Abstract and Applied Analysis

Volume 2013 (2013), Article ID 608578, 7 pages

http://dx.doi.org/10.1155/2013/608578

Research Article

Differences of Composition Operators Followed by Differentiation between Weighted Banach Spaces of Holomorphic Functions

Cui Chen, Ren-Yu Chen, and Ze-Hua Zhou

Department of Mathematics, Tianjin University, Tianjin 300072, China

Received 25 May 2013; Accepted 2 October 2013

Academic Editor: Aref Jeribi

Copyright © 2013 Cui Chen et al. 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.

We characterize the boundedness and compactness of differences of the composition operators followed by differentiation between weighted Banach spaces of holomorphic functions in the unit disk. As their corollaries, some related results on the differences of composition operators acting from weighted Banach spaces to weighted Bloch type spaces are also obtained.

1. Introduction

Let and denote the class of holomorphic functions and analytic self-maps on the unit disk of the complex plane of , respectively. Let be a strictly positive continuous and bounded function (weight) on .

The weighted Bloch space is defined to be the collection of all that satisfy

Provided we identify the functions that differ by a constant, becomes a norm and a Banach space.

The endowed with the weighted sup-norm is referred to as the weighted Banach space. In setting the so-called associated weight plays an important role.

For a weight , its associated weight is defined as follows:

where denotes the point evaluation at . By [1] the associated weight is continuous, , and for every we can find with such that .

We say that a weight is radial if for every . A positive continuous function is called normal if there exist three positive numbers and , such that for every with ,

A radial, nonincreasing weight is called typical if . When studying the structure and isomorphism classes of the space , Lusky [2, 3] introduced the following condition (renamed after the author) for radial weights:

which will play a great role in this paper. In case is a radial weight, if it is also normal, then it satisfies the condition . Moreover, the radial weights with are essential (e.g., see [4]); that is, we can find a constant such that

Let ; the composition operator induced by is defined by

This operator has been studied for many years. Readers interested in this topic are referred to the books [5–7], which are excellent sources for the development of the theory of composition operators, and to the recent papers [8, 9] and the references therein.

By differentiation we are led to the linear operator , , which is regarded as the product of the composition operator and the differentiation operator denoted by , . The product operators have been studied, for example, in [10–16] and the references therein.

Recently, there has been an increasing interest in studying the compact difference of composition operators acting on different spaces of holomorphic functions. Some related results on differences of the composition operators or weighted composition operators on weighted Banach spaces of analytic functions, Bloch-type spaces, and weighted Bergman spaces can be found, for example [17–27]. More recently, Wolf [28, 29] characterized the boundedness and compactness of differences of composition operators between weighted Bergman spaces or weighted Bloch spaces and weighted Banach spaces of holomorphic functions in the unit disk. The same problems between standard weighted Bergman spaces were discussed by Saukko [30].

For each and in , we are interested in the operators , and we characterize boundedness and compactness of the operators between weighted Banach spaces of holomorphic functions in terms of the involved weights as well as the inducing maps. As a corollary we get a characterization of boundedness and compactness about the differences of composition operators acting from weighted Banach spaces to weighted Bloch type spaces.

Throughout this paper, we will use the symbol to denote a finite positive number, and it may differ from one occurrence to another. And for each , denotes a function in with such that . The existence of this function is a consequence of Montel's theorem as can be seen in [1].

2. Background and Some Lemmas

Now let us state a couple of lemmas, which are used in the proofs of the main results in the next sections. The first lemma is taken from [14].

Lemma 1. Let be a radial weight satisfying condition . There is a constant (depending only on the weight ) such that for all ,

for every .

In order to handle the differences, we need the pseudohyperbolic metric. Recall that for any point , let , . It is well known that each is a homeomorphism of the closed unit disk onto itself. The pseudohyperbolic metric on is defined by

We know that is invariant under automorphisms (see, e.g., [5]).

Lemma 2. Let be a radial weight satisfying condition . There is a constant such that for all ,

for all .

Proof. For , let , by Lemma 1, we obtain , so by Lemma 3.2 in [31] and Lemma 1, there is a constant such that

for each . This completes the proof.

Remark 3. From Lemma 2, it is not hard to see that for any , then

for any , where .

The following result is well known (see, e.g., [32]).

Lemma 4. Assume that is a normal weight. Then for every the following asymptotic relationship holds:

Here and below we use the abbreviated notation to mean for some inessential constant .

The following lemma is the crucial criterion for compactness, and its proof is an easy modification of that of Proposition 3.11 of [5].

Lemma 5. Suppose that and . Then the operator is compact if and only if whenever is a bounded sequence in with uniformly on compact subsets of , and then , as .

