譜理論簡明教程

前言

　　This book presents the basic tools of modern analysis within the context of what might be called the fundamental problem of operator theory： to calculate spectra of specific operators on infinite-dimensional spaces， especially operators on Hilbert spaces. The tools are diverse， and they provide the basis for more refined methods that allow one to approach problems that go well beyond the computation of spectra; the mathematical foundations of quantum physics， noncommutative K-theory， and the classification of simple C*-algebras being three areas of current research activity that require mastery of the material presented here.　　The notion of spectrum of an operator is based on the more abstract notion of the spectrum of an element of a complex Banach algebra. After working out these fundamentals we turn to more concrete problems of computing spectra of operators of various types. For normal operators， this amounts to a treatment of the spectral theorem. Integral operators require the development of the Riesz theory of compact operators and the ideal L2 of Hilbert Schmidt operators. Toeplitz operators require several important tools; in order to calculate the spectra of Toeplitz operators with continuous symbol one needs to know the theory of Fredholm operators and index， the structure of the Toeplitz C*-algebra and its connection with the topology of curves， and the index theorem for continuous symbols.　　I have given these lectures several times in a fifteen-week course at Berkeley （Mathematics 206）， which is normally taken by first- or second-year graduate students with a foundation in measure theory and elementary functional analysis. It is a pleasure to teach that course because many deep and important ideas emerge in natural ways. My lectures have evolved sig-nificantly over the years， but have always focused on the notion of spectrum and the role of Banach algebras as the appropriate modern foundation for such considerations. For a serious student of modern analysis， this material is the essential beginning.

內(nèi)容概要

本書以作者提供的具備測度論和基礎(chǔ)泛函分析的一二年級研究生十五周課程為基礎(chǔ)，為了計算無限維空間中特殊算子譜，特別是Hilbert空間中的算子，書中在算子理論基本問題的內(nèi)容框架內(nèi)講述了現(xiàn)代分析的基本工具。工具眾多，提供了解決超越譜計算之外問題的更加具體方法的基礎(chǔ)，這些問題如量子物理數(shù)學(xué)基礎(chǔ)，非交換K理論，簡單C*代數(shù)的分類。目次：譜理論和Banach代數(shù)；Hilbert空間上的算子；漸進：緊擾動和Fredholm理論；方法和應(yīng)用。

書籍目錄

Preface Chapter 1. Spectral Theory and Banach Algebras 　1.1. Origins of Spectral Theory 　1.2. The Spectrum of an Operator 　1.3. Banach Algebras: Examples 　1.4. The Regular Representation 　1.5. The General Linear Group of A 　1.6. Spectrum of an Element of a Banach Algebra 　1.7. Spectral Radius 　1.8. Ideals and Quotients 　1.9. Commutative Banach Algebras 　1.10. Examples: C(X) and the Wiener Algebra 　1.11. Spectral Permanence Theorem 　1.12. Brief on the Analytic Functional Calculus Chapter 2. Operators on Hilbert Space 　2.1. Operators and Their C*-Algebras 　2.2. Commutative C*-Algebras 　2.3. Continuous Functions of Normal Operators 　2.4. The Spectral Theorem and Diagonalization 　2.5. Representations of Banach *-Algebras 　2.6. Borel Functions of Normal Operators 　2.7. Spectral Measures 　2.8. Compact Operators 　2.9. Adjoining a Unit to a C*-Algebra 　2.10. Quotients of C*-Algebras Chapter 3. Asymptotics: Compact Perturbations and Fredholm Theory 　3.1. The Calkin Algebra 　3.2. Riesz Theory of Compact Operators 　3.3. Fredholm Operators 　3.4. The Fredholm Index Chapter 4. Methods and Applications 　4.1. Maximal Abelian yon Neumann Algebras 　4.2. Toeplitz Matrices and Toeplitz Operators 　4.3. The Toeplitz C*-Algebra 　4.4. Index Theorem for Continuous Symbols 　4.5. Some H2 Function Theory 　4.6. Spectra of Toeplitz Operators with Continuous Symbol 　4.7. States and the GNS Construction 　4.8. Existence of States: The Gelfand-Naimark Theorem Bibliography Index

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