The Bishop–Phelps–Bollobás point property

In this article, we study a version of the Bishop–Phelps–Bollobás property. We investigate a pair of Banach spaces (X,Y)(X,Y) such that every operator from X into Y   is approximated by operators which attain their 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)(X,Y) which have and fail this property. Some stability results are obtained about ℓ1ℓ1 and ℓ∞ℓ∞ sums of Banach spaces and we also study this property for bilinear mappings.