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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 408168, 8 pages
Commuting Quasihomogeneous Toeplitz Operator and Hankel Operator on Weighted Bergman Space
Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China
Received 26 January 2013; Accepted 1 July 2013
Academic Editor: Giovanni P. Galdi
Copyright © 2013 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.
We characterize the commuting Toeplitz operator and Hankel operator with quasihomogeneous symbols. Also, we use it to show the necessary and sufficient conditions for commuting Toeplitz operator and Hankel operator with ordinary functions.
Let denote Lebesgue area measure on the unit disk , normalized so that the measure of equals 1. For , we denote by the measure . For , the space is a Banach space. The weighted Bergman space is the closed subspace of analytic functions in the Hilbert space . For each , the application is continuous and can be represented as , where This means that, if is the orthogonal projection from onto , then can be defined by For a function , we define the Toeplitz operator with symbol by It is well known that
Let be the unitary operator defined by , where belongs to . Let be in ; we define a bounded linear operator on as follows: Then we can define the small Hankel operator as follows: as .
The study of commuting Toeplitz operators on the Bergman and Hardy spaces over various domains and related operator algebras has a long lasting history. On the Hardy space of the unit disk, Brown and Halmos  first showed that two Toeplitz operators are commuting if and only if either both symbols of these operators are analytic, or both symbols of these operators are coanalytic or a nontrivial linear combination of the symbols of these operators is constant. On the Bergman space, the situation is more complicated. Axler and Čučković obtained the analogous result for Toeplitz operators with bounded harmonic symbols on the Bergman space of the unit disk . The problem of characterizing commuting Toeplitz operators with arbitrary bounded symbols seems quite challenging and is not fully understood until now. In , Čučković and Rao used the Mellin transform to characterize all Toeplitz operators on which commute with for . Later in  Louhichi and Zakariasy gave a partial characterization of commuting Toeplitz operators on with quasihomogeneous symbols. Recently, Lu and Zhang [5, 6] characterized the commuting Toeplitz operators and Hankel operators with quasihomogeneous symbols. There are also many other important results [7–13]. Motivated by those works, we study commuting Toeplitz operator and Hankel operator on the weighted Bergman space. In this paper, we obtain the necessary and sufficient conditions for commuting Toeplitz operator and Hankel operator.
An operator that will arise in our study of Toeplitz operators is the Mellin transform, defined for any function ; from the formula, is the Mellin transform as follows: which is a bounded holomorphic function in the half plane .
Let be a radial function; that is, suppose that , . In fact, if we define the function on by , then a direct calculation shows that so that if , Thus, is a diagonal operator on with coefficient sequence as follows: This makes it relatively simple to work with the product of two operators with such radial symbols.
Now, we define the “radialization” of a function by the following: It is clear that a function is a radial if and only if .
Let be the space of weighted square integrable radial functions on . By using that, trigonometric polynomials are dense in and that, for , is orthogonal to , we see that is
Definition 1. Let be a function in which is of the form , where is a radial function. Then one says that is a quasihomogeneous function of quasihomogeneous degree .
A direct calculation gives the following lemmas which we will use often.
Lemma 2. Let and let be an integrable radial function. Then,
Lemma 3. Let be an integrable radial function. Then, for , and for ,
2. Commuting of Toeplitz Operator and Hankel Operator
Theorem 4. Let be a bounded function of quasihomogeneous degree and . Then if and only if the following conditions holds(1), if and ;(2) , if and .
Proof. For ,
If , then we have
which is equivalent to
From the aforementioned we get the following.
Case 1. For and , Case 2. For and , As a special case of Theorem 4, we can have the following corollary.
Corollary 5. Let be a bounded radial function and . Then if and only if for and .
Theorem 6. Let be a bounded function of quasihomogeneous degree and . Then if and only if the following conditions holds(1), if and ;(2) , if and .
Proof. For , we have
Then one has the following.
Case 1. For ,
Case 2. For , If , then we have the following.
Case 1. For , Case 2. For , That is one has the following.
Case 1. For , Case 2. For ,
Theorem 7. Let be a bounded function of quasihomogeneous degree and . Then if and only if or (1), if and ;(2) , if and ;(3), if and .
Proof. For , we have the following.
Case 1. For , Case 2. For , Applying , we have the following.
Case 1. For , Case 2. For , If , then the equation holds.
Otherwise , we have the following.
Case 1. For , Case 2. For , that is one has the following.
Case 1. For , Case 2. For , Then we get the following.
Case 1. For , , .
Case 2. For , Case 3. For ,.
Corollary 8. Let be a bounded radial function and . Then if and only if (1);(2) and , for .
Finally, we will investigate the situation that both functions are ordinary functions.
Theorem 9. Let and . Then if and only if for and .
Proof. For , we have
From the aforementioned we get, for and ,
If , then we get
for and .
The converse is easy to get.
The author would like to thank the referees for their valuable comments and suggestions, which helped to improve the paper. This work is supported by the National Natural Science Foundation of China (no. 11126061), Innovation Program of Shanghai Municipal Education Commission (no. 13YZ090) and the Science & Technology Program of Shanghai Maritime University (no. 20120098).
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