New York Journal of Mathematics

New York J. Math. 13 (2007) 299–316.

Schatten class Toeplitz operators on weighted Bergman spaces of the unit ball

Kehe Zhu

Abstract. For positive Toeplitz operators on Bergman spaces of the unit ball, we determine exactly when membership in the Schatten classes S_{p} can be characterized in terms of the Berezin transform.

Contents

1. Introduction 299 2. Preliminaries 302 3. Characterization by μ̂_{r} 305 4. Characterization by μ̃ 312 5. Further generalizations 314 References 316

1. Introduction

Let ℂ^{n} be the n-dimensional complex Euclidean space. For any two points z = (z_{1}, ⋯, z_{n}) and w = (w_{1}, ⋯, w_{n}) in ℂ^{n} we write

⟨z, w⟩ = z_{1}w_{1} + ⋯ + z_{n}w_{n},

and

|z| = \sqrt{⟨z, z⟩} = \sqrt{|z_{1}|^{2} + ⋯ + |z_{n}|^{2}}.

The set

𝔹_{n} = {z ∈ ℂ^{n} : |z| < 1}

is the open unit ball in ℂ^{n}. We denote by dv the usual Lebesgue volume measure on 𝔹_{n}, normalized so that the volume of 𝔹_{n} is 1.

It is well-known that, for a real parameter α, we have

∫_{𝔹_{n}}(1 − |z|^{2})^{α} dv(z) < ∞

299^{ISSN 1076-9803/07}

Received July 25, 2006.

Mathematics Subject Classification. 47B35 and 32A36.

Key words and phrases. Bergman spaces, Toeplitz operators, the Berezin transform, Schatten classes.

Research partially supported by the National Science Foundation.

if and only if α > −1. Throughout this paper we fix a real parameter α with α > −1 and write

dv_{α}(z) = c_{α}(1 − |z|^{2})^{α} dv(z),

where c_{α} is a positive constant chosen so that v_{α}(𝔹_{n}) = 1. The precise value of c_{α} is computable in terms of the gamma function, but it is not important for us.

Let H(𝔹_{n}) denote the space of all holomorphic functions in 𝔹_{n}. The space

A_{α}^{2} = L^{2}(𝔹_{n}, dv_{α}) ∩ H(𝔹_{n})

is called a weighted Bergman space of 𝔹_{n}. It is a Hilbert space with the following inner product inherited from L^{2}(𝔹_{n}, dv_{α}):

⟨f, g⟩ = ∫_{𝔹_{n}} f(z) g(z) dv_{α}(z).

This will be the only inner product we use on A_{α}^{2} and the associated norm on A_{α}^{2} will be denoted by ∥f∥.

The orthogonal projection

P : L^{2}(𝔹_{n}, dv_{α}) → A_{α}^{2}

is an integral operator. In fact,

P f(z) = ∫_{𝔹_{n}} \frac{f(w) dv_{α}(w)}{(1 − ⟨z, w⟩)^{n+1+α}} , f ∈ L^{2}(𝔹_{n}, dv_{α}).

This formula enables us to extend the domain of the operator P, which is usually called a Bergman projection, to the larger space L^{1}(𝔹_{n}, dv_{α}). Thus P is an integral operator that maps L^{1}(𝔹_{n}, dv_{α}) into H(𝔹_{n}).

More generally, if μ is a complex Borel measure on 𝔹_{n} with |μ|(𝔹_{n}) < ∞, we can densely define an integral operator

T_{μ} : A_{α}^{2} → H(𝔹_{n})

by

T_{μ}f(z) = ∫_{𝔹_{n}} \frac{f(w) dμ(w)}{(1 − ⟨z, w⟩)^{n+1+α}} .

This is usually called the Toeplitz operator on A_{α}^{2} with symbol μ. In the case when

dμ(z) = φ(z) dv_{α}(z), φ ∈ L^{1}(𝔹_{n}, dv_{α}),

we write T_{φ} instead of T_{μ}. Naturally, T_{φ} is also called the Toeplitz operator on A_{α}^{2} with symbol φ.

A particularly interesting case is when φ ∈ L^{∞}(𝔹_{n}). In this case we can write

T_{φ}(f) = P (φf), f ∈ A_{α}^{2}.

Since P is an orthogonal projection on L^{2}(𝔹_{n}, dv_{α}), the operator T_{φ} is bounded on A_{α}^{2} with ∥T_{φ}∥ ≤ ∥φ∥_{∞}.

It is well-known that the operator T_{φ}, and more generally, the operator T_{μ}, can turn out to be a bounded linear operator on A_{α}^{2} even when φ or μ does not have any boundedness property. More specifically, we say that T_{μ} is bounded on A_{α}^{2} if there exists a positive constant C such that ∥T_{μ}f∥ ≤ C∥f∥ for all f ∈ H^{∞}(𝔹_{n}), where H^{∞}(𝔹_{n}) consists of all bounded holomorphic functions in 𝔹_{n}. It is easy to check that H^{∞}(𝔹_{n}) is dense in A_{α}^{2}.

When μ is nonnegative, the boundedness, and the compactness as well, of the operator T_{μ} on A_{α}^{2} can be characterized in terms of the following two averaging functions.

For z ∈ 𝔹_{n} we define

μ̃(z) = ∫_{𝔹_{n}} \frac{(1 − |z|^{2})^{n+1+α}}{|1 − ⟨z, w⟩|^{2(n+1+α)}} dμ(w).

The function μ̃ is usually called the Berezin transform of μ. For z ∈ 𝔹_{n} and r > 0 we define

μ̂_{r}(z) = \frac{μ(D(z, r))}{v_{α}(D(z, r))},

where D(z, r) is the Bergman metric ball at z with radius r.

The following result was proved in [5] (for the case of bounded symmetric domains).

Theorem 1. Suppose μ ≥ 0, p ≥ 1, and r > 0. Then:

(a) T_{μ} is bounded on A_{α}^{2} if and only if μ̃ is bounded on 𝔹_{n} if and only if μ̂_{r} is bounded on 𝔹_{n}.

(b) T_{μ} is compact on A_{α}^{2} if and only if μ̃ ∈ ℂ_{0}(𝔹_{n}) if and only if μ̂_{r} ∈ ℂ_{0}(𝔹_{n}).

