Home New sequence spaces derived by using generalized arithmetic divisor sum function and compact operators
Article
Licensed
Unlicensed Requires Authentication

New sequence spaces derived by using generalized arithmetic divisor sum function and compact operators

  • Taja Yaying , Nipen Saikia and Mohammad Mursaleen EMAIL logo
Published/Copyright: March 5, 2024
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 1

Abstract

Define an infinite matrix D α = ( d n , v α ) by

d n , v α = { v α σ ( α ) ⁢ ( n ) , v ∣ n , 0 , v ∤ n ,

where σ ( α ) ⁢ ( n ) is defined to be the sum of the 𝛼-th power of the positive divisors of n ∈ N , and construct the matrix domains ℓ p ⁢ ( D α ) ( 0 < p < ∞ ), c 0 ⁢ ( D α ) , c ⁢ ( D α ) and ℓ ∞ ⁢ ( D α ) defined by the matrix D α . We develop Schauder bases and determine 𝛼-, 𝛽- and 𝛾-duals of these new spaces. We characterize some matrix transformation from ℓ p ⁢ ( D α ) , c 0 ⁢ ( D α ) , c ⁢ ( D α ) and ℓ ∞ ⁢ ( D α ) to ℓ ∞ , 𝑐, c 0 and ℓ 1 . Furthermore, we determine some criteria for compactness of an operator (or matrix) from X ∈ { ℓ p ⁢ ( D α ) , c 0 ⁢ ( D α ) , c ⁢ ( D α ) , ℓ ∞ ⁢ ( D α ) } to ℓ ∞ , 𝑐, c 0 or ℓ 1 .

Keywords: Matrix domains; Arithmetic divisor sum function; Schauder basis; 𝛼-; 𝛽-; 𝛾-duals; matrix transformation; compactness
MSC 2020: 46A45; 46B45; 40C05; 47B07; 47B37

Funding source: Science and Engineering Research Board

Award Identifier / Grant number: EEQ/2019/000082

Funding statement: The work of Dr. Taja Yaying is funded by SERB, New Delhi, India, under the grant EEQ/2019/000082.

  1. Communicated by: Freydoon Shahidi

References

[1] T. M. Apostol, Introduction to Analytic Number Theory, Undergrad. Texts Math., Springer, New York, 1976. 10.1007/978-1-4757-5579-4Search in Google Scholar

[2] M. Ayman Mursaleen, A note on matrix domains of Copson matrix of order 𝛼 and compact operators, Asian-Eur. J. Math. 15 (2022), no. 7, Article ID 2250140. 10.1142/S1793557122501406Search in Google Scholar

[3] F. Başar, Summability Theory and its Applications, Bentham Science, Oak Park, 2012. 10.2174/97816080545231120101Search in Google Scholar

[4] F. Başar and B. Altay, On the space of sequences of 𝑝-bounded variation and related matrix mappings, Ukraïn. Mat. Zh. 55 (2003), no. 1, 108–118. Search in Google Scholar

[5] F. Başar and Y. Sever, The space L q of double sequences, Math. J. Okayama Univ. 51 (2009), 149–157. Search in Google Scholar

[6] S. Demiriz and S. Erdem, Domain of Euler-totient matrix operator in the space L p , Korean J. Math. 28 (2020), no. 2, 361–378. Search in Google Scholar

[7] S. Demiriz, M. İlkhan and E. E. Kara, Almost convergence and Euler totient matrix, Ann. Funct. Anal. 11 (2020), no. 3, 604–616. 10.1007/s43034-019-00041-0Search in Google Scholar

[8] S. Demiriz and A. Şahin, 𝑞-Cesàro sequence spaces derived by 𝑞-analogue, Adv. Math. 5 (2016), no. 2, 97–110. Search in Google Scholar

[9] S. Erdem and S. Demiriz, A new RH-regular matrix derived by Jordan’s function and its domains on some double sequence spaces, J. Funct. Spaces 2021 (2021), Article ID 5594751. 10.1155/2021/5594751Search in Google Scholar

[10] K.-G. Grosse-Erdmann, Matrix transformations between the sequence spaces of Maddox, J. Math. Anal. Appl. 180 (1993), no. 1, 223–238. 10.1006/jmaa.1993.1398Search in Google Scholar

[11] M. İlkhan, Matrix domain of a regular matrix derived by Euler totient function in the spaces c 0 and 𝑐, Mediterr. J. Math. 17 (2020), no. 1, Paper No. 27. 10.1007/s00009-019-1442-7Search in Google Scholar

[12] M. İlkhan and E. E. Kara, A new Banach space defined by Euler totient matrix operator, Oper. Matrices 13 (2019), no. 2, 527–544. 10.7153/oam-2019-13-40Search in Google Scholar

[13] M. İlkhan, E. E. Kara and F. Usta, Compact operators on the Jordan totient sequence spaces, Math. Methods Appl. Sci. 44 (2021), no. 9, 7666–7675. 10.1002/mma.6537Search in Google Scholar

[14] M. İlkhan, N. Şimşek and E. E. Kara, A new regular infinite matrix defined by Jordan totient function and its matrix domain in ℓ p , Math. Methods Appl. Sci. 44 (2021), no. 9, 7622–7633. 10.1002/mma.6501Search in Google Scholar

[15] V. A. Khan and U. Tuba, On paranormed ideal convergent sequence spaces defined by Jordan totient function, J. Inequal. Appl. 2021 (2021), Paper No. 96. 10.1186/s13660-021-02634-7Search in Google Scholar

