Integral representations of positive definite kernels

Authors

  • Yurii Bokhonov Educational and Scientific Institute for Applied System Analysis of the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine https://orcid.org/0000-0002-3355-008X

DOI:

https://doi.org/10.20535/SRIT.2308-8893.2023.1.10

Keywords:

Hilbert space, scalar product, symmetric operator, self-adjoint operator, positive definite kernel, defect index, operator extension

Abstract

The paper proposes proof of the possibility of an integral representation of a positive definite kernel of two pairs of variables. Using this kernel, we use the technique of constructing a new Hilbert space in which symmetric differential operators formally commute. In this case, the kernel satisfies a system of differential equations with partial derivatives. It is known that a kernel given in a subdomain of the real plane, generally speaking, does not always imply an extension to the entire plane. This possibility is related to the problem of the existence of a commuting self-adjoint extension of symmetric operators. The author applies his own results related to a commuting self-adjoint extension in a wider Hilbert space. The resulting representation in the form of an integral of elementary positive-definite kernels with respect to the spectral measure generated by the resolution of the identity of the operators allows us to extend the positive-definite kernel to the entire plane.

Author Biography

Yurii Bokhonov, Educational and Scientific Institute for Applied System Analysis of the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv

Candidate of Physical and Mathematical Sciences (Ph.D.), an associate professor at the Department of Mathematical Methods of System Analysis of Educational and Scientific Institute for Applied System Analysis of the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine.

References

Yu. Bokhonov, “Commuting self-adjoint extensions of systems of Hermitian operators,” Ukr. Math. Journ., vol. 40, no. 2, pp. 149–153, 1988.

Yu. Bokhonov, “On self-adjoint extensions of commuting Hermitian operators,” Ukr. Math. Journ., vol. 42, no. 5, pp. 695–697, 1990.

Yu. Berezansky, Expansion in Eigenfunctions of Self-Adjoint Operators. Kyiv: Naukova Dumka, 1965, 798 p.

Yu. Berezansky, Self-adjoint operators in spaces of functions of an infinite number of variables. Kyiv: Naukova Dumka, 1978, 360 p.

Yu. Berezansky and Yu. Kondratiev, Spectral methods in infinite-dimensional analysis. Kyiv: Naukova Dumka, 1988, 680 p.

A.D. Manov, “On the uniqueness of the extension of one function to a positive definite one,” Math. Notes, 107:4, pp. 639–652, 2020.

A.G. Sergeev, “Lectures on functional analysis,” Lekts. courses of the SEC of the MIAS, vol. 23, pp. 3–101, 2014.

Published

2023-03-30

Issue

Section

Mathematical methods, models, problems and technologies for complex systems research