} \frac{1}{|η|}, \overline{∂}_{b}ϕ) + L(\overline{∂}_{b}ψ, \overline{∂}_{b}ϕ)}{t} + \overline{ϕ}ψQ(1, 1),

Now σ_{−ν}(Z_{kj}) = Q(\overline{ρ}_{k}, ρ_{j}). We claim that

(30)  L(\overline{∂}_{b}ρ_{j}, \overline{∂}_{b}ρ_{k}) = r_{j\overline{k}} + qr_{j}r\overline{k}

with some q ∈ C^{∞}(∂Ω). To see this, assume we are at a point of ∂Ω where e.g. r_{1} ≠ 0, so that

R_{j} := ∂_{j} − \frac{r_{j}}{r_{1}} ∂_{1}, \overline{R}_{k} := \overline{∂}_{k} − \frac{r\overline{k}}{r\overline{1}} \overline{∂}_{1}, j = 2, …, d,

form a basis for T′ and T′′, respectively. By a simple computation,

L′(R_{k}, R_{n}) = R_{k}r\overline{n} − \frac{r\overline{n}}{r\overline{1}} R_{k}r\overline{1} =: ℓ_{k\overline{n}}.

Let ℓ^{\overline{m}k} be the inverse matrix of ℓ_{k\overline{n}} (which is positive definite by the strict pseudoconvexity). Thus (employing the summation convention)

(31)  ℓ^{\overline{k}m}\overline{R}_{k}r_{j} = δ_{j}^{m} + \frac{r_{j}}{r_{1}} ℓ^{\overline{k}m}\overline{R}_{k}r_{1}.

Remembering the definition of the dual Levi form, we have L(\overline{∂}_{b}ρ_{j}, \overline{∂}_{b}ρ_{k}) = \overline{∂}_{b}ρ_{j}(\overline{X}) = \overline{X}ρ_{j} where \overline{X} ∈ T′′ is characterized by L′′(\overline{R}_{n}, \overline{X}) = \overline{∂}_{b}ρ_{k}(\overline{R}_{n}) = \overline{R}_{n}ρ_{k} for all n = 2, …, d. Writing temporarily X = ∑_{j} c_{j}\overline{R}_{j} this yields, since L′′(\overline{R}_{n}, \overline{R}_{j}) = \overline{L′(R_{j}, R_{n})} = ℓ_{j\overline{n}},

\overline{c}_{j}ℓ_{j\overline{n}} = \overline{R}_{n}ρ_{k},

or, using (31),

\overline{c}_{m} = ℓ^{\overline{n}m}\overline{R}_{n}ρ_{k} = −ℓ^{\overline{n}m}\overline{R}_{n}r_{k} = −δ_{k}^{m} − \frac{r_{k}}{r_{1}} ℓ^{\overline{n}m}\overline{R}_{n}r_{1}.

Hence, using (31) one more time,

L(\overline{∂}_{b}ρ_{j}, \overline{∂}_{b}ρ_{k}) = c_{m}\overline{R}_{m}ρ_{j} = (δ_{k}^{m} + \frac{r\overline{k}}{r\overline{1}} ℓ^{\overline{m}n}R_{n}r\overline{1})\overline{R}_{m}r_{j}

= \overline{R}_{k}r_{j} + \frac{r\overline{k}}{r\overline{1}} (R_{n}r\overline{1})(δ_{j}^{n} + \frac{r_{j}}{r_{1}} ℓ^{\overline{m}n}\overline{R}_{m}r_{1})

= (\overline{R}_{k}r_{j} + \frac{r\overline{k}}{r\overline{1}} R_{j}r\overline{1}) + \frac{r\overline{k}r_{j}}{r_{1}r\overline{1}} ℓ^{\overline{m}n}(R_{n}r\overline{1})(\overline{R}_{m}r_{1})

= r_{j\overline{k}} + \frac{r\overline{k}r_{j}}{r_{1}r\overline{1}} [ℓ^{\overline{m}n}(R_{n}r\overline{1})(\overline{R}_{m}r_{1}) − r_{1\overline{1}}]

= r_{j\overline{k}} + r_{j}r\overline{k}q,

with

(32)  q := \frac{1}{r_{1}r\overline{1}} [ℓ^{\overline{m}n}(R_{n}r\overline{1})(\overline{R}_{m}r_{1}) − r_{1\overline{1}}]

(on the piece of ∂Ω where r_{1} ≠ 0; note that q is real-valued). Thus (30) holds. Inserting (30) into (29), noting that \frac{[ν−2]}{t} = \frac{[ν−1]}{ν−1}, we finally get (33)

σ_{−ν}(Z_{kj}) = Q(\overline{ρ}_{k}, ρ_{j})

= \frac{[ν − 1]}{ν − 1}(r\overline{k} \frac{|η|}{ν}L(\overline{∂}_{b}r_{j}, \overline{∂}_{b} \frac{1}{|η|}) + r_{j} \frac{|η|}{ν}L(\overline{∂}_{b} \frac{1}{|η|}, \overline{∂}_{b}r_{k}) + r_{j\overline{k}} + qr_{j}r\overline{k})

+ r_{j}r\overline{k}Q(1, 1),

completing the proof of the proposition (with c = q + \frac{ν−1}{[ν−1]}Q(1, 1)). □

Throughout the rest of this section, we will assume that on ∂Ω

(34)  [r_{j\overline{k}}]_{j,k=1}^{d} is positive definite, and |η| = 1.

