Note on a kind of Bishop-Phelps-Bollob\'as property for operators

TL;DR

This paper investigates a Bishop-Phelps-Bollobás type property for Banach space operators, providing characterizations of the local property in strictly convex domains and establishing its implications for the domain space’s convexity.

Contribution

It characterizes the local Bishop-Phelps-Bollobás property for operators with strictly convex domains and compactness, linking it to strong convexity properties of the domain space.

Findings

The local property implies strong convexity of the domain space.

Characterization of the local property in strictly convex domain spaces.

Connection between the local property and convexity properties.

Abstract

We study a Bishop-Phelps-Bollob\'as type property for Banach space operators introduced by Dantas (2017). In that paper there is a local and a global version of a natural property which is somewhat similar but simpler compared to the Bishop-Phelps-Bollob\'as type property for operators studied in Acosta et al. (2008). Here we characterize the mentioned local property in the setting with strictly convex domain spaces and compact operators. We show that the local property implies that the domain space has strong convexity properties.