SELECTED LECTURES ON FUNCTIONAL ANALYSIS

The main aim of this book is to provide an advanced textbook on functional analysis, focusing on the functional calculus of operators. Based on basic knowledge of functional analysis on metric spaces and linear bounded operators, this book systematically introduces the spectral theory of operators and the theory of operator semigroups, including Banach algebras, spectral theories of both bounded and unbounded linear operators, as well as semigroups of operators. They play important roles in various areas of modern mathematics such as harmonic analysis, partial differential equations, probability and statistics, engineering science, quantum physics, and statistical mechanics.

This book provides rich examples and exercises, emphasizing the combination of the theory and examples to enhance readers' ability to apply functional analysis theory to solve related problems. Compared with many other functional analysis textbooks, this book focuses on the spectral theory and the operator semigroup theory, serving as an appropriate connection for functional analysis between undergraduate and graduate levels, making it an ideal textbook for both. It will also be a good reference book for researchers who are interested in modern analysis, such as functional analysis, harmonic analysis, and partial differential equations.

Contents:

• Preface
• About the Authors
• Spectral Theory of Compact Operators:
  • Preliminary
  • Spectrums of Bounded Linear Operators
  • Compact Operators
  • Spectral Theory of Compact Operators
  • Hilbert–Schmidt Theorem
• Banach Algebras:
  • Algebraic Preliminaries
  • Banach Algebras
  • Examples and Applications
  • C*-Algebras
  • Normal Operators on Hilbert Spaces
  • Hilbert–Schmidt Operators and Trace Operators
• Unbounded Operators:
  • Closed Operators
  • Cayley Transform and Spectral Decomposition of Self-Adjoint Operators
  • Spectral Decomposition of Unbounded Normal Operators
• Semigroups of Operators:
  • Strongly Continuous Semigroups of Bounded Linear Operators and Their Infinitesimal Generators
  • Examples of Infinitesimal Generators
  • One-Parameter Groups of Unitary Operators and Stone Theorem
• References
• Index

Readership: Advanced undergraduates and graduate students with a basic knowledge of functional analysis, as well as university or college teachers.