Abstract

We construct a function in which is unbounded on any neighborhood of each boundary point of such that Toeplitz operator is a Schatten -class operator on Dirichlet-type space . Then, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Dirichlet-type space . We determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. We investigate the zero-product problem for several Toeplitz operators with radial symbols. Furthermore, the corresponding commuting problem of Toeplitz operators whose symbols are of the form is studied, where , , and is a radial function.

1. Introduction

Let represent the open unit ball in several complex spaces . The Sobolev space consists of the functions which satisfy where denotes the normalized Lebesgue volume measure on . is a Hilbert space with inner product where denotes the inner product in . The Dirichlet-type space is the subspace of all analytic functions in with . Then, is a closed subspace of the . Let be the orthogonal projection from onto . is an integral operator represented by where is the reproducing kernel of . By computation, we know where , , , and .

In recent years, the Dirichlet-type space has received a lot of attention from mathematicians in the areas of modern analysis and probability and statistical analysis. Many mathematicians are interested in function theory and operator theory on the Dirichlet-type space (See [1, 2]). In [3, 4], for the Dirichlet-type space of one complex variable, that is, , Rochberg and Wu defined the Toeplitz operator with nonnegative measure on as follows: by Rochberg and Wu discussed the boundedness and compactness. Lu and Sun define Toeplitz operators on a Dirichlet-type space of several variables in [5].

Definition 1. Suppose that is a finite measure on . Toeplitz operators on the Dirichlet-type space with symbol are defined as follows: If , we write and is called the Toeplitz operator with symbol .

2. Schatten -Class Toeplitz Operators with Unbounded Symbols

In general, if , the space of essentially bounded functions on and then is densely defined only. In the case of Hardy space, it is well known that is bounded if and only if is essentially bounded, and is compact if and only if (see Douglas [6] and Davie and Jewell [7]). However, there are indeed bounded and compact Toeplitz operators with unbounded symbols on Bergman spaces of one complex variable; in fact, Miao and Zheng [8] have introduced a class of functions, called , which contains , for ; is compact on if and only if the Berezin transform of vanishes on the unit circle. Zorboska [9] has proved that if belongs to the hyperbolic BMO space, the is compact if and only if the Berezin transform of vanishes on the unit circle. Cima and Cuckovic [10] construct a class of unbounded functions over a Cantor set; the Toeplitz operators with these functions are compact. Essentially, if the values of the function vanish rapidly near the unit circle in the sense of measure , then will be compact. Cao [11] also construct compact Toeplitz operators and trace class Toeplitz operators with unbounded symbols on Bergman space of several complex variables.

In this section, we construct the compact Toeplitz operators and the Schatten -class Toeplitz operators for with unbounded symbols on Dirichlet-type space of several complex variables.

For preparation, we introduce a special set in , which is important for building our main results in this section.

For and , the boundary of , let then it is obvious that is a domain in , which is called circular-like cone with vertex , because it looks like a circular cone. For , set to be the ball with radius . We use to denote the area measure on , for ; is the normalized area measure on ; then there holds and . For , because of the property of , there exits proper such that, for any , where is a constant number that is independent of and . For convenience, we write .

Lemma 2 (Poincaré’s inequality). Suppose that , is a bounded domain in and belongs to Sobolev space ; then where is a constant that depends on .

Lemma 3. Suppose that , , and . For any , let ; we use to represent the characteristic function of the set . Then, introduces a compact Toeplitz operator on .

Proof. Take a sequence from such that weakly and ; it is enough to obtain that From Definition 1, we find that For , set ; then By Hölder’s and Poincaré’s inequality, where is constant. Then where are constants. Thus, for any , there exists a such that for .
On the other hand, because , the sequence converges uniformly to 0 in by its weak convergence (see Zhu [12]). Then, for any , there is a such that, for any , we find that , . Thus
This shows that .

Using Lemma 3, we can construct a compact Toeplitz operator with a symbol that is unbounded on any neighborhood of every point in unit surface.

Theorem 4. There exists a function which is unbounded on any neighborhood of every point in unit surface (i.e., for any and , where ; is a compact operator on .

