Toeplitz Operators on the Dirichlet Space of 𝔹_𝑛

Abstract

We study the algebraic properties of Toeplitz operators on the Dirichlet space of the unit ball 𝔹_𝑛. We characterize pluriharmonic symbol for which the corresponding Toeplitz operator is normal or isometric. We also obtain descriptions of conjugate holomorphic symbols of commuting Toeplitz operators. Finally, the commuting problem of Toeplitz operators whose symbols are of the form 𝑧^𝑝𝑧^𝑞𝜙(|𝑧|²) is studied.

1. Introduction

For any integer 𝑛≥1, let 𝔹_𝑛={𝑧∈ℂ^𝑛∶|𝑧|<1} denote the open unit ball of ℂ^𝑛 and 𝑑𝑚 denote the normalized Lebesgue measure on 𝔹_𝑛. The Sobolev space 𝑤^1,2 is defined to be the completion of smooth functions on 𝔹_𝑛 which satisfy 𝑓²=_𝔹𝑛𝑓𝑑𝑚²+^𝑛_{𝑖=1𝔹𝑛}𝜕𝑓𝜕𝑧_𝑖²+𝜕𝑓𝜕𝑧_𝑖²𝑑𝑚<∞. (1.1) The inner product ⟨⋅,⋅⟩ on 𝑤^1,2 is defined by 𝑓,𝑔=_𝔹𝑛𝑓𝑑𝑚_𝔹𝑛𝑔𝑑𝑚+^𝑛_{𝑖=1𝔹𝑛}𝜕𝑓𝜕𝑧_𝑖𝜕𝑔𝜕𝑧_𝑖+𝜕𝑓𝜕𝑧_𝑖𝜕𝑔𝜕𝑧_𝑖𝑑𝑚,∀𝑓,𝑔∈𝑤^1,2. (1.2) The Dirichlet space 𝒟 of 𝔹_𝑛 is the closed subspace consisting of all holomorphic functions in 𝑤^1,2. It is easily verified that each point evaluation is a bounded linear functional on 𝒟. Hence, for each 𝑧∈𝔹_𝑛, there exists a unique reproducing kernel 𝐾_𝑧(𝑤)∈𝒟 such that 𝑓(𝑧)=𝑓(𝑤),𝐾_𝑧(𝑤),∀𝑓∈𝒟. (1.3) Actually, it can be calculated that 𝐾_𝑧(𝑤)=1+∑_𝛼∈ℤ^+𝑛(((|𝛼|+𝑛−1)!/|𝛼|𝑛!𝛼!)𝑤^𝛼𝑧^𝛼), where 𝛼=(𝛼₁,…,𝛼_𝑛) is a multi-index, 𝛼_𝑖∈ℤ⁺, |𝛼|=∑^𝑛_𝑖=1𝛼_𝑖 and 𝑧^𝛼=𝑧^𝛼1₁⋯𝑧^𝛼𝑛_𝑛. For multi-indexes 𝛼 and 𝛽, the notation 𝛼≽𝛽 means that 𝛼_𝑖≥𝛽_𝑖,𝑖=1,…,𝑛 (1.4) and 𝛼≻𝛽 means that 𝛼≽𝛽 and 𝛼≠𝛽.

Let 𝑃 be the orthogonal projection from 𝑤^1,2 onto 𝒟. By the explicit formula for 𝐾_𝑧(𝑤), we have 𝑃𝜓(𝑧)=𝑃𝜓,𝐾_𝑧=𝜓,𝐾_𝑧=_𝔹𝑛𝜓𝑑𝑚_𝔹𝑛𝐾_𝑧𝑑𝑚+^𝑛_{𝑖=1𝔹𝑛}𝜕𝜓𝜕𝑤_𝑖𝜕𝐾_𝑧𝜕𝑤_𝑖𝑑𝑚(𝑤),∀𝜓∈𝑤^1,2. (1.5) Let Ω={𝜑∈𝑤^1,2∶𝜑,𝜕𝜑/𝜕𝑧_𝑖,𝜕𝜑/𝜕𝑧_𝑖∈𝐿^∞(𝔹_𝑛)}. Given 𝜑∈Ω, the Toeplitz operator 𝑇_𝜑 with symbol 𝜑 is the linear operator on 𝒟 defined by 𝑇_𝜑𝑓=𝑃𝜑𝑓,∀𝑓∈𝒟. (1.6) It is easy to verify that the Toeplitz operator 𝑇_𝜑∶𝒟→𝒟 is always bounded, whenever 𝜑∈Ω.

The algebric properties of Toeplitz operators on the classical Hardy spaces and Bergman spaces have been well studied, for example, as in [1–5].

On the Hardy space of the unit circle, a well-known theorem of Brown and Halmos [1] has shown that two Toeplitz operators with bounded symbols commute if and only if one of the followings holds: (i) both symbols are holomorphic; (ii) both symbols are antiholomorphic; (iii) a nontrivial linear combination of the symbols is constant.

On the Bergman space, the commuting problem is more complicated. Axler and C̆‌uc̆‌ković [2] proved that Brown-Halmos Theorem also holds for Toeplitze operators with bounded harmonic symbols. However, the corresponding problem for Toeplitz operator with general symbol remains open.

In recent years, more and more attention has been paid to the Toeplitz operators on Dirichlet spaces. The algebric properties of the Toeplitz operators on the classical Dirichlet spaces of the unit disc have been investigated intensively in [6–13]. Cao considered Fredholm properties of Toeplitz operators with 𝐶¹(𝔻) symbols in [6]. Lee showed in [8] that Brown-Halmos's result with harmonic symbols remains vaild on the Dirichlet space of the unit disc. In [12], Duistermaat and Lee gave the following characterizations of the harmonic symbols for which the associated Toeplitz operators are commuting, self-adjoint, or isometric: (1) for a harmonic symbol 𝑢∈Ω^′={𝑢∈𝐶¹(𝔻)∶𝑢,𝜕𝑢/𝜕𝑧,𝜕𝑢/𝜕𝑧∈𝐿^∞(𝔻,𝑑𝐴)}, 𝑇_𝑢 is self-adjoint if and only if 𝑢 is a real constant function; (2) for a harmonic symbol 𝑢∈Ω^′, 𝑇_𝑢 is an isometry if and only if 𝑢 is a constant function of modulus 1; (3) for two harmonic symbols 𝑢, 𝑣∈Ω^′, 𝑇_𝑢 and 𝑇_𝑣 commute if and only if either 𝑢 and 𝑣 are holomorphic or a nontrivial linear combination of 𝑢 and 𝑣 is constant on 𝔻. In [13], the corresponding problems have been investigated on the polydisc Dirichlet spaces and similar results have been obtained.

Motivated by the work of [12, 13], we study the corresponding problems on the Dirichlet spaces of 𝔹_𝑛. In Section 2, we give the characterizations of the pluriharmonic symbol for which the associated Toeplitz operator is self-adjoint or an isometry. In Section 3, we discuss when two Toeplitz operators with conjugate holomorphic symbols commute. At last, we concern with the commuting Toeplitz operators with symbols 𝑧^𝑝𝑧^𝑞𝜙(|𝑧|²).

2. Characterization of Normality and Isometry

In this section, we will give the condition under which Toeplitz operators with pluriharmonic symbols are self-adjoint or isometric on 𝒟. Before doing this, we first exhibit some properties of Toeplitz operators on 𝒟.

Lemma 2.1. Let 𝑓=∑_𝛽∈ℤ^+𝑛𝑓_𝛽𝑧^𝛽∈Ω be holomorphic. Then the following statements hold:(1)𝑇_𝑓1=𝑓(0); (2)𝑇^∗_𝑓1=𝑓(0); (3)𝑇^∗_𝑓1=∫_𝔹𝑛𝑓𝐾_𝑧𝑑𝑚=𝑓(0)+∑_|𝛽|>0((𝑓_𝛽/|𝛽|(𝑛+|𝛽|))𝑧^𝛽).

Proof. By the definition of the Toeplitz operators and the properties of the reproducing kernel, we obtain that 𝑇_𝑓1=⟨𝑇_𝑓1,𝐾_𝑧⟩=𝑃𝑓,𝐾_𝑧=𝑓(0),𝐾_𝑧=𝑓(0),𝑇^∗_𝑓1=𝑇^∗_𝑓1,𝐾_𝑧=1,𝑓𝐾_𝑧=_𝔹𝑛𝑓𝐾_𝑧𝑑𝑚=_𝔹𝑛𝑓𝐾_𝑧𝑑𝑚=𝑓(0),𝑇^∗_𝑓1=𝑇^∗_𝑓1,𝐾_𝑧=1,𝑓𝐾_𝑧=_𝔹𝑛𝑓𝐾_𝑧=𝑑𝑚_{𝔹𝑛𝛽∈ℤ𝑛+}𝑓_𝛽𝑤^𝛽_{𝛼∈ℤ+𝑛}(|𝛼|+𝑛−1)!|𝛼|𝑧𝑛!𝛼!^𝛼𝑤^𝛼(0)+𝑑𝑚=𝑓_|𝛽|>0𝑓_𝛽𝛽+𝑛−1!𝛽𝑧𝑛!𝛽!^𝛽_𝔹𝑛𝑤^𝛽2(0)+𝑑𝑚=𝑓_𝛽>0𝑓_𝛽𝛽+𝑛−1!𝛽𝑧𝑛!𝛽!^𝛽⋅𝑛!𝛽!𝛽!(0)𝑛+=𝑓+_|𝛽|>0𝑓_𝛽𝛽𝛽𝑧𝑛+^𝛽. (2.1)

Lemma 2.2. Let 𝑢=𝑓+𝑔∈Ω, where 𝑓 and 𝑔 are holomorphic. If 𝑇_𝑢 is normal, then ^𝑛_{𝑖=1𝔹𝑛}𝜕𝑓𝜕𝑧_𝑖²𝑑𝑚=^𝑛_{𝑖=1𝔹𝑛}𝜕𝐺𝜕𝑧_𝑖²𝑑𝑚, (2.2) where 𝐺=𝑇^∗_𝑔(1).

