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 a
lgebraic Riccati equation (ARE) associated with T(z). Conversely, it is pro
ved that the existence of a (not necessarily stabilizing) solution of the a
ssociated ARE implies bounded realness of T(z). Inertia results are obtaine
d and a condition for the existence of negative semidefinite solutions of t
he associated ARE is given.