(c) T_{μ} belongs to the Schatten class S_{p} if and only if μ̃ ∈ L^{p}(𝔹_{n}, dλ) if and only if μ̂_{r} ∈ L^{p}(𝔹_{n}, dλ).

Here ℂ_{0}(𝔹_{n}) is the space of complex-valued continuous functions f on 𝔹_{n} such that f(z) → 0 as |z| → 1^{−}, and

dλ(z) = \frac{dv(z)}{(1 − |z|^{2})^{n+1}}

is the Möbius invariant volume measure on 𝔹_{n}.

On the other hand, in the case of the unit disk 𝔻 in the complex plane ℂ, Daniel Luecking proved the following result in [2].

Theorem 2. Suppose μ ≥ 0, r > 0, and p > 0. Then T_{μ} belongs to the Schatten class S_{p} if and only if {μ̂_{r}(a_{k})} ∈ l^{p}, where {a_{k}} is any hyperbolic lattice in the unit disk.

It was observed in [3] that the condition {μ̂_{r}(a_{k})} ∈ l^{p} is equivalent to μ̂_{r} ∈ L^{p}(𝔻, dλ), although no details were given there.

A natural problem arises now, namely, does part (c) in Theorem 1 remain valid for 0 < p < 1 in the context of the unit ball? We completely settle the problem in this paper. It turns out that the answer is affirmative in the case of the averaging function μ̂_{r}; this is not surprising in view Luecking’s theorem above for the unit disk. However, the answer is trickier in the case of the Berezin transform μ̃; this part of the solution is even new for the unit disk.

We state the main result of the paper as follows.

Theorem 3. Suppose μ ≥ 0, r > 0, and 0 < p < 1. Then:

(a) T_{μ} belongs to the Schatten class S_{p} if and only if μ̂_{r} ∈ L^{p}(𝔹_{n}, dλ).

(b) If p > n/(n + 1 + α), then T_{μ} belongs to the Schatten class S_{p} if and only if μ̃ ∈ L^{p}(𝔹_{n}, dλ).

(c) The cut-off point n/(n + 1 + α) above is best possible.

In the next secton we gather the necessary technical results that will be needed for the proof of our main result. Section 3 is devoted to the generalization of Luecking’s theorem to the unit ball. The characterization of Schatten class Toeplitz operators in terms of the Berezin transform is obtained in Section 4. We conclude the paper in Section 5 with an extension of our results to products of balls, and in particular, to the polydisk in ℂ^{n}.

We are grateful to Boo Rim Choe and Hyungwoon Koo for detecting a mistake in the first draft of the paper and for showing us how to correct the problem.

2. Preliminaries

All results in this section are known; we include them here for convenience of reference. The references given here are not necessarily the original ones. We begin with the following integral estimate which has become indispensible in this area of analysis.

Lemma 4. Suppose t > −1 and

I(z) = ∫_{𝔹_{n}} \frac{(1 − |w|^{2})^{t} dv(w)}{|1 − ⟨z, w⟩|^{n+1+t+s}} .

If s > 0, then there exists a positive constant C such that

I(z) ≤ \frac{C}{(1 − |z|^{2})^{s}} , z ∈ 𝔹_{n}.

If s < 0, then there exists a positive constant C such that I(z) ≤ C for all z ∈ 𝔹_{n}.

Proof. See [4] or [7]. □

Let Aut(𝔹_{n}) denote the automorphism group of the unit ball. It is well-known that Aut(𝔹_{n}) is generated by two special classes of automorphisms: the unitary transformations and the involutive automorphisms. Nothing needs to be said about the unitary mappings. For every z ∈ 𝔹_{n} there exists a unique automorphism φ_{z} of 𝔹_{n} such that φ_{z}(0) = z and φ_{z} ∘ φ_{z}(w) = w for all w ∈ 𝔹_{n}. These maps φ_{z} are called involutions of 𝔹_{n}, or involutive automorphisms. Explicit formulas are known for them; see [4] or [7]. For example, in the case of the unit disk, we have

φ_{z}(w) = \frac{z − w}{1 − zw} .

Recall that the Bergman metric on the unit ball is given by

β(z, w) = \frac{1}{2} log \frac{1 + |φ_{z}(w)|}{1 − |φ_{z}(w)|}.

For any z ∈ 𝔹_{n} and r > 0 we introduce the Bergman metric ball

D(z, r) = {w ∈ 𝔹_{n} : β(w, z) < r}.

It is well-known that if r is fixed, then the weighted volume v_{α}(D(z, r)) is comparable to (1 − |z|^{2})^{n+1+α}. See [7] for example.

A sequence {a_{k}} in 𝔹_{n} is called an r-lattice in the Bergman metric if the following conditions are satisfied:

(a) The unit ball is covered by the Bergman metric balls {D(a_{k}, r)}.

(b) β(a_{i}, a_{j}) ≥ r/2 for all i and j with i ≠ j. If {a_{k}} is an r-lattice in 𝔹_{n}, then it also has the following properties:

(c) For any R > 0 there exists a positive integer N_{1} (depending on r and R) such that every point in 𝔹_{n} belongs to at most N_{1} sets in {D(a_{k}, R)}.

(d) For any R > 0 there exists a decomposition of {a_{k}} into a finite number of sequences {a_{jk}}, 1 ≤ j ≤ N_{2}, such that β(a_{jk}, a_{jm}) ≥ R for all k ≠ m.

There are elementary constructions of r-lattices in 𝔹_{n}.

The following result is usually referred to as the atomic decomposition for Bergman spaces. See [1] and [7].

Theorem 5. For any b > n + (α + 1)/2 there exists a positive constant r_{0} with the following property: if {a_{k}} is any r-lattice in 𝔹_{n} with r < r_{0}, then the Bergman space A_{α}^{2} consists exactly of functions of the form

(1)  f(z) = ∑_{k=1}^{∞} c_{k} \frac{(1 − |a_{k}|^{2})^{b−(n+1+α)/2}}{(1 − ⟨z, a_{k}⟩)^{b}} ,

where {c_{k}} ∈ l^{2}. Moreover, ∥f∥^{2} is comparable to

∥f∥_{∗}^{2} = inf {∑_{k=1} |c_{k}|^{2} : {c_{k}} satisfies (1)} .

