Book contents
- Frontmatter
- Preface
- Contents
- Dedication
- Chapter 1 Abstract spectral theory in Hilbert spaces
- Chapter 2 Spectral theory of differential operators
- Chapter 3 Second order elliptic expressions on manifolds
- Chapter 4 Essential self-adjointness of the Minimal Operator
- Chapter 5 C*-Comparison algebras
- Chapter 6 Minimal comparison algebra and wave front space
- Chapter 7 The secondary symbol space
- Chapter 8 Comparison algebras with non-compact commutators
- Chapter 9 Hs-Algebras; higher order operators within reach
- Chapter 10 Fredholm theory in comparison algebras
- Appendix A Auxiliary results concerning functions on manifolds
- Appendix B Covariant derivatives and curvature
- Appendix C Summary of the conditions (xj) used
- List of symbols used
- References
- Index
- Frontmatter
- Preface
- Contents
- Dedication
- Chapter 1 Abstract spectral theory in Hilbert spaces
- Chapter 2 Spectral theory of differential operators
- Chapter 3 Second order elliptic expressions on manifolds
- Chapter 4 Essential self-adjointness of the Minimal Operator
- Chapter 5 C*-Comparison algebras
- Chapter 6 Minimal comparison algebra and wave front space
- Chapter 7 The secondary symbol space
- Chapter 8 Comparison algebras with non-compact commutators
- Chapter 9 Hs-Algebras; higher order operators within reach
- Chapter 10 Fredholm theory in comparison algebras
- Appendix A Auxiliary results concerning functions on manifolds
- Appendix B Covariant derivatives and curvature
- Appendix C Summary of the conditions (xj) used
- List of symbols used
- References
- Index
Summary
The main purpose of this volume is to introduce the reader to the concept of comparison algebra, defined as a type of C*-algebra of singular integral operators, generally on a noncompact manifold, generated by an elliptic second order differential expression, and certain classes of multipliers and ‘Riesz-operators’ As for singular integral operators on ℝn or on a compact manifold the Fredholm properties of operators in such an algebra are governed by a symbol homomorphism. However, for noncompact manifolds the symbol is of special interest at infinity. In particular the structure of the symbol space over infinity is of interest, and the fact, that the symbol no longer needs to be complex-valued there.
The first attempts of the author to make a systematic presentation of this material happened at Berkeley (1966) and at Lund (1970/71). Especially the second lecture exists in form of (somewhat ragged) notes [CS]. The cases of the Laplace comparison algebra of Rn and the half-space were presented in [C1].
In the course of laying out theory of comparison algebras we had to develop in details spectral theory of differential operators, as well as many of the basic properties of elliptic second order differential operators. This was done in the first four chapters. Comparison algebras (in L2-spaces and L2-Sobolev spaces) are discussed in chapters V to IX. Finally, in chapter X we recall the basic facts of theory of Fredholm operators, partly without proofs.
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- Publisher: Cambridge University PressPrint publication year: 1987