  Upcoming experiments on the interaction of electrons with intense laser
fields are envisaged to become more and more accurate, which calls for
theoretical computations of rates and probabilities with correspondingly higher
precision. In strong-field QED this requires the knowledge of radiative
corrections to be added to leading-order results. Here, we first derive the
mass operator in momentum space of an off-shell electron in the presence of an
arbitrary plane wave. By taking the average of the mass operator in momentum
space over an on-shell electron state, we obtain a new representation,
equivalent to but more compact than the known one computed in [Sov. Phys. JETP
\textbf{42}, 400 (1975)]. Moreover, we use the obtained mass operator to
determine the electron mass shift in an arbitrary plane wave, which generalizes
the already known expression in a constant crossed field. The spin-dependent
part of the electron mass shift can be related to the anomalous magnetic moment
of the electron in the plane wave. We show that within the locally constant
field approximation it is possible to conveniently define a local expression of
the electron anomalous magnetic moment, which reduces to the known expression
in a constant crossed field. Beyond the locally constant field approximation,
however, the interaction between the electron and the plane wave is non-local
such that it is not possible to conveniently introduce an electron anomalous
magnetic moment.
