A Theorem of Nehari Type on Weighted Bergman Spaces of the Unit Ball

Abstract

This paper shows that if 𝑆 is a bounded linear operator acting on the weighted Bergman spaces 𝐴²_𝛼 on the unit ball in ℂ^𝑛 such that 𝑆𝑇_𝑧𝑖=𝑇_𝑧𝑖𝑆(𝑖=1,…,𝑛), where 𝑇_𝑧𝑖=𝑧_𝑖𝑓 and 𝑇_𝑧𝑖=𝑃(𝑧_𝑖𝑓); and where 𝑃 is the weighted Bergman projection, then 𝑆 must be a Hankel operator.

1. Introduction

Let 𝐵_𝑛 be the open unit ball in the complex vector space ℂ^𝑛. For 𝑧=(𝑧₁,…,𝑧_𝑛),𝑤=(𝑤₁,…,𝑤_𝑛)∈ℂ^𝑛, let ⟨𝑧,𝑤⟩=𝑧₁𝑤₁+⋯+𝑧_𝑛𝑤_𝑛, where 𝑤_𝑘 is the complex conjugate of 𝑤_𝑘, and |𝑧|=⟨𝑧,𝑧⟩. For a multi-index 𝑚=(𝑚₁,…,𝑚_𝑛) and 𝑧=(𝑧₁,…,𝑧_𝑛)∈ℂ^𝑛, we also write𝑧^𝑚=𝑧^𝑚1₁…𝑧^𝑚𝑛_𝑛. (1.1)

Let 𝑑𝑉 be the volume measure on 𝐵_𝑛, normalized so that 𝑉(𝐵_𝑛)=1. For 𝛼>−1, the weighted Lebesgue measure 𝑑𝑉_𝛼 is defined by𝑑𝑉_𝛼(𝑧)=𝑐_𝛼(1−|𝑧|²)^𝛼𝑑𝑉(𝑧), (1.2)where𝑐_𝛼=Γ(𝑛+𝛼+1)𝑛!Γ(𝛼+1) (1.3)is a normalizing constant so that 𝑑𝑉_𝛼 is a probability measure on 𝐵_𝑛.

For 𝑝≥1 and 𝛼>−1, the weighted Bergman space 𝐴^𝑝_𝛼 consists of holomorphic functions 𝑓 in 𝐿^𝑝(𝐵_𝑛,𝑑𝑉_𝛼), that is,𝐴^𝑝_𝛼=𝐿^𝑝(𝐵_𝑛,𝑑𝑉_𝛼)∩𝐻(𝐵_𝑛). (1.4)When 𝛼=0,𝐴^𝑝_𝛼 is the standard (unweighted) Bergman spaces, which is simply denoted by 𝐴^𝑝.

The weighted Bergman space 𝐴^𝑝_𝛼 is a closed subspace of 𝐿^𝑝(𝐵_𝑛,𝑑𝑉_𝛼) and the set of all polynomials is dense in 𝐴^𝑝_𝛼. See, for example, [1].

With the norm‖𝑓‖_𝑝=(_𝐵𝑛|𝑓(𝑧)|^𝑝𝑑𝑉_𝛼(𝑧))^1/𝑝, (1.5)𝐿^𝑝(𝐵_𝑛,𝑑𝑉_𝛼) and 𝐴^𝑝_𝛼 become Banach spaces. 𝐿²(𝐵_𝑛,𝑑𝑉_𝛼) is a Hilbert space whose inner product will be denoted by ⟨⋅,⋅⟩_𝛼. Some other properties of Bergman spaces as well as some recent results on the operators on them, can be found, for example, in [2–13] (see, also the references therein).

For 𝜑∈𝐿^∞(𝐵_𝑛), the Hankel operator 𝐻_𝜑 is defined on 𝐴²_𝛼 by𝐻_𝜑(𝑓)=𝑃(𝐽(𝜑𝑓)), (1.6)where 𝐽 is the unitary operator defined on 𝐿²(𝐵_𝑛,𝑑𝑉_𝛼) by𝐽(𝑓(𝑧))=𝐽(𝑓(𝑧₁,…,𝑧_𝑛))=𝑓(𝑧)=𝑓(𝑧₁,…,𝑧_𝑛), (1.7)and 𝑃 is the weighted Bergman projection from 𝐿²(𝐵_𝑛,𝑑𝑉_𝛼) onto 𝐴²_𝛼.

