Not yet reviewed by Pith; the record is open.

We consider highly radiating ultra-relativistic electrons in a strong external electromagnetic field. High intensity radiative losses and consequent $e^+e^-$-pair production, appearing in the frame of quantum electrodynamics, determine indirectly mass operator of electrons in the strong field. We calculate a leading term of the mass operator at asymptotically high energies of electrons in a strong constant field. We do not use any perturbation theory based on a low ratio of this mass operator to the electron mass $m_e$, but only the original small parameter of quantum electrodynamics, the fine structure constant, $\alpha$. The analogous %but less nontrivial result is obtained for the polarization operator for photon propagator. These results resolve at least partially a very long-lasted controversy originating from traditional and straightforward application of perturbative approaches for description of highly radiating ultra-relativistic charged particles in a very strong electromagnetic fields.