Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 414201, 16 pages

doi:10.1155/2012/414201

Research Article

Toeplitz Operators with Quasihomogeneous Symbols on the Bergman Space of the Unit Ball

Bo Zhang¹ and Yufeng Lu²

¹College of Science, Dalian Ocean University, Dalian 116023, China

²School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received 24 May 2012; Revised 6 August 2012; Accepted 30 August 2012

Academic Editor: Nikolai M. Vasilevski

Copyright © 2012 Bo Zhang and Yufeng Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider when the product of two Toeplitz operators with some quasihomogeneous symbols on the Bergman space of the unit ball equals a Toeplitz operator with quasihomogeneous symbols. We also characterize finite-rank semicommutators or commutators of two Toeplitz operators with quasihomogeneous symbols.

1. Introduction

Let denote the Lebesgue volume measure on the unit ball of normalized so that the measure of equals 1. The Bergman space is the Hilbert space consisting of holomorphic functions on that are also in . Hence, for each , there exists an unique function with the following property:

for all . As is well known that the reproducing kernel is given by

Let be the orthogonal projection from onto . Given a function , the Toeplitz operator : is defined by the formula

for . Since the Bergman projection has norm , it is clear that Toeplitz operators defined in this way are bounded linear operators on and .

We now consider a more general class of Toeplitz operators. For , in analogy to (1.3) we define an operator on by

Since the Bergman projection can be extended to , the operator is well defined on , the space of bounded analytic functions on . Hence, is always densely defined on . Since is not bounded on , it is well known that can be unbounded in general. In [1], Zhou and Dong gave the following definitions, which are based on the definitions on the unit disk in [2].

Definition 1.1. Let .

(a) We say that is a T-function if (1.4) defines a bounded operator on .(b) If is a T-function, we write for the continuous extension of the operator (it is defined on the dense subset of ) defined by (1.4). We say that is a Toeplitz operator if and only if is defined in this way.(c) If there exists an , such that is (essentially) bounded on the annulus , then we say is “nearly bounded.”

On the Bergman space of the unit ball, Grudsky et al. [3] gave necessary and sufficient conditions for boundedness of Toeplitz operators with radial symbols. These conditions give a characterization of the radial functions in which correspond to bounded operators and furthermore show that the T-functions form a proper subset of which contains all bounded and “nearly bounded” functions.

We denote the semicommutator and commutator of two Toeplitz operators and by

In the setting of the classical Hardy space, Brown and Halmos [4] gave a complete characterization for the product of two Toeplitz operators to be a Toeplitz operator. On the Bergman space of the unit disk, Ahern and Čučković [5] and Ahern [6] obtained a similar characterization for Toeplitz operators with bounded harmonic symbols. For general symbols, the situation is much more complicated. Louhichi et al. [2] gave necessary and sufficient conditions for the product of two Toeplitz operators with quasihomogeneous symbols to be a Toeplitz operator and Louhichi and Zakariasy made a further discussion in [7]. But it remains open to determine when the product of two Toeplitz operators is a Toeplitz operator on the Bergman space.

The problem of determining when the semicommutator or commutator on the Bergman space has finite-rank seems to be far from solution. The analogous problem on the Hardy space has been completely solved (see [8, 9]). Guo et al. [10] completely characterized the finite-rank semicommutator or commutator of two Toeplitz operators with bounded harmonic symbols on the Bergman space of the unit disk and Luecking [11] showed that finite-rank Toeplitz operators on the Bergman space of the unit disk must be zero. Recently, Čučković and Louhichi [12] studied finite-rank semicommutators and commutators of Toeplitz operators with quasihomogeneous symbols and obtained different results from the case of harmonic Toeplitz operators.

Motivated by recent work of Čučković and Louhichi, Zhang et al., and Zhou and Dong (see [1, 12, 13]), we discuss the finite-rank commutator (semicommutator) of Toeplitz operators with more general symbols on the unit ball in this paper. Let and be two multi-indexes. A function is called a quasihomogeneous function of quasihomogeneous degree if is of the form for all in the unit sphere and some function defined on the interval .

Let and be two nonconstant quasihomogeneous functions (with certain restrictions on their quasihomogeneous degree). In this paper, we investigate the following problems:

(1)Under what conditions does hold for some quasihomogeneous function ?(2)Under what conditions does the semicommutator have finite rank?(3)Under what conditions does the commutator have finite rank?

2. The Mellin Transform and Mellin Convolution

Main tool in this paper will be the Mellin transform. Recall that the Mellin transform of a function is defined by the equation:

It is easy to check that

where and .

For convenience, we denote by when the form of is complicated. It is clear that is well defined on and analytic on . It is well known that the Mellin transform is uniquely determined by its values on , where and . The following classical theorem is proved in [14, page 102].

