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Compact operators on the Motzkin sequence space $c_0(\mathcal{M})$

Yıl 2024, Cilt: 13 Sayı: 2, 109 - 118, 31.08.2024
https://doi.org/10.54187/jnrs.1517251

Öz

The concept of non-compactness measure is extremely beneficial for functional analysis in theories, such as fixed point and operator equations. Apart from these, the Hausdorff measure of non-compactness also has some applications in the theory of sequence spaces which is an interesting topic of functional analysis. One of these applications is to obtain necessary and sufficient conditions for the matrix operators between Banach coordinate (BK) spaces to be compact. In line with these explanations, in this study, the necessary and sufficient conditions for a matrix operator to be compact from the Motzkin sequence space $c_0(\mathcal{M})$ to the sequence space $\mu\in\{\ell_{\infty},c,c_0,\ell_1\}$ are presented by using Hausdorff measure of non-compactness.

Etik Beyan

No approval from the Board of Ethics is required.

Kaynakça

  • B. Altay, F. Başar, On some Euler sequence spaces of nonabsolute type, Ukrainian Mathematical Journal 57 (1) (2005) 1-17.
  • F. Başar, M. Kirişci, Almost convergence and generalized difference matrix, Computers & Mathematics with Applications 61 (3) (2011) 602-611.
  • T. Yaying, B. Hazarika, M. İlkhan, M.Mursaleen, Poisson-like matrix operator and its application in $p-$summable space, Mathematica Slovaca 71 (5) (2021) 1189-1210.
  • G. C. Hazar, M. A. Sarıgöl, Absolute Cesaro series spaces and matrix operators, Acta Applicandae Mathematicae 154 (1) (2018) 153-165.
  • M. Candan, Domain of the double sequential band matrix in the classical sequence spaces, Journal of Inequalities and Applications 2012 (2012) Article Number 281 15 pages.
  • S. Demiriz, A. Şahin, $q$-Cesaro sequence spaces derived by $q$-analogue, Advances in Mathematics 5 (2) (2016) 97-110.
  • S. Erdem, S. Demiriz, On the new generalized block difference sequence space, Applications and Applied Mathematics: An International Journal 14 (5) (2019) 68-83.
  • H. B. Ellidokuzoğlu, S. Demiriz, $[\ell_p]_{e.r}$ Euler-Riesz difference sequence spaces, Analysis in Theory and Applications 37 (4) (2021) 557-571.
  • M. İlkhan, P. Z. Alp, E. E. Kara, On the spaces of linear operators acting between asymmetric cone normed spaces, Mediterranean Journal of Mathematics 15 (2018) Article Number 136 12 pages.
  • M. Stieglitz, H. Tietz, Matrix transformationen von folgenraumen eine ergebnisbersicht, Mathematische Zeitschrift 154 (1977) 1-16.
  • F. Başar, Summability theory and its applications, 2nd Edition, CRC Press, Taylor Francis Group, Boca Raton-London-New York, 2022.
  • J. Boos, Classical and modern methods in summability, Oxford Science Publications, Oxford University Press, 2000.
  • M. Mursaleen, F. Başar, Sequence spaces: Topic in modern summability theory, CRC Press, Taylor Francis Group, Series: Mathematics and its applications, Boca Raton, London, New York, 2020.
  • E. E. Kara, M. Başarır, An application of Fibonacci numbers into infinite Toeplitz matrices, Caspian Journal of Mathematical Sciences 1 (1) (2012) 43-47.
  • M. C. Dağlı, A novel conservative matrix arising from Schröder numbers and its properties, Linear and Multilinear Algebra 71 (8) (2023) 1338-1351.
  • M. C. Dağlı, Matrix mappings and compact operators for Schröder sequence spaces, Turkish Journal of Mathematics 46 (6) (2022) 2304-2320.