Such defining functions exist in abundance: indeed, it is standard that there exists a (negatively signed) defining function r′ for Ω such that r′ is strictly plurisubharmonic, i.e. [r′_{j\overline{k}}]_{j,k=1}^{d} is positive definite, on \overline{Ω}. If g ∈ C^{∞}(\overline{Ω}) and r = r′g, then on ∂Ω

|η|^{2} = r_{k}r\overline{k} = |g|^{2}r′_{k}r′ ; \overline{k},

and

r_{j\overline{k}} = gr′_{j\overline{k}} + g_{j}r′ ; \overline{k} + r′_{j}g\overline{k} = gr′_{j\overline{k}} + (\frac{∂g}{∂n} ⊗ \frac{∂r′}{∂n} )_{j\overline{k}}

(where ∂/∂n stands for the normal derivative; note that the tangential derivatives of r′ vanish). Thus taking any positive g with

g = (r′_{k}r′ ; \overline{k})^{−1/2} and \frac{∂g}{∂n} = 0

on ∂Ω produces an r satisfying (34).

Corollary 10. Assume that (34) holds. Then

σ_{−ν}(Z_{kj}) = \frac{[ν − 1]}{ν − 1} (r_{j\overline{k}} + cr_{j}r\overline{k}), c ∈ C^{∞}(∂Ω).

Proof. If |η| = 1, then \overline{∂}_{b} \frac{1}{|η|} = 0. □

Proposition 11. Assume that ν > 1 and r satisfies (34). Then

(i) the inverse of the d × d matrix of gTo’s Q = [Q_{kj}]_{j,k=1}^{d} is given by S = [S_{jm}]_{j,m=1}^{d}, where S_{jm} are gTo’s of order ν;

(ii) the inverse of the (d+1)×(d+1) matrix of gTo’s κ^{∗}κ is a matrix all of whose entries are gTo’s of order ν, except the bottom right entry which is a positive selfadjoint elliptic gTo of order ν + 1.

Proof. Denote, for brevity, R_{j} := T_{ρ_{j}} and Ř = \frac{1}{ν} T_{[ν−2]} + \frac{1}{ν−1}T_{c[ν−1]}, with the c from (25). By the last proposition and its corollary, Ỹ_{kj} := Q_{kj} − R_{k}^{∗}ŘR_{j} is a gTo of order −ν with σ_{−ν}(Ỹ_{kj}) = \frac{[ν−1]}{ν−1}r_{j\overline{k}}. Since the matrix [r_{j\overline{k}}]_{j,k=1}^{d} is positive definite by hypothesis, there exists a matrix S_{1} of (finite rank) smoothing operators such that [Y_{kj}] := [Ỹ_{kj}] + S_{1} is positive selfadjoint on ⊕^{d}H^{2}(∂Ω). (Sketch of proof: looking at the symbols shows that (ν − 1)[Ỹ_{kj}] = T_{[ν−1]}^{1/2}[T_{r_{j\overline{k}}}]^{1/2}(I + K)[T_{r_{j\overline{k}}}]^{1/2}T_{[ν−1]}^{1/2} with K of order −1, i.e. K is a compact selfadjoint operator. Thus K has at most finitely many eigenvalues, each of finite multiplicity, in the interval (−∞, −\frac{1}{2}); the corresponding spectral projection P_{(−∞,− \frac{1}{2} )} is thus a finite rank smoothing operator.

Take S_{0} := P_{(−∞,− \frac{1}{2} )}KP_{(−∞,− \frac{1}{2} )} (a finite rank smoothing operator too, with

I + K + S_{0} ≫ \frac{1}{2}I) and S_{1} := \frac{1}{ν−1}T_{[ν−1]}^{1/2}[T_{r_{j\overline{k}}}]^{1/2}S_{0}[T_{r_{j\overline{k}}}]^{1/2}T_{[ν−1]}^{1/2} (a finite rank

smoothing operator, with [Ỹ_{kj}] + S_{1} ≫ \frac{1}{2} \frac{1}{ν−1}T_{[ν−1]}^{1/2}[T_{r_{j\overline{k}}}]T_{[ν−1]}^{1/2} ≫ 0).) Consequently, by the (matrix variant of the) property (P8) in §2.2, the powers Y^{z} of Y = [Y_{kj}]_{j,k=1}^{d}, with any z ∈ C, are gTo’s, with entries of order −zν; in particular, the inverse Y^{−1} = W, W = [W_{jm}]_{j,m=1}^{∞}, is matrix of gTo’s, and so are the square roots Y^{1/2} and W^{1/2} = Y^{−1/2}. For the matrix Q = [Q_{kj}] above, we can then write, in the obvious block matrix notation (R is to be viewed as a row vector)

(35) Q = Y + R^{∗}ŘR − S_{1} = Y^{1/2}(I + z^{∗}Řz − S_{2})Y^{1/2},  z := RW^{1/2},

with S_{2} = W^{1/2}S_{1}W^{1/2} smoothing. Recall from (21) that

(36)  κ^{∗}κ = [ \begin{matrix}T_{Λ[\overline{Ψ}_{k}wΨ_{j}]} & T_{Λ[\overline{Ψ}_{k}w]} \\ T_{Λ[wΨ_{j}]} & T_{Λ[w]} \end{matrix} ] ≡ [ \begin{matrix}A & B \\ C & D \end{matrix} ] .

Note that D = T_{[ν]}, as well as all of κ^{∗}κ, are positive selfadjoint, so have an (unbounded) inverse. Using the decomposition

(37)  [ \begin{matrix}A & B \\ C & D \end{matrix} ] = [ \begin{matrix}I & BD^{−1} \\ 0 & I \end{matrix} ] [ \begin{matrix}A − BD^{−1}C & 0 \\ 0 & D \end{matrix} ] [ \begin{matrix}I & 0 \\ D^{−1}C & I \end{matrix} ] ,

we thus see that A − BD^{−1}C = Q must also be positive selfadjoint. Hence, by (35), so must be (I + z^{∗}Řz − S_{2}). The operator

