Algebraic Properties of Quasihomogeneous and Separately Quasihomogeneous Toeplitz Operators on the Pluriharmonic Bergman Space

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 ℂ^𝑛. 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 𝑛≥1, let ℂ^𝑛 be the cartesian product of 𝑛 copies of ℂ. For any points 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑛) and 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) in ℂ^𝑛, we use the notions ⟨𝑧,𝑤⟩=𝑧₁𝑤₁+𝑧₂𝑤₂+⋯+𝑧_𝑛𝑤_𝑛 and |𝑧|=|𝑧₁|²+|𝑧₂|²+⋯+|𝑧_𝑛|² for the inner product and the associated Euclidean norm. Let 𝔹_𝑛 denote the open unit ball which consists of points 𝑧∈ℂ^𝑛 with |𝑧|<1 and let 𝕊_𝑛 denote the unit sphere. Let 𝑑𝑉 be the normalized Lebesgue volume measure on 𝔹_𝑛 and let 𝑑𝜎 be the normalized surface area measure on 𝕊_𝑛. 𝐿²(𝔹_𝑛,𝑑𝑉) is the Hilbert space consisting of all Lebesgue square integrable functions on 𝔹_𝑛 with the inner product ⟨𝑓,𝑔⟩=_𝔹𝑛𝑓(𝑧)𝑔(𝑧)𝑑𝑉(𝑧). (1)

The Bergman space 𝐿²_𝑎(𝔹_𝑛) is the closed subspace consisting of the analytic functions in 𝐿²(𝔹_𝑛,𝑑𝑉). Let 𝑃 be the orthogonal projection from 𝐿²(𝔹_𝑛,𝑑𝑉) onto 𝐿²_𝑎(𝔹_𝑛), then 𝑃 can be expressed by 𝑃𝑓(𝑧)=⟨𝑓,𝐾_𝑧⟩=_𝔹𝑛𝑓(𝑤)1(1−⟨𝑧,𝑤⟩)^𝑛+1𝑑𝑉(𝑤), (2) where 𝐾_𝑧(𝑤)=1/(1−⟨𝑤,𝑧⟩)^𝑛+1 is the Bergman reproducing kernel.

A function 𝑓 is said to be pluriharmonic if and only if 𝑓 satisfies that 𝐷_𝑗𝐷_𝑘𝑓=0(𝑗,𝑘=1,2,…,𝑛), where 𝐷_𝑗=𝜕/𝜕𝑧_𝑗 and 𝐷_𝑗=𝜕/𝜕𝑧_𝑗 (see page 9 of [1]). The pluriharmonic Bergman space, denoted by 𝐿²_ℎ(𝔹_𝑛), is the closed subspace of 𝐿²(𝔹_𝑛,𝑑𝑉) consisting of all the pluriharmonic functions on 𝔹_𝑛. It is well known that 𝐿²_ℎ(𝔹_𝑛) is also a Hilbert space. We will write 𝑄 for the orthogonal projection from 𝐿²(𝔹_𝑛,𝑑𝑉) onto 𝐿²_ℎ(𝔹_𝑛). It is easy to verify that each point evaluation is a bounded linear functional on 𝐿²_ℎ(𝔹_𝑛). It follows that 𝐿²_ℎ(𝔹_𝑛) is also a reproducing function space with reproducing kernel: 𝑅_𝑧(𝑤)=𝐾_𝑧(𝑤)+𝐾_𝑧(𝑤)−1,𝑧,𝑤∈𝔹_𝑛,𝑄𝑓(𝑧)=𝑃𝑓(𝑧)+𝑃𝑓(𝑧)−𝑃𝑓(0),𝑓∈𝐿²𝔹_𝑛,𝑑𝑉. (3)

For a function 𝜙∈𝐿^∞(𝔹_𝑛,𝑑𝑉), we define the Toeplitz operator 𝑇_𝜙∶𝐿²_ℎ(𝔹_𝑛)→𝐿²_ℎ(𝔹_𝑛) with symbol 𝜙 by 𝑇_𝜙𝑓=𝑄𝜙𝑓,𝑓∈𝐿²_ℎ𝔹_𝑛. (4)

For product problem, on the Hardy space, Brown and Halmos [2] showed that if 𝑓 and 𝑔 are bounded functions on the unit circle, then 𝑇_𝑓𝑇_𝑔 is another Toeplitz operator if and only if either 𝑓 or 𝑔 is analytic. In the setting of the Bergman space, the condition that either 𝑓 or 𝑔 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 𝐿²_𝑎(𝔹_𝑛). 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 𝐻²(𝔻^𝑛). 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 𝑓,𝑔∈𝐿^∞(𝕋) such that 𝑇_𝑓𝑇_𝑔=0, then one of the symbols must be the zero function. Motivated by this result, Guo [13] showed that 𝑇_𝜙1𝑇_𝜙2⋯𝑇_𝜙𝑛=0 implies that 𝜙_𝑖=0 for some 𝑖, when 𝑛=5. After that, Gu [14] proved that for 𝑛=6, 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 𝑇_𝑓𝑇_𝑔=0 implies 𝑓=0 or 𝑔=0, where 𝑓 is arbitrary bounded and 𝑔 is radial in [16]. In 2003, Čučković [17] proved that if 𝑓∈𝐿^∞(𝔻) such that 𝑇_𝑓𝑇_{𝑧𝑗−𝑧}^𝑙=0, where 𝑗,𝑙 are both positive integers, then 𝑓=0. Later in [18], Louhichi et al. considered the zero-product problem for 𝑓,𝑔∈𝐿^∞(𝔻,𝑑𝐴) with 𝑔=∑^𝑁_𝑘=−∞𝑒^𝑖𝑘𝜃𝑔_𝑘, where 𝑔_𝑘 is a bounded radial function and 𝑁 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 𝑇_𝜙 and 𝑇_𝜓 commute if and only if (1) both 𝜙 and 𝜓 are analytic, (2) both 𝜙 and 𝜓 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 𝐿²_𝑎(𝔻) and described those operators which commute with 𝑇_{𝑒𝑖𝑝𝜃𝑟𝑚} for (𝑝,𝑚)∈ℕ×ℕ. Later in [22], Louhichi and Zakariasy characterized commuting Toeplitz operators on 𝐿²_𝑎(𝔻) 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 ℕ denote the set of all nonnegative integers. For any 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑛)∈ℕ^𝑛, for any point 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑛)∈𝔹_𝑛, we write |𝛼|=𝛼₁+𝛼₂+⋯+𝛼_𝑛,𝛼!=𝛼₁!𝛼₂!,…,𝛼_𝑛𝑧!,_𝛼=𝑧^𝛼1₁𝑧^𝛼2₂,…,𝑧^𝛼𝑛_𝑛. (5)

For two multi-indexes 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑛), 𝛽=(𝛽₁,𝛽₂,…,𝛽_𝑛)∈ℕ^𝑛, the notations 𝛼≽𝛽 and 𝛼⟂𝛽 mean that 𝛼_𝑖≥𝛽_𝑖 for every 𝑖 and 𝛼₁𝛽₁+𝛼₂𝛽₂+⋯+𝛼_𝑛𝛽_𝑛=0. Let 𝛼−𝛽 denote (𝛼₁−𝛽₁,𝛼₂−𝛽₂,…,𝛼_𝑛−𝛽_𝑛). Moreover, if 𝛼≽𝛽, |𝛼−𝛽|=|𝛼|−|𝛽|.

For a function 𝜙∈𝐿¹(𝔹_𝑛,𝑑𝑉), 𝜙 is said to be radial if and only if 𝜙(𝑈𝑧)=𝜙(𝑧) for any unitary transformation 𝑈 of ℂ^𝑛; 𝜙 is said to be separately radial if and only if 𝜙(𝑈𝑧)=𝜙(𝑧) for any unitary transformation 𝑈 of ℂ^𝑛 with a diagonal matrix. This implies that a radial function has a form 𝜙(𝑧)=𝜙(|𝑧|) and a separately radial function has a form 𝜙(𝑧)=𝜙(|𝑧₁|,…,|𝑧_𝑛|).

