Operators satisfying some forms of Bishop-Phelps-Bollobas type properties for norm and numerical radius

Abstract

In this paper we study a weaker form of the property $\text{\textbf{L}}_{o,o}$ called the weak $\text{\textbf{L}}_{o,o}$ and its uniform version called the weak $\text{BPB}_{\text{op}}$ which is again a weaker form the property $\text{BPB}_{\text{op}}$ for a pair of Banach spaces. We prove that a Banach space $X$ is reflexive and weakly uniformly convex if and only if the pair $(X,\mathbb{R})$ has the property weak $\text{BPB}_{\text{op}}$. We further investigate the class of all bounded linear operators from a Banach space to another Banach space satisfying the property weak $\text{\textbf{L}}_{o,o}$. Finally we introduce and study similar properties for numerical radius of a bounded linear map.