Algebraic Properties of Toeplitz Operators on the Pluriharmonic Bergman Space

We study some algebraic properties of Toeplitz operators with radial or quasihomogeneous symbols on the pluriharmonic Bergman space. We first give the necessary and sufficient conditions for the product of two Toeplitz operators with radial symbols to be a Toeplitz operator and discuss the zero-product problem for several Toeplitz operators with radial symbols. Next, we study the finite-rank product problem of several Toeplitz operators with quasihomogeneous symbols. Finally, we also investigate finite rank commutators and semicommutators of two Toeplitz operators with quasihomogeneous symbols.

For a function  ∈  ∞ (  , ), the Toeplitz operator   :  2 ℎ (  ) →  2 ℎ (  ) with symbol  is defined by   () =  () () = ∫    ()  ()   () () .(8) In the setting of the classical Hardy space, for  and  in  ∞ (), Brown and Halmos [1] proved that     =  ℎ if and only if one of the following conditions holds: (a)  is analytic, (b)  is analytic.They also showed that, in both cases ℎ = .On the Bergman space of the unit disk, conditions (a) and (b) are only sufficient but not necessary.Ahern and Čučković [2] showed that a Brown-Halmos-type result holds for Toeplitz operators with harmonic symbols under certain conditions.Later, Ahern gave the necessary and sufficient conditions for the product of two Toeplitz operators with harmonic symbols to be a Toeplitz operator in [3].For general symbols, the product problem for Toeplitz operators remains open.Louhichi et al. [4] gave the necessary and sufficient conditions for the product of two quasihomogeneous Toeplitz operators to be a Toeplitz operator.
On the Hardy or Bergman space of several complex variables, the situation is much more complicated.Ding [5] characterized when the product of two Toeplitz operators with bounded symbols on the Hardy space  2 (  ) is still a Toeplitz operator.Motivated by the results of Ahern and Čučković [2], Choe et al. solved the product problem for Toeplitz operators with pluriharmonic symbols on the Bergman space of the polydisk [6].On the Bergman space of the unit ball, Zhou and Dong [7] gave the necessary and sufficient conditions for the product of two Toeplitz operators with radial symbols to be equal to a Toeplitz operator.In [8], they only gave the necessary condition for the product problem of two separately quasihomogeneous Toeplitz operators.Recently, Lu and Zhang characterized when the product of two quasihomogeneous Toeplitz operators is equal to another quasihomogeneous Toeplitz operator in [9].
For the so-called "zero-product" problem, Brown and Halmos [1] easily deduced that if     = 0 on the Hardy space  2 () then either  or  must be identically zero.A natural and interesting problem on the classical Hardy space of one complex variable is the following zero-product problem: If   1   2 ⋅ ⋅ ⋅    = 0, whether one of these symbols is identically zero.Guo [10] proved that if   1   2 ⋅ ⋅ ⋅   5 = 0, then there exists some  such that   = 0.In [11], Gu showed that   1   2 ⋅ ⋅ ⋅   6 = 0, then there exists some  such that   = 0. Aleman and Vukotić [12] completely solved the zero-product problem for several Toeplitz operators on the Hardy space.In [2], Ahern and Čučković proved that the result is analogous to that in [1] for two Toeplitz operators with harmonic symbols on the Bergman space of unit disk.Moreover in [13] they proved that if     = 0, where  is arbitrary bounded and  is radial, then either  ≡ 0 or  ≡ 0. Čučković and Louhichi [14] studied finite rank product of several quasihomogeneous Toeplitz operators on the Bergman space of the unit disk.On the Bergman space of the unit ball, Choe and Koo [15] gave a result which is analogous to that in [2] with an assumption about the continuity of the symbols on an open subset of the boundary and solved the zero-product problem for several Toeplitz operators with harmonic symbols that have Lipschitz continuous extensions to the whole boundary.Also, analogous results were proved on polydisk in [16].Recently, those zero product results have been generalized to finite rank product results in [17].Le [18] discussed finite rank products of several Toeplitz operators on the weighted Bergman space of the unit ball.Dong and Zhou [8] investigated the zeroproduct problem of two Toeplitz operators, one of whose symbols is separately quasihomogeneous and the other is arbitrary bounded.In [7] they studied the zero-product problem for several Toeplitz operators with radial symbols.Ding [5] considered the problem of two Toeplitz operators with symbols in  2 (  ) on the Hardy space of the polydisk.
For two Toeplitz operators   and   we define the commutator and semicommutator, respectively, by On the Hardy space the problem of determining when the commutator or semicommutator has finite rank has been completely solved (see [19,20]).For the Bergman space the problem seems to be far from solution.Guo et al. [21] completely characterized the finite rank commutator and semicommutator of two Toeplitz operators with bounded harmonic symbols on the Bergman space of the unit disk.Luecking [22] showed that finite rank Toeplitz operators on the Bergman space of the unit disk must be zero.Čučković and Louhichi [14] studied the finite rank semi-commutators and commutators of Toeplitz operators with quasihomogeneous symbols on the Bergman space of the unit disk and obtained different results from the case of harmonic Toeplitz operators.Lu and Zhang [9,23] characterized finite rank commutators and semicommutators of two quasihomogeneous Toeplitz operators on the Bergman space of the unit ball and the polydisk, respectively.
The fact that the product of two harmonic functions is no longer harmonic adds some mystery in the study of operators on the harmonic Bergman space.The theory of Toeplitz operators on the harmonic Bergman space is quite different from that on  2  .For example, Choe and Lee [24] showed that two analytic Toeplitz operators on  2 ℎ commute only when their symbols and the constant function 1 are linearly dependent, but analytic Toeplitz operators always commute on  2  .The following question was raised by Choe and Lee in [24]: If an analytic Toeplitz operator and a coanalytic Toeplitz operator on the harmonic Bergman space commute, then is one of their symbols a constant?To solve this problem, they proved in [25] that if ,  ∈  ∞ and suppose one of them is noncyclic, then     =     if and only if either  or  is constant.On the pluriharmonic Bergman space of the unit ball, Lee and Zhu [26] characterized commuting Toeplitz operators with holomorphic symbols and obtained the necessary and sufficient condition for the product of two Toeplitz operators with pluriharmonic symbols to be equal to a Teoplitz operator.Furthermore, they gave a complete description of holomorphic symbols for which the associated Toeplitz operators have zero semicommutator.On the pluriharmonic Bergman space of the polydisk, Choe and Nam [27] obtained results which are parallel to those of [26].Dong and Zhou [28] studied the problem of when the product of two Toeplitz operators with quasihomogeneous symbols is a Toeplitz operator on the harmonic Bergman space of the unit disk.
Motivated by the recent work of Čučković, Louhichi [14], Zhou and Dong [7,28], and Lee and Zhu [26] we study Toeplitz operators with radial or quasihomogeneous symbols on the pluriharmonic Bergman space of the unit ball.
The present paper is assembled as follows: In Section 2, we introduce some basic properties of the Mellin transform and Mellin convolution.In Section 3, we first give the necessary and sufficient condition for the product of two Toeplitz operators with radial symbols to be a Toeplitz operator.Then we investigate the zero-product problem for several Toeplitz operators with radial symbols.In Section 4, we study the finite rank product problem for several Toeplitz operator with quasihomogeneous symbols.In Sections 5 and 6, we discuss the finite rank commutators and semicommutators of two Toeplitz operators with quasihomogeneous symbols on the pluriharmonic Bergman space.