Proof. By assumption, we have 𝑇^∗_𝑓+𝑔𝑇_𝑓+𝑔=𝑇_𝑓+𝑔𝑇^∗_𝑓+𝑔. In particular, 𝑇^∗_𝑓+𝑔𝑇_𝑓+𝑔1,1=𝑇_𝑓+𝑔𝑇^∗_𝑓+𝑔1,1. (2.3) That is, 𝑇_𝑓+𝑔1,𝑇_𝑓+𝑔1=𝑇^∗_𝑓+𝑔1,𝑇^∗_𝑓+𝑔1. (2.4) It follows from Lemma 2.1 that 𝑓+𝑔(0),𝑓+𝑔(0)=𝑓(0)+𝐺,𝑓(0)+𝐺. (2.5) Hence, 𝑓(0)+𝑔(0)²+^𝑛_{𝑖=1𝔹𝑛}𝜕𝑓𝜕𝑧_𝑖²𝑑𝑚=𝑓(0)+𝐺(0)²+^𝑛_{𝑖=1𝔹𝑛}𝜕𝐺𝜕𝑧_𝑖²𝑑𝑚. (2.6) On the other hand, by the reproducing property and Lemma 2.1, we have 𝐺(0)=𝑇^∗_𝑔1(0)=𝑇^∗_𝑔1,𝐾₀=𝑇^∗_𝑔1,1=𝑔(0). (2.7) Then, ^𝑛_{𝑖=1𝔹𝑛}𝜕𝑓𝜕𝑧_𝑖²𝑑𝑚=^𝑛_{𝑖=1𝔹𝑛}𝜕𝐺𝜕𝑧_𝑖²𝑑𝑚. (2.8) This completes the proof.

The next lemma shows there is only trivial normal Toeplitz operator with holomorphic (or antiholomorphic) symbols.

Lemma 2.3. Let 𝑓=∑_𝛽∈ℤ^+𝑛𝑓_𝛽𝑧^𝛽∈Ω be holomorphic. Then the following statements are equivalent:(1)𝑇_𝑓 is normal;(2)𝑇_𝑓 is normal;(3)𝑇_𝑓+𝑓 is normal;(4)𝑓 is a constant function on 𝔹_𝑛.

Proof. (1)⇒(4) By Lemma 2.2 (with 𝑔=0), we have that ^𝑛_{𝑖=1𝔹𝑛}𝜕𝑓𝜕𝑧_𝑖²𝑑𝑚=^𝑛_{𝑖=1𝔹𝑛}𝜕𝐺𝜕𝑧_𝑖²𝑑𝑚=0. (2.9)
This along with the fact that ^𝑛_{𝑖=1𝔹𝑛}𝜕𝑓𝜕𝑧_𝑖²𝑑𝑚=𝑓−𝑓(0)²=_|𝛽|>0𝑓_𝛽²𝑧^𝛽2 (2.10) proves that 𝑓_𝛽=0, for |𝛽|>0. This shows that 𝑓 is a constant.
(2)⇒(4) Since 𝑇_𝑓 is normal, it follows from Lemma 2.2 that ^𝑛_{𝑖=1𝔹𝑛}𝜕𝐺𝜕𝑧_𝑖²𝑑𝑚=^𝑛_{𝑖=1𝔹𝑛}𝜕𝑓𝜕𝑧_𝑖²𝑑𝑚=0, (2.11) which implies that 𝐺 is a constant function for 𝐺 is holomorphic.
On the other hand, Lemma 2.1 ensures that 𝐺=𝑇^∗_𝑓1=𝑓(0)+_𝛽>0𝑓_𝛽𝛽𝛽𝑧𝑛+^𝛽. (2.12) It follows that 𝑓_𝛽=0, for all |𝛽|>0. Hence 𝑓 is a constant, as desired.
(3)⇒(4) Suppose 𝑇_𝑓+𝑓 is normal. Using Lemma 2.2, ^𝑛_{𝑖=1𝔹𝑛}𝜕𝑓𝜕𝑧_𝑖²𝑑𝑚=^𝑛_{𝑖=1𝔹𝑛}𝜕𝐺𝜕𝑧_𝑖²𝑑𝑚, (2.13) where 𝐺=𝑇^∗_𝑓(1). We conclude that 𝑓_𝛽=0, for |𝛽|>0, since by direct computation and Lemma 2.1^𝑛_{𝑖=1𝔹𝑛}𝜕𝑓𝜕𝑧_𝑖²𝑑𝑚=_|𝛽|>0𝑓_𝛽²𝑧^𝛽2,_{𝑛𝑖=1𝔹𝑛}𝜕𝐺𝜕𝑧_𝑖²𝑑𝑚=_|𝛽|>0𝑓_𝛽²𝛽²𝑛+𝛽²𝑧^𝛽2. (2.14) Hence 𝑓 is a constant.
The converse implications are clear. The proof is complete.

Since ‖𝑇_𝑓1‖≥‖𝑇^∗_𝑓1‖ for hyponormal Toeplitz operator 𝑇_𝑓, using Lemma 2.1, 𝑇_𝑓 is hyponormal if and only if 𝑓 is a constant. Consequently, normality of 𝑇_𝑓 can be replaced by hyponormality in Lemma 2.3.

On the Hardy space and the Bergman space, we always have 𝑇^∗_𝑢=𝑇_𝑢. So it is easy to see that 𝑇_𝑢 is self-adjoint (i.e., 𝑇_𝑢=𝑇^∗_𝑢) if and only if 𝑢 is a real-valued function. However, on the Dirichlet space of disc and polydisc, the situations are different because 𝑇^∗_𝑢 is not equal to 𝑇_𝑢 in both cases. In the following, we will study the adjoint of Toeplitz operators with pluriharmonic symbols on the Dirichlet space of 𝔹_𝑛.

Theorem 2.4. Let 𝑢=𝑓+𝑔∈Ω, where 𝑓 and 𝑔 are holomorphic. Then 𝑇^∗_𝑢=𝑇_𝑢 if and only if u is a constant function.

Proof. First, assume that 𝑢=∑_𝛽∈ℤ^+𝑛𝑎_𝛽𝑧^𝛽 is holomorphic. Since 𝑇^∗_𝑢=𝑇_𝑢, for each multi-index 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑛), we have 𝑇^∗_𝑢𝑧^𝛼=𝑇_𝑢𝑧^𝛼,∀|𝛼|>0. (2.15) Moreover, 𝑇^∗_𝑢𝑧^𝛼,1=𝑇_𝑢𝑧^𝛼,1. (2.16) In fact, for |𝛼|>0, 𝑇^∗_𝑢𝑧^𝛼,1=𝑧^𝛼,𝑇_𝑢1=𝑧^𝛼,𝑢=𝑎_𝛼𝑧^𝛼2=𝑎_𝛼⋅|𝛼|𝑛!𝛼!(|𝛼|)!.𝑛+−1 (2.17) On the other hand, 𝑇_𝑢𝑧^𝛼,1=𝑢𝑧^𝛼,1=_𝔹𝑛𝑢𝑧^𝛼𝑑𝑚=_𝔹𝑛𝑎_𝛼𝑧^𝛼2𝑑𝑚=𝑎_𝛼⋅𝑛!𝛼!(|𝛼!.𝑛+|) (2.18) Note that |𝛼|𝑛!𝛼!(|𝛼|)!−𝑛+−1𝑛!𝛼!(𝑛+|𝛼|)!>0, (2.19) we conclude that, 𝑎_𝛼=0, for |𝛼|>0.
Second, assume that 𝑢=𝑓+𝑔 is the general pluriharmonic symbol and 𝑇^∗_𝑢=𝑇_𝑢. In particular, we have 𝑇^∗_𝑓+𝑇^∗_𝑔1=𝑇_𝑓1+𝑇_𝑔1. (2.20) By Lemma 2.1, we get that 𝑓(0)+_𝔹𝑛𝑔𝐾_𝑧𝑑𝑚=𝑓(0)+𝑔. (2.21) Since _𝔹𝑛𝑔𝐾_𝑧𝑑𝑚=𝑔(0)+_|𝛽|>0𝑔_𝛽𝛽𝛽𝑧𝑛+^𝛽, (2.22) where 𝑔=∑_𝛽∈ℤ^+𝑛𝑔_𝛽𝑧^𝛽, it follows that _|𝛽|>0𝑔_𝛽𝛽𝛽𝑧𝑛+^𝛽=_|𝛽|>0𝑔_𝛽𝑧^𝛽. (2.23) Equivalently, _𝛽>011−𝛽𝛽𝑔𝑛+^𝛽𝑧^𝛽=0. (2.24) This implies that 𝑔_𝛽=0, for |𝛽|>0. So 𝑢=𝑓+𝑔(0) is holomorphic. The desired result follows immediately from the previous holomorphic case.
The converse implication is clear. The proof is complete.

