Compact Operators, Fredholm Operators, and Index Theory

Abstract

In the preceding chapter we studied operators on Hilbert space and obtained, in particular, the spectral theorem for normal operators. As we indicated this result can be viewed as the appropriate generalization to infinite-dimensional spaces of the diagonalizability of matrices on finite-dimensional spaces. There is another class of operators which are a generalization in a topological sense of operators on a finite-dimensional space. In this chapter we study these operators and a certain related class. The organization of our study is somewhat unorthodox and is arranged so that the main results are obtained as quickly as possible. We first introduce the class of compact operators and show that this class coincides with the norm closure of the finite rank operators. After that we give some concrete examples of compact operators and then proceed to introduce the notion of a Fredholm operator. We begin with a definition.