This is an announcement for the paper “The Bishop-Phelps-Bollobás point property” by Sheldon Dantas, Sun Kwang Kim, Han Ju Lee.

Abstract: In this article, we study a version of the Bishop-Phelps-Bollob'as 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-Bollob'as 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.

The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.00245