Open Access

# Poisson measure as a spectral measure of a family of commuting selfadjoint operators, connected with some moment problem

### Abstract

It is proved that the Poisson measure is a spectral measure of some family of commuting selfadjoint operators acting on a space constructed from some generalization of the moment problem.

Key words: Spectral measure, Poisson measure, Kondratiev–Kuna convolution.

### Article Information

 Title Poisson measure as a spectral measure of a family of commuting selfadjoint operators, connected with some moment problem Source Methods Funct. Anal. Topology, Vol. 22 (2016), no. 4, 311-329 MathSciNet MR3591083 Milestones Received 20/07/2016; Revised 16/09/2016 Copyright The 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

### Citation Example

Yu. M. Berezansky, Poisson measure as a spectral measure of a family of commuting selfadjoint operators, connected with some moment problem, Methods Funct. Anal. Topology 22 (2016), no. 4, 311-329.

### BibTex

@article {MFAT912,
AUTHOR = {Berezansky, Yu. M.},
TITLE = {Poisson measure as a spectral measure of a family of commuting selfadjoint operators, connected with some moment problem},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {22},
YEAR = {2016},
NUMBER = {4},
PAGES = {311-329},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=912},
}

### References

1. S. Albeverio, Yu. G. Kondratiev, and M. Rockner, Analysis and geometry on configuration spaces, J. Funct. Anal. 154 (1998), no. 2, 444-500.  MathSciNet CrossRef
2. Yu. M. Berezansky, Spectral approach to white noise analysis, Dynamics of complex and irregular systems (Bielefeld, 1991), World Sci. Publ., River Edge, NJ, 1993, pp. 131-140.  MathSciNet
3. Yu. M. Berezansky, Commutative Jacobi fields in Fock space, Integral Equations Operator Theory 30 (1998), no. 2, 163-190.  MathSciNet CrossRef
4. Yu. 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
5. Yu. M. Berezansky, The generalized moment problem associated with correlation measures, Funct. Anal. Appl. 37 (2003), no. 4, 311-315.  MathSciNet CrossRef
6. Yu. M. Berezansky and Y. G. Kondratiev, Spectral methods in infinite-dimensional analysis. Vol. 1, Kluwer Academic Publishers, Dordrecht, 1995.  MathSciNet CrossRef
7. 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
8. 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
9. 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.
10. D. L. Finkelshtein, Stochastic dynamics of continuous systems, Ph.D. thesis, Institute of Mathematics of NAS of Ukraine, Kyiv, 2014.
11. Yoshifusa Ito and Izumi Kubo, Calculus on Gaussian and Poisson white noises, Nagoya Math. J. 111 (1988), 41-84.  MathSciNet
12. 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
13. 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
14. Eugene Lytvynov, The projection spectral theorem and Jacobi fields, Methods Funct. Anal. Topology 21 (2015), no. 2, 188-198.  MathSciNet
15. M. J. Oliveira, Configuration space analysis and Poissonian white noise analysis, Ph.D. thesis, University of Lisbon, Lisbon, 2002.
16. K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967.  MathSciNet