The Bishop-Phelps-Bollobás and approximate hyperplane series properties

We study the Bishop-Phelps-Bollobás property for operators between Banach spaces. Sufficient conditions are given for generalized direct sums of Banach spaces with respect to a uniformly monotone Banach sequence lattice to have the approximate hyperplane series property. This result implies that Bishop-Phelps-Bollobás theorem holds for operators from ℓ1 into such direct sums of Banach spaces. We also show that the direct sum of two spaces with the approximate hyperplane series property has such property whenever the norm of the direct sum is absolute.