Proof. Suppose that , , and is the function defined in Lemma 3. Let be a countable dense subset of ; then, for every , the introduces a compact Toeplitz operator by Lemma 3. Furthermore, for any , by H lder’s and Poincarè’s inequality, where is a constant and is defined in (8); thus . Set ; then is compact, and, for any and , we find Equation (19) shows that . Then converges to a compact operator in norm. It is obvious that and ; thus converges to function in norm. For any holomorphic polynomial , namely, which is the Toeplitz operator with symbol . Hence, for any and , where , because of the density of set in unit surface.

Following the above theorem, we construct Schatten -class Toeplitz operators for all whose symbols are also unbounded on any neighborhood of every point in unit surface.

Lemma 5 (See [5]). Let , , Then

Theorem 6. There is a function which is unbounded on any neighborhood of each boundary point of   (i.e., for any and , where ; is a Schatten -class operator on Dirichlet-type space for .

Proof. For every multi-index , we have (see Rudin [13]). Thus forms the orthonormal base in . Choose a countable dense subset of ; let , and set for any , where . Then where is a constant independent of . Integrating by parts, we find where is a constant that depends on . Let then is compact operator by Theorem 4. Note that is positive operator; thus, for , where . Changing the order of summation, we have Note that, for any continuous function on , we have where , . Then, by induction, we see obviously that Further, there are constants such that Namely, is a -class operator for .
For , let ; let be the operator defined in (27). By the above proof, we have By the convergence in (33), we know that there exists a such that, for , ; thus, we find . Hence, That is, is a -class operator for .

3. Toeplitz Operators with Radial Symbols

Is the product of two Toeplitz operators equal to a Toeplitz operator? In general, the answer is negative, but Brown and Halmos [14] showed that two bounded Toeplitz operators and commute on the Hardy space if and only if (I) both and are analytic, (II) both and are analytic, or (III) one is a linear function of the other. For more details on the same question for Toeplitz operators on the Bergman spaces of one variable, see Cuckovic et al. [15, 16] and Louhichi et al. [17, 18].

For the case of the Bergman spaces of several variable, Zheng [19] studied commuting the Toeplitz operators with pluriharmonic symbols on the unit ball in . Recently, Quiroga-Barranco and Vasilevski [20, 21] gave the description of many (geometrically defined) classes of commuting Toeplitz operators on the unit ball. Zhou and Dong [22] discussed commuting Toeplitz operators with radial symbols on the unit ball.

In this section, we discuss the same questions for the Dirichlet-type space on the unit ball. The rest of this section is organized as follows. First, we introduce some basic properties of the Mellin transform and Mellin convolution which will be needed later. Second, we discuss when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. Then, the zero-product problem for several Toeplitz operators with radial symbols on the Dirichlet-type space is investigated. Finally, the corresponding commuting problem of Toeplitz operators with quasihomogeneous symbols is studied.

3.1. Mellin Transform and Mellin Convolution

Mellin transform, the most useful tool we use later, is defined as follows.

Definition 7. The Mellin transform of a function is

It is known that is a bounded analytic function in the half plane . It is important and helpful to know that the Mellin transform is uniquely determined by its values on an arithmetic sequence of integers. In fact, we have the following classical theorem (see [23]).

Theorem 8. Let be a bounded analytic function on which vanishes at the pairwise distinct points , where (1)inf   ,(2) . Then vanishes identically on .

Remark 9. We will often use this theorem to show that if and if there exists a sequence such that then for all and so .

When considering the product of two Toeplitz operators, we need a known fact about the Mellin convolution of their symbols. If and are defined on , then their Mellin convolution is defined by The Mellin convolution theorem states that and that if and are in , then so is .

3.2. Products of Toeplitz Operators with Radial Symbols

For convenience, we use to denote The notations and mean, respectively, It is trivial that when ,

A function on is called the radial function, if depends only on . It is obvious that is radial if and only if for any unitary transform of . Then, for each radial function , we define the function on [0, 1) by where is a unit vector in . It is trivial that is well defined. In the following, we will often identify an integrable radial function on the unit ball with the corresponding function defined on the interval .

In the following, some basic results concerning Toeplitz operators with radial symbols on the Dirichlet-type space of the unit ball are obtained.

