Rapid developments in multivariable spectral theory have led to important and fascinating results which also have applications in other mathematical disciplines. In this book, various concepts from function theory and complex analytic geometry are drawn together to give a new approach to concrete spectral computations and give insights into new developments in the spectral theory of linear operators. Classical results from cohomology theory of Banach algebras, multidimensional spectral theory, and complex analytic geometry have been freshly interpreted using the language of homological algebra. The advantages of this approach are illustrated by a variety of examples, unexpected applications, and conceptually new ideas that should stimulate further research among mathematicians.

• Preface


• 1: Review of spectral theory


• 2: Analytic functional calculus via integral representations


• 3: Topological homology


• 4: Analytic sheaves


• 5: Fréchet modules over Stein algebras


• 6: Bishop's condition ( ) and invariant subspaces


• 7: Applications to function theory


• 8: Spectral analysis on Bergmann spaces


• 9: Finiteness theorems in analytic geometry


• 10: Multidimensional index theory


• Appendices:


• Locally convex spaces


• Homological algebra


• K-Theory and Riemann-Roch theorems


• Sobolev spaces


• References