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Some topologic and geometric properties of new Catalan sequence spaces

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Abstract

The essential target of this study is to define a novel conservative matrix, including Catalan and Motzkin numbers, and to study its matrix domain in the absolutely p-summable sequence spaces and bounded sequence spaces. Besides, \(\alpha \)-, \(\beta \)-, \(\gamma \)-duals of Banach spaces introduced here are presented and certain matrix classes are characterized. Finally, certain geometric properties containing uniform convexity, strict convexity and superreflexivity are discussed.

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References

  1. Altay, B., Başar, F.: The matrix domain and the fine spectrum of the difference operator \(\Delta \) on the sequence space \(\ell _{p}\), \((0<p<1)\). Commun. Math. Anal. 2(2), 1–11 (2007)

    MATH  Google Scholar 

  2. Altay, B., Başar, F.: Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space. J. Math. Anal. Appl. 336, 632–645 (2007)

    MATH  Google Scholar 

  3. Altay, B., Başar, F., Mursaleen, M.: On the Euler sequence spaces which include the spaces \(\ell _{p}\) and \(\ell _{\infty }\) I. Inf. Sci. 176(10), 1450–1462 (2006)

    MATH  Google Scholar 

  4. Başar, F.: Summability Theory and Its Applications. Bentham Science Publishers, İstanbul (2012)

    MATH  Google Scholar 

  5. Başarir, M., Kara, E.E.: On compact operators on the Riesz \(B(m)\)-difference sequence spaces. Iran. J. Sci. Technol. Trans. A Sci. 35, 279–285 (2011)

    MATH  Google Scholar 

  6. Başarir, M., Kara, E.E.: On some difference sequence spaces of weighted means and compact operators. Ann. Funct. Anal. 2, 114–129 (2011)

    MATH  Google Scholar 

  7. Başarir, M., Kara, E.E.: On the \(B\)-difference sequence space derived by generalized weighted mean and compact operators. J. Math. Anal. Appl. 391, 67–81 (2012)

    MATH  Google Scholar 

  8. Candan, M.: Domain of the double sequential band matrix in the spaces of convergent and null sequences. Adv. Differ. Equ. 2014, 163 (2014)

    MATH  Google Scholar 

  9. Candan, M.: Almost convergence and double sequential band matrix. Acta Math. Sci. 34(2), 354–366 (2014)

    MATH  Google Scholar 

  10. Candan, M.: A new sequence space isomorphic to the space \(\ell \left( p\right) \) and compact operators. J. Math. Comput. Sci. 4(2), 306–334 (2014)

    Google Scholar 

  11. Candan, M.: Some characteristics of matrix operators on generalized Fibonacci weighted difference sequence space. Symmetry 14, 1283 (2022)

    Google Scholar 

  12. Candan, M.: A new aspect for some sequence spaces derived using the domain of the matrix \(\hat{{\hat{B}}}\). Fundam. J. Math. Appl. 5(1), 51–62 (2022)

    Google Scholar 

  13. Chidume, C.: Geometric Properties of Banach Spaces and Nonlinear Iterations. Lecture Notes in Mathematics, vol. 1965. Springer, London (2009)

    MATH  Google Scholar 

  14. Dağlı, M.C.: A novel conservative matrix arising from Schröder numbers and its properties. Linear Multilinear Algebra (2022). https://doi.org/10.1080/03081087.2022.2061401

    Article  Google Scholar 

  15. Dağlı, M.C.: A new almost convergent sequence space defined by Schröder matrix. Linear Multilinear Algebra (2022). https://doi.org/10.1080/03081087.2022.2084499

    Article  Google Scholar 

  16. Dağlı, M.C.: Matrix map**s and compact operators for Schröder sequence spaces. Turk. J. Math. 46(6), 2304–2320 (2022)

    MATH  Google Scholar 

  17. Diestel, J.: Geometry of Banach Spaces-Selected Topics. Lecture Notes in Mathematics, vol. 485. Springer, Berlin (1976)

  18. Ercan, S., Bektaş, C.A.: Some topological and geometric properties of a new BK-space derived by using regular matrix of Fibonacci numbers. Linear Multilinear Algebra 65(5), 909–921 (2017)

    MATH  Google Scholar 

  19. Et, M.: On some difference sequence spaces. Turk. J. Math. 17, 18–24 (1993)

    MATH  Google Scholar 

  20. Et, M., Başarır, M.: On some new generalized difference sequence spaces. Period. Math. Hung. 35(3), 169–175 (1997)

    MATH  Google Scholar 

  21. Et, M., Çolak, R.: On some generalized difference sequence spaces. Soochow J. Math. 21(4), 377–386 (1995)

    MATH  Google Scholar 

  22. Gökçe, F., Sarıgöl, M.A.: Generalization of the space \(\ell \left( p\right) \) derived by absolute Euler summability and matrix operators. J. Inequal. Appl. 2018, 133 (2018)

    MATH  Google Scholar 

  23. Güngör, N.: Some geometric properties of the non-Newtonian sequence spaces. Math. Slovaca 70, 689–696 (2020)

    MATH  Google Scholar 

  24. İlkhan, M.: Matrix domain of a regular matrix derived by Euler totient function in the spaces \(c_{0}\) and \(c\). Mediterr. J. Math. 17(1), 27 (2020)

