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Abstract and Applied Analysis
Volume 2008, Article ID 538573, 7 pages
http://dx.doi.org/10.1155/2008/538573
Research Article

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

Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China

Received 13 June 2008; Revised 12 October 2008; Accepted 20 November 2008

Academic Editor: Stevo Stevic

Copyright © 2008 Yufeng Lu and Jun Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper shows that if is a bounded linear operator acting on the weighted Bergman spaces on the unit ball in such that , where and ; and where is the weighted Bergman projection, then must be a Hankel operator.

1. Introduction

Let be the open unit ball in the complex vector space . For let where is the complex conjugate of and For a multi-index and , we also write

Let be the volume measure on , normalized so that For the weighted Lebesgue measure is defined bywhereis a normalizing constant so that is a probability measure on

For and , the weighted Bergman space consists of holomorphic functions in , that is,When is the standard (unweighted) Bergman spaces, which is simply denoted by .

The weighted Bergman space is a closed subspace of and the set of all polynomials is dense in . See, for example, [1].

With the norm and become Banach spaces. is a Hilbert space whose inner product will be denoted by . Some other properties of Bergman spaces as well as some recent results on the operators on them, can be found, for example, in [213] (see, also the references therein).

For , the Hankel operator is defined on bywhere is the unitary operator defined on byand is the weighted Bergman projection from onto .

The Toeplitz operator with the symbol is defined on by

Toeplitz operators have the following properties: if and are complex numbers, and and , then moreover, if then and

The symbol will denote the th coordinate function ().

It is easy to see that . Thus, the Hankel operators are particular solutions of the operator equationwhere is a bounded linear operator on .

It is well known that on the classical Hardy space , 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 [1422].

On the Hardy space , Nehari [19] proved that if is a bounded linear operator such that then for some ; moreover, can be chosen such that 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 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

In this paper, we take the weighted Bergman space 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 , which plays an important role in this paper. It is shown that every function in the weighted Bergman space 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 . The following lemma is Theorem 2.30 in [1]. Lemma 2.1. Suppose , andThen there exists a sequence in such that consists exactly of functions of the formwhere belongs to the sequence space and the series converges in the norm topology of .Remark 2.2. By the proof of Theorem 2.30 in [1], it can be seen that the sequence in Lemma 2.1 is chosen independent of , and Remark 2.3. The proof of Theorem 2.30 in [1] tells us that for any , we can choose a sequence in Lemma 2.1 so thatwhere is a positive constant independent of .

The following lemma follows immediately from Lemma 2.1.

Lemma 2.4. Suppose is a sequence as in Lemma 2.1, , and LetThen, consists exactly of the functions of the formwhere belongs to the sequence space and the series converges in the norm topology of .

From now on, we assume that is fixed and and are defined as in Lemma 2.4.

The following two lemmas follow immediately from Theorem 1.12 in [1]. Lemma 2.5. Let , , then for every , one haswhere is a constant which only depends on Lemma 2.6. There exists a constant such that for every ,where is independent of and Theorem 2.7. Let be a bounded linear operator acting on the weighted Bergman space such that . Then, there exists such that .Proof. Define the linear functional on by . Clearly, is a bounded linear functional on . Note that . From Lemma 2.4 and Remark 2.3, given , there exists in such that converges in and , where is a positive constant independent .
For , let . From (1.9), it is easy to see that . If , then we haveHence, we establish that where and are polynomials in .
Since the set of all polynomials is dense in , there are sequences of polynomials and such thatFurthermore,
Sinceby using the boundedness of and the continuity of the scalar product, it follows that
Given , from Lemma 2.5, converges in . Thus, with , we see thatNote thatand consequently
Therefore,Consequently, it follows from Lemma 2.6 thatbut in . Thus, by the continuity of it follows that for some constant . Since is dense in , it follows that is extended by continuity to an element of , and consequently, by the Hahn-Banach theorem to an element of . Therefore, there exists such thatSinceand by using the fact that , where , are polynomials in , it follows thatHence, , 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).

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