Unbounded Banach-Saks Operators and Unbounded Grothendieck Operators on Banach Lattices

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/p1448-4815-0941-k Unbounded Banach-Saks Operators and Unbounded Grothendieck Operators on Banach Lattices Sabbagh, R. , Zabeti, O. Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 4. Abstract: Recently, there has been much attention to the ordered structures beyond Banach lattices.Moreover, we have many nice properties in Banach spaces that can be transformed naturally into Banach lattice cases. Therefore, combining these notions with order structure can produce nicer notions, as well. Suppose \(E\) is a Banach lattice. Recently, there have been some motivating contexts regarding the known Banach-Saks property and the Grothendieck property from an order point of view. In this paper, we establish these results for operators that enjoy different types considered for the Banach-Saks property. We characterize order continuity and reflexivity of the underlying Banach lattices in terms of the corresponding operator versions related to the Banach-Saks properties. Moreover, we consider different notions related to the Grothendieck property from an ordered attitude; then, we investigate operator versions of these concepts, as well. In particular, beside other results, we characterize order continuity and reflexivity of the underlying Banach lattices in terms of the corresponding bounded linear operators defined on the corresponding Banach lattices, as well. Keywords: unbounded Banach-Saks operators, unbounded Grothendieck operators, the Banach--Saks property, the Grothendieck property, Banach lattice Language: English Download the full text For citation: Sabbagh, R. and Zabeti, O. Unbounded Banach-Saks Operators and Unbounded Grothendieck Operators on Banach Lattices, Vladikavkaz Math. J., 2024, vol. 26, no. 4, pp. 137-143. DOI 10.46698/p1448-4815-0941-k + References 1. Deng Y., O'Brien M. and Troitsky, V. G. Unbounded Norm Convergence in Banach Lattices, Positivity, 2017, vol. 21(3), pp. 963-974. DOI: 10.1007/s11117-016-0446-9. 2. Erkursun-Ozcan, N., Gezer, N. A. and Zabeti, O. Unbounded Absolutely Weak Dunford-Pettis Operators, Turkish Journal of Mathematics, 2019, vol. 43, no. 6, pp. 2731-2740. DOI: 10.3906/mat-1904-27. 3. Machrafi, N., El Fahri, K., Moussa, M. and Altin, B. A Note on Weak Almost Limited Operators, Hacettepe Journal of Mathematics and Statistics, 2019, vol. 48, no. 3, pp. 759-770. 4. Zabeti, O. The Grothendieck Property from an Ordered Point of View, Positivity, 2022, vol. 26, article no. 17. DOI: 10.1007/s11117-022-00893-2. 5. Zabeti, O. Unbounded Continuous Operators and Unbounded Banach-Saks Property in Banach Lattices, Positivity, 2021, vol. 25, no. 1, pp. 1989-2001. DOI: 10.1007/s11117-021-00858-x. 6. Aliprantis, C. D. and Burkinshaw, O. Positive operators, Springer, 2006. 7. Nieberg, P. Banach Lattices, Berlin, Springer-Verlag, 1991. 8. Gao, N., Troitsky, V. G. and Xanthos, F. Uo-convergence and Its Applications to Cesaro Means in Banach Lattices, Israel Journal of Mathematics, 2017, vol. 220, pp. 649-689. DOI: 10.1007/s11856-017-1530-y. 9. Zabeti, O. Unbounded Absolute Weak Convergence in Banach Lattices, Positivity, 2018, vol. 22(1), pp. 501-505. DOI: 10.1007/s11117-017-0524-7. 10. Dodds, P. G. and Fremlin, D. H. Compact Operators in Banach Lattices, Israel Journal of Mathematics, 1979, vol. 34(4), pp. 287-320. DOI: 10.1007/BF02760610. 11. Domanski, P., Lindstrom, M. and Schluchtermann, G. Grothendieck Operators on Tensor Products, Proceedings of the American Mathematical Society, 1997, vol. 125, no. 8, pp. 2285-2291. DOI: 10.1090/S0002-9939-97-03763-5. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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