A Generalized Form of Mercer's Theorem

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http://urn.fi/URN:NBN:fi:hulib-202012094771
Title: A Generalized Form of Mercer's Theorem
Author: Salonen, Ella
Contributor: University of Helsinki, Faculty of Science
Publisher: Helsingin yliopisto
Date: 2020
Language: eng
URI: http://urn.fi/URN:NBN:fi:hulib-202012094771
http://hdl.handle.net/10138/322530
Thesis level: master's thesis
Discipline: Matematiikka
Abstract: In this thesis we prove a generalized form of Mercer's theorem, and go through the underlying mathematics involved in the result. Mercer's theorem is an important result in the theory of integral equations, as it can be used as a tool in solving the trace of integral operators. With certain assumptions on a topological space X and measure space (X,dµ), the generalized theorem states that the trace of a positive and self-adjoint bounded integral operator on L^2(X,dµ) with a continuous kernel can be determined by integrating the diagonal of the kernel function. The integral operator being trace class depends then on whether the value of the integral is finite or not. We start the thesis by introducing the general settings we have for the theorem, and provide wider background for the main assumptions. We assume that X is a locally compact Hausdorff space that is σ-compact, and µ is a Radon measure on X with support equal to X. We also need the following technical assumption. Since X is σ-compact, then there exists an increasing sequence of compact subsets C_n with union equal to X. We assume that for each C_n there exists a sequence of increasingly fine partitions, compatible with the measure µ. We then go through the basics on Banach spaces, and we introduce the L^p spaces. Theory on Hilbert spaces is represented in greater detail. We introduce some classes of bounded linear operators on Hilbert spaces, including self-adjoint and positive operators. Some spectral theory is considered, first for Banach algebras in general, and then for the Banach algebra of bounded linear operators on a complex Banach space. The space of bounded linear operators on a Hilbert space can be seen as a C^*-algebra, and results for the spectrum of different kind of Hilbert space operators are given. Compact operators are first defined on Banach spaces. We prove that they form a closed, two-sided ideal in the algebra of bounded linear operators on a Banach space. We also consider compact operators on a Hilbert space, and of special interest are the Hilbert-Schmidt integral operators on the space L^2, which are proven to be compact. The existence of the canonical decomposition for compact operators is proven as this property is used in several proofs of the thesis. In the final chapter we focus on the theory of Hilbert-Schmidt operators and trace class operators on Hilbert spaces. We show that operators in these classes are compact. Considering the Hilbert-Schmidt operators on the space L^2, we prove that they then correspond to the Hilbert-Schmidt integral operators. A trace is first defined for a positive operator, and then for a trace class operator. Finally, in the last section, we construct a proof for the generalized form of Mercer's theorem. As a result, we find a way to determine the trace of an integral operator that satisfies the assumptions described in the first paragraph.
Subject: Hilbert spaces
compact operators
Hilbert-Schmidt integral operators
trace
self-adjointness


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