On Random Normal Operators and Their Spectral Measures

  • Păstorel GaşparEmail author


The main aim of this paper is to introduce and study the subclass of not necessarily continuous, normal random operators, establishing connections with other subclasses of random operators, as well as with the existing concept of random projection operator-valued measure. Hence, after recalling some basic facts regarding random operators on a complex separable Hilbert space, theorems about transforming the class of not necessarily continuous decomposable random operators into the class of purely contractive random operators are proved. These are applied to obtain integral representations for not necessarily continuous normal or self-adjoint random operators on a Hilbert space with respect to the corresponding random projection operator-valued measures.


Stochastic mappings Random operator Normal random operator Self-adjoint random operator Random projection operator-valued measure Integral representation 

Mathematics Subject Classification

60G60 46F12 42B10 



I express hereby my gratitude to the referees for their careful reading of the manuscript and their comments and suggestions, which led to considerable improvement of the paper. This work was partially supported by the European Union through the European Regional Development Fund (ERDF) under the Competitiveness Operational Program (COP) through a project entitled Novel Bio-inspired Cellular Nano-architectures, Grant POC-A1.1.4-E-2015 nr. 30/01.09.2016.


  1. 1.
    Azzouz, A., Messirdi, B., Djellouli, G.: New results on the closedness of the product and sum of closed linear operators. Bull. Math. Anal. Appl. 3(2), 151–158 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Gaşpar, P., Popa, L.: Stochastic mappings and random distribution fields. A correlation approach correlation approach. Monatshefte für Mathematik.
  3. 3.
    Gaşpar, P., Popa, L.: Stochastic mappings and random distribution fields II. Stationarity. Mediterr. J. Math. 13(4), 2229–2252 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gaşpar, P., Popa, L.: Stochastic mappings and random distribution fields III. Module propagators and uniformly bounded linearly stationarity. J. Math. Anal. Appl. 435(2), 1229–1240 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gaşpar, P., Sida, L.: Periodically correlated multivariate second order random distribution fields and stationary cross correlatedness. J. Funct. Anal. 267, 2253–2263 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gustafson, K., Mortad, M.H.: Unbounded products of operators and connections to Dirac-type operators. Bul. Sci. Math. 138(5), 626–642 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hackenbroch, W.: Random Operators (M. Megan, N. Suciu, eds.), The National Conference on Mathematical Analysis and Applications,West University of Timişoara, pp. 135–146 (2000)Google Scholar
  8. 8.
    Hackenbroch, W.: Point localization and spectral theory for symmetric random operators. Arch. Math. 92, 485–492 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kakihara, Y.: Multidimensional Second Order Stochastic Processes. World Scientific Publ. Comp, River Edge (1997)CrossRefGoogle Scholar
  10. 10.
    Kaufmann, W.E.: Closed operators and pure contractions in Hilbert space. Proc. AMS 87(1), 83–87 (1983)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lennon, M.J.J.: On sums and products of unbounded operators in hilbert space. Trans. AMS 198, 273–285 (1974)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)zbMATHGoogle Scholar
  13. 13.
    Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory, Operator Theory: Advances and Applications, vol. 37. Springe, Basel (1990)CrossRefGoogle Scholar
  14. 14.
    Schmüdgen, K.: Unbounded Self-adjoint Operators on Hilbert space, Graduate Texts in Mathematics, Vol. 265. Springer, Dodrecht (2012)Google Scholar
  15. 15.
    Skorohod, A.V.: Random Linear Operators. D. Reidel Publ. Comp, Dardrecht (1984)CrossRefGoogle Scholar
  16. 16.
    Sz.-Nagy, B., Foiaş, C., Bercovici, H., Kérchy, L.: Harmonic Analysis of Operators on Hilbert Space. Springer, New York, Revised and Enlarged Edition, (2010)Google Scholar
  17. 17.
    Thang, D.H.: Random operators in Banach space. Probab. Math. Stat. 8, 155–157 (1987)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Thang, D.H.: The adjoint and the composition of random operators. Stoch. Stoch. Rep. 54, 53–73 (1995)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Thang, D.H.: Series and spectral representations of random stable mappings. Stoch. Stoch. Rep. 64, 33–49 (1998)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Thang, D.H.: Random mappings on infinite dimensional spaces. Stoch. Stoch. Rep. 64, 51–73 (1998)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Thang, D.H., Ng. Thinh, : Random bounded operators and their extension. Kyushu J. Math. 58(25), 257–276 (2004)Google Scholar
  22. 22.
    Thang, D.H.: Transforming random operators into random bounded operators. Random Oper. Stoch. Equ. 16, 293–302 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Thang, D.H., Thinh, N.: Generalized random operators on a Hilbert space. Stoch: An Intern. J. Probab. Stoch. Proc. 85(6), 1040–1059 (2013)CrossRefGoogle Scholar
  24. 24.
    Thang, D.H., Thinh, N., Quy, T.X.: Generalized random spectral measures. J. Theor. Probab. 27, 576–600 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Thang, D.H., Quy, T.X.: On the Spectral Theorem for Random Operators. Southeast Asian Bull. Math. 41, 271–286 (2017)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Faculty of Exact Sciences“Aurel Vlaicu” UniversityAradRomania

Personalised recommendations