Open Access

The investigation of Bogoliubov functionals by operator methods of moment problem


Abstract

The article is devoted to a study of Bogoliubov functionals by using methods of the operator spectral theory being applied to the classical power moment problem. Some results, similar to corresponding ones for the moment problem, where obtained for such functionals. In particular, the following question was studied: under what conditions a sequence of nonlinear functionals is a sequence of Bogoliubov functionals.

Key words: Projection spectral theorem, Kondratiev--Kuna convolution, Lenard transform, Bogolyubov functional


Full Text





Article Information

TitleThe investigation of Bogoliubov functionals by operator methods of moment problem
SourceMethods Funct. Anal. Topology, Vol. 22 (2016), no. 1, 1-47
MathSciNet MR3522860
zbMATH 06630281
MilestonesReceived 21/04/2011; Revised 03/08/2015
CopyrightThe Author(s) 2016 (CC BY-SA)

Authors Information

Yu. M. Berezansky
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

V. A. Tesko
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine


Google Scholar Metrics

Citing articles in Google Scholar
Similar articles in Google Scholar

Export article

Save to Mendeley



Citation Example

Yu. M. Berezansky and V. A. Tesko, The investigation of Bogoliubov functionals by operator methods of moment problem, Methods Funct. Anal. Topology 22 (2016), no. 1, 1-47.


BibTex

@article {MFAT593,
    AUTHOR = {Berezansky, Yu. M. and Tesko, V. A.},
     TITLE = {The investigation of Bogoliubov functionals by operator methods of moment problem},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {22},
      YEAR = {2016},
    NUMBER = {1},
     PAGES = {1-47},
      ISSN = {1029-3531},
  MRNUMBER = {MR3522860},
 ZBLNUMBER = {06630281},
       URL = {http://mfat.imath.kiev.ua/article/?id=593},
}


