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ABSTRACT RANDOM LINEAR OPERATORS ON PROBABILISTIC UNITARY SPACES

  • Tran, Xuan Quy (Department of Mathematics College of Science Thai Nguyen University) ;
  • Dang, Hung Thang (Department of Mathematics Hanoi University of Sciences) ;
  • Nguyen, Thinh (Department of Mathematics Hanoi University of Sciences)
  • 투고 : 2015.01.14
  • 발행 : 2016.03.01

초록

In this paper, we are concerned with abstract random linear operators on probabilistic unitary spaces which are a generalization of generalized random linear operators on a Hilbert space defined in [25]. The representation theorem for abstract random bounded linear operators and some results on the adjoint of abstract random linear operators are given.

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참고문헌

  1. A. A. Dorogovstev, On application of a Gaussian random operator to random elements, Theory Probab. Appl. 30 (1986), no. 4, 812-814.
  2. H. W. Engl, M. Z. Nashed, and M. Zuhair, Generalized inverses of random linear operators in Banach spaces, J. Math. Anal. Appl. 83 (1981), no. 2, 582-610. https://doi.org/10.1016/0022-247X(81)90143-8
  3. H. W. Engl and W. Romisch, Approximate solutions of nonlinear random operator equations: Convergence in distribution, Pacific J. Math. 120 (1985), no. 1, 55-77. https://doi.org/10.2140/pjm.1985.120.55
  4. T. Guo, Module homomorphisms on random normed modules, Northeast. Math. J. 12 (1996), no. 1, 102-114.
  5. T. Guo, Relations between some basic results derived from two kinds of topologies for a random locally convex module, J. Funct. Anal. 258 (2010), no. 9, 3024-3047. https://doi.org/10.1016/j.jfa.2010.02.002
  6. T. Guo and G. Shi, The algebraic structure of finitely generated $L^0$(${\cal{F}}$, K)-modules and the Helly theorem in random normed modules, J. Math. Anal. Appl. 381 (2011), no. 2, 833-842. https://doi.org/10.1016/j.jmaa.2011.03.069
  7. T. Guo and Y. Yang, Ekelands variational principle for an $L^-0$-valued function on a complete random metric space, J. Math. Anal. Appl. 389 (2012), no. 1, 1-14. https://doi.org/10.1016/j.jmaa.2011.11.025
  8. Wu. Mingzhu, The Bishop-Phelps theorem in complete random normed modules endows with the (${\varepsilon},{\lambda}$)-topology, J. Math. Anal. Appl. 391 (2012), 648-652. https://doi.org/10.1016/j.jmaa.2012.02.037
  9. M. Z. Nashed and H. W. Engl, Random generalized inverses and approximate solution of random equations, In: A. T. Bharucha-Reid (Ed.) Approximate Solution of random equations, pp. 149-210, Elsevier /North-Holland, New York-Amsterdam, 1979.
  10. H. Olga and P. Endre, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, 2001.
  11. B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier, New York, 1983.
  12. N. Shahzad, Random fixed points of K-set and pseudo-contractive random maps, Nonlinear Anal. 57 (2004), no. 2, 173-181. https://doi.org/10.1016/j.na.2004.02.006
  13. N. Shahzad, Random fixed point results for continuous pseudo-contractive random maps, Indian J. Math. 50 (2008), no. 2, 331-337.
  14. N. Shahzad and N. Hussain, Deterministic and random coincidence point results for f-nonexpansive maps, J. Math. Anal. Appl. 323 (2006), no. 2, 1038-1046. https://doi.org/10.1016/j.jmaa.2005.10.057
  15. A. V. Skorokhod, Random Linear Operators, Reidel Publishing Company, Dordrecht, 1984.
  16. D. H. Thang, Random Operator in Banach spaces, Probab. Math. Statist. 8 (1987), 155-157.
  17. D. H. Thang, The adjoint and the composition of random operators on a Hilbert space, Stoch. Stoch. Rep. 54 (1995), no. 1-2, 53-73. https://doi.org/10.1080/17442509508833998
  18. D. H. Thang, Random mappings on infinite dimensional spaces, Stoch. Stoch. Rep. 64 (1998), no. 1-2, 51-73. https://doi.org/10.1080/17442509808834157
  19. D. H. Thang, Series and spectral representations of random stable mappings, Stoch. Stoch. Rep. 64 (1998), no. 1-2, 33-49. https://doi.org/10.1080/17442509808834156
  20. D. H. Thang, Transforming random operators into random bounded operators, Random Oper. Stoch. Equ. 16 (2008), no. 3, 293-302.
  21. D. H. Thang and P. T. Anh, Random fixed points of completely random operators, Random Oper. Stoch. Equ. 21 (2013), no. 1, 1-20. https://doi.org/10.1515/rose-2013-0001
  22. D. H. Thang and T. N. Anh, On random equations and applications to random fixed point theorems, Random Oper. Stoch. Equ. 18 (2010), no. 3, 199-212. https://doi.org/10.1515/ROSE.2010.011
  23. D. H. Thang and T. M. Cuong, Some procedures for extending random operators, Random Oper. Stoch. Equ. 17 (2009), no. 4, 359-380.
  24. D. H. Thang and Ng. Thinh, Random bounded operators and their extension, Kyushu J. Math. 58 (2004), no. 2, 257-276. https://doi.org/10.2206/kyushujm.58.257
  25. D. H. Thang and Ng. Thinh, Generalized random linear operators on a Hilbert space, Stochastics 85 (2013), no. 6, 1040-1059. https://doi.org/10.1080/17442508.2012.736995
  26. D. H. Thang, Ng. Thinh, and Tr. X. Quy, Generalized random spectral measures, J. Theoret. Probab. 27 (2014), no. 2, 576-600. https://doi.org/10.1007/s10959-012-0461-0