  • M. Karakaş, M. C. Dağlı, Some topologic and geometric properties of new Catalan sequence spaces, Advances in Operator Theory 8 (2023) Article Number 14 15 pages.
  • E. E. Kara, Some topological and geometrical properties of new Banach sequence spaces, Journal of Inequalities and Applications 2013 (2013) Article Number 38 15 pages.
  • M. Karakaş, H. Karabudak, An application on the Lucas numbers and infinite Toeplitz matrices, Cumhuriyet Science Journal 38 (3) (2017) 557-562.
  • M. Karakaş, A. M. Karakaş, New Banach sequence spaces that is defined by the aid of Lucas numbers, Iğdır University Journal of the Institute of Science and Technology 7 (4) (2017) 103-111.
  • T. Yaying, B. Hazarika, S. A. Mohiuddine, Domain of Padovan $q-$difference matrix in sequence spaces $\ell_p$ and $\ell_\infty$, Filomat 36 (3) (2022) 905-919.
  • T. Yaying, B. Hazarika, O. M. Kalthum S. K. Mohamed, A. A. Bakery, On new Banach sequence spaces involving Leonardo numbers and the associated mapping ideal, Journal of Function Spaces 2022 (2022) Article ID 8269000 21 Pages.
  • M. İlkhan, A new conservative matrix derived by Catalan numbers and its matrix domain in the spaces $c$ and $c_0$, Linear and Multilinear Algebra 68 (2) (2020) 417-434.
  • M. İlkhan, E. E. Kara, Matrix transformations and compact operators on Catalan sequence spaces, Journal of Mathematical Analysis and Applications 498 (1) (2021) Article Number 124925 17 pages.
  • M. Karakaş, M. C. Dağlı, Some topological and geometrical properties of new Catalan sequence spaces, Advances in Operator Theory 8 14 (2023) 1-15.
  • M. Karakaş, On the sequence spaces involving Bell numbers, Linear and Multilinear Algebra 71 (14) (2022) 2298-2309.
  • S. Erdem, S. Demiriz, A. Şahin, Motzkin sequence spaces and Motzkin core, Numerical Functional Analysis and Optimization 45 (4) (2024) 1-21.
  • S. Demiriz, S. Erdem, Mersenne matrix operator and its application in $p$-summable sequence space, Communications in Advanced Mathematical Sciences 7 (1) (2024) 42-55.
  • T. Motzkin, Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products, Bulletin of the American Mathematical Society 54 (4) (1948) 352-360.
  • M. Aigner, Motzkin numbers, European Journal of Combinatorics 19 (1998) 663-675.
  • E. Barrucci, R. Pinzani, R. Sprugnoli, The Motzkin family, PUMA Series A 2 (3-4) (1991) 249-279.
  • R. Donaghey, L.W. Shapiro, Motzkin numbers, Journal of Combinatorial Theory, Series A 23 (3) (1977) 291-301.
  • E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence spaces and measure of noncompactness, Zbornik Radova 17 (2000) 143-234.
  • M. Mursaleen, A. K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Analysis 73 (8) (2010) 2541-2557.
  • M. Mursaleen, A. K. Noman, Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means, Computers & Mathematics with Applications 60 (5) (2010) 1245-1258.
  • A. Wilansky, Summability through functional analysis, Amsterdam, New York, Oxford: North-Holland Mathematics Studies, 1984.
  • F. Başar, E. Malkowsky, The characterization of compact operators on spaces of strongly summable and bounded sequences, Applied Mathematics and Computation 217 (2011) 5199-5207.
  • 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.
Yıl 2024, Cilt: 13 Sayı: 2, 109 - 118, 31.08.2024
https://doi.org/10.54187/jnrs.1517251

