Abstract
In this paper, we first study the mean ergodicity of random linear operators using some techniques of measure theory and L 0-convex analysis. Then, based on this, we give a characterization for a complete random normed module to be mean ergodic.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Albanese, A. A., Bonet, J., Ricker, W. J.: C 0-semigroups and mean ergodic operators in a class of Fréchet spaces. J. Math. Anal. Appl., 365, 142–157 (2010)
Beck, A., Schwartz, J. T.: A vector-valued random ergodic theorem. Proc. Amer. Math. Soc., 8, 1049–1059 (1957)
Diestel, J., Uhl Jr., J. J.: Vector Measures, Amer. Math. Soc., Rhodo Island: Providence, 1977
Dunford, N., Schwartz, J. T.: Linear Operators (I), Interscience, New York, 1957
Fonf, V. P., Lin, M., Wojtaszczyk, P.: Ergodic characterizations of reflexivity of Banach spaces. J. Funct. Anal., 187, 146–162 (2001)
Guo, T. X.: Random Metric Theory and Its Applications. Ph.D thesis, Xi’an Jiaotong University, Xi’an, 1992
Guo, T. X.: A new approach to random functional analysis. Proceedings of the first China doctoral academic conference. The China National Defense and Industry Press, Beijing, 1993
Guo, T. X.: Some basic theories of random normed linear spaces and random inner product spaces. Acta Anal. Funct. Appl., 1(2), 160–184 (1999)
Guo, T. X.: Relations between some basic results derived from two kinds of topologies for a random locally convex module. J. Funct. Anal., 258(9), 3024–3047 (2010)
Guo, T. X., Li, S. B.: The James theorem in complete random normed modules. J. Math. Anal. Appl., 308, 257–265 (2005)
Guo, T. X., Zhang, X.: Stone’s representation theorem of a group of random unitary operators on complete complex random inner product modules (in Chinese). Sci. Sin. Math., 42(3), 181–202 (2012)
Guo, T. X.: Recent progress in random metric theory and its applications to conditional risk measures. Sci. China Ser. A, 54(4), 633–660 (2011)
Guo, T. X., Zhu, L. H.: A characterization of continuous module homomorphisms on random seminormed modules and its applications. Acta Math. Sin., Engl. Ser., 19(1), 201–208 (2003)
Guo, T. X.: The relation of Banach–Alaoglu theorem and Banach–Bourbaki–Kakutani–Šmulian theorem in complete random normed modules to stratification structure. Sci. China Ser. A, 51(9), 1651–1663 (2008)
Guo, T. X., Zhao, S. E., Zeng, X. L.: The relations among the three kinds of conditional risk measures. Sci. China Math., 57(8), 1753–1764 (2014)
Guo, T. X., Zhao, S. E., Zeng, X. L.: Random convex analysis (I): separation and Fenchel–Moreau duality in random locally convex modules (in Chinese). Sci. Sin. Math., 45(12), 1961–1980 (2015)
Guo, T. X., Zhao, S. E., Zeng, X. L.: Random convex analysis (II): continuity and subdifferentiability theorems in L 0-pre-barreled random locally convex modules (in Chinese). Sci. Sin. Math., 45(5), 647–662 (2015)
Guo, T. X., Zhang, E. X., Wu, M. Z., et al.: On random convex analysis. J. Nonlinear Conv. Anal., accepted, arXiv:1603.07074 (2016)
Guo, T. X.: On some basic theorems of continuous module homomorphisms between random normed modules. J. Funct. Space Appl., Article ID 989102, 13 pages (2013)
Schweizer, B., Sklar, A.: Probabilistic Metric Spaces, Elsevier, New York, 1983; reissued by Dover Publications, New York, 2005
Skorohod, A. V.: Random Linear Operators. Holland: D. Reidel Publishing Company, 1984
Wu, M. Z.: The Bishops–Phelps theorem in complete random normed modules endowed with the (ε, λ)-topology. J. Math. Anal. Appl., 391(2), 648–952 (2012)
Wu, M. Z.: Farkas’ lemma in random locally convex modules and Minkowski–Weyl type results in L 0(F,R n). J. Math. Anal. Appl., 404(2), 300–309 (2013)
Wu, M. Z., Guo, T. X.: A counterexample shows that not every locally L 0-convex topology is necessarily induced by a family of L 0-seminorms. arXiv:1501.04400v1 (2015)
Zapata, J. M.: On characterization of locally L 0-convex topologies induced by a family of L 0-seminorms. J. Convex Anal., 24(1), to appear (2017)
Zeng, X. L.: Various expressions for modulus of random convexity. Acta Math. Sin., Engl. Ser., 29(2), 263–280 (2013)
Zhang, X.: On mean ergodic semigroups of random linear operators. Proc. Japan Acad. Ser. A, 88(4), 53–58 (2012)
Zhang, X.: On conditional mean ergodic semigroups of random linear operators. J. Inequal. Appl., 150, 1–10 (2012)
Zhang, X., Liu, M.: On almost surely bounded semigroups of random linear operators. J. Math. Phy., 54(5), 1–10 (2013)
Zhang, X., Guo, T. X.: The mean ergodic theorem on random reflexive random normed modules. Adv. Math. Sinica, 41(1), 21–30 (2012)
Acknowledgements
The authors would like to express his sincere gratitude to Professor Guo Tiexin for his constant care and help. The first author would also like to express his sincere gratitude to associate professor Zhao Shien for his invaluable suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China (Grant Nos. 11301380, 11401399 and 11301568) and the Higher School Science and Technology Development Fund Project in Tianjin (Grant No. 20131003)
Rights and permissions
About this article
Cite this article
Zhang, X., Liu, M. A characterization for a complete random normed module to be mean ergodic. Acta. Math. Sin.-English Ser. 33, 899–910 (2017). https://doi.org/10.1007/s10114-017-6444-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-017-6444-4