We now characterize pluriharmonic symbols inducing self-adjoint Toeplitz operators.

Theorem 2.5. Let 𝑢=𝑓+𝑔∈Ω, where 𝑓, 𝑔 are holomorphic. Then 𝑇_𝑢=𝑇^∗_𝑢 if and only if 𝑢 is a real constant function.

Proof. The “if” part is clear. Suppose 𝑓=∑_𝛽∈ℤ^+𝑛𝑓_𝛽𝑧^𝛽 and 𝑔=∑_𝛽∈ℤ^+𝑛𝑔_𝛽𝑧^𝛽. It follows from Lemma 2.1 that 𝑇_𝑢1=𝑇_𝑓+𝑔1=𝑓+𝑔(0)=𝑓(0)+𝑔(0)+_|𝛽|>0𝑓_𝛽𝑧^𝛽,𝑇^∗_𝑢1=𝑇^∗_𝑓+𝑔1=𝑇^∗_𝑓1+𝑇^∗_𝑔1=𝑓(0)+𝑔(0)+_|𝛽|>0𝑔_𝛽𝛽𝛽𝑧𝑛+^𝛽. (2.25) Since 𝑇_𝑢1=𝑇^∗_𝑢1, by comparing the coefficients of the above two equations, we have that 𝑔_𝛽𝛽𝑛+𝛽=𝑓_𝛽,𝛽>0,𝑓(0)+𝑔(0)=𝑓(0)+𝑔(0). (2.26)(2.27)
Let 𝑤_𝑖=𝑤^𝑒𝑖, where 𝑒_𝑖=(0,0,…,0,1_𝑖,0,…,0). Then we have 𝑔𝑤_𝑖,𝐾_𝑧(𝑤)=_𝔹𝑛𝑔𝑤_𝑖𝑑𝑚+^𝑛_{𝑗=1𝔹𝑛}𝜕𝑔𝑤_𝑖𝜕𝑤_𝑗𝜕𝐾_𝑧(𝑤)𝜕𝑤_𝑗=𝑑𝑚_{𝔹𝑛𝛽∈ℤ+𝑛}𝑔_𝛽𝑤^𝛽𝑤_𝑖𝑑𝑚+_𝔹𝑛𝑔𝜕𝐾_𝑧(𝑤)𝜕𝑤_𝑖=𝑑𝑚𝑔_𝑒𝑖⋅_𝔹𝑛𝑤^𝑒𝑖2𝑑𝑚+_{𝔹𝑛𝛽∈ℤ+𝑛}𝑔_𝛽𝑤^𝛽⋅_{𝛼≽𝑒𝑖}(|𝛼|+𝑛−1)!|𝛼|𝛼𝑛!𝛼!^𝑖𝑤^{𝛼−𝑒𝑖}𝑧^𝛼=𝑑𝑚𝑔_𝑒𝑖+𝑛+1𝑔(0)⋅𝑧^𝑒𝑖. (2.28) This shows that 𝑇_𝑓+𝑔𝑤_𝑖(𝑧)=𝑇_𝑓𝑤_𝑖(𝑧)+𝑇_𝑔𝑤_𝑖(𝑧)=_{𝛽∈ℤ+𝑛}𝑓_𝛽𝑧^{𝛽+𝑒𝑖}+𝑔_𝑒𝑖+𝑛+1𝑔(0)⋅𝑧^𝑒𝑖. (2.29) On the other hand, if ℎ=∑_𝛽∈ℤ^+𝑛ℎ_𝛽𝑤^𝛽 is holomorphic, then _𝔹𝑛𝜕ℎ𝜕𝑤_𝑖𝑑𝑚=_{𝔹𝑛𝛽≽𝑒𝑖}ℎ_𝛽𝛽_𝑖𝑤^{𝛽−𝑒𝑖}𝑑𝑚=ℎ_𝑒𝑖. (2.30) Therefore, _𝔹𝑛𝜕𝑓𝐾_𝑧(𝑤)𝜕𝑤_𝑖𝑑𝑚=𝑓_𝑒𝑖+𝑓(0)⋅𝑧^𝑒𝑖. (2.31) A direct computation shows that _𝔹𝑛𝑔⋅𝜕𝐾_𝑧(𝑤)𝜕𝑤_𝑖𝑑𝑚=_{𝔹𝑛𝛼∈ℤ+𝑛}𝑔_𝛼𝑤^𝛼⋅_{𝛼≽𝑒𝑖}𝛽+𝑛−1!𝛽𝛽𝑛!𝛽!^𝑖𝑤^{𝛽−𝑒𝑖}𝑧^𝛽𝑑𝑚=_{𝛼∈ℤ+𝑛}𝑔_𝛼𝛼+𝑒_𝑖+𝑛−1!𝛼+𝑒_𝑖𝑛!𝛼+𝑒_𝑖!𝛼_𝑖+1𝑧^{𝛼+𝑒𝑖}⋅_𝔹𝑛𝑤^𝛼2=𝑑𝑚_{𝛼∈ℤ+𝑛}𝑔_𝛼𝑧^{𝛼+𝑒𝑖}|𝛼|.+1 (2.32) Hence, 𝑇^∗_𝑓+𝑔𝑤_𝑖(𝑧)=𝑇^∗_𝑓+𝑔𝑤_𝑖,𝐾_𝑧(𝑤)=𝑤_𝑖,𝑓+𝑔𝐾_𝑧(𝑤)=_𝔹𝑛𝜕𝑓+𝑔𝐾_𝑧(𝑤)𝜕𝑤_𝑖=𝑑𝑚_𝔹𝑛𝜕𝑓𝐾_𝑧(𝑤)𝜕𝑤_𝑖+𝑔⋅𝜕𝐾_𝑧(𝑤)𝜕𝑤_𝑖=𝑑𝑚𝑓_𝑒𝑖+𝑓(0)⋅𝑧^𝑒𝑖+_{𝛼∈ℤ+𝑛}𝑔_𝛼𝑧^{𝛼+𝑒𝑖}|𝛼|.+1 (2.33)
Comparing the expressions of (𝑇_𝑓+𝑔𝑤_𝑖)(𝑧) and (𝑇^∗_𝑓+𝑔𝑤_𝑖)(𝑧), we obtain 𝑔_𝛽𝛽+1=𝑓_𝛽,∀𝛽>0. (2.34)
It follows from (2.26) and (2.34) that 𝑓_𝛽=𝑔_𝛽=0,for𝛽>0,𝑛>1, (2.35) which, according to (2.27), implies that 𝑢=𝑓(0)+𝑔(0) is a real constant function. This complete the proof of the theorem.

Note that for Theorem 2.5 the assumption “𝑢=𝑓+𝑔∈Ω, where 𝑓, 𝑔 are holomorphic” can not be removed. For example, let 𝑢=1−|𝑧|², that is 𝑢=1−(𝑧²₁+𝑧²₂+⋯+𝑧²_𝑛), then by the below Theorem 4.5  𝑇_𝑢=𝑇^∗_𝑢=0. However, 𝑢 is not a constant function.

Corollary 2.6. Let 𝑢=𝑓+𝑔∈Ω, where 𝑓 and 𝑔 are holomorphic. Then 𝑇_𝑢 is a projection operator if and only if 𝑢=1 or 𝑢=0.

Proof. The “if” part is clear. If 𝑇_𝑢 is a projection, then 𝑇_𝑢=𝑇^∗_𝑢. Theorem 2.5 implies that 𝑢=𝑐 where 𝑐 is a real. Since 𝑇_𝑢=𝑇²_𝑢, we see that 𝑐²=𝑐. This proves 𝑢=1 or 𝑢=0.

Next, we will characterize pluriharmonic symbols for which the corresponding Toeplitz operator is an isometry.

Theorem 2.7. Let 𝑢=𝑓+𝑔∈Ω, where 𝑓 and 𝑔 are holomorphic. Then the following statements are equivalent:(1)𝑇_𝑢 is unitary;(2)𝑇_𝑢 is isometric;(3)𝑢 is a constant function of modulus 1.