The Toeplitz operator with the symbol 𝜑∈𝐿^∞(𝐵_𝑛) is defined on 𝐴²_𝛼 by𝑇_𝜑𝑓=𝑃(𝑓𝜑),𝑓∈𝐴²_𝛼. (1.8)

Toeplitz operators have the following properties: if 𝑎 and 𝑏 are complex numbers, and 𝜑 and 𝜓∈𝐿^∞(𝐵_𝑛), then 𝑇_{𝑎𝜑+𝑏𝜓}=𝑎𝑇_𝜑+𝑏𝑇_𝜓,𝑇^∗_𝜑=𝑇_𝜑; moreover, if 𝜑∈𝐻^∞(𝐵_𝑛), then 𝑇_𝜓𝑇_𝜑=𝑇_𝜑𝜓 and 𝑇_𝜑𝑇_𝜓=𝑇_𝜑𝜓.

The symbol 𝑧_𝑖 will denote the 𝑖th coordinate function (𝑖=1,…,𝑛).

It is easy to see that 𝐻_𝜑𝑇_𝑧𝑖=𝑇_𝑧𝑖𝐻_𝜑. Thus, the Hankel operators 𝐻_𝜑 are particular solutions of the operator equation𝑆𝑇_𝑧𝑖=𝑇_𝑧𝑖𝑆,𝑖=1,…,𝑛, (1.9)where 𝑆 is a bounded linear operator on 𝐴²_𝛼.

It is well known that on the classical Hardy space 𝐻², Toeplitz operators and Hankel operators are of the same status, and present different operators classes. The authors of [14] regarded Hankel operators as an essential part of Toeplitz operator theory, and many authors studied Hankel operators and their related problems in [14–22].

On the Hardy space 𝐻², Nehari [19] proved that if 𝑆 is a bounded linear operator such that 𝑆𝑇_𝑧=𝑇_𝑧𝑆, then 𝑆=𝐻_𝜑 for some 𝜑∈𝐿^∞; moreover, 𝜑 can be chosen such that ‖𝐻_𝜑‖=‖𝜑‖. Faour [20] proved a theorem of Nehari type on the Bergman spaces of the unit disk. In [21], the authors gave the characterization of Hankel operators on the generalized 𝐻² spaces, which is also similar to the Nehari theorem on the Hardy space.

The motivation for this paper is the question whether solutions of the operator (1.9) must be the Hankel operator on the Bergman space 𝐴²_𝛼.

In this paper, we take the weighted Bergman space 𝐴²_𝛼 as our domain and prove a Nehari-type theorem. While our method is basically adapted from [20, 21], substantial amount of extra work is necessary for the setting of the weighted Bergman spaces on the unit ball.

2. Nehari-Type Theorem

To establish a Nehari-type theorem on the weighted Bergman spaces on the unit ball, we recall the atomic decomposition of the weighted Bergman space 𝐴^𝑝_𝛼, which plays an important role in this paper. It is shown that every function in the weighted Bergman space 𝐴^𝑝_𝛼 can be decomposed into a series of nice functions called atoms. These atoms are defined in terms of kernel functions and in some sense act as a basis for 𝐴^𝑝_𝛼. The following lemma is Theorem 2.30 in [1]. Lemma 2.1. Suppose 𝑝>0,𝛼>−1, and1𝑏>𝑛max(1,𝑝)+𝛼+1𝑝. (2.1)Then there exists a sequence {𝑎_𝑘} in 𝐵_𝑛 such that 𝐴^𝑝_𝛼 consists exactly of functions of the form𝑓(𝑧)=^∞_𝑘=1𝑐_𝑘(1−|𝑎_𝑘|²)^{(𝑝𝑏−𝑛−1−𝛼)/𝑝}(1−⟨𝑧,𝑎_𝑘⟩)^𝑏,𝑧∈𝐵_𝑛, (2.2)where {𝑐_𝑘} belongs to the sequence space 𝑙^𝑝 and the series converges in the norm topology of 𝐴^𝑝_𝛼.Remark 2.2. By the proof of Theorem 2.30 in [1], it can be seen that the sequence {𝑎_𝑘} in Lemma 2.1 is chosen independent of 𝑝,𝛼, and 𝑏.Remark 2.3. The proof of Theorem 2.30 in [1] tells us that for any 𝑓∈𝐴^𝑝_𝛼, we can choose a sequence {𝑐_𝑘} in Lemma 2.1 so that_𝑘|𝑐_𝑘|^𝑝≤𝐶_𝐵𝑛|𝑓(𝑧)|^𝑝𝑑𝑉_𝛼(𝑧), (2.3)where 𝐶 is a positive constant independent of 𝑓.

The following lemma follows immediately from Lemma 2.1.

