Local forms of Bishop-Phelps-Bollobás type properties for bilinear maps

In this paper, we characterize the so called property ${\mathbf{\text{L}}}_{o,o}$ as defined by Dantas and Rueda Zoca, for compact, weak-weak continuous bilinear maps. Motivated by this we weaken this property by defining the weak ${\mathbf{\text{L}}}_{o,o}$ for bilinear maps. We provide equivalence of the weak ${\mathbf{\text{L}}}_{o,o}$ property of $(X{\widehat{\otimes }}_{\pi }Y,\mathbb{R})$ for linear functionals and that of $(X,Y;\mathbb{R})$ for bilinear forms under certain conditions on $X$ and $Y$. Moreover, we have also established that under certain preassigned conditions, if a bilnear map $T$ belongs to a class which satisfies the property ${\mathbf{\text{L}}}_{o,o}$ (resp. the weak ${\mathbf{\text{L}}}_{o,o}$) for bilinear maps, then $T^{\ast }$ is a member of a class of operators which satisfy the property ${\mathbf{\text{L}}}_{o,o}$ (resp. the weak ${\mathbf{\text{L}}}_{o,o}$) for operators.