Proof. That (1) implies (2) follows from the fact that unitary operator is isometric.
To prove that (2) implies (3), we denote 𝑓=∑_𝛽∈ℤ^+𝑛𝑓_𝛽𝑧^𝛽 and 𝑔=∑_𝛽∈ℤ^+𝑛𝑔_𝛽𝑧^𝛽. Recalling the proof of Theorem 2.5, we have that 𝑇_𝑓+𝑔1=𝑓(0)+𝑔(0)+_|𝛽|>0𝑓_𝛽𝑧^𝛽,𝑇_𝑓+𝑔𝑤_𝑖(z)=_{𝛽∈ℤ+𝑛}𝑓_𝛽𝑧^{𝛽+𝑒𝑖}+𝑔_𝑒𝑖+𝑛+1𝑔(0)⋅𝑧^𝑒𝑖,𝑇^∗_𝑓+𝑔1=𝑓(0)+𝑔(0)+_|𝛽|>0𝑔_𝛽𝛽𝛽𝑧𝑛+^𝛽,𝑇^∗_𝑓+𝑔𝑤_𝑖(𝑧)=𝑓_𝑒𝑖+𝑓(0)⋅𝑧^𝑒𝑖+_{𝛽∈ℤ+𝑛}𝑔_𝛽𝑧^{𝛽+𝑒𝑖}𝛽.+1 (2.36) Calculating the norms of the above items, it follows that 𝑇_𝑓+𝑔1²=𝑓(0)+𝑔(0)²+_|𝛽|>0𝑓_𝛽²𝑧^𝛽2,𝑇_𝑓+𝑔𝑤_𝑖(𝑧)²=_|𝛽|>0𝑓_𝛽²𝑧^{𝛽+𝑒𝑖2}+𝑔_𝑒𝑖𝑛+1²+𝑓(0)+𝑔(0)²,𝑇^∗_𝑓+𝑔1²=𝑓(0)+𝑔(0)²+_|𝛽|>0𝑔_𝛽𝛽𝑛+𝛽²𝑧^𝛽2,(𝑇^∗_𝑓+𝑔𝑤_𝑖)(𝑧)²=𝑓_𝑒𝑖²+𝑓(0)+𝑔(0)²+_|𝛽|>0𝑔_𝛽𝛽+1²𝑧^{𝛽+𝑒𝑖2}. (2.37)(2.38)(2.39)(2.40) By the assumption, (2.37), (2.38), (2.39), and (2.40) are all equal to 1 since 𝑇_𝑢 as well as 𝑇^∗_𝑢 is an isometry.
Note that ‖𝑧^𝛽‖²/[|𝛽|(𝑛+|𝛽|)]²<‖𝑧^{𝛽+𝑒𝑖}‖²/(1+|𝛽|)², for |𝛽|>0 and 𝑛>1. Comparing (2.39) and (2.40), we obtain that |𝑓(0)+𝑔(0)|=1 and 𝑔_𝛽=0, for |𝛽|>0 and 𝑛>1. Then (2.37) implies that 𝑓_𝛽=0,for𝛽>0,𝑛>1. (2.41) Therefore, 𝑢=𝑓(0)+𝑔(0) and |𝑢|=|𝑓(0)+𝑔(0)|=1.
Finally, if 𝑢=𝑐 is a constant function, by Theorem 2.4, we have 𝑇^∗_𝑢=𝑇_𝑢. The desired implication (3)⇒(1) follows from the fact that 𝑇^∗_𝑢𝑇_𝑢=𝑇_𝑢𝑇_𝑢=𝑀_|𝑢|²=1 and 𝑇_𝑢𝑇^∗_𝑢=𝑇_𝑢𝑇_𝑢=𝑀_|𝑢|²=1.

3. Commuting Toeplitz Operators with Conjugate Holomorphic Symbols

In this section, we will study the commuting problems of Toeplitz operators with conjugate holomorphic symbols. By the definition of 𝑇_𝜙, if 𝜙∈Ω is holomorphic, then 𝑇_𝜙=𝑀_𝜙. Therefore, for two holomorphic symbols 𝑓, 𝑔∈Ω, 𝑇_𝑓, and 𝑇_𝑔 commute. It is natural to ask when 𝑇_𝑓 and 𝑇_𝑔 commute, The following theorem shows that 𝑇_𝑓 commutes with 𝑇_𝑔 only in the trival case. In this section, we may always assume 𝑓=∑_𝛽∈ℤ^+𝑛𝑓_𝛽𝑧^𝛽 and 𝑔=∑_𝛽∈ℤ^+𝑛𝑔_𝛽𝑧^𝛽.

Theorem 3.1. Let 𝑓∈Ω and 𝑔∈Ω be holomorphic. Then 𝑇_𝑔𝑇_𝑓=𝑇_𝑓𝑇_𝑔 if and only if for 𝛼, 𝛽, 𝛾≻0, _{𝛾+𝛽=𝛼}𝑔_𝛾𝑓_𝛽𝛾−𝛽𝑛+𝛽𝑛+𝛾=0. (3.1)

Proof. Suppose reproducing kernel 𝐾_𝑧(𝑤)=1+∑_|𝛼|>0(((|𝛼|+𝑛−1)!/|𝛼|𝑛!𝛼!)𝑤^𝛼𝑧^𝛼) = 1+∑_|𝛼|>0𝑐_𝛼𝑤^𝛼𝑧^𝛼. Without loss of generality, we may assume 𝑓(0)=𝑔(0)=0.
Note that for 𝛼≻𝛽≻0, 𝑃𝑧^𝛽𝑧^𝛼=𝑑𝛼,𝛼−𝛽𝑧^𝛼−𝛽, (3.2) where 𝑑(𝛼,𝛼−𝛽)=(𝛼!/(𝑛+|𝛼|−1)!)/((𝛼−𝛽)!/(𝑛+|𝛼−𝛽|−1)!). It follows that 𝑇_𝑔𝑧^𝛼=𝑃𝑔𝑤^𝛼=𝑔_𝛼𝑧^𝛼2₂+_𝛾≺𝛼𝑔_𝛾𝑑𝛼,𝛼−𝛾𝑧^𝛼−𝛾. (3.3) Therefore, we have 𝑇_𝑓𝑇_𝑔𝑧^𝛼=𝑔_𝛼𝑧^𝛼2₂𝑇_𝑓(1)+_𝛾≺𝛼𝑔_𝛾𝑑𝛼,𝛼−𝛾𝑇_𝑓𝑧^𝛼−𝛾=𝑔_𝛼𝑓(0)𝑧^𝛼2₂+_𝛾≺𝛼𝑔_𝛾𝑓_𝛼−𝛾𝑑𝛼,𝛼−𝛾𝑧^𝛼−𝛾2₂+_{𝛾≺𝛼𝛽≺𝛼−𝛾}𝑔_𝛾𝑓_𝛽𝑑𝛼,𝛼−𝛾𝑑𝛼−𝛾,𝛼−𝛾−𝛽𝑧^{𝛼−𝛾−𝛽}=_{𝛾+𝛽=𝛼}𝑔_𝛾𝑓_𝛽𝑛!𝛼!𝑛+𝛽(𝑛+|𝛼|−1)!+_{𝛾+𝛽≺𝛼}𝑔_𝛾𝑓_𝛽𝑑𝛼,𝛼−𝛾−𝛽𝑧^{𝛼−𝛾−𝛽}. (3.4) Similarly, we have that 𝑇_𝑔𝑇_𝑓𝑧^𝛼=_{𝛾+𝛽=𝛼}𝑓_𝛾𝑔_𝛽𝑛!𝛼!𝑛+𝛽(𝑛+|𝛼|−1)!+_{𝛾+𝛽≺𝛼}𝑓_𝛾𝑔_𝛽𝑑𝛼,𝛼−𝛾−𝛽𝑧^{𝛼−𝛾−𝛽}=_{𝛾+𝛽=𝛼}𝑔_𝛾𝑓_𝛽𝑛!𝛼!𝑛+𝛾(𝑛+|𝛼|−1)!+_{𝛾+𝛽≺𝛼}𝑔_𝛾𝑓_𝛽𝑑𝛼,𝛼−𝛾−𝛽𝑧^{𝛼−𝛾−𝛽}. (3.5) Observe that (3.4) and (3.5) have the same coefficients of 𝑧^{𝛼−𝛾−𝛽}, it follows that 𝑇_𝑓𝑇_𝑔𝑧^𝛼=𝑇_𝑔𝑇_𝑓𝑧^𝛼 (3.6) if and only if _{𝛾+𝛽=𝛼}𝑔_𝛾𝑓_𝛽𝛾−𝛽𝛽𝛾⋅𝑛+𝑛+𝑛!𝛼!(𝑛+|𝛼|−1)!=0. (3.7) Since 𝑛!𝛼!/(𝑛+|𝛼|−1)!>0, the desired result is obtained.

Corollary 3.2. If 𝑓_𝛽=𝑔_𝛽=0 for |𝛽|≠𝑘₀, where 𝑘₀ is a positive integer, then 𝑇_𝑓𝑇_𝑔=𝑇_𝑔𝑇_𝑓.

Proof. Since each item 𝑔_𝛾𝑓_𝛽(|𝛾|−|𝛽|) equals to 0, (3.1) is satisfied. Thus the desired result follows by Theorem 3.1.

