Espacios twisted Hilbert y transformaciones de espacios quasinormados

Abstract

This thesis deals with how to continuously transform the unit ball from one quasinorm to another. Chapter 1 deals with the space Q of quasinorms in finite-dimensional spaces, which is a Banach space. Chapter 2 develops the idea of continuous transformation between the quasinorm q1 and the quasinorm q2 by setting a continuous curve in Q between q1 and q2. This idea have its origin in interpolation theory: an interpolation scale of extremes X and Y is a continuous transformation between X and Y in which the “intermediate states” are the spaces of the scale. When a space X is obtained by complex interpolation of a family of Banach spaces, a centralizer is generated in X. In Chapter 3, we study the relationship between different configurations of equidistributed spaces on arcs of the unit circle and the centralizers they generate. A process of fragmentation and amalgamation of interpolation scales is also studied. A twisted Hilbert space is a Banach space X that has a Hilbert subspace H such that X/H is also a Hilbert space. Understanding twisted Hilbert spaces involves constructing twisted Hilbert spaces that are very different or very similar to Hilbert space. In the last chapter we can find three new examples of these spaces: an asymptotically Hilbert non-weak Hilbert space, a non-asymptotically Hilbert space, and a space all whose subspaces have the Approximation Property.