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

Definition 1. Let 𝑝,𝑠∈ℕ^𝑛 with 𝑝⟂𝑠 and 𝑓∈𝐿¹(𝔹_𝑛,𝑑𝑉).
(I) 𝑓 is called a quasihomogeneous function of degree (𝑝,𝑠) if 𝑓 has the following decomposition: 𝑓(|𝑟|𝜉)=𝜉^𝑝𝜉^𝑠𝜙(|𝑟|) (6) for any 𝜉∈𝕊_𝑛, 𝑟=(𝑟₁,𝑟₂,…,𝑟_𝑛) and |𝑟|=|𝑟₁|²+|𝑟₂|²+⋯+|𝑟_𝑛|², where 𝑟_𝑖=|𝑧_𝑖| for 1≤𝑖≤𝑛 and 𝜙 is a radial function. In this case, 𝑇_𝑓 is called quasihomogeneous Toeplitz operator of degree (𝑝,𝑠).
(II) 𝑓 is called a separately quasihomogeneous function of degree (𝑝,𝑠) if 𝑓 has the following decomposition: 𝑓(|𝑟|𝜉)=𝜉^𝑝𝜉^𝑠𝜙(𝑟) (7) for any 𝜉∈𝕊_𝑛, where 𝜙 is a separately radial function. In this case, 𝑇_𝑓 is called separately quasihomogeneous Toeplitz operator of degree (𝑝,𝑠).
We now recall some useful results from [34]. Denote by 𝜏(𝔹_𝑛) the base of 𝔹_𝑛, that is, 𝜏𝔹_𝑛=𝑟=𝑟₁,𝑟₂,…,𝑟_𝑛=𝑧₁,𝑧₂,…,𝑧_𝑛∶𝑧=𝑧₁,𝑧₂,…,𝑧_𝑛∈𝔹_𝑛. (8) If 𝜙 is a bounded separately radial function, we get that _𝔹𝑛𝜙(𝑧)𝑑𝑉(𝑧)=2^𝑛𝑛!_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟𝑑𝑟, (9) where 𝑟𝑑𝑟=∏^𝑛_𝑖=1𝑟_𝑖𝑑𝑟_𝑖.
Let ℛ={𝜙∶𝔹_𝑛→ℂisseparatelyradial∶∫_{𝜏(𝔹𝑛)}|𝜙(𝑟)|²𝑟𝑑𝑟<∞}. Dong and Zhou [10] showed that for 𝑓∈𝐿²(𝔹_𝑛,𝑑𝑉), 𝑓 has the decomposition 𝑓(|𝑟|𝜉)=_{𝑝⟂𝑠,𝑝,𝑠∈ℕ𝑛}𝜉^𝑝𝜉^𝑠𝑓_𝑝,𝑠(𝑟),𝑓_𝑝,𝑠∈ℛ. (10) They also proved the following result.

Lemma 2. Let 𝑓(𝑧)=∑_{𝑝⟂𝑠,𝑝,𝑠∈ℕ}^𝑛𝜉^𝑝𝜉^𝑠𝑓_𝑝,𝑠(𝑟)∈𝐿^∞(𝔹_𝑛,𝑑𝑉); then 𝜉^𝑝𝜉^𝑠𝑓_𝑝,𝑠(𝑟) is bounded on 𝔹_𝑛 for multi-indexes 𝑝,𝑠∈ℕ^𝑛 with 𝑝⟂𝑠.

In order to get our main results, we need to introduce the Mellin transform, which is defined for any function 𝜙∈𝐿¹([0,1],𝑟𝑑𝑟) by the formula 𝜙(𝑧)=¹₀𝜙(𝑟)𝑟^𝑧−1𝑑𝑟. (11) It is well known that 𝜙 is well defined on the right half-plane {𝑧∶Re𝑧≥2} and is analytic on {𝑧∶Re𝑧>2}. 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 𝜙 is a bounded analytic function on {𝑧∶Re𝑧>0} which vanishes at the pairwise distinct points 𝑧₁,𝑧₂,…, where(I)inf{|𝑧_𝑛|}>0, and(II)∑_𝑛≥1Re(1/𝑧_𝑛)=∞. Then 𝜙 vanishes identically on {𝑧∶Re𝑧>0}.

Remark 4. We will often use Lemma 3 to show that if 𝜙∈𝐿¹([0,1],𝑟𝑑𝑟) and if there exists a sequence {𝑛_𝑘}⊂ℕ such that 𝜙𝑛_𝑘=0,_𝑘≥01𝑛_𝑘=∞, (12) then 𝜙(𝑧)=0 for all 𝑧∈{𝑧∶Re𝑧>2} and so 𝜙=0.

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

Definition 5. Let 𝐸 be a subset of ℤ²₊; one says that 𝐸 satisfies condition (I) if the following statement holds:
(I) there exists a sequence {𝛼_𝑖⁽¹⁾}^∞_𝑖=1 such that ∑^∞_𝑖=1=1/𝛼_𝑖⁽¹⁾=∞, and for every fixed 𝛼_𝑖⁽¹⁾, there also exists a sequence {𝛼⁽²⁾_𝑗(𝑖)}^∞_𝑗(𝑖)=1 such that ∑^∞_𝑗(𝑖)=11/𝛼⁽²⁾_𝑗(𝑖)=∞ and {(𝛼_𝑖⁽¹⁾,𝛼⁽²⁾_𝑗(𝑖))∶𝑗(𝑖)=1,2,…}⊂𝐸.

Remark 6. It follows from Definition 5 that for a multi-index 𝛿∈ℕ², if 𝐸 is a subset of {𝛼∈ℤ²₊∶𝛼≽𝛿} and if 𝐸^𝑐 is the complement of 𝐸 in {𝛼∈ℤ²₊∶𝛼≽𝛿}, then either 𝐸 or 𝐸^𝑐 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 𝜙 be an integrable radial function on 𝔹_𝑛 such that 𝑇_𝜙 is a bounded operator then for any multi-index 𝛼, 𝑇_𝜙𝑧^𝛼=2(𝑛+|𝛼|)!𝜙(2𝑛+2|𝛼|)𝑧^𝛼,𝑇_𝜙𝑧^𝛼=2(𝑛+|𝛼|)!𝜙(2𝑛+2|𝛼|)𝑧^𝛼. (13)

Lemma 8. Let 𝑝,𝑠 be two multi-indexes and let 𝜙 be an integrable radial function on 𝔹_𝑛 such that 𝑇_𝜉^𝑝_𝜙, 𝑇_𝜉^𝑠_𝜙 and 𝑇_𝜉𝑝𝜉^𝑠_𝜙 are bounded operators. Then for any multi-index 𝛼, 𝑇_𝜉𝑝𝜉^𝑠_𝜙𝑧^𝛼=2𝑝+𝛼!𝑛+𝑝+|𝛼|−|𝑠|!!|𝛼|+𝑝!𝜙×|𝛼|+𝑝−|𝑠|𝑧𝑝+𝛼−𝑠𝑛−1+2𝑛+2^{𝑝+𝛼−𝑠}|𝑠|−|𝛼|−𝑝!𝑝+𝛼≽𝑠,2𝑠!𝑛+!(|𝑠!𝜙×|𝑠|−𝑝𝑠−𝛼−𝑝𝑛−1+|)2𝑛+𝑧^{𝑠−𝛼−𝑝}𝑠≽𝑝+𝛼,𝑠≠𝑝+𝛼,0𝑠⪰^╱𝑝+𝛼,𝑝+𝛼^╱𝑇⪰𝑠;_{𝜉𝑝𝜉𝑠𝜙}𝑧^𝛼=2(𝑠+𝛼)!𝑛+|𝑠|+|𝛼|−𝑝!!(|𝛼|+|𝑠!𝜙×|𝛼|+|𝑠|−𝑝𝑠+𝛼−𝑝𝑛−1+|)2𝑛+2𝑧^{𝑠+𝛼−𝑝}𝑝−|𝛼|−|𝑠|!𝑠+𝛼≽𝑝,2𝑝!𝑛+!𝑝!𝜙×𝑝−|𝑠𝑧𝑝−𝛼−𝑠𝑛−1+2𝑛+|^{𝑝−𝛼−𝑠}𝑝≽𝑠+𝛼,𝑝≠𝑠+𝛼,0𝑝⪰^╱𝑠+𝛼,𝑠+𝛼⪰^╱𝑝. (14) In particular, if 𝑝,𝑠 are two nonzero multi-indexes with 𝑝⟂𝑠, one has 𝑇_𝜉𝑝𝜉^𝑠_𝜙𝑧^𝛼=2𝑝+𝛼!𝑛+𝑝+|𝛼|−|𝑠|!!|𝛼|+𝑝!𝜙×|𝛼|+𝑝−|𝑠|𝑧𝑝+𝛼−𝑠𝑛−1+2𝑛+2^{𝑝+𝛼−𝑠}𝛼≽𝑠,0𝛼^╱𝑇⪰𝑠;_{𝜉𝑝𝜉𝑠𝜙}𝑧^𝛼=2(𝑠+𝛼)!𝑛+|𝑠|+|𝛼|−𝑝!!(|𝛼|+|𝑠!𝜙×|𝛼|+|𝑠|−𝑝𝑠+𝛼−𝑝𝑛−1+|)2𝑛+2𝑧^{𝑠+𝛼−𝑝}𝛼≽𝑝,0𝛼⪰^╱𝑝. (15)

Lemma 9. Let 𝛼,𝛽∈ℕ^𝑛 with 𝛼≠𝛽 and let 𝑡∈[0,1). If 𝜙 is a bounded separately radial function on 𝔹_𝑛, then _𝕊𝑛𝜙(𝑡𝜉)𝜉^𝛼𝜉^𝛽𝑑𝜎(𝜉)=0. (16)

Lemma 10. Let 𝑝,𝑠∈ℕ² and let 𝑔(𝑟) be a bounded function on 𝜏(𝔹₂). If the set 𝐸=𝛼∈ℤ²₊∶𝛼≽𝑠,_𝜏(𝔹2)𝑔(𝑟)𝑟^{2𝛼+𝑝−𝑠}𝑟𝑑𝑟=0 (17) satisfies condition (I), then 𝑔=0.

