Functions of class II

We consider the class II of operator functions f(z), meromorphic for ¦z¦1, which admit a representation in the form f(z)=f ₂ ^–1 (z)f₁(z), where f₁(z) is an operatorvalued holomorphic function and f₂(z) a scalar-valued bounded holomorphic function, such that the strong limits and coincide a.e. on the unit circle ¦¦=1. We prove that any function of class II can be represented as a block of some J-inner function of class II. We describe all such representations. The results found are applied to the question of realizations of functions of class II as transfer functions of linear systems.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta.im. V. A. Steklova AN SSSR, Vol. 135, pp. 5–30, 1984.