References

  1. N. I. Akhiezer, The classical moment problem and some related questions in analysis, Translated by N. Kemmer, Hafner Publishing Co., New York, 1965.  MathSciNet
  2. Yu. M. Berezanskii, Generalization of Bochners theorem to expansions according to eigenfunctions of partial differential equations, Dokl. Akad. Nauk SSSR (N.S.) 110 (1956), 893-896.  MathSciNet
  3. Ju. M. Berezanskii, A generalization of a multidimensional theorem of Bochner, Dokl. Akad. Nauk SSSR 136 (1961), 1011-1014.  MathSciNet
  4. Ju. M. Berezans′kii, Expansions in eigenfunctions of selfadjoint operators, Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968.  MathSciNet
  5. Yu. M. Berezanskii, Selfadjoint operators in spaces of functions of infinitely many variables, Translations of Mathematical Monographs, vol. 63, American Mathematical Society, Providence, RI, 1986.  MathSciNet
  6. Yurij M. Berezansky, Commutative Jacobi fields in Fock space, Integral Equations Operator Theory 30 (1998), no. 2, 163-190.  MathSciNet CrossRef
  7. Yurij M. Berezansky, Some generalizations of the classical moment problem, Integral Equations Operator Theory 44 (2002), no. 3, 255-289.  MathSciNet CrossRef
  8. Yu. M. Berezanskii, The generalized moment problem associated with correlation measures, Funktsional. Anal. i Prilozhen. 37 (2003), no. 4, 86-91.  MathSciNet CrossRef
  9. Y. M. Berezansky and Y. G. Kondratiev, Spectral methods in infinite-dimensional analysis. Vol. 1, Mathematical Physics and Applied Mathematics, vol. 12/1, Kluwer Academic Publishers, Dordrecht, 1995.  MathSciNet CrossRef
  10. Yuri M. Berezansky, Yuri G. Kondratiev, Tobias Kuna, and Eugene Lytvynov, On a spectral representation for correlation measures in configuration space analysis, Methods Funct. Anal. Topology 5 (1999), no. 4, 87-100.  MathSciNet MFAT Article
  11. Yu. M. Berezansky, V. O. Livinsky, and E. W. Lytvynov, A generalization of Gaussian white noise analysis, Methods Funct. Anal. Topology 1 (1995), no. 1, 28-55.  MathSciNet MFAT Article
  12. Yurij M. Berezansky and Dmytro A. Mierzejewski, The construction of the chaotic representation for the gamma field, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), no. 1, 33-56.  MathSciNet CrossRef
  13. Yu. M. Berezansky and D. A. Mierzejewski, The investigation of a generalized moment problem associated with correlation measures, Methods Funct. Anal. Topology 13 (2007), no. 2, 124-151.  MathSciNet MFAT Article
  14. Yurij M. Berezansky and Mykola E. Dudkin, On the complex moment problem, Math. Nachr. 280 (2007), no. 1-2, 60-73.  MathSciNet CrossRef
  15. Yurij M. Berezansky and Mykola E. Dudkin, The complex moment problem and direct and inverse spectral problems for the block Jacobi type bounded normal matrices, Methods Funct. Anal. Topology 12 (2006), no. 1, 1-31.  MathSciNet MFAT Article
  16. Y. M. Berezansky, Z. G. Sheftel, and G. F. Us, Functional analysis. Vol. II, Operator Theory: Advances and Applications, vol. 86, Birkhauser Verlag, Basel, 1996.  MathSciNet
  17. Yu. M. Berezanskyi and V. A. Tesko, Spaces of test and generalized functions related to generalized translation operators, Ukrainian Math. J. 55 (2003), no. 12, 1907–1979.  MathSciNet CrossRef
  18. N. N. Bogolyubov, Problemy dinamicheskoj teorii v statisticheskoj fizike, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad,], 1946.  MathSciNet
  19. D. L. Finkelshtein, On convolutions on configuration spaces. II. Spaces of locally finite configurations, Ukrainian Math. J. 64 (2013), no. 12, 1919-1944.  MathSciNet CrossRef
  20. I. Gelfand, D. Raikov, and G. Shilov, Commutative normed rings, Translated from the Russian, with a supplementary chapter, Chelsea Publishing Co., New York, 1964.  MathSciNet
  21. Yuri G. Kondratiev and Tobias Kuna, Harmonic analysis on configuration space. I. General theory, SFB 256 Preprint no. 626, University of Bonn, Bonn, 1999.
  22. Yuri G. Kondratiev and Tobias Kuna, Harmonic analysis on configuration space. I. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002), no. 2, 201-233.  MathSciNet CrossRef
  23. Yuri G. Kondratiev, Tobias Kuna, and Maria Joao Oliveira, On the relations between Poissonian white noise analysis and harmonic analysis on configuration spaces, J. Funct. Anal. 213 (2004), no. 1, 1-30.  MathSciNet CrossRef
  24. Yuri G. Kondratiev, Tobias Kuna, and Maria Joao Oliveira, Holomorphic Bogoliubov functionals for interacting particle systems in continuum, J. Funct. Anal. 238 (2006), no. 2, 375-404.  MathSciNet CrossRef
  25. A. G. Kostjucenko and B. S. Mitjagin, Positive-definite functionals on nuclear spaces, Trudy Moskov. Mat. Ob\v s\v c. 9 (1960), 283-316.  MathSciNet
  26. M. Krein, On a general method of decomposing Hermite-positive nuclei into elementary products, C. R. (Doklady) Acad. Sci. URSS (N.S.) 53 (1946), 3-6.  MathSciNet
  27. M. G. Krein, On Hermitian operators with directed functionals, Akad. Nauk Ukrain. RSR. Zbirnik Prac Inst. Mat. 1948 (1948), no. 10, 83-106.  MathSciNet
  28. Tobias Kuna, Studies in configuration space analysis and applications, Bonner Mathematische Schriften [Bonn Mathematical Publications], 324, Universitat Bonn, Mathematisches Institut, Bonn, 1999.  MathSciNet
  29. A. Lenard, Correlation functions and the uniqueness of the state in classical statistical mechanics, Comm. Math. Phys. 30 (1973), 35-44.  MathSciNet
  30. A. Lenard, States of classical statistical mechanical systems of infinitely many particles. I, Arch. Rational Mech. Anal. 59 (1975), no. 3, 219-239.  MathSciNet
  31. A. Lenard, States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures, Arch. Rational Mech. Anal. 59 (1975), no. 3, 241-256.  MathSciNet
  32. G. I. Nazin, Method of the generating functional, J. Soviet Math. 31 no. 2, 2859-2886. CrossRef
  33. M. J. Oliveira, Configuration space analysis and Poissonian white noise analysis, University of Lisbon, PhD Thesis, Lisbon, 2002.
  34. David Ruelle, Cluster property of the correlation functions of classical gases, Rev. Modern Phys. 36 (1964), 580-584.  MathSciNet
  35. Volodymyr Tesko, One generalization of the classical moment problem, Methods Funct. Anal. Topology 17 (2011), no. 4, 356-380.  MathSciNet MFAT Article
  36. V. S. Vladimirov, Obobshchennye funktsii v matematicheskoi fizike, Izdat. ``Nauka'', Moscow, 1976.  MathSciNet


All Issues