H := zz^{∗} = RW R^{∗}

is an elliptic gTo on H^{2}(∂Ω) of order ν, with σ_{ν}(H) = |[r_{j\overline{k}}]^{−1/2}[r\overline{k}]|^{2} > 0. Being of the form zz^{∗}, it is also automatically nonnegative; by a similar argument as above, it follows that the projection H_{0} onto Ker H is a finite rank smoothing operator, and H + H_{0} is a positive selfadjoint elliptic gTo of order ν; hence H^{1/2} = (H + H_{0})^{1/2} − H_{0} =: h is a gTo (of order \frac{ν}{2} ), and so is h_{−} := (H + H_{0})^{−1/2} − H_{0} (of order −\frac{ν}{2} ), with hh_{−} = h_{−}h = I − H_{0}. Taking polar decomposition of z^{∗}, we obtain

z^{∗} = Vh

with some V a column matrix of gTo’s of order 0 and a partial isometry from H^{2}(∂Ω) into ⊕^{d}H^{2}(∂Ω), with V^{∗}V = I − H_{0}, and z^{∗}Řz = V^{∗}hŘhV. Again, the operator I + hŘh is a selfadjoint gTo of order 1 with symbol σ_{1}(hŘh) = \frac{[ν−2]}{ν}σ_{ν}(H) > 0, so there exists a finite rank smoothing operator S_{3} such that I + hŘh + S_{3} ≫ 0; and by (P8), (I + hŘh + S_{3})^{−1/2} is an elliptic gTo of order −\frac{1}{2}. Set

b := h_{−}^{2} − h_{−}(I + hŘh + S_{3})^{−1/2}h_{−},

this is an elliptic demi-classical gTo of order −ν (the first summand is elliptic of order −ν, the second summand is of order −ν − \frac{1}{2}); and

X′′ : = I − VhbhV^{∗}
= I − V(I − H_{0})[I − (I + hŘh + S_{3})^{−1/2}](I − H_{0})V^{∗},

which differs by a finite-rank smoothing operator from

I − V[I − (I + hŘh + S_{3})^{−1/2}]V^{∗} = (I − VV^{∗}) + V(I + hŘh + S_{3})^{−1/2}V^{∗}.

The last operator is selfadjoint, equals I on Ker V^{∗}, while on (Ker V^{∗})^{⊥} = Ran V it is unitarily equivalent to V^{∗}V(I+hŘh+S_{3})^{−1/2}V^{∗}V, which differs by a finiterank smoothing operator from (I + hŘh + S_{3})^{−1/2}, an elliptic gTo of order −\frac{1}{2}. It follows one more time that there exists a finite-rank smoothing operator S_{4} such that

X′ := X′′ + S_{4} = I − VhbhV^{∗} + S_{4}

is a matrix of demi-classical gTo’s of order 0 which is also positive selfadjoint as an operator on ⊕^{d}H^{2}(∂Ω). Now

X^{′2} ∼ (I − VhbhV^{∗})^{2}

= I + Vh(−2b + bhV^{∗}Vhb)hV^{∗}

∼ I + Vh(−2b + bh^{2}b)hV^{∗}

= I + V[(I − hbh)^{2} − I]V^{∗}

∼ I + V[\underbrace{(I + hŘh + S_{3})^{−1} − I}_{=:G}]V^{∗},

and so

X^{′2}(I + z^{∗}Řz − S_{2}) = X^{′2}(I + VhŘhV^{∗} − S_{2})
∼ (I + VGV^{∗})(I + VhŘhV^{∗})
= I + V(G + hŘh + GV^{∗}VhŘh)V^{∗}
∼ I + V(G + hŘh + GhŘh)V^{∗}
= I + V[(I + G)(I + hŘh) − I]V^{∗}
∼ I + V[(I + G)(I + hŘh + S_{3}) − I]V^{∗} = I.

Consequently,

X^{′2}(I + z^{∗}Řz − S_{2}) = I + S_{5}

with some smoothing operator S_{5}. Since we have managed that both terms on the left-hand side are positive, hence invertible, so must be the right-hand side, thus (I + S_{5})^{−1} is a gTo, and

(I + S_{5})^{−1}X^{′2}(I + z^{∗}Řz − S_{2}) = I,

i.e. (I + S_{5})^{−1}X^{′2} is a left inverse of (I + z^{∗}Řz − S_{2}). Taking adjoints, we see that X^{′2}(I + S_{5})^{−1} is a right inverse, hence

(38)  S′ := (I + S_{5})^{−1}X^{′2} = X^{′2}(I + S_{5})^{−1}

is the two-sided inverse for (I + z^{∗}Řz − S_{2}); thus finally by (35),

S := W^{1/2}S′W^{1/2}

is the inverse of Q, proving part (i). Note that although X′ is only demi-classical, the operator G above, and thus also X^{′2}, is classical, hence so are also S′ and S. For (ii), recall from (37) that

(κ^{∗}κ)^{−1} = [ \begin{matrix}I & 0 \\ −D^{−1}C & I \end{matrix} ] [ \begin{matrix}Q^{−1} & 0 \\ 0 & D^{−1} \end{matrix} ] [ \begin{matrix}I & −BD^{−1} \\ 0 & I \end{matrix} ]

= [ \begin{matrix}Q^{−1} & −Q^{−1}BD^{−1} \\ −D^{−1}CQ^{−1} & D^{−1} + D^{−1}CQ^{−1}BD^{−1} \end{matrix} ] .