As a matter of fact, if {a_{k}} is any r-lattice in the Bergman metric, and if {c_{k}} is any sequence in l^{2}, then the function f defined in (1) belongs to A_{α}^{2} and

∥f∥^{2} ≤ C ∑_{k=1}^{∞} |c_{k}|^{2}

for some positive constant C that is independent of {c_{k}}. This part of Theorem 5 does not require r to be small.

We will also need the following estimate for the Bergman kernel function.

Lemma 6. Suppose R is a positive radius and b is any real number. Then there exists a positive constant C such that

|\frac{(1 − ⟨z, u⟩)^{b}}{(1 − ⟨z, v⟩)^{b}} − 1| ≤ Cβ(u, v)

for all z, u, and v in 𝔹_{n} with β(u, v) ≤ R.

As a consequence, we easily see that for any positive R there exists a positive constant C such that

(2)  C^{−1} ≤ \frac{|1 − ⟨z, u⟩|}{|1 − ⟨z, v⟩|} ≤ C

for all z, u, and v in 𝔹_{n} with β(u, v) ≤ R.

Lemma 7. Suppose p > 0 and r > 0. Then there exists a positive constant C such that

|f(z)|^{p} ≤ \frac{C}{v(D(z, r))} ∫_{D(z,r)} |f(w)|^{p} dv(w)

for all f ∈ H(𝔹_{n}) and all z ∈ 𝔹_{n}.

We will also need a few results concerning Schatten class operators on a separable Hilbert space. Recall that if T is a positive, compact operator on a separable Hilbert space H, then there exists an orthonormal set {e_{k}} in H and a sequence {λ_{k}} that decreases to 0 such that

T x = ∑_{k} λ_{k}⟨x, e_{k}⟩e_{k}

for all x ∈ H. This is called the canonical decomposition of T and the numbers λ_{k} are called the singular values of T. A positive operator T belongs to the Schatten class S_{p}, where p > 0, if the sequence {λ_{k}} of its singular values belongs to the sequence space l^{p}. In this case, we write

∥T ∥_{p} = [∑_{k} λ_{k}^{p}]^{1/p} .

More generally, a compact (not necessarily positive) operator T on H belongs to the Schatten class S_{p} if the positive operator |T | = (T^{∗}T)^{1/2} belongs to S_{p}. In this case, we define ∥T ∥_{p} = ∥|T |∥_{p}.

Lemma 8. Suppose A is a bounded surjective operator on H and T is any bounded linear operator on H. Then T ∈ S_{p} if and only if A^{∗}T A ∈ S_{p}.

Proof. Each Schatten class S_{p} is an ideal in the full algebra of bounded linear operators on H. Therefore, if T ∈ S_{p}, then the operator S = A^{∗}T A is also in S_{p}. On the other hand, since A is surjective, it has bounded right inverse. Thus there exists a bounded linear operator B on H such that AB = I, the identity operator on H. It follows that T = B^{∗}SB, so S ∈ S_{p} implies T ∈ S_{p}. □

Lemma 9. Suppose T is a bounded linear operator on H and {e_{k}} is any orthonormal basis of H. Then for any 0 < p ≤ 2 we have

∥T ∥_{p}^{p} ≤ ∑_{i=1}^{∞} ∑_{j=1}^{∞} |⟨T e_{i}, e_{j}⟩|^{p}.

Proof. See [2] for example. □

Lemma 10. Suppose T is a positive operator on H and {e_{k}} is an orthonormal basis for H. If 0 < p < 1 and

∑_{k=1}^{∞}⟨T e_{k}, e_{k}⟩^{p} < ∞,

then T belongs to the Schatten class S_{p}.

Proof. See [6] for example. □

In the rest of the paper we use C to denote a positive constant whose value may change from one occurence to another.

3. Characterization by μ̂_{r}

In this section we characterize Schatten class Toeplitz operators T_{μ} on A_{α}^{2} with positive symbols μ based on the averaging function μ̂_{r}. This was already done in [5] for the unit ball when p ≥ 1, and in [2] for the unit disk when p > 0. So the important case here is when 0 < p < 1. The ideas of this section are clearly from [2].

Lemma 11. Suppose μ ≥ 0, r > 0, 0 < p < 1, and μ̂_{2r} ∈ L^{p}(𝔹_{n}, dλ). Then T_{μ} ∈ S_{p}.

Proof. Fix an r-lattice {a_{k}} in the Bergman metric of 𝔹_{n}. If z ∈ D(a_{k}, r), then D(a_{k}, r) ⊂ D(z, 2r) by the triangle inequality. Since v_{α}(D(a_{k}, r)) is comparable to v_{α}(D(z, 2r)) whenever z ∈ D(a_{k}, r), we can find a positive constant C, independent of k, such that

μ̂_{r}(a_{k}) = \frac{μ(D(a_{k}, r))}{v_{α}(D(a_{k}, r))} ≤ C \frac{μ(D(z, 2r))}{v_{α}(D(z, 2r))} = Cμ̂_{2r}(z)

for all z ∈ D(a_{k}, r). Also, for any fixed r > 0, there exists a constant C > 0 such that

C^{−1} ≤ λ(D(z, r)) ≤ C

for all z ∈ 𝔹_{n}. It follows that there exists a constant C > 0 such that

μ̂_{r}(a_{k})^{p} ≤ C ∫_{D(a_{k},r)} μ̂_{2r}(z)^{p} dλ(z)

for all k. Recall that every point of 𝔹_{n} belongs to at most N of the sets D(a_{k}, r). So

∑_{k=1}^{∞} μ̂_{r}(a_{k})^{p} ≤ C ∑_{k=1}^{∞} ∫_{D(a_{k},r)} μ̂_{2r}(z)^{p} dλ(z)

≤ CN ∫_{𝔹_{n}} μ̂_{2r}(z)^{p} dλ(z) < ∞.

Fix a positive constant b > n+(n+1+α)/2. If 0 < r_{1} < r_{2} and μ̂_{r_{2}} ∈ L^{p}(𝔹_{n}, dλ), then clearly μ̂_{r_{1}} ∈ L^{p}(𝔹_{n}, dλ). So by shrinking r if necessary, we may assume that atomic decomposition for A_{α}^{2} (Theorem 5) already holds on the lattice {a_{k}}.