3. The Product of Toeplitz Operators with Separately Quasihomogeneous Symbols

We start this section with the following result.

Lemma 11. Let 𝜙 be a bounded separately radial function on 𝔹_𝑛; then for any multi-index 𝛼∈ℕ^𝑛, 𝑇_𝜙𝑧^𝛼=2^𝑛(𝑛+|𝛼|)!∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝛼𝑟𝑑𝑟𝑧𝛼!^𝛼,𝑇_𝜙𝑧^𝛼=2^𝑛(𝑛+|𝛼|)!∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝛼𝑟𝑑𝑟𝛼!𝑧^𝛼. (18)

Proof. For multi-indexes 𝛼,𝛽∈ℕ^𝑛, it is well known that ⟨𝑧^𝛼,𝑧^𝛽⟩=0𝛼≠𝛽,𝑛!𝛼!(𝑛+|𝛼|)!𝛼=𝛽. (19) According to (9) and Lemma 9, we get that 𝑇_𝜙𝑧^𝛼,𝑧^𝛽=𝜙𝑧^𝛼,𝑧^𝛽=_𝔹𝑛𝜙(𝑧)𝑧^𝛼𝑧^𝛽(𝑧)=2𝑑𝑉0𝛼≠𝛽,_𝑛𝑛!∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝛼𝑟𝑑𝑟𝛼=𝛽. (20) Similarly, if 𝛽≽0, 𝛽≠0, we obtain ⟨𝑇_𝜙𝑧^𝛼,𝑧^𝛽⟩=0=⟨𝑧^𝛼,𝑧^𝛽⟩. Since {𝑧^𝛽}_𝛽≽0∪{𝑧^𝛽}_{𝛽≽0,𝛽≠0} is a basis of the pluriharmonic Bergman space, we have 𝑇_𝜙𝑧^𝛼=2^𝑛(𝑛+|𝛼|)!∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝛼𝑟𝑑𝑟𝑧𝛼!^𝛼. (21) By a similar argument, one can deduce that 𝑇_𝜙(𝑧^𝛼)=((2^𝑛(𝑛+|𝛼|)!∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝛼𝑟𝑑𝑟)/𝛼!)𝑧^𝛼. This completes the proof.

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

Theorem 12. Let 𝑓 be a bounded function on 𝔻. Then the following conditions are equivalent:(a)for any 𝛼∈ℕ^𝑛, there exists 𝜆_𝛼∈ℂ such that 𝑇_𝑓(𝑧^𝛼)=𝜆_𝛼𝑧^𝛼;(b)for any 𝛼∈ℕ^𝑛, there exists 𝜆_𝛼∈ℂ such that 𝑇_𝑓(𝑧^𝛼)=𝜆_𝛼𝑧^𝛼.

Proof. Assume (a) holds; that is, 𝑇_𝑓(𝑧^𝛼)=𝜆_𝛼𝑧^𝛼. For any multi-index 𝛽∈ℕ^𝑛, it follows from (19) that 𝑇_𝑓𝑧^𝛼,𝑧^𝛽=𝑓𝑧^𝛼,𝑧^𝛽=𝑓𝑧^𝛽,𝑧^𝛼=𝑇_𝑓𝑧^𝛽,𝑧^𝛼=𝜆_𝛽𝑧^𝛽,𝑧^𝛼=𝜆0𝛼≠𝛽,_𝛼𝑧^𝛼,𝑧^𝛼𝛼=𝛽=𝜆_𝛼𝑧^𝛼,𝑧^𝛽,𝑇_𝑓𝑧^𝛼,𝑧^𝛽=𝑓𝑧^𝛼,𝑧^𝛽=𝑓𝑧⁰,𝑧^𝛼+𝛽=𝑇_𝑓𝑧⁰,𝑧^𝛼+𝛽=𝜆₀𝑧⁰,𝑧^𝛼+𝛽=𝜆0𝛼≠0or𝛽≠0,₀𝛼=𝛽=0=𝜆_𝛼𝑧^𝛼,𝑧^𝛽. (22){𝑧^𝛽}_𝛽≽0∪{𝑧^𝛽}_𝛽≽0 which is a basis of the pluriharmonic Bergman space implies that 𝑇_𝑓(𝑧^𝛼)=𝜆_𝛼𝑧^𝛼.
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 𝜙∈𝐿^∞(𝔹_𝑛,𝑑𝑉). Then the following statements are equivalent:(a)for any 𝛼∈ℕ^𝑛, there exists 𝜆_𝛼∈ℂ such that 𝑇_𝜙(𝑧^𝛼)=𝜆_𝛼𝑧^𝛼;(b)𝜙 is a separately radial function.

Proof. It is easy to show that (b) implies (a) by Lemma 11.
Now suppose (a) holds. That is, 𝑇_𝜙(𝑧^𝛼)=𝜆_𝛼𝑧^𝛼. Then for any unitary transformation 𝑈 of ℂ^𝑛 with diagonal matrix and 𝑧∈𝔹_𝑛, we get 𝑈⁻¹𝑧∈𝔹_𝑛. Hence, one can obtain 𝑇_𝜙(𝑈⁻¹𝑧)^𝛼=𝜆_𝛼(𝑈⁻¹𝑧)^𝛼. A direct calculation shows that 𝑃𝜙(𝑈𝑧)𝑧^𝛼(𝑤)=_𝔹𝑛𝜙(𝑈𝑧)𝑧^𝛼(1−⟨𝑤,𝑧⟩)^𝑛+1(𝑧)=𝑑𝑉_𝔹𝑛𝜙(𝑧)𝑈⁻¹𝑧^𝛼1−𝑤,𝑈⁻¹𝑧^𝑛+1(𝑧)=𝑑𝑉_𝔹𝑛𝜙(𝑧)𝑈⁻¹𝑧^𝛼(1−⟨𝑈𝑤,𝑧⟩)^𝑛+1(𝑧)𝜙(𝑧𝑈𝑑𝑉=𝑃)⁻¹𝑧^𝛼(𝑈𝑤). (23) Similarly, we get that 𝑃𝜙(𝑈𝑧)𝑧^𝛼(𝑤)=𝑃𝜙(𝑧)𝑈⁻¹𝑧^𝛼),𝑃𝜙()𝑧(𝑈𝑤𝑈𝑧^𝛼(0)=𝑃𝜙(𝑧)𝑈⁻¹𝑧^𝛼(0). (24) It follows that 𝑇_𝜙∘𝑈𝑧^𝛼(𝑤)=𝑃𝜙(𝑈𝑧)𝑧^𝛼𝑤)+(𝑃𝜙(𝑈𝑧)𝑧^𝛼(𝑤)−𝑃𝜙(𝑈𝑧)𝑧^𝛼0)𝜙(𝑧)𝑈(=𝑃⁻¹𝑧^𝛼(𝑈𝑤)+𝑃𝜙(𝑧)𝑈⁻¹𝑧^𝛼×(𝑈𝑤)−𝑃𝜙(𝑧)𝑈⁻¹𝑧^𝛼(0)=𝑇_𝜙𝑈⁻¹𝑧^𝛼(𝑈𝑤)=𝜆_𝛼𝑈⁻¹𝑧^𝛼(𝑈𝑤)=𝜆_𝛼𝑤^𝛼=𝑇_𝜙𝑧^𝛼(𝑤). (25) By Theorem 12, we have 𝑇_𝜙∘𝑈(𝑧^𝛼)=𝜆_𝛼𝑧^𝛼=𝑇_𝜙(𝑧^𝛼). Consequently, one can get that 𝑇_𝜙∘𝑈=𝑇_𝜙 and so 𝜙∘𝑈=𝜙. Then 𝜙 is a separately radial function. This completes the proof.

