The Bishop-Phelps-Bollobás point property

Abstract

In this article, we study a version of the Bishop-Phelps-Bollobás property. We investigate a pair of Banach spaces $(X,Y)$ such that every operator from $X$ into $Y$ is approximated by operators which attains its norm at the same point where the original operator almost attains its norm. In this case, we say that such a pair has the Bishop-Phelps-Bollobás point property (BPBpp). We characterize uniform smoothness in terms of BPBpp and we give some examples of pairs $(X,Y)$ which have and fail this property. Some stability results are obtained about ${\ell }_1$ and ${\ell }_{\infty }$ sums of Banach spaces and we also study this property for bilinear mappings.