For example, 𝑇_𝑧^(1,1)𝑇_𝑧^(2,0)=𝑇_𝑧^(2,0)𝑇_𝑧^(1,1) since |(1,1)|=|(2,0)|=2. On the Dirichlet space of the unit disc or polydisc, Dusitermaat, Lee, Geng, and Zhou prove that for holomorphic functions 𝑓 and 𝑔, 𝑇_𝑔𝑇_𝑓=𝑇_𝑓𝑇_𝑔 if and only if 𝑓, 𝑔, and 1 are dependent, see [12, 13]. However, this is not true on the unit ball Dirichlet space by Corollary 3.2. Indeed, the condition that 𝑓, 𝑔, and 1 are linearly dependent is sufficient but not necessary for the commuting of 𝑇_𝑓 and 𝑇_𝑔.

Next, we will discuss when Toeplitz operator with holomorphic symbol and Toeplitz operator with conjugate holomorphic symbol will commute.

Theorem 3.3. Let 𝑓∈Ω and 𝑔∈Ω be holomorphic. Then 𝑇_𝑔𝑇_𝑓=𝑇_𝑓𝑇_𝑔 if and only if 𝑓 or 𝑔 is a constant function.

Proof. The “if” part implication is obvious. Now suppose 𝑇_𝑔𝑇_𝑓=𝑇_𝑓𝑇_𝑔. For each multi-index 𝛼, we have 𝑇_𝑓𝑇_𝑔𝑧^𝛼=𝑓⋅𝑇_𝑔𝑧^𝛼=_𝛽≽0𝑓_𝛽𝑧^𝛽⋅𝑔_𝛼𝑧^𝛼2+_𝛾≺𝛼𝑔_𝛾𝑑𝛼,𝛼−𝛾𝑧^𝛼−𝛾=𝑔_𝛼𝑧^𝛼2⋅𝑓(0)+𝑔_𝛼𝑧^𝛼2_𝛽≻0𝑓_𝛽𝑧^𝛽+_{𝛾≺𝛼𝛽≽0}𝑔_𝛾𝑓_𝛽𝑑𝛼,𝛼−𝛾𝑧^{𝛼−𝛾+𝛽},𝑇_𝑔𝑇_𝑓𝑧^𝛼=𝑇_𝑔𝑓𝑧^𝛼=_{𝜉=𝛽+𝛼}𝑔_𝜉𝑓_𝛽𝑤^𝜉2+_{𝜉≺𝛼+𝛽𝛽≽0}𝑔_𝜉𝑓_𝛽𝑑𝛼+𝛽,𝛼+𝛽−𝜉𝑧^{𝛼+𝛽−𝜉}. (3.8)(3.9) Assume that 𝑓 is not a constant function. Hence there exists 𝛽₀≻0 such that 𝑓_𝛽0≠0.
For (3.8), let 𝛽=𝛽₀ and 𝛾=𝛼≻0, the coefficient of 𝑧^𝛽0 is 𝑔_𝛼𝑓_𝛽0𝑛!𝛼!(|𝛼!.𝑛+|) (3.10) On the other hand, if we let 𝛽=𝛽₀ and 𝜉=𝛼≻0 in (3.9), then the coefficient of 𝑧^𝛽0 is 𝑔_𝛼𝑓_𝛽0𝑑𝛼+𝛽₀,𝛽₀. (3.11) Since 𝑛!𝛼!(𝑛+|𝛼|)!≠𝑑𝛼+𝛽₀,𝛽₀ (3.12) and 𝑓_𝛽0≠0, we deduce that 𝑔_𝛼=0,for|𝛼|>0, (3.13) which implies that 𝑔 is a constant function. The proof is complete.

4. Commuting Toeplitz Operators with Symbols 𝑧^𝑝𝑧^𝑞𝜙(|𝑧|²)

Zhou and Dong [14] discussed the commuting and zero product problems of Toeplitz operators on the Bergman space of the unit ball in ℂ^𝑛 whose symbols are of the form 𝜉^𝑘𝜙 where 𝜙 is a radial function. In [15], they generalized the case of the radial symbols to that of the separately quasi-homogeneous symbols. In [16], Grudsky et al. considered weighted Bergman spaces on the unit ball in ℂ^𝑛. In terms of the Wick symbol of a Toeplitz operator, the complete information about the operator with radial symbols was given. Vasilevski [17] studied the Toeplitz operators with the quasi-radial quasi-homogeneous symbol. For the case of Dirichlet spaces, Chen et al. [18, 19] studied the quasi-radial Toeplitz operaors on the disk. However, little work has been done in the unit ball case. The commuting problem on it is subtle and no general answer is known. Dong and Zhou [15] have shown that any function 𝑓 in 𝐿²(𝔹_𝑛,𝑑𝑚) has the decomposition 𝑓(𝑧)=∑_𝑘∈ℤ𝜉^𝑘𝑓_𝑘(𝑟), where 𝑓_𝑘(𝑟) is separately radial. In this section, the commuting and zero product problems of Toeplitz operators 𝑇_𝑧𝑝𝑧^𝑞_𝜙(|𝑧|²₎,𝑝,𝑞≽0 will be concerned, which may be helpful to the further study of the commuting Toeplitz operators with general symbols. We denote Σ={𝜙∶𝜙,𝜙^′∈𝐿¹([0,1])} and Σ^′={𝜙 is absolutely continuous on [0,1)∶𝜙,𝜙^′∈𝐿¹([0,1))}. In the remaining part of this paper, we will always assume 𝜙∈Σ. A direct calculation gives the following lemma.

Lemma 4.1. Let 𝑝≽0 and 𝜙(|𝑧|²)∈Σ be radial functions. Then 𝑇_{𝑧𝑝𝜙(|𝑧|}²₎𝑧^𝛼=𝑛+𝑝+𝛼−1𝜙𝑛+𝑝+𝛼−1𝑧^𝑝+𝛼,𝑝+𝛼≻0,𝑛∫¹₀𝜙(𝑟)𝑟^𝑛−1𝑑𝑟,𝑝=𝛼=0, (4.1) where 𝜙(𝑧)=∫¹₀𝑟^𝑧−1[𝜙+∫¹_𝑟𝜙^′(𝑡)𝑑𝑡]𝑑𝑟.

Proof. To simiplify the statement, we denote reproducing kernel 𝐾_𝑧(𝑤) by ∑_|𝛾|≥0𝑐_𝛾𝑤^𝛾𝑧^𝛾. Notice that (𝜕/𝜕𝑤_𝑖)(𝑤^𝑝+𝛼𝜙(|𝑤|²))=(𝑝_𝑖+𝛼_𝑖)𝜙𝑤^{𝑝+𝛼−𝑒𝑖}+𝑤^𝑝+𝛼𝜙^′𝑤_𝑖. For 𝑝+𝛼≻0, with integration in polar coordinates we have 𝑇_{𝑧𝑝𝜙(|𝑧|}²₎𝑧^𝛼=𝑤^𝑝+𝛼𝜙,𝐾_𝑧(𝑤=_{𝑛𝑖=1𝔹𝑛})𝜕𝜕𝑤_𝑖𝑤^𝑝+𝛼𝜙𝜕𝐾_𝑧(𝑤)𝜕𝑤_𝑖=𝑑𝑚_{𝑛𝑖=1𝔹𝑛}𝑝_𝑖+𝛼_𝑖²𝑐_𝑝+𝛼𝜙𝑤^{𝑝+𝛼−𝑒𝑖2}+𝑑𝑚_𝔹𝑛𝑝_𝑖+𝛼_𝑖𝑐_𝑝+𝛼𝜙^′𝑤^𝑝+𝛼2𝑑𝑚𝑧^𝑝+𝛼=𝑛+𝑝+𝛼−1¹₀𝑡^{𝑛+|𝑝+𝛼|−2}𝜙(𝑡)𝑑𝑡+¹₀𝑡^{𝑛+|𝑝+𝛼|−1}𝜙^′𝑑𝑡𝑧^𝑝+𝛼. (4.2) Since ∫¹₀𝑡^{𝑛+|𝑝+𝛼|−1}𝜙^′𝑑𝑡=(𝑛+|𝑝+𝛼|−1)∫¹₀𝑡^{𝑛+|𝑝+𝛼|−2}[∫¹_𝑡𝜙^′𝑑𝑟]𝑑𝑡 with integration by part, the desired result is obvious.
For 𝑝=𝛼=0, it is easy to see that 𝑇_{𝑧𝑝𝜙(|𝑧|}²₎𝑧^𝛼=𝜙,𝐾_𝑧(𝑤)=_𝔹𝑛𝜙𝑑𝑚=𝑛¹₀𝜙(𝑟)𝑟^𝑛−1𝑑𝑟. (4.3) The proof is complete.

We now characterize the commuting Toeplitz operators whose symbols are of the form 𝑧^𝑝𝜙(|𝑧|²), where 𝑝≻0.

Theorem 4.2. Let 𝑝, 𝑞≻0, 𝜙, 𝜓∈Σ. 𝑇_{𝑧𝑝𝜙(|𝑧|}²₎𝑇_{𝑧𝑞𝜓(|𝑧|}²₎=𝑇_{𝑧𝑞𝜓(|𝑧|}²₎𝑇_{𝑧𝑝𝜙(|𝑧|}²₎ if and only if (𝑛+|𝑞+𝛼|−1)𝜓(𝑛+|𝑞+𝛼|−1)𝜙(𝑛+|𝑝+𝑞+𝛼|−1)=(𝑛+|𝑝+𝛼|−1)𝜙(𝑛+|𝑝+𝛼|−1)𝜓(𝑛 + |𝑝+𝑞+𝛼|−1) holds for any multi-index 𝛼≽0.

