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Composition Operator on Bergman-Orlicz Space

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Abstract

Let Open image in new window denote the open unit disk in the complex plane and let Open image in new window denote the normalized area measure on Open image in new window . For Open image in new window and Open image in new window a twice differentiable, nonconstant, nondecreasing, nonnegative, and convex function on Open image in new window , the Bergman-Orlicz space Open image in new window is defined as follows Open image in new window Let Open image in new window be an analytic self-map of Open image in new window . The composition operator Open image in new window induced by Open image in new window is defined by Open image in new window for Open image in new window analytic in Open image in new window . We prove that the composition operator Open image in new window is compact on Open image in new window if and only if Open image in new window is compact on Open image in new window , and Open image in new window has closed range on Open image in new window if and only if Open image in new window has closed range on Open image in new window .

Keywords

Composition Operator Bergman Space Blaschke Product Closed Range Carleson Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Let Open image in new window be the open unit disk in the complex plane and let Open image in new window be an analytic self-map of Open image in new window . The composition operator Open image in new window induced by Open image in new window is defined by Open image in new window for Open image in new window analytic in Open image in new window . The idea of studying the general properties of composition operators originated from Nordgren [1]. As a sequence of Littlewood's subordinate theorem, each Open image in new window induces a bounded composition operator on the Hardy spaces Open image in new window for all Open image in new window ( Open image in new window ) and the weighted Bergman spaces Open image in new window for all Open image in new window ( Open image in new window ) and for all Open image in new window ( Open image in new window ). Thus, boundedness of composition operators on these spaces becomes very clear. Nextly, a natural problem is how to characterize the compactness of composition operators on these spaces, which once was a central problem for mathematicians who were interested in the theory of composition operators. The study of compact composition operators was started by Schwartz, who obtained the first compactness theorem in his thesis [2], showing that the integrability of Open image in new window over Open image in new window implied the compactness of Open image in new window on Open image in new window . The work was continued by Shapiro and Taylor [3], who showed that Open image in new window was not compact on Open image in new window whenever Open image in new window had a finite angular derivative at some point of Open image in new window . Moreover, MacCluer and Shapiro [4] pointed out that nonexistence of the finite angular derivatives of Open image in new window was a sufficient condition for the compactness of Open image in new window on Open image in new window but it failed on Open image in new window . So looking for an appropriate tool of characterizing the compactness of Open image in new window on Open image in new window was difficult at that time. Fortunately, Shapiro [5] developed relations between the essential norm of Open image in new window on Open image in new window and the Nevanlinna counting function of Open image in new window , and he obtained a nice essential norm formula of Open image in new window in 1987. As a result, he completely gave a characterization of the compactness of Open image in new window in terms of the function properties of Open image in new window .

Another solution to the compactness of Open image in new window on Open image in new window was done by the Aleksandrov measures which was introduced by Cima and Matheson [6]. It is well known that the harmonic function Open image in new window can be expressed by the Possion integral

for each Open image in new window . Cima and Matheson applied Open image in new window the singular part of Open image in new window to give the following expression:

They showed that Open image in new window was compact on Open image in new window if and only if all the measures Open image in new window were absolutely continuous.

The study of compactness of composition operators is also an important subject on other analytic function spaces, and we have chosen two typical examples above, and for more related materials one can consult [7, 8]. Another natural interesting subject is the composition operator with closed range. Considering angular derivatives of Open image in new window , it is known that Open image in new window is compact on Open image in new window if and only if Open image in new window fails to have finite angular derivatives on Open image in new window , in this case, Open image in new window does not have closed range since Open image in new window is not a finite rank operator. And if Open image in new window has finite angular derivatives on Open image in new window , then Open image in new window is necessarily a finite Blaschke product and hence one can easily verify that Open image in new window has closed range on Open image in new window . Zorboska has given a necessary and sufficient condition for Open image in new window with closed range on Open image in new window , and she also has done on Open image in new window [9]. Luecking [10] considered the same question on Dirichlet space after Zorboska's work. Recently, Kumar and Partington [11] have studied the weighted composition operators with closed range on Hardy spaces and Bergman spaces.

This paper will study the compactness of composition operator on Bergman-Orlicz space. We are mainly inspired by the following results.

(i)Liu et al. [12] showed that composition operator was bounded on Hardy-Orlicz space. Lu and Cao [13] also showed that composition operator was bounded on Bergman-Orlicz space.