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

Lemma 14. Let 𝑝,𝑠 be two multi-indexes and let 𝜙∈ℛ and 𝜉^𝑝𝜉^𝑠𝜙∈𝐿^∞(𝔹_𝑛,𝑑𝑉). Then for any multi-index 𝛼∈ℕ^𝑛, 𝑇_𝜉𝑝𝜉^𝑠_𝜙𝑧^𝛼=2^𝑛𝑛+|𝛼|+𝑝−|𝑠|!×∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝛼+2𝑝×|𝑟|^{−(|𝑝|+|𝑠|)}×!𝑟𝑑𝑟𝑝+𝛼−𝑠⁻¹𝑧^{𝑝+𝛼−𝑠}2𝑝+𝛼≽𝑠,_𝑛𝑛+|𝑠|−|𝛼|−𝑝!×∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝑠×|𝑟|^{−(|𝑝|+|𝑠|)}×!𝑟𝑑𝑟𝑠−𝛼−𝑝⁻¹𝑧^{𝑠−𝛼−𝑝}𝑠≽𝑝+𝛼,𝑠≠𝑝+𝛼,0𝑠⪰^╱𝑝+𝛼,𝑝+𝛼^╱𝑇⪰𝑠;_{𝜉𝑝𝜉𝑠𝜙}𝑧^𝛼=2^𝑛𝑛+|𝛼|+|𝑠|−𝑝!×∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝛼+2𝑠×|𝑟|^{−(|𝑝|+|𝑠|)}×!𝑟𝑑𝑟𝑠+𝛼−𝑝⁻¹𝑧^{𝑠+𝛼−𝑝}2𝑠+𝛼≽𝑝,_𝑛𝑛−|𝛼|+𝑝−|𝑠|!×∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝑝×|𝑟|^{−(|𝑝|+|𝑠|)}×!𝑟𝑑𝑟𝑝−𝛼−𝑠⁻¹𝑧^{𝑝−𝛼−𝑠}𝑝≽𝑠+𝛼,𝑝≠𝑠+𝛼,0𝑝⪰^╱𝑠+𝛼,𝑠+𝛼⪰^╱𝑝. (26) Furthermore, if 𝑝,𝑠 are two nonzero multi-indexes with 𝑝⟂𝑠, one has 𝑇_𝜉𝑝𝜉^𝑠_𝜙𝑧^𝛼=2^𝑛𝑛+|𝛼|+𝑝−|𝑠|!×∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝛼+2𝑝×|𝑟|^{−(|𝑝|+|𝑠|)}×!𝑟𝑑𝑟𝑝+𝛼−𝑠⁻¹𝑧^{𝑝+𝛼−𝑠}𝑝+𝛼≽𝑠,0𝑝+𝛼^╱𝑇⪰𝑠;_{𝜉𝑝𝜉𝑠𝜙}𝑧^𝛼=2^𝑛𝑛+|𝛼|+|𝑠|−𝑝!×∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝛼+2𝑠×|𝑟|^{−(|𝑝|+|𝑠|)}×!𝑟𝑑𝑟𝑠+𝛼−𝑝⁻¹𝑧^{𝑠+𝛼−𝑝}𝑠+𝛼≽𝑝,0𝑠+𝛼⪰^╱𝑝. (27)

Proof. Here, we only prove that (26) and (27) hold. For any multi-index 𝛽∈ℕ^𝑛, if 𝑝+𝛼⪰^╱𝑠, then there exists 𝑖 such that 𝛼_𝑖+𝑝_𝑖<𝑠_𝑖. Hence, 𝑝+𝛼≠𝛽+𝑠. It follows from Lemma 9 that 𝑃𝜉^𝑝𝜉^𝑠𝜙𝑧^𝛼,𝑧^𝛽=_𝔹𝑛𝜉^𝑝𝜉^𝑠𝜙(𝑧)𝑧^𝛼𝑧^𝛽𝑑𝑉(𝑧)=¹₀2𝑛|𝑟|^2𝑛−1|𝑟|^{|𝛼|+|𝛽|}𝑑|𝑟|×_𝕊𝑛𝜙(|𝑟|𝜉)𝜉^𝑝+𝛼𝜉^𝑠+𝛽𝑑𝜎(𝜉)=0. (28) For 𝛼+𝑝≽𝑠, using Lemma 9, (9), and (19), we get 𝑃𝜉^𝑝𝜉^𝑠𝜙𝑧^𝛼,𝑧^𝛽=_𝔹𝑛𝜉^𝑝𝜉^𝑠𝜙(𝑧)𝑧^𝛼𝑧^𝛽(𝑧)=2𝑑𝑉0𝛽≠𝑝+𝛼−𝑠,_𝑛𝑛!∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝛼+2𝑝×|𝑟|^{−(|𝑝|+|𝑠|)}𝑧𝑟𝑑𝑟𝛽=𝑝+𝛼−𝑠,_{𝑝+𝛼−𝑠},𝑧^𝛽=!0𝛽≠𝑝+𝛼−𝑠,𝑛!𝑝+𝛼−𝑠𝑛+|𝛼|+𝑝−|𝑠|!𝛽=𝑝+𝛼−𝑠. (29) Moreover, for 𝛽≽0,𝛽≠0, we have ⟨𝑃[𝜉^𝑝𝜉^𝑠𝜙𝑧^𝛼],𝑧^𝛽⟩=0. It follows that for 𝑝+𝛼≽𝑠, ⟨𝑃[𝜉^𝑝𝜉^𝑠𝜙𝑧^𝛼],𝑧^𝛽⟩=⟨𝑧^{𝑝+𝛼−𝑠},𝑧^𝛽⟩=0. Thus, one can obtain that 𝑃𝜉^𝑝𝜉^𝑠𝜙𝑧^𝛼=2^𝑛𝑛+|𝛼|+𝑝−|𝑠|!×∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝛼+2𝑝×|𝑟|^{−(|𝑝|+|𝑠|)}×!𝑟𝑑𝑟𝑝+𝛼−𝑠⁻¹𝑧^{𝑝+𝛼−𝑠}𝑝+𝛼≽𝑠,0𝑝+𝛼⪰^╱𝑠. (30) Note that 𝜙 is still a separately radial function; by a similar argument, we get that 𝑃𝜉^𝑝𝜉^𝑠𝜙𝑧^𝛼=2^𝑛𝑛+|𝑠|−|𝛼|−𝑝!×∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝑠×|𝑟|^{−(|𝑝|+|𝑠|)}×!𝑟𝑑𝑟𝑠−𝛼−𝑝⁻¹𝑧^{𝑠−𝛼−𝑝}𝑠≽𝑝+𝛼,0𝑠⪰^╱𝑝+𝛼. (31) It follows from above two equations that 𝑇_𝜉𝑝𝜉^𝑠_𝜙𝑧^𝛼=2^𝑛𝑛+|𝛼|+𝑝−|𝑠|!×∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝛼+2𝑝×|𝑟|^{−(|𝑝|+|𝑠|)}×!𝑟𝑑𝑟𝑝+𝛼−𝑠⁻¹𝑧^{𝑝+𝛼−𝑠}2𝑝+𝛼≽𝑠,_𝑛𝑛+|𝑠|−|𝛼|−𝑝!×∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝑠×|𝑟|^{−(|𝑝|+|𝑠|)}×!𝑟𝑑𝑟𝑠−𝛼−𝑝⁻¹𝑧^{𝑠−𝛼−𝑝}𝑠≽𝑝+𝛼,𝑠≠𝑝+𝛼,0𝑠⪰^╱𝑝+𝛼,𝑝+𝛼⪰^╱𝑠. (32) Furthermore, if 𝑝⟂𝑠 and 𝑝,𝑠 are nonzero multi-indexes, there exists 𝑖∈{1,2,…,𝑛} such that 𝑝_𝑖>0 and 𝑠_𝑖=0. This implies that there does not exist 𝛼∈ℕ^𝑛 such that 𝑠≽𝑝+𝛼. It follows that (27) holds. This completes the proof.

Theorem 15. Let 𝑝,𝑠∈ℕ^𝑛 be two nonzero multi-indexes with 𝑝⟂𝑠 and let 𝜙 be a bounded function on 𝔹_𝑛. Then the following conditions are equivalent:(a)for any 𝛼∈ℕ^𝑛, there exists 𝜆_𝛼∈ℂ such that 𝑇_𝜙𝑧^𝛼=0𝑝+𝛼^╱𝜆⪰𝑠,_𝛼𝑧^{𝑝+𝛼−𝑠}𝑝+𝛼≽𝑠; (33)(b)𝜙 is a separately quasihomogeneous function of degree (𝑝,𝑠).

Proof. It is obvious that (b) implies (a). Now assume (a) holds. For any multi-index 𝛽∈ℕ^𝑛, we have 𝑇_𝑧^𝑝_𝑧𝑠𝜙𝑧^𝛼,𝑧^𝛽=𝜙𝑧^𝛼+𝑠,𝑧^𝑝+𝛽=𝑇_𝜙𝑧^𝛼+𝑠,𝑧^𝑝+𝛽=𝜆_𝛼+𝑠𝑧^𝛼+𝑝,𝑧^𝑝+𝛽=𝜆0𝛼≠𝛽,_𝛼+𝑠𝑛!𝛼+𝑝!𝑛+|𝛼|+𝑝!𝛼=𝛽. (34) Similarly, for 𝛽≽0 and 𝛽≠0, we get that ⟨𝑇_𝑧^𝑝_𝑧𝑠𝜙(𝑧^𝛼),𝑧^𝛽⟩=0=⟨𝑧^𝛼,𝑧^𝛽⟩. In light of (19), one can deduce that 𝑇_𝑧^𝑝_𝑧𝑠𝜙𝑧^𝛼=𝜆_𝛼+𝑠𝛼+𝑝!(𝑛+|𝛼|)!|𝛼|+𝑝!𝑧𝛼!𝑛+^𝛼. (35) It follows from Theorem 13 that 𝑧^𝑝𝑧^𝑠𝜙 is a separately radial function. Let 𝑧^𝑝𝑧^𝑠𝜙(𝑧)=𝜓(𝑧), where 𝜓 is a separately radial function on 𝔹_𝑛. It follows that 𝜙(𝑧)=𝜓(𝑟)𝑟^{−(2𝑝+2𝑠)}|𝑟|^{(|𝑝|+|𝑠|)}𝜉^𝑝𝜉^𝑠, (36) which implies that 𝜙 is a separately quasihomogeneous function of degree (𝑝,𝑠).