Proof. For any multi-index 𝛼≽0, by Lemma 4.1, it follows that 𝑇_{𝑧𝑝𝜙(|𝑧|}²₎𝑇_{𝑧𝑞𝜓(|𝑧|}²₎𝑧^𝛼=𝑛+𝑞+𝛼−1𝜓𝑛+𝑞+𝛼−1𝑛+𝑝+𝑞+𝛼−1𝜙𝑛+𝑝+𝑞+𝛼−1𝑧^{𝑝+𝑞+𝛼},𝑇_{𝑧𝑞𝜓(|𝑧|}²₎𝑇_{𝑧𝑝𝜙(|𝑧|}²₎𝑧^𝛼=𝑛+𝑝+𝛼−1𝜙𝑛+𝑝+𝛼−1𝑛+𝑝+𝑞+𝛼−1𝜓𝑛+𝑝+𝑞+𝛼−1𝑧^{𝑝+𝑞+𝛼}. (4.4) Since (𝑛+|𝑝+𝑞+𝛼|−1)>0, the result is followed.

A particular case of the above theorem is 𝜙=𝜓. In this case 𝑇_{𝑧𝑝𝜙(|𝑧|}²₎𝑇_{𝑧𝑞𝜙(|𝑧|}²₎=𝑇_{𝑧𝑞𝜙(|𝑧|}²₎𝑇_{𝑧𝑝𝜙(|𝑧|}²₎ (4.5) if and only if 𝜙−𝜙⋅𝜙𝑛+𝑞+𝛼−1𝑛+𝑞+𝛼−1𝑛+𝑝+𝛼−1𝑛+𝑝+𝛼−1𝑛+𝑝+𝑞+𝛼−1=0. (4.6) Thus we immediately have the following result.

Corollary 4.3. Let 𝑝, 𝑞≻0, 𝜙∈Σ. If |𝑝|=|𝑞|, then 𝑇_{𝑧𝑝𝜙(|𝑧|}²₎𝑇_{𝑧𝑞𝜙(|𝑧|}²₎=𝑇_{𝑧𝑞𝜙(|𝑧|}²₎𝑇_{𝑧𝑝𝜙(|𝑧|}²₎.

If 𝜙 is absolutely continuous on [0,1), integrating by parts, one has 𝑚𝜙(𝑚)=lim_𝑟→1⁻𝜙(𝑟)=𝜙(1⁻), for any positive integer 𝑚. Thus, using Lemma 4.1 one can get the following lemma which will be often used in the sequel.

Lemma 4.4. Let 𝑝≽0, 𝜙∈Σ^′, one has 𝑇_{𝑧𝑝𝜙(|𝑧|}²₎𝑧^𝛼=𝜙(1⁻)𝑧^𝑝+𝛼,𝑝+𝛼≻0;𝑛∫¹₀𝜙(𝑟)𝑟^𝑛−1𝑑𝑟,𝑝=𝛼=0. (4.7)

By Lemma 4.4, a regular argument shows the results below.

Theorem 4.5. Let 𝑝_𝑖≻0 and 𝜙_𝑖∈Σ^′. Then the followings hold.(1)𝑇_{𝑧𝑝1𝜙1}𝑇_{𝑧𝑝2𝜙2}=𝑇_{𝑧𝑝2𝜙2}𝑇_{𝑧𝑝1𝜙1}=𝑇_{𝑧𝑝12+𝑝𝜙1𝜙2}. (2)𝑇_{𝑧𝑝1𝜙1}×⋯×𝑇_𝑧^𝑝_𝑘𝜙𝑘=0 if and only if 𝜙₁(1⁻)×⋯×𝜙_𝑘(1⁻)=0.(3)Let 𝑝_𝑖≠𝑝_𝑗 for 𝑖≠𝑗, 𝑇_{𝑧𝑝1𝜙1}+⋯+𝑇_𝑧^𝑝_𝑘𝜙𝑘=0 if and only if each 𝜙_𝑖(1⁻)=0,1≤𝑖≤𝑘.

Before discussing the commutivity of Toeplitz operator with symbols 𝑧^𝑞𝜙(|𝑧|²), one needs the following lemma which can be obtained by direct computation.

Lemma 4.6. Let multi-index 𝑞≽0 and 𝜙∈Σ. Then 𝑇_𝑧^𝑞_𝜙(|𝑧|²₎𝑧^𝛼=𝑑𝛼,𝛼−𝑞(𝑛+|𝛼|−1)𝜙(𝑛+|𝛼|−1)𝑧^𝛼−𝑞,𝛼≻𝑞;𝑛!𝑞!𝑞!∫𝑛+−1¹⁰𝑟^{𝑛+|𝑞|−1}𝜙(𝑟)𝑑𝑟,𝛼=𝑞;0,𝛼≽^╱𝑞, (4.8) where 𝑑(𝛼,𝛼−𝑞)=(𝛼!/(𝑛+|𝛼|−1)!)/((𝛼−𝑞)!/(𝑛+|𝛼−𝑞|−1)!) and 𝛼≽^╱𝑞 means that there exists 𝑖₀ such that 𝛼_𝑖0<𝑞_𝑖0.

Proof. For 𝛼≻𝑞, We get that 𝑇_𝑧^𝑞_𝜙(|𝑧|²₎𝑧^𝛼=𝑤^𝛼𝑤^𝑞𝜙,𝐾_𝑧(𝑤=_{𝑛𝑖=1𝔹𝑛})𝜕𝜕𝑤_𝑖𝑤^𝛼𝑤^𝑞𝜙𝜕(𝐾_𝑧(𝑤))𝜕𝑤_𝑖=𝑑𝑚_{𝑛𝑖=1𝔹𝑛}𝛼_𝑖𝑤^{𝛼−𝑒𝑖}𝜙+𝑤^𝛼𝑤_𝑖𝜙^′𝑤^𝑞⋅_𝑟𝑐_𝑟𝑟_𝑖𝑤^{𝑟−𝑒𝑖}𝑧^𝑟=𝑑𝑚_{𝑛𝑖=1𝔹𝑛}𝛼_𝑖𝜙𝑤^{𝛼−𝑒𝑖2}𝑐_𝛼−𝑞𝛼_𝑖−𝑞_𝑖+𝑤^𝛼2𝜙^′𝑐_𝛼−𝑞𝛼_𝑖−𝑞_𝑖𝑑𝑚𝑧^𝛼−𝑞=𝑐_𝛼−𝑞𝑧^𝛼−𝑞𝛼−𝑞𝑛!𝛼!(|𝛼|)!|𝛼|)𝑛+−1(𝑛+−1¹₀𝑡^{𝑛+|𝑞+𝛼|−2}𝜙𝑑𝑡+¹₀𝑡^{𝑛+|𝑞+𝛼|−1}𝜙^′(|𝛼|)𝜙(|𝛼|)𝑧𝑑𝑡=𝑑𝛼,𝛼−𝑞𝑛+−1𝑛+−1^𝛼−𝑞. (4.9)
For 𝛼=𝑞, we have 𝑇_𝑧^𝑞_𝜙(|𝑧|²₎𝑧^𝛼=𝑤^𝛼𝑤^𝑞𝜙,𝐾_𝑧(𝑤)=_𝔹𝑛𝑧^𝑞2𝜙|𝑤|²=𝑑𝑚𝑛!𝑞!𝑛+𝑞−1!¹₀𝑡^{𝑛+|𝑞|−1}𝜙(𝑡)𝑑𝑡. (4.10)
If there exists 1≤𝑖≤𝑛 such that 𝛼_𝑖<𝑞_𝑖, then 𝑇_𝑧^𝑞_𝜙(|𝑧|²₎𝑧^𝛼=𝑤^𝛼𝑤^𝑞𝜙,𝐾_𝑧(𝑤)=0. (4.11) Thus the proof is complete.

Note that if 𝜙∈Σ^′, then (𝑛+|𝛼|−1)𝜙(𝑛+|𝛼|−1)=𝜙(1⁻). It follows that 𝑇_𝑧^𝑞_𝜙(|𝑧|²₎𝑧^𝛼=𝑑𝛼,𝛼−𝑞𝜙(1⁻)𝑧^𝛼−𝑞,for𝛼≻𝑞. (4.12)

The following theorem gives some properties of the Toeplitz operator with symbols 𝑧^𝑝𝜙(|𝑧|²).