Theorem 10 (see Lu and Sun [5]). Let be a radial function and . Then, (1) is bounded if and only if ;(2) is compact if and only if .

Theorem 11. Let be a radial function in which is bounded on ; then, for any ,

Proof. By the definition of Toeplitz operator, we have From [13], we know the unique nonzero item in (43) is A direct computation shows that

Theorem 11 shows that the Toeplitz operator with a radial symbol on the Dirichlet-type space of the unit ball acts in a very simple way. On the other hand, the Toeplitz operators of Theorem 11 must be Toeplitz operators with radial symbols.

Theorem 12. Let . For each multi-index , if there exists which depends only on such that , then is a radial function.

Proof. Suppose ; then, for any unitary transform of with there holds where are multi-indexes. By the definition of Toeplitz operator, note that ; we have which shows that . Thus, and is a radial function.

Corollary 13. Let and be two radial functions on such that are bounded. If , then is a radial function and is a bounded operator.

Proof. Using (42) to calculate , we obtain From Theorem 12, is a radial function. Moreover, is obviously a bounded operator.

Grudsky et al. [24] gave a particular answer to this question considering the Toeplitz operators with radial symbols. In [22], Zhou and Dong discussed the same question about the Toeplitz operators with radial symbols; they gave a different way to characterize when the product of two Toeplitz operators is equal to a Toeplitz operator. Let and be two radial functions on which induce bounded Toeplitz operators, and let further

The formal construction (inverse Fourier-Laplace transform) defines a holomorphic function in the upper half plane which coincides on the real axis with the inverse Fourier transform of the function . Theorem 3.7 of [24] shows that if the function belongs to Wiener ring of the inverse Fourier transforms of summable functions, then there exists a Toeplitz operator with the radial symbol such that . The following theorem will give the condition for the product of two Toeplitz operators with radial symbols to be a Toeplitz operator on Dirichlet-type space.

Theorem 14. Let and be two radial functions on which induce bounded Toeplitz operators. Then, is equal to the Toeplitz operator if and only if

Proof. For any , it follows from (42) and (49) that if and only if A direct computation gives where is well defined if . By Remark 9, (56) is equivalent to (53).

For some products of Toeplitz operators, the following fun result is obtained.

Corollary 15. Let and be two real numbers greater than or equal to . Then,

Proof. By Theorem 14, is equal to the Toeplitz operator if and only if A direct calculation shows that The desired result follows from (59).

Axler and Cuckovic [16] and Choe and Koo [25] study, respectively, the zero-product problem for two Toeplitz operators with harmonic symbols on unit disk and on the Bergman spaces of the unit ball. In [22], Zhou and Dong discuss the same question about the Toeplitz operators with radial symbols. In the following theorem, we will solve the zero-product problem for several Toeplitz operators with radial symbols acting on the Dirichlet-type space of the unit ball.

Theorem 16. Suppose that are radial functions on that induce bounded Toeplitz operators on Dirichlet-type space. If , then for some .

Proof. If , then, for any multi-index , by Theorem 11, there holds Set Since , there exists an such that then, by Remark 9, .

By Theorem 16, we can show that the only idempotent Toeplitz operators with radial symbols are and .

Corollary 17. If is a radial function on , then if and only if either or .

Proof. If , then , and, by Theorem 16, or . The converse implication is trivial.

3.3. Commuting Toeplitz Operators with Quasihomogeneous Symbols on Dirichlet-Type Space

In this subsection, commuting Toeplitz operators with bounded quasihomogeneous symbols on the Dirichlet-type space of the unit ball are discussed. The definition of the quasihomogeneous function on the unit disk has been given in [17, 26], and the definition on the unit ball has been given in [22].

Definition 18. Let and . 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 .

Remark 19. It is obvious that any can be uniquely written as , where and are two multi-indexes such that . Thus, in this paper, we always define the function for any .

In the following lemma, a result which we will use often is given.

Lemma 20. Let be two multi-indexes and let be a bounded radial function on . Then, for any multi-index ,

Proof. We just need to prove (67). For any , By calculation, we get the desired results.

