Introduction

This part of the book is intended as an invitation to the subject of local spectral theory. It contains the basics and some indications of the way the subject has developed. I would like to thank Garth Dales and Michael Neumann for their numerous good comments and suggestions. The entire story of the fascinating subject that Chapters 21–25 deal with may be found in Laursen and Neumann (2000), and I hope that after having been through these chapters you will want to go for more in that book, which also contains a full bibliography.

The phrase local spectral theory carries many connotations. Among the ones that are appropriate here you should expect to find concepts such as spectral subspaces, that is, invariant subspaces on which the restricted operator has a spectrum consisting of a chunk of the original spectrum. The archetypal conceptual framework is provided by the spectral theorem for normal operators on a Hilbert space, which specifies how this decomposition of the underlying space and of the spectrum is supposed to look. Another similar example is provided by the spectral theorem for compact operators on a Banach space.

Both of these examples may be traced back to what is often a high point of a first course in linear algebra, namely a result on diagonalizing symmetric matrices such as the following. (A symmetric matrix [ai j] satisfies the relations ai j = aji for all i, j, while for a symmetric operator T on a finite-dimensional, real inner-product space V with inner product [·, ·], it is true that we have [T x, y] = [x, T y] for all x, y ∈ V.)

This volume is based on a collection of lectures intended for graduate students and others with a basic knowledge of functional analysis. It surveys several areas of current research interest, and is designed to be suitable preparatory reading for those embarking on graduate work. The volume consists of five parts, which are based on separate sets of lectures, each by different authors. Each part provides an overview of the subject that will also be useful to mathematicians working in related areas. The chapters were originally presented as lectures at instructional conferences for graduate students, and we have maintained the styles of these lectures.

The sets of lectures are an introduction to their subjects, intended to convey the flavour of certain topics, and to give some basic definitions and motivating examples: they are certainly not comprehensive accounts. References are given to sources in the literature where more details can be found.

The chapters in Part I are by H. G. Dales. These are an introduction to the general theory of Banach algebras, and a description of the most important examples: B(E), the algebra of all bounded linear operators on a Banach space E; L1(G), the group algebra of a locally compact group G, taken with the convolution product; commutative Banach algebras, including Banach algebras of functions on compact sets in ℂ and radical Banach algebras. Chapters 3–6 cover Gelfand theory for commutative Banach algebras, the analytic functional calculus, and, in a chapter on ‘automatic continuity’, the lovely results that show the intimate connection between the algebraic and topological structures of a Banach algebra.