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

Theorem 16. Let 𝜙₁,𝜙₂ be two bounded separately radial functions on 𝔹_𝑛. If there exists a bounded function ℎ such that 𝑇_𝜙1𝑇_𝜙2=𝑇_ℎ, then ℎ is also a separately radial function on 𝔹_𝑛.

Proof. It follows from Lemma 11 that for 𝛼∈ℕ^𝑛𝑇_ℎ𝑧^𝛼=𝑇_𝜙1𝑇_𝜙2𝑧^𝛼=2^𝑛(𝑛+|𝛼|)!∫_{𝜏(𝔹𝑛)}𝜙₂(𝑟)𝑟^2𝛼𝑟𝑑𝑟⋅2𝛼!_𝑛(𝑛+|𝛼|)!∫_{𝜏(𝔹𝑛)}𝜙₁(𝑟)𝑟^2𝛼𝑟𝑑𝑟𝑧𝛼!^𝛼. (37) In virtue of Theorem 13, we get that ℎ is a separately radial function. This completes the proof.

Theorem 17. Let 𝑝,𝑠∈ℕ^𝑛 be two nonzero multi-indexes with 𝑝⟂𝑠 and let 𝜙₁,𝜙₂∈ℛ such that 𝜉^𝑠𝜙₁,𝜉^𝑝𝜙₂∈𝐿^∞(𝔹_𝑛,𝑑𝑉). If there exists a bounded function ℎ such that 𝑇_𝜉^𝑠_𝜙1𝑇_𝜉^𝑝_𝜙2=𝑇_ℎ, then ℎ is a separately quasihomogeneous function of degree (𝑝,𝑠).

Proof. For any multi-index 𝛼∈ℕ^𝑛, using Lemma 14, one can obtain that 𝑇_ℎ𝑧^𝛼=𝑇_𝜉^𝑠_𝜙1𝑇_𝜉^𝑝_𝜙2𝑧^𝛼=𝜆_𝛼𝑧^{𝛼+𝑝−𝑠}𝛼+𝑝≽𝑠,0𝛼+𝑝⪰^╱𝑠, (38) where 𝜆_𝛼=(2^𝑛(𝑛+|𝛼|+|𝑝|−|𝑠|)!∫_{𝜏(𝔹𝑛)}𝜙₁(𝑟)𝑟^2𝛼+2𝑝|𝑟|^−|𝑠|𝑟𝑑𝑟/(𝑝+𝛼−𝑠)!)⋅(2^𝑛∫(𝑛+|𝛼|+|𝑝|)!_{𝜏(𝔹𝑛)}𝜙₂(𝑟)𝑟^2𝛼+2𝑝|𝑟|^−|𝑝|𝑟𝑑𝑟/(𝑝+𝛼)!). It follows from Theorem 15 that ℎ is a separately quasihomogeneous function of degree (𝑝,𝑠).

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

Theorem 18. Let 𝑝,𝑠∈ℕ^𝑛 be two nonzero multi-indexes with 𝑝⟂𝑠 and let 𝜙₁,𝜙₂∈ℛ such that 𝜙₁,𝜉^𝑝𝜉^𝑠𝜙₂∈𝐿^∞(𝔹_𝑛,𝑑𝑉). If there exists a bounded function ℎ such that 𝑇_𝜙1𝑇_𝜉𝑝𝜉^𝑠_𝜙2=𝑇_ℎ, then ℎ is a separately quasihomogeneous function of degree (𝑝,𝑠).

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 𝑝_𝑖,𝑠_𝑖(1≤𝑖≤𝑚)∈ℕ^𝑛 be nonzero multi-indexes with 𝑝_𝑖⟂𝑠_𝑖 and let 𝜙₁,𝜙₂,…,𝜙_𝑚−1 be square integrable radial functions and 𝜙_𝑚∈ℛ such that 𝜉^𝑝𝑖𝜉^𝑠𝑖𝜙_𝑖∈𝐿^∞(𝔹_𝑛,𝑑𝑉). Then 𝑇_{𝜉𝑝1𝜉}^𝑠_1𝜙1𝑇_{𝜉𝑝2𝜉}^𝑠_2𝜙2…𝑇_{𝜉𝑝𝑚𝜉}^𝑠𝑚_𝜙𝑚=0 (39) if and only if 𝜙_𝑖=0 for some 𝑖.

Proof. Suppose 𝑇_{𝜉𝑝1𝜉}^𝑠_1𝜙1𝑇_{𝜉𝑝2𝜉}^𝑠_2𝜙2…𝑇_{𝜉𝑝𝑚𝜉}^𝑠𝑚_𝜙𝑚=0. Then for multi-index 𝛼∈ℕ^𝑛, 𝑇_{𝜉𝑝1𝜉}^𝑠_1𝜙1𝑇_{𝜉𝑝2𝜉}^𝑠_2𝜙2…𝑇_{𝜉𝑝𝑚𝜉}^𝑠𝑚_𝜙𝑚𝑧^𝛼𝑇=0,_{𝜉𝑝1𝜉𝑠1𝜙1}𝑇_{𝜉𝑝2𝜉}^𝑠_2𝜙2…𝑇_{𝜉𝑝𝑚𝜉}^𝑠𝑚_𝜙𝑚𝑧^𝛼=0. (40) It follows from Lemmas 8 and 14 that 𝑇_{𝜉𝑝1𝜉}^𝑠_1𝜙1𝑇_{𝜉𝑝2𝜉}^𝑠_2𝜙2…𝑇_{𝜉𝑝𝑚𝜉}^𝑠𝑚_𝜙𝑚𝑧^𝛼=𝐶_𝛼𝑧^{𝛼+∑𝑚−1𝑗=0𝑝𝑚−𝑗−∑𝑚−1𝑗=0𝑠𝑚−𝑗}𝛼+𝑝_𝑚≽𝑠_𝑚,…,𝛼+∑^𝑚−1_𝑗=0𝑝_𝑚−𝑗≽∑^𝑚−1_𝑗=0𝑠_𝑚−𝑗,0others, (41) where 𝐶_𝛼=2^𝑛𝑛+|𝛼|+𝑝_𝑚−𝑠_𝑚!×_{𝜏(𝔹𝑛)}𝜙_𝑚(𝑟)𝑟^{2𝛼+2𝑝𝑚}|𝑟|^{−(|𝑝𝑚|+|𝑠𝑚|)}×𝑝𝑟𝑑𝑟_𝑚+𝛼−𝑠_𝑚!⁻¹⋅⋅^𝑚−1_𝑖=12𝛼+^𝑖_𝑗=0𝑝_𝑚−𝑗−^𝑖−1_𝑗=0𝑠_𝑚−𝑗!×𝑛+|𝛼|+^𝑖_𝑗=0𝑝_𝑚−𝑗−^𝑖_𝑗=0𝑠_𝑚−𝑗!×𝛼+^𝑖_𝑗=0𝑝_𝑚−𝑗−^𝑖_𝑗=0𝑠_𝑚−𝑗!×𝑛−1+|𝛼|+^𝑖_𝑗=0𝑝_𝑚−𝑗−^𝑖−1_𝑗=0𝑠_𝑚−𝑗!⁻¹×𝜙_𝑚−𝑖2𝑛+2|𝛼|+2^𝑖−1_𝑗=0𝑝_𝑚−𝑗−2^𝑖−1_𝑗=0𝑠_𝑚−𝑗+𝑝_𝑚−𝑖−𝑠_𝑚−𝑖. (42) If 𝑇_{𝜉𝑝1𝜉}^𝑠_1𝜙1𝑇_{𝜉𝑝2𝜉}^𝑠_2𝜙2…𝑇_{𝜉𝑝𝑚𝜉}^𝑠𝑚_𝜙𝑚(𝑧^𝛼)=0, then, for any multi-index 𝛼∈ℕ^𝑛 such that 𝛼+𝑝_𝑚≽𝑠_𝑚,…,𝛼+∑^𝑚−1_𝑗=0𝑝_𝑚−𝑗≽∑^𝑚−1_𝑗=0𝑠_𝑚−𝑗, we have _{𝜏(𝔹𝑛)}𝜙_𝑚(𝑟)𝑟^{2𝛼+2𝑝𝑚}|𝑟|^{−(|𝑝𝑚|+|𝑠𝑚|)}⋅𝑟𝑑𝑟_{𝑚−1𝑖=1}𝜙_𝑚−𝑖2𝑛+2|𝛼|+2^𝑖−1_𝑗=0𝑝_𝑚−𝑗−2^𝑖−1_𝑗=0𝑠_𝑚−𝑗+𝑝_𝑚−𝑖−𝑠_𝑚−𝑖=0. (43) Now we are ready to show that 𝜙_𝑖=0 for some 𝑖. For the sake of simplicity, we will consider the case of 𝑛=2. Let 𝐸={𝛼≽∑^𝑚−1_𝑗=0𝑠_𝑚−𝑗∶∫_{𝜏(𝔹𝑛)}𝜙_𝑚(𝑟)𝑟^{2𝛼+2𝑝𝑚}|𝑟|^{−(|𝑝𝑚|+|𝑠𝑚|)}𝑟𝑑𝑟=0}. If 𝐸 satisfies condition (I), then 𝜙_𝑚=0 by Lemma 10. Otherwise, let 𝐸^𝑐 denote the complement of 𝐸 in {𝛼∈ℤ²₊∶𝛼≽∑^𝑚−1_𝑗=0𝑠_𝑚−𝑗}, and it follows from Remark 6 that 𝐸^𝑐 satisfies condition (I). Now let 𝑀=𝛼+𝑝_𝑚≽𝑠_𝑚,…,𝛼+^𝑚−1_𝑗=0𝑝_𝑚−𝑗≽^𝑚−1_𝑗=0𝑠_𝑚−𝑗∶_{𝜏(𝔹𝑛)}𝜙_𝑚(𝑟)𝑟^{2𝛼+2𝑝𝑚}×|𝑟|^{−(|𝑝𝑚|+|𝑠𝑚|)}𝑟𝑑𝑟≠0. (44) It is obvious that 𝑀⊃𝐸^𝑐, which implies that 𝑀 satisfies condition (I). Furthermore, we get that _𝛼∈𝑀1|𝛼|=∞. (45) Set 𝑀_𝑖=𝛼+𝑝_𝑚≽𝑠_𝑚,…,𝛼+^𝑚−1_𝑗=0𝑝_𝑚−𝑗≽^𝑚−1_𝑗=0𝑠_𝑚−𝑗∶𝜙_𝑖2𝑛+2|𝛼|+2^{𝑚−𝑖−1}_𝑗=0×𝑝_𝑚−𝑗−2^{𝑚−𝑖−1}_𝑗=0𝑠_𝑚−𝑗+𝑝_𝑖−𝑠_𝑖=0 (46) for 1≤𝑖≤𝑚−1. According to (43), we obtain that 𝑀⊂∪^𝑚−1_𝑖=1𝑀_𝑖. Hence there exists some 𝑖 such that _{𝛼∈𝑀𝑖}1|𝛼|=∞. (47) By Remark 4, one can deduce that 𝜙_𝑖=0 for some 𝑖(1≤𝑖≤𝑚−1). Moreover, if 𝜙_𝑖=0 for some 𝑖(1≤𝑖≤𝑚), then 𝑇_{𝜉𝑝1𝜉}^𝑠_1𝜙1𝑇_{𝜉𝑝2𝜉}^𝑠_2𝜙2…𝑇_{𝜉𝑝𝑚𝜉}^𝑠𝑚_𝜙𝑚(𝑧^𝛼)=0.
Conversely, it is obvious that if 𝜙_𝑖=0 for some 𝑖, then 𝑇_{𝜉𝑝1𝜉}^𝑠_1𝜙1𝑇_{𝜉𝑝2𝜉}^𝑠_2𝜙2…𝑇_{𝜉𝑝𝑚𝜉}^𝑠𝑚_𝜙𝑚=0. (48) This completes the proof.

