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Abstract and Applied Analysis
Volume 2009, Article ID 741920, 14 pages
http://dx.doi.org/10.1155/2009/741920
Research Article

Products of Composition and Differentiation Operators from Spaces to Bloch-Type Spaces

Department of Mathematics and Physics, Hunan Institute of Engineering, Xiangtan 411104, China

Received 25 March 2009; Accepted 16 April 2009

Academic Editor: Stevo Stevic

Copyright © 2009 Weifeng 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.

Abstract

We study the boundedness and compactness of the products of composition and differentiation operators from spaces to Bloch-type spaces and little Bloch-type spaces.

1. Introduction

Let be the open unit disk in the complex plane and the class of all analytic functions on . The -Bloch space () on is the space of all analytic functions on such that

Under the above norm, is a Banach space. When , is the well-known Bloch space. Let denote the subspace of consisting of those for which

as This space is called the little -Bloch space.

Assume that is a positive continuous function on , and there exist positive numbers and and such that

then is called a normal function ([1]).

An is said to belong to the Bloch-type space , if (see, e.g, [25])

is a Banach space with the norm (see [3]). When , the induced space becomes the -Bloch space .

Throughout this paper, we assume that is a nondecreasing continuous function. Assume that , . A function is said to belong to (see [6]) if

where   denotes the normalized Lebesgue area measure in (i.e., ) and is the Green function with logarithmic singularity at , that is, ( is a conformal automorphism defined by for ). If , , the space equals to , which is introduced by Zhao in [7]. Moreover (see [7]) we have that, and for , and for , and , , and . When , is a Banach space under the norm

From [6], we know that , if and only if

Moreover, (see in [6, Theorem  2.1] or [8, Lemma  2.1]). Throughout the paper we assume that (see [6])

since otherwise consists only of constant functions.

Let denote a nonconstant analytic self-map of . Associated with is the composition operator defined by for . The problem of characterizing the boundedness and compactness of composition operators on many Banach spaces of analytic functions has attracted lots of attention recently, see, for example, [9, 10] and the reference therein.

Let be the differentiation operator on , that is, For , the products of composition and differentiation operators and are defined, respectively, by

The boundedness and compactness of on the Hardy space were investigated by Hibschweiler and Portnoy in [11] and by Ohno in [12]. The case of the Bergman spaces was studied in [11], while the case of the Hilbert-Bergman space was studied by Stević in [13]. In [14], Li and Stević studied the boundedness and compactness of the operator on -Bloch spaces, while in [15] they studied these operators between and -Bloch spaces. The boundedness and compactness of the operator from mixed-norm spaces to -Bloch spaces was studied by Li and Stević in [16]. Norm and essential norm of the operator from -Bloch spaces to weighted-type spaces were studied by Stević in [17]. Some related operators can be also found in [1821]. For some other papers on products of linear operators on spaces of holomorphic functions, mostly integral-type and composition operators, see, for example, the following papers by Li and Stević: [5, 2230].

Motivated basically by papers [14, 15], in this paper, we study the operators and from space to and spaces. Some sufficient and necessary conditions for the boundedness and compactness of these operators are given.

Throughout this paper, constants are denoted by , they are positive and may differ from one occurrence to the other. The notation means that there is a positive constant such that .

2. Main Results and Proofs

In this section we give our main results and proofs. For this purpose, we need some auxiliary results. The following lemma can be proved in a standard way (see, e.g, in [9, Proposition  3.11]). A detailed proof, can be found, for example, in [31].

Lemma 2.1. Let be an analytic self-map of . Suppose that is normal, , . Then 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 The following lemma can be proved similarly as [32], one omits the details (see also [2, 4]).

Lemma 2.2. A closed set K in is compact if and only if it is bounded and satisfies

Now one is in a position to state and prove the main results of this paper.

