Realizations of Non-commutative Rational Functions Around a Matrix Centre, II: The Lost-Abbey Conditions

In a previous paper the authors generalized classical results on minimal realizations of non-commutative (nc) rational functions, using nc Fornasini–Marchesini realizations which are centred at an arbitrary matrix point. In particular, it was proved that the domain of regularity of a nc rational function is contained in the invertibility set of the corresponding pencil of any minimal realization of the function. In this paper we prove an equality between the domain of a nc rational function and the domain of any of its minimal realizations. As for evaluations over stably finite algebras, we show that the domain of the realization w.r.t any such algebra coincides with the so called matrix domain of the function w.r.t the algebra. As a corollary we show that the domain of regularity and the stable extended domain coincide. In contrary to both the classical case and the scalar case—where every matrix coefficients which satisfy the controllability and observability conditions can appear in a minimal realization of a nc rational function—the matrix coefficients in our case have to satisfy certain equations, called linearized lost-abbey conditions, which are related to Taylor–Taylor expansions in nc function theory.