Fix an orthonormal basis {e_{k}} for A_{α}^{2} and define an operator

A : A_{α}^{2} → A_{α}^{2}

by

A (∑_{k=1}^{∞} c_{k}e_{k}) = ∑_{k=1}^{∞} c_{k}h_{k},

where

h_{k}(z) = \frac{(1 − |a_{k}|^{2})^{b−(n+1+α)/2}}{(1 − ⟨z, a_{k}⟩)^{b}} .

By Theorem 5, A is a bounded and surjective operator on A_{α}^{2}. According to Lemma 8, the Toeplitz operator T_{μ} will be in S_{p} if we can show that the operator T = A^{∗}T_{μ}A belongs to S_{p}. By Lemma 10, we just need to verify the following condition:

(3)  S = ∑_{k=1}^{∞}⟨T e_{k}, e_{k}⟩^{p} < ∞.

It is clear that

⟨T e_{k}, e_{k}⟩ = ⟨T_{μ}h_{k}, h_{k}⟩ = ∫_{𝔹_{n}} |h_{k}(z)|^{2} dμ(z).

Since {D(a_{j}, r)} is an open cover of 𝔹_{n}, we have

⟨T e_{k}, e_{k}⟩ ≤ ∑_{j=1}^{∞} ∫_{D(a_{j},r)} |h_{k}(z)|^{2} dμ(z).

By Lemma 6, there exists a positive constant C such that

⟨T e_{k}, e_{k}⟩ ≤ C ∑_{j=1}^{∞} |h_{k}(a_{j})|^{2}μ(D(a_{j}, r)).

Since 0 < p < 1, an application of Hölder’s inequality gives

⟨T e_{k}, e_{k}⟩^{p} ≤ C ∑_{j=1}^{∞} |h_{k}(a_{j})|^{2p}μ(D(a_{j}, r))^{p}.

Recall that v_{α}(D(a_{j}, r)) is comparable to (1 − |a_{j}|^{2})^{n+1+α}. So there is another constant C > 0 such that

⟨T e_{k}, e_{k}⟩^{p} ≤ C ∑_{j=1}^{∞}(1 − |a_{j}|^{2})^{p(n+1+α)}|h_{k}(a_{j})|^{2p} μ̂_{r}(a_{j})^{p}.

By Fubini’s theorem, we have

S ≤ C ∑_{j=1}^{∞}(1 − |a_{j}|^{2})^{p(n+1+α)} μ̂_{r}(a_{j})^{p} ∑_{k=1}^{∞} |h_{k}(a_{j})|^{2p}.

For each j ≥ 1 we consider the sum

S_{j} = ∑_{k=1}^{∞} |h_{k}(a_{j})|^{2p} = ∑_{k=1}^{∞} \frac{(1 − |a_{k}|^{2})^{p(2b−n−1−α)}}{|1 − ⟨a_{j}, a_{k}⟩|^{2pb}} .

By Lemma 7, there is a constant C > 0 such that

\frac{1}{|1 − ⟨a_{j}, a_{k}⟩|^{2pb}} ≤ \frac{C}{v(D(a_{k}, r))} ∫_{D(a_{k},r)} \frac{dv(z)}{|1 − ⟨a_{j}, z⟩|^{2pb}}

for all j and k. Since v(D(a_{k}, r)) is comparable to (1 − |a_{k}|^{2})^{n+1}, and 1 − |z|^{2} is comparable to 1 − |a_{k}|^{2} for z ∈ D(a_{k}, r), we obtain

S_{j} ≤ C ∑_{k=1}^{∞} ∫_{D(a_{k},r)} \frac{(1 − |z|^{2})^{p(2b−n−1−α)−(n+1)}}{|1 − ⟨a_{j}, z⟩|^{2pb}} dv(z).

Since every point of 𝔹_{n} belongs to at most N of the sets D(a_{k}, r), we obtain

S_{j} ≤ CN ∫_{𝔹_{n}} \frac{(1 − |z|^{2})^{p(2b−n−1−α)−(n+1)}}{|1 − ⟨a_{j}, z⟩|^{2pb}} dv(z).

By Lemma 4, there is a positive constant C such that

S_{j} ≤ \frac{C}{(1 − |a_{j}|^{2})^{p(n+1+α)}}

for all j ≥ 1. It follows that

S ≤ C ∑_{j=1}^{∞} μ̂_{r}(a_{j})^{p} < ∞.

This completes the proof of Lemma 11. □

Lemma 12. Suppose T_{μ} ∈ S_{p}, 0 < p < 1, and {a_{k}} is an r-lattice in the Bergman metric of 𝔹_{n}. Then the sequence {μ̂_{r}(a_{k})} is in l^{p}.

Proof. Fix a sufficiently large positive radius R and partition the lattice {a_{k}} into N subsequences such that the Bergman metric between any two points in each subsequence is at least R. Let {ζ_{j}} be such a subsequence and define a measure ν on 𝔹_{n} as follows:

dν(z) = ∑_{k=1}^{∞} χ_{k}(z) dμ(z),

where χ_{k} is the characteristic function of D(ζ_{k}, r). We assume that R > 2r, so that the Bergman metric balls {D(ζ_{k}, r)} are disjoint.

Since T_{μ} ∈ S_{p} and 0 ≤ ν ≤ μ, we must also have T_{ν} ∈ S_{p}. In fact, 0 ≤ T_{ν} ≤ T_{μ} implies 0 ≤ T_{ν}^{p} ≤ T_{μ}^{p}, which in turn implies that ∥T_{ν}∥_{p} ≤ ∥T_{μ}∥_{p}.

Fix an orthonormal basis {e_{k}} for A_{α}^{2} and define an operator A on A_{α}^{2} by

A [∑_{k=1}^{∞} c_{k}e_{k}] = ∑_{k=1}^{∞} c_{k}h_{k},

where b is a sufficiently large positive constant and

h_{k}(z) = \frac{(1 − |ζ_{k}|^{2})^{(2b−n−1−α)/2}}{(1 − ⟨z, ζ_{k}⟩)^{b}} .

Since T_{ν} ∈ S_{p}, we also have T = A^{∗}T_{ν}A ∈ S_{p} with

∥T ∥_{p} ≤ ∥A∥^{2}∥T_{ν}∥_{p}.

Here the boundedness of A on A_{α}^{2} follows from the remarks after Theorem 5; all that is needed is the fact that {ζ_{k}} is separated in the Bergman metric. In fact, we can find a positive constant C (that only depends on r, b, and α, but not on the particular subsequence {ζ_{k}} of {a_{k}} being used) such that

∥T ∥_{p}^{p} ≤ C∥T_{μ}∥_{p}^{p}.