The following result is a partial answer to the zero-product problem for two Toeplitz operators on 𝐿²_ℎ(𝔹_𝑛).

Theorem 20. Let 𝑓(𝑧)=∑_{𝑝,𝑠∈ℕ}^𝑛_{,𝑝≠0,𝑠≠0,𝑝⟂𝑠}𝜉^𝑝𝜉^𝑠𝑓_𝑝,𝑠(𝑟)∈𝐿^∞(𝔹_𝑛,𝑑𝑉), where 𝑓_𝑝,𝑠(𝑟)∈ℛ. Let 𝑔=𝜉^𝑝∗𝜉^𝑠∗𝜙(𝑟)∈𝐿^∞(𝔹_𝑛,𝑑𝑉), where 𝑝^∗,𝑠^∗ are two nonzero indexes with 𝑝^∗⟂𝑠^∗ and 𝜙∈ℛ. Then 𝑇_𝑓𝑇_𝑔=0 if and only if either 𝑓=0 or 𝑔=0.

Proof. By Lemma 2, we get that 𝜉^𝑝𝜉^𝑠𝑓_𝑝,𝑠(𝑟) is bounded for any multi-indexes 𝑝,𝑠∈ℕ^𝑛,𝑝≠0,𝑠≠0,𝑝⟂𝑠.
If 𝑇_𝑓𝑇_𝑔=0, then for any multi-index 𝛼∈ℕ^𝑛,  𝑇_𝑓𝑇_𝑔(𝑧^𝛼)=0 and 𝑇_𝑓𝑇_𝑔(𝑧^𝛼)=0. It follows from (27) that 𝑇_𝜉𝑝𝜉^𝑠_{𝑓𝑝,𝑠(𝑟)}𝑧^𝛼=𝑐𝑝,𝑠,𝛼_{𝜏(𝔹𝑛)}𝑓_𝑝,𝑠(𝑟)𝑟^2𝛼+2𝑝×|𝑟|^{−(|𝑝|+|𝑠|)}𝑟𝑑𝑟𝑧^{𝛼+𝑝−𝑠}, (49) where 𝑐𝑝,𝑠,𝛼=0𝑝+𝛼^╱2⪰𝑠,_𝑛𝑛+|𝛼|+𝑝−|𝑠|!𝑝+𝛼−𝑠!𝑝+𝛼≽𝑠 (50) and consequently,𝑇_𝑓𝑧^𝛼=_{𝑝,𝑠∈ℕ𝑛,𝑝≠0,𝑠≠0,𝑝⟂𝑠}𝑐×𝑝,𝑠,𝛼_{𝜏(𝔹𝑛)}𝑓_𝑝,𝑠(𝑟)𝑟^2𝛼+2𝑝|𝑟|^{−(|𝑝|+|𝑠|)}𝑟𝑑𝑟𝑧^{𝛼+𝑝−𝑠}. (51)𝑇_𝑓𝑇_𝑔(𝑧^𝛼)=0 together with (27) implies that for any multi-indexes 𝑝,𝑠∈ℕ^𝑛, 𝑝≠0, 𝑠≠0, and 𝑝⟂𝑠, _{𝜏(𝔹𝑛)}𝑓_𝑝,𝑠(𝑟)𝑟^{2𝛼+2𝑝∗−2𝑠∗+2𝑝}|𝑟|^{−(|𝑝|+|𝑠|)}⋅𝑟𝑑𝑟_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^{2𝛼+2𝑝∗}|𝑟|^{−(|𝑝∗|+|𝑠∗|)}𝑟𝑑𝑟=0 (52) for all 𝛼≽2𝑠^∗+𝑠. Next we will prove that either 𝜙=0 or 𝑓_𝑝,𝑠=0 for 𝑝,𝑠∈ℕ^𝑛, 𝑝≠0, 𝑠≠0, and 𝑝⟂𝑠. Similarly as the proof of Theorem 19, we will prove it in the case of 𝑛=2. Let 𝐸_𝑝,𝑠=𝛼∈ℤ²₊∶𝛼≽2𝑠^∗+𝑠,_{𝜏(𝔹𝑛)}𝑓_𝑝,𝑠(𝑟)𝑟^{2𝛼+2𝑝∗−2𝑠∗+2𝑝}×|𝑟|^{−(|𝑝|+|𝑠|)}𝑟𝑑𝑟=0. (53) If 𝐸_𝑝,𝑠 satisfies condition (I), noting that 𝑓_𝑝,𝑠(𝑟)𝑟^𝑝+𝑠|𝑟|^{−(|𝑝|+|𝑠|)}=𝜉^𝑝𝜉^𝑠𝑓_𝑝,𝑠(𝑟)<∞, (54) then 𝑓_𝑝,𝑠=0 by Lemma 10. Otherwise, let 𝐸^𝑐_𝑝,𝑠 denote the complement of 𝐸_𝑝,𝑠 in {𝛼∈ℤ²₊∶𝛼≽2𝑠^∗+𝑠}; then 𝐸^𝑐_𝑝,𝑠 satisfies condition (I) by Remark 6. It follows from (52) that _{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^{2𝛼+2𝑝∗}|𝑟|^{−(|𝑝∗|+|𝑠∗|)}𝑟𝑑𝑟=0,∀𝛼∈𝐸^𝑐_𝑝,𝑠. (55) Using Lemma 10 again, we obtain that 𝜙=0. Moreover, if either 𝑓=0 or 𝜙=0, then 𝑇_𝑓𝑇_𝑔(𝑧^𝛼)=0. Hence, 𝑇_𝑓𝑇_𝑔=0 implies that either 𝑓=0 or 𝜙=0.
The converse implication is obvious. This completes the proof.

