  We develop a transfer-matrix formulation of the scattering of electromagnetic
waves by a general isotropic medium which makes use of a notion of
electromagnetic transfer matrix $\mathbf{M}$ that does not involve slicing of
the scattering medium or discretization of some of the position- or
momentum-space variables. This is a linear operator that we can express as a
$4\times 4$ matrix with operator entries and identify with the S-matrix of an
effective nonunitary quantum system. We use this observation to establish the
composition property of $\mathbf{M}$, obtain an exact solution of the
scattering problem for a non-magnetic point scatterer that avoids the
divergences of the Green's function approaches, and prove a general
invisibility theorem. The latter allows for an explicit characterization of a
class of isotropic media ${\cal M}$ displaying perfect broadband invisibility
for electromagnetic waves of arbitrary polarization provided that their
wavenumber $k$ does not exceed a preassigned critical value $\alpha$, i.e.,
${\cal M}$ behaves exactly like vacuum for $k\leq\alpha$. Generalizing this
phenomenon, we introduce and study $\alpha$-equivalent media that, by
definition, have identical scattering features for $k\leq\alpha$.
