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