Evenly positive definite function of Hilbert space and some algebraic relationship

  • O. V. Lopotko National Forestry and Wood Technology University of Ukraine
Keywords: : integral representation; bounded evenly positive definite functions; bounded self-adjoint operators


A generalization of P. A. Minlos, V. V. Sazonov’s theorem is proved in the case of bounded evenly positive definite function given in a Hilbert space. The integral representation is obtained for a family of bounded commutative self-adjoint operators which are connected by algebraic relationship.


Berezansky Yu.M. Generalization of Bochner theorem on expansions in eigenfunctions of partial differ-

ential operators// Dokl. AN SSSR. – 1956. – V.110, no.6. – P. 893–896.

Berezansky Yu.M. Expansions in eigenfunctions of self-adjoint operators. (Translations of Mathematical

Monographs V.17), Providence, R.I.: Am. Math. Soc., 1968, 809 p.

Berezansky Yu.M. Self-adjoint operators in space of functions of infinitely many varibles. – Kyiv: Naukova

dumka, 1978. – 360 p.

Berezansky Yu.M., Kondratiev Yu.G. Spectral methods in infinite-dimensional analysis. – Kyiv: Naukova

dumka, 1988. – 679 p. Engl.transl.: Springer, Dordrecht. 1995, doi: 10.1007/978-94-011-0509-5

Krein M.G. On a general method on decomposition of Hermite positive definite nuclei into elementary

products// Dokl. AN SSSR. – 1946. – V.53(1). – P. 3–6.

Kurepa S.A. A cosine functional equation in Hilbert space // Canadian J. Math. – 1960. – V.12. – P. 45–50.

Minlos R.A. Generalized random processes and their extension in measure// Trudy Moskov. Mat. Obsc.

– 1959. – V.8. – 497–518. (in Russian)

Shilov G.E., Fan Dyk Tin. Integral, measure and derivative on linear spaces. – M.: Science, 1967. – 192 p.

Sazonov V.V. Remark on characteristic functionals// Theory of Probability and its Applications. – 1958.

– V.3, no.2. – P. 188–192. doi: 10.1137/1103018

How to Cite
Lopotko OV. Evenly positive definite function of Hilbert space and some algebraic relationship. Mat. Stud. [Internet]. 2021Mar.6 [cited 2021Oct.16];55(1):85-3. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/59