
This dissertation is devoted to the study of the BishopPhelpsBollobás property in different contexts.
In Chapter 1 we give a historical resume and the motivation behind this property as the classics BishopPhelps and BishopPhelpsBollobás theorems. We define the BishopPhelpsBollobás property (BPBp) and we comment on some important current results.
In Chapter 2 we study similar properties to the BPBp. First, we define the BishopPhelpsBollobás point property (BPBpp). The BPBpp is stronger than the BPBp. We study it for bounded linear operators and then for bilinear mappings. After that, we study two more similar properties: properties 1 and 2. Property 2 is just the dual property of the BPBpp. We observe that is not so easy to get positive results for this property and we comment some differences between this property and the BPBpp since, although they are very similar at first sight, they have completely different behavior from each other. On the other hand, property 1, which is defined similarly but depending on a fixed norm one bounded linear operator, has some positive examples. We finish this chapter studying the BPBpp version for numerical radius on complex Hilbert spaces and the BPBp for absolute norms.
We dedicate Chapter 3 to the study of the BPBp for compact operators which is defined analogously to the BPBp but now considering just this type of operator. Our strategy is to study the conditions that the Banach spaces must satisfy to get a BPBp for compact operators and use technical results to pass the BPBp for compact operators from sequences to functions spaces. We apply these results for both domain and range situations. For example, if the pair (c_0; Y) has the property so does the pair (C_0(L); Y) for every locally compact Hausdorff topological space L. We also prove that we can pass the BPBp for compact operators from the pair (X; \ell_p(Y)) to the pair (X; L_p(\mu,Y)). Moreover, if Y has a certain geometric property, then the pairs (X; L_{\infty} (\nu,Y)) and (X; C(K,Y)) have the BishopPhelpsBollobás property for compact operators.
In the last chapter the BPBp is extended to the multilinear version. We discuss when it is possible to pass some known results about the BPBp for operators to the multilinear case. We give some results for symmetric multilinear mappings and homogeneous polynomials. Still on this chapter, we study the numerical radius on the set of all multilinear mappings defined in L_1. We prove that, in this case, the numerical radius and the norm of a multilinear mapping coincide. We also study the BPBp for numerical radius (BPBpnu) for multilinear mappings. It is shown that if X is finite dimensional, then X satisfies this property. On the other hand, L_1 fails it although L_1 has the analogously property for bounded linear operators. We also prove that if a c_0 or a \ell_1sum satisfies it, then each component of the direct sum also satisfies the BPBpnu for multilinear mappings.
We finish the dissertation by presenting a list of open problems with the intention to expand new horizons. Also we present tables which summary the pairs of classical Banach spaces satisfying the BishopPhelpsBollobás property with the purpose to put the reader in the current scenario on this topic.
