On the numerical radius of Lipschitz operators in Banach spaces

We study the numerical radius of Lipschitz operators on Banach spaces via the Lipschitz numerical index, which is an analogue of the numerical index in Banach space theory. We give a characterization of the numerical radius and obtain a necessary and sufficient condition for Banach spaces to have Lipschitz numerical index 1. As an application, we show that real lush spaces and $C$-rich subspaces have Lipschitz numerical index 1. Moreover, using the G$\hat{a}$teaux differentiability of Lipschitz operators, we characterize the Lipschitz numerical index of separable Banach spaces with the RNP. Finally, we prove that the Lipschitz numerical index has the stability properties for the $c_0$-, $l_1$-, and $l_\infty$-sums of spaces and vector-valued function spaces. From this, we show that the $C(K)$ spaces, $L_1(\mu)$-spaces and $L_\infty(\nu)$ spaces have Lipschitz numerical index 1.