In this paper we present some results on the dispersionless limit
of the Toda lattice equations viewed as the semiclassical limit of an
equation involving certain Toeplitz operators. We consider both
nonperiodic and periodic boundary conditions. For the nonperiodic case
the phase space is the Riemann sphere, while in the periodic case it
is the torus $\mathbb {C}/\mathbb {Z}\sp 2$. In both cases we prove
precise estimates on the dispersionless limit. In addition, we show
that the Toda equations, although they are nonlinear, propagate a
Toeplitz operator into an operator arbitrarily close to a Toeplitz
operator as long as the Toda partial differential equation (PDE)
(dispersionless limit) admits smooth solutions.