Optimización, differentiability and convexity in Banach spaces

Zusammenfassung

This thesis deals with several topics in the framework of Functional Analysis: the search for extensions of James weak compactness theorem where the set of functionals attaining its supremum is restricted, the study of the concept of spear operator and its relation with classical properties of Banach spaces, and the analysis of properties of the Hardy spaces of vector-valued Dirichlet series. Concerning the methodology, the research work has been mainly based upon the study of books and articles on the subject in order to learn techniques and tools that may be useful. This has been combined with attendance to conferences and research stays in other institutions, allowing to interact with other researchers and specialists on these topics. The thesis consists of three chapters and one appendix. In chapter 1 we study extensions of results related to James theorem where hypothesis are restricted to sets of functionals satisfying a certain separating condition. The main result establishes that for a Banach space X whose dual ball is weak* convex block compact, if A and B are two bounded convex and closed subsets strictly separated such that every with sup(x*, A) < inf(x*, B) attains its supremum on A and its infimum on B, then both A and B are weakly compact. In chapter 2 we carry out a systematic study of the concept of spear operator inspiring on the theory of Banach spaces with numerical index one. We develop a new approach based on the concepts of spear vector and set which let us unify previously known results and to prove new ones. We introduce and extensively study three new classes of operators (aDP, target and lush) for which we provide with geometrical characterizations and examples in concrete spaces, as well as applications to the study of spear vectors in spaces of Lipschitz functions. Chapter 3 is focused on Dirichlet series with coefficients in a Banach space. The first part of the chapter is aimed at the study of the Bohr transform as a tool to establish isometric isomorphism between Hardy spaces of Dirichlet series, spaces of Bochner integrable functions on the infinite-dimensional torus, spaces of cone absolutely summing operators and spaces of holomorphic functions in infinitely many variables. In the second part we provide with asymptotic estimations of the best constant comparing the p-norm and the q-norm of a Dirichlet polynomial in terms of the degree. In the appendix, we describe results obtained in relation to other topics which have been tackled, namely indexes and quantification of Banach space properties (RNP and existence of copies of c_0), Baire ideals, analysis of functions on the Boolean cube and operations with SCD sets in Banach spaces.