On Banach spaces with the approximate hyperplane series property

TL;DR

This paper introduces a new sufficient condition for Banach spaces to possess the approximate hyperplane series property (AHSP), explores its stability in vector-valued function spaces, and characterizes AHSP in dual spaces via $w^*$-continuous operators.

Contribution

It provides a unifying condition for AHSP, extends stability results to vector-valued spaces, and offers a new dual space characterization related to the Bishop-Phelps-Bollobás theorem.

Findings

A sufficient condition for AHSP covering all known examples.

Stability of AHSP in vector-valued integrable function spaces.

Characterization of AHSP in dual spaces via $w^*$-continuous operators.

Abstract

We present a sufficient condition for a Banach space to have the approximate hyperplane series property (AHSP) which actually covers all known examples. We use this property to get a stability result to vector-valued spaces of integrable functions. On the other hand, the study of a possible Bishop-Phelps-Bollob\'as version of a classical result of V. Zizler leads to a new characterization of the AHSP for dual spaces in terms of $w^*$-continuous operators and other related results.