The 𝔧‐eigenfunctions and s ‐numbers

It is a truth universally acknowledged, that a compact linear map between Hilbert spaces has an excellent structure that can be described by projections on eigenmanifolds. However, until comparatively recently there were no similar results when the action takes place between Banach spaces. The focus of this paper is on these new developments. Let X and Y be uniformly convex and uniformly smooth Banach spaces, and let T : X → Y be a compact linear map. Denote by J x and J Y normalized duality mappings on X and Y , respectively. We describe a geometric approach for obtaining a “new” class of eigenfunctions and eigenvalues for non‐linear equations of the form S * J Y Sx = λJ x x , where S denotes the restriction of T to subspaces generated by James orthogonality. Our method, which seems to be more direct than the Lusternik‐Schnirelmann method, is based on a procedure developed recently by Evans, Harris, and one of the authors together with the use of James (otherwise called Birkhoff) orthogonality as a decomposition tool. Using the Hardy operator, for which we prove that “classical” eigenvalues and eigenvalues obtained by the Lusternik‐Schnirelmann method and all “strict” s ‐numbers are same, we give numerical computations indicating that these new eigenvalues lie outside the family of s ‐numbers and that the eigenfunctions are different from classical eigenfunctions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)