Orlicz Spaces and Their Hyperbolic Composition Operators

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The article is well written. There is no comments for authors.

Comments for author File: Comments.pdf

Author Response

The paper under review is devoted to composition operators on Orlicz spaces. As we know,
the Orlicz spaces are a generalization of L
p
-spaces. There are a numerous function spaces
between L
p
-spaces and Orlicz spaces. In reference [9] hyperbolicity and shadowing property
of composition operators on L
p
-spaces are studied. In the present paper, the authors have
written the definitions 2.6.3 and 2.6.4 of [9] for Orlicz spaces and they did it very well. In the
sequel they obtained and proved the Proposition 2.7 in Orlicz spaces, that is a generalization
of proposition 2.6.5 [9] and it is the main tool in this generalization. Then they have provided
some equivalent conditions for composition operators to have the shadowing property on the
Orlicz space L
Φ(µ). Additionally, they have obtained that under some conditions for composition operators on Orlicz spaces, the notions of generalized hyperbolicity and the shadowing
property coincide. The results of this paper is a real generalization of the results of [9] to
Orlicz spaces.
As far as I can see the paper is well written and the theorems and their proofs are mathematically correct. Based on the above observations I recommend it for publication.
Regards.

 

Response to reviewers: Thank you for your deep reading the paper.

I have changed the address of some authors.

  

Reviewer 2 Report

Comments and Suggestions for Authors

The authors of this interesting and well-written paper provide equivalent conditions for composition operators to have the shadowing property on certain Orlicz spaces. In addition, they also show that for composition operators on Orlicz spaces, the important concepts of generalized hyperbolicity and the shadowing property are equivalent. Since composition operators have been and continue to be of considerable research interest, my recommendation is that this paper be accepted for publication in the journal "Mathematics".

Author Response

The authors of this interesting and well-written paper provide equivalent conditions for composition operators to have the shadowing property on certain Orlicz spaces. In addition, they also show that for composition operators on Orlicz spaces, the important concepts of generalized hyperbolicity and the shadowing property are equivalent. Since composition operators have been and continue to be of considerable research interest, my recommendation is that this paper be accepted for publication in the journal "Mathematics".

 

Respanse to reviwers: thank you for your comments.

Reviewer 3 Report

Comments and Suggestions for Authors

Please see the attachment.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

The english should be revised.

Author Response

This article explores the dynamic properties of composition operators on Orlicz
spaces, with a particular focus on the relationship between generalized hyperbolicity
and the shadowing property, and proposes their equivalence conditions. The authors
extend existing results from Lp spaces to Orlicz spaces, which are more widely applied in mathematics and finance, and prove that generalized hyperbolicity and the
shadowing property are equivalent in these spaces. Through rigorous mathematical
derivations, the article provides a new theoretical framework for linear dynamical
systems in Orlicz spaces, deepening the understanding of the dynamic behavior of
composition operators in these spaces, which is of significant theoretical importance.
This article excels in theoretical innovation, application importance, mathematical rigor, and academic contribution. Therefore, the article should be accepted.
However, there are some aspects and details that need further improvement. The
following are suggestions for addressing these issues.
1. The abstract could more clearly convey the core contributions of the article. It
is recommended to explicitly highlight the specific innovations in the abstract,
such as detailing how the results from Lp spaces were successfully extended to
Orlicz spaces and discussing the specific significance or potential applications of
these extensions in mathematical or applied contexts. By further emphasizing
these key contributions, the abstract will better help readers quickly grasp the
academic value of the article.
2. It is recommended that the introduction more clearly articulate the specific
problem or challenge that the research aims to address. By explicitly stating
1
these aspects, the introduction will help readers better understand the motivation and objectives of the study.
3. It would be beneficial to right-align the equation numbering, in line with the
formatting used in other articles, to better distinguish the main text from the
equation numbers. This adjustment will enhance the readability and presentation of the formulas in the manuscript.
4. In the 8th line of the main text on page 6, there is an extra comma after ’and’.
It is suggested to remove this comma for correct punctuation.
5. Some formulas and the text following them are missing appropriate punctuation
marks. For instance, a period is recommended after the last formula on page
7, and the conclusion (2) of Proposition 2.15 on page 8 also requires a period.
It is suggested to review the entire manuscript for similar issues and ensure
consistent punctuation throughout.
6. Please confirm whether there is a writing error in the first equation on page 9,
as it appears that ’H
N(Φ)(C
t+s
φ (χW))
N(Φ)(Cs
φ(χW)
)
’ should be replaced with ’H
N(Φ)(C
t+s
φ (χW))
N(Φ)(Cs
φ(χW)) ’.
Kindly review and make the necessary correction if applicable.
7. Please verify if there is a need to change ’H
N(Φ)(C
−(k+j)
φ )(χW)
N(Φ)(C
−(k+j+n)
φ(χW)
)
’ to ’H
N(Φ)(C
−(k+j)
φ )(χW)
N(Φ)(C
−(k+j+n)
φ (χW))
’
in the second equation on page 9. It is recommended to review the entire
manuscript for similar issues.
8. The punctuation usage is incorrect, such as in the second and third formulas
on page 9 where periods should be used instead of other punctuation marks.
It is recommended to review the entire manuscript for similar issues to ensure
consistency.
9. Please check the use of ’χw’ in several places on pages 7 and 10, particularly
in Proposition 2.13. However, in subsequent text, this expression is written
as ’χW’. It is recommended to ensure consistency throughout these instances
and consider standardizing the format.
10. In the proof of Theorem 2.25, reference is made to ’Theroem RN’. It is suggested
to include the article containing ’Theroem RN’ as reference [21]. This reference
can then be cited as [6, 21].
11. It would be beneficial to add a conclusion section to summarize the main
findings and methodologies of the study. This will enhance the completeness
of the article and aid readers’ understanding. 

Response to reviewer:  First of all I would like to thank the reviewers for reading the paper carefully and giving useful comments. We have edited the paper as the reviewers asked as follows:

1. P-1, I have added some more details to abstract as the reviewer asked.
2.  P-1,2, I have added some more information to the introduction to convenience readers as the reviewer asked.
3.  p-6, I have removed the comma after 'and' in page 6 as the reviewer asked.
4.  p-7, I have added a dot to the end of last equation in the proof of Proposition 4 and also I have added a space between two last lines in the proof of Proposition 4, as the reviewer asked.
5.  P-8, I have added a bracket in Proposition 6 item two around f_1+f_2 .
6.  P-9, I have made change in the formulas in the first and second inequalities of the proof of the Proposition 7, as the reviewer asked.
7.  P-7, I have corrected the characteristic function in Proposition 4 as the reviewer asked. 
8.  P-14, I have corrected the sentences in the proof of  Theorem 3 as the reviewer asked.