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Representations of compact linear operators in Banach spaces and nonlinear eigenvalue problems
Author(s) -
Edmunds D. E.,
Evans W. D.,
Harris D. J.
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm083
Subject(s) - mathematics , eigenvalues and eigenvectors , banach space , linear subspace , pure mathematics , dual polyhedron , countable set , mathematical analysis , operator theory , approximation property , reflexive space , operator (biology) , nonlinear system , discrete mathematics , functional analysis , interpolation space , physics , quantum mechanics , biochemistry , chemistry , repressor , transcription factor , gene
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.