In this paper the bounded real lemma for discrete-time systems is extended in several directions. It is shown that an H-infinity-norm bound for a (not necessarily stable) transfer matrix T(z) combined with controllability of unimodular eigenvalues yields the existence of an unmixed solution of the algebraic Riccati equation (ARE) associated with T(z). Conversely, it is proved that the existence of a (not necessarily stabilizing) solution of the associated ARE implies bounded realness of T(z). Inertia results are obtained and a condition for the existence of negative semidefinite solutions of the associated ARE is given.