(ii)A composition operator was compact on the Nevanlinna class Open image in new window if and only if it was compact on Open image in new window [14].

(iii)If a composition operator was compact on Open image in new window for some Open image in new window , then it was compact on Open image in new window for all Open image in new window [3]. Moreover, paper [15] compared the compactness of composition operators on Hardy-Orlicz spaces and on Hardy spaces. All these results lead us to wonder whether there is a equivalence for the compactness of Open image in new window on Open image in new window and on the Bergman-Orlicz space, and whether there is a equivalence for the closed range of Open image in new window on Open image in new window and on the Bergman-Orlicz space. In this paper, we are going to give affirmative answers for the proceeding questions.

2. Preliminaries

Let Open image in new window denote the space of all analytic functions on Open image in new window . Let Open image in new window denote the normalized area measure on Open image in new window , that is, Open image in new window . Let Open image in new window denote the class of strongly convex functions Open image in new window , which satisfies

(i) Open image in new window , Open image in new window as Open image in new window ,

(ii) Open image in new window exists on Open image in new window ,

(iii) Open image in new window for some positive constant Open image in new window and for all Open image in new window .

For Open image in new window and Open image in new window the Bergman-Orlicz space Open image in new window is defined as follows:

where Open image in new window . Although Open image in new window does not define a norm in Open image in new window , it holds that the Open image in new window defines a metric on Open image in new window , and makes Open image in new window into a complete metric space. Obviously, the inequalities

and the fact that Open image in new window is nondecreasing convex function imply that

Then Open image in new window if and only if

or if and only if

Throughout this paper, constants are denoted by Open image in new window , they are positive and may differ from one occurrence to the other. The notation Open image in new window means that there is a positive constant Open image in new window such that Open image in new window . Moreover, if both Open image in new window and Open image in new window hold, we write Open image in new window and say that Open image in new window is asymptotically equivalent to Open image in new window .

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

Lemma 2.1.

where Open image in new window is Laplacian and Open image in new window .

Proof.

By the Green Theorem, if Open image in new window , where Open image in new window is a domain in the plane with smooth boundary, then
Integrating equality (2.10) with respect to Open image in new window from Open image in new window to Open image in new window , we obtain

the proof is complete.

Let Open image in new window be an analytic self-map of Open image in new window . The generalized Nevanlinna counting function of Open image in new window is defined by

Lemma 2.2 (see [9]).

If Open image in new window is an analytic self-map of Open image in new window and Open image in new window is a nonnegative measurable function in Open image in new window , then

Lemmas 2.1 and 2.2(see [9])can lead to the following corollary.

Corollary 2.3.

We will end this section with the following lemma, which illustrates that the counting functional Open image in new window is continuous on Open image in new window .

Lemma 2.4.

Proof.

By the subharmonicity of map Open image in new window , we get
Since Open image in new window is convex and increasing, we have

that is, Open image in new window . Thus Open image in new window .

3. Compactness

In this section, we are going to investigate the equivalence between compactness of composition operator on the Bergman-Orlicz space Open image in new window and on the weighted Bergman space Open image in new window . The following lemma characterizes the compactness of Open image in new window on Open image in new window in terms of sequential convergence, whose proof is similar to that in [7, Proposition 3.11].

Lemma 3.1.

Let Open image in new window be an analytic self-map of Open image in new window , bounded operator Open image in new window is compact on Open image in new window if and only if whenever Open image in new window is bounded in Open image in new window and Open image in new window uniformly on compact subsets of Open image in new window , then Open image in new window as Open image in new window .

In order to characterize the compactness of Open image in new window , we need to introduce the notion of Carleson measure. For Open image in new window and Open image in new window we define Open image in new window . A positive Borel measure Open image in new window on Open image in new window is called a Carleson measure if Open image in new window . Moreover, if Open image in new window satisfies the additional condition Open image in new window , Open image in new window is called a vanishing Carleson measure (see [16] for the further information of Carleson measure). The following result for the compactness of Open image in new window on Open image in new window is useful in the proof of Theorem 3.3.

Lemma 3.2 (see [14, 17]).

Let Open image in new window be an analytic self-map of Open image in new window . Then the following statements are equivalent:

(i) Open image in new window is compact on Open image in new window , (ii) Open image in new window , and (iii) the pull measure Open image in new window is a vanshing Carleson measure on Open image in new window .

