Natural Lacunae Method and Schatten–Von Neumann Classes of the Convergence Exponent
Round 1
Reviewer 1 Report
Report: Natural lacunae method and Schatten-von Neumann classes of the convergence exponent
In this article some families of non-self-adjoint operators with compact resolvents are studied. The author obtains several asymptotic formulas for the eigenvalues and applies them to the existence of solutions of a particular Cauchy Problem.
The article needs a complete proofread in english and the mathematical presentation must be revised. More clear Definitions are needed overall and, in particular, the operator W that appears many times should have a clear definition and all the results involving it, or at least the main Theorems, should point at it for easy reference.
In general I am in favour of proposing this article for publication but the presentation must be throughly and carefully revised. I do not doubt of the results but their correctness is the entire responsibility of the author.
Typos: This list is longer, but I stopped noting them as there were too many.
- First sentence of the Introduction. ...fact that the _eigenvalues_ of the compact selfadjoint operator form... Should be eigenfunctions.
- Introduction line 9: _they_ considered
- Preliminaries Line 9: def T, nul T should be def L and nul L respectively.
- Lemma 2: if at _list_ one: Should be least
- After equation 5: Stilted -> Stieltjes. Several ocurrences in the text.
Author Response
Dear referee, I am sincerely grateful to you for a number of valuable remarks. However let us consider them carefully and consistently.\\
Referee.\\
In this article some families of non-self-adjoint operators with compact resolvents are studied. The author obtains several asymptotic formulas for the eigenvalues and applies them to the existence of solutions of a particular Cauchy Problem.
The article needs a complete proofread in english and the mathematical presentation must be revised. More clear Definitions are needed overall and, in particular, the operator $W$ that appears many times should have a clear definition and all the results involving it, or at least the main Theorems, should point at it for easy reference.\\
Dear referee, thank you for the remark, I italicized the places in the text where the operator $W$ is defined and the related theorems are given. Moreover, I have add more references providing the concrete examples of the operator $W.$ I have also made a more precise clarification of the auxiliary results devoted to the decomposition on the root vector system. Thus, I hope that it is sufficient to satisfy your remark.\\
Referee.\\
In general I am in favour of proposing this article for publication but the presentation must be throughly and carefully revised. I do not doubt of the results but their correctness is the entire responsibility of the author.\\
Dear referee, thank you very much for the comment and the remark. Frankly speaking the paper results were discussed in the framework of the well-known seminars, I care a great deal on the results correctness and not only.
As for thorough revision, generally, I think it is useful at any rate and there is no case when we could claim that the paper has been complete. More precisely, I have considered more than three variants of the results presentation and sure that the current one is sufficiently convenient for the reader. At the same time, I perceived with gratitude your remarks regarding the necessity of clarification of some notions.\\
Referee.\\
Typos: This list is longer, but I stopped noting them as there were too many.
- First sentence of the Introduction. ...fact that the "eigenvalues" of the compact selfadjoint operator form... Should be eigenfunctions.\\
Dear referee, thank you very much for the valuable finding, frankly speaking such things happen, I mean constant missing the places with terrible misprints while proofreading again and again . I have replaced it with "eigenvectors" for this terminology is definitely being used in terms of the abstract Hilbert space.\\
Referee.\\
- Introduction line 9: "they" considered\\
Dear referee, frankly speaking, I cannot see here any mistake or misprint, for the extract ", for instance in the mentioned papers there considered a domain of the parabolic type containing the spectrum of the operator." contains the grammar expression "there considered" that is quite correct (be so kind produce your arguments if you do not agree). At the same time, I honor you wish and tried to reformulate the sentence on the whole to avoid the controversial question. Let it be "A reasonable question that appears is about minimal conditions that guaranty the desired result, for instance in the mentioned papers the authors considered a domain of the parabolic type containing the spectrum of the operator."\\
Referee.\\
- Preliminaries Line 9: def T, nul T should be def L and nul L respectively.\\
Dear referee, thank you very much for the remark, I have made the necessary changes.\\
Referee.\\
- Lemma 2: if at "list" one: Should be least\\
Dear referee, thank you very much for the remark, I have made the necessary changes.\\
Referee.\\
- After equation 5: "Stilted" -- should be Stieltjes. Several occurrences in the text.\\
Dear referee, thank you for the remarks,
my writing has undergone a careful revision.
I have made the necessary changes.\\
Author Response File: 
Author Response.pdf
Reviewer 2 Report
In the paper, the author investigate the decomposition on the root vector system of the non-selfadjoint operator and introduce so-called Shatten-von Neumann class of the convergence exponent. The obtained results clarify and improved the results given by Lidskii.
The paper is well written, all statements are clear, the list of references is sufficient.
In my opinion, the paper contains new interesting results, and I'm recommending it for publication in the journal Mathematics.
Author Response
Referee.\\
In the paper, the author investigate the decomposition on the root vector system of the non-selfadjoint operator and introduce so-called Shatten-von Neumann class of the convergence exponent. The obtained results clarify and improved the results given by Lidskii.
The paper is well written, all statements are clear, the list of references is sufficient.
In my opinion, the paper contains new interesting results, and I'm recommending it for publication in the journal Mathematics.\\
Dear referee, I highly appreciate your credibility and support, they inspired me to improve the paper significantly and work in the direction. I sincerely grateful to you for your fair opinion and I am ready for any kind of collaboration!
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Best wishes!
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Ph.D Maksim V. Kukushkin
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The changes made by the author have improved the presentation.
English proofreading is still necessary and in particular there are new typos in the newly added paragraphs.
Author Response
Dear referee, thank you very much for your critical contribution to the paper, I have tried to proofread the paper one more time. If you are so kind produce please a short list of typos. Thank you very much for your attantion.