We split the operator T as T = D + E, where D is the diagonal operator on A_{α}^{2} defined by

Df = ∑_{k=1}^{∞}⟨T e_{k}, e_{k}⟩⟨f, e_{k}⟩e_{k}, f ∈ A_{α}^{2},

and E = T − D. By the triangle inequality, we have

(4)  ∥T ∥_{p}^{p} ≥ ∥D∥_{p}^{p} − ∥E∥_{p}^{p}.

Since D is a positive diagonal operator, we have

∥D∥_{p}^{p} = ∑_{k=1}^{∞}⟨T e_{k}, e_{k}⟩^{p} = ∑_{k=1}^{∞}⟨T_{ν}h_{k}, h_{k}⟩^{p}

= ∑_{k=1}^{∞} [∫_{𝔻} |h_{k}(z)|^{2} dν(z)]^{p}

≥ ∑_{k=1}^{∞} [∫_{D(ζ_{k},r)} |h_{k}(z)|^{2} dν(z)]^{p}

≥ C ∑_{k=1}^{∞} ν̂_{r}(ζ_{k})^{p}.

The last inequality follows from (2). In particular, the constant C only depends on r, b, and α. Since ν = μ on each D(ζ_{k}, r), we obtain

(5)  ∥D∥_{p}^{p} ≥ C_{1} ∑_{k=1}^{∞} μ̂_{r}(ζ_{k})^{p}.

On the other hand, according to Lemma 9, we have

∥E∥_{p}^{p} ≤ ∑_{k=1}^{∞} ∑_{j=1}^{∞} |⟨Ee_{j}, e_{k}⟩|^{p} = ∑_{j≠k} |⟨T_{ν}h_{j}, h_{k}⟩|^{p}

= ∑_{j≠k} |∫_{𝔹_{n}} h_{j}(z)h_{k}(z) dν(z)|^{p}

≤ ∑_{j≠k} [∫_{𝔹_{n}} |h_{j}(z)h_{k}(z)| dν(z)]^{p} .

For any j and k we write

I_{jk} = ∫_{𝔹_{n}} |h_{j}(z)h_{k}(z)| dν(z).

Then

I_{jk} = ∑_{i=1}^{∞} ∫_{D(ζ_{i},r)} |h_{j}(z)h_{k}(z)| dμ(z),

and by Lemma 6, there exists a positive constant C (depending on r, b, and α) such that

I_{jk} ≤ C ∑_{i=1}^{∞} |h_{j}(ζ_{i})h_{k}(ζ_{i})|μ(D(ζ_{i}, r))

≤ C ∑_{i=1}^{∞}(1 − |ζ_{i}|^{2})^{n+1+α}|h_{j}(ζ_{i})h_{k}(ζ_{i})|μ̂_{r}(ζ_{i}).

Here in the last inequality we used the fact that v_{α}(D(ζ_{i}, r)) is comparable to (1 − |ζ_{i}|^{2})^{n+1+α}.

Schatten class Toeplitz operators 309

Since 0 < p < 1, we apply Hölder’s inequality to get

|I_{jk}|^{p} ≤ C ∑_{i=1}^{∞}(1 − |ζ_{i}|^{2})^{p(n+1+α)}|h_{j}(ζ_{i})h_{k}(ζ_{i})|^{p}μ̂_{r}(ζ_{i})^{p}.

Combining this with Fubini’s theorem, we obtain

∥E∥_{p}^{p} ≤ C ∑_{i=1}^{∞}(1 − |ζ_{i}|^{2})^{p(n+1+α)}μ̂_{r}(ζ_{i})^{p}I_{i},

where

I_{i} = ∑_{j≠k} |h_{j}(ζ_{i})h_{k}(ζ_{i})|^{p}

= ∑_{j≠k} \frac{[(1 − |ζ_{j}|^{2})(1 − |ζ_{k}|^{2})]^{pb−p(n+1+α)/2}}{[|1 − ⟨ζ_{j}, ζ_{i}⟩||1 − ⟨ζ_{k}, ζ_{i}⟩|]^{pb}}

for all i ≥ 1. Let

Ω = ⋃_{j≠k} D(ζ_{n}, r) × D(ζ_{k}, r) ⊂ 𝔹_{n} × 𝔹_{n}.

Since the union above is a disjoint one, we can find a positive constant C (depending on r, b, and α) such that

I_{i} ≤ C∫∫_{Ω} \frac{[(1 − |z|^{2})(1 − |w|^{2})]^{pb−n−1−p(n+1+α)/2}}{[|1 − ⟨z, ζ_{i}⟩||1 − ⟨w, ζ_{i}⟩|]^{pb}} dv(z) dv(w).

By assumption, we have

Ω ⊂ G_{R} = {(z, w) ∈ 𝔹_{n} × 𝔹_{n} : β(z, w) ≥ R − 2r}.

Therefore,

I_{i} ≤ C∫∫_{G_{R}} \frac{[(1 − |z|^{2})(1 − |w|^{2})]^{pb−p(n+1+α)/2}}{[|1 − ⟨z, ζ_{i}⟩||1 − ⟨w, ζ_{i}⟩|]^{pb}} dλ(z) dλ(w).

Making the change of variables z = φ_{ζ_{i}}(u) and w = φ_{ζ_{i}}(v), we obtain

I_{i} ≤ C(1 − |ζ_{i}|^{2})^{−p(n+1+α)}∫∫_{G_{R}}F (u, v) dv(u) dv(v),

where

F (u, v) = \frac{[(1 − |u|^{2})(1 − |v|^{2})]^{pb−n−1−p(n+1+α)/2}}{[|1 − ⟨u, ζ_{i}⟩||1 − ⟨v, ζ_{i}⟩|]^{pb−p(n+1+α)}} .

We can assume b is large enough so that

pb ≥ n + 1 + p(n + 1 + α)/2.

Since

(1 − |u|^{2})(1 − |v|^{2}) ≤ 4|1 − ⟨u, ζ_{i}⟩||1 − ⟨v, ζ_{i}⟩|,

there exists another positive constant C (independent of R) such that

I_{i} ≤ C(1 − |ζ_{i}|^{2})^{−p(n+1+α)}∫∫_{G_{R}} \frac{dv(u) dv(v)}{[|1 − ⟨u, ζ_{i}⟩||1 − ⟨v, ζ_{i}⟩|]^{n+1−p(n+1+α)/2}} .

