PDF(531 KB)
PDF(531 KB)
PDF(531 KB)
完备随机赋范模上的连续模同态半群
On Semigroups of Continuous Module Homomorphisms on Complete Random Normed Modules
本文首先利用完备随机赋范模的层次结构研究了一致连续模同态半群与其无穷小生成元之间的关系，并进一步给出几乎处处有界半群的指数刻画.在此基础上，建立几乎处处有界半群的微分和积分公式，推广了经典的结论.同时，用反例说明要求上述半群几乎处处有界的条件是必要的.
In this paper, using the stratification structures of complete random normed modules, we first study some relations between the uniformly continuous semigroups of continuous module homomorphisms and their infinitesimal generators, and further give an exponential characterization of such almost surely bounded semigroups. Then, based on this, we establish the differential and integral formulas for such almost surely bounded semigroups, which generalize the classical case. Meantime, the counterexample also shows that it is necessary to require the almost surely bounded assumption for such semigroups.
L0-Lipschitz / 随机赋范模 / 连续模同态半群 / 指数刻画 {{custom_keyword}} /
L0-Lipschitz / random normed module / semigroup of continuous module homomorphisms / exponential characterization {{custom_keyword}} /
[1] Dunford N., Schwartz J. T., Linear Operators (I), Interscience, New York, 1957.
[2] Guo T. X., The Theory of Probabilistic Metric Spaces and Its Applications to Random Functional Analysis (in China), Master's thesis, Xi'an Jiaotong University, Xi'an, 1989.
[3] Guo T. X., Random Metric Theory and Its Applications (in China), Ph. D. thesis, Xi'an Jiaotong University, Xi'an, 1992.
[4] Guo T. X., A new approach to random functional analysis, In:Proceedings of the first China postdoctral academic conference, The China National Defense and Industry Press, 1993, Beijing, pp. 1150-1154.
[5] Guo T. X., Relations between some basic results derived from two kinds of topologies for a random locally convex module, J. Funct. Anal., 2010, 258:3024-3047.
[6] Guo T. X., Recent progress in random metric theory and its applications to conditional risk measures, Sci. China Ser. A, 2010, 54(4):633-660.
[7] Guo T. X., On some basic theorems of continuous module homomorphisms between random normed modules, J. Funct. Space. Appl., 2013, Article ID 989102, 13 pp.
[8] Guo T. X., Zhao S. E., Zeng X. L., The relations among the three kinds of conditional risk measures, Sci. China Math., 2014, 57(8):1753-1764.
[9] Guo T. X., Zhang E. X., Wang Y. C., et al., Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations, J. Math. Anal. Appl., 2020, 483(2):123644, 30 pp.
[10] Guo T. X., Zhang E. X., Wu M. Z., et al., On random convex analysis, J. Nonlinear and Convex Anal., 2017, 18(11):1967-1996.
[11] Guo T. X., Zhang X., Stone's representation theorem of a group of strongly continuous unitary operators on complex complete random inner product modules, Sci. Sin. Math., 2012, 42:181-202.
[12] Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, World Publishing Company, Beijing, 2006.
[13] Schweizer B., Sklar A., Probabilistic Metric Spaces, Elsevier, New York, 1983; Dover Publications, New York, 2005.
[14] Tang Y. H., Random spectral theorems of self-adjoint random linear operators on complete complex random inner product modules, Linear and Multilinear Algebra, 2013, 61(3):409-416.
[15] Tang Y. H., The Wintner theorem in unital complete random normed algebras, Bull. Korean Math. Soc., 2013, 50(6):1973-1979.
[16] Thang D. H., Son T. C., Thinh Ng., Semigroups of continuous module homomorphisms on complex complete random normed modules, Lith. Math. J., 2019, 59(2):229-250.
[17] Wu M. Z., The Bishops-Phelps theorem in complete random normed modules endowed with the (ε, λ)-topology, J. Math. Anal. Appl., 2012, 391:648-652.
[18] Wu M. Z., Farkas' lemma in random locally convex modules and Minkowski-Weyl type results in L0(F, Rn), J. Math. Anal Appl., 2013, 404:300-309.
[19] Wu M. Z., Guo T. X., A counterexample shows that not every locally L0-convex topology is necessarily induced by a family of L0-seminorms, 2015, arXiv:1501.04400v1.
[20] Zhang X., On mean ergodic semigroups of random linear operators, Proc. Japan Acad. Ser. A, 2012, 88:53-58.
[21] Zhang X., Liu M., A characterization for a complete random normed module to be mean ergodic, Acta Math. Sin. Engl. Ser., 2017, 33(7):899-910.
[22] Zhang X., Liu M., On almost surely bounded semigroups of random linear operators, J. Math. Phy., 2013, 54(5):053517, 10 pp.
[23] Zhang X., Liu M., Guo T. X., The Hille-Yosida generation theorem for almost surely bounded C0-semigroups of continuous module homomorphisms, J. Nonlinear and Convex Anal., 2020, 18(11):1995-2009.
[24] Zhao S. E., Guo T. X., The random subreflexivity of complete random normed modules, Internat. J. Math., 2012, 23(3):14 pp.
[25] Zhao S. E., Shi G., A geometric form of the Hahn-Banach extension theorem for L0-linear functions and the Goldstine-Weston theorem in random normed modules, Sci. Sin. Math., 2011, 41(9):827-836.
国家自然科学基金资助项目（11301380）；教育部人文社科基金资助项目（20YJC790174）；天津市自然科学基金资助项目（18JCYBJC18900）
/
| 〈 |
|
〉 |
