This is an announcement for the paper “The strong Bishop-Phelps-Bollobás property” by Sheldon Dantas.
Abstract: In this paper we introduce the strong Bishop-Phelps-Bollob$'$as property (sBPBp) for bounded linear operators between two Banach spaces $X$ and $Y$. This property is motivated by a Kim-Lee result which states, under our notation, that a Banach space $X$ is uniformly convex if and only if the pair $(X, N)$ satisfies the sBPBp. Positive results of pairs of Banach spaces $(X, Y)$ satisfying this property are given and concrete pairs of Banach spaces $(X, Y)$ failing it are exhibited. A complete characterization of the sBPBp for the pairs $(\ell_p, \ell_q)$ is also provided.
The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.01461