- Research Article
- Open Access

# Weighted Composition Operators from Logarithmic Bloch-Type Spaces to Bloch-Type Spaces

- Stevo Stević
^{1}Email author and - Ravi P. Agarwal
^{2}

**2009**:964814

https://doi.org/10.1155/2009/964814

© S. Stević and R. P. Agarwal. 2009

**Received:**13 April 2009**Accepted:**3 July 2009**Published:**17 August 2009

## Abstract

The boundedness and compactness of the weighted composition operators from logarithmic Bloch-type spaces to Bloch-type spaces are studied here.

## Keywords

- Compact Subset
- Holomorphic Function
- Unit Ball
- Unit Disk
- Composition Operator

## 1. Introduction

Let be the unit disc in the complex plane , the normalized Lebesgue area measure on , the class of all holomorphic functions on , and the space of bounded holomorphic functions on with the norm

When , becomes the -Bloch space . For -Bloch and other Bloch-type spaces, see, for example, [1–9], as well as the related references therein. For , is the logarithmic Bloch space, which appeared in characterizing the multipliers of the Bloch space (see [3, 9]).

The following theorem summarizes the basic properties of the logarithmic Bloch-type spaces. Here, as usual, for fixed

Theorem 1 A (see [1]).

The following statements are true.

(a)The logarithmic Bloch-type space is Banach with the norm given in (1.2).

(b) is a closed subset of

(c)Assume Then if and only if

(d)The set of all polynomials is dense in .

A positive continuous function
on
is called *weight*.

the Bloch-type space becomes a Banach space.

It is of interest to provide function-theoretic characterizations for when and induce bounded or compact weighted composition operators on spaces of holomorphic functions. For some classical results mostly on composition operators, see, for example, [10]. For some recent related results, mostly in or related to Bloch-type or weighted-type spaces, see, for example, [4, 10–46] and the references therein.

Here we study the boundedness and compactness of the weighted composition operator from the logarithmic Bloch-type space and the little logarithmic Bloch-type space to the Bloch-type or the little Bloch-type space.

In 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 . We say that , if both and hold.

## 2. Auxiliary Results

In this section we quote several auxiliary results which will be used in the proofs of the main results.

Lemma 2.1.

Assume , then the following statements are true.

is increasing on the interval

is increasing on the interval

from which this statement follows.

The proof of (b) is similar, hence it is omitted.

Lemma 2.2.

for some independent of

The proof of the following lemma is similar to [25, Lemma 2.1], so we omit it.

Lemma 2.3.

Remark 2.4.

If in Lemma 2.3 we assume that
is not closed, then the word *compact* can be replaced by *relatively compact*.

The next characterization of compactness is proved in a standard way (see, e.g., the proofs of the corresponding lemmas in [10, 30, 47–49]). Hence we omit it.

Lemma 2.5.

Some concrete examples of the functions belonging to logarithmic Bloch-type spaces can be found in the next lemma.

Lemma 2.6.

The following statements are true.

where and is a nonconstant function belonging to

where and is a nonconstant function belonging to

where and is a nonconstant function belonging to

where in (2.13) we have used that and in (2.14) we have used the fact that the function in (2.1) is increasing on the interval .

where in (2.16) we have used the assumption , while in (2.17), as in (a), we have used the fact that the function in (2.1) is increasing on the interval .

where we have used the assumption and the fact that function (2.1) is increasing on .

as that is,

we finish the proof of the lemma.

Remark 2.7.

Note that from Lemmas 2.2 and 2.6 the functions defined in (2.9)–(2.11) have maximal growths in the corresponding logarithmic Bloch-type spaces.

## 3. Boundedness and Compactness of the Operator

This section studies the boundedness and compactness of the weighted composition operator .

Case 1.

Theorem 3.1.

Proof.

Applying (3.1) and (3.2) in (3.4), the boundedness of follows.

and as an easy consequence of Lemma 2.6(a), and for each

Hence, (3.11) and (3.12) imply (3.2).

for each , .

condition (3.1) follows.

Theorem 3.2.

Proof.

from which (3.18) follows.

for some positive . From (3.23) and (3.24), equality (3.17) follows.

whenever .

Since is an arbitrary positive number it follows that the last limit is equal to zero. Applying Lemma 2.5, the implication follows.

Theorem 3.3.

Proof.

First assume that is bounded. Then, it is clear that is bounded, and as usual by taking the test functions and and using the fact , we obtain (3.30) and (3.31).

Conversely, assume that the operator is bounded, and condition (3.31) holds.

as Hence and consequently is bounded.

Remark 3.4.

Note that Theorem 3.3 holds for all and

Theorem 3.5.

Proof.

If is compact, then it is bounded so that conditions (3.30) and (3.31) hold. On the other hand, is compact, which implies that (3.17) and (3.18) hold.

when , and where is the function in Lemma 2.1(b).

Combining (3.39) and (3.40), we obtain (3.36). Similarly, from (3.17) and (3.30) is obtained (3.35), as claimed.

which is a contradiction with (3.35).

is bounded in .

from which along with Lemma 2.3 the compactness of the operator follows.

Case 2.

Theorem 3.6.

Proof.

The proof of the theorem is similar to the proof of Theorem 3.1. The sufficiency follows by using the triangle inequality in (3.3) and then the third inequality in Lemma 2.2 and the definition of the space .

which belong to (for the functions in (3.48) and (3.49) it easily follows by Lemma 2.6(b), where is the function in (2.10). We omit the details.

The proofs of the following two theorems are similar to the proofs of Theorems 3.2 and 3.5, where the test functions in (3.48) and (3.49) are used as well as the lemmas in Section 2. Hence their proofs are omitted.

Theorem 3.7.

Theorem 3.8.

Case 3.

The following results were proved in [15]. Hence we quote them for the benefit of the reader, and without any proof.

Theorem 3.9.

Theorem 3.10.

Theorem 3.11.

Case 4.

Here we consider the cases , or and .

Theorem 3.12.

Assume that , or and , , is a weight, and is a holomorphic self-map of Then is bounded if and only if and condition (3.2) holds.

Proof.

The sufficiency follows by using the first inequality in Lemma 2.2 and the definition of the space in (3.3).

and similar to Lemma 2.6(b), and for each

from which along with (3.5) and the assumption , easily follows (3.2) in this case.

When , condition (3.2) follows as in Theorem 3.1, by using the test functions in (3.8).

Theorem 3.13.

and condition (3.18) holds.

Proof.

The proof is similar to the corresponding parts of the proofs of Theorems 3.2 and 3.7, so is omitted.

Remark 3.14.

Note that if , or and and is compact, then condition (3.18) is proved as in Theorems 3.2 and 3.7, by using the test functions in (3.8) and (3.48). If then condition (3.58) is vacuously satisfied. At the moment, we are not sure if the compactness implies condition (3.58) in the case . Hence for the interested readers we leave this as an open problem.

The following theorem is proved as the corresponding part of Theorem 3.5.

Theorem 3.15.

Assume that , or and , , is a weight, and is a holomorphic self-map of . Then the operator is compact if and condition (3.36) holds.

Remark 3.16.

Note that if is compact, then clearly .

## Authors’ Affiliations

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