  This is the third installment in a series of papers concerning the Bounded
Real Lemma for infinite-dimensional discrete-time linear input/state/output
systems. In this setting, under appropriate conditions, the lemma characterizes
when the transfer function associated with the system has contractive values on
the unit circle, expressed in terms of a Linear Matrix Inequality, often
referred to as the Kalman-Yakubovich-Popov (KYP) inequality. Whereas the first
two installments focussed on causal systems with the transfer functions
extending to an analytic function on the disk, in the present paper the system
is still causal but the state operator is allowed to have nontrivial dichotomy
(the unit circle is not contained in its spectrum), implying that the transfer
function is analytic in a neighborhood of zero and on a neighborhood of the
unit circle rather than on the unit disk. More generally, we consider bicausal
systems, for which the transfer function need not be analytic in a neighborhood
of zero. For both types of systems, by a variation on Willems' storage-function
approach, we prove variations on the standard and strict Bounded Real Lemma. We
also specialize the results to nonstationary discrete-time systems with a
dichotomy, thereby recovering a Bounded Real Lemma due to
Ben-Artzi--Gohberg-Kaashoek for such systems.
