The computation of eigenvalues and eigenvector of a completely continuous self-adjoint operator

摘要

If A is a completely continuous self-adjoint operator on a Hilbert space its eigenvalues are the values of the inner product <(Ax, x> at stationary points on the unit sphere. Gradient procedures can be used to determine eigenvectors and eigenvalues provided that certain regularity conditions hold at the eigenvectors. It is proven that these conditions are satisfied at any eigenvector belonging to an eigenvalue of multiplicity one.