Nonlinear Spectral Theory
In view of the eminent importance of spectral theory of linear operators in many fields of mathematics and physics, it is not surprising that various attempts have been made to define and study spectra also for nonlinear operators. This book provides a comprehensive and self-contained treatment of the theory, methods, and applications of nonlinear spectral theory.
The first chapter briefly recalls the definition and properties of the spectrum and several subspectra for bounded linear operators. Then some numerical characteristics for nonlinear operators are introduced which are useful for describing those classes of operators for which there exists a spectral theory. Since spectral values are closely related to solvability results for operator equations, various conditions for the local or global invertibility of a nonlinear operator are collected in the third chapter. The following two chapters are concerned with spectra for certain classes of continuous, Lipschitz continuous, or differentiable operators. These spectra, however, simply adapt the corresponding definitions from the linear theory which somehow restricts their applicability. Other spectra which are defined in a completely different way, but seem to have useful applications, are defined and studied in the following four chapters. The remaining three chapters are more application-oriented and deal with nonlinear eigenvalue problems, numerical ranges, and selected applications to nonlinear problems.
The only prerequisite for understanding this book is a modest background in functional analysis and operator theory. It is addressed to non-specialists who want to get an idea of the development of spectral theory for nonlinear operators in the last 30 years, as well as a glimpse of the diversity of the directions in which current research is moving.
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Chapter 1 Spectra of Bounded Linear Operators
Chapter 2 Some Characteristics of Nonlinear Operators
Chapter 3 Invertibility of Nonlinear Operators
Chapter 4 The Rhodius and Neuberger Spectra
Chapter 5 The Kachurovskij and Dörfner Spectra
Chapter 6 The FuriMartelliVignoli Spectrum
Chapter 7 The Feng Spectrum
Chapter 9 Other Spectra
Chapter 10 Nonlinear Eigenvalue Problems
Chapter 11 Numerical Ranges of Nonlinear Operators
Chapter 12 Some Applications
List of Symbols
Chapter 8 The Väth Phantom