Thus the matrix of gTo’s

[ \begin{matrix}S & −SBD^{−1} \\ −D^{−1}CS & D^{−1} + D^{−1}CSBD^{−1} \end{matrix} ] (note that C = B^{∗})

is the inverse for κ^{∗}κ. Now D^{−1}CS is a gTo of order (ν + 1) + (−ν) + ν = ν + 1, with symbol

σ_{ν+1}(D^{−1}CS) = σ_{ν+1}(D^{−1}CW^{1/2}X^{′2}W^{1/2})
= σ_{ν+1}(D^{−1})σ_{−ν/2}(CW^{1/2}X′)σ_{ν/2}(X′W^{1/2}),

with

(39) \begin{aligned}σ_{−ν/2}(CW^{1/2}X′) & = σ_{−ν}(C)σ_{ν/2}(W^{1/2})σ_{0}(X′) \\ & = [ν − 1]R · σ_{ν/2}(W^{1/2})σ_{0}(I − VhbhV^{∗}) \\ & = [ν − 1]σ_{ν/2}(z)σ_{0}(I − VhbhV^{∗}) \\ & = [ν − 1]σ_{ν/2}(hV^{∗})σ_{0}(I − VhbhV^{∗}) \\ & = [ν − 1]σ_{ν/2}(h)σ_{0}(V^{∗} − V^{∗}VhbhV^{∗}) \\ & = [ν − 1]σ_{ν/2}(h)σ_{0}((I − hbh)V^{∗})  (since V^{∗}V ∼ I) \\ & = [ν − 1]σ_{ν/2}(h)σ_{0}(I − hbh)σ_{0}(V^{∗}) \\ & = 0, \end{aligned}

since I − bhb ∼ (I + hŘh + S_{3})^{−1/2} is of order −\frac{1}{2}. Thus D^{−1}CS is in fact of order ν, and similarly for SBD^{−1}, while D^{−1}CSBD^{−1} = (D^{−1}CS)Q(SBD^{−1}) has order ν + (1 − ν) + ν = ν + 1, same as D^{−1}. Consequently, (κ^{∗}κ)^{−1} is a

(d + 1) × (d + 1) matrix of gTo’s with orders [ \begin{matrix}ν & ν \\ ν & ν + 1 \end{matrix} ], as asserted.

Finally, as C = B^{∗},

σ_{ν+1}(D^{−1} + D^{−1}CSBD^{−1})
= σ_{ν+1}(D^{−1} + D^{−1}CW^{1/2}X^{′2}W^{1/2}C^{∗}D^{−1})
= σ_{ν+1}(D^{−1}) + |σ_{ν+1}(D^{−1})σ_{−ν/2−1/2}(CW^{1/2}X′)|^{2} > 0,

showing the ellipticity of the bottom right corner and thus completing the proof of part (ii). □

Proposition 12. Under the same hypothesis as in Proposition 11, there exists an isometry V on ⊕^{d+1}H^{2}(∂Ω) such that V (κ^{∗}κ)^{−1/2} is a (d + 1) × (d + 1) matrix of demi-classical gTo’s.

Proof. Keeping the notations from the previous proof, we have from the commutativity of the two factors on the right-hand side of (38) that I + S_{5} is positive and

S′ := (I + S_{5})^{−1/2}X^{′2}(I + S_{5})^{−1/2}.

Thus the operator

X := X′(I + S_{5})^{−1/2}W^{1/2}

satisfies

X^{∗}X = W^{1/2}S′W^{1/2} = S = Q^{−1}.

Finally, with the notation (37), set

(40)  Z := [ \begin{matrix}X & −XBD^{−1} \\ 0 & D^{−1/2} \end{matrix} ] .

Then

Z^{∗}Z = [ \begin{matrix}X^{∗} & 0 \\ −D^{−1}CX^{∗} & D^{−1/2} \end{matrix} ] [ \begin{matrix}X & −XBD^{−1} \\ 0 & D^{−1/2} \end{matrix} ]

= [ \begin{matrix}S & −SBD^{−1} \\ −D^{−1}CS & D^{−1} + D^{−1}CSBD^{−1} \end{matrix} ]

is precisely the matrix of (κ^{∗}κ)^{−1} from the preceding proposition. Taking polar decomposition, it follows that Z = V (κ^{∗}κ)^{−1/2} with V a partial isometry with initial space \overline{Ran(κ^{∗}κ)^{−1/2}} = (Ker(κ^{∗}κ)^{−1/2})^{⊥} = 0^{⊥}, i.e. an isometry on ⊕^{d+1}H^{2}(∂Ω). □

We pause to note what are the orders of the entries in Z: while those of X have orders \frac{ν}{2} , the bottom right entry is a positive selfadjoint elliptic gTo of order \frac{ν+1}{2} with symbol [ν]^{−1/2}. For XBD^{−1} = (D^{−1}CX^{∗})^{∗}, a brute count gives order \frac{ν}{2} + 1; however, D^{−1}CX^{∗} ∼ D^{−1}CW^{1/2}X′ and we have seen in (39) that σ_{−ν/2}(CW^{1/2}X′) = 0 so that CW^{1/2}X′ is actually of order −\frac{ν}{2} − \frac{1}{2}; thus XBD^{−1} is actually of order \frac{ν}{2} + \frac{1}{2}, i.e. the same as D^{−1/2}.

Proof of Theorem 2. We have seen in (20) that the Toeplitz operator T_{ϕ} on C_{2,w} is unitarily equivalent to (κ^{∗}κ)^{−1/2}κ^{∗}ϕκ(κ^{∗}κ)^{−1/2} on ⊕^{d+1}H^{2}(∂Ω). By the last proposition, the latter is in turn equivalent to (modulo smoothing operators)

V^{∗}Zκ^{∗}ϕκZ^{∗}V ≅ Zκ^{∗}ϕκZ^{∗}

with Z from (40). Writing now (21) as κ^{∗}ϕκ = [ \begin{matrix}A_{ϕ} & B_{ϕ} \\ C_{ϕ} & D_{ϕ} \end{matrix} ] (similarly to (36))

we have by (40) (noting again that B^{∗} = C)

Zκ^{∗}ϕκZ^{∗} = [ \begin{matrix}X & −XBD^{−1} \\ 0 & D^{−1/2} \end{matrix} ] [ \begin{matrix}A_{ϕ} & B_{ϕ} \\ C_{ϕ} & D_{ϕ} \end{matrix} ] [ \begin{matrix}X^{∗} & 0 \\ −D^{−1}CX^{∗} & D^{−1/2} \end{matrix} ]

which equals

[ \begin{matrix}{X(A_{ϕ} − BD^{−1}C_{ϕ} − B_{ϕ}D^{−1}C+ \atop +BD^{−1}D_{ϕ}D^{−1}C)X^{∗}} & X(B_{ϕ} − BD^{−1}D_{ϕ})D^{−1/2} \\ D^{−1/2}(C_{ϕ} − D_{ϕ}D^{−1}C) & D^{−1/2}D_{ϕ}D^{−1/2} \end{matrix} ] .

