On Collectively \(\sigma\)-Levi Sets of Operators

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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/y6929-3405-2251-o On Collectively \(\sigma\)-Levi Sets of Operators Emelyanov, E. Yu. Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 1. Abstract: The Levi operators are operator abstractions of the Levy property of Banach lattices. Such operators have been studied recently by several authors. The present paper deals with the collective properties of the Levi operators of several kinds: \(\sigma\)-Levi operators; quasi c-\(\sigma\)-Levi operators; and quasi \(\sigma\)-Levi operators. A notion of collectively \(\sigma\)-Levi set generalizes the notion of a single \(\sigma\)-Levi operator to the families of operators. Working with families of sequences of elements of a vector lattice requires the notion of the collective order convergence. This notion that is introduced and studied in the present paper may have its own interest and further possible applications. Various relations of the collectively quasi \(\sigma\)-Levi sets to the collectively compact sets are investigated. The domination problem for the collectively quasi \(\sigma\)-Levi sets is studied. In this study a special notion of a set of operators dominated by another set of operators is used. Keywords: vector lattice, normed lattice, collective order convergence, collectively \(\sigma\)-Levi set, collectively compact set Language: English Download the full text Download JATS XML For citation: Emelyanov, E. Yu. On Collectively \(\sigma\)-Levi Sets of Operators, Vladikavkaz Math. J., 2025, vol. 27, no. 1, pp. 36-43. DOI 10.46698/y6929-3405-2251-o + References 1. Alpay, S., Emelyanov, E. and Gorokhova, S. \(o\tau\)-Continuous, Lebesgue, KB, and Levi Operators Between Vector Lattices and Topological Vector Spaces, Results in Mathematics, 2022, vol. 77, no. 3, article no. 117, pp. 1-25. DOI: 10.1007/s00025-022-01650-3. 2. Emelyanov, E. On KB and Levi Operators in Banach Lattices, arXiv:2312.05685v2 [math.FA], 2023. DOI: 10.48550/arXiv.2312.05685. 3. Gorokhova, S. G. and Emelyanov, E. Y. On Operators Dominated by Kantorovich-Banach Operators and Levy Operators in Locally Solid Lattices, Vladikavkaz Math. J. , 2022, vol. 24, no. 3, pp. 55-61 (in Russian). DOI: 10.46698/f5525-0005-3031-h. 4. Zhang, F. and Chen, Z. Some Results of (\(\sigma\)-)Levi Operators in Banach Lattices, Positivity, 2022, vol. 26, article no. 49. DOI: 10.1007/s11117-022-00903-3. 5. Aliprantis, C. D. and Burkinshaw, O. Locally Solid Riesz Spaces with Applications to Economics: Second Edition, Mathematical Surveys and Monographs, vol. 105, Providence, RI, American Mathematical Society, 2003. 6. Anselone, P. M. and Palmer, T. W. Collectively Compact Sets of Linear Operators, Pacific Journal of Mathematics, 1968, vol. 25, no. 3, pp. 417-422. 7. Kusraev, A. G. Dominated Operators, Dordrecht, Kluwer, 2000. 8. Meyer-Nieberg, P. Banach Lattices, Berlin, Springer-Verlag, Universitext, 1991. 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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