We study the large time behavior of solutions to two-dimensional Euler and Navier-Stokes equations linearized about shearflows of the mixing layer type in the unbounded channel T×R.  Under a simple spectral stability assumption on a self-adjoint operator, we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity. We also prove a local form of the enhanced viscous dissipation that takes place at times of order ν^(−1/3), ν being the small viscosity. To prove these results, we use a Hamiltonian approach, following the conjugate operator method developed in the study of Schroedinger operators, combined with a hypocoercivity argument to handle the viscous case.