Choose t ∈ (1, ∞) and s ∈ (1, ∞) (independent of R) such that

A = t [n + 1 − \frac{p(n + 1 + α)}{2}] < n + 1, \frac{1}{t} + \frac{1}{s} = 1.

By Hölder’s inequality, the integral

∫∫_{G_{R}} \frac{dv(u) dv(v)}{[|1 − ⟨u, ζ_{i}⟩||1 − ⟨v, ζ_{i}⟩|]^{n+1−p(n+1+α)/2}}

is less than or equal to

[∫_{𝔹_{n}} ∫_{𝔹_{n}} \frac{dv(u) dv(v)}{|1 − ⟨u, ζ_{i}⟩|^{A}|1 − ⟨v, ζ_{i}⟩|^{A}} ]^{\frac{1}{t}} [v^{2}(G_{R})]^{\frac{1}{s}} ,

where

v^{2}(G_{R}) = ∫∫_{G_{R}} dv(u) dv(v).

Since A < n+1, we apply Lemma 4 to find another positive constant C (independent of R) such that

I_{i} ≤ C(1 − |ζ_{i}|^{2})^{−p(n+1+α)} [v^{2}(G_{R})]^{\frac{1}{s}} .

We conclude that there exists a constant C_{2} > 0 (independent of R) such that

∥E∥_{p}^{p} ≤ C_{2} [v^{2}(G_{R})]^{\frac{1}{s}} ∑_{i=1}^{∞} μ̂_{r}(ζ_{i})^{p}.

Combining this with (4) and (5), we obtain

∥T ∥_{p}^{p} ≥ (C_{1} − C_{2} [v^{2}(G_{R})]^{\frac{1}{s}} ) ∑_{i=1}^{∞} μ̂_{r}(ζ_{i})^{p},

where C_{1} and C_{2} are positive constant independent of R. If we chose R large enough so that

C_{1} − C_{2} [v^{2}(G_{R})]^{\frac{1}{s}} > 0,

then we could find a positive constant C (independent of μ) such that

∑_{i=1}^{∞} μ̂_{r}(ζ_{i})^{p} ≤ C∥T_{μ}∥_{p}^{p}.

Since this holds for each one of the N subsequences of {a_{n}}, we obtain

(6)  ∑_{n=1}^{∞} μ̂_{r}(a_{n})^{p} ≤ CN∥T_{μ}∥_{p}^{p}

for all positive Borel measures μ such that

∑_{n=1}^{∞} μ̂_{r}(a_{n})^{p} < ∞.

An easy approximation argument then shows that (6) actually holds for all positive Borel measures μ. This completes the proof of Lemma 12. □

Lemma 13. Suppose μ is a positive Borel measure on 𝔹_{n} and r > 0. If for every 2r-lattice {a_{k}} in the Bergman metric we have

∑_{k=1} μ̂_{2r}(a_{k})^{p} < ∞,

then μ̂_{r} ∈ L^{p}(𝔹_{n}, dλ).

Proof. Fix an r-lattice {a_{k}} in the Bergman metric and partition it into N subsequences {a_{jk}}, 1 ≤ j ≤ N, such that each subsequence is a 2r-lattice in the Bergman metric. We have

∑_{k=1}^{∞} μ̂_{2r}(a_{k})^{p} = ∑_{j=1}^{N} ∑_{k=1}^{∞} μ̂_{2r}(a_{jk})^{p} < ∞,

because each subsequence {a_{jk}} is a 2r-lattice. We now show that the integral

I = ∫_{𝔹_{n}} μ̂_{r}(z)^{p} dλ(z)

is finite.

Since {D(a_{k}, r)} is an open cover of 𝔹_{n}, we have

I ≤ ∑_{k=1}^{∞} ∫_{D(a_{k},r)} μ̂_{r}(z)^{p} dλ(z)

≤ ∑_{k=1}^{∞} λ(D(a_{k}, r)) sup{μ̂_{r}(z)^{p} : z ∈ D(a_{k}, r)}.

There exists a positive constant C such that λ(D(a_{k}, r)) ≤ C for all k ≥ 1. Also, if z ∈ D(a_{k}, r), then by the triangle inequality, D(z, r) ⊂ D(a_{k}, 2r). Combining this with the fact that v_{α}(D(a_{k}, 2r)) and v_{α}(D(z, r)) are comparable whenever z ∈ D(a_{k}, r), we can find another positive constant C such that

I ≤ C ∑_{k=1}^{∞} μ̂_{2r}(a_{k})^{p} < ∞.

This completes the proof of Lemma 13. □

As a consequence of the three lemmas above, we have proved the main result of the section.

Theorem 14. Suppose μ is a positive Borel measure on 𝔹_{n}, 0 < p < 1, and r > 0. Then the following conditions are equivalent.

(a) T_{μ} ∈ S_{p}.

(b) μ̂_{r} ∈ L^{p}(𝔹_{n}, dλ).

(c) {μ̂_{r}(a_{k})} ∈ l^{p} for some r-lattice {a_{k}}.

(d) {μ̂_{r}(a_{k})} ∈ l^{p} for every r-lattice {a_{k}}.

Since the condition T_{μ} ∈ S_{p} does not involve r, we see that μ̂_{r} ∈ L^{p}(𝔹_{n}, dλ) for some r > 0 if and only if μ̂_{r} ∈ L^{p}(𝔹_{n}, dλ) for every r > 0. The same remark applies to the characterizations in (c) and (d) above.

4. Characterization by μ̃

In this section we consider the problem of characterizing membership of T_{μ} in the Schatten classes by integral properties of the Berezin transform μ̃. It turns out that this cannot be done for the full range 0 < p < 1. We will exhibit an obvious obstruction, and we will then show that this obstruction is the only one.

First observe that if μ is a positive Borel measure with compact support in 𝔹_{n}, then the function μ̂_{r}(z) is also compactly supported in 𝔹_{n}, so the condition

∫_{𝔹_{n}} μ̂_{r}(z)^{p} dλ(z) < ∞

is satisfied for all p > 0. By Theorem 14, the operator T_{μ} belongs to the Schatten class S_{p} for every p > 0.

