Coupling operators, wedderburn-forney spaces, and generalized inverses

Abstract

The pole-zero structure of a transfer function has been analyzed by two distinct approaches: module-theoretic (coordinate-free) and analytic (basis-dependent). The former (the Wyman-Sain-Conte-Perdon school) emphasizes pole modules, zero modules, exact sequences of maps, and abstract realization theory; applications developed include the model-matching problem, zero structure of one-sided inverses, and connections with geometric systems theory. The latter (the Gohberg school) works with pole chains, zero chains, matrix equations, and concrete factorization theory (minimal, Wiener-Hopf, inner-outer) and interpolation (zero-pole, Lagrange- Sylvester, Nevanlinna-Pick). In this paper we make explicit the connection between the two approaches. In particular, we describe the zero-pole exact sequence of a transfer function without a pole or zero at infinity in the coordinate system provided by a global left null-pole triple. An important tool for us, and an object of interest in its own right, is a generalized inverse G × of a transfer function G. We relate the poles of G × with the zeros of G. We also show that if the pole module of G × is isomorphic to the zero module of G, then left (right) pole pairs of G × are left (right) null pairs of G.