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Statistically order compact operators on Riesz spaces

Year 2024, Volume: 53 Issue: 3, 628 - 636, 27.06.2024
https://doi.org/10.15672/hujms.1223922

Abstract

This research paper introduces and establishes the concept of compact operators in the context of Riesz spaces, specifically considering statistical order convergence. We define statistical order compact operators as operators that map statistical order bounded sequences to sequences with statistical order convergent subsequences. Additionally, we define statistical $M$-weakly compact operators. By utilizing these non-topological concepts, we derive some new results pertaining to these operators.

References

  • [1] C. D. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs Centrum, 2003.
  • [2] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, Dordrecht, 2006.
  • [3] A. Aydın, The statistically unbounded $\tau$-convergence on locally solid vector lattices, Turkish J. Math. 44 (3), 949-956, 2020.
  • [4] A. Aydın, The statistical multiplicative order convergence in vector lattice algebras, Fact. Univ. Ser.: Math. Infor. 36 (2), 409-417, 2021.
  • [5] A. Aydın, E. Emelyanov and S. G. Gorokhova, Full lattice convergence on Riesz spaces, Indagat. Math. 32 (3), 658-690, 2021.
  • [6] A. Aydın, E. Emelyanov and S. G. Gorokhova, Multiplicative order continuous operators on Riesz algebras, https://arxiv.org/abs/2201.12095v1.
  • [7] A. Aydın, E. Y. Emelyanov, N.E. Özcan and M. A. A. Marabeh, Compact-like operators in lattice-normed spaces, Indagat. Math. 29 (2), 633-656, 2018.
  • [8] A. Aydın, S. Gorokhova, R. Selen and S. Solak, Statistically order continuous operators on Riesz spaces, Maejo Int. J. Sci. Tech. 17 (1), 1-9, 2023.
  • [9] Y. Azouzi, Completeness for vector lattices, J. Math. Anal. Appl. 472 (1), 216-230, 2019.
  • [10] Y. Azouzi, M. A. B. Amor, On Compact Operators Between Lattice Normed Spaces, Positivity and its Applications, Birkhäuser, 2021.
  • [11] H. Fast, Sur la convergence statistique, Colloq. Math. 2, 241-244, 1951.
  • [12] S. G. Gorokhova, Intrinsic characterization of the space $c_0(A)$ in the class of Banach lattices, Math. Notes 60, 330-333, 1996.
  • [13] W. A. J. Luxemburg, A. C. Zaanen, Vector Lattices I, North-Holland Pub. Co. Amsterdam, 1971.
  • [14] I. J. Maddox, Statistical convergence in a locally convex space, Math. Proc. Cambr. Phil. Soc. 104 (1), 141-145, 1988.
  • [15] O. V. Maslyuchenko, V. V. Mykhaylyuk and M. M. Popov, A lattice approach to narrow operators, Positivity, 13 (3), 459495, 2009.
  • [16] N. E. Özcan, N. A. Gezer, . E. Özdemir and . M. Geyikçi, Order compact and unbounded order compact operators, Turkish J. Math. 45 (2), 634-646, 2021.
  • [17] B. de Pagter, f-Algebras and Orthomorphisms, Ph. D. Dissertation, Leiden, 1981.
  • [18] F. Riesz, Sur la Décomposition des Opérations Fonctionelles Linéaires, Bologna, Atti Del Congresso Internazionale Dei Mathematics Press, 1928.
  • [19] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2, 73-74, 1951.
  • [20] C. Şençimen, S. Pehlivan, Statistical order convergence in Riesz spaces, Math. Slovac. 62 (2), 557-570, 2012.
  • [21] B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, Wolters- Noordhoff Ltd, Groningen, 1967.
  • [22] A.C. Zaanen, Riesz Spaces II, North-Holland Publishing C., Amsterdam, 1983.
Year 2024, Volume: 53 Issue: 3, 628 - 636, 27.06.2024
https://doi.org/10.15672/hujms.1223922

