Spectral Families and Geometry Of Banach Spaces

Abstract

The principal objects of study in this thesis are arbitrary spectral families, E, on a complex Banach space X. The central theme is the relationship between the geometry of X and the p-variation of E. We show that provided X is super- reflexive, then given any E, there exists a value 1 · p < 1, depending only on E and X, such that var p(E) < 1. If X is uniformly smooth we provide an explicit range of such values p, which depends only on E and the modulus of convexity of X*, delta X*(.).