Abstract
In this paper we find conditions for boundedness of self-map composition operators on weighted spaces of holomorphic functions on the upper half-plane for two kinds of weights which are of moderate growth.
MSC(2010):
Public Interest Statement
The present paper is devoted to the problem of continuity of composition operators on the weighted spaces of holomorphic functions on the upper half-plane equipped with sup-norms. These spaces of holomorphic functions (on the unit disc) with controlled growth are natural classes studied by Shields and Williams in seventies and later by large variety of authors. Composition operators are very natural operators and their study is by now a true industry which is interesting and worth studying.
1. Introduction
Different properties of composition operators between weighted spaces of holomorphic functions on the unit disc or upper half-plane have been the subject of many papers in recent decades (Ardalani, Citation2014; Ardalani & Lusky, Citation2011,Citation2012a,Citation2012b; Bonet, Citation2003; Bonet, Fritz, & Jorda, Citation2005; Bonet, Domanski, & Lindstrom, Citation1998,Citation1999; Cowen, Citation1995; Madigan, Citation1993; Shapiro, Citation1987; Zhu, Citation1990). In Theorem 2.3 of Bonet et al. (Citation1998), authors have characterized boundedness of self-map composition operators on weighted spaces of holomorphic functions on the unit disc in terms of associated weight which satisfies well-known growth condition that is used by Lusky (Citation1995). Indeed they have found a condition under which all self-map composition operators on weighted spaces of holomorphic functions on the unit disc are bounded. In this paper we intend to find conditions for boundedness of all self-map composition operators on weighted spaces of holomorphic functions on the upper half-plane for standard weights in the sense of Ardalani (Citation2014), Ardalani and Lusky (Citation2011,Citation2012a) and for a new type of weights on the upper half-plane which we call it type(II) weights. For standard weights we use the results of Ardalani and Lusky (Citation2012b) in order to prove Theorem 2.1. For weights of type(II) we make an isomorphism between weighted spaces of holomorphic functions on the unit disc and weighted spaces of holomorphic functions on the upper half-plane. Then we use this isomorphism and Theorem 2.3 of Bonet et al. (Citation1998) to obtain a sufficient condition for boundedness of self-map composition operators on weighted spaces of holomorphic functions on the upper half-plane. This isomorphism is constructed under a certain growth condition which we call it throughout this paper. Although, we use the concept of associated weight to prove Theorem 3.2, the associated weight does not appear in the assertion of Theorem 3.2 and that is important because it is difficult to calculate an associated weight. We continue with the preliminaries which are required in the rest of this paper.
The sets and stand for the unit disc and upper half-plane, respectively.
Definition 1.1
A continuous function is called a standard weight if (i.e depends only on the imaginary part), when , and
Definition 1.2
A continuous function is called a type(II) if , whenever , and there is a constant such that for any
Note that Standard weights are increasing thorough the imaginary axis while type(II) weights are increasing on the imaginary axis whenever the imaginary part is in (0, 1]. Also type(II) weights has symmetric property which makes them interesting (see Remark 1.13).
Example 1.3
Define and for some and are type(II) weights. For examples of standard weights see Ardalani and Lusky (Citation2012a,Citation2012b).
Remark 1.4
Note that weights and of Example 1.3 are type(II) weights which are not standard weights.
Definition 1.5
Let be a standard weight on G.
(i) | satisfies condition if there are constants such that | ||||
((ii) | satisfies condition (**) if there are constants such that |
For examples of weights which satisfy both and or satisfy but not (see Ardalani & Lusky, Citation2012a or Citation2012b).
Definition 1.6
Let be a type(II) weight on G. We say satisfies condition if there are constants s.t whenever and
Condition is really condition which is restricted to the intersection of the unit disc and upper half-plane. Evidently, condition implies condition . We have also proved condition is equivalent to (Ardalani & Lusky, Citation2012a). Similarly, we have:
Lemma 1.7
Let be a continuous weight on G which depends only on the imaginary part and satisfy the following property:
whenever Then satisfies
In particular any type(II) weight satisfies
Proof
suppose is arbitrary. Put and , . Now, since satisfies , there exist and such that . Therefore, .
Let with and be given. We can find n and such that and Then where . Now, with we have . The last assertion of the theorem is clear.
Example 1.8
and of Example 1.3 are type(II) weights which satisfy condition . Indeed, and satisfy condition .
Definition 1.9
Let O be an open subset of . For a function we define the weighted sup-norm
and the space
Throughout this paper we deal with the cases or .
Remark 1.10
(a) | According to a result of Stanev (Citation1999), if and only if there are constants such that Note that if standard (type(II)) weight satisfies then . | ||||
(b) | For a weight defined from into , we always assume is radial (i.e ), continuous and non-increasing weight with respect to and |
Definition 1.11
Let O be an open subset of . Also, suppose is a weight. Corresponding to , the associated weight is defined as follows.
