Statistical Ergodic Theorem in Symmetric Spaces for Infinite Measures

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Abstract

Let (Ω,μ) be a measurable space with σ -finite continuous measure, μ(Ω)=∞. A linear operator T:L1(Ω)+L(Ω)→L1(Ω)+L(Ω) is called the Dunford-Schwartz operator if ||T(f)||1<||f||1 (respectively, ||T(f)||<||f||) for all f∈L1(Ω) (respectively, f∈L(Ω)).  {Tt}t>0 is a strongly continuous in L1(Ω) semigroup of Dunford-Schwartz operators, then each operator At(f)=1t0tTs(f)dsL1(Ω){{{A_t(f)} ={\frac{1}{t}} {\int_0^t} {T_s(f)} ds \in L_1(\Omega)}} has a unique extension to the Dunford-Schwartz operator, which is also denoted by At, t>0. It is proved that in the completely symmetric space of measurable functions on (Ω,μ) the means At converge strongly as t→+∞ for each strongly continuous in L1(Ω) semigroup {Tt}t>0 of Dunford-Schwartz operators if and only if the norm ||.||E(Ω) is order continuous.

About the authors

A. S. Veksler

Institute of Mathematics of the Academy of Sciences of Uzbekistan

Author for correspondence.
Email: aleksandr.veksler@micros.uz
Tashkent, Uzbekistan

V. I. Chilin

Institute of Mathematics of the Academy of Sciences of Uzbekistan

Email: vladimirchil@gmail.com
Tashkent, Uzbekistan

References

  1. Векслер А. С. Эргодическая теорема в симметричных пространствах// Сиб. мат. ж. - 1985. - 26, № 4. - С. 189-191.
  2. Векслер А. С. Статистическая эргодическая теорема в несепарабельных симметричных пространствах функций// Сиб. мат. ж. - 1988. -29, № 3. - С. 183-185.
  3. Векслер А. С. Статистические эргодические теоремы в перестановочно-инвариантных пространствах измеримых функций. - Beau Bassin: Lambert Academic Publishing, 2018.
  4. Векслер А. С., Федоров А. Л. Симметрические пространства и статистические эргодические теоремы для автомофизмов и потоков. - Ташкент: ФАН, 2016.
  5. Канторович Л. В., Акилов Г. П. Функциональный анализ. - М.: Наука, 1977.
  6. Bennett C., Sharpley R. Interpolation of operators. - Boston, etc.: Academic Press Inc., 1988.
  7. Chilin V., Comez¨ D., Litvinov S. Individual ergodic theorems for infinite measure// ArXiv. - 2019. - 1907.04678v1 [math.FA].
  8. Chilin V., Litvinov S. Noncommutative weighted individual ergodic theorems with continuous time// ArXiv. - 2018. - 1809.01788v1 [math.FA].
  9. Chilin V., Litvinov S. Almost uniform and strong convergences in ergodic theorems for symmetric spaces// Acta Math. Hungar. - 2019. -157, № 1. - С. 229-253.
  10. Chilin V., Litvinov S. Noncommutative weighted individual ergodic theorems with continuous time// Infin. Dimens. Anal. Quantum Probab. Relat. Top. - 2020. -23, № 2. - 2050013.
  11. Chilin V. I., Veksler A. S. Mean ergodic theorem in function symmetric spaces for infinite measure// Uzb. Math. J. - 2018. - № 1. - С. 35-46.
  12. Dodds P. G., Dodds T. K., Pagter B. Noncommutative Kothe duality// Trans. Am. Math. Soc. - 1993. -¨ 339. - С. 717-750.
  13. 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.
  14. Dunford N., Schwartz J. T. Linear operators. Part I: General theory. - New York, etc.: John Willey & Sons, 1988.
  15. Garsia A. Topics in almost everywhere convergence. - Chicago: Markham Publishing Company, 1970.
  16. Krein S. G., Petunin Ju. I., Semenov E. M. Interpolation of linear operators. - Providence: Am. Math. Soc., 1982.
  17. Krengel U. Ergodic theorems. - Berlin-New York: Walter de Gruyter, 1985.
  18. 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.
  19. Sukochev F., Veksler A. The Mean Ergodic Theorem in symmetric spaces// C.R. Acad. Sci. Paris. Ser. I. - 2017. -355. - С. 559-562.
  20. Sukochev F., Veksler A. The Mean Ergodic Theorem in symmetric spaces// Studia Math. - 2019. -245, № 3. - С. 229-253.
  21. Vladimirov D. A. Boolean algebras in analysis. - Dordrecht: Kluwer Academic Publishers, 2002.
  22. Yosida K. Functional analysis. - Berlin-Gottingen-Heidelberg: Springer Verlag, 1965.¨

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