Técnicas de extensão de operadores multilineares em espaços de Banach

Abstract

The main purpose of this work is to study several techniques of extending continuous multilinear operators between Banach spaces. The general problem is as follows: Given subspaces G_1, ... , G_n of the Banach spaces E_1, ... ,E_n and a continuous n-linear operator A: G_1 x ... x G_n -> F, does there exist a continuous n-linear operator of E_1 x ... x E_n in F which extends A? Is there a norm preserving extension? Is this extension, if any, unique? The first aim it show that there is no multilinear version of the Hahn-Banach Theorem. Motivated by this fact we show that continuous multilinear operators can be extended to (i) the closure of each subspace, (ii) when the subspaces are complemented, (iii) when the subspaces are subspaces of a Hilbert space. Another problem we deal with, a more delicate one, concerns the extension of continuous multilinear operators to the bidual. In this direction we provide a detailed study of the so-called Aron-Berner extensions, including a complete study of when they coincide, of when they are separetely weak-star-weak-star continuouos and of when they take values in the original target space. Finally we conclude this work presenting some examples of Arens-regular spaces and other situations where all continuous linear operators are weakly compact.