However, from (21) we see that σ(A_{ϕ}) = ϕσ(A), so A_{ϕ} = T_{ϕ}A + A′_{ϕ} with A′_{ϕ} of order 1 less than A i.e. −ν; and similarly B_{ϕ} = BT_{ϕ} + B′_{ϕ}, C_{ϕ} = T_{ϕ}C + C′_{ϕ} and D_{ϕ} = DT_{ϕ} + D′_{ϕ}. It follows that

D^{−1/2}D_{ϕ}D^{−1/2} = D^{1/2}T_{ϕ}D^{−1/2} + D^{−1/2}D′_{ϕ}D^{−1/2}
= T_{ϕ} + ([D^{1/2}, T_{ϕ}]D^{−1/2} + D^{−1/2}D′_{ϕ}D^{−1/2}),

where the term in the parentheses is of order −1. Next,

R_{ϕ} : = X(B_{ϕ} − BD^{−1}D_{ϕ})D^{−1/2}
= X[(BT_{ϕ} + B′_{ϕ}) − BD^{−1}(DT_{ϕ} + D′_{ϕ})]D^{−1/2}

= X[B′_{ϕ} − BD^{−1}D′_{ϕ}]D^{−1/2}, (41)

which is of order −\frac{1}{2}; similarly for D^{−1/2}(C_{ϕ} − D_{ϕ}D^{−1}C)X^{∗} = R∗ ; \overline{ϕ}. Finally,

X(A_{ϕ} − BD^{−1}C_{ϕ} − B_{ϕ}D^{−1}C + BD^{−1}D_{ϕ}D^{−1}C)X^{∗}

= X[(T_{ϕ}A + A′_{ϕ}) − BD^{−1}(T_{ϕ}C + C′_{ϕ}) − (BT_{ϕ} + B′_{ϕ})D^{−1}C
+ BD^{−1}(DT_{ϕ} + D′_{ϕ})D^{−1}C]X^{∗}

= X[(T_{ϕ}A + A′_{ϕ}) − BD^{−1}(T_{ϕ}C + C′_{ϕ}) − B′_{ϕ}D^{−1}C + BD^{−1}D′_{ϕ}D^{−1}C]X^{∗}

= X[T_{ϕ}A + A′_{ϕ} − [BD^{−1}, T_{ϕ}]C − T_{ϕ}BD^{−1}C − BD^{−1}C′_{ϕ}
− B′_{ϕ}D^{−1}C + BD^{−1}D′_{ϕ}D^{−1}C]X^{∗}

= X[T_{ϕ}(A − BD^{−1}C)+

\underbrace{A′_{ϕ} − [BD^{−1}, T_{ϕ}]C − BD^{−1}C′_{ϕ} − B′_{ϕ}D^{−1}C + BD^{−1}D′_{ϕ}D^{−1}C}_{=:W′_{ϕ}}]X^{∗}

= X[T_{ϕ}Q + W′_{ϕ}]X^{∗}
= [X, T_{ϕ}]QX^{∗} + T_{ϕ}[X, QX^{∗}] + T_{ϕ}QX^{∗}X + XW′_{ϕ}X^{∗}
(42) = T_{ϕ} + [X, T_{ϕ}]QX^{∗} + T_{ϕ}[X, QX^{∗}] + XW′_{ϕ}X^{∗},

since QX^{∗}X = I by the construction of X. The last line is a d × d matrix of gTo’s of order 0, and their symbols — M(ϕ) = [M_{kj}(ϕ)]_{j,k=1}^{d}, say — depend linearly on derivatives of ϕ of at most first order, i.e. are of the form

M_{kj}(ϕ) = a_{kj}ϕ + L_{kj}ϕ

with some a_{kj} ∈ C^{∞}(∂Ω) and smooth vector-fields L_{kj} (not necessarily tangential) on ∂Ω. We thus conclude that

(43)  T_{ϕ} ≅ Zκ^{∗}ϕκZ^{∗} = [ \begin{matrix}T_{M(ϕ)} & 0 \\ 0 & T_{ϕ} \end{matrix} ] + lower order term,

where the “lower order term” is a matrix of gTo’s of order −\frac{1}{2}. Now, first of all, T_{1} = I, which implies that a_{kj} = δ_{kj}1. Secondly, remembering that, by (18), T_{ϕf} = T_{ϕ}T_{f} for f holomorphic, we see that for all such f,

(ϕI + Lϕ)(fI + Lf) = ϕfI + L(ϕf),

or, using the Leibniz rule, (Lϕ)(Lf) = 0. Taking ϕ = \overline{f} shows that Lf = 0 for all holomorphic f. Since T_{ϕ}^{∗} = T\overline{ϕ}, we similarly get L\overline{g}=0 for all holomorphic g. Thus L contains neither holomorphic nor anti-holomorphic derivatives, i.e. L = 0. This completes the proof. □

Remark. An alternative proof of L = 0 can be given by brute force computation from (42) using (39). □

