The projection spectral theorem and Jacobi fields

Abstract

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

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Article Information

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

Authors Information

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

Citation Example

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

BibTex

@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 = {http://mfat.imath.kiev.ua/article/?id=769},
}

References

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