Corollary 21. Let 𝑓(𝑧)=𝜉^𝑝𝜉^𝑠𝑓_𝑝,𝑠(𝑟)∈𝐿^∞(𝔹_𝑛,𝑑𝑉), where 𝑝,𝑠∈ℕ^𝑛, 𝑝≠0, 𝑠≠0, 𝑝⟂𝑠, and 𝑓_𝑝,𝑠(𝑟)∈ℛ. Then 𝑇²_𝑓=𝑇_𝑓 implies that either 𝑓=0 or 𝑓=1.

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 𝑝,𝑠∈ℕ^𝑛 be two nonzero multi-indexes with 𝑝⟂𝑠 let 𝜓∈ℛ such that 𝜉^𝑝𝜉^𝑠𝜓∈𝐿^∞(𝔹_𝑛,𝑑𝑉), and let 𝜙 be a bounded radial function on 𝔹_𝑛. If 𝜙 is nonconstant, then 𝑇_𝜙𝑇_𝜉𝑝𝜉^𝑠_𝜓=𝑇_𝜉𝑝𝜉^𝑠_𝜓𝑇_𝜙 if and only if either |𝑝|=|𝑠| or 𝜓=0.

Proof. It follows from Lemmas 7 and 14 that 𝑇_𝜙𝑇_𝜉𝑝𝜉^𝑠_𝜓=𝑇_𝜉𝑝𝜉^𝑠_𝜓𝑇_𝜙 if and only if _{𝜏(𝔹𝑛)}𝜓(𝑟)𝑟^2𝛼+2𝑝|𝑟|^{−(|𝑝|+|𝑠|)}×|𝛼|𝑝|𝑠|𝜙×|𝛼|𝑝|𝑠|=𝑟𝑑𝑟2𝑛+2+2−22𝑛+2+2−2_{𝜏(𝔹𝑛)}𝜓(𝑟)𝑟^2𝛼+2𝑝|𝑟|^{−(|𝑝|+|𝑠|)}×(|𝛼𝜙(|𝛼𝑟𝑑𝑟2𝑛+2|)2𝑛+|) (56) for 𝑝+𝛼≽𝑠 and _{𝜏(𝔹𝑛)}𝜓(𝑟)𝑟^2𝛼+2𝑠|𝑟|^{−(|𝑝|+|𝑠|)}×|𝛼||𝑠|𝑝𝜙×|𝛼||𝑠|𝑝=𝑟𝑑𝑟2𝑛+2+2−22𝑛+2+2−2_{𝜏(𝔹𝑛)}𝜓(𝑟)𝑟^2𝛼+2𝑠|𝑟|^{−(|𝑝|+|𝑠|)}×(|𝛼𝜙(|𝛼𝑟𝑑𝑟2𝑛+2|)2𝑛+|) (57) for 𝑠+𝛼≽𝑝. As in the proof of Theorem 5.1 in [10], for 𝑝+𝛼≽𝑠, (56) and the property that 𝜙 is nonconstant imply that either |𝑝|=|𝑠| or 𝜓=0. Furthermore, if either |𝑝|=|𝑠| or 𝜓=0, it is easy to show that (57) holds for 𝑠+𝛼≽𝑝. Hence, 𝑇_𝜙 commutes with 𝑇_𝜉𝑝𝜉^𝑠_𝜓 implies either |𝑝|=|𝑠| or 𝜓=0.
Conversely, if either |𝑝|=|𝑠| or 𝜓=0, one can get that (56) and (57) hold. That is, 𝑇_𝜙𝑇_𝜉𝑝𝜉^𝑠_𝜓=𝑇_𝜉𝑝𝜉^𝑠_𝜓𝑇_𝜙. This completes the proof.

As for 𝑛=1, 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 𝑛≥2. It is given in [29] that a Toeplitz operator on 𝐿²_ℎ(𝔻) 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 𝐿²_ℎ(𝔹_𝑛)(𝑛≥2).

Theorem 23. For 𝑛≥2, let 𝜓(𝑧) be a bounded separately radial function and let 𝜙(𝑧) be a bounded radial function on 𝔹_𝑛. Then 𝑇_𝜓 and 𝑇_𝜙 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 𝑝,𝑠∈ℕ^𝑛 be two nonzero multi-indexes with 𝑝⟂𝑠,|𝑝|≠|𝑠| and let 𝜙,𝜓 be two square integrable radial function on 𝔹_𝑛 such that 𝜉^𝑝𝜉^𝑠𝜙,𝜉^𝑝𝜉^𝑠𝜓∈𝐿^∞(𝔹_𝑛,𝑑𝑉). Then 𝑇_𝜉𝑝𝜉^𝑠_𝜙𝑇_𝜉𝑝𝜉^𝑠_𝜓=𝑇_𝜉𝑝𝜉^𝑠_𝜓𝑇_𝜉𝑝𝜉^𝑠_𝜙 if and only if 𝜙=𝐶𝜓 for some constant 𝐶.

Proof. It follows from Lemma 8 that 𝑇_𝜉𝑝𝜉^𝑠_𝜙𝑇_𝜉𝑝𝜉^𝑠_𝜓=𝑇_𝜉𝑝𝜉^𝑠_𝜓𝑇_𝜉𝑝𝜉^𝑠_𝜙 (58) if and only if 𝜙|𝛼|𝑝|𝑠|𝜓|𝛼|+𝑝−|𝑠|=𝜓|𝛼|𝑝|𝑠|𝜙×|𝛼|+𝑝−|𝑠|2𝑛+2+3−32𝑛+22𝑛+2+3−32𝑛+2 (59) for all 𝛼+2𝑝≽2𝑠 and 𝜙|𝛼||𝑠|𝑝𝜓|𝛼|+|𝑠|−𝑝=𝜓|𝛼||𝑠|𝑝𝜙×|𝛼|+|𝑠|−𝑝2𝑛+2+3−32𝑛+22𝑛+2+3−32𝑛+2 (60) for all 𝛼+2𝑠≽2𝑝. If |𝑝|>|𝑠|, according to Lemma 3, then (59) holds which implies that 𝜙𝑝|𝑠|𝜓(𝑧)=𝜓𝑝|𝑠|𝜙(𝑧),{}.𝑧+2−2𝑧+2−2𝑧∈𝑧∶Re𝑧>0 (61) It follows from Lemma 6 in [36] that 𝜙=𝐶𝜓. Furthermore, if 𝜙=𝐶𝜓, then (60) holds. Otherwise, if |𝑝|<|𝑠|, by a similar argument, (60) implies that 𝜙=𝐶𝜓 and so (59) holds.
The converse implication is obvious. This completes the proof.

Theorem 25. Let 𝑓(𝑧)=∑_𝑠∈ℕ^𝑛𝜉^𝑠𝑓_𝑠(𝑟)∈𝐿^∞(𝔹_𝑛,𝑑𝑉), where 𝑓_𝑠(𝑟)∈ℛ. Let 𝜙 be a bounded separately radial function on 𝔹_𝑛. Then 𝑇_𝑓𝑇_𝜙=𝑇_𝜙𝑇_𝑓 if and only if 𝑇_𝜉^𝑠_{𝑓𝑠(𝑟)}𝑇_𝜙=𝑇_𝜙𝑇_𝜉^𝑠_{𝑓𝑠(𝑟)} for every multi-index 𝑠∈ℕ^𝑛.

