• Part I. A Spectral Theory of Matrix Polynormials: 1. Matrix polynomials; 2. Spectral triples for matrix polynomials; 3. Monic matrix polynomials; 4. Further results; Part II. Manifolds and Vector Bundles: 5. Manifolds and vector bundles; 6. Differential forms; Part III. Pseudo-Differential Operators and Elliptic Boundary Value Problems: 7. Pseudo-differential operators on Rn; 8. Pseudo-differential operators on a compact manifold; 9. Elliptic systems on bounded domains in Rn; Part IV. Reduction Of A Boundary Value Problem To An Elliptic System On The Boundary: 10. Understanding the L-condition; 11. Applications to the index; 12. BVPs for ordinary differential operators and the connection with spectral triples; 13. Behaviour of a pseudo-differential operator near a boundary; 14. The Main Theorem revisited; Part V. An Index Formula For Elliptic Boundary Problems In The Plane: 15. Further results on the Lopatinskii Condition; 16. The index in the plane; 17. Elliptic systems with 2 x 2 real coefficients.

'It is the strength of this book that, for the first time, the theory of (elliptic) systems is presented on the level of recent research theory of (scalar) pseudodifferential operators … the authors put new life into the classical method of shifting boundary value problems in a domain to its boundary.' Hans Triebel, Bulletin of the London Mathematical Society