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Book DetailsGraduate Studies in MathematicsVolume: 226; 2022; 472 ppMSC: Primary 47; 34; 35; 46;
The central topic of this book is the spectral theory of bounded and unbounded selfadjoint 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 selfadjoint operators. These can be viewed as infinitedimensional analogues of Hermitian matrices; the infinitedimensional setting leads to a richer theory which goes beyond eigenvalues and eigenvectors and studies selfadjoint 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: selfadjoint operators, their spectral measures, and Herglotz functions, which are complex analytic functions mapping the upper halfplane to itself. Selfadjoint 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 onedimensional Schrödinger 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.ReadershipGraduate students and advanced undergraduates interested in functional analysis and operator theory.

Table of Contents

Chapters

Measure theory

Banach spaces

Hilbert spaces

Bounded linear operators

Bounded selfadjoint operators

Measure decompositions

Herglotz functions

Unbounded selfadjoint operators

Consequencse of the spectral theorem

Jacobi matrices

Onedimensional Schrödinger operators


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The central topic of this book is the spectral theory of bounded and unbounded selfadjoint 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 selfadjoint operators. These can be viewed as infinitedimensional analogues of Hermitian matrices; the infinitedimensional setting leads to a richer theory which goes beyond eigenvalues and eigenvectors and studies selfadjoint 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: selfadjoint operators, their spectral measures, and Herglotz functions, which are complex analytic functions mapping the upper halfplane to itself. Selfadjoint 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 onedimensional Schrödinger 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.
Graduate students and advanced undergraduates interested in functional analysis and operator theory.

Chapters

Measure theory

Banach spaces

Hilbert spaces

Bounded linear operators

Bounded selfadjoint operators

Measure decompositions

Herglotz functions

Unbounded selfadjoint operators

Consequencse of the spectral theorem

Jacobi matrices

Onedimensional Schrödinger operators