Domain of Jordan Totient Type Matrix Operator in the Spaces of Convergent and Null Sequences

Year 2025, Volume: 11 Issue: 4, 395 - 411, 31.12.2025

Emrah Evren Kara , Hüseyin İdin , Merve İlkhan Kara

https://doi.org/10.28979/jarnas.1788558

Abstract

Motivated by $\Phi$-summability, novel matrix operators are produced by the aid of
fascinating arithmetical functions. One of them is the Riesz-Jordan totient matrix operator,
introduced very recently. The primary purpose of this study is to use this matrix to obtain
new spaces as the matrix domain of it in some classical sequence spaces. After presenting
basic topological properties and identifying dual spaces, the characterization of certain
matrix transformations is studied, defined on the resulting spaces. In the final section, the
compactness of the operators on the domain of Jordan totient type matrices in the space of
null sequences is researched.

Keywords

Jordan totient function , sequence space , matrix transformation , compact operator

Ethical Statement

No ethical approval was required for this study.

Supporting Institution

Not applicable.

References

• B. Altay, F. Başar, M. Mursaleen, On the Euler sequence spaces which include the spaces ℓᵖ and ℓ∞ I, Information Sciences 176 (10) (2006) 1450–1462.
• F. Başar, B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Mathematical Journal 55 (1) (2003) 136–147.
• T. Yaying, B. Hazarika, On sequence spaces defined by the domain of a regular Tribonacci matrix, Mathematica Slovaca 70 (3) (2020) 697–706.
• T. Yaying, B. Hazarika, M. Mursaleen, On sequence space derived by the domain of q-Cesàro matrix in ℓᵖ space and the associated operator ideal, Journal of Mathematical Analysis and Applications 493 (1) (2021) 124453.
• P. Zengin Alp, A new paranormed sequence space defined by Catalan conservative matrix, Mathematical Methods in the Applied Sciences 44 (9) (2021) 7651–7658.
• P. Zengin Alp, E. E. Kara, New Banach spaces defined by the domain of Riesz–Fibonacci matrix, Korean Journal of Mathematics 29 (4) (2021) 665–677.
• F. Başar, Summability theory and its applications, Bentham Science Publishers, İstanbul, 2012.
• E. Malkowsky, Recent results in the theory of matrix transformations in sequence spaces, Matematički Vesnik 49 (1997) 187–196.
• M. Mursaleen, A. K. Noman, On some new difference sequence spaces of non-absolute type, Mathematical and Computer Modelling 52 (3–4) (2010) 603–617.
• M. C. Dağlı, Matrix mappings and compact operators for Schröder sequence spaces, Turkish Journal of Mathematics 46 (6) (2022) 2304–2320.
• M. C. Dağlı, T. Yaying, Some results on matrix transformation and compactness for fibonomial sequence spaces, Acta Mathematica Scientia 89 (3–4) (2023) 593–609.
• F. Gökçe, Characterizations of matrix and compact operators on BK spaces, Universal Journal of Mathematics and Applications 6 (2) (2023) 76–85.
• M. Başarır, E. E. Kara, On some difference sequence spaces of weighted means and compact operators, Annals of Functional Analysis 2 (2011) 114–129.
• M. Başarır, E. E. Kara, On the B-difference sequence space derived by generalized weighted mean and compact operators, Journal of Mathematical Analysis and Applications 391 (2012) 67–81.
• M. Başarır, E. E. Kara, On compact operators on the Riesz Bm-difference sequence spaces II, Iranian Journal of Science and Technology, Transaction A 36 (2012) 371–376.
• E. E. Kara, M. Başarır, On compact operators and some Euler B(m)-difference sequence spaces, Journal of Mathematical Analysis and Applications 379 (2) (2011) 499–511.
• M. Mursaleen, A. K. Noman, The Hausdorff measure of noncompactness of matrix operators on some BK spaces, Operators and Matrices 5 (2011) 473–486.
• M. Mursaleen, H. Roopaei, Sequence spaces associated with fractional Copson matrix and compact operators, Results in Mathematics 76 (3) (2021) 134.
• B. Altay, F. Başar, Some paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bulletin of Mathematics 30 (4) (2006) 591–608.
• M. İlhan, E. E. Kara, A new Banach space defined by Euler totient matrix operator, Operators and Matrices 13 (2) (2019) 527–544.
