The Bishop-Phelps-Bollob\'as point property

TL;DR

This paper introduces and characterizes the Bishop-Phelps-Bollobás point property (BPBpp) for pairs of Banach spaces, exploring its relation to smoothness, providing examples, and analyzing stability and bilinear mappings.

Contribution

It defines BPBpp, characterizes uniform smoothness via this property, and investigates its stability and behavior in bilinear mappings.

Findings

BPBpp characterized by uniform smoothness.

Examples of pairs with and without BPBpp provided.

Stability results for sums of Banach spaces and bilinear mappings obtained.

Abstract

In this article, we study a version of the Bishop-Phelps-Bollob\'as property. We investigate a pair of Banach spaces $(X, Y)$ such that every operator from $X$ into $Y$ is approximated by operators which attains its 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-Bollob\'as point property (BPBpp). We characterize uniform smoothness in terms of BPBpp and we give some examples of pairs $(X, Y)$ which have and fail this property. Some stability results are obtained about $\ell_1$ and $\ell_\infty$ sums of Banach spaces and we also study this property for bilinear mappings.