Weighted composition operators as Daugavet centers

We investigate the norm identity ‖uCφ+T‖=‖u‖∞+‖T‖ for classes of operators on C(S), where S is a compact Hausdorff space without isolated point, and characterize those weighted composition operators which satisfy this equation for every weakly compact operator T:C(S)→C(S). We also give a characterization of such weighted composition operator acting on the disk algebra A(D).