Proof. Using Lemmas 11 and 14, one can get that for any multi-index 𝛼∈ℕ^𝑛𝑇_𝑓𝑇_𝜙𝑧^𝛼=𝜇_{𝛼𝑠∈ℕ𝑛}𝑇_𝜉^𝑠_{𝑓𝑠(𝑟)}𝑧^𝛼=𝜇_{𝛼𝑠∈ℕ𝑛,𝛼≽𝑠}𝑇_𝜉^𝑠_{𝑓𝑠(𝑟)}𝑧^𝛼+_{𝑠∈ℕ𝑛,𝑠≽𝛼,𝑠≠𝛼}𝑇_𝜉^𝑠_{𝑓𝑠(𝑟)}𝑧^𝛼+_{𝑠∈ℕ𝑛,𝛼╱⪰𝑠,𝑠╱⪰𝛼}𝑇_𝜉^𝑠_{𝑓𝑠(𝑟)}𝑧^𝛼=𝜇_{𝛼𝑠∈ℕ𝑛,𝛼≽𝑠}2^𝑛(|𝛼|−|𝑠!×𝑛+|)_{𝜏(𝔹𝑛)}𝑓_𝑠(𝑟)𝑟^2𝛼|𝑟|^−|𝑠|×)!)𝑟𝑑𝑟((𝛼−𝑠⁻¹𝑧^𝛼−𝑠+_{𝑠∈ℕ𝑛,𝑠≽𝛼,𝑠≠𝛼}2^𝑛(|𝑠|−|𝛼!×𝑛+|)_{𝜏(𝔹𝑛)}𝑓_𝑠(𝑟)𝑟^2𝑠×|𝑟|^−|𝑠|×)!)𝑟𝑑𝑟((𝑠−𝛼⁻¹𝑧^𝑠−𝛼, (62) where 𝜇_𝛼=2^𝑛(𝑛+|𝛼|)!∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝛼𝑟𝑑𝑟/𝛼!. It follows that 𝑇_𝑓𝑇_𝜙𝑧^𝛼,𝑧^𝛽=0𝛼^╱2⪰𝛽,_𝑛𝜇_𝛼𝑛+𝛽!∫_{𝜏(𝔹𝑛)}𝑓_𝛼−𝛽(𝑟)𝑟^2𝛼|𝑟|^{|𝛽|−|𝛼|}𝑟𝑑𝑟×𝑧𝛽!_𝛽,𝑧^𝛽𝛼≽𝛽 (63) for 𝛽≽0 and 𝑇_𝑓𝑇_𝜙𝑧^𝛼,𝑧^𝛽=2^𝑛𝜇_𝛼𝑛+𝛽!∫_{𝜏(𝔹𝑛)}𝑓_𝛼+𝛽(𝑟)𝑟^2𝛼+2𝛽|𝑟|^{−(|𝛽|+|𝛼|)}𝑟𝑑𝑟×𝑧𝛽!_𝛽,𝑧^𝛽 (64) for 𝛽≽0 and 𝛽≠0.
By a similar argument, one can deduce that 𝑇_𝜙𝑇_𝑓𝑧^𝛼,𝑧^𝛽=0𝛼^╱2⪰𝛽,_𝑛𝜈_𝛽𝑛+𝛽!∫_{𝜏(𝔹𝑛)}𝑓_𝛼−𝛽(𝑟)𝑟^2𝛼|𝑟|^{|𝛽|−|𝛼|}𝑟𝑑𝑟𝛽!×⟨𝑧^𝛽,𝑧^𝛽⟩𝛼≽𝛽 (65) for 𝛽≽0 and ⟨𝑇_𝜙𝑇_𝑓𝑧^𝛼,𝑧^𝛽⟩=2^𝑛𝜈_𝛽𝑛+𝛽!∫_{𝜏(𝔹𝑛)}𝑓_𝛼+𝛽(𝑟)𝑟^2𝛼+2𝛽|𝑟|^{−(|𝛽|+|𝛼|)}𝑟𝑑𝑟𝛽!×⟨𝑧^𝛽,𝑧^𝛽⟩ (66) for 𝛽≽0 and 𝛽≠0, where 𝜈_𝛽=2^𝑛(𝑛+|𝛽|)!∫_{𝜏(𝔹𝑛)}𝜙(𝑟)𝑟^2𝛽𝑟𝑑𝑟/𝛽!.
Now suppose that 𝑇_𝑓𝑇_𝜙=𝑇_𝜙𝑇_𝑓. It follows that 𝑇_𝑓𝑇_𝜙(𝑧^𝛼)=𝑇_𝜙𝑇_𝑓(𝑧^𝛼) and 𝑇_𝑓𝑇_𝜙(𝑧^𝛼)=𝑇_𝜙𝑇_𝑓(𝑧^𝛼) for any multi-index 𝛼∈ℕ^𝑛. Using Lemmas 11 and 14 again, we have ⟨𝑇_𝜉^𝑠_𝑓𝑠𝑇_𝜙𝑧^𝛼,𝑧^𝛽⟩=2^𝑛𝜇_𝛼𝑛+𝛽!×∫_{𝜏(𝔹𝑛)}𝑓_𝛼−𝛽(𝑟)𝑟^2𝛼×|𝑟|^{|𝛽|−|𝛼|}×𝑟𝑑𝑟𝛽!⁻¹⟨𝑧^𝛽,𝑧^𝛽=⟩𝛽=𝛼−𝑠,0others⟨𝑇_𝑓𝑇_𝜙𝑧^𝛼,𝑧^𝛽=⟩𝛽=𝛼−𝑠,0others⟨𝑇_𝜙𝑇_𝑓𝑧^𝛼,𝑧^𝛽=2⟩𝛽=𝛼−𝑠,0others_𝑛𝜈_𝛽𝑛+𝛽!×∫_{𝜏(𝔹𝑛)}𝑓_𝛼−𝛽(𝑟)𝑟^2𝛼×|𝑟|^{|𝛽|−|𝛼|}×𝑟𝑑𝑟𝛽!⁻¹⟨𝑧^𝛽,𝑧^𝛽⟩𝛽=𝛼−𝑠,0others=⟨𝑇_𝜙𝑇_𝜉^𝑠_𝑓𝑠𝑧^𝛼,𝑧^𝛽⟩ (67) for any multi-indexes 𝛽∈ℕ^𝑛 and 𝑠∈ℕ^𝑛 and ⟨𝑇_𝜉^𝑠_𝑓𝑠𝑇_𝜙𝑧^𝛼,𝑧^𝛽⟩=2^𝑛𝜇_𝛼𝑛+𝛽!×∫_{𝜏(𝔹𝑛)}𝑓_𝛼+𝛽(𝑟)𝑟^2𝛼+2𝛽×|𝑟|^{−(|𝛽|+|𝛼|)}×𝑟𝑑𝑟𝛽!⁻¹⟨𝑧^𝛽,𝑧^𝛽=⟩𝛽=𝑠−𝛼,0others⟨𝑇_𝑓𝑇_𝜙𝑧^𝛼,𝑧^𝛽=⟩𝛽=𝑠−𝛼,0others⟨𝑇_𝜙𝑇_𝑓𝑧^𝛼,𝑧^𝛽=2⟩𝛽=𝑠−𝛼,0others_𝑛𝜈_𝛽𝑛+𝛽!×∫_{𝜏(𝔹𝑛)}𝑓_𝛼+𝛽(𝑟)𝑟^2𝛼+2𝛽×|𝑟|^{−(|𝛽|+|𝛼|)}×𝑟𝑑𝑟𝛽!⁻¹⟨𝑧^𝛽,𝑧^𝛽=𝑇⟩𝛽=𝑠−𝛼,0others_𝜙𝑇_𝜉^𝑠_𝑓𝑠𝑧^𝛼,𝑧^𝛽 (68) for any multi-indexes 𝛽∈ℕ^𝑛, 𝛽≠0, and 𝑠∈ℕ^𝑛. By the same technique, we get that 𝑇_𝜉^𝑠_𝑓𝑠𝑇_𝜙𝑧^𝛼,𝑧^𝛽=𝑇_𝜉^𝑠_𝑓𝑠𝑇_𝜙𝑧^𝛼,𝑧^𝛽 (69) for any multi-indexes 𝛽∈ℕ^𝑛 and 𝑠∈ℕ^𝑛, and 𝑇_𝜉^𝑠_𝑓𝑠𝑇_𝜙𝑧^𝛼,𝑧^𝛽=𝑇_𝜉^𝑠_𝑓𝑠𝑇_𝜙𝑧^𝛼,𝑧^𝛽 (70) for any multi-indexes 𝛽∈ℕ^𝑛, 𝛽≠0, and 𝑠∈ℕ^𝑛. Hence 𝑇_𝜉^𝑠_𝑓𝑠𝑇_𝜙=𝑇_𝜙𝑇_𝜉^𝑠_𝑓𝑠.
Conversely, if 𝑇_𝜉^𝑠_𝑓𝑠𝑇_𝜙=𝑇_𝜙𝑇_𝜉^𝑠_𝑓𝑠, it is easy to show that 𝑇_𝑓𝑇_𝜙=𝑇_𝜙𝑇_𝑓. This completes the proof.

It is well known that 𝑇^∗_𝜙=𝑇_𝜙, so one can easily get the following result.

Theorem 26. Let 𝑓(𝑧)=∑_𝑝∈ℕ^𝑛𝜉^𝑝𝑓_𝑝(𝑟)∈𝐿^∞(𝔹_𝑛,𝑑𝑉), where 𝑓_𝑝(𝑟)∈ℛ. Let 𝜙 be a bounded separately radial function on 𝔹_𝑛. Then 𝑇_𝑓𝑇_𝜙=𝑇_𝜙𝑇_𝑓  if and only if 𝑇_𝜉^𝑝_{𝑓𝑝(𝑟)}𝑇_𝜙=𝑇_𝜙𝑇_𝜉^𝑝_{𝑓𝑝(𝑟)} for every multi-index 𝑝∈ℕ^𝑛.

Acknowledgment

This research is supported by the National Natural Science Foundation of China (Grant no. 11271059).