4 - Compact operators

Summary

Introduction

This chapter is the core of the book. We have seen that if the function k is sufficiently well-behaved then the operator K defined by is a compact operator from L₂(a, b) to itself, so results about compact operators will produce corresponding information about integral equations. Just as for general matrices, the amount that can be said about general compact operators is limited, and the most substantial conclusions follow if the operator has some symmetry, symmetry in this case being self-adjointness. It turns out that there is a powerful classification theorem for compact self-adjoint operators which describes the action of the operators in terms of their eigenvalues and eigenvectors.

The benefit of the theory of this chapter is that it expresses compact self-adjoint operators in a standard form, which is known to exist for each operator. This reduces many problems to one of evaluation of information about parameters already known to exist, and the techniques can be used to provide specific information about integral operators. We shall, in fact, return frequently to the structure theorem for compact self-adjoint operators to find the basis for a variety of techniques in subsequent chapters.

Many of the results given here are true in greater generality than we shall state them. In particular, the theory can be extended to compact operators on Banach spaces, although the additional complication is considerable and the results on self-adjointness do not extend satisfactorily.