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内容大纲
本书是反映20世纪初数学家所发现的一种新的看待传统素材的工具——巴拿赫空间与希尔伯特空间的算子理论的英文版专著，中文书名可译为《算子理论问题集》。作者穆罕默德·希赫姆·莫尔塔德是阿尔及利亚数学家，任阿尔及利亚奥兰大学教授。 -
作者介绍
穆罕默德·希赫姆·莫尔塔德，他是阿尔及利亚数学家，任阿尔及利亚奥兰大学教授。 -
目录
Preface
Notation and Terminology
0.1 Notation
0.2 Terminology
Part 1 EXERCISES ET AL
Chapter 1 Normed Vector Spaces.Banach Spaces
1.1 Basics
1.1.1 Definitions and Examples
1.1.2 Operations on Banach Spaces
1.1.3 Convex Sets
1.1.4 LP-Spaces
1.2 True or False: Questions
1.3 Exercises with Solutions
1.4 Tests
1.5 More Exercises
Chapter 2 Bounded Linear Operators on Banach Spaces
2.1 Basics
2.1.1 Basic Definitions
2.1.2 Dual Spaces
2.1.3 The Fourier Transform
2.1.4 Invertibility
2.1.5 Series in Banach Spaces
2.1.6 Neumann Series
2.1.7 Operator-Valued Analytic Functions
2.1.8 The Four Pillars of Functional Analysis
2.2 True or False: Questions
2.3 Exercises with Solutions
2.4 Tests
2.5 More Exercises
Chapter 3 Hilbert Spaces
3.1 Basics
3.1.1 Definitions and Examples
3.1.2 Modes of Convergence of Sequences
3.1.3 Orthogonality
3.1.4 The Closest Point Property and the Projection Theorem
3.1.5 Orthonormal Sets and Basy ana tne rro
3.1.6 Orthogonal Polynomials
3.1.7 Linear Operators on a Hilbert Space
3.1.8 Operator Topologies
3.2 True or False: Questions
3.3 Exercises with Solutions
3.4 Tests
3.5 More Exercises
Chapter 4 Classes of Linear Operators on Hilbert Spaces 99
4.1 Basics
4.1.1 Operator Theory Basics
4.1.2 Cartesian Form
4.1.3 Operator Matrices
4.1.4 A Word on the Invariant Subspace Problem
4.2 True or False: Questions
4.3 Exercises with Solutions
4.4 Tests
4.5 More Exercises
Chapter 5 Positive Operators.Square Root
5.1 Basics
5.1.1 Positive Operators
5.1.2 Invertibility Revisited
5.1.3 Square Root of Positive Operators
5.2 True or False: Questions
5.3 Exercises with Solutions
5.4 Tests
5.5 More Exercises
Chapter 6 Absolute Value.Polar Decomposition of an Operator
6.1 Basics
6.1.1 Definitions and Properties
6.1.2 The Absolute Value and the Product
6.1.3 The Absolute Value and the Sum
6.1.4 The Absolute Value and Other Inequalities
6.1.5 Polar Decomposition
6.2 True or False: Questions
6.3 Exercises with Solutions
6.4 Tests
6.5 More Exercises
Chapter 7 Spectrum of an Operator
7.1 Basics
7.1.1 Definitions and Properties
7.1.2 The Resolvent Function
7.1.3 Subsets of the Spectrum
7.1.4 Spectrum of the Product of Operators
7.1.5 On The Operator Equation AX - XB = C
7.1.6 Polynomial Calcul-yuawva A
7.2 True or False: Questions
7.3 Exercises with Solutions
7.4 Tests
7.5 More Exercises
Chapter 8 Spectral Radius Numerical Range
8.1 Basics
8.1.1 Spectral Radius
8.1.2 Numerical Range
8.2 True or False: Questions
8.3 Exercises with Solutions
8.4 Tests
8.5 More Exercises
Chapter 9 Compact Operators
9.1 Basics
9.1.1 Compact Operators
9.1.2 Hilbert-Schmidt Operators
9.1.3 Spectrum and Compact Operators
9.2 True or False: Questions
9.3 Exercises with Solutions
9.4 Tests
9.5 More Exercises
Chapter 10 Closed Operators
10.1 Basics
10.1.1 Spectrum
10.1.2 Closedness of Products and Sums
10.1.3 A Word on Distributional Derivatives
10.2 True or False: Questions
10.3 Exercises with Solutions
10.4 Tests
10.5 More Exercises
Chapter 11 Functional Calculi
11.1 Basics
11.1.1 Functional Calculus for Self-adjoint Operators
11.1.2 Functional Calculus for Normal Operators
11.1.3 A Little Digression
11.1.4 The Fuglede-Putnam Theorem Revisited
11.1.5 The Beck-Putnam Theorem
11.2 True or False: Questions
11.3 Exercises with Solutions
11.4 Tests
11.5 More Exercises
Chapter 12 Hyponormal Operators
12.1 Basics
12.2 True or False: Questions
12.3 Exercises with Solutions
12.4 Tests
12.5 More Exercises
Chapter 13 Similarities of Operators
13.1 Important Theorems
13.1.1 The Berberian Theorem
13.1.2 The Williams Theorem
13.1.3 The Embry Theorem
13.2 Exercises with Solutions
13.3 Tests
13.4 More Exercises
Part 2 SOLUTIONS
Chapter 1 Normed Vector Spaces Banach Spaces
1.2 True or False: Answers
1.3 Solutions to Exercises
1.4 Hints/Answers to Tests
Chapter 2 Bounded Linear Operators on Banach Spaces
2.2 True or False: Answers
2.3 Solutions to Exercises
2.4 Hints/Answers to Tests
Chapter 3 Hilbert Spaces
3.2 True or False: Answers
3.3 Solutions to Exercises
3.4 Hints/Answers to Tests
Chapter 4 Classes of Linear Operators on Hilbert Spaces 377
4.2 True or False: Answers
4.3 Solutions to Exercises
4.4 Hints/Answers to Tests
Chapter 5 Positive Operators Square Root
5.2 True or False: Answers
5.3 Solutions to Exercises
5.4 Hints/Answers to Tests
Chapter 6 Absolute Value Polar Decomposition of an Operator
6.2 True or False: Answers
6.3 Solutions to Exercises
6.4 Hints/Answers to Tests
Chapter 7 Spectrum of an Operator
7.2 True or False: Answers
7.3 Solutions to Exercises
7.4 Hints/Answers to Tests
Chapter 8 Spectral Radius Numerical Range
8.2 True or False: Answers
8.3 Solutions to Exercises
8.4 Hints/Answers to Tests
Chapter 9 Compact Operators
9.2 True or False: Answers
9.3 Solutions to Exercises
9.4 Hints/Answers to Tests
Chapter 10 Closed Operators
10.2 True or False: Answers
10.3 Solutions to Exercises
10.4 Hints/Answers to Tests
Chapter 11 Functional Calculi
11.2 True or False: Answers
11.3 Solutions to Exercises
11.4 Hints/Answers to Tests
Chapter 12 Hyponormal Operators
12.2 True or False: Answers
12.3 Solutions to Exercises
12.4 Hints/Answers to Tests
Chapter 13 Similarities of Operators
13.2 Solutions to Exercises
13.3 Hints/Answers to Tests
Bibliography
Index
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