7.1 Banach Algebras

In a first course in linear algebra, one encounters a variety of interesting ideas surrounding linear operators on finite dimensional spaces. The central object of this discussion is the concept of an eigenvalue or characteristic value. The eigenvalues of an operator togetherwith their eigenvectors help us build a collection of ideas such as the Cayley–Hamilton Theorem, the question of diagonalizability of an operator, the theory of canonical forms and much much more.

In this chapter, we will revisit the idea of an eigenvalue in the context of operators on infinite dimensional spaces. Here, the set of eigenvalues needs to be replaced by the spectrum of an operator. This is a compact subset of the complex plane that carries a great deal of information about the operator and is the object we wish to study.

Note. Before we get going, wemake one important assumption.Henceforth all vector spaces will be over C. The precise reason for this will be explained in due course, but it is related to the Fundamental Theorem of Algebra.