Öz

Kaynakça

  • B. Altay, F. Başar, On some Euler sequence spaces of nonabsolute type, Ukrainian Mathematical Journal 57 (1) (2005) 1-17.
  • F. Başar, M. Kirişci, Almost convergence and generalized difference matrix, Computers & Mathematics with Applications 61 (3) (2011) 602-611.
  • T. Yaying, B. Hazarika, M. İlkhan, M.Mursaleen, Poisson-like matrix operator and its application in $p-$summable space, Mathematica Slovaca 71 (5) (2021) 1189-1210.
  • G. C. Hazar, M. A. Sarıgöl, Absolute Cesaro series spaces and matrix operators, Acta Applicandae Mathematicae 154 (1) (2018) 153-165.
  • M. Candan, Domain of the double sequential band matrix in the classical sequence spaces, Journal of Inequalities and Applications 2012 (2012) Article Number 281 15 pages.
  • S. Demiriz, A. Şahin, $q$-Cesaro sequence spaces derived by $q$-analogue, Advances in Mathematics 5 (2) (2016) 97-110.
  • S. Erdem, S. Demiriz, On the new generalized block difference sequence space, Applications and Applied Mathematics: An International Journal 14 (5) (2019) 68-83.
  • H. B. Ellidokuzoğlu, S. Demiriz, $[\ell_p]_{e.r}$ Euler-Riesz difference sequence spaces, Analysis in Theory and Applications 37 (4) (2021) 557-571.
  • M. İlkhan, P. Z. Alp, E. E. Kara, On the spaces of linear operators acting between asymmetric cone normed spaces, Mediterranean Journal of Mathematics 15 (2018) Article Number 136 12 pages.
  • M. Stieglitz, H. Tietz, Matrix transformationen von folgenraumen eine ergebnisbersicht, Mathematische Zeitschrift 154 (1977) 1-16.
  • F. Başar, Summability theory and its applications, 2nd Edition, CRC Press, Taylor Francis Group, Boca Raton-London-New York, 2022.
  • J. Boos, Classical and modern methods in summability, Oxford Science Publications, Oxford University Press, 2000.
  • M. Mursaleen, F. Başar, Sequence spaces: Topic in modern summability theory, CRC Press, Taylor Francis Group, Series: Mathematics and its applications, Boca Raton, London, New York, 2020.
  • E. E. Kara, M. Başarır, An application of Fibonacci numbers into infinite Toeplitz matrices, Caspian Journal of Mathematical Sciences 1 (1) (2012) 43-47.
  • M. C. Dağlı, A novel conservative matrix arising from Schröder numbers and its properties, Linear and Multilinear Algebra 71 (8) (2023) 1338-1351.
  • M. C. Dağlı, Matrix mappings and compact operators for Schröder sequence spaces, Turkish Journal of Mathematics 46 (6) (2022) 2304-2320.
  • M. Karakaş, M. C. Dağlı, Some topologic and geometric properties of new Catalan sequence spaces, Advances in Operator Theory 8 (2023) Article Number 14 15 pages.
  • E. E. Kara, Some topological and geometrical properties of new Banach sequence spaces, Journal of Inequalities and Applications 2013 (2013) Article Number 38 15 pages.
  • M. Karakaş, H. Karabudak, An application on the Lucas numbers and infinite Toeplitz matrices, Cumhuriyet Science Journal 38 (3) (2017) 557-562.
  • M. Karakaş, A. M. Karakaş, New Banach sequence spaces that is defined by the aid of Lucas numbers, Iğdır University Journal of the Institute of Science and Technology 7 (4) (2017) 103-111.
  • T. Yaying, B. Hazarika, S. A. Mohiuddine, Domain of Padovan $q-$difference matrix in sequence spaces $\ell_p$ and $\ell_\infty$, Filomat 36 (3) (2022) 905-919.
  • T. Yaying, B. Hazarika, O. M. Kalthum S. K. Mohamed, A. A. Bakery, On new Banach sequence spaces involving Leonardo numbers and the associated mapping ideal, Journal of Function Spaces 2022 (2022) Article ID 8269000 21 Pages.
  • M. İlkhan, A new conservative matrix derived by Catalan numbers and its matrix domain in the spaces $c$ and $c_0$, Linear and Multilinear Algebra 68 (2) (2020) 417-434.
  • M. İlkhan, E. E. Kara, Matrix transformations and compact operators on Catalan sequence spaces, Journal of Mathematical Analysis and Applications 498 (1) (2021) Article Number 124925 17 pages.
  • M. Karakaş, M. C. Dağlı, Some topological and geometrical properties of new Catalan sequence spaces, Advances in Operator Theory 8 14 (2023) 1-15.
  • M. Karakaş, On the sequence spaces involving Bell numbers, Linear and Multilinear Algebra 71 (14) (2022) 2298-2309.
  • S. Erdem, S. Demiriz, A. Şahin, Motzkin sequence spaces and Motzkin core, Numerical Functional Analysis and Optimization 45 (4) (2024) 1-21.
  • S. Demiriz, S. Erdem, Mersenne matrix operator and its application in $p$-summable sequence space, Communications in Advanced Mathematical Sciences 7 (1) (2024) 42-55.
  • T. Motzkin, Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products, Bulletin of the American Mathematical Society 54 (4) (1948) 352-360.
  • M. Aigner, Motzkin numbers, European Journal of Combinatorics 19 (1998) 663-675.
  • E. Barrucci, R. Pinzani, R. Sprugnoli, The Motzkin family, PUMA Series A 2 (3-4) (1991) 249-279.
  • R. Donaghey, L.W. Shapiro, Motzkin numbers, Journal of Combinatorial Theory, Series A 23 (3) (1977) 291-301.
  • E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence spaces and measure of noncompactness, Zbornik Radova 17 (2000) 143-234.
  • M. Mursaleen, A. K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Analysis 73 (8) (2010) 2541-2557.
  • M. Mursaleen, A. K. Noman, Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means, Computers & Mathematics with Applications 60 (5) (2010) 1245-1258.
  • A. Wilansky, Summability through functional analysis, Amsterdam, New York, Oxford: North-Holland Mathematics Studies, 1984.
  • F. Başar, E. Malkowsky, The characterization of compact operators on spaces of strongly summable and bounded sequences, Applied Mathematics and Computation 217 (2011) 5199-5207.
  • 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.
Toplam 38 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Operatör Cebirleri ve Fonksiyonel Analiz
Bölüm Articles
Yazarlar

