The Bishop-Phelps-Bollob{\'a}s property for numerical radius of operators on $L_1(\mu )$

In this paper, we introduce the notion of the Bishop-Phelps-Bollob'as property for numerical radius (BPBp-$\nu$) for a subclass of the space of bounded linear operators. Then, we show that certain subspaces of $\mathcal{L}(L_1(\mu ))$ have the BPBp-$\nu$ for every finite measure $\mu$. As a consequence we deduce that the subspaces of finite-rank operators, compact operators and weakly compact operators on $L_1(\mu )$ have the BPBp-$\nu$.