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Integrals and Operators

TO THE SECOND EDITION Since publication of the First Edition several excellent treatments of advanced topics in analysis have appeared. However, the concentration and penetration of these treatises naturally require much in the way of technical preliminaries and new terminology and notation. There c...

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Bibliographic Details
Main Authors: Segal, I.E., Kunze, R.A. (Author)
Format: eBook
Language:English
Published: Berlin, Heidelberg Springer Berlin Heidelberg 1978, 1978
Edition:2nd ed. 1978
Series:Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
Subjects:
Mathematical Analysis
Analysis
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • I. Introduction
  • 1.1 General preliminaries
  • 1.2 The idea of measure
  • 1.3 Integration as a technique in analysis
  • 1.4 Limitations on the concept of measure space
  • 1.5 Generalized spectral theory and measure spaces
  • Exercises
  • II. Basic Integrals
  • 2.1 Basic measure spaces
  • 2.2 The basic Lebesgue-Stieltjes spaces
  • Exercises
  • 2.3 Integrals of step functions
  • Exercises
  • 2.4 Products of basic spaces
  • 2.5* Coin-tossing space
  • Exercises
  • 2.6 Infinity in integration theory
  • Exercises
  • III. Measurable Functions and Their Integrals
  • 3.1 The extension problem
  • 3.2 Measurability relative to a basic ring
  • Exercises
  • 3.3 The integral
  • Exercises
  • 3.4 Development of the integral
  • Exercises
  • 3.5 Extensions and completions of measure spaces
  • Exercises
  • 3.6 Multiple integration
  • Exercises
  • 3.7 Large spaces
  • Exercises
  • IV. Convergence and Differentiation
  • 4.1 Linear spaces of measurable functions
  • Exercises
  • 4.2 Set functions
  • XIV. The Trace as a Non-Commutative Integral
  • 14.1 Introduction
  • 14.2 Elementary operators and the trace
  • Exercises
  • 14.3 Hilbert algebras
  • Exercises
  • Selected references
  • Exercises
  • 4.3 Differentiation of set functions
  • Exercises
  • V. Locally Compact and Euclidean Spaces
  • 5.1 Functions on locally compact spaces
  • Exercises
  • 5.2 Measures in locally compact spaces
  • Exercises
  • 5.3 Transformation of Lebesgue measure
  • Exercises
  • 5.4 Set functions and differentiation in euclidean space
  • Exercises
  • VI. Function Spaces
  • 6.1 Linear duality 152 Exercises
  • Exercises
  • 6.2 Vector-valued functions
  • Exercises
  • VII. Invariant Integrals
  • 7.1 Introduction
  • 7.2 Transformation groups
  • Exercises
  • 7.3 Uniform spaces
  • Exercises
  • 7.4 The Haar integral
  • 7.5 Developments from uniqueness
  • Exercises
  • 7.6 Function spaces under group action
  • Exercises
  • VIII. Algebraic Integration Theory
  • 8.1 Introduction
  • 8.2 Banach algebras and the characterization of function algebras
  • Exercises
  • 8.3 Introductory features of Hilbert spaces
  • Exercises
  • 8.4 Integration algebras
  • Exercises
  • IX. Spectral Analysis in Hilbert Space
  • 9.1 Introduction
  • 9.2 The structure of maximal Abelian self-adjoint algebras
  • Exercises
  • X. Group Representations and Unbounded Operators
  • 10.1 Representations of locally compact groups
  • 10.2 Representations of Abelian groups
  • Exercises
  • 10.3 Unbounded diagonalizable operators
  • Exercises
  • 10.4 Abelian harmonic analysis
  • Exercises
  • XI. Semigroups and Perturbation Theory
  • 11.1 Introduction
  • 11.2 The Hille-Yosida theorem
  • 11.3 Convergence of semigroups
  • 11.4 Strong convergence of self-adjoint operators
  • 11.5 Rellich-Kato perturbations
  • Exercises
  • 11.6 Perturbations in a calibrated space
  • Exercises
  • XII. Operator Rings and Spectral Multiplicity
  • 12.1 Introduction
  • 12.2 The double-commutor theorem
  • Exercises
  • 12.3 The structure of abelian rings
  • Exercises
  • XIII. C*-Algebras and Applications
  • 13.1 Introduction
  • 13.2 Representations and states
  • Exercises

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