Some Bishop–Phelps–Bollobás type properties in Banach spaces with respect to minimum norm of bounded linear operators

Abstract

In this paper, we study a Bishop-Phelps-Bollobás type property called the property o,o of a pair of Banach spaces. Getting motivated by this, we introduce the notion of Approximate minimizing property (AMp) of a pair of Banach spaces and characterize finite dimensionality of Banach spaces with respect to this property. We further introduce the notion of approximate minimum norm attainment set of a bounded linear operator and characterize the AMp with the help of Hausdorff convergence of the sequence of approximate minimum norm attainment sets of bounded linear operators. We also investigate sufficient conditions for the holding of some weaker forms of the AMp for a pair of Banach spaces. Finally, we define and study uniform -approximation of a bounded linear operator in terms of its minimum norm.