  We make a spectral analysis of discrete Schroedinger operators on the
half-line, subject to complex Robin-type boundary couplings and complex-valued
potentials. First, optimal spectral enclosures are obtained for summable
potentials. Second, general smallness conditions on the potentials guaranteeing
a spectral stability are established. Third, a general identity which allows to
generate optimal discrete Hardy inequalities for the discrete Dirichlet
Laplacian on the half-line is proved.
