On Density and Bishop–Phelps–Bollobás-Type Properties for the Minimum Norm

We study the set \({\text {MA}}(X,Y)\) of operators between Banach spaces X and Y that attain their minimum norm, and the set \({\text {QMA}}(X,Y)\) 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 \({\text {MA}}(X,Y)\) and \({\text {QMA}}(X,Y)\). 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–Bollobás 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.