This book examines the theory of boundary value problems for elliptic systems of partial differential equations, a theory which has many applications in mathematics and the physical sciences. The aim is to 'algebraize' the index theory by means of pseudo-differential operators and methods in the spectral theory of matrix polynomials. This latter theory provides important tools that will enable the student to work efficiently with the principal symbols of the elliptic and boundary operators on the boundary. This book is ideal for use in graduate-level courses on partial differential equations, elliptic systems, pseudo-differential operators and matrix analysis. Since many new methods and results are introduced and used throughout the book, all the theorems are proved in detail, and the methods are well illustrated through numerous examples and exercises.

‘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 Source: Bulletin of the London Mathematical Society

‘The book can be recommended both as a textbook for graduate students and as a handbook for researchers.’

T. Weidl Source: Proceedings of the Edinburgh Mathematical Society

‘… certainly of great interest for specialists and can be used for advanced lectures or seminars in this field.’

Source: Monatshefte für Mathematik