#### Abstract

We study some algebraic properties of Toeplitz operator with quasihomogeneous or separately quasihomogeneous symbol on the pluriharmonic Bergman space of the unit ball in . We determine when the product of two Toeplitz operators with certain separately quasi-homogeneous symbols is a Toeplitz operator. Next, we discuss the zero-product problem for several Toeplitz operators, one of whose symbols is separately quasihomogeneous and the others are quasi-homogeneous functions, and show that the zero-product problem for two Toeplitz operators has only a trivial solution if one of the symbols is separately quasihomogeneous and the other is arbitrary. Finally, we also characterize the commutativity of certain quasihomogeneous or separately quasihomogeneous Toeplitz operators.

#### 1. Introduction

For , let be the cartesian product of copies of . For any points and in , we use the notions and for the inner product and the associated Euclidean norm. Let denote the open unit ball which consists of points with and let denote the unit sphere. Let be the normalized Lebesgue volume measure on and let be the normalized surface area measure on . is the Hilbert space consisting of all Lebesgue square integrable functions on with the inner product

The Bergman space is the closed subspace consisting of the analytic functions in . Let be the orthogonal projection from onto , then can be expressed by where is the Bergman reproducing kernel.

A function is said to be pluriharmonic if and only if satisfies that , where and (see page 9 of [1]). The pluriharmonic Bergman space, denoted by , is the closed subspace of consisting of all the pluriharmonic functions on . It is well known that is also a Hilbert space. We will write for the orthogonal projection from onto . It is easy to verify that each point evaluation is a bounded linear functional on . It follows that is also a reproducing function space with reproducing kernel:

For a function , we define the Toeplitz operator with symbol by

For product problem, on the Hardy space, Brown and Halmos [2] showed that if and are bounded functions on the unit circle, then is another Toeplitz operator if and only if either or is analytic. In the setting of the Bergman space, the condition that either or is analytic is still sufficient, but it is no longer necessary. Ahern and Čučković [3] showed that a Brown-Halmos type result holds for Toeplitz operators with harmonic symbols on . In [4], Ahern characterized when the product of two Toeplitz operators with harmonic symbols is a Toeplitz operator. Later in [5], Louhichi et al. gave the necessary and sufficient conditions for the product of two quasihomogeneous Toeplitz operators to be a Toeplitz operator. Recently, Dong and Zhou [6] characterized when the product of quasihomogeneous Toeplitz operators is a Toeplitz operator on the harmonic Bergman space of the unit disk.

The situation is more complicated on the Hardy, Bergman, and harmonic Bergman spaces of several complex variables. In 2003, Ding [7] discussed the product problem for two Toeplitz operators with bounded symbols on the Hardy space . After that, Choe et al. [8] solved the product problem for pluriharmonic Toeplitz operators on the Bergman space of the polydisk. On the Bergman space of the unit ball, Zhou and Dong [9] determined when the product of two radial Toeplitz operators is a Toeplitz operator. Later in [10], they discussed the product problem for two separately quasihomogeneous Toeplitz operators. In Zhang and Lu’s paper [11], they characterized the product problem for two Toeplitz operators with quasihomogeneous symbols. On the pluriharmonic Bergman space, Yang et al. [12] gave the necessary and sufficient conditions for the product of two radial Toeplitz operators to be a Toeplitz operator.

For zero-product problem, on the Hardy space, Brown and Halmos [2] proved that if such that , then one of the symbols must be the zero function. Motivated by this result, Guo [13] showed that implies that for some , when . After that, Gu [14] proved that for , the result in [13] is also true. Recently, Aleman and Vukotić [15] completely solved the zero-product problem for several Toeplitz operators on the Hardy space. On the Bergman space of the unit disk, Ahern and Čučković [3] obtained that the result is analogous to that in [2] for the zero-product problem of two harmonic Toeplitz operators. Furthermore, they got that implies or , where is arbitrary bounded and is radial in [16]. In 2003, Čučković [17] proved that if such that , where are both positive integers, then . Later in [18], Louhichi et al. considered the zero-product problem for with , where is a bounded radial function and is a positive integer. On the Bergman space of the unit ball, Dong and Zhou [10] investigated the zero-product problem of two Toeplitz operators, one of whose symbols is separately quasihomogeneous and the other is arbitrary bounded. Bauer and Vasilevski [19] considered the zero-product problem and a more general problem of zero finite sum of finite products of Toeplitz operators. Recently, Yang et al. [12] discussed the zero-product problem for several radial Toeplitz operators on the pluriharmonic Bergman space of the unit ball.