Remark. We have been somewhat silent about the log terms in the various ψdo’s and gTo’s, so here we spell them out: clearly Ř, c and R_{j} contain no log terms, while Q_{kj} ∈ Ψ_{log}^{1−ν} have log terms starting at order 1 − ν − M (i.e. at distance M from the leading order). The operators Y_{kj} ∈ Ψ_{log}^{−ν} have log terms starting at distance M − 1 from the leading order already (since they were basically fabricated from Q_{kj} by cancelling the leading order term); and similarly for W, z, H, S, b, X and Z, as well as for the right-hand side of (43). □

Remark. It should be noted that, in general, even if a Toeplitz operator T_{ϕ} is positive selfadjoint on H^{2}(∂Ω) — hence, having an inverse there with the same properties — then this inverse need not be a gTo in general (i.e. if T_{ϕ} is not in addition elliptic); hence, in particular, the effort needed for proving Proposition 11. An example is ϕ = |z_{1}|^{2} on the unit ball in C^{d}, d > 1: its inverse would have to be of order 0 with symbol 1/|z_{1}|^{2}, but T_{1/|z_{1}|^{2}} is not a well defined operator. Note that, in fact, for the ball the entries in the matrix (36) for κ^{∗}κ are precisely of the above type: for instance, the (1,1)-entry in (36) is then T_{\overline{z}_{1}}T_{[ν−2]}T_{z_{1}}, hence with symbol [ν − 2]|z_{1}|^{2} of the kind as above. It is noteworthy that although each entry is non-manageable in this way, the entire matrix κ^{∗}κ is nonetheless invertible as a gTo by Proposition 11. Similar remarks apply to the square root T_{ϕ}^{1/2} and Proposition 12 (see also the remark at the very end of the next section). □

Remark. Note that the only place where (34) was used was in the proof of Proposition 11, to ensure the existence of powers of [Y_{kj}]. One can get this even if the second part of (34), i.e. |η| = 1, is dropped, as soon as ν is sufficiently large: namely, thanks to the factors of \frac{|η|}{ν} in (33), the matrix

[r_{j\overline{k}} + \frac{|η|}{ν}(r\overline{k}L(\overline{∂}_{b}r_{j}, \overline{∂}_{b} \frac{1}{|η|}) + r_{j}L(\overline{∂}_{b} \frac{1}{|η|}, \overline{∂}_{b}r_{k}))]

will be positive whenever [r_{j\overline{k}}] is and ν gets large enough, because the second summand becomes negligible compared to [r_{j\overline{k}}]. We are not sure what happens when (34) is dropped completely (i.e. if r is not strictly-PSH near ∂Ω), though we expect that at least Theorem 2 will still remain in force. □

Remark. We have left aside the case of dimension d = 1. In that case, many things simplify considerably: namely, the operator Q = A − BD^{−1}C in (37) is then an elliptic positive selfadjoint gTo, so it follows immediately that so is also its inverse Q^{−1} and inverse square root Q^{−1/2} = X; the constructions in the proofs of Propositions 11 and 12 are thus not needed, while the statements of these two propositions remain in force, and so does that of Theorem 2. □

Remark. We pause to remark that the function q from (32) occurs on at least one more interesting occasion. Namely, denote by E the holomorphic vector field on ∂Ω given by

Ef = \frac{1}{i}E_{⊥}f for holomorphic f,

where E_{⊥} is the Reeb vector field from §2.4; in other words, by (8),

Ef = {t, f} for holomorphic f.

(This operator makes appearance e.g. when one computes the symbol of operators like the B′_{ϕ}, D′_{ϕ} in the proof of Theorem 2.) Then one can show that

Er\overline{k} = −qr\overline{k}, k = 1, …, d.

The function q seems to be an interesting object from the point of view of complex geometry of strictly pseudoconvex domains, and even more so its normalized version |η|q which does not depend on the choice of the defining function r. □ Finally, it should be clear how to proceed for analogues of Theorem 2 for the spaces C_{m,w} with general m > 2.

5. The case of the ball

In this final section, we work out the situation from Theorem 2, i.e. the Toeplitz operators on C_{2,w}, more explicitly for the unit ball Ω = B^{d} of C^{d}, with the standard weights w = ρ^{ν}, ρ(z) = 1 − |z|^{2}. (Note that this defining function satisfies the hypothesis (34).) In particular, we show that in this case the inverse square root (κ^{∗}κ)^{−1/2} is itself a gTo (so the isometry V in Proposition 12 is not needed). We are unable to prove whether this is also the case for general strictly pseudoconvex domains Ω and their defining functions from Section 4. For multiindices α and β, the familiar formula for integration over the unit sphere

(44)  ⟨z^{α}, z^{β}⟩_{H^{2}(∂B^{d})} = ∫_{∂B^{d}} z^{α}\overline{z}^{β} dσ(z) = \frac{2π^{d}δ_{αβ}α!}{Γ(d + |α|)}

(where dσ stands for the surface measure on ∂B^{d}) implies

(45)  ∫_{B^{d}} ρ(z)^{ν}z^{α}\overline{z}^{β} dz = \frac{α!δ_{αβ}π^{d}}{(ν + 1)_{d+|α|}} .

Here (ν)_{k} := ν(ν +1) … (ν +k −1) is the Pochhammer symbol. It follows that

T_{Λ[ρ^{ν}]}z^{α} = ∑_{β}⟨T_{Λ[ρ^{ν}]}z^{α}, z^{β}⟩\frac{z^{β}}{⟨z^{β}, z^{β}⟩}

= ∑_{β} ( ∫_{B^{d}} ρ^{ν}z^{α}\overline{z}^{β} dz)\frac{Γ(d + |β|)}{2π^{d}β!}z^{β} by (44)

= \frac{Γ(d + |α|)}{2(ν + 1)_{d+|α|}} by (45). (46)

Similarly, we compute how the operators in the matrix (36) act on the basis {z^{α}}:

