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arxiv: 2606.22912 · v1 · pith:S7WSDQBZnew · submitted 2026-06-22 · 🧮 math.FA

Convolution Operators on Weighted Hahn Spaces

Sayan Saha , Arnab Patra This is my paper

Pith reviewed 2026-06-26 06:51 UTC · model grok-4.3

classification 🧮 math.FA
keywords convolution operatorsweighted Hahn spacesspectrumfine spectrumboundednesscompactnessmultiplier algebras
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The pith

Convolution operators on weighted Hahn sequence spaces have their spectra and fine spectra fully characterized.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines convolution operators on weighted Hahn sequence spaces. It determines the conditions under which these operators are bounded or compact and identifies the multiplier algebras of the space and its dual. The central result is a complete description of the spectrum and fine spectrum of the operators, supported by examples. Introducing weights produces new multiplier and spectral properties compared to the unweighted case.

Core claim

The paper obtains a complete characterization of the spectrum and fine spectrum of convolution operators on weighted Hahn sequence spaces, together with boundedness and compactness conditions and multiplier algebras of the space and its dual. The weighted framework leads to the emergence of new multiplier and spectral properties.

What carries the argument

Convolution operators defined on weighted Hahn sequence spaces, together with the spectral theory and multiplier algebra techniques applied to them.

If this is right

  • Explicit conditions determine when convolution operators are bounded or compact.
  • The multiplier algebras of the weighted Hahn space and its dual are identified.
  • The spectrum and fine spectrum receive complete descriptions with concrete examples.
  • The weighted setting produces new spectral and multiplier properties absent in the unweighted case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The characterizations may extend to related weighted sequence spaces such as Lorentz or Orlicz spaces.
  • The weighting parameter could be used to tune spectral properties for applications in approximation or signal processing.
  • Duality relations between the space and its dual multipliers might yield further structural results.

Load-bearing premise

The weighted Hahn spaces admit a well-defined convolution operation to which standard sequence space techniques apply directly.

What would settle it

An explicit convolution operator on a weighted Hahn space whose spectrum or fine spectrum falls outside the characterized sets would disprove the main claim.

read the original abstract

This paper studies the convolution operator on weighted Hahn sequence spaces. The boundedness and compactness of these operators, together with the multiplier algebras of the weighted Hahn space and its dual, are investigated. A complete characterization of the spectrum and fine spectrum is obtained, with illustrative examples. The introduction of the weighted framework leads to the emergence of new multiplier and spectral properties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript studies convolution operators on weighted Hahn sequence spaces. It derives conditions for boundedness and compactness of these operators, identifies the multiplier algebras of the space and its dual, and provides a complete characterization of the spectrum and fine spectrum, accompanied by illustrative examples. The weighted setting is claimed to produce new multiplier and spectral properties beyond the unweighted case.

Significance. If the characterizations are rigorously established, the work extends classical results on Hahn sequence spaces to the weighted case, supplying explicit spectral descriptions and multiplier algebras that may prove useful in operator theory on sequence spaces. The inclusion of examples aids in verifying the distinctions introduced by the weights.

minor comments (1)
  1. The abstract and introduction would benefit from an explicit statement of the norm on the weighted Hahn space and the precise definition of the convolution product used throughout the paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript on convolution operators on weighted Hahn sequence spaces. The provided summary accurately reflects the paper's scope, including boundedness, compactness, multipliers, and spectral characterizations in the weighted setting. No specific major comments were enumerated in the report, so we offer no point-by-point responses at this time. We stand ready to address any additional questions or clarifications the referee may have.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract and description indicate the paper applies standard sequence-space techniques to characterize boundedness, compactness, multipliers, spectrum, and fine spectrum of convolution operators on weighted Hahn spaces. No equations, definitions, or self-citations are supplied that reduce any claimed result to a fitted input, self-definition, or prior author work by construction. The derivation chain is presented as an extension of existing theory and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.1-grok · 5565 in / 928 out tokens · 24295 ms · 2026-06-26T06:51:45.108631+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

28 extracted references

  1. [1]

    Altay, B., Ba¸ sar, F.: On the fine spectrum of the generalized difference operator B(r, s) over the sequence spacesc 0 andc. Int. J. Math. Math. Sci.2005(18), 3005–3013 (2005)

    2005

  2. [2]

    Hokkaido Math

    Furkan, H., Bilgi¸ c, H., Kayaduman, K.: On the fine spectrum of the generalized difference operatorB(r, s) over the sequence spaceℓ 1 andbv. Hokkaido Math. J. 35(4), 893–904 (2006)

    2006

  3. [3]

    Nonlinear Anal.68(3), 499–506 (2008)

    Bilgi¸ c, H., Furkan, H.: On the fine spectrum of the generalized difference operator B(r, s) over the sequence spacesℓ p andbv p,(1< p <∞). Nonlinear Anal.68(3), 499–506 (2008)

    2008

  4. [4]

    Bilgi¸ c, H., Furkan, H.: On the fine spectrum of the operatorB(r, s, t) over the sequence spacesℓ 1 andbv. Math. Comput. Modelling45(7-8), 883–891 (2007)

    2007

  5. [5]

    Furkan, H., Bilgi¸ c, H., Ba¸ sar, F.: On the fine spectrum of the operatorB(r, s, t) 25 over the sequence spacesℓ p andbv p,(1< p <∞). Comput. Math. Appl.60(7), 2141–2152 (2010)

