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

Received 13 June 2008; Revised 12 October 2008; Accepted 20 November 2008 Recommended by Stevo Stevic This paper shows that if S is a bounded linear operator acting on the weighted Bergman spaces Aα on the unit ball in C n such that STzi TziS i 1, . . . , n , where Tzi zif and Tzi P zif ; and where P is the weighted Bergman projection, then Smust be a Hankel operator. Copyright q 2008 Y. Lu and J. 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.


Introduction
Let B n be the open unit ball in the complex vector space C n .For z z 1 , . . ., z n , w w 1 , . . ., w n ∈ C n , let z, w z 1 w 1 • • • z n w n , where w k is the complex conjugate of w k , and |z| z, z .For a multi-index m m 1 , . . ., m n and z z 1 , . . ., z n ∈ C n , we also write z m z m 1 1 . . .z m n 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 by where is a normalizing constant so that dV α is a probability measure on B n .
where S is a bounded linear operator on A 2 α .It is well known that on the classical Hardy space H 2 , 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 H 2 , Nehari 19 proved that if S is a bounded linear operator such that ST z T z S, 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 H 2 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 A 2 α .In this paper, we take the weighted Bergman space A 2 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.

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 .

2.1
Then there exists a sequence {a k } in B n such that A p α consists exactly of functions of the form where C is a positive constant independent of f.
The following lemma follows immediately from Lemma 2.1.

2.4
Then, A 1 α B n consists exactly of the functions of the form where {c k } belongs to the sequence space l 1 and the series converges in the norm topology of A 1 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 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 , where C is independent of a and r.
Theorem 2.7.Let S be a bounded linear operator acting on the weighted Bergman space A 2 α such that ST z i T z i S i 1, . . ., n .Then, there exists ϕ ∈ L ∞ B n such that S H ϕ .

2.8
Hence, we establish that S pq , 1 α Sp, q α , where p and q are polynomials in z z 1 , . . ., z n .
Since the set of all polynomials is dense in A 2 α , there are sequences of polynomials p n z and q n z such that Furthermore, q n − l 1/2 a k 2 → 0. Since S p n q 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

2.12
Note that ϕ z p z q z dV α z H ϕ p, 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 ϕ p, q α .

2.19
Hence, S H ϕ , finishing the proof of the theorem.

Lemma 2 . 1 .
Suppose p > 0, α > −1, and b > n max 1 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 A 2 n , dV α and A p α become Banach spaces.L 2 B n , dV α is a i will denote the ith coordinate function i 1, . . ., n .It is easy to see that H ϕ T z i T z i H ϕ .Thus, the Hankel operators H ϕ are particular solutions of the operator equation 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.
2.2where {c k } belongs to the sequence space l p and the series converges in the norm topology of A p α .Remark 2.2.
Sf r , 1 α ≤ Cβ S f 1 ; 2.16 but f r → f in A 2 α B n .Thus, by the continuity of G it follows that |G f | ≤ γ f 1 for some constant γ.Since A 2 α is dense in A 1 α , it follows that G is extended by continuity to an element of A 1 α * , and consequently, by the Hahn-Banach theorem to an element of L 1 B n * L ∞ B n .Therefore, there exists ϕ ∈ L ∞ B n such that