This volume studies linear operators on Banach and Hilbert spaces, with emphasis on spectral properties, structure, and applications in analysis and...
This volume studies linear operators on Banach and Hilbert spaces, with emphasis on spectral properties, structure, and applications in analysis and physics.
Part I. Foundations
Chapter 1. Linear Operators
1.1 Definitions and examples 1.2 Domains and ranges 1.3 Bounded vs unbounded operators 1.4 Operator norms 1.5 Basic properties
Chapter 2. Classes of Operators
2.1 Bounded operators 2.2 Compact operators 2.3 Self-adjoint operators 2.4 Normal operators 2.5 Examples
Chapter 3. Operator Topologies
3.1 Norm topology 3.2 Strong operator topology 3.3 Weak operator topology 3.4 Convergence concepts 3.5 Applications
Part II. Spectral Theory
Chapter 4. Spectrum of Operators
4.1 Definitions 4.2 Point, continuous, residual spectrum 4.3 Resolvent set 4.4 Spectral radius 4.5 Examples
Chapter 5. Spectral Theorem
5.1 Self-adjoint operators 5.2 Normal operators 5.3 Projection-valued measures 5.4 Functional calculus 5.5 Applications
Chapter 6. Compact Operator Spectrum
6.1 Eigenvalue structure 6.2 Spectral decomposition 6.3 Fredholm alternative 6.4 Applications 6.5 Examples
Part III. Operator Algebras
Chapter 7. Banach Algebras of Operators
7.1 Definitions 7.2 Spectrum in algebras 7.3 Ideals 7.4 Applications 7.5 Examples
Chapter 8. C*-Algebras
8.1 Definitions 8.2 Representations 8.3 Gelfand–Naimark theorem (overview) 8.4 Applications 8.5 Examples
Chapter 9. Von Neumann Algebras
9.1 Definitions 9.2 Commutants 9.3 Factors 9.4 Applications 9.5 Examples
Part IV. Unbounded Operators
Chapter 10. Unbounded Operators
10.1 Definitions 10.2 Domains 10.3 Closable operators 10.4 Examples 10.5 Applications
Chapter 11. Self-Adjoint Extensions
11.1 Symmetric operators 11.2 Extensions 11.3 Deficiency indices 11.4 Applications 11.5 Examples
Chapter 12. Spectral Theory for Unbounded Operators
12.1 Functional calculus 12.2 Spectral measures 12.3 Applications 12.4 Examples 12.5 Connections
Part V. Special Operator Classes
Chapter 13. Fredholm Operators
13.1 Definitions 13.2 Index theory 13.3 Stability 13.4 Applications 13.5 Examples
Chapter 14. Integral Operators
14.1 Kernel operators 14.2 Compactness 14.3 Spectral properties 14.4 Applications 14.5 Examples
Chapter 15. Differential Operators
15.1 Operators on function spaces 15.2 Boundary conditions 15.3 Spectral analysis 15.4 Applications 15.5 Examples
Part VI. Functional Models and Decompositions
Chapter 16. Operator Decompositions
16.1 Jordan decomposition (overview) 16.2 Polar decomposition 16.3 Singular value decomposition 16.4 Applications 16.5 Examples
Chapter 17. Functional Models
17.1 Model spaces 17.2 Invariant subspaces 17.3 Applications 17.4 Examples 17.5 Connections
Chapter 18. Dilation Theory (Overview)
18.1 Definitions 18.2 Applications 18.3 Examples 18.4 Connections 18.5 Emerging ideas
Part VII. Applications
Chapter 19. Quantum Mechanics
19.1 Operators as observables 19.2 Spectral theory 19.3 Hilbert space framework 19.4 Applications 19.5 Examples
Chapter 20. PDE and Analysis
20.1 Operator methods for PDEs 20.2 Semigroup theory 20.3 Applications 20.4 Examples 20.5 Connections
Chapter 21. Computational Aspects
21.1 Numerical operator theory 21.2 Approximation methods 21.3 Stability 21.4 Software tools 21.5 Applications
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Non-self-adjoint operators 22.2 Random operators 22.3 Operator theory in geometry 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Spectral classification 23.2 Invariant subspace problem 23.3 Computational challenges 23.4 Analytical limits 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of operator theory 24.2 Key contributors 24.3 Evolution of ideas 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Operator identities reference B. Spectral properties summary C. Proof techniques checklist D. Numerical methods reference E. Cross-reference to other MSC branches
This volume develops operator theory as a central framework for analyzing linear transformations in infinite-dimensional spaces. It emphasizes spectral behavior, structure, and applications.