    MATH  Google Scholar 

  25. İlkhan, M.: A new conservative matrix derived by Catalan numbers and its matrix domain in the spaces \(c\) and \(c_{0}\). Linear Multilinear Algebra 68(2), 417–434 (2020)

    MATH  Google Scholar 

  26. İlkhan, M., Kara, E.E.: A new Banach space defined by Euler totient matrix operator. Oper. Matrices 13(2), 527–544 (2019)

    MATH  Google Scholar 

  27. İlkhan, M., Kara, E.E., Usta, F.: Compact operators on the Jordan totient sequence spaces. Math. Methods Appl. Sci. 44(9), 7666–7675 (2020)

    MATH  Google Scholar 

  28. İlkhan, M., Simsek, N., Kara, E.E.: A new regular infinite matrix defined by Jordan totient function and its matrix domain in \(\ell _{p}\). Math. Methods Appl. Sci. 44(9), 7622–7633 (2020)

    MATH  Google Scholar 

  29. James, C.R.: Super reflexive spaces with bases. Pac. J. Math. 41, 409–419 (1972)

    MATH  Google Scholar 

  30. Kara, E.E.: Some topological and geometrical properties of new Banach sequence spaces. J. Inequal. Appl. 2013(38) (2013)

  31. Kara, E.E., Başarır, M.: On some Euler \(B^{\left( m\right) }\) difference sequence spaces and compact operators. J. Math. Anal. Appl. 379, 499–511 (2011)

    MATH  Google Scholar 

  32. Kara, E.E., İlkhan, M.: Some properties of generalized Fibonacci sequence spaces. Linear Multilinear Algebra 64(11), 2208–2223 (2016)

    MATH  Google Scholar 

  33. Kara, M.İ., Kara, E.E.: Matrix transformations and compact operators on Catalan sequence spaces. J. Math. Anal. Appl. 498(1) (2021). Article no. 124925

  34. Karakaş, M.: On the sequence spaces involving bell numbers. Linear Multilinear Algebra (2022). https://doi.org/10.1080/03081087.2022.2098225

    Article  Google Scholar 

  35. Karakaş, M., Çinar, M., Et, M.: Some geometric properties of a new sequence space. J. Comput. Anal. Appl. 15, 23–31 (2013)

    MATH  Google Scholar 

  36. Kirişçi, M., Başar, F.: Some new sequence spaces derived by the domain of generalized difference matrix. Comput. Math. Appl. 60, 1299–1309 (2010)

    MATH  Google Scholar 

  37. Meng, J., Mei, L.: A generalized fractional difference operator and its applications. Linear Multilinear Algebra 68(9), 1848–1860 (2020)

    MATH  Google Scholar 

  38. Meng, J., Mei, L.: The matrix domain and the spectra of a generalized difference operator. J. Math. Anal. Appl. 470(2), 1095–1107 (2019)

    MATH  Google Scholar 

  39. Mursaleen, M., Noman, A.K.: On some new difference sequence spaces of non-absolute type. Math. Comput. Model. 52(3–4), 603–617 (2010)

    MATH  Google Scholar 

  40. Oğur, O.: Superposition operators on sequence spaces \(\ell _{p}\left( F\right) \) derived by using matrix of Fibonacci numbers. Linear Multilinear Algebra 68(10), 2087–2098 (2020)

    MATH  Google Scholar 

  41. Sarıgöl, M.A.: Spaces of series summable by absolute Cesàro and matrix operators. Commun. Math. Appl. 7(1), 11–22 (2016)

    Google Scholar 

  42. Savas, E.: Matrix transformations between some new sequence spaces. Tamkang J. Math. 19(4), 75–80 (1988)

    MATH  Google Scholar 

  43. Şimşek, N., Savaş, E., Karakaya, V.: On geometrical properties of some Banach spaces. Appl. Math. Inf. Sci. 7, 295–300 (2013)

    Google Scholar 

  44. Stanley, R.P.: Catalan Numbers. Cambridge University Press, New York (2015)

    MATH  Google Scholar 

  45. Stieglitz, M., Tietz, H.: Matrix transformationen von folgenraumen eine ergebnisbersicht. Math. Z. 154, 1–16 (1977)

    MATH  Google Scholar 

  46. Yaying, T., Hazarika, B.: On sequence spaces defined by the domain of a regular Tribonacci matrix. Math. Slovaca 70(3), 697–706 (2020)

    MATH  Google Scholar 

  47. Yaying, T., Hazarika, B., Esi, A.: Geometric properties and compact operator on fractional Riesz difference space. Kragujev. J. Math. 47, 545–566 (2023)

    Google Scholar 

  48. Yaying, T., Kara, M.İ: On sequence spaces defined by the domain of tribonacci matrix in \(c_{0}\) and \(c\). Korean J. Math. 29(1), 25–40 (2021)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Bitlis Eren University, Bitlis, Turkey

    Murat Karakas

  2. Department of Mathematics, Akdeniz University, 07058, Antalya, Turkey

    Muhammet Cihat Dağlı

Authors
  1. Murat Karakas
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  2. Muhammet Cihat Dağlı
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Correspondence to Muhammet Cihat Dağlı.

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Communicated by Timur Oikhberg.

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Karakas, M., Dağlı, M.C. Some topologic and geometric properties of new Catalan sequence spaces. Adv. Oper. Theory 8, 14 (2023). https://doi.org/10.1007/s43036-022-00243-9

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