Theorem 2.1. Assume that is a bounded analytic function on which vanishes at the pairwise distinct points , , where

(i) and(ii).

Then vanishes identically on .

Remark 2.2. We will often use this theorem to show that if and if there exists a sequence such that

then for all and so .

If and are defined on the interval , then their Mellin convolution is defined by

The Mellin convolution theorem states that

and that, if and are in , then so is .

3. Products of Toeplitz Operators with Quasihomogeneous Symbols

For any multi-index , where each is a nonnegative integer, we will write

for .

For and , the notation means that

and means that

We also define

and obtain

It is known that is radial if and only if for any unitary transform of . So we have for , and . That is, depends only on . In this case, we denote by for convenience. The definition of quasihomogeneous function on the unit disk has been given in many papers (see [2] or [7]), and a similar definition on the unit ball will be given in the following.

Definition 3.1. Let . A function is called a quasihomogeneous function of quasihomogeneous degree if is of the form where is a radial function, that is,

for any in the unit sphere and .

The following lemma is from [1] and we will use it often.

Lemma 3.2. Let , be two multi-indexes and let be a bounded radial function on . Then for any multi-index ,

Proposition 3.3. Let be multi-indexes and let be bounded radial functions on . If the product is of finite rank, then there exists such that

Proof. Denote by the product of Toeplitz operators and let rank . For multi-indexes , we have

where

It follows that , where and is a constant dependent on , and . Thus the set contains at most elements. Let , then there exists such that

Proposition 3.4. , are defined as in Proposition 3.3. The product is of finite rank if and only if for some .

Proof. Using Proposition 3.3 and Theorem 2.1, we can easily get the result.

This result is analogous to Theorem 3.2 in [15], but we get it in a different way.

Similar to the proof of Proposition 3.3, we can get a result about finite-rank commutators (semicommutators).

Proposition 3.5. Let , and be multi-indexes and let , be bounded radial functions on . If the commutator (or the semicommutator ) is of finite rank, then there exists such that (or ) for .

Now we are in a position to characterize when the product of two Toeplitz operators with some quasihomogeneous symbols equals a Toeplitz operator with quasihomogeneous symbols.

When , if , then or . But when , there exist nonzero multi-indexes and such that . In this case, we have the following theorem.

Theorem 3.6. Suppose and are two nonzero multi-indexes with . Let and be bounded radial functions on . If there exists a bounded radial function such that , then must be a solution of the equation

Proof. Obviously, the equality holds for each monomial with the multi-index .

Since , is equivalent to . By Lemma 3.2, it is easy to check that

where

Since , we have . So (3.13) implies that

As , we have

A direct calculation gives that for . Then we have

or equivalently,

Combining the above equality with Remark 2.2, we get the conclusion.

In the following, we give some explicit examples in which Theorem 3.6 is applied.

Example 3.7. Suppose , , , , is a bounded radial function such that . Using Theorem 3.6, we can get that .

Example 3.8. Suppose , is a bounded radial function such that , then must be a solution of the equation

where is the characteristic function of the set . For example, suppose , , , is a bounded radial function such that , then it follows that and .

Louhichi et al. [2] showed that there exist two nontrivial quasihomogeneous Toeplitz operators on the Bergman space of the unit disk such that the product of those Toeplitz operators is also a nontrivial Toeplitz operator, for example . On weighted Bergman space of the unit ball , Vasilevski [16, 17] showed that there exist parabolic quasihomogeneous (It is clear that the quasihomogeneous function is also a parabolic quasihomogeneous function) symbol Toeplitz operators such that the finite product of those Toeplitz operators is also a Toeplitz operator of this type. However, on the unit ball , if and are two nonzero multi-indexes which are not orthogonal, we can get that there exist no nontrivial and such that .

Theorem 3.9. Let , be two nonzero multi-indexes which are not orthogonal. Given , and let and be two bounded radial functions on . If there exists a bounded radial function such that , then or .

Proof. If , as in Theorem 3.6, we can get

where

We claim that there exists such that

Since is not orthogonal to , without loss of generality, we can suppose .

Case 1 (). We denote , . For , let and .

Let . Then

where .

Therefore, the equation has two solutions at most. It means that there exists such that for any . Thus for each , we have and .

Case 2  . Given a multi-index , let and for . As in Case 1, we can find an integer with such that and for any .

Using (3.20), we have

for . As and , it is easy to see that (3.22) holds.

Let and and and . Then . Since

we know that at least one of the series and diverges. Hence it follows from Remark 2.2 that or .

4. Finite-Rank Semicommutator

On the unit ball , we will show that the semicommutator of two Toeplitz operators with some quasihomogeneous symbols is of finite rank if and only if it is zero.

