6.1 Iterative Methods of Solving the Fredholm Equations of the Second Kind at a Characteristic Value

The aim of this chapter is to provide in abstract setting some results which justify the iterative processes given in Chapter 2.

1. Let A be a linear compact operator on a Hilbert space H, ? _n, ? _n its characteristic values and eigenelements, ? _n = ? _nA ? _n, ? ₁ < ? ₂ ? ? ₃ ? . Let G ₁ ? { ? : ( I - ? ₁ A ^*) ? = 0} and be its orthogonal complement in H. The equation

(6.1)

is solvable if and only if .

(6.2)

This means that the pole ? = ? ₁ of the resolvent ( I - ? A) ^-1 is simple. This also means that the root subspace of A corresponding to ? ₁ coincides with the eigensubspace of A corresponding to ? ₁. The root subspace is defined as follows. Let ? = ? ₁ A ?. Consider the equations

(6.3)

Only a finite number r of these equations are solvable ([44]). If (6.3) has no solution for j = 0 then ? ₁ is semisimple. If (6.3) is solvable for 0 ? j ? r and is not...