On the Essential Decreasing of the Summation Order in the Abel-Lidskii Sense

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please find the attached file.

Comments for author File: Comments.pdf

Author Response

Dear referee, I highly appreciate your attention and very grateful to you for the remarks which allow me to see the matter from another point of view and in this way to  improve the paper significantly. I have made the changes corresponding to the remarks 2-5, as for the remark 1, i.e.  references  reorganizing, if you do not mind I leave it for the  proofreading process since some additional remarks in this regard  from the editorial board may appear.   I warmly thank you for the hard work you have done and believe that your positive opinion brings  me  good  in  further investigation!

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper presents interesting results on the Abel-Lidskii summation and its applications to evolution equations. However, there are several areas where the paper could be improved,

1. Page 19 Lemma 7 the proof of Lemma 7 involves asymptotic estimates for the counting function nk(r)nk​(r). The transition from (19) to (20) seems to rely on the assumption that Δj≤CϵNνϵΔj​≤Cϵ​Nνϵ​, but this assumption is not explicitly justified. The author should provide more details on how this inequality is derived, especially since it plays a crucial role in the proof.
2. Page 22 Lemma 9 the proof of Lemma 9 involves the estimation of the Fredholm determinant Δ(λ)Δ(λ). The transition from (22) to the final estimate seems to rely on the assumption that the resolvent norm ∥(I−λB)−1∥∥(I−λB)−1∥can be bounded by the product ∏n=1∞(1+∣λμn(B)∣)∏n=1∞​(1+∣λμn​(B)∣). However, this step is not fully justified, and the author should provide more details on how this bound is derived.
3. Page 25  Theorem 1 the proof of Theorem 1 involves splitting the operator BBinto a sequence of operators BkBk​ and estimating their resolvents. The transition from (24) to (25) relies on the assumption that hk(r)=o(rs)hk​(r)=o(rs), but this is not explicitly proven. The author should provide more details on how this estimate is obtained. 
4. The introduction provides a historical review of the problem, but it could benefit from a clearer statement of the main results and their significance. The author should explicitly state what new contributions this paper makes compared to previous work.

Author Response

Dear referee, I am sincerely grateful to you for the brilliant remarks. However, let us consider them consistently. See the atteched file.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

In this paper, the authors investigate the problem of decreasing the summation order in the Abel-Lidskii sense, a problem with a significant history dating back to 1962. As the main result, the authors highlight that the summation order can be reduced from values greater than the convergence exponent, in accordance with Lidskii’s result, to an arbitrarily small positive number. Moreover, the paper contains many important findings.

The introduction provides a detailed historical overview of the problem, followed by preliminary results essential for understanding the main contributions. The key results are presented in Section 3.

I would only suggest clarifying the notation for the real and imaginary parts, which are first introduced on page 3 and then used throughout the text. There are two variants: \text{Re} and \mathcal{Re}, as well as \text{Im} and \mathcal{Im}. Consistency in notation would improve readability.

Given the extensive research presented, I find the paper to be well-written, precise, and of high quality. The only minor issue I noticed is that Eqs. (29) and (33) are numbered but not referenced in the text, so their numbering should be removed.

Author Response

Dear referee, I highly appreciate your attention and sincerely   grateful  to you for the invaluable support! indeed while writing this paper for three years I lost my mind and   it is a grate pleasure for me to realize that somebody can appreciate the result.

Author Response File: Author Response.pdf