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.