On the other hand, if μ is any positive Borel measure on 𝔹_{n} with μ(𝔹_{n}) > 0, then an elementary estimate shows that

μ̃(z) = (1 − |z|^{2})^{n+1+α} ∫_{𝔹_{n}} \frac{dμ(w)}{|1 − ⟨z, w⟩|^{2(n+1+α)}}

≥ \frac{μ(𝔹_{n})}{4^{n+1+α}} (1 − |z|^{2})^{n+1+α}.

It follows that

∫_{𝔹_{n}} μ̃(z)^{p} dλ(z) = ∞

whenever p(n + 1 + α) ≤ n. Therefore, in the range 0 < p ≤ n/(n + 1 + α), it is not possible to characterize the membership of T_{μ} in S_{p} in terms of the Berezin transform μ̃. Our next result shows that this is the only obstruction.

Theorem 15. Suppose μ is a positive Borel measure on 𝔹_{n} and

n/(n + 1 + α) < p < 1.

Then T_{μ} ∈ S_{p} if and only if μ̃ ∈ L^{p}(𝔹_{n}, dλ).

Proof. Fix any positive radius r. For any z ∈ 𝔹_{n} we have

μ̃(z) = ∫_{𝔹_{n}} \frac{(1 − |z|^{2})^{n+1+α}}{|1 − ⟨z, w⟩|^{2(n+1+α)}} dμ(w)

≥ ∫_{D(z,r)} \frac{(1 − |z|^{2})^{n+1+α}}{|1 − ⟨z, w⟩|^{2(n+1+α)}} dμ(w).

By Lemma 6, there exists a positive constant C such that

\frac{(1 − |z|^{2})^{n+1+α}}{|1 − ⟨z, w⟩|^{2(n+1+α)}} ≥ \frac{C}{v_{α}(D(z, r))}

for all z ∈ 𝔹_{n} and all w ∈ D(z, r). It follows that Cμ̂_{r}(z) ≤ μ̃(z) for all z ∈ 𝔹_{n}. So the condition μ̃ ∈ L^{p}(𝔹_{n}, dλ) implies μ̂_{r} ∈ L^{p}(𝔹_{n}, dλ), which, in view of Theorem 14, implies that T_{μ} ∈ S_{p}. This argument works regardless of the range of p.

Next we suppose that T_{μ} ∈ S_{p}. Fix an r-lattice {a_{k}} in the Bergman metric and estimate the Berezin transform μ̃ as follows.

μ̃(z) = ∫_{𝔹_{n}} \frac{(1 − |z|^{2})^{n+1+α}}{|1 − ⟨z, w⟩|^{2(n+1+α)}} dμ(w)

≤ ∑_{k=1}^{∞} ∫_{D(a_{k},r)} \frac{(1 − |z|^{2})^{n+1+α}}{|1 − ⟨z, w⟩|^{2(n+1+α)}} dμ(w)

≤ C ∑_{k=1}^{∞} \frac{(1 − |z|^{2})^{n+1+α}}{|1 − ⟨z, a_{k}⟩|^{2(n+1+α)}} μ(D(a_{k}, r)).

The last step above follows from Lemma 6. Since v_{α}(D(a_{k}, r)) is comparable to (1 − |a_{k}|^{2})^{n+1+α}, we obtain another positive constant C such that

μ̃(z) ≤ C ∑_{k=1}^{∞} \frac{(1 − |z|^{2})^{n+1+α}(1 − |a_{k}|^{2})^{n+1+α}}{|1 − ⟨z, a_{k}⟩|^{2(n+1+α)}} μ̂_{r}(a_{k})

= C ∑_{k=1}^{∞} (1 − |φ_{a_{k}}(z)|^{2})^{n+1+α} μ̂_{r}(a_{k}).

Here we have used the well-known identity

1 − |φ_{a}(z)|^{2} = \frac{(1 − |a|^{2})(1 − |z|^{2})}{|1 − ⟨z, a⟩|^{2}}, a, w ∈ 𝔹_{n},

which can be found in [4] and [7] for example.

When 0 < p < 1, an application of Hölder’s inequality leads to

μ̃(z)^{p} ≤ C ∑_{k=1}^{∞} (1 − |φ_{a_{k}}(z)|^{2})^{p(n+1+α)} μ̂_{r}(a_{k})^{p}.

It follows that

∫_{𝔹_{n}} μ̃(z)^{p} dλ(z) ≤ C ∑_{k=1}^{∞} μ̂_{r}(a_{k})^{p} ∫_{𝔹_{n}}(1 − |φ_{a_{k}}(z)|^{2})^{p(n+1+α)} dλ(z).

Since λ is invariant under the action of automorphisms, we have

∫_{𝔹_{n}}(1 − |φ_{a_{k}}(z)|^{2})^{p(n+1+α)} dλ(z) = ∫_{𝔹_{n}}(1 − |z|^{2})^{p(n+1+α)} dλ(z)

for all k. The last integral above can be written as

∫_{𝔹_{n}}(1 − |z|^{2})^{p(n+1+α)−(n+1)} dv(z),

which is finite because of the assumption that p(n + 1 + α) > n.

Therefore, there exists a positive constant C such that

∫_{𝔹_{n}} μ̃(z)^{p} dλ(z) ≤ C ∑_{k=1}^{∞} μ̂_{r}(a_{k})^{p}.

This combined with Lemma 12 shows that the condition T_{μ} ∈ S_{p} implies the condition μ̃ ∈ L^{p}(𝔹_{n}, dλ). The proof of Theorem 15 is now complete. □

Once again, we recall that Theorem 15 above was shown in [5] to hold for all p ≥ 1 as well. We have now completed the proof of our main result which was stated as Theorem 3 in the introduction.

5. Further generalizations

We can combine the main results of [5], [2], and the previous sections as follows.

Theorem 16. Suppose μ is a positive Borel measure on 𝔹_{n}, 0 < p < ∞, and 0 < r < ∞. Then the following conditions are equivalent.

(a) T_{μ} ∈ S_{p}.

(b) μ̂_{r} ∈ L^{p}(𝔹_{n}, dλ).

(c) {μ̂_{r}(a_{k})} ∈ l^{p} for every r-lattice {a_{k}}.

