#### Abstract

We say if a constant could be found, such that Under the assumption , boundedness and compactness of weighted composition operators on the Zygmund-Orlicz space are investigated in this paper.

#### 1. Introduction

The composition operator, defined by an analytic self-map of the unit disk , could be defined as where is, as usual, the collection of all analytic functions on . For , denotes its iterates of times, . Specially, turns to the identity mapping. Moreover, appears when the analytic self-map is invertible.

For a given analytic function , the multiplication operator can be induced. Then, the weighted composition operator is induced by

Theory of (weighted) composition operators has been establishing since the last century. Properties including boundedness, compactness, essential norm, and spectral properties are always the highlights of research of composition operators. To study the composition operators on classical spaces of analytic functions, book [1] is a good reference. Moreover, this theory is also established on the basis of theory of analytic functions (on the unit disk), which is basically a convenient tool.

Recall that Zygmund space is defined as

Zygmund space is a Banach space of holomorphic functions endowed with the norm .

Generally, for a positive continuous function on , Zygmund space is defined by

becomes a Banach spaces with .

It is well-known that Hardy space, Bergman space, and Bloch space are important in the history of spaces of analytic functions. Motivated by the classical Orlicz space, the Hardy-Orlicz space, Bergman-Orlicz space, and Bloch-Orlicz space are studied in the recent years (see, e.g., [2–11]).

Motivated by the same approach, for a Young’s function , Zygmund-Orlicz space is defined as where is a positive number depending of .

By the similar discussion of with the function , we conduct that becomes a Banach space with the norm where

Specially, we say if there exists a constant such that

Note that this also implies that

Under the hypothesis , the boundedness and compactness of weighted composition operators on the Zygmund-Orlicz space are investigated, respectively.

#### 2. Auxiliary

Basic properties of are given in this section. Since most of them are similar arguments with the ones of Bloch-Orlicz space(see [3]), we omit the proofs.

Proposition 1. *For each*,

Proposition 2. *The Zygmund-Orlicz space is isometrically equal to the -Zygmund space, where
*

In other words, holds for each .

The proposition above is of vital significance, which states that is isometrically equal to -Zygmund space.

For each , by Proposition 2, we have that

It follows the following.

Proposition 3. *The evaluation functional is continuous on , where is fixed.*

Moreover, the equivalent condition below is easily conducted.

Corollary 4. *The equivalent condition
holds for each .*

Specially, for two real numbers and , we say if there exists a constant such that .

#### 3. Boundedness of Weighted Composition Operator on

Under the assumption , the boundedness of on the Zygmund-Orlicz is investigated in this section.

Basic properties of the testing functions entailed to the proof of the boundedness of the composition operator are described in Lemma 5.

Lemma 5. *For a fixed such that , suppose that
where . Then,
*(i)* and , where *(ii)*, , and .*

*Proof. *Observing that
Moreover,

Theorem 6. *For , is bounded on if and only if
*

*Proof. *Suppose that . For any , we have that
where is a number which satisfies and (15), (14), and (13) are employed. Note that, by (13), (14), and (15), , and

Then, the boundedness of the on is guaranteed by and the above estimations.

Conversely, if is bounded on , then there exists a constant such that for each . It follows by Proposition 1 that holds for each , which is equivalent with

Taking into the above equality, we have that

Taking into it, we have that

This implies that

Taking into the same equality again, we have that

This leads to

Since , we obtain that

This leads to

Note that where (24) is used in the last inequality. Combining the two equality above, we prove that

Since , we obtain that

Note that

Combing the above equality and (24), we obtain that

Moreover, by the fact , and some similar estimations, we have that

This completes the proof.

#### 4. Compactness of Weighted Composition Operator on

Theorem 7. *For , the weighted composition operator is compact on if and only if is bounded on ,
*

*Proof. *Suppose that is bounded on with (36), (37), and (38) hold. Observe that , , and (defined in the proof of Theorem 6) by the boundedness of .

For every , there exists a such that for ,

Choose a sequence which satisfies and converges to zero uniformly on any compact subsets of the unit disk as , where is a fixed constant. Then by the compactness of , we only should conduct that .

Note that implies that uniformly on any compact subsets of , . It follows that where the second inequality is calculated by (13), (14), and (15).

Then, we conclude that is compact on .

Conversely, suppose that is compact on . Then, is bounded on . Let be a sequence in which satisfies (If such sequence does not exist, then the proof is completed.)

Note that is bounded uniformly in and uniformly converges to 0 on any compact subset of . Thus, we have

By the similar calculations shown in the proof of boundedness of , we conclude that

Note that is bounded uniformly in and uniformly converges to zero on any compact subset of the unit disk as . Thus, we have

Note that is bounded uniformly in and uniformly converges to zero on any compact subset of the unit disk as . Thus, we have

This completes the proof.

#### 5. Conclusions

In this paper, the boundedness and compactness of weighted composition operators on the Zygmund-Orlicz space are investigated whenever , which are concluded in the following. (i)For , is bounded on if and only if

For , the weighted composition operator is compact on if and only if is bounded on ,

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

I would like express my gratitude to the support from Guangzhou College of Commerce.