For commuting problem, Brown and Halmos [2] firstly considered the commutativity of two Toeplitz operators on the Hardy space. They showed that two bounded Toeplitz operators and commute if and only if (1) both and are analytic, (2) both and are coanalytic, or (3) one is a linear function of the other. On the Bergman space of the unit disk, Axler and Čučković [20] obtained that the same result is also true for Toeplitz operators with bounded harmonic symbols. In [21], Čučković and Rao used the Mellin transform to study the commutativity of two Toeplitz operators on and described those operators which commute with for . Later in [22], Louhichi and Zakariasy characterized commuting Toeplitz operators on with quasihomogeneous symbols. On the Bergman space of the unit ball, Zheng [23] studied commuting Toeplitz operators with pluriharmonic symbols. Recently, extending Vasilevski's results in [24, 25], Quiroga-Barranco and Vasilevski gave the description of many (geometrically defined) classes of commuting Toeplitz operators of the unit ball in [26, 27]. Zhou and Dong [9] studied the commuting problem for quasihomogeneous Toeplitz operators. In 2012, Dong and Zhou [28] and Louhichi and Zakariasy [29] characterized the commuting Toeplitz operators with radial or quasihomogeneous symbols on the harmonic Bergman space of the unit disk. In papers [19, 30–33], the authors studied the wide classes of (nongeometrically defined) commutative Banach algebras generated by Toeplitz operators of the Bergman spaces on the unit ball.

Motivated by recent results of the unit ball in [9, 10, 12], in this paper, on the pluriharmonic space of the unit ball, we first characterize the product of two Toeplitz operators with certain separately quasihomogeneous symbols to be a Toeplitz operator. Next, we solve the zero-product problem for several Toeplitz operators when one of the symbols is separately quasihomogeneous and the others are quasihomogeneous and show that zero-product problem for two Toeplitz operators with certain symbols has only a trivial solution. At last, the commutativity of certain (separately) quasihomogeneous Toeplitz operators is also discussed.

#### 2. Preliminaries

Let denote the set of all nonnegative integers. For any , for any point , we write

For two multi-indexes , , the notations and mean that for every and . Let denote . Moreover, if , .

For a function , is said to be radial if and only if for any unitary transformation of ; is said to be separately radial if and only if for any unitary transformation of with a diagonal matrix. This implies that a radial function has a form and a separately radial function has a form .

Using radial function and separately radial function, we give the following definition.

*Definition 1. *Let with and .

(I) is called a quasihomogeneous function of degree if has the following decomposition:
for any , and , where for and is a radial function. In this case, is called quasihomogeneous Toeplitz operator of degree .

(II) is called a separately quasihomogeneous function of degree if has the following decomposition:
for any , where is a separately radial function. In this case, is called separately quasihomogeneous Toeplitz operator of degree .

We now recall some useful results from [34]. Denote by the base of , that is,
If is a bounded separately radial function, we get that
where .

Let . Dong and Zhou [10] showed that for , has the decomposition
They also proved the following result.

Lemma 2. *Let ; then is bounded on for multi-indexes with .*

In order to get our main results, we need to introduce the Mellin transform, which is defined for any function by the formula It is well known that is well defined on the right half-plane and is analytic on . It is helpful that the Mellin transform is uniquely determined by its value on an arithmetic sequence of integers. In fact, we have the following result (see [35, page 102]).

Lemma 3. *Suppose that is a bounded analytic function on which vanishes at the pairwise distinct points , where*(I)*, and*(II)*. **Then vanishes identically on . *

*Remark 4. *We will often use Lemma 3 to show that if and if there exists a sequence such that
then for all and so .