Theorem 3.3.

Let Open image in new window be an analytic self-map of Open image in new window , then Open image in new window is compact on Open image in new window if and only if Open image in new window is compact on Open image in new window .

Proof.

First we assume that Open image in new window is compact on Open image in new window . Choose a sequence Open image in new window that is bounded by a positive constant Open image in new window in Open image in new window and converges to zero uniformly on compact subsets of Open image in new window . By Lemma 3.1, it is enough to show that Open image in new window as Open image in new window . Let Open image in new window , we can find Open image in new window such that Open image in new window for all Open image in new window Since Open image in new window uniformly on compact subsets of Open image in new window as Open image in new window , so is Open image in new window . Thus we can choose Open image in new window such that Open image in new window and Open image in new window on Open image in new window , whenever Open image in new window . Hence for such Open image in new window we have
We first prove that the first term in previous equality is bounded by a constant multiple of Open image in new window .
Now, we show that the previous second term above is also bounded by a constant multiple of Open image in new window
Conversely, we assume that Open image in new window is compact on Open image in new window . By Lemma 3.2, we need to verify that Open image in new window is a vanishing Carleson measure. For Open image in new window and Open image in new window we write Open image in new window and Open image in new window . Then Open image in new window . Put Open image in new window . Open image in new window is well defined, beacuse Open image in new window is nondecreasing on range of Open image in new window . Since Open image in new window is concave, there is a constant Open image in new window such that Open image in new window for enough big Open image in new window . Thus we get Open image in new window . Set Open image in new window . Since Open image in new window , it means that Open image in new window . Let Open image in new window . Then clearly Open image in new window uniformly on compact subsets of Open image in new window as Open image in new window . Moreover,

For the compactness of Open image in new window , we know that Open image in new window as Open image in new window , which means that Open image in new window uniformly for Open image in new window . This means that Open image in new window is a vanishing Carleson measure. By Lemma 3.2, Open image in new window is compact on Open image in new window .

For special case Open image in new window , the Bergman-Orlicz space Open image in new window is called the area-type Nevanlinna class and we write Open image in new window .

Corollary 3.4.

Let Open image in new window be an analytic self-map of Open image in new window , then Open image in new window is compact on Open image in new window if and only if Open image in new window is compact on Open image in new window .

Remark 3.5.

Theorem 3.3 may be not true if Open image in new window does not satisfy the given conditions in this paper. For example, if Open image in new window is a nonnegative function on Open image in new window such that Open image in new window as Open image in new window , and Open image in new window is nondecreasing but Open image in new window for some Open image in new window . Then the compactness of Open image in new window on the Bergman space Open image in new window (i.e., Open image in new window ) is different from that on Open image in new window . Here Open image in new window is defined as follows:

If we take Open image in new window for Open image in new window , and Open image in new window for Open image in new window , then Open image in new window is Open image in new window . We know that Open image in new window is compact on Open image in new window if and only if Open image in new window (consult [2]). But MacCluer and Shapiro constructed an inner function Open image in new window in [4] such that Open image in new window was compact on Open image in new window .

4. Closed Range

In this section we will develop a relatively tractable if and only if condition for the composition operator on Open image in new window with closed range. Considering that any analytic automorphism of Open image in new window has the form Open image in new window , where Open image in new window and Open image in new window . By [13], we have the following lemma.

Lemma 4.1.

If one of Open image in new window , Open image in new window , Open image in new window has closed range on Open image in new window , so have the other two.

Now that Open image in new window is a closed subspace of Open image in new window and Open image in new window , the following lemma is easily proved.

Lemma 4.2.

Let Open image in new window be an analytic self-map of Open image in new window , then Open image in new window has closed range on Open image in new window if and only if Open image in new window has closed range on Open image in new window .

Recall that the pseudohyperbolic metric Open image in new window , Open image in new window is given by

For Open image in new window and Open image in new window we define Open image in new window . For Open image in new window we put Open image in new window and Open image in new window . We say that Open image in new window satisfies the Open image in new window -reverse Carleson measure condition if there exists a positive constant Open image in new window such that

where Open image in new window is analytic in Open image in new window and Open image in new window .

Theorem 4.3.