Abstract

References

  • [1] C. D. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs Centrum, 2003.
  • [2] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, Dordrecht, 2006.
  • [3] A. Aydın, The statistically unbounded $\tau$-convergence on locally solid vector lattices, Turkish J. Math. 44 (3), 949-956, 2020.
  • [4] A. Aydın, The statistical multiplicative order convergence in vector lattice algebras, Fact. Univ. Ser.: Math. Infor. 36 (2), 409-417, 2021.
  • [5] A. Aydın, E. Emelyanov and S. G. Gorokhova, Full lattice convergence on Riesz spaces, Indagat. Math. 32 (3), 658-690, 2021.
  • [6] A. Aydın, E. Emelyanov and S. G. Gorokhova, Multiplicative order continuous operators on Riesz algebras, https://arxiv.org/abs/2201.12095v1.
  • [7] A. Aydın, E. Y. Emelyanov, N.E. Özcan and M. A. A. Marabeh, Compact-like operators in lattice-normed spaces, Indagat. Math. 29 (2), 633-656, 2018.
  • [8] A. Aydın, S. Gorokhova, R. Selen and S. Solak, Statistically order continuous operators on Riesz spaces, Maejo Int. J. Sci. Tech. 17 (1), 1-9, 2023.
  • [9] Y. Azouzi, Completeness for vector lattices, J. Math. Anal. Appl. 472 (1), 216-230, 2019.
  • [10] Y. Azouzi, M. A. B. Amor, On Compact Operators Between Lattice Normed Spaces, Positivity and its Applications, Birkhäuser, 2021.
  • [11] H. Fast, Sur la convergence statistique, Colloq. Math. 2, 241-244, 1951.
  • [12] S. G. Gorokhova, Intrinsic characterization of the space $c_0(A)$ in the class of Banach lattices, Math. Notes 60, 330-333, 1996.
  • [13] W. A. J. Luxemburg, A. C. Zaanen, Vector Lattices I, North-Holland Pub. Co. Amsterdam, 1971.
  • [14] I. J. Maddox, Statistical convergence in a locally convex space, Math. Proc. Cambr. Phil. Soc. 104 (1), 141-145, 1988.
  • [15] O. V. Maslyuchenko, V. V. Mykhaylyuk and M. M. Popov, A lattice approach to narrow operators, Positivity, 13 (3), 459495, 2009.
  • [16] N. E. Özcan, N. A. Gezer, . E. Özdemir and . M. Geyikçi, Order compact and unbounded order compact operators, Turkish J. Math. 45 (2), 634-646, 2021.
  • [17] B. de Pagter, f-Algebras and Orthomorphisms, Ph. D. Dissertation, Leiden, 1981.
  • [18] F. Riesz, Sur la Décomposition des Opérations Fonctionelles Linéaires, Bologna, Atti Del Congresso Internazionale Dei Mathematics Press, 1928.
  • [19] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2, 73-74, 1951.
  • [20] C. Şençimen, S. Pehlivan, Statistical order convergence in Riesz spaces, Math. Slovac. 62 (2), 557-570, 2012.
  • [21] B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, Wolters- Noordhoff Ltd, Groningen, 1967.
  • [22] A.C. Zaanen, Riesz Spaces II, North-Holland Publishing C., Amsterdam, 1983.
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Abdullah Aydın

Early Pub Date August 15, 2023
Publication Date June 27, 2024
Published in Issue Year 2024 Volume: 53 Issue: 3

Cite

APA Aydın, A. (2024). Statistically order compact operators on Riesz spaces. Hacettepe Journal of Mathematics and Statistics, 53(3), 628-636. https://doi.org/10.15672/hujms.1223922
AMA Aydın A. Statistically order compact operators on Riesz spaces. Hacettepe Journal of Mathematics and Statistics. June 2024;53(3):628-636. doi:10.15672/hujms.1223922
Chicago Aydın, Abdullah. “Statistically Order Compact Operators on Riesz Spaces”. Hacettepe Journal of Mathematics and Statistics 53, no. 3 (June 2024): 628-36. https://doi.org/10.15672/hujms.1223922.
EndNote Aydın A (June 1, 2024) Statistically order compact operators on Riesz spaces. Hacettepe Journal of Mathematics and Statistics 53 3 628–636.
IEEE A. Aydın, “Statistically order compact operators on Riesz spaces”, Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 3, pp. 628–636, 2024, doi: 10.15672/hujms.1223922.
ISNAD Aydın, Abdullah. “Statistically Order Compact Operators on Riesz Spaces”. Hacettepe Journal of Mathematics and Statistics 53/3 (June 2024), 628-636. https://doi.org/10.15672/hujms.1223922.
JAMA Aydın A. Statistically order compact operators on Riesz spaces. Hacettepe Journal of Mathematics and Statistics. 2024;53:628–636.
MLA Aydın, Abdullah. “Statistically Order Compact Operators on Riesz Spaces”. Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 3, 2024, pp. 628-36, doi:10.15672/hujms.1223922.
Vancouver Aydın A. Statistically order compact operators on Riesz spaces. Hacettepe Journal of Mathematics and Statistics. 2024;53(3):628-36.