A first course in spectral theory
著者
書誌事項
A first course in spectral theory
（Graduate studies in mathematics, 226）
American Mathematical Society, c2022
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注記
Includes bibliographical references (p. 459-465) and index
内容説明・目次
内容説明
The central topic of this book is the spectral theory of bounded and unbounded self-adjoint operators on Hilbert spaces. After introducing the necessary prerequisites in measure theory and functional analysis, the exposition focuses on operator theory and especially the structure of self-adjoint operators. These can be viewed as infinite-dimensional analogues of Hermitian matrices; the infinite-dimensional setting leads to a richer theory which goes beyond eigenvalues and eigenvectors and studies self-adjoint operators in the language of spectral measures and the Borel functional calculus. The main approach to spectral theory adopted in the book is to present it as the interplay between three main classes of objects: self-adjoint operators, their spectral measures and Herglotz functions, which are complex analytic functions mapping the upper half-plane to itself. Self-adjoint operators include many important classes of recurrence and differential operators; the later part of this book is dedicated to two of the most studied classes, Jacobi operators and one-dimensional Schrodinger operators.
This text is intended as a course textbook or for independent reading for graduate students and advanced undergraduates. Prerequisites are linear algebra, a first course in analysis including metric spaces, and for parts of the book, basic complex analysis. Necessary results from measure theory and from the theory of Banach and Hilbert spaces are presented in the first three chapters of the book. Each chapter concludes with a number of helpful exercises.
目次
Measure theory
Banach spaces
Hilbert spaces
Bounded linear operators
Bounded self-adjoint operators
Measure decompositions
Herglotz functions
Unbounded self-adjoint operators
Consequencse of the spectral theorem
Jacobi matrices
One-dimensional Schrodinger operators
Bibliography
Notation index
Index
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