On norm attaining operators and multilinear maps

Abstract

This thesis is divided into three chapters. In the first one we do a summary of the state of the art about norm attaining linear forms. We will introduce the Bishop-Phelps and Bishop-Phelps-Bollobás Theorems, the study of whose extensions to different contexts is the main point of interest in this dissertation. The second chapter is devoted to the study of operator versions of Bishop-Phelps and Bishop-Phelps-Bollobás Theorems. In Section 2.2 we will study the extension of these results to the operator case from the point of view of attaining the numerical radius to conclude in Section 2.3.1 that the space L1 satisfy the Bishop-Phelps-Bollobás Property for Numerical Radius. To finish, we will present the Lindenstrauss' result about norm attaining extensions of operator, which will be the motivation of our study from Section 3.2 to Section 3.6 in the next chapter. In the third chapter, we extend the theory of norm attaining linear forms to the non-linear case. Focusing on the line of work initiated by Lindenstrauss, our main point of interest is to study whether the extensions of multilinear maps to the bidual are norm attaining, with special interest on multilinear forms over the space l1, see Sections 3.4 and 3.5. To finish, in Section 3.6 we will study the dependence of the Lindenstrauss-Bollobás Theorems introduced by Carando, Lassalle and Mazzitelli, and the n-linear version of Bishop-Phelps-Bollobás Theorem for spaces M-embedded or L-embedded in the bidual.