On density and Bishop-Phelps-Bollob\'as type properties for the minimum norm

TL;DR

This paper investigates properties related to operators that attain or nearly attain their minimum norm between Banach spaces, characterizes the Radon-Nikodym property in this context, and introduces Bishop-Phelps-Bollobás type properties for the minimum norm.

Contribution

It characterizes the Radon-Nikodym property via minimum norm attainment and explores Bishop-Phelps-Bollobás properties for minimum norm in Banach spaces.

Findings

Characterization of Radon-Nikodym property through minimum norm operators

Existence of Banach spaces where not all operators quasi attain minimum norm

Development of Bishop-Phelps-Bollobás type properties for minimum norm

Abstract

We study the set $\operatorname{MA}(X,Y)$ of operators between Banach spaces $X$ and $Y$ that attain their minimum norm, and the set $\operatorname{QMA}(X,Y)$ of operators that quasi attain their minimum norm. We characterize the Radon-Nikodym property in terms of operators that attain their minimum norm and obtain some related results about the density of the sets $\operatorname{MA}(X,Y)$ and $\operatorname{QMA}(X,Y)$. We show that every infinite-dimensional Banach space $X$ has an isomorphic space $Y$ such that not every operator from $X$ to $Y$ quasi attains its minimum norm. We introduce and study Bishop-Phelps-Bollob\'as type properties for the minimum norm, including the ones already considered in the literature, and we exhibit a wide variety of results and examples, as well as exploring the relations between them.