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Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 925901, 6 pages

http://dx.doi.org/10.1155/2013/925901

## A New Characterization of Generalized Weighted Composition Operators from the Bloch Space into the Zygmund Space

^{1}Department of Mathematics, Henan Normal University, Xinxiang 453007, China^{2}Department of Mathematics, JiaYing University, Meizhou, Guangdong 514015, China

Received 10 April 2013; Accepted 16 June 2013

Academic Editor: Jiecheng Chen

Copyright © 2013 Hao Li and Xiaohong Fu. 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

A new criterion for the boundedness and the compactness of the generalized weighted composition operators from the Bloch space into the Zygmund space is given in this paper.

#### 1. Introduction

Let be the space of analytic functions on the unit disk . An is said to belong to the Bloch space, denoted by , if Let be the little Bloch space, which consists of those such that , as . Denote by the set of all such that where the supremum is taken over all and . In fact, if and only if by Theorem 5.3 of [1] and the Closed Graph Theorem. The class with the norm is a Banach space. We call the Zygmund space. For more information on the Zygmund space, see, for example, [1, 2].

Let denote the set of analytic self-map of . Associated with is the composition operator , which is defined by for and . We refer the readers to the book [3] for the theory of the composition operator on various function spaces.

Let . The weighted composition operator, denoted by , is defined by

Let be a nonnegative integer. The generalized weighted composition operator, denoted by , is defined as follows (see [4–7]): When , then becomes the weighted composition operator. If and , then . If and , then . If and , then . The operators and were studied, for example, in [8–15].

Composition operators, weighted composition operators, and related operators on the Bloch space were studied in [10, 12–22], while composition operators and weighted composition operators between the Zygmund space and some other spaces were studied in [2, 12, 23–28].

Recently, many researchers studied the generalized weighted composition operator on various spaces; see, for example, in [4–7, 25, 26, 29–32]. In [26], Stević has studied the operator from the Bloch space to the th weighted-type space, which includes the Zygmund space. Among other things, he obtained the following result (see [26]).

Theorem A. *Let , , an analytic self-map of and a positive integer. Then the following assertions hold.*(a)*The operator is bounded if and only if
*(b)*Suppose that the operator is bounded then is compact if and only if
*

Here we give a new criterion for the boundedness or compactness of the operator ; namely, we use three families of functions to characterize the operator .

Throughout the paper, denotes a positive constant which may differ from one occurrence to the other. The notation means that there exists a positive constant such that .

#### 2. Main Results

In this section we give our main results and proofs. For , set Next, we will use these three families of functions to characterize generalized weighted composition operators .

Theorem 1. *Let , be an analytic self-map of , and a positive integer. Then the following conditions are equivalent:*(a)*the operator is bounded;*(b)*the operator is bounded;*(c)*, , ,
*

*Proof. *(a)*⇒*(b). This implication is obvious.

(b)*⇒*(c). Assume that is bounded. Taking the functions , , and and using the boundedness of , we see that , and and are finite. For each , it is easy to check that . Moreover , are bounded by constants independent of . By the boundedness of , we get
as desired.

(c)*⇒*(a). Suppose that , and and and are finite. To prove this implication, we only need to show that these conditions imply (6). A calculation shows that
For the simplicity, we denote by . From (11), for , we have
Multiplying (12) by and then adding (13), we get
Multiplying (12) by and (14) by 2, respectively, we obtain
Multiplying (15) by , we get
Subtracting (17) from (16), we obtain
which implies that
From (16) and (18), we obtain
which implies that
By (12), (18), and (22), we have
which implies that
Fix . If , then from (21) we obtain
On the other hand, if , we get
From (30) and (31) we see that is finite. Similarly, from (25) and (29) we can obtain that and are finite as well. The proof of this theorem is finished.

To get the characterization of the compactness of , we need the following criterion, which follows from standard arguments similar to those outlined in Proposition 3.11 of [3].

Lemma 2. *Let , , an analytic self-map of and be a positive integer. The operator is compact if and only if is bounded, and for any bounded sequence in which converges to zero uniformly on compact subsets of , one has as .*

Theorem 3. *Let , be an analytic self-map of , and a positive integer. Suppose that the operator is bounded; then the following conditions are equivalent:*(a)*the operator is compact;*(b)*the operator is compact;*(c)*. *

*Proof. *(a)*⇒*(b). This implication is clear.

(b)*⇒*(c). Assume that is compact. Let be a sequence in such that (if such a sequence does not exist, then the limits in (c) automatically hold). Since the sequences , , are bounded in and converge to 0 uniformly on compact subsets of , by Lemma 2, we get , as , which means that (c) holds.

(c)*⇒*(a). Suppose that the limits in are 0. To prove this implication, we only need to show that (7) hold. Using the inequality (28), we get
as .

Using inequality (24), we get
as .

Using inequality (20), we get
as . The desired result follows. The proof of this theorem is complete.

#### Acknowledgment

The first author is supported by National Natural Science Foundation of China (no. 11126284).

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