Contemporary Mathematics. Fundamental DirectionsContemporary Mathematics. Fundamental Directions2413-36392949-0618Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)3007710.22363/2413-3639-2021-67-4-654-667Research ArticleStatistical Ergodic Theorem in Symmetric Spaces for Infinite MeasuresVekslerA. S.aleksandr.veksler@micros.uzChilinV. I.vladimirchil@gmail.comInstitute of Mathematics of the Academy of Sciences of Uzbekistan3012202167465466724012022Copyright © 2022, Contemporary Mathematics. Fundamental Directions2022<p style="text-align: justify;">Let <em>(Ω,μ)</em> be a measurable space with σ -finite continuous measure, <em>μ(Ω)=∞</em>. <em>A linear operator T:L<sub>1</sub>(Ω)+L<sub>∞</sub>(Ω)→L<sub>1</sub>(Ω)+L<sub>∞</sub>(Ω)</em> <em>is called the Dunford-Schwartz operator if ||T(f)||<sub>1</sub>||f||<sub>1</sub></em> (respectively,<em> ||T(f)||<sub>∞</sub>||f||<sub>∞</sub></em>) <em>for all f∈L<sub>1</sub>(Ω)</em> (respectively, <em>f∈L<sub>∞</sub>(Ω)</em>). <em>{T<sub>t</sub>}<sub>t0</sub></em><em>is a strongly continuous in L<sub>1</sub>(Ω)</em> semigroup of Dunford-Schwartz operators, then each operator <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><msub><mi>A</mi><mi>t</mi></msub><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>t</mi></mfrac><msubsup><mo>∫</mo><mn>0</mn><mi>t</mi></msubsup><mrow><msub><mi>T</mi><mi>s</mi></msub><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mi>d</mi><mi>s</mi><mo>∈</mo><msub><mi>L</mi><mn>1</mn></msub><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><annotation encoding="LaTeX">{{{A_t(f)} ={\frac{1}{t}} {\int_0^t} {T_s(f)} ds \in L_1(\Omega)}}</annotation></semantics></math> <em>has a unique extension to the Dunford-Schwartz operator, which is also denoted by A<sub>t</sub>, t0</em>. <em>It is proved that in the completely symmetric space of measurable functions on (Ω,μ)</em> <em>the means A<sub>t</sub></em> <em>converge strongly as t→+∞</em> <em>for each strongly continuous in L<sub>1</sub>(Ω)</em> <em>semigroup {T<sub>t</sub>}<sub>t0</sub></em> <em>of Dunford-Schwartz operators if and only if the norm ||.||<sub>E(Ω)</sub> </em>is order continuous.</p>[Векслер А. С. Эргодическая теорема в симметричных пространствах// Сиб. мат. ж. - 1985. - 26, № 4. - С. 189-191.][Векслер А. С. Статистическая эргодическая теорема в несепарабельных симметричных пространствах функций// Сиб. мат. ж. - 1988. -29, № 3. - С. 183-185.][Векслер А. С. Статистические эргодические теоремы в перестановочно-инвариантных пространствах измеримых функций. - Beau Bassin: Lambert Academic Publishing, 2018.][Векслер А. С., Федоров А. Л. Симметрические пространства и статистические эргодические теоремы для автомофизмов и потоков. - Ташкент: ФАН, 2016.][Канторович Л. В., Акилов Г. П. Функциональный анализ. - М.: Наука, 1977.][Bennett C., Sharpley R. Interpolation of operators. - Boston, etc.: Academic Press Inc., 1988.][Chilin V., Comez¨ D., Litvinov S. Individual ergodic theorems for infinite measure// ArXiv. - 2019. - 1907.04678v1 [math.FA].][Chilin V., Litvinov S. Noncommutative weighted individual ergodic theorems with continuous time// ArXiv. - 2018. - 1809.01788v1 [math.FA].][Chilin V., Litvinov S. Almost uniform and strong convergences in ergodic theorems for symmetric spaces// Acta Math. Hungar. - 2019. -157, № 1. - С. 229-253.][Chilin V., Litvinov S. Noncommutative weighted individual ergodic theorems with continuous time// Infin. Dimens. Anal. Quantum Probab. Relat. Top. - 2020. -23, № 2. - 2050013.][Chilin V. I., Veksler A. S. Mean ergodic theorem in function symmetric spaces for infinite measure// Uzb. Math. J. - 2018. - № 1. - С. 35-46.][Dodds P. G., Dodds T. K., Pagter B. Noncommutative Kothe duality// Trans. Am. Math. Soc. - 1993. -¨ 339. - С. 717-750.][Dodds P. G., Dodds T. K., Sukochev F. A. Banach-Saks properties in symmetric spaces of measurable operators// Studia Math. - 2007. -178. - С. 125-166.][Dunford N., Schwartz J. T. Linear operators. Part I: General theory. - New York, etc.: John Willey & Sons, 1988.][Garsia A. Topics in almost everywhere convergence. - Chicago: Markham Publishing Company, 1970.][Krein S. G., Petunin Ju. I., Semenov E. M. Interpolation of linear operators. - Providence: Am. Math. Soc., 1982.][Krengel U. Ergodic theorems. - Berlin-New York: Walter de Gruyter, 1985.][Rubshtein B. A., Muratov M. A., Grabarnik G. Ya., Pashkova Yu. S. Foundations of symmetric spaces of measurable functions. Lorentz, Marcinkiewicz and Orlicz spaces. - Cham: Springer, 2016.][Sukochev F., Veksler A. The Mean Ergodic Theorem in symmetric spaces// C.R. Acad. Sci. Paris. Ser. I. - 2017. -355. - С. 559-562.][Sukochev F., Veksler A. The Mean Ergodic Theorem in symmetric spaces// Studia Math. - 2019. -245, № 3. - С. 229-253.][Vladimirov D. A. Boolean algebras in analysis. - Dordrecht: Kluwer Academic Publishers, 2002.][Yosida K. Functional analysis. - Berlin-Gottingen-Heidelberg: Springer Verlag, 1965.¨]