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Abstract and Applied Analysis
Volume 2008 (2008), Article ID 279691, 11 pages
http://dx.doi.org/10.1155/2008/279691
Research Article

Essential Norms of Weighted Composition Operators from the -Bloch Space to a Weighted-Type Space on the Unit Ball

Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 36/III, 11000 Beograd, Serbia

Received 8 May 2008; Accepted 5 September 2008

Academic Editor: Simeon Reich

Copyright © 2008 Stevo Stević. 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 finds some lower and upper bounds for the essential norm of the weighted composition operator from -Bloch spaces to the weighted-type space on the unit ball for the case .

1. Introduction

Let be the open unit ball in , be the class of all holomorphic functions on the unit ball, and let be the class of all bounded holomorphic functions on with the normLet and be points in and . For a holomorphic function , we denote

A positive continuous function on the interval is called normal (see [1]) if there is and and , such thatFrom now on, if we say that a function is normal, we will also assume that it is radial, that is,

The weighted space consists of all such thatwhere is normal on the interval . For , we obtain the weighted space (see, e.g., [2, 3]).

The -Bloch space is the space of all such thatWith the normthe space is a Banach space ([46]).

The little -Bloch space is the subspace of consisting of all such that

Let and be a holomorphic self-map of the unit ball. Weighted composition operator on , induced by and is defined byThis operator can be regarded as a generalization of a multiplication operator and a composition operator. It is interesting to provide a function theoretic characterization when and induce a bounded or compact weighted composition operator between some spaces of holomorphic functions on . (For some classical results in the topic see, e.g., [5]. For some recent results on this and related operators, see, e.g., [24, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] and the references therein.)

In [18], Ohno has characterized the boundedness and compactness of weighted composition operators between and the Bloch space on the unit disk. In the setting of the unit polydisk , we have given some necessary and sufficient conditions for a weighted composition operator to be bounded or compact from to the Bloch space in [12] (see, also [21]). Corresponding results for the case of the unit ball are given in [14]. Among other results, in [14], we have given some necessary and sufficient conditions for the compactness of the operator , which we incorporate in the following theorem.

Theorem 1. Let be a holomorphic self-map of and .(a)If , then the following statements are equivalent:(a1) is a compact operator,(a2) is a compact operator,(a3), and(b)If , then the following statements are equivalent:(b1) is a compact operator,(b2) is a compact operator,(b3), and

We would also like to point out that if , then the boundedness of and are equivalent (see [22] for the case , and the proof of Theorem 3 in [14]).

The essential norm of an operator is its distance in the operator norm from the compact operators. More precisely, assume that and are Banach spaces and is a bounded linear operator, then the essential norm of , denoted by , is defined as follows:where denotes the operator norm. If , it is simply denoted by (see, e.g., [5, page 132]). If is an unbounded linear operator, then clearly

Since the set of all compact operators is a closed subset of the set of bounded operators, it follows that an operator is compact if and only if

Motivated by Theorem A, in this paper, we find some lower and upper bounds for the essential norm of the weighted composition operator , when

Throughout this paper, constants are denoted by , they are positive and may differ from one occurrence to another. The notation means that there is a positive constant such that . If both and hold, then we say that .

2. Auxiliary Results

In this section, we quote several auxiliary results which we need in the proofs of the main results in this paper. The following lemma should be folklore.

Lemma 2.1. Let Then, for some independent of .

The proof of the lemma for the case can be found, for example, in [26]. The formulation of the corresponding estimate in [26], for the case , is slightly different. In this case, Lemma 2.1 follows from the following estimate:

The next lemma can be proved in a standard way (see, e.g., the proofs of the corresponding results in [5, 2729]).

Lemma 2.2. Assume , , is normal, and is an analytic self-map of . Then, is compact if and only if is bounded and for any bounded sequence in converging to zero uniformly on compacts of as , one has as .

Lemma 2.3. Let where and Then,

Proof. We have from which it easily follows that Set Then, Hence, the points are stationary for the function Since , it follows that attains its maximum on the interval at the point By some long but elementary calculations, it follows that From this and since , (2.4) follows.

Remark 2.4. Note that

3. Estimates of the Essential Norm of

In this section, we prove the main results in this paper. Before we formulate and prove these results, we prove another auxiliary result.

Lemma 3.1. Assume , is normal, is a holomorphic self-map of such that , and the operator is bounded. Then, is compact.

Proof. First note that since is bounded and , it follows that . Now, assume that is a bounded sequence in converging to zero on compacts of as . Then, we have as , since is contained in the ball which is a compact subset of , according to the assumption, . Hence, by Lemma 2.2 , the operator is compact.

Theorem 3.2. Assume , is normal, , is a holomorphic self-map of , and is bounded. Then, for some positive constant .

Proof. Since is bounded, recall that . If , then, from Lemma 3.1, it follows that is compact which is equivalent with , and, consequently, On the other hand, it is clear that in this case the condition is vacuous, so that inequalities in (3.2) are vacuously satisfied.
Hence, assume . Let be a sequence in such that and be fixed. Set By Lemma 2.3, it follows that , moreover, it is easy to see that for every and uniformly on compacts of as . Then, by [6, Theorem 7.5], it follows that converges to zero weakly as . Hence, for every compact operator , we have as
We have for every compact operator .
Letting in (3.4) and using the definition of , we obtain where is the quantity in (2.12).
Taking in (3.5) the infimum over the set of all compact operators , then letting in such obtained inequality, and using (2.13), we obtain from which the first inequality in (3.2) follows.
Since the second inequality in (3.2) is obvious, we only have to prove the third one. By Lemma 3.1, we have that for each fixed the operator is compact.
Let be fixed, and let be a sequence of positive numbers which increasingly converges to 1, then for each fixed , we have
By the mean-value theorem, we have as
Moreover, by Lemma 2.1 (case ), and known inequality where , , we have for some positive constant Replacing (3.10) in (3.7), letting in such obtained inequality , employing (3.8), and then letting , the third inequality in (3.2) follows, finishing the proof of the theorem.

Corollary 3.3. Assume , is normal, , is a holomorphic self-map of , and the operator is bounded. Then, is compact if and only if

Theorem 3.4. Assume is normal, is a holomorphic self-map of , and is bounded. Then, for a positive constant

Proof. Clearly, If , then the result follows as in Theorem 3.2.
Hence, assume . We use the following family of test functions
We have when
From this and since for we have that
Assume is a sequence in such that as . Note that is a bounded sequence in (moreover in ) converging to zero uniformly on compacts of Then, by [6, Theorem 7.5], it follows that converges to zero weakly as . Hence, for every compact operator , we have
On the other hand, for every compact operator , we have Using (3.15), letting in (3.17), and applying (3.16), it follows that

Taking in (3.18) the infimum over the set of all compact operators , we obtain from which the first inequality in (3.12) follows.
As in Theorem 3.2, we need only to prove the third inequality in (3.12).
Recall that for each , the operator is compact. Let the sequence be as in Theorem 3.2. Note that inequality (3.7) and relationship (3.8) also hold for . Hence, we should only estimate the quantity On the other hand, by Lemma 2.1 (case ) applied to the function , which belongs to the Bloch space for each , and inequality (3.9) with , we have
From (3.21), by letting in (3.7) and using (3.8) (with ), and letting in such obtained inequality, we obtain as desired.

Corollary 3.5. Assume is normal, is a holomorphic self-map of , and the operator is bounded. Then, is compact if and only if

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