Theorem 4.1. Let , be two multi-indexes, and let , be two integrable radial functions on such that , , and are T-functions. If the semicommutator has finite rank, then it is equal to zero.

Proof. Let be the semicommutator . If is finite rank, using Proposition 3.5, there exists such that

Therefore, Lemma 3.2 implies that

for all . Since

the above equation is equivalent to

Note that and are both analytic on the right half-plane and the sequence is arithmetic. Then Remark 2.2 implies that

Hence, . The proof is complete.

Next we will consider when the semicommutator of two quasihomogeneous Toeplitz operators is a finite-rank operator.

Remark 4.2. If the semicommutator is of finite rank, following the same process as in Theorem 4.1, we can prove that it must be zero.

On the unit disk, Čučković and Louhichi [12] gave an example to show that there exists a nonzero finite rank semicommutator , where and are radial functions. However, the situation on the unit ball is different. Let , be two integrable radial functions on and , be two multi-indexes. Then we will prove that is a finite-rank operator if and only if . Now, we begin with the case that and are not orthogonal.

Theorem 4.3. Let be two multi-indexes which are not orthogonal, and let be two integrable radial functions on such that , and are T-functions. If the semicommutator has finite rank, then or .

Proof. Let be the semicommutator . If is of finite rank, using Proposition 3.5, we can get that there exists such that

Lemma 3.2 gives that

for all .

Since is not orthogonal to , from the proof of Theorem 3.9 and using (4.7), we can get that there exists such that

Analogous to the proof of Theorem 3.9, it is easy to get or .

Next, we will show that there exists no nontrivial finite-rank semicommutator in the case that .

Theorem 4.4. Let , be two multi-indexes with , and let and be two integrable radial functions on such that , and are T-functions. The semicommutator has finite rank if and only if .

Proof. We only need to prove the necessity. Let be of finite rank. Since , it is easy to see that if and only if for multi-indexes . By Lemma 3.2, the following statements hold:

(i) if , then ;(ii) if , then

Combining (i) and (ii) with the assumption that has finite rank, we get that there exists such that

To finish the proof, we will prove that for .

Note that for and . Then for each , (ii) implies that if and only if

that is,

Since , the preceding equality is equivalent to

for all .

By (4.10), we obtain that the equality (4.13) holds for all . It is easy to see that is arithmetic. Therefore, by Remark 2.2, we have

for all .

In particular if , with , we have

It follows that the equality (4.13) holds for all . So for all . Hence, the proof is complete.

Example 4.5. Let , be two multi-indexes, is a bounded radial function and , . If is of finite rank, using Theorem 4.1, we obtain that . If is not orthogonal to and is of finite rank, so it follows from Theorem 4.3 that or . But if , there exist and such that . In particular, suppose , , , a direct calculation gives that and , that is, .

5. Finite-Rank Commutators

In this section, let be two integrable radial functions on . We now pass to investigate the commutator of two quasihomogeneous Toeplitz operators and consider when , , or have finite rank, respectively.

Theorem 5.1. Let be two multi-indexes with , and let be two integrable radial functions on such that and are T-functions. If is not a constant, then is of finite rank if and only if or .

Proof. Let be the commutator . By Lemma 3.2, if and only if

for . If is of finite rank, using Proposition 3.5, there exists such that

Since , by Theorem 2.1 and following the same process as in Theorem 4.4, we get . Using Theorem 4.4 in [1], we have if and only if or .

Conversely, if or , then we can easily show that for each multi-index , which implies that and commute.

Remark 5.2. The same as in Theorem 5.1, we can easily prove if the commutator is of finite rank, then it must be a zero operator.

Next, we give some examples.

Example 5.3. Suppose that , where and . Let be a T-function, where is a radial function. Then is of finite rank if and only if and commute. By Theorem 4.9 in [1], we can also get is of finite rank if and only if is a monomial.

On the unit disk, if the commutator has finite rank , then is at most equal to the quasihomogeneous degree and a nonzero finite rank commutator has been given in [12]. On the unit ball , we will show that the commutator has finite rank if and only if commutes with if and only if or .

Theorem 5.4. Let , be two nonzero multi-indexes, and let , be two integrable radial functions on such that and are T-functions. If the commutator has finite rank, then or .

Proof. Let denote the commutator . Applying Lemma 3.2, we get

for . If is finite rank, using Proposition 3.5, there exists such that

If , then we have . Combining (5.4) and (5.3), we have

for . Analogous to the proof of Theorem 4.8 in [1], it is not difficult to get that or .

On the other hand, if is not orthogonal to , then (5.3) and (5.4) imply that

for , where

Following the same process as in Theorem 3.9, we get or , as desired.

Acknowledgment

The authors thank the referee for several suggestions that improved the paper. This research is supported by NSFC, Item Number: 10971020.

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