Sezer Erdem 0000-0001-9420-8264

Erken Görünüm Tarihi 30 Ağustos 2024
Yayımlanma Tarihi 31 Ağustos 2024
Gönderilme Tarihi 16 Temmuz 2024
Kabul Tarihi 19 Ağustos 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 13 Sayı: 2

Kaynak Göster

APA Erdem, S. (2024). Compact operators on the Motzkin sequence space $c_0(\mathcal{M})$. Journal of New Results in Science, 13(2), 109-118. https://doi.org/10.54187/jnrs.1517251
AMA Erdem S. Compact operators on the Motzkin sequence space $c_0(\mathcal{M})$. JNRS. Ağustos 2024;13(2):109-118. doi:10.54187/jnrs.1517251
Chicago Erdem, Sezer. “Compact Operators on the Motzkin Sequence Space $c_0(\mathcal{M})$”. Journal of New Results in Science 13, sy. 2 (Ağustos 2024): 109-18. https://doi.org/10.54187/jnrs.1517251.
EndNote Erdem S (01 Ağustos 2024) Compact operators on the Motzkin sequence space $c_0(\mathcal{M})$. Journal of New Results in Science 13 2 109–118.
IEEE S. Erdem, “Compact operators on the Motzkin sequence space $c_0(\mathcal{M})$”, JNRS, c. 13, sy. 2, ss. 109–118, 2024, doi: 10.54187/jnrs.1517251.
ISNAD Erdem, Sezer. “Compact Operators on the Motzkin Sequence Space $c_0(\mathcal{M})$”. Journal of New Results in Science 13/2 (Ağustos 2024), 109-118. https://doi.org/10.54187/jnrs.1517251.
JAMA Erdem S. Compact operators on the Motzkin sequence space $c_0(\mathcal{M})$. JNRS. 2024;13:109–118.
MLA Erdem, Sezer. “Compact Operators on the Motzkin Sequence Space $c_0(\mathcal{M})$”. Journal of New Results in Science, c. 13, sy. 2, 2024, ss. 109-18, doi:10.54187/jnrs.1517251.
Vancouver Erdem S. Compact operators on the Motzkin sequence space $c_0(\mathcal{M})$. JNRS. 2024;13(2):109-18.


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