Now, we can discuss the commuting problem of Toeplitz operators with quasihomogeneous symbols on Dirichlet-type space.

Theorem 21. Let be two multi-indexes and let and be two bounded radial functions on . If is not equal to 0, then if and only if either or .

Proof. If (69) holds, then for any multi-index such that ; by Theorem 11 and Lemma 20, we have Assume that ; without loss of generality, we can also assume that , for otherwise we could take the adjoins. Set It will show later that , which implies that by Remark 9.
If , we will induce a contradiction. Let where is the complement of in ; then On the other hand, for any , (70) gives Denote then is analytic and bounded on since is bounded. Moreover, (74) implies that According to Theorem 8, must be zero; thus, For any integer greater than , the above equation gives that If we denote by the constant , we obtain Multiply two sides of (79) by , which leads to where denotes the constant function with value one, for any , by Remark 9 again, and, then, clearly, is constant.
By calculation, we know that then, derive (81) with respect to variable ; we get and, then, clearly, is , which is a contradiction. Thus, we conclude that either or .
Conversely, if or , then we can easily show that (70) holds, and, consequently, for each multi-index , which implies and commute.

It was shown in [27] that a Toeplitz operator with a radial symbol on the Bergman spaces of the unit disk may only commute with another such operator with a radial symbol, but it is not true in higher dimensions by the theorem above. It is known that every function has the decomposition where are square integrables in with respect to the measure . More details can be found in [27]. Similarly, let Denote for . Each is a subspace of since, for , and It is also clear that if . Obviously, there exist many square-integrable functions on which do not have the decomposition , but we still might study a function of that form. In the one-dimensional case, the function is exactly the same as (84). Moreover, if then, for each , and so the functions are bounded on .

Lemma 22. Let be a multi-index and let be a bounded radial function on . If then

Proof. Suppose that ; then, for any multi-index , a direct calculation by Lemma 20 gives that If and commute, the equality of the above two series implies that Recall that is equal to zero if , then, for each multi-index and for all , Thus, , . The converse implication is clear.

Corollary 23. Let be a nonzero bounded radial function on . If then if and only if for almost all and .

Proof. It follows from Lemma 22 that commutes with if and only if for all . Suppose that , where and are two multi-indexes such that ; then, by Theorem 21, the above equation is equivalent to Obviously, (96) is equivalent to Therefore, commutes with if and only if for almost all and .

Remark 24. In the one-dimensional case, a function satisfying is exactly a radial function, so this corollary coincides with Theorem 6 of [27].

There are lots of examples of functions of the form , which are the symbols of commuting Toeplitz operators (see [27]), but the following theorem will show that two Toeplitz operators with quasihomogeneous symbols of degrees and , respectively, commute only in the trivial case.

Theorem 25. Suppose that and are two nonzero multi-indexes, and let and be two bounded radial functions on . If then, or .

Proof. For any multi-index , it follows from Lemma 20 that if , then and, if , then .
Similarly, if , then and, if , then .
For two nonzero multi-indexes and , to prove this theorem, we need to consider two cases.
Case  1. Suppose that and for some . If and commute, then(a) , if and ;(b) , if ,where Consider the multi-index ; then and , since and . Denote ; then, it follows from (a) that If , we will let ; then, a direct calculation from (b) shows that So we can find a sequence , which is defined by or , such that It is clear that . Let and . Since we know that at least one of the series and diverges; then it follows from Remark 9 that or .
Case  2. Suppose that either or for all . Without loss of generality, we can also assume that . Obviously, for any multi-index , the fact that either or for all implies Thus, if and commute, it follows from (b) that As in the proof of Theorem 21, the above equation implies that for . Let thus, the above equation can be written as Next, denote Obviously, is analytic on . It follows from (110) and that which implies that is a periodic function with period on . Thus, the function can be extended to whole plane , so we can think of the function as an entire function. By the definition of the Mellin transform, we get Noting that , we obtain , which implies or .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Jin Xia was supported by the China NNSF Grant 11301101, Xiaofeng Wang was supported by the Guangzhou Higher Education Science and Technology Projection no. 2012A018, and Guangfu Cao was supported by the China NNSF Grant 11271092.