A propriedade de Bishop-Phelps-Bollobás

Abstract

This work aims to study the Bishop-Phelps-Bollobás Theorem for operators between Banach spaces. We will show that the set of norm-attaining operators from $X$ to $Y$ is norm dense in $\mathcal{L}(X,Y)$ when $ Y$ has property $\beta$ for any Banach space $X$. We will see that the pair $(\ell_1,Y)$ has the Bishop-Phelps-Bollobás property for operators if and only if the Banach space $Y$ has the property $AHSP$. Besides that, if $Y$ is a uniformly convex space, we will see that the pair $(\ell^n_\infty,Y)$ satisfies the Bishop-Phelps-Bollobás property for operators. Furthermore, we will show that if $X$ is a uniformly convex space, the pair $(X,Y)$ has the Bishop-Phelps-Bollobás property for operators for any Banach space $Y$.