SPECTRAL PROPERTIES OF DIFFERENTIAL AND INTEGRAL OPERATORS

Abstract

This dissertation explores the spectral properties of differential and integral operators in infinite-dimensional vector spaces, especially Hilbert spaces. It starts by introducing fundamental concepts to understand how these opera tors behave in infinite dimensions. An overview of spectral theory is provided, focusing on the importance of eigenvalues, eigenvectors, and spectral decom positions, with a discussion of the spectral theorem and its significance. Subsequently, the spectral properties of linear differential and integral opera tors are explored, with particular emphasis on their role in solving differential equations in Hilbert spaces. Various differential operators, including Sturm Liouville operators, are analyzed to understand their spectral behavior and implications. The practical importance of these theoretical results is illustrated through numerous applications in physics, engineering, and other scientific fields. In stances include the application of spectral properties in quantum mechanics, signal processing, and structural analysis, demonstrating the wide-ranging real life applications of these ideas.