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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 365286, 10 pages
http://dx.doi.org/10.1155/2013/365286
Research Article

Weighted Composition Operators from Hardy to Zygmund Type Spaces

Department of Mathematics, Fujian Normal University, Fuzhou 350007, China

Received 20 January 2013; Accepted 24 March 2013

Academic Editor: Yansheng Liu

Copyright © 2013 Shanli Ye and Zhengyuan Zhuo. 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 aims at studying the boundedness and compactness of weighted composition operator between spaces of analytic functions. We characterize boundedness and compactness of the weighted composition operator from the Hardy spaces to the Zygmund type spaces and the little Zygmund type spaces in terms of function theoretic properties of the symbols and .

1. Introduction

Let be the open unit disk in the complex plane and its boundary, and denotes the set of all analytic functions on . An analytic self-map induces the composition operator on , defined by for analytic on . It is a well-known consequence of Littlewood’s subordination principle that the composition operator is bounded on the classical Hardy    spaces, Bergman    spaces, and Bloch spaces (see, e.g., [14]).

Let be a fixed analytic function on the open unit disk. Define a linear operator on the space of analytic functions on , called a weighted composition operator, by , where is an analytic function on . We can regard this operator as a generalization of a multiplication operator and a composition operator. In recent years the weighted composition operator has received much attention and appears in various settings in the literature. For example, it is known that isometries of many analytic function spaces are weighted composition operators (see [5], for instance). Their boundedness and compactness have been studied on various Banach spaces of analytic functions, such as Hardy, Bergman, BMOA, Bloch-type, and Zygmund spaces; see, for example, [611]. Also, it has been studied from one Banach space of analytic functions to another; one may see [1223].

The purpose of this paper is to consider the weighted composition operators from the Hardy space    to the Zygmund type spaces . Our main goal is to characterize boundedness and compactness of the operators from to in terms of function theoretic properties of the symbols and .

Now we give a detailed definition of these spaces. For , , we set For , the Hardy space consists of those functions , for which It is well known that with norm (2) the space is a Banach space if , for , space is a nonlocally convex topological vector space, and is a complete metric for it. For more information about the space, one may see these books, for example, [24, 25].

For the -Bloch space consists of all analytic functions defined on such that The space consists of all analytic functions defined on such that When , it is called the Zygmund space. From a theorem by Zygmund (see [26, vol. I, p. 263] or [24, Theorem 5.3]), we see that if and only if is continuous in the close unit disk and the boundary function such that When , from Proposition 8 of [27], we know that . Then the space is called a Zygmund type space if . However, all results in this paper are valid for all spaces (). An analytic function is said to belong to the little Zymund type space which consists of all satisfying . It can be easily proved that is a Banach space under the norm And the polynomials are norm-dense in closed subspace . For some other information on this space and some operators on it, see, for example, [2831].

Throughout this paper, constants are denoted by , , they are positive, and are only depending on and may differ from one occurrence to the another.

2. Auxiliary Results

In order to prove the main results of this paper. We need some auxiliary results. The first lemma is well known.

Lemma 1 (see [24, p. 65]). For , there exists a constant such that

Lemma 2. Suppose that , ; then for every and all nonnegative integer .

Proof. We use induction on . The case holds because it is Exercise  5 in [25, p. 85]. Assume the case holds. Fix and let . Then is in , and . It follows that Let ; we have Then the case holds. Hence (8) holds.

Lemma 3. For , suppose is a bounded operator. Then is a bounded operator.

This is obvious.

3. Boundedness of from to and

In this section we characterize bounded weighted composition operators from the Hardy space    to the Zygmund spaces .

Theorem 4. Let ,  ,  and be an analytic function on the unit disc and an analytic self-map of . Then is a bounded operator from to the Zygmund spaces if and only if the following are satisfied:

Proof . Suppose is bounded from to the Zygmund spaces . Then we can easily obtain the following results by taking and in , respectively: By (14) and the boundedness of the function , we get Let in again; in the same way we have Using these facts and the boundedness of the function again, we get Fix ; we take the test functions for . From Lemma 1 we obtain that and with a direct calculation. Since ,  ,  and , it follows that, for all with , we have Let ; it follows that For all with , by (17), we have Hence (12) holds.
Next, fix ; we take another test functions for . From Lemma 1 we obtain that and with a direct calculation. Since ,  ,  and , it follows that, for all with , we obtain that For all with , by (15), we have Hence (13) holds.
Finally, fix , and, for all , let From Lemma 1 we obtain that and with a direct calculation. Since ,  ,  and , it follows that, for all , we obtain that Then (11) holds.
Conversely, suppose that (11), (12), and (13) hold. For , by Lemma 2, we have the following inequality: This shows that is bounded. This completes the proof of Theorem 4.

