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Über dieses Buch

Written by an author who was at the forefront of developments in multi-variable spectral theory during the seventies and the eighties, this guide sets out to describe in detail the spectral mapping theorem in one, several and many variables. The basic algebraic systems – semigroups, rings and linear algebras – are summarised, and then topological-algebraic systems, including Banach algebras, to set up the basic language of algebra and analysis.

Spectral Mapping Theorems is written in an easy-to-read and engaging manner and will be useful for both the beginner and expert. It will be of great importance to researchers and postgraduates studying spectral theory.



Chapter 1. Algebra

The fundamentals of abstract algebra are introduced: semigroups, rings and linear algebras, as the arena in which various kinds of invertibility and singularity occur, and in particular “exactness”. Homomorphisms between such things give rise to an abstract version of Fredholm theory.
Robin Harte

Chapter 2. Topology

“Point set topology” is the abstract theory of limits and continuity; the ideas of compactness and connectedness are introduced. The more “quantitative” ideas of metric spaces and completeness are developed here.
Robin Harte

Chapter 3. Topological Algebra

When algebra and topology meet, they should be “compatible”, in the sense that the basic algebraic operations are continuous. In this context we meet the spectral topology of a ring, and the fundamental ideas of Banach algebra, duality and enlargement.
Robin Harte

Chapter 4. Spectral Theory

The spectrum of a linear algebra element collects those numbers which give scalar perturbations which fail to be invertible, and transfers the discussion of invertibility and singularity to the screen which is the complex plane.
Robin Harte

Chapter 5. Several Variables

For finite sequences of operators the “spectrum” becomes a joint spectrum, living now in the hyper space of sequences of numbers. The spectral mapping theorem, now for several-variable polynomials, is fundamental.
Robin Harte

Chapter 6. Many Variables

The passage from finite sequences to infinite systems of operators and of linear algebra elements brings with it the possibility of algebra and topology on the indexing material, and of systems which respect that.
Robin Harte

Erratum to: Spectral Mapping Theorems

Robin Harte


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