COUPLING OPERATORS, WEDDERBURN-FORNEY SPACES, AND GENERALIZED INVERSES

The pole-zero structure of a transfer function has been analyzed by tw o distinct approaches: module-theoretic (coordinate-free) and analytic (basis-dependent). The former (the Wyman-Sain-Conte-Perdon school) em phasizes pole modules, zero modules, exact sequences of maps, and abst ract realization theory, applications developed include the model-matc hing problem, zero structure of one-sided inverses, and connections wi th geometric systems theory. The latter (the Gohberg school) works wit h pole chains, zero chains, matrix equations, and concrete factorizati on theory (minimal, Wiener-Hopf, inner-outer) and interpolation (zero- pole, Lagrange-Sylvester, Nevanlinna-Pick). In this paper we make expl icit the connection between the two approaches. In particular, we desc ribe the zero-pole exact sequence of a transfer function without a pol e or zero at infinity in the coordinate system provided by a global le ft null-pole triple. An important tool for us, and an object of intere st in its own right, is a generalized inverse G(x) of a transfer funct ion G. We relate the poles of G(x) with the zeros of G. We also show t hat if the pole module of G(x) is isomorphic to the zero module of G, then left (right) pole pairs of G(x) are left (right) null pairs of G.