3. The Boundedness of

In this section we will characterize the boundedness of . For this purpose, we consider the following three conditions:

Theorem 6. Suppose that is an arbitrary weight and that is a normal and radial weight. Then the following statements are equivalent.

(i) is bounded.(ii)The conditions (12) and (14) hold.(iii)The conditions (13) and (14) hold.

Proof. First, we prove the implication (i) (ii). Assume that is bounded. Fixing , we consider the function defined by

Next prove that . In fact,

By Lemma 4 we have

thus , and . Note that , and . So by the boundedness of , it then follows that

for any . Since is an arbitrary element, then from (18) and (4), we can obtain (12).

Next we prove (14). For given , we consider the function

Like for above, we can show that with . One sees that . Then

where

By Lemma 2 and (12), we conclude that , which combines with (20), and we obtain that

for all ; thus (14) holds.

(ii) (iii). Assume that (12) and (14) hold, we need only to show that (13) holds. In fact,

from which, using (12) and (14), the desired condition (13) holds.

(iii) (i). Assume that (13) and (14) hold. By Lemmas 1 and 2, for any , we have

from which it follows that is bounded. The whole proof is complete.

Corollary 7. Suppose that is an arbitrary weight and that is a normal and radial weight satisfying condition . Then the following statements are equivalent.

(i) is bounded.(ii)The conditions (12) and (14) hold.(iii)The conditions (13) and (14) hold.

4. The Compactness of

In this section, we turn our attention to the question of compact difference. Here we consider the following conditions:

Theorem 8. Suppose that is an arbitrary weight and that is a normal and radial weight. Then is compact if and only if is bounded and the conditions (25)–(27) hold.

Proof. First we suppose that is bounded and the conditions (25)–(27) hold. Then the conditions (12)–(14) hold by Theorem 6. From (25)–(27), it follows that for any , there exists such that

Now, let be a sequence in such that (constant) and uniformly on compact subsets of . By Lemma 5 we need only to show that as . A direct calculation shows that

where

We divide the argument into a few cases.

Case 1 ( and ). By the assumption, note that converges to zero uniformly on as ; using (14) and Cauchy's integral formula, it is easy to check that uniformly for all with .

On the other hand, it follows from Remark 3 after Lemma 2 and (12) that

Case 2 ( and ). As in the proof of Case 1, uniformly as . On the other hand, using Lemma 2 and (28) we obtain .

Case 3 ( and ). For sufficiently large, by Lemma 2 and (28) we obtain that . Meanwhile, by Lemma 1 and (30).

Case 4 ( and ). We rewrite

where

The desired result follows by an argument analogous to that in the proof of Case 2. Thus, together with the above cases, we conclude that

for sufficiently large . Employing Lemma 5 we obtain the compactness of .

For the converse direction, we suppose that is compact. From which we can easily obtain the boundedness of . Next we only need to show that (25)–(27) hold.

Let be a sequence of points in such that as . Define the functions

Clearly, converges to uniformly on compact subsets of as and with for all . Moreover,

By the compactness of and Lemma 5, it follows that . On the other hand, using (38) we have

Letting in (39), it follows that (25) holds. The condition (26) holds for the similar arguments.

Now we need only to show the condition (27) holds. Assume that is a sequence in such that and as . Define the function

It is easy to check that converges to uniformly on compact subsets of as and with for all . Note that , then , and it follows from Lemma 5 that , . On the other hand we obtain that

where

By Lemma 2 and the condition (25) that has been proved, we get , . This combines with (41), and we obtain , . This shows that (27) is true. The whole proof is complete.

Corollary 9. Suppose that is an arbitrary weight and that is a normal and radial weight satisfying condition . Then is compact if and only if is bounded and the conditions (25)–(27) hold.

5. Examples

In this final section we give an example of function for which the operator between the weighted Banach spaces to show that the condition in Theorem 8 that is bounded is necessary.

Example 1. In this example we will show that there exist weight (normal, radial) and , analytic self-maps on the unit disk such that the conditions (25)–(27) in Theorem 8 are satisfied while is not compact.

Let

and , where .

Since for , we have so belongs to , as well as . Moreover, and can never tend to 1 for any , which means that conditions (25)–(27) hold trivially.

Now we will show that is not bounded, and then not compact. Let , and then it is easy to check that and as . So

However, since , then as . Thus choose and , and we can obtain

Hence does not map boundedly into by Theorem 6.

Acknowledgment

This work was supported in part by the National Natural Science Foundation of China (Grant nos. 11371276, 11301373, and 11201331).

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