# 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.

## 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).

(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.

from which this statement follows.

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

Lemma 2.2.

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 .

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).

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.

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).

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.

## Authors’ Affiliations

## References

- Stević S:
**On new Bloch-type spaces.***Applied Mathematics and Computation*2009,**215**(2):841–849. 10.1016/j.amc.2009.06.009MathSciNetView ArticleMATHGoogle Scholar - Anderson JM, Clunie J, Pommerenke Ch:
**On Bloch functions and normal functions.***Journal für die reine und angewandte Mathematik*1974,**270:**12–37.MathSciNetMATHGoogle Scholar - Brown L, Shields AL:
**Multipliers and cyclic vectors in the Bloch space.***The Michigan Mathematical Journal*1991,**38**(1):141–146.MathSciNetView ArticleMATHGoogle Scholar - Clahane DD, Stević S:
**Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball.***Journal of Inequalities and Applications*2006,**2006:**-11.Google Scholar - Li S, Stević S:
**Weighted-Hardy functions with Hadamard gaps on the unit ball.***Applied Mathematics and Computation*2009,**212**(1):229–233. 10.1016/j.amc.2009.02.019MathSciNetView ArticleMATHGoogle Scholar - Stević S:
**On an integral operator on the unit ball in**.*Journal of Inequalities and Applications*2005,**2005**(1):81–88. 10.1155/JIA.2005.81MATHGoogle Scholar - Stević S:
**On Bloch-type functions with Hadamard gaps.***Abstract and Applied Analysis*2007,**2007:**-8.Google Scholar - Yamashita S:
**Gap series and****-Bloch functions.***Yokohama Mathematical Journal*1980,**28**(1–2):31–36.MathSciNetMATHGoogle Scholar - Zhu K:
*Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics*.*Volume 226*. Springer, New York, NY, USA; 2005:x+271.Google Scholar - Cowen CC, MacCluer BD:
*Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics*. CRC Press, Boca Raton, Fla, USA; 1995:xii+388.Google Scholar - Fang Z-S, Zhou Z-H:
**Differences of composition operators on the space of bounded analytic functions in the polydisc.***Abstract and Applied Analysis*2008,**2008:**-10.Google Scholar - Fu X, Zhu X:
**Weighted composition operators on some weighted spaces in the unit ball.***Abstract and Applied Analysis*2008,**2008:**-8.Google Scholar - Galanopoulos P:
**On**to**pullbacks.***Journal of Mathematical Analysis and Applications*2008,**337**(1):712–725. 10.1016/j.jmaa.2007.02.049MathSciNetView ArticleMATHGoogle Scholar - Gu D:
**Weighted composition operators from generalized weighted Bergman spaces to weighted-type spaces.***Journal of Inequalities and Applications*2008,**2008:**-14.Google Scholar - Krantz SG, Stević S:
**On the iterated logarithmic Bloch space on the unit ball.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(5–6):1772–1795. 10.1016/j.na.2009.01.013View ArticleMATHGoogle Scholar - Li S, Stević S:
**Composition followed by differentiation between Bloch type spaces.***Journal of Computational Analysis and Applications*2007,**9**(2):195–205.MathSciNetMATHGoogle Scholar - Li S, Stević S:
**Weighted composition operators from Bergman-type spaces into Bloch spaces.***Proceedings of Indian Academy of Sciences: Mathematical Sciences*2007,**117**(3):371–385. 10.1007/s12044-007-0032-yView ArticleMATHGoogle Scholar - Li S, Stević S:
**Weighted composition operators from**-Bloch space to**on the polydisc.***Numerical Functional Analysis and Optimization*2007,**28**(7–8):911–925. 10.1080/01630560701493222MathSciNetView ArticleMATHGoogle Scholar - Li S, Stević S:
**Weighted composition operators from****to the Bloch space on the polydisc.***Abstract and Applied Analysis*2007,**2007:**-13.Google Scholar - Li S, Stević S:
**Generalized composition operators on Zygmund spaces and Bloch type spaces.***Journal of Mathematical Analysis and Applications*2008,**338**(2):1282–1295. 10.1016/j.jmaa.2007.06.013MathSciNetView ArticleMATHGoogle Scholar - Li S, Stević S:
**Weighted composition operators between**and**-Bloch spaces in the unit ball.***Taiwanese Journal of Mathematics*2008,**12**(7):1625–1639.MathSciNetMATHGoogle Scholar - Li S, Stević S:
**Products of integral-type operators and composition operators between Bloch-type spaces.***Journal of Mathematical Analysis and Applications*2009,**349**(2):596–610. 10.1016/j.jmaa.2008.09.014MathSciNetView ArticleMATHGoogle Scholar - Lindström M, Wolf E:
**Essential norm of the difference of weighted composition operators.***Monatshefte für Mathematik*2008,**153**(2):133–143. 10.1007/s00605-007-0493-1View ArticleMATHGoogle Scholar - MacCluer BD, Zhao R:
**Essential norms of weighted composition operators between Bloch-type spaces.