Theorem 5. Let ,  , and be an analytic function on the unit disc and an analytic self-map of . Then is a bounded operator provided that the following are satisfied: Conversely, if is a bounded operator, then , (11), (12), and (13) hold, and the following are satisfied:

Proof. Assume that (28), (29), and (30) hold. Then for any , there is a constant ,  , such that implies Then, for any , from Lemma 2 we obtain that Hence for all . On the other hand, (25), (28), and (29) imply that (11), (12), and (13) hold; then is bounded by Theorem 4. So is bounded.
Conversely, assume that is bounded from to the little Zygmund type space . Then . Also ; thus Since and , we have . Hence (32) holds.
Similarly, ; then By (32), , and , we get that ; that is, (31) holds.
On the other hand, from Lemma 3 and Theorem 4, we obtain that (11), (12), and (13) hold.

4. Compactness of

In order to prove the compactness of from to the Zygmund spaces , we require the following lemmas.

Lemma 6. Let ,  , and be an analytic function on the unit disc and an analytic self-map of . Suppose that is a bounded operator from to . Then is compact if and only if, for any bounded sequence in which converges to uniformly on compact subsets of , one has as .

The proof is similar to that of Proposition  3.11 in [32]. The details are omitted.

Theorem 7. Let , be an analytic function on the unit disc and an analytic self-map of . Then is a compact operator from to if and only if is a bounded operator and the following are satisfied:

Proof. Suppose that is compact from to the Zygmund type space . Let be a sequence in such that as . If such a sequence does not exist, then (37) are automatically satisfied. Without loss of generality we may suppose that for all . We take the test functions By a direct calculation, we may easily prove that converges to uniformly on compact subsets of and . Then is a bounded sequence in which converges to uniformly on compact subsets of . Then by Lemma 6. Note that It follows that Then
Next, let By a direct calculation we obtain that    on compact subsets of and . Consequently, is a bounded sequence in which converges to uniformly on compact subsets of . Then by Lemma 6. Note that , and ; it follows that Then .
Finally, let By a direct calculation we obtain that    on compact subsets of and . Consequently, is a bounded sequence in which converges to uniformly on compact subsets of . Then by Lemma 6. Note that ,  , and ; it follows that Then . The proof of the necessary is completed.
Conversely, Suppose that (37) hold. Since is a bounded operator, from Theorem 4, we have Let be a bounded sequence in with and uniformly on compact subsets of . We only prove by Lemma 6. By the assumption, for any , there is a constant ,  , such that implies Let . Note that is a compact subset of . Then from Lemma 2 it follows that As , Hence is compact. This completes the proof of Theorem 7.

In order to prove the compactness of on the little Zygmund spaces , we require the following lemma.

Lemma 8. Let . Then is compact if and only if it is closed, bounded and satisfies

The proof is similar to that of Lemma  1 in [1], but we omit it.

Theorem 9. Let , ,  be an analytic function on the unit disc and an analytic self-map of . Then is compact from to the little Zygmund type spaces if and only if (28), (29), and (30) hold.

Proof. Assume that (28), (29), and (30) hold. By Theorem 5, we know that is bounded from to the little Zygmund type spaces . Suppose that with . From Lemmas 1 and 2 we obtain that thus and it follows that Hence is compact by Lemma 8.
Conversely, suppose that is compact.
Firstly, it is obvious is bounded, from Theorem 5 we have , and (31), (32) hold. On the other hand, we have for some by Lemma 6.
Next, note that the proof of Theorem 4 and the fact that the functions given in (18) are in and have norms bounded independently of ; we obtain that for . However, if , by (31), we easily have
Similarly, note that the functions given in (22) and (25) are in and have norms bounded independently of , we obtain that for . However, if , from and (32), we easily have This completes the proof of Theorem 9.

Remark 10. From Theorems 5 and 9, we conjecture that is compact if and only if is bounded.

Taking from Theorems 4, 7, and 9, we obtain the following results about the characterization of the boundedness and compactness of the composition operator .

Corollary 11. Let ,  ,  and   be an analytic self-map of . Then is a bounded operator if and only if the following are satisfied:

Corollary 12. Let ,  ,  and   be an analytic self-map of . Then is a compact operator if and only if is bounded and the following are satisfied:

Corollary 13. Let ,  ,  and   be an analytic self-map of . Then is a compact operator if and only if

In the formulation of corollary, we use the notation on defined by for . Taking from Theorems 4, 5, 7, and 9, we obtain the following results about the characterization of the boundedness and compactness of pointwise multiplier .

Corollary 14. Let ,  . Then the pointwise multiplier is a bounded operator if and only if(i) if ;(ii) if ;(iii) if .

Corollary 15. Let ,  . Then the pointwise multiplier is a bounded operator if and only if is a compact operator if and only if is a compact operator if and only if(i) if ,(ii) if .

Acknowledgments

The research was supported by Special Fund of Colleges and Universities in Fujian Province (no. JK2012010) and Natural Science Foundation of Fujian Province, China (no. 2009J01004).

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