Introduction

We have already defined a non-negative operator K to be one which is self-adjoint and such that (Kφ, φ), ≥ 0 for all φ. A positive operator is self-adjoint and satisfies the stronger condition (Kφ, φ), > 0 for all φ ≠ 0. In Mercer's Theorem we saw that if the operator generated by a continuous kernel k(x, t) (0 ≤ x, t ≤ 1) is non-negative, then k(x, t) can be expanded in a uniformly convergent series and the eigenvalues of the operator are such that. If we know that all of these eigenvalues are positive then each of them does not exceed the value of the integral, which is therefore an upper bound for the largest eigenvalue.

Given an arbitary operator T in a Hilbert space, we can construct the operator T*T which is non-negative since (T*Tφ, φ) = ∥Tφ∥∥2 ≥ 0. We are therefore well supplied with non-negative operators. In practice, however, we usually need to determine whether a given operator is non-negative and this is a difficult issue to resolve. We give it some attention in this chapter, because there are many techniques, especially in approximation theory, which apply only for non-negative or positive operators. The eigenvalue bound referred to above is a simple example.

As we intend to apply our results to integral equations, we shall restrict attention to compact operators and those closely related to them.