***The Rocky Mountain Journal of Mathematics*2003,**33**(4):1437–1458. 10.1216/rmjm/1181075473MathSciNetView ArticleMATHGoogle Scholar - Madigan K, Matheson A:
**Compact composition operators on the Bloch space.***Transactions of the American Mathematical Society*1995,**347**(7):2679–2687. 10.2307/2154848MathSciNetView ArticleMATHGoogle Scholar - Montes-Rodríguez A:
**Weighted composition operators on weighted Banach spaces of analytic functions.***Journal of the London Mathematical Society*2000,**61**(3):872–884. 10.1112/S0024610700008875MathSciNetView ArticleMATHGoogle Scholar - Ohno S:
**Weighted composition operators between****and the Bloch space.***Taiwanese Journal of Mathematics*2001,**5**(3):555–563.MathSciNetMATHGoogle Scholar - Ohno S, Zhao R:
**Weighted composition operators on the Bloch space.***Bulletin of the Australian Mathematical Society*2001,**63**(2):177–185. 10.1017/S0004972700019250MathSciNetView ArticleMATHGoogle Scholar - Shi J, Luo L:
**Composition operators on the Bloch space of several complex variables.***Acta Mathematica Sinica*2000,**16**(1):85–98. 10.1007/s101149900028MathSciNetView ArticleMATHGoogle Scholar - Stević S:
**Composition operators between**and**-Bloch spaces on the polydisc.***Zeitschrift für Analysis und ihre Anwendungen*2006,**25**(4):457–466.View ArticleMATHGoogle Scholar - Stević S:
**Weighted composition operators between mixed norm spaces and****spaces in the unit ball.***Journal of Inequalities and Applications*2007,**2007:**-9.Google Scholar - Stević S:
**Essential norms of weighted composition operators from****-Bloch space to a weighted-type space on the unit ball.***Abstract and Applied Analysis*2008,**2008:**-11.Google Scholar - Stević S:
**Norm of weighted composition operators from Bloch space to****on the unit ball.***Ars Combinatoria*2008,**88:**125–127.MathSciNetMATHGoogle Scholar - Stević S:
**On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball.***Discrete Dynamics in Nature and Society*2008,**2008:**-14.Google Scholar - Stević S:
**Essential norms of weighted composition operators from the Bergman space to weighted-type spaces on the unit ball.***Ars Combinatoria*2009,**91:**391–400.MathSciNetMATHGoogle Scholar - Stević S:
**On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball.***Journal of Mathematical Analysis and Applications*2009,**354**(2):426–434. 10.1016/j.jmaa.2008.12.059MathSciNetView ArticleMATHGoogle Scholar - Stević S:
**Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball.***Applied Mathematics and Computation*2009,**212**(2):499–504. 10.1016/j.amc.2009.02.057MathSciNetView ArticleMATHGoogle Scholar - Ueki S-I:
**Composition operators on the Privalov spaces of the unit ball of**.*Journal of the Korean Mathematical Society*2005,**42**(1):111–127.MathSciNetView ArticleMATHGoogle Scholar - Ueki S-I, Luo L:
**Compact weighted composition operators and multiplication operators between Hardy spaces.***Abstract and Applied Analysis*2008,**2008:**-12.Google Scholar - Ueki S-I, Luo L:
**Essential norms of weighted composition operators between weighted Bergman spaces of the ball.***Acta Scientiarum Mathematicarum*2008,**74**(3–4):829–843.MathSciNetMATHGoogle Scholar - Wolf E:
**Compact differences of composition operators.***Bulletin of the Australian Mathematical Society*2008,**77**(1):161–165.MathSciNetView ArticleMATHGoogle Scholar - Wolf E:
**Weighted composition operators between weighted Bergman spaces and weighted Bloch type spaces.***Journal of Computational Analysis and Applications*2009,**11**(2):317–321.MathSciNetMATHGoogle Scholar - Yang W: Weighted composition operators from Bloch-type spaces to weighted-type spaces. to appear in Ars Combinatoria to appear in Ars CombinatoriaGoogle Scholar
- Ye S:
**Weighted composition operator between the little****-Bloch spaces and the logarithmic Bloch.***Journal of Computational Analysis and Applications*2008,**10**(2):243–252.MathSciNetMATHGoogle Scholar - Zhu X:
**Generalized weighted composition operators from Bloch type spaces to weighted Bergman spaces.***Indian Journal of Mathematics*2007,**49**(2):139–150.MathSciNetMATHGoogle Scholar - Zhu X:
**Weighted composition operators from**spaces to**spaces to spaces.***Abstract and Applied Analysis*2009,**2009:**-14.Google Scholar - Li S, Stević S:
**Riemann-Stieltjes-type integral operators on the unit ball in**.*Complex Variables and Elliptic Equations*2007,**52**(6):495–517. 10.1080/17476930701235225MathSciNetView ArticleMATHGoogle Scholar - Stević S:
**Boundedness and compactness of an integral operator on a weighted space on the polydisc.***Indian Journal of Pure and Applied Mathematics*2006,**37**(6):343–355.MathSciNetMATHGoogle Scholar - Stevich S:
**Boundedness and compactness of an integral operator in a mixed norm space on the polydisc.***Sibirskiĭ Matematicheskiĭ Zhurnal*2007,**48**(3):559–569.MathSciNetGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.