  We present a new theorem concerning a sufficient condition for a symmetric
operator acting in a complex Hilbert space to be essentially self-adjoint. By
applying the theorem, we prove that the Dirac Maxwell Hamiltonian, which
describes a quantum system of a Dirac particle and a radiation field minimally
interacting with each other, is essentially self-adjoint. Our theorem covers
the case where the Dirac particle is in the Coulomb type potential.
