  The Maskawa-Nakajima equation has attracted considerable interest in
elementary particle physics. From the viewpoint of operator theory, we study
the Maskawa-Nakajima equation in the massless abelian gluon model. We first
show that there is a nonzero solution to the Maskawa-Nakajima equation when the
parameter $\lambda$ satisfies $\lambda>2$. Moreover, we show that the solution
is infinitely differentiable and strictly decreasing. We thus conclude that the
massless abelian gluon model generates the nonzero quark mass spontaneously and
exhibits the spontaneous chiral symmetry breaking when $\lambda>2$. We next
show that there is a unique solution $0$ to the Maskawa-Nakajima equation when
$0<\lambda<1$, from which we conclude that each quark remains massless and that
the model realizes the chiral symmetry when $0<\lambda<1$.
