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

In this paper we study a weaker form of the property ${\mathbf{\text{L}}}_{o,o}$ called the weak ${\mathbf{\text{L}}}_{o,o}$ 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 ${\mathbf{\text{L}}}_{o,o}$. Finally we introduce and study similar properties for numerical radius of a bounded linear map.