Theorem 2.3. Let be an analytic self-map of . Suppose that is normal, , , and is a nonnegative nondecreasing function on such that where denote the characteristic function of the set . Then is bounded if and only if

Proof. Suppose that the conditions in (2.3) hold. Then for any and , where we have used the fact that as well as the following well-known characterization for -Bloch functions (see, e.g., [33]) Taking the supremum in (2.4) for , then employing (2.3) we obtain that is bounded.
Conversely, suppose that is bounded, that is, there exists a constant such that for all . Taking the functions and , which belong to , we get From (2.6), (2.7), and the boundedness of the function , it follows that For , let By some direct calculation we have that
From [8], we know that for each , moreover there is a positive constant such that . Hence, we have for . Therefore, we obtain Next, for , let Then from [8], we see that and . Since we have
Thus Inequality (2.8) gives Therefore, the first inequality in (2.3) follows from (2.16) and (2.17). From (2.12) and (2.15), we obtain Equations (2.6) and (2.18) imply Inequality (2.19) together with (2.20) implies the second inequality of (2.3). This completes the proof of Theorem 2.3.

Theorem 2.4. Let be an analytic self-map of . Suppose that is normal, , and is a nonnegative nondecreasing function on such that (2.2) holds. Then is compact if and only if is bounded,

Proof. Suppose that is bounded and (2.21) holds. Let be a sequence in such that and converges to uniformly on compact subsets of as By the assumption, for any , there exists a such that when . Since is bounded, then from the proof of Theorem 2.3 we have Let . Then, we have The assumption that as on compact subsets of along with Cauchy's estimate give that and as on compact subsets of . Letting in (2.24) and using the fact that is an arbitrary positive number, we obtain Applying Lemma 2.1, the result follows.
Now, suppose that is compact. Then it is clear that is bounded. Let be a sequence in such that as (if such a sequence does not exist then condition (2.21) is vacuously satisfied). Let Then, and converges to uniformly on compact subsets of as . Since is compact, by Lemma 2.1 we have On the other hand, from (2.11) we have
which implies that if one of these two limits exists.
Next, for , set Then is a sequence in . Notice that ,
and converges to uniformly on compact subsets of as . Since is compact, we have On the other hand, we have Therefore This along with (2.27) implies From the last two equalities, the desired result follows.

Theorem 2.5. Let be an analytic self-map of . Suppose that is normal, , and is a nonnegative nondecreasing function on such that (2.2) holds. Then is compact if and only if

Proof. Sufficiency. Let . By the proof of Theorem 2.3 we have Taking the supremum in (2.34) over all such that then letting , we get
From which by Lemma 2.2 we see that the operator is compact. Necessity. Assume that is compact. By taking the function given by and using the boundedness of , we get
From this, by taking the test function and using the boundedness of it follows that If from (2.36) and (2.37), we obtain that from which the result follows in this case.
Assume that Let be a sequence such that From the compactness of we see that is compact. From Theorem 2.4 we get
From (2.36) and (2.40), we have that for every , there exists an such that when and there exists a such that when Therefore, when and , we have On the other hand, if and , we obtain Inequality (2.42) together with (2.43) gives the second equality of (2.33). Similarly to the above arguments, by (2.37) and (2.39) we get the first equality of (2.33). The proof is completed.

From the above three theorems, we get the following corollary (see [14]).

Corollary 2.6. Let be an analytic self-map of . Then the following statements hold. (i) is bounded if and only if (ii) is compact if and only if is bounded, (iii) is compact if and only if

Similarly to the proofs of Theorems 2.32.5, we can get the following result. We omit the proof.

Theorem 2.7. Let be an analytic self-map of . Suppose that is normal, , and is a nonnegative nondecreasing function on such that (2.2) holds. Then the following statements hold. (i) is bounded if and only if (ii) is compact if and only if is bounded and (iii) is compact if and only if

From Theorem 2.7 we get the following corollary.

Corollary 2.8. Let be an analytic self-map of . Then the following statements hold. (i) is bounded if and only if (ii) is compact if and only if is bounded and (iii) is compact if and only if

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