Representations of compact linear operators in Banach spaces and nonlinear eigenvalue problems

Let X and Y be reflexive Banach spaces with strictly convex duals, and let T be a compact linear map from X to Y. It is shown that a certain nonlinear equation, involving T and its adjoint, has a normalised solution (an 'eigenvector') corresponding to an 'eigenvalue', and that the same is true for each member of a countable family of similar equations involving the restrictions of T to certain subspaces of X. The action of T can be described in terms of these 'eigenvectors'. There are applications to the p-Laplacian, the p-biharmonic operator and integral operators of Hardy type.