Lemma 2.4. Suppose {𝑎_𝑘} is a sequence as in Lemma 2.1, 𝛼>−1, and 𝑏>𝑛+𝛼+1. Let𝑙_𝑎(𝑧)=(1−|𝑎|²)^{𝑏−𝑛−1−𝛼}(1−⟨𝑧,𝑎⟩)^𝑏. (2.4)Then, 𝐴¹_𝛼(𝐵_𝑛) consists exactly of the functions of the form𝑓(𝑧)=^∞_𝑘=1𝑐_𝑘𝑙_𝑎𝑘,𝑧∈𝐵_𝑛, (2.5)where {𝑐_𝑘} belongs to the sequence space 𝑙¹ and the series converges in the norm topology of 𝐴¹_𝛼.

From now on, we assume that 𝑏>2(𝑛+𝛼+1) is fixed and {𝑎_𝑘} and 𝑙_𝑎(𝑧) are defined as in Lemma 2.4.

The following two lemmas follow immediately from Theorem 1.12 in [1]. Lemma 2.5. Let 𝛼>−1, 0<𝑟<1, then for every 𝑎∈𝐵_𝑛, one has‖𝑙_𝑎(𝑟𝑧)‖₂≤𝑘(𝑟), (2.6)where 𝑘(𝑟) is a constant which only depends on 𝑟.Lemma 2.6. There exists a constant 𝐶 such that for every 𝑎∈𝐵_𝑛,𝑟∈(0,1),‖𝑙_𝑎(𝑟𝑧)‖₁≤𝐶, (2.7)where 𝐶 is independent of 𝑎 and 𝑟.Theorem 2.7. Let 𝑆 be a bounded linear operator acting on the weighted Bergman space 𝐴²_𝛼 such that 𝑆𝑇_𝑧𝑖=𝑇_𝑧𝑖𝑆(𝑖=1,…,𝑛). Then, there exists 𝜑∈𝐿^∞(𝐵_𝑛) such that 𝑆=𝐻_𝜑.Proof. Define the linear functional 𝐺 on 𝐴²_𝛼 by 𝐺(𝑓)=⟨𝑆𝑓,1⟩_𝛼. Clearly, 𝐺 is a bounded linear functional on 𝐴²_𝛼. Note that 𝐴²_𝛼⊂𝐴¹_𝛼. From Lemma 2.4 and Remark 2.3, given 𝑓∈𝐴²_𝛼, there exists {𝑐_𝑘} in 𝑙¹ such that 𝑓=∑_𝑘𝑐_𝑘𝑙_𝑎𝑘 converges in 𝐴¹_𝛼 and ∑|𝑐_𝑘|≤𝛽‖𝑓‖₁, where 𝛽 is a positive constant independent 𝑓.
For 𝑓∈𝐴²_𝛼, let 𝑓⁺(𝑧)=𝑓(𝑧)∈𝐴²_𝛼. From (1.9), it is easy to see that 𝑆𝑇^𝑘_𝑧𝑖=𝑇^𝑘_𝑧𝑖𝑆(𝑖=1,…,𝑛;𝑘=1,2,…,). If 𝑝=𝑎𝑧^𝑘_𝑖,𝑞=𝑏𝑧^𝑚_𝑗, then we have⟨𝑆𝑝,𝑞⟩_𝛼=𝑎𝑏⟨𝑆𝑇^𝑘_𝑧𝑖1,𝑇^𝑚_𝑧𝑗1⟩_𝛼=𝑎𝑏⟨𝑇^𝑚_𝑧𝑗𝑆𝑇^𝑘_𝑧𝑖1,1⟩_𝛼=𝑎𝑏⟨𝑆𝑇^𝑚_𝑧𝑗𝑇^𝑘_𝑧𝑖1,1⟩_𝛼=𝑎𝑏⟨𝑆(𝑧^𝑚_𝑗𝑧^𝑘_𝑖),1⟩_𝛼=⟨𝑆(𝑝𝑞⁺),1⟩_𝛼. (2.8)Hence, we establish that ⟨𝑆(𝑝𝑞⁺),1⟩_𝛼=⟨𝑆𝑝,𝑞⟩_𝛼, where 𝑝 and 𝑞 are polynomials in 𝑧=(𝑧₁,…,𝑧_𝑛).
Since the set of all polynomials is dense in 𝐴²_𝛼, there are sequences of polynomials 𝑝_𝑛(𝑧) and 𝑞_𝑛(𝑧) such that‖𝑝_𝑛−𝑙_𝑎^1/2_𝑘‖₂→0,‖𝑞_𝑛−(𝑙_𝑎^1/2_𝑘)⁺‖₂→0(as𝑛→∞). (2.9)Furthermore, ‖𝑞⁺_𝑛−𝑙_𝑎^1/2_𝑘‖₂→0.