    2010

  6. [6]

    Furkan, H., Bilgi¸ c, H., Altay, B.: On the fine spectrum of the operatorB(r, s, t) overc 0 andc. Comput. Math. Appl.53(6), 989–998 (2007)

    2007

  7. [7]

    Complex Anal

    Birbonshi, R., Srivastava, P.: On some study of the fine spectra ofn-th band triangular matrices. Complex Anal. Oper. Theory11(4), 739–753 (2017)

    2017

  8. [8]

    Complex Anal

    Patra, A., Birbonshi, R., Srivastava, P.: On some study of the fine spectra of triangular band matrices. Complex Anal. Oper. Theory13(3), 615–635 (2019)

    2019

  9. [9]

    Ricker, W.J.: Convolution operators in discrete Ces` aro spaces. Arch. Math. 112(1), 71–82 (2019)

    2019

  10. [10]

    Curbera, G.P., Ricker, W.J.: Convolution in dual Ces` aro sequence spaces. J. Math. Anal. Appl.519(2), 126838 (2023)

    2023

  11. [11]

    RAC- SAM Rev

    Ricker, W.J.: Convolution operators in the Fr´ echet sequence spaceω=C N. RAC- SAM Rev. R. Acad. Cienc. Exactas F` ıs. Nat. Ser. A Mat.113(4), 3069–3088 (2019)

    2019

  12. [12]

    Nikol’skii, N.K.: Spaces and algebras of toeplitz matrices operating inℓ p. Sib. Math. J.7(1), 118–126 (1966)

    1966

  13. [13]

    Monatshefte f¨ ur Mathematik und Physik32(1), 3–88 (1922)

    Hahn, H.: ¨Uber folgen linearer operationen. Monatshefte f¨ ur Mathematik und Physik32(1), 3–88 (1922)

    1922

  14. [14]

    Journal of Mathematics2022(1), 6343084 (2022)

    Tu˘ g, O.: The Generalized Difference Operator ∆3 i of Order Three and Its Domain in the Sequence Spacesℓ 1 andbv. Journal of Mathematics2022(1), 6343084 (2022)

    2022

  15. [15]

    Linear Multilinear Algebra70(22), 7433–7451 (2022)

    Tuˇ g, O., Rakoˇ cevi´ c, V., Malkowsky, E.: Domain of generalized difference operator ∆3 i of order three on the hahn sequence spacehand matrix transformations. Linear Multilinear Algebra70(22), 7433–7451 (2022)

    2022

  16. [16]

    Malkowsky, E., Milovanovi´ c, G.V., Rakoˇ cevi´ c, V., Tu˘ g, O.: The roots of polyno- mials and the operator ∆ 3 i on the Hahn sequence spaceh. Comput. Appl. Math. 40(6), 222 (2021)

    2021

  17. [17]

    Malkowsky, E., Rakoˇ cevi´ c, V., Tu˘ g, O.: Compact operators on the Hahn space. Monatsh. Math.196(3), 519–551 (2021)

    2021

  18. [18]

    John Wiley & Sons, New Jersey (1988)

    Dunford, N., Schwartz, J.T.: Linear Operators, Part I: General Theory. John Wiley & Sons, New Jersey (1988)

    1988

  19. [19]

    Wilansky, A.: Summability Through Functional Analysis vol. 85. North-Holland, Amsterdam, New York, Oxford (2000) 26

    2000

  20. [20]

    Rao, K.C.: The Hahn sequence space. Bull. Calcutta Math. Soc82, 72–78 (1990)

    1990

  21. [21]

    Goes, G.: Sequences of bounded variation and sequences of Fourier coefficients. II. J. Math. Anal. Appl.39(2), 477–494 (1972)

    1972

  22. [22]

    CRC Press, Taylor & Francis Group, Boca Raton (2019)

    Malkowsky, E., Rakoˇ cevi´ c, V.: Advanced Functional Analysis. CRC Press, Taylor & Francis Group, Boca Raton (2019)

    2019

  23. [23]

    Toledano, J.M.A., Benavides, T.D., Acedo, G.L.: Measures of Noncompactness in Metric Fixed Point Theory vol. 99. Birkh¨ auser Verlag, Basel, Boston, Berlin (1997)

    1997

  24. [24]

    Bana´ s, J.: On measures of noncompactness in Banach spaces. Comment. Math. Univ. Carolin.21(1), 131–143 (1980)

    1980

  25. [25]

    In: Advances in Nonlinear Analysis Via the Concept of Measure of Noncompactness, pp

    Malkowsky, E., Rakoˇ cevi´ c, V.: On some results using measures of noncom- pactness. In: Advances in Nonlinear Analysis Via the Concept of Measure of Noncompactness, pp. 127–180. Springer, Singapore (2017)

    2017

  26. [26]

    Malkowsky, E., Rakoˇ cevi´ c, V.: An introduction into the theory of sequence spaces and measures of noncompactness. Zb. Rad. (Beogr.)9(17), 143–234 (2000)

    2000

  27. [27]

    Nonlinear Anal.73(8), 2541–2557 (2010)

    Mursaleen, M., Noman, A.K.: Compactness by the Hausdorff measure of noncom- pactness. Nonlinear Anal.73(8), 2541–2557 (2010)

    2010

  28. [28]

    McGrawHill, New York (1966) 27

    Goldberg, S.: Unbounded Linear Operators: Theory and Applications. McGrawHill, New York (1966) 27

    1966