Banach Limits Invariant Under Dilation 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/z6430-9873-2568-v Banach Limits Invariant Under Dilation Operators Zvolinskii, R. E. , Semenov, E. M. Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 1. Abstract: This paper considers sets of Banach limits invariant under dilation operators. It is known that the set of these limits is a non-empty and convex subset of the set of Banach limits. However, the union of all such subsets is non-convex. This paper provides a necessary and sufficient condition for the convexity of finite unions of such subsets. The obtained criterion provides a complete answer to the question regarding the convexity of finite unions of sets of Banach limits invariant under dilation operators. At the same time, the question of a similar criterion for infinite unions remains open: the authors have found only necessary and, separately, sufficient conditions for convexity. Keywords: Banach limits, dilation operators, convex subsets. Funding: The study was carried out with financial support from the Russian Science Foundation, project No. 24-21-00220. Language: Russian Download the full text Download JATS XML For citation: Zvolinskii, R. E. and Semenov, E. M. Banach Limits Invariant Under Dilation Operators, Vladikavkaz Math. J., 2026, vol. 28, no. 1, pp. 68-72 (in Russian). DOI 10.46698/z6430-9873-2568-v + References 1. Mazur, S. O Metodach Sumowalnosci, Ann. Soc. Polon. Math. (Supplement), 1929, pp. 102-107. 2. Banach, S. Theory of Linear Operations, Dover Publications, 2009, 254 p. 3. Eberlein, W. F. Banach-Hausdorff Limits, Proceedings of the American Mathematical Society, 1950, vol. 1, no. 5, pp. 662-665. DOI: 10.1090/S0002-9939-1950-0038009-0. 4. Semenov, E. M. and Sukochev, F. A. Invariant Banach Limits and Applications, Journal of Functional Analysis, 2010, vol. 259, no. 6, pp. 1517-1541. DOI: 10.1016/j.jfa.2010.05.011. 5. Semenov, E. M., Sukochev, F. A. and Usachev, A. S. Geometry of Banach Limits and Their Applications, Russian Mathematical Surveys, 2020, vol. 75, no. 4, pp. 725-763. DOI: 10.1070/RM9901. 6. Avdeev, N. N., Semenov, E. M. and Usachev, A. S. Banach Limits: Extreme Properties, Invariance and the Fubini Theorem, St. Petersburg Mathematical Journal, 2022, vol. 33, no. 4, pp. 607-618. DOI: 10.1090/spmj/1717. 7. Alekhno, E. A., Semenov, E. M., Sukochev, F. A. and Usachev, A. S. Order and Geometric Properties of the Set of Banach Limits, St. Petersburg Mathematical Journal, 2017, vol. 28, no. 3, pp. 299-321. DOI: doi.org/10.1090/spmj/1452. 8. Zvolinskii, R. E. and Semenov, E. M. Invariant Banach Limits and Their Convex Subsets, Mathematical Notes, 2022, vol. 112, no. 6, pp. 881-884. DOI: 10.1134/S0001434622110220. 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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