Theorem 4.7. Let 𝑝, 𝑞, 𝑝_𝑖≻0, 𝑝_𝑖≠𝑝_𝑗 for 𝑖≠𝑗 and 𝜙, 𝜓, 𝜙_𝑖∈Σ^′. Then the following assertions hold.(1)𝑇_𝑧^𝑝_𝜙(|𝑧|²₎𝑇_𝑧^𝑞_𝜓(|𝑧|²₎=𝑇_𝑧^𝑞_𝜓(|𝑧|²₎𝑇_𝑧^𝑝_𝜙(|𝑧|²₎ if and only if 𝜓(1⁻)∫¹₀𝑟^{𝑛+|𝑝|−1}𝜙(𝑟)𝑑𝑟 = 𝜙(1⁻)∫¹₀𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟.(2)𝑇_𝑧^𝑝_1𝜙1×⋯×𝑇_𝑧^𝑝_𝑘𝜙𝑘=0 if and only if one of the following holds:(i)𝜙₁(1⁻)=0 and ∫¹₀𝑟^{𝑛+|𝑝1|−1}𝜙₁(𝑟)𝑑𝑟=0;(ii)There exists 𝑖₀ where 2≤𝑖₀≤𝑘 such that 𝜙_𝑖0(1⁻)=0.(3)𝑇_𝑧^𝑝_1𝜙1+⋯+𝑇_𝑧^𝑝_𝑘𝜙𝑘=0 if and only if 𝜙_𝑖(1⁻)=0 and ∫¹₀𝑟^{𝑛+|𝑝𝑖|−1}𝜙_𝑖(𝑟)𝑑𝑟=0 for each 𝑖, 1≤𝑖≤𝑘.

Proof. Assertions (2) and (3) are the direct consequence of Lemma 4.6. We only need to prove assertion (1). By Lemma 4.6, for ℎ≻𝑝+𝑞, since 𝜙, 𝜓∈Σ^′ we have 𝑇_𝑧^𝑝_𝜙(|𝑧|²₎𝑇_𝑧^𝑞_𝜓(|𝑧|²₎𝑧^ℎ=𝑑ℎ,ℎ−𝑞𝑑ℎ−𝑞,ℎ−𝑞−𝑝𝜙(1⁻)𝜓(1⁻)𝑧^{ℎ−𝑞−𝑝}=𝑑ℎ,ℎ−𝑞−𝑝𝜙(1⁻)𝜓(1⁻)𝑧^{ℎ−𝑞−𝑝},𝑇_𝑧^𝑞_𝜓(|𝑧|²₎𝑇_𝑧^𝑝_𝜙(|𝑧|²₎𝑧^ℎ=𝑑ℎ,ℎ−𝑝𝑑ℎ−𝑝,ℎ−𝑞−𝑝𝜙(1⁻)𝜓(1⁻)𝑧^{ℎ−𝑞−𝑝}=𝑑ℎ,ℎ−𝑞−𝑝𝜙(1⁻)𝜓(1⁻)𝑧^{ℎ−𝑞−𝑝}. (4.13) It is obvious that 𝑇_𝑧^𝑝_𝜙(|𝑧|²₎𝑇_𝑧^𝑞_𝜓(|𝑧|²₎𝑧^ℎ=𝑇_𝑧^𝑞_𝜓(|𝑧|²₎𝑇_𝑧^𝑝_𝜙(|𝑧|²₎𝑧^ℎ (4.14) holds for ℎ≻𝑝+𝑞.
For ℎ=𝑝+𝑞, we obtain 𝑇_𝑧^𝑝_𝜙(|𝑧|²₎𝑇_𝑧^𝑞_𝜓(|𝑧|²₎𝑧^ℎ=𝑑𝑝+𝑞,𝑝𝜓(1⁻)𝑛!𝑝!𝑛+𝑝−1!¹₀𝑟^{𝑛+|𝑝|−1}𝜙(𝑟)=!𝑑𝑟𝑛!𝑝+𝑞!𝜓(1𝑛+𝑝+𝑞−1⁻)¹₀𝑟^{𝑛+|𝑝|−1}𝜙(𝑟)𝑇𝑑𝑟,_{𝑧𝑞𝜓(|𝑧|2)}𝑇_𝑧^𝑝_𝜙(|𝑧|²₎𝑧^ℎ=𝑑𝑝+𝑞,𝑞𝜙(1⁻)𝑛!𝑞!𝑛+𝑞−1!¹₀𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)=!𝑑𝑟𝑛!𝑝+𝑞!𝜙(1𝑛+𝑝+𝑞−1⁻)¹₀𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟. (4.15) Since 𝑛!(𝑝+𝑞)!/(𝑛+|𝑝+𝑞|−1)!>0, then the desired result is obvious.

In the assertion (1) of Theorem 4.7, if 𝜙=𝜓=1, then we get 𝜓(1⁻)¹₀𝑟^{𝑛+|𝑝|−1}𝜙(𝑟)1𝑑𝑟=𝑝(1𝑛+,𝜙⁻)¹₀𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)1𝑑𝑟=𝑞.𝑛+ (4.16)

Therefore, it is easy to get the following corollary.

Corollary 4.8. Let 𝑝, 𝑞≻0. Then 𝑇_𝑧^𝑝𝑇_𝑧^𝑞=𝑇_𝑧^𝑞𝑇_𝑧^𝑝 if and only if |𝑝|=|𝑞|.

It is well known that 𝑇_𝜙𝑇_𝜓=𝑇_ℎ on the Hardy space if and only if either 𝜙 or 𝜓 is holomorphic. However, Lemma 4.6 and Theorem 4.7 implies that a similar result does not hold on the Dirichlet space of the unit ball. Indeed, by the computation in Lemma 4.6 and Theorem 4.7, it is easy to verify that for any 𝑝, 𝑞≻0, 𝑇_𝑧^𝑝𝑇_𝑧^𝑞𝑧^𝑝+𝑞≠𝑇_𝑧^𝑝+𝑞𝑧^𝑝+𝑞.

Theorem 4.9. Let 𝑝, 𝑞≻0 and 𝜙, 𝜓∈Σ^′. Then 𝑇_{𝑧𝑝𝜙(|𝑧|}²₎𝑇_𝑧^𝑞_𝜓(|𝑧|²₎=𝑇_𝑧^𝑞_𝜓(|𝑧|²₎𝑇_{𝑧𝑝𝜙(|𝑧|}²₎ if and only if 𝜙(1⁻)=0 or 𝜓(1⁻)=0 and ∫¹₀𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟=0.

Proof. For each multi-index ℎ≽0, by Lemmas 4.4 and 4.6, we have that 𝑇_{𝑧𝑝𝜙(|𝑧|}²₎𝑇_𝑧^𝑞_𝜓(|𝑧|²₎𝑧^ℎ=𝑑ℎ,ℎ−𝑞𝜓(1⁻)𝜙(1⁻)𝑧^{ℎ+𝑝−𝑞},ℎ≻𝑞𝑛!𝑞!𝑞!∫𝑛+−1¹⁰𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟𝜙(1⁻)𝑧^𝑝,ℎ=𝑞0,others,𝑇_𝑧^𝑞_𝜓(|𝑧|²₎𝑇_{𝑧𝑝𝜙(|𝑧|}²₎𝑧^ℎ=𝑑ℎ+𝑝,ℎ+𝑝−𝑞𝜓(1⁻)𝜙(1⁻)𝑧^{ℎ+𝑝−𝑞},𝑝+ℎ≻𝑞𝑛!𝑞!𝑞!∫𝑛+−1¹⁰𝑟^{𝑛+|𝑞|−1}𝜙(𝑟)𝑑𝑟𝜙(1⁻),𝑝+ℎ=𝑞0,others. (4.17) Consequently, for ℎ≻𝑞, we conclude 𝑑ℎ,ℎ−𝑞𝜓(1⁻)𝜙(1⁻)𝑧^{ℎ+𝑝−𝑞}=𝑑ℎ+𝑝,ℎ+𝑝−𝑞𝜓(1⁻)𝜙(1⁻)𝑧^{ℎ+𝑝−𝑞}. (4.18) Since 𝑝, 𝑞≻0, it is clear that 𝑑(ℎ,ℎ−𝑞) can not always equal to 𝑑(ℎ+𝑝,ℎ+𝑝−𝑞) for all ℎ≻𝑞. Thus, we get 𝜓(1⁻)𝜙(1⁻)=0. (4.19)
On the other hand, for ℎ=𝑞, we have 𝑛!𝑞!𝑛+𝑞−1!¹₀𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟𝜙(1⁻)=𝑑ℎ+𝑝,ℎ+𝑝−𝑞𝜓(1⁻)𝜙(1⁻). (4.20) Combining (4.19) and (4.20), we obtain the desired result.

Notice the assertion (3) of Theorem 4.5 and the assertion (3) of Theorem 4.7, Theorem 4.9 above shows that 𝑇_{𝑧𝑝𝜙(|𝑧|}²₎𝑇_𝑧^𝑞_𝜓(|𝑧|²₎=𝑇_𝑧^𝑞_𝜓(|𝑧|²₎𝑇_{𝑧𝑝𝜙(|𝑧|}²₎ if and only if 𝑇_{𝑧𝑝𝜙(|𝑧|}²₎=0 or 𝑇_𝑧^𝑞_𝜓(|𝑧|²₎=0 holds. That is, 𝑇_{𝑧𝑝𝜙(|𝑧|}²₎ commutes with 𝑇_𝑧^𝑞_𝜓(|𝑧|²₎ only in the trival case.

