On Bishop-Phelps-Bollobas Property in Banach Spaces

Abstract

In this paper, we establish Bishop-Phelps-Bollobas Property (BPBp) for finite rank operators (fro) between Banach spaces (BS). We prove that $BPBp$ for $fro$ holds in several settings including when a Banach space $X$ is of finite dimension or uniformly convex. We also extend these results and show that this property also holds on $BS$ with geometrical properties.
Moreover, we characterize the numerical radius (nr) of $fro$ via the $BPBp$. We establish the extent to which $fro$ satisfy $BPBp$ with respect to $nr$. We show that this property holds in $BS$ settings which include when $X$ is reflexive.