Solvability of Unbounded Operator Equation AX - XB = C

Abstract

This study investigates the role of Toeplitz operators in mathematical analysis, focusing on their application within functional analysis and factor theory. It highlights the behavior of infinite, linear, and symmetric operators in solving matrix equations, emphasizing the development and testing of efficient methods for solving operator equations in linear algebra. The study considers practical applications across various scientific fields, analyzing matrix equations involving subtraction. It explores conventional solutions and transformations of matrices with distinct eigenvalues using operator theory and spectral analysis. The research also examines finite linear operators and their applications in practical contexts, aiming to deepen the understanding of matrix equations in areas such as transport theory, theoretical physics, and group theory, thus contributing to a more comprehensive grasp of complex systems.