  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.