• M. İlhan, Matrix domain of a regular matrix derived by Euler totient function in the spaces c₀ and c, Mediterranean Journal of Mathematics 17 (2020) 27.
• S. Demiriz, M. İlhan, E. E. Kara, Almost convergence and Euler totient matrix, Annals of Functional Analysis 11 (3) (2020) 604–616.
• M. İlhan, S. Demiriz, E. E. Kara, A new paranormed sequence space defined by Euler totient matrix, Karaelmas Science and Engineering Journal 9 (2) (2019) 277–282.
• S. Demiriz, S. Erdem, Domain of Euler–totient matrix operator in the space Lᵖ, Korean Journal of Mathematics 28 (2) (2020) 361–378.
• G. C. Hazar Güleç, M. İlhan, A new paranormed series space using Euler totient means and some matrix transformations, Korean Journal of Mathematics 28 (2) (2020) 205–221.
• M. İlhan, G. C. Hazar Güleç, A study on absolute Euler totient series space and certain matrix transformations, Muğla Journal of Science and Technology 6 (1) (2020) 112–119.
• G. C. Hazar Güleç, M. İlhan, A new characterization of absolute summability factors, Communications in Optimization Theory (2020) 15.
• S. Erdem, S. Demiriz, 4-dimensional Euler–totient matrix operator and some double sequence spaces, Mathematical Sciences and Applications E-Notes 8 (2) (2020) 110–122.
• S. Demiriz, S. Erdem, 4D first order q-Euler matrix operator and its domain in the space LS, Advances in Operator Theory 8 (2023) 68.
• I. Schoenberg, The integrability of certain functions and related summability methods, The American Mathematical Monthly 66 (1959) 361–375.
• M. İlhan, N. Şimşek, E. E. Kara, A new regular infinite matrix defined by Jordan totient function and its matrix domain in ℓᵖ, Mathematical Methods in the Applied Sciences 44 (9) (2021) 7622–7633.
• E. E. Kara, N. Şimşek, M. İlhan Kara, On new sequence spaces related to domain of the Jordan totient matrix, in: S. A. Mohiuddine, B. Hazarika (Eds.), Sequence Space Theory with Applications, Chapman and Hall/CRC, New York, 2022.
• S. Erdem, S. Demiriz, A new RH-regular matrix derived by Jordan’s function and its domains on some double sequence spaces, Journal of Function Spaces 2021 (2021), Article ID 5594751.
• S. Erdem, S. Demiriz, Some results related to new Jordan totient double sequence spaces, Turkish Journal of Mathematics and Computer Science 14 (2) (2022) 271–280.
• M. İlhan Kara, G. Örnek, Domain of Jordan totient matrix in the space of almost convergent sequences, Mathematical Sciences and Applications E-Notes 10 (4) (2022) 199–207.
• V. A. Khan, M. Et, M. Faisal, Some new neutrosophic normed sequence spaces defined by Jordan totient operator, Filomat 37 (26) (2023) 8953–8968.
• V. A. Khan, Z. Rahman, Kamal M. A. S. Alshlool, On I-convergent sequence spaces defined by Jordan totient operator, Filomat 35 (11) (2021) 3643–3652.
• V. A. Khan, T. Umme, Topological properties of Jordan intuitionistic fuzzy normed spaces, Mathematica Slovaca 73 (2) (2023) 439–454.
• M. İlhan Kara, M. A. Bayrakdar, A study on matrix domain of Riesz–Euler totient matrix in the space of p-absolutely summable sequences, Communications in Advanced Mathematical Sciences 4 (1) (2021) 14–25.
• U. Devletli, M. İlhan Kara, New Banach sequence spaces defined by Jordan totient function, Communications in Advanced Mathematical Sciences 6 (4) (2023) 211–225.
• A. M. Jarrah, E. Malkowsky, Ordinary, absolute and strong summability and matrix transformations, Filomat 17 (2003) 59–78.
• M. Stieglitz, H. Tietz, Matrix transformationen von Folgenräumen – eine Ergebnisübersicht, Mathematische Zeitschrift 154 (1977) 1–16.
• B. Altay, F. Başar, Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space, Journal of Mathematical Analysis and Applications 336 (2007) 632–645.
• E. Malkowsky, V. Rakočević, An introduction into the theory of sequence spaces and measure of noncompactness, Zbornik Radova 9 (17) (2000) 143–234.
• B. De Malafosse, The Banach algebra B(X), where X is a BK space and applications, Matematički Vesnik 57 (1–2) (2005) 41–60.
• V. Rakočević, Measures of noncompactness and some applications, Filomat 12 (2) (1998) 87–120.
• M. Mursaleen, A. K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Analysis 73 (8) (2010) 2541–2557.