Since⟨𝑆(𝑝_𝑛𝑞⁺_𝑛),1⟩_𝛼=⟨𝑆𝑝_𝑛,𝑞_𝑛⟩_𝛼, (2.10)by using the boundedness of 𝑆 and the continuity of the scalar product, it follows that⟨𝑆𝑙_𝑎^1/2_𝑘,(𝑙_𝑎^1/2_𝑘)⁺⟩_𝛼=⟨𝑆𝑙_𝑎𝑘,1⟩_𝛼. (2.11)
Given 𝑟∈(0,1), from Lemma 2.5, 𝑓(𝑟𝑧)=∑_𝑘𝑐_𝑘𝑙_𝑎𝑘(𝑟𝑧) converges in 𝐴²_𝛼. Thus, with 𝑓_𝑟(𝑧)=𝑓(𝑟𝑧), we see that⟨𝑆𝑓_𝑟,1⟩_𝛼=⟨𝑆(_𝑘𝑐_𝑘𝑙_𝑎𝑘(𝑟𝑧)),1⟩_𝛼=_𝑘𝑐_𝑘⟨𝑆(𝑙_𝑎𝑘(𝑟𝑧)),1⟩_𝛼=_𝑘𝑐_𝑘⟨𝑆(𝑙_𝑎^1/2_𝑘(𝑟𝑧)),(𝑙_𝑎^1/2_𝑘(𝑟𝑧))⁺⟩_𝛼. (2.12)Note that‖𝑙_𝑎^1/2_𝑘(𝑟𝑧)‖₂=(_𝐵𝑛|(𝑙_𝑎𝑘(𝑟𝑧))|𝑑𝑉_𝛼(𝑧))^1/2,‖(𝑙_𝑎^1/2_𝑘(𝑟𝑧))⁺‖₂=(_𝐵𝑛|(𝑙_𝑎𝑘(𝑟𝑧))|𝑑𝑉_𝛼(𝑧))^1/2=‖𝑙_𝑎^1/2_𝑘(𝑟𝑧)‖₂, (2.13)and consequently‖𝑙_𝑎^1/2_𝑘(𝑟𝑧)‖₂‖(𝑙_𝑎^1/2_𝑘(𝑟𝑧))⁺‖₂=‖𝑙_𝑎𝑘(𝑟𝑧)‖₁. (2.14)
Therefore,|⟨𝑆𝑓_𝑟,1⟩_𝛼|≤_𝑘|𝑐_𝑘|‖𝑆‖⋅sup_𝑎𝑘‖𝑙_𝑎𝑘(𝑟𝑧)‖₁. (2.15)Consequently, it follows from Lemma 2.6 that|⟨𝑆𝑓_𝑟,1⟩_𝛼|≤𝐶𝛽‖𝑆‖‖𝑓‖₁; (2.16)but 𝑓_𝑟→𝑓 in 𝐴²_𝛼(𝐵_𝑛). Thus, by the continuity of 𝐺 it follows that |𝐺(𝑓)|≤𝛾‖𝑓‖₁ for some constant 𝛾. Since 𝐴²_𝛼 is dense in 𝐴¹_𝛼, it follows that 𝐺 is extended by continuity to an element of (𝐴¹_𝛼)^∗, and consequently, by the Hahn-Banach theorem to an element of (𝐿¹(𝐵_𝑛))^∗=𝐿^∞(𝐵_𝑛). Therefore, there exists 𝜑∈𝐿^∞(𝐵_𝑛) such that⟨𝑆𝑓,1⟩_𝛼=⟨𝜑𝑓,1⟩_𝛼=⟨𝐽(𝜑𝑓),1⟩_𝛼=⟨𝑃(𝐽(𝜑𝑓)),1⟩_𝛼=⟨𝐻_𝜑𝑓,1⟩_𝛼. (2.17)Since⟨𝐻_𝜑(𝑝𝑞⁺),1⟩_𝛼=_𝐵𝑛𝜑(𝑧)𝑝(𝑧)𝑞(𝑧)𝑑𝑉_𝛼(𝑧)=⟨𝐻_𝜑𝑝,𝑞⟩_𝛼, (2.18)and by using the fact that ⟨𝑆(𝑝𝑞⁺),1⟩_𝛼=⟨𝑆𝑝,𝑞⟩_𝛼, where 𝑝, 𝑞 are polynomials in 𝑧, it follows that⟨𝑆𝑝,𝑞⟩_𝛼=⟨𝐻_𝜑𝑝,𝑞⟩_𝛼. (2.19)Hence, 𝑆=𝐻_𝜑, finishing the proof of the theorem.

Acknowledgments

The authors would like to express their sincere thanks to the referees whose comments considerably improved the original version of the paper. This research was also supported by NSFC (Item no.10671028).