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Infinite Matrices and their Finite Sections
An Introduction to the Limit Operator Method
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This book is an introduction to a fascinating topic at the interface of functional analysis, algebra and numerical analysis, written for a broad audience of students, researchers and practitioners. It is concerned with the study of infinite matrices and their approximation by matrices of finite size. Our framework includes the simplest, important case where the matrix entries are numbers, but also the more general case where the entries are bounded linear operators. This ensures that examples of the class of operators studied - band-dominated operators on Lebesgue function spaces and sequence spaces - are ubiquitous in mathematics and physics. The main items and concepts studied are band-dominated operators, invertibility at infinity, Fredholmness, the method of limit operators, and the stability and convergence of finite matrix approximations. Concrete examples are used to illustrate the results throughout, including discrete Schrodinger operators and integral and boundary integral operators arising in mathematical physics and engineering.
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