  We consider a hydrogen-like atom in a quantized electromagnetic field which
is modeled by means of a no-pair operator acting in the positive spectral
subspace of the free Dirac operator minimally coupled to the quantized vector
potential. We prove that the infimum of the spectrum of the no-pair operator is
an evenly degenerate eigenvalue. In particular, we show that the bottom of its
spectrum is strictly less than its ionization threshold. These results hold
true, for arbitrary values of the fine-structure constant and the ultra-violet
cut-off and for all Coulomb coupling constants less than the critical one of
the Brown-Ravenhall model. For Coulomb coupling constants larger than the
critical one, we show that the quadratic form of the no-pair operator is
unbounded below. Along the way we discuss the domains and operator cores of the
semi-relativistic Pauli-Fierz and no-pair operators, for Coulomb coupling
constants less than or equal to the critical ones.