Let Open image in new window be an analytic self-map of Open image in new window . Then Open image in new window has closed range on Open image in new window if and only if there exists Open image in new window such that Open image in new window satisfies the Open image in new window -reverse Carleson measure condition.

Proof.

We first assume that there exists Open image in new window such that Open image in new window satisfies the Open image in new window -reverse Carleson measure condition. If Open image in new window , then
where Open image in new window is the zero point set of Open image in new window and Open image in new window is a partition of Open image in new window into at most countably many semiclosed polar rectangles such that Open image in new window is univalent on each Open image in new window . Let Open image in new window . Then by the change of variables involving Open image in new window , the last line above becomes

So we show that Open image in new window has closed range on Open image in new window .

Conversely, by Lemma 4.2, we need to prove that Open image in new window has closed range on Open image in new window . Suppose that there does not exist Open image in new window such that Open image in new window satisfies Open image in new window -reverse Carleson measure condition. We can choose a sequence Open image in new window in Open image in new window such that Open image in new window for all Open image in new window and yet Open image in new window as Open image in new window , where Open image in new window and Open image in new window . Now
Since Open image in new window is an analytic self-map of Open image in new window . The Nevanlinna counting function Open image in new window satisfies
as Open image in new window . Using (4.6) and decompositions of the disk into polar rectangles [8], one can find a positive constant Open image in new window such that

as Open image in new window . Evidently, Open image in new window as Open image in new window , though Open image in new window for all Open image in new window . It follows that Open image in new window does not have closed range on Open image in new window .

We have offered a criterion for the composition operator with closed range on Open image in new window , but it seems that it is difficult to check whether or not Open image in new window satisfies the Open image in new window -reverse Carleson measure condition.

Theorem 4.4.

The composition operator Open image in new window has closed range on Open image in new window if and only if there are positive constants Open image in new window , and Open image in new window such that Open image in new window for all Open image in new window .

Proof.

We first assume that Open image in new window has closed range on Open image in new window . Then there is a constant Open image in new window such that Open image in new window satisfies the Open image in new window -reverse Carleson measure condition. Thus, applying the proceeding constructed function Open image in new window to the Open image in new window -reverse Carleson condition gives
Since Open image in new window , it allows to choose a fixed constant Open image in new window such that
Changing Open image in new window in (4.10) gives
Combing (4.11) gives
The integral in the left of (4.12) is dominated by

The converse can be derived from modification of [18], so we omit it here.

Remark 4.5.

From [18], we find that the composition operator has closed range on the weighted Bergman space Open image in new window if and only if there are positive constants Open image in new window and Open image in new window such that Open image in new window for all Open image in new window . Thus, we have the following fact.

The composition operator Open image in new window has closed range on Open image in new window if and only if Open image in new window has closed range on Open image in new window .

Let us further investigate the Open image in new window -reverse Carleson measure condition, which can be formulated as follows.

The space Open image in new window is a closed subspace of Open image in new window if and only if there exists a constant Open image in new window such that Open image in new window satisfies Open image in new window -reverse Carleson measure condition.

From the perspective of closed subspace, we will see the following special setting. Let Open image in new window be a Open image in new window -sequence in Open image in new window . That is, there is Open image in new window with Open image in new window for every Open image in new window . We also assume that Open image in new window is Open image in new window separated for some fixed Open image in new window , that is, Open image in new window for all Open image in new window . Using the subharmonicity of Open image in new window for analytic function Open image in new window , it is easy to see that

Since Open image in new window is convex and increasing, we have

Moreover, the formula Open image in new window allows us to write

Since Open image in new window are disjoint, we obtain

Hence, the map Open image in new window takes Open image in new window into Open image in new window , where Open image in new window is a measure on Open image in new window that assigns Open image in new window to the mass Open image in new window and space Open image in new window . Of course, the map Open image in new window may be one to one. If the map Open image in new window is one to one, the map Open image in new window has closed range if and only if

Notes

Acknowledgments

The authors are extremely thankful to the editor for pointing out several errors. This work was supported by the Science Foundation of Sichuan Province (no. 20072A04) and the Scientific Research Fund of School of Science SUSE.

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Copyright information

© Z. Jiang and G. Cao. 2009

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.

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceGuangzhou UniversityGuangzhouChina
  2. 2.Department of MathematicsSichuan University of Science and EngineeringZigongChina

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