[16] C. G. Lascarides and I. J. Maddox, Matrix transformations between some classes of sequences, Proc. Cambridge Philos. Soc. 68 (1970), 99–104. 10.1017/S0305004100001109Search in Google Scholar

[17] E. Malkowsky and V. Rakočević, An introduction into the theory of sequence spaces and measures of noncompactness, Zb. Rad. (Beogr.) 9(17) (2000), 143–234. Search in Google Scholar

[18] M. Mursaleen and F. Başar, Sequence Spaces: Topics in Modern Summability Theory, Math. Appl., CRC Press, New York, 2020. 10.1201/9781003015116Search in Google Scholar

[19] M. Mursaleen and A. K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal. 73 (2010), no. 8, 2541–2557. 10.1016/j.na.2010.06.030Search in Google Scholar

[20] M. Mursaleen and A. K. Noman, The Hausdorff measure of noncompactness of matrix operators on some B ⁢ K spaces, Oper. Matrices 5 (2011), no. 3, 473–486. 10.7153/oam-05-35Search in Google Scholar

[21] I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, John Wiley & Sons, New York, 1991. Search in Google Scholar

[22] W. H. Ruckle, Arithmetical summability, J. Math. Anal. Appl. 396 (2012), no. 2, 741–748. 10.1016/j.jmaa.2012.06.048Search in Google Scholar

[23] M. Stieglitz and H. Tietz, Matrixtransformationen von Folgenräumen. Eine Ergebnisübersicht, Math. Z. 154 (1977), no. 1, 1–16. 10.1007/BF01215107Search in Google Scholar

[24] A. Wilansky, Summability Through Functional Analysis, North-Holland Math. Stud. 85, North-Holland, Amsterdam, 1984. Search in Google Scholar

[25] T. Yaying and B. Hazarika, On arithmetical summability and multiplier sequences, Nat. Acad. Sci. Lett. 40 (2017), no. 1, 43–46. 10.1007/s40009-016-0525-2Search in Google Scholar

[26] T. Yaying and B. Hazarika, Lacunary arithmetic statistical convergence, Nat. Acad. Sci. Lett. 43 (2020), no. 6, 547–551. 10.1007/s40009-020-00910-6Search in Google Scholar

[27] T. Yaying and N. Saikia, On sequence spaces defined by arithmetic function and Hausdorff measure of noncompactness, Rocky Mountain J. Math. 52 (2022), no. 5, 1867–1885. 10.1216/rmj.2022.52.1867Search in Google Scholar
Received: 2023-04-15
Revised: 2023-12-30
Published Online: 2024-03-05
Published in Print: 2025-01-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Some properties of extended frame measure
  3. Orthogonal separation of variables for spaces of constant curvature
  4. Fundamental properties of Cauchy–Szegő projection on quaternionic Siegel upper half space and applications
  5. Uniform bounds for Kloosterman sums of half-integral weight with applications
  6. q-supercongruences from Watson's 8φ7 transformation
  7. Submodules of normalisers in groupoid C*-algebras and discrete group coactions
  8. Explicit bounds for the solutions of superelliptic equations over number fields
  9. The existence of optimal solutions for nonlocal partial systems involving fractional Laplace operator with arbitrary growth
  10. A Kollár-type vanishing theorem for k-positive vector bundles
  11. New sequence spaces derived by using generalized arithmetic divisor sum function and compact operators
  12. On the Iwasawa main conjecture for generalized Heegner classes in a quaternionic setting
  13. A p-adic analog of Hasse--Davenport product relation involving ϵ-factors
  14. On arithmetic quotients of the group SL2 over a quaternion division k-algebra
  15. Euler’s integral, multiple cosine function and zeta values
  16. Paley inequality for the Weyl transform and its applications
  17. Homogeneous ACM and Ulrich bundles on rational homogeneous spaces
  18. Arithmetic progression in a finite field with prescribed norms
https://doi.org/10.1515/forum-2023-0138
Keywords for this article
Matrix domains; Arithmetic divisor sum function; Schauder basis; 𝛼-; 𝛽-; 𝛾-duals; matrix transformation; compactness

Articles in the same Issue

  1. Frontmatter
  2. Some properties of extended frame measure
  3. Orthogonal separation of variables for spaces of constant curvature
  4. Fundamental properties of Cauchy–Szegő projection on quaternionic Siegel upper half space and applications
  5. Uniform bounds for Kloosterman sums of half-integral weight with applications
  6. q-supercongruences from Watson's 8φ7 transformation
  7. Submodules of normalisers in groupoid C*-algebras and discrete group coactions
  8. Explicit bounds for the solutions of superelliptic equations over number fields
  9. The existence of optimal solutions for nonlocal partial systems involving fractional Laplace operator with arbitrary growth
  10. A Kollár-type vanishing theorem for k-positive vector bundles
  11. New sequence spaces derived by using generalized arithmetic divisor sum function and compact operators
  12. On the Iwasawa main conjecture for generalized Heegner classes in a quaternionic setting
  13. A p-adic analog of Hasse--Davenport product relation involving ϵ-factors
  14. On arithmetic quotients of the group SL2 over a quaternion division k-algebra
  15. Euler’s integral, multiple cosine function and zeta values
  16. Paley inequality for the Weyl transform and its applications
  17. Homogeneous ACM and Ulrich bundles on rational homogeneous spaces
  18. Arithmetic progression in a finite field with prescribed norms
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 20.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2023-0138/html
Scroll to top button