FACTORIZATIONS OF GENERALIZED SCHUR FUNCTIONS AND PRODUCTS OF PASSIVE SYSTEMS

Factorizations of Pontryagin space operator-valued generalized Schur functions are studied. Main tools are products of contractive operator colligations, or cascade connections of passive discrete-time systems. The well-known notion of regular factorizations of ordinary Schur functions is extended to the generalized Schur class functions by using canonical reproducing kernel Pontryagin space models. Factorizations stronger than the regular factorization are also introduced to obtain characterizations in the case where the products of observable co-isometric (controllable isometric) systems preserve the observability (controllability). These factorizations are related to backwards shift invariant regular subspaces of de Branges-Rovnyak spaces, and they can alternatively be viewed as regular factorizations of generalized Schur functions with certain extreme properties. Moreover, their properties are linked with how the optimality is preserved under the product of optimal passive systems.