Composition Operator on Bergman-Orlicz Space

Abstract

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

1. Introduction

Let be the open unit disk in the complex plane and let be an analytic self-map of . The composition operator induced by is defined by for analytic in . The idea of studying the general properties of composition operators originated from Nordgren [1]. As a sequence of Littlewood's subordinate theorem, each induces a bounded composition operator on the Hardy spaces for all () and the weighted Bergman spaces for all () and for all (). 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 over implied the compactness of on . The work was continued by Shapiro and Taylor [3], who showed that was not compact on whenever had a finite angular derivative at some point of . Moreover, MacCluer and Shapiro [4] pointed out that nonexistence of the finite angular derivatives of was a sufficient condition for the compactness of on but it failed on . So looking for an appropriate tool of characterizing the compactness of on was difficult at that time. Fortunately, Shapiro [5] developed relations between the essential norm of on and the Nevanlinna counting function of , and he obtained a nice essential norm formula of in 1987. As a result, he completely gave a characterization of the compactness of in terms of the function properties of .

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

(1.1)

for each . Cima and Matheson applied the singular part of to give the following expression:

(1.2)

They showed that was compact on if and only if all the measures 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 , it is known that is compact on if and only if fails to have finite angular derivatives on , in this case, does not have closed range since is not a finite rank operator. And if has finite angular derivatives on , then is necessarily a finite Blaschke product and hence one can easily verify that has closed range on . Zorboska has given a necessary and sufficient condition for with closed range on , and she also has done on [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 if and only if it was compact on [14].

(iii)If a composition operator was compact on for some , then it was compact on for all [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 on and on the Bergman-Orlicz space, and whether there is a equivalence for the closed range of on and on the Bergman-Orlicz space. In this paper, we are going to give affirmative answers for the proceeding questions.

2. Preliminaries

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

(i), as ,

(ii) exists on ,

(iii) for some positive constant and for all .

For and the Bergman-Orlicz space is defined as follows:

(2.1)

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

(2.2)

and the fact that is nondecreasing convex function imply that

(2.3)

Then if and only if

(2.4)

or if and only if

(2.5)

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 . Moreover, if both and hold, we write and say that is asymptotically equivalent to .

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.

If , then

(2.6)

where is Laplacian and .

Proof.

By the Green Theorem, if , where is a domain in the plane with smooth boundary, then

(2.7)

Let , , , and . Since , by (2.7) we have

(2.8)

Since is bounded near to , we get

(2.9)

Let in (2.8), we have

(2.10)

Integrating equality (2.10) with respect to from to , we obtain

(2.11)

Thus

(2.12)

Since ,

(2.13)

the proof is complete.

Let be an analytic self-map of . The generalized Nevanlinna counting function of is defined by

(2.14)

Lemma 2.2 (see [9]).

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

(2.15)

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

Corollary 2.3.

Let be an analytic self-map of and , then

(2.16)

We will end this section with the following lemma, which illustrates that the counting functional is continuous on .

Lemma 2.4.

Let , then

(2.17)

Proof.

By the subharmonicity of map , we get

(2.18)

Since is convex and increasing, we have

(2.19)

Since , we get

(2.20)

that is, . Thus .

3. Compactness

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

Lemma 3.1.

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

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

Lemma 3.2 (see [14, 17]).

Let be an analytic self-map of . Then the following statements are equivalent:

(i) is compact on , (ii), and (iii) the pull measure is a vanshing Carleson measure on .

Theorem 3.3.

Let be an analytic self-map of , then is compact on if and only if is compact on .

Proof.

First we assume that is compact on . Choose a sequence that is bounded by a positive constant in and converges to zero uniformly on compact subsets of . By Lemma 3.1, it is enough to show that as . Let , we can find such that for all Since uniformly on compact subsets of as , so is . Thus we can choose such that and on , whenever . Hence for such we have

(3.1)

As as and as , we only need to verify that as . Now

(3.2)

We first prove that the first term in previous equality is bounded by a constant multiple of .

(3.3)

Now, we show that the previous second term above is also bounded by a constant multiple of

(3.4)

Conversely, we assume that is compact on . By Lemma 3.2, we need to verify that is a vanishing Carleson measure. For and we write and . Then . Put . is well defined, beacuse is nondecreasing on range of . Since is concave, there is a constant such that for enough big . Thus we get . Set . Since , it means that . Let . Then clearly uniformly on compact subsets of as . Moreover,

(3.5)

On the other hand, if for some fixed , where , that is, , we have

(3.6)

Hence, for we have

(3.7)

Thus, for we obtain

(3.8)

So, for all and we get

(3.9)

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

For special case , the Bergman-Orlicz space is called the area-type Nevanlinna class and we write .

Corollary 3.4.

Let be an analytic self-map of , then is compact on if and only if is compact on .

Remark 3.5.

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

(3.10)

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

4. Closed Range

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

Lemma 4.1.

If one of , , has closed range on , so have the other two.

Now that is a closed subspace of and , the following lemma is easily proved.

Lemma 4.2.

Let be an analytic self-map of , then has closed range on if and only if has closed range on .

Recall that the pseudohyperbolic metric , is given by

(4.1)

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

(4.2)

where is analytic in and .

Theorem 4.3.

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

Proof.

We first assume that there exists such that satisfies the -reverse Carleson measure condition. If , then

(4.3)

where is the zero point set of and is a partition of into at most countably many semiclosed polar rectangles such that is univalent on each . Let . Then by the change of variables involving , the last line above becomes

(4.4)

So we show that has closed range on .

Conversely, by Lemma 4.2, we need to prove that has closed range on . Suppose that there does not exist such that satisfies -reverse Carleson measure condition. We can choose a sequence in such that for all and yet as , where and . Now

(4.5)

Since is an analytic self-map of . The Nevanlinna counting function satisfies

(4.6)

as . Using (4.6) and decompositions of the disk into polar rectangles [8], one can find a positive constant such that

(4.7)

as , and

(4.8)

as . Evidently, as , though for all . It follows that does not have closed range on .

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

Theorem 4.4.

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

Proof.

We first assume that has closed range on . Then there is a constant such that satisfies the -reverse Carleson measure condition. Thus, applying the proceeding constructed function to the -reverse Carleson condition gives

(4.9)

Since , it allows to choose a fixed constant such that

(4.10)

Changing in (4.10) gives

(4.11)

Combing (4.11) gives

(4.12)

The integral in the left of (4.12) is dominated by

(4.13)

Since for , we get

(4.14)

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 if and only if there are positive constants and such that for all . Thus, we have the following fact.

The composition operator has closed range on if and only if has closed range on .

Let us further investigate the -reverse Carleson measure condition, which can be formulated as follows.

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

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

(4.15)

Since is convex and increasing, we have

(4.16)

Moreover, the formula allows us to write

(4.17)

Since are disjoint, we obtain

(4.18)

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

(4.19)