DOI: https://doi.org/10.36719/2789-6919/56/107-115
Gunay Salmanova
Ganja State University
PhD in Mathematics
https://orcid.org/0009-0002-4502-7349
gunay-salmanova@mail.ru
Arbitrary Placement of the Spectrum of the Leading Operator in Polynomial Operator Pencils
Abstract
Scientific work is devoted to the study of the arbitrary placement of the spectrum of the leading operator in polynomial operator pencils. The spectral theory of operator pencils is one of the important areas of functional analysis and has wide applications in the theory of differential equations and mathematical physics. The spectrum of an operator, which consists of its eigenvalues, is one of the fundamental concepts determining the structural and dynamic properties of operators. In this work, the spectral properties of polynomial operator pencils are investigated and sufficient conditions for the arbitrary placement of the spectrum of the leading operator in the complex plane are established. The study examines the behavior of eigenvalues and eigenvectors, spectral decomposition, and the dependence of spectral characteristics on parameters within operator pencils.
It is shown that under certain conditions the spectrum of polynomial operator pencils can be located in previously specified subsets of the complex plane. The obtained results contribute to the development of spectral theory of operator pencils and can also be applied in the spectral analysis of differential operators and various models of mathematical physics.
The results obtained in this research may be used in operator theory, the theory of differential equations, and in solving various problems arising in mathematical physics.
Keywords: polynomial operator pencil, leading operator, spectrum, spectral analysis, eigenvalues, operator theory