SPECTRAL PROPERTIES OF SELF-ADJOINT PARTIAL INTEGRAL OPERATORS WITH KERNELS BELONGING TO A SPECIFIC FUNCTION CLASS

Abstract

Spectral theory has become one of the fundamental branches of functional analysis due to its wide range of applications in mathematics, physics, engineering, and computational sciences. The concept of spectrum provides valuable information about the behavior of linear operators and offers effective analytical tools for solving differential and integral equations.Among various classes of linear operators, self-adjoint operators occupy a central position because of their remarkable mathematical properties. Their spectra consist entirely of real numbers, and the associated eigenfunctions form orthogonal systems in Hilbert spaces. These characteristics simplify both theoretical investigations and numerical computations.Partial integral operators represent a natural generalization of classical integral operators and frequently appear in multidimensional mathematical models. They arise in the study of heat transfer, wave propagation, elasticity theory, quantum mechanics, image reconstruction, and stochastic processes. Their mathematical analysis requires a combination of operator theory, measure theory, and functional analysis.