Abstract and Applied Analysis

Abstract and Applied Analysis / 2008 / Article

Research Article | Open Access

Volume 2008 |Article ID 605807 | https://doi.org/10.1155/2008/605807

Xiaohong Fu, Xiangling Zhu, "Weighted Composition Operators on Some Weighted Spaces in the Unit Ball", Abstract and Applied Analysis, vol. 2008, Article ID 605807, 8 pages, 2008. https://doi.org/10.1155/2008/605807

Weighted Composition Operators on Some Weighted Spaces in the Unit Ball

Academic Editor: Stevo Stevic
Received18 May 2008
Accepted28 May 2008
Published23 Jun 2008

Abstract

Let be the unit ball of , the space of all holomorphic functions in . Let and be a holomorphic self-map of . For , the weigthed composition operator is defined by The boundedness and compactness of the weighted composition operator on some weighted spaces on the unit ball are studied in this paper.

1. Introduction

Let be the unit ball of , the space of all holomorphic functions in , and the space of all bounded holomorphic functions in the unit ball. For , letbe the radial derivative of .

A positive continuous function on is called normal if there exist positive numbers and and such that (see, e.g., [1, 2])

An is said to belong to the weighted-type space, denoted by , ifwhere is normal on (see [3]). is a Banach space with the norm .

The little weighted-type space, denoted by , is the subspace of consisting of those such that When , the induced spaces and become the (classical) weighted spaces and respectively.

An is said to belong to the logarithmic-type space ifIt is easy to see that becomes a Banach space under the norm , and that the following inclusions hold:where is the Bloch space defined byFor some information on the Bloch and related spaces see, for example, [413] and the references therein. For some information on the space in the unit disk see [14].

Let , and let be a holomorphic self-map of . For , the weighted composition operator is defined byThe weighted composition operator can be regarded as a generalization of a multiplication operator and a composition operator, which is defined by The work in [15] contains much information on this topic.

In the setting of the unit ball, Zhu studied the boundedness and compactness of the weighted composition operator between Bergman-type spaces and in [16]. More general results can be found in [17, 18]. Some necessary and sufficient conditions for the weighted composition operator to be bounded and compact between the Bloch space and are given in [19]. In the setting of the unit polydisk, some necessary and sufficient conditions for a weighted composition operator to be bounded and compact between the Bloch space and are given in [20, 21] (see also [22] for the case of composition operators). Other related results can be found, for example, in [3, 2332].

In this paper, we study the weighted composition operator from to the spaces and . Some necessary and sufficient conditions for the weighted composition operator to be bounded and compact are given.

Throughout the paper, constants are denoted by ; they are positive and may not be the same in every occurrence.

2. Main Results and Proofs

In this section, we give our main results and their proofs. Before stating these results, we need some auxiliary results, which are incorporated in the lemmas which follow.

Lemma 2.1. Assume that , is a holomorphic self-map of , and is a normal function on . Then, is compact if and only if is bounded, and for any bounded sequence in which converges to zero uniformly on compact subsets of as , one has as

The proof of Lemma 2.1 follows by standard arguments (see, e.g., [15, Proposition 3.11] as well as the proofs of the corresponding results in [7, 22, 33, 34]). Hence, we omit the details.

Lemma 2.2. Assume that is normal. A closed set in is compact if and only if it is bounded and satisfies

The proof of Lemma 2.2 is similar to the proof of Lemma 1 in [35]. We omit the details.

Now, we are in a position to state and prove our main results.

Theorem 2.3. Assume that , is a holomorphic self-map of , and is normal on . Then, is bounded if and only if

Proof. Assume that is bounded. For , setIt is easy to see that and .
For any , we havewhich implies (2.2).
Conversely, assume that (2.2) holds. Then, for any , we haveTaking the supremum in (2.5) over and using condition (2.2), the boundedness of the operator follows, as desired.

Theorem 2.4. Assume that , is a holomorphic self-map of , and is a normal function on . Then, is compact if and only if and

Proof. Assume that is compact. Then, it is obvious that is bounded. Taking the function , we see that . Let be a sequence in such that . SetIt is easy to see that . Moreover, uniformly on compact subsets of as . By Lemma 2.1,We havewhich together with (2.8) implies thatThis proves that (2.6) holds.
Conversely, assume that and (2.6) holds. From this, it follows that (2.2) holds; hence is bounded. In order to prove that is compact, according to Lemma 2.1, it suffices to show that if is a bounded sequence in converging to 0 uniformly on compact subsets of , thenLet be a bounded sequence in such that uniformly on compact subsets of as By (2.6), we have that for any , there is a constant such thatwhenever . Let . Equation (2.12) along with the fact that impliesObserve that is a compact subset of so thatWith the aid of the above inequality, we can deduce thatby letting . Since is an arbitrary positive number, it follows that the last limit is equal to zero. Therefore, is compact. The proof is complete.

Theorem 2.5. Assume that , is a holomorphic self-map of , and is a normal function on . Then, is compact if and only if and

Proof. Assume that is compact. Then, it is clear that is compact, and hence (2.16) holds. In addition, taking the function given by , we get .
Conversely, suppose that and (2.16) holds. In the proof of the implication we follow the lines, for example, of the proof of Lemma 4.2 in [24]. From (2.16), it follows that for every , there exists a such thatwhen From the assumption , we have that for the above , there exists an such that when ,Therefore, if and , we obtainIf and we have thatCombining (2.19) with (2.20), we get On the other hand, from (1.5) we have thatTaking the supremum in the above inequality over all such that then letting , by (2.21) it follows thatFrom this and by employing Lemma 2.2, we see that is compact. The proof is complete.

Similar to the proofs of Theorems 2.3 and 2.4, we easily get the following two results. We also omit their proofs.

Theorem 2.6. Assume that and is a holomorphic self-map of . Then, the following statements hold.
(a) is bounded if and only if
(b) is compact if and only if and

Theorem 2.7. Assume that and is a holomorphic self-map of . Then, the following statements hold.
(a) is bounded if and only if
(b) is compact if and only if and

Acknowledgment

X. Fu is supported in part by the Natural Science Foundation of Guangdong Province (no. 73006147).

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Copyright © 2008 Xiaohong Fu and Xiangling Zhu. 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.


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