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. Fund name: 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. ← Contents of issue


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