Representation of Bounded Linear Operators

Abstract

In this final chapter we make a start on the difficult problem of representing linear maps (between Banach spaces) that are merely bounded. The magnitude of the task is underscored by the fact that even in a Hilbert space context, really satisfactory results are available only for normal operators. Moreover, the methods used in the spectral analysis of self-adjoint or normal maps are totally different from those brought into play to handle compact maps between Banach spaces. Much is left to be done, but we hope that the tentative and incomplete material to be presented may be of some use in the construction of a satisfactory theory. As in the case of compact operators, the presence of nonlinear duality maps is a major source of difficulty, and the resulting problems have not yet been resolved to any great extent. It remains to be seen whether or not concepts and results from the classical Hilbert space theory, which is developed solely by linear means, provide the best guide when studying the Banach space theory. Of course they do indicate ways in which to proceed, but it is quite possible that, to some extent, the classical techniques may serve as distractions and that what is really needed is a radically different, fundamentally nonlinear, approach. We give below some preliminary work that may help to arrive at such an approach.