Finally, we will discuss when Toepitze operaor 𝑇_𝜙(|𝑧|²₎ commute with 𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎.

Theorem 4.10. Let 𝑝, 𝑞≻0 and 𝜙, 𝜓∈Σ^′. Then the following assertions hold.(1)If 𝑝≻𝑞, 𝑇_𝜙(|𝑧|²₎𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎=𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎𝑇_𝜙(|𝑧|²₎ if and only if 𝜓(1⁻)=0 or 𝜙(1⁻)=𝑛∫¹₀𝑟^𝑛−1𝜙(𝑟)𝑑𝑟.(2)If 𝑞≻𝑝, 𝑇_𝜙(|𝑧|²₎𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎=𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎𝑇_𝜙(|𝑧|²₎ if and only if ∫¹₀𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟=0 or 𝑛∫¹₀𝑟^𝑛−1𝜙(𝑟)𝑑𝑟=𝜙(1⁻).(3)If 𝑝≽^╱𝑞 and 𝑞≽^╱𝑝 or 𝑝=𝑞, 𝑇_𝜙(|𝑧|²₎𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎=𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎𝑇_𝜙(|𝑧|²₎.

Proof. For each multi-index ℎ≽0, by Lemmas 4.4 and 4.6 we have 𝑇_𝜙(|𝑧|²₎𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎𝑧^ℎ=𝑑𝑝+ℎ,𝑝+ℎ−𝑞𝜓(1⁻)𝜙(1⁻)𝑧^{ℎ+𝑝−𝑞},ℎ+𝑝≻𝑞𝑛!𝑞!𝑛+𝑞−1!<?displaystyle?>∫¹₀𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟⋅𝑛∫¹₀𝑟^𝑛−1𝜙(𝑟)𝑇𝑑𝑟,ℎ+𝑝=𝑞,_{𝑧𝑝𝑧𝑞𝜓(|𝑧|2)}𝑇_𝜙(|𝑧|²₎𝑧^ℎ=𝑑𝑝+ℎ,𝑝+ℎ−𝑞𝜓(1⁻)𝜙(1⁻)𝑧^{ℎ+𝑝−𝑞}𝜙(1,ℎ≻0,ℎ+𝑝≻𝑞₋)𝑛!𝑞!𝑞!∫𝑛+−1¹⁰𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟,ℎ≻0,ℎ+𝑝=𝑞0,ℎ≻0,ℎ+𝑝≽^╱𝑞𝑛∫¹₀𝑟^𝑛−1𝜙(𝑟)𝑑𝑟𝑑𝑝,𝑝−𝑞𝜓(1⁻)𝑧^{ℎ+𝑝−𝑞}𝑛∫,ℎ=0,𝑝≻𝑞₁₀𝑟^𝑛−1𝜙(𝑟)𝑑𝑟𝑛!𝑞!𝑞!∫𝑛+−1¹⁰𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟,ℎ=0,𝑝=𝑞0,ℎ=0,𝑝≽^╱𝑞. (4.21)Case 1. Suppose 𝑝≻𝑞. We have 𝑇_𝜙(|𝑧|²₎𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎𝑧^ℎ=𝑑𝑝+ℎ,𝑝+ℎ−𝑞𝜓(1⁻)𝜙(1⁻)𝑧^{ℎ+𝑝−𝑞},𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎𝑇_𝜙(|𝑧|²₎𝑧^ℎ=𝑑𝑝+ℎ,𝑝+ℎ−𝑞𝜓(1⁻)𝜙(1⁻)𝑧^{ℎ+𝑝−𝑞},ℎ≻0𝑛∫¹₀𝑟^𝑛−1𝜙(𝑟)𝑑𝑟𝑑𝑝,𝑝−𝑞𝜓(1⁻)𝑧^{ℎ+𝑝−𝑞},ℎ=0. (4.22)𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎ commutes with 𝑇_𝜙(|𝑧|²₎ if and only if 𝑑𝑝,𝑝−𝑞𝜓(1⁻)𝜙(1⁻)=𝑑𝑝,𝑝−𝑞𝜓(1⁻)𝑛¹₀𝑟^𝑛−1𝜙(𝑟)𝑑𝑟, (4.23) which is equivalent to 𝜓(1⁻)=0 or 𝜙(1⁻)=𝑛∫¹₀𝑟^𝑛−1𝜙(𝑟)𝑑𝑟.
Case 2. Suppose 𝑞≻𝑝. Note that ℎ+𝑝≻𝑞 if and only if ℎ≻𝑞−𝑝. We have 𝑇_𝜙(|𝑧|²₎𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎𝑧^ℎ=𝑑𝑝+ℎ,𝑝+ℎ−𝑞𝜓(1⁻)𝜙(1⁻)𝑧^{ℎ+𝑝−𝑞},ℎ≻𝑞−𝑝𝑛!𝑞!𝑞!∫𝑛+−1¹⁰𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟⋅𝑛∫¹₀𝑟^𝑛−1𝜙(𝑟)𝑑𝑟,ℎ=𝑞−𝑝0,ℎ^╱𝑇≽𝑞−𝑝,_{𝑧𝑝𝑧𝑞𝜓(|𝑧|2)}𝑇_𝜙(|𝑧|²₎𝑧^ℎ=𝑑𝑝+ℎ,𝑝+ℎ−𝑞𝜓(1⁻)𝜙(1⁻)𝑧^{ℎ+𝑝−𝑞}𝜙(1,ℎ≻𝑞−𝑝₋)𝑛!𝑞!𝑞!∫𝑛+−1¹⁰𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟,ℎ=𝑞−𝑝0,ℎ≽^╱𝑞−𝑝. (4.24) It follows that 𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎ commutes with 𝑇_𝜙(|𝑧|²₎ if and only if ¹₀𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟⋅𝑛¹₀𝑟^𝑛−1𝜙(𝑟)𝑑𝑟=¹₀𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟𝜙(1⁻), (4.25) which is equivalent to ∫¹₀𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟=0 or 𝑛∫¹₀𝑟^𝑛−1𝜙(𝑟)𝑑𝑟=𝜙(1⁻).
Case 3. Suppose 𝑝≽^╱𝑞 and 𝑞≽^╱𝑝. Let ℎ^′={ℎ_𝑖}, where ℎ_𝑖=max{𝑞_𝑖−𝑝_𝑖,0} for 1≤𝑖≤𝑛. Then for ℎ≽0, ℎ+𝑝≻𝑞 if and only if ℎ≻ℎ^′. Thus, 𝑇_𝜙(|𝑧|²₎𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎𝑧^ℎ=𝑑𝑝+ℎ,𝑝+ℎ−𝑞𝜓(1⁻)𝜙(1⁻)𝑧^{ℎ+𝑝−𝑞},ℎ≽ℎ^′0,ℎ≽^╱ℎ^′,𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎𝑇_𝜙(|𝑧|²₎𝑧^ℎ=𝑑𝑝+ℎ,𝑝+ℎ−𝑞𝜓(1⁻)𝜙(1⁻)𝑧^{ℎ+𝑝−𝑞},ℎ≽ℎ^′0,ℎ≽^╱ℎ^′. (4.26) It is obvious that 𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎ commutes with 𝑇_𝜙(|𝑧|²₎.
Case 4. Suppose 𝑝=𝑞. We have 𝑇_𝜙(|𝑧|²₎𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎𝑧^ℎ=𝑑𝑝+ℎ,𝑝+ℎ−𝑞𝜓(1⁻)𝜙(1⁻)𝑧^{ℎ+𝑝−𝑞},ℎ≻0𝑛!𝑞!𝑞!∫𝑛+−1¹⁰𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟⋅𝑛∫¹₀𝑟^𝑛−1𝜙(𝑟)𝑇𝑑𝑟,ℎ=0,_{𝑧𝑝𝑧𝑞𝜓(|𝑧|2)}𝑇_𝜙(|𝑧|²₎𝑧^ℎ=𝑑𝑝+ℎ,𝑝+ℎ−𝑞𝜓(1⁻)𝜙(1⁻)𝑧^{ℎ+𝑝−𝑞},ℎ≻0𝑛∫¹₀𝑟^𝑛−1𝜙(𝑟)𝑑𝑟𝑛!𝑞!𝑞!∫𝑛+−1¹⁰𝑟^{𝑛+|𝑞|−1}𝜓(𝑟)𝑑𝑟,ℎ=0. (4.27) It is easy to see that 𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎ commutes with 𝑇_𝜙(|𝑧|²₎. This completes the proof.

Corollary 4.11. Let 𝑝⟂𝑞 and 𝜙, 𝜓∈Σ^′. Then 𝑇_𝜙(|𝑧|²₎𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎=𝑇_𝑧𝑝𝑧^𝑞_𝜓(|𝑧|²₎𝑇_𝜙(|𝑧|²₎.

Proof. Note that 𝑝⟂𝑞 implies 𝑝≽^╱𝑞 and 𝑞≽^╱𝑝. The desired result is immediately followed by Theorem 4.10.

Acknowledgments

This research is supported by the NSFC (Grant no. 10971020 and Grant no. 11271059) and Research Fund for the Doctoral Program of Higher Education of China.