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AIMS Mathematics

2024, Volume 9, Issue 9: 24193-24212. doi: 10.3934/math.20241177
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Research article

Compact operators on the new Motzkin sequence spaces

  • Department of Basic Engineering Sciences, Malatya Turgut Özal University, Malatya, Turkey
  • Received: 12 June 2024 Revised: 21 July 2024 Accepted: 26 July 2024 Published: 15 August 2024

  • MSC : 11B83, 46A45, 46B45, 47B07, 47B37

  • This study aims to construct the BK-spaces $ \ell_p(\mathcal{M}) $ and $ \ell_{\infty}(\mathcal{M}) $ as the domains of the conservative Motzkin matrix $ \mathcal{M} $ obtained by using Motzkin numbers. It investigates topological properties, obtains Schauder basis, and then gives inclusion relations. Additionally, it expresses $ \alpha $-, $ \beta $-, and $ \gamma $-duals of these spaces and submits the necessary and sufficient conditions of the matrix classes between the described spaces and the classical spaces. In the last part, the characterization of certain compact operators is given with the aid of the Hausdorff measure of non-compactness.

    Citation: Sezer Erdem. Compact operators on the new Motzkin sequence spaces[J]. AIMS Mathematics, 2024, 9(9): 24193-24212. doi: 10.3934/math.20241177

    Related Papers:

  • This study aims to construct the BK-spaces $ \ell_p(\mathcal{M}) $ and $ \ell_{\infty}(\mathcal{M}) $ as the domains of the conservative Motzkin matrix $ \mathcal{M} $ obtained by using Motzkin numbers. It investigates topological properties, obtains Schauder basis, and then gives inclusion relations. Additionally, it expresses $ \alpha $-, $ \beta $-, and $ \gamma $-duals of these spaces and submits the necessary and sufficient conditions of the matrix classes between the described spaces and the classical spaces. In the last part, the characterization of certain compact operators is given with the aid of the Hausdorff measure of non-compactness.



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    [1] M. Aigner, Motzkin numbers, Europ. J. Combinatorics, 19 (1998), 663–675. https://doi.org/10.1006/eujc.1998.0235 doi: 10.1006/eujc.1998.0235
    [2] B. Altay, F. Başar, Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space, J. Math. Anal. Appl., 336 (2007), 632–645. https://doi.org/10.1016/j.jmaa.2007.03.007 doi: 10.1016/j.jmaa.2007.03.007
    [3] F. Başar, E. Malkowsky, The characterization of compact operators on spaces of strongly summable and bounded sequences, Appl. Math. Comput., 217 (2011), 5199–5207. https://doi.org/10.1016/j.amc.2010.12.007 doi: 10.1016/j.amc.2010.12.007
    [4] F. Başar, Summability theory and its applications, 2nd ed., New York: CRC Press/Taylor & Francis Group, 2022. https://doi.org/10.1201/9781003294153
    [5] J. Boos, Classical and modern methods in summability, Oxford University Press, 2000.
    [6] M. C. Dağlı, A novel conservative matrix arising from Schröder numbers and its properties, Linear Mult. Algebra, 71 (2023), 1338–1351. https://doi.org/10.1080/03081087.2022.2061401 doi: 10.1080/03081087.2022.2061401
    [7] M. C. Dağlı, Matrix mappings and compact operators for Schröder sequence spaces, Turkish J. Math., 46 (2022), 2304–2320. https://doi.org/10.55730/1300-0098.3270 doi: 10.55730/1300-0098.3270
    [8] S. Demiriz, S. Erdem, Mersenne matrix operator and its application in $p$-Summable sequence space, Commun. Adv. Math. Sci., 7 (2024), 42–55. https://doi.org/10.33434/cams.1414791 doi: 10.33434/cams.1414791
    [9] S. Erdem, S. Demiriz, On the new generalized block difference sequence space, Appl. Appl. Math., 5 (2019), 68–83.
    [10] S. Erdem, S. Demiriz, A. Şahin, Motzkin sequence spaces and motzkin core, Numer. Funct. Anal. Optim., 45 (2024), 283–303. https://doi.org/10.1080/01630563.2024.2333250 doi: 10.1080/01630563.2024.2333250
    [11] M. İlkhan, P. Z. Alp, E. E. Kara, On the spaces of linear operators acting between asymmetric cone normed spaces, Mediterr. J. Math., 15 (2018). https://doi.org/10.1007/s00009-018-1182-0
    [12] M. İlkhan, A new conservative matrix derived by Catalan numbers and its matrix domain in the spaces $c$ and $c_0$, Linear Mult. Algebra, 68 (2019), 417–434. https://doi.org/10.1080/03081087.2019.1635071 doi: 10.1080/03081087.2019.1635071
    [13] M. İlkhan, E. E. Kara, Matrix transformations and compact operators on Catalan sequence spaces, J. Math. Anal. Appl., 498 (2021), 124925. https://doi.org/10.1016/j.jmaa.2021.124925 doi: 10.1016/j.jmaa.2021.124925
    [14] E. E. Kara, M. Başarır, An application of Fibonacci numbers into infinite Toeplitz matrices, Caspian J. Math. Sci., 1 (2012), 43–47.
    [15] E. E. Kara, Some topological and geometrical properties of new Banach sequence spaces, J. Inequal. Appl., 2013. https://doi.org/10.1186/1029-242X-2013-38
    [16] M. Karakaş, M. C. Dağlı, Some topologic and geometric properties of new Catalan sequence spaces, Adv. Oper. Theory, 8 (2023). https://doi.org/10.1007/s43036-022-00243-9
    [17] M. Karakaş, H. Karabudak, An application on the Lucas numbers and infinite Toeplitz matrices, Cumhuriyet Sci. J., 38 (2017), 557–562. https://doi.org/10.17776/csj.340510 doi: 10.17776/csj.340510
    [18] M. Karakaş, A. M. Karakaş, New Banach sequence spaces that is defined by the aid of Lucas numbers, Iğdır Univ. J. Inst. Sci. Tech., 7 (2017), 103–111.
    [19] M. Karakaş, On the sequence spaces involving Bell numbers, Linear Mult. Algebra, 71 (2022), 2298–2309. https://doi.org/10.1080/03081087.2022.2098225 doi: 10.1080/03081087.2022.2098225
    [20] E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence spaces and measure of noncompactness, Zbornik Radov., 9 (2000), 143–234.
    [21] T. Motzkin, Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for nonassociative products, Bull. Amer. Math. Soe., 54 (1948), 352–360. https://doi.org/10.1090/S0002-9904-1948-09002-4 doi: 10.1090/S0002-9904-1948-09002-4
    [22] M. Mursaleen, A. K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal., 73 (2010), 2541–2557. https://doi.org/10.1016/j.na.2010.06.030 doi: 10.1016/j.na.2010.06.030
    [23] M. Mursaleen, A. K. Noman, Applications of the Hausdorffmeasure of noncompactness in some sequence spaces of weighted means, Comput. Math. Appl., 60 (2010), 1245–1258. https://doi.org/10.1016/j.camwa.2010.06.005 doi: 10.1016/j.camwa.2010.06.005
    [24] M. Mursaleen, F. Başar, Sequence spaces: Topic in modern summability theory, CRC Press, 2020. https://doi.org/10.1201/9781003015116
    [25] V. Rakocevic, Measures of noncompactness and some applications, Filomat, 12 (1998), 87–120.
    [26] M. Stieglitz, H. Tietz, Matrix transformationen von folgenraumen eine ergebnisbersicht, Math Z., 154 (1977), 1–16. https://doi.org/10.1007/BF01215107 doi: 10.1007/BF01215107
    [27] A. Wilansky, Summability through functional analysis, Elsevier, 1984.
    [28] T. Yaying, B. Hazarika, S. A. Mohiuddine, Domain of Padovan $q$-difference matrix in sequence spaces $\ell_p$ and $\ell_\infty$, Filomat, 36 (2022), 905–919. https://doi.org/10.2298/FIL2203905Y doi: 10.2298/FIL2203905Y
    [29] T. Yaying, B. Hazarika, O. M. K. S. K. Mohamed, A. A. Bakery, On new Banach sequence spaces involving Leonardo numbers and the associated mapping ideal, J. Funct. Spaces, 2022 (2022), 1–21. https://doi.org/10.1155/2022/8269000 doi: 10.1155/2022/8269000
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