(d) {μ̂_{r}(a_{k})} ∈ l^{p} for some r-lattice {a_{k}}. Moreover, if p > n/(n + 1 + α), then the above conditions are also equivalent to (e) μ̃ ∈ L^{p}(𝔹_{n}, dλ).

Proof. Everything has been proved except conditions (c) and (d) for p ≥ 1 in the case of higher dimensions. But this follows from exactly the same arguments used in [2] together with the high-dimensional preliminaries included in Section 2. We leave the details to the interested reader. □

Our main result remains valid for certain other domains in ℂ^{n}. In particular, our result holds for the polydisk in ℂ^{n}. We will now make this precise.

Suppose n = n_{1} + ⋯ + n_{m}, where each n_{k} is a positive integer. Let

Ω = ∏_{k=1}^{m} 𝔹_{n_{k}}

be the product of m unit balls. When each n_{k} = 1, the resulting domain is the polydisk in ℂ^{n}.

Suppose α_{k} > −1 for each 1 ≤ k ≤ m. Consider the measure

dv_{α}(Z) = dv_{α_{1}}(Z_{1}) ⋯ dv_{α_{m}}(Z_{m}),

where a point Z ∈ ℂ^{n} is written as Z = (Z_{1}, ⋯, Z_{m}), with each Z_{k} ∈ ℂ^{n_{k}}, and dv_{α_{k}} is the normalized volume measure on 𝔹_{n_{k}} defined in the introduction.

We define a weighted Bergman space on Ω by

A_{α}^{2}(Ω) = H(Ω) ∩ L^{2}(Ω, dv_{α}),

where H(Ω) is the space of holomorphic functions in Ω.

If μ is a finite Borel measure on Ω, then the Toeplitz operator

T_{μ} : A_{α}^{2}(Ω) → H(Ω)

is densely defined by

T_{μ}f(Z) = ∫_{Ω} K_{α}(Z, W)f(W) dμ(W),

where

K_{α}(Z, W) = ∏_{k=1}^{m} K_{α_{k}}(Z_{k}, W_{k}),

and each K_{α_{k}}(Z_{k}, W_{k}) is the reproducing kernel of A_{α_{k}}^{2}(𝔹_{n_{k}}).

The averaging function μ̂_{r} with respect to the Bergman metric is defined exactly as before, and the Berezin transform of μ is simply defined by

μ̃(Z) = ∫_{Ω} \frac{|K_{α}(Z, W)|^{2}}{K_{α}(Z, Z)} dμ(W), Z ∈ Ω.

We can now state the corresponding theorems for Ω.

Theorem 17. Suppose μ is a positive Borel measure on Ω, p > 0, and r > 0. Then:

(a) T_{μ} is bounded on A_{α}^{2}(Ω) if and only if μ̂_{r} ∈ L^{∞}(Ω).

(b) T_{μ} is compact on A_{α}^{2}(Ω) if and only if μ̂_{r}(Z) → 0 as Z approaches the full boundary of Ω.

(c) T_{μ} is in the Schatten class S_{p} if and only if μ̂_{r} ∈ L^{p}(Ω, dλ), where

dλ(Z) = ∏_{k=1}^{m} \frac{dv(Z_{k})}{(1 − |Z_{k}|^{2})^{n_{k}+1}}

is the Möbius invariant volume measure on Ω.

Theorem 18. Suppose μ is a positive Borel measure on Ω, p > 0, and {a_{k}} is an r-lattice in the Bergman metric of Ω. Then:

(a) T_{μ} is bounded on A_{α}^{2}(Ω) if and only if {μ̂_{r}(a_{k})} ∈ l^{∞}.

(b) T_{μ} is compact on A_{α}^{2}(Ω) if and only if {μ̂_{r}(a_{k})} ∈ c_{0}, where c_{0} is the space of sequences that tend 0.

(c) T_{μ} is in the Schatten class S_{p} if and only if {μ̂_{r}(a_{k})} ∈ l^{p}.

Conditions (a) and (b) here warrant some comments, because I believe they have not appeared explicitly before. First, if T_{μ} is bounded (or compact), then by [5], μ̂_{r} is in L^{∞}(Ω) (or ℂ_{0}(Ω)). So {μ̂_{r}(a_{k})} ∈ l^{∞} (or c_{0}).

On the other hand, if {μ̂_{r}(a_{k})} ∈ l^{∞} (or c_{0}) for some r-lattice {a_{k}}, then {μ̂_{2r}(a_{k})} ∈ l^{∞} (or c_{0}) as well. This is because we can find a positive integer N, indepedent of j, such that every set D(a_{j}, 2r) can be covered by at most N of the sets in {D(a_{k}, r)}. If z ∈ D(a_{k}, r), then D(z, r) ⊂ D(a_{k}, 2r). Since v_{α}(D(z, r)) is comparable to v_{α}(D(a_{k}, 2r)) whenever z ∈ D(a_{k}, r), we can find a constant C > 0, independent of k, such that μ̂_{r}(z) ≤ Cμ̂_{r}(a_{k}, 2r) for all k and all z ∈ D(a_{k}, r). This shows that μ̂_{r} ∈ L^{∞}(Ω) (or ℂ_{0}(Ω)).

Theorem 19. Suppose μ is a positive Borel measure on Ω and

(7)  p > max (\frac{n_{k}}{n_{k} + 1 + α_{k}} : 1 ≤ k ≤ m) .

Then:

(a) T_{μ} is bounded on A_{α}^{2}(Ω) if and only if μ̃ ∈ L^{∞}(Ω).

(b) T_{μ} is compact on A_{α}^{2}(Ω) if and only if μ̃(Z) → 0 as Z approaches the full boundary of Ω.

(c) T_{μ} ∈ S_{p} if and only if μ̃ ∈ L^{p}(Ω, dλ). Moreover, part (c) above becomes false if p does not satisfy condition (7).

The main obstacle to a generalization of our results to arbitrary bounded symmetric domains is the validity of Lemma 4, which is known to be false for general bounded symmetric domains. When p ≥ 1, the approach taken in [5] does not 316 Kehe Zhu

involve the usage of Lemma 4. In conclusion, it is still an open problem whether part (c) of Theorems 17, 18, and 19, in the case 0 < p < 1, can be extended to domains more general than products of balls.

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Department of Mathematics, University at Albany, Albany, NY 12222, USA kzhu@math.albany.edu

This paper is available via http://nyjm.albany.edu/j/2007/13-15.html.