Abstract: The paper investigates general averaging operators defined on the space of all bounded real sequences. It proves sufficient conditions for these operators to map the space of almost convergent sequences to itself and to the subspace of convergent sequences. As a corollary of these results, it obtains sufficient conditions for the existence of Banach limits invariant with respect to these averaging operators. It studies the properties of subsets of such Banach limits. In particular, it computes their radii. For several specific averaging operators the paper calculates a functional characteristic. In all cases it turns out to be continuous, but distinct from that of Cesaro invariant Banach limits. This shows that the sets of Banach limits invariant with respect to these averaging operators are pairwise disjoint and with the set of Cesaro invariant Banach limits. A result in the opposite direction establishes sufficient conditions for the coincidence of the sets of Banach limits invariant with respect to distinct averaging operators.
For citation: Zvolinskii, R. E. and Usachev, A. S. General Averaging Operators and Banach Limits, Vladikavkaz Math. J., 2026, vol. 28, no. 2, pp. 80-90 (in Russian). DOI 10.46698/d4220-2436-3949-r
1. Mazur, S. O Metodach Sumowalnosci, Annales Polonici Mathematici (Supplement), 1929, pp. 102-107.
2. Banach, S. Theorie des Operations Lineaires, Editions Jacques Gabay, Sceaux, 1993. Reprint of the 1932 original, 237 p.
3. Lorentz, G. G. A Contribution to the Theory of Divergent Sequences, Acta Mathematica,
1948, vol. 80, pp. 167-190. DOI: 10.1007/BF02393648.
4. Sucheston, L. Banach Limits, The American Mathematical Monthly,
1967, vol. 74, no. 3, pp. 308-311. DOI: 10.2307/2316038.
5. 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.
6. Semenov, E., Sukochev, F., Usachev, A. and Zanin, D.
Invariant Banach Limits and Applications to Noncommutative Geometry,
Pacific Journal of Mathematics, 2020, vol. 306, no. 1, pp. 357-373.
DOI: 10.2140/pjm.2020.306.357.
7. Alekhno, E., Semenov, E., Sukochev, F. and Usachev, A. Order and Geometric Properties
of the Set of Banach Limits, St. Petersburg Mathematical Journal, 2017,
vol. 28, no. 3, pp. 299-321. DOI: 10.1090/spmj/1452.
8. Semenov, E., Sukochev, F., Usachev, A. and Zanin, D. Dilation Invariant Banach Limits,
Indagationes Mathematicae, 2020, vol. 31, no. 5, pp. 885-892.
DOI: 10.1016/j.indag.2019.12.003.
9. Semenov, E. M., Sukochev, F. A. and Usachev, A. S.
The Main Classes of Invariant Banach Limits,
Izvestiya: Mathematics, 2019, vol. 83, no. 1, pp. 124-150. DOI: 10.1070/IM8704.
10. Kalton, N., Sedaev, A. and Sukochev, F. Fully Symmetric Functionals on a Marcinkiewicz Space
are Dixmier Traces, Advances in Mathematics, 2011, vol. 226, no. 4, pp. 3540-3549.
DOI: 10.1016/j.aim.2010.09.025.
11. Agnew, R. P. On Deferred Cesaro Means, Annals of Mathematics,
1932, vol. 33, no. 3, pp. 413-421. DOI: 10.2307/1968524.
12. Alekhno, E., Semenov, E., Sukochev, F. and Usachev, A. Invariant Banach Limits and Their Extreme Points,
Studia Mathematica, 2018, vol. 242, no. 1, pp.79-107. DOI: 10.4064/sm8785-5-2017.
13. Bennett, G. and Kalton, N. J. Consistency Theorems for Almost Convergence,
Transactions of the American Mathematical Society, 1974, vol. 198, pp. 23-43.
DOI: 10.1090/S0002-9947-1974-0352932-X.
14. Aliprantis, C. D. and Border, K. C. Infinite Dimensional Analysis.
A hitchhiker's Guide, Berlin, Springer, 2006, 703 p. DOI: 10.1007/3-540-29587-9.
15. Usachev, A. S. Transformations in the Space of Almost Convergent Sequences,
Siberian Mathematical Journal, 2008, vol. 49, no. 6, pp. 1136-1137. DOI: 10.1007/s11202-008-0110-0.
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