In this paper, we will need a similar result in higher dimensions. Now we give the following definition.

*Definition 5. *Let be a subset of ; one says that satisfies condition (I) if the following statement holds:

(I) there exists a sequence such that , and for every fixed , there also exists a sequence such that and .

*Remark 6. *It follows from Definition 5 that for a multi-index , if is a subset of and if is the complement of in , then either or satisfies condition (I).

In this paper, we will often use Lemmas 4 and 12 in [12] and Lemma 2.1 and Corollary 2.7 in [10] which can be stated as follows.

Lemma 7. *Let be an integrable radial function on such that is a bounded operator then for any multi-index ,
*

Lemma 8. *Let be two multi-indexes and let be an integrable radial function on such that , and are bounded operators. Then for any multi-index ,
**
In particular, if are two nonzero multi-indexes with , one has
*

Lemma 9. *Let with and let . If is a bounded separately radial function on , then
*

Lemma 10. *Let and let be a bounded function on . If the set
**
satisfies condition (I), then .*

#### 3. The Product of Toeplitz Operators with Separately Quasihomogeneous Symbols

We start this section with the following result.

Lemma 11. *Let be a bounded separately radial function on ; then for any multi-index ,
*

*Proof. *For multi-indexes , it is well known that
According to (9) and Lemma 9, we get that
Similarly, if , , we obtain . Since is a basis of the pluriharmonic Bergman space, we have
By a similar argument, one can deduce that . This completes the proof.

The following theorem is crucial for us to get our main results.

Theorem 12. *Let be a bounded function on . Then the following conditions are equivalent:*(a)*for any , there exists such that ;*(b)*for any , there exists such that .*

*Proof. *Assume (a) holds; that is, . For any multi-index , it follows from (19) that
which is a basis of the pluriharmonic Bergman space implies that .

By a similar argument, one can show that (b) implies (a), which completes the proof.

Using Theorem 12, we give the necessary and sufficient condition when a bounded function is a separately radial function.

Theorem 13. *Let . Then the following statements are equivalent:*(a)*for any , there exists such that ;*(b)* is a separately radial function.*

*Proof. *It is easy to show that (b) implies (a) by Lemma 11.

Now suppose (a) holds. That is, . Then for any unitary transformation of with diagonal matrix and , we get . Hence, one can obtain . A direct calculation shows that
Similarly, we get that
It follows that
By Theorem 12, we have . Consequently, one can get that and so . Then is a separately radial function. This completes the proof.

A direct calculation gives the following lemma, which we will use often.

Lemma 14. *Let be two multi-indexes and let and . Then for any multi-index ,
**
Furthermore, if are two nonzero multi-indexes with , one has
*

*Proof. *Here, we only prove that (26) and (27) hold. For any multi-index , if , then there exists such that . Hence, . It follows from Lemma 9 that
For , using Lemma 9, (9), and (19), we get
Moreover, for , we have . It follows that for , . Thus, one can obtain that
Note that is still a separately radial function; by a similar argument, we get that
It follows from above two equations that
Furthermore, if and are nonzero multi-indexes, there exists such that and . This implies that there does not exist such that . It follows that (27) holds. This completes the proof.

Theorem 15. *Let be two nonzero multi-indexes with and let be a bounded function on . Then the following conditions are equivalent:*(a)*for any , there exists such that
*(b)* is a separately quasihomogeneous function of degree . *

*Proof. *It is obvious that (b) implies (a). Now assume (a) holds. For any multi-index , we have
Similarly, for and , we get that . In light of (19), one can deduce that
It follows from Theorem 13 that is a separately radial function. Let , where is a separately radial function on . It follows that
which implies that is a separately quasihomogeneous function of degree .

Now we discuss when the product of two Toeplitz operators with certain symbols is a Toeplitz operator.

Theorem 16. *Let be two bounded separately radial functions on . If there exists a bounded function such that , then is also a separately radial function on .*

*Proof. *It follows from Lemma 11 that for
In virtue of Theorem 13, we get that is a separately radial function. This completes the proof.

