Approximating eigenfunctions of Fredholm operators in Banach spaces (Q1079165)

Let A be a bounded linear Fredholm operator of index zero with finite dimensional null space N(A). Let L be a (specific) finite rank operator such that \(B=A-L\) is invertible. The authors show how \(B^{-1}\) can be used to compute bases for N(A) and more generally \(N(A^ m)\). If \(\lambda\) \(\neq 0\) is a real eigenvalue of an integral operator K, taking \(A=\lambda I-K\) allows computation of the eigenfunctions of A. If \({\tilde \lambda}\) is an approximation to \(\lambda\), by replacing B by \(B+({\tilde \lambda}-\lambda)I\), approximations to the eigenfunctions of A are obtained which are of the same order of accuracy as that of the eigenvalue approximation. The method is illustrated by some numerical examples.