Table of Contents

Introduction.

2. Banach Algebras.

2.1 Algebras.

2.1.1 General Results.

2.1.2 Invertible Elements.

2.1.3 The Spectrum.

2.1.4 Standard Examples.

2.1.5 Complexification of Algebras.

Exercises.

2.2 Normed Algebras.

2.2.1 General Results.

2.2.2 The Standard Examples.

2.2.3 The Exponential Function and the Neumann Series.

2.2.4 Invertible Elements of Unital Banach Algebras.

2.2.5 The Theorems of Riesz and Gelfand.

2.2.6 Poles of Resolvents.

2.2.7 Modules.

Exercises.

2.3 Involutive Banach Algebras.

2.3.1 Involutive Algebras.

2.3.2 Involutive Banach Algebras.

2.3.3 Sesquilinear Forms.

2.3.4 Positive Linear Forms.

2.3.5 The State Space.

2.3.6 Involutive Modules.

Exercises.

2.4 Gelfand Algebras.

2.4.1 The Gelfand Transform.

2.4.2 Involutive Gelfand Algebras.

2.4.3 Examples.

2.4.4 Locally Compact Additive Groups.

2.4.5 Examples.

2.4.6 The Fourier Transform.

Exercises.

3. Compact Operators.

3.1 The General Theory.

3.1.1 General Results.

3.1.2 Examples.

3.1.3 Fredholm Operators.

3.1.4 Point Spectrum.

3.1.5 Spectrum of a Compact Operator.

3.1.6 Integral Operators.

Exercises.

3.2 Linear Differential Equations.

3.2.1 Boundary Value Problems for Differential Equations.

3.2.2 Supplementary Results.

3.2.3 Linear Partial Differential Equations.

Exercises.

Name Index. Subject Index. Symbol Index.