The Bishop-Phelps-Bollob\'{a}s and approximate hyperplane series properties

TL;DR

This paper investigates the Bishop-Phelps-Bollobás property for operators between Banach spaces, providing conditions under which certain direct sums of Banach spaces possess the approximate hyperplane series property, extending known theorems.

Contribution

It introduces new sufficient conditions for the approximate hyperplane series property in generalized direct sums of Banach spaces, broadening the applicability of Bishop-Phelps-Bollobás results.

Findings

Bishop-Phelps-Bollobás theorem holds for operators from ℓ₁ into certain direct sums.

Direct sum of two spaces with the property retains the property if the norm is absolute.

Provides conditions for generalized direct sums to have the approximate hyperplane series property.

Abstract

We study the Bishop-Phelps-Bollob\'as 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-Bollob\'as theorem holds for operators from $\ell_1$ 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.