A Primer on Spectral Theory

This short chapter is intended as an aide-mémoire of some fundamental results in functional analysis that will be used in the rest of this book. Consequently we shall give no proofs as they can be easily found in all the standard textbooks (for instance [7], [8]), with two exceptions: the proof of Milman’s theorem (Theorem 1.1.12) and the nice and simple proof of Machado’s theorem (Theorem 1.1.16) which is a strong extension of the Stone-Weierstrass theorem.

Let X be a finite-dimensional vector space. We know that all norms on X are equivalent and this implies in particular that all linear mappings T from X into X are continuous. Indeed if ‖·‖ is a norm on X and if e₁,...,e _n is a basis of X, then we have

A complex algebra is a vector space A over the complex field ℂ, with a multiplication satisfying the following properties:

Let A be a Banach algebra. A linear functional ϰ on A is called a character of A if it is multiplicative and not identical to 0 on A. This last condition is equivalent to saying that ϰ(1) = 1 because ϰ(x) = ϰ(x)ϰ(1). If ϰ is a character of A it is easy to verify that ϰ(x)∈ Sp(x), for all x ∈ A, because (x - ϰ(x)1)y = y(x - ϰ(x)1) = 1 leads to an absurdity. Consequently \(|\chi (x)| \leqslant \rho (x) \leqslant \parallel x\parallel\) so a character is continuous and of norm one.

The powerful technique of subharmonic functions which we introduced in Chapter III, §4, has a great number of applications in spectral theory.

Among Banach algebras there are very interesting ones called Banach algebras with involution. A mapping x → x^⋆ from a Banach algebra A into itself is called an involution on A if it satisfies the following properties for all x,y ∈ A and λ ∈ ℂ:

As we explained in the Preface this chapter will be a quick incursion, as the crow flies, into the important new field of analytic multifunctions. A complete treatment would need a full book. In particular it would be necessary to recall the great number of results we need in the theory of functions of several complex variables. Of course we are not able to do this, in such a limited number of pages, so we shall suppose that the reader is familiar with all these prerequisites, which are contained for instance in [5,9] or in many other books on the field, and which are summarized in the Appendix, §2. In the last ten years the use of subharmonic functions in spectral theory and in the theory of uniform algebras has given a lot of interesting new results (see Chapters III and V for spectral theory; Exercises VII.8–9–10 and [10] for the theory of uniform algebras).