Theorem 17. *Let be two nonzero multi-indexes with and let such that . If there exists a bounded function such that , then is a separately quasihomogeneous function of degree . *

*Proof. *For any multi-index , using Lemma 14, one can obtain that
where . It follows from Theorem 15 that is a separately quasihomogeneous function of degree .

Using the same technique, we give the following result and omit the proof.

Theorem 18. *Let be two nonzero multi-indexes with and let such that . If there exists a bounded function such that , then is a separately quasihomogeneous function of degree . *

#### 4. The Zero-Product Problem of Toeplitz Operators with Quasihomogeneous and Separately Quasihomogeneous Symbols

In this section, we will study the zero-product problem for several Toeplitz operators when one of the symbols is separately quasihomogeneous and the others are homogeneous, and show that the zero-product problem for two Toeplitz operators with certain symbols has only a trivial solution on the pluriharmonic Bergman space of the unit ball.

Theorem 19. *Let be nonzero multi-indexes with and let be square integrable radial functions and such that . Then
**
if and only if for some . *

*Proof. *Suppose . Then for multi-index ,
It follows from Lemmas 8 and 14 that
where
If , then, for any multi-index such that , we have
Now we are ready to show that for some . For the sake of simplicity, we will consider the case of . Let . If satisfies condition (I), then by Lemma 10. Otherwise, let denote the complement of in , and it follows from Remark 6 that satisfies condition (I). Now let
It is obvious that , which implies that satisfies condition (I). Furthermore, we get that
Set
for . According to (43), we obtain that . Hence there exists some such that
By Remark 4, one can deduce that for some . Moreover, if for some , then .

Conversely, it is obvious that if for some , then
This completes the proof.

The following result is a partial answer to the zero-product problem for two Toeplitz operators on .

Theorem 20. *Let , where . Let , where are two nonzero indexes with and . Then if and only if either or .*

*Proof. *By Lemma 2, we get that is bounded for any multi-indexes .

If , then for any multi-index , and . It follows from (27) that
where
and consequently, together with (27) implies that for any multi-indexes , , , and ,
for all . Next we will prove that either or for , , , and . Similarly as the proof of Theorem 19, we will prove it in the case of . Let
If satisfies condition (I), noting that
then by Lemma 10. Otherwise, let denote the complement of in ; then satisfies condition (I) by Remark 6. It follows from (52) that
Using Lemma 10 again, we obtain that . Moreover, if either or , then . Hence, implies that either or .

The converse implication is obvious. This completes the proof.

Corollary 21. *Let , where , , , , and . Then implies that either or . *

#### 5. The Commutativity of Toeplitz Operators with Quasihomogeneous and Separately Quasihomogeneous Symbols

In this section, we will consider the commuting problem for two Toeplitz operators with certain symbols.

Theorem 22. *Let be two nonzero multi-indexes with let such that , and let be a bounded radial function on . If is nonconstant, then if and only if either or .*

*Proof. *It follows from Lemmas 7 and 14 that if and only if
for and
for . As in the proof of Theorem 5.1 in [10], for , (56) and the property that is nonconstant imply that either or . Furthermore, if either or , it is easy to show that (57) holds for . Hence, commutes with implies either or .

Conversely, if either or , one can get that (56) and (57) hold. That is, . This completes the proof.

As for , a separately radial function is a radial and it follows that Toeplitz operators with separately radial symbols commute, there is no contradiction with an extension of a result in [29] to the case of . It is given in [29] that a Toeplitz operator on with radial symbol commutes with another Toeplitz operator if that operator also has a radial symbol. The following theorem will show that this result is not true on .

Theorem 23. *For , let be a bounded separately radial function and let be a bounded radial function on . Then and commute.*

*Proof. *The proof is similar to that of Theorem 22, so we omit it.

Next, we will give a description of commuting quasihomogeneous Toeplitz operators the with same degree.

Theorem 24. *Let be two nonzero multi-indexes with and let be two square integrable radial function on such that . Then *