T_{Λ[z_{k}ρ^{ν−2}\overline{z}_{j}]}z^{α} = ∑_{β} \frac{(α + e_{k})!δ_{α+e_{k},β+e_{j}}π^{d}}{(ν − 1)_{d+|α|+1}} \frac{Γ(d + |β|)}{2π^{d}β!} z^{β}

= \frac{(α_{j} + δ_{kj})Γ(d + |α|)}{2(ν − 1)_{d+|α|+1}} z^{α+e_{k}−e_{j}},

T_{Λ[ρ^{ν−2}\overline{z}_{j}]}z^{α} = \frac{α_{j}Γ(d + |α| − 1)}{2(ν)_{d+|α|}}z^{α−e_{j}},
T_{Λ[z_{k}ρ^{ν−2}]}z^{α} = \frac{α_{j}Γ(d + |α| + 1)}{2(ν)_{d+|α|+1}}z^{α+e_{k}},

Documenta Mathematica 22 (2017) 1081–1116

Higher Cauchy-Riemann Spaces 1113

where e_{k} = (0, …, 0, 1, 0, …, 0) with 1 at k-th position. It follows that

T_{Λ[z_{k}ρ^{ν−2}]}T_{Λ[ρ^{ν}]}^{−1}T_{Λ[ρ^{ν−2}\overline{z}_{j}]}z^{α} = \frac{α_{j}}{ν} \frac{Γ(d + |α|)}{2(ν)_{d+|α|}} z^{α+e_{k}−e_{j}}

and so Q_{kj} = T_{Λ[z_{k}ρ^{ν−2}\overline{z}_{j}]} − T_{Λ[z_{k}ρ^{ν−2}]}T_{Λ[ρ^{ν}]}^{−1}T_{Λ[ρ^{ν−2}\overline{z}_{j}]} satisfies

Q_{kj}z^{α} = \frac{Γ(d + |α|)}{2(ν)_{d+|α|}}[\frac{α_{j} + δ_{kj}}{ν − 1} − \frac{α_{j}}{ν} ] z^{α+e_{k}−e_{j}}.

On the other hand, T_{\overline{z}_{j}}z^{α} = \frac{α_{j}}{d+|α|−1}z^{α−e_{j}}, whence

T_{z_{k}}T_{Λ[ρ^{ν−2}]}T_{\overline{z}_{j}}z^{α} = \frac{α_{j}}{d + |α| − 1} \frac{Γ(d + |α| − 1)}{2(ν − 1)_{d+|α|−1}} z^{α+e_{k}−e_{j}}.

Thus for the operator Z_{kj} = Q_{kj} − \frac{1}{ν} T_{z_{k}}T_{[ν−2]}T_{\overline{z}_{j}} from (24) we get

Z_{kj}z^{α} = \frac{Γ(d + |α|)}{2(ν)_{d+|α|}}[\frac{δ_{kj}}{ν − 1} − \frac{2ν − 1}{ν(ν − 1)} \frac{α_{j}}{d + |α| − 1} − \frac{α_{j}}{(d + |α| − 1)^{2}} ] z^{α+e_{k}−e_{j}}.

Consequently,

Z_{kj} = \frac{δ_{kj}}{ν − 1}T_{[ν−1]} − T_{z_{k}}DT_{\overline{z}_{j}}

where the operator

D : z^{α} ↦ [\frac{2ν − 1}{ν(ν − 1)} + \frac{1}{d + |α|}]\frac{Γ(d + |α| + 1)}{2(ν)_{d+|α|+1}}z^{α}

satisfies

D = \frac{2ν − 1}{ν(ν − 1)}(T_{[ν−1]} − T_{[ν]}) + \frac{1}{ν} T_{[ν]} = \frac{2ν − 1}{ν(ν − 1)}T_{[ν−1]} − \frac{1}{ν − 1}T_{[ν]}.

Thus, in full accordance with Proposition 9,

σ_{−ν}(Z_{kj}) = \frac{[ν − 1]}{ν − 1}δ_{kj} − \frac{2ν − 1}{ν(ν − 1)}[ν − 1] \overline{z}_{j}z_{k}.

Indeed, r_{j\overline{k}} = δ_{kj}, while, by a routine computation, q = −1 and

(\frac{ν−1}{ν}T_{[ν−2]} − T_{[ν−1]}T_{[ν]}^{−1}T_{[ν−1]})z^{α}

= \frac{Γ(d + |α|)}{2}[\frac{ν − 1}{ν} \frac{1}{(ν − 1)_{d+|α|}} − \frac{(ν + 1)_{d+|α|}}{(ν)_{d+|α|}^{2}} ]z^{α}

= −\frac{1}{ν} \frac{Γ(d + |α|)}{2(ν)_{d+|α|}}z^{α} = −\frac{1}{ν} T_{[ν−1]}z^{α},

Documenta Mathematica 22 (2017) 1081–1116

so Q(1, 1) = σ_{−ν}(\frac{ν−1}{ν}T_{[ν−2]} − T_{[ν−1]}T_{[ν]}^{−1}T_{[ν−1]}) = −\frac{[ν−1]}{ν} and

c = \frac{ν − 1}{[ν − 1]}Q(1, 1) + q = −\frac{2ν − 1}{ν}.

