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Recommended by Stevo Stevic
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China
Received 13 June 2008; Revised 12 October 2008; Accepted 20 November 2008
1. Introduction
Let Bn be the open unit ball in the complex vector space ...n . For z=(z1 ,...,zn ), w=(w1 ,...,wn )∈...n , let ...z,w...=z1w¯1 +...+znw¯n , where w¯k is the complex conjugate of wk , and |z|=...z,z.... For a multi-index m=(m1 ,...,mn ) and z=(z1 ,...,zn )∈...n , we also write [figure omitted; refer to PDF]
Let dV be the volume measure on Bn , normalized so that V(Bn )=1. For α>-1, the weighted Lebesgue measure dVα is defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a normalizing constant so that dVα is a probability measure on Bn .
For p≥1 and α>-1 , the weighted Bergman space Aαp consists of holomorphic functions f in Lp (Bn ,dVα ) , that is, [figure omitted; refer to PDF] When α=0, Aαp is the standard (unweighted) Bergman spaces, which is simply denoted by Ap .
The weighted Bergman space Aαp is a closed subspace of Lp (Bn ,dVα ) and the set of all polynomials is dense in Aαp . See, for example, [1].
With the norm [figure omitted; refer to PDF] Lp (Bn ,dVα ) and Aαp become Banach spaces. L2 (Bn ,dVα ) is a Hilbert space whose inner product will be denoted by ...[sm middot],[sm middot]...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 [straight phi]∈L∞ (Bn ) , the Hankel operator H[straight phi] is defined on Aα2 by [figure omitted; refer to PDF] where J is the unitary operator defined on L2 (Bn ,dVα ) by [figure omitted; refer to PDF] and P is the weighted Bergman projection from L2 (Bn ,dVα ) onto Aα2 .
The Toeplitz operator with the symbol [straight phi]∈L∞ (Bn ) is defined on Aα2 by [figure omitted; refer to PDF]
Toeplitz operators have the following properties: if a and b are complex numbers, and [straight phi] and ψ∈L∞ (Bn ) , then Ta[straight phi]+bψ =aT[straight phi] +bTψ , T[straight phi]* =T[straight phi]¯ ; moreover, if [straight phi]∈H∞ (Bn ), then TψT[straight phi] =T[straight phi]ψ and T[straight phi]¯Tψ =T[straight phi]¯ψ .
The symbol zi will denote the i th coordinate function (i=1,...,n ).
It is easy to see that H[straight phi]Tzi =Tz¯iH[straight phi] . Thus, the Hankel operators H[straight phi] are particular solutions of the operator equation [figure omitted; refer to PDF] where S is a bounded linear operator on Aα2 .
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 STz =Tz¯ S, then S=H[straight phi] for some [straight phi]∈L∞ ; moreover, [straight phi] can be chosen such that ||H[straight phi] ||=||[straight phi]||. 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 Aα2 .
In this paper, we take the weighted Bergman space Aα2 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 Aαp , which plays an important role in this paper. It is shown that every function in the weighted Bergman space Aαp 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 Aαp . The following lemma is Theorem 2.30 in [1].
Lemma 2.1.
Suppose p>0, α>-1 , and [figure omitted; refer to PDF] Then there exists a sequence {ak } in Bn such that Aαp consists exactly of functions of the form [figure omitted; refer to PDF] where {ck } belongs to the sequence space lp and the series converges in the norm topology of Aαp .
Remark 2.2.
By the proof of Theorem 2.30 in [1], it can be seen that the sequence {ak } 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∈Aαp , we can choose a sequence {ck } in Lemma 2.1 so that [figure omitted; refer to PDF] where C is a positive constant independent of f .
The following lemma follows immediately from Lemma 2.1.
Lemma 2.4.
Suppose {ak } is a sequence as in Lemma 2.1, α>-1 , and b>n+α+1. Let [figure omitted; refer to PDF] Then, Aα1 (Bn ) consists exactly of the functions of the form [figure omitted; refer to PDF] where {ck } belongs to the sequence space l1 and the series converges in the norm topology of Aα1 .
From now on, we assume that b>2(n+α+1) is fixed and {ak } and la (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∈Bn , one has [figure omitted; refer to PDF] where k(r) is a constant which only depends on r.
Lemma 2.6.
There exists a constant C such that for every a∈Bn , r∈(0,1) , [figure omitted; refer to PDF] where C is independent of a and r.
Theorem 2.7.
Let S be a bounded linear operator acting on the weighted Bergman space Aα2 such that STzi =Tz¯i S (i=1,...,n) . Then, there exists [straight phi]∈L∞ (Bn ) such that S=H[straight phi] .
Proof.
Define the linear functional G on Aα2 by G(f)=...Sf,1...α . Clearly, G is a bounded linear functional on Aα2 . Note that Aα2 ⊂Aα1 . From Lemma 2.4 and Remark 2.3, given f∈Aα2 , there exists {ck } in l1 such that f=∑kcklak converges in Aα1 and ∑|ck |≤β||f||1 , where β is a positive constant independent f .
For f∈Aα2 , let f+ (z)=f(z¯)¯∈Aα2 . From (1.9), it is easy to see that STzi k =Tz¯i k S (i=1,...,n; k=1,2,...,) . If p=azik ,q=bzjm , then we have [figure omitted; refer to PDF] Hence, we establish that ...S(pq+ ),1...α =...Sp,q...α , where p and q are polynomials in z=(z1 ,...,zn ) .
Since the set of all polynomials is dense in Aα2 , there are sequences of polynomials pn (z) and qn (z) such that [figure omitted; refer to PDF] Furthermore, ||qn+ -lak 1/2||2 [arrow right]0.
Since [figure omitted; refer to PDF] by using the boundedness of S and the continuity of the scalar product, it follows that [figure omitted; refer to PDF]
Given r∈(0,1) , from Lemma 2.5, f(rz)=∑kcklak (rz) converges in Aα2 . Thus, with fr (z)=f(rz) , we see that [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] and consequently [figure omitted; refer to PDF]
Therefore, [figure omitted; refer to PDF] Consequently, it follows from Lemma 2.6 that [figure omitted; refer to PDF] but fr [arrow right]f in Aα2 (Bn ) . Thus, by the continuity of G it follows that |G(f)|≤γ||f||1 for some constant γ . Since Aα2 is dense in Aα1 , it follows that G is extended by continuity to an element of (Aα1 )* , and consequently, by the Hahn-Banach theorem to an element of (L1 (Bn ))* =L∞ (Bn ) . Therefore, there exists [straight phi]∈L∞ (Bn ) such that [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] and by using the fact that ...S(pq+ ),1...α =...Sp,q...α , where p , q are polynomials in z , it follows that [figure omitted; refer to PDF] Hence, S=H[straight phi] , 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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Abstract
This paper shows that if S is a bounded linear operator acting on the weighted Bergman spaces [superscript]Aα2[/superscript] on the unit ball in [superscript]...n[/superscript] such that S[subscript]T[subscript]zi[/subscript] [/subscript] =[subscript]T[subscript]z¯i[/subscript] [/subscript] S (i=1,...,n) , where [subscript]T[subscript]zi[/subscript] [/subscript] =[subscript]zi[/subscript] f and[subscript]T[subscript]z¯i[/subscript] [/subscript] =P([subscript]z¯i[/subscript] f) ; and where P is the weighted Bergman projection, then S must be a Hankel operator.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer