Czechoslovak Mathematical Journal, first online, pp. 1-22
$(y,z)$-Catalan sequence spaces: Algebraic, topological, and geometric investigations
Serkan DEMİRİZ, Sezer ERDEM, Muhammet Cihat DAĞLI
Received December 21, 2025. Published online October 13, 2026.
Abstract: We introduce and investigate the sequence spaces $\mathcal{A} _p$ and $\mathcal{A}_{\infty}$ generated by an infinite matrix operator $\mathcal{A} $ derived from $(y,z)$-Catalan numbers. The main aim is to explore the topological and geometric structures of these newly defined sequence spaces. We establish the linear isomorphism of $\mathcal{A}_p$ and $\mathcal{A} _{\infty}$ with their classical counterparts, and we construct Schauder basis for the space $\mathcal{A}_p$. Moreover, the inclusion relations and duals of these spaces are characterized. In addition, we present essential geometric properties such as rotundity, smoothness, the approximation property, the Dunford-Pettis property, the Radon-Riesz property, and the Hahn-Banach extension property within the context of the newly expressed sequence spaces.
References: [1] N. I. Akhiezer, M. G. Kreĭn: Some Questions in the Theory of Moments. Translations of Mathematical Monographs 2. AMS, Providence (1962). MR 0167806 | Zbl 0117.32702
[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. DOI 10.1016/j.jmaa.2007.03.007 | MR 2348531 | Zbl 1152.46003
[3] F. Başar: Summability Theory and its Applications. CRC Press, Boca Raton (2022). DOI 10.1201/9781003294153 | MR 4600803 | Zbl 1495.40001
[4] J. Boos: Classical and Modern Methods in Summability. Oxford Mathematical Monographs. Oxford University Press, New York (2000). DOI 10.1093/oso/9780198501657.001.0001 | MR 1817226 | Zbl 0954.40001
[5] R. A. Brualdi: Introductory Combinatorics. Prentice Hall, Upper Saddle River (2010). MR 2655770 | Zbl 0915.05001
[6] M. Candan: Some new sequence spaces defined by a modulus function and an infinite matrix in a seminormed space. J. Math. Anal. 3 (2012), 1-9. MR 2966272 | Zbl 1312.46007
[7] M. Candan: Vector-valued FK-spaces defined by a modulus function and an infinite matrix. Thai J. Math. 12 (2014), 155-165. MR 3194916 | Zbl 1310.46009
[8] M. Candan: A new perspective on paranormed Riesz sequence space of non-absolute type. Global J. Math. Anal. 3 (2015), 150-163. DOI 10.14419/gjma.v3i4.5573
[9] M. C. Dağlı: Explicit formulas and recurrence relations for generalized Catalan numbers. Turk. J. Math. 45 (2021), 2171-2179. DOI 10.3906/mat-2106-59 | MR 4316915 | Zbl 1493.05015
[10] M. C. Dağlı: Matrix mappings and compact operators for Schröder sequence spaces. Turk. J. Math. 46 (2022), 2304-2320. DOI 10.55730/1300-0098.3270 | MR 4453923 | Zbl 1510.47045
[11] M. C. Dağlı: A novel conservative matrix arising from Schröder numbers and its properties. Linear Multilinear Algebra 71 (2023), 1338-1351. DOI 10.1080/03081087.2022.2061401 | MR 4584288 | Zbl 1531.46005
[12] S. Demiriz, S. Erdem: Mersenne matrix operator and its application in $p$-summable sequence space. Commun. Adv. Math. Sci. 7 (2024), 42-55. DOI 10.33434/cams.1414791
[13] H. B. Ellidokuzoğlu, S. Demiriz, S. Erdem: New sequence spaces generated via a matrix defined by Schröder and Catalan numbers. Sahand Commun. Math. Anal. 23 (2026), 123-143. DOI 10.22130/scma.2025.2066829.2265 | Zbl 08162474
[14] S. Erdem: Compact operators on the Motzkin sequence space $c_0(\mathcal{M})$. J. New Results Sci. 13 (2024), 109-118. DOI 10.54187/jnrs.1517251
[15] S. Erdem: Compact operators on the new Motzkin sequence spaces. AIMS Math. 9 (2024), 24193-24212. DOI 10.3934/math.20241177 | MR 4787186
[16] S. Erdem: Schröder-Catalan matrix and compactness of matrix operators on its associated sequence spaces. Symmetry 16 (2024), Article ID 1317, 16 pages. DOI 10.3390/sym16101317
[17] S. Erdem, S. Demiriz: On the new generalized block difference sequence space. Appl. Appl. Math., Spec. Iss. 5 (2019), 68-83. MR 3998164 | Zbl 1465.46006
[18] S. Erdem, S. Demiriz: Oresme numbers and associated BK-sequence spaces. Iran. J. Sci. 49 (2025), 725-732. DOI 10.1007/s40995-024-01734-5 | MR 4905129
[19] S. Erdem, S. Demiriz, A. Şahin: Motzkin sequence spaces and Motzkin core. Numer. Funct. Anal. Optim. 45 (2024), 283-303. DOI 10.1080/01630563.2024.2333250 | MR 4770517 | Zbl 1560.46005
[20] S. Erdem, H. B. Ellidokuzoğlu, S. Demiriz, M. İ. Kara: Application of Jordan-type matrix operator in the theory of sequence spaces. Palest. J. Math. 15 (2026), 654-672. MR 5042071
[21] A. Grothendieck: Sur les applications linéaires faiblement compactes d'espaces du type $C(K)$. Can. J. Math. 5 (1953), 129-173. (In French.) DOI 10.4153/CJM-1953-017-4 | MR 0058866 | Zbl 0050.10902
[22] T.-X. He: Parametric Catalan numbers and Catalan triangles. Linear Algebra Appl. 438 (2013), 1467-1484. DOI 10.1016/j.laa.2012.10.001 | MR 2997825 | Zbl 1257.05003
[23] M. İlkhan: A new conservative matrix derived by Catalan numbers and its matrix domain in the spaces $c$ and $c_0$. Linear Multilinear Algebra 68 (2020), 417-434. DOI 10.1080/03081087.2019.1635071 | MR 4048065 | Zbl 1486.46005
[24] 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), Article ID 136, 12 pages. DOI 10.1007/s00009-018-1182-0 | MR 3808563 | Zbl 1454.46019
[25] M. İlkhan Kara, E. E. Kara: Matrix transformations and compact operators on Catalan sequence spaces. J. Math. Anal. Appl. 498 (2021), Article ID 124925, 18 pages. DOI 10.1016/j.jmaa.2021.124925 | MR 4199799 | Zbl 1471.46018
[26] E. E. Kara: Some topological and geometrical properties of new Banach sequence spaces. J. Inequal. Appl. 2013 (2013), Article ID 38, 15 pages. DOI 10.1186/1029-242X-2013-38 | MR 3027694 | Zbl 1286.46008
[27] E. E. Kara, M. Basarir: An application of Fibonacci numbers into infinite Toeplitz matrices. Casp. J. Math. Sci. 1 (2012), 43-47. Zbl 1412.46035
[28] M. Karakas: On the sequence spaces involving Bell numbers. Linear Multilinear Algebra 71 (2023), 2298-2309. DOI 10.1080/03081087.2022.2098225 | MR 4628312 | Zbl 1535.46006
[29] M. Karakas, M. C. Dağlı: Some topologic and geometric properties of new Catalan sequence spaces. Adv. Oper. Theory 8 (2023), Article ID 14, 15 pages. DOI 10.1007/s43036-022-00243-9 | MR 4531705 | Zbl 1515.46012
[30] R. E. Megginson: An Introduction to Banach Space Theory. Graduate Texts in Mathematics 183. Springer, New York (1998). DOI 10.1007/978-1-4612-0603-3 | MR 1650235 | Zbl 0910.46008
[31] M. Mursaleen, F. Başar: Sequence Spaces: Topic in Modern Summability Theory. Mathematics and its Applications. CRC Press, Boca Raton (2020). DOI 10.1201/9781003015116 | MR 4779682 | Zbl 1468.46003
[32] R. S. Phillips: On linear transformations. Trans. Am. Math. Soc. 48 (1940), 516-541. DOI 10.1090/S0002-9947-1940-0004094-3 | MR 0004094 | Zbl 0025.34202
[33] S. Shah, B. Hazarika: Sequence spaces and operator ideals induced by the $q$-bronze Leonardo-Lucas matrix. Filomat 40 (2026), 975-993. DOI 10.2298/FIL2603975S | MR 5034617
[34] R. P. Stanley: Enumerative Combinatorics. Volume 1. Cambridge Studieds in Advanced Mathematics 49. Cambridge University Press, Cambridge (1997). DOI 10.1007/978-1-4615-9763-6 | MR 0847717 | Zbl 0889.05001
[35] M. Stieglitz, H. Tietz: Matrixtransformationen von Folgenräumen: Eine Ergebnisübersicht. Math Z. 154 (1977), 1-16. (In German.) DOI 10.1007/BF01215107 | MR 0446785 | Zbl 0331.40005
[36] A. Wilansky: Summability Through Functional Analysis. North-Holland Mathematics Studies 85. Nort-Holland, Amsterdam (1984). DOI 10.1016/s0304-0208(08)x7133-6 | MR 0738632 | Zbl 0531.40008
[37] T. Yaying: A study of novel telephone sequence spaces and some geometric properties. J. Inequal. Appl. 2024 (2024), Article ID 141, 19 pages. DOI 10.1186/s13660-024-03222-1 | MR 4823221 | Zbl 08034420
[38] T. Yaying, B. Hazarika: On sequence spaces defined by the domain of a regular Tribonacci matrix. Math. Slovaca 70 (2020), 697-706. DOI 10.1515/ms-2017-0383 | MR 4104357 | Zbl 1505.46013
[39] 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. J. Funct. Spaces 2022 (2022), Article ID 8269000, 21 pages. DOI 10.1155/2022/8269000 | MR 4410788 | Zbl 1498.46023
[40] 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. DOI 10.2298/FIL2203905Y | MR 4424052
[41] T. Yaying, M. İ. Kara: On sequence spaces defined by the domain of Tribonacci matrix in $c_0$ and $c$. Korean J. Math. 29 (2021), 25-40. DOI 10.11568/kjm.2021.29.1.25 | MR 4272747 | Zbl 1481.46005
[42] T. Yaying, E. Savaş, M. Mursaleen: Compact matrix operators on the sequence spaces involving Riordan numbers. J. Inequal. Appl. 2026 (2026), Article ID 34, 18 pages. DOI 10.1186/s13660-026-03444-5 | MR 5029091
[43] Y. Yılmaz: Characterizations of bounded linear operators on Banach space-valued Fibonomial sequence spaces. Asian Res. J. Math. 21 (2025), 101-110. DOI 10.9734/arjom/2025/v21i8974
[44] Y. Yılmaz, A. O. Akdemir: On some topological and geometric properties of some $q$-Cesáro sequence spaces. Symmetry 15 (2023), Article ID 791, 12 pages. DOI 10.3390/sym15040791
[45] Y. Yılmaz, H. Bozkurt: On Euler sequence spaces. Open J. Math. Sci. 8 (2024), 128-136. DOI 10.30538/oms2024.0230
[46] Y. Yılmaz, A. N. Tuncer, S. Yalçın: On structral properties of some Banach space-valued Schröder sequence spaces. Symmetry 17 (2025), Article ID 977, 12 pages. DOI 10.3390/sym17070977
