Projections on Convex Sets in Hilbert Space and Spectral Theory: Part I. Projections on Convex Sets: Part II. Spectral Theory

Abstract

This chapter discusses projections on convex sets in Hilbert Space and Spectral Theory. It consists of two separate parts of which the second is an outgrowth of the first and, to a certain extent, its motivation. A general study of projections is presented on convex sets in Hilbert space without any particular aim in mind. Projections appear as one of the simplest instances of nonlinear mappings that can be defined in abstract terms. It was in the study of the algebra of projections that the similarities with the linear counterpart became obvious, especially after the specialization to projections on convex cones. All obstacles were removed at once and the way toward a spectral theory was opened. The notion of a spectral resolution made up of projections on convex cones makes sense; from it, a spectral measure can be built and a spectral integration theory of real valued functions can be developed. The resulting integrals are operators in Hilbert space, nonlinear in general, generalizing linear selfadjoint ones whose properties they mimic.