Abstract: | Let X and Y be reflexive Banach spaces with strictly convexduals, and let T be a compact linear map from X to Y. It isshown that a certain nonlinear equation, involving T and itsadjoint, has a normalised solution (an eigenvector)corresponding to an eigenvalue, and that the sameis true for each member of a countable family of similar equationsinvolving the restrictions of T to certain subspaces of X. Theaction of T can be described in terms of these eigenvectors.There are applications to the p-Laplacian, the p-biharmonicoperator and integral operators of Hardy type. |