  The kinematic evolution of axisymmetric magnetic fields in rotating
magnetospheres of relativistic compact objects is analytically studied, based
on relativistic Ohm's law in stationary axisymmetric geometry. By neglecting
the poloidal flows of plasma in simplified magnetospheric models, we discuss
self-excited dynamos due to the frame-dragging effect (originally pointed out
by Khanna & Camenzind), and we propose alternative processes to generate
axisymmetric magnetic fields against ohmic dissipation. The first process
(which may be called induced excitation) is caused by the help of a background
uniform magnetic field in addition to the dragging of inertial frames. It is
shown that excited multipolar components of poloidal and azimuthal fields are
sustained as stationary modes, and outgoing Poynting flux converges toward the
rotation axis. The second one is self-excited dynamo through azimuthal
convection current, which is found to be effective if plasma rotation becomes
highly relativistic with a sharp gradient in the angular velocity. In this case
no frame-dragging effect is needed, and the coupling between charge separation
and plasma rotation becomes important. We discuss briefly the results in
relation to active phenomena in the relativistic magnetospheres.
