Spectral theory of Compact Operators and its Applications
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Date
2025-08
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Addis Ababa University
Abstract
This Thesis presents a systematic investigation into the spectral theory of compact linear operators and demonstrates its profound utility across several key areas of mathematics. The central motivation for this work is the challenge of extending the well-understood spectral properties of matrices in finite-dimensional linear algebra to operators on infinite-dimensional Banach and Hilbert spaces. While general bounded operators can exhibit complex spectral behavior, compact operators form a special class whose spectral properties are remarkably tractable and analogous to the finite-dimensional case. The theoretical core of the work is dedicated to establishing the foundational structure theorems for this class of operators. We develop the Riesz-Schauder theorem, which asserts that the spectrum of a general compact operator on a Banach space is a discrete set of eigenvalues with finite multiplicity that can only accumulate at zero. For the more specialized case of compact self-adjoint operators on a Hilbert space, we present the Spectral Theorem, a powerful result guaranteeing the existence of a complete orthonormal basis of eigenvectors. This theorem provides an infinite-dimensional analogue of matrix diagonalization, which is the key to its wide-ranging applicability. The utility of this abstract framework is demonstrated through three cornerstone applications. First, in the theory of integral equations, we show that Fredholm integral operators are compact and apply the Fredholm Alternative to derive a comprehensive theory for the existence and uniqueness of solutions. Second, we address Sturm-Liouville eigenvalue problems for unbounded differential operators by analyzing their compact, self-adjoint inverse operators (Green's functions), thereby proving the existence of a discrete spectrum and a complete basis of eigenfunctions.
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Spectral, Compact Operators, Applications