Remark 1.12
Define by An easy computation shows that . Hence, . Put Then we have Thus and
Remark 1.13
Let be a standard weight. By definition of standard weight there exists a constant such that and . Thus(1.1) (1.1)
Obviously, . Hence, inserting in (1.1) we have
Definition 1.14
Let be an analytic function. Put . For any the composition operator defined by
Here we recall the Theorem 2.3 of Bonet et al. (Citation1998) in the following lemma.
Lemma 1.15
Let be a radial weight on . Then the following are equivalent:
(i) | All the operators are bounded. | ||||
(ii) |
Remark 1.16
Following example shows that there are standard weights satisfying , but not all composition operators are bounded. Therefore, the situation on the upper half-plane is essentially different from the unit disc.
Example 1.17
For any in G, define and . Then we have
while
2. Boundedness of composition operators for standard weights
Although Remark 1.15 and Example 1.17 show that we cannot expect to obtain a result similar to Lemma 1.15 for standard weights but we are able to characterize all the analytic maps such that the self-map composition operators on the upper half-plane are bounded.
Theorem 2.1
Let be a standard weight which satisfy and .
Composition maps are bounded if and only if
Proof
By Corollary 1.5 of Ardalani and Lusky (Citation2012b) maps are bounded if and only if(2.1) (2.1)
Since satisfies (2.2) (2.2)
Since satisfies (2.3) (2.3)
Now relations (2.1), (2.2) and (2.3) prove the theorem.
Following example shows that Theorem 2.1 is not true if does not satisfy condition .
Example 2.2
Let be a bounded standard weight (so does not satisfy , see Ardalani & Lusky, Citation2012a) and put for some . Certainly is bounded but .
3. Main results
We begin this section with Lemma 3.1 which makes an isomorphism between weighted spaces of holomorphic functions on the upper half-plane (for type(II) weights) and weighted spaces of holomorphic functions on the unit disc. This isomorphism is our main tool to prove Theorem 3.2.
Lemma 3.1
Let be a type(II) weight on G which satisfies . Put and define by for all and all . Then is radial weight on and T is an onto isomorphism.
Proof
First assertion of the lemma is obvious. By Remark 1.13 there is a constant such that(3.1) (3.1)
Consider a fixed . Firstly, assume . Since , we have
.
Thus(3.2) (3.2)
Now, relation and the fact satisfies imply that there exists a and such that
Hence(3.3) (3.3)
Now relations (3.1), (3.2) and (3.3) imply that(3.4) (3.4)
whenever and
If , then . Using relations (3.1), (3.2) and (3.3) we have
.
Therefore(3.5) (3.5)
whenever and
Relations (3.4) and (3.5) show that weights and are equivalent on . Hence T is well defined and if and only if . This proves the lemma.
Now we present the following theorem:
Theorem 3.2
Let be a type(II) weight on G. If satisfies , then all composition operators are bounded operators.
Proof
Consider the following diagram.
where T and are as in Lemma 3.1 and . For any we have
This means our diagram is commutative. Therefore, is bounded if and only if is bounded. Using Lemma 1.15 is bounded if and only if(3.6) (3.6)
To end the proof it is enough to show that relation (3.6) holds. we have (Bonet et al., Citation1999, Lemma 5, p. 145)(3.7) (3.7)
Since satisfies , . Thus . Now by inserting this relation in relation (3.7) we have which implies that(3.8) (3.8)
Also it is well known that (Bonet et al., Citation1998). Therefore
Obviously, . satisfies implies that there exist and such that . It is easy to see that is an increasing sequence which converges to . Hence, . Therefore
Corollary 3.3
Let be a standard weight on G which satisfies , then
Proof
is a bounded operator is a bounded operator which is equivalent to
As in the proof of Theorem 3.2, we have and (relation (3.8). Hence, .
But . Since and is increasing, . Thus
Relation and condition imply that . But, is a decreasing sequence which converges to . Therefore
which completes the proof.
Additional information
Funding
Notes on contributors
Mohammad Ali Ardalani
Mohammad Ali Ardalani is a faculty member of the Mathematics Department at the University of Kurdistan, Sanadaj, Iran. His fields of specialty include Complex Analysis and Functional Analysis. During 2007–2010, he completed his PhD in Pure Mathematics (Complex and Functional Analysis) and worked as Faculty of computer science, Electrical Engineering and Mathematics, University of Paderborn, Paderborn Germany. During 2001–2007, he worked as Faculty member of the Mathematics Department at the University of Kurdistan. During 1998–2001, he completed Msc in Pure Mathematics (Functional Analysis) in the Department of Mathematics of Shiraz University, Shiraz, Iran. In 1997, he completed Bsc in Pure mathematics in the Department of Mathematics of Isfahan University, Isfahan, Iran and was a member of Iranian Mathematical Society.
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