The Mellin Transform and Mellin Convolution
One of the most useful tools in the following calculations is the Mellin transform.The Mellin transform φ of a function  ∈  1 ([0, 1],  ) is defined by It is clear that φ is well defined on the right half-plane { : Re  ≥ 2} and analytic on { : Re  > 2}.It is important and helpful to know that the Mellin transform φ is uniquely determined by its value on an arithmetic sequence of integers.
In fact, we have the following classical theorem [29, page 102].
In the following discussion we need a known fact about the Mellin convolution of two functions.If  and  are defined on [0, 1), then their Mellin convolution is defined by The Mellin convolution theorem states that and that, if  and  are in  1 ([0, 1],  ) then so is  *  .

Products of Toeplitz Operators with Radial Symbols
For any multi-index  = ( 1 , . . .,   ), where each   is a nonnegative integer, we write For two multi-indexes  = ( 1 , . . .,   ) and  = ( 1 , . . .,   ), the notation  ⪰  means that   ≥   ,  = 1, 2, . . ., and  ⊥  means that It is known that  is radial if and only if () = () for any unitary transformation  of   .That is, () only depends on ||.Then for each radial function , we define φ on [0, 1) by φ() = () where  is a unit vector in   .It is obvious that φ is welldefined.In the following, we will identify an integrable radial function  on the unit ball with the corresponding function φ defined on the interval [0, 1).Lemma 3. Let  ∈  2 (  , ),   is a Toeplitz operator densely defined on  2 ℎ (  ), then the symbol map  →   is one to one.
Proof.Assume   = 0, we have A direct calculation gives the following lemma, which will be used constantly in this paper.

Lemma 4.
Let  be an integrable radial function on   , such that   is a bounded operator, then for any multi-index , Proof.For any multi-index , we have which implies If  ≻ 0, we have ⟨    ,   ⟩ = 0 = ⟨  ,   ⟩.This shows that Since {  } ⪰0 ⋃{  } ≻0 is a basis for the pluriharmonic Bergman space, we have By a similar argument, we have The following theorem is very simple but essential.
(1) For each multi-index , there exists (2)  is a radial function.
First assume   (  ) =  ||   .For any unitary transformation  of   and  ∈   , we have  −1  ∈   .Thus, By (7), we have A direct calculation shows that Similarly, we have Then we can imply Proof.According to (14) we have It follows from Theorem 5 that  is a radial function.
where 1 denotes the constant function with value one.
Proof.For each multi-index , it follows from ( 14) that are equivalent to A direct calculation shows 1(2 + 2||) = 1/(2 + 2||), so ( 27) is equivalent to Example 8. Let  and  greater than or equal to −1.Then In [18] Le discussed finite rank product of Toeplitz operators with general symbols on the weighted Bergman space of the unit ball.Similarly, we will use Mellin transform and Remark 2 to study zero-product problem of Toeplitz operators with radial symbols on pluriharmonic Bergman space.

Product of 𝑛 Toeplitz Operator
In this section, we will show that if the product of several quasihomogeneous Toeplitz operators has finite rank, then at least one of the symbols is equal to zero.This result is analogous to Theorem 3.2 in [18], but we get it in a different way.
Definition 11.Let ,  ⪰ 0. A function  ∈  1 (  , ) is called a quasihomogeneous function of degree (, ) if  is of the form     , where  is a radial function, that is, for any  in the unit sphere   and  ∈ [0, 1).
Similar to the proof of Theorem 4.4 in [7], we can imply  = 0 or || = ||.It follows from ( 52) and (53) that Similar to the discussions for (  ), we can imply that (  ) = 0 for all  ⪰ 0. In conclusion, the rank of  is equal to zero.