We also pause to note that the Reeb vector field is given simply by E_{⊥} = \frac{1}{i}E = \frac{1}{i}(\overline{z}_{j}\overline{∂}_{j} − z_{j}∂_{j}), and E = z_{j}∂_{j} is just the holomorphic radial derivative. Furthermore, ΠE = EΠ and T_{E} = −E. GTo’s that are invariant under the action of the unitary group U(d) of rotations of B^{d} are precisely the diagonal operators

(47)  T_{s} : z^{α} ↦ s_{|α|}z^{α}

on H^{2}(∂B^{d}), with eigenvalue sequence s = (s_{k})_{k=0}^{∞} possessing an asymptotic expansion

s_{k} ∼ ∑_{j=0}^{∞} c_{j}(k + 1)^{m−j} as k → +∞,

with some c_{j} ∈ C, j = 0, 1, 2, …, where m is the order of T_{s} and σ_{m}(T_{s}) = c_{0}t^{m}; alternatively, T_{s} ∼ ∑_{j=0}^{∞} c_{j}(I + E)^{m−j}. One has

(48)  T_{s}T_{z_{k}} = T_{z_{k}}T_{s′},

where T_{s′} is again a rotation-invariant gTo with eigenvalue sequence s′_{k} = s_{k+1}, i.e. s′ = S^{∗}s where S^{∗} is the “backward shift” operator on sequences; alternatively, T_{s′} = ∑_{k=1}^{d} T_{\overline{z}_{k}}T_{s}T_{z_{k}}. In particular, this applies to T_{[ν]} = T_{s_{[ν]}} where, by (46),

(49)  s_{[ν]} = (\frac{Γ(d + k)}{2(ν + 1)_{d+k}} )_{k=0}^{∞},

and so

Q_{kj} = Z_{kj} + \frac{1}{ν} T_{z_{k}}T_{[ν−2]}T_{\overline{z}_{j}} = \frac{δ_{kj}}{ν − 1}T_{[ν−1]} + T_{z_{k}}(\frac{1}{ν} T_{[ν−2]} − D)T_{\overline{z}_{j}}

= \frac{1}{ν − 1}T_{[ν−1]}^{1/2}(δ_{kj}I + T_{z_{k}}T_{s}T_{\overline{z}_{j}})T_{[ν−1]}^{1/2},

with

(50)  s = (ν − 1)(S^{∗}s_{[ν−1]})^{−1}(\frac{1}{ν} s_{[ν−2]} − \frac{2ν−1}{ν(ν−1)}s_{[ν−1]} + \frac{1}{ν−1}s_{[ν]})

(the multiplication and inverse of sequences being understood pointwise; note that any two operators of the form (47) commute). From the simple formula

(51) (δ_{kj}I + T_{z_{k}}AT_{\overline{z}_{j}})(δ_{jm}I + T_{z_{j}}BT_{\overline{z}_{m}}) = δ_{km}I + T_{z_{k}}(A + B + AB)T_{\overline{z}_{m}}

(because ∑_{j=1}^{d} T_{\overline{z}_{j}}T_{z_{j}} = T_{∑_{j} |z_{j}|^{2}} = T_{1} = I) we thus see that the matrix Q = [Q_{kj}]_{j,k=1}^{d} has the inverse S = [S_{jm}]_{j,m=1}^{d} given by the gTo’s

(52)  S_{jm} = (ν − 1)T_{[ν−1]}^{−1/2}(δ_{jm}I − T_{z_{j}}T_{x}T_{\overline{z}_{m}})T_{[ν−1]}^{−1/2}

where x = s/(1 + s) (i.e. x_{k} = s_{k}/(1 + s_{k}) for all k); furthermore, S = X^{∗}X for X = [X_{jm}]_{j,m=1}^{d} where

X = (ν − 1)^{1/2}(δ_{jm}I − T_{z_{j}}T_{y}T_{\overline{z}_{m}})T_{[ν−1]}^{1/2}

with y_{k} = 1 − (1 − x_{k})^{1/2} = 1 − (1 + s_{k})^{−1/2} (note that, by a direct check from (50) and (49), s_{k} > 0 for all k). This offers explicit expressions for the various operators constructed in Propositions 11 and 12. One can even do a little better and show that the positive square root S^{1/2} of the matrix S is a matrix of gTo’s. Namely, using again (48) we can rewrite (41) as

S_{jm} = (ν − 1)(δ_{jm}T_{[ν−1]}^{−1} − T_{z_{j}}T_{w}T_{\overline{z}_{m}})

where w = (S^{∗}s_{[ν−1]})^{−1}x. By (48) and the same computation as in (51),

(δ_{kj}T_{a} − T_{z_{k}}T_{b}T_{\overline{z}_{j}})(δ_{jm}T_{a} − T_{z_{j}}T_{b}T_{\overline{z}_{m}}) = δ_{km}T_{a}^{2} − T_{z_{k}}T_{2(S^{∗}a)b−b^{2}}T_{\overline{z}_{m}}.

Thus taking a = s_{[ν−1]}^{−1/2} and b = S^{∗}a − \sqrt{(S^{∗}a)^{2} − w} we see that the ma-

trix of gTo’s T = (ν − 1)^{1/2}[δ_{jm}T_{[ν−1]}^{−1/2} − T_{z_{j}}T_{b}T_{\overline{z}_{m}}]_{j,m=1}^{d} satisfies T^{2} = S. (Note that (S^{∗}a)^{2} − w = (S^{∗}s_{[ν−1]})^{−1}/(1 + s) is a sequence with positive elements; similarly one checks from T = (ν − 1)^{1/2}T_{[ν−1]}^{−1/4}[δ_{jm} − T_{z_{j}}T_{c}T_{\overline{z}_{m}}]T_{[ν−1]}^{−1/4} with c = (S^{∗}s_{[ν−1]})^{1/2}b = 1 − (1 + s)^{−1/2} ∈ (0, 1) that T is positive selfadjoint as an operator on ⊕^{d}H^{2}(∂B^{d}).)

Remark. In general, it is not true that if T is a positive selfadjoint gTo (not necessarily elliptic), then its positive square root T^{1/2} is also a gTo. A counterexample is furnished by T = T_{\overline{z}_{1}}T_{z_{1}} = T_{|z_{1}|^{2}} on H^{2}(∂B^{d}), d > 1. Namely, if T^{1/2} =: T_{P}, then T_{P} has to be of order 0 with σ_{0}(P)^{2} = |z_{1}|^{2}, which has no solutions in C^{∞}(∂Ω). □

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Miroslav Engliš

Mathematics Institute

Silesian University in 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