Open Access

The projection spectral theorem and Jacobi fields


We review several applications of Berezansky's projection spectral theorem to Jacobi fields in a symmetric Fock space, which lead to L\'evy white noise measures.

Key words: Free L´evy white noise, Gaussian measure, Jacobi field, L´evy white noise, Poisson measure, projection spectral theorem

Full Text

Article Information

TitleThe projection spectral theorem and Jacobi fields
SourceMethods Funct. Anal. Topology, Vol. 21 (2015), no. 2, 188–198
MathSciNet   3407910
zbMATH 06533476
Milestones  Received 09/01/2015
CopyrightThe Author(s) 2015 (CC BY-SA)

Authors Information

Eugene Lytvynov
Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K.

Export article

Save to Mendeley

Citation Example

Eugene Lytvynov, The projection spectral theorem and Jacobi fields, Methods Funct. Anal. Topology 21 (2015), no. 2, 188–198.


@article {MFAT769,
    AUTHOR = {Lytvynov, Eugene},
     TITLE = {The projection spectral theorem and Jacobi fields},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {21},
      YEAR = {2015},
    NUMBER = {2},
     PAGES = {188–198},
      ISSN = {1029-3531},
  MRNUMBER = {3407910},
 ZBLNUMBER = {06533476},
       URL = {},


  1. Ju. M. Berezans′kii, Expansions in eigenfunctions of selfadjoint operators, American Mathematical Society, Providence, R.I., 1968.  MathSciNet 
  2. Yu. M. Berezanskii, The projection spectral theorem, Uspekhi Mat. Nauk 39 (1984), no. 4(238), 3-52.  MathSciNet 
  3. Yu. M. Berezanskii, On the projection spectral theorem, Ukrain. Mat. Zh. 37 (1985), no. 2, 146-154, 269.  MathSciNet 
  4. Yu. M. Berezanskii, Selfadjoint operators in spaces of functions of infinitely many variables, American Mathematical Society, Providence, RI, 1986.  MathSciNet 
  5. Yu. M. Berezansky, Spectral approach to white noise analysis, in: Dynamics of complex and irregular systems (Bielefeld, 1991), World Sci. Publ., River Edge, NJ, 1993.  MathSciNet 
  6. Yurij M. Berezansky, Poisson measure as the spectral measure of Jacobi field, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), no. 1, 121-139.  MathSciNet  CrossRef
  7. Y. M. Berezansky, Y. G. Kondratiev, Spectral methods in infinite-dimensional analysis. Vol. 1, Kluwer Academic Publishers, Dordrecht, 1995.  MathSciNet  CrossRef
  8. Ju. M. Berezans′kii, V. D. Kovsmanenko, Axiomatic field theory in terms of operator Jacobi matrices, Teoret. Mat. Fiz. 8 (1971), no. 2, 175-191.  MathSciNet 
  9. Yu. M. Berezanskii, V. O. Livinskii, E. V. Litvinov, A spectral approach to the analysis of white noise, Ukrain. Mat. Zh. 46 (1994), no. 3, 177-197.  MathSciNet  CrossRef
  10. Marek Bo\.zejko, Eugene Lytvynov, Meixner class of non-commutative generalized stochastic processes with freely independent values. I. A characterization, Comm. Math. Phys. 292 (2009), no. 1, 99-129.  MathSciNet  CrossRef
  11. Marek Bo\.zejko, Eugene Lytvynov, Meixner class of non-commutative generalized stochastic processes with freely independent values II. The generating function, Comm. Math. Phys. 302 (2011), no. 2, 425-451.  MathSciNet  CrossRef
  12. S. Das, Orthogonal Decompositions for Generalized Stochastic Processes with Independent Values, PhD Dissertation, Swansea University, 2012.
  13. I. M. Gelfand, Ya. N. Vilenkin, Generalized functions. Vol. 4: Applications of harmonic analysis, Academic Press, New York - London, 1964, 1964.  MathSciNet 
  14. Takeyuki Hida, Hui-Hsiung Kuo, Jurgen Potthoff, Ludwig Streit, White noise, Kluwer Academic Publishers Group, Dordrecht, 1993.  MathSciNet  CrossRef
  15. Yoshifusa Ito, Izumi Kubo, Calculus on Gaussian and Poisson white noises, Nagoya Math. J. 111 (1988), 41-84.  MathSciNet  CrossRef
  16. Olav Kallenberg, Random measures, Akademie-Verlag, Berlin; Academic Press, London-New York, 1976.  MathSciNet 
  17. Y. Kondratiev, E. Lytvynov, A. Vershik, Laplace operators on the cone of Radon measures, in preparation.
  18. E. W. Lytvynov, Multiple Wiener integrals and non-Gaussian white noises: a Jacobi field approach, Methods Funct. Anal. Topology 1 (1995), no. 1, 61-85.  MathSciNet 
  19. Eugene Lytvynov, Fermion and boson random point processes as particle distributions of infinite free Fermi and Bose gases of finite density, Rev. Math. Phys. 14 (2002), no. 10, 1073-1098.  MathSciNet  CrossRef
  20. Eugene Lytvynov, Determinantal point processes with $J$-Hermitian correlation kernels, Ann. Probab. 41 (2013), no. 4, 2513-2543.  MathSciNet  CrossRef
  21. Roland Speicher, Free probability theory and non-crossing partitions, S\em. Lothar. Combin. 39 (1997), Art.\ B39c, 38 pp.\ (electronic).  MathSciNet 
  22. Natalia Tsilevich, Anatoly Vershik, Marc Yor, An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process, J. Funct. Anal. 185 (2001), no. 1, 274-296.  MathSciNet  CrossRef

All Issues