The major goal of this book is to make the theory of elliptic boundary problems accessible to mathematicians and physicists working in global analysis and operator algebras. The book is about operators of Dirac type on manifolds with boundary....what's? Front Cover......Page 1 Title......Page 4 Copyright......Page 5 Contents......Page 6 Preface......Page 8 List of Notations......Page 16 Part I. CLIFFORD ALGEBRAS AND Dirac OPERATORS ......Page 20 1. Clifford Algebras and Clifford Modules ......Page 22 2. Clifford Bundles and Compatible Connections ......Page 29 3. Dirac Operators ......Page 38 4. Dirac Laplacian and Connection Laplacian ......Page 45 5. Eudidean Examples ......Page 48 6. The Classical Dirac (Atiyah-Singer) Operators on Spin Manifolds ......Page 55 7. Dirac Operators and Chirality ......Page 59 8. Unique Continuation Property for Dirac Operators ......Page 62 9. Invertible Doubles......Page 69 10. Clueing Constructions. Relative Index Theorem ......Page 78 PART II. ANALYTICAL AND TOPOLOGICAL TOOLS......Page 84 11. Sobolev Spaces on Manifolds with Boundary ......Page 86 12. Calderon Projector for Dirac Operators ......Page 94 13. Existence of of Null Space Elements ......Page 114 14. Spectral Projections of Dirac Operators ......Page 124 15. Pseudo-Differential Grassmannians ......Page 130 A. Elementary Decompositions and Deformations ......Page 146 B. The Homotopy Groups of C. ......Page 152 A. Continuity of Eigenvalues ......Page 157 B. The Spectral Flow on Loops in F. ......Page 159 C. Spectral Flow and Index ......Page 164 D. Non-Vanishing Spectral Flow ......Page 176 Part III. APPLICATIONS......Page 180 18. Elliptic Boundary Problems and Pseudo-Differential Projections ......Page 182 19. Regularity of Solutions of Elliptic Boundary Problems ......Page 199 20. Fredhoim Property of the Operator AR ......Page 207 21. Exchanges on the Boundary: Type Formulas and the Cobordism Theorem for Dirac Operators . ......Page 224 A. Preliminary Remarks ......Page 230 B. Heat Kernels on the Cylinder ......Page 233 C. Duhamel's principle. Heat Kernels on Manifolds with Boundary ......Page 250 D. Proof of the Index Formula ......Page 258 E. L2-Reformulation ......Page 261 F. The Odd-Dimensional Case. A Three-Dimensional Example ......Page 267 23. Some R2marks on the Index of Generalized Atiyah-Patodi-Singer Problems ......Page 272 24. Bojarski's Theorem. General Linear Conjugation Problems ......Page 281 25. Cutting and Pasting of Elliptic Operators ......Page 295 26. Dirac Operators on the Two-Sphere ......Page 301 Bibliography ......Page 308 Index ......Page 322 Back Cover......Page 327 Elliptic boundary problems have enjoyed interest recently, espe cially among C• -algebraists and mathematical physicists who want to understand single aspects of the theory, such as the behaviour of Dirac operators and their solution spaces in the case of a non-trivial boundary. However, the theory of elliptic boundary problems by far has not achieved the same status as the theory of elliptic operators on closed (compact, without boundary) manifolds. The latter is nowadays rec ognized by many as a mathematical work of art and a very useful technical tool with applications to a multitude of mathematical con texts. Therefore, the theory of elliptic operators on closed manifolds is well-known not only to a small group of specialists in partial dif ferential equations, but also to a broad range of researchers who have specialized in other mathematical topics. Why is the theory of elliptic boundary problems, compared to that on closed manifolds, still lagging behind in popularity? Admittedly, from an analytical point of view, it is a jigsaw puzzle which has more pieces than does the elliptic theory on closed manifolds. But that is not the only reason. Clifford Algebras And Dirac Operations -- Analytical And Topological Tools -- Applications. Bernhelm Booss- Bavnbek, Krzysztof P. Wojciechowski. Includes Bibliographical References (p. [289]-302) And Index.
