  We present results concerning resolvent estimates for the linear operator
associated with the system of differential equations governing perturbations of
the Couette flow. We prove estimates on the L_2 norm of the resolvent of this
operator showing this norm to be proportional to the Reynolds number R for a
region of the unstable half plane. For the remaining region, we show that the
problem can be reduced to estimating the solution of a homogeneous ordinary
differential equation with non-homogeneous boundary conditions. Numerical
approximations indicate that the norm of the resolvent is proportional to R in
the whole region of interest.
