Norm of Hilbert Operator’s Commutants

Round 1

Reviewer 1 Report

See attached.

Comments for author File: Comments.pdf

Author Response

Dear Professor

I am very grateful for your perfect comments that caused the improvement of the paper. I have revised my paper based on your suggestions and applied all of them through this study. 

I hope this new version can satisfy your expectations.

Best Regards

Hadi Roopaei  

Author Response File: Author Response.pdf

Reviewer 2 Report

1) Abstract in the current form is not enough. Must Revise.

 

2) Add the ``paper construction" paragraph in the last of introduction section.

 

3) Section 1 is not enough. More recent results/development should be presented to support this research. Some applications are also needed in Section 1. 

 

4) Add the section ``Preliminaries" after introduction section and mentioned all the published definitions or work which are used throughout the paper.

 

5) Highlight the aim and novelty of your work.

 

6) Why authors are doing this work? What is the hurdle encountered by this study? 

 

7) No clear motivation and reason to study such type of operator. so provide the advantages of Hilbert operator in the field of applied sciences.? 

 

8)The authors should also check for any grammatical errors throughout the paper.

 

9) Remove the typos errors throughout the paper. 

 

10) Add '' Conclusion'' section. Scope of the study and future directions are to be add in this Section. 

 

11) Remove the unnecessary equation numbers throughout the paper.

 

12) Add the word ''References'' in the line 219.

 

Author Response

Dear Professor

I have revised my paper many times and consider your precious comments through the latest version. I hope this new version covers all your suggestions.

Many thanks for your kind and helpful comments.

Best Regards

Hadi Roopaei

Author Response File: Author Response.pdf

Reviewer 3 Report

Hilbert type operators have numerous applications in approximation theory, functional analysis, representation theory, and noncommutative geometry. The study of the continuity properties of this operator in various spaces and the estimates of the norm of this operator are of considerable interest and has a long history.

The proposed article is a continuation of the author's early works, in which results on the factorization of the Hilbert operator in the form of a product of Cesaro-type operators are obtained, and operators commuting with the Hilbert operator are described in this language. In this paper, the author proves norm separating theorems for the commutants. This makes it possible to calculate the norms of commutants explicitly. The author has obtained very elegant formulas for the norms of commutants.

Author Response

Dear Professor

I am very thankful for the time to spend reading my paper. Thanks for your kind review.

Best Regards

Hadi Roopaei

Round 2

Reviewer 1 Report

Alterations successfully implemented.