CHAPTER 1 - Elementary Spectral Theory

Abstract

This chapter presents the basic results of spectral theory. The most important of these are the non-emptiness of the spectrum, Beurling's spectral radius formula, and the Gelfand representation theory for commutative Banach algebras. The chapter also discusses compact and Fredholm operators and describes their elementary theory. Important concepts are the essential spectrum and the Fredholm index. The ground field for all vector spaces and algebras is the complex field C. A complete normed algebra is called a Banach algebra. A complete unital normed algebra is called a unital Banach algebra. One thinks of the spectrum as simultaneously a generalization of the range of a function and the set of eigenvalues of a finite square matrix. According to Gelfand theorem, if a is an element of a unital Banach algebra A, then the spectrum σ (a) of a is nonempty. There are algebras in which not all elements have nonempty spectrum.