A first course in spectral theory

Author(s)

• Lukić, Milivoje

Bibliographic Information

A first course in spectral theory

Milivoje Lukić

（Graduate studies in mathematics, 226）

American Mathematical Society, c2022

Note

Includes bibliographical references (p. 459-465) and index

Description and Table of Contents

Description

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.

Table of Contents

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

by "Nielsen BookData"

Details

• NCID: BC17221964
• ISBN:
  • 9781470466565
• LCCN: 2022028354
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: Providence, Rhode Island
• Pages/Volumes: xvi, 472 p.
• Size: 27 cm
• Classification:
  • LCC : QC20.7.S64
  • DC23 : 515/.7222
• Subject Headings:
  • LCSH : Spectral theory (Mathematics) -- Textbooks
  • FREE : Operator theory -- Special classes of linear operators -- Hermitian and normal operators (spectral measures, functional calculus, etc.). msc
  • FREE : Operator theory -- Special classes of linear operators -- Symmetric and selfadjoint operators (unbounded). msc
  • FREE : Operator theory -- Special classes of linear operators -- Jacobi (tridiagonal) operators (matrices) and generalizations. msc
  • FREE : Ordinary differential equations -- Ordinary differential operators -- Particular operators (Dirac, one-dimensional Schrödinger, etc.). msc
  • FREE : Partial differential equations -- Elliptic equations and systems -- Schrödinger operator. msc
  • FREE : Functional analysis -- Inner product spaces and their generalizations, Hilbert spaces -- Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product). msc
• Parent Bibliography ID:
  • BA20542214