  Passive discrete-time systems with Hilbert spaces as an incoming and outgoing
space and a Pontryagin space as a state space are investigated. A geometric
characterization when the index of the transfer function coincides with the
negative index of the state space is given. In this case, an isometric
(co-isometric) system has a product representation corresponding to the left
(right) Krein-Langer factorization of the transfer function. A new criterion,
based on the inclusion of reproducing kernel spaces, when a product of two
isometric (co-isometric) systems preserves controllability (observability), is
obtained. The concept of the defect function is expanded for generalized Schur
functions, and realizations of generalized Schur functions with zero defect
functions are studied.
