Localized Bishop-Phelps-Bollobás type properties for minimum norm and Crawford number attaining operators

Abstract

In this paper, we study the approximate minimizing property (AMp) for operators, a localized Bishop-Phelps-Bollobás type property with respect to the minimum norm. Given Banach spaces X and Y we define a new class \(\mathcal{A}\mathcal{M}(X,Y)\) of bounded linear operators from X to Y for which the pair (X, Y) satisfies the AMp. We provide a necessary and sufficient condition for non-injective operators from X to Y to be in the class \(\mathcal{A}\mathcal{M}(X,Y)\). We also prove that X is finite dimensional if and only if for every Banach space Y, (X, Y) has the AMp for all minimum norm attaining operators from X to Y if and only if for every Banach space Y, (Y, X) has the AMp for all minimum norm attaining operators from Y to X. We also study the AMp with respect to Crawford number called AMp-c for operators.