Extension of the Bessmertnyi Realization Theorem for Rational Functions of Several Complex Variables

We prove a realization theorem for rational functions of several complex variables which extends the main theorem of Bessmertnyi (in: Alpay, Gohberg, Vinnikov (eds) Interpolation Theory, Systems Theory and Related Topics. Operator Theory Advances and Applications vol 134. Birkhauser Verlag, Basel, pp 157-185, 2002). In contrast to Bessmertnyi's approach of solving large systems of linear equations, we use an operator theoretical approach based on the theory of Schur complements. This leads to a simpler and more "natural" construction to solving the realization problem as we need only apply elementary algebraic operations to Schur complements such as sums, products, inverses, and compositions. A novelty of our approach is the use of Kronecker product as opposed to the matrix product in the realization problem. As such our synthetic approach leads to a solution of the realization problem that has potential